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

TL;DR

This study investigated alpha-condensation signatures in light nuclei through high-energy nuclear reactions, finding no direct evidence for alpha-condensation but revealing complex decay mechanisms involving multi-particle breakups.

Contribution

It provides experimental insights into alpha-clustered states and decay mechanisms, highlighting the limitations in observing alpha-condensation signatures due to Coulomb barriers.

Findings

Measured alpha-particle multiplicity distributions exceeding sequential decay predictions.

Better agreement with Fermi break-up model over extended Hauser-Feshbach calculations.

No evidence of alpha-condensation or highly clustered states in the studied nuclei.

Abstract

Method: To examine signatures of this alpha-condensation, a compound nucleus reaction using 160, 280, and 400 MeV 16O beams impinging on a carbon target was used to investigate the 12C(16O,7a) reaction. This permits a search for near-threshold states in the alpha-conjugate nuclei up to 24Mg. 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 8Be and 12C(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 16O, 20Ne, and 24Mg 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 16O was observed from (28Si*,12C(0_2+))16O(0_6+) when reconstructing the Hoyle state from 3 alpha-particles. Therefore, no experimental signatures for alpha-condensation were observed.

Tables

Table 1. Table 1: Final yields as predicted from the EHF calculation for the three different beam energies (160, 280, and 400 MeV). The α 𝛼 \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	$3.79\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟑}}$	$3.27\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟑}}$	$2.74\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟑}}$
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	$3.69\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟏}}$	$1.19\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟐}}$	$7.36\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟏}}$
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	$7.91\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟐}}$	$7.01\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟐}}$	$2.14\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟐}}$
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	$8.75\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟐}}$	$1.32\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟐}}$	$5.27\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟏}}$
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	$1.02\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟐}}$	$9.91\times {\mathrm{𝟏𝟎}}^{\mathrm{𝟎}}$	$8.52\times {\mathrm{𝟏𝟎}}^{-\mathrm{𝟐}}$

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

Decay path	Break-up
	$𝜶$-particles
24Mg + $\alpha$	2	1
20Ne + ${}_{}{}^8\mathrm{Be}$	2	2
${}_{}{}^{16}O$+${}_{}{}^{12}C$	2	0
${}_{}{}^{16}O$+${}_{}{}^{12}C\left(0_2^{+}\right)$	2	3
20Ne+2$\alpha$	3	2
${}_{}{}^{16}O$+${}_{}{}^8\mathrm{Be}$+$\alpha$	3	3
${}_{}{}^{12}C$+${}_{}{}^{12}C$+$\alpha$	3	1
${}_{}{}^{12}C\left(0_2^{+}\right)$+${}_{}{}^{12}C\left(0_2^{+}\right)$+$\alpha$	3	7
${}_{}{}^{12}C$+${}_{}{}^8\mathrm{Be}$+${}_{}{}^8\mathrm{Be}$	3	4
${}_{}{}^{12}C\left(0_2^{+}\right)$+${}_{}{}^8\mathrm{Be}$+${}_{}{}^8\mathrm{Be}$	3	7
${}_{}{}^{12}C$+${}_{}{}^8\mathrm{Be}$+2$\alpha$	4	4
${}_{}{}^{12}C\left(0_2^{+}\right)$+${}_{}{}^8\mathrm{Be}$+2$\alpha$	4	7
${}_{}{}^8\mathrm{Be}$+${}_{}{}^8\mathrm{Be}$+${}_{}{}^8\mathrm{Be}$+$\alpha$	4	7
${}_{}{}^{16}O$+3$\alpha$	4	3
${}_{}{}^{12}C$+4$\alpha$	5	4
${}_{}{}^{12}C\left(0_2^{+}\right)$+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

Table 3. Table 3: Different decay paths through α 𝛼 \alpha -gas type states until the system has decayed to 8 Be or an α 𝛼 \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 {\to }^{20}{\mathrm{Ne}}^{\star }+\alpha \to$${}_{}{}^{16}O_{}^{\star }+\alpha \to$${}_{}{}^{12}C_{}^{\star }+\alpha \to$${}_{}{}^8\mathrm{Be}+\alpha$
2	${}_{}{}^{24}\mathrm{Mg}_{}^{\star }+\alpha \to$${}_{}{}^{20}\mathrm{Ne}_{}^{\star }+\alpha \to$${}_{}{}^{16}O_{}^{\star }+\alpha \to$${}_{}{}^8\mathrm{Be}+$${}_{}{}^8\mathrm{Be}$
3	${}_{}{}^{24}\mathrm{Mg}_{}^{\star }+\alpha \to$${}_{}{}^{20}\mathrm{Ne}_{}^{\star }+\alpha \to$${}_{}{}^{12}C_{}^{\star }{+}^8\mathrm{Be}\to$${}_{}{}^8\mathrm{Be}+\alpha$
4	${}_{}{}^{24}\mathrm{Mg}_{}^{\star }+\alpha \to$${}_{}{}^{16}O_{}^{\star }+$${}_{}{}^8\mathrm{Be}\to$12C${}_{}{}^{\star }+\alpha \to$${}_{}{}^8\mathrm{Be}+\alpha$
5	${}_{}{}^{24}\mathrm{Mg}_{}^{\star }+\alpha \to$${}_{}{}^{16}O_{}^{\star }+$${}_{}{}^8\mathrm{Be}\to$${}_{}{}^8\mathrm{Be}{+}^8\mathrm{Be}$
6	${}_{}{}^{24}\mathrm{Mg}_{}^{\star }+\alpha \to$${}_{}{}^{12}C_{}^{\star }+$${}_{}{}^{12}C_{}^{\star }\to$${}_{}{}^8\mathrm{Be}{+}^8\mathrm{Be}$
7	${}_{}{}^{20}\mathrm{Ne}_{}^{\star }{+}^8\mathrm{Be}\to$${}_{}{}^{16}O_{}^{\star }+\alpha \to$${}_{}{}^{12}C_{}^{\star }+\alpha {\to }^8\mathrm{Be}+\alpha$
8	${}_{}{}^{20}\mathrm{Ne}_{}^{\star }{+}^8\mathrm{Be}\to$${}_{}{}^{16}O_{}^{\star }+\alpha \to$${}_{}{}^8\mathrm{Be}+$${}_{}{}^8\mathrm{Be}$
9	${}_{}{}^{20}\mathrm{Ne}_{}^{\star }{+}^8\mathrm{Be}\to$${}_{}{}^{12}C_{}^{\star }{+}^8\mathrm{Be}{\to }^8\mathrm{Be}+\alpha$
10	${}_{}{}^{16}O_{}^{\star }{+}^{12}$C${}_{}{}^{\star }\to {(}^{12}{C}^{\star }+\alpha )+{(}^8\mathrm{Be}+\alpha )\to$${}_{}{}^8\mathrm{Be}+\alpha$
11	${}_{}{}^{16}O_{}^{\star }{+}^{12}{C}^{\star }\to {(}^8\mathrm{Be}+$${}_{}{}^8\mathrm{Be})+{(}^8\mathrm{Be}+\alpha$)

Table 4. Table 4: Experimental and theoretical branching ratios for the 7- α 𝛼 \alpha break-up modes for the different beam energies. The theoretical branching ratios for the 7- α 𝛼 \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 ∼ similar-to \sim 1 % percent \% 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	${}_{}{}^{12}C\left(0_2^{+}\right)$ + ${}_{}{}^{12}C\left(0_2^{+}\right)$ + $\alpha$	0.0(0.0)	0.0(0.0)	0.2(0.2)	7.1	0.3	0.0
II	${}_{}{}^{12}C\left(0_2^{+}\right)$ + ${}_{}{}^8\mathrm{Be}$ + ${}_{}{}^8\mathrm{Be}$	0.0(0.0)	0.0(0.0)	0.0(0.0)	1.0	0.3	0.1
III	${}_{}{}^{12}C\left(0_2^{+}\right)$ + ${}_{}{}^8\mathrm{Be}$ + 2$\alpha$	1.8(1.3)	2.8(1.1)	1.1(0.5)	43.3	11.5	5.0
IV	${}_{}{}^{12}C\left(0_2^{+}\right)$ + 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

Equations

$$t_0=d\sqrt{\frac{m}{2E}}-t,$$

$$R_{if}=\frac{2\pi }{\hslash }|A_{if}{|}^2{\omega }_f,$$

$${\omega }_f=$$

$$\mathbf{N}=$$

$$=$$

$${ϵ}_{i,j}=\left(\genfrac{}{}{0ex}{}{j}{i}\right)p^i(1-p{)}^{j-i},$$

$$\mathbf{M}=ϵ{\mathbf{M}}^{\prime },$$

$${\mathbf{M}}^{\prime }={ϵ}^{-1}\mathbf{M}.$$

$$E_x=\sum _i{E}_i-\frac{({\sum }_i\overrightarrow{p_i}{)}^2}{2{\sum }_i{m}_i}-Q.$$

Full text

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

††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.

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.

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:

[TABLE]

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.

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.

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.

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):

[TABLE]

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):

[TABLE]

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.