In terms of the nn center-of-mass coordinates defined in equation we can rewrite equation as
The polarization observables defined by equation can not be easily related to the separate tensor components of the scattering matrix, . Therefore, it is convenient to define a new set of spin observables each of which depend on individual components of the scattering matrix [BBW82].
The new set of observables can be expressed as linear combinations of the polarization transfer observables defined above:
These observables are not independent quantities and are subject to the constraint
and are all non-negative: , where . Using the definition of the polarization transfer coefficients (eqn. ) and the standard expression for the scattering matrix (eqn. ) one can derive expressions for the nn partial differential cross sections which relate the spin observables, , to the amplitudes of the M-matrix [IcK92]:
where I is the unpolarized differential cross section in the center-of-mass frame. By inspection one can see that is related solely to the piece of the interaction in the direction, or the piece in which the spin is parallel to the direction of the momentum transfer (referred to as the longitudinal direction). is related exclusively to a piece of the interaction in the spin-transverse direction. is also dependent on a spin-transverse part of the reaction but not as simply as is the case with . The result is that a comparison of the spin-longitudinal and spin-transverse parts of the nn interaction will come down to a comparison of and .