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and accurate for describing the nuclear precession in the presence of an external field B₀.

On a macroscopic scale, one can find several analogs for the precessional motion of the nuclear spin. For example, a spinning top (Figure 2.3c) can have a precessional motion when the axis of rotation does not pass through the top’s center of gravity (i.e., the top is not standing up perfectly vertical), which yields a torque and an angular momentum for the top that induces it into a precessional motion. In a spinning top, the gravitational force mg that points vertically downward plays the same role of the magnetic field B0 in NMR. Since a top is a macroscopic object, the tipping angle of a spinning top can vary from 0˚ to a large angle continuously, while the tipping angle of a nuclear precessional motion is fixed and cannot be varied. A second common example is the precessional motion of our planet Earth in the solar system, which is tipped at a constant angle of 23.4˚ from an “axis” in space, with a period of the precessional motion of 26,000 years. The torque on Earth is exerted by the sun and the moon.

      2.4 MACROSCOPIC MAGNETIZATION

      Any practical sample, no matter how small, contains an enormous number of nuclei (remember Avogadro’s constant). The macroscopic (or bulk) magnetization M is a spatial density of magnetic moments and can be written as

       (2.6)

      where N is the number of spins in the volume. Since M is a vector, we can therefore, in general, write M in the component form as

(2.7)

      where i, j, and k are the unit vectors along x, y, and z axes, respectively, in the usual 3D Cartesian coordinate system. Equation (2.7) can also be grouped into the transverse and longitudinal forms, as

(2.8)

      where M_⊥ = M_xi + M_yj, and M_|| = M_zk.

In the absence of any external magnetic field, the ensemble average of the nuclear spins in any practical sample should cause the value of magnetization to be averaged to zero, due to the random directions of the magnetic dipoles of the nuclei (since most nuclei act independently).

In the presence of an external field B₀, however, the magnetization M of a sample with non-zero spin will be non-zero (after the specimen has been immersed in B₀ for a sufficient time for the spin system to reach a thermal equilibrium with the environment). This non-zero spin is formed according to Boltzmann’s distribution, which describes the probability distribution function of a particle in an energy state E, as exp(–E/(kBT)), where kB is the Boltzmann constant (1.380649 × 10^–23 m² kg s–² K^–1) and T is the absolute temperature of the system. Since the spin-up nucleus is at a lower energy state than a spin-down nucleus, the number of nuclei along the direction of the external magnetic field will be more than the nuclei against the direction of the field (Figure 2.4b). Therefore, a non-zero net magnetization, which can be written in magnitude as M₀, occurs due to the population difference between the spin states at the two spin levels.

      Figure 2.4 (a) A single nucleus in an external magnetic field B₀. (b) A nuclear ensemble that is the collection of a large number of nuclei. (c) The vector average of the nuclear ensemble is represented by a macroscopic magnetization M.

      For spin-1/2 particles such as protons, the net magnetization can be visualized as an ordinary vector, aligned in the direction of B₀, as shown in Figure 2.4c. Since any practical sample has an enormous number of nuclear spins, it is easy to see that at thermal equilibrium the net magnetization has no transverse component (i.e., Mx = My = 0); the only component of the net magnetization is along the direction of B₀ (i.e., Mz = M₀).

      Note that in the classical description, the individual magnetic moment µ undergoes precessional motion in an external magnetic field B₀ (as in Figure 2.4a), which is commonly represented graphically by vectors on the surface of a cone (as in Figure 2.4b). The ensemble average of µ in any practical sample is M, which is represented by a stationary vector that aligns with the direction of B₀ (as in Figure 2.4c). At equilibrium, M itself does not precess graphically as µ around B₀; M should be represented by a vector in parallel with B0, never on the surface of any cone.

      2.5 ROTATING REFERENCE FRAME

      The usefulness of M is not its equilibrium state but its time evolution after M is tipped away by an external perturbation (another radio-frequency field) from its thermal equilibrium along the z axis. The evolution of M produces the NMR signal, which reveals the environment of the molecules. In a classical description, the time evolution of the macroscopic magnetization in the presence of a magnetic field can follow the same approach that we used before in deriving Eq. (2.4). By equating the torque to the rate of change of the angular momentum of the macroscopic magnetization M, we have

(2.9)

      Equation (2.9) is a vector equation, which states that the rate of the change of the magnetization has a direction that is at right angles to both the magnetization vector and the magnetic field vector (by the right-hand rule in the cross product of vector analysis, Appendix A1.1). When B is parallel with the z axis as B₀k, the above equation has the same solution as given before in Eq. (2.5), which corresponds to a precessional motion about k at the rate ω₀.

A time evolution of M can be introduced by the application of a small linearly polarized radio-frequency (rf) field B₁(t) that is oscillating in the transverse plane. This B₁ field is actually a superposition of two counter-rotating fields in the transverse plane (Figure 2.5):

Figure 2.5 Two counter-rotating fields (right) can form a B₁ field precessing in the transverse plane of the laboratory reference frame (left).

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