Abstract
Rotational excitation in collisions of structureless atoms and diatomic rigid rotor molecules interacting by a rigid potential shell is considered in classical mechanics. The double differential cross sectionJ(u *,θ) for final (over initial) relative velocityu *=ν′/ν and deflection byθ is analytically related to the shell form in the case of vanishing initial molecular rotation.J(u *,θ) exhibits the strong structure of “bulge” scattering or “orientational rainbows” which has been observed in the K−N2 and K-CO systems and is expected to occur in rotationally inelastic collisions of many nonreactive systems under appropriate scattering conditions. The present results elucidate the nature of the sensitive and direct relation of bulge scattering to the anisotropy of the intermolecular potential.
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The scattering would of course not be rotation symmetric aboutp, had the definedj component and therebyα been referred to a different axis
For a spherical shell\((R \times \hat n) = q = B = 0\); (13) yieldsu *=1 and, noting that\((\hat n \cdot \hat p) = - \surd \overline {1 - b^2 /a^2 } \), one recovers from (14) the cosθ/2=b/a deflection law of the rigid sphere of radiusa
The absolute cross section results of [4] should be multiplied by 2 which factor was erroneously omitted in that calculation
ForB}>1 and depending onθ theB}-ν * relation is no longer one-to-one, and theν * interval of non-vanishingJ(ν *,θ=const) covers the whole recoil velocity scale 0≦u *≦1. Simulating real systems, B>1 will be untypical, although by no means excluded
This feature of rotationally inelastic scattering has been obtained before more or less clearly in various calculations using realistic potentials, e.g. Barg, G.D.: Diplomarbeit, Göttingen, 1973; Preston, R.K., Pack, R.T.: J. Chem. Phys.66, 2480 (1977); Loesch, H.J.: Chem. Phys.18, 431 (1976). Seemingly, however, its relation to the interaction has not been spelt out
The integrability is seen by approximatingB=B K +β K(X1 −X 1K )2 in the neighbourhood of the extremumK, so thatB X1=±2trβ K (B}−B K ) and using (24) to findB−B K =(u *-u * K )f(u *,θ) withfne0 atu *=u * K .
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For an interaction represented by a ridig bar there is no bulge. As a consequence the inelastic singularities of the cross section are missing in this case. For another example of no bulge of a lever arm see Fig. 5
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For this model small contributions due to masking and double collisions would have to be considered in a cross section calculation
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Pack, R.T.: private communication
Schinke, R.: private communication
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Dedicated to Prof. Otto Osberghaus on the occasion of his 60. birthday
We express our gratitude to Prof. Osberghaus whose interest in this work has been challenge and support to it.
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Beck, D., Ross, U. & Schepper, W. Rotationally inelastic, classical scattering from an anisotropic rigid shell potential of rotation symmetry. Z Physik A 293, 107–117 (1979). https://doi.org/10.1007/BF01559752
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DOI: https://doi.org/10.1007/BF01559752