Up: Polarization for pair annihilation

# 3 Total annihilation cross-sections

Integrating the differential rate over the azimuth angle, , and dividing the resulting rate by the current density ( ) we obtain the one-photon annihilation cross-section (for any pz):

 (20)

In the above  cm is the Compton wavelength of the electron. Summing the above expression over the polarization of the photon yields the same result as given by Wunner et al. (1986).

If we suppose that and are found in the lowest Landau state ( ) and , so , then the photon is emitted perpendicular to the magnetic field ( ). We allow any polarization of the photon. Then the above expression for the cross-section reduces to:

 (21)

Then using Table 2, one obtains:
 = 0 = = = (22)

So the radiation obtained in 1 photon annihilation from and in the ground state with has linear polarization. The result for summing over polarization agrees with the result of Wunner (1979).

If we take the case of an electron with zero longitudinal momentum (pz = 0), and sum over polarization, then we find:

 = (23)

which has been obtained previously by Wunner et al. (1986).

From the conservation laws: and , we find the expression for the longitudinal component of the momentum:

 = (24)

However, if or , the following applies:
 = (25)

with the upper sign for , the lower sign for .

If we average over the polarization of the and in Eq. (20) above, we obtain:

 (26)

If we sum over polarization of the photon in the above equation and take pz=0 (i.e. zero longitudinal momentum of the electron) we obtain:
 (27)

This differs by a factor of 2 in the last two terms from Eq. (7) of Wunner et al. (1986), and corrects an error in their expression.

Up: Polarization for pair annihilation

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