the individual spins in the transverse plane. This decay in phase coherence describes the process in which the spins come to a thermal equilibrium among themselves in the transverse plane, which results in signal loss since NMR and MRI measure the net transverse magnetization. T2 in biological tissues is usually in the range of tens or hundreds of milliseconds.
The solution to Eq. (2.15b) becomes
The motion of magnetization in spin-spin relaxation is shown schematically in Figure 2.10, where again t = 0 marks the moment that the B1 field is turned off.
Figure 2.10 The motion of the magnitude of the transverse magnetization after a 90˚ B1 field/pulse, where a slow decay leads to a long T2 value. Note that if Figure 2.9 and Figure 2.10 are plotted together on one graph (i.e., share the same scale in the horizontal axis t), the transverse magnetization would decay to zero much faster than the return of the longitudinal magnetization to its thermal equilibrium (i.e., the maximum), since T2 is commonly much shorter than T1.
The decay of a time-domain signal in the transverse plane leads to a spectral broadening in the frequency domain (cf. Section 2.8 and Appendix A1.2 for Fourier transform). The line broadening due to the relaxation processes is named homogeneous and is inherently irreversible. In practice, the signal decays faster than the intrinsic rate due to the T2 relaxation. Other contributions to the signal decay include, for example, the inhomogeneity of the magnetic field B0 or low-frequency molecular motions in the specimens (or even a field gradient; cf. Chapter 11). The line broadening due to non-uniformity of the magnetic field is named inhomogeneous and can be eliminated by using an appropriate rf pulse sequence (provided the molecules do not move during the measurement time). T2 is the time constant that describes the homogeneous broadening, while the term T2* is used when the decay process contains both T2 and other (in principle) removable factors. T2* is always shorter than T2 (cf. Chapter 7.3 for T2 and T2*).
2.7 BLOCH EQUATION
When it can be assumed that the change of the magnetization following excitation is independently caused by external magnetic fields and relaxation processes, the equation of motion of M can be written by combining Eq. (2.14) and Eq. (2.15), in the laboratory frame, as
This is the well-known Bloch equation [7]. The first term is due to precessional motion and the second term is due to relaxation. While a precise evaluation of the spin system requires a quantum mechanical treatment, the Bloch equation provides a classical, phenomenological description for liquids and liquid-like systems where the Hamiltonian is of a simple magnetic (vector) form, for example, protons of water molecules in non-viscous liquids and many biological tissues.
Now we are ready to solve the Bloch equation under various conditions. First rewrite the vector equation into the component form, as
The above equations have the usual setup for the magnetic fields as
Thermal equilibrium ensures the initial condition of M in Eq. (2.19) as
Note that as soon as M is tipped away from its thermal equilibrium state, relaxation processes start. In most analyses, however, we consider only one event at a time – that is, when we use the B1 field to tip the magnetization, we do not consider the relaxation of the magnetization during the tipping process.
In order to better examine the solution of the Bloch equation (more precisely, to examine the spectral shapes of the waveform solutions), we will describe the magnetization in a rotating frame with an angular velocity ω about the z axis. In this x′y′z′ frame, we have the component u in the direction of x′ and v in the direction of y′. We can use the common rotation matrix in linear algebra to rewrite the transform matrix as
Note that Eq. (A1.23) is used in this clockwise rotation, which is consistent with the convention specified in Figure 1.3. (For a counterclockwise rotation, keep both terms
