Attenuation of the NMR signal in a field gradient due to stochastic dynamics with memory

Abstract

The attenuation function S(t) for an ensemble of spins in a magnetic-field gradient is calculated by accumulation of the phase shifts in the rotating frame resulting from the displacements of spin-bearing particles. The found S(t), expressed through the particle mean square displacement, is applicable for any kind of stationary stochastic motion of spins, including their non-markovian dynamics with memory. The known expressions valid for normal and anomalous diffusion are obtained as special cases in the long time approximation. The method is also applicable to the NMR pulse sequences based on the refocusing principle. This is demonstrated by describing the Hahn spin echo experiment. The attenuation of the NMR signal is also evaluated providing that the random motion of particle is modeled by the generalized Langevin equation with the memory kernel exponentially decaying in time.

Fractional Langevin Equation Model for Characterization of Anomalous Brownian Motion from NMR Signals

“…(42) [10]. The limit of (3) for a free particle was used in [12] to calculate the NMR signals from human neuronal tissues, however, with an incorrect spin-echo attenuation S (t) (see [3]). The basic formulas used to calculate S (t) are presented in the following section.…”

“…The magnetization of spins is modulated by a field gradient g(t). General formulas for the attenuation function S (t) of the observed NMR signal that are applicable to any kind of stochastic motion of spins, including their dynamics with memory, have been obtained in [3,7]. In the evaluation of S (t) it was only assumed that the studied random processes are stationary in the sense that the autocorrelation function of a fluctuating dynamical variable x at times t and t depends only on the time difference, t − t :…”

“…It seems natural to apply NMR methods to follow such unusual BM. However, as discussed in [3][4][5], the mathematical description of suitable NMR experiments in the literature is valid only for long times when the particles are in diffusion regime (normal or anomalous). The only exception is the memoryless Langevin model for which correct interpretations of the NMR experiments have been proposed [6].…”

“…The aim of our recent papers [3][4][5]7] was to overcome this limitation and to calculate the all-time attenuation functions of the NMR signals from spin-bearing particles resulting from the BM with memory. We have considered the Brownian particles in Maxwell fluids [3,7] and the BM with hydrodynamic memory [4]. Here we focus on a popular model of fractional BM [8][9][10][11].…”

“…The results for the mean square displacement (MSD) of the particles are then applied to calculate the attenuation function S (t) for an ensemble of spins in a magnetic-field gradient. S (t) is used in the form obtained by the method of accumulation of phases in the frame rotating with the resonance frequency [3] and applicable for any time and kind of the stochastic motion od spins. Although our work was mainly aimed to contribute to adequate interpretations of the NMR experiments on the BM in complex fluids, in the limit of free particles coupled to a fractal heat bath our results correct the description and give favorable comparison with experiments acquired in human neuronal tissues [12].…”