Transverse relaxation: moments

Relaxation in liquids

The previous chapter was devoted to a consideration of solid systems, where the spin magnetic moments were immobile, located at fixed positions. We examined the transverse relaxation which occurred in such cases and we saw that the relaxation profile could become quite complex. In the present chapter we turn our attention to fluid systems. Here the spins are moving and as a general rule the relaxation profile becomes much simpler. Our fundamental task will be to examine the effect of the particle motion upon the relaxation.

There is an important distinction, from the NMR point of view, between liquids and gases. In a gas the atoms or molecules spend the majority of their time moving freely, only relatively occasionally colliding with other particles. The atoms of a liquid, however, are constantly being buffeted by their neighbours. This distinction is relevant for relaxation mediated by interparticle interactions; clearly their effects will be much attenuated in gases. On the other hand, when considering relaxation resulting from interactions with an inhomogeneous external magnetic field, interparticle collisions are unimportant except insofar as they influence the diffusion coefficient of the fluid. However, since diffusion can be quite rapid in gases, some motional averaging of the inhomogeneity of the NMR magnet's Bo field can then occur. As a consequence, the usual spin-echo technique will no longer recover the transverse magnetisation lost due to the imperfect magnet.

In molecules there are two types of interparticle interaction which must be considered. Interactions between nuclear spins in the same molecule are averaged away relatively inefficiently by molecular motion.

Basic principles

Spatial encoding

Since the NMR resonance frequency of a spin is proportional to the magnetic field it experiences, it follows that in a spatially varying field spins at different positions will resonate at different frequencies. The spectrum from such a system will give an indication of the number of spins experiencing the different fields.

In a uniform magnetic field gradient the precession frequency is directly proportional to displacement in the direction of the gradient; there is a direct linear mapping from the spatial co-ordinates to frequency. Thus the absorption spectrum yields the number of spins in ‘slices’ perpendicular to the gradient. In Figure 10.1 we show how the spectrum would be built up from such slices.

We have already encountered the concept of spatial encoding of spins in Section 4.5 where we considered diffusion and the way it can be measured using spin echoes. There the important point was that whereas spins in a field gradient with their corresponding spread of precession frequencies suffer a decay of transverse magnetisation, this can be recovered, to a large extent, by the time-reversing effect of a 180° pulse. However, if the particles are defusing then, because of the field gradient, their motion will take them to regions of differing precession frequencies. The resultant additional dephasing cannot be recovered by a 180° pulse, which thus permits the diffusion coefficient to be measured.

In imaging one is concerned with the main dephasing effect of the gradient field. Compared with this the diffusive effects are small, and in our initial treatment we shall assume that the resonating spins are immobile.

Calcium fluoride lineshape

Why calcium fluoride?

Although the calculation of the NMR absorption lineshape in a solid is a well-defined problem, the discussions of Chapter 6 have indicated that a complete and general solution is difficult and indeed unlikely. From the practical point of view one would like to explain/understand the characteristic features of transverse decays and lineshape as reflecting details of internal structure and interactions, while from the theoretical point of view the interest is the possibility of treating a ‘relatively’ simple many-body dissipative system.

In this respect it is worthwhile to consider the solvable models considered in Chapter 6. The model of Section 6.2 actually arose from discussions in the seminal paper on calcium fluoride by Lowe and Norberg in 1957. The essential point was that the time evolution generated by the dipole interaction is complicated because of non-commutation of the various spin operators. In the solvable models there is only an IzIz part of the interspin interaction. Since the Zeeman interaction also involves only Iz this means that all operators commute and the time evolution can be calculated purely classically. It is only in this case that the evolution of each spin can be factored giving a separate and independent contribution from every other spin. In other words it is only in this case that each spin can be regarded as precessing in its own static local field – and the problem is solved trivially. Each many-body eigenstate is simply a product of single-particle eigenstates. However, once transverse components of the interspin interaction are admitted then everything becomes coupled together and a simple solution is no longer possible. The many-body eigenstates no longer factorise.

Transverse relaxation: rigid lattice lineshape

Introduction

The calculation (or at least the attempt at calculation) of the dipolar-broadened NMR absorption lineshape in solids has been one of the classical problems in the theory of magnetic resonance. Of course, the lineshape is the Fourier transform of the free precession decay so that a calculation of one is equivalent, formally, to a calculation of the other.

The method for performing such calculations was pioneered by Waller in 1932 and Van Vleck in 1948. However, to the present date no fully satisfactory solution has been found, despite the vast number of publications on the subject and the variety of mathematical techniques used. Nor is there likely to be. General expressions for transverse relaxation were given in the previous chapter. Restriction to a rigid lattice solid: the absence of a motion Hamiltonian, results in a considerable simplification of the equations, as we shall see. Nevertheless it is still a many-body problem of considerable complexity.

The various attempts at solving the problem of the transverse relaxation profile in solids have usually been based on the use of certain approximation methods whose validity is justified a posteriori by the success (or otherwise) of their results. We shall be examining some of these; none is really satisfactory. Conversely, and it may come as a surprise to discover, the more complicated case of a fluid system often permits approximations to be made which are well justified and with such approximations the resulting equations may be solved. This will be treated in the following chapter, although we have had a foretaste of this in Chapter 4.

Man's use of materials, both as a craft and more recently as a science, depends on his ability to produce a particular microstructure with desirable properties in the material when it has been fabricated into a useful object. Such a microstructure occurs, for example, in a steel crankshaft heat-treated for maximum strength, a glass lens heat-treated for fracture resistance or a small crystal of silicon containing non-uniform distributions of solute acting as a complex electronic circuit. Such microstructures are almost always thermodynamically unstable. This situation arises since for any alloy there is only one completely stable structure and there is an infinite number of possible unstable microstructures. The one with the best properties is therefore almost always one of the unstable ones. The desired structure is usually produced by some combination of heat-treatment, solute diffusion and deformation, in the course of which the transformation is arrested, normally by cooling to room temperature, at the right time to obtain the optimum structure. The success of these processes, many of which were derived from craft skills, to give materials with good strength, toughness, electrical properties, etc., is an essential part of current technology. There is, however, a price to be paid in that all these structures are potentially unstable, so that the structures can, and frequently do, transform with time into less desirable forms, especially if used at elevated temperatures.