A simple mathematical formalism is presented which allows narrow pulses. In particular, it is assumed that not only is the closed form expressions for the echo attenuation, E(q), in spin duration d of the pulses much smaller than their separation echo diffusion experiments, for practically all gradient waveforms D but that the distance diffused during the pulses is small and for the case of restricted diffusion in enclosing pores, with compared with characteristic dimensions of the pore space or without wall relaxation. The method, which derives from the morphology. Because the wave vector amplitude q depends multiple propagator approach of A. Caprihan et al. (1996, J. on the time integral of the pulse, this restriction to narrow Magn. Reson. A 118, 94), depends on the representation of the pulses represents a constraint on the maximum available gradient waveform by a succession of sharp gradient impulses. It scattering wave vector. Furthermore the need to ensure that leads to E(q) being expressed as a product of matrix operators the diffusion distance is small during the pulses further concorresponding quite naturally to the successive sandwich of phase strains the distance scales which can be probed using this evolution and Brownian migration events. Simple expressions are experiment. In consequence it would be helpful to find a given for the case of the finite width gradient pulse PGSE expericonvenient mathematical treatment which is applicable to ment, the CPMG pulse train used in frequency-domain modulated gradient spin echo NMR, and the case of a sinusoidal waveform. the case of finite pulse durations. An additional reason to The finite width gradient pulse PGSE and CPMG pulse trains consider such a treatment concerns applications where the are evaluated for the case of restricted diffusion between parallel applied gradient is constant, for example, as in stray field reflecting planes. The former results agree precisely with published experiments. Consequently a number of authors have recomputer simulations while the latter calculation provides useful cently addressed the issue of q-space diffraction under condiinsight regarding the spectral density approach to impeded tions of finite gradient pulse widths, both in qualitative terms Brownian motion. แญง 1997 Academic Press (14) and by means of computer simulation (6,15). A significant and successful analytic treatment of the finite pulse problem has been demonstrated by Caprihan et al. (16).