Lorentzian line shape liquids

Fig. 3.3. Lorentzian line shape (solid line) and experimental CARS data (points) of liquid nitrogen (T — 77 K) from [136].

While cooling, when the limit xa = 1 /, (cf. Eq. 15) is reached, the central Lorentzian NMR line, which is characteristic of a liquid (t-,8, solid-state spectrum, in the case of 2H NMR the Pake spectrum. The breadth of the solid-state spectra makes it difficult to measure the corresponding (short) free induction decay (FID), so that it is necessary to use echo-techniques (cf. Section II.D.2). Figure 38 (left) shows solid-echo 2H NMR spectra of glycerol-. The crossover from a Lorentzian line to the Pake spectrum is observed some 20% above Tg. Below Tg the spectrum is independent of temperature. In Fig. 37 (right), the corresponding 31P NMR spectra of m-tricresyl phosphate (m-TCP) are displayed. The characteristic spectral shape is now determined by the anisotropic chemical... 

Most lines in liquids arise from systems where there is interaction between nuclear spins, and are not Lorentzian in shape. Natural line shapes are closer to a Gaussian shape in the tails of the intensity distribution in liquids. The relationship between peak width at half height and T2 is still retained and gives values in agreement with more rigorous transient methods (Hahn, 1950). 

Theoretical analyses of spin-lattice relaxation have been performed by Kubo and Tomita 420), Redfield (579), and Kivelson (390). This mechanism is applicable to solids and liquids. The line shape for this type of relaxation is Lorentzian. 

The amplitude of the oscillation is a maximum when the driving frequency equals the natural resonance frequency and the width of the resonance line depends on the degree of damping of the system. If the liquid in which the system is immersed is not viscous, the line will be narrow. On the other hand, the line will be quite broad for a viscous liquid. The line-shape produced by such a damped harmonic oscillator is a Lorentzian lineshape. The full width at half height of such a... 

An ESR line is not infinitely sharp it has a shape and width due to spin relaxation. The equations of motion for Mx, My, and M in the presence of an applied field Ho and including the spin relaxation processes discussed above are called the Bloch equations. The solution to these equations predicts a Lorentzian line with a halfwidth at halfheight of Lorentzian lineshapes are indeed often found for free radicals in liquids. In this case T2 can be determined from the linewidth. The Bloch equations also predict how the ESR signal intensity will vary with increasing microwave power. The ESR signal increases, reaches a maximum, and then decreases with increasing microwave power this behavior is called power saturation. From an analysis of the power saturation curve of ESR intensity versus microwave power, it is possible to determine Ti. 

In liquids the interactions between neighboring molecules are considerably more complicated than in gases. The resultant broadening obliterates the fine line structure seen in gas spectra, leaving only broad band profiles. There are many possible contributors to this broadening. In some cases, adequate approximation is obtained by assuming that the band contour is established by collisions. Ramsay (1952) has noted that substitution of appropriate molecular density and collision diameter numbers in the collision broadening formula results in realistic band widths for certain liquid-phase systems. In such systems, the bands typically show an approximately Lorentzian profile. Approximate deconvolution of inherently broadened liquid-phase spectra may therefore be obtained on the basis of the assumption of Lorentzian shape (Kauppinen et al., 1981). 

This short correlation time regime corresponds to that found in liquids, where narrow resonance lines with a Lorentzian shape are observed. 

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