Configuration space theory of ‘truly three-body’ scattering rates

Abstract

The configuration space theory of (non-relativistic) three-body scattering is reviewed, with two main objectives in mind: (i) to derive from a very different approach, wherein comparatively little use is made of questionable mathematical manipulations (e.g. operator techniques or the representation of functions by infinite integrals) the general expressions for reaction rates customarily deduced via momentum space procedures; (ii) to determine the ‘physical5 three-particle transition operator T 4 , to be distinguished from the conventional T = F — VGV , where G is the Green function. The matrix elements ⟨ f | T t | i ⟩ yield the reaction coefficient w ¯ ( i → f ) expressing the probability of True5 three-body reactions; contained in the matrix elements ⟨ f | T | i ⟩ are terms representing, for example, purely two-body scattering events. Although the possibility of inelastic processes is fully taken into account, for simplicity the detailed analysis is limited to those transition amplitudes representing elastic scattering under the influence of short-range forces; however, it is reasonable to suppose the results obtained are relevant to broader classes of reactions and forces. In essence, the analysis concentrates on the δ -functions occurring in transition amplitudes, as well as in expressions for the solution ψ i ( + ) ( E ) to Schrodinger’s equation presumably satisfying the boundary conditions at real energy E for specified incident wave ψ i = exp { i ( k 1 . r 1 + k 2 . r 2 + k 3 . r 3 ) } It is found that these δ -functions—in a configuration space formulation—always are associated with (and in effect signal) previous illegitimate mathematical operations, e.g. unjustified interchange of order of integration and limit r → ∞ , or improper computation of the limit ∈→ 0 in expressions for ψ i ( E + i ∈ ) . This last assertion does not negate the fact that the δ -functions so produced often are physically interpretable and indeed desirable, as, for example, the customary total momentum conserving δ ( K f − K i ) factor in laboratory system transition amplitudes. On the other hand, such δ -functions, when on-shell (as can be, for example, either the aforementioned δ ( K f − K i ) or the δ -functions associated with single—i.e. not multiple two-body scattering events), yield meaninglessly infinite reaction rates unless reinterpreted in terms of the (large) volume τ within which the three particles 1, 2, 3 are reacting. Moreover, the ‘physical’ three-body amplitudes ⟨ f | T t | i ⟩ will contain no δ -functions other than the ever-present δ K f − K i . Thus, the presence of non-three-body contributions to ⟨ f | T | i ⟩ is also signalled by anomalous τ -dependence of reaction rates inferred therefrom. In particular, the δ -function contributions to ⟨ f | T | i ⟩ from two successive purely two-body scatterings, if retained, would result in predicted three-body scattering rates proportional to τ 4 3 , whereas the true three-body rate should be proportional to τ . A mathematically correct derivation of ⟨ f | T t | i ⟩ , in which these double scattering δ -functions would be wholly avoided, seems very difficult; however, it is possible to subtract these δ -functions from the divergent integral which—in the configuration space formalism—represents the contributions to ⟨ f | T t | i ⟩ associated with double scattering events. In this fashion it is concluded that ⟨ f | T t | i ⟩ is the sum of all contributions from n ≥ 3 successive purely binary collisions, plus the off-shell contributions from double scattering ( n = 2) processes. The configuration space and momentum space results for ⟨ f | T | i ⟩ agree, as do the configuration space and momentum space expressions for ⟨ f | T t | i ⟩ , provided it is granted—as is not apparent from momentum space procedures—that ⟨ f | T t | i ⟩ should include the off-shell double scattering contributions. Including these off-shell double scattering contributions keeps finite the predicted three-body elastic scattering rate observed with fixed counters arranged so as to exclude actual physical (on-shell) double scattering events, but makes infinite the total three-body elastic scattering rate obtained from integration over all counter positions which exclude on-shell double scattering as well as single scattering. Our analysis also relates the τ -dependence to the behaviour of ψ i ( + ) ( E ) at large distances, and examines off-shell δ -function contributions in certain (not all) formulas for ψ i ( + ) ( E ) , whose presence apparently is typically associated with the existence of bound states. In large part, the text is an amplification (often essentially a correction) of assertions concerning configuration space three-body scattering theory which previously were inferred somewhat offhandedly from conclusions carefully derived for two-body reactions only. The Faddeev equations are mentioned, but the problem with which these equations are mainly concerned—namely the reformulation of Schrodinger’s equation as an integral equation permitting solution by Fredholm’s method—is not seriously considered in the present work. Setting aside its purely formal implications for scattering theory, the considerations of this publication will be most relevant and least dispensable in the theory of three-body reactions which actually produce three outgoing products; such 4 three-three ’ reactions of actual interest are not uncommon in the field of chemistry. In a sense, therefore, this publication is a first step in the direction of deducing correct formal expressions for important often measurable three-three chemical reaction rates.