  The general theoretical description of spin self-diffusion under nonlinear
gradient is proposed, which extends the effective phase diffusion method for
linear gradient field. Based on the phase diffusion, the proposed method
reveals the general features of phase evolutions in non-nonlinear gradient
fields. There are three types of phase evolutions: phase diffusion, float phase
evolution, and shift based on the starting position. For spin diffusion near
the origin of the nonlinear field, these three phase evolutions significantly
affect the NMR signal. The traditional methods have difficulties in handling
these three-phase evolutions. Notably, the phase from float phase evolution is
missed or misplaced in traditional methods, which leads to incorrect NMR signal
attenuation or phase shift. The method here shows that the diffusing and float
phase evolutions come from the first and second derivatives of the gradient
field. Based on these three phase evolutions, the phase variance and
corresponding NMR signal attenuation are obtained, demonstrated by calculating
the phase diffusions under the parabolic and cubic fields. The results indicate
that signal attenuation obeys Gaussian attenuation for a short time, then
changes to Lorentzian or Mittag Leffler function attenuations when time
increases, significantly different from Gaussian attenuation. For spins
starting diffusion far away from the origin, the signal attenuation is
Gaussian, but the float phase still has an important effect on the total phase
shift of even-order gradient fields, which could be used to measure the
diffusion coefficient directly. Random walk simulations are performed, which
support the obtained theoretical results. The obtained general theoretical
expressions can handle random order nonlinear gradient field. The results could
help develop advanced experimental techniques for NMR and MRI.
