Separated spin-up and spin-down quantum hydrodynamics of degenerated electrons: Spin-electron acoustic wave appearance

The quantum hydrodynamic (QHD) model of charged spin-1/2 particles contains physical quantities defined for all particles of a species including particles with spin-up and with spin-down. Different populations of states with different spin directions are included in the spin density (the magnetization). In this paper I derive a QHD model, which separately describes spin-up electrons and spin-down electrons. Hence electrons with different projections of spins on the preferable direction are considered as two different species of particles. It is shown that the numbers of particles with different spin directions do not conserve. Hence the continuity equations contain sources of particles. These sources are caused by the interactions of the spins with the magnetic field. Terms of similar nature arise in the Euler equation. The $z$ projection of the spin density is no longer an independent variable. It is proportional to the difference between the concentrations of the electrons with spin-up and the electrons with spin-down. The propagation of waves in the magnetized plasmas of degenerate electrons is considered. Two regimes for the ion dynamics, the motionless ions and the motion of the degenerate ions as the single species with no account of the spin dynamics, are considered. It is shown that this form of the QHD equations gives all solutions obtained from the traditional form of QHD equations with no distinction of spin-up and spin-down states. But it also reveals a soundlike solution called the spin-electron acoustic wave. Coincidence of most solutions is expected since this derivation was started with the same basic equation: the Pauli equation. Solutions arise due to the different Fermi pressures for the spin-up electrons and the spin-down electrons in the magnetic field. The results are applied to degenerate electron gas of paramagnetic and ferromagnetic metals in the external magnetic field. The dispersion of the spin-electron acoustic waves in the partially spin-polarized degenerate neutron matter are also considered.

Figure 1

Dependence of dimensionless frequency square in units of the square of Langmuir frequency $\xi ={\omega }^2/{\omega }_{\mathrm{Le}}^2$ on the dimensionless wave vector $k/\sqrt[3]{n_0}$ and the spin polarization $\eta =\Delta n/n_0$ for two longitudinal waves existing in the spin-up–spin-down degenerate electron quantum plasmas. The figure is obtained for the waves propagating parallel to the external magnetic field. Ions are assumed to be motionless. (a) shows the dispersion of the Langmuir wave, (b) presents the dispersion properties of the spin-electron acoustic wave. This figure is obtained for $n_0={10}^{21}{\mathrm{cm}}^{-3}$. Hence the Langmuir frequency ${\omega }_{\mathrm{Le}}$, in our case, is equal to $1.5\times {10}^{15}$ radians per second. Estimation for maximal wave vector is $k<{10}^7{\mathrm{cm}}^{-1}$.

Figure 2

The dispersion of the ion acoustic wave and the spin-electron acoustic wave at rather large spin polarization $\eta =0.98$. The branch with the largest phase velocity $\omega /k$ (the red branch) presents the spin-electron acoustic wave. Lower curve presents the ion acoustic wave. Here and in other figures below we draw the horizontal line presenting the ion Langmuir frequency. The frequency is presented in units of the electron Langmuir frequency $\varpi =\omega /{\omega }_{\mathrm{Le}}$, and the wave vector is presented in units of the inverse electron Debye radius $\kappa =k{r}_{\mathrm{De}}$, with $r_{\mathrm{De}}=v_{\mathrm{Fe}}/{\omega }_{\mathrm{Le}}$.

Figure 3

We present all three longitudinal waves propagating parallel to the external magnetic field at $\eta =0.8$. We have the following waves in order of the frequency decrease: the Langmuir wave, the spin-electron acoustic wave, and the ion-acoustic wave.

Figure 4

Comparison of this figure with Fig. 3 shows the decrease of the frequency of the spin-electron acoustic wave with the increase of spin polarization up to $\eta =0.9998$. Figures 3 and 4 show this decrease in compare with the frequency of the ion acoustic wave. Distribution of the dispersion branches in this figure corresponds to their distribution in Fig. 3.

Figure 5

Continuation of the comparison of the dispersion branch positions presented in Figs. 3 and 4. Increasing the spin polarization we decrease the frequency of the spin-electron acoustic waves. At $\eta =999998$ and $\eta =9999998$ the frequencies of the spin-electron acoustic wave and the ion-acoustic wave are comparable.

Figure 6

Figure 5 shows hybridization of the ion-acoustic and spin-electron acoustic waves and area of splitting of the hybrid waves. Detailed picture of the splitting of the hybrid waves is presented in this figure.

Figure 7

Comparison of dispersion dependence of the spin-electron acoustic wave and the ion-plasmas wave at large spin polarization of electrons $n_{0u}/n_0={10}^{-6},\eta =0.999998$, when the frequency of the spin-electron acoustic wave is rather small.

Figure 8

Comparison of dispersion dependence of the spin-electron acoustic wave and the ion-plasmas wave at larger spin polarization of electrons $n_{0u}/n_0={10}^{-9},\eta =0.999999998$ than in Fig. 7. The upper figure shows dispersion branches in a wide range of the wave vectors. The lower figure shows area of splitting of the hybrid branches. Comparing this figure with Fig. 6 we see that at larger spin polarization the splitting of the branches becomes smaller.