Last post on optical model, we did not include the spin. to introduced the spin, we just have to modify the wave function. For spin-½ case.
where the M is a matrix:
the f is for the spin-Independence part of the wave function. For the incident wave and the scattered plane wave.
where and are the angle of spin . not the detector angle.
after calculation by routine algebra, we have the scattered spherical wave.
The expectation of the wavefunction, or the intensity of the spherical part will be:
the beam polarization should be equal the intensity and normalized polarization.
Thus, we have the induced polarization when incident beam is unpolarized:
for a beam of many particle and formed an ensemble, the is the average.
and Analyzing power, which is a short term for Polarization Analyzing Power , or the spin asymmetry, is given by
Therefore, in order to get the spin asymmetry, we have to use 2 polarized beams, one is up-polarized, and another is down-polarized, to see the different between the scattering result.
However, to have 100% polarized beam is a luxury. in most cases, we only have certain polarization. thus, the spin-asymmetry is not equal to the analyzing power. the spin-asymmetry is from the yield measurement.
since f and g only depend on the detector angle. and we can assume they are symmetry. Thus
the P is the polarization of the target.