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.

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