Dispersive properties of a magnetized degenerate electron gas

Jeanette I. Weise

Phys. Rev. E 78, 046408 – Published 31 October 2008

Abstract

Accurate dispersion relations are fundamental in the evaluation of many astrophysical processes such as, for example, heat loss by neutrino emission. In this paper, the parallel propagating wave modes of a magnetized, degenerate, electron gas are evaluated exactly using the $S$ -matrix form of the linear response tensor (for a relativistic quantum magnetized electron gas) and the vertex formalism. The modes are then discussed in terms of the dissipative regions in which they occur. In addition to analogous modes to the unmagnetized case, there also exist doublet modes due to the logarithmic singularites that occur in the dissipative regimes of both the longitudinal and transverse dispersion relations and modes above the gyromagnetic thresholds for one of the transverse components. These additional modes only appear over a restricted range of wave numbers and are unique to the magnetized case.

Received 30 July 2008

DOI:https://doi.org/10.1103/PhysRevE.78.046408

Authors & Affiliations

Jeanette I. Weise^*

School of Physics, University of Sydney, NSW 2006, Australia

^*weise@physics.usyd.edu.au

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Vol. 78, Iss. 4 — October 2008

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Figure 1
(a): The boundaries of the different regions in the $ω − k_{z}$ plane for $α^{33}$ , parameters $p_{F} = 2.2 m$ , $B = B_{cr}$ , and $n = 1$ . The solid and dotted lines are the $ω = k_{z}$ and $ω = {ω_{PC} ∣}_{n = 1}$ lines, respectively. The dashed-double-dotted and short-dashed lines, labeled $ω_{P 1}$ and $ω_{P 2}$ , are the $ω = {(ε_{F}^{2} \pm 2 p_{1} k_{z} + k_{z}^{2})}^{1 ∕ 2} + ε_{F}$ lines, respectively. The long-dashed and dashed-dotted lines, labeled $ω_{G 1}$ and $ω_{G 2}$ , are the $ω = ∣ {(ε_{F}^{2} \pm 2 p_{1} k_{z} + k_{z}^{2})}^{1 ∕ 2} - ε_{F} ∣$ lines, respectively. (b) The boundaries of the different numbered regions in the GA dissipative regime of $α^{11}$ and $i α^{12}$ , parameters $p_{F} = 2.2 m$ , $B = B_{cr}$ , and $η = 1$ for $I_{A}$ $(η = n)$ and $I_{B}$ $(η = n^{'})$ , polarizations $e_{\mp}^{μ}$ , respectively. The solid and long-dashed lines are the $ω = k_{z}$ and $ω = {ω_{GA} ∣}_{η = 1}$ lines, respectively. The short-dashed and dashed-double-dotted lines are the $ω = ∣ {(ε_{F}^{2} \pm 2 p_{1} k_{z} + k_{z}^{2} + 2 e B)}^{1 ∕ 2} - ε_{F} ∣$ lines, respectively. (c) As for (b), but for $I_{D}$ $(η = n)$ and $I_{C}$ $(η = n^{'})$ , polarizations $e_{\mp}^{μ}$ , respectively, with the short-dashed and dashed-double-dotted lines here being the $ω = ∣ {(ε_{F}^{2} \pm 2 p_{1} k_{z} + k_{z}^{2} - 2 e B)}^{1 ∕ 2} - ε_{F} ∣$ lines, respectively. (d) As for (b), but for $η = 2$ .Reuse & Permissions
Figure 2
(a): The left-hand side of the longitudinal dispersion relation, $1 - 4 π Re {(α^{33})}_{total} ∕ ω^{2} = 0$ , is plotted as a function of $ω$ , $0 < ω < k_{z}$ , for the parameters $p_{F} = 2.2 m$ , $B = B_{cr}$ , and $k_{z} = 0.1 m$ . Peaks and troughs occur at the logarithmic singularities when one has $ω = ∣ {(ε_{F}^{2} \mp 2 p_{n} k_{z} + k_{z}^{2})}^{1 ∕ 2} - ε_{F} ∣$ , respectively. (b) The different longitudinal modes for the parameters $p_{F} = 2.2 m$ and $B = B_{cr}$ . The different lines are discussed in the text.Reuse & Permissions
Figure 3
(a): The gyromagnetic absorption modes are plotted as a function of $k_{z}$ for the transverse component $Re (α_{-})$ with polarization $e_{-}^{μ}$ , for the parameters $p_{F} = 2.2 m$ and $B = B_{cr}$ . The solid line represents the mode $ω_{-}$ , the dotted lines labeled $C$ and $D$ are the $n = 0$ and $n = 1$ values, respectively, of ${ω ∣}_{n}$ in (29). The long-dashed lines labeled $A$ and $B$ are the $n^{'} = 1$ and $n^{'} = 2$ values, respectively, of (30). (b) The left-hand side of the transverse dispersion relation $ω^{2} - k_{z}^{2} - 4 π Re (α_{+}) = 0$ is plotted as a function of $ω$ $(0 < ω < ω_{GA})$ for the parameters $p_{F} = 2.2 m$ , $B = B_{cr}$ , $Δ ω = 10^{- 4} m$ , and $k_{z} = 0.1 m$ . (c) As for (b) but for $k_{z} = 0.5 m$ . (d) The gyromagnetic absorption modes are plotted as a function of $k_{z}$ for the transverse component $Re (α_{+})$ with polarization $e_{+}^{μ}$ , for the parameters $p_{F} = 2.2 m$ and $B = B_{cr}$ . The different lines are defined in the text. (e) Transverse pair modes above the pair creation thresholds for the parameters $p_{F} = 2.2 m$ and $B = B_{cr}$ . See the text for a description of the different lines.Reuse & Permissions

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