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