Analysis of Carr—Purcell Spin-Echo NMR Experiments on Multiple-Spin Systems. II. The Effect of Chemical Exchange

Abstract

A general matrix formulation for the treatment of Carr—Purcell spin‐echo (CPSE) nuclear magnetic resonance experiments on systems undergoing chemical exchange between r sites is presented. Indirect spin coupling is ignored. A simple similarity transformation is given, which makes the matrix L (a basic quantity in the theory) Hermitian, and clearly exposes its positive definite character. As a consequence, it is shown that the general solution for the complex transverse magnetization in the rotating frame is, at the nth echo, a sum of r terms of the form [A+(−1) n B] exp (−nt CP R) , where t CP is the interval between 180° pulses, and R is a real positive function of t CP. A computer program calculates all these quantities. In many cases of interest, such as fast exchange, one exponential term dominates in this sum. For fast exchange, a second‐order perturbation treatment can be used to derive an explicit equation for the dominant decay time constant R 1. This equation reduces to the Luz—Meiboom equation in the special case considered by those authors. The second‐order treatment also gives an expression for the steady‐state high‐resolution linewidth. This equation reduces to the Piette—Anderson equation under the same special conditions for which the Luz—Meiboom equation holds. Detailed calculations which illustrate the use of the second‐order equations are exhibited for the cases of the hydride shift in the 2‐norbornyl cation and the degenerate Cope rearrangement in bullvalene. The second‐order equation takes a particularly simple form when applied to the study of a nucleus A coupled to a quadrupolar nucleus X. The result shows that one can extract both the magnitude of the coupling constant J AX and the relaxation timeT X from the CPSE experiment on A, even when the high‐resolution spectrum shows a single line.