Bivariate distributions in statistical spectroscopy studies: III. Non interacting particle strength densities for one-body transition operators
In statistical spectroscopy, it was shown by French et al. (Ann. Phys., N.Y. 181, 235 (1988)) that the bivariate strength densities take a convolution form with the non interacting particle (NIP) strength density being convoluted with a spreading bivariate Gaussian due to interactions. Leaving aside the question of determining the parameters of the spreading bivariate Gaussian, one needs good methods for constructing the NIP bivariate strength densitiesIOh(E,E′) (h is a one-body hamiltonian andO is a transition operator) in large shell model spaces. A formalism for constructingIOh is developed for one-body transition operators by using spherical orbits and spherical configurations. For rapid construction and also for applying the statistical theory in large shell model spacesIOh is decomposed into partial densities defined by unitary orbit configurations (unitary orbit is a set of spherical orbits). Trace propagation formulas for the bivariate momentsMrs with r+s ≤2 of the partial NIP strength densities, which will determine the Gaussian representation, are derived. In a large space numerical example with Gamow-Tellerβ− transition operator, the superposition of unitary orbit partial bivariate Gaussian densities is shown to give a good representation of the exact NIP strength densities. Trace propagation formulas forMrs with r+<—4 are also derived inm-particle scalar spaces which are useful for many purposes.