Validity limits of Gaussian approximation in cumulant expansion for diffusion attenuation of spin echo

Cumulant expansion is a way to average out the spin frequency #uctuations, caused by molecular random migration in nonuniform magnetic "eld, in order to get spin echo attenuation. Since numerous spins have a share in NMR induction, the cumulative yield of frequency #uctuations features Gaussian randomness. Thus the cumulant expansion can be terminated by the second term giving the spin echo attenuation related to the time-correlation of molecular motion (Stepis\ nik, Physica B 104 (1981) 350) and to the time-space-correlation in the case of restricted self-di!usion. The validity limits of this approximation is tested by considering the convergence of the cumulant series. The estimate of high-order velocity correlations displays that the gap between the magnetization grating (spin-phase structure) caused by applied gradient "eld has to be much larger than the free path of moving spins. With the spins in con"nement, the spin phase structure can be written as composition of plane waves (Stepis\ nik, J. Phys. 39 (1978) 689; Stepis\ nik, J. Magn. Res. 131 (1998) 339). The cumulant expansion in Gaussian approximation gives the spin echo attenuation as a discord of magnetization grating that can exhibit the di!usive di!raction patterns of porous structure (Coy, Callaghan, J. Chem. Phys. 101 (1994) 4599). Advantage of method is its ability to be implemented with any general gradient pulse sequence, i.e., the gradient pulses can violate the short pulse approximation that is required with the propagator method. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 33.25.#f; 76.60.Lz