Regularities and symmetries in atomic structure and spectra

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 k I0 and 2 k+1 I0, I0 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 spinadapted space (LL coupling) and the odd-even staggering in the distribution of quantum numbers, pointed out by Bauche and Cossé.

Distribution of the total angular momentum in relativistic configurations

“…(II. 19) One notes that the same recurrence relations formally hold for P (M ) and Q(J). Of course the solutions for P (M ) and Q(J) differ because the initial values (N = 1 or j = 0) are different for P and Q.…”

“…As shown in Ref. [19] and in Appendix A the present generating-function formalism allows for a simpler derivation of the excess evaluation by computing G (j; N ; −1). Moreover, other sum rules may also be obtained by computing this generating function at other points on the trigonometric circle.…”

“…C. Application to the analysis of even-odd staggering in angular-momentum distribution It has been shown by Bauche and Cossé [18] and Pain [19] that when the number of electrons N in a subshell is even, the levels with even angular momentum J even Q(J) outnumber the levels with odd angular momentum J odd Q(J). This effect was proven to exist for non-relativistic as well as relativistic configurations.…”

Simple Explanation for the Observed Power Law Distribution of Line Intensity in Complex Many-Electron Atoms

“…Pain recently reviewed this power law dependence problem and presented a discussion regarding fractal dimension and quantum chaos [6]. According to his discussion, the line strength distribution evaluated under the fully quantum-chaos assumption does not explain Learner's law.…”

“…According to his discussion, the line strength distribution evaluated under the fully quantum-chaos assumption does not explain Learner's law. As presented in his review [6], as well as in the book [7], the origin of this power law is still not understood, despite almost 40 years passing since the first report.…”