Spontaneous symmetry breaking in rotating nuclei

Stefan Frauendorf

doi:10.1103/RevModPhys.73.463

Spontaneous symmetry breaking in rotating nuclei

Stefan Frauendorf

Rev. Mod. Phys. 73, 463 – Published 13 June 2001

Abstract

The concept of spontaneous symmetry breaking is applied to the rotating mean field of nuclei. The description is based on the tilted-axis cranking model, which takes into account that the rotational axis can take any orientation with respect to the deformed density distribution. The appearance of rotational bands in nuclei is analyzed, focusing on weakly deformed nuclei at high angular momentum. The quantization of the angular momentum of the valence nucleons leads to new phenomena. Magnetic rotation represents the quantized rotation of the anisotropic current distribution in a near spherical nucleus. The restricted amount of angular momentum of the valence particles causes band termination. The discrete symmetries of the mean-field Hamiltonian provide a classification scheme of rotational bands. New symmetries result from the combination of the spatial symmetries of the density distribution with the vector of the angular momentum. The author discusses in detail which symmetries appear for a reflection-symmetric density distribution and how they show up in the properties of the rotational bands. In particular, the consequences of rotation about a nonprincipal axis and of breaking the chiral symmetry are analyzed. Also discussed are which symmetries and band structures appear for non-reflection-symmetric mean fields. The consequences of breaking the symmetry with respect to gauge and isospin rotations are sketched. Some analogies outside nuclear physics are mentioned. The application of symmetry-restoring methods to states with large angular momentum is reviewed.

DOI:https://doi.org/10.1103/RevModPhys.73.463

Authors & Affiliations

Stefan Frauendorf

Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556
Institute for Nuclear and Hadronic Physics, Research Center Rossendorf, PB 51 01 19, D-01314 Dresden, Germany

^*This definition of the quadrupole operators corresponds to $Q_{0} {= r}^{2} P_{2} (\cos ϑ),$ with $P_{2}$ being the Legendre polynomial.
^†We use the definition of the $D$ functions byBohr and Mottelson (1975).
^‡The reader may find a more extended discussion in the textbook ofLandau and Lifshitz (1985).
^§We use the notation introduced byBengtsson and Frauendorf (1979b). Another convention, which followsBohr and Mottelson (1975), uses the signature quantum number ${r = e}^{- i α π} .$ In order to avoid long-winded formulations we shall adopt the somewhat loose but common terminology calling α simply signature.
^∥The labeling of the Nilsson states in a deformed axial potential is explained in standard textbooks like those ofBohr and Mottelson (1975) orRing and Schuck (1980).
^¶See for example,Stephens, 1975;de Voigt et al., 1983;Szymanski, 1983;Bengtsson and Garrett, 1984;Garrett et al., 1986;Åberg et al., 1990;Nilsson and Ragnarsson, 1995.
^**The first acronym stands for ground-state band. When the back bending irregularity was discovered its nature was unclear, and the vague name superband was coined, which became s band with frequent use.
^††The name alludes to two characteristics: The t band is similar to the s band (large $j_{1}$ ) but tilted (large $j_{3}$ ).
^‡‡The term “magnetic rotation” is also used in optics, solar physics, and nuclear magnetic resonance physics in very different contexts.
^§§The subscript on the parenthesis gives the angular momentum for stretched coupling of the particles within the subshell.
^∥∥We use the notation introduced byFrauendorf and Pashkevich (1984).Nazarewicz et al. (1984) introduced another, frequently used convention, which followsBohr and Mottelson (1975). They call simplex ${s = e}^{- i σ π} .$
^¶¶SeeBlaizot and Ripka (1986). A coherent state is a wave packet for which the product $Δ p Δ x$ of the momentum and coordinate takes a minimum. It behaves much like a classical object as permitted by the laws of quantum mechanics.