Converged values of second-order core-polarization diagrams with orthogonalized-plane-wave intermediate states

Abstract

We carry out an extended study of the Vary-Sauer-Wong effect on the second-order core-polarization diagrams ${G}_{3\mathrm{p}1\mathrm{h}}$ and ${G}_{3\mathrm{p}1\mathrm{h}}^{T}$ in the effective interaction between two valence nucleons. As in Vary-Sauer-Wong, we first calculate ${G}_{3\mathrm{p}1\mathrm{h}}$ using a harmonic oscillator propagator for the intermediate particle states $p$, including particle-hole excitations with energies up to $22\ensuremath{\hbar}\ensuremath{\omega}$. Results are in close agreement with those of Vary-Sauer-Wong. We then calculate ${G}_{3\mathrm{p}1\mathrm{h}}^{T}$ where a free particle propagator for $p$ is used. This is obtained by including the $\ensuremath{-}U$, the oscillator one-body potential, insertions of all orders to $p$ of ${G}_{3\mathrm{p}1\mathrm{h}}$. Although the resulting matrix elements are generally smaller in magnitude than those of Vary-Sauer-Wong, the qualitative feature of the Vary-Sauer-Wong effect is clearly maintained. Namely there are strong cancellations between the contributions from the low and high energy $p$ states. This makes the net effect from the core polarization diagrams significantly weaker than from ${G}_{3\mathrm{p}1\mathrm{h}}$ calculated with $2\ensuremath{\hbar}\ensuremath{\omega}$ excitations alone. We also study some intermediate choices for the propagator of $p$, where the free particle propagator is used only for high energy valence states. The Brueckner reaction matrix elements in a mixed representation where one particle is in a harmonic oscillator state and the other in a plane wave state are needed in our calculations. By using the vector transformation brackets of Wong and Clement and of Balian and Brezin and the Tsai-Kuo treatment of the Pauli exclusion operator, we have developed a technique for accurately calculating these matrix elements.NUCLEAR STRUCTURE Contributions to effective interaction for $A=18$ nuclei calculated from converged values of lowest order core polarization diagrams as function of intermediate state propagator using new momentum space techniques.