Super-Lorentzian

Some swollen crosslinked polymer gels, such as styrene-divinylbenzene copolymers, exhibit high resolution spectra with so-called super-Lorentzian (SL) line shapes... 

A significant difference between TDAE-C60 and TDAE-C70 is also seen in the 270-MHz proton NMR spectra at 6.3 T. In powdered TDAE-C70 only a single proton resonance line of super-Lorentzian shape is observed at room temperature,... 

Fig. 9. The solid curve depicts the super-Lorentzian lineshape g(u>) for a single spin species in vesicles according to Eq. (57) for Q -= 100. The frequency scale is given in units of I/T2. The wide skirts on which the sharp line is superimposed are illustrated by the plot of 10g(aj) versus u> (dashed curve). A Lorentzian line (x 10) of width 100/72 is shown for comparison (small-dashed curve). Reproduced with permission from M. Bloom, Chem. Phys. Lipids, 1975, 14, 107. 1975 American Chemical Society.

The cholesteric phase line shape is similar to those found in smectic phases and could also be called super-Lorentzian. It is not believed, however, that the origins of the line shapes in the two kinds of phases are the same since the smectic phase has never been shown to have any preferred orientation in a magnetic field (27). It is shown below that the smectic line shape can be changed significantly by changing the molecular structure and that it appears to have its origin in the molecules rather than in the over-all orientation of the phase. 

Spectra of Neat and Middle Phases. Typical spectra of the neat and middle phases are shown in Figures 12 and 13. These spectra were obtained at 100°C. from systems of 70% surfactant (neat phase) and 40% surfactant (middle phase) in D20. The small peak, or shoulder, visible on the low-field side of the spectra arises from the residual HDO in the solvent. Shown also in the figures, by means of dots, are calculated Lorentzian lines having the same heights and widths as the experimental lines. The experimental lines have the super-Lorentzian shapes discussed above. 

The unique super-Lorentzian line shapes found in the lyotropic neat and middle phases can be explained in terms of a distribution of correlation times of the surfactant chain protons similar to the conditions in the waxy phases (17). This explanation can be related to the structures proposed by Luzzati (22) for these phases, shown in Figure 11, if it is assumed that the chain protons near the hydrophilic groups are more restricted in their motion than those near the ends of the hydrocarbon chains. Several lines of evidence support this hypothesis. 

Besides the composite proton NMR line shape which is observed just below on cooling (Fig. 1(E)), we can observe at least three different line shapes in the solid state of the polymer DDA9 namely, one below Tg, and two others, particularly meaningful, between and the cold crystallization temperature and above this temperature, respectively. The cold crystallization temperature, of DDA9 occurs about 50 °C below The proton NMR spectra observed between and have a super-Lorentzian shape, i.e. one in which the wings spread out more than in the true Lorentzian, and consequently the ratio he line widths... 

In theoretical studies, one usually deals with two simple models for the solvent relaxation, namely, the Debye model with the Lorentzian form of the frequency dependence, and the Ohmic model with an exponential cut-off [71, 85, 188, 203]. The Debye model can work well at low frequencies (long times) but it predicts nonanalytic behavior of the time correlation function at time zero. Exponential cut-off function takes care of this problem. Generalized sub- and super-Ohmic models are sometimes considered, characterized by a power dependence on CO (the dependence is linear for the usual Ohmic model) and the same exponential cut-off [203]. All these models admit analytical solutions for the ET rate in the Golden Rule limit [46,48]. One sometimes includes discrete modes or shifted Debye modes to mimic certain properties of the real spectrum [188]. In going beyond the Golden Rule limit, simplified models are considered, such as a frequency-independent (strict Ohmic) bath [71, 85, 203], or a sluggish (adiabatic)... 

In the absence of significant molecular motion the spectra can be calculated simply as a powder-like super-imposition of the individual molecular static lines of Lorentzian shape from all over the sample. These lines are then positioned into the spectrum according to Eq. (5) as in Ref. [6j. To include also dynamic effects, such as fluctuations of molecular long axes (defining the scalar order parameter S and the director n) and translational molecular diffusion, it is convenient to use a semi-classical approach with the time-dependent deuteron spin Hamiltonian [25] where the H NMR line shape I cj) is calculated as the Fourier transform of... 

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