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A variational wave function for the ground state of He³, and its application to the D(p,y)He³ capture reaction Banville, Marcel Roland

Abstract

The present work proposes trial wave functions for the three-body problem in nuclear physics taking into account the group theoretical classification of the states given by Derrick and Blatt and by Verde. We start from the Schroedinger equation in the internal variables (the interparticle distances) obtained by Derrick from a summation over the matrix elements for kinetic energy and potential energy extended over all variables except the internal variables. An “equivalent" Schroedinger equation is set up using a potential due to Eckart. This equation has the same form as the original Schroedinger equation in the region outside the range of the nuclear forces. The variables in this equation can be separated in a hyperspherical coordinate system and the resulting separate equations can be solved. Then using a superposition principle the solutions of the original equation are expanded in terms of solutions to the "equivalent" equation. The Rayleigh-Ritz variational procedure is used to determine the coefficients of the expansions with a given potential. Because of the computational labor involved significant approximation is made in allowing only the leading terms in the angular variables to appear in the expansions while keeping a sufficient number of radial terms to insure convergence. The present functions with a radial variable R = [formula omitted] give less than 1/2 of the binding energy predicted by Blatt, Derrick and Lyness (1962) who used a radial variable R = r₁₂ + r₂₃ + r₃₁. This shows that our approximation with the former radial variable is indeed too crude to predict a reliable value for the binding energy and that more angular terms must be included in the expansions, at least for the preponderent symmetric S-state. Wave functions derived by the Rayleigh-Ritz variational principle are used to calculate cross sections for the reaction D(p, γ)He³. The electric dipole cross section depends very sensitively on the potential used to derive the wave function and a comparison with experimental data provides a test of the various model assumptions used to describe the nuclear interaction. A realistic potential must contain a tensor potential plus a hard core in the central potential. The tensor interaction couples the S and D states and is necessary to explain the quadrupole moment of He³ while the hard core produced the required mixed-symmetry S-state. The experimentally observed isotropic component of the gamma ray yield is attributed to a magnetic dipole transition between a continuum quartet S-state and the mixed-symmetry component of the ground state wave function. For a range of the variable parameter used in the calculation comparison with experiment requires a 5% admixture of the mixed-symmetry S-state in the ground state wave function.

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