Effective phase diffusion for spin phase evolution under random nonlinear magnetic field

TL;DR

This paper extends the phase diffusion method to nonlinear magnetic fields in NMR, revealing complex phase evolutions and their effects on signal attenuation, with implications for improved experimental techniques.

Contribution

It introduces a generalized phase diffusion approach for nonlinear gradients, capturing float and shift phases, and provides analytical expressions for phase variance and signal attenuation.

Findings

Signal attenuation transitions from Gaussian to Lorentzian or Mittag Leffler functions over time.

Float phase evolution significantly impacts total phase shift in even-order gradient fields.

Theoretical results are supported by random walk simulations.

Abstract

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.