Temperature and Density Conditions for Alpha Clustering in Excited Self-Conjugate Nuclei

Starting from experimental studies on alpha-clustering in excited self-conjugate nuclei (from $^{16}$O to $^{28}$Si), temperature and density conditions for such a clustering are determined. Measured temperatures have been found in the range of 5.5 - 6.0 MeV whereas density values of 0.3 - 0.4 times the saturation density are deduced, i.e., 0.046 to 0.062 $fm^{-3}$. Such a density domain is also predicted by constrained self-consistent mean field calculations. These results constitute a benchmark for alpha clustering from self-conjugate nuclei in relation to descriptions of stellar evolution and supernovae.


Introduction
The knowledge of the composition of warm nuclear matter at low density is of paramount importance for a better understanding of the description of the core-collapse of supernovae as well as for the formation and static properties of proto-neutron stars [1,2]. In this context, cluster formation is one of the fundamental aspects with, in particular, the role of α-particles which are predicted to be present due to the instability of nuclear matter against cluster formation [3,4,5,6,7]. Related to this, the formation of α-particle clustering from excited expanding self-conjugate nuclei was also revealed in two different constrained self-consistent mean field calculations [8,9,10]. On the experimental side, cluster formation and their in-medium effects have been probed by heavy-ion experiments [11,12,13,14] and alpha-clustering was observed in excited expanding self-conjugate nuclei [15,16,17]. The aim of the present paper is to give a benchmark for the temperature and density needed to observe alpha-clustering in self-conjugate nuclei. The information is derived from experimental data (temperature, freeze-out volume/density) and densities compared with theoretical expectations. The paper is organized as follows. In section 2 the experimental context and the event selection will be presented. Section 3 is dedicated to alpha-particle clustering (evidence and deduced temperature-density information). Finally, in section 3 density information from the theoretical side is discussed.

Experiment and event selection
The experiment was performed at INFN, Laboratori Nazionali del Sud in Catania, Italy. The chosen experimental strategy was to use the reaction 40 Ca + 12 C at an incident energy (25 MeV per nucleon) high enough to produce some hot expanding fragmentation products [18,19] associated to a high granularity large solid angle particle array to precisely reconstruct the directions of the velocity vectors. The beam impinging on a thin carbon target (320 µg/cm 2 ) was delivered by the Superconducting Cyclotron and the charged reaction products were detected by the CHIMERA 4π multi-detector [20]. The beam intensity was kept around 10 7 ions/s to avoid pile-up events and random coincidences, which is mandatory for high multiplicity studies. CHIMERA consists of 1192 telescopes (∆E silicon detectors 200-300 µm thick and CsI(Tl) stopping detectors) mounted on 35 rings covering 94% of the solid angle, with very high granularity at forward angles. Details on A and Z identifications and on the quality of energy calibrations can be found in Refs. [20,21,22,15]. One can just underline that careful identifications and selections were allowed by a complete exclusive detection in A and Z of all reaction products and by the excellent forward granularity of CHIMERA; energy resolution was better than 1% for silicon detectors and varies between 1.0 and 2.5% for alpha particles stopped in CsI(Tl) crystals.
As a first step in our event selection procedure, we want to exclude poorly-measured events. Without making any hypothesis about the physics of the studied reaction one can measure the total detected charge Z tot (neutrons are not measured). Due to their cross-sections and to the geometrical efficiency of CHIMERA, the well detected reaction products correspond to projectile fragmentation (PF) [18,19] with Z tot = 19-20 (target reaction products not detected) and to incomplete/complete fusion with Z tot = 21-26 [23]. At this stage we can have the first indication of the multiplicity of α-particles, M α , emitted per event for well identified mechanisms (Z tot ≥ 19 -see figure 1 from Ref. 15). M α extends up to thirteen, which means a deexcitation of the total system into α-particles. Moreover, a reasonable number of events exhibit M α values up to about 6-7.
The goal is now to tentatively isolate, in the detected events, reaction products emitting α-particles only. Refs. [19,24] have shown that, at incident energies close to ours, 20 Ne or 32 S PF is dominated by alpha-conjugate reaction products. Based on this, and expecting the same for 40 Ca, we restrict our selection to completely detected PF events (Z tot = 20) composed of one projectile fragment and α-particles. Charge conservation imposes Z f rag = 20 -2M α . An example of the mass distribution of the single fragment can be seen in Ref. 15.
After this double selection, the question is: from which emission source are the α-particles emitted? Several possibilities are present and further selections must be made before restricting our study to alpha-sources emitting exclusively the M α observed (called N α sources in what follows). Possibilities that we must examine are the following: I) considering the incident energy of the reaction and the forward focusing of reaction products, it is important to identify the possible presence of preequilibrium (PE) α-particles in our selected PF events. With the hypothesis that all the α-particles are emitted from their center-of-mass reference frame, we noted an energy distribution which resembles a thermal one with the presence of a high energy tail starting at 40 MeV, which signs PE emission (see figure 3 from Ref. 15). To prevent errors on alpha emitter properties, it is necessary to remove events in which such PE emission can be present; an upper energy limit of 40 MeV found irrespective of M α was imposed to the α-particle energy. At this stage 6.9 to 9.2% of events were excluded.
II) α-particles can be emitted from deexcitation of PF events via unbound states of 12 C, 16 O, 20 Ne and not directly from excited expanding N α sources. We want, for instance, to exclude from the selection an event composed of two fragments ( 24 Mg and 12 C*) and one α-particle finally producing one single fragment ( 24 Mg) and four αparticles. Multi-particle correlation functions [25,26] were used to identify unbound states α-particle emitters and to exclude a small percentage of events (1.6-3.9%).
III) it must be verified that the fragments associated with M α are not the evaporation residues of excited Ca projectiles that have emitted sequentially α-particles only.
As far as the two first items are concerned the effect was to suppress from 8.5 to 12.8% of previously selected events; more details can be found in Ref. 15. The last item will be discussed in the following section.

Evidence for alpha-particle clustering
Before discussing different possible deexcitation mechanisms involved in the retained events, information on the projectile fragmentation mechanism is needed. Global features of PF events are reproduced by a model of stochastic transfers [27]. The main characteristics for primary events with Z tot =20 are the following: i) excitation energy extends up to about 200 MeV, which allows the large excitation energy domain (20-150 MeV) measured for N α sources when associated to a single fragment (see Figure 1); ii) angular momenta extend up to 24h, which gives an upper spin limit for Ca projectiles or N α sources. We recall that excitation energies of N α sources are equal to the sum of kinetic energies of particles in the N α reference frame plus the reaction Q value.
Are α-particles emitted sequentially or simultaneously? To answer the question α-energy spectra have been compared to simulations. For excited Ca projectiles and N α sources, experimental velocity and excitation energy distributions as well as distributions for spins were used as inputs. The results of simulations were then filtered using a software replica of the multi-detector including all detection and identification details. The simulated spectra are normalized to the area of experimental spectra. For sequential emission the GEMINI++ code [28] was used.
Before discussing decays of N α sources, the possible evaporation from Ca projectiles, as stated previously, was considered. Excitation energy for projectiles is deduced from E * =E * (N α)+E rel +Q. E rel is the relative energy between the N α source and the associated fragment (evaporation residue). An example of the comparison between simulation and experimental energy spectra of α-particles is displayed in Figure 2 (from 17); see also Figure 6 of Ref. 15 for N α 4 and 6. They show a rather poor agreement indicating that the hypothesis of sequential evaporation of alpha particles is not correct. Note that no more 24 Mg, 20 Ne or 16 O evaporation residues associated to N α from 4 to 7 are produced in simulations for 40 Ca spin distributions centered at values larger than 25h. Now, considering now sequential deexcitation of N α sources it appears, as shown in Figure 5 of Ref. 15, that the agreement between data and simulations becomes poorer and poorer when N α value decreases. Moreover an important disagreement between data and simulations is observed for the percentages of N α sources which de-excite via 8 Be emission [15].
For simultaneous emission from N α sources, a dedicated simulation was made that mimics a situation in which α clusters are formed early on while the N α source is expanding [8,9] due to thermal pressure. By respecting E (MeV) 5   the experimental excitation energy distributions of N α sources shown in Figure 1, a distribution of N α events is generated as starting point of the simulation. Event by event, the N α source is first split into α's. Then the remaining available energy (E * + Q) is directly randomly shared among the α-particles such as to conserve energy and linear momentum [29]. As an example, the histogram in Figure 3 (from 17) is the result of such a simulation for N α = 5 ( 20 Ne*), which shows a good agreement with data; see also   Figure 3. Alpha particle spectrum from the N α=5 system ( 20 Ne*). Black dots with statistical error bars correspond to experimental data (same as in Fig.2). Histogram superimposed on data corresponds to filtered simulation of a simultaneous decay process (see text).
sequential emission was not able to reproduce experimental data whereas a remarkable agreement is obtained when an α-clustering scenario is assumed. However one cannot exclude that a few percent of N α sources, those produced with lower excitation energies, sequentially de-excite.

Alpha clustering: temperature and density information
Heavy-ion reactions at energies around the Fermi energy have been shown good tools to create expanding nuclei and, more generally, low density nuclear matter. It is why, in the past years, various methods were used or developed to measure temperature and density in such a nuclear matter [11,14,30,31]. As we will see in what follows, we are in our study in a simple case to extract temperature and density information. The excitation energy distributions of selected excited self-conjugate nuclei (N α sources) and the corresponding kinetic energy spectra in their reference frame are displayed in Figure 1. Excitation energy thresholds for total deexcitation into α-particles vary from 20 to 60 MeV when N α moves from 4 to 7 whereas mean excitation energy per nucleon is rather constant around 3.3-3.5 MeV. Note that for the excitation spectrum of 28 Si*, even if the statistics are limited, two peaks around 110 and 125 MeV could be present which were also observed in Ref. 32 and possibly related to toroidal high-spin isomers. Kinetic energy spectra exhibit a thermal Maxwellian shape. We recall that events with emitted preequilibrium α-particles were removed in our event selection (see Figure 3 of Ref. 15), which explains our kinetic energy limit of 40 MeV. From these spectra, temperature, T , and the Coulomb correction, Table 1. Alpha-clustering for self-conjugate nuclei. Excitation energy information: mean value < E * > and standard deviation σ E * . Parameters from Maxwellian fits to energy spectra: temperature T and Coulomb correction Cc. Densities normalized to saturation density ρ 0 have been deduced from simulations (see text). Statistical errors are within parentheses. C c , are extracted from a fit procedure using a thermal Maxwellian formula: with a volume pre-exponential factor, which is used for simultaneous break-up such as alpha clustering [33]. Curves in Figure 1 correspond to Maxwellian fits and Table 1  which qualitatively indicates rather low densities for expanding excited nuclei at freeze-out due to thermal pressure. Note that with a surface pre-exponential factor (E − B c ) , which is used for sequential emissions, the best fits to the same data are of lower quality and can only be achieved by allowing negative values for the Coulomb barrier B c . Such negative values have no physical meaning and the necessity to use a volume pre-exponential factor further confirms the simultaneous emission of α-particles which characterizes the clustering.
To derive quantitative density information, the Coulomb corrections parameters deduced from the fits, C c , have been used. As mentioned in the previous subsection dedicated simulations were made to be compared with kinetic energy spectra. By imposing temperatures deduced from the fits (see Table 1) within a simulation containing an intermediate freeze-out volume stage it was easy to determine freeze-out volumes by also imposing an agreement with the most probable value of kinetic energy spectra which is equal to T/2 + C c . Freeze-out volumes values are in the range 2.7 -3.0V 0 , V 0 being the volumes of self-conjugate nuclei at the saturation density. The corresponding normalized densities ρ/ρ 0 are indicated in Table 1.

Discussion
As mentioned in the introduction, two self-consistent mean field calculations have been performed by imposing constrained deformation with a restriction to spherical symmetric configurations [8,9]. By gradually increasing the nuclear radius, the density of self-conjugate nuclei is decreased, and at a certain density value, the formation of clusters appears. The first calculation was made with the constrained Hartree-Fock-Bogoliubov approach using the Gogny D1S interaction, and in the second, the self-consistent relativistic Hartree-Bogoliubov model with the effective interaction DD-ME2 has been employed. For a comparison of density needed for α-clustering one can refer in calculations to the clustering radius normalized to the ground state radius r c /r g.s. . In the first simulation this ratio is found around 1.8 for 16 O and 24 M g, whereas in the second, a smaller value around 1.3 is obtained for 16 O. The explanation given in Ref. 9 is the fact that the single-nucleon localization is more pronounced with the relativistic functional, which facilitates the formation of α clusters in excited states. Directly translated into densities, these ratios correspond to ρ/ρ 0 around 0.17 [8] and 0.45 [9]. However, it is important to stress that calculations contain their own spurious center of mass energy which should be removed. In Ref. 8 an estimate of the correction needed was made starting from a calculation performed on 8 Be constraining the distance between the two nascent α-particles. About 14 MeV were found to be missing to get twice the binding energy of a single particle in the asymptotic limit. Thus, a correction of 7 MeV per α-particle was made, whatever the self-conjugate nucleus (from 16 O to 28 Si) and as a consequence the density for the appearance of α-clustering was increased to ρ/ρ 0 ∼ 1/3, which is close to our results. However, regarding these calculations, an important point one must underline is the fact that they correspond to zero temperature.
In Ref. 7 a generalized relativistic mean field model is used to calculate α-particle fractions in symmetric matter at subsaturation density for different temperatures. In particular this work describes the sudden decrease of the α-particle fraction at high densities by the vanishing of α-particle binding energy due to the Pauli blocking that leads to the Mott effect, i.e., the dissolution of α-particles in the medium. Thus, the maximum cluster abundance is reached around the Mott density. At a temperature of 6 MeV this density is also ∼ ρ 0 /3.
To conclude we can say that α-clustering in self-conjugate nuclei, experimentally deduced from 16 O to 28 Si, occurs around T = 5.5-6.0 MeV and densities in the range ρ/ρ 0 = 0.3-0.4. At present, constrained self consistent mean field calculations at zero temperature are in qualitative agreement for densities but even more realistic calculations are needed for a valuable comparison with data.