Thomas-Fermi theory of the breathing mode and nuclear incompressibility

Abstract

A Thomas-Fermi theory with a linear scaling assumption is proposed for the breathing mode of nuclear collective motion. It leads to a general result ${K}_{A}=〈K(\ensuremath{\rho},\ensuremath{\delta})〉{+K}_{\mathrm{GD}}\ensuremath{-}{2E}_{C}/A$ which states that the incompressibility ${K}_{A}$ of a finite nucleus $A$ mainly equals the nuclear matter incompressibility $K(\ensuremath{\rho},\ensuremath{\delta})$ averaged over the nucleon density distribution $\ensuremath{\rho}(\mathbf{r})$ of nucleus $A$, added to a term ${K}_{\mathrm{GD}}$ contributed from the gradients of nucleon densities, with twice the Coulomb energy per nucleon ${E}_{C}/A$ subtracted. The nuclear matter equation of state given by the Thomas-Fermi statistical model with a Seyler-Blanchard-type interaction is employed to calculate the nuclear matter incompressibility $K(\ensuremath{\rho},\ensuremath{\delta})$ and a localized approximation of the Seyler-Blanchard-type interaction, which is shown to be similar to the Skyrme-type interaction, is developed to calculate the value of ${K}_{\mathrm{GD}}.$ ${K}_{\mathrm{GD}}$ and $\ensuremath{-}{2E}_{C}/A$ contribute about 20--10 % and 1--5 %, respectively, to the nuclear incompressibility ${K}_{A},$ from the light to the heavy nuclei. The shell and the even-odd effects are discussed by a scaling model which shows that these effects can be neglected for medium and heavy nuclei. The anharmonic effect is shown to be significant only for light nuclei. The leptodermous expansion of ${K}_{A}$ is obtained and the contribution from the curvature term proportional to ${A}^{\ensuremath{-}2/3}$ is discussed. The calculated isoscalar giant monopole resonance energy ${E}_{M}$ for a variety of nuclei are shown to be in agreement with experimental measurements.