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# The Hoyle and associated excited states from the viewpoint of
pocket resonances in $\alpha$ \+ 8Be reactions

Teck-Ghee Lee<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Physics, Auburn University, Auburn, Alabama 36849, USA.  Orhan Bayrak
<EMAIL_ADDRESS>Department of Physics, Akdeniz University, Antalya
07058, Turkey.  Ian J. Thompson<EMAIL_ADDRESS>Lawrence Livermore
National Laboratory, L-414, Livermore, California 94550, USA.  Cheuk-Yin Wong
<EMAIL_ADDRESS>Physics Division, Oak Ridge National Laboratory1, Oak Ridge,
Tennessee 37831, USA.

###### Abstract

We examine the production of the Hoyle and associated excited states from the
viewpoint of pocket resonances in the reaction of an $\alpha$-particle on a
ground state prolate 8Be nucleus within the optical model coupled-channel
framework. The predicted reaction cross sections, as a function of the center-
of-mass energy $E_{\rm cm}$, show prominent resonances, including the Hoyle
resonance. The positions and widths of these resonances are sensitive to the
target deformation ($\beta_{2}$ parameter) and the parity of the nuclear
surface potential $-$ deeper for the even-parity $L$ partial waves relative to
those for the odd-parity $L$ partial waves at the surface region because of
the Bose-Einstein exchange of the $\alpha$-bosons. Decomposing the reaction
cross sections to different partial waves, we find that the resonance energies
and widths reasonably agree with the available experimental data and previous
hyperspherical calculations for the $0_{2}^{+}$ (Hoyle state), $0_{3}^{+}$,
$1_{1}^{-}$ and $3_{1}^{-}$ states of 12C, except for the narrow theoretical
width of the $2_{2}^{+}$ state. Analyzing the wavefunctions and the resonance
widths, we identify the narrow and sharp $0_{2}^{+}$, $3_{1}^{-}$ and
$2_{2}^{+}$ resonances as pocket resonances – resonances which occur below the
potential barrier, while the broad $0_{3}^{+}$ and $1_{1}^{-}$ resonances as
above-the-barrier resonances. For astrophysical applications, we also evaluate
the astrophysical $S(E_{\rm cm})$-factor for $E_{\rm cm}$ $<$ 1.0 MeV, for the
fusion of $\alpha$+8Be into the 12C$(2^{+})$ state based on our estimated
$s$-wave $\alpha$+8Be reaction cross section and the associated $\gamma$\- and
$\alpha$-decay widths for the decay of 12C excited states in the potential
pocket.

###### pacs:

25.70.Ef, 24.10.Eq, 24.10.Ht, 26.35.+c, 26.20.+f

##  I Introduction

A potential pocket with a barrier, depicted in Fig. 5 of Ford and Wheeler [1],
and Fig.1(a) below, is ubiquitous in quantum mechanics. Here, a pocket
resonance is defined as a narrow resonance that appears in the continuum at an
energy below the potential barrier. It is an extension of the bound states
into the continuum when the appropriate Bohr-Sommerfeld-type quantization
condition in the pocket is satisfied, as demonstrated in the case of
$\bar{n}$+heavy-nuclei collisions [2]. Such resonance occurs in atomic
collisions [3], molecular collisions [4], and, as will be demonstrated here,
also in nucleus-nucleus collisions. As the number of degrees of freedom gets
smaller, the more prominent will be the appearance of the pocket resonance and
the larger will be the oscillations of the reaction cross section.
Consequently, from the nuclear reaction viewpoint, we expect such resonances
to occur commonly in the reactions of $\alpha$-conjugate nuclei,
$\alpha^{n_{1}}$+$\alpha^{n_{2}}$, with $\\{n_{1},n_{2}\\}$=1,2,3…. The most
prominent among these may be the the $\alpha$ +8Be reaction with the
production of the 7.65-MeV resonant state and its associated excited states in
12C [5, 6, 7, 8, 9] as pocket resonances.

The production of 12C through the Hoyle resonance is critical to the synthesis
of elements with masses $A$ $>$ 8 [10, 11, 12] for a star moving off the main
sequence and ending up as a red giant. Inside the stellar core, gravitational
collapse can raise the temperature to $T~{}\gtrsim$ 108 K ($\sim$ 8.6 keV) and
the density to $\sim$103 to 105 g/cc. Under these conditions, Salpeter [13]
showed that in a thermodynamical equilibrium state, a concentration ratio of
8Be nuclei to $\alpha$ particles of order $\sim$ 10-9 is sufficient to produce
the 12C by a two-stage process [13, 14]. That is, there occurs first (i) the
fusion of two alpha-particles to form a metastable 8Be ground state (g.s) or
8Be(0+) nucleus, i.e., $\alpha+\alpha$ $\leftrightharpoons$
$\rm{{}^{8}Be}_{\rm g.s}$, and there then follows by (ii) the fusion of the
$\rm{{}^{8}Be}_{\rm g.s}$ nucleus with an additional $\alpha$ particle,
leading to the production of either the 12C ground state, that is,
12C($0_{1}^{+}$), or the first 12C excited state, 12C($2_{1}^{+})$, that
subsequently decays into the 12C ground state, i.e.,
$\alpha+\rm{{}^{8}Be}_{\rm
g.s}\leftrightharpoons{\rm{}^{12}C}^{*}\rightarrow{\rm{}^{12}C_{\rm
g.s}}+\gamma$.

In a thermalized system of alpha-particles, there is always a finite
concentration of 8Beg.s nuclei in the system that can be the target of
$\alpha$ \+ 8Beg.s reactions, leading to the eventual synthesis of the 12C
nucleus. By detailed consideration of the reaction rates and the resulting
relative abundances of even heavier elements, Hoyle predicted that the
$\alpha$+8Be fusion process for 12C synthesis must proceed through a 12C∗
resonance within the range of energies at which the $\alpha$+8Be reaction
occurs. Hoyle predicted resonance was located at $E_{\rm cm}$ $\approx$ 0.33
MeV was corresponding to an excited state of 12C at $E_{x}$ $\approx$ 7.70 MeV
[9]. Soon after, such resonance was duly confirmed by experiment [8], and
hence bears the name of the Hoyle state. The production of the 12C by the
Salpeter-Hoyle two-stage nucleosynthesis process [9] is important because the
Hoyle state lies close to the $\alpha$+8Be fusion threshold of $E_{x}$=7.367
MeV.

The 8Be(0+) nucleus has a half-life $t_{1/2}\sim$ 10-16 s or $\sim$ 107 fm/$c$
is very short-lived, so it is impossible to fabricate 8Be as the target for
fusion cross section measurements. Consequently, the quantitative use of the
cross section for the fusion of $\alpha$ and 8Be to 12C at different energies
in nucleosynthesis studies must rely on theoretical predictions of which there
are few in number, as summarized, for instance, in Descouvemont and Baye [15],
and Nguyen et al [16]. Nevertheless, the resonant and non-resonant
$\alpha$+8Be fusion cross sections remain missing in the database for
theoretical nucleosynthesis studies.

In contrast, there are arrays of extensive experimental and theoretical
investigations on the structural properties of the Hoyle family of states. For
instance, Freer $et~{}al$ [17, 18] have summarized the experimental and
theoretical results for the Hoyle and associated states, standing mainly from
the viewpoint of nuclear structure. On theoretical ground, of particulars are
the cluster approximations such as the 3$\alpha$-cluster hyperspherical
methods of Jensen and coworkers [19, 20, 21, 22], and of Descouvemont and
coworkers [15, 23, 24, 25] where the theoretical energy levels and
$\alpha$-decay widths for a subset of 12C resonant states have been shown to
agree with the experimental data. There is also the microscopic $R$-matrix
analysis of the 12C states by Neff and Feldmeier [26]. Furthermore, it has
been shown recently by an ab initio few-body lattice calculations [27] that
the configurations for the 0${}_{2}^{+}$ Hoyle state of 12C and the second
excited $2_{2}^{+}$ state resemble a bent-arm triangle in which two $\alpha$
particles are near each other and further away from the third $\alpha$. Other
calculations have also yielded a similar conclusion on their “molecular”
structure of 12C [29, 19, 28, 30].

In the present work, we study the Salpeter-Hoyle reaction,
$\alpha+^{8}$Be${}_{\rm
g.s}\leftrightharpoons{\rm{}^{12}C}^{*}\rightarrow{\rm{}^{12}C}+\gamma$, using
a standard optical model (OM) coupled-channel (CC) analysis by treating the
8Be target as a prolate nucleus with a deformation parameter $\beta_{2}$ that
approximately depicts its 2$\alpha$ structure [31]. Accordingly, we examine
the reaction cross section $\sigma_{R}$ and the fusion cross section
$\sigma_{F}$ into 12C in the collision of $\alpha$ and 8Be as a function of
the center-of-mass energy $E_{\rm cm}$, corresponding to the production of a
12C nucleus at an excitation energy of $E_{x}=E_{\rm cm}-7.367$ MeV. The
0${}_{2}^{+}$ Hoyle excited state at $E_{x}=7.648$ MeV and other associated
excited states will show up as pocket resonances in a two-body interaction
through our analysis. Our first objective is to show how some of the 12C∗
excited states, including the Hoyle state, can be described as pocket
resonances in the reaction of an alpha-particle with the 8Beg.s nucleus.
Success in such a study will indicate that a subset of 12C∗ states contains
important $\alpha+^{8}$Beg.s reaction channel components, in complement of the
studies of these states from the nuclear structure viewpoint. Having said
that, it requires a proper description of the fused system of the 12C nucleus
that the parameters of the optical model be so chosen as to lead to the
appropriate properties of the Hoyle and its family of 12C excited states. The
agreement of such properties is necessary as it benchmarks and provides the
confidence for the extrapolation, theoretically with the optical model, to the
unknown reaction cross section and 12C-fusion cross sections in the low-energy
region, for which experimental measurements are impossible. Our second
objective is to take advantage of the reaction framework to evaluate the
astrophysical $S(E_{\rm cm})$-factor for the fusion of a low-energy alpha-
particle with the 8Beg.s nucleus into a 12C nucleus, for astrophysical
applications. Moreover, the access to these cross sections and understanding
of the underlying mechanisms that lead to the 12C formation, from the heavy-
ion collisions perspective, may be valuable to the recent laboratory search
for analogous states to the Hoyle state in the $A>$ 12 nuclei [32]. All told
the salient reaction-physics viewpoints introduced here complement the
alternative structure-physics viewpoints presented in Refs.[17, 18].

The 8Be nucleus is unstable in both the 8Be$(0^{+})$ with a half-life is of
order $10^{7}$ fm/$c$ and the first excited 8Be$(2^{+})$ state at
$E_{x}({}^{8}{\rm Be}(2^{+}))$ = 3.03 MeV, with a half-life of order
$\hbar/$(1.5 MeV) $\sim$ 131 fm/$c$. In the Salpeter-Hoyle’s $\alpha$+8Be
reaction, it is relevant to inquire whether the instability of 8Be needs to be
taken into account in the fusion of $\alpha+^{8}$Be. The 8Beg.s half-life is
of order $10^{7}$ fm/$c$ is long in comparison with the time scale for the
fusion of $\alpha$+8Be [33]. It is therefore reasonable to ignore the decay of
8Beg.s in those 12C-fusion reactions in which the 8Be($2^{+})$ state is not
excited. The 8Be(2+) is excited only for $\alpha+^{8}$Be collisions at $E_{\rm
cm}$ greater than 3 MeV and the concern of the decay of 8Be(2+) in the
Salpeter-Hoyle reaction process above $E_{\rm cm}>4.0$ MeV will be addressed
later.

The paper is organized as follows. In Sec. II, we describe the effective
optical model potential for the coupling of many partial-wave $L$ channels
suitable for the interaction between the $\alpha$-particle and the 8Be nuclei.
The reaction cross section has a complex structure with resonances and off-
resonance regions. In Sec. III, we present the phenomenological description of
the complex structure of the reaction cross section obtained in the optical
model coupled-channel calculations for the $\alpha$ \+ 8Be reaction. In Sec.
IV, we compare the coupled-channel calculated resonance energies and widths
with the available results from the experiment and hyperspherical calculation.
In Sec. V, we calculate the $s$-wave fusion cross section and its
corresponding $S$-factor for the $\alpha$+8Be $\buildrel\gamma\over{\to}$
12C(2${}_{1}^{+}$) reaction by evaluating the $\gamma$\- and $\alpha$-decay
widths of the excited compound nucleus 12C nucleus in the potential pocket.
Sec. VI concludes our present discussions. Unless ambiguity arises, for the
brevity of notation, we shall also use the symbol $E$ for $E_{\rm cm}$
interchangeably throughout the rest of the manuscript.

##  II The optical model potential in the coupled-channel calculation for the
$\alpha$+8Be reaction

The optical model coupled-channel method (and calculation) has been a standard
and successful theoretical tool for analyzing low-energy nuclear reactions
between two atomic nuclei. Thus, we shall not delve into the theoretical
details of the CC approach as one can find them in Refs. [34, 35, 36]. We
shall employ the CC analysis utilizing the FRESCO code [35] to study the
reaction between the projectile $\alpha$ and the deformed rotational target
8Be nucleus. For the low-energy reaction in question, it is reasonable to
consider only the channels of the ground and the first excited $(2^{+})$ state
of the target 8Be nucleus and to consider the effective interaction in the
scattering partial $L$-wave among the many coupled $L$-channels of the
incident $\alpha$ as

$\displaystyle V_{\rm eff}^{(L)}(r)$ $\displaystyle=$
$\displaystyle-\left(\frac{V^{(L)}_{o}}{(1+e_{v})}+\frac{iW_{o}}{(1+e_{w})}\right)$
(1)
$\displaystyle+\frac{Z_{p}Z_{t}e^{2}}{2R_{c}}\left(3-\frac{r^{2}}{R^{2}_{c}}\right)\Theta(R_{c}-r)+\frac{Z_{p}Z_{t}e^{2}}{r^{2}}\Theta(r-R_{c})$
$\displaystyle+\frac{L(L+1)}{2\mu
r^{2}}+V_{\pi}^{(L)}(r)+<Y_{L0}|{\mathbb{V}}^{(2)}_{\rm def}({\bf
r})|Y_{L0}>_{\Omega},$

where

$\displaystyle V_{\pi}^{(L)}(r)$ $\displaystyle=-\frac{(-1)^{L}\times 4V_{\pi
0}\,e_{\pi}}{(1+e_{\pi})^{2}},$ (2)
$\displaystyle{\mathbb{V}}^{(\lambda)}_{\rm def}({\bf r})$
$\displaystyle=\sum_{\lambda=2,4,...}\left(V_{n}^{(\lambda)}(r)+V_{c}^{(\lambda)}(r)\right)D_{0,0}^{(\lambda)}Y_{\lambda
0}(\Omega),$ (3) $\displaystyle V_{n}^{(\lambda)}(r)$
$\displaystyle=-\int^{+1}_{0}d(\cos\theta^{\prime})\left(\frac{V_{o}}{1+e^{\prime}_{v}}+\frac{iW_{o}}{1+e^{\prime}_{w}}\right)Y_{\lambda
0}(\theta^{\prime}),\hskip 8.5359pt$ (4) $\displaystyle V^{(\lambda)}_{c}(r)$
$\displaystyle=M_{\lambda}\left(\frac{r^{\lambda}\Theta(R_{c}-r)}{R_{c}^{2\lambda+1}}+\frac{\Theta(r-R_{c})}{r^{\lambda+1}}\right),$
(5) $\displaystyle M_{\lambda}$
$\displaystyle=\frac{3Z_{t}\beta_{\lambda}R_{c}^{\lambda}}{4\pi}\left(\frac{\sqrt{4\pi}e^{2}}{2\lambda+1}\right),$
(6) $\displaystyle e_{(v,w,\pi)}$
$\displaystyle=\exp([r-R_{(v,w,\pi)}]/a_{(v,w,\pi)}),$ (7) $\displaystyle
e^{\prime}_{(v,w)}$
$\displaystyle=\exp([r-R_{(v,w)}\\{1+\beta_{2}Y_{20}(\Omega^{\prime})\\}]/a_{(v,w)}).\hskip
8.5359pt$ (8)

In Eq. (1), $Z_{\\{p,t\\}}$ are the nuclear charges for a projectile $p$ and
target $t$, respectively, $V_{\pi}^{(L)}(r)$ is the parity- and $L$-dependent
surface potential (proportional to the first derivative of the Woods-Saxon
nuclear potential), $V^{(\lambda)}_{\\{n,c\\}}$ are the channel-coupling
potentials for nuclear ($n$) and Coulomb ($c$) deformations, respectively,
where $R_{\\{c,v,w,\pi\\}}$ = $r_{\\{c,v,w,\pi\\}}A_{t}^{1/3}$, $A_{t}$ is the
target mass number, $\mu$ is the reduced mass of the collision pair. The polar
angles $\Omega^{\prime}$ $\equiv$ $(\theta^{\prime},\phi^{\prime})$ and
$\Omega$ $\equiv$ $(\theta,\phi)$ refer to the body-fixed and space-fixed
systems, respectively, $Y_{\lambda\,0}$ is the spherical harmonic,
$D_{m\,m^{\prime}}^{(\lambda)}(\theta_{i})$ is a rotation $D$-matrix and
$\theta_{i}$ stands for the Euler angles between the body-fixed and the space-
fixed coordinates. The reaction cross section $\sigma_{R}(E)$ can be
decomposed into the sum of the partial reaction cross section
$\sigma_{R}(E,L)$ for the $L$-partial waves in the standard way.

To find the suitable potential depth $V_{o}$ in Eq. (1) for the $\alpha$ \+
8Be reaction, we must first estimate the nuclear and Coulomb geometrical
parameters as many of the OM parameters are currently unknown. We searched by
performing multiple $s$-wave scattering calculations using a spherical 8Be
nucleus fixed at the Hoyle energy $E$ = 0.28 MeV. With many trials, we
eventually obtained a series of central potential well-depth $V_{o}$ = 4.4,
11, 22, 38, 59 MeV, and so on.

Figure 1:  (a) Re($V_{\rm eff}^{(L)}(r))$ curves for various $L$-wave for
$\alpha$+8Be system. The horizontal dashed lines mark the corresponding $E$
resonance energies of the experimental $J^{\pi}$ states of 12C [51]. (b)
Coupled-channel (FRESCO) calculated partial-wave reaction cross section
$\sigma_{R}(E_{\rm cm},L)$ that takes into account the channel couplings
between the 8Be(0+) and 8Be(2+) states. (c) The open circle, plotted in a log-
log scale, is the $\sigma_{R}(E_{\rm cm})$ of the CC calculation, and the
solid curve is the phenomenological reaction cross section from the analysis
in Section III using Eqs. (16) to (21).

Deep, local potential nuclear structure calculations for two-body $\alpha$
+8Be interaction have been carried out by Buck $et~{}al$ [37]. Using $V_{o}$ =
144 MeV in their nuclear potential of Eq. (2) together with the Coulomb
potential in their cluster-core Hamiltonian, numerically solving the
Schrödinger equation they obtained from their eigensolutions the excitation
energy levels, widths, and charge radii for the first few states of the Hoyle
band in 12C. Such a deep 144-MeV potential has the artifact that in addition
to the two Pauli-blocked states occupied by the two $\alpha$ particles of the
target 8Be nucleus in the potential, there are many unoccupied
single-$\alpha$-particle bound states that can lead to spurious high-energy
gamma transitions upon the formation of a compound system of the $\alpha$
particle with 8Be inside the potential pocket. To choose the appropriate
starting point for our potential depth search, we, therefore, take the view
that our OM potential is an extension of the bound-state potential of a single
$\alpha$ particle in the continuum. Therefore, of the series of well-depths
found above, we chose $V_{o}$ $\sim$ 11 MeV as the starting point for our
potential parameters search. This $V_{o}$ $\sim$ 11 MeV also happens to be
compatible with the well-depth of attractive mean-field potential for an
alpha-particle reported in [20, 21, 23] in their treatment of the 12C nucleus
as a 3$\alpha$ system, where the repulsive interactions among the $\alpha$
particles have been taken into account.

Next, we consider the deformed 8Be nucleus in our scattering calculations. We
fine-tuned and obtained $\beta_{2}$ = 0.6 [31] and other OM parameters through
systematic iterations of fixing $V_{o}$ (first) to constrain the geometric
parameters, and vice versa, in the CC calculations, until we obtained a
handful of decent resonances for $E<$ 4 MeV region. But, we cannot reproduce
the experimental data unless we introduce a parity-dependent surface potential
at large radial separations, $V_{\pi}^{(L)}(r)$ of Eq. (1), with the potential
deeper for the even parity, even-$L$, states than it is for the odd-parity,
odd-$L$, states, as shown in Fig.1(a). This parity-dependent potential is
necessary because the effect of the Bose-Einstein exchange of bosons in a
spatially extended configuration; the lowest-energy state of an extended
system of bosons is symmetrical for the exchanging of identical bosons, and
such exchange leads to a lower potential energy for even-parity interactions
in the $\alpha$+8Be potential. The parity-dependent potential considered here
is analogous to the variations of the strengths of the phenomenological
effective three-body interaction $S$ extracted from the Hyperspherical-Fadeev-
Equation (HFE) treatment of 12C as a 3$\alpha$ system in Ref. [20], where $S$
parametrizes the depth of their 3-body interaction,
$V_{3b}(\rho)=S\exp(\rho^{2}/b^{2})$. As shown in Table 2 of Ref. [20], the
attractive $S$ is deeper for even-parity states compared to those for odd-
parity states with $S=S_{0}+(-1)^{L}S_{1}$ for $S_{0}\approx-11$ MeV and
$S_{1}\approx-9$ MeV. Note that this idea of the parity-dependent potential
has been used before to analyze the differential cross sections of $\alpha$
scattering of a light nucleus [38, 39, 40].

The calculation of the resonance energies and widths are numerically delicate,
so the OM parameters must be further systematically fine-tuned to achieve a
reasonable description of the low-lying Hoyle resonances with potentials for
different $L$ states depicted in Fig.1(a). The values of the OM parameters are
listed in the caption of Table IV. In addition, experimental $J^{\pi}$ states’
energies are marked in Fig.1(a) to see whether a potential $L$-resonance may
be a sharp pocket resonance [2] if it lies below the potential barrier or a
broad above-the-barrier (Ramsauer-type) resonance [41, 42, 43] if it lies
nearly at or above the top of the potential barrier. To capture the resonances
(see Table IV) as shown in the partial reaction cross sections as functions of
energy depicted in Fig.1(b) for $E<$ 4 MeV, in our coupled-channel
calculation, we have taken into account the 8Be(0+) and 8Be(2+) states of the
target and the channel couplings between these states at a matching radius of
50-fm. An unevenly distributed 22000-point energy grid with a very fine step
size was also required to resolve these very narrow resonances. From a fit to
the reaction cross section with either a Breit-Wigner-Fano [44, 45] or a
Breit-Wigner form as will be described in detail in the next section, we
extract the theoretical resonance energies and widths as obtained in the
present CC analysis. Among all the resonances appear in Fig.1(b), the most
important benchmark for our coupled-channel calculation is the Hoyle resonance
at the excitation energy $E_{x}$ = 7.648 MeV (relative to the 12Cg.s) and the
0${}_{3}^{+}$ resonance which has a broad width relative to the Hoyle state at
$E_{x}$ = 10.16 MeV.

##  III Phenomenological description of the coupled-channel cross sections for
the $\alpha$ \+ 8Be reaction

The reaction cross sections of $\alpha+^{8}$Be obtained from the coupled-
channel calculation in Fig.1 in the last section arise from the numerical
solution of the quantum mechanical problem in the scattering of an $\alpha$
particle with a 8Be ground-state nucleus. As a function of the center-of-mass
energy $E_{\rm cm}$ of $\alpha$+8Be or the equivalent excitation energy
$E_{x}$ of 12C∗, it has a complex structure which requires a comprehensive
physical description in terms of the reaction both on 12C resonances and off
the 12C resonances. The CC calculation also reaches its lowest energy limit,
below which no numerical CC calculation can be performed because of the
finiteness of the spatial integration region and the finite integration grid
size. This is, however, an important energy region in astrophysical studies,
for which we would like to evaluate the 12C fusion cross section. It is
desirable to represent numerical CC cross sections in analytical
phenomenological forms (down to the lowest energy region) for their
theoretical interpretation as well as for their subsequent applications.

We therefore carry out a phenomenological analysis of the CC reaction cross
sections for many purposes. Firstly, from the complex structure of the
$\alpha$+8Be reaction cross sections, we wish to extract valuable information
on those 12C states, which have large overlaps with the $\alpha$+8Be channels.
Some of these states may be identified as a pocket resonance beneath the
potential barrier while others may be an above-the-barrier (Ramsauer-type)
resonance [41, 42, 43]. The extracted information on the positions and widths
of the resonances in the $\alpha$+8Be reaction can be compared directly with
the corresponding experimental data and previous hyperspherical calculations
[22]. Secondly, we wish to extend our previous analysis of the reaction cross
section of Ref. [46] on barrier penetration without resonances to new cases of
barrier penetration in the presence of resonances. In particular, we can study
the resonance profile to examine the mutual interference between the resonance
and its background barrier penetration amid a potential barrier, similar to
the interference between potential pocket resonances and the barrier
penetration over a double-hump barrier [47]. Thirdly, in addition to
extracting reaction information at resonances and their vicinities, we can
also extract reaction information on the off-resonance continuum states. Such
extraction of the off-resonance CC data provides the necessary tool to
determine the barrier penetration for the calculations of the alpha decay
width off-resonance continuum states, which may be difficult to obtain by
other means. Finally, the analytical description of the reaction cross
sections in the lowest energy region allows an analytical extrapolation to
extend the reaction cross sections to the lowest energy region which cannot be
reached by coupled-channel or other methods.

Figure 2: The solid curve is the $s$-wave reaction cross section at the Hoyle
peak obtained by the CC calculation. The dashed curve is the Breit-Wigner
parametrization of the $s$-wave reaction cross section as
$\sigma_{R}(E,0)=(\pi/k^{2})(2/\pi)P_{\rm
res}(E,0)/[(E-E_{j})^{2}/(\Gamma_{j}/2)^{2}+1]$, where $E_{j}=E_{\rm Hoyle}$ =
0.2809338737 MeV, $\Gamma_{j}=8.0$ eV and $P_{\rm res}(E_{j},0)=$1.

The reaction cross section in Fig. 1(b) for different partial $L$-waves
exhibit resonance structure on top of a relatively smooth variation, as a
function of the reaction energy. The energy region $\cal R$ under
consideration can be conveniently and conceptually partitioned into the
resonance regions $\cal R_{\rm res}$ about the neighborhood of the resonances
and the off-resonance continuum regions $\cal R_{\rm contin}$ in between
resonances. Different physics are at play for these two different regions and
we can partition the reaction cross section judiciously.

In the resonance energy region, ${\cal R}_{\rm res}$, we envisage that there
can be sharp resonances in the reaction cross section associated with pocket
resonances – states in the potential pocket under the potential barrier which
are eigenstates inside the optical potential pocket in the continuum,
satisfying the Bohr-Sommerfeld quantization rule for a quasi-bound state
inside the pocket. Of course, there can also be resonances with broad widths
which are above-the-barrier Ramsauer-type resonances, arising from the
interference of the waves passing through the nucleus and the wave passing
around the nucleus [41, 42, 43].

In the off-resonance continuum region ${\cal R}_{\rm contin}$ which we label
with the subscript “contin”, the compound nucleus 12C state lies in the
continuum and the off-resonance reaction cross sections in between the
resonances depend not only on the penetrability through the potential barrier
but also on the matching of the wavefunction inside the pocket, at the
continuum state. The reaction cross section therefore depends on the barrier
and the pocket shapes of the potential as a function of the internuclear
separation $r$ as shown in Fig.1(a). We envisage that the resonance regions
$\cal R_{\rm res}$ disperse themselves in the continuum background regions of
$\cal R_{\rm contin}$.

The reaction cross section at an energy $E(=E_{\rm cm})$ in a nucleus-nucleus
collision can be phenomenologically written in the general form of [46]

$\displaystyle\sigma_{R}(E)=\sum_{L=0}\sigma_{R}^{(L)}(E)=\sum_{L=0}\frac{(2L+1)\pi}{k^{2}(E)}P(E,L),$
(9)

where $P(E,L)$ represents the probability for the incident particle with
partial wave $L$ to commence from the external entrance point beyond the
barrier, to penetrate the potential barrier, and reach the interior potential
pocket region of the target nucleus. We can decompose the contributions from
states in the entire energy regions as an integral over individual states
specified by the delta function at different energies $E$ as

$\displaystyle\sigma_{R}(E)=\sum_{L=0}\int_{\cal
R}dE^{\prime}\delta(E-E^{\prime})\frac{(2L+1)\pi}{k^{2}(E^{\prime})}P(E^{\prime},L).$
(10)

Because of the presence of resonances and the off-resonance continuum region
with $\cal R$=$\cal R_{\rm res}$+$\cal R_{\rm contin}$, we have

$\displaystyle\sigma_{R}(E)\\!\\!=\\!\\!\sum_{L=0}\biggl{\\{}\int_{{\cal
R}_{\rm
res}}\\!dE^{\prime}\delta(E-E^{\prime})\frac{(2L+1)\pi}{k^{2}(E^{\prime})}P_{\rm
res}(E^{\prime},L)$ $\displaystyle+\int_{{\cal R}_{\rm
contin}}dE^{\prime}\delta(E-E^{\prime})\frac{(2L+1)\pi}{k^{2}(E^{\prime})}P_{\rm
contin}(E^{\prime},L)\biggr{\\}},~{}~{}$ (11)

where we have introduced $P_{\rm res}(E,L)$ the phenomenological penetrability
in the resonance region, and $P_{\rm contin}(E,L)$ in the off-resonance
continuum region. The penetrability $P_{\rm res}(E,L)$ is expected to exhibit
a resonance structure with its energy position depending predominantly on the
pocket potential satisfying the Bohr-Sommerfeld quantization rule, while its
width depends on the potential barrier which must be penetrated to reach from
the potential pocket to the exterior. The penetrability $P_{\rm contin}(E,L)$
for a continuum state is expected to exhibit a smooth behavior as it depends
on the penetration over the potential barrier from the pocket region to the
exterior. However, in the region near a pocket resonance, there may be
interference from the presence of the pocket resonances, as the barrier
penetrability $P_{\rm contin}(E,L)$ will also depend on the matching of the
wavefunction at the boundary of the potential pocket.

For the off-resonance continuum region, the integral $\int_{{\cal R}_{\rm
contin}}\\!\\!dE^{\prime}...$ can be simply integrated out to yield the
partial reaction cross section for the $L$-partial wave for low-energy
$\alpha$+8Be(0+)$\rightarrow$12C∗ reactions,

$\displaystyle\sigma_{R}^{(\rm contin)}(\alpha+{}^{8}{\rm
Be}(0^{+})\to{}^{12}{\rm C}^{*},E,L)$ $\displaystyle=\int_{{\cal R}_{\rm
contin}}dE^{\prime}\delta(E-E^{\prime})\frac{(2L+1)\pi}{k^{2}(E^{\prime})}P_{\rm
contin}(E^{\prime},L)$ $\displaystyle=\frac{(2L+1)\pi}{k^{2}(E)}P_{\rm
contin}(E,L).$ (12)

An example of $P_{\rm contin}(E,L)$ is the Hill-Wheeler penetrability [48] for
an inverted parabola as given in Eqs. (1) and (2) of Ref. [46]. For our case
with the CC calculations, the penetrability will depend not only on the
potential barrier beyond the potential pocket at large $r$, but also on the
matching of the wavefunction inside the potential pocket.

In the resonance region ${\cal R}_{j}$ for the $j$-th resonance at a resonance
$E_{j}$ and a width $\Gamma_{j}$ of the $L$ partial wave, the delta function
distribution $\delta(E-E^{\prime})$ with $E^{\prime}=E_{j}$ in the integral
$\int_{{\cal R}_{\rm res}}dE^{\prime}...$ in Eq. (11) need to be modified to
become a Breit-Wigner distribution $\Delta_{\rm BW}(E-E^{\prime})$ with a
width $\Gamma_{j}$,

$\displaystyle\delta(E-E^{\prime})\bigg{|}_{E^{\prime}\to E_{j}}$
$\displaystyle\to$ $\displaystyle\Delta_{\rm BW}(E-E_{j}),$

where

$\displaystyle\Delta_{\rm BW}(E-E_{j})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{2}{\pi\Gamma_{j}}\frac{1}{[(E-E_{j})/(\Gamma_{j}/2)]^{2}+1}.$
(13)

The Breit-Wigner distribution $\Delta_{\rm BW}(E-E_{j})$ has been normalized
such that

$\displaystyle\int\\!\\!dE^{\prime}\Delta_{\rm BW}(E-E^{\prime})$
$\displaystyle=$
$\displaystyle\frac{2}{\pi\Gamma_{j}}\\!\\!\int\\!\\!\\!\frac{dE^{\prime}}{[(E-E_{j})/(\Gamma_{j}/2)]^{2}+1}$
(14) $\displaystyle=$ $\displaystyle 1.$

The above normalization suggests that we can choose the resonance region
$\Theta({\cal R_{\rm res}})$ in two equally useful ways. For resonances with a
broad width, we can choose Eq. (13) to represent the resonance at all energies
and allow $\Theta(\cal R_{\rm res})$ to extend over the whole energy region
without restriction. Because the amplitude in Eq. (13) decreases rapidly
beyond a few units of the resonance width $\Gamma_{j}$, such an extension will
not incur large errors. For very narrow resonance such as the Hoyle resonance,
we can alternatively choose ${\cal R}_{\rm res}$ to be only at the bin at
$E_{\rm res}$ with the width $\Gamma_{j}$, with the resonance amplitude given
by Eq. (13), and $\Theta({\cal
R}_{j})=\Theta(E_{j}+\Gamma_{j}/2)-\Theta(E_{j}-\Gamma_{j}/2)$. Thus in the
neighborhood of the resonance $E_{j}$ for a partial wave $L$, the reaction
cross section is therefore

$\displaystyle\sigma_{R}^{({\rm res})}(\alpha+{}^{8}{\rm
Be}(0^{+})\to{}^{12}{\rm C}^{*},E,L)$ $\displaystyle=\int_{{\cal
R}_{j}}dE^{\prime}\delta(E-E^{\prime})\frac{(2L+1)\pi}{k^{2}(E^{\prime})}P_{\rm
res}(E^{\prime},L)$ $\displaystyle=\Gamma_{j}\left[\Delta_{\rm
BW}(E-E^{\prime})\frac{(2L+1)\pi}{k^{2}(E^{\prime})}P_{\rm
res}(E^{\prime},L)\right]_{E^{\prime}=E_{j}}\Theta({\cal R}_{j})$
$\displaystyle=\frac{(2L+1)\pi}{k^{2}(E)}\left(\frac{2}{\pi}\right)\frac{P_{\rm
res}(E_{j},L)\Theta({\cal R}_{j})}{[(E-E_{j})/(\Gamma_{j}/2)]^{2}+1},$ (15)

where we have introduced the phenomenological parameter $P_{\rm res}(E_{j},L)$
to describe the CC theoretical reaction cross section data.

Figure 2 shows how Eq.(15) with $P_{\rm res}(E,L$=$0)$ = 1 and the normalized
Breit-Wigner distribution describes well the CC calculated $s$-wave reaction
cross section in the vicinity of the Hoyle peak, locally at the scale of eV
energies, suggesting that the $s$-wave resonance without substantial
absorption. An analogous and telling example of the penetrability in pocket
resonances can be found in Fig. 1 of Ref. [47], where the potential between a
double-hump barrier leads to pocket pockets and pocket resonances with
resonance structures in the penetrability at the pocket resonance energies.

Figure 2 also clearly indicates at the Hoyle resonance, the $\alpha$ width
$\Gamma_{\alpha}\approx\Gamma_{j}$ = 8.0 eV is close to the most recent
measured value of 8.5 eV [18]. Of course, further fine-tuning the two-body
interaction potential could further improve this figure to better match the
experimental width.

When all the resonances in ${\cal R}_{\rm res}$ are included, the reaction
cross section is

$\displaystyle\sigma_{R}^{({\rm res})}(\alpha+{}^{8}{\rm
Be}(0^{+})\to{}^{12}{\rm C}^{*},E)$
$\displaystyle=\sum_{j}\frac{(2L_{j}+1)\pi}{k^{2}(E)}\left(\frac{2}{\pi}\right)\frac{P_{\rm
res}(E_{j},L_{j})\Theta({\cal R}_{j})}{[(E-E_{j})/(\Gamma_{j}/2)]^{2}+1}.$
(16)

Table 1 tabulates the resonances energies $E_{j}$, resonance width
$\Gamma_{j}$, and partial-wave $L$, and the penetrability parameter $P_{{\rm
res}}(E_{j},L)$ extracted from the CC calculated reaction cross sections.

Table 1:  The values for $P_{\rm res}(E_{j},L=J)$, $\Gamma_{j}$ and $E_{j}$ in Eq. (16) in the different resonance regions. For display purposes, the calculated $E_{\rm Hoyle}$ = 0.2809338737 MeV is truncated to $E_{1}$ = 0.2809 MeV.  $j$ |  $J^{\pi}$ |  $P_{\rm res}(E_{j},L=J)$ |  $E_{j}$/MeV |  $\Gamma_{j}$/keV
---|---|---|---|---
1 |  0${}_{2}^{-}$ |  1.0 |  0.2809 |  8.0 eV
2 |  3${}_{1}^{-}$ |  1.4206$\times 10^{-3}$ |  2.165 |  38.0
3 |  2${}_{2}^{+}$ |  0.5363 |  2.770 |  0.026
4 |  0${}_{3}^{+}$ |  2.3$\times 10^{-5}$ |  2.780 |  300
5 |  1${}_{1}^{-}$ |  1.443$\times 10^{-5}$ |  3.33 |  409
6 |  4${}_{1}^{+}$ |  5.093$\times 10^{-4}$ |  2.279 |  12.4
Figure 3: (Color online) (a) Reaction cross section for the $s$-wave as a
function of energy in a semi-log scale. (b) The solid curve gives the
phenomenological penetrability $P(E,0)$ extracted from the CC calculations as
a function of energy in a semi-log scale, for the region near the Hoyle state.

While the Breit-Wigner distribution works well for describing the resonances
locally on the energy scale of eV as shown in Fig.2, over a larger energy
scale of sub-MeVs, there are however important effects arising from the mutual
interference of the resonance and the underlying continuum barrier
penetration. The situation becomes even more acute because there is an overall
increase in the penetrability as the potential barrier is reduced in size when
the collision energy increases. Consequently, the reaction cross section is
significantly modified further away from the resonance when comparing the
result of the Breit-Wigner formula against the CC calculated $s$-wave reaction
cross section. Albeit the Breit-Wigner expression predicts a symmetric
distribution relative to $E-E_{\rm Hoyle}$ with a narrow 8-eV width as
illustrated in Fig. 2, the distribution from the CC calculation broadens
substantially with a low penetrability and is not symmetric, with a greater
penetrability in the higher collision energy region relative to the lower
collision energy region. As a result, in the energy range of 0.26
$\leq$$E_{\rm cm}$ $\leq$ 0.32 MeV, the $P(E,L)$ probability of the
$\sigma_{R}(E,0)=\pi P(E,0)/{k^{2}}$ reaction cross section has to be modified
to the Breit-Wigner-Fano form [44, 45], with the introduction of an additional
energy scale $q\Gamma_{j}/2$ with dimensionless $q$ to write the reaction for
the $s$-wave reaction cross section in the vicinity of the Hoyle state as

$\displaystyle\sigma_{R}^{({\rm res})}(\alpha+{}^{8}{\rm
Be}(0^{+})\to{}^{12}{\rm C}^{*},E,L=0)$
$\displaystyle\approx\\!\frac{\pi}{k^{2}(E)}\\!\left(\\!\frac{2}{\pi}\\!\right)\\!\frac{P_{\rm
res}(E_{j},L=0)\Theta({\cal
R}_{j})}{[(E-E_{j})/(\Gamma_{j}/2)]^{2}+1}\\!\\!\left[\frac{E-E_{j}}{q\Gamma_{j}/2}+1\right]^{2}\\!\\!\\!\\!,~{}~{}$
(17)

where $E_{\rm Hoyle}=$ 0.2809338737 MeV, $\Gamma_{\rm Hoyle}=8$ eV,
$q^{2}=1.3\times 10^{8}$ and $P_{\rm res}(E_{\rm Hoyle},L$=$0)$ $\approx$ 1\.
This finding could also explain why there are substantial non-perturbative
effects in penetrability arising from the presence of a pocket resonance.
Therefore, it is necessary to introduce an energy-scale parameter
$q\Gamma_{\rm Hoyle}$$\sim 10^{4}$ eV in the modified Breit-Wigner-Fano
distribution of Eq.(17) to describe fully the interference of the resonance
and the continuum states under the barrier substantiated by the CC calculation
in in the energy range of 0.26 $\leq$$E_{\rm cm}$ $\leq$ 0.32 MeV.

For all other resonances of $j$ = 2, 3, 4,… in Table 1, the resonance shapes
are nearly symmetrical and there is no need to modify the Breit-Wigner
distribution to include the Fano modification. Furthermore, the cross sections
of the resonance region decrease so rapidly in comparison with the continuum
background that the total reaction can just be treated as adding the resonance
contributions onto the background distribution without incurring significant
differences. Therefore, beyond the Hoyle resonance region, we carry out the
Breit-Wigner fits of the other resonances from 1.77 to 4 MeV.

Next, we describe how the CC calculated results in the off-resonance continuum
region ${\cal R}_{\rm contin}$ can be described by an analytical formula.
Because the reaction cross section decreases precipitously as $E$ tends to
zero, due to the Coulomb repulsion, it is well-known that the low-energy cross
section can be expressed by factorizing the cross section into the
contributions by nuclear and Coulomb interaction as [49]

$\displaystyle\sigma_{R}^{({\rm contin})}(E)=\frac{S(E)\exp(-2\pi\eta)}{E},$
(18)

where $\eta=\alpha Z_{p}Z_{t}\sqrt{\mu/2E}$, $\mu$ is the reduced mass of the
collision-pair in the units of MeV, $\alpha$ = $e^{2}/4\pi\epsilon_{o}\hbar c$
$\approx$ 1/137 is the fine structure constant, $S(E)$ is the astrophysical
$S$-factor and $\exp(-2\pi\eta)$ is the Gamow-Sommerfeld barrier
penetrability.

Equation (18) is the classic Gamow cross section in which the penetrability
$P_{\rm contin}(E)=2\mu\hbar^{2}\exp(-2\pi\eta)/\pi$ with $S(E$) = 1. Its
energy dependence plotted against the CC calculated reaction cross section is
shown in Fig. 3(a) to illustrate their degree of agreement and one sees the CC
reaction cross section approximately equal to the Gamow cross section at $E$
$\approx$ 0.8 MeV and 0.1 MeV. Now it is clear that to better describe the CC
result for all energies, Eq. (18) must account for the channels coupling
presence in the scattering dynamic by phenomenologically modifying the $S(E)$
= 1 to a smooth function of energy given by

$\displaystyle S(E)=\frac{1}{(s_{0}E+s_{1}E^{2}+s_{2}E^{3}+s_{3}E^{4})}$ (19)

for $E$ $\leq$ 0.26 MeV, where $s_{0}$ = 1.28205 MeV-1, $s_{1}$ = $-$11.763
MeV-2, $s_{2}$ = 35.068 MeV-3 and $s_{3}$ = $-$33.612 MeV-4 coefficients are
necessary to take care of the peculiar shape of the potential we have for the
$\alpha$-particle to penetrate the “Coulomb-modified-by-nuclear” barrier. And
Fig. 3(a) shows the modified Gamow cross section with Eq. (19) matches the
results of the CC calculation.

Now, away from the Hoyle resonance region where the $s$-wave contribution
remains dominance in the range of 0.32- to 1.77-MeV off-resonance region, in
the light of Eq. (1) of Wong [46, 50], we introduce a new formula by
amalgamating the probability functions of Hill-Wheeler [48] and Wong [46, 50]
to

$\displaystyle P_{\rm contin}(E)$
$\displaystyle=\left(\frac{1}{2}\right)\frac{c_{0}}{[1+\exp(2\pi(B_{0}-E)/\hbar\omega_{0})]}$
(20) $\displaystyle+c_{1}\ln\\{1+\exp(2\pi(E-B_{1})/\hbar\omega_{1})\\},$

where $c_{0}$ = 5.6$\times 10^{-6}$, $c_{1}$ = 2$\times 10^{-7}$, $B_{0}$ =
0.97 MeV, $B_{1}$ = 1.12 MeV, $\hbar\omega_{0}$ = 0.8607 MeV and
$\hbar\omega_{1}$ = 0.2805 MeV, so that the subsequent reaction cross section

$\displaystyle\sigma_{R}^{({\rm contin})}(E)$
$\displaystyle=\left(\frac{\pi}{k^{2}}\right)\left(\frac{1}{2}\right)\frac{c_{0}}{[1+\exp(2\pi(B_{0}-E)/\hbar\omega_{0})]}$
(21)
$\displaystyle+\left(\frac{\pi}{k^{2}}\right)c_{1}\ln\\{1+\exp(2\pi(E-B_{1})/\hbar\omega_{1})\\}$

matches the results of the CC calculation.

Notice that we have inserted an additional 1/2 factor in the first term of Eq.
(20) to “complete” the Wong integral formula, as well as the exact expression
of the Hill-Wheeler formula [48], because the numerical integration of the
original Eq. (1) of Wong integral formula is effectively founded on the
Simpson integration rule in which its first term for the $s$-wave in the
summation of the Simpson rule only considered a 1/2 factor [46], not a factor
of 1. Note that Eq. (20), in a sense, is a generalization of the Hill-Wheeler
[48] and Wong [46, 50] formulas as it accounts for, in addition to the off-
resonance, the non-perturbative channel-coupling and resonance contributions.
The solid curve in Fig. (3) gives the phenomenological penetrability
$P(E,L=0)$ extracted from the CC calculations as a function of energy in a
semi-log scale, for the region near the Hoyle state. The penetrability in the
Gamow model is shown as the dashed curve. The differences in the two curves
show that there is important interference between the resonance and the
underlying potential barrier penetration.

##  IV Comparison of 12C states from the CC calculations with experiments and
the hyperspherical calculation

We show in Table II and Fig. 4 the 12C energy levels $E_{x}$ relative to the
12C ground state, the $\alpha$-widths $\Gamma_{\alpha}$, and the corresponding
resonance energies $E$ in the $\alpha$ +8Be reaction, with $E$ =
$E_{x}$$-$7.367 MeV. As shown in Table I, the CC obtained energy levels
$E_{x}$ for the $0_{2}^{+}$(Hoyle), $0_{3}^{+}$, 1${}_{1}^{-}$, $2_{2}^{+}$,
and $3_{1}^{-}$ states are in reasonable agreement with the experimental data
[51]. These resonant energies also agree approximately with those of the HFE
method [20], including the $2_{3}^{+}$ resonance which is yet to be observed.

tableComparison of the experimental 12C energy level $E_{x}$ and the
$\alpha$-width $\Gamma_{\alpha}$ [51], with the resonance energy $E$ in the
$\alpha$ +8Be reaction from the CC and HFE calculations [20]. The $\alpha$
+8Be OMP parameters are $\beta_{2}$ = 0.6 (nuclear and coulomb) [31],
$V^{(L=0,2,4,...)}_{o}$ = 9.698 MeV, $V^{(L=1,3,5,...)}_{o}$ = 9.860 MeV,
$r_{v}$ = 1.469 fm, $a_{v}$ = 0.18 fm, $W_{o}$ = 1.5 eV, $r_{w}$ = 1.47 fm,
$a_{w}$ = 0.4 fm, $V_{\pi 0}$ = 2.5 MeV, $r_{\pi}$ = 2.26 fm, $a_{\pi}$ = 0.37
fm, $r_{c}$ = 1.01 fm. For displaying, $E_{\rm Hoyle}$ = 0.2809338737 MeV is
truncated to $E$ = 0.281 MeV.  State Expt.[51] CC result HFE [20] $J^{\pi}$
$E_{x}$ $\Gamma_{\alpha}$ $E_{x}$ $E$ $\Gamma_{\alpha}$ $E_{x}$
$\Gamma_{\alpha}$ (MeV) (keV) (MeV) (MeV) (keV) (MeV) (keV) 0${}_{2}^{+}$
7.654 0.0093 7.648 0.281 0.008 7.655 0.0625 0${}_{3}^{+}$ 10.3 3000 10.159
2.780 300 11.225 1000 1${}_{1}^{-}$ 10.847 273 10.697 3.33 409 10.885 475
2${}_{2}^{+}$ 9.87 850 10.138 2.770 0.026 8.655 132 2${}_{3}^{+}$ 11.204 3.837
217 11.755 1086 3${}_{1}^{-}$ 9.641 46 9.530 2.165 38.0 9.605 68 4${}_{1}^{+}$
9.646 2.279 12.4 10.525 396

For the $0_{2}^{+}$(Hoyle), $0_{3}^{+}$, 1${}_{1}^{-}$, and $3_{1}^{-}$ states
the theoretical widths $\Gamma_{\alpha}$ obtained from the fusion excitation
function in CC calculations as shown in Table II and Fig. 4 are in approximate
agreement (within the the same order of magnitude) with the experimental
widths. However, for the $2_{2}^{+}$ state at theoretical $E$ = 2.771 MeV, the
CC method gives a $\Gamma_{\alpha}$ of 26 eV, whereas the experimental
$\Gamma_{\alpha}$ width of the $2_{2}^{+}$ state extracted in Ref. [52] is 850
keV, obtained by using the spherical $R$-matrix fit, which however missed the
8Be(2+) and thus missed other partial waves that couple the 8Be states and
$\alpha$ to the 2+, to the excitation function of
12C($\gamma$,$\alpha$)8Be(0+). It should be noted that $\Gamma_{\alpha}$ is a
physical quantity that depends sensitively on the resonance energy, $E$ of the
$2_{2}^{+}$ resonance, relative to the top of the $L=2$ effective total
nuclear + Coulomb + centrifugal barrier. The large $\Gamma_{\alpha}$ with the
assumed $\Gamma_{\alpha}/\Gamma_{\rm tot}\sim 1$ of Ref. [52] can come only
from the description of the $2_{2}^{+}$ state, not as a narrow pocket
resonance below the barrier, but as a broad Ramsauer-type resonance [41, 42,
43], considerably above the barrier for the $L=2$ wave. Whether the
$2_{2}^{+}$ state at $E_{x}$ = 9.87 MeV (experimental $E$ = 2.50 MeV) is
indeed considerably above the total nuclear + Coulomb \+ centrifugal barrier
for $L=2$ needs further investigations (see footnote [53] and Fig. 2(b) in
Ref. [54]). It should be further noted that, in the procedure to extract the
$\Gamma_{\alpha}$ width in Ref. [52], there may be a high degree of
uncertainty in the extraction of the value of $\Gamma_{\alpha}$ from the
spherical $R$-matrix theory for which the penetrability of the $\alpha$
particle in a strongly prolate 8Be nucleus depends sensitively on the
deformation and the orientation of the 8Be nucleus Ref. [46]. There may also
be the question of the ambiguity in extracting the width $\Gamma_{\alpha}$ in
which only the product $\Gamma_{\alpha}\Gamma_{\gamma}$ are in effect measured
by the photo-disintegration excitation function of 12C into $\alpha$ and 8Be.
Further, Kirsebom et al [55], in their low-energy 11B(3He,d)12C* reaction with
the subsequent 12C* decaying into 3$\alpha$ particles, have observed many
states in their spectrum, i.e., the 0${}_{2}^{+}$ state at $E_{x}$ = 7.65 MeV,
3${}_{1}^{-}$ state at 9.64 MeV, the 1${}_{1}^{-}$ state at 10.84 MeV, the 2-
state at 11.83 MeV, except the 2${}_{2}^{+}$ at $E_{x}$ = 9.87 MeV. Therefore,
an independent physical determination of the $\Gamma_{\alpha}$ of the
12C($2_{2}^{+}$) state will be of great interest to ascertain whether the
$2_{2}^{+}$ is indeed substantially above the barrier with a broad width or as
narrow as indicated by the optical model couple-channels analysis here.

Figure 4: (Color online) Comparison of the experimental energy levels $E_{x}$
of 12C, relative to the 12C ground state energy, and their widths, with those
obtained from the present CC calculations, for states in the $\alpha+^{8}$Be
continuum. The experimental data are from [51] and note that the most recent
updated Hoyle state width is 8.5 eV. Figure 5: (Color online) Real and
imaginary parts of the radial wavefunction corresponds to individual $J^{\pi}$
state listed in Table II.

Now we examine the locations of the $0_{2}^{+}$ and $0_{3}^{+}$ state energies
relative to the $L=0$ potential in Fig. 1(a). The $0_{2}^{+}$ state lies in
the potential pocket and thus, it is a sharp pocket resonance. On the other
hand, the $0_{3}^{+}$ state lies above the potential and thus, it is a broad
above-the-barrier (or Ramsauer-type) resonance. However, to better identify
these two types of resonances, it is useful to examine the wavefunctions of
these states. Fig.5(a) shows remarkable variations in wavefunctions at
$0_{2}^{+}$ and $0_{3}^{+}$ resonances. Because the $0_{2}^{+}$ resonant lies
way below the broad $L=0$ potential barrier, its wavefunction displays a
strong tunneling character with appreciable internal amplitude in the
potential pocket. Contrast this with the $0_{3}^{+}$ state, its wavefunction
shows weak to zero tunneling characters and, oscillates with near equal,
internal and external amplitudes. So, by knowing these, in Fig.5(b)-(d), we
can identify the $2_{2}^{+}$ and $3_{1}^{-}$ states as pocket resonances and
the $2_{3}^{+}$ and $1_{1}^{-}$ states as above-the-barrier resonances.
Furthermore, it is surprising to see that the $0_{2}^{+}$ state wavefunction
has a node at $r$ $\sim$ 3.8 fm in the $L$ = 0 pocket and the $2_{2}^{+}$
state wavefunction has a subtle node at $r$ $\sim$ 5.7 fm in the $L$ = 2
pocket. This seems to imply that the $0_{2}^{+}$ state is the second state in
continuation of the bound state into the continuum in the series of $J^{\pi}$
= $0^{+}$ state, and likewise for the $2_{2}^{+}$ resonant. All said, it is
remarkable to see our 3${}_{1}^{-}$ state energy and width are compatible with
those from the experiment and the HFE calculation, which should be of interest
to the recent experiment that suggested the 3${}_{1}^{-}$ state may enhance
the 3$\alpha$ reaction rate [56] even though its contribution has been widely
accepted in the past to be insignificant. In addition, we also predicted a
$4_{1}^{+}$ state near the 3${}_{1}^{-}$ state, which requires more
investigations.

##  V Evaluation of the cross section for $\alpha$+8Be to fuse into a
12C($2_{1}^{+})$ nucleus

We can enumerate the reaction channels in the collision of an alpha-particle
with the 8Beg.s nucleus at an energy $E_{\rm cm}\lesssim 3$ MeV, below the
energy for the excitation of the 8Be($2^{+}$) state. The first is the elastic
channel in which both the $\alpha$ nucleus and the 8Be nucleus remain in their
ground states. The second is the reaction channels associated with the $L$
partial-wave, in which $\alpha$ and 8Be nuclei form a 12C*($J^{\pi}$) compound
nucleus in either a resonance of a continuum state with angular momentum $L$
with a total reaction cross section $\sigma_{{}_{R}}(E,L)$.

The reaction cross section can be further divided into partial reaction cross
sections for the compound 12C system with angular momentum $L$ to decay
electromagnetically, that is, either cascading to the ground ${}^{12}{\rm
C}(0_{1}^{+})$ state with a width $\Gamma_{\gamma}({}^{12}{\rm
C}^{*}\buildrel\gamma\over{\to}{}^{12}{\rm C}(0_{1}^{+}))$ or to the excited
${}^{12}{\rm C}(2_{1}^{+})$ state with a width $\Gamma_{\gamma}({}^{12}{\rm
C}^{*}\buildrel\gamma\over{\to}{}^{12}{\rm C}(2_{1}^{+}))$. For $E_{\rm
cm}\lesssim 1$ MeV, the dominant contribution comes from the $s$-wave., with
$L=0$. In this case, as Ref. [18] pointed out that $\Gamma_{\gamma}(^{12}{\rm
C}(0_{2}^{+})\buildrel\gamma\over{\to}{}^{12}{\rm
C}(2_{1}^{+})\gg\Gamma_{\gamma}(^{12}{\rm
C}(0_{2}^{+})\buildrel\gamma\over{\to}{}^{12}{\rm C}(0_{1}^{+})$, so it is
reasonable to ignore the radiative ${}^{12}{\rm
C}(0^{+*})\buildrel\gamma\over{\to}{}^{12}{\rm C}(0_{1}^{+}))$ transition in
comparison with the ${}^{12}{\rm
C}(0^{+*})\buildrel\gamma\over{\to}^{12}\\!\\!{\rm C}(2_{1}^{+})$ E2
transition. The total radiative width of the 12C nucleus in the continuum,
$\Gamma_{\gamma}(E_{\rm cm})$, is just given approximately by
$\Gamma_{\gamma}(0^{+*}\buildrel\gamma\over{\to}2_{1}^{+})$. Additional to the
radiative decay, the compound 12C nucleus can also decay via the emission of
an alpha to return to the state of $\alpha$+8Beg.s nucleus with a width
$\Gamma_{\alpha}({}^{12}{\rm C}^{*}\buildrel\alpha\over{\to}{}^{8}{\rm
Be}+\alpha,E_{i})$.

The cross section $\sigma_{F}(E)$ for the $\alpha$ particle in the $L=0$
partial-wave state to fuse with the 12Beg.s nucleus into a 12C nucleus is
equal to the reaction cross section $\sigma_{R}(E_{\rm cm})$ multiplied by the
probability $F(E_{\rm cm},L)$ for the compound 12C nucleus to decay
radiatively to the ground state band of the 12 nucleus,

$\displaystyle\sigma_{F}(E_{\rm cm},L=0)=F(E_{\rm cm})\sigma_{R}(E_{\rm
cm},L=0),$ (22)

where the fusion fraction probability $F(E_{\rm cm})$ is

$\displaystyle F(E_{\rm cm})=F(E_{\rm cm},^{12}{\rm
C}(0^{+*})\buildrel\gamma\over{\to}{}^{12}{\rm C}(2_{1}^{+}))$
$\displaystyle=\frac{\Gamma_{\gamma}({}^{12}{\rm
C}(0^{+*})\buildrel\gamma\over{\to}{}^{12}{\rm
C}(2_{1}^{+}),E)}{\Gamma_{\gamma}({}^{12}{\rm
C}(0^{+*})\buildrel\gamma\over{\to}{}^{12}{\rm
C}(2_{1}^{+}),E)+\Gamma_{\alpha}({}^{12}{\rm
C}(0^{+*})\buildrel\alpha\over{\to}{}^{8}{\rm Be}+\alpha,E).}$ (23)

The 12C nucleus in the excited $2_{1}^{+}$ state will eventually decay down to
its ground state, with a fused stable 12C nucleus. The above quantity
$\sigma_{F}(E_{\rm cm},L=0)$ is therefore the $s$-wave 12C fusion cross
section, in the collision of the $\alpha$-particle with 8Beg.s.

In heavy-ion collisions the Bethe model of strong absorption in-going wave
boundary approximation [57] assumes $F_{\gamma}$ = 1, to reach the commonly
accepted equivalent assumption of $\sigma_{F}=\sigma_{R}$ [46] for the
following reason. In the context of the heavy-ion collisions, because the
masses of the colliding nuclei are large and the probability of tunneling of
large-mass nuclei below the barrier is low compared to the probability for
radiative decay to lower energies states, which are numerous in number, the
tunneling probability of a fused two heavy ion system is small in comparison
to the probability of radiative decay, resulting the fusion fraction $F(E_{\rm
cm})$ is close to unity, and the reaction cross section is approximately equal
to the fusion cross section. However, this may not be the case for light
nuclei with few states with a “large” fusion barrier, meaning the fusion
fraction could vary and therefore one must evaluate the above
$\Gamma_{\gamma}$ and $\Gamma_{\alpha}$ as a function of energy for the states
near the Hoyle resonance. In other words, the in-going wave approximation is
reasonable for heavy-ion collisions with large heavy-ion mass because
tunneling for a massive projectile through a barrier is difficult, but may not
be for a light-ion projectile such as an $\alpha$-particle.

To calculate the energy-dependent $\Gamma_{\gamma}$, we use the energy
dependence of the gamma-ray for E2 transition by surmising that the values of
the E2 matrix elements are more or less the same for the Hoyle state and those
of nearby states close to the Hoyle because they are in the same neighborhood,
plus the energy difference between them is small compared to the energy
difference for the $\gamma$ decay $E_{x}-E_{x}({}^{12}{\rm C}(2_{1}^{+}))$,
where $E_{x}=E_{\rm cm}+Q_{3\alpha}$. Therefore,

$\displaystyle\Gamma_{\gamma}(E_{i})=\frac{4(E_{x}-E_{x}(^{12}{\rm
C}(2_{1}^{+})))^{5}\pi}{15}B({\rm E2};2_{1}^{+}\to 0_{2}^{+}),$ (24)

where $Q_{3\alpha}$ = 7.2747 MeV, $E_{x}(^{12}{\rm C}(2_{1}^{+}))$ = 4.4398
MeV and $B({\rm E2};2_{1}^{+}\to 0_{2}^{+})$ = 2.7 $e^{2}$ fm4 is the
experimental E2 strength from [58] which we think would be more reliable
because according to Table I of Ref. [25], the theoretical $B({\rm
E2};2_{1}^{+}\to 0_{2}^{+})$ values can vary from $\sim$1 to 10 $e^{2}$ fm4,
depending on the theoretical frameworks and approximations made within the
calculations.

To calculate the energy-dependent $\Gamma_{\alpha}$, we assume the
penetrability $P_{\alpha}(E)$, for $\alpha$-particle returning to the entrance
channel is the same as for the $\alpha$-particle to tunnel into the potential
barrier obtained in the CC calculations, therefore we can express our
$\Gamma_{\alpha}$ for $\alpha$ escaping back to the exit channel as

$\displaystyle\Gamma_{\alpha}(E)=\nu(E)P_{\alpha}(E),$ (25)

where $\nu$ is the frequency of attack, assaulting the potential wall and
$P_{\alpha}(E)$ is the penetrability which can be extracted from the reaction
CC cross sections, whose energy dependence is shown in Fig. 3(b). That is, for
our case when we limit our attention to $L=0$ wave, $P_{\alpha}(E)$ is $P_{\rm
res}(E,L=0)$ of Eq. (17) extracted from the CC calculation results at the
resonance shown in Fig. 3(b), and $P_{\alpha}(E,L=0)$ is $P_{\rm contin}(E)$
of Eq. (12) for energies of the off-resonance in the continuum region
extracted from the CC reaction cross sections of Fig. 3(b).

Figure 6: (a) Energy-dependent widths for $\alpha$-particle
($\Gamma_{\alpha}$) and $\gamma$ ($\Gamma_{\gamma}$) decays in a semi-log
scale. (b) Fusion factor $F$ as a function of energy.

The $s$-wave potential curve has a peculiar shape as a function of $r$
requires us to obtain the assault frequency $\nu$ through the tunneling time
$T(E)$, in units of fm/$c$, given by

$\displaystyle
T(E)=\frac{1}{\nu(E)}=\int_{R_{T}(E)}^{0}\frac{2dr}{\sqrt{2(E-V(r))/\mu}},$
(26)

where $R_{T}$, in units of fm, is the turning point satisfying $E=V(R_{T})$
with the reduced mass $\mu$ in units of MeV. Here, it is probably reasonable
to assume the tunneling time is instantaneous on the ground of the
“wavefunction connection” at the barrier boundary. Furthermore, various
theories have estimated a range of tunneling time that varies from zero to
hundreds of attoseconds, and no consensus has yet been reached on which single
theoretical estimate was correct. One reason for such disagreements boils down
to the fundamental concept of time in quantum mechanics [59, 60].

Figure 7: (Color online) Energy-dependent cross section for $\alpha+{}^{8}{\rm
Be}\to{}^{12}{\rm C}(2_{1}^{+})$ fusion reaction in a semi-log scale.

Figure 6(a) shows the $\Gamma$(Ecm) results of Eqs. (24) and (25); the energy
dependence of the semi-empirical $\Gamma_{\gamma}$ shows a monotonous increase
as $E_{\rm cm}$ increases in contrast to the $\Gamma_{\alpha}$ which rises
more rapidly, and intersects the $\Gamma_{\gamma}$ at $E_{\rm cm}$ $\approx$
0.227 MeV. At this point, we must emphasize that the result of Eq.(25) is only
valid for a broad continuum.

The changes in the $\Gamma_{\alpha}$ width are most drastic in the vicinity of
the Hoyle state as shown in the inset of Fig. 6(a) where we observe, away from
the Hoyle resonance, the values of $\Gamma_{\alpha}$ are on the order of
$\sim$ 1.0 MeV, whereas, in the proximity of the Hoyle state governed by the
Breit-Wigner distribution, the value of $\Gamma_{\alpha}$ reduces to $\sim$
8.0 eV. Here, we use a simple “flat” function to qualitatively represent the
distribution of the Hoyle width in the energy range between 0.280927 MeV and
0.280938 MeV to simplify our proceeding calculation for the fusion cross
section.

Figure 6(b) shows the fusion fraction result $F(E_{\rm cm})$ of Eq. (23) as a
function of energy. The 12C fusion factor corresponds to the energy variation
of $\Gamma_{\alpha}$ and $\Gamma_{\gamma}$. For energies below 0.239 MeV, the
plot shows $F_{\gamma}$ is near a maximum probability of 1.0, indicating the
$\gamma$-decay process plays a dominant role. At $E$ $\approx$ 0.227 MeV, the
probability of the $\alpha$-decay almost equals to the $\gamma$-decay. For
higher $E$, the $F_{\gamma}$ factor continues to drop until it reaches the
Hoyle resonance. At the Hoyle resonance, $F_{\gamma}$ increases again to
several orders of magnitudes as illustrated by the inset of Fig. 6(b). As $E$
further increases, away from the Hoyle resonance, the $F_{\gamma}$, and again,
decreases to a minute value of $\sim$10-8. Note that, to make sure our
calculation is reasonable, we also checked our ratio
$\Gamma_{\gamma}/\Gamma_{\alpha}\sim 3.98\times 10^{-4}$ by comparing to the
ratio $\Gamma_{\gamma}/\Gamma_{\alpha}=4\times 10^{-4}$ reported in Ref. [18]
at the Hoyle energy.

Figure 8: (Color online) Energy-dependent $S(E)$ factor for $\alpha+{}^{8}{\rm
Be}\to{}^{12}{\rm C}(2_{1}^{+})$ fusion reaction in log-log scale.

Evaluating Eq. (22) we obtained the cross section for the fusion of
$\alpha+{}^{8}{\rm Be}\to{}^{12}{\rm C}(2_{1}^{+})$ as a function of energy
shown in Fig. 7. One can see on the plot that the Hoyle resonance manifested
as a sharp spike “piggybacking” on the broad Gamow-like background. Focusing
on the Hoyle region, the inset in the figure indicates $\Gamma_{\alpha}$
$\sim$ 9.0 eV, which is consistent with the earlier noted width of 8.0 eV,
which is, in turn, within the error bar of the experimental value of 8.5 eV.

Figure 9: (Color online) Fusion cross section for a sample of $A\leq$ 12
systems in log-log scale. The recommended fusion cross section data are from
the IAEA’s ENDF [64], except for the $p+^{11}$B reaction from the JANIS [63]
database.

We can use Eqs. (22) and (18) to extract the corresponding astrophysical
$S(E)$ factor which is shown in Fig. 8. Note that its energy dependence is
significantly different from the one estimated by Descouvemont and Baye [15]
using the Breit-Wigner form (see Fig. 2 therein).

It is also informative to compare our estimated fusion cross section for
$\alpha+{}^{8}{\rm Be}$ with the five well-known fusion reactions in Fig. 9.
It is shown that our fusion cross section resembles, in particular, the
general behavior of either the D+T or D+3He, or p+11B reactions, and it is by
far the smallest among the samples considered here.

We return next to the question of whether the breakup of the target from the
excitation and the decay of the 8Be($2^{+}$) nucleus could play any role in
our reaction cross sections at high energies. The answer is “negligible” for
$E_{\rm cm}\gtrsim 4$MeV for the following reason: During the collisions, the
$2^{+}$ state can be excited virtually from the 8Be ground state and then back
to the ground state, leading to a polarization potential through the second-
order perturbation theory [61]. Such a polarization potential has been taken
into account in the coupled-channel calculation. The virtual excitation, on
the one hand, does not involve the disintegration of the 8Be state into two
$\alpha$ particles; the real excitation of the 8Be, on the other, does lead to
the population of the unstable 2+ state, which will subsequently disintegrate
into two $\alpha$ particles by 1.513 MeV [62]. However, the 8Be($2^{+}$)
channel remains energetically close and insignificant for the projectile
energy in question with respect to 3.03-MeV, the 8Be($2^{+}$) energy.
Nevertheless, we must check to be sure. Thus, using the coupled-channel
calculated inelastic cross section $\sigma(E)=\sigma^{\rm
tot}_{R}(E)+\sigma_{\rm ex}(E)$ and 8Be$(2^{+})$ excitation cross section
$\sigma_{\rm ex}(E)$ with $\Gamma(^{8}{\rm Be}(2^{+}\rightarrow
0_{1}^{+}),\gamma)$ = 8.3 meV, $\Gamma(^{8}{\rm Be}(2^{+})\rightarrow
2\alpha)$ = 1.5 MeV [62] and ${\Gamma(^{8}{\rm Be}(2^{+})\rightarrow
2\alpha)}/{\Gamma_{\rm total}(^{8}{\rm Be}(2^{+}))}$ $\approx$ 1.0, where
$\Gamma_{\rm total}(^{8}{\rm Be}(2^{+}))$ is the total width for $\alpha$ and
$\gamma$ decays, we evaluated the 8Be($2^{+}$) breakup fraction given by

$\displaystyle F_{2^{+}{\rm breakup}}=\left(\frac{\sigma_{\rm
ex}(E)}{\sigma(E)}\right)\frac{\Gamma(^{8}{\rm Be}(2^{+})\rightarrow
2\alpha)}{\Gamma_{\rm total}(^{8}{\rm Be}(2^{+}))}$ (27)

and found them to be nominal for collision energies up to 4.0 MeV. For
example, the fraction $F_{2^{+}{\rm breakup}}$ is of the order of 10-2 at
$E_{\rm cm}$ = 4 MeV, and such a small value will not affect the physics we
have discussed here.

##  VI Summary and Conclusions

The quest for understanding the structure of Hoyle state in 12C began with
Fred Hoyle, in 1954, who solved Salpeter’s puzzle of 12C production via the
2-stage 3$\alpha$ reaction process in stars, by the prediction of the must-
exist resonance 0+ state in the 12C near the 3$\alpha$ breakup threshold. The
Hoyle and associated states of 12C are fundamentally interesting because these
states possess exotic triangular-molecular structures with extended size.
Moreover, knowing the structure of the Hoyle state could help reveal how it
gives rise to carbon, oxygen, nitrogen, and other light chemical elements that
compose the complex molecules of living things $-$ the synthesis of these
elements drives the stellar evolutions which ultimately enables the genesis of
life in the Universe. Having said that, the extensive experimental and
theoretical investigations on the properties of the Hoyle and family of states
stem from the spectroscopic viewpoints.

In this work, we introduced an alternative founded on the viewpoint of pocket
resonances, within a optical model coupled-channel framework, to examine the
production of the Hoyle and associated excited states in the nuclear reaction
of $\alpha$-particle on a prolate 8Be nucleus. To do so, we adopted an optical
model potential suited for our scattering problem, permitting us to describe
and identify the origins of, the narrow and broad resonances present in the
reaction cross section in the Hoyle state energy region. While developing the
model potential, we discovered that the energies and widths of the resonances,
particularly for $L$ $>$ 0 partial waves, are extremely sensitive to (i) the
target deformation $\beta_{2}$ and (ii) the parity of the nuclear surface
potential, that is, more attractive for the even $L$ partial waves than those
of the odd ones owing to the Bose-Einstein exchange of the $\alpha$ bosons.
Partial-wave decomposition analysis of the reaction cross sections and in
comparison with the results from experiments and the 3-body hyperspherical
calculations, we found that although the comparison between our theoretical
results and these data is not perfect, our optical model coupled-channel
calculation has captured the essential elements of the physics in connection
to the reaction of the $\alpha$ particle on a 8Be nucleus. Further improvement
upon our calculations can be carried, of course, by introducing a $\beta_{4}$
deformation parameter that will improve the description of the 2$\alpha$
structure of the 8Be nucleus. Nonetheless, the reasonable agreement between
our results and the experimental data, as well as the hyperspherical
calculations $-$ in terms of the energies and the widths, provides us the
confidence that allows for a reasonable estimate of the fusion cross section
for $\alpha$ +8Be reaction, not only at the resonances but also away from the
resonances, for which reaction theory can serve well. The important new
element here is that at energies away from the resonances and substantially
below the barrier, the penetrability of $\alpha$ on 8Be depends on the shape
of the interaction potential under which the $\alpha$ particle needs to
tunneling through. Such detailed structure can only be obtained by an optical
model analysis, as carried out here. Consequently, guided by such principles,
we parametrize the reaction cross section $\sigma_{R}$ for the $\alpha$ +8Be
reaction up to $E$ = 4.0 MeV. Furthermore, given the future astrophysical
interest, we further examine the fundamental relationship between the reaction
and fusion cross section by calculating the $\gamma$\- and $\alpha$-decay
widths for $\alpha$ particles captured in the potential pocket and thus
evaluate the $s$-wave cross-section for $\alpha$+8Be fusion into 12C for $E$
$<$ 1.0 MeV.

## Acknowledgment

CYW’s work is supported, in part, by the Division of Nuclear Physics, U.S.
Department of Energy under Contract DE-AC05-00OR22725. IJT’s work was
performed under the auspices of the U.S. Department of Energy by Lawrence
Livermore National Laboratory under Contract DE-AC52-07NA27344.

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