Normal modes in magnetized two-fluid spin quantum plasmas

ABSTRACT

We extend the classical two-fluid magnetohydrodynamic (MHD) formalism to include quantum effects such as electron Fermi pressure, Bohm pressure, and spin couplings. At scales smaller than the electron skin-depth, the Hall effect and electron inertia must be taken into account, and can overlap with the quantum effects. We write down the full set of two-fluid quantum MHD (QMHD) and analyse the relative importance ofthese effects in the high-density environments of neutron star atmospheres and white dwarf interiors, finding that for a broad range of parameters all these effects are operative. Of all spin interactions we analyse only the spin-magnetic coupling, as it is linear in ℏ and consequently it is the strongest spin effect. We re-obtain the classical two-fluid MHD dispersion relations corresponding to the magnetosonic and Alfvén modes, modified by quantum effects. In the zero-spin case, for propagation parallel to the magnetic field, we find that the frequency of the fast mode is due to quantum effects modified by electron inertia, while the frequency of the Alfvén-slow sector has no quantum corrections. For perpendicular propagation, the fast-mode frequency is the same as for the parallel propagation plus a correction due only to classical two-fluid effects. When spin is considered, a whistler mode appears, which is due to two-fluid effects plus spin-magnetic interaction. There are no modifications due to spin for parallel propagation of magnetosonic and Alfvén waves, while for perpendicular propagation a dispersive term due to spin arises in the two-fluid expression for the fast magnetosonic mode.

MHD, plasmas, stars: neutron, white dwarfs

1 INTRODUCTION

Magnetohydrodynamics (MHD) can be described as a ‘coarse grained’ formalism, suitable to study magnetized plasmas at scales larger than the ion inertial length, λ_i = c/ω_pi, with c the speed of light and ω_pi the ion plasma frequency. At those scales, for example, hydrodynamic and MHD turbulence display the same power-law spectrum for the energy cascade, i.e. a k^−5/3 Kolmogorov spectrum (Matthaeus & Goldstein 1982; Smith et al. 1982; Leamon et al. 2000). However, while at shorter scales the hydrodynamic turbulence still displays a Kolmogorov spectrum all the way down to the dissipation scale, MHD turbulence does not. At scales smaller than λ_i, a regime known as Hall–MHD is established, in which the energy power spectrum becomes somewhat steeper (Goldstein, Roberts & Fitch 1994; Ghosh et al. 1996; Smith et al. 2006). At scales much smaller than λ_i, an approximate description known as electron MHD (EMHD) has been proposed, which assumes the ions to be static (because of their much larger mass) and consistently the electric currents are fully carried by the electrons. Biskamp et al. (1999) studied numerically the EMHD turbulence and confirmed the steeper behaviour of the energy spectrum at small scales. Recently, Andrés et al. (2014a,b) wrote down a complete two-fluid MHD model which includes the Hall and electron inertia effects. Within this description, it is possible to study classical plasma effects all the way from scales as large as the size of the system down to the electron inertial scale λ_e = c/ω_pe, and this includes MHD, Hall–MHD, and EMHD as asymptotic limits in the appropriate range of scales. At scales below λ_i, as a result of the Hall current term, ions are no longer frozen-in to the magnetic field lines, while electrons still remain frozen. Therefore, at these scales, the motion of electrons decouple from the one of ions, even though the dynamics can be properly described by Hall–MHD. At scales below λ_e, electrons decouple from magnetic field lines as well, and a proper description of the dynamics at these scales requires the full two-fluid MHD description.

All the plasma effects discussed thus far are classical, and quantum effects will also become relevant at sufficiently small scales. More specifically, quantum effects should have to be taken into account whenever the thermal de Broglie wavelength of the plasma particles |$\lambda _\mathrm{ B}=\hbar /\sqrt{k_{\rm B} T m}$| (k_B: Boltzmann’s constant, ℏ = h/2|$\pi$|⁠, T: temperature and m: mass) becomes of the order of the interparticle separation, i.e. |$\lambda _\mathrm{ B} \gt n_0^{-1/3}$| with n₀ the average particle density of the plasma. Interparticle separations smaller than λ_B may be found in extremely dense astrophysical plasmas, such as e.g. white dwarfs, magnetars, or neutron stars. For example in a magnetar atmosphere we have T ≃ 10⁹ K and therefore λ_B ≃ 9 × 10⁻¹¹ cm which is of the order of the interparticle separation. For a white dwarf, T ≃ 10⁴, and λ_B ≃ 3 × 10⁻⁸ cm which is again of the order of |$n_0^{-1/3}$|⁠. Therefore, it is expected that quantum effects might play a non-negligible role in these extreme astrophysical environments.¹

The formalism to study MHD with quantum effects is known as quantum magnetohydrodynamics (QMHD; Haas 2005, 2011; Marklund & Brodin 2007; Brodin & Marklund 2007b). In it, the equations of classical MHD are extended to include terms that take into account the quantum nature of the charge carriers. The paradigmatic model of a quantum fluid is that of a Fermi gas, with pressure |$p_\mathrm{ F}=(2/5)n_0 E_\mathrm{ F}=(3\pi ^2)^{2/3} (\hbar ^2/5m)n_0^{5/3}$|⁠, where E_F is the Fermi energy. Besides the Fermi pressure, more complete descriptions include a ‘quantum force’ whose origin is the Bohm potential due to the overlap of wavefunctions as well as spin effects. These spin effects are mainly due to three sources: a spin–spin coupling due to spin gradients, a spin-density coupling due to spin and density gradients and, in the presence of an external magnetic field |$\boldsymbol {B}$|⁠, a |$\mathrm{ spin}-\boldsymbol {B}$| interaction due to the coupling of the spins to gradients of the magnetic field. In the presence of inhomogeneous magnetic fields, the most intense effect is the |$\mathrm{ spin}-\boldsymbol {B}$| one, because it is of order ℏ while the others are of order ℏ².

From what was said in the previous paragraphs, it seems apparent that there might be cases in which the Hall effect and electron inertia can be as important quantum effects. Therefore, in this manuscript we extend the two-fluid formalism developed by Andrés et al. (2014b) by including quantum effects such as Fermi pressure, Bohm pressure, and spin interactions. Our final aim is to find a theory that describes as accurately as possible small-scale effects in dense magnetized plasmas.

One possible approach to assess how the different effects mentioned in the previous paragraphs affect the dynamical properties of multispecies plasmas is to obtain the dispersion relations for the propagation of linear perturbations of the different quantities that enter in the problem (e.g. density, magnetic field, spin). For two-fluid plasmas, this study was done by Andrés et al. (2014b). For QMHD we may mention the studies on the propagation of linear sonic waves (Brodin & Marklund 2007a; Marklund & Brodin 2007; Shukla 2007; Asenjo 2012; Andreev 2015) and of low-frequency waves (Haas et al. 2003; Shukla & Stenflo 2006; Saleem et al. 2008). Non-linear phenomena such as shock waves (Misra & Ghosh 2008; Masood, Karim & Shah 2010) and non-linear waves (Shukla et al. 2006; Ali et al. 2007) were also analysed. Moreover, the effect of radiative processes on quantum plasmas was also addressed (Cross, Reville & Gregory 2014). This list of references is, of course, not exhaustive.

We consider an electrically neutral plasma composed by two fermionic fluids of equal modulus and opposite sign charges at temperatures of the order or below the Fermi temperature. To visualize more clearly the role of the different effects, we neglect kinematic viscosity as well as electrical resistivity. In order to analyse the relative importance of each term we rewrite the equations in non-dimensional form by defining several dimensionless parameters. This procedure has the advantage of making the analysis independent of the unit system, avoids spurious over- or underestimations of the different effects and also allows to directly rescale between completely different systems as, e.g. the astrophysics and laboratory plasmas (see Cross et al. 2014 for a discussion of this procedure).

We write down the system of dimensionless two-fluid QMHD equations and linearize them around an equilibrium configuration. We find expressions for the dispersion relations of the Alfvén and magnetosonic waves that generalize the results found previously in the literature on classical MHD (Landau & Lifshitz 1999) and QMHD (Brodin & Marklund 2007a) and two-fluid MHD (Andrés et al. 2014b). Moreover, due to the presence of a spin-magnetic coupling, we obtain a new dispersion relation that corresponds to a whistler mode (Stenzel 1999). This mode arises in the two-fluid approach considered, and because of the spin effect it becomes dispersive also at wavelengths well larger than the interparticle separation.

The manuscript is organized as follows: In Section 2, we obtain the two-fluid QMHD equations and analyse the applicability of each effect in the parameter space of astrophysical compact objects given by (n₀, B₀), with B₀ a mean magnetic field. In Section 3, we obtain the generalized dispersion relations for the cases without spin (Section 3.1) and with spin (Section 3.2). In Section 4, we draw our main conclusions. In the appendix, we detail the procedure to turn the equations non-dimensional.

We work in c.g.s. units, where |$\hbar=1.0546\times 10^{-27}\, \mathrm{cm}^2\mathrm{s}^{-1}\, \mathrm{g}$|⁠, |$c=3\times 10^{10}\, \mathrm{cm}\ \mathrm{s}^{-1}$|⁠, |$m_p=1.67\times 10^{-24}\, \mathrm{g}$|⁠, |$m_e=9.1\times 10^{-28}\, \mathrm{g}$|⁠, and obtain the electric charge from the fine structure constant α, i.e. e² ≃ ℏc/137. Finally, summation over repeated indices is assumed.

2 QUANTUM MHD

In this section, we derive the QMHD equations for a two-fluid ion-electron plasma in an external magnetic field |$\bar{B}$|⁠, starting from the equations for each individual species, and analyse the relative importance of each term.

As stated in Section 1, we consider an electrically neutral degenerate plasma composed by two species with charges q_s = ±e, particle masses m_s and spin 1/2. The Fermi pressure in three dimensions for a gas of particle mass m_s and particle density n_s in the limit T_s → 0 is

\begin{eqnarray*} p_{\mathrm{ s}}=\frac{2}{5}n_\mathrm{ s}E_{\mathrm{ Fs}}= \left(3\pi ^2\right)^{2/3}\frac{\hbar ^2}{5m_\mathrm{ s}}n_\mathrm{ s}^{5/3}, \end{eqnarray*}

(1)

where E_Fs is the Fermi energy of species s, given by E_Fs = (ℏ²/2m_s)(3|$\pi$|²n_s)^2/3. The equations for each species were considered elsewhere (Haas 2005; Marklund & Brodin 2007; Brodin & Marklund 2007b; Mahajan & Asenjo 2008; Andrés et al. 2014b) and read

\begin{eqnarray*} \partial _\mathrm{ t} n_\mathrm{ s} + \bar{\nabla }\cdot \left(n_\mathrm{ s}\bar{u}_\mathrm{ s}\right) = 0, \end{eqnarray*}

(2)

\begin{eqnarray*} \partial _\mathrm{ t} \bar{u}_\mathrm{ s} + \left(\bar{u}_\mathrm{ s}\cdot \bar{\nabla }\right) \bar{u}_\mathrm{ s} = \frac{q_s}{m_\mathrm{ s}}\bar{E} + \frac{q_s}{m_\mathrm{ s}c}\bar{u}_\mathrm{ s}\times \bar{B} -\frac{1}{m_\mathrm{ s} n_\mathrm{ s}}\bar{\nabla }p_\mathrm{ s} + \frac{\hbar ^2}{2m_\mathrm{ s}^2}\bar{\nabla }\left(\frac{\nabla ^2 n_\mathrm{ s}^{1/2}}{n_s^{1/2}}\right) + \frac{\hbar q_\mathrm{ s}}{2m_\mathrm{ s}^2c}S^s_j\bar{\nabla }\hat{\bar{B}}^\mathrm{ s}_j + \frac{\hbar ^2}{2m_\mathrm{ s}^2}\bar{\nabla }\left(\partial _jS^\mathrm{ s}_i\partial _j\mathrm{ S}^\mathrm{ s}_i\right), \end{eqnarray*}

(3)

\begin{eqnarray*} \left(\partial _t + \bar{u}_\mathrm{ s}\cdot \bar{\nabla }\right)\bar{S}^\mathrm{ s} = \frac{q_\mathrm{ s}}{m_\mathrm{ s} c}\bar{S}^\mathrm{ s}\times \hat{\bar{B}}^\mathrm{ s}, \end{eqnarray*}

(4)

with c the speed of light and where we defined

\begin{eqnarray*} \hat{\bar{B}}^\mathrm{ s}=\bar{B} + \frac{\hbar c}{2q_\mathrm{ s} n_\mathrm{ s}} \partial _j\left(n_\mathrm{ s} \partial _j \bar{S}^\mathrm{ s}\right). \end{eqnarray*}

(5)

In the previous expressions the average spin vector field for species s, |$\bar{S}^s$|⁠, satisfies |$\bar{S}^s\cdot \bar{S}^s=1$|⁠. From equations (3) and (5) we see that the spin introduces three forces, |$S^i\bar{\nabla }\bar{B}^s_i$|⁠, |$S^s_i\partial _j(n_s \partial _j S^s_i)$|⁠, and |$\bar{\nabla }(\partial _i S^s_j\partial _i S^s_j)$|⁠, which arise after the passage from particle to fluid description (Holland 1993). The first is due to the interaction of the spins with an external, inhomogeneous magnetic field |$\bar{B}$|⁠, the second is caused by an inhomogeneous magnetization created by the spins themselves. The other term, ∂_iS_j∂_iS_j can be interpreted as a spin pressure, that vanishes if the spin distribution is homogeneous, i.e. spins are completely aligned (see Ref. Mahajan & Asenjo 2008 for an analysis of the importance of this term). These equations must be supplemented with Maxwell equations, which for neutral, non-relativistic systems read

\begin{eqnarray*} \bar{\nabla }\cdot \bar{E} = 0, \end{eqnarray*}

(6)

\begin{eqnarray*} \skew5\bar{J}=\frac{c}{4\pi }\bar{\nabla }\times \bar{B} = \sum _\mathrm{ s} q_\mathrm{ s} n_s \bar{u}_\mathrm{ s}. \end{eqnarray*}

(7)

To analyse the relative importance of each term we turn to dimensionless variables. The calculations are done in detail in the appendix. Here, we only quote the different parameters: μ = m_e/M, |$\beta _0=\left(3\pi ^2\right)^{2/3}\left(\lambda _0 n_0^{1/3}\right)^{2}/5$| (quantum plasma β₀), M = m_e + m_p, |$V_A=B_0/\sqrt{4\pi M n_0}$|⁠, λ₀ = ℏ/(MV_A) (de Broglie-Alfvén length), ℓ = λ₀/L₀, ε = c/(ω_ML₀), |$\omega _M=\sqrt{4\pi e^2 n_0/M}$| (plasma frequency). L₀ is an arbitrary length scale that we introduced to make lengths non-dimensional. It can be interpreted as the resolution with which we look at the system. We choose it as a multiple of the particle separation, namely |$L_0=10^q n_0^{-1/3}$| with q ≥ 0. Note that ℓ and β₀ track quantum effects, while μ tracks electron inertia. In the specific case of a proton-electron plasma, the dimensionless equations of motion for each species become

\begin{eqnarray*} \mu \frac{{\rm d}\bar{u}_e}{{\rm d}t} = -\frac{1}{\varepsilon }\left(\bar{E} + \bar{u}_e\times \bar{B}\right) - \beta _0\frac{\bar{\nabla }p_e}{n} + \frac{\ell ^2}{\mu } \bar{\nabla }\left(\frac{\nabla ^2 n^{1/2}}{2n^{1/2}}\right) - \frac{\ell }{2\varepsilon \mu }S_j^e\bar{\nabla }\hat{B}^e_j + \frac{\ell ^2}{2\mu }\bar{\nabla }\left(\partial _jS^e_i\partial _jS^e_i \right), \end{eqnarray*}

(8)

\begin{eqnarray*} \left(1 -\mu \right) \frac{{\rm d}\bar{u}_p}{{\rm d}t} = \frac{1}{\varepsilon }\left(\bar{E} + \bar{u}_p\times \bar{B}\right) - \beta _0\frac{\bar{\nabla }p_p}{n} + \frac{\ell ^2}{\left(1-\mu \right)} \bar{\nabla }\left(\frac{\nabla ^2 n^{1/2}}{2n^{1/2}}\right) + \frac{\ell }{2\varepsilon (1-\mu)}S_j^p\bar{\nabla }\hat{B}^p_j + \frac{\ell ^2}{2(1-\mu)}\bar{\nabla }\left(\partial _jS^p_i\partial _jS^p_i \right), \end{eqnarray*}

(9)

\begin{eqnarray*} \left(\partial _t + \bar{u}_e\cdot \bar{\nabla }\right)\bar{S}^e = - \frac{1}{\mu \varepsilon }\bar{S}^e\times \hat{\bar{B}}^e, \end{eqnarray*}

(10)

\begin{eqnarray*} \left(\partial _t + \bar{u}_p\cdot \bar{\nabla }\right)\bar{S}^p = \frac{1}{(1-\mu)\varepsilon }\bar{S}^p\times \hat{\bar{B}}^p, \end{eqnarray*}

(11)

and

\begin{eqnarray*} \hat{\bar{B}}_{e,p} = \bar{B} \pm \frac{\ell \varepsilon }{2n}\partial _i\left(n\partial _i\bar{S}_{e,p}\right), \end{eqnarray*}

(12)

\begin{eqnarray*} \mu p_e = \left(1-\mu \right) p_p=n^{5/3}, \end{eqnarray*}

(13)

\begin{eqnarray*} \skew5\bar{J} = \frac{n}{\varepsilon }\left(\bar{u}_p - \bar{u}_e\right). \end{eqnarray*}

(14)

2.1 Two-fluid QMHD equations

To describe the system in terms of single fluid variables, we begin by defining the hydrodynamic velocity field |$\bar{u}$| in the usual way, namely

\begin{eqnarray*} \bar{u} = \left(1 - \mu \right)\bar{u}_p + \mu \bar{u}_e, \end{eqnarray*}

(15)

\begin{eqnarray*} \frac{\varepsilon }{n}\skew5\bar{J} = \bar{u}_p - \bar{u}_e, \end{eqnarray*}

(16)

from where we obtain

\begin{eqnarray*} \bar{u}_p = \bar{u} + \frac{\varepsilon \mu }{n}\skew5\bar{J}, \end{eqnarray*}

(17)

\begin{eqnarray*} \bar{u}_e = \bar{u} - \left(1-\mu \right)\frac{\varepsilon }{n}\skew5\bar{J}. \end{eqnarray*}

(18)

The continuity equation is obtained by adding equation (2) for the two species, i.e.

\begin{eqnarray*} \partial _t n + \bar{\nabla }\cdot \left(n\bar{u}\right)=0. \end{eqnarray*}

(19)

Adding equations (8) and (9) we obtain the evolution equation for |$\bar{u}$|⁠, namely

\begin{eqnarray*} \frac{{\rm d}\bar{u}}{{\rm d}t} &=& \frac{\skew5\bar{J}}{n}\times \left[\bar{B} + \left(1-\mu \right)\mu \varepsilon ^2\bar{\nabla }\times \left(\frac{\skew5\bar{J}}{n} \right)\right] - \bar{\nabla }\left[\left(1-\mu \right)\mu \varepsilon ^2\frac{J^2}{2n^2}\right] -\frac{5}{2}\frac{\beta _0}{\mu \left(1-\mu \right)}\bar{\nabla }n^{2/3} \nonumber \\&&+ \, \frac{\ell ^2}{\mu \left(1-\mu \right)} \bar{\nabla }\left(\frac{\nabla ^2\sqrt{n}}{2\sqrt{n}} \right) + \mathrm{`spin \ forces^{\prime }}, \end{eqnarray*}

(20)

where by ‘spin forces’ we mean the sum of all spin dependent terms. Unlike the other non-linear terms, they cannot be written in a compact form. Notwithstanding, this fact will not be a drawback as the main spin effects are due to electrons. The remaining equation is equation (8) with the replacement |$\bar{E}=-\partial _t\bar{A} - \bar{\nabla }\phi$|⁠. It reads

\begin{eqnarray*} \partial _t\bar{A} +\bar{\nabla }\phi = \bar{u}_e\times \bar{B} + \varepsilon \mu \frac{{\rm d}\bar{u}_e}{{\rm d}t} + \beta _0\varepsilon \frac{\bar{\nabla }n^{5/3}}{n} - \frac{\ell ^2\varepsilon }{\mu }\bar{\nabla }\left(\frac{\nabla ^2\sqrt{n}}{2\sqrt{n}} \right) + \frac{\ell }{2\varepsilon \mu }S_j^e\bar{\nabla }\hat{B}^e_j - \frac{\ell ^2}{2\mu }\bar{\nabla }\left(\partial _jS^e_i\partial _jS^e_i \right). \end{eqnarray*}

(21)

In the rhs of equation (21), the first term is the Hall effect on the electrons, the second term represents electron inertia, the third term is the force due to Fermi pressure, the fourth term is the Bohm force, the fifth is the force due to the spin-magnetic field coupling and the sixth is the force due to spin–spin couplings. Note that the sixth term is quadratic in ℓ. Due to the definition of |$\hat{B}$| (equation 5), the fifth term consists of two contributions: one due to the coupling of the spin with the external magnetic field which is linear in ℓ, and another due to the coupling of the spin with density and spin gradients which is quadratic in ℓ. This means that the main spin contribution comes from the coupling between an external, inhomogeneous magnetic field and spin, unless spin gradients are strong enough to compensate for the smallness of ℏ. Taking the curl of equation (21) eliminates all ‘gradient’ forces and gives rise to a generalized induction equation:

\begin{eqnarray*} \partial _t\bar{B}=\bar{\nabla }\times \left(\bar{u}_e\times \bar{B} \right) + \varepsilon \mu \frac{\rm d}{{\rm d}t} \left(\bar{\nabla }\times \bar{u}_e\right) + \frac{\ell }{2\varepsilon \mu }\bar{\nabla }S_j^e\times \bar{\nabla }\hat{B}^e_j. \end{eqnarray*}

(22)

The first two terms are the classic ones of MHD (actually Hall MHD, since the Hall term is included), the second term in the rhs represents a battery effect due to electron inertia while the last one is a spin electromotive force.

2.2 Parameters space

Due to their much smaller mass, it is likely that the electrons will be responsible for the main dynamical effects. Therefore, in order to assess the relative importance of the different terms in equation (21), we shall plot the coefficient of the different terms as functions of the background magnetic field B₀ and background density n₀. We recall that |$L_0=10^q n_0^{-1/3}$| with q ≥ 0, represents the resolution with which we observe the plasma. The number of particles in each volume element of linear size L₀ is then |$\mathcal {N}=n_0L_0^3=10^{3q}$|⁠. Our parameter space is (n₀, B₀).

2.2.1 Hall effect

It is described by log ε = 7.4 − q − (1/6) log n₀. The region in parameter space such that log ε < 0 means negligible Hall effect in comparison to the reference terms, while for log ε ≥ 0 i.e. when log n₀ ≤ 6(7.4 − q), it must be taken into account.

2.2.2 Electron inertia

It must be taken into account when log (με²) = 11.5 − 2q − (1/3) log n₀ ≥ 0, i.e. when log n₀ ≤ 3(11.5 − 2q).

2.2.3 Fermi pressure

It is important for log (β₀ε) ≥ 0, which means log B₀ ≤ −11.55 − q/2 + (3/4) log n₀.

2.2.4 Bohm pressure

It is not negligible when log (ℓ²ε/μ) ≥ 0, or equivalently when log B₀ ≤ −9.25 − (3/2)q + (3/4) log n₀.

2.2.5 Spin forces

As mentioned above, spin forces are due to two interactions: one with the external magnetic field, |$S_j\bar{\nabla }B^j$| with weight ℓ/2μ, and the other due to spin and density gradients, |$S_j\bar{\nabla }[\partial _i(n\partial _iS^j)]$| and |$\bar{\nabla }(\partial _jS_i\partial _jS_i)$| both with weight ℓ²ε/2μ. The latter is important when log (ℓ²ε/2μ) ≥ 0 which is equivalent to log B₀ ≤ −19.98 − (3/2)q + (3/4) log n₀. The former will play a non-negligible role if log (ℓ/εμ) > 0, which after replacing the different expressions gives log B₀ < −17 log (3.19) + log (n₀).

In Fig. 1, we have plotted the curves that we have just described, for q = 0, i.e. the resolution scale is the interparticle separation. The horizontal axis corresponds to log n₀ and the vertical axis is log B₀. The dashed rectangle shows the parameter space region that corresponds to white dwarfs, while the dotted rectangle indicates the parameter’s region corresponding to magnetar atmospheres. The thin vertical line corresponds to the Hall effect: to the left of this line the Hall effect must be taken into account. To the left of the thick vertical line, electron inertia is important. We see that for compact astrophysical objects the Hall effect is in general non-negligible, while electron inertia will be important for the whole range of densities at scales of the order of the interparticle separation. For poorer resolution, i.e. for q > 0 both vertical lines will be displaced to the left. Below the short-dashed oblique line, Fermi pressure must be taken into account. Below the long-dashed line Bohm pressure plays a non-negligible role and below the dashed-dotted line pure spin forces (i.e. due to the term ∂_iS_j∂_iS_j) must be considered. The dotted line indicates the spin-magnetic interaction, i.e. S_j∂_iB_j and |$S_j\bar{\nabla }[\partial _i(n\partial _iS^j)]$|⁠. Below this line, these terms must be taken into account. Above the continuous oblique line the Alvén velocity becomes larger than the speed of light, indicating that relativistic effects become important. The position of the different lines below the continuous one indicates that the non-relativistic treatment is adequate for the parameters interval considered in this work. For astrophysical compact objects we see that the terms of equation (21) that should in principle be considered are the Hall effect, spin − B interaction, Fermi pressure, and Bohm pressure while second order spin terms are important for weak magnetic fields.

Figure 1.

Parameter Space: Particle density in cm⁻³ and magnetic field in Gauss. Resolution is of the order of the interparticle separation (q = 0). The dashed rectangle shows the region that corresponds to white dwarfs, while the dotted rectangle delimits the parameter’s region corresponding to magnetar atmospheres. To the left of the vertical thin line the Hall effect is important, while to the left of the thick vertical line electron inertia must be taken into account. Above the continuous oblique line the Alfvén velocity becomes larger than c, indicating the breakdown of the non-relativistic treatment. Therefore, all the effects considered are reasonably well described by the non-relativistic formulation of quantum plasmas. Below the dotted oblique line spin-B coupling is important, below the short-dash line Fermi pressure is not negligible, below the long-dash line Bohm pressure must be taken into account and below the dashed-dotted line spin-spin coupling becomes important.

At this point it is important to calculate at which densities the electron skin-depth is smaller than the interparticle separation. Replacing the figures quoted at the end of the Introduction in λ_e = c/ω_pe, we find that |$\lambda _e \lesssim n_0^{-1/3}$| for n₀ ≳ 1.09 × 10²⁷ cm⁻³. From Fig. 1 we see that for most of the density range, this relation is satisfied, thus confirming that the two-fluid MHD treatment is indeed correct.

3 NORMAL MODES IN TWO-FLUID QMHD

To find the normal modes we linearize equations (19), (20), and (22) around a homogeneous equilibrium configuration and transform Fourier in space and time. In order to have a better understanding of how Fermi and Bohm pressures modify the behaviour of the standard normal modes, we disregard spin effects in a first analysis. We shall take them into account in Section 3.2 and compare the differences they introduce in the behaviour of the spinless modes. We shall consider only the effect of the coupling of spin to an external magnetic field, in view of the fact that it is linear in ℏ. Moreover, due to the high non-linearity of the other spin effects, of order ℏ², their main contribution will be on modes of extremely short wavelength, where the fluid assumption might eventually break down.

3.1 Normal modes without spin forces

Without loss of generality, we consider an equilibrium configuration given by |$\langle \bar{B}\rangle=\check{z}$|⁠, |$\langle \bar{u}\rangle=0$| and 〈n〉 = 1 and wave vector |$\bar{k}=k\left(\sin \theta , 0, \cos \theta \right)$|⁠. Linear perturbations around the equilibrium configuration are |$\bar{b}=b_{\perp }\left(\cos \theta , 0, -\sin \theta \right) + b_y\left(0, 1, 0\right)$| and |$\bar{u}=u_{\perp }\left(\cos \theta , 0, -\sin \theta \right) + u_y\left(0, 1, 0\right) + u{\shortparallel}\left(\sin \theta , 0, \cos \theta \right)$|⁠. Note that |$\bar{k}\cdot \bar{b}=0$|⁠. Replacing in equations (19), (20), and (22) and defining |$v$| = ω/k, we write the resulting set of equations in matrix form to help to visualize the structure of the modes:

\begin{eqnarray*} \left({\begin{array}{cccccc}v & \quad -1 & \quad 0 & \quad 0 & \quad 0 & \quad 0 \\-\beta \left(k\right) & \quad v & \quad 0 & \quad 0 & \quad 0 & \quad \sin \theta \\0 & \quad 0 & \quad v & \quad \cos \theta & \quad 0 & \quad 0 \\0 & \quad 0 & \quad \cos \theta & \quad v \left[ 1 + \mu \left(1-\mu \right)\varepsilon ^2 k^2 \right] & \quad -i\varepsilon k\mu v & \quad -i\varepsilon \left(1-\mu \right) k\cos \theta \\0 & \quad 0 & \quad 0 & \quad 0 & \quad v & \quad \cos \theta \\0 & \quad \sin \theta & \quad i\mu \varepsilon k v & \quad i\left(1 - \mu \right) \varepsilon k \cos \theta & \quad \cos \theta & \quad v \left[ 1 + \mu \left(1-\mu \right)\varepsilon ^2 k^2 \right] \end{array}} \right) \left({\begin{array}{c}n\\u{\shortparallel}\\u_y\\b_y\\u_{\perp }\\b_{\perp } \end{array}} \right)=0 \end{eqnarray*}

(23)

where

\begin{eqnarray*} \beta \left(k\right)=\frac{(5/3)\beta _0 + \ell ^2k^2/4}{\mu \left(1-\mu \right)}. \end{eqnarray*}

(24)

Equations (23) constitute a linear set for the unknowns n, u∥, u_⊥, u_y, b_⊥, b_y. These equations contain electron inertia (through μ) and quantum effects (through β₀ and ℓ) and thus extend the standard derivation for the classical magnetosonic and Alfvén modes down to scales smaller than the ion skin-depth. Observe that (u∥, n) correspond to the fast mode, while (u_y, b_y, u_⊥, b_⊥) correspond to the Alfvén-slow sector. For any arbitrary direction of propagation θ with respect to the equilibrium magnetic field, the determinant of the square matrix in equation (23) gives a sixth order polynomial, a₆|$v$|⁶ + a₄|$v$|⁴ + a₂|$v$|² + a₀ = 0 with

\begin{eqnarray*} a_6 = \left[1+\mu \left(1-\mu \right)\varepsilon ^2 k^2\right]^2, \end{eqnarray*}

(25)

\begin{eqnarray*} a_4 = -\cos ^2\left(\theta \right)\left[2+k^2\varepsilon ^2\left(1-2\mu +2\mu ^2\right)\right] -\left[1+\mu \left(1-\mu \right)\varepsilon ^2 k^2\right]^2\beta \left(k\right) -\sin ^2\left(\theta \right)\left[1+\mu \left(1-\mu \right)k^2\varepsilon ^2 \right], \end{eqnarray*}

(26)

\begin{eqnarray*} a_2 = \cos ^2\theta \left\lbrace 1 + \beta \left(k\right)\left[ 2 + k^2\varepsilon ^2\left(1-2\mu + 2\mu ^2\right) \right] \right\rbrace, \end{eqnarray*}

(27)

\begin{eqnarray*} a_0 = -\beta \left(k\right)\cos ^4\theta. \end{eqnarray*}

(28)

In order to better understand how the various effects modify the behaviour of the different modes, we analyse the asymptotic configurations of parallel and perpendicular propagation.

3.1.1 Parallel propagation

This case is characterized by |$\bar{k} \parallel \bar{B}_0$|⁠, i.e. θ = 0. Note that the fast mode decouples from the Alfvén-slow sector. This fast mode is described by

\begin{eqnarray*} \left({\begin{array}{cr}v & \quad -1 \\-\beta \left(k\right) & \quad v \end{array}} \right)\left({\begin{array}{c}n\\u{\shortparallel} \end{array}} \right)=0, \end{eqnarray*}

(29)

which corresponds to an acoustic mode with propagation speed |$v^2_{\parallel f}=\beta (k)$| given by equation (24) above. Due to Bohm pressure (i.e. |$\ell \not=0$|⁠), this mode is dispersive for wavenumbers such that ℓ²k²/4 ≳ 5β₀/3, which leads to wavelengths λ = 2|$\pi$|/k satisfying |$\lambda \lesssim \left(3\pi ^2 \right)^{1/6}10^{-2q} n_0^{-1/3}\sim 1.7 \times 10^{-2q}n_0^{-1/3}$|⁠, i.e, about the mean particle separation or smaller. This is consistent with the scale given by the thermal de Broglie wavelength, indicating that quantum effects are operative at those scales. For the Alfvén-slow sector, we have

\begin{eqnarray*} \left({\begin{array}{cccc}v & \quad 1 & \quad 0 & \quad 0 \\1 & \quad v\left[ 1 + \mu \left(1 - \mu \right)\varepsilon ^2 k^2\right] & \quad - i\mu \varepsilon k v & \quad -i\left(1-\mu \right)\varepsilon k \\0 & \quad 0 & \quad v & \quad 1 \\i\mu \varepsilon k v & \quad i\left(1-\mu \right)\varepsilon k & \quad 1 & \quad v\left[ 1 + \mu \left(1 - \mu \right)\varepsilon ^2 k^2\right] \end{array}} \right) \left({\begin{array}{c}u_y\\b_y\\u_{\perp }\\b_{\perp } \end{array}} \right)=0, \end{eqnarray*}

(30)

which gives the following dispersion relation

\begin{eqnarray*} v^4 \left[ 1 + \mu \left(1 - \mu \right)\varepsilon ^2 k^2\right]^2 - 2v^2 \left[ 1 + \frac{\mu ^2 +\left(1 - \mu \right)^2}{2}\varepsilon ^2 k^2\right] + 1=0. \end{eqnarray*}

(31)

Note that this equation has no quantum effects. The corresponding solutions were found in Andrés et al. (2014a) and we refer the reader to that reference for the detailed analysis of the corresponding modes. It is clear that the difference with the MHD dispersion relation is due to the two-fluid effects considered. The two main roots of equation (31) are shown in Fig. 2. At high wavenumbers, the MHD Alfvén frequency gives rise to two modes, a shear ion cyclotron mode (lower branch) and a whistler mode (upper branch). The latter saturates at the electron cyclotron frequency, while the former does so at the proton cyclotron frequency (Andrés et al. 2014a).

Figure 2.

No-spin Alfvén-slow sector: Lower branch corresponds to shear ion cyclotron waves while upper branch to whistler waves. Middle straight line corresponds to the MHD Alfvén mode, which is shown for reference.

3.1.2 Perpendicular propagation

For |$\bar{k}\perp \bar{B}_0$|⁠, i.e. θ = π/2, we see from expr. (23) that the subspace (b_y, u_⊥) becomes degenerate with (u_y, b_⊥). For the remaining variables we have

\begin{eqnarray*} \left({\begin{array}{crcc}v & \quad -1 & \quad 0 & \quad 0 \\-\beta \left(k\right) & \quad v & \quad 0 & \quad 1\\0 & \quad 0 & \quad v & \quad 0\\0 & \quad 1 & \quad i\mu \varepsilon k v & \quad v\left[ 1 + \mu \left(1 - \mu \right)\varepsilon ^2 k^2\right] \end{array}} \right)\left({\begin{array}{c}n \\u{\shortparallel}\\u_y\\b_{\perp } \end{array}} \right)=0, \end{eqnarray*}

(32)

which leads to the following dispersion relation

\begin{eqnarray*} v^2\left[1 + \mu \left(1 - \mu \right)\varepsilon ^2 k^2\right] - \left[ 1 + \beta \left(k\right)\left(1 + \mu \left(1-\mu \right)\varepsilon ^2 k^2\right) \right]=0 \end{eqnarray*}

(33)

that reduces to the following propagation speed for the fast mode

\begin{eqnarray*} v^2_{\perp f}=\frac{1}{1 + \mu \left(1-\mu \right)\varepsilon ^2 k^2} + \frac{5\beta _0/3 + \ell ^2 k^2/4}{\mu \left(1-\mu \right)}. \end{eqnarray*}

(34)

We see that the acoustic fast mode dispersion relation is modified by electron inertia (με²) and by quantum effects (β₀, ℓ²). In view of the discussion on the fast mode made in the previous subsection, we know that the effect of the Bohm pressure is important for scales of the order of the interparticle separation or smaller. To estimate the importance of the first term in (34) for scales larger than |$n_0^{1/3}$| we neglect the correction ∝ℓ² in expr. (34). Hence, the first term will surpass the second one for modes such that k² < [3/5β₀ − 1/μ(1 − μ)]/ε². However, it is easy to check that for the most part of the parameter’s space of white dwarfs and magnetars the term between square brackets is negative. Only for a small region in the left lowest corner of the parameter space, Fig. 1, i.e. low densities and weak magnetic fields, electron inertia can be operative at any scale. We therefore conclude that for scales larger than the interparticle separation Fermi pressure is the dominant effect for perpendicular propagation.

3.2 Inclusion of spin effects

Let us now take into account the effects of spin. We consider only the term proportional to |$\bar{B}$| in equations (20) and (21) because it is linear in ℓ (equivalently in ℏ) while the terms proportional to ∂_l(n∂_lS_j) and to (∂_jS_i)² are both of order ℓ², and consequently are expected to play a weaker role.

From the equations of motion for each species (i.e. equations 8 and 9), we see that the main contribution to the spin forces is the one of the electrons, because they are proportional to μ⁻¹ ≫ (1 − μ)⁻¹. This fact justifies neglecting the ion spin in equation (20). Moreover, as spin forces are gradients, they disappear from the induction equation, i.e. the curl of equation (21). We must therefore solve the system

\begin{eqnarray*} \frac{{\rm d}\bar{u}}{{\rm d}t} &=& \frac{\skew5\bar{J}}{n}\times \left[\bar{B} + \left(1-\mu \right)\mu \varepsilon ^2\bar{\nabla }\times \left(\frac{\skew5\bar{J}}{n} \right)\right] - \bar{\nabla }\left[\left(1-\mu \right)\mu \varepsilon ^2\frac{J^2}{2n^2}\right] -\frac{5}{2}\frac{\beta }{\mu \left(1-\mu \right)}\bar{\nabla }n^{2/3}\nonumber \\&&+ \, \frac{\ell ^2}{\mu \left(1-\mu \right)} \bar{\nabla }\left(\frac{\nabla ^2\sqrt{n}}{2\sqrt{n}} \right) - \frac{\ell }{2\varepsilon \mu }S^e_j\bar{\nabla }\left[ B_j + \frac{\ell \varepsilon }{2}\partial _l\left(n\partial _lS^e_j\right)\right] \end{eqnarray*}

(35)

\begin{eqnarray*} \left(\partial _t + \bar{u}_e\cdot \bar{\nabla }\right) \bar{S}^e = \frac{1}{\mu \varepsilon }\bar{S}^e \times \left[\bar{B} + \frac{\ell \varepsilon }{2}\partial _j\left(n\partial _j \bar{S}^e\right)\right]. \end{eqnarray*}

(36)

We now linearize these additional terms around an equilibrium configuration for the spin given by

\begin{eqnarray*} \bar{S}^e=\tanh \left(\frac{\mu B_0}{k_{\rm B} T_e}\right)\hat{z} + \left({\begin{array}{c}s_x\\s_y\\0 \end{array}} \right), \end{eqnarray*}

(37)

i.e. we consider deviations in a plane perpendicular to a homogeneous spin configuration along |$z$|⁠. The function tanh (μB₀/k_BT_e) is the Brillouin function that describes a spin distribution in a magnetic field, in thermodynamic equilibrium at temperature T_e. For the present analysis it will be considered constant and equal to 1. The evolution equations for the velocity perturbations now read

\begin{eqnarray*} \omega \left({\begin{array}{c}u{\shortparallel}\sin \theta + u_{\perp }\cos \theta \\u_y\\u{\shortparallel}\cos \theta - u_{\perp }\sin \theta \end{array}} \right)=\dots -\frac{\ell b_{\perp }k\sin \theta }{2\varepsilon \mu }\left({\begin{array}{c}\sin \theta \\0\\\cos \theta \end{array}} \right), \end{eqnarray*}

(38)

where with ‘…’ we refer to the terms without spin, already discussed in the previous sections. Equation (36) leads to

\begin{eqnarray*} -i\omega \left({\begin{array}{c}s_x\\s_y\\0 \end{array}} \right)=\frac{1}{\varepsilon \mu }\left({\begin{array}{c}-b_y + \left(1 + \ell \varepsilon k^2/2\right)s_y\\b_{\perp }\cos \theta - \left(1 + \ell \varepsilon k^2/2\right)s_x\\0 \end{array}} \right). \end{eqnarray*}

(39)

The linear problem is now extended from a 6 × 6 system to an 8 × 8 one, to accommodate the new spin variables s_x, s_y. The full system now is

\begin{eqnarray*} \left({\begin{array}{cccccccc}v & \quad -1 & \quad 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0\\-\beta & \quad v & \quad 0 & \quad 0 & \quad 0 & \quad \left(1+ \ell k/2\varepsilon \mu \right)\sin \theta & \quad 0 & \quad 0\\0 & \quad 0 & \quad v & \quad \cos \theta & \quad 0 & \quad 0 & \quad 0 & \quad 0 \\0 & \quad 0 & \quad \cos \theta & \quad v \left[ 1 + \mu \left(1-\mu \right)\varepsilon ^2 k^2 \right] & \quad -i\varepsilon k\mu v & \quad -i\left(1-\mu \right)\varepsilon k\cos \theta & \quad 0 & \quad 0\\0 & \quad 0 & \quad 0 & \quad 0 & \quad v & \quad \cos \theta & \quad 0 & \quad 0\\0 & \quad \sin \theta & \quad i\mu \varepsilon k v & \quad i\left(1 - \mu \right) \varepsilon k \cos \theta & \quad \cos \theta & \quad v \left[ 1 + \mu \left(1-\mu \right)\varepsilon ^2 k^2 \right] & \quad 0 & \quad 0 \\0 & \quad 0 & \quad 0 & \quad 1/\varepsilon \mu & \quad 0 & \quad 0 &-iv & \quad -\left(1 + \ell \varepsilon k^2/2\right)/\varepsilon \mu \\0 & \quad 0 & \quad 0 & \quad 0 & \quad 0 & \quad - 1/\varepsilon \mu & \quad \left(1 + \ell \varepsilon k^2/2\right)/\varepsilon \mu & \quad -iv \end{array}} \right) \left({\begin{array}{c}n\\u{\shortparallel}\\u_y\\b_y\\u_{\perp }\\b_{\perp }\\s_x\\s_y \end{array}} \right)=0 \nonumber \\ \end{eqnarray*}

(40)

The determinant of this 8 × 8 matrix reduces to

\begin{eqnarray*} \left[v^2 - \left(\frac{1+\ell \varepsilon k^2/2}{\varepsilon \mu }\right)^2\right] \mathbf {\Delta }_6=0, \end{eqnarray*}

(41)

where |$\mathbf {\Delta }_6$| is the determinant of the minor solved in the previous subsection, with the only difference with respect to the non-spin case of ‘(1 + ℓk/2εμ)sin θ’ in the (u∥, b_⊥) coupling. The new factor in equation (41) corresponds to a whistler wave (Stenzel 1999). The speed of propagation for this whistler is

\begin{eqnarray*} v_{ws}=\pm \left(\frac{1+\ell \varepsilon k^2/2}{\varepsilon \mu }\right) \end{eqnarray*}

(42)

and is exclusive of the spin degrees of freedom. This new mode is independent of the one described in the no-spin case, for parallel propagation. It is a highly dispersive mode at frequencies higher than the electron cyclotron one. Note that if we set ℓ = 0 this mode continues to exist, but it is not dispersive and only represents the transport of a ‘spin-label’ due to the spiraling of the electrons around the magnetic lines and not a real spin effect. This spin effect becomes important for |$k\gt \sqrt{2/\ell \varepsilon }$|⁠. In Fig. 3, we show this wavenumber as a function of B₀ for three different values of n₀. We see that for weak background magnetic fields and high densities, this mode can be present at long wavelengths. In Fig. 4, we show the dependence of this wavenumber with n₀ for three different values of B₀. Once again, we see that the mode can be macroscopic for weak fields and high densities, consistent with the previous plot. According to (Fig. 1), magnetars would be good candidates for the propagation of this mode.

Figure 3.

Log-log plot of |$k=\sqrt{2/\ell \varepsilon }$| as a function of B₀ for three values of the background density. Wavenumbers higher than the ones indicated by the curves experience spin effects.

Figure 4.

Log–log plot of |$k=\sqrt{2/\ell \varepsilon }$| as a function of n₀ for three values of the background magnetic field. Wavenumbers higher than the ones indicated by the curves experience spin effects.

The dispersion relation for the other three normal modes in QMHD is again of the form d₆|$v$|⁶ + d₄|$v$|⁴ + d₂|$v$|² + d₀ = 0, with

\begin{eqnarray*} d_6 = \left[1+\mu \left(1-\mu \right)\varepsilon ^2 k^2\right]^2. \end{eqnarray*}

(43)

\begin{eqnarray*} d_4 = -\cos ^2\left(\theta \right)\left[2+k^2\varepsilon ^2\left(1-2\mu + 2\mu ^2\right)\right] - \left[1+\mu \left(1-\mu \right)k^2\varepsilon ^2\right]^2\beta \left(k\right) - \sin ^2\left(\theta \right)\left[1+\mu \left(1-\mu \right)k^2\varepsilon ^2\right]\left(1+\frac{\ell k}{2\varepsilon \mu }\right). \end{eqnarray*}

(44)

\begin{eqnarray*} d_2 = \cos ^2\left(\theta \right) \left\lbrace 1 + \beta \left(k\right)\left[2 + k^2\varepsilon ^2\left(1-2\mu + 2\mu ^2\right) \right] + \frac{\ell k}{2\varepsilon \mu }\sin ^2\left(\theta \right)\right\rbrace. \end{eqnarray*}

(45)

\begin{eqnarray*} d_0 = - \beta \left(k\right)\cos ^4\theta. \end{eqnarray*}

(46)

3.2.1 Parallel and perpendicular propagation

As for the non-spin case, we analyse the parallel and perpendicular propagation. By simple inspection of expressions (43)–(46), we see that there is no spin contribution for parallel propagation and consequently the normal modes of this sector coincide with those found in Section 3.1.1.

For perpendicular propagation, d₂ = d₀ = 0 and again the only surviving mode is the fast mode, modified by the spin-magnetic coupling, The dispersion relation in this case is

\begin{eqnarray*} v_{\perp s}^2=\beta \left(k\right) +\frac{1+\ell k/2\varepsilon \mu }{1+\mu \left(1-\mu \right)\varepsilon ^2 k^2}. \end{eqnarray*}

(47)

The spin correction will be larger than one if k > 2εμ/ℓ, or equivalently k > (cm_e/2πeℏ)(B₀/n₀) ∼ 1.8 × 10²⁸(B₀/n₀). For the values of B₀ and n₀ corresponding to astrophysical compact objects (see Fig. 1) this corresponds again to wavelengths shorter than the interparticle separation. To physically interpret the quantum correction we write

\begin{eqnarray*} \frac{\ell }{2\varepsilon \mu }=\frac{\mu _B B_0 n_0}{B_0^2/4\pi }, \end{eqnarray*}

(48)

where μ_B = eℏ/2m_ec is the Bohr magneton. Expression (48) then represents the potential energy density of all the electron spins embedded in the external field B₀ relative to the magnetic energy density of B₀.

4 CONCLUSIONS

In this paper, we extended the two-fluid MHD description of a magnetized plasma developed by Andrés et al. (2014b) by including quantum effects such as Fermi pressure, Bohm potential, and spin interactions. The motivation behind this study is that the mentioned quantum effects are operative at scales that may overlap with the ones where two-fluid effects must be taken into account, i.e. scales shorter than the ion-skin depth. At those small scales it is known that the MHD is not an appropriate formalism to describe, for example, the turbulent energy spectrum, while the inclusion of two-fluid effects such as the Hall effect and electron inertia does account for the small scales features observed in magnetized plasma turbulence.

We wrote down the complete set of two-fluid QMHD equations and, after linearizing them, we obtained the dispersion relations that generalize previous results found in the literature on two-fluid MHD description of linear waves in plasmas (Andrés et al. 2014b). We separated our study into spinless and spin plasmas and, in each of these cases we analysed parallel and perpendicular propagation of the linear perturbations.

In the absence of spin effects, we found that for parallel propagation the frequency in the fast magnetosonic sector is determined by the quantum effects (Fermi pressure corrected by Bohm forces), weighed by the ‘non-quantum’ two fluid effects. In contrast, the frequencies of the Alfvén-slow sector depend only on electron inertia and Hall effect, i.e. they are not affected by quantum effects. The features of this sector were analysed in detail by Andrés et al. (2014a), who showed that due to the two-fluid effects, at high wavenumbers the MHD Alfvén mode separates into whistler and ion cyclotron modes. For perpendicular propagation on the other side, we only have the fast mode, and we found that its velocity is the one obtained for parallel propagation plus a term that depends only on two-fluid effects. For the parameter space of compact objects, this ‘non-quantum’ correction, however, is in general not operative, except for very small densities and weak magnetic fields.

We considered spin effects only at the level of the spin-B coupling, as it is linear in ℏ. The other spin interactions, being second order in ℏ are expected to have weaker effects. In this case, irrespective of the propagation direction, a whistler mode appears, which is exclusive of the spin degrees of freedom. To the extent of our knowledge, until now this mode was not described in the literature. It arises only when two-fluid effects are considered (Andrés et al. 2014a,b) and, due to the electron spin, it is highly dispersive at frequencies higher than the electron cyclotron one. For high densities, the corresponding wavelengths can be well larger than the interparticle separation thus having potentially observable effects.

For the modes in the fast and Alfvén-slow sectors, the presence of spin does not modify parallel propagation of linear modes, which retain the same features as they had without spin. For perpendicular propagation, however, the spin-B interaction modifies the fast mode frequency by introducing a dispersive, spin-dependent correction in the pure two-fluid term which, for the densities and magnetic field ranges of compact objects, would manifest at scales well smaller than the interparticle separation.

In summary, we extended the two-fluid MHD formalism to include quantum effects and studied the propagation of linear waves. A next step would be to analyse non-linear effects, as e.g. shock waves and turbulence, to investigate the modifications that the two-fluid together with quantum effects introduce in those phenomena.

ACKNOWLEDGEMENTS

DOG acknowledges financial support from grants UBACyT 20020130100629BA to the Department of Physics of FCEyN-UBA and PICT 1007 to IAFE. AK thanks the Physics Department of Facultad de Ciencias Exactas y Naturales - UBA for kind hospitality during the development of part of this work, and also financial support from UESC, BA, Brazil.

Footnotes

1

Other plasma systems where quantum effects might become important are those extremely small, so that the classical transport models become invalid. Examples of such systems are nanoscale electronic devices (Cui & Lieber 2001), thin metal films (Su et al. 2010), and high energy lasers (Ridgers, Ahmad & Khan 2017).

APPENDIX: MAKING THE SYSTEM OF MHD EQUATIONS NON-DIMENSIONAL

Here, we put the equations in dimensionless form. We begin with

\begin{eqnarray*} \partial _t \bar{u}_\mathrm{ s} + \left(\bar{u}_\mathrm{ s}\cdot \bar{\nabla }\right) \bar{u}_\mathrm{ s} = \frac{q_\mathrm{ s}}{m_\mathrm{ s}}\bar{E} + \frac{q_\mathrm{ s}}{m_\mathrm{ s}c}\bar{u}_\mathrm{ s}\times \bar{B} -\frac{1}{m_\mathrm{ s} n_\mathrm{ s}}\bar{\nabla }p_\mathrm{ s} + \frac{\hbar ^2}{2m_\mathrm{ s}^2}\bar{\nabla }\left(\frac{\nabla ^2 n_\mathrm{ s}^{1/2}}{n_\mathrm{ s}^{1/2}}\right) +\frac{\hbar q_\mathrm{ s}}{2m_\mathrm{ s}^2c}S^\mathrm{ s}_j\bar{\nabla }\hat{\bar{B}}^\mathrm{ s}_j + \frac{\hbar ^2}{2m_\mathrm{ s}^2}\bar{\nabla }\left(\partial _jS^\mathrm{ s}_i\partial _jS^\mathrm{ s}_i\right) \end{eqnarray*}

(A1)

We consider the following fiducial quantities (to be properly defined later) to get rid of units: n₀, u₀, L₀, E₀, and B₀. Sⁱ is already dimensionless. Time and spatial derivatives are then written

\begin{eqnarray*} \nabla &\rightarrow& \frac{1}{L_0}\nabla\\\frac{\partial }{\partial t} &\rightarrow& \frac{u_0}{L_0}\frac{\partial }{\partial t}. \end{eqnarray*}

(A2),(A3)

We keep the same letters for the dimensionless variables for simplicity. We also write |$m_s=M \tilde{m}_s$|⁠. So equation (A1) becomes

\begin{eqnarray*} \frac{u_0^2}{L_0} \left[\partial _t \bar{u}_\mathrm{ s} + \left(\bar{u}_\mathrm{ s}\cdot \bar{\nabla }\right) \bar{u}_\mathrm{ s}\right] &=& \frac{q_\mathrm{ s} E_0}{M\tilde{m}_\mathrm{ s}}\bar{E} + \frac{q_\mathrm{ s} u_0 B_0}{M\tilde{m}_\mathrm{ s}c}\bar{u}_\mathrm{ s}\times \bar{B} -\frac{p_0}{M\tilde{m}_\mathrm{ s} n_0 \tilde{n}_\mathrm{ s} L_0}\bar{\nabla }\tilde{p}_\mathrm{ s} + \frac{\hbar ^2}{2 M^2 \tilde{m}_\mathrm{ s}^2L_0^3}\bar{\nabla }\left(\frac{\nabla ^2 \tilde{n}_\mathrm{ s}^{1/2}}{\tilde{n}_\mathrm{ s}^{1/2}}\right)\nonumber \\&&+ \, \frac{\hbar q_\mathrm{ s}}{2M^2 \tilde{m}_\mathrm{ s}^2cL_0}S^\mathrm{ s}_j\bar{\nabla }\hat{\bar{B}}^\mathrm{ s}_j + \frac{\hbar ^2}{2 M^2\tilde{m}_\mathbf{ s}^2 L_0^3}\bar{\nabla }\left(\partial _jS^\mathrm{ s}_i\partial _jS^\mathrm{ s}_i\right) \end{eqnarray*}

(A4)

We rewrite it as

\begin{eqnarray*} \tilde{m}_s\left[\partial _t \bar{u}_\mathrm{ s} + \left(\bar{u}_\mathrm{ s}\cdot \bar{\nabla }\right) \bar{u}_\mathrm{ s}\right] &=& \frac{L_0}{u_0^2} \frac{q_\mathrm{ s} E_0}{M}\bar{E} +\frac{L_0}{u_0} \frac{q_\mathrm{ s} B_0 }{Mc}\bar{u}_\mathrm{ s}\times \bar{B} -\frac{L_0}{u_0^2} \frac{p_0}{M n_0 \tilde{n}_\mathrm{ s} L_0}\bar{\nabla }\tilde{p}_\mathrm{ s} + \frac{L_0}{u_0^2} \frac{\hbar ^2}{2 M^2 \tilde{m}_sL_0^3}\bar{\nabla }\left(\frac{\nabla ^2 \tilde{n}_\mathrm{ s}^{1/2}}{\tilde{n}_\mathrm{ s}^{1/2}}\right)\nonumber \\&&+ \,\frac{L_0}{u_0^2} \frac{\hbar q_\mathrm{ s} B_0}{2M^2 \tilde{m}_\mathrm{ s}cL_0}S^\mathrm{ s}_j\bar{\nabla }\hat{\bar{B}}^\mathrm{ s}_j + \frac{L_0}{u_0^2} \frac{\hbar ^2}{2 M^2\tilde{m}_\mathrm{ s} L_0^3}\bar{\nabla }\left(\partial _jS^\mathrm{ s}_i\partial _jS^\mathrm{ s}_i\right). \end{eqnarray*}

(A5)

For a proton–electron plasma, we define

\begin{eqnarray*} M = m_e + m_p \end{eqnarray*}

(A6)

\begin{eqnarray*} \tilde{m}_e = \mu=\frac{m_e}{M} \end{eqnarray*}

(A7)

and besides

\begin{eqnarray*} E_0 = \frac{u_0}{c}B_0 \end{eqnarray*}

(A9)

\begin{eqnarray*} u_0 = V_A =\frac{B_0}{\sqrt{4\pi M n_0}} \end{eqnarray*}

(A10)

\begin{eqnarray*} p_0 = \frac{\left(3\pi ^2\right)^{2/3}\hbar ^2}{5 M}n_0^{5/3} \end{eqnarray*}

(A11)

\begin{eqnarray*} \omega _M = \sqrt{\frac{4\pi e^2 n_0}{M}} \end{eqnarray*}

(A12)

\begin{eqnarray*} \lambda _0 = \frac{\hbar }{MV_A} \end{eqnarray*}

(A13)

\begin{eqnarray*} \ell = \frac{\lambda _0}{L_0} \end{eqnarray*}

(A14)

Replacing in (A5) we have for the electron fluid

\begin{eqnarray*} \mu \left[\partial _t \bar{u}_s + \left(\bar{u}_s\cdot \bar{\nabla }\right) \bar{u}_s\right] &=& -\frac{L_0 e B_0}{u_0 c M} \left[\bar{E} +\bar{u}_s\times \bar{B}\right] -\frac{1}{u_0^2} \frac{p_0}{M n_0}\frac{\bar{\nabla }\tilde{p}_s}{\tilde{n}_s} + \frac{1}{u_0^2} \frac{\hbar ^2}{2 M^2 \mu L_0^2}\bar{\nabla }\left(\frac{\nabla ^2 \tilde{n}_s^{1/2}}{\tilde{n}_s^{1/2}}\right)\nonumber \\&&- \, \frac{1}{u_0^2} \frac{\hbar e B_0}{2M^2 \mu c}S^s_j\bar{\nabla }\hat{\bar{B}}^s_j + \frac{1}{u_0^2} \frac{\hbar ^2}{2 M^2\mu L_0^2}\bar{\nabla }\left(\partial _jS^s_i\partial _jS^s_i\right) \end{eqnarray*}

(A15)

The coefficients of the different terms can be cast as

\begin{eqnarray*} \frac{L_0 e B_0}{u_0 c M} = \frac{L_0 e \sqrt{4\pi M n_0 }}{ c M}=\frac{L_0}{c}\omega _M \equiv \frac{1}{\varepsilon } \end{eqnarray*}

(A16)

\begin{eqnarray*} \frac{1}{u_0^2} \frac{p_0}{M n_0} = \frac{1}{V_\mathrm{ A}^2} \frac{1}{M n_0} \frac{\left(3\pi ^2\right)^{2/3}\hbar ^2}{5 M}n_0^{5/3}=\frac{\left(3\pi ^2\right)^{2/3}}{5 }\left(\lambda _0 n_0^{1/3}\right)^2 \equiv \beta _0 \end{eqnarray*}

(A17)

\begin{eqnarray*} \frac{1}{u_0^2} \frac{\hbar ^2}{2 M^2 \mu L_0^2} = \frac{\hbar ^2}{V_\mathrm{ A}^2 M^2} \frac{1}{2 \mu L_0^2}=\frac{\lambda _0^2}{2\mu L_0^2}=\frac{\ell ^2}{2\mu } \end{eqnarray*}

(A18)

\begin{eqnarray*} \frac{1}{u_0^2} \frac{\hbar e B_0}{2M^2 \mu c} &=& \frac{\hbar }{V_A M} \frac{B_0 e}{2M V_\mathrm{ A} \mu c}=\frac{\lambda _0}{L_0}\frac{L_0 \sqrt{4\pi e^2M n_0}}{2M \mu c}=\ell \frac{L_0\omega _M}{2\mu c}=\frac{\ell }{2\varepsilon \mu }. \end{eqnarray*}

(A19)

Substituting all these expressions into equation (A15), we obtain

\begin{eqnarray*} \mu \left[\partial _t \bar{u}_\mathrm{ s} + \left(\bar{u}_\mathrm{ s}\cdot \bar{\nabla }\right) \bar{u}_\mathrm{ s}\right] = -\frac{1}{\varepsilon } \left[\bar{E} +\bar{u}_\mathrm{ s}\times \bar{B}\right] -\beta \frac{\bar{\nabla }\tilde{n}_\mathrm{ s}^{5/3}}{\tilde{n}_\mathrm{ s}} + \frac{\ell ^2}{2\mu } \bar{\nabla }\left(\frac{\nabla ^2 \tilde{n}_s^{1/2}}{\tilde{n}_\mathrm{ s}^{1/2}}\right) -\frac{\ell }{2\varepsilon \mu } S^\mathrm{ s}_j\bar{\nabla }\hat{\bar{B}}^\mathrm{ s}_j + \frac{\ell ^2}{2\mu }\bar{\nabla }\left(\partial _jS^\mathrm{ s}_i\partial _jS^\mathrm{ s}_i\right). \end{eqnarray*}

(A20)