Energy Spectrum of the Nuclear Spin Hamiltonian in High-Field NMR Spectroscopy

The determination of the energy spectrum of the nuclear spin Hamiltonian requires one to find the eigenvalues and eigenvectors of the spin Hamiltonian. In solids this may be an impossible task to carry out by direct methods due to the prohibitive number of interacting spins, and approximation methods become the only option. In soft matter particularly liquids and also gases the molecules undergo rapid rotational and translational diffusion motions; the perturbing Hamiltonian H₁ = H_a + Hj + H_D + Hq observed with an NMR experiment is the result of an average over these fast motions and H₁ is replaced in these systems by H₁

(Emsley, 1983). Two terms in Hi₁ include couplings between pairs of spins, Hij and Hi_D, one consequence of the averaging process resulting from the fast motions is that the interactions between two spins belonging to two different molecules are averaged to zero. Each molecule is still a complicated spin system, but Hij and Hi_D are inversely proportional to r cube, where r is the distance between two interacting spins, this fact enables one to consider only small groups of interacting spins in each molecule. Each molecule will have in general several of these groups of interacting spins with the identical molecules equivalent to each other. The study of the energy spectrum of the Hamiltonian for a N spin system in soft matter systems is performed by studying the Hamiltonian for each one of the n subsystems of interacting spins in each molecule disregarding the interactions between the n subsystems (BOS et al., 1980; HSI et al., 1978). The total Hamiltonian is a sum of the Hamiltonians for each subsystem. The interactions between the n subsystems themselves and the subsystems and the lattice can be neglected in calculating the energy spectrum of the Hamiltonian because they only contribute by broadening the energy levels and hence the resonance lines. On the other hand these neglected interactions are responsible for bringing the spin system to equilibrium with the lattice at a certain temperature T (Abragam, 1961; Slichter, 1992). In the study of relaxation processes, these interactions play a dominant role.