Читать книгу Essential Concepts in MRI - Yang Xia - Страница 36

From the energy eigenvalue equation,

(3.8)

one can obtain the energy eigenvalues of the Zeeman Hamiltonian ℋ, which are the energy levels (called the Zeeman levels):

(3.9)

Therefore, the energy difference ΔE between any two adjacent eigenstates of a spin system, known as the Zeeman splitting, is

(3.10)

As indicated in Eq. (3.9), a spin-1/2 system (I = 1/2) has only two eigenstates, corresponding to m = +1/2 (spin-up) and m = –1/2 (spin-down) states. Its two energy levels are therefore given by

(3.11a)

(3.11b)

Schematically, these two energy levels, which were shown once before briefly in Figure 2.2a, are now shown more completely in Figure 3.1. Note that for the spin-1/2 system, the “spin-up” and “spin-down” states are stationary states of the system and exist only in a time-independent magnetic field, that is, when ℋ is time-independent. Note also that the schematic in Figure 3.1, where E(–1/2) has a higher energy than E(+1/2), is based on a positive γ.

Figure 3.1 The quantities in a spin-1/2 system. The application of an external magnetic field B₀ introduces two energy levels for the spin-1/2 system. The population difference between the two states, determined by the Boltzmann distribution, results in a macroscopic magnetization, M, pointing along the same direction as the external magnetic field.

The terms “spin-up” and “spin-down” refer to the z-component mħ of the angular momentum. The actual spin vector has a magnitude of ℏI(I+1), which is greater than mħ. Hence, a spin vector I cannot lie graphically along any fixed axis in space. This is the reason that the precessional motion of a nucleus spin in a classical description is tilted at a fixed angle (Figure 2.3b). Since Ix, Iy, and Iz do not commute, we cannot specify or measure any two quantities simultaneously. Only the z-component Iz and the magnitude of I are known with certainty as Iz = 1/2 and I² = 3/4, which can be used to determine the fixed angle of the cone in Figure 2.3b and Figure 2.4b.

Instead of visualizing a vector µ precessing on the surface of a cone, a spin vector in the stationary state can be thought of as uniformly smeared out over the surface of a cone, similar to the advanced concept in modern physics that visualizes an electron as a cloud around a nucleus instead of a point charge in an orbit around the nucleus. In addition, quantum mechanically, a nuclear spin in a stationary state does not precess, since the probability density and expectation values are independent of time. Since I is uniformly smeared out as described and cannot be specified to lie completely along any axis, we should only be concerned with the components of I, not I itself. Therefore, the spin-up state can be thought of as a spin vector lying along the z axis, parallel with the field direction.