next up previous contents index
Next: Subroutine Package for Nonlinear Up: Incoherent Pair Production Previous: Breit-Wheeler Process

Virtual (almost real) photon approximation


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:

where `tex2html_wrap_inline7172' is a virtual photon.

Let the electron energy be tex2html_wrap_inline10720 (tex2html_wrap_inline10722). The number of virtual photons with energy tex2html_wrap_inline10028 and transverse momentum tex2html_wrap_inline10726 is given by
where tex2html_wrap_inline7492 is the fine structure constant. For given tex2html_wrap_inline10028, the typical transverse momentum is very small, tex2html_wrap_inline10732, so that it is not important in collision kinematics but, instead, the finite transverse extent tex2html_wrap_inline10734 can bring about significant effects. In the (transverse) configuration space, the above expression becomes
where tex2html_wrap_inline10736 is the transverse coordinate with respect to the parent electron, tex2html_wrap_inline10738, and tex2html_wrap_inline10740 the modified Bessel function.

The transverse momentum cut off tex2html_wrap_inline10742 (or tex2html_wrap_inline10744) 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 tex2html_wrap_inline10746 of the integration over tex2html_wrap_inline10028 is, in our case, determined by the pair creation threshold. Let us introduce dimensionless variables tex2html_wrap_inline10750, tex2html_wrap_inline10752, and tex2html_wrap_inline10754. The total number of the virtual photons is given bygif


When tex2html_wrap_inline10760, the total number is
where tex2html_wrap_inline10762 is Euler's constant. At very high energies the number of virtual photons per electron is O(1), in spite of the small factor tex2html_wrap_inline10766, due to the factor tex2html_wrap_inline10768.


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 tex2html_wrap_inline10772 for small y, the spectrum becomes almost flat if one chooses tex2html_wrap_inline10776 as the primary variable. To account for relatively large y too, CAIN adopts the variable tex2html_wrap_inline7944 instead of y:
Here, c>0 is introduced so that the function tex2html_wrap_inline10786 defined later, is finite. It is chosen to be 0.2 but is almost arbitrary provided tex2html_wrap_inline10788. The maximum tex2html_wrap_inline7944 is

Figure 11: Function tex2html_wrap_inline10786 defined in eq.(125). It is close to unity because only large tex2html_wrap_inline7944 region is important. G(0) is finite and depends on the parameter c. G(0)<1 if tex2html_wrap_inline10802. Here, c=0.2 is adopted.

Now, the spectrum with respect to tex2html_wrap_inline7944 is
For tex2html_wrap_inline10808, tex2html_wrap_inline10810 and close to 1 except for the small tex2html_wrap_inline7944 region which is umimportant in practice. Thus,
For given tex2html_wrap_inline7944 (or y) the distribution of x is proportional to dV(x)/V(y) and, therefore, can be random-generated by using inverse function tex2html_wrap_inline10822.

The algorithm is as follows.

From the given parameters, compute tex2html_wrap_inline8962, tex2html_wrap_inline7882 and

where w is the weight of virtual photon to be created. tex2html_wrap_inline10114 is the expected number of macro-virtual photons. If tex2html_wrap_inline10114 is not small enough (say, >0.1), divide it by an integer N and repeat the following steps N times.
Generate a uniform random number tex2html_wrap_inline10658. Reject if tex2html_wrap_inline10130. Otherwise redefine tex2html_wrap_inline9572 by tex2html_wrap_inline10846.
Generate a random number tex2html_wrap_inline10848, define tex2html_wrap_inline10850 and calculate tex2html_wrap_inline10786 from a table. Reject if tex2html_wrap_inline10854. (The probability to be rejected here is small because G is close to unity.) Otherwise, accept.
Calculate y using eq.(122) and tex2html_wrap_inline10860. If LOCAL option is specified, stop here and return tex2html_wrap_inline10862. Otherwise, calculate the value of V(y) from G using eq.(125).
Generate a random number tex2html_wrap_inline10868 and solve the equation tex2html_wrap_inline10870 with respect to x. This is done by using a table of inverse function of V.
Compute tex2html_wrap_inline10876, tex2html_wrap_inline10878 being the Compton wave length.
Generate a random number tex2html_wrap_inline10880 and compute the photon coordinate

next up previous contents index
Next: Subroutine Package for Nonlinear Up: Incoherent Pair Production Previous: Breit-Wheeler Process

Toshiaki Tauchi
Thu Dec 3 17:27:26 JST 1998