Gaussian profiles

All molecules of a given component do not move at an identical velocity. The centre of the elution band represents the average retention time of the analyte. Some molecules travel slower, some faster. This variation in velocity is a result of the number of sorption and desorption of the analytes and is what gives the detected signal its Gaussian profile. 

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Assuming a Gaussian profile, the extent of band broadening is measured by the variance or standard deviation of a chromatographic peak. The height of a theoretical plate is defined as the variance per unit length of the column... 

Recently the effect of intrinsic traps on hopping transport in random organic systems was studied both in simulation and experiment [72]. In the computation it has been assumed that the eneigy distribution of the traps features the same Gaussian profile as that of bulk states. 

The globally optimal laser field for this example is presented in Fig. 2. The field is relatively simple with structure at early times, followed by a large peak with a nearly Gaussian profile. Note that the control formalism enforces no specific structure on the field a priori. That is, the form of the field is totally unconstrained during the allotted time interval, so simple solutions are not guaranteed. Also shown in Fig. 2 is the locally optimal... 

Pettinger et al. observed TERS with a sharp Au tip for MGITG dye on Au(l 11) with a side illumination [28]. They studied the bleaching of the dye and fitted the data by taking into account the radial varying intensity distribution of the field as a Gaussian profile instead of a Heaviside profile. For the former profile, the TERS radius is smaller by a factor of 1/2 than for the latter profile. They obtained a TERS enhancement factor of 6.25 x 10. The radius of the enhanced field Rfieid is about 50 nm, which results in a TERS radius of 25 nm, which is smaller than the radius of the tip apex (about 30 nm). 

The Voigt function is a convolution product ( ) between L and G. As the convolution is expensive from a computational point of view, the pseudo-Voigt form is more often used. The pseudo-Voigt is characterized by a mixing parameter r], representing the fraction of Lorentzian contribution, i.e. r] = 1(0) means pure Lorentzian (Gaussian) profile shape. Gaussian and Lorentzian breadths can be treated as independent parameters in some expressions. 

Since the parameter y is non-vanishing, the wave packet will disperse with time as indicated by equation (1.28). For a gaussian profile, the absolute value of the wave packet is given by equation (1.31) with y given by (1.43). We note that y is proportional to m, so that as m becomes larger, y becomes smaller. Thus, for heavy particles the wave packet spreads slowly with time. 

The radial velocities were estimated by identifying a few lines (3 to 6, depending on the quality of the spectra) and fitting a Gaussian profile to each line to find the line core. We estimated the average radial velocity, and we present here only the stars with standard deviation of the mean less than 7 km s 1. 

As normal peaks have a Gaussian profile, which approximates to an isosceles triangle, their area can be estimated by multiplying the height by... 

The measurements of Rouse, Yih and Humphries (1952) [1] helped to generalize the temperature and velocity relationships for turbulent plumes from small sources, and established the Gaussian profile approximation as adequate descriptions for normalized vertical velocity (w) and temperature (7), e.g. 

Properties are assumed uniform across the plume at any elevation, z. This is called a top-hat profile as compared to the more empirically correct Gaussian profile given in Equation (10.1). 

Solutions for a Gaussian profile, with Equation (10.1) for an axisymmetric point source, and for a line source... 

Table 10.2 gives correlation results based on Gaussian profiles with [3 selected as 1 for the axisymmetric and line-fire plumes [12]. It is indeed remarkable that the local ... 

Emission Asymmetric, often tailing to long- Symmetric, Gaussian-profile,... 

Gaussian profiles are also utilized to approximate peak shapes observed in different types of spectroscopy. Again, we need to stress that the actual molecular processes behind a spectroscopically observed transition are very complex and do not strictly follow Gaussian curves. However, here too, Gaussian curves can serve as useful approximations. 

We use the function gauss.m in several examples, not only for the generation of chromatographic concentration profiles, but also for the generation of absorption spectra since these can often be approximated by a combination of Gaussian profiles. For example ... 

Under an extended area of irradiation, Tc is only a function of z. Under spot irradiation the lateral coordinates also have to be treated as variables. For simplicity we use cylindrical Gaussian profiles and //o = 1. Then... 

Resolution, on the other hand, is a more technical term. It refers to the distance between adjacent bands relative to their bandwidths and acknowledges the fact that proteins are distributed in Gaussian profiles with overlapping distributions. The numerical expression for resolution is obtained by dividing the distance between the centers of adjacent bands by some measure of their average bandwidths. It expresses the distance between band centers in units of bandwidth and gives a measure of the overlap between two adjacent bands. For preparative applications, when maximal purity is desired, two proteins to be isolated should be separated by at least a bandwidth. In many applications it is sufficient to be able to simply discern that two bands are distinct. In this case bands can be less than a bandwidth apart. 

Figure 5.7 Profiles of the time-averaged concentration at four downstream locations. The profiles are self-similar and the solid line corresponds to a Gaussian profile shape.

This is illustrated in Figure 4.2. The peak capacities of the two dimensions are shown as the number of adjacent Gaussian profiles that can be packed into the space along the respective separation coordinates. The separation plane is divided into rectangular boxes that represent the resolution units in the 2D plane. The total peak capacity 2d is therefore approximately equal to the number of such boxes. 

The most recent calculations, however, of the photoemission final state multiplet intensity for the 5 f initial state show also an intensity distribution different from the measured one. This may be partially corrected by accounting for the spectrometer transmission and the varying energy resolution of 0.12, 0.17, 0.17 and 1,3 eV for 21.2, 40.8, 48.4, and 1253.6 eV excitation. However, the UPS spectra are additionally distorted by a much stronger contribution of secondary electrons and the 5 f emission is superimposed upon the (6d7s) conduction electron density of states, background intensity of which was not considered in the calculated spectrum In the calculations, furthermore, in order to account for the excitation of electron-hole pairs, and in order to simulate instrumental resolution, the multiplet lines were broadened by a convolution with Doniach-Sunjic line shapes (for the first effect) and Gaussian profiles (for the second effect). The same parameters as in the case of the calculations for lanthanide metals were used for the asymmetry and the halfwidths ... 

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