On the Nuclear Two-, Three- and Four-Body Problems

The simplest nuclear Hamiltonian satisfying all present requirements includes a Majorana-Heisenberg interaction ${(1\ensuremath{-}g)P+gPQ}V(r)$ between unlike particles and an attractive singlet interaction between like particles which is equal to that for unlike particles. The experimental mass defects of ${\mathrm{H}}^{2}$ and ${\mathrm{H}}^{3}$ together with the cross section $\ensuremath{\sigma}$ for slow neutron-proton scattering will determine the range $b$ and depth $B$ of the triplet well and the proportion $g$ of Heisenberg force (we use throughout the potential $B{e}^{\ensuremath{-}2\frac{r}{b}}$). An exact analytic expression relating $\ensuremath{\sigma}$, $b$, $B$ and $g$ is derived for this potential and $g$ is found to be very insensitive to $\ensuremath{\sigma}$. An exact solution of ${\mathrm{H}}^{2}$ gives the relation between $B$ and $b$. The final relation which fixes the parameters is furnished by a Ritz-Hylleraas variational treatment of ${\mathrm{H}}^{3}$ with the above Hamiltonian and the wave function: $\ensuremath{\psi}={2}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\alpha}}_{1}({\ensuremath{\alpha}}_{2}{\ensuremath{\beta}}_{3}\ensuremath{-}{\ensuremath{\alpha}}_{3}{\ensuremath{\beta}}_{2}){\ensuremath{\varphi}}_{1}+{6}^{\ensuremath{-}\frac{1}{2}}({\ensuremath{\alpha}}_{1}({\ensuremath{\alpha}}_{2}{\ensuremath{\beta}}_{3}+{\ensuremath{\alpha}}_{3}{\ensuremath{\beta}}_{2})\ensuremath{-}2{\ensuremath{\beta}}_{1}{\ensuremath{\alpha}}_{2}{\ensuremath{\alpha}}_{3}){\ensuremath{\varphi}}_{2}$ where ${\ensuremath{\varphi}}_{1}$ and ${\ensuremath{\varphi}}_{2}$ each represents an exponential times a power series in the interparticle distances of proper symmetry ${\ensuremath{\varphi}}_{2}$ is brought in by the Heisenberg term; the Breit-Feenberg operator is used for the small triplet like-particle interaction). The convergence of energies obtained from successive improvements in $\ensuremath{\psi}$ is rapid and the eigenvalue may be closely estimated. After a relativistic correction is made we obtain: $b=1.73\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm; $B=242 m{c}^{2}$ and $g=0.215$. The binding energy of ${\mathrm{He}}^{3}$ is obtained by the same method and the ${\mathrm{H}}^{3}$---${\mathrm{He}}^{3}$ difference is found to be $1.48 m{c}^{2}$, agreeing well with experiment. The proton-proton scattering depth is checked to within 1 percent. When applied to ${\mathrm{He}}^{4}$, our potential gives approximately 20 percent too much binding energy. Parallel calculations with the Gaussian and Morse curves lead to essentially the same result. No reasonable modification of the experimental data can explain more than a small fraction of the discrepancy.