Investigation of Nuclear Three- and Four-Body Systems with Soft-Core Nucleon-Nucleon Potentials

The nuclear three- and four-body systems are investigated with a variational method and various soft-core nucleon-nucleon potentials. A Monte Carlo method is used to compute the multidimensional integrals involved. The results show that (i) a soft-core potential and a hard-core potential which give the same effective-range parameters and $S$-wave phase shifts yield very nearly the same values for the binding energies and rms radii of ${\mathrm{H}}^3$ and ${\mathrm{He}}^4$, and (ii) as far as the ground-state properties of ${\mathrm{H}}^3$ and ${\mathrm{He}}^4$ are concerned, the fitting of the high-energy ($\gtrsim 100$-MeV) $S$-wave phase shifts need not be seriously considered in choosing a nucleon-nucleon potential. Also, it is shown that by paying less attention to the higher-energy ($\gtrsim 50$-MeV) nucleon-nucleon scattering data, a soft-core potential can be found which yields nearly correct values for the binding energies, rms radii, and form factors up to about 10 ${\mathrm{F}}^{-2\mathrm{}}$ for both ${\mathrm{H}}^3$ and ${\mathrm{He}}^4$. The soft-core potentials proposed by Eikemeier and Hackenbroich and by Volkov have also been briefly considered. Here the results indicate that the mathematical approximations used by Hackenbroich et al. to investigate the properties of light nuclei are inaccurate, while the binding energy of ${\mathrm{He}}^4$ calculated by Volkov is too small by about 2 MeV.