Relaxation and Dynamic Processes

2 Although multiple pulse and multi-dimensional NMR techniques permit generation of off-diagonal density matrix elements and observation of complex coherence transfer processes, eventually the density operator returns to an equilibrium state in which all coherences (off-diagonal elements of the density operator) have decayed to zero and the populations of the energy levels of the system (diagonal elements of the density operator) have been restored to the Boltzmann distribution. Analogously with similar phenomena in other areas of spectroscopy, the process by which an arbitrary density operator returns to the equilibrium operator is called nuclear magnetic, or spin, relaxation. The following sections will describe the general features of spin relaxation and important consequences of spin relaxation processes for multi-dimensional NMR experiments. In addition, other dynamic processes, such as chemical reactions and conformational exchanges, that transfer nuclei between magnetic environments can affect the NMR experiment; these processes also are discussed. As relaxation is one of the fundamental aspects of magnetic resonance, an extensive literature on theoretical and experimental aspects of relaxation has developed since the earliest days of NMR spectroscopy (see (1) and references therein). At one level, relaxation has important consequences for the NMR experiment: the relaxation rates of single quantum transverse operators determine the linewidths of the resonances detected during the acquisition period of an NMR experiment; the relaxation rates of the longitudinal magnetization and off-diagonal coherences generated by the pulse sequence determine the length of the recycle delay needed between acquisitions; and the relaxation rates of 3 operators of interest during multi-dimensional experiments determine the linewidths of resonances in the indirectly detected dimensions and affect the overall sensitivity of the experiments. At a second level, relaxation affects quantitative measurement and interpretation of NMR experimental parameters, including chemical shifts and scalar coupling constants and At a third level, relaxation provides experimental information on the physical processes governing relaxation, including molecular motions and intramolecular distances. In particular, cross-relaxation gives rise to the nuclear Overhauser effect (NOE) and makes possible the determination of three-dimensional molecular structures by NMR spectroscopy. Additionally, a variety of chemical kinetic processes can be studied through effects manifested in the NMR spectrum; in many cases, such phenomena can be studied while the molecular system remains in chemical equilibrium. Because the theoretical formalism describing relaxation is more complicated mathematically than the product operator formalism, the present treatment will emphasize application of semi-classical relaxation theory to cases of practical interest, rather …