[Paper Review] Generation of longitudinal electric current by the transversal electromagnetic field in classical and quantum degenerate plasma

А. В. Латышев, A. A. Yushkanov|arXiv (Cornell University)|Mar 30, 2015

Dust and Plasma Wave Phenomena9 references3 citations

TL;DR

This paper demonstrates that nonlinear interactions between transverse electromagnetic fields and degenerate collisionless plasmas—both classical and quantum—generate a longitudinal electric current orthogonal to the conventional transverse current. Using Vlasov and Wigner kinetic equations with second-order perturbation expansions, the authors derive analytical expressions showing that the longitudinal current arises due to quadratic nonlinearities in the field intensity (classical) or vector potential (quantum), with identical low-wave-number behavior in both regimes.

ABSTRACT

The analysis of nonlinear interaction of transversal electromagnetic field with degenerate collisionless classical and quantum plasmas is carried out. Formulas for calculation electric current in degenerate collisionless classical and quantum plasmas are deduced. It has appeared, that the nonlinearity account leads to occurrence of longitudinal electric current directed along a wave vector. This second current is orthogonal to the known transversal current, received at the classical linear analysis. Graphic comparison of density of electric current for classical degenerate Fermi plasmas and Fermi-Dirac plasmas (plasmas with any degree of degeneration of electronic gas) is carried out. Graphic comparison of density of electric current for classical and quantum degenerate plasmas is carried out. Also comparison of dependence of density of electric current of quantum degenerate plasmas from dimensionless wave number at various values of dimensionless frequency of oscillations of electromagnetic field is carried out.

Motivation & Objective

• To investigate nonlinear interactions between transverse electromagnetic fields and degenerate collisionless plasmas.
• To derive analytical expressions for electric current in both classical and quantum degenerate plasmas under nonlinear conditions.
• To identify and quantify the emergence of a longitudinal current component orthogonal to the standard transverse current.
• To compare the behavior of longitudinal current in classical Fermi–Dirac plasmas versus quantum degenerate plasmas across varying wave numbers and frequencies.
• To establish the equivalence of longitudinal current expressions in the small-wave-number limit between classical and quantum regimes.

Proposed method

• Employed the Vlasov equation with perturbative expansion up to second order in electric field intensity for classical degenerate plasmas.
• Applied the Wigner integral equation with second-order expansion in vector potential for quantum degenerate plasmas.
• Used decomposition of distribution functions and electromagnetic fields in powers of field intensity (classical) or potential (quantum).
• Calculated current density components: transverse (linear in E) and longitudinal (quadratic in E, aligned with wave vector k).
• Derived analytical expressions for longitudinal current density in both classical and quantum cases using Fermi–Dirac statistics and phase-space integrals.
• Performed graphical analysis of real and imaginary parts of dimensionless current density as functions of dimensionless wave number q and frequency Ω.

Experimental results

Research questions

• RQ1 Does nonlinear interaction of transverse electromagnetic fields with degenerate plasmas generate a longitudinal electric current?
• RQ2 How does the longitudinal current in classical degenerate plasma compare quantitatively to that in quantum degenerate plasma?
• RQ3 What is the behavior of the longitudinal current at low wave numbers (k → 0) in both classical and quantum regimes?
• RQ4 How do the real and imaginary parts of the longitudinal current vary with dimensionless wave number q and dimensionless frequency Ω in quantum plasma?
• RQ5 Is there a regime where classical and quantum longitudinal current expressions converge?

Key findings

• A longitudinal electric current is generated in both classical and quantum degenerate plasmas due to second-order nonlinearities in the electromagnetic field.
• The longitudinal current is orthogonal to the transverse current and proportional to the square of the electric field intensity in classical plasma.
• In the small-wave-number limit (k → 0), the longitudinal current in classical and quantum plasmas is described by the same analytical expression: $ j_x^{ ext{class}} = j_x^{ ext{quant}} = -rac{ ho_0 v_F}{8 au} rac{1}{ u^3} $, where $ u $ is the dimensionless frequency.
• Graphical analysis shows that at small q, the real parts of the longitudinal current in classical and quantum plasmas are nearly identical and converge in the limit q → 0.
• As q increases, both real and imaginary parts of the longitudinal current in quantum plasma approach those in classical plasma.
• The imaginary part of the quantum plasma current exhibits a minimum across all Ω values, while the real part shows a minimum at q < Ω and a maximum at q > Ω.