Super Transition Arrays: A Tool for Studying Spectral Properties of Hot Plasmas

Abstract

For the theoretical study of X and extreme-UV spectra of ions in plasmas, quantum mechanics brings more detailed results than statistical physics. However, it is impossible to handle individually the billions of levels that must be taken into account in order to properly describe hot plasmas. Such levels can be gathered into electronic configurations or superconfigurations (groups of configurations) and the corresponding calculations rely on appropriate statistical methods, for local or non-local thermodynamic equilibrium plasmas. In this article we present the basic principles of the Super-Transition-Array approach as well as its practical implementation. During the last decades, calculations performed with the SCO code (Superconfiguration Code for Opacity) have been compared to opacity measurements. The code includes static screening of ions by plasma and is well suited for studying plasma density effects (for example pressure ionization) on opacity and equation of state. The recently developed SCO-RCG code (Superconfiguration Code for Opacity combined with Robert Cowan’s “G” subroutine) combines statistical methods from SCO and fine-structure (detailed-level-accounting) calculations using subroutine RCG from Cowan’s code. SCO-RCG enables us to obtain very detailed spectra and to significantly improve the interpretation of experimental spectra. The Super-Transition-Array formalism is still the cornerstone of several opacity codes, and new ideas are emerging, such as the Configurationally Resolved-Super-Transition-Array approach or the extension of the Partially Resolved-Transition-Array concept to the superconfiguration method.

A superconfiguration calculation of opacity with consistent bound and continuum electron treatments using green’s functions

“…For concision, we will not repeat the detailed expressions here, as these have been the focus of previous works by others (e.g. [31,38,46]). Where possible, we use the same notation as in Bar-Shalom et al's original STA work and we apply Hartree atomic units throughout.…”

“…Making this type of substitution means that only one structure calculation is required for each SC, but the fact that each configuration has a unique set of integer subshell occupation numbers means that we would still need to explicitly calculate the electron-electron contribution to the configuration-average energy for each configuration. Once again, this means possibly summing over millions of configurations even for simple systems [31,46].…”

Multi-Configuration Calculation of Ionization Potential Depression

“…A superconfiguration [51,52] is an ensemble of configurations. The superconfiguration approximation consists in partitioning the ensemble of configurations, i.e., in gathering them into groups.…”

“…The superconfiguration approximation consists in partitioning the ensemble of configurations, i.e., in gathering them into groups. A superconfiguration represents a number of configurations of the plasma and is composed by an ensemble of supershells (i.e., groups of subshells) and their respective populations [51,52]. For example, superconfiguration Ξ = (1s2s2p) 10 (3s3p3d) 18 (4s4p4d4 f 5s5p5d) 19 (5 f 5g) 3 represents all the configurations with shells n = 1, n = 2 and n = 3 full, a total of 19 electrons distributed in all possible ways (consistent with the Pauli exclusion principle) in the subshells 4s, 4p, 4d, 4 f , 5s, 5p and 5d (the third supershell consists of 7 orbitals 4s to 5d), and 3 electrons in the last supershell made of subshells 5 f and 5g only.…”

Number of populated electronic configurations in a hot dense plasma

“…In the STA method [28,33,[46][47][48][49][50][51][52][53], a large number of configurations C are grouped into super-configurations (SCs), commonly denoted by Ξ = Π σ σ Qσ , which are defined as sets of configurations which have Q σ electrons in each 'supershell' σ, which is a group of shells. The total occupation of a super configuration is naturally:…”