A laboratory beam-plasma experiment has a background electron density and a warm…

PHYSICS · Subjective · Fermi University Physics · Plasma Physics · Kinetic theory, collisions and transport · Kinetic theory, collisions and transport

Plasma Physics, Ch Hot plasmas and free energy: beam instabilities, cyclotron damping and Bernstein waves

Problem

A laboratory beam-plasma experiment has a background electron density ne=1.0×1016 m−3n_e = 1.0\times10^{16}\,\mathrm{m^{-3}} and a warm electron beam of relative density nb/ne=1.0×10−4n_b/n_e = 1.0\times10^{-4}, mean speed vb=6.0×106 m s−1v_b = 6.0\times10^{6}\,\mathrm{m\,s^{-1}} and Gaussian spread (standard deviation) Δv=1.0×106 m s−1\Delta v = 1.0\times10^{6}\,\mathrm{m\,s^{-1}}, so that the normalised beam contribution to the distribution is f^0,beam(v)=nb/neΔv2πexp⁡ ⁣[−(v−vb)22Δv2].\hat f_{0,\mathrm{beam}}(v) = \frac{n_b/n_e}{\Delta v\sqrt{2\pi}}\exp\!\left[-\frac{(v-v_b)^2}{2\Delta v^2}\right]. The bulk plasma is cold enough (vte≪vbv_{te}\ll v_b) that its slope is negligible at speeds near vbv_b. Take the wave phase speed to be one spread below the beam mean, vφ=vb−Δv=5.0×106 m s−1v_\varphi = v_b - \Delta v = 5.0\times10^{6}\,\mathrm{m\,s^{-1}}, where the beam distribution is rising, and use the weak-growth formula γ=π2 ωpe vφ2 f^0′(vφ).\gamma = \frac{\pi}{2}\,\omega_{pe}\,v_\varphi^2\,\hat f_0'(v_\varphi). Calculate the growth rate γ\gamma, and check that it is small compared with ωpe\omega_{pe} as the formula assumes.

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