The neutron strength function and the shell model

Abstract

The projection operator-equivalent potential method is employed for the study of neutron widths and the neutron strength function. Assuming that the residual potential, V R, consists of a sum of two particle potentials it can be shown with the aid of another plausible assumption that the width can be written as a product of two factors. One of these is the square of the matrix element of V R between the open channel wave function and a “shell model” two particle-one hole state. This factor which plays a crucial role measures the probability that in the first scattering of the incident particle a transition to the two particle-one hole state occurs. It contains the fluctuations in the width which accompany changes in the target nucleus. The second factor, giving the probability that the compound nucleus wave function contains the two particle-one hole state, decreases as the difference between the energy of the compound nucleus and the energy of the two particle-one hole state increases. This factor has a Lorentzian form with a half-width Δ corresponding to the lifetime of the two particle-one hole state. These results are generalized to include the case where the energies of several two particle-one hole states fall close to each other. Here it is found that the giant resonance for the widths will now have a substructure which should be observable. The strength function is found to depend upon the average of the transition probabilities to the two particle-one hole states, the number of these states, and Δ. The number is sharply limited by the requirements that angular momentum and parity be conserved in the transition and that energy be conserved to within an error equal to Δ. If the residual potential is expanded in multipoles, keeping terms up to and including the quadripole, it is found that for the most part in the region 40 ≦ A ≦ 64 the only possible transitions involve the monopole; between A = 68 and 85, both dipole and quadripole terms are effective as well, while between 85 and 130 only the quadripole term can induce transitions. This behavior enables one to obtain a fit to the data by means of three parameters: the average matrix elements squared for the monopole, dipole, and quadripole parts of the residual potential. The resulting fit is excellent providing a description not only of the deviations of the strength function from the optical model result but also of the fluctuations in the strength function as the target nucleus is changed. Predictions of the dependence of the strength function for various isotopes become possible. The success of these calculations demonstrates the essential importance of the “first” collision which the incident particle can undergo.