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Polarization effects have been added since 2.1. In order to describe polarizations, let us introduce an orthonormal basis vector (tex2html_wrap_inline10376, tex2html_wrap_inline10378, tex2html_wrap_inline10380). Here, tex2html_wrap_inline10380 is the unit vector along the initial electron velocity, tex2html_wrap_inline10376 the unit vector along the direction of the transverse component of acceleration, and tex2html_wrap_inline10386. If the acceleration is due to a transverse magnetic field, tex2html_wrap_inline10388 is the unit vector along the magnetic field (times the sign of charge). The transition rate from the initial electron polarization tex2html_wrap_inline9394 to the final polarization tex2html_wrap_inline10392 with the photon Stokes parameter tex2html_wrap_inline7924 (based on the basis vector (tex2html_wrap_inline10376, tex2html_wrap_inline10378, tex2html_wrap_inline10380)) is


where tex2html_wrap_inline10300 is defined in eq.(85) and the argument of the Bessel functions is z. We omitted the terms involving tex2html_wrap_inline10392 and tex2html_wrap_inline7924 simultaneously, which means to ignore the correlation of polarization between the final electron and photon. (See Sec.5.2.2 for the meaning of bars on tex2html_wrap_inline10392 and tex2html_wrap_inline7924.)

The radiation energy spectrum summed over the final polarization is given by eq.(85) with tex2html_wrap_inline10300 replaced by
Since the function tex2html_wrap_inline10316 with tex2html_wrap_inline10300 replaced by tex2html_wrap_inline10420 has still the above mentioned property, the same algorithm of generating the photon energy can be used. (tex2html_wrap_inline10316 is slightly larger when tex2html_wrap_inline10424 but still tex2html_wrap_inline10426.)

For the given radiation energy tex2html_wrap_inline10428, the polarizations of the final electron and photon are calculated by the prescription described in Sec.5.2.2. Thus,

In the case when the event generation is rejected, the polarization of the electron must be changed according to eq.(12):

In storage rings, the electron polarization builds up slowly along the direction of the magnetic field. This effect comes from the difference between the coefficient of tex2html_wrap_inline9394 and tex2html_wrap_inline10432 (see eq.(14)):
When tex2html_wrap_inline9648 is small (tex2html_wrap_inline10436 in storage rings), each term on the left hand side is proportional to tex2html_wrap_inline9648 whereas the right hand side is tex2html_wrap_inline10440 because of cancellation. CAIN cannot reproduce such slow buildup, even if the computing time allows, because the approximate polynomials adopted do not have that accuracy. They are enough, however, for beam-beam problems.

next up previous contents index
Next: Enhancement factor of the Up: Beamstrahlung Previous: Algorithm of event generation

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