The Theory of NMR

In this chapter we discuss the basic physics underlying NMR. A few fundamental concepts in both classical and quantum mechanics are assumed. Standard texts in these areas should be consulted if additional background is needed.

2.1 NUCLEAR SPIN AND MAGNETIC MOMENT

We know from basic quantum mechanics that angular momentum is always quantized in half-integral or integral multiples of, ħ, where ħ is Planck’s constant divided by 2π. For the electron spin, the multiple (or spin quantum number) is ½, but the value for nuclear spin differs from one nuclide to another as a result of interactions among the protons and neutrons in the nucleus. If we use the symbol I to denote this nuclear spin quantum number (or, more commonly, just nuclear spin), we can write for the maximum observable component of angular momentum

=Iℏ=Ih/2π

We can classify nuclei, then, according to their nuclear spins. There are a number of nuclei that have I = 0 and hence possess no angular momentum. This class of nuclei includes all those that have both an even atomic number and an even mass number; for example, the isotopes 12C, 16O, and 32S. These nuclei, as we shall see, cannot experience magnetic resonance under any circumstances. The spins of a few of the more common nuclei are:

=12:1H,3H,13C,15N,19F,31Ρ

=1:2HD,14N

>1:10B,11B,17O,23Na,27Al,35Cl,59Co

The nuclei that have I > ½ have a nonspherical nuclear charge distribution and hence an electric quadrupole moment Q. We shall consider the effect of the quadrupole moment later. Our present concern is with all nuclei that have I ≠ 0, because each of these possesses a magnetic dipole moment, or a magnetic moment μ. We can think of this moment qualitatively as arising from the spinning motion of a charged particle. This is an oversimplified picture, but it nevertheless gives qualitatively the correct results that (1) nuclei that have a spin have a magnetic moment and (2) the magnetic moment is collinear with the angular momentum vector. We can express these facts by writing.

=γp

(We use the customary boldface type to denote vector quantities.) The constant of proportionality γ is called the magnetogyric ratio and is different for different nuclei, because it reflects nuclear properties not accounted for by the simple picture of a spinning charged particle. (Sometimes γ is termed the gyromagnetic ratio.) While p is a simple multiple of ħ, μ and hence γ are not and must be determined experimentally for each nuclide (usually by an NMR method). Properties of most common nuclides that result from nuclear spin are given in Appendix A.

2.2 THEORETICAL DESCRIPTIONS OF NMR

The theory of NMR can be approached in several ways, each of which has both advantages and disadvantages. We will emphasize several apparently different theoretical strategies in explaining various aspects of NMR. Actually, these approaches are entirely consistent but utilize differing degrees of approximation to give simple pictures and intuitively appealing explanations that are correct within their limits of validity.

Transitions between Stationary State Energy Levels

We have seen that NMR is a quantum phenomenon, and to some extent we can develop a theoretical framework by using the same sort of treatment used for other branches of spectroscopy: We first find the eigenvalues (energy levels) of the quantum mechanical Hamiltonian operator that describes the nuclear spin system and then use time-dependent perturbation theory to predict the probability of transitions among these energy levels. This procedure provides the frequencies and relative intensities that characterize an NMR spectrum. As we see in the following section and in more detail in Chapter 6, this approach works very well in explaining many basic NMR phenomena, and it accounts quantitatively for many spectra illustrated in Chapter 1. However, because this approach is based on the time-independent Schrödinger equation, it cannot account for the time-dependent phenomena that occur as nuclear spins respond to perturbations such as application of short radio frequency pulses.

Classical Mechanical Treatment

By using the time-dependent Schrödinger equation, we can follow the behavior of a nuclear spin in the presence of an applied magnetic field, and we show in Section 2.6 that the results are consistent with an appealing vector picture. Moreover, we shall show that the summation of nuclear magnetic moments over the whole ensemble of molecules that constitutes a real sample leads to a macroscopic magnetization that can be treated according to simple laws of classical mechanics. Many NMR phenomena can be quite well understood in terms of such simple classical treatments of magnetization vectors; indeed, in the first full paper on NMR, Bloch used classical mechanics to describe many important features of the technique. We shall employ this vector picture extensively, particularly in Chapters 2, 9, and 10.

Yet, it is clear that the classical mechanics approach is inadequate simply because it ignores quantum effects. To some extent, these features can be arbitrarily grafted onto a classical picture, but as we shall see, many of the newer and now most important NMR studies cannot be understood this way.

The Density Matrix

Fortunately there is a simple mathematical formalism that gives us the best of the quantum and classical approaches. By recasting the time-dependent Schrödinger equation into a form using a so-called density operator, physicists have long been able to follow the development of a quantum system with time. This formalism preserves all quantum features but permits a natural explanation of time-dependent phenomena, so that effects of phase and coherence can be understood. For NMR it turns out to be rather easy to set up the necessary theory in terms of a matrix—the density matrix—and to use simple matrix manipulations to follow the time course of the nuclear spin system.

Why, then, do we not ignore the preceding approaches and go directly to the density matrix? The answer is that the density matrix method, while conceptually simple, becomes very tedious and gives very little physical insight into processes that occur. We will come back to the density matrix in Chapter 11, but only after we have developed a better physical feeling for what happens in NMR experiments.

Product Operators

In Chapter 11 we shall also introduce the product operator formalism, in which the basic ideas of the density matrix are expressed in a simpler algebraic form that resembles the spin operators characteristic of the steady-state quantum mechanical approach. Although there are some limitations in this method, it is the general approach used to describe modern multidimensional NMR experiments.

2.3 STEADY-STATE QUANTUM MECHANICAL DESCRIPTION

As with other branches of spectroscopy, an explanation of many aspects of NMR requires the use of quantum mechanics. Fortunately, the particular equations needed are simple and can be solved exactly.

The Hamiltonian Operator

We are interested in what happens when a magnetic moment μ, interacts with an applied magnetic field B0—an interaction commonly called the Zeeman interaction. Classically, the energy of this system varies, as illustrated in Fig. 2.1a, with the cosine of the angle between μ. and B0, with the lowest energy when they are aligned. In quantum theory, the Zeeman appears in the Hamiltonian operator H,

=−μ·Β0