Computer simulation of the spin-echo spatial distribution in the case of restricted self-diffusion.

This article concerns the question of a proper stochastic treatment of the spin-echo self-diffusion attenuation of confined particles that arises when short gradient pulse approximation fails. Diffusion is numerically simulated as a succession of random steps when motion is restricted between two perfectly reflecting parallel planes. With the magnetic field gradient perpendicular to the plane boundaries, the spatial distribution of the spin-echo signal is calculated from the simulated trajectories. The diffusion propagator approach (Callaghan, "Principles of Nuclear Magnetic Resonance Microscopy," Oxford Univ. Press, Oxford, 1991), which is just the same as the evaluation of the spin-echo attenuation by the method of cumulant expansion in the Gaussian approximation, with Einstein's approximation of the velocity correlation function (VCF) (delta function), agrees with the results of simulation only for the particle displacements that are much smaller than the size of the confinement. A strong deviation from the results of the simulation appears when the bouncing rate from the boundaries increases at intermediate and long gradient sequences. A better fit, at least for intermediate particle displacements, was obtained by replacing the VCF with the Oppenheim--Mazur solution of the Langevin equation (Oppenheim and Mazur, Physica 30, 1833--1845, 1964), which is modified in a way to allow for spatial dependence of particle displacements. Clearly, interplay of the correlation dynamics and the boundary conditions is taking place for large diffusion displacements. However, the deviation at long times demonstrates a deficiency of the Gaussian approximation for the spin echo of diffusion inside entirely closed pores. Here, the cumulants higher than the second one might not be negligible. The results are compared with the experiments on the edge enhancement by magnetic resonance imaging of a pore.

Effect of natural convection in a horizontally oriented cylinder on NMR imaging of the distribution of diffusivity

This paper describes the influence of natural convection on NMR measurement of a self-diffusion constant of fluid in the earth's magnetic field. To get an estimation of the effect, the Lorenz model of natural convection in a horizontally oriented cylinder, heated from below, is derived. Since the Lorenz model of natural convection is derived for the free boundary condition, its validity is of a limited value for the natural no-slip boundary condition. We point out that even a slight temperature gradient can cause significant misinterpretation of measurements. The chaotic nature of convection enhances the apparent self-diffusion constant of the liquid.

MRI edge enhancement as a diffusive discord of spin phase structure.

The enhancement of magnetic resonance image intensity near impermeable boundaries can be nicely described by a new approach where the diffusional spin echo attenuation is linked to the correlation function of molecular motion. In this method the spin phase structure created by the applied gradient is considered to be a composition of plane waves with the wave vectors representing feasible momentum states of a particle in confinement. The enhancement of edges on the magnetic resonance images (MRI) comes out as a discord of plane waves due to particle motion. It results from the average of the wave phase by using the cumulant expansion in the Gaussian approximation. The acquired analytical expression describes the MRI signal space distribution where the enhancement of edges depends on the intensity and the duration of gradient sequence as well as on the length of the mean squared particle displacement in restricted geometry. This new method works well with gradients of general waveform and is, therefore, suitable for imaging sequences where finite or even modulated gradients are usually used.

Self-diffusion imaging by spin echo in Earth's magnetic field.

The NMR of the Earth's magnetic field is used for diffusion-weighted imaging of phantoms. Due to a weak Larmor field, care needs to be taken regarding the use of the usual high field assumption in calculating the effect of the applied inhomogeneous magnetic field. The usual definition of the magnetic field gradient must be replaced by a generalized formula valid when the strength of a nonuniform magnetic field and a Larmor field are comparable (J. Stepisnik, Z. Phys. Chem. 190, 51-62 (1995)). It turns out that the expression for spin echo attenuation is identical to the well-known Torrey formula only when the applied nonuniform field has a proper symmetry. This kind of problem may occur in a strong Larmor field as well as when the slow diffusion rate of particles needs an extremely strong gradient to be applied. The measurements of the geomagnetic field NMR demonstrate the usefulness of the method for diffusion and flow-weighted imaging.