Frequency of Zero Distortion

This computes the zero of the distortion

[Graphics:SoftComptongr431.gif]

The frequency of  zero distortion in the classical S-Z effect is

[Graphics:SoftComptongr432.gif]

[Graphics:SoftComptongr433.gif]

[Graphics:SoftComptongr434.gif]

[Graphics:SoftComptongr435.gif]

while the derivative of the zero frequency wrt temperature (at zero temperature) is

[Graphics:SoftComptongr436.gif]

[Graphics:SoftComptongr437.gif]

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

[Graphics:SoftComptongr438.gif]

[Graphics:SoftComptongr439.gif]

Rephaeli  (1995) who has done the full collision integral says the following ratio is close to unity

[Graphics:SoftComptongr440.gif]

[Graphics:SoftComptongr441.gif]

and says this number should be 3.94

[Graphics:SoftComptongr442.gif]

[Graphics:SoftComptongr443.gif]

[Graphics:SoftComptongr444.gif]

[Graphics:SoftComptongr445.gif]

Document converted by Mathematica of Wolfram Research