# The impact of 17O + $\alpha$ reaction rate uncertainties on the s-process in
rotating massive stars

J. Frost-Schenk,1, P. Adsley,2,3,4,5 A.M. Laird,1 R. Longland,6,7 C. Angus,1,8
C. Barton,1 A. Choplin,9C. Aa. Diget,1 R. Hirschi,10,11 C. Marshall,6,7 F.
Portillo Chaves,6,7 K. Setoodehnia,6,7
1Department of Physics, University of York, York, YO10 5DD, UK,
2School of Physics, University of the Witwatersrand, Johannesburg 2050, South
Africa,
3iThemba LABS, National Research Foundation, PO Box 722, Somerset West 7129,
South Africa
4Cyclotron Institute, Texas A&M University, College Station, Texas 77843, USA
5Department of Physics and Astronomy, Texas A&M University, College Station,
Texas 77843, USA
6Department of Physics North Carolina State University, Raleigh, North
Carolina 27695-8202, USA
7Triangle Universities Nuclear Laboratory, Durham, North Carolina, 27708-0308,
USA
8TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3
9Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles, CP
226, B-1050 Brussels, Belgium
10Astrophysics Group, Lennard-Jones Laboratories, Keele University, Keele ST5
5BG, UK
11Kavli IPMU (WPI), The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

###### Abstract

Massive stars are crucial to galactic chemical evolution for elements heavier
than iron. Their contribution at early times in the evolution of the Universe,
however, is unclear due to poorly constrained nuclear reaction rates. The
competing 17O($\alpha,\gamma$)21Ne and 17O($\alpha,n$)20Ne reactions strongly
impact weak s-process yields from rotating massive stars at low metallicities.
Abundant 16O absorbs neutrons, removing flux from the s-process, and producing
17O. The 17O($\alpha,n$)20Ne reaction releases neutrons, allowing continued
s-process nucleosynthesis, if the 17O($\alpha,\gamma$)21Ne reaction is
sufficiently weak. While published rates are available, they are based on
limited indirect experimental data for the relevant temperatures and, more
importantly, no uncertainties are provided. The available nuclear physics has
been evaluated, and combined with data from a new study of astrophysically
relevant 21Ne states using the 20Ne($d,p$)21Ne reaction. Constraints are
placed on the ratio of the ($\alpha,n$)/($\alpha,\gamma$) reaction rates with
uncertainties on the rates provided for the first time. The new rates favour
the ($\alpha,n$) reaction and suggest that the weak s-process in rotating low-
metallicity stars is likely to continue up to barium and, within the computed
uncertainties, even to lead.

###### keywords:

nuclear reactions – nucleosynthesis – stars:rotation

††pagerange: The impact of 17O + $\alpha$ reaction rate uncertainties on the
s-process in rotating massive stars–References

##  1 Introduction

Massive stars are key contributors to the abundance of chemical elements,
producing elements up to the iron group via charged-particle reactions during
their evolution and subsequent explosion in core-collapse supernovae, and
synthesising elements heavier than iron via neutron-capture reactions. The
weak s-process during core-helium (Langer et al., 1989; Prantzos et al., 1990;
Baraffe et al., 1992) and to a smaller extent during shell-carbon burning
(Raiteri et al., 1991; The et al., 2007) and possibly the weak r-process in
the supernova explosion (Thielemann et al., 2011) produce elements up to the
strontium peak in standard, non-rotating, models.

Rotation in massive star models significantly boosts the efficiency of the
weak s-process, especially at low metallicity (Pignatari et al., 2008),
enhancing production of elements above strontium. With rotation, the helium
core contribution to the s-process was shown to increase at the expense of the
carbon-burning shell. In the models of Frischknecht et al. (2016), the carbon-
burning shell contribution is less than 10 % at sub-solar metallicities.
During the core helium burning phase, rotation-induced mixing transports 12C
and 16O from the He-core to the H-shell, leading to substantial overproduction
of 13C and 14N through the CNO cycle. These isotopes are later engulfed by the
growing convective He-burning core, leading to a more efficient activation of
neutron sources through the
14N($\alpha,\gamma$)18F($\beta+$)18O($\alpha,\gamma$)22Ne($\alpha,n$)25Mg
chain and the 13C($\alpha,n$) reaction (Pignatari et al., 2008; Frischknecht
et al., 2016; Choplin et al., 2018; Limongi & Chieffi, 2018; Banerjee et al.,
2019). Specifically, Cescutti et al. (2013) have shown that the Sr/Ba scatter
at low metallicity of the observed Milky Way halo stars can be reproduced if
the contribution of fast rotating massive stars is included.

The contributions of rotating massive stars to chemical evolution are subject
to stellar and nuclear uncertainties. One key nuclear uncertainty is the ratio
between the 17O($\alpha,n$)20Ne and 17O($\alpha,\gamma$)21Ne reaction rates
(Frischknecht et al., 2012; Choplin et al., 2018). At all metallicities, 16O
is abundant in both the helium-burning core and the shell carbon burning. It
is a strong neutron absorber, producing copious amounts of 17O and loss of
neutron flux. The neutrons absorbed by 16O may be recovered via the
17O($\alpha,n$)20Ne reaction. Alternatively, the 17O($\alpha,\gamma$)21Ne
reaction permanently absorbs the neutron, preventing it from contributing to
s-process nucleosynthesis. The 17O($\alpha,n$)20Ne to 17O($\alpha,\gamma$)21Ne
ratio, therefore, determines the fraction of neutrons released and the
strength of the s-process.

At He-core burning temperatures (0.2-0.3 GK), the 17O+$\alpha$ reactions are
dominated by resonant contributions from states between $E_{x}=7600$ and
$8100$ keV ($E_{r}\approx 250-750$ keV) in 21Ne. However key properties of the
most important states have not been measured. These unknown properties lead to
large uncertainties in the reaction rates though no previous rate estimates
have provided uncertainties (Best et al., 2013; Caughlan & Fowler, 1988).
Here, a new calculation of the 17O$+\alpha$ reaction rates is presented,
providing realistic uncertainties for the first time. The rates are derived
from a rigorous evaluation of excited states in 21Ne based on a new
measurement of the 20Ne($d,p$)21Ne reaction, together with an evaluation of
existing literature.

##  2 Existing literature on 21Ne levels

Some experimental data on levels in 21Ne are available from studies using a
variety of populating reactions. As these works are used in the present paper
to constrain the properties of observed levels, they are briefly summarised.

Direct measurements of 17O($\alpha,\gamma$)21Ne (Best et al., 2011; Taggart et
al., 2019; Williams et al., 2022) and 17O($\alpha,n$)20Ne (Best et al., 2013)
have been performed but were not able to observe all states within the Gamow
window due to the prohibitively low cross-sections. The
17O($\alpha,\gamma$)21Ne and 17O($\alpha,n$)20Ne measurements in forward
kinematics of Best et al. (2011, 2013) used anodised tantalum-oxide targets
with enriched 17O and a germanium detector. Measurements of the
17O($\alpha,\gamma$)21Ne radiative-capture reaction were performed in inverse
kinematics with the DRAGON recoil separator by Taggart et al. (2019) and
Williams et al. (2022).

The 17O($\alpha,\gamma$)21Ne measurement of Best et al. (2011) observed three
resonances at $E_{r}=811$, $1122$, and $1311$ keV. The subsequent study of the
same reaction by Taggart et al. (2019) observed resonances at $E_{r}=633$,
$721$, $810$ and $1122$ keV. Some of the resonance strengths were revised by
Williams et al. (2022) based on new DRAGON measurements with higher beam
intensities, including an upper limit for the strength of the $E_{r}=612$-keV
resonance in 17O($\alpha,\gamma$)21Ne.

Best et al. (2013) also measured the 17O($\alpha,n_{\mathrm{tot}}$)20Ne
reaction using tantalum-oxide targets. The neutrons were detected in 20 3He
counters within a polyethylene moderator. The 17O($\alpha,n_{1}\gamma$)20Ne
reaction was measured by observation of $\gamma$ rays depopulating the first-
excited state of 20Ne. The excitation function was measured between
$E_{cm}=650$ and $1860$ keV. A number of resonance structures were observed
and an $R$-matrix analysis was performed. Rates of the
17O($\alpha,\gamma$)21Ne and 17O($\alpha,n$)20Ne reactions were reported.
These rates included estimated contributions from unmeasured resonances below
the region scanned in the excitation function, making assumptions about
unknown spins and parities in combination with known information about levels
in 21Ne (Firestone, 2015).

A subsequent uncertainty analysis of the measured 17O($\alpha$,n)20Ne
excitation functions presented in the literature was performed by Mohr (2017),
which raised some doubts about the consistency of the data presented in Best
et al. (2013) in comparison to historical literature. However, the study found
the largest discrepancy was in the high-energy cross section affecting the
reaction rate at temperatures above 1 GK. At those energies, Mohr (2017)
suggested that the cross section presented in Best et al. (2013) should be
lowered by a factor of 2-3. At lower energies, though, their data appears to
be in better agreement with the literature (see Fig. 5 of Mohr (2017)).
Furthermore, at the temperatures of interest in the present work (0.2-0.3 GK),
the reaction rate is dominated by un-observed resonances that we treat in a
rigorous manner and were not considered in Mohr (2017), which focused on
applying the statistical model to compute reaction rates.

Additional key information about the excited states has been extracted from
earlier measurements, including scattering and transfer studies. Cohn & Fowler
(1959) performed a neutron resonance-scattering experiment. Neutrons were
produced by bombarding a zirconium tritide and 7Li targets with protons. The
incident neutron energies were varied by changing the bombarding energy of the
proton beam. Information on excitation energies, widths, and spins and
parities for 21Ne levels above the neutron threshold was determined. The
uncertainty in the neutron energy scale was less than 5 keV and the neutron
energy resolution was around 13 keV. Resonances with widths as small as
$\Gamma=6(2)$ keV were observed in this experiment.

Various $\gamma$-ray studies using reactions such as 18O($\alpha,n\gamma$)21Ne
(Rolfs et al., 1972; Hoffman et al., 1989), 12C(13C,$\alpha$)21Ne
(Andritsopoulos et al., 1981; Hallock et al., 1975) and 16O(${}^{7}Li,np$)21Ne
(Thummerer et al., 2003; Wheldon, C. et al., 2005) provided additional
information. Spins and parities of excited levels were assigned on the basis
of observed decay branching and angular distributions. The observation of
$\gamma$-ray decays from neutron-unbound levels in 21Ne can be used in
combination with other observables to rule out certain spins and parities
since the neutron partial width of a state cannot exceed the $\gamma$-ray
partial width by a significant factor if a $\gamma$-ray transition is observed
depopulating the state.

The 20Ne($d,p$)21Ne single-neutron transfer reaction has previously been
studied by Stanford & Quin (1980). In this experiment, the differential cross
sections and analysing powers of eight states in 21Ne were measured using a
vector polarised deuteron beam on an enriched 20Ne gas cell target. The
resolution was around 100 keV, largely due to the significant energy loss
through the target for the deuteron beam. Large deviations between the
calculated differential cross sections from the DWBA and the experimental data
were observed at higher angles. This is likely due to the strong deformation
of the 20Ne core. Inelastic excitation of deformed nuclei means that treatment
with the DWBA may no longer accurately reflect the observed differential cross
section (e.g. the 20Ne($d,^{3}$He)19F study of Dudek & Edens (1971). For this
reason, analysis of the strength of the 20Ne($d,p$)21Ne single-neutron
transfer reaction should be limited to centre-of-mass angles below around 30
degrees.

##  3 Experimental details

The experiment was performed at the Triangle Universities Nuclear Laboratory
(TUNL). Deuterons were accelerated to 14 MeV through the tandem accelerator
with an energy precision of better than 1 keV. Typical currents recorded with
an electron-suppressed beamstop downstream from the target were 300-575 nA,
except at the most forward angle where currents were limited to around 90 nA.

The target consisted of 20Ne implanted into a 44 $\mu$g$/$cm2 carbon foil with
a Ne/C abundance ratio of 4.3$\pm$0.3$\%$ determined by Rutherford
Backscattering Spectrometry. The neon content was monitored with deuteron
elastic scattering at $\theta_{lab}$ = 25° after collecting 20Ne($d,p$)21Ne
data at each angle to account for any target degradation. Uncertainties in
neon target areal density were of the order of 7$\%$.
Reaction products entered the TUNL Enge split-pole spectrograph through a
0.54$\pm$0.01 msr aperture and were momentum analyzed at the focal plane. The
focal-plane detector comprised of two position-sensitive gas avalanche
counters, a $\Delta$E proportional counter, and a scintillation counter
(Marshall et al., 2018). Protons were identified using a $\Delta$E-position
cut. Data were collected at five laboratory angles: 10°, 15°, 20°, 25° and
38°. Additional data for background characterisation were collected using a
natural carbon target at each angle.

A quadratic internal calibration using the 21Ne states at
$E_{\mathrm{x}}=6609(1)$, $7420(1)$, $8069(2)$ and $8189(2)$ keV (Firestone,
2015) was used to convert focal-plane position to excitation energy. The
Bayesian method outlined by Marshall et al. (2018) gave realistic
$E_{\mathrm{x}}$ uncertainties, explicitly including the statistical
uncertainties in the fitted peak centroids and the systematic uncertainties
from the focal-plane calibration. For peaks observed at multiple angles, our
recommended energy was obtained from a weighted average of individual
measurements. The reported energy uncertainty was conservatively constrained
to be no smaller than the uncertainty at a single angle.

An example focal-plane spectrum for the astrophysically important region is
shown in Fig. 1.

Figure 1: A proton focal-plane position spectrum from the 20Ne($d,p$)21Ne
reaction at $\theta_{lab}$ = 38°. Peaks are labelled with their corresponding
excitation energy in 21Ne (keV). The carbon spectrum (arbitrary scale) is
shown in red, highlighting the location of the 17O background contamination
peak.

To extract peak intensities and positions, the focal plane was divided into
regions based on the behaviour of the background reactions determined from the
carbon target. For more details on the peak fitting see Frost-Schenk (2020).
Angular distributions were extracted from the yields for each state observed
at each angle accounting for beam on target, aperture solid angle, target
content and dead time, which was typically below 9$\%$.

##  4 Calculation of neutron widths

The code fresco (Thompson, 1988) was used to calculate differential cross-
sections under the assumption of a single-step reaction, in order to assign
the transferred angular momentum, $\ell_{n}$, and extract the neutron width,
$\Gamma_{n}$. Calculations were performed using the first-order distorted-wave
Born approximation since at these low energies the breakup effects on the
deuteron in the entrance channel are minimal (Johnson & Soper, 1970). Past
20Ne($d,p$)21Ne transfer reactions reported in Stanford & Quin (1980) have
observed significant deviation between the expected differential cross section
and the observed data at angles higher than $\theta_{lab}=25$° degrees. For
this reason, $\theta_{lab}=38$° data was not used to assign $\ell_{n}$. The
$\theta_{lab}=38$° data were used to determine $E_{\mathrm{x}}$ and constrain
the total width for observed states.

The optical model potentials for 20Ne$+d$, 21Ne$+p$, 20Ne$+n$, $n+p$ and
20Ne$+p$ were taken from An & Cai (2006), Varner et al. (1991), Madland
(1997), Yahiro et al. (1986) and Menet et al. (1971), respectively.
Spectroscopic factors ($C^{2}S$) relating the DWBA and experimental cross
sections:

$\frac{d\sigma}{d\Omega}_{\mathrm{exp}}=C^{2}S\frac{d\sigma}{d\Omega}_{\mathrm{DWBA}}$
(1)

were found by normalising the calculations to the experimental data. Example
differential cross sections for four states are shown in Fig. 2. Two states
with known $J^{\pi}$ (the $J^{\pi}=\nicefrac{{3}}{{2}}^{-}$,
$E_{\mathrm{x}}=7981$-keV state and $J^{\pi}=\nicefrac{{3}}{{2}}^{+}$,
$8068$-keV state) have been included to demonstrate that the DWBA calculations
reproduce the data well. Two states with inconclusive $J^{\pi}$ assignments
are also shown.

Figure 2: Experimental differential cross sections for some of the relevant
states discussed in the present work. The $7981$\- and $8068$-keV states have
known $J^{\pi}$ (Cohn & Fowler, 1959; Firestone, 2015) while the $7602$\- and
$7820$-keV states have unknown $J^{\pi}$. Note that, while the 38° data are
included in the plot, they are not used in the DWBA analysis.

The wave-functions from fresco, $\phi(r)$, were used to compute the neutron
widths, $\Gamma_{n}$ as in Iliadis (1997):

$\Gamma_{n}=2P_{\ell}(E_{n},a)\frac{\hbar^{2}a}{2\mu}\,C^{2}S\,|\phi(a)|^{2},$
(2)

where $P_{\ell}(E_{n},a)$ is the penetrability of a neutron of energy $E_{n}$
and orbital angular momentum $\ell$ evaluated at the radius $a$; $\mu$ is the
reduced mass. The radius, $a$, was chosen to be where the 20Ne$+n$ wave-
function is at 99% of the asymptotic value (Meyer et al., 2020) and varied for
different binding energies and $\ell_{n}$. We used the weak-binding
approximation, calculating at various positive binding energies and
extrapolating with a quadratic function to the negative, physical neutron
binding energy (Meyer et al., 2020). The uncertainties resulting from the
calculations are much smaller than the uncertainty in the absolute
normalisation from other sources.

##  5 Evaluation of 17O$+\alpha$ reaction rates

Table 1 summarises the spectroscopic information on relevant states in 21Ne
above the $\alpha$-particle threshold. We discuss below the spin-parity
assignments of astrophysically important states from which we produce
physically-motivated reaction rates, with uncertainties for the first time. A
comprehensive discussion on all of the states observed in this work will be
presented in a forthcoming paper.

The $E_{\mathrm{x}}=7559$-, $7820$-, $8146$\- and $8189$-keV states all have
differential cross sections which are consistent with $\ell_{n}=1$ or $2$
assignments. For $\ell_{n}=1$ the $\Gamma_{n}$ are, for
$J^{\pi}=\nicefrac{{1}}{{2}}^{-}$, $25(1)$, $20(3)$, $19(5)$ and $130(13)$,
respectively (for $J^{\pi}=\nicefrac{{3}}{{2}}^{-}$, $\Gamma_{n}=14(1)$,
$11(2)$, $11(3)$, and $74(7)$ keV, respectively). Resonances with $\Gamma_{n}$
of this size would both have been observed in the neutron-scattering study
reported in Cohn & Fowler (1959) and would additionally result in visible
broadening of the states in the focal-plane spectrum. Since neither of these
are observed, $\ell_{n}=1$ is ruled out for all of these states.

The differential cross section of the $E_{\mathrm{x}}=7602$-keV state is
consistent with $\ell_{n}=2-4$. This state has been observed in $\gamma$-ray
data of Rolfs et al. (1972). An $\ell_{n}=2$ assignment results in a neutron
width of more than 100 eV. Based on lifetimes reported in that work, this
neutron width greatly exceeds realistic $\gamma$-ray partial widths. It is,
therefore, unlikely that $\gamma$-ray decay from this state would be observed
if the $\ell_{n}=2$ assignment were made. We therefore assign this level
$\ell_{n}=3$ or $\ell_{n}=4$.

Close to $E_{\mathrm{x}}=7982$ keV are two states, one narrow at
$E_{\mathrm{x}}=7982.1(6)$ keV (Firestone, 2015; Taggart et al., 2019), likely
not observed in this experiment, and a broader one at $E_{\mathrm{x}}=7981(2)$
keV, likely the state populated in the present work. The neutron width
($\Gamma_{n}=6(2)$ keV), spin and parity ($J^{\pi}=\nicefrac{{3}}{{2}}^{-}$)
of the $E_{\mathrm{x}}=7981(2)$-keV state are known from 20Ne$+n$ resonance
scattering (Cohn & Fowler, 1959). From the current data, we determine
$\Gamma_{n}=14(5)$ keV.

Table 1: Spectroscopic information for relevant states above the
$\alpha$-particle threshold in 21Ne. Previous excitation energies from other
sources are given in the second column. The neutron partial widths
($\Gamma_{n}$) are those determined in the present experiment. Widths given in
bold are measured or experimentally constrained values. The
$\Gamma_{\gamma}=0.20(14)$ eV are taken from the average of measured lifetimes
in Rolfs et al. (1972) except where noted in the final column to preserve the
$\Gamma_{n}/\Gamma_{\gamma}$ ratio of Best et al. (2013) for resonances for
which no updated information is available. Resonance information on higher
resonances is taken from Best et al. (2013).

$E_{\mathrm{x}}$ [keV] |  Previous $E_{\mathrm{x}}$ [keV] |  $E_{r,\alpha}$ [keV] | $2J^{\pi}$ | $\ell_{\alpha}$ | $\ell_{n}$ |  $\Gamma_{\alpha}$ [eV] |  $\Gamma_{n}$ [eV] | Comments
---|---|---|---|---|---|---|---|---
$7420.4(15)^{*}$ | $7420.3(10)$ | $72.5(15)$ | $(5,7)^{-}$ | $1$ | $3$ | $1.2\times 10^{-33}$ | $\mathbf{14(1),11(1)}$ |
$7470(2)$ | $7465(10)$ | $122(2)$ | $(1,3)^{-}$ | $3,1$ | $1$ |  $7.9\times 10^{-24}$, $3.9\times 10^{-22}$ | $200(140)$ |  Adopt $\frac{\Gamma_{\gamma}}{\Gamma_{n}}=10^{-3}{}^{a}$
$7559.1(15)$ | $7547(10)$ | $211.2(15)$ | $(3,5)^{+}$ | $2,0$ | $2,2$ |  $2.4\times 10^{-14}$, $2.5\times 10^{-13}$ |  $\mathbf{570(30)}$, $\mathbf{420(20)}$ |
$7602.0(15)$ | $7600(5)$ | $254.1(15)$ |  $(5,7)^{-}$ $(7,9)^{+}$ |  $1$ $2$ |  $3$ $4$ |  $2.6\times 10^{-11}$ $5.6\times 10^{-12}$ |  $\mathbf{8(2)}$, $\mathbf{6(2)}$ $\mathbf{0.4(1)}$, $\mathbf{0.3(1)}$ |
$7619.9(10)$ | $7628(10)$ | $272.0(10)$ | $3^{-}{}^{f}$ | $1$ | $1$ | $1.7\times 10^{-10}$ | $\mathbf{8000(1000)}$ |
$7655.7(22)$ | $7648(2)$ | $307.8(22)$ | $7^{+}{}^{h}$ | $2$ | $4$ | $9.8\times 10^{-10}$ | $0.10(7)$ |  Adopt $\frac{\Gamma_{\gamma}}{\Gamma_{n}}=2^{a}$
$7748.8(17)$ | $7740(10)$ | $400.9(17)$ | $5^{+}{}^{a}$ | $0$ | $2$ | $5.2\times 10^{-6}$ | $200(140)$ |  Adopt $\frac{\Gamma_{\gamma}}{\Gamma_{n}}=10^{-3}{}^{a}$
$7820.1(15)$ | $7810(10)$ | $472.2(15)$ | $(3,5)^{+}$ | $2,0$ | $2$ |  $1.8\times 10^{-5}$, $1.7\times 10^{-4}$ |  $\mathbf{560(90)}$, $\mathbf{400(60)}$ |
$7960(2)$ | $7960.9(13)$ | $612(2)$ | $11^{-}{}^{g}$ | $3$ | $5$ | $5.3\times 10^{-8}$ | $0.10(7)$ |  Adopt $\frac{\Gamma_{\gamma}}{\Gamma_{n}}=2^{a}$
$7981(2)$ | $7980(10)$ | $633(2)$ | $3^{-}{}^{f}$ | $1$ | $1$ | $1.9\times 10^{-2}$ | $\mathbf{14000(5000)}$ |
$7982.1(7)$ | $7982.1(6)$ | $634.2(7)$ |  |  |  | $7.5(15)\times 10^{-6}$ |  |  From $\omega\gamma_{(\alpha,\gamma)}=4.98(97)$ $\mu$eVe
$8008(2)$ | $8009(10)$ | $660(2)$ | $1^{-}{}^{f}$ | $3$ | $1$ | $1.2\times 10^{-3}$ | $0.20(14)$ |  $\Gamma_{\gamma}=2.0(14)\times 10^{-4}$ eV
$8069(1)^{*}$ | $8069(2)$ | $721(1)$ | $3^{+}{}^{a}$ | $2$ | $2$ | $\mathbf{46.2(46)\times 10^{-3}}$ | $\mathbf{1600(200)}$ |  $\Gamma_{\gamma}=0.54(35)$ eVd
$8146(1)$ | $8146(2)$ | $798(1)$ | $3^{+}{}^{a}$ | $2,0$ | $2$ | $\mathbf{54.7(55)\times 10^{-3}}$ |  $\mathbf{550(150)}$, $\mathbf{400(100)}$ |
$8159(2)^{b}$ | $8155.0(10)$ | $811(2)$ | $9^{+}{}^{g}$ | $2$ | $4$ |  |  |  $\omega\gamma_{(\alpha,,\gamma)}=7.72(55)^{c}$ meV
$8160(2)$ | $8160(2)$ | $812(2)$ | $5^{+}{}^{a}$ | $0$ | $2$ | $\mathbf{1.6(2)\times 10^{-3}}$ | $\mathbf{23000(2300)}$ |  Adopted from Ref. Williams et al. (2022)

${}^{*}:$ used in calibration
$a:$ Best et al. (2013)
$b:$ Best et al. (2011)
$c:$ Weighted average of Williams et al. (2022) and Best et al. (2011).
$d:$ To preserve the measured $\omega\gamma_{(\alpha,\gamma)}$ from Ref.
Williams et al. (2022).
$e:$ Williams et al. (2022)
$f:$ Cohn & Fowler (1959)
$g:$ Thummerer et al. (2003)
$h:$ Hoffman et al. (1989)

Reaction rates were calculated using the RatesMC Monte Carlo code Longland et
al. (2010); Sallaska et al. (2013). Data on directly measured
17O($\alpha,\gamma$)21Ne and 17O($\alpha,n$)20Ne resonances were taken from
Best et al. (2011, 2013); Williams et al. (2022). The weighted mean with
inflated uncertainties described by Longland et al. (2012) was used for the
17O($\alpha,\gamma$)21Ne resonance strengths to account for systematic
uncertainties. For many of the resonances listed in Best et al. (2013) there
are no listed uncertainties. These were arbitrarily assumed to be 10% for all
widths. For any resonances for which all partial widths could be estimated,
the reaction rate was numerically integrated.

For resonances with no measured $\Gamma_{\alpha}$, the reduced
$\alpha$-particle width was sampled from a Porter-Thomas distribution (see
Sallaska et al., 2013; Longland et al., 2012, for details). The neutron widths
were, where available, taken from the present work. $\Gamma_{\gamma}=0.20(14)$
eV was used, from the average of the lifetimes in Rolfs et al. (1972). For
resonances where the $\Gamma_{n}$ and $\Gamma_{\gamma}$ are unknown, the ratio
of these widths is adopted from Best et al. (2013). Importantly, we adopt the
tentative $J^{\pi}=\nicefrac{{7}}{{2}}^{+}$ assignment for the $E_{r}=308$-keV
resonance from Firestone (2015) which reduces the contribution of this state
to the 17O($\alpha,\gamma$)21Ne reaction rate compared to Best et al. (2013)
in which a $J^{\pi}=\nicefrac{{5}}{{2}}^{+}$ assignment was used. We
furthermore note that a $\nicefrac{{5}}{{2}}^{+}$ assignment is inconsistent
with systematic trends in 21Ne in which no other
$J^{\pi}=\nicefrac{{5}}{{2}}^{+}$ states at a similar excitation energy decay
by $\gamma$-ray emission.

The $E_{\mathrm{x}}=7749$-keV state was only observed at one angle due to 16O
contamination. The peak observed in the present experiment does not appear in
the background spectrum and was therefore assumed to be a 21Ne state, but
confirmation of this state with additional measurements is required. An
angular distribution could not be extracted and the spin remains
unconstrained. For the calculation of the rate we adopted the assumptions of
Best et al. (2013).

Figure 3: The ratios of the 17O($\alpha,n$)20Ne and 17O($\alpha,\gamma$)21Ne
reaction rates. The solid black line shows the ratio of the median rates in
the current calculation and the dashed red lines show the ratios of the upper
(lower) 17O($\alpha,n$)20Ne rate to the lower (upper) 17O($\alpha,\gamma$)21Ne
reaction rates. The green dotted line shows the ratio of the rates from Best
et al. (2013) The purple dashed-dotted line shows the ratio of the rates from
Caughlan & Fowler (1988).

For the 17O($\alpha,\gamma$)21Ne reaction, the Monte-Carlo calculations show
that the dominant contributions are from the resonances at
$E_{\mathrm{r}}=308$, $634$ and $811$keV ($E_{\mathrm{x}}=7656$, $7961$,
$7982$ and $8159$ keV). The $634$\- and $811$-keV resonances have measured
strengths (Williams et al., 2022; Taggart et al., 2019; Best et al., 2011).
For the 17O($\alpha,n$)20Ne reaction, the dominant contributions are from the
$E_{\mathrm{r}}=401$-, $472$\- and $721$-keV resonances
($E_{\mathrm{x}}=7749$, $7820$ and $8069$ keV) with a small contribution from
the $E_{\mathrm{r}}=633$-keV resonance. The dominant contributions to the
uncertainty in the reaction-rate ratio are the unknown $\Gamma_{\alpha}$
partial widths.

The 17O($\alpha,n$)20Ne to 17O($\alpha,\gamma$)21Ne reaction-rate ratio for
our median rates is presented in Fig. 3. Also shown are $1\sigma$ ‘high’ and
‘low’ ratios from our work, and those from Caughlan & Fowler (1988) and Best
et al. (2013). The present ratio is significantly higher than that of Best et
al. between 0.25 and 0.7 GK for a number of reasons. There are some
inconsistencies in the data in Table II of Best et al. (2013) (e.g. the
$E_{r}=308$-keV resonance state), for which the listed resonance strengths are
in disagreement with the $\Gamma_{n}$ and $\Gamma_{\gamma}$ ratio (Best,
2021). We have, additionally, changed spin-parity assignments where
appropriate resulting in some changes in contributions of states to the
reaction rates. Lastly, we have utilised the direct 17O($\alpha,\gamma$)21Ne
measurements from DRAGON (Williams et al., 2022).

Contribution plots for the two reactions are shown in Fig. 4 (see caption for
details). For the 17O($\alpha,\gamma$)21Ne reaction, the main contribution
within the astrophysically relevant region is the $E_{r}=308$-keV resonance,
for which no estimate of the $\alpha$-particle width is yet available. More
resonances can potentially contribute to the 17O($\alpha,n$)20Ne reaction,
since for many of these resonances the neutron partial width is known to be
much larger than the $\gamma$-ray partial width. For most resonances,
therefore, the $\gamma$-ray decay is vanishingly small and these states cannot
meaningfully contribute to the flux of abundances through the
17O($\alpha,\gamma$)21Ne reaction.

Figure 4: Contribution plots for (top) the 17O($\alpha,\gamma$)21Ne reaction
and (bottom) the 17O($\alpha,n$)20Ne reaction indicating which resonances
dominate the reaction rate within the Gamow window (vertical lines). The
coloured bands show the uncertainty in the contribution of each resonance to
the total rate.

The 17O($\alpha,\gamma$)21Ne and 17O($\alpha,n$)20Ne reaction rates are shown
in Tables 2 and 3, respectively.

Table 2: The 17O($\alpha,\gamma$)21Ne reaction rate. The lower, median and upper columns correspond to the 32%, 50% and 68% intervals for the computed reaction rate. The reaction rates are in units of cm3 mol-1 s-1. $T$ [GK] | Lower | Recommended | Upper
---|---|---|---
0.100 | 2.18$\times$10-22 | 1.20$\times$10-21 | 5.33$\times$10-21
0.110 | 3.67$\times$10-21 | 2.05$\times$10-20 | 9.19$\times$10-20
0.120 | 3.91$\times$10-20 | 2.21$\times$10-19 | 1.01$\times$10-18
0.130 | 2.94$\times$10-19 | 1.68$\times$10-18 | 7.95$\times$10-18
0.140 | 1.68$\times$10-18 | 9.59$\times$10-18 | 4.66$\times$10-17
0.150 | 7.71$\times$10-18 | 4.38$\times$10-17 | 2.18$\times$10-16
0.160 | 3.00$\times$10-17 | 1.67$\times$10-16 | 8.43$\times$10-16
0.170 | 1.01$\times$10-16 | 5.46$\times$10-16 | 2.81$\times$10-15
0.180 | 2.99$\times$10-16 | 1.59$\times$10-15 | 8.22$\times$10-15
0.200 | 2.13$\times$10-15 | 1.02$\times$10-14 | 5.13$\times$10-14
0.250 | 3.23$\times$10-13 | 5.84$\times$10-13 | 1.71$\times$10-12
0.300 | 4.54$\times$10-11 | 5.53$\times$10-11 | 6.94$\times$10-11
0.350 | 2.43$\times$10-09 | 2.70$\times$10-09 | 3.02$\times$10-09
0.400 | 5.24$\times$10-08 | 5.68$\times$10-08 | 6.18$\times$10-08
0.450 | 5.76$\times$10-07 | 6.21$\times$10-07 | 6.69$\times$10-07
0.500 | 3.90$\times$10-06 | 4.19$\times$10-06 | 4.50$\times$10-06
0.600 | 6.74$\times$10-05 | 7.21$\times$10-05 | 7.71$\times$10-05
0.700 | 5.03$\times$10-04 | 5.37$\times$10-04 | 5.72$\times$10-04
0.800 | 2.23$\times$10-03 | 2.37$\times$10-03 | 2.53$\times$10-03
0.900 | 7.04$\times$10-03 | 7.48$\times$10-03 | 7.95$\times$10-03
1.000 | 1.76$\times$10-02 | 1.87$\times$10-02 | 1.98$\times$10-02
Table 3: As Table 2 but for the 17O($\alpha,n$)20Ne reaction. $T$ [GK] | Lower | Recommended | Upper
---|---|---|---
0.100 | 7.04$\times$10-21 | 2.52$\times$10-20 | 8.75$\times$10-20
0.110 | 9.98$\times$10-20 | 3.24$\times$10-19 | 1.05$\times$10-18
0.120 | 1.01$\times$10-18 | 3.00$\times$10-18 | 9.13$\times$10-18
0.130 | 7.92$\times$10-18 | 2.22$\times$10-17 | 6.40$\times$10-17
0.140 | 5.02$\times$10-17 | 1.38$\times$10-16 | 3.91$\times$10-16
0.150 | 2.66$\times$10-16 | 7.30$\times$10-16 | 2.14$\times$10-15
0.160 | 1.20$\times$10-15 | 3.40$\times$10-15 | 1.07$\times$10-14
0.170 | 4.75$\times$10-15 | 1.42$\times$10-14 | 4.78$\times$10-14
0.180 | 1.70$\times$10-14 | 5.30$\times$10-14 | 1.91$\times$10-13
0.200 | 1.70$\times$10-13 | 5.58$\times$10-13 | 2.23$\times$10-12
0.250 | 3.63$\times$10-11 | 7.30$\times$10-11 | 2.35$\times$10-10
0.300 | 4.72$\times$10-09 | 5.78$\times$10-09 | 9.44$\times$10-09
0.350 | 1.92$\times$10-07 | 2.15$\times$10-07 | 2.57$\times$10-07
0.400 | 3.24$\times$10-06 | 3.54$\times$10-06 | 3.95$\times$10-06
0.450 | 2.99$\times$10-05 | 3.23$\times$10-05 | 3.54$\times$10-05
0.500 | 1.80$\times$10-04 | 1.94$\times$10-04 | 2.10$\times$10-04
0.600 | 2.90$\times$10-03 | 3.08$\times$10-03 | 3.28$\times$10-03
0.700 | 2.46$\times$10-02 | 2.59$\times$10-02 | 2.72$\times$10-02
0.800 | 1.49$\times$10-01 | 1.55$\times$10-01 | 1.61$\times$10-01
0.900 | 7.29$\times$10-01 | 7.56$\times$10-01 | 7.84$\times$10-01
1.000 | 3.03$\times$10+00 | 3.13$\times$10+00 | 3.25$\times$10+00

##  6 Astrophysical Implications

We tested the impact of the new 17O($\alpha,n$)20Ne and
17O($\alpha,\gamma$)21Ne rates on s-process nucleosynthesis using a simplified
one-zone nucleosynthesis code mimicking core helium burning. Details on this
code can be found in Choplin et al. (2016). The code was also used in Placco
et al. (2020) to make comparisons with an observed star enriched in trans-iron
elements. We follow the central temperature and density profiles obtained from
a complete rotating 25 $M_{\odot}$ stellar model at a metallicity of $10^{-3}$
in mass fraction, computed with the Geneva stellar evolution code (Eggenberger
et al., 2008). The initial composition of the one-zone code is extracted from
the core of this stellar model, at core helium-burning ignition. To mimic
rotation, 13C and 14N are injected (cf. Sect. 1) at a constant rate (expressed
in $M_{\odot}$ yr-1) during the nucleosynthesis calculation (Choplin et al.,
2016). During injection, $100$ times more 14N as 13C is injected as a typical
value in full stellar models (e.g. Fig. 9 in Choplin et al., 2018). This
factor of $\sim 100$ corresponds to the CNO 14N/13C equilibrium ratio at
$T\sim 80$ MK, which is found at the bottom of the H-burning shell in massive
stars. The injection rate was calibrated so as to reproduce the central
abundances of a full rotating stellar model at the end of the core helium-
burning phase. For this calibration, we used the same rates as in the full
stellar model, namely the rates of Best et al. (2013) for
17O($\alpha,\gamma$)21Ne and 17O($\alpha,n$)20Ne, the rates of Longland et al.
(2012) for 22Ne($\alpha,\gamma$)26Mg and 22Ne($\alpha,n$)25Mg and the rate
from Guo et al. (2012) for 13C($\alpha,n$)16O. The adopted standard injection
rate is $2.5\times 10^{-7}M_{\odot}$ yr-1 and produces the overproduction
factors111The overproduction factors are expressed as $X_{\rm f}/X_{\rm i}$
where $X_{\rm{i/f}}$ represent the initial/final mass fractions. shown by the
solid red line in Fig 5. For most elements, it deviates by less than 5 % from
the full model (green pattern). Cases with higher (red dotted line) and lower
(red dashed line) injection rates are shown for comparison. Without injection,
we obtain a typical weak s-process pattern from non-rotating massive stars
(black pattern).

In the left panel of Fig. 6, all one-zone models were computed with the
standard injection rate of $2.5\times 10^{-7}M_{\odot}$ yr-1 obtained from the
calibration discussed previously. The only differences between the models
shown in Fig. 6 (left panel) and the model shown by the solid red line in Fig.
5 are the rates of 17O($\alpha,n$)20Ne and 17O($\alpha,\gamma$)21Ne. The
recommended reaction rates were used (fs21), as well as the limiting cases of
the minimum (fs21_min) and maximum (fs21_max) ($\alpha,n$)/($\alpha,\gamma$)
ratio. These results show that the s-process in rotating massive stars is
likely to continue at least to barium, and potentially up to lead for the
largest ($\alpha,n$)/$(\alpha,\gamma$) reaction-rate ratio. The scatter for
elements with atomic number $Z>40$ goes up to about 2 dex. The bg13 and
fs21_min sets have the lowest 17O($\alpha,n$)20Ne/17O($\alpha,\gamma$)21Ne
ratios (Fig. 3) hence giving the lowest yields (green and red dotted pattern).
The cf88 and fs21 sets with higher ($\alpha,n$)/($\alpha,\gamma$) ratios
substantially produce elements with $Z>55$ and the fs21_max set shows the
highest yields, as expected from the high ($\alpha,n$)/($\alpha,\gamma$)
ratio.

As an estimate of the impact of the new rate at very low metallicity, we
include in the right panel of Fig. 6 a one-zone calculation similar to the
fs21 model (black pattern) but computed with an initial composition
corresponding to a metallicity of $10^{-5}$ in mass fraction (red pattern).
This shows that a lower initial metallicity combined with a similar neutron
production leads to a stronger overproduction of elements heavier than atomic
number $Z\sim 50$ at the expense of lighter elements. This is even more
visible if considering a higher injection rate (green pattern) which would
correspond to a more efficient rotational mixing in full stellar models. A
higher injection rate at lower metallicity is not unrealistic since rotational
mixing is expected to be more efficient with decreasing metallicity (e.g.
Maeder & Meynet, 2001). We note that contrary to the fs21 rates, the bg13
rates at lower metallicity do not lead to significant changes in the
overproduction factors (compare the green patterns in the two panels of Fig.
6) Full stellar models would be required to get a more accurate estimate of
the overproduction factors.

Figure 5: Overproduction factors of a one zone nucleosynthesis model mimicking
the core helium burning phase of a rotating massive star at low metallicity.
Different injection cases (see text for details) are considered: no injection
(black pattern) and injection at three different rates (red patterns). The
green pattern shows the outputs of the central layer of a full stellar model
at the end of the core helium-burning stage. Here all models were computed
with the rates of Best et al. (2013) for 17O($\alpha,\gamma$) and
17O($\alpha,n$). The 4 vertical lines highlight the elements Fe, Sr, Ba and
Pb.

Figure 6:  Overproduction factors of a one zone nucleosynthesis model
mimicking the core helium burning phase of a rotating massive star at low
metallicity. Left panel: the five sets of rates shown in Fig. 3 are tested:
cf88 (Caughlan & Fowler, 1988), bg13 (Best et al., 2013), the recommended
rates derived in this work (fs21) plus the two limiting cases shown in Fig. 3.
The 4 vertical lines highlight the elements Fe, Sr, Ba and Pb. Right panel:
the fs21 model at a metallicity of $10^{-3}$ in mass fraction is shown again
(black pattern) together with the same model but with initial abundances
corresponding to a metallicity of $10^{-5}$ (red pattern). The blue model is
computed like the red one but with an injection rate 4 times higher. The bg13
model at a metallicity of $10^{-5}$ is also shown.

##  7 Conclusions

For the first time, available data on the energies, spins, parities and
partial widths of excited states in 21Ne have been thoroughly evaluated
including a careful consideration of their ambiguities and uncertainties. In
addition, states in 21Ne have been populated via the 20Ne($d,p$)21Ne reaction,
using an implanted 20Ne target. Angular distributions and neutron widths for
states within the Gamow window for massive-star He-core burning were
extracted. By combining these data with the evaluated data, reaction rates for
17O($\alpha,n$)20Ne and 17O($\alpha,\gamma$)21Ne have been calculated using
updated excitation energies, $J^{\pi}$ assignments and experimentally derived
neutron widths. Using the RatesMC Monte Carlo code, uncertainties have been
estimated consistently for the first time. Our recommended rates indicate
enhanced s-process abundances between Sr and Pb. Production of these elements
via the enhanced weak s-process in massive stars significantly shortens the
timescale for the production of Pb (otherwise only produced via the main
s-process in low-mass stars with much longer lifetimes) and provides an
alternative to the r-process for producing elements between Fe and Ba in the
early Universe. Experimental constraints on the $\alpha$-widths of the key
states in 21Ne are thus crucial to allow the production of elements above
barium in such massive stars, and so for the evolution of elements heavier
than iron in the early Universe to be understood.

## acknowledgements

The authors thank Nick Keeley and Andreas Best for useful discussions. JFS,
AML, CB and CD thank the UK Science and Technology Facilities Council (STFC).
PA thanks the trustees and staff of Claude Leon Foundation for support in the
form of a Postdoctoral Fellowship. RL, CM, FP, and KS thank the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics, for
support under Grant No. DE-SC0017799 and Contract No. DE-FG02-97ER41041. This
work was supported by the Fonds de la Recherche Scientifique-FNRS under Grant
No IISN 4.4502.19. This paper is based upon work from the ‘ChETEC’ COST Action
(CA16117), supported by COST (European Cooperation in Science and Technology).
RH acknowledges support from the IReNA AccelNet Network of Networks, supported
by the National Science Foundation under Grant No. OISE-1927130 and from the
World Premier International Research Centre Initiative (WPI Initiative), MEXT,
Japan. RH also acknowledges funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No 101008324 (ChETEC-
INFRA).

## Data Availability

The nuclear physics data used in the evaluation are available from NNDC
(https://www.nndc.bnl.gov) via ENSDEF and EXFOR. The reaction rates in tabular
form will be available as part of the Starlib reaction rate library
(https://github.com/Starlib/Rate-Library). Other data arising from the present
work are available on reasonable request to the corresponding author.

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