Peak shape simulation

Better peak shape simulations take into account the basic van Deemter-Golay equations to compute the degree of peak broadening that would occur under the set of isothermal conditions, or with each simulation step for temperature programming. A more intensive and accurate approach uses the Giddings-Golay equation (22,23), which includes additional compensation terms for carrier-gas expansion. In either case, ultimately the chromatographer must transfer an optimized set of conditions into an instrument and evaluate the efficacy of the optimization procedure. 

Amatore et al. developed a theoretical framework to describe the electrochanical responses of ultramicroelectrode ensemble and NEEs by considering mass transport for assemblies of microdisk and microband electrodes. Lee et al. used finite element simulation to solve 3D diffusion equations and found that a collection of 10 pm diameter microdisk electrodes required a separation distance of more than 40R to exhibit a sigmoidal simulated CV response typical for radial diffusion. " CV response typical of reversible linear diffusion at macroelectrodes was observed when the separation distance was less than 6R . Assemblies of microelectrodes for which the separation distances were between 6R and 4QR exhibited peak-shaped simulated CVs indicative of a mixture of radial and linear diffusion behavior. Thus, 12/ seems to be too small a separation distance for the design of ideal microelectrode arrays. 

Comparison of the measured peak shape with simulations based on Equations (2-5) and (2-6) reveals that a nucleation and growth model best describes the reduction... 

Experimental and simulated cyclic voltammograms for a solution that was 5 [iM in TMAFc and 0.5 mM in supporting electrolyte (sodium nitrate) are shown in Fig. 7 [25], The experimental data were obtained at a lONEE. In agreement with the above discussion, the experimental voltammograms are peak shaped, and peak current increases with the square root... 

A third problem with simulation of high resolution diffraction data is that there is no unique instrament function. In the analysis of powder diffraction data, the instalment function can be defined, giving a characteristic shape to all diffraction peaks. Deconvolution of these peaks is therefore possible and fitting techniques such as that of Rietveld can be used to fit overlapping diffraction peaks. No such procedure is possible in high resolution diffraction as the shape of the rocking curve profile depends dramatically on specimen thickness and perfection. Unless you know the answer first, you cannot know the peak shape. 

When a particular component eluting at a certain retention volume is to be estimated, this approach can be outlined as follows. Since SEC is extremely reproducible, the peak shape, peak width and peak height are dependent on the amount of the species in the sample volume injected, sample volume and retention time. From these factors the SEC peaks can be simulated or elution pattern of any species within the separation range can be plotted as a function of mass vs. retention volume. The analysis data supplies the concentration of this particular species over two or more 0.5 ml intervals. A match-up computer program has to be developed so that it can pick up the peak shape and concentration based on 3 or 4 data points at known Intervals. 

Fig. 11. High-resolution 29Si MAS NMR spectra of synthetic zeolites Na-X and Na-Y at 79.80 MHz (58). Experimental spectra are given in the left-hand columns Si(nAl) signals are identified by the n above the peaks. Computer-simulated spectra based on Gaussian peak shapes and corresponding with each experimental spectrum are given in the right-hand columns. Individual deconvoluted peaks are drawn in dotted lines.

Comparison of the measured peak shape with simulations based on Eqs. (2-5) and (2-6) reveals that a nucleation and growth model describes the reduction of Fe304 to Fe best. Thus, the formation of metallic iron nuclei at the surface of the particles is the difficult step. Once these nuclei have formed, they provide the site where molecular hydrogen dissociates to yield atomic hydrogen, which takes care of further reduction. The studies of Wimmers and co-workers [8] show nicely that TPR allows for detailed conclusions on reduction mechanisms, albeit in favorable cases only. 

There are several ways of detecting peaks in such noisy signals. The Wiener-Hopf filter minimizes the expectation value of the noise power spectrum and may be used to optimally smooth the original noisy profile [19]. An alternative approach described by Hindeleh and Johnson employs knowledge of the peak shape. It synthesizes a simulated diffraction profile from peaks of known width and shape, for all possible peak amplitudes and positions, and selects that combination of peaks that minimizes the mean square error between the synthesized and measured profiles [20], This procedure is illustrated... 

The effect of various degrees of reduction on a 2-D pattern is shown on Figure 5.6. The pattern is taken from a simulated 2-D NMR spectrum (ref. 8). The peaks are drawn as contours at different levels to show the consequence of the reduction on the original intensities. Comparing the results of the two transformations, the advantage of FHT is lesser noise, while the FFT gives a more faithful reproduction of original peak shapes. 

A good refinement of the data with Rpr 14.5 percent has been obtained in Pm3m (Figure 7). Most of the discrepancies between the observed and calculated profiles stem from our inability to simulate the peak shapes for the low angle reflections, a problem that arises from the large vertical acceptance angle of the counters on the D1A diffractometer. Since we have now established that the superlattice is absent, this experiment could usefully be repeated with a shorter neutron wavelength. 

The computer models described provide a functional simulation of SEC-viscometry-LS analysis of linear polymers. The results for the Flory-Schulz MWD are in qualitative agreement with previous results for the Wesslau MWD. Both models emphasize the importance of determining the correct volume offset between the concentration detector and molecular weight-sensitive detectors. For the Flory-Schulz model, the peak shape, as well as the peak elution volume, can provide information about molecular weight polydispersity. Future work will extend the model to incorporate peak skew and polymer branching. 

With this boundary condition, the Navier-Stokes equations for longitudinal F, and radial V, components of EOF velocity were solved numerically, and the calculated EOF profiles were used to simulate the solute peak shapes. The situation where the part of the capillary length was modified to the zero value of the zeta potential and the part of capillary was not modified + 0)... 

Figure 6.16. A. Observed and simulated O MAS NMR centre band spectrum of crystalline diopside (CaMgSi206) showing the individual fitted components. From Timken et al. (1987) by permission of the American Chemical Society. B. Three-quantum MAS NMR spectrum of forsterite showing resolution of the three non-bridging oxygen sites. C. Cross-sections parallel to the p2 axis of the MQMAS spectrum in B from which the xq and t values were derived by computer simulation of these peak shapes. From Ashbrook et al. (1999) by permission of the Mineralogical...

Moment analysis offers only two global parameters to characterize the peak while the exact peak shape is not taken into account, which makes this method more sensitive to signal distortions. By fitting of either analytical equations (Section 6.5.33) or simulation results to the peak shape this drawback may be overcome. It also allows an easy comparison between the calculated peak and the measured concentration profile. 

Figure 17 Crosses show oh/od obtained by fitting simulated data with non-Gaussian peak shapes for H and D, with the standard fitting programs, which assume that the peak shapes are Gaussian. The circles show values of oh /on obtained by fitting experimental data.

IL and ML represent already hybrid functions between G and L and simulate rather well the shape of a symmetrical X-ray reflection. If a variable slope width is introduced as a 4th parameter one gets the two following, widely used peak shape functions ... 

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