High-order moments of spin-orbit energy in a multielectron configuration

In order to analyze the energy-level distribution in complex ions such as those found in warm dense plasmas, this paper provides values for high-order moments of the spin-orbit energy in a multielectron configuration. Using second-quantization results and standard angular algebra or fully analytical expressions, explicit values are given for moments up to 10th order for the spin-orbit energy. Two analytical methods are proposed, using the uncoupled or coupled orbital and spin angular momenta. The case of multiple open subshells is considered with the help of cumulants. The proposed expressions for spin-orbit energy moments are compared to numerical computations from Cowan's code and agree with them. The convergence of the Gram-Charlier expansion involving these spin-orbit moments is analyzed. While a spectrum with infinitely thin components cannot be adequately represented by such an expansion, a suitable convolution procedure ensures the convergence of the Gram-Charlier series provided high-order terms are accounted for. A corrected analytical formula for the third-order moment involving both spin-orbit and electron-electron interactions turns out to be in fair agreement with Cowan's numerical computations.

Figure 1

Gram-Charlier distribution for the spin-orbit energy of the $3d^6$ configuration in ${\mathrm{Au}}^{55+}$ with an Ar-like core. The various curves correspond to the expansions truncated at various indexes $p$. For comparison purpose one has plotted the data from Cowan's code where each energy level is arbitrarily represented by a Lorentzian of 2 eV FWHM not accounted for in the moment computation. The Gram-Charlier curves are normalized to the degeneracy $g=210$. The ordinates for Cowan's data have been divided by a suitable factor in order to allow comparison.

Figure 2

Gram-Charlier distribution for the spin-orbit energy of the $3d^6$ configuration in ${\mathrm{Au}}^{55+}$ with an Ar-like core convolved by a Gaussian profile with standard deviation $\tau =20\text{eV}$. The Gram-Charlier curves and Cowan's data are normalized to the degeneracy $g=210$.

Figure 3

Absolute value of the Gram-Charlier term series $c_n{\text{He}}_n(X/\sigma )$ and $c_n^{\prime }{\text{He}}_n(X/\upsilon )$ as a function of $n$ for the $d^6$ configuration in ${\mathrm{Au}}^{+55}$. The standard deviation on energy before convolution is $\sigma =2\xi =79.7\text{eV}$. The original distribution has been convolved with a Gaussian profile with standard deviation $\tau =20\text{eV}$. See main text for more details.