Phonon and multi-phonon excitations in rotational nuclei by exact diagonalization of the Bohr Hamiltonian

Exact numerical diagonalization of the Bohr Hamiltonian by SU(1, 1) × SO(5) methods is used to obtain detailed quantitative predictions for single-phonon and multi-phonon excitations in well-deformed rotor nuclei. Dynamical γ deformation is found to significantly influence the predictions through its coupling to the rotational motion. Basic signatures for the onset of rigid triaxial deformation are obtained. PACS: 21.60.Ev, 21.10.Re The Bohr collective Hamiltonian has served as a conceptual benchmark for the interpretation of quadrupole collective dynamics in nuclei for several decades [1, 2]. A tractable scheme for numerical diagonalization of the Bohr Hamiltonian, the algebraic collective model (ACM) [3–7], has recently been proposed, based on SU(1, 1)× SO(5) algebraic methods. The need for such an approach arises since the conventional approach to numerical diagonalization of the Bohr Hamiltonian, in a fivedimensional oscillator basis [8–10], is slowly convergent and requires a large number of basis states to describe a general deformed rotor-vibrator nucleus. Consequently, it has been necessary to apply varying degrees of approximation in addressing the dynamics of transitional and deformed nuclei, as in the classic rotation-vibration model [11] and rigid triaxial rotor [12] treatments of the Bohr Hamiltonian, or in more recent studies of critical phenomena [13–16]. The ACM scheme, in conjunction with recent progress in construction of the relevant SO(5) ⊃ SO(3) Clebsch-Gordan coefficients [7], now permits the diagonalization of the Bohr Hamiltonian for potentials of essentially arbitrary stiffness, as considered in this letter. The Bohr Hamiltonian can consequently be applied, without approximation, to the full range of nuclear quadrupole rotational-vibrational structure, from spherical oscillator to axial rotor to triaxial rotor. Specifically, the direct product basis obtained from an optimally chosen set of SU(1, 1) β wave functions [17] and the SO(5) ⊃ SO(3) spherical harmonics ΨvαLM (γ,Ω) [4] provides an exceedingly efficient basis for numerical solution of the Bohr Hamiltonian [5]. For application to transitional and deformed nuclei, the method yields order-of-magnitude reductions in the basis size needed for convergence, as compared to diagonalization in a fivedimensional oscillator basis. The SU(1, 1) × SO(5) algebraic structure of the basis facilitates construction of matrix elements for a wide variety of potential and kinetic energy operators. In this letter, detailed quantitative predictions for singlephonon and multi-phonon excitations in deformed rotor nuclei are established by exact numerical diagonalization of the Bohr Hamiltonian, making use of newly-calculated SO(5) ⊃ SO(3) Clebsch-Gordan coefficients [7]. In the past, interpretation of rotational phonon states within the Bohr description has largely been at a schematic level (e.g., Refs. [18–22]): adiabatic separation of the rotational and vibrational degrees of freedom is assumed, the β and γ excitations are taken to be harmonic, and phonon selection rules are assumed for electric quadrupole transitions. These predictions are then adjusted by spin-dependent band mixing [23] with ad hoc mixing parameters. Here, instead, we explore the actual predictions of the Bohr Hamiltonian. The signatures for the onset of rigid triaxial deformation within the Bohr framework are also considered. Preliminary results were presented in Ref. [24]. The Bohr Hamiltonian [2] is given, in terms of the quadrupole deformation variables β and γ and Euler angles Ω, by