Physics Letters B 300 ( 1993 ) 199-204 North-Holland PHYSICS LETTERS B Cranking anharmonic gamma vibrations E.R. Marshalek and R.G. Nazmitdinov L Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA Received 28 September 1992; revised manuscript received 3 December 1992 Anharmonic collective vibrations that carry nonzero angular momentum along an equilibrium symmetry axis can be described by the self-consistent cranking model. This concept is illustrated for y vibrations of a deformed nucleus using two examples: the interacting boson model (IBM) in the SU ( 3 ) limit, and the volume-conserving oscillator shell model. In both models, the crank- ing solutions describe a sequence of bandheads with spin projections K= 2, 4, 6, ... along the symmetry axis, which happen to lie along a backbending trajectory. The results for the oscillator shell model suggest that the anharmonicity of the isoscalar giant quadrupole resonance is quite small. The question of anharmonicity of nuclear collec- tive vibrations has long been one of the most chal- lenging in nuclear structure, both experimentally and theoretically. The most direct experimental evidence for anharmonicity, provided by vibrational states with two or more photons, has been rather elusive. How- ever, eighteen candidates have recently been pro- posed for K= 4 two-phonon ~ vibrations of deformed nuclei [ 1 ], as well as some evidence for a two-phonon octupole level in 2°8pb [2] and even a suggestion of two-phonon giant resonances in spherical nuclei [ 3 ]. While the theorist's toolkit contains a plethora of dif- ferent microscopic methods for calculating anhar- monic corrections [ 4 ], each method has its pros and cons and none is easy to implement. The main pur- pose of this note is to call attention to an especially simple method for finding periodic solutions of mean- field equations that is applicable to collective vibra- tions carrying nonzero angular momentum for small or large amplitudes. This method employs the self- consistent cranking model (SCCM), which has played a central role in elucidating rotational states of high angular momentum. However, the proposal here is to use the SCCM to describe states that are This work was supported in part by the Department of Energy under grant DE-FG02-91 ER40640. i Permanent address: Joint Institute for Nuclear Research, Lab- oratory of Theoretical Physics, 141980 Dubna, Russian Federation. normally regarded as vibrational rather than rota- tional in character. In particular, the SCCM will be applied for the first time to models of vibrating de- formed nuclei. The idea of applying the SCCM to vibrational modes was introduced over twenty years ago by Mar- shalek and Sabato [ 5,6 ], who showed that the SCCM could be used not only to derive the RPA for spheri- cal nuclei, but also the leading-order perturbative an- harmonic corrections to the excitation energies as well as the static quadrupole moments and B(E2)'s for the aligned n-phonon quadrupole excitations with spins I=2n. Since the essential ingredient of this spherical cranking model is the presence of an axis of symmetry in the uncranked mean field, the model can be extended to axially symmetric deformed nuclei, thereby providing a new tool with some special ad- vantages for calculating the n-phonon 7-vibrational bandheads with spin projections K=2n on the axis of symmetry and, possibly, other bandheads as well. This will be illustrated via two examples: the inter- acting boson model (IBM) and SU (3) limit and the volume-conserving harmonic oscillator shell model, which can be interpreted as a rough model of giant quadrupole resonances. Unlike the earlier treatment, the present application avoids perturbation theory, which often leads to problems with small resonance denominators. To motivate our approach, we recall the physical 0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved. 199