# [paper] The energy-weighted sum rule and the nuclear radius

* The energy-weighted sum rule and the nuclear radius*

Hans Peter Schröder

doi: 10.1140/epja/i2015-15109-9

The energy-weighted integrated cross-section for photon absorption –known as sum rule σ−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 σ−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 σ−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 σ−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)).