Consider cold ions in a uniform field

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

Plasma Physics, Ch Waves that need pressure or tension: Langmuir, ion-acoustic, Alfven and magnetosonic

Problem

Consider cold ions in a uniform field B0=B0z^\mathbf{B}_0 = B_0\hat{\mathbf{z}}, with electrons in Boltzmann equilibrium along B0\mathbf{B}_0, ne=ne0exp⁡(eϕ/Te)n_e = n_{e0}\exp(e\phi/T_e), and quasineutrality ni≈nen_i\approx n_e. Take k\mathbf{k} nearly perpendicular to B0\mathbf{B}_0, with a small parallel component so the Boltzmann response applies, and perturbations ∝ei(k⋅x−ωt)\propto e^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)}. ((a)) Starting from the linearised ion continuity and momentum equations (with the Lorentz force qi(E+ui×B0)q_i(\mathbf{E}+\mathbf{u}_i\times\mathbf{B}_0) and cold ions, so no ion pressure term), the linearised Boltzmann electron density, and quasineutrality, derive the electrostatic ion cyclotron dispersion relation ω2=Ωi2+k2cs2\omega^2 = \Omega_i^2 + k^2c_s^2, where cs2=Te/mic_s^2=T_e/m_i. ((b)) A deuterium plasma (mi=2mpm_i=2m_p) sits in B0=0.40 TB_0 = 0.40\,\mathrm{T} with Te=8.0 eVT_e = 8.0\,\mathrm{eV}. For a wave with k=500 m−1k = 500\,\mathrm{m^{-1}} (nearly across B0\mathbf{B}_0), find Ωi\Omega_i, csc_s and the wave frequency ω\omega.

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