Monopole moments and nuclear compressibility

The nuclear compressibility has a role in nuclear physics in several ways. Its relationship to the giant monopole is well known and has been subject of much theoretical work. Less well known is its affect on in nuclear structure, namely monopole transitions between low-lying states in the spectrum. Here I revisit the relationship between monopole and quadrupole moments in deformed nuclei. It has often been assumed without any microscopic justification that the nucleus can be treated as an incompressible fluid. An analytic formula is derived here for the resulting relationship between monopole and quadrupole moments. The formula is shown to be well satisfied within self-consistent mean-field theory calculated with several energy functionals. The formula can be tested when both monopole and quadrupole matrix elements of band-to-band transitions are known. Data is available for the decay of the first excited $K^\pi = 0^+$ band to the ground-state band in $^{156}$Gd, and the extracted matrix elements are found to be consistent with the incompressible fluid picture.