# Generating series of classes of exotic un-ordered configuration spaces

Sabir M. Gusein-Zade The work was supported by the grant 21-11-00080 of the
Russian Science Foundation. Keywords: configuration spaces, generating series,
Grothendieck ring of complex quasiprojective varieties. Math. Subject
Classification: 55R80, 18F30, 32S35.

###### Abstract

A notion of exotic (ordered) configuration spaces of points on a space $X$ was
suggested by Yu. Baryshnikov. He gave equations for the (exponential)
generating series of the Euler characteristics of these spaces. Here we
consider un-ordered analogues of these spaces. For $X$ being a complex
quasiprojective variety, we give equations for the generating series of
classes of these configuration spaces in the Grothendieck ring
$K_{0}({\rm{Var}_{\mathbb{C}}})$ of complex quasiprojective varieties. The
answer is formulated in terms of the (natural) power structure over the ring
$K_{0}({\rm{Var}_{\mathbb{C}}})$. This gives equations for the generating
series of additive invariants of the configuration spaces such as the
Hodge–Deligne polynomial and the Euler characteristic.

##  1 Introduction

The configuration space of sequences of $k$ (different) points of a
topological space $X$ (or the $k$th (ordered) configuration space of $X$) is

${\rm Conf}(X,k)=\\{(x_{1},\ldots,x_{k})\in X^{k}:x_{i}\neq x_{j}\mbox{ for
}i\neq j\\}\,,$

The configuration space of $k$-point subsets of a (topological) space $X$ (or
the $k$th unordered configuration space of $X$) is

${\rm UConf}(X,k)=\\{(x_{1},\ldots,x_{k})\in X^{k}:x_{i}\neq x_{j}\mbox{ for
}i\neq j\\}/S_{k}\,,$

where $S_{k}$ is the group of permutations on $k$ elements. One has the well-
known Macdonald type equation:

$1+\sum_{k=1}^{\infty}\chi({\rm Conf}(X,k))\cdot t^{k}=(1+t)^{\chi(X)}\,.$ (1)

Here $\chi(\cdot)$ is the additive Euler characteristic defined as the
alternating sum of the ranks of the cohomology groups with compact support.
This Euler characteristic is the universal additive topological invariant (on
some classes of topological spaces, say, on locally closed unions of cells in
finite CW-complexes). If one considers spaces with additional structure, one
may have other addititve invariants: generalized Euler characteristics. For
example, on the set of complex quasiprojective varieties the universal
adiitive invariant is the class of a variety in the Grothendieck ring
$K_{0}({\rm{Var}_{\mathbb{C}}})$ of complex quasi-projective varieties. Other
additive invariants are reductions of this universal one under (well-defined)
group homomorphisms from $K_{0}({\rm{Var}_{\mathbb{C}}})$ to the groups of
values of the invariants. If an additive invariant is at the same time
multiplicative, the corresponding homomorphism is a ring one. An invariant of
this sort (additive and multiplicative) is the Hodge–Deligne polynomial of a
complex quasiprojective variety. Thus it is obtained via a ring homomorphism
from the the Grothendieck ring $K_{0}({\rm{Var}_{\mathbb{C}}})$ to the ring
${\mathbb{Z}}[u,v]$ of polynomials in two variables. An equation for the
classes of complex quasiprojective varieties gives (through this homomorphism)
an equation for their Hodge–Deligne polynomials.

A generalization of the notion of the configuration space ${\rm Conf}(X,k)$
called exotic configuration spaces (see below) was suggested by Yu.
Baryshnikov in [1]. Here we consider a natural version of this notion to the
case of un-ordered subsets of points, i.e. a generalization of the notion of
the unordered configuration space ${\rm UConf}(X,k)$. We give Macdonald type
equations for their Euler characteristics and (for the case when $X$ is a
complex quasiprojective variety) their generalized versions for the generating
series of classes of these configuration spaces in the Grothendieck ring
$K_{0}({\rm{Var}_{\mathbb{C}}})$. This gives a similar equatoion for the
generating series of the Hodge–Deligne polynomials of these configuration
spaces.

##  2 The power structure over the Grothendieck ring
$K_{0}({\rm{Var}_{\mathbb{C}}})$ and the Hodge–Deligne polynomial

The notion of the power structure over a ring was introduced in [6]. A power
structure over a ring $R$ is a method to give sense to an expression of the
form $(1+a_{1}t+a_{2}t^{2}+\ldots)^{m}$ with $a_{i}$ and $m$ from $R$ as an
element of $1+b_{1}t+b_{2}t^{2}+\ldots\in 1+tR[[t]]$ satisfying some natural
properties. In [6] there was constructed a natural power structure over the
Grothendieck ring $K_{0}({\rm{Var}_{\mathbb{C}}})$ of complex quasiprojective
varieties, i.e., for $a_{i}$ and $m$ being the classes $[A_{i}]$ and $[M]$ of
complex quasiprojective varieties the coefficients $b_{i}$ were described as
classes of quasiprojective varieties (or rather as sums of them) as well: see
Equation (2) below

A power structure over a ring $R$ (defined as a map $(1+tR[[t]])\times R\to
1+tR[[t]]$) permits to give sense to a similar expression with the series
depending on several variables $t_{1}$, …, $t_{r}$: see [8].

The ring ${\mathbb{Z}}$ of integers carries a natural power structure defined
by the usual (say, in calculus) exponent of a series.

A complex quasiprojective variety is the difference of two complex projective
varieties (a projective variety minus a projective variety). The Grothendieck
ring $K_{0}({\rm{Var}_{\mathbb{C}}})$ of complex quasiprojective varieties is
the abelian group generated by the isomorphism classes $[X]$ of
quasiprojective varieties modulo the relation $[X]=[Y]-[X\setminus Y]$ for a
Zariski closed subset $Y\subset X$. (The multiplication in
$K_{0}({\rm{Var}_{\mathbb{C}}})$ is defined by the Cartesian product.) The
(natural) power structure over the Grothendieck ring
$K_{0}({\rm{Var}_{\mathbb{C}}})$ is defined by the equation (see [6])

$\displaystyle(1+[A_{1}]t+[A_{2}]t^{2}+\ldots)^{[M]}=$ (2) $\displaystyle=$
$\displaystyle
1+\sum_{k=1}^{\infty}\left(\sum_{\\{k_{i}\\}:\sum_{i}ik_{i}=k}\left[\left.\left(\left(M^{\sum_{i}k_{i}}\setminus\Delta\right)\times\prod_{i}A_{i}^{k_{i}}\right)\right/\prod_{i}S_{k_{i}}\right]\right)\cdot
t^{k}\,,$

where $A_{i}$, $i=1,2,\ldots$, and $M$ are complex quasi-projective varieties,
$[A_{i}]$ and $[M]$ are their classes in $K_{0}({\rm{Var}_{\mathbb{C}}})$,
$\Delta$ is the “large diagonal” in $M^{\sum_{i}k_{i}}$, that is the set of
(ordered) collections of $\sum_{i}k_{i}$ points from $M$ with at least two
coinciding ones, the group $S_{k_{i}}$ of permutations on $k_{i}$ elements
acts by simultaneous permutations on the components of the corresponding
factor $M^{k_{i}}$ in $M^{\sum_{i}k_{i}}=\prod_{i}M^{k_{i}}$ and on the
components of the factor $A_{i}^{k_{i}}$.

An additive invariant of a complex quasiprojective variety $X$ is the
Hodge–Deligne polynomial $e_{X}(u,v)\in{\mathbb{Z}}[u,v]$. It is (completely)
defined by the following two properties:

  1. 1)

for a smooth projective manifold $X$

$e_{X}(u,v)=\sum_{p,q}(-1)^{p+q}h^{p,q}(X)u^{p}v^{q}\,,$

where $h^{p,q}(X)$ are Hodge numbers of the manifold $X$;

  2. 2)

if $Y$ is a Zariski closed subvariety of $X$, one has
$e_{X}(u,v)=e_{Y}(u,v)+e_{X\setminus Y}(u,v)$.

The existence of the Hodge–Deligne polynomial was proved in [4].

A power structure over the ring ${\mathbb{Z}}[u,v]$ of polynomials in two
variables was defined in [7] through a pre-$\lambda$-structure on this ring.
For the case when all the coefficients $a_{i}$ are integers (that is
polynomials do not depanding on $u$ and $v$; this is just the case in Section
4), the series $(1+a_{1}\underline{t}+a_{2}\underline{t}^{2}+\ldots)^{p(u,v)}$
($a_{i}\in{\mathbb{Z}}$, $p(u,v)=\sum p_{ij}u^{i}v^{j}$) can be described in
the following way. The series
$1+a_{1}\underline{t}+a_{2}\underline{t}^{2}+\ldots$ has a unique
representation of the form $\prod_{\underline{m}\in{\mathbb{Z}}_{\geq
0}^{r}\setminus{\overline{0}}}(1-\underline{t}^{\underline{m}})^{s_{\underline{m}}}$
with $s_{\underline{m}}\in{\mathbb{Z}}$. Then

$(1+a_{1}\underline{t}+a_{2}\underline{t}^{2}+\ldots)^{p(u,v)}=\prod_{\underline{m}\in{\mathbb{Z}}_{\geq
0}^{r}\setminus{\overline{0}}}\prod_{i,j}(1-u^{i}v^{j}\underline{t}^{\underline{m}})^{-s_{\underline{m}}p_{ij}}\,.$
(3)

The Hodge–Deligne polynomial determines a ring homomorphism
$e_{\bullet}:K_{0}({\rm{Var}_{\mathbb{C}}})\to{\mathbb{Z}}[u,v]$. Moreover,
$e_{\bullet}$ is a power structure homomorphism, that is

$e_{\bullet}\left((1+[A_{1}]t+[A_{2}]t^{2}+\cdots)^{[M]}\right)=\left(e_{\bullet}(1+[A_{1}]t+[A_{2}]t^{2}+\cdots)\right)^{e_{M}(u,v)}\,.$

(The homomorphism $e_{\bullet}$ acts on a series applying to its
coefficients.) This is a reformulation of a result from [2].

The Euler characteristic defines a power structure homomorphism
$\chi:K_{0}({\rm{Var}_{\mathbb{C}}})\to{\mathbb{Z}}$.

##  3 Exotic configuration spaces

The notion of exotic (ordered) configuration spaces was introduced in [1].

Assume that we consider $r$ different colors numbered from $1$ to $r$.
Elements of ${\mathbb{Z}}_{\geq 0}^{r}$ are called color counts. For an
element $a=(c_{1},\ldots,c_{r})\in{\mathbb{Z}}_{\geq 0}^{r}$, let
$c_{i}(a):=c_{i}$. Let ${\cal I}$ be a subset of ${\mathbb{Z}}_{\geq 0}^{r}$
containing $\overline{0}$. (In [1] one considers only the case when ${\cal I}$
is a so-called ideal in ${\mathbb{Z}}_{\geq 0}^{r}$. In the discussion below
this restriction will not be used.) An ${\cal I}$-permitted collection of
colored points (${\cal I}$-collection for short) in $X$ is a finite subset $K$
of $X$ with a map $\psi:K\to{\cal I}$. The (multi-)degree of a collection
$(K,\psi)$ is $\sum_{x\in K}\psi(x)$. Let ${\rm UConf}_{{\cal
I}}(X,\underline{k})$ be the configuration space of ${\cal I}$-collections in
$X$ of multi-degree $\underline{k}$. (If $\underline{k}=\overline{0}$, ${\rm
UConf}_{{\cal I}}(X,\underline{k})$ is a one-point space.) If $X$ is a
topological space, one has a natural topological structure on the parts of
${\rm UConf}_{{\cal I}}(X,\underline{k})$ corresponding to fixed collections
$\\{\psi(x)|x\in K\\}$. There are somewhat different possibilities to define
the topology on the union of this parts, i.e., on ${\rm UConf}_{{\cal
I}}(X,\underline{k})$, however, the the Euler characteristic (being additive)
does not depend on the choice of this topology. If $X$ is a complex
quasiprojective variety, the configuration space ${\rm UConf}_{{\cal
I}}(X,\underline{k})$ in a natural way is the union of complex quasiprojective
varieties.

Examples. 1. If $r=1$ and ${\cal I}=\\{0,1\\}$, ${\rm UConf}_{{\cal I}}(X,k)$
is the (unordered) configuration space ${\rm UConf}(X,k)$.
2. If $r=1$ and ${\cal I}={\mathbb{Z}}_{\geq 0}$, ${\rm UConf}_{{\cal I}}(X,k)$ is the symmetric product $S^{k}X=X^{k}/S^{k}$.
3. If $r=1$ and ${\cal I}=\\{k:0\leq k<m$, ${\rm UConf}_{{\cal I}}(X,k)$ is the unordered no-$m$-equal configuration space in the sense of [1]. (For $m=2$ it coincides with the usual configuration space ${\rm UConf}(X,k)$ from Example 1.)
4. If, for an arbitrary $r$, ${\cal I}$ is the union of the coordinate axes that is consists of all points of ${\mathbb{Z}}^{r}_{\geq 0}$ with not more than one coordinate different from zero (the “apartheid” ideal in the terms of [1]), the configuration space ${\rm UConf}_{{\cal I}}(X,k)$ parametrizes collections of colored points where it is forbidden points of different colors to collide, but the collision of any number of points of the same color is allowed.
5. If ${\cal I}=\\{\underline{c}\in{\mathbb{Z}}^{r}_{\geq 0}:c_{1}\leq c_{2}\leq\ldots\leq c_{r}\\}$, one can speak about the configuration space of nested subsets (a sort of a reduction of nested Hilbert schemes from [3]).

##  4 Generating series of classes of configuration spaces and of their
invariants

Now we give equations for the generating series of the classes in
$K_{0}({\rm{Var}_{\mathbb{C}}})$ of the configuration spaces ${\rm
UConf}_{{\cal I}}(X,\underline{k})$ (in the case when $X$ is a complex
quasiprojective variety) and of their invariants. For a subset ${\cal
I}\subset{\mathbb{Z}}_{\geq 0}^{r}$, $\overline{0}\in{\cal I}$, let

$C_{{\cal I}}(\underline{t}):=\sum_{\underline{c}\in{\cal
I}}\underline{t}^{\underline{c}}\in{\mathbb{Z}}[[t_{1},\ldots,t_{r}]]\subset
K_{0}({\rm{Var}_{\mathbb{C}}})[[t_{1},\ldots,t_{r}]]\,.$

###### Theorem 1

For a quasiprojective variety $X$ one has

$\sum_{\underline{k}\in{\mathbb{Z}}_{\geq 0}^{r}}[{\rm UConf}_{{\cal
I}}(X,\underline{k})]\cdot\underline{t}^{\underline{k}}=\left(C_{{\cal
I}}(\underline{t})\right)^{[X]}\,.$ (4)

###### Proof.

The statement follows directly (more or less tautologically) from the
following version of the description of the coefficients at the monomials
$\underline{t}^{\underline{k}}$ in

$\left(1+\sum_{\underline{k}\in{\mathbb{Z}}_{\geq
0}^{r}\setminus{\overline{0}}}[A_{\underline{k}}]\cdot\underline{t}^{\underline{k}}\right)^{[X]}$
(5)

($A_{\underline{k}}$ and $X$ are complex quasiprojective varieties) formulated
in the classical case ($r=1$) by E. Gorsky in [5]. Assume that, on the variety
$X$, there live particles equiped with $r$ non-negative integer numbers (say,
of different sorts of charges; colors in our case). A particle with charges
$k_{1}$, …, $k_{r}$ has a space of internal states parametrized by points of a
quasiprojective variety $A_{\underline{k}}$
($\underline{k}=(k_{1},\ldots,k_{r})$). Then the coefficient at
$\underline{t}^{\underline{k}}$ in (5) is the configuration space of tuples of
particles on $X$ with the total charges $k_{1}$, …, $k_{r}$. (In our case each
of the varieties $A_{\underline{k}}$ is either empty or consists of one
point.) ∎

###### Corollary 1

One has

$\sum_{\underline{k}\in{\mathbb{Z}}_{\geq 0}^{r}}e_{{\rm UConf}_{{\cal
I}}(X,\underline{k})}(u,v)\cdot\underline{t}^{\underline{k}}=\left(C_{{\cal
I}}(\underline{t})\right)^{e_{X}(u,v)}$

in the sense of the power structure over the polynomial ring
${\mathbb{Z}}[u,v]$ given by (3).

###### Remark 1

In Examples 1-3 in Section 3 the series $C_{{\cal I}}(t)$ has the
decomposition $(1-t)^{-1}(1-t^{2})$, $(1-t)^{-1}$, and $(1-t)^{-1}(1-t^{k})$
respectively. In Example 4 with $r=2$ one has $C_{{\cal
I}}(t)=(1-t_{1})^{-1}(1-t_{2})^{-1}(1-t_{1}t_{2})$.

###### Corollary 2

One has

$\sum_{\underline{k}\in{\mathbb{Z}}_{\geq 0}^{r}}\chi({\rm UConf}_{{\cal
I}}(X,\underline{k}))\cdot\underline{t}^{\underline{k}}=\left(C_{{\cal
I}}(\underline{t})\right)^{\chi(X)}\,.$ (6)

As a consequence of Theorem 1, Corollary 2 is valid for complex
quasiprojective varieties. Let us show that Equation (6) holds for a wider
class of spaces, namely for spaces homeomorphic to locally closed unions of
cells in finite CW-complexes.

###### Theorem 2

Let $X$ be a locally closed union of cells in a CW-complex $W$. Then one has

$\sum_{\underline{k}\in{\mathbb{Z}}_{\geq 0}^{r}}\chi({\rm UConf}_{{\cal
I}}(X,\underline{k}))\cdot\underline{t}^{\underline{k}}=\left(C_{{\cal
I}}(\underline{t})\right)^{\chi(X)}\,.$ (7)

###### Proof.

Let us denote the left hand side of Equation (7) by $\lambda_{X}^{{\cal
I}}(\underline{t})$. Assume that $Y$ is also a union of cells of $W$, closed
in $X$. Obviously one has

${\rm UConf}_{{\cal
I}}(X,\underline{k})=\bigsqcup_{\underline{\ell}\leq\underline{k}}{\rm
UConf}_{{\cal I}}(Y,\underline{\ell})\times{\rm UConf}_{{\cal I}}(X\setminus
Y,\underline{k}-\underline{\ell})$

($\underline{\ell}=(\ell_{1},\ldots,\ell_{r})\leq\underline{k}=(k_{1},\ldots,k_{r})$
iff $\ell_{i}\leq k_{i}$ for all $i$). This implies that

$\lambda_{X}^{{\cal I}}(\underline{t})=\lambda_{Y}^{{\cal
I}}(\underline{t})\lambda_{X\setminus Y}^{{\cal I}}(\underline{t})\,.$

Since $X$ is a disjoint union of (open) cells, it is sufficient to prove
Equation (7) for $X$ being a cell $\sigma^{d}$ of dimention $d$. One has
$\chi(\sigma^{d})=(-1)^{d}$. Thus it is necessary to show that

$\lambda_{\sigma^{d}}^{{\cal I}}(\underline{t})=\left(C_{{\cal
I}}(\underline{t})\right)^{(-1)^{d}}.$ (8)

For $d=0$ ($\sigma^{0}$ being a point) Equation (8) is obvious. Assume that it
is proved for cells of dimension less than $d$. In an obvious way the cell
$\sigma^{d}$ can be decomposed into two $d$-dimensional and one
$(d-1)$-dimensional cells. Therefore

$\lambda_{\sigma^{d}}^{{\cal
I}}(\underline{t})=\left(\lambda_{\sigma^{d}}^{{\cal
I}}(\underline{t})\right)^{2}\lambda_{\sigma^{d-1}}^{{\cal I}}(\underline{t})$

and thus $\lambda_{\sigma^{d}}^{{\cal
I}}(\underline{t})=\left(\lambda_{\sigma^{d}}^{{\cal
I}}(\underline{t})\right)^{-1}$, what proves the statement. ∎

## References

  * [1] Yu. Baryshnikov. Euler characteristics of exotic configuration spaces. Sém. Lothar. Combin. 84B, Art.20 (2020).
  * [2] J. Cheah. On the cohomology of Hilbert schemes of points. J. Algebraic Geom. v.5, no.3, 479–511 (1996).
  * [3] J. Cheah. The virtual Hodge polynomials of nested Hilbert schemes and related varieties. Math. Z. v.227, no.3, 479–504 (1998).
  * [4] P. Deligne. Théorie de Hodge III. Inst. Hautes Etudes Sci. Publ. Math. v.44, 5–77 (1974).
  * [5] E. Gorsky. Adams operations and power structures. Mosc. Math. J. v.9, no.2, 305–323 (2009).
  * [6] S.M. Gusein-Zade, I. Luengo, A. Melle-Hernández. A power structure over the Grothendieck ring of varieties. Math. Res. Lett. v.11, no.1, 49–57 (2004).
  * [7] S.M. Gusein-Zade, I. Luengo, A. Melle-Hernández. Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points. Michigan Math. J. v.54, no.2, 353–359 (2006).
  * [8] S.M. Gusein-Zade, I. Luengo, A. Melle-Hernández. On the power structure over the Grothendieck ring of varieties and its applications. Proc. Steklov Inst. Math. v.258, no.1, 53–64 (2007).

Moscow State University, Faculty of Mechanics and Mathematics,
Moscow, GSP-1, 119991, Russia
& National Research University “Higher School of Economics”,
Usacheva street 6, Moscow, 119048, Russia.
E-mail<EMAIL_ADDRESS>


# Spin-parities of sub-threshold resonances in the 18F(p, $\alpha$)15O
reaction

F. Portillo  R. Longland<EMAIL_ADDRESS>Department of Physics,
North Carolina State University, Raleigh, NC 27695, USA  Triangle Universities
Nuclear Laboratory, Durham, NC 27708, USA  A. L. Cooper  Present Address: Los
Alamos National Laboratory, Los Alamos, NM 87545, USA  S. Hunt  Department of
Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel
Hill, NC 27599, USA  Triangle Universities Nuclear Laboratory, Durham, NC
27708, USA  A. M. Laird  Department of Physics, University of York, York YO10
5DD, UK  C. Marshall  Present address: Institute of Nuclear & Particle
Physics, Department of Physics & Astronomy, Ohio University, Athens, Ohio
45701, USA  K. Setoodehnia  Present address: Facility for Rare Isotope Beams,
Michigan State University, East Lansing, MI 48824, USA  Department of Physics,
North Carolina State University, Raleigh, NC 27695, USA  Triangle Universities
Nuclear Laboratory, Durham, NC 27708, USA

###### Abstract

The 18F(p, $\alpha$)15O reaction is key to determining the 18F abundance in
classical novae. However, the cross section for this reaction has large
uncertainties at low energies largely caused by interference effects. Here, we
resolve a longstanding issue with unknown spin-parities of sub-threshold
states in 19Ne that reduces these uncertainties. The 20Ne(3He, 4He)19Ne
neutron pick-up reaction was used to populate 19Ne excited states, focusing on
the energy region of astrophysical interest ($\approx$ 6 - 7 MeV). The
experiment was performed at the Triangle Universities Nuclear Laboratory using
the high resolution Enge split-pole magnetic spectrograph. Spins and parities
were found for states in the astrophysical energy range. In particular, the
state at 6.133 MeV ($E_{r}^{\text{c.m.}}=-278$ keV) was found to have spin and
parity of $3/2^{+}$ and we confirm the existence of an unresolved doublet
close to 6.288 MeV ($E_{r}^{\text{c.m.}}=-120$ keV) with Jπ = $1/2^{+}$ and a
high-spin state. Using these results, we demonstrate a significant factor of
two decrease in the reaction rate uncertainties at nova temperatures.

##  I Introduction

Classical novae are the observational signature of thermonuclear explosions
that occur in binary systems consisting of a white dwarf and typically a Main
Sequence star (see Ref. Hernanz (2005) for an overview). When $\approx$ 10-5
M⊙ of hydrogen-rich material is accreted by the white dwarf at a low rate
(e.g., 10-11 M⊙yr-1), the base of the envelope reaches the conditions
necessary to generate a thermonuclear explosion, where the temperature rises
dramatically from 0.1 - 0.4 GK a thermonuclear runaway occurs. This explosion
releases an energy of $\approx$ 1045 ergs and material containing elements up
to mass $A=40$ Bode and Evans (2008). The nucleosynthesis path proceeds close
to the valley of stability following break-out from the carbon-nitrogen-oxygen
(CNO) cycle in a series of (p,$\gamma$) and (p,$\alpha$) reactions, and
$\beta^{+}$ decays.

Among the elements synthesized in classical novae, the radioactive 18F
($\tau_{1/2}=$110 min) decays by $\beta^{+}$ decay, thus emitting 511 keV
$\gamma$ rays. This is of particular interest because the decay is expected to
occur shortly after the explosion when the remnant becomes transparent to
gamma rays. However, to date no $\gamma$ rays have been observed at these
energies, in part due to the low sensitivity of the instruments used for their
detection (e.g, INTEGRAL Hernanz and José (2004)) and in part because these
$\gamma$ rays are emitted before the nova visual peak José et al. (2006).
Detection of these low energy $\gamma$ rays would help improve nova
theoretical models, but will be an unreliable constraint unless uncertainty in
the nuclear reactions producing fluorine is significantly reduced. The 18F(p,
$\alpha$)15O reaction rate is key among these.

The source of the uncertainty in the 18F(p, $\alpha$)15O reaction is mostly
due to interference effects between low energy resonances close to the proton
threshold and higher energy resonances. Many of the resonances in the
18F(p,$\alpha$)15O and 18F(p,$\gamma$)19Ne reactions were first identified and
compared with the mirror 19F in 1998 by Utku et al. Utku et al. (1998).
Following that, a series of measurements have further constrained the reaction
rates through direct cross section measurements (Refs. Bardayan et al. (2001,
2002); de Séréville et al. (2009); Beer et al. (2011)), differential cross
sections Murphy et al. (2009); Mountford et al. (2012), and particle transfer
or charge exchange reactions Adekola et al. (2011); Laird et al. (2013);
Parikh et al. (2015); Bardayan et al. (2017); Kahl et al. (2019); Riley et al.
(2021), including some coincident $\gamma$-ray measurements Hall et al. (2019,
2020). These results have been thoroughly compiled previously by Nesaraja et
al. 2007 Nesaraja et al. (2007) and most recently by Kahl et al. 2021 Kahl et
al. (2021). However, despite these efforts, the reaction rate of the 18F(p,
$\alpha$)15O reaction contains large uncertainties.

In a recent result, Ref. Kahl et al. (2019) found that the Jπ quantum numbers
of the state below the 19Ne proton threshold at 6.133 MeV
($E_{r}^{\text{c.m.}}=-278$ keV resonance) should be either $1/2^{+}$ or
$3/2^{+}$, confirming the presence of an s-wave resonance close to and below
the proton threshold. The other 19Ne excited state, close to the proton-
threshold, predicted to correspond to an s-wave resonance, has been proposed
to exist around the 6.29 MeV energy region ($E_{r}^{\text{c.m.}}=-120$ keV
resonance). The energy and the spin and parity of this state have been much
debated, where assignments of low spin Bardayan et al. (2015) or high spin
(e.g., Laird et al. (2013), Hall et al. (2019), Riley et al. (2021)) have been
proposed. Parikh et al. Parikh et al. (2015) suggested that this state could
be either a doublet or a broad state. These ambiguities lead to large
uncertainties arising from strongly different interference effects in the
astrophysical energy region, as explored by the study of Kahl et al. 2021 Kahl
et al. (2021).

In this work, we present results that significantly constrain the interference
possibilities by clarifying the spin-parities of the sub-threshold resonances
at $E_{r}^{\text{c.m.}}=-278$ keV and $E_{r}^{\text{c.m.}}=-121$ keV. These
were obtained using the neutron pick up reaction 20Ne(3He,$\alpha$)19Ne to
populate 19Ne excited states in order to determine their energies and spins
and parities. In Section II experimental details of this work are described.
Section III describes our analysis, for which the results for 19Ne states are
discussed in section IV. The impact of these findings on the cross section and
their impact on the 18F(p, $\alpha$)15O reaction rates at classical nova
temperatures is presented in Sec. V. A summary and conclusions are given in
Section VI.

##  II Experiment Details

The implanted 20Ne transmission targets used in the experiment were fabricated
using an electron cyclotron resonance ion source Cooper et al. (2018) to
implant 20Ne at an energy of 25 keV directly into a 40 $\pm 4$ $\mu$g/cm2
thick carbon foil. The foil was mounted on a rectangular glass slide (3.7 cm x
2.5 cm) from which a total of four targets were obtained.

Figure 1: RBS spectrum. The red and blue lines are the experimental data and
SIMNRA Mayer (1999) simulation, respectively. The analysis done shows that,
apart from the natural carbon substrate, the target contains a 3.6 $\pm$ 0.4
$\mu$g/cm2 thick 20Ne layer and an 16O contamination layer 0.22 $\pm$ 0.02
$\mu$g/cm2 thick located at the front.

To characterize the targets, we used the Rutherford Backscattering
Spectrometry technique (RBS) on one of the targets by accelerating a 2 MeV
4He++ beam and placing a silicon surface barrier detector at 165$\,{}^{\circ}$
, with respect to the incident beam, to detect the back-scattered
$\alpha$-particles. Figure 1 shows the RBS spectrum for one of the 20Ne
targets. Based on analysis using the software package SIMNRA Mayer (1999), the
targets were determined to consist of carbon, with a 3.6 $\pm$ 0.4 $\mu$g/cm2
thick region of 20Ne, located close to the center, a relatively thin 0.22
$\pm$ 0.02 $\mu$g/cm2 thick 16O contamination layer located at the front. The
16O contamination was also present on unimplanted carbon foils, indicating
that it has a common source.

The neutron pick up reaction 20Ne(3He,4He)19Ne at a beam energy of 21 MeV was
used to populate excited states in 19Ne. This beam energy was chosen to ensure
the direct nature of the pick-up reaction. The Q-value of this reaction is
3712.32 (16) keV Wang et al. (2017). The experiment was performed at the
Triangle Universities Nuclear Laboratory (TUNL) by using its 10 MV FN Tandem
Van de Graaff accelerator to accelerate a 3He++ beam to an energy of 21 MeV
with a mean intensity on target of 600 pnA. The dead-time registered while
collecting data was less than 1% at a rate of $\approx$ 200 Hz. Before
impinging on target, the beam was tuned through a 1-mm-diameter collimator.
Data were collected for both implanted 20Ne and a natural carbon target used
for background reference. Reaction products entered the TUNL Enge split-pole
magnetic spectrograph Setoodehnia et al. (2017) whose magnetic field was set
at B = 0.81 T to focus reaction products into the detector’s focal plane. The
spectrograph was set at six different laboratory angles ($\theta_{lab}$=
9$\,{}^{\circ}$ , 12$\,{}^{\circ}$ , 20$\,{}^{\circ}$ , 22$\,{}^{\circ}$ ,
25$\,{}^{\circ}$ , and 27$\,{}^{\circ}$ ) and its entrance aperture subtended
a solid angle with respect to the target of 1 msr for all angles. The beam was
on target for about 10 hours for each angle. Target degradation was monitored
hourly using the beam current normalized intensity of the peak corresponding
to the strongly populated 6.0143 (11) MeV excited state. Degradation was
corrected for during analysis.

Figure 2:  Two-Dimensional energy loss vs front position spectrum collected at
$\theta_{Lab}=$ 20∘ from signals coming from these sections of the focal plane
detector. Bands of events coming from protons, deuterons, 3He, and 4He are
indicated, with the software gate used for $\alpha$-particles shown as a black
polygon.

A position-sensitive focal plane detector was used to detect the outgoing
reaction products that entered the spectrograph. This detector consists of two
position sensitive sections, one energy loss section, and a total energy
section. The position sensitive sections are avalanche counters consisting of
five equally spaced high voltage wires placed in the isobutane gas filled
region across the detector. Their position resolution corresponds to an energy
resolution of about 20 keV for the present experiment. These two sections are
separated by an energy loss section, which is a proportional gas counter
consisting of a single high voltage wire also within an isobutane gas filled
region. Finally, the total energy section is a plastic scintillator attached
to a photomultiplier tube and is used to register the particles’ residual
energy. Details of the focal plane detector can be found in Marshall et al.
(2019).

##  III Analysis

Alpha particles were selected by setting a software gate on the energy loss vs
front position spectrum. An example gathered at $\theta_{Lab}=$20∘ is shown in
Fig. 2. By setting an off-line software gate to data in the 4He band we
generated the 1-dimensional spectrum shown in Fig. 3. Peak positions and
intensities for the $\alpha$-particle spectra at each angle were obtained by
fitting Gaussian peaks with a linear function to describe the local
background. In areas with many peaks, several Gaussian peaks were fitted with
a common width to help constrain the fit.

Figure 3:  Energy calibrated $\alpha$-particle gated spectrum at
$\theta_{\text{lab}}=20^{\circ}$. The peaks corresponding to 19Ne excited
states are labeled with the energies (in MeV) in column 2 of Tab. 2. The peaks
with a red label correspond to excited states coming from reactions involving
the substrate or contaminants. The inset figure highlights the region around
the sub-threshold resonances, which strongly influence the interference
effects at high energies. The vertical dashed blue line denotes the proton
separation energy.

To determine the energies of the observed 19Ne excited states, the code SPANC
Visser (2009) was employed to perform an internal calibration from known
calibration states in 19Ne . A second degree polynomial model was used to fit
particle position on the focal plane as a function of the centroid channel of
calibration peaks. To account for energy loss in the target, SPANC includes
angle-dependent entrance and exit particle energy loss estimation. From the
polynomial fits, excitation energies at each spectrograph measurement angle
could be extracted. Our reported excitation energies in the third column of
Tab. 2 were then determined through a weighted average over all angles. Due to
systematic effects, the uncertainty in the weighted average is constrained to
be no smaller than the smallest single-angle energy uncertainty. To account
for an estimated detector response uncertainty, the weighted average energies
are summed in quadrature with an additional 2 keV uncertainty.

The spin and parities of states analyzed in this work were constrained by
comparing the experimental angular distribution for each state with
theoretical angular distributions. Theoretical angular distributions were
computed assuming that the pick up reaction was a one-step transfer reaction.
We used the Distorted Wave Born Approximation (DWBA) finite-range code FRESCO
Thompson (1988) with the global optical potential parameters shown in Tab. 1.
The depth of the potentials used for the neutron binding was varied to
reproduce the neutron binding energy.

Table 1:  Global optical potential parameters used in the DWBA analysis.

Channel | rC(fm) | VR (MeV) | rR (fm) | aR (fm) | VI (MeV) | rI (fm) | aI (fm) | VD (MeV) | rD (fm) | aD (fm) | Vso (MeV) | rso (fm) | aso (fm)
---|---|---|---|---|---|---|---|---|---|---|---|---|---
3He+20Neb | 1.3 | 148.33 | 1.2 | 0.72 | 34.77 | 1.4 | 0.88 | - | - | - | 2.5 | 1.2 | 0.72
4He+19Nec | 1.35 | 160.12 | 1.34 | 0.66 | - | 1.43 | 0.56 | 23.38 | 1.29 | 0.64 | - | 1.27 | 0.85
3He + 19Neb | 1.3 | 160.12 | 1.34 | 0.66 | 32.45 | 1.43 | 0.56 | - | - | - | - | - | -
19Ne+n | 1.2 | a | 1.12 | 0.79 | - | - | - | - | - | - | 3.0 | 1.27 | 0.75
3He+n | 2 | a | 1.17 | 0.69 | - | - | - | - | - | - | 6.0 | 1.17 | 0.69

00footnotetext: Adjusted to produce binding energy00footnotetext: From Perey
and Perey 1976 Perey and Perey (1976)

00footnotetext: From Su and Han 2015 Su and Han (2015)

##  IV Results and Discussion

Table 2: 19Ne excited states energies (in MeV) observed in this work. The columns show, from left to right, the energies reported by NNDC Tilley et al. (1995) compilation; our compiled energies since the NNDC values were published (1997-2020), i.e., Refs. Utku et al. (1998), Tan et al. (2009), Dalouzy et al. (2009), Murphy et al. (2009), Laird et al. (2013), Bardayan et al. (2015), Bardayan et al. (2017), Kahl et al. (2019), Hall et al. (2019), andKahl et al. (2021); and the energies determined in this work (see text for details). The compiled energies marked with an asterisk (*) and in italics were used for calibrations in this work. NNDC | Compilation | Present Work
---|---|---
1.5360 (4) | 1.535 (3) | 1.532 (3)
1.6156 (5) | 1.6156 (4) | 1.613 (3)
2.7947 (6) | *2.7944 (5) | -
4.0329 (24) | 4.0344 (6) | 4.0347 (23)
4.140 (4) | 4.1431 (7) | 4.1421 (22)
4.1971 (24) | *4.1998 (7) | -
4.549 (4) | *4.5477 (9) | -
4.600 (4) | 4.6021 (6) | 4.6017 (25)
4.635 (4) | 4.6341 (9) | 4.633 (3)
4.712 (10) | *4.7088 (17) | -
5.092 (6) | 5.0906 (23) | 5.0908 (25)
5.351 (10) | 5.353 (5) | 5.351 (3)
5.424 (7) | 5.424 (5) | 5.424 (3)
5.539 (9) | 5.535 (7) | 5.543 (3)
5.832 (9) | *5.828 (7) | -
6.013 (7) | *6.0147 (15) | -
6.092 (8) | 6.0998 (9) | 6.092 (3)
6.149 (20) | 6.1327 (23) | 6.133 (3)
6.288 (7) | 6.2883 (14) | 6.284 (3)
6.699 (3) | 6.699 (3) | Unobserved
6.742 (7) | 6.7420 (12) | 6.738 (3)
6.861 (7) | 6.8619 (15) | 6.861 (4)
7.067 (9) | *7.0747 (9) | -
7.531 (15) | 7.558 (3) | 7.549 (5)
7.616 (16) | 7.617 (3) | 7.605 (5)
7.700 (10) | *7.706 (17) | -
7.788 (10) | *7.788 (9) | -
7.994 (15) | 7.982 (6) | 7.970 (4)
8.069 (12) | 8.069 (9) | 8.057 (5)
8.236 (10) | *8.236 (10) | -

The energies are reported in Tab. 2. For ease of comparison, we first list the
energies as compiled in the National Nuclear Data Center (NNDC) Tilley et al.
(1995). The second column corresponds to an average of the NNDC values and all
following energies reported in the literature from the years 1995 to 2021
(Utku et al. (1998), Kozub et al. (2005), Chae et al. (2006), Nesaraja et al.
(2007), Dalouzy et al. (2009), Murphy et al. (2009), Tan et al. (2009), Laird
et al. (2013), Bardayan et al. (2015), Bardayan et al. (2017), Hall et al.
(2019), Kahl et al. (2019), Kahl et al. (2021)). The third column contains
energies obtained in this work using the procedure described above.

Each of the observed levels is discussed individually below for the energy
region relevant for classical novae (6 - 7 MeV). A tentative analog state
assignment in 19F is made for most of the observed 19Ne excited states by
assuming isospin symmetry and by considering their energies. This follows the
same procedure as other recent works (Refs. Bardayan et al. (2015); Hall et
al. (2020); Kahl et al. (2021), for example). Mirror pairs were assigned below
6 MeV by systematically compiling the levels in each mirror nuclei comparing
the Jπ values of the analog pair states. In compiling mirror assignments, we
find an average mirror energy difference (MED) of 70 $\pm$ 50 keV between the
pairs, which we use as a guide in our discussion below. However, we should
stress that the information is not complete. There could be intruder states,
or missing states from the mirror, 19F , that may complicate the picture, and
some well-known states still do not have an assigned mirror partner (see
below). Future work should concentrate on a full accounting of mirror states
between 19Ne and 19F, and their applicability to calculating the 18F(p,
$\alpha$)15O cross section.

6.0147 (15) MeV, $\mathbf{E_{r}^{\textbf{c.m.}}=-395}$ keV, Jπ = (1/2-, 3/2-):
This state was suggested by Ref. Riley et al. (2021) to be an unresolved
doublet consisting in two states approximately around 6.008 MeV and a 6.014
MeV. However, the 6.008 MeV state was proposed to be very broad
($\Gamma=124(25)$ keV). We were not able to resolve this lower-energy state,
and furthermore observed no broadening of the peak in our spectra. The angular
distribution obtained for this state indicates that only a single state was
strongly populated. The angular momentum transfer is clearly described by
$\ell=1$, so the spin and parity of this state could be either Jπ = $1/2^{-}$
or Jπ = $3/2^{-}$. The average MED suggests that the analog state in 19F could
be the 6.088 MeV level, which has Jπ = $3/2^{-}$. There is no mirror state
with a Jπ = $1/2^{-}$ within the average MED (the closest is at 6.429 MeV with
an MED=414 keV), but we nevertheless maintain a tentative Jπ =
$(1/2^{-},3/2^{-})$.

6.0998 (9) MeV, $\mathbf{E_{r}^{\textbf{c.m.}}=-310}$ keV, Jπ = (5/2-,
7/2-,7/2+, 9/2+): The peak corresponding to this state appeared as a member of
a resolved doublet with the 6.133 MeV state. Fig. 4 shows the experimental
angular distribution for this state together with the theoretical curves for
the possible $\ell$-values that correspond to the Jπ values of close energy
analog states within the average MED, namely, 6.070 ($7/2+$), 6.088 ($3/2-$),
6.100 ($9/2-$), and 6.161 ($7/2-$) MeV. The angular distribution for this
state is very flat, suggesting a high-spin state. The experimental data are
best described by the $\ell=3$ curve and to a lesser extent to the one
corresponding to $\ell=4$. This is in slight tension with the results of Laird
et al. Laird et al. (2013).

6.1327 (23) MeV, $\mathbf{E_{r}^{\textbf{c.m.}}=-277}$ keV, Jπ = 3/2+: This
state corresponds to the 18F+p sub-threshold resonance at
$E_{r}^{\text{c.m.}}=-277$ keV. The spin-parity of this state has long been
discussed, and was recently narrowed down by Kahl et al. Kahl et al. (2019).
In their work they populated 19Ne states resulting from Gamow-Teller
transitions in the 19F (3He,t)19Ne reaction. They observed a state at 6.133
MeV that must have Jπ = $1/2^{+}$ or $3/2^{+}$. As will be discussed in
Section V, the spin and parity of this state impacts the 18F(p, $\alpha$)15O
reaction cross section uncertainty at low energies due to interference
effects.

This state was observed in our work as one of the members of a resolved
doublet whose other member is the state at 6.100 MeV discussed above. Fig. 4
presents the experimental angular distribution for this state together with
curves corresponding to the two angular momentum transfer possibilities
established by Ref. Kahl et al. (2019) (i.e., $\ell=0$ and $\ell=2$). The
angles between $\theta_{c.m.}$ = 11∘ \- 18∘ and $\theta_{c.m.}$ = 30∘ \- 36∘
are where the two theoretical angular distributions differ appreciably (by
more than one order of magnitude in the cross section). Consequently, data at
$\theta_{c.m.}\approx$ = 14∘ and $\theta_{c.m.}\approx$ = 32∘ are key to
determining that $\ell=2$. This is in clear contrast to the state at 5.353 MeV
corresponding to $\ell=0$, where only an upper limit could be obtained at
$\theta_{c.m.}=14^{\circ}$. In combination with Ref. Kahl et al. (2019), we
can confirm that this state has a spin-parity of Jπ = $3/2^{+}$. We could not,
however, locate a mirror level in 19F within the calculated average MED $70\pm
50$ keV, corresponding to a state in 19F at about 6.2 MeV. The closest are
either at 5.501 MeV (MED=630 keV) or the pair at 6.497 and 6.527 MeV (MED=365
and 394 keV, respectively). This finding suggests that there may be an
undiscovered Jπ 3/2+ state in the mirror nucleus. A full re-accounting for
mirror states should be undertaken to assign mirror pairs, but this is outside
the scope of the present work.

Ex= 6.288 MeV ($\mathbf{E_{r}^{\textbf{c.m.}}=-122}$ keV) region: A state in
this region corresponds to the $E_{r}^{\text{c.m.}}=-122$ keV sub-threshold
resonance closest to the proton threshold. The spin-parity of this state has
been discussed extensively owing to its impact on interference patterns in the
cross section. Refs. Parikh et al. (2015) and Hall et al. (2020) suggested
that there is likely an unresolved doublet of high and low spin states at this
energy.

The measured angular distribution for a peak corresponding to this energy
region is shown in Fig. 4. Unfortunately owing to the presence of contaminant
peaks in the spectrum, data were only extracted at higher angles for this
state. They are not described by a single transferred angular momentum
($\ell=0$, $\ell=2$, or $\ell=4$), but rather support the hypothesis that
there is a doublet. Since the expected separation of $\approx 10$ keV is far
below the resolution of our detector, we do not see any obvious broadening of
the peak. However, the data are best described by the combination of
$\ell=0+4$ in the proportion 0.40 (10) + 0.60 (10), although all angular
momenta with $\ell\geq 4$ for the second member are supported by our data.
This supports the results of Refs. Parikh et al. (2015) and Hall et al. (2020)
that a closely-spaced doublet is present at this energy. The latter of these
results placed a likely value of Jπ = $11/2^{+}$ from observed $\gamma$-ray
decays. However, the nearest mirror state in 19F is at 6.500 MeV with MED=212
keV, placing it in tension with other mirror assignments. We therefore adopt
an assignment of Jπ $=7/2^{+}$ in our cross section calculations to
demonstrate the maximum effect of this level. The Jπ = $1/2^{+}$ component of
this doublet has the most impact on the reaction rates for the 18F(p,
$\alpha$)15O reaction, since it interferes with the broad 1.4 MeV resonance
Bardayan et al. (2015). The Jπ $\geq$ $7/2^{+}$ component has negligible
impact since it implies a proton orbital angular momentum of $l_{p}\geq 2$
i.e., a higher angular momentum barrier in the 18F(p, $\alpha$)15O reaction.
If a spin-parity of Jπ $=11/2^{+}$ were assumed, as suggested by Ref. Hall et
al. (2020), the contribution would be even less.

Based on the calculated average MED and on the Jπ values found, the only two
analog states in 19F that have these Jπ quantum numbers and that fall within
the 6.2 - 6.3 MeV energy region are the 6.255 MeV ($1/2^{+}$ with MED=33 keV)
and 6.330 MeV ($7/2^{+}$ with MED=42 keV) levels.

Ex= 6.416 – 6.459 MeV ($\mathbf{E_{r}^{\textbf{c.m.}}=6-49}$ keV): Structure
corresponding to states in this region was only observed at 25 and 27 degrees.
We were also unable to extract any parameters for these states corresponding
to $E_{r}^{\text{c.m.}}=6-49$ keV owing to their close proximity to each
other. We adopt the following from a combination of Refs. Kahl et al. (2021),
Hall et al. (2020), and references therein. The 6.416 MeV state was assigned a
spin-parity of $3/2^{+}$ by Kahl et al. Kahl et al. (2021), which is in
tension with the assignments of Refs. Adekola et al. (2011); Laird et al.
(2013); Hall et al. (2020). We adopt the 6.439 MeV, $1/2^{-}$ state parameters
from Ref. Bardayan et al. (2015). Hall et al. proposed two $3/2^{+}$ states at
6.423 MeV and 6.441 MeV Hall et al. (2020). As noted by Ref. Kahl et al.
(2021), this number of $3/2^{+}$ states near the proton threshold exceeds the
number of known $3/2^{+}$ states in the mirror 19F. As with the 6.132 MeV
state, this suggests that there may be undiscovered $3/2^{+}$ states in 19F.
We adopt the upper limit values assigned by Ref. Hall et al. (2020) for the
6.441 MeV state. Finally, we adopt the 6.459 MeV, $5/2^{-}$ state parameters
from Ref. Laird et al. (2013).

Attempts were made to fit multiple peaks to the peak structure in the region.
By constraining the peak widths using neighboring states, one or more peaks
were fit to the spectrum at 25 degrees, where the 6.4 MeV region is free of
background contamination. No more than a single peak at 6.452 MeV is
consistent with our data. It is possible that the selectivity of the (3He,4He)
reaction only populates a single state, or perhaps the number of counts
obtained for these states is not high enough to observe the other states in
the region.

Ex= 6.699 (3) MeV, $\mathbf{E_{r}^{\textbf{c.m.}}=289}$ keV, Jπ = $(5/2^{+})$:
The state corresponding to the narrow $E_{r}^{\text{c.m.}}=289$ keV resonance
was not observed in this study due to an overlapping background line from
carbon. We adopt the resonance parameters obtained by Bardayan et al. Bardayan
et al. (2015).

Ex= 6.7420 (12) MeV, $\mathbf{E_{r}^{\textbf{c.m.}}=332}$ keV, Jπ = $3/2^{-}$:
This state corresponds to the $E_{r}^{\text{c.m.}}=332$ keV resonance in the
18F + p reaction. It was observed at only three angles due to the presence of
a background carbon peak in this region of the focal plane. The angular
distribution for this state corresponds to $\ell=1$, indicating that it could
have spin and parity Jπ = $1/2^{-}$ or Jπ = $3/2^{-}$. Visser et al. Visser et
al. (2004) found that this state has Jπ quantum numbers of $3/2^{-}$, later
confirmed by Refs. Nesaraja et al. (2007), Kahl et al. (2019), and Hall et al.
(2019). Based on this Jπ value and on the calculated average MED, the analog
level in 19F assigned to this state is likely the one at 6.787 MeV (Jπ =
$3/2^{-}$).

Ex= 6.8619 (15) MeV, $\mathbf{E_{r}^{\textbf{c.m.}}=452}$ keV, Jπ = $7/2^{-}$:
This state corresponds to a $E_{r}^{\text{c.m.}}=452$ keV resonance in the 18F
+ p system. It was one of the most strongly populated states in the spectra
and was observed at almost all of the measured angles. Fig. 4 shows that data
for this level is best described by the curve corresponding to $\ell=3$,
confirming the literature assignment of Jπ $=7/2^{-}$ (Refs. Tilley et al.
(1995) and Laird et al. (2013)). The high spin of this state creates a high
angular momentum barrier, reducing its contribution to the 18F(p, $\alpha$)15O
reaction. Based on the calculated average MED, the analog state in 19F could
be the one at 6.927 MeV.

Ex=7.0747 (9) MeV, $\mathbf{E_{r}^{\textbf{c.m.}}=665}$ keV, Jπ = $3/2^{+}$:
This state corresponds to the $E_{r}^{\text{c.m.}}=665$ keV resonance in the
18F + p system. The resonance has a reported low spin of Jπ = $3/2^{+}$ and is
broad ($\Gamma_{\alpha}=23.8$ keV) Bardayan et al. (2001). Although the width
of this broad resonance should be resolvable in our experiment, the peak was
in close proximity to a strong background peak, making a robust analysis
challenging. This state corresponds to one of the key resonances in 18F(p,
$\alpha$)15O at nova temperatures since it strongly interferes with other
$3/2^{+}$ resonances, and the sign of that interference is unknown. In this
experiment, the state was observed at just three angles due to the presence of
a carbon peak at the focal plane. However, the measured angular distribution
shown in Fig. 4 supports the literature Jπ = $3/2^{+}$ assignment (e.g.,
Tilley et al. (1995), Bardayan et al. (2001), Bardayan et al. (2017)). The
mirror energy level for this state has not been assigned, and the only Jπ =
$3/2^{+}$ in the mirror is at 7.262 MeV: a narrow state.

Ex=7.788 (9) MeV, $\mathbf{E_{r}^{\textbf{c.m.}}=1378}$ keV, Jπ =
$\mathbf{1/2^{+}}$: The broad state predicted by Ref. Dufour and Descouvemont
(2007) corresponding to a resonance at about $E_{r}^{\text{c.m.}}=1400$ keV
was only observed at two angles in this work owing to the presence of
background peaks from oxygen contamination in the target. It was also not
observed by Murphy et al Murphy et al. (2009), but was at least consistent
with the cross section measured by Mountford et al. Mountford et al. (2012).
More recently, however, Kahl et al. Kahl et al. (2019) firmly observed an
$\ell=0$ state at $E_{x}=7.790$ MeV, albeit with a much narrower width than
that predicted in Ref.Dufour and Descouvemont (2007). More discussion of the
impact of this resonance can be found in Sec. V.

E${}_{x}>$7.1 MeV: Although they are not expected to have a large impact on
the reaction rate at astrophysical energies, we include resonances from
Mountford et al. Mountford et al. (2012) with the exception of their broad,
inferred resonance at $E_{x}=1.455$ MeV, which we replace by the $E_{x}=7.788$
MeV state as discussed above.

Figure 4: Angular distributions obtained from the 20Ne(3He, 4He)19Ne
experiment for 19Ne excited states with energies of astrophysical interest (6
- 8 MeV). The curves were calculated with FRESCO. Each panel shows the 19Ne
level energy together with the angular momentum transfers that were found to
best describe the data. The downward-facing arrow for the state at 5.351 MeV
indicates an upper limit.

##  V R-Matrix analysis

In order for interference effects to significantly affect the reaction rate,
the interfering resonances must have the same spin, parity, and be broad
enough to overlap significantly. The R-Matrix formalism Lane and Thomas (1958)
provides a convenient method for investigating a reaction cross section in the
presence of many broad, interfering resonances. In this formalism, the total
cross section is given by de Séréville et al. (2009):

$\sigma(E)=\frac{\pi}{k^{2}}\sum_{J\pi}\frac{2J+1}{(2J_{0}+1)(2J_{1}+1)}\left|U^{J\pi}(E)\right|^{2},$
(1)

where $J$, $J_{0}$, and $J_{1}$ are the spins of the resonance, projectile,
and target, respectively. $U^{J\pi}(E)$ is the collision matrix as described
in Ref. Lane and Thomas (1958). To visualize more easily the effect of
resonances in the cross section, it is more useful to plot the so-called
astrophysical S-factor. Using the notation above and expanding the collision
matrix to resonance energies ($E_{\lambda}$) and partial widths,
($\Gamma_{\lambda}$), the astrophysical S-factor can be written as Adekola
(2009):

$S(E)=\frac{\pi\,E\,e^{2\pi\eta}}{k^{2}}\sum_{J\pi}\frac{2J+1}{(2J_{0}+1)(2J_{1}+1)}\left|\sum_{\lambda}\frac{\pm\sqrt{\Gamma^{J\pi}_{\lambda
p}(E)\,\,\Gamma^{J\pi}_{\lambda\alpha}(E)}}{E^{J\pi}_{\lambda}-E-i\Gamma^{J\pi}_{\lambda}(E)}\right|^{2},$
(2)

where $\eta$ is the Sommerfeld Parameter. The phenomenon of interference
becomes clear in the sum over energy levels, $\lambda$, where for fixed Jπ
quantum numbers each term could have either a positive or negative relative
sign. We will refer to the cases where the cross section increases or
decreases between two resonances as constructive and destructive interference.
Note, however, that these terms do not correspond to higher and lower reaction
rates, respectively, at all temperatures. Table LABEL:table-resonances details
the R-matrix input parameters used in this work to calculate the 18F(p,
$\alpha$)15O cross section.

The uncertainty in the reaction rates for the 18F(p, $\alpha$)15O reaction in
classical nova at low temperatures (e.g., $T_{9}<80MK$) is dominated by
interference between resonances near the proton threshold and high energy
resonances (e.g., 665, 1462 keV). To study the interference effects on this
reaction rate, with the spin and parities found in this work for the sub-
threshold resonances at -278 and -120 keV, we used the R-matrix code AZURE2
Azuma et al. (2010), with resonance parameters defined in Tab. LABEL:table-
resonances. In order to remain consistent with previous studies and in light
of the fact that most of the reaction rate uncertainty arises from the
interference effects, we make two simplifications: (i) no uncertainties in
resonance parameters are considered, and (ii) upper limit values are used as-
is. The rate is likely overestimated due to the latter simplification, but the
effect is minor at the temperatures of interest here.

Although new excitation energies were determined here with comparable
uncertainties to the literature, they are not used for our R-Matrix
calculations. This is to ensure better consistency with the literature when
comparing rates. Resonance energies are determined using the compiled
resonance energies and the proton separation energy of S${}_{p}=6410.0(5)$ keV
Wang et al. (2017). Future work should concentrate on finalizing the resonance
energies and their uncertainties to investigate their effect on the 18F(p,
$\alpha$)15O reaction rate uncertainty.

Table 3: List of the resonances and their properties used in the R-matrix analysis.  Energy (MeV) |  $E_{R}$ (keV) | Jπ | ANC (fm-1/2)/$\Gamma_{p}$ (keV) |  $\Gamma_{\alpha}$(keV)
---|---|---|---|---
6.1327 (23) | -277 | 3/2+ | 6111From Kahl et al. 2019 Kahl et al. (2019) | 0.74a
6.2883 (14) | -122 | 1/2+ \+ 7/2+777Maximizing spin-parity (see text for details) | 83.5444From Laird et al. 2013 Laird et al. (2013) & 16222Upper Limit | 11.6 555From Bardayan et al. 2015 Bardayan et al. (2015) & 2.51
6.416 (4) 333From Hall et al. 2020 Hall et al. (2020) | 6 | 3/2- | 4.7$\times$ 10-50 | 0.5
6.439 (3) 333From Hall et al. 2020 Hall et al. (2020) | 29 | 1/2- |  $<$ 3.8$\times$ 10-19 | 220
6.441 (3) 333From Hall et al. 2020 Hall et al. (2020) | 31 | 3/2+ |  $<$ 8.4$\times$ 10-18 | 1.3
6.459 (5) 333From Hall et al. 2020 Hall et al. (2020) | 49 | 5/2- | 8.4$\times$ 10-14 | 5.5
6.699 (3) 333From Hall et al. 2020 Hall et al. (2020) | 289 | 5/2+ | 2.4$\times$ 10-5 | 1.2
6.7420 (12) | 332 | 3/2- | 2.2$\times$ 10-3 555From Bardayan et al. 2015 Bardayan et al. (2015) | 5.2 555From Bardayan et al. 2015 Bardayan et al. (2015)
6.8619 (15) | 452 | 7/2- | 1.1$\times$ 10-5 444From Laird et al. 2013 Laird et al. (2013) | 1.2 444From Laird et al. 2013 Laird et al. (2013)
7.0747 (9) | 665 | 3/2+ | 15.2 555From Bardayan et al. 2015 Bardayan et al. (2015) | 23.8 555From Bardayan et al. 2015 Bardayan et al. (2015)
7.170 (20) 666From Mountford et al. 2012 Mountford et al. (2012) | 759 | 3/2+ | 1.6 (5) | 2.4 (6)
7.507 (11) 666From Mountford et al. 2012 Mountford et al. (2012) | 1096 | 5/2+ | 3 (1) | 54 (12)
7.571 (34) 666From Mountford et al. 2012 Mountford et al. (2012) | 1160 | 3/2+ | 2.3 (6) | 1.9 (6)
7.629 (22) 666From Mountford et al. 2012 Mountford et al. (2012) | 1219 | 3/2- | 21 (3) | 0.1 (1)
7.745 (6) 666From Mountford et al. 2012 Mountford et al. (2012) | 1335 | 3/2+ | 65 (8) | 26 (4)
7.788 (9) 111From Kahl et al. 2019 Kahl et al. (2019) | 1378 | 1/2+ | $83^{+56}_{-82}$ | $47^{-46}_{+92}$
7.982 (13) 666From Mountford et al. 2012 Mountford et al. (2012) | 1571 | 5/2+ | 1.7 (4) | 12 (3)

Since no values have been measured for alpha decay partial widths,
$\Gamma_{\alpha}$ or proton Asymptotic Normalization Coefficients (ANCs)
reported for the 6.288 MeV state with Jπ $\geq 7/2^{+}$, they were estimated
from the mirror nucleus. To investigate the maximizing contribution of this
high-spin resonance, we adopt the minimum Jπ $=7/2^{+}$ spin-parity in our
cross-section calculation. For the $\alpha$-particle width, the reported value
of $\Gamma_{\alpha}$ = 3.36 keV Bardayan et al. (2005) for the analogous state
in 19F at 6.330 MeV, Jπ = 7/2+ was scaled using the equation Nesaraja et al.
(2007):

$[\Gamma_{\alpha}]^{{}^{19}\text{Ne}}=\left[\frac{\rho}{F^{2}+G^{2}}\right]^{{}^{15}\text{N}+\alpha}\left[\frac{F^{2}+G^{2}}{\rho}\right]^{{}^{15}\text{O}+\alpha}[\Gamma_{\alpha}]^{{}^{19}\text{F}},$
(3)

where $\rho$ is the product of the wave number and the interaction radius, and
F and G are the regular and irregular Coulomb wave functions, respectively.
The proton ANC was determined using the following equation derived from
Pogrebnyak et al. (2013) and Mukhamedzhanov and Shubhchintak (2017):

$\text{ANC}=\left(\frac{2\theta^{2}}{W^{2}R}\right)^{1/2}.$ (4)

In this equation R is the interaction radius between a single proton and a 18F
core ($R=1.25(A_{T}^{1/3}+A_{p}^{1/3})=4.53$ fm), $W$ is the Whittaker
function for the proton, and $\theta^{2}$ is the dimensionless reduced width.
To evaluate the maximum contribution of the 7/2+ component of the 6.288 MeV
state, its ANC is assigned a value of $\theta^{2}$ = 1 to calculate the upper
limit. The obtained values for the $\Gamma_{\alpha}$ and the ANC are shown in
the corresponding column in Table LABEL:table-resonances.

Figure 5: Astrophysical S-factor for this work (black lines), Kahl et al. 2021
Kahl et al. (2021) set VIII (purple dot-dashed lines) and set XVI (solid green
lines), and Hall et al. 2020 Hall et al. (2020) (orange dashed lines). To
calculate interference, relative signs for each resonance are entered into
AZURE2. The signs used to obtain the maximum and minimum rates correspond to
the $1/2^{+}$ resonances (-120, 1380 keV) and the $3/2^{+}$ resonances (-278,
31, 665 keV). For all rates, the maximum is obtained using relative signs
(+,-) for the $1/2^{+}$ resonances and (-,-,+) for the $3/2^{+}$ resonances.
The minimum “destructive” rate is obtained by assigning the same sign to all
resonances: (+,+) and (+,+,+).

Figure 5 shows the astrophysical S-factors obtained in this work (black
lines), the results of Hall et al. 2020 Hall et al. (2020) (orange) and the
calculations presented in Kahl et al. 2021 Kahl et al. (2021) set VIII
(purple) and set XVI (green lines). The S-factor is shown over a broad energy
range. Although the higher-energy region does not directly affect the reaction
rate at astrophysically-interesting temperatures, we include the parameters of
higher-lying resonances to highlight the fact that the unknown interference
terms cannot be determined trivially by measuring the high-energy cross
section as one might assume from extrapolating the S-factors from the
Hall/Kahl parameters.

The S-factor determined in Hall et al. 2020 is similar at low energies to our
results, with only minor differences arising from the width and location of
the $\approx$1400 keV resonance. Here, we adopt the $E_{c.m.}$ = 1380 keV
resonance parameters from Ref. Kahl et al. (2019) and Kahl et al. (2021)
rather than the $E_{c.m.}$ = 1461 keV resonance assumed by Refs. Bardayan et
al. (2017), Hall et al. (2020), and references therein. This result is in
disagreement with the theoretical prediction of a broader state at this energy
Dufour and Descouvemont (2007) and subsequent inferred resonance in Ref.
Mountford et al. (2012). However, Ref. Kahl et al. (2019) clearly observed a
narrower resonance at $E_{c.m.}$ = 1380 keV in their study, confirming earlier
indications from Ref. Murphy et al. (2009). The proton and $\alpha$-particle
partial widths are computed using the $\Gamma_{p}/\Gamma_{\alpha}$ branching
ratio from Ref. Nesaraja et al. (2007).

Figure 6: 18F(p, $\alpha$)15O reaction rates ratios between constructive
interference and this work’s destructive interference for Kahl et al. Kahl et
al. (2021), set XVI (green shade) and set VIII (purple shade), Hall et al.
2020 Hall et al. (2020) (orange shade), and this work (grey hatched region).
The solid black line at unity represents the case were both interference
possibilities are the same. The figure shows that unknown interference causes
a factor of about 2 uncertainty with our present rates at 150 MK in contrast
to factors of 3 and 4 in the Hall et al. 2019 and Kahl et al. 2020 set XVI
rates, respectively.

To more readily compare the reaction rates, Fig. 6 displays their ratios to a
common norm. We follow a similar procedure to Ref. Bardayan et al. (2015),
normalizing the different rate possibilities to our minimum reaction rate.
Over the temperature region of interest, this coincides with our destructive
interference case shown in Fig. 5. The left-hand panel of Fig. 6 compares the
Hall et al. 2020 rates to the present rates, while the right-hand panel
displays the ratio of the Kahl et al. 2021 Kahl et al. (2021) sets VIII and
XVI rates. Our determined rates are displayed as a grey hatched region in both
panels.

Clearly, in comparison to the Hall et al. 2020 reaction rates, the uncertainty
from unknown interference is significantly reduced. At 150 MK, the uncertainty
decreases from a factor of almost 3 to a factor of 2. This is because the
spin-parity assignments for the resonances at $E_{r}^{\text{c.m.}}=-278$ keV
(Jπ = $3/2^{+}$) and $E_{r}^{\text{c.m.}}=-120$ keV (Jπ = $1/2^{+}$)
determined in this work reduce the effect of interference. At higher
temperatures, the uncertainty is reduced for two reasons: (i) including
higher-energy resonances reduces the impact of interference effects, and (ii)
the 1.4 MeV resonance adopted here is narrower than the one assumed by Ref.
Hall et al. (2020), slightly reducing the span of cross sections that
interference produces.

The reaction rates presented in Kahl et al. 2021 Kahl et al. (2021) are
arguably a much more exhaustive study that considered the combinations of
resonances allowed by the literature. Their Set XVI was reported to produce
the maximum constructive interference effect. The sub-threshold spin-parity
results of our work now rule out that possibility, thus significantly reducing
the range of reaction rates by almost a factor of 2 at 150 MK. The large
difference arises almost entirely because of the order of the sub-threshold
$1/2^{+}$ and $3/2^{+}$ resonances, which is reversed between their
calculation and our experimental results. Conversely, the interference effects
computed using our resonance parameters are larger than those predicted using
the Kahl et al. set VIII parameters. Our reaction rate is considerably faster
than theirs below 100 MK. Although their set VIII parameters most closely
resemble our results, both of these effects are explained by the additional
low energy resonances included here at $E_{x}=6.416-6.459$ MeV.

##  VI Conclusions

A 20Ne(3He, 4He)19Ne neutron pick up reaction at 21 MeV was used to populated
19Ne excited states relevant for the 18F(p, $\alpha$)15O reaction at
astrophysical energies. It was carried out at the Triangle Universities
Nuclear Laboratory (TUNL) using the Enge split-pole magnetic spectrograph. The
experiment employed targets of 20Ne implanted on carbon foils, and a focal
plane detector was used to collect the reaction products. A total of 29 19Ne
excited states were populated and their energy determined. Angular
distributions were used to extract the spin and parity of 19Ne excited states
in the astrophysical energy range (6 - 7 MeV). Crucially, the state at 6.133
MeV corresponding to the $E_{r}^{\text{c.m.}}=-278$ keV sub-threshold
resonance was found to have a Jπ value of $3/2^{+}$, and the 6.288 MeV state
($E_{r}^{\text{c.m.}}=-120$ keV sub-threshold resonance) was observed as an
unresolved doublet with a combination of Jπ quantum numbers: $1/2^{+}$ and
$\geq 7/2^{+}$.

Previous studies of this reaction (Refs. Hall et al. (2020); Kahl et al.
(2021) being the most recent) showed that ambiguous spin-parity assignments to
states around the proton threshold lead to large uncertainties in the 18F(p,
$\alpha$)15O cross section due to unknown interference. R-matrix calculations
were performed to investigate the impact of our spin-parity assignments on
these effects. The cross sections and reaction rates derived here are compared
with those of Hall et al. 2020 Hall et al. (2020) and Kahl et al. 2021 Kahl et
al. (2021) (sets VIII and XVI). We find that our spin-parity assignments
significantly reduce the uncertainty in the 18F(p, $\alpha$)15O cross section
arising from interference effects by up to a factor of two.

In light of our findings, the remaining questions regarding the 18F(p,
$\alpha$)15O reaction are becoming clearer. A full re-analysis and accounting
for mirror states between 19Ne and 19F should be performed to understand the
issue of unmatched states around the proton threshold. A statistical treatment
of the remaining ambiguities and uncertainties in the nuclear structure of
19Ne would then be possible. Furthermore, complete nucleosynthesis studies may
elucidate where, or if, further experimental information is required.

##  VII Acknowledgements

We would like to thank Nicolas de Séréville and Faïrouz Hammache for their
very insightful discussion on some aspects of this work. This material is
based upon work supported by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, under Award Numbers DE-SC0017799 and under Contract
No. DE-FG02-97ER41041. This paper is also based upon work from the ‘ChETEC’
COST Action (CA16117), supported by COST (European Cooperation in Science and
Technology)

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