Solids and slow motion

When the reorientation is slow compared to the inverse strength of the dipole-dipole interaction, the line-narrowing assumption of the BPP theory is no longer applicable. Explicit motional mechanisms and nuclear spin interactions have to be taken into account in the calculation of the NMR signal decay [Aill, Spil]. 

This expression can be evaluated under two assumptions. The first one is, that the distribution of Q t) is Gaussian. Then s t) is Gaussian as well. The second assumption is that the local field fluctuation can be described by a correlation function (r) of the frequency offset 2, 

In the short-time limit and with an exponential correlation function the FID s(t) is approximated by 

Note that the use of moments derives from the days of CW NMR, when Fourier transformation was not standard, and the information about the FID had to be derived from the spectrum. The moments are defined by an expansion of the exponential in the Fourier transform of the FID, 

For symmetric lineshapes integrals with odd powers in t vanish, and the moments are given by 

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