The Approximate Solution of Nuclear Three and Four Particle Eigenvalue Problems

The problem of determining intranuclear forces from the mass defects of the hydrogen and helium isotopes is investigated under the assumption that the interaction potentials are proportional to a function $f\left(\alpha r^2\right)$ having the general form of a potential well and possessing the power series expansion $f\left(\alpha r^2\right)=1-\alpha r^2+\frac{c_1{\left(\alpha r^2\right)}^2}{2!}-\frac{c_2{\left(\alpha r^2\right)}^3}{3!}+\cdots$ about the origin. With this assumption the Rayleigh-Schroedinger perturbation theory is applicable to the two, three and four particle eigenvalue problems. The perturbation calculation yields small corrections to the eigenvalues given by the "equivalent" two particle method. The corrections are checked very satisfactorily in a special case by means of a complicated variational calculation. Numerical results are given for two extreme forms of the neutronproton model: Model I—Interaction between unlike particles only, Model II—Equal interactions between all pairs of particles. These results put close upper and lower bounds on the strength of the interaction between like particles in the model, intermediate between (I) and (II), which corresponds most closely to the experimental facts.