Consider the calculation for an incident beam of polarized electrons. The Coulomb scattering of an electron incident with momentum and spin , where , and summed over final spin states , is given by

(7.336) |

In order to take advantage of the trace technique we introduce the spin projection operator

(7.337) | |||

(7.338) | |||

(7.339) |

We have

(7.340) |

We could also introduce twice, both into matrix element and into adjoint, but this is unnecessary. The trace becomes

(7.341) |

where and we have used in the last two terms. The additional trace involving the spin vector vanishes and we once again return to the Mott formula. Our result, that the differential cross-section is the same for a polarized incident beam as for an unpolarized incident beam is a special consequence of the use of lowest order perturbation theory only and is not true generally. In other words, Coulomb scattering of electrons does not lead to a polarization of the beam to lowest order.

We now develop some useful relationships for the polarization vector of an electron. The polarization vector satisfies

(7.342) |

or

(7.343) | |||

(7.344) | |||

(7.345) |

Therefore

(7.346) | |||

(7.347) | |||

(7.348) | |||

(7.349) |

where is a unit vector along .

For the electron spin polarization lined up along , denoting a right-handed electron with polarization , we have

(7.350) |

and

(7.351) |

Similarly, for spin polarizaton antiparallel to , denoted a left-handed electron with , we have

(7.352) | |||

(7.353) | |||

(7.354) |

Similar formulas apply to ingoing and outgoing scattered electrons. The eigenstates of with are known as positive- and negative-helicity eigenstates.

We now consider an incident electron with spin lined up along its direction of motion and compute the polarization of the scattered electron as a function of the scattering angle. The polarizaton of scattered electrons is measured by

(7.355) |

where denotes the number emerging with positive helicity (or polarized right-handed) and the number with negative helicity. are generally functions of the scattering energy and angle.

The polarization for Coulomb scattering of a right-handed electron, , is given by

(7.356) |

Since we have just shown that the trace involving a single vanishes, we have

The trace in the denominator has been evaluated before

(7.358) |

The trace in the numerator becomes

The second term becomes

(7.360) |

where . The first term in equation 7.360 is a rather involved calculation and we have used FORM to evaluate it. The result is

(7.361) |

The total trace in the numerator of equation 7.358 now becomes

(7.362) |

The polarization thus becomes

(7.363) |

In the relativistic, limit or and we find . Thus no de-polarization of incident electrons occurs in the high energy limit of Coulomb scattering.

In the nonrelativistic limit, and this reduces to

(7.364) |

This is just the geometric overlap between the initial and final quantization axis

(7.365) |

and indicates, that the spin is not influenced at all by the collision, when viewed from a fixed system.

For an incident electron beam that is not completely polarized, but only partially polarized along its direction of motion, we define as the fraction with positive helicity and the fraction with negative helicity. Using to denote the polarization of the incident electrons, we can show that . Using

(7.366) |

and inserting this identity into the expression for gives

(7.367) |

For the special case of we see that any initially unpolarized beam of electrons remains unpolarized in Coulomb scattering.

2004-03-18