Theory of Chemical Exchange Effects in Magnetic Resonance

The Anderson—Weiss formalism is used to develop a general theory for calculating the effects of chemical exchange and spin coupling on echo amplitudes in Carr—Purcell pulse sequences on spin systems in the liquid phase. The theory is shown to involve the matrix integral solution of the Hahn—Maxwell—McConnell equations, generalized to include spin coupling, and of Alexander's equations for the relevant density matrix elements, subject to the boundary conditions imposed by the pulses. Some specific systems are treated in detail, and closed formulas are given for the decay of the echo train in several situations, including the coupled and uncoupled AB systems, and the coupled ABX and ABXq systems. We show that if the natural relaxation rates of nuclei in exchanging sites are identical, then in general effects of both exchange and spin coupling can be removed by pulsing sufficiently rapidly. Modulation of the echo train by spin coupling is removed by rapid exchange between the coupled sites. The Carr—Purcell experiment on an equivalent set of spins affords effective decoupling from another equivalent set unaffected by the pulses, in the absence of exchange between the sets. An analogy is drawn between chemical exchange and quadrupolar relaxation, and a method is outlined which in principle should provide an indirect determination of quadrupolar or other fast relaxation rates. We illustrate the applicability of the theory to the calculation of line shapes in steady-state spectra by treating the exchanging coupled AB system in detail, and show that in the fast exchange limit the theory predicts weak broad resonances not previously described.