| In studying the structure of a porous medium and in typing of the pore-filling fluid using nuclear magnetic resonance (NMR), one faces the complex problem of spin dynamics in a restricted geometry. Additional complications arise due to the presence of nonuniform-gradient magnetic fields, surface relaxation, and radio-frequency (RF) pulses that vary with the location in the sample. We have developed a general theoretical framework that accounts for all these effects. Our approach connects the Torrey-Bloch equation, the master equation governing the evolution of the nuclear magnetization in fluids, with the Gaussian phase approximation (GPA) for the phases of the diffusing spins. It treats the interaction of the spins with the magnetic field as a scattering problem and naturally incorporates arbitrary pulse sequences by separating their effects from diffusional relaxation. In the short-time and long-time limits, we have derived explicit model-independent expressions for the signal attenuation, valid both in closed geometries, such as an isolated pore or cell, and open geometries, such as a connected porous medium. We discuss applications to internal fields (due to susceptibility differences between the fluid and the surrounding matrix), to grossly inhomogeneous fields (arising in ex-situ NMR as in bore-hole tools), and to the measurement of the time-dependent diffusion coefficient. We also study the breakdown of the GPA for particular magnetic field profiles and the effects of inter-pore connectivity on higher eigenmodes of the diffusion equation to better understand the multi-exponential decay of magnetization in inhomogeneous-field systems. Finally, we examine in detail the interplay of diffusion, bulk relaxation, and pulse rotation angle in the steady-state free precession (SSFP) pulse sequence in which a stream of identical pulses is applied with a uniform-gradient field in the background. |
