Quasidynamical symmetries in the backbending of chromium isotopes

Background: Symmetries are a powerful way to characterize nuclear wave functions. A true dynamical symmetry, where the Hamiltonian is block-diagonal in subspaces defined by the group, is rare. More likely is a quasidynamical symmetry: states with different quantum numbers (i.e., angular momentum) nonetheless sharing similar group-theoretical decompositions.

Purpose: We use group-theoretical decomposition to investigate backbending, an abrupt change in the moment of inertia along the yrast line, in ${}^{48,49,50}\mathrm{Cr}$: prior mean-field calculations of these nuclides suggest a change from strongly prolate to more spherical configurations as one crosses the backbending and increases in angular momentum.

Methods: We decompose configuration-interaction shell-model wave functions using the SU(2) groups $L$ (total orbital angular momentum) and $S$ (total spin), and the groups SU(3) and SU(4). We do not need a special basis but only matrix elements of Casimir operators, applied with a modified Lanczos algorithm.

Results: We find quasidynamical symmetries, albeit often of a different character above and below the backbending, for each group. While the strongest evolution was in SU(3), the decompositions did not suggest a decrease in deformation. We point out with a simple example that mean-field and SU(3) configurations may give very different pictures of deformation.

Conclusions: Persistent quasidynamical symmetries for several groups allow us to identify the members of a band and to characterize how they evolve with increasing angular momentum, especially before and after backbending.

Figure 1

Backbending in ${}^{48-50}\mathrm{Cr}$, as signaled by the evolution of $E_{\gamma }\left(I\right)=E\left(I\right)-E(I-2)$. The distinct shapes/colors represent, to the best of our ability to identify, different configurations along the yrast as discussed in detail in the text: (red) solid squares for the lower subband, (blue) dotted triangles for the upper subband, and a black “$\times$” and (green) striped circle for upper and lower “intruder” levels, respectively. The calculated values are in good agreement with experiment (not shown).

Figure 2

Calculated spectrum of ${}^{48}\mathrm{Cr}$. The $x$ axis (angular momentum $I$) is scaled as $I(I+1)$ so as to emphasize rotational bands. The labeling of levels, i.e., (red) squares, (blue) triangles, and (green) circles, correspond to the same (initial) state as in panel (a) of Fig. 1. According to our decompositions, the yrast state at $I=10$, marked by as “$\times$,” belongs to neither the lower nor upper subbands. Bars indicate levels found in our calculation but which we do not decompose.

Figure 3

Decomposition of wave functions of ${}^{48}\mathrm{Cr}$ into components of total $L$ (orbital angular momentum). The fill (and color) scheme is matched to the levels shown in Fig. 2, i.e., (red) solid bars: lower subband; (blue) dotted: upper subband; and (black) cross-hatched and (green) striped: intruder levels. Here and throughout we superimpose levels which have the same $I$ but which belong to different subbands.

Figure 4

Decomposition of wave functions of ${}^{48}\mathrm{Cr}$ into components of total $S$ (spin). The fill (and color) scheme is the same as in Fig. 3.

Figure 5

Decomposition of wave functions of ${}^{48}\mathrm{Cr}$ into SU(3) irreps, labeled by eigenvalues of the two-body SU(3) Casimir (see text for definition). The fill (and color) scheme is the same as in Fig. 3.

Figure 6

Decomposition of wave functions of ${}^{48}\mathrm{Cr}$ into SU(4) irreps, labeled by eigenvalues of the two-body SU(4) Casimir (see text for definition). The fill (and color) scheme is the same as in Fig. 3.

Figure 7

Calculated spectrum of ${}^{49}\mathrm{Cr}$. The $x$ axis (angular momentum $I$) is scaled as $I(I+1)$ to emphasize rotational bands. The labeling of levels, i.e., red squares, blue triangles, and green circles, corresponds to the same (initial) state as in panel (b) in Fig. 1. Bars indicate levels found in our calculation but which we do not decompose.

Figure 8

Decomposition of wave functions of ${}^{49}\mathrm{Cr}$ into components of total $L$ (orbital angular momentum). Much like Fig. 3, the fill (and color) scheme is matched to the levels shown in Fig. 7, i.e., (red) solid bars: lower subband; (blue) dotted: upper subband; and (green) striped: the lowest $I=1/2,3/2$, which technically are not part of the yrast line.

Figure 9

Decomposition of wave functions of ${}^{49}\mathrm{Cr}$ into components of total $S$ (spin). The fill (and color) scheme is the same as in Fig. 8.

Figure 10

Decomposition of wave functions of ${}^{49}\mathrm{Cr}$ into SU(3) irreps. See text for the definition of the SU(3) Casimir. The fill (and color) scheme is the same as in Fig. 8.

Figure 11

Decomposition of wave functions $\mathrm{of}{}^{49}\mathrm{Cr}$ into SU(4) irreps. See text for the definition of the SU(4) Casimir. The fill (and color) scheme is the same as in Fig. 8.

Figure 12

Calculated spectrum of ${}^{50}\mathrm{Cr}$. The $x$ axis (angular momentum $I$) is scaled as $I(I+1)$ to emphasize rotational bands. The labeling of levels, i.e., (red) squares, (blue) triangles, and (green) circles, correspond to the same (initial) state as in panel (a) of Fig. 1. Bars indicate levels found in our calculation but which we do not decompose.

Figure 13

Decomposition of wave functions of ${}^{50}\mathrm{Cr}$ into components of total $L$ (orbital angular momentum). Decomposition of wave functions of ${}^{50}\mathrm{Cr}$ into components of total $L$ (orbital angular momentum). Much like Fig. 3, the fill (and color) scheme is matched to the levels shown in Fig. 12, i.e., (red) solid bars: lower subband; (blue) dotted: upper subband; and (green) striped: “intruder,” that is, outside of the ${\left(0f_{7/2}\right)}^{10}$ configuration space. Here and throughout we superimpose levels which have the same $I$ but which belong to different subbands.

Figure 14

Decomposition of wave functions of ${}^{50}\mathrm{Cr}$ into components of total $S$ (spin). The fill (and color) scheme is the same as that of Fig. 13.

Figure 15

Decomposition of wave functions of ${}^{50}\mathrm{Cr}$ into SU(3) irreps. The fill (and color) scheme is the same as that of Fig. 13.

Figure 16

Decomposition of wave functions of ${}^{50}\mathrm{Cr}$ into SU(4) irreps. The fill (and color) scheme is the same as that of Fig. 13.