2 Model: NLPE with spin–orbit interaction

The spin- $1/2$ QHD of plasmas can be directly derived from the many-particle Pauli equation (Kuz’menkov et al. Reference Kuz’menkov, Maksimov and Fedoseev2001a ; Kuz’menkov, Maksimov & Fedoseev Reference Kuz’menkov, Maksimov and Fedoseev2001b ). This method allows us to include the spin–orbit interaction (Andreev & Kuzmenkov Reference Andreev and Kuzmenkov2011; Andreev & Kuz’menkov Reference Andreev and Kuz’menkov2012) and other relativistic effects (Andreev & Kuz’menkov Reference Andreev and Kuz’menkov2007; Ivanov et al. Reference Ivanov, Andreev and Kuz’menkov2014; Andreev Reference Andreev2015a ; Ivanov et al. Reference Ivanov, Andreev and Kuz’menkov2015). Next, the QHD equations can be represented in the form of the nonlinear Schrödinger equation or the nonlinear Pauli equation (Kuz’menkov & Maksimov Reference Kuz’menkov and Maksimov1999; Kuz’menkov et al. Reference Kuz’menkov, Maksimov and Fedoseev2001a ; Andreev & Kuz’menkov Reference Andreev and Kuz’menkov2012; Andreev Reference Andreev2014, Reference Andreev2015a ).

The set of QHD equations can be presented as the nonlinear Schrödinger equation (NLSE) (Kuz’menkov & Maksimov Reference Kuz’menkov and Maksimov1999). This can be done for spinless particles (Kuz’menkov & Maksimov Reference Kuz’menkov and Maksimov1999) and spinning particles (Kuz’menkov et al. Reference Kuz’menkov, Maksimov and Fedoseev2001a ). In the latter case we have the nonlinear Pauli equation (Kuz’menkov et al. Reference Kuz’menkov, Maksimov and Fedoseev2001a ). NLSEs for three-dimensional and two-dimensional degenerate electron gases with Coulomb exchange interactions are derived in Andreev (Reference Andreev2014). The nonlinear Pauli equation for an electron gas with spin–orbit interaction is presented in Andreev & Kuz’menkov (Reference Andreev and Kuz’menkov2012). The set of two nonlinear Pauli equations for spin- $1/2$ electron–positron plasmas with the annihilation interaction is derived in Andreev (Reference Andreev2015a ). NLSEs, or more exactly, nonlinear Klein–Fock–Gordon and nonlinear Feynman–Gell-Mann equations are presented in Mahajan & Asenjo (Reference Mahajan and Asenjo2014). A discussion of the nonlinear Pauli equation can also be found in Mahajan & Asenjo (Reference Mahajan and Asenjo2014).

We want to present a new model and thus we apply a simplified method of derivation. Hence, readers can focus on new effects instead of rather large equations existing in the many-particle derivation. A derivation of the SSE model from the many-particle Pauli equations is presented in Andreev (Reference Andreev2016b ), where it is done for the SSE quantum kinetics. We use the nonlinear Pauli equation (NLPE) derived in Andreev & Kuz’menkov (Reference Andreev and Kuz’menkov2012) for spin- $1/2$ quantum plasmas with spin–orbit interaction. Following Andreev & Kuz’menkov (Reference Andreev and Kuz’menkov2015c ) we generalize the NLPE to include the spinor pressure. We apply the generalized NLPE to derive the SSE-QHD with the spin–orbit interaction and the Fermi spin current.

The NLPE equation appears as follows

(2.1) $$\begin{eqnarray}\imath \hbar \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6F7}(\boldsymbol{r},t)=\left(\frac{1}{2m}\widehat{\boldsymbol{D}}^{2}+q_{e}\unicode[STIX]{x1D711}+\widehat{\unicode[STIX]{x03C0}}-\unicode[STIX]{x1D6FE}_{e}\widehat{\unicode[STIX]{x1D748}}\boldsymbol{B}-\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}(\widehat{\unicode[STIX]{x1D748}}\boldsymbol{\cdot }(\boldsymbol{E}\times \hat{\boldsymbol{D}}))\right)\unicode[STIX]{x1D6F7}(\boldsymbol{r},t),\end{eqnarray}$$

where $\boldsymbol{D}=\boldsymbol{p}-q_{e}\boldsymbol{A}/c$ and $\unicode[STIX]{x1D6FE}_{e}$ is the magnetic moment of the particles. Equation (2.1) is coupled to the electro-magneto-static Maxwell equations.

In this equation, the wave function is the spinor function and we present its explicit form (see § 56 in Landau & Lifshitz Reference Landau and Lifshitz1977): $\unicode[STIX]{x1D6F7}=\left(\!\begin{smallmatrix}\unicode[STIX]{x1D6F7}_{u}\\ \unicode[STIX]{x1D6F7}_{d}\end{smallmatrix}\!\right)$ . Each of the functions $\unicode[STIX]{x1D6F7}_{u}$ and $\unicode[STIX]{x1D6F7}_{d}$ can be presented as $\unicode[STIX]{x1D6F7}_{s}=a_{s}\text{e}^{\imath \unicode[STIX]{x1D719}_{s}}$ . The last three terms in the NLPE (2.1) contain the Pauli matrices $\widehat{\unicode[STIX]{x1D70E}}^{\unicode[STIX]{x1D6FC}}$ . The third term on the right-hand side describes the spinor pressure contribution $\widehat{\unicode[STIX]{x03C0}}=\left(\!\begin{smallmatrix}\unicode[STIX]{x03C0}_{u} & 0\\ 0 & \unicode[STIX]{x03C0}_{d}\end{smallmatrix}\!\right)$ , which is a diagonal second rank spinor. It can be represented in term of the Pauli matrixes $\widehat{\unicode[STIX]{x03C0}}=\unicode[STIX]{x03C0}_{u}(\hat{\text{I}}+\widehat{\unicode[STIX]{x1D70E}}_{z})/2+\unicode[STIX]{x03C0}_{d}(\hat{\text{I}}-\widehat{\unicode[STIX]{x1D70E}}_{z})/2$ , where $\hat{\text{I}}$ is the unit second rank spinor $\hat{\text{I}}=\left(\!\begin{smallmatrix}1 & 0\\ 0 & 1\end{smallmatrix}\!\right)$ . The explicit form of $\unicode[STIX]{x03C0}_{s}$ is determined by the equation of state. In this paper, we consider the degenerate electrons. Hence, $\unicode[STIX]{x03C0}_{s}$ is determined by the Fermi pressure $\unicode[STIX]{x03C0}_{s}={(6\unicode[STIX]{x03C0}^{2}n_{s})^{2/3}\hbar }^{2}/2m$ . Here, we consider one particle in a quantum state, instead of two particles with different spin directions in each state (see § 57 in Landau & Lifshitz Reference Landau and Lifshitz1980).

Equation (2.1) is coupled with the following equations of the field $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}=0$ , $\unicode[STIX]{x1D735}\times \boldsymbol{E}=0$ ,

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{E}=4\unicode[STIX]{x03C0}(en_{i}-e\unicode[STIX]{x1D6F7}^{\dagger }\unicode[STIX]{x1D6F7}),\end{eqnarray}$$

and

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D735}\times \boldsymbol{B}=\frac{2\unicode[STIX]{x03C0}q_{e}}{mc}(\unicode[STIX]{x1D6F7}^{\dagger }\boldsymbol{D}\unicode[STIX]{x1D6F7}+(\boldsymbol{D}\unicode[STIX]{x1D6F7})^{\dagger }\unicode[STIX]{x1D6F7})+4\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}_{e}\unicode[STIX]{x1D735}\times (\unicode[STIX]{x1D6F7}^{\dagger }\unicode[STIX]{x1D748}\unicode[STIX]{x1D6F7}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6F7}^{\dagger }$ is the Hermitian conjugate of the spinor wave function. We consider the field acting on the particle to be caused by the weakly relativistic motion of other particles. The assumed structure of the electromagnetic field is discussed in appendix A.

The spin–orbit interaction in the model described by (2.1)–(2.3) is presented in accordance with the semi-relativistic limit of the Dirac equation and the Breit Hamiltonian (Berestetskii et al. Reference Berestetskii, Lifshitz and Pitaevskii1982). The Breit Hamiltonian (Berestetskii et al. Reference Berestetskii, Lifshitz and Pitaevskii1982) or the Darwin Lagrangian in the classic regime (Landau & Lifshitz Reference Landau and Lifshitz1975) describes the semi-relativistic interaction of particles. Therefore, model (2.1)–(2.3) is constructed for the semi-relativistic description of the medium and the corresponding interaction of the fluid elements.

Considering quantum plasmas with spin–orbit interaction is to work in the weakly relativistic regime. This regime requires relatively large values of the parameters. If a degenerate electron gas is under consideration, then concentration is one of major parameters. The equilibrium concentration of electrons determines the Fermi velocity $v_{\text{Fe}}=\sqrt[3]{3\unicode[STIX]{x03C0}^{2}n_{0e}}\hbar /m_{e}$ . If the Fermi velocity is close to the speed of light  $c$ , the system of electrons is in the relativistic regime. If $v_{\text{Fe}}\approx 0.1c$ we get the weakly relativistic regime. If the Fermi velocity is even smaller $v_{\text{Fe}}\leqslant 0.01c$ then the weakly relativistic corrections might be negligibly small. However, this depends on the required accuracy. The relation between the Coulomb interaction and the spin–orbit interaction is governed by the concentration also. It requires concentrations to be of the same order as follows from the estimation of the Fermi velocity. Estimation $v_{\text{Fe}}\approx 0.1c$ gives the following value for the concentration $n_{e}\sim 10^{27}~\text{cm}^{-3}$ .

Condition $\unicode[STIX]{x1D735}\times \boldsymbol{E}=0$ presented above is not an obvious condition for the weakly relativistic approach if the Hamiltonian is derived from the single-particle Dirac equation. However, the properties of weakly relativistic interparticle interaction follow in more detail from the derivation of the Breit Hamiltonian (in the quantum case) or the Darwin Lagrangian (in the classic case). They explicitly show that the electric field is a potential field.

We are going to derive the SSE-QHD from the NLPE (2.1). To this end, it is useful to present the explicit form of the NLPE via the wave functions of spin-up and spin-down electrons:

(2.4) $$\begin{eqnarray}\displaystyle \imath \hbar \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6F7}_{u} & = & \displaystyle \left(\frac{\left({\displaystyle \frac{\hbar }{\imath }}\unicode[STIX]{x1D735}-{\displaystyle \frac{q_{e}}{c}}\boldsymbol{A}\right)^{2}}{2m}+q_{e}\unicode[STIX]{x1D711}-\unicode[STIX]{x1D6FE}_{e}B_{z}+\unicode[STIX]{x03C0}_{u}-\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}(E_{x}D_{y}-E_{y}D_{x})\right)\unicode[STIX]{x1D6F7}_{u}\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D6FE}_{e}(B_{x}-\imath B_{y})\unicode[STIX]{x1D6F7}_{d}-\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}((E_{y}D_{z}-E_{z}D_{y})-\imath (E_{z}D_{x}-E_{x}D_{z}))\unicode[STIX]{x1D6F7}_{d},\end{eqnarray}$$

and

(2.5) $$\begin{eqnarray}\displaystyle \imath \hbar \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6F7}_{d} & = & \displaystyle \left(\frac{\left({\displaystyle \frac{\hbar }{\imath }}\unicode[STIX]{x1D735}-{\displaystyle \frac{q_{e}}{c}}\boldsymbol{A}\right)^{2}}{2m}+q_{e}\unicode[STIX]{x1D711}+\unicode[STIX]{x1D6FE}_{e}B_{z}+\unicode[STIX]{x03C0}_{d}+\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}(E_{x}D_{y}-E_{y}D_{x})\right)\unicode[STIX]{x1D6F7}_{d}\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D6FE}_{e}(B_{x}+\imath B_{y})\unicode[STIX]{x1D6F7}_{u}-\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}((E_{y}D_{z}-E_{z}D_{y})+\imath (E_{z}D_{x}-E_{x}D_{z}))\unicode[STIX]{x1D6F7}_{u}.\end{eqnarray}$$

These equations describe the evolution of the spin-up electrons and the spin-down electrons independently. We can represent this evolution in terms of observables. This representation leads to the hydrodynamic equations.

The semi-relativistic or weakly relativistic approximation assumes that the Coulomb interaction dominates over the spin–orbit (SO) interaction. If we reach conditions where the SO interaction is comparable with the Coulomb interaction, we have reached a relativistic regime. This means that extra effects will occur. These extra effects do not have any trace in the weakly relativistic regime, but they are comparable with the SO interaction and the Coulomb interaction in the fully relativistic regime.

3 Hydrodynamic equations: general form

3.1 Continuity equations

To take the first step in the derivation of the SSE-QHDs we define the concentrations of spin-up and spin-down electrons $n_{u}=\unicode[STIX]{x1D6F7}_{u}^{\ast }\unicode[STIX]{x1D6F7}_{u}$ and $n_{d}=\unicode[STIX]{x1D6F7}_{d}^{\ast }\unicode[STIX]{x1D6F7}_{d}$ and differentiate them with respect to time. Applying (2.4) and (2.5) for the time derivatives of the partial wave functions $\unicode[STIX]{x1D6F7}_{u}$ and $\unicode[STIX]{x1D6F7}_{d}$ , we derive the continuity equations

(3.1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}n_{s}+\unicode[STIX]{x1D735}\boldsymbol{j}_{s}=\pm \frac{\unicode[STIX]{x1D6FE}_{e}}{\hbar }\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}S_{\unicode[STIX]{x1D6FD}}B_{\unicode[STIX]{x1D6FE}}\mp \frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar c}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}E^{\unicode[STIX]{x1D6FC}}[\unicode[STIX]{x1D6F7}_{s^{\prime }\neq s}^{\ast }r_{s}^{\unicode[STIX]{x1D708}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}_{s}+\text{c.c.}]/2m,\end{eqnarray}$$

where $\boldsymbol{r}_{u}=\{1,\imath ,1\}$ , $\boldsymbol{r}_{d}=\{1,-\imath ,-1\}$ , $q_{e}=-e$ for electrons and c.c. means complex conjugate.

The spin densities presented in the continuity equations have the following definitions: $S_{x}=\unicode[STIX]{x1D6F7}^{\ast }\widehat{\unicode[STIX]{x1D70E}}_{x}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D6F7}_{d}^{\ast }\unicode[STIX]{x1D6F7}_{u}+\unicode[STIX]{x1D6F7}_{u}^{\ast }\unicode[STIX]{x1D6F7}_{d}=2a_{u}a_{d}\cos \unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ , $S_{y}=\unicode[STIX]{x1D6F7}^{\ast }\widehat{\unicode[STIX]{x1D70E}}_{y}\unicode[STIX]{x1D6F7}=\imath (\unicode[STIX]{x1D6F7}_{d}^{\ast }\unicode[STIX]{x1D6F7}_{u}-\unicode[STIX]{x1D6F7}_{u}^{\ast }\unicode[STIX]{x1D6F7}_{d})=-2a_{u}a_{d}\sin \unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ , where $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{u}-\unicode[STIX]{x1D719}_{d}$ .

The sum of $[\unicode[STIX]{x1D6F7}_{d}^{\ast }r_{u}^{\unicode[STIX]{x1D6FC}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}_{u}+\text{c.c.}]/2m$ and $[\unicode[STIX]{x1D6F7}_{u}^{\ast }r_{d}^{\unicode[STIX]{x1D6FC}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}_{d}+\text{c.c.}]/2m$ gives the non-relativistic part of the spin current tensor $J_{\text{n.r.}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}=(\unicode[STIX]{x1D6F7}^{+}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D6FC}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}+\text{h.c.})/2m$ , where n.r. means non-relativistic and h.c. means Hermitian conjugation.

During the derivation of continuity equation (3.1), we find the explicit forms of the particle current: $j_{s}^{\unicode[STIX]{x1D6FC}}=(1/2m)(\unicode[STIX]{x1D6F7}_{s}^{\ast }D^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6F7}_{s}+\text{c.c.})+(-1)^{i_{s}}\unicode[STIX]{x1D6FE}_{e}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}n_{s}E_{\unicode[STIX]{x1D6FD}}/mc$ , where $s=u$ or $d$ and $i_{s}$ is a number that is different for spin-up and spin-down electrons: $i_{u}=2$ , $i_{d}=1$ . Below, we use the non-relativistic part of the particle current $j_{0s}^{\unicode[STIX]{x1D6FC}}=(1/2m)(\unicode[STIX]{x1D6F7}_{s}^{\ast }D^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6F7}_{s}+\text{c.c.})$ along with $j_{s}^{\unicode[STIX]{x1D6FC}}$ . We introduce the velocity fields $\boldsymbol{v}_{s}$ via the particle currents $\boldsymbol{j}_{s}\equiv n_{s}\boldsymbol{v}_{s}$ , with the following explicit form of the velocities $\boldsymbol{v}_{s}=(\hbar /m)\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{s}-(q_{e}/mc)\boldsymbol{A}+(-1)^{i_{s}}\unicode[STIX]{x1D6FE}_{e}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}E_{\unicode[STIX]{x1D6FD}}/mc$ , where $\unicode[STIX]{x1D719}_{s}$ is the phase of the partial wave function $\unicode[STIX]{x1D6F7}_{s}=a_{s}\text{e}^{\imath \unicode[STIX]{x1D719}_{s}}$ .

Below, we show that the right-hand sides of the continuity equations contain traditional hydrodynamic variables after the introduction of the velocity fields.

Summing up the partial concentrations $n_{s}$ , we obtain the full concentration of electrons $n_{e}=n_{u}+n_{d}$ , which should satisfy the continuity equation with zero right-hand side. However, directly summing the continuity equations (3.1), we find a non-zero right-hand side of the continuity equation for  $n_{e}$ :

(3.2) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}n_{e}+\unicode[STIX]{x1D735}(\boldsymbol{j}_{u}+\boldsymbol{j}_{d})=\frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar c}E^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}[(\unicode[STIX]{x1D6F7}_{d}^{\ast }r_{u}^{\unicode[STIX]{x1D708}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}_{u}+\text{c.c.})/2m-(\unicode[STIX]{x1D6F7}_{u}^{\ast }r_{d}^{\unicode[STIX]{x1D708}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}_{d}+\text{c.c.})/2m],\end{eqnarray}$$

where the right-hand side is caused by the spin–orbit interaction. Considering the right-hand side of (3.2), we find that it can be presented as the divergence of a vector $-\unicode[STIX]{x0394}\boldsymbol{j}$ . During this calculation, we have

(3.3) $$\begin{eqnarray}[\unicode[STIX]{x1D6F7}_{d}^{\ast }r_{u}^{\unicode[STIX]{x1D708}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}_{u}+\text{c.c.}]-[\unicode[STIX]{x1D6F7}_{u}^{\ast }r_{d}^{\unicode[STIX]{x1D708}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}_{d}+\text{c.c.}]=\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D708}z\unicode[STIX]{x1D706}}\hbar \unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}S_{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

Using $\unicode[STIX]{x1D735}\times \boldsymbol{E}=0$ , the right-hand side of (3.3) can be written as the divergence of a vector $\unicode[STIX]{x0394}\boldsymbol{j}$ . This vector $\unicode[STIX]{x0394}j^{\unicode[STIX]{x1D6FD}}$ is a part of the particle current $\unicode[STIX]{x0394}j^{\unicode[STIX]{x1D6FD}}=-(2\unicode[STIX]{x1D6FE}_{e}/\hbar c)E^{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D708}z\unicode[STIX]{x1D706}}\hbar S_{\unicode[STIX]{x1D706}}$ which can be rewritten as

(3.4) $$\begin{eqnarray}\unicode[STIX]{x0394}j^{\unicode[STIX]{x1D6FD}}=\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}E^{\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D707}}S_{\unicode[STIX]{x1D707}}-\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}z}S_{z}).\end{eqnarray}$$

Consequently, the full concentration $n_{e}$ satisfies the continuity equation

(3.5) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}n_{e}+\unicode[STIX]{x1D735}\boldsymbol{j}=0,\end{eqnarray}$$

where the right-hand side is equal to zero, $\boldsymbol{j}=\boldsymbol{j}_{u}+\boldsymbol{j}_{d}+\unicode[STIX]{x0394}\boldsymbol{j}$ , with $\boldsymbol{j}$ is the flux of all electrons.

The current of all electrons contains the non-relativistic part and the spin–orbit part. The SSE exists such that a part of the flux of all electrons $\boldsymbol{j}$ becomes the flux for each subspecies of electrons $\boldsymbol{j}_{s}$ and a part of the full flux $\unicode[STIX]{x0394}\boldsymbol{j}$ separates to give the $z$ projection of spin–orbit torque. So, we have a linear decomposition of the current. This separation is not made on purpose, but it automatically appears in the derivation.

The difference of the partial concentrations $n_{s}$ is the $z$ -projection of the spin density $S_{z}=n_{u}-n_{d}$ . Applying the continuity equations (3.1), we find an equation for  $S_{z}$

(3.6) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}S_{z}+\unicode[STIX]{x1D735}(\boldsymbol{j}_{u}-\boldsymbol{j}_{d})=\frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar }\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}S_{\unicode[STIX]{x1D6FD}}B_{\unicode[STIX]{x1D6FE}}-\frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar c}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}E^{\unicode[STIX]{x1D6FC}}J^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FD}},\end{eqnarray}$$

where

(3.7) $$\begin{eqnarray}J^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}=\frac{1}{2m}(\unicode[STIX]{x1D6F7}^{+}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D6FC}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}+\text{h.c.})+\frac{\unicode[STIX]{x1D6FE}_{e}}{2mc}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}n_{e}E_{\unicode[STIX]{x1D6FE}}\end{eqnarray}$$

is the full spin current. The last term in (3.6) presents the $z$ -projection of the spin torque caused by the spin–orbit interaction. The full expression on the right-hand side of (3.6) is the $z$ -projection of the full spin torque.

3.2 Euler equations

Application of the NLPE (2.1) to the time evolution of the momentum density of the spin-up electrons $j_{u}^{\unicode[STIX]{x1D6FC}}$ gives the following Euler equations for the spin-s electrons:

(3.8) $$\begin{eqnarray}\displaystyle m\unicode[STIX]{x2202}_{t}j_{s}^{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F1}_{s}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}} & = & \displaystyle q_{e}n_{s}E^{\unicode[STIX]{x1D6FC}}+\frac{q_{e}}{c}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}j_{0s}^{\unicode[STIX]{x1D6FD}}B^{\unicode[STIX]{x1D6FE}}+F_{\text{SO}s}^{\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & & \displaystyle \pm \,\unicode[STIX]{x1D6FE}_{e}n_{s}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}B_{z}+\frac{\unicode[STIX]{x1D6FE}_{e}}{2}(S_{x}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}B_{x}+S_{y}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}B_{y})\pm \frac{m\unicode[STIX]{x1D6FE}_{e}}{\hbar }\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}J^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}}B^{\unicode[STIX]{x1D6FE}},\end{eqnarray}$$

where the force field of the spin–orbit interaction $F_{\text{SO}u}^{\unicode[STIX]{x1D6FC}}$ has the following form:

(3.9) $$\begin{eqnarray}\displaystyle F_{\text{SO}s}^{\unicode[STIX]{x1D6FC}} & = & \displaystyle \pm \frac{\unicode[STIX]{x1D6FE}_{e}}{mc}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x2202}_{t}(n_{s}E^{\unicode[STIX]{x1D6FD}})\mp \frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar c}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}E^{\unicode[STIX]{x1D6FD}}j^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{q_{e}}{2mc}\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}E^{\unicode[STIX]{x1D6FD}}B^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}}(S_{x}\unicode[STIX]{x1D700}^{x\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}+S_{y}\unicode[STIX]{x1D700}^{y\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\pm 2n_{s}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}})+\frac{\unicode[STIX]{x1D6FE}_{e}}{2m^{2}c}(\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}E^{\unicode[STIX]{x1D6FD}})\nonumber\\ \displaystyle & & \displaystyle \times \,[\pm \unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6F7}_{s}^{\ast }D^{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6F7}_{s}+\unicode[STIX]{x1D700}^{x\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6F7}_{s}^{\ast }D^{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6F7}_{s^{\prime }\neq s}\mp \unicode[STIX]{x1D700}^{y\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6F7}_{s}^{\ast }\imath D^{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6F7}_{s^{\prime }\neq s}+\text{c.c.}],\end{eqnarray}$$

where

(3.10) $$\begin{eqnarray}j^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}=\frac{1}{4m^{2}}(\unicode[STIX]{x1D6F7}^{+}D^{\unicode[STIX]{x1D6FE}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6F7}+\text{h.c.})\end{eqnarray}$$

is a part of the spin current flux. The full expression for the non-relativistic spin current flux is

(3.11) $$\begin{eqnarray}J^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}=\frac{1}{4m^{2}}(\unicode[STIX]{x1D6F7}^{+}D^{\unicode[STIX]{x1D6FE}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6F7}+(D^{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6F7})^{+}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6F7}+\text{h.c.}).\end{eqnarray}$$

In (3.9), we have dropped the term proportional to $\unicode[STIX]{x1D735}\times \boldsymbol{E}$ , since it is equal to zero in the semi-relativistic approach.

The momentum current for the spin-s electrons appears during our derivation of (3.8) in the following form:

(3.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{s}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}} & = & \displaystyle \frac{1}{4m}(\unicode[STIX]{x1D6F7}_{s}^{\ast }D^{\unicode[STIX]{x1D6FC}}D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}_{s}+(D^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6F7}_{s})^{\ast }D^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6F7}_{s}+\text{c.c.})\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}}E^{\unicode[STIX]{x1D6FE}}[\unicode[STIX]{x1D6F7}_{s^{\prime }\neq s}^{\ast }r_{s}^{\unicode[STIX]{x1D707}}D^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6F7}_{s}+\text{c.c.}]/2m+n_{s}\unicode[STIX]{x1D735}\unicode[STIX]{x03C0}_{s}.\end{eqnarray}$$

The last term in the Euler equation (3.8) describes the spin flipping contribution in the momentum evolution.

3.3 Spin evolution equation

In the SSE-QHD we need to have equations for the evolution of the $x$ and $y$ projections of the spin density. If we include the contribution of the spin–orbit interaction they are obtained as follows:

(3.13) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}S_{j}+\Im _{j}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}J^{j\unicode[STIX]{x1D6FD}}=\frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar }\unicode[STIX]{x1D700}^{j\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}S^{\unicode[STIX]{x1D6FD}}B^{\unicode[STIX]{x1D6FE}}+T_{\text{SO}j},\end{eqnarray}$$

where $j=x,y$ , and

(3.14) $$\begin{eqnarray}T_{\text{SO}}^{\unicode[STIX]{x1D6FC}}=-\frac{\unicode[STIX]{x1D6FE}_{e}}{\hbar c}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}}E_{\unicode[STIX]{x1D6FE}}J^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FD}}\end{eqnarray}$$

is the spin torque caused by the SO interaction corresponding to the earlier works on the single-fluid model of electrons (Andreev & Kuz’menkov Reference Andreev and Kuz’menkov2009, Reference Andreev and Kuz’menkov2012).

The spin current is an important characteristic of a medium in spintronics (Sun & Xie Reference Sun and Xie2005; An et al. Reference An, Liu, Lin and Liu2012; Sinova et al. Reference Sinova, Valenzuela, Wunderlich, Back and Jungwirth2015). In our model, the many-particle spin current naturally appears in the spin evolution equation. It also appears in the Euler equation in terms describing non-conservation of the particle number when we account for the spin–spin interaction. If we account for the spin–orbit interaction, the spin current appears in the force field of the spin–orbit interaction and in the spin torque caused by the spin–orbit interaction. Parts of the spin current also exist in the continuity equation.

Modification of the distribution of electrons in momentum space and the influence of this on the equation of state under the influence of a strong magnetic field are described in the literature (see for instance Strickland, Dexheimer & Menezes Reference Strickland, Dexheimer and Menezes2012). In contrast with these works, we focus on the different occupation of quantum states by the spin-up and spin-down electrons.

3.4 Discussion of the Euler equations

Let us compare the obtained results with the Euler equation found for the single-fluid model of electrons considered in Kuz’menkov et al. (Reference Kuz’menkov, Maksimov and Fedoseev2001a ), Brodin & Marklund (Reference Brodin and Marklund2007), Marklund & Brodin (Reference Marklund and Brodin2007), Shukla & Eliasson (Reference Shukla and Eliasson2011), Koide (Reference Koide2013) and especially Andreev & Kuzmenkov (Reference Andreev and Kuzmenkov2011), Andreev & Kuz’menkov (Reference Andreev and Kuz’menkov2012) concerning the spin–orbit interaction. To this end, we need to consider the sum of two Euler equations to find the Euler equation in terms of the single-fluid model.

Combining Euler equations (3.8), we find the following equation

(3.15) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}(j_{u}^{\unicode[STIX]{x1D6FC}}+j_{d}^{\unicode[STIX]{x1D6FC}})+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F1}_{u}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}+\unicode[STIX]{x1D6F1}_{d}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}})=q_{e}n_{e}E^{\unicode[STIX]{x1D6FC}}+\frac{q_{e}}{c}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}(j_{0u}^{\unicode[STIX]{x1D6FD}}+j_{0d}^{\unicode[STIX]{x1D6FD}})B^{\unicode[STIX]{x1D6FE}}+\unicode[STIX]{x1D6FE}_{e}S^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}B^{\unicode[STIX]{x1D6FD}}+F_{\text{SO}}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

where

(3.16) $$\begin{eqnarray}F_{\text{SO}}^{\unicode[STIX]{x1D6FC}}=\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x2202}_{t}(S_{z}E^{\unicode[STIX]{x1D6FD}})+\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}E^{\unicode[STIX]{x1D6FD}}\cdot \unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}}J^{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}}-\frac{q_{e}}{mc}\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}}E^{\unicode[STIX]{x1D6FD}}B^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D707}}S^{\unicode[STIX]{x1D707}}.\end{eqnarray}$$

Equation (3.16) is not the complete spin–orbit interaction force for the single-fluid model. Above we showed that the full particle current consists of three terms (3.5). If we consider the time evolution of the full particle current $\boldsymbol{j}=\boldsymbol{j}_{u}+\boldsymbol{j}_{d}+\unicode[STIX]{x0394}\boldsymbol{j}$ we need to include the time evolution of vector $\unicode[STIX]{x0394}\boldsymbol{j}$ . Its explicit form is given by (3.4). Differentiating (3.4) with respect to time and combining the result with the Euler equation (3.16) we obtain the time derivative of the full particle current $\unicode[STIX]{x2202}_{t}j^{\unicode[STIX]{x1D6FC}}$ on the left-hand side. On the right-hand side we have the change in the force field of the spin–orbit interaction. The second term in $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x0394}j^{\unicode[STIX]{x1D6FC}}$ cancels the first term in the force field (3.16). Thus, we have that the first term in $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x0394}j^{\unicode[STIX]{x1D6FC}}$ replaces the first term in (3.16) and we obtain the final form for the spin–orbit interaction as:

(3.17) $$\begin{eqnarray}\bar{F}_{\text{SO}}^{\unicode[STIX]{x1D6FC}}=\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\unicode[STIX]{x2202}_{t}(E^{\unicode[STIX]{x1D6FD}}S^{\unicode[STIX]{x1D6FE}})+\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}E^{\unicode[STIX]{x1D6FD}}\cdot \unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}}J^{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}}-\frac{q_{e}}{mc}\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}}E^{\unicode[STIX]{x1D6FD}}B^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D707}}S^{\unicode[STIX]{x1D707}}.\end{eqnarray}$$

Before we present a comparison of our result with the previously obtained equations in the single-fluid model (Andreev & Kuz’menkov Reference Andreev and Kuz’menkov2012; Ivanov et al. Reference Ivanov, Andreev and Kuz’menkov2015), we need to mention that the first term in (3.17) can be represented in the following way. If we take the time derivative we find two terms $(\unicode[STIX]{x1D6FE}_{e}/mc)\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x2202}_{t}E^{\unicode[STIX]{x1D6FD}})S^{\unicode[STIX]{x1D6FE}}+(\unicode[STIX]{x1D6FE}_{e}/mc)\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}E^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x2202}_{t}S^{\unicode[STIX]{x1D6FE}}=(\unicode[STIX]{x1D6FE}_{e}/mc)\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x2202}_{t}E^{\unicode[STIX]{x1D6FD}})S^{\unicode[STIX]{x1D6FE}}+(\unicode[STIX]{x1D6FE}_{e}/mc)\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}E^{\unicode[STIX]{x1D6FD}}((2\unicode[STIX]{x1D6FE}_{e}/\hbar )\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}S^{\unicode[STIX]{x1D707}}B^{\unicode[STIX]{x1D708}}-\Im ^{\unicode[STIX]{x1D6FE}}-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}J^{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D707}})$ . Other terms can be found in Andreev & Kuzmenkov (Reference Andreev and Kuzmenkov2011), Andreev & Kuz’menkov (Reference Andreev and Kuz’menkov2012) except for the last term in (3.17). It can be shown that the last term in (3.17) can be found in the single-fluid model. However, it was not reported in the mentioned papers since it was lost during the derivation.

4 Spin–orbit interaction contribution in the spectrum of longitudinal waves

It has been recently demonstrated that longitudinal spin waves, called spin-electron acoustic waves, can exist in a degenerate electron gas (Andreev Reference Andreev2015b ). The oblique propagation of longitudinal waves reveals the existence of the second or upper SEAW (Andreev & Kuz’menkov Reference Andreev and Kuz’menkov2015b ). Partial spin polarization is required for the existence of the SEAWs. It is also necessary to apply the SSE-QHD. The SSE-QHD is generalized in this paper. The explicit analytical contribution of the spin–orbit interaction in the spectrum of the Langmuir waves propagating perpendicular to the external magnetic field (the upper hybrid wave) was found in Ivanov et al. (Reference Ivanov, Andreev and Kuz’menkov2015). It was shown that the spin–orbit interaction gives a contribution to the spectrum of the longitudinal waves. This contribution arises from the following term: $F_{1}^{\unicode[STIX]{x1D6FC}}=-(\unicode[STIX]{x1D6FE}_{e}/mc)S_{0z}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}z}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FF}E^{\unicode[STIX]{x1D6FD}}$ . In this paper, we generalize the force field of the spin–orbit interaction including the following term: $F_{2}^{\unicode[STIX]{x1D6FC}}=-(q_{e}/mc)(\unicode[STIX]{x1D6FE}_{e}/mc)S_{0z}B_{0}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}z}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}z}\unicode[STIX]{x1D6FF}E^{\unicode[STIX]{x1D6FD}}$ . These do not equal to each other. However, in the linear approximation they cancel the contribution of each other in the dispersion curves of the longitudinal waves. Therefore, we do not expect any contribution of the spin–orbit interaction in the spectra of oblique propagating longitudinal waves: Langmuir, Trivelpiece–Gould, lower and upper SEAWs. However, the spin–orbit interaction affects these waves if we do not apply the restriction of considering longitudinal waves. In the general case they are longitudinal–transverse waves and the spin–orbit interaction gives a contribution via the transverse part.

5 Hydrodynamic equations with the velocity field

After the introduction of the velocity field in the equations derived above for SSE-QHDs with the spin–orbit interaction we find a closed set of hydrodynamic equations. The parts of the spin currents found in the terms describing the spin–orbit interaction are found via the concentrations and velocity fields of the spin-up and spin-down electrons. Next, we obtain the following forms of the continuity equations:

(5.1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}n_{s}+\unicode[STIX]{x1D735}(n_{s}\boldsymbol{v}_{s})=(-1)^{i_{s}}\frac{\unicode[STIX]{x1D6FE}_{e}}{\hbar }\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}z}S^{\unicode[STIX]{x1D6FC}}B^{\unicode[STIX]{x1D6FD}}-\frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar c}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}E^{\unicode[STIX]{x1D6FC}}\left(\frac{1}{2}v_{s}^{\unicode[STIX]{x1D6FD}}S^{\unicode[STIX]{x1D708}}-(-1)^{i_{s}}\frac{\hbar }{4m}\frac{\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FD}}n_{s}}{n_{s}}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FF}}S^{\unicode[STIX]{x1D6FF}}\right).\end{eqnarray}$$

The last term in the continuity equation is caused by the spin–orbit interaction. The introduction of the velocity field modifies the right-hand side of the continuity equation, so it contains the hydrodynamic variables.

The introduction of the velocity field transforms the Euler equations to a form closer to the traditional form:

(5.2) $$\begin{eqnarray}\displaystyle & & \displaystyle mn_{s}(\unicode[STIX]{x2202}_{t}+\boldsymbol{v}_{s}\unicode[STIX]{x1D735})\boldsymbol{v}_{s}+\unicode[STIX]{x1D735}p_{s}-\frac{\hbar ^{2}}{4m}n_{s}\unicode[STIX]{x1D735}\left(\frac{\triangle n_{s}}{n_{s}}-\frac{(\unicode[STIX]{x1D735}n_{s})^{2}}{2n_{s}^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle \quad =q_{e}n_{s}\left(\boldsymbol{E}+\frac{1}{c}[\boldsymbol{v}_{s},\boldsymbol{B}]\right)+\boldsymbol{F}_{\text{SS}s}+\tilde{\boldsymbol{F}}_{\text{SO}s},\end{eqnarray}$$

with the thermal or Fermi pressure $p_{s}$ , the force field of spin–spin interaction

(5.3) $$\begin{eqnarray}\boldsymbol{F}_{\text{SS}s}=(-1)^{i_{s}}\unicode[STIX]{x1D6FE}_{e}n_{u}\unicode[STIX]{x1D735}B_{z}+\frac{\unicode[STIX]{x1D6FE}_{e}}{2}(S_{x}\unicode[STIX]{x1D735}B_{x}+S_{y}\unicode[STIX]{x1D735}B_{y})+(-1)^{i_{s}}\frac{m\unicode[STIX]{x1D6FE}_{e}}{\hbar }\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}z}(\boldsymbol{J}_{(M)\unicode[STIX]{x1D6FD}}B^{\unicode[STIX]{x1D6FE}}-\boldsymbol{v}_{s}S^{\unicode[STIX]{x1D6FD}}B^{\unicode[STIX]{x1D6FE}}),\end{eqnarray}$$

and the force field of spin–orbit interaction acting on spin-s electrons

(5.4) $$\begin{eqnarray}\displaystyle \tilde{F}_{\text{SO}s}^{\unicode[STIX]{x1D6FC}} & = & \displaystyle \pm \frac{\unicode[STIX]{x1D6FE}_{e}}{mc}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x2202}_{t}(n_{s}E^{\unicode[STIX]{x1D6FD}})\mp \frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar c}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}E^{\unicode[STIX]{x1D6FD}}j^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{q_{e}}{2mc}\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}E^{\unicode[STIX]{x1D6FD}}B^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}}(S_{x}\unicode[STIX]{x1D700}^{x\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}+S_{y}\unicode[STIX]{x1D700}^{y\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\pm 2n_{s}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{2}\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}(\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}E^{\unicode[STIX]{x1D6FD}})[\unicode[STIX]{x1D700}^{x\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}v_{s^{\prime }\neq s}^{\unicode[STIX]{x1D6FE}}S^{x}+\unicode[STIX]{x1D700}^{y\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}v_{s^{\prime }\neq s}^{\unicode[STIX]{x1D6FE}}S^{y}+\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}v_{s}^{\unicode[STIX]{x1D6FE}}S^{z}]\nonumber\\ \displaystyle & & \displaystyle \pm \,\frac{\hbar }{4m}\frac{\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FE}}n_{s^{\prime }\neq s}}{n_{s^{\prime }\neq s}}\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}(\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}E^{\unicode[STIX]{x1D6FD}})\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}}S^{\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

A presentation of tensor $j^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}}$ in terms of the velocity fields is discussed in appendix C.

In terms of the velocity field the spin current tensor is found as follows

(5.5) $$\begin{eqnarray}J_{j\unicode[STIX]{x1D6FC}}=\frac{1}{2}(v_{u}^{\unicode[STIX]{x1D6FC}}+v_{d}^{\unicode[STIX]{x1D6FC}})S_{j}-\frac{\hbar }{4m}\unicode[STIX]{x1D700}^{j\unicode[STIX]{x1D6FD}z}\left(\frac{\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}n_{u}}{n_{u}}-\frac{\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}n_{d}}{n_{d}}\right)S_{\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$

The relativistic part of the spin current tensor is hidden in the definition of velocity fields.

In the Euler equations (5.2) we have used a reduced form of the spin current $\boldsymbol{J}_{(M)x}$ and $\boldsymbol{J}_{(M)y}$ which means $J^{x\unicode[STIX]{x1D6FC}}$ and $J^{y\unicode[STIX]{x1D6FC}}$ correspondingly. Here, the bold symbols indicate a vector related to the second index in  $J^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ .

The existence of the quantum part of the spin current or, in other words, the quantum Bohm potential contribution in the spin evolution equation was demonstrated by Takabayasi in Takabayasi (Reference Takabayasi1955a ). The contribution of this effect to the wave properties of quantum plasmas is considered in Trukhanova (Reference Trukhanova2013a ), where it is applied for the spin-plasma waves.

After the introduction of the velocity field, the spin evolution equations have the following form:

(5.6) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}S_{j}+\frac{1}{2}\unicode[STIX]{x1D735}[S_{j}(\boldsymbol{v}_{u}+\boldsymbol{v}_{d})]-\frac{\hbar }{4m}\unicode[STIX]{x1D700}^{j\unicode[STIX]{x1D6FD}z}\unicode[STIX]{x1D735}\left(S^{\unicode[STIX]{x1D6FD}}\left(\frac{\unicode[STIX]{x1D735}n_{u}}{n_{u}}-\frac{\unicode[STIX]{x1D735}n_{d}}{n_{d}}\right)\right)+\Im _{j}=\frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar }\unicode[STIX]{x1D700}^{j\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}S^{\unicode[STIX]{x1D6FD}}B^{\unicode[STIX]{x1D6FE}}+T_{\text{SO}j},\end{eqnarray}$$

where $j=x,y$ .

The electric and magnetic fields entering SSE-QHD equations (5.1)–(5.6) satisfy the quasi-static Maxwell equations ( $\unicode[STIX]{x1D735}\times \boldsymbol{E}=0$ and (2.3)), where sources of fields are presented in terms of hydrodynamic variables: $\unicode[STIX]{x1D6F7}^{\dagger }\unicode[STIX]{x1D6F7}=n_{eu}+n_{ed}$ , $(1/2m)(\unicode[STIX]{x1D6F7}^{\dagger }\boldsymbol{D}\unicode[STIX]{x1D6F7}+(\boldsymbol{D}\unicode[STIX]{x1D6F7})^{\dagger }\unicode[STIX]{x1D6F7})=n_{eu}\boldsymbol{v}_{eu}+n_{ed}\boldsymbol{v}_{ed}$ , and $\unicode[STIX]{x1D6F7}^{\dagger }\unicode[STIX]{x1D748}\unicode[STIX]{x1D6F7}=\{S_{ex},S_{ey},(n_{eu}-n_{ed})\}$ . We consider motionless ions.

The SSE-QHD equations (5.1)–(5.6) are derived from model (2.1)–(2.3) for the motion of a medium with semi-relativistic interactions of particles. This model does not describe the propagation of the electromagnetic radiation in the medium and the action of the radiation on the medium since we have incomplete Maxwell equations. Next, we promote our model to include the radiation propagation. To this end, we restore $\unicode[STIX]{x2202}_{t}\boldsymbol{E}$ and $\unicode[STIX]{x2202}_{t}\boldsymbol{B}$ in the Maxwell equations. So, the SSE-QHD equations (5.1)–(5.6) are coupled with the following Maxwell equations $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}=0$ , $\unicode[STIX]{x1D735}\times \boldsymbol{E}=-(1/c)\unicode[STIX]{x2202}_{t}\boldsymbol{B}$ ,

(5.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{E}=4\unicode[STIX]{x03C0}(en_{i}-en_{eu}-en_{ed}), & \displaystyle\end{eqnarray}$$

(5.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\times \boldsymbol{B}=\frac{1}{c}\unicode[STIX]{x2202}_{t}\boldsymbol{E}+\frac{4\unicode[STIX]{x03C0}q_{e}}{c}(n_{eu}\boldsymbol{v}_{eu}+n_{ed}\boldsymbol{v}_{ed})+4\unicode[STIX]{x03C0}\unicode[STIX]{x1D735}\times \boldsymbol{M}_{e}. & \displaystyle\end{eqnarray}$$

Since we consider degenerate electrons we use the Fermi pressures for spin-up and spin-down electrons as equations of state: $p_{s}={(6\unicode[STIX]{x03C0}^{2})^{2/3}\hbar }^{2}n_{s}^{5/3}/5m$ .

Vector $\boldsymbol{\Im }$ in the spin evolution equations (5.6) is the divergence of the thermal part of the spin current tensor $\Im ^{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}J_{\text{th}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ . In accordance with the NLPE equation, as it was demonstrated in Andreev & Kuz’menkov (Reference Andreev and Kuz’menkov2015c ), we have

(5.9) $$\begin{eqnarray}\boldsymbol{\Im }=(\unicode[STIX]{x03C0}_{u}-\unicode[STIX]{x03C0}_{d})\unicode[STIX]{x1D6FE}_{e}[\boldsymbol{S},\boldsymbol{e}_{z}]/\hbar .\end{eqnarray}$$

Due to its nature we can also call this the Fermi spin current.

Our model is bound to an independent hydrodynamic description of spin-up and spin-down electrons which was originally introduced in Kuzmenkov & Harabadze (Reference Kuzmenkov and Harabadze2004). However, this model appears to be incomplete (Andreev Reference Andreev2015b ). Before, the final version of the model had been presented in 2015 (Andreev Reference Andreev2015b ) some works based on Kuzmenkov & Harabadze (Reference Kuzmenkov and Harabadze2004) had been presented. Let us discussed some of them for a better perspective in the understanding of our results. In § II A of Zamanian et al. (Reference Zamanian, Stefan, Marklund and Brodin2010) the authors discussed a two-fluid form of the quantum hydrodynamic equations. The spin–orbit interaction was not discussed. Equations were not demonstrated explicitly, but the authors suggested that ‘However, since the kinetic equation is linear in  $f$ , it is straightforward to divide the electron fluid into two parts depending on the initial spin state, e.g., up or down relative to the magnetic field’. However, our analysis shows that the division of hydrodynamic equations into two fluids is not straightforward even if the spin–orbit interaction is dropped. Otherwise, the Euler equation presented in Zamanian et al. (Reference Zamanian, Stefan, Marklund and Brodin2010) for the single-fluid description of the electrons is the traditional Euler equation for the spin- $1/2$ quantum plasmas (Kuz’menkov et al. Reference Kuz’menkov, Maksimov and Fedoseev2001a ).

Considering a kinetic model for spin- $1/2$ quantum particles, Stefan & Brodin (Reference Stefan and Brodin2013) started their description with the single-fluid model of electrons, where the spin was described via an extra argument of the quasi-distribution function. An argument describing spin is a vector with a fixed module, but it can rotate in three-dimensional space and its evolution is reduced to a two-dimensional evolution described by angles in spherical coordinates. It seems that the coherent states for the spin degree of freedom were used. Hence, the values at $\unicode[STIX]{x1D703}=0$ or $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ do not directly correspond to spin-up and spin-down states. Next, the authors presented an equilibrium distribution function $f_{0}(s,p)$ and considered the small amplitude transverse perturbations. Waves propagating parallel to the external field directed parallel to $Oz$ axes was considered there. An integral form of the dispersion relation for transverse waves with left and right circular polarization was obtained. An algebraic form of this equation was obtained after integration was performed in the long wavelength limit. The long wavelength limit presented by equation (14) of Stefan & Brodin (Reference Stefan and Brodin2013) does not contain the wave vector. So, there are probably limit values for the frequencies for two waves at $k=0$ . Judging by the fact that some terms contain $(\unicode[STIX]{x1D707}/mc^{2})^{2}$ , we conclude that the calculation and result are presented beyond the limit required by the weakly relativistic regime mentioned before equation (1) of Stefan & Brodin (Reference Stefan and Brodin2013).

Equation (14) in Stefan & Brodin (Reference Stefan and Brodin2013) is a huge formula with no numerical analysis. Thus comparison of the contribution of the spin–orbit interaction with our result is not possible.

6 Linearized SSE-QHD equations

Analysis of wave propagation can be made by a consideration of the linearized set of SSE-QHD equations:

(6.1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FF}n_{s}+n_{0s}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}\boldsymbol{v}_{s}=0,\end{eqnarray}$$

(6.2) $$\begin{eqnarray}\displaystyle mn_{0s}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FF}v_{s}^{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FF}p_{s} & = & \displaystyle q_{e}n_{0s}\unicode[STIX]{x1D6FF}E^{\unicode[STIX]{x1D6FC}}\pm \unicode[STIX]{x1D6FE}_{e}n_{0s}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FF}B_{z}\nonumber\\ \displaystyle & & \displaystyle +\,q_{e}n_{0s}\frac{1}{c}B_{0}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}z}\unicode[STIX]{x1D6FF}v_{s}^{\unicode[STIX]{x1D6FD}}\mp \frac{q_{e}}{mc}\frac{\unicode[STIX]{x1D6FE}_{e}}{mc}n_{0s}B_{0}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}z}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}z}\unicode[STIX]{x1D6FF}E^{\unicode[STIX]{x1D6FD}}\nonumber\\ \displaystyle & & \displaystyle \mp \,\frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar c}\unicode[STIX]{x1D700}^{z\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D707}}j_{0}^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FF}E^{\unicode[STIX]{x1D6FD}}\pm \frac{\unicode[STIX]{x1D6FE}_{e}}{mc}n_{0s}\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}z}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FF}E^{\unicode[STIX]{x1D6FD}},\end{eqnarray}$$

with the upper sign for spin-up electrons and the lower sign for spin-down electrons, and

(6.3) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FF}S_{j}+\unicode[STIX]{x1D6FF}\Im _{j}=\frac{2\unicode[STIX]{x1D6FE}_{e}}{\hbar }(\unicode[STIX]{x1D700}^{j\unicode[STIX]{x1D6FD}z}B_{0z}\unicode[STIX]{x1D6FF}S_{\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D700}^{j\unicode[STIX]{x1D6FD}z}S_{0z}\unicode[STIX]{x1D6FF}B_{\unicode[STIX]{x1D6FD}})\end{eqnarray}$$

for the spin evolution, and $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{B}=0$ , $\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6FF}\boldsymbol{E}=-\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FF}\boldsymbol{B}/c$ ,

(6.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{E}=-4e\unicode[STIX]{x03C0}(\unicode[STIX]{x1D6FF}n_{eu}+\unicode[STIX]{x1D6FF}n_{ed}), & \displaystyle\end{eqnarray}$$

(6.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6FF}\boldsymbol{B}=\frac{1}{c}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FF}\boldsymbol{E}-\frac{4\unicode[STIX]{x03C0}e}{c}(n_{0u}\unicode[STIX]{x1D6FF}\boldsymbol{v}_{u}+n_{0d}\unicode[STIX]{x1D6FF}\boldsymbol{v}_{d})+4\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}_{e}\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6FF}\boldsymbol{S}_{e}. & \displaystyle\end{eqnarray}$$

We neglect the quantum Bohm potential since it gives a noticeable contribution at large wave vectors close to $k_{\text{max}}=n_{0}^{1/3}$ , $S_{0z}=n_{0u}-n_{0d}$ and $\unicode[STIX]{x1D6FF}\boldsymbol{\Im }=(\unicode[STIX]{x03C0}_{0u}-\unicode[STIX]{x03C0}_{0d})\unicode[STIX]{x1D6FE}_{e}[\unicode[STIX]{x1D6FF}\boldsymbol{S},\boldsymbol{e}_{z}]/\hbar$ , with $\unicode[STIX]{x03C0}_{0s}={(6\unicode[STIX]{x03C0}^{2}n_{0s})^{2/3}\hbar }^{2}/2m$ . The quantum Bohm potential becomes noticeable at $kv_{\text{Fe}}/\unicode[STIX]{x1D714}_{\text{Le}}\geqslant 1$ . It corresponds to $\unicode[STIX]{x1D705}\geqslant 10^{3}$ , but all analyses are made for $\unicode[STIX]{x1D705}\leqslant 1$ , where $\unicode[STIX]{x1D705}=kc/\unicode[STIX]{x1D714}_{\text{Le}}$ .

The spin–orbit interaction describes the force acting on the magnetic moment moving in an external electric field. It exists even for static electric fields. Therefore, the spin–orbit interaction affects the evolution of electrostatic waves in spin- $1/2$ plasmas, since electrons possess the magnetic moments affected by the electric field caused by the charges of the surrounding electrons. We consider high frequency perturbations and assume that the ions are motionless. Hence, on this time scale, ions generate an equilibrium electric field compensated by the equilibrium part of the electric field of the electrons. Relatively small temperatures are considered. Thus, there is a distribution of spins due to the temperature effects, but the deviations from the average value are small.

Traditional derivation of the Vlasov equation performed by N. N. Bogoliubov is made for particles with Coulomb interaction. Hence, the corresponding hydrodynamic equations contain the Coulomb interaction only. This means that the right-hand side of the Euler equation contains $qn\boldsymbol{E}$ plus the force field caused by the external field, where $\boldsymbol{E}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ , which satisfies the Poisson equation. Similar analysis in terms of the many-particle quantum hydrodynamics (MPQHD) is given in Kuz’menkov & Maksimov (Reference Kuz’menkov and Maksimov1999). In spite the fact that the derivations are made for the Coulomb interaction only, the Vlasov equation, the classic hydrodynamic equations and the quantum hydrodynamics are applied for regimes where the full set of Maxwell equations is used. We also need to mention that an averaging of the single-particle equations, which can be obtained in the literature, is not a satisfactory justification. Such models do not consider the interparticle interaction at all. These models avoid existing problem. The same assumption were used in Asenjo et al. (Reference Asenjo, Zamanian, Marklund, Brodin and Johansson2012) and Stefan & Brodin (Reference Stefan and Brodin2013), but without any discussion.

Our goal is to study the structure of the spin–orbit interaction in the hydrodynamic equations with separate spin evolution. This can be done in the weakly relativistic regime. However, we want to study transverse waves. Hence, the Poisson equation is replaced by the full set of Maxwell equations. This can be partially justified by the analysis of external electromagnetic field propagation through plasmas in which the motion of the charges modifies properties of the electromagnetic field.

7 Dispersion dependence of waves propagating perpendicular to the external magnetic field

Let us present the dispersion equation for the waves propagating perpendicular to the external magnetic field

(7.1) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D714}^{2}-k^{2}c^{2}-\mathop{\sum }_{s=u,d}\frac{\unicode[STIX]{x1D714}_{Ls}^{2}}{\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D6FA}_{e}^{2}-k^{2}U_{\text{Fs}}^{2}}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,(2\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{\text{Le}}^{2}-k^{2}c^{2}-k^{2}U_{\text{Fs}}^{2}+\unicode[STIX]{x1D6FC}_{e}\unicode[STIX]{x1D6FA}_{e}(2\unicode[STIX]{x1D714}_{\text{Le}}^{2}+k^{2}U_{\text{Fs}}^{2}))=0\end{eqnarray}$$

found in this paper as a solution of (6.1)–(6.5) in the linear approximation for small perturbations of the equilibrium state of magnetized plasmas, where $\unicode[STIX]{x1D6FC}_{e}=\unicode[STIX]{x1D6FE}_{e}/q_{e}c$ . In (7.1) $U_{\text{Fs}}=(6\unicode[STIX]{x03C0}^{2}n_{0s})^{1/3}\hbar /m$ is the Fermi velocity for spin-s electrons.

The SSE affects the ordinary electromagnetic wave and spin-plasma wave having an electric field oscillating in the direction of the external magnetic field $\boldsymbol{B}_{\text{ext}}=B_{0}\boldsymbol{e}_{z}$ via the Fermi spin current. This effect is described in Andreev & Kuz’menkov (Reference Andreev and Kuz’menkov2015c ). Therefore, we do not discuss the ordinary electromagnetic wave and spin-plasma wave in this paper.

7.1 Numerical analysis

For the numerical analysis and presentation of our results we use the following dimensionless parameters $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{\text{Le}}$ , $\unicode[STIX]{x1D705}=kc/\unicode[STIX]{x1D714}_{\text{Le}}$ and the following expression for the equilibrium spin polarization $\unicode[STIX]{x1D702}=\tanh (\unicode[STIX]{x1D707}_{B}B_{0}/\unicode[STIX]{x1D700}_{\text{Fe}})$ , where $\unicode[STIX]{x1D700}_{\text{Fe}}=T_{\text{Fe}}/k_{B}={(3\unicode[STIX]{x03C0}^{2}n_{0e})^{2/3}\hbar }^{2}/2m$ is the Fermi energy, and $k_{B}$ is the Boltzmann constant.

The spin–orbit interaction is a relativistic effect modelled in the semi-relativistic approach. To include its contribution we need to consider relatively large equilibrium concentrations $n_{0e}\sim 10^{27}~\text{cm}^{-3}$ . As was demonstrated and discussed earlier, the exchange interaction plays a considerable role if the concentration is below ${\approx}10^{25}~\text{cm}^{-3}$ (Andreev & Ivanov Reference Andreev and Ivanov2015; Andreev Reference Andreev2016a ).

Usually, the dispersion equation for longitudinal–transverse waves propagating perpendicular to an external field and having an electric field in the plane perpendicular to the external magnetic field gives two solutions, called extraordinary waves. However, the dispersion equation (7.1) has three solutions due to the SSE. The dispersion equation (7.1) also contains the spin–orbit interaction contribution.

The third solution of (7.1) appears due to the SSE. The SSE lead to the appearance of SEAWs if we consider the regime of longitudinal waves (Andreev Reference Andreev2015b ; Andreev & Kuz’menkov Reference Andreev and Kuz’menkov2015b ). Therefore, the third solution is called the extraordinary SEAW.

Next, we need to study the dispersion curves of three extraordinary waves and the influence of the spin–orbit interaction on their dispersion dependencies. Therefore, we consider a degenerate quantum plasma with $n_{0e}=1.4\times 10^{27}~\text{cm}^{-3}$ and $B_{0}=10^{11}$  G. The cyclotron frequency is smaller than the Langmuir frequency in this regime.

The presented regime of parameters corresponds to temperatures below $10^{7}$  K. Required conditions can be found for electrons in neutron stars, where concentrations and magnetic fields are relatively high and temperatures are below the presented regime.

We consider this value of the particle concentration for the following reasons. The spin–orbit interaction is a semi-relativistic effect. Hence, the characteristic velocity $\tilde{v}$ should be comparable to the speed of light $\tilde{v}\approx 0.1c$ . The Fermi velocity is the characteristic velocity for a degenerate electron gas. This problem can be addressed more explicitly. The Hamilton function for the spin–orbit interaction is $\unicode[STIX]{x1D6FE}Ep/mc=e\hbar Ep/2m^{2}c^{2}$ . It should be noticeable in comparison with the Coulomb interaction $e\unicode[STIX]{x1D711}$ , but is smaller than the Coulomb interaction $e\hbar Ep/2m^{2}c^{2}\approx 0.1e\unicode[STIX]{x1D711}$ . Estimating the electric field as $\boldsymbol{E}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}\approx n^{1/3}\unicode[STIX]{x1D711}$ . So the presented estimation is correct if $\hbar pn^{1/3}/m^{2}c^{2}$ is comparable to 0.1. Estimating the momentum $p$ as the Fermi momentum $p_{\text{Fe}}$ , we have that $(\hbar n^{1/3}/mc)^{2}$ is comparable to 0.1. Both estimations lead to the considered concentrations.

To consider the influence of the spin–orbit interaction on the extraordinary waves we present figures 1 and 2. The increase of frequency of the upper extraordinary wave and the decrease of frequency of the lower extraordinary wave in the regime of relatively small wave vectors are demonstrated in figures 1 and 2, respectively.

Figure 1. The figure shows the upper extraordinary wave. The dashed line presents a wave under the influence of the SSE. The continuous line includes the effect of the spin–orbit interaction in addition to the SSE. This figure is plotted for $n_{0e}=1.4\times 10^{27}~\text{cm}^{-3}$ , $B_{0}=10^{11}$  G,  $\unicode[STIX]{x1D702}_{e}=0.11$ and temperature $T\ll T_{\text{Fe}}=5.9\times 10^{7}$  K. The following notations are used in figures $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{\text{Le}}$ and $\unicode[STIX]{x1D705}=kc/\unicode[STIX]{x1D714}_{\text{Le}}$ .

Figure 2. The figure shows the lower extraordinary wave. The dashed line presents SSE without spin–orbit interaction. The continuous line presents SSE with spin–orbit interaction. Parameters are as in figure 1.

The contribution of the spin–orbit interaction in the chosen parameter regime $B_{0}=10^{11}~\text{G}\ll B_{cr}=m_{e}^{2}c^{3}/(e\hbar )=4.41\times 10^{13}$  G corresponding to $\hbar \unicode[STIX]{x1D6FA}\leqslant mc^{2}$ and $\unicode[STIX]{x1D700}_{\text{Fe}}=0.01mc^{2}$ is relatively small. The contribution of the spin–orbit interaction increases with the increase of concentration. However, we work in the regime of parameters corresponding to the semi-relativistic approach in accordance with the area of applicability of our equations.