Direct Evidence for Octupole Deformation in $^{146}$Ba and the Origin of Large $E1$ Moment Variations in Reflection-Asymmetric Nuclei

TL;DR

This paper provides direct evidence of octupole deformation in $^{146}$Ba, showing strong octupole correlations and explaining large variations in $E1$ moments in reflection-asymmetric nuclei through experimental and microscopic analysis.

Contribution

It unambiguously confirms octupole deformation in $^{146}$Ba and links electric dipole moments to neutron orbital occupancy using microscopic calculations.

Findings

Octupole correlations in $^{146}$Ba are as strong as in $^{144}$Ba.

Measured $B(E3)$ value is approximately 48 W.u.

Electric dipole moments depend on neutron orbital occupancy.

Abstract

Despite the more than one order of magnitude difference between the measured dipole moments in $^{144}$Ba and $^{146}$Ba, the strength of the octupole correlations in $^{146}$Ba are found to be as strong as those in $^{144}$Ba with a similarly large value of $B(E3;3^- \rightarrow 0^+)$ determined as 48($^{+21}_{-29}$) W.u. The new results not only establish unambiguously the presence of a region of octupole deformation centered on these neutron-rich Ba isotopes, but also manifest the dependence of the electric dipole moments on the occupancy of different neutron orbitals in nuclei with enhanced octupole strength, as revealed by fully microscopic calculations.

Tables

Table 1. Table 1: The experimental | ⟨ I f π ∥ M ^ λ ∥ I i π ⟩ | quantum-operator-product superscript subscript 𝐼 𝑓 𝜋 subscript ^ 𝑀 𝜆 superscript subscript 𝐼 𝑖 𝜋 |\langle I_{f}^{\pi}\|\hat{M}_{\lambda}\|I_{i}^{\pi}\rangle| matrix elements ( e ⋅ b λ / 2 ⋅ 𝑒 superscript 𝑏 𝜆 2 e\cdot b^{\lambda/2} ) based on the gosia fit along with new symmetry-conserving configuration-mixing calculations (see text and Ref.  Bernard  et al. ( 2016 ) for details).

$I_i^{\pi }\to I_f^{\pi }$	$E\lambda$	Exp.	SCCM
$0^{+}\to 1^{-}$	$E1$	0.000223(${}_{-8}{}^{}{}_{}{}^{10}$) 111primarily determined by previous lifetime and/or branching ratio data Peker and Tuli (1997)	0.00474
$1^{-}\to 3^{-}$	$E2$	1.2(5)	1.6
$0^{+}\to 2^{+}$	$E2$	1.17(2)${}^a$	1.14
$2^{+}\to 4^{+}$	$E2$	1.97(14)	1.90
$4^{+}\to 6^{+}$	$E2$	2.35(${}_{-24}{}^{}{}_{}{}^{+20}$)	2.43
$6^{+}\to 8^{+}$	$E2$	2.17(${}_{-33}{}^{}{}_{}{}^{+65}$)	2.90
$0^{+}\to 3^{-}$	$E3$	0.65(${}_{-20}{}^{}{}_{}{}^{+14}$)	0.54
$2^{+}\to 5^{-}$	$E3$	1.01(${}_{-20}{}^{}{}_{}{}^{+61}$)	0.87
$4^{+}\to 7^{-}$	$E3$	1.25(${}_{-34}{}^{}{}_{}{}^{+85}$)	1.11
$6^{+}\to 9^{-}$	$E3$	1.5(${}_{-12}{}^{}{}_{}{}^{+8}$)

Full text

Unlike the electrons in atoms, protons and neutrons are closely bound together in nuclei by the strong nuclear force, occupying quantum levels that can result in different nuclear shapes because of sizeable long range multipole-multipole interactions. The studies of these shapes, and of the associated nuclear moments, facilitate our understanding of the origin of simple patterns in such complex quantum many-body systems. Certain isotopes are thought to develop octupole deformation due to strong octupole-octupole interactions present when both types of valence nucleons occupy pairs of single-particle orbitals near the Fermi surface with orbital ($\ell$) and total ($j$) angular momenta differing by 3$\hbar$ Butler and Nazarewicz (1996). There is now experimental evidence to suggest that nuclei with a low-lying negative-parity band of states interleaved with the ground-state positive-parity band and linked by fast $E1$ transitions between the two sequences result from strong octupole correlations. However, because $E3$ transitions are fundamentally hindered in the electromagnetic decay of nuclear states when competing with $E1$ and $E2$ transitions, the presence of strong octupole correlations is often inferred from the observation of large $E1$ transition probabilities. The latter are related to the intrinsic electric dipole moment and are typically obtained from $E1/E2$ intensity ratios, with the $E2$ transition probabilities then being estimated from lifetime measurements of low-spin states or from systematics. A direct experimental determination of the electric octupole moment requires the use of the Coulomb excitation process for the nuclei of interest.

The neutron-deficient radium isotopes around 224Ra and the neutron-rich barium isotopes centered at 146Ba have been predicted to belong to the two regions with the strongest octupole correlations Butler (2016). However, large fluctuations, by as much as two orders of magnitude for a given spin, in the value of the intrinsic electric dipole moment have been well documented in these two regions Ahmad and Butler (1993); Butler and Nazarewicz (1996), even though other spectroscopic features; i.e., negative-parity bands located at comparably low excitation energies, strongly suggest the presence of similar octupole strengths. Classically, octupole-deformed nuclei should be characterized by large electric dipole moments proportional to the strength of the octupole correlations Strutinsky (1956); Bohr and Mottelson (1957) because of the redistribution of the mass and charge of the protons and neutrons. Interestingly, in the radium isotopes, a minimum occurs in the value of the intrinsic electric dipole moment for 224Ra, but the magnitude of octupole strength in this nucleus, as recently quantified through Coulomb excitation with a 224Ra radioactive beam, is one of the largest in the region Gaffney et al. (2013). Neutron-rich barium nuclei form another interesting set in terms of studying the relationship between the intrinsic electric dipole and octupole moments Butler and Nazarewicz (1996); Phillips et al. (1986); Mach et al. (1990). Specifically, between 144Ba88 and 146Ba90, the electric dipole moments are observed to drop suddenly by more than an order of magnitude Phillips et al. (1986); Mach et al. (1990); Urban et al. (1997), but the value quickly returns to an enhanced level in 148Ba92 Urban et al. (1997). Furthermore, it has been pointed out in Ref. Urban et al. (1997) that the octupole strength in 146Ba may in fact be quenched as this could account for the presence of a particle alignment at moderate angular momentum (I^{\pi}$$\sim12+) in the ground-state band. Clarification of these issues requires a direct measurement of the $E3$ strength in this nucleus, a challenge until recently because of its short half-life ($T_{1/2}=2.2$ s Peker and Tuli (1997)).

Despite the many experimental challenges associated with measurements of the electric octupole moments in these nuclei, there is also additional motivation in fundamental physics to understand the relation between the intrinsic electric dipole and octupole moments: The existence of an atomic electric dipole moment (EDM) has important implications for $CP$-violation in the early universe that could possibly be responsible for the observed asymmetry between matter and anti-matter Liu et al. (2007), herewith signifying new physics beyond the Standard Model Engel et al. (2013). In diamagnetic atoms, a measureable EDM could be induced by the so-called Schiff moment, a quantity that can be enlarged by orders of magnitude by a sizeable octupole moment and is sensitive to details of the charge distribution Auerbach et al. (1996); Spevak et al. (1997). Moreover, the contribution of the nuclear intrinsic electric dipole moment to the Schiff moment is not negligible Spevak et al. (1997). Hence, it is important to recognize the origin and magnitude of nuclear intrinsic moments, especially the electric dipole moment as it is closely associated with the nuclear mass and charge distributions. An accurate estimate of Schiff moments for different nuclei is required to evaluate the precision of calculations of this quantity and to compare the limits on $P$ and $T$ violation reported by various experiments involving them.

To determine the octupole strength in 146Ba, a Coulomb excitation experiment was performed similar to the one carried out recently for 144Ba Bucher et al. (2016). The beam of 146Ba ions was produced from 252Cf fission in the CARIBU facility Savard et al. (2008, 2015) along with the isobaric contaminants 146La and 146Ce, and was charge bred to $q$=28+. The $A$=146 beam was accelerated through the ATLAS accelerator to 659 MeV and was focused onto a 1.1 mg/cm2 208Pb target (99.86% enriched). The average 146Ba beam intensity was 3$\times 10^{3}$ ions per second over 12 days. Additional stable contaminants (with the same $A/q$) included 94Mo18+, 94Zr18+, 120Sn23+, 193Ir37+, and 198Hg38+, but all were readily separated from the $A$=146 beam, based on time-of-flight (TOF) and scattering angle data recorded in the CHICO2 heavy-ion counter Wu et al. (2016) (Fig. 1). This allowed for a clean $A$=146 $\gamma$-ray spectrum resulting from Coulomb excitation (Fig. 2).

In the spectrum of Fig. 2, measured with the GRETINA $\gamma$-ray tracking array Paschalis et al. (2013), several transitions from 146Ba are apparent, especially those belonging to positive-parity levels in the ground-state band which are excited (and decay) by $E2$ transitions. The negative-parity levels are populated less frequently, but the excitation occurs predominantly through $E3$ transitions and their decay yields provide a measurement of the corresponding $E3$ matrix elements.

The $\gamma$-ray yields were extracted for two separate scattering angle ($\theta$) ranges: 30∘–40∘ and 40∘–75∘. At lower angles, it is difficult to isolate the $A$=146 ions from other beam contaminants in the TOF spectrum while, at higher $\theta$ values, statistics are insufficient (the cross sections fall off with the $1/\sin^{4}(\frac{\theta}{2})$ Rutherford angular dependence). The data for each set of angles were analyzed with the Coulomb excitation least-squares search code, gosia Czosnyka et al. (1983); *GosiaManualURL. Yields were determined for levels up to 10$\hbar$ in the ground-state band and 9$\hbar$ in the negative-parity sequence. The energies of all observed $\gamma$ rays, along with several branching ratios and level lifetimes, were known from previous works Peker and Tuli (1997); Urban et al. (1997); Mach et al. (1990); Phillips et al. (1986). The latter information proved useful for constraining the gosia fit. The sets of $E1$, $E2$, and $E3$ matrix elements between levels with no previously known lifetimes were coupled according to the rigid-rotor prescription Bohr and Mottelson (1975) governed by the individual intrinsic $E\lambda$ moments (see also Refs. Wollersheim et al. (1993); Bucher et al. (2016); Phillips et al. (1986)). Once the $\chi^{2}$ minimum was found, the rigid-rotor constraint was removed to properly determine the associated uncertainties, including correlations between matrix elements. In most cases, the uncertainty was primarily limited by the lack of statistics in the measured yields due to the low radioactive beam intensity.

As anticipated, the extracted $E1$ matrix elements did not display much sensitivity to the data; as mentioned above, the dipole strength was known to be small from earlier work Phillips et al. (1986); Mach et al. (1990); Urban et al. (1997) and, in fact, the only observed $\gamma$ rays from $E1$ decays in the present measurement came from the 3- and 5- states. Moreover, the relative sign between the intrinsic $E1$ and $E3$ moments was found to also be insensitive to the data. On the other hand, a number of new $E2$ and $E3$ matrix elements were well-determined from the data (Table 1).

The most significant result obtained here is the ground-state $E3$ matrix element $|\langle 3_{1}^{-}\|\hat{M}_{E3}\|0_{1}^{+}\rangle|$; it is determined to be 0.65(${}^{+14}_{-20}$) $e$b3/2 and reflects the amplitude of octupole deformation present in the ground state Butler and Nazarewicz (1996). This value corresponds to a $B(E3;3^{-}\rightarrow 0^{+})$ reduced transition probability of 48(${}^{+21}_{-29}$) W.u. which is essentially the same as the value of 48(${}^{+25}_{-34}$) W.u. reported recently for 144Ba Bucher et al. (2016). This new result supports the long standing prediction that 146Ba is indeed one of the isotopes with strong octupole collectivity foo .

The persistence of this strong collectivity between 144Ba88 and 146Ba90 confirms that the drastic reduction in electric dipole moment between the two isotopes is not the result of quenched octupole strength, as suggested by the high-spin behavior of 146Ba. The sudden band alignments in 146Ba pointed out in Ref. Urban et al. (1997), are then most likely the result of a crossing between yrast and yrare bands predicted in this mass region sometime ago Bengtsson and Frauendorf (1979); Bengtsson et al. (1978). It should be noted that alternative explanations have also been proposed. These include a transition to a more reflection-symmetric shape at moderate spin Nazarewicz and Tabor (1992), and a description in terms of a condensate of rotationally-aligned octupole phonons Frauendorf (2013, 2008). Concerning the latter, however, it should be mentioned that while the interleaved sequences of opposite parity are consistent with the proposed picture, the absence of strong $E1$ linking transitions associated with multi-octupole phonon excitations at higher spins is not.

Over the past three decades, extensive theoretical efforts have been devoted to understanding the variation of the $E1$ transition strengths observed in nuclei near 146Ba and 224Ra Leander et al. (1986); Hamamoto et al. (1989); Peker and Hamilton (1979); Egido and Robledo (1989); Butler and Nazarewicz (1991); Egido and Robledo (1990, 1992); Robledo et al. (2010); Dorso et al. (1986). It is generally believed that the observations are the result of the relation between octupole collectivity and the non-uniform distribution of protons and neutrons. This was first shown within the framework of a macroscopic-microscopic approach where the experimental $E1$ transition strengths could be described Leander et al. (1986); Butler and Nazarewicz (1991). Early self-consistent Hartree-Fock-Bogoliubov (HFB) calculations with the Gogny interaction were also able to reproduce the very low values of the dipole moment $D_{0}$ in 224Ra and 146Ba Egido and Robledo (1989, 1990). However, all of these models predicted that the nuclei under study are reflection-asymmetric, and argued that this is at the core of the observed variations. Thus, the recently measured strong octupole collectivity in 144Ba Bucher et al. (2016) and 146Ba (Table 1) provides an important validation of this interpretation.

More recently, microscopic self-consistent methods have been improved by including beyond-mean-field correlations. These developments provide an explanation of the microscopic origin of octupole collectivity and study the impact of octupole correlations on both ground-state properties and electromagnetic transitions. To explore in 146Ba this phenomenon of a strong octupole collectivity accompanied by a much reduced electric dipole moment, a theoretical model based on mean field HFB intrinsic wave functions has been used with a symmetry-conserving configuration-mixing method (SCCM). The model assumes that only the axially symmetric quadrupole ($Q_{20}$) and octupole ($Q_{30}$) degrees of freedom are relevant (collectively referred to as $\mathbf{Q}$). A set of constrained HFB states $|\mathbf{Q}\rangle$, subsequently projected onto good angular momentum, parity and particle number (the corresponding states are denoted as $|\Phi^{J,\pi}(\mathbf{Q})\rangle$) is used as a variational subspace. Linear combinations of the above states $|\Psi^{J,\pi}_{\sigma}\rangle=\sum_{\mathbf{Q}}f^{J,\pi}_{\sigma}(\mathbf{Q})|\Phi^{J,\pi}(\mathbf{Q})\rangle$ are used in the spirit of the generator coordinate method (GCM) to obtain the low-lying collective spectrum (see Ref. Bernard et al. (2016) for a recent account and an application to 144Ba). The interaction generating the intrinsic states is the well-known Gogny D1S force Berger et al. (1984). The physics of the low-lying quadrupole and octupole states is contained in the collective amplitudes $F^{J,\pi}_{\sigma}(\mathbf{Q})$ defined in Eq (5) of Ref. Bernard et al. (2016) in terms of the GCM amplitudes $f^{J,\pi}_{\sigma}(\mathbf{Q})$ (the latter are obtained by solving the Hill-Wheeler (HW) equations of the GCM).

The HFB potential energy surface (PES) as a function of the deformation parameters $\beta_{2}$ and $\beta_{3}$ is given in Fig 3 for 146Ba. Note that the potential energy is symmetric under the change of sign of $\beta_{3}$ due to the parity symmetry of the nuclear interaction. A reflection-asymmetric, absolute minimum is obtained at $\beta_{2}=0.21$ and $\beta_{3}=\pm 0.1$. The shape of the $F^{J,\pi}$ collective amplitudes is mainly driven by the intrinsic potential energy surface. As a consequence, the $F^{J,\pi}$ amplitudes for 146Ba are concentrated around those minima as seen in Fig 4 where they are compared with those for 144Ba and 148Ba. In the three nuclei, the $1^{-}$ and $0^{+}$ collective amplitudes have a very large overlap, characteristic of strong octupole correlations. The energy of the $1^{-}$ state is well-reproduced, however the ground-state and negative-parity sequences are characterized by a smaller moment of inertia than observed, due to limitations discussed in Ref. Bernard et al. (2016).

The electromagnetic transition strengths have been computed without invoking effective charges or uncontrolled approximations. The calculated values are compared with the experimental data in Table 1. There is fair agreement between the calculated and measured $E\lambda$ matrix elements, including the strong $B(E3)$ strengths between the $3^{-}$ and $0^{+}$ and the much quenched $E1$ between the $1^{-}$ and $0^{+}$ states. In the present microscopic framework, the $B(E1)$ transition strength is proportional to the square of the overlap of the dipole operator between the initial $1^{-}$ and final $0^{+}$ states. There are two basic ingredients entering the required overlap (see Eq (4) of Ref. Bernard et al. (2016)). One is the structure of the collective wave functions $F^{J,\pi}_{\sigma}(\mathbf{Q})$, and the other is the overlap between the projected intrinsic states $|\Phi^{J,\pi}(\mathbf{Q})\rangle$. Because the values of $F^{J,\pi}_{\sigma}(\mathbf{Q})$, as shown in Fig 4, exhibit little variation with neutron number in the three Ba isotopes, the sudden drop of $B(E1)$ in 146Ba has to be associated with the overlap of the dipole operator between the projected intrinsic states $|\Phi^{J,\pi}(\mathbf{Q})\rangle$. Specifically, the calculations indicate that the dipole moment changes from positive values in 144Ba, to nearly zero values in 146Ba, and finally to negative values in 148Ba. Hence, the changes in $E1$ strengths with neutron number are associated with changes in the intrinsic dipole moment linked to the evolving mean field. A similar conclusion was reached in Ref. Egido and Robledo (1990).

The behavior of the dipole moment with neutron number in these Ba isotopes can further be traced back to the occupation of specific single-particle states near the Fermi surface. Considering the evolution of the single-particle energies with $\beta_{3}$ in Fig 5, three neutron orbitals are of interest with $K^{\pi}$ quantum numbers $3/2^{-}$, $5/2^{-}$ and $1/2^{+}$. These are of $h_{9/2}$, $f_{7/2}$, and $i_{13/2}$ spherical parentage, respectively. The three states are empty in 144Ba, but have significant respective occupancies of $v^{2}=0.4$, $v^{2}=0.27$, and $v^{2}=0.20$ in 146Ba. As visualized in Fig. 4 of Ref. Egido and Robledo (1990), their occupation results in a contribution to the dipole moment which almost cancels that by the protons. As the occupancies increase further with two additional neutrons in 148Ba, the (total) dipole moment changes sign and returns to a sizeable value.

To summarize, the $E3$ strength in short-lived 146Ba was measured directly by multi-step Coulomb excitation with GRETINA and CHICO2. The longstanding prediction of an enhanced octupole collectivity was verified. The data also provide firm experimental evidence that the large drop of $B(E1)$ value is not the result of quenched octupole collectivity in 146Ba. Such a collectivity is well-reproduced by the SCCM model with the Gogny energy density functional, and the variation in $E1$ strength between isotopes is associated with changes in the neutron occupancy of high-$j$, low-$K$ orbitals located near the Fermi surface. The present results help validate the general character of the microscopic origin of large variations in electric dipole moments in the reflection-asymmetric nuclear potential, and they represent an important confirmation of such effects in the Ba region of neutron-rich nuclei.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Contract No. DE-AC02-06CH11357 (ANL). Work at LLNL and INL is supported by the U.S. DOE under respective Contracts No. DE-AC52-07NA27344 and No. DE-AC07-05ID14517. GRETINA was funded by the U.S. DOE-Office of Science, Office of Nuclear Physics by the ANL contract No. above and by Contract No. DE-AC02-05CH11231 (LBNL). This research used resources of ANL’s ATLAS facility, which is a DOE Office of Science User Facility. The work of LMR was supported by the Spanish grants FIS2012-34479-P MINECO, FPA2015-65929-P MINECO and FIS2015-63770-P MINECO, and TRR by the Spanish grants FIS-2014-53434-P MINECO and Programa Ramón y Cajal 2012 No. 11420.