Deflection function

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... 

Substitute the deflection function, Equation (3.95), the shear strain expression, Equation (3.130), and the stress-strain relation. Equation (3.131), in Equation (3.132) to get... 

Flexural member design requires the determination of (1) the design blast loads, (2) the initial design cross-section, (3) an idealized resistance deflection function, (4) the calculated response (maximum deflection) and, (5) allowable ultimate deflection and (6) design for shear. 

Resistance-Deflection Function. The resistance-deflection function establishes the dynamic resistance of the trial cross-section. Figure 4a shows a typical design resistance-deflection function with elastic stiffness, Kg (psi/in), elastic deflection limit, Xg (in) and ultimate resistance, r.. (psi). The stiffness is determined from a static elastic analysis using the average moment of inertia of a cracked and uncracked cross-section. (For design... 

Calculation of many trajectories at different impact parameters for each given incident energy yields the energy-dependent deflection function and energy loss, which can then, through equation (1), be used to calculate the stopping cross section. 

At the terminus of each trajectory, we obtain the final momentum of the projectile, hk (Ep, b), which defines the deflection function for the projectile when projected on the initial momentum, Hki, i.e.,... 

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... 

Here, Jo(x) is the Bessel function of zero order, r7- kf-ki is the momentum transfer, which depends on the scattering angle 6, and 6 b) is the semiclassical phase shift, which is given in terms of the deflection function as b) = dd b)Hk , db. 

Fig. 2.3. Computed classical deflection function, x(b), of He-Ar pairs at a low (/), two intermediate (i) and a high (h) translational speed.

In a classical picture of Penning ionization,24 the molecules approach along a trajectory on the initial A + B (real) potential V0(r). Ionization occurs at a specified (but random) value of the internuclear distance, r/( and the products then complete their trajectories on an ion-molecule potential V+(r) for A + B +. Neglecting the momentum of the ejected electron, deflection functions can be computed according to whether the ionization occurs on the incoming or out going part of the V0(r) trajectory. These are... 

The discussion of interferences in ground-state atom-atom scattering relies heavily on the classical deflection function, which can be calculated from equation (II.4) if the potential is known. For He -He scattering,... 

The other oscillations can be understood with the help of Fig. 28, which shows schematically a typical deflection function including the effect of... 

Figure 25. Quantal deflection functions for He (2 S) + He at 42 meV. Small splitting at large / causes damping of symmetry oscillations.

Figure 26. Quantal deflection functions at higher energies for He (2 5) + He. Orbiting spikes result from trajectories that spiral into inner minimum of ungerade potential.

Figure 28. Typical deflection functions for energies above ungerade barrier. Dashed lines give contributions from exchange scattering.

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... 

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