# Direct evidence of α-particle condensation for the Hoyle state

Ad.R. Raduta, B. Borderie, E. Geraci, N. Le Neindre, P. Napolitani, M.F. Rivet

► To cite this version:

Ad.R. Raduta, B. Borderie, E. Geraci, N. Le Neindre, P. Napolitani, et al. Direct evidence of α-particle condensation for the Hoyle state. International Workshop on multifragmentation and Related Topics - IWM 2011, Nov 2011, Caen, France. pp.00034, 10.1051/epjconf/20123100034. in2p3-00730977

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<figure><img src="image_1.png" /></figure>

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# Direct evidence of α-particle condensation
for the Hoyle state

AD. R. RADUTA<fnref n="1" />,<fnref n="2" />, B. BORDERIE<fnref n="1" />, E. GERACI<fnref n="3" />, N. LE NEINDRE<fnref n="1" />,<fnref n="4" />,
P. NAPOLITANI<fnref n="1" /> and M. F. RIVET<fnref n="1" />
for the ISOSPIN COLLABORATION

## Abstract

The fragmentation of quasi-projectiles from the nuclear reaction
⁴⁰Ca+¹²C at 25 MeV/nucleon was used to produce excited states can-
didates to α-particle condensation. Complete kinematic characteriza-
tion of individual decay events, made possible by the CHIMERA high-
granularity 4π charged particle multi-detector, reveals that 7.5±4.0 %
of the Hoyle state particle decays correspond to direct decays in three
equal-energy α-particles. Moreover, events with increased kinetic en-
ergy dispersion in the ¹²C center of mass, which amount to 9.5±4.0 %,
point toward the occurrence of a second competing molecular configu-
ration, of a linear α-chain.

Bose-Einstein condensation is known to occur in weakly and strongly
interacting systems such as dilute atomic gases and liquid ⁴He. During
the last decade it was theoretically shown that dilute symmetric nuclear
matter may also experience particle condensation [1-3]. This new possible
phase of nuclear matter may have its counterpart in low-density states of

<footnote n="1">¹Institut de Physique Nucléaire, CNRS/IN2P3, Université Paris-Sud 11,
Orsay, France</footnote>

<footnote n="2">²National Institute for Physics and Nuclear Engineering, POB-MG6,
077125 Bucharest-Măgurele, Romania</footnote>

<footnote n="3">³INFN-Catania and Dipartimento di Fisica e Astronomia, Università di
Catania, Italy</footnote>

<footnote n="4">⁴LPC, CNRS/IN2P3, ENSICAEN, Université de Caen, Caen, France</footnote>
self conjugate lighter nuclei, in the same way as superfluid nuclei are the
finite-size counterpart of superfluid nuclear and neutron matter.

Convincing pleading in favour of α-particle condensation in nuclei is
provided by the excellent theoretical description of the Hoyle state (i.e. the
first 0⁺ excited state at 7.654 MeV of ¹²C) and of the 0<sub>6</sub>⁺ state at 15.097
MeV of ¹⁶O in terms of condensate type wave functions [4, 5]. A note of
caution has to be added though: according to recent Fermionic Molecular
Dynamics (FMD) calculations [6] and pioneering works of Uegaki *et al.* [7],
the Hoyle state has rather to be regarded as a mixture of configurations
where exotic linear chains may exist among others.

From the experimental side, the detection of an α-particle condensate
expected to occur in the vicinity of the Nα-decay threshold [8] is a chal-
lenging task because of the few hundred-keV kinetic energies of emitted
particles in the emitter center of mass which make the detection difficult. If,
in order to overcome this drawback, one considers fast moving emitters as
those obtained in projectile fragmentation, it is clear that high-granularity
large-solid-angle-coverage detectors are necessary.

In this conference we report on recent results obtained out of the
⁴⁰Ca+¹²C at 25 MeV/nucleon nuclear reaction performed in 2003 at LNS-
Catania using the CHIMERA multi-detector [9]. The beam impinging on
a thin carbon target (320 µg/cm²) was delivered by the Superconducting
Cyclotron and its intensity was kept around 10⁷ ions/s to avoid pile-up
events. While standard procedures (E − ΔE and shape identification) have
been used for most reaction product identifications and energy calibration,
dedicated energy calibrations based on TOF information were realized for
Z = 1, 2 nuclei in order to achieve the best-possible precision. The average
energy resolution is of about 2%. α − α and α − d correlations used to check
the spectroscopic performances of our multi-detector revealed a satisfactory
accuracy of few tens of keV in determining the centroids of the peaks corre-
sponding to the ground and/or first exited states of ⁸Be and, respectively,
⁷Li [10]. While this result may be certainly considered encouraging, we note
that both correlated spectra and correlated functions manifest, because of
finite angular and energy resolution, a peak broadening which gets more
severe when the excitation energy increases.

Complete kinematical characterization of the reaction products allows
one to identify the particle emitting sources using invariant velocity plots
in the γβ⊥ vs. 1/2 log [(E + p‖) / (E − p‖)] plane and show the dominant
binary character of the collisions with the formation of a quasi-projectile
(QP) and a quasi-target (QT). Here E, p‖, p⊥ stand for the energy and
momentum components along and, respectively, perpendicular to the beam axis, $\beta = v/c$ is the reduced velocity and $\gamma = 1/\sqrt{1-\beta^2}$. The QP source may be easily isolated according to a rapidity of reaction products larger than $y_{proj}/2$, where $y_{proj}$ is the projectile rapidity. Hereafter, we shall focus exclusively on QP decay products with an $\alpha$ multiplicity $m_\alpha = 3$.

<figure><img src="image_2.png" /><figcaption>Figure 1: 3-$\alpha$ correlation function as a function of excitation energy. The arrows correspond to centroids of Breit-Wigner distributions (solid lines). Inset: zoom of the correlated distribution in the energy domain of the Hoyle state.</figcaption></figure>

Similar to two light-particle correlations, information on the $\alpha$-particle unstable excited states of $^{12}\text{C}$ populated by the $^{40}\text{Ca}+^{12}\text{C}$ at 25 MeV/nucleon reaction may be extracted from the 3$\alpha$-correlations. The 3$\alpha$ correlated spectrum and the correlation function (CF) manifest two peaks. As one can see in fig. 1, in the case of CF the peaks are centered at $E_{ex}=7.61$ MeV ($\Gamma=0.33$ MeV) and $E_{ex}=9.64$ MeV ($\Gamma=1.14$ MeV) thus showing that by projectile fragmentation we have populated the Hoyle state ($E_{ex}^{exp}=7.654$ MeV, $\Gamma^{exp}=8.5$ eV) and the complex excited region characterized by the strong $E_{ex}^{exp}=9.64$ MeV ($\Gamma^{exp}=34$ keV), $3^-$ state and by the broad $E_{ex}^{exp}=10.3$ MeV, $0^+$ state submerging a possible $2^+$ state at 9.7 MeV [11, 12]. We remind that the CF are defined as the ratio between the correlated (physical) yield $Y_{corr}$ and the product of single particle yields, generically termed as uncorrelated spectrum $Y_{uncorr}$, measured under the same conditions,

$$1 + R(X) = \frac{Y_{corr}(X)}{Y_{uncorr}(X)}. \qquad (1)$$

$Y_{uncorr}$ can also be built by mixing particles from different events as it is done in this work. The correlated spectrum allows one to estimate the ratio of background events (particles in coincidence originating from physically uncorrelated events) with respect to genuine three-particle decays. For the excitation energy domain 7.37-7.97 MeV and $m_\alpha=3$ this ratio is 1.

The question which arises now is whether some of the 3$\alpha$-decays of the Hoyle state do correspond to an $\alpha$-particle condensate. Taking into account that according to the present understanding the existence of a condensate
<figure><img src="image_3.png" /><figcaption>Figure 2: Three-α correlation function (b) expressed as a function of average kinetic energy - RMS of α particles corresponding to 1072 experimental events with $7.37 \le E_{ex} \le 7.97$ MeV. The uncorrelated yield is built such as to allow for decay through $^8$Be. The dotted line marks the maximum RMS compatible with momentum conservation. $<E_\alpha>$ (a) and RMS (c) spectra (normalized to 1072 events) of simulated DDE (solid lines), SD (dashed lines) and DDL (dot-dashed lines) decays of the Hoyle state after filtering through the detector replica with $R_E=2\%$. Panel (c) presents also the RMS projection of $Y_{corr}(<E_\alpha>, RMS)/Y_{uncorr}(<E_\alpha>, RMS)$ (thick lines).</figcaption></figure>

may be judged upon the occurrence of simultaneous decays with equal kinetic energy particles, our task is to find such direct decay events and estimate their branching ratio. This may be done by comparing the experimental results with numerical simulations filtered by the multi-detector replica. Several decay channels have been considered: direct emission of three α-particles with equal energies (DDE), sequential decays proceeding via the g.s. of $^8$Be with isotropic emission of the two α-particles from $^8$Be (SD)
<figure><img src="image_4.png" /><figcaption>Figure 3: RMS($E_\alpha$) spectrum of simulated events corresponding to the best $\chi^2$ and $90 \le E_\alpha \le 150$ keV (solid line) in comparison with the corresponding experimental data (solid symbols). Dashed and dotted histograms illustrate simulation results if only sequential decay (60%) and, respectively, sequential decay (55.5%) + DDE (4.5%) are considered. In all cases, background events amount to 40%.</figcaption></figure>

and direct decay from a linear chain (DDL). Limitation of direct decays (DD), which in principle assume random energy sharing among the emitted particles, to the particular sub-class of DDE is due to the predicted condensate nature of the Hoyle state. In each case and, for sequential decay, at each decay step the available energy ($E_{ex} + Q$) is shared among the reaction products such as to conserve the linear and angular momenta and, eventually, obey to the postulated particularities of the decay. No preferential orientation between emitted particle velocities and source boost exists. The thus obtained reaction products are assumed to freely propagate toward the detector. Once the detector is reached, the events are filtered: if one or more particles hit a dead detector area or a detector which is out of work, the event is suppressed. If all particles have been correctly identified, the velocity vectors are altered such as to account for finite angular and energy resolution. After a first estimation of detection efficiencies ranging between 41% and 49%, we have analyzed each decay pattern as a function of average kinetic energy and RMS in the $^{12}$C c.m.. The spectra illustrated in the panels (a) and (c) of fig. 2 show that while the different decay mechanisms are practically indiscernable from the $<E_\alpha>$ point of view, they may be discriminated using the RMS values. More precisely, for our average energy resolution of 2% $Y_{corr}(RMS)$ of DDE, SD and DDL are peaked at 30, 70 and 90 keV. As a consequence, we adopt the best agreement between experimental and simulated $Y_{corr}(<E_\alpha>, RMS)$ as the procedure to quantify each decay channel. For an optimum description of the experimental data, we add to pure simulated events the same proportion of background events that exists in the data. These are produced by partial event mixing (PEM) of experimental events; the technique of PEM which assumes that two particles belong to the same event while the third one comes from a different
event is adopted in order to account for $^8$Be decays. The comparison re-
stricted to the excitation energy domain $7.4 \le E_{ex} \le 7.8$ MeV in order to
reduce the importance of background events to 40% gets the best $\chi^2$ for
$7.5 \pm 4.0\%$ DDE and $9.5 \pm 4.0\%$ DDL. This best $\chi^2$ value has been obtained
after correcting for efficiencies; the error bars are estimated by taking into
account statistical, $\chi^2$ and background errors. The RMS($E_\alpha$) spectrum of
simulated events (solid line) corresponding to the best $\chi^2$ and the energy
domain $90 \le \langle E_\alpha \rangle \le 150$ keV is represented in fig. 3 in comparison with
the corresponding experimental data (solid symbols). Note that a limitation
to DDE and SD (dotted histogram) as possible decays leads to a significant
increase of $\chi^2_{min}$ without changing the percentage of DDE. The existence of
the direct decay channel was firstly investigated in Ref. [13] and its contri-
bution to the total alpha decay (dominated by the $^8$Be$_{g.s.}$ + $\alpha$ sequential
decay) was estimated to be at most 4%. The existence of such a decay
channel is also of the greatest importance for nuclear astrophysics, due to
the crucial role played by the Hoyle state for the synthesis of $^{12}$C in the
universe [14].

To visualize the results, fig. 2(b) presents the intra-event correlation
function expressed as a function of $\langle E_\alpha \rangle$ and RMS corresponding to the
selected 1072 events with $7.37 \le E_{ex} \le 7.97$ MeV. Uncorrelated events are
built by PEM but full event mixing (FEM) leads to a similar image. The
relative separation of the peaks in terms of RMS allows us to identify the
peak at low RMS$\le 25$ keV as stemming from DDE, the broad peak at RMS$\approx$
$70$ keV as being due to SD and the high RMS ($\approx 90$keV) peak at $\langle E_\alpha \rangle$
$= 120-140$ keV as coming from the direct decay of a linear chain with an $\alpha$
at rest and an equal sharing of the available energy between the other two
$\alpha$-particles. We mention that the traditional way of identifying competing
three-particle decays, that is the Dalitz plot, is not discriminative enough in
the case of our data, because of high background contamination. Though
the CF shows, as expected, a certain sensitivity to the event selection or
event mixing recipe, the two observed peaks appear to be stable.

The same analyses has been performed for the much higher statistics
region centered at 9.6 MeV and no indication in favour of DDE has been
obtained. In what regards the $0_6^+$ state at 15.097 MeV of $^{16}$O, the statistics
is to poor to allow unambiguous interpretation of low-RMS events.

In conclusion, the nuclear reaction $^{40}$Ca+$^{12}$C at 25 MeV/nucleon bombarding energy was used to produce states theoretically predicted as $\alpha$-particle condensate states. Supposing that equal values of kinetic energy of the emitted $\alpha$-particles represent a sufficient criterion for deciding in favour of $\alpha$-particle condensation, we found that $7.5 \pm 4.0\%$ of events correspond-
ing to the Hoyle state decay fulfil this criterion. To our knowledge, this is
the first direct experimental indication of α-particle condensation in nuclei.
This study also evidenced the presence of direct alpha decays from a linear
α-chain. An experiment with higher statistics is planned to study the $^{16}$O
case.

Acknowledgments

The authors are indebted to P. Schuck for numerous discussions and one
of the authors Ad. R. R. acknowledges the partial financial support from
ANCS, Romania, under grant Idei nr. 267/2007.

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