This computes the zero of the distortion
The frequency of zero distortion in the classical S-Z effect is
while the derivative of the zero frequency wrt temperature (at zero temperature) is
To compute the 2nd derivative would require knowledge of S30.
Here is plotted the variation of the zero frequency as a function of temperature for our level of approximation
Rephaeli (1995) who has done the full collision integral says the following ratio is close to unity
and says this number should be 3.94
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