Magnetic Neutron Scattering and Spin-Polarized Neutrons

Use trick to simplify this expression:1dp ip rcos22 eipr d p e d cosp0 1 2 sin p r 22 d p p r r0Here, p is an auxiliary variable without physical meaning.1 1 dp iprse 2 2 se re 2p1 ipr ˆ2 pse pˆe dp2 1 1 k s k dre dp p s p er 24Qˆs ˆe Q iQriprf e i ˆ 2 eˆs e The last line follows by doing the r -integration first and using 1 ipQr dr e 3 p Q2Q s esecollect all prefactors:2 2 mn 2 0 2 2 2 2NB 4 r024 15where r 02.810m is the “classical electron radius” that also appeared in theThompson cross section for x-ray scattering.d2 2r0 mf semidFor an unpolarized neutron beam, one has to average over the neutron spinstates m . For convenience, take the neutron spin quantization axis ẑ to beparallel to se. Then m s m s m mf ei ef z iseif mf mi 0 otherwiseThe cross section for a single electron at rest for an unpolarized neutron beamis therefored 2 r20sedwith the projection of the electron spin perpendicular to Q .se2