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Physical Review

Exchange Scattering in a Three-Body Problem

H. E. Moses

Published Jul 1, 1953• 8 citations
Publisher

Abstract

It is proved that in the three-body scattering problem, the prototype of which is the scattering of an electron by a hydrogen atom, the coefficient corresponding to exchange scattering behaves like a radially outgoing wave. The essential conjecture used by Mott and Massey in their treatment of the problem is thus verified.
DOI
10.1103/physrev.91.185

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For the problem of the scattering of an electron by a hydrogen atom, the complete equivalence is shown of the treatment of Trainor and Wu based on symmetry considerations and that of Altshuler giving a direct proof of the method of Mott and Massey. A generalization of the treatment of Altshuler to scattering processes involving non-identical particles or other rearrangement collisions is made. Also, the usual treatment involving matrix elements for transitions between states belonging to two different Hamiltonians is discussed and justified. Finally, the recent treatment by Dirac of rearrangement collisions is discussed, and it is shown that a correct formulation of the method and interpretation of the result give exactly the same result as in the usual treatment.
Abstract A hydrogen atom in the ground state scatters an electron with kinetic energy too small for inelastic collisions to occur. The wave function Ψ(r1; r2) of the system has boundary conditions at infinity which must be chosen to allow correctly for the possibilities of both direct and exchange scattering. The expansion Ψ = Σ ψ,(r1)Fy(r2) of the total wave function in y terms of a complete set of hydrogen atom wave functions ψy(r1) includes an integration over the continuous spectrum. It is si own that the integrand contains a singularity. The explicit form of this singularity and its connexion with the boundary conditions are examined in detail. The symmetrized functions Y* may be represented by expansions of the form Σ {ψy(r1) Gy±(r2) ±ψy(r2) y Gy±(r1)}, where the integrand in the continuous spectrum does not involve singularities. Finally, it is shown that because all the states ψy of the hydrogen atom are included in the expansion, the equation satisfied by F1, the coefficient of the ground state, contains a polarization potential which behaves like — a/2r4 for large r and is independent of the velocity of the incident electron.
The electron-electron correlation term in two-atomic collision is the most important, most difficult term to obtain the effective interatomic potential. Generally the H and H collision is a four center problem. It is extremely difficult to compute the electron-electron correlation term to include the effect of exchange or antisymmetry between two system electrons exactly. All the two-atomic collision related theoretical data differ from each other due to its difference in approximating the electron-electron correlation term. I invent a trick to evaluate the term exactly. Earlier the positronium (Ps) and H system was easily approximated as a three center problem due to the light mass of Ps. My new code for H-H collision using the ab-initio and exact static-exchange model (SEM) can reproduce exactly the same data of Ps and H system just by using the appropriate atomic parameters. The success of the present trick makes the foundation of a big monument in cold and low energy atomic collision physics. The Feshbach resonances are observed in H(1s)-H(1s) elastic scattering in both the singlet and triplet channels. One can use the trick to solve any four center problem including excitation and ionization processes. Again one can use the idea in larger systems too.
We derive some new equations for the wavefunction for the three-body scattering problem using the ``partial-wavefunction approach.'' All scattering processes are solutions of inhomogeneous coupled integral equations for the scattered parts of the total wavefunction. We show how the Pauli principle is incorporated when some or all of the particles are identical. This leads to a reduction in the number of independent scattered partial wavefunctions when some of the particles are identical. The kernels in the integral equations contain no δ functions but each scattered partial wavefunction does have a self-coupling term. We apply the method to the scattering of three identical bosons interacting through zero-range pair potentials. The method gives the same equation for this problem as obtained by the Faddeev T-matrix approach.
For NNN particles with equal mass, interacting with repulsive or attractive $\ensuremath{\delta}$-function interaction of the same strength, the SSS matrix is explicitly given and shown to be symmetrical and unitary. The incoming and outgoing states may consist of bound compounds as well as single particles. The momenta of the particles and compounds are not changed in the scattering, but particles are exchanged, such as $ABC+DE\ensuremath{\rightarrow}ADC+BE$. Only distinguishable particles are considered.