Energy Levels ofPb210from Shell Theory

The present calculations for ${\mathrm{Pb}}^{210}$ have been based upon the double-closed-shell-core model with harmonic-oscillator wave functions. All two-neutron configurations of the $2g_{\frac{9}{2}}$, $1i_{\frac{11}{2}}$, $1j_{\frac{15}{2}}$, $3d_{\frac{5}{2}}$, $4s_{\frac{1}{2}}$, $2g_{\frac{7}{2}}$, $3d_{\frac{3}{2}}$ main shell are included. A singlet-even Gaussian effective interaction with parameters determined by low-energy proton-proton scattering and zero triplet-odd effective interaction give, within 50 keV, the binding energy of the two outer nucleons determined from experimental data and the energies of four excited states which have been experimentally identified. An empirical relation between matrix elements for slightly different parameters is described. It indicates that the interaction and the harmonic-oscillator parameter $\nu$ used here also give the general agreement with experiment which had been obtained for ${\mathrm{Pb}}^{206}$ with similar parameters. Calculations with a Yukawa singlet-even interaction and with additional attractive and repulsive central and central+tensor triplet-odd effective interactions were made for ${\mathrm{Pb}}^{210}$; they all lead to somewhat worse agreement. None of these interactions is, however, definitely excluded by the pressent data. Level schemes, tables of level energies and of some wave functions, and detailed formulas for the calculation of $\gamma$-ray transition probabilities are given. A model for the 2.15-MeV ${\mathrm{½}}^{-}$ state of ${\mathrm{Pb}}^{209}$ is also discussed.