Perturbation Theory of Light Nuclei:He4andLi6

Abstract

Perturbation theory is applied to nuclei in the centrally symmetric representation in which the neutrons and protons have the wave functions of three-dimensional harmonic oscillators. In first order the particles are independent, but the second-order calculation removes this over-simplification. In order to calculate the kinetic energy correctly, a transformation to the coordinate system in which the center of gravity is at rest is introduced. The question of convergence has two aspects: whether the successive contributions, first, of more and more highly excited states, and second, of the successive higher orders, diminish rapidly. One of the many interaction assumptions which are equivalent for ${\mathrm{He}}^4$ is used in calculating the binding energy of ${\mathrm{He}}^4$, with a result only slightly less than that given by the equivalent two-body method, a satisfactory proof of both methods. The calculation of the ${\mathrm{Li}}^6$ binding energy with one form of interaction is carried far enough to include the second-order contribution of the sextuply excited states and the third-order contribution of the doubly excited states, the convergence being apparently sufficiently rapid that further contributions would be negligible. The change in some of the smallest of these contributions effected by altering the interaction assumption is also neglected, in calculating the ${\mathrm{Li}}^6$ binding energy with other forms of interaction. All forms of interaction considered have a radial dependence resembling the error curve, and all but one treat like-particle and unlike-particle interactions symmetrically. Of these, the only forms which satisfy the demands of scattering and of the ${\mathrm{H}}^2$ and ${\mathrm{He}}^4$ energies, and which also give enough binding energy for ${\mathrm{Li}}^6$, involve combinations of all types of permutation operators with rather large positive and negative coefficients. The influence of the second order on the calculation of nuclear mechanical and magnetic moments, in particular those of ${\mathrm{Li}}^6$, is also discussed.