2 Basic equations

In this paper, we consider a zero-temperature plasma composed of ions and electrons. Since the mass of ions is much more than that of electrons, ions are treated as a stationary neutralizing background, and only the motion of electrons is considered. The quantum magnetohydrodynamics model for spin-1/2 electrons is composed of the continuity equation:

(2.1)\begin{equation} \frac{\partial n}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}(n\boldsymbol{u})=0, \end{equation}

and the momentum equation (Brodin & Marklund Reference Brodin and Marklund2007; Marklund & Brodin Reference Marklund and Brodin2007):

(2.2)\begin{align} mn\left(\frac{\partial}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\boldsymbol{u}={-}en\left(\boldsymbol{E}+\frac{\boldsymbol{u}}{c}\times \boldsymbol{B}\right)-\boldsymbol{\nabla} P+\frac{\hbar^{2}n}{2m}\boldsymbol{\nabla}\left(\frac{\nabla^{2}\sqrt{n}}{\sqrt{n}}\right)+\frac{2n\mu_{B}}{\hbar}\boldsymbol{\nabla}(\boldsymbol{S}\boldsymbol{\cdot}\boldsymbol{B}), \end{align}

where $n$ is the number density of electrons, $\boldsymbol {u}$ is the fluid velocity of electrons, $e$ is the charge of the electron, $m$ is the electron mass, $\mu _{B}=e\hbar /2mc$ is the Bohr magneton and $\hbar$ is Planck's constant divided by $2{\rm \pi}$. Here, $P$ denotes the relativistic electron degeneracy pressure in dense plasmas, which can be written as (Shukla & Eliasson Reference Shukla and Eliasson2011)

(2.3)\begin{equation} P=\frac{{\rm \pi} m^{4}_{e}c^{5}}{3h^{3}}f(\xi), \end{equation}

where $f(\xi )=\xi (2\xi ^{2}-3)(1+\xi ^{2})^{1/2}+3\sinh ^{-1}(\xi )$, $\xi =p/mc$ and $p=(3{\rm \pi} ^{2}n)^{1/3}\hbar$ is the momentum of an electron on the Fermi surface. Expanding (2.3) around the unperturbed density of electrons $n_{0}$ by the Taylor series expansion and neglecting the higher order terms, we have (Maroof et al. Reference Maroof, Ali, Mushtaq and Qamar2015; El-Shamy Reference El-Shamy2015)

(2.4)\begin{equation} P=P_{0}+\frac{mv^{2}_{Fe}}{3\gamma_{0}}n_{1}, \end{equation}

where $n_{1}$ denotes the perturbed electron number density, $v_{Fe}=(3{\rm \pi} ^{2}n_{0})^{1/3}\hbar /m$ is the Fermi velocity of electrons, and $\gamma _{0}=1/\sqrt {1-\xi ^{2}_{0}}$ with $\xi _{0}=p_{0}/mc$, where $p_{0}=(3{\rm \pi} ^{2}n_{0})^{1/3}\hbar$ is the Fermi momentum of electrons.

Neglecting the spin–thermal coupling terms and the nonlinear spin fluid contribution, the spin vector $\boldsymbol {S}$ in (2.2) satisfies the evolution equation

(2.5)\begin{equation} \frac{\textrm{d}\boldsymbol{S}}{\textrm{d}t}=\frac{2\mu_{B}}{\hbar}\boldsymbol{B}\times\boldsymbol{S}. \end{equation}

Neglecting the spin inertia, the spin vector is determined from $\boldsymbol {B}\times \boldsymbol {S}=0$, which has a solution

(2.6)\begin{equation} \boldsymbol{S}={-}\frac{\hbar}{2}\eta\left(\frac{\mu_{B} B}{k_{B}T_{Fe}}\right)\boldsymbol{\hat{B} }, \end{equation}

where $B$ denotes the magnitude of the magnetic field, and $\boldsymbol {\hat {B}}$ is a unit vector in the direction of the magnetic field. Here, $\eta (\alpha )=\tanh \alpha$ is the Brillouin function, where $\alpha =\mu _{B}B_{0}/(k_{B}T_{F})$.

The QED vacuum polarization effect is formulated by the Heisenberg–Euler Lagrangian density of electromagnetic fields, which is expressed as (Heisenberg & Euler Reference Heisenberg and Euler1936; Marklund & Shukla Reference Marklund and Shukla2006)

(2.7)\begin{equation} \mathcal{L}=\frac{1}{8{\rm \pi}}(E^{2}-B^{2})+\frac{\xi}{8{\rm \pi}}[(E^{2}-B^{2}) +(\boldsymbol{E}\boldsymbol{\cdot}\boldsymbol{B})^{2}], \end{equation}

where $\xi =\hbar e^{4}/(45{\rm \pi} m^{4}c^{7})$, and $c$ is the speed of light in vacuum. The first term of (2.7) is the classical Lagrangian density, and the second term is the correction term originating from the vacuum polarization effect. The effective polarization and magnetization vectors of the vacuum derived from Lagrangian density can be expressed as (Shen, Yu & Wang Reference Shen, Yu and Wang2003; Lundin et al. Reference Lundin, Stenflo, Brodin, Marklund and Shukla2007)

(2.8)\begin{equation} \boldsymbol{P}=\frac{\xi}{4{\rm \pi}}[2(E^{2}-B^{2})\boldsymbol{E} +7(\boldsymbol{E}\boldsymbol{\cdot}\boldsymbol{B})\boldsymbol{B}], \end{equation}

and

(2.9)\begin{equation} \boldsymbol{M}=\frac{\xi}{4{\rm \pi}}[{-}2(E^{2}-B^{2})\boldsymbol{B} +7(\boldsymbol{E}\boldsymbol{\cdot}\boldsymbol{B})\boldsymbol{E}]. \end{equation}

Assuming that the amplitude of oscillation is small, we can solve the system by using linearized equations. The plasma equilibrium is assumed as $\boldsymbol {E}_{0}=0$, $\boldsymbol {u}_{0}=0$, therefore, the linearized continuity equation is derived as

(2.10)\begin{equation} \frac{\partial n_{1}}{\partial t} +n_{0}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}_{1}=0, \end{equation}

and the linearized momentum equation is obtained as

(2.11)\begin{equation} \displaystyle\frac{\partial\boldsymbol{u}_{1}}{\partial t}={-}\frac{e}{m}\left(\boldsymbol{E}_{1}+\frac{\boldsymbol{u}_{1}}{c}\times \boldsymbol{B}_{0}\right)-\frac{v^{2}_{Fe}}{3n_{0}\gamma_{0}}\boldsymbol{\nabla} n_{1} +\frac{\hbar^{2}}{4m^{2}n_{0}}\boldsymbol{\nabla}\nabla^{2}n_{1}+\frac{2\mu_{B}}{m\hbar}\boldsymbol{\nabla}(\boldsymbol{S}\boldsymbol{\cdot}\boldsymbol{B}_{1}). \end{equation}

The linearized Maxwell equations modified by the vacuum polarization effect and spin effect can be presented as

(2.12)\begin{gather} \boldsymbol{\nabla}\times \boldsymbol{E}_{1}={-}\frac{1}{c}\frac{\partial \boldsymbol{B}_{1}}{\partial t}, \end{gather}

(2.13)\begin{gather} \boldsymbol{\nabla}\times \boldsymbol{B}_{1}=\frac{1}{c}\frac{\partial \boldsymbol{E}_{1}}{\partial t}+\frac{4{\rm \pi}}{c}(\boldsymbol{J}_{e}+\boldsymbol{J}_{M}+\boldsymbol{J}_{\textrm{vac}}), \end{gather}

(2.14)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{E}_{1}=4{\rm \pi}(\rho_{e}+\rho_{\textrm{vac}}), \end{gather}

where $\rho _{e}=-en_{1}$ and $\boldsymbol {J}_{e}=-en_{0}\boldsymbol {u}_{1}$ are the charge and current density of electrons, respectively. Then $\boldsymbol {J}_{M}=-c\boldsymbol {\nabla }\times (2n_{0}\mu _{B}\boldsymbol {S}/\hbar )$ is the magnetization spin current density, $\rho _{\textrm {vac}}=-\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {P}$ and $\boldsymbol {J}_{\textrm {vac}}=\partial \boldsymbol {P}/\partial \boldsymbol {t}+c\boldsymbol {\nabla }\times \boldsymbol {M}$ are the effective vacuum charge and current density, respectively.

3 Dispersion relation of upper-hybrid wave

We choose the external magnetic field as $\boldsymbol {B}_{0}=B_{0}\hat {z}$ with respect to the propagation direction determined by the wavenumber $\boldsymbol {k}=k\hat {y}$ of the wave. Since the upper-hybrid wave is an electrostatic wave, the first-order electromagnetic fields are chosen as $\boldsymbol {E}_{1}=E_{1}\hat {y}$ and $\boldsymbol {B}_{1}=0$, as shown in figure 1. It should be noted that $E^{2}$ as well as $B^{2}$ in (2.8) are constant and that $\boldsymbol {E}\boldsymbol {\cdot }\boldsymbol {B}=0$. The effective vacuum charge density in (2.14) can be written as

(3.1)\begin{equation} \rho_{\textrm{vac}}={-}\frac{\xi}{2{\rm \pi}}(E^{2}-B^{2})\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{E}_{1}. \end{equation}

Inserting (3.1) into (2.14), we derive a new Poisson equation modified by the vacuum polarization correction as

(3.2)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{E}_{1}=4{\rm \pi}\rho_{e}(1-\beta)^{{-}1}, \end{equation}

where

(3.3)\begin{equation} \beta=2\xi(B_{0}^{2}-E_{1}^{2})\sim\frac{2\alpha B_{0}^{2}}{45{\rm \pi} B_{c}^{2}}. \end{equation}

Here $E_{1}\ll B_{0}$ is the amplitude of the firs-order electric field, $\alpha =e^{2}/\hbar c$ is the fine structure constant, and $B_{c}= m^{2}c^{3}/\hbar e\sim 4.44\times 10^{13} {\rm Gs}$ is the Schwinger critical magnetic field.

Figure 1. Cartesian coordinate system, chosen such that $\boldsymbol {B}_{0}$ is along $\hat {z}$ and $\boldsymbol {E}_{1}$ is along $\hat {y}$.

Supposing the perturbations are proportional to $\exp [\textrm {i}(ky-\omega t)]$, (2.10) and (2.11) become

(3.4)\begin{equation} \displaystyle-\textrm{i}\omega\boldsymbol{u}_{1}={-}\frac{e}{m}\left(\boldsymbol{E}_{1}+\frac{\boldsymbol{u}_{1}}{c}\times \boldsymbol{B}_{0}\right)-\frac{\textrm{i}kv^{2}_{Fe}n_{1}}{3\gamma_{0}n_{0}}\hat{y} -\frac{\textrm{i}\hbar^{2}k^{3}n_{1}}{4m^{2}n_{0}}\hat{y} \end{equation}

and

(3.5)\begin{equation} n_{1}=\frac{ku_{1y}}{\omega}n_{0}. \end{equation}

By solving (3.4) and (3.5), we have the component $u_{1y}$ of the fluid velocity as

(3.6)\begin{equation} \displaystyle u_{1y}={-}\frac{\textrm{i}eE_{1}}{\omega m}\left(1-\frac{\omega^{2}_{c}} {\omega^{2}}-\varDelta\right)^{{-}1}, \end{equation}

where $\omega _{c}=eB_{0}/mc$ is the electron cyclotron frequency and $\varDelta ={k^{2}v^{2}_{Fe}}/{3\gamma _{0}\omega ^{2}} +{\hbar ^{2}k^{4}}/{4m^{2}\omega ^{2}}$ is the quantum correction.

Inserting (3.6) into (3.2), we derive the dispersion relation of upper-hybrid wave as

(3.7)\begin{equation} \omega_{h}^{2}=\frac{\omega_{p}^{2}}{1-\beta}+\omega^{2}_{c} +\frac{k^{2}v^{2}_{Fe}}{3\gamma_{0}}+\frac{\hbar^{2}k^{4}}{4m^{2}}. \end{equation}

When setting $\hbar \rightarrow 0$ and $\beta \rightarrow 0$, (3.7) can be degenerated to the frequency of upper-hybrid oscillation in the classical cold plasmas, and it indicates that an upper-hybrid oscillation can propagate in cold plasmas due to quantum effects.

4 Dispersion relation of an extraordinary wave

We choose the external magnetic field as $\boldsymbol {B}_{0}=B_{0}\hat {z}$ with respect to the propagation direction determined by the wavenumber $\boldsymbol {k}=k\hat {y}$ of the wave. Since we investigate the propagation of an extraordinary wave, the first-order electromagnetic fields are particularly chosen as $\boldsymbol {E}_{1}=E_{1x}\hat {x}+E_{1y}\hat {y}$ and $\boldsymbol {B}_{1}=B_{1}\hat {z}=-kcE_{1x}/\omega \hat {z}$, as shown in figure 2. It should be noted that $E^{2}$ as well as $B^{2}$ are constant and that $\boldsymbol {E}\boldsymbol {\cdot }\boldsymbol {B}=0$. This means that

(4.1)\begin{equation} \boldsymbol{J}_{\textrm{vac}}={-}\frac{\xi}{2{\rm \pi}}(E^{2}-B^{2})\left(\boldsymbol{\nabla}\times \boldsymbol{B}_{1}-\frac{1}{c}\frac{\partial \boldsymbol{E}_{1}}{\partial t}\right). \end{equation}

Inserting (4.1) into (2.13) we derive a new equation

(4.2)\begin{equation} \boldsymbol{\nabla}\times \boldsymbol{B}_{1}=\frac{1}{c}\frac{\partial \boldsymbol{E}_{1}}{\partial t}+\frac{4{\rm \pi}}{c}(\boldsymbol{J}_{e}+\boldsymbol{J}_{M})(1-\chi)^{{-}1}, \end{equation}

where

(4.3)\begin{equation} \chi=\frac{2\alpha}{45{\rm \pi} B_{c}^{2}}[(n^{2}-1)E_{1}^{2}+B_{0}^{2}]. \end{equation}

Here $n=kc/\omega$ is the index of refraction and $E_{1}$ is the amplitude of the first-order electric field.

Figure 2. Cartesian coordinate system, chosen such that $\boldsymbol {B}_{0}$ and $\boldsymbol {B}_{1}$ are along $\hat {z}$, and $\boldsymbol {E}_{1}$ is in the plane of $xoy$.

The spin magnetization current density in (2.13) is calculated as

(4.4)\begin{equation} \boldsymbol{J}_{M}=\frac{\textrm{i}\mu_{B}\eta(\alpha)n_{0}k^{2}v_{1y}}{\omega}\hat{x}. \end{equation}

Supposing the perturbations are proportional to $\exp [\textrm {i}(ky-\omega t)]$, (2.10) and (2.11) become

(4.5)\begin{equation} \displaystyle-\textrm{i}\omega\boldsymbol{u}_{1}={-}\frac{e}{m}\left(\boldsymbol{E}_{1}+\frac{\boldsymbol{u}_{1}}{c}\times \boldsymbol{B}_{0}\right)-\frac{\textrm{i}kv^{2}_{Fe}n_{1}}{3\gamma_{0}n_{0}}\hat{y} -\frac{\textrm{i}\hbar^{2}k^{3}n_{1}}{4m^{2}n_{0}}\hat{y}+\frac{\textrm{i}\mu_{B}\eta(\alpha)k^{2}c E_{1x}}{m\omega}\hat{y} \end{equation}

and

(4.6)\begin{equation} n_{1}=\frac{ku_{1y}}{\omega}n_{0}. \end{equation}

By solving (4.5) and (4.6), we have the two components $u_{1x}$ and $u_{1y}$ of the fluid velocity as

(4.7)\begin{equation} {\begin{array}{*{1}c} \displaystyle u_{1x}=\frac{e}{\omega m}\left\{-iE_{1x}-\left[\frac{i\omega^{2}_{c}}{\omega^{2}}\left(1-\frac{\omega}{\omega_{c}}S\right)E_{1x}+\frac{\omega_{c}}{\omega}E_{1y}\right]\left(1-\frac{\omega^{2}_{c}}{\omega^{2}}-\Delta\right)^{-1}\right\},\\ \displaystyle u_{1y}=\frac{e}{\omega m}\left[\frac{\omega_{c}}{\omega}\left(1-\frac{\omega}{\omega_{c}}S\right)E_{1x}-iE_{1y}\right]\left(1-\frac{\omega^{2}_{c}}{\omega^{2}}-\Delta\right)^{-1},\\ \end{array}} \end{equation}

where $S=({\mu _{B}\eta (\alpha )k^{2}c})/{e\omega }$ is the spin correction term.

From the linearized Maxwell's equations (2.12) and (4.2), we have

(4.8)\begin{equation} (\omega^{2}-k^{2}c^{2})E_{1x}={-}4{\rm \pi} \textrm{i}\omega (J_{ey}+J_{M})(1-\chi)^{{-}1}, \end{equation}

and

(4.9)\begin{equation} \omega^{2}E_{1y}={-}4{\rm \pi} \textrm{i}\omega J_{ex}(1-\chi)^{{-}1}, \end{equation}

where $J_{ex}=-en_{0}u_{1x}$ and $J_{ey}=-en_{0}u_{1y}$ are the two components of electron current density $\boldsymbol {J}_{e}$, and $J_{M}=({\textrm {i}\mu _{B}\eta (\alpha )n_{0}k^{2}v_{1y}})/{\omega }$ is the magnitude of the spin magnetization current density.

According to (4.8) and (4.9), the dispersion equation can be obtained as

(4.10)\begin{align} \left| \begin{array}{cc} \displaystyle-\textrm{i}\left(\dfrac{\omega_{c}}{\omega}-S\right)\omega_{p}^{2} & \displaystyle\omega^{2}\left(1-\dfrac{\omega^{2}_{c}}{\omega^{2}}-\varDelta\right)(1-\chi)-\omega_{p}^{2}\\ \displaystyle(\omega^{2}-k^{2}c^{2})\left(1-\dfrac{\omega^{2}_{c}}{\omega^{2}}-\varDelta\right)(1-\chi)-\omega_{p}^{2}\left(1-\varDelta-S^{2}\right) & \displaystyle \textrm{i}\left(\dfrac{\omega_{c}}{\omega}+S\right)\omega_{p}^{2} \\ \end{array} \right|=0, \end{align}

where $\omega _{p}^{2}=4{\rm \pi} n_{0}e^{2}/m$ is the plasma frequency.

Solving (4.10), the dispersion relation of the extraordinary wave in spin quantum magnetoplasmas with the vacuum polarization effect is derived as

(4.11)\begin{equation} \frac{k^{2}c^{2}}{\omega^{2}}=1-\frac{\varOmega^{2}_{p}}{\omega^{2}} \frac{\omega^{2}(1-\varDelta-S^{2})-\varOmega_{p}^{2}}{\omega^{2}-\tilde{\omega}_{h}^{2}}, \end{equation}

where

(4.12)\begin{equation} \tilde{\omega}_{h}^{2}=\varOmega^{2}_{p}+\omega^{2}_{c}+\frac{k^{2}v^{2}_{Fe}}{3\gamma_{0}}+\frac{\hbar^{2}k^{4}}{4m^{2}} \end{equation}

is the dispersion relation of an upper-hybrid wave, and $\varOmega ^{2}_{p}=\omega _{p}^{2}/(1-\chi )$.

In the absence of quantum effects and the vacuum polarization effect we have $\varDelta =0, S=0$ and $\chi =0$, so (4.11) and (4.12) will reproduce to the dispersion relation of an extraordinary wave and the frequency of upper-hybrid oscillation in the classical cold plasmas.