N.M.R. dipolar echoes in solids containing spin-1/2 pairs

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Abstract
The proton echo responses to resonant 90°-τ-β90° (XY) and 90-τ-β0° (XX) pulse sequences in powdered crystalline hydrates are reported. The echo produced by the XY sequence consists of two components: one is proportional to sin2 β and the other to sin2 β cos2 β; the former component decays much faster than the latter on increasing the pulse spacing τ. In contrast, the XX sequence produces a single component echo of the form - sin2 β cosβ. The maximum echo amplitudes for the sequences 90°-τ-90°90° and 90°-τ-54°44′0° exhibit a gaussian dependence on τ2 over at least 95 per cent of their decays. The decay constant for the 90-τ-90°90° echo corresponds to M 2(inter) = 5/6 M 2 vv(inter), where M 2 vv(inter) is the interpair second moment calculated by the van Vleck procedure. These observations can be explained in terms of a simple model consisting of a planar arrangement of two spin-1/2 pairs provided the interpair dipolar hamiltonian is truncated so that [ℋo a(intra) + ℋo, t d(inter), ℋz] = 0 and [ℋo a(intra), ℋo, t d(inter)] = 0. The model predicts the echo behaviour only if the spin-1 character of the eigenfunctions of ℋo d(intra) + ℋz is preserved in the presence of the interpair interactions. It is shown that the XX echo and the sin2 β cos2 β components of the XY sequence originate solely in the interpair interactions and contain no contributions from the intrapair interactions. The decay of the maximum echo amplitude with increasing τ is caused by the incomplete refocusing of the interpair interactions by the XX and XY sequences; the correct decay is only determined provided the interpair dipolar hamiltonian is correctly truncated. The model also accounts for the proton echo behaviour in solid hydrogen reported by Metzger and Gaines. Interestingly, the N.M.R. behaviour observed for these spin-1/2 pair systems is largely determined by the eigenfunctions of the spin Hamiltonian with Mz = 0.