Gaussian-approximation formalism for evaluating decay of NMR spin echoes

We present a formalism for evaluating the amplitude of the NMR spin echo and stimulated echo as a function of pulse spacings, for situations in which the nuclear spins experience an effective longitudinal magnetic field ${\mathit{h}}_{\mathit{z}}$(t) resulting from an arbitrary number of independent sources, each characterized by its own arbitrary time correlation function. The distribution of accumulated phase angles for the ensemble of nuclear spins at the time of the echo is approximated as a Gaussian. The development of the formalism is motivated by the need to understand the transverse relaxation of ${}_{}{}^{89}{}_{}{}^{}\mathrm{Y}$ in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$, in which the ${}_{}{}^{89}{}_{}{}^{}\mathrm{Y}$ experiences ${}_{}{}^{63,65}{}_{}{}^{}\mathrm{Cu}$ dipolar fields which fluctuate due to ${}_{}{}^{63,65}{}_{}{}^{}\mathrm{Cu}$ ${\mathit{T}}_1$ processes. The formalism is applied successfully to this example, and to the case of nuclei diffusing in a spatially varying magnetic field. Then we examine a situation in which the approximation fails—the classic problem of chemical exchange in dimethylformamide, where the methyl protons experience a chemical shift which fluctuates between two discrete values. In this case the Gaussian approximation yields a monotonic decay of the echo amplitude with increasing pulse spacing, while the exact solution yields distinct ‘‘beats’’ in the echo height, which we confirm experimentally. In light of this final example the limits of validity of the approximation are discussed. © 1996 The American Physical Society.