Electrons in a one-dimensional

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Plasma Physics · Kinetic theory, collisions and transport

Plasma Physics, Ch The Vlasov equation and Landau damping

Problem

Electrons in a one-dimensional, unmagnetised plasma have the equilibrium 'waterbag' velocity distribution (normalised to unit area)

\hat f_{0e}(v) = 1/(2v_0)

for

|v|\le v_0

and

\hat f_{0e}(v)=0

for

|v|>v_0

, where

v_0

is a constant speed. The ions are fixed and form a uniform neutralising background. Consider small electrostatic perturbations

\propto e^{i(kx-\omega t)}

with real

k>0

and a real phase speed satisfying

\omega/k > v_0

. ((a)) Write

\hat f_{0e}'(v)

as a combination of delta functions at

v=\pm v_0

, and use it to evaluate

\displaystyle\int_{-\infty}^{\infty}\frac{\hat f_{0e}'(v)}{v-\omega/k}\,\mathrm{d}v

. ((b)) Substitute into the linearised Vlasov-Poisson dispersion relation

1-(\omega_{pe}^2/k^2)\int \hat f_{0e}'(v)/(v-\omega/k)\,\mathrm{d}v=0

and show that

\omega^2=\omega_{pe}^2+k^2v_0^2

. ((c)) Explain, in terms of resonant particles, why this root has no imaginary part for any real

k