1.1 Nuclear Magnetism

Nuclear magnetic resonances in bulk condensed phase were reported for the first time in 1946 by Bloch et al. (6) and by Purcell et al. (7). Nuclear magnetism and NMR spectroscopy are manifestations of nuclear spin angular momentum. Consequently, the theory of NMR spectroscopy is largely the quantum mechanics of nuclear spin angular momentum, an intrinsically quantum mechanical property that does not have a classical analog. The physical origins of the nuclear spin angular momentum are complex, but have been discussed in review articles (8, 9). The spin angular momentum is characterized by the nuclear spin quantum number, I. Although NMR spectroscopy takes the nuclear spin as a given quantity, certain systematic features can be noted: (i) nuclei with odd mass numbers have half-integral spin quantum numbers, (ii) nuclei with an even mass number and an even atomic number have spin quantum numbers equal to zero, and (iii) nuclei with an even mass number and an odd atomic number have integral spin quantum numbers. Because the NMR phenomenon relies on the existence of nuclear spin, nuclei belonging to category (ii) are NMR inactive. Nuclei with spin quantum numbers greater than 1/2 also possess electric quadrupole moments arising from nonspherical nuclear charge distributions. The lifetimes of the magnetic states for quadrupolar nuclei in solution normally are much shorter than are the lifetimes for nuclei with I = 1/2. NMR resonance lines for quadrupolar nuclei are correspondingly broad and can be more difficult to study. Relevant properties of nuclei commonly found in biomolecules are summarized in Table 1.1. For NMR spectroscopy of biomolecules, the most important nuclei with I = 1/2 are 1H, 13C, 15N, 19F, and 31P; the most important nucleus with I = 1 is the deuteron (2H).

The nuclear spin angular momentum, I, is a vector quantity with magnitude given by

[1.1]

in which I is the nuclear spin angular momentum quantum number and ħ is Planck’s constant divided by 2π. Due to the restrictions of quantum mechanics, only one of the three Cartesian components of I can be specified simultaneously with I2 ≡ I • I. By convention, the value of the z-component of I is specified by the following equation:

[1.2]

in which the magnetic quantum number m = (−I, −I + 1, …, I −1, I). Thus, Iz has 2I + 1 possible values. The orientation of the spin angular momentum vector in space is quantized, because the magnitude of the vector is constant and the z-component has a set of discrete possible values. In the absence of external fields, the quantum states corresponding to the 2I + 1 values of m have the same energy, and the spin angular momentum vector does not have a preferred orientation.

Nuclei that have nonzero spin angular momentum also possess nuclear magnetic moments. As a consequence of the Wigner—Eckart theorem (10), the nuclear magnetic moment, μ, is collinear with the vector representing the nuclear spin angular momentum vector and is defined by

[1.3]

in which the magnetogyric ratio, γ, is a characteristic constant for a given nucleus (Table 1.1). Because angular momentum is a quantized property, so is the nuclear magnetic moment. The magnitude of γ, in part, determines the receptivity of a nucleus in NMR spectroscopy. In the presence of an external magnetic field, the spin states of the nucleus have energies given by

[1.4]

in which B is the magnetic field vector. The minimum energy is obtained when the projection of μ onto B is maximized. Because |I| > Iz, μ cannot be collinear with B and the m spin states become quantized with energies proportional to their projection onto B. In an NMR spectrometer, the static external magnetic field is directed by convention along the z-axis of the laboratory coordinate system. For this geometry, [1.4] reduces to

[1.5]

in which B0 is the static magnetic field strength. In the presence of a static magnetic field, the projections of the angular momentum of the nuclei onto the z-axis of the laboratory frame results in 2I + 1 equally spaced energy levels, which are known as the Zeeman levels. The quantization of Iz is illustrated by Fig. 1.1.

At equilibrium, the different energy states are unequally populated because lower energy orientations of the magnetic dipole vector are more probable. The relative population of a state is given by the Boltzmann distribution,

[1.6]

in which Nm is the number of nuclei in the mth state and N is the total number of spins, T is the absolute temperature, and kB is the Boltzmann constant. The last two lines of [1.6] are obtained by expanding the exponential functions to first order using Taylor series, because at temperatures relevant for solution NMR spectroscopy, mħγB0/kBT « 1. The populations of the states depend both on the nucleus type and on the applied field strength. As the external field strength increases, the energy differences between the nuclear spin energy levels become larger and the population differences between the states increase. Of course, polarization of the spin system to generate a population difference between spin states does not occur instantaneously upon application of the magnetic field; instead, the polarization, or magnetization, develops with a characteristic rate constant, called the spin-lattice relaxation rate constant (see Chapter 5).

The bulk magnetic moment, M, and the bulk angular momentum, J, of a macroscopic sample are given by the vector sum of the corresponding quantities for individual nuclei, μ and I. At thermal equilibrium, the transverse components (e.g., the x- or y-components) of μ and I for different nuclei in the sample are uncorrelated and sum to zero. The small population differences between energy levels give rise to a bulk magnetization of the sample parallel (longitudinal) to the static magnetic field, M = M0k, in which k is the unit vector in the z-direction.

Using [1.2], [1.3], and [1.6], M0 is given by

[1.7]

By analogy with other areas of spectroscopy, transitions between Zeeman levels can be stimulated by applied electromagnetic radiation. The selection rule governing magnetic dipole...