Linear Compton Scattering

  When the parameter NPH=0 is specified in LASERQED command, the formulas of linear Compton scattering are used.

Let us define the following variables in the rest frame of the initial electron:

,

Photon Stokes parameters before and after collision as defined in[3](page 361).

,

Initial (laser) and final energies of the photon.

,

Initial (laser) and final momenta of the photon.

,

Polar and azimuthal scattering angle of the photon.

Solid angle .

The range of is given by

The Compton relation is

The crosssection is given by eq(87.22) in [3]. :




 


See Sec.5.2.2 for the meaning of the bars on and . The omitted terms are products of three and four among , , , and . (Actually, we need the terms and but they are not found in literature.) The functions introduced in the above expression are:




  


These formulas are used in their exact forms in CAIN.

Summation over the final polarization and the azimuthal angle gives the differential crosssection with respect to the final photon energy . Introducing the variables inplace of by







we write the differential crossection as

where







Note that does not appear here because it is based on the scattering plane and, therefore, disappears after integration over the azimuthal angle. The function satisfies for any and and is O(1) except when h is close to +1 and is extremely large. The Function is plotted in Fig.5.

  
Figure 5: Function for for various values of

The total crosssection for given initial momenta and polarizations is given by

  
Figure 6: Function for as a function of . is less than unity and is O(1) unless h is close to +1 and is extremely large.

Let us briefly describe the algorithm of event generation.

1. Compute the total event rate in the given time interval using without the factor . Since , this is an over estimation of the rate. If is too large, divide the time interval by an integer N and repeat the following procedure N times.
2. Generate a random number uniform in (0,1). Reject if .
3. Compute and multiply it to . Reject if still . Otherwise accept. Note that the Lorentz transformation of is not needed for the computation of h because is Lorentz invariant. Also note that input is defined already in the rest frame of electron. Only the Lorentz transformation of is needed.
4. Generate two random numbers and in (0,1). Repeat this step until is satisfied. Once or twice repetition is normally enough unless h is close to +1 and is very large.
5. Compute from . Generate the azimuthal angle, compute the final polarization if needed, and go back to the laboratory frame. In this step many Lorentz transformations are needed.