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Linear Compton Scattering

  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).gif
Initial (laser) and final energies of the photon.
Initial (laser) and final momenta of the photon.
tex2html_wrap_inline10036, tex2html_wrap_inline8612
Polar and azimuthal scattering angle of the photon.
Solid angle tex2html_wrap_inline10042.

The range of tex2html_wrap_inline9996 is given by
The Compton relation is
The crosssection is given by eq(87.22) in [3]. gif:


See Sec.5.2.2 for the meaning of the bars on tex2html_wrap_inline9290 and tex2html_wrap_inline10014. The omitted terms are products of three and four among tex2html_wrap_inline9290, tex2html_wrap_inline10054, tex2html_wrap_inline10012, and tex2html_wrap_inline10058. (Actually, we need the terms tex2html_wrap_inline10060 and tex2html_wrap_inline10062 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 tex2html_wrap_inline8612 gives the differential crosssection with respect to the final photon energy tex2html_wrap_inline9996. Introducing the variables tex2html_wrap_inline10068 inplace of tex2html_wrap_inline9996 by


we write the differential crossection as


Note that tex2html_wrap_inline7246 does not appear here because it is based on the scattering plane and, therefore, disappears after integration over the azimuthal angle. The function tex2html_wrap_inline10074 satisfies tex2html_wrap_inline10076 for any tex2html_wrap_inline10068 and tex2html_wrap_inline7928 and is O(1) except when h is close to +1 and tex2html_wrap_inline7928 is extremely large. The Function tex2html_wrap_inline10074 is plotted in Fig.5.

Figure 5: Function tex2html_wrap_inline10074 for tex2html_wrap_inline10094 for various values of tex2html_wrap_inline7928

The total crosssection for given initial momenta and polarizations is given by


Figure 6: Function tex2html_wrap_inline10098 for tex2html_wrap_inline10094 as a function of tex2html_wrap_inline7928. tex2html_wrap_inline10104 is less than unity and is O(1) unless h is close to +1 and tex2html_wrap_inline7928 is extremely large.

Let us briefly describe the algorithm of event generation.

  1. Compute the total event rate tex2html_wrap_inline10114 in the given time interval using tex2html_wrap_inline10116 without the factor tex2html_wrap_inline10104. Since tex2html_wrap_inline10120, this is an over estimation of the rate. If tex2html_wrap_inline10114 is too large, divide the time interval by an integer N and repeat the following procedure N times.
  2. Generate a random number tex2html_wrap_inline9572 uniform in (0,1). Reject if tex2html_wrap_inline10130.
  3. Compute tex2html_wrap_inline10104 and multiply it to tex2html_wrap_inline10114. Reject if still tex2html_wrap_inline10130. Otherwise accept. Note that the Lorentz transformation of tex2html_wrap_inline10012 is not needed for the computation of h because tex2html_wrap_inline7244 is Lorentz invariant. Also note that input tex2html_wrap_inline9290 is defined already in the rest frame of electron. Only the Lorentz transformation of tex2html_wrap_inline9316 is needed.
  4. Generate two random numbers tex2html_wrap_inline10068 and tex2html_wrap_inline9574 in (0,1). Repeat this step until tex2html_wrap_inline10152 is satisfied. Once or twice repetition is normally enough unless h is close to +1 and tex2html_wrap_inline7928 is very large.
  5. Compute tex2html_wrap_inline9996 from tex2html_wrap_inline10068. 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 up previous contents index
Next: Quantum Electrodynamics Involving a Up: Laser Previous: Laser Geometry

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