of M is not its equilibrium state but its time evolution after M is tipped away by an external perturbation (another radio-frequency field) from its thermal equilibrium along the z axis. The evolution of M produces the NMR signal, which reveals the environment of the molecules. In a classical description, the time evolution of the macroscopic magnetization in the presence of a magnetic field can follow the same approach that we used before in deriving Eq. (2.4). By equating the torque to the rate of change of the angular momentum of the macroscopic magnetization M, we have
Equation (2.9) is a vector equation, which states that the rate of the change of the magnetization has a direction that is at right angles to both the magnetization vector and the magnetic field vector (by the right-hand rule in the cross product of vector analysis, Appendix A1.1). When B is parallel with the z axis as B0k, the above equation has the same solution as given before in Eq. (2.5), which corresponds to a precessional motion about k at the rate ω0.
A time evolution of M can be introduced by the application of a small linearly polarized radio-frequency (rf) field B1(t) that is oscillating in the transverse plane. This B1 field is actually a superposition of two counter-rotating fields in the transverse plane (Figure 2.5):
Figure 2.5 Two counter-rotating fields (right) can form a B1 field precessing in the transverse plane of the laboratory reference frame (left).
One of the two B1(t) components (which rotates in the same direction as ω0) will have strong influence on the nuclei despite its small magnitude compared with B0, while the other component, which rotates in the opposite direction, will have negligible effect on the nuclei (provided |B1| ≪ B0).
It will be very useful to introduce a rotating frame of reference, as shown in Figure 2.6, where the xyz frame is the laboratory frame of reference (stationary) and the x′y′z′ frame is the rotating frame of reference at an angular frequency ω′ in the laboratory frame, in which z and z′ are parallel. In addition, one of the B1(t) components appears stationary in the rotating frame; let’s set that stationary component of B1(t) along the x′ axis.
Figure 2.6 (a) A magnetic moment µ precessing in the stationary (xyz) and rotating (x’y’z’) frames. (b) A macroscopic magnetization vector M in the two frames.
When a magnetic moment is precessing at a rate of ω0 in the xyz frame, a stationary observer in the x′y′z′ frame should see the magnetic moment precessing at a reduced rate ω0 – ω′. Since ω = γB, a reduced precession rate implies that the magnetic moment is experiencing a reduced B0, as
In the presence of both B0 and B1(t) fields, the total vector field Btotal is the sum of all fields and the effective field Beff varies with the frequency of B1(t), as
and
where Eq. (2.12) is for the stationary xyz coordinates and Eq. (2.13) is for the rotating x′y′z′ coordinates. The directions of these vector fields are shown schematically in Figure 2.7.
Figure 2.7 In the rotating frame that has a frequency ω′, the external magnetic field B0 appears to be reduced in magnitude, where the reduction is a function of ω′. This reduction results in the tipping of Beff towards the x’ axis, which is in parallel with B1.
In a typical NMR system, |B1| is several orders of magnitude smaller than B0, which means that Btotal is almost along the z axis. However, the magnitude and the direction of Beff will drastically depend upon the frequency of the rotating frame ω′. When ω′ approaches the value of ω0, the effective magnetic field Beff tips more towards the transverse plane. When ω′ = ω0 (i.e., the frequency of B1(t), ω′, equals the Larmor frequency ω0), the second term in Eq. (2.13) becomes zero. Hence B1(t) becomes the only field in Beff to interact with the nuclei. M will therefore respond to the effect of B1(t), which is set along the x′ axis (Figure 2.7). This condition, ω′ = ω0, is termed as the resonance condition.
Under this resonance condition, the magnetization will be tipped away from its equilibrium orientation towards the transverse plane under the influence of B1(t), as ω1 = γB1 (Figure 2.8). Since the effective component of B1(t) has been set along the +x′
