Hydrodynamics of quantum corrections to the Coulomb interaction via the third rank tensor evolution equation: Application to the Langmuir waves and the spin-electron-acoustic waves

TL;DR

This paper develops an advanced quantum hydrodynamic model including third rank tensor evolution equations to study quantum corrections to Coulomb interactions, revealing new effects on plasma wave spectra beyond traditional models.

Contribution

The paper introduces a quantum hydrodynamic framework with higher-rank tensor equations, extending beyond the 13-moments approximation to incorporate quantum corrections to Coulomb interactions.

Findings

Quantum corrections modify Langmuir wave spectra, increasing the pressure contribution from $(3/5)v_{Fe}^{2}$.

New wave solutions emerge due to quantum effects, affecting spin-electron-acoustic wave behavior.

The model distinguishes between single-fluid and spin-separated electron regimes, impacting plasma wave dynamics.

Abstract

If we study the quantum effects in plasmas in terms of traditional hydrodynamics via the continuity and Euler equations we find the quantum Bohm potential and the force of spin-spin interaction. However, if we extend the set hydrodynamic equations beyond the 13-moments approximation, and include the third rank tensor evolution equation along with the pressure evolution equation, we obtain the quantum corrections to the Coulomb interaction. It is found in contrast with the fact that hydrodynamic equations for the higher rank tensors do not contain interaction in the classic plasmas studied in the selfconsistent (meanfield) approximation. Therefore, we present the quantum hydrodynamic model, where the quantum effects are studied beyond the quantum Bohm potential. Developed model is considered in two regimes: all electrons in plasmas are considered as the single fluid, and the separate spin evolution regime, where electrons with different spin projections are considered as two different fluids. To illustrate the fundamental meaning of found quantum effects we demonstrate their contribution in the spectrum of the Langmuir waves and the spin-electron-acoustic waves. It is worth to mention that the application of the pressure evolution equation ensures that the contribution of pressure in the Langmuir wave spectrum is proportional to $(3/5)v_{Fe}^{2}$, unlike $(1/3)v_{Fe}^{2}$ appearing from hydrodynamics based on the continuity and Euler equations, where $v_{Fe}$ is the Fermi velocity. Same correction corresponds on other plasmas phenomena like the speed of sound for spin-electron-acoustic waves. Moreover, it is found that novel quantum effects provide the novel wave solutions.