Optical thickness of the second harmonic cyclotron motion in a plasma

[eoghan]

Homework Statement

Let's consider a Tokamak with major radius $R=1m$ and minor radius $a=0.3m$, magnetic field $B=5T$ with a deuterium plasma with central density $10^{20}m^{-3}$, central temperature $1keV$ and parabolic temperature and density profiles $\propto (1-r^2/a^2)$


a) Find the electronic cyclotron frequency for the second harmonics

b) Verify that the emission in the second extraordinary harmonic in a direction perpendicular to the magnetic field is optically thick

Homework Equations

$$\omega_c=\frac{\Omega}{\gamma}=\frac{eB_0}{m_e\gamma}$$
$$\omega_m=\frac{m\omega_c}{1-\beta_{//}\cos\theta}$$
$$\tau=\int\!\!ds\,\alpha(\nu)$$

The Attempt at a Solution

a) I just apply the formula for $\omega_m$ with $m=2$
b) I have no idea... please give me some hint... I tried to calculate the cutoff frequencies for the second harmonic in the extraordinary mode, but the second harmonic frequency doesn't fall in the cutoff and it is not absorbed. I think I have to apply the integral and find $\tau>>1$ but I don't know how to apply that integral. I don't want the solution, just an hint

Thank you very much

 

[eoghan]

The problem has been taken from exercise 5.4, chapter 5, Hutchinson - Principles of Plasma Diagnostic