MReferences

# Orbital perspective on high-harmonic generation from solids

###### Abstract

High-harmonic generation in solids allows probing and controlling electron
dynamics in crystals on few femtosecond timescales, paving the way to
lightwave electronics. In the spatial domain, recent advances in the real-
space interpretation of high-harmonic emission in solids allows imaging the
field-free, static, potential of the valence electrons with picometer
resolution. The combination of such extreme spatial and temporal resolutions
to measure and control strong-field dynamics in solids at the atomic scale is
poised to unlock a new frontier of lightwave electronics. Here, we report a
strong intensity-dependent anisotropy in the high-harmonic generation from
ReS2 that we attribute to angle-dependent interference of currents from the
different atoms in the unit cell. Furthermore, we demonstrate how the laser
parameters control the relative contribution of these atoms to the high-
harmonic emission. Our findings provide an unprecedented atomic perspective on
strong-field dynamics in crystals and suggest that crystals with a large
number of atoms in the unit cell are not necessarily more efficient harmonic
emitters than those with fewer atoms.

Álvaro Jiménez-Galán1,2, Chandler Bossaer1, Guilmot Ernotte1, Andrew M.
Parks3, Rui E.F. Silva4, David M. Villeneuve1, André Staudte1, Thomas Brabec3,
Adina Luican-Mayer3 and Giulio Vampa1.

Joint Attosecond Science Laboratory, National Research Council of Canada and
University of Ottawa, Ottawa, ON K1A 0R6, Canada.

Max-Born-Institute, Max-Born Strasse 2A, D-12489, Berlin, Germany.

Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, Canada.

Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de
Investigaciones Científicas (CSIC), Sor Juana Inés de la Cruz 3, 28049 Madrid,
Spain.

The foundational concept underpinning attosecond physics, and high-harmonic
generation in gas-phase atoms and molecules in particular, is the energetic
recollision of an electron ionized and accelerated by a strong laser field
with the parent ion [1, 2, 3, 4]. This dynamic real-space framework is
instrumental to link the characteristics of the emitted harmonic radiation
(amplitudes, phases and polarization) to sub-laser-cycle dynamics of atomic
and molecular orbitals [5, 6, 7, 8, 9]. In solids, high-harmonic generation
(HHG) is understood using a similar framework, albeit exchanging the real-
space perspective for one in reciprocal space, where electron-hole pairs
accelerate and recombine across energy bands in the Brillouin zone of the
crystal [10, 11, 12, 13]. This reciprocal-space approach has been paramount in
virtually all investigations of solid-state high-harmonics: from revealing the
role of electron-hole recollisions in the emission process [14], to
reconstructing the band structure of a ZnO crystal [15], to explaining the
multiple plateaus observed in the HHG spectrum [16] and to map regions of
crystal momenta where the electron-hole velocity vanishes [17], among others
[18, 19, 20, 21, 22, 23, 24].

Despite the success of the reciprocal-space picture, a real-space approach
offers a more intuitive framework, in particular in complex materials with
many narrowly spaced and overlapping bands. The advantages of using a real-
space perspective to understand HHG from solids are quickly starting to become
apparent [25, 26, 27], for example, in interpreting spatially-displaced
electron-hole recollision processes [27, 28] or as means to directly
reconstruct the field-free (static) valence electron potential at the
picometer scale [29]. The possibility to link features of the high-harmonic
spectrum to dynamics occurring at specific orbitals in the lattice remains,
however, largely unexplored. Here, we demonstrate this possibility through
angle-resolved measurements of HHG in ReS2. We measure a strong, intensity-
dependent anisotropy of the HHG emission and trace it back to the interplay
between the currents generated by each individual atom in the unit cell.
Simulating the laser-matter interaction using a basis constructed from
maximally-localized Wannier orbitals, we show that by changing the laser
parameters (intensity and polarization), one can activate or suppress the
contribution of specific atoms to the HHG emission and interfere the atomic
currents differently, increasing or decreasing the high-harmonic emission
efficiency.

Figure 1: Monolayer ReS2. (a) Crystal structure, composed of 4 rhenium atoms
and 8 sulfur atoms in a distorted octohedral structure. The unit cell is
delimited by the parallelogram. (b-d) Band structure of the monolayer along
(a) $\theta=0^{\circ}$, (b) $\theta=60^{\circ}$ and (c) $\theta=120^{\circ}$
(see panel a for definition of $\theta$). Circular markers across the bands in
panels (b-d) highlight vertical transitions resonant with the 11th harmonic
(H11). Monolayer and bulk (not shown) forms of ReS2 are both inversion
symmetric and display a very similar electronic band structure, with a nearly
identical direct band gap of 1.4eV at the $\Gamma$ point [30, 31].

ReS2 is a layered semiconductor that crystallizes in a distorted octahedral
(T) phase [30, 32, 33, 34]. Figure 1a illustrates the unit cell of the
monolayer, formed by 4 rhenium atoms and 8 sulfur atoms. The 4 Re clusters are
linked in a chain oriented along $\theta=120^{\circ}$ (see panel a). While the
anisotropy of the crystal structure is clear (the crystal symmetry group is
P-1), the band structure is similar along different angles and is very dense
(see Figure 1b and Supplementary Note 1), and with a density of states near
the Fermi energy significantly higher than other prototypical materials used
in HHG spectroscopy, such as MgO or ZnO [35]. Going from the monolayer limit
to bulk, these features remain, and the band structure changes only slightly
[30, 31]. In such dense band diagram, associating an individual harmonic with
reciprocal space trajectories of charge carriers in a particular set of bands,
according to the reciprocal-space method, is hardly straightforward (see
circular markers in Fig. 1b-d), and is unlikely to provide much insight into
the carrier dynamics. On the other hand, the small bandwidth indicates that
the electrons are very localized in the individual atoms of the lattice,
making it ideally suited for a real-space or orbital-based framework.

The first question we want to address is if high harmonics generated from ReS2
reflect the strong anisotropy apparent in real space or rather the weak
angular dependence of its band structure. We generate high harmonics from bulk
ReS2 with a linearly-polarized mid-infrared pulse with a duration of 80 fs and
a center wavelength of 3.5 $\mu$m (see Methods). Figure 2a shows the high-
harmonic spectrum measured for a laser intensity of 0.64 TW/cm2 and
polarization along $\theta=120^{\circ}$ (see inset). We observe odd harmonics
extending up to the 13${}^{\text{th}}$ order, while even harmonics are absent
as expected from the inversion symmetry of ReS2. We measure the orientation
dependence of the harmonics by rotating the polarization of the linear pulse
with respect to the crystal. The results, shown in Figure 2b-d, display a
clear anisotropy for all harmonic orders. Furthermore, the anisotropy depends
strongly on the laser intensity.

Figure 2: Measured orientation-dependent HHG from ReS2. (a) High-harmonic
spectrum for a laser intensity of 0.64 TW/cm2 and polarization along
$\theta=120^{\circ}$ (parallel to the rhenium chains). The inset shows an
optical micrograph of the bulk ReS2 flake using a CMOS camera and white-light
illumination, with the longest edge corresponding to the rhenium chains. (b-d)
Orientation dependence of (b) H9, (c) H11 and (d) H13 for three different
intensities: 0.25 TW/cm2 (blue), 0.64 TW/cm2 (green) and 0.76 TW/cm2 (red).

In order to understand the origin of this anisotropy, we perform time-
dependent simulations in a basis constructed from 44 maximally-localized
Wannier orbitals (see Methods for details) [36]. The similarity between the
monolayer and bulk forms in the case of ReS2 [30, 31], allows us to reduce the
computational complexity and simulate the monolayer system. The orientation
dependence of H9 and H11 obtained from the numerical simulations is shown in
Figure 3a,b. While the uncertainty of the experimental intensities and the
differences between the monolayer and bulk forms do not allow for a
quantitative experiment-theory comparison (e.g., of the exact position of the
harmonic maxima), our simulations clearly display the strong intensity-
dependent anisotropy observed in the experiment. As a result, the simulations
in monolayer ReS2 can provide valuable insight for the origin of this effect.

Figure 3: Calculated orientation-dependent HHG from ReS2. (a,b) Full
calculation of harmonics (a) H9 and (b) H11 for different intensities: 0.1
TW/cm2 (blue), 0.5 TW/cm2 (green) and 0.6 TW/cm2 (red). (c) Calculation
neglecting the Fourier phase (solid lines) $\varphi_{n}$ of the orbital
current for H11 for 0.1 TW/cm2 (blue) and 0.6 TW/cm2 (red). For comparison,
the full calculation curves of panel (b) are shown in (c) with dashed, faint
lines.

The high-harmonic spectrum is given by the Fourier components of the time-
dependent current that is generated by the laser-induced oscillating dipole of
the medium (see Methods),

$I(\omega)=\sum_{\alpha}\left|\mathcal{F}[J_{\alpha}(t)](\omega)\right|^{2},$
(1)

where $J_{\alpha}(t)$ is the total current along direction
$\alpha=(\parallel,\perp)$, corresponding to the components parallel and
perpendicular to the electric field, respectively. The total current can be
expressed as a sum of currents from all the orbitals in the lattice,
$J_{\alpha}(t)=\sum_{n}^{N_{\text{orb}}}J^{\text{(W)}}_{n,\alpha}(t)$, where
$J^{\text{(W)}}_{n,\alpha}(t)$ represents the contribution to the total
current of the changing population of orbital $n$ and its coherence with all
other orbitals (see Methods). The subscript (W) indicates that such orbital
currents are defined in the Wannier gauge and, even if they are not
observable, provide a unique real-space perspective into the HHG process.
Expressed in terms of the individual orbital currents, the high-harmonic
spectrum is

$I_{\alpha}(\omega)=\left|\sum_{n}^{N_{\text{orb}}}\mathcal{F}[J^{(W)}_{n,\alpha}(t)](\omega)\right|^{2}=\sum_{n}^{N_{\text{orb}}}\left[|A_{n,\alpha}(\omega)|^{2}+|A_{n,\alpha}(\omega)|\sum_{m\neq
n}|A_{m,\alpha}(\omega)|\cos(\varphi_{m,\alpha}(\omega)-\varphi_{n,\alpha}(\omega))\right],$
(2)

where $A_{n,\alpha}$ and $\varphi_{n,\alpha}$ are, respectively, the spectral
amplitude and phase of the current of orbital $n$ along direction $\alpha$.

Equation 2 allows us to distinguish features that arise from the interference
of different orbital currents. The incoherent sum of the individual currents,
$I^{\text{incoh}}_{\alpha}(\omega)=\sum_{n}^{N_{\text{orb}}}\left|\mathcal{F}[J^{(W)}_{n,\alpha}(t)](\omega)\right|^{2}$,
will be absent of such interference. In Figure 3c we compare the angle-
dependent harmonic yield of H11 for $I^{\text{incoh}}_{\alpha}$ (solid lines)
and the observable signal $I_{\alpha}$ (faint dashed lines). A similar
analysis for H9 is made in Supplementary Note 2. The angular variation is
stronger for $I_{\alpha}$, with near-complete suppression of various secondary
maxima that are present in $I^{\text{incoh}}_{\alpha}$ (most notably near
60$\degree$), strongly modifying the orientation dependence. Thus, orbital
phase interference is an important factor determining the orientation
dependence.

Since the electrons are well localized on each atomic site (see Extended
Data), we can group together the currents of the $m$ orbitals belonging to the
same atom $A$ into an atomic current,
$J^{\text{(W)}}_{A,\alpha}(t)=\sum_{m}J^{\text{(W)}}_{m,\alpha}(t)$.
Furthermore, due to the inversion symmetry of ReS2, each atom is related to
one other by an inversion operation, for example, Re1 and Re3 or S1 and S6
(see Fig. 1a). Both of the atoms in the pair give rise to the same Fourier
amplitudes and phases, so that the total harmonic spectrum in Eq. 2 reduces to
the sum of the Fourier amplitudes and phases of six atomic (inversion-related)
pairs.

Figure 4: Atomic contributions to harmonic emission in ReS2. The circle colors
represent the six different atomic pairs, the size of the circle is
proportional to the Fourier amplitude $|A_{n}|$ and the Fourier phase
$\varphi_{n}$ is given in the vertical axis. The panels display the Fourier
amplitudes $|A_{n}|$ and phases $\varphi_{n}$ of the six atomic pairs
(inversion-symmetric partners) as a function of the laser polarization angle
for H11. Two driver intensities are shown: (a) 0.1TW/cm2 and (b) 0.6 TW/cm2.
The results shown are for the harmonic polarization $\alpha$ that is parallel
to the electric field.

Figures 4a,b show the Fourier amplitudes and phases of the six atomic pairs,
indicated with different colors, for H11 along $\alpha=\parallel$ and for two
intensities: 0.1 TW/cm2 and 0.6 TW/cm2. At both low and high intensities
(Figure 4a-b respectively) emission is spread over a wide range of phases at
any given angle. For the lowest intensity (Figure 4a), every atomic pair
contributes a similar amplitude to the emission near $\theta=40-60^{\circ}$,
but their phases are spread equally over $\pi$ rad, thus leading to the near-
perfect destructive interference seen in Fig. 3b (blue curve) at these angles.
On the other hand, for angles close to $\theta=100^{\circ}$, the Fourier
phases from the different atomic sites are similar, leading to constructive
interference in Eq. 2 and therefore to the peak observed in Figure 3b (blue
curve). At $\theta=100^{\circ}$, the atomic pair Re2-Re4, which contributes
the most to H11 at low intensity, is largely suppressed at large intensity
(compare size of orange circle in Figure 4a,b). This analysis shows that atoms
that do not contribute to the generation of a particular harmonic order for
one driver intensity, can be activated for other intensities, and vice versa,
suggesting that laser intensity could be used as a mechanism to control the
relative weight of atomic orbitals in HHG. An analogous analysis can be made
for the rest of harmonic orders, along both $\alpha=\parallel,\perp$
directions (see Supplementary Note 2), where we observe a larger spread of the
Fourier phases for increasing harmonic orders. This leads to sharper changes
in the angle-resolved spectrum for higher orders, as also seen in the
experiment.

In conclusion, we identify how the nonlinear currents residing on each of the
twelve atoms in the unit cell of a ReS2 crystal are responsible for the
strongly anisotropic and intensity-dependent emission of high-order harmonics.
Our orbital analysis based on maximally localized Wannier functions reveals
that each atomic contribution depends strongly on the polarization angle and
intensity of the driving field, paving the way to characterizing and
controlling electron dynamics at the picometer-scale in solids on sub-laser-
cycle timescales. Moreover, we show that interference between atoms in the
unit cell of a crystal is key to determine the macroscopic high-harmonic
emission, a critical factor to consider in the route towards developing
efficient harmonic emitters.

## Methods

###  0.1 Experimental methods

ReS2 flakes were mechanically exfoliated from an extracted section of a bulk
sample using tape, then dispersed across the tape by folding over itself to
reduce thickness and produce generally flat flakes. The ReS2 crystal was
transferred from the tape to a PDMS stamp, then transferred from the stamp to
the substrate at 80∘. The substrate consists of a two-side polished,
10x10x0.5mm, (100)-cut MgO single crystal that is cleaned with acetone and
isopropanol. The PDMS stamp was peeled off to leave the bulk ReS2 flakes on
the MgO substrate.

The sample is imaged in-situ with a white-light source as well as the laser
source, allowing sample areas of interest to be located and crystallographic
orientation to be measured.

The laser source consists of a YB:KGW laser (LightConversion Carbide CB3)
delivering 200fs pulses at a center wavelength of 1030nm with a repetition
rate of 100kHz and average power of 80W. A portion of this power (60W) pumps a
commercial optical parametric amplifier (LightConversion Orpheus-MIR),
generating 60fs mid-infrared pulses at a wavelength of 3.5$\mu$m. An Ag off-
axis parabolic mirror focuses the mid-infrared beam onto the sample, producing
high harmonics. The generated high harmonics are collected in transmission
geometry and focused on the input slit of a Princeton Instruments IsoPlane
spectrometer with an Al spherical mirror of 15cm focal length. A half-wave
plate is positioned between the parabolic mirror and the sample to rotate the
linear laser polarization with respect to the crystal axis.

###  0.2 Numerical methods

The field-free Hamiltonian and dipole couplings of monolayer ReS2 were
calculated with the electronic structure code Quantum Espresso [37] on a
Monkhorst-Pack (MP) grid of 12x12x1 points using a norm-conserving Perdew-
Burke-Ernzerhof (PBE) exchange correlation functional. The field-free
Hamiltonian used in the time-dependent propagation was constructed by
projecting the Bloch states onto a set of maximally-localized Wannier
functions using the Wannier90 code [36]. In particular, we projected onto the
$d$ orbitals of the four rhenium atoms and the $p$ orbitals of the six sulfur
atoms, totalling 44 bands. The Hamiltonian in the basis constructed from
Wannier functions was then propagated in the presence of the electric field
using the density matrix formalism with the code described in Ref. [26]. The
large size of the unit cell allowed us to obtain convergence with a modest MP
grid of 50x50 $k$-points along the $b_{1}$ and $b_{2}$ reciprocal lattice
vectors. The time step was set to 0.2 a.u. and the dephasing time was chosen
to be $T_{2}=10$ fs.

The time-dependent current along direction $\alpha$, used to extract the high
harmonic spectrum, is defined as

$J_{\alpha}(t)=-\frac{|e|}{N_{k}}\sum_{\mathbf{k}}\text{Tr}\left[\hat{\mathbf{v}}_{\alpha}(\mathbf{k})\cdot\rho(\mathbf{k},t)\right].$
(3)

Above, $e$ is the electron charge, $N_{k}$ is the number of crystal momenta
included in the calculation, $\hat{\mathbf{v}}$ is the velocity operator, and
$\rho$ is the density matrix. In the Wannier gauge, the density matrix
$\rho^{\text{(W)}}$ contains the orbital populations and coherences in its
diagonal and off-diagonal terms, respectively. In the Wannier gauge, we may
define a (real) current from an individual orbital $n$ along direction
$\alpha$ as

$J^{\text{(W)}}_{n,\alpha}(t)=-\frac{|e|}{N_{k}}\operatorname{\operatorname{Re}}\\{\sum_{\mathbf{k}}\sum_{m}^{N_{\text{orb}}}\left[\hat{v}^{\text{(W)}}_{nm,\alpha}\cdot\rho^{\text{(W)}}_{mn}\right]\\},$
(4)

such that the sum of the currents from all orbitals equals the total current,

$J_{\alpha}(t)=\sum_{n}^{N_{\text{orb}}}J^{\text{(W)}}_{n,\alpha}(t).$ (5)

For clarity, we give an example for a two-orbital model, although we point out
that our analysis is only relevant for multi-orbital crystals as the one
presented in this work. In the two-orbital case,

$\begin{split}&J_{1,\alpha}(t)=\operatorname{\operatorname{Re}}(\\{v_{11,\alpha}\rho_{11}+v_{12,\alpha}\rho_{21}\\}\\\
&J_{2,\alpha}(t)=\operatorname{\operatorname{Re}}\\{v_{22,\alpha}\rho_{22}+v_{21,\alpha}\rho_{12}\\},\end{split}$
(6)

where the subscripts $1$ and $2$ identify the orbital and
$\alpha=\parallel,\perp$ the direction of current emission. Since both the
velocity and density matrices are hermitian, the current of an individual
orbital is composed of a term associated to the population change of that
orbital, plus exactly half of the contribution of the coherence between that
orbital and the rest. Thus, this approach offers a way of quantifying the
contribution of individual orbitals, and their interference, to the high-
harmonic generation.

###  0.3 Data availability

The data that support the plots within this paper and other findings of this
study are available from the corresponding author upon reasonable request.

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naturemag

###  0.4 Acknowledgements

We thank David Crane, Ryan Kroeker and Andrei Naumov for continued technical
support. A. J. G. acknowledges support from the European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie Grant
Agreement (101028938), from Comunidad de Madrid through TALENTO Grant
2022-T1/IND-24102 and from the Joint Center for Extreme Photonics. G. V.
acknowledges support from the Joint Center for Extreme Photonics and the
National Research Council’s Quantum Sensing program. D.V. acknowledges support
from the Joint Center for Extreme Photonics. R.E.F.S. acknowledges support
from the fellowship LCF/BQ/PR21/11840008 from “La Caixa” Foundation (ID
100010434).

###  0.5 Authors contribution

C. B. and G. E. performed the experiments; A. J. G. performed the numerical
analysis and calculations; A. P. and R.E.F.S. performed ancillary
calculations; R.E.F.S. developed the numerical code; D. M. V., A. S., T. B.,
A. L. M. and G. V. supervised the work. A. J. G. and G. V. wrote the
manuscript with contributions from all co-authors.

###  0.6 Corresponding authors

please address all correspondence to G. V. (gvampa@uottawa.ca).

###  0.7 Competing interests

The authors declare no competing interests.


# The stellar $\beta$-decay rate of 134Cs and its impact on the Barium
nucleosynthesis in the $s$ process

Kuo-Ang Li  CAS Key Laboratory of High Precision Nuclear Spectroscopy,
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 73000,
People’s Republic of China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 100049, People’s Republic
of China  Chong Qi  Department of Physics, Royal Institute of Technology,
Stockholm, Sweden  Maria Lugaro  Konkoly Observatory, Research Centre for
Astronomy and Earth Sciences, Eötvös Loránd Research Network (ELKH), Konkoly
Thege Miklós út 15-17, H-1121 Budapest, Hungary  ELTE Eötvös Loránd
University, Institute of Physics, Budapest 1117, Pázmány Péter sétány 1/A,
Hungary  School of Physics and Astronomy, Monash University, VIC 3800,
Australia  Andrés Yagüe López  Konkoly Observatory, Research Centre for
Astronomy and Earth Sciences, Eötvös Loránd Research Network (ELKH), Konkoly
Thege Miklós út 15-17, H-1121 Budapest, Hungary  Amanda I. Karakas  School of
Physics and Astronomy, Monash University, VIC 3800, Australia  ARC Centre of
Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Canberra, ACT
2611, Australia  Jacqueline den Hartogh  Konkoly Observatory, Research Centre
for Astronomy and Earth Sciences, Eötvös Loránd Research Network (ELKH),
Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary  Bing-Shui Gao  CAS
Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China  School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China  Xiao-Dong Tang  CAS
Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China  School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China  Joint department for
nuclear physics, Institute of Modern Physics and Lanzhou University, Lanzhou,
China, 730000

###### Abstract

We have calculated the stellar $\beta$-decay rate of the important s-process
branching point 134Cs based on the state of the art shell model calculations.
At typical $s$-process temperatures ($T\sim$ 0.2-0.3 GK), our new rate is one
order of magnitude lower than the widely-used rate from Takahashi and Yokoi
(hereafter TY87). The impact on the nucleosynthesis in AGB stars is
investigated with various masses and metallicities. Our new decay rate leads
to an overall decrease in the 134Ba/136Ba ratio, and well explains the
measured ratio in meteorities without introducing the $i$ process. We also
derive the elapsed time from the last AGB nucleosynthetic event that polluted
the early Solar System to be $>$28 Myr based on the 135Cs/133Cs ratio, which
is consistent with the elapsed times derived from 107Pd and 182Hf. The
$s$-process abundance sum of 135Ba and 135Cs is found to increase, resulting
in a smaller $r$-process contribution of 135Ba in the Solar System.

Nuclear Reactions, Nucleosynthesis, Abundances, Stars: AGB and Post-AGB

††journal: ApJL

##  1 Introduction

The 134Cs stellar $\beta-$decay rate is crucial to understand the origin of
the Ba isotopes. The competition between the neutron capture and the
$\beta$-decay of 134Cs determines the relative abundances of 134Ba and 136Ba,
two pure $s$-process isotopes shielded by their stable xenon isobars, and
influences the $s$-process nucleosynthesis of 135Ba and 135Cs. The
astrophysical sites for the production of these two Ba isotopes in the Galaxy
are asymptotic giant branch (AGB) stars with initial masses predominantly in
the range 2-4 M⊙. The main neutron source is the 13C($\alpha$,n)16O reaction
activated in a thin region of the He-rich shell at $T\sim 0.1$GK (Gallino et
al., 1998; Travaglio et al., 1999; Karakas & Lattanzio, 2014).The
22Ne($\alpha$,n)25Mg reaction is also activated during the thermal
instabilities associated with He burning in these stars at $T\sim 0.3$GK. It
provides a peak neutron density orders of magnitude higher than that generated
by the 13C source: up to $10^{12-13}$ cm-3 (22Ne source) compared to
$10^{7-8}$ cm-3 (13C source). As a result, the branching point 134Cs is much
more efficiently activated in the neutron flux provided by the 22Ne than by
the 13C neutron source(Lugaro et al., 2003; Fishlock et al., 2014; Bisterzo et
al., 2015). A typical $s$-process path is shown in Figure 1.

The rich variety of observations of the Ba and Cs isotopes in the Universe,
which related to this branching, offers a number of opportunities to probe the
nucleosynthesis at various sites: (i)It was found that two nominally
mainstream SiC grains with strongly negative $\delta$(134Ba/136Ba) values
could not be explained by any of the current AGB model calculations,
suggesting that such negative values were consistent with the intermediate
neutron capture process ($i$-process), whose neutron density is much higher
than that of s-process, during the post-AGB stage(Liu et al., 2014).
(ii)135Cs($T_{1/2}$=2.3 Myr) is a long-lived isotope produced by AGB stars. It
can be used to derive the elapsed time from the last AGB nucleosynthetic event
that polluted the pre-solar system matter to the birth of the Sun(Brennecka &
Kleine, 2017). (iii)135Ba is produced in the $s-$ and $r-$processes. A
reliable prediction of the abundance sum of 135Ba and its progenitor nucleus,
135Cs, in AGB stars is important to determine the $r$-process contribution in
the Solar System.

Figure 1: The s-process path at the vicinity of 134Cs at $T=$ 0.35 GK($kT$ =
30 keV) generated by the NUCNET code(Bojazi & Meyer, 2014). The red lines
present $\beta$-decay, while the black lines present the neutron capture. Line
width indicates the flow current in linear scale. The stable nuclei are marked
with yellow color.

A reliable $\beta$-decay rate of 134Cs is essential for describing
quantitatively the branching ratio and interpret correctly these
nucleosynthetic outcomes. In stellar environments, the low-lying states of
134Cs are thermally populated and contribute to the total $\beta$-decay rates.
Takahashi and Yokoi calculated the beta decay rate of 134Cs based on the
empirical log$ft$ values extracted from the outdated nuclear structure data in
the 1980s (Takahashi & Yokoi, 1987). By artificially varying the transition
strengths, the uncertainty of the 134Cs $\beta$-decay has been estimated to be
of one order of magnitude (Goriely, 1999). As the neutron capture cross
sections on 134Cs and 134,136Ba are determined with errors better than 10%
(Patronis et al., 2004; KADoNis, v0.3), the uncertainty of the $\beta$-decay
rate is the major limitation in the studies of the $s$ process.

In this paper, we present a new stellar $\beta$-decay rate of 134Cs deduced
from the Gamow-Teller (GT) transition strength [$B$(GT)] obtained from large-
scale shell-model calculations. With the new $\beta$-decay rate, we
investigate its impact on the nucleosynthesis using AGB stellar models
covering various masses and metallicities.

##  2 Calculation of the $\beta-$decay rate of 134Cs in the stellar
environment

Figure 2: The stellar $\beta$-decay scheme of 134Cs. The weak $\beta^{+}$/EC
channel is not shown. Only allowed transitions of excited states below 200keV
are listed. The important transitions to low-lying states of 134Ba are
highlighted with red lines. The tentative $J^{\pi}$ assignments are shown with
parentheses.

The terrestrial decay rate of 134Cs has been well studied in the laboratory
with a half-life of 2.0652(4) years. It is dominated by the $\beta^{-}$ decay.
Only a tiny fraction of the decay channels, $3\times 10^{-6}$, is
$\beta^{+}$/electron-capture(EC) decay, which is negligible. In stellar
environments, $\beta$-decays from thermally populated excited states could
make a significant contribution to total decay rates. However, even at the $s$
process temperatures, $\beta^{+}$/EC decays contribute less than 1% of the
total decay rate according to TY87. Due to its minor contribution, in the
present work we focus on the $\beta^{-}$-decay channel and adopt TY87 rate for
the $\beta^{+}-$/EC channel.

Fig 2 shows the stellar $\beta$-decay scheme of 134Cs. The terrestrial 134Cs
$\beta$-decay is dominated by transitions from the 134Cs $4^{+}$ ground state
to the two $4^{+}$ states of 134Ba with excitation energies of 1401 and 1970
keV, respectively. Due to the small decay energies (658 and 89 keV), the
terrestrial decay rate is relatively slow. However, with the thermal
population of the excited states, the stellar $\beta$-decay rate could be
significantly enhanced due to contributions from the two transitions,
($3^{+}$, 60 keV)$\rightarrow$($2^{+}$, 605 keV) and ($1^{+}$, 177
keV)$\rightarrow$($0^{+}$, g.s.), in the $s$-process environment, due to the
combination with high population possibility, high transition strength and the
large decay phase space.

In the TY87 rate, the unknown transition strengths of the excited states were
estimated from the analogous transitions in the existing experimental data,
i.e., the transitions from the terrestrial $\beta$-decay of neighbouring
nuclei. Nowadays with the progress in computational nuclear physics, the
transition strength can be calculated more precisely by using the large-scale
nuclear shell model.

We have applied the large-scale shell model (hereafter SM) to calculate the
transition strengths of the excited states and updated the stellar
$\beta$-decay rate of 134Cs. The large-scale shell model is a full
configuration interaction approach that takes into account all possible
couplings and correlations among valence nucleons within one or a few major
shells. We assume 100Sn as the inert core and consider the neutron and proton
orbitals between the shell closures $N$ (and $Z$) $=50$ and 82, comprising
$0g_{7/2}$, $1d_{5/2}$, $1d_{3/2}$, $2s_{1/2}$ and $0h_{11/2}$ orbitals. The
starting point of our calculation is the realistic CD-Bonn nucleon-nucleon
potential (Machleidt, 2001). The interaction was renormalized using the
perturbative G-matrix approach to take into account the core-polarization
effects (Hjorth-Jensen et al., 1995). The $T=1$ component of the monopole
interaction was optimized by fitting the low-lying states of all Sn isotopes
between 101Sn and 132Sn (see Ref. Qi & Xu (2012) for details). Our
calculations in the present work reproduced the spectrum of 134Ba and all the
lying states of 134Cs in Fig. 2 with recommended spin-parity assignments
within ${\pm}200$ keV, though the orders can be slightly different. It is,
however, not a major issue for the present work because the levels and their
corresponding wave functions are identified based on their Jπ values. The wave
functions of 134Cs and 134Ba are calculated to be dominated by the coupling of
protons in nearly degenerate $g_{7/2}d_{5/2}$ orbitals and neutrons in the
$s_{1/2}d_{3/2}h_{11/2}$ orbitals. Those are consistent with the low-lying
spectra of neighbouring odd-A 133,135Cs and 133,135Ba. As a result, the GT
strength mostly come from the transition ${\pi}d_{5/2}$ to ${\nu}d_{3/2}$.

With the updated log$ft$ values calculated from the shell model, we obtained a
new stellar $\beta^{-}$-decay rate of 134Cs which is shown in Fig 3(a) along
with TY87 rate. The new rate is about one order of magnitude slower than TY87
at the $s$-process temperatures.

The individual contributions of the major transitions in the total stellar
$\beta^{-}$-decay rate of 134Cs is presented in Fig 3(b). The ($3^{+}$, 60
keV)$\rightarrow$($2^{+}$, 605 keV) transition (log$ft=$ 7.8(SM) vs.
6.5(TY87)) dominates, and the transition ($1^{+}$, 177
keV)$\rightarrow$($0^{+}$, g.s.) contributes only a minor fraction (log$ft=$
7.6(SM) vs. 5.5(TY87)). To estimate the uncertainty of the transition strength
calculated by the shell model calculation, we have done several calculations
by slightly varying nucleon excitations from $g_{7/2}d_{5/2}$ to
$s_{1/2}d_{3/2}h_{11/2}$ orbitals and the single-particle energies of the
neutron $d_{3/2}$ orbital. In those calculations the calculated GT strength
for the mostly relevant $3^{+}$ to $2^{+}$ decay vary mostly within a factor
of two. As the most important transition ($3^{+}$, 60
keV)$\rightarrow$($2^{+}$, 605 keV) is between two short lived excited states,
our approach is the best way at present to obtain the strength.

Figure 3: 134Cs stellar $\beta^{-}-$decay rate of TY87 and the present work
obtained with the shell model (SM)(a), and the decay rates of individual
important transitions (b). The terrestrial rate is also presented.

One major source of the uncertainty of the 134Cs decay rate comes from the
spin-parity ($J^{\pi}$) assignments of the excited states. In our shell model
calculations and TY87, the level scheme is taken from NNDC (2021). However,
due to the limitations of experimental data, the $J^{\pi}$ assignments of some
states are still debated. $J^{\pi}$ = $3^{+}$ is suggested for the 60-keV
state in which case the transition to the ($2^{+}$, 605keV) state in 134Ba is
allowed. However, other $J^{\pi}$ values such as $4^{+}$ could not be excluded
(NNDC, 2021; Bogdanović et al., 1987) resulting in a forbidden transition with
a significant reduction in the total decay rate. A similar situation also
exists for the 177-keV state. The transition from this state to the 134Ba
ground state will be of forbidden nature if its $J^{\pi}$ is not $1^{+}$. Such
ambiguity could be eliminated by further experiment such as the in-beam
$\gamma$ spectroscopy.

##  3 Impact of new 134Cs $\beta-$decay rate on nucleosynthesis

To test the impact of the new decay rate presented here on the production of
the Ba isotopes in the Galaxy, we have run several selected models of AGB
nucleosynthesis for masses in the range 2 to 4.5 M⊙and metallicities from
1/140 to twice solar metallicity (Lugaro et al., 2012; Karakas, 2014; Karakas
& Lugaro, 2016; Karakas et al., 2018). A solar metallicity of 0.014 (Asplund
et al., 2009) is adopted in the present work. As described in more detail in,
e.g., Karakas & Lugaro (2016), our computational method is based on the post-
processing nucleosynthesis code developed by Cannon (1993), where the changes
in the abundances due to nucleosynthesis and mixing within convective regions
are solved simultaneously. This means that we can treat the temperature
dependence of the decay rate of 134Cs together with mixing during the thermal
instability where the 22Ne neutron source is activated. In any case, our
results appear generally robust considering that for our 134Ba/136Ba ratios
are remarkably similar (within 10%) to those produced by the FRUITY models
(Cristallo et al., 2009, 2011) for the same stellar masses and metallicities.
The FRUITY models were calculated by interpolating the TY87 decay rates over
the temperature of interest (i.e., not using an average value of the rate),
but differently to our computations they solved separately the effects of
nucleosynthesis and of the mixing.

Table 1: Stellar surface abundances by number (in units of $10^{-11}$) and isotopic ratios for selected Ba and Cs isotopes at the end of the evolution for our set of AGB models. Values in roman font are calculated using the 134Cs decay rate by Takahashi & Yokoi (1987), while values given in italics are calculated using the decay rate from the present study, with the change indicated in brackets.  Mass (M⊙) |  134Ba |  135Ba |  135Ba+135Cs | (134Ba/136Ba)/$\odot$ |  135Cs
---|---|---|---|---|---
Z = 0.03
2.5 | 10.2, 9.53 (-7%) | 4.86, 4.67 (-4%) | 5.15, 6.00 (+16%) | 1.38, 1.25 | 0.29, 1.33 ($\times$4.6)
2.75 | 10.6, 9.53 (-10%) | 5.06, 4.74 (-6%) | 5.55, 6.63 (+20%) | 1.33, 1.17 | 0.49, 1.89 ($\times$3.8)
3 | 9.29, 7.93 (-15%) | 4.57, 4.13 (-10%) | 5.21, 6.46 (+24%) | 1.28, 1.08 | 0.64, 2.33 ($\times$3.6)
3.5 | 4.67, 3.75 (-20%) | 2.87, 2.58 (-10%) | 3.47, 4.32 (+25%) | 1.22, 0.97 | 0.60, 1.75 ($\times$2.9)
4 | 3.48, 2.46 (-29%) | 2.40, 2.00 (-17%) | 2.99, 3.73 (+25%) | 1.15, 0.82 | 0.71, 1.79 ($\times$2.5)
4.5 | 1.07, 0.88 (-18%) | 1.59, 1.51 (-5%) | 1.73, 1.93 (+11%) | 1.12, 0.93 | 0.17, 0.42 ($\times$2.4)
Z = 0.014
2 | 10.5, 9.20 (-12%) | 4.43, 4.01 (-10%) | 4.97, 6.19 (+25%) | 1.28, 1.12 | 0.54, 2.18 ($\times$4.0)
3 | 14.0, 10.6 (-24%) | 5.67, 4.65 (-18%) | 7.86, 10.7 (+37%) | 1.18, 0.90 | 2.19, 6.10 ($\times$2.8)
3.5 | 7.84, 5.50 (-30%) | 3.34, 2.59 (-22%) | 5.52, 7.56 (+37%) | 1.16, 0.82 | 2.18, 4.97 ($\times$2.3)
4 | 6.93, 4.18 (-40%) | 2.95, 1.83 (-38%) | 6.97, 9.37 (+34%) | 1.16, 0.71 | 4.01, 7.54 ($\times$1.9)
4.5 | 2.32, 1.48 (-36%) | 1.40, 1.04 (-25%) | 2.72, 3.49 (+29%) | 1.11, 0.71 | 1.32, 2.45 ($\times$1.8)
Z = 0.007
2.5 | 14.8, 11.0 (-26%) | 5.77, 4.64 (-35%) | 8.50, 12.0 (+41%) | 1.08, 0.80 | 2.72, 7.32 ($\times$2.7)
3 | 15.8, 10.3 (-20%) | 6.01, 4.48 (-25%) | 11.8, 16.6 (+41%) | 1.05, 0.69 | 5.78, 12.1 ($\times$2.1)
Z = 0.0028
3 | 4.92, 2.84 (-42%) | 1.93, 1.23 (-36%) | 5.67, 7.45 (+31%) | 1.04, 0.61 | 3.74, 6.22 ($\times$1.7)
Z = 0.001
2.5 | 1.90, 1.07 (-36%) | 0.75, 0.47 (-37%) | 2.01, 2.70 (+34%) | 1.06, 0.61 | 1.26, 2.23 ($\times$1.8)
Z = 0.0001
2.5 | 0.19, 0.11 (-42%) | 0.08, 0.05 (-37%) | 0.26, 0.33 (+28%) | 0.94, 0.17 | 0.18, 0.28 ($\times$1.6)

The results of our models are presented in Table 1. Because the new decay rate
is typically lower than the TY87 rate, e.g., roughly by a factor of ten at 270
MK (23 keV), the neutron capture of 134Cs is more effectively activated in the
models calculated with the new rate. As the 22Ne reaction is dependent on the
temperature in the He shell, which in turn increases with increasing stellar
mass and decreasing metallicity, the significance of the impact of the new
rate also increases with increasing stellar mass and decreasing metallicity.
In summary, we obtained a significant decrease in the abundance of 134Ba and
135Ba by up to roughly 40%, a strong increase in 135Cs (by roughly a factor 2
to 4), and no significant changes in the 136Ba abundance, which indicates that
the branching path rejoins the standard s-process path at this isotope, in
other words, there is no further effect of the branching chain through the Cs
isotopes on 136Ba and 137Ba. Interestingly, since 135Cs eventually decays into
135Ba, the final s-process 135Ba abundance, i.e., the abundance sum of 135Ba
and 135Cs, increases by up to 40%.

Figure 4: Comparison between data for single stardust SiC grains (black dots
with 2$\sigma$ error bars, from Liu et al., 2014) and AGB model predictions of
different masses, metallicities, and decay rates of 134Cs, as indicated by the
labels, where Z = 0.014 is the solar metallicity, TY87 is the rate from
Takahashi & Yokoi (1987) and “Present work” refers to the new rate presented
in this paper. The data are plotted using the standard $\delta$ notation,
where $\delta$134,135Ba/136Ba represents the permil variation of the ratio
with respect to solar. The model predictions start at solar composition
($\delta=0$) and follow the coloured lines until the envelope becomes C-rich
(the condition for the formation of SiC) and open symbols are plotted. This
means that the SiC data points should be compared exclusively to the open
symbols. Circles represent models where the extent in mass $M_{\rm PMZ}$ of
the mixing of protons leading to the formation of the 13C neutron source was
taken to be the standard reported in Table 1 of Lugaro et al. (2018a). Squares
represents models where $M_{\rm PMZ}$ was reduced by a factor of 10 in the
Z=0.014 models and by a factor of 2 in the Z=0.03 models.

The decrease in 134Ba results in an overall decrease of the 134Ba/136Ba
isotopic ratio, with the geometric average of all the models shown in Table 1
changing from 16% higher than solar using the old rate, to 16% lower than
solar using the new rate. However, a proper comparison with the solar
abundances can only be accomplished by running full galactic chemical
evolution models.

For the final $s$-process 135Ba abundance, the geometric average of all the
models shown in Table 1 is higher by 30% when using the new rate, relatively
to using the TY87 rate, although a full galactic chemical evolution
calculation is needed to derive the accurate number. Such variation results in
a small decrease of roughly 12% in the $r$-process residual abundance of
135Ba. When added to the present error bars of roughly 9%, based on
uncertainties on the neutron-capture cross section and the solar abundance
(Arlandini et al., 1999), the value of the $r$-process residual abundance of
135Ba may be up to 20% lower than used usually. Considering, for example,
Figure 6 of Vockenhuber et al. (2007), this decrease would move the
$r$-process residual abundance of 135Ba to just below that of 137Ba, while one
would expect 135Ba to be above 137Ba. A full analysis of the branching points
along the Cs isotopes leading also to the production of 137Ba is needed to
evaluate and resolve this possible tension. If the second $r$-process peak (at
$A\sim 130$) results in being reshaped and potentially narrowed by the lower
135Ba residual derived using our new rate, this will have implications on our
understating of the fission recycling during the $r$ process (Kajino et al.,
2019; Mumpower et al., 2018).

In Figure 4 we present the comparison between our model predictions and the
composition of single stardust SiC grains measured with Resonant Ionisation
Mass Spectrometry (RIMS, see Liu et al., 2014, and references therein). We
focus in this figure on the models with solar and higher-than-solar
metallicity, as these has been reported previously to provide the best match
to the SiC data (Lugaro et al., 2018a; Liu et al., 2018, e.g.,). They are the
same models used by Lugaro et al. (2018a) and more details on their structural
features can be found in Table 1 of that paper. As discussed above, the models
calculated with the new 134Cs decay rate result in lower 134Ba/136Ba ratios,
which, when compared to the grain data, provide a general better fit to the
observed spread. The new models reach down to cover, within the error bars,
the two grains most strongly depleted in 134Ba which were previously
unexplained and potentially attributed to the operation of the intermediate
neutron capture ($i$ process) in post-AGB stars (Liu et al., 2014), suggesting
that the two grains could still be explained by the $s$ process alone. Due to
its being already significantly lower than solar, the predicted lines for the
135Ba/136Ba ratio are only marginally shifted to the left due to the new rate.
The grain data indicates a spread of 135Ba/136Ba ratio with values towards the
solar value. This spread can be predicted when changing the free parameter in
the models which represents the extent in mass of the region of partial mixing
($M_{\rm PMZ}$) leading to the formation of the 13C neutron source (Karakas &
Lugaro, 2016; Lugaro et al., 2018a). In the case of the solar metallicity
models we needed to divide $M_{\rm PMZ}$ by a factor of 10, in the case of the
models of twice-solar metallicity it was enough to reduce this mass by a
factor of two to derive a better match to the measured 135Ba/136Ba ratios. The
difference is due to the models of lower metallicity being more efficient in
producing the $s$-process isotopes, such as 136Ba, because of the lower number
of Fe seeds (Clayton, 1988). Finally we note that for this comparison we
considered the 135Ba abundance alone, without the contribution of 135Cs. This
assumes that SiC condenses in AGB envelopes within a few Myr, i.e., before a
significant amount of 135Cs has decayed into 135Ba, and that Cs does not
condense into SiC, since it is a relatively volatile element. Our models
support this assumptions, which was already discussed in Lugaro et al. (2003)
(see their Figure 16), and even more strongly when considering the predictions
obtained using the new decay rate of 134Cs.

While our new rate has direct and indirect implications on the $s$\- and
$r$-process, respectively, it is not expected to lead to significant
modification of the Ba isotopic abundances produced by the $i$ process, with
neutron densities of the order of $10^{13}-10^{15}$ cm-3, in-between those of
the $s$ and $r$ processes. This is because during the $i$ process the Ba
isotope most produced is 135Ba, resulting from the decay of the neutron magic
unstable 135I (see, e.g., Figure 1 of Hampel et al. (2016)). The behavior of
the Cs branching points represents a less significant effect relative to this
major feature of the $i$ process.

There are also potential implications of our results for the early Solar
System. Long-lived radioactive isotopes have been found to be present at the
time of the formation of the first solids at the birth of the Sun, however,
for 135Cs/133Cs ratio only an upper limit of $2.8\times 10^{-6}$ (Brennecka &
Kleine, 2017) is available. This means that from this upper limit we can only
derive a lower limit for the time elapsed from the last AGB nucleosynthetic
event that polluted the pre-solar system matter to the birth of the Sun
(Lugaro et al., 2014). For consistency with (Lugaro et al., 2014, 2018b) we
consider the 3 M⊙, Z=0.014 model as typical, and calculated the time that
elapsed from the last AGB events using Eq. 15 of Lugaro et al. (2018b) with
the Galactic chemical evolution parameter K=2.3 (Côté et al., 2019a) and with
the time interval between polluting events of 50 Myr to ensure that we are in
the regime where it is only one event that contributed to the early Solar
System abundances of the long-lived isotopes produced by AGB stars (Côté et
al., 2019b). We derive elapsed times of the order of 32-38 Myr from 107Pd and
182Hf. For 135Cs we derive an elapsed time $>$24 Myr and $>$28 Myr when
considering the 135Cs/133Cs ratios predicted using the 134Cs decay rate by
TY87 and the present work, respectively. In both cases agreement is found with
the other two long-lived isotopes.

Finally, we examined the impact of the new decay rate on the ratio $f_{\rm
odd}$ between the abundances of the odd-A and even-A isotopes of Ba. In most
of our models $f_{\rm odd}$ is very close to traditional $s$-process reported
value of $f_{\rm odd}$ = 0.11 (Arlandini et al., 1999). The changes are at
most a increase of 15% when considering the new decay rate.

##  4 Summary

The stellar $\beta-$decay rate of the branching point 134Cs is crucial to the
understanding of $s$-process nucleosynthesis at the vicinity of 134-136Ba.
Large-scale shell-model calculations were performed to determine the GT
strengths of the important transitions from the 134Cs excited states to the
excited and ground states of 134Ba. At typical $s$-process temperatures, the
new decay rate from our calculation is significantly lower than the TY87 rate
estimated with empirical log$ft$ values. With the new $\beta-$decay rate, we
preformed nucleosynthesis calculations using AGB stellar models with various
masses and metallicities. Our result shows an overall decrease in the $s$-only
134Ba/136Ba isotopic ratio and well explains the 134Ba/136Ba isotopic ratios
in meteorites without introducing the $i$ process at the post-AGB phase. We
also derive the elapsed time from the last AGB nucleosynthetic event that
polluted the early Solar System to be $>$28 Myr based on the 135Cs/133Cs
ratio, consistent with the elapsed times derived from 107Pd and 182Hf. The
abundance sum of 135Ba and 135Cs is found to increase, resulting in a smaller
$r$-process contribution of 135Ba to the Solar System. The remaining
uncertainty of the 134Cs stellar decay rate mainly comes from the tentative
spin-parity assignment of some low-lying states. The spin-parity assignments
are essential to match shell model levels with experimental levels, and
incorrect assignments would yield incorrect matching. Experimental
investigation of the spin-parity structure of 134Cs is therefore key to a
precise understanding of multiple astrophysical nucleosynthesis sites in this
mass region.

This project is supported by the Strategic Priority Research Program of
Chinese Academy of Sciences, Grant No. XDB34020000, the national key research
and development program (MOST 2016YFA0400501), the ERC Consolidator Grant
(Hungary) funding scheme (Project RADIOSTAR, G.A. n. 724560), the ISSI-Beijing
project ”Radioactive Nuclei in the Cosmos and in the Solar System” and the
Chinese and Hungarian Academy of Sciences visitor exchange program. C.Q. was
supported by the Swedish Research Council (VR) under grant Nos. 621-2012-3805,
and 621-2013-4323 and the Göran Gustafsson foundation. The calculations were
performed on resources provided by the Swedish National Infrastructure for
Computing (SNIC) at PDC at KTH, Stockholm. A.I.K. was supported by the
Australian Research Council Centre of Excellence for All Sky Astrophysics in 3
Dimensions (ASTRO 3D), through project number CE170100013. X.T. acknowledges
support from the National Natural Science Foundation of China under Grant No.
11021504, 11321064, 11475228 and 11490564, 100 talents Program of the Chinese
Academy of Sciences. X.T. thanks Hayden Campbell for proofreading.

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