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:

- Bethe-Heitler
- Landau-Lifshitz
- Bremsstrahlung

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

with

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 .

1ex

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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).
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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:
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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

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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.
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Now, the spectrum with respect to is
with
For , and close to 1 except for the small
region which is umimportant in practice. Thus,
For given (or __

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The algorithm is as follows.
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__(a)____From the given parameters, compute , and__

where*w*is the weight of virtual photon to be created. is the expected number of macro-virtual photons. If is not small enough (say, >0.1), divide it by an integer*N*and repeat the following steps*N*times.__(b)____Generate a uniform random number . Reject if . Otherwise redefine by .____(c)____Generate a random number , define and calculate from a table. Reject if . (The probability to be rejected here is small because__*G*is close to unity.) Otherwise, accept.__(d)____Calculate__*y*using eq.(122) and . If LOCAL option is specified, stop here and return . Otherwise, calculate the value of*V*(*y*) from*G*using eq.(125).__(e)____Generate a random number and solve the equation with respect to__*x*. This is done by using a table of inverse function of*V*.__(f)____Compute , being the Compton wave length.____(g)____Generate a random number and compute the photon coordinate__

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Thu Dec 3 17:27:26 JST 1998