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