B 0 period"/> (2.3)
By multiplying the above equation on both sides with γ, we have
which is shown in Figure 2.3a. In classical mechanics, this equation describes a precessional motion [6] (Figure 2.3b) of µ around B0 at an angular frequency ω0, known as the Larmor frequency,1
Figure 2.3 (a) Precessional motion in classical mechanics. (b) The Larmor precession of a single nucleus with the aid of classical mechanics. (c) A spinning top can have a precessional motion.
where ω is expressed as an angular frequency in rad/s, which can be converted to the temporal/linear frequency f in Hz by noting ω = 2πf. Note that when γ is positive (which it is for proton [the nucleus of 1H] and many other nuclei), the rotation will be clockwise, as shown in Figure 2.3b. γ can also be negative (e.g., 3He, 15N), where the rotation becomes counterclockwise.
Although the equation for this Larmor precession seems simple, it is the fundamental equation of the NMR phenomenon. The equation states that (a) the frequency of the nuclear precession is proportional to the externally applied magnetic field B0, and (b) the proportionality constant is γ. Equation (2.4) and Eq. (2.5) are useful 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
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
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
where M⊥ = Mxi + Myj, and M|| = Mzk.
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
