In order to factor out the expression for total nA cross section into the product of several independent terms such as the nn cross section, response function, and other corrections (eqn. ) the assumption must be made that the nn amplitudes depend only on the incident energy and the momentum transfer, q. However, in general, the stuck nucleon will have its own momentum, , due to Fermi motion which enters the problem and can break down that simple factorization. Instead of integrating over the struck nucleon's Fermi momentum the process becomes much simpler if the nn amplitudes can be factored out by evaluating them in some ``optimal'' frame where the struck nucleon has a constant value, . This procedure has been discussed in [Gur86], [Smi88], and in detail in [IcK92]. For inelastic scattering, in the non-relativistic case, the optimal momentum was worked out to be [Gur86]
Notice, the optimal momentum will be zero for free scattering where .
The scattering matrix in the optimal frame, , is related to the nn scattering matrix, M (eqn. ) by [Che93]
where is the incident proton energy, is the Möller (kinematic) factor, and is the matrix which performs the frame rotation as defined in [IcK92]. The new scattering matrix can be written as
with the spin-dependent pieces being defined
where the superscript 0 indicates the incoming proton and the superscript i denotes a sum over A nucleons. The amplitudes (, , etc.) are related to those in equation by equation . From the equations above it can be seen that () does not excite the purely longitudinal(transverse) mode in the optimal frame. Therefore, and are no longer related directly to the spin-longitudinal and spin-transverse nuclear response functions. Instead, the nA center-of-mass partial cross sections, 's, given by
where is a kinematic factor defined in [Che93] and is the distortion factor.
Obviously the choice of the optimal reference frame somewhat distorts the simple relationship between the center-of-mass spin observables, 's, and the nuclear response functions. However, it is the only way to deal with amplitudes consistently over a range of energy transfers, such as in the quasifree region, without violating energy conservation in each nn (nucleon-nucleon) interaction.