Monopole moments and the \( \beta\)-vibration in deformed nuclei
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I revisit the theory of the \( \beta\)-vibration in deformed nuclei focusing on the transition matrix elements to the ground band. The relation between monopole and quadrupole matrix elements is derived from an incompressibility assumption and is validated by self-consistent mean-field theory with several energy density functionals. With estimates of the deformation-dependent mean-field potential energy function, it is found that the excited \( 0^+\) band in 156Gd comes fairly close to meeting the criteria to be classified as a \( \beta\)-vibration.