11institutetext: School of Physics & Astronomy, James Clerk Maxwell Building,
University of Edinburgh, Edinburgh EH9 3FD, UK 22institutetext: Extreme Light
Infrastructure – Nuclear Physics, Horia Hulubei National Institute for R&D in
Physics and Nuclear Engineering (IFIN-HH), 077125 Bucharest-Măgurele, Romania
22email<EMAIL_ADDRESS>33institutetext: Departament de Física,
Universitat Politècnica de Catalunya, EEBE, 08019 Barcelona, Spain
44institutetext: Institut d’Estudis Espacials de Catalunya (IEEC), E-08034
Barcelona, Spain

# Uncertainties in the 18F(p, $\alpha$)15O reaction rate in classical novae

D. Kahl 1122 J. José 3344 P. J. Woods 11

(Last updated: June 4, 2021)

###### Abstract

Context.  Direct observation of $\gamma$-ray emission from the decay of 18F
ejected in classical nova outbursts remains a major focus of the nuclear
astrophysics community. However, modeling the abundance of ejected 18F, and
thus the predicted detectability distance of a $\gamma$-ray signal near 511
keV emitted from these transient thermonuclear episodes, is hampered by
significant uncertainties in our knowledge of the key 18F(p, $\alpha$)
reaction rate.

Aims.  We analyze uncertainties in the most recent nuclear physics
experimental results employed to calculate the 18F(p, $\alpha$) reaction rate.
Our goal is to determine which uncertainties have the most profound influence
on the predicted abundance of 18F ejected from novae, in order to guide future
experimental works.

Methods.  We calculated a wide range of 18F(p, $\alpha$) reaction rates using
$R$-Matrix formalism, allowing us to take into account all interference
effects. Using a selection of 16 evenly-spaced rates over the full range, we
performed 16 new hydrodynamic nova simulations.

Results.  We performed one of the most thorough theoretical studies of the
impact of the 18F(p, $\alpha$) reaction in classical novae to date. The 18F(p,
$\alpha$) rate remains highly uncertain at nova temperatures, resulting in a
factor $\sim\\!10$ uncertainty in the predicted abundance of 18F ejected from
nova explosions. We also found that the abundance of 18F may be strongly
correlated with that of 19F.

Conclusions.  Despite numerous nuclear physics uncertainties affecting the
18F(p, $\alpha$) reaction rate, which are dominated by unknown interference
signs between $1/2^{+}$ and $3/2^{+}$ resonances, future experimental work
should focus on firmly and precisely determining the directly measurable
quantum properties of the subthreshold states in the compound nucleus 19Ne
near 6.13 and 6.29 MeV.

###### Key Words.:

Stars: novae, cataclysmic variables – Gamma rays: stars – Stars: abundances –
Stars: white dwarfs

##  1 Introduction

Classical novae are thermonuclear explosions that take place in the envelopes
of accreting white dwarfs in stellar, short orbital period, binary systems
(see Starrfield et al. (2008); José & Shore (2008); José (2016) for reviews).
They have been systematically observed in all wavelengths, ranging from radio
to $\gamma$-rays. However, while novae have been predicted to emit
$\gamma$-rays of specific energies around 1 MeV, associated to electron-
positron annihilation (e.g., from 18F$(\beta^{+}\nu)$) and to nuclear decay
(e.g., 22Na at 1.275 MeV and 26Al at 1.809 MeV), they have only been detected
in $\gamma$-rays for energies exceeding 100 MeV. In symbiotic binary systems,
such as V407 Cyg, this high-energy emission has been attributed to shock
acceleration in the ejected shells after interaction with the dense wind of
the red giant companion. The emission reported from several classical novae
involving less evolved stellar companions (e.g., V1324 Sco, V959 Mon, V339
Del, & V1369 Cen (Ackermann et al., 2014) has been attributed to internal
shocks in the ejecta. In this scenario, $\gamma$-rays are produced either
through a hadronic process, in which accelerated protons collide with matter
to create neutral pions that decay into $\gamma$-ray photons, or through a
leptonic process, in which visible or infrared photons reach high energies in
the interaction with high-energy electrons. While the hadronic process seems
to be favored, the exact nature of the mechanism responsible for $\gamma$-ray
production remains unknown.

The most intense $\gamma$-ray signal predicted for classical novae corresponds
to the 511-keV line due to electron-positron annihilation in the expanding
ejecta, together with a lower energy continuum (powered both by positronium
decay and by Comptonization of 511-keV photons) with a cut-off at $\sim$20–30
keV due to photoelectric absorption (Clayton & Hoyle, 1974; Leising & Clayton,
1987; Gomez-Gomar et al., 1998; Hernanz et al., 1999). As the expanding
envelope begins to become transparent to $\gamma$ radiation after around
$\sim\\!2$ hrs, the main contributor to the bulk of positrons is 18F, which
decays with a characteristic half-life, $t_{1/2}$, of about 110 min.
Therefore, all nuclear processes involved in the synthesis and destruction of
18F are of paramount importance to predict the expected amount of 18F that
survives the explosion, and powers the predicted prompt 511-keV line and the
lower-energy continuum. Synthesis of 18F in novae is driven by the CNO-cycle
reaction 16O(p, $\gamma$)17F, which is either followed by 17F(p,
$\gamma$)18Ne($\beta^{+}$)18F or by 17F($\beta^{+}$)17O(p, $\gamma$)18F.
Because of the relatively large half-life of 18F, its dominant destruction
channel is mainly 18F(p, $\alpha$)15O, plus a minor contribution from 18F(p,
$\gamma$)19Ne. The most uncertain nuclear process involved in the creation and
destruction of 18F for nova conditions is 18F(p, $\alpha$)15O (José et al.
(2006); José (2016); and references therein).

We begin in Sect. 2 with a review and analysis of the present nuclear physics
uncertainties affecting the 18F(p, $\alpha$) reaction, including the latest
experimental results. Using these evaluated experimental nuclear
uncertainties, in Sect. 3 we calculate a range of reaction rates and vary
these within new hydrodynamic nova simulations. Subsequently, in Sect. 2 we
discuss and interpret all observed differences in our model outputs. In Sect.
5, we conclude by providing a roadmap to guide future nuclear experiments.

##  2 Nuclear physics uncertainties

The 18F(p, $\alpha$)15O reaction plays a critical role in nova explosions, yet
insufficient experimental information is available to calculate a reliable,
precise rate for this reaction. A quarter of a century ago, the first
pioneering direct measurements were performed using 18F radioactive beams
(Coszach et al., 1995; Rehm et al., 1995). However, the cross section at the
lowest temperatures achieved in novae ($T_{8}\approx$ 1, where $T_{8}$ is in
units of $10^{8}$ K) remains highly uncertain, because it is too small for
direct measurement, even with presently available radioactive beam
intensities. Consequently, the structure of the 19Ne compound nucleus has also
been extensively studied in order to indirectly estimate the 18F(p, $\alpha$)
stellar reaction rate; for previous evaluations of the 18F(p, $\alpha$) rate,
see, e.g., Nesaraja et al. (2007), Iliadis et al. (2010b), La Cognata et al.
(2017), Kahl et al. (2019), Hall et al. (2019), and Hall et al. (2020). As our
understanding of excited states near the proton threshold in 19Ne continues to
progress, it is necessary to address the latest findings and any
discrepancies. Several new experimental works were published recently (Kahl et
al., 2019; Hall et al., 2019; La Cognata et al., 2019), and it is timely to
review the status of nuclear physics uncertainties to examine which ones most
strongly influence the 18F(p, $\alpha$) rate in classical novae. Here we
provide the first self-consistent evaluation of the 18F(p, $\alpha$) reaction
rate combining all these latest results in a hydrodynamic nova model.

The 18F(p, $\alpha$) reaction rate at the highest nova temperatures
($T_{8}\approx 4$) is dominated by a $J^{\pi}=3/2^{-}$ (orbital angular
momentum $\ell=1$) state at a resonance energy $E_{r}=332$ keV and a broad
$3/2^{+}$ ($\ell=0$) resonance at 665 keV (Bardayan et al., 2002, 2001); we
adopt the precisely measured properties of these states in the present work.
At lower nova temperatures ($T_{8}\lesssim 3$), $1/2^{+}$ and $3/2^{+}$
($\ell=0$) states predicted near the proton threshold in 19Ne, including
subthreshold resonances, can significantly influence the reaction rate by
interfering with higher-lying, broad $1/2^{+}$ and $3/2^{+}$ states (Dalouzy
et al., 2009; Chae et al., 2006). A broad $1/2^{+}$ state was reported, but
not resolved, near $E_{r}\approx 1.5$ MeV in several works (Dalouzy et al.,
2009; Adekola et al., 2012; Mountford et al., 2012). Most recently, an
$\ell=0$ state with a total width of $\Gamma=130(10)$ keV was firmly observed
at 1.38(3) MeV by Kahl et al. (2019).

Figure 1: Partial level scheme of 19Ne, with emphasis on s-wave states
affecting the 18F(p, $\alpha$) reaction rate. See Table 1 for details.  Table
1:  Resonances parameters and uncertainties used in the 18F(p, $\alpha)$
reaction rate calculations.

$E_{\rm ex}$ | $E_{\rm c.m.}$ | $J^{\pi}$ | ANC or $\Gamma_{\rm p}$ | $\Gamma_{\alpha}$
---|---|---|---|---
(MeV) | (keV) |  | (fm-1/2 or keV) | (keV)
6.132(5) aa$a$Laird et al. (2013) | $-278(5)$ |  $1/2^{+}$, $3/2^{+}$ |  $\leq$8 bb$b$Kahl et al. (2019), $\leq$6 bb$b$Kahl et al. (2019) |  $8.6\leavevmode\nobreak\ \leavevmode\nobreak\ \sim 16$cc$c$Bardayan et al. (2005); Nesaraja et al. (2007) dd$d$Torresi et al. (2017), $0.74\leavevmode\nobreak\ \leavevmode\nobreak\ \sim 8.5$cc$c$Bardayan et al. (2005); Nesaraja et al. (2007) ee$e$La Cognata et al. (2019)
$6.286(4)$ ${}^{f,\,g}$ | $-124(4)$ |  $3/2^{+}$, $1/2^{+}$ | 59 hh$h$Adekola et al. (2011a, b), 83.5 hh$h$Adekola et al. (2011a, b) |  $1.00\leavevmode\nobreak\ \leavevmode\nobreak\ \sim 11.6$cc$c$Bardayan et al. (2005); Nesaraja et al. (2007) ee$e$La Cognata et al. (2019), $11.7\leavevmode\nobreak\ \leavevmode\nobreak\ \sim 16$cc$c$Bardayan et al. (2005); Nesaraja et al. (2007) dd$d$Torresi et al. (2017)
6.416 bb$b$Kahl et al. (2019), 6.423 gg$g$Hall et al. (2019) | 6, 13 | (3/2+) |  $\leq$4.2$\times 10^{-45}$ b,\,hb,\,hfootnotemark: $b,\,h$, $\leq$3.9$\times 10^{-29}$ g,\,hg,\,hfootnotemark: $g,\,h$ |  $\leq$0.5 cc$c$Bardayan et al. (2005); Nesaraja et al. (2007)
6.742 i | 332 | 3/2- | 2.22$\times$10-3 ii$i$Bardayan et al. (2002) | 5.2 cc$c$Bardayan et al. (2005); Nesaraja et al. (2007)
7.0747 j | 664.7 | 3/2+ | 15.2 jj$j$Bardayan et al. (2001) | 23.8 jj$j$Bardayan et al. (2001)
7.79(3) b | 1380(30) | $1/2^{+}$ | 83${}^{+56}_{-82}$ b,\,kb,\,kfootnotemark: $b,\,k$ | 47${}^{-46}_{+92}$ b,\,kb,\,kfootnotemark: $b,\,k$

111 The asymptotic normalization coefficient (ANC) is used for subthreshold
states ($E_{\rm c.m.}<0$) and the proton partial width, $\Gamma_{\rm p}$,
otherwise. Values tabulated firmly were not varied in the present work. In the
case of values separated by a comma, corresponding quantities are tabulated in
the same order throughout a row. See the text and Fig. 1.
$f$$f$footnotetext: Parikh et al. (2015) , $k$$k$footnotetext: Adekola et al.
(2012)

To explore the magnitude of quantum interference effects on the 18F(p,
$\alpha$) cross section, we carefully evaluate uncertainties in our knowledge
of s-wave states in 19Ne close to the proton separation energy $S_{\rm
p}=6.4100(5)$ MeV (Wang et al., 2017). Because 19F (the mirror nucleus of
19Ne) is stable, its excitation structure has been comparatively well
understood (see, e.g., Tilley et al. (1995)), including a comprehensive, high-
resolution experimental and theoretical investigation of electron inelastic
scattering (Brown et al., 1985). Near the proton threshold energy in 19Ne,
there is one $1/2^{+}$ state and two $3/2^{+}$ states known in 19F located at
6.255, 6.497, and 6.528 MeV, respectively. We therefore focus our attention on
the location and properties of three such s-wave states in 19Ne. The values
adopted in the present work are shown in Fig. 1 & Table 1 and discussed
presently.

Most recently, Hall et al. (2019, 2020) considered the astrophysical impact of
uncertainties in the resonance properties on the 18F(p, $\alpha$) reaction
rate by deriving attributes of multiple resonances in 19Ne from the same
analog state in 19F. Unfortunately, such an approach can lead to unphysical
results. In the present work, all calculations are self-consistent, i.e., they
are in agreement with the overall known structure and properties of 19F. For
example, we only pair the properties of the known 6.528 MeV state in 19F with
just one candidate analog state in 19Ne for each calculation, whereas these
properties are applied simultaneously to four resonances in 19Ne by Hall et
al. (2020). This, for example, impacts on the potential importance of the
recently observed $\ell=0$ subthreshold resonance at 6.13 MeV in 19Ne by Kahl
et al. (2019).

Despite tabulated mirror pairings,222Hall et al. (2020) show a firm value for
$\Gamma_{\alpha}$ of the 19F state at 6.497 MeV in their Table 1, where there
is only an upper limit (see Nesaraja et al. (2007) cited therein). They also
do not consider the recent value of $\Gamma_{\alpha}=13.9$ keV for the
6.528-MeV state in 19F (La Cognata et al., 2019). Hall et al. (2020) clearly
obtained the alpha partial width, $\Gamma_{\alpha}$, for both their suggested
$(3/2^{+})$ states in 19Ne from the broader 6.528 MeV state in 19F. They
further adopted a $(5/2^{-})$ resonance with $\Gamma_{\alpha}$ and the proton
partial width, $\Gamma_{\rm p}$, derived from the same 6.528 MeV state in 19F
(see Appendix A & Laird et al. (2013)). Hall et al. (2020) then evaluated
subsequent ‘inclusion’ of the 6.13 MeV state in 19Ne (with its properties
derived from the same 6.528-MeV state in 19F) has a less than $\pm 7.5\%$
influence on their reaction rate limits over nova temperatures. Those authors
justified not ‘including’ this $\ell=0$ state because “no $\gamma$-ray
transitions were observed from this state, and no significant strength to
populate this state was observed in the 18F(d, n)19Ne measurements (Adekola et
al., 2011a),” yet they adopted other states which do not satisfy the same
criteria. We demonstrate in Fig. 4 that the sum of four $\ell>0$ resonances
included by Hall et al. (2019, 2020) exhibit a smaller influence than 7.5%
near peak nova temperatures, resulting in an inconsistent methodology.

In the present work, for the neighboring $3/2^{+}$ states at 6.497 and 6.528
MeV in 19F, one of which has a measured alpha strength and one just an upper
limit from experiment, we only allow one of these values per candidate analog
state in 19Ne. For near- and sub-threshold resonances in the 18F(p, $\alpha$)
reaction, $\Gamma$ is dominated by $\Gamma_{\alpha}$. We find the values of
$\Gamma_{\alpha}$ chosen for a given state strongly affect the magnitude of
interference with higher-lying, broad resonances and thus can significantly
influence the 18F(p, $\alpha$) reaction rate. In contrast, Hall et al. (2019,
2020) find $\Gamma_{\alpha}$ to be negligible by application of the isolated,
narrow-resonance condition for the reaction A($i$,$j$)B as

$\gamma\equiv\frac{\prod_{i}\Gamma_{i}}{\sum_{i}\Gamma_{i}}\approx\Gamma_{j}\iff\Gamma_{j}\ll\Gamma_{i\neq
j},$ (1)

where $\gamma$ is the reduced width (Rolfs & Rodney, 1988). However, the
simplification of Eq. 1 is inapplicable to the case of interference with a
broad resonance of the same $J^{\pi}$.

Kahl et al. (2019) unambiguously identified an $\ell=0$ state at 6.130(5) MeV
in 19Ne using a model-independent approach, and based on the systematics of
Coulomb energy differences strongly favored pairing this state with the
$1/2^{+}$ 6.255-MeV state in 19F. This conclusion is supported by the
significant strength from a $1/2^{+}$ resonance in 19Ne observed near $\sim
6.1$ MeV (Laird, 2000; Laird et al., 2002) with a magnitude comparable to
population of the $1/2^{+}$ 6.225-MeV state in analogous studies of 19F (Green
et al., 1970; Schmidt & Duhm, 1970). Torresi et al. (2017) observed the
signature of a $1/2$ state of ambiguous parity in 19Ne at 6.197(8)(50)
[statistical and systematic uncertainties] with a large $\Gamma_{\alpha}$ of
16 keV. In the present work, we adopt the excitation energy of 6.132(5) MeV
from the highest resolution studies to observe this low-spin state (Laird et
al., 2013; Parikh et al., 2015). As a possible $3/2^{+}$ assignment for this
state cannot be definitively ruled out, we compare and contrast the
implications of the $J$ assignment of the 6.132-MeV state. Table 1 shows that
the resulting uncertainty in $\Gamma_{\alpha}$ of the 6.132-MeV state spans
considerably different ranges depending upon the assumed $J$.

A strong $\ell=0$ component was observed by Adekola et al. (2011a, b) at 6.289
MeV, but subsequent studies led to the conflicting conclusions that the state
was not low spin (Laird et al., 2013) and that the state is $1/2^{+}$
(Bardayan et al., 2015). There is now compelling evidence of a doublet in this
region. Parikh et al. (2015) observed a structure broader than their
experimental resolution ($\Gamma\approx 16$ keV) and identified the
possibility of a doublet with peaks at 6.282 and 6.294 MeV. Subsequently, Kahl
et al. (2019) demonstrated that the structure at $6.288(5)$ MeV cannot be
attributed to a single, pure $\ell=0$ resonance. Finally, Hall et al. (2019)
observed a state at 6291.6(9) keV they suggested to be ($11/2^{+}$). Although
such a high spin, subthreshold state would not contribute to the astrophysical
reaction rate, its presence near a subthreshold $\ell=0$ state continues to
pose an experimental challenge to simultaneously and unambiguously resolve.

The mirror partner of the remaining $\ell=0$ state in 19F is likely located
above the proton threshold in 19Ne, but the situation is complicated as half a
dozen states may exist within just 50 keV (see, e.g., Nesaraja et al. (2007);
Laird et al. (2013)). Hall et al. (2019) tentatively suggest $(3/2^{+})$
assignments for two states above the proton threshold: a newly-proposed state
at 6.423(3) MeV and a previously-identified state at 6.441(3) MeV. Clearly
three $\ell=0$ states in 19F cannot be paired with four states in 19Ne, and
one of these tentative $(3/2^{+})$ pairings should be rejected. Kahl et al.
(2019) observed a peak at 6.421(10) MeV which could include an $\ell=0$
component near the centroid, but an $\ell=0$ state at higher excitations is
incompatible with that study and Laird et al. (2013). The $3/2^{+}$ state at
6.528 MeV in 19F has the larger $\Gamma_{\alpha}$, with values 4, 1.2, and
13.9 keV reported (Smotrich et al., 1961; Bardayan et al., 2005; La Cognata et
al., 2019); its mirror partner in 19Ne is thus unlikely to be observed in the
low-statistics $\gamma$-ray spectroscopy experiment of Hall et al. (2019). We
therefore assume the mirror partner of the $3/2^{+}$ state at 6.497 MeV in 19F
may be near 6.42 MeV in 19Ne, which is supported by perfect consistency in the
observed $\gamma$ transitions (Tilley et al., 1995; Hall et al., 2019).

As for the possible contribution of $\ell>0$ resonances near the proton
threshold, none have firm values for all their properties—besides many
$J^{\pi}$ ambiguities, at least one of the partial widths for each resonance
is an experimental upper limit (see, e.g., Nesaraja et al. (2007); Laird et
al. (2013)). Nonetheless, we consider the possible impact of such $\ell>0$
states on the 18F(p, $\alpha$) reaction rate in novae (see Table 4).

##  3 Reaction rate and stellar model

We calculated a wide range of 18F(p, $\alpha$) stellar reaction rates with the
$R$-Matrix code AZURE2 (Azuma et al., 2010; Mountford et al., 2014) based on
the nuclear physics uncertainties presented in Table 1. Our goal is to assess
which specific nuclear quantities have the largest impact on nova observables
in our model to guide future experimental measurements. Based on our
evaluation in Sect. 2, we focused on, but did not limit ourselves to,
variation of the following parameters within their uncertainties:

  * •

Interference sign between the two $1/2^{+}$ states

  * •

Interference signs between up to three $3/2^{+}$ states

  * •

$J$ of the $\ell=0$ state at 6.13 MeV

  * •

Unknown proton ANC of the 6.13 MeV state

  * •

$J$ and $E_{\rm ex}$ of the $\ell=0$ state near 6.29 MeV

  * •

$\Gamma_{\alpha}$ for the mirror of the $3/2^{+}$, 6.528-MeV state in 19F

  * •

$\Gamma_{\alpha}$ of the $1/2^{+}$ subthreshold state

  * •

$E_{\rm ex}$ of a narrow $(3/2^{+})$ state just above $S_{\rm p}$

  * •

Influence of $\ell>0$ states near $S_{\rm p}$

  * •

$\Gamma$ as well as the $\Gamma_{\rm p}/\Gamma_{\alpha}$ ratio of the
$1/2^{+}$ state near 7.8 MeV

We found variation in other properties to be implausible or inconsequential.
Further details of the $R$-Matrix calculations are provided in Appendix A.

Considering this large set of codependent parameters, a Monte-Carlo approach,
such as employed in RatesMC (Longland et al., 2010; Iliadis et al., 2010b, a)
is quite attractive. However, the case of 18F(p, $\alpha$) presents a
particular challenge owing to the possible influence of the interference
between up to three resonances of the same $J^{\pi}$, namely the $3/2^{+}$
states. For this reason, the previous evaluation of Iliadis et al. (2010b)
considered only the interference between a pair of $3/2^{+}$ states, and
RatesMC uses the simple analytic formula of Rolfs (Longland, priv. comm.).
Here we aim to improve upon the previous works by fully considering the latest
experimental results using an $R$-matrix approach to handle multiple
interference effects over a large and internally-consistent parameter space. A
comparison of the range of the present rates against those of Iliadis et al.
(2010b) is shown in Fig. 2; the corresponding rates from the present work are
listed in Table LABEL:tab:rates. We recommend the median rates (Sets XIII &
IX) obtained in the present work (see Fig. 5). After careful evaluation of the
impact of each noted uncertainty on the stellar reaction rate, we selected a
representative sample of 16 possible rates over the full range for a further
analysis.

Figure 2: Range of reaction rates explored in the present work (see Tables 1 &
LABEL:tab:rates) normalized to the median rate of Iliadis et al. (2010b); the
low and high rates of Iliadis et al. (2010b) are also shown. The recommended
(median) rates from the present work are shown in magenta for Sets VIII & IX,
where the subthreshold spin ordering is $3/2^{+},1/2^{+}$ or
$1/2^{+},3/2^{+}$, respectively. The highest rate (XVI) represents a case of
maximally constructive interference, while the lowest rates (I & II) are
obtained by maximally destructive interference.

The astrophysical impact of these new 18F(p, $\alpha$) rates has been assessed
through a series of 16 new hydrodynamic models of nova explosions. The
simulations have been performed with the hydrodynamic, Lagrangian, finite-
difference, time-implicit code SHIVA (José & Hernanz, 1998; José, 2016). SHIVA
relies on the standard set of differential equations of stellar evolution and
has been extensively used in the characterization of nova outbursts, Type-I
X-ray bursts and sub-Chandrasekhar supernova explosions, for more than 20
years. SHIVA uses a general equation of state that includes contributions from
the degenerate electron gas, the ion plasma, and radiation. Coulomb
corrections to the electron pressure are taken into account, and both
radiative and conductive opacities are considered in the energy transport.
Nuclear energy generation is obtained by means of a reaction network that
contains 120 species (from 1H to 48Ti), linked through 630 nuclear processes,
with updated rates from the STARLIB database (Sallaska et al. (2013) and
Iliadis, priv. comm.). The accreted matter from the stellar companion, at a
typical rate of $2\times 10^{-10}$ M⊙ yr-1, is assumed to mix with material
from the outer layers of the underlying white dwarf to a characteristic level
of 50%. In all the hydrodynamic simulations performed in this work, the mass
of the (ONe) white dwarf hosting the explosion has been assumed to be 1.25
solar masses (and its initial luminosity $10^{-2}$ solar luminosities), and
are identical, differing only in the specific prescription adopted for the
18F(p, $\alpha$) rate.

##  4 Results

Table 2:  Mean, mass-averaged abundances of 18,19F in the nova ejecta.

Set | Abundance | Rate ($T_{8}$)
---|---|---
|  18F aa$a$1 hour after $T_{\rm peak}$. |  19F | 1 | 2.5 | 4
I | 1.1$\times 10^{-5}$ | 1.8$\times 10^{-7}$ | 1.22$\times 10^{-5}$ | $0.466$ | $96.5$
II | 9.3$\times 10^{-6}$ | 1.2$\times 10^{-7}$ | 2.63$\times 10^{-6}$ | $0.664$ | $115$
III | 8.6$\times 10^{-6}$ | 1.1$\times 10^{-7}$ | 3.69$\times 10^{-6}$ | $0.685$ | $116$
IV | 7.9$\times 10^{-6}$ | 1.4$\times 10^{-7}$ | 2.31$\times 10^{-5}$ | $0.493$ | $95.7$
V | 7.8$\times 10^{-6}$ | 9.9$\times 10^{-8}$ | 6.57$\times 10^{-6}$ | $0.702$ | $116$
VI | 7.0$\times 10^{-6}$ | 8.9$\times 10^{-8}$ | 8.83$\times 10^{-6}$ | $0.740$ | $118$
VII | 2.9$\times 10^{-6}$ | 4.0$\times 10^{-8}$ | 6.05$\times 10^{-5}$ | $1.16$ | $132$
VIII | 2.8$\times 10^{-6}$ | 3.8$\times 10^{-8}$ | 6.46$\times 10^{-5}$ | $1.21$ | $134$
IX | 2.6$\times 10^{-6}$ | 3.1$\times 10^{-8}$ | 6.15$\times 10^{-5}$ | $1.48$ | $154$
X | 2.4$\times 10^{-6}$ | 2.9$\times 10^{-8}$ | 7.03$\times 10^{-5}$ | $1.55$ | $157$
XI | 1.9$\times 10^{-6}$ | 2.3$\times 10^{-8}$ | 1.10$\times 10^{-4}$ | $1.82$ | $162$
XII | 1.8$\times 10^{-6}$ | 2.2$\times 10^{-8}$ | 1.17$\times 10^{-4}$ | $1.89$ | $165$
XIII | 1.6$\times 10^{-6}$ | 2.0$\times 10^{-8}$ | 1.36$\times 10^{-4}$ | $2.05$ | $170$
XIV | 1.4$\times 10^{-6}$ | 1.6$\times 10^{-8}$ | 1.58$\times 10^{-4}$ | $2.51$ | $198$
XV | 1.3$\times 10^{-6}$ | 1.6$\times 10^{-8}$ | 1.77$\times 10^{-4}$ | $2.53$ | $193$
XVI | 1.1$\times 10^{-6}$ | 1.2$\times 10^{-8}$ | 2.30$\times 10^{-4}$ | $3.20$ | $225$
333 Calculated for each set of 19Ne resonance parameters considered for the
18F(p, $\alpha$) reaction rate, $N_{A}\langle\sigma v\rangle$, in cm3mol-1s-1.
Samples of the rates are provided at the representative temperatures of
$T_{8}$=1, 2.5, and 4. The sets are enumerated according to the 18F abundance
for convenience.

Figure 3:  Mass fractions of selected isotopes within the expanding envelope
at various stages of the explosion using our newly recommended 18F(p,
$\alpha$) reaction rate Set IX. The horizontal axes correspond to a relative
mass coordinate where the origin is set at the surface of the envelope and
where the arrows show the limit of the ejected material. The panels are
labeled from top to bottom and from left to right and correspond to
temperatures at the base of the accreted envelope of approximately $5\times
10^{7}$ (Panel 1), $7\times 10^{7}$ (Panel 2), $1\times 10^{8}$ (Panel 3),
$2\times 10^{8}$ (Panel 4), $T_{\rm max}=2.4\times 10^{8}$ K (Panel 5),
respectively, and to the last phases of the evolution, when the nova envelope
has already expanded to a size of $R_{\rm wd}\sim 10^{9}$ (Panel 6), $10^{10}$
(Panel 7) and $10^{12}$ cm (Panel 8), respectively. We plot our results in a
manner almost identical to Fig. 2 in Coc et al. (2000) to facilitate
comparison.

The main difference in our output nova model parameters is centered, as
expected, around 18F. As the energetics of the explosion are not affected by
variations in the 18F(p, $\alpha$) reaction rate within current uncertainties,
the results for other model quantities are identical, such as the total
ejected mass, the peak temperature, $T_{\rm peak}$, achieved, etc. The values
shown in Table 2 correspond to mass-averaged (mean value) abundances of 18,19F
in mass fractions, $X$, in the ejecta and representative temperature samples
of the employed reaction rate. The ejected abundances of all other isotopes
examined varied between the models by at most a relative shift of $\pm 1$%,
which we adopt as the precision of sensitivity achieved in the present work.
While classical novae are thought to be a major source of 15N, 17O, and to
some extent 13C (José et al. (2006); José (2016); and references therein), we
observed no significant variation in the abundances of any of these isotopes
in the present study. For example, the contribution of radiogenic 15N from the
18F(p, $\alpha$)15O reaction is negligible, as our model consistently ejected
15N with an abundance of $3.39\times 10^{-2}$, about four orders of magnitude
larger than our predicted $\pm 5\times 10^{-6}$ abundance shift in 18F. It can
be seen in Table 2 that the ejected abundances of 18F (1 hour after $T_{\rm
peak}$) and 19F covary by about an order of magnitude and are inversely
correlated with the respective 18F(p, $\alpha$) reaction rate. As the
observational detectability distance goes as the square root of the ejected
mass fraction, our results indicate a factor of $\sim\\!3$ uncertainty in the
detectability distance of $\gamma$-rays from 18F decay in classical novae from
the 18F(p $\alpha$) reaction rate alone.

The strong correlation of 18F and 19F abundances over a wide range of 18F(p,
$\alpha$) reactions rates is consistent with its predominance over the 18F(p,
$\gamma$)19Ne reaction in novae. 19Ne is very short-lived ($t_{1/2}\approx 18$
s) and decays to the only stable isotope of fluorine, 19F, which was indeed
identified in the spectra of Nova Mon 2012 (Shore et al., 2013). The abundance
shifts in 19F can be understood, as the 18F(p, $\alpha$) rate limits the
available 18F by flushing away material faster than the 18F(p,
$\gamma$)19Ne($\beta^{+}$)19F sequence can occur. We emphasize that the true
18F(p, $\alpha$) rate remains quite uncertain.

Snapshots of the evolution of a number of representative isotopes of the O-Ne
group, relevant in the synthesis and destruction of 18F (i.e., 16,17,18O,
17,18,19F, and 18,19Ne) are depicted in Fig. 3. The initial mass fractions of
these species throughout the envelope, at the onset of the accretion stage,
are as follows: $X(^{16}{\rm O})=0.13$, $X(^{17}{\rm O})=2.1\times 10^{-6}$,
$X(^{18}{\rm O})=1.2\times 10^{-5}$, and $X(^{19}{\rm F})=3.1\times 10^{-7}$.
As matter transferred from the secondary star piles up on the surface of the
white dwarf star, compressional heating and nuclear reactions increase the
temperature (and pressure) of the accreted envelope. When the temperature at
the base of the envelope achieves $T_{\rm base}\sim 5\times 10^{7}$ K (Fig. 3,
Panel 1), the main nuclear activity in the O-Ne mass region is dominated by
the chain 16O(p, $\gamma$)17F($\beta^{+}$)17O(p, $\alpha$)14N. The main
effect, at this stage, is the increase of 17O, whose abundance rises by two
orders of magnitude at the base of the envelope, with respect to its initial
value. However, since the temperature achieved is not high enough to fuse a
significant fraction of 16O, changes on most of the species are moderate. This
mostly affects the build up of 17,18F, whose mass fractions reach $\sim
10^{-7}$. Proton capture reactions onto 17,18F result also in a very modest
increase of both 18,19Ne ($<10^{-10}$, by mass). Convective transport is
scarcely present at this stage, which explains the inhomogeneous chemical
abundance pattern of the envelope (see, for instance, the larger abundance of
18F near the core-envelope interface in Panel 1).

A similar behavior is observed when $T_{\rm base}\sim 7\times 10^{7}$ K (Fig.
3, Panel 2), with the same chain of reactions described in Panel 1 still
dominating the nuclear activity in the envelope. Convection has already
extended throughout larger areas of the envelope, which translates into a
smoother chemical profiles across the envelope, for species such as 18F, which
is synthesized close to the base of the envelope and efficiently carried away
to the outer, cooler layers of the envelope. At this stage, 18F $\sim 10^{-6}$
(17F $\sim 10^{-5}$) at the base of the envelope. Aside from 16O, the second
most abundant isotope of the O-Ne group at this stage is still 17O.

When $T_{\rm base}$ reaches $10^{8}$ K (Fig. 3, Panel 3), most of the accreted
envelope becomes convective. This tends to homogenize the envelope, as shown
by the nearly flat profiles of most species, particularly 16,17O, and
17,18,19F. The abundance of 16O is being reduced by (p, $\gamma$) reactions.
17O shows an increase, with a mass fraction ranging from $6.3\times 10^{-4}$
(inner envelope) to $8.7\times 10^{-4}$ (outer envelope). Here, the chain
16O(p, $\gamma$)17F($\beta^{+}$)17O dominates the destruction channel 17O(p,
$\alpha$)14N. Both 17,18F have notoriously increased with respect to the
previous panel, through 16,17O(p, $\gamma$)17,18F reactions, while 19F has
been severely depleted, mostly through 19F(p, $\alpha$)16O. 18,19Ne continue
to rise in the innermost layers of the envelope by means of proton captures
onto 17,18F.

When $T_{\rm base}$ reaches $2\times 10^{8}$ K (Fig. 3, Panel 4), 16O
experiences a significant decrease by proton-capture reactions, which reduce
its abundance by nearly a factor of two in the innermost layers of the
envelope. This, in turn, induces a reduction in 17O (with 17O(p, $\alpha$)14N
reactions dominating over 17F($\beta^{+}$)17O). With regard to 18F, its
abundance is mostly dominated by the destruction channel 18F(p, $\alpha$)15O
at the hotter, inner regions of the envelope, while 17O(p, $\gamma$)18F and
18Ne($\beta^{+}$)18F contribute to rise its mass fraction in the outer, cooler
regions. As a result, 18F begins to show a steeper profile which, except for
the outermost layers of the envelope, increases outward. At this stage, 18O
and 19F have been severely reduced by proton captures, while a remarkable
increase in 18,19Ne by several orders of magnitude, mainly driven by proton-
capture reactions onto 17,18F, which at these temperature become faster than
the corresponding $\beta^{+}$-decays.

Shortly thereafter, the base of the envelope achieves a maximum temperature of
$2.38\times 10^{8}$ K (Fig. 3, Panel 5). At this stage, the entire envelope
has become fully convective. The 16O abundance continues to decrease by proton
captures, and for the first time, a different species, 17F, becomes the most
abundant O-Ne group element in most of the envelope. The nuclear path is still
dominated by the 16O(p, $\gamma$)17F($\beta^{+}$)17O(p, $\alpha$)14N chain. At
this stage, the abundances of the different species reflect a competition
between different destruction and creation modes, mostly governed by (p,
$\gamma$) and (p, $\alpha$) reactions in the inner regions of the envelope,
and by $\beta^{+}$-decays in the outer regions. With the exception of 16O, the
other O-Ne group isotopes shown in Panel 5 experience a net increase.

A few minutes after the envelope attains peak temperature, the sudden release
of energy from the very abundant, short-lived species 13N, 14,15O, and 17F,
forces the envelope to expand. Hereafter, proton-capture reactions become
confined to the innermost regions of the envelope, following a dramatic
temperature decrease in the outer layers (Fig. 3, Panels 6 & 7).
Simultaneously, convection begins to recede from the outer envelope shells.

The final stages of the outburst (Fig. 3, Panel 8), when a significant
fraction of the envelope has already achieved escape velocity, are dominated
by the release of nuclear energy by the $\beta^{+}$-decay processes
17,18F($\beta^{+}$)17,18O. The mean 18F abundance in the ejecta for this
Model, one hour after peak temperature, turns out to be $X(^{18}{\rm
F})=2.6\times 10^{-6}$. The most abundant isotopes of the O-Ne group in the
ejecta are, however, $X(^{16}{\rm O})=9.2\times 10^{-3}$ and $X(^{17}{\rm
O})=7.9\times 10^{-3}$. In addition, some amounts of 18O ($1.1\times 10^{-6}$,
by mass) and 19F ($3.1\times 10^{-8}$, by mass), have been found for this
Model, assuming that all radioactive isotopes have already decayed into the
corresponding daughter nuclei.

We confirmed that the largest variation among the 18F(p, $\alpha$) reaction
rates, and thus the ejected 18F, arises from the unknown interference signs
between the $\ell=0$ resonances. As for directly measurable quantum
properties, we found that $\Gamma_{\alpha}$ of a broad $3/2^{+}$ subthreshold
state has the most profound influence on both the maximum and minimum reaction
rates. In the case of maximally destructive interferences, the lowest rates
are always obtained when the 6.13 MeV state is assumed to be $1/2^{+}$.
Conversely, the maximum rates are similar for either spin ordering for the
$\ell=0$ subthreshold states, depending on the other assumptions. Our minimum
rate is sensitive to the proton asymptotic normalization coefficient (ANC) of
the 6.13 MeV state when it is $1/2^{+}$; here, a lower (but non-zero) value of
4 fm-1/2 compared with the presumed experimental upper limit of 8 fm-1/2
surprisingly leads to the largest effects, by shifting the dip of the
destructive interference into the center of the astrophysical energy region.
Although the broad $1/2^{+}$ state near 7.8 MeV is found to strongly affect
the reaction rate by interference, we found that the rate is not sensitive to
the adopted $\Gamma_{\rm p}/\Gamma_{\alpha}$ ratio nor the uncertainty in
$E_{\rm ex}$ for this state. Finally, a $3/2^{+}$ state above the proton
threshold can have a modest impact on the rate, depending on its exact energy
and if $\Gamma_{\alpha}$, $\Gamma_{\rm p}$, and $E_{\rm ex}$ are
simultaneously taken to be near the maximal values consistent with
experimental measurements, although we consider such a scenario to be
unlikely.

We found that the possible impact of $\ell>0$ states (e.g., the ones assumed
in Hall et al. (2019)) near $S_{\rm p}$ to the maximum 18F(p, $\alpha$) rate
is in general trivial in relation to other nuclear uncertainties, the
magnitude of their effect on the reaction rate being comparable to the least-
influential uncertainties of $\ell=0$ resonances we considered (see Appendix
A). For the minimum rate, setting a given partial width with only an upper
limit to an arbitrarily small value is compatible with experimental results.

##  5 Conclusions

We performed a thorough analysis of the impact of the presumed 18F(p,
$\alpha$) reaction rate on the predicted abundance of the astronomical
observable 18F produced in classical novae. The reaction rates were obtained
using the $R$-Matrix method, where the uncertainties in the nuclear input
parameters were derived from a self-consistent evaluation of all available
nuclear physics data, including the latest experimental results. An advantage
of our approach was the ability to include all quantum interference effects on
the reaction rate. The range of reaction rates we explored turned out to be
quite large, and we selected 16 evenly-spaced rates over our full range for
further analysis. Using these new rates, we performed 16 (otherwise identical)
hydrodynamic nova simulations, and we presented and analyzed the only
differences in model results we observed. The uncertainty in the observational
distance arising from the decay of ejected 18F from novae is about a factor of
three. We conclude by emphasizing the extant nuclear physics quantities which
need to be known more precisely to significantly reduce this astronomical
uncertainty:

  * •

Our lack of precise knowledge of the properties of the $\ell=0$ states near
6.13 & 6.29 dominate the uncertainties.

  * •

The proton ANC of the 6.13 MeV state.

  * •

More definitive spin assignments on the $\ell=0$ states near 6.13 & 6.29 MeV,
ideally with $<5$ keV resolution.

  * •

A stronger upper-limit on $\Gamma_{\alpha}$, and thus $\Gamma$, of the
putative $3/2^{+}$ state near 6.42 MeV.

In the future, it will be interesting to investigate the uncertainties in
other nuclear reactions affecting 19F abundances in novae to determine if it
can be used as an observational tracer for the 18F(p, $\alpha$) reaction.

###### Acknowledgements.

D.K. and P.J.W. are appreciative of funding from the UK STFC. J.J.
acknowledges partial support through the Spanish MINECO grant AYA2017-86274-P,
E.U. FEDER funds, and through the AGAUR/Generalitat de Catalunya grant
SGR-661/2017. This article benefited also from discussions within the “ChETEC”
COST Action (CA16117).

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##  Appendix A Details of $R$-Matrix calculations

We calculated the 18F(p, $\alpha$) stellar reaction rate using $R$-Matrix
formalism with AZURE2 (Azuma et al. 2010; Mountford et al. 2014). We fixed the
channel radius $a=5.25$ fm, similar to the values adopted in previous works of
5.5, 5.0, and 5.2 fm, respectively (de Séréville et al. 2009; Beer et al.
2011; Hall et al. 2019). The sensitivity of the $R$-Matrix calculations on the
assumed channel radius was previously explored over the range 4.5–6.5 fm (see
Fig. 6 of de Séréville et al. (2009)), where a larger channel radius was seen
to push a destructive interference dip to lower energies, and vice versa, all
other parameters being equal. We separate our employed 19Ne resonance
parameters into three categories, as follows:

  * •

Strongly influential resonances used in all calculations where variation of
their properties within experimental uncertainties has a negligible influence
on the reaction rate, shown in Table 3.

  * •

$\ell>0$ resonances with poorly-known properties which nevertheless exhibit a
minimal influence on the present uncertainty in the reaction rate, shown in
Table 4.

  * •

$\ell=0$ resonances that dominate the present uncertainty in the reaction
rate, shown in Table 5.

The possible contribution of the $\ell>0$ resonances in Table 4 is shown in
Fig. 4. We compare the sum of these four resonances against the maximum rate
(Set XVI), as well as the case when the 6.13 MeV state is $1/2^{+}$ and has
$\Gamma_{\alpha}=16$ keV (Torresi et al. 2017) rather than 8.6 keV (as assumed
in Set XVI). Our analysis showed the largest resonant contribution in Table 4
is from the 6.459 MeV state in 19Ne, which was tentatively assigned as
($5/2^{-}$) by Laird et al. (2013), who did not observe any $3/2^{+}$ states
in 19Ne in this region. As there seemed to be no corresponding $5/2^{-}$ state
in 19F near this energy, Laird et al. (2013) scaled the properties of the
6.528 MeV $3/2^{+}$ state in 19F given by Nesaraja et al. (2007). As our
calculations already assign a mirror partner to the 6.528 MeV state in 19F, we
infer that the maximum contribution of the 6.459 MeV state in 19Ne is
significantly less than Fig. 4 suggests.

For the $(3/2)^{-}$ state at 6.416 MeV, Hall et al. (2019) have the value of
$\Gamma_{\rm p}$ shown in Table 4, while Hall et al. (2020) claim $\Gamma_{\rm
p}=2.8\times 10^{-27}$ keV for a 1 keV resonance energy shift (inconsistent
with Bardayan et al. (2015)). We instead found $\Gamma_{\rm p}=2.3\times
10^{-45}$ keV for such a $(3/2)^{-}$ resonance at $E_{\rm c.m.}=6$ keV. We
tested all of these values for $\Gamma_{\rm p}$, and even with an erroneous
increase of $10^{18}$ in $\Gamma_{\rm p}$, the contribution of this state to
the maximum reaction rate of Set XVI here remains negligible. We explore the
energies near 6.416 vs. 6.423 MeV for a $(3/2^{+})$ state in Table 5, with
appropriate shifts in $\Gamma_{\rm p}$ and consistent use of $\Gamma_{\alpha}$
so that readers can evaluate the impact of the state newly suggested by Hall
et al. (2019); this is not the same as considering a 6.416-MeV, negative
parity state the same as the mirror pair of a $3/2^{+}$ state, but rather a
previously unresolved doublet.

The astrophysical S-factor, $S(E)$, is shown in Fig. 5 and compared with
experimental results. We show the exact $S(E)$ without taking into account the
individual experimental target thicknesses, which differ from one another by
more than a factor of two. Considering the scatter in the experimental data,
the agreement is reasonably good, except at 250 keV, where only two events
(and no background) were observed by Beer et al. (2011), allowing for the
possibility that the true $S(E)$ could be significantly smaller as explored in
the present work.

Table 3:  Resonance parameters used in all of the 18F($p$, $\alpha$)
astrophysical calculations.

$E_{\rm ex}$ | $E_{\rm c.m.}$ | $2J^{\pi}$ | $\Gamma_{\rm p}$ | $\Gamma_{\alpha}$
---|---|---|---|---
(MeV) | (keV) |  | (keV) | (keV)
6.742 aa$a$All level parameters taken from Bardayan et al. (2002) | 332 | $3^{-}$ | $2.22\times 10^{-3}$ | 5.2
7.0747 bb$b$All level parameters taken from Bardayan et al. (2001) | 664.7 | $3^{+}$ | 15.2 | 23.8
7.79(3) cc$c$All level parameters taken from Kahl et al. (2019) | 1380 | $1^{+}$ | 83${}^{+56}_{-82}$ | 47${}^{-46}_{+92}$
444

Table 4:  Resonance parameters added to the maximum rate (Set XVI) exhibiting
minimal influence.

$E_{\rm ex}$ | $E_{\rm c.m.}$ | $2J^{\pi}$ | $\Gamma_{\rm p}$ | $\Gamma_{\alpha}$
---|---|---|---|---
(MeV) | (keV) |  | (keV) | (keV)
6.416(4) | 6 | $(3)^{-}$ |  $4.7\times 10^{-50}$ aa$a$Differs from Hall et al. (2020). See the text. | $<0.5$
6.439(3) | 29 | $1^{-}$ | $\leq 3.8\times 10^{-19}$ | 220
6.459(5) | 49 | $(5^{-})$ |  $8.4\times 10^{-14}$ bb$b$Laird et al. (2013), shifted from the $3/2^{+}$ 6.528-MeV state in 19F. Provided for illustrative purposes only. See the text. | 5.5 bb$b$Laird et al. (2013), shifted from the $3/2^{+}$ 6.528-MeV state in 19F. Provided for illustrative purposes only. See the text.
6.669(3) | 289 | $(5^{+})$ | $<2.4\times 10^{-5}$ | 1.2
555 All level parameters are taken from Hall et al. (2019), modified here to
show limit signs and tentative assignment notation provided in the original
references therein.

Figure 4: Comparison of the potential contribution of $\ell>0$ resonances in
Table 4 to the 18F(p, $\alpha$) rates utilized in the astrophysical
calculations in the present work (normalized to our Set IX). We demonstrate
that the potential influence of the sum of those four resonances is not much
more than, e.g., the uncertainty in $\Gamma_{\alpha}$ of the 6.13 MeV state
observed by Kahl et al. (2019).  Figure 5: Astrophysical $S(E)$ for the 18F(p,
$\alpha$) reaction. See Tables 1, 3 and 5 for details. Experimental works are
from Bardayan et al. (2001, 2002); de Séréville et al. (2009); Beer et al.
(2011).  Table 5:  Resonance parameters varied in the calculation of the
18F(p, $\alpha$) reaction rate and $S(E)$.

Set | Interference Signs |  Properties of $\ell=0$ state |  Properties of $\ell=0$ state |  Properties of ($\ell=0$) state
---|---|---|---|---
| $J^{\pi}$ | $J^{\pi}$ | $E_{\rm ex}$ | $J^{\pi}$ |  ANCp | $\Gamma_{\alpha}$ | $E_{\rm ex}$ | $J^{\pi}$ |  ANCp | $\Gamma_{\alpha}$ | $E_{\rm ex}$ | $J^{\pi}$ | $\Gamma_{\rm p}$ | $\Gamma_{\alpha}$
| $1/2^{+}$ | $3/2^{+}$ | (MeV) |  |  (fm-1/2) | (keV) | (MeV) |  |  (fm-1/2) | (keV) | (MeV) |  | (keV) | (keV)
I | $(++)$ | $(++)$ | 6.132 | $1/2^{+}$ | 4 | 8.6 | 6.289 | $3/2^{+}$ | 59 | 11.6 | — | — | — | —
II | $(++)$ | $(+++)$ | 6.132 | $1/2^{+}$ | 2.5 | 8.6 | 6.289 | $3/2^{+}$ | 59 | 1 | 6.423 | $3/2^{+}$ | $<3.9\times 10^{-29}$ | $<0.5$
III | $(++)$ | $(+++)$ | 6.132 | $1/2^{+}$ | 2.5 | 8.6 | 6.289 | $3/2^{+}$ | 59 | 1 | 6.416 | $3/2^{+}$ | $<4.2\times 10^{-45}$ | $<0.5$
IV | $(++)$ | $(+++)$ | 6.132 | $1/2^{+}$ | 8 | 8.6 | 6.289 | $3/2^{+}$ | 59 | 11.6 | 6.423 | $3/2^{+}$ | $<3.9\times 10^{-29}$ | $<0.5$
V | $(++)$ | $(+++)$ | 6.132 | $1/2^{+}$ | 8 | 8.6 | 6.289 | $3/2^{+}$ | 59 | 1 | 6.416 | $3/2^{+}$ | $<4.2\times 10^{-45}$ | $<0.5$
VI | $(++)$ | $(++)$ | 6.132 | $1/2^{+}$ | 8 | 8.6 | 6.289 | $3/2^{+}$ | 59 | 1 | — | — | — | —
VII | $(++)$ | $(+++)$ | 6.132 | $3/2^{+}$ | 6 | 0.74 | 6.289 | $1/2^{+}$ | 83.5 | 11.7 | 6.416 | $3/2^{+}$ | $<4.2\times 10^{-45}$ | $<0.5$
VIII | $(++)$ | $(++)$ | 6.132 | $3/2^{+}$ | 6 | 0.74 | 6.289 | $1/2^{+}$ | 83.5 | 11.7 | — | — | — | —
IX | $(-+)$ | $(-+)$ | 6.132 | $1/2^{+}$ | 8 | 8.6 | 6.289 | $3/2^{+}$ | 59 | 1 | — | — | — | —
X | $(-+)$ | $(--+)$ | 6.132 | $1/2^{+}$ | 8 | 8.6 | 6.289 | $3/2^{+}$ | 59 | 1 | 6.416 | $3/2^{+}$ | $<4.2\times 10^{-45}$ | $<0.5$
XI | $(-+)$ | $(-+)$ | 6.132 | $3/2^{+}$ | 6 | 0.74 | 6.289 | $1/2^{+}$ | 83.5 | 11.7 | — | — | — | —
XII | $(-+)$ | $(--+)$ | 6.132 | $3/2^{+}$ | 6 | 0.74 | 6.289 | $1/2^{+}$ | 83.5 | 11.7 | 6.416 | $3/2^{+}$ | $<4.2\times 10^{-45}$ | $<0.5$
XIII | $(-+)$ | $(-+)$ | 6.132 | $3/2^{+}$ | 6 | 0.74 | 6.289 | $1/2^{+}$ | 83.5 | 16 | — | — | — | —
XIV | — | $(-+)$ | — | — | — | — | 6.289 | $3/2^{+}$ | 59 | 11.6 | — | — | — | —
XV | $(-+)$ | $(--+)$ | 6.132 | $3/2^{+}$ | 6 | 8.5 | 6.289 | $1/2^{+}$ | 83.5 | 16 | 6.423 | $3/2^{+}$ | $<3.9\times 10^{-29}$ | $<0.5$
XVI | $(-+)$ | $(--+)$ | 6.132 | $1/2^{+}$ | 8 | 8.6 | 6.282 | $3/2^{+}$ | 59 | 11.6 | 6.423 | $3/2^{+}$ | $<3.9\times 10^{-29}$ | $<0.5$
666 See Table 1.

##  Appendix B Reaction rate table