(a) A large fusion device measures the line-averaged electron density along an chord…

PHYSICS · Subjective · Fermi University Physics · Plasma Physics · Waves in plasmas · Laboratory, industrial and space plasmas

Plasma Physics, Ch Plasma oscillations and light in an unmagnetised plasma

Problem

(a) A large fusion device measures the line-averaged electron density along an

L=8.0\,\mathrm m

chord using a

\lambda=195\,\mu\mathrm m

far-infrared laser interferometer. The measured phase shift relative to vacuum is

\Delta\phi=132.0\,\mathrm{rad}

. Using

\Delta\phi=r_e\lambda\int n_e\,\mathrm dl

with

r_e=2.82\times10^{-15}\,\mathrm m

, and a uniform density along the chord, find the line-averaged electron density. Compute the critical density

n_c=1.1\times10^{27}\lambda_{\mu\mathrm m}^{-2}\,\mathrm m^{-3}

for this wavelength and check that

n_e\ll n_c

. (b) A

2.0\,\mathrm{mm}

microwave interferometer is proposed as a cheaper replacement on the same chord. For a uniform density the exact phase shift is

\Delta\phi=k_0L[1-(1-x)^{1/2}]

, with

x=n_e/n_c

and

k_0=2\pi/\lambda

. Show that it exceeds the first-order value

r_e\lambda n_eL

by the factor

R(x)=[1-(1-x)^{1/2}]/(x/2)

, and that

R\approx1+x/4

for small

x

. Evaluate

R

for the

195\,\mu\mathrm m

and the

2.0\,\mathrm{mm}

beams at the density found in part (a), find the density at which the exact

2.0\,\mathrm{mm}

phase is 10 per cent larger than the first-order value (

R=1.10

), and say which instrument to keep if the density is expected to rise to

1\times10^{20}\,\mathrm{m^{-3}}