Statistical properties of transition arrays

Abstract

A radiative transition is the quantum jump between two electronic states, associated with the emission or absorption of a photon. It can be of various electromagnetic modes, among which the more intense are denoted E1, E2, and M1. The transition array is the denomination for the ensemble of transitions between two configurations. Compact formulas are obtained for the number of lines, for the total intensity, and for the first two moments (average wavenumber σ and width v) of the energy distribution function of the array. This description can be refined by using its third- and fourth-order moments. It is also useful to compute the strength-weighted average wavenumber and width of the level distribution, for determining the emissive zone, i.e., the levels which emit (or absorb) most. All these results are exact results: they account for the intermediate coupling, without resorting to any matrix diagonalization.

The configuration-interaction effects on the σ quantity are generally small, but, in some specific cases, they can be spectacular. Then, the vanishing of some whole arrays can be predicted. For heavy elements, the occurrence of large spin-orbit integrals breaks the band of radiative lines into two or three narrow bands, for which the σ and v values are computed. Many spectra of heavy elements are presented.

Eventually, statistical laws are given for explaining other aspects of the spectral structure: first, a correlation between the upper and lower energies of the lines; secondly, a correlation between the line wavenumbers and intensities; thirdly, the extended J-file sum rule, the application of the Porter-Thomas law, the occurrence of scars of symmetries, and fractal structures.