To this point in the formalism only nucleon-nucleon (nn) scattering has been considered. In order to relate this to the nucleon-nucleus (nA) scattering of the experiment it is necessary to introduce the idea of nuclear response functions. The cross section for the nA collision can be expressed in terms of the nn cross section [KMT59] [BeS82]:
where is the free nn cross section, is the effective number of nucleons, which for reactions would be neutrons, that can participate in the reaction. This is also sometimes called the distortion factor [Che93]. The effective number of nucleons will, in general, be less than the full number of nucleons in the nucleus because the projectile nucleon will be strongly attenuated as it travels through the bulk of the nucleus--exactly how much is a function of energy--and therefore a portion of the possible target nucleons will be shadowed.
The Response Functions, , to a given projectile scattering operator, , are defined as [IcK92] [Bro95]:
where is the nuclear wavefunction. The operators, , can be defined as corresponding to the spin operators , where j = 0,n,p,q:
The operators, , are then defined as
where sum is over the number of neutrons, N, and the normalization means that the response is given as the response per neutron. In the limit of no Pauli Blocking ( orthogonal to all ) [Che93] [BeS82]
Just as the total cross section for nn scattering can be defined as the sum of the partial cross sections defined in equations to
one can use the response functions to define the nA scattering cross section. As can be inferred from equation the nA cross section is given by
where the 's are still the partial nn cross sections.