A classical description of relaxation of interacting pairs of unlike spins: Extension to T1 ϱ, T2, and T1 ϱoff, including contact interactions

Abstract

A novel derivation of the equations that describe the spin-lattice magnetic relaxation of nuclear spin moments, in liquids, resulting from magnetic dipolar interactions with neighboring paramagnetic ions, the Solomon-Bloembergen-Morgan equations was previously presented (S. H. Koenig, J. Magn. Reson. 31, 1 (1978)). The derivation involves a computation of the dissipative energy flow from the nuclear spins to the lattice rather than a computation of the lattice-produced fluctuations of the local field at the nuclear spins. Two advantages accrue: (1) the spectral densities that enter into the relaxation expressions can be directly related to well-defined absorption transitions and relaxation processes of the paramagnetic ions, clarifying the physical processes that produce relaxation, and (2) the derivation can be readily generalized to paramagnetic ions with arbitrary spin Hamiltonian, and to deviations of their susceptibility from Curie law behavior. The derivation is extended to include relaxation in liquids in the rotating frame: the on resonance T 1 ϱ which reduces to T 2 for small amplitude radiofrequency fields; and the off resonance T 1 ϱoff , which reduces to T 1. The results, which are given for contact as well as dipolar interactions, also describe relaxation of 13C and 15N nuclei by protons under conditions of proton-decoupling, a situation becoming increasingly important in the study of biological macromolecules by high-resolution NMR spectroscopy.