Nuclear incompressibility: An analytical study on leptodermous expansion

A comparative study of the liquid-drop model (LDM) type expansions of energy $E$ and compression modulus $K_A$ is made within the energy density formalism using Skyrme interactions. As compared to the energy expansion, it is found that, in the pure bulk mode of density vibration, the LDM expansion of $K_A$ shows an anomalous convergence behaviour due to {\it pair \ effect}. A least squares fit analysis is done to estimate the minimum error, one would expect even with synthetic data due to the inherent nature of the LDM expansion of $K_A$ as well as the narrow range of accessible mass number $A$, in the values of the various coefficients. The dependence of the higher-order coefficients like curvature and Gauss curvature on the coupling $\beta_c$ between the bulk and surface parts of the monopole vibrations is analytically studied. It is shown that the $K_A -$ expansion including the dynamical effect ( $A-$ dependence of $\beta_c$ ) shows an `up-turn' behaviour below mass number about 120, suggesting the inapplicability of the LDM expansion of $K_A$ over this mass region.