To treat the Bethe-Heitler , Landau-Lifshitz and the Bremsstrahlung processes, the so-called almost-real-photon approximation, or equivalent photon approximation, or Weizacker-Williams approximation, is employed. An electron is accompanied by virtual photons which look like real photons at ultra-relativistic limit. They interact with on-coming (real or virtual) photons incoherently. Thus, the Bethe-Heitler and Landau-Lifshitz processes above are reduced to the Breit-Wheeler process and the Bremsstrahlung to the Compton process:
Let the electron energy be (). The number of
virtual photons with energy and transverse momentum
is given by
where is the fine structure constant. For given , the typical transverse momentum is very small, , so that it is not important in collision kinematics but, instead, the finite transverse extent can bring about significant effects. In the (transverse) configuration space, the above expression becomes
where is the transverse coordinate with respect to the parent electron, , and the modified Bessel function.
The transverse momentum cut off (or ) is somewhat umbiguous. It should depend on the momentum transfer of the whole process. This dependence is ignored in CAIN because the virtual photons are generated independently from the following processes and because it does not much affect the low energy pairs.
The lower limit of the integration over is, in our case,
determined by the pair creation threshold.
Let us introduce dimensionless variables ,
, and . The total number of
the virtual photons is given by
When , the total number is
where is Euler's constant. At very high energies the number of virtual photons per electron is O(1), in spite of the small factor , due to the factor .
When Bethe-Heitler and/or Landau-Lifshitz processes are specified by PPINT command, CAIN generates virtual photons in each longitudinal slice at each time step and counts them in the same mesh as that generated by the LUMINOSITY command. The number of macro-virtual photons is somewhat arbitrary. In the present version it is determined such that the weight of the macro-virtual photons is equal to the maximum weight of the electrons in the on-coming beam (not equal to the weight of each parent electron in order to prevent low-weight electrons from generating many photons).
Since the y (energy) spectrum is approximately proportional to
for small y, the spectrum becomes almost flat if one chooses
as the primary variable. To account for relatively large y too, CAIN
adopts the variable instead of y:
Here, c>0 is introduced so that the function defined later, is finite. It is chosen to be 0.2 but is almost arbitrary provided . The maximum is
Figure 11: Function defined in eq.(125). It is close to unity because only large region is important. G(0) is finite and depends on the parameter c. G(0)<1 if . Here, c=0.2 is adopted.
Now, the spectrum with respect to is
For , and close to 1 except for the small region which is umimportant in practice. Thus,
For given (or y) the distribution of x is proportional to dV(x)/V(y) and, therefore, can be random-generated by using inverse function .
The algorithm is as follows.