Cross section average

Probably the easiest place to begin is with a consideration of the product 6 A. As defined by Fig. 2.8, 5 is the average diameter of a hole and A is its average cross section. 

If the critical separation is determined for a large number of relative geometries of the electron and molecule it is possible to obtain a three-dimensional picture of the probability of ionization as a function of the orientation of the molecule. Effectively, the idea of an ionization cross section, the area the target molecule presents to the electron, is extended to a three-dimensional object defined by the critical distances, with ionization occurring when the electron penetrates the surface enclosing this volume. The volume enclosed by the electron impact ionization surface may be used to obtain an estimate for the cross section (volume averaged cross section) ... 

Ionization surfaces calculated for several of the molecules listed in Table 1 are shown in Figures 10 and 11. The volume averaged cross sections determined from the volumes enclosed by these surfaces for nonspherical molecules such as C02 are in much better accord with experiment, consistent with the idea that the poor performance of Cartesian averaging for molecules such as C02 is due to the large departure from a spherical shape. Improved agreement with experiment is also... 

A quite different aspect of local kinetics is that having to do with changes of charge state, e.g., between H+ and H° or H° and H. Such changes require emission or absorption of electrons or holes. Since the mean free paths of these carriers are large compared with atomic dimensions, it is customary (see for example Lax, 1960) to use a velocity-averaged cross section a as the key descriptor of the rate of a capture reaction such as H+ + e— H°. Explicitly, we write, for this case,... 

The average cross-sectional area (S) of the repeating unit can be calculated according to the following equation ... 

TABLE 4.9 Average Cross Sections (Base e) for N204u b... 

In most experiments and observations we are confronted with a collection of very many particles unless special pains are taken to align the particles, or in the absence of a known alignment mechanism, we may reasonably assume that they are randomly oriented. Under these conditions the quantities of interest are the average cross sections (Cabs) and (Csca), which are independent of the polarization of the incident light provided that the particles are not intrinsically optically active. Let p(Cl) dti be the probability that one of the axes fixed relative to a particle, the x axis, say, lies within a solid angle dSl around the... 

We have discussed intrinsically anisotropic particles—ones with anisotropy originating in their optical constants rather than their shape—in previous chapters. In Section 5.6 we gave the solution to the problem of scattering by an anisotropic sphere in the Rayleigh approximation. From the results of that section and Section 5.5 it follows that the average cross section (C) (scattering or absorption) of a collection of randomly oriented, sufficiently small, anisotropic spheres is... 

The average cross section of identical, but randomly oriented ellipsoids will, in general, exhibit three peaks in the frequency range between co, and o)t. An example of this is given in Fig. 12.7, where Cabs for a silicon carbide ellipsoid is shown as a function of frequency. 

The average cross-sectional area of rubber in filled samples has been the subject of much discussion. In a filled system which swells to a foam-... 

Cross sections are defined for various kinds of collisions, such as elastic, inelastic, superelastic, reactive, etc. We are interested here mostly in the elastic collisions of atoms and molecules, often in the presence of light. If bulk properties are considered, averages of the cross sections over the velocity distribution are important the averaged cross sections become functions of temperature. 

These latter measurements led only to relative cross-section values. However, by comparison with absolute values of velocity-averaged cross sections, they can be put on an absolute scale. To do this, the absolute values obtained in FA measurements were used because here the velocity distribution is exactly known—a Maxwellian distribution /(t>, T) with the temperature of the buffer gas. Denoting the velocity-dependent relative total ionization cross section, obtained in the beam experiment, by oKl(v) and the absolute total ionization rate constant obtained in the FA experiment by R(T), then a normalization k may be determined by... 

Meyerhof, W.E. (1963). Threshold effects in average cross sections according to R-matrix theory. Phys. Rev. 129 692-702. 

Fig. 7a-c shows the time-averaged cross-sectional liquid saturation distribution in the column at superficial gas velocities of 0, 5 and 10 cm/s, respectively. For each gas velocity, the liquid saturation distribution is shown at three axial positions and three liquid velocities. Also provided are the... 

Fig. 7. Time averaged cross-sectional liquid saturation distribution in structured bed at (a) Ug Ocm/s (b) Ug 5 cm/s (c) Ug 10 cm/s. In the table, sy shows the liquid saturation and

Figure 7.13 illustrates the relation between resolution in the energy and in the time domains for the dissociation of CH30N0(Si). The right-hand side shows the modified autocorrelation function S(t) = S(t) t(t) for four windows At and the left-hand side depicts the corresponding averaged cross sections dtot(E). The spectrum with the lowest resolution is structureless and resembles the kind of spectrum obtained for fast and direct dissociation without any recurrence. S(t) merely shows the rapid... 

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