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When the parameter NPH=0 is specified in LASERQED command,
the formulas of linear Compton scattering are used.
Let us define the following variables in the rest frame of the initial
electron:
- ,
- Photon Stokes parameters
before and after collision as defined in[3](page 361).
- ,
- Initial (laser) and final energies of the photon.
- ,
- Initial (laser) and final momenta of the photon.
- ,
- Polar and azimuthal scattering angle of the photon.
-
- Solid angle
.
The range of is given by
The Compton relation is
The crosssection is given by eq(87.22) in [3].
:
See Sec.5.2.2 for the meaning of the bars on and
. The omitted terms are products of three and four among ,
, , and .
(Actually, we need the terms and
but they are not found in literature.)
The functions introduced in the above expression are:
These formulas are used in their exact forms in CAIN.
Summation over the final polarization and the azimuthal angle
gives the differential crosssection with respect to the final photon energy
. Introducing the variables inplace of by
we write the differential crossection as
where
Note that does not appear here because it is based on the scattering plane
and, therefore, disappears after integration over the azimuthal angle.
The function satisfies for any and
and is O(1) except when h is close to +1 and is extremely large.
The Function is plotted in Fig.5.
Figure 5: Function for for various values of
The total crosssection for given initial momenta and polarizations is given by
Figure 6: Function for as a function of .
is less than unity and is O(1) unless h is close to +1 and
is extremely large.
Let us briefly describe the algorithm of event generation.
- Compute the total event rate
in the given time interval using
without the factor . Since , this is an over estimation
of the rate. If is too large, divide the time interval by an integer N
and repeat the following procedure N times.
- Generate a random number uniform in (0,1). Reject if .
- Compute and multiply it to . Reject if still .
Otherwise accept. Note that the Lorentz transformation of
is not
needed for the computation of h because
is Lorentz invariant. Also note that input is defined
already in the rest frame of electron. Only the Lorentz transformation of
is needed.
- Generate two random numbers and in (0,1). Repeat this step
until is satisfied. Once or twice repetition is
normally enough unless h is close to +1 and is very large.
- Compute from . Generate the azimuthal angle,
compute the final polarization if needed, and go back to the
laboratory frame. In this step many Lorentz transformations are needed.
Next: Quantum Electrodynamics Involving a
Up: Laser
Previous: Laser Geometry
Toshiaki Tauchi
Thu Dec 3 17:27:26 JST 1998