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Homework Help: Optical thickness of the second harmonic cyclotron motion in a plasma

  1. Jan 25, 2014 #1
    1. The problem statement, all variables and given/known data
    Let's consider a Tokamak with major radius [itex]R=1m[/itex] and minor radius [itex]a=0.3m[/itex], magnetic field [itex]B=5T[/itex] with a deuterium plasma with central density [itex]10^{20}m^{-3}[/itex], central temperature [itex]1keV[/itex] and parabolic temperature and density profiles [itex]\propto (1-r^2/a^2)[/itex]

    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

    2. Relevant equations
    [tex] \omega_c=\frac{\Omega}{\gamma}=\frac{eB_0}{m_e\gamma}[/tex]
    [tex] \omega_m=\frac{m\omega_c}{1-\beta_{//}\cos\theta}[/tex]
    [tex] \tau=\int\!\!ds\,\alpha(\nu)[/tex]

    3. The attempt at a solution
    a) I just apply the formula for [itex]\omega_m[/itex] with [itex]m=2[/itex]
    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 [itex]\tau>>1[/itex] but I don't know how to apply that integral. I don't want the solution, just an hint

    Thank you very much
  2. jcsd
  3. Jan 25, 2014 #2
    The problem has been taken from exercise 5.4, chapter 5, Hutchinson - Principles of Plasma Diagnostic
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