Equation (3.17) is important because it may be shown that any state of the density matrix (defined in Appendix A2.6) for an ensemble of non-interacting spin-1/2 particles can be described using the macroscopic magnetization defined in this manner, thus permitting a classical description of simple spin systems.
In the absence of an external magnetic field, the ensemble average of the magnetization vector should be zero due to the random directions of the magnetic dipoles of the nuclei.
If a sample is immersed in an external field and in thermal equilibrium, the density operator associated with this magnetization vector is given by
The transverse component of M is zero due to the even distribution of the azimuthal phase angles of the precessing nuclei in the transverse plane. This corresponds to phase incoherence leading to the zero value of the off-diagonal elements of ρ,
The z component of the magnetization M arises from the difference in populations between the upper and lower energy states. At room temperature, the magnitude of this magnetization in the equilibrium state, M0, can be derived as
For a spin-1/2 system at room temperature, the population difference between the spin-up (m=+1/2) state and the spin-down (m=−1/2) state can be calculated from the diagonal elements of ρ, as
For protons at B0 = 1.4 T (60 MHz), it is equal to about 5 × 10-6, a small value resulting from the small value of γħB0 (Zeeman splitting) compared to kBT (Boltzmann energy). It is this small magnetization of nuclei at room temperature that limits NMR detection sensitivity and leads to resolution limitations in MRI experiments [7, 8].
3.6 RESONANT EXCITATION
When both B0 and B1(t) are present and perpendicular to each other (B1 in the transverse plane), we can write down the Hamiltonian in the laboratory frame as
In the rotating frame, the Hamiltonian becomes
At ω = ω0, we have
Since Ix=12(I++I−), where I+ and I- are the raising and lowering operators defined in Appendix A2.4, the time evolution of the spin system corresponds to an inter-conversion of each spin between |1/2> and |–1/2> at a rate of γB1 (an oscillation).
3.7 MECHANISMS OF SPIN RELAXATION
Spin relaxation is truly fundamental and central to the theory of NMR and MRI; the influence of spin relaxation on both NMR and MRI measurements is wide, deep, and quite often subtle. It is therefore worth taking some time to learn to appreciate the subtleties of spin relaxation.
A nucleus in a liquid experiences a fluctuating field, due to the magnetic moments of nuclei in other molecules as they undergo thermal motions. (This experience of a fluctuating field is actually true for all environments, not just a nucleus in a liquid.) This fluctuating field may be resolved by Fourier analysis into a series of terms that are oscillating at different frequencies, which may be further divided into components parallel to B0 and perpendicular to B0. The component parallel to B0 could influence the steadiness of the static field B0, while the components perpendicular to the static field at the Larmor frequency could induce transitions between the levels in a similar way to B1. These influences give rise to a non-adiabatic (or non-secular) contribution to relaxation of both the longitudinal and transverse components of M.
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.
