On Tensor Forces and the Theory of Light Nuclei

The quadrupole moment of the deuteron indicates the existence of non-central tensor forces in nuclei which destroy the constancy of the total orbital angular momentum. With simple operational representations of the wave functions, the influence of two-body tensor forces on the ground state eigenfunctions of the light nuclei ${\mathrm{H}}^3$ and ${\mathrm{He}}^4$ has been calculated. In ${\mathrm{H}}^3$ the tensor forces directly couple to the fundamental ${}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state a ${}_{}{}^4{}_{}{}^{}D_{\frac{1}{2}}^{}{}_{}{}^{}$ state, which in turn interacts with ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^4{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$. To the fundamental ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ state of ${\mathrm{He}}^4$ is admixed a ${}_{}{}^5{}_{}{}^{}D_0^{}{}_{}{}^{}$ state which is coupled by the tensor forces with ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$. All states consistent with the total angular momentum and parity conservation rules occur in the ground state eigenfunctions, and these nuclei therefore constitute the simplest examples of the complete break-down of spin and orbital angular momentum conservation laws. Rarita and Schwinger have satisfactorily accounted for the properties of the deuteron by including the tensor force in a simple interaction represented by a rectangular well potential. With this interaction to describe the forces between all pairs of nuclear particles, the binding energies of ${\mathrm{H}}^3$ and ${\mathrm{He}}^4$ have been estimated by a variation method The trial functions are of the form ${}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}+{}_{}{}^4{}_{}{}^{}D_{\frac{1}{2}}^{}{}_{}{}^{}$ for ${\mathrm{H}}^3$ and ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}+{}_{}{}^5{}_{}{}^{}D_0^{}{}_{}{}^{}$ for ${\mathrm{He}}^4$, with Gaussian radial functions. The calculations yield 32 and 50 percent of the binding energy for ${\mathrm{H}}^3$ and ${\mathrm{He}}^4$, respectively, while a similar test calculation for the deuteron gives 54 percent of the binding energy. The probability that these nuclei are in a $D$ state is found to be 4 percent for all three nuclei, in agreement with the exact deuteron computations. Improvement of the radial dependence of the trial functions increases the estimated binding energy of the deuteron to 76 percent of the known value but does not materially affect either the estimated binding energies of ${\mathrm{H}}^3$ and ${\mathrm{He}}^4$, or the amount of $D$ state admixture of the three nuclei. An analysis of the results shows that the tensor forces, which produce all the binding in the deuteron, are relatively ineffective in binding ${\mathrm{H}}^3$ and ${\mathrm{He}}^4$. This apparently indicates that the assumption of ordinary and tensor forces of the same range is not adequate to represent the properties of ${\mathrm{H}}^3$ and ${\mathrm{He}}^4$.