Calculation of the Thomson Scattering Cross Section - llnl/Thomson-Scattering-Cross-Section-Calculator GitHub Wiki

The spectral flux Φ_(e,λ) (W nm^-1) that should be collected by a detector, due to scattering from a plasma is calculated using a model for the scattered power:

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Where E_e is the average flux density of the probe beam (power per unit area), λ_in is the probe wavelength, λ_out is the scattered wavelength, Ω_C is the solid angle collected by the diagnostic, V_S is the scattering volume and r_e is the classical electron radius. S(k,ω) is the dynamic structure factor, describing the spectral density of thermally excited electron density fluctuations of wavevector k and frequency ω.

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In these and the following equations, subscript e is used to label the parameters of the electron population, j the two ion populations and subscript s is used to indicate that the equation may be used for either type of population. Parameters n_j and Z_j are the density and charge state of each ion species making up the plasmas, and f_e0, f_j0 are the velocity distributions of each plasma species. For arbitrary distribution functions the electric susceptibilities χ_s and the total permittivity ϵ are calculated:

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Where ω_ps is the species-specific plasma frequency.

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Making the simplifying assumption that the distribution functions are Maxwellian.

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Here v_Ts is the thermal velocity and v_ϕs is the species-specific Doppler-corrected phase velocity, which accounts for the bulk flow velocity v_s of the species with respect to the reference frame in which the scattering is measured. The mean drift velocity of the electron population v_e is calculated based on the weighted mean v_j and the plasma current density J:

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For Maxwellian distributions the susceptibilities can be reduced to the following expression:

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Where Z' is the derivative of the plasma dispersion function, α_S is the species-specific, dimensionless scattering parameter and λ_Ds is the species-specific Debye length.

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The plasma dispersion function can be calculated using the Faddeeva function:

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References: J. Sheffield, D. Froula, S.H. Glenzer, N.C. Luhmann, and Jr., Plasma Scattering of Electromagnetic Radiation: Theory and Measurement Techniques, 2nd ed. (Academic Press, 2010).