A laboratory ion source produces a cold

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

Plasma Physics, Ch Cold magnetised plasma waves: the dielectric tensor, cutoffs, resonances and the CMA diagram

Problem

A laboratory ion source produces a cold, uniform deuterium plasma (deuteron mass

m_i = 2\times1.67\times10^{-27}\,\mathrm{kg}

, singly charged) with equal electron and ion densities

n_e = n_i = 2.0\times10^{18}\,\mathrm{m^{-3}}

, immersed in a steady field

B_0 = 0.15\,\mathrm{T}

along

\hat{\mathbf{z}}

. A wave of frequency

f = 3.0\,\mathrm{MHz}

propagates parallel to

\mathbf{B}_0

(

\theta = 0

). The ions are cold and at rest apart from their response to the wave. ((a)) Starting from the linearised cold equation of motion

-i\omega m_sv_s = q_s(\mathbf{E}+\mathbf{v}_s\times\mathbf{B}_0)

of a species with charge

q_s

and mass

m_s

, and working in the rotating basis

E_\pm = E_x\pm iE_y

, show that including both the electrons and the deuterons gives

\begin{aligned} R &= 1-\frac{\omega_{pe}^2}{\omega(\omega+\Omega_e)}-\frac{\omega_{pi}^2}{\omega(\omega+\Omega_i)},\\ L &= 1-\frac{\omega_{pe}^2}{\omega(\omega-\Omega_e)}-\frac{\omega_{pi}^2}{\omega(\omega-\Omega_i)} \end{aligned}

with

\Omega_e < 0 < \Omega_i

. ((b)) Using

S=(R+L)/2

D=(R-L)/2

and the

\theta=0

wave-equation matrix, show that the dispersion relation factors into

P(n^2-R)(n^2-L)=0

. ((c)) Evaluate

R

and

L

numerically for this plasma at

3.0\,\mathrm{MHz}