Gaussian lineshapes

Normalized Gaussian lineshape function as a function of frequency... 

Random broadening effects can dominate the lineshape and lead to a lineshape that is almost Gaussian. This is often the case in MRS.32 The Gaussian lineshape is also a convenient approximation to the lineshape for some solids.10 The normalized Gaussian lineshape function in the frequency domain is given by... 

Fig. 1. The Voigt lineshape is plotted for three different values of the Voigt parameter, a - WL/WG, namely a = 0 (Gaussian, dotted line), a = 1 (solid line, with the component Lorentzian and Gaussian lineshapes having an equal width given by tl/(, W 0.6107 H-) and a x (Lorentzian, dashed line). The frequency scale is given in units of the FWHM of the Voigt lineshape, W.

To describe symmetric lineshapes, Maltempo80 defined linear combinations of normalized Lorentzian and Gaussian lineshapes (hybrid lineshapes) as the function... 

The rational function approximations to the Lorentzian and Gaussian lineshapes were written as... 

The full width at half-maximum (FWHM) of the Lorentzian curve of Fig. 11.54 is FWHM/ = 3.1905 for the Gaussian of equal peak height, FWHMg = 2.3549 at equal peak heights, a Gaussian lineshape is narrower, while the Lorentzian is broader—that is, has more intensity far from the peak. In practice, NMR or EPR lineshapes can also be intermediate between Lorentzian and Gaussian. 

Avj/2 can be approximated by (nT2)1 for Lorentzian lineshapes and (2.13 T2) 1 for Gaussian lineshapes.94 Therefore, for solid-state materials with Gaussian... 

The width (fwhm) of the electronic origin peak at 23,836 cm is 6.5 cm" at T = 1.2 K. Its lineshape can be well approximated by a Lorentzian Hneshape function, while a fit to a Gaussian lineshape is unsuccessful [74]. This indicates that the peak is homogeneously broadened [103,104]. Thus, the width of 6.5 cm" can be related to the lifetime of this Sj state to r(Si) = 8 x 10 s. (Compare Eq. (2).) Eurther, the high phosphorescence intensity that is observed from the Tj state when the Sj state is excited, indicates that the lifetime of this Sj state is largely determined by the intersystem crossing process from Sj to Tj. This would correspond to the rate of kjs = 1/r(Si) = 10 s. A value of the same order of magnitude has also been determined for Pd-phthalocyanine [144]. 

Simulated NMR spectra are generated by summing four Gaussian lineshapes. Two peaks of equal intensity at 152.6 and 148.4ppm represent contributions from etherified units and two peaks of equal intensity at 148.4 and 145.0 ppm represent contributions from nonetherified units. The width of each component should be about 210 Hz at half height, although some variation may be observed between spectrometers. Relative contributions are adjusted until the result fits the experimental spectrum across the range 143 to 156 ppm. The procedure is illustrated in Fig. 4.5.9 for spruce wood (spectrum as dotted in Fig. 4.5.8). The best fit was obtained for 60% etherification. 

Figure 7.6. A. Na MAS NMR spectrum of the mullite-structured compound NaAl90 4 acquired at 8.45 T. B. Na DOR spectrum of the same compound acquired at 8.45 T. C. Observed and simulated 8.45 T MAS NMR spectrum of NaAl90 4 showing the separate simulated lineshapes of the two crystalline Na sites. The Gaussian lineshape on the left is from the residual amorphous precursor gel phase. From MacKenzie et al. (2001), by permission of the copyright owner.

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