A deuterium plasma has

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

B_0 = 3\,\mathrm{T}

n_e = n_i = 1\times10^{19}\,\mathrm{m^{-3}}

and

T_e = T_i = 100\,\mathrm{eV}

. Take the deuteron mass as

m_i = 3.343\times10^{-27}\,\mathrm{kg}

and the single-fluid adiabatic index

\gamma_{\rm ad} = 5/3

. ((a)) The one-dimensional adiabatic closure used for the electrons in the Bohm-Gross wave sets

\gamma_e

from the number of degrees of freedom

N

that take part in the compression, through

\gamma = (N+2)/N

(the ideal-gas rule for the adiabatic index in terms of

N

). State which value of

N

is appropriate for a plane electron-density wave, use the formula to recover

\gamma_e = 3

, and give, in terms of

\omega/k

and

v_{te}

, the condition on the wave's phase speed that makes this one-dimensional treatment sensible. ((b)) For the plasma above, compute the plasma beta

\beta = 2\mu_0 p/B_0^2

with

p = n_eT_e + n_iT_i

, the Alfven speed

v_A

, and the sound speed

c_s = (2\gamma_{\rm ad}T/m_i)^{1/2}

(using

T = T_e = T_i

). Hence evaluate the fast magnetosonic speed at

\theta = 90^\circ

(propagation exactly across

B_0

v_f = (v_A^2+c_s^2)^{1/2}

, and give the fractional amount by which it exceeds

v_A

. State which wave of Chapter 9's cold-plasma theory this low-

\beta

, perpendicular fast wave reduces to, and why the cold theory could not have produced the correction you found.