An R-Matrix Approach to Electron-Molecule Collisions
The R-matrix formalism has a long and venerable history. The method was introduced into nuclear physics by Wigner1 and Wigner and Eisenbud2 in the late 1940’s to enable a unified treatment of nuclear reactions dominated by compound state formation. However, there are earlier sources,3–4 which developed quite similar approaches to resonant nuclear reactions. All of these theories utilize the short-range character of the nuclear force to define a reaction zone of finite radius but differ in the mathematical details of the treatment of the wavefunction within that reaction zone. By enclosing the scattering partners within this sphere of radius r = a (the R-matrix surface), where a is chosen to be the range of the nuclear force, it should be possible to characterize the system using energies and wavefunctions computed within the sphere. By matching to the known asymptotic solutions, which in the nuclear problem are simply free waves, one can easily extract the relevant scattering parameters. The connection between the internal and external solutions is provided by the R-matrix, which is a sum over quantities related to the overlap integrals (level widths) of the internal and external wavefunctions evaluated on the surface of the sphere, and the energies of the internal states.5–8 Since the low-energy nuclear scattering problem is dominated by the formation of resonances which can be identified fairly easily with the internal states, the method is a natural one for the parametrization of nuclear cross sections. Thus the R-matrix method becomes a systematic framework for understanding and characterizing large amounts of data in terms of energy levels and widths obtained from experimental measurement. In addition, once these energies and level widths are obtained, the R-matrix provides a vehicle for predicting new results which may be difficult or impossible to obtain from experiment.
KeywordsExternal Region Schrodinger Equation Hamiltonian Matrix Fixed Boundary Condition Basis Ofthe
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