Rearrangement Collisions

We now proceed to set up equations for a binary rearrangement collision in which molecules a and b meet to form molecules c and 

He and Ha are the Hamiltonians of the separated molecules Ted is the relative kinetic energy expressed as 

Continuing with our arbitrary choice to solve Eq. (151) in terms of the complete orthonormal set of final states fenV v hereafter abbreviated as we may expand 

As in Section IV-B (3), the Born approximation solution for Eq. (157) entails the replacement of y,- with a particular initial-state wave function, which may be written 

A practical criterion for the validity of this approximation is difficult to develop, but it is usually suitable when the relative velocity is large compared with the internal motions. One may demonstrate that the asymptotic form of E p(rcd) is 

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Positronium formation involves the capture by an incident positron of one of the target electrons, to form the bound state Ps. This is one of the simplest examples of a rearrangement collision and accordingly it has attracted considerable attention, both experimental and theoretical. 

Omidvar, K. (1975). Asymptotic form of the charge-exchange cross section in three-body rearrangement collisions. Phys. Rev. A 12 911-926. 

We present here a summary of our work on the collision dynamics of three Interacting atoms, (O adding new developments on the theory of resonances in rearrangement collisions. We begin by showing how to go from a description In terms of electrons and nuclei to one... 

Most of the reviewed applications deal with atom-diatom rearrangement collisions. This is not a big restriction, however, because the dynamics of three atoms includes most of the challenges of larger systems. Also, most of the discussed reactions involve only ground electronic states, i.e. they are adiabatic. The amount of work done so far on non-adiabatic transitions in chemical reactions is small, and we shall only briefly refer to it. 

The important role of distortion in rearrangement collisions has been emphasized in work by Gelb and Suplinskas (1970). These authors treated the interna] motions within the perturbed-stationary state approximation, i.e. in an adiabatic fashion, and relative motions within both the Born (no distortion) and the distortion approximations. Calculations performed for Ar+ + HD with a model switching potential showed that distortion of the final relative motion was essential to obtain the correct product translational energies, because these are determined by the potential drop in the product valley. Isotope effect were, however, incorrect in both cases. 

For a rearrangement collision the differential scattering cross-section is... 

Here we wish to solve an equation such as Eq. (145) or Eq. (157), but without the Born approximation, which is not ordinarily valid for slow collisions. Equation (145) is for particles without internal structure and Eq. (157) is for rearrangement coUisions. It is instructive at this point to introduce the analogous equation for particles with internal structure, but without a rearrangement collision. Clearly, such an equation may be evolved by regarding the final products in Eq. (157) as a and b instead of c and d. We would then have... 

IV-C (2)-(4). The methods may be generalized to include rearrangement collisions for chemical reactions. 

Here, i and stand for initial states, k and I for final states and a i(gfi,4>) dQ, defined for a rearrangement collision in Eq. (161) has been substituted for the classical 2nb db. Equation (230) may be derived from the quantum mechanical Boltzmann equation of Uehling and Uhlenbeck with the assumption of Boltzmann statistics, i.e., that the low-temperature symmetry effects of Fermi-Dirac or Bose-Einstein statistics are negligible, but that the quantum mechanical collision cross-section should be retained. We should note that although one may heuristically introduce the quantum mechanical cross-section as in Eq. (230), the Uehling-Uhlenbeck or a similar equation is strictly derived for special interactions only. In this connection it is interesting that the method given later in Section V-C yields the same result [Eq. (330)] as that of this section [Eq. (251)] only when an approximation equivalent to the Bom approximation is made. 

One, suggested by Husain and Norrish (8), involves a rearrangement collision to form NO, i.e.. 

The leptonic annihilation is nevertheless observable in hydrogen-antihydrogen collisions. Its main appearance is, however, due to indirect processes, namely the intermediate formation of positronium as a consequence of rearrangement collisions. In that case the leptons are likely to annihilate with a certain time delay after hadronic annihilation. This is because regardless the final state of positronium its life time is longer than that of the most probable final state of protonium with N = 23. 

In conclusion, it may be observed that the study of chemistry at ultra-high energies is still at a very rudimentary stage. It is of definite interest that atomic-rearrangement collisions may still occur at collision energies... 

J. C. Light and J. Horrocks, Molecular rearrangement collisions at high impact energies, Proc. Phys. Soc. 84, 527-530 (1964). 

By neglecting the two internal rotational (or bending) degrees of freedom, the mathematical description of the rearrangement collision event is simplified so extensively that the computational treatment of reaction dynamics within this model is routinely possible. This computational simplication arises because the rotational motion of the line of collision is treated analytically by a partial wave expansion of the scattering wavefunction. Consequently, the computational effort reduces to that of a family of colllnear reactive scattering calculations, one for each partial wave term in the wavefunction expansion. 

The formal DW theory of rearrangement collisions Is a well established topic in scattering theory. Detailed derivations can be found in books concerned with collision theory (for example. Messiah (591. Rodberg and Thaler (711) or the theory of direct nuclear reactions (for example. Austern (31. Glendenning (431. Satchler (721). see also Choi et al. (191. 

R. W. Emmons and S. H. Suck. Distorted-wave Born-approximation study of angular distributions for state-to-state rearrangement collisions Role of orbital angular momentum. Phys. Rev.. 25A. 178-86 (1982). 

S.H. Suck and R.W. Emmons. Effect of partial wave interference on angular distributions and sideways scattering in rearrangement collisions. Chem. Phys. Lett.. 79. 93-6 (1981). 

S.H. Suck. Theory of atom-diatom rearrangement collisions based on... 

S.H. Suck and R.W. Emmons. Importance of relative angular momentum coherence in state-to-state rearrangement collisions (reactive scattering). The 9th Symposium of Korean Science and Technology. 3-6 July 1984. Seoul. Korea. 1. 46-9 (1984). 

D-fH2 rearrangement collision Effects of vibrational excitation. Phys. Rev. Lett.. 44. 1211-4 (1980). 

Photoassociation experiments involving alkali-metal systems have stimulated considerable interest in chemical reactivity in alkali-metal dimer-alkali-metal atom collisions at ultracold temperatures [14]. Rearrangement collisions in identical particle alkali-metal trimer systems occur without energy barrier, and recent studies have... 

We shall concentrate in this contribution on energy transfer in electronically adiabatic phenomena involving collisions of atoms with diatomics and with polyatomics. We shall not deal with collisions involving electronic excitations. The formalism can be written down for these cases but not much has yet been done to develop the computational methods required in applications. This is in great part due to the lack of information on interaction-potential energies of electronically excited states and on their couplings due to nuclear motions, for polyatomic systems. Similarly, the formalism can be extended to include rearrangement collisions. Little is known however about interaction potentials for reactions... 

The APH method is applicable to any three particle rearrangement collision for which the potential is known. A good example of this is a problem from atomic physics, a positron scattering with a hydrogen atom. Positrons are antipartides, positively charged electrons. Besides the usual elastic and inelastic scattering processes a rearrangement process also... 

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