Peak shape models Gaussian

Most GPC columns are provided with vendor estimates of the plate count of the column and a chromatogram of a series of test peaks. These plate count estimates are usually obtained using small molecule analytes that elute at the total permeation volume (Vp) of the column. The Gaussian peak shape model... 

The success of this modeling can be ascertained by the ability to replicate the observed peak shapes using Gaussian peaks centered at peak positions suggested by the modeling (Fig. 3). For the case where n = 3/2, five peaks were needed 1 for the monomeric A1 and 2 peaks each for each of the two dinuclear A1 species. These peaks were combined to successfully replicate the observed NMR peaks recorded for this sample. When n = 2, the data were reproduced using only 4 peaks, two each for each of the two dinuclear A1 species. Our earlier predictions (11) showed that HCl could combine with either of the dinuclear Al-species in three different positions, which showed different acid strengths. 

It is often convenient modeling the peak shape assuming some analytical functions [25]. The most commonly used functions are, at present, the Voigt and pseudo-Voigt functions, a combination of a Gaussian (G) and a Lore-ntzian (L) function centered at 20(y. An expression for Gaussian and Lorentzian contributions is ... 

A chromatogram without noise and drift is composed of a number of approximately bell-shaped peaks, resolved and unresolved. It is obvious that the most realistic model of a single peak shape or even the complete chromatogram could be obtained by the solution of mass transport models, based on conservation laws. However, the often used plug flow with constant flow velocity and axial diffusion, resulting in real Gaussian peak shape, is hardly realistic. Even a slightly more complicated transport equation... 

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 most common performance indicator of a column is a dimensionless, theoretical plate count number, N. This number is also referred to as an efficiency value for the column. There is a tendency to equate the column efficiency value with the quality of a column. However, it is important to remember that the column efficiency is only part of the quality of a column. The calculation of theoretical plates is commonly based on a Gaussian model for peak shape because the chromatographic peak is assumed to result from the spreading of a population of sample molecules resulting in a Gaussian distribution of sample concentrations in the mobile and stationary phases. The general formula for calculating column efficiency is... 

The majority of chromatographic separations as well as the theory assume that each component elutes out of the column as a narrow band or a Gaussian peak. Using the position of the maximum of the peak as a measure of retention time, the peak shape conforms closely to the equation C = Cjjjg, exp[-(t -1] ) The modelling of this process, by traditional descriptive models, has been extensively reported in the literature. 

The DV-Xa calculations were made with C, symmetry for models A, C, and D, and without symmetry for model B. Numerical atomic orbitals of lx to 5p for Cu, and lx to 2p for C, N and O, and lx for H were used as a basis set for the DV-Xa calculations in the ground and transition states. The sample points used in the numerical integration were taken up to 30000 for each calculation. Self-consistency within 0.001 electrons was obtained for the final orbital populations. Transition probabilities calculated for each model were convoluted by a Gaussian function with a half-width at half-height (HWHH) of 1.0 eV to make transition peak shapes comparable with experimental XANES spectra. 

Many spectrometric peaks can be described reasonably well in terms of one of two model peak shapes, those of a Gaussian and of a Lorentzian (or Cauchy) peak. Leaving out the normalizing factors as immaterial in the present context, the Gaussian peak can be described as... 

The models used to predict peak shape, based on Gaussian distributions, have the advantage of using very intuitive parameters, related to properties which can directly be measured on the chromatograms (position and height of the maxima, and width of the peaks). The equation describing a pure Gaussian peak is ... 

Another approach for data processing involves simulation of pure spectra. These model spectra are then taken for a quantitative description of the mixture spectra. This procedure is referred to as indirect hard modelling (IHM). Obviously, changes in line shape, line width, and chemical shift may occur as function of concentration and due to system imperfections which are taken into account by IHM. The peaks are modelled by Voigt-functions with variable Gaussian to exponential ratio. The main advantage of IHM is that it allows a limited physical interpretation of the models. Further, unlike PLS based methods, IHM only requires reference spectra of the pure compounds, reducing the calibration effort drastically. 

The Thompson-Cox-Hastings function is often used to refine profiles with broad diffraction peaks because it is the more appropriate model for line-broadening analysis where the Lorentzian and Gaussian contributions for crystallite size and for microstrains are weighted. So in this case, the peak shape is simulated by the pseudo-Voigt function, which is a Unear combination of a Gaussian and a Lorentzian function (Table 8.5). 

Several of the standard statistical distributions are described by Hahn Shapiro (Statistical Models in Engineering, 1967) with mention of their applicability. The most useful models are the Gamma (or Erlang) and the Gaussian and some of their minor modifications. As an illustration of something different the Weibull distribution is touched on in problem P5.02.18. These distributions usually are representable by only a few parameters that define the asymmetry, the peak and the shape in the vicinity of the peak. The moments are such parameters. 

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