A helium-4 processing plasma contains singly ionised He ions of mass and electrons of mass

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Waves in plasmas · Waves in plasmas

Plasma Physics, Ch Plasma oscillations and light in an unmagnetised plasma

Problem

A helium-4 processing plasma contains singly ionised He+^+ ions of mass mHe=6.642×10−27 kgm_{\rm He}=6.642\times10^{-27}\,\mathrm{kg} and electrons of mass me=9.109×10−31 kgm_e=9.109\times10^{-31}\,\mathrm{kg}, with ni=nen_i=n_e (quasineutral, ion charge +e+e). Both species are cold and collisionless. ((a)) Starting from the linearised, collisionless, cold two-fluid momentum equations for electrons and ions separately (no pressure term, both species free to move), find the current perturbation of each species for a mode ∝ei(k⋅x−ωt)\propto e^{i(\mathbf k\cdot\mathbf x-\omega t)} and show that the total dielectric function is ϵ(ω)=1−ωpe2ω2−ωpi2ω2,ωps2=nsqs2ϵ0ms.\epsilon(\omega)=1-\frac{\omega_{pe}^2}{\omega^2}-\frac{\omega_{pi}^2}{\omega^2},\qquad \omega_{ps}^2=\frac{n_sq_s^2}{\epsilon_0m_s}. ((b)) Set ϵ(ω)=0\epsilon(\omega)=0 (the longitudinal branch of the wave equation of an isotropic medium) and show that the electrostatic oscillation frequency is ω=(ωpe2+ωpi2)1/2\omega=(\omega_{pe}^2+\omega_{pi}^2)^{1/2}, independent of kk, so the group velocity is again zero. ((c)) Expand ω\omega to first order in me/mHem_e/m_{\rm He} and evaluate the fractional shift of ω\omega above ωpe\omega_{pe} numerically.

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