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

will be

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.

Tue Apr 1 08:52:10 EST 1997