# Experimental investigation of $\alpha$-condensation in light nuclei

J. Bishop  Current address: School of Physics & Astronomy and Cyclotron
Institute, Texas A&M University, College Station, 77843, TX, USA
<EMAIL_ADDRESS>School of Physics $\&$ Astronomy, University of
Birmingham, UK  Tz. Kokalova  School of Physics $\&$ Astronomy, University of
Birmingham, UK  M. Freer  School of Physics $\&$ Astronomy, University of
Birmingham, UK  L. Acosta  INFN, Sezione di Catania, Via Santa Sofia, 62,
95123, Catania, Italy  Instituto de Física, Universidad Nacional Autónoma de
México, AP 20-364, Cd.Mx. 01000, Mexico  M. Assié  Institut de Physique
Nucléaire, CNRS/IN2P3, Université Paris-Sud 11, France  S. Bailey  School of
Physics $\&$ Astronomy, University of Birmingham, UK  G. Cardella  INFN,
Sezione di Catania, Via Santa Sofia, 62, 95123, Catania, Italy  N. Curtis
School of Physics $\&$ Astronomy, University of Birmingham, UK  E. De Filippo
INFN, Sezione di Catania, Via Santa Sofia, 62, 95123, Catania, Italy  D.
Dell’Aquila  INFN Sezione di Napoli $\&$ Dipartimento di Fisica, Universitá
Federico II, Napoli, Italy  Rudjer Bošković Institute, Zagreb, Croatia  S. De
Luca  INFN, Sezione di Catania, Via Santa Sofia, 62, 95123, Catania, Italy
Dipartimento di Scienze MIFT - Università di Messina, Italy  L. Francalanza
INFN Sezione di Napoli $\&$ Dipartimento di Fisica, Universitá Federico II,
Napoli, Italy  B. Gnoffo  INFN, Sezione di Catania, Via Santa Sofia, 62,
95123, Catania, Italy  Dipartimento di Fisica e Astronomia “Ettore Majorana” -
Università di Catania, Italy  G. Lanzalone  INFN LNS, Catania, Italy
Università Kore, Enna, Italy  I. Lombardo  INFN, Sezione di Catania, Via Santa
Sofia, 62, 95123, Catania, Italy  N. S. Martorana  Dipartimento di Fisica e
Astronomia “Ettore Majorana” - Università di Catania, Italy  INFN LNS,
Catania, Italy  S. Norella  INFN, Sezione di Catania, Via Santa Sofia, 62,
95123, Catania, Italy  Dipartimento di Scienze MIFT - Università di Messina,
Italy  A. Pagano  INFN, Sezione di Catania, Via Santa Sofia, 62, 95123,
Catania, Italy  E. V. Pagano  INFN LNS, Catania, Italy  M. Papa  INFN, Sezione
di Catania, Via Santa Sofia, 62, 95123, Catania, Italy  S. Pirrone  INFN,
Sezione di Catania, Via Santa Sofia, 62, 95123, Catania, Italy  G. Politi
INFN, Sezione di Catania, Via Santa Sofia, 62, 95123, Catania, Italy
Dipartimento di Fisica e Astronomia “Ettore Majorana” - Università di Catania,
Italy  F. Rizzo  INFN LNS, Catania, Italy  Dipartimento di Fisica e Astronomia
“Ettore Majorana” - Università di Catania, Italy  P. Russotto  INFN, Sezione
di Catania, Via Santa Sofia, 62, 95123, Catania, Italy  L. Quattrocchi  INFN,
Sezione di Catania, Via Santa Sofia, 62, 95123, Catania, Italy  Dipartimento
di Scienze MIFT - Università di Messina, Italy  R. Smith  School of Physics
$\&$ Astronomy, University of Birmingham, UK  I. Stefan  Institut de Physique
Nucléaire, CNRS/IN2P3, Université Paris-Sud 11, France  A. Trifirò  INFN,
Sezione di Catania, Via Santa Sofia, 62, 95123, Catania, Italy  Dipartimento
di Scienze MIFT - Università di Messina, Italy  M. Trimarchì  INFN, Sezione di
Catania, Via Santa Sofia, 62, 95123, Catania, Italy  Dipartimento di Scienze
MIFT - Università di Messina, Italy  G. Verde  Institut de Physique Nucléaire,
CNRS/IN2P3, Université Paris-Sud 11, France  INFN, Sezione di Catania, Via
Santa Sofia, 62, 95123, Catania, Italy  M. Vigilante  INFN Sezione di Napoli
$\&$ Dipartimento di Fisica, Universitá Federico II, Napoli, Italy  C. Wheldon
School of Physics $\&$ Astronomy, University of Birmingham, UK

###### Abstract

Background

Near-threshold $\alpha$-clustered states in light nuclei have been postulated
to have a structure consisting of a diffuse gas of $\alpha$-particles which
condense into the 0s orbital. Experimental evidence for such a dramatic phase
change in the structure of the nucleus has not yet been observed.

Method

To examine signatures of this $\alpha$-condensation, a compound nucleus
reaction using 160, 280, and 400 MeV ${}^{16}\mathrm{O}$ beams impinging on a
carbon target was used to investigate the
${}^{12}\mathrm{C}(^{16}\mathrm{O},7\alpha)$ reaction. This permits a search
for near-threshold states in the $\alpha$-conjugate nuclei up to
${}^{24}\mathrm{Mg}$.

Results

Events up to an $\alpha$-particle multiplicity of 7 were measured and the
results were compared to both an Extended Hauser-Feshbach calculation and the
Fermi break-up model. The measured multiplicity distribution exceeded that
predicted from a sequential decay mechanism and had a better agreement with
the multi-particle Fermi break-up model. Examination of how these $7\alpha$
final states could be reconstructed to form ${}^{8}\mathrm{Be}$ and
${}^{12}\mathrm{C}(0_{2}^{+})$ showed a quantitative difference in which decay
modes were dominant compared to the Fermi break-up model.

No new states were observed in ${}^{16}\mathrm{O}$, ${}^{20}\mathrm{Ne}$, and
${}^{24}\mathrm{Mg}$ due to the effect of the N-$\alpha$ penetrability
suppressing the total $\alpha$-particle dissociation decay mode.

Conclusion

The reaction mechanism for a high energy compound nucleus reaction can only be
described by a hybrid of sequential decay and multi-particle breakup. Highly
$\alpha$-clustered states were seen which did not originate from simple binary
reaction processes. Direct investigations of near-threshold states in
N-$\alpha$ systems are inherently impeded by the Coulomb barrier prohibiting
the observation of states in the N-$\alpha$ decay channel. No evidence of a
highly clustered 15.1 MeV state in ${}^{16}\mathrm{O}$ was observed from
$(^{28}\mathrm{Si}^{\star},{{}^{12}\mathrm{C}(0_{2}^{+})})^{16}\mathrm{O}(0_{6}^{+})$
when reconstructing the Hoyle state from 3 $\alpha$-particles. Therefore, no
experimental signatures for $\alpha$-condensation were observed.

$\alpha$-condensation, Compound nucleus reaction, $\alpha$-clustering, High-
multiplicity, detector array keywords, Nuclear reactions, Nuclear structure

###### pacs:

21.60.-n,24.10.-i,25.70.Gh,25.85.Ge

††preprint: APS/123-QED

##  I Introduction

The role of the $\alpha$-particle in the structure of light nuclei has been
well explored for over 80 years. Observation of the binding energy per nucleon
shows that $\alpha$-conjugate nuclei are much more strongly bound and can be
described by a model of a tightly packed geometry of $\alpha$-particles
Hafstad and Teller (1938). Since this initial investigation, the importance of
the $\alpha$-particle in clustering, by virtue of its inert nature and own
high binding energy, has been demonstrated experimentally and theoretically.
The lightest (non-trivial) $\alpha$-conjugate system, ${}^{8}\mathrm{Be}$, has
been shown to have a structure comprising of a dumbbell configuration of
$\alpha$-particles Arickx  _et al._ (1979) with a large $\alpha$-$\alpha$
separation distance of 4.4 fm Tohsaki  _et al._ (2017); Freer  _et al._
(2018). This dilute $\alpha$-particle behavior is then extended to the Hoyle
state in carbon-12 (${}^{12}\mathrm{C}(0_{2}^{+})$), a resonance above the
3$\alpha$ threshold of astrophysical importance, which has been the subject of
extensive studies Dunbar  _et al._ (1953); Navrátil  _et al._ (2000); Morinaga
(1966); Suzuki  _et al._ (1972); Uegaki  _et al._ (1977); Raduta  _et al._
(2011). Recent experimental efforts have provided evidence for the structure
of ${}^{12}\mathrm{C}$ as an equilateral triangle formation of 3
$\alpha$-particles Marín-Lámbarri  _et al._ (2014) but a
${}^{8}\mathrm{Be}{+}\alpha$ configuration is favored by some theoretical
works Epelbaum  _et al._ (2012); Kanada-En’yo (2007).

This idea of describing the structure of $\alpha$-conjugate nuclei in terms of
a dilute arrangement of $\alpha$-particles has been accompanied by a study of
the nuclear equation of state in infinite systems. It was demonstrated Röpke
_et al._ (1998) that a nuclear liquid can undergo a phase change associated
with the scattering length of the $\alpha$-particle when the density of a
nuclear system $\rho$, drops below a specific value relative to the nuclear
saturation density $\rho_{0}$, the current estimation is $\frac{\rho_{0}}{3}$
to $\frac{\rho_{0}}{5}$ Röpke  _et al._ (1998); Sogo  _et al._ (2009, 2010);
Schuck (2013). Below this value, the system is no longer described by the
fermionic interactions of the nucleons but the dominant degrees of freedom are
those of the $\alpha$-particles. Their bosonic nature allows for a macroscopic
occupation of the ground-state. This system is analogous to a Bose-Einstein
Condensate which has been well studied in atomic systems London (1938);
Anderson  _et al._ (1995). This work has been furthered in the nuclear realm
by several experiments involving high-multiplicity particle decays Borderie
_et al._ (2016); Morelli  _et al._ (2019); Akimune  _et al._ (2013) and
experiments to observe an $\alpha$-gas via “Coulomb explosion” of
${}^{40}\mathrm{Ca}$ appear possible from a theoretical perspective Yamada and
Schuck (2004). Such an exotic state of matter is therefore postulated to
describe the well-studied light N-$\alpha$ clustered states
(${}^{8}\mathrm{Be}$(g.s) and ${}^{12}\mathrm{C}(0_{2}^{+})$) and is known as
an $\alpha$-condensate. This presents a unique probe into understanding the
nuclear force and generates two questions. Firstly, how can such an unusual
state of matter be detected? Secondly, can these N-$\bm{\alpha}$ states behave
as an $\bm{\alpha}$-condensate?

##  II Signatures of $\alpha$-condensation

To understand the signatures associated with condensation, it is important to
first comprehend the differences in the properties of an $\alpha$-condensate
and a clustered state. When discussing $\alpha$-condensation, the system being
described is one whereby the bosonic degrees of freedom are dominant Tohsaki
_et al._ (2001). An additional degree of selectivity, necessary to describe
the phase change of the system as mentioned above, is the need for a more
dilute structure. In this case, the individual $\alpha$-particles have a large
average separation where the underlying fermionic structure of the
$\alpha$-particles is not resolved and the system behaves like an
$\alpha$-gas. When choosing to describe the observed N-$\alpha$ states, a
large separation is required and therefore $\alpha$-gas is chosen as an apt
description for the resonances of interest.

Figure 1: Schematic representation of the effect a condensate state has on
modifying the Coulomb barrier for $\alpha$-gas emission and therefore the
yield of the decay products at low kinetic energies compared to a clustered
state. The near-threshold resonance has a width which is spread around the
Coulomb barrier. The observed yield of such a decay through the Coulomb
barrier is shown to the left for the two situations of a clustered and a
condensate state. The shift in the energy of the decaying products between the
two different types of state can be a signature of $\alpha$-condensation.

Such a change in the size of a nuclear system can have a profound effect on
the decay properties. Previous investigations into this phenomena Kokalova
_et al._ (2006) have demonstrated $\alpha$-gas states will exhibit an
increased preference for emission of $\alpha$-gas states ($\alpha$,
${}^{8}\mathrm{Be}$(g.s), and ${}^{12}\mathrm{C}(0_{2}^{+})$) as a consequence
of the modification to the Coulomb barrier for these light clusters. This
effect is demonstrated in Fig. 1 where a more dilute nuclear structure
modifies the penetrability factor. Such an enhancement in the yield of these
states is therefore indicative of a more dilute system congruent with an
$\alpha$-gas structure. This can be compared to predictions from statistical
decay models. The reduction of the barrier in proximity to the state also
produces decay products with lower kinetic energies than those expected from a
geometric cluster configuration Kokalova  _et al._ (2006). Finally, the direct
observation of states in the $\alpha$-conjugate nuclei in proximity to the
N-$\alpha$ threshold and their decay modes can also be compared to theoretical
predictions. Due to the nature of the $\alpha$-gas structure, the non
$\alpha$-gas decay modes should be extremely inhibited and as such, their
reduced widths should be extremely small due to the small overlap in the
wavefunction.

##  III Experimental investigation

To examine the signatures mentioned in Section II, an experiment was performed
at Laboratori Nazionali del Sud, Catania, Italy. An ${}^{16}\mathrm{O}$ beam
at lab energies of 160, 280, and 400 MeV was provided by the K800 cyclotron
with an average beam current of 100 pA. This beam was then impinged on a
natural carbon target of thickness 58 $\mu\mathrm{g/cm}^{2}$ for the 280 and
400 MeV beam energies and 92 $\mu\mathrm{g/cm}^{2}$ for the 160 MeV beam
energy. A gold target of thickness 174 $\mu\mathrm{g/cm}^{2}$ was also used
for calibration purposes to provide elastic scattering events. The
${}^{12}\mathrm{C}(^{16}\mathrm{O}$,${}^{28}\mathrm{Si}^{\star}$) reaction was
then investigated. To measure the reaction products, two detector arrays were
combined to provide a large solid angle coverage, high granularity system with
good particle identification (PID) and energy range. These two arrays used
were the CHIMERA Pagano  _et al._ (2004) and FARCOS Pagano  _et al._ (2016);
Acosta  _et al._ (2016) detector systems, which combined 1192 Si-CsI(Tl)
telescopes with 4 DSSD-DSSD-CsI(Tl) telescopes placed at forward angles. The
combined system provided an excellent combination of high angular resolution
at small angles and a high detector coverage. An overall solid angle coverage
of 28$\%$ (of the full 4$\pi$) was achieved.

###  III.1 CHIMERA

The CHIMERA (Charged Heavy Ion Mass and Energy Resolving Array) detector
Pagano  _et al._ (2004) is comprised of two sections, the first being the
forward rings. This section consists of 18 rings (i.e. full annular azimuthal
coverage) of detectors covering lab scattering angles from
$1^{\circ}\rightarrow 30^{\circ}$ at varying distances with the smallest
angles the furthest away. The second section is the backwards ball component
which has rings covering from $30^{\circ}\rightarrow 176^{\circ}$ at a
constant distance from the target of 40 cm. The rings consist of a series of
individual telescopes which allow for high dynamic-energy range PID and
momentum measurements. A single telescope is made from two detectors, the
first is a 300 $\mu$m (nominal thickness) n-type planar Si detector (25 cm2)
behind which is a CsI(Tl) scintillator read out via a silicon photodiode.

Figure 2: Particle identification plot from CHIMERA showing the signal in the
silicon stage against that in the cesium iodide stage for a single telescope.
The different loci corresponding to nuclei of interest are labeled. The
${}^{4}\mathrm{He}$ can be separated from the ${}^{3}\mathrm{He}$ locus with
99.4$\%$ purity.

Both of these detectors in the telescope utilize a dual-gain setup which
allows for a high energy-range while avoiding ADC (Analogue-to-Digital
Converter) discretization effects. The silicon detectors were calibrated using
a combination of elastic scattering data from the beam impinging on a gold
target, a triple-alpha calibration source (${}^{239}\mathrm{Pu}$,
${}^{241}\mathrm{Am}$, and ${}^{244}\mathrm{Cm}$), a pulser calibration and
time-of-flight (TOF) information. A timing signal was also taken from the
silicon detectors in reference to the cyclotron radiofrequency signal. This
energy and timing information allowed for particle identification across a
wide energy range. For particles which have sufficient energy to penetrate the
300 $\mu$m nominal thickness of silicon, the energy deposited in the silicon
($\Delta$E) was plotted against that left in the scintillator ($E$). This then
allowed for PID via the observation of different loci according to the
expected energy loss $\delta E$, through a small distance $\delta x$, given by
the Bethe-Bloch formula $\frac{\delta E}{\delta x}\propto mZ^{2}/E$ Neindre
_et al._ (2002). This is the $\Delta$E-E PID method. These loci are shown in
Fig. 2. For those particles where the energy was insufficient to be identified
this way ($E\lesssim 24$ MeV for an $\alpha$-particle traversing 300 $\mu$m of
silicon), the timing signal $t$, was used in addition to the energy deposited
in the silicon $E$, to identify the mass of the particle using the time-of-
flight particle identification - TOF PID. To do this, the interaction time
$t_{0}$, was calculated assuming the hit corresponded to an $\alpha$-particle
($m=4$) by:

$t_{0}=d\sqrt{\frac{m}{2E}}-t,$ (1)

for a detector of distance $d$ from the target. The result from this TOF PID
method is shown in Fig. 3 to identify $\alpha$-particles. Those events which
lay outside of the gate shown undergo the same procedure assuming $m=$ 6, 7,
12, and 16. The timing and energy resolution were insufficient to
differentiate other mass nuclei. These can however still be identified using
the $\Delta$E-E method.

To determine the energy of the particle where $\Delta$E-E PID was possible,
this value was calculated via the signal in the silicon stage due to the non-
linear response of the CsI(Tl) crystal as well as its mass and charge signal
dependence Amorini  _et al._ (2012). This gives an energy resolution of
approximately 400 keV for a 30 MeV $\alpha$-particle increasingly
approximately linearly to 1000 keV for a 60 MeV $\alpha$-particle.

Figure 3: Interaction time plot for an example CHIMERA ring 9I
($24^{\circ}<\theta<27^{\circ}$). The two peaks (with the second peak arising
from events where the original beam pulse was misidentified at $\Delta t$ =
62.5 ns) are labeled with a red arrow and correspond to correct identification
of $\alpha$-particles whereas those outside the pair of dashed magenta lines
are not classified as such and other masses are tested.

###  III.2 FARCOS

The FARCOS (Femtoscope ARray for COrrelation and Spectroscopy) detector Pagano
_et al._ (2016) is a high granularity, high resolution detector system mainly
designed for heavy-ion collisions and studies of light-clustering. The FARCOS
system comprises of three detector stages, the first being a nominal 300
$\mu$m DSSD (double-sided silicon strip detector) with 32 strips in both the
vertical and horizontal direction. The second stage is a thicker nominal 1500
$\mu$m DSSD of the same strip number. Behind this is a set of four 6 cm-thick
CsI(Tl) crystals with a Si PIN diode read-out arranged in a 2 $\times$ 2 grid.
The overall system is 64 $\times$ 64 mm in extent giving an angular resolution
of around $0.1^{\circ}$ for a single strip at a distance of $1$ m. Timing
information was not available from these detectors therefore particle
identification in the FARCOS detector relied solely on the $\Delta$E-E method
from the energy deposited in the 300 and 1500 $\mu$m silicon detectors as
discussed above for CHIMERA.

The presence of the FARCOS detectors provided excellent angular and energy
resolution ($\delta\theta\approx 0.1^{\circ}$, $\delta E/E<1\%$ at 5.5 MeV) at
low angles ($2^{\circ}\rightarrow 8^{\circ}$) to study the different reaction
mechanisms, which will be covered in Section V.1.

##  IV Theoretical models of expected reaction mechanisms

In order to demonstrate the observed yields in the experiment performed were
congruent with an $\alpha$-gas state, theoretical input was needed to provide
a benchmark. To understand the reaction, the sequential decay mechanism,
multi-particle break-up and the 7-$\alpha$ decay through predicted
$\alpha$-gas states were modeled to compare to the experimental data.

###  IV.1 Sequential decay

To model the sequential decay mechanism, an Extended Hauser-Feshbach (EHF)
code was used Matsuse  _et al._ (1997) to model the p, n, $\alpha$, and
$\gamma$ decay from the compound system following fission. The compound
nucleus is populated with a range of angular momenta values as given by the
input channel’s masses and energy. In this code, the fission yields are
calculated from the density of states at the scission point then sequential
emission is modeled until all decay paths are exhausted.

A calculation was performed for each of the three beam energies and provided a
number of observables. The first of these, the final yields for different
nuclei are summarized in Table 1 where the dominance of the $\alpha$-conjugate
nuclei can clearly be seen for the lowest beam energy of 160 MeV. At higher
beam energies, the non $\alpha$-conjugate nuclei become more readily populated
as their larger Q-values are less inhibited by virtue of the increased
excitation energy in the compound nucleus.

An important prediction to extract from the calculation is the expected
$\alpha$-particle multiplicity. As discussed in Section II, from an
$\alpha$-gas structure, the modification of the Coulomb barrier would create a
higher multiplicity of $\alpha$-particles in a decay than normally expected.
The EHF code models sequential $\alpha$-decay by summing the total cross
section for $n$ sequential $\alpha$-decays for different nuclei following
fission, hence the expected multiplicity from the sequential decay model can
be extracted. Decay chains where $n$ sequential decays were interspersed with
a proton or neutron decay were also examined but shown to have a negligible
contribution to the total cross section. These values can be seen in Fig. 4
where the behavior for different beam energies can be observed to vary
significantly. As the excitation energy in the compound nucleus increases
because of a higher beam energy, the cross section for high multiplicity
decays can be seen to dramatically decrease. Examining the dynamics of the
sequential decay demonstrated this reduction is due to the decreased
preference for $\alpha$-decay as the effect of the Q-value for each decay
stage becomes less important. As such, the system becomes to more equally
prefer proton, neutron and $\alpha$-decay as the beam energy increases.
Consequently, the largest multiplicity one might expect at a beam energy of
400 MeV is 5. In comparison, at the lowest beam energy of 160 MeV, the smaller
amount of energy in the system means that when a nucleus still has sufficient
energy to decay, the preferred path is that of $\alpha$-decay. This results in
a still sizeable cross section for multiplicity 7 decays. It should also be
noted that the cross section for multiplicity 6 events is $<0.1$ mb. This is a
consequence of the need to break apart an $\alpha$-particle to have a final
state which is comprised of 6 $\alpha$-particles and d+d or
${}^{3}\mathrm{He}$+n etc.

Table 1: Final yields as predicted from the EHF calculation for the three different beam energies (160, 280, and 400 MeV). The $\alpha$-conjugate nuclei masses are shown in bold. Decay | Final cross section (mb)
---|---
product mass | 160 MeV | 280 MeV | 400 MeV
1 | 1.92 $\times 10^{3}$ | 3.69 $\times 10^{3}$ | 4.78 $\times 10^{3}$
2 | 2.58 $\times 10^{2}$ | 6.56 $\times 10^{2}$ | 1.10 $\times 10^{3}$
3 | 1.67 $\times 10^{2}$ | 3.21 $\times 10^{2}$ | 5.30 $\times 10^{2}$
4 | $\mathbf{3.79\times 10^{3}}$ | $\mathbf{3.27\times 10^{3}}$ | $\mathbf{2.74\times 10^{3}}$
5 | 8.31 $\times 10^{1}$ | 1.52 $\times 10^{2}$ | 2.41 $\times 10^{2}$
6 | 7.82 $\times 10^{1}$ | 2.07 $\times 10^{2}$ | 2.71 $\times 10^{2}$
7 | 3.60 $\times 10^{1}$ | 1.23 $\times 10^{2}$ | 1.27 $\times 10^{2}$
8 | $\mathbf{3.69\times 10^{1}}$ | $\mathbf{1.19\times 10^{2}}$ | $\mathbf{7.36\times 10^{1}}$
9 | 7.64 $\times 10^{0}$ | 4.43 $\times 10^{1}$ | 6.29 $\times 10^{1}$
10 | 7.81 $\times 10^{1}$ | 2.34 $\times 10^{2}$ | 1.31 $\times 10^{2}$
11 | 2.47 $\times 10^{1}$ | 1.06 $\times 10^{2}$ | 4.76 $\times 10^{1}$
12 | $\mathbf{7.91\times 10^{2}}$ | $\mathbf{7.01\times 10^{2}}$ | $\mathbf{2.14\times 10^{2}}$
13 | 4.46 $\times 10^{1}$ | 1.93 $\times 10^{2}$ | 1.23 $\times 10^{3}$
14 | 2.12 $\times 10^{2}$ | 3.35 $\times 10^{2}$ | 1.51 $\times 10^{2}$
15 | 1.51 $\times 10^{2}$ | 1.32 $\times 10^{2}$ | 5.14 $\times 10^{1}$
16 | $\mathbf{8.75\times 10^{2}}$ | $\mathbf{1.32\times 10^{2}}$ | $\mathbf{5.27\times 10^{1}}$
17 | 6.87 $\times 10^{1}$ | 2.20 $\times 10^{2}$ | 1.00 $\times 10^{1}$
18 | 1.57 $\times 10^{2}$ | 3.71 $\times 10^{1}$ | 9.05 $\times 10^{0}$
19 | 2.59 $\times 10^{1}$ | 5.02 $\times 10^{1}$ | 1.42 $\times 10^{0}$
20 | $\mathbf{1.02\times 10^{2}}$ | $\mathbf{9.91\times 10^{0}}$ | $\mathbf{8.52\times 10^{-2}}$
Figure 4: Predicted $\alpha$-particle multiplicity from the EHF calculation
for the three beam energies. The higher beam energies can be seen to
suppressed alpha decay cross sections.

Finally, the distribution of $\alpha$-particle energies was also extracted
according to the initial population of different nuclei. This can be observed
in Fig. 5. The figure shows that when populating ${}^{24}\mathrm{Mg}$, the
kinetic energy distribution is a smooth continuum whereas when lighter
$\alpha$-conjugate nuclei are populated (e.g. ${}^{16}\mathrm{O}$), peaks can
be seen which correspond to the population of discrete resonances which are
included automatically in the code. This demonstrates that discrete levels in
nuclei heavier than ${}^{16}\mathrm{O}$ are not expected to be strongly
populated above the strength of the continuum. The behavior with increasing
beam energy also demonstrates that the continuum becomes increasingly dominant
as the magnitudes of the discrete peaks become negligible in comparison to the
smooth statistical decay contribution. In the experimental data, the larger
beam energies should therefore correspondingly be almost entirely dominated by
uncorrelated $\alpha$-particles.

Figure 5: Distribution of $\alpha$-particle energies in the center-of-mass
system from the EHF for a beam energy of a) 160 MeV, b) 280 MeV and c) 400
MeV. The yields from the population of different initial nuclei are shown in
different colors with ${}^{12}\mathrm{C}$ (red solid with circles),
${}^{16}\mathrm{O}$ (blue short-dashed with triangles), ${}^{20}\mathrm{Ne}$
(green dotted with diamonds) and ${}^{24}\mathrm{Mg}$ (magenta dot-dashed with
squares) with the lines guiding the eye between the data points.

###  IV.2 Multi-particle break-up

To model the dissociation of the compound nucleus into a large number of
particles, the Fermi break-up model Carlson  _et al._ (2012) was also
employed. This calculation is driven by modeling the decay according to
Fermi’s golden rule Fermi (1950):

$R_{if}=\frac{2\pi}{\hbar}|A_{if}|^{2}\omega_{f},$ (2)

with $|A_{if}|^{2}$ corresponding to the transition matrix element between the
initial state $i$ and the final state $f$. The important factor in the Fermi
break-up calculations originates from the $\omega_{f}$ term which describes
the density of states. The phase space available for a decay of the system of
mass number $A$ into $n$ fragments is given by Block (1956):

$\displaystyle\omega_{f}=$
$\displaystyle\prod_{b=1}^{n}(2s_{b}+1)\prod_{j=1}^{k}\frac{1}{n_{j}!}\left(\frac{r^{3}_{0}A}{6\pi^{2}\hbar^{3}}\right)^{n-1}$
(3)
$\displaystyle\left(\frac{1}{\sum_{b=1}^{n}m_{b}}\prod_{b=1}^{n}m_{b}\right)^{\frac{3}{2}}\frac{(2\pi)^{3(n-1)/2}}{\Gamma(3(n-1)/2)}E_{\mathrm{kin}}^{\frac{3n}{2}-\frac{5}{2}},$

with $s_{b}$ being the spin of particle $b$ and $n_{j}$ is the number of
particles of type $j$. There is a very strong dependence therefore on $n$,
particularly in the term
$\left(\frac{r^{3}_{0}A}{6\pi^{2}\hbar^{3}}\right)^{n-1}$ which describes the
partition of the phase space of the decay. Additionally, the energy available
in the decay also provides a very large density of states for high $n$ decays.
For a break-up of a system into 7-$\alpha$, one generates an 8th order power
dependence on the kinematically available energy. This is usually offset by a
correspondingly smaller energy available for higher multiplicity decays. In
our case, for a decay to provide 1 $\alpha$-particle, 10.0 MeV is needed to
break-up the system whereas for 7-$\alpha$ particles one requires 38.5 MeV.
Therefore, for a state with an excitation energy of $\sim$ 40 MeV, the small
available energy is therefore prohibitive to the 7-$\alpha$ break-up and
preferential towards a smaller $n$ break-up.

To investigate the dynamics of this break-up mode, the phase space for the
dissociation into different $\alpha$-conjugate nuclei was investigated, both
in their ground state and for the predicted $\alpha$-gas states above the
N-$\alpha$ threshold. The $\alpha$-gas states in ${}^{8}\mathrm{Be}$ and
${}^{12}\mathrm{C}$ have been given energies of 0.0 and 7.65 respectively.
These reaction modes are summarized in Table IV.2. Due to the experiment being
a high excitation energy compound nucleus reaction, there is a large amount of
energy present which means the system prefers a higher $n$ body break-up. To
demonstrate this, the sum of the binary fission modes to the ground state
(i.e. p+${}^{27}\mathrm{Al}$, ${}^{11}\mathrm{C}$+${}^{17}\mathrm{O}$ etc.)
was compared to the sum of those modes in Table IV.2. These binary fission
modes were demonstrated to only constitute 9.6$\%$, 0.2$\%$ and 0.02$\%$ for
the 160, 280, and 400 MeV beam energies respectively. For the lowest beam
energy, the binary fission can be observed to be important however for the
higher beam energies, the majority of the phase space is associated with $n>2$
decays, a fact which will be important later.

Table 2: Fermi break-up modes of $\alpha$-conjugate and the associated number
of break-up particles. The final column details the number of
${\alpha}$-particles such a decay would yield in the final state (due to
unbound resonances).

Decay path |  Break-up |
---|---|---
|  $\boldsymbol{\alpha}$-particles |
24Mg + $\alpha$ | 2 | 1
20Ne + ${}^{8}\mathrm{Be}$ | 2 | 2
${}^{16}\mathrm{O}$+${}^{12}\mathrm{C}$ | 2 | 0
${}^{16}\mathrm{O}$+${}^{12}\mathrm{C}(0_{2}^{+})$ | 2 | 3
20Ne+2$\alpha$ | 3 | 2
${}^{16}\mathrm{O}$+${}^{8}\mathrm{Be}$+$\alpha$ | 3 | 3
${}^{12}\mathrm{C}$+${}^{12}\mathrm{C}$+$\alpha$ | 3 | 1
${}^{12}\mathrm{C}(0_{2}^{+})$+${}^{12}\mathrm{C}(0_{2}^{+})$+$\alpha$ | 3 | 7
${}^{12}\mathrm{C}$+${}^{8}\mathrm{Be}$+${}^{8}\mathrm{Be}$ | 3 | 4
${}^{12}\mathrm{C}(0_{2}^{+})$+${}^{8}\mathrm{Be}$+${}^{8}\mathrm{Be}$ | 3 | 7
${}^{12}\mathrm{C}$+${}^{8}\mathrm{Be}$+2$\alpha$ | 4 | 4
${}^{12}\mathrm{C}(0_{2}^{+})$+${}^{8}\mathrm{Be}$+2$\alpha$ | 4 | 7
${}^{8}\mathrm{Be}$+${}^{8}\mathrm{Be}$+${}^{8}\mathrm{Be}$+$\alpha$ | 4 | 7
${}^{16}\mathrm{O}$+3$\alpha$ | 4 | 3
${}^{12}\mathrm{C}$+4$\alpha$ | 5 | 4
${}^{12}\mathrm{C}(0_{2}^{+})$+4$\alpha$ | 5 | 7
${}^{8}\mathrm{Be}$+${}^{8}\mathrm{Be}$+3$\alpha$ | 5 | 7
${}^{8}\mathrm{Be}$+5$\alpha$ | 6 | 7
7$\alpha$ | 7 | 7
Figure 6: Phase space for the different break-up partitions in Table IV.2
showing how modifying the beam energy (160 MeV - green inverted triangles, 280
MeV - red triangles, 400 MeV - blue squares) changes the preference for a
different number of break-up particles.

This high preference for larger $n$ particle decay can also be seen in Fig. 6
where the phase space against the number of break-up particles is shown for
the three beam energies. The largest phase space is associated with $n$ = 6
for the 280 and 400 MeV beam energy but a smaller $n$ = 4 is preferred at the
lowest beam energy of 160 MeV.

An underlying assumption of the $\alpha$-condensate wave function is that the
dissociation of an $\alpha$-gas into a set of sub-units which are also
$\alpha$-gas states has the same transition matrix Kokalova  _et al._ (2006).
This can therefore be applied to the Fermi golden rule to compare the expected
strength of the different break-up paths with a final state comprising of 7
$\alpha$-particles, i.e. those where the sub-units are expected to be
$\alpha$-gas states. This prediction can then be compared to the experimental
data as a probe of the $\alpha$-condensate. In order to accurately compare the
predictions from the Fermi break-up model to those from the experimental data,
detector effects and the efficiency of these different channels needed to be
well understood to make a meaningful comparison on the efficiency-corrected
data. For this purpose, a Monte Carlo simulation was performed to bridge the
connection between the predictions of the EHF and the Fermi break-up models to
the data.

###  IV.3 Monte Carlo simulations

As discussed above, the detector effects (resolution, detector coverage and
particle pileup) play an important part in the comparison between experimental
and theoretical data. To model the particle break-up, the RESOLUTION8.1 code
Curtis  _et al._ (1995, 1996) was used which models sequential break-up
reactions. The output from the simulation allows for “truth-level” (i.e. the
true values that were generated from the Monte Carlo) momentum vectors for the
different reaction products to be generated. These truth level vectors were
then passed through a wrapper which simulates the CHIMERA and FARCOS position
and energy resolution which was then fed through the analysis code. As well as
verifying the rigor of the analysis code, which requires care in this type of
high-multiplicity environment, this also allows for an investigation into the
reaction mechanisms seen in the data and the efficiency correction needed to
compare to the 7-$\alpha$ decay channels calculated in Section IV.2 using the
Fermi break-up model. It is worth noting that the Monte Carlo simulation only
models sequential decay, whereas the Fermi break-up is a single stage decay.
The kinematics of the two systems are comparable as both produce an isotropic
spray of particles in the center of mass system. There are 11 different decay
paths through the $\alpha$-gas states which were used to compare to the data
which are tabulated in Table 3.

Table 3:  Different decay paths through $\alpha$-gas type states until the system has decayed to 8Be or an $\alpha$-particle. Only the decay of the excited states marked with a star are shown at each subsequent stage. Label | Path
---|---
1 |  ${}^{24}\mathrm{Mg}^{\star}{+}\alpha{\rightarrow}^{20}\mathrm{Ne}^{\star}{+}\alpha{\rightarrow}$${}^{16}\mathrm{O}^{\star}{+}\alpha{\rightarrow}$${}^{12}\mathrm{C}^{\star}{+}\alpha{\rightarrow}$${}^{8}\mathrm{Be}{+}\alpha$
2 |  ${}^{24}\mathrm{Mg}^{\star}{+}\alpha{\rightarrow}$${}^{20}\mathrm{Ne}^{\star}{+}\alpha{\rightarrow}$${}^{16}\mathrm{O}^{\star}{+}\alpha{\rightarrow}$${}^{8}\mathrm{Be}{+}$${}^{8}\mathrm{Be}$
3 |  ${}^{24}\mathrm{Mg}^{\star}{+}\alpha{\rightarrow}$${}^{20}\mathrm{Ne}^{\star}{+}\alpha{\rightarrow}$${}^{12}\mathrm{C}^{\star}{+}^{8}\mathrm{Be}{\rightarrow}$${}^{8}\mathrm{Be}{+}\alpha$
4 |  ${}^{24}\mathrm{Mg}^{\star}{+}\alpha{\rightarrow}$${}^{16}\mathrm{O}^{\star}{+}$${}^{8}\mathrm{Be}{\rightarrow}$12C${}^{\star}{+}\alpha{\rightarrow}$${}^{8}\mathrm{Be}{+}\alpha$
5 |  ${}^{24}\mathrm{Mg}^{\star}{+}\alpha{\rightarrow}$${}^{16}\mathrm{O}^{\star}{+}$${}^{8}\mathrm{Be}{\rightarrow}$${}^{8}\mathrm{Be}{+}^{8}\mathrm{Be}$
6 |  ${}^{24}\mathrm{Mg}^{\star}{+}\alpha{\rightarrow}$${}^{12}\mathrm{C}^{\star}{+}$${}^{12}\mathrm{C}^{\star}{\rightarrow}$${}^{8}\mathrm{Be}{+}^{8}\mathrm{Be}$
7 |  ${}^{20}\mathrm{Ne}^{\star}{+}^{8}\mathrm{Be}{\rightarrow}$${}^{16}\mathrm{O}^{\star}{+}\alpha{\rightarrow}$${}^{12}\mathrm{C}^{\star}{+}\alpha{\rightarrow}^{8}\mathrm{Be}{+}\alpha$
8 |  ${}^{20}\mathrm{Ne}^{\star}{+}^{8}\mathrm{Be}{\rightarrow}$${}^{16}\mathrm{O}^{\star}{+}\alpha{\rightarrow}$${}^{8}\mathrm{Be}{+}$${}^{8}\mathrm{Be}$
9 |  ${}^{20}\mathrm{Ne}^{\star}{+}^{8}\mathrm{Be}{\rightarrow}$${}^{12}\mathrm{C}^{\star}{+}^{8}\mathrm{Be}{\rightarrow}^{8}\mathrm{Be}{+}\alpha$
10 |  ${}^{16}\mathrm{O}^{\star}{+}^{12}$C${}^{\star}{\rightarrow}(^{12}\mathrm{C}^{\star}{+}\alpha){+}(^{8}\mathrm{Be}{+}\alpha){\rightarrow}$${}^{8}\mathrm{Be}{+}\alpha$
11 |  ${}^{16}\mathrm{O}^{\star}{+}^{12}\mathrm{C}^{\star}{\rightarrow}(^{8}\mathrm{Be}{+}$${}^{8}\mathrm{Be}){+}(^{8}\mathrm{Be}{+}\alpha$)

##  V Experiment

By virtue of the high incident energy involved in this experiment, the
probability of compound nucleus formation is reduced and other direct reaction
mechanisms may become prevalent. To separate and quantify the effect of these
reactions, the FARCOS detector, due to its low angle and high angular
resolution, was used to investigate whether particles hitting this detector
originate from a direct reaction or sequential decay.

Figure 7: Yields in the FARCOS detector for $E_{b}$ = 400 MeV with the
different loci labeled. Figure 8: Yields in the FARCOS detector for $E_{b}$ =
280 MeV with the different loci labeled. Figure 9: Yields in the FARCOS
detector for $E_{b}$ = 160 MeV with the different loci labeled.

###  V.1 FARCOS particle yields

The three beam energies show different particle yields (as seen in Figs. 7, 8
and 9), mainly as a result of the differing excitation energy of the compound
nucleus due to the change in beam energy. Two-body decays from a higher
initial excitation energy are much more likely to produce daughter nuclei with
an excitation energy above their particle decay thresholds therefore
undergoing secondary decay processes. Additionally, as the energy of the
system increases, the phase space for 3 body decays becomes significantly
larger. The FARCOS detectors give a (polar) coverage between $2^{\circ}$ and
$8^{\circ}$ which allows for the low angle behavior to be investigated where
direct reactions are expected to be more dominant. To examine this, one can
firstly examine the data taken with a beam energy ($E_{b}$) of 400 MeV (Fig.
7).

These data show a large swathe of particles produced including nuclei heavier
than the beam demonstrating these results are not solely due to beam break-up.
In particular, the ${}^{4}\mathrm{He}$ locus is observed to be extremely
strong with large contributions also arising from ${}^{12}\mathrm{C}$ and
${}^{16}\mathrm{O}$. By reconstructing the missing momentum, the possibility
that these particles originate from a binary fission mode to the ground-state
nuclei can be investigated by examining the Q-value for the reaction. As
suggested by the Fermi break-up calculations, this mode is not seen in the
data apart from a small component for
${}^{12}\mathrm{C}(^{16}\mathrm{O}{,}^{10}\mathrm{B}){{}^{18}}\mathrm{F}$
(with around $1/3$ of the measured ${}^{18}\mathrm{F}$ reproducing the
expected Q-value of -17.7 MeV) and the elastic scattering channel which can
clearly be observed where the ${}^{16}\mathrm{O}$ scattered beam is labeled in
the figure.

The data from the 280 MeV beam energy (Fig. 8) also show a very similar
behavior with a strong yield in the $\alpha$-conjugate nuclei. The beam energy
here is insufficient for the reaction products heavier than the beam to be
detected by $\Delta$E-E (they are stopped in the silicon 300 $\mu$m stage). As
with the 400 MeV beam energy data, there is also a small component for the
${}^{12}\mathrm{C}(^{16}\mathrm{O}{,}^{10}\mathrm{B}){{}^{18}}\mathrm{F}$
break-up measured from the ${}^{10}\mathrm{B}$ although this corresponds to a
much smaller fraction of $<5\times 10^{-3}\%$ of the total measured
${}^{10}\mathrm{B}$. Other direct break-up components are also absent in these
data.

In the 160 MeV beam energy data (Fig. 9), the degree to which the
${}^{4}\mathrm{He}$ yield dominates is largely increased. Despite the lower
beam energy meaning that isotopes heavier than carbon cannot be identified
using this method, there is a very small contribution from ${}^{12}\mathrm{C}$
($<0.03\%$) with the yield from ${}^{6/7}\mathrm{Li}$ becoming more dominant
to constitute $\sim 3\%$ of the total counts.

These results mirror very strongly the predictions from the EHF calculations
where the $\alpha$-decay mode was seen to be more important at the lowest beam
energies where the proton and neutron decay modes are “frozen-out” because of
the larger Q-value required. The Fermi break-up model also predicted a
reasonable contribution for direct 2-body decay at a beam energy of 160 MeV
but, as with the higher beam energies, no such decay channel is observed.

In addition to the decay of the compound nucleus, the possibility of these
particles originating from the beam break-up was investigated by calculating
the Q-value. These Q-value plots also demonstrate that the detected particles
do not originate from the break-up of the beam. This is therefore indicative
that the observed reaction mechanism is related to a higher multiplicity
reaction such as that of sequential decay or direct multi-particle break-up.

##  VI CHIMERA particle yields

As well as investigating the low angle particle yields, the high coverage of
CHIMERA can be utilized to provide information about how the yield varies as a
function of angle. This can give a much better signature for the dominance of
compound nucleus formation which, as mentioned previously, will give an
isotropic spray of particles in the center-of-mass. The CHIMERA detector
covers from $6^{\circ}$ and the forward rings were used to study how the
relative yields change up to $30^{\circ}$. The results for the three beam
energies show the same features. Therefore, only the results for the 400 MeV
data are shown in Figs. 10 and 11 where missing telescopes in the CHIMERA
rings were corrected for by considering the azimuthal solid angle coverage.
For the lower beam energies there is a threshold effect observed. This occurs
where the nuclei are created with a lower average energy in the compound
nucleus decay and therefore may have insufficient energy to pass through the
300 $\mu$m silicon stage to be detected using $\Delta$E-E in CHIMERA. No
correction is made for this effect.

Figure 10: Yields per solid angle in the CHIMERA detector for $E_{b}$ = 400
MeV as a function of angle for the non $\alpha$-conjugate nuclei in comparison
to ${}^{4}\mathrm{He}$. Figure 11: Yields per solid angle in the CHIMERA
detector for $E_{b}$ = 400 MeV as a function of angle for the
$\alpha$-conjugate nuclei.

When examining the angular dependence of the non $\alpha$-conjugate nuclei in
Fig. 10, it can clearly be seen that all the nuclei follow the same behavior
with a small decrease in yield with increasing angle. The amount of
${}^{4}\mathrm{He}$ measured far exceeds that of lithium, beryllium, and boron
which all have roughly the same yields. This similar situation is also
mirrored when examining the contributions from the $\alpha$-conjugate nuclei
and their associated excited states are discussed in Section VIII. The
${}^{8}\mathrm{Be}$(g.s) is observed to be measured at a level consistent with
the ${}^{12}\mathrm{C}$ when reconstructed from 2 $\alpha$-particles. When the
efficiency of measuring both these $\alpha$-particles (compared to needing to
detect only one particle for ${}^{12}\mathrm{C}$ rather than reconstructing
${}^{8}\mathrm{Be}$ from 2 $\alpha$-particles), the strength of
${}^{8}\mathrm{Be}$(g.s) exceeds that of ${}^{12}\mathrm{C}$(g.s).
Reconstruction of events where 3 $\alpha$-particles are detected, the strength
of contributions from the Hoyle state (${}^{12}\mathrm{C}(0_{2}^{+})$) and a
broad contribution at 9.6 MeV (${}^{12}\mathrm{C}(3_{1}^{-})$) also show the
same fall-off with angle as the ground state nuclei detected showing the
reaction mechanism for their population is common.

The total strength of these yields in CHIMERA were then compared to the
predictions from the EHF calculations by normalizing the experimental yield to
the mass-4 EHF cross section. The results for this are seen in Figs. 12, 13
and 14. The influence of the threshold effect can be observed here for the 160
MeV beam energy data (Fig. 12) where the experimental yield for the heavier
mass nuclei is lower than expected for mass 9, 10, and 11 where the TOF PID
was not possible. Aside from this discrepancy due to experimental effects,
there is a good agreement between the predictions from the EHF and the
experimental results for the three beam energies although the yield for
${}^{8}\mathrm{Be}$ can clearly be seen to exceed the predictions for all beam
energies. When taking into account the reduced efficiency for this path by
virtue of needing to reconstruct ${}^{8}\mathrm{Be}$ from 2
$\alpha$-particles, this increase over the sequential decay predictions is
even larger showing an enhancement inconsistent with the EHF calculations.
This effect has been observed previously in other experiments Kokalova  _et
al._ (2005); Kokalova (2003) in the
${}^{13}\mathrm{C}(^{18}\mathrm{O},{{}^{8}\mathrm{Be}})$ and
${}^{24}\mathrm{Mg}(^{28}\mathrm{Si},^{12}\mathrm{C}^{\star})$ reactions
employing charged-particle and gamma-decay spectroscopy in tandem. In the
latter of these, the $\gamma$-rays associated with ${}^{36}\mathrm{Ar}$ were
observed corresponding to the removal of 4 $\alpha$-particles. This 4
$\alpha$-particle removal was not predicted to be observed in the EHF model
for the ${}^{52}\mathrm{Fe}$ compound nucleus decay at that energy.

Figure 12: Yields in the CHIMERA detector for $E_{b}$ = 160 MeV in comparison
to the EHF calculation predictions. Errors appear in the center of the bars
but may be too small to be visible. Figure 13: Yields in the CHIMERA detector
for $E_{b}$ = 280 MeV in comparison to the EHF calculation predictions. Errors
appear in the center of the bars but may be too small to be visible. Figure
14: Yields in the CHIMERA detector for $E_{b}$ = 400 MeV in comparison to the
EHF calculation predictions. Errors appear in the center of the bars but may
be too small to be visible.

Overall, the majority of the reaction products can be seen to originate via
the compound nucleus formation. For the
${}^{12}\mathrm{C}+{{}^{16}\mathrm{O}}$ exit channel, there will undoubtedly
be a component of inelastic scattering and other direct reaction mechanisms
which cannot be easily disentangled from the compound nucleus formation. The
angular coverage of CHIMERA + FARCOS to large lab angles means that the
contribution of this mechanism to the data can be assigned as negligible to
the first-order. However, care must be taken to ensure that the origins of any
states observed in the $\alpha$-conjugate nuclei (i.e. did they arise from
direct/compound nucleus formation) is taken into consideration to validate the
assumptions made in Sec. IV. This will be examined later in Sec. IX where an
attempt to understand the reaction mechanism used to generate the events of
interest is made.

##  VII Multiplicity of alpha-particles

It was shown in Section II that an enhancement in the production of
$\alpha$-particles in comparison with that expected from theory is a potential
signature of an $\alpha$-gas due to the modification to the Coulomb barrier.
The calculations from the EHF code showed a large decrease is expected as the
beam energy (and therefore excitation energy of the compound nucleus) is
increased. Conversely, the Fermi break-up model showed that as the energy in
the system increases, large $n$-body break-ups are preferred as a consequence
of the increased phase space. To firstly demonstrate the compound nucleus
description of the reaction was correct, the energy distribution of the
$\alpha$-particles measured in the experiment was examined, the results of
which are visible in Fig. 15. It can clearly be seen that a smooth
distribution of energies is seen, a key signature that the mechanism seen is
indeed due to compound nucleus formation. Additionally, the discrete peaks
predicted from the EHF code in the center of mass are not visible. This is
either: conversion to the lab frame smoothing these discrete peaks or the
statistical emission of $\alpha$-particles having a larger effect than that
predicted within the sequential decay framework. There are a few peaks which
are visible on top of the broad continuum, most noticeably around 5 MeV, just
above the threshold. By separately analyzing these events, it was demonstrated
that (despite a larger impurity in the TOF PID around this energy due to the
inclusion of particles from previous beam pulses) this increase corresponds to
genuine events where particles are produced at backwards angles in the center
of mass.

Figure 15: Kinetic energy distribution of the $\alpha$-particles measured in
CHIMERA and FARCOS. The discontinuities around 24 MeV are due to a gap in the
TOF PID and the $\Delta$E-E PID for varying thicknesses of the silicon
($\Delta$E) detector.

As well as observing the distribution of the $\alpha$-particle energies, the
more important probe is that of the $\alpha$-particle multiplicities. This
involves the $\alpha$-particles observed in FARCOS (via $\Delta$E-E PID) and
CHIMERA (via $\Delta$E-E and TOF PID), the results of which can be seen in
Fig. 16. For all three beam energies, a very similar behavior is seen. As the
multiplicity increases, the yield decreases by roughly a factor of 10 for each
additional $\alpha$-particle. The full multiplicity events are seen for all
beam energies and the data from the 400 MeV beam energy also demonstrate
multiplicity 8 events. There are three possible reasons for this extraneous
yield: firstly, it is possible that contaminants in the target are also
causing a contribution from the
${}^{16}\mathrm{O}(^{16}\mathrm{O},$${}^{32}\mathrm{S}^{\star})$ reaction.
This is unlikely as any contaminant is expected to constitute a very small
percentage of the target and is therefore insufficient to provide the observed
level of 8 $\alpha$-particle events. A second explanation is the impurity of
the PID classifying non $\alpha$-particle events incorrectly as
$\alpha$-particles. At low energies, there is an observed impurity in the TOF
PID (Fig. 3) although as the event multiplicity is increased, the effect of
this impurity is decreased therefore suggesting this is not the dominant
effect. The final possibility is that event pileup has a small effect
sufficient to modify the normalized single event-multiplicity $\mathbf{M}$ to
the observed raw event multiplicity $\mathbf{N}$ in Fig. 16. By calculating
the probability of pileup for a single event, this effect can be removed.

Figure 16: Raw $\alpha$-particle multiplicity for the three beam energies of
160 (red), 280 (blue), and 400 (green) MeV.

The single event-multiplicity $\mathbf{M}$ can be related to the raw event
multiplicity $\mathbf{N}$ by the pileup probability $\gamma_{\mathrm{pileup}}$
by:

$\displaystyle\mathbf{N}=$
$\displaystyle\mathbf{N}_{\mathrm{nopileup}}+\mathbf{N}_{\mathrm{pileup}},$
(4) $\displaystyle=$
$\displaystyle(1-\gamma_{\mathrm{pileup}})\mathbf{M}+\gamma_{\mathrm{pileup}}\sum_{ij}^{7}\delta_{i+j,k}\left(\mathbf{M}_{i}\mathbf{M}_{j}\right).$
(5)

The Kronecker delta selects combinations of events of multiplicity $i$ and $j$
that combine to make an event of total multiplicity $k$. To obtain
$\mathbf{M}$ from $\mathbf{N}$, a fit was performed with $\mathbf{N}$ being a
free parameter which effectively inverts Eq. 5. The pileup probabilities are
extracted from the cross section given from the EHF code, the beam current
used for each beam energy, the target thickness and the cyclotron time period.
These probabilities are $4.0\times 10^{-4}$, $8.0\times 10^{-5}$ and
$3.2\times 10^{-4}$ for the beam energies of 160, 280, and 400 MeV
respectively. These values are low, albeit sufficient, to reproduce the
observed 8 $\alpha$-particle events for the 400 MeV beam energy. The
modification of $\mathbf{M}$ from $\mathbf{N}$ is very small, particularly for
the smaller multiplicities. To compare the experimental multiplicity to the
predictions from the EHF and Fermi break-up calculations, the detector
efficiency must also be taken into consideration. For a compound nucleus
formation, the products are emitted isotropically in the center of mass,
therefore, the probability to detect $i$ $\alpha$-particles in an event where
$j$ $\alpha$-particles are emitted is given by the binomial probability:

$\displaystyle\epsilon_{i,j}={{j}\choose{i}}p^{i}(1-p)^{j-i},$ (6)

with the probability of detection $p$ taking the value of 0.28 which best
reproduces the data and is also the solid angle coverage fraction. The single
event $\mathbf{M}$ is then:

$\displaystyle\mathbf{M}=\mathbf{\epsilon}\mathbf{M^{\prime}},$ (7)

with the elements of the efficiency matrix $\mathbf{\epsilon}$, being those
given in Eq. 6 and $\mathbf{M^{\prime}}$ being the true multiplicity.

Inversion of the efficiency matrix then gives the true multiplicity as:

$\displaystyle\mathbf{M^{\prime}}=\mathbf{\epsilon}^{-1}\mathbf{M}.$ (8)

The result of this true multiplicity inversion is given in Fig. 17. This
inversion is such that there is no restriction of the elements of
$\mathbf{M^{\prime}}$ being positive. This only occurs for the 400 MeV data
where the multiplicity for 6 $\alpha$-particles has a small negative value
($-2.6\times 10^{-3}$).

Figure 17: True $\alpha$-particle multiplicity for the three beam energies of
160 (red), 280 (blue), and 400 (green) MeV. For each beam energy the total
number of counts is normalized to 1.

For all beam energies, the multiplicity can be seen to well exceed that
predicted by the sequential decay model (Fig. 4) with the multiplicities
peaking around 2 or 3 $\alpha$-particles with a reasonable fraction of events
generating a 7 $\alpha$-particle final state. The events with 0
$\alpha$-particles still constitute the largest fraction of events for all
three energies. The most important result here is that the multiplicity for
the three beam energies show a very similar behavior, far in contrast to the
predictions from the EHF which demonstrated a strong dependence on the beam
energy.

##  VIII Reconstruction

In order to understand the reaction, the excited states populated during the
decay process can be examined. As discussed in Section IV.2, a signature of an
$\alpha$-gas state can be extracted from the 7-$\alpha$ break-up channels.
Therefore, in events with sufficient multiplicity, the N-$\alpha$ excitation
energies were calculated from reconstruction of the $N$ $\alpha$-particles.

$E_{x}=\sum_{i}E_{i}-\frac{(\sum_{i}\vec{p_{i}})^{2}}{2\sum_{i}m_{i}}-Q.$ (9)

The results from these excitation spectra are separated by the nucleus
studied.

###  VIII.1 Beryllium-8

The 2-$\alpha$ excitation function is shown in Fig. 18 where the ground-state
energy can be clearly seen at 92 keV (the Q-value of the break-up is omitted
here in order to clearly see the ground-state). There is also a broad
contribution at $\sim$ 3 MeV which may be ascribed to the first excited state,
a $2_{1}^{+}$ resonance at 3.03 MeV with a large width of $\Gamma$ = 1.5 MeV.
In order to understand whether this yield is correctly designated as arising
from this broad state, event-mixing was used Charity  _et al._ (1995) which
sought to separate the contribution from correlated and uncorrelated events.
Here, the scale of the event-mixed events was set such that the total integral
of the spectra were the same. Event-mixing takes particles from different
events and merges them into a new event where they pass through the same cuts
and reconstruction as the non-mixed data. By taking $\alpha$-particles from
different events, one can therefore understand the importance of uncorrelated
events (whether from statistical decay processes from the continuum or from
mixing of $\alpha$-particles from different parent nuclei).

Figure 18: Excitation function for ${}^{8}\mathrm{Be}$ (red) and the
contribution from the event-mixing (blue) for a beam energy of 160 MeV.

This contribution above 2 MeV can be seen to be well reproduced by the event-
mixing process demonstrating the contribution from the first excited state of
${}^{8}\mathrm{Be}$ is not the cause of this bump. The three beam energies of
160, 280, and 400 MeV all show this identical behavior with the ground-state
being well populated and any higher energy bumps described by uncorrelated
$\alpha$-particles. The dip between 0.5 MeV and 1.5 MeV can be understood as
arising from a ‘Coulomb hole’ where $\alpha$-particles with a small relative
energy are suppressed by the presence of the Coulomb barrier (see Charity  _et
al._ (1995)) which cannot be accounted for by mixing.

###  VIII.2 Carbon-12

Taking events where the $\alpha$-particle multiplicity is greater than or
equal to 3, resonances in ${}^{12}\mathrm{C}$ can be reconstructed. To further
reduce the background, events were taken where 2 of these 3 $\alpha$-particles
can also be reconstructed to form ${}^{8}\mathrm{Be}$(g.s). The results from
these two different possibilities are shown in Fig. 19 for the three beam
energies.

Figure 19: Excitation function for ${}^{12}\mathrm{C}$ showing the results for
the three beam energies of 160, 280, and 400 MeV as well as those events where
the intermediate decay product corresponds to the ${}^{8}\mathrm{Be}$(g.s).

Once again, the results for the three beam energies are very similar. There is
a very strong population of two resonances above the triple-alpha threshold.
The first of these is at 7.65 MeV which is the well known Hoyle state, a
resonance that is of interest as discussed above. This is well populated in
all three beam energies and is easily separable from the higher energy peak
seen at 9.7 MeV. This peak is much broader than the Hoyle state suggesting
this contribution is not solely due to the $3_{1}^{-}$ state at 9.64 MeV which
has a measured width of 48 keVKokalova  _et al._ (2013a). Using the Monte
Carlo simulations to study the expected resolution response as a function of
excitation energy (and scaling the expected response to the experimental width
of the Hoyle state), the width of this state averaged over the three beam
energies is $\Gamma=600(60)$ keV. This is a value which is incompatible with
the $3_{1}^{-}$ and may therefore correspond to a contribution from the
rotational excitation of the Hoyle state, $2_{2}^{+}$, which has been measured
to have an excitation energy of 9.75(15) MeV with a width of 750(150) keV
Freer  _et al._ (2009, 2012) from inelastic scattering experiments. Reanalysis
of the photodissociation breakup reaction ${}^{12}\mathrm{C}(\gamma,3\alpha)$
Zimmerman  _et al._ (2013) where the $2^{+}$ was unambiguously identified gave
a much larger value of 1.6(1) MeV Freer  _et al._ (2018).

As with ${}^{8}\mathrm{Be}$, event-mixing can be used to examine the smooth
contribution above these two resonances which shows a shift in excitation
energy with increasing beam energy. Figure 20 shows the result of this process
indicating the broad contribution can be described by event mixing. While the
resonances are not reproduced as one would expect (showing they are due to
correlated $\alpha$-particles), the double humped continuum is well matched by
the event mixing contribution apart from a small excess around 25-30 MeV.

Figure 20: Event mixing (blue) applied to the 3-$\alpha$ excitation function
for a beam energy of 160 MeV scaled to give the same total area as the data.
The smooth continuum in ${}^{12}\mathrm{C}$ seen in the data (red) can be
described by uncorrelated $\alpha$-particles as demonstrated by event mixing.

###  VIII.3 Oxygen-16

Following the formulations described for ${}^{8}\mathrm{Be}$ and
${}^{12}\mathrm{C}$, taking events with a multiplicity greater than or equal
to 4, the excited states in ${}^{16}\mathrm{O}$ which decay via the 4$\alpha$,
${}^{8}\mathrm{Be}$+2$\alpha$, ${}^{12}\mathrm{C}(0_{2}^{+}){+}\alpha$, and
${}^{8}\mathrm{Be}{+}^{8}\mathrm{Be}$ channels were examined. While the
experimental resolution in this channel is expected to be $\sim$ few 100 keV
(depending on the exact excitation energy and the decay path used to produce
it), any strong resonances should stick out above the smooth background. The
first two of these decay channels can be seen in Figure IV.2. A broad double
humped structure can be seen for all the three beam energies (albeit at
different excitation energies) in the 4-$\alpha$ channel however no narrow
peaks are evident in the data. The bottom part of Fig. IV.2 shows the
excitation function near the 4-$\alpha$ barrier where the counts can be seen
to very quickly approach zero around 15 MeV with no obvious enhancement in
this region. One would not expect to see any strong ‘traditionally’
$\alpha$-clustered states until over the Coulomb barrier at $\sim$ 18 MeV. To
understand the yield around this important area, event mixing was employed to
demonstrate the importance of uncorrelated $\alpha$-particles.

Figure 21: Excitation for ${}^{16}\mathrm{O}$ for the three beam energies
showing all decays to 4 $\alpha$-particles in lin-lin (top) for a wide
excitation range and log-lin (bottom) for near the 4-$\alpha$ threshold. Also
shown (top) is the contribution where a pair of the 4 $\alpha$-particles form
${}^{8}\mathrm{Be}$(g.s).

Figure 22 shows the event mixing for the 160 MeV beam energy. This can
describe the seen distribution extremely well, particularly at lower
excitation energies where the smooth decrease in counts towards the 4-$\alpha$
threshold is well reproduced. The second peak in the distribution is not
reproduced by this mixing suggesting this corresponds to a correlated
contribution to the total excitation spectrum. This is reproduced for all
three beam energies suggesting the observed enhancement is not a particularly
well populated state but a manifestation of the break-up mechanism as the
excitation energy changes with the different beam energy.

Figure 22: Event-mixing (blue) for ${}^{16}\mathrm{O}$ for a beam energy of
160 MeV in comparison to the experimental data (red). The event-mixing
describes the majority of the structures seen apart from the secondary peak in
the data and the shape of the high energy tail.

To better understand the role of the continuum against genuine 4-$\alpha$
resonances, the ${}^{12}\mathrm{C}(0_{2}^{+}){+}\alpha$ and
${}^{8}\mathrm{Be}{+}^{8}\mathrm{Be}$ paths were also investigated where the
contribution from mismatching and uncorrelated $\alpha$-particles is expected
to be greatly reduced. The results from this can be seen in Fig. 23. In these
decay channels, evidence of structure starts to manifest. For the data from
beam energies of 160 and 400 MeV, there is evidence of a broad structure
around 19 MeV in the ${}^{12}\mathrm{C}(0_{2}^{+}){+}\alpha$ decay channel.
This region corresponds to well known states which have been demonstrated to
have a reasonable strength in several reactions employing different population
methods Curtis  _et al._ (2016). The resonance, which has a possible $6^{+}$
spin-parity assignment Curtis  _et al._ (2013), may therefore be
preferentially populated by the $(2J+1)$ spin dependency of the cross section.
Additionally, there is evidence of a broader component at 23-24 in the data
from the 280 and 400 MeV beam energies. The statistics above the background
here are bolstered by the occurrence of peaks at the same excitation energy
with two different beam energies which provides an increased statistical
significance. Therefore, there is evidence of population of resonances (or
most probably a collection of resonances which cannot be experimentally
resolved) in addition to the dominance of the non-correlated
$\alpha$-particles which constitute the larger fraction of the total
4-$\alpha$ events.

Figure 23: Excitation function in ${}^{16}\mathrm{O}$ for the clustered
${}^{12}\mathrm{C}(0_{2}^{+}){+}\alpha$ (solid lines) and
${}^{8}\mathrm{Be}{+}^{8}\mathrm{Be}$ decay paths (dashed) for the 160
(red/magenta), 280 (blue/cyan), and 400 (green/black) MeV data for the
(${}^{12}\mathrm{C}(0_{2}^{+}){+}\alpha$/${}^{8}\mathrm{Be}{+}^{8}\mathrm{Be}$)
channel. The violet arrows indicate a possible increased yield in the
${}^{12}\mathrm{C}(0_{2}^{+})+\alpha$ channel.

###  VIII.4 Neon-20

When increasing the size of the N-$\alpha$ system, the reduction in the
significance of resonant states as well as the dominance of uncorrelated
$\alpha$-particles and the continuum is noticeable. By examining the
5-$\alpha$ events corresponding to ${}^{20}\mathrm{Ne}$ in Fig. 24, it is
apparent this trend continues.

Figure 24: Excitation function of ${}^{20}\mathrm{Ne}$ for states decaying to
5 $\alpha$-particles.

The low excitation energy region for ${}^{20}\mathrm{Ne}$ demonstrates a
difference from the previously investigated nuclei however. The yield near the
5-$\alpha$ threshold at 19.2 MeV is not present. This region is (as with the
15-18 MeV region in ${}^{16}\mathrm{O}$) below the Coulomb barrier, hence, the
yield extends no lower than $\sim 24$ MeV, a result consistent with previous
experiments of the 5-$\alpha$ channel where the lowest energy state seen was
at 24.5 MeV in the
${}^{16}\mathrm{O}(\alpha,{{}^{12}\mathrm{C}(0_{2}^{+})})^{8}\mathrm{Be}$
channel Kokalova  _et al._ (2013b). Calculations from the EHF showed a large
population was expected in this region suggesting the quenching of the
5-$\alpha$ decay channel for states in this region, instead preferring other
decay paths such as ${}^{16}\mathrm{O}+\alpha$.

Figure 25: Excitation function of ${}^{20}\mathrm{Ne}$ for states decaying to
${}^{12}\mathrm{C}(0_{2}^{+}){+}^{8}\mathrm{Be}$.

The decays via highly clustered (potential $\alpha$-gas) decay channels, e.g.
${}^{12}\mathrm{C}(0_{2}^{+}){+}^{8}\mathrm{Be}$, also mirror this result with
only a small fraction of the 5-$\alpha$ events belonging to this decay
channel. The excitation energy for this decay path can be seen in Fig. 25. The
160, 280, and 400 MeV data show a possible broad contribution from one or
several states around 38 MeV matching a doublet of states seen previously
Freer  _et al._ (2005) although the statistics are limited here. An additional
possible state is visible (in the 160 MeV data only) with an energy of around
50 MeV. There is also a much larger contribution from higher excitation
energies outside of these peaks which is perhaps indicative that the
${}^{12}\mathrm{C}(0_{2}^{+})$ and ${}^{8}\mathrm{Be}$ do not originate mainly
from the decay of the ${}^{20}\mathrm{Ne}$ system but instead arise from
separate sequential decay stages or from a multi-particle break-up as
described by the Fermi break-up model. The three excitation functions for the
different beam energies can be seen to essentially be a stretching of the 160
MeV structure to higher energies corresponding to a larger phase space.

##  IX Origin of $\alpha$-clustered states

In order to better understand the reaction mechanisms which generate the
observed $\alpha$-clustered states, it is important to study where these
resonances originate from. To do this, the role of the two-body break-up of
the compound nucleus can be studied to understand the possible role of direct
reactions. By reconstructing the missing momentum following the measurement of
a given $\alpha$-conjugate state, the Q-value for the reaction can be
examined. These will be briefly discussed for each nucleus.

###  IX.1 Origins of beryllium-8

By selecting events where 2 $\alpha$-particles can be reconstructed to form
${}^{8}\mathrm{Be}$(g.s) and then using the missing momentum to form
${}^{20}\mathrm{Ne}$, the total Q-value for the reaction is calculated. For a
simple binary decay from the compound nucleus, the difference between the
expected Q-value and the observed Q-value is therefore due to the excitation
energy of the ${}^{20}\mathrm{Ne}$ which can therefore be observed without
measuring its decay. For ease of visualization, the negative Q-value is
plotted so excitation energy is congruent with the Q-value. The results for
this are shown in Fig. 26. An obvious path that would be a strong signature of
$\alpha$-condensation here is evidence of a peak at $\sim$ 20 MeV,
corresponding to an $\alpha$-gas state in ${}^{20}\mathrm{Ne}$ (which are
predicted to exist just above the N-$\alpha$ breakup thresholds). As discussed
above, a good signature that a state is an $\alpha$-condensate states is the
decay into two sub-units which are also $\alpha$-condensates. Such an
enhancement is clearly absent for the three beam energies which show an
extremely similar form for the different energies albeit translated by virtue
of the modification to the initial beam momentum. Such results are highly
indicative that there is no ${}^{8}\mathrm{Be}$(g.s.) +
${}^{20}\mathrm{Ne}^{\star}$ path with only $2\times 10^{-3}\%$ of the
measured ${}^{8}\mathrm{Be}$ reproducing the correct Q-value for the
${}^{12}\mathrm{C}(^{16}\mathrm{O},^{8}\mathrm{Be}$(g.s))${}^{20}\mathrm{Ne}$(g.s.)
break-up of $-$2.6 MeV for the 160 MeV beam energy (and no events for the 280
and 400 MeV beam energy data). The absence of the mutual ground state break-up
mirrors well the results of the EHF calculations which show this decay path is
expected to be very weak.

Figure 26: (Negative) Q-value for the break-up of the compound nucleus into
${}^{8}\mathrm{Be}$(g.s) + ${}^{20}\mathrm{Ne}$ by reconstructing the
${}^{20}\mathrm{Ne}$ from the measurement of the ${}^{8}\mathrm{Be}$(g.s) from
2 $\alpha$-particles.

###  IX.2 Origins of carbon-12

It was demonstrated in Section VIII.2 that a large number of the observed
${}^{8}\mathrm{Be}$(g.s) events corresponded to the decay of states in
${}^{12}\mathrm{C}$, therefore, examination of the Q-value to study the binary
fission mode ${}^{12}\mathrm{C}^{\star}{+}^{16}\mathrm{O}^{\star}$ (which is
also shown to be dominant in the EHF calculations) may indicate a large
fraction of the ${}^{12}\mathrm{C}(0_{2}^{+})$ and
${}^{12}\mathrm{C}(3_{1}^{-})$ originates from this decay mode. One may also
determine the amount of inelastic scattering (where the beam is left in its
ground state) in this way. The results of the (negative) Q-value from
reconstructing the missing ${}^{16}\mathrm{O}^{\star}$ from measuring the
Hoyle state from 3 $\alpha$-particles are shown in Fig. 27.

Figure 27: (Negative) Q-value for the break-up of the compound nucleus into
${}^{12}\mathrm{C}(0_{2}^{+})$ \+ ${}^{16}\mathrm{O}$ by reconstructing the
${}^{16}\mathrm{O}$ from the measurement of the ${}^{12}\mathrm{C}(0_{2}^{+})$
from 3 $\alpha$-particles.

For the three beam energies, two components can be seen. The lowest component
is a broad peak centered around 0 MeV with a width of 1.2 MeV for the 160 MeV
beam energy data but increasing to 12 MeV for a beam energy of 400 MeV. For
the 400 MeV data in particular, this lower energy component can be seen to
correspond well with the clustered 4-$\alpha$ yields seen in Fig. 23 which
terminate around 40 MeV. This is therefore highly indicative of a
${}^{12}\mathrm{C}(0_{2}^{+}){+}^{16}\mathrm{O}^{\star}$ component to the
compound nucleus decay or alternatively a contribution from inelastic
scattering. These events are clearly a small fraction of the total events in
the plot for each beam energy therefore show that (even if all of these events
are inelastic scattering rather than compound nucleus decay) the direct
contribution is small in comparison to the other observed paths. The events
that lie in the secondary peak, which has a smoothly increasing structure
followed by an abrupt drop-off (at the largest value seen for each beam
energy), correspond to paths which are not
${}^{12}\mathrm{C}(0_{2}^{+}){+}^{16}\mathrm{O}^{\star}$ and therefore cannot
be inelastic scattering. These have a form well reproduced by Monte Carlo
simulation of other reactions paths such as the sequential $\alpha$-decay via
${}^{24}\mathrm{Mg},^{20}\mathrm{Ne},^{16}\mathrm{O},{{}^{12}\mathrm{C}(0_{2}^{+})}$
or multi-particle break-up.

The same formulation can be used to study the mechanism of
${}^{12}\mathrm{C}(3_{1}^{-})$ formation, as shown in Fig. 28. The form is
extremely similar to that seen for the Hoyle state albeit with a slightly
smaller component corresponding to the direct decay and a larger component
from more sequential paths. There is, despite being slightly weaker than the
${}^{12}\mathrm{C}(0_{2}^{+}){+}^{16}\mathrm{O}^{\star}$ component, a
reasonable strength ${}^{12}\mathrm{C}(3_{1}^{-}){+}^{16}\mathrm{O}^{\star}$
path present in the data for all three beam energies.

Figure 28: (Negative) Q-value for the break-up of the compound nucleus into
${}^{12}\mathrm{C}(3_{1}^{-})$ \+ ${}^{16}\mathrm{O}^{\star}$ by
reconstructing the ${}^{16}\mathrm{O}$ from the measurement of the
${}^{12}\mathrm{C}(3_{1}^{-})$ from 3 $\alpha$-particles.

These data therefore match the EHF calculations which predict a large
${}^{12}\mathrm{C}^{\star}{+}^{16}\mathrm{O}^{\star}$ strength. There is no
evidence of the predicted 15.1 MeV $\alpha$-gas state Funaki  _et al._ (2009)
in the data as measured from the ${}^{12}\mathrm{C}(0_{2}^{+})$ and a large
portion of the seen ${}^{12}\mathrm{C}(0_{2}^{+})$ and
${}^{12}\mathrm{C}(3_{1}^{-})$ events do not originate from binary scission
but instead arise from more sequential decays or multi-particle break-up.

###  IX.3 Origins of oxygen-16

While there are only a few resonances (although none of them near-threshold)
in ${}^{16}\mathrm{O}$ observed in the 4-$\alpha$ breakup channel (and their
contributions are weak), the same formulation can still be applied to the
4-$\alpha$ states observed to examine the two-body break-up contribution.
Rather than selecting events within a certain excitation range in
${}^{16}\mathrm{O}$, the excitation energy in ${}^{16}\mathrm{O}$ in the
4$\alpha$ channel was plotted against the (negative) Q-value to give an
indication of the excitation energy present in ${}^{12}\mathrm{C}$. This is
shown in Fig. 29 for the three beam energies. For the 400 MeV data, an
enhancement can be seen corresponding to the
${}^{12}\mathrm{C}(0_{2}^{+}){+}^{16}\mathrm{O}(50\rightarrow
90\,\mathrm{MeV})$ and
${}^{12}\mathrm{C}(14\,\mathrm{MeV}){+}^{16}\mathrm{O}(50\rightarrow
90\,\mathrm{MeV})$ where the excitation energy projection between the red
lines can be seen in Fig. 30. While the resolution and statistics are limited
so as not to allow for a more precise breakdown of the contribution of states
here, there is some evidence of resonances populated in this region. This
enhanced yield is also visible in the 280 MeV beam energy data although the
statistics are limited here that a definite assertion cannot be made.
Examination of the 160 MeV beam data shows no significant peak here, partially
due to the obscuring of the signal by the strong diagonal seen at all three
beam energies which can be ascribed to non
${}^{12}\mathrm{C}^{\star}{+}^{16}\mathrm{O}^{\star}$ break-up. This agrees
with the previous conclusion that the secondary peak in ${}^{16}\mathrm{O}$
was not described well by event-mixing demonstrating it had a correlated
$\alpha$-particle component.

The majority of events for ${}^{16}\mathrm{O}$ are seen to lie on the diagonal
loci in Fig. 29 where, as verified by the Monte Carlo simulation,
contributions from non-binary decays of ${}^{28}\mathrm{Si}$ lie involving
multi-particle breakup and sequential decay. This also suggests that the
contribution of inelastic scattering is not dominant over compound nucleus
formation in this experiment for all beam energies.

Figure 29: (Negative) Q-value for
${}^{12}\mathrm{C}^{\star}{+}^{16}\mathrm{O}^{\star}$ against the excitation
energy in ${}^{16}\mathrm{O}$ from the 4-$\alpha$ channel. The results are
shown for the three beam energies of 160 (a), 280 (b), and 400 (c) MeV. Red
annotations denote the areas associated with excess yields in the data which
are mentioned in the text.  Figure 30: Projection of the region between the
red lines in Fig. 29c showing the excitation energy in ${}^{12}\mathrm{C}$ as
determined from the measurement of the 4-$\alpha$ breakup with excitation
energies between 50 and 95 MeV in ${}^{16}\mathrm{O}$. The two red lines guide
the eye to the locations of the enhancements at 7.65 MeV and 14 MeV which may
be as a consequence of the highly-clustered states at those excitation
energies. A small component corresponding to the ${}^{12}\mathrm{C}$(g.s) can
also be seen at 0 MeV.

###  IX.4 Origins of neon-20

The EHF calculations also expect a large decay path via the intermediate
$\alpha$-decay from ${}^{24}\mathrm{Mg}$ to ${}^{20}\mathrm{Ne}$. To verify
this conclusion, the missing ${}^{8}\mathrm{Be}^{\star}$ can also be
reconstructed from the measured ${}^{20}\mathrm{Ne}^{\star}$ from the decay to
5 $\alpha$-particles. Agreeing with the previous results, Fig. 31 shows the
results of this where the expected Q-value is not seen in the data for any of
the three beam energies. Additionally, there is no component corresponding to
higher excitations of ${}^{8}\mathrm{Be}$. This demonstrates these 5-$\alpha$
events either originate from sequential decay through ${}^{24}\mathrm{Mg}$ or
from multi-particle break-up, both of which correspond to a Q-value not
located along the expected Q-value labeled on Fig. 31.

Figure 31: Q-value for ${}^{20}\mathrm{Ne}^{\star}{+}^{8}\mathrm{Be}^{\star}$
while reconstructing the missing ${}^{8}\mathrm{Be}$ from the measured
${}^{20}\mathrm{Ne}$ from 5 $\alpha$-particle events.

###  IX.5 Origins of magnesium-24

The excitation energy spectrum for ${}^{24}\mathrm{Mg}$ shows no resonances in
any determinable channel (6-$\alpha$,
${}^{12}\mathrm{C}(0_{2}^{+})+{{}^{12}\mathrm{C}(0_{2}^{+})}$,
${}^{8}\mathrm{Be}+{{}^{8}\mathrm{Be}}+{{}^{8}\mathrm{Be}}$) with the primary
decay mode being sequential $\alpha$-particle emission from
${}^{28}\mathrm{Si}$ to the continuum. Taking multiplicity 6 events and
reconstructing for a missing $\alpha$-particle, the Q-value can be formulated,
the results of which are seen in Fig. 32.

To verify that the 6 $\alpha$-particles measured correspond to a complete
7-$\alpha$ decay, the majority of events lie around the expected Q-value
showing the majority of these multiplicity 6 events are multiplicity 7 events
with the final $\alpha$-particle being undetected. This matches well with the
EHF calculations which demonstrated the multiplicity 6 yield is expected to be
very small by virtue of the need to break apart an $\alpha$-particle (only
becoming more common at a beam energy of 400 MeV, agreeing with the result
seen here). This also demonstrates the multiplicity 6 events seen are genuine
and not a result of pileup, matching well with the work in Section VII.

Figure 32: (Negative) Q-value for
${}^{12}\mathrm{C}(^{16}\mathrm{O},{{}^{24}\mathrm{Mg}^{\star}}$) while
reconstructing the missing $\alpha$ from ${}^{24}\mathrm{Mg}^{\star}$.

##  X Experimental signatures results

The dominance of the uncorrelated $\alpha$-particles, particularly in the
heavier N-$\alpha$ systems and the emergence of the near-threshold resonant
states in ${}^{12}\mathrm{C}$ and ${}^{8}\mathrm{Be}$ where the known
N-$\alpha$ structures appear, can be understood from the data. Aspects of the
decay can be well described by sequential $\alpha$-decay predominantly through
${}^{24}\mathrm{Mg}$ or by multi-particle break-up as demonstrated by Fig. 32.
The EHF predictions for such a sequential $\alpha$-decay model however
demonstrate that the predicted multiplicities are inconsistent with the
experimental data and the Fermi break-up model better describes the higher
multiplicities seen via multi-particle break-up. An important signature of
$\alpha$-condensation is the equivalency of the different decay paths into
$\alpha$-gas substructures. Taking this, the phase spaces for the different
partitions can therefore be ascribed an equal transition probability so the
experimental yields can be directly compared to these phase spaces. To account
for the experimental effects, most notably the efficiency, the different final
states from the Fermi break-up model were efficiency-corrected using the Monte
Carlo code to give a predicted experimental yield for the 7-$\alpha$ final
state. The comparison between these predictions and the experimental data is
shown in Table 4.

Table 4: Experimental and theoretical branching ratios for the 7-$\alpha$ break-up modes for the different beam energies. The theoretical branching ratios for the 7-$\alpha$ break-up modes are calculated with the Fermi break-up model and efficiency-corrected using the Monte Carlo simulations. The error on the theoretical values is estimated at $\sim$ 1$\%$ of the total branching ratio from the uncertainty in the MC efficiency. Decay Path | Exp. branching ratio $\%$ | Theor. branching ratio $\%$
---|---|---
Label | Constituents | 160 MeV | 280 MeV | 400 MeV | 160 MeV | 280 MeV | 400 MeV
I |  ${}^{12}\mathrm{C}(0_{2}^{+})$ \+ ${}^{12}\mathrm{C}(0_{2}^{+})$ \+ $\alpha$ | 0.0(0.0) | 0.0(0.0) | 0.2(0.2) | 7.1 | 0.3 | 0.0
II |  ${}^{12}\mathrm{C}(0_{2}^{+})$ \+ ${}^{8}\mathrm{Be}$ \+ ${}^{8}\mathrm{Be}$ | 0.0(0.0) | 0.0(0.0) | 0.0(0.0) | 1.0 | 0.3 | 0.1
III |  ${}^{12}\mathrm{C}(0_{2}^{+})$ \+ ${}^{8}\mathrm{Be}$ \+ 2$\alpha$ | 1.8(1.3) | 2.8(1.1) | 1.1(0.5) | 43.3 | 11.5 | 5.0
IV |  ${}^{12}\mathrm{C}(0_{2}^{+})$ \+ 4$\alpha$ | 4.5(2.0) | 2.8(1.1) | 1.5(0.6) | 11.7 | 16.7 | 10.8
V |  ${}^{8}\mathrm{Be}$ \+ ${}^{8}\mathrm{Be}$ \+ ${}^{8}\mathrm{Be}$ \+ $\alpha$ | 3.6(1.8) | 0.4(0.4) | 0.4(0.3) | 20.7 | 5.2 | 2.3
VI |  ${}^{8}\mathrm{Be}$ \+ ${}^{8}\mathrm{Be}$ \+ 3$\alpha$ | 33.0(6.3) | 13.8(2.5) | 10.1(1.5) | 7.0 | 8.4 | 4.6
VII |  ${}^{8}\mathrm{Be}$ \+ 5$\alpha$ | 45.5(7.7) | 37.8(4.5) | 32.8(3.0) | 9.2 | 57.3 | 76.7
VIII | 7$\alpha$ | 11.6(10.4) | 42.5(5.4) | 53.9(3.5) | 0.0 | 0.3 | 0.5

###  X.1 Comparison of 7-$\alpha$ events with Fermi break-up predictions

As the beam energy increases in the Fermi break-up calculation, the dominant
decay modes change as the energy dependency becomes more important than the
permutation factor for the ${}^{8}\mathrm{Be}{+}5\alpha$ (VII) path. This
decay mode is suppressed by a factor of $(5!=120)$ in comparison to the
${}^{12}\mathrm{C}(0_{2}^{+}){+}^{8}\mathrm{Be}{+}2\alpha$ (III) path which
only has a suppression factor of 2. A similar situation can be seen for the
7$\alpha$-particle (VIII) decay which, while a 7-body decay, has a high power
dependence on the energy, is suppressed by a factor of $(7!=5040)$. The lower
beam energies therefore correspondingly favor a small $n$-body decay.

It is apparent from Table 4 that the decay modes seen experimentally for
$E_{b}=160$ MeV heavily favor the ${}^{8}\mathrm{Be}$ \+ ${}^{8}\mathrm{Be}$
(VI), single ${}^{8}\mathrm{Be}$ (VII) and 7-$\alpha$ (VIII) decay modes far
in excess of those predicted theoretically. Part of this discrepancy can be
attributed to modeling additional excited states in ${}^{12}\mathrm{C}$ above
7.65 MeV. While these states will have an increasingly reduced phase space as
a consequence of the decrease in the kinematically available energy, they may
also have a large spin component which partially compensates this effect as in
Eq. 3. This angular momentum consideration and all aspects of penetrability
are omitted in the Fermi break-up model. A large phase space to higher
excitations in ${}^{12}\mathrm{C}$ would therefore manifest itself as
replacing the ${}^{12}\mathrm{C}(0_{2}^{+})$ strength with
${}^{8}\mathrm{Be}+\alpha$ and 3-$\alpha$ strength. Repositioning the
branching ratio from the ${}^{12}\mathrm{C}(0_{2}^{+}){+}^{8}\mathrm{Be}$ and
single-${}^{12}\mathrm{C}(0_{2}^{+})$ to these decay modes (i.e. III and IV
move to VI and VII) then creates a better agreement with the experimentally
observed data. It is not possible to include these additional contributions in
the calculations as one cannot be certain of the density and spin of states
once one enters the continuum. Additionally the transition matrix elements may
start to deviate drastically from the constant value used under the assumption
of an $\alpha$-condensate Kokalova  _et al._ (2006).

An additional explanation for the discrepancy between the experimental and
theoretical values is that the Hoyle state is not well described as an
$\alpha$-condensate and as such, the transition matrix to break-up into the
Hoyle state and $\alpha$-condensed systems is reduced or the mechanism is
better described by a series of sequential decays rather than a direct break-
up into constituent $\alpha$-particles. This latter explanation has been
demonstrated to be unlikely as the experimental data suggest a reaction
mechanism that differs from the EHF calculations due to the measured
$\alpha$-particle multiplicities and the disagreement over the strength of
binary scission modes which were shown to be extremely weak experimentally.
The reaction mechanism has been demonstrated to be described well by aspects
of both the Fermi break-up model and the statistical decay model showing the
importance of both of these decay paths.

The 280 and 400 MeV data show a very similar pattern to those at 160 MeV
albeit with a movement of strength in the
${}^{8}\mathrm{Be}{+}^{8}\mathrm{Be}$ (VI) channel to the 7-$\alpha$ channel
(VII). Even directly assigning the strength from the channels with excitations
in ${}^{12}\mathrm{C}$ to the 7-$\alpha$ path, the strength is still in excess
of that expected. This suggests that some of the yield seen in the single
${}^{8}\mathrm{Be}$ is therefore also sent to the 7-$\alpha$ channel. This
could perhaps be by virtue of populating higher excitations in
${}^{8}\mathrm{Be}$ (i.e. the $\mathrm{{}^{8}\mathrm{Be}}(2_{1}^{+}$)) where
limited statistics and resolution mean this contribution is difficult to
observe although there is no evidence of such a strong contribution in the
data. Again, this channel is difficult to calculate as the assumption of an
identical transition matrix is no longer valid.

Overall, the Fermi break-up description can be seen to show a reasonable
agreement with the experimental data provided the points above regarding
modeling additional higher excitations are taken into consideration.

###  X.2 Decay mechanisms

By modeling the decay processes via sequential decay using the EHF and
modeling multi-particle break-up with the Fermi break-up model, the
experimentally observed $\alpha$-particle multiplicities are seen to be in
greater agreement with the Fermi break-up model. This can also be seen by the
consistency of the population of the resonances in ${}^{12}\mathrm{C}$ (Fig.
19) where as in Fig. 5, as the beam energy increases, the EHF code predicts
the dominance of sequential $\alpha$-particle emission rather than the
population of discrete states. Additionally, the relative agreement between
the 7-$\alpha$ channels discussed above in Section X.1 with the discussed
improvements to the model demonstrate the $\alpha$-particle decays are mainly
driven by the multi-particle break-up decay mode. The sequential decay model
does however have some success in describing the binary fission products
extremely well and a lack of describing the importance of light cluster
emissions in the form of ${}^{8}\mathrm{Be}$ and
${}^{12}\mathrm{C}(0_{2}^{+})$ (which are shown to be very important in the
Fermi break-up model) means this model fails to describe the high
$\alpha$-particle multiplicity features seen experimentally. There is no
evidence of an enhancement in the $\alpha$-gas multiplicity compared to those
predicted by a non $\alpha$-condensate state. Previous investigations Akimune
_et al._ (2013) which have stated such an enhancement over that predicted from
the statistical decay processes require further examination and theoretical
input into the exact decay mechanisms which incorporate many-particle break-up
effects and light $\alpha$-conjugate cluster emission (i.e.
${}^{8}\mathrm{Be}$).

###  X.3 Measurement of near-threshold states

Another key signature of $\alpha$-gas states is a highly clustered near-
threshold state. While the previously-known highly clustered states were seen
in ${}^{8}\mathrm{Be}$ and ${}^{12}\mathrm{C}$, no enhancement was seen in
${}^{16}\mathrm{O}$. This absence may be largely due to the decay mechanisms
associated with such a state. The predicted energy of an $\alpha$-gas state in
${}^{16}\mathrm{O}$ is 15.1 MeV $(0_{6}^{+})$ however the proximity of this
state to the 4-$\alpha$ threshold at 14.44 MeV means that the preferential
decay is via the $\alpha_{0}$ and $\alpha_{1}$ decay paths with the
$\alpha_{2}$ “signature-decay” having a predicted branching ratio of $\sim
10^{-9}$ Funaki  _et al._ (2013). This calculation omits any modification to
the Coulomb barrier by the mechanisms discussed in Section II, but, however,
demonstrates this state cannot be observed in the 4-$\alpha$ decay channel in
the current experiment. Instead, this state was investigated by looking for
the decay of the compound nucleus ${}^{28}\mathrm{Si}$ into
${}^{12}\mathrm{C}(0_{2}^{+}){+}^{16}\mathrm{O}(0_{6}^{+})$ by measurement of
the Hoyle state and reconstructing the ${}^{16}\mathrm{O}$ where the decay
mode is no longer inhibitive. Figure 27 showed no such state populated in
connection to the Hoyle state. Additionally, there was no evidence of resonant
structure in the ${}^{12}\mathrm{C}$(g.s) + $\alpha$ excitation function
demonstrating such a state is not well populated in this type of reaction
despite the beryllium-8 ground state, the Hoyle state and its (possible)
rotational excitation all being clearly visible above the continuum
contribution.

In ${}^{20}\mathrm{Ne}$, a quenching of the 5-$\alpha$ strength was clearly
evident in proximity to the 5-$\alpha$ barrier in agreement with previous
experiments. This can be attributed to the dominance of the
${}^{16}\mathrm{O}+\alpha$ decay channels here mirroring the situation in
${}^{16}\mathrm{O}$ where such a near-threshold state cannot be detected in
the 5-$\alpha$ decay channel until one is well over the Coulomb barrier ($\sim
24$ MeV). Indirect measurement of this state via measurement of
${}^{8}\mathrm{Be}$(g.s) and reconstruction of ${}^{20}\mathrm{Ne}$ also
demonstrated no increased yield above the 5-$\alpha$ threshold where such a
state would be expected to manifest and the limitations of measuring the
characteristic 5-$\alpha$ decay are not required.

##  XI Summary

In summary, the reaction mechanism of the
${}^{12}\mathrm{C}(^{16}\mathrm{O},^{28}\mathrm{Si}^{\star})$ channel was
studied at three beam energies of 160, 280, and 400 MeV. This experiment was
performed to search for $\alpha$-gas states in ${}^{12}\mathrm{C}$,
${}^{16}\mathrm{O}$, ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$ and
${}^{28}\mathrm{Si}$. No such states were found for systems heavier than
${}^{12}\mathrm{C}$. In the case of heavier isotopes, there were various
reasons discussed in this work in detail which could have obscured the
observation of those states. While the Hoyle state
(${}^{12}\mathrm{C}(0_{2}^{+})$) is the best candidate for such an
$\alpha$-gas state (from theoretical predictions), following other recent
experiments (e.g. of the direct 3$\alpha$-decay of the Hoyle state Smith  _et
al._ (2017); Dell’Aquila  _et al._ (2017)), we can conclude that although
theoretically this state fits well with an $\alpha$-gas description,
experimentally it appears that the underlying fermionic structure of the
$\alpha$-particles is sufficient to prevent this $\alpha$-gas nature and none
of the expected signatures for $\alpha$-condensation were seen in the current
analysis.

This type of high-energy reaction requires input from multi-particle breakup
models to describe the observed $\alpha$-particle multiplicities which are in
excess of those predicted from Hauser Feshbach calculations. The enhancements
are not sufficient however to support the observation of $\alpha$-condensate
states.

##  XII Acknowledgments

The authors would like to thank the operators and technicians at Catania for
their support and for providing a stable beam during the experiment. J.B.
would also like to thank LNS for financial support during his attendance at
the experiment.

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