Hard sphere scattering

Figure 3.12. Inelastic scattering of Ar from Pt(lll) at the various input energies listed in the figure and for an initial angle of incidence 0, = 45° and Ts = 800 K. Results are plotted as EfIE vs. the final scattered angle . Points are the experimental results and the lines marked adjacently in the label are results of molecular dynamics simulations on an empirical PES. The long dot-dashed curve is the prediction of a cube model of energy transfer, while the dashed curve is the prediction from hard sphere scattering. From Ref. [135].

Shvartsburg, A.A. Jarrold, M.R, An exact hard-spheres scattering model for the mobilities of polyatomic ion, Chem. Phys. Lett. 1996, 261, 86-91. 

The problem is often treated in terms of the collision density n(E), whidi is the number of collisions per unit energy at energy E. For classical hard-sphere scattering with a target at rest the collision density at energies well below the initial energy has the simple form... 

The quantities Fi,Gi,F, j.es defined in Sect. 9- The separation into hard sphere scattering (phase shift (pi) and nuclear resonant scattering (phase shift i) is a mathematically convenient way of distinguishing between potential scattering... 

Hard sphere scattering, Streuung an barter Kugel 36. 

Using the Effective Hard Spheres Scattering (EHSS) method within MOBCAL,( ) theoretical cross sections were produced using the published structures for comparison... 

Figure 4.4 Collision trajectories at different impact parameters for a given collision energy. The ordinate is the reduced impact parameter b —b/Rrn where fim is the equilibrium distance of the well in the potential. The resulting deflection function is shown on the left-hand side. Note the qualitative difference between hard-sphere scattering and scattering by a realistic potential for hard spheres there is no attractive part of the potential so going from the deflection xlWto bis single valued x(b) is defined uniquely by the value of b. But in the presence of a well in the potential there can be more than one value of bthat results in scattering into a given value of x(b).

We are finally ready to use the angular distribution as a probe for the potential. As usual, we begin with the rigid-sphere model. Using Eq. (4.8) for the deflection function needed in Eq. (4.9), Eq. (4.10) yields, for the hard-sphere scattering,... 

Figure 4.10 The phase shift S/ computed for a realistic interatomic potential vs. /. The computation is for the realistic value A= kx = 00, meaning that many partial waves contribute to the scattering a is the range of the potential and k= p/ft is the wave vector). Note the steep variation of the phase shift at lower k that is due to the repulsive core of the potential (the initial decline is with a slope of n/2, which is what we expect for a hard-sphere scattering). The stationary point occurs at the glory impact parameter, /g= kbg.

For expUcit results we need, however, to adopt specific models. One such model is that of rebound reactions. Here one assumes tiiat reaction only occurs on close collisions when the reactants are subject to the short-range repulsive part of the intermolecular potential. The rearrangement thus takes place at close quarters and the newly formed products recede under the influence of the short-range repulsion. Hence, the net deflection is that typical of hard-sphere scattering. 

For hard-sphere scattering we have seen that the angular distribution is a constant, independent of the collision energy, (lizbdb/dico) = d jd. f d is the radius at which reaction takes place, we therefore have for reboimd reactions... 

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