Nuclear Spin Hamiltonian for NMR of Anisotropic Fluid Systems
The relevant nuclear spin Hamiltonian for NMR of anisotropic fluid systems includes beyond the Zeeman term Hz the four terms introduced in the previous chapter, Section 4.4; these terms are the chemical shift Ha, the indirect spin-spin coupling Hj , the direct dipolar interaction HD, and the quadrupolar interaction Hq, present for spins with I > 1. As discussed in the previous chapter, Section 4.4.1, in high-field NMR spectroscopy, the energy levels of the nuclear spin Hamiltonian can be determined in first- order perturbation theory, considering the Zeeman term Hz as the unperturbed Hamiltonian Ho = Hz and the remaining terms as the perturbation H1 = Ha + Hj + HD + Hq. Due to the rapid molecular reorientation and diffusional motions occurring in anisotropic fluid systems the contribution of H1 for the system's Hamiltonian is actually replaced by a time average of this quantity Hu As discussed in Section 4.4.4 the molecular motions decouple the spins from different molecules in terms of energy levels of the Hamiltonian and due to the 1/r3 dependence of the Hamiltonian terms involving the interaction of two spins r apart, it is frequently possible to separate the spins of a molecule in several groups of few interacting spins each and analyze each group separately. The total Hamiltonian is the sum of the Hamiltonians from each spin group in the molecules with the different molecules equivalent to each other. From Eq. 4.27 the perturbing Hamiltonian terms arising from a group of n spins is
Al these terms have a similar form, they are constituted by the product of spin operators by the zz component or the trace of phenomenological second rank tensors related to system properties, these tensor components are
The time averages of the zz components of the different second rank tensorial quantities associated with the specific molecular properties contain information on molecular structure and orientational order and can be given in terms of the principal values of the corresponding second rank tensorial quantities as follows
The quantities Vk,xx,, Vk,yy, Vk,are the principal values of the tensor Vk, the primed frame [x', y', z/] is the principal frame of that tensor and the angles в and ф are the polar and azimuthal angles defining the orientation of the B0 field in the principal frame. Introducing the asymmetry parameter n for the tensor Vk,
and considering that the tensor Vk is traceless, Vk,zz can be written in a more compact form as
A corresponding development applies to the other tensorial quantities in Eq. 5.1. All those quantities can also be related to a set of orientational order parameters characterizing the anisotropic fluid.
The two terms and Jki are scalars and carry information on the molecular structure, they are the only nonzero terms in isotropic systems and are discussed in 4.4.2. The quantities listed in Eq. 5.2 can in principle be obtained from the energy spectrum of the spin Hamiltonian.