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The differential crosssection with respect to the scattering angle
of the final electron in the center-of-mass frame is given by

with

where
, p
- Energy and momentum of final electron
in the center-of-mass frame.
- c
- Cosine of the scattering angle
of the final electron in the center-of-mass frame.
- h
- Product of circular polarizations of the two initial photons.
The total crosssection is

where

where

Events are generated by the following algorithm using inverse function.
- (a)
- Compute
, p, a, b, G and
for given
initial parameters (reject if
, i.e., below threshold)
and calculate the event probability for the given time step

where
and
are the weights of initial photons
(number of real photons divided by that of macro photons), w the
weight of the pair to be created,
the time interval
and V the volume in which the macro phtons are located.
- (b)
- If P is too large (say, >0.1), divide the interval
(and P) by an integer
,
and repeat the following procedure
times.
- (c)
- Generate a random number
.
Reject if
.
- (d)
- Generate another random number
and solve the equation

with respect to z.
Here z is defined by
(
). The left hand side is the integral
of f from 0 to
. The sign of
is determined by the sign of
.
- (e)
- Generate another random number
and
compute the transverse component of electron momentum by

where
and
are arbitrary unit vectors
perpendicular to
, the unit vector along the initial photon
momentum in the center-of-mass frame. The latter is given by

where
are the energy momentum of the
photons in the original frame.
The value of
should be computed from

rather than from
because the latter is usually very
close to unity when
is much larger than
the electron rest mass.
- (f)
- Then, the momentum of the electron in the original frame
is calculated by

where
.
Note that
must be computed from
in order to avoid round off errors.
- (g)
- The momentum of positron is computed from the momentum
conservation.
Next: Virtual (almost real) photon
Up: Incoherent Pair Production
Previous: Incoherent Pair Production
Toshiaki Tauchi
Thu Dec 3 17:27:26 JST 1998