Cold ions ( ) of mass and charge move in a uniform plasma with Boltzmann electrons of…

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Waves in plasmas · Waves in plasmas

Plasma Physics, Ch Waves that need pressure or tension: Langmuir, ion-acoustic, Alfven and magnetosonic

Problem

Cold ions (

T_i\to0

) of mass

m_i

and charge

+e

move in a uniform plasma with Boltzmann electrons of temperature

T_e

, equilibrium density

n_0

. A one-dimensional electrostatic perturbation of potential

\phi_1e^{i(kx-\omega t)}

is imposed. ((a)) Starting from the linearised ion continuity and momentum equations (no ion pressure term), the linearised electron Boltzmann relation

n_{e1}=n_0e\phi_1/T_e

, and Poisson's equation, derive

\omega^2 = \frac{k^2c_s^2}{1+k^2\lambda_{De}^2}, \qquad c_s^2=T_e/m_i.

((b)) For hydrogen ions (

m_i=1.673\times10^{-27}\,\mathrm{kg}

) with

n_0=1.0\times10^{18}\,\mathrm{m^{-3}}

and

T_e=10\,\mathrm{eV}

, and a wave with

k\lambda_{De}=0.50

, find

\omega

and the ratio of the phase velocity to

c_s

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