Reduced Spin Hamiltonian in High-Field NMR Spectroscopy of Isotropic Liquids and Gases
When the spin system under study belongs to a gas or an isotropic fluid, another very important simplification of the spin Hamiltonian arises. In these cases, the fast molecular motions average to zero the anisotropic interactions and in spectroscopy analysis where the energy levels of the total Hamiltonian are being probed, only the isotropic chemical shift and the isotropic indirect spin-spin interaction remain leading to
Selection Rules and Transition Probabilities
In modern NMR spectroscopy analysis the resonance frequencies of the spin system are probed by RF pulse techniques, where an oscillating magnetic induction with a frequency meeting the Bohr condition is applied perpendicular to the static Bo field for short time periods. This gives rise to a time-dependent perturbation term in the spin Hamiltonian amounting to:
where it was considered that Bi = B1ex. This term includes the spin operators Ik,x that only have nonzero expectation values < faIk,xfb > between states [fa > and > differing by
and consequently only the transitions obeying 4.30 are induced by H2. Equation 4.30 is the selection rule for nuclear spin transitions induced by a high-frequency induction field meeting the Bohr condition. The transition probability per unit time induced by the high-frequency field may be obtained from time-dependent perturbation theory and is given by (Abragam, 1961)
where ю, is the Larmor frequency for the spin i. In practice the delta function in 4.31 becomes replaced by a shape function arising from the finite width of the resonance lines. We have gathered all the ingredients required for a first determination of the NMR spectra associated with a given nuclear spin system and experimentally observable either with a pulsed Fourier transform NMR experiment using a nonselective high-frequency pulse or a CW NMR experiment.