Cold argon ions (

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 argon ions (

m_i=39.95\,\mathrm{u}

, charge

+e

) sit in a uniform field

B_0\hat{\mathbf{z}}

, with Boltzmann electrons of temperature

T_e

maintaining quasineutrality along

\mathbf{B}_0

. A wave has

\mathbf{k}

almost across

\mathbf{B}_0

(write

k

for its perpendicular magnitude) with a small parallel component that lets the electrons equilibrate along the field. ((a)) Starting from the linearised ion equation of motion

m_i\dot{\mathbf u}_i=e(\mathbf{E}+\mathbf{u}_i\times\mathbf{B}_0)

for the perpendicular components, the ion continuity equation, and quasineutrality with the electron Boltzmann relation, derive

\omega^2=\Omega_i^2+k^2c_s^2, \qquad c_s^2=T_e/m_i,\ \ \Omega_i=eB_0/m_i.

((b)) Evaluate

\omega

and

f=\omega/2\pi

for argon with

B_0=0.15\,\mathrm{T}

T_e=3.0\,\mathrm{eV}

and

k=135\,\mathrm{m^{-1}}

, and compare with

\Omega_i/2\pi

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