Chaos and regularity in the spectra of the low-lying dipole excitations of ${}^{50,52,54}\mathrm{Cr}$

Recent high-resolution nuclear resonance fluorescence experiments performed on the even-even Chromium isotopes ${}^{50,52,54}\mathrm{Cr}$ have lead to the identification (energy, spin, parity, and transition strength) of altogether 108 nuclear levels of spin $J=1$ (70 levels with $J^{\pi }=1^{-}$ and 38 with $J^{\pi }=1^{+}$) at excitation energies $E_x$ ranging roughly from 4.5 to 9.7 MeV. In this region just above the orbital magnetic-dipole scissors mode, sizable spin-flip magnetic-dipole strength as well as electric-dipole strength belonging to the pygmy dipole resonance (PDR) is expected. Using statistical measures for short- and long-range correlations, we perform an analysis of the fluctuation properties in the subspectra of the energy levels and also of the distributions of their respective dipole transition strengths. We compare the results with those of a random matrix ensemble interpolating between Poisson statistics generally describing the fluctuation properties in the energy spectra of many-body systems with collective, i.e., regular motion of the particles and the Gaussian orthogonal ensemble (GOE) for complex (i.e., chaotic) behavior. This comparison reveals that the spectral properties of the $1^{+}$ states are close to the GOE results while those of the $1^{-}$ states are closer to Poisson. This is confirmed by an analysis of the spectral fluctuations based on the method of Bayesian inference and corroborated by large-scale shell-model and quasiparticle-phonon model calculations, respectively. The nearly Poissonian behavior of the $1^{-}$ levels suggests a sizable collectivity of the PDR indeed.

Figure 1

Transition strength $B\left(E1\right)$ for the $1^{-}$ states. (a)–(c) Experimentally determined strengths [1, 2, 3]. (d)–(f) Calculated strengths as described in the main text. The black dashed lines indicate the experimental threshold for detectability.

Figure 2

Transition strength $B\left(M1\right)$ for the $1^{+}$ states. (a)–(c) Experimentally determined strengths [1, 2, 3]. (d)–(f) Calculated strengths as described in the main text. The black dashed lines indicate the experimental threshold for detectability.

Figure 3

Nearest-neighbor spacing distributions of the experimental (a) $J^{\pi }=1^{-}$ and (b) $J^{\pi }=1^{+}$ energy levels of ${}^{54}\mathrm{Cr}$ (histograms) in comparison with the Poisson (dashed lines) and the GOE (full line) results.

Figure 4

Spectral statistics of the experimental $1^{-}$ energy levels of ${}^{50}\mathrm{Cr}$, ${}^{52}\mathrm{Cr}$, and ${}^{54}\mathrm{C}$ (up-triangles and histograms) as indicated in the insets in comparison to the Poissonian (dashed lines) and the GOE (full lines) results. Shown are the nearest-neighbor spacing distribution $P\left(s\right)$, the integrated nearest-neighbor spacing distribution $I\left(s\right)$, the number variance ${\mathrm{\Sigma }}^2\left(L\right)$, and the ratio distribution $P\left(r\right)$.

Figure 5

Same as Fig. 4 for the experimental $1^{+}$ energy levels of ${}^{50}\mathrm{Cr}$, ${}^{52}\mathrm{Cr}$, and ${}^{54}\mathrm{C}$ as indicated in the insets.

Figure 6

Comparison of the averaged spectral fluctuation properties of the experimental $J^{\pi }=1^{-}$ energy levels of ${}^{50}\mathrm{Cr},{}^{52}\mathrm{Cr}$, and ${}^{54}\mathrm{Cr}$ with the Poisson (dashed lines) and the GOE (full lines) results and those obtained from QPM calculations (red down-triangles and dash-dotted lines).

Figure 7

Same as Fig. 6. Comparison of the results for the experimental $J^{\pi }=1^{+}$ energy levels with those obtained from shell-model calculations (red down-triangles and dash-dotted lines).

Figure 8

Comparison of the averaged spectral fluctuation properties of the experimental $1^{+}$ and $1^{-}$ energy levels of ${}^{50}\mathrm{Cr},{}^{52}\mathrm{Cr}$, and ${}^{54}\mathrm{Cr}$ with the Poisson (dashed lines) and the GOE (full lines) results and with those obtained from missing-level statistics [35] (red circles and dash-dotted lines). Best agreement was found for a fraction of $\phi =0.75$ detected ones, corresponding to $25\%$ for the $1^{+}$ states and $\phi =0.4$ corresponding to $60\%$ missing levels for the $1^{-}$ states.

Figure 9

Same as Fig. 8. Comparison with the spectral statistics of a random matrix ensemble interpolating between Poisson ($\lambda =0$) and GOE [25, 27, 36, 37] ($\lambda =1$) (red down triangles and dash-dotted lines). Best agreement was found for $\lambda =0.8$ for the $1^{+}$ states and $\lambda =0.3$ for the $1^{-}$ states.

Figure 10

Nearest-neighbor spacing distribution of the experimental energy levels with (a) $J^{\pi }=1^{-}$ and (b) $J^{\pi }=1^{+}$ in comparison with the NNSDs of the calculated ones for (c) $J^{\pi }=1^{-}$ and (d) $J^{\pi }=1^{+}$ obtained by taking into account only those levels with transition strengths above the experimental minimum value. They are compared with the Poisson (dashed line) and the GOE (full line) distributions. The dash-dotted curves in red were determined with the method of Bayesian inference. The resulting values for the chaoticity parameter $\overline{f}$ are given in the insets.

Figure 11

(a), (b) Averaged distribution of the unfolded transition strengths $y_i=B_i/B_{av,i}$ and of (c), (d) $z_i={log}_{10}\left(y_i\right)$ of the experimental results (histograms) for the (a), (c) $J^{\pi }=1^{-}$ and (b), (d) $J^{\pi }=1^{+}$ states in comparison with a truncated Porter–Thomas distribution $P\left(y\right)$ with $y\ge y_0$ and $y_0$ denoting the smallest experimentally observed transition strength (dashed and dash-dotted lines).

Figure 12

Averaged spectral statistics of the $1^{+}$ and $1^{-}$ energy levels obtained from shell-model calculations using the effective KB3G interaction and from QPM calculations (red dash-dotted lines and histograms), respectively. Here, the number of energy levels taken into account for each nucleus was chosen to be similar to those of the corresponding experimental energy levels.

Figure 13

(a), (b) Averaged distribution of the unfolded transition strengths $y_i=B_i/B_{av,i}$ and (c), (d) $z_i={log}_{10}\left(y_i\right)$ obtained from QPM and shell-model calculations including the KB3G interaction (histograms) for the (a), (c) $J^{\pi }=1^{-}$ and (b), (d) $J^{\pi }=1^{+}$ states, respectively, in comparison with the Porter–Thomas distribution (dashed lines).