Quantum deflection functions

In a recent paper (16) we explored the effects of resonances on the phase behavior of Individual elements of the S matrix, with particular attention paid to the quantum deflection function, defined In Equation 7. Following In the spirit of Child s (35) analysis, we assume the partial wave dependence of the resonance energy to be... 

Equations 18-20 resemble the single-channel case, but differ by a factor of two, since we are concerned here only with off-dlagonal elements. Note that It also follows from Equation 9 that the resonant part of the phase shift Is Independent of the channel labels v and V. Equations 16-20 then lead to the following expression for the behavior of the quantum deflection function near a resonance. 

Figure 2. Quantum deflection function, 0(f), as defined by Equation 7, for the reaction F+H2(v 0) HF(v 2)+H at total...

The END trajectories for the simultaneous dynamics of classical nuclei and quantum electrons will yield deflection functions. For collision processes with nonspherical targets and projectiles, one obtains one deflection function per orientation, which in turn yields the semiclassical phase shift and thus the scattering amplitude and the semiclassical differential cross-section... 

From the deflection function we calculate the differential cross section which is needed in equation (1). We note that there could be several different trajectories (two different impact parameters) that produce the same scattering angle, leading to quantum mechanical interference of their nuclear wave functions. We thus... 

This RE is radially unstable if j / 2mr ) + V r) is a maximum, radially stable if it is a minimum. If an unstable RE occurs, the deflection function 0/ =/(h,), [41,76], displays rainbows (0/ is the final angle of exit of the particle in the inertial frame, h,- is the initial impact parameter). The structure of these rainbows is well known in the classical or quantum cases [77]. For such an integrable Hamiltonian like equation (45), there are as many singularities (rainbows) of the deflection function as integer numbers each singularity is characterized by an increase by 1 of k = mod(0/, 2ti). There is one impact parameter b such that... 

In Equation 6, 6(b) Is the classical deflection function, and specifies the angle 6 at which the particles separate after a collision at an Initial Impact parameter b. The deflection function has the quantum analog (35)... 

The theoretical description of the total cross section is easier in the semi classical limit since the angular range is indeed restricted to 9 = 0 so that the transition approximation gives results which are in quantitative agreement with those calculated by quantum mechanics. Using the parabola approximation for the phase shifts in the maximum or the straight line approximation for the deflection function at the zero point we have... 

Once the deflection function and the phase shift have been determined for a collisional system, the direct differential cross section is obtained as the classical direct differential cross section when neglecting quantum interference effects [equation (13) with Pfo = 1] or by introducing semiclassical corrections using the Schiff approximation [equation (20)]. 

For molecules with central finite attractive and repulsive forces (Fig. 2-4c), we may take S v ) = n9 Xmm where b Xmin) is the impact parameter corresponding to a minimum angle of deflection selected as an arbitrary cutoff to prevent S Vr) from going to infinity as x goes to zero when classical collision theory is used. The specific dependence of h(Xmin) on will vary with the magnitude of the parameters s and t or a and b in the empirical potential-energy functions. A realistic calculation for this model, i.e., one which avoids an arbitrary cutoff Xmiw must be carried out quantum mechanically. 

Semiconductor cluster polarizabilities have been the subject of some very important experimental studies via beam-deflection techniques (Backer 1997 Schlecht et al. 1995 Schnell et al. 2003 Schafer et al. 1996 Kim et al. 2005) while they have been extensively studied using quantum chemical and density functional theory. In this research realm, one of the areas intensively discussed is the evolution of the cluster s polarizabilities per atom (PPA) with the cluster size. The PPA is obtained by dividing the mean polarizability of a given system by the number of its atoms. Such property offers a straightforward tool to compare the microscopic polarizability of a given cluster with the polarizability of the bulk (see O Fig. 20-16) as the latter is obtained by the hard sphere model with the bulk dielectric constant via the Clausius-Mossotti relation ... 

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