Kuchel PW, Lennon AJ, Durrant C
Department of Biochemistry, University of Sydney, New South Wales, 2006, Australia
J Magn Reson B. 1996 Jul;112(1):1-17. doi: 10.1006/jmrb.1996.0103.
Nuclear spins (in molecules) are considered to be diffusing in a sphere in a linearly inhomogeneous magnetic field (field gradient) that is imposed during a spin-echo NMR experiment. Relaxation of magnetization both in the bulk medium and on the inner surface of the sphere is assumed to occur. Analytical solutions were obtained for the relevant modified diffusion (partial differential) equation by using separation of variables with a Green's function (propagator) and three different boundary conditions. Neuman [J. Chem. Phys. 60, 4508 (1974)] analyzed the same physical system, but with no relaxation, to obtain an expression that relates the NMR spin-echo signal intensity to the magnitude of the magnetic field gradient, the spin-echo time, and the intrinsic molecular diffusion coefficient. The present analysis was based on that originally used by Neuman and, like the latter, it employed the assumption of a Gaussian distribution of phases of the spin magnetizations. This assumption, while rendering a tractable solution, nevertheless limits its range of applicability; this aspect, and the convergence properties of the series solutions were investigated in conjunction with numerical simulations made with diffusion modeled as a three-dimensional random (Monte Carlo) walk. A novel prediction for spheres with finite surface relaxation and a given radius is the presence of two minima in a graph of the normalized spin-echo signal intensity versus the reciprocal of the diffusion coefficient.
在自旋回波核磁共振实验中施加的线性非均匀磁场（场梯度）中，（分子中的）核自旋被认为在一个球体中扩散。假定在主体介质和球体的内表面都发生磁化弛豫。通过使用格林函数（传播子）的变量分离和三种不同的边界条件，得到了相关修正扩散（偏微分）方程的解析解。诺伊曼[《化学物理杂志》60, 4508 (1974)]分析了相同的物理系统，但没有考虑弛豫，以获得一个将核磁共振自旋回波信号强度与磁场梯度大小、自旋回波时间和分子固有扩散系数联系起来的表达式。目前的分析基于诺伊曼最初使用的方法，并且与后者一样，采用了自旋磁化相位的高斯分布假设。这个假设虽然给出了一个易于处理的解，但限制了其适用范围；结合将扩散建模为三维随机（蒙特卡罗）游走的数值模拟，研究了这一方面以及级数解的收敛性质。对于具有有限表面弛豫和给定半径的球体，一个新的预测是归一化自旋回波信号强度与扩散系数倒数的关系图中存在两个最小值。
