  Pulsed field gradient (PFG) has been increasingly employed to study anomalous
diffusions in Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging
(MRI). However, the analysis of PFG anomalous diffusion is complicated. In this
paper, a fractal derivative model based modified Gaussian phase distribution
method is proposed to describe PFG anomalous diffusion. By using the phase
distribution obtained from the effective phase shift diffusion method based on
fractal derivatives, and employing some of the traditional Gaussian phase
distribution approximation techniques, a general signal attenuation expression
for free fractional diffusion is derived. This expression describes a stretched
exponential function based attenuation, which is distinct from both the
exponential attenuation for normal diffusion obtained from conventional
Gaussian phase distribution approximation, and the Mittag-Leffler function
based attenuation for anomalous diffusion obtained from fractional derivative.
The obtained signal attenuation expression can analyze the finite gradient
pulse width (FGPW) effect. Additionally, it can generally be applied to all
three types of PFG fractional diffusions classified based on time derivative
order alpha and space derivative order beta. These three types of fractional
diffusions include time-fractional diffusion, space-fractional diffusion, and
general fractional diffusion. The results in this paper are consistent with
reported results based on effective phase shift diffusion equation method and
instantaneous signal attenuation method. This method provides a new, convenient
approximation formalism for analyzing PFG anomalous diffusion experiments.
