positive and use a minus sign for the second term of u; see Appendix A1.1).
By writing the Bloch equation in this rotating frame and by setting the time derivatives in the equation equal to zero, we can solve the Bloch equation in the rotating frame. The solutions are
2.8 FOURIER TRANSFORM AND SPECTRAL LINE SHAPES
Before we proceed to analyze the characteristics of the magnetization motion expressed in Eq. (2.23), let us pause for a moment to briefly mention two concepts that are essential to modern NMR, Fourier transform (FT) and spectral line shapes.
2.8.1 Fourier Transform
Fourier transform (defined mathematically in Appendix A1.2) utilizes the properties of sine and cosine functions and allows their periods to approach infinity. An operation by FT mathematically decomposes a generic function (often a function of time) into a complex-valued function of frequency, where each frequency can have a certain amplitude. When discussing the FT, one commonly uses the term domain to refer to the “space” where two functions are associated by FT, such as the time-domain function and its frequency domain counterpart. Figure 2.11 shows a sinusoidal oscillation in time and a delta function in frequency as a pair of Fourier functions. Both functions carry the same amount of information, where the frequency (f0) of the delta function is determined by the period (T0) of the oscillation, f0 = 1/T0. One can perform the FT for a multidimensional function (2D, 3D, …); one can also carry out an inverse FT that mathematically restores the original time-domain function from its frequency-domain representation. Table 2.2 lists a few functions and their FT products.
Figure 2.11 Two equivalent functions associated by a Fourier transformation: (a) a sinusoidal oscillation and (b) a delta function. Note that the oscillation frequency in the time-domain function is f0 in the frequency domain.
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.
