The CW method

Frequency and time domains

We have seen in Chapter 1 that the essential features of a resonance phenomenon may be investigated by looking at the system's response to a small sinusoidal disturbance of varying frequency. This is called the continuous wave (CW) method and is the technique traditionally used in most branches of spectroscopy. Alternatively, as we also saw in Chapter 1, we may look at the time response to a transient excitation, this being called the pulse method in NMR.

Results of the two methods have been shown to be equivalent, one response being the Fourier transform of the other. However, if we look at the history of the subject we find that studying the frequency response (CW) was the method used almost exclusively for the first ten years of NMR, i.e. 1946–56. In Andrew's book (Andrew, 1955) for instance 27 pages are devoted to CW experimental methods while pulse methods are treated in a page and a half. Erwin Hahn is usually thought of as the father of pulsed NMR due to his important paper in 1950. Other important contributors include Carr and Purcell (1954) and Torrey (1952).

Since Andrew's book was written, pulsed NMR has to a large extent eclipsed CW NMR. There are two main reasons for this. Firstly, it is much easier to measure the relaxation times T1 and T2 using pulse methods. In fact the long T2 values found in some liquids can only be measured by the pulse technique because magnet inhomogeneity masks the weaker effects of spin–spin coupling. This point will be taken up again in the following chapter.

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.

One of the central questions of science is: how are complex things made from simple things? In many cases larger systems are more complicated than their smaller subsystems. In biology and chemistry the issue is how to understand large molecules in terms of atoms. In atomic physics one may strive to understand properties of many electron systems in terms of single electron properties. The general theme is interdependency of subsystems, or ‘correlation’.

Correlation may be regarded as a conceptual bridge from properties of individuals to properties of groups or families. In atoms and molecules correlation occurs because electrons interact with one another – the electrons are interdependent. This electron correlation determines much of the structure and dynamics of many electron systems, i.e., how complex electronic systems are made from single electrons. Complexity is the more significant idea, but complexity may be seldom, if ever, understood. Correlation is the key to complexity.

Understandably, much has been done on the correlation of static systems. There are many excellent methods and computer codes to evaluate energies and wavefunctions for complex atomic and molecular systems. However, the dynamics of these many electron systems is less well understood. Hence, the dynamics of electron correlation is a central theme in this book.

The dynamics of electron correlation may affect single electron transitions. However, this effect is sometimes difficult to separate from other effects.

This introductory chapter begins with a review of uncorrelated classical probabilities and then extends these concepts to correlated quantum systems. This is done both to establish notation and to provide a basis for those who are not experts to understand material in later chapters.

Probability of a transition

Seldom does one know with certainty what is going to happen on the atomic scale. What can be determined is the probability P that a particular outcome (i.e. atomic transition) will result when many atoms interact with photons, electrons or protons. A transition occurs in an atom when one or more electrons jump from their initial state to a different final state in the atom. The outcome of such an atomic transition is specified by the final state of the atom after the interaction occurs. Since there are usually many atoms in most systems of practical interest, we can usually determine with statistical reliability the rate at which various outcomes (or final states) occur. Thus, although one is unable to predict what will happen to any one atom, one may determine what happens to a large number of atoms.

1 Single particle probability

A simple basic analogy is tossing a coin or dice. Tossing of a coin is analogous to interacting with an atom. In the case of a simple coin there are two outcomes: after the toss one side of the coin (‘heads’) will either occur or it will not occur.