experimental study of the reaction li + sm at 7 mev/u ... · saci-pererê spectrometer zully elisa...

Experimental study of the reaction63Li + 154

62Sm at 7 MeV/u using theSaci-Pererê spectrometer

Zully Elisa Johanna Guevara Maldonado

Universidad Nacional de ColombiaFacultad de CienciasDepartamento FísicaBogotá, Colombia

2016

Experimental study of the reaction63Li + 154

62Sm at 7 MeV/u using theSaci-Pererê spectrometer

Zully Elisa Johanna Guevara Maldonado

Master’s thesis submitted in partial fulfillment of the requirements for the degree of:Magister en Física

Advisor: PhD. Edna Carolina Pinilla BeltránCo-advisor: PhD. Diego Alejandro Torres Galindo

Research area: Nuclear reactionsResearch group: Grupo de FÃsica Nuclear de la Universidad Nacional de Colombia

Universidad Nacional de ColombiaFacultad de CienciasDepartamento FísicaBogotá, Colombia

2016

With deep gratitudeto God, to my parents and sister.

“Often when you think you are at theend of something, you are at the beginningof something else.”

Fred Rogers

Aknowledgments

Al finalizar esta etapa de mi vida deseo expresar mi más profundo agradecimiento a Dios porguiar cada paso, y gracias por todas las personas que pusiste en mi camino sin las cuales no habríasido posible culminar este proceso.

Es un placer para mi dar gracias a mis supervisores. Al Dr. Diego Torres por toda su experiencia,confianza, por su constante apoyo y entusiasmo. A la Dra. Carolina Pinilla por desear hacer partede este proceso, por toda su dedicación y experiencia. En general, a mis dos supervisores por susinvaluables sugerencias, aportes y por toda su paciencia.

Al Grupo de Física Nuclear, al profesor Fernando Cristancho, por sus contribuciones y granexperiencia. A todos mis compañeros de oficina y amigos: A Fitzgerald, a William, a Andrés, aJuansebastian, a Frank, a Eduardo, a Jeshua, a Ana, a Katherine, a Andrea y a David. En particularme gustaría agradecer a Wilmar por las discusiones y soporte durante toda la maestría.

Al profesor José Oliveira por su interés en el desarrollo de esta investigación y por toda su ayuda.Al comité evaluador: Maurizio De Sanctis, Ramiro Cardona y Justo López por sus comentarios

y aportes sobre la tesis.A la Universidad Nacional de Colombia y a Colciencias por ser el soporte financiero, sin el cual no

habría sido posible culminar este proceso. Específicamente al proyecto: “Exploración del surgimientode la colectividad nuclear en las capas de protones 1f7/2, 1g3/2, 1p1/2, 1f5/2 y de neutrones 0f5/2,1p3/2, 1p1/2, 0g9/2, 1d5/2 por medio de la medición de momentos magnéticos y eléctricos en estadosnucleares excitados” y al proyecto de jóvenes investigadores: “Efectos de la interacción espín-órbitaen colisiones elásticas a altas energías”.

Y por último, pero no menos importante, a mis amados padres y hermana, por su infinito amor;por llevarme en su mente, en su corazón y en sus oraciones. A mis amigas: a Diana, a María, aMaureen y a Caori por su sincera amistad y por sus oraciones.

Preliminary results of the present work were presented in the conferences:

1. Euro-school on exotic beams (August 28th - September 3rd, 2016)

2. III Uniandes Detector School (September 26th - September 30th, 2016)

The investigations performed during my Master resulted in the following publications:

1. Z.E. Guevara, and D.A. Torres, Lifetime and g-factor measurements of excited states usingCoulomb excitation and alpha transfer reactions, AIP Conference Proceedings, 1753, 030006(2016) 10.1063/1.4955347.

2. G. J. Kumbartzki, N. Benczer-Koller, K.-H. Speidel, D. A. Torres, J. M. Allmond, P. Fallon,I. Abramovic, L. A. Bernstein, J. E. Bevins, H. L. Crawford, Z. E. Guevara, G. Gurdal,A. M. Hurst, L. Kirsch, T. A. Laplace, A. Lo, E. F. Matthews, I. Mayers, L. W. Phair, F.Ramirez, S. J. Q. Robinson, Y. Y. Sharon, and A. Wiens, Z = 50 core stability in 110Sn frommagnetic-moment and lifetime measurements. Phys. Rev. C 93, (2016) 044316-1 044316-7.

3. N. Benczer-Koller, G. J. Kumbartzki, K.-H. Speidel, D. A. Torres, J. M. Allmond, P. Fallon,I. Abramovic, L. A. Bernstein, J. E. Bevins, H. L. Crawford, Z. E. Guevara, G. Gurdal, A. M.Hurst, L. Kirsch, T. A. Laplace, A. Lo, E. F. Matthews, I. Mayers, L. W. Phair, F. Ramirez,S. J. Q. Robinson, Y. Y. Sharon, and A. Wiens, Magnetic moment and lifetime measurementsof coulomb-excited states in 106Cd, Phys. Rev. C 94, (2016) 034303-1 034303-6.

Note: Copy of the articles are presented at the appendices of this document.

vii

Abstract

In this work we analyze experimental data of the nuclear reaction between 63Li + 154

62Sm . The experi-mental data come from the gamma-particle coincidence experimental technique using Saci-Pererêγ-ray spectrometer and Pelletron accelerator located at the Instituto de Física da Universidade deSão Paulo, Brazil. The Saci-Pererê spectrometer is designed to study nuclear reaction processessuch as inelastic scattering, transfer, break-up and fusion (complete and incomplete). The aim ofthe experiment is to study the different reaction mechanisms of the 6Li + 154Sm nuclear reaction.In particular, to measure inelastic differential cross section of the excited states of the 154Sm. Theprojectile 6Li is weakly bound and the 154Sm target is quadrupole deformed in its ground state.Little information about weakly bound nuclei interacting with deformed nuclei is known. In thosekind of reactions the continuum states of the weakly bound projectile are expected to play an im-portant role. The projectile 6Li was accelerated above the coulomb barrier energy at 42 MeV (VB ≈28.7 MeV), thus various kind of reactions are produced.

The inelastic scattering of 154Sm is observed in coincidence with Z = 3 nuclei in a ∆E Vs.E spectrum. The γ-ray spectrum shows only the first exited state of the 154Sm. However, theexperimental results do not provide enough statistics to measure the angular-particle distributionand to compare with the theoretical predictions. The others identified products are influenced bythe easy breaking of 6Li. The first product identified by the breaking up of 6Li is 156Eu. Thisnucleus is formed after the absorption of a deuteron by the target (incomplete fusion process). Thesecond identified product by breaking of 6Li is 157Gd. This result is confirmed by the coincidenceof gamma-rays with Z = 2 nuclei. 157Gd is formed after of the absorption of an alpha particle bythe target (incomplete fusion process). Gamma-rays in coincidence with Z = 1 nuclei suggest theemission of one neutron.

Keywords: Nuclear reaction, weakly bound nucleus, deformed nucleus and gamma-particle coin-cidence experimental technique.

viii

Resumen

En este trabajo analizamos los datos experimentales provenientes de la reacción nuclear entre 63Li

+ 15462Sm. Los datos experimentales vienen de la técnica experimental de coincidencia γ-patícula

usando el espectrómetro de rayos gamma Saci-Pererê y el acelerador de Pelletron localizados enel Instituto de Física da Universidade de São Paulo, Brazil. El espectrómetro Saci-Pererê está dise-ñado para estudiar los procesos de reacción nuclear, tales como dispersión inelástica, transferencia,rompimiento (break up) y fusión (completa o incompleta). El objetivo del experimento es estudiarlos diferentes mecánismos de la reacción nuclear 6Li + 154Sm. En particular, medir la sección eficazdiferencial de los estados excitados del 154Sm. El proyectil 6Li es dédilmente ligado y el blanco 154Smpresenta una deformación cuadrupolar en su estado base. Poca información acerca de la interacciónde los núcleos débilmente ligados con núcleos deformados es conocida. En esta clase de reaccionesse espera que los estados del contínuo del proyectil débilmente ligado jueguen un papel importante.El proyectil 6Li fue acelerado por encima de la energía de la barrera de Coulomb a 42 MeV (VB ≈28.7 MeV), esto da lugar a varios tipos de reacciones.

La dispersión inelástica del 154Sm es observado en coincidencia con los núcleos de Z = 3 enun espectro ∆E Vs. E. El espectro de rayos γ muestra solamente el primer estado excitado del154Sm. Sin embargo, los resultados experimentales no proveen suficiente estadística para medir ladistribición angular de partículas y comparar con las predicciones teóricas. Los otros productosidentificados son influenciados por el fácil rompimiento de 6Li. El primer producto identificado porel rompimiento del 6Li es 156Eu. Este núcleo se forma después de la absorción de un deuterón porel blanco (proceso de fusión incompleta). El segundo producto identificado por el rompimiento del6Li es 157Gd. Este resultado es confirmado por la coincidencia de los rayos gamma con los núcleosZ = 2. 157Gd se forma después de la absorción de una partÃcula alfa por el blanco (proceso defusión incompleta). Los rayos gamma en coincidencia con los núcleos de Z = 1 sugieren la emisiónde un neutrón.

Palabras clave: Reacción nuclear, núcleo débilmente ligado, núcleo deformado y técnica experi-mental de coincidencia gamma-partícula.

Table of Contents

Aknowledgments v

Abstract vii

1. Introduction 1

2. Theoretical fundamentals 62.1. Fundamentals of nuclear reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2. Elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3. Continuum discretized coupled channels (CDCC) method . . . . . . . . . . . . . . . 152.4. Nuclear deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3. Experimental methods 213.1. Spectrometer Saci-Pererê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2. Pelletron accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3. Detection system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4. Data analysis and results 294.1. Data format for the nuclear reaction 6Li+154Sm . . . . . . . . . . . . . . . . . . . . . 294.2. Reduction of data and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3. Products of the nuclear reaction 6Li+154Sm . . . . . . . . . . . . . . . . . . . . . . . 37

5. Conclusions and perspectives 44

Bibliography 45

A. Appendix: Two-dimensional spectrum ∆E−E and coincidence γ spectra for scin-tillator 2 48

B. Appendix: Subroutine in PRETSCAN to analyse raw data of the Saci-Pererê spec-trometer 53

C. Appendix: Contributions to Physical Review C 61

Chapter 1

Introduction

The existence of indivisible particles that form matter has been one of the concerns that human-kind has had throughout history. The first person who postulated the existence of the atom wasDemocritus, Greek philosopher who lived in the fifth century BC. Although his hypothesis on theexistence of atoms was a great idea, it was only based on logical reasoning and not on experimentalfacts. In 1803 John Dalton gave the basis on the atomic model with the measurements, for the firsttime, of the relative weights of the chemical elements [1]. Subsequently, several atomic models wereproposed (Thomson in 1904 [2], Rutherford in 1911 [3], Bohr in 1913 [4]), as a representation ofthe structure of the atom. In 1911 Rutherford discovered the existence of the atomic nucleus whenbombarded a gold foil with α particles. Nowadays, the fundamental structure of matter and itsinteractions form a broad and exciting field of research [5].

Nuclear physics is a field that studies the structure and properties of the atomic nuclei, aswell as the interactions among its constituents. Thus, some of the tools commonly utilized by anuclear physicists are the measurement and calculation of quantities such as masses, shapes, sizesand decays of stable and radioactive nuclei, among others. The atomic nucleus is a quantum systemthat is constituted by protons and neutrons called nucleons. An important concept in nuclear physicsis the binding energy (B), which is the energy needed to separate all the nucleons in a nucleus. Inaverage the binding energy is approximately 8 MeV per nucleon. The Chart of the nuclides is agraphic that locates all known nuclei in a plane of Protons versus Neutrons, see Figure 1.1. Eachisotope is represented by a colored box according to its predominant decay mode from the groundstate. Stable isotopes are represented by black squares, these stable isotopes are found naturally inthe universe. The set of stable nuclei form the so called stability line. Stable nuclei are characterizedby large binding energies and long lifetimes, longer than the lifetime of the solar system (about4.6 billion of years). Outside the stability line are unstable or radioactive nuclei. Radioactive nucleidecay successively to populate nuclei in the stability line. Magic numbers (N or Z=2, 8, 20, 28, 82and 126) are indicated by a rectangle on the chart. According to the nuclear shell model, the lastorbit of such nuclei is complete filled by either protons or neutrons, and form nuclei with largerbinding energy compared with its neighbor nuclei, something similar to what happens with noblegases in atoms, whose last energy level is full of electrons.

The interaction between two nucleons is mediated by the strong nuclear force, which has not acomplete theoretical explanation yet [5]. Some questions for nuclear physics are: Why do nucleons

2

Figure 1.1. The chart of nuclides. Each nucleus is placed at the positions (Z, N), where Z is thenumber of protons and N is the number of neutrons. The way nuclei decay is shown by the colorreference. Figure edited from Ref. [6].

stay together in the nucleus? How many neutrons can I add in a nucleus? Why does nuclei rotate?Scattering experiments aim at providing answers to most of these questions [7, 8]. These experi-ments collide nuclei for the detection and characterization of reaction products. In this way, theinteractions among elementary particles are investigated to probe the internal structure of complexmicroscopic systems. The quantum scattering theory is the appropriate framework to study scatte-ring experiments, since nuclei are quantum systems. The predictions for the outcome of scatteringexperiments require the knowledge of the solutions of the Schrödinger equation that describes thedynamics of the collision partners. This is a challenging task, because we do not know an exactanalytical form of the nucleon-nucleon interaction. However, approximate models are developed.

When two nuclei collide different products are produced. For example, the reaction 12C+d has avariety of mass rearrangements that may be possible, such as 13C+p, 14N+γ, 12C+n+p, etc. Theseproducts in different energy levels or possibilities of output are called channels 1 [9, 10].

1The concept of channel is used to define the asymptotic state of the collision system far away from the region ofinteraction, i.e. before or after the collision. A particular mass rearrangement of the total number of nucleons betweenthe collision partners plus a set of quantum numbers defines an arrangement channel. A particle transfer during acollision then causes a rearrangement.

3

The elastic scattering process is the simplest channel, so it can be considered as the startingpoint for the understanding of more complicated reaction channels. It is well known that at energiesclose to the coulomb barrier, the elastic scattering process between heavy ions (it is called heavyion to nuclei with atomic number greater than four: A > 4) is quite predominant, in terms of crosssections [1]. The most widely used model for the analysis of experimental data of elastic scatteringis the optical model. In this model, the nuclear potential is described by a real part and a imaginarypart. The real part describes elastic particle scattering. Thus, the imaginary part is zero when theonly outgoing channel is the elastic channel (flux is conserved). The imaginary part is related tothe other reaction channels (it simulates the absorption of the scattering flux).

In the last decades great efforts have been made to improve knowledge about the interactionbetween nuclei [11, 12, 13], providing important advances in various areas of nuclear physics, suchas: heavy ion reactions, nucleon-nucleon collisions, reactions involving exotic nuclei, etc. The SãoPaulo Potential is a theoretical model which attempts to describe a mean nuclear potential of twointeracting nuclei [13]. This potential has had great success in describing the elastic and inelasticscattering, transfer and fusion processes for heavy-ion reactions [14, 11]. The São Paulo Potentialis a potential inspired in the double-folding potential, which depends on the density of the collisionpartners and on a local velocity term (that arises from the Pauli non-locality in the exchange ofnucleons between the collision partners).

Some stable and light nuclei can be described through an association between a core and afragment. When the binding energy between the core and the fragment is small, between 0.1 and2.5 MeV, it is said that this nucleus is a weakly bound nucleus [15]. Therefore, in this type ofnuclei the most likely channel is break up (the projectile fragments between its constituents). Someexamples of weakly bound nuclei are: The 6Li which can be thought as a core of 4He and deuterium(energy of separation 1.5 MeV), or 7Li which can be described as an association between 4He andtritium (energy of separation 2.5 MeV), or 9Be which is interpreted as a system of three bodies,composed by two alpha particles and one neutron (energy of separation 1.6 MeV).

Samarium isotopes have been used in the investigation of fusion for a long time. The reason is thatthere are several stable Samarium isotopes, from the almost spherical as 144Sm until the deformedones as 154Sm. The influence of deformation in sub-barrier fusion reactions has been the main topicinvestigated since 1970’s. The pioneer work in this field was done by Stokstad et al. [16, 17], who

Then, if we consider the reaction: x+A→ y +B, where x, A are the nuclei in the entrance channel, and y, B arethe nuclei in the exit channel. A particular arrangement channel consisting of the two nuclei (x, A), is labeled by

α = (x,A). (1.1)

To a characterization of a two-body channel a, we introduce the complete set of quantum numbers of the channelwhich is

a = α; ax, aA;ka, (1.2)

where ax = εx, Ix,Mx,Tx,MTx , πx and aA = εA, IA,MA,TA,M

TA, πA are the internal quantum numbers of

the fragments x and A: εi is the intrinsic energy, Ii the spin with z-component Mi, Ti the isospin with z-componentMTi , and πi is the parity of the nucleus i (where i = x,A), and finally ka is the relative momentum of the collision

partners x and A. Indeed, formally speaking (but outside of the scope of this thesis) a channel a must be characterizedby a complete and orthonormal basis: S basis or Jx basis. The S basis is defined as S = Ix + IA, JTotal = L + S,where L is the relative angular momentum. The Jx basis is defined as Jx = L + Ip, JTotal = Jx + IA.

4

studied sub-barrier fusion reactions using the double-magic nucleus 16O with 144,148,150,152,154Sm.Some years later, di Gregorio et al. [18, 19], also in collaboration with Stokstad, complementedthese investigations by measuring sub-barrier fusion with 144,147,149Sm. More recently, the fusionand quasi fission of 40,48Ca+144,154Sm were also measured [20, 21, 22].

In recent years, fusion of weakly bound nuclei with Samarium isotopes has been intensivelyinvestigated, both experimentally and theoretically [23, 24]. The influence of the breakup of thesenuclei on the fusion cross section was the main topic under study. Measurements of complete andincomplete fusion cross sections for 6Li+154Sm have been performed at energies above the Coulombbarrier [23].

On the other hand, theoretical calculations of the break up cross section of weakly bound nucleihave some difficulties. Initially the interaction potentials between the projectile constituents (coreand fragment) and the target are unknown. Besides, the break up process must connect boundstates of a projectile with states in the continuum of the projectile constituents. This leads todivergences of the continuum-continuum couplings due to the oscillatory asymptotic behavior ofthe continuum states. Currently a numerical method is used to calculate the break up process. Thismethod is called Continuum Discretized Coupled Channels method, or simply CDCC [15]. As itsname indicates, this method discretize the continuum to square integrable states. The main goal ofthe CDCC method is to solve the Schrödinger equation for reactions where the projectile exhibitsa cluster structure as a weakly bound nuclei which are constituted by a core and fragment with alow dissociation energy [25].

This thesis studies the nuclear reaction between the weakly bound projectile 6Li and the deformedtarget 154Sm. The 6Li projectile was accelerated above the Coulomb barrier energy at 42 MeV (VB ≈28.7 MeV). This gives rise to several reaction products.

The γ-particle coincidence experimental technique is employed with the aim of measuring theparticle angular distribution of the excited states of the 154Sm [27, 28] to contrast the experimentalresults with the theoretical predictions (see Figure 1.2) [26]. The experiment was performed usingthe Saci-Pererê gamma-ray spectrometer and the Pelletron accelerator located at the Institutode Física Universidade de São Paulo, Brazil. The Saci-Pererê spectrometer originally designed fornuclear structure studies has been adapted to study nuclear reaction channels such as inelastic scat-tering, transfer, break-up and fusion (complete and incomplete). This technique has been previouslyimplemented to study the inelastic scattering of the first excited states of the 110Pd and 120Sn, inthe 18O+110Pd and 7Li+120Sn reactions [27, 28, 11].

The inelastic scattering of 154Sm is observed in coincidence with Z = 3 nuclei in a ∆E Vs.E spectrum. However, the experimental results do not provide enough statistics to measure theangular particle distribution and to make the theoretical comparison. The others identified productsare influenced by the easy breaking of 6Li. The first identified product by breaking up of 6Li is156Eu. This nucleus is formed after the absorption from a deuteron by the target (incomplete fusionprocess). These results are provided by gamma-rays in coincidence with Z = 2 nuclei provide thisresults. The second identified product by breaking up of 6Li is 157Gd. This nucleus formed afterof the absorption of an alpha particle by the target (incomplete fusion process), gamma-rays incoincidence with Z = 1 nuclei suggest the emission of one neutron.

The present work is organized as follows: Chapter 2 aims at presenting the theoretical basis:The São Paulo potential (SPP) and the Continuum Discretized Coupled Channels (CDCC) method.

5

Figure 1.2. CDCC inelastic angular distribution for the different excited states of the 154Sm. Thecolor reference shows the angular distribution of different excited states as a function of the scatteringangle (from center of mass). Figure taken from Ref. [26].

Chapter 3 is devoted to the description of the experimental setup (Pelletron accelerator and detec-tion system). Chapter 4 shows some results of the data analysis and it explains the gamma-particlecoincidence experimental technique (which is employed to identify the reactions channels). Finally,in Chapter 5 we present the concluding remarks and outlook.

Chapter 2

Theoretical fundamentals

This chapter aims at showing some of the theoretical basis relevant for the understanding of thiswork. In Section 2.1 we present a brief introduction of nuclear reactions. Section 2.2 is devotedto explain elastic scattering, the optical potential, and the São Paulo potential (SPP). Then, inSection 2.3 we present the formalism of the Continuum Discretized Coupled Channels (CDCC)method. Finally, in Section 2.4 we explain the nuclear deformation.

2.1. Fundamentals of nuclear reactions

A nuclear reaction takes place when a nucleus called projectile interacts with another nucleus calledtarget. The way to notate a typical nuclear reaction is

X(a, b)Y, (2.1)

where a is the accelerated projectile, X is the target (usually stationary in the laboratory frame),and Y and b are the reaction products. The final nuclei of a nuclear reaction can be different fromthe initial ones. A nuclear reaction can be classified by the outgoing products and by the mechanismthat governs the process. These classifications are explained in the Tables 2.1 and 2.2.

Table 2.1. Classification of nuclear reactions according to the mechanism that governs the process.

Direct reactions In this process the projectile interacts with the surface of the target. Therefore,the interaction is only with some nucleons. The time that the projectile typicallytakes to interact with the target is 10−22 s. In this kind of reaction it is possibleto remove or to insert nucleons between the projectile and the target.

Compound nucleusreaction

The projectile is completely captured by the target forming a compound nu-cleus. The time that typically the projectile takes to interact with the targetis of the order of 10−18 s to 10−16 s (although interacting time of 10−14 s hasbeen found). The beam energy is shared and re-shared among the nucleons ofthe target until it reaches a state of statistical equilibrium.

2.1 Fundamentals of nuclear reactions 7

Table 2.2. Classification of nuclear reactions by the outgoing products.

Elastic scattering This is the simplest reaction. Kinetic energy is conserved.The projectile and the target stay in their ground state (a = b and X = Y ).Technically is not a reaction.

Inelastic scattering The projectile or the target is left in a excited state which generally decays quicklyby γ emission.

Break up Pick up: The target takes a (some) nucleon(s) of the projectile.Knockout: The projectile takes a (some) nucleon(s) of the target.

Transfer A fragment is transferred between the projectile and the target.Fusion Evaporation: After the reaction, there is an emission of particles and/or γ emission.

Complete: The projectile is combined completely with the target.Incomplete: Some nucleons of the projectile are combined with the target.

Cross section

The collision or interaction between two nuclei is generally described in terms of the cross section.This quantity gives a measure of the probability for a reaction to occur. Formally, the cross sectionis defined in the following manner. Consider a beam of particles incident upon a target as is shownin Figure 2.1. Assume that the beam is much broader than the target, and the particles in the beamare uniformly distributed in space and time. Then, we can speak of a flux F of particles incidents.The flux of the incident beam is defined as the number of particles crossing a unit area perpendicularto the beam per unit time.

Figure 2.1. An incident beam collides on a target.

If the beam is uniform and contains n′ particles per unit volume, moving with velocity v withrespect to the stationary target, the flux is given by

F = n′v. (2.2)

2.1 Fundamentals of nuclear reactions 8

Figure 2.2. An incident beam is scattered by a scattering center. The scattered particles are detectedin a solid angle dΩ.

In most calculations, the number of incident particles is normalized to one particle per volumeV . The number of reaction per unit time is given by

number of reactions∆t

= σ(NF ), (2.3)

number of reactions∆t

= σ(nAδx)F. (2.4)

Where:

σ is the effective area of each scattering center.

F is the flux, i.e. number of particles per unit area per unit time.

n is the number of scattering centers per unit volume.

A is the area of target.

N = nAδx is the total number of scattering center.

Aδx is the volume of the target.

The coefficient of proportionality is σ which is called total cross section. A dimensional analysisof the Eq. (2.4) is given by

number of reactions[T ]

= σnumber of reactions

[L3][L2][L]

incident particles[T ][L2]

, (2.5)

from the dimensional analysis done in Eq. (2.5) we obtain

σ =[L2]

Incident particles. (2.6)

2.2 Elastic scattering 9

Roughly speaking σ is a measure of the probability that the reaction happens. σ has dimensionsof area per nucleus and its unit is the barn (1 b = 100 fm2). Despite σ is a quantity which hasdimension of area, it is proportional to the reaction probability.

If the detector defines a small solid angle dΩ then not all the outgoing particles can be detected.Only a small fraction scattered nuclei is detected and then only a fraction of the cross section dσis deduced. In general, the outgoing particles are not emitted uniformly in all directions. Then wemeasure

dσ

dΩ(θ, φ) ≡ differential cross section. (2.7)

The quantity dσ/dΩ is called the differential cross section1, and it gives information about theprobability to detect in a solid angle dΩ scattered nuclei. The solid angle is measured in steradians.The units of differential cross section are barns/steradian. The cross section σ can be found byintegrating dσ/dΩ over all θ, φ

σ =

∫dσ

dΩdΩ. (2.8)

We might modify the definition of cross section to give the probability to observe the ejectile bin the angular range of dΩ and in the energy range dEb. This gives the so-called doubly differentialcross section d2σ/dEbdΩ. Table 2.3 illustrates different types of cross sections that can be measuredand/or calculated.

Table 2.3. Types of reaction cross sections.

Cross Sections Symbol MeaningTotal σ Integrated over all angles and all energies of b or all excited

states of Y.Differential (Angular) dσ/dΩ Observe b at (θ, φ). Do not observe Y.Differential (Energy) dσ/dE Observe excitation of Y by subsequent γ emission. It means,

study of decay of excited states of Y . Do not observe b.Doubly differential d2σ/dEbdΩ Observe b at (θ, φ) at a specific energy. In other words, infor-

mation about the excited states of Y by the measurementof angular distribution of b.

2.2. Elastic scattering

Let us consider a collision of two interacting nuclei through a potential. Calling rP and rT the posi-tion vectors of the projectile and the target in the laboratory frame respectively, and pP and pT theirassociated momenta. The coordinate representation is shown in the Figure 2.3. The transformationfrom these coordinates to the center of mass frame (CM) is

1This amount is also called angular distribution.


Rc.m. =mP rP +mT rT

M; r = rP − rT , (2.9)

and

Pc.m. = pP + pT ; p =mTpP −mPpT

M, (2.10)

where M = mP +mT and mP , mT are the projectile and the target masses respectively.

Figure 2.3. Relative and center of mass coordinates of two interacting nuclei.

The total scattering Hamiltonian separates into the center of mass and the relative. This termis given by

Htot = Hc.m.(Rc.m.,Pc.m.) +Hrel(r,p). (2.11)

The Hamiltonian of the center of mass of the projectile-target system is written as

Hc.m. =P 2c.m.

2M, (2.12)

equation (2.12) gives the direct dependence with the total moment Pc.m. (vector sum of themoments of each nuclei Eq.(2.10)).

The Hamiltonian of the relative motion of the projectile-target system is given by

Hrel(r,p) =p2

2µ+ V (r), (2.13)

where µ = mPmT /M is the reduced mass, and V (r) is the total interaction potential. The totalpotential of a nuclear reaction is usually composed of two terms. One representing the short-rangenuclear interaction VN (r) and the other that represents the long-range Coulomb interaction, denotedby VC(r). Therefore, V (r) = VN (r) + VC(r).

Elastic scattering is a process in which the ingoing channel and outgoing channel are the same.Although this is the simplest process that can occur, it can be one of the most important becauseit provides information about the potentials and mechanisms that may be involved in the reaction.Also it is the dominant process for low energies with respect to Coulomb barrier energy, is the


dominant process. In general, the eigenfunction of the Hamiltonian given by Eq. (2.11) is of thetype

ψ(rP , rT ) = eiK·Rc.m.ψ(r), (2.14)

in which Kc.m. = Pc.m./~ and Rc.m. represents the center of mass vector of the system. The partthat depends on ψ(r) is related to the relative motion of the nuclei.

In order to obtain the part of the wave function related to the relative motion of the nuclei,given by ψ(r), it is necessary to solve the Schrödinger equation. We can then write the Schrödingerequation related to the relative motion of the nuclei as follows(

−∇2 +2µ

~2[VC(r) + VN (r)]− k2

)ψ(r) = 0, (2.15)

where k2 = 2µErel/~2 and µ is the reduced mass. The resulting wave function of Eq.(2.15) canbe separated into one solution for the nuclear part and another for the Coulomb part, can be writtenas ψ(r) = ψC(r) + ψN (r). Usually the Coulomb part ψC(r) of the wave function is composed by theincident wave (it is included the scattering because of Coulomb potential), while the nuclear partψN (r) of the wave function is composed only by the scattered wave of the nuclear potential.

After extensive calculations, which can be found in [7], the Coulomb part of the wave function(r > R) is

ψc(r, z)→ eikz+η ln[k(r−z)] + fc(θ)ei[kr−η ln(2kr)]

rfor r →∞, (2.16)

being η the Sommerfeld parameter η = Z1Z2e2/~ν, ν = ~kµ and µ is the reduced mass among

scattered nuclei, and fC(θ) is the Coulomb scattering amplitude, given by

fc(θ) = − η

2k sin2(θ/2)e−iη ln(sin

2(θ/2))+2iσ0 with σ0 = arg(Γ (1 + iη)), (2.17)

the expression (2.16) can be expanded in functions called partial waves [7], making that thefunction of Coulomb wave has the form

ψC(r, θ) =1

kr

∞∑l=0

(2l + 1)ileiσlFl(η, kr)Pl(cos(θ)), (2.18)

where the term σl is Coulomb phase difference, given by σl = arg(Γ (l+ 1 + iη)), and Fl(η, kr) aCoulomb regular function, whose asymptotic behavior (r →∞) is Fl(η, kr)→ sin[kr − ηIn(2kr)−π2 l + σl].

Regarding nuclear part of the wave function, it must be represented only by a scattering function,with this its asymptotic form can be written as

ψN (r)→ fN (θ)ei[kr−η ln(2kr)]

rfor r →∞, (2.19)

where fN (θ) is the scattering nuclear amplitude 2. To find the nuclear scattering amplitude, it isenough to compare the expansion of the total wave function with the expansion of the Coulomb part

2This is only an approximation since there is a possibility that fN is also affected by the Coulomb interaction.


of the wave function, given by Eq. (2.18). Because of the total wave function can be written moregenerally, it not only depends on the regular Coulomb function Fl(η, kr), but also on the irregularCoulomb function Gl(η, kr). The behavior of the irregular part of the function when (r → ∞) isthe type Gl(η, kr)→ cos[kr − ηIn(2kr)− π

2 l + σl].The total wave function can be expanded (again this expression can be seen in detail in the

reference [7]) as:

ψ(r) = ψC(r, θ) +1

kr

∞∑l=0

(2l + 1)ileiσlfNl H(+)l (η, kr)Pl(cos(θ)), (2.20)

in which H+l (η, kr) = Gl(η, kr)+iFl(η, kr) and fNl = 1

2i(e2iσNl −1) where σNl is the nuclear phase

difference. Observing (2.19) and comparing with (2.20) (remembering that ψ(r) = ψC(r) + ψN (r))finally obtains the expression of fN (θ) that can be written as

fN (θ) =1

2ik

∞∑l=0

(2l + 1)e2iσl(e2iσNl − 1)Pl(cos(θ)). (2.21)

The total wave function ψ(r) can then be written as the sum of the Coulomb and nuclear part,or in the last analysis as a superposition of a plane incident wave (in the k = p/~ direction) witha scattered spherical wave. This total wave function can be written asymptotically as

ψ(r)→ eik·r + f(Ω)eikr

rfor r →∞, (2.22)

in which Ω is the angular part of the variable r. Where f(Ω) is the total scattering amplitude,given by the sum of Coulomb scattering amplitude of equation (2.17) with the nuclear scatteringamplitude given by (2.21).

The differential cross section is defined as

dσ(Ω) =Probability current in dΩ in the direction of ΩProbability current density of the incident wave

. (2.23)

Remembering that the probability current density is given by the expression j = ~2µi(ψ

∗∇ψ −ψ∇ψ∗), and this same quantity can be obtained for the wave function given by Eq. (2.22), and thenreplacing j in the Eq. (2.23). With this, it is found that

dσ

dΩ= |f(Ω)|2, (2.24)

it is possible to write the total scattering amplitude as the sum of the Coulomb and nuclearscattering amplitude and then substitute that sum in Eq. (2.24), finding the following equation

dσ

dΩ= |f(θ)|2 = |fC(θ)|2 + 2Re[f∗C(θ)fN (θ)] + |fN (θ)|2, (2.25)

which contains a characteristic Coulomb-nuclear interference term.Given that the interaction potential between the nuclei V (r) is a term that plays a central role

in solving the Schrödinger equation it is necessary to discuss these issues in what follows.


Optical potential

When only the elastic scattering channel is taken into account the total scattering flux is conserved.To include in the nuclear scattering theory the effects of other channels of absorption, a negativeimaginary part must be added to the nuclear potential. The inclusion of that term causes an ab-sorption of the scattering flux. Therefore, the total scattering outgoing flux is reduced. The realand imaginary part of the nuclear potential is called the optical potential. That is so called to makean analogy with the refraction index in optics, where adding an imaginary part accounts for theprocesses in which light is absorbed by a medium. The optical potential is given by

VN (r) = VN,Real(r) + iVN,Imag(r). (2.26)

To verify that the imaginary part provides a loss of the incident flux, it is sufficient to study thedivergence of the probability current density, which can be written as j = ~

2µi(ψ ∗∆ψ−ψ∆ψ∗). Itsdivergence results in

∇ · j(r) =2

~|ψ|2VN,Imag(r). (2.27)

The Eq. (2.27) is known as the equation of stationary continuity. The mathematical interpreta-tion of the absorption flux is described by ∇ · j(r) < 0. Thus the imaginary part must be negativeto decrease the flux (VN,Imag(r) < 0).

São Paulo potential

A model that predicts the nuclear interaction is the São Paulo Potential (SPP). In general, theSPP is a theoretical model which attempts to describe a mean nuclear potential of two interactingnuclei [13]. The SPP is a potential inspired in the double-folding potential, which depends on thedensity of the collision partners and on a local velocity term. The local velocity arises from the Paulinon-locality in the exchange of nucleons between the collision partners. This potential is given by

VSPP (R12, Erel) = VFold(R12)e−4v2/c2 , (2.28)

where c is the speed of light, v is the speed of the relative motion between the nuclei, VFold is thedouble-folding potential, and R12 (whose magnitude is R12) is the relative coordinate between thecenter of mass of the projectile and the target (see Figure 2.4). The speed of the relative motion isgiven by

v(R12, Erel) =

√2[Erel − VC(R12)− VSPP (R12, Erel)]

µ, (2.29)

where Erel is the bombarding energy in the center of mass frame, VC is the Coulomb potential,and µ is the reduced mass.

In Eq. (2.28) VFold comes from the double-folding potential

VDouble−Fold(R12) =

∫ρ1(r1)ρ2(r2)vNN (R12 − r1 + r2)dr1dr2, (2.30)


Figure 2.4. Coordinates used in folding calculations between two nuclei of mass number A1 and A2.

after considering that the nucleon-nucleon interaction is short-range such as

vNN (R12 − r1 + r2) ≈ Voδ(R12 − r1 + r2), (2.31)

thus,

VFold(R12) = Vo

∫ρ1(r1)ρ2(r2)δ(R12 − r1 + r2)dr1dr2, (2.32)

where Vo = −456 MeVfm3 and ρi(ri) are the matter densities of the interacting nuclei. The SPPis obtained by solving iteratively Eq. (2.28) and Eq. (2.29).

The SPP has been successfully used to describe heavy-ion reactions for many systems in a verywide energy-region [14]. Over recent years the SPP has proved to be efficient in the description ofnuclear reactions in various energy and mass regions. Some advantages of the SPP is the absenceof free parameters, since the SPP depends only on the distance between the nuclei, energy and therelative velocity between the nuclei.

Coulomb potential

If we consider a nucleon interacting with a nucleus, we can approximate the Coulomb potentialas the one of a charge sphere of charge Zte interacting with a point charge Zpe, which is a goodapproximation for the system in question. This potential is written as

VC =

ZtZpe2

r if r > rC

ZtZpe2

2RC(3− r2

R2C

) if r ≤ rC ,(2.33)

where Zt and Zp are the atomic numbers of the nuclei involved in the reaction, e is the funda-mental unit of charge, and RC = rC(A

1/3t ) is the Coulomb radius, where At is the total number

2.3 Continuum discretized coupled channels (CDCC) method 15

nucleons of each nucleus. Typically rC = 1.2 fm.

Historically, the Coulomb potential in Eq. (2.33) was extended to study nucleus-nucleus collisions,approximating the problem to the collision of a point nucleus of charge Zpe with a Coulomb sphereof charge Zte and radius R = rC(A

1/3p + A

1/3t ). This approximation has been successful in fitting

experimental elastic scattering cross sections.

2.3. Continuum discretized coupled channels (CDCC) method

Some light nuclei can be seen as the union of two particles: a core and a fragment. When the bindingenergy between the core and the fragment is small, between 0.1 MeV and 2.5 MeV, it is said thatthis nucleus is a weakly bound nucleus (even if it is stable) [15]. Some examples of weakly boundnuclei are: The 6Li which can be thought as a core of 4He and deuterium (separation energy of 1.5MeV), or 7Li which can be seen as the union between 4He and tritium (separation energy of 2.5MeV), or 9Be which is interpreted as a system of three bodies, composed of two alpha particles andone neutron (separation energy of 1.6 MeV).

Figure 2.5. Schematic representation of a projectile, which is a weakly bound nucleus composedof a core and a fragment. Coordinates describe the collision between a target and a two-fragmentprojectile are shown.

A difference between reactions involving weakly bound nuclei, with those reactions with stronglybound nuclei is the strong influence to the cross sections of the break up channel.

The breakup process can result in different final configurations. To simplify the explanationof such configurations we consider a nucleus composed of only two fragments. It is important toemphasizes that only the processes resulting from the breakup are explained. However, other typesof reactions (such as inelastic excitations) can occur without the nucleus being fragmented.

The nuclear break up occurs when the target or the projectile are divided and possibly excited.When the interaction is due to mainly the Coulomb field of the target, mainly this interaction iscalled Coulomb break up. If the interaction is due to the nuclear force is called nuclear break up.


Finally, if both terms contribute to the occurrence of break up, this process is theoretically calledinterference break up.

The calculation of the break up cross section of weakly bound nuclei has some difficulties. Initiallythe interaction potentials between the projectile constituents (core and fragment) and the targetare unknown. Besides, the break up process must connect bound states of a projectile with statesin the continuum of the projectile constituents, given that the final state is unbound. This leads todivergences of the continuum-continuum couplings due to the oscillatory asymptotic behavior of thecontinuum states. Currently a numerical method is used to calculate the break up cross sections.This method is called Continuum Discretized Coupled Channels method, or simply CDCC [15]. Asits name indicates, this method discretize the continuum to square integrable states (there are threemethods to discretize it). The main goal of the CDCC method is to solve the Schrödinger equationfor reactions where the projectile exhibits a cluster structure as a weakly bound nuclei which areconstituted by a core and fragment with a low dissociation energy [25]. Such a method is describedbelow.

In the process of breaking up the nucleus, it is natural that the angular momentum between thefragments can vary from zero to infinity. To overcome this difficulty and to perform the numericalcalculations, it is necessary to make a truncation of the values of angular and linear momentum.In addition to that, it is necessary to choose the maximum value of the radius of interaction,within which the process of breaking can occur. These approximations are known as the ModelSpace Approximation. Such truncation constitutes the most basic hypothesis implied by CDCCcalculations.

Once all these observations have been made, the explanations of the CDCC equations can beshown. Initially, the coordinate system used in such calculations are discussed. In what follows, wetalk about the CDCC wave functions, the method that discretizes the continuous using the CDCCequations.

Jacobi coordinates

CDCC calculations usually involve the treatment of a problem of at least three bodies, two belon-ging to the projectile. Therefore, the breakup of a projectile p into a core plus a fragment (C + F )

due to an interaction with a target T can be described in a three-body model. There is a specialcoordinate system that facilitates the treatment, the Jacobi coordinate system. The Jacobi coor-dinates are defined by the relative distance between the two bodies (r1) of the projectile, and thedistance of the third body of interaction in relation to the center of mass of the first two (R1). Apossible configuration can be seen in the Figure 2.6.

Wave functions of the three bodies

The projectile (C + F ) can have bound states. Imagine that a bound state is given by the wavefunction φ(1)o (r1), written as a function of the coordinate system of the Figure 2.6. Then the incidentwave function of the projectile is given by expression of type Ψ (1) = φ

(1)o (r1)e

iK1R1 . Besides thebound states of the projectile (C + F ), others bound states of the projectile and the target arepossible as (T +F ) and (T +C). When there are bound states of the (F +T ) or (C+T ) pairs, then


Figure 2.6. Jacobi coordinates that can describe a three-body system. The relative distance betweentwo bodies is (r1), and the distance the third body from the center of mass of the first two is (R1).

wave function Ψ (1)(r1,R1) is not suitable to describe them, because they depend on r2 (coordinatefor the configuration T + F ) and r3 (coordinate for the configuration T + F ), and Ψ (1)(r1,R1)

cannot be suitably factorized into products of those bound states and spherical outgoing waves.Therefore, we need to write the full system wave function as a sum of components. A componentexpressed in each of the Jacobi coordinate pairs that can be written as

Ψ =

3∑i=1

Ψ (n)(rn,Rn). (2.34)

The Hamiltonian of the three bodies is given by

H3b = Tr + TR + VFC + VFT + VCT , (2.35)

where Tr and TR are the kinetic energy operator for each of Jacobi coordinates and V theinteraction potentials for each pair described above. The Schröndiger equation of this system iswritten as

[H3b − E]Ψ = 0. (2.36)

In the case of CDCC, each ψ(n)(rn,Rn) contains not only the description of the bound states butalso the wave functions responsible for describing the continuum. There are situations that reducethe number of Jacobi coordinates. The first possible situation is that there are no bound states forthe F-T or C-T pairs. The second possible situation is one in which the transfer channels are notimportant and are suppressed of Eq. (2.34). When one of these two cases occurs, the resulting newSchröndiger equation has the form

[H3b − E]Ψ (1)(r1,R1) = 0. (2.37)


Then you can expand the wave function in terms of the complete set of eigenstates of the internalHamiltonian of the projectile (C + F ), with this you have

Ψ(1)Ko

(r1,R1) =

nb∑p=1

φp(r1)ψp(R1) +

∫dkφk(r1)ψ

kK(R1). (2.38)

In the Eq. (2.38), k is the momentum of internal motion between C+F and K is the momentumof the relative motion between the projectile and the target. The first term on the right hand siderefers to the bound states while the second term refers to the continuum states. The wave functionφp represents an internal wave function of the projectile while the wave function ψ represents theprojectile-target relative motion. The momentum k is related to the momentum K through energyconservation

Ec.m. + ε0 = E =~2k2

2µCF+

~2K2

2µ(CF )T, (2.39)

where E is the center-of-mass energy of the three-body system, which is the sum of the projectile-target center-of-mass energy Ec.m. and the (negative) projectile binding energy ε0. The µFC is thereduced mass for F + C relative motion, and µ(FC)T is the reduced mass of the projectile-targetmotion. The projectile eigenstates φp(r) are eigensolutions of the Hamiltonian Hp = Tr(r)+VFC(r)

for energies εp < 0, and the continuum states φk(r) are the eigensolutions of the same Hamiltonian,for scattering energies εk = ~2k2

2µV C> 0 .

The radial part of the internal wave functions are denoted for now on as up,k(r), in which p refersto the bound states of the projectile, while k refers to the states of the continuum. Besides both aresolutions of the radial equation for the projectile[

− ~2

2µFC

(d2

dr2− l(l + 1)

r2

)+ VFC − ε

]u(p,k)(r) = 0. (2.40)

The bound states with εp < 0 exponentially decay in the asymptotic limit. While that the con-tinuum states with energy εk > 0 oscillate to infinity.

Discretization of the continuum

There are three methods to discretize the continuum, they are:

1. Another method is the pseudostates method which are simply the eigenstates of the internalHamiltonian Hint on some convenient square-integrable basis. The basis can be harmonicoscillator states, transformed harmonic oscillators, or a large set of Gaussians, among others.These pseudostates decay to zero at large distances.

2. The second method is continuum bin method which consists in making a mean of the radialwave function uk(r) and weigh it by the moment p which is determined by the bin. This methodgenerates wave functions for the bins, as superpositions of the eigenstates of scattering weightby k.

2.4 Nuclear deformation 19

CDCC equations and couplings

We can rewrite the CDCC Schrödinger equation as

(H3b − E)ΨCDCC(r,R) = 0, (2.41)

where the ΨCDCC(r,R) wave function is

ΨCDCC(r,R) =N∑p=0

φp(r)ψp(R). (2.42)

The CDCC wave function uses just one of the Jacobi coordinates since just study inelasticexcitations, some of them to the continuum. Multiplying Eq. (2.41) on the left by the conjugateprojectile wave functions, and using the expansion of Eq. (2.42), we obtain coupled-channel equations∑

p=0

〈φp′(r)|H − E|φp(r)〉ψp(R) = 0. (2.43)

These equations couple projectile bound states with the continuum states.The CDCC method basically consists in obtaining the wave functions of the projectile in the

continuum, calculating the coupling potentials and solving the system of coupled equations. TheSchrödinger equation then becomes a system of equations where there are potential matrix elementsrepresenting the couplings between different channels. The exact way of obtaining this scatteringamplitude is much more laborious (see [25]).

2.4. Nuclear deformation

The nuclear deformation is associated with the rotation of a quantum object, this concept wasintroduced by Niels Bohr in 1936 [1]. It is important to understand that a perfect spherical nucleuscan not rotate (i.e. if a perfect spherical nucleus is rotating it is not possible realize it), only adeformed nucleus can rotate (i.e. rotational motion can be observed only in nuclei with non-sphericalequilibrium shapes). The deformation is the key to describe rotations in a quantum object.

The deformed nuclei can be found in the mass range 150 < A < 190 and A > 220. The nuclearsurface can be described using the coordinates R(θ, φ) which give the position of a point in thesurface of a nucleus with an arbitrary shape. It is given by

R(θ, φ) = R0

(1 +

∞∑λ=0

µ=λ∑µ=−λ

αλ,µYλ,µ(θ, φ)), (2.44)

where R0 is the radius of a spherical nucleus with the same volume as the deformed nucleus. It iscustomary (although not quite exact) to take R0 = rA1/3 where r = 1.2 fm, Yλ,µ(θ, φ) is a sphericalharmonic, and αλ,µ is the deformation parameter.

A common representation of these nuclei is through an ellipsoid of revolution. For λ = 2

R(θ, φ) = R0

(1 + αY2,0(θ, φ)

), (2.45)

2.4 Nuclear deformation 20

here α is related with the eccentricity (is a parameter associated to every conic section)

α =4

3

√π

5

∆R

R0, (2.46)

where ∆R is the difference between the semi-major and semi-minor axes of the ellipse. Whenα > 0 the nucleus has the elongate form of a prolate ellipsoid; when α < 0, the nucleus has theflattened form of an oblate ellipsoid.

Figure 2.7. Schematic representation of the equilibrium shapes of nuclei with permanent deforma-tions.

One indicator of the stable deformation of a nucleus is a large electric quadruple moment whichindicates the charge distribution. The relationship between the deformation parameter and thequadrupole moment is

Q0 =3√5πR2

0Zα(1 + 0.16α). (2.47)

The quadrupole moment Q0 is known as the intrinsic quadrupole moment and it would only beobserved in a frame of reference in which the nucleus were at rest.

The quantities that describes the nuclear rotation are: energy, angular momentum. rotationalfrequency and moment of inertia. The kinetic energy of a rotating object is 1

2Iω2, where I is the

moment of inertia. In terms of the angular momentum l = Iω, the energy is l2/2I. Taking thequantum mechanical value of l2, and letting I represent the angular momentum quantum number,gives

E =~2

2II(I + 1). (2.48)

For energies of a rotating object in quantum mechanics, increasing the quantum number I co-rresponds to adding rotational energy to the nucleus, and the nuclear exited states form a sequenceknown as a rotational band. Exited states in molecules also form rotational bands, corresponding torotations of the molecule about its center of mass [7].

Chapter 3

Experimental methods

In this Chapter we revise several aspects related to the experimental setup used to study thenuclear reaction 6Li+154Sm at 7 MeV/u. In Section 3.1 we explain the Spectrometer Saci-Pererê,which is placed at the Instituto de Física da Universidade de São Paulo (IFUSP). Afterwards, inSection 3.2 we describe the functioning of the Pelletron accelerator used to produce the beam of6Li at 7 MeV/u. Finally, Section 3.3 is devoted to measurement system. The measurement systemconsists of the detectors responsible of the particle detection (scintillators) called SACI, and in thedetectors responsible of the Gamma-rays detection (Hyper Pure Germanium detector or HPGe)called PERERÊ.

3.1. Spectrometer Saci-Pererê

The spectrometer Saci-Pererê [15, 29] consists of four Germanium detectors for the detectionof Gamma rays. Two of them with 60% of efficiency and two of 20% efficiency, coupled to BGOCompton suppressors [30]. In addition to that, it is formed by eleven plastic scintillators for detectionof particles. The scintillators are of the type Phoswich distributed around the target, forming aregular dodecahedron. This covers a solid angle of about 76% of 4π.

Only three Gamma-ray detectors and nine particle detectors were used for this experiment. Table3.1 presents the location of the Germanium detectors in relations with the beam line. The Table3.2 presents the same information for the ∆E − E detectors with the solid angles defined by thecollimating hole. The experiment made use of a beam of 6Li at an energy of 7 MeV/u. The beamwas accelerated using the Pelletron Accelerator at University of São Paulo. The target was a thinself-supported foil of 154Sm with a thickness of 400 mg/cm2. The experimental setup is shown inthe Figures 3.1 (side view) and 3.2 (top view).

The Saci-Pererê spectrometer is designed to study nuclear reaction processes such as inelasticscattering, transfer, break-up and fusion (complete and incomplete). The good energy resolutionof Gamma-ray detectors allows the identification populated excited states in any nuclear reaction.However, Gamma-ray detectors have low efficiency. The scintillators compensate the low efficiencyof the Gamma-ray detectors. The advantage of using this particular spectrometer consists of itssystem of scintillators composed by eleven plastic phoswich scintillators allow measurements ateleven different scattering angles at the same time. The main objective of this work is to implement

3.1 Spectrometer Saci-Pererê 22

Figure 3.1. Scheme of the detection system (Side view). The inclined plane of the Gamma-raydetectors is shown in relation to the horizontal plane. The detectors responsible of the particledetection are called SACI, and the detectors responsible of the Gamma-rays detection are calledPERERÊ. Figure taken from Ref. [15].

Figure 3.2. Scheme of the detection system (Top view). The angles of the Gamma ray detectors areshowing relation to the incident beam. The detectors responsible of the particle detection are calledSACI, and the detectors responsible of the Gamma-rays detection are called PERERÊ. Figuretaken from Ref. [15].

3.2 Pelletron accelerator 23

a technique that allows us to identify the reaction products as well as to measure different scatteringangular distributions using the Gamma-spectrometer Saci-Pererê.

Table 3.1. Polar (θ) and azimuthal angle (φ) of each γ-ray detector.

Detector θ(o) φ(o)G1 37 180G2 101 0G3 101 35.3G4 37 20.9

Table 3.2. Polar (θ) and azimuthal angle (φ) of each collimating aperture and its solid angle (∆Θin msr).

Detector θ(o) φ(o) ∆Θ (msr)C1 30 0 0.64C2 43 108 2.00C3 53 36 2.72C4 65 324 4.71C5 71 252 6.78C6 80 180 8.67C7 99 8 8.63C8 107 297 14.96C9 118 234 16.68C10 127 164 24.69C11 135 94 37.84

3.2. Pelletron accelerator

The Pelletron accelerator is a particle accelerator which uses electromagnetic fields to acceleratecharged particles. The Pelletron 8-UD accelerator is a type of electrostatic particle acceleratorsimilar to a Van de Graaff generator, whose terminal voltage is 8 MV [31].

The initial point is an ion source. The 6Li beam is obtained from the ion source calledMulticathodeSource of Negative Ions by Cesium Sputtering (MC-SNICS) from NEC1 installed at the eighth floorof the Instituto de Física da Universidade de São Paulo (building called LAFN). The ion sourceMC-SNICS is composed of a reservoir of Cesium and of the material or chemical element of whichyou want to extract the beam.

When a pre-acceleration occurs, the beam traverses a electronic analyzer (ME-20) at an angle of90o, changing its course from horizontal to vertical. We have a beam of charged particles with chargeZe, mass m and energy E, under the action of a magnetic field of intensity B. The equation that

1National Electrostatics Corporation, Middleton, Wisconsin, EUA

3.2 Pelletron accelerator 24

Figure 3.3. Schematic representation of the path made by the Pelletron accelerator from the ionsource to the detection system . Figure taken from Ref. [29].

describes the circular path of radius r of a charged particle applied in the perpendicular directionto the plane of motion, it is given by

mv2

r= (Ze)vB → mE

Z2=

1

2(Bre)2. (3.1)

Therefore, the ME-20 selects the ions of interest according to their mass, cleaning the beamfrom contaminants. The electronic ME-20 is able to deflect the beam to an upper limit such thatthe product mE/Z2 is equal to 20 MeV times the unit of atomic mass. The Figure 3.3 showsschematically the trajectory of the beam from the production in ion source MC-SNICS to thedetection system.

3.3 Detection system 25

During the whole trajectory, the beam passes through electromagnetic lens such as: electrostaticquadrupole triplet, quadrupole electrostatic double, magnetic triplet, magnetic double and switchingmagnet. The electromagnetic lens is a device that assists in the transport of charged particles.Analogous to what happens with optical lens that assists in the transport of light in an opticalinstrument. Systems of electromagnetic lenses can be designed in the same way as optical lenses,so electromagnetic lenses easily magnify or converge the beam trajectories. An electromagnetic lenscan also be used to focus an ion beam.

After the ME-20, the negative beam is injected to the accelerator. The Pelletron 8UD is anaccelerator of type Tamdem with maximum voltage at the terminal of 8 MV. The manufacture isfrom NEC, the machine has been installed in LAFN since 1972 [29]. The negative beam is acceleratedtoward the charge terminal of the accelerator tube, which has a positive voltage (Vt), whose valueis controlled as a function of the energy to be supplied to the ions of interest.

The accelerator tube is filled with the SF6 gas, at a pressure sufficient to prevent sparking andsubsequent loss of the electrical charge produced at the terminal. At the center of the accelerationtube, the beam passes through a thin sheet of carbon (thickness 5µg/cm2), where an exchange ofcharge occurs (stripping). The beam, now positive, may have varying charge states (+qe), dependingon the amount of electron the beam loses as it traverses the carbon sheet. After passage throughthe carbon sheet, the ions are again accelerated towards the exit of the accelerator tube by thesame potential of the terminal Vt, so that the total energy with which the beam emerges from theaccelerator tube is given by

ET = eVi + (q + 1)eVt, (3.2)

where Vi is the beam extraction potential plus the potential applied to the pre-accelerator tube(typically Vi=105 kV).

The next step in transporting the beam towards the target includes a new selection. This selectionis performed through a second selector electrode called ME-200. The deflection limit for the mass-energy product is now 200 MeV times the atomic mass unit. A selected charge state of the beamis selected adjusting the mechanical field of the ME-200. The ME-200, deflecting the beam at 90o,directs it again to the horizontal. Finally, the beam arrives to detection system.

3.3. Detection system

Gamma-ray Germanium detectors

Germanium and silicon are crystalline semiconductor materials, which are the base of semiconductordetectors [30]. These devices provided the first high-resolution detectors for energy measurementand were quickly adopted in nuclear physics research for charged particle detection and Gammaspectroscopy.

Semiconductors are crystalline materials whose outer shell atomic levels exhibit an energy bandstructure. Figure 3.4 schematically illustrates the basic structure of a valence band, a “forbidden”energy gap and a conduction band.


Figure 3.4. Scheme of the structure in bands of semiconductors.

The energy bands are actually regions of many discrete levels which are so closely spaced thatthey may be considered as a continuum, while the “forbidden” energy gap is a region in which thereare no available energy levels at all.

The np semiconductor junction and depletion depth

The functioning of all semiconductor detectors depends on the formation of a semiconductor jun-ction. The semiconductor junction is called the np junction it is formed by the juxtaposition of ap-type semiconductor with an n-type material.

The formation of a np-junction creates a special zone which is the interface between the twomaterials. This is illustrated in Figure 3.5. Because of the difference in the concentration of electronsand holes between the two materials, there is an initial diffusion of holes towards the n-region anda similar diffusion of electrons towards the p region. As a consequence, the diffusing electrons fillup holes in the p-region while the diffusing holes capture electrons on the n-side. Since the p-regionis injected with extra electrons it thus becomes negative while the n-region becomes positive. Thiscreates an electric field gradient across the junction.

The charge density and the corresponding electric field are schematically in Figure 3.5. Dueto electric field there is a potential difference across the junction. This is known as the contactpotential. The contact potential generally is on the order of 1V.

The region of changing potential is known as the depletion zone or space charge region and hasthe special property of being empty of all mobile charge carriers.

An electron created in this zone is detected by the electric field. This characteristic of the de-pletion zone is particularly attractive for radiation detection. When ionizing radiation entering thiszone an electron-hole pair is liberated. After the electron-hole pair is detected by the electric field.Finally, a current signal proportional to the ionization it is detected.


Figure 3.5. Schematic diagram of a np junction, emphasizing the number of positive and negativecharge carriers on each side. Figure from Ref. [15].

Plastic scintillators

Scintillators are materials (solids, liquids, gases) that produce sparks or scintillations of light whenionizing radiation passes through them. In other words, a scintillator is a material capable of con-verting part of the kinetic energy of an incident charged particle into light [32, 31].

The amount of light produced in the scintillator is very small. The light must be amplifiedbefore it can be recorded as a pulse. The amplification or multiplication of the light of scintillatoris achieved with a device known as the photo-multiplier tube (or photo-tube). As its name denotes,its function is to accept a small amount of light to amplify it many times to deliver a strong pulseto an output signal. Amplifications are of order of the 106.

The operation of a scintillation detector may be divided into broad steps:

1. Absorption of incident radiation energy by the scintillator and production of photons in thevisible part of the electromagnetic spectrum.

2. Amplification of the light by the photo-multiplier tube and the production of the output pulse.

A photo-multiplier consists of an vacuum glass tube with a photo-cathode at its entrance andseveral dynodes in the interior. The anode, located at the end of a series of dynodes serves as thecollector of electrons. The photons produced in the scintillator enter the photo-tube and they hit thephoto-cathode. Then they are guided with the help of an electric field, toward the first dynode. Thisdynode is covered with a substance that emits secondary electrons if electrons collide with it. Thesecondary electrons from the first dynodes move toward the second, then toward the third, and so on.


Phoswich detector

A phoswich detector is a combination of two scintillators, a thin scintillator and a thick scintillator,close to a photo-multiplier tube (PMT). The thick scintillator has a fast-response time, while the thinscintillator has a long-response time (see Figure 3.6). This arrangement also known as a ∆E − Earrangement, where ∆E is the loss energy and E is the residual energy. This allows discerningbetween different types of charged particles and their respective energy.

Figure 3.6. Scintillator ∆E−E. They are used for detection and identification of charged particles.

The first fast scintillator has thickness of 0.1 mm and a time of 2.4 ns for the acquisition ofthe pulse ∆E. The slow scintillator has a thickness of 10 mm and a decay time of 264 ns for theacquisition of the pulse E. These coupled detector produce an analog pulse, what gives rise to anenergy value deposited in each scintillator (∆E − E) can be measured.

From the combination ∆E−E values it is possible to distinguish a detected particle. The energyloss of charged particles in matter can be described by the reduced Bethe-Block equation [33]

dE

dx∝ mZ2

E, (3.3)

where m is the mass of the incoming particle, Z2 its proton number, E is its energy and dE

its energy loss over the distance dx. This kind of arrangement produces a typical two-dimensionalspectrum E-∆E (as the spectrum of the Figure 4.8). Different particles appear in different regions ofthe histogram due to their mass, charge and energy. The detection of charged particles is particularlyuseful in Gamma spectroscopy techniques. The formed regions in this spectra are the residual nucleiproceeding from a specific nuclear reaction.

Chapter 4

Data analysis and results

In this Chapter we analyze the experimental data coming from the γ-particle coincidence experimentthat uses a 6Li projectile colliding on a154Sm target at 7 MeV/u. This analysis is divided in differentsteps: i) To generate particle and Gamma energy spectra. ii) To calibrate the resulting spectra inenergy. iii) To substract the background. iv) To generate the particle-γ coincidence matrices. v) Tointerpret the information contained in the matrices to analyze the influence of the large probability ofthe break up of 6Li (weak bound nucleus) on the deformed nucleus 154Sm. This Chapter is organizedas follows. In Section 4.1 we explain how the experimental data is stored and read. Section 4.2 showshow the spectra is constructed. In section 4.3 we analyze the coincidence matrices to identify thedifferent reaction products.

4.1. Data format for the nuclear reaction 6Li+154Sm

The experiment is performed using the Saci-Pererê γ-ray spectrometer and the Pelletron accele-rator located at the Instituto de Física of the Universidad de São Paulo, Brazil. The data that comesfrom the Saci-Pererê acquisition system is called raw data. All the information from the experimentis saved in files called run, and they are read in a data format called event-by-event. In total thereare 31 runs. Each run is conformed by blocks and in turn each block is conformed by words. Eachblock weights 32 kbytes and it is sub-divided in words of 16 bits.

Table 4.1. Relation between the hexadecimals and the decimal numbers.

Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Hexadecimal 0 1 2 3 4 5 6 7 8 9 a b c d e f

A certain set of words registered in a time interval of 400 ns is called event. Each word is written inan hexadecimal (hex) format. This means that each word has four letters, or better, four hex-digits(see Table 4.1). In a typical experiment several hundreds of Gbytes are usually produced. Theycorrespond to a few billion of events. As the electronics cannot register all the Gamma and particleevents in coincidence, only a small fraction of them will be processed during the data analysis.

4.1 Data format for the nuclear reaction 6Li+154Sm 30

Those are called good events.In Figure 4.1 several events in the Saci-Pererê data format are shown, each word corresponds

to a set of four hexadecimal characters (006d for example). Enclosed in a box is the word ffffwhich is used to separate events. Each registered event contains the time and energy of the γ-ray, incoincidence with the energy lost, ∆E, and the residual energy, E, of the detected ejectile. The dataformat defines different words to identify which of the available Germanium (particle detectors)registers a nuclear transition (kinetic energy of the ejectiles) and in which time. Those words arewhat we called identifiers (ID). The IDs of the experiment are shown in Tables 4.2 and 4.3.

8000 0000 8001 0067 8003 021b 8007 02648017 0234 00e5 801d 0053 000f 005b 0018ffff 8001 0079 006d 8009 0133 8005 07d908b4 8017 0593 00c4 801d 0053 0013 005c0018 ffff 8001 012a 8005 0de8 800f 025a0665 801d 0052 001e 005b 0026 ffff 80010183 8005 0464 8011 02e0 0099 801d 00530020 005a 002b ffff 8002 06ba 8006 08a0800d 006b 005a 801d 0052 001f 005b 002bffff 8001 0185 0829 8005 0a89 8013 012a03b8 801d 0053 001d 005b 0026 ffff 8001

Figure 4.1. Example of several events written in the Saci-Pererê data format coming from the6Li+154Sm Gamma-particle coincidence experiment.

Each registered event contains the time and energy of the γ-ray, in coincidence with the energylost, ∆E, and the residual energy, E, of the detected ejectile. The data format defines differentwords to identify which of the available Germanium (particle detectors) registers a nuclear transition(kinetic energy of the ejectiles) and in which time. Such as words are what we called identifiers (ID).Those IDs are shown in Tables 4.2 and 4.3.

Table 4.2. Identifiers of energy (ID-Eγ) and time (ID-tγ) for the three γ detectors in the Saci-Pererê data format.

Germanium detector 1 2 3ID-Eγ 8001 8002 8003ID-tγ 8005 8006 8007

In Table 4.4, we show the order in which an event should be stored according with the Saci-Pererê data format. In this table EγGi indicates when a signal is registered in the ith γ-detector,tγGi is the time in which this signal is registered. The ∆Escii-Escii indicates when a particle isdetected.

Most of the data obtained from the experiment studied in this work do not follow the standard

4.1 Data format for the nuclear reaction 6Li+154Sm 31

Table 4.3. Identifiers of the energy lost (ID-∆E) and residual energy (ID-E) of the 11 particledetectors in the Saci-Pererê data format.

Scintillator 1 2 3 4 5 6 7 8 9 10 11ID-∆E 800D 800E 800F 8010 8011 8012 8013 8014 8015 8016 8017ID-E 8018 8019 801A 801B 801C 801D 801E 801F 8020 8021 8022

Table 4.4. Standard Saci-Pererê data format.

ffff ID-Eγ EγG1 EγG2 EγG3 ID-tγ tγG1 tγG2 tγG3 ID-∆E or ID-E ∆Esci1-Esci1... ∆Esci11-Esci11

Table 4.5. Examples of several types of events in the Saci-Pererê data format. All of them aregood events.

Example 1 Example 2 Example 3ffff ffff ffff ffffID-Eγ 8001 8002 8003EγG1 0079EγG2 006d 06baEγG3 0186ID-tγ 8005 8006 8007tγG1 07d9tγG2 08b4 08a0tγG3 0a89

ID-∆E or ID-E 8017 800d 8013∆Esci 0593 006b 012aEsci 00c4 005a 03b8

ID-∆E or ID-E 801d 801d∆Esci 0053 0052Esci 0013 001f∆Esci 005c 005bEsci 0018 002b

structure shown in Table 4.4. For this reason we have to classify the events into good and bad. Forgood events there is a one by one correspondence between the γ-ray and the time signals. Otherwise,they are classified as bad events. In Tables 4.5 and 4.6 we show some examples of good and badevents.

The Gamma-particle coincidence is a technique which is very complex to implement in thelaboratory. The experimental setbacks did not let most of the data be registered. Therefore few

4.2 Reduction of data and analysis 32

Table 4.6. Examples of several types of events in the Saci-Pererê data format. All of them arebad events. The identifier 8009 is neglected, because of this identifier does not belong to time andenergy.

Example 1 Example 2 Example 3ffff ffff ffff ffffID-Eγ 8001 8002 8003EγG1 0079EγG2 006d 06baEγG3 0186ID-tγ 8005 8009tγG1 08a0tγG2

tγG3

ID-∆E or ID-E 8017 800d∆Esci 0593 006bEsci 00c4 005a

ID-∆E or ID-E 801d 801d∆Esci 0053 0052Esci 0013 001f∆Esci 005c 005bEsci 0018 002b

Table 4.7. Structure of the coincidence matrix. The subindex i and j indicates the Gamma andparticle detectors, respectively.

Counts Eγi tγi ∆Ej Ej

coincidence events were stored. Aiming at increasing the events in coincidence for the analysis asoftware to read the data was written by us. The code that reads the raw data (with all variationsin event types) is shown in Appendix B.

4.2. Reduction of data and analysis

The γ-particle coincidence experimental technique aims at establishing a proper method to analyzethe data and to identify the reaction channels involved [27, 28, 34, 35]. The technique makes use ofa time window to register the signal of arrival of a particle and the arrival of a γ-ray. In this way, toestablish the origin of the nuclear transition. If both signals are produced in the same time window,around 400 ns, a coincidence can be established. A coincidence is an event and it is originated fromthe same process. A illustration of the γ-particle coincidence experimental technique is shown inFigure 4.2.


Figure 4.2. Illustration of the γ-particle coincidence experimental technique. The nuclear reactionis A + B → C + D∗ where A is the beam, B is the target, C is the charged particle, and D is aexited nucleus which decays emitting γ-rays. The technique makes use of a time window to registerthe signal of arrival of a particle and the arrival of a γ-ray.

♠♥♠ ♠♥♠

r♦♦ts

♠♥♠ ♠♥♠

❱

r♦♦ts

Figure 4.3. Total γ-ray spectrum obtained after reading all the experimental data for the Germaniumdetectors 1 and 2. These spectra are with no coincidences. Left: this spectrum is not calibrated.Right: this spectra is calibrated, the calibration is done with 60Co and 133 Ba.

We obtain two types of spectra of the experiment:

1. A single or with no coincidence spectrum is a spectrum that contains all the events detectedby the Germanium detectors. They are not related to the particle detectors, i.e. they are notrelated to a specific region of the spectrum ∆E−E. The use of these spectra allows the initialidentification of the reaction products.

2. A gated or with coincidence spectrum contains all the events detected by the Germaniumdetectors or by just one Germanium detector. They are related to the particle detectors, i.e.they are related to a specific region of the ∆E−E spectrum and under a time condition. Theuse of these spectra allows a more detailed identification of the reaction products.


The with no coincidence spectra (type 1) are shown in this Section. The with coincidence spectra(type 2) are shown in Section 4.3. In order to obtain the with no coincidence spectra (see Figure4.3), the first step is to do the data reduction. The data reduction consists of reading the raw datato generate the spectra after the energy calibration. In this experiment the calibration is performedby using the γ-ray transition energies of 60Co (Eγ =1173.2 keV and Eγ =1332.5 keV) and 133Ba(Eγ =80.9 keV, Eγ =276.3 keV, Eγ =302.8 keV, Eγ =356.0 keV, Eγ =383.8 keV). The calibrationis linear with the sources mentioned above.

♠♥♠ ♠♥♠

rtrr② ts

♦♦

♠♥♠ ♠♥♠

rtrr② ts

♦♦

Figure 4.4. Time spectra for the two Germanium detectors 1 and 2. Left: The spectrum is notcalibrated. Right: The spectrum is calibrated.

Figure 4.5. Schematic representation of the time signal recorded in the Germanium detector.

The total γ-ray spectra for the Germanium detector two is shown in Figure 4.3. The spectrapresent well defined peaks of the order of nuclear transitions. The big challenge for this work is theclear identification of the isotopes that are responsible for the creation of such peaks. In the spectrait is expected to observe γ rays coming from several excited isotopes. The excited isotopes are theproducts of the reaction under study. The identified products are: 156Eu and 181Ta.

As we mentioned in the Section 4.1 an event is considered good when the event has energyinformation and time information. If the event does not have time information, the event is bad.


♠♥♠

❱

r♦♦ts

♠♥♠

❱

r♦♦ts

Figure 4.6. Total γ-ray spectra of the Germanium detector 2. This is a spectrum with no coincidence,calibrated and with background. Left: from 35 to 430 keV. Right: from 430 to 900 keV.

♠♥♠

❱

r♦♦ts

♠♥♠

❱

r♦♦ts

Figure 4.7. Total γ-ray spectra of the Germanium detector 2. This is a spectrum with no coincidence,calibrated and without background. Left: from 35 to 430 keV. Right: from 430 to 900 keV.

Since the 6Li beam is a beam pulsed and not continuous we need to know when the beam isinteracting with the target. This information is given by the time signal of each Germanium detector.A schematic representation of the time signal recorded in the Germanium detector can be seen inFigure 4.5. In Figure 4.4 shows the time spectra for the two Germanium detectors. Initially, thespectra are not calibrated.

The time signal (tγ) is the time between a 6Li beam pulse and the Germanium energy detectionsignal. The time spectra (see Figure 4.4) of the Germanium detectors are used to guarantee a timeinterval between a detection of particles and Gamma rays. The time interval serves for the elimina-tion of noise signals from electronic components or any other source of noise that can eventually beinterpreted as a good event through the assembled electronics.

The spectra presented in Figure 4.6 are a detailed view of the total γ-ray for the Germaniumdetector 2. The distinctive 511 keV transition coming from pair production is notoriously prominentin the spectrum at the right hand side. The spectra presented in Figure 4.7 are a detailed view of the


Figure 4.8. Regions selected in the time spectrum to subtract background in all spectra. Thisprocedure is done in coincidence: time-energy.

♠♥♠

❱

r♦♦ts

♠♥♠

❱

r♦♦ts

Figure 4.9. Backgrounds of the total γ-ray spectrum. Left: background of the total γ-ray spectrumin coincidence with region A of the time spectrum (see Figure 4.8). Right: background of the totalγ-ray spectrum in coincidence with region B of the time spectrum (see Figure 4.8).

total γ-ray for the Germanium detector 2 as Figure 4.6 but without background. Note the decreaseof counts at the most peaks.

Two different regions of background are shown in Figure 4.9. They are used to improve theidentification of the products. The events that are in coincidence with the background regions aresuppressed. The net effect when no background subtraction is applied can be observed in Figure 4.7.Note that the energies that correspond to transitions between states of 181Ta identified in Figure4.9 are still present in the spectra of Figure 4.7. Therefore, this shows a strong evidence that thebeam collide mostly with 181Ta, that is the frame that holds the 154Sm target, than with the targetitself (see Figure 4.10).

4.3 Products of the nuclear reaction 6Li+154Sm 37

Figure 4.10. Schematic representation of the effective area of interaction (yellow circle) between the6Li beam and the 154Sm target. The frame of the target is of 181Ta.

4.3. Products of the nuclear reaction 6Li+154Sm

The big challenge of this work is the clear identification of the isotopes created in the reaction6Li+154Sm at 7 MeV/u. As we mentioned before, the reduction of data consists of two stages. Thefirst, is the identification of nuclear transitions in the total spectrum without coincidences (Section4.2). The second is to perform a time correlation between the γ-rays and charged particles, whichis the main theme of this Section.

Figure 4.11. Two-dimensional spectrum for ∆E − E particle detectors located at θ = 30o, φ = 0o

(scintillator 1).

The selected regions on the spectrum of Figure 4.11 are attributed to different particles. 6Liparticles transfer more energy to the fast scintillator (∆E) than α particles. Thus, 6Li region appearsabove the α one, and α region appears above the proton one. The same happens with neutrons andγ. The X region indicates charged particles with energy less than the necessary to cross the ∆Edetector, mostly α particles and protons in the case of these spectra.

The probability that a particle and a γ ray is detected in the direction (θγ , φγ), into the time win-dow that guarantees a coincidence between the two events, is determined by the directional emissionprobability (EDP) W (E, θp, θγ , φγp) where E is the energy of the beam. Then the differential cross


section approximately is given by

d2σ

dΩpdΩγ≈ dσ

dΩp(E, θp)W (E, θp, θγ , φγp), (4.1)

where dσ/dΩp(E, θp) is the diferencial cross-section as a function of the particle scattering angle,φγp = φγ−φp is the polar angle difference between the γ-ray and the particle direction of detection.A Illustration of the procedure carried out for the identification of reaction products is shown inFigure 4.12.

Figure 4.12. Illustration of the procedure carried out for the identification of reaction products. Aregion ∆E − E spectrum is selected (Figure a), subsequently a time window (Figure b) is selectedin order to generate the γ-ray spectra (Figure c). In this case the selected region corresponds to 6Lidetected in the scintillator 2 (located at θ = 43o, φ = 108o). The peak at 81 keV corresponds to thefirst excited state of 154Sm. The level scheme of 154Sm is shown in Figure d.

♠♥♠

❱

r♦♦ts

Figure 4.13. Coincidence between particles (scintillator 1) and γ-ray (Germanium 2) in the regionZ = 3. Left: selection of events in Z = 3 of the ∆E−E spectrum . Right: γ-ray spectrum obtainedin coincidence with the Z = 3 particles.


♠♥♠

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Figure 4.14. Gate between particles and γ-ray of the scintillator 1, region 2. Left: selection of eventsin Z = 2 of the two-dimensional spectrum ∆E −E. Right: γ-ray spectrum obtained in coincidencewith the particles of the region 2.

♠♥♠

❱

r♦♦ts


The Figure 4.13 presents a ∆E − E spectrum for the region 1 of the first particle detector, andits corresponding γ ray spectra in coincidence with such region. Peaks are very well defined and itis expected to identify some of the products of the reaction. Figures 4.14, 4.15 and 4.16 presents asimilar situation with different region. One important result is that we obtain different γ-ray spectrafor different for different particle regions which is a good signal that the coincidence are working.

Gamma-ray spectrum in Figure 4.13 shows clearly the transitions coming from the first excitedstates of 181Ta (135, 164 and 301 keV). On the other, hand transitions comming from first the excitedstates of 154Sm are not clearly identified. The first excited state of the 154Sm is only found in theparticle detector number 2 located at θ = 43o, φ = 108o (see Appendix A, Figure 6.3). This confirmthat the beam collides mostly with the 181Ta and not with 154Sm. Adittionally in the Gamma-rayspectra of Figures 4.14, 4.15, 4.16 the 181Ta peaks at energies of 164 and 301 keV disappear. Thelater shows that the coincidence is working properly.


♠♥♠

❱

r♦♦ts


The posible channels reaction that occurred during the experiment, as a result of the presentanalysis, are presented in Figure 4.17 (see also Table 4.9). The identified products are 154Sm, productof Coulomb excitation, 156Eu due to the capture of a deuteron, 157Gd product from the alphacapture. Table 4.10 presents all peaks which are in coincidence with different regions, region 1corresponds to Z = 3, region 2 to Z = 2, region 3 to Z = 1, and region 4 to Z = 0. Lines of 181Tafoils are also observed in the spectra (136, 165, 301 keV). The excited states of the 181Ta as productof the nuclear reaction between 6Li+181Ta are shown in Table 4.9.

Level schemes of the nuclei identify: 154Sm, 156Eu, 157Gd and 181Ta are shown in Figures 4.18and 4.19 [36]. The energies enclosed in a color box are the energies identified in the experiment.

Table 4.8. Residual channels of the collision between 6Li+154Sm identified in this work. They areexplained in the text.

Region ∆E-E Residual channels Eγ Transitions(keV)

Z = 3 154Sm 81(1) 2+ → 0+

Z = 2 156Eu 56(1) 3+ → 2+

Z = 2 156Eu 57(1) 4+ → 3+

Z = 1 157Gd 65(1) 72

− → 52

−

Z = 1 157Gd 170(1) 92

− → 52

−


Table 4.9. Residual channels of the collision between 6Li+181Ta identified in this work. They areexplained in the text.

Residual channels Eγ Transitions(keV)

181Ta 135(1) 9/2+ → 7/2+

181Ta 163(1) 11/2+ → 9/2+

181Ta 300(1) 11/2+ → 7/2+

Figure 4.17. Schematic representation of the main reaction channels identified in the reaction6Li+154Sm. The first reaction channel is the inelastic excitation of 154Sm∗. The other reactionchannels are influenced by break up of 6Li. 156Eu is obtained from the capture of a deuteron. 157Gdis obtained from capture of an alpha and subsequent emission of neutron.


Table 4.10. The first column shows all the energies identified in the total γ-ray spectrum (Figure4.7). The other columns show all energies found in coincidence with each region of the ∆E − Espectrum for the scintillators one and two. The energies in blue are unique energies found in eachregion. The energies in red were found after doing background subtraction.

Energy Region 1 Region 1 Region 2 Region 2 Region 3 Region 3 Region 4 Region 4Z = 3 Z = 3 Z = 2 Z = 2 Z = 1 Z = 1 Z = 0 Z = 0

∆E-Esci1 ∆E-Esci2 ∆E-Esci1 ∆E-Esci2 ∆E-Esci1 ∆E-Esci2 ∆E-Esci1 ∆E-Esci2(keV) (keV) (keV) (keV) (keV) (keV) (keV) (keV) (keV)57(1) 56(1) 57(1) 56(1) 57(1) 57(1) 56(1)65(1) 65(1) 65(1) 65(1) 65(1) 65(1) 65(1)74(1)

81(1)127(1)135(1) 135(1) 135(1) 134(1) 134(1) 136(1) 135(1)163(1) 165(1) 164(1)171(1) 171(1) 172(1) 170(1) 170(1)180(1) 179(1) 179(1)228(1) 228(1) 228(1)236(1)263(1) 264(1) 263(1) 263(1) 263(1) 263(1)270(1)291(1)297(1)300(1) 300(1) 300(1)349(1) 350(1) 349(1) 349(1) 350(1) 349(1)364(1) 364(1) 364(1)389(1) 389(1) 390(1) 388(1) 389(1) 389(1)410(1) 408(1) 408(1)

417(1)463(1)473(1)500(1) 500(1) 500(1)510(1) 510(1) 510(1) 510(1) 510(1) 510(1) 510(1)537(1) 537(1) 537(1)596(1)602(1) 602(1) 602(1) 603(1) 603(1) 602(1)671(1) 670(1) 670(1)717(1) 717(1) 717(1)

701(1)742(1)813(1)845(1) 841(1) 845(1) 843(1)885(1) 886(1)

960(1)


Figure 4.18. Level schemes of the nuclei identify: 154Sm and 156Eu. Figures are taken from Ref. [36].The energies enclosed in a color box are the energies measured in the experiment.

Figure 4.19. Level schemes of the nuclei identify: 157Gd and 181Ta. Figures are taken from Ref. [36].The energies enclosed in a color box are the energies measured in the experiment.

Chapter 5

Conclusions and perspectives

The Gamma-particle coincidence experimental technique is employed with the aim of measuringthe particle angular distribution of the excited states of the 154Sm to contrast the experimental re-sults with the theoretical predictions, i. e. with CDCC inelastic angular distribution for the differentexcited states of the 154Sm. However, the experimental results do not provide enough statistics tomeasure the angular-particle distribution and to compare with the theoretical predictions.

The big challenge of this work is the clear identification of the isotopes created in the reaction6Li+154Sm at 7 MeV/u. The possible channels reaction that occurred during the experiment, as aresult of the present analysis are 154Sm, product of Coulomb excitation, 156Eu due to the captureof a deuteron (incomplete fusion process) and 157Gd product from the alpha capture (incompletefusion process). The inelastic scattering of 154Sm is observed in coincidence with Z = 3 nuclei in a∆E Vs. E spectrum. The Gamma-ray spectrum shows only the first exited state of the 154Sm. Theothers identified products are influenced by the easy breaking of 6Li. The first product identifiedis 156Eu influenced by the breaking up of 6Li . The second identified product is 157Gd influencedby breaking up of 6Li . The last result is confirmed by the coincidence of gamma-rays with Z = 2

nuclei.The data obtained could serve as reference to study again the inelastic scattering of 154Sm. Given

that it was not possible to contrast the experimental results with the theoretical predictions, it isproposed to perform again the experiment with a more bound beam. For instance, the 18O is astrong bound nucleus which could replace 6Li. Therefore, we propose the nuclear reaction between18O+154Sm to study the inelastic angular distribution for the different excited states of the 154Sm.

The Gamma-ray spectra show clearly the transitions coming from the first excited states of 181Ta(135, 164 and 301 keV). This confirms that the beam collides mostly with the 181Ta (frame) andnot with 154Sm (target). The data analysis of the experiments should continue in order to performan experimental characterization of the reaction products and to explore other effects.

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Appendix A

Appendix: Two-dimensional spectrum∆E − E and coincidence γ spectra forscintillator 2

This appendix shows the two-dimensional spectrum ∆E − E (Figure 1.1), the selected regions onthe spectrum are attributed to different particles. Subsequently, the γ-ray spectra in coincidencewith each region of the two-dimensional spectrum ∆E − E are shown.

Figure 1.1. Two-dimensional spectrum ∆E −E for particle detectors scintillator 2. In the directionθ = 43o, φ = 108o. The selected regions on the spectrum are attributed to different particles. 6Liparticles transfer more energy to the fast scintillator (∆E) than α particles. Thus, 6Li region appearsabove the α one, and α region appears above the proton one. The same happens with neutrons andγ. The X region indicates charged particles with energy less than the necessary to cross ∆E, mostlyα particles and protons in the case of this spectra.

49

(a)

t ♥ t♠ t♦t r♦♥t ♥ t♠ t r♦♥

❱

s

(b)

Figure 1.2. Gate between particles and γ-ray of the scintillator 2, region 1. (a) Selection of events inZ = 3 of the two-dimensional spectrum ∆E − E (b) γ-ray spectrum obtained in coincidence withthe particles of the region 1 (with and without background).

♠

♠♥♠

❱

r♦♦ts

(a)

♠♥♠

❱

r♦♦ts

(b)

Figure 1.3. Enlargement of the total γ-ray spectrum (Figure 1.2). The peak 81 correspond to firstexited state of 154Sm. Gate on scintillator 2 and region 1. These spectra are without backgroundand with time condition. (a). From 0 to 1000 keV. (b). From 1000 to 2000 keV.

50

(a)

t ♥ t♠ t♦t r♦♥t ♥ t♠ t r♦♥

❱

s

(b)


♠♥♠

❱

r♦♦ts

(a)

♠♥♠

❱

r♦♦ts

(b)

Figure 1.5. Enlargement of the total γ-ray spectrum (Figure 1.4). These spectra are without back-ground and with time condition. (a). From 0 to 1000 keV. (b). From 1000 to 2000 keV.

51

(a)

t ♥ t♠ t♦t r♦♥t ♥ t♠ t r♦♥

❱

s

(b)


♠♥♠

❱

r♦♦ts

(a)

♠♥♠

❱

r♦♦ts

(b)


52

(a)

t ♥ t♠ t♦t r♦♥t ♥ t♠ t r♦♥

❱

s

(b)


♠♥♠

❱

r♦♦ts

(a)

♠♥♠

❱

r♦♦ts

(b)


Appendix B

Appendix: Subroutine in PRETSCAN toanalyse raw data of the Saci-Pererêspectrometer

The data analysis was done using PRETSCAN which is a preprocessor for TSCAN [37]. This programallows to read raw data from a file creating matrices and repacking events can be done at the sametime. The full code that performs the data analysis is an ordinary routine in C (also can be written inFORTRAN) except that it contains preprocessing header for PRETSCAN. The preprocessing header isused to define the matrix type and size. All the routine is divided in three big interface subroutines:userin, usereof, usersub.

The subroutine userin is used to initialize some parameters and to interact with TSCAN by user-defined commands. The subroutine usereof is called by TSCAN at the end of a file reading. Thesubroutine usersub reads and analyses event by event. This last subroutine was adaptted to readand analyses the type data of the Saci-Pererê spectrometer and is presented below. The currentversion has also been used to sort another types of data, such as the HHIRF L002 format, LBLHERA format, ANL Daphne format, and Gammasphere format [38].

Listing B.1. Subroutine adapted to read and analyse data of the Saci-Pererê spectrometer.1 void usersub ( evlen , i bu f )2 i n t ev l en ;3 I2 ∗ i bu f ;4 5 /∗Dec larac ión de v a r i a b l e s ∗/67 GEDATA ge [MAX_GES] ;8 SIDATA s i [MAX_SI ] ;9 DATA data [ 3 0 ] ;

10 I2 ∗ current ,∗ end ,∗ s t a r t ;11 I2 bad_code ;12 I2 word , ev_length , good_word ;13 I2 ev_label , ev_tag , recordnumber , runnumber ;14 I2 idx , idy , idxx , idyy , idz , ix , iy , ixx , iyy ;

54

15 I2 iz , iw , ix1 , ix2 , idp ;2116 I2 idb1 , idb2 , idb3 , idb4 , idb5 , idb6 , idb7 ;17 I2 ge_count , ge_start , nge ;18 I2 si_count , s i_star t , n s i ;19 I2 ne_count , ne_start , nne , ne_td i f f ;20 I2 nprot , nalph , npart , nneut ,ngamm;21 I2 obuf [ 1 0 24 ] ,∗ ox ;22 I2 good0 , good1 , good2 , good3 ;23 shor t record l ength , header length , tapenumber , r e co rd id ;24 i n t i , j , k ,w, l , n , i i ;25 double y , t , t_di f f , e_detected , e_lab , e_com ;26 i n t l ou t ;272829 /∗ I n i c i a l i z a c i ó n de v a r i a b l e s ∗/30 counter=0;31 ge_count = 0 ;32 si_count = 0 ;3334 /∗Aquí entra e l a rch ivo . f i l y l o l e e por b loques ∗/35 cur rent = ibu f ;36 end = current + num_word ;37 word = ∗ cur rent ;3839 /∗Aquí l e e l a s pa labras de cada bloque ∗/4041 whi le ( cur rent < end ) /∗Empieza e l c i c l o BLOQUE∗/4243 f o r ( i =0; i <3; i++)ge [ i ] . id = ge [ i ] . energy = ge [ i ] . time = ge [ i ] . f l a g = ge [ i ] . tag

= ge [ i ] . enca l = ge [ i ] . t imeca l = 0 ;4445 f o r ( i =0; i <3; i++) s i [ i ] . id = s i [ i ] . denergy = s i [ i ] . energy = s i [ i ] . f l a g = s i [ i ] .

tag = s i [ i ] . time = 0 ;4647 word = ∗ cur rent ;48 whi l e (word!=EVSEP) /∗Cic lo que permite buscar e l primer f f f f ∗/49 ∗ cur rent++;50 word = ∗ cur rent ;51 counter++;52 t_counter++;53 5455 i f (word==EVSEP) /∗ Ing re sa e l primer f f f f y empieza a l e e r e l evento ∗/56 ∗ cur rent++;57 word = ∗ cur rent ;58 counter++;59 t_counter++;60 ev_counter++;61 6263 f o r ( i =0; i <50; i++)64 WordsEvent [ i ]=0x0000 ;

55

65 IdsEvent [ i ]=0x0000 ;66 IdSc i [ i ]=0x0000 ;67 68 ev_length = 0 ;6970 whi le ( ( word!=EVSEP)&&(counter < num_word) ) //Empieza e l c i c l o EVENTO71 WordsEvent [ ev_length ]=word ;72 ∗ cur rent++;73 word = ∗ cur rent ;74 counter++;75 t_counter++;76 ev_length++;77 /∗ F ina l i z a c i c l o EVENTO∗/7879 f o r ( i =0; i<ev_length ; i++)// I d e n t i f i c a l o s IDS de ene rg í a y tiempo80 i f (WordsEvent [ i ]==0x8001 | | WordsEvent [ i ]==0x8002 | |81 WordsEvent [ i ]==0x8003 | | WordsEvent [ i ]==0x8005 | |82 WordsEvent [ i ]==0x8006 | | WordsEvent [ i ]==0x8007 | |83 WordsEvent [ i ]==0x8009 | | WordsEvent [ i ]==0x8000 )84 IdsEvent [ i ]=WordsEvent [ i ] ;85 86 8788 f o r ( i =0; i<ev_length ; i++)89 i f ( IdsEvent [ i ] !=0 x0000 ) 90 91 9293 f o r ( i =0; i<ev_length ; i++)94 i f ( (WordsEvent [ i ] == 0x800D) | | ( WordsEvent [ i ] == 0x800E ) | |95 (WordsEvent [ i ] == 0x800F ) | | ( WordsEvent [ i ] == 0x8010 ) | |96 (WordsEvent [ i ] == 0x8011 ) | | ( WordsEvent [ i ] == 0x8012 ) | |97 (WordsEvent [ i ] == 0x8013 ) | | ( WordsEvent [ i ] == 0x8014 ) | |98 (WordsEvent [ i ] == 0x8015 ) | | ( WordsEvent [ i ] == 0x8016 ) | |99 (WordsEvent [ i ] == 0x8017 ) | | ( WordsEvent [ i ] == 0x8018 ) | |100 (WordsEvent [ i ] == 0x8019 ) | | ( WordsEvent [ i ] == 0x801A) | |101 (WordsEvent [ i ] == 0x801B) | | ( WordsEvent [ i ] == 0x801C) | |102 (WordsEvent [ i ] == 0x801D) | | ( WordsEvent [ i ] == 0x801E ) | |103 (WordsEvent [ i ] == 0x801F ) | | ( WordsEvent [ i ] == 0x8020 ) | |104 (WordsEvent [ i ] == 0x8021 ) | | ( WordsEvent [ i ] == 0x8022 ) )105 IdSc i [ i ]=WordsEvent [ i ] ; 106 107108 /∗Condición para i d e n t i f i c a r eventos buenos y malos∗/109110 good0=good1=good2=good3=0;111 f o r ( i =0; i<ev_length ; i++)112 i f ( IdsEvent [ i ]==0x8001 ) good0=1;good2=1;113 i f ( IdsEvent [ i ]==0x8002 ) good0=1;good2=1;114 i f ( IdsEvent [ i ]==0x8003 ) good0=1;good2=1;115116 i f ( IdsEvent [ i ]==0x8005 ) good1=1;good2=1;

56

117 i f ( IdsEvent [ i ]==0x8006 ) good1=1;good2=1;118 i f ( IdsEvent [ i ]==0x8007 ) good1=1;good2=1;119120 i f ( IdsEvent [ i ]==0x8009 ) good3=1;121 i f ( IdsEvent [ i ]==0x8000 ) good3=1;122 123124 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/125 //k : Recorre e l a r r e g l o WordsEvent [ k ]126 // j : Recorre l o s a r r e g l o s de GEDATA ( ge . energy y ge . time ) .127 // l : Reasigna l o s id de GEDATA128 //w: Recorre l o s a r r e g l o s de SIDATA129 //n : Contador u t i l i z a d o para e l a n á l i s i s de l o s s c i n t i l l a t o r s130 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/131132 l =0,k=0, j =0,w=0,n=0;133 ix=0, i y=0, i z =0, iw=0, ixx=0, iyy=0, ix1=0, ix2=0;134 idx=0, idy=0, idxx=0, idyy=0, idz =0;135136 i f ( good1==1 && good0==1 && good2==1 && good3==0)//Empiezan buenos eventos137 // i f ( good2==1 && good3==0)//Condición para c a l i b r a c i ó n138 good_counter++;139 cgood_event ;140 good_word=WordsEvent [ k ] ;141142 /∗ENERGIA∗/143144 i f ( ( good_word==0x8001 ) | | ( good_word==0x8002 ) | | ( good_word==0x8003 ) ) 145 i f ( good_word==0x8001 ) j =0;146 i f ( good_word==0x8002 ) j =1;147 i f ( good_word==0x8003 ) j =2;148 ge [ j ] . id = (good_word & FIRST) ; //Extrae e l número 1 de 8001149 idx=ge [ j ] . id ;150 151152 k++;good_word=WordsEvent [ k ] ; //Avanzo y guardo153 ww[ k ]=(good_word & LAST) ; //Extrae l a primera l e t r a de l a palabra154155 whi le (ww[ k ] !=0 x8000 && k<ev_length ) 156 ge [ j ] . energy = good_word ;157 ix=ge [ j ] . energy ;158 i f (k<ev_length ) k++;good_word=WordsEvent [ k ] ;ww[ k ]=(good_word & LAST) ; 159 i f (ww[ k ] !=0 x8000 && k<ev_length ) idx=idx+1; ge [ j ] . id=idx ; j++;160 161162 /∗Correcc ión para e l caso 8001−8003−8007∗/163164 i f ( good_word==0x8003 ) 165 j =2;166 ge [ j ] . id = (good_word & FIRST) ;167 idx=ge [ j ] . id ;168 k++;good_word=WordsEvent [ k ] ;

57

169 ge [ j ] . energy = good_word ;170 ix=ge [ j ] . energy ;171 k++;172 good_word=WordsEvent [ k ] ;173 174175 /∗TIEMPO∗/176177 i f ( ( good_word==0x8005 ) | | ( good_word==0x8006 ) | | ( good_word==0x8007 ) ) 178 i f ( good_word==0x8005 ) idy=1; j =0;179 i f ( good_word==0x8006 ) idy=2; j =1;180 i f ( good_word==0x8007 ) idy=3; j =2;181 ge [ j ] . id=idy ; 182183 k++;good_word=WordsEvent [ k ] ;184 ww[ k ]=(good_word & LAST) ;185186 whi le (ww[ k ] !=0 x8000 && k<ev_length ) 187 ge [ j ] . time = good_word ;188 iy=ge [ j ] . time ;189 i f (k<ev_length ) k++;good_word=WordsEvent [ k ] ;ww[ k ]=(good_word & LAST) ; 190 i f (ww[ k ] !=0 x8000 && k<ev_length ) idy=idy+1; ge [ j ] . id=idy ; j++;191 192193 /∗CALIBRACIÓN TIEMPO∗/194195 f o r ( i =0; i <3; i++)196 l=l +1;197 ge [ i ] . id=l ; 198199 f o r ( i =0; i <3; i++)200 y=0.0;201 idyy=ge [ i ] . id ;202 y=(double ) ge [ i ] . time ;203 i f ( i p e r t >= MaxPert ) i p e r t =0;204 y=y∗ getga in [ i ]+ g e t o f f [ i ] ;205 ge [ i ] . t imeca l=(I2 )y ;206 iyy=ge [ i ] . t imeca l ;207 i f ( ge [ i ] . time !=0) inc_geca l t ime ( idyy−1, iyy ) ; 208209 /∗CALIBRACIÓN ENERGÍA∗/210211 f o r ( i =0; i <3; i++)212 y=0.0;213 idxx=ge [ i ] . id ;214 y=(double ) ge [ i ] . energy ;215 i f ( i p e r t >= MaxPert ) i p e r t =0;216 y = (y∗y∗y∗gecub [ i ]+y∗y∗ gesqr [ i ]+y∗ gega in [ i ]+ g e o f f s e t [ i ] ) ;217 ge [ i ] . enca l=(I2 )y ;218 ixx=ge [ i ] . enca l ;219 i f ( ge [ i ] . energy != 0) inc_geca lenergy ( idxx−1, ixx ) ; 220

58

221 i f ( ge [ 1 ] . t imecal >1944 && ge [ 1 ] . t imecal <2068)222 idxx=ge [ 1 ] . id ;223 ixx=ge [ 1 ] . enca l ;224 inc_get ime in te rva l ( idxx−1, ixx ) ; 225226 /∗PARTICULAS∗/227228 i f ( good_word==0x800D) w=0; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 229 i f ( good_word==0x800E ) w=0; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 230 i f ( good_word==0x800F ) w=1; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 231 i f ( good_word==0x8010 ) w=1; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 232 i f ( good_word==0x8011 ) w=2; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 233 i f ( good_word==0x8012 ) w=2; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 234 i f ( good_word==0x8013 ) w=3; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 235 i f ( good_word==0x8014 ) w=3; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 236 i f ( good_word==0x8015 ) w=4; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 237 i f ( good_word==0x8016 ) w=4; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 238 i f ( good_word==0x8017 ) w=5; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 239 i f ( good_word==0x8018 ) w=5; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 240 i f ( good_word==0x8019 ) w=6; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 241 i f ( good_word==0x801A) w=6; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 242 i f ( good_word==0x801B) w=7; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 243 i f ( good_word==0x801C) w=7; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 244 i f ( good_word==0x801D) w=8; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 245 i f ( good_word==0x801E ) w=8; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 246 i f ( good_word==0x801F ) w=9; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 247 i f ( good_word==0x8020 ) w=9; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 248 i f ( good_word==0x8021 ) w=10; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 249 i f ( good_word==0x8022 ) w=10; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 250251 k++;good_word=WordsEvent [ k ] ;252 ww[ k ]=(good_word & LAST) ;253254 whi le (ww[ k ] !=0 x8000 && k<ev_length ) 255 s i [w ] . energy=good_word ;256 i f (ww[ k ] !=0 x8000 && k<ev_length ) k++;good_word=WordsEvent [ k ] ;257 ww[ k ]=(good_word & LAST) ; idz=idz+0x0001 ; 258 s i [w ] . denergy=good_word ;259 i f (ww[ k ] !=0 x8000 && k<ev_length ) k++;good_word=WordsEvent [ k ] ;260 ww[ k ]=(good_word & LAST) ;w++; idz=idz+0x0001 ; 261 262263 i f ( good_word==0x800D) w=0; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 264 i f ( good_word==0x800E ) w=0; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 265 i f ( good_word==0x800F ) w=1; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 266 i f ( good_word==0x8010 ) w=1; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 267 i f ( good_word==0x8011 ) w=2; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 268 i f ( good_word==0x8012 ) w=2; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 269 i f ( good_word==0x8013 ) w=3; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 270 i f ( good_word==0x8014 ) w=3; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 271 i f ( good_word==0x8015 ) w=4; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 272 i f ( good_word==0x8016 ) w=4; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ;

59

273 i f ( good_word==0x8017 ) w=5; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 274 i f ( good_word==0x8018 ) w=5; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 275 i f ( good_word==0x8019 ) w=6; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 276 i f ( good_word==0x801A) w=6; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 277 i f ( good_word==0x801B) w=7; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 278 i f ( good_word==0x801C) w=7; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 279 i f ( good_word==0x801D) w=8; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 280 i f ( good_word==0x801E ) w=8; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 281 i f ( good_word==0x801F ) w=9; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 282 i f ( good_word==0x8020 ) w=9; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 283 i f ( good_word==0x8021 ) w=10; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 284 i f ( good_word==0x8022 ) w=10; s i [w ] . id=(good_word & FIRSTWO) ; idz=s i [w ] . id ; 285286 k++;good_word=WordsEvent [ k ] ;287 ww[ k ]=(good_word & LAST) ;288289 whi le (ww[ k ] !=0 x8000 && k<ev_length ) 290 s i [w ] . energy=good_word ;291 i f (ww[ k ] !=0 x8000 && k<ev_length ) k++;good_word=WordsEvent [ k ] ;292 ww[ k ]=(good_word & LAST) ; idz=idz+0x0001 ; 293 s i [w ] . denergy=good_word ;294 i f (ww[ k ] !=0 x8000 && k<ev_length ) k++;good_word=WordsEvent [ k ] ;295 ww[ k ]=(good_word & LAST) ;w++; idz=idz+0x0001 ; 296 297298 /∗ ANÁLISIS SCINTILLATORS∗/299300 ix1=ge [ 0 ] . enca l ;301 ix2=ge [ 1 ] . enca l ;302303 f o r ( i =0; i <11; i++)304 i f ( ( s i [ i ] . energy !=0) && ( s i [ i ] . denergy !=0) ) 305 i f ( i==0)306 s i [ i ] . energy=4.0∗ s i [ i ] . energy ;307 s i [ i ] . denergy=4.0∗ s i [ i ] . denergy ;308 309310 i z=iw=0;311 i z=s i [ i ] . energy ;312 iw=s i [ i ] . denergy ;313 inc_ede ( i , iw , i z ) ;314315 f o r ( j =0; j <8; j++)316 i f ( bantes t ( j , iw , i z ) ) 317 inc_rede ( j , iw , i z ) ;318319 // s o l o se e s ta u t i l i z a ndo e l germanio 2320 i f ( ge [ 1 ] . t imecal >1944 && ge [ 1 ] . t imecal <2068)inc_ge2A ( j , ix2 ) ; 321 i f ( ge [ 1 ] . t imecal >1882 && ge [ 1 ] . t imecal <1944) inc_ge2B ( j , ix2 ) ; 322 i f ( ge [ 1 ] . t imecal >2068 && ge [ 1 ] . t imecal <2130) inc_ge2C ( j , ix2 ) ; 323324 i f ( ge [ 1 ] . t imecal >1944 && ge [ 1 ] . t imecal <2068)inc_ge2D ( j , ix2 ) ;

60

325 i f ( ge [ 1 ] . t imecal >1882 && ge [ 1 ] . t imecal <1924)dec_ge2D( j , ix2 ) ; 326 i f ( ge [ 1 ] . t imecal >2088 && ge [ 1 ] . t imecal <2130)dec_ge2D( j , ix2 ) ; 327 328 329 330 331332 // F ina l i z a buenos eventos333334 e l s e 335 cbad_event ;336 bad_counter++;337 338339 /∗ F ina l i z a c i c l o BLOQUE∗/340 /∗End o f usersub ∗/

Appendix C

Appendix: Contributions to PhysicalReview C

The investigations performed during my Master resulted in the following publications:

1. Z.E. Guevara, and D.A. Torres, Lifetime and g-factor measurements of excited states usingCoulomb excitation and alpha transfer reactions, AIP Conference Proceedings, 1753, 030006(2016) 10.1063/1.4955347.

2. G. J. Kumbartzki, N. Benczer-Koller, K.-H. Speidel, D. A. Torres, J. M. Allmond, P. Fallon,I. Abramovic, L. A. Bernstein, J. E. Bevins, H. L. Crawford, Z. E. Guevara, G. Gurdal, A. M.Hurst, L. Kirsch, T. A. Laplace, A. Lo, E. F. Matthews, I. Mayers, L. W. Phair, F. Ramirez,S. J. Q. Robinson, Y. Y. Sharon, and A. Wiens, Phys. Rev. C 93, (2016) 044316-1 044316-7.

3. G. J. Kumbartzki, N. Benczer-Koller, K.-H. Speidel, D. A. Torres, J. M. Allmond, P. Fallon,I. Abramovic, L. A. Bernstein, J. E. Bevins, H. L. Crawford, Z. E. Guevara, G. Gurdal, A. M.Hurst, L. Kirsch, T. A. Laplace, A. Lo, E. F. Matthews, I. Mayers, L. W. Phair, F. Ramirez,S. J. Q. Robinson, Y. Y. Sharon, and A. Wiens, Phys. Rev. C 94, (2016) 034303-1 034303-6.

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Lifetime and g-factor measurements of Excited States usingCoulomb Excitation and Alpha Transfer Reactions

Z.E. Guevara1,a) and D.A. Torres1,b)

1Physics Department, Universidad Nacional de Colombia, Bogota D.C., Colombia

a)Corresponding author: [email protected])[email protected]

Abstract. In this contribution the challenges in the use of a setup to simultaneously measure lifetimes and g-factor values willbe presented. The simultaneous use of the transient field technique and the Doppler Shift Attenuation Method, to measure magneticmoments and lifetimes respectively, allows to obtain a complete characterization of the currents of nucleons and the deformation inexcited states close to the ground state. The technique is at the moment limited to Coulomb excitation and alpha-transfer reactions,what opens an interesting perspective to consider this type of experiments with radioactive beams. The use of deep-inelastic andfusion-evaporation reactions will be discussed. An example of a setup that makes use of a beam of 106Cd to study excited states of110Sn and the beam nuclei itself will be presented.

INTRODUCTION

Model independent measurements are the key to contrast fundamental models in physics. In nuclear structure, twoquantities represent one of the best examples of this fact, lifetimes, which provide information of the nuclear de-formation, and magnetic moments, which allow to know the distribution of proton and neutron currents inside thenucleus. For excited states in the range of picoseconds two experimental techniques can be simultaneously utilized,the Doppler Shift Attenuation Method (DSAM) and the Transient Field (TF) technique for lifetimes and magneticmoments respectively.

In this contribution, some comments about the simultaneous use of the DSAM and the TF techniques to obtaininformation of the wave function that governs nuclear excited states will be presented. The extension of the techniqueto use with reactions di↵erent from coulomb excitation and alpha-transfer reactions is one of the challenges that willbe addressed. Some experimental examples will be also presented.

EXPERIMENTAL SETUP

Several articles can be consulted for a detailed description of the DSAM [1, 2] and the TF techniques [3, 4]. Here wewill focus in how these two experimental techniques can be utilized simultaneously. The main motivation is the factthat g-factor and lifetime measurements for a given state are intimately related in the di↵erential formula for thespin precession, , utilized in the TF technique

= g · µN

~·Z tout

tinB [v(t),Z] · et/dt, (1)

where µN is the nuclear magneton and B [v(t),Z] is a function that describes the hyperfine magnetic field responsiblefor the precession of the nuclear spin of the state, g is the nuclear g factor, v(t) is the speed of the ion inside theferromagnetic material, Z is the number of protons of the ion, is the lifetime of the preccesing state, and tin and toutare the in and out time during which the precession of the magnetic moment, and the slowing down of the nuclei, takes

Latin American Symposium on Nuclear Physics and ApplicationsAIP Conf. Proc. 1753, 030006-1–030006-5; doi: 10.1063/1.4955347

Published by AIP Publishing. 978-0-7354-1411-2/$30.00

030006-1 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 168.176.40.64 On: Tue, 02 Aug 2016 14:59:30

63

place inside the ferromagnetic material, which is also used as a stopper material as will be presented below. A simul-taneously measurement of the lifetime and the magnetic moment permits not only more precise results, also allowsto obtain a complete set of quantum numbers during a single experiment, this is pivotal to contrast the experimentalresults with theoretical models.

A very important point in the use of formula (1) is that the measured precession is model independent, this isalso the case for lifetime measurements using the DSAM. The term B [v(t),Z] describes the spin-orbit interaction thatoccurs in a hydrogen-like isotope. The description of this interaction from first principles is one of the challenges toaddress in the forthcoming years.

FIGURE 1. Experimental setup utilized in DSAM and TF experiments. Four HPGe clover detectors are located in a plane at twospecific angles. Figure to the left a nucleus with a long-live state is completely stopped inside the target and emits at rest and noDoppler shift in the -ray is registered. Figure to the right a nucleus with a short lifetime state decays in flight inside the target,a Doppler shift is registered in the detectors according to the positions in the setup. Figure 2 presents an example of such Dopplershift.

In DSAM experiments, the ion beam shown in Figure 1 is focused on a target (green) to populate the states, andsubsequently the ion goes to a second target (blue) that is suciently thick to slow down and even stop the beam. Anadditional layer is located behind the stopper material to stop other beam-like reaction products. For the case of analpha-transfer process the reaction occurs in inverse kinematics, i.e., nuclei in the beam are heavier than target nuclei.To use DSAM the stopping time must be in the same order of the lifetime of the state under study. Two phenomenaoccur simultaneously during the passing time in the stopping material, the precession of the nuclear spin due to theinteraction of the TF with the magnetic moment of the state, and the slowing down of the isotope. The -ray decaymay occur in-motion while this two process happen. Due to the velocity of the de-exciting nucleus, relative to the-ray detectors which located in the lab frame, the observed energy, Eobs

, will be Doppler shifted in relation with thereal energy, E0

, according to the formula

Eobs = E0

p1 (t)2

1 (t) cos , (2)

where (t) = v(t)c is the speed of the excited nucleus in the target, and it can be defined as a function of the time after

the population. It can be estimated using Monte Carlo simulations. Figure 2 shows an example of the Doppler shiftregistered in a typical DSAM experiment.

There are several aspects to consider for the use of the DSAM and the TF technique in the same experiment, herewe will focus on three of those aspects: the reaction mechanism for the population of the states, the target utilizedduring the experiment and the general setup.

Lifetime MeasurementsThe reaction. Several reactions have been utilized in conjunction with the DSAM, being the main consideration thespeed of the nucleus inside the stopper material, a good number is around 6% the speed of light.


64

FIGURE 2. Spectra observed in a DSAM experiment. Figure to the left is registered at forward detectors while Figure to the rightis registered by a detector at backward angles. The Doppler shift depends on the -ray energy, E0

, and the location of the detectorsrelative to the beam line ().

the reaction should be fast enough to allow the population of the states under study and the slowing down ofthe nucleus during the decay process. No special requirement for the spin alignment is needed, but for the case offusion-evaporation reactions the fact that the upper states are populated with a high degree of alignment is used tomeasure the angular distribution of the ray radiation.

The target. The target is the heart of DSAM and TF experiments. A first layer is responsible for the reaction andwhile traversing a second layer the populated states decay and the line shape of the -ray emission can be correlatedwith the lifetime of the state.

Figure 3 presents a example of a mutilayered target where two reactions can be utilized simultaneously. A firstcarbon layer is used to excite via Coulomb-excitation or alpha-transfer reactions, and a second layer of Gadoliniumworks as a stopper. A final layer of Tantalum and Cooper are responsible for the stopping of heavy ions or nuclei withlonger lifetimes. This reaction is very interesting because allow the use of a beam and a target to study two di↵erentnuclear species.

FIGURE 3. An example of a reaction and a multilayer target that can be simultaneously use for Doppler Shift Attenuation andTransient Field experiments. A beam of 106Cd is accelerated and impact a first layer of Carbon. Excited states of 106Cd are popu-lated via Coulomb reactions. When beam energies around and above the Coulomb barrier are utilized, excited states of 110Sn arepopulated.

The Setup. During the last decades DSAM experiments received a boost thanks to the use of fusion-evaporationexperiments in conjunction with the evolution of High Purity Germanium (HPGe) detectors, with Compton suppres-sion, utilized in large experimental setups such as Gammasphere [5] and ancillary detectors such as Microball [6].Figure 1 shows an experimental setup that can be simultaneously utilized in DSAM and TF experiments.

Magnetic Moment MeasurementsThe reaction. Magnetic moment measurements are more restrictive than lifetimes measurements. The spin of thestate under study must be as aligned as possible. This has limited the reaction utilized to Coulomb excitation andalpha-transfer. Some attempts to use fusion-evaporation reactions for 194,194Hg [7, 8], 152Dy [9] and 162,163,164Hf [10]


65

have presented limited results, mainly the average g-factor value in deform bands. Coulomb excitation reactions ininverse kinematics presents the best possible option for TF experiments, and in recent years the use of alpha-transferexperiments have become important after the existence of a limited number of experiments that can be performedusing Coulex experiments.

Reference [11] presents an example of a TF experiment using an alpha-transfer reaction. The small spin align-ment obtained is the main source of the large errors obtained.

The target. The interaction of nuclei in the beam with the first carbon layer on the target populates nuclear statesof the ions via either Coulomb excitation or alpha-transfer reactions, see Figure 3. Right after this excitation nucleitravel through a stopper material where decay with the subsequent -ray emission. A stopper of Tantalum is one of themost used in DSAM experiments. For the case of TF experiments, Tantalum is replaced by ferromagnetic materialsof Gadolinium or Iron [4]. Gadolinium needs to be cooled with liquid nitrogen, this adds an additional complexity tothe setup. An example of the composition of a multilayer target for this type of experiment is shown in Table 1.

TABLE 1. Composition of multilayer target. All thicknesses are in mg/cm2.

C Gd Ta Cu

0.636 8.34 1.1 5.40

The setup. Figure 4 presents a typical experimental setup for a combined DSAM and TF experiment. The seg-mentation HPGe Clover detectors permits to obtain detailed information of the angular distribution and the change ofthe angular distribution with the change of an external magnetic field.

FIGURE 4. Photography of a typical experimental setup utilized at the Lawrence Berkeley National Laboratory (LBNL) forsimultaneously measurements of nuclear lifetimes and g factors.

DISCUSSION

The DSAM and the TF technique can be utilized to measure the mean lifetime and the magnetic moment µ ofan excited state in nuclei ranging between 1015 and 1012 s, before a de-excitation via -ray emission takes place.During the last decades a systematic study of this two quantities have been performed in several mass regions [12].The use of Coulomb excitation reactions have been successfully utilized for DSAM and the TF experiments duringthe last decades. The use of fusion-evaporation reactions for lifetimes measurements have been also implemented withgreat success, this has been possible due to our knowledge in the stopping power of the material, in conjunction withadvance Monte Carlo simulations techniques. These makes possible to obtain a detailed reconstruction of the decayhistory of the nuclei, a very important information to use correctly equation 2.

The use of the TF technique with fusion-evaporation reactions has been a challenge. The main problem is thefact that the spin precession of a upper state is inherited by lower states, and the correction of that process through aset of states, for example a rotational band, is a dicult task that increase the error bars of the experiments. The use of


66

fusion-evaporation reactions can still be used to measure the average g-factor of rotational bands, and can be utilizedto probe the existence of magnetic rotations reported in certain nuclei [13].

Alpha transfer reactions can be seen as an intermediate reaction between Coulomb excitation and fusion-evaporation [11], and recent experiments have been performed to populate states in nuclear species that cannot beproduced in the present radioactive beam facilities. The detail study of the ↵-transfer reaction process is worth tostudy, because it can provide a firm ground to extend the results to fusion-evaporation reactions.

ACKNOWLEDGMENTS

This work has been partially supported by the Colombian agency for the foment of science and technology, COL-CIENCIAS, under contract FP44842-019-2015.

REFERENCES

[1] R. Stokstad, I. Fraser, J. Greenberg, S. Sie, and B. D.A., Nuclear Physics A 156, 145 – 169 (1970).[2] D. Branford and I. Wright, Nuclear Instruments and Methods 106, 437 – 443 (1973).[3] N. Benczer-Koller and G. J. Kumbartzki, Journal of Physics G: Nuclear and Particle Physics 34, p. R321

(2007).[4] G. Kumbartzki, Nuclear magnetic moments, a guide to transient field experiments. cited October 2015.[5] I.-Y. Lee, Nuclear Physics A 520, c641 – c655 (1990).[6] D. Sarantites, P.-F. Hua, M. Devlin, L. Sobotka, J. Elson, J. Hood, D. LaFosse, J. Sarantites, and M. Maier,

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectorsand Associated Equipment 381, 418 – 432 (1996).

[7] R. H. Mayer, G. Kumbartzki, L. Weissman, N. Benczer-Koller, C. Broude, J. A. Cizewski, M. Hass,J. Holden, R. V. F. Janssens, T. Lauritsen, I. Y. Lee, A. O. Macchiavelli, D. P. McNabb, and M. Satteson,Phys. Rev. C 58, R2640–R2643Nov (1998).

[8] L. Weissman, R. Mayer, G. Kumbartzki, N. Benczer-Koller, C. Broude, J. Cizewski, M. Hass, J. Holden,R. Janssens, T. Lauritsen, I. Lee, A. Macchiavelli, D. McNabb, and M. Satteson, Physics Letters B 446, 22– 27 (1999).

[9] M. Hass, N. Benczer-Koller, G. Kumbartzki, T. Lauritsen, T. L. Khoo, I. Ahmad, M. P. Carpenter, R. V. F.Janssens, E. F. Moore, F. L. H. Wolfs, P. Benet, and K. Beard, Phys. Rev. C 44, 1397–1404Oct (1991).

[10] L. Weissman, M. Hass, and C. Broude, Phys. Rev. C 57, 621–627Feb (1998).[11] D. A. Torres, G. J. Kumbartzki, Y. Y. Sharon, L. Zamick, B. Manning, N. Benczer-Koller, G. Gurdal, K.-

H. Speidel, M. Hjorth-Jensen, P. Maier-Komor, S. J. Q. Robinson, T. Ahn, V. Anagnostatou, M. Elvers,P. Goddard, A. Heinz, G. Ilie, D. Radeck, D. Savran, and V. Werner, Phys. Rev. C 84, p. 044327Oct (2011).

[12] N. Stone, “Table of nuclear magnetic dipole and electric quadrupole moments,” Tech. Rep. (Research spon-sored by the IAEA Nuclear Data Section, Vienna International Centre, 1400 Vienna, Austria, 2011).

[13] D. A. Torres, F. Cristancho, L.-L. Andersson, E. K. Johansson, D. Rudolph, C. Fahlander, J. Ekman, R. du Ri-etz, C. Andreoiu, M. P. Carpenter, D. Seweryniak, S. Zhu, R. J. Charity, C. J. Chiara, C. Hoel, O. L.Pechenaya, W. Reviol, D. G. Sarantites, L. G. Sobotka, C. Baktash, C.-H. Yu, B. G. Carlsson, and I. Rag-narsson, Phys. Rev. C 78, p. 054318Nov (2008).


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PHYSICAL REVIEW C 93, 044316 (2016)

Z = 50 core stability in 110Sn from magnetic-moment and lifetime measurements

G. J. Kumbartzki,1,* N. Benczer-Koller,1 K.-H. Speidel,2 D. A. Torres,3 J. M. Allmond,4 P. Fallon,5 I. Abramovic,6

L. A. Bernstein,5,6,7 J. E. Bevins,6 H. L. Crawford,5 Z. E. Guevara,3 G. Gurdal,8 A. M. Hurst,5 L. Kirsch,6 T. A. Laplace,7,6

A. Lo,6 E. F. Matthews,6 I. Mayers,6 L. W. Phair,5 F. Ramirez,3 S. J. Q. Robinson,8 Y. Y. Sharon,1 and A. Wiens5

1Department of Physics and Astronomy, Rutgers University, New Brunswick, New Jersey 08903, USA2Helmholtz-Institut fur Strahlen- und Kernphysik, Universitat Bonn, D-53115 Bonn, Germany

3Departamento de Fısica, Universidad Nacional de Colombia, Carrera 30 No 45-03, Bogota D.C., Colombia4Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

5Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA6Department of Nuclear Engineering, University of California, Berkeley, California 94720, USA

7Lawrence Livermore National Laboratory, Livermore, California 94551, USA8Physics Department, Millsaps College, Jackson, Mississippi 39210, USA

(Received 21 December 2015; revised manuscript received 8 March 2016; published 18 April 2016)

Background: The structure of the semimagic 50Sn isotopes were previously studied via measurements ofB(E2; 21

+ → 01+) and g factors of 21

+ states. The values of the B(E2; 21+) in the isotopes below midshell at

N = 66 show an enhancement in collectivity, contrary to predictions from shell-model calculations.Purpose: This work presents the first measurement of the 21

+ and 41+ states’ magnetic moments in the unstable

neutron-deficient 110Sn. The g factors provide complementary structure information to the interpretation of theobserved B(E2) values.Methods: The 110Sn nuclei have been produced in inverse kinematics in an α-particle transfer reaction from12C to 106Cd projectiles at 390, 400, and 410 MeV. The g factors have been measured with the transient fieldtechnique. Lifetimes have been determined from line shapes using the Doppler-shift attenuation method.Results: The g factors of the 21

+ and 41+ states in 110Sn are g(21

+) = +0.29(11) and g(41+) = +0.05(14),

respectively. In addition, the g(41+) = +0.27(6) in 106Cd has been measured for the first time. A line-shape

analysis yielded τ (110Sn;21+) = 0.81(10) ps and a lifetime of τ (110Sn;31

−) = 0.25(5) ps was calculated from thefully Doppler-shifted γ line.Conclusions: No evidence has been found in 110Sn that would require excitation of protons from the closedZ = 50 core.

DOI: 10.1103/PhysRevC.93.044316

I. INTRODUCTION

The tin isotopes are recognized as one of the best environ-ments for studying nuclear structure and the shell model inthe intermediately heavy nuclei. Sn is the isotope chain withthe largest number of stable isotopes and potentially can bestudied experimentally over the region between the unstabledoubly magic 100Sn (N = 50) and 132Sn (N = 82) and perhapsbeyond. The Sn nuclei, with Z = 50, are semimagic and thusexhibit a closed proton shell. In this particular case, mainlyvalence neutrons are expected to determine the structure ofthe energy levels and the transitions between them. In a simpleshell-model picture, the nuclei near the doubly magic numbersof protons and neutrons should exhibit single-particle charac-teristics, while in midshell they should show collective aspects.

Extensive spectroscopic measurements of energy levels,lifetimes, and/or B(E2) reduced transition probabilities, andelectromagnetic moments, µ and Q, have been carried outin the Sn isotopic chain. The reduced transition probabilitydata in 104−134Sn (Refs. [1–10] and references therein), followonly above midshell (A = 116) the parabola-like curve of theshell-model expectations [6]. For nuclei below midshell theB(E2) values are larger than expected but finally decrease at

*[email protected]

N = 54, as the doubly closed shell Z = N = 50 is approached.It has been suggested that proton excitations across the Z = 50shell gap have to be invoked as an explanation of the enhancedcollectivity.

Magnetic moments of the 21+ states were previously

measured in the even-even stable 112−124Sn isotopes [9,11–13] as well as in radioactive 126−128Sn [14,15]. However,experiments using transient field (TF) or recoil-in-vacuumtechniques and beam energies both below and above theCoulomb barrier yielded results which challenge comparisonswith theoretical calculations.

The measurements of the magnetic moments offer a specifichandle in the determinations of the nuclear structure by distin-guishing between possible neutron and proton configurations.The g-factor measurement in the radioactive 110Sn could shedlight on the role of protons in the interpretation of the transitionprobabilities for the neutron-deficient Sn isotopes.

In the isotopes lighter than 112Sn, the neutrons may occupythe orbitals g7/2 and d5/2, for which the magnetic momentsshould, respectively, be positive or negative. Proton excitationsout of the closed core into the g7/2 orbital would yield largepositive g factors. In fact, 110Sn lies between 109Sn (5/2+

gs ;g < 0) and 111Sn(7/2+

gs ; g > 0) and a determination of even thesign of its magnetic moment would yield significant structureinformation.

2469-9985/2016/93(4)/044316(7) 044316-1 ©2016 American Physical Society

68

G. J. KUMBARTZKI et al. PHYSICAL REVIEW C 93, 044316 (2016)

Several theoretical approaches have focused on g factorsin this region. These include, among others, the quasiparticlerandom-phase approximation (QRPA) [16] and the relativisticQRPA (RQRPA) [17] approach. Shell-model calculationsbased on the nucleon-pair approximation, as well as cal-culations using a low-momentum interaction and the high-precision CD-Bonn free nucleon-nucleon potential, have alsobeen carried out [13,18–20].

While the main focus of this project is the measurementof magnetic moments, the lifetime of the 21

+ state in 110Snhas also been determined from the same data when they areanalyzed via the Doppler-shift-attenuation method (DSAM).In addition, this experiment yields new data on the magneticmoment and lifetime of the 41

+ state in Coulomb-excited106Cd beam nuclei.

II. THE EXPERIMENT

The experiment was performed at the Lawrence BerkeleyNational Laboratory (LBNL) 88-Inch cyclotron. The 110Snwas produced via an α-particle transfer to a beam ofisotopically pure 106Cd ions impinging on a carbon layer atthe front of a multilayer target. The specific reaction is 12C(106Cd, 8Be) 110Sn. The 8Be nuclei decay spontaneously intotwo α particles.

The experiment and setup were similar to those describedin Ref. [21]. The components of the multilayer target for thisexperiment are shown in Table I.

Beam energies of 390, 400, and 410 MeV were employedto find the best α-transfer yield. The bulk of the data wastaken at 410 MeV. At this energy the beam has lost about20 MeV when reaching the middle of the carbon layer andits energy there is close to that of the Coulomb barrier. Asfound in Ref. [21], the α-transfer reaction is optimal near theCoulomb barrier between the projectile and carbon. The newlycreated 110Sn and Coulomb-excited 106Cd recoils traverse thegadolinium layer where they experience the TF. These nucleiare stopped in the copper backing. Their decay γ rays aredetected in coincidence with forward-scattered particles.

The target was mounted between the pole tips (8-mm gap)of a liquid-nitrogen-cooled magnet. The gadolinium layer ofthe target was magnetized by a field of 0.07 T. Its direction wasreversed every 150 s during the measurements. The particledetector was a 300-mm2 Si surface-barrier detector (CanberraPIPS). The detector was placed 25 mm downstream of thetarget at 0 to the beam direction. Its opening angle was±20. The detector was covered by a 5.6 mg/cm2 thick copperfoil, which stopped the noninteracting beam particles passingthrough the target but was transparent to the light particlesresulting from the reactions. The different particle groups werewell separated in the 300-µm-thick detector as is shown inFig. 1.

TABLE I. Composition of multilayer target. All thicknesses arein mg/cm2.

C Gd Ta Cu

0.636 8.34 1.1 5.40

0 20 40 60 80 100 120 140 160Energy (MeV)

20000

40000

60000

Sing

les

0 20 40 60 80 100 120 140 160Energy (MeV)

100

200

300

400

500

600

110 Sn

γ-G

ated

0 20 40 60 80 100 120 140 160Energy (MeV)

8Be

12C

FIG. 1. Single-particle spectrum observed in the Si detector. Alsoshown are particles in coincidence with gates set on time and γ

rays corresponding to 110Sn. The two peaks labeled 8Be relate to thedetection of only one or both α particles (2α) from the 8Be breakup.

The γ rays were recorded in four clover HPGe detectorsfrom the ORNL and LBNL inventories. These were located125 mm away from the target at angles of θ = ±60 and±120 with respect to the beam direction. At that distance theindividual elements of the clover detectors subtend angles of±8 with respect to the center of the clover enclosure.

The preamplifier output signals of all detectors weredigitized using a PIXIE-4 system [22]. Their time stamps andenergies were written to disk. The data handling and analysiswere performed as described in greater detail in Ref [21].

Typical particle-γ coincidence spectra gated on the 12C andthe 2α peaks, displaying respectively the decay of the excitedenergy levels of 106Cd and 110Sn, are shown in Fig. 2. For 110Snonly three prominent γ lines are seen in the spectrum referringto the (21

+ → 01+) 1212 keV, the (41

+ → 21+) 985 keV, and

the (61+ → 41

+) 280 keV transitions. This picture clearly

400 800 12000

5000

10000

15000

Coun

ts

400 800 1200Energy (keV)

0200400600800

1000

Coun

ts

985

632

1212

632

511

280

861

2+1 0+

1

4+1 2+

1

6+1 4+

14+

1 2+1

2+1 0

+1

106Cd

110Sn

611 10 x

FIG. 2. Coincidence γ spectra. The 106Cd spectrum was gated onthe 12C peak in Fig. 1 while the 110Sn spectrum was gated on the 2α

peak of the 8Be breakup.

044316-2

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Z = 50 CORE STABILITY IN 110Sn FROM . . . PHYSICAL REVIEW C 93, 044316 (2016)

TABLE II. The kinematic information related to the transient fieldmeasurement at 410 MeV. ⟨E⟩in, ⟨E⟩out, ⟨v/v0⟩in, and ⟨v/v0⟩out arethe energies, in MeV, and velocities of the excited probe ions as theyenter into, and exit from, the gadolinium layer; v0 = e2/! is the Bohrvelocity. Teff is the effective time the transient field acts on the ionstraversing the ferromagnetic layer.

Nucleus ⟨E⟩in ⟨E⟩out ⟨ vv0

⟩in ⟨ vv0

⟩out Teff (fs)

106Cd 232 46 9.4 4.2 715110Sn 252 54 9.6 4.5 438

demonstrates the direct and highly selective nature of theα-transfer reaction in contrast to a fusion reaction. In theCoulomb excitation of 106Cd mainly the 21

+ and the 41+ states

are seen via their E2 transitions in the spectrum.

A. Precession measurement

The magnetic properties of the 106Cd and the 110Sn nucleiwere measured simultaneously. The g factor of the 21

+ statein 106Cd was measured previously [23]. Its value is used asa check on the experiment and also serves to calibrate thetransient field strength.

In a TF measurement the spin precession of the alignednuclei as they pass through the magnetized ferromagneticlayer causes a rotation of the angular distribution of the decayγ radiation. The precession angle is derived from countingrate changes in the stationary γ detectors when the polarizingfield, which is perpendicular to the detection plane of the γdetectors, is reversed. The so-called rate effect ϵ, as describedin many publications (e.g., Ref. [24]), is calculated from peakintensities in the spectra of four γ detectors. Together with thelogarithmic slope, S(θγ ) = [1/W (θγ )]dW/dθγ of the angularcorrelation relevant for the precession, the precession angle

%θ = ϵ

S(θγ )= g

µN

!

! tout

tin

BTF[v(t), Z]e−t/τdt

is obtained. In this expression g is the g factor of the excitedstate, µN is the nuclear magneton, and BTF is the effectivetransient field acting on the nucleus during the time interval(tout − tin) spent by the ions in the gadolinium layer. Theexponential factor accounts for the nuclear decay duringthe transit time of the ions through the gadolinium layer.The relevant kinematic information for the transient-fieldcalculation is summarized in Table II.

B. Angular correlations

The angular correlations for the states in both 106Cd and110Sn were also derived from the precession data. The peakintensities of the 21

+ → 01+ and 41

+ → 21+ transitions in

the spectra of each clover crystal, summed over both fielddirections and corrected for relative efficiencies, are plotted inFig. 3. The relative efficiencies were measured with a 152Eusource in the target position before and verified immediatelyafter the experiment using the activity of the target. The solidlines represent fits to the angular correlation function restrained

0 30 60 90 120 150 180

0.5

1

1.5

2

W(θ

γ )

0 30 60 90 120 150 180

0.5

1

1.5

2

0 30 60 90 120 150 180θγ (deg)

0.5

1

1.5

2

W(θ

γ )

0 30 60 90 120 150 180θγ (deg)

0.5

1

1.5

2

106Cd 110Sn

2+1 0

+1 2+

1 0+1

4+1 2+

1 4+1 2

+1

FIG. 3. Angular particle-γ correlations measured at a beamenergy of 410 MeV.

to only physical parameters for the correlation coefficients

W (θγ ) = 1 + A2Q2P2(cos θγ ) + A4Q4P4(cos θγ ).

Here the Pk(cos θγ ) are the Legendre polynomials, the Ak

are the experimental angular-correlation coefficients, whichdepend on the multipolarity of the γ -ray transition, and theQk are attenuation coefficients accounting for the finite solidangle of the γ detectors.

C. Lifetimes

Because of the high recoil velocities of the ions, the linesof the short-lifetime transitions exhibit prominent lineshapes(Fig. 2) suitable for lifetime analysis using the DSAMtechnique. The LINESHAPE [25] code was used. The Doppler-broadened shape of the γ lines was fitted to the reactionkinematics by applying stopping powers [26] to Monte Carlosimulations and including second-order Doppler effects as wellas the finite size and energy resolution of the γ detectors. Asample fit for the 1212-keV line in 110Sn is shown in Fig. 4.The short-lived 21

+ state is fed about 40% from the long-lived61

+ and 41+ states, which accounts for nearly all of the stopped

components in the lineshape. Also excited in 110Sn is a 31−

state at 2458.4 keV which decays into the 21+, contributing

about 8% to its intensity. The lifetime of the 31− state was

unknown, but from the fully Doppler-shifted peak positionsat 52 and 68, together with the reaction kinematics andenergy loss in the multilayer target, a lifetime of 0.25(5) pswas derived. Unfortunately, this 31

− → 21+ fully shifted and

Doppler-broadened γ transition of 1246.4 keV also fallsunder the lineshape of the 1212-keV line at backward angles.By including various scenarios for the feeding contributionsand using the detectors at different angles, a number of fitsproduced lifetimes close to the literature value calculated fromB(E2) measurements. Overall, in this work, the lineshape fitsgive an average τ (110Sn; 21

+) = 0.81(10) ps, which is slightlylarger than the NNDC value of 0.69(6) ps [27].

044316-3

70


1150 1200 1250Energy (keV)

0

500

1000

1500

Coun

ts

1212

.0

1243

.3

1257

QFN = 1.22

FIG. 4. LINESHAPE fit of the 21+ → 01

+ γ line in 110Sn asobserved in two clover segments at 128. The greyed-out areasrepresent feeding intensities. QFN represents the normalized χ2.

D. Magnetic moments

The Coulomb excitation of the 21+ state in 106Cd provided a

solid check of the precession measurement. At a beam energyof 400 MeV, for which feeding is the lowest, the knowng-factor value was reproduced. In later runs with various beamintensities, the Cd value was taken to monitor and determinethe magnetization (temperature) of the target spot. Indeed, astrong correlation between the beam current, represented bythe measured single-particle rate, and the precession effect ofthe 21

+ → 01+ transition in 106Cd was observed.

The results are summarized in Table III. The g factor ofthe 21

+ of 110Sn was analyzed from γ spectra in coincidencewith the 2α-peak (Fig. 1). Because of considerable feeding bythe long-lived states in the 61

+ → 41+ → 21

+ cascade, andto avoid feeding corrections, only the Doppler-shifted part ofthe 1212-keV γ line was used in the precession analysis. Thisprocedure reduces the counting statistics but avoids even larger

errors caused by the error propagation in the feeding correc-tions [31]. The very short lifetime of the state affects the effec-tive transit time in the gadolinium layer of the target and the gfactor. The slightly shorter literature lifetime of 0.69 ps wouldincrease the value of the g factor quoted in Table III by 7%. Theg factors of the 41

+ and 61+ states were determined from γ

spectra gated on both α peaks in Fig. 1. Both values are withinthe errors close to zero. For the 41

+ state a lifetime of 4 ps wasused, which is long enough to not affect the quoted g factor.

The g factor of the 41+ state in 106Cd was measured for the

first time. The state has a short lifetime and is fed by a second4+. The literature value [29] is τ (106Cd; 41

+) = 1.26(15) ps. Alineshape analysis of the current data favors a longer lifetimeof τ = 2.5(2) ps. The longer lifetime would reduce the g-factorvalue quoted in Table III by ∼25%. More details of the 106Cdresults will be presented in a forthcoming paper.

III. DISCUSSION AND THEORY

None of the theoretical approaches shown in Fig. 5 candescribe the g(21

+) factors of all the Sn isotopes. Although,the common trend from positive to negative g-factor valuesis reproduced. No shell-model calculation explicitly breakingthe Z = 50 shell was published.

The results of Ansari and Ring [17] agree with the measuredg(21

+) of the present paper but not with most of the measuredvalues for other Sn isotopes. The other two calculations inFig. 5 are within 1.7 standard deviations but lower than themeasured value.

Jiang et al. [19] calculated the magnetic moments of thefirst 2+ states in the Sn isotopes within the framework ofthe nucleon-pair approximation of the shell model. Theircalculation yields a g factor for 110Sn about three times smallerthan the measured value. Furthermore, the authors used aneffective gνl = +0.09, rather than the generally accepted value,gνl = −0.1 [32]. This sign inversion could alter considerablythe final result.

TABLE III. Experimental results for states in 110Sn and 106Cd. Also included are the slopes for full clovers and precession angles.&θ (g = 1) was calculated using the Rutgers parametrization [28]. The lifetimes are taken from the National Nuclear Data Center (NNDC)database [27,29].

EBeam Iπi Eγ τ &θ (g = 1) |S(60)| &θ g

(MeV) (keV) (ps) (mrad) (mrad −1) (mrad) This work Others

110Sn410 21

+ 1212.0 0.81(10)a 65.8 0.384(37) 18.8(68) +0.29(11)41

+ 984.0 >4.0b 92.6 0.463(55) 4.7(130) +0.05(14)61

+ 280.2 8.1(4)·103 103.0 0.565(120) 0.8(191) +0.01(19) +0.012(3)c

106Cd400 21

+ 632.7 10.49(12) 98.5 1.76(3)d 39.14(94) +0.398(22) +0.393(31)e

410 41+ 861.2 1.26(16) 73.5 0.66(3) 19.6(40) +0.27(6)

aThis work. The NNDC value is τ (21+) = 0.69(6) ps [27].

bThe lifetime is unknown. The adopted value agrees with systematics in neighboring isotopes. The lineshape shows no discernible Doppler-shifted component.cReference [30].d|S(67)|eReference [23].

044316-4

71


100 104 108 112 116 120 124 128 132 136Atomic Number

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

g(2+ 1) f

acto

r

Hass et al. (1980)East et al. (2008)Walker et al. (2011)Kumbartzki et al. (2012)Allmond et al. (2013)Allmond et al. (2015)This work

Ekström (2015)

Jiang et al. (2014)

Ansari et al. (2007)

LSSM

RQRPA

FIG. 5. Compilation of experimental g(21+) factors from

Refs. [9,11–15]. The LSSM calculation is based on the 100Sn corecalculation of Ref. [6] as published in Ref. [13] and extended byEkstrom [33] to the neutron-deficient Sn isotopes.

The shell-model calculation by Ekstrom [33] based on astable 100Sn core as described in Ref. [13] also underpredictsthe present 110Sn result. This calculation predicts a downturnof the g factors for the proton-rich isotopes and overall agreesbest with the experimental data.

In the present work, large-scale shell model (LSSM)calculations were carried out for 110Sn adopting the SN100PNinteraction [34]. A 100Sn core was employed and protonexcitations were excluded. The neutron orbital space includedg7/2, d5/2, d3/2, and s1/2 with and without h11/2. The bestagreement with the experimental data was obtained withoutthe h11/2 orbital, gνl = +0.1 and gνs = 0.75gfree

s , valuessimilar to those used in Ref. [19]. The resulting g factors wereg(2+) = +0.15, g(4+) = +0.12, and g(6+) = +0.1. Withgνl = −0.1, all three g factors turn negative. Furthermore,the inclusion of the h11/2 orbital considerably reduced theseresults.

If the wave functions of the states of interest in 110Sn wereof purely neutron nature, in the spirit of the seniority scheme,at a minimum two neutrons not coupled to zero are requiredin the available (g7/2) and (d5/2) orbitals near the Fermi levelto form excited states.

Therefore, in a simple single-particle approach, the caseof two neutrons in the (g7/2)2 and the mixed [(g 7/2)1,(d5/2)1]configurations is considered. The expression

gj = gl ± (gs − gl)(2l + 1)

with j = l ± 1/2

gives gcal(g7/2) = +0.208 and gcal(d5/2) = −0.654 using ef-fective values for the neutron spin and orbital g factors,geff

s = −3.826 × 0.75 and geffl = −0.1. Thus the configuration

(g7/2)2 yields g(21+) = g(41

+) = g(61+) = +0.208. Then,

the expression [35]

g(j1 ⊗ j2,I ) = (g1+g2)2

+ (g1−g2)2

j1(j1+1)−j2(j2 + 1)I (I + 1)

,

100 104 108 112 116 120 124 128 132Atomic Number

0

0.1

0.2

0.3

B(E2

;0+ 1

2+ 1) (e2 b2 )

NNDCORNLGSIMSUISOLDERIKENJungclausThis work

FIG. 6. Compilation of measured B(E2; 0+gs → 21

+) values ofthe even tin isotopes. The majority of the data [1,2,4–6,8,9,36]are from Coulomb-excitation cross-sectional measurements. TheJungclaus data [10] and this work (circles) are from DSAM lifetimemeasurements. The solid and dashed curves show shell-modelcalculations from Ref. [6] using a 100Sn and 90Zr closed core,respectively. Also shown are calculations using the quasiparticlephonon model from Ref. [37].

where j1 = 7/2, j2 = 5/2, and I , the total angular momentum,yields the effective g factors for the mixed configuration

gcal(21+)=+0.28, gcal(41

+)=−0.07, gcal(61+)=−0.151.

The results from the two configurations, when averaged, yieldvalues of

gcal(21+)=+0.24, gcal(41

+)=+0.07, gcal(61+)=+0.03.

These values are in very good agreement with the observedg factors of the present work (Table III). Any inclusion ofthe (d5/2)2 configuration leads to smaller or negative g factorsin contrast to the experimental results. In this context it isnoteworthy to mention that any contribution from protonsexcited from the core would significantly increase these valuesdue to the corresponding large positive g values of the relevantproton orbitals.

The B(E2) value deduced from the present lifetime valueof the 21

+ state:

B(E2; 01+ → 21

+)exp = 0.192(24)e2b2.

agrees with shell model calculations based on the doubly magic100Sn core and 90Zr core [6] (see Fig. 6). The result is slightlylower than the two previous Coulomb-cross-section-basedvalues of 0.220(22) e2b2 [5] and 0.240(20) e2b2 [4].

The g(21+) factor is positive but not large

enough to require proton excitation from theZ = 50 core, an observation additionally supported bythe small g factors for the 41

+ and 61+ states.

This conclusion is also supported by a very recent lifetimemeasurement of the 21

+ state of the isotonic 112Te isotope [38].This nucleus has the same neutron number as 110Sn, N = 60,with two additional protons outside the magic proton core.In spite of an expected polarizing effect of the two valence

044316-5

72


protons on the proton core, the deduced B(E2) value is wellexplained by shell model calculations based on a 100Sn core.The authors also expect this robustness of the Z = 50 core forall 52Te isotopes.

IV. SUMMARY

The current work presents the first measurements of theg(21

+) and g(41+) factors in the unstable neutron-deficient

even Sn isotopes. Altogether, the present data are in agreementwith the classical seniority scheme of the shell model. Thesedata can be understood without proton excitation from theZ = 50 core. In view of several B(E2) measurements on lightneutron-deficient Sn isotopes claiming proton core excitationsand in view of the present observations and results, furthermeasurements of magnetic moments and lifetimes for otherneutron-deficient Sn isotopes with radioactive Sn ion beamsare highly desirable.

ACKNOWLEDGMENTS

The authors thank the Berkeley 88-Inch Cyclotron stafffor their help in setting up the experiment and pro-viding the cadmium beam. The target was prepared byP. Maier-Komor at the Technische Universitat Munich,Germany. The authors are grateful to L. Zamick for manydiscussions and suggestions about the theoretical interpre-tation of the g factor results. K.-H.S. acknowledges sup-port by the Deutsche Forschungsgemeinschaft under GrantNo. SP190/18-1. D.A.T., Z.E.G., and F.R. acknowledgesupport by Colciencias under Contract No. 110165842984-2015. Y.Y.S. acknowledges support from Stockton Uni-versity under a Research and Professional Developmentaward. The work has been supported in part by the USNational Science Foundation and by the US Departmentof Energy under Contracts No. DE-AC02-05CH11231 andNo. DE-AC52-07NA27344.

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PHYSICAL REVIEW C 94, 034303 (2016)

Magnetic moment and lifetime measurements of Coulomb-excited states in 106Cd

N. Benczer-Koller,1,* G. J. Kumbartzki,1 K.-H. Speidel,2 D. A. Torres,3 S. J. Q. Robinson,4 Y. Y. Sharon,1 J. M. Allmond,5

P. Fallon,6 I. Abramovic,7 L. A. Bernstein,6,7,8 J. E. Bevins,7 H. L. Crawford,6 Z. E. Guevara,3 A. M. Hurst,6,7 L. Kirsch,7

T. A. Laplace,7,8 A. Lo,7 E. F. Matthews,7 I. Mayers,7 L. W. Phair,6 F. Ramirez,3 and A. Wiens6

1Department of Physics and Astronomy, Rutgers University, New Brunswick, New Jersey 08903, USA2Helmholtz-Institut fur Strahlen- und Kernphysik, Universitat Bonn, D-53115 Bonn, Germany

3Departamento de Fısica, Universidad Nacional de Colombia, Carrera 30 No 45-03, Bogota D.C., Colombia4Physics Department, Millsaps College, Jackson, Mississippi 39210, USA

5Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA6Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

7Department of Nuclear Engineering, University of California, Berkeley, California 94720, USA8Lawrence Livermore National Laboratory, Livermore, California 94551, USA

(Received 1 August 2016; published 6 September 2016)

Background: The Cd isotopes are well studied, but experimental data for the rare isotopes are sparse. At energiesabove the Coulomb barrier, higher states become accessible.Purpose: Remeasure and supplement existing lifetimes and magnetic moments of low-lying states in 106Cd.Methods: In an inverse kinematics reaction, a 106Cd beam impinging on a 12C target was used to Coulomb excitethe projectiles. The high recoil velocities provide a unique opportunity to measure g factors with the transient-fieldtechnique and to determine lifetimes from lineshapes by using the Doppler-shift-attenuation method. Large-scaleshell-model calculations were carried out for 106Cd.Results: The g factors of the 2+

1 and 4+1 states in 106Cd were measured to be g(2+

1 ) = +0.398(22) and g(4+1 ) =

+0.23(5). A lineshape analysis yielded lifetimes in disagreement with published values. The new results areτ (106Cd; 2+

1 ) = 7.0(3) ps and τ (106Cd; 4+1 ) = 2.5(2) ps. The mean life τ (106Cd; 2+

2 ) = 0.28(2) ps was determinedfrom the fully-Doppler-shifted γ line. Mean lives of τ (106Cd; 4+

3 ) = 1.1(1) ps and τ (106Cd; 3−1 ) = 0.16(1) ps

were determined for the first time.Conclusions: The newly measured g(4+

1 ) of 106Cd is found to be only 59% of the g(2+1 ). This difference cannot

be explained by either shell-model or collective-model calculations.

DOI: 10.1103/PhysRevC.94.034303

I. INTRODUCTION

The Cd isotopes with Z = 48 are very close to the magicproton Z = 50 shell closure. The two-proton-holes configura-tion is expected to contribute significantly to the nuclear wavefunctions. This aspect differentiates the Cd isotopes from theneighboring Sn isotopes, where the stability of the Z = 50 corerestricts the nuclear structure to the valence neutrons. Indeed,the heavier Cd isotopes exhibit collective properties and the112,114,116Cd isotopes have long been examples of sphericalvibrational nuclei.

However, experiments on Cd isotopes carried out by Garrett[1,2], Ekstrom [3], and Stuchbery [4], among others, suggest amore complex nuclear structure for some Cd nuclei, includingthe existence of deformation with consequent rotationalmotion.

In both the light Sn and Cd isotopes, the B(E2; 2+1 →

0+1 ) values show an increase over the values calculated in the

shell model [3]. These discrepancies can be attributed to avariety of causes, ranging from the possible nonequivalenceof B(E2) values determined either from Coulomb excitation orfrom lifetime measurements, or to actual structure differencescaused by the two valence proton holes.

*[email protected]

The recent measurements of lifetimes in 110Sn [5] did notexhibit this enhancement of the B(E2) values. Furthermore,the measured g factors in 110Sn were found to be in goodagreement with excitations of neutrons and a stable Z = 50core.

The data presented in this paper stem from that particularexperiment on 110Sn. The radioactive 110Sn nucleus wasproduced by the capture of an α particle by the nuclei in a 106Cdbeam impinging on a 12C target. Simultaneously, the beamions were Coulomb excited by the target C nuclei, allowingthe measurements of lifetimes and g factors of several statesin 106Cd.

The details of the experiment and analysis are extensivelydiscussed in Ref. [5]. Only the analysis results leading to thenew information about 106Cd are described in the present paper.Shell-model calculations were carried out in a frameworksimilar to that presented in Ref. [5].

The main interest of the present work is to obtain moredetailed information about the nuclear structure of 106Cd bysearching for single-particle aspects in the nuclear propertiesof 106Cd.

It should be noted that in many previous studies the2+

1 states of several Cd isotopes have been discussedwithin the framework of collective models with littleattention paid to the single-particle structure (see, e.g.,Ref. [6]).

2469-9985/2016/94(3)/034303(6) 034303-1 ©2016 American Physical Society

75

N. BENCZER-KOLLER et al. PHYSICAL REVIEW C 94, 034303 (2016)

0 20 40 60 80 100 120 140 160Energy (MeV)

carbon

100000

50000

150000

Coun

ts

FIG. 1. Single-particle spectrum. At the beam energy of 410 MeV,light particles dominate. The carbon peak is a result of Coulombscattering of the beam projectiles in the carbon layer of the target.

II. THE EXPERIMENT

The experiment was performed at the Lawrence BerkeleyNational Laboratory (LBNL) 88-Inch cyclotron.

The experiment was primarily designed to measure gfactors of low-lying states in 110Sn via an α-particle transfer tothe 106Cd-beam nuclei [5]. In this experiment, additional dataon 106Cd have been obtained.

The multilayer target, front to back, consisted of 0.636 C,8.34 Gd, 1.10 Ta, and 5.40 Cu (mg/cm2). The beam energywas 410 MeV, close to the Coulomb barrier of 106Cd on 12C(390 MeV). The Coulomb excitation of the beam particlesin the first target layer is established by measuring γ rays incoincidence with forward-scattered carbon ions.

The target was mounted between the pole tips of a liquid-nitrogen-cooled magnet. The gadolinium layer of the targetwas magnetized by a field of 0.07 T. Its direction was reversedevery 150 s during the measurements. The particle detectorwas a 300 mm2 Si surface-barrier detector (Canberra PIPS)placed 25 mm downstream of the target at 0 with respect tothe beam direction. The beam was stopped in a 5.6-mg/cm2-thick copper foil, which was placed in front of the particledetector. Only the carbon ions and light particles resulting fromreactions reached the detector. The carbon particles were wellseparated in the 300-µm-thick detector, as shown in Fig. 1.

The γ rays were observed in four clover HPGe detectorsfrom the Oak Ridge National Laboratory (ORNL) and LBNLinventories. These were located 125 mm away from the targetat angles of θ = ±60 and ±120 with respect to the beamdirection. At that distance, the individual elements of the cloverdetectors subtended angles of ±8 with respect to the centerof the clover enclosure.

The preamplifier output signals of all detectors weredigitized by using a PIXIE-4 system [8]. Their time stamps andenergies were written to disk. The data handling and analysiswere performed as described in greater detail in Ref. [9].

Particle-γ coincidence spectra gated on the 12C peak,obtained at a beam energy of 410 MeV, are shown in Fig. 2.

The low-lying levels of 106Cd that were identified in thisexperiment are shown in Fig. 3.

500 1000 1500 200010

100

1000

10000

500 1000 1500 2000Energy (keV)

10

100

1000

10000

511 61

163

3

811

861

998

1084 14

72

1746

128o

52o

Coun

ts

2+1 0+

1

4+2 4+

1 2+2 2+

14+

2 2+1

3-1 2

+1

4+3 4+

1

4+1 2+

1

2+2 0+

16+1 4

+1

1716

FIG. 2. Coincidence γ spectra gated on the carbon peak in Fig. 1.The spectra show the Doppler-broadened and -shifted lines, includingthe distinct lineshapes observed in a backward- and in a forward-positioned detector segment at the indicated angle θ with respect tothe beam direction.

A. Precession measurement

The g factor of the 2+1 state in 106Cd was measured

previously by the transient field technique (TF) [6]. Its valuewas used as a check on the experiment and also served tocalibrate the transient-field strength.

In a TF measurement, the spin precession of the alignednuclei traversing the magnetized ferromagnetic layer causes arotation of the angular distribution of the decay γ radiation.The precession angle is derived from counting-rate changesin the stationary γ detectors when the polarizing magneticfield at the target, which is perpendicular to the detectionplane of the γ detectors, is reversed. The so-called rate

632.

6

632.6

1493.81716.5

2104.5

2+1

2+2

4+1

4+2

3--1

1716

.510

84.2

861.

2

610.

8

106Cd

2304.9

0+1

1745

.8

2378.52491.7

811.

1

997.

9

4+3

6+1

1471

.9

τ (ps)

0.16(1)

0.28(2)2.5(2)

1.1(1)

7.0(3)

> 10

E (keV)

FIG. 3. Partial level scheme indicating the states in 106Cd thatwere excited in this experiment. The energies are taken from theNational Nuclear Data Center (NNDC) [7]. The lifetime columnshows the newly determined mean lives.

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MAGNETIC MOMENT AND LIFETIME MEASUREMENTS OF . . . PHYSICAL REVIEW C 94, 034303 (2016)

TABLE I. The kinematic information related to the transient-fieldmeasurement at a beam energy of 410 MeV. ⟨E⟩in, ⟨E⟩out, ⟨v/v0⟩in,and ⟨v/v0⟩out are the average energies, in MeV, and velocities, in unitsof v0 = e2/!, the Bohr velocity, of the excited probe ions as they enterinto, and exit from, the gadolinium layer. Teff is the effective time thetransient field acts on the ions traversing the ferromagnetic layer.

Nucleus ⟨E⟩in ⟨E⟩out ⟨v/v0⟩in ⟨v/v0⟩out Teff (fs)

106Cd 232 46 9.4 4.2 715

effect ϵ, as described in many publications (e.g., Ref. [10]),is calculated from peak intensities in the spectra of fourγ detectors. Together with the logarithmic slope, S(θγ ) =[1/W(θγ )]dW/dθγ of the angular correlation relevant for theprecession, the precession angle

$θ = ϵ

S(θγ )= g

µN

!

! tout

tin

BTF(v(t),Z)e−t/τdt

is obtained. In the above expression, g is the g factor ofthe excited state and µN is the nuclear magneton. BTF isthe effective transient field acting on the nucleus during thetime interval (tout − tin) spent by the ions in the gadoliniumlayer. The exponential factor accounts for the nuclear decayduring the transit time of the ions through the gadoliniumlayer. The relevant kinematic information for the transient-fieldcalculation is summarized in Table I.

The angular correlations for the states were also derivedfrom the precession data. The peak intensities of the 2+

1 → 0+1

and 4+1 → 2+

1 transitions in the spectra of each clover crystal,summed over both field directions and corrected for relativeefficiencies, were fit to the angular-correlation function

W (θγ ) = 1 + A2Q2P2(cos θγ ) + A4Q4P4(cos θγ ).

Here the Pk(cos θγ ) are the Legendre polynomials, the Ak

are the experimental angular-correlation coefficients, whichdepend on the multipolarity of the γ -ray transition, and theQk are attenuation coefficients accounting for the finite solidangle of the γ detectors. Representative fits are shown in Fig. 3of Ref. [5].

B. Lifetimes

On average, the cadmium ions exit the carbon foil witha velocity of 6.86% c. In Fig. 2, the γ lines of the 2+

1 →0+

1 , 4+1 → 2+

1 , and 4+3 → 4+

1 transitions show prominentlineshapes, while the 2+

2 → 2+1 and 3−

1 → 2+1 transitions are

fully shifted and Doppler broadened. The shifted 2+2 → 0+

1transition is mostly hidden in the 1745.8 keV γ line of the3−

1 → 2+1 transition. The 4+

2 → 4+1 , 610.8 keV, and 4+

2 → 2+1 ,

1471.9 keV, transitions exhibit sharp γ lines indicating nodecay in flight. Therefore, the mean life of the 4+

2 state can beestimated to be longer than 10 ps, in contrast to the NNDCreport of t1/2 ! 2 ps.

Each of the 16 HPGe crystals in the four clovers can be usedfor the DSAM lifetime analysis. The LINESHAPE [12] codewas used. In the first step, by using a Monte Carlo simulationand Ziegler’s stopping powers [13], energy-loss cascades werecalculated for the reaction kinematics in the multilayer target.

600 620 640 660 680100

1000

10000

820 840 860 880 900Energy (keV)

100

1000

632.

6

861.

2

τ = 7.0(3) ps

τ = 2.5(2) ps

21%

610.

8

Cou

nts

(a)

(b)

FIG. 4. Simultaneous LINESHAPE fit of (a) the 2+1 → 0+

1 γ lineand (b) 4+

1 → 2+1 γ line in 106Cd as observed in a clover segment

at 68. The shaded area represents the feeding intensity from the4+

2 → 4+1 γ line of 610.8 keV seen in panel (a).

In the second step, the cascades relevant for each detectorgeometry were selected. The Doppler-broadened shapes of theγ lines were then fit to the corresponding data sets. Sample fitsare shown in Figs. 4 and 5. The results in Table II are averagedresults of fits to lines in forward and backward detectors. Theerrors were enlarged to reflect uncertainties in feeding and thespread of the fit results in various detectors.

All the lifetimes reported in this paper have been measuredfor the first time by the DSAM lineshape technique and dis-agree with the literature values [7] determined from Coulomb-excitation cross-section B(E2) measurements. Notably, themean life of the 2+

1 state is shorter by 33%, while the mean lifeof the 4+

1 state is twice the literature value. The mean life ofthe 2+

2 state is shorter by 38% and the mean life of the 4+2 state

is much longer. The mean lives of the 4+3 and the 3−

1 stateshave not been measured previously.

775 800 825 850Energy (keV)

100

1000

Coun

ts

811

861

τ= 1.1(1) ps

τ= 2.5(2) ps

FIG. 5. LINESHAPE fit of the 811.1 keV 4+3 → 4+

1 γ line and the861.2 keV 4+

1 → 2+1 γ line in 106Cd as observed in a clover segment

at 112.

034303-3

77


TABLE II. Experimental results for states in 106Cd. Also included are the slopes for full clovers and the precession angles. !θ (g = 1) wascalculated by using the Rutgers parametrization [11]. The literature values of the mean lives are taken from the NNDC data base [7].

EBeam Iπi Eγ τ (ps) !θ (g = 1) |S(60)| !θ g

(MeV) (keV) This work NNDCa (mrad) (mrad−1) (mrad) This work Others

400 2+1 632.6 7.0(3) 10.49(12) 98.5 1.76(3)b 39.14(94) +0.398(22) +0.393(31)c

410 4+1 861.2 2.5(2) 1.26(16) 85.7 0.66(3) 19.6(40) +0.23(5)

2+2 1084.2 0.28(2) 0.45(7)

1716.54+

2 610.8 >10 !2.91471.9

4+3 811.1 1.1(1)

3−1 1745.8 0.16(1)

aThe NNDC publications quote half-lives.b|S(67)|.cReference [6].

C. Magnetic moments

The Coulomb excitation of the 2+1 state in 106Cd would be

best measured below the Coulomb barrier of projectile andtarget nuclei. At a beam energy of 400 MeV, the adoptedg(2+

1 ) value of +0.393(31) [6] was reproduced by usingthe Rutgers parametrization [11]. In runs at 410 MeV withvarious beam intensities; this g factor was taken to monitor themagnetization, which is a sensitive function of the beam-spottemperature. Indeed, a strong correlation between the beamcurrent, represented by the measured singles particle rate, andthe precession rate effect of the 2+

1 → 0+1 transition in 106Cd

was observed [14].The g factor of the 4+

1 state in 106Cd was measured for thefirst time. This state has a short lifetime and is fed by another4+ state. The literature value [7] is τ (106Cd; 4+

1 ) = 1.26(16) pswhich leads to the value g(106Cd; 4+

1 ) = +0.27(6) quoted inRef. [5]. A lineshape analysis of the current data yielded a newmean life of 2.5(2) ps, and a g factor g(106Cd; 4+

1 ) = +0.23(5).The results are summarized in Table II.

III. DISCUSSION AND THEORY

In the present work, large-scale shell-model (LSSM)calculations were carried out for 106

48 Cd58. The G-matrixinteraction jj45pna was used. This interaction is included in theshell-model code NUSHELLX [15] and can be used for protonnumbers below Z = 50 and neutron numbers above N = 50.

A 7828Ni50 core was employed. The two proton valence holes

below the Z = 50 magic number were always permitted to beanywhere in the f5/2, p3/2, p1/2, and g9/2 orbital space. Twodifferent spaces were considered for the eight valence neutronsbeyond the N = 50 core. Space 1 included the g7/2, d5/2, d3/2,and s1/2 neutron orbitals. Space 2 encompassed only theg7/2, d5/2, and d3/2 orbitals. The shell-model calculations showthat, in both spaces, the occupancies of the various orbitals areessentially the same for each of the 0+

1 , 2+1 , and 4+

1 states in106Cd. The proton holes are largely in the g9/2 orbital and theneutrons are primarily in the d5/2 and the g7/2 orbitals.

In the B(E2) calculations, two different sets of effectivecharges (ep,en) were utilized: (1.75e,0.75e) and (2.0e,1.0e).

In Table III the two corresponding calculated B(E2) resultsare presented.

Two sets of nucleon g factors were used in each of the twospaces for the g-factor calculations. The first set involved thebare g factors [glp = 1, gsp = 5.581, gln = 0, gsn = −3.826].The second set included effective nucleon g factors [glp =1.1, gsp = 4.186, gln = −0.1, gsn = −2.870]. In each casethe two calculated g-factor results are presented in Table III,first with bare and then with effective nucleon g factors.

Table III shows that the calculated excitation energiesE(2+

1 ) and E(4+1 ) in Space 2 are closer to the experimental

values.Experimentally, the g(2+

1 ) is about twice the g(4+1 ).

However, the present shell-model calculations always predictvalues that are very close to each other.

The larger g(2+1 ) value is best predicted with the bare

nucleon g factors in Space 2. The smaller g(4+1 ) value is

well accounted for in both spaces with the effective nucleon g

TABLE III. Large-scale shell-model results for 106Cd. The config-urations used in the calculations for Space 1 and Space 2 are identifiedin the text. The two results quoted for the B(E2) values correspondto different choices of effective charges, (ep,en), as discussed in thetext. Similarly, the two results for the calculated g factors correspondto choices of either bare or effective nucleon g factors, as describedin the text.

Expt. Space 1 Space 2

E(2+1 ) 632.6 keV 493 685

E(4+1 ) 1493.8 keV 1216 1357

B(E2; 2+1 → 0+

1 ) 0.115(8) e2b2 0.061 0.0520.097 0.083

B(E2; 4+1 → 2+

1 ) 0.069(4) e2b2 0.083 0.0550.132 0.087

g(2+1 ) +0.398(22) +0.320a +0.371a

+0.211b +0.253b

g(4+1 ) +0.23(5) +0.339a +0.346a

+0.214b +0.204b

aCalculation done with bare nucleon g factors.bCalculation done with effective nucleon g factors.

034303-4

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MAGNETIC MOMENT AND LIFETIME MEASUREMENTS OF . . . PHYSICAL REVIEW C 94, 034303 (2016)

factors. The calculation using effective g factors always leadsto predicted 106Cd g-factor values that are about 70% of thosepredicted by the calculations using bare g factors.

In Ref. [6], tidal wave calculations predict for106Cd g(2+

1 ) = +0.314 and g(4+1 ) = +0.327.

The corresponding calculated B(E2) values, with any oneset of (ep,en) values, are always larger in Space 1 (whichincludes the s1/2 orbital). For the 2+

1 → 0+1 transition the

results of the B(E2) calculations even with ep = 2.0 anden = 1.0 are only about 70%–80% of the experimental value.For the B(E2; 4+

1 → 2+1 ) the calculated results agree with

the experimental value best for ep = 1.75, en = 0.75. Similarlarge effective charges were used in this region [3,16]. Anothercalculation with smaller (ep,en) = (1.5,0.5) led to B(E2)results much smaller than the experimental ones and are notincluded in Table III.

The need for large (ep,en) effective charges to explain theB(E2) data indicates the presence of some collectivity in106Cd. Yet that collectivity is limited since this nucleus isonly two proton holes away from the Z = 50 magic number.

It should be noted that simple collective models donot account for several properties of 106Cd, as detailedbelow.

The observed ratio of the excitation energies E(4+1 )/E(2+

1 )is 2.36; the pure vibrational model predicts 2.00 for this ratiowhile the pure rotational model predicts 3.33. The vibrationalmodel predicts a degenerate 0+

2 , 2+2 , 4+

1 triplet at an excitationenergy of twice E(2+

1 ) or at 1266 keV. Experimentally, nolow-lying 0+

2 was observed in this experiment, the 4+1 state lies

at 1493.8 keV, and the 2+2 state is at 1716.5 keV.

The observed ratio B(E2; 4+1 → 2+

1 )/B(E2; 2+1 → 0+

1 ) =0.599(54). This ratio is predicted to be 2.00 in the vibrationalmodel and 1.43 in the rotational model.

Collective models predict identical values for g(2+1 ) =

g(4+1 ) = Z/A = +0.453. Greiner [17] suggested corrections

which reduce these values. The measured g(2+1 ) in the present

work can be explained by Greiner’s approach, but the g(4+1 )

is still too low. A ratio of g(2+1 )/g(4+

1 ) = 1.70(39) wasobserved here for 106Cd. The highest theoretical value forg(2+

1 )/g(4+1 ) = 1.24, was obtained from the LSSM calculation

in Space 2 with effective nucleon g factors.

IV. SUMMARY

The mean lives of the 4+3 and 3−

1 states in 106Cd weremeasured for the first time. The current investigation alsoremeasured the mean lives of the 2+

1 , 2+2 , 4+

1 , and 4+2 levels

in 106Cd. In all four of these cases, the new values disagreesignificantly with the literature values.

The current experiments also measured for the first time theg(4+

1 ) value in 106Cd and fully reproduced the literature valueof the g(2+

1 ). The g factor of the 4+1 state is about 59% that

of the 2+1 state. This large difference cannot be explained by

simple collective models, or within the framework of a tidalwave model [6]. These models predict g(4+

1 ) values that arevery close to g(2+

1 ). The shell-model Space 2 calculations,with effective nucleon g factors, do yield g(2+

1 ) > g(4+1 ),

in agreement with experiment. But while these calculationsare in agreement with the experimental g(4+

1 ) value theyunderpredict the g(2+

1 ) value. Overall, unlike some heavier Cdisotopes, 106Cd is somewhat better described in the shell modelbased on specific single proton and neutron orbitals near thedoubly magic N = Z = 50 shell closure. The experimentaldiscrepancies in the lifetimes should be resolved by futureCoulomb excitation and dedicated DSAM measurements.

ACKNOWLEDGMENTS

The authors thank the Berkeley 88-Inch Cyclotron stafffor their help in setting up the experiment and providingthe cadmium beam. The target was prepared by P. Maier–Komor at the Technische Universitat Munich, Germany. Theauthors are grateful to L. Zamick for many discussions andsuggestions about the theoretical interpretation of the g-factor results. K.-H.S. acknowledges support by the DeutscheForschungsgemeinschaft under SP190/18-1. D.A.T., Z.E.G.,and F.R. acknowledge support by Colciencias under contract110165842984 - 2015. Y.Y.S. acknowledges a StocktonUniversity Research and Professional Development award.The work has been supported in part by the U.S. NationalScience Foundation and by the US Department of Energy,Office of Science, Office of Nuclear Physics under contractsNo. DE-AC02-05CH11231, No. DE-AC52-07NA27344 andNo. DE-AC05-00OR22725(ORNL).

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