After all the selection criteria have been applied to the data the neutron polarizations can be extracted and then used to calculate the polarization observables. The normally polarized beam is used to extract the observables , P, and . The polarization transfer coefficients , , , are obtained from data gathered with the sideways and longitudinally polarized beams.
Analyzing power, , can be determined without finding the neutron polarization. From equation it is obvious that to determine the analyzing power it is only necessary to determine the angular dependence of the cross-section. Therefore if is the total counts in the detector when the proton polarization is in its (+) state and, similarly, the counts during with a (-) polarization state then
is the average proton polarization, and is the difference in the magnitudes of the (+) and (-) proton polarization states. Because is, for all intents and purposes, zero the term in parentheses in equation will not contribute significantly.
The polarization of the sideways and longitudinally polarized neutrons is obtained from the detector asymetry by
Then the observables obtained from the polarized proton beam are
Those obtained from the polarized proton beam are
For the and type proton beams the incident polarization is taken to be negative. This is a result of the coordinate systems used. The coordinate system of the neutrons has pointing downward as a result of the conventional definition
This is illustrated in figure . However the convention for the proton beam at IUCF is different with (+), or normal, being defined as upwards. The longitudinal direction is the same for both neutrons and protons -- along the particle flux.
For the observables determined from type beam, aside from , it is necessary to define new variables dependent on the proton spin state, (+) or (-).
where the first superscript refers to the proton beam state and the second denotes a right-handed (+) or left-handed (-) precession in the neutron solenoid. The polarization for proton spin (+) and (-) will be
respectively. The average and differences in the polarization are then given by
respectively. The induced polarization, P, and the polarization transfer coefficient can be defined in terms of average polarization values and differences between the (+) and (-) gated polarizations . An expansion in terms of the polarization differences gives, to first order [Tad85]:
where the primed quantities are, as before, the outgoing neutron polarizations and the unprimed are the incident proton polarizations. As before assumed to be negligible.