JUNA Collaboration

#  Deep underground laboratory measurement of 13C($\alpha$,$n$)16O in the
Gamow windows of the s- and i-processes

B. Gao  Joint department for nuclear physics, Lanzhou University and Institute
of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  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  T.Y. Jiao  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  Y.T. 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  H. Chen  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  W.P. Lin  Key
Laboratory of Radiation Physics and Technology of the Ministry of Education,
Institute of Nuclear Science and Technology, Sichuan University, Chengdu
610064, China  Z. An  Key Laboratory of Radiation Physics and Technology of
the Ministry of Education, Institute of Nuclear Science and Technology,
Sichuan University, Chengdu 610064, China  L.H. Ru  CAS Key Laboratory of High
Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 73000, People’s Republic of China  Z.C. Zhang  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  X.D. Tang
<EMAIL_ADDRESS>Joint department for nuclear physics, Lanzhou University
and Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  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  X.Y. Wang
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  N.T. Zhang  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  X. Fang  Sino-French
Institute of Nuclear Engineering and Technology, Sun Yat-sen University,
Zhuhai, Guangdong 519082, People’s Republic of China  D.H. Xie Key Laboratory
of Radiation Physics and Technology of the Ministry of Education, Institute of
Nuclear Science and Technology, Sichuan University, Chengdu 610064, China
Y.H. Fan 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, Lanzhou
University, Lanzhou 730000, China  L. Ma Key Laboratory of Radiation Physics
and Technology of the Ministry of Education, Institute of Nuclear Science and
Technology, Sichuan University, Chengdu 610064, China  X. Zhang Key Laboratory
of Radiation Physics and Technology of the Ministry of Education, Institute of
Nuclear Science and Technology, Sichuan University, Chengdu 610064, China  F.
Bai Key Laboratory of Radiation Physics and Technology of the Ministry of
Education, Institute of Nuclear Science and Technology, Sichuan University,
Chengdu 610064, China  P. Wang Key Laboratory of Radiation Physics and
Technology of the Ministry of Education, Institute of Nuclear Science and
Technology, Sichuan University, Chengdu 610064, China  Y.X. Fan Key Laboratory
of Radiation Physics and Technology of the Ministry of Education, Institute of
Nuclear Science and Technology, Sichuan University, Chengdu 610064, China  G.
Liu Key Laboratory of Radiation Physics and Technology of the Ministry of
Education, Institute of Nuclear Science and Technology, Sichuan University,
Chengdu 610064, China  H.X. Huang  China Institute of Atomic Energy, Beijing
102413, People’s Republic of China  Q. Wu  Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou 730000, China  Y.B. Zhu  Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  J.L. Chai
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  J. Q. Li  Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China  L. T. Sun  Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 730000, China  S. Wang  Shandong Provincial Key
Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute
of Space Sciences, Shandong University, Weihai 264209, China  J.W. Cai  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  Y.Z. 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  J. Su  China
Institute of Atomic Energy, Beijing 102413, People’s Republic of China  Key
Laboratory of Beam Technology and Material Modification of Ministry of
Education, College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China  H. Zhang  China Institute of Atomic Energy,
Beijing 102413, People’s Republic of China  Z. H. Li  School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
100049, People’s Republic of China  China Institute of Atomic Energy, Beijing
102413, People’s Republic of China  Y. J. Li  China Institute of Atomic
Energy, Beijing 102413, People’s Republic of China  E. T. Li  College of
Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060,
China  C. Chen  China Institute of Atomic Energy, Beijing 102413, People’s
Republic of China  Y. P. Shen  China Institute of Atomic Energy, Beijing
102413, People’s Republic of China  G. Lian  China Institute of Atomic Energy,
Beijing 102413, People’s Republic of China  B. Guo  China Institute of Atomic
Energy, Beijing 102413, People’s Republic of China  X. Y. Li  Key Laboratory
of Beam Technology and Material Modification of Ministry of Education, College
of Nuclear Science and Technology, Beijing Normal University, Beijing 100875,
China  L. Y. Zhang  Key Laboratory of Beam Technology and Material
Modification of Ministry of Education, College of Nuclear Science and
Technology, Beijing Normal University, Beijing 100875, China  J. J. He  Key
Laboratory of Beam Technology and Material Modification of Ministry of
Education, College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China  Y. D. Sheng  Key Laboratory of Beam
Technology and Material Modification of Ministry of Education, College of
Nuclear Science and Technology, Beijing Normal University, Beijing 100875,
China  Y. J. Chen  Key Laboratory of Beam Technology and Material Modification
of Ministry of Education, College of Nuclear Science and Technology, Beijing
Normal University, Beijing 100875, China  L. H. Wang  Key Laboratory of Beam
Technology and Material Modification of Ministry of Education, College of
Nuclear Science and Technology, Beijing Normal University, Beijing 100875,
China  L. Zhang  China Institute of Atomic Energy, Beijing 102413, People’s
Republic of China  F. Q. Cao  China Institute of Atomic Energy, Beijing
102413, People’s Republic of China  W. Nan  China Institute of Atomic Energy,
Beijing 102413, People’s Republic of China  W. K. Nan  China Institute of
Atomic Energy, Beijing 102413, People’s Republic of China  G. X. Li  China
Institute of Atomic Energy, Beijing 102413, People’s Republic of China  N.
Song  China Institute of Atomic Energy, Beijing 102413, People’s Republic of
China  B. Q. Cui  China Institute of Atomic Energy, Beijing 102413, People’s
Republic of China  L. H. Chen  China Institute of Atomic Energy, Beijing
102413, People’s Republic of China  R. G. Ma  China Institute of Atomic
Energy, Beijing 102413, People’s Republic of China  Z. C. Zhang  College of
Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060,
China  S. Q. Yan  China Institute of Atomic Energy, Beijing 102413, People’s
Republic of China  J. H. Liao  China Institute of Atomic Energy, Beijing
102413, People’s Republic of China  Y. B. Wang  China Institute of Atomic
Energy, Beijing 102413, People’s Republic of China  S. Zeng  China Institute
of Atomic Energy, Beijing 102413, People’s Republic of China  D. Nan  China
Institute of Atomic Energy, Beijing 102413, People’s Republic of China  Q. W.
Fan  China Institute of Atomic Energy, Beijing 102413, People’s Republic of
China  N. C. Qi  Yalong River Hydropower Development Company, Chengdu 610051,
China  W. L. Sun  Yalong River Hydropower Development Company, Chengdu 610051,
China  X. Y. Guo  Yalong River Hydropower Development Company, Chengdu 610051,
China  P. Zhang  Yalong River Hydropower Development Company, Chengdu 610051,
China  Y. H. Chen  Yalong River Hydropower Development Company, Chengdu
610051, China  Y. Zhou  Yalong River Hydropower Development Company, Chengdu
610051, China  J. F. Zhou  Yalong River Hydropower Development Company,
Chengdu 610051, China  J. R. He  Yalong River Hydropower Development Company,
Chengdu 610051, China  C. S. Shang  Yalong River Hydropower Development
Company, Chengdu 610051, China  M. C. Li  Yalong River Hydropower Development
Company, Chengdu 610051, China  S. Kubono  RIKEN Nishina Center, Wako, Saitama
351-0198, Japan  Center for Nuclear Study, University of Tokyo, Wako, Saitama
351-0198, Japan  W. P. Liu<EMAIL_ADDRESS>China Institute of Atomic Energy,
Beijing 102413, People’s Republic of China  R.J. deBoer  Department of Physics
and the Joint Institute for Nuclear Astrophysics, University of Notre Dame,
Notre Dame, IN 46556, USA  M. Wiescher  Department of Physics and the Joint
Institute for Nuclear Astrophysics, University of Notre Dame, Notre Dame, IN
46556, USA  Wolfson Fellow of Royal Society, School of Physics and Astronomy,
University of Edinburgh, King’s Buildings, Edinburgh EH9 3FD, United Kingdom
M. Pignatari  Konkoly Observatory, Research Centre for Astronomy and Earth
Sciences (CSFK), Eötvös Loránd Research Network (ELKH), Konkoly Thege Miklós
út 15-17, H-1121 Budapest, Hungary  CSFK, MTA Centre of Excellence, Budapest,
Konkoly Thege Miklós út 15-17, H-1121, Hungary  E. A. Milne Centre for
Astrophysics, Department of Physics and Mathematics, University of Hull, HU6
7RX, United Kingdom  NuGrid Collaboration, http://nugridstars.org

###### Abstract

The 13C($\alpha$,$n$)16O reaction is the main neutron source for the slow-
neutron-capture (s-) process in Asymptotic Giant Branch stars and for the
intermediate (i-) process. Direct measurements at astrophysical energies in
above-ground laboratories are hindered by the extremely small cross sections
and vast cosmic-ray induced background. We performed the first consistent
direct measurement in the range of $E_{\rm c.m.}=$0.24 MeV to 1.9 MeV using
the accelerators at the China Jinping Underground Laboratory (CJPL) and
Sichuan University. Our measurement covers almost the entire i-process Gamow
window in which the large uncertainty of the previous experiments has been
reduced from 60% down to 15%, eliminates the large systematic uncertainty in
the extrapolation arising from the inconsistency of existing data sets, and
provides a more reliable reaction rate for the studies of the s- and
i-processes along with the first direct determination of the alpha strength
for the near-threshold state.

††preprint: APS/123-QED

Low-mass Asymptotic Giant Branch (AGB) stars are the primary sources of
elements above iron in the Galaxy via the activation of the slow-neutron-
capture process (s-process) Kobayashi  _et al._ (2020). In these stars, the
bulk of the s-process abundances is created by neutrons from the the
13C($\alpha$,$n$)16O reaction at temperatures around $T_{9}$=0.1 in the
radiative 13C-pocket, located in the helium-rich layers, just below the
stellar envelope Gallino  _et al._ (1998); Bisterzo  _et al._ (2017). Here
$T_{9}$ is defined as a temperature divided by 109 Kelvin. The subsequent
neutron capture and $\beta$-decay processes transmute lighter elements into
heavier ones, with a major production efficiency between Sr and Pb depending
on the initial metallicity of the star Käppeler  _et al._ (2011). In some AGB
simulations, it was also found that part of the 13C is still alive in the
13C-pocket at the onset of the convective thermal pulse: the remaining 13C is
mixed at the bottom of the He intershell region activating
13C($\alpha$,$n$)16O at He burning temperatures of approximately
$T_{9}$=0.2-0.25 Guo  _et al._ (2012); Cristallo  _et al._ (2018). This
scenario tends to be favored by a low 13C($\alpha$,$n$)16O rate Ciani  _et
al._ (2021), and the anomalous activation of the 13C($\alpha$,$n$)16O reaction
together with the 22Ne($\alpha$,$n$)25Mg reaction at the bottom of the
convective thermal pulse. This may affect the s-process isotopic pattern near
the active s-process branching points Cristallo  _et al._ (2018); Ciani  _et
al._ (2021).

The 13C($\alpha$,$n$)16O reaction is also the main neutron source of the
intermediate process (i-process ) Cowan and Rose (1977), which matches the
puzzling abundances observed in some post AGB stars Herwig  _et al._ (2011),
of a subset of carbon-enhanced metal poor (CEMP) stars Hampel  _et al._
(2016), of presolar grains Jadhav  _et al._ (2013) and stars in young open
clusters Mishenina  _et al._ (2015). The i-process can be activated in
different stellar environments, including among other low-mass AGB stars
Cristallo  _et al._ (2009); Hampel  _et al._ (2016), super AGB stars Jones
_et al._ (2016) and post AGB stars Herwig  _et al._ (2011), massive stars
Clarkson  _et al._ (2018); Banerjee  _et al._ (2018) and rapidly-accreting WDs
Denissenkov  _et al._ (2019). In those models, a small amount of hydrogen is
ingested into the convective helium-burning zone underneath the envelope.
Hydrogen reacts with the primary product of He-burning 12C, making 13N that
will decay to 13C. The i-process is generated by the 13C($\alpha$,$n$)16O
activated at He-burning temperatures of around $T_{9}$=0.2 or above Herwig
_et al._ (2011). This results in a neutron density of around 10${}^{14}\sim$
1016 cm-3, which is significantly higher than typical values of the s-process
(10${}^{6}\sim$ 1010 cm-3). At least for one-dimensional models of metal-poor
low-mass AGB models, Cristallo  _et al._ (2018) showed that after the hydrogen
ingestion, the 13C($\alpha$,$n$)16O reaction also becomes a relevant energy
source in the He shell: in their calculations even a factor of two variation
of the 13C($\alpha$,$n$)16O rate changes the i-process production by orders of
magnitude. Such an impact will need to be confirmed for different types of
stars by multi-dimensional hydrodynamics simulations, providing guidance for
how one-dimensional models should behave once hydrogen has been ingested in
the hotter He-burning regions Herwig  _et al._ (2014); Denissenkov  _et al._
(2019); Clarkson and Herwig (2021).

It is clear that the 13C($\alpha$,$n$)16O reaction rate is a fundamental
ingredient in the s- and i-process models, determining the neutron density and
the final isotopic production. A solid understanding of the reaction cross
section is needed at the associated Gamow energies of about
$E_{\mathrm{c.m.}}$ = 0.15 to 0.3 MeV and 0.2 to 0.54 MeV, respectively. The
reaction cross sections in these energy regions are strongly influenced by the
$\alpha$ cluster state near the separation threshold according to the theory
of Ikeda Ikeda  _et al._ (1972). Descouvemont made a theoretical prediction of
the level structure using a microscopic generator-coordinate method (GCM) and
concluded that the theoretical S-factor below $E_{\mathrm{c.m.}}$ = 0.3 MeV
increases rapidly with respect to the extrapolations that ignored this
threshold state Descouvemont (1987). Reliable experimental measurements down
to energies below 0.3 MeV are called for to confirm the prediction.

Considerable efforts have been devoted to push the direct measurement of the
13C($\alpha$,$n$)16O reaction cross section ($\sigma$) Sekharan  _et al._
(1967); Bair and Haas (1973); Davids (1968); Drotleff  _et al._ (1993); Heil
_et al._ (2008) down to the stellar energies where $\sigma$ becomes extremely
small. Due to the vast cosmic background, direct measurements performed in
laboratories on the Earth’s surface stopped at energies above
$E_{\mathrm{c.m.}}$ = 0.27 MeV with a lower limit of $\sigma$ = 6(4)$\times$
$10^{-11}$ barn Drotleff  _et al._ (1993), unable to effectively constrain the
crucial threshold state and provide a reliable extrapolation down to stellar
energies. Besides that, the extrapolation accuracy is further limited by large
discrepancies among those measurements deBoer  _et al._ (2020); Brown  _et
al._ (2018).

Recent breakthrough in the direct measurement of this reaction was reported by
the LUNA collaboration Ciani  _et al._ (2021) at $E_{\mathrm{c.m.}}$ =
0.23-0.30 MeV, the upper range of the s-process Gamow window. However, they
had to rely on other existing data at higher energies, ANC of the threshold
state and the R-matrix analysis to determine the S-factor in the range of 0.15
to 0.5 MeV, most of which is inaccessible with the current LUNA facility. The
large discrepancies between Harissopulos  _et al._ (2005) and other
measurements Drotleff  _et al._ (1993); Heil  _et al._ (2008) result in
$\sim$50% differences in their recommended upper and lower limits for the
reaction rate at $T_{9}$=0.1-0.3, leading to significant uncertainties in the
production yields of several important isotopes, such as 60Fe and 205Pb, by
using their AGB model.

In this paper, we report the first consistent direct measurement of the
13C($\alpha$,$n$)16O reaction over a wider energy range of $E_{\rm c.m.}$ =
0.24-1.9 MeV with improved precision. Our measurement reduces the 60% large
uncertainty down to 15% at the center of the Gamow window of the i-process,
provides the first direct determination of the alpha strength for the near
threshold state, and eliminates the large systematic uncertainty in the
extrapolation incurred by the discrepancy of the existing experiments. A new
reliable reaction rate is recommended based on our measurement.

The underground experiment was performed in the A1 hall of the China JinPing
underground Laboratory (CJPL) Cheng  _et al._ (2017); Liu  _et al._ (2016);
Zhang  _et al._ (2021). High-intensity 4He+ and 4He2+ ions were extracted from
2.45- and 14.5-GHz electron-cyclotron-resonance (ECR) sources, respectively,
and accelerated by a 400-kV platform called Jinping Underground Nuclear
Astrophysics experimental facility (JUNA). The highest beam energy of 800 keV
was achieved by using 4He2+ ions, allowing comparisons with previous
measurements. The acceleration voltage was calibrated using the
12C(p,$\gamma$)13N, 27Al(p,$\gamma$)28Si, 11B(p,$\gamma$)12C and
14N(p,$\gamma$)15O reactions Wang (2021). The absolute beam energy was
determined to an accuracy of 0.5 keV with an energy spread of less than 0.2
keV Wang (2021). A 90∘ dipole magnet with a mass resolution of 250 was used
together with a set of analyzing slits to eliminate the H${}_{2}^{+}$/D+
contamination in the 4He2+ beam Chen  _et al._ (2018). A clear separation of
4He2+ from the other impurities was observed at the slit position and the
count ratio of the inner and outer rings of the 3He detector array indicated
that no neutrons came from the deuterium impurity.

To avoid the source of systematic uncertainty incurred by target deterioration
in traditional thin target experiments, we used 2-mm thick 13C enriched
targets with a purity of 97%. The target was installed on a water-cooled
copper backing. On-target beam intensity of up to 2.5 particle mA, the highest
$\alpha$ beam intensity among the deep underground laboratories, was achieved.
The thick targets turned out to be very stable and only two targets were used
for the whole experiment. A cold trap was installed to reduce the natural
carbon buildup on the targets. For the 4He+ runs, the analyzing slits used in
the 4He2+ runs were removed to allow maximum transmission efficiency and
achieve higher beam intensities.

Neutrons from the 13C($\alpha$, $n$)16O reaction were detected by an array
consisting of 24 3He-filled proportional counters. By placing 35-cm thick 7%
borated polyethylene blocks and 1-mm thick cadmium sheets around the detection
array, a background of 4.7(2) events/hour was achieved, compared to 1238(11)
events/hour measured on the Earth’s surface. The detection efficiency of the
array was determined to be 26% for 2.5-MeV neutrons using the 51V(p,n)51Cr
reaction together with Geant4 simulations Li  _et al._ (2022).

The thick target yield ${\rm Y(E)}$ of the 13C($\alpha$,$n$)16O reaction was
measured with beam energies of 0.3 $<$ $E_{\alpha}$ $<$ 0.785 MeV. The beam-
induced neutron background (BINB) was estimated by measuring the ${\rm Y(E)}$
at $E_{\alpha}$ = 0.25 MeV where the cross section of the 13C($\alpha$,$n$)16O
reaction is negligibly small and all neutrons detected above the environmental
background level should be attributed to the BINB. BINB was determined to be
0.05(8) events/Coulomb, consistent with zero.

The cross sections and the corresponding effective energies were extracted by
differentiating the thick target yield Spillane  _et al._ (2007); Notani  _et
al._ (2012). We repeatedly checked the neutron yields at 17 energy points and
the reproducibility was found to be 8% and 3% for the 4He+ and 4He2+ data
sets, respectively. This random error likely originates from the beam tuning
and the potential carbon build up. Therefore, we added this error
quadratically together with the statistical error.

Another thick target measurement was performed in the range of $E_{\rm c.m.}$
= 0.75 MeV to 1.9 MeV using the 4He+ beam from the 3 MV Tandetron at Sichuan
University Han  _et al._ (2018) to resolve the discrepancies among the
S-factors at higher energies in the previous works Harissopulos  _et al._
(2005); Bair and Haas (1973); Drotleff  _et al._ (1993). The same detection
setup was used to minimize extra systematic uncertainties. The beam energy was
calibrated using the 7Li(p,n)7Be reaction, and confirmed by the narrow
resonances of the 13C($\alpha$,$n$)16O at $E_{\alpha}$=1055.6, 1334.7 and
1338.8 keV. A thin target measurement was also carried out in the range of
$E_{\mathrm{c.m.}}$=1.6 to 1.9 MeV using a 3.2$\mu$g/cm2-thick 13C target. The
thin-target data is normalized to the thick-target data. The reproducibility
of the thick-target and thin-target measurements are estimated to be 3% and
2%, respectively. This uncertainty is included with the statistical error as
discussed above.

By adopting a compiled angular distribution Brune (2021); Walton  _et al._
(1957) in the Geant4 simulation, our efficiency has been corrected for angular
distribution effects. These effects were found to change the efficiency by
$\pm$2% at $E_{\mathrm{c.m.}}<$0.6 MeV. However, the efficiency at
$E_{\mathrm{c.m.}}\sim$0.9 MeV deviates from the nominal efficiency with an
isotropic distribution by $\sim$5%, which is larger than the statistical
uncertainties in the previous measurements Harissopulos  _et al._ (2005); Bair
and Haas (1973). This deviation becomes even larger at the narrow resonances
Li  _et al._ (2022). Such an effect was overlooked in these previous works.

Systematic uncertainties of our measurements at CJPL and SCU are estimated to
be 11%, which includes contributions from the beam current integration(5%),
detection efficiency (7%), angular distribution (2% for JUNA underground
measurement and 4% for SCU experiment), and stopping power (6%) Ziegler  _et
al._ (2010).

Our S-factor is converted into the bare S-factors after correcting for the
screen effect using our fitted screening potential of $U_{e}$ = 0.78 keV
together with the previous measurements Drotleff  _et al._ (1993); Heil  _et
al._ (2008); Harissopulos  _et al._ (2005); Ciani  _et al._ (2021); Kellogg
_et al._ (1989). The results are presented in Fig. 1 . It can be seen that our
underground data cover the energy range from $E_{\rm c.m.}$ = 0.24 to 0.59
MeV, greatly overlapping with the astrophysical region of $E_{\rm c.m.}$ =
0.15 to 0.5 MeV with a statistical uncertainty better than 15%.

With the unique energy range and ultra low neutron background in the deep
underground lab, we are able to precisely measure the S-factor in the range of
astrophysical interest for i-process nucleosynthesis. The center of the Gamow
window for the i-process is located at 0.35 MeV, beyond the accessible energy
range of LUNA. The two extrapolation scenarios of LUNA using either the
normalization of Heil, or Drotleff Drotleff  _et al._ (1993); Heil  _et al._
(2008); Bair and Haas (1973) or that of Harissopulos  _et al._ (2005) resulted
in their so-called best fit and “low LUNA” fit, respectively. To be on the
safe side, they defined the “low-LUNA” fit by taking the 95% confidence level
of the lower limit of the fit with the original Harissopulos data. These two
fits differ from each other by a factor of 2 at 0.35 MeV. Such a large
systematic uncertainty in their extrapolation is eliminated by our consistent
measurement, which rules out the lower normalization of Harissopulos  _et al._
(2005). Drotleff  _et al._ (1993) was the best measurement before ours at the
energy around 0.35 MeV. While our data above 0.4 MeV is in good agreement with
that of Drotleff, our data around 0.27 MeV are about 50% lower and disagree
with the upturning trend in this data set. The nearly 60% uncertainty in Ref.
Drotleff  _et al._ (1993) within the Gamow window has been reduced to 15%.

Figure 1: The S-factor of the 13C($\alpha$,$n$)16O reaction. The uncertainties
from the fit to the JUNA+SCU data are indicated by dotted lines. The best fit
and lower limit recommended by LUNA Ciani  _et al._ (2021) are shown as black
and blue dashed lines, respectively. The S-factors have been corrected with
the screening potential $U_{e}$ = 0.78 keV. The temperatures in $T_{9}$ on the
top correspond to the center energy of the Gamow window on the bottom.

The S-factor at $E_{\rm c.m.}<$0.24 MeV was obtained using an $R$-matrix
analysis Lane and Thomas (1958) in the range of $E_{\rm c.m.}$=0.24 to 1.9 MeV
using the code AZURE2 Azuma  _et al._ (2010); Uberseder and deBoer (2015). In
our analysis, we only included our measurements of the 13C($\alpha$,$n$)16O
cross section, to eliminate the systematic uncertainty of the inconsistent
data sets, and the 16O+$n$ total cross section Fowler  _et al._ (1973).

Our best fit is shown together with its estimated uncertainty in Fig. 1. The
screening potential($U_{e}$) is fitted to be 0.78$\pm$0.43 keV. It agrees with
the theoretical prediction of $U_{e}$=0.937 keV using the adiabatic limit
while ruling out the larger prediction of $U_{e}$=2 keV Trippella and Cognata
(2017). Our fit is about 15% systematically higher than the LUNA measurement
Ciani  _et al._ (2021). The reduced-$\chi^{2}$ of the LUNA data is 25 by using
their best fit. It drops to 1.02 with our fit after the normalization and
excluding the point at $E_{\rm c.m.}$=0.29 MeV, which is 5$\sigma$ lower than
our best fit.

Although the LUNA measurement agrees with ours within the quoted errors, the
inconsistency between the measurement of Harissopulos  _et al._ (2005) and
other measurements at higher energies leads to a $\sim$50% difference between
the upper and the lower limits of the reaction rate recommended by LUNA at
$T_{9}$=0.1-0.3. This demonstrates a key limitation of the LUNA measurement,
that its limited energy range did not allow for a direct comparison with
higher energy data. Using our consistent measurement over a board energy
range, the uncertainty of our fit are reliably constrained to the level
of$<$16% at the Gamow windows of s- and i-processes.

The extrapolated S-factor towards lower energy is dominated by the $\alpha$
reduced width $\gamma_{\alpha}$ or the Coulomb renormalized asymptotic
normalization coefficient ($\tilde{C}^{2}$) of the 1/2+ threshold state. The
R-matrix analysis performed in previous works involved fixing the ANC of the
threshold state to values obtained from indirect measurements. However, the
uncertainties in these ANCs often suffer from difficulties to quantify
systematic uncertainties from the models used to obtain them. The lower and
higher limits of the measured $\tilde{C}^{2}$ differ from each other by a
factor of $\sim$5 deBoer  _et al._ (2020). These systematic uncertainties have
been eliminated in our fit by treating the $\Gamma_{\alpha}$ of this state as
a free parameter. The reduced widths $\gamma_{\alpha}$ obtained from our best
$R$-matrix analysis is -0.14(2) MeV1/2 with a channel radius of 6.684 fm and
$E_{x}$=6.3772 MeV, corresponding to an ANC of $\tilde{C}^{2}$=2.1(5) fm-1
with $E_{x}$=6.356 MeV Mukhamedzhanov  _et al._ (2017); Brune (2020). Our
value is slightly lower than the indirect measurements of 3.6(7) fm-1 Avila
_et al._ (2015) and agree with 2.7(8) fm-1 Guo  _et al._ (2012); Shen  _et
al._ (2019) and 4.5(2.2)Pellegriti  _et al._ (2008). For the first time, we
not only validate the $\alpha$ width of the threshold state obtained with the
indirect method using the direct measurement, but also determine the
interference pattern in the $R$-matrix analysis. As LUNA used the higher
$\tilde{C}^{2}$ from Avila  _et al._ (2015) to constrain their extrapolation
towards lower energies, our best fit is 23% lower than their best fit at
$E_{\rm c.m.}$=0.19 MeV, the center of the Gamow window for $T_{9}$=0.1 (see
Fig.2). At the same energy, with the combination of a larger reduced width
Avila  _et al._ (2015) and the cross section of Harissopulos  _et al._ (2005),
the “low LUNA” fit is 11% lower than our best fit.

Figure 2: The Gamow function of 13C($\alpha$,$n$)16O at $T_{9}$ = 0.1 and 0.2.
Color coding is identical to Fig.1

The 13C($\alpha$,$n$)16O reaction rate is calculated by numerical integration
of the standard reaction rate equation Rolfs and Rodney (1988):

$\langle\sigma
v\rangle=\Big{(}\frac{8}{\pi\mu}\Big{)}^{1/2}\frac{1}{{kT}^{3/2}}\int_{0}^{\infty}\sigma(E)E\exp\Big{(}-\frac{E}{kT}\Big{)}\mathrm{d}{\rm
E}$ (1)

To highlight the important stellar energy range for a typical helium burning
temperature $T_{9}$ = 0.1 and 0.2, the integrand of Eq. (1) (Gamow function)
is computed and shown in Fig. 2. At $T_{9}$ = 0.1, the temperature of the 13C
pocket in the AGB model, our extrapolation is lower than the best fit of LUNA
and tends to agree better with their “low LUNA” fit. At $T_{9}$ = 0.2, which
is of importance for both the i-process and s-process nucleosynthesis in the
thermal pulse in the AGB model, our measurement covers nearly the entire Gamow
function with significantly improved uncertainties. This is a substantial
improvement compared to previous measurements as the ground-based measurements
from Ref. Drotleff  _et al._ (1993); Heil  _et al._ (2008) covered only the
upper part of the Gamow window with large uncertainties while the LUNA
extrapolation suffered from the inconsistencies in the absolute cross section
of the higher energy measurements. At the center of the Gamow window of
$T_{9}$ = 0.2, our result agrees with the best fit of LUNA within our quoted
uncertainty, but rules out the “low LUNA” fit, reflecting a significant
difference in the shape of our extrapolation from that of LUNA.

The reaction rate is calculated with the JUNA fit shown in Fig. 1. Comparisons
of the reaction rates are shown in Fig. 3. Our reaction rate agrees well with
previous Reaclib compilations based on the ANC method Guo  _et al._ (2012) and
NACRE-II Xu  _et al._ (2013) at $T_{9}\geq$ 0.1. The nearly 50% difference
between the upper limit and lower limit (“low LUNA”) of LUNA and the even
larger uncertainty in NACRE-II have been improved significantly. At
$T_{9}$=0.1-0.3, the typical temperatures for s- and i-processes, we have
reached an uncertainty of 13%-16%.

Figure 3: Selected reaction rates normalized by the rate determined in this
work. The uncertainties of our new rate and the LUNA rate, based on their best
fit, are indicated by the red hatched area and yellow shaded area,
respectively. For comparison, we also show the rates from NACRE-II Xu  _et
al._ (2013), LUNA Ciani  _et al._ (2021), and JINA Reaclib Guo  _et al._
(2012); Cyburt  _et al._ (2010).

It has been shown that the “low-LUNA” rate increases the survivability of 13C
in the 13C-pocket of an AGB star, and that it burns at a high temperature in
the subsequent thermal pulse Ciani  _et al._ (2021). Compared with the “low-
LUNA” rate, our recommended rate is slightly higher at temperatures typical of
the 13C-pocket within 15$\%$, and about 30$\%$-40% higher at the thermal pulse
temperatures. Therefore, we expect our rate to produce effects similar to
those discussed by Ciani  _et al._ (2021), although a follow up, detailed
study, on AGB stellar models is needed. Concerning the impact on the i-process
nucleosynthesis, future models based on the next generation of multi-
dimensional hydrodynamics simulations will be more predictive thanks to the
more reliable reaction rate uncertainties provided by this work.

In summary we have performed a direct measurement of the 13C($\alpha$,$n$)16O
reaction cross section over the range of $E_{\rm c.m.}$ = 0.24-1.9 MeV using
the most intense $\alpha$ beam available in the deep underground laboratories
with the highest precision to date. Our consistent measurement, covering a
wide energy range, reduce the large uncertainty in the reaction rate down to
13% to 16% for i- and s-process nucleosynthsis. Our reaction rate is similar
to the “low LUNA” rate at the typical 13C pocket, favoring the release of more
neutrons from the 13C($\alpha$,$n$)16O reaction during the thermal pulse
phase. Our direct measurement eliminates an important systematic uncertainty
in the R-matrix extrapolation by resolving the inconsistency among the data
sets at higher energies Bair and Haas (1973); Harissopulos  _et al._ (2005);
Kellogg  _et al._ (1989). For the first time, we determines the ANC of the
threshold state using the direct measurement, fix the interference pattern and
determine the screening potential using R-matrix analysis.

###### Acknowledgements.

The authors would like to thank Tsinghua University and Prof. Jianping Cheng
for the support of Lab infrastructure and the China National Nuclear
Corporation for the finial support, CDEX and PandaX collaborations for their
kind helps, L. He, Y. Wang, P. Wang and J.K. Liu for their helps in setting up
the experimental terminal and Prof. Z.G. Wang for developing the instruments
which was used to develop the thick 13C targets. The authors also acknowledge
the assistance of Carl Brune in estimating neutron angular distribution
corrections. X.T. thanks A. Best and D. Rapagnani for their helpful
discussions throughout the project and providing their original data sets.
This work was supported by the National Natural Science Foundation of China
under Grants No. 11490564, the Strategic Priority Research Program of Chinese
Academy of Sciences, grant No. XDB34020000, the national key research and
development program (MOST 2016YFA0400501). S. W. was supported by the National
Natural Science Foundation of China under Grants No. 11775133, 11405096. B. G.
was supported by the National Natural Science Foundation of China under Grants
No. 12125509. W.P. L. was supported by the National Natural Science Foundation
of China under Grant No. 11805138. X. F. was supported by the National Natural
Science Foundation of China under Grants No. 11875329. R.J.D. acknowledges the
use of resources from the Notre Dame Center for Research Computing. R.J.D. and
M.W. were supported by the National Science Foundation through Grant No.
Phys-2011890, and the Joint Institute for Nuclear Astrophysics through Grant
No. PHY-1430152 (JINA Center for the Evolution of the Elements). MP
acknowledges support to NuGrid from STFC (through the University of Hull’s
Consolidated Grant ST/R000840/1) and ongoing access to viper, the University
of Hull High Performance Computing Facility. MP thanks the support by the ERC
Consolidator Grant (Hungary) funding scheme (Project RADIOSTAR, G.A. n.
724560), the ChETEC COST Action (CA16117), supported by the European
Cooperation in Science and Technology, the IReNA network supported by NSF
AccelNet, and the European Union ChETEC-INFRA (project no. 101008324).

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