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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 R-Matrix theory of Wigner and Eisenbud and the S-Matrix 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 S-Matrix approach and to use expansions other than the Mittag-Leffler which was used in the development of the general theory. We discuss two alternate expansions. With the modification to the S-Matrix 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 R-Matrix interpretation is fully satisfactory. The S-Matrix 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. S-Matrix theory correctly locates the bound state but R-Matrix theory does not. For threshold resonances one R-Matrix level is involved but two S-Matrix poles (except for the 1-S state) give rise to the resonance cross section. The physical interpretations are consolidated by describing the cross section for n-p, n-l60 and n-208Pb 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. S-Matrix 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 R-Matrix and S-Matrix 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 R-Matrix collision matrix is demonstrated for all approximations. It is possible to satisfy the unitarity requirements explicitly in the S-Matrix 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 S-Matrix theory is poor. Except for isolated resonances far from threshold the R-Matrix and S-Matrix theories give quite different values for the parameters of resonance levels.

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