Abstract

A Search for Spontaneous Positron Production in Heavy Ion - Atom Collisions

John Edward Schweppe

Yale University

1985

An experimental search has been conducted for the spontaneous production of

positrons in heavy-ion collisions at the Coulomb barrier. This previously

undetected QED process would signal the decay of the electron-positron vacuum

in an overcritical electromagnetic field.

The experiments were conducted at GSI Darmstadt using a positron

spectrometer based on the axial focussing property of a solenoidal magnetic Held

for charged particles together with a baffle system and an annihilation-radiation

detector to provide a large detection efficiency for positrons while suppressing the

gammarray and electron backgrounds. Two position-sensitive particle detectors

determined the kinematic parameters for each collision, allowing the isolation of

interesting events. The background of positrons from internal pair conversion of

excited nuclear states was calculated from the gamma-ray flux monitored in

Nal(Tl) detectors. The collision systems measured were U+Cm at a bombarding

energy of 6.05 MeV/amu, U-fU at 5.9 MeV/amu, and U+Pb at 5.9 MeV/amu.

General agreement exists between theoretical calculations of the total dynamical

positron production for Rutherford scattering and the overall features of the

measured positron distributions. A well-defined peak structure, however, has

been found in the positron energy spectrum of the heaviest, U-f-Cm, system at

316 ± 10 keV, associated with a particular particle scattering-angle region. The

narrow width of ~ 70 keV, consistent with the minimum Doppler broadening,

implies a source living > 10-20 sec.

An analysis of the lineshape and angular distribution of the structure and of the

simultaneously measured gamma-ray spectrum substantially rules out the internal

pair conversion of an excited nuclear state as a trivial source of the peak. The

posibility of explaining the structure within the context of spontaneous positron

production, enhanced by the formation of giant, metastable, nuclear complexes at

the Coulomb barrier, is discussed.

A Search for Spontaneous Positron Production in Heavy Ion - Atom Collisions

A Dissertation

Presented to the Faculty of the Graduate School

of

Yale University

in Candidacy for the Degree of

Doctor of Philosophy

by

John Edward Schweppe

May, 1985

Acknowledgements

I wish to acknowledge the leading role that my advisor, Jack- S. Greenberg,

played in every aspect of this thesis project. In addition to Professor Greenberg, I

would like to thank D. Allan Bromley, the director of A W. Wright Nuclear

Structure Laboratory, and the other members of my committee, Vernon Hughes,

Peter Parker, and Peter Mohr, for their careful reading of my thesis and their

comments.

The experiments described in this thesis were carried out by the EPOS

collaboration at GSI Darmstadt, and clearly reflect the combined efforts of the

many members. In addition to my advisor and other fellow Yalies Paul Vincent

and Tom Cowan, this collaboration has included Dirk Schwalm, Klaus Bethge,

Helmut Bokemeyer, Roland Schule, Axel Gruppe, Hans Grein, S. Matsuki, Andre

Baianda, M. Waldschmidt, Shin Ito, Hartmut Backe, Kurt Stiebing, Michaela

Kltiver, Marieluise Begemann, Helmut Folger, and N. Trautmann,

I would also like to thank Walter Greiner, Berndt Muller, Udo Muller,

J. Reinhardt, G. Soff, Theo de Reus, and their fellow workers at the Theoretische

Institut of the Universitat Frankfurt both for valuable discussions and the use of

unpublished calculations.

I must thank Sara Batter, Mary Ann Schulz, and Rita Bonito at Yale University

for their help, in particular for seeing to it that stipend checks were promptly

forwarded and countless forms were properly filled out, no matter were on earth I

happened to be. In addition, I thank Sandy Sicignano for her excellent work in

preparing the illustrations for this thesis.

We are grateful to the staff at GSI Darmstadt for their assistance in carrying

out the measurements. I would like to single out R. Kliiber, Traudel Eisold, and

Isia Busch, who were particularly helpful and kind to me.

We are also indebted to the Transplutonium Program of the U.S. Department of

Energy for the loan of the 248Cm isotope material.

My research was supported in part by U.S. Department of Energy Contract

Number DE-AC02-76ERO3074, the Bundesministerium fur Forschung und

Technologie of the Federal Republic of Germany, and the Deutscher

Akademischer Austauschdienst.

There simply isn’t enough room to mention all of the other friends who in

addition to those above have made life so interesting on both sides of the Atlantic

during during my years as a graduate student (Judy, Steve, Gary, Buzz, Gabi,

Peer and Mina, Ray and Mary Lynn, Rebecca, Zoe, ...). I would, however, like to

give special mention to Karl-Heinz and Ute Hechler, who under the guise of

teaching me to speak German, made me feel right at home in a foreign land.

Table of Contents

L ist of F igures viiL ist of Tables ^1. Introduction 1

1.1. Historical Background 21.1.1. Early Theoretical Work 31.1.2. Recent Theoretical Work 61.1.3. Experiments with Heavy Ions 10

1.2. Present Experiment 151.3. Organization of the Thesis 19

2. Theory 212.1. Spontaneous Positron Production 24

2.1.1. Klein Paradox 242.1.2. Dirac Equation for a Point Nucleus 302.1.3. Dirac Equation for a Finite Nucleus 32

2.1.3.1. Calculations for Z < Z 34c r

2.1.3.2. Calculations for Z > Z 37CT2.1.3.3. Projection Operator Method 422.1.3.4. Effective Potential Method 45

2.1.4. Two-Center-Dirac Equation 492.1.4.1. Multipole Expansion of the TCD Equation 512.1.4.2. Results of the Calculations 53

2.2. Dynamic Positron Production 582.2.1. For Rutherford Trajectories 63

2.2.1.1. Description of the Scattering Nuclei 632.2.1.2. Description of the Electronic States 662.2.1.3. Spontaneous Component of the Positron Production 702.2.1.4. Description of the Positron Production 732.2.1.5. Results of the Calculations 74

2.2.2. With Nuclear Interactions 772.2.2.1. Implications for Spontaneous Positron Production 812.2.2.2. Nucleus-Nucleus Potential in Heavy-Ion Collisions 832.2.2.3. Quantum Mechanical Treatment 882.2.2.4. Semiclassical Treatment 91

2.3. Nuclear Positron Production 982.3.1. Calculation of the EPC Coefficients 1012.3.2. Results of the Calculations 106

iii

iv

3. Experimental Apparatus 1003.1. Experimental Apparatus 116

3.1.1. Target Mounting System 1173.1.2. Positron Detection System 118

3.1.2.1. The Apparatus. 1183.1.2.2. Positron Detection Efficiency. 1243.1.2.3. Doppler Broadening of the Detected Line Shape. 137

3.1.3. Particle Detectors 1393.1.3.1. Description of the Detectors 1393.1.3.2. Scattering Kinematics 1433.1.3.3. Operating Characteristics. 1463.1.3.4. Effect of the Solenoidal Magnetic Field. 146

3.1.4. Gamma Ray Detectors 1523.2. Data Collection System 153

3.2.1. Electronics System 1553.2.1.1. Types of Signals. 1553.2.1.2. Types of Events. 1643.2.1.3. Scaledowns. 166

3.2.2. Computer Interface 1683.2.2.1. DMA Data Transfer. 1683.2.2.2. DMI Data Transfer. 169

3.2.3. Online Computer System 1703.2.3.1. Hardware. 1713.2.3.2. Software. 172

4. Calibration of the Apparatus 1754.1. Detection of Positrons 176

4.1.1. Si(Li) Positron Detector Calibration 1774.1.1.1. Energy Calibration 1774.1.1.2. Response Function for Electrons 1774.1.1.3. Response Function for Positrons 178

4.1.2. Nal Annihilation-Radiation Detector Calibration 1854.1.2.1. Energy Calibration 1854.1.2.2. Detection Efficiency 186

4.1.3. Positron Detection Efficiency 1874.1.3.1. Transport Efficiency 1884.1.3.2. Detection Efficiency 189

4.2. Detection of Gamma Rays 1934.2.1. Energy Calibration 1944.2.2. Response Function 1954.2.3. Detection Efficiency 195

4.3. Detection of Particles 1984.3.1. Angle Calibration 1984.3.2. Detection Efficiency 200

V

5.1. Systems Measured 2045.1.1. Measurement of Atomic Positron Production 2045.1.2. Measurement of Nuclear Positron Background 2115.1.3. Scattered-Particle Angle Calibration 213

5.2. Parameters Measured 2145.3. Online Analysis 2145.4. Special Considerations 216

5.4.1. Beam Condition. 2165.4.2. Target Condition. 218

6. Data Analysis 2216.1. Event-Type Identification 223

6.1.1. Scattered-Particle Identification 2276.1.2. Gamma-Ray Identification 2336.1.3. Positron Identification 241

6.2. Analysis of the Chance-Coincidence Rate 2526.2.1. Scattered-Particle Event 2526.2.2. Gamma-Ray Event 2536.2.3. Positron Event 254

6.3. Annihilation-Radiation Sum-Energy Background Subtraction 2596.4. Correction for Detector Response and Normalization 265

6.4.1. Lineshape Correction 2666.4.1.1. Gamma-Ray Lineshape Correction 2666.4.1.2. Positron Lineshape Correction 275

6.4.2. Detection Efficiency 2766.4.3. Scaledown and Deadtime Correction 2776.4.4. Normalization to Scattered Particles 278

6.5. Calculation of the Nuclear Positron Background 2796.5.1. Extrapolation from Low-Z Systems 2796.5.2. Determination of the Effective IPC Coefficient 2806.5.3. Nuclear Positron Background for High-Z Systems 288

6.6. Production of the Final Spectra 2946.6.1. Gross Features of the Data 2946.6.2. Fine Features of the Data 298

7. Results and Discussion 3077.1. Gross Features of Positron Production 3077.2. Fine Features of Positron Production 315

7.2.1. Description of the Structure 3167.2.2. Possible Sources of Structure 3257.2.3. Spontaneous Positron Production 332

7.3. Summary and Outlook 337

6. Data Collection 203

vi

Appendix A. ESC Kinematic Equations 345A.l. Rutherford-Scattering Kinematics 345A.2. Rutherford-Scattering Cross Section 348

Appendix B. Abbreviations Used in the Thesis 351References 353

List of Figures

Figure 1-1: Pb+Pb positron production. 11Figure 1-2: Positron to gamma ray ratio. 13Figure 1-3: Pb+Pb, U+Pb, and U+U positron production. 14Figure 1-4: Schematic diagram of the experimental apparatus. 16Figure 1-6: U+Cm positron production R min dependence. 17Figure 2-1: The Klein Paradox. 25Figure 2-2: Scattering coefficients for the Klein Paradox. . 26Figure 2-3: Three cases of scattering for the Klein Paradox. 27Figure 2-4: Potentials considered for the Klein Paradox. 29Figure 2-5: Electronic energy eigenvalues of the point nucleus. 33Figure 2-6: Comparison of nuclear Coulomb potentials. 35Figure 2-7: Isa energy eigenvalues. 36Figure 2-8: Comparison of nuclear charge distributions. 37Figure 2-9: Electronic energy eigenvalues for 100 < Z < 173. 38Figure 2-10: Energy and radial size of electronic states. 39Figure 2-11: Mean radius of the 1 so state. 40Figure 2-12: 1 so resonance shape. 41Figure 2-13: Charge density distribution of the 1 so state. 42Figure 2-14: Electron density and potential well for Z — 184. 43Figure 2-15: Coulomb and effective potential at Z — Z . 47Figure 2-16: Electronic energy eigenvalues for 0 < Z < 250. 48Figure 2-17: Mixing of e* and e+ states in a strong potential. 49Figure 2-18: Two-center Coulomb potential. 54Figure 2-19: Quasimolecular state correlation diagrams. 55Figure 2-20: Quasimolecular wave functions. 56Figure 2-21: Quasimolecular matrix elements. 57Figure 2-22: Quasimolecular 1 so energy and mean radius. 58Figure 2-23: ls<r state resonance energy and decay width. 59Figure 2-24: ls<7 state critical nuclear separation. 60Figure 2-25: Atomic sources of positrons in HI collisions. 62Figure 2-26: Induced and spontaneous coupling time dependence. 71Figure 2-27: Contribution of spontaneous positron emission. 72Figure 2-28: 1 so ionization probability. 75Figure 2-29: Positron energy and Rmin dependence. 76Figure 2-30: Positron energy spectra. 77Figure 2-31: Positron R min dependence. 78

vii

viii

Figure 2-32: Positron angular dependence. 79Figure 2-33: Positron energy spectra. 80Figure 2-34: Energy and matrix elements with nuclear sticking. 82Figure 2-35: Nuclear charge distribution and shape for U+Cm. 84Figure 2-36: Nuclear charge distribution and shape for U+U. 85Figure 2-37: Nuclear charge distribution and shape for U+Pb. 86Figure 2-38: Nucleus-nucleus potentials. 87Figure 2-30: Quantum mechanical calculations. 91Figure 2-40: Quantum mechanical calculations. 92Figure 2-41: Positron spectra for a single sticking time. 94Figure 2-42: Positron emission for a single sticking time. 95Figure 2-43: Exponentially distributed nuclear sticking times. 97Figure 2-44: Gaussian distribution of nuclear sticking times. 98Figure 2-45: Internal conversion processes. 100Figure 2-46: Internal conversion diagrams. 101Figure 2-47: Differential internal pair conversion probability. 106Figure 2-48: Transition-energy dependence of EPC coefficients. 107Figure 2-49: Z-dependence of internal pair conversion. 108Figure 3-1: GSI UNILAC and experimental hall. 110Figure 3-2: i, e-, and e+ emission in heavy-ion collisions. 112Figure 3-3: The experimental apparatus. 113Figure 3-4: Baffle. 121Figure 3-5: Suppression of electrons by the baffle. 122Figure 3-6: Diagram of the solenoid positron transport system. 126Figure 3-7: Positron-transport efficiency for a solenoid. 128Figure 3-8: Calculated baffle transmission. 131Figure 3-9: The detection of annihilation radiation. 133Figure 3-10: Calculated positron transport efficiency. 136Figure 3-11: Doppler broadening of the positron line shape. 140Figure 3-12: Placement of the particle detectors. 141Figure 3-13: Schematic diagram of the particle detector. 142Figure 3-14: U+Pb angular distributions. 145Figure 3-15: Angular resolution for heavy-ion detection. 147Figure 3-16: Effect of solenoid field on particle detection. 150Figure 3-17: Effect of solenoid field on particle detection. 151Figure 3-18: Placement of the gamma-ray detectors. 153Figure 3-19: Schematic diagram of the experimental set-up. 154Figure 3-20: Schematic diagram of the electronics system. 156Figure 3-21: Schematic diagram of the positron-detection electronics. 158Figure 3-22: Schematic diagram of the annihilation-radiation- 159

detection electronics.Figure 3-23: Schematic diagram of the gamma-ray-detection 160

electronics.Figure 3-24: Schematic diagram of the particle-detection electronics. 161

Figure 3-25: Schematic diagram of the event-type-selection 162electronics.

Figure 3-26: Schematic diagram of the ADC/TDC electronics 163Figure 3-27: Diagram of signals in the scaledown unit. 167Figure 4-1: Measured line shape in the Si(Li) detector. 179Figure 4-2: e- and e+ line shape in the Si(Li) detector. 180Figure 4-3: Second annihilation gamma-ray absorbtion. 181Figure 4-4: Response function of the Si(Li) detector. 184Figure 4-5: Annihilation-radiation detection efficiency. 187Figure 4-6: Electron transport efficiency. 190Figure 4-7: Comparison of positron spectra. 191Figure 4-8: Measured positron detection efficiency. 192Figure 4-0: Measured gamma-ray line shapes. 196Figure 4-10: Gamma-ray detection efficiency. 197Figure 4-11: Particle detection efficiency. 202Figure 5-1: U+Cm kinematic plots. 206Figure 5-2: U+U kinematic plots. 207Figure 5-3: U+Pb kinematic plots. 208Figure 5-4: U+Sm kinematic plots. 212Figure 5-5: Beam profile. 217Figure 6-1: Event-type analysis program. 225Figure 6-2: Positron-event analysis program. 226Figure 6-3: Particle angle-angle scatter plot. 228Figure 6-4: Particle energy scatter plot: upper detector. 229Figure 6-5: Particle energy scatter plot: lower detector. 231Figure 6-6: Particle time-of-flight scatter plot. 232Figure 6-7: U+Cm particle isometric plot. 234Figure 8-8: U+U particle isometric plot. 235Figure 6-0: U+Pb particle isometric plot. 236Figure 6-10: Gamma-ray energy spectrum before analysis. 237Figure 6-11: Particle - gamma-ray time difference. 238Figure 6-12: Gamma-ray energy spectrum after analysis. 239Figure 6-13: Gamma ray isometric plots. 240Figure 6-14: Positron energy spectrum before analysis. 242Figure 6-15: Positron front - rear detector time difference. 244Figure 6-16: Scattered particle - positron time difference. 246Figure 6-17: Positron • annihilation radiation time difference. 247Figure 6-18: Annihilation radiation energy spectra. 249Figure 6-10: Positron energy spectrum after analysis. 250Figure 6-20: Positron isometric plot. 251Figure 6-21: Annihilation-radiation background comparison. 260Figure 6-22: Annihilation-radiation background spectra. 261Figure 6-23: Annihilation-radiation-background calculation. 265Figure 6-24: Measured total positron energy spectra. 267

ix

X

Figure 6-25: U+Cm positron scatter plot. 268Figure 6-26: U+U positron scatter plot. 269Figure 6-27: U+Pb positron scatter plot. 270Figure 6-28: U+Cm scatter plots. 271Figure 6-20: U+U scatter plots. 272Figure 6-30: U+Pb scatter plots. 273Figure 6-31: Comparison of forward and rear gamma-ray detectors 282Figure 6-32: Positron-to-gamma-ray ratio. 283Figure 6-33: Positrons from nuclear processes. 284Figure 6-34: U+Ho P(Hmin) distribution. 286Figure 6-35: U+Sm energy and Rmin spectra. 287Figure 6-36: Gamma-ray spectra for various collision systems. 289Figure 6-37: Gamma-ray spectra for various Rmin regions. 290Figure 6-38: Gamma-ray spectra for various angular regions. 291Figure 6-30: U+U gamma-ray energy spectra. 292Figure 6-40: Positron production from nuclear processes. 293Figure 6-41: Total measured positron energy spectra. 295Figure 6-42: Total measured positron angular spectra. 297Figure 6-43: Positron analysis angular regions. 299Figure 6-44: U+Cm positron energy spectra. 301Figure 8-45: U+Cm positron energy spectrum. 302Figure 6-46: U+Cm scatter plot of the peak energy. 303Figure 6-47: U+Cm AS spectrum at the peak energy. 304Figure 6-48: U+U positron energy spectra. 305Figure 6-40: U+Pb positron energy spectra. 306Figure 7-1: Quasimolecular positron energy spectra. 310Figure 7-2: Quasimolecular positron angular distributions. 312Figure 7- 3: U+Pb P(Rmin). 313Figure 7-4: U+Cm positron energy peak. 317Figure 7-5: U+Cm positron energy peak. 319Figure 7-6: U+Cm beam excitation function. 320Figure 7-7: U+Cm positron emitter velocity. 321Figure 7-8: U+Cm angular-correlation diagram. 323Figure 7-0: Nuclear sources of positrons. 326Figure 7-10: DPC fit to the U+Cm peak. 328Figure 7-11: Pair conversion calculated for gamma rays. 330Figure 7-12: Pair conversion calculated for electrons. 333Figure 7-13: Monoenergetic CM fit to the U+Cm peak. 334Figure 7-14: U+U positron energy spectrum. 338Figure 7-15: Z-dependence of the positron peak energy. 340Figure 7-18: Equal separation bombarding energies. 343

List of Tables

Table 2*1: Coulomb Scattering Phase Analysis Results 41Table 3-1: Symbols used in Figures 3-20 to 3-26 157Table 4-1: Electron line source energies. 178Table 4-2: Annihilation-radiation-detector calibration sources. 185Table 4-3: Gamma-ray detector calibration sources. 194Table 4-4: Particle detector angle calibration. 199Table 5-1: Average projectile energies. 205Table 5-2: Kinematic parameters: high-Z systems. 210Table 5-3: Kinematic parameters: low-Z systems. 213

xi

Chapter 1

Introduction

In recent years an opportunity has developed to examine experimentally the

possibility of a fundamental change of the quantum-electrodynamic (QED)

vacuum state in strong electrostatic fields. The Dirac theory describing the

electronic states of an atom predicts that for a nuclear charge Z > 173, the

binding energy of the most tightly bound electrons exceeds 2mc2. Overcritical

binding, however, makes the electron-positron vacuum unstable to a new decay

process. The presence of a vacancy in the overcritically bound electronic state

makes it energetically favorable for the state to spontaneously decay to a new

vacuum state by the creation of an electron-positron pair. The electron is

captured into the vacant state and the positron is repelled away from the nucleus.

The mass of the created pair is provided by the binding energy made available in

the process. The detection of the spontaneously emitted positron serves as the

signature of this new, predicted rearrangement of the QED vacuum.

Although atoms with a large enough nuclear charge for overcritical binding do

not occur naturally, the effect can possibly be realized in the quasimolecules

formed transiently in collisions of accelerated heavy ions with heavy atoms. Since

relative velocities of ~ 0.1 c are enough to bring two very heavy nuclei, such as

uranium plus uranium, together to the Coulomb barrier, the quasimolecular

electronic orbitals can adjust almost adiabatically to the changing nuclear charge

configuration. Collisions of the heaviest nuclei are predicted theoretically to

produce overcritical binding of the innermost quasimolecular electronic states.

I

2

The construction of heavy ion accelerators such as the UNELAC at the

Gesellschaft fiir Schwerionenforschung (GSI1) in Darmstadt, West Germany,

capable of accelerating uranium ions to the U+U Coulomb barrier, has opened

the door to experimental study of the predicted spontaneous decay of the QED

vacuum state.

This thesis describes an experimental search for spontaneous positron production

in heavy-ion - atom collisions at the Coulomb barrier. The experiments described

here were conducted at GSI Darmstadt in 1981 as part of an ongoing research

program. The positron production was measured in three heavy-ion collision

systems which straddle the predicted critical binding point: 238U + 248Cm at a

bombarding energy of 6.05 MeV/amu, + 238U at 5.9 MeV/amu, and 238U +

^ P b at 5.9 MeV/amu.

The following sections of this introductory chapter will give a brief history of

the development of the concept of spontaneous positron production, describe in

more detail the present experiment, and outline the organization of this thesis.

1.1. Historical Background

The concept of spontaneous positron production in overcritical electrostatic

fields has developed principally in three stages [Rafelski et al. 78a, Zel’dovich and

Popov 71, Greiner 83a]. During the first period, stretching from Dirac’s

publication of his relativistic electron theory in 1928 until the late 1960’s, the

general mathematical properties of strong electromagnetic fields and overcritical

binding were examined primarily as a theoretical exercise, with little thought

toward the possibility of experimental verification. Serious interest began,

however, around 1969 with the nearly simultaneous realization that overcritical

binding might be achieved in heavy-ion - atom collisions and that an overcritically

*The abbreviations used in this thesis are collected in Appendix B.

3

bound vacant state would lead to the experimentally detectable process of

spontaneous positron emission [Pieper and Greiner 69, Gershtein and Zel’dovich

69a, Gershtein and Zel’dovich 69b]. This initiated from several theoretical groups

more detailed calculations of the properties of the overcritically bound state and

of the dynamics of heavy-ion - atom collisions, as described more thoroughly

below. The completion of the very heavy-ion accelerator at GSI Darmstadt

opened the final period with the first experiments in the late 1970’s to look for

spontaneous positron production in heavy-ion - atom collisions near the Coulomb

barrier. This thesis is an extension of the earlier efforts to search for experimental

verification of the spontaneous decay of the electron-positron vacuum in

overcritical electromagnetic fields.

1.1.1. Early Theoretical Work

Since spontaneous positron production is an inherently relativistic, quantum

mechanical process, its discussion begins with the Dirac theory of the relativistic

electron published in 1928 [Dirac 28a, Dirac 28b]. A modification of the existing

nonrelativistic quantum mechanics to have the correct relativistic relationship

between energy and momentum, Dirac’s theory also serendipitously pointed the

way to antimatter states and the existence of the positron, later verified

experimentally by Anderson (Anderson 33].

The same year, Darwin [Darwin 28] and Gordan [Gordan 28] solved the new

Dirac Equation for the Coulomb potential of a hydrogen-like atom (a point

nucleus plus a single electron). When they calculated the energy levels of the

electron states, they recovered the Sommerfeld fine-structure formula which

Sommerfeld had derived in an ad hoc manner in the old Bohr quantum

mechanics. The formula agreed remarkably with the measured spectrum of

hydrogen, but broke down mathematically for nuclear charge Z greater than

a~* ~ 137. (For some time it was conjectured that this marked a natural limit

4

to the periodic table of the elements.) This was the first indication that new

behavior appears for the electron near a very heavy nucleus.

Immediately following this, Klein (Klein 28] discovered and Sauter [Sauter

31a, Sauter 31b] expanded upon the anomolous behavior of an electron scattering

off of a one dimensional potential with height greater than 2mc2, a behavior that

came to be known as Klein’s Paradox. The paradox was resolved by, among

others, Hund in 1940 with the introduction of the idea of the stimulated creation

of electron-positron pairs at a large enough potential [Hund 40]. Though no

longer a paradox, this was a further indication of interesting behavior of the

electron-positron field in the presence of a large potential.

During the following two decades four theoretical papers appeared which dealt

with the problem of overcritical fields. In 1940, Schiff, Snyder, and Weinberg

solved the Klein-Gordan Equation, for bosons, for the case of a square well

potential [Schiff et al. 40]. They calculated the dependence of the energy of the

states as a function of the well depth and found no overcritical binding. Based on

a schematic calculation, they speculated that the Dirac Equation would show the

same lack of overcritical binding.

In 1945 a pivotal paper by Pomeranchuk and Smorodinsky

appeared [Pomeranchuk and Smorodinsky 45]. They calculated for the first time

the electronic energy levels in the Dirac theory for a superheavy nucleus with

Z > 137. The key to extending calculations past Z = 137 was the assumption of

a finite extent for the nucleus to remove the singularity in the Coulomb potential

for a point nucleus, V(r) = —Ze2/r, at the origin, r = 0. They predicted that the

binding energy would reach 2mc2 for a high enough nuclear charge Z . Although

inadequate mathematical approximations lead to a first estimate for the critical

nuclear charge of Z' 250 which later calculations showed to be too high, they

did qualitatively predict the general features of critical binding in superheavy

5

In 1950, Case published a general study of singular potentials [Case 50],

including the Dirac Equation for a point nucleus. He showed that the the

ambiguity caused by the singular nature of the potential could be removed by

specifying the phase of the wave scattering at the location of the singularity. For

the case of the point nucleus, this is a mathematical alternative to the physical

solution of using a nucleus with finite extent.

In 1958, Werner and Wheeler solved the Dirac Equation for extended

superheavy nuclei [Werner and Wheeler 58]. They found critical binding for the

nuclear charge Z' cz 170, a value more in line with recent calculations.

They, as the others before them, did not speculate, however, on what happened

beyond Z' . That did not come until 1960 and the first calculation of the energy

levels of a superheavy nucleus for Z > Z , by Voronkov and

Kolesnikov [Voronkov and Kolesnikov 60]. This paper was the first to predict the

overcritical binding of an electron in the Coulomb field of a large enough nucleus.

Furthermore, and just as important for the problem of experimental verification,

they were the first to realize the possibility of a new phenomenon beyond Z : that a vacancy in an overcritically bound state could be filled by spontaneous pair

production with the emission of the positron. Unfortunately, as perhaps other

theoretical works which have had the misfortune to appear too soon before the

possibility of their experimental verification, the paper went unnoticed.

In 1963, Beck et al. in a paper about the energy levels of a square-well

potential [Beck et al. 63] also mentioned the possibility of spontaneous pair

AA footnote in a later Russian paper [Popov 70a] claims that Pomeranchuk had also speculated

in 1945 on the possibility of producing critical binding in collisions of heavy nuclei, but never published his private musings. This foreshadowed predictions of the possibility of experimentally producing critical binding, discussed below, by over two decades.

nuclei2.

production when the binding energy of an electronic state in the well reaches

2mc2. It is not clear, however, from their paper whether they envisioned this as

an actual physical process or as just a mathematical construct.

1.1.2. Recent Theoretical Work

In the late 1960’s, two ideas changed the study of overcritical electrostatic fields

from an academic, theoretical pursuit to an experimentally verifiable theory. As

has been often the case, both ideas were discovered independently and nearly

simultaneously by two groups, one working at the Universitat Frankfurt in West

Germany and the other at Moscow in the U.S.S.R. The first idea was the

(re)discovery that, in the Dirac theory for the electronic states of an extended

nucleus, the binding energy of the electrons can continue to increase beyond 2mc2

for Z > Z' and that a overcritically bound vacant state can decay by

spontaneous pair production with the emission of an experimentally detectable

positron. Both groups described this situation in papers published in 1969:

Pieper and Greiner from the Frankfurt school [Pieper and Greiner 69] and

Gershtein and Zel’dovich from the Moscow school [Gershtein and Zel’dovich

69a, Gershtein and Zel’dovich 69b]. According to Greiner [Greiner 83a], both

groups were unaware of the earlier paper of Voronkov and Kolesnikov.

The second important idea was the realization that superheavy quasimolecules

could be formed during collisions of heavy nuclei. This idea is based on the

observation that heavy nuclei could be brought together to the Coulomb barrier

by a relative velocity of only about 1/10 of the speed of light. Since this is

relatively small compared to the velocity of the highly relativistic, tightly bound

inner electrons, these electrons could have the time to adiabatically readjust their

orbitals to the joint field of both nuclei as the two nuclei move past each other.

Thus, although large enough atoms do not exist to provide the necessary critical

charge Z' ~ 170, the transiently formed electrostatic field of two colliding

7

heavy ions, such as uranium plus uranium, could be strong enough to produce

overcritical binding. Experimental evidence for overcritical binding in

electromagnetic fields was to be sought as spontaneous positron emission in

collisions of heavy nuclei. This idea was promoted by Greiner during GSI-

Seminars in 1969 and 1970 (according to footnote ** in [Muller et al. 72a] and

footnote 6 in. [Muller et al. 73a]) and was published by the Frankfurt group

in [Rafelski et al. 71]. It appeared in the same papers from the Moscow school

mentioned above [Gershtein and Zel’dovich 69a, Gershtein and Zel’dovich 69b].

These two concepts implied that spontaneous positron production could identify

overcritical binding and spurred further theoretical activity in these two groups to

study the phenomenom. The Moscow school concentrated on the study of the

properties of the electronic states near the point of critical binding in a

superheavy atom using analytical techniques. Early work was based on an

effective Schroedinger potential and included calculations of the energy

eigenvalues and the form of the wave functions as Z —*• Z , of Z itself, of the

spontaneous positron production rate near Z , and the important observation

that the electronic state remains localized as Z —*■ Z ^ [Popov 70b, Popov

70c, Popov 70a]. These were followed by further calculations of the form of the

electronic state for Z > Z [Popov 71a, Zel’dovich and Popov 71, Migdal et al.

71], a demonstration of the applicability of the WKB approximation [Krainov 71],

and variational calculations of Z [Popov 71b, Popov 72a, Popov 72b, Popov and

Rozhdestvenskaya 71, Perelomov and Popov 73].

The Frankfurt school also started with the study of the supercritical atom, using

numerical techniques to extend the calculations beyond Z , where they treated

the overcritically bound state as a resonance in the negative energy continuum.

They developed a description of the resonant state based on the autoionization

formalism, and determined values for the energy and decay width and of the form

8

of the wave functions for Z > Z by a numerical Coulomb-phase-shift analysis of

the negative energy continuum states [Muller el al. 72b, Muller et al. 72c, Muller

et al. 72a, Muller et al. 73b]. A study of limiting field Lagrangians showed that

such theories could not prevent critical binding [Rafelski el al. 71, Muller et al. 72c, Rafelski et at. 73j.

The earlier calculations were confined, for the most part, to the problem of a

static, superheavy atom. The study of positron production in the quasimolecular

field of two heavy nuclei started after B. Muller’s thesis work [Muller 74] on the

solution of the Two-Center-Dirac Equation [Muller et al. 73a, Rafelski and Muller

76a, Rafelski and Muller 76b, Muller and Greiner 76]. In addition to obtaining

more accurate values for the quasimolecular energy eigenvalues and wave

functions [Soff et at. 74, Soff et al. 78a] and for R^ [Soff et al. 74, Rafelski and

Muller 76b, Wietschorke et al. 79], the Frankfurt school used the solutions of this

equation as basis states to attacked the full dynamic problem of positron

production in collision of heavy nuclei. A study was made of all the elements in

heavy-ion collisions. In addition to the spontaneous production of positrons [Peitz

et al. 73] and the formation of the necessary K-sheil vacancies [Betz et al. 76],

three competing sources of positron production in heavy ion collisions were also

examined:

• induced pair production by excitation of negative energy continuum electrons into vacant quasimolecular bound states [Smith et al. 74],

• direct pair formation by excitation of negative energy continuum electrons into positive energy continuum states [Soff et al. 77],

• and the internal pair conversion of excited nuclear states created by Coulomb and nuclear interaction during the collision [Oberacker et al.76a].

Further results used time-dependent perturbation theory to describe the total

positron production in heavy-ion collisions [Smith et al. 74, Muller 76, Reinhardt

et at. 78, Soff et al. 79]. General reviews of this earlier work can be found

9

in (Muller 76, Reinhardt and Greiner 77, Rafelski et al. 78a, Greiner and Peitz

78, Brodsky and Mohr 78, Fulcher et al. 79, Reinhardt et al. 80a).

Coupled-channel calculations followed in 1979 [Reinhardt et al. 79], and were

found to have a large effect through multistep processes on the absolute, size of

the predicted inner shell ionization [Soff et al. 80, Soff et al. 81a, Soff et al.

81b, Muller et al. 83] and the resultant positron emission [Reinhardt et al.

81a, Reinhardt et al. 81b]. Reviews of this later work can be found in [Reinhardt

et al. 80b, Reinhardt et al. 80c, Reinhardt et al. 83a, Soffel et al. 82, Muller et al.

82a, Muller et al. 82b, Muller 83a, Soff et al. 83]. Similar but independent

coupled-channel calculations of the positron production in heavy-ion collisions

were carried out by [Tomoda and Weidenmiiller 82, Tomoda 82].

At the same time, the Russian work concentrated on effective-potential

calculations of the positron production in heavy-ion collisions [Gershtein and

Popov 73, Marinov and Popov 73, Marinov and Popov 74a, Popov 73a, Popov

73b, Popov 74] and of the form of the critically bound state [Popov and Mur

73, Migdal 76, Migdal 77, Popov et al. 76, Migdal et al. 77, Eletskii and Popov

77], more refined variational computations of R [Marinov et al. 74, Marinov and

Popov 75a, Marinov and Popov 76], and applications of the WKB approximation

to overcritical binding [Marinov and Popov 74b, Marinov and Popov 75b, Eletskii

et al. 77, Mur and Popov 78, Mur et al. 78a, Mur et at. 78b, Popov et al. 79].

Recent theoretical work has focussed on the possible enhancement of

spontaneous positron production by nuclear reactions [Rafelski et al.

78b, Reinhardt et al. 81b, Muller et al. 82a, Muller et al. 82b, Muller et al.

83, Reinhardt et al. 83b, Tomoda and Weidenmiiller 83] and its rigorous quantum

mechanical description [Heinz et al. 83a, Heinz et al. 83b].

10

1.1.3. Experiments with Heavy Ions

The acceleration of beams of uranium ions to the U+U Coulomb barrier with

the UNILAC heavy-ion accelerator at GSI Darmstadt in 1976 opened the way for

experimental investigations of what had been up to this point a theoretical

exercise. The first experimental results from GSI bearing on spontaneous positron

production in heavy-ion - atom collisions, carried out by a collaboration of Yale

University, the Technische Hochschule Darmstadt, and GSI, concerned the very

feasibility of the proposed experiments. As stated above, the spontaneous

emission of a positron depends on the presence of a vacancy in the overcritically

bound electronic state. [Greenberg et al. 77] analysed the Doppler broadening of

characteristic X-rays emitted at 0 ’ to the beam direction to determine

experimentally that the ionization probability in heavy ion collisions for the most

tightly bound, ls<7, state is both relatively large ( ~ 10-2 —10-1) and

concentrated at small impact parameters. The size of the vacancy production,

about three orders of magnitude larger than extrapolations from lighter systems

had predicted, guaranteed reasonable experimental counting rates. The

concentration at small impact parameter was also advantageous for studying the

the most strongly bound states.

This was followed shortly by the first measurement of the positron production in

heavy-ion collisions by [Backe et al. 78], who measured the collision system ^ P b

+ ^ P b at bombarding energies between 3.6 and 5.6 MeV/amu. This system was

chosen because the relatively simple nuclear excitation spectrum of 208Pb

(dominated by the 2.6 MeV 3* first excited state) permitted a straightforward

determination of background positrons from the internal pair conversion (IPC) of

nuclear states excited during the collision. Figure 1-1 (reproduced from [Backe et al. 78]) shows the measured total and differential cross sections as a function of

the distance of closest approach in a head-on collision, 2a, for the collision system

11

Figure 1-1: Pb+Pb positron production.The total (part (a)) and differential (part (b)) positron production cross sections are plotted as a function of the distance of closest approach in a head-on collision, 2a, for the 208Pb + ^ P b collision system. In part (c), the differential cross section is plotted as a function of the laboratory scattering angle. The solid circles are the measured yields after subtraction of the nuclear positron background. The open circles are the measured positron yield from the decay of the 3* state in ^P b. The dot-dashed lines are calculations of Reinhardt et al. (Reproduced from [Backe et al. 78].)

12

^ P b + ^ P b . The open circles correspond to positrons from the IPC of the 3’

excited state; the closed circles to the positron yield after the subtraction of

background positrons from nuclear excitations. The subtraction of this

background by comparison with a 212Pb source provided the first evidence for the

production of positrons in heavy-ion collisions by atomic QED processes in excess

of positrons from nuclear excitations.

With the addition of the Technische Universitat Munchen to the collaboration,

the measurements were extended by [Kozhuharov et al. 79] to the heavier collision

systems 238U + 208Pb and 238U + 238U. In order to determine the background of

positrons from nuclear excitations in these more complicated systems, a study was

made of the relationship of positron to gamma-ray production over a range of

heavy-ion collision systems. Figure 1-2 [Greenberg 77] shows the ratio of

measured positrons to gamma rays as a function of the nuclear charge Z of the

target bombarded by 238U at an energy of 5.9 MeV/amu. The simple behavior

evident in this ratio, i.e. constant up to (Z 4-Z.) « 170 and rising sharply butP *smoothly above, formed the basis of an empirical method for determing the

background of positrons from nuclear excitations by extrapolation from low-Z

collision systems. It also provided further dramatic indication of the production

of atomic QED positrons in heavy-ion collision systems in excess of nuclear

positrons. The measured differential probability for positron production per

scattered particle is shown in Figure 1-3 (reproduced from [Kozhuharov et al. 79])

as a function of the center of mass (CM) scattering angle of the projectile for the

three collision systems 208Pb + 208Pb, 238U + ^ P b , and 238U + 238U. Using the

method outlined above, the positrons from nuclear excitations have already been

subtracted.

These early experiments at GSI in the late 1970’s had found convincing evidence

for atomic positron production, i.e. positron production not accountable for by

13

Figure 1-2: Positron to gamma ray ratio.The measured ratio of positrons to gamma rays (with an arbitrary scale) is plotted as a function of the nuclear charge Z of the target for bombardment with 238U at 5.9 MeV/amu. (From [Greenberg 77j.)

nuclear processes alone and evidently coming from the formation of the

quasimolecular states in the collisions. No unambiguous evidence could be found,

however, within the limitations of positron energy measurement, particle angle

measurement, and statistical uncertainty for spontaneous positrou production.

Spontaneous positron production is strongly suppressed in heavy-ion collisions at

the Coulomb barrier because the time that electronic states remain overcritically

bound during the collision is typically only about 10-2 of the lifetime for

spontaneous decay of a vacancy. The competing positron production processes,

14

Figure 1-3: Pb+Pb, U+Pb, and U+U positron production.The differential probability per scattered particle for the production of positrons with kinetic energy in the interval 440 keV to 510 keV is plotted as a function of the CM scattering angle of the projectile for the collision systems U+U, U+Pb, and Pb+Pb. The background of positrons from nuclear excitations has been subtacted. The solid lines are the calculations of Reinhardt et al. 1978. The insert shows the total measured positron probability (upper curve) and the calculated nuclear background (lower curve) for the U+U system. (Reproduced from [Kozhuharov et al. 79].)

mentioned above, of induced pair production, direct pair production, and the

internal pair conversion of excited nuclear states dominate the positron emission.

Furthermore, the production amplitudes for induced and direct pair production

add coherently to that for spontaneous positron production because all have the

same initial and final positron states. As a result, no dramatic signal can be

expected in the total positron production as the transition past the critical nuclear

charge Z is made. Against this background, a new generation of experiments

15

was planned to take a closer look at positron production processes in heavy-ion

collisions.

1.2. Present Experiment

The present experiment was planned as an expansion of the previous efforts

with respect to three of the experimentally accessible parameters by increasing the

combined nuclear charge of the quasimolecule formed of projectile and target

nuclei and by making a more detailed measurement of the kinetic energy of the

emitted positrons and of the scattering angles of the colliding nuclei.

AiQThe procurement of Cm targets extended the list of experimentally observed

collision systems to include U+Cm, heavier by four charge units than U+U, the

heaviest system measured to that point. The higher combined nuclear charge was

theoretically predicted to increase the decay width for spontaneous positron

production by a factor of four and the average kinetic energy of the

spontaneously emitted positrons by a factor of two over U+U. Both changes

would greatly ease the detection of the sought spontaneous component of the

positron emission.

For these experiments a new positron spectrometer was built at GSI Darmstadt.

A schematic diagram of the important elements of the experimental apparatus is

shown in Figure 1-4. The spectrometer uses a solenoidal magnetic field to

transport positrons away from the target area and the overwhelming backgrounds

of gamma rays and delta electrons emitted in heavy-ion collisions at the Coulomb

barrier. The kinetic energy of the positrons is measured in an axially mounted,

cylindrical Si(Li) detector with an intrinsic resolution of ~ 10 keV. The axial

focussing property of the solenoid for charged particles together with a baffle

system against electrons provide a large transport efficiency approaching 20% for

positrons in the energy range from 100 keV to 1 MeV while suppressing the

16

gamma-ray and electron backgrounds. Backgrounds are further suppressed by

the detection of the characteristic positron annihilation radiation in a cylindrical

geometry of Nal(Tl) detectors positioned around the positron detector.

8 fold

Originally, a plastic scintillator had been used to measure the kinetic energy of

the positrons, instead of the present Si(Li) detector. The poorer energy resolution

(30%-50% over the range 100 keV to 1 MeV) was not expected to play a

significant role because theory predicted smooth positron distributions from

heavy-ion collisions below the Coulomb barrier. The first measurements of the

U+Cm collision system [Balanda et al. 80, Balanda et al. 80, Bokemeyer et al. 81]

with the plastic scintillator indicated, however, that the positron energy would be

an important parameter. Figure 1-5 (reproduced from [Greenberg 83]) shows the

measured probability for positron production per scattered particle as a function

17

of the distance of closest approach, R in, during the collision for the collision

system U+Cm at a bombarding energy of 6.05 MeV/amu. Two broad positron

kinetic energy regions have been picked out by coincidence with the plastic

scintillator. The squares represent positrons in the energy interval of 300 to 500

keV, while the circles are positrons from 600 to 900 keV. The dashed lines are

calculations of [Reinhardt et al. 83a]. This first indication of an excess positron

production at lower positron kinetic energies in the overcritically bound U+Cm

system prompted the switch to the Si(Li) detector with its much better energy

resolution in order to study this effect.

a8U-2*eCm

Figure 1-5: U+Cm positron production R min dependence.The probability of positron production per scattered particle is plotted as

• a function of the distance of closest approach, Rm|n, for the collision system 238U + 248Cm at a bombarding energy of 6.05 MeV/amu. The squares are positrons in the kinetic energy interval 350 keV to 500 keV; the circles are 600 keV to 900 keV. (Reproduced from [Greenberg 83].)

The kinematic parameters for each positron-producing collision were determined

by measuring the scattering angles of both colliding nuclei in the laboratory

angular region from 20 ° to 700. Essentially 100% detection efficiency was

18

obtained with two position-sensitive, parallel-plate avalanche counters with delay-

line readout mounted symmetrically about the beam axis. The determination of

the scattering angles of both nuclei allowed a kinematic separation of the

quasielestic scattering events from background scatterings off lower-Z backings

and target contaminants, and in general permitted the isolation of any interesting

scattering events.

In order to calculate the background of positrons from the internal pair

conversion of excited nuclear states, the gamma-ray flux was monitored in two

3*X3" Nal(Tl) detectors mounted at 45° to the beam axis. One detector was

mounted forward and the other to the rear of the target, in order to allow a

consistency check of the Doppler shift and solid angle transformation.

The experiment reported in this thesis was performed in January and February

of 1981 at GSI Darmstadt. At the time, the UNILAC at GSI was the only

accelerator capable of providing beams of very heavy ions up to the Coulomb

barrier energy. A beam of uranium ions was used to study the positron

production in three collision systems straddling the critical nuclear charge: 238U

+ 248Cm at a bombarding energy of 6.05 MeV/amu, + 238U at 5.9

MeV/amu, and + 208Pb at 5.9 MeV/amu. In addition, the lower-Z collision

system -I- 154Sm was also measured at a bombarding energy of 5.9 MeV/amu

to study the background production of positrons from nuclear processes.

Data were collected and written event-by-event onto magnetic tape using three

different triggers. A positron event was defined by a coincidence between the

positron detector, the annihilation-radiation detector, and both of the scattered-

particle detectors. A gamma-ray event was signalled by a coincidence between

either of the gamma-ray detectors and both of the particle detectors. These two

event types were normalized to particle events, triggered by a coincidence

between the two particle detectors. The subsequent data analysis was done on

the IBM-370 computer at GSI.

19

1.3. Organization of the Thesis

The thesis is divided into five parts: introduction, theory, experiment, data

analysis, and conclusions. Following the introduction to the thesis in this first

chapter, chapter two discusses the theory behind this experimental thesis project.

Chapters three through five describe the experiment itself. Chapter three

describes the experimental apparatus, chapter four the calibration of this

apparatus, and chapter five the collection of data. Chapter six outlines the

analysis of the data collected during the experiment and presents the analysed

results. The conclusions of this thesis project are then presented in the seventh

and final chapter.

Chapter 2

Theory

A theoretical description will be given in this chapter of the three major sources

of positron production expected in heavy-ion - atom collisions near the Coulomb

barrier: spontaneous, dynamic, and nuclear processes. Due to the extensive use

of theoretical results in the interpretation of our experiments, a lengthy summary

is presented in order to bring all the appropriate points together in one place.

Those already familiar with the theory can proceed to the next chapter without

loss of continuity.

As described in Chapter 1, spontaneous positron emission results when a

vacancy occurs in an electronic state which is critically bound to a charge center,

i.e. when the binding energy of the vacant state exceeds the rest mass 2mc2 of an

electron-positron pair. This is fundamentally a static process which depends only

on the magnitude and size of the nuclear charge distribution and would serve as a

signature of the decay of the electron-positron vacuum in overcritical

electromagnetic fields.

The two other sources of positrons in heavy-ion - atom collisions near the

Coulomb barrier come about because of the inherently dynamic nature of the

collision process. Dynamic positron production is due to the relative motion of

the two nuclei during the collision which constantly changes the charge

distribution of the quasimolecule formed by the two moving nuclei. The

continuously varying electromagnetic field produced by the two nuclei induces

21

22

transitions of electrons from the negative energy continuum to higher energy

states, leaving holes behind which appear as positrons. The process is similar to

spontaneous positron production in that both can be viewed as transitions of

electrons in the negative energy continuum to bound states and the positive

energy continuum. The important difference is that the dynamic positron

production is inherently linked to the relative motion of two charge centers, and

disappears in the static limit, while spontaneous positron production is a static

process which depends ultimately only on the magnitude aud density of the

positive charge distribution. It can be present not only in collision systems but

remains also in the static limit.

The final source of positron production, nuclear processes, is also the result of

the relative motion of the colliding nuclei. Positrons are produced by the internal

pair conversion of excited nuclear states with an energy greater than

2mc2 ~ 1.02 MeV formed during the collision.

A significant portion of the theoretical description in this chapter deals with the

assumptions and simplifications made to treat the problem of positron production

in heavy-ion collisions since the fully stated theory is mathematically intractable

with present techniques. The problem is inherently relativistic, field theoretical,

and time-dependent. The electronic states must be described relativistically

because the binding energies for the interesting inner shells are on the order of

twice the electron’s rest mass. The involvement of positron creation immediately

implies that the creation and annihilation operators of field theory are necessary,

as opposed to simple relativistic quantum mechanics. Since the source of the

quasimolecular charge configuration producing the electromagnetic field which

causes positron production is constantly changing, the problem to be solved is

overtly time-dependent. The adoption of reasonable assumptions has been

necessary in order to simplify the problem. For the most part, the treatment of

23

W. Greiner and coworkers at the Universitat Frankfurt is followed (see the

references in Section 1.1 and in the following sections).

As the result of these assumptions and simplifications, the full problem has been

reduced to the following scenario. The nuclear motion is assumed to be described

by nonrelativistic, classical mechanics, so that the positive charge distribution is

given as a function of time by the motion of two spherical charges moving past

each other on Coulomb trajectories. In this approximation, the effects on the

nuclear trajectory of nuclear recoil, retardation effects, and the transfer of nuclear

kinetic energy to the electron-positron field are neglected. Furthermore, it is

assumed that the nuclei move slowly enough compared to the electrons that the

electron clouds carried by the two nuclei have time to constantly readjust

themselves adiabatically to the changing positive charge configuration. This

allows the full, time-dependent problem for the transition amplitudes during a

heavy-ion collision to be solved by expanding the time-dependent wave functions

in a basis consisting of the time-independent solutions. These have been found by

solving the Dirac Equation for the static charge distribution as a function of the

internuclear separation. In the Hamiltonian describing the dynamic electron

states, it is assumed that the magnetic field is negligibly small compared to the

electric field, the rotational coupling between electronic states is small compared

to the radial coupling, and that the monopole part of the two-center Coulomb

potential dominates the other multipole terms. Relativistic field theory provides

the prescription for determining the positron and electron annihilation and

creation operators from the transition amplitudes. The total positron production

is calculated by integrating the probability for the creation of positrons over all

possible collision trajectories. The calculation relies on nonrelativistic, classical

mechanics to describe the nuclei, relativistic quantum mechanics to describe the

electronic states, and relativistic quantum field theory to describe the production

of positrons. The discussion and results which follow are presented within the

context of these approximations.

24

2.1. Spontaneous Positron Production

The static case of spontaneous positron production from a single nuclear charge

distribution will be discussed in this section. In this connection, the description of

the Klein Paradox, the unusual behavior of the reflection and transmission

coefficients for a Dirac electron scattering on a one-dimensional potential step,

provides some interesting insights. It was historically the first indication of

positron production in strong fields and is conceptually a good starting point for

describing this behavior. Additional important elements in an understanding of

spontaneous positron emission are the solutions of the Dirac Equation for the

Coulomb field of a point nucleus as a function of the nuclear charge, the

modification to the case of a more realistic finite-sized nuclear charge distribution,

and the solutions of the two-center Dirac equation. These aspects are reviewed

below.

2.1.1. Klein Paradox

Shortly after Dirac published his relativistic quantum mechanics [Dirac

28a, Dirac 28b], Klein discovered the perplexing behaviour in the reflection and

transmission coefficients, calculated in the new theory for an electron striking a

step potential with a height greater than V = 2me2, that became known as

Klein’s Paradox [Klein 28]. The situation considered by Klein was the simplest,

one dimensional case of electron scattering, in which an electron impinges from

the left on a potential step with height V, as shown in Figure 2-1. The Dirac

Equation for this electron with four-momentum p = (E/c,p) in an external

electromagnetic field described by the four-potential = ($/c,A) is:

1 /(P M + ^ M) - m c ] * = ° (2.1)

with the usual definition for ^ = (P,0a) in terms of the 4X4 matrices aQ = 0

and a = (a1,a2,a3) which obey the commutation relations:V * + V n = 2 6 n u for » ' v — ° ’1’2’3 (2*2)

For the case Klein considered:

25

rO for r<0 A = 0, e<P(x) = <

\Vfor 0<x(2.3)

V -

a;

♦ i p .x / f i r\ f \ f \ +

r \ f \ P - ♦ip.x/fi

0X

Figure 2-1: The Kleiu Paradox.An electron represented by the plane wave exp(+ip_x/ft) impinges from the left on a step potential of height V The electron is partially reflected as the plane wave exp(—ip_x/h) and partially transmitted as the plane wave exp(+tp+z/ft).

In order to determine the one-dimensional reflection and transmission

coefficients for the potential step, Klein followed the procedure which had been

worked out for the nonrelativistic Schrodinger equation. As shown in Figure 2-1,

he matched an incident and a reflected wave on the left of the barrier to a

transmitted wave on the right to obtain the reflection coefficient R and the

transmission coefficient T:1 — k 0 4 k

b = ( t t ) ' T : <2-4)l + K (1 + *)

with (following [Hansen and Ravndal 81[) the definition:P - E + me2K ------------------ (2.5)P + E - V + m c 2

26

As required, R + T = 1. R and T are shown as a function of the potential height

Vin Figure 2-2 (for the arbitrarily chosen energy E = 2mc2).

4c .2*

*o£ 2a> o o

0w0)

o - 2o c~t n

0 I 2 3 4eV/mc2

Figure 2-2: Scattering coefficients for the Klein Paradox.The reflection (solid line) and transmission (dotted line) coefficients for

A

an electron of total energy 2mc striking a step potential is plotted as a function of the potential height V

Three cases are obvious:1) V< (E-mc2)2) {E-mc2) < V < {E+mc2)3) (E+mc2) < V

These are shown diagramatically in Figure 2-3. The shaded area in each part

indicates the continuum region, \E[ > me2, classically accessible to a free electron.

The first two cases are as expected. In part (a), the small increase of the potential

at x — 0 causes partial reflection of the impinging electron flux. In part (b),

transmission into the classically forbidden region is not allowed, and complete

reflection results (R = 1 and T = 0).

27

>O<TLlJZUJ

0POSITION

Figure 2-3: Three cases of scattering for the Klein Paradox.The dashed line indicates the energy E of the scattering electron. In part (a), V < (E—mc2), in part (b), (E—mc2) < V < (E+mc2), and in part (c), (E+mc2) < V.

The third case, shown in part (c), is Klein’s Paradox. Here, since the relativistic

total energy E > me2 then (E+mc2) < Vimplies that V > 2mc2, k < 0 and thus

R > 1 and T < 0. There are apparently more particles being reflected than

originally striking the potential step and there are a negative number of particles

passing through the potential step (whatever that should mean) so as to

compensate for the excess of reflected particles. This is the paradox.

Shortly after, Sauter [Sauter 31a, Sauter 31b] checked whether the effect was

caused by either the abruptness of the potential step or the sharp corners. In the

first case, he replaced the step potential of Klein, shown again in Figure 2-4{a),

with a straight line:

28

e#(x) =

/ I a V for x < — —

2 2x a a-V for — - < x < + —a 2 2

+ ^ for + < x (2.6)

as shown in Figure 2-4(b). In the second case, shown in Figure 2-4(c), he used a

rounded version of this potential:1 x

e#(x) = - V tanh - (2.7)' ’ 2 a

In both cases he found that Klein’s Paradox reappeared if:V me2“ > — (2-8)a a

that is, if the potential was steep enough that it changed by an amount me2

within the quantum mechanical size of an electron, given by the electron’s

Compton wavelength X = hfmc.

The resolution of the paradox was soon found in the phenomenon of electron-

positron pair creation in the second quantization formalism of quantum field

theory. (Early attempts within the framework of the Dirac single particle theory

include [Sommerfeld 53, Hund 40]. More modern explanations can be found

in [Stueckelberg 41, Feynman 49, Nikishov 69, Nikishov 70, Narozhnyi and

Nikishov 70, Damour 77, Hansen and Ravndal 81]. Indeed, the paradox arises in

the Dirac theory because it is a single-particle theory which can describe either

electrons or positrons, but not their creation and annihilation in pairs. The

quantization of the electromagnetic photon field as well as the of electron in field

theory leads to the necessary pair creation and annihilation. In field theory, the

scenario of electron scattering from a potential of arbitrary strength can be

described in a natural, consistent manner, leaving no hint of the supposed

paradox.

Even though the Dirac theory is inadequate to describe electron scattering in the

29

x/aF ig u re 2-4: Potentials considered for the Klein Paradox.

P art (a) shows the simple step function considered by Klein. Parts (b) and (c) show the two potentials considered by Sauter, as described in the text.

case of arbitrarily large potentials, the resolution found in quantum field theory is

hinted at. Within the Dirac theory it is roughly as follows. The electron coming

from the left, striking the potential step, and scattering back to the left from the

potential larger than 2mc2, stimulates the creation of electron-positron pairs with

their rest mass being supplied by the energy of the potential. The created

30

electrons go back to the left also, as extra electrons, making the reflection

coefficient R greater than one. The excess is balanced by the created positrons

going forward to the right as electrons with negative energy, producing the

negative transmission. The paradox, resolved by the concept of pair production,

was perhaps the first hint of the possibility of positron production in strong

electromagnetic fields.

2.1.2. Dirac Equation for a Point Nucleus

A similar problem appears in the solution of the Dirac Equation for a hydrogen­

like atom with the Coulomb potential of a point nuclear charge Ze, first worked

out by Darwin [Darwin 28] and Gordan [Gordan 28]. The difficulties arise as the

charge Z, and thus the strength of the electric field around the point nucleus, is

increased to the critical point Za = 1.

The Dirac Equation of motion for a single electron in an external

electromagnetic field described by the four-potential A = (#/c,A) was given

above in Equation (2.1). For the case of the static, Coulomb potential:Ze2

A = 0, e* = V[r) = ------- (2.9)rthe equation can be written in the Hamiltonian form:

H * = E * (2.10)

with the Hamiltonian:H = cap — V[r) + 0mc2 (2.11)

The problem can be solved by well known techniques (e.g. [Gordan 28, Darwin

28, Rose 61, Akhiezer and Berestetskii 65, Bethe and Salpeter 77]) which are

reviewed here, since they will be mimicked in the solution of the Dirac Equation

for a finite-sized nucleus and the two-center Dirac Equation below.

Since the Coulomb potential is spherically symmetric, the wave function tf(r)

can be separated into the product of a part dependent on the radius r and a part

dependent on the angular variables 6 and # and spin coordinate <r.

31

* { t ) =

(2.12)

where <7K(r) and f K(r) are the radial parts of the upper and lower bispinor

components of the wave function V, and are spinor spherical

harmonics (Rose 61, Akhiezer and Berestetskii 65]. The two quantum numbers k

and (i are the eigenvalues of the relativistic spin-orbit coupling operator

k — 0{o-\ -f ft) and of the r-component j of the total angular momentum

J = L + S, respectively. The wave functions gK and f satisfy the following

coupled radial equations:

g M - IE + me2 - \ \ r ) ]H r) = 0d K + 17r + ~d K — 1dr

f K(r) + [E — me - V(r)]a (r) = 0 (2.13)

and can be expressed in terms of confluent hypergeometric functions which

depend on the total energy E.

The boundary conditions that the wave functions behave in a physically

reasonable manner at both r = 0 and r —► oo to allow normalization of the wave

function as:

r°°

/ o '( I M * + l«MI2) r2 dr = 1 (2.14)

gives a condition for E which is solved to yield the Sommerfeld Fine-structure

Formula for the energy of the electronic states of a hydrogen-like atom:Za

E — mc~ 1 + f - 1 / 2 (2.15)

- and, n — |k | + Vk 2 — (Za)2/ .

The electronic states are specified by the quantum numbers n = 1,2,3,

k = + l,+ 2 ,+ 3 , • • • + (n—1),—n. For the lowest energy state, the state

n = — k = 1, this formula reduces to:

= mc2\ / l — (Za)2 (2.16)' U 1/2

32

Because of the term — (Za)2, the last two equations break down for

Za > |k|. This is shown in Figure 2-5, which depicts the energy levels for the 6

lowest energy states as a function of the nuclear charge Z. As can be seen, at

Zz=. l/a ~ 137 the slope of the energy curve of the states corresponding to

|k| = 1 (i.e. curves 1, 2, and 4) becomes infinite:i E K— -ft -0 0 as {Za) — M (2.17)

The mathematical breakdown at Z ~ 137 is also present in the behavior of the

Wave functions. The limiting form of the radial wave function close to the point

nuclear charge is, i.e. for r —*• 0, can be written as:/ 2 2

r<Kr) ~ r ~ (Za) (2.18)

For Za > |k| , the wave function starts to oscillate near the origin r = 0 as:

nrir) ~ cos \J{Za)2 - (C2 In r] (2.19)

and is no longer normalizable. As pointed out in [Case 50], this can be traced to

the fact that the Hamiltonian is no longer self-adjoint in this situation.

In the simple case of a point nucleus, the nuclear charge Z = 1/a ~ 137 marks

a transition point in the theory, just as 2mc2 did for one-dimensional

scattering in the Klein Paradox described in the last section.

2.1.3. Dirac Equation for a Finite Nucleus

As seen above, the wave functions and energy eigenvalues in the Dirac theory

for an electron bound to a point nuclear charge become undefined for Za > |*|.

In particular, the lowest energy, 1«^2 state> reaches a minimum total energy of

E — 0 at Z = l/a cz 137 and is undefined for larger Z. The indeterminacy is

removed [Pomeranchuk and Smorodinsky 45] by replacing the singular Coulomb

potential of a point charge with the potential of a charge distribution of finite

extent. The infinity in the potential at the origin caused by confining a finite

ENER

GY

keV

33

6 0 0

4 0 0

200i

j

0

-200

- 4 0 0

“ 6 0 0 0 4 0 8 0 1 2 0 1 6 0 2 0 0

NUCL EAR CHARGE NUMBERFigure 2-5: Electronic energy eigenvalues of the point nucleus.

The energy of the six lowest electronic states of the hydrogen-like atom calculated with the Dirac Equation and the Coulomb potential of a point nucleus are plotted as a function of the nuclear charge Z.

H I I I I I I I I H h H 1 1 1 ^ 1 E = + mc 2 I I J

Curve n I/Cl1 l 12 2 13 2 24 3 15 3 26 3 3

E = - me2

Ij Z= I/a

i i i i i i i i i i i 1 ' ii i i i i i

34

amount of charge to an infinitesimally small region of space disappears with the

assumption of a more realistic nuclear charge distribution of finite extent. For all

the electronic states, the wave functions remain well defined and the energy levels

decrease smoothly and monotonically past Za = |/c| down to E — —me2.

2.1.3.1. Calculations for Z < ZC T

This new behavior can be demonstrated by a simple example, in which the

Coulomb point potential —Ze2/r is replaced within a radius R corresponding to

the radius of the nuclear charge distribution with a finite function:s2

V(r) = ' ' ' — ' - (2 2Q)( ZeL

— /(»•) for r < R

I S*2^------ for R < rCalculations have been done for the two choices /(r) = 1, corresponding to a

spherical shell of charge [Pomeranchuk and Smorodinsky 45, Voronkov and

Kolesnikov 60, Rein 69], and f[r) = - [3 — ( — )2], describing a uniformly charged2 nsphere (Werner and Wheeler 58, Pieper and Greiner 69]. These are compared to

the point Coulomb potential for Z = 137 in Figure 2-6 (reproduced from [Rafelski

et al. 78a]). The nuclear radius R for these calculations is determined by:R = RQ A1/3, RQ = 1.2 fm (2.21)

and A is extrapolated to superheavy elements (Pieper and Greiner 69] as:A(Z) = 0.00733 Z2 + 1.3 Z + 63.6 (2.22)

The technique of solution is the same in both cases. For r>R, the radial

solutions of the Dirac Equation are the same as for a point charge as described in

the last section. For r < R, where the potential is a polynomial:

= E K"r” <2-23)n = 0

solutions can be found in terms of power series in r.

n f r ) = f , ' M = E v +' (2-24>n = 0 n = 0

35

Figure 2-6: Comparison of nuclear Coulomb potentials.The Coulomb potentials for a point charge (solid line), a uniformly charged sphere (dashed line), and a charged spherical shell (dot-dashed line) are plotted as a function of the radial distance r for a nucleus with Z — 137. A nucler radius of 8.7 fm is assumed for the second and third curves. (Reproduced from [Rafelski et al. 78a].)

If the solutions f < and g< for r < R regular at r = 0 and the solutions f > and

g> for r > R which decrease exponentially as r —► oo are taken, the energy

eigenvalues E of the electronic states are fixed by the boundary condition at

r = R:/<(*) />(*) = -------- (2.25)? < ( * ) * > (* ) 1 }

This gives in both cases a transcendental equation in E which can be solved

numerically. An example of this calculation for the lowest energy, 1 s ^ 2 state as a

function of the nuclear charge Z for a spherical charge shell (solid line) and a

uniformly charged sphere (dashed line) is shown in Figure 2-7 (reproduced

from [Rafelski et al. 78a]).

More sophisticated calculations [Fricke and Soff 77] have been done assuming a

two-parameter Fermi distribution for the nuclear charge distribution:

n(r) --------- j — r- (2.26)1 + efr-c)/*

36

Figure 2-7: ls<r energy eigenvalues.The energy eigenvalue of the 1 so electronic state calculated assuming the nucleus to be a charged spherical shell (solid line) or a uniformly charged sphere (dashed line) are plotted as a function of the nuclear charge Z. (Reproduced from [Rafelski et al. 78a].)

Here, a = t/(4ln 3), where t = 2.2 fm is the 10%-to-90% skin thickness, c is the

half density radius, and the overall magnitude pQ is fixed by the requirement that:

Ze = 4*JQ p{t) f2 (2.27)

This charge distribution is compared to a uniformly charged sphere for a uranium

nucleus in Figure 2-8. In addition, electron-electron interactions have been

included by solving the Dirac Equation in the Dirac-Fock-Slater (DFS)

approximation (a relativistic Hartree-Fock formalism with the Slater'

approximation for the electronic exchange term). The resulting energy

eigenvalues are shown in Figure 2-9 (reproduced from [Fricke and Soff 77]).

Several features are worth pointing out. As seen in Figure 2-10(a) (reproduced

from [Pieper and Greiner 69]), where the energy of the lower electronic states is

shown as a function of the nuclear charge Z, the energy of the Is state continues

smoothly past Z = 137. At Z ~ 150 the binding energy of the Is state reaches

me2. At Z = Z^ cc 173 the binding energy reaches 2mc2. The next state,

2Pj/2, reaches the critical binding energy at Z ~ 185. As seen in Figure 2-10(b)

37

x [fm]F ig u re 2-8: Comparison of nuclear charge distributions.

The nuclear charge distribution of U is plotted as a function of the radial distance for the case of a two-parameter Fermi distribution (solid line) and a uniformly charged sphere (dashed line).

(reproduced from [Soff et al. 74]), where the maximum of the radial density of the

Isa wave function is plotted with respect to Z, the wave function of the Isa state

becomes concentrated closer and closer to the nucleus as Z increases. This can

also be seen in Figure 2-11 (reproduced from [Brodsky and Mohr 78], and based

on calculations by [Popov 70c]), where the mean radius °f the ls<7

electronic state is shown as a function of the energy eigenvalue «.

2.1.3.2. C a lcu la tio n s fo r Z > Zc r

A qualitative change in the spectrum occurs at Z = Zcr ~ 173. At this point

the total energy of the lowest, l s ^ i electronic state reaches —me*’, and the

bound level disappears. It has been suggested by [Popov 70b, Muller et al. 72b],

that the state becomes part of the negative energy continuum. A Coulomb phase

shift analysis of the negative energy continuum states [Muller et al. 72b, Muller et

38

Atom ic N u m b e r —

F ig u re 2-9: Electronic energy eigenvalues for 100 < Z < 173.The energy eigenvalues of the inner electrons are plotted as a function of the nuclear charge. The calculation assumed a Fermi distribution for the nuclear charge and a Dirac-Fock-Slater formalism for the electron electron interaction. (Reproduced from [Fricke and Soff 77].)

al. 72c, Muller 76] shows that the bound state enters the continuum as a

resonance. Their numerical calculations show that the energy and wave functions

of the resonant 1 state for Z > Za appear to be a smooth continuation of

those of the subcritical, bound 1 s ^ 2 state.

Figure 2-12 (reproduced from [Muller et al. 72a]) shows the calculated shape of

the resonance in the negative energy continuum for the overcritically bound Is. ,1/ Istate in a superheavy atom with Z = 184. sin2 5 is plotted as a function of the

total electron state energy , where S is the Coulomb phase shift due to the

admixture of the 1 s ^ 2 state into the continuum, and is related to the expansion

amplitude a{E) of the 1 s l j 2 state by:

39

max l? r l2 [fm )

F ig u re 2-10: Energy and radial size of electronic states.P art (a) shows the energy eigenvalues E of the inner bound electronic states as a function of the nuclear charge Z (reproduced from [Pieper and Greiner 69].) P art (b) shows the position of the maximum of the electronic distribution of the ls<r state as a function of the nuclear charge (reproduced from [Soff et al. 74].)

40

E/mc2

Figure 2-11: Mean radius of the Isa state.The mean radius f of the Isa electronic state (in units of the Compton wavelength of the electron) is plotted as a function of the energy e of the state (in units of the rest mass of the electron), as calculated by [Popov 70c]. (Reproduced from [Brodsky and Mohr 78].)

|a(£)l = “7 sin2* (2-28)They report that the shape can be parameterized as a Breit-Wigner resonance:

T2/ 4

(E-Erf + r2/ 4sin2 (2.29)

)2 + r 2

By numerical calculation of the resonance maximum EQ as a function of Z, they

found the following dependence for the total energy of the two lowest states, ls^ 2

and 2pjy2> which both decrease smoothly beyond Z^ as:Eu ^ -me2 - iX(Z-ZJ - ,y.(Z-ZJ- (2.30)

The decay width r of the resonances increase for (Z—Z ) > 3 as:

rul/2 - i X l Z - Z j (2.31)The parameters found for the ls^ 2 and the 2p^2 [Muller et al. 73b[ are given in

Table 2-1.

41

Figure 2-12: ls<r resonance shape.The square of the sine of the Coulomb phase of the ls<r resonant state bound to an oversized nucleus with Z = 184 is plotted as a function of the continuum state energy. (Reproduced from [Muller et al. 72a].)

State 6 r . i(keV) (keV) (keV)

171.5 29.0 0.33 0.041 (N

1 w*cx 185.5 37.8 0.22 0.08

(from [Muller et al. 73b])

Table 2-1: Coulomb Scattering Phase Analysis Results

The spatial distribution of the resonance was studied by defining the charge

density of the resonant l s ^ Par the continuum as:

^ Je - A r {* E ,Z * E ,Z ~ *E , Z ^ * E , z J d E (2 32)where Er is the energy of the resonance determined from the phase shift analysis,

and Ar is a characteristic width of the resonance. Again, numerical calculations

show that the charge density of the ls ^ 2 state f°r Z > Z^ looks very much like

that for Z < Z , and even continues to contract with increasing Z as for

Z < Z , as shown in Figure 2-13 (reproduced from [Muller et al. 72a]).

42

Figure 2-13: Charge density distribution of the Isa state.The density is plotted as a function of the radial distance for a undercritical bound state at Z = 169 in part (a), and for the overcritical resonant state at Z — 184 in part (b). (Reproduced from [Muller et at. 72a].)

2.1.3.3. Projection Operator Method

For Z > Z , the presence of the l*j^2 resonance in the negative energy

continuum leads to difficulties in the numerical calculation of matrix elements

involving the continuum states. The similarity of the spatial charge distribution

in the under- and overcritical fields, seen above, has suggested an adaption of the

projection operator method to remove the resonance from the

continuum [Reinhardt 79, Reinhardt et al. 80a, Reinhardt et al. 80c, Reinhardt et

al. 81a, Reinhardt et al. 81b]. The negative energy continuum wave function

l* E r> at the energy Ef of the resonance looks like the undercritically bound 1 s^ 2

wave function |#. ) except for a small, oscillating tail at large r which indicates1 / 2

the possibility of tunneling through the classically forbidden gap between

\E—V(r)] + me2 and \E— V(r)] — me2. Both the ls<r resonance wave function and

the potential gap are displayed for Z — 184 in Figure 2-14 (reproduced

43

from [Schwalm 83], based on [Reinhardt et al. 81a]) in parts (a) and (b),

respectively. Damping out this tail by solving the Dirac Equation with a modified

potential V\r) , which is held constant outside (approximately) the classical outer

turning point, they obtain a resonance wave function, |#^), as a smooth

continuation of the undercritical, bound state.

F ig u re 2-14: Electron density and potential well for Z = 184.P art (b) shows the electrostatic potential gap as a function of the radial distance for a U+U quasimolecule with an internuclear separation of 16 fm. P art (a) shows the resulting Isa resonance wave function. (Reproduced from [Schwalm 83], adapted from [Reinhardt et al. 81a].)

The negative energy continuum states \$E ) satisfy the Dirac Equation:

( H - E _ ) \ * e ) = 0 (2.33)

If the resonance state |#^) is normalized:

44

(*R \*R) = 1 (2 -3 4 )

and is orthagonal to the bound states |$n) and the positive energy continuum

states |#£ ):

= 0 = ( * £ + l * f i ) (2 .3 5 )

then a new negative energy continuum \&E ) orthagonal to the resonance state

1 )•= 0 <2 36>

can be defined by the projection operators:

Q = J r f £ + l * £ + X * E + l + £ l * „ X M + l * * X * * ln - -

P = l - Q = J dE_\fB KfE_\ ( 2 .3 7 )

The operator Q projects out the positive energy continuum, the bound states, and

the resonance. P projects out the modified negative energy continuum states

which exclude the resonance. The new negative energy continuum obeys a

modified Dirac Equation:(PHP-EJ \*>E ) = 0 (2 .3 8 )

which can be rewriten in the form:( f f - E J = (#f l | t f K £ _ ) | * R ) (2 .3 9 )

The modified continuum states, freed of the singular resonance, are more suitable

for the numerical calculations made to solve the Dirac Equation for realistic

nuclear charge distributions.

In addition to facilitating calculations, the projection operator method also helps

identify the spontaneous positron production component of the theory for

Z > Z . Because of the inhomogenous term on the right of the Dirac Equation

above for the modified continuum states, and since neither the resonance |0 ) nor

the modified continuum states \<&E ) are eigenstates of the Hamiltonian H, the

coupling matrix elements between the 1 ^ / 2 s a e aQd the negative energy

continuum acquire a second component for Z > Z :

45 \

Even in the static case, there remains a coupling between the l s ^ resonance and

the positron continuum. A vacancy in the resonance state will decay

exponentially in time with a decay width depending on the strength of the

coupling:

r = j \(^e _\h \*r )\2 (2.4i)

The transfer of a hole from the resonance state to the negative energy, positron

continuum appears as the creation of an electron-positron pair. The electron is

bound to the nucleus in the resonance state and the positron escapes in a

continuum state. The vacant, overcritically bound electronic state is filled,

accompanied by spontaneous positron emission.

2.1.3.4. Effective Potential Method

A complimentary view of spontaneous positron emission is offered by the

effective potential method of Popov [Popov 70c, Popov 70a, Popov

71b, Zel’dovich and Popov 71]. The system of coupled differential equations for

the two radial components of the Dirac bispinor, Equation 2.20 above, is reduced

by the elimination of the lower component J{r) to yield a single, second order

differential equation for the upper component g(r) = G (r)/r.

G" + j ^ =y \ G ' + ^ G \ + UE - v f - l - ^ l \ G = 0 (2.42)

The substition:

G(r) = J l + E - \\r) #(r) (2.43)

reduces this equation to the form of the Schroedinger Equation:

+ k2* = 0 (2.44)

a s Z — Z ^ (2.40)

with:£ & _ j

k2 = 2(t — U), e = — — (2.45)

and the effective potential:

46

2 *(*+1) U = \ E V — — +

i r 1V " 3 V ' 0 2 kV ' ,

+ » ( 7 T S r T , ) - r T T T T T T T (2 -4 6 )1 + E -Y 2 v l + £ - V ' r ( l + E - V )(The first term is the effective potential of the Klein Gordan Equation and the

second term contains the additional spin effects of the Dirac Equation.) For the

l$ l/2 state and Z = Z^ , then #c = —1 and E = —me2 and the effective potential

reduces to:

1 o ti2—u’ u .V = j [ l - ( I + K ) - ] + [ — + - ] (2.47)

with u = V /2 V . The first term dominates. For small r, the potential is

attractive:

V 2U[r) — as r —► 0 (2.48)

2For large r (with \ \r) < 0 and E < 0), the potential is repulsive:

U[r) — E V = + \ E V \ as r — oo (2.49)

In between is a Coulomb barrier. The Dirac potential V(r) and the corresponding

effective potential Ulr) for the critical case Z = Z , E. = —me2, are sketchedcr i s 1f 2

in Figure 2-15 (reproduced from [Zel’dovich and Popov 71]).

In the interpretation of Popov and co-workers, the existence of the Coulomb

barrier in the effective potential U{r) has the following consequences. If a filled

l s ^ state becomes critically bound, the Coulomb barrier ensures that the charge

distribution of the quasistationary 1 s l j 2 state that forms does not become

delocalized at Z = Z (as verified in the Coulomb phase shift analysis described

above). If the state has a vacancy, it can be filled when an electron-positron pair

is created near the nucleus (where the potential |C/(r)| > 2mc2) and the positron

tunnels through the Coulomb barrier to escape. At Z = Z ' , E = —me2 so that

t = 0 and the effective barrier is infinitely long. Spontaneous positron emission «

only sets in for Z > Z .

The probability for tunneling through the barrier provides an estimate of the

spontaneous positron rate for 0 < (Z—Z^) Z ^ [Zel’dovich and Popov 71]:

47

r

F ig u re 2-15: Coulomb and effective potential at Z = Z^.The Coulomb potential V in the Dirac problem and the effective potential U in the corresponding Schroedinger problem are plotted as a function of the radial distance r at Z — Z . (Reproduced from [Zel’dovich and Popov 71].)

The function 6 ( /cr) is determined numerically, with 6 ~ 1.73 for Z ^ = 170.

By means of the Coulomb phase shift analysis, the projection operator method,

or the effective potential techniques described above, the energy eigenvalues and

wave functions of the electronic states in superheavy atoms can be continued past

from [Muller 76]). The energy eigenvalues E ' of the inner electronic states, as

as a function of the nuclear charge Z of a superheavy atom for 0 < Z < 250.

(2.51)

the critical binding energy 2mc2. This is depicted in Figure 2-16 (reproduced

bound states for E > —me2 and as resonant states for E < —me2, are plotted

48

F ig u re 2-16: Electronic energy eigenvalues for 0 < Z < 250.The energy eigenvalues of the inner electronic states are plotted as a function of the nuclear charge. The solid lines indicate the region of known elements and the dotted lines, the calculated extension to superheavy atoms. For E < —me2, the energy of the resonant state is given. (Reproduced from [Muller 76].)

As a final note, the connection between the anomolous scattering behavior of

Klein’s Paradox and spontaneous positron production in an overcritical Coulomb

potential can be made. Both can be viewed as the mixing of particle and

antiparticle states by a strong electrostatic potential. The two cases are shown

schematically in Figure 2-17 (reproduced from [Rafelski et al. 78a]). P art (a)

depicts a potential step, like Sauter’s smoothed potential for Klein’s Paradox in

Figure 2-4(c) above. P art (b) shows a finite range potential, like the Coulomb

potential of a finite nuclear charge distribution, as already seen in Figure 2-14. In

both cases, positron production is possible when the potential is strong enough to

bring positive and negative energy continuum states to the same energy level, and

steep enough to allow quantum tunnelling through the classically forbidden gap

between the two continua (cf. Sauter’s condition for the Klein Paradox, Equation

(2.8) above).

49

P A R TIC LE CON TINUUM

E'

m *V e T P AR TICLE C O N TINU UM'o

(a )

A N TIP A R TIC LE

Q

CO N TIN U UM

(b )A N TIP A S TIC L E c o n t i n u u m 2

z

F ig u re 2-17: Mixing of e" and e+ states in a strong potential.The mixing of particle and antiparticle states is shown schematically for the case of a strong electrostatic potential step in part (a) and for a finite range potential in part (b). (Reproduced from [Rafelski et al. 78a].)

2 .1.4. T w o -C en te r-D irac E q u a tio n

Since the nuclear charge necessary for overcritical binding of electronic states is

to be assembled in collisions of heavy ions with atoms, the solutions of the Dirac

Equation with two charge centers are ultimately required. The solutions of the

Two-Center-Dirac (TCD) Equation provide a qualitative description in the

adiabatic limit of the evolution of the electronic energy levels and wave functions

during the collision, indicating the maximum internuclear separation R at which

critical binding occurs. In addition, as described in Section 2.2.1 below, the wave

functions determined as a function of the separation R between the two nuclei

will serve as basis states for the full solution of the dynamic positron production

during a heavy-ion - atom collision.

The TCD Equation is the same as Equations (2.10) and (2.11) above, except that

the potential V(r) is:(2.52)

50

v y ,,R ) = - V / | r _ ^ - A , ,2.53)

R is the separation between the two nuclei, r is the position of the electron with

respect to the center of mass of the two nuclei, r i is the position of the element of

nuclear charge within the i th nucleus with respect to the center of the nucleus,

and p / r {) is the charge distribution of the t th nucleus. In addition, /ij =

and n2 — —p / M 2, where p is the reduced mass of the two nuclei, and Af. is the

mass of the t4*1 nucleus.

In contrast to the Dirac Equation for one (spherically symmetric) nucleus

discussed in the last two sections, and even the non-relativistic two-center

problem, no known methods exist to separate the solution of the TCD Equation in

an orthogonal coordinate system, since the specification of two charge centers

breaks the spherical symmetry. The total angular momentum operator J 2 does

not commute with the Hamiltonian H TCD, and no other operator (as opposed to

the non-relativistic case) has been found to take the place of J 2.

If the two charge distributions V ^r^R ) and V^rg.R) are both spherically

symmetric, then the Hamiltonian H TCD is cylindrically symmetric with respect to

the axis through the centers of the two charge distributions. In this case, the

component m z of the total angular momentum with respect to this axis is still a

good quantum number, and the dependence on the azimuthal angle # can be

separated from the other two variables. m z is found to take the values

+ 1 /2 , +3/2 , +5/2 , • • • , where the energy eigenvalues for the two signs are

degenerate. The corresponding states are designated by the Greek leters

<r, ?r, S, • • • , to mimic the atomic spectral designations a,p, d, • • • . The

identification of the solutions of the TCD Equation is completed by specifying the

quantum numbers of the united atom states they become in the limit as the

51

intemuclear separation R —* 0. Thus the lowest energy states are designated

^ 8 l / 2 f f’ ^1/2^’ 2si/2<t’ ‘

2.1.4.1. Multipole Expansion of the TCD Equation

Early approaches to solve the TCD Equation for Za > 1 include a

diagonalization method from the Frankfurt school [Muller et al. 73a, Muller

74, Muller and Greiner 76] and approximation techniques and variational

calculations by the Moscow school [Popov 71b, Popov and Rozhdestvenskaya

71, Popov 72b, Popov 73b] and [Marinov and Popov 73, Marinov and Popov

74b, Marinov and Popov 74a, Marinov et al. 74, Marinov and Popov 75a]. The

most recent calculations, better suited for numerical evaluation, are based on

multipole expansions of the radial functions and the two-center potential [Rafelski

and Muller 76a, Rafelski and Muller 76b].

In the last method, the solution Hr) is assumed to be expanded in spinor

spherical harmonics x^{9,$,ci):

*<r> = E *«|r| = EK , m

i ' '/:> ) x : > , m j (2 .5 4 )to mimic Equation (2.12) above for the one-nucleus problem. A multipole

expansion is also made of the two-center potential:

VTC(r,R) = £ v ” (r,R) y ” (e,#) (2.55)l , m

where Yjm(©,#) are the spherical harmonics and V™(r,R) are determined by:

V™{r,R) = J * sin 9 de J**d* Vtr,R) Y /V ,# ) (2.56)

With these two expansions, the Dirac Equation yields the following doubly infinite

set of coupled differential equations (cf. Equation (2.13) above):

52

+ ~ ^ E + mc2^ K + 23 Am( - K’l - K') vi f K> = °

\d —

Lir r .

id,I

f K + [E — mc2\gK - ^ Am(+/e,/,+*') VJ ^ = 0id,I

(2.57)

where the coefficients:

A m(KM = (x> /" |x™ > (2.58)

are determined entirely by angular momentum algebra. The sums over k and /

are truncated at a sufficiently large total angular momentum j max- This reduces

the set to 2(,;' +1) coupled differential equations, which have been solved by

numerical integration [Rafelski and Muller 76a, Rafelski and Muller 76b].

Electron-electron interactions can be included by the Dirac-Fock-Slater (DFS)

approximation, as for the one-nucleus problem discussed above. The finite size of

the nucleus is generally accounted for by assuming two uniformly charged spheres,

in which case the two-center potential can be written [Kirsch et al. 81] as:

Vrc(r,R) = Ir+XgRe

3Z2e

2R,1 -

Ir-Xj/feJ

for

—Z2e2r —XjJfeJ < R 2

2 Ir+XgRe3Zje

|r—XjReJ 2 R n1 -

for |

-Zje2p + XjJ&J < R x

otherwise- Z 2e2

^ I r + X ^ e j Ir -X jJ fc J ......................... (2‘59)with X . = M i/(M1 + M 2). Furthermore, investigations [Soff et al. 78a, Soff et

Aal. 79] have shown that for R < 10 fm, the eigenvalues and the eigenfunctions of

the TCD Equation can be calculated to within several percent with just the

monopole term of this potential:

53

7 Ze2 R v 0(r,R) = - — for ( - + R n) < r

for i - - R ) < r < ( - + R ) ' o n — — 2 n

(2.60)

(where here R^ = R^ — R n is the nuclear radius and = Z2 = Z the nuclear

charge). The monopole term is compared to the full potential in Figure 2-18

(reproduced from [Rafelski et al. 72]) for the Pb+ Pb system for an internuclear

separation of 200 fm.

The last point has important consequences, since no techniques exist as

yet [Reinhardt et al. 81a] (and cf. [Schliiter et al. 83a, Wietschorke et al. 83]) to

extend the solution of the TCD Equation to the continuum states except with the

monopole approximation. The monopole term of the two-center potential is

spherically symmetric, and solutions can be found for the continuum states with

the techniques described in the last section.

2.1.4.2. Results of the Calculations

The following figures illustrate the results of the calculations made for the TCD

Equation. Figure 2-19 shows energy level diagrams calculated for the U+Cm,

U+U, and U +Pb systems, (reproduced from [Soff et al. 82a, Muller 76, Soff et al.

78b], respectively). The binding energy of several of the lower states is plotted as

a function of the internuclear separation R.

Figure 2-20 reproduced from [de Reus et al. 83]) typifies calculations of the wave

54

rifm )

F ig u re 2-18: Two-center Coulomb potential.The two-center Coulomb potential (solid line) and the monopole part (dashed line) for two lead nuclei separated by 200 fm is plotted as a function of the radial distance r. (Reproduced from [Rafelski et al. 72].)

functions. In parts (a) and (b), the calculated radial density |r0j2 of several

states is plotted as a function of the radial distance for the U+Cm quasimolecule

with an internuclear separation of 43 fm. Calculations with (solid lines) and

without (dashed lines) the inclusion of electron electron interactions in a DFS

approximation are presented. Typical matrix elements calculated for the same

collision system are presented in Figure 2-21 (also reproduced from [de Reus et al.

83]). P art (a) shows the spontaneous coupling between the ls<r resonance and the

modified continuum states for two internuclear separations as a function of the

energy of the continuum state. P art (b) shows the radial coupling between the

same two states as a function of the internuclear separation R for several energies

of the continuum states (indicated in units of me2).

55

F ig u re 2-19: Quasimolecular state correlation diagrams.The energy eigenvalues E of the inner electronic states are plotted as a function of the internuclear separation R for the quasimolecules U+Cm (reproduced from [Soff et al. 82a]) in part (a), U+U (reproduced from [Muller 76]) in part (b), and U+Pb (reproduced from [Soff et al. 78b]) in part (c).

56

■nr35

4

3

2

1

0

V f e ) A

' T »

U + Cmr \ R -4 3 fm

1 \ ----------- HFS '

\ ..................Coul.

/ 2sdt \

-

- ( 3 ) J A )i \/ / 3 s o - \ /

i — ^

10° 101 102 103 r[fm ]

1.0

■ W 2J U-CmR-43fm , x —1 , E^-I.IfiieC2

HFS Coul.

F ig u re 2-20: Quasimolecular wave functions.Electronic densities for the quasimolecule U+Cm with an internuclear separation of 43 fm are plotted as a function of the radial distance r for several ns<r bound states in parts (a) and (b) and for continuum states at E + = 1.1 me2 in part (c) and E + = 3.0 me2 in part (d). (Reproduced€ cfrom [de Reus et al. 83].)

The systematic behavior of the most tightly bound, l s ^ * state is diagrammed

in Figure 2-22 (reproduced from [Soff et al. 83]). In part (a), the energy E lsa of

the 1 level is displayed as a function of the internuclear separation R for

several combined nuclear charges (Zp + Zt). In part (b), the average radius {rl3(T)

of the stat e is shown as a function of R for the same (Zp + Zt).

57

F ig u re 2-21: Quasimolecular matrix elements.In part (a) the interaction matrix element between the Isa resonance and the modified continuum is shown as a function of the energy E of the continuum state for the U+Cm quasimolecule at two different internuclear separations R. P art (b) shows the radial coupling matrix element between the same two states as a function of the internuclear separation for several continuum states. (Reproduced from [de Reus et a l 83].)

In Figure 2-23 (reproduced from [Reinhardt et a l 81b]), the results of calculation

also using the DFS approximation but assuming an electronic ionization state of

50+ (as could be expected in heavy-ion collisions) for the three quasimolecular

systems U+Cm, U+U, and U +Th are shown. Part (a) depicts the energy and

part (b) the decay width of the resonance as a function of the internuclear

separation R.

Finally, the separation R ^ at which the state becomes critically bound,

58

F ig u re 2-22: Quasimolecular Isa energy and mean radius.P art (a) shows the energy E and par t (b ) the average radius (r) of the Isa electronic state as a function of the internuclear separation R for various combined nuclear charges, calculated from the Two-Center-Dirac Equation. (Reproduced from [Soff et al. 83].)

calculated for uniformly charged spheres using the DFS approximation, is showm

in Figure 2-24 (reproduced from [Wietschorke et al. 79]) as a function of

(Zp + Zt), assuming Zp = Zf.

2 .2 . D y n a m ic P o s i t r o n P r o d u c t io n

Following the introduction to spontaneous positron emission as a static process

above,, the theory of dynamic positron production in heavy-ion collisions will be

briefly described in this section. In addition to providing the nuclear charge

concentration necessary for the study of spontaneous positron production, the

dynamic collision system also produces many new elements not found in the static

case of a superheavy atom described in the last section. From the standpoint of

the search for spontaneous positron production, some of these elements are

welcome, while others are not so.

On the positive side, the dynamic nature of the collision serves to create

59

0 5 t) 15 20 25 30 Rllm]

400

600

'1--------7-------u - c .

:— Ipvj

/

' u-CmIsa Resonance Energy

SCF-Calculation

■ (a) 50* Ions

Figure 2-23: Isa state resonance energy and decay width.The energy E—2mc2 (part (a)) and the decay width r (part (b)) of the Isa resontant state of three selected quasimolecules (U+Cm, U+U, and U+Th) is plotted as a function of the internuclear separation R. (Reproduced from [Reinhardt et al. 81b].)

vacancies in the most tightly bound electronic states. This happens despite the

tight binding because the wave functions of these states are strongly contracted

by the large combined nuclear charge and overlap with the moving nuclei. The

overlap allows the relatively efficient transfer of nuclear kinetic energy to ionize

these inner orbitals. The two effects of increased binding energy and contracting

wave functions with increasing nuclear charge compete to determine how easily

the inner electrons can be ionized. Around the critical nuclear charge Z , the

shrinking of the wavefunctions wins out and the inner electrons are ejected in

about 5% to 10% of the collisions. As pointed out in Chapter 1 , just such a

vacancy is required for spontaneous positron production.

Other new elements which are not so beneficial for the experimental

determination of spontaneous positron production include the following. First is

60

zp +ZT

Figure 2-24: Isa state critical nuclear separation.The internuclear separation R at which the Isa state becomes criticallybound to two nuclear charge centers is plotted as a function of thecombined charge (Z + Z.) of the two nuclei for several electronic

P fionization states d. (Reproduced from [Wietschorke et al. 79].)

simply the short time of the collision. If heavy-ion collisions below the Coulomb

barrier are used to avoid the large backgrounds produced by nuclear interactions,

then the time that the ls^ 2a electronic state remains overcritically bound is

typically calculated to be ~ 10~21 seconds (see Figures 5.1(b) and 5.2(b)). The

calculated lifetime of an overcritically bound 1 slj2<r vacancy, however, as seen in

the last section (cf. Figure 2-23(b)), is typically ~ 10“ 19 seconds, or about a

factor of 100 larger. Only about 1% of the overcritically bound vacancies would

have time to decay by spontaneous positron emission during the time that the

electronic state is predicted to be overcritically bound.

61

The second problem is caused by the fact that the colliding nuclei are moving

charge centers. Even below the Coulomb barrier, because the electrostatic

repulsion is a long range interaction, the changing joint electromagnetic field can

induce electron-positron pair production. The electron may be captured into a

vacant state or escape into a positive-energy continuum state. The positron

escapes, as a major background to the detection of spontaneous positron

production.

The positron production processes occuring in heavy-ion collisions are shown

schematically in Figure 2-25 (reproduced from [Rafelski 78]). The energy level of

the most tightly bound state is plotted as a function of time during the collision.

Part (a) shows the direct production of electron-positron pairs and part (b) the

induced production of positrons. As can be seen, these two dynamic processes

occur independently of critical binding. Part (c) depicts spontaneous positron

production from a vacant, critically bound electronic state. In the last two parts,

the formation of the vacancy necessary for the process is also indicated.

This section will describe the production during heavy-ion collisions of atomic

positrons, i.e. positrons produced by the changing joint electromagnetic field of

both nuclei passing by each other, as opposed to the weaker fields of the

individual separated nuclei. This will be done for two cases. First the production

of positrons from two nuclei moving past each other without touching on

Rutherford trajectories will be described. This is appropriate for our experiment

because, based on systematics of heavy ion-atom collisions, the collision systems

described in this thesis are nominally below the Coulomb barrier. It appears,

however, from our data that in some small part ( ~ 10 ) of the collisions,

nuclear interactions may be distorting the normal collision trajectories, causing

the two nuclei to stick together for a short time ( ~ 10-2° seconds) before

separating and proceeding on nearly Rutherford trajectories again. Since a time

62

Figure 2-25: Atomic sources of positrons iu HI collisions.Schematic diagrams of the production of positrons during heavy-ion collisions are shown for direct pair production in part (a), induced positron production in part (b), and spontaneous positron production in part (c). (Reproduced from [Rafelski 78].)

delay could enhance the spontaneous positron production of positrons by

increasing the collision time so that it becomes more comparable to the lifetime of

a critically bound vacancy, the probability of the production of positrons in

collision systems with a time delay due to nuclear interactions will be described in

the second subsection.

63

2.2.1. For Rutherford Trajectories

As stated in the introduction to this chapter, the full field theoretical calculation

of the positron production during the collision of a heavy ion and a heavy atom is

a formidable computational task which has not yet been completed. The adoption

of approximations, however, has allowed a calculation of positron production

which will be seen to show reasonable agreement with the general features of our

measured positron distributions for collision energies below the Coulomb barrier.

The most extensive calculations to date have been carried out by W. Greiner and

co-workers at the Universitat Frankfurt, and the description in this section follows

their methods, primarily as reported in [Reinhardt et al. 81a, Reinhardt et al. 81b, de Reus et al. 83].

2.2.1.1. Description of the Scattering Nuclei

The first major assumption made is that for heavy-ion collisions below the

Coulomb barrier, the motion of the nuclei during the collision can be described by

nonrelativistic, classical mechanics. The first part of this assumption can be

justified by a calculation of the projectile velocity for heavy-ion collisions at the

Coulomb barrier [Popov 71b]. Equating the minimum distance of closest

approach 2a during Rutherford scattering:A + A t Z Z /

2a = -------------- :— , e2 ~ 1.44 MeV-fm (2.61)A n A . E / A

p t p 1 pwith the internuclear separation r-nt at which the nuclei begin to interact:

<4/3 + 4 /3> <2-62>where R{nt ~ 1.34 fm for both 80 < Zt,Zp < 100 [Wilcke et al. 80], gives an

equation for the projectile energy per nucleon (£pMp)in*:Z Z . A + A .

p t A lp/3 + A j /32

En = ----- ~ 1.07 MeV/amu (2.63)R i n t

64

The projectile velocity u^which just brings the nuclei to the interaction

separation is then:

v i n t y / ( 2 + t ) f / — = ----------- « v2 £

C 1 + £

' - K i t { 2 M )p i p

where M /A ~ 931.5 MeV/amu. For a typical heavy-ion collision system, 238Up i p

+ 238U, this yields (Ep/Ap)int cz 6.2 MeV/amu and vint ~ 0.1 1 c. Since

relativistic corrections in general enter as a factor 7 = [1 — (t//c)2] ^ 2, the

projectile velocity of 0.1 1 c would lead to only 1% corrections, which can be

ignored.

The second part of the assumption, that the nuclear motion can be treated by

classical as opposed to quantum mechanics, follows from the largeness of the

Sommerfeld parameter 7 [Eisenberg and Greiner 70, Muller et at. 72a] for heavy-

ion collisions at the Coulomb barrier. <7 is the ratio of half the distance of closest

approach a to the reduced Compton wavelength X of the projectile nucleus: a

„ = - = — (2.65)

and thus is a measure of the ratio of the classical to quantum length scales

associated with the collision. For the typical U+U case considered above,

r\ ~ 5.4X102. q 2 > 1, as is here the case, signifies that quantum effects can be

ignored.

Two additional observations combine with the approximations just discussed to

completely decouple the nuclear motion from the electronic motion. First, the

recoil velocity of the combined nuclei is small compared to the average velocity of

the electrons (cf. [Muller et al. 72c]. This can be computed by combining the

conservation of momentum in the center of mass for the system of two nuclei and

an electron:

65

recm N VN m e - V (2.66)

J \ - ( v ^ / c f

with a determination of the average electron velocity (v ) based on the virial

theorem (i.e., pp. 69-71 of [Goldstein 50]):

/ m e- c 2 \B=(T) = ( ■ ................... - m^c2 \ (2.67)

V l ~ ( v eJ c )2where B is the binding energy of the system and () indicates a time average, to

obtain the following equations for {v /vg_) (and {vg_/c)):

M =

recVN M 771 e—

V 7 l - (1 - N p ) ( V 'J c ) 2 m N

Ve— V B ' { B ' + 2) B = ----7T,------- B = ------------- (2.68)c B'+ 1 ’ m c2e—

For the U+U system and B — 2mc2, this yields {vN/vg_) ~ 3.5 X10-6 . Nuclear

recoil and the resultant retardation effects (cf. [Kolb et al. 78]) can be ignored for

the strongly bound levels.

In addition, the energy scale for induced positron production, which is on the

order of 2mc2 & 1 MeV, is small compared to the kinetic energy of the projectile,

which in the example above is ~ 1.5X10 MeV. In this case, the loss of nuclear

kinetic energy to the electron-positron field during the collision can be

ignored [Popov 71b, Miiller et al. 72a]. Together these assumptions imply that

the positive charge distribution provided by the colliding nuclei can to good

approximation be treated as a given, external potential, decoupled from the

electronic motion.

Below the Coulomb barrier, the nuclear charge distribution is given by two

charged spheres (generally approximated as uniformly charged spheres) with

66

charges Zpe and Zfe, respectively, moving past each other on the classically

calculated hyperbolic trajectories of Coulomb scattering. Equations for the

trajectories are given in Appendix A. Numerical inversion of the transcendental

relation between t and R (cf. Equation 5.5) gives the separation R between the

two nuclei as a function of time.

2.2.I.2. Description of the Electronic States

With the prescription of an external potential, the equation governing the

relativistic states of electrons and positrons during the collision is the time

dependent two-center-Dirac Equation for the state labelled i:

= (2.60)

The two-center-Dirac Hamiltonian is given by:

HTCD(R{t)) = - ihca-V- VT(J[R[t)) + pmc2 (2.70)

where the electric potential V^c is the Coulomb potential of two charged spheres

separated by a distance R(t), as given in Section 2.1.4 in Equations (2.52) and

(2.53). The magnetic interaction between the electron and the current of moving

nuclear charge has been ignored because the effective magnetic coupling constant,

(t//c)ZjZ2e2, is an order of magnitude smaller than the electrostatic coupling

constant Z Z e2 (cf. [Rafelski and Muller 76a, Soff et al. 81a, Soff 82j).

The second major assumption made is that the relative nuclear motion is slow

enough compared to the average velocity of the most strongly bound electrons so

that these electronic levels continually adjust adiabatically during the collision to

the changing nuclear charge configuration [Gershtein and Zel’dovich 69a, Rafelski

et al. 71, Popov 71b, Theis et al. 79], and see footnote 6 in [Muller et al. 73a]).

The adiabatic approximation is justified by comparing the nuclear and electronic

velocities. As seen in the U+U system considered above, the relative velocity of

the colliding nuclei is typically v ^ ~ 0.11c. The average velocity of the most

tightly bound electrons in the center of mass system of electron and nuclei was

given above in Equation (2.68) based on the virial theorem. For the same U+U

system, it varies from {v ) cl: 0.58c for the separated U atoms with binding

energy E. cl: 116 keV to (u ) ~ 0.94c for the united atom where 1/2 e_

B lg g « 2mc2. Thus vg_ fv N varies from about 5 to 9 during the collision. The 1/2

nearly order of magnitude difference between the electronic motion and the

relative nuclear motion should guarantee to good approximation that at least the

most tightly bound electronic levels develop adiabatically during the collision.

The adiabatic approximation implies that the electronic states of the collision

system approximately follow the solutions of the static TCD Equation solved for

each internuclear separation R(t). To take advantage of this, the wave function

tJ[t) which solves the time-dependent Dirac Equation (Equation (2.69) above), is

expanded in the complete set of quasimolecular basis functions:

*,(') = * £ a i k m *o> i2-71'

^ indicates a summation over bound states and an integration over positive and

negative energy continuum states. These #f{R(t)) are the solutions to the static

two-center Dirac equation described in Section 2.1.4. As indicated, the static,

quasimolecular basis depends on t through the time dependence of the nuclear

trajectory R(t). The phase factor x^(0 is chosen to be:

x k w = i (2-72>to conveniently eliminate off diagonal matrix elements in the following

calculations. Electron-electron interactions are accounted for in that the basis

wavefunctions were found using a Dirac-Fock-Slater formalism.

The insertion of this expansion for $. back into the Dirac Equation gives the

following infinite set of coupled differential equations:d , . d * ■/- - v

68

These are solved for the expansion, or occupation, amplitudes aik, using the basis

wave functions $k described in Section 2.1.4 above.

The appropriate boundary conditions for this equation were taken to be the

following. The two colliding atoms start out with normal electron states, so

aik(—o°) = Sik. Because of the use of the quasimolecular basis, which is a good

representation of the two atom system only when they are close together, the

basis wave functions #. were corrected by translation factors beyond R > 1000 fm

to bring them in line with the separated atom wavefunctions (cf. [Heinz et al. 81]).

For the same reason, the matrix elements aik all are given a smooth cutoff at

about R £& 2000 fm. Multiple collisions of the beam atom with more than one

target atom are ignored.

At R = R ^ , the l s l j 2 bound electron state joins the negative energy continuum

as a resonance. As explained above in Section 2.1.3, this added resonance

behavior of the continuum wavefunctions make the numerical calculation of

expansion amplitudes involving continuum states near the resonant energy very

difficult. For this reason the projection method described in Section 2.1.3 is used

to modify the basis states for R < R .

The coupled Equations (2.73) can be rewritten by a formal integration in the

form:

a it(/=°o) = r y y % (o (o jJ ;+ j >

j f ^ k

(2.74)

and solved by numerical integration in a truncated basis. For the study of

positron production in heavy-ion collisions, the basis has generally been taken to

include the lower energy bound states, the higher energy states in the negative

energy continuum, and the lower energy states in the positive energy continuum.

These states contribute most strongly to the positron production.

69

A third important approximation must be made in order to calculate the matrix

elements coupling to the continuum states. Since, as pointed out in Section 2.1.4

above, solutions of the Dirac Equation for the continuum states are only known

for spherically symmetric potentials, the basis functions t-(R(t)) of the continuum

are found using the spherically symmetric part of the two-center potential

VjJ^R{t)). This is the monopole (/ = 0) term of the multipole expansion given in

Equation (2.55). As mentioned, though, the monopole term reproduces the

eigenvalues and eigenfunctions of the bound states to an accuracy of several

percent in the region R < 10 fm, where essentially all the positron production

occurs. For the sake of consistency, the monopole approximation is then used for

all of the basis states and in the dynamic Hamiltonian HTCD{R(t)).

The partial derivative with respect to time, which appears in the matrix

elements, can be split up into a radial and an angular part:

The radial term dominates the time dependence for positron production [Soff et

al. 79]. The lower bound states, which contribute most strongly to the positron

production, are o-states with their angular momentum aligned along the

internuclear axis, and thus do not couple rotationally because of angular

momentum selection rules. Furthermore, the rotational couplings in general scale

with the moment arm R and decrease for the small-i? part of the collision, where

most of the positron production occurs. Finally, rotational couplings disappear

completely in the spherically symmetric case of the monopole approximation. For

these reasons, only the dominant radial term was used to calculate the matrix

elements.

70

2.2.1.3. Spontaneous Component of the Positron Production

Equation (2.74) shows the role that spontaneous positron emission plays in the

positron production during heavy-ion collisions. In the modified basis for

-R < R ^ , the extracted resonance state couples to the modified negative

energy continuum through the matrix element:

The first term is the normal radial part of the time dependence (where the

smaller angular part has, as noted above, been ignored), and is the same form as

for all other matrix elements. The second term is, however, unique to this matrix

element. For other matrix elements, a term such as this would identically vanish,

since the state on the right would be an eigenstate of the Hamiltonian and

orthogonal to the state on the left. It remains here because the normal basis was

modified to extract the l« 1y2 resonance from the negative energy continuum.

Thus neither the resonance nor the new positron continuum are

eigenstates of the Hamiltonian This term is independent of the nuclear

motion R and represents the spontaneous decay of the l s ^ 2 resonance. In the

static limit of completely adiabatic nuclear motion, so that R —* 0 and the first

term in Equation (2.76) does not cause couplings first, a vacancy in the state

would decay with an exponential time dependence and a decay width r given by:

r = f i I2-77)

In a heavy-ion collision, however, the first term, d/dt = R (d /dR ), does not

vanish. It leads to the dynamically induced positron production which actually

dominates the positron emission in these collisions. Furthermore, the spontaneous

and the dynamic positron production add coherently. As a result, there is no

dramatic threshhold effect in the positron production as R passes through R ^ .

Figure 2-26 (reproduced from (Reinhardt et al. 81a|) compares the time

71

dependence of the spontaneous and the dynamic radial couplings during a

collision. The case chosen is a head-on collision of U+Cf with a distance of closest

approach 2a = 18 fm. The matrix elements coupling the Isa resonant state to

the continuum state with E = —2me2 is shown as a function of time during the

collision. Of particular importance for the following section is the combination of

the increase of the spontaneous component and the disappearence of the dynamic

component at the turning point of the collision, when the internuclear separation

is smallest.

Figure 2-26: Induced and spontaneous coupling time dependence.The matrix elements for induced and spontaneous coupling are plotted as a function of time during a collision in a central collision of U+Cf with 2a = 18 fm (solid line) and the radial coupling in the Pb+Pb collision system (dashed line). (Reproduced from [Reinhardt et al. 81a].)

An approximate indication of the size of the spontaneous component of the

positron production in heavy-ion collisions is given in Figure 2-27 (reproduced

from [Reinhardt et al. 80a]). The differential probability for positron production

is plotted in parts (a) and (b) as a function of the positron energy for head-on

collisions of the two collision systems U+U with 2a = 16 fm and U+Cf with

72

2a = 17 fm, respectively. The solid line is a full coupled channel calculation for

the channels with k = —1, while the dashed line is the same calculation without

the spontaneous coupling. Although the contributions add coherently, the

difference between the two curves serves as an indication of the spontaneous

contribution. P art (c) shows the same calculations as a function of the projectile-

nucleus CM scattering angle.

Figure 2-27: Contribution of spontaneous positron emission.The differential probability for positron emission dPjdE as a function of the positron energy E is shown with (solid line) and without (dashed line) the spontaneous coupling for the collision systems U+U in part (a) and U+Cf in part (b). P art (c) shows the same comparison as a function of the center of mass projectile scattering angle. (Reproduced from [Reinhardt et al. 80a].)

Studies of the field theory corrections to the single particle scenario described in

this section indicate that they could not significantly change the results.

Calculations of the two main corrections, the vacuum polarization [Soff et al. 74]

and the self energy of the electron [Soff et al. 82b], show that both change the

approximately 1 MeV binding energy of a critically bound state by about only

1%, and moreover in opposite directions so as to nearly cancel.

73

2.2.I.4. Description of the Positron Production

In the prescription of quantum field theory in the independent particle

approximation (since the correlations between emmited positrons and electrons

are not being sought here) the transitions between states of this many electron

system is given [Reinhardt et al. 81a, Reinhardt et al. 81b] by an incoherent sum

of single electron transition probabilities, la^oo)!2. If the Fermi level F is taken

to be the atomic level up to which the quasimolecular electron levels are initially

filled with electrons before the collision process, then the number of particles

created in a state i > F above the Fermi level during the collision is:

N i “ E K.-<‘=o°)i2 (2-78)k < F

Similarly, the number of positrons created in a state * < F below the Fermi level

is:

N i = E n e_ = E K,e_<*=00>l2 <2-70)k > F k> F

This last expression yields the differential positron production probability

dP (n)/dE for a given Rutherford scattering trajectory, defined by the CM-6 r C T

scattering angle D. dP J d E depends primarily on the combined nuclear chargeC T C T

(Z + Z.) and the distance of closest approach J? •_ of the two nuclei achieved' p t 771*71during the collision.

The differential positron production probability for a given collision system is

obtained by integrating dP (n)/dE ,. over all possible scattering trajectories,Ct vtweighted with the Rutherford-scattering cross section doj^n)/dn for each

trajectory:

d P J n ) d ,^ n )

dP'+ J i n ' iE i n= . 2 (2.80)

d E < * . i , ^ n )

dn/■ dn

In the case of symmetric or nearly symmetric collision systems, where the

74

projectile and target nuclei cannot be differentiated, the symmetrized probability

will be measured at each scattering angle:

d-Pe+(rtp) d<Jf ^ n p ) d P e+ i n ^ d<Td n t)

dP*el m(n) d E e+ d o + d E e+ d o

dEe+ ~ dad ° p ) dcrd nt) (2’81)d o + d o

Expressions for the Rutherford-scattering cross section daRjdO are given in

Appendix A.

2.2.1.5. Resalts of the Calculations

The results of these calculations are shown in Figures 2-28 to 2-33. Figure 2-28

(reproduced from [de Reus et al. 83]) shows the probability for the production of

vacancies in the Isa state, a prerequisite for spontaneous positron production, for

several collision systems at a bombarding energy of 4.7 MeV/amu as a function of

the impact parameter b. (The solid curves include the effects of electron-electron

interactions, within a DFS approximation; the dashed curves do not.) Figure 2-29

(reproduced from (Reinhardt et al. 81a]) shows the Z-dependence of positron

production for a bombarding energy of 5.9 MeV/amu. The positron production

probability in head-on collisions of U+Cf, U+U, Pb+U , and Pb+ P b are given in

part (a) as a function of the kinetic energy i?e+ of the emitted positron and in

part (b) as a function of the distance of closest approach R min of the two nuclei

during the collision. (Electron-electron interactions were not considered in this

calculation.) Although the first two collisions systems experience overcritical

binding and the second two do not, there is evidently no drastic difference in the

spectra to indicate the transition to an overcritically bound system.

Figure 2-30 (reproduced from [Tomoda 82]) shows the results of similar

calculations by [Tomoda 82, Tomoda and Weidenmiiller 82] to those in Figure

2-29(a), for U+U and P b+ P b head-on collisions at a bombarding energy of 5.9

MeV/amu. These calculations, done independently of the Frankfurt group, follow

75

F ig u re 2-28: ls<r ionization probability.The ls<7 ionization probability P is plotted as a function of the impact parameter b for several combined nuclear charges Z = (Zp + Zt). (Reproduced from [de Reus et al. 83].)

a similar overall method, but rely more on analytical rather than numerical

techniques.

Figures 2-31 to 2-33 give the results of recent calculations [Miiller 83b, Muller

and de Reus 83] of the positron production which include the effects of electron-

electron interactions in a DFS approximation. Three collision systems are

considered: U+Cm at a bombarding energy of 5.8 MeV/amu (solid line), U+U at

5.9 MeV/amu (dotted line), and U +Pb at 5.9 MeV/amu (dashed line). Figure

2-31 shows the probability Pg+ for the production of positrons with kinetic energy

between 100 keV and 1 MeV as a function of the distance of closest approach

i?m ,n between the two colliding nuclei during the collision [Muller and de Reus

83]. Figure 2-32 shows the same calculation as a function of the projectile nucleus

76

F ig u re 2-29: Positron energy and R min dependence.The probability of positron emission in heavy-ion collisions with a bombarding energy of 5.9 MeV/amu is plotted as a function of the positron energy E g+ in part (a) and of the distance of closest approach R min in part (b) for several collision systems. (Reproduced from [Reinhardt et al. 81a].)

laboratory scattering angle, symmetrized as in Equation (2.81) for the case that

the projectile and target nuclei are not differentiated. Figure 2-33 shows the

differential probability dPg /d E e+ for the production of positrons in coincidence

with particles scattered into the angular region 25 ° < < 6 5 0 as a function

of the kinetic energy E of the emitted positron [Muller 83b]. These curves are

compared to our measured data in Chapter 7.

77

E_ ( m e 2 )

Figure 2-30: Positron energy spectra.The probability dP/dE of positron production in head-on collisions of U+U and Pb+ Pb with a bombarding energy of 5.9 MeV/amu is plotted as a function of the positron energy E . (Reproduced from [Tomoda 82].)

2.2.2. With Nuclear Interactions

The experiments described in this thesis were conducted using collision systems

below the Coulomb barrier to limit the background of positrons produced in

nuclear interactions. Our data, however, indicate the possibility that in someA

small fraction of the collisions (perhaps ~ 10 ), the two nuclei might interact.

Specifically, the width of the peak observed in the U+Cm collision system

(FWHM < 40 keV in the CM sytem) implies by the uncertainty principle a

source that lives for more than 2X10-20 seconds. This is an order of .magnitude

longer than the time of overcritical binding for a Rutherford collision trajectory.

78

F ig u re 2-31: Positron R min dependence.The probability for positron production per collision for positrons with kinetic energy between 100 keV and 1 MeV is plotted as a function of the distance of closest approach R min for the three collision systems 5.8

MeV/amu 238U + 248Cm, 5.9 MeV/amu 238U + ^ U , and 5.9 MeV/amu 238U + 208Pb. The coupled-channeled calculations (Muller and de Reus 83] include the effects of electronic screening.

79

e r i i deg]Figure 2-32: Positron angular dependence.

Same as Figure 2-31, but plotted as a function of the laboratory

projectile scattering angle assuming the projectile and target nuclei are not differentiated [Muller and de Reus 83].

80

E e+ [keV]

F ig u re 2-33: Positron energy spectra.The differential probability dP/dE for positron production per collision in coincidence with particles scattering into the laboratory angular region from 250 to 6 5 0 is plotted as a function of the kinetic energy i?e+

of the emitted positron for the three collision systems 5.8 MeV/amu 238U + 248Cm, 5.9 MeV/amu 238U + 238U, and 5.9 MeV/amu + 208Pb. The coupled channeled calculations include the effects of electronic screening [Muller 83b].

81

The formation of nuclear molecules in collisions near the Coulomb barrier is

known from lighter collision systems (see, for example, the review articles

of [Bromley 78a, Bromley 78b]. The possibility that nuclear reactions could be

causing the two nuclei to stick together for this amount of time long compared to

the Rutherford scattering time, and the effect this would have on the positron

production, will be explored in this section.

2.2.2.1. Implications for Spontaneous Positron Production

The possibility of two nuclei sticking together is important for spontaneous

positron production for two reasons [Rafelski et al. 78b]. First, the lifetime of

about I0~19 seconds of a 1 vacancy in the negative energy continuum is two

orders of magnitude larger than the time of about 2X10~21 seconds that the

collision system U+Cm is critical for Rutherford trajectories. Thus only a small

fraction of the critically bound vacancies have time to spontaneously decay into

electron-positron pairs before the two nuclei fly apart again. Holding the two

nuclei together at less than the critical separation increases the fraction of

vacancies that have time to decay, producing more spontaneous positrons. The

situation is depicted in Figure 2-34(a) (reproduced from [Rafelski et at. 78b]). The

energy levels of the innermost electrons are shown schematically as a function of

time during a collision where the nuclei stick together during the time interval

from to <2.

Furthermore, the ratio of spontaneous positron production to dynamical

positron production increases when the two nuclei remain stuck together for a

time. The spontaneous positron production depends only on the internuclear

separation i?, and will occur as long as R < R ^ . The other, dynamical forms of

positron production depend on the relative motion dR/dt of the two nuclei and

are thus reduced if the two nuclei stick together at a fixed separation R. This can

be seen in the relative behavior of the spontaneous and the radial couplings,

82

0.10Coupling (ten Etanents

005

-3— ^ -r Y 1 i 2 4 5„ t|10 seel/. 005• (-2IHI®.)

-am- (b )

F ig u re 2-34: Energy and matrix elements with nuclear sticking.P art (a) (reproduced from [Rafelski et al. 78b]) shows schematically the lowest electronic energy levels and part (b) (reproduced from [Reinhardt et al. 83a]) the matrix elements of the radial coupling and the spontaneous decay between the Isa resonance and the continuum state with energy E = —2me2 as a function of time during a collision involving nuclear sticking.

shown schematically in Figure 2-34(b) (reproduced from [Reinhardt et al. 83a]) as

a function of time during the collision. The matrix elements depicted are for a

head-on collision of U+Cf with the distance of closest approach 2a = 18 fm. It

has furthermore been assumed that the nuclei stick together at this separation for

2X10-21 sec. This picture should be compared to Figure 2-26 above, which is the

same collision system without any nuclear interactions.

This section will describe three aspects of nuclear interactions during heavy ion-

83

atom collisions relevant to spontaneous positron production. First, a double

folding model calculation of the potential between heavy ions will be described

which suggests the possible existence of pockets in the effective nucleus-nucleus

potential at the Coulomb barrier which could lead to a sticking effect. Then, a

quantum mechanical treatment of heavy ion-atom collisions based on a potential

of this same form is outlined which predicts the formation of peaks in the positron

energy distribution. Finally, a schematic, classical, perturbation theory

calculation is presented which relates in a simple way the sticking time of the two

nuclei to the width of the peak observed in the positron energy distribution.

2.2.2.2. Nucleus-Nucleus Potential in Heavy-ion Collisions

Although the possibility of nuclear interactions below the Coulomb barrier

might seem a contradiction of the definition of Coulomb barrier, it is possible

because of the non-spherical shape of some heavy ions. The determination of the

Coulomb barrier from systematics of elastic and inelastic scattering experiments,

which average over all relative orientations of the nuclear shape, implicitly

assumes that the colliding nuclei are spherical. Among the collision system

partners that we studied, uranium and curium nuclei, for example, are not

spherical, but rather prolate (or football) shaped [Bemis et al. 73]. As a result,

favourable orientations of two colliding nuclei can lead to a significant overlap of

the nuclear densities, even below the Coulomb barrier.

The three collision systems measured for this thesis, U+Cm, U+U, and U+Pb,

are pictured in Figures 2-35 to 2-37, respectively. In all three figures the nuclei

are oriented for maximum overlap with the long axes of the nuclei head-to-head.

The separation between the nuclei corresponds to the minimum separation in a

head-on collision at the average bombarding energy of our experiment (see

Chapter 5). In each figure, part (a) shows the profile of the nuclear charge

distributions [Hofstadter and Collard 67] and part (b) the shape of tlie nuclei at

84

the half density point [Ronningen et al. 81, Bemis et al. 73]. Although all three

collision systems are well below the Coulomb barrier [Wilcke et al. 80],the nuclear

densities do overlap for these special orientations.

x [fm]

F ig u re 2-35: Nuclear charge distribution and shape for U+Cm.In part (a) the nuclear charge distribution is plotted as a function of distance x along the internuclear axis and in part (b) the nuclear profile at the half-density point is plotted in the plane containing both long internuclear axes for a head-on collision of the system U+Cm with a internuclear separation of 18.3 fm.

Recent calculations [Rhoades-Brown et al. 83] of U+U collisions show that,

indeed, special orientations of the two colliding nuclear shapes can lead to pockets

in the nucleus-nucleus potential. In those rare collisions where the two nuclei line

up in an advantageous manner, sticking together of the two nuclei for short

periods can occur. The calculations are based on the double folding model of

nuclear interactions. A nucleus-nucleus interaction potential U[R) was obtained

by averaging a nucleon-nucleon interaction potential V(pi2) over static

nucleon distribution of both colliding nuclei, and p^ t^ \

85

x [fm]

F ig u re 2-36: Nuclear charge distribution and shape for U+U.Same as Figure 2-35, but for the collision system U+U with a internuclear separation of 17.6 fm.

CTR) = f A , J d \ ,,(!■,) Vtr12) , , ( r 2) (2.82)

where r 12 = R + r 2 — r 1? R is the internuclear separation, and is the position

of the nucleon in the itl1 nucleus relative to the center of the nucleus. The local,

two-body, nucleon-nucleon interaction V(r12) consisted of a Coulomb part due to

the electromagnetic interaction of the protons in each nucleus and a nuclear part

due to the strong force interaction between all of the nucleons in the two nuclei:

,2 (2.83)

The Coulomb interaction between protons is:

CVC(r12) = + — (2.84)r12

The nuclear interaction is the so-called MY3 force which has the form [Bertsch et

al. 77, Satchler and Love 79]:

V G» r C C N - t r N NPl V p 2 == pi V p2 I pti

86

x [ fm ]

F ig u re 2-37: Nuclear charge distribution and shape for U+Pb.Same as Figure 2-35, but for the collision system U +Pb with a internuclear separation of 16.7 fm.

12'c ar12ar,

- B6-^12br. - C S ( r l2) (2.85)

12 w ,12

where the parameters A = 7.999 MeV, a = 4 fm-1 , B = 2.134 MeV, 6 = 2.5

fm - 1 , and C = 262 MeV, have been fixed from medium heavy ion-atom elastic

scattering experiments [Satchler and Love 79]. The charge and matter

distributions are assumed to have the deformed two-parameter Fermi distribution:

P(r,0) j _ e[r-c(©)|/aC(e) = c0 [1 + 52Y20(e ) + *4Y40(e)] (2.86)

with parameters determined by electron scattering experiments. The integral in

Equation (2.82) is evaluated numerically.

The results for several relative orientations of the two scattering nuclei are

shown in Figure 2-38(a) (reproduced from [Greiner 83b]). The arrows indicate the

87

points at which the nuclei overlap at half densities to add up to normal nuclear

density. At smaller than this separation, the potential is unrealistically based on a

nuclear mass distribution with greater than normal nuclear matter density as the

two nuclei start to compress each other. If this effect is renormalized with a

surface thickness correction [Greiner 83b], then the curves in Figure 2-38(b) and

(c) result. These curves show a pocket whose depth, height, and location depend

on the relative orientation of the two deformed nuclei. The lowest barrier occurs

for head to head collisions. Quasistable states within this pocket could lead to

metastable configurations of the two nuclei, so that the two nuclei could stick

together for a short time during the collision.

F ig u re 2-38: Nucleus-nucleus potentials.The nucleus-nucleus potential in the double folding model for various relative orientations of the two nuclei is shown as a function of the internuclear separation r in part (a) for the Collision system U+U, and with additional surface thickness corrections in part (b) for U+U and in part (c) for U+Cm. (Reproduced from [Greiner 83b].)

The rest of this section will deal with the attempts made so far to calculate the

positron production from collisions with nuclear reactions leading to sticking

together of the nuclei for a time. First a quantum mechanical treatment will be

outlined, then a semiclassical treatment which relates the shape of the peak to the

sticking time will be described.

88

2.2.2.3. Quantum Mechanical Treatment

A complete description of positron production from heavy ion-atom collisions

near the Coulomb barrier but nonetheless with nuclear interactions leading to a

time delay in scattering, or sticking together of the two nuclei, requires a

quantum mechanical treatment [Heinz et al. 83a, Heinz et al. 83b, Heinz et al.

83c, Heinz et al. 84, Reinhardt et al. 83b]. This is necessary to account properly

for the interplay between atomic and nuclear production processes, particularly at

the point of closest approach of the two nuclei, where the electronic K-sbell is not

much larger than the nuclear molecule. Such a treatment would involve solving

the following stationary scattering problem [Heinz et al. 83a]:H * = E * (2.87)

with the Hamiltonian H:

P 2 ZH = — + Vc (R) + VN (/?) + ■ £ / / ^ 0(r,R)

1

+ 5a ( E v ' ,2 + i E l/te ,r0')N i = i

A+ J 2 H in t ( X ’R ) + H ra< tX ’T 'R ) (2-88)

m = 1P is the nuclear momentum, Vc is the Coulomb potential and the nuclear

potential between the two nuclei, is the two center Dirac Hamiltonian for

the Ith electron, v. is the velocity of the ith electron, Vgg is the electron-electron

interaction potential, H int is the Hamiltonian describing the internal nuclear

degrees of freedom which can be excited during the collision, and H rad is the

Hamiltonian describing the interaction of the photons, which couple to both the

nuclear (proton) and electronic currents. Both nuclear and electronic terms must

be explicitely included because interference effects between atomic and nuclear

positron production processes will be important.

A complete quantum mechanical treatment has not been accomplished, however,

89

for two reasons. Quite fundamentally, one of the most important terms, the

nuclear potential is not known. More practically, the solution of the above

equation, even after granting the use of the best possible guesses for the many

terms shown, is a challenging computational task for a system as complicated as

heavy ion-atom collisions with ~ 480 nucleons and — 140 electrons. The task

has been approached by [Heinz et al. 83a, Heinz et al. 83b], who start with the

following simplifications. For VN they take a nuclear optical potential with a

shape like that described above. The pocket provides the possibility of sticking

and nuclear scattering resonances. They ignore the other nuclear degrees of

freedom, so H i t = 0. For positron production, they also ignore the radiation

t e r m H r a d -

They simplified the problem further by expanding the wavefunction * in a basis

which diagonalizes the electronic, internal nuclear, and angular (with respect to

the internuclear axis) eigenstates. This produces a set of coupled differential

equations for the channel wavefunctions depending only on the relative nuclear

motion and separation R. These channel wavefunctions were expanded in basis

sets of ingoing and outgoing elastic scattering wavefunctions to produce a set of

coupled channel equations for the occupation amplitudes. Using the distorted

wave Born approximation and dividing the scattering trajectories into a part with

only atomic positron production processes and a part with only nuclear positron

production processes, the positron production was calculated.

Typical results are shown in Figure 2-39 (reproduced from [Heinz et al. 84]) for

the U+Cm collision system at a bombarding energy of 6.2 MeV/amu. A model

based on an internuclear potential like that in Figure 2-38 with 10 rotational

bands of molecular resonances starting at 1,2, • • • ,10 MeV below the top of the

Coulomb barrier was used to obtain semiquantitative results [Heinz et al. 84[.

P arts (a) and (b) display the delayed (non-Rutherford) nuclear scattering cross

90

section as a function of the delay time for the two nuclear scattering angles 60 °

and 90 ° , respectively. Parts (c) and (d) show the positron energy spectra

calculated from the time distributions in parts (a) and (b), respectively. (Dynamic

-positron production from Rutherford scattering events was not included.) In both

cases, a peak appears in the emmited positron energy distribution at the position

of the kinetic energy equal to the binding energy of the lSjy2 electron level minus

2mc2. Its width depends on the scattering width r and its intensity depends on

what fraction of the scattering events enter the potential pocket.

[Tomoda and Weidenmiiller 83] obtained very similar results starting with the

autocorrelation function of the nuclear scattering S-matrix. They use a form for

the function derived from general statistical considerations which is very similar

to that deduced from the nuclear optical potential described above (cf. (Reinhardt

et al. 83b]). The results of their calculations are shown in Figure 2-40 (reproduced

from {Tomoda and Weidenmiiller 83]) for the three collision systems U+Cm,

U+U, and Pb+Pb. The probability of positron production is plotted as a

function of the total energy of the positron state (in units of me2). As above, the

curves include only positrons from the delayed nuclear scattering events. The

different curves correspond to various values of the width r of the auto

correlation function of the fluctuating part of the nuclear scattering matrix.

According to [Reinhardt et al. 83b], this is equivalent to an exponential

distribution of nuclear sticking times T:

f ( T ) = r ce - r cr (2.89)

An interesting outcome of these calculations, as above, is the appearance of a

peak at the critical binding energy in the two cases of overcritically bound

electronic states, parts (b) and (c) of Figure 2-40.

01

nwr™s) Tic

500 1000 500 1000EWV)F ig u re 2-30: Quantum mechanical calculations.

The results of semiquantitative calculations for the U+Cm collision system at a bombarding energy of 6.2 MeV/amu (reproduced from [Heinz et al. 84]). In parts (a) and (b) the delayed (non-Rutherford) nuclear scattering cross section is plotted as a function of the delay time for the two nuclear scattering angles 60° and 90", respectively. Parts (c) and (d) show the positron energy spectra calculated from the time distributions in parts (a) and (b), respectively. (Dynamic positron production from Rutherford scattering events have not been included.)

2 .2 .2 .4 . Sem iclassical T re a tm e n t

The preliminary quantum mechanical calculations outlined above suggest that

the formation of a narrow peak in the positron energy spectrum is possible if

spontaneous positron production is enhanced with respect to elastic scattering by

quasielastic resonant scattering. A more intuitive picture can be obtained by

looking at a simple semiclassical calculation. [Reinhardt et al. 81b] have suggested

92

10'-3 P b ♦ P bE |0b= 7 M eV/nucl«on e =90-

co m

U . UElo b r5 9 M e W n u e l» o n

6 :90**V 3 cont.

3 E .3 E .

10'3 P b ♦ P bE |0 b : 7 M e W n u c lto n < W 9 0 ‘P v i c o n l

10-7

~r"\ t 'r ~rn~

U . UE|0b s5.9MeVZr*jd«one tm .= 9 0 *Pi/2 com .

w-

ui 3 IQ "3

U*CmE ia b = 5 .8 M ^ n u c l» o netm=120*Pvjeont.

r|'t rt .o j r , .a o i

F ig u re 2-40: Quantum mechanical calculations.The probability of positron production is plotted as a function of the total energy of the positron state (in units of me2) for the three collisions systems U+Cm at a bombarding energy of 5.8 MeV/amu, U+U at 5.9 MeV/amu, and U +Pb at 7 MeV/amu. (Only positrons from delayed nuclear scattering events have been considered.) The different curves correspond to the indicated widths r of the auto correlation function of the fluctuating part of the nuclear scattering matrix. (Reproduced from [Tomoda and Weidenmiiller 83].)

93

the following scenario. The ingoing and outgoing parts of the scattering

trajectory are handled exactly as in Section 2.2.1 for the case of pure Rutherford

scattering. At the point of closest approach, however, it is imagined that the

radial motion of the two nuclei is frozen. The two nuclei remain at a constant

separation R = R Q for a time T.

The results of coupled channel calculations using this schematic nuclear

trajectory for several heavy ion-atom collision systems and various nuclear

sticking times T are shown in Figure 2-41 (reproduced from [Muller et al. 83]).

The probability of positron production is plotted as a function of the kinetic

energy of the emitted positron for head-on collisions at a bombarding energy of

6.2 MeV/amu of the four collision systems Pb+Pb, Pb+U, U+U, and U+Cm.

The different curves correspond to various nuclear sticking times T, as indicated.

For the subcritical cases shown in parts(a) and (b), oscillations in the positron

energy spectrum caused by interference between positron production during the

ingoing and the outgoing parts of the nuclear trajectories separated by a time

delay T are apparent. The oscillatory interference pattern has the period

A E = 2nh/T. The supercritical cases in parts (c) and (d) show, in addition, the

emergence of a peak at the positron energy equal to the supercritical binding

energy minus 2mc2- The intensity of the peak increases and width decreases as

the sticking time T grows. A rough estimate from time-dependent perturbation

theory gives the full width at half maximum (FWHM) width of the spontaneous

positron peak as:

1/2

's .5 6 h 3.66 X10-21 MeV seci cv ■ ■!T — T

r T for-—- < 3

Ar T

r f o r - - - > 10 (2.90)

where r is the natural decay width of the supercritically bound l s ^ state (cf.

Figure 2-23(b)).

94

Pb*uL -U O IW /vt»0

F* 3

----T.fl-----1-3 •I'1----T.13;

« qb

Q6

QA

a2

500 1000 Er (keV)

A / \ 1 I U*Cm

i i i ii 1

•UOMtf/u b • 0 -F-3

' / 1i \

* t 1----------- 3*3— ......... T.6----- T.u

• tf* *

i i‘ ? !

i - i

' 1 i \-

- ' i r ' X1 * f ' \w \

- /,v v,»/ r v

500 1000 E^keV)

F ig u re 2-41: Positron spectra for a single sticking time.The differential probability for positron production in head-on collisions at a bombarding energy of 6.2 MeV/amu where the nuclei stick together for a time T is plotted as a function of the positron kinetic energy E g for four collision systems and four nuclear sticking times. (Reproduced from [Muller et al. 83].)

95

Figure 2-42 (also reproduced from [Muller et al. 83]) shows, the total probability

for positron production as a function of the nuclear sticking time T for the same

four collision systems. The solid lines are the contributions of the s states and the

dashed lines that of the p ^ 2 states. For the overcritically bound states, the total

positron production increases linearly with T.

F ig u re 2-42: Positron emission for a single sticking time.The total probability for positron production in head-on collisions at a bombarding energy of 6.2 MeV/amu where the nuclei stick together for a time T is plotted as a function of T for the s-state (solid lines) and the p-state (dashed lines) contributions for four collision systems. (Reproduced from [Muller et al. 83].)

These results assumed a single, well-defined sticking time T for all collisions.

The assumption of a perhaps more realistic distribution of sticking times changes

the results somewhat. As noted above, an exponential distribution of times would

96

correspond to the resonant decay process assumed for the quantum mechanical

treatment outlined above. The results of coupled channel calculations [Reinhardt

et al. 83b] with an exponential distribution of times (heavy lines) is shown in

Figure 2-43 for U+Cm. For comparison, the calculation for a single sticking time

T (thin lines) is also shown.

There are two major changes. In the subcritical case, the oscillations seen for a

single delay time T are completely washed out by an exponential distribution of

times. In the supercritical case, the peak is even narrower for the exponential

distribution of times with decay constant T than for. a single time T. This is due to

the significant presence of long sticking times in the exponential time distribution.

The width AE of the spontaneous positron peak is related to the decay constant T

of the exponential time distribution by:

„ fW /M 1.32 X 1 0 '21 MeVsee ___*'1/2 ~ 7 “ ------------- T ------------ ,2 9 l)

Similar results were obtained for a guassian distribution of sticking times, as

shown in Figure 2-44 (reproduced from [Muller et al. 83]). The subcritical

oscillations are damped out, but the supercritical peak remains.

All these approximate calculations show that a time delay between the ingoing

and the outgoing part of the scattering trajectories caused by the momentary

sticking together of the two nuclei enhances the spontaneous production of

positrons. It leads to the emergence of a peak in the positron energy distribution

at an energy equal to the supercritical binding energy minus 2mc2. The intensity

of the peak increases and the width of the peak decreases as the sticking time

grows larger.

97

F ig u re 2-43: Exponentially distributed nuclear sticking times.The differential probability for positron production in head-on collisions of the system U+Cm at a bombarding energy of 6.2 MeV/amu where the nuclei stick together for a time described by an exponential distribution (thick lines) or a single time (thin lines) is plotted as a function of the positron kinetic energy E g+. (Reproduced from [Reinhardt et al. 83b].)

98

F ig u re 2-44: Gaussian distribution of nuclear sticking times.The differential probability for positron production in head-on collisions at a bombarding energy of 6.2 MeV/amu where the nuclei stick together for a time described by a Guassian distribution with mean 15X10' “21 sec and width 0 sec (dashed lines) or 2.5 X10-21 sec (solid lines) is plotted as a function of the positron kinetic energy for the Pb+Pb collision system in part (a) and for U+U in part (b). (Reproduced from [Muller et al. 83].)

2.3. Nuclear Positron Production

As pointed out in Chapter 1, the positrons from the internal pair conversion of

excited nuclear states formed during heavy-ion collisions constitute a major

background to the positron production. The determination of this background

requires knowledge of the internal pair conversion coefficient 0, which is defined

as the ratio of the probability Pe+e_ for the production of an electron-positron

pair to the probability P' of gamma ray emission, for the de-excitation of the

nuclear state:P

e + e -t - - p - (2.92)

1

99

Alternately, for the case of a nuclear EO transition where gamma-ray emission is

impossible (as described below), P is replaced by the probability P _ for the7 ®internal conversion of an electron to form the ratio ij for monopole transitions:

Pe + e -

v = - J — (2.93)e—

In both cases, the differential internal pair conversion coefficient is defined as:i f <‘Pe+tJ < ‘E t+ dv i P ' + J i E ^

dE» P , ' dE t+ - Pe_(2.94)

If a vacant electronic state is available, a third process is possible. The electron

can be captured into the vacant bound state, leaving the positron with a definite

kinetic energy. (This process will be used in the discusion in Chapter 7.) In this

case, the monoenergetic internal pair conversion coefficients <»e+^ and £ are

defined in a matter analogous to 0 and above as:P Pe+K e+K

a I = - 7 7 - (2.95)e+K p ’ * p7 e -

The three processes involved are shown in Figure 2-45 (reproduced from [Soff et

al. 81b]), where the quantum numbers of the initial and final states are indicated.

Process (a) is internal pair conversion, (b) is internal electron conversion, and (c)

is monoenergetic pair conversion. This section will outline the calculation of these

coefficients and present their dependence on the important parameters: the

energy w and multipolarity L of the nuclear transition, and the nuclear charge Z.

The first calculations of the internal pair conversion coefficient were done in the

1930’s [Nedelsky and Oppenheimer 33, Rose and Uhlenbeck 35, Thomas 40] and

standard treatments abound (e.g. [Akhiezer and Berestetskii 65]). The

presentation in this section will follow that of the Frankfurt school [Oberacker et

al. 76b, Schliiter et al. 78, Schliiter et al. 81, Soff et al. 81b, Schliiter et al. 83b],

since this work was done with heavy-ion collisions in mind and the actual

computer program used to reduce our data was provided by this group.

100

J f=J ,

Tc'= -1

E, = E f *w • E'= £ .0 )

■E’=ES W

EO

E,

a) b)* - -1

c)

F ig u re 2-45: Internal conversion processes.Possible internal conversion processes are shown schematically. P art (a) depicts internal pair conversion, part (b) internal conversion of an electron, and part (c) monoenergetic internal pair conversion. (Reproduced from [Soff et al. 81b].)

The calculations were done within the framework of S-matrix theory and the

interaction picture of QED, where the scattering matrix, 5, transforms the initial

state ]*) = |$ ( /= —oo)) of a system into the final state \ f ) = |#(f=+oo)) by

multiplication:I f ) = S 10 (2.96)

The scattering amplitude to go from an initial state |») to a final state \ f ) is the

matrix element:Si f = (f\S\i) (2.97)

and the probability P y for the transition is:

P ,J X> IS , . / (2.98)

101

2.3 .1 . C a lcu la tio n o f th e EPC C oefficients

The processes involved in the calculation of the internal pair conversion

coefficients 0 and ij are shown in Figure 2-46 to lowest non-vanishing order in

5-matrix theory. P art (a) shows the emission of a gamma ray from a nucleus, a

first order process involving only one vertex. P art (b) shows the internal

conversion of an electron-positron pair and part (c) the internal conversion of an

electron. In the 5-matrix theory, these last two diagrams are second order

processes involving a virtual photon. The internal pair conversion coefficient was

calculated to the lowest nonvanishing order, as shown in these diagrams. An

important point is that, to this order, the dependence on the nuclear wave

functions is almost the same in the two processes described in the numerator and

the denominator, and nearly cancels out when the ratio is taken. In this order of

approximation, a detailed knowledge of the nuclear wave functions is not

necessary.

F ig u re 2-46: Internal conversion diagrams.Lowest order diagrams of the processes involved in internal conversion are displayed. P art (a) is gamma-ray emission from an excited nucleus. Parts (b) and (c) show internal pair conversion and internal electron conversion, respectively, of an excited nuclear state.

102

The probability P' of the emission of a gamma ray from the de-exciting nucleus

is calculated from the first order 5-matrix:

4 '1 = - i t f i 4z / '(* ) , 2M)

The current j is the nuclear current and A ^ is the photon field. By taking the

nuclear and the photon wave functions to be eigenfunctions of momentum and

parity, separating out a sinusoidal time dependence, forming the matrix element

of 5 between the initial state |i) and the final state |/} , doing the time integration

to produce an energy 5-function, and finally averaging over the nuclear magnetic

substates of the initial state and summing over the nuclear magnetic substates and

photon states of the final state, they obtained the following form for the

probability of gamma-ray emission:+ / • +J# , j

8/raw * J Z i „ (1 ) 9

pt = u n E E E E E 11' <2I00>‘ M. = - J. Mj = -J j L = 1 M = -L t = e,m

with the matrix element:

vf l = j f <2101>

MThe transition current and the transition field are defined by:

= ( / | i nW |i)

A f K ) « +irf = ( / |A " (* ) |i) (2.102)

and the energy requirement w = E i — Ej. comes from the 5-function mentioned

above.

Both the probability of producing an electron-positron pair and that of

internally converting an electron following the de-excitation of the nucleus are

calculated from the second-order S-matrix:

s<2) = J i h f S v T y j x ) A H * ) )> ) (2.103)

The photon operators AJ^x) and A v{y) are defined as above. For this term,

however, the current operator has both a nuclear and an electronic part:

103

j(Sx) = j f c ) + j f c ) (2.104)

Although an exact form for the nuclear current is not known, it will be seen below

that its effect cancels out of the ratios P . IP and P . IP .Ct C r C " C

Using the solutions iP(x) of the Dirac Equation for the electronic part of the

current / m(*)» and performing operations similar to those outlined above for the

determination of P , they calculated the differential probability for pair emission:

dFe+e- 2n +J' +0° +0°d E ^ 2 /+1 E E E E

e+ * M. = M r = - J . , „ w» i / / k = —oo Ar = —ookj&O

* E E 17+i-l2 (2105)P = - j P1 — - j ‘

with the matrix element:

(2) roo /*oo e ,wlpnuy> = - a dr / r f r j ( r ) . j ( r ) - -------- r (2.106)e+ e- J Q n J Q e J n ' e> ^ - r j V ’

and the probability for electron emission:+J. +/ .2^ * / +oo +oo

~ 2J+ 1 E E E E' Mi “ Mf = - Jf k = -ooV = -oo

Jfc/0

x E E U dE^ u ^ <21071/ • i vp = - j p — - j

with a similar definition for the matrix element as fore— e+e—

The further evaluation of the ratios d0/dEg+ and dr)/dEg+ was facilitated by

making a multipole expansion of the term [exp (tw|rn—:r el)l/lp„—r eI appearing in

i P in terms of products of Bessel or Hankel functions and spherical harmonics,

which form a complete set described by the angular momentum L and its

z-component M of the transition from initial to final state. Because of the parity

properties of the spherical harmonics, the expansion of and breaks up

naturally in the following way:

104

oo +Lv M = V mp(L = 0 )+ Y

L = 1 M = - L oo +L

( j l2) = U MP(L = 0 )+ Y 5 3 [Ue {L,M) + U m{L,M)\ (2.108)L = 1M=-L

The first term corresponds to a transition which does not change the angular

momentum, an EO transition. Since a real (as opposed to a virtual) photon has

only transverse degrees of freedom and must carry away at least one unit of

angular momentum, such a transition cannot be effected by a real photon. An EO

transition will not appear in the gamma-ray spectra. The other two types of

terms are referred to as magnetic and electric, respectively, because of the parity

properties of the transition. The parity of the final state is related to the parity

of the initial state as:

Pf = P . ( - 1)L+P (2.109)

where p = 0 for "electric* transitions and p = 1 for "magnetic" transitions.

This expansion is helpfull because these two remaining types of terms can both

be written in the form:U = 4jriau/V M (2.110)

where contains most of the dependence on the essentially unknown nuclear

current j n. This cancels out in the ratio of terms of the same multipolarity. In

addition, it is easier to find M separately for the two types of terms, U M and U E.

In both cases, M can be written as:M = M 3 + M d ~ M s (2.111)

M , named the static contribution, contains only electronic wavefunctions, while

M j, the so-called dynamic term, contains the remaining dependence on the

nuclear currents. According to [Schliiter et al. 81], M d should be small compared

to M because only that part of the electron wave functions inside the nucleus 8

contributes to M ^ In the point nucleus approximation, M^ vanishes identically.

The calculations of [Schliiter et al. 81, Soff et al. 81b], therefore, ignore these

terms.

105

Integrating out the angular dependence and performing the averaging over

initial states and the summing over final states indicated above led them to the

following form for the differential internal pair conversion coefficient: dp(EL) _T E ~ - E Ak / U'L) x

e+ k,kf| L (R,+R2+R3-R J + (*-*0(R3+R4) I2

iAML) _________ ,I F - ^ E V * " ’1 ) lR 5+ R 6l (2112)

C+ kThe coefficients A k ki{w,L) and B k fJ(u,L) depend on the energy u> and multipolarity

L of the nuclear transition, and on the angular momentum k and k1 of the

electron and the positron, respectively. The expressions J?. are integrals involving

only the radial wave functions of the electron and the positron and spherical

Hankel functions [Schliiter et al. 81].

In a similar manner, the coefficient for monopole transitions could be reduced to

the form:

E |c+ -12d” * = ±1 (2.113)

i E ' + y . 'lc - i 2m

The coefficients Cg+e_ and Cg__ depend only on the radial wave functions of the

electron and the positron, and of the converted electron, in the limit r -* 0 [Soff

et al. 81b].

In the point nucleus approximation, the internal pair conversion coefficients can

be evaluated analytically. For finite sized nuclei, numerical integration is

neccessary. The finite size of the nucleus becomes important for small E g+ and

large Z.

106

The results of these calculations are shown in Figures 2-47 to 2-49. Figure 2-47

(reproduced from [Soff et al. 81b]) shows the characteristic triangular shape of the

energy spectra of positrons produced by internal conversion of excited states of

heavy nuclei. The calculation was done for uranium. The differential internal

conversion coefficient is plotted as a function of the kinetic energy of the emitted

positron for an EO transition in part (a), and E l transition in part (b), and an E2

transition in part (c).

2.3.2. R esults o f th e C alculations

100 200 300 400 900 t t>

F ig u re 2-47: Differential internal pair conversion probability.The differential internal pair conversion coefficient is plotted as a function of the positron kinetic energy for nuclear EO transitions (part (a)), E l transitions (part (b)), and E2 transitions (part (c)) in a uranium nucleus. The curves correspond to nuclear transition energies of 1323 keV, 1423 keV, 1523 keV, and 1623 keV. (Reproduced from [Soff et al. 81b].)

Figure 2-48 displays the nuclear transition energy dependence of the coefficient

for transitions in uranium. The internal conversion coefficient is plotted as a

function of the transition energy for E0 transitions in part (a) (reproduced

from [Soff et al. 81b]) and for several higher multipolarities in part (b)

(reproduced from [Schliiter et al. 81]).

107

F ig u re 2-48: Transition-energy dependence of IPC coefficients.The total internal pair conversion coefficient is plotted as a function of the nuclear transition energy w in a uranium nucleus for several multipolarities of the nuclear transition. (Reproduced from [Soff et al.81b, Schliiter et al. 81).)

Finally, Figure 2-49 gives the Z-dependence of the coefficient. The internal

conversion coefficient is plotted as a function of the nuclear charge Z for E0

transitions in part (a) (reproduced from [Soff et al. 81b]) and for several higher

multipolarities in parts (b) through (e) (reproduced from [Schliiter et al. 81]).

The monoenergetic internal pair conversion coefficients ag+ R and £ are

determined in exactly the same manner as 0 and rj, respectively, with the

exception that the final state of the electron is different. The bound state

solutions #(x) of the Dirac Equation are used for the electron of the pair to£

determine the electronic current i^(x) above, instead of the continuum solutions.

Examples of the calculation of these coefficients can be found in [Schliiter et al.

83b].

108

P-10 E 1 VI)

-------------- --5' (b ) •

30 20 (0 I 60 80 100

■ p-io‘

■ (d )E 2

VR

:

, -

0 20 (0 7 60 80 100

The total internal pair conversion coefficient is plotted as a function of the nuclear charge Z for several multipolarities of the nuclear transition and several nuclear transition energies, indicated in units of m £_c2. (Reproduced from [Soff et al. 81b, Schliiter et al. 81].)

Chapter 3

Experimental Apparatus

The experimental appparatus set up to measure the positron production in

heavy ion-atom collisions will be described in this chapter . In doing so, the

following sections will show how the apparatus constructed addresses the major

experimental problems:

• the extraction of a small positron flux from a huge background of gamma rays and delta electrons,

• the determination of the kinematic parameters of each positron- producing collision,

• and the determination of the major background of positrons from nuclear processes.

Since Z it cz. 173, the formation of critically bound electronic systems requires

both beams and targets of the heaviest stable elements. At the time of this

experiment, the only laboratory capable of providing intense beams of the

heaviest elements up to uranium at Coulomb barrier energies was the heavy-ion

linear accelerator UNILAC (for f/niversal Linear Accelerator) at the Gesellschaft

fur Schwerionenforschung (Association for Heavy-ion Research) in Darmstadt,

Federal Republic of Germany (GSI Darmstadt). As shown in Figure 3-l(a), the

UNILAC consists of an ion source, pre-accelerator, four Wideroe-type drift-tube

3Additional descriptions can be found in [Balanda et al. 80] and [Bokemeyer et al. 83].

109

110

( b ) ^

11 Irra d ia tio n Eiperlnents12 Scattering CheaterU /4 Set-ups fo r Atonic Physics XS/C Set-ups fo r Atonic Physics X6 Gas-Jet

-y j 17 T-T-Correlatlon Neesurenents] , X8/9 - Spectrometers

XK Charge Changing ProcessesXO Irra d ia tio n F a c ility

T3 Chenlcal Separationst 1 T5 On-line Isotope Separator^ T7 Velocity Separator ‘ SHIP’

'v?*r; fi‘,1 jj ^, M v.

20 T*T*Corre1at1on Measurenents21 Magnetic Spectrograph 72 e * . Spectruneter23 TOF Telescope24 Many P a rtic le Telescope ZS Varling Cquipnent26 tinena tlc Coincidence Telescope

4*1 “

j ^

Sun «Muni

■ *

afrMrftifMw't wawiUHaaa w s u h m auunat "Ban*-'

F ig u re 3-1: GSI UNILAC and experimental hall.P art (a) is a diagram of the UNILAC heavy-ion accelerator at GSI Darmstadt (reproduced from [Angert and Schmelzer 77]). P art (b) shows the layout of the experimental hall.

I l l

accelerator tanks, two Alvarez-type cavity resonator tanks4, and ten single-gap

cavities for fine-tuning of the beam energy between 2 and 10 MeV/amu. P art (b)

shows the layout of the main experimental hall at GSI. The apparatus described

here was set up at site Z2.

The heart of the apparatus is a system for the detection of positrons. It was

designed to measure positron emission immersed in the overwhelming background

of gamma rays and delta electrons produced during the interaction of two

colliding nuclei. The extent of the problem is shown in Figure 3-2, which sketches

the energy distributions of the competing processes for a typical case, the collision

system 238U + 238U at a bombarding energy of 5.9 MeV/amu. As shown in the

figure, heavy ion-atom collisions produce on the average about 10'4 positrons per

collision with more than 10s gamma rays and delta electrons emitted for each

positron.

The combination of a large positron detection efficiency with good separation of

the positrons from the gamma rays and the delta electrons is accomplished with a

solenoid transport system to move the positrons away from the target area and

the backgrounds (cf. [Greenberg and Deutsch 56, Burginyon and Greenberg 66]).

Figure 3-3 shows a diagram of the experimental apparatus. The target is placed

in the middle and on the axis of a magnetic solenoid. The positrons are separated

from the gamma rays by the axial focussing property of a solenoid for charged

particles. Positrons emitted in a collision of a beam ion and a target atom spiral

down the solenoid on a helical path. Since the starting point of the positron’s

trajectory is in the target on the solenoid axis, the positron’s helical path returns

repeatedly to the axis (i.e. does not circle the solenoid axis). The positrons are

detected in a thin cylindrical Si(Li) detector placed along the solenoid axis at a

4Since the experiments described in this thesis took place, the UNILAC has been upgraded by the addition of two new Alvarez tanks, to produce beam energies up to 20 MeV/amu [Bohne 82].

112

10'

10°

10*

10*

5^ I0 '3 ui

10 I0*4

I0*5

10*

\\

s . \

I

\\

V\

\

\ N■ U + U \ 5 .9 MeV/amu x\

\

. - - — Gamma Rays \ Delta Electrons *

Atomic Positrons \ — - — -Nuclear Positrons “

E [MeV]

F ig u re 3-2: 7, e-, and e+ emission in heavy-ion collisions.The probability for the production of gamma rays (dashed line), delta electrons (dot-dashed line, scaled from [Backe et al. 83]), positrons from atomic processes (solid line), and positrons from nuclear processes (dotted line) in coincidence with particles scattered into the angular region 25 ° < 9 j h < 65 ° is sketched as a function of kinetic energy for the

collision system 238U + at a bombarding energy of 5.9 MeV/amu.2 3 8

B /

Tesl

a

113

Solenoid e* Spectrometer

EPO S

F ig u re 3-3: The experimental apparatus.P art (a) is a diagram of the experimental apparatus. P art (b) shows the strength of the magnetic field produced by the solenoid along the

solenoid axis.

114

distance from the target. The detection efficiency for positrons is high because

the positrons pass repeatedly through the solenoid axis as they travel down the

solenoid and most strike the Si(Li) detector. On the other hand, the detection

efficiency for gamma rays is low because only the end of the cylinder is facing the

target. The positrons which start out from the target moving along the solenoid

away from the Si(Li) detector are reflected back in the proper direction by a

higher magnetic field on that side which creates a magnetic mirror.

The solenoid magnetic field also transports electrons, but they are mechanically

blocked from reaching the Si(Li) positron detector by a fan-shaped baffle system,

which capitalizes on the fact that the helicity of the helical trajectory of the

electron in the magnetic field of the solenoid is opposite that of the positron. The

blades of the baffle are pitched to allow the positrons to pass through, but to stop

the electrons. In addition, the axial focussing property of the solenoid, described

above, ensures that particles (electrons or positrons) scattered from the baffle or

chamber walls at an off-axis position have a reduced chance of returning to the

axis and being detected.

The identification of positrons immersed in the gamma-ray and delta-electron

background is further enhanced by the detection of the characteristic annihilation

radiation emitted when the positrons stop in the material of the Si(Li) positron

detector. The two 511-keV gamma rays emitted back to back from the point of

annihilation, are detected in a cylindrical array of eight Nal(Tl) segments placed

around the Si(Li) positron detector.

Since the first generation of positron experiments (a Darmstadt-Yale-GSI

collaboration [Backe et al. 78], and a Darmstadt-Yale-Miinchen-GSI

collaboration [Kozhuharov et al. 79]) had not found conclusive evidence for

spontaneous positron production in the gross features of the positron production,

it appeared necessary to make a more differential measurement of the positron

115

production. In addition to a measurement of the positron energy with the Si(Li)

detector mentioned above, the scattering angle of both colliding nuclei is

measured using two position-sensitive parallel-plate avalanche counters. The

scattering angle is determined from the time delay between the signals induced in

the anode foil and in the cathode shifted-meander delay line located at the base of

the counter. This method combines coverage of a large solid angle with a simple

readout scheme. Measuring the scattering angles of both nuclei allows the

determination of the kinematic parameters of each collision, giving information on

the distance of closest approach of the two colliding nuclei for Rutherford

scattering, as well as any energy loss, mass transfer, and atomic charge

redistribution during the collision. The subsequent data analysis confirmed the

importance of this information in order to preferentially choose collisions

associated with interesting modes of positron production.

The earlier experiments had also shown that a large fraction of the positrons

produced in heavy ion-atom collisions come from nuclear processes, primarily the

internal pair conversion of nuclear states excited by the Coulomb and nuclear

interactions of the colliding nuclei. This can be seen above in Figure 3-2 for the

5.9 MeV/amu 238U + 238U system, where the positrons from both atomic and

nuclear processes are shown. To determine this large positron background, two

Nal(Tl) gamma-ray detectors were mounted near the target. The positrons from

nuclear processes are determined by folding theoretical pair conversion coefficients

with the measured gamma ray flux. Two detectors were used, one forward of the

target and one backward, to experimentally check the Doppler-shift correction

and the amount of contamination from neutrons entering the forward detector.

The time and energy signals from the detectors are processed before being

passed on to the computer for storage on magnetic tape. To pick out the

interesting events, triggers are made from the time signals of the detectors. Three

types of events were chosen for storage on tape:

116

• a positron-producing collision, indicated by a coincidence of time signals from the positron detector, the annihilation-radiation detector, and both particle detectors,

• a gamma-ray-producing collision, signalled by a coincidence of time signals from either one of the gamma-ray detectors and both particle detectors,

• a collision event for normalizing the other two event types, signalled by coincidence between the two particle detectors.

For each event all the detector signals were written onto magnetic tape. An

online analysis monitored the progress of the experiment and the condition of the

beam, the target, and the apparatus.

The following sections of this chapter will describe in greater detail the parts of

the apparatus assembled for the experiment. The first section is a description of

the apparatus already outlined above: the positron detection system, the particle

detectors, and the gamma-ray detectors. This is folowed by a description of the

electronics and computer system used to gather, store, and analyse the signals

received from these detectors during the experiment.

3.1. Experimental Apparatus

The experimental apparatus was assembled in the main experimental hall of GSI

Darmstadt at site Z2, beginning in 1978. As described in the introduction to this

chapter, the apparatus has three main parts: the positron-detection system,

particle detectors, and gamma-ray detectors. In addition, a target mounting

system provided for the proper positioning of targets and of radioactive sources

for calibrations. All four parts are mounted on a stand to allow for positioning

and adjustment . The four parts of the apparatus will be described in turn in the

following sections.

117

3.1.1. Target Mounting System

The target chamber can be seen in Figure 3-3 above. It is cylindrical in shape

and fabricated from aluminum. It has ports for the incoming beam and outgoing

beam dump, a cryogenic pump below, and to the right and the left the two arms

of the solenoid positron transport system. Above is a smaller, rectangular

chamber for changing targets, separated from the main target chamber by a valve

to allow the changing of targets and the mounting of radioactive sources without

disrupting the vacuum in the main target chamber.

The targets are all mounted on thin frames pressed from 0.3 mm aluminum

sheet. The targets have an area of 10 mm X 10 mm within the frame, which is

about twice the size of the beam spot in order to minimize the amount of target

material that the positrons must pass through yet prevent the beam from striking

the frame. The frames are likewise made as thin as possible to be mechanically

stable yet provide as small and as thin an obstruction as possible for the positrons

emitted from the target.

The target frame is attached with a spring mount to an aluminum rod which

can be slid up and down through O-rings to move the target between the main

target chamber and the smaller chamber for changing. The entire small target

chamber with the target mounting rod can be moved with respect to the main

chamber in the directions parallel and perpendicular to the beam axis by micro

screws. In addition, the rod can be moved up and down with respect to both

chambers by a screw. This system allows the target to be positioned (with an

accuracy of about ±0.5 mm) with respect to the beam axis and the axis of the

solenoidal positron transport system, which extends perpendicular to the beam

axis.

The target was positioned horizontally with respect to the beam axis using an

118

optical transit sighting along the beam axis. Since this direction of placement is

not so critical for defining either the particle or the positron trajectories, this

adjustment was made only once. The horizontal placement of the target along

the beam and the vertical placement was made by sighting through a transit

telescope aimed down the axis of the solenoid. Because this placement is critical

for both particle and positron trajectories, it was repeated each time a target or

radioactive source was mounted in the chamber.

In addition to the cryogenic pump directly below the main target chamber,

vacuum is maintained in the target chamber and the solenoid positron' transport

system by a forepump-and-turbomolecular-pump combination attached to the

beam dump, and by additional cryogenic pumps connected to the incoming

beamline before the target chamber and to the chamber containing the Si(Li)

positron detector. The vacuum in the target chamber and positron transport

system was maintained at better than 10*5 Torr during the measurements. Since

the range (from [Page et al. 72])of a positron of 1010 cm in this vacuum is 108

times the distance from the target to the Si(Li) detector, collisions of positrons

with air molecules in the positron transit system can be completely neglected.

3.1.2. Positron Detection System

3.1.2.1. The Apparatus.

The positron detection system is diagramed above in Figure 3-3. Its three major

parts are a solenoidal positron transport system, a Si(Li) positron detector, and a

Nal(Tl) annihilation-radiation detector. These are described in turn below.

Positrons are transported away from the target area by a solenoidal magnetic

field created by 15 coils arranged along an axis perpendicular to the beam

direction. Each coil has an inner diameter of 27 cm, is water cooled, and (with

one exception, described below) is operated at a current of 450 ampere. The

119

magnetic field generated by each of the coils was calculated from the geometry of

the wires in the coil (see, for example, [Jackson 75], pp.177-178), and the form of

the calculated magnetic field was fit to measurements made with a Hall probe.

solenoid is shown in Figure 3-3.

The target was positioned in the middle of the soleniod on the solenoid axis.

and angle of emission of the positron with respect to the solenoid axis. Since the

positrons are emitted on the solenoid axis, their helical paths return repeatedly to

the axis, tracing out a helix parallel to and adjacent to the solenoid axis, as they

move down either arm of the solenoid.

field in this arm (about 2.15 times the field strength at the target position). All

positrons with pitch angle less than the critical angle 9 ifT given by:

are reflected back toward the Si(Li) positron detector. For our apparatus,

B. . = 0.184 Tesla and £ ■__ = 0.395 Tesla, so that 9 ■ = 43.0°, As atarget mirror mirror

result, about 70% of the half of the positrons heading toward the mirror are

reflected back.

A fan-shaped baffle, positioned about halfway between the target and the Si(Li)

positron detector, mechanically stops electrons emitted from collisions in the

target from reaching the detector. Since electrons move along the solenoid on

helical trajectories that have the opposite helicity as positron trajectories, the 22

The position of the coils and the resulting magnetic field on the axis of the

The field at this position was 0.184 Tesla. Positrons emitted from collisions in the

target move along the solenoid axis on helical trajectories whose diameter and

pitch angle are determined by the magnetic field strength and by the momentum

As shown in Figure 3-3, a magnetic mirror is created in the arm of the solenoid

opposite that where the Si(Li) positron detector is located by the larger magnetic

(3.1)

120

blades of the baffle are pitched (at 570 ) to allow positrons to pass through but to

block electrons. A diagram of the baffle is shown in Figure 3-4. Figure 3-5 shows

the suppression factor of the baffle for electrons as a function of the electron

kinetic energy. The suppression is greatest at low electron energy, matching the

expected delta electron spectrum as shown in Figure 3-2. The baffle reduced the

electron counting rate in the Si(Li) detector to the point that the pile up of

electrons with positrons was less than a few percent3.

The polarity of two of the 15 coils producing the solenoid field is opposite that

of the other coils. One is located at the position of the baffle and is also run with

35% less current than the other coils. This reduces the magnetic field at the

baffle to about 80% of the Held on both sides, and accomplshes two goals. In the

smaller magnetic field at the baffle position the pitch angle of all the positron

trajectories decreases. The flattened-out positron trajectories pass better through

the spaces between the baffle blades. In addition, the diameter of the positron’s

helical trajectory increases in the lower magnetic field at the baffle, allowing more

of the lower energy positrons to spiral past the central hub of the baffle. Both

changes increase the transmission of positrons through the baffle.

The last coil on the Si(Li) positron detector side of the solenoid also has reversed

polarity. This acts as a bucking field to decrease the solenoidal magnetic field at

the end of the solenoid so that the photomultiplier tubes for the segments of the

annihilation-radiation detector can be attached to the Nal(Tl) crystals with as

short lengths of light pipe as possible.

Positrons are detected in a cylindrically-shaped, lithium-drifted-silicon (Si(Li))

detector located on the axis of the solenoid positron transport system, about 83

3The design of this baffle, made before measurements of the electron production in heavy-ion collisions were available, was was found to be quite conservative. Latter designs [Cowan 85] relaxed the suppression of electrons and achieved 2-3 times the transmission of positrons.

121

S id e View

F i g u r e 3 - 4 : B a ff le .

T h e fa n -s h a p e d b a ff le used fo r th e suppression o f e le c tro n s in th e

s o le n o id a l p o s itro n tra n s p o r t system .

c m fro m th e ta rg e t . T h e d e te c to r , b u i lt b y S eph S o c ie te d ’e tu d e P h y s iq u e o f

P a r is , F ra n c e , has tw o p a rts , each a h o llo w c y lin d e r 5 3 m m lo n g w ith an in n e r

d ia m e te r o f 5 .0 m m a n d an o u te r d ia m e te r o f 1 0 .3 m m . T h e h o llo w c y lin d r ic a l

fo rm o f each p iece w as ach ie v e d b y a c o u s tic a l d r i l l in g o f th e c e n tra l h o le in a

c y lin d r ic a l p iece o f s ilico n . L i th iu m w as d r if te d fro m th e in n e r s u rface to th e

o u te r a n d c o n ta c t is m a d e th ro u g h a m ic ro le a d to th e in n e r su rfa c e , w h ic h is

c o a te d w i th in d iu m . B o th pieces a re m o u n te d on a c y lin d r ic a l c o o lin g fin g e r

122

Ee- [hev]Figure 3-5: S up press ion o f e le c tro n s b y th e b a ffle .

T h e r a t io o f th e n u m b e r o f e le c tro n s d e te c te d w ith th e so le n o id a l

m a g n e tic f ie ld p o le d to a llo w th e tra n s m is s io n o f e le c tro n s th ro u g h th e

b a ff le to th e n u m b e r d e te c te d w i th th e o p p o s ite p o la r i ty is p lo tte d as a

fu n c tio n o f th e e le c tro n e n e rg y .

w h ic h p ro v id e s m e c h a n ic a l s u p p o rt a n d is c o n n e c te d to a liq u id n itro g e n re s e v o ir

to m a in ta in th e S i(L i ) d e te c to r a t its o p e ra t in g te m p e r a tu re o f -1 6 0 ° C .

T h e S i(L i) d e te c to r a n d th e c o o lin g sy s te m a re m o u n te d in a v a c u u m c h a m b e r

s e p a ra te d f r o m th e re s t o f th e so len o id p o s itro n t r a n s p o r t sys tem b y an

a lu m in iz e d M y la r fo il . A c ry o g e n ic p u m p m a in ta in s a v a c u u m in th is c h a m b e r o f

less th a n 1 X 1 0 - 7 T o r r , w h ic h is necessary to p re v e n t th e c o n d e n s a tio n o f

m a te r ia l o n th e c o ld s u rfa c e o f th e S i(L i) d e te c to r . A v a lv e lo c a te d b e tw e e n th e

123

d e te c to r a n d th e M y la r v a c u u m fo il iso lates th e S i(L i) v a c u u m c h a m b e r fro m th e

re s t o f th e so len o id p o s itro n tra n s p o r t sys tem w h e n th e la t te r m u s t be o p e n e d .

T h e S i(L i) d e te c to r w as o p e ra te d a t -2 5 0 v o lts . T h e d e te c to r is e le c tr ic a lly

is o la te d fro m th e re s t o f th e s p e c tro m e te r b y a T e f lo n m o u n t in g r in g , w h ic h ho lds

th e d e te c to r a n d th e c o o lin g sys tem to th e d e te c to r v a c u u m c h a m b e r, a n d b y an

a lu m in u m g r id p la c e d a ro u n d th e d e te c to r w ith in th e v a c u u m c h a m b e r . T h e

d e te c to r is also th e r m a lly is o la te d b y a th in a lu m in u m s leeve lo c a te d ju s t ins ide

th e w a lls o f th e v a c u u m c h a m b e r. T h e tw o pieces o f th e d e te c to r a re c o n n e c te d

b y m ic ro le a d s to s e p a ra te p re -a m p lif ie rs lo c a te d a t th e end o f th e so len o id , w h e re

th e s ig n a l f ro m each h a lf o f th e S i(L i) d e te c to r is d iv id e d in to a fa s t t im in g s ig n a l

a n d a lin e a r e n e rg y s ig n a l.

S in ce th e S i(L i) d e te c to r has a d ia m e te r o f 1 c m a n d is s e p a ra te d fro m th e ta r g e t

b y a d is ta n c e o f 8 3 c m , i t s u b ten d s a so lid an g le o f o n ly 1 .2 X 1 0 - 4 s te rra d ia n s

w ith re s p e c t to th e ta rg e t . T h u s th e S i(L i) d e te c to r in te rc e p ts o n ly 9 . 6 X 1 0 o f

th e ro u g h ly is o tro p ic g a m m a ra y f lu x fro m th e ta r g e t . T h is g a m m a ra y f lu x a t

th e S i(L i ) d e te c to r is fu r th e r re d u c e d , p a r t ic u la r ly a t lo w energ ies w h e re i t is

la rg e s t (see F ig u re 3 -2 ) , b y a 1 m m th ic k , 10 m m d ia m e te r d isk o f lead g lu ed to

th e ta r g e t s ide o f th e c e n tra l h u b o f th e b a ffle . T h is p lu g lies d ire c t ly b e tw e e n th e

ta r g e t a n d th e S i(L i) p o s itro n d e te c to r , s h a d o w in g th e en d o f th e d e te c to r fro m

th e b e a m s p o t in th e ta rg e t . T h e re d u c tio n o f th e g a m m a ra y f lu x re a c h in g th eo

p o s itro n d e te c to r b y th e le a d d isk ranges fro m a fa c to r o f 2 .7 X 10 a t 100 k e V to

0 .9 2 a t 1 M e V [V e ig e le 7 3 ]. T h e p lu g is encased in a p la s tic cap to t r a p an y

p o s itro n s fro m p a ir p ro d u c tio n o f g a m m a ra y s in th e ta n ta lu m d isk .

A fu r th e r id e n t if ic a t io n o f p os itro n s is m a d e b y d e te c tin g th e c h a ra c te r is tic

a n n ih ila t io n ra d ia t io n , tw o b a c k -to -b a c k 5 1 1 -k e V g a m m a rays , e m it te d a f te r a

p o s itro n loses a ll its k in e t ic en e rg y in th e S i(L i) p o s itro n d e te c to r a n d s tr ik e s an

e le c tro n . T h e g a m m a rays a re d e te c te d in a c y lin d r ic a lly shaped N a l ( T l ) d e te c to r

124

p la c e d a ro u n d th e S i(L i ) p o s itro n d e te c to r . T h e d e te c to r consists o f e ig h t

seg m en ts , each 5 0 .8 m m th ic k a n d 1 9 .5 c m lo n g , f i t te d to g e th e r to fo rm a c y lin d e r

w ith a h in n e r d ia m e te r o f 1 1 .5 c m . T h e c y lin d r ic a l d e te c to r s u b ten d s a b o u t 7 0 %

o f th e so lid a n g le in to w h ic h th e a n n ih ila t io n p h o to n s c o m in g fro m th e S i(L i)

d e te c to r ca n b e e m itte d .

E a c h o f th e e ig h t seg m en ts is co n n e c te d to a s e p a ra te p h o to m u lt ip l ie r tu b e b y a

l ig h t p ip e , w h ic h a llo w s th e p la c e m e n t o f th e p h o to m u lt ip l ie r tu b e o u ts id e th e

m a g n e tic f ie ld o f th e so len o id . A s d escrib ed a b o v e , th e m a g n e tic fie ld a t th e en d

o f th e s o len o id is re d u c e d b y a b u c k in g co il, a n d is fu r th e r re d u c e d b y a c y lin d e r

o f m u -m e ta l m a g n e tic s h ie ld in g p la c e d a ro u n d th e p h o to m u lt ip l ie r tu b es .

M a g n e t ic fie ld s u p to tw ic e as la rg e as th e n o rm a l o p e ra tin g f ie ld w e re fo u n d to

h a v e n e g lig ib le e ffe c t on th e s igna ls o f th e seg m en ts o f th e a n n ih ila t io n -ra d ia t io n

d e te c to r . T h e fa s t t im e s ig n a l fo r each se g m e n t is ta k e n f r o m th e a n o d e o f th e

p h o to m u lt ip l ie r tu b e w h ile th e lin e a r e n e rg y s ig n a l com es fro m a d y n o d e .

3.1.2.2. Positron Detection Efficiency.

T h e d e te c tio n o f p o s itro n s is a ffe c te d b y th re e fac to rs :

• th e t r a n s p o r t e ff ic ie n c y o f th e so le n o id a l p o s itro n t ra n s p o r t system ,

• th e response fu n c tio n o f th e S i(L i) p o s itro n d e te c to r ,

• a n d th e d e te c tio n e ff ic ie n c y o f th e N a l ( T l ) a n n ih ila t io n -ra d ia t io n

d e te c to r .

T h e s o le n o id a l p o s itro n t r a n s p o r t sys tem acts as a b ro a d b andpass sys tem , w i th a

p e a k tr a n s p o r t e ff ic ie n c y o f a b o u t 1 5 % , d ro p p in g d o w n to 1 /1 0 o f th is v a lu e a t

100 k e V a n d 8 0 0 k e V . T h e d e te c tio n e ff ic ie n c y o f th e S i(L i) p o s itro n d e te c to r is

a p p r o x im a te ly c o n s ta n t, b u t o u ts c a tte r in g a n d in c o m p le te c o lle c tio n e ff ic ie n c y

cause th e fo rm a t io n o f e x p o n e n tia l ta ils in th e lin e s h a p e o f th e d e te c to r . T h e

s u b te n d e d so lid a n g le a n d th e in tr in s ic d e te c tio n e ff ic ie n c y o f th e N a l ( T l )

a n n ih ila t io n -r a d ia t io n d e te c to r fo r th e a n n ih ila t io n g a m m a ra y s reduces th e

125

p o s itro n d e te c tio n e ff ic ie n c y b y a fa c to r o f a b o u t 0 .6 . T h e s e e ffec ts w i l l be

d escrib ed in d e ta il in th e fo llo w in g p a ra g ra p h s .

T h e e ff ic ie n c y o f th e s o le n o id a l p o s itro n tra n s p o r t sys tem can be u n d e rs to o d b y

f irs t c o n s id e rin g a s im p lif ie d v e rs io n [G re e n b e rg a n d D e u ts c h 5 6 , B u rg in y o n an d

G re e n b e rg 66 ]. A s sh o w n in F ig u re 3 -6 , th e im p o r ta n t aspects o f th e a p p a ra tu s

can b e p o r tra y e d b y a c y lin d r ic a l v a c u u m c h a m b e r o f ra d iu s J?2 lo c a te d in a

u n ifo rm m a g n e tic fie ld B, w ith th e ax is o f th e a p p a ra tu s p a ra lle l to th e f ie ld . A

p o in t source o f p o s itro n s a n d a c y lin d r ic a l d e te c to r o f le n g th D a n d ra d iu s R v

w h ic h is se n s itiv e o n ly a lo n g its s ide, a re p la c e d on th e axis o f th e so len o id ,

s e p a ra te d b y a d is ta n c e L. P o s itro n s le a v e th e p o in t source w ith m o m e n tu m p a t

an a n g le 0 to th e ax is o f th e so len o id . A s d escrib ed a b o v e , th e t r a je c to r y o f a

p o s itro n in th e c o n s ta n t m a g n e tic f ie ld is a h e lix , p a ra lle l to a n d a d ja c e n t to th e

axis . T h e d ia m e te r o f th e h e lix is

2 pcX = — — sin 0 (3 .2 )

eBT h e d is ta n c e b e tw e e n a x ia l crossings is

2 pcZ = —— jt cos 0 (3 .3 )

eB

A p o s itro n w i l l re a c h th e d e te c to r o n ly i f th e d ia m e te r X o f its h e lic a l t r a je c to r y

is la rg e en ou g h so th a t th e p o s itro n passes o v e r th e d e a d fr o n t end o f th e d e te c to r

a n d s m a ll en ou g h so th a t i t does n o t s tr ik e th e w a l l o f th e v a c u u m c h a m b e r. T h is

re q u ire s th a t

R l < X < R 2 (3 .4 )

In a d d it io n , th e p o s itro n w i l l be d e te c te d o n ly i f i t s tr ik e s th e side o f th e p o s itro n

d e te c to r , w h ic h h ap p en s o n ly i f th e h e lic a l t r a je c to r y o f th e p o s itro n crosses th e

axis a t th e p o s itio n o f th e d e te c to r . T h is re q u ire s th a t fo r som e in te g e r n :

L < n Z < ( L + D) fo r n = 1 ,2 ,3 , • • • (3 .5 )

I t is n o t necessary to ch eck a ll in te g e rs n because i f th e d is ta n c e b e tw e e n a x ia l

crossings is s m a lle r th a n th e le n g th o f th e d e te c to r D, th e n th e p o s itro n w il l

126

B e + D E T E C T O R -

7►-------------------------------

■ ' ‘ ‘' - “ T IT«--------------- l -------- + D - IK| R

S O U R C E

•0

Figure 3-6: D ia g r a m o f th e so len o id p o s itro n t r a n s p o r t system .

c e r ta in ly cross th e ax is s o m e w h e re a lo n g th e p o s itro n d e te c to r . T h is occurs fo r

0 < Z < D, w h ic h is t r u e w h e n e v e r n > (L+D)/D in th e expression ab o ve .

T h u s i f

L + DN = th e la rg e s t in te g e r in

D(3 .6 )

th e re q u ire m e n t th a t th e p o s itro n s tr ik e th e s ide o f th e d e te c to r can be w r i t te n as:

0 < N Z < {L + D) o r

L < n Z < (L + D) fo r a n y n = 1 ,2 ,3 , • • • ,{N— 1) (3 .7 )

C o m b in in g th is w i th th e re q u ire m e n t a b o v e fo r th e d ia m e te r o f th e h e lix , s o lv in g

fo r 9, a n d d e fin in g th e v a r ia b le

127

a = a(p) =2pc

~eB

~ 0 .6 6 7 2 cmp [M e V /c ]

B [T es la ](3.8)

w h ic h has th e u n its o f le n g th , g ives th e fo llo w in g :

RnR \ “ 2 sin 1 ( — ) < 9 < sin - 1 ( — )

aa n d

i + Ocos ( ——— ) < 9 < COS

Nirao r

cos- 1 (i L + D _ . L

~rT— ) < © < cos * ( -------- ) fo r n = 1 ,2 ,INira nira

T h e va lu e s o f p an d 9 w h ic h sa tis fy these re q u ire m e n ts a re th e shaded re g io n in

F ig u re 3 -7 (a ) .

T h e d e te c tio n e ff ic ie n c y fo r th e p o s itro n s e m itte d fro m th e p o in t source is fo u n d

b y in te g ra t in g f{9,4>) sin 9 d 9 d $ o v e r th e lim its g iven a b o v e , w h e re f[9,<t>) is th e

a n g u la r d is tr ib u t io n o f th e e m it te d p os itro n s , a n d d iv id e d b y th e m a x im u m

possib le:

r2it rnd* sin edef{e,<p)Jo Jo

F o r is o tro p ic em ission , / ( © , * ) = 1, a n d th e re s u lt in g e ff ic ie n c y is:

£( a ) = e ^ a ) - e2(a ) - £3(0)

w h e re th e th re e te rm s a re g iven b y :

4(«) =

(3 .1 0 )

(3 .1 1 )

an d

£3 ( a ) =

'0 fo r a < i ?1

fo r R l< a < R 2

A x - A 2 fo r R 2< a

<AX - B ( 0 ) fo r £>1(0 ) < o < D 2(0 )

\a 1 - a 2 fo r D 2(0 ) < a

A l ~ B[n) fo r £>1( n ) < a < £ ' 1(n )

B{n) - C(n ) fo r E 1( n ) < a < D 2(n)

C ( n ) - A 2 fo r D 2( n ) < a < E 2(n)

i f E^n) < a < Z>2(n ) o r

(3 .1 2 )

(3 .1 3 )

(3.14)

128

9 0 1 * 1 i i; — z = r 2 -

Z-R, (a) -

:----- nX= L

600 1000 E . + [keV]

1400

Figure 3-7: P o s itro n -tra n s p o r t e ff ic ie n c y fo r a so len o id .

T h e res u lts o f c a lc u la tio n s fo r th e so le n o id a l a r ra n g e m e n t o f F ig u re 3 -6 .

T h e s h ad ed re g io n in p a r t (a ) in d ic a te s th e em ission angles o f p o s itro n s

f ro m th e ta r g e t w h ic h s tr ik e th e d e te c to r . P a r t (b ) show s th e re s u lt in g

t r a n s p o r t e ff ic ie n c y as a fu n c tio n o f th e k in e t ic e n e rg y o f th e e m it te d

p o s itro n .

129

A l - B(n) fo r D l{ n ) < a < D 2(n)

«3(a ) = < A x - A 2 f o r D 2{ n ) < a < E l(n)

^ C ( n ) - A 2 fo r J5’1( n ) < a < E 2(n ) (3 .1 5 )

i f D 2 < a < E y T h e fo llo w in g d e fin itio n s h a v e been used:

A.= /l-(/?./a)2 .=1,2S + D

5C(n) = —

Tina

R iDJn) = -------------- — ------------ t = l ,2

sin ( ta n [ (n+l)*R./(S+D ) ] )

R {

E{n) = ---------------— 1----------------- i = l ,2 (3 .1 6 )sin [ ta n ( n j r ^ / 5 ) ]

F o r o u r a p p a ra tu s , R l = 0 .5 1 5 c m , R 2 = 4 .3 5 c m , L — 8 3 .5 c m , D = 1 0 .5 cm ,

a n d B = 0 .1 8 4 T e s la . T h e re s u lt in g e ffic ie n c y fo r th e h a lf o f th e p o s itro n s le a v in g

th e p o in t source m o v in g to w a r d th e d e te c to r is sh ow n in F ig u re 3 -7 (b ) .

P a r t o f th e h a lf o f th e p o s itro n s w h ic h a re e m it te d a lo n g th e so leno id a w a y fro m

th e d e te c to r a re re f le c te d b a c k to w a rd th e p o s itro n b y ra is in g th e m a g n e tic fie ld

on th is s ide re la t iv e to th e f ie ld a t th e source . A l l p o s itro n s a re re f le c te d b a c k

w ith angles in th e reg io n

. - i l l t a r g e t

* Sin \ B .f mirrorjr/ 2 < e < jr — sin 1 y J ~ r9et (3 .1 7 )

F o r o u r a p p a ra tu s , B target = 0 .1 8 4 T e s la a n d ^ m irro r = 0 .3 9 5 T e s la , so those

p o s itro n s in th e ra n g e

9 0 ° < e < 1 3 7 .0 °

a re re f le c te d b a c k . T h is a m o u n ts to

f l i t r\3>l ’I d* I s in 6 de

J o J 90*------------------------------------------ ~ 0 .7 3 1 (3 .1 8 )

r2ir rl

Jo “I*r2n ,180'

sin 9 de! 90*

130

of th e p o s itro n s e m it te d a w a y fro m th e d e te c to r . U n d e r th e as s u m p tio n th a t th e

re f le c te d p o s itro n s a c t as a n a d d it io n a l so urce o f p o s itro n s m o v in g to w a r d th e

p o s itro n d e te c to r f ro m th e so urce p o s itio n , th e m ir r o r f ie ld has th e e ffe c t o f

m u lt ip ly in g th e p o s itro n d e te c tio n e ff ic ie n c y c a lc u la te d a b o v e fo r ju s t o n e s ide o f

th e s o len o id b y th e fa c to r

( 0 . 7 3 1 X 0 . 5 ) + 0 .5 0 .8 6 6~ 1 .7 3 (3 .1 9 )

0 .5 0 .5

T h e e ffe c t o f th e b a ff le can b e c a lc u la te d in a s im p le w a y b y c o n s id e rin g th e

g e o m e tr ic a l o p e n in g b e tw e e n th e b lad es seen b y th e p o s itro n s in th e ir h e lic a l

p a th s . F ig u r e 3 -8 (a ) show s th e p ro je c tio n o f th e b lad es o f th e b a ff le o n to th e

lo n g itu d in a l a n g le 9go{ o f th e so len o id . A s sh ow n in F ig u re 3 -8 (b ) , th e g e o m e tr ic a l

o p e n in g seen b y th e p o s itro n can b e p a ra m e te r iz e d s im p ly in te rm s o f th e ta n g e n t

o f th e p o s itro n em ission a n g le 9. T h e m a x im u m tra n s m is s io n is fo r 9 e q u a l to th e

57 ° -p itc h a n g le o f th e b a ff le b lad es , a n d th e a m o u n t o f tra n s m is s io n a t th is a n g le

is th e r a t io o f th e p ro je c te d o p e n in g b e tw e e n th e b la d e s to th e o p e n in g -p lu s -b la d e .

In te g r a t in g o v e r th is tra n s m is s io n c u rv e fo r a l l ang les 0 ° < 9 < 9 0 ° g ives th e

tra n s m is s io n th ro u g h th e b a ffle :

/ I f (© ) s in 9 d9

P baffle ~

r rJo d*J*12

sin 9 d 9

2 ^ (0 .2 3 4 )

“ ~ i y r - 0234 |3-20)T h e to ta l p o s itro n tra n s m is s io n th ro u g h t th e sy s te m o f so len o id , m ir r o r f ie ld , an d

b a ff le is th e c u rv e a b o v e in F ig u re 3 -7 (b ) tim e s th e in crease d u e to th e m ir r o r fie ld

a n d th e re d u c tio n d u e to th e b a ff le . T h is is th e so lid c u rv e in F ig u re 3 -1 0 .

T h e p o s itro n s s tr ik in g th e S i(L i) p o s itro n d e te c to r lose e n e rg y in th e d e te c to r

m a te r ia l . A lo w e r l im i t on th e th ic k n e s s o f S i(L i) m a te r ia l th e p o s itro n passes

131

T a n ( 0 e + )

F i g u r e 3 - 8 : C a lc u la te d b a ff le tran sm iss io n .

P a r t (a ) show s th e p ro je c tio n o f th e b a ff le b lades o n to th e c y lin d r ic a l

c o o rd in a te s Z aol a n d 9 gol o f th e so len o id . P a r t (b ) show s th e c a lc u la te d

tra n s m is s io n o f p o s itro n s th ro u g h th e b a ff le as a fu n c tio n o f th e em ission

an g le o f th e p o s itro n w ith resp ec t to th e so leno id axis.

132

th ro u g h is th e r a d ia l th ic k n e s s 2 .6 5 m m o f th e d e te c to r . T h is th ickn ess

co rresp o n d s to th e ra n g e o f a p o s itro n w i th a k in e t ic e n e rg y o f 1100 k e V ( [P a g e

et al. 7 2 ]) , so e s s e n tia lly a l l p o s itro n s w ith k in e t ic e n e rg y u p to 1100 k e V w h ic h

re a c h th e p o s itro n d e te c to r sh o u ld s to p in th e d e te c to r . T h is covers th e re g io n o f

in te re s t fo r o u r e x p e r im e n t , so th e p ro b a b il i ty fo r a p o s itro n s to p p in g in th e

d e te c to r a f te r i t has h i t i t w i l l be ta k e n to b e u n ity .

T h e tw o b a c k -to -b a c k 5 1 1 -k e V a n n ih ila t io n r a d ia t io n g a m m a ra y s a re d e te c te d

in th e c y lin d r ic a l N a l ( T l ) d e te c to r p o s itio n e d a ro u n d th e S i(L i) p o s itro n d e te c to r ,

as sh o w n in F ig u re 3 -9 (a ) . T h e th ickn ess o f N a l ( T l ) m a te r ia l tra v e rs e d b y a

g a m m a r a y le a v in g th e m id d le o f th e a n n ih ila t io n -ra d ia t io n d e te c to r is g iv e n b y :

' i> fo r 0 < 9 < ta n ----------(1 /2)(L/2 ) R i _ j R 2------------------: fo r ta n . . . v < & < ta n ■ -cos e sin 9 (L/2) (L/2)

R 2 “ R l R 2 R 2((© ) = ^ — :-------------- fo r ta n - 1 ■ ■ ■ < © < it — ta n - 1 -----------

sin 9 (Zy/2 ) (L/2)

( L / 2 ) R \ , R 2 , R i----------------- :------ fo r it—ta n 777 - -<9<it— ta n —cos 9 s in 9 (E/2) (L/2)

-l R *0 fo r * - ta n 1 < 9 < it (3 .2 1 )

F o r o u r a p p a ra tu s , L/2 = 9 .7 5 c m , R l = 5 .7 5 c m , a n d f ?2 = 1 0 .8 3 c m . T h e

th ic k n e s s t is sh o w n as a fu n c tio n o f th e em ission a n g le r e la t iv e to th e so len o id

ax is in F ig u re 3 -9 (b ) . In F ig u re 3 -9 (c ) , th e to ta l a b s o rb tio n p r o b a b il i ty P > l fo r

d e te c tin g a t le a s t o n e o f th e tw o 5 1 1 -k e V g a m m a ra y s in t h a t th ickn ess o f N a l ( T l )

m a te r ia l is p lo t te d . T h e in te g ra l o f th is c u rv e o v e r a ll angles is 0 .5 5 .

T h e p ro d u c t o f th is e ff ic ie n c y fo r d e te c tin g a n n ih ila t io n r a d ia t io n , th e

e n e rg y d e p e n d e n t a b s o rb tio n c o e ffic ie n t fo r d e te c tin g th e p o s itro n in th e

S i(L i ) d e te c to r , a n d th e p o s itro n t r a n s p o r t e ff ic ie n c y tgol d escrib ed a b o v e is th e

to ta l p o s itro n d e te c tio n e ff ic ie n c y t g + :

£e+ = €eol £Si(L«) *ARD (3*22)

133

F i g u r e 3 - 9 : T h e d e te c tio n o f a n n ih ila t io n ra d ia t io n .

P a r t (a ) show s th e p o s itio n o f th e a n n ih ila t io n -ra d ia t io n d e te c to r w ith

re s p e c t to th e p o s itro n d e te c to r a n d th e p a th o f tw o a n ih ila t io n g a m m a

ra y s e m itte d fro m th e c e n te r o f th e a n n ih ila t io n -ra d ia t io n d e te c to r . In

p a r t (b ) , th e th ickn ess o f N a l ( T l ) m a te r ia l tra v e rs e d b y each a n n ih ila t io n

g a m m a ra y is p lo tte d as a fu n c tio n o f th e em ission a n g le 9. P a r t (c)

d e p ic ts th e p r o b a b il i ty o f d e te c tio n o f a t le as t o ne o f th e tw o g a m m a s as

a fu n c tio n o f th e em ission a n g le 9.

134

T h is is 0 .5 5 t im e s th e so lid c u rv e in F ig u re 3 -1 0 , a c c o rd in g to th is s im p le m o d e l

c a lc u la tio n .

A m o re s o p h is tic a te d c a lc u la t io n o f th e p o s itro n d e te c tio n e ff ic ie n c y m u s t ta k e

in to a c c o u n t th e v a r ia t io n o f th e m a g n e tic f ie ld a lo n g th e le n g th o f th e so len o id .

T h e s tre n g th o f th e m a g n e tic f ie ld a lo n g th e so len o id ax is w a s sh ow n a b o v e in

F ig u r e 3 - 6 . Ajq a p p r o p r ia te a p p ro a c h is th e a d ia b a tic a p p ro x im a tio n fo r th e

e le c tro m a g n e tic f ie ld , v a lid h e re because th e m a g n e tic f ie ld changes s lo w ly

c o m p a re d to th e d im en s io n s o f th e p o s itro n tr a je c to r ie s [Jackson 7 5 , C o w a n 85 ],

T h e d im en s io n s o f th e p o s itro n tra je c to r ie s m u s t th e n be w r i t te n as a fu n c tio n o f

th e p o s itio n z a lo n g th e so len o id axis , a n d ta k e th e fo llo w in g fo rm [C o w a n 85 ]:

i [B 0 2ff/?7»nc2 , / B{z) I

, ms in [6 (z )] = y - g - s in eQ (3 .2 3 )

S in c e th e a d ia b a t ic c h a n g e in th e m a g n e tic fie ld s tre n g th a lo n g th e so len o id axis

te n d s to erase th e p hase re la t io n s h ip b e tw e e n th e source o f p o s itro n s a n d th e

p o s itro n d e te c to r , o n e can in te g ra te o v e r a ll phases. T h e p o s itro n d e te c tio n

e ff ic ie n c y th e n c o n ta in s th e sam e e le m e n ts as th e s im p le r c a lc u la tio n a b o v e w ith

tw o a d d itio n s . S im ila r to th e d e r iv a t io n a b o v e ,' th e p r o b a b il i ty o f passing th ro u g h

th e v a c u u m c h a m b e r w ith o u t s tr ik in g th e in n e r w a lls goes as:

P chamber = 6 I R 2 ~ X ^Zm in) 1 (3 -2 4 )

w ith th e u s u a l d e f in it io n fo r th e s tep fu n c tio n :

(1 fo r x > 09 = { (3 .2 5 )

lo fo r x < 0T h e p ro b a b i l i ty o f s tr ik in g th e p o s itro n d e te c to r , i.e . o f passing o v e r th e d e a d end

a t th e f r o n t en d o f th e d e te c to r a n d s tr ik in g th e sides o f th e d e te c to r , is [C o w a n

85]:

135

P det ~ 1 sin1t

-ll ''dead

1X Z

det! Jdet2 Z

Z detdet

sin-11deti

Jdead (3.26)

w ith th e d e fin it io n s

X * s X ^det) Z det = (3 .2 7 )

T h e p ro b a b ilit ie s t h a t th e p o s itro n is re f le c te d in th e m ir r o r fie ld an d th a t i t

passes th ro u g h th e b a ff le b lad es can b e ta k e n o v e r d ire c t ly f ro m th e c a lc u la tio n

a b o v e . In a d d it io n , th e re is a s m a ll m a g n e tic b o t t le a t th e p o s itio n o f th e source

a n d a p lu g in th e m id d le o f th e b a ff le t h a t m u s t b e co ns id ered . T h e m a g n e tic

b o t t le , l ik e th e m a g n e tic m ir ro r , tra p s those p o s itro n s w ith a n g le g re a te r th a n :

<w = sm (3 .2 8 )bottle

F o r o u r e x p e r im e n t, B target = 0 .1 8 4 T e s la a n d B bottle = 0 .1 8 6 T e s la , so

9 bottle — 8 3 2

T h e p lu g in th e m id d le o f th e b a ff le acts in m u c h th e sam e w a y as th e dead

f r o n t end o f th e p o s itro n d e te c to r . T h e p ro b a b ili ty th a t p o s itro n s pass b y is:

P , = 1 ---------sin 1 -plug it \ X

2 t ( F p lug \ ^plug

w ith th e d e fin itio n splug} Jplug

X plug X ^Zplug}Z plug Z ^2plug}

(3 .2 9 )

(3 .3 0 )

T h e p r o b a b il i ty o f a p o s itro n re a c h in g th e d e te c to r is th e n th e p ro d u c t o f th e

(3 .3 1 )

in d iv id u a l p a rts

P =

a n d is p lo tte d in F ig u re 3 -1 0 as th e dashed c u rv e .

p — p P P P Pchamber det mirror bottle plug

In a d d it io n , a M o n te C a r lo c a lc u la tio n o f th e p o s itro n tra n s p o rt e ffic ie n c y w as

136

0 400 800 1200

Ee+ [keV]Figure 3 - 1 0 : C a lc u la te d p o s itro n t ra n s p o r t e ff ic ie n c y .

T h e p o s itro n t ra n s p o r t e ff ic ie n c y is p lo tte d as a fu n c tio n o f th e k in e tic

e n e rg y o f th e e m it te d p o s itro n fo r th re e m e th o d s o f c a lc u la tio n : th e

s o le n o id a l a p p ro x im a tio n (so lid lin e ) , th e a d ia b a tic a p p ro x im a tio n

(d ash ed lin e ) , a n d a fu ll M o n te C a r lo c a lc u la tio n (c irc les ).

d o n e [C o w a n 8 5 ]. T h e tra je c to r ie s o f p o s itro n s o f g iv e n e n e rg y a n d em ission an g le

w ith re s p e c t to th e so len o id axis w e re tra c e d th ro u g h th e s p e c tro m e te r b y

re p e a te d ly s o lv in g M a x w e l l ’s E q u a t io n fo r c losely spaced in te rv a ls a lo n g th e

p o s itro n ’s p a th . A t each s tep , a p ro g ra m c a lc u la te s th e fo rc e on th e p o s itro n fro m

th e m a g n e tic H e ld a t t h a t p o in t, c a lc u la te s th e su bseq u en t m o tio n o f th e p o s itro n

as a re s u lt o f th e m a g n e tic f ie ld , a n d checks to see i f th e p o s itro n s tr ik e s e ith e r

137

th e b o u n d a rie s o f th e s o le n o id a l tra n s p o r t sys tem (a n d is e lim in a te d ) o r th e S i(L i)

d e te c to r (a n d is d e te c te d ). S p e c ia l s u b ro u tin e s h a n d le th e b o u n d a ry p ro b le m

w h e n th e p o s itro n is in th e re g io n o f th e ta r g e t fra m e o r th e b a ffle . C h e c k s w e re

m a d e to d e te rm in e th e o p t im u m s tep size fo r th e p a th c a lc u la tio n a n d th e n u m b e r

o f s a m p le d em ission angles necessary to e lim in a te f lu c tu a t io n s in th e e ffic ie n c y

e n e rg y d is tr ib u t io n . T h e c a lc u la tio n w as p e rfo rm e d a t 2 3 e v e n ly spaced p o s itro n

k in e t ic energ ies fro m 100 k e V to 1 2 0 0 k e V , an d 1000 d if fe re n t em ission angles

w e re co ns id ered fo r each e n e rg y . T h e resu lts o f th is c a lc u la tio n a re th e c irc les

sh ow n in F ig u re 3 -1 0 .

3.1.2.3. Doppler Broadening o f the Detected Line Shape.

S in ce th e p o s itro n s a re e m it te d fro m a co llis ion sys tem w h ic h is m o v in g w ith

re s p e c t to th e la b o ra to ry a n d th e p o s itro n d e te c to r sys tem , th e s p e c tra m e a s u re d

in th e la b o ra to ry a re D o p p le r -s h ifte d . T h e axis o f th e s o le n o id a l p o s itro n

d e te c tio n sy s te m is p e rp e n d ic u la r to th e b e a m axis a n d th e m o tio n o f th e co llis ion

s y s te m , a n d th e p o s itro n em ission f ro m th e q u a s im o le c u la r sys tem is e x p e c te d to

b e ro u g h ly is o tro p ic in th e C M sy s te m , so th e m a in e ffe c t is a b ro a d e n in g o f th e

p o s itro n lin e sh ape . T h e c a lc u la tio n o f th e D o p p le r b ro a d e n in g o f th e d e te c te d

p o s itro n lin e sh ape fo r o u r s p e c tro m e te r is tr e a te d in d e ta il in [C o w a n 8 5 ], a n d is

o u t lin e d h ere .

T h e p o s itro n (k in e t ic ) e n e rg y s p e c tru m d N / d T ^ m e a s u re d in th e la b o ra to ry is

re la te d to th e a n g u la r d is tr ib u t io n dN/d(cos o f em ission in th e c e n te r o f

m ass o f th e c o llid in g n u c le i b y th e tra n s fo rm a tio n [V in c e n t 8 1 , C o w a n 85]:

d N d N rf( c o s e C M >

I t — = Ti-a— i ------------------£( 0 >7 ' ) (3 .3 2 )Lab ^ C0S GCm) Lab

T h e re la t iv is t ic tra n s fo rm a tio n f r o m th e C M system to th e L a b sys tem is

c o n ta in e d in th e th ir d te rm :

d(cos e CM> 1dT Lab

(3.33)

138

w h e re th e c e n te r-o f-m a s s m o m e n tu m is re la te d to th e e m it te d k in e t ic en erg y

t c m by:

Pc M = T CM + 2 m e+TCM (3-34)a n d 0 = v/c a n d 7 = ( 1—02 )-1/2 d escrib e th e v e lo c ity o f th e e m it t in g ( C M )

sy s te m r e la t iv e to th e la b o ra to ry s ys tem . T h e C M a n g u la r d is tr ib u t io n is

m o d if ie d b y th e a n g le -d e p e n d e n t e ff ic ie n c y c(9,T) fo r th e tr a n s p o r t an d d e te c tio n

o f p o s itro n s , in te g ra te d o v e r a ll

e(9,T) = J*\{9,*,T)d* (3 .3 5 )

T h is is ju s t th e p o s itro n d e te c tio n e ff ic ie n c y d escrib ed a b o v e , tra n s fo rm e d to th e

so len o id c o o rd in a te sy s te m b y th e re la tio n :

sin 9U b s in *U b = cos 9 Sol (3 .3 6 )

so th a t

/•jt/2 d*Lab£(9Lab'TLab\ = / ff/2_ e d 9 Sol <6SoI'Tl J 4^ (3 3 7 )

w h e re

d*Lab ~ sin 9 Sol

d e S °l ^ sin 2 9 Lab ~ COs2 9 Sol

(3 .3 8 )

F ig u re 3 -1 1 show s th e e ffe c t o f th e D o p p le r b ro a d e n in g on em ission f ro m th e

c e n te r o f m ass. T h e d if fe r e n t ia l p o s itro n y ie ld d N / d T ^ m e a s u re d in th e

la b o ra to ry sy s te m is p lo tte d as a fu n c tio n o f th e la b o ra to ry k in e t ic e n e rg y T^ab o f

th e p o s itro n fo r m o n o e n e rg e tic p o s itro n em iss ion in th e c e n te r o f m ass o f th e

+ 248C m c o llis io n sys tem a t a b o m b a rd in g e n e rg y o f 6 .0 5 M e V /a m u

(0 = 0 .0 5 5 7 , 7 = 1 .0 0 1 5 6 ) . T h e C M k in e t ic e n e rg y o f th e e m it te d p o s itro n is 3 1 5

k e V 6 . P a r t (a ) show s th e re s u lt in g lin e sh ap e fo r t{9,7) — 1 . In th is s im p le case

o f p e r fe c t p o s itro n d e te c tio n e ff ic ie n c y , th e la b o ra to ry lin e sh ap e is a b ox o f

c e n tro id

AThis example is chosen because of its relevance to the analysis of our data in Chapter 7 below.

139

<r i J = T CM + (-> - ! ) W a n + ”«?) ~ 3 1 6 3 k e V (3 3 »)

a n d w id th

A T = 207pC M ~ 7 2 .6 k e V (3 .4 0 )

T h e D o p p le r s h ift o f (7— l ^ T ^ + m c 2) = + 1 . 3 k e V is s m a ll a n d , as s ta te d ab o ve ,

th e m a in e ffe c t is th e D o p p le r b ro a d e n in g o f 7 3 k e V a t 3 1 6 k e V . P a r t (b ) shows

th e re s u lt in g lin e shape fo r o u r p o s itro n d e te c tio n sys tem , c a lc u la te d w ith th e

e ffic ie n c y d escrib ed a b o v e in th e a d ia b a tic a p p ro x im a tio n [C o w a n 8 5 ]. T h e d ip in

th e m id d le o f th e lin e sh ape is d u e to th e re d u c e d d e te c tio n e ff ic ie n c y fo r

p o s itro n s w i th m in im u m D o p p le r s h ift th a t a re e m it te d p a ra lle l to th e ax is o f th e

so len o id (a n d th u s a t r ig h t a n g le to th e b e a m axis ).

3.1.3. Particle Detectors

T h e s c a tte r in g angles o f b o th o f th e p ro je c ti le a n d ta r g e t n u c le i a re m easu red

w ith tw o p o s itio n -s e n s itiv e , p a ra lle l-p la te a v a la n c e co u n te rs ( P P A C ) m o u n te d in

th e ta r g e t c h a m b e r . A s sh ow n in F ig u re 3 -1 2 , o ne d e te c to r is m o u n te d a b o v e th e

b e a m ax is a n d th e o th e r b e lo w , each fo rm in g an a n g le o f 2 5 0 w ith th e b e a m axis.

T h e a n g u la r a c c e p ta n c e o f each d e te c to r is 18 ° < < 7 2 0 and

—3 0 0 < < + 3 0 0 , as d e fin e d b y an a lu m in u m m a s k m o u n te d on th e f r o n t o f

each d e te c to r .

3.1.3.1. Description of the Detectors

T h e c o n s tru c tio n o f th e p a r t ic le d e te c to rs is sh ow n s c h e m a tic a lly in F ig u re 3 -1 3 .

T h e s c a tte r in g a n g le o f th e p a r t ic le passing th ro u g h each c o u n te r is d e te rm in e d

f r o m th e t im e d iffe re n c e b e tw e e n th e e le c tr ic s ignals in d u c e d in th e an o d e a n d th e

c a th o d e . T h e an o d e is a n a lu m in iz e d , 3 .5 jxm th ic k M y la r fo il. T h e c a th o d e is a

s h if te d -m e a n d e r d e la y lin e . T h e d e la y lin e consists o f a 0 .1 5 m m th ic k a n d 0 .5

m m w id e s tr ip e o f c o p p e r e tc h e d on b o th sides o f a 0 .1 5 m m th ic k fiberg lass -

7A detailed description of the particle detectors can be found in [Gruppe 84].

140

E e+ [keV]F i g u r e 3 - 1 1 : D o p p le r b ro a d e n in g o f th e p o s itro n lin e shape.

T h e d if fe r e n t ia l p o s itro n y ie ld d N / d T ^ is p lo tte d as a fu n c tio n o f th e

p o s itro n k in e t ic e n e rg y TU b fo r m o n o e n e rg e tic p o s itro n em ission in th e

c e n te r o f m ass o f th e 238U + 248C m co llis ion sy s te m a t a b o m b a rd in g

e n e rg y o f 6 .0 5 M e V /a m u . T h e C M k in e t ic e n e rg y o f th e e m itte d

p o s itro n is 3 1 5 k e V . P a r t (a ) show s th e re s u lt in g lin e sh ape fo r

t(e,7) — 1 a n d p a r t (b ) th e lin e sh ape fo r o u r p o s itro n d e te c tio n sys tem .

re in fo rc e d T e f lo n s h ee t b y p r in te d -c ir c u it te c h n iq u e s a n d m o u n te d o n a S te s a lit

base. T h e d e la y lin e m e a n d e rs b a c k a n d fo r th across th e base o f th e d e te c to r ,

fo llo w in g lin es o f c o n s ta n t $Lab s e p a ra te d b y 0 .5 m m , as sh o w n in F ig u re 3 -1 3 .

T h e le n g th o f th e d e la y lin e on each a rc o f c o n s ta n t v a r ie s as a fu n c tio n o f

9Lab so ^ a t ^me d e la y r b e tw e e n th e s ig n a l in th e an o d e a n d th e s ig n a l in th e

c a th o d e is d ire c t ly p ro p o r t io n a l to th e s c a tte r in g a n g le &Lab-

141

S ID E V I E W D O W N S T R E A M

V IE W

F i g u r e 3 - 1 2 : P la c e m e n t o f th e p a r t ic le d e te c to rs .

e Lab = K' T ( 3 4 1 )

T h e e n t ire d e la y lin e is 12.8 m lo n g an d co rresponds to a to ta l t im e d e la y o f a b o u t

16 6 nsec fo r 1 3 ° to 7 7 ° , so th a t k ~ 2 .6 n s e c /d e g re e . T h e p o s itio n -s e n s itiv e

d e la y lin e a llo w s a la rg e so lid an g le to be co vered w ith a s im p le re a d o u t.

A s sh ow n in F ig u re 3 -1 3 , th e an o d e a n d c a th o d e a re s e p a ra te d b y 5 m m . A

v a c u u m fo il o f a lu m in iz e d , 3 .5 p m th ic k M y la r is p la c e d 12 m m in f r o n t o f th e

a n o d e fo il so t h a t th e sam e pressure can be m a in ta in e d w i th in th e p a rtic le

d e te c to r on b o th sides o f th e an o d e fo il to p re v e n t d is to rt io n o f th e e le c tr ic fie ld

b e tw e e n an o d e a n d c a th o d e . Is o b u ty le n e is c irc u la te d th ro u g h th e d e te c to rs a t a

p ressure o f 1 3 .6 T o r r a n d th e an o d e fo il is m a in ta in e d a t + 8 5 0 v o lts w ith respect

to th e c a th o d e d e la y lin e .

142

TOP V IE W

S A N G L E M A S K

Icm

1 ^V A C U U M F O IL A N O D E F O I L ^ FR A M E — —

J *

B A S E D ELA Y L IN E

CRO SS S E C T IO N

F i g u r e 3 - 1 3 : S c h e m a tic d ia g ra m o f th e p a r t ic le d e te c to r .

T h e an o d e a n d c a th o d e s igna ls a re passed th ro u g h is o la te d fe e d th ro u g h s in th e

ta r g e t ch am b eT w a l l to p re -a m p lif ie rs o u ts id e th e c h a m b e r. T h e rise t im e o f th e

a n o d e s ig n a l is a b o u t 3 nsec w h ile th a t o f th e c a th o d e is a b o u t 12 nsec. T o

p re v e n t g ro u n d lo o p s b e tw e e n th e d e te c to rs a n d th e res t o f th e s p e c tro m e te r , th e

tw o d e te c to rs a re m o u n te d on a brass r in g w h ic h is is o la te d fro m th e ta rg e t

c h a m b e r b y p la s tic w ash ers . E le c t r ic a l p ic k u p in th e d e te c to rs o f th e h ig h

143

fre q u e n c y s igna ls fro m th e G S I U N I L A C a n d o th e r e le c tr ic a l e q u ip m e n t in th e

e x p e r im e n ta l h a ll has been re d u c e d b y c o a tin g th e p la s tic d e te c to r bod ies w ith

s ilv e r - im p re g n a te d c o n d u c tiv e p a in t an d c o n n e c tin g th e c o a tin g th ro u g h iso la ted

fe e d th ro u g h s to th e d e te c to r e le c tro n ic s g ro u n d .

3.1.3.2. Scattering Kinematics

T h e k in e m a t ic p a ra m e te rs o f each io n -a to m co llis ion a re d e te rm in e d fro m th e

m e a s u re d s c a tte r in g angles o f th e tw o c o llid in g n u c le i. F o r R u th e r fo rd s c a tte r in g ,

th e d is ta n c e o f closest a p p ro a c h R min d u r in g a co llis ion is re la te d to th e s c a tte r in g

angles €? an d 9* o f th e p ro je c ti le a n d ta r g e t n u c le i, re s p e c tiv e ly , by:

R min = “ [ c s c ( ~ 7 ^ ) + 1 ] = 0 [ s e c ( - ^ ) + 1 ] <3 ' 4 2 )

w h e re th e c e n te r-o f-m a s s s c a tte r in g angles a re re la te d to th e la b o ra to ry s c a tte r in g

angles b y :

e C M = 9 Lab + sin~ 1 ^ sineL b )

e C M ~ <2,GLab (3 .4 3 )

T h e m in im u m d is ta n c e o f closest a p p ro a c h , 2 a , in a h e a d -o n co llis ion is g iv e n by:

2 a = - | ------- (3 .4 4 )e c m

a n d th e o th e r p a ra m e te rs a re d e fin e d by:

M p E p M p M t

C = ~ M ' E c m = “ m ' ' '“ j u Mt P P te2 ~ 1 .4 4 M e V f m (3 .4 5 )

In g e n e ra l, th e s c a tte r in g a n g le o f th e ta r g e t nucleus is re la te d to th a t o f th e

p ro je c t i le by:

- ( 1+ p ) c o t 9 ^ ± ( 1+ p ) J \ - p2 + c o t2 9*ab

co t 9 r . = ----------------------------------------------------------------------------- (3 46 )Lab . 21 — p

T h e m e a s u re m e n t o f th e s c a tte r in g angles o f both o f th e c o llid in g n u c le i a llo w s a

d e te rm in a t io n o f th e k in e m a t ic p a ra m e te rs fo r quas ie leas tic as w e ll as e las tic

144

s c a tte r in g . T h e c o rre la t io n b e tw e e n th e tw o s c a tte r in g angles p ro v id e s

in fo rm a t io n on e n e rg y loss a n d m ass tra n s fe r d u r in g th e co llis io n . A llo w in g fo r

m ass tra n s fe r b y u s in g p rim e s to in d ic a te th e m asses a n d energ ies a f te r th e

c o llis io n a n d d e fin in g th e e n e rg y loss Q = E — (E ’+EJ), th e re la tio n s h ip b e tw e e n

th e tw o s c a tte r in g angles is g e n e ra liz e d to :

- ( l +ppPt) c o t e[ab ± (p+pt) J\ - p2 + c o t2 e[ab

c o t e Lab -------------------------------------- ;---------5------------------------------------- (3 4 ? )P P

w ith th e a d d it io n a l d e fin itio n s :

M ' M l

PP = A / / R ’ Pt = n T ' R M E

R ^ M tE + M Q ' M = M p + M t = M ; + M l (3 .4 8 )t p

T h e a n g u la r d is tr ib u tio n s p re s e n te d in th is thesis a re in g e n e ra l p lo tte d w ith

re s p e c t to 0 ^ a n d 0^, th e la b o ra to ry s c a tte r in g angles o f th e tw o n u c le i d e te c te d

in th e u p p e r a n d th e /o w e r p a r t ic le d e te c to rs , re s p e c tiv e ly . A ty p ic a l a n g u la rq o o ooa

d is tr ib u t io n is sh o w n in F ig u re 3 -1 4 (a ) fo r th e U + ^uoP b co llis ion sys tem a t a

b o m b a rd in g e n e rg y o f 5 .9 M e V /a m u . In a d d it io n , w e fo u n d i t u se fu l to d e fin e a

n e w set o f s c a tte r in g v a r ia b le s A 6 a n d E 6 by :

A 0 = eu - eL

E 0 = ev + eL (3 .4 9 )

P a r t (b ) o f F ig u r e 3 -1 4 show s th e sam e d a ta as p a r t (a ) , b u t p lo tte d w ith resp ect

to th e tw o n e w v a r ia b le s . T h e t ra n s fo rm a tio n to th e n e w v a r ia b le s a m o u n ts to a

c o u n te r -c lo c k w is e ro ta t io n b y 4 5 ° . B ecause th e e n e rg y -m o m e n tu m c o n s e rv a tio n

la w s on w h ic h th e k in e m a t ic re la tio n s a b o v e a re based a re s y m m e tr ic w i th resp ect

to th e in te rc h a n g e o f 6p a n d 9t fo r b o th e la s tic a n d q u as ie las tic s c a tte r in g [B a ld in

et al. 6 1 ], a l l co llis ions a re s y m m e tr ic a b o u t A 9 = 0 . In a d d it io n , exp an s io n o f th e

E 9 scale w i th re s p e c t to th e A 9 scale a llo w s a m o re d e ta ile d lo o k a t th e n e a rly

s y m m e tr ic c o llis io n system s s tu d ie d in th is thesis.

145

94

90

g* 86,T 3 .

CDW 82

78

-60 -40 -20 0 20 40 60A 9 [deg]

F i g u r e 3 - 1 4 : U + P b a n g u la r d is tr ib u tio n s .

S c a tte r p lo t o f s c a tte re d p a r t ic le e ven ts fo r th e 238U + 20aP b co llis ion

sys tem a t a b o m b a rd in g e n e rg y o f 5 .9 M e V /a m u . E a c h e v e n t is p lo tte d

w ith resp ec t to Gu a n d in p a r t (a ) , a n d w ith resp ec t to AG a n d EG in

p a r t (b ) .

146

3.1.3.3. Operating Characteristics.

T h e o p e ra t in g c h a ra c te r is tic s o f th e d e te c to rs w e re d e te rm in e d th ro u g h tests

w ith a lp h a -p a r t ic le s ( f ro m a 241 A m so urce) a n d h e a v y ions. T h e in tr in s ic s p a tia l

re s o lu tio n o f th e d e te c to r w as m easu red b y p la c in g a m a s k w ith s m a ll holes inA40 A4Q

f r o n t o f o n e o f th e d e te c to rs a n d m e a s u rin g th e U + U c o llis io n sy s te m a t a

b o m b a rd in g e n e rg y o f 5 .9 M e V /a m u . U n fo ld in g th e e ffe c t o f th e f in i te h o le s ize

f r o m th e re s u lt in g d is tr ib u t io n g a v e th e s p a tia l re s o lu tio n as ~ 1 .5 m m o v e r th e

e n t ir e a n g u la r ra n g e o f th e d e te c to r . T h is tra n s la te s in to an a n g u la r re s o lu tio n o f

0 .3 9 a t 2 0 9 a n d 0 . 9 9 a t 709. T h e in tr in s ic d e te c to r re s o lu tio n is d e te rm in e d

p r im a r i ly b y th e s ig n a l-to -n o is e r a t io in th e d e la y lin e [G ru p p e 8 4 ].

T h e to ta l a n g u la r re s o lu tio n fo r th e d e te c tio n o f h e a v y ions w as d e te rm in e d

f r o m th e m e a s u re m e n t o f th e 238U + ^ P b co llis io n sy s te m sh o w n a b o v e in

F ig u r e 3 -1 4 , w h e re th e p ro je c t i le a n d ta r g e t n u c le i e n te r in g each d e te c to r can be

d if fe re n t ia te d b y k in e m a tic s . T h e fu l l -w id th -a t -h a lf -m a x im u m ( F W H M ) in th e

r e d i r e c t i o n fo r th e d e te c tio n o f b o th s c a tte re d p a rtic le s in c o in c id e n c e is p lo tte d

as a fu n c tio n o f A & in F ig u r e 3 -1 5 .

A d d it io n a l tes ts c o n f irm e d th e l in e a r ity o f th e re la tio n s h ip b e tw e e n th e d e la y

t im e b e tw e e n th e a n o d e a n d c a th o d e a n d th e s c a tte r in g a n g le to b e t te r th a n 0 . 1 9 .

T h e o p t im u m gas p ressu re a n d v o lta g e fo r o p e ra t in g th e c o u n te rs , g iv e n a b o v e ,

w a s d e te rm in e d b y a s tu d y o f th e e ffe c t o f these tw o p a ra m e te rs on th e pulse

h e ig h t a n d rise t im e o f th e an o d e a n d c a th o d e s igna ls (see [G ru p p e 84] fo r d e ta ils ).

3.1.3.4. Effect o f the Solenoidal Magnetic Field.

A f in a l im p o r ta n t c o n s id e ra tio n w a s th e e ffe c t o f th e m a g n e tic f ie ld o f th e

s o le n o id a l p o s itro n tr a n s p o r t sy s te m o n th e d e te c tio n o f p o s itro n s . A s sh ow n

a b o v e in F ig u re 3 -1 2 , th e d e te c to rs a re p o s itio n e d w ith in th e ~ 0 .1 8 T e s la fie ld

o f th e so len o id . T h e m a g n e tic f ie ld w as fo u n d to h a v e th re e m a in effects:

• th e tr a je c to r ie s o f th e tw o ions b e tw e e n th e ta r g e t a n d th e p a r t ic le

c o u n te rs a re c u rv e d

147

a eF i g u r e 3 - 1 5 : A n g u la r re s o lu tio n fo r h e a v y -io n d e te c tio n .

T h e a n g u la r re s o lu tio n ( F W H M ) in th e EG -d ire c t io n fo r th e c o in c id e n t

d e te c tio n o f b o th s c a tte re d n u c le i in th e 238U + 208P b co llis ion sys tem a t

a b o m b a rd in g en e rg y o f 5 .9 M e V /a m u is p lo tte d as a fu n c tio n o f AG. T h e c irc les a re th e m e a s u re d d a ta . T h e so lid lin e is a leas t-sq uares

q u a d ra t ic f i t to th e d a ta , y ie ld in g th e resu lt:

F W H M r f l = 1 .52 - 8 .O 6 X 1 O “ 3( 4 0 ) + 6 .4 8 X l O ” ^ AG)1-4/

• th e a v a la n c h e o f e le c tro n s w ith in th e c o u n te r is s y s te m a tic a lly s h ifte d

• th e s ig n a l p u lse h e ig h t f ro m th e co u n te rs is decreased .

A s th e ions m o v e th ro u g h th e n e a r ly c o n s ta n t m a g n e tic fie ld o f th e so len o id a l

t r a n s p o r t s ys tem , th e ir p a th s a re ch ang ed fro m s tr a ig h t lines to arcs o f c irc les

w ith a ra d iu s r g iven by:

mcv

qB(3.50)

148

T h e p a th d ep en d s o n th e m ass m , v e lo c ity v, a n d c h a rg e s ta te q o f th e io n , as

w e ll as th e a v e ra g e s tre n g th B o f th e m a g n e tic f ie ld . S in ce th e d e la y lin e o f th e

p a r t ic le c o u n te rs w e re des igned fo r no m a g n e tic f ie ld , a n d th e c u rv e d tra je c to r ie s

caused d e v ia tio n s in th e p o in t o f c o n ta c t w ith th e c o u n te rs on th e o rd e r o f a

d e g re e , c o rre c tio n s w e re necessary . C o m p a r is o n s w i th h e a v y -io n co llis ions sh ow ed

t h a t th e c h a rg e s ta te o f th e ions e x it in g fro m th e ta r g e t is g iven to good

a p p ro x im a tio n b y th e s e m i-e m p ir ic a l e q u ilib r iu m c h a rg e -s ta te d is tr ib u t io n

o f [N ik o la e v a n d D m it r ie v 68 ]:

- ' [ - f e n - 1 < 3 -5 , )

w ith th e ir p a ra m e te rs a = 0 .4 5 , k = 0 .6 , a n d v ' = 3 . 6 X 1 0 8 c m /s e c . In S ec tio n

7 .2.2 b e lo w , th e p o s s ib ility th a t a s m a ll fra c t io n o f th e o b served co llis ion even ts

c o rre s p o n d to d if fe r e n t v a lu e s o f m , v, o r q w i l l be discussed.

T h e .e le c tro n a v a la n c h e caused b y th e passage o f th e h e a v y ions th ro u g h th e gas

o f th e c o u n te rs is also d e fle c te d b y th e s o le n o id a l m a g n e tic f ie ld as i t m o ves fro m

th e a n o d e fo il to th e c a th o d e d e la y lin e . In a d d it io n to th e m a g n e tic fie ld

(B cs 0 .1 8 T e s la ) , th e e le c tro n c lo u d also m oves in th e e le c tr ic f ie ld of th e

c o u n te r (E c s 1 .6 X 1 0 s v o lts /m ) . T h e tw o fie ld s a re p e rp e n d ic u la r , an d

\E\ > \B\, so th e m o tio n is h y p e rb o lic . T h e d is p la c e m e n t Ay is g iv e n [G ru p p e 84]

b y :

^ vd B

*» = — E (3M)w h e re h = 5 m m is th e s e p a ra tio n b e tw e e n a n o d e a n d c a th o d e , vd is th e d r i f t

v e lo c ity o f th e e le c tro n s , a n d k = 0 .7 5 com p ensates fo r a M a x w e ll ia n d is tr ib u t io n

o f e le c tro n v e lo c itie s . A n e s tim a te o f vi ~ 1 0 5 m /s e c fro m th e rise t im e o f th e

d e te c to r e le c tr ic a l s igna ls g ives Ay ~ 0 .8 m m . M e a s u re m e n ts o f th e 238U +

c o llis io n sy s te m a t a b o m b a rd in g e n e rg y o f 5 .9 M e V /a m u m a d e w ith a m a s k w ith

s m a ll holes p la c e d in f r o n t o f one o f th e c o u n te rs a t 0 a n d 0 .1 8 T e s la y ie ld e d a

v a lu e o f Ay za 0 .9 m m fo r th e e n t ire c o u n te r . T h is tra n s la te s in to a n a n g u la r

s h ift o f 0 .2 5 ° a t 2 0 ° a n d 0 .5 ° a t 7 0 ° .

149

F ig u re 3 -1 6 show s th e e ffe c t o f th e s o le n o id a l m a g n e tic fie ld on th e p a r t ic le

d e te c tio n . T h e s c a tte r in g -a n g le v a r ia b le E 9 is p lo tte d as a fu n c tio n o f A S fo r th eA 4 Q A A A

U + C m co llis ion sys tem a t a b o m b a rd in g e n e rg y o f 5 .9 M e V /a m u in p a r t

(a ) a n d fo r th e + 208U sy s te m a t 5 .8 M e V /a m u in p a r t (b ) . T h e d o tte d lin e

rep re s e n ts th e k in e m a t ic re la tio n s h ip b e tw e e n th e tw o s c a tte r in g angles fo r th e

case o f no m a g n e tic f ie ld . T h e dashed lin e in c lu d es th e e ffe c t o f th e m a g n e tic

f ie ld on th e tra je c to r ie s o f th e ions, a n d th e so lid lin e show s th e e ffe c t o f th is plus

th e d e fle c tio n o f th e e le c tro n ic a v a la n c h e w ith in th e c o u n te r .

F ig u re 3 -1 7 show s th e e ffe c t o f th e ch arg e s ta tes o f th e s c a tte r in g ions on th eA A A

d e te c te d s c a tte r in g angles . A g a in , L 9 is p lo tte d as a fu n c tio n o f A 6 fo r th e U

+ ^ C m co llis io n sys tem a t a b o m b a rd in g e n e rg y o f 5 .9 M e V /a m u in p a r t (a ) and

fo r th e + 208U sy s te m a t 5 .8 M e V /a m u in p a r t (b ) . T h e so lid lin e re p resen ts

th e k in e m a t ic re la tio n s h ip b e tw e e n th e tw o s c a tte r in g angles u n d e r th e

a s s u m p tio n o f e q u ilib r iu m c h a rg e s ta tes fo r b o th th e p ro je c ti le an d th e ta r g e t ions,

as d escrib ed a b o v e . T h e s e v e lo c ity -d e p e n d e n t c h a rg e s ta tes a re in g e n e ra l

d if fe r e n t fo r th e tw o s c a tte re d ions. T h e e q u ilib r iu m c h a rg e s ta tes w e re fo u n d to

d escrib e th e gross fe a tu re s o f th e co llis ion system s s tu d ie d fo r th is thesis, w h ic h

w e re a ll b e lo w th e C o u lo m b b a r r ie r . T h e s h o rt-d a s h e d , lo n g -d ash ed , a n d d o t-

d ash ed lin es a re c a lc u la te d w ith th e a s s u m p tio n th a t b o th ions h a v e th e sam e

c h a rg e s ta te o f 6 0 + , 7 5 + , a n d 9 0 + , re s p e c tiv e ly . T h e s e la s t c a lc u la tio n s w i l l be

im p o r ta n t fo r th e d iscussion in C h a p te r 7 b e lo w .

T h e c u rv a tu re o f th e e le c tro n a v a la n c h e t r a je c to r y in th e so len o id a l m a g n e tic

f ie ld w i th in th e p a r t ic le d e te c to rs also has th e e ffe c t o f re d u c in g th e p u lse h e ig h t

o f th e e le c tr ic a l s ignals fro m th e co u n te rs . T h e s id ew ays d e fle c tio n o f th e

e le c tro n s reduces th e m o tio n in th e d ire c t io n o f th e e le c tr ic fie ld b e tw e e n

co llis ions , le a d in g to re d u c e d a m p lif ic a t io n o f th e a v a la n c h e b y a b o u t

20 % [G ru p p e 8 4 ].

150

‘ 6 0 -4 0 "20 0 20 A8 [deg]

Figure 3-16: E ffe c t o f so len o id f ie ld on p a r t ic le d e te c tio n .

E 9 is p lo t te d as a fu n c tio n o f A 9 fo r th e + ^ C m co llis io n sys tem

a t a b o m b a rd in g e n e rg y o f 5 .9 M e V /a m u in p a r t (a ) a n d fo r th e 238U +

208U sy s te m a t 5 .8 M e V /a m u in p a r t (b ) . T h e d o tte d lin e is fo r no

m a g n e tic f ie ld , th e d ashed lin e in c lu d es th e e ffe c t o f th e f ie ld on th e ion

t r a je c to r ie s , a n d th e so lid lin e in c lu d es b o th th is a n d th e e ffe c t on th e

e le c tro n ic a v a la n c h e w ith in th e c o u n te r .

151

o»ft).T3.Q>W

- 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0

A 0 [ d e g ]

F i g u r e 3 - 1 7 : E f fe c t o f so leno id fie ld on p a r t ic le d e te c t io n ..

L 9 is p lo tte d as a fu n c tio n o f A 9 fo r th e 238U + ^ C m co llis io n sys temey o q

a t a b o m b a rd in g e n e rg y o f 5 .9 M e V /a m u in p a r t (a ) a n d fo r th e U +

208U sy s te m a t 5 .8 M e V /a m u in p a r t (b ) . T h e so lid lin e is fo r

e q u ilib r iu m ch arg e s ta tes , th e s h o rt-d a s h e d lin e fo r eq u a l 6 0 + c h a rg e

s ta tes , th e lo n g -d ash ed lin e fo r e q u a l 7 5 + c h a rg e s ta tes , a n d th e d o t-

dashed lin e fo r e q u a l 9 0 + c h a rg e states,

152

T h e g a m m a ra y p ro d u c tio n o f th e h e a v y io n - a to m co llis ions w as m o n ito re d

w ith tw o 3 " X 3 " N a l ( T l ) g a m m a -ra y d e te c to rs m o u n te d o u ts id e th e ta r g e t

c h a m b e r . T h e d e te c to rs w e re lo c a te d a b o u t 3 7 c m fr o m th e ta r g e t a t a n a n g le o f

4 5 ° to th e b e a m axis . A s sh ow n in F ig u re 3 -1 8 , o n e w a s m o u n te d in th e fo rw a rd

'd ir e c t io n w i th resp ec t to th e b e a m a n d b e h in d th e u p p e r p a r t ic le d e te c to r , a n d

th e o th e r w a s m o u n te d in th e b a c k w a rd d ire c t io n a n d b e lo w th e b e a m ax is . In

b o th cases, th e N a l ( T l ) c ry s ta l, a H a rs h a w T y p e 1 2 M B 1 2 /3 A - X , w a s c o n n e c te d b y

a 7 .6 c m lo n g , 7 .6 c m d ia m e te r l ig h tp ip e to a m a tc h e d 3 " p h o to m u lt ip l ie r tu b e .

T lu ^ e n t ir e assem b ly is enclosed in a m u -m e ta l sh ie ld so t h a t th e d e te c to rs c o u ld

be p la c e d close to th e s o le n o id a l m a g n e tic f ie ld .

T h e la rg e f lu x o f lo w -e n e rg y g a m m a ra y s fro m th e ta r g e t w as re d u c e d b y

p la c in g 2 .6 m m o f t in a n d 4 m m o f le a d in f r o n t o f b o th d e te c to rs . A d d it io n a l

a b s o rb tio n o f th e g a m m a -ra y f lu x c a m e f r o m th e a lu m in u m ta r g e t c h a m b e r w a ll

in f r o n t o f b o th d e te c to rs . A p a r t ic le d e te c to r , w h ic h consists p r im a r i ly o f a 5 m m

th ic k S te s a lit b o a rd co vered a p p ro x im a te ly 5 0 % w ith 0 .1 5 m m c o p p e r a n d tw o

a lu m in iz e d 3 .5 m m M y la r fo ils , is lo c a te d in f r o n t o f th e fo rw a rd g a m m a -ra y

c o u n te r , b u t causes n e g lig ib le a b s o rb tio n o f g a m m a ra y s in th e re g io n o f in te re s t

a b o v e 1 M e V . A c y lin d e r o f 3 .0 m m t in , 2 .5 m m ta n ta lu m , a n d 1 .0 m m le a d w as

p la c e d a ro u n d th e sides o f th e N a l ( T l ) c ry s ta ls to re d u c e th e f lu x o f g a m m a ra y s

w h ic h C o m p to n s c a tte r f ro m th e s p e c tro m e te r in to th e sides o f th e c ry s ta l,

p a r t ic u la r ly f r o m th e n e a rb y c o p p e r so len o id coils . F o r th e 1 .3 3 -M e v g a m m a -ra y

l in e o f ^ C o , th e re s o lu tio n o f th e d e te c to rs has been m e a s u re d to b e 5 .2 % .

3.1.4. G am m a R ay D etectors

153

F i g u r e 3 - 1 8 : P la c e m e n t o f th e g a m m a -ra y d e te c to rs .

3 .2 . D a ta C ollection System

T h e la s t sec tio n d escrib ed th e v a rio u s d e te c to rs o f o u r e x p e r im e n ta l a p p a ra tu s

u p to th e g e n e ra tio n o f s ignals a t th e ir p re -a m p lif ie rs . T h is sec tion w il l describe

h o w these s igna ls a re ta k e n fro m th e p re -a m p lif ie rs a n d ch ang ed in to th e p ro p e r

d ig it iz e d n u m b e rs re c o rd e d on m a g n e tic ta p e fo r la te r an a lys is a n d in te r p r e ta t io n .

T h is process is d ia g ra m e d in F ig u re 3 -1 9 a n d has th re e steps:

• F ir s t th e e le c tro n ic s a m p lifie s an d shapes th e p re -a m p lif ie r s igna ls and

m a k e s log ic decisions based on co inc idences b e tw e e n t im e signals fro m

gro up s o f d e te c to rs to fo rm e v e n t typ es fo r re c o rd in g on ta p e .

154

Figure 3-19: S c h e m a tic d ia g ra m o f th e e x p e r im e n ta l se t-u p .

155

• T h e n th e C A M A C in te r fa c e tra n s fe rs b o th th e d e te c to r s ignals an d th e

log ic decisions to th e c o m p u te r sys tem fo r fu r th e r w o rk .

• F in a l ly th e o n lin e c o m p u te r sys tem accepts th e signals fro m o u r

e le c tro n ic s , reco rd s a ll o f th e signals on m a g n e tic ta p e fo r s to rag e an d

fu r th e r an a lys is , a n d p re -a n a ly s e s th e signals o n lin e to a llo w c o n s ta n t

c o n tro l o f th e e x p e r im e n t.

T h e s e th re e steps w i l l be d escrib ed in g re a te r d e ta il in th e fo llo w in g subsections.

3.2.1. Electronics System

T h e fo llo w in g seven d ia g ra m s , F ig u re s 3 -2 0 to 3 -2 6 , show th e e le c tro n ic s system

used d u r in g th e J a n u a ry /F e b ru a ry - , 1 9 8 1 , e x p e r im e n t. T h e f irs t d ia g ra m , F ig u re

3 -2 0 , show s th e m a in c o m p o n e n ts o f th e e le c tro n ic s sys tem a n d th e ir

in te rc o n n e c tio n s . T h e six m a in c o m p o n e n ts a re th e fo u r typ es o f d e te c to r

system s, an e v e n t ty p e log ic c irc u it , an d th e A D C / T D C system . T h e f in a l six

d ia g ra m s , F ig u re s 3 -2 1 th ro u g h 3 -2 6 , d is p la y each o f these c o m p o n en ts in g re a te r

d e ta il . T a b le 3 -1 d e fin es th e sym b o ls used in F ig u re s 3 -2 0 to 3 -2 6 .

3.2.1.1. Types of Signals.

A s can be seen in F ig u re s 3 -2 1 th ro u g h 3 -2 4 , th e fo u r d e te c to r system s p ro d u c e

tw o ty p e s o f s ignals . T h e f irs t ty p e a re t im e a n d e n e rg y s ignals , w h ic h a re ro u te d

th ro u g h a s ta n d a rd fa s t-s lo w e le c tro n ic s s e t-u p . T h e t im e s ignals a re fa s t pulses

fo rm e d in a c o n s ta n t fra c t io n t im in g a m p lif ie r f ro m th e fa s t p re -a m p lif ie r signals.

T h e e n e rg y in fo rm a tio n is c a rr ie d b y s lo w e r s ignals p ro p o rt io n a l to th e en erg y

d e p o s ite d in th e d e te c to rs a n d sen t th ro u g h lin e a r p re -a m p lif ie rs an d a m p lif ie rs .

T h e s e s igna ls a re passed on to th e A D C ’s a n d T D C ’s a t th e in p u t to th e

e le c tro n ic s /c o m p u te r in te r fa c e . T h e e n e rg y s ignals go to th e s ig n a l in p u ts o f th e

A D C ’s a n d th e t im e s igna ls go to th e s to p in p u ts o f th e T D C ’s. T h e s e d e te c to r

en erg ies a n d tim e s a re th e p r im a r y in fo rm a tio n g a th e re d d u r in g d u r in g th e

e x p e r im e n t.

T h e second ty p e o f s igna ls a re also t im e s ignals , b u t in s te a d o f g o in g to T D C ’s,

156

Event Type Trigger

Figure 3-20: Schematic diagram of the electronics system.(The symbols are defined in Table 3-1.)

157

A A m p lif ie r

A D C A n a lo g -to -d ig ita l c o n v e r te rA D C A A n a lo g -to -d ig ita l c o n v e r te r a d a p te r

B S B o re r sca ler

C C o in c id e n c eC D C h a rg e d ig it iz e rC F C o n s ta n t f ra c tio n

C S C A M A C sca ler

D D is c r im in a to r

F D F re q u e n c y d iv id e r

F I F a n - in

F O F a n -o u t

G G G a te g e n e ra to r

L A L e v e l a d a p te r

P A P r e a m p lif ie r

P U P a t te r n u n it

R M R a te m e te r

R M S R a te m e te r se lec to r

T D C T im e - to -d ig i t a l c o n v e r te r

T F A T im in g f i l te r a m p lif ie r

T a b l e 3 - 1 : S y m b o ls used in F ig u re s 3 -2 0 to 3 -2 6

th e y a re ro u te d to a log ic t im e -c o in c id e n c e c irc u it . T h e t im e signals a re fa s t logic

pulses p ic k e d o f f th e d is c r im in a to r fo llo w in g each c o n s ta n t fra c t io n u n it a n d a re

th u s id e n t ic a l to th e t im e signals w h ic h go to th e T D C ’s. T h e y p ro v id e a

h a rd w a re t im e c o in c id en ce b e tw e e n in te re s tin g groups o f d e te c to rs to a c t as th e

t r ig g e r fo r th e c o lle c tin g a n d tra n s fe r o f d a ta ( th e f irs t ty p e o f signals, describ ed

a b o v e ) to th e c o m p u te r th ro u g h th e c o m p u te r in te r fa c e . T h is a llo w s th e se lec tio n

o f o n ly those e ven ts fo r s to ra g e o n m a g n e tic ta p e a n d an a lys is w h ic h fu lf i l l

m in im u m re q u ire m e n ts as in te re s tin g ev e n ts , w h ic h is a necessary re q u ire m e n t

because th e d e te c to rs ’ s ingles c o u n tin g ra te s o f 104 to 103 H z w e re m u c h to o h igh

to re c o rd a ll e ven ts on m a g n e tic ta p e . T h e m a x im u m d a ta tra n s fe r ra te w ith

158

Logic

Figure 3-21: Schematic diagram of the positron-detection electronics.(The symbols are defined in Table 3-1.)

159

.Annihilation _ “ Radiation - -Detectors- -

| Segment I \Segment 2 f

Segment 3 \Segment 4 f

Segment 5 [

Segment 6 \Segment 7~ )

J Segment 8

ADC/TDC

TLogic

ADC/TDC

Segment I or 2or . . . or-8

Energy

Nal DynodePA

Detector AnodePDTime

• A

CF - PD • D I GG

ADC

TDC

Logic

Figure 3-22: Schematic diagram of theannihilation-radiation-detection electronics.

(The symbols are defined in Table 3-1.)

160

Logic n\

\\

No I Deteetor

Logic

Figure 3-23: Schematic diagram of the gamma-ray-detection electronics.(The symbols are defined in Table 3-1.)

161

Logic \

Figure 3-24: Schematic diagram of the particle-detection electronics.(The symbols are defined in Table 3-1.)

162

Figure 3-25: S c h e m a tic d ia g ra m o f th e e v e n t-ty p e -s e le c tio n e lec tro n ics .

(T h e sym b o ls a re d e fin e d in T a b le 3 -1 .)

163

Positron Energy

Positron TimeAnnotation Rodiotion Energy

r H U h E rI ^ g g H la

GG

■RdcH a a]—

■ p H S " ]8x

TTDCf

CAMACInterface

r i meIhh

u g g .

Tr igger

Figure 3-26: Schematic diagram of the ADC/TDC electronics(The symbols are defined in Table 3-1.)

164

a b o u t 10 -20 % d e a d t im e , d u e m a in ly to tra n s fe r t im e , w as a b o u t 102 H z fo r th e 3 2

p a ra m e te rs w e re c o rd e d .

3.2.1.2. Types of Events.

F o r th is e x p e r im e n t w e w e re in te re s te d in re c o rd in g th re e ty p e s o f events:

• p o s itro n s , w h ic h w e re th e p r im a r y in te re s t,

• g a m m a ra y s , necessary fo r c a lc u la t in g th e n u c le a r p o s itro n

b a c k g ro u n d ,

• a n d s c a tte re d p a rtic le s , used to n o rm a liz e th e p o s itro n a n d th e

g a m m a -ra y ev e n ts , w h ic h w e re b o th m e a s u re d in co in c id en ce w ith

p a rtic le s .

T h e m in im u m re q u ire m e n ts fo r a p o s itro n e v e n t w e re ta k e n to be a t im e

co in c id e n c e b e tw e e n a t le as t o n e o f th e tw o h a lv e s o f th e p o s itro n d e te c to r , a t

le a s t o n e o f th e e ig h t seg m en ts o f th e a n n ih ila t io n -ra d ia t io n d e te c to r , a n d b o th

p a r t ic le d e te c to rs . T h is p ro d u c e d a c o u n tin g ra te o f 0.1 — 2 H z , d e p e n d in g on th e

s y s te m m e a s u re d . A g a m m a -ra y e v e n t w a s ta k e n to b e a t im e co in c id en ce

b e tw e e n o n e o f th e tw o g a m m a -ra y d e te c to rs a n d b o th p a r t ic le d e te c to rs . T h e

re a r g a m m a -ra y e v e n t h a d a c o u n tin g ra te o f 10 — 3 0 H z ; th e fo rw a rd h ad a

h ig h e r c o u n tin g r a te o f 5 0 — 100 H z because o f th e n e u tro n s g o in g fo rw a rd . T h e

n o r m a liz in g s c a tte re d -p a r t ic le e v e n t w as a c o in c id e n c e b e tw e e n th e tw o p a r t ic le

d e te c to rs . T h is h a d a c o u n tin g r a te o f 2 — 4 X 104 H z .

T h e t im e co in c id en ces necessary to fo rm these e v e n t ty p e tr ig g e rs a re m a d e in

tw o steps. F ir s t th e p ro p e r lo g ic s ig n a l is e x tra c te d f r o m each d e te c to r system :

• th e t im e s igna ls f r o m th e tw o h a lv e s o f th e p o s itro n d e te c to r a re fed

in to a fa n - in (F ig u re 3 -2 1 ) ,

• th e t im e s igna ls f r o m th e e ig h t seg m en ts o f th e a n n ih ila t io n ra d ia t io n

d e te c to r a re fe d in to a n o th e r fa n - in (F ig u re 3 -2 2 ) ,

• a n d th e t im e s igna ls fro m th e a n o d e fo ils (w h ic h w e re m u c h s h a rp e r

t im e s igna ls th a n those f r o m th e c a th o d e d e la y lin es ) a re fed in to a

c o in c id e n c e u n i t (F ig u re 3 -2 4 )

165

T h is p ro d u ces a fa s t t im e s ig n a l c o rre s p o n d in g to an or o f th e tw o h a lves o f th e

p o s itro n d e te c to r as w e ll as o f th e e ig h t segm ents o f th e a n n ih ila t io n ra d ia t io n

d e te c to r , a n d an and s ig n a l fo r th e tw o p a r t ic le d e te c to rs .

In th e n e x t s tep , sh ow n in F ig u re 3 -2 5 , th e or s ig n a l f ro m th e p o s itro n d e te c to r

h a lv e s , th e or s ig n a l f ro m th e a n n ih ila t io n -ra d ia t io n d e te c to r segm ents , a n d th e

and s ig n a l f ro m th e p a r t ic le d e te c to rs ’ anodes is fed in to a co in c id en ce u n it (a n

and g a te ) to fo rm th e t im e s ig n a l c o rre s p o n d in g to a p o s itro n e v e n t. T h e t im e

s ig n a l f ro m th e fo rw a rd g a m m a -ra y d e te c to r is fed in to a c o in c id e n c e u n it w ith

th e and s ig n a l f ro m th e tw o p a r t ic le d e te c to rs ’ anodes to fo rm th e fo rw a rd

g a m m a -ra y e v e n t. T h e re a r g a m m a -ra y e v e n t is fo rm e d in th e sam e w a y w ith th e

t im e s ig n a l f ro m th e re a r g a m m a -ra y d e te c to r . F in a l ly , th e and s ig n a l f ro m th e

p a r t ic le d e te c to rs ’ anodes is fed a lo n e in to a co in c id en ce u n it to p ro d u c e th e t im eA

s ig n a l c o rre s p o n d in g to th e n o rm a ly z in g s c a tte re d -p a r t ic le e v e n t type® .

T h e o u tp u ts f ro m th e fo u r co in c id en ce u n its , c o rre s p o n d in g to th e fo u r ty p e s o f

d es ired even ts , go to a p a t te r n u n i t in th e c o m p u te r in te r fa c e to re c o rd w h ic h

e v e n t ty p e s a re p re s e n t each t im e d a ta is tra n s fe re d a n d a re also fed in to a fa n -in

(F ig u re 3 -2 5 ) . T h e o u tp u t o f th is or g a te is th e n th e tr ig g e r fo r th e tra n s fe r o f

d a ta to th e c o m p u te r . A s sh ow n in F ig u re 3 -2 6 , i t is th e g a te fo r a l l o f th e A D C ’s

a n d th e tw o c h a rg e d ig it iz e rs fo r th e p a r t ic le d e te c to rs , th e c o m m o n s ta r t fo r a ll

o f th e T D C ’s, a n d th e g a te fo r th e p a t te r n u n it re c o rd in g w h ic h e v e n t ty p e s w e re

p re s e n t. I t in d ic a te s th e p resence o f a co in c id e n c e o f a t le a s t o n e in te re s tin g set o f

d e te c to rs .

QThe coincidence unit is used for the scattered particle event with only one input so that the

resulting time signal will have the same shape and delay as the other three event types.

166

S in c e th e m a x im u m to ta l c o u n tin g ra te w h ic h th e c o m p u te r in te r fa c e co u ld

t ra n s fe r w i th as m a n y p a ra m e te rs as w e h a d (3 2 ) a n d a rea s o n a b le d e a d t im e

( < 2 0 % ) w a s a b o u t 102 H z , i t w as necessary to scale d o w n th e fo rw a rd g a m m a -

ra y e v e n t ty p e b y a fa c to r o f a b o u t 2 a n d th e n o r m a liz in g s c a tte re d -p a r t ic le e v e n t

ty p e b y a fa c to r o f a b o u t 2 0 0 0 . T h is m a d e th e c o u n t r a te fo r th e tw o g a m m a ra y

e v e n t ty p e s a b o u t e q u a l a n d re d u c e d th e p a r t ic le e v e n t ty p e to th e p o in t w h e re

th e s u m o f a l l fo u r e v e n t ty p e c o u n tin g ra te s w a s a b o u t 102 H z . T h e c o u n tin g

ra te s fo r th e fo u r e v e n t ty p e s a f te r sc a le d o w n w as a b o u t 1 H z fo r th e p o s itro n

e v e n t ty p e a n d a b o u t 2 0 — 3 0 H z fo r each o f th e o th e r th re e e v e n t ty p e s . T h u s

w e c o lle c te d as m a n y p o s itro n s as possib le a n d m o re th a n a n o rd e r o f m a g n itu d e

m o re o f th e g a m m a ra y s n eed ed fo r th e b a c k g ro u n d c a lc u la tio n a n d th e p a rtic le s

n eed ed fo r n o r m a liz a t io n .

T h e s c a lin g d o w n o f th e e v e n t ty p e s is d o n e b e tw e e n th e fo rm a tio n o f th e e v e n t

ty p e s in th e la s t c o in c id e n c e u n its a n d th e fa n - in w h ic h fo rm s th e tr ig g e r fo r th e

c o m p u te r in te r fa c e , as sh o w n in F ig u re 3 -2 5 . T h e s c a lin g d o w n is m a d e in a

s ta t is t ic a l m a n n e r b y m a k in g a n o v e r la p c o in c id e n c e b e tw e e n th e e v e n t ty p e fa s t

lo g ic s ig n a l a n d a sq u a re w a v e . T h e sq uare w a v e is g e n e ra te d w ith a fre q u e n c y o f

1 M H z a n d a w id th e q u a l to o n e -h a lf o f th e c y c le t im e . T h is w a v e fo rm goes

th ro u g h a fre q u e n c y d iv id e r w h ic h e lim in a te s cycles b e fo re b e in g fed in to th e

o v e r la p c o in c id e n c e u n it . T h e s e tt in g o f th e fre q u e n c y d iv id e r d e re rm in e s h o w

o fte n th e s q u are w a v e is p re s e n t a t th e c o in c id e n c e u n it . S in ce th e n a r ro w e v e n t

ty p e s ig n a l com es ra n d o m ly , th is m e th o d a llo w s ra n d o m e v e n t ty p e signals

th ro u g h w i th a n a v e ra g e s c a le d o w n d e te rm in e d b y th e in te g e r s e tt in g o n th e

fre q u e n c y d iv id e r . A n e x a m p le is sh o w n in F ig u re 3 -2 7 .

A f in a l c o n s id e ra tio n is th e t im in g o f th e c o in c id e n c e tr ig g e rs . T h e t im in g

c h a ra c te r is tic s o f th e fo u r ty p e s o f d e te c to rs a re q u ite d if fe re n t . In a d d it io n , th e

3.2.1.3. Scaledowns.

167

i mhz riS 0 . . J ]

Events

Scale

ru2 Down

TJUUnit

TJ1Jla

SD Events

— T i m e - *

F i g u r e 3 - 2 7 : D ia g ra m o f s ignals in th e sca led o w n u n it .

G S I U N I L A C is a p u lsed a c c e le ra to r w ith a m a c ro p u ls e fre q u e n c y o f 27 M H z an d

th e c o m p u te r in te r fa c e , as d escrib ed in th e n e x t sec tion , re q u ire s a b o u t 3 3 0 psec

to tra n s fe r th e d a ta o f one e v e n t to th e c o m p u te r . T h e c o m p ro m is e s o lu tio n to

these v a r io u s t im in g d e m a n d s w as to m a k e th e h a rd w a re t im e co in c id en ce w in d o w

3 0 0 nsec w id e b y m a k in g an o v e r la p co in c id en ce fo r th e e v e n t ty p e s b e tw e e n fas t

lo g ic pulses 150 nsec w id e . T h is w id th in c lu d es a b o u t e ig h t b e a m m acrop u lses .

O f course, th e s o ftw a re t im e c o in c id en ce w in d o w s used la te r d u r in g th e d a ta

an a lys is w e re m u c h n a rro w e r , on th e o rd e r o f 2 0 to 3 0 nsec, a n d c o m p a ra b le to

th e t im e b e tw e e n b e a m m acrop u lses .

168

O n c e th e t im e a n d e n e rg y s igna ls fro m th e d e te c to r system s h a v e been sen t to

T D C ’s a n d A D C ’s, re s p e c tiv e ly , a lo n g w ith a tr ig g e r s ig n a l in d ic a t in g th e p resence

o f a n in te re s tin g e v e n t, b y th e e le c tro n ic s s ys tem , th is in fo rm a tio n is t ra n s fe rre d

to th e c o m p u te r s y s te m fo r s to ra g e on m a g n e tic ta p e a n d o n lin e an a lys is b y th e

in te r fa c e b e tw e e n th e e le c tro n ic s a n d th e c o m p u te r system s. A s ta n d a rd

C A M A C [ E S O N E 7 2 , I E E E 7 5 , IE E E 76 ] in te r fa c e w i th a p a ra lle l b ra n c h d r iv e r

d r iv in g tw o C A M A C c ra te s w as used.

3 .2 .2 .I. DMA Data Transfer.

T h e p r im a r y p ro b le m o f th is in te r fa c e is t h a t o f t im in g : m a tc h in g th e t im in g o f

th e e le c tro n ic s to t h a t o f th e c o m p u te r . T o th is e n d , tw o m odes o f d a ta c o lle c tio n

w e re used s im u lta n e o u s ly . T h e f irs t m o d e consists o f th e tra n s fe r o f a l l o f th e

v a lu e s s to re d in A D C ’s a n d T D C ’s a n d th e p a t te r n u n it as a lis t o f n u m b e rs to a

b u f fe r in th e c o m p u te r m e m o ry co re each t im e a tr ig g e r fro m th e e v e n t- ty p e log ic

u n it in d ic a te d th a t an in te re s tin g e v e n t ty p e h a d been d e te c te d . T h is e v e n t-b y -

e v e n t re c o rd in g o f a l l m e a s u re d v a lu e s a llo w s a la te r , m o re c o m p le te , o ff l in e

an a ly s is o f th e e x p e r im e n t l im ite d o n ly b y th e gross h a rd w a re c o in c id e n c e o f 3 0 0

nsec.

T h e t im in g fo r th is c o m p lic a te d tra n s fe r w a s h a n d le d b y a i f a n d o m S c a n M o d e

(R A S M O ) m o d u le w h ic h s tee red th e A D C ’S, T D C ’s, th e p a t te r n u n it , a n d th e

C A M A C s ys tem . T h e R A S M O m o d u le is a s low co in c id e n c e u n it . W h e n e v e r a

t r ig g e r fro m th e e v e n t ty p e log ic c ir c u it a rr iv e s , in d ic a t in g a g ro u p o f t im e signals

a r r iv in g w i th in th e 3 0 0 nsec c o in c id e n t t im e w in d o w , i t s ta r ts th e process o f d a ta

tra n s fe r . I t f irs t w a its 4 nsec, to a llo w a n y o th e r c o in c id e n t t im e a n d en e rg y

signals to a r r iv e a t th e A D C ’s a n d T D C ’s, th e n sends in h ib it s ignals to a ll A D C ’s

a n d T D C ’s to p re v e n t la te r a p p e a r in g t im e a n d e n e rg y s igna ls fro m o v e rw r it in g

th e s igna ls w h ic h a r r iv e d d u r in g th e p ro p e r c o in c id e n t t im e in te r v a l. F o llo w in g

3.2.2. C om puter In terface

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th is , i t w a its 100 //sec fo r a ll th e T D C ’s a n d A D C ’s to c o m p le te th e process of

c o n v e r t in g th e a n a lo g s ignals to d ig ita l s ignals , a n d th e n sends a ■ L o o k -a t -M e "

( L A M ) re a d y s ig n a l to th e p a t te r n u n it . T h is signals th e C A M A C system th a t th e

d a ta tra n s fe r can b eg in .

A lis t o f th e lo c a tio n s w ith in th e C A M A C c ra te o f th e A D C ’s a n d T D C ’s and

th e p a t te r n u n it to b e re a d in to th e c o m p u te r fo r each e v e n t is s to re d ( f ro m th e

in i t ia l iz a t io n p ro g ra m ) in th e co re o f th e c o m p u te r . T h e C A M A C sy s te m fo llo w s

th is lis t, tra n s fe r in g th e v a lu e s to re d in each m o d u le in tu r n th ro u g h th e c ra te

c o n tro lle r a n d b ra n c h d r iv e r ( C A l l - C ) d ire c t ly to a b u f fe r in th e c o m p u te r . T h e

t ra n s fe r re q u ire s 7 //sec fo r each p a ra m e te r . W h e n th e la s t C A M A C m o d u le in

th e lis t has been re a d , i t sends a s ig n a l to th e R A S M O m o d u le , w h ic h in tu rn

sends a rese t s ig n a l to a ll C A M A C m o d u les , p re p a r in g th e in te r fa c e to a w a it th e

n e x t e v e n t. T h e e n t ire tra n s fe r process ta k e s 3 2 8 //sec, o f w h ic h th e g re a te s t p a r t

(2 2 4 //sec) is th e s e ria l tra n s fe r o f 3 2 p a ra m e te rs fro m C A M A C m o d u les to th e

c o m p u te r b u ffe r . T h is D ir e c t M e m o r y A ccess ( D M A ) a p p ro a c h by-passes th e

c o m p u te r c o n tro l p ro g ra m a n d th u s th e d a ta ac q u is itio n can p ro ceed

s im u lta n e o u s ly w ith a n d in d e p e n d e n tly o f th e o n lin e an a lys is .

3.2.2.2. DMI Data Transfer.

In th e second m o d e o f d a ta tra n s fe r , s ingles s p e c tra fro m in d iv id u a l d e te c to rs

a re fo rm e d d ire c t ly in a p a r t o f th e m e m o ry core o f th e o n lin e c o m p u te r ,

in d e p e n d e n tly o f th e e v e n t-b y -e v e n t s to ra g e o f th e e n t ire A D C / T D C lis t described

a b o v e . T h is is h a n d le d b y th e D ir e c t M e m o r y In c re m e n t ( D M I ) system . S e lected

signals a re each fed th ro u g h an A D C ( th e t im e s ignals g o in g f irs t th ro u g h a T A C )

in to an A D C a d a p te r in th e second C A M A C c ra te 9 . T h e va lu e s reg is te re d in each

A D C w e re tra n s fe re d to th e c o m p u te r as fa s t as possib le a n d in d e p e n d e n tly o f th e

9An ADC adapter was necessary because standard CAMAC ADC’s were not fast enough for the4 5high singles rates on the order of 10 — 10 Hz being recorded.

170

e v e n t-b y -e v e n t tra n s fe r d escrib ed ab o v e , a n d in c re m e n te d th e p ro p e r c h a n n e l o f a

s p e c tru m s to re d in m e m o ry . T h is m o d e o f d a ta tra n s fe r p ro v id e d e ig h t

m u lt ic h a n n e l a n a ly s e rs , w i th th e a d d e d a d v a n ta g e th a t th e re s u lt in g singles

s p e c tra w e re s to re d in th e c o m p u te r sy s te m a n d c o u ld b e eas ily h a n d le d an d

tra n s fe rre d to m a g n e tic tap es .

T h e fo llo w in g s igna ls w e re m o n ito re d :

• th e e n e rg y s igna ls o f b o th h a lv e s o f th e p o s itro n d e te c to r ,

• th e e n e rg y s ig n a l o f o ne o f th e segm ents o f th e a n n ih ila t io n -ra d ia t io n

d e te c to r ,

• th e e n e rg y s igna ls o f b o th g a m m a -ra y d e te c to rs ,

• th e t im e s igna ls o f th e d e la y lines o f b o th p a r t ic le d e te c to rs ,Il

• a n d th e e n e rg y s ig n a l o f one o f tw o s u rfa c e b a r r ie r d e te c to rs p la c e d in

th e ta r g e t c h a m b e r to m o n ito r th e c o n d itio n o f th e b e a m a n d th e

ta r g e t .

T h e s e e ig h t s ing les s p e c tra w e re used to m o n ito r th e c o n d itio n o f th e d e te c to r

system s a n d in th e case o f th e p a r t ic le d e te c to r a n d th e s u rfa c e b a r r ie r d e te c to rs

th e c o n d itio n o f th e ta r g e t a n d th e b e a m .

3.2.3. Online Computer System

T h e th ir d a n d f in a l p a r t o f th e d a ta a c q u is itio n sy s te m is th e o n lin e c o m p u te r

sy s te m , w h ic h serves tw o im p o r ta n t ro les. I t is used to a n a ly s e th e in c o m in g d a ta

o n lin e in o rd e r to m o n ito r th e progress o f th e e x p e r im e n t, a n d i t reco rd s th e

m e a s u re d d a ta on m a g n e tic ta p e fo r la te r an a lys is a n d s tu d y . T h e o n lin e an a lys is

w a s necessary to m o n ito r d u r in g th e e x p e r im e n t th e w o rk in g c o n d itio n o f th e

b e a m , ta r g e t , v a c u u m sys tem , a n d d e te c to rs , a n d to d e te rm in e w h e n a

m e a s u re m e n t h a d a c c u m u la te d en o u g h d a ta to b e s ta t is t ic a lly s ig n if ic a n t an d

c o u ld b e e n d e d . T h e c o m p le x ity o f th e re q u ire d an a lys is , h o w e v e r, p re c lu d e d a

f in a l an a lys is o n lin e . R e c o rd in g th e e x p e r im e n t e v e n t b y e v e n t on ta p e m a d e

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possib le a re p e a te d an a lys is o f th e e x p e r im e n t in o rd e r to s tu d y d if fe re n t and

o c c a s io n a lly u n a n t ic ip a te d aspects o f th e m e a s u re m e n t.

T h e c o m p u te r sys tem used fo r o u r e x p e r im e n t w as th e .E x p e r im e n ta l D a t a

A c q u is it io n S y s te m (E D A S ) d e v e lo p e d a t G S I fo r th e tw o c o m p u te rs a v a ila b le

th e re fo r d a ta a c q u is itio n a n d an a lys is , th e D E C P D P -11 a n d th e IB M / 3 7 0 .

3.2.3.1. Hardware.

A D E C P D P -11 c o m p u te r w as used fo r th e d ire c t a c q u is itio n o f d a ta fro m th e

e x p e r im e n ta l a p p a ra tu s th ro u g h th e C A M A C in te r fa c e , a n d fo r s im p le o n lin e

an a lys is . T h e P D P -11 accesses th e d a ta th ro u g h th e B ra n c h D r iv e r ( D E C C A l l -

c ) o f th e C A M A C in te r fa c e . In th e case o f th e D M I m o d e o f d a ta tra n s fe r ( th e

d ire c t fo rm a tio n o f m u lt ic h a n n e l- l ik e singles s p e c tra , d escrib ed a b o v e ), s p e c tra

s to re d in th e P D P - l l ’s m e m o ry co re a re in c re m e n te d b y th e va lu e s tra n s fe rre d

f r o m th e A D C ’s. In th e case o f th e D M A m o d e o f d a ta tra n s fe r ( th e e v e n t-b y -

e v e n t w r i t in g o f a l l A D C a n d T D C v a lu e s ), th e tra n s fe rre d lis t o f va lu e s is s to re d

in a b u f fe r in co re . W h e n th e b u f fe r is fu ll , i t is tra n s fe rre d to a P D P ta p e d r iv e

a n d w r i t te n o n to s ta n d a rd 8 0 0 -b y te s -p e r- in c h (b p i) m a g n e tic ta p e . T h e tra n s fe r

o f d a ta to ta p e is g iven th e h ig h e s t p r io r i ty . S im u lta n e o u s ly , b u t w ith a lo w e r

p r io r i ty , th e P D P -11 c o m p u te r read s as m a n y o f th e e ven ts w r it t e n in to th e b u ffe r

as possib le a n d ana lys is th e m w ith th e o n lin e an a lys is p ro g ra m . T h is m o d e o f

E D A S w as used fo r o u r c a lib ra t io n ru n s b e fo re an d a f te r th e e x p e r im e n t a n d w as

th e b a c k u p sys tem d u r in g th e e x p e r im e n t.

F o r e x p e rim e n ts such as ours w h ic h re q u ire m o re s to ra g e a n d an a lys is p o w e r fo r

th e o n lin e an a lys is th a n o ffe re d b y th e P D P -11 c o m p u te r , th e P D P -11 is jo in e d to

a la rg e r c o m p u te r , G S I ’s IB M /3 7 0 - 1 6 8 , th ro u g h an IB M S y s te m 7 L in k . In th is

case th e P D P -11 acts as an ex ten s io n o f th e c o m p u te r in te r fa c e a n d tra n s fe rs d a ta

to th e S y s te m 7 L in k in s te a d o f th e P D P ta p e d r iv e . T h e r e , as describ ed ab o ve

fo r th e case o f th e P D P -11 s ta n d in g a lo n e , th e d a ta is sen t w ith h ig h es t p r io r i ty to

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a n I B M ta p e d r iv e to be w r it t e n o n to s ta n d a rd 6 2 5 0 -b p i m a g n e ic ta p e . W i t h a

lo w e r p r io r i ty , as m u c h o f th is d a ta as possib le is m a d e a v a ila b le to th e I B M / 3 7 0

fo r o n lin e a n a ly s is . T h e P D P -11 is th e n used s im p ly to c o lle c t s ingles s p e c tra an d

th e sc a le r v a lu e s w h ic h a re necessary fo r m e a s u rin g th e sc a le d o w n a n d d e a d t im e

c o rre c tio n . A s an a d d it io n a l bonus, th e IB M tap es h e ld a lm o s t e ig h t t im e s as

m u c h d a ta as th e P D P tap es d u e to th e h ig h e r w r i t in g d e n s ity o f IB M ta p e d riv e s .

T h is m e a n t c h a n g in g tap es , a 10 m in u te p e r io d o f d o w n tim e , o n ly o n e -e ig h th as

o fte n d u r in g th e e x p e r im e n t.

3 . 2 . 3 . 2 . S o f t w a r e .

T h e s o ftw a re p a r t o f th e d a ta acq u is is tio n sy s te m E D A S is a spec ia l c o m p u te r

la n g u a g e d e v e lo p e d a t G S I c a lle d S IM P L E (S y s te m In d e p e n d e n t A /a c ro

/P ro g ra m in g L a n g u a g e fo r E D A S ) . S IM P L E is a p ro b le m -o r ie n te d , h ig h -le v e l

p ro g ra m in g la n g u a g e based on th e a n a ly s e r c o n c e p t. I t uses G S I-w r i t t e n m acros ,

o r s p e c ia l c o m m a n d s , w h ic h a re tu rn e d in to in lin e c o m p u te r code b y a S P IT B O L

p re -c o m p ile r to m a n ip u la te d a ta in th e fo rm o f a n a lysers , o r s p e c tra , w h ic h can be

c re a te d , in c re m e n te d , a n d m a n ip u la te d .

T o m a tc h th e h a rd w a re , S IM P L E has tw o p a rts , a n d consists o f tw o v e ry s im ila r

p ro g ra m in g lan g u ag es , o ne fo r th e P D P -11 a n d o n e one fo r th e IB M / 3 7 0 .

S IM P L E is ru n on th e P D P -11 u s in g G O L D A ( G S I O n /in e D a t a A c q u is it io n ),

w h ic h is based on th e p ro g ra m in g la n g u a g e P L l l , d e v e lo p e d a t C E R N fo r th e

P D P - 11 . T h e second p a r t o f S IM P L E , fo r th e IB M / 3 7 0 , is S A T A N (S y s te m to

A n a ly s e T re m e n d o u s A m o u n ts o f N u c le a r D a ta ) , a n d is based on th e IB M -

d e v e lo p e d p ro g ra m in g la n g u a g e P L / I .

S in c e o u r e x p e r im e n t w as c o n d u c te d w ith th e P D P -11 c o m p u te r lin k e d to th e

IB M / 3 7 0 , an a ly s is p ro g ra m s w e re w r i t t e n fo r b o th m ac h in e s . T h e p ro g ra m fo r

th e P D P - 11 , w r i t t e n in G O L D A , h a d th re e p a r ts . T h e f irs t p a r t is an

in i t ia l iz a t io n p a r t w h ic h c o n ta in s a lis t o f th e lo c a tio n s in o u r tw o C A M A C cra tes

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o f a ll th e A D C ’s, T D C ’s, p a t te r n u n its a n d scalers. T h is lis t is used b y th e

C A M A C in te r fa c e fo r b o th th e D M A a n d D M I m odes o f d a ta ac q u is itio n ,

d escrib ed a b o v e . T h e second p a r t co n ta in s a s im p le o n lin e an a lys is ro u tin e . S ince

i t o n ly h a d to a u g m e n t a n d check th e m a in o n lin e an a lys is d o n e w ith th e

IB M / 3 7 0 , i t s im p ly g e n e ra te d singles s p e c tra fo r m o n ito r in g th e e x p e r im e n ta l

a p p a ra tu s a n d th e c o n d itio n o f th e b e a m a n d ta r g e t . T h e th ir d p a r t o f th e

p ro g ra m w as a sp ec ia l ro u tin e execu ted o n ly w h e n th e a c q u is itio n o f d a ta w as

s ta r te d o r s to p p e d . A t each s ta r t , th is ro u t in e c le a re d C A M A C scalers used to

re c o rd th e sing les c o u n tin g ra te s o f a ll th e d e te c to rs fo r n o rm a liz a t io n purposes.

A t each h a lt o f d a ta a q c u is it io n , i t tra n s fe rre d th e c o n te n ts o f th e C A M A C scalers,

to a n a n a ly s e r in th e m e m o ry o f th e P D P - 11 . In a d d it io n to th e th re e p a rts , th e

an a ly s is p ro g ra m also a u to m a tic a lly c o lle c te d th e D M I singles s p e c tra in to

a n a ly s e rs . A t th e end o f each m e a s u re m e n t, a ll o f these s p e c tra w e re w r it t e n o n to

th e IB M ta p e as sing les s p e c tra .

T h e p ro g ra m fo r th e E B M /3 7 0 , w r it te n in S A T A N , c o n ta in e d th e m a in o n lin e

an a ly s is ro u tin e . T h e o n lin e an a lys is p ro g ra m w as a s tre a m lin e d v e rs io n o f th e

o ff l in e an a lys is , o m it t in g such tim e -c o n s u m in g a n d less im p o r ta n t p ro c e d u re s as

a b s o lu te c a lib ra t io n o f th e A D C a n d T D C signals , c h e c k in g o f tw o d im e n s io n a l

w in d o w s to d e te rm in e t im e a n d a n g u la r co inc idences, a n d c o rre c tio n o f th e t im e

s p e c tra fo r d r i f t . S in ce th e o ffl in e an a lys is is d escrib ed in d e ta il in C h a p te r 6 , th e

o n lin e an a lys is p ro g ra m w i l l be o n ly b r ie f ly o u t lin e d h ere .

T h e p ro g ra m analyses each e v e n t as a llo w e d b y th e h ig h e r p r io r i ty o f w r it t in g

th e e v e n t-b y -e v e n t d a ta o n to m a g n e tic ta p e . F o r each e v e n t, s p e c tra fo r each o f

th e th re e e v e n t ty p e s a re p ro d u c e d :

• fo r th e s c a tte re d -p a r t ic le e v e n t, p lo ts o f th e p a r t ic le in te n s ity versus

th e s c a tte r in g angles o f b o th s c a tte re d p a rtic le s , as d e te c te d in th e tw o

p a r t ic le co un ters ;

• fo r each o f th e tw o g a m m a -ra y even ts , c o rre s p o n d in g to th e fo rw a rd

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a n d re a r g a m m a -ra y d e te c to rs ,re s p e c tiv e ly , a p lo t o f th e d e te c te d

g a m m a -ra y in te n s ity versus g a m m a -ra y e n e rg y a n d th e s c a tte r in g

angles o f b o th s c a tte r in g p a rtic le s ;

• fo r th e p o s itro n e v e n t ty p e , (a f te r c h e c k in g th a t a t le as t o n e 5 1 1 -k e V

a n n ih ila t io n g a m m a ra y h a d been d e te c te d in t im e co in c id en ce w ith

th e p o s itro n ), a p lo t o f th e d e te c te d p o s itro n in te n s ity versus th e

p o s itro n e n e rg y a n d th e s c a tte r in g angles o f b o th c o llid in g p a rtic le s .

As d e s c rib e d b e lo w in C h a p te r 5 , these s p e c tra w e re used d u r in g th e e x p e r im e n t

to m o n ito r th e q u a lity o f th e b e a m a n d ta r g e t , to d e te rm in e th e a m o u n t o f d a ta

c o lle c te d , a n d even to m a k e a p re lim in a r y search fo r in te re s tin g s tru c tu re s in th e

p o s itro n e n e rg y a n d a n g u la r d is tr ib u tio n s .

Chapter 4

Calibration of the Apparatus

T h e d e te c to r system s o f th e a p p a ra tu s , d escrib ed in th e la s t c h a p te r , w e re

c a lib ra te d to p ro d u c e ab s o lu te y ie ld s , to c o m p a re w ith th e o re tic a l c a lc u la tio n s ,

a n d to c h eck w i th each o th e r fo r co ns is tency . A n en erg y c a lib ra t io n o f th e

p o s itro n d e te c to r , th e a n n ih ila t io n -ra d ia t io n d e te c to r , a n d th e g a m m a -ra y

d e te c to rs w as m a d e us ing ra d io a c tiv e sources. A n a n g le c a lib ra t io n o f th e p a r t ic le

d e te c to rs w as o b ta in e d b y c o m p a r in g th e m easu red s c a tte r in g -a n g le c o rre la tio n s

in s e v e ra l c o llis io n system s w ith c a lc u la te d k in e m a tic s . In a d d it io n , th e response

fu n c tio n a n d th e d e te c tio n e ff ic ie n c y o f th e p o s itro n a n d th e g a m m a -ra y d e te c to rs

w e re m e a s u re d w ith ra d io a c tiv e sources w h ic h p ro d u c e d s ing le o r a fe w w e ll-

s e p a ra te d lines.

T h e c a lib ra tio n s re p ro d u c e d th e a c tu a l m e a s u re m e n t co n d itio n s o f th e

e x p e r im e n t as c los ily as possib le in u s in g th e sam e p h y s ic a l se t-u p an d e le c tro n ics

a n d in b e in g d o n e as close in t im e to th e e x p e r im e n t as co u ld be. T h e e n e rg y an d

a n g le c a lib ra tio n s , w h ic h w e re th e m o s t se n s itiv e to d r if t in g o f th e e lec tro n ics ,

w e re d o n e d u r in g an d im m e d ia te ly a f te r th e a c tu a l e x p e r im e n t. T h e an g le

c a lib ra t io n o f th e p a r t ic le d e te c to rs w as d o n e d u r in g th e e x p e r im e n t, u s in g th e

s a m e -e n e rg y u ra n iu m b e a m an d ta rg e ts o f s im ila r th ickn ess m o u n te d on ta rg e t

fra m e s lik e those used fo r th e e x p e r im e n t. T h e e n e rg y c a lib ra tio n s o f th e p o s itro n

d e te c to r , th e a n n ih ila t io n -ra d ia t io n d e te c to r , a n d th e g a m m a -ra y d e te c to rs w e re

d o n e im m e d ia te ly a f te r th e e x p e r im e n t us ing ra d io a c tiv e sources o f a b o u t th e

sam e d ia m e te r as th e b e a m sp ot, m o u n te d on th in n o n -m a g n e tic b ack in g s g lu ed to

175

176

ta r g e t fra m e s also lik e those used fo r th e e x p e r im e n t. T h e less s en s itive

m e a s u re m e n ts o f lin e shapes a n d d e te c to r e ff ic ie n c y w e re d o n e in th e w eeks

im m e d ia te ly fo llo w in g th e e x p e r im e n t.

4 .1 . D etec tio n o f P ositron s

A p r im a r y g o a l o f th e e x p e r im e n t w as th e e x tra c t io n o f th e c e n te r-o f-m a s s ( C M )

e n e rg y d is tr ib u t io n o f p o s itro n s f ro m th e m e a s u re d d is tr ib u t io n . T h is sec tio n

d escribes th e c a lib ra t io n s m a d e to d e te rm in e th e e ffe c t o f th e m e a s u re m e n t

p ro c e d u re o n th e p o s itro n d is tr ib u t io n so th a t its e ffe c t c o u ld be re m o v e d , as

d e s c rib e d in C h a p te r 6 , b y th e an a lys is . T h e c a lib ra tio n s a re d escrib ed in th e

o rd e r th a t th e an a ly s is rem o v e s th e m , w h ic h is th e reverse o f th e o rd e r in w h ic h

th e ch anges o c c u r to th e p o s itro n s .

A f t e r a n e n e rg y c a lib ra t io n o f th e S i(L i) p o s itro n d e te c to r w ith e le c tro n lin e

sources, th e response fu n c tio n o f th e S i(L i) d e te c to r fo r e le c tro n s w as d e te rm in e d

fr o m m e a s u re m e n ts o f th e lin e sh ape o f s in g le e le c tro n lines . T h e e le c tro n

lin e s h a p e w a s m o d ifie d to b e used fo r p o s itro n s b y e x a m in in g c o n tin u o u s p o s itro n

b e ta -d e c a y s p e c tra . T h e e ff ic ie n c y o f a n n ih ila t io n -ra d ia t io n d e te c tio n in

c o in c id e n c e w ith p o s itro n s w as m e a s u re d b y c o m p a r in g c o n tin u o u s p o s itro n b e ta -

d e c a y s p e c tra m e a s u re d in th e S i(L i ) p o s itro n d e te c to r w i th a n d w ith o u t

c o in c id e n c e to th e a n n ih ila t io n -ra d ia t io n d e te c to r . F in a l ly , th e p o s itro n d e te c tio n - •

e ff ic ie n c y , w h ic h re fle c ts p r im a r i ly th e p o s itro n tra n s p o r t-e ff ic ie n c y a n d th e

a n n ih ila t io n -r a d ia t io n d e te c tio n -e ff ic ie n c y , w as d e te rm in e d b y a c o m p a ris o n o f

u n fo ld e d , m e a s u re d p o s itro n b e ta -d e c a y s p e c tra w i th b e ta -d e c a y th e o ry . T h e s e

steps w i l l b e d escrib ed in th e fo llo w in g th re e sections.

177

4.1.1. Si(Li) Positron Detector Calibration

4.1.1.1. Energy Calibration

T h e e n e rg y c a lib ra t io n o f th e S i(L i) p o s ito n d e te c to r w as m a d e im m e d ia te ly

fo llo w in g th e e x p e r im e n t w ith th e e le c tro n lin e sources 113Sn a n d ^ B i m o u n te d

o n M y la r b a c k in g s g lu e d to fra m e s lik e th o se used fo r th e ta rg e ts a n d p la c e d a t

th e ta r g e t p o s itio n . E le c tr o n sources h a d to be used because o f th e la c k o f

m o n o e n e rg e tic , ra d io a c t iv e p o s itro n sources. A d d it io n a l checks d u r in g th e

e x p e r im e n t its e lf o b s e rv in g th e C o m p to n edges fro m g a m m a -ra y sources p laced

o u ts id e th e s p e c tro m e te r n e a r th e S i(L i) p o s itro n d e te c to r h a d show n th e en e rg y

c a lib ra t io n to be s ta b le w i th in 1 k e V d u r in g th e w e e k -lo n g ru n . T a b le 4 -1 , show n

b e lo w , lis ts th e energ ies o f th e in te r n a l co nvers io n e le c tro n s ( f ro m [L e d e re r an d

S h ir le y 7 8 ]) v is a b le in th e 113Sn a n d ^ B i s p e c tra . T h e e ig h t en e rg y c a lib ra t io n

p o in ts ra n g e d fro m 3 6 4 k e V to 1061 k e V a n d w e re f i t us ing a least-squares

m e th o d w ith a l in e a r c a lib ra t io n fo r b o th h a lv e s o f th e S i(L i) p o s itro n d e te c to r .

4.1.1.2. Response Function for Electrons

T h e response fu n c tio n o f th e S i(L i ) d e te c to r fo r e lec tro n s w as d e te rm in e d b y

m e a s u r in g th e lin e shapes o f e n e rg y d e p o s itio n in th e S i(L i) d e te c to r o f th e

e le c tro n s fro m 113Sn a n d 207B i. S in ce th e 113S n source has o n ly one g a m m a ra y ,

th e lin e sh ape o f its in te r n a l ly c o n v e rte d e le c tro n w as m easu red as a singles

s p e c tru m . T h e m e a s u re d lin e sh ape is sh ow n in F ig u re 4 -1 . T h e in te r n a l

0(\ 7co n v e rs io n e le c tro n lines c o rre s p o n d in g to th e tw o s tro n g g a m m a ra y s o f B i

w e re s e p a ra te d b y m e a s u rin g th e e le c tro n s c o rre s p o n d in g to each g a m m a ra y in

c o in c id e n c e w ith th e o th e r g a m m a ra y d e te c te d in a 3 * X 3 " N a l ( T l ) g a m m a -ra y

d e te c to r . A s seen in th e f ig u re , th e c o n tr ib u t io n f ro m K -s h e ll e lec tro ns d o m in a te s ,

a n d th is p a r t w a s used to d e te rm in e th e response fu n c tio n .

T h e response fu n c tio n o f th e S i(L i) d e te c to r fo r e le c tro n s w as o b ta in e d fro m

these th re e m e a s u re d lin e shapes. A fu n c tio n a l fo rm w a s fo u n d th a t d escrib ed

178

E e.S o u rce E e- e- Weighted

(k e V ) (k e V ) S h e ll A v e ra g e(k e V )

113Sn 3 9 1 .6 9 3 6 3 .7 5 K 3 6 8 .4

3 8 7 .7 L207B i 5 6 9 .6 5 4 8 1 .6 5 K 5 0 2 .6

5 5 5 .0 L

5 6 6 .5 M207B i 1 0 6 3 .6 3 9 7 5 .6 3 K 9 9 5 .3

1 0 4 8 .9 L

1 0 6 0 .5 M

( A l l v a lu e s f ro m [L e d e re r a n d S h ir le y 7 8 j. )

T a b l e 4 - 1 : E le c tr o n lin e source energ ies.

each o f th e th re e m e a s u re d lin e shapes. I t consists b a s ic a lly o f a G u ass ian

fu n c t io n fo r th e p e a k , a s teep e x p o n e n tia l to d escrib e th e le f t s ide o f th e p e a k , a

s h a llo w e x p o n e n tia l to d e s c rib e th e ta i l , a n d an e x p o n e n tia l c u t -o f f fu n c tio n to

d es c rib e th e lo w -e n e rg y en d o f th e ta i l . T h e e le c tro n lin e sh ape is s k e tc h e d in

F ig u re 4 -2 (a ) .

4.1.1.3. Response Function for Positrons

T h e e le c tro n lin e sh ape w as used as th e basis fo r d e te rm in in g th e p o s itro n lin e

s h a p e . T h e u n a v a i la b i l i ty o f m o n o e n e rg e tic p o s itro n lin e sources has a lre a d y been

m e n tio n e d a b o v e . F o r th e p o s itro n lin e sh ape , tw o a d d it io n a l fe a tu re s h a d to be

a d d e d to th e e le c tro n lin e shape: a b o x -s h a p e d fu n c tio n e x te n d in g to h ig h

en erg ies fro m th e G a u s ia n p e a k , a n d a fa c to r m u lt ip ly in g th e tw o e x p o n e n tia l- ta il

fu n c tio n s . T h e s e a re e x p la in e d in th e fo llo w in g p a ra g ra p h s .

S in c e th e p o s itro n tr ig g e r re q u ire s th e d e te c tio n o f e ith e r o n e o r b o th o f th e

179

CO

c:z>o

Ee- [keV]

F i g u r e 4 - 1 : M e a s u re d lin e sh ape in th e S i(L i) d e te c to r .1 1A

T h e m e a s u re d in te n s ity o f e lec tro ns fro m a Sn source m o u n te d a t th e

ta r g e t p o s itio n is p lo tte d as a fu n c tio n o f c h a n n e l n u m b e r . T h e tw o

v is a b le p eaks c o rresp o n d to e lec tro ns o f k in e t ic e n e rg y 3 6 4 k e V ( f ro m K -

sh e ll c o n v e rs io n ) a n d 3 8 8 k e V (f ro m L -s h e ll c o n vers io n ).

a n n ih ila t io n g a m m a ra y s in co in c id en ce w ith th e p o s itro n *0 , th e second,

u n d e te c te d g a m m a ra y can C o m p to n s c a tte r in th e S i(L i) d e te c to r fo llo w in g th e

p o s itro n ’s a n n ih ila t io n . T h e en erg y d ep o s ited b y th e g a m m a ra y in th e S i(L i)

d e te c to r p iles u p w ith th e e n e rg y o f th e p o s itro n , so a b o x -sh ap ed ta i l o f le n g th

3 4 1 k e V ( th e C o m p to n edge fo r a 5 1 1 -k e V g a m m a r a y ) is a d d e d to th e r ig h t side

o f th e p e a k o f th e e le c tro n lin e shape. T h e h e ig h t o f th e box w as d e te rm in e d by

c o m p a r in g th e p o s itro n b e ta -d e c a y s p e c tru m o f 22N a m e a s u re d w ith th e n o rm a l

*°The positron trigger is described in Section 6.1 below.

180

CDrrTLJ

U l

> -*c_03L.

JQL.<

COc13OLJ

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 E / E 0

Figure 4-2: e- a n d e + lin e sh ap e in th e S i(L i) d e te c to r .

T h e lin e sh ap e m e a s u re d in th e S i(L i) d e te c to r fo r e le c tro n s is s k e tc h e d

in p a r t (a ) a n d fo r p o s itro n s in p a r t (b ) as a fu n c tio n o f th e ra t io o f th e

m e a s u re d e n e rg y E to th e e m it te d e n e rg y E Q.

181

o n e -g a m m a -ra y tr ig g e r w ith th a t ta k e n w i th a tr ig g e r on d e te c tin g b o th

a n n ih ila t io n g a m m a rays w ith fu ll e n e rg y . T h e m e a s u re d s p e c tra a re sh ow n in

F ig u re 4 -3 , w h e re th e p ile -u p on th e r ig h t s ide o f th e s p e c tru m ta k e n w ith th e

o n e -g a m m a -ra y tr ig g e r is c le a r ly v is a b le . T h e re la t iv e in te n s ity o f th e box w as

a d ju s te d so th a t a c o n v o lu tio n o f th e s p e c tru m ta k e n w i th th e tw o -g a m m a -ra y

t r ig g e r w i th th e box fu n c tio n re p ro d u c e d th e sh ape o f th e s p e c tru m w ith th e

n o r m a l t r ig g e r . T h e p ile -u p o c c u re d in a b o u t 1 5 % o f th e d e te c te d p o s itro n

ev e n ts . T h e c o rre c tio n fo r these e ven ts w a s c le a r ly p re fe r ra b le to th e a lte rn a t iv e

o f u s in g th e tw o -g a m m a -r a y tr ig g e r , w h ic h re d u c e d th e p o s itro n d e te c tio n

e fffic ie n c y b y m o re th a n a fa c to r o f th re e .

CHANNELSF i g u r e 4 - 3 : S econd a n n ih ila t io n g a m m a -ra y a b s o rb tio n .

T h e m e a s u re d in te n s ity o f p o s itro n s f r o m a 22N a source is p lo tte d as a

fu n c tio n o f c h a n n e l n u m b e r; th e e n d p o in t en e rg y o f th e d ecay is 5 4 5

k e V . T h e u p p e r c u rv e w as tr ig g e re d in th e s ta n d a rd m a n n e r b y a

co in c id e n c e w ith a t le a s t o n e a n n ih ila t io n g a m m a ra y (see S ec tio n 6 .1 ) ,

w h ile th e lo w e r c u rv e w as tr ig g e re d on b o th a n n ih ila t io n g a m m a rays .

182

B ecause o f th e d if fe re n t a b s o rb tio n p ro p e r tie s o f e le c tro n s a n d p o s itro n s in S i(L i)

m a te r ia l a n d because o f th e p o s itro n tr ig g e r , th e p e a k - to - ta il r a t io w as e x p e c te d to

b e la r g e r fo r th e p o s itro n lin e sh ape th a n fo r th e e le c tro n lin e shape.

M e a s u re m e n ts m a d e b y th e m a n u fa c tu re r h a d sh ow n t h a t th e c h a rg e c o lle c tio n

e ff ic ie n c y o f th e d e te c to r w as es s e n tia lly u n ity in th e e n e rg y reg io n o f in te re s t

w h ile e le c tro n -lin e -s h a p e m e a s u re m e n ts m a d e w ith th e b a ff le re m o v e d h a d sh ow n

t h a t th e lo w -e n e rg y ta i l does n o t co m e fro m th e s c a tte r in g o f p a rtic le s o f f th e

b a ff le . T o g e th e r these o b s e rv a tio n s im p ly th a t th e ta i l o f th e lin e sh ape m u s t

co m e fro m b a c k s c a tte r in g a t th e s u rfa c e o f th e S i(L i) d e te c to r .

T h e e ffe c t is re d u c e d b y th e so le n o id a l m a g n e tic f ie ld , w h ic h b rin g s som e o f th e

b a c k s c a tte re d p a rtic le s b a c k to th e S i(L i) d e te c to r . S ince th e e le c tro n s and

p o s itro n s p re fe r e n t ia l ly s tr ik e th e end o f th e S i(L i) d e te c to r n e a re s t th e ta rg e t ,

h o w e v e r , those w h ic h b a c k s c a tte r a w a y f ro m th e S i(L i ) d e te c to r an d m o v e b ack

u p th e so len o id to w a r d th e ta r g e t can b e lo s t b e fo re d e p o s itin g fu ll e n e rg y in th e

S i(L i ) d e te c to r . N e v e rth e le s s , s ince th e p o s itro n s a re a lw a y s d e te c te d in

co in c id e n c e w i th th e ir a n n ih ila t io n ra d ia t io n , in d ic a t in g th a t th e y h a v e co m e to

re s t in th e S i(L i ) d e te c to r , th e d e te c tio n o f p o s itro n s w h ic h h a v e o n ly d ep o s ited

p a r t o f th e ir e n e rg y in th e d e te c to r b e fo re b a c k s c a tte r in g a w a y sh o u ld b e re d u c e d

c o m p a re d to th e e le c tro n s . F u r th e r m o r e , a n u n fo ld in g o f p o s itro n b e ta -d e c a y

s p e c tra m e a s u re d w i th th e tw o -a n u ih i la t io n -g a m m a tr ig g e r to e lim in a te th e h ig h

e n e rg y ta i l d e s c rib e d a b o v e , b u t u s in g th e e le c tro n lin e sh ape , p ro d u c e d n e g a tiv e

c o u n ts in th e lo w e s t c h an n e ls in d ic a t in g th a t to o la rg e a lo w -e n e rg y ta i l w as b e in g

u n fo ld e d . T h e lo w -e n e rg y t a i l w as m u lt ip lie d b y a fa c to r w h ic h w as a d ju s te d so

t h a t th e u n fo ld in g o f th e p o s itro n b e ta -d e c a y s p e c tra y ie ld e d a lo w e n e rg y p a r t

w h ic h d ecreased s m o o th ly to ze ro . T h e necessary fa c to r w as a b o u t 0 .7 .

T h e p o s itro n lin e sh ape is s k e tc h e d in F ig u re 4 -2 (b ) a n d has th e fo llo w in g fo rm :

183

U E ,E 0) = A (E ) [B (E ,E 0) + C (E ,E 0) + D (E ,E 0) + F ( E ,E 0j\ for E > 0 keV (4.1)

where E is the measured energy, EQ is the emitted positron (or electron) energy,

and the functions are defined by:

fME) = 1 — exp

2 < Y 2 Jfor 0 < E < Oj

1 for*

< E

B(E,EQ) =E

,(£:„) 6l(£ 0)exp|—

0 for E Q< E

for 0 < E < En

C(E,EQ) = {g (£0) Cj(E0) exp [ j ^ - j ] for 0 < E < EQ

for E q < E

D{E,E0) = dJEg) exp

F(E,E0) = ( f ( E Q)

1°

1 ( g - g 0)2

2 l ^ l ^ o ) ! 2 ]for 0 < E

for Eq < E < E0+341 keVfor E0 < E < E q + 3 4 1 keV (4 .2 )

Values of the electron line shape parameters a, a0, b E'), b^E’), c^E'), c2(E'),

d^E1), and d2(E') were found for the three measured electron line shapes

separately with the positron line shape parameters set identically to f(E') = 0

and g(E') = 1. These values were plotted versus E' and a polynomial fitted

through each to give a smooth interpolation of the line shape for E ' in the energy

region of 0 keV < E' < 1500 keV. The positron parameters /(E ') and g{E') were found with positron sources as described above. The resulting function

L{E,E') is the response function of the Si(Li) detector for positrons for energies

between 0 keV and 1500 keV. It is shown in Figure 4*4.

184

E [keV]Figure 4-4: Response function of the Si(Li) detector.

The response function of the Si(Li) detector for positrons is plotted as a function of the kinetic energy E q of the emitted positron and the measured kinetic energy E.

185

4.I.2.I. Energy Calibration

The energy calibration of the eight segments of the Nal annihilation-radiation

detector was made with two types of sources. 511-keV annihilation gamma rays,

the line being sought, were provided by a 22Na positron beta-decay source

mounted at the target position. In addition, since the data analysis (described in

Section 6.1) requires a careful addition of the energy signals of all eight segments

to minimize gamma-ray backgrounds, additional calibration points were obtained11o 007by placing Sn and Bi gamma ray sources outside each of the eight segments

of the annihilation-radiation detector in turn. Table 4-2 shows the four

calibration points used, covering the energy range from 392 keV to 1064 keV. A

linear calibration was made for each of the eight segments.

4.1.2. N al A nnihilation-R adiation D etector C alib ration

E Source(keV)

391.69 113Sn511.00 22Na569.65 207Bi

1063.63 207Bi

(All values from [Lederer and Shirley 78].)

Table 4-2: Annihilation-radiation-detector calibration sources.

186

The detection efficiency for annihilation radiation in coincidence with positrons

was measured with a 22Na positron beta-decay source mounted at the target

position. The beta-decay energy distribution was measured in the Si(Li) positron

detector twice, once with and once without the normal coincidence (described in

Section 6.1) between the positron detector and the annihilation-radiation detector.

In order to suppress electronic noise in the low-energy part of the positron energy

distribution, both measurements were also made in coincidence with the 1275-keV

gamma ray which follows the beta decay of 22Na, detected in a S^XS" Nal(Tl)

gamma-ray detector. The positron detection efficiency, the 1275-keV gamma-ray

detection efficiency, the activity of the source, and the branching ratio for the

decay to a gamma ray and beta decay divide out in the ratio of these two

measurements, and the detection efficiency for the detection of annihilation

radiation in coincidence with positrons in the Si(Li) detector can be written as: NJE) Sj Tx

t{E) = N2(E) s2 t2 (43)where t is the coincident annihilation-radiation detection efficiency, E is the

positron energy, N is the detected yield of positrons, Sn is the electronic

scaledown for the event type n, and Tn is the elapsed time of the measurement n.

The annihilation-radiation detection efficiency, determined in this way, is shown

in Figure 4-5. The efficiency is essentially flat above the constant fraction

discriminator from about 100 keV up nearly to the endpoint energy of the beta

decay at 546 keV. The value of the integral detection efficiency is 0.634. This

can be understood as the product of the solid angle subtended by the annihilation-

radiation detector with respect to the positron detector, which is about (0.7)4jt,

and the intrinsic detection efficiency of the approximately 5 cm thick segments of

Nal material for a 511-keV gamma ray, which is about 0.8.

4 .I.2 .2 . D etection Efficiency

187

1.0

0.8

+'QJ

"Z. 0.6 \

£ 0.4OJ

Z

0.2

00 100 200 300 400 500 600

Ee+ [keV]

Figure 4-5: Annihilation-radiation detection efficiency.The detection efficiency for annihilation radiation determined with a 22Na 0+ decay source is plotted as a function of the measured kinetic energy of the emitted positron.

4.1.3. Positron Detection Efficiency

The positron detection efficiency has three parts: a

• the efficiency of the solenoid transport system for transporting positrons from the target to the Si(Li) detector,

• the detection efficiency of the Si(Li) detector itself,

• and the detection efficiency for the coincident annihilation radiation.

From the design of the Si(Li) detector and measurements made with the detector

by the manufacturer it was determined that the charge collection efficiency of the

1 1 1 i r " T '

- -

• • •••«•

-2 2 ,

-

______1_____ _____ 1 _____L........... L.......... 1. i

188

Si(Li) detector is close to one over the energy region of interest, from 100 keV to

1500 keV. The calculations described above in Section 3.2.2 predict, however,

that the solenoid transport efficiency is on the order of 15% from 200 keV to 800

kev, falling off outside these bounds, and that the annihilation-radiation detection

efficiency is about 60%. Thus the shape of the positron detection efficiency is

determined largely by the positron transport efficiency, while the height is

determined by the transport efficiency and the annihilation-radiation detection

efficiency.

The positron detection efficiency was determined by comparing measured,

unfolded, continuous, positron beta-decay spectra with theory. As an additional

check, measurements of the electron transport efficiency made with electron line

sources were compared to the calculated transport efficiency. These steps are

described in more detail in the following paragraphs.

4.1.3.1. Transport Efficiency

The calculation of the positron transport efficiency, described above in Section

3.2.2, was checked with electron line sources. The sources used were the same as

for the energy and line shape calibration, 113Sn and 207Bi. The one electron line

of 113Sn was measured as a singles measurement. The efficiency was calculated

by comparing the measured electron yield with that expected from a calculation

based on the known activity of the source, the length of time of the measurement,

the branching ratio of the electron, and the measured scaledown factor. The

electron-detection efficiency e is:M S

A T Bwhere M is the measured electron yield, 5 is the scaledown of the measurement

electronics, A is the activity of the source, T is the time of the measurement, and

B is the branching ratio for electrons in the decay.

The two lines of 207Bi were separated, as discribed above, by measuring in

189

coincidence with a gamma-ray detector. This had the added advantage of

allowing a calculation of the efficiency without knowledge of the absolute activity

of the 2°7Bi source. The efficiency for detecting electrons was determined by

comparing the ratio of the yield of electrons and gamma rays detected in

coincidence to the yield of gamma rays detected alone with the known branching

ratio for electrons per gamma ray. The activity of the source, the time of the

measurement, and the detection efficiency for gamma rays cancel out in the ratio.

The efficiency can be written as:M S I Be - ,7 e - , 7 ' e - ,7

(4.5)M S / £7 7' 7

where Be—,7" indicates an electron and gamma ray in coincidence and ■7" a

gamma ray alone. The calculated electron-detection efficiencies for the 368-keV

electron of 113Sn and the 503-keV and 995-keV electrons of ^ B i are displayed in

Figure 4-6, were they are plotted with the calculated positron transport efficiency.

As can be seen, the measured points agree with the calculated detection efficiency.

4.1.3.2. Detection Efficiency

The positron detection efficiency was determined with the continuous positron

beta-decay sources 22Na and ^G e. The positron beta-decay spectrum of 22Na

has an endpoint energy of 546 keV while that of Ge (which decays first by

electron capture to 68Ga) is 1899 keV [Lederer and Shirley 78]. The shape of the

two beta decay spectra are similar to the expected dynamic positron spectra (cf.

Figure 2-33) which lies between the two in its energy range. This is illustrated in

Figure 4-7. As with the electron line sources, the two positron sources were

mounted on target frames, and the 22Na source was mounted on a Mylar backing.

Because of the chemistry of germanium, however, it could not be mounted on

Mylar and had to be mounted on a nickel backing. The backing was magnetic,

but a check with a Hall probe showed that the distortion of the magnetic field at

the target position due to the metallic backing was insignificantly small.

190

20

1055l_4

, 5a>CO

2

10 200 400 600 800 1000 1200 U00

E e. [keV]

F ig u re 4-6: Electron transport efficiency.114

The transport efficiency for electrons measured with a Sn source (diamonds) and a 207Bi source (circles and squares) is plotted as a function of the kinetic energy of the emitted electron. The solid line is a Monte Carlo calculation of the transport efficiency described in Section 3.2.2 (multiplied by a factor of 1.1).

nnThe measured positron beta-decay spectra for Na and Ge were corrected for

the energy calibration and unfolded with the response function measured above11.

Rather than divide the unfolded spectra by the detection efficiency to obtain the

beta-decay spectrum for comparison with the theoretically calculated beta-decay

t | i |-------1-------1-------1-------1-------1-------1-------1-------1-------r

..A-**-®-*?'o-x\ 00/ \ <v

<*' \

0 * ‘

O't

f

o '

- o 113Sn (scaled)0 207 Bi (502 keV)

- □ 207 Bi (995 keV)------ Monte Carlo * 1.10

\ 0 V\\ O

J I i I i I i I 1 I 1 *1 ■

11The unfolding procedure is described below in Section 6.4.1

191

0 400 800 1200 1600 2000Ee. [keV]

F ig u re 4-7: Comparison of positron spectra.The dynamic positron production in the collision system U+U at a bombarding energy of 5.9 MeV/amu (solid line, from [Muller 83a]) is compared to the positron production from the 0+ decay sources 22Na (dotted line) and Ge (dashed line). The differential positron production probability dP/dE is plotted as a function of the kinetic energy E g+ of the emitted positron.

spectrum, the oposite was done: the unfolded spectra were divided by the

theoretically calculated beta-decay spectrum to obtain the detection efficiency

distribution. The theoretical beta-decay spectrum is:

B [ E ) = ^ N p E { E Q- E ) 2 F{E) (4.6)

where E is the positron energy, EQ is the endpoint energy of the beta decay, N is

a normalizing factor, determined by numerical integration to give unit area, and

F(E) is the Fermi function, cubically interpolated from the tables in [Behren and

Jaenecke 69].

The results are shown below in Figure 4-8. The detection efficiency distribution

calculated from 22Na extends only up to the cutoff energy of the 22Na beta decay

at 546 keV. The cutoff energy of 68Ge at 1899 keV is well above the end of the

detection efficiency distribution of the apparatus under the usual operating

conditions. The accuracy of the calculated detection efficiency distributions is

limited primarily by the reliability of the unfolding of the detector response

function.

102

E f fkeV ]

Figure 4-8: Measured positron detection efficiency.The detection efficiency for positrons as measured with the 0+ decay sources 22Na (circles) and 68Ge (squares) is plotted as a function of the kinetic energy of the emitted positron.

193

As seen in the figure, the distribution calculated from Ge agrees quite well in

shape with the calculation above about 300 keV. The distributions obtained fromOO fiftboth Na and Ge fall off faster than the calculation at low energies below

about 300 keV, due primarily to the sensitivity of low energy positron detection

efficiency to the alignment of the Si(Li) detector, baffle, and target, and at very

low energies to the efficiency of the coincidence electronics. The more reliable of

the two contributions at low energy is that from 22Na, since the beta decay

spectrum does not extend to as nearly high an energy as that from 68Ge and thus

is not nearly so susceptible to cumulative errors in the unfolding of the high

energy part of the spectrum. Therefore the low energy (E < 300 keV) part of the

distribution obtained from the measurement of 22Na was used to determine the

low energy part of the detection efficiency, and the high energy (E > 300 keV)

part of the efficiency determined with the Ge measurement was used for the

high energy part of the detection efficiency. The two were smoothly joined ateyty

about 300 keV. The height was determined from the calibration of the Na

source. The resulting positron detection efficiency is the solid line shown in

Figure 4-8.

4.2. Detection of Gamma Rays

The calibrations of the two 3"X 3" Nal(Tl) gamma-ray detectors used to

determine the positron background from nuclear processes are described in this

section. The energy calibration, response function, and detection efficiency of the

detectors was determined with standard gamma-ray sources, mounted at the

target position on target frames. The sources were either open sources on thin

Mylar backings or encased in plastic. Neither encasement method had a

measurable effect on the gamma-ray spectra needed for these calibrations in the

energy range of interest, above 1 MeV.

fifi

194

The energy calibration of the gamma ray detectors was done immediately

following the experiment with the gamma ray sources shown in Table 4-3,

covering the energy range from 511 keV to 6130 keV. The first four are standard

sources. The last two high energy sources are combinations of an alpha-particle

emitter and a low-Z nucleus which reacts with the alpha particles to emit

neutrons and high-energy gamma rays. A linear energy calibration was

determined for each detector by a least-squares fit to the position of the centroids

of the gamma-ray peaks.

4.2.1. E nergy C alib ration

Source(keV)

511.00 22Na569.65 ^ B i898.02 88y

1063.63 207Bi

1173.21 60Co1274.55 22Na1332.47 “ Co1770.22 207Bi

1836.08 88y

4439.1 241 Am + °Be6130.4 ^ P u + 13C

(All values from [Lederer and Shirley 78].)

T a b le 4-3: Gamma-ray detector calibration sources.

195

4.2.2. Response Function

The response function of each of the two Nal(Tl) detectors was determined by

measuring the the line shape of gamma rays from the following sources with

widely separated strong lines: ^C o , 22Na, ®®Y, and 212Pb. These lines covered

the energy range from 1 MeV to 3 MeV, which is the source of most of the

positrons from nuclear processes. Since the two Nal(Tl) detectors used are

essentially identical with respect to their internal configuration, shielding, and

placement with respect to the target, the line shapes measured for the two

detectors are the same. The measured line shapes for the forward detector are

shown in Figure 4-9. The main features of the line shape are a Gausian-shaped

full-energy peak, a tail due to Compton scattering of gamma rays in the Nal(Tl)

material, and the single and double escape peaks for Nal(Tl). A response matrix

for each detector was constructed to reproduce the measured line shapes.

4.2.3. Detection Efficiency

The gamma ray detection efficiency was determined by comparing the gamma

ray yields measured from each of the four calibrated sources used above for the

line shape measurement with that calculated from the known activity of the

source. The gamma ray detection efficiency cis given by:N S ■

£ = -------- 4.7A T B V

where N is the measured gamma ray yield, 5 is the electronic scaledown, A is the

source activity, T is the elapsed time of the measurement, and B is the branching

ratio for the measured transition.

A measurement of the gamma-ray peak-detection efficiency was checked with a

calculation of the peak-detection efficiency based on the intrinsic gamma-ray total

detection efficiency of a 3" X3" Nal(TI) crystal [Marion and Young 68, Neiler and

Bell 65], the peak to total ratio for a 3*X3* Nal(Tl) crystal [Marion and Young

106

t/)CZ3O

Et (MeV]

F ig u re 4-0: Measured gamma-ray line shapes.The line shape for the detection of various gamma rays in the forward gamma-ray detector is plotted as a function of the measured gamma-ray energy E .

197

68], the distance from the target to the detector ( ~ 37 cm), and the attenuation

of gamma rays due to the shielding placed in front of the detector (Storm and

Israel 70] (described in Section 3.2.4). The result of the calculation was found to

be in agreement with the measured points and was adopted for the shape of the

peak-efficiency curve. The gamma-ray peak-detectiou efficiency is shown as the

dashed line in Figure 4-10, along with two measured points obtained immediatelyfinafter the experiment with a calibrated Co source as a check.

F ig u re 4-10: Gamma-ray detection efficiency.The detection efficiency for gamma rays is plotted as a function of the gamma-ray energy E . The circles are a measurement of peak detection

efficiency with a 88Co source. The dashed line is a calculation of the peak detection efficiency and the solid line is the. total detection efficiency determined from this (see the text).

Since the total gamma-ray detection efficiency was needed for the data analysis,

an experimental peak-to-total ratio was determined from the measured line shapes

described above. This ratio included the low-energy cutoff at 047 keV. The total

gamma-ray detection efficiency is the quotient of the peak-detection efficiency

and the peak-to-total ratio, and is shown in Figure 4-10 as the solid line.

4.3. Detection o f Particles

An angular calibration of the two parallel-plate avalanche counters used to

measure the scattering angles of both colliding nuclei was made by examining the

kinematics of several collision systems measured during the experiment. In

addition, the relative detection efficiency of the two particle detectors was

determined from a measurement of the symmetric collision system, + 238U.

These two calibrations are described in this section.

4.3.1. Angle Calibration

The two particle detectors were calibrated for scattering angle by measuring the

scattering-angle distribution of 238U on the following targets: 248Cm, 238U,

^ P b , 197Au, 154Sm, 124Sn, and 109Ag. The beam energy was 5.9 MeV/amu in

all cases except for the curium target, when the beam energy 6.05 Mev/amu (to

compensate for a 640 //g/cm2 backing of titanium). The distribution of collisions

was measured as a two dimensional function of the scattering angle in both

detectors. The assymetric collision systems were used to obtain a first-order angle

calibration for both detectors. For the assymmetric collision systems, i.e. for all

targets except uranium, the kinematic plots of scattering angle measured in one

detector versus scattering angle measured in the other show a crossing point

e „ . which is given by:CTvoo

I I™= C0S_1 < J V A ^ + 1 ) ( « )

where M p is the mass of the projectile nucleus and M t is the mass of the target

nucleus.

108

For the systems where the target is lighter than the uranium projectile, which is

all targets measured except uranium and curium, the maximum scattering angle

&maz or heavier uranium projectile is given by:

M t

0 maz = sin J f (+»>P

Table 4-4, shown below, lists the crossing points and maximum scattering angles

for the systems measured. A first-order angle calibration of both particle

detectors, accurate to about 2 ° , was obtained from these points by a linear least-

squares fit. These angle calibrations were then checked for consistency using the

symmetric system uranium on uranium, where the sum of the two scattering

angles is constrained by kinematics to be 90 *. An iterative proceure was used.

In each iterative step, the angular calibration of each detector was used in

conjunction with the kinematic constraint to determine the calibration of the

other detector. The new calibrations were averaged with the old to obtain the

starting point for the next step. The method converged after three iterations to

yield an angular calibration for both particles detectors accurate to about 1 ° in

the absolute scattering angle.

109

ojectile Target e cross(deg)

®max(deg)

238jj 248Cm 45.58238 j 238jj 45.00 -

2 3 8 u MSpb 42.93 60.922 3 8 u 197 Au 42.01 55.872 3 8 u 154Sm 37.09 40.322 3 8 u 124Sn 31.32 31.40238u 109Ag 26.86 27.26

T a b le 4-4: Particle detector angle calibration.

200

A measurement has been made of the angular dependence of the particle

detection efficiency ep(0), which enters into calculations of the positron or gamma-

ray yield as a relative weighting function when integrating over finite angular

regions:

4.3.2. D etection Efficiency

r e d a jj<&)

I — tnie ) deJ e l a a de pPn = ------------------------------------------- for a = e + ,K (4.10)

t (e)de

a

ex de pThis was done by comparing the measured angular distribution for the collision

system 238U + 238U at a bombarding energy of 5.9 MeV/amu, with a calculation

of Rutherford scattering. This system is below the Coulomb barrier so that the

majority of the collisions do indeed correspond to Rutherford scattering.

The Rutherford cross section for Coulomb scattering of the projectile nucleus in

the laboratory system is:

da da d n C M d n Lab(4.11)

where:d e Lab d!2C M dn Lab d 6 Lab

da a 2

d n C M sin4(0^A^ /2)

dnCM [ i + t 2 + 27cos e vCM \z!2

d tiLab 1 + 7 COS e c M

d0l b= 2* sin e[ab, 9cM = eLb + s 'm !( 7 sin )

d* LZiZ2e2 M t

a = ——----- , 7 = ——, e2 ~ 1.44 MeV-fm (4-12)2 £ c m M p

The cross section for scattering of the target nucleus is:

201

da da d n C M d ° C M d°Lab

d e Lab d n C M dfiC M d° U b d S Lab

(4 .13)

with the additional terms:

d n p , d n l‘C M "“ C M t= 1 - = 4 cos &Lab

d n C M dn Lab

M l kLao t n i= 2n sin eLab , QC U ~ 2e Lah + * (4*14)

d 9 LabSince in a symmetric collision it is impossible to differentiate projectile from

target nucleus, what is measured in the particle detectors corresponds to the sum

of the contributions from the detection of the projectile at a given angle and the

detection of the target nucleus at that same angle. This is:

— - - ■ « E T > + - ^ 9L - <4-15>

d s Z r i e L i s LFor a symmetric collision system, 7 = 1, and this expression for the measured

Rutherford cross section reduces to: da

= «2 s in (2 e -7 )m ea s.

. r n e a sd e u t

• 4 jm e a s 4 rneas sm eLab cos e Lab

(4.16)

This curve was plotted over the measured angular distribution, normalized in

height at 45°. The result is shown in Figure 4-11. The aggreement is good

between the limits of 250 and 650 labratory angle. This indicates that in this

region the particle detection efficiency is a constant, falling off outside of these

limits due to scattering in the vacuum foils of the detectors and to mismatch of

the angular masks which define constant 4 acceptance.

Since the positron and gamma-ray events were measured in coincidence with the

particle detectors, and then normalized to Rutherford scattering by dividing by

the number of particle events, a constant particle detection efficiency t^{e) = «

divides out in the determination of the production probability P.

202

roO

COI -zZ>Oo

#Lab [deg]F ig u re 4-11: Particle detection efficiency.

The measured intensity of particles is plotted as a function of the laboratory scattering angle 9 ^ detected in the upper particle detector

rtqo noQfor the collision system U + U at a bombarding energy of 5.9 MeV/amu. The dotted curve is the Rutherford cross section d<r/d9Lab for the projectile nucleus entering the upper counter; the dashed curve is the same for the target nucleus. The solid curve is the symmetrized sum of the two. The three calculated curves have a common normalization, determined by the measured data at 9^ = 458.

dtxR(9)

dOde

r 6 d9d9

for <„(») = »=«+ (4.17)

The angular cuts made for this thesis are all confined to the angular region

258 < 9 ^ < 6 58, and thus no correction for particle detection efficiency is

necessary.

Chapter 5

Data Collection

As related in Chapter 1, the experimental search for spontaneous positron

production in heavy ion-atom collisions has been in progress using the heavy-ion

accelerator at GSI Darmstadt since the late 1970’s. This thesis predominantly

describes one in this series of experiments, an experiment conducted at GSI from

January 30 to February 7, 1981, using a beam of approximately 6 MeV/amu.

The heart of the experiment was a measurement of the positron production in the

three collision systems 238U + 248Cm, 238U + ^ U , and + ^ P b at

bombarding energies just below the Coulomb barrier. These collision systems

bracket the region of combined nuclear charge where spontaneous positron

production should set in. Measurements of other, lower-Z collision systems were

made to check the the production of positrons from nuclear processes and to

establish the angle calibration of the two particle detectors.

In addition to a description of the collision systems studied in this experiment,

the following sections will also describe the parameters measured, the online

analysis, and those matters which required special consideration during the

experiment.

203

204

5.1. Systems Measured

5.1.1. Measurement of Atomic Positron Production

The three collision systems studied were 238U + 248Cm, 238U + 238U, and 238U

+ 208Pb. The combined nuclear charge in each of the systems is 92 + 96 = 188,

92 + 92 = 184, and 92 + 82 = 174, respectively. In all three cases the beam was

uranium-238.

The curium-248 target was 500 Mg/cm2 of Cm2Oa mounted on a backing of 640

/ig/cm2 Ti. The backing was necessary for the electroplating procedure used in

the fabrication of the target [Trautmann et al. 82]. The target also had a 38

jig/cm2 layer of carbon on the downstream side to prevent the highly radioactive

curium from boiling off the surface and contaminating the target chamber.

The uranium and lead targets were self-supporting, i.e. no backing, since a

backing could introduce unwanted backgrounds in the form of neutrons entering

the gamma-ray detectors and fission particles entering the particle detectors. The

uranium-238 target was 750 #xg/cm2 thick and was covered on both sides with a

carbon foil, 20 /xg/cm2 thick on the upstream side and 30 /xg/cm2 and on the

downstream side. The lead target was 530 /xg/cm2 thick and had no carbon foil

cover.

The bombarding energy of the uranium beam was 6.05 MeV/amu for the

curium target, 5.91 MeV/amu for the uranium target, and 5.92 MeV/amu for the

lead target. Data was collected from the U+Cm system for 36 hours (actual beam

on target time), from the U+U system for 13 hours, and from the U +Pb system

for 11 hours. Table 5-1 shows the calculated energy loss [Geissel 82, Hubert et al.

80] of the beam in the backing material and in the first half of the target material

for each of the targets, and the resulting average beam energy in the middle of

the target material-. This average projectile energy is used in all following

205

calculations for the gross features of the data12.

Material (pg/cm) BeamEnergy

Energy Loss MeV/amu

Aver.Proj.

Backing Target (MeV/amu) inBacking

in 1/2 Target

Energy(MeV/amu)

640 Ti 500 Cm20 3 6.05 0.27 0.06 5.7220 C 750 U 5.91 0.02 0.07 5.82

— 530 Pb 5.92 — 0.05 5.87— 750 Sm 5.91 — 0.08 5.82— 950 Au 6.04 — 0.10 5.94— 980 Sn 6.04 — 0.13 5.91— 970 Ag 6.04 — 0.13 5.91

(Energy losses based on [Hubert et al. 80].)

T a b le 5-1: Average projectile energies.

All three collision systems correspond to collisions just below the Coulomb

barrier. The distance of closest approach R min in Rutherford scattering is given

as a function of scattering angle 9 P of the projectile and 9 l of the target in

Equations (3.42) to (3.45) of Section 3.1.3 above. This relationship is shown in

part (a) of Figures 5-1 to 5-3 for the three studied collision systems U+Cm, U+U,

and U+Pb, using the average projectile energy defined above.

As can be seen, the minimum distance of closest approach, 2a, for these three

systems is 18.3 fm for U+Cm, 17.6 fm for U+U, and 16.7 fm for U+Pb. These

values are in each case larger than the nuclear interaction distance [Wilcke et al.

80] for the collision system: 16.7 fm for U+Cm, 16.6 fm for U+U, and 16.3 fm for

12In Chapter 7, the possibility that deterioration of the Ti backing caused the beam energy in the Cm target to be higher will be discussed.

206

- 2 - 1 0 I 2

Time [l0~2 , s]

F ig u re 5-1: U+Cm kinematic plots.Linematic plots for the collision system U+Cm at a bombarding energy

< f 5.72 MeV/amu. P art (a) shows the distance of closest approach R min

s a function of the scattering angle 9 ^ for detection of the projectile (solid line) or the target (dashed line) nucleus. P art (b) shows the nuclear separation K as a function of time for various laboratory

scattering angles, 9^ab, of the recoiling target nucleus.

207

Tim e [ l0 2 ls]

F ig u re 5-2: U+U kinematic plots.

Same as Figure 5-1, but for the collision system U+U at a bombarding

energy of 5.82 MeV/amu.

U +Pb.13

^T he value for U+Cm is extrapolated from the values for U+U and U+Pb in the table using

the form = (1-34 fm) (A^1/ 3 + A ^ ^ ) -

208

- 2 - 1 0 I 2

Time [lO*21 s ]

F ig u re 5-3: U +Pb kinematic plots.

Same as Figure 5-1, but for the collision system U +Pb at a bombarding

energy of 5.87 MeV/amu.

The Isa electron becomes critically bound in the U+Cm system when the

internuclear separation is about 33 fm, and in the U+U system when the

separation is about 27 fm (from [Wietschorke et al. 79], assuming a charge state of

209

about 50+). The U +Pb would require a separation of less than 15 fm, and thus

will not become critical for collisions below the Coulomb barrier. As a result, for

these bombarding energies the Iso electron in the U +Pb system is not criticaly

bound, while in the U+U system it is critically bound for the closest 9 fm of

approach, and in the U+Cm system for the closest 15 fm of approach.

Since the scattering angle is related to the distance of closest approach by:

,p

e L t = U n 1s m e C M

+ cos OrCM

e C M ~ 2 Sm \ R . e U b ~ COS ' ( R . _ „ ) I5 '*)' mxn ' x mm

the ls<7 electronic state is critically bound in the U+Cm system for projectile

scattering angles greater than 24°, and in the U+U system for projectile

scattering angles greater than 3 1 0. Since the measured scattering-angle region is

from 25 ° to 650 in the laboratory reference frame, the three systems chosen

bracket the critical point with a system which is not critical, a system which is

critical over a part of the measured scattering-angle region, and a system which is

critical over the entire measured scattering-angle region.

The length of time that each system can be -critically bound is shown in part (b)

of Figures 5-1 to 5-3 for Rutherford scattering. The separation R between the

nuclei is shown as a function of time during the central part of the collision. In

Rutherford scattering, the time T for the nuclei to move from a separation

R > R min in to a separation f?min and then back out to a separation R again is

given (as can be calculated by integrating Equation 3-18 on page 63 of [Goldstein

50], with a Coulomb potential) by:

210

T{R) = 2 Aj (R+Ry/l-a-p2-

R v l —a—p + a ln l --------' m i n

I m no , I MeV/amuA . y - a 7.1987x 10“ sec/fm /

P ' P2a 6

' - 5 (5-2»

where the impact parameter b is given by:

6 = a cot{e£M /2) = a tan 9 ^ (5.3)

and R min is given above in Equation (3.42). The longest possible time for a given

beam bombarding energy and separation R is the case of a head-on collision,

where 6 = 0, R min = 2a, 9t = 0 0, and 9p = 180 ° , in which case the expression

above for the time reduces to:

n = 0( * ) = 4aA[ - r - + v ^These kinematic parameters are summarized in Table 5-2.

(5.4)

Beam Aver. 2aSys.

^ c o mEner. Proj. (fm)

Ener.(MeV/amu)

U+Cm 188 6.05 5.72 18.3U+U 184 5.91 5.82 17.6

U +Pb 174 5.92 5.87 16.7U+Sm 154 5.91 5.82 15.1

(1) [Wilcke et al. 80](2) [Wietschorke et al. 79]

R int R cr e pcp e lcp Tcr(fm) (fm) (6=0)

(deg) (deg) (10-21(1) (2) sec)

16.7 33 23 67 2.216.6 27 29 61 1.616.3 <15 — — —15.7 <15 — — —

T able 5-2: Kinematic parameters: high-Z systems.

211

As a check of the calculation of the background of positrons from nuclear

processes, the collision system + 154Sm was measured. The combined

nuclear charge of the U+Sm collision system, 92+62=154, is small enough that

the rate of dynamically produced positrons is negligible compared to the positrons

produced by pair production of gamma rays from nuclear transitions (as can be

seen in Figure 1-2). Essentially all the measured positrons are produced in

nuclear processes, and should correspond to the nuclear positron background

calculation.

The bombarding energy of the uranium beam was 5.91 MeV/amu. The self-

supporting sumarium-154 target was 750 /xg/cm2 of 154Sm with a downstream

cover consisting of a 40 /xg/cm2 carbon foil. The energy loss of the beam in the

first half of the target and the resulting average projectile energy are shown in

Table 5-1. The kinematic parameters for this collision system are given in Table

5-2, and Figure 5-4 diagrams the relationship between the distance of closest

approach and the scattering angle in part (a) and between the internuclear

separation and time during the collision in part (b). The measurement required

about two hours of beam on target. The minimum distance of closest approach at

this bombarding energy was 15.1 fm and the interaction distance [Wilcke et al.

80] is 15.7 fm. Since this interaction distance corresponds to a target scattering

angle of 9 — 220, the interactions occur for scattering angles outside the range of

measurement.

6.1.2. M easurem ent o f N uclear P ositron Background

212

0 15 30 45 60 75 90 ..

0|_ob [ d e 9 ]

T i m e

F ig u re 5-4: U+Sm kinematic plots.Same as Figure 5-1, but for the collision system U+Sm at a bombarding

energy of 5.91 MeV/amu.

213

In order to make the angle calibration of the two particle detectors, several

other targets were placed in a 6.04 MeV/amu uranium beam. The targets were

950 /jg/cm2 197Au, 980 /ig/cm2 124Sn, and 970 pg/cm2 109Ag. The average

projectile energy in the middle of each target is given in Table 5-1. The

kinematic parameters for these collision systems are given in Table 5-3.

5.1.3. S ca ttered -P artic le Angle C alib ration

System Beam Aver. Energy Proj.

Ener.(MeV/amu)

2a(fm)

R int

(fm)(1)

«-Pi%nt

(deg)

e !tnt

(deg)

U+AuU+SnU+Ag

6.04 5.946.04 5.916.04 5.91

16.313.714.1

16.215.215.0

30.727.0

35.427.6

(1) [Wilcke et al. 80].

Table 5-3: Kinematic parameters: low-Z systems.

Only the particle event type was run for these measurements, since the

calibration was based on looking at the crossing points and maximum scattering

angles of the kinematic plots of scattering angle detected in one detector against

the scattering angle detected in the other detector, as described above in Section

4.3.

214

5.2. Parameters Measured

As described above in Section 3.3.1, 32 parameters were measured and recorded

event by event onto magnetic tape during the January-February, 1981,

experiment. The three types of parameters were time signals from TDC’s, energy

signals from ADC’s, and pattern words from a pattern box. One time and one

energy signal were recorded for each of the two halves of the Si(Li) positron

detector, the eight segments of the Nal(Tl) annihilation-radiation detector, the

two Nal(Tl) gamma-ray detectors, and the anode foil of each of the two particle

detectors. A time signal alone was recorded for the cathode delay line of each of

the particle detectors. Two pattern words were also recorded with each event:

one logging which event types had triggered the record and the other which

ADC’s and TDC’s had converted signals.

In addition to the parameters described above which were written onto tape

event by event, a group of parameters was automatically written into an analyser

each time the experiment was halted. These parameters, one for each event type,

were the number of events before the scaledown circuit recorded in each event

type since the last time the experiment had been started, and were used to

determine the scaledown and deadtime correction.

5.3. Online Analysis

An online analysis of the incoming data was performed during the experiment

using both a PDP-11 and an IBM/370 computer, as described above in Section

3.3.3. The analysis served to monitor the condition of the experimental

apparatus, target, and beam, to determine when enough data had been collected

for each experiment, and to make a preliminary online search for interesting

structures in the positron production distributions.

In addition to producing positron distributions, the condition of the

215

experimental apparatus and all the detector systems was monitored by generation

of the time and energy plots for intermediate levels of the data analysis. Singles

spectra were made in the smaller, PDP-11 computer of all 32 parameters

measured event by event, and for selected detectors the singles spectra were

recorded directly into the computer memory (see Section 3.3.3). The IBM/370

was used for the more complicated spectra. Time-difference spectra monitored

the time coincidences between pairs of detectors to ensure that the timing

electronics and coincidence circuits functioned properly. Energy spectra for all

the detectors were made at each level of time coincidence to monitor the overall

performance of the experimental apparatus. Monitoring the condition of the

beam and the target with the online analysis is described in the next section.

The online analysis program also performed a simplified version of the final

offline analysis. It produced positron distributions as a function of the positron

energy and of the scattering angles of the projectile and the target, gamma-ray

distributions as a function of the gamma-ray energy and the two particle

scattering-angles, and particle distributions as a function of the two scattering

angles. These distributions were used to determine when enough data were taken

for each experiment.

In addition, the online analysis also permitted a search for interesting structures

in the positron production distributions. Previous experiments with a plastic

scintillator positron detector, which had much worse energy resolution than the

Si(Li) detector used for this experiment (i.e. 30-50% versus 8-10 keV in the range

of 100-1000 keV), had hinted at the possibility of structures in the positron energy

distributions correlated to particular regions of particle scattering-

angles [Bokemeyer et al. 81, Bokemeyer et al. 83, Greenberg 83). A preliminary

search was made online for these correlations. Positron energy distributions were

projected out of the three dimensional energy-versus-angle-versus-angle

216

distibutions for various combinations of scattering angles. Possible structures

were checked by the reverse operation of projecting out the angular distributions

for given energy regions. These searches formed the basis for the later offline

analysis.

5.4. Special Considerations

5.4.1. Beam Condition.

The position and focus of the beamspot were monitored with a retractable wire-

grid probe [Angert and Schmelzer 77], which could be pneumatically inserted in

the beam directly in front of the target. The probe consisted of a grid of 0.1 mm

tungsten-rhenium alloy wires 1.5 mm apart, in which the passing beam induced a

charge. The amount of charge for each horizontal and for each verticle wire were

displayed separately on two oscilliscope traces, giving a horizontal and vertical

projection of the beam position and shape. The beam position and focus were

checked at the begining of each run, and about once an hour on the average

during each run. The beam was positioned with steering magnets and focussed

with quadrapole magnets so that the horzontal and vertical profiles had the

approximate shapes shown schematically below in Figure 5-5. These profiles were

checked with the probe to insure that the beam position and focus were set

properly before each experiment and that they remained stable during the entire

run.

The position of the beam spot on the target was monitored carefully because of

its direct affect on the stability of the angle calibration of the two particle

detectors. The angle calibration of each detector is determined by the position of

the meander delay line on each detector with respect to the point of collision on

the target. Any movement of the beamspot changes this relationship, and hence

the angular calibration. Calculations showed that movements of the beamspot on

217

■ > i ■ v I IVert ical Profi le

t r

_ 1.0 OJn>ui/l^0.5m<_

i n

< 0 > s

J) .0a/

ES 0.5CO

(a )

■I 1----- 1----- 1----- H— 1

Horizontal Profi le

( b )

- U - 2 0 +2Position [mm]

F ig u re 6-5: Beam profile.The beam intensity is plotted as a function of position as measured by the current in the grid wires of the beam probe. P art (a) shows the vertical profile and part (b) the horizontal profile.

the order of 1 mm were critical, causing measurable changes in the angular

calibrations of the particle detectors of ~ 0.1 ° .

The focus of the beam was monitored because of its affect on the positioning of

the beam and to prevent intensity problems, due to too sharp a focussing of the

beam within the beam spot. This could lead not only to too high a counting rate

for the electronics and the data transfer system, but also to deterioration or

destruction of the target. Beam intensity was especially critical for the

radioactive curium target.

218

Obtaining a usable measurement was found to depend critically on maintaining

a stable target, as changes in the target affect in a complicated way the conditions

of the entire experiment. The heating of the targets in the beam was carefully

monitored because of possible changes in the target structure: local melting and

recoalescing into globules that could increase the local thickness of the target, or

sputtering of target material from either face that would decrease the thickness of

the target. Either change in the target thickness affects the average energy loss of

the beam in the target, and thus the average energy of the projectile nuclei in

collision with the target nuclei. In addition, buckling or folding of the metal

targets due to uneven heating of the target could change the position of the beam

spot with respect to the particle detectors, and affect the particle scattering-angle

calibrations, as mentioned above. In the most drastic case, the destruction of the

target by uneven beating would stop the experiment and, in the case of the

radioactive curium target, necessitate a messy and difficult cleanup of the

contaminated target chamber.

Thermal shock to the curium targets was prevented by careful preheating of the

target each time the target was placed in beam. This was done by checking the

beam with the target removed for proper position and focus - in particular that

the focus was not too sharp. Then with the target in place, the intensity of the

beam was gradually increased to the full amount over a time span of about five

minutes.

The primary method of monitoring the target condition was visual checks.

Periodically during the beamtime the target was inspected for signs of

deterioration by means of a glass port at the end of the mirror-field arm of the

spectrometer. Sighting through the port with a telescope while turning the target

holder allowed a view of both sides of the target.

5.4.2. T a rg e t C ondition.

219

An additional online inspection of both the beam and the target was possible

using the detected scattered particles. Two surface barrier detectors were

mounted in the target chamber, at an angle of 45" to the beam axis. The energy

spectrum of the particles scattered into the detectors were recorded as singles

spectra channeled directly into the PDP-11 computer. The spectra were erased

and started over periodically to check for changes in the tailing structure of the

energy distribution or shifting of the energy of the scattered particles, indicating a

change in the target.

In addition, a real time scatter plot of the position of each event in a graph of

particle scattering-angle in one detector against that in the other was made

several times an hour during the run. This plot was found to be very sensitive to

both the beam position and the target condition. When the curium target with

its thick titanium backing deteriorated, this was immediately visible in the scatter

plot as the ratio of the U+Cm elastic scattering events to U+Ti events changed.

As any of the targets deteriorated, the angular resolution visible in the scatter

plot also worsened. Continual monitoring of this angle-angle scatter plot

indicated when the target was damaged or significantly altered and had to be

replaced. Moreover, movements of the beam caused a shift of the entire

kinematic curve, and thus could also be monitored with this plot.

On the whole, it was found that the need to limit the beam current on the

curium target to no more than about 30 nA current ( ~ 3/4 particle-nA for the

charge state 40+ uranium beam) to prevent overheating of the target was the

overall limiting factor for the data-taking rate in the U+Cm measurement. For

the other experiments, the counting rate in the low scattering-angle end of the

particle detectors set the limiting rate for the experiment.

Chapter 6

Data Analysis

The analysis of the data can be divided into three major parts:

• the identification of positron, gamma-ray, and scattered-particle events,

• the correction for backgrounds and for the influence of the detector systems,

• and the study of the correlations between positron energy spectra and the scattering angles of the colliding nuclei.

The initial event-type identification stage of the analysis serves to identify the

positron, gamma-ray, and scattered-particle events from the recorded detector

signals. The data is played back from magnetic tapes and for each event:

• the event type or types recorded are determined,

• the presence of all associated detector signals is confirmed,

• the time coincidences between detectors is checked and the chance- coincidence rates are calculated,

• and, finally, energy and angle plots for positrons, gamma rays, and scattered particles are generated.

Following this initial event-type identification analysis, the following corrections

to the generated spectra are made.

• The annihilation-radiation sum-energy spectrum is used to subtract a background from the positron energy spectra.

• The intrinsic line shape of the Si(Li) positron detector is unfolded and the positron detection efficiency divided out of all positron spectra.

221

222

• The recorded scaledowns and normalization particle spectra are used to determine the absolute magnitude of the measured positron yields.

The gamma-ray energy distributions have also been corrected for the detector

response and efficiency. The background of positrons from nuclear processes is

calculated from the resulting gamma-ray energy spectra.

The corrected probability distributions for positron production have been

studied as a function of positron energy and the scattering angles of the two

colliding nuclei. For the kinematically-separable systems, the probability

distribution with respect to the distance of closest approach during the collision,

has also been constructed. Spectra typifing the gross features of the data,

integrated over the entire measured angular or energy range, respectively, are

presented. These show the smooth energy and angular dependence expected for

collision systems dominated by dynamically produced positrons. A study of the

fine features of the data, however, has revealed the existence of prominent peaked

structure in the positron-energy distribution correlated to specific scattering nuclei

angular regions for the heaviest system measured, U+Cm.

The computer-based part of the analysis was done on the IBM/370 computer at

GSI Darmstadt. The initial data analysis programs for event-type identification

are written in the GSI-developed language SATAN, a PLI-based language for

event-by-event analysis described above in Section 3.3.3. Following the event-

type-identification analysis and the correction for the annihilation-radiation sum-

energy background, the spectra were transferred to GAMMEL, an analysis system

written at GSI in FORTRAN for spectra manipulation, for further analysis and

plotting. In addition, some of the plots were produced with the SATANGD

graphics routines developed at GSI (by K.-H. Schmidt).

223

6.1. Event-Type Identification

The initial part of the offline data analysis is event-type identification: choosing

the true positron, gamma-ray, and scattered-nuclei events from the time and

energy signals recorded event by event on magnetic tape during the experiments.

It is primarily based on an analysis of the timing relationships between the

detector signals and produces the following distributions:

• positron spectra (with proper annihilation-radiation identification),

• annihilation-radiation spectra (for background subtraction in the positron spectra),

• gamma-ray spectra (for the calculation of background positrons from nuclear processes),

• and scattered-particle spectra (for normalization).

The spectra are generated in forms suitable for a study of the relationships

between the measured parameters: in particular, of the correlations between the

positron energy spectra and the angles of the scattered nuclei. This section

describes the internal logic of the initial analysis program for event-type

identification and the form of the generated spectra. The chance-coincidence

subtraction will be outlined here, and described in more detail in the next section.

Because of the large amount of computer time required to perform the event-A

type-identification analysis (approximately 3X10 CPU seconds for each collision

system) and the need to analyse the data often in order to study different aspects,

the initial analysis has been done in two steps. In the first step, the data, is read

event by event from magnetic tapes, pre-analysed, and written as an abreviated

event list onto a disk file. The pre-analysis consists of all features of the data

analysis that did not vary from analysis to analysis:

• checking that all time and energy signals are within allowed bounds,

• calibrating all time and energy signals,

224

• generating the time difference spectra between all detectors,

• correcting all the time signals and the energy signals of the particle detectors and Nal(Tl) annihilation-radiation detectors for gain shifts,

• ascertaining whether each time difference signal comes in a prompt or a chance coincidence window,

• summing the energy signals from the two halves of the Si(Li) positron detector,

• and summing the energy signals from the eight segments of the annihilation-radiation detector.

The abreviated event list written onto a disk included only the calibrated energy

signals for positrons, the annihilation-radiation sum, and gamma rays, the

calibrated angles of the two scattered particles, and logic words describing which

event types were present and whether the various time coincidences between

detectors had occured in prompt- or chance- coincidence windows. This pre­

analysis accounted for about 90% of the total required computer time for event-

type identification in the analysis of each collision system and needed to be done

only once.

The second step of the particle identification analysis could then be done much

more quickly and repeated as often as necessary for the subsequent study of the

data. The abreviated event list is read event by event from the disk file. The

logic words describing the recorded event types and the types of time coincidences

are then used to construct from the recorded energies and angles the final

positron, annihilation-radiation, gamma-ray, and scattered-particle distributions.

The resulting one-, two-, and three-dimensional spectra can be plotted directly or

stored for transfer to other analysis programs for further work.

Figures 6-1 and 6-2 show the internal logic structure of the event-type

identification analysis program in more detail. As can be seen from the diagrams,

225

the analysis is divided into four parts. In the first part, the scattered-particle

spectra, which are common to all event types, are examined. In the other three

parts, the three types of events (positrons, gamma rays, and scattered particles)

are separately analysed. All but the final step of each of the last three parts,

where the final spectra are built, corresponds to the pre-analysis described above.

P0 3 I t r o n e v ent ?

3TTsCont i nue p o s i t r o n

event ( F i g u r e

6 .2 )

lEnterl, *|Read event l

XP o s i t r o n or gamma-ray

or p a r t i c l e event?♦ Yea

Are a l l s i g n a l s f rom both p a r t i c l e d e t e c t o r s pr esent?

^ YesAre e n e r g i e s f o r both p a r t i c l e

d e t e c t o r s w i t h i n po l ygons?* Yes

Are the tuo p a r t i c l e d e t e c t o r s _______ In t ime co i nc i de nc e ?_______

True Chance lExft l

Gamma-ray HEx 1 tl P a r t i c l eevent? No event? No

fesAre a l l s i g n a l s ■ *|ex i tl Generate p a r t i c l e

from Nai pr esent ? No n o r ma l i z a t i o nilr Yes spec t ra

I s Nai in t ime co i n c i d e n c e wi th a r t i c l e d e t e c t o r s ?

e ^ | ChanceGenerate gamma-

parTru

ray s pe c t r a

F ig u re 6-1: Event-type analysis program.Flow diagram of the analysis program for event-type identification.

226

j c ont l nue p o s i t r o n event| *A r e a l l s i g n a l s f rom at l e a s t one h a l f o f the S i ( L l ) and at l e a s t one segment o f the 8 - f o l d Nal p r esent ?

♦ YesCheck 3 - f o l d t ime c o i n c i d e n c e : S i ( L i ) f r o n t - S i ( L l ) r e a r ,

S i ( L l ) - p a r t i c l e d e t e c t o r s , and S i ( L i ) - 8 - f o l d Nal ------------J l

Are the two ha l v e s o f the S i ( L i ) in t ime

c o i nc i de n c e ? NoX es

P o s i t r o n or gamma In S i ( L l )

f r o n t h a l f ?

XI s S i ( L i ) in t ime c o i n c i ­

dence wi th p a r t i c l e

d e t e c t o r s ?True

No

Chance

I s S i ( L i ) in t ime c o i nc i de n c e wi th

at l e a s t one segment o f the

8 - f o l d Nal?True

Which h a l f o f the S i ( L l ) has a s i g n a l ?

Front Rear

No

Chance

I s S i ( L l ) in t rue t ime co i n c i d e n c e wi th at l e a s t one segment o f

the 8 - f o l d Nal?Yes

Add two Use Use Useenergy r e a r f r o n t rears i g n a l s energy energy energyt o g e t he r s i g n a l s 1gnal s i g na l

No

Bui ld segment sum energy

spectrum wi th sum o f energy s i g n a l s f rom on l y segments in t rue t ime

c o i nc i de n c e wi th S i ( L i ) and energy

s i g n a l w i th i n proper l i m i t s

Bui ld segment sum energy

spectrum wi th sum o f energy s i g n a l s f rom onl y segments in chance t ime

c o i nc i d e nc e wi th S i ( L i ) and energy

s i g n a l w i th i n proper l i m i t s

I s segment I s segmentsum energy sum energy

wi t h i n No wi th i nproper r properl i m i t s ? lExi t l l i m i t s ?

No

Yes Yes

Generate p o s i t r o n spec t ra , ' us ing the s e l e c t e d energy s i g n a l and combining the t rue and chance t ime c o i n c i d e n c e s from

S1(L1 ) - p a r t i c l e d e t e c t o r s and S i ( L l ) - 8 - f o l d Nal ___________ t ime c o i nc i de n c e s by the MxU mat r i x method

F ig u re 8-2: Positron-event analysis program.Flow diagram of the part of the event-type analysis program dealing with positron events.

227

6.1.1. Sc&ttered-Particle Identification

The first part of the analysis, triggered by the presence of at least one of the

four event types and all the signals from the particle detectors, examines the

scattered-particle spectra, which are common to all event types. As explained

above in Section 3.2.2, each of the two particle detectors produces three signals: a

time signal each from the anode foil and the cathode delay line (whose difference

is proportional to the laboratory, angle of the scattered particle) and a rough

energy signal from the anode. The scattering angles of the two particles is

determined from the difference of the two time signals in each detector (based on

the calibration described in Section 4.3) In addition, the time difference between

the anode-foil signals of the two detectors is obtained, based on the time

calibration of the TDC14.

These scattered-particle parameters are inspected to verify that they correspond

to particles coming from elastic (or quasielastic) collisions. The correlations

between the scattering angles of the two particles, between the energy and

scattering angle of each particle, and between the time difference of the two

particles and their scattering angles provide triggers for the subsequent analysis.

These correlations are shown in Figures 6-3 to 6-6 for the 238U + 238U collision

system at a bombarding energy of 5.9 MeV/amu15.

The measured angular correlations for scattered-particle events are shown as a

scatter plot in Figure 6-3. Each event is plotted in a plane with respect to the

angular variables A9 = 6U — &L and E9 = 9 y + 9L (defined in Section 3.2.4

14The TDC calibration had been checked with the 37.5 nsec separation between consecutive beam pulses (the beam macrostructure), visable in the scattered-particle chance coincidences. It was found to be accurate at the percent level and thus adequate for our purposes.

15These and most of the examples found in this chapter are taken from the U+U collision system. A similar analysis was separately performed for the U+Cm, U+Pb, and the U+Sm collision systems.

228

1 0 0

9 6

O'<D

uH, 9 2

Q>W

8 8

8 4

8 0- 6 0 -4 0 - 2 0 0 2 0 4 0 6 0

A 9 [deg]

F ig u re 6-3: Particle angle-angle scatter plot.

Scatter plot of scattered particle events for the + 238U collision system at a bombarding energy of 5.9 MeV/amu. Each event is plotted with respect to AS and L9. The solid line indicates the polygonal trigger used to set the other three polygonal windows (cf. Figures 6-4 to 6-6). The dashed line indicates the trigger used for the subsequent particle- identification analysis.

2 3 8 y + 2 3 8 jj

5.9 MeV/amu

229

1 0 0

CJ

ooCO

>Naw

15w.

OUJ

8 0

6 0

4 0

20

23 8 u + 238 y

5.9 MeV/amu,i— i , i i— t— r — i— m — r

’ l . - . r i - ’ ' . f I-■ • ' '.i. •< •. ‘ • . • . .•* " l t * r ' . i - i 'i I . i 1 . • •

;i ‘hi!' Yii:!'.:!"!!'- v . • ' • • ‘l ! S i i M ! / -• - ’ - ’ •

" Upper DetectorJ L J I L

10 2 0 3 0 4 0 5 0 6 0

#U [deg]7 0 8 0

F ig u re 6-4: Particle energy scatter plot: upper detector.AOQ AAQ

Scatter plot of scattered particle events for the U + U collision system at a bombarding energy of 5.9 MeV/amu. Each event is plotted with respect to the laboratory scattering angle and the energy E y of the particle detected in the upper counter. The solid line indicates the trigger used for the particle-identification analysis.

230

above). In Figures 6-4 and 6-5, the energy of each scattered particle is plotted

against the scattering angle for particles enetering the upper and the lower

particle detector , respectively16. Finally, the time between the anode foils of the

two detectors is shown as a scatter plot, graphed as a function of A9, in Figure

6-6. A two-dimensional plot against the variable A9 is necessary (as opposed to a

one-dimensional plot) because the time difference between the two anode signals

has a slight angular dependence, due to the angular dependence of the velocities

(and thus the time of flight) of the two nuclei.

Since the collision systems studied for this work were all below the Coulomb

barrier, elastic (and quasielastic) scattering events dominate, as is readily apparent

in these four scatter plots. Two-dimensional, polygonal windows, indicated in the

plots by solid lines, are placed around around the intense part of the distribution

to pick out these events for the subsequent parts of the event-type-identification

analysis17.

The time difference between the anode foils of the two detectors also serves to

test whether the two detectors are in time coincidence. The events falling within

the polygonal window in Figure 6-6 are taken to be the prompt coincidences

between the two particle detectors. The same polygonal window, moved by twice

the distance between the pulses of the beam macrostructure

(2 X 37.5 nsec = 75 nsec), measures the chance coincidences between the two

1 fiThe particle-energy curves bend over for smaller scattering angles because of the magnetic

field of the solenoidal positron transport system, primarily due to distortion of the electron avalanche cloud within the particle detectors. In the absence of the solenoidal magnetic field, this curve is straighter, reflecting the expected kinematical relationship between energy and scattering angle.

17To facilitate the setting of the gates on the scattered-particle energy and time-of-flight distributions, the smaller window indicated by the solid line in Figure 6-3 was first placed around the elastic-scattering kinematic region. After setting the windows in Figures 6-4 to 6-6, the smaller window was replaced by the larger window indicated by the dashed line for the subsequent analysis, to allow freedom in choosing angular cuts later.

231

Q)

OOCO>%wOw

J 5w<

-JLlI

1 0 0

2 3 8 u + 2 3 8 u

5 .9 MeV/amu

3 0 4 0 5 0 6 0

0L [deg]

F ig u re 6-5: Particle energy scatter plot: lower detector.Same as Figure 6-4, but plotted for the lower particle detector.

particle detectors18. This was found to be less than 0.2% of the total count rate

for all systems measured, which is small enough to be ignored in the subsequent

18An integral multiple of the beam pulse separation has to be used since no scattered particles are detected between pulses. Shifting by two beam pulses guaranteed that the polygonal gates would not overlap.

232

2 3 8 u + 2 3 8 y

5 . 9 MeV/amu

A 6 [deg]

F ig u re 6-6: Particle time-of-flight scatter plot.

Scatter plot of scattered particle events for the collisionsystem at a bombarding energy of 5.9 MeV/amu. Each event is plotted with respect to the time difference between the anode-foil signals of the two particle detectors and to A9. The solid line indicates the trigger used for the particle-identification analysis.

233

Following the test of the particle spectra common to all event types, the further

analysis of the positron, gamma-ray, and scattered-particle events proceeds

separately in order to use the scaledown information recorded with each event20.

The presence of the scattered-particle event-type bit, set by a 300-nsec,

hardware coincidence between the anode signals of the two particle detectors (as

described in Section 3.3.1), triggers the final part of the scattered-particle

analysis. The particle spectra necessary to verify the two-dimensional gates

shown in Figures 6-3 to 6-6) above and to normalize the positron and gamma-ray

spectra, as described in Section 6.4 below, are generated. As an example,

isometric plots of the scattered-particle events are displayed in Figures 6-7 to 6-9.A4Q AiO

for the three collision systems studied in this work: U + Cm at a

bombarding energy of 6.05 MeV/amu, + 238U at 5.9 MeV/amu, and 238U +

208Pb at 5.9 MeV/amu. Intensity is plotted as a function of A9 and E6.

6.1.2. Gamma-Ray Identification

The identification of gamma-rays is slightly more complicated than that of

scattered-particles, as described above, because of the additional time coincidence

between the gamma-ray detector and the two particle detectors.

The analysis of this event type is prompted by the presence of all the signals

from the gamma-ray detector21 and of the gamma-ray event-type bit. This bit

analysis19.

19The analysis of the chance coincidences is described in greater detail in Section 6.2.

20The two separate gamma-ray event types, one for the forward detector and one for the backward detector, are analysed identically, so their description will be combined into one for the following.

21The particle-detector signals have, of course, already been verified in the first part of the analysis.

234

mcdoo

A 9 [deg]F ig u re 6-7: U+Cm particle isometric plot.

Isometric plot of scattered particle events for the + 248Cm collision system at a bombarding energy of 6.05 MeV/amu. Intensity is plotted against A d and EO.

indicates a 300-nsec, hardware time coincidence between the gamma-ray detector

and the anode of the upper particle detector as well as between the anodes of the

two particle detectors. Figure 6-10 shows the energy distribution of the signals

present in the two gamma-ray detectors triggered only by the 300-nsec, hardware

time coincidence. In addition to steep, exponentially shaped distribution of

gamma rays visable at lower energies, there is in both detectors a high energy

component due to chance coincidences which fall within the wide hardware

window. The forward detector also has an additional high-energy component due

to neutrons produced in nuclear reactions which move preferentially forward in

the beam direction.

2 3 8 u + 2 4 8 C m

6.05 MeV/amu

- 4 0 . *2 0 0 2 0 4 0

235

4 0 0 -co 2 0 0 -

6 0 0 - 238 y + 238y

5.9 MeV/amu

-4 0 -2 0 0 20 4 0A 6 [deg]

F ig u re 6-8: U+U particle isometric plot.

Same as Figure 6-7, but for the 238U + 238U collision system at a bombarding energy of 5.9 MeV/amu.

Gamma rays are identified by time coincidence between the gamma-ray detector

and the anode foil of the upper particle-detector. This is shown as a scatter plot

in Figure 6-11. As in Figure 6-6 above, the time difference between the upper

particle detector and the gamma-ray detector is portrayed in a two-dimensional

plot as a function of A9 because of the angular dependence of the time of flight of

the scattered-particles. P art (a) shows the time coincidence between the upper

particle detector and the forward gamma-ray detector. Part (b) shows the same

for the rear gamma-ray detector22. Polygonal gates, indicated by solid lines, are

2°“Differences between the two plots are due primarily to the forward-moving neutron contamination preferentially entering the forward gamma-ray detector.

236

6 0 0 - 2 3 8 ( j + 2 0 8 p b

~ 4 0 0 - 5 * ® M e V / a m u

- 4 0 - 2 0 0 2 0 4 0

A6 [deg]

F ig u re 6-9: U +Pb particle isometric plot.

Same as Figure 6-7, but for the ^ U + ^ P b collision system at a bombarding energy of 5.9 MeV/amu.

placed around the prompt coincidences for each detector. The same gate, shifted

in time by two beam pulses (75 nsec), determined the chance coincidences.

Three-dimensional distributions of the gamma-ray yield, describing the

measured intensity as a function of the gamma-ray energy and scattering-angles of

the two particles (or the equivalent angular parameters E 9 and A&) are generated,

using the time coincidence spectrum to subtract chance coincidences, as described

in Section 6.2 below. In these forms the gamma-ray spectra can be checked for an

angular dependence of the energy distribution and used to calculate the

background of positrons from nuclear processes corresponding to any scattered-

particle angular region. Examples of these distributions for the U+U collision

237

F ig u re 6-10: Gamma-ray energy spectrum before analysis.The yield of gamma-ray events as a function of channel number detected in the forward (upper curve) or the rear (lower curve) gamma-ray counters for the collision system at a bombarding energy of5.9 MeV/amu. The events are triggered only by the hardware coincidence of the gamma-ray event type.

system at a bombarding energy of 5.9 MeV/amu are shown in Figures 6-12 and

6-13. Figure 6-12 shows the energy distribution of gamma rays in coincidence

with particles scattered into the angular region 20 ° < 0 ^ < 70 ° for both the

forward and the rear gamma-ray detectors. A comparison with Figure 6-10 shows

the effect of the gamma-ray-identification analysis. Figure 6-13 shows isometric

plots of the scattering angles of the particles in coincidence with gamma rays

detected in the forward and the rear detectors in parts (a) and (b), respectively.

238

23 8 u + 23 8 y

5 .9 M eV/am u

o<D(/)C

ocroEEo

01a>

aa.

i-<

- 6 0 -4 0 -2 0

Ad [deg]

F ig u re 6-11: Particle - gamma-ray time difference.

Scatter plot of gamma ray events for the + 238U collision system at a bombarding energy of 5.9 MeV/amu. Each event is plotted with respect to A9 and to the time difference between the anode foil of the upper particle detector and the gamma-ray detector. P art (a) corresponds to the forward gamma ray detector and part (b) to the rear detector. The solid lines indicate the triggers used for the gamma-ray identification analysis.

01010202000001024800010202019153000102020000900101020000010102

000013021000530202029100000101020001010202000001010100010102020000

^

239

F ig u re 6-12: Gamma-ray energy spectrum after analysis.Measured yield of gamma rays as a function of the energy of the gamma ray detected in the forward (upper curve) or the rear (lower curve) counter for the + 238U collision system at a bombarding energy of 5.9 MeV/amu after the gamma-ray-identification analysis (cf. Figure 6- 10).

240

A 6 [deg]

A 6 [deg]

F ig u re 6-13: Gamma ray isometric plots.Isometric plot of gamma-ray events detected in the forward gamma-ray counter in part (a) and in the rear counter in part (b) for the 238U + 238U collision system at a bombarding energy of 5.9 MeV/amu. Intensity is plotted against A9 and E9. For comparison, the same distribution with the particle trigger alone can be seen in Figure 6-20(b).

241

Intensity is plotted with respect to A9 and E9. (The last figure can be compared

with a similar plot gated only on the scattered-particle trigger shown in Figure

6-20(b).)

6.1.3. Positron Identification

The positron event type involves the most complicated analysis, as can be seen

from Figures 6-1 and 6-2. The positron event type entails at least a triple

coincidence. As outlined above in Section 3.2.2, the detection of a positron

includes a coincidence between the two particle detectors, the Si(Li) positron

detector, and the annihilation-radiation detector. The situation is further

complicated by the fact that the positron detector is divided into two parts and

the annihilation-radiation detector into eight segments. The following method is

adopted to identify a positron event.

The positron-identification analysis begins with the presence of the time and

energy signals from at least one of the two halves of the positron detector and at

least one of the eight segments of the annihilation-radiation detector (in addition

to the particle detector signals already confirmed above) and the positron event-

type bit. This bit represents the presence within the 300-nsec, hardware

coincidence window of time signals from the anodes of both scattered particle

detectors, at least one of the two halves of the positron detector, and at least one

of the eight segments of the annihilation-radiation detector.

Figure 6-14 shows the preliminary energy distributions recorded in the two

halves of the positron detector with only the hardware time coincidence. P art (a)

shows the energy distribution for the front half of the Si(Li) detector. In addition

to the positrons, a large, low-energy, exponentially shaped background consisting

primarily of gamma rays and delta electrons falling within the 300-nsec

coincidence window is clearly visable. Part (b) shows the distribution for the rear

242

120

toh- 0zz> 25oo 20

15

10

5

04 0 0 8 0 0 1200

C H A N N E L

1600 2 0 0 0

F ig u re 6-14: Positron energy spectrum before analysis.The yield of positron events as a function of channel number detected in the front (part (a)) or the rear (part (b)) halves of the positron counter for the 238U + 238U collision system at a bombarding energy of 5.9 MeV/amu. The events are triggered only by the hardware coincidence of the positron event type.

243

half. The low-energy background is not so prominent here because the rear half

of the detector is shielded from low-energy gamma rays and delta electrons by the

front half.

The identification of positron events is based on the inspection of three time

coincidences23:

• between the two halves of the positron detector,

• between the positron detector and the particle detectors,

• and between the positron detector and the annihilation-radiation detector.

In the 15-20% of positron events when both halves of the positron detector

register signals, the time difference spectrum between the two halves, shown in

Figure 6-15, determines how signals are interpreted. A study of the energy signals

from the two halves of the positron detector indicated that two classes of events

are present.

About 90% of the coincidence events come in a prompt peak, corresponding to

the window labelled 1 in Figure 6-15. The sum of the energy signals from the two

halves form a bell-shaped spectrum like that of the 75-80% of the positron events

which register in only one of the two halves, whereas the signals from either half

alone do not. These events therefore are assumed to correspond to a positron

which outscatters from the front half of the Si(Li) positron detector, spirals back

in the solenoidal positron transport system, and is absorbed in the rear half. In

this case, the energy signals from both halves are added together and the time

signal from the front half is used for checking the coincidences to the particle

detectors and the annihilation-radiation detectors below. Approximately 15% of

all the positron events fall into this class.

A4The fourth time coincidence between the two particle detectors has already been checked in

the first part of the analysis.

244

A t ( Front e+ -R e a r e+ ) [nsec]

F ig u re 6-15: Positron front - rear detector time difference.The yield of positron events detected in both halves of the positron detector is plotted as a function of the time difference between the two halves for the 238U + 238U collision system at a bombarding energy of 5.9 MeV/amu. Windows 1 and 2 are explained in the text.

The remaining coincidence events appear in a shoulder on the prompt peak,

corresponding to Window 2 in Figure 6*15. With respect to Window 1, the time

signal in the front half comes about 4 nsec before the signal in the rear half for

these events. In this case, the energy signals from the rear half form a bell-shaped

spectrum like that of the other positron events, while the signals from the front

half have a low-energy, exponential shape. It is assumed that these relatively rare

events correspond to a positron in the rear detector half and a gamma ray in the

front half. In this case, only the signals from the rear half are used.

The time coincidence between the positron detector and the upper particle

245

detector is handled much as the corresponding coincidence for the gamma-ray

events, already described. A scatter plot of the time difference between the signal

from the anode foil of the upper particle detector and the proper half of the

positron detector, as determined above, is shown as a function of AO in Figure

6-16. A polygonal window (the solid line in the figure) was placed around the

prompt coincidences. The same gate, shifted by two beam pulses (75 nsec),

measured the chance coincidences.

The final time coincidence, that between the positron detector and the eight

segments of the annihilation-radiation detector, is demonstrated by Figure 6-17

which shows the spectrum of the time difference between the positron detector

and one of the eight segments of the annihilation-radiation detector. P art (a)

shows coincidences with the front half of the positron detector, part (b) with the

rear half. In both cases, the shaded region represents the prompt time peak used

in the analysis24.

In order to extract the maximum detection efficiency for annihilation radiation

from the experimental set-up, the energy signals from the eight segments of the

annihilation-radiation detector are summed to produce a single spectrum

containing both annihilation quanta. Proper analysis of the chance coincidences

(describes in Section 6.2 below) dictates the following procedure. In those cases

where at least one segment has a prompt time coincidence with the positron

detector, the energy signals from all the segments which are in prompt time

coincidence with the positron detector (and only those segments) are summed to

form the prompt annihilation-radiation sum-energy. If no segments are in

prompt coincidence with the positron detector, then the energy signals from all

24The large exponential tail on the left side of the prompt peak in each spectrum consists largely of chance coincidences with gamma rays in the positron detector and is excluded, as described in greater detail in Section 6.3.

246

co0

CL1

0)

a -20

h<

- 4 0

- 6 0

2 3 8 y + 2 3 8 y

5 .9 MeV/amu

- ih i: . i i f l i : • * » « 1 1•• • I t * Y »* • *. •••••?•!•* ^ i* r — I 1 ii

• - •?; I r*f ffi+'ft'V’ . .** .1 .• ■■ :! !l:i:iX.:.- '.n ::-:'::<!itf:-rY u:u:v 'ii!J ;r*:n iljl.r»lTi«,.::.-..i.:3^'.' I:."

: - i . r ■■•:«..•••■ I . • . i .1 • V I.. •■••••. • i . *1• I’,.. : . . •. •• - r . . •• •• *.ri , r _. - I . « .• ... .: . i.. *. .1 * •. . I .

I .. :i" : i . : I.':-. . '" ' . .. . " J ' . • ** I3*' 1i. ; rMj - i-T-

j 1 1 , i - ■ f t hL'livii'-i1

- 6 0 *4 0 -2 0 20 4 0 60

Ad [deg]

Figure 6-16: Scattered particle - positron time difference.AOO AAQ

Scatter plot of positron events for the U + U collision system at a bombarding energy of 5.9 MeV/amu. Each event is plotted with respect to A9 and to the time difference between the anode foil of the upper particle detector and the front half of the positron detector. The solid line indicates the trigger used for the positron-identificatiou analysis.

247

I—i—i—r—i—i—i—i—i—i—i—i—i—i—i—r

A t (e+-Ann. Rad) [nsec]Figure 6-17: Positron - annihilation radiation time difference.

The yield of positron events detected in both the positron detector and one of the eight segments of the annihilation-radiation detector is plotted as a function of the time difference between the positron detector and the segment for the 238U + 238U collision system at a bombarding energy of 5.9 MeV/amu. Part (a) shows coincidence with the front half of the positron detector and part (b) with the rear half. The shaded regions indicate the prompt time peak used for the positron- identification analysis.

248

those segments whose signals fall in the chance coincidence gate are summed to

form the chance sum-energy.

Figure 6-18 illustrates the construction of the sum-energy for the U+U collision

system. Part (a) shows the energy distribution of annihilation radiation detected

in one of the eight segments. A single 511-keV annihilation gamma ray line is

visable, with the line shape due to Compton scattering expected for a Nal(Tl)

detector. Part (b) shows the distribution of the prompt-coincidence sum-energy

signals. In this distribution, both the detection of a single annihilation gamma ray

and the detection of both gamma rays summing to 1022 keV are apparent.

The prompt or the chance sum-energy, as appropriate, is required to fall within

a window extending from 440 keV (just below the single annihilation-radiation

peak) to 1040 keV (just above the double annihilation-radiation peak). This is tire-

shaded region in Figure 6- 18(b)25. By summing the signals from the segments, the

energy of annihilation quanta which Compton scatter out of the first segment

they enter can be partially recovered. This method provides approximately 3

times the detection efficiency of the the more obvious procedure of requiring full

511-keV signals in opposite segments.

The positron spectra are generated with the energy signals from one half or

from both halves of the positron detector added together, as determined by the

analysis above. Chance coincidences are subtracted by a method described in

Section 6.2. As was the case for the gamma-ray spectra, the final positron spectra

are three-dimensional distributions describing measured intensity as a function of

the positron energy and the two particle scattering-angles (or the equivalent

angular parameters AO and EO). In this form, positron energy distributions for

25The region below 440 keV was not included because it is largely contaminated with low- energy gamma rays from the target, as described in greater detail in Section 6.3.

249

COH23OV

1600

r [keV]

Figure 6-18: Annihilation radiation energy spectra.The intensity of annihilation radiation is plotted as a function of themeasured energy of the annihilation gamma rays for the 238U + ’" 8U collision system at a bombarding energy of 5.9 MeV/amu. Part (a) shows the energy distribution for one of the eight segments and part (b) shows the sum energy distribution. The shaded area is the window used for the positron-identification analysis.

238t

250

any angular region and conversely angular distributions for any energy window

can be easily generated.

Examples of the resulting positron distributions for the U+U collision system are

displayed in Figures 6-19 and 6-20. Figure 6-19 shows the energy distribution in

coincidence with particles scattered into the angular region 20 ° < &Lab < 70 ° .

This distribution can be compared with Figure 6-14 to see the effect of the

positron-identification analysis. Figure 6-20(a) shows as an isometric plot the

scattering angles of the particles in coincidence with positrons. Intensity is

plotted with respect to A9 and E6. For comparison, Figure 6-20(b) shows a

similar plot, gated only on the scattered-particle trigger.

25i— i— i— i— I— i— i— i— — i— i— i— |— i— i— r

0 4 0 0 8 00 1200 1600

Ee+ [keV]

Figure 6-19: Positron energy spectrum after analysis. Measured yield of positrons as a function of the positron kinetic energy for the 238U + 238U collision system at a bombarding energy of 5.9 MeV/amu after the positron-identification analysis (cf. Figure 6-14).

251

2 3 8 U + 2 3 8 u

5 . 9 M e V / a m u6 0 1

(a )

col -z3Oo

Positron+ParticleTrigger

- 6 0 * 4 0 - 2 0 0 20 4 0 6 0

(b )Partic le Trigger

- 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0A d [deg]

Figure 6-2 0 : Positron isometric plot.Part (a) shows an isometric plot of positron events for the U + U collision system at a bombarding energy of 5.9 MeV/amu. Intensity is plotted against A9 and E9. For comparison, part (b) shows the same distribution with the scattered-particle trigger alone.

252

The method of chance-coincidence subtraction used for the three different event

types will be presented in this section. Separate schemes are employed for the

three event types because of differences in the relative size of the chance

coincidence component, the. complexity of the required coincidence scheme, and

the correlations between the detectors in each case. As an orientation, the

amount of chance coincidences found in each collision system for the various time

coincidences is shown in Table 6-1 .

6.2 . A n a lysis o f th e C h ance-C oin cidence R ate

System Particle Particle Positron Particle Particle Particle

Particle Positron Annih.Radiat.

Positron

Annih.Radiat.

ForwardGamma

RearGamma

U+Cm 0.0091 4.20 0.17 4.46 14.96 9.057U+U 0.011 3.19 0.17 3.36 5.370 3.127U+Pb 0.026 1.9 0.09 2.0 2.530 1.85U+Sm 0.20 3.9 0.0 3.9 3.59 3.00

Table 6-1: Chance Coincidences (% of total rate).

6.2.1. Scattered-Particle Event

The correction for the chance coincidence rate in the scattered-particle event

type is the simplest possible, namely none. This event type involves only a single

coincidence between the two particle detectors. An investigation of the spectrum

of time-difference between the two detectors, shown in Figure 6-6 above,

indicated that the chance coincidence rate is less than 0.2% of the total counting

rate for all the measured collision systems. The chance rate is determined by

comparing the count rate in the prompt, two-dimensional polygonal gate with

253

that in the same gate shifted by 75 nsec (twice the separation of 37.5 nsec

between pulses of the beam macrostructure).

The reasons for this small chance rate are threefold:

• at the beam energies below the Coulomb barrier at which this experiment was made, Rutherford scattering completely dominates the particle counting rate,

• the particle detectors cover kinematically complimentary regions of space,

• and the detection efficiency of both detectors within this region is essentially one.

This small chance rate is more than an order of magnitude smaller than the

chance-coincidence rate between the particle detectors and either the positron

detector or the gamma-ray detectors, and indeed much smaller than the statistical

accuracy of the final positron and gamma-ray spectra. As a result, the chance

coincidences between the two particle detectors is ignored in this event type and

also in the positron and gamma-ray event types, described below. This greatly

simplifies the subsequent analysis, since the two particle detectors can be treated

as a single unit in the following chance-coincidence subtraction schemes.

6.2.2. Gamma-Ray Event

The gamma-ray event type involves a triple coincidence between the two

scattered-particle detectors and a gamma-ray detector. Treating the two particle

detectors as a single unit by ignoring the small chance rate between them, as

described above, reduces this to a single coincidence between the particle

detectors, handled together, and a gamma-ray detector. For this scheme, the

particle-detector unit is represented by the time signal of the anode of the upper

detector.

The polygonal-shaped gate which picks out the prompt time coincidences

254

between the upper particle detector and gamma-ray detector is shown above in

Figure 6-1 1 . The chance rate is measured by shifting the gate by two beam pulses

(75 nsec). For the forward gamma-ray detector, the chance window holds 3-15%

as many counts as the prompt window, depending on the collision system, while

for the rear detector, it holds 2-10% as many .

The correlation between the scattered particle detectors and the gamma-ray

detectors is small, due to the relatively small geometric plus intrinsic detection

efficiencies of both detectors ( ~ 15% of the forward hemisphere for the particle

detectors and ~ 0.1% for the gamma ray detectors). Since the chance-

coincidence rate itself is also small, the particle-detector unit and the gamma-ray

detector can be treated as uncorrelated counters. The gamma-ray spectra are

corrected for chance coincidences by simply subtracting the counts which fall in

the chance window from those in the prompt time window.

6.2.3. Positron Event

The analysis of the chance-coincidence rate for the positron events is more

complicated because this event requires a quadruple time coincidence between

three types of detectors, each of the detector systems had multiple parts, and the

events registered in the detectors were correlated in a complicated manner. A

study of the correlations between the various detectors, described in the following

paragraphs, indicates how each of the three types of detectors can be treated as a

single unit, which time difference spectra provide the most straightforward

coincidence information, and how the measured spectra must be handled to

properly account for the chance coincidences.

The identification of a positron event is based on a coincidence between both

26The higher rate in the forward counter was due to contamination in this counter from forward-moving neutrons.

255

scattered-particle detectors, at least one of the two halves of the positron detector,

and at least one of the eight segments of the annihilation-radiation detector. In

all, twelve detectors are involved. Since, however, the required coincidence is not

simply that all twelve detectors register signals simultaneously, but rather only

certain combinations, an examination of the correlations among the expected

signals has simplified the arrangement considerably.

As has already been pointed out above, the two scattered-particle detectors can

be treated as a single unit by ignoring the negligible chance-coincidence rate

between them. In a similar manner, the correlations between the two halves of

the positron detector show how they also can be treated as a single unit.

As described in the last section, each positron event can be established as the

detection of either a positron in the front half, in the rear half, or in both halves

(i.e., a positron which outscattered from the front half and spiralled back in the

magnetic field of the solenoidal transport system to strike the rear half). In the

last case, the energy signals from both halves are summed and the time signal

from the front half is used to represent both halves. The positron counting rate

(roughly 1 Hz in the front half and about 1/10 as large in the rear half) is too

small compared to the time coincidence window ( ~ 10 nsec) for significant

chance coincidences between separate positrons in both halves.

Chance coincidences between a gamma ray in the front half of the positron

detector and a positron in the rear half can be picked out of the time difference

spectrum between the two halves of the detector and eliminated because of the

longer time of flight of the positron than the gamma ray in the transport system

from the target to the positron detector. The other possible types of chance

coincidences between the two halves of the positron detector are negligible

because the counting rates are so small. The chance rate for a positron in the

front half and a gamma ray in the rear half is small because the front half of the

256

positron detector shields the rear half from gamma rays from the target. Any

combination of gamma rays and electrons in the two halves of the positron

counter is suppresed by the hardware coincidence to the annihilation-radiation

detector. As a result, a single time and a single energy signal can be

unambiguously assigned to the positron detector for each positron event, so that

the two halves of the positron detector can be handled together as a single unit.

The eight segments of the annihilation-radiation detector are correlated in a

complicated manner because each positron produces two annihilation gamma rays

and because the gamma rays can Compton scatter from one segment to another.

Adding the energy signals together, however, as described in Section 6.1 above,

produces a sum-energy spectrum containing all the events corresponding to the

detection of either one or both of the annihilation gamma rays. The chance-

coincidence rate between the segments of the annihilation-radiation detector is

negligible because the sum-energy is formed only with those segments whose time

signals fall in a narrow ( ~ 10 nsec wide) prompt gate with respect to the positron

detector (Cf. Figure 6-17), and the total counting rate in each segments is only

about 103-104 Hz. Using this sum allows the annihilation-radiation detector also

to be treated as a single unit. This reduces the system of twelve detectors to a

triple coincidence between the three groups of detectors.

The correlations between the three detector systems depend primarily on the

relative detection efficiencies of the three groups of detectors, described above in

Chapters 3 and 4.

• The detection efficiency of the scattered-particle counters, determined by the subtended solid angle, is approximately 15% of the forward hemisphere.

• The detection efficiency for positrons in the Si(Li) detector, reflecting primarily the transport efficiency, peaks at about 18%.

• The annihilation-radiation detector has a detection efficiency, mainly

257

limited by solid angle, of approximately 60% for the annihilation radiation from a positron striking the positron detector. It thus has a detection efficiency relative to the scattered particle detectors of about 1 1 % for a positron event.

The strongest correlation, by a factor of three, is between the positron detector

and the annihilation-radiation detector. Since the chance-coincidence rate is small

(<5% in all systems), insignificant error is allowed by ignoring the correlations

between the other two pairs of detectors in comparison to this coincidence. Thus

the particle detectors are treated as uncorrelated to the other two detector groups

in the analysis of the triple coincidence between the three groups of detectors.

Since the annihilation-radiation detector has only a 60% efficiency for the

detection of the annihilation gamma rays emitted from the Si(Li) detector, only

this fraction of the counts in the chance-coincidence window actually correspond

to a chance coincidence between a positron in the positron detector and an

annihilation gamma ray in the annihilation-radiation detector. The correlation

between the positron detector and the annihilation-radiation detector can be

accounted for to good approximation for the small chance-coincidence rates

involved here (a few percent of the total rate) by reducing the counts in the

chance-coincidence window by a factor of 60% before subtracting them from the

counts in the prompt window.

This is done in each event for each of the eight segments of the annihilation-

radiation detector by counting a chance event only after checking that the other

seven out of the eight segments do not have a prompt coincidence to the positron

detector. This uses the other segments of the detector to empirically reduce the

chance rate in each segment by 7/8 of 60%. As described in the last section, the

prompt sum-energy is formed by adding together the energy signals from only

those segments in prompt time coincidence to the positron detector. The chance

sum-energy spectrum is in turn made by adding together those segments in

258

chance coincidence to the positron detector, but only after checking that none of

the segments are in true time coincidence for that event. In both cases, the sum

energy has to be within the window enclosing the single and double annihilation

radiation events.

Since the correlation between the positron detector and the annihilation-

radiation detector is already accounted for, the two time coincidences between the

particle detectors and the positron detector and between the particle detector and

the annihilation-radiation detector are combined by the standard method for a

triple coincidence of uncorrelated counters27. A prompt gate and a chance gate

are set on both time difference spectra. The prompt gate containes both true and

chance events and the chance window contains only chance events. The four

possible combinations of the two sets of two windows are shown in Table 6-2.

PI = Tl + Cl

Cl

Tl • T2

P2 = Tl • C2 Cl • T2

T2 + C2 Cl • T2 Cl • C2

Cl • C2

C2 Tl • C2

Cl • C2

Cl C2

Table 6-2: Chance coincidence subtraction method.

The correct function of these combinations which picks out the true

coincidences in both time spectra is:

27The third possibility, the time difference between the scattered-particle detectors and the annihilation-radiation detector, provides no independent information and is not used.

259

PlP2 PlC2 C\P2+ C\C2= (T^CJiTz+CJ - {TX+C\C2

- C ^ + C J + c x- c 2

- [(T^+iC^)]- [(C'1-7'2)h-(C71-C>2)]

+ [(CyC,)]= TxT2 (6 .1 )

where P indicates a prompt coincidence, T a true coincidence, C a chance

coincidence, and 1,2 indicate the two time difference spectra.

This scheme is applied to the positron events to determine the chance-

coincidence rate, which accounted for 2 to 5% of the total counting rate in the

positron event type for the various collision systems, as listed in Table 6-1 (and

also in the summary of positron backgrounds in Table 6-3, in Section 6.3 below).

0.3. Annihilation-Radiation Sum-Energy Background Subtraction

A comparison of the annihilation-radiation sum-energy spectra for the collision

systems measured during the experiment with that obtained from the radioactive00 fio/?+ sources Na and Ge shows clearly the existence of a low-energy,

exponentially shaped background in the collision system measurements not found

in the source measurements. The two cases are shown in Figure 6-2 1 . Since the

chance-coincidence subtraction described in the last section has already been

made, the remaining background must come from a real coincidence between the

two scattered-particle detectors, the positron detector, and the annihilation-

radiation detector. This section describes the determination of the nature of this

background, its reduction, and the method used for the subtraction of the

remaining part.

As part of the investigation of the background, Figure 6-22 shows the

260

Et [keV]Figure 6-21: Annihilation-radiation background comparison.

The intensity of the annihilation-radiation energy is plotted as a function of the measured energy E' of the annihilation gamma ray for the + 23«u collision system at a bombarding energy of 5.9 MeV/amu in part (a) and a 22Na 0* source in part (b).

261

Et fkeV]

Ee+ [keV]

Figure 6-22: Annihilation-radiation background spectra.Part (a) shows the annihilation-radiation sum-energy distribution for the 238U + 238U collision system at a bombarding energy of 5.9 MeV/amu. The shaded regions are energy gates for positron distributions (region 1

is the normal positron gate). Part (b) is the yield of positrons as a function of the positron kinetic energy produced by gating on each of the two windows in part (a), as indicated.

262

relationship between the annihilation-radiation sum-energy spectrum and positron

spectra gated on it for the 238U + 238U collision system ay a bombarding energy

of 5.9 MeV/amu. Gate 1 in part (a) is the normal positron gate described in the

last two sections. Spectrum 1 in part (b) is the result of coincidence with these

annihilation gamma rays: the bell-shaped curve expected for dynamically-

produced positrons. Gate 2 in part (a), at low energies in the annihilation-

radiation sum-energy spectrum where the background clearly dominates, produces

spectrum 2. This has a completely different, low-energy, exponential shape,

which confirms that the background in the annihilation-radiation detector

corresponds to a background in the positron detector also.

The two most obvious sources for this background are the gamma rays and

delta electrons which are produced copiously in heavy-ion collisions and which

both would be expected to have the observed low-energy, exponentially shaped

energy distributions (see Figure 3-2) found in both the positron and the

annihilation-radiation detector. The two sources are differentiated by their time

of flight from the target to the positron or annihilation-radiation detector, since

the delta electrons spiral down the solenoid as the positrons do. As a result, two

different methods are used to correct for the background.t

Because of the time of flight of a positron from the target to the positron

detector, the part of the background due to a real coincidence between a gamma

ray in both the positron detector and the annihilation-radiation detector,

irregardless of their sources, has a different time relationship than that between a

positron detected in the positron counter and its annihilation gamma ray detected

in the annihilation-radiation detector, as in a positron event. This part of the

background can be seen in the time difference spectra between the positron

detector and the segments of the annihilation-radiation detector, gated again on

the low-energy part of the sum-energy spectrum (gate 2 in Figure 6-22(a)). The

263

exponentially shaped tail visable on the left side of the prompt time peak in

Figure 6-17 above is greatly enhanced in the collision-system measurements

compared to the source measurements.

A study of the shape of the positron energy spectra gated on this exponential

part of the time-difference distribution for both source and collision-system

measurements demonstrated that the exponential tail on the side of the prompt

time peak in the time-difference spectra between the positron detector and the

segments of the annihilation-radiation detector is due almost entirely to gamma-

gamma coincidences between these two detectors, and does not correspond to

positron events. This contribution from gamma-gamma coincidences is reduced as

much as possible without affecting the true positron events by fixing the gates

used in these time-difference spectra on the basis of calibration measurements

made with a 22Na 0+ source directly after the experiment. The gates are made as

narrow as possible, with FWHM ~ 10 nsec, as shown by the shaded regions in

Figure 6-17. This reduces the background from approximately 10% of the total

counting rate to less than 5%.

Since on the other hand, the part of the background involving a positron or an

electron in the positron detector has the same time relationship as the true

positron events (except for a negligible time of flight difference for a gamma ray

originating from somewhere other than the positron detector), the narrow

coincidence gates described above do not prevent it, so the following method is

used to remove it. The background is assumed to have an exponential shape in

the annihilation-radiation sum-energy spectrum. The sum of the lineshape of the

annihilation-radiation sum-energy spectrum measured with a 22Na source plus

an exponential function is fitted to the sum-energy spectrum measured for each

collision system separately. An example of this fit is shown in Figure 6-23 for the

U+U collision system. The amount of background in each case is taken to be the

264

area of the exponential function inside the sum-energy window used as a gate for

the positron events. The shape of the positron background to be subtracted for

each collision system is determined by gating a positron spectrum on the low-

energy part of the sum-energy spectrum, from 200 keV to 390 keV, as shown in

Figure 6-22(a), where this background dominates. This shape, scaled to the

proper area to match the area of the fitted exponential function, is then

subtracted from the positron energy distribution. The amount of background

subtracted in each collision system is shown in Table 6-3, where it is seen to vary

from 5 to 12% of the total positron counting rate. This dependence mirrors the

relative increase of the ratio of positrons from quasimolecular processes to

background gamma rays and delta electrons, since the former increases sharply

with the combined nuclear charge of the collision system ( x>(Zp+Zt) ~ 20) while

the later do not.

Positron distributions corrected for chance coincidences and the annihilation-

radiation-sum energy background are displayed in Figures 6-24 to 6-30. The first

figure, Figure 6-24 shows the positron energy distributions in coincidence with

particles scattered into the angular region 20 ° < 9 ^ < 70 ° for the three

collision systems studied: 238U + 248Cm, 238U + 238U, and 238U + 208Pb.

Figures 6-25 to 6-30 show as scatter plots the angular distributions of the

scattered particles, both with and without coincidence to positrons for the same

three collision systems. The first three figures are plotted with respect to 9y and

Qj, while the last three show the same three distributions plotted with respect to

AS and E9. In each figure, part (a) displays the angular distributions gated on

both positrons and scattered particles while part (b) shows the same distributions

gated on the scattered particles alone.

265

I02CO

c3O

L J I 0 I

1 0 ° 0 200 4 0 0 600 8 0 0 1000 1200

Er [keVlFigure 6-23: Annihilation-radiation-background calculation.

Determination of the annihilation-radiation sum-energy background fornOQ A4A

the U + U collision system at a bombarding energy of 5.9 MeV/amu. The histogram is the measured distribution. The fitted distribution (dashed line) is the sum of the annihilation-radiation sum- energy measured with a 22Na source (Figure 6-2 1(b)) and an exponentially shaped background (dot-dashed line). The shaded region marks the background within the sum-energy trigger window.

6.4. Correction for Detector Response and Normalization

The final steps in preparing the positron, gamma-ray, and scattered-particle

spectra are to correct the analysed spectra for the intrinsic line shape and

detection efficiency of the respective detection systems, for the scaledown (and

deadtime) factors used to control the analysis rate, and for the binsize and the

particle normalization.

266

System Chance Ann. Rad. NuclearCoinc. Sum Processes(%) {%) (%)

U+Cm 4.46 4.59 25.9U+U 3.36 5.06 32.9U+Pb 1.96 7.43 47.9U+Sm 3.91 12.3 100

Table 6-3: Positron Backgrounds

6.4.1. Lineshape Correction

The positron and gamma-ray spectra are corrected for the intrinsic lineshape of

the respective detection systems while the particle spectra are not. The lineshape

for the detection of positrons in the Si(Li) counter is shown in Figure 4-2(b) in

Section 4.1.1 above. The lineshape for the Nal(Tl) gamma-ray detectors is

described in Section 4.2 and illustrated by Figure 4-9. In both cases an analytical

form was determined from the calibration data for the detected lineshape as a

function of the impinging particle energy. The unfolding of the detector

lineshape, however, is done using different methods for the positron and the

gamma-ray spectra. The lineshape of the particle detectors is not unfolded since

both the positron and the gamma-ray distributions are measured in coincidence

with scattered particles and then normalized to the particle singles rate.

6.4.1.1. Gamma-Ray Lineshape Correction

The intrinsic lineshape of the Nal(Tl) gamma-ray detector is unfolded from the

measured gamma-ray energy distributions using an iterative procedure. The

measured energy distribution P ' ( E ' ) is the convolution of the emitted distribution

P { E ) and the lineshape matrix L { E ' , E ):

267

Figure 6-24: Measured total positron energy spectra.The measured total positron yields in coincidence with particles scattering into the laboratory angular region from 20° to 70° are plotted as a function of channel number for the three collision systems: 238U + 248Cm at a bombarding energy of 6.05 MeV/amu, + 238U at 5.9 MeV/amu, and 238U + ^ P b at 5.9 MeV/amu.

268

10 20 3 0 40 50 60 70 800 U [deg]

Figure 6-25: U+Cm positron scatter plot.qqo

Scatter plots for the U + Cm collision system at a bombarding energy of 6.05 MeV/amu. Each event is plotted with respect to By and &L. Part (a) shows positron events and part (b) the scattered-particle events.

269

0 u[deg]

Figure 6-26: U+U positron scatter plot.Same as Figure 6-25, but for the 3®U + ^ U collision system at a bombarding energy of 5.9 MeV/amu.

270

10 20 3 0 4 0 50 60 70 80#u [deg]

Figure 6-27: U+Pb positron scatter plot.Same as Figure 6-25, but for the 238U + ^ P b collision system at bombarding energy of 5.9 MeV/amu.

271

238u + 248 Cm

• 6 . 0 5 M e V /o m u

• • • • ¥ * * *.* * • •* • • • • *

_................ jS§S& & i;a

• . V A1 ; fv- . F V ?^;v.Vi.*»wr.rM; f ; -•• «• • 7l* .1 •»• • •, „ • %••?• " J A • • • • . • / . . «• • • • • • ’ w ► , %r. * • * . • • • • . * ■ . • . . • • • • •• *• • j* » . * • • • • * • • • • •* * *

O'<u8 6 -

( o ) P O S I T R O N + PART.ICLE T R I G G E R

Q>W 98

94

90

86

_L_I I I I J l . J — LI I I I ! I I T 7 T

* • ;.»• •»/• • * . . • • • •• •• s f • * * *\ s' t ,

'•v i3s*q. .. •-_/ *5

•. .• “- ’ .; \ •' - : • • . :> V v- •:* • • • •

( b )* P A R T I C L E TR IG G E R

-60 -40 -20 0 2 0 40 6 0A 6 [deg]

Figure 6-28: U+Cm scatter plots.Scatter plots for the 238U + 248Cm collision system at a bombarding energy of 6.05 MeV/amu. Each event is plotted with respect to AS and 270. Part (a) shows positron events and part (b) the scattered-particle events. The dashed lines indicate the calculated angular correlation for Rutherford scattering.

T.Q

[de?

j]

272

9 6 -

i—i. i ■ i i—i—i i i—i—i i i ■ i i i .i i i 1 i -i—r r 238y ^ 238 y

5 .9 M e V / a m u . . .

92

8 8 -

84

m • * • • • • • • * • •• V.,«.• • . • / • •••**•. • * • •• i i• .. Vi .Vi^ . v * * . . • • . • • • ; . 1 *•*.>. '.V . *.•

. .?..*/ ”~'/:rv!,;v. •. !. :. ;.o •:>. •• •..•<.• * \ • . V ‘ . T i V V 4 . V * ; • ' ’

(a ) POSITRON + PARTICLE TRIGGERI M l i t"! < 1 1 1 i ' l l 1 I 1 I ! 1 I 1 I

9 6 -

•s »i •;' •Vv92 -

w

’ ./• • \s\ I*. . ?/. ,:m•* • • •

88

84

•.•••. ... .... ..-• - ‘•.*•*•; • • . • >• *#.%••>L 1 AV.in ..* •% . »; f- fc|», •• • 1* . *:

(b). ‘ PARTICLE TRIGGER» i i i .i i i i i i i i i i i i i i j - j i , i i

■60 -40 -20 20 40 6 0A 6 [deg]

Figure 6-29: U+U scatter plots.Same as Figure 6-28, but for the ^ U + ^ U collision system at a bombarding energy of 5.9 MeV/amu.

10

[de

g]

273

t—i i—i—i—i—i—r~i—i—i—i—i—i—i—i—i—i—i—i—r2 3 8 U + 2 0 8 p b5 . 9 M e V /a m u

■* , »» • » * • -

v - i -

8 2 -

- ( a ) - • POSITRON+ 'PARTICLE TRIGGER1 H - l I 1 1 I I I I H . I ' I I. V H -

78 ■ (b ) PARTICLE’ TRIGGER■ • i «• 1 1 » ■ J I I I I L_L J L

- 6 0 -4 0 -2 0 0 20A 8 [deg]

4 0 60

Figure 6-30: U+Pb scatter plots.Same as Figure 6-28, but for the 238U + 208Pb collision system at a bombarding energy of 5.9 MeV/amu.

274

F ' ( E ' )= = ^ d E L ( E ' , E ) P { E ) (6.2)

The response function L { E ' , E ) of the Nal(Tl) counters was reduced to a 156

channel X 156 channel matrix, with both columns and rows covering the energy

range from 0 MeV to 8.44 MeV. Each column of the matrix corresponds to the

detector lineshape as a function of the measured energy E ’ for a given emitted

energy E . For the finite number of energy bins in the matrix, the convolution

integral above becomes matrix multiplication:156

P 'n = E V n Pm («-3)m=l

or simplyP \ E ,) = L { E \ E ) X P { E ) (6.4)

P { E ) is determined by the following iterative procedure of chosing an initial

guess P q(E), folding it with the lineshape matrix L { E ' ,E ) , comparing the result to

the measured distribution P ,(I ,), and forming a new guess by the difference

method:/>,(£) = P0(E] + { P ( E ' ) — [L{ E \ E ) X r„(£)]> (6.5)

The initial guess is simply the measured spectrum itself:P Q(E) = P \ E ' ) (6.6)

The procedure is repeated until the iterations converge, i.e. until P n[E) and

Pn_j(P) for some step were the same:156

i E W - i < ■ <8-7>&=1

The procedure converged for the gamma ray spectra in all cases after 5 to 10

iterations.

1

275

The positron spectra are corrected for the intrinsic lineshape of the Si(Li)

detector by the different method of subtracting the low and high energy tails of

the lineshape (see Figure 4-4) to leave just the peak. Since the two types of tails

are formed independently by different physical processes, they can be subtracted

separately. The positron striking the Si(Li) detector forms the low-energy tail

first, primarily through incomplete charge collection and out-scattering from the

detector. The annihilation radiation exiting the positron detector, however, can

also deposit energy in the Si(Li) detector by Compton scattering, and this

annihilation-radiation energy can pile-up with the positron signal to form a high

energy tail. The two tails are correcting for in turn by working backwards. The

high energy tail subtraction is done for the entire spectrum first, then the low

energy tail subtraction,

The subtraction of the high energy tail takes advantage of the fact that this tail

extends only to energies higher than the energy E deposited by the positron ii self.

The subtraction procedure starts at the lowest energy channel, n = 1 , and

assumes that this channel contains no contributions of high energy tails from

lower channels. Under this assumption that the contents dP[E^)ldE’ of this

channel give the correct size of the peak at this energy, the corresponding high

energy tail components can be calculated for all higher channels and subtracted

from the measured spectrum:F /(E ') = PQ'(dE') - P^E '= £ /) X L (E ' ,£ = £ /) (6.8)

where Pq{E') = P\E') is the measured spectrum. The procedure is repeated

for the next higher channel, now with the assumption that this channel has been

corrected in the last step for the high energy tail. The procedure is repeated for

each higher channel in turn from the lowest energy channel n = 1 up to the high

energy limit n = N of the measured energy distribution:

6.4.1.2. Positron Lineshape Correction

276

Pn'(E ') = P,n-l(dE') ~ Pn'(E - En) X 1 (E '^=^„')for n = 1 to N

The unfplded spectrum is then:P[E) = Pn\E')

(6.9)

(6.10)

Following the subtraction of the high energy tail from the entire spectrum, the

low energy tail is subtracted by the same procedure. In this case, however, the

subtraction proceeds in the opposite direction, from the high-energy end n = N of

the measured spectrum to the low energy limit n = 1 . The unfolded spectrum is

then:P(E) = P1(E>) (6.1 1 )

The completion of both subtraction procedures leaves a spectrum consisting of

only the peak contributions to the positron energy distributions.

6.4.2. E etection Efficiency

Following the lineshape correction, the positron and gamma-ray spectra are

corrected for the measured efficiency «(E') of the detection systems, described in

Sections 4.1.3.2 and 4.2.3 (see Figures 4-8 and 4-10), respectively. The detection

efficiencies are divided out of the measured (and unfolded) positron and the

gamma- 1

P'lE') (6 .12)

Because

spectra.

Section

ay energy spectra P ’(E') channel by channel: />'(£')

P(E) = £(E')the gamma ray spectra have had the tails of the lineshape unfolded, i.e.

shifted ilnto the peaks, while the positron spectra have had the tails subtracted,

the toL.l lineshape detection efficiency is used for the gamma-ray energy

distributions while the peak lineshape detection efficiency is used for the positron

The positron peak detection efficiency was displayed in Figure 4-8 in

4.1.3 above, and the gamma-ray total detection efficiency in Figure 4-10

in Section 4.2. Care is taken to use a consistent definition of the peak and of the

total lin jshape for both the lineshape and the efficiency correction.

277

In Section 4.3 above it was shown that the detection efficiency for the scattered-

particle detectors is flat for the angular region 250 < © ^ < 65 ° . Since both

the positron and gamma-ray spectra are measured in coincidence with the

scattered particles and are normalized to the particle singles rate, it is not

necessary to correct for the scattered-particle detection efficiency within this

angular region.

6.4.3. Scaledown and Deadtime Correction

The positron, gamma-ray, and scattered-particle spectra are corrected for the

effects of the scaledown circuits and the electronic and computer deadtimes. As

described previously in Sections 3.3.1 and 5.4, the scaledowns reduce the

measurement rate of the particle and gamma-ray event types to prevent a large

deadtime due to the ~ 330 psec necessary to transfer each event to the

computer. The particle events are scaled down by a factor of about 2000 and the

forward gamma ray detector events are scaled down by a factor of about 2. The

rear gamma ray detector events and the positron events are not scaled down at

aU.

The scaledown factor and deadtime are measured together by comparing the

number of analysed events for each event type with the number of actual events

enetering the scaledown circuits, as recorded in scalers28. The scaledown

deadtime factor is calculated for each event type and for each collision sys

and the factors are listed in Table 6-4. As a check, the scaledown and dead

factors have also been calculated for each individual magnetic-tape file made

during the experiment. In all cases, the correction factors remain constant di ring

ao ^The deadtime of the coincidence circuits and the scalers themselves was much less than 1%

and completely negligible compared to the 10-20% deadtime due to the data transfer for event.

and

tern,

time

each

278

each run29. In addition, all calculated scaledowns are consistent with the

approximate scaledown factors indicated by the settings of the scaledown circuits.

The spectra produced for each event type are multiplied by the ratio

5 = N act/Nanal of actual events to analysed events for the event type to give

the actual counting rate:P.JE) = x S <6 l3 >

System PositronEvent

Forward Gamma Ray

Event

Rear Gamma Ray

Event

ScatteredParticleEvent

U+Cm 1.193 1.971 1.199 2064U+U 1.274 2.155 1.295 2011

U+Pb 1.056 1.794 1.064 1649U+Sm 1.113 1.829 1.109 1728

Table 6-4: Scaledown Factors.

6.4.4. Normalization to Scattered Particles

The final correction in the production of the positron, gamma-ray, and

scattered-particle spectra is to scale each spectrum with the chosen binsize and to

scale the positron and gamma ray spectra with the scattered-particle singles rate.

The energy distributions are scaled with the energy range per bin and the

positron and the gamma-ray distributions are divided by the number of particles

in the corrected scattered-particle distribution corresponding in each case to the

same scattered-particle angular region. This produced positron energy

distributions with the absolute units of "keV- 1 per scattered particle" and

correspondingly gamma-ray energy distributions labeled "MeV- 1 per scattered

particle. ■

29Small variations in the deadtime are seen, due to changes in the beam intensity which alter

the overall counting rate.

279

In order to obtain the probability distribution for the production of positrons

from atomic processes, the positrons produced in nuclear processes must be

subtracted from the total positron production rate. This background, due to the

internal pair conversion (BPC) of excited nuclear states formed during the collision

of the two nuclei, ranges from ~ 1/4 for the total measured positron rate in the

heaviest collision system measured, U+Cm, to ~ 1/2 in U+Pb, and increases to

essentially 100% of the positron rate in lower-Z systems.

6.5.1. Extrapolation from Low-Z Systems

A direct experimental separation of the positrons from atomic and nuclear

sources is not presently available. Even though the decay of excited nuclear1 9states is delayed by typically ~ 10 sec with respect to the atomic emission

from the quasimolecule that exists during the collision, the nuclei move only

(0.1 c) (10“ 13 sec) 10-4 mm during this time. Instead, the backgrounds of

positrons from nuclear processes (hereafter referred to as nuclear positrons) in the

collision systems U+Cm, U+U, and U+Pb measured for this work are calculated

from the gamma-ray spectra measured simultaneously with the positron

distributions using an effective internal conversion coefficient determined by a

semi-empirical extrapolation from low-/ systems.

The method [Meyerhof et al. 77, Greenberg 77, Kozhuharov et al. 79, Greenberg

80] is based on the observation that nuclear sources of positrons dominate in

low-Z collision systems and on the assumption that the extrapolation to high-/

systems can be described in the following simple form [Kozhuharov et al. 79, Greenberg 80]:

N . = N?°m + N ™ d = P N + C N (6.14)e+ e+ e+ p 7P

P and C are coefficients describing the systematics of positron production from

atomic and nuclear sources, respectively.

6.5 . C alcu la tion o f th e N uclear P ositron B ackground

280

The atomic positron production, P N p, proportional to the number of scattered

particles, b described for Rutherford scattering in Section 2.2.1 above. The

coefficient P describes the atomic positron production probability and depends

most strongly on the combined nuclear charge Zcom = (Zp+Zt) and minimum

separation R • (determined by the scattering angle 9) of the collbion system:Pm~ P(Zcom,e) (6.15)

The nuclear positron production, C N , b proportional to the gamma-ray

production. The internal pair conversion of excited nuclear states b described iu

Section 2.3 above. C b an effective internal conversion coefficient for heavy-ion

collbion systems that has been found experimentally [Kozhuharov et al. 79, Greenberg 80] to be relatively independent of Zcom and the detaib of the

gamma-ray production. The determination of C from low-Z systems assumes that

the general features of nuclear excitation spectra are approximately the same for

all deformed heavy ions, and in particular that the admixture of transition

multipolarities b nearly the same. The extrapolation procedure determines an

effective, experimental admixture of multipolarities, described by the effective

IPC coefficient C.

6.5.2. Determination of the Effective IPC Coefficient

As described previously in Section 3.2.4, gamma ray yields were measured

during the experiment in two 3" X3" Nal(Tl) detectors located outside the target

chamber, as shown in Figure 3-18. Both detectors were mounted at 450 to the

beam direction, with one forward and the other to the rear of the target.

As a check of the various corrections made to the gamma-ray spectra before the

calculation of the background of positrons from nuclear processes, the spectra

from the two counters have been compared [Vincent 81]. A comparison which

allows a simple calculation of the Doppler correction b that with the symmetric

281

rtoo nogcollision system U + U for gamma rays in coincidence with particles

scattering into 45 ° (specifically: 40 ° < 9^ < 50 °) in each counter. At a

and the intrinsic lineshape and relative detection efficiency of the two Nal(Tl)

detectors, as described in Section 6.4. The resulting spectra are compared in

Figure 6-31. They can be seen to agree at the 15% level, verifying the correction

procedures that are applied to the gamma-ray spectra for the nuclear background

calculation.

The effective EPC coefficient C, or equivalently the admixture of nuclear

transition multipolarities, is fixed by a comparison of the positron production in

lower-Z collision systems where positrons from nuclear processes dominate the

positron production with that calculated from the gamma-ray spectra. The

theory behind this calculation is described above in Section 2.4. The actual

calculations for the heavy ion collision systems used here followed

specifically [Schliiter et al. 78] (cf. the tables in [Schliiter and Soff 79]).

Figure 6-32(a), taken from [Greenberg 80], (and see also Figure 1-2 above) shows

the ratio of the measured positron production to the measured gamma-ray

production as a function of the combined nuclear charge Zcom of the collision

system. Part (b), reproduced from [Backe et al. 83], shows similar data plotted as

the ratio of measured positrons to the positron yield calculated from the measured

gamma-ray distribution assuming nuclear transitions of El multipolarity. In both

cases two regions of positron production are evident. Below Zcom 160, the

particle scattering angle of 45 * in our experimental set-up (Figure 3-18), it can be

assumed that half of the gamma rays have no Doppler shift and half have the

maximum. The spectrum from each detector was corrected for Doppler shift E = E '7(1 — p cos 9) |(6.16)

(6.17)

282

Figure 8-31: Comparison of forward and rear gamma-ray detectorsThe measured yield of gamma rays is plotted as a function of the

r tArt AAA

gamma-ray energy for the U + U collision system at a bombarding energy of 5.9 MeV/amu. The gamma rays are in coincidence with both particles scattering into the angular range 40 ° < e ^ < 500. The circles are measured with the forward gamma-ray detector and the squares with the rear detector. Three different angular distributions for the CM emission are shown.

ratio is nearly constant, indicating that the positron production is entirely of

nuclear origin. Above ZCQm zz 160, however, the ratio climbs steeply, signaling

the onset of a second source: atomic positron production.

283

Ne*pN p

0.004

0.003

0.002

0.001

0

Ne*p (ZuA,8>p£ (2ua. 9)-Np + C. N/p

0.44 MeV< Ee*S 0.55 MeV Ey> 1.46 MeV

(a)

— & (0.1 ± 0 .6 )

•10*

U-Lo — li—

il i

(V 13.5*-13.7- OI6.I*-20.2* • 20.6*-26.3* 026.9*- 32J *

Tf U'.^VU*Tp U-Au g-Pb u-u150 160

z.+z,170 180

Figure 6-32: Positron-to-gamma-ray ratio.In part (a) the ratio of the measured positron yield to the measured gamma-ray yield is plotted as a function of the combined nuclear charge Zp + Zt of the projectile and the target nuclei for a uranium beam on various targets. (Reproduced from [Greenberg 80].) In part (b) the ratio of the measured positron yield to the positron yield calculated from the measured gamma-ray yield assuming nuclear transitions of El multipolarity is plotted as a function of the combined nuclear charge Zu. (Reproduced from [Backe et al. 83].)

A systematic study [Ito 81] of collision systems in this region around

^com ~ mac e ear er w^h a plastic scintillator as positron detector (with

much poorer energy resolution than the present Si(Li) detector), had shown that

the positron production from nuclear processes is consistent with a mixture of El

and E2 transitions. Some of the measured positron energy distributions are shown

in Figure 6-33 (but note that the response function of the plastic scintillator has

not been unfolded from the data here). It is evident that for the lower-Z systems

(parts (a) and (b)), the positron production dP^/dE^ falls between that

calculated from the simultaneously measured gamma ray spectra dP /dE assuming all El or all E2 transitions, respectively, using:

284

Figure 6-33: Positrons from nuclear processes.The measured yield of positrons per scattered particle is plotted as a function of the measured positron kinetic energy in the four collision systems 238U + 144Sm (part (a)), + 154Sm (part (b)), + 165Ho(part (c)), and 238U + 181Ta (part (d)), at a bombarding energy of 5.9 MeV/amu. The circles are the measured data. The response function of the positron detection system has not been unfolded. The dashed line is the spectrum of positrons calculated from the measured gamma-ray yield assuming all transitions are El and folded with the positron detection response function. The dot-dashed line shows the same assuming all transitions are E2.

where 0Z M(Ee+,E ) is the internal pair conversion coefficient described in Section

2.4. For the higher-Z systems (parts (c) and (d)), the increase of positrons from

quasimolecular processes begins to emerge. More specifically, the positron

production from nuclear processes seems consistent with all El transitions above a

positron energy of about 800 keV and with a mixture of El and E2 below about

600 keV, making a smooth transition inbetween.

Figure 6-34 shows the positron production probability in the + 165Ho

collision system plotted as a function of the distance of closest approach R min

during the collision (normalized to a head-on collision where R min = 2a) as

calculated kinematically from the particle scattering angles. It demonstates that

the admixture of multipolarities used to calculate the nuclear positrons (solid line)

from the measured gamma rays also reproduces the measured positron R min

distribution (triangles).

Figure 6-35 shows the measured positron production probability in coincidence

with particles scattering into the angular region 25 ° < 9 ^ < 65 ° for the

collision system 238U + 154Sm at a bombarding energy of 5.9 MeV/amu, the

heaviest collision system of deformed nuclei in which positrons from nuclear

processes dominate. The dotted line shows the positron distribution expected

from nuclear processes as calculated from the gamma ray distribution assuming

all El transitions, and the dot-dashed line that assuming all E2 transitions. The

weak Z-dependence is accounted for by using the average of calculations done for

Z = 62 and Z = 92. The Doppler shift and relativistic solid angle transformation

corrections are approximated by averaging the calculated spectra for the forward

and the rear gamma-ray detectors [Vincent 81]. The dashed line markes the

286

Rmin/2a

Figure 6-34: U+Ho P(Rmin) distribution.The probability of positron production per scattered particle is plotted asa function of the distance of closest approach R . in the collisionr mtitsystem + 165Ho at a bombarding energy of 5.8 MeV/amu. The triangles are the measured positron production probability, while the solid curve is the spectrum of positrons calculated from the measured gamma-ray yield according to the prescription described in the text.

287

>o>

'o

Q-Iuj*T3 "O

2 5 °< 0 LAB<651— i— r

238U+ l54Sm5.9 MeV/amu

ii / —L- I Kri

! / / ./ E lx f

E2J__L J L

0 200 4 0 0 6 0 0 800 1000 1200 1400

Ee+ [keV]

10r4

: » 10r5

10.*6

u I II i i i

p (b)

i i i 236

i r i

U+,54Sm 5.9 MeV/amu

r r i "A

100 keV< Ee+<l MeVJ u 1 1 1 J_L I I I n 1—L-L—

1.0 1.2 1.4 R

1.6 1.8 2.0

mm /2 a

Figure 6-35: U+Sm energy and Rmio spectra.The probability of positron production per scattered particle is plotted as a function of the kinetic energy E^ of the emitted positron in part (a) and of the distance of closest approach Rmin in part (b) for the collisionsystem 238U + 154Sm at a bombarding energy of 5.9 MeV/amu. The cirles are the measured positron-production probability, and the curves are positron distributions calculated from the measured gamma-ray yield according to the prescription described in the text.

288

effective admixture, taken to be an energy-dependent factor, f{Ee+), times the

positron distribution calculated from the gamma-ray spectrum assuming all El

transitions. The same factor / (Eg+) is used for all of the following calculations of

the higher-Z collision systems U+Cm, U+U, and U+Pb.

6.5.3. Nuclear Positron Background for High-Z Systems

Because of the weak Z-dependence and the assumption of the same

multipolarity mixture for all cases, the calculation of the positron background

from nuclear processes for the three collision systems U+Cm, U+U, and U+Pb

depends primarily on the energy dependence dP /dE of the measured gamma-

ray yield. A systematic study of the gamma-ray energy distributions for the

measured collision systems and different angular regions showed that the shape of

the gamma-ray energy distribution, an exponential decreasing with the gamma-

ray energy, is essentially the same under all conditions. Figure 6-36 shows the

gamma-ray spectra integrated over the total measured particle scatttering-angle

region for the four collision systems U+Cm, U+U, U+Pb, U+Sm. With the

exception of a small bump in the U+Pb spectrum at 2.6 MeV due to the decay of

the 3", first excited state of 208Pb, all the spectra show the same exponential

shape. Figure 6-37 shows spectra for the U+Pb collision system corresponding to

three different distances of closest approach Rmi during the collision, as

calculated from the scattering angles of the colliding nuclei. The spectra are seen

to have the same shape. Finally, Figure 6-38 shows 25 gamma-ray spectra from

the U+U collision system in coincidence with all combinations of five A 9 and five

E 9 angular regions covering the entire measured elastic-scattering region. Again

the spectra all have essentially the same shape.

Since the shape of the gamma-ray spectra is essentially independent of the

collision system and the particle scattering-angle region, the shape of the positron

background from nuclear processes calculated from the gamma-ray distribution is

289

Figure 6-36: Gamma-ray spectra for various collision systems.The measured yield of gamma rays per scattered particle is plotted as a

1)00function of the gamma-ray energy for the four collision systems " U + 248Cm (circles), 238U + 238U (triangles), 238U + ^ P b (squares), and 238U + 154Sm (diamonds). The bombarding energy was 6.05 MeV/amu for U+Cm and 5.9 MeV/amu for the other three systems. The gamma rays were measured with the rear gamma ray counter in coincidence with particles scattered into the angular region 200 < 9 ^ < 70 ° .

290

LL)_IOoEg?x:co

idoCJ

10,-6

10

10

10'

-6

10<■8

10,-9

, < ti320

a 126 < < U32a

■ U3 < 1£0'••• 2a

VA a a

* aaA

■■ ■

• u - ^ P b59 MeV/u

uo 20 20 CO

y - Energy / MeV

Figure 6-37: Gamma-ray spectra for various Rmin regions.The measured yield of gamma rays per scattered particle is plotted as a

AOO OOQfunction of the gamma-ray energy in the U + Pb collision system at a bombarding energy of 5.9 MeV/amu for three different regions of distance of closest approach, as indicated.

determined from a single calculation and then scaled to fit each positron

distribution taken from the U+Cm, U+U, and U+Pb collision systems. The

calculation is done with the gamma-ray distribution for the entire measured

angular range of the middle collision system measured, U+U. Since this is a

symmetric system, the Doppler shift and the relativistic solid angle correction can

201

CZ3O

UJ

I 2 I 2 I 2 I 2 I 2 3E t [ M e V ]

Figure 6-38: Gamma-ray spectra for various angular regions.The measured yield of gamma rays is plotted as a function of the gamma-ray energy for the 238U + collision system at a bombarding energy of 5.9 MeV/amu for five AS and five E9 regions:

Part (a): —509 < A 9 < —30 ° 84° <270<88°Part (b): —30° <4© <—1 0 ° 8 8 ° <r©<89°Part (c): - 1 0 ° <4S<+10° ▲: 89° <E9<90°Part (d): +10° <4$<+30° 90° <r©<91 °Part (e): +30 ° < A9<.+50 ° 91° <T©<96°

be approximated by calculating the positron background separately for the

gamma-ray spectra obtained with the forward and the rear gamma-ray counters,

and then averaging the two results [Vincent 81]. Figure 6-39 shows the two

gamma-ray spectra, corrected for lineshape, detection efficiency, scaledown, and

~+~1—I—I—£( a ) : : ( b )

i— i— r --F “i— i— r~--E i— i— n

- ( c )

T* 4a ■ " 4»"0 3 l * > ,A ^ V A0 3

o 2

o'0 °

%

J I L

( d ) ' i ( e )• •• :i •

% • . . \ -

\ V . V■ V W

*AV : : 4a ,* ■ A a

\

a - ■

k mm— +l \ % 4

4a ,A ■

% :: t V

J I L

. . ♦♦ jA: r ♦ "z

* \ Aft A\ * 1

VJ I L J I L

292

“ I 1-----1----- 1----- 1----- r238 y 4.2 38 y

5.9 MeV/amu

r 1 "Ti><D

» «

CL! • Forward Detector

id 31—Rear Detector

= 2 O ° < 0 l a b < 7 O °

i d 4i i i J L

•_ * V

J L

I

E y [M ev]

Figure 6-39: U+U gamma-ray energy spectra.The probability of gamma-ray production per scattered particle in coincidence with particles scattering into the laboratory angular region from 20° to 70* is plotted as a function of the kinetic energy of theemitted gamma ray in the ^ U + °°U collision system at a bombarding energy of 5.9 MeV/amu. Neither curve has been corrected for Doppler shift. The circles are the measurement made with forward gamma-ray detector and the squares are the measurement with the rear detector.

238

293

normalized to scattered particles. Figure~6-40 shows the positron background

calculated from these gamma-ray spectra.

Z - 5i

* * 4oo'O

+o>UJT3 2S:*o

, | | r !_T I i I - r - r - r - y | i i4- •

• • •U + U '

5 .9 MeV/amu_ • •

«• —

• i i•

• •

i

—• • • 4 —

_• 2 O ° < 0 l a b < 7 O °i i i i i i i 1 i i t \

1 1 14 0 0 8 0 0 1200

Ee+ [keV]1600 2 0 0 0

Figure 6-40: Positron production from nuclear processes.The differential probability for positron production from nuclear processes is plotted as a function of the positron kinetic energy for the 23®U + 238U collision system at a bombarding energy of 5.9 MeV/amu in coincidence with particles scattered into the angular region 200 < 9 ^ <10°, as calculated from the gamma-ray spectra in Figure 6-39.

The positron background spectra for all other positron distributions obtained

from the U+Cm, U+U, and U+Pb collision systems are determined by scaling the

calculated positron background with the ratio of the measured gamma-ray yield

in coincidence with each positron distribution to the yield used for the calculation:

O e + >

d E e + * f c a l c d E e +

Calculations show that most of the positron background comes from gamma rays

294

with energies between 1.5 and 2.5 MeV. Therefore this energy region in the

gamma-ray distributions is used to scale the positron background.

0.6. Production o f the Final Spectra

Following the description in the first five sections of this chapter of the various

steps in the data analysis, the resulting plots will be presented in this final section.

The presentation is done in two steps, with those spectra representing the gross

features of the data being presented first, followed by the spectra describing the

fine features of the data.

6.6.1. Gross Features of the Data

The gross features of the data measured for the three heavy ion collision systems

U+Cm, U+U, and U+Pb, are exemplified by positron energy distributions in

coincidence with the entire measured range of scattered particles and conversely

by scattered-particle angular distributions in coincidence with the entire range of

measured positron energies. These two types of distributions are shown in Figures

6-41 and 6-42, respectively.

The first figure, Figure 6-41, displays the total positron energy distributions for

U+Cm, U+U, and U+Pb. The measurements are integrated over the angular

range of 25 ° < 6 ^ < 65 ° , as measured in both scattered particle counters. In

addition, all three measurements are limited to the A9 and E9 windows listed in

Table 6-5 (which also correspond to the limits of the total displayed angular

regions in Figure 6-43 in the next section). These last windows confine the

measurement in each case to the elastic (and quasielastic) scattering events.

The data in each part is indicated by the circles. The measured data has been

corrected as described in this chapter for chance coincidences between the

detector systems, the lineshape of the Si(Li) positron detector, the detection

295

0 200 400 600 800 1000 1200 U00

Ee. [ keV ]

Figure 6-41: Total measured positron energy spectra.The measured differential probability (circles) for the total (atomic plus nuclear) positron production in coincidence with particles scattering into the angular region 25 ° < < 65 ° is plotted as a function of thepositron kinetic energy for the three collision systems U + Cm at a bombarding energy of 6.05 MeV/amu, + 238U at 5.9 MeV/amu,and 238U + 208Pb at 5.9 MeV/amu. The dashed curve is thecontribution of positrons from nuclear processes calculated from the measured gamma-ray yield.

296

System *U *L A9 EQ

U+Cm 250 - 650 259 - 659 -60° - +60° 8 6 ° -98°U+U 259 - 659 259 - 659 -609 - +609 849 - 969U+Pb 259 - 659 259 - 659 -609 - +60° 76° - 94°U+Sm 259 - 659 259 - 659 -609 - +609 459 - 929

Table 6-5: Total measured angular regions.

efficiency of the positron detection system, measurement scaledowns and

deadtime, and normalized to the scattered-particle rate. The errors indicated

reflect both the statistical uncertainty in the original measured data as well as the

propagation of these errors through the lineshape unfolding procedure. The

dashed line in each case shows the calculation of the background of positrons from

nuclear processes (which has not been subtracted here).

Figure 6-42 shows the same data plotted as a function of the scattering angle of

the particle detected in the upper particle counter. The data is confined to the

same angular region described above, and in addition is limited for definiteness to

positron kinetic energies in the range 100 keV < E < 1 MeV (which is both

roughly the limits of the expected positron distribution and the bandpass limits of

the spectrometer). Again the data, represented by circles, has been corrected as

described above, and the dashed line gives the calculation of the nuclear positron

background component, which has not been subtracted.

297

l 1 I ' i— i— i— i— i— i— |— 100 keV < < 1 MeV 238U + 2i8Cm

| ♦ ♦ 6.05 M e V /a m u. (a) 4 4

1 -

> 0 03 J*

ts' 2 O

1 -

1-►—I ------ 1— --- 1— --- 1--- I—+ 238y + 238u

(J j) ^ 4 5.9 M eV /am u

♦* ♦

(C )

238 + 208pb

♦ ♦ ^ ♦ 5.9 M eV /am u1 -

N1 I I I I I I ■ I I

20 30 40 50 60 70

©tab 1 deg ]

Figure 6-42: Total measured positron angular spectra.The measured probability (circles) for the total (atomic plus nuclear)production of positrons with kinetic energy 100 keV < Eg+ < 1 MeV is plotted as a function of the particle scattering angle 6Lab in the upper detector for the three collision systems: + 248Cm at a bombardingenergy of 6.05 MeV/amu, at 5.9 MeV/amu, and 238U +208Pb at 5.9 MeV/amu. The dashed curve is the contribution of positrons from nuclear processes calculated from the measured gamma- ray yield.

298

In this section the fine features of the data will be presented. These include

distributions in one or more of the measured parameters for selected regions of

the other parameters. The prime motivation for this was the discovery of

structure in positron energy distributions correlated with particular scattered-

particle angular regions. This is in contrast to the relatively smooth, structureless

distributions presented in the last section. These fine features include positron

distributions for selected regions of the particle scattering angle-angle plane and,

conversely, angle-angle plots for selected positron energy regions.

An extensive study was made of the positron energy distributions as a function

of position in the scattered particle angle-angle correlation plane for the three

systems measured in this experiment: U+Cm, U+U, and U+Pb. In general a

complicated picture emerged of the correlation between structure in the positron

energy distribution and position in the angle-angle plane, a picture complicated

even more by the limits of statistics. The following spectra do not necessarily

exhaust the examples of structure to be found in these systems. They do however

represent the best defined examples and also seem to display a systematic

behavior which has a possible explanation in the spontaneous production of

positrons. Thus rather than attempt to catalog all occurences of structure in

these measurements, a rather meaningless enterprise given the statistical

limitations of the data presented in this thesis , the emphasis will be on the

following examples which are both the most prominent and seem to have

relevence for the topic of this thesis.

Figure 6-43 shows the calculated scattered-particle angular correlations as a

function of A9 and EQ for Rutherford scattering in the ^ U + 248Cm, 238U +

30Additional data acquired in recent experiments appear to corraborate the results quoted here [Cowan 85].

6.6.2. Fine Features o f the Data

260

92

9 6

88

94

,— , 90 o»9i*o86

QDw

92

88

84

80

76-60 -4 0 -20 0 20 40 60

A 9 [deg]

Figure 6-43: Positron analysis angular regions.The position in the scattered-particle angular-correlation plane of the

AAA OAfigates used for the analysis of the collision system U + Cm is shown in part (a), for 238U + 238U in part (b), and for 238U + 208Pb in part (c). E9 is plotted against A9. The dashed lines indicate the calculated angular correlations for Rutherford scattering.

■ 1 1 - ( a )

1

i i i i i i

^ 3 *

23

■ i i i

- (b)1

238u +238 u . — — —

- (c)

i//

/i i

1

•m

238U+208p b

>—

2

\\ \1 1 _

<

2i

///f...........

300

238U, and 238U + 208Pb collision systems in parts (a), (b), and (c), respectively.

Several angular regions are indicated for the three collision systems.

In Figures 6-44 and 6-45 the positron energy spectra for the U+Cm collision

system in coincidence with particles scattering into the angular regions indicated

in Figure 6-43(a) are shown. Figure 6-44(a) corresponds to region 1 in Figure

6-43(a), and Figure 6-44(b) to region 2. A well defined peak is evident in the

positron energy spectra associated with region 1 . It is absent in the spectrum

associated with the nearby region 2 . Indeed, region 1 was chosen to maximize the

presence of this peak.

The second figure, Figure 6-45 shows the energy distribution associated with

Region 3 in Figure 6-43(a). This spectrum also shows a similar peaked structure,

and is associated with an angular region roughly symmetric to Region 1 with

respect to A9 (a symmetry expected from the up-down symmetry of the two

particle detectors with respect to the beam axis: the slight asymmetry with

respect to A0 is discussed below in Section 7.2.1.)

Figures 6-46 and 6-47 show the reverse operation: particle scattering-angle plots

associated with only those positrons in the energy region 280 keV < Eg+ < 360

keV of the peak in Figure 6-44(a). Figure 6-46 is a scatter plot of the angle-angle

correlations of the particles associated with the peak energy region (including

underlying dynamic spectrum) plotted with respect to A9 and L9. Figure 6-47 is

the projection of this two dimensional plot onto the A9 axis, and then divided by

the same plot gated on the entire measured positron energy region 100

keV < E < 1000 keV as a normalization.6T

Figure 6-47, in particular, demonstrates that the positrons of the peak energy

have a different angular behavior than the majority of the positrons produced in

the U+Cm collision system, which are dominated by dynamically produced

301

*0)•

>-co(0o

0 2 0 0 4 00 6 0 0 8 00 1000 1200

Ee+ [ keV ]Figure 6-44: U+Cm positron energy spectra.

The measured yield of positrons is plotted as a function of the positron kinetic energy for the 238U + 248Cm collision system at a bombarding energy of 6.05 MeV/amu. Part (a) shows positrons in coincidence with particles scattered into the angular region 1 in Figure 6-43(a) while part (b) shows positrons in coincidence with region 2 of Figure 6-43(a).

302

TJ0)

• m u m

>Co

</>s.

Ee+ [k«V]Figure 6-45: U+Cm positron energy spectrum.

The measured yield of positrons is plotted as a function of the positron kinetic energy for the U + Cm collision system at a bombarding energy of 6.05 MeV/amu in coincidence with particles scattered into the angular region 3 in Figure 6-43(a).

positrons from Rutherford scattering events. It must be stressed, however, that

6-47 does not represent an angular distribution for the peak positrons, since it has

been normalized to a distribution which also has an angular dependence. Rather

it indicates where the ratio of the peak positrons to the dynamically produced

positrons is most favourable for selecting the first type of event.

Figures 6-48 and 6-49 show positron energy distributions gated on similar

angular regions for the U+U and the U+Pb collision systems, respectively. The

scattered particle angular regions picked out are marked in Figure 6-43(b) for the

303

9 4

9 3

95

9 2<D.“P.

05 91 w

t---- r— i-------r2 3 8 y + 2 4 8 C m .

6.05 MeV/amu 280 < Ee+ •< 360.

it* I\ ' 1

v •I I k* * * I ' • » I MI S* I• ji. '►Ill■ - I Ij I

3. / -

i i • 11 /■ i i . . .yi..

. ii'. -i i -i-x ✓■ • i • \i • 1 1 i • • • * i .i.jjjj.'.lfci'ii*i*1 ■1 ■1" 'it**' • 'i' *■ I •■ in- -i i .iv)v.|tj|,fi?[i|ii i ■ ii

■ ......................... i . i'irrM-'Ji■]• i i I' •■ •■ ■ i ■ i 11 • | i ■ ■ | | i >-i i • | T ■*■ * j,

|79 0

8 9

88

• I ! • 11 ■ i • i iI| . j ... II. t . .

■i i ■J-i• i • 11 ■ ■ ii i ................... ...

i .

J L

- 4 0 -2 0 0

A 6 [deg]

20 4 0

Figure 6-46: U+Cm scatter plot of the peak energy.Scatter plot of the positron events in the positron kinetic energy interval280 keV < Eg+ < 360 keV for the ^ U + x18Cm collision system at a bombarding energy of 6.05 MeV/amu. Each event is plotted with respect to AG and LG.

248/

U+U system and in Figure 6-43(c) for U+Pb, and were chosen to mimic the

regions found for the U+Cm collision system: they exclude the region of

maximum dynamic positron production around Ad = 0 and are rotated in the

angular-correlation plane with respect to the Rutherford-scattering events.

Figure 6-48(a) shows the positron energy distribution for the U+U collision

system associated with both Regions 1 in Figure 6-43(b) while 6-48(b) shows that

304

2 3 8 u + 2 4 8 Cm

6 . 0 5 M eV/am u

*0tOfO1O00C\i♦a>

0 .28

| 0 . 2 4ooT 0.20OO

0 .1 6

0.12

- 6 0 * 4 0 - 2 0 0 2 0 4 0 6 0

A0[deg]Figure 6-47: U+Cm AS spectrum at the peak energy.

The yield of positrons in the energy interval 280 keV < E < 360 keV divided by the yield in the interval 100 keV < Eg+ < 1000 keV is

OOO AiOplotted as a function of AS for the U + Cm collision system at a bombarding energy of 6.05 MeV/amu.

associated with the two Regions 2. Again the first shows some structure while the

second is relatively smooth in comparison.

Figure 6-49(a) shows the positron energy distribution for the final system,

U+Pb, in coincidence with the Regions 1 in Figure 6-43(c) and Figure 6-49(b) for

the Regions 2. In this case (within the constraints of limited statistics), both plots

show a relative lack of structure.

305

POSITRON ENERGY IN KeV

Figure 6-48: U+U positron energy spectra.The measured yield of positrons is plotted as a function of the positron kinetic energy for the 238U + 238U collision system at a bombarding energy of 5.9 MeV/amu. Part (a) shows positrons in coincidence with particles scattered into the angular regions 1 in Figure 6-43(b) while part (b) shows positrons in coincidence with regions 2 of Figure 6-43(b).

306

Figure 6»40: U+Pb positron energy spectra.The measured yield of positrons is plotted as a function of the positron

AAA QftQkinetic energy for the U + Pb collision system at a bombarding energy of 5.9 MeV/amu. Part (a) shows positrons in coincidence with particles scattered into the angular regions 1 in Figure 6-43(c) while part (b) shows positrons in coincidence with regions 2 of Figure 6-43(c).

Chapter 7

Results and Discussion

As shown at the end of the last chapter, the analysed data can be examined

with respect to two aspects, representing the gross and the fine features of the

positron production, respectively. The two features of the data have a different

character and will be seen to offer complementary information. The first section

discusses the gross features of the data and the comparison of theoretical

calculations with them. The gross features of the data, the positron distributions

for each measured parameter integrated over all the other measured parameters,

are characterized by smooth distributions. In contrast, the fine features, spectra

for selected scattering particle angular regions or positron energy regions, show

definite structure. The second section describes the structures found in the data,

discusses and analyses possible sources of the structure, and discusses the

implications of the observed structures to the search for spontaneous positron

production. The third and final section summarizes the conclusions and presents

suggestions for future work in this field.

7.1. Gross Features of Positron Production

The gross features of the positron production refer to the positron distributions

obtained with respect to each of the measured parameters of this experiment

when integrated over all the other measured parameters. Since the kinetic energy

of the positron was measured in coincidence with the scattering angles of both of

the two colliding nuclei, the positron energy distributions integrated over all

307

308

particle scattering angles and the angular distributions integrated over all positron

energies will be presented. These spectra can be expected to reflect the general

features of dynamic positron production in heavy-ion collision systems below the

Coulomb barrier, including the influence of spontaneous positron production. The

theoretical calculations described above in Section 2.2.1 for collision systems

below the Coulomb barrier with no nuclear interactions [de Reus et al. 83] will be

compared to these general spectra.

Integrated spectra were presented at the end of the last chapter. The total,

measured positron production was shown in Figures 6-41 and 6-42. For

comparison, the background positron production from nuclear processes, as

calculated from the measured gamma-ray production, was also displayed. In

order to allow a more direct comparison of the theoretical calculations to the

measured positron distributions, Figures 7-1 and 7-2 display the same data with

the background of positrons from nuclear processes subtracted. What remains is

expected to be the positrons from quasimolecular processes, both dynamic and any

spontaneous positron production.

As described in Section 6.5 above, the determination of the background of

positrons from nuclear processes is based on an extrapolation (the dashed line in

Figure 6-32(b)) from lower-Z systems where nuclear positrons dominate. The

accuracy of the method depends on the validity of the underlying assumption that

the general features of the nuclear reactions occuring in heavy-ion collisions can

be parameterized by the single (positron-energy-dependent) coefficient C in

Equation (6.14). Although an independent verification of this assumption is not

presently available, the constant dependence shown in Figure 6-32 is supported by

the systematic studies of the positron production as a function of Z typified by

Figures 6-32 and 6-33.

The nuclear component is calculated to constitute 26%, 33%, and 48% of the

309

total positron production in the measured U+Cm, U+U, and U+Pb collision

systems, respectively. The uncertainty (both relative and absolute) in the

determination of the nuclear positron background is dominated by the statistical

uncertainty in the measured U+Sm positron distribution (Figure 6-35(a)) used to

fix the absolute size and the positron-energy dependence of the coefficient.

Combining the U+Sm measurement with the results of an earlier, systematic

study of low-Z collision systems, mentioned above in Section 6.5, leads to an

estimate that the subtraction introduces a respective error of about 8%, 10%, and

14% to the U+Cm, U+U, and U+Pb systems, respectively.

Figure 7-1 shows the total integrated positron energy distributions for the three

measured systems: U+Cm at a bombarding energy of 6.05 MeV/amu, U+U at

5.9 MeV/amu, and U+Pb at 5.9 MeV/amu. The circles are the measured

differential probability for positron production as a function of the positron

kinetic energy. The positrons were detected in coincidence with both nuclei, each

scattering into the angular region 25 * < eLaf) < 65 ° . The solid curve in each

case is the theoretical calculation of [Muller 83b].

The theoretical calculations are based on the average beam energy in the middle

of the target, as described in Chapter 5 and given in Table 5-1. For all three

collision systems, the energy loss of the beam in the primary target material

causes a variation in the beam energy of less than ±0.07 MeV/amu about this

average value, which translates [de Reus 83] into an uncertainty in the expected

yield for positrons of about ±7%.

In the case of the U+Cm collision system, however, an additional uncertainty is

introduced by the titanium backing for the curium target. The energy loss of the

uranium beam in this backing is calculated to be 0.27 Mev/amu. The titanium

backing was found to deteriorate noticeably during the course of the

measurement. As a result, the beam energy in parts of the curium material may

310

0 200 400 600 800 1000 1200 U00

Ee. [ keV 1

Figure 7-1: Quasimolecular positron energy spectra.The measured differential probability (circles) for quasimolecular positron production in coincidence with both nuclei scattering into the angular region 259 < 9 ^ < 659 is plotted as a function of the

positron kinetic energy E for the three collision systems: +

248Cm at a bombarding energy of 6.05 MeV/amu, at 5.9MeV/amu, and 238U + 208Pb at 5.9 MeV/amu. The positrons from nuclear processes (cf. Figure 6-41 have been subtracted. The solid curves are the calculation of [Muller 83b].

311

have ranged up to 6.05 MeV/amu. This would be consistent with later

measurements [Cowan 85] described in more detail below, which indicate a

resonance-like behavior for the production of the peak seen in Figure 6-44, above

~ 6 Mev/amu. Even with this uncertainty, the theoretical calculations can be

compared to the measured data at the 30% level.

Figure 7-2 shows the total positron production probability P(9) as a function of

the particle scattering angle 9^ for the same three systems. The angle 9^ab is

the laboratory scattering angle of the particle, either projectile or target nucleus,

detected in the upper particle detector. As pointed out in Chapter 6 , this variable

is used because the projectile and target nuclei cannot be diffentiated in

symmetric (as U+U) or nearly symmetric (as U+Cm) collision systems. The

positron production probability is integrated over the positron energy region 1 0 0

keV < £ e+ < 1000 keV, which is the region of significant positron production

and, for our apparatus, detection efficiency. The dashed line represents the

theoretical calculation of [Muller and de Reus 83].

The system U+Pb is assymetric enough in mass that an identification of the

projectile and target nucleus in each collision is possible from the scattering angles

based on kinematic considerations. This allows a calculation under the

assumption of Rutherford scattering of the distance of closest approach Rmin of

the two nuclei during the collision (using Equation 5-1 above). Since most of the

positron production during the collision occurs at this point of minimal separation,

this is the most physically meaningful variable for studying the angular

dependence. In Figure 7-3 the total positron production probability as a function

of the normalized minimum distance of closest approach R min/2a during the

collision is shown for the kinematically separable system U+Pb. As above, the

production probability is for the positron energy region 1 0 0 keV < E g+ < 1 0 0 0

keV. The squares are the total measured positron production and the circles are

312

20 30 40 50 60 70

©lab t deg ]

Figure 7-2: Quasimolecular positron angular distributions.The measured probability (circles) for the quasimolecular production of positrons with kinetic energy 1 0 0 keV < E < 1 MeV is plotted as a function of the particle scattering angle 9^ in the upper detector for

the three collision systems: 238U + 248Cm at a bombarding energy of 6.05 MeV/amu, 238U + 238U at 5.9 MeV/amu, and 238U + 208Pb at 5.9 MeV/amu. The positrons from nuclear processes (cf. Figure 6-42) have been subtracted. The solid curves are the calculation of (Muller and de Reus 83].

313

the quasimolecular component left after the subtraction of the positrons from

nuclear processes. The dashed curve is the theoretical calculation of [Muller and

de Reus 83].

'>o>

+<uQ_

R m i n ^ 0

Figure 7-3: U+Pb P(RmJ .The probability for the production of positrons with kinetic energy between 1 0 0 keV and 1 MeV is plotted as a function of the normalized

QOQdistance of closest approach R mij 2 a for the collision system U +

208Pb at a bombarding energy of 5.9 MeV/amu. The squares are the measured total (quasimolecular plus nuclear) positron production and the circles are the quasimolecular positron production. The dot-dashed line is the calculation of [Muller and de Reus 83].

314

As in Chapter 6 , the errors shown in Figures 7-1 to 7-3 correspond to the

relative uncertainty of the points. This is dominated by the statistical uncertainty

in the original data and its propagation by the lineshape unfolding correction. In

addition, there is estimated to be a systematic 2 0 % error in the absolute yields,

due primarily to uncertainties in the calibration of the positron detection

efficiency. This has been determined by a consideration of the consistency among

the calibrations done with electron sources, with positron sources, and the Monte

Carlo calculations of the positron detection efficiency, as described in Section

4.1.3.

The most striking aspect of Figures 7-1 to 7-3 is perhaps the overall agreement

of the theoretical calculations with all of the experimental results, both as a

function of the positron kinetic energy and of the angular variables, and of the

combined nuclear charge. This is strong evidence for the basic validity of the

theoretical model of positron production in heavy-ion collisions, and particularly

for the adiabatic nuclear-quasimolecule picture which underlies these calculations.

It should be noted that this agreement has been achieved without any overt

adjustment of parameters from either the experimental or the theoretical side.

The experimental curves are obtained as absolute positron yields from

measurements of the collision systems and calibrations of the detectors, as

described in Chapters 4 and 5. The theoretical calculations, as described in

Chapter 2, also contain no free parameters (except for the Fermi level31). No

parameters have been fitted to the final data. This should serve to emphasize the

overall understanding which has been achieved of the gross features of positron

production in heavy-ion collisions.

31The Fermi level is not known exactly for heavy-ion collisions, but according to [Reinhardt et

al. 79, Soft et al. 80, Reinhardt et a l. 81a], reasonable choices (above the Isa level) causes only small (<10% ) changes.

315

Although spontaneous positron production has been assumed in these

calculations, it would constitute only a small part of the total positron production.

Since, as pointed out in Section 2 .2 .1 , any spontaneous positron component adds

coherently to the dynamic positron production, and since Rutherford collision

times are short compared to the natural decay time of an overcritically bound,

vacant electronic state, the spontaneous contribution is expected to be difficult to

isolate in the gross features of the data at the present level. As a result, these

spectra can provide no unambiguous signature of spontaneous positron production

beyond a general sense of confidence in the acquired understanding of the positron

production processes in heavy ion-atom collisions.

7.2 . F in e F eatu res o f P ositron P rod u ction

A finer study of the data is possible because the positron energy was measured

in coincidence with the scattering angle of both colliding nuclei in this experiment.

The detailed relationships between these three parameters gives a closer look at

the positron production process. The fine features of the data, then, refer to

positron energy spectra for selected scattering particle angular regions and,

conversely, angular plots gated on positron energy regions.

The fine features of the data reveal a more complicated story. In contrast to

the smooth spectra shown in the last section, describing the gross features of

positron production, definite structure have been found in these spectra, a

structure which hints at new physics. A striking relationship has been found

between the positron energy distribution and the scattering-angle plane for the

collision systems U+Cm and U+U.

The following sections will concentrate on the most prominent structure found,

that in the U+Cm collision system. In addition, data which has been obtained

since the experiments described in this thesis were conducted, but which are

316

directly relevant to the topics discussed here, will also be introduced. This later

data, and the calculations involving them, will be described in greater detail

in [Cowan 85]. They are properly a part of this later thesis, and only the results

will be quoted here in order to provide a more complete description of the

structure first found in this present work.

7.2.1. Description o f the Structure

The most prominent structure was found in the heaviest system measured,

U+Cm. As shown in Figure 6-44(a), a well-defined peak centered around an

energy of 316 ± 1 0 keV and with a width of ~ 70 keV, appears in the positron

energy distribution measured in coincidence with a selected region of the

scattered-particle angular-correlation plane, designated by Region 1 in Figure

6 -4 3 (a). This region was chosen empirically, expressly to maximize the peak

structure.

Because of the symmetry about A9 = 0 expected for scattering, explained above

in Section 3.2.3, a search was made for similar structure in the region AS > 0.

The best example found was shown in Figure 6-45 above, corresponding to

coincidence with Region 3 in Figure 6-43(a). The implications of the near but not

perfect symmetry will be discussed below.

That the peak is not of some trivial instrumental origin is demonstrated by its

absence in a nearby angular region. Figure 6-44(a) above shows the positron

energy distribution in coincidence with Region 2 in Figure 6-43(a).

The peak is superimposed on the smooth distribution expected for dynamic

positron production. This is demonstrated in Figure 7-4, which repeats the

measured data shown in Figure 6-44 above. The underlying smooth distribution

for part (a) has been calculated by [Cowan 85] as a fit to the data in the nearby

angular region of part (b) as follows. Using the prescription described in Section

317

0 200 400 600 800 1000 1200

Ee+ [ keV ]

Figure 7-4: U+Cm positron energy peak.Same as Figure 6-44. The additional dashed lines are the sum of positrons from nuclear and from dynamic production processes, based on a fitting procedure [Cowan 85] described in the text.

318

6 .5 , the background distribution of positrons from nuclear processes for part (b)

was calculated from the gamma-ray distribution measured simultaneously in

coincidence with the same scattering-angle region as gated the positron

distribution of part (b). The sum of this nuclear positron background dP nucl/d E

and the calculated dynamic positron distribution dP iynjd E of [Muller 83b] was fit

by a least-squares method to the measured data dP meaa/d E in part (b):jpmeas ^pnucl d p I/n

7 ( 7 1 )Only the intensity of the dynamic component (the constant C) was varied. This

produced the dashed curve shown in part (b). The intensity of the dynamic

component determined by the fit agreed with that predicted theoretically for the

scattering-angle region selected, within the experimental uncertainty in the

determination of the admixture of near and far collisions possible at those angles.

This curve was scaled with the number of scattered particles to produce the

dashed curve in part (a).

The measurement of the U+Cm collision system was repeated and, as shown in

Figure 7-5 (taken from this later measurement [Cowan 85]) the structure is

reproducible.

. Among the interesting features associated with this peak structure is its narrow

width of about 70 keV (FWHM). This is consistent with the minimum Doppler

broadening possible for the measured collision system, as described above in

Section 3.2.2, and implies that the actual width must be much smaller.

In addition, the peak appears for only a narrow range of projectile energies

corresponding to collisions near the Coulomb barrier. This was seen in a later

measurement of the beam energy excitation function [Cowan 85] and is shown in

Figure 7-6. The ratio of positrons per scattered particle (using an arbitrary scale)

is plotted as a function of the projectile energy for two positron kinetic energy

319

73%> -

Co

oQ_

■e+ [ keV ]

Figure 7-5: U+Cm positron energy peak.U+Cm positron energy peak. The measured yield of positrons is plotted as a function of the kinetic energy of the emitted positrons in coincidence with a selected particle-scattering-angle region [Cowan 85].

regions. The circles correspond to the energy interval 280 keV < E e+ < 360

keV around the peak in Figure 7-5. The dot-dashed line shows the trend of a

similar interval, 400 keV < E < 480 keV, located just above the peak in

energy. Since the width of the bump evident in the distribution of peak events is

comparable to the energy loss of the beam in the target, the actual range of beam

energies leading to an enhancement of the peak with respect to the dynamic

positron production could be narrower.

It is the data in this plot, along with the observed deterioration of the titanium

backing for the curium target, mentioned above, which argue that the uranium

320

9

0) 80

« 7>tw1 6

10

oCL

Z

z

1 a 8 U + 248Cm4 280 keV< Ee+ < 360 keV

400keV< Ee* < 480 keV

. - P y - f Z -r -

5.9 6 .0 6.1

Eproj. [MeV/amu J6.2 6.3

Figure 7-6: U+Cm beam excitation function.The ratio of positrons to scattered particles for the U+Cm collision system is plotted as a function of the bombarding energy (from [Cowan85]). The circles represent positrons with kinetic energy in the interval280 keV < E g+ < 360 keV and the dot-dashed line those with energy400 keV < E g+ < 480 keV.

beam energy must have been nearly the full beam energy of 6.05 MeV/amu in at

least part of the curium target material for the data of Figure 7-4 above.

The positrons associated with the peak structure also appear to be emitted from

a system moving with the velocity of the center of mass of the two colliding nuclei

rather than the velocity of either of the two separate nuclei. This was

determined [Cowan 85] from a measurement of the angular dependence of the

Doppler-broadened width of the peak, and is shown in Figure 7-7. The calculated

321

a>

♦a>LU^3

■QD

a

25 30 35 40

*iab I d e g ]45 50

Figure 7-7: U+Cm positron emitter velocity.The width of the positron distribution due to Doppler broadening is plotted as a function of the laboratory particle-scattering-angle for the U+Cm collision system at a bombarding energy of 6.05 MeV/amu. Calculations [Cowan 85] are shown for emission of the positron from the slower of the two nuclei (dotted line) scattering to a given angle, from the faster nucleus (dot-dashed line), or from the center of mass of the two nuclei. The circle is the measurement of this thesis. The square is the later measurement of [Cowan 85].

322

width due to Doppler broadening of the positron peak, described in Section 3.2.2,

is plotted as a function of the laboratory scattering angle for emission of the

positron from either the slower (dotted line) or the faster (dot-dashed line)

scattered nucleus at each angle. The constant width, independent of scattering

angle, for emission from the center of mass of the two nuclei is indicated by the

dashed line. The circle is the measured width of the peak in the U+Cm collision

system from the experiments done for this thesis. The square is a later

measurement of the U+Cm system [Cowan 85].

Finally, the scattering events producing the peak seem to be associated with a

different particle-scattering kinematics than that of the Rutherford scattering

which dominates in these collisions just below the Coulomb barrier. This can be

seen by a careful comparison of the scatter plot of positron events as a function of

the particle scattering angles, shown in Figure 6-46 with that in Figure &-28(a).

The first plot shows the events in coincidence with the positrons having the

energy of the peak; the second is gated on all positron energies. The events

corresponding to the peak appear to be rotated slightly in the angular-correlation

plane with respect to the distribution for all the positrons.

The difference between the scattering behavior associated with the two positron

energy regions is made more evident in Figure 7-8. The centroid in the E9 direction of the scattering events is plotted as a function of A9 [Cowan 85]. The

dashed line is in coincidence with positrons in the energy region 400

keV < E e+ < 800 keV above the peak, which contains mostly dynamically

produced positrons from the Rutherford-scattering events that dominate. The

circles represent coincidence with the positrons in the region 280 keV

< E ^ < 360 keV around the peak, and show a somewhat different angular

correlation.

Scattering kinematics alone could not produce a rotation in the angular-

323

1— I— T

91 .8

91 .4

O '<D

, " 0 , 9 1 .0

ct>W

9 0 .6

9 0 . 2

T"" | 1 | 1 1 I ""I I | 238u + 248C m

6 . 0 5 M e V /am u

_ 4 2 8 0 keV < E e + < 3 6 0 keV

4 0 0 k « V < E e* < 8 0 0 keV

« I I- 4 0 -20

1 i i i 1 l I I0

A 6 [deg]20 4 0

Figure 7-8: U+Cm angular-correlation diagram.

The centroid of the positron distribution parallel to the E 9 axis is

plotted as a function of A 9 for the collision system U+Cm at a

bombarding energy of 6.05 MeV/amu [Cowan 85], The circles

correspond to coincidence to positrons with kinetic energy in the interval

280 keV < E g+ < 360 keV and the dashed line to positrons with

energy 400 keV < E g+ < 800 keV.

correlation plane. As pointed out in Section 3.2.3, the up-down symmetry of the

two particle counters with respect to the beam axis implies a symmetry in the

variable A 9 about A 9 = 0 for any scattering event, even those involving the

transfer or loss of mass or energy. The scattered ions pass through the magnetic

field of the solenoidal positron transport system, however, before striking the

particle detectors. The magnetic field breaks the up-down symmetry of the

scattering chamber, and allows asymmetrical scattering correlations.

324

The correlation between £& and 4 0 for the scattering events gated on the upper

positron energy, above the peak, can be explained in terms of the two scatttering

ions having acquired the different, velocity-dependent, equilibrium charge states

of heavy ions moving through foils, as described above in Section 3.2.3. This

corresponds to the solid line in Figure 3-17. On the other hand,

calculations [Gruppe 84] have indicated that the observed angular correlations of

the scattering events gated on the positron peak energy is more consistent with

ions having acquired equal mass and charge states. The results of some of these

calculations are also displayed in Figure 3-17.

The calculations show that a clockwise rotation of the angular-correlation curve

and perhaps a slight shift toward negative 4 0 are possible. These shifts with

respect to the dominate Rutherford scattering events help to explain the position

of the windows in Figure 6-43 chosen to enhance the peak structure, and in

particular the apparent asymmetry between windows 1 and 3. These windows

can be seen to be those which maximize the ratio of the peak-producing events to

the underlying distribution of positrons from nuclear and dynamic atomic

processes. The angular correlations for these scattering events seems to be

indicative of the formation of the peak in some type of special events which differ

from the dominant Rutherford-scattering events. In addition, the fact that the

peak-producing events appear to fall farther away from the Rutherford-scattering

events on the negative-40 side could explain the apparently greater ratio of the

peak to the smooth dynamic distribution for Figure 6-44(a) in coincidence with

Region 1 in Figure 6-43 in comparison to Figure 6-45 in coincidence with Region

3.

325

An obvious source of positrons in heavy-ion collisions is the internal pair

conversion of excited nuclear states formed during the collision. The existence of

a single or a few closely grouped, strongly excited nuclear state could (in contrast

to the smooth nuclear positron distributions calculated in Section 6.5 above from

the structureless, exponential gamma-ray distributions) lead to the appearance of

structure in positron energy distributions. The possibility of a nuclear origin for

the observed structure will be explored in this section. The details of the

calculations presented can be found in [Cowan 85]; the important results are

described here because of their relevance to the question of the origin of the

observed structure.

Figure 7-9 presents schematically (based on [Schliiter et al. 83b]) the two

possibilities to be considered. Part (a) shows the characteristic triangular shape of

the positron energy distribution resulting from the normal internal pair conversion

(IPC) process in heavy atoms. The differential IPC coefficient df}/dEe , described

in Section 2.3 above, is plotted as a function of the kinetic energy of the emitted

positron for an E l nuclear transition in uranium of energy u = 1.34 MeV (chosen

so that the maximum of the positron energy distribution matches the observed

peak energy). Part (b) shows the much rarer process of monoenergetic internal

pair conversion (MIPC). In this case, the electron is captured into a vacant bound

state, leaving the positron'in a final state of well-defined energy, indicated by the

S -function at 316 keV in part (b). Capture into the K-shell dominates, since by

virtue of being the smallest atomic shell it couples most strongly to the nucleus.

The second process clearly depends on the availabiliy of an atomic vacancy at

the time of the nuclear transition to capture the electron. Indeed, this

requirement leads to a very large suppression of MIPC, due to the mismatch of

the atomic and nuclear time scales. Since the time for filling a vacancy in the K-

7.2.2. Possible Sources of Structure

326

0 200 400 600Ee+ [keV]

Figure 7-9: Nuclear sources of positrons.The differential pair conversion coefficient is plotted as a function of the kinetic energy of the emitted positron for an E l transition in uranium assuming IPC in part (a) and MIPC in part (b) (based on [Schliiter et al. 83b]).

shell of a heavy ion is of order 1 0 - 1 7 sec [Anholt and Rasmussen 74] while nuclear1 9 ________

transition times are typically 1 0 sec, the MIPC process is suppressed by four

orders of magnitude with respect to the IPC process. Unless a mechanism exists

to prolong the lifetime of a K-vacancy in a heavy ion by several orders of

magnitude, the MIPC process should be completely negligible. The energy

distribution for the competing IPC process is also indicated in Figure 7-9(b). It

327

extends up to a maximum energy which is lower than the monoenergetic positron

kinetic energy by an amount equal to the binding energy of the vacant state.

Several experimental observations, however, are inconsistent with an

explanation for the positron peak found in the U+Cm collision system based on

the IPC of excited nuclear states. To begin with, normal EPC produces a positron

lineshape which is too wide for the observed structure. This can already be seen

in Figure 7-9(a) above, and is demonstrated more quantitatively in Figure 7-10.

This figure shows the data points of Figure 7-4(a) for the U+Cm positron peak.

In the same manner as was described in the last section for the calculation of the

dashed curve in Figure 7-4(a), a background curve was determined. A least-

squares fit by [Cowan 85] of the sum of positrons from nuclear and dynamic

processes, folded with the response function of the detection system, was made to

the spectrum, excluding the region of the peak from 280 keV to 360 keV. A peak

lineshape, calculated for an assumed EO IPC transition in the uranium nucleus

and folded with the Doppler broadening integrated over the measured scattering-

angle region and with the response function of the positron-detection system, was

added to this smooth distribution. The least-squares fit within the peak region,

made by varying only the nuclear transition energy and the intensity of the peak,

is shown as the dashed line. The energy of the nuclear transition determined by

the fit is 1392 ± 20 keV and the reduced-x2 is 2.41.

Because of the intrinsically large width of the IPC lineshape, a better match to

the data is not possible. Similar results are obtained for emission from the curium

nucleus and for all other multipolarities since the shape of the positron

distribution depends only weakly on the multipolarity of the nuclear transition or

the charge Z of the nucleus.

Furthermore, the emission of positrons in an IPC or MIPC process from either

one of the nuclei after the collision implies a particle scattering-angle dependence

328

Ee+ [kev]

Figure 7-10: IPC fit to the U+Cm peak.The positron yield is plotted as a function of the kinetic energy of the emitted positron. The circles repeat the data of Figure 7-4. The dashed line represents positrons from nuclear processes, the dot-dashed line positrons from dynamic processes, and the dotted line positrons due to the assumed IPC of a nuclear EO transition in uranium. The curves have been fit to the data [Cowan 85], as described in the text.

for the Doppler-broadened width of the peak structure. As shown already in

Figure 7-7 above, however, the measured width of the peak is nearly constant as a

function of angle and consistent in size with emission from the center of mass ofithe two nuclei rather than from either of the separate nuclei.

Two additional pieces of information about nuclear processes are provided by

329

the gamma-ray and (from a later measurement [Cowan 85]) electron spectra

measured simultaneously with the positron spectrum. The connection between

these three spectra is given by the IPC and MIPC coefficients described in Section

2.3, which relate the probability of positron emission by pair conversion processes

to the gamma-ray and electron production (cf. Equations (2.92) to (2.95)). The

typical size of these coefficients, as shown in Figures 2-47 to 2-49, requires on theO A

order of 1 0 — 1 0 as many gamma rays or electrons as positrons if the positrons

are being produced by nuclear pair conversion processes. The existence of a peak

in the positron energy distribution thus implies a corresponding structure in either

the electron spectrum (for an E0 transition) or the gamma-ray spectrum (for all

other transitions). The size of the associated gamma-ray or electron structure can

be calculated from the measured intensity P^ of the positron peak and the

appropriate coefficient.

For an E0 transition, the number of associated electrons is given by [Schluter et

at. 83b]:

{ x 1(1 - - ) - P . for IPC

2 J « (7.2)

( j - l ) 7 Ps+ for MIPC

For any other multipolarity, the number of associated gamma rays is given by:n- P . for IPC

0 e+ (7.3)2 1 V ' P . for MIPC

- x a e+K *+x is the number of vacancies in the K-shell (so 0 < x < 2) and the other

coefficients were defined above in Section 2.3.

Figure 7-11 shows the measured yield of gamma rays (histogram) as a function

of the gamma-ray energy E ' for the U+Cm collision system at a bombarding

energy of 6.05 MeV/amu. (The spectrum has not been corrected for the response

function of the gamma-ray detection system.) The spectrum was measured at the

p i = l

330

5000 1-----1----- 1-----1-----r -2 3 8 u + 2 4 8 Cm

/ \ K 6 .05 MeV/amu• /i »

(a ) :

1000

5 0 0 .// \ l Unuc

i > V V — -

\ I PC Transition in = nucleus —

UJ>

<Ea:i<

<o

100

5 0

1000

5 0 0

100

5 0assumed: >

I k-vacancyT k>>,Ttrans. a

1200 1400 1600 1800 2 0 0 0

E y TkeV l

Figure 7-11: Pair conversion calculated for gamma rays.The gamma-ray yield is plotted as a function of the gamma-ray energy E ' for the U+Cm collision system at a bombarding energy of 6.05 MeV/amu gated on Region 1 of Figure 6-43(a). The gamma-ray structure (calculated by [Cowan 85]) is shown which would accompany the positron peak in Figure 7-4(a) in the case of an E l (dot-dashed line) or an E 2 (dashed line) nuclear transition in uranium, assuming the positron peak is caused by IPC in part (a) or MIPC in part (b).

331

same time and gated on the same particle scattering angle region as the positron

distribution in Figure 7-4(a), where the peak appears.

The dot-dashed curve is a calculation by [Cowan 85] of the gamma-ray line

which must appear in this spectrum under the assumption that the positron peak

in Figure 7-4(a) is due to the internal pair conversion of an E l transition in the*%ty

uranium nucleus . The calculation includes the effects of Doppler shift and the

resolution and efficiency of the gamma-ray detector. The double-humped form

results from the different Doppler shifts with respect to the gamma-ray detector

of the forward- and backward-scattered uranium nuclei detected in the particle

scattering-angle region considered. The dashed line shows the same calculation

for an E2 transition. Higher multipolarity transitions would require

correspondingly more gamma rays because the EPC coefficients decrease with

multipolarity (see Figure 2-49). Large enough structure appears to be absent.

In part (b) the results of a similar calculation for the case of MIPC are

displayed. Again curves are presented for E l and E2 transitions. Since vacancies

are required for MIPC, the calculations were done under the assumption that one

K-shell vacancy is still present at the time of the nuclear transition, despite the

fact that the considerations of the lifetime of a K-shell vacancy, mentioned above,

indicate that ~ 10- 4 would be a more reasonable number. Even so, the required

structures are too large. The absence of large enough structure in the measured

gamma-ray distributions virtually rules out a nuclear origin for the peak involving

nuclear transitions of multipolarity E l or higher.

In order to check the only remaining possibility, an EO transition, a later

measurement of the U+Cm collision system included a simultaneous measurement

4AAn essentially identical plot results for a transition in the curium nucleus because of the weak

^-dependence of the IPC coefficients: cf. Figure 2-49

332

of the emitted electrons [Cowan 85]. In Figure 7-12, plotted as a function of the

electron kinetic energy E g_, is the yield of electrons gated on the same particle-

scattering-angle region which produced the positron distribution with the peak in

Figure 7-5. (The electron spectrum has not been corrected for the response

function of the electron detection system.) Calculations [Cowan 85] similar to

those described above for the gamma-ray spectra yielded the two curves. The

dashed line shows the size of the structure required assuming the IPC of an EO

transition in the uranium nucleus, and the dot-dashed line assuming MIPC. In

both cases it has been assumed that a vacancy still exists in the uranium K-shell

at the time of the nuclear transition, even though, as already mentioned, the EO

transition time is about four orders of magnitude longer than the lifetime of a K-

shell vacancy. Again, the measured spectrum does not contain large enough

structure to account for the size of the observed positron peak.

In general, nuclear processes do not seem to provide an explanation for the peak

structure found in the U+Cm system. The next section will discuss another

possible source for the peak: spontaneous positron production.

7.2.3. Spontaneous Positron Production

Because of the apparent failure of nuclear pair conversion processes to explain

the observed peak structure in the U+Cm collision system, it is necessary to

consider other possible sources. With the possible exception of spontaneous

positron production, however, it is difficult to imagine other sources of nearly

monoenergetic positrons from heavy-ion collisions with the experimental

properties mentioned above in Section 7.2.1. Moreover, for the data presented in

this thesis, as will be shown in this section, the present experimental observations

on the U+Cm collision system could be consistent with a scenario based on

spontaneous positron production.

333

co(0

LU>4-o*o

Ee. tkeV]

Figure 7-12: Pair conversion calculated for electrons.The electron yield is plotted as a function of the electron kinetic energy for the U+Cm collision system at a bombarding energy of 6 .1

MeV/amu [Cowan 85]. The electron structure associated with the positron peak in Figure 7-5 is shown for IPC (dashed line) or MIPC (dot- dashed line) assuming the positron peak is caused by an EO nuclear transition in uranium.

To begin with, the assumption of spontaneous positron production produces an

excellent fit to the the lineshape of the measured peak, as shown in Figure 7-13.

The fit was generated in the same manner as that shown in Figure 7-10

above [Cowan 85]. In Figure 7-13, however, the source of the peak was assumed

to be monoenergetic positron emission from the center of mass of the two colliding

nuclei, i.e. from the U+Cm quasimolecule. The position and intensity of the peak

component were varied for the fit, and the reduced-x2 obtained is 0.81. The CM

energy of positron emission was determined to be 316 ± 1 0 keV. This value is

consistent with calculations [Reinhardt et al. 81b] (and Figure 2-23(a) above) of

334

the positron resonance energy in the U+Cm quasimolecule frozen at the distance

of closest approach achieved in these collisions.

2 0 0 4 0 0 6 0 0 8 0 0 1000 1200

Ee+ [kev]

Figure 7-13: Monoenergetic CM fit to the U+Cm peak.The positron yield is plotted as a function of the kinetic energy E of the emitted positron. The circles repeat the data of Figure 7-4. The solid line represents positrons from nuclear processes, the dot-dashed line positrons from dynamic processes, and the dashed line positrons due to assumed monoenergetic emission in the center of mass of the U+Cm system. The curves have been fit to the data [Cowan 85), as described in the text.

The good fit to the measured lineshape of the peak with an assumption of

335

monoenergetic CM emission underscores the statement made above the the

observed width of the peak is almost entirely due to Doppler broadening alone.

Replacing the 5-function of monoenergetic emission with a Gaussian CM energy

distribution determined the CM width (FWHM) of the positron emission to be

<40 keV at the 95% confidence level [Cowan 85].

As has already been noted above and exhibited in Figure 7-7, the the Doppler-

broadened width is nearly constant with respect to particle scattering angle, and

consistent in size with emission from the quasimolecule moving with the CM

velocity, as opposed to the velocity of either of the two separate nuclei.

The narrow width indicated by the fit implies an unexpectedly long-lived source

for the peak positrons. By the Uncertainty Principle, a width of 40 keV implies a

source living at least 2 X 10"” 20 sec. This, however, is a factor of ten longer than

the overcritical binding time for the 1 so state during normal Rutherford

scattering in the U+Cm collision system at a bombarding energy of 6.05

MeV/amu, as shown in Figure 5-l(a) above.

The experimental observations describe the apparent emission of the positrons of

the peak from a system moving with the velocity of the U+Cm center of mass,

living at least 1 0 times longer than the overcritical binding time provided by

Rutherford scattering, and with a positron kinetic energy matching that of

spontaneous positron emission from a U+Cm quasimolecule frozen at the

m in im um separation achieved during the collisions. Together these observations

suggest the possibility that the peak could be produced by spontaneous positron

emission enhanced by the formation of some sort of metastable nuclear complex

which causes the nuclei to stick together briefly at the turning point of the

collision.

It is important to note that this hypothesis would be consistent with the

336

appearence of the peak for only a narrow range of projectile energies near the

Coulomb barrier, as seen in the beam energy excitation function of the peak

shown above in Figure 7-6. It could also account for the non-Rutherford

scattering kinematics associated with the peak shown in the scattering-angle

correlation diagram of Figure 7-8. The formation of a long-lived quasimolecule

might provide the mechanism for equalizing the mass and ionization of the

colliding ions suggested by the analysis of Figure 7-8 described in Section 7.2.1.

The schematic, quasiclassical model described at the end of Section 2 .2 .2 which

incorporates a single nuclear sticking time into Rutherford scattering has been

employed by [Muller et al. 83] in a coupled-channel calculation to determine thatQft § O

a nuclear sticking time of 6.5X10 sec in 1.8X10 of the collisions could

account for the observed width and intensity, respectively, of the peak in the

U+Cm collision system shown in Figure 7-4(a). The small fraction of collisions

required to produce the peak has two implications. As a rare event, the possible

formation of metastable nuclear complexes at the Coulomb barrier could have

escaped detection until now. Furthermore, the detection of positrons could serve

as a sensitive probe for this rare exit channel of heavy-ion collisions.

In addition, this schematic model also provides a possible though highly

speculative explanation for the structure observed in the U+U system in Figure

6-48(a). If a similar quasimolecule were formed in the U+U collision system at the

distance of closest approach in these collions, the primary energy of spontaneous

positron emmision would be expected to come at about 190 keV with an intensity

approximately 1/4 of that of the U+Cm peak [Reinhardt et al. 81b] (and Figure

2-23 above). Due to the reduced positron detection efficiency of the apparatus at

this lower energy, a peak this small would not be readily observable. The three

structures visable at ~ 400 keV, ~ 550 keV, and ~ 700 keV could, however,

represent the interaction of spontaneous positron production with the postulated

337

nuclear interaction that enhances it. The metastable nuclear complex could be

expected to have internal degrees of freedom in the form of rotational the

vibrational states, and energies on the order of several hundred keV are not

unreasonable. The observed structures might represent the sum of the

spontaneous positron energy and the excitation energy of some state of the

nuclear complex which has been carried off by a spontaneously emitted positron.

Of course, similar structure is also possible in the U+Cm system in addition to the

large peak, but could be lost in statistical fluctuations. It should be stressed

again, however, that this is entirely speculation.

As a final note, additional measurements of the 238U + 238U collision system

made after the experiment reported in this thesis [Kido 83] and the results

of [Clemente et al. 84] indicate that peak structures are also present in this

system, although apparently at a somewhat lower bombarding energy than

studied here. In particular, Figure 7-14 shows the positron energy distribution for

the U+U system at a bombarding energy of 5.8 MeV/amu, gated on a scattered-

particle angular region similar to those described above [Kido 83].

7.3 . Sum m ary and O utlook

The purpose of this experiment was a search for the spontaneous production of

positrons due to the overcritical binding of the l a ^ electronic state in the

quasimolecular electromagnetic fields transiently formed during collisions of heavy

ions and atoms at the Coulomb barrier. The experiment yielded two major

results. The gross features of the positron energy spectra and the coincident

scattered-particle angular distributions produced in heavy-ion collisions below the

Coulomb barrier are described by smooth distributions which can be reproduced

by theoretical calculations of the dynamic (including spontaneous) positron

production during Rutherford scattering. This implies the overall validity of the

theoretical understanding of positron production processes in heavy-ion collisions

338

Ee* [ keV 3

Figure 7-14: U+U positron energy spectrum.The measured yield of positrons is plotted as a function of the positron kinetic energy E g+ for the 238U + 238U collision system at a bombarding energy of 5.8 MeV/amu. The positrons are in coincidence with a selected particle-scattering-angle region [Kido 83].

below the Coulomb barrier. In addition, structures have been found in the

overcritically bound U+Cm and U+U collision systems in the positron energy

distributions correlated with selected particle scattering angles. In particular, a

well-defined peak in the U+Cm system has been observed.

An analysis of the lineshape of the peak in the U+Cm positron energy

distribution, of the angular dependence of the Doppler-broadened width of the

peak, and of the gamma-ray and electron spectra measured simultaneously under

the same kinematic conditions as the positron spectrum appears to rule out the

339

internal pair conversion of excited nuclear states formed during the collision as a

possible source of the peak structure. On the other hand, the energy, the

lineshape, and the size and angular dependence of the Doppler-broadened width of

the peak are found to be consistent with theoretical predictions for the

spontaneous emission of positrons from the U+Cm quasimolecule. Furthermore,

the narrowness of the width of the peak, the resonance-like production of the

peak in only a limited range of bombarding energies near the Coulomb barrier,

and the association of the peak with scattering events which differ in their

angular correlation from the dominant Rutherford scattering events suggest the

possibility of the enhancement of spontaneous positron production by the

formation of metastable nuclear complexes.

Before trying to reach any definite conclusions, however, on the basis of the

systems studied here, several open experimental questions remain to be addressed.

QED theory makes definite predictions for the behavior of overcritical states

(assuming the nuclear charge distribution is known) and the data presented above

suggest several experimental avenues which need to be explored. The immediate

future of the study of the possible spontaneous production of positrons in heavy

ion-atom collisions must deal primarily with improving and extending the

measurements with respect to the four experimentally accessible parameters.

These are the total nuclear charge Zcom = (Zp+Zt) of the quasimolecule, the

bombarding energy of the collision leading to the formation of a quasimolecule,

the angular distribution of the coincident scattered particles, and the differential

energy spectra of the emitted positrons.

The list of critically bound systems which have been studied to date (Schweppe

et al. 83, Clemente et al. 84] must be expanded to include the other available

combinations of heavy ions in order to look for the predicted strong Z-dependence

of the spontaneous positron production mechanisms. Figure 7-15 shows this

340

combined nuclear charge (Z + Z.) of the projectile and target. (The figure isP 1

based on the calculations of [Reinhardt et al. 81b] and the assumption that the

line is emitted from a quasimolecule with the same internuclear separation as

produces the observed line at ~ 316 keV in the U+Cm system.) The nearly

linear relationship in the experimentally accessible region from

180 < Z < 190 can be written as:^ com

E ^ k as (31.3 keV) (Z + z t- 177.9) (7.4)

dependence for the energy E^ak of the positron peak as a function of the

z , * z 2Figure 7-15: Z-dependence of the positron peak energy.

The calculated kinetic energy E ^ ak of the positron peak is plotted as a function of the combined nuclear charge (Zp + Zt) of the projectile and target nuclei (based on [Reinhardt et al. 81b]).

In view of the possibility of the formation of structure in positron energy

distributions due to excitations of the uranium nucleus, it particularly important

341

to study systems without the uranium nucleus. At GSI, for example, it is possible

to produce a beam of the thorium isotope 232Th. Targets of the plutonium

isotope 244Pu are also obtainable, although they are radioactive. With these

additions, possible overcritical combinations of beam and target to be studied in

the future include U+Pu with Z = 186, Th+Pu with Z = 184, andcom ' com ’

Th+Cm with = 18633.com

If indeed metastable nuclear complexes are being formed in heavy ion-atom

collisions at or near the Coulomb barrier, then their formation can be expected to

be dependent on the energy of the collision: too low an energy and the two nuclei

do not come close enough together to stick, too high an energy and the collision is

too violent for sticking. A careful study of the intensity and position of the peak

in the positron energy spectrum as a function of the beam energy could provide

information on the shape of the effective nucleus-nucleus potential at the

Coulomb barrier and on any possible sticking mechanism. If this is the case, then

the detection of positrons seems to be a particularly sensitive and well-suited

probe of any formation of nuclear complexes at the Coulomb barrier.

To this end, the simple scenario described above of two nuclei sticking together

at a separation determined by the distance of closest approach during a collision

provides a starting point for extending the search for peaked structure to other

overcritical collision systems. One simple model to predict how the conjectured

metastable nuclear complexes might form in other systems is to assume that the

formation scales with the distance d between the equivalent hard sphere surfaces

of radius r = (1.2 fm) A1/ 3 of the two nuclei. The projectile bombarding energy

E /A such that the intersurface distance d is attained at the point of closestp i p

approach is given by:

ooSince the completion of the experiments for this thesis, measurements have been made on the

Th+Cm collision system at bombarding energies around 6.0 MeV/amu [Cowan 85].

where rQ cz 1.2 fm and e2 c l 1.44 MeV fm. It is plotted in Figure 7-16 as a

function of the combined nuclear charge Zegm for several separations d, as

indicated. The solid line is for a uranium projectile and the dashed line for a

thorium projectile.

As seen above, the angle-angle correlation plot of the two scattering nuclei also

seems to contain interesting information on the interaction of the two nuclei

during the collisions which lead to the formation of the structure in the positron

energy spectra. This plot should be measured with greater statistical accuracy to

allow a more detailed analysis of the apparent deviation from normal Rutherford

scattering during these rare collisions. By eventually combining this with the last

point, a study of the scattering angle-angle correlations as a function of the beam

energy could provide insight into any possible formation of nuclear complexes.

Finally, careful studies of the energy distribution with high statistics could give

information about the source of the structure, since the lineshape of the structure

appears to be dominated by Doppler broadening due to the motion of the emitting

system. As in the data of Figure 7-7 above, the velocity of the emitting system

can be determined from a careful analysis of the lineshape of the structure,

particularly as a function of the particle scattering angles. In addition, a search

should be made for evidence of the continuum positron distribution that must

accompany the monoenergetic pair conversion process that has been suggested as

a possible source of the peak in the U+Cm collision system.

The data presented in this thesis indicate the existence of an unexpected mode

of positron production in heavy ion-atom collisions. In view of the apparent

absence to date of a trivial explanation for the observed peak in the U+Cm

343

180 182Z | + Z 2

Figure 7-16: Equal separation bombarding energies.The bombarding energy of a uranium (solid line) or a thorium (dashed line) projectile which leads to the indicated separation between the equivalent hard surfaces r = (1 .2 fm) A1/ 3 of the projectile and target nuclei is plotted as a function of the combined nuclear charge (Zj+Z2) of the two nuclei.

344

collision system, the determination of the source of this structure, even when not

spontaneous positron emission, can be expected to provide interesting information

on the physics of heavy ion-atom collisions.

A ppendix A

HIC K inem atic Equations

This appendix collects for reference the kinematic relations valid in heavy-ion

collisions near the Coulomb barrier. Since the relative velocity of the two nuclei

is typically v/c ~ 0 .1 , nonrelativistic relations can be used, and since the

Sommerfield parameter i\ ~ 500 is large, classical mechanics applies.

A .I . R u th erford -S catter in g K in em atics

For Rutherford scattering, the potential is:k 9

V(r) = - with k = ZpZte (A l)

and the trajectories of the two nuclei have the form of the hyperbola

p 2 Eb2- = — 1 + £ cos 6 with p = —■— (A2)r k

which can be parameterized in terms of a variable ( as:

r = a(« cosh £ + 1 ) t — pa?/k (e sinh £ + {)

x = a(cosh f + e) y = a V t2—l sinh ( (A3)

In these equations, Zp and Zt are the nuclear charge, and Mp and M( the mass of

the projectile and the target, respectively,

ecM=*TT <A 4 >P

is the CM energy of the colliding nuclei,

ti = ---- (A5)M p + M t

is the reduced mass of the collision system,

345

346

. - / n (p/t>r (A6)

is the eccentricity of the orbit, k

2 a ~ T ~C M

is the distance of closest approach during a head-on collision,

6 = a cot(0 PM / 2 ) = a tan 9^

is the impact parameter, and e2 cz 1.44 MeV-fm.

(A.7)

(A.8)

The distance of closest approach Rmin during a collision is related to the

scattering angles 9 P and 6* of the projectile and target nuclei, respectively, by:

(A.9)mm( c m \ ] K m ) 1

= a CSC \ —- — / + 1 = a sec 1—— /+ 1V 2 / \ 2 /

where the center-of-mass scattering angles are related to the laboratory scattering

angles by:

e C M = e h b + s i T r l ( pS'n e L b )

& C M = 2 0 L a b (A. 10)and p = A/p/A/j. Conversely, the scattering angle is related to the distance of

closest approach by:

,p

e L a b s s t u r l

sin 9.C M

(M / M t) + cos

e ’ M = 2sin • ( — — , 9 ^ = 003-*mm ,/? ■ — a ,\ mm /

(All)

In general, the scattering angle of the target nucleus is related to that of the

projectile by:

— (1 +p) cot 9Lah ± (1 +p)cot 6 U b =

p* + cot 9Lab

1 - /(A. 12)

Defining the energy loss Q = E — (E ' + E t') and allowing for mass transfer by

347

using primes to indicate the masses and energies after the collision, the

relationship between the two scattering angles can be generalized to:

cot 9 r Lab

- ( J + / y t ) c o t e [ a b ± (pp+ p t ) ~ pl + COt2 elab(A 13)

with the additional definitions:M J

_ Mp'pp ~ m ; ’ Pt ~ m p , r

M E

R s Mt E + M Q ’(A 14)

In Rutherford scattering, the time T for the nuclei to move from a separation

R > R m in in to a separation £ • and then back out to a separation R again is

given by:

T[R) = 2A/ n (r + R \ / l —a— —

v l —a—0 + a In 1------------------------\ R m in ~ a

I rn / MeV/amu4 - / t E “ 7 I987X1° S'c/fn /

p/ p 2 a b

O S — , , S - (A15)

The longest possible time for a given beam bombarding energy and separation R

is the case of a head-on collision, where 6 = 0 , R . = 2 a, 9. = 0 ° , and1 1 mm ' i '9p = 180 ° , in which case the expression above for the time reduces to:

T b = 0(R ) = 4 a AY l - a I 1 + Y l - a

+ In -------------s T *

(AX6)

For assymmetric collision systems, the kinematic relationship of scattering angle

measured in one detector versus scattering angle measured in the other shows a

crossing point © „ which is given by: cross

' M n9 ___ = cos- 1 I t . / t t + 1cross

12

(A 17)

For systems where the target nucleus is lighter than the projectile nucleus, the

maximum scattering angle ©max for the projectile is given by:

348

M t8 m a , = sin ' 1 J f <A 1 8 >

P

A .2 . R u th erford -S catter in g Cross Section

The Rutherford cross section for Coulomb scattering of the projectile nucleus in

the laboratory system is:

where:

da da d n CM dl1Lab

d8L d n CM dnLab d0Lab

da a2 1

4 . „in CM s in (eCM/2)

dnCM [ 1 + 72 + 2 7 c o s e vCM

dnL 1 + 7 c o s e vCM

dnL ^ v2k s m &Lab

d 8 l a i

]3/2

8 C M “ 8 L a i + sin ' < 1 s in 8 Lab > ' 1 ~ J f <A 2 0 >P

The cross section for scattering of the target nucleus is:

M t

da da d n CM d°C M d°Lab

with:d e Lab d n C M d tiCM dnLab d S Lab

d° C M d n C M t = i , = 4 cos eLab

d° C M dnLab

(A19)

(A 2 1 )

d n L ab t p t— 2* sm — L a i + - (A 2 2 )

d8L

349

In a nearly symmetric collision system where it is impossible to differentiate

projectile from target nucleus, the measured cross section corresponds to the sum

of the contributions from the detection of the projectile at a given angle and the

detection of the target nucleus at that same angle:

(A23)da da da

. m e a s , » , tde, . de; , de.' Lab

Lab Lab LabFor a symmetric collision system, 7 = 1 , and the expression for the measured

Rutherford cross section reduces to: da ■z • m ^ n e a s .

= a sm(20La6 )demeas

'Lab• 4 ja ita s

SU1 0 u t4 meas

cos Lab

(A 24)

The probability Pg+ for the production of positrons (and similarly for gamma

rays) from collisions leading to scattering into an angular region 0 j < e < ©2 given:

d a p i e )

p » w — d 9

p » =(A25)

e 0 d° d e )de

de

Again, if the projectile and target nuclei can not be differentiated, then the

measured positron production probability is a weighted average of contributions

from the detection of the projectile and the target nuclei:

_ meas __e+

f a i

d° d V d a ^ e t )

dede

(A26)

/ .e A d<Td e v) | dad e ty

e l I de dede

A ppendix B

A bbreviations Used in the Thesis

CAMAC Computer Automated A/easurement and Control

CM Center of Mass

DFS Dirac-Fock-Slater

DMA Direct A/emory Access

DMI Direct A/emeory Increment

EDAS Experimental Data Acquisition System

EPOS Electron-Positron Spectrometer

GAMMEL Gamma Ana/ysis

GSI Gesellschaft fur Schwerionenforschung .

GOLDA GSI On-Line Data Acquisition System

m e Heavy-ion Collisions

IPC Internal Fair Conversion

LAM Look-At-Me

MIPC Afonoenergetic Internal Fair Conversion

QED Quantum Electrodynamics

PPAC Parallel-Plate Avalanche Counter

RASMO Random Scan Mode/Module

SATAN System to Analyse Tremendous Amounts of Nuclear Data

SIMPLE System-Independent Macro Processing Language for EDAS

351

TCD

UNILAC

Two-Center Dirac

Universal Linear Accelerator

352

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