The energy-weighted sum rule and the nuclear radius

Abstract

The energy-weighted integrated cross-section for photon absorption --known as sum rule \( \sigma_{-1}\) -- is under certain conditions proportional to the mean square nuclear radius (Levinger, Bethe (Phys. Rev. 78, 115 (1950))). Due to the energy weight factor the low-energy absorption components are emphasized and the dipole transitions in the region of giant resonances contribute enhanced at \( \sigma_{-1}\) . Thus, the cross-section of the full interaction can be replaced in good approximation by the dipole cross-section. Under these aspects, we have calculated \( \sigma_{-1}\) and the radii of various gg-nuclei. For our purpose, we have chosen a simple shell model where the integrals can be solved analytically, and the contributions of uncorrelated functions and correlation corrections can be shown explicitly. The mean square radius as a function of \( \sigma_{-1}\) differs by a factor of 1.5/0.87 from the previous result of Levinger and Kent (Phys. Rev. 95, 418 (1954)) without correlation corrections. Plotting the function of the correlation corrections \( g(A)\) and the uncorrelated function \( f(A)\) as a ratio it shows that \( g(A)/f(A)\) tends towards a limit. Finally, our results for the radii of gg-nuclei are in good agreement with recent experiments (I. Angeli, K.P. Marinova, At. Data Nucl. Data Tables 99, 69 (2013)).