  Through the use of Heisenberg spin-spin interactions, we provide analytical
representations for inelastic neutron scattering eigenstates and excitation
cross-sections of the general $S_1$-$S_2$ spin dimeric systems. Using an exact
diagonalization approach to the spin Hamiltonian, we analyze various spin
coefficients to provide general representations for the neutron scattering
cross-sections of two interacting spins. We also detail a generalized method
for the determination of $S_z$ polarized excitations, which provide an
approximation for the excitations within an applied $z$-axis magnetic field.
These calculations provide a general understanding of the interactions between
two individual or compound spin systems, which can help provide insight into
condensed matter systems like molecular magnets, quantum dots, and spintronic
systems, as well as particle physics investigations into quark matter and meson
interactions.
