Spin induced nonlinearities in the electron MHD regime

We consider the influence of the electron spin on the nonlinear propagation of whistler waves. For this purpose a recently developed electron two-fluid model, where the spin up- and down populations are treated as different fluids, is adapted to the electron MHD regime. We then derive a nonlinear Schrodinger equation for whistler waves, and compare the coefficients of nonlinearity with and without spin effects. The relative importance of spin effects depend on the plasma density and temperature as well as the external magnetic field strength and the wave frequency. The significance of our results to various plasmas are discussed.


I. INTRODUCTION
The dynamics of dense ionized matter, cold matter in strong magnetic fields, and nonlinear degenerate plasmas has applications to both laboratory and naturally occurring systems. Many such matter states goes under the collective notation of quantum plasmas. A multitude of studies devoted to quantum plasma effects can be found in the literature, much of it inspired by works like e.g. Refs. [1,2]. In the last decade there has been somewhat of a surge in the interest of quantum plasmas [3,4,5,6,7,8,9,10,11,12]. Interesting applications of this field can be found in for example plasmonics [13,14], quantum wells [15] and ultracold plasmas [16]. Common to such applications are rather "extreme" parameters, compared to most laboratory and space plasmas. More specifically, the plasma densities needs to be very high and/or the temperatures correspondingly low. For astrophysical plasmas, the situation is somewhat different since the strong magnetic fields [17,18] may induce various types of quantum effects. In most of the above studies, the spin effects plays little or no dynamic role. The inclusion of collective spin dynamics [19,20]gives rise to new modes in plasma, both at fluid [9,21] and kinetic scale [12]. Indeed, even dusty plasmas can show interesting magnetization effects [22,23]. In Ref. [11] the picture outlined above, concerning the necessary parameter space for quantum effects to be important, was to some extent modified, as it was shown that the spin properties of electrons can be important in plasmas even outside the high density/low temperature regime, also for moderate magnetic field strengths. Moreover, a recent focus on the nonlinear regime in quantum plasmas [24,25,26] makes the question of magnetization nonlinearities interesting.
Motivated by the above, we will in the present work further extend the analysis put forward in Ref. [11]. In that work, the electrons were described using a two-fluid model, where the spin-up and spin-down populations relative to the magnetic field were treated as different fluids in the standard magnetohydrodynamic (MHD) regime. Here we will extend that treatment to cover the electron-MHD (EMHD) regime. In particular we will study weakly nonlinear whistler waves, and derive a nonlinear Schrödinger (NLS) equation for the slowly varying amplitude, both using classical weakly relativistic theory and the recent two-fluid spin model, adapted for the EMHD regime. By comparing the nonlinear coefficients in the dif-ferent models and their dependence on the plasma parameters (temperature, density external magnetic field strength), the relative importance of the electron spin effects in various regimes can be deduced. The result that electron spin effects can be important in other regimes, as compared to certain much studied quantum effects [such as the Bohm-de Broglie potential (see also Ref. [27] for a discussion) and the Fermi pressure], is confirmed. Finally we compare the relative importance of electron spin effects in the EMHD regime with that in the standard MHD regime.

II. TWO-FLUID MODEL WITH SPIN
The purpose of our work is to compare the nonlinearities from classical and quantum effects, respectively, in the EMHD regime. As a starting point, we follow Ref. [28] and derive the EMHD model. We assume that the time-scale of interest is small enough that the ion motion, due to their large mass compared to the electrons, can be neglected. We also neglect the displacement current in Ampères law. Furthermore by assuming an isothermal pressure model and no dissipation, the governing equation can be written as where B is the magnetic field, d 2 e = c 2 /ω 2 pe , ω 2 pe = q 2 e n/m e ε 0 being the electron plasma frequency, and α = 1/nq e µ 0 . Studying the linear modes of this equation propagating parallel to the magnetic field one finds whistler waves with dispersion relation ω(k) = ω ce c 2 k 2 /ω 2 pe . However, since this model does not allow any density fluctuations, an attempt to derive an NLS-equation shows that the model do not give rise to any cubic nonlinearities. To still be able to do the intended comparison, we still use the assumptions corresponding to the EMHD regime but perform a more general treatment, allowing for relativistic particle velocities and using a multifluid model that permits density perturbations.
The quantum model used in the comparison is obtained formally by starting from the Pauli Hamiltonian as is done in Ref. [8], using ensemble averaging to obtain the fluid equations the sign depending on the spin orientation S ± = ±(nh/2)B relative to the magnetic field. Thus, assuming a two-electron fluid model, for which the distinction between the fluids is through their relative spin orientation, we have where the subscript ± denotes spin orientation parallel or anti-parallel to the external magnetic field respectively, and B = |B|. In case the up-and down spin populations are not equal, there will be a net magnetization and a corresponding magnetization current. Thus within this model the total current density to be used in Ampere's law is written where the last term is the magnetization current due to the spin, andB = B/B is a unit vector in the direction of the magnetic field.

III. LINEAR THEORY
Linearizing the momentum and fluid equations around a constant magnetic field B 0 = B 0ẑ , and assuming transversal waves propagating parallel to this external magnetic field we can deduce v =σ E, from Eq. (5), wherê is the conductivity tensor, and ω cs = q s B 0 /m s is the cyclotron frequency for particle species s, and we let the vectors here just contain the parts perpendicular toẑ. Note that since the variations of B is nonlinear in the amplitude for parallel propagation, the spin effects do not enter here. Furthermore, when the thermal energy k B T is much larger than the energy difference between the spin states, µ B B 0 , the difference between the number density of the spin-up and down populations, n 0+ − n 0− , in the thermodynamic ground state is exponentially small (proportional to exp(−µ B B 0 /k B T )), and hence we can omit the linearized part of the magnetization current. Thus in this approximation no quantum effects remains in the linearized theory. From Maxwell equations, we then obtain (10) Thus, we obtain the general dispersion relation for this geometry To obtain the EMHD limit, we disregard the displacement current in Ampères law and regard the ions as fixed. This corresponds to neglecting the unit matrix in Eq. (9) and letting ω ce ≪ ω. By this procedure we get the dispersion relation for small amplitude whistler waves propagating parallel to the external magnetic field. The dispersion relation (12) was obtained by starting from the two-fluid model and then taking the EMHD limit. If one would start directly from the EMHD plasma equation, the corresponding result would be This difference is due to the fact that for the EMHD assumptions to apply, the parallel (to the external magnetic field) wavenumber k must obey k ≪ ω p /c. Thus in our case with parallel propagation (k =k ) we must approximate the denominator with unity, in which case the different expressions agree. As a side note, for general directions of propagation the factor (1+c 2 k 2 /ω 2 p ) −1 appears correctly from EMHD theory, but since only the perpendicular part of the wavenumber k ⊥ is allowed to be comparable to the inverse skin depth (i.e. As usual, by using an ansatz of a weakly modulated amplitude in the EMHD model Eq. (1), neglecting nonlinear terms and higher order dispersion we obtain the linear part of the one-dimensional NLS equation.
Here v g is the group velocity and v ′ g = dv g /dk is the group velocity dispersion. Since the spin effects do not enter linear theory, these coefficients are the same in our classical and quantum mechanical models.

IV. CLASSICAL NONLINEAR THEORY
To explore nonlinearities due to relativistic effects, the momentum equation is modified by letting v s → γv s in the left hand side, and it thus reads where γ = 1/(1 − v 2 /c 2 ) and we have introduced the thermal velocity v st = (k B T /m s ) 1/2 for species s. The γ-factor can be Taylor expanded to first order and will thus result in purely cubic nonlinearity in the velocity. Including this nonlinearity only, it is straightforward to deduce the NLS equation (16) The nonlinear term above will be complemented by nonlinear density modifications induced by the ponderomotive force. Within a model that only includes the electron dynamics, the density modifications will be limited due to the general tendency of charge neutrality. However, for sufficiently long pulses the low frequency ion dynamics will start to contribute to the nonlinear behavior of the electrons, and it turns out that a fair comparison between quantum and classical nonlinearities must include this effect. To capture the ponderomotive nonlinearities, we start with Eq. (15) (for simplicity omitting the relativistic contribution, that we already know), for a twofluid ion-electron model . Again neglecting the displacement current in Ampères law, linearizing as previously and using the Maxwell equations we obtain the system for the low-frequency variables, wherê and we can read off that The coefficients κ e and κ i are determined by solving the corresponding differential equation, obtained from Eq. (17), using Greens function techniques, and the result is: Now that the low frequency perturbations have been determined, the back reaction on the original time scale can be calculated. We note that on this fast time scale we return to neglecting ion motion. Then, we obtain where the subscript x indicates thex-component of the vector. Combining this result with the linear theory we obtain an NLS equation that reads

V. FULLY NONLINEAR THEORY
As opposed to the case of a normal EMHD plasma, in the quantum case we have one equation for each electron spin direction, and the extra term ±µ B n ± ∇B due to the spins influence on the magnetization. One can note that if the spin populations are exactly equal in density, when adding the two force equations this term will vanish, and this corresponds to the classical case. However, if there is a slight difference in number density, nonlinear fluctuations in the magnetic field will be induced. To explore this effect, we try to derive an EMHD model , but now using the two-fluid spin model. Eq.
(1) is then replaced by where the average number of electrons N = (n + + n − )/2 tend to be deviate little from the unperturbed density (due to charge neutrality), but the difference between the electron species n = (n + − n − )/2 can vary more. Furthermore we introduce the We here point out that in the approximation considered, where the unperturbed density difference difference is neglected, similar as before the linear treatment give agreement with the classical case. Next, to calculate the low frequency perturbations of n, we need the difference between Eqs. (5) for the two species, which is written Filtering out the low frequency time scale we obtain: and where we have introduced the electron thermal velocity, v te = (k B T /m e ) 1/2 . Due to the reduced geometry of the problem, with parallel propagating circularly polarized modes, all second harmonic density perturbations can be neglected, and all second order nonlinearities in the magnetic field also vanish. Furthermore, N lf ≪ n lf as a consequence of the system tending towards charge neutrality, and thus only n lf needs to be considered here.
Inserting this now in Eq. (24) and considering the original time scale the only nonvanishing contribution to the nonlinear constant is Thus, the full NLS equation including all the effects discussed above (relativistic nonlinearity, classical density perturbations induced by the ponderomotive force, and a spin dependent density modification, driven by the nonlinear magnetic dipole force) will be (cf. Eq. (23)) This equation is the main result of this paper. From the magnitude of the nonlinear coefficient, one can determine the regimes in which the spin terms can dominate and be responsible for e.g. soliton formation.

VI. DISCUSSION
In the present paper we have studied weakly nonlinear whistler waves propagating along the magnetic field. A nonlinear Schrödinger equation has been derived for the case of classical nonlinerities (see Eq. (29)). The nonlinear coefficient than gets two contributions; from relativistic effects and from low-frequency density modifications induced by the ponderomotive force. Taking spin effects into account, within in a electron two-fluid spin model, it is found that the lowfrequency part of the magnetic dipole force separates the spin up and spin down populations. Due to the different magnetization currents from the two populations, a spin contribution to the nonlinear coefficient then arises. Firstly, comparing the spin contribution to the nonlinear coefficient with the relativistic contribution, we see that the former is larger provided that Here we have used the lowest frequency allowed by the model ω ∼ ω ci to get a condition that is relatively easy to fulfill. However, we must also compare the spin induced nonlinearity against the contribution from the nonlinear density oscillations induced by the ponderomotive force. It is found that the former dominates when where we have used that the maximum value of kc is roughly ω pe , due to the limitations imposed by the geometry in combination with the EMHD approximation. The factor hω ce /m i C 2 A is the condition for nonlinear spin effects to dominate, when a similar comparison is made in the standard MHD regime, according to Ref. [11]. To get a more favorable comparison (i.e. a condition that is easier to reach under laboratory conditions) than in these previous works, the second factor, hω ce /m i v 2 ti , must be larger than unity. This is unfortunately not the case for the parameters usually found in laboratory conditions. However, astrophysical plasmas with parameters fulfilling both conditions (30) and (31) can be found, e.g. in the vicinity of pulsars or magnetars [17], and thus we note that effects associated with the electron spin can be more important than the classical relativistic and ponderomotive nonlinearities in such environments.
The present study has focused on the EMHD regime. While it is shown that spin effect certainly can be important during e.g. astrophysical plasma conditions, our study suggest the standard MHD regime [11] can be more affected by the electron spin properties during laboratory conditions . However, much more work remains to be done in order for this conclusion to be settled, as the picture may change when a more general geometry is considered, or when kinetic effects [12] are taken into account.