N.M.R. solid echoes in systems of dipolar-coupled inequivalent spins-1 or dipolar-coupled inequivalent pairs of spins-1/2

Proton solid-echo transverse relaxation functions for many thermotropic and lyotropic mesophases, mapped by measurement of the echo amplitude S_yx (t′ = τ) as a function of τ using a P_y (90°)-τ-P_x (90°)-t′ sequence, yield gaussian behaviour of the form exp -½M ^E ₂τ²] for decays up to 20–30 per cent of the value at τ = 0. M ^E ₂, the second moment for the dipolar interactions between the spin-½ pairs, is related to the van Vleck second moment M ^VV ₂ through a factor f. Whilst experiments suggested a value of 0·70–0·72 for f = M ^E ₂/M ^VV ₂, simple models that ignored the non-equivalence of the dipolar-coupled spin-pairs had predicted f = 0·65. In this paper we derive an exact analytic expression for the spin response of a model of two dipolar-coupled inequivalent spins-1 to the pulse sequence P_y (90°)-τ-P _α(β)-t′, and show that the present model, with the quenching of the spin-flip terms of the dipolar hamiltonian, resolves the afore-mentioned discrepancy. We also reconcile the differences between the experimental and the earlier predicted values of f for deuteron N.M.R. spin echoes in perdeuterated solids.