Exchange of Interacting Nuclear Spins in Nuclear Magnetic Resonance. II. Chemical Exchange

Abstract

The study of the effect of exchange on high-resolution nuclear magnetic resonance spectra is extended to chemical exchange. A Boltzmann equation for the nuclear spin density matrix is derived. This equation is nonlinear because two uncorrelated molecules participate in each exchange process. For temperatures high compared to the nuclear Zeeman energy the equations can, however, be linearized. In particular the steady-state (slow-passage) solutions can be obtained from a set of linear algebraic equations for the elements of the density matrix. Explicit approximation formulas for the slow- and fast-exchange limits are derived. As an example the spectrum of a molecule with two nonequivalent spin ½ nuclei, one of which is exchanging, is calculated explicitly for arbitrary exchange rates and coupling strengths, and the behavior of the spectrum is discussed in detail.