owned and operated as a joint venture by a consortium of canadian universities via a contribution...

Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada

Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada

Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire

et en physique des particules

Accelerating Science for Canada

Un accélérateur de la démarche scientifique canadienne

Low energy reaction theory: The forsaken realm

Lothar Buchmann

TRIUMF







What kind of nuclear reactions do we encounter?

Direct measurements: Low energy scattering

Continuous cross section Narrow resonances

Wide resonances

Direct reactions

Extrapolation

Low energy reaction theory

No extrapolation

Nuclear structure theory







Endangered species act

While with the resurgence of low energy physics in the context of radioactive beams facilities the experimental data are growing, there are very few low energy reaction theorists left. In 10-20 years there may be none left.

Experi-menters muddle through







Why do we need low energy scattering theory:

1. Data need to be interpreted. While models may fail the test of being strictly reality, it always has been the goal of physics, to unify observations by simple, all encompassing models. Typically, a first reduction of scattering data is to obtain nuclear potentials and eigenstates of those potentials.

2. The need for predictions, or in nuclear astrophysics, extrapolations. Frequently nuclear cross sections are needed where they cannot be measured. Sometimes other quantities, even at zero energy are also important to characterize a reaction. The task is to make extrapolations as reliable as possible. While there are many approaches to it no king’s way has been determined yet for this problem. Worse, errors on extrapolations can only be derived within a certain approximation used for extrapolation, but the possible errors of the model can not be computed.







An amateur view on nuclear scattering

Most things we do in nuclear physics is scattering: the view with the microscope.`

Nuclei are quantum systems: typically very few (low) angular moments are selected in nuclear reactionsphaseshifts.

0)())](2

)1(([

2

22

2

2

ryrVm

rk

dr

d

Radial Schrödinger equation







Introducing the jargon: PhaseshiftsFor elastic scattering: energy (in cm) is conserved for scattered particle, thus the (de Broglie) wavelength is conserved. The only effect of elastic scattering is a shift in phase in the periodic wave function. Phaseshifts are intimately connected to nuclear potentials and eigenstates. For non charged particles is (spin 0):

2

02

sin)12(4

ktot

This sum breaks down typically to a few or even one angular momentum.Thus maximum cross section at 90 deg resonanceeigenstate.

Typical 1- resonances in 12C(α,ɣ)16O.

Nota bene: Phaseshifts for charged particles contain Coulomb dependent parts.

mEk 21







The scattering length

The concept of the scattering length comes from neutron physics, where phaseshifts can be easily derived at zero energy. For non charged particles is (for the s wave):

20

20

00

40

2

11

a

kra

k s

)(

.....cot

With a0 being the scattering length and r0 being the interaction radius.A positive value of a0 corresponds to an infinite (attractive) square potential of the same radius, a negative a0 corresponds to an infinite (repulsive) square potential of the same radius. For charged particles a Coulomb correction is necessary:

.....1

.....2

11

)(2cot)(),(

200

20

0

0200

Era

kra

hkkCl s

Thus l is a linear function of the energy.

v

eZZ

221

Sommerfeld parameter







The transition matrix:

Typically, there are many channels open to a reaction, even if only elastic scattering is allowed as reaction channels. This is due to the fact that in many cases non zero spins are present, and each spin combination presents its own separate channel. However, as processes are coherent, mixed terms between different channel combinations are also possible. Mixing can be described e.g. by the concept of mixing angles. Due to different angular moments, the resulting cross sections typically show an angular dependence beyond Rutherford scattering.

So far scattering matrices are an abstract concept and need to be brought in a form that allows relatively simple terms for input.

These are (here): The phaseshift approach The R-matrix approximation

Example: 7Be(3/2-)+p(1/2+) Channel spin: s=1 and s=2 (π=-1).







Nuclear part of the collision matrix:

with the mixing coefficients (7Be+p case):

Formalism by R.G. Seyler:

c, p order parameters.

Phaseshift analysis:

7Be case:Number of angular moments included.

Phases are eigenphases for each angular momentum and channel spin combination.







The collision matrix can be expressed in R-matrix terms as:

}2{ 2/1'

2/1'

)( 'c

JJc

Jcccc

iJc PAiPeU cc

The summation runs over states. It is

)arctan(c

chccc G

F

And the inverse of the state (pole) matrix:

)()()( 1ccc

Jc

c

Jc

JJ iPBSEEA

hardsphere phaseshift

R-matrix approach:

Reduced width (state) amplitudes Energy independent.







Example: 12C(α,α)12C

Excitation ratio: 89o to 58o.

12C+α leads to the tightly bound 16O. It shows a series of narrow or medium width resonances. The scattering is well described by the R-matrix formalism (see fit) including a hard sphere potential. The s wave follows essentially hard sphere.

However, the use of a hard sphere potential is not always justified.







Example: 7Be(p,p)7Be:Theoretical s-wave prediction:

Descouvement

Navratil (2010)

In general, these s waves cannot be fitted by R-matrix expressions.

8B is a very weakly bound nucleus!

Hard sphere phaseshift







Derivation of phaseshifts in 7Be(p,p)7Be:

A phaseshift analysis using all 41 parameters resulting from above expressions will result in a very good fit to the data, if done for individual points. However, neighbouring points in energy and angle can deviate considerably in phases and mixing angles. As both of them are expected to show a smooth angle and angular dependence, a global fit to all data with less parameters is required. However, it can be shown in single fits that the mixing angle parameters and the imaginary phaseshifts are largely irrelevant to describe the cross section.

For our global fits, three forms of the phaseshift are chosen:

)( 33

2210 EbEbEbbPh

Js

Polynomial approach. For the s wave it can be shown that b0~a0.

Fits Navratil s-wave perfectly well.

EE

ER

BESER

EPEE

JJs

Js

hJs

2

1

)(

))()(

)(arctan()()(

,

,

OrResonance approach. (i) ɣλ, Eλ real;

(ii) ɣλ, Eλ complex.







Poles and Resonances

The R-matrix approach produces poles of the function

))()(

)(arctan()()(

,BESER

EPEE

Js

hJs

1

for 0;90)(

dE

dE

JsoJ

s

In contrast, e.g. to the K matrix, where

Ee

g

pp πη)(πJ

s

2

122

K

;arctan( exp K);

Describes a resonance as

090 dE

dE

JsoJ

s

;)( i.e. the definitions are not equal.

J. Humblet et al. (1998) also find that in neither theory poles necessarily correspond to resonances, even with ɣ2>0 for R-matrix terms.







Phaseshift for 7Be(p,p)7Be for Oak Ridge data:







Scattering lengths in 7Be(p,p)7Be (discussion):

The l-function for the previous (s wave) phaseshifts.

Correlation between a0 and b0 for a penetration-polynomial fit.







Scattering lengths in 7Be(p,p)7Be (results):

Least squares dependence of a2-,0.

a2- and a1- correlation.







Experimental considerations:

Elastic scattering data are in general folded by the finite experimental conditions: These include target thickness, angular resolution, detector energy and angle resolution, finite beam resolution in space and energy, and straggling effects throughout the target and in the detectors.

While those effects can be modeled or treated by Monte Carlo simulations, the experimental effects are best studied at narrow resonances. There these effects are most pronounced which typical leads to several iterations in the folding model between modeled cross section and experimental yields.

One other interesting point is, how many parameters to be used, i.e., how many partial waves and mixing ratios. The fits shown are restricted in this respect as there are typically more parameters than necessary for fitting.







Conclusions:

1. A phaseshift analysis of typical scattering data is always possible in a general approach.

2. R-matrix analysis with real poles is frequently possible. However, description of background, i.e., non pole terms is sometimes insufficient.

3. With introducing as many poles as necessary to fit data in R-matrix, it is not guaranteed that such poles correspond to eigenstates of the compound nucleus. On the hand, R-matrix analysis is relatively robust against fluctuations in the data.

4. A phaseshift analysis shows the general physics trends of the data. However, it is not robust against fluctuations in the data and the influence of other partial waves. The result is that frequently poles with no justification are produced in the fit. Therefore dense and very precise data are required.

5. A K-matrix analysis is as robust as the R-matrix one. Background terms are more flexible than in R-matrix, for better or for worse.

The analysis of scattering data in (light) nuclei is a fascinating area requiring specialized knowledge. This knowledge is leaving us now, because no faculty is hired to replace outgoing specialists in the field. Books and other publications are just not good enough to guarantee a thorough and valid analysis of data,

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