Gaussian bands

On the basis of these assumptions, the position of a band on the chromatogram is defined by measuring the distance between the origin and the geometrical center of the band. Despite the considerable imprecision of this definition for skewed (i.e., non-Gaussian) bands, its two feamres are of great importance ... 

Figure 54-1, however, still shows a number of characteristics that reveal the behavior of derivatives. First of all, we note that the first derivative crosses the X-axis at the wavelength where the absorbance peak has a maximum, and has maximum values (both positive and negative) at the point of maximum slope of the absorbance bands. These characteristics, of course, reflect the definition of the derivative as a measure of the slope of the underlying curve. For Gaussian bands, the maxima of the first derivatives also correspond to the standard deviation of the underlying spectral curve. 

In summary, for displaced oscillators, absorption and emission spectra show a mirror image relation and for the strong coupling case, a(oo) will exhibit a Gaussian band shape, absorption maximum independent of temperature, and bandwidth increasing with temperature. It should be noted that the distortion effect and Duschinsky effect have not been considered in this chapter, but these effects can be treated similarly. 

Tables 5 and 6 gives the temperature variations of the parameters of the component Gaussian bands in the stretching mode regions of HaO(as), D20(as), and HD0/D20(as). As is evident from the entries, the band centers and half-widths of the component Gaussians remain nearly constant.

Little is known about the fluorescence of the chla spectral forms. It was recently suggested, on the basis of gaussian curve analysis combined with band calculations, that each of the spectral forms of PSII antenna has a separate emission, with Stokes shifts between 2nm and 3nm [133]. These values are much smaller than those for chla in non-polar solvents (6-8 nm). This is due to the narrow band widths of the spectral forms, as the shift is determined by the absorption band width for thermally relaxed excited states [157]. The fluorescence rate constants are expected to be rather similar for the different forms as their gaussian band widths are similar [71], It is thought that the fluorescence yields are also probably rather similar as the emission of the sj tral forms is closely approximated by a Boltzmann distribution at room temperature for both LHCII and total PSII antenna [71, 133]. 

For PSII it has been demonstrated by gaussian band analysis of room temperature absorption spectra that long wavelength bands are not concentrated in the chl-protein complexes of the core antenna [71]. 

The Cj/Dj ratios are generally obtained by fitting the observed MCD and absorption spectra to a series of Gaussian band-shape functions under the assumption that all of the MCD intensity is due to temperature-dependent MCD (164-166). Following the fit, eight transitions were identified in the spectrum of each protein. 

An infinitely sharp zone of solute is placed at the center of a column at time t = 0. After diffusion for time the standard deviation of the Gaussian band is 1.0 mm. After 20 min more, at time f2, the standard deviation is 2.0 mm. What will be the width after another 20 min. at time r3 ... 

Gaussian curves (normal distribution functions) can sometimes be used to describe the shape of the overall envelope of the many vibrationally induced subbands that make up one electronic absorption band, e.g., for the absorption spectrum of the copper-containing blue protein of Pseudomonas (Fig. 23-8) Gaussian bands are appropriate. They permit resolution of the spectrum into components representing individual electronic transitions. Each transition is described by a peak position, height (molar extinction coefficient), and width (as measured at the halfheight, in cm-1). However, most absorption bands of organic compounds are not symmetric but are skewed... 

Figure 23-8 Resolution of the visible circular dichroism (ellipticity) spectrum (A) and absorption spectrum (B) of the Pseudomonas blue protein into series of overlapping Gaussian hands (—). The numbers 1 to 6 refer to hands of identical position and width in both spectra. Absorption envelopes resulting from the sum of the set of overlapping Gaussian bands (—) correspond within the error of the measurement to the experimental spectra. The dashed part of the CD envelope above 700 nm was completed by a curve fitter with the use of a band in the position of hand 1 of the absorption spectrum. From Tang et al.68...

Figure B3.5.7 CD spectrum of D-(+)-10-camphorsulfonic acid (CSA) in water. The vertical bars represent variations of 1.5% and 5%. The broken line represents the extrapolation of a gaussian band. Commercial CSA was twice recrystallized. (From Chen and Yang, 1977.)...

The separation of compounds by their differential partition between two immiscible phases is the basis for partition chromatography. The system consists of a stationary liquid phase coated on an inert solid support, and an immiscible mobile phase. Chromatographic separations are based on the different equilibrium distributions of the samples between these two phases. The greater the quantity of substance in the stationary phase at equilibrium the dower is the migration. For analyses, this equilibrium must remain constant over a suitable concentration range. Thus an increase in the concentration of solute results in a linear increase in the concentration of solute in the mobile and stationary phase, respectively. Under these conditions, the retention time, tR, is independent of the amount of sample chromatographed and a symmetrical peak (gaussian band) is observed. 

Figure 41 Left panel calculated 62 first asymmetric peak (- - -) and its Gaussian fit (—) for the (a) Sip]-SiC>2, (b) Si[2]-SiC>2 and (c) S1O2 superlattices. The letter I indicates the interface Gaussian band while the letter Q indicates the bulk-like Gaussian band. Right panel PL spectra of c-Si/Si02 single quantum wells under 488 nm laser excitation at 2 K (a) 1.7 nm, (b) 1.3 nm and (c) 0.6 nm thickness. The asymmetric PL spectra can be fitted by two Gaussian bands, the weak Q band and the strong I band [51],...

Figure 4.3 Left - 10 three-component spectra having a single Gaussian band for each component. Right - First four loading spectra produced from 20 two-component spectra.

Figure 1. CD spectrum of SBA in the (a) far-UV and (b) near-UV regions. The lectin concentration was 0.046 mg/ml in the far-UV region and 0.48 mg/ml in the near-UV region. The light path length was 1 mm below 250 nm and 1 cm above 250 nm. The solid curves were constructed from four recordings each. The far-UV region was resolved into gaussian bands using the curve resolver. Solvent, CMF-PBS, pH 7.5. Bars indicate maximum deviation from mean. Comparable CD patterns were obtained using SBA, prepared by the method of Lotan et al.

It is possible to model the vibronic bands in some detail. This has been done, for example, by Liu et al. (2004) forthe 6d-5f emission spectrum of Pa4+ in Cs2ZrCl6, which is analogous to the emission spectrum of Ce3+. However, most of the simulations discussed in this chapter approximate the vibronic band shape with Gaussian bands. The energy level calculations yield zero-phonon line positions, and Gaussian bands are superimposed on the zero-phonon fines in order to reproduce the observed spectra. Peaks of the Gaussian band are offset from the zero phonon fine by a constant. Peak offset and band widths, which are mostly host-dependent, may be determined from examination of the lowest 5d level of the Ce3+ spectrum, as they will not vary much for different ions in the same host. It is also common to make the standard... 

The emission spectra for LiYF4 Pr3+ are shown in fig. 7 as an illustration, where the top and bottom panels present the calculated and experimental spectra, respectively. As can be seen from this figure, the calculations correctly reproduce the relative intensities of the emission bands. The calculated spectra are produced by superimposing a Gaussian band that is offset from the zero-phonon line by 600 cm-1 on the calculated zero-phonon lines. The FWHM is set to 1000 cm-1 for the Gaussian (phonon) bands, and 20 cm-1 for the zero-phonon lines. The zero-phonon lines for the 4f5d - 3H4 emission are not observed, most likely due to resonant reabsorption. 

The calculations do not attempt to model detailed vibronic structure. The simulated spectra are produced by superimposing a Gaussian band on the zero-phonon line, rather than calculating the individual vibronics. For the emission spectra of lanthanides in LiYF4, individual vibronic features are not clearly resolved (see fig. 7), and the calculated broad vibronic bands give a reasonably good description. By contrast, for lanthanides in YPO4, the offset between... 

Fig. 8. Calculated and measured emission spectra of YP04 Pr3+ from Peijzel et al. (2005a). The bars in the upper spectrum give information on the positions and intensities calculated for the zero-phonon lines, while the spectrum is obtained by superimposing a Gaussian band (offset 600 cm-1, width 1000 cm-1) on the zero-phonon lines.

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