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LANGMUIR WAVES - THE BOHM-GROSS DISPERSION RELATION

LANGMUIR WAVES - THE BOHM-GROSS DISPERSION RELATION

LANGMUIR WAVES - THE BOHM-GROSS DISPERSION

LANGMUIR WAVES - THE BOHM-GROSS DISPERSION RELATION Kalman Knizhnik Here I will derive the Bohm-Gross dispersion relation for electron plasma waves, also known as Langmuir waves. Electron waves in plasma are caused by small electron density perturbations. If I create a slight density disturbance in the plasma, a potential will be set up that will try to pull the electrons back to the equilibrium density distribution. However, due to their inertia, the electrons will overshoot the equilibrium position, in the same way that an oscillating spring overshoots its equilibrium position. As a result, these electrons will oscillate with a frequency that will be derived below. We start with the equations of continuity and momentum conservation for electrons: ∂n e ∂t + ∇ · (n ev) = 0 (1) mn e [ ∂v ∂t + (v · ∇)v] = −en e(E + v × B) − ∇p (2) The first thing we will do (even before we linearize these equations), is assume that we are dealing with an unmagnetized plasma, and that our electrons are isothermal electrons, we have p = γnk B T and ∇p = γk B T ∇n. By the way, we don’t deal with ion motions because we are assuming that the ions in the plasma are so massive that they cannot respond to perturbations sufficiently quickly. Our momentum equation now is: mn e [ ∂v ∂t + (v · ∇)v] = −en eE − γk B T ∇n e (3) Now we assume that the equilibrium in the plasma has been perturbed, so that there is a first order correction to each equilibrium value: v = v 0 + v 1 , n e = n e0 + n e1 , E = E 0 + E 1 . (4) Without loss of generality, we can set the equilibrium flow velocity v 0 to 0 by going into that frame, and since the plasma is uniform and neutral, E 0 = 0. Thus, v = v 1 , n e = n e0 + n e1 , E = E 1 . (5) Plugging these perturbed values into the continuity and momentum equations: ∂(n e0 + n e1 ) ∂t + ∇ · ((n e0 + n e1 )v 1 ) = 0 ⇒ ∂n e1 ∂t + ∇(n e0 v 1 ) + ∇(n e1 v 1 ) = 0 (6) m(n e0 + n e1 )[ ∂v 1 ∂t + (v 1 · ∇)v 1 ] = −e(n e0 + n e1 )E 1 − γk B T ∇(n e0 + n e1 ) (7) In equation 6, the time derivative of the equilibrium density has vanished since it is assumed to be uniform. In addition, we can neglect the second order quantity n e1 v 1 , which is assumed to be small, to come up with our linearized continuity equation: ∂n e1 ∂t + ∇ · (n e0 v 1 ) = 0 (8) Performing a similar simplification on the perturbed momentum equation (7) yields mn e0 ∂v 1 ∂t = −en e0 E 1 − γk B T ∇n e1 (9) The electric field E 1 is determined through Gauss’s Law: ∇ · E 1 = −en e1 /ɛ 0 . At this point we have our linearized equations, and we will solve them by Fourier transforming them. The general technique is to assume plane wave solutions for all varying quantities, i.e. n e0 , n e1 ∼ e i(k·x−ωt) (10)

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