Gaussian-like profile

The width of the ridge is related to the spatial extension of quantum coherence. Compared to the half width at half maximum (HWHM) of 6 A 1 of the Gaussian-like profile for scattering by a single slit, the observed HWHM of 0.20 A-1 reveals quantum coherence extending itself over more than 30 double slits (more than 300 A-1 along the x direction). This is certainly an underestimate since the instrumental resolution (A Q / Q 1%) is not negligible. 

As a solute is eluted, the distribution of its molecules along the longitudinal axes of the column changes, generating a Gaussian-like profile a band broadening effect whose maximum is known as time of retention as depicted in Fig. 1. This parameter, corrected for the time of retention of a solute, that is, not retained by the column (dead time, is related to the solute capacity factor k) by the relation ... 

Studied channels with circular cross-sections. However, in microfluidics, one frequently encounters other shapes. One example, shown in Figure 2.6(a,b), is the Gaussian-like profile that results from producing microchannels by laser ablation in the surface of a piece of the polymer poly methyl methacrylate (PMMA). The heat from the laser beam cracks the PMMA into monomer, MMA, which by evaporation leaves the substrate. A whole network of microchannels can then be created by sweeping the laser beam across the substrate in a well-defined pattern. The channels are sealed by placing and bonding a polymer lid on the top of the structure. 

Figure 2.6 Fabrication of tnicrochannels by laser ablation of polymer, PMMA (a) schematic diagram of the laser beam, the laser ablated groove, including the molten PMMA and the vaporized hemispherical cloud of MMA leaving the cut-zone, (b) Scanning electron microscope (SEM) micrograph of the cross-section of an actual microchannel showing the resulting Gaussian-like profile...

We examine a solute zone migrating under linear conditions. For the moment we confine our attention to the solute in the mobile phase. If complete equilibrium between phases existed at all points, the mobile-phase solute would form a Gaussian-like concentration profile as indicated by the shaded profile in Figure 10.7. However the actual profile for the mobile phase, shown by the dashed line, is shifted ahead of the shaded (equilibrium) profile due to solute migration. The basis of the profile shift is explained as follows. 

For the indication of absolute laser fluence values, e.g., for ablation, modification, melting etc. of a material, the determination of the spot size is crucial. Fluence values allow the comparison of laser treatment with different types of lasers showing various spatial beam characteristics like a square (e.g., excimer laser) or a Gaussian beam profile (e.g., Tksapphire laser). In the following, the determination of the Gaussian beam radius is described. 

Provided that the sample zone is initially formed as a sharp front of the substances analyzed, after a certain time t, due to diffusion the result will be a bell-like (Gaussian) concentration profile. This profile can be... 

The shape of the angular correlation distribution profile is a superposition of two contributions. Annihilation of the positron with the conduction electrons results in a contribution that has the approximate shape of an inverted parabola, while the contribution of the high-momentum core electrons is reflected in a broad, Gaussian-like distribution. When the core electron density is reduced, as is the case for a positron trapped in a vacancy, the contribution of the conduction electrons is enhanced and a narrower distribution is measured. Therefore, changes in the shape of the angular correlation distribution profile reflect a reduction in annihilation with core electrons [127]. 

A sine-shape has side lobes which impair the excitation of a distinct slice. Other pulse envelopes are therefore more commonly used. Ideally, one would like a rectangular excitation profile which results from a sine-shaped pulse with an infinite number of side lobes. In practice, a finite pulse duration is required and therefore the pulse has to be truncated, which causes oscillations in the excitation profile. Another frequently used pulse envelope is a Gaussian frmction ... 

It seen that the de-convolution is likely to be successful as the position of the peak maximum, and the peak width, of the major component is easily identifiable. This would mean that the software could accurately determine the constants in the Gaussian equation that would describe the profile of the major component. The profile of the major component would then be subtracted from the total composite peak leaving the small peak as difference value. This description oversimplifies the calculation processes which will include a number of iteration steps to arrive at the closest fit for the two peaks. 

The third key case for diffusion occurs when the solute is originally present as a very sharp pulse, like that shown in Fig. 3. The total amount of material in the pulse is M and the area across which the pulse is spreading perpendicular to the direction of diffusion is A. Under these cases the concentration profile is Gaussian ... 

Analytical self-consistent mean field theories were developed independently by Zhulina el al.29 30 and Milner et al.31,32 They are based on the assumption that for large stretchings of the grafted chains with respect to their Gaussian dimension, one can approximate the set of conformations of a stretched grafted chain by a set of most likely trajectories, and predict for such cases a parabolic density profile. In the calculations of the interactions between the two brushes, the interdigitation between the chains was ignored. 

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