Chiral Bands, Dynamical Spontaneous Symmetry Breaking, and the Selection Rule for Electromagnetic Transitions in the Chiral Geometry

A model for a special configuration in triaxial odd-odd nuclei is constructed which exhibits degenerate chiral bands with a sizable rotation, a manifestation of dynamical spontaneous symmetry breaking. A quantum number obtained from the invariance of the model Hamiltonian, which characterizes observable states, is given and selection rules for electromagnetic transition probabilities in chiral bands is derived in terms of this quantum number. The degeneracy of the lowest two bands is indeed obtained in the numerical diagonalization of the Hamiltonian at an intermediate spin range, over which electromagnetic transitions follow exactly the selection rule expected for the chiral geometry.

Figure 1

Sketch of the idealized chiral geometry. The core, proton, and neutron angular momenta denoted by $\overrightarrow{R}$, $\overrightarrow{j_p}$, and $\overrightarrow{j_n}$, respectively, are almost parallel to the principal axes of the triaxial body, which are labeled by the numbers inside parentheses. The right-handed system is denoted by $R$, while the left-handed one by $L$.

Figure 2

Selection rule expected for both interbands and intrabands $E2$ and $M1$ transitions, when chiral geometry is realized in the present model. Allowed $E2$ transitions (with $\Delta I=1$ and 2) and stronger $M1$ transitions (with $\Delta I=1$) are indicated by thick arrows. Allowed but weaker $\Delta I=1$ $M1$ transitions are indicated by thinner arrows. Solid lines represent states of $A=+1$, while the dashed lines those of $A=-1$. Two chiral pair bands are denoted by $f$ and $u$.

Figure 3

Calculated level scheme for a pair of $\Delta I=1$ bands. Arrows represent transitions of strong $B(E2)$ or $B(M1)$. The calculated $E2$ ($M1$) transitions, which are not shown in the figure, are totally forbidden (negligibly weak). Solid and dashed lines represent levels with $A=+1$ and $-1$, respectively, or vice versa. See text for detail.