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Rotational bands are commonly used in the analysis of the spectra of atomicnuclei. The early version of the interacting boson model of Arima and Iachellohas been foundational to the description of rotations in nuclei. The model isbased on a unitary spectrum generating algebra $U(6)$ and an orthogonal(angular momentum) symmetry algebra $SO(3)$. A solvable limit of the modelcontains $SU(3)$ in its dynamical symmetry chain. The corresponding Hamiltonianis written as a linear combination of linear and quadratic Casimir invariantsof all the algebras in the chain. Prompted by these facts, a Hamiltoniancontaining the $SU(3)$ quadratic Casimir squared is proposed to evaluate itseffects on rotational bands. The additional term yields three undeterminedparameters into the theory, which need be obtained from experiment. The lack ofdata does not allow one to perform a detailed numerical analysis, but a ratherrestricted one in terms of a one-parameter fit. Nevertheless, these additionalterms provide a good description of rotational bands of two nuclei of interest,$^{156}\mathrm{Ga}$ and $^{234}\mathrm{U}$.

Contribution of $SU(3)$ quadratic Casimir squared to rotational bands in the interacting boson model