In deriving the plasma frequency from a displaced slab

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · The plasma state and single-particle motion · The plasma state and single-particle motion

Plasma Physics, Ch What makes an ionised gas a plasma: shielding, the plasma parameter and the natural frequencies

Problem

In deriving the plasma frequency from a displaced slab, the ions were held fixed. Suppose instead a slab of fully ionised deuterium (singly charged deuterons, mass mi=3.344×10−27 kgm_i = 3.344\times10^{-27}\,\mathrm{kg}) has both its electrons and its ions free to move, displaced from a common equilibrium by ξe(t)\xi_e(t) and ξi(t)\xi_i(t) respectively, with equal unperturbed densities ne=ni=nn_e=n_i=n far from the slab. Ignore thermal motion, collisions and any magnetic field. ((a)) Write down Gauss's law for the electric field produced by the relative displacement ζ=ξe−ξi\zeta=\xi_e-\xi_i, and Newton's second law for each species in that field. Show that ζ\zeta oscillates at ω2=ωpe2+ωpi2,ωps2=ne2ϵ0ms.\omega^2 = \omega_{pe}^2+\omega_{pi}^2, \qquad \omega_{ps}^2=\frac{ne^2}{\epsilon_0m_s}. ((b)) For n=5.0×1019 m−3n=5.0\times10^{19}\,\mathrm{m^{-3}}, evaluate fpef_{pe} and fpif_{pi}, and find the fractional difference between the combined frequency f=ω/2πf=\omega/2\pi and fpef_{pe} alone.

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