Spin-Echo Experiment with NMR Spectra only Partially Narrowed by an Insufficiently Fast Motion

Abstract

The spin-echo experiment on a spin system with only partial motional narrowing and an exponential field autocorrelation function is considered. The pattern of the intensity decrease in the echo spectra depends on the ratio τ/τc of the time delay τ in the echo experiment to the correlation time τc of the narrowing motion. With the large ratios (fast motion), the decrease is the same as in the case of extreme narrowing; to obtain undistorted T2 values, the ratio should be at least several units in the single-echo experiment and at least few decades in the multiple-echo experiment. With the small ratios (slow motion), the logarithmic decrease depends non-linearly on τ, and the T2 value found by the linear least-squares adjustment is much longer than that obtained from the extreme narrowing approximation. At very small ratios, the multiple echo yields about 3τc/(ωpτ)2 for T2 as compared with 1/(ωp2rc) obtained from the extreme narrowing approximation; ωp2 is the second moment of the Gaussian line being narrowed. The expression for T2 in the multiple spin echo is similar to that previously found for T2e in the solid multiple spin echo. The echo experiment changes the line shape, which at large τ/τc approaches the Lorentzian one. The case of a multiexponential field autocorrelation function is also briefly considered.