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

Abstract

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 population of states with different spin direction is included in the spin density (magnetization). In this paper we derive a QHD model, which separately describes spin-up electrons and spin-down electrons. Hence we consider electrons with different projection of spin on the preferable direction as two different species of particles. We show that numbers of particles with different spin direction do not conserve. Hence the continuity equations contain sources of particles. These sources are caused by the interactions of spins with magnetic field. Terms of similar nature arise in the Euler equation. We have that z-projection of the spin density is no longer an independent variable. It is proportional to difference between concentrations of electrons with spin-up and electrons with spin-down. In terms of new model we consider propagation of waves in magnetized plasmas of degenerate electrons and motionless ions. We show that new form of QHD equations gives all solutions obtained from traditional form of QHD equations with no distinguish of spin-up and spin-down states. But it also reveals a sound-like solution we call the spin-electron acoustic wave. Coincidence of most solutions is expected since we started derivation with the same basic equation.