Multifragmentation of reconstructed quasi-projectiles in the mass region A ∼ 30

In nucleus–nucleus collisions with energies around 10–100 MeV/nucleon there is a strong interaction between a large number of nucleons which may lead to a fast equilibration of the nuclear systems. The thermal multifragmentation reaction, i.e. when a nuclear system breaks nearly instantaneously into multiple fragments, is especially interesting. In this case, we can address the properties of nuclear fragments in close proximity with other nuclear species existing in a low-density freeze-out volume, where thermalization takes place. On the other hand, this condition corresponds to a state of clusterized nuclear matter at subnuclear densities which occurs in many astrophysical processes [1, 2], e.g., in supernova explosions and the crusts of neutron stars. Until now, only theoretical predictions of the properties of such matter were possible. However, these predictions vary widely depending on assumptions for the nuclear forces and approximations for solving the nuclear many-body problem.

Multifragmentation reactions provide a unique possibility to study clusterized nuclear matter in terrestrial experiments, and, therefore, obtain more reliable predictions. In particular, nuclear composition and in-medium properties of the nuclear clusters are important for calculating the weak reaction rates in stellar matter, which are crucial for the determination of the electron fraction in this matter [2, 3]. Abundances of chemical elements and nucleosynthesis processes in various astrophysical environments also depend on these properties.

In this paper we present and analyze detailed data on the multifragmentation of highly excited nuclear systems with mass A∼30 near the N = Z line. The interaction of ³²S (45 MeV/nucleon) + ¹¹²Sn was used to produce a wide variety of hot quasi-projectile sources that underwent multifragmentation. After reconstruction, the quasi-projectiles ³⁰S, ³²S and ³⁴S were selected and their disintegration was studied in detail. The study addresses the dependence of the multifragmentation on the source composition and probes the influence of the symmetry energy of the produced primary clusters. We remind that within the nuclear physics community, two aspects of the symmetry energy are discussed. First, the density dependence of the symmetry energy of uniform nuclear matter from subnuclear to supranuclear densities (see, e.g., [4] and references therein) and, second, the symmetry energy of individual fragments produced at subnuclear density, which is addressed in this work. We believe that such studies can stimulate improvements in the equation of state used currently in hydrodynamical simulations of supernova explosions. For such calculations, the Lattimer–Swesty [5] and Shen [6] models are primarily used. However, they use only properties of isolated single heavy nuclei, even if medium modifications are implemented in [5]. Efforts to describe the equation of state taking into account the ensemble of all possible nuclear species and their mutual interactions are currently underway (see, e.g., [2, 7, 8], and references therein).

The experimental work was performed at the Cyclotron Institute of Texas A&M University. A beam of ³²S ions of energy 45 MeV/nucleon was delivered by the K500 superconducting cyclotron and interacted with an isotopically enriched ¹¹²Sn target. Fragments produced in peripheral and semi-peripheral collisions were detected by the FAUST multi-detector array [9]. FAUST is comprised of 68 Si-CsI(Tl) telescopes arranged in five concentric rings. This configuration allows for a solid angular coverage of 90% from 2.3° to 33.6°, 71% from 1.6° to 2.3° and 25% from 33.6° to 44.8°. Each Si-CsI(Tl) telescope is comprised of a 300 μm thick silicon detector followed by a CsI(Tl) crystal with thickness of either 3.0 or 2.25 cm read out by a photodiode. For further information about the construction of the array, the reader is directed to references [9–11]. The isotopic resolution in the experiment was achieved up to Z = 8 throughout the array and up to Z=13 in several detectors. The experimental setup was not designed to be sensitive to neutrons or photons; only charged particles were measured. Due to the design characteristics of the FAUST detectors, Z = 1 particles have a high-energy cut-off where they punch through the Si-CsI(Tl) telescope. The value of this cut-off for the 3.0 cm CsI(Tl) detectors is 95, 130 and 153 MeV for protons, deuterons and tritons, respectively. Similarly, for the 2.25 cm CsI(Tl) detectors, the values are 81, 110 and 130 MeV, respectively.

The excited quasi-projectile source was reconstructed from the detected fragments on an event-by-event basis. Events were considered where the total charge of the reconstructed quasi-projectile was equal to that of the beam, Z = 16. This constraint has been shown to produce a single well-defined equilibrated source [12–14]. For the study presented in this paper, further constraints were placed on the source selection for better comparison with the theoretical model framework DIT+SMM (deep-inelastic transfer and statistical multifragmentation model, see below). First, it was required that all fragments in the event be isotopically identified. Second, the event must be comprised only of particles with Z ⩽ 8. By choosing events where all particles are less than half the size of the emitting source, we allow for preferential selection of events not associated with the compound nucleus, which is usual for evaporation-like processes at low excitation energies. This selection guarantees that the majority of events have multiplicity M > 2 and thus they correspond to multifragmentation events at high excitation energy. With this selection, nuclear fragments are produced in the vicinity of other nuclear species of similar sizes and lighter particles and/or nucleons. This physical picture corresponds to clusterized nuclear matter at low densities expected in astrophysical processes.

The experimental set up allows an efficient reconstruction of these events [13]. As demonstrated in figure 1, the Z distribution of fragments from the reaction ³²S (45 MeV/nucleon)+ ¹¹²Sn calculated via the DIT+SMM model framework (without the FAUST detector filtering) deviates from the experimental data for fragments larger than Z = 8. This effect is essentially due to the FAUST detector acceptance. (The reason is that large fragments predominantly have relatively low excitation energy and deexcite mainly via neutron evaporation.) The Z distribution resulting after filtering of the calculated distribution is in good agreement with the data. Thus, the filtering only slightly affects small and intermediate fragments with Z = 1–8, which is the group of fragments selected for the present analysis, as mentioned above.

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The excitation energy of the reconstructed quasi-projectile was estimated according to the equation:

$$\begin{eqnarray} &&E_{\rm app}^{*}=\sum _{i}^{}\big(T^{{\rm QP}}_{i}+\Delta M_{i}\big)-\Delta M_{{\rm QP}} \end{eqnarray} \tag{ 1 }$$

where T^QP_i are the kinetic energies of the detected charged particles in the center-of-mass system of the quasi-projectile source, ΔM_i are their mass excesses and ΔM_QP is the mass excess of the quasi-projectile source. This estimate of the quasi-projectile excitation energy represents an apparent excitation energy, since the emitted neutrons were not taken into account. These constraints provide for a well-defined, equilibrated, quasi-projectile source for further analysis. Figure 2 shows the distribution of perpendicular velocity v_⊥ versus parallel velocity v_∥ for fragments with Z = 2 (left panel) and Z = 3–8 (right panel). The velocity of each fragment was transformed into the center-of-mass of the quasi-projectile source. In this frame, the coordinate system was rotated such that the parallel direction (z-axis) coincides with the velocity direction of the quasi-projectile. These distributions demonstrate that, indeed, a well-defined equilibrated source is obtained. The v_⊥ versus v_∥ distributions are isolated in space as suggested by the absence of other populated regions which could indicate a target-like source or a neck-like region. These features are essential in constraining the source properties for further comparisons to theoretical models.

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The calculations presented are based on a two-stage Monte Carlo approach. The dynamical stage of the collision was described by the DIT code [15] simulating stochastic nucleon exchange in peripheral and semiperipheral collisions. This model has been successful in describing the isospin, excitation energy and kinematic properties of excited quasiprojectiles in a variety of recent studies [16–20]. We remind that dynamical models used for the description of the initial stage of a heavy-ion reaction indicate that after a time interval of a few tens of fm c⁻¹, when fast particles have left the system, the character of the process in the remaining nuclear residue changes. The remaining hot nuclear system expands and breaks-up into primary fragments. For the peripheral collisions analyzed in this work, the use of the DIT model for the primary interaction stage is thus supported. For the subsequent de-excitation stage, the SMM was employed. SMM is based upon the assumption of statistical equilibrium between the produced fragments in a low-density freeze-out volume [21]. All breakup channels (partitions) composed of nucleons and excited fragments are considered and the conservation of mass, charge, momentum and energy is taken into account. An advantage of the model is that the formation of a compound nucleus is included as one of the channels. This allows for a smooth transition from decays via evaporation and fission at low excitation energies [22] to multifragmentation at high excitation. In the microcanonical treatment, the statistical weight of the decay channel j is given by W_j∝exp S_j, where S_j is the entropy of the system in channel j which is a function of the excitation energy E*, mass number A₀, charge Z₀ and other parameters of the source. After break-up in the freeze-out volume, the fragments propagate independently in their mutual Coulomb field and undergo secondary decays. The de-excitation of the hot primary fragments proceeds via evaporation, fission or Fermi breakup (for A < 16) [23].

In SMM, light fragments with mass number A ⩽ 4 are considered as stable particles (nuclear gas) with masses and spins taken from the nuclear tables. Only translational degrees of freedom of these particles contribute to the entropy of the system. Fragments with A > 4 are treated as heated nuclear liquid drops, and their individual free energies F_AZ are parameterized as a sum of the bulk, surface, Coulomb and symmetry energy contributions:

$$\begin{equation} F_{AZ}=F^{B}_{AZ}+F^{S}_{AZ}+E^{C}_{AZ}+E^{\rm sym}_{AZ}. \end{equation} \tag{ 2 }$$

In this standard expression F^B_AZ = ( − W₀ − T²/₀)A is the bulk energy term, where the parameter ₀ is related to the level density, and W₀ = 16 MeV is the binding energy of infinite nuclear matter; $F^{S}_{AZ}=B_0A^{2/3}\left(\frac{T^2_c-T^2}{T^2_c+T^2}\right)^{5/4}$ is the surface energy term, where B₀ = 18 MeV is the surface coefficient, and T_c = 18 MeV is the critical temperature of infinite nuclear matter; E^sym_AZ = C_sym(A − 2Z)²/A is the symmetry energy term, where C_sym = 25 MeV is the standard value for the symmetry energy parameter. These parameters are those of the Bethe–Weizsacker formula and correspond to the assumption of isolated fragments with normal density in the freeze-out configuration, an assumption found to be quite successful in many applications. Nevertheless, in a more realistic treatment, primary fragments in the freeze-out volume should be considered with modifications as a result of a residual nuclear interaction between them. These effects can be accounted for in the fragment free energies by changing the corresponding liquid-drop parameters, if such modifications are indicated by the experimental data. The secondary evaporation model was adapted to describe secondary de-excitation taking into account new modified parameters [24].

The present version of SMM [25] is based on generating a Markov chain of partitions which is representative of the whole partition ensemble. Individual partitions are generated and selected into the chain by applying the Metropolis algorithm and taking into account that fragments with the same mass A and charge Z are indistinguishable. Due to the high efficiency of this method, the Coulomb interaction energy is directly calculated for each spatial configuration of primary fragments at the break-up. Also, the moments of inertia for individual configurations are calculated and angular momentum conservation is taken into account. In this way, the correlations between positions of the primary fragments and their Coulomb and rotational energies that influence the partition probabilities are taken into account. Furthermore, the influence of the target–projectile Coulomb energy on the multifragmentation of the projectile can be considered in this version of SMM [25]. However, it was not the focus of the present work. The present version of SMM is consistent with the previous one [21] based on the Wigner–Seitz approximation: for an isolated thermal source, the mean parameters of the produced fragments obtained in these two versions agree with each other reasonably well.

We believe that for relatively small systems, the Markov-chain SMM is a better choice for the analysis of nuclear multifragmentation data concerning the isospin degree of freedom, as angular momentum may be important. In the calculations presented in this work, the freeze-out volume was taken to be six times larger than the normal nuclear volume. We have verified that the variation of the freeze-out volume within reasonable values (e.g., three to six times the normal nuclear volume) does not influence the isotopic distributions of the produced fragments. We point out that the whole ensemble of events generated by the DIT code was furnished as input to the SMM code. We note that the quasiprojectiles generated by DIT are at normal density. When furnished to SMM, they are treated as already expanded at the assumed density (see above). In the present treatment we assumed that the energy spent for the expansion is negligible. Furthermore, to compare the results of the DIT/SMM calculations with the experimental data, the theoretical results were filtered with a software representation of the FAUST detector array (FAUST filter). The FAUST filter takes into account the geometry of the FAUST array, the energy thresholds of each element and isotope for each detector telescope, as well as the elemental and isotopic resolution in each detector telescope, as determined by the experimental data (see relevant discussion in section 2, in conjunction with figure 1). This filtering procedure allows for the theoretical calculations to be treated in exactly the same way as the experimental data.

For the ³²S + ¹¹²Sn system, reconstructed quasi-projectile sources in a wide mass range are observed. Figure 3 (upper panel) shows the reconstructed sulfur (Z = 16) mass distributions sorted by excitation energy. In this figure, five excitation energy ranges are considered (see figure caption) and the quasi-projectile yields were normalized to the total number of events. We see that the widths of the source mass distributions are large. Also, from figure 3 (lower panel), we see that the average excitation energy of Z = 16 sources is around 4–5 MeV/nucleon.

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We should note that the free neutrons are not taken into account in the determination of the mass number of these sources. Only charged particles are included. We have estimated the number of free neutrons from the DIT+SMM model (see also [13, 14]). Obviously, this number increases with increasing mass and/or N/Z of the source. However, this uncertainty does not influence the following analysis, since, as already mentioned, the theoretical calculations are filtered and, thus, selected in the same way as the experimental data.

Previous work [26] has shown that the elemental cross sections of projectile fragments exhibited a more pronounced odd–even behavior when their isospin T_z (defined here, as usual, as T_z = (N − Z)/2) was equal to zero, when compared to those of projectiles with T_z = 1. In the data of reference [26], the T_z = 1 projectiles produce a rapid, nearly exponential, decrease. Recent works on odd–even staggering [27, 28] indicated that the effect is independent of the first stages of the reaction and mostly appears in the last stage (or stages) of the deexcitation path. Presented in figure 4 are the fragment charge distributions of the reconstructed sources ²⁷S to ³⁷S from the present data. The source isospin spans the range from T_z = −5/2 (neutron-poor) to T_z = 5/2 (neutron-rich). The neutron-poor sources exhibit a strong odd–even behavior which diminishes progressively as the neutron content of the source increases. A exponential-like drop-off occurs for the fragment yields in the case of the most neutron-rich sources.

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The present work will now focus on the fragmentation of nuclei around N = Z, where the experimental statistics has been adequate for detailed studies. Specifically, it will focus on the three reconstructed sources ³⁰S, ³²S and ³⁴S from the reaction ³²S + ¹¹²Sn at 45 MeV/nucleon. First, we investigate the excitation energy dependence of the fragment charge distribution for the neutron-poor and neutron-rich sources. Figure 5 shows the evolution of ³⁰S in the left column and ³⁴S in the right column for the five excitation energy bins. The DIT+SMM calculations were performed with two symmetry energy parameters: the standard one C_sym = 25 MeV and the reduced one C_sym = 8 MeV, since this reduction has been suggested by previous analyses [2, 29–32]. Each point is normalized to the total number of reconstructed quasi-projectiles produced in the given energy bin.

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As shown in figure 5, in both the ³⁰S and ³⁴S cases, the fragment charge distribution changes shape with the increase of the source excitation energy. The general trend in both cases is the transition from a U-shaped charge distribution at low excitation energy to an exponential-like one at high energy, which is well known from previous multifragmentation studies. For the ³⁰S case, there is a more pronounced odd–even effect for the lowest energy bins than in the ³⁴S case. One can see that there is only a slight increase in the normalized yield for the Z = 2 element as the source excitation energy increases. The DIT+SMM calculations behave slightly differently for the Z = 2 element, as being lower than the experimental data at low excitation energy, but increasing to meet the experimental data as the source excitation increases. This is related to the secondary production of α-particles with the Fermi break-up model [23], which takes into account the phase space but not the matrix elements of the break-up channels which can become important at low excitations. Both of the DIT+SMM calculations provide similar results, we thus conclude that the fragment charge distributions are not sensitive to the variation of the symmetry energy of the hot primary fragments.

After studying the charge distributions, the next step is the analysis of the isospin characteristics of the multifragmenting quasi-projectile sources. In figure 6, we show the average neutron-to-proton ratio 〈N/Z〉 of the produced fragments versus Z for the three sources under consideration. The horizontal lines indicate the apparent N/Z of the corresponding QP sources. There is a correlation of the neutron richness of the fragments with the source isospin. Also by moving from the ³⁰S to the ³⁴S source, we see that the 〈N/Z〉 versus Z moves from a strong odd–even behavior (³⁰S) to a more flat distribution (³⁴S). In all cases the theoretical calculations provide a reasonable estimate of the observed system behavior. However, for the most neutron-rich source considered here, the DIT+SMM calculations do not relax the odd–even behavior as much as necessary to match the experimental data, as the calculation does for the ³⁰S and ³²S sources.

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It is instructive to investigate the evolution of the fragment isospin with the excitation energy of the sources. Figure 7 expands the results shown in figure 6 by demonstrating their excitation-energy dependence. Each column corresponds to a source, ^30–34S, and each row represents a given excitation energy range according to the caption of the figure. For the neutron-poor source of ³⁰S, a fairly strong odd–even effect is present at each excitation energy bin. As the neutron content of the system increases, the odd–even effect diminishes at each excitation energy bin. As shown for all three cases, the increase of excitation energy does not appreciably affect the odd–even (or lack thereof) character of the 〈N/Z〉 versus Z correlation. The calculation agrees with the data very well, especially for the ³⁰S and ³²S sources. The error bars reflect the statistics of the calculations. Comparing figures 6 and 7, it can be concluded that the 〈N/Z〉 versus Z correlation is influenced by the N/Z of the emitting source and the source excitation energy has a relatively small effect in most cases. Furthermore, it is rather difficult to distinguish the effect of the symmetry energy for these fragments: the calculations with different C_sym parameters lead to nearly similar average values.

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Apart from the N/Z distributions of the fragments, we also investigated the behavior of the light mirror nuclei ³H and ³He. In figure 8, the evolution of their yield ratio Y_3H/Y_3He as a function of source excitation energy for the three sources is presented. As one can see, this ratio behaves qualitatively differently for the neutron-poor ³⁰S source (showing an increasing trend with energy) to the neutron-rich ³⁴S source (showing a decreasing trend with energy). The DIT+SMM calculations give the same qualitative behavior, though in some cases they deviate in absolute values. Moreover, one can see that the calculations cannot differentiate between the two values of C_sym. The reason may be again that these light particles are produced mostly as evaporated particles or after the Fermi break-up and, therefore, their sensitivity to the symmetry energy of the primary fragments appears to be reduced.

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For further investigation of the isospin effects, the isotopic distributions of produced fragments will be studied in detail. In figure 9, the yields of isotopes with Z > 2 emitted from the reconstructed ³⁰S and ³⁴S quasi-projectile sources are presented. Comparing the isotope distributions from ³⁰S and ³⁴S, we see, as expected, that the more neutron-rich source produces more neutron-rich fragments. Correspondingly, there is a reduction in the yield of the neutron-poor isotopes emitted by the neutron-rich source. Furthermore, when comparing the DIT+SMM calculations to the experimental data, the calculations with the two C_sym values of 25 and 8 MeV do rather well in reproducing the data. A closer examination reveals that the isotopic distributions of most of the elements appear to be better reproduced by the calculation with the reduced value C_sym = 8 MeV of the symmetry energy. Moreover, there are discrepancies which can not be totally eliminated by changing the value of C_sym parameter. These discrepancies may be related to low-energy structure effects and, to some extent, to the partitioning of the free neutrons which are not accounted for in the present analysis. The observed weak sensitivity to the symmetry energy of the primary fragments may be, in part, due to the fact that the light intermediate-mass fragments are produced in the calculations mainly after secondary Fermi break-up [23], which is not sensitive to modifications of the symmetry energy. On the contrary, heavier intermediate-mass fragments, as in the case of the nitrogen (Z = 7) isotopes, are produced from the primary fragments mainly after the secondary evaporation [23, 24], which takes into account the evolution of the symmetry energy during the de-excitation cascade. Therefore, these heavier fragments are expected to be more sensitive indicators of the symmetry energy variation.

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In the case of the nitrogen (Z = 7) fragments presented in figure 9, whereas the DIT+SMM calculation with the reduced symmetry energy C_sym = 8 MeV does indeed a good job to describe the data from the neutron rich source ³⁴S, it fails to do so for the data from the neutron-poor ³⁰S source. Interestingly, for the ³⁰S source, it is the calculation with the standard value of C_sym = 25 MeV that describes the data. (A separate calculation with C_sym = 14 MeV demonstrates that with an intermediate value of C_sym, the calculation cannot adequately describe the data from either source.)

Thus, the present data indicate that the isotopic distributions of the lighter intermediate-mass fragments with Z = 3–6 favor a reduced value of C_sym= 8 MeV for both the neutron-rich, and the neutron-poor source. Nevertheless, the nitrogen isotopes provide some hint that the symmetry energy of the heavy primary fragments may be lower only in the case of neutron-rich multifragmenting nuclear system. We note that this result does not corroborate a possible explanation based on the influence of the surface symmetry energy term [33, 34]. This surface term for large neutron-rich primary fragments (with the same charge Z) would lead to an increase of their total symmetry energy, i.e. to an opposite result from that obtained. A plausible interpretation may point to the residual interaction between the fragments in the freeze-out volume which depends on the number of surrounding neutrons. Microscopic dynamical calculations are necessary to shed light to this interesting issue. Nevertheless, this observation needs to be further explored experimentally by high-resolution data on heavier intermediate-mass fragments and from heavier sources. Based on the above results and discussion, we wish to point out that the heavier fragments produced in multifragmentation may constitute a sensitive experimental probe of the symmetry energy characterizing the primary fragments (see also [33–35]) and warrant further detailed investigation.

We have selected experimental data on the multifragmentation of reconstructed quasi-projectile sources with considerable excitation energy. The data show a dependence of the fragmentation pattern (both with respect to mass and charge) on the neutron content of these sources. This behavior can in general be adequately described by the DIT+SMM model framework, suggesting that the decay occurs in a statistically equilibrated state. Multifragmentation events from the decay of reconstructed quasi-projectile sources were selected for the analysis. According to the physical picture of the freeze-out volume in the SMM, these events represent an ensemble of nuclear fragments in close proximity to each other. This picture corresponds to clusterized nuclear matter at low density which also exists in stellar matter of supernovae and neutron stars (see, e.g., [2, 8] and references therein). We have investigated the symmetry energy of these primary fragments by comparing the DIT+SMM calculations with the experimental data. A good overall description of the experimental distributions of fragments is obtained by the calculations involving both the standard (C_sym = 25 MeV), as well as a reduced symmetry energy of C_sym = 8 MeV, even if a slightly better description of the isotopic distributions is obtained in most cases with the reduced value of C_sym. This is in agreement with the observation of a reduced symmetry energy of primary fragments from a variety of multifragmentation studies of heavier systems (e.g., [29–31, 33, 35]).

As demonstrated in this work, it is possible to experimentally reconstruct and select sources with a wide range of isospin content and excitation energies and, thus, probe the symmetry energy and other properties of primary fragments in close proximity with other clusters and/or nucleons. This approach can accurately model the physical conditions expected in a number of astrophysical environments, such as core-collapse supernova and the crusts of neutron stars.

We wish to thank the Cyclotron Institute staff for excellent beam quality. This work was supported in part by the Robert A Welch Foundation through grant no. A-1266, and the Department of Energy through grant no. DE-FG03-93ER40773. We also wish to thank L Tassan-Got for using his DIT code. AB thanks the Cyclotron Institute of TAMU for support and hospitality.