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Comparison of nuclear reaction theories Tindle, Christopher Thomas
Abstract
The two theories of low energy nuclear reactions which are mainly used for the interpretation of experimental data are compared. The two theories of interest are the RMatrix theory of Wigner and Eisenbud and the SMatrix theory of Humblet and Rosenfeld. The two approaches to resonance reactions are quite different and the differences are discussed with reference to a variety of specific examples. A simple soluble model  the threshold resonances of scattering by a square potential well  is analysed in detail using the two approaches. The approximate formulae are then compared numerically with the exact solution. It proves necessary to modify the usual SMatrix approach and to use expansions other than the MittagLeffler which was used in the development of the general theory. We discuss two alternate expansions. With the modification to the SMatrix theory both approaches give very accurate approximate formulae. The theories give different interpretations of the position and width of the threshold level. If the level is unbound the RMatrix interpretation is fully satisfactory. The SMatrix interpretation is unsatisfactory because the level has the characteristics of a bound state but none exists. If the threshold level is bound the position is reversed. SMatrix theory correctly locates the bound state but RMatrix theory does not. For threshold resonances one RMatrix level is involved but two SMatrix poles (except for the 1S state) give rise to the resonance cross section. The physical interpretations are consolidated by describing the cross section for np, nl60 and n208Pb scatterings. The slow neutron cross section of ¹³⁵Xe is discussed using both formalisms. This is an example of a narrow compound nucleus resonance very close to a channel threshold. The theories fit the data with different parameters and very near threshold they give quite different shapes to the cross section. The origin of this difference is traced to unitarity. SMatrix theory, in this situation, fails to give the cross section the correct behaviour very near threshold, because its approximation to the collision matrix is not unitary. Two level interference is discussed. Artificial cross sections are constructed to illustrate the very different interpretations that the two approaches may give to an interference cross section. The (p, y) and (p, n) cross sectionsof ¹⁴C are analysed using both RMatrix and SMatrix formalisms. ¹⁵N* has two very wide ½+ levels near neutron threshold. Both approaches fit the data to very good accuracy. The level positions and widths are quite different but the partial widths are similar. An analytic method of relating the parameters of the two theories by a transformation is given with the necessary approximations noted. The accuracy of the method is confirmed by application to the ¹⁴C+p cross section parameters. The transformation is used to discuss some theoretical points. Unitarity is discussed and the unitarity of the RMatrix collision matrix is demonstrated for all approximations. It is possible to satisfy the unitarity requirements explicitly in the SMatrix theory in only the simplest situations and with poor approximations and the reasons for this are discussed. It is concluded that in most situations both theories are capable of fitting experimental data. The only situation in which there is a measurable (though small) difference is very near threshold. If one requires that unitarity be satisfied for all approximate formulae the SMatrix theory is poor. Except for isolated resonances far from threshold the RMatrix and SMatrix theories give quite different values for the parameters of resonance levels.
Item Metadata
Title  Comparison of nuclear reaction theories 
Creator  Tindle, Christopher Thomas 
Publisher  University of British Columbia 
Date Issued  1970 
Description 
The two theories of low energy nuclear reactions which are mainly used for the interpretation of experimental data are compared. The two theories of interest are the RMatrix theory of Wigner and Eisenbud and the SMatrix theory of Humblet and Rosenfeld. The two approaches to resonance reactions are quite different and the differences are discussed with reference to a variety of specific examples.
A simple soluble model  the threshold resonances of scattering by a square potential well  is analysed in detail using the two approaches. The approximate formulae are then compared numerically with the exact solution. It proves necessary to modify the usual SMatrix approach and to use expansions
other than the MittagLeffler which was used in the development of the general theory. We discuss two alternate expansions. With the modification to the SMatrix theory both approaches give very accurate approximate formulae. The theories give different interpretations of the position and width of the threshold level. If the level is unbound the RMatrix interpretation is fully satisfactory. The SMatrix interpretation is unsatisfactory because the level has the characteristics of a bound state but none exists. If the threshold level is bound the position is reversed. SMatrix theory correctly locates the bound state but RMatrix theory does not. For threshold resonances one RMatrix level is involved but two SMatrix poles (except for the 1S state) give rise to the resonance cross section. The physical interpretations are consolidated by describing the cross section for np, nl60 and n208Pb scatterings.
The slow neutron cross section of ¹³⁵Xe is discussed using both formalisms. This is an example of a narrow compound nucleus resonance very close to a channel threshold. The theories fit the data with different parameters
and very near threshold they give quite different shapes to the cross section. The origin of this difference is traced to unitarity. SMatrix theory, in this situation, fails to give the cross section the correct behaviour very near threshold, because its approximation to the collision matrix is not unitary.
Two level interference is discussed. Artificial cross sections are constructed to illustrate the very different interpretations that the two approaches may give to an interference cross section. The (p, y) and (p, n) cross sectionsof ¹⁴C are analysed using both RMatrix and SMatrix formalisms. ¹⁵N* has two very wide ½+ levels near neutron threshold. Both approaches fit the data to very good accuracy. The level positions and widths are quite different but the partial widths are similar.
An analytic method of relating the parameters of the two theories by a transformation is given with the necessary approximations noted. The accuracy of the method is confirmed by application to the ¹⁴C+p cross section parameters. The transformation is used to discuss some theoretical points. Unitarity is discussed and the unitarity of the RMatrix collision matrix is demonstrated for all approximations. It is possible to satisfy the unitarity
requirements explicitly in the SMatrix theory in only the simplest situations and with poor approximations and the reasons for this are discussed.
It is concluded that in most situations both theories are capable of fitting experimental data. The only situation in which there is a measurable
(though small) difference is very near threshold. If one requires that unitarity be satisfied for all approximate formulae the SMatrix theory is poor. Except for isolated resonances far from threshold the RMatrix and SMatrix theories give quite different values for the parameters of resonance levels.

Subject  Nuclear reactions 
Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20110531 
Provider  Vancouver : University of British Columbia Library 
Rights  For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. 
DOI  10.14288/1.0084794 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Physics 
Affiliation  Science, Faculty of; Physics and Astronomy, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
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For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.