Variance of the distributions of energy levels and of the transition arrays in atomic spectra

Abstract

Formulas are derived for the mean values and variances of the energy distributions of the levels of an atomic configuration and of the radiative transitions between the levels of two configurations (in intermediate coupling). The variance ${\sigma }^2$ of the distribution of the eigenstate energies belonging to a given configuration is considered first: ${\sigma }^2$ is expressed as a linear combination of squares and cross products of the usual Slater electrostatic and spin-orbit radial integrals. It is shown how this expression can be used to check the numerical matrices of energy-integral coefficients. Then expressions are derived for the mean value and for the variance of the weighted distribution of the transition energies between two configurations (the weight of each transition being its strength) in the $n{l}^{N+1\mathrm{}}-n{l}^N{n}^{\prime }{l}^{\prime }$ and $n{l}^N{n}^{\prime }{l}^{\prime }-n{l}^N{n}^{\prime \prime }{l}^{\prime \prime }$ cases. This derivation is based on the second-quantization formalism. An extension is made to the case of complementary configurations. For transitions $n{l}^{N+1\mathrm{}}-n{l}^N{n}^{\prime }{l}^{\prime }$, an explicit formula is obtained for the shift between the mean energy of the transition array and the difference of the mean energies of the configurations. Numerical tables of the angular coefficients appearing in ${\sigma }^2$ are given for most cases where $l$, $l^{\prime }$, $l^{\prime \prime }\le 3$. The main application presented here concerns highly ionized spectra of molybdenum, with transitions between $3d^{N+1\mathrm{}}$ and $3d^{N}4p$, $3d^{N}4f$, $3d^{N}5p$, and $3d^{N}5f$. The agreement between experimental and theoretical (ab initio) mean wave numbers and variances is good. A discussion of the physical conditions of applicability of the results to experimental situations is given.