Regularities and symmetries in atomic structure and spectra

TL;DR

This paper explores statistical regularities and symmetries in atomic spectra, revealing fractal properties, Benford's law adherence, and selection rules that deepen understanding of complex quantum systems.

Contribution

It introduces the fractal nature of line intensity distributions and identifies new symmetries and regularities in atomic spectra not previously characterized.

Findings

Line strength distribution follows Benford's law.

The fractal dimension of line intensity regularity is calculated.

Additional selection rules and symmetries are identified.

Abstract

The use of statistical methods for the description of complex quantum systems was primarily motivated by the failure of a line-by-line interpretation of atomic spectra. Such methods reveal regularities and trends in the distributions of levels and lines. In the past, much attention was paid to the distribution of energy levels (Wigner surmise, random-matrix model...). However, information about the distribution of the lines (energy and strength) is lacking. Thirty years ago, Learner found empirically an unexpected law: the logarithm of the number of lines whose intensities lie between 2^kI_0 and 2^{k+1}I_0, I_0 being a reference intensity and k an integer, is a decreasing linear function of k. In the present work, the fractal nature of such an intriguing regularity is outlined and a calculation of its fractal dimension is proposed. Other peculiarities are also presented, such as the fact that the distribution of line strengths follows Benford's law of anomalous numbers, the existence of additional selection rules (PH coupling), the symmetry with respect to a quarter of the subshell in the spin-adapted space (LL coupling) and the odd-even staggering in the distribution of quantum numbers, pointed out by Bauche and Coss\'e.