A comprehensive view of nuclear shapes, rotations and vibrations from fully quantum mechanical perspectives

TL;DR

The paper presents a fully quantum mechanical, projection-based framework for low-energy quadrupole collectivity in heavy nuclei, challenging the long-standing axial, rigid-body view. Using MCSM/QVSM, it demonstrates that most strongly deformed nuclei are triaxial and that rotational bands arise from angular-momentum projection, with $J(J+1)$-like spacings driven by binding-energy changes rather than rotational kinetic energy. It identifies two robust mechanisms for triaxiality— symmetry restoration via $K$ projection and specific NN force components— and shows that the $K$ quantum number is practically conserved, making $K$ the natural organizing principle for bands, with the $2^+_ abla$ (gamma) band corresponding to a $K=2$ rotation and the $4^+_ abla$ state to a $K=4$ rotation. The work renews the traditional picture of heavy-nucleus quadrupole motion, offering a simpler, more natural quantum description and revealing vibrational modes that emerge on top of the rotational structure.

Abstract

The nuclear quadrupole collective states at low excitation energies are described in a novel, fully quantum mechanical and systematic manner as compared to traditional pictures initiated by Aage Bohr. The ellipsoidal shapes are shown to be triaxial in virtually all strongly deformed nuclei, in contrast to the Ansatz of axially symmetric shapes. The rotational bands of such triaxially deformed nuclei are described in a fully quantum mechanical way, i. e., without resorting to quantized free rotation of rigid body. The excitation energies within a rotational band, exhibiting the $J(J+1)$ dependence on angular momentum $J$, are shown to basically represent the change of binding energies due to nuclear forces. This differs from the interpretation á la Aage Bohr as rotational kinetic energies. The $K$ quantum numbers are shown to be practically conserved for triaxial ellipsoids, which turned out to be a real but positive surprise to many people in the field. The so-called $γ$ bands are shown to be $K$=2$^+$ rotations rather than $γ$-vibrations, leading to a nice description of the so-called $γγ$ 4$^+$ state as a $K$=4$^+$ rotation. Vibrational modes are also shown to emerge in this study. Thus, the whole picture of low-energy quadrupole collective motion of heavy nuclei has been renewed in a fully quantum mechanical fashion, which differs from the traditional picture but appears to be simpler and more natural.

Figures (7)

• Figure 1: Schematic illustrations of the rotations of atomic nuclei in ellipsoidal shapes. a. axially symmetric and b. triaxial nuclear shapes, with associated rotations $\vec{R}$ and $\vec{K}$. The upper part is side views, while the lower part top views. Intuitive images, rugby ball and almond, are shown in both sides.
• Figure 2: PES and T-plots for $^{166}$Er. a Legend. b PES. c T-plots of ground and low-lying states for the red-square region of b. Based on Fig. 5 of epja with kind permission of The European Physical Journal (EPJ).
• Figure 3: Schematic illustrations of K-projected states. Taken from Fig. 6 of epja with kind permission of The European Physical Journal (EPJ).
• Figure 4: Schematic views and descriptions of the rotation. a. Classical mechanical view. b. Quantization of freely rotating rigid body. c. View of the quantum mechanical system composed of many constituents with angular momentum $\hbar J$. The J(J+1)-K$^2$ rule arises. The green arrow indicates similarity in the wave function, but the energy comes from different origins. Taken from Fig. 24 of epja with kind permission of The European Physical Journal (EPJ).
• Figure 5: a, b Schematic illustration of the practical conservation of $K$ quantum number at the limit of strong deformation. Rods represent strongly deformed triaxial intrinsic states projected onto $K$ or $K'$. c Schematic illustration of the $K$ mixing in weakly deformed nuclei expressed by ellipsoids. Taken from Fig. 14 of epja with kind permission of The European Physical Journal (EPJ).