Wing parameter

As most experimenters want to know not the absolute value of the linewidth but how it changes as a function of physical parameters, these ratios have been taken up as the simplest way of describing the linewidth. C/T is called the S (sharpness) parameter, and (A+E)/T is W, the wing parameter. As we can induce from Figure 3.7, S and W should be sensitive to changes in the momentum density of lower- and higher-momentum electrons, respectively. Positron annihilation in open volume defects thus typically leads to an increase in 5 and a decrease in W. 

W wing parameter of the annihilation line Tb bulk lifetime... 

The quasi-classical description of the Q-branch becomes valid as soon as its rotational structure is washed out. There is no doubt that at this point its contour is close to a static one, and, consequently, asymmetric to a large extent. It is also established [136] that after narrowing of the contour its shape in the liquid is Lorentzian even in the far wings where the intensity is four orders less than in the centre (see Fig. 3.3). In this case it is more convenient to compare observed contours with calculated ones by their characteristic parameters. These are the half width at half height Aa)i/2 and the shift of the spectrum maximum ftW—< > = 5a>+A, which is usually assumed to be a sum of the rotational shift 5larger scale A determined by vibrational dephasing. 

In Figs. 24 and 25 we show the measured double differential cross sections for electron emission at zero degrees in collisions of 100-keV protons with He and H2 [39] compared to CDW-EIS predictions [39]. Uncertainties associated with the experimental results vary from 1% near the electron capture to the continuum peak to about 15% near the extreme wings of the distribution. These results have been scaled to provide a best fit with CDW-EIS calculations. In both cases there is satisfactory agreement between the CDW-EIS calculations and experiment, particularly with excellent agreement for electrons with velocities greater than v, where v is the velocity of the projectile. For lower-energy electrons the eikonal description of the initial state may have its limitations, especially for lower-impact parameters. 

The determination of relative configurations in saturated four-membered ring systems requires a conformational analysis because they exist in noncoplanar wing-shaped conformations461,462, meaning that experimental NMR parameters may be weighted averages. 

With the full Arrhenius rate law, an extra unfolding parameter y is introduced. Even then, however, the appropriate stationary-state condition and its derivatives for the winged cusp cannot be satisfied simultaneously (at least not for positive values of the various parameters). Thus we do not expect to find all seven patterns. 

In an attempt to model the spectral functions of rare gas mixtures, Fig. 3.2, it was noted that a Gaussian function with exponential tails approximates the measurements reasonably well [75], about as well as the Lorentzian core with exponential tails. Two free parameters were chosen such that at the mending point a continuous function and a continuous derivative resulted the negative frequency wing was again chosen as that same curve, multiplied by the Boltzmann factor, to satisfy Eq. 3.18. Subsequent work retained the combination of a Lorentzian with an exponential wing and made use of a desymmetrization function [320],... 

It is a straightforward matter to fit various model profiles to realistic, exact computed profiles, selecting a greater or lesser portion near the line center of the exact profile for a least mean squares fit. In this way, the parameters and the root mean square errors of the fit may be obtained as functions of the peak-to-wing intensity ratio, x = G(0)/G(comax)- As an example, Fig. 5.8 presents the root mean square deviations thus obtained, in units of relative difference in percent, for two standard models, the desymmetrized Lorentzian and the BC shape, Eqs. 3.15 and 5.105, respectively. 

Numerical problems arising from the use of the Lorentzian for fitting spectra have also been reported [69]. These are related to the non-dif-ferentiable profile at the points where the exponential wings are attached to the Lorentzian core. Partial derivatives with respect to the line shape parameters are usually needed in least mean squares fitting procedures. 

Computations of the EPR line shape made in ref. 71 with the help of eqn. (18) for the model distribution functions f(R) that are frequently used in radiation chemistry, have shown that the shape of the wings of the EPR lines is far more sensitive to changes in the distribution functions of radical pairs over the distances than to changes in such a conventionally used parameter as the line width between the points of the maximum slope, A//p. Thus, to estimate the distances of tunneling and their variations in the course of a reaction it is necessary to analyze the shape of the wings of the EPR lines. 

Reaction-diffusion systems provide a means to subdivide successively a domain at a sequence of critical parameter values due to size, shape, diffusion constants, or other parameters. The chemical patterns that arise are the eigenfunctions of the Laplacian operator on that geometry. The succession of eigenfunctions on geometries close to the wing, leg, haltere, and genital discs yield sequential nodal lines reasonably similar to the observed sequence and symmetries and geometries of the observed com-... 

The slope and position of the cut line also depend on the solubility parameter of the precipitating solvent as well as the temperature of operation. For paraffin solvents, the overall solubility parameter of the solvent is to the left of the composition map and the cut line is the right-hand wing of the solubility bell curve as it cuts through the composition map. 

The cut line for extraction with polar solvents such as phenol, cresol, n-methylpyrrollidone, and furfural is slanted in the opposite direction from solvent precipitation, because the solubility parameter of the polar solvent lies to the right of the composition map. Thus, the cut line is the left-hand wing of the solubility bell curve of the polar solvent. It moves up and down and left and right as the solubility parameter of the polar solvent is changed. Furthermore, it is also sensitive to the operating conditions of the extraction, such as temperature and solvent-to-oil ratio. 

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