The Nuclear Five-Body Problem

Perturbational and variational calculations with single-particle Legendre functions have been made for the nuclear five-body problem of ${\mathrm{He}}^5$. The symmetric Hamiltonian is employed; the values of the nuclear parameters chosen favor high binding. With a symmetric interaction, ${\mathrm{Li}}^5$ and ${\mathrm{He}}^5$ differ only in the Coulomb energy, states of the former lying about one Mev above those of the latter. The calculations show that there is no state of ${\mathrm{He}}^5$ which is stable against dissociation into an alpha-particle and a neutron. There are two low-lying virtual levels, one of ${}_{}{}^2{}_{}{}^{}P$, the other of ${}_{}{}^2{}_{}{}^{}S$ symmetry; according to the Hartree model, the energies of these states are -11.7 and -6.5 Mev, respectively. Second-order perturbation calculations diminish the interval ${}_{}{}^2{}_{}{}^{}S-{}_{}{}^2{}_{}{}^{}P$ radically and locate both the states in the neighborhood of -19.0 Mev. The last two results are roughly confirmed by variational calculations with two variation parameters (alpha-particle model), and -20 Mev may be set as a probable lower bound to the position of either state. These estimated values fall some seven Mev short of the experimental energy.