Symmetry Energy from Systematic of Isobaric Analog States
P. D. Danielewicz 1*, J. Lee 2
1 National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA
2 RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
Excitation energies to isobaric states, that are analogs of ground states, are dominated by contributions from the symmetry energy. This opens up a possibility of investigating the symmetry energy on nucleus-by-nucleus basis. Upon correcting energies of measured nuclear levels for shell and pairing effects, we find that the lowest energies for a given isospin rise in proportion to the square of isospin, allowing for an interpretation of the coefficient of proportionality in terms of a symmetry coefficient for a given nucleus. In the (A,Z) regions where there are enough data, we demonstrate a Z-independence of that coefficient. We further concentrate on the A-dependence of the coefficient, in order to learn about the density dependence of symmetry energy in uniform matter, given the changes of the density in the surface region. In parallel to the analysis of data, we carry out an analysis of the coefficient for nuclei calculated within the Skyrme-Hartree-Fock (SHF) approach, with known symmetry energy for uniform matter. While the data from isobaric analog states suggest a simple interpretation for the A-dependent symmetry coefficient, in terms of the surface and volume symmetry coefficients, the SHF results point to a more complicated situation within the isovector sector than in the isoscalar, with much stronger curvature effects in the first. We exploit the SHF results in estimating the curvature contributions to the symmetry coefficient. That assessment is hampered by instabilities of common Skyrme parameterizations of nuclear interactions. Nonetheless, in the end, we find that we can arrive at the value of symmetry energy at normal density with an uncertainty of just few percent. However, the uncertainty in our estimate of the slope of symmetry energy is much larger.