If the fluctuating field manages to alter the populations of the states, the populations would evolve immediately until they reach the values predicted by the Boltzmann equations for the temperature of the Brownian motion (lattice temperature). This process is described by T1 and results in the relaxation of the longitudinal component of M. The contribution of the fluctuating field to T2 can be seen from the following argument. According to Eq. (3.9)–Eq. (3.11), the Zeeman energy levels are known precisely (Figure 3.1), which implies the resonance frequency associated with the transition between two neighboring Zeeman levels should have a unique value, that is, a delta function at a singular ω0 (Figure 3.2a). In reality, however, the resonant peak even in simple liquids is broadened due to the fluctuation of the Zeeman levels (Figure 3.2b), caused by the distributions of local interactions in their environment experienced by the nuclear spins. The line width of the resonant peak (excluding the effect of inhomogeneity in B0), which is inversely proportional to T2, is a measure of the uncertainty in the energies between two neighboring Zeeman levels. This uncertainty can also be traced back to the fluctuating fields due to the magnetic moments of nuclei in other molecules as they undergo thermal motions.
Figure 3.2 (a) A precise value of the Zeeman energy difference between the two states in a spin-½ system should imply a single value in the transition, hence a delta function in the frequency distribution. (b) In reality, a wider line shape such as a Lorentzian or Gaussian suggests an uncertainty in the difference between the energy levels. For simple liquids, the line shape is a Lorentzian in the frequency domain, which corresponds to the exponential decay in the time-domain FID, shown in (c). A fast decay of the FID (e.g., short blue dash) implies a short T2 and a wide line shape; a slow decay (e.g., red solid line) implies a long T2 and a narrow line shape. A precise value of the energy difference as in (a) would imply a sinusoidal oscillation without any decay in the time domain (as shown in Figure 2.15d).
The Bloch equation [Eq. (2.18)] contains a phenomenological term leading to exponential relaxation. This classical description is quite accurate for spins in rapidly tumbling molecules but breaks down when the motions of molecules become slow or complex, such as in the case of internal motion in macromolecules. In the following sections, we first explain the relaxation mechanisms in terms of quantum transitions between eigenstates of operators Ix, Iy, and Iz; then, we briefly describe the results of the random field model of relaxation.
3.7.1 Relaxation Mechanism in Terms of Quantum Transitions
For spin-1/2 particles, the relaxation mechanism can be understood with a set of equations and analysis in terms of quantum transition [9, 10]. In this approach, the spin populations (the occupancies of the eigenstates of Iz with eigenvalues m=±1/2) are defined as
We also define the total population N0 and the population difference n as
Hence, the macroscopic magnetization M is proportional to the population difference n. Using Eq. (3.21), the z component of the magnetization at time = 0 can be written as
Since the population is at equilibrium with the environment according to the Boltzmann distribution, the population ratio is
To consider the dynamics of the two populations, we define w+- as the probability of transition of a spin from |+> state to |–> state per spin per second, and w-+ as the probability of transition of a spin from |–> state to |+> state per spin per second. At equilibrium, we have
Combining Eq. (3.28) and Eq. (3.29), we have
With this equation, the changes of the spins with time can be defined as
Each