Question of dynamic chirality in nuclei: The case of ${}^{134}\mathrm{Pr}$

Lifetimes of exited states in ${}^{134}\mathrm{Pr}$ were measured by means of the recoil distance Doppler-shift and Doppler-shift attenuation techniques. The branching ratios and the electric or magnetic character of the transitions were also investigated. The experiments were performed at IReS, Strasbourg, using the EUROBALL IV spectrometer, in conjunction with the inner bismuth germanate ball and the Cologne coincidence plunger apparatus. Exited states in ${}^{134}\mathrm{Pr}$ were populated in the fusion-evaporation reaction ${}^{119}\mathrm{Sn}$(${}^{19}\mathrm{F}$, 4$n$)${}^{134}\mathrm{Pr}$. The possible chiral interpretation of twin bands was investigated in the two-quasiparticle triaxial rotor and interacting boson-fermion-fermion models. The analysis of the wave functions has shown that the possibility for the angular momenta of the proton, neutron, and core to find themselves in the favorable, almost orthogonal geometry, is present but is far from being dominant. The structure is characterized by large β and γ fluctuations. The existence of doublets of bands in ${}^{134}\mathrm{Pr}$ can be attributed to weak chirality dominated by shape fluctuations.

Figure 1

Partial level scheme of ${}^{134}\mathrm{Pr}$ from Ref. [26]. Two nearly degenerate positive-parity bands, candidates for chiral partner bands, are indicated as Band 1 and Band 2.

Figure 2

Gated γ-ray spectra of ${}^{134}\mathrm{Pr}$ taken at four different distances. Summed spectra from all detectors at a backward angle of ${137.60}^{°}$ with respect to the beam axis are shown. On the left-hand side, both shifted and unshifted peaks of the ${13}_1^{+}\to {12}_1^{+}$ transition are presented. On the right-hand side, the shifted and unshifted peaks of the ${13}_2^{+}\to {12}_2^{+}$ transition are shown. At the distance 41.6 μm, the 323.6 keV transition (${13}_1^{+}\to {12}_2^{+}$) is fully shifted, whereas the 416.2 keV transition (${13}_1^{+}\to {12}_1^{+}$) has only a shifted component already at the distance of 20.6 μm.

Figure 3

Example of the line-shape analysis of the 594 keV γ-ray transition and determination of the lifetime of the $I^{\pi }=({12}^{+})$ level of the Band 1. On the left-hand side, spectra measured with the detectors of ring 7 and the fit (full line) are shown at three distances. The DSA portion of the fit is represented by a dash-dotted line, whereas the unshifted peak and the background are presented with a dashed line. The target-to-stopper distances and the corresponding mean end-of-flight times $\langle t_{f_f}\rangle$ are also indicated. The reduced ${\chi }^2$ of the fit at all 15 distances is 1.08. On the right-hand side are displayed the τ curve (top), the numerator (middle), and the denominator (bottom) in Eq. (3). The straight line fitting the τ curve and its uncertainty limits are drawn through the sensitivity region. The derived lifetime and its statistical error are also displayed. See also the text.

Figure 4

Line-shape analysis of the 1027 keV transition in the Band 1 and determination of the lifetime of the $I^{\pi }=({17}_1^{+})$ level according to DDCM [24]. On the left-hand side are shown fits of the line shapes measured at the angles of ${130.4}^{°}$, ${137.6}^{°}$, and ${163.5}^{°}$ with respect to the beam axis. The decay functions yielding the fits displayed in the left part are presented in the middle with their statistical uncertainties. The decay function of the feeding transition of 521 keV is also shown as determined from independent fits at the different detector angles. Its area is normalized to that of the decay function of the 1027 keV transition. The τ curves and their statistical uncertainties calculated with the decay functions shown in the middle of the figure are presented on the right-hand side. The lifetimes ${\tau }_{\theta }$ derived at the three detector angles are also displayed.

Figure 5

Comparison of the polarization experimental data for the ($7_1^{+})\to (6_1^{-}$), (${13}_1^{+})\to ({12}_1^{+}$), (${13}_1^{+})\to ({11}_1^{+}$) transitions.

Figure 6

Comparison of the experimental bands in ${}^{134}\mathrm{Pr}$ with the lowest two bands calculated in the TQPTR. The thickness of the arrows corresponds to the relative γ intensity in each branch.

Figure 7

Comparison of the experimental bands in ${}^{134}\mathrm{Pr}$ with the lowest two bands calculated in the IBFFM. The thickness of the arrows corresponds to the relative γ intensity in each branch.

Figure 8

Distributions of the orientation parameter σ calculated by TQPTR. For the ${14}^{+}$ states a finer binning (0.01 instead of 0.1) is used to display details of the distribution.

Figure 9

Distributions of the orientation parameter σ calculated by TQPTR assuming axial shape. For the ${14}^{+}$ states a finer binning (0.01 instead of 0.1) is used to display details of the distribution.

Figure 10

Distribution of the deformation parameters β and γ for the ground-state band $6_1^{+}$ and γ-band $6_2^{+}$ state in the core ${}^{134}\mathrm{Ce}$ nucleus. The axes in the β-γ plane have $\gamma =0^{°}$ and $\gamma ={60}^{°}$, whereas the middle line in the β-γ plane marks $\gamma ={30}^{°}$.

Figure 11

Distributions of the parameters characterizing the ${10}_1^{+}$ and ${10}_2^{+}$ states in ${}^{134}\mathrm{Pr}$. The upper panels present the deformation parameters β and γ and orientation parameter σ. The axes in the β-γ plane have $\gamma =0^{°}$ and $\gamma ={60}^{°}$, whereas the middle line in the β-γ plane marks $\gamma ={30}^{°}$. The lower panels present the angles $\psi (j_{\pi }R),\xi (j_{\nu }R)$, and $\zeta (j_{\pi }{j}_{\nu })$.

Figure 12

Distributions of the parameters characterizing the ${13}_1^{+}$ and ${13}_2^{+}$ states in ${}^{134}\mathrm{Pr}$. The upper panels present the deformation parameters β and γ and orientation parameter σ. The axes in the β-γ plane have $\gamma =0^{°}$ and $\gamma ={60}^{°}$, whereas the middle line in the β-γ plane marks $\gamma ={30}^{°}$. The lower panels present the angles $\psi (j_{\pi }R),\xi (j_{\nu }R)$, and $\zeta (j_{\pi }{j}_{\nu })$.

Figure 13

Distributions of the orientation parameter σ for the ${14}_1^{+}$ and ${14}_2^{+}$ states in ${}^{134}\mathrm{Pr}$ calculated by IBFFM.

Figure 14

The excitation energies in both chiral candidate bands are plotted relative to a rigid rotor. The value of the normalization parameter is choosen 0.0107 as in the work [21] to obtain a smooth behavior of the curves. In all panels curves for even-even spin difference $\Delta I=2$ are presented by the full line, whereas for the odd-odd spin difference $\Delta I=2$ are shown with a dashed line. In such a way relative energies of the partner bands calculated in the IBFFM and TQPTR are compared to the experimental ones.

Figure 15

Experimentally determined and theoretically calculated values for in-band $B(E2)$ and $B(M1)$ values in ${}^{134}\mathrm{Pr}$. In the upper panels experimental $B(E2)$ and $B(M1)$ values in Band 1 and Band 2 are presented. In the middle panel the predictions of the TQPTR are presented. In the panel on the bottom the calculations of the IBFFM are shown. The spin assignments for band members above spin 15 in TQPTR calculations have been arranged in a different way in respect to the assignments in Ref. [20]. See also the text.

Figure 16

Experimentally determined and theoretically calculated interband $B(E2)$ values in ${}^{134}\mathrm{Pr}$. In the upper panel experimental $B(E2)$ values from Band 1 to Band 2 and from Band 2 to Band 1 are presented. With arrows are presented limits for the $B(E2)$ values for the states with $I^{\pi }={13}^{+}$ to ${16}^{+}$. In the middle panel the predictions of the IBFFM are presented. In the panel on the bottom the predictions of the TQPTR are presented. See also text.

Figure 17

Experimentally determined limits and theoretically calculated values for interband $B(M1)$ values in ${}^{134}\mathrm{Pr}$. In the upper panel experimental $B(M1)$ limits from Band 1 to Band 2 and from Band 2 to Band 1 are presented. In the middle panel the predictions of the IBFFM are presented. In the panel on the bottom the calculations of the TQPTR are shown. See also text.