Collision Matrices for the Compound Nucleus

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The collision matrix found by Newton to satisfy at all excitation energies the requirement that it describe an excited nucleus whose decay is independent of the mode of formation is shown to imply the vanishing of the absorption cross sections at high energies where the levels overlap and therefore does not describe a compound nucleus in the usual sense. The essential characteristic of this matrix is the high degree of the correlations of the signs of the square roots ${\ensuremath{\gamma}}_{\ensuremath{\lambda}c}$ of the reduced level widths for the various levels $\ensuremath{\lambda}$ and channels $c$. On the other hand, a collision matrix, which is similar to one first considered by Bethe and involves ${\ensuremath{\gamma}}_{\ensuremath{\lambda}c}$ whose signs are uncorrelated, implies energy average decays that are independent of the formation mode and absorption cross sections that are of the order of magnitude of the nuclear area at high energies. These matrices are derived and discussed by using the $R$-matrix theory of Wigner, Eisenbud, and Teichmann. It is shown that the Bethe form of the collision matrix, which is valid only if all of the partial level widths are much less than the spacings, may be modified by means of the Teichmann-Wigner channel elimination procedure so that it is also valid in situations where some of the partial widths exceed the spacings. The form of the compound-nucleus collision matrix thus obtained is similar to one deduced by Weisskopf by considerations involving the compound-nucleus hypothesis and the reciprocity theorem. The pole strength functions ${s}_{c}$, which are the averages of the ratios of reduced widths ${{\ensuremath{\gamma}}_{\ensuremath{\lambda}c}}^{2}$ to level spacings, and their Stieltjes energy transforms are decisive in the determination of the behavior of these collision matrices and their associated cross sections. The $s$ functions and their transforms are presented and discussed in the cases of the strong-coupling and complex square well potential representations of the particle-nuclei interactions. The latter representation with an additional surface absorption is also considered. The present theory indicates that the imaginary part of the complex square well potential should increase with the absorption width, and it suggests a "giant resonance" interpretation of the average cross-section behaviors. The effect of the compound-nucleus on non-compound-nucleus processes such as stripping and pickup is also mentioned.