Statistical analysis of excitation energies in actinide and rare-earth nuclei

Statistical analysis of distributions of the collective states in actinide and rare-earth nuclei is performed in terms of the nearest-neighbor spacing distribution (NNSD). Several approximations, such as the linear approach to the level repulsion density and that suggested by Brody to the NNSDs were applied for the analysis. We found an intermediate character of the experimental spectra between the order and the chaos for a number of rare-earth and actinide nuclei. The spectra are closer to the Wigner distribution for energies limited by 3 MeV, and to the Poisson distribution for data including higher excitation energies and higher spins. The latter result is in agreement with the theoretical calculations. These features are confirmed by the cumulative distributions, where the Wigner contribution dominates at smaller spacings while the Poisson one is more important at larger spacings, and our linear approach improves the comparison with experimental data at all desired spacings.

Figure 1

The location and the $(p,t)$ strength of $0^{+}$ states in ${}^{228}\mathrm{Th}$, ${}^{230}\mathrm{Th}$, ${}^{232}\mathrm{U}$, and ${}^{240}\mathrm{Pu}$. Horizontal lines indicate limitations in the investigation energy.

Figure 2

Histogram of the cumulative number of states, $N\left(E\right)$, for the $0^{+}$ energy spacings in ${}^{230}\mathrm{Th}$ and its fitting by two smooth polynomials, Eq. (5), present the method of extraction of the normalized effective NNSD from the experimental data.

Figure 3

Nearest neighbor spacing distributions $p\left(s\right)$ as functions of a dimensionless spacing variable $s$ for $0^{+}$ states in the rare-earth nuclei and fits by the LWD approximation (red solid lines) and the Brody approach (blue dashed lines): (a) for a number of rare-earth nuclei up to the energy 3 MeV (see the text); (b) for the ${}^{158}\mathrm{Gd}$ and ${}^{168}\mathrm{Er}$ nuclei up to about 4 MeV. A sampling interval of ${\gamma }_s=0.2$ was used.

Figure 4

The same as in Fig. 3 but for different states in the actinide nuclei: (a)–(d) for $0^{+}$, $2^{+}$, $4^{+}$, and $6^{+}$ states, respectively. The same fits by the LWD (red solid lines) and the Brody (blue dashed lines) approach as in Fig. 3 are shown.

Figure 5

Comparison of the NNSD between the experimental data (a) and the theoretical quasiparticle-phonon model results (b) in the energy interval up to 3 MeV in the ${}^{228,230}\mathrm{Th}$ and ${}^{232}\mathrm{U}$ actinide nuclei, and those (c) up to 4 MeV. Other notations are the same as in Figs. 3 and 4.

Figure 6

Histograms of the cumulative nearest-neighbor spacing distributions (20) for the $0^{+}$ (a), (e), $2^{+}$ (b), (f), $4^{+}$ (c), (g), and $6^{+}$ (d), (h) states in the actinide nuclei discussed in Fig. 4. (a)–(d) Closely dotted and dashed lines are the Poisson [Eq. (21)] and Wigner [Eq. (22)] cumulative distribution limits, respectively. (e)–(h) Solid and sparsely dotted lines are the LWD [Eq. (23)] and Brody [Eq. (24)] cumulative NNSD.