Prolate shape of ${}^{140}$Ba from a first combined Doppler-shift and Coulomb-excitation measurement at the REX-ISOLDE facility

Background: Quadrupole moments of excited nuclear states are important observables for geometrically interpreting nuclear structure in terms of deformed shapes, although data are scarce and sometimes ambiguous, in particular, in neutron-rich nuclides.

Purpose: A measurement was performed for determining the spectroscopic quadrupole moment of the $2_1^{+}$ state of ${}^{140}$Ba in order to clarify the character of quadrupole deformation (prolate or oblate) of the state in its yrast sequence of levels.

Method: We have utilized a new combined technique of lifetime measurement at REX-ISOLDE and MINIBALL using the Doppler-shift attenuation method (DSAM) and a reorientation analysis of Coulomb-excitation yields.

Results: On the basis of the new lifetime of $\tau \left(2_1^{+}\right)=10.4_{-0.8}^{+2.2}$ ps the electric quadrupole moment was determined to be $Q\left(2_1^{+}\right)=-0.52\left(34\right)$ $e$b, indicating a predominant prolate deformation.

Conclusions: This finding is in agreement with beyond-mean-field calculations using the Gogny D1S force and with results from the Monte Carlo shell-model approach.

Figure 1

Background-subtracted particle-$\gamma$ coincidence spectra applying Doppler correction with respect to the (a) projectile or (b) recoiling target nuclei showing the only observed transitions, namely the $2_1^{+}\to 0_1^{+}$ transitions in ${}^{140}$Ba at 602 keV and in ${}^{96}$Mo at 778 keV.

Figure 2

Panels (a) and (b) display the stopped $0_1^{-}\to 1_1^{-}$ transition in ${}^{140}$La from $\beta$ decay of ${}^{140}$Ba. Low-energy tails of the $\gamma$-ray lines appear due to partial neutron damage of the detectors. Dashed lines show a symmetric Gaussian peak shape compared to the fitted shape accurately, reproducing the described effect (solid). The Doppler-broadened lineshapes of the $2_1^{+}\to 0_1^{+}$ transition in ${}^{140}$Ba for (c) 49${}^{\circ }$ and (d) 133${}^{\circ }$ are shown after background subtraction. The spectra are shifted by $+$20 counts per channel for better visibility on the logarithmic scale. The full fits are represented by solid lines; the corresponding peak shapes without the effect of neutron damage are dashed. At forward angles the Doppler lineshapes are not significantly disturbed by any low-energy tailing due to partial neutron damage of the Ge detectors. Panel (e) shows the ${\chi }^2$ values for the combined fit to the lineshapes detected in 18 crystals as a function of the lifetime of the $2_1^{+}$ state (${\chi }_{\mathrm{red},\min }^2$ $=$ 0.92).

Figure 3

Differential Coulomb-excitation cross sections for the excitation of the $2_1^{+}$ state in ${}^{140}$Ba as function of the scattering angle in the laboratory frame for diagonal matrix elements $M_{22}=0$ $e$b (solid curve) and for $M_{22}=\pm 0.832e\text{b}=\pm 1.195M_{20}$ which represent the rigid rotational limit for the $M_{20}$ matrix element derived from the present lifetime. Each of the vertical bars (gray) illustrates the angular coverage of the corresponding rings of the CD detector. The relative sensitivity on $Q$ is denoted by $S$ [Eq. (2)].

Figure 4

The bands for the different scattering-angle regimes show the agreement of the calculations with the experimental yields. The lifetime measurement provides the horizontal band as the allowed region of the transitional matrix element $\langle 2_1^{+}\left|\right|E2\left|\right|0_1^{+}\rangle$ (dashed line for the most likely value). Vertical dashed lines give the rotational model values ($M_{22}$ $=$ $\pm$ 1.195$M_{20}$) of a pure prolate or oblate rotor.

Figure 5

Systematics of excitation energy and $B\left(E2\right)$ value in W.u. (a) and $Q$ moment (b) for the first excited $2_1^{+}$ states in the even Ba isotopes with $130\le A\le 146$. The ratio $E(2_1^{+};82-X_N)/E(2_1^{+};82+X_N)-1$ is shown (c) and analogously $B(E2;82+X_N)/B(E2;82-X_N)-1$ (d) to highlight the specific mirror symmetry of the transition strengths around $N=82$. All $B\left(E2\right)$ values are given in W.u. in order to account for a simple size effect.

Figure 6

(a) Energy levels calculated with the Gogny D1S density functional including beyond-mean-field effects. (b) Experimental level energies taken from Ref. [37] and $B\left(E2\right)\downarrow$ from this work. (c) Exact shell-model (SM) calculations using the $P+QQ$ interaction from the MCSM [4].

Figure 7

Distribution of probability in the triaxial $({\beta }_2,\gamma )$ plane ($\gamma$ in degrees) for the states belonging to the first two bands calculated with the Gogny D1S density functional including beyond-mean-field effects.