# Thermonuclear 17O($n$,$\gamma$)18O reaction rate and its astrophysical
implications

Li-Yong Zhang11affiliationmark:  22affiliationmark:  , Jian-Jun
He11affiliationmark:  22affiliationmark:  , Motohiko
Kusakabe33affiliationmark:  , Zhen-Yu He33affiliationmark:  , Toshitaka
Kajino33affiliationmark:  44affiliationmark:  55affiliationmark:
11affiliationmark:  Key Laboratory of Beam Technology of Ministry of
Education, College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China  22affiliationmark:  Beijing Radiation
Center, Beijing 100875, China  33affiliationmark:  School of Physics, and
International Research Center for Big-Bang Cosmology and Element Genesis,
Beihang University, Beijing 100191, China  44affiliationmark:  Department of
Astronomy, School of Science, the University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo, 113-0033, Japan  55affiliationmark:  National Astronomical Observatory
of Japan 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan  Corresponding author
emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A new thermonuclear 17O($n$,$\gamma$)18O rate is derived based on a complete
calculation of the direct-capture (DC) and resonant-capture contributions, for
a temperature region up to 2 GK of astrophysical interest. We have firstly
calculated the DC and subthreshold contributions in the energy region up to 1
MeV, and estimated the associated uncertainties by a Monte-Carlo approach. It
shows that the present rate is remarkably larger than that adopted in the JINA
REACLIB in the temperature region of 0.01 $\sim$ 2 GK, by up to a factor of
$\sim$80\. The astrophysical impacts of our rate have been examined in both
$s$-process and $r$-process models. In our main $s$-process model which
simulates flash-driven convective mixing in metal deficient asymptotic giant
branch stars, both 18O and 19F abundances in interpulse phases are enhanced
dramatically by factors of $\sim 20$–$40$ due to the new larger
17O($n$,$\gamma$)18O rate. It shows, however, that this reaction hardly
affects the weak $s$-process in massive stars since the 17O abundance never
becomes significantly large in the massive stars. For the $r$-process
nucleosynthesis, we have studied impacts of our rate in both the collapsar and
neutron burst models, and found that the effect can be neglected, although an
interesting “loophole” effect is found owing to the enhanced new rate, which
significantly changes the final nuclear abundances if fission recycling is not
involved in the model, however, these significant differences are almost
completely washed out if the fission recycling is considered.

nuclear reactions, nucleosynthesis, abundances — stars: abundances — stars:
AGB and post-AGB — stars: massive

††software: RADCAP (Bertulani, 2003), MESA (Paxton et al., 2011), NucNet Tools
(Meyer, 2012)

##  1 Introduction

In He shell flash models, both 16O and 17O are on the pathway to heavier
elements in the nucleosynthesis process, and the nuclear reaction
17O($n$,$\gamma$)18O is expected to play a crucial role. Yamamoto et al.
(2010) studied this reaction and demonstrated how the 17O($n$,$\gamma$)18O
rate changes the synthesis paths to resultant heavier elements. Because there
were no experimental data available for this ($n$,$\gamma$) cross section at
that time, they assumed two limits for the rate at $kT$ = 30 keV, i.e., the
lower limit (Data-A) of $N_{\mathrm{A}}$$\langle\sigma v\rangle$ =
1.3$\times$103 cm3s-1mol-1 and the upper limit (Data-B) of
$N_{\mathrm{A}}$$\langle\sigma v\rangle$ = 1.5$\times$106 cm3s-1mol-1, and
also assumed the constant reaction rate w.r.t. temperature in their
calculations. It was shown that the abundances predicted for 5 $<Z<$ 16
elements, the number abundance of neutron, as well as the synthesis pathway
from C to Ne, can be largely affected by such rate changes. In fact, this
lower limit corresponds to the 18O($n$,$\gamma$)19O rate at $kT$ = 30 keV, and
the upper limit corresponds to that compiled in the Brussels Nuclear Library
for Astrophysics Applications (BRUSLIB)111http://www.astro.ulb.ac.be/bruslib/
(Goriely & Khan, 2002; Arnould & Goriely, 2006; Xu et al., 2013). Later on,
Nishimura et al. (2009) studied the role of 17O neutron capture reactions in
the metal-free and extremely metal-poor asymptotic giant branch (AGB) stars,
by assuming the 17O($n$,$\gamma$)18O rate equal to the 16O($n$,$\gamma$)17O
rate. The orders of magnitude uncertainty previously assumed for the
17O($n$,$\gamma$)18O rate is indeed too large to constrain the stellar
evolution models.

Massive stars with initial mass $M>8M_{\sun}$ experience the weak slow neutron
capture process ($s$-process) during the He- and C-burning stages (Lamb et
al., 1977; Raiteri et al., 1991, 1993; The et al., 2007). 16O is one of the
most important neutron poisons in the weak $s$-process (e.g., Mohr et al.,
2016), and 17O nuclei are produced via the reaction 16O($n$,$\gamma$)17O. This
reaction significantly consumes neutrons produced via 22Ne($\alpha$,$n$)25Mg,
and abundances of light nuclei C to Mg evolve sensitively to the reaction rate
in the weak $s$-process (He et al., 2020). The subsequent reaction
17O($n$,$\gamma$)18O on the product 17O is also involved in a reaction network
of the light nuclei that determine the neutron budget during the weak
$s$-process. In addition, this neutron capture reaction is also involved bf in
the main $s$-process nucleosynthesis in AGB stars (Käppeler et al., 2011) and
rapid neutron capture process ($r$-process) nucleosynthesis (Rauscher et al.,
1994). Therefore, a more reliable rate for the 17O($n$,$\gamma$)18O reaction
is highly desirable for the nuclear astrophysics community.

About twenty years ago, Koehler & Graff (1991) derived an analytic reaction
rate formula for the 17O($n$,$\gamma$)18O reaction. It was based on a thermal
($n$,$\gamma$) cross section of $\sigma_{\mathrm{th}}$ = (0.538$\pm$0.065) mb
and two known resonances at $E_{\mathrm{R}}$ = 0.169, 0.238 MeV (in the
center-of-mass frame). This experimental thermal $\sigma_{\mathrm{th}}$ value
was originally reported by Lone et al. (1979) in a Conference proceedings and
later compiled into a book by Mughabghab et al. (1981). This reaction rate was
utilized in the study of heavy element production in inhomogeneous cosmologies
(Rauscher et al., 1994), and referred to as _bb92_ in the JINA
REACLIB222http://groups.nscl.msu.edu/jina/reaclib/db (Cyburt et al., 2010).
Recently, a more precise value of $\sigma_{\mathrm{th}}$ = (0.67$\pm$0.07) mb
has been obtained experimentally (Firestone & Revay, 2016). Pritychenko et al.
(2010) calculated the Maxwellian-averaged cross sections (MACSs) and
astrophysical reaction rates using the evaluated nuclear reaction data
libraries, where the MACSs and rates of 17O($n$,$\gamma$)18O were listed in
tables at 15 data points in an energy/temperature range of $kT$ = 1 $\sim$
1000 keV, with ENDF/B-VII.0 (Chadwick et al., 2006) & B-VI.8 (CSEWG-
Collaboration, 2020) and JEFF-3.1 (Jacqmin et al., 2002) libraries,
respectively. In fact, these three libraries gave the exactly same results.
Furthermore, in BRUSLIB, the thermonuclear reaction rates for nuclei with $Z$
= 8–109 were estimated using the Hauser-Feshbach statistical model. However,
it has been shown that this statistical model is highly applicable to heavy
nuclei which have large nuclear level densities at excitation energies of a
few MeV. Actually, overestimations in the BRUSLIB theoretical values can be
seen when the mass number decreases below $A<$ 50\. For instance, the
17O($n$,$\gamma$)18O rates estimated in BRUSLIB is about two orders of
magnitude larger than those rates mentioned above.

In this work, we have obtained the thermonuclear 17O($n$,$\gamma$)18O rate
based on theoretical calculation of the direct capture (DC) as well as the
resonance contributions, for a temperature up to $T_{9}$ = 2 of general
astrophysical interest. Three rates based on the recent evaluated nuclear
reaction data libraries are also calculated and compared. It shows
significantly large discrepancies between different rates. For the higher
temperature region of $T_{9}$ = 2 $\sim$ 10, the 17O($n$,$\gamma$)18O rate is
calculated by using the high-energy evaluated data in ENDF/B-VIII with a
normalized factor of 1.45. The impacts of our new rate have been investigated
in stellar $s$-process and $r$-process models, and many interesting characters
have been revealed for the present rate.

##  2 Present rates

To calculate the thermonuclear reaction rate, we have adopted the method
described by Pritychenko et al. (2010), which will be introduced simply as
follows. The MACS is defined as

$\displaystyle\sigma^{\mathrm{MACS}}(kT)=\frac{\langle\sigma
v\rangle}{v_{T}},$ (1)

where $v$ is the relative velocity of neutron with respect to a target
nuclide, and $v_{T}$ is the mean thermal velocity given by

$\displaystyle v_{T}=\sqrt{\frac{2kT}{\mu}},$ (2)

where $\mu$ is the reduced mass of the target nucleus and neutron. Here,
nuclear masses for neutron and 17O are $m_{n}$ = 1.008665 u and
$m\mathrm{(^{17}O)}$ = 16.994743 u (with u in the atomic mass unit), based on
AME2016 (Wang et al., 2017), respectively. The MACS can be calculated as

$\displaystyle\sigma^{\mathrm{MACS}}(kT)=\frac{2}{\sqrt{\pi}}\times\frac{1}{(kT)^{2}}\int^{\infty}_{0}\sigma(E)E\mathrm{exp}\left(-\frac{E}{kT}\right)dE,$
(3)

where $E$ is the energy of relative motion of the neutron with respect to the
target, i.e., the center-of-mass energy $E_{\mathrm{c.m.}}$. It should be
noted that a format of neutron energy $E_{n}$ $v.s.$ $\sigma$ is adopted in
all the evaluated nuclear reaction data libraries, and hence one should be
careful in using Eq. 3. The astrophysical reaction rate, $R$, is defined as
$R=N_{A}\langle\sigma v\rangle$, with $N_{A}$ the Avogadro number. Finally,
the rate can be calculated by

$\displaystyle R(T_{9})=10^{-24}N_{A}\sigma^{\mathrm{MACS}}(kT)v_{T},$ (4)

with $R$ in units of cm3s-1mol-1, $v_{T}$ in units of cm/s, and
$\sigma^{\mathrm{MACS}}$ in units of barn. The temperature $kT$ in units of
energy (e.g., MeV) is related to that in Kelvin as $T_{9}$ = 11.605 $kT$
(here, $T_{9}$ = 1 means $T$ = 109 K). Therefore, once we obtain the cross
section data in the energy region of astrophysical interest, the rates can be
numerically calculated by using the equations introduced above.

In this work, we have calculated the cross section of 17O($n$,$\gamma$)18O
reaction in an $E_{\mathrm{c.m.}}$ energy region up to 1 MeV, which are
sufficient to account for thermonuclear reaction rate up to a temperature of
$T_{9}$ = 2. The total neutron capture rate is the sum of resonant- and
direct-capture (DC) on the ground state and thermally excited states in target
nucleus, weighted with their individual population factors (Rolfs & Rodney,
1988; Lam et al., 2016). In the present case, such a thermal excitation effect
can be neglected for the temperature below $T_{9}$ = 2 because the first-
excited state in target nucleus of 17O is quite high ($E_{x}$ = 870.73 keV).
Therefore, we calculate here only the contributions of the ground-state
capture.

###  2.1 DC contribution

We have calculated the DC cross sections of 17O($n$,$\gamma$)18O with a RADCAP
code (Bertulani, 2003). The Woods-Saxon nuclear potential (central + spin
orbit) and a Coulomb potential of uniform-charge distribution were utilized in
the calculation. The nuclear central potential $V_{0}$ was determined by
matching the bound-state energies. The spectroscopic factors ($C^{2}S$) and
the orbital ($nlj$) information were taken from Li et al. (1976). The optical-
potential parameters adopted from Huang et al. (2010) are:
$R_{0}=R_{\mathrm{s.o.}}=R_{C}=r_{0}(1+A_{T})^{1/3}$ (with $r_{0}$ = 1.25 fm,
$A_{T}$ = 17 for the present 17O case), $a_{0}=a_{\mathrm{s.o.}}=0.65$ fm,
with a spin-orbit potential depth of $V_{\mathrm{s.o.}}=-10$ MeV. Here,
$R_{0}$, $R_{\mathrm{s.o.}}$, and $R_{C}$ are the radii of central potential,
the spin-orbit potential and the Coulomb potential, respectively; $a_{0}$ and
$a_{\mathrm{s.o.}}$ are the corresponding diffuseness parameters in the
central and spin-orbit potentials, respectively. Similarly to He et al.
(2014), the parameter ($r_{0}$, $a$ and $V_{\mathrm{s.o.}}$) dependence on
cross sections has been studied, and the associated uncertainties are
estimated to be no more than 50% in the energy region studied.

Figure 1 shows the level scheme of the 17O($n$,$\gamma$)18O reaction. We have
calculated twelve DC captures into the ground state and the excited states up
to $E_{x}$ = 6.351 MeV, if $C^{2}S$ factors are available. These captures
include three $s$-wave, eight $p$-wave and one $d$-wave, which are indicated
by three different colored arrows. The calculation results are shown in Fig.
2, where the major contributions from three $s$\- and six $p$-wave captures
are shown separately. The remaining one $d$\- and two $p$-wave contributions
are negligibly small and not shown in the figure. It can be seen that $s$\-
and $d$-wave captures contribute equivalently to the cross sections around 3
keV, but $s$\- and $d$-wave become dominate below and above this energy,
respectively. It should be noted that there are no any $d$-wave calculations
available in the literature. Therefore, we have made the $d$-wave calculations
for the first time, and found that its contribution can not be neglected.

Figure 1:  Level scheme of the 17O($n$,$\gamma$)18O reaction. Three $s$-wave,
eight $p$-wave and one $d$-wave captures are indicated by different colored
arrows; two resonances at $E_{x}$ = 8.213 and 8.282 MeV (i.e.,
$E_{\mathrm{R}}$ = 168 and 237 keV), and one subthreshold state at $E_{x}$ =
8.038 MeV (i.e., $E_{\mathrm{R}}$ = $-$7.6 keV), are marked in red font. The
location of $\alpha$ threshold is also shown. The data are adopted from Tilley
et al. (1995). Figure 2:  Major DC contributions calculated for the
17O($n$,$\gamma$)18O reaction by a RADCAP code. Thereinto, $s$-wave and
$p$-wave are categorized.

It is well-known that the cross section of the $s$-wave neutron capture
follows the 1/$v$ law. Accordingly, we predicted a thermal value of
$\sigma_{\mathrm{th}}$ = 1.91 mb. Compared to the recent experimental value of
$\sigma_{\mathrm{th}}$ = (0.67$\pm$0.07) mb (Firestone & Revay, 2016), our
predicted one is about a factor of 3 larger. This can be explained by the
possible uncertainties in the experimental $C^{2}S$ factors derived in Li et
al. (1976), where no errors were quoted explicitly. All experimental $C^{2}S$
factors for three $s$-wave captured states (at $E_{x}$ = 4.456, 6.198, 6.341
MeV) are 0.03, a very small quantity, and hence their errors may be large. The
weak-coupling model predicted $C^{2}S$ = 2$\times$10-4 and 0.013, respectively
for the $E_{x}$ = 4.456 and 6.341 MeV states, which are about a factor of 3 or
more smaller than the experimental ones. If we use weak-coupling predictions
for these three states, a value of $\sigma_{\mathrm{th}}$ = 0.58 mb is then
obtained, which is consistent with the experimental result. Therefore, we
normalized the $s$-wave results to the experimental value of
$\sigma_{\mathrm{th}}$ = 0.67 mb at the thermal energy. Figure 3 shows the
summed $s$-wave (normalized) and $p$-wave contributions, as well as the total
DC contributions, respectively.

Figure 3:  Major DC contributions calculated for the 17O($n$,$\gamma$)18O
reaction by a RADCAP code. Thereinto, $s$-wave and $p$-wave are categorized,
and the calculated $s$-wave contribution is normalized to the experimental
value of $\sigma_{\mathrm{th}}$ = 0.67 mb. The corresponding uncertainties
indicated as the error bands are calculated by a Monte-Carlo method, see text
for details.

###  2.2 Resonant contribution

A general Breit-Wigner formula is utilized to calculate cross sections of the
resonances (Rolfs & Rodney, 1988; Iliadis, 2007),

$\displaystyle\sigma(E)=\pi\lambdabar^{2}\omega\frac{\Gamma_{n}(E)\Gamma_{\gamma}(E)}{(E-E_{\mathrm{R}})^{2}+[\Gamma_{\mathrm{tot}}(E)/2]^{2}},$
(5)

with the de Broglie wavelength $\lambda$ divided by 2$\pi$, i.e.,
$\lambdabar=\hbar/\sqrt{2\mu E}$, and the statistical factor
$\omega=(2J_{\mathrm{R}}+1)/[(2J_{n}+1)(2J_{\mathrm{{}^{17}O}}+1)]$. Here
$J_{\mathrm{R}}$ is the spin of the resonance in the compound nucleus 18O, and
the spins of projectile and target nuclei are $J_{n}$ = 1/2,
$J_{\mathrm{{}^{17}O}}$ = 5/2, respectively. All energies and widths are in
the center-of-mass system, the partial widths $\Gamma_{n}(E)$ and
$\Gamma_{\gamma}(E)$, and thus the total width $\Gamma_{\mathrm{tot}}(E)$, are
all energy dependent. In this work, one subthreshold state and two resonances
have been considered, and their resonance parameters are summarized in Table
1.

The cross section at the resonant energy $E_{\mathrm{R}}$ is
$\sigma(E_{\mathrm{R}})=\sigma(E=E_{\mathrm{R}})=4\pi\lambdabar_{\mathrm{R}}^{2}\omega\Gamma_{n}\Gamma_{\gamma}/\Gamma_{\mathrm{tot}}^{2}$,
with the total width
$\Gamma_{\mathrm{R}}=\Gamma_{\mathrm{tot}}(E=E_{\mathrm{R}})$, and thus the
resonance Eq. 5 can be expressed as,

$\displaystyle\sigma(E)=\sigma(E_{\mathrm{R}})\frac{E_{\mathrm{R}}}{E}\frac{\Gamma_{n}(E)}{\Gamma_{n}(E_{\mathrm{R}})}\frac{\Gamma_{\gamma}(E)}{\Gamma_{\gamma}(E_{\mathrm{R}})}\frac{(\Gamma_{\mathrm{R}}/2)^{2}}{(E-E_{\mathrm{R}})^{2}+[\Gamma_{\mathrm{tot}}(E)/2]^{2}}.$
(6)

Here, the total width includes three terms, i.e.,
$\Gamma_{\mathrm{tot}}(E)=\Gamma_{\alpha}(E)+\Gamma_{n}(E)+\Gamma_{\gamma}(E)$.
The energy dependence of these three terms will be explained below: (1) for
the dominant alpha width, the ratio can be calculated by
$\Gamma_{\alpha}(E)/\Gamma_{\alpha}(E_{\mathrm{R}})=P_{\ell_{\alpha}}(E)/P_{\ell_{\alpha}}(E_{\mathrm{R}})$
with the penetration factor $P_{\ell_{\alpha}}$ for the $\alpha$ emission; (2)
for the low energy neutrons, the ratio can be calculated by
$\Gamma_{n}(E)/\Gamma_{n}(E_{\mathrm{R}})=(E/E_{\mathrm{R}})^{\ell_{n}+1/2}$;
(3) for the gamma width, the ratio can be calculated by
$\Gamma_{\gamma}(E)/\Gamma_{\gamma}(E_{\mathrm{R}})=[(E+Q-E_{f})/(E_{\mathrm{R}}+Q-E_{f})]^{2L+1}$,
where $Q$ is the $Q$ value of ($n$,$\gamma$) reaction, and $E_{f}$ is the
energy of the final state in the compound nucleus. $L$ is the multipolarity of
the electromagnetic transition, $L$ = 1 for the major transitions of the
present $E1$ or $M1$ case. The calculations show that the energy dependence of
$\Gamma_{\alpha}$ and $\Gamma_{\gamma}$ is not very large, while that of
$\Gamma_{n}$ is quite significant. Anyway, we have included all the energy
dependence in the calculations. It is worthy of noting that we have derived
the following equation that is used for the subthreshold state (with a
negative value of $E_{\mathrm{R}}$) calculation,

$\displaystyle\sigma(E)=\sigma(|E_{\mathrm{R}}|)\frac{|E_{\mathrm{R}}|}{E}\frac{\Gamma_{n}(E)}{\Gamma_{n}(|E_{\mathrm{R}}|)}\frac{\Gamma_{\gamma}(E)}{\Gamma_{\gamma}(|E_{\mathrm{R}}|)}\frac{(2E_{\mathrm{R}})^{2}+(\Gamma_{\mathrm{R}}/2)^{2}}{(E-E_{\mathrm{R}})^{2}+[\Gamma_{\mathrm{tot}}(E)/2]^{2}}.$
(7)

Table 1:  Relevant nuclear levels in the compound nucleus 18O, including one
subthreshold state and two resonances. Most of the parameters are adopted from
Schatz et al. (1993) unless otherwise specified. The reaction $Q$ value of
$S_{n}$ = 8.045 MeV is taken from AME2016 (Wang et al., 2017).

$E_{x}$ (keV) |  $E_{\mathrm{R}}$ (keV) | $J^{\pi}$ |  $\Gamma_{\mathrm{tot}}$ (eV) |  $\Gamma_{\alpha}$ (eV)a |  $\Gamma_{n}$ (eV)b |  $\Gamma_{\gamma}$ (eV)c
---|---|---|---|---|---|---
8037.8d |  $-$7.6$\pm$0.7 |  1- |  2400e |  2300 [$\ell_{\alpha}$ = 1] |  98 [$\ell_{n}$ = 1] |  1.15$\pm$0.15 [E1]
8213 |  168$\pm$4 |  2+ |  1280$\pm$1000 |  1200$\pm$800 [$\ell_{\alpha}$ = 2] |  14$\pm$9 [$\ell_{n}$ = 0] |  0.41$\pm$0.09 [E1+M1]
8282 |  237$\pm$3 |  3- |  8000$\pm$1000 |  6900$\pm$400 [$\ell_{\alpha}$ = 3] |  830$\pm$490 [$\ell_{n}$ = 1] |  0.49$\pm$0.13 [E1]

a: The transferred $\ell_{\alpha}$ value is listed in the square brackets; b:
The transferred $\ell_{n}$ value is listed in the square brackets; c: The
major $\gamma$-ray multipolarity is listed in the square brackets; d: The
uncertainties in $\Gamma_{\mathrm{tot}}$, $\Gamma_{\alpha}$ and $\Gamma_{n}$
(eV) are estimated to be 20% for this subthreshold state; e: An upper limit of
2500 eV was set by Ajzenberg-Selove (1987); Tilley et al. (1995).

Figure 3 shows the contributions owing to the two resonances and one
subthreshold state listed in Table 1. We have made the first calculation for
the subthreshold state at $E_{\mathrm{R}}$ = $-$7.6 keV, and found that its
contribution dominates the total cross sections in the energy range of
$\sim$0.1–20 keV.

###  2.3 Total cross section

The total cross sections have been calculated by summing the contributions of
DC and resonances (including subthreshold state) discussed above. The present
total cross section is shown as a black solid line in Fig. 3, where respective
contributions are illustrated separately. Figure 4 shows a comparison between
present results with those evaluated in various nuclear reaction data
libraries. The typical libraries are, ENDF/B-VIII.0 in USA (Brown et al.,
2018), JEFF3.3 in Europe (Koning et al., 2011), ROSFOND2010 in Russia
(Zabrodskaya et al., 2007), where the data are available from the website:
https://www.nndc.bnl.gov/exfor/endf00.jsp. We find that the evaluated
17O($n$,$\gamma$)18O cross section data are almost the same among various
ENDF/B series libraries (i.e., B-V, B-VI, B-VII, B-VIII). A very large thermal
value of $\sigma_{\mathrm{th}}$ = 3.83 mb was given in various ENDF/B
libraries, whose header information in the 17O($n$,$\gamma$)18O data file
stated as “THERMAL CROSS SECTIONS N,GAMMA = 3.8 MB+/-20PC CALCULATED BY
S.F.MUGHABGHAB (MU78) USING METHOD OF LANE + LYNN (SEE DOCUMENT)”. The cited
reference MU78 was a BNL report (Mughabghab, 1978), that was later published
as Mughabghab (1979) in which no any information about the
17O($n$,$\gamma$)18O reaction was discussed. However, a value of
$\sigma_{\mathrm{th}}$ = 0.38 mb was given in the book of Mughabghab et al.
(1981). As explained by Pritychenko & Mughaghab (2012), “The ENDF/B-VII.1
thermal capture cross section, 3.8 mb, for 17O is attributed to a misquoted
direct capture cross section calculated by Mughabghab et al. (1981)”. Now,
it’s clear that all ENDF/B series gave a wrong $\sigma_{\mathrm{th}}$ value
due to this misquotation. Therefore, the thermal value and associated header
information should be corrected in the future evaluations. ROSFOND2010
normalized the thermal cross section to the old experimental value of 0.538 mb
(Lone et al., 1979) for the $s$-wave dominated energy region below $E_{n}\sim$
0.1 MeV, and adopted the high-energy data as in the ENDF/B libraries. The
header of ROSFOND2010 data file for 17O($n$,$\gamma$)18O also included the
same information of the thermal cross section as mentioned above, although it
adopted an experimental thermal value. JEFF3.3, which is based on a software
system built round the nuclear model code TALYS (Koning & Rochman, 2012), gave
a value of $\sigma_{\mathrm{th}}$ = 0.556 mb, and two resonant contributions
were also involved. It shows that the discrepancies between present results
and the evaluated ones are remarkable.

The associated uncertainties in the present total cross sections and the
reaction rates have been calculated by the Monte-Carlo techniques described by
Longland et al. (2010). Although we normalized our $s$-wave contribution to
the experimental thermal value of $\sigma_{\mathrm{th}}$ = (0.67$\pm$0.07) mb,
it deviates from the old value of (0.538$\pm$0.065) mb by about 20%, and thus
we estimated conservatively the uncertainties in the $s$-wave contribution to
be 20%. The uncertainties in the $p$-wave contribution is estimated to be 50%
as discussed in the previous DC section. The uncertainties in the resonance
contributions (including the subthreshold state) are determined by the
parameter uncertainties as listed in Table 1. The uncertainties of the present
results are shown as the colored error bands in Figs. 3–5.

Figure 4:  Cross sections of 17O($n$,$\gamma$)18O calculated in this study as
a function of incident neutron energy. The typical evaluated data are shown
for comparison. The uncertainties of the present cross section indicated as an
error band are calculated by a Monte-Carlo method.

###  2.4 Total reaction rate

Based on the presently calculated and evaluated cross sections shown in Fig.
4, we have numerically calculated the thermonuclear 17O($n$,$\gamma$)18O rates
using the approach discussed above. The reaction rates and the associated
ratios between the present results and others are shown in Fig. 5,
respectively. If the misquoted thermal value in ENDF/B-VIII is corrected, the
corresponding rate should be almost the same as that of ROSFOND2010.
Therefore, our new rate is remarkably (up to 2 orders of magnitude) larger
than those calculated by using the cross-section data evaluated in the current
ENDF/B, JEFF and ROSFOND libraries, in the temperature range up to $T_{9}$ =
2. However, we think that the BRUSLIB overestimated this rate significantly
because of the present light target system of 17O. The JINA _bb92_ rate, which
is the exact one derived by Koehler & Graff (1991), is also small, because the
DC $p$-wave capture and subthreshold state contributions were not taken into
account in their calculations. In the temperature region of $T_{9}$ = 0.1
$\sim$ 2, the present rate is about 35$\sim$80 times larger than the _bb92_
rate, which is frequently utilized in the nucleosynthesis network
calculations. The numerical values of the present reaction rate and the
associated lower and upper limits are listed in Table 2, where other rates are
also listed for comparison.

Figure 5:  Thermonuclear 17O($n$,$\gamma$)18O rates (in units of cm3s-1mol-1)
as a function of $T_{9}$: the upper panel for reaction rates, and the lower
panel for ratios between the present results and others. The uncertainties of
the present rates indicated as the error band are calculated by a Monte-Carlo
method.

For completeness, the 17O($n$,$\gamma$)18O rate in the high temperature region
of $T_{9}$ = 2 $\sim$ 10 has been treated below. As shown in Fig. 4, the
present cross sections are about 1.45 times smaller than those of ENDF/B-VIII
in the energy region of 0.5–1 MeV. Therefore we adopt the ENDF/B-VIII data in
the energy region of 1.0–20 MeV by dividing a normalization factor of 1.45.
The 17O($n$,$\gamma$)18O rate is then calculated numerically based on the
presently calculated cross section data ($\leq$1 MeV) and the normalized
ENDF/B-VIII data (1–20 MeV), by using the Monte-Carlo approach. The high
temperature rates are also listed in Table 2. The present 17O($n$,$\gamma$)18O
(Mean) rate can be parameterized by the standard format of Rauscher &
Thielemann (2000),

$\displaystyle N_{A}\langle\sigma v\rangle_{(n,\gamma)}$ $\displaystyle=$
$\displaystyle\mathrm{exp}(11.1633-\frac{0.0400988}{T_{9}}+\frac{4.56334}{T_{9}^{1/3}}-6.40963T_{9}^{1/3}+0.317629T_{9}-0.0163527T_{9}^{5/3}+4.25928\ln{T_{9}})$
(8) $\displaystyle+$
$\displaystyle\mathrm{exp}(55.7181-\frac{0.184283}{T_{9}}+\frac{17.6063}{T_{9}^{1/3}}-76.4751T_{9}^{1/3}+10.0633T_{9}-1.1705T_{9}^{5/3}+21.7074\ln{T_{9}})\,,$

with a fitting error of less than 0.2% over the temperature region of $T_{9}$
= 0.01 $\sim$ 10.

Table 2:  Presently derived thermonuclear 17O($n$,$\gamma$)18O rate, together with the lower and upper limits (in units of cm3s-1mol-1). The ENDF/B-VIII, JEFF3.3 and ROSFOND2010 rates are calculated based on the cross sections evaluated in the respective libraries, and the rates from JINA REACLIB and BURSLIB are also listed for comparison. $T_{9}$ | Present(Mean) | Lower limit | Upper limit | ENDF/B-VIII | JEFF3.3 | ROSFOND2010 | JINA REACLIB | BRUSLIB
---|---|---|---|---|---|---|---|---
0.01 | 1.56E+03 | 1.14E+03 | 2.18E+03 | 5.03E+02 | 5.24E+01 | 7.06E+01 | 7.12E+01 | 8.74E+04
0.02 | 2.31E+03 | 1.70E+03 | 3.18E+03 | 5.02E+02 | 4.13E+01 | 7.06E+01 | 7.12E+01 | 6.64E+04
0.03 | 2.77E+03 | 2.06E+03 | 3.77E+03 | 5.02E+02 | 3.43E+01 | 7.05E+01 | 7.12E+01 | 5.72E+04
0.04 | 3.08E+03 | 2.32E+03 | 4.15E+03 | 5.02E+02 | 2.94E+01 | 7.05E+01 | 7.12E+01 | 5.17E+04
0.05 | 3.31E+03 | 2.51E+03 | 4.42E+03 | 5.02E+02 | 2.57E+01 | 7.05E+01 | 7.12E+01 | 4.80E+04
0.06 | 3.50E+03 | 2.66E+03 | 4.65E+03 | 5.02E+02 | 2.29E+01 | 7.05E+01 | 7.12E+01 | 4.52E+04
0.07 | 3.66E+03 | 2.79E+03 | 4.83E+03 | 5.02E+02 | 2.07E+01 | 7.05E+01 | 7.12E+01 | 4.31E+04
0.08 | 3.80E+03 | 2.89E+03 | 4.98E+03 | 5.02E+02 | 1.88E+01 | 7.05E+01 | 7.12E+01 | 4.14E+04
0.09 | 3.92E+03 | 2.98E+03 | 5.11E+03 | 5.02E+02 | 1.73E+01 | 7.05E+01 | 7.12E+01 | 4.00E+04
0.10 | 4.04E+03 | 3.06E+03 | 5.23E+03 | 5.02E+02 | 1.61E+01 | 7.05E+01 | 7.12E+01 | 3.88E+04
0.11 | 4.15E+03 | 3.13E+03 | 5.37E+03 | 5.02E+02 | 1.50E+01 | 7.05E+01 | 7.12E+01 | 3.78E+04
0.12 | 4.26E+03 | 3.19E+03 | 5.49E+03 | 5.02E+02 | 1.42E+01 | 7.05E+01 | 7.12E+01 | 3.69E+04
0.13 | 4.36E+03 | 3.25E+03 | 5.63E+03 | 5.02E+02 | 1.35E+01 | 7.05E+01 | 7.12E+01 | 3.61E+04
0.14 | 4.46E+03 | 3.31E+03 | 5.76E+03 | 5.02E+02 | 1.31E+01 | 7.05E+01 | 7.12E+01 | 3.54E+04
0.15 | 4.56E+03 | 3.36E+03 | 5.89E+03 | 5.03E+02 | 1.31E+01 | 7.06E+01 | 7.12E+01 | 3.47E+04
0.16 | 4.65E+03 | 3.41E+03 | 6.02E+03 | 5.03E+02 | 1.36E+01 | 7.07E+01 | 7.12E+01 | 3.42E+04
0.18 | 4.85E+03 | 3.51E+03 | 6.29E+03 | 5.06E+02 | 1.76E+01 | 7.11E+01 | 7.13E+01 | 3.31E+04
0.20 | 5.04E+03 | 3.59E+03 | 6.58E+03 | 5.10E+02 | 2.76E+01 | 7.18E+01 | 7.15E+01 | 3.23E+04
0.25 | 5.53E+03 | 3.81E+03 | 7.32E+03 | 5.37E+02 | 1.01E+02 | 7.68E+01 | 7.28E+01 | 3.06E+04
0.30 | 6.05E+03 | 4.03E+03 | 8.09E+03 | 5.98E+02 | 2.75E+02 | 9.13E+01 | 7.71E+01 | 2.93E+04
0.35 | 6.59E+03 | 4.27E+03 | 8.90E+03 | 7.05E+02 | 5.70E+02 | 1.25E+02 | 8.59E+01 | 2.82E+04
0.40 | 7.14E+03 | 4.51E+03 | 9.76E+03 | 8.69E+02 | 9.74E+02 | 1.89E+02 | 1.00E+02 | 2.74E+04
0.45 | 7.71E+03 | 4.78E+03 | 1.07E+04 | 1.10E+03 | 1.46E+03 | 2.97E+02 | 1.21E+02 | 2.66E+04
0.50 | 8.30E+03 | 5.06E+03 | 1.16E+04 | 1.39E+03 | 2.01E+03 | 4.60E+02 | 1.46E+02 | 2.60E+04
0.60 | 9.55E+03 | 5.64E+03 | 1.35E+04 | 2.19E+03 | 3.16E+03 | 9.76E+02 | 2.05E+02 | 2.49E+04
0.70 | 1.08E+04 | 6.26E+03 | 1.54E+04 | 3.25E+03 | 4.26E+03 | 1.77E+03 | 2.70E+02 | 2.40E+04
0.80 | 1.21E+04 | 6.89E+03 | 1.73E+04 | 4.56E+03 | 5.24E+03 | 2.84E+03 | 3.31E+02 | 2.32E+04
0.90 | 1.34E+04 | 7.52E+03 | 1.93E+04 | 6.07E+03 | 6.05E+03 | 4.16E+03 | 3.86E+02 | 2.25E+04
1.00 | 1.47E+04 | 8.15E+03 | 2.12E+04 | 7.76E+03 | 6.72E+03 | 5.70E+03 | 4.32E+02 | 2.19E+04
1.25 | 1.80E+04 | 9.80E+03 | 2.61E+04 | 1.26E+04 | 7.80E+03 | 1.03E+04 | 5.12E+02 | 2.06E+04
1.50 | 2.12E+04 | 1.14E+04 | 3.10E+04 | 1.79E+04 | 8.30E+03 | 1.55E+04 | 5.50E+02 | 1.96E+04
1.75 | 2.45E+04 | 1.31E+04 | 3.58E+04 | 2.34E+04 | 8.46E+03 | 2.10E+04 | 5.62E+02 | 1.89E+04
2.00 | 2.77E+04 | 1.48E+04 | 4.05E+04 | 2.89E+04 | 8.42E+03 | 2.67E+04 | 5.58E+02 | 1.82E+04
2.50 | 3.40E+04 | 1.85E+04 | 4.94E+04 | 3.99E+04 | 8.08E+03 | 3.79E+04 | 5.27E+02 | 1.74E+04
3.00 | 4.02E+04 | 2.27E+04 | 5.76E+04 | 5.05E+04 | 7.64E+03 | 4.87E+04 | 4.86E+02 | 1.70E+04
4.00 | 5.23E+04 | 3.25E+04 | 7.21E+04 | 7.06E+04 | 6.81E+03 | 6.91E+04 | 4.09E+02 | 1.73E+04
5.00 | 6.41E+04 | 4.34E+04 | 8.46E+04 | 8.94E+04 | 6.20E+03 | 8.83E+04 | 3.48E+02 | 1.93E+04
6.00 | 7.58E+04 | 5.46E+04 | 9.64E+04 | 1.07E+05 | 5.79E+03 | 1.06E+05 | 3.01E+02 | 2.32E+04
7.00 | 8.70E+04 | 6.54E+04 | 1.08E+05 | 1.25E+05 | 5.55E+03 | 1.24E+05 | 2.66E+02 | 3.02E+04
8.00 | 9.83E+04 | 7.59E+04 | 1.19E+05 | 1.41E+05 | 5.43E+03 | 1.41E+05 | 2.39E+02 | 4.24E+04
9.00 | 1.09E+05 | 8.61E+04 | 1.31E+05 | 1.58E+05 | 5.43E+03 | 1.57E+05 | 2.17E+02 | 6.39E+04
10.00 | 1.20E+05 | 9.59E+04 | 1.43E+05 | 1.73E+05 | 5.55E+03 | 1.73E+05 | 1.99E+02 | 1.03E+05

##  3 Astrophysical implications

The main origin of the trans-iron elements is believed to be either
$s$-process or $r$-process which proceeds under the neutron-rich conditions.
In this section, we examine the impacts of our new 17O($n$,$\gamma$)18O rate
on the both nucleosynthetic processes. The stellar models and the calculated
results are described in the following subsections in details. Thereinto, the
previous 17O($n$,$\gamma$)18O rate used in the models refers to the _bb92_
rate compiled in the JINA REACLIB.

###  3.1 s-process

The astrophysical site for the production of the $s$-process elements whose
masses are lighter than $A<$ 90 is most likely the massive star, and this
nucleosynthetic process is called weak $s$-process. On the other hand, those
of heavier mass $A>$ 90 is believed to be produced in the light-to-
intermediate mass AGB stars, which is called the main $s$-process. (Käppeler
et al., 2011).

####  3.1.1 Weak $s$-process in massive stars

In this subsection we study the impact of the present 17O($n$,$\gamma$)18O
rate on the evolution of light element abundances in massive stars. We adopt a
multi-zone nucleosynthesis calculation (He et al., 2020) for a model star with
$M$ = 25 $M_{\sun}$ from the Modules for Experiments in Stellar Astrophysics
(MESA) (Paxton et al., 2011, 2015, 2018, 2019). There is no discernable
difference in nuclear abundances when a result for the new rate is compared
with that of the JINA rate for both core He and shell C burning stages (see
Figures 6 and 7 in He et al. (2020)). However, the change in the reaction rate
leads to a small change of 18O abundance which is maximized $\sim 10$ yr
before the onset of Si burning. Figure 6 shows ratios of mole fractions
$Y_{i}$ of $i$ = 17,18O at the Lagrangian mass coordinate $M_{r}=2M_{\sun}$,
for example, as a function of the time to the start of core Si burning, i.e.,
$\Delta t\equiv t_{\rm Si}-t$. When the new rate is utilized, the 18O
abundance increases by up to $\sim 4$% at $\Delta t\sim 10$ yr. At $\Delta
t\sim 10^{4}$ yr in the last stage of He burning, a neutron exposure from
22Ne($\alpha$,$n$) enhances the 18O abundances via the reaction
17O($n$,$\gamma$)18O. In this epoch, the 18O production dominantly proceeds
via 18F($\beta^{+}$)18O, 14C($\alpha$,$\gamma$)18O, and 17O($n$,$\gamma$)18O.
Then, an increase of the last reaction rate results in a slight increase of
the 18O abundance. At $\Delta t\sim 10$ yr in the C burning stage, the neutron
exposure is again gradually increasing due to $\alpha$ particles provided from
the C+C fusion. Also in this epoch, the 18O production occurs through
18F($\beta^{+}$)18O, 14C($\alpha$,$\gamma$)18O, and 17O($n$,$\gamma$)18O, and
an increase of 18O abundance results again. At $\Delta t\lesssim 10$ yr, the
rate of the reaction 17O($n$,$\gamma$)18O is significantly smaller than the
other 2 reactions for the 18O production. Then, the effect of changing the
17O($n$,$\gamma$)18O rate on the 18O abundance becomes smaller. We note that
the effect of increasing 18O abundance is cancelled at high temperatures since
the other pathway of 17O($n$,$\alpha$)14C($\alpha$,$\gamma$)18O also
contributes to the 18O yield when enough $\alpha$ particles are supplied.
Since the 17O abundance never becomes significantly large in non-rotating
massive stars, the neutron capture reaction 17O($n$,$\gamma$)18O hardly
affects the weak $s$-process.

Figure 7 shows rates for main production and destruction reactions of 18O as a
function of $t_{\rm Si}-t$ during the C burning. The new rate of the
17O($n$,$\gamma$)18O reaction results in a 18O production via the reaction
much more efficient than the old rate does. However, because the $\beta$-decay
of 18F and 14C($\alpha$,$\gamma$)18O reaction are predominant in the 18O
production, the sensitivity of 18O abundance to the 17O($n$,$\gamma$)18O
reaction rate is still small. It can be seen that three reactions of
18O($p$,$\alpha$)15N, 14C($\alpha$,$\gamma$)18O and 18O($\alpha$,$n$)21Ne can
most effectively influence the 18O abundance, except for the 18F
$\beta$-decay. At the same time, the impact of our new rate on heavier nuclear
abundances involved in the present $s$-process network has been investigated,
and such impacts are quite small, only up to a level of
${\mathcal{O}}(10^{-3})$.

Figure 6:  Deviations from unity of ratios of mole fractions of 17,18O in the
cases of new and JINA rates as a function of the time to the start of core Si
burning, i.e., $t_{\rm Si}-t$ during the shell C burning in a star with
initial mass $M=25M_{\sun}$ at $M_{r}$ = 2 $M_{\odot}$. See text for details.
Figure 7:  Reaction rates for production and destruction of 18O as a function
of $t_{\rm Si}-t$ during the C burning in the same stellar model as in Fig. 6.

####  3.1.2 Main $s$-process in AGB stars

Utilizing a similar reaction network where the new and old rates of the
reaction 17O($n$,$\gamma$)18O are adopted, we investigate effects of new rate
on elemental abundances during the main $s$-process in metal-poor stars in
this subsection. In our present model, we set up environmental conditions for
nucleosynthesis in C- and $s$-enhanced metal-poor stars (MPSs), i.e., LP625-44
and CS31062-012 (equivalent to LP706-7). Observational data on abundances of
$s$-elements are adopted from Aoki et al. (2006, 2001) for LP625-44 with [C/H]
$=-0.43$333[A/B]
$\equiv\log[(n_{\mathrm{A}}/n_{\mathrm{B}})/(n_{\mathrm{A}}/n_{\mathrm{B}})_{\sun}]$,
where $n_{i}$ is the number density of element $i$ and the mark $\sun$
indicates the solar value. and [Fe/H] $=-2.70$, and Aoki et al. (2008, 2001)
for CS31062-012 with [C/H] $=-0.43$ and [Fe/H] = $-2.74$. Theoretical
calculations of the main $s$-process are performed assuming repeated neutron
capture episodes during interpulse phases of AGB stars (e.g., Iwamoto et al.,
2004). Solar abundances are taken from Lodders et al. (2009). For these two
MPSs, a parametric study (Aoki et al., 2001) based upon a simple neutron
injection model (Howard et al., 1986) demonstrates that the final yields only
depend on the neutron exposure, i.e., $\tau_{\mathrm{exp}}=\int n_{n}v_{T}dt$,
where $n_{n}$ is the neutron number density, and $v_{T}$ is the thermal
neutron velocity at temperature $T$. Following this suggestion, we vary the
neutron exposure assuming some specific setting in the present consistent
network calculations.

The current $s$-process model assumes repeated episodes of flash-driven
convective mixing (Iwamoto et al., 2004) followed by an interpulse phase that
occurs under a radiative $s$-process condition with fixed temperature and
density of $T_{9}=0.1$ and $\rho=500$ g cm-3 (Iben, 1975). It is assumed that
10% of the material that has experienced the $s$-process in $n$-th interpulse
phase undergoes the $s$-process in the $(n+1)$-th interpulse phase, i.e., the
overlap fraction $r=0.1$. Initial abundances in the $s$-process region are
adopted so that they satisfy observed data, i.e., 12C/13C isotopic ratios
(Aoki et al., 2001, 2002) and abundance ratios of C, N, and O (Aoki et al.,
2001, 2002, 2008; Boesgaard et al., 2011): for LP625-44,
$Y(^{12}\mathrm{C})=7.9\times 10^{-3}$, $Y(^{13}\mathrm{C})=4.0\times
10^{-4}$, $Y(^{14}\mathrm{N})=1.5\times 10^{-4}$, and
$Y(^{16}\mathrm{O})=7.9\times 10^{-3}$; for CS31062-012,
$Y(^{12}\mathrm{C})=7.8\times 10^{-4}$, $Y(^{13}\mathrm{C})=7.8\times
10^{-4}$, $Y(^{14}\mathrm{N})=1.5\times 10^{-4}$, and
$Y(^{16}\mathrm{O})=3.6\times 10^{-3}$. Note that we need some adjustment for
CS31062-012 to fit the heavy elemental abundance pattern. As a result, the
total CNO abundance is rather small, and 13C abundance has been amplified by a
factor of 15 to reproduce a large enough neutron exposure. In addition, the
22Ne mole fraction is set to the total mole fraction of CNO nuclei at the star
formation (Iben, 1975). Since we fix the initial 13C abundance, the duration
of interpulse phase is taken as a free parameter that determines the amount of
neutron injection during the $s$-process. Effects of the
22Ne($\alpha$,$n$)25Mg reaction during pulse phases are not considered.
Although this reaction partially contributes to the $s$-process near the solar
metallicity condition (Reifarth et al., 2014) the 22Ne abundance is small in
the low metallicity condition and the effect is expected to be negligibly
small.

Initial nuclear abundances before the H-burning are assumed as follows. Below
Fe-peak, abundances satisfy [X/Fe] = 0.3 (McWilliam, 1997) except for H and He
that are taken at solar values because of their primordial origin. For Fe-peak
elements, scaled solar abundances (Lodders et al., 2009) of [X/Fe] = 0 are
taken. For nuclei heavier than the Fe-peak elements, scaled $r$-elemental
abundances are adopted, i.e., [r/Fe] = 0. Relative contributions from the
$s$\- and $r$-processes to solar abundances are adopted from Bisterzo et al.
(2014). The initial $s$-elemental abundances are set to zero since there is no
enough time for evolution of low and intermediate mass stars that provide
$s$-nuclei before the formation of such low-metallicity stars at very early
cosmic time. Although enrichment of $r$-nuclei in the early epoch is uncertain
(Yamazaki et al., 2021), results of $s$-process do not depend on the initial
heavy element abundances (Aoki et al., 2001).

We have derived the best fit abundance patterns for the $s$-nuclei by a chi-
squared analysis. The parameters are the interpulse duration $\Delta t$ and
[s/C], which is the overall normalization factors for all $s$-nuclei applied
to the calculated results of nucleosynthesis. In the $s$-process region, the
hydrogen abundance is very small, and the use of the ratio [X/H] is in
convenient. Instead, the $s$-process in AGB stars is associated with abundant
C and is known as a partial C producer in Galactic chemical evolution (Henry
et al., 2000). Therefore, we take the ratio [X/C] of the calculated results
and make comparisons with observed abundance ratios.

Figure 8 shows results of fittings for the two MPSs. The upper panel shows the
reduced $\chi^{2}$ at the best fit normalization factor as a function of the
neutron exposure. The fitted normalization factor is shown in the middle
panel. Both stars have local minima at $\tau_{\mathrm{exp}}\lesssim 1$ mb-1
where the normalization factors are minimal. When the $\tau_{\mathrm{exp}}$
value of LP625-44 is smaller or larger, the required normalization factor
increases, indicating that those cases do not correspond to the
nucleosynthesis in LP625-44. Because of the minimum normalization factor close
to no need for abundance shifts relative to C abundance, i.e., [s/C] $\sim 0$,
it is apparent that the $\chi^{2}$ minimum is the best fit case in the adopted
model. For CS31062-012, the large $|$[s/C]$|$ value indicates the diffuculty
in reproduction of the $s$-process in that star with the one-zone model. The
best cases correspond to the following values: $\tau_{\mathrm{exp}}=0.75$
mb-1, [s/C]$=-0.13$, and $\Delta t=16$ kyr for LP625-44;
$\tau_{\mathrm{exp}}=0.85$ mb-1, [s/C]$=-1.7$, and $\Delta t=7.6$ kyr for
CS31062-012.

Figure 8: Reduced $\chi^{2}$ versus the neutron exposure for the best fit
cases (top panel) and fitted normalization factor versus the neutron exposure
(middle panel). The normalization factor was added to logarithmic abundance
ratio of [X/C] to fit the observed abundances. Also shown are fractional
differences in the neutron exposure between cases with the new and old rates
of 17O($n$,$\gamma$)18O (bottom panel). Results for LP625-44 and CS31062-012
are shown.

Figure 8 (botom panel) shows differences in $\tau_{\mathrm{exp}}$ caused by
the use of the different reaction rates for 17O($n$,$\gamma$)18O. Within the
parameter region of the metal-poor stars searched in this study,
${\mathcal{O}}(1)$ % of differences are found. This difference sensitively
depends on the neutron exposure (horizontal axis) and initial nuclear
abundances. Therefore, it is important to determine reaction rates that have
been unmeasured in order to understand the main $s$-process under metal-poor
conditions.

Figure 9 shows nuclear flows in the $s$-process via main reactions in the best
case for LP625-44 with the new rate. The thickness of each arrow reflects the
logarithmic amplitude of the flow, i.e., $\Delta Y_{ij}$ for nuclei $i$ and
reaction $j$. The dominant neutron source is 13C($\alpha$,$n$)16O ($\Delta
Y=2.1\times 10^{-3}$). The dominant neutron poison is 14N via 14N($n$,$p$)14C
($\Delta Y=1.1\times 10^{-3}$). There is a significant flow via
17O($n$,$\gamma$)18O ($\Delta Y=1.1\times 10^{-5}$). Because of protons
produced abundantly via 14N($n$,$p$)14C, 18O is mainly destroyed via
18O($p$,$\alpha$)15N ($\Delta Y=1.1\times 10^{-5}$). 15N nuclei tend to be
destroyed via 15N($p$,$\alpha$)12C ($\Delta Y=6.5\times 10^{-5}$). In metal-
poor stars the dominant neutron-poison is the light nuclei, not the minor
heavier nuclei including Fe-peak elements. Thus, a large change in the
17O($n$,$\gamma$)18O rate influences the neutron budget among the light
nuclei, and the neutron exposure is noticeably revised as seen in Fig. 8
(botom panel). We note that 19F is produced in the interpulse phase
predominatly via 18O($p$,$\gamma$)19F ($\Delta Y=6.4\times 10^{-8}$), not via
18O($n$,$\gamma$)19O ($\Delta Y=8.5\times 10^{-10}$) followed by the
$\beta^{-}$ decay.

Figure 9: Nuclear flows in the $s$-process via main reactions in the best case
for LP625-44 with the new rate. Arrows correspond to reactions of
($n$,$\gamma$), ($n$,$p$), ($n$,$\alpha$), ($\alpha$,$n$), ($p$,$\alpha$),
($p$,$\gamma$) and $\beta$-decays. Each arrow is shown for the flow, i.e.,
$\Delta Y_{ij}$ for nuclei $i$ and reaction $j$ if $\Delta Y_{ij}>10^{-8}$
with the thickness proportional to $\log(\Delta Y_{ij}/10^{-8})$. Stable
nuclei are shown by the yellow shaded boxes.

Figure 10 shows final yields of the $s$-process in the best cases as a
function of mass number. The final yields are identified as the yields after
the first interpulse phase after which abundances of $s$-only nuclei reproduce
those after the previous interpulse. The new rate of 17O($n$,$\gamma$)18O
corresponds to the solid circles, and the old rate corresponds to the open
circles. Figure 11 shows ratios for the final yields of the $s$-process as a
function of mass number. Noticeable changes are observed only in the 18O and
19F abundances. The newer rate leads to the dramatically enhanced abundances
of 18O and 19F, by factors of 41 and 40, respectively, for LP625-44 and by a
factor of 24 for CS31062-012.

Figure 10: Nuclear abundances after realization of converging abundances of
$s$-only nuclei in the $s$-process for metallicities of LP625-44 (upper panel)
and CS31062-012 (lower), respectively. Solid and open circles correspond to
the new and old rates for the reaction 17O($n$,$\gamma$)18O, respectively, in
the best fit models.  Figure 11: Ratios of nuclear yields of the $s$-process
with the new and old reaction rates of 17O($n$,$\gamma$)18O, for metallicities
of LP625-44 (solid circles) and CS31062-012 (open triangles), respectively.

Figure 12 shows time evolution of nuclear abundances in the best fit model for
LP625-44 for the new (thin solid lines) and old (thick dashed lines) reaction
rates. The convergence of $s$-process yields is obtained after four interpulse
episodes. In the old rate case, production of 18O is very small, and the 19F
abundance decreases after the mixing of material until the 18O abundance
increases enough. However, in the new rate case, 18O abundance increases
significantly via the larger 17O($n$,$\gamma$)18O rate, and accordingly, the
19F abundance is more resistant to destruction due to the production via the
18O($p$,$\gamma$)19F. These enhancements result in the differences as shown in
Fig. 10.

Figure 12: Time evolution of nuclear abundances in the best fit model for
LP625-44. Thin solid and thick dashed lines correspond to the new and old
rates for the reaction 17O($n$,$\gamma$)18O, respectively.

Figure 13 shows abundance ratios [A/C] of $s$-nuclei for calculated best fit
abundances (lines) and observational data (points with error bars) for
LP625-44 (upper panel) and CS31062-012 (lower panel). The fittings are fairly
good as indicated by small reduced $\chi^{2}$ values (Fig. 8). The density,
temperature, and initial nuclear abundances were fixed in the current
calculation. Nevertheless, the best fit cases do not require any significant
shifts of the calculated abundances of $s$-nuclei with respect to C abundance,
i.e. [s/C]$\sim 0$. This is a piece of evidence that the environmental
condition of the $s$-process in LP625-44 is well modeled without any
normalization or artificial change in adopted physical quantities.

Figure 13: Calculated best fit abundances of $s$-nuclei (lines) and
observational data (points with error bars) as a function of mass number for
LP625-44 (upper panel) and CS31062-012 (lower). Observational data are adopted
from Aoki et al. (2001) (solid circles), Aoki et al. (2008) (open circles),
and Aoki et al. (2006) (open triangles).

We note that in rotating stellar models effective diffusions produce abundant
primary 14N nuclei that induce a burst of neutrons via the
14N($\alpha$,$\gamma$)18F($\beta^{+}$)18O($\alpha$,$\gamma$)22Ne($\alpha$,$n$)25Mg
sequence (Limongi & Chieffi, 2018). Therefore, the larger rate of
17O($n$,$\gamma$)18O may affect yields of 19F and $s$-process isotopes in
rotating stars.

###  3.2 $r$-process

After the discovery of gravitational waves (GWs) from the binary neutron star
merger (NSM) GW170817 and its associated kilonova GRB170817 (Abbott et al.,
2017a, b), there is a growing consensus that the NSM could be a dominant
astrophysical site for $r$-process elements as well as CCSNe (Shibagaki et
al., 2016; Kajino et al., 2019). In addition, collapsar also has been
suggested to be another viable site for the $r$-process abundances in the
Milky Way (Siegel et al., 2019). NSM has an unavoidable time delay of
cosmological time scale from the birth of progenitor stars until they merge
due to very slow GW radiation of losing orbital energy. Collapsar and CCSNe,
on the other hand, quickly evolve in the stellar evolution time scale
$\sim$106 y and contribute to enriching the interstellar materials. Taking
account of all these candidate sites, recent theoretical study of Galactic
chemical evolution of $r$-process elements has proved that the collapsars and
CCSNe could be the dominant contributor to the $r$-process abundances of the
Milky Way during entire history of Galactic evolution, while the NSM
contribution has delayed and arrived recently (Yamazaki et al., 2021).

From the nuclear physics viewpoint, it is widely accepted concept of the
$r$-process nucleosynthesis that the outflows from any explosive events
contain only protons, neutrons and a small amount of alpha particles as the
initial composition. This does not apply to the r-process in dynamical ejecta
from NSMs. Alpha particles and nucleons combine to form carbon, nitrogen,
oxygen and heavier nuclei through the $\alpha$-process coupled with efficient
neutron-capture flow followed by $\beta$-decays. We therefore discuss the
impact of the 17O($n$,$\gamma$)18O new rate on the $r$-process nucleosynthesis
in stellar evolution of massive stars and explosion based on codes in NucNet
Tools (Meyer, 2012).

####  3.2.1 $r$-process in collapsar

Collapsar is a single massive star which literally culminates its evolution in
collapsing into a black hole instead of neutron star in ordinary core-collapse
supernova, i.e. neutrino-driven wind or magneto-hydrodynamically driven jet
supernova. In view of the progress in understanding the origin of $r$-process
elements in the Milky Way, we first demonstrate the effect of our new reaction
rate for 17O($n$,$\gamma$)18O on the collapsar $r$-process in this subsection.

The collapsar model which we use in our calculation is based on Harikae et al.
(2009), whose progenitor mass, rotational velocity, and metallicity are 35
$M_{\sun}$, $v_{\phi}$ = 380 km/s, and 0.1 $Z_{\sun}$, respectively. We use
the temperature and density profiles of one trajectory as shown in Fig. 14
from Nakamura et al. (2015) for $r$-process nucleosynthesis. This trajectory
was used also in Famiano et al. (2020) for their theoretical studies of the
effects of strong magnetic field on the $r$-process nucleosynthesis because it
is the most effective and typical flow among many trajectories to reach the
fissile region. In the following calculations, an initial electron mole
fraction value of $Y_{e}$ = 0.05 (Nakamura et al., 2015) has been utilized
unless noted otherwise.

Figure 14:  Temperature and density profiles used in the present $r$-process
collapsar model. Figure 15:  Time evolution of carbon isotopic abundances
predicted in the $r$-process collapsar model ($Y_{e}$ = 0.05, without fission
recycling effect). (Upper panel) abundances calculated with the present (solid
lines) and previous (dashed lines) 17O($n$,$\gamma$)18O rates, respectively;
(Lower panel) the corresponding abundance ratios calculated using two rates.
Figure 16:  Time evolution of nitrogen isotopic abundances for the two
17O($n$,$\gamma$)18O rates as in Fig. 15. Figure 17:  Time evolution of oxygen
isotopic abundances for the two 17O($n$,$\gamma$)18O rates as in Fig. 15.
Figure 18:  Time evolution of nuclear flows for 17O($n$,$\gamma$)18O and
18O($n$,$\gamma$)19O reactions in the $r$-process collapsar model ($Y_{e}$ =
0.05, without fission recycling effect). Here, old and new denote the
calculations using the previous and present 17O($n$,$\gamma$)18O rates,
respectively. Figure 19:  Final abundances of the $r$-process nucleosynthesis
in the collapsar model calculated by using two 17O($n$,$\gamma$)18O rates with
and without the fission recycling effect (with $Y_{e}$ = 0.05). The indexes,
new and old, denote the calculated results by using the present and previous
rates, respectively. The very small difference in the fission recycling case
is invisible in the figure. Figure 20:  The nuclear flows in the collapsar
$r$-process model at $t\approx$ 3000 s when the largest difference in 18O
abundance is observed ($Y_{e}$ = 0.05, without fission recycling effect). The
thickness of the arrow is proportional to the amplitude of the flow.

All nucleosynthetic products at earlier times $t<$ 0.1 s of collapsar event
are once reset by the shock passage at $t\sim$ 0.1 s (see Figs. 8–11). At
later times, the temperature is still high enough ($T_{9}\sim$ 5) for charged-
particle reactions to operate. Then, $\alpha$-induced reactions
3H($\alpha$,$\gamma$)7Li and 8Li($\alpha$,$n$)11B as well as neutron capture
flow play the important roles in the production of carbon, nitrogen, oxygen,
and even heavier neutron-rich nuclei up to $A\sim$ 80 (Sasaqui et al., 2005).
Figures 9–11 show the time evolution of carbon, nitrogen, and oxygen isotopic
abundances calculated with the present (solid lines) and previous (dashed
lines) 17O($n$,$\gamma$)18O rates. Significant differences are seen between
the two cases in the 15C, 17N and 18,19O abundances during the
“$\alpha$-process” within 0.25 s $<t<$ 1 s, corresponding to a temperature
region of 5 $>$ $T_{9}$ $>$ 2; also remarkable differences in 18,19O
abundances are seen after the “$\alpha$-process” at $t$ $>$ 1000 sec,
corresponding to a temperature region of $T_{9}$ $<$ 0.1. Compared to the
abundances calculated with the previous 17O($n$,$\gamma$)18O rate, the 18O and
19O abundances calculated with the present rate are larger by almost the same
factor of $\sim$20\. This enhancement is made by the interplay between the
strong neutron-capture flow under the condition of extremely high neutron-
number density and the $\alpha$-induced reactions in collapsar $r$-process,
where the $\alpha$-particles are continuously provided by the ($n$,$\alpha$)
reactions during the $r$-process. Once 3H($\alpha$,$\gamma$)7Li reaction
produces 7Li, subsequent light nuclear reactions
7Li($n$,$\gamma$)8Li($\alpha$,$n$)11B and
8Li($n$,$\gamma$)9Li(e${}^{-}\nu$)9Be cross over the mass gap of unstable $A$
= 8 nuclear systems, and radiative neutron-capture flow followed by the
$\beta$-decay is established in the neutron-rich Be-B-C-N isotopes (Terasawa
et al., 2001). 18O is thus predominantly produced via the $\beta$-decay of 18N
in the neutron-capture flow at $t$ $<$ 1000 s. Therefore, one cannot see any
difference in 18O abundance until $t$ $\approx$ 1000 s. However, as the time
passes by, neutrons are gradually exhausted and their number density rapidly
decreases at around $t$ $\approx$ 1000 s. Resultantly, the light nuclear
reactions do not provide neutron-rich Be-B-C-N isotopes any more. After this
epoch of changing the environmental condition, 18O is now produced via
16O($n$,$\gamma$)17O($n$,$\gamma$)18O reaction sequence (see Fig. 17) by
residual neutrons. Since the destruction rate of 18O via 18O($n$,$\gamma$)19O
is unchanged, the produced 18O remains at higher abundance level in the case
using larger cross section for 17O($n$,$\gamma$)18O. Accordingly, 19O
abundance produced via 18O($n$,$\gamma$)19O also stays at higher level by the
same factor. Figure 19 shows the time evolution of nuclear reaction flows
through 17O($n$,$\gamma$)18O and 18O($n$,$\gamma$)19O, $|dY_{i}/dt|$ for $i$ =
17O and 18O, which supports this production mechanism.

We have found an interesting “loophole” effect in the present model. Compared
to the 17O($n$,$\gamma$)18O rate, the 17O($n$,$\alpha$)14C rate is
tremendously larger (see Appendix B), and this results in a “loop” of nuclear
reaction network in which the abundance flows back to the lighter-mass nucleus
14C, yet taking away appreciable abundance flow out of the 17O-14C-17O loop
due to the enhanced new rate (or cross section) of the 17O($n$,$\gamma$)18O
reaction, named “loophole” effect. The enhancement of the 17O($n$,$\gamma$)18O
rate significantly changes the final nuclear abundances if fission recycling
is not included in the model, as shown in Fig. 19 (black and red solid lines).
The present 17O($n$,$\gamma$)18O rate results in an overall increase by one
order of magnitude in the abundances of nuclei in almost the whole mass region
of 20 $<$ $A$ $<$ 275\. It can be explained by the fact that
17O($n$,$\gamma$)18O works as a “loophole” in the
17O($n$,$\alpha$)14C($n$,$\gamma$)15C($\beta^{-}$)15N($n$,$\gamma$)16N($\beta^{-}$)16O($n$,$\gamma$)17O
cycle (see Fig. 20 which shows nuclear flows at $t\approx$ 3000 sec when the
largest difference in 18O abundance is observed). Neutrons are exhausted
through the “loophole” mechanism without producing new heavier nuclei.
Although a competing 17O($n$,$\alpha$)14C reaction takes a huge flux of
abundance which is about two orders of magnitude larger than that via the
17O($n$,$\gamma$)18O reaction, the present larger 17O($n$,$\gamma$)18O rate
broadens a flow path to produce more seed nuclei for the subsequent radiative
neutron-capture process. Resultant increase in 18O and 19O abundances occur
efficiently in accordance with the enhanced $r$-process nuclei when the
temperature of the system becomes as low as $T_{9}$ $<$ 0.1 at $t$ $>$ 300
sec. It is worthy of noting that although the maximum temperature can reach up
to $T_{9}$ $\sim$ 9 in the collapsar model (see Fig. 14), all the abundance
enhancements illustrated here mainly emerge from the present
17O($n$,$\gamma$)18O rate in the temperature region below $T_{9}$ $\sim$ 2.

We have calculated the $r$-process nucleosynthesis including a fission
recycling based on the model of Famiano et al. (2020), in which fission was
implemented for the Cf isotopic chain, i.e., 270-311Cf, and the fission rates
were assumed to be 100 s-1 for all californium isotopes. As displayed in Fig.
19, the calculated final abundances are almost identical for the previous
(green solid line) and present (blue solid line) 17O($n$,$\gamma$)18O rates.
The resulting difference is so small (less than 1%) that it is invisible in
the figure. The final abundance is redistributed due to the fission recycling,
especially in the mass region 100 $<A<$ 275\. Therefore, the difference in the
final abundances (at the order of magnitude of 10-15) observed in the case
without fission is almost completely washed out by the fission recycling
effect. The fission modes, i.e. spontaneous, $\beta$-delayed and neutron
capture-induced fissions, and the fission fragment distribution which were
used in the present model are taken from the theoretical calculation used in
Shibagaki et al. (2016). Laboratory experiments are still far to reach
significant fissile region of such superheavy nuclei with large neutron-number
excess (Ahmad et al., 2021). Further studies of the fission and the fission
recycling effects on collapsar $r$-process are highly desirable both
experimentally and theoretically.

To study the influence of less neutron-rich conditions on the above
$r$-process calculations, we have performed calculations with values of
$Y_{e}$ = 0.1, 0.2, 0.3 and 0.4, respectively. When fission is included in the
network for either of the four $Y_{e}$ values, similarly to the above case
with $Y_{e}$ = 0.05, the differences caused by the new 17O($n$,$\gamma$)18O
rate are invisible in the final abundance pattern. However, the patterns vary
between the cases by using the new and old rates if the fission recycling is
switched off artificially. As shown in Fig. 21, the differences in abundances
can be seen for $Y_{e}$ = 0.2 but not for $Y_{e}$ = 0.3. The results for
$Y_{e}$ = 0.1 and 0.2, and those for 0.3 and 0.4, are respectively analogous
to each other. Therefore, we only focus on the cases with $Y_{e}$ = 0.2 and
0.3 as representative values in the following discussion. Figure 22 shows the
time evolution of free neutron abundances in the system with different initial
$Y_{e}$ values and 17O($n$,$\gamma$)18O rates. It shows that $Y_{e}$ plays a
more critical role in determining the evolution of neutron abundances than the
reaction rate does. The abundant nuclei in the system after $\alpha$-process
are those having larger binding energies corresponding to the initial $Y_{e}$
value (Meyer, 1994). Since $Y_{e}$ = 0.3 is closer to the proton-to-mass ratio
$Z$/$A$ of the seed nucleus 78Ni for the $r$-process, most neutrons are
exhausted by the time when 78Ni is abundantly produced after the
$\alpha$-process. As a result, the number of neutrons decreases drastically
after $t\sim$ 10 s, which suppresses the subsequent neutron-capture reactions.
On the contrary, for the $Y_{e}$ = 0.2 case, plenty of free neutrons still
remain in the system even at later times and they could synthesize more
abundant heavy nuclei in the region $A$ $>$ 305 as shown in Fig. 21. The
nuclear flow through the 17O($n$,$\gamma$)18O reaction, i.e., $\rho
N_{A}\langle\sigma v\rangle Y_{\mathrm{{}^{17}O}}Y_{n}$, is obviously enhanced
due to the larger new rate in both cases as displayed in Fig. 23. However,
this enhancement in the $Y_{e}$ = 0.3 case is at the level of about four
orders of magnitude smaller than that in the $Y_{e}$ = 0.2 case after $t\sim$
10 s. Thus, for $Y_{e}$ = 0.3, the neutron deficiency eliminates the influence
of the new 17O($n$,$\gamma$)18O rate on the final nucleosynthetic yield, as
shown in Fig. 20. Combined with the fact that nuclear abundances in the mass
region 20 $<A<$ 250 remain high in this case, the effect of new
17O($n$,$\gamma$)18O rate is small.

Figure 21:  Final abundances of the $r$-process nucleosynthesis in the
collapsar model calculated by using two 17O($n$,$\gamma$)18O rates and two
different initial $Y_{e}$ values (without fission recycling effect). Figure
22:  Time evolution of free neutron abundances in two different initial
$Y_{e}$ cases in the collapsar $r$-process model (without fission recycling
effect). The differences between the two cases using new and old
17O($n$,$\gamma$)18O rates are hardly distinguishable. Figure 23:  Time
evolution of nuclear flows for 17O($n$,$\gamma$)18O reaction in the collapsar
model calculated by using two 17O($n$,$\gamma$)18O rates and two different
initial $Y_{e}$ values (without fission recycling effect).

####  3.2.2 $r$-process in neutron bursts

We also study the impact of new 17O($n$,$\gamma$)18O reaction rate on a
neutron burst model (Bojazi et al., 2014), which is developed for supernova
nucleosynthesis of massive stars (Rauscher et al., 2002). The model is based
on the explosion of initially 15$M_{\sun}$ stellar model s15a28 for explosion
energy $E$ = 1.0$\times 10^{51}$ erg $\equiv$ 1.0 B (Bojazi et al., 2014).
Having confirmed that pressure waves in the post-shock material tend to
homogenize the energy density dominated by the radiation (Woosley & Weaver,
1995), the post-shock temperature $T$ and the other quantities are solved by
the NucNet Tools $simple\\_snII$ (Meyer, 2012).

In order to find a most promising site where the new 17O($n$,$\gamma$)18O rate
could make the biggest effect on the 18O production, let us assume that the
system is approximately in the steady state. We consider here the following
four possible reaction sequences as the main pathways for the production of
18O: 14N($\alpha$,$\gamma$)18F($n$,$p$)18O,
16O($n$,$\gamma$)17O($n$,$\gamma$)18O, 14N($n$,$p$)14C($\alpha$,$\gamma$)18O,
17O($n$,$\alpha$)14C($\alpha$,$\gamma$)18O. We must include of course the
competing destruction reactions of 18F and 14N: 18F($n$,$\alpha$)15N and
14N($n$,$\alpha$)17F. We denote the 18F abundance upon this condition as
$\tilde{Y}_{{}^{18}\mathrm{F}}$. The first main reaction pathway
14N($\alpha$,$\gamma$)18F($n$,$p$)18O and 18F($n$,$\alpha$)15N in the steady
state leads to

$\displaystyle\rho N_{\mathrm{A}}Y_{{}^{14}\mathrm{N}}Y_{\alpha}\langle\sigma
v\rangle_{{}^{14}\mathrm{N}(\alpha,\gamma)}=\rho
N_{\mathrm{A}}Y_{\mathrm{n}}\tilde{Y}_{{}^{18}\mathrm{F}}\langle\sigma
v\rangle_{{}^{18}\mathrm{F}(n,\alpha)}+\rho
N_{\mathrm{A}}Y_{\mathrm{n}}\tilde{Y}_{{}^{18}\mathrm{F}}\langle\sigma
v\rangle_{{}^{18}\mathrm{F}(n,p)}\,,$ (9)

and $\tilde{Y}_{{}^{18}\mathrm{F}}$ is easily solved to be

$\displaystyle\tilde{Y}_{{}^{18}\mathrm{F}}=\frac{Y_{{}^{14}\mathrm{N}}Y_{\alpha}}{Y_{\mathrm{n}}}\frac{\langle\sigma
v\rangle_{{}^{14}\mathrm{N}(\alpha,\gamma)}}{\langle\sigma
v\rangle_{{}^{18}\mathrm{F}(n,\alpha)}+\langle\sigma
v\rangle_{{}^{18}\mathrm{F}(n,p)}}.$ (10)

Assuming that 17O and 14C are also in the steady state, their abundances can
be solved similarly,

$\displaystyle\tilde{Y}_{{}^{17}\mathrm{O}}=Y_{{}^{16}\mathrm{O}}\frac{\langle\sigma
v\rangle_{{}^{16}\mathrm{O}(n,\gamma)}}{\langle\sigma
v\rangle_{{}^{17}\mathrm{O}(n,\alpha)}+\langle\sigma
v\rangle_{{}^{17}\mathrm{O}(n,\gamma)}},$ (11)

$\displaystyle\tilde{Y}_{{}^{14}\mathrm{C}}=Y_{\mathrm{n}}\frac{Y_{{}^{14}\mathrm{N}}\langle\sigma
v\rangle_{{}^{14}\mathrm{N}(n,p)}+\tilde{Y}_{{}^{17}\mathrm{O}}\langle\sigma
v\rangle_{{}^{17}\mathrm{O}(n,\alpha)}}{Y_{\alpha}\langle\sigma
v\rangle_{{}^{14}\mathrm{C}(\alpha,\gamma)}}.$ (12)

We now define time evolution of the yield of 18O in question through the
reaction $A$($a$,$b$) as,

$\displaystyle\left.\frac{dY_{{}^{18}\mathrm{O}}}{dt}\right|_{A(a,b)}=\rho
N_{\mathrm{A}}Y_{A}Y_{a}\langle\sigma v\rangle_{A(a,b)},$ (13)

in terms of the steady state abundances of 18F, 17O and 14C. Combining these
Eqs.(10)-(13), we can determine time evolution of the yields of 18O through
18F($n$,$p$)18O, 17O($n$,$\gamma$)18O, and 14C($\alpha$,$\gamma$)18O,
respectively,

$\displaystyle\left.\frac{dY_{{}^{18}\mathrm{O}}}{dt}\right|_{{}^{18}{\mathrm{F}(n,p)}}=\rho
N_{\mathrm{A}}Y_{{}^{14}\mathrm{N}}Y_{\alpha}\frac{\langle\sigma
v\rangle_{{}^{14}\mathrm{N}(\alpha,\gamma)}\cdot\langle\sigma
v\rangle_{{}^{18}\mathrm{F}(n,p)}}{\langle\sigma
v\rangle_{{}^{18}\mathrm{F}(n,\alpha)}+\langle\sigma
v\rangle_{{}^{18}\mathrm{F}(n,p)}}$ (14)

$\displaystyle\left.\frac{dY_{{}^{18}\mathrm{O}}}{dt}\right|_{{}^{17}{\mathrm{O}(n,\gamma)}}=\rho
N_{\mathrm{A}}Y_{{}^{16}\mathrm{O}}Y_{\mathrm{n}}\frac{\langle\sigma
v\rangle_{{}^{16}\mathrm{O}(n,\gamma)}\cdot\langle\sigma
v\rangle_{{}^{17}\mathrm{O}(n,\gamma)}}{\langle\sigma
v\rangle_{{}^{17}\mathrm{O}(n,\alpha)}+\langle\sigma
v\rangle_{{}^{17}\mathrm{O}(n,\gamma)}}$ (15)

$\displaystyle\left.\frac{dY_{{}^{18}\mathrm{O}}}{dt}\right|_{{}^{14}{\mathrm{C}(\alpha,\gamma)}}=\rho
N_{\mathrm{A}}Y_{\mathrm{n}}\left(Y_{{}^{14}\mathrm{N}}\langle\sigma
v\rangle_{{}^{14}\mathrm{N}(n,p)}+Y_{{}^{16}\mathrm{O}}\frac{\langle\sigma
v\rangle_{{}^{16}\mathrm{O}(n,\gamma)}\cdot\langle\sigma
v\rangle_{{}^{17}\mathrm{O}(n,\alpha)}}{\langle\sigma
v\rangle_{{}^{17}\mathrm{O}(n,\alpha)}+\langle\sigma
v\rangle_{{}^{17}\mathrm{O}(n,\gamma)}}\right).$ (16)

From these equations, we can find a most promising site where the distinct
difference in the 18O abundance could arise from our larger
17O($n$,$\gamma$)18O rate. It is obvious that such an ideal condition should
satisfy the higher initial abundances of neutron and 16O than those of 4He and
14N so that the reaction sequence 16O($n$,$\gamma$)17O($n$,$\gamma$)18O makes
predominant contribution to the production of 18O. Zone 564 at $M_{r}$ = 2.82
$M_{\sun}$ and initial temperature $T_{9}$ = 1.12 in Fig. 24 is thus
identified to be a most promising site to see the effect of new
17O($n$,$\gamma$)18O rate.

In the above discussions, we confine ourselves to several limited number of
most effective nuclear reactions on the production of 18O. However, once we
identify the zone 564, the $r$-process nucleosynthesis calculation is
performed by using exactly the same full nuclear reaction network adopted in
collapsar r-process as discussed in the previous subsection.

The calculated final abundances in this zone are shown in Fig. 25, where the
impact of the new 17O($n$,$\gamma$)18O rate is hardly seen. The final
abundances of the key isotopes of carbon, nitrogen, oxygen, and fluorine are
listed in Table 3. The yield of 18O is enhanced by only 0.2% due to the new
larger 17O($n$,$\gamma$)18O rate, and less than 1% difference is found in the
yields of the other isotopes. This result is explained by defining the
integrated current for the reaction $r$ that links reactant $j$ and product
$i$ as

$\displaystyle
I[r]\left(t,t_{0}\right)=\int_{t_{0}}^{t}\left[\left(\frac{dY_{i}}{dt}\right)_{r:j\rightarrow
i}-\left(\frac{dY_{i}}{dt}\right)_{r:i\rightarrow j}\right]dt^{\prime}.$ (17)

Fig. 26 displays the significance of nuclear reactions where the thickness of
each arrow is proportional to the magnitude of the integrated current for that
reaction ${r:j\rightarrow i}$. It is indicated that the integrated current for
a competing reaction 17O($n$,$\alpha$)14C takes about two orders of magnitude
larger current to produce 14C than the current for 17O($n$,$\gamma$)18O.
Subsequently, the condition of high temperature and sufficiently abundant
alpha particles facilitates 14C to capture $\alpha$ particles. Even in the
ideal zone 564 which is most favorable for the production of 18O through
16O($n$,$\gamma$)17O($n$,$\gamma$)18O, the 14C($\alpha$,$\gamma$)18O reaction
makes the largest contribution to the yield of 18O, thus the change of
17O($n$,$\gamma$)18O rate is highly suppressed. We conclude that impact of the
present new larger 17O($n$,$\gamma$)18O rate on the neutron burst model is
marginal.

Figure 24:  Mass fraction of some key species as a function of Lagrangian coordinate $M_{r}$ after the $E$ = 1.0 B explosion of s15a28. Figure 25:  Final abundances of the $r$-process nucleosynthesis with fission recycling switched off in zone 564 at 106 seconds after the $E$ = 1.0 B explosion of s15a28 by using two 17O($n$,$\gamma$)18O rates. Table 3:  Final abundances of key species at 106 seconds after the $E$ = 1.0 B explosion of s15a28 by using two 17O($n$,$\gamma$)18O rates. |  12C |  13C |  14N |  15N
---|---|---|---|---
new rate | 1.7104E-02 | 3.4205E-09 | 7.3949E-08 | 1.8553E-07
old rate | 1.7104E-02 | 3.4209E-09 | 7.4000E-08 | 1.8543E-07
ratio(new/old) | 1.0000E+00 | 9.9988E-01 | 9.9931E-01 | 1.0005E+00
|  16C |  17O |  18O |  19F
new rate | 4.3387E-02 | 4.3715E-06 | 1.5002E-06 | 3.7045E-08
old rate | 4.3387E-02 | 4.3721E-06 | 1.4999E-06 | 3.7024E-08
ratio(new/old) | 1.0000E+00 | 9.9985E-01 | 1.0002E+00 | 1.0006E+00
Figure 26:  Integrated current diagram at 106 seconds after the $E$ = 1.0 B
explosion of s15a28 using the new 17O($n$,$\gamma$)18O rate.

##  4 Summary and outlook

In this work, we have obtained a new thermonuclear 17O($n$,$\gamma$)18O rate
based on theoretical calculation of the DC as well as the resonance
contributions, for a temperature up to $T_{9}$ = 2 of general astrophysical
interest. The associated uncertainties of this rate are also estimated based
on reasonable parameter variations in the Monte-Carlo calculations. It shows
that the present rate is larger than that adopted in the JINA REACLIB library
in the temperature region of $T_{9}$ = 0.01 $\sim$ 2 significantly, by up to a
factor of $\sim$80\. In addition, for completeness, the 17O($n$,$\gamma$)18O
rate in the high temperature region of $T_{9}$ = 2 $\sim$ 10 has been obtained
by normalizing the ENDF/B-VIII data in the energy region of 1.0–20 MeV to the
presently calculated cross section data around 1 MeV.

The impacts of the change in the 17O($n$,$\gamma$)18O rate on the evolution of
light element abundances in AGB stars and massive stars have been investigated
in both main and weak $s$-process models. In the main $s$-process model for
metal deficient stars, 18O and 19F abundances are found to be enhanced by
factors of $\sim 40$ and $\sim 20$ in LP625-44 ([Fe/H] $=-2.70$) and
CS31062-012/LP706-7 ([Fe/H] $=-2.58$), respectively, due to new larger
17O($n$,$\gamma$)18O rate, still reproducing the observed abundances of heavy
$s$-process elements fairly well. The new rate also results in
${\mathcal{O}}(1)$ % modifications of neutron exposure under metal-poor
conditions. However, as for the weak $s$-process in massive stars, there is no
discernable difference in nuclear abundances when a result for the new rate is
compared with that of the JINA rate for both of core He and shell C burning
stages. The new rate of the 17O($n$,$\gamma$)18O reaction results in 18O
production via the reaction much more efficient than the old rate does.
However, because the $\beta$-decay of 18F and 14C($\alpha$,$\gamma$)18O
reaction are predominant in the 18O production, the sensitivity of 18O
abundance to the 17O($n$,$\gamma$)18O reaction rate is still very small.
Furthermore, the impact of the present 17O($n$,$\gamma$)18O rate on the
$r$-process nucleosynthesis in the collapsar model has been investigated. It
shows that when the fission recycling is not included, the 18O and 19O
abundances calculated with the present rate are larger by almost a factor of
$\sim$20 during and after the “$\alpha$-process” phase, compared to those
calculated with the previous JINA rate. An interesting “loophole” effect is
found owing to the enhanced new 17O($n$,$\gamma$)18O rate, which may
significantly change the final nuclear abundances if fission recycling is not
included in the model. In this case, an overall increase by one order of
magnitude was observed in the abundances of nuclei in almost the whole mass
region of 20 $<A<$ 275\. However, such significant differences are almost
completely washed out by the fission recycling effect, with which the final
abundances are redistributed. The effect of new 17O($n$,$\gamma$)18O rate on
$r$-process in neutron burst model is also studied. In this model, the
condition of high temperature and sufficiently abundant alpha particles makes
17O(n,$\alpha$)14C($\alpha$,$\gamma$)18O much more efficient than
17O($n$,$\gamma$)18O for the production of 18O. It shows that
14C($\alpha$,$\gamma$)18O contributes overwhelmingly to the yield of 18O, and
the impact of new 17O($n$,$\gamma$)18O rate is highly suppressed.

So far, there are no experimental data available for the 17O($n$,$\gamma$)18O
reaction except for the thermal $\sigma_{\mathrm{th}}$ value. Therefore, the
present calculated cross sections above the $E_{n}$ = 10-5 MeV energy region
need further experimental clarification. Besides the direct neutron beam
experiments, the time-reversal photo-dissociation of 18O experiments may be
also an option. In addition, the 14C($\alpha$,$\gamma$)18O reaction rate still
bears very large uncertainties in the AGB temperature regime (up to a factor
of $\sim$10; Iliadis et al. (2010)), and hence the precise measurement of this
reaction is strongly required in the low energy region, to better constrain
the 18O abundance in the $s$-process models.

This work was financially supported by the National Natural Science Foundation
of China (Nos. 11825504, 12075027, 11850410441, 11961141004), and by Grants-
in-Aid for Scientific Research of Japan Society for the Promotion of Science
(Nos. 20K03958, 17K05459).

Figure 27:  Comparison of MACSs for 17O($n$,$\gamma$), 16O($n$,$\gamma$) and
18O($n$,$\gamma$). The MACSs for the latter two reactions are taken from
KADoNiS. In addition, the calculated MACSs for the 17O($n$,$\gamma$)18O
reaction by using the ENDF/B-VIII, JEFF3.3 and ROSFOND2010 libraries are
indicated by the solid lines, respectively. See text for details. Figure 28:
Ratios of 17O($n$,$\alpha$)14C and 17O($n$,$\gamma$)18O rates. The label
“Schatz93” indicates the Schatz et al. (1993) rate, and “JINA (bb92)”
indicates the _bb92_ rate compiled by JINA REACLIB.

##  Appendix A MACS of 17O($n$,$\gamma$)18O

The Maxwellian-averaged cross sections (MACSs) of 17O($n$,$\gamma$)18O are
calculated by using the presently calculated cross sections, as well as the
evaluated data compiled in the recent ENDF/B-VIII, JEFF3.3 and ROSFOND2010
libraries. The resulting MACSs are shown in Fig. 27. Also the MACSs for
16O($n$,$\gamma$) and 18O($n$,$\gamma$) which are compiled in the
KADoNiS444Karlsruhe Astrophysical Database of Nucleosynthesis in Stars
(KADoNiS), https://www.kadonis.org/ (Bao et al., 2000; Dillmann et al., 2006,
2009), are shown for comparison. It shows that the present MACSs of
17O($n$,$\gamma$) are close to those of 16O($n$,$\gamma$) in the temperature
region of $kT>$ 40 keV, below which they are quite different. The
17O($n$,$\gamma$) MACSs are significantly larger than those of
18O($n$,$\gamma$)19O in the whole temperature region, by a factor of 2–16. In
addition, the presently calculated MACSs based on the evaluated libraries
(indicated by three solid lines) are close to those of 18O($n$,$\gamma$)19O.
In Section 1, we mentioned that Nishimura et al. (2009) assumed the
17O($n$,$\gamma$)18O rate equaled to that of 16O($n$,$\gamma$)17O in the
temperature region of AGB stars, and here we think this assumption is
basically reasonable. But the predictions made by Yamamoto et al. (2010) can
be constrained better with the present more reliable rate.

##  Appendix B 17O($n$,$\alpha$)14C rate

For the 17O($n$,$\alpha$)14C reaction, Koehler & Graff (1991) measured its
cross section from thermal energy to approximately 1 MeV and derived a
reaction rate in a temperature region up to $T_{9}$ = 5, together with the
experimental data of 14C($\alpha$,$n$)17O reaction measured by Sanders (1956).
Later on, Schatz et al. (1993) measured the 17O($n$,$\alpha$)14C cross section
in the neutron energy range from 10 to 250 keV, and corrected the rate of
Koehler & Graff (1991) by about 10% owing to the anisotropy effect. Here, we
have fitted two rates graphically shown in the previous work in the standard
format of Rauscher & Thielemann (2000) as shown below. I) The rate of Koehler
& Graff (1991) shown in their Fig. 5 (labelled as a solid line) can be fitted
by,

$\displaystyle N_{A}\langle\sigma v\rangle_{(n,\alpha)}$ $\displaystyle=$
$\displaystyle\mathrm{exp}(27.9671+\frac{0.00030507}{T_{9}}+\frac{0.29888}{T_{9}^{1/3}}-24.2927T_{9}^{1/3}+25.4217T_{9}-34.5487T_{9}^{5/3}+2.55092\ln{T_{9}})$
(B1) $\displaystyle+$
$\displaystyle\mathrm{exp}(72.2274-\frac{0.055398}{T_{9}}+\frac{18.496}{T_{9}^{1/3}}-80.6603T_{9}^{1/3}+6.9012T_{9}-0.57146T_{9}^{5/3}+28.1568\ln{T_{9}})$

with a fitting error of less than 1% over the temperature region of $T_{9}$ =
0.001 $\sim$ 5; II) The rate of Schatz et al. (1993), which was shown as a
solid line in their Fig. 6, can be fitted by,

$\displaystyle N_{A}\langle\sigma v\rangle_{(n,\alpha)}$ $\displaystyle=$
$\displaystyle\mathrm{exp}(-2145.6-\frac{100.742}{T_{9}}+\frac{1904.1}{T_{9}^{1/3}}+683.59T_{9}^{1/3}-400.627T_{9}+75.8261T_{9}^{5/3}+581.563\ln{T_{9}})$
(B2) $\displaystyle+$
$\displaystyle\mathrm{exp}(3038.36+\frac{46.9075}{T_{9}}-\frac{2990.02}{T_{9}^{1/3}}-3039.63T_{9}^{1/3}+5760.5T_{9}-3140.75T_{9}^{5/3}-1672.1\ln{T_{9}})$
$\displaystyle+$
$\displaystyle\mathrm{exp}(1029.25+\frac{9.41958}{T_{9}}-\frac{1003.42}{T_{9}^{1/3}}-1351.56T_{9}^{1/3}+2794.69T_{9}-1732.65T_{9}^{5/3}-622.95\ln{T_{9}})$

with a fitting error of less than 2% over the temperature region of $T_{9}$ =
0.1 $\sim$ 2.

In fact, the discrepancies between the available experimental ($n$,$\alpha$)
cross section data (Koehler & Graff, 1991; Schatz et al., 1993; Sanders, 1956)
are still significant in the energy region of $E_{n}\sim$ 40–600 keV, and the
resulting rates can be very different, by up to a factor of 2. Therefore,
further experimental verification is needed to clarify such discrepancies. In
addition, it should be noted that there are two 17O($n$,$\alpha$)14C rates
(i.e., _kg91_ and _bb92_) collected in the JINA REACLIB. The _kg91_ rate is
actually the analytic one expressed by Eq. 4 of Koehler & Graff (1991), which
was simply based on the thermal $\sigma_{\mathrm{th}}$($n$,$\alpha$) value and
the parameters for the $E_{\mathrm{R}}$ = 169, 238 keV resonances (Weinman &
Silverstein, 1958). This analytic rate deviated greatly from the numerical
results based on the experimental ($n$,$\alpha$) data, as shown in their Fig.
5. Therefore, this analytic rate should not be utilized in the nucleosynthesis
models, and it should be removed from the library for safety. In JINA REACLIB,
the source of _bb92_ rate was cited as Rauscher et al. (1994), who did not
give an analytic equation for the 17O($n$,$\alpha$)14C reaction rate. We found
that the _bb92_ rate deviates from the Koehler & Graff (1991) numerical one up
to $\sim$20%, and hence the _bb92_ rate should be replaced by the well fitted
rate with Eq. B1, which is actually used in the present $r$-process model. In
addition, we suggest the rate of Schatz et al. (1993), which is well fitted by
Eq. B2, should be compiled into the library in the future (but the temperature
range is limited to $T_{9}$ $\leq$ 2).

For the 17O($n$,$\gamma$)18O and 17O($n$,$\alpha$)14C reactions, their
corresponding reaction rate ratios are shown in Fig. 28. Owing to the present
large ($n$,$\gamma$) rate, the ($n$,$\alpha$)/($n$,$\gamma$) ratio reduced by
factors of 35 $\sim$ 80, and the minimal ratio reduced to $\sim$90 from the
previous value of $\sim$7000 around $T_{9}$ = 0.3.

## References

  * Abbott et al. (2017a) Abbott, B.P., et al., 2017, Phys. Rev. Lett., 119, 161101
  * Abbott et al. (2017b) Abbott, B.P., et al., 2017, ApJ, 848, L13
  * Ahmad et al. (2021) Ahmad, N.S., et al., 2021, AAPPS (Association of Asia Pacific Physical Societies) Bulletin, to be published.
  * Ajzenberg-Selove (1987) Ajzenberg-Selove, F. 1987, Nucl. Phys. A, 475, 1
  * Aoki et al. (2008) Aoki, W., Beers, T.C., Sivarani, T., et al. 2008, ApJ, 678, 1351
  * Aoki et al. (2006) Aoki, W., Bisterzo, S., Gallino, R., et al. 2006, ApJ, 650, L127
  * Aoki et al. (2002) Aoki, W., Norris, J. E., Ryan, S. G., et al. 2002, PASJ, 54, 933. doi:10.1093/pasj/54.6.933
  * Aoki et al. (2001) Aoki, W., Ryan, S.G., Norris, J.E., et al. 2001, ApJ, 561, 346
  * Arnould & Goriely (2006) Arnould, M. & Goriely, S. 2006, Nucl. Phys. A, 777, 157
  * Bao et al. (2000) Bao, Z.Y., et al., 2000, At. Data Nucl. Data Tables, 76, 70
  * Bertulani (2003) Bertulani, C.A. 2003, Comput. Phys. Commun., 156, 123
  * Bisterzo et al. (2014) Bisterzo, S., Travaglio, C., Gallino, R., et al. 2014, ApJ, 787, 10. doi:10.1088/0004-637X/787/1/10
  * Boesgaard et al. (2011) Boesgaard, A. M., Rich, J. A., Levesque, E. M., et al. 2011, ApJ, 743, 140. doi:10.1088/0004-637X/743/2/140
  * Bojazi et al. (2014) Bojazi, M.J. & Meyer, B.S. 2014, Phys. Rev. C, 89, 025807
  * Brown et al. (2018) Brown, D.A., Chadwick, M.B., Capote, R., et al., 2018, Nucl. Data Sheets, 148, 1
  * Chadwick et al. (2006) Chadwick, M.B., et al., 2006, Nucl. Data Sheets, 107, 2931
  * CSEWG-Collaboration (2020) CSEWG-Collaboration, Evaluated Nuclear Data File ENDF/B-VI.8. Available from: http://www.nndc.bnl.gov/endf/
  * Cyburt et al. (2010) Cyburt, R.H., et al., 2010, ApJS, 189, 240
  * Dillmann et al. (2006) Dillmann, I., et al., 2006, AIP Conf. Proc., 819, 123
  * Dillmann et al. (2009) Dillmann, I., Plag, R., Käppeler, F., Rauscher, T. 2009, in Proc. EFNUDAT (European Facilities for Nuclear Data Measurements) Scientific Workshop on Neutron Measurements, Theory and Applications, ed. F.-J. Hambsch (Luxembourg: Publications Office of the European Union (2009), 55
  * Famiano et al. (2020) Famiano, M., et al., 2020, ApJ, 898, 163
  * Firestone & Revay (2016) Firestone, R.B. & Revay, Zs. 2016, Phys. Rev. C, 93, 044311
  * Goriely & Khan (2002) Goriely, S. & Khan, E. 2002, Nucl. Phys., 706, 217
  * Harikae et al. (2009) Harikae, S., Takiwaki, T., & Kotake, K. 2009, ApJ, 704, 354
  * He et al. (2014) He, J.J., et al., 2014, Phys. Rev. C, 89, 035802
  * He et al. (2020) He, M., et al., 2020, ApJ, 899, 133
  * Henry et al. (2000) Henry, R.B.C., Edmunds, M.G., & Köppen, J. 2000, ApJ, 541, 660
  * Howard et al. (1986) Howard, W.M., Mathews, G.J., Takahashi, K., & Ward, R.A. 1986, ApJ, 309, 633
  * Huang et al. (2010) Huang, J.T., Bertulani, C.A., Guimar$\tilde{a}$es, V. 2010, At. Data Nucl. Data Tables, 96, 824
  * Iben (1975) Iben, I. 1975, ApJ, 196, 525
  * Iliadis (2007) Iliadis, C. 2007, Nuclear Physics of Stars (Wiley, Weinheim)
  * Iliadis et al. (2010) Iliadis, C., Longland R, Champagne, A.E., Coc, A. 2010, Nucl. Phys. A, 841, 323
  * Jacqmin et al. (2002) Jacqmin, R, Forrest, R.A., Rowlands, J., Nouri, A., Kellett, M. 2002, J. Nucl. Sci. Technol. 39, Suppl. 2, 54
  * Kajino et al. (2019) Kajino, T., Aoki, W., Balantekin, A.B., et al., 2019, Prog. Part. Nucl. Phys., 107, 109
  * Koehler & Graff (1991) Koehler, P.E. & Graff, S.M. 1991, Phys. Rev. C, 44, 2788
  * Käppeler et al. (2011) Käppeler, F., Gallino, R., Bisterzo, S., Aoki, W., 2011, Rev. Mod. Phys., 83, 157
  * Koning et al. (2011) Koning, A.J., Bauge, E., Dean, C.J., et al., 2011, J. Korean Phys. Soc., 59, No. 2, 1057; See website, https://www.oecd-nea.org/dbdata/jeff/jeff33/index.html
  * Koning & Rochman (2012) Koning, A.J. & Rochman, D. 2012, Nucl. Data Sheets, 113, 2841
  * Lam et al. (2016) Lam, Y.H., et al., 2016, ApJ, 818, 78
  * Lamb et al. (1977) Lamb, S.A., Howard, W.M., Truran, J.W., et al., 1977, ApJ, 217, 213
  * Li et al. (1976) Li, T.K., Dehnhard, D., Brown, R.E., Ellis, P.J. 1976, Phys. Rev. C, 13, 55
  * Limongi & Chieffi (2018) Limongi, M. & Chieffi, A. 2018, ApJS, 237, 13
  * Lodders et al. (2009) Lodders, K., Palme, H., Gail, H.-P. 2009, Landolt Börnstein, 4B, 712
  * Lone et al. (1979) Lone, M.A. & Inglis, W.M. 1979, in Neutron Capture Gamma-Ray Spectroscopy, Chrien, R.E. & Kane, W.R., Eds., (Plenum Press, New York, 1979) p. 678 (Proc. Intern. Symp. Neutron Capture Gamma Ray Spectroscopy and Related Topics, 3rd, BNL, Upton, NY, 1978)
  * Longland et al. (2010) Longland, R, Iliadis, C., Champagne, A.E., et al., 2010, Nucl. Phys. A, 841, 1
  * Iwamoto et al. (2004) Iwamoto, N., Kajino, T., Mathews, G.J., et al. 2004, ApJ602, 377
  * McWilliam (1997) McWilliam, A. 1997, ARA&A, 35, 503
  * Meyer (1994) Meyer, B.S. 1994, Annu. Rev. Astron. Astrophys., 32, 153
  * Meyer (2012) Meyer, B.S. 2012, PoS (NIC XII) 096
  * Mohr et al. (2016) Mohr, P., Heinz, C., Pignatari, M. et al., 2016, ApJ, 827, 29
  * Mughabghab (1978) Mughabghab, S.F. 1978, BNL-NCS25086R (to be published in Phys. Lett.)
  * Mughabghab (1979) Mughabghab, S.F. 1979, Phys. Lett. B, 81, 93
  * Mughabghab et al. (1981) Mughabghab, S.F., Divadeenam, M., Holden, N.E. 1981, Neutron Cross Sections (Academic, New York), Vol. 1
  * Nakamura et al. (2015) Nakamura, K., Kajino, T., Mathews, G.J., Sato, S., Harikae, S. 2015, A&A, 582, A34
  * Nishimura et al. (2009) Nishimura, T., Aikawa, M., Suda, T., Fujimoto, M.Y. 2009, Publ. Astron. Soc. Jpn., 61, 909
  * Pritychenko et al. (2010) Pritychenko, B., Mughaghab, S.F., Sonzogni, A.A. 2010, At. Data Nucl. Data Tables, 96, 645
  * Paxton et al. (2011) Paxton, B., et al., 2011, ApJS, 192, 3
  * Paxton et al. (2015) Paxton, B., et al., 2015, ApJS, 220, 15
  * Paxton et al. (2018) Paxton, B., et al., 2018, ApJS, 234, 34
  * Paxton et al. (2019) Paxton, B., et al., 2019, ApJS, 243, 10
  * Pritychenko & Mughaghab (2012) Pritychenko, B. & Mughaghab, S.F. 2012, Nucl. Data Sheets, 113, 3120
  * Raiteri et al. (1991) Raiteri, C.M., Busso, M., Gallino, R., et al., 1991, ApJ, 371, 665
  * Raiteri et al. (1993) Raiteri, C.M., Gallino, R., Busso, M., et al., 1993, ApJ, 419, 207
  * Rauscher et al. (1994) Rauscher, T, Applegate, J.H., Cowan, J.J., Thielemann, F.-K., Wiescher, M. 1994, ApJ, 429, 499
  * Rauscher & Thielemann (2000) Rauscher, T. & Thielemann, F.-K. 2000, At. Data Nucl. Data Tables, 75, 1
  * Rauscher et al. (2002) Rauscher, T, Heger, A, Hoffman, R.D., Woosley, S.E. 2002, ApJ, 576, 323
  * Reifarth et al. (2014) Reifarth, R., Lederer, C., Käppeler, F. 2014, J. of Phys. G: Nucl. Phys., 41, 053101
  * Rolfs & Rodney (1988) Rolfs, C.E. & Rodney, W.S. 1988, Cauldrons in the Cosmos (Univ. of Chicago Press, Chicago)
  * Sanders (1956) Sanders, R.M. 1956, Phys. Rev., 104, 1434
  * Sallaska et al. (2013) Sallaska, A.L., et al., 2013, ApJS, 207, 18
  * Sasaqui et al. (2005) Sasaqui, T., Kajino, T., Mathews, G.J., Otsuki, K., Nakamura, T. 2005, ApJ, 634, 1173
  * Schatz et al. (1993) Schatz, H, Käppeler, F., Koehler, P.E., Wiescher, M., Trautvetter, H.-P. 1993, ApJ, 413, 750
  * Shibagaki et al. (2016) Shibagaki, S., et al., 2016, ApJ, 816, 79
  * Siegel et al. (2019) Siegel, D.M., Barnes, J, Metzger, B.D., 2019, Nature, 569, 241
  * Terasawa et al. (2001) Terasawa, M, Sumiyoshi, K., Kajino, T., Tanihata, I., Mathews, G.J. 2001, ApJ562, 470
  * The et al. (2007) The, L.-S., El Eid, M.F., Meyer, B.S., 2007, ApJ, 655, 1058
  * Tilley et al. (1995) Tilley, D.R., Weller, H.R., Cheves, C.M., Chasteler, R.M. 1995, Nucl. Phys. A, 595, 1
  * Wang et al. (2017) Wang, M., et al., 2017, Chin. Phys. C, 41, 030003
  * Weinman & Silverstein (1958) Weinman, J.A. & Silverstein, E.A. 1958, Phys. Rev., 111, 277
  * Xu et al. (2013) Xu, Y, Goriely, S., Jorissen, A., Chen, G., Arnould, M. 2013, A&A, 549, A106
  * Yamamoto et al. (2010) Yamamoto, K., Kato, K., Wada, T., Ohta, M. 2010, Prog. Theor. Phys., 123, 1029
  * Yamazaki et al. (2021) Yamazaki, Y., Kajino, T., Mathews, G.J., He, Z.Y., Famiano, M.A., Tang, X.D., Shi, J.R. 2021, arXiv:2102.05891
  * Zabrodskaya et al. (2007) Zabrodskaya, S.V., Ignatyuk, A.V., Koscheev, V.N., et al., 2007, VANT, Nuclear Constants 1, 3
  * Woosley & Weaver (1995) Woosley, S.E. & Weaver, T.A. 1995, ApJS, 101, 181