t divided by T subscript italic 2 right parenthesis comma"/> (2.32)
where S0 is proportional to M0, and ϕ is a spectrometer parameter called the receiver phase.
With the use of Fourier transformation, we can derive the signal in the frequency domain in both real (Re) and imaginary (Im) parts [2], as
When ϕ = 0, the above equations become
These two equations can be plotted as in Figure 2.14c–d, where the real part is called the absorption signal, while the imaginary part is called the dispersion signal. By comparing Eq. (2.34) with Eq. (2.23) in the CW NMR experiment, we find that the results are of the same form as the absorption and dispersion components. However, the results in Eq. (2.34) are not centered at the Larmor frequency ω0, but at the difference Δω, the offset frequency.
When ϕ ≠ 0, which is common in practice and means that M is not along any axis in the transverse plane, the real and imaginary parts of the signal contain a mixture of absorption and dispersion components. We call the spectrum “out of phase.” We can correct this phase by multiplying the signal by exp(–iϕ); that is, we apply a 2D rotation matrix to the signal, as we did in Eq. (2.22). This process is termed to “phase” the spectrum in NMR experiments (cf. Chapter 6.10), which illustrates that in actual NMR experiments, the phase of the signal detector can be adjusted continuously.
2.13 PHASES OF THE NMR SIGNAL
In classical physics, a variety of phenomena (e.g., the analysis of Hooke’s law in mechanics) can be characterized as simple harmonic motion, which is often visualized with the aid of a rotating disc. Plotting the trajectory of a fixed point on the rotating disc (or in the current context, the trajectory of the tip of a vector M in the 2D xy plane) as a function of time leads to a sinusoidal wave (Figure 2.15a and b). This type of circular motion can be described by a few fundamental equations,
Figure 2.15 (a) A vector M rotates in the xy plane at a constant angular frequency ω and with a constant amplitude M, where the tip of the vector draws a sinusoidal wave in a plot of the amplitude vs. time, as in (b). (c) Four different initial orientations of the rotating vector give rise to four different phases of the wave, as shown in (d). The constant-amplitude oscillation implies no attenuation of the vector length. (e) When the oscillation is attenuated by an exponential decay, the motion of the rotating vector generates a damped oscillatory signal in the transverse plane (the FID). The FID signal will have the same decay envelope, regardless of which of the four initial orientations of the rotating vector shown in (c).
where m is the amplitude of the motion, M is the maximum amplitude (i.e., the length of the vector), θ is the rotation angle in radian (rad), ω is the angular frequency in rad/s, T is the period of the wave (i.e., the time to complete one cycle) in seconds (s), and f is the frequency of the wave in hertz (Hz).
If no friction slows down the rotation of the disc and nothing changes the length of the vector, the disc will rotate at a constant frequency indefinitely and the wave will oscillate between +M and –M continuously as the function of time without any attenuation to its amplitude, as shown in Figure 2.15b. If the vector M in Figure 2.15a does not start the motion from being parallel with the x axis but with another axis, the oscillation will have exactly the same frequency and period, only with an extra phase shift (Figure 2.15c and d).
Similar visualization has in fact been used in the illustration of the motion of the magnetization in the laboratory and rotating frames (cf. Figure 1.3, Figure 2.14), except the FID signal has an attenuation term, which modulates the oscillation amplitude in the time domain. So instead of the constant amplitude sinusoidal waves as in Figure 2.15d, the amplitude of the FID oscillation will decay with time, as in Figure 2.15e, which generates the NMR signal. The only difference among the four NMR signals in Figure 2.15e is the four initial orientations of the magnetization vectors in the transverse plane (Figure 2.15c), that is, the initial phases of the magnetization. Even when M does not start from being parallel with any axis in the transverse plane, the real and imaginary parts of the FID both start with some initial values but still oscillate and decay in the same manner as in Figure 2.15e.
With this in mind, we can make some further comments on the laboratory and rotating frames. In the laboratory (stationary) frame, the z axis is firmly defined by the direction of B0 (as shown in Figure 1.2), while the x and y axes are commonly defined by the typical 3D Cartesian coordinates in the 2D transverse plane, which is perpendicular to the z axis (Figure 1.3). In the rotating frame, the direction of the x′ axis is quite arbitrary, defined at the instant when the B1(t) field is switched on. For a single B1(t) field, whether B1 is along the x′ or y′ or –x′ or –y′ axis does not matter at all, since the variation of the B1 direction only results in
