1
Introduction

Structure determination of almost any organic or biological molecule, as well as that of many inorganic molecules, begins with nuclear magnetic resonance () spectroscopy. During its existence of more than half a century, NMR spectroscopy has undergone several internal revolutions, repeatedly redefining itself as an increasingly complex and effective structural tool. Aside from X-ray crystallography, which can uncover the complete molecular structure of some pure crystalline materials, NMR spectroscopy is the chemist's most direct and general tool for identifying the structure of both pure compounds and mixtures, as either solids or liquids. The process often involves performing several NMR experiments to deduce the molecular structure from the magnetic properties of the atomic nuclei and the surrounding electrons.

1.1 Magnetic Properties of Nuclei

The simplest atom, hydrogen, is found in almost all organic compounds and is composed of a single proton and a single electron. The hydrogen atom is denoted as 1H, in which the superscript signifies the sum of the atom's protons and neutrons, that is, the atomic mass of the element. For the purpose of NMR, the key aspect of the hydrogen nucleus is its angular momentum properties, which resemble those of a classical spinning particle. Because the spinning hydrogen nucleus is positively charged, it generates a magnetic field and possesses a magnetic moment µ, just as a charge moving in a circle creates a magnetic field (Figure 1.1). The magnetic moment µ is a vector, because it has both magnitude and direction, as defined by its axis of spin in the figure. In this context, boldface symbols connote a vectorial parameter; when only the magnitude is under consideration, the symbol is depicted without boldface, as µ. The NMR experiment exploits the magnetic properties of nuclei to provide information on the molecular structure.

Figure 1.1Analogy between a charge moving in a circle and a spinning nucleus.

The spin properties of protons and neutrons in the nuclei of heavier elements combine to define the overall spin of the nucleus. When both the atomic number (the number of protons) and the atomic mass (the sum of the protons and neutrons) are even, the nucleus has no magnetic properties, as signified by a zero value of its spin quantum number, I (Figure 1.2). Such nuclei are considered not to be spinning. Common nonmagnetic (nonspinning) nuclei are carbon (12C) and oxygen (16O), which therefore are invisible to the NMR experiment. When either the atomic number or the atomic mass is odd, or when both are odd, the nucleus has magnetic properties that correspond to spin. For spinning nuclei, the spin quantum number can take on only certain values, which is to say that it is quantized. Those nuclei with a spherical shape have a spin I of ½, and those with a nonspherical, or quadrupolar, shape have a spin of 1 or more (in increments of ½).

Figure 1.2 Three classes of nuclei.

Common nuclei with a spin of ½ include 1H, 13C, 15N, 19F, 29Si, and 31P. Thus, many of the most common elements found in organic molecules (H, C, N, P) have at least one isotope with I = ½ (although oxygen does not). The class of nuclei with I = ½ is the most easily examined by the NMR experiment. Quadrupolar nuclei (I > ½) include 2H, 11B, 14N, 17O, 33S, and 35Cl.

The magnitude of the magnetic moment produced by a spinning nucleus varies from atom to atom in accordance with the equation µ = ??I (see Appendix A for a derivation of this equation). The quantity ? is Planck's constant h divided by 2p, and ? is a characteristic of the nucleus called the gyromagnetic or the magnetogyric ratio. The larger the gyromagnetic ratio, the larger is the magnetic moment of the nucleus. Nuclei that have the same number of protons, but different numbers of neutrons, are called isotopes (1H/2H, 14N/15N). The term nuclide generally is applied to any atomic nucleus.

To study nuclear magnetic properties, the experimentalist subjects nuclei to a strong laboratory magnetic field B0 with units of tesla, or T (1 T = 104 Gauss, or G). In the absence of the laboratory field, nuclear magnets of the same isotope have the same energy. When the B0 field is turned on along a direction designated as the z-axis, the energies of the nuclei in a sample are affected. There is a slight tendency for magnetic moments to move along the general direction of B0 (+z) rather than the opposite direction (-z). (This motion will be more fully described presently.) Nuclei with a spin of ½ assume only these two modes of motion. The splitting of spins into specific groups has been called the Zeeman effect.

The interaction is illustrated in Figure 1.3. At the left is a magnetic moment with a +z component, and at the right is one with a -z component. The nuclear magnets are not actually lined up parallel to the +z or -z direction. Rather, the force of B0 causes the magnetic moment to move in a circular fashion about the +z direction in the first case and about the -z direction in the second. In terms of vector analysis, the B0 field in the z-direction operates on the x component of µ to create a force in the y-direction (Figure 1.3, inset in the middle). The force F is the cross, or vector, product between the magnetic moment µ and the magnetic field B (a vector with magnitude only in the z-direction at this stage with value B0), that is, F = µ × B. The nuclear moment then begins to move toward the y-direction. Because the force of B0 on µ, is always perpendicular to both B0 and µ (according to the definition of a cross product), the motion of µ describes a circular orbit around the +z or the -z-direction, in complete analogy to the forces present in a spinning top or gyroscope. This motion is termed precession.

Figure 1.3 Interaction between a spinning nucleus and an external magnetic field B0.

As the process of quantization allows only two directions of precession for a spin-½ nucleus (Figure 1.3), two assemblages or spin states are created, designated as Iz = +½ for those precessing with the field (+z) and Iz = -½ for those precessing against the field (-z) (some texts refer to the quantum number Iz as mI). The assignment of signs (+ or -) is entirely arbitrary. The designation Iz = +½ is given to the slightly lower energy. In the absence of B0, the precessional motions are absent, and all nuclei have the same energy.

The relative proportions of nuclei with + z and -z precession in the presence of B0 is defined by Boltzmann's law (Eq. 1.1),

in which n is the population of a spin state, k is Boltzmann's constant, T is the absolute temperature in kelvin (K), and ?E is the energy difference between the spin states. Figure 1.4a depicts the energies of the two states and the difference ?E between them.

Figure 1.4 (a) The energy difference between spin states. (b) The energy difference as a function of the external field B0.

The precessional motion of the magnetic moment around B0 occurs with angular frequency ?0, called the Larmor frequency, whose units are radians per second (rad s-1). As B0 increases, so does the angular frequency, that is, ?0 ? B0, as is demonstrated in Appendix A. The constant of proportionality between ?0 and B0 is the gyromagnetic ratio ?, so that ?0 = ?B0. The natural precession frequency can be expressed as linear frequency in Planck's relationship ?E = h?0, or as angular frequency ?E = ??0 (?0 = 2p?0). In this way, the energy difference between the spin states is related to the Larmor frequency by the formula of Eq. 1.2.

Thus, as the B0 field increases, the difference in energy between the two spin states increases, as illustrated in Figure 1.4b. Appendix A provides a complete derivation of these relationships.

The foregoing equations indicate that the natural precession frequency of a spinning nucleus (?0 = ?B0) depends only on the nuclear properties contained in the gyromagnetic ratio ? and on the laboratory-determined value of the magnetic field B0. For a proton in a magnetic field B0 of 7.05 T, the frequency of precession is 300 MHz, and the...