Distribution functions. Energy levels

Abstract

The frequency distribution functions are universal quantities for describing the statistics of large ensembles. These functions are generally represented by their distribution moments, of various orders. In atomic physics, such moments are computed by means of the tensor-operator formalism, as sums of products of Wigner n-j coefficients. When the summation problem appears to be untractable, two methods may bring a decisive help: the second-quantization formalism developed in atomic physics by Judd, and the graphical methods elaborated by Jucys and his team. After a (limited) number of moment values have been obtained, one enters them into the distribution function of the “best” statistical model, which is a matter of choice (due to the limitation in the number of moments).

The simplest example computed is the statistics of the J values, from which the numbers of levels of the configurations are deduced. The Gram-Charlier distribution generally yields the best results, but it ought to be replaced by a more complicated model for some peculiar types of configurations. In contrast, the statistics of the degeneracy-weighted level energies follows very well the Gram-Charlier model.