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
where
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
where
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
and
respectively. The average and differences in the polarization are then given by
and
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]:
and
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.
