A Beam of Electrons in an Isotropic Photon Background

Take the differential cross-section and integrate over &phgr; which  is either the azimuthal angle n about n' or the azimuthal angle of n' about n.

[Graphics:SoftComptongr732.gif]

[Graphics:SoftComptongr733.gif]

Obtaining  [Graphics:SoftComptongr734.gif]

To get [Graphics:SoftComptongr735.gif] we must multiply by &egr;(1-&bgr; &mgr;e) and a &dgr;-function with the following argument:; and then integrate over &mgr;e'

[Graphics:SoftComptongr736.gif]

[Graphics:SoftComptongr737.gif]

Integrating through the &dgr;-function eliminates [Graphics:SoftComptongr738.gif] with the folloing replacement

[Graphics:SoftComptongr739.gif]

[Graphics:SoftComptongr740.gif]

and one picks up a factor ofrom the integration over the  &dgr;-function:

[Graphics:SoftComptongr741.gif]

[Graphics:SoftComptongr742.gif]

we have added the extra factor -1 since it is the absolute value which is required

[Graphics:SoftComptongr743.gif]

[Graphics:SoftComptongr744.gif]

It is sort of obvious that this is just (1-&bgr; [Graphics:SoftComptongr745.gif])dSigmadDelta[&dgr;,&bgr;,[Graphics:SoftComptongr746.gif]] which we now verify:

[Graphics:SoftComptongr747.gif]

[Graphics:SoftComptongr748.gif]

[Graphics:SoftComptongr749.gif]

[Graphics:SoftComptongr750.gif]

Obtaining [Graphics:SoftComptongr751.gif]

To get [Graphics:SoftComptongr752.gif] we instead integrate  [Graphics:SoftComptongr753.gif] through the &dgr;-function which eliminates  [Graphics:SoftComptongr754.gif] with the following replacement

[Graphics:SoftComptongr755.gif]

[Graphics:SoftComptongr756.gif]

and one picks up a different factor from the integration over the  &dgr;-function:

[Graphics:SoftComptongr757.gif]

[Graphics:SoftComptongr758.gif]

(no -1 required here).

[Graphics:SoftComptongr759.gif]

[Graphics:SoftComptongr760.gif]

It is not so obvious why this is just (1+&Dgr;)(1-&bgr; [Graphics:SoftComptongr761.gif])dSigmadDelta[[Graphics:SoftComptongr762.gif]]  , but it is true:

[Graphics:SoftComptongr763.gif]

[Graphics:SoftComptongr764.gif]

[Graphics:SoftComptongr765.gif]

[Graphics:SoftComptongr766.gif]

Document converted by Mathematica of Wolfram Research