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Anomalous diffusion exists widely in polymer and biological systems. Pulsedfield gradient (PFG) techniques have been increasingly used to study anomalousdiffusion in NMR and MRI. However, the interpretation of PFG anomalousdiffusion is complicated. Moreover, there is not an exact signal attenuationexpression based on fractional derivatives for PFG anomalous diffusion, whichincludes the finite gradient pulse width effect. In this paper, a new method, aMainardi-Luchko-Pagnini (MLP) phase distribution approximation, is proposed todescribe PFG fractional diffusion. MLP phase distribution is a non-Gaussianphase distribution. From the fractional diffusion equation based on fractionalderivatives in both real space and phase space, the obtained probabilitydistribution function is a MLP distribution. The MLP distribution leads to aMittag-Leffler function based PFG signal attenuation rather than theexponential or stretched exponential attenuation that is obtained from aGaussian phase distribution (GPD) under a short gradient pulse approximation.The MLP phase distribution approximation is employed to get a complete signalattenuation expression E{\alpha}(-Dfb*{\alpha},\b{eta}) that includes thefinite gradient pulse width effect for all three types of PFG fractionaldiffusion. The results obtained in this study are in agreement with the resultsfrom the literature. These results provide a new, convenient approximationformalism to interpret complex PFG fractional diffusion experiments.

Analyzing signal attenuation in PFG anomalous diffusion via a non-Gaussian phase distribution approximation approach by fractional derivatives