Table 2.2 Some functions and their FT representations.
| Function in time | FT of the function in frequency |
|---|---|
| A sine or cosine function [e.g., sin(t)] | A delta function at f |
| A constant [a DC offset with an amplitude] | A spike at the origin |
| A square/rectangular pulse | A sinc function [i.e., sin(θ)/θ] |
| A Lorentzian | An exponential |
| A Gaussian | A Gaussian |
Due to the use of sine and cosine functions in FT, there are various symmetries in the two functions associated by FT. For example, an even/odd time-domain function will keep its even/odd property in the frequency-domain function, except when the time-domain function is odd, the frequency-domain function, which is also odd, will exchange the real/imaginary channels [8]. This and other symmetry relationships are very useful in practical applications such as NMR spectroscopy. In addition, modern science and technology use digital computers, where the original analytical function of time needs to be sampled into a digital, that is, discrete form (see Appendix A1.2). To find out how a computer program calculates the FT, one should also consult a highly useful book titled Numerical Recipes [9]. (There are several versions of this book, where each version has the recipes written in a particular programming language, such as Fortran, C, Pascal.)
While the Fourier transform may be regarded as a purely mathematical operation, it plays an important role in many branches of modern science and technology. For example, a waveform (optical, electrical, or acoustical) and its spectrum are in fact an FT pair, which are appreciated equally as physically picturable and measurable entities. In electronics, the signal V(t) is a single-valued real function of time t while the spectrum S(f) is the frequency version of V(t). Specific filters can be designed so that the output of the amplifier only contains a certain range of frequencies, which are used extensively in electrical power-line and hi-fi audio electronics. In the current context, the NMR signal, known as the free induction decay (FID) in the time domain, and an NMR spectrum in a frequency representation are an FT pair.
2.8.2 Spectral Line Shapes – Lorentzian and Gaussian
Spectral line shapes in NMR describe features of the energy exchange in an atomic system. As shown in Eq. (2.2), a nuclear transition is associated with a specific amount of energy, which would imply an extremely sharp spectral line in NMR. However, the spectral line as measured in NMR is not sharp but broadened considerably. The factors that broaden the spectral line include some fundamental physics principles as well as instrumentation factors.
The fundamental physics principles that leads to the line broadening in high-resolution NMR spectroscopy of solutions is the relaxation process, which causes the NMR signal to decay (hence the naming of the NMR signal as the free induction decay). This decay in the NMR signal of liquids is approximately exponential, so the spectral line shape in high-resolution NMR spectroscopy is Lorentzian. The longer the lifetime of the excited states, the narrower the spectral line width (cf. Chapter 3.7). The line shape in spectra of crystallized solids could be a Gaussian or a mixture of both Gaussian and Lorentzian, due to additional nuclear interactions such as dipolar interaction (see Chapter 4 ).
Table 2.3 summarizes the features of Lorentzian and Gaussian line shapes. Each can be characterized by three parameters, the peak position (x0), the peak amplitude, and the full width at the half maximum amplitude (FWHM). These two functions can also be normalized; they have the integrals of f1(x) and g1(x) equal to 1. When Lorentzian and Gaussian are plotted with the same FWHM (Figure 2.12), a Gaussian curve is a bit wider than a Lorentzian above the half maximum amplitude but drops more rapidly towards the tails/wings below the half maximum amplitude.
Figure 2.12 Comparison between a Lorentzian and a Gaussian with the same FWHM.
Table 2.3 Some features of Lorentzian and Gaussian (A, B, a are constants).
| Lorentzian | Gaussian | |
|---|---|---|
| Line-shape equation | f(x)=A(1+B2(x−x0)2 | g(x)=e−(x−x0)2a2 |
| Line width (FWHM) | 2/B | 2aIn(2)= 1.66511a |
| Normalized expression | f1(x)=(B/πA)f(x) | g1(x)=1xa2g(x) |
| Fourier transform | Exponential | Gaussian |
2.9 CW NMR
The earliest NMR experiments ran in a continuous-wave (CW) mode, where the spectrometer is tuned to observe the component of M, which is 90˚ out of phase to the rotating field B1, the so-called absorption mode signal. (Earlier in Section 2.7, we set both B1(t) and u in the direction of x′, and v in the direction of y′ in the rotating frame.) During an experiment, the magnetic field B0 is swept slowly through the resonance frequency. As each chemically identical spin group comes into resonance, it undergoes nuclear induction and a voltage is induced in the pick-up coil (cf. the three peaks of ethanol in Figure 1.4). This approach is called the CW method, where the signal of the specimen is recorded continuously on an oscilloscope. Provided that this field sweep is done sufficiently slow, the absorption mode signal at each frequency corresponds to the steady state value of v when M has come to rest in the rotating coordinate system. Hence it is also called the slow passage experiment. Since neither the resonance frequency nor the number of the equivalent groups in a specimen is known, doing an NMR experiment using the CW method could take a long time.
By examining the Bloch equation in the rotating frame [Eq. (2.23)], the following observations can be made:
1 When we are far from the resonance (i.e., |ω0 – ω| is large), we have u = v = 0 and Mz = M0. The non-zero values of u and v appear only in a small interval around ω0, that is, when there is a resonance.
2 Where T1T2(γB1)2 ≪ 1 (i.e., the rf power applied is sufficiently low so that the saturation does not occur), v can be simplified as (2.24)By comparing Eq. (2.24) with the line-shape functions in Table 2.3, we see that v is a Lorentzian centered at ω0 with a line width at half maximum of 1/(πT2). Hence, in principle, the FWHM of the resonant peak can be used to determine the T2 relaxation time.
3 When T1T2(γB1)2 is not sufficiently smaller than 1, we can have these situations:when T1T2(γB1)2 < 1, the spins are below saturation, and the signal ∝