Two equal Maxwellians of density each

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

Two equal Maxwellians of density n/2n/2 each, drifting at ±u\pm u with u=1.1vtu = 1.1v_t (vtv_t the thermal speed of each), are superposed to form a single electron distribution f0(v)f_0(v) of total density nn in an otherwise uniform, unmagnetised plasma. Someone argues: "A dip appears at v=0v=0 because u>vtu > v_t. Any distribution with two peaks and a dip in between is unstable, because a wave with phase velocity just above v=0v=0 sees f^0′(v)>0\hat f_0{}'(v)>0 there (the rising flank towards the beam at +u+u), and the weak-growth formula γ∝f^0′(ω/k)\gamma \propto \hat f_0{}'(\omega/k) then gives a growing root. Hence the plasma radiates Langmuir waves for ω/k\omega/k just above zero." Find the error in this reasoning and state the correct stability verdict for u=1.1vtu = 1.1v_t.

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