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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
![equation5313](img664.gif)
with
![eqnarray5319](img665.gif)
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
![equation5323](img666.gif)
where
![equation5325](img667.gif)
where
![displaymath5327](img668.gif)
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
![displaymath5329](img670.gif)
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
![equation5331](img677.gif)
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
![equation5333](img684.gif)
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
![displaymath5335](img686.gif)
where
are the energy momentum of the
photons in the original frame.
The value of
should be computed from
![displaymath5337](img689.gif)
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
![equation5339](img691.gif)
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