Gaussian curves distorted

There have been several attempts to modify the Gaussian curve in order to better model distorted chromatographic concentration profiles. This might be useful if model-based fitting is applied to asymmetrical chromatograms. The following skewed Gaussian function is often suitable. 

On the chromatogram, a represents the half-width of the peak at 60.6 per cent of its height and the retention time of the compound, and a should be measured in the same units (time, distances or eluted volumes if the flow is constant). If a is expressed in units of volume (using the flow), then 4cr corresponds to the volume of the peak , that contains around 95 per cent of the injected compound. By consequence of the properties of the Gaussian curve w = 4cr and Wi 2 — 2.35cr), Equation 1.9 results. However, because of the distortion of most peaks at their base, expression 1.9 is rarely used and finally Equation 1.10 is preferred. 

The most important parameter of the detection system is its response function. We have studied this extensively in Monte Carlo and other calculations. The calculated time-spectrum response to monoenergetic neutrons is composed of a Gaussian timing curve (2.97-ns FWHM), a trapezoidal contribution from detector thickness and non-axial paths, and an exponential tail, calculated by Monte Carlo, from multiple scattering in the neutron scintillator. (Spectrum distortion due to neutrons multiply scattered by structural and other parts of the apparatus and arriving at the neutron... 

Tables 1 and 2 gives the numerical data for a series of vanadium (II), chromium (III), manganese (IV), molybdenum (III), rhenium (IV), iridium (VI), cobalt (II), and nickel (II) complexes. The first spin-allowed absorption band, caused by an internal transition in the partly filled shell, has the wavenumber equal to A. If spin-forbidden transitions are superposed on this band, a certain distortion from the usual shape of Gaussian error curve can be observed, and one takes the centre of gravity of intensity as the corrected wavenumber ai. One has to be careful not to confuse electron transfer or other strong bands with the internal transitions discussed here. Obviously, one has also to watch for absorption due to other coloured species, produced e. g. by oxidation or hydrolysis of the solutions. In the case of certain octahedral nickel (II), and nearly all tetrahedral cobalt (II) complexes, the first band has not actually been...

The present treatment of solvent broadening is of course an oversimplification. It does not take into account the dynamic nature of the interaction or the contributions of solvent excitations to the resonant state. On a more elementary level, the representation of A by a Lorentzian is probably unrealistic since solvent-broadened lines often approximate a Gaussian shape. Our choice of a Lorentzian is made to obtain analytical expressions for the cross section a Gaussian lineshape requires numerical procedures, but presents no fundamental difficulties. It turns out (O. Sonnich Mortensen, unpublished) that, for a given halfwidth A, a Gaussian tends to distort REPs and polarization dispersion curves more strongly than a Lorentzian. 

The slit function can be observed well in the imaging of line sources when the inlet and outlet slits (and the middle slit if present) are moved synchronously. If the slit is too wide the line of a mercury lamp does not appear as the expected Gaussian-shaped curve intensity distribution, but as a triangle. If the inlet and exit slits are different, a trapezoid is obtained. Triangular shapes in the spectrum indicate defective adjustment of slit widths in the equipment. These effects are observed mainly in spectrometers in which only a small number of preset slit widths can be selected. Distortion of a spectral band is negligible only if... 

The explicit solution of the above equation is rather complicated,but the distortion of a normal Gaussian peak can be clearly illustrated from curves obtained from the above equation by the use of a computer. Consider a column, 25 cm long and 4.6 mm in diameter having a dead volume (Vq) of 3 ml, an efficiency n of 12,000 theoretical plates and operated at a flow rate of 1 ml/min (Q). From the plate theory the standard deviation of the dead volume peak in ml of mobile phase will be,... 

Fig. 15-2 (a) Plots of the smoothed-out profiles of Eq. (15-9), where m = 0 is the Gaussian profile and m = oo is the step profile, and (b) the solid curves are the approximation of Table 15-2 for the distortion parameter, while the dashed curves are numerical solutions of the scalar wave equation [7]. 

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