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7 for more description on spin manipulation under various pulse sequences. The motions of the magnetization in the laboratory frame (xyz) are also shown in Figure 2.8, which graphically is a spiraling vector away from its equilibrium position over the envelope of a dome. When the magnetization M reaches the transverse plane, the longitudinal component (Mz) of M becomes zero.

In general, the equation of motion for M in the presence of B₁(t) is given by

StartFraction d upper M Over d t EndFraction equals gamma bold upper M times upper B Subscript e f f (2.14a)

StartFraction d upper M Over d t EndFraction equals gamma bold upper M times left-parenthesis left-parenthesis upper B 0 en-dash omega prime slash gamma right-parenthesis k plus upper B 1 bold i right-parenthesis (2.14b)

2.6 SPIN RELAXATION PROCESSES

After M has been tipped to the transverse plane, if we switch off the B₁(t) field and sit there watching, what happens to the spin system? As soon as B₁ is switched off, two processes will happen to M, which would eventually lead to the return of M to thermal equilibrium (i.e., Mz = M₀, and the zero transverse components of M).

The processes that return the magnetization M to the thermal equilibrium are termed as relaxation, which may be described by two time-constants in the following equations:

StartFraction d upper M Subscript z Baseline Over d t EndFraction equals minus StartFraction upper M Subscript z Baseline minus upper M 0 Over upper T 1 EndFraction (2.15a)

and

StartFraction d upper M Subscript up-tack Baseline Over d t EndFraction equals minus StartFraction upper M Subscript up-tack Baseline Over upper T 2 EndFraction comma (2.15b)

where M⊥ refers to the transverse component, defined by Eq. (2.8).

T₁ is known as the spin-lattice (or longitudinal) relaxation time because the relaxation process involves an energy exchange between the spin system and its surrounding thermal reservoir, known as the “lattice.” The term “longitudinal” comes from the fact that this relaxation process restores the disturbed magnetization to its thermal equilibrium, being along the longitudinal direction k. T₁ in simple liquids is usually in the range of several seconds.

When the B₁ field rotates the magnetization entirely to the transverse plane (i.e., Mz = 0), the magnetization is said to be rotated by 90˚ (i.e., π/2). The solution to Eq. (2.15a) becomes

upper M Subscript z Baseline left-parenthesis t right-parenthesis equals upper M 0 left-parenthesis 1 minus e Superscript minus StartFraction t Over upper T 1 EndFraction Baseline right-parenthesis period (2.16a)

If the B₁ field is sufficiently powerful or its duration is sufficiently long, the magnetization can be inverted (i.e., Mz = –M₀). Such a B₁ field is said to rotate the magnetization by 180˚ (a π pulse). Under this condition, the solution to Eq. (2.15a) becomes

upper M Subscript z Baseline left-parenthesis t right-parenthesis equals upper M 0 left-parenthesis 1 minus 2 e Superscript minus StartFraction t Over upper T 1 EndFraction Baseline right-parenthesis period (2.16b)

Figure 2.9 shows schematically the motion of the longitudinal magnetization with these two different initial conditions, where t = 0 marks the moment that the B₁ field is turned off.

Figure 2.9 The motion of the longitudinal magnetization after it has been tipped by 90˚ (a) and 180˚ (b). The B₁ fields that are capable of tipping M by 90˚ and 180˚ are called a 90˚ B₁ field/pulse and a 180˚ B₁ field/pulse, respectively.

The second relaxation process is characterized by a time constant T₂, which is called the spin-spin (or transverse) relaxation time since it describes the decay in phase coherence between 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

upper M Subscript up-tack Baseline left-parenthesis t right-parenthesis equals upper M Subscript up-tack Baseline left-parenthesis 0 right-parenthesis e Superscript minus StartFraction t Over upper T Baseline 2 EndFraction Baseline period (2.17)

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 T_1.

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

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