Damping mechanisms and order-to-chaos transition in the warm rotating ${}^{163}\mathrm{Er}$ nucleus

The γ decay in the quasicontinuum is used to study the order-to-chaos transition in the thermally excited ${}^{163}\mathrm{Er}$ nucleus. The experimental analysis is performed on high-statistics EUROBALL data, focusing on the spin region $I\approx (20–40)\hslash$, and internal excitation energy up of $\approx 2.5\mathrm{MeV}$. The results are compared to cranked shell model calculations for this nucleus, taking into account the dependence on the K quantum number. Two main topics are investigated. First, the validity of the selection rules associated with the K quantum number are studied as a function of the internal energy U above yrast. K-selection rules are found to be obeyed in the decay along discrete unresolved rotational bands up to $U\approx 1.2\mathrm{MeV}$, whereas in the interval $U\approx 1.2–2.5\mathrm{MeV}$, where the order-to-chaos transition is expected to take place, selection rules are found to be only partially valid. Second, the line-shape analysis of $\gamma -\gamma$ coincidence spectra provides a direct experimental measurement of the rotational and compound damping widths (${\Gamma }_{\mathrm{rot}}$ and ${\Gamma }_{\mu }$), yielding values of 200 and 20 keV, respectively, in good agreement with theory.

Figure 1

(Color online) Schematic illustration of the fragmentation of the rotational $E2$ strength from a state at spin I to a number of final states at spin ($I-2$), as a consequence of the complex nature of the compound nucleus states. In the figure, the ${\mu }_i$'s are np-nh unperturbed bands, which are mixed to form a state at spin I. The dispersion $\Delta \omega$ in rotational frequency from spin I to spin ($I-2$) arises from the different response of each unperturbed band to the rotation. (Adapted from Ref. 12.)

Figure 2

Excitation energy versus spin of the most strongly populated bands of ${}^{163}\mathrm{Er}$, relative to a rigid reference with dynamic moment of inertia $J^{(2)}=63.6{\mathrm{MeV}}^{-1}\hslash {}^2$. The upper scale of panel (b) refers to the corresponding γ-ray energy $E_{\gamma }$. Panel (a) refers to positive-parity bands, whereas panel (b) to negative-parity bands. The gates used to construct the 2D spectra in coincidence with the different intrinsic configurations are indicated by filled symbols; the transitions of band A used to construct the matrix collecting the total decay flow of ${}^{163}\mathrm{Er}$ are marked by crosses.

Figure 3

(Color online) Double-gated spectra of the most intense low-K bands of ${}^{163}\mathrm{Er}$, labeled according to Ref. 20. The spectra have been obtained by setting the additional gate, shown by arrows in the figure, on the corresponding 2D gate-selected matrix and subtracting the gate-related background. Transitions marked by circles occur in band A, to which the specific configuration finally decays.

Figure 4

Double-gated spectra of the most intense high-K bands of ${}^{163}\mathrm{Er}$, labeled according to Ref. 20. The spectra have been obtained by setting the additional gate, shown by arrows in the figure, on the corresponding 2D gate-selected matrix and subtracting the gate-related background. The groups of $M1$ and $E2$ transitions, characteristic of the high-K band structure are indicated by arrows.

Figure 5

60-keV-wide projections perpendicular to the $E_{{\gamma }_1}=E_{{\gamma }_2}$ diagonal of experimental and simulated 2D spectra of ${}^{163}\mathrm{Er}$ (left and right panels, respectively), at the average transition energy $\langle E_{\gamma }\rangle =900$ keV. The spectra collect either the total γ-decay flow (a and b) or the γ decay in coincidence with low-K (c and d) or high-K (e and f) configurations. In the simulation, a state is defined as low K (high K) if $K\le 8$ ($K>$ 8). The ridge-valley structure typical of rotational nuclei is seen in all spectra, with a separation between the two most inner ridges equal to 8$\hslash {}^2/J^{(2)}$, as indicated by the arrows. The reduced intensity observed in the $E_{{\gamma }_1}\ge E_{{\gamma }_2}$ region of the spectra is mostly due to the subtraction of all discrete lines known from the level scheme, in the case of the experimental data, and of the yrast and first excited bands, in the case of the simulation. (Adapted from Ref. 13.)

Figure 6

The experimental total number of discrete paths populating the ridge structure of ${}^{163}\mathrm{Er}$ (symbols) and the prediction from the CSM of Ref. 15 (solid line). For comparison we also show calculations in which no $J_z^2$ term (dotted line) or no residual interaction (dashed line) is included in the cranking Hamiltonian.

Figure 7

(a) The number of decay paths extracted from the fluctuation analysis of the ridge structure of ${}^{163}\mathrm{Er}$. The open circles (squares) refer to the number of rotational bands populating the first ridge of $\gamma -\gamma$ matrices gated by low-K (high-K) configurations; the filled triangles give the number of discrete paths obtained from the total matrix. The corresponding values for simulated spectra are shown by dotted, dashed, and full lines (low K, high K, and total, respectively). (b) The effective number of paths among strongly interacting bands obtained from the fluctuation analysis of the valley region is shown by open circles (squares) for low-K (high-K) gated spectra; the filled triangles show the results obtained from the total $\gamma -\gamma$ matrix. The dashed and solid lines give the results for total and high-K cascades, as obtained from simulated spectra. (Adapted from Ref. 13.)

Figure 8

Spectra of ${\mu }_1$ (left) and ${\mu }_2$ (middle) obtained from experimental $\gamma -\gamma$ coincidence matrices gated by three different types of structures (named in each panel). The spectra have been obtained by projecting the $E_{{\gamma }_1}\times$ $E_{{\gamma }_2}$ spectra perpendicularly to the $E_{{\gamma }_1}=E_{{\gamma }_2}$ diagonal, with a width of 60 keV and an average transition energy ($E_{{\gamma }_1}+E_{{\gamma }_2}$)$/2=960$ keV. The right part of the figure shows the covariance between pairs of gated spectra, also evaluated in two dimensions before projection.

Figure 9

Spectra of ${\mu }_1$ (left) and ${\mu }_2$ (middle) obtained from simulated $\gamma -\gamma$ coincidence matrices gated by the three different type of structures (named in each panel). The spectra have been obtained by projecting the $E_{{\gamma }_1}\times$ $E_{{\gamma }_2}$ spectra perpendicularly to the $E_{{\gamma }_1}=E_{{\gamma }_2}$ diagonal, with a width of 60 keV and an average transition energy ($E_{{\gamma }_1}+$ $E_{{\gamma }_2}$)$/2=960$ keV. The right part of the figure shows the covariance between pairs of gated spectra, also evaluated in two dimensions before projection.

Figure 10

The results of the covariance analysis on ridge (bottom panels) and valley (top panels) structures of ${}^{163}\mathrm{Er}$. Panels (a) and (c) show by open squares the correlation coefficient r obtained experimentally by averaging over pairs of $\gamma -\gamma$ spectra gated by low-K configurations; the correlation coefficient obtained from the experimental analysis of the low-K versus the high-K matrices is shown by filled circles in panels (b) and (d). The theoretical values, as obtained from the covariance analysis of simulated spectra, are represented by solid lines in all the panels; the dashed lines indicate the two extreme limits for full mixing (1) and no mixing (0). (Adapted from Ref. 13.)

Figure 11

(Color online) Illustration of the line-shape analysis technique applied and tested on simulated spectra. (a) An example of the calculated strength function for two consecutive $E2$ γ rays in the $E_{{\gamma }_1}$ × $E_{{\gamma }_2}$ plane. (b) Schematic of the analytic two-component function S($E_{{\gamma }_1},E_{{\gamma }_2}$), which is found to reproduce well the microscopically calculated $E2$ strength. (c) Projection on the $E_{{\gamma }_1}-E_{{\gamma }_2}$ axis of the $E2$ strength shown in (a). In the figure, the full drawn line is the projection of the two-component Gaussian function schematically shown in panel (b), which contains wide and narrow distributions of width ${\Gamma }_{\mathrm{wide}}\approx \sqrt{2}{\Gamma }_{\mathrm{rot}}$ and ${\Gamma }_{\mathrm{nar}}\approx 2{\Gamma }_{\mu }$, respectively. The dark histogram represents the discrete bands distribution, which is obtained by requiring a branching number $n_b$ smaller than 2. (d) A projection of a $\gamma -\gamma$ spectrum calculated for several successive $E2$ decays. The full drawn line is the parametrization, which incorporates the two-component function, with ${\Gamma }_{\mathrm{rot}}=180$ keV, ${\Gamma }_{\mathrm{nar}}=44$ keV, and $I_{\mathrm{nar}}=0.3$; the dashed line has no narrow component and a width ${\Gamma }_{\mathrm{rot}}=150$ keV.

Figure 12

Panels (a), (b) and (c) show 60-keV-wide projections on simulated spectra of ${}^{163}\mathrm{Er}$ collecting the $E2$ strengths from discrete rotational bands (bottom spectra) and damped transitions (top spectra), at the average transition energies $\langle E_{\gamma }\rangle =900$, 960, and 1020 keV. The interpolation of the damped spectrum by the two-component function is given by the solid line. Panels (d) and (e) show the calculated values of ${\Gamma }_{\mathrm{rot}}$ and ${\Gamma }_{\mathrm{nar}}$, respectively. The lines refer to the values calculated directly from the CSM bands at three different excitation energies: for $\langle U\rangle$ < 1 MeV (in which case the bands branch out to less than two final states, $n_b<$ 2), $\langle U\rangle$ ≈ 1.4 MeV, and $\langle U\rangle$ ≈ 2 MeV. The open (filled) stars refer to the values extracted from the simulated discrete (damped) spectra using the analytic parameterization. Finally, the open circles refer to the experimental data extracted from the spectra presented in Fig. 13.

Figure 13

60-keV-wide projections on the $E_{{\gamma }_1}-E_{{\gamma }_2}$ axis of experimental matrices of ${}^{163}\mathrm{Er}$, at the average transition energies $E_{\gamma }=900$ and 960 keV. Panels (a) and (b) show projections on the total $\gamma -\gamma$ matrix, whereas panels (c) and (d) and (e) and (f) correspond to spectra gated on low-K and high-K configurations of ${}^{163}\mathrm{Er}$. The (−150,150) keV interval shown here is equivalent to the one adopted in Figs. 12a, 12b, and 12c (namely (−200,200) keV), considering the different values of $8\hslash {}^2/J^{(2)}$ in the simulated and experimental spectra (≈150 and 120 keV, respectively).

Figure 14

Experimental values of ${\Gamma }_{\mathrm{nar}}$ and ${\Gamma }_{\mathrm{rot}}$, as extracted from the spectral shape analysis of experimental low-K (a and c) and high-K (b and d) $\gamma -\gamma$ coincidence spectra of ${}^{163}\mathrm{Er}$. Predictions from CSM calculations [15] for average excitation energies of 1.4 and 2 MeV are shown by solid and dashed lines, respectively.

Figure 15

The level density of ${}^{163}\mathrm{Er}$ as a function of internal energy U at spin $I=40\hslash$, $41\hslash$, calculated according to the model of Ref. 15. The solid line refers to the total number of states, the dashed (dotted) lines correspond to the states with $\langle K\rangle \le 8\hspace{1em}(\langle K\rangle >8)$.

Figure 16

The relation between the transition energy $E_{\gamma }$ and the spin I, for the most intense experimental low-K bands (open symbols). The average among the experimental data (solid line) is also compared with the corresponding values from the CSM of Ref. 15 (filled diamonds).

Figure 17

Comparison of the fraction of the total intensity populating the ridge structure of ${}^{163}\mathrm{Er}$ in the experimental $\gamma -\gamma$ spectrum (symbols) and in the simulated matrix obtained using the hindrance factors $h_{E1}=0.6$ , 0.4, and 0.2.

Figure 18

Comparison of the measured and calculated intensities of the discrete γ transitions of the strongest populated bands in ${}^{163}\mathrm{Er}$ as a function of spin, relative to the strongest yrast transition.

Figure 19

Projections on the $E_{{\gamma }_1}-E_{{\gamma }_2}$ axis of the strength function for two consecutive $E2$ γ rays, obtained by microscopic CSM calculations of ${}^{163}\mathrm{Er}$ [8, 12], for the levels 11 to 100 for each $I^{\pi }$. The calculations are performed at spin $I=30\hslash$, $31\hslash$ (a), $I=40\hslash$,$41\hslash$ ( b), and $I=50\hslash$,$51\hslash$ (c). The full drawn line is the two-component function, which contains wide and narrow Gaussian distributions of width ${\Gamma }_{\mathrm{wide}}~\sqrt{2}{\Gamma }_{\mathrm{rot}}$ and ${\Gamma }_{\mathrm{nar}}~2{\Gamma }_{\mu }$, respectively. The discrete bands distribution (defined by the relation $n_b<2$) is also shown by histograms.

Figure 20

Values of ${\Gamma }_{\mathrm{rot}}$ and ${\Gamma }_{\mathrm{nar}}$, as a function of spin, extracted from the FWHM of the wide and narrow component of the two-step distribution shown in Fig. 19. The calculations are performed for the sets of levels 11 to 100 (dashed lines) and 101 to 300 (solid lines), corresponding to the average internal energy $\langle U\rangle \approx 1.4$ and 2 MeV, respectively. The dotted line gives the FWHM of the distribution of discrete bands (shown by histograms in Fig. 19 and defined by the condition $n_b<$ 2).

Figure 21

The same as shown in Fig. 19 for (a) low-K ($K\le 8$), and (b) high-K ($K>$ 8) states, separately. In both cases, the calculated distribution is well accounted for by a two-component function containing wide and narrow Gaussian distributions of width ${\Gamma }_{\mathrm{wide}}~\sqrt{2}{\Gamma }_{\mathrm{rot}}$ and ${\Gamma }_{\mathrm{nar}}~2{\Gamma }_{\mu }$, respectively. The discrete bands distribution (defined by the relation $n_b<2$) is also shown by histograms.

Figure 22

(a) Projections of 2D parametrized spectra for several successive $E2$ decays. The calculations are performed for $I_{\mathrm{nar}}=15$, 30, and 60%, assuming constant values for ${\Gamma }_{\mathrm{nar}}=50$ keV and ${\Gamma }_{\mathrm{rot}}=200$ keV. (b) Results obtained for different values of ${\Gamma }_{\mathrm{rot}}$ (i.e., 100, 200, and 300 keV), if no narrow component is included ($I_{\mathrm{nar}}=0$).

Figure 23

Projections of 2D parametrized spectra for several successive $E2$ decays, obtained assuming a 20% constant value for $I_{\mathrm{nar}}$. In panel (a) the calculations are performed for ${\Gamma }_{\mathrm{rot}}=200$ keV and ${\Gamma }_{\mathrm{nar}}=20$, 40, and 100 keV. In panel (b) ${\Gamma }_{\mathrm{nar}}$ is 50 keV whereas the rotational damping width assumes the values ${\Gamma }_{\mathrm{rot}}=100$, 200, and 300 keV, respectively.