Frequency moments of the Coulomb dynamic structure factor and related integrals

TL;DR

This paper derives explicit formulas for the moments of the Coulomb plasma's dynamic structure factor, linking them to dielectric properties, and extends these results to various plasma degeneracy regimes.

Contribution

It provides explicit polynomial-based formulas for the moments of the structure factor up to n=20, including generalizations to degenerate plasmas.

Findings

Explicit formulas for moments up to n=20

Generalized results for degenerate plasmas

Connections between moments and dielectric function

Abstract

This report addresses the moments, ${\mathfrak{G}_n}\left( {\mathbf{q}} \right) = \int_{ - \infty }^{ + \infty } {{\omega ^n}S\left( {{\mathbf{q}},\omega } \right)\mathrm{d}\omega },\,n \in \mathbb{N},\,n \geq - 1$, of the quantum mechanical dynamic structure factor $S\left( {{\mathbf{q}},\omega } \right)$ for a one-component Coulomb plasma in thermodynamic equilibrium. The Fluctuation Dissipation Theorem relates these moments to integrals involving the imaginary part of the inverse longitudinal dielectric function, with the odd moments in particular being equivalent to the odd moments of the imaginary part of the inverse dielectric function. Application of the Generalized Plasmon Pole Approximation arXiv:1508.05606 [physics.plasm-ph] to a weakly-coupled non-degenerate plasma, leads to general formulae expressed in terms of polynomial functions. Explicit forms of these functions are given for $n \leq 20$. These formulae are generalized to degenerate and partially degenerate plasmas, in small-$\mathbf{q}$ (long-wavelength) regimes.