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Inelastic Collisions – A First Overview

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Atoms, Molecules and Optical Physics 2

Part of the book series: Graduate Texts in Physics ((GTP))

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Abstract

We introduce some characteristic questions about inelastic and reactive collisions and approaches to answer them for several important examples. We start in Sect. 7.1 with very simple models. The general trends for excitation processes as a function of the relative kinetic energy are presented in Sect. 7.2. Specifically, in Sect. 7.2.7 we focus on the threshold region. In Sect. 7.3 we introduce multichannel theory, and discuss the alternative adiabatic and diabatic viewpoints. In Sect. 7.4 we extend the semiclassical methods already employed in the elastic case. In Sect. 7.5 we make a short excursions into the world of collision processes with highly charged ions. Finally, we address reactive scattering processes in Sect. 7.6.

In the previous chapter we have introduced potential scattering. Even though the concepts discussed there describe elastic heavy particle scattering very well (and in some cases even elastic electron scattering), we had to exclude so far completely the very important field of atomic and molecular excitation by collisions, as well as reactions: quite generally, atomic collisions are many body problems, and whenever changes of the internal states of the collision partners are possible, one has to account for these degrees of freedom.

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Notes

  1. 1.

    Excitation cross sections as a function of collision energy are called excitation functions.

  2. 2.

    Often the computation can be dramatically simplified by distinguishing between active valence electrons and the passive core electrons and by using correspondingly pseudopotentials rather than summing over all \(\mathcal{N}\). The generalization to molecules as targets is for this ansatz without problems, however, when solving the problem in detail, much more complicated.

  3. 3.

    The antisymmetrization necessary in the case of electron scattering is, however, not yet included and will have to be added in Sect. 8.1.

  4. 4.

    One finds slightly different notations in the literature, which differ by i or i/ħ in the definition of the coupling element G jj′.

  5. 5.

    As in the elastic case, the classical deflection angle Θ has a well defined sign, in contrast to the scattering angle θ.

  6. 6.

    We recall that we use ℓ (a vector) and ℓ (a number) for the nuclear angular momentum and its quantum number, respectively (to be distinguished from the electronic orbital angular momentum L).

  7. 7.

    For simplicity of the derivation we consider here only radial coupling, for which the Landau-Zener model is typically used. Allan and Korsch (1985) have shown, however, that the formalism may also be applied to rotational coupling.

  8. 8.

    Inversely, the probability to remain in the initial state during the overall process is \(w_{ba}w_{ba}+ ( 1-w_{ba} ) ( 1-w_{ba} ) =1-w_{ba}^{\mathrm{tot}}\).

  9. 9.

    Crossing (D) is important for the dependence of the process on polarization – which we cannot describe here.

  10. 10.

    This is a consequence of the rotation of the internuclear axis during the collision (rotational coupling). We note here in passing, that this splitting corresponds directly to lambda-type doubling in molecular spectroscopy (Sect. 3.6.6), a splitting of energy levels into Λ + and Λ − states for higher rotational quantum numbers due to the coupling of the electronic angular momentum Λ with the nuclear rotation N.

  11. 11.

    W pot is the sum of all ionization potentials W I (q′) for q′≤q.

  12. 12.

    The scenarios sketched in Fig. 7.27 correspond to just this reaction for q=18 (i.e. for the naked Ar nucleus). Not completely correct but illustrative we have shown in these schematics also the reduction of the states at R>R th – according to (7.86) and (7.88).

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Authors and Affiliations

  1. Max-Born-Institut für Nichtlineare Optikund Kurzzeitspektroskopie, Berlin, Germany

    Ingolf V. Hertel & Claus-Peter Schulz

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  1. Ingolf V. Hertel
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Acronyms and Terminology

a.u.:

‘atomic units’, see Sect. 2.6.2 in Vol. 1.

BO:

‘Born Oppenheimer’, approximation, the basis when solving the Schrödinger equation for molecules (see Sect. 3.2).

CC:

‘Close-coupling’, calculations, computation of scattering cross sections by solving (part of) the coupled integro-differential equations (see Sect. 8.1.1).

CCC:

‘Convergent close-coupling’, calculations, special solutions of the coupled integro-differential equations for collisions (see Sect. 8.1).

CCD:

‘Charge coupled device’, semiconductor device typically used for digital imaging (e.g. in electronic cameras).

CM:

‘Centre of mass’, coordinate system (or frame) (see Sect. 6.2.2).

COLTRIMS:

‘Cold target recoil ion momentum spectroscopy’, see Appendix B.4.

CW:

‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc.

DCS:

‘Differential cross section’, see Sect. 6.2.1.

E1:

‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation.

EBIS:

‘Electron beam ion source’, source for highly charged ion beams see Sect. 7.5.

EBIT:

‘Electron beam ion trap’, source for highly charged ion beams see Sect. 7.5.

ECR:

‘Electron cyclotron resonance’, used e.g. in sources for highly charged ion beams see Sect. 7.5.

FC:

‘Franck-Condon’, introduced an important approximation for optical transition between electronic states (see Sect. 5.4.1).

FBA:

‘First order Born approximation’, approximation describing continuum wave functions by plane waves; used in collision theory and photoionization (see Sects. 6.6 and 5.5.2, Vol. 1, respectively).

HCI:

‘Highly charged ions’, see Sect. 7.5.

HF:

‘Hartree-Fock’, method (approximation) for solving a multi-electron Schrödinger equation, including exchange interaction.

HOMO:

‘Highest occupied molecular orbital’.

JWKB:

‘Jeffreys-Wentzel-Kramers-Brillouin’, semiclassical method to determine scattering phases.

MCP:

‘Multi channel plate’, electron multiplier with many amplifying elements.

MD:

‘Molecular dynamics’, classical trajectory computations for molecular systems.

MP2:

‘Møller-Plesset correction of 2nd order’, perturbative approach to correct HF energies for contributions from non-spherical repulsive potentials.

ODE:

‘Ordinary differential equation’.

QED:

‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction.

RCCC:

‘Relativistic convergent close-coupling’, relativistic version of CCC calculations (including spin orbit interaction).

RMPS:

‘R-matrix with pseudo-states method’, advanced quantum mechanical theory for electron scattering.

SEC:

‘Single electron capture’, see Sect. 7.5.1.

VMI:

‘Velocity map imaging’, experimental method for registration (and visualization) of particle velocities as a function of their angular distribution (see Appendix B).

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Hertel, I.V., Schulz, CP. (2015). Inelastic Collisions – A First Overview. In: Atoms, Molecules and Optical Physics 2. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54313-5_7

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