The M-matrix, or scattering matrix, gives all the necessary information about the nucleon-nucleon interaction. In order to define the form of the M-matrix first one must choose the frame of reference for the reaction. The nucleon-nucleon (nn) center-of-mass frame is the most convenient. In terms of the ingoing and outgoing momenta we can define the nn center-of-mass coordinates:
Invariance arguments (i.e. parity, time reversal, rotation, ) reduce the possible number of independent terms in the M-matrix to five [Wol56]. In the nn center-of-mass coordinates the M-matrix has the standard form [KMT59]:
where and are the Pauli spin matrices for the incident and target nucleons, respectively. The scattering coefficients A, B, C, E, and F are functions of energy, momentum transfer , and isospin. In fact, for a given coefficient
where and are the isospin of the incident and target nucleons, respectively, and and are linear combinations of the coefficients of the isospin triplet and singlet states. For the charge-exchange reaction only the will contribute, and the coefficients in equation (e.g. A) will be understood to refer to the isospin = 1 piece (e.g. ).