Читать книгу Essential Concepts in MRI - Yang Xia - Страница 26

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