Classical cross sections

Figure 2. Quantum classical cross-sections for the reaction D-I-Ha (r — l,j — 1) DH (v — l,/)-l-H at 1.8-eV total energy as a function of /. The solid line indicates results obtained without including the geometric phase effect. Boxes show the results with geometric phase effect included using either a complex phase factor (dashed) or a vector potential (dotted).

Pack R. T. Close coupling test of classical cross-sections for rotationally inelastic Ar-N2 collisions, J. Chem. Phys. 62, 3143-8 (1975). 

The curve marked ion-dipole is based on the classical cross-section corresponding to trajectories which lead to intimate encounters (9, 13). The measured cross-sections differ more dramatically from the predictions of this theory than previously measured cross-sections for exothermic reactions (7). The fast fall-off of the cross-section at high energy is quite close to the theoretical prediction (E 5 5) (2) based on the assumption of a direct, impulsive collision and calculation of the probability that two particles out of three will stick together. The meaning of this is not clear, however, since neither the relative masses of the particles nor the energy is consistent with this theoretical assumption. This behavior is, however, probably understandable in terms of competition of different exit channels on the basis of available phase space (24). 

Physical exercise has been investigated in a number of studies. The relationship between exercise and cognitive performance has been studied in both older adults (Kramer et al., 2000 Wood et al., 1999 Laurin et al., 2001) and young, healthy volunteers. The general consensus is of beneficial effects. In a classic cross-sectional study Spirduso and Clifford (1978) demonstrated that reaction time among older adults... 

A comparison with classical results (Saxon and Light, 1972c) shows that classical cross sections are lower near threshold, which was attributed to tunnelling. At higher energies classical results are larger, unlike the relation... 

The amplitude of a single contribution is just the classical cross section, so that the classical result is obtained for the cross section. If more than one point of stationary phase exists the cross section displays characteristic interference oscillations which arise from the different classical paths ... 

The classical cross section in the forward direction which is proportional to a 9 (s+2)/s for a potential of the asymptotic form V =... 

Differential cross section Deflection function. First we describe methods which take advantage of the close relationship between semi-classical cross sections and deflection function as outlined in Section III. A procedure which uses nearly all measurable quantities has been proposed and applied by Buck (1971). In order to unfold the multivalued character of 6(9), the deflection function is separated into monotonic functions g/[b) such that 0(6) = . gj(6)and6 = g (9). The g are represented by the usual functional approximations made in the semiclassical scattering theory ... 

For knock-on collisions, the classical cross-section K(de/e2) can be used. It can be shown, in this case, that... 

These belts can be used on the same applications as the classical belts, but allow for a hghter, more compact drive. Three cross sectious—3V, 5V, and 8V—replace the five classical cross sections (Fig. 5.75). Also available are the bandless molded notch cross sections—3VX, 5VX. These belts range in width from % in for the 3V to 1 in for the 8V. 

Within certain limits, V belts are well suited to drives that must run at varying input or output speeds. These are common in the air-moving industry and are referred to as variable-pitch drives. These drives must incorporate special sheaves. Speed ratio on these drives is controlled by moving one sheave sidewall relative to the fixed sidewall so that the belt can ride at different pitch diameters. Variable-pitch drives using a single variable-pitch sheave and classical cross-section belts will yield only about 1.4 1 overall speed variation. 

One of the most important examples of this is the semi-classical approximation to quantum mechanics in 1959 Ford and Wheeler [65] described the explicit sequence of approximations by which the rigorous quantum cross section of (2.1)-(2.3) degenerates to the completely classical cross section,... 

The problem of atoms colliding with molecules is in practice considerably more complicated due to the additional degrees of freedom, The simplest example is to approximate a diatomic molecule as a rigid linear rotor and consider collisions with a spherically symmetric atom (S electronic state), In this case the classical cross section is given by... 

By rainbow hereafter, I mean the classical rainbow — points where the classical cross section is singular. These points do not, of course, coincide with the points where the maxima in the scattering intensity are actually observed, as shown in Figs. 2 and 3. They are nonetheless a fundamental quantity in the location and interpretation of the wave mechanical rainbows. For example, the Airy functions of semiclassical theory depend on the deviation from the classical rainbow. Therefore, the maxima in the semiclassical scattering intensity can never be located any more accurately than have been the classical rainbows. 

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