Читать книгу Essential Concepts in MRI - Yang Xia - Страница 20

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.