Exciton level densities with spin and parity based on random matrix model

Abstract

Level densities for fixed J~ and the exciton number are evaluated for closed shell nuclei 4~ and z~ The single particle spectra and wave functions are generated by Woods-Saxon potentials. The effects of the residual interaction are taken into account statistically by the method of generating function and Grassmann integral. The matrix elements for the residual interaction are assumed to be random variables with Gaussian distributions whose second moments are calculated by using a zero range interaction. The second moments are evaluated for fixed J~ by ignoring the Pauli principle between active nucleons and the spectator. This approximation is shown numerically to be very good. The partial level densities are calculated using the second moments as well as independent particle model spectra. The resulting level densities spread over wider energy ranges, have a smoother energy dependence and are enhanced at low energies compared with the independent particle model densities, although the total level densities do not differ by much.

Advances in nuclear reaction calculations by incorporating information from nuclear mean-field theories

“…However, such calculation does not include particle-hole interactions. An approximation to the shell model level density based on the random matrix theory was proposed by Pluhař and Weidenmüller [10,25], then applied to practical calculations [11,12]. In this technique the Hamiltonian for the nuclear system has the singleparticle part H 0 and the residual interaction V ,…”

“…M m,m is for the scattering process, and M m,m+2 is for the particle-hole creation process. Although this is basically repeating the calculations of Sato, Takahashi, and Yoshida [11], the current calculation has a much larger model space, and the wave-functions are taken from FRDM, instead of the spherical Woods-Saxon potential. An extension to the deformed nucleus case is straightforward.…”

“…We calculate the microscopic nuclear level densities that include the residual interaction of Eq. (1) based on the random matrix theory [10][11][12]. This microscopic level density model was originally developed for calculating partial level densities with a fixed number of excitons in the quantum mechanical multistep direct reactions.…”

Approximation for shell-model level densities: Ergodicity

“…The method which was used to calculate the ensemble-averaged level densities employs a representation of the Green's function of a random Hamiltonian in terms of a Grassmann integral, and a saddle-point approximation. In a more complex but also * On leave flom the Charles University, Prague, Czech Republic more realistic framework, this approach has meanwhile been used for practical calculations [5,6]. In principle, the approach can be used for estimating the level densities of general Hamiltonians which act in a finite-dimensional Hilbert space.…”

“…The desired density estimates follow by taking the Grassmann integral (6) in the saddle-point approximation. For thejth partial density, the explicit result is (to simplify the notation, we write ~cj and ~j without denoting their dependence on E, and use the same symbol for the exact density and for its saddle-point approximation)…”