Quantum theory of electron gas plasma oscillations in a magnetic field

• Published in: Annals of physics 1965, Vol.31 (1), p.1-63
• Main Author: Horing, Norman J
• Format: Article
• Language: English
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/0003-4916(65)90231-9

The response function (inverse dielectric function) for an electron gas in an external magnetic field is investigated by the method of Green's functions using the random phase approximation. The spectrum of plasma oscillations, and associated damping, are obtained. In particular, for wave vector p perpendicular to the magnetic field the dispersion relation is studied as a power series in p 2, and is shown to contain oscillatory terms of the de Haas-van Alphen type in the degenerate case, and quantum dynamical corrections through the parameter ℏ ω c β in the nondegenerate case. A Mittag-Leffler expansion of this dispersion relation shows that integral multiples of the cyclotron frequency are forbidden, and there are an infinite number of plasma modes (resonances), one in each interval [ nω c , ( n + 1) ω c ]. A semiclassical model is worked through in detail, and the frequencies of these undamped plasma modes are calculated when p 2 (ζ or 1 β ) mω c 2 ⪢ 1 (low magnetic field). The gaps in the frequency spectrum are calculated. Likewise the excitation amplitudes of the various modes are calculated. “Phase mixing” of the modes (resonances) is considered for small ω c , and is seen to result in a single damped plasma mode at Ω ∼ ω p , as one should expect. The dispersion relation for wave vector p having arbitrary direction with respect to the magnetic field is studied as a power series in p 2, and again oscillatory terms of the de Haas-van Alphen type are found in the degenerate case. Known nondegenerate results bearing quantum dynamical corrections through the parameter ℏ ω c β are verified. The semiclassical model is employed to calculate angular corrections to the plasma modes (resonances) in the intervals [ nω c , ( n + 1) ω c ] for small angular departures from the case of perpendicular propagation, θ ⪡ 1, when p 2 (ζ or 1 β ) mω c 2 ⪢ 1 . The excitation amplitudes of all the plasma modes considered are calculated. Detailed formulas for natural damping, which take full account of the influence of the magnetic field, are given for all the plasma modes considered. These reduce to known results in the nondegenerate case and the field free degenerate case. They show clearly the effects of the field on damping in the degenerate case, and the role of oscillatory terms in the de Haas-van Alphen sense is investigated.