I = 0, then the nucleus has no spin and cannot be observed by NMR (e.g., there is no NMR for 12C, even though each 12C nucleus contains six protons). If I > 0, the nucleus will have an associated magnetic dipole moment µ, given by
where γ is called the gyromagnetic ratio, a characteristic constant for each nuclear species (γ = 2.675 × 108 rad s-1 T-1 or 42.576 MHz T-1 for protons), and ħ is the Planck’s constant (6.62607015 × 10−34 J s) divided by 2π. (To convert γ between rad s-1 T-1 to MHz T-1, consider rad/s as the angular velocity equal to 2π times the linear frequency.) The unit of µ is Joule per Tesla (J T-1). Since γ, ħ, and I are all known constants, the magnitude of µ can be determined accurately.
Note that Eq. (2.1) could be also written as γ = µ/(ħI), which illustrates the fact that γ is the magnetic dipole moment divided by the angular momentum; hence, γ should be more properly named the magnetogyric ratio, not the gyromagnetic ratio that implies the inverse of the two quantities. The term magnetogyric ratio has indeed been used in some books and papers and also recommended by the 2001 International Union of Pure and Applied Chemistry (IUPAC) nomenclature [1]. In modern literature, however, γ is commonly known as the gyromagnetic ratio. Both magnetogyric ratio and gyromagnetic ratio refer to the same value.
The total number of the allowed spin states for a given nucleus is a discrete value, 2I + 1, which ranges from –I, –I + 1, –I + 2, …, to I. (It is another quantum mechanical concept that we need to cite in the classical description of nuclear magnetic resonance.) These values can be written as mI, the azimuthal quantum number. Hence, a nucleus with
I = 1/2 has two spin states, mI = –1/2 and mI = 1/2. These two spin states are commonly illustrated in quantum mechanics by a two-level energy diagram as in Figure 2.2a, which we will discuss more in Chapter 3. Since there are only two spin states, we may use an arrow to describe the spin-1/2 particles. The mI = –1/2 state is called spin down (↓) while the mI = 1/2 state is called spin up (↑). The spin-down state has higher energy than the spin-up state, hence is the upper level in the energy level diagram (Figure 2.2a).Figure 2.2 The application of an external magnetic field B0 causes (a) a spin-1/2 system to have two discrete energy levels, and (b) a spin-1 system to have three energy levels. Each energy level can be labeled by the individual spin state.
I > 1/2 has nuclear quadrupole moments that produce splitting of the resonant lines or line-broadening effects. For example, a deuteron (2H) has I = 1, which would have three spin states, corresponding to mI = –1, 0, 1. These three spin states can be used to label a three-level energy diagram as in Figure 2.2b. A quantum mechanical description must be used to understand the behavior of any spin with an angular momentum larger than 1/2.
Note that I is defined in this book as a dimensionless angular momentum; hence, a reduced Planck’s constant has been explicitly included in Eq. (2.1). This use of a dimensionless angular momentum can be found in many books, including those by Callaghan [2] and Hennel and Klinowski [3]. In contrast, I can also be defined as an angular momentum with in its definition [4]. With this definition, Eq. (2.1) would be written as µ = γI.
Table 2.1 lists some fundamental properties of several common nuclei (more lengthy tables can be found in many books and papers [1, 5]). On paper, the bigger γ yields better sensitivity (the column of relative sensitivity, which is defined for the equal numbers of nuclei at constant field.). In practice, one must consider the normal or natural concentration of a nucleus in the specimen (i.e., its availability). For example, the relative sensitivities of 1H and 19F are similar (1 vs. 0.834); however, a human body is about 60% water (~75% in a newborn, 60–65% in men, and 55–60% in women), while the amount of 19F in human or other biological systems is many orders of magnitude smaller by comparison. This explains the challenges and difficulty in detecting the signals other than protons in biological experiments by NMR and MRI.
Table 2.1 Properties of common nuclei.
| Isotope | Abundance (%) | Spin | γ (108 rad s-1 T-1) | Relative sensitivity | Resonance frequency f0 at 1T (MHz) |
|---|---|---|---|---|---|
| 1H | 99.9844 | 1/2 | 2.6752 | 1.00 | 42.577 |
| 2H | 0.0156 | 1 | 0.4107 | 0.00964 | 6.536 |
| 13C | 1.108 | 1/2 | 0.6726 | 0.0159 | 10.705 |
| 19F | 100 | 1/2 | 2.5167 | 0.834 | 40.055 |
| 31P | 100 | 1/2 | 1.0829 | 0.0664 | 17.235 |
2.3 THE TIME EVOLUTION OF NUCLEAR MAGNETIC MOMENT
Now look at the situation when one places a single nucleus in an externally applied magnetic field B0. Because of the dipole moment, the nucleus (which is represented by a magnetic moment µ) will interact with the external field B0, from which the energy of the nucleus in the field will be given by a dot product
Since the nucleus also has an angular momentum, it experiences a torque, which can be expressed as a cross product, µ × B0. Since this torque is equal to the time derivative of the angular momentum, by quoting Newton’s second law, we have
