A Monte Carlo approach to study neutron and fragment emission in heavy-ion reactions

Advances in Space Research 40 (2007) 1350–1356 www.elsevier.com/locate/asr

A Monte Carlo approach to study neutron and fragment emission in heavy-ion reactions M.V. Garzelli

a,b,*

, P.R. Sala b, F. Ballarini c, G. Battistoni b, F. Cerutti d, A. Ferrari d, E. Gadioli a,b, A. Ottolenghi c, L.S. Pinsky e, J. Ranft f a

c

University of Milano, Department of Physics, via Celoria 16, I-20133 Milano, Italy b INFN, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy University of Pavia, Department of Theoretical and Nuclear Physics, and INFN, Sezione di Pavia, via Bassi 6, I-27100 Pavia, Italy d CERN, CH-1211 Geneva, Switzerland e University of Houston, Department of Physics, 617 Science & Research Building 1, Houston, TX 77204-5005, USA f Siegen University, Fachbereich 7 – Physik, D-57068 Siegen, Germany Received 1 November 2006; received in revised form 6 July 2007; accepted 18 July 2007

Abstract Quantum Molecular Dynamics models (QMD) are Monte Carlo approaches targeted at the description of nucleon–ion and ion–ion collisions. We have developed a QMD code, which has been used for the simulation of the fast stage of ion–ion collisions, considering a wide range of system masses and system mass asymmetries. The slow stage of the collisions has been described by statistical methods. The combination of both stages leads to final distributions of particles and fragments, which have been compared to experimental data available in the literature. A few results of these comparisons, concerning neutron double-differential production cross-sections for C, Ne and Ar ions impinging on C, Cu and Pb targets at 290–400 MeV/A bombarding energies and fragment isotopic distributions from Xe + Al at 790 MeV/A, are shown in this paper.  2007 Published by Elsevier Ltd on behalf of COSPAR. Keywords: Heavy-ion collisions; QMD; Fragmentation; Evaporation

1. Introduction Since several years, Quantum Molecular Dynamics models (QMD) have been introduced to simulate heavyion collisions (see e.g. Aichelin, 1991 and references therein, Niita et al., 1995). They are semi-classical approaches (Boal and Glosli, 1988), derived from the Classical Molecular Dynamics ones, by adding a few quantum features. The main characteristic of QMD is that each nucleon is described by a coherent state, and is thus identified by gaussian spatial coordinate and momentum distributions, centered around mean values, and characterized by gaussian widths the product of which minimizes *

Corresponding author. E-mail address: [email protected] (M.V. Garzelli).

0273-1177/$30  2007 Published by Elsevier Ltd on behalf of COSPAR. doi:10.1016/j.asr.2007.07.026

the Heisenberg uncertainty relationship. A second quantum feature is the fact that nucleon–nucleon scattering cross-sections taken from the experiments are used to describe nucleon–nucleon collisions occurring in the fast stage of the reactions between ions, when nuclei can overlap, depending on the impact parameter value. The impact parameters are randomly chosen, according to a Monte Carlo procedure. Thus both very peripheral reactions and more central ones can occur, on an event by event basis. Actually the measured free nucleon–nucleon crosssections are used when a nucleon–nucleon collision occurs in these reactions, whereas the effects on each nucleon of all other nucleons present both in the projectile ion and in the target one are taken into account by means of effective potentials, included in the interaction part of the Hamiltonian, during the whole simulation. Another quantum

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feature is the fact that the collisions which lead a nucleon in an already occupied phase-space region are forbidden, according to the Pauli blocking principle. This is especially effective at non-relativistic energies, where nucleon–nucleon potentials are increasingly more important than nucleon–nucleon collisions to determine the fragmentation process, since most nucleon–nucleon collisions are blocked. Finally, a few authors have also developed fully antisymmetrized versions of QMD models (e.g. Fermionic Molecular Dynamics by Feldmeier (1990); Feldmeier and Schnack (2000), and Antisymmetrized Molecular Dynamics by Ono et al. (1992a); Ono et al. (1992b)), by fully antisymmetrizing the nuclear wave-functions, which in the original QMD versions are merely given by the product of nucleon wave-functions. At present, these advanced versions are successfully used in very specific problems, such as the study of exotic nuclear structure, mainly for light nuclei due to the complexity of the underlying approaches, and their intensive CPU requirement. We limit to the original approximation for the nuclear wave-functions, instead of using a more advanced approach, since our final aim is developing a code to be used real-time to microscopically describe ion–ion collisions in general applicative problems, which can involve even the heaviest ions and composite targets. To accomplish this aim, a preliminary version of our QMD model has been interfaced to the general purpose Monte Carlo FLUKA code, allowing the simulation of fragment formation and the study of the emission of fragmentation products, in thin and thick targets, even in complex geometries. Nevertheless, to characterize and investigate the nuclear reaction processes better, only simulations concerning thin target experiments are shown in this paper. 2. Theoretical elements of our simulations A QMD code has recently been developed by our collaboration and interfaced to the FLUKA code. FLUKA (Fasso` et al., 2005) is a particle transport and interaction code, used since several years in many fields of Physics. It is currently developed according to an INFN–CERN official agreement and is made available on the web (http://www.fluka.org). It is based, as far as possible, on theoretical microscopic models, which are continuously updated and improved, and benchmarked with respect to newly available experimental data. Nucleon–nucleus collisions are simulated in FLUKA by means of the module called PEANUT (PreEquilibrium Approach to NUclear Thermalization, Ferrari and Sala, 1998), which contains also a de-excitation sub-module. This sub-module has also been used to describe the further nuclear de-excitation which may occur at the end of the overlapping stage of ion–ion collisions, which has been simulated in this work by means of our QMD code. In particular, the QMD code has been used to follow the ion–ion system evolution for a few hundreds of fm/c, whereas de-excitation processes can

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occur on a time scale far larger (up to 107 fm/c), and have been simulated by means of the statistical models contained in the FLUKA de-excitation sub-module. Deexcitation is provided by different mechanisms, in competition among themselves. Therefore the sequence of steps leading to a ground-state nucleus is non-deterministic and can involve more than one process (evaporation of different particles, in competition with fission and fragmentation, followed by c emission). A Monte Carlo procedure is used to describe each nuclear de-excitation step, by choosing which particle to emit and its energy, on the basis of relative widths computed in the framework of the Weisskopf–Ewing formalism with modifications, taking into account (A, Z) and energy dependent level density parameters, pairing energies, and fission barriers. The evaporation model is an extended version of previous ones (Ferrari et al., 1996a,b, 1998), implemented in FLUKA since a few years with many refinements (Fasso` et al., 2001; Ballarini et al., 2005) including sub-barrier effects and a full energy dependence of the nuclear level densities. The evaporation/fragmentation algorithm in its extended version can treat the emission of more than 600 different fragments with A 6 24, in their ground or excited states. For light residuals (A 6 17), evaporation is substituted by statistical fragmentation into up to 6 fragments according to phase-space considerations as implemented in a Fermi break-up model. The fission algorithm, recently revised, is computing the fission widths, Cfiss, taking into account barriers from Myers and Swiatecki (1999) and the customary enhancement of the level density at the saddle point with respect to the one used for neutron and charged particle emission. The latter is now supposed to wash out with excitation energy as it should be, resulting in a much improved mass distribution of residuals from proton–nucleus interactions in the actinide region. Finally, de-excitation by photon emission is also included in FLUKA according to statistical considerations (Ferrari et al., 1996a; Ferrari and Sala, 1998). In the FLUKA implementation used to do the work presented in this paper, c de-excitation always follows the other processes, for simplicity. However the algorithm can optionally describe c emission in competition with other particle emission channels, albeit with some CPU penalties which are usually not worthwhile, given the low probability for this channel. A correct description of de-excitation is crucial, since it deeply modifies the particle and fragment pattern present at the end of the fast stage of ion–ion collisions. In fact, while peripheral collisions often give rise to slightly excited projectile-like and target-like fragments, with masses very similar to the projectile and target ones, respectively, very central collisions are capable of generating intermediate mass fragments (IMF), characterized by lower masses and higher excitation energies: hot fragments in turn decay, deeply affecting the final fragment and particle emission distributions. A few features of our QMD code are

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presented in Section 2.1, while the results of our simulations are shown in Section 3.

3. Results of the simulations and comparisons with experimental data

2.1. QMD specific features

As already mentioned, the results presented in this paper have been obtained by the interface between our QMD code and the FLUKA nuclear de-excitation routines, based on the models described in Section 2. In Section 3.1 doubledifferential neutron production cross-sections for several projectile–target systems at energies 290 and 400 MeV/A are compared to available experimental data, while in Section 3.2 fragment isotopic distributions from Xe projectiles impinging on an aluminium target at 790 MeV/A are shown. QMD + FLUKA has also been tested at energies around and below 100 MeV/A. A few results of these lower-energy tests have been presented in Garzelli et al. (2006a,b,c).

As explained in Section 1, several QMD versions and codes have already been developed, characterized by some common features, and some specific differences, depending on the working group and the intended application. Here some features of the code we have developed are briefly sketched. The Hamiltonian is made by a kinetic and an effective interaction part. Our code can switch between non-relativistic and relativistic kinematics, according to a user initial choice, whereas nucleon–nucleon potentials are treated classically. A fully relativistic version, including relativistic potentials, is also under development thanks to the cooperation with the University of Houston and NASA, and is the object of a separate work (Zapp et al., 2006). Neutrons and protons are distinguished both for their interaction and for their mass. The initial nuclear states, used in the ion–ion collision simulation processes, are prepared in advance, stored off-line and exactly fulfill the experimental constraints on binding energies. Binding energies are calculated by summing the kinetic energy and the potential energy of all nucleons in the nucleus. The potential energy is given by a superposition of different terms, due to the interaction of each nucleon with all other nucleons present in the ion. In particular, Skyrme-type two-body and three-body interaction terms are included, as well as two-body symmetry and surface terms. The symmetry term helps taking into account the difference between neutrons and protons, by assigning different weights to the interactions between nucleons with the same isospin and nucleons with different isospin. Furthermore, a Coulomb term is included, which acts only on protons. Further details on the effect of each term in determining the total potential energy of each ion can be found in Garzelli et al. (2006b). When two ions collide, each nucleon is subject to the effects of both the nucleons in the same nucleus and those in the other one. At ion–ion distances exceeding a few fm, only the Coulomb force is important, whereas nuclear interaction among nucleons of different nuclei becomes significant only at lower distances, i.e. during the ion–ion overlapping process. Also nucleon–nucleon collisions can occur in this stage, which are implemented in our code taking into account free n–p and p–p (n–n) reaction cross-sections. As an approximation, this process is assumed to be isotropic, i.e. the angular dependence of the cross-sections is neglected, as well as the pion production channel. Further extensions of the model will include these elements. Ion–ion system evolution is followed for a time of 150 200 fm/c after the overlapping stage. At the end, hot excited fragments may be present, as well as emitted neutrons and protons. The de-excitation of hot fragments has been simulated thanks to the interface to the FLUKA de-excitation module, as explained at the beginning of Section 2.

3.1. Double-differential neutron production cross-sections Several papers presenting neutron double-differential production cross-section data have been published. Most of them consider thick targets. Nevertheless, at the aim of investigating the reliability and the prediction capabilities of theoretical ion–ion reaction models, it is better disentangling transport effects, thus data from thin target experiments are more suited. Iwata et al. (2001) present double-differential neutron production cross-sections for many systems, mostly considering thin targets. Therefore, the experimental data presented by these authors have been considered for our simulations. The targets used in their experiment are made of carbon, copper and lead. The results of our simulations for a carbon beam at a 290 MeV/A bombarding energy and neon and argon beams at 400 MeV/A, hitting carbon, copper and lead ions, are shown in Fig. 1, and compared to the experimental data. Different histograms refer to different emission angles. In particular, from the top to the bottom of each panel one can distinguish neutrons emitted at 5, 10, 20, 30, 40, 60 and 80 angles with respect to the incoming beam direction in the laboratory frame. The results at different angles have been multiplied by the same factors used in Iwata et al. (2001) for display purposes. The results of the theoretical simulations show an overall reasonable agreement with the experimental data, especially at emission angles P20. On the other side, at forward emission angles (5–10), the simulations give rise to neutron emission peaks in some case lower than those experimentally observed. The largest underestimations are seen in the case of the Pb target, and can amount even up to a factor 3– 3.5, while for lighter targets, when occurring, they stay within a factor 1.3–2.2. In the case of carbon projectiles at 290 MeV/A these peaks are located at energies slightly higher than the experimentally measured ones. Our simulations have been performed in the infinitely thin target approximation, neglecting all target geometry effects, i.e. each target is merely given by a single ion. Ion energy losses in the target spread and reduce the effective interaction

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Fig. 1. Double-differential neutron production cross-section for C (top), Ne (center) and Ar (bottom) projectiles impinging on C (left), Cu (center) and Pb (right) at 290, 400, 400 MeV/A bombarding energies, respectively. The results of the theoretical simulations made by QMD + FLUKA are shown by solid histograms, while the experimental data measured by Iwata et al. (2001) are shown by filled circles. In each panel, distributions at 5, 10, 20, 30, 40, 60 and 80 angles with respect to the incoming beam direction in the laboratory frame, have been multiplied by decreasing even powers of 10, for display purposes.

energy by about one-half of the average energy loss. For the experiments at 290 MeV/n under consideration, the energy loss is of the order of 5% (C and Pb targets) and 10% (Cu target). Therefore the inclusion of the target thickness in the simulations (as in Iwase et al., 2002) marginally

improves the results (shifting the small angle peak by typically few percent). The same targets used in the experiments with carbon projectiles at 290 MeV/A have also been used by Iwata et al. (2001) with neon projectiles at 400 MeV/A. In this

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case, the agreement of the results of the theoretical simulations with the experimental data is more satisfactory, both regarding the location of the peaks and their height. Finally, as far as argon projectiles at 400 MeV/A are concerned, the energy location of the experimental peaks is reproduced, but the theoretical simulation fails in predicting the width of the distributions, underestimating the peak heights and neutron emission tails at forward angles at energies above the bombarding one. One source of discrepancy could be ascribed to the fact that isotropic n–p and p–p cross-sections have been used in the simulations, instead of considering the angular dependence really observed in nucleon–nucleon scattering experiments. The inclusion of proper anisotropic nucleon–nucleon scattering could further improve the agreement of the theoretical simulations with the experimental data. As far as intermediate emission angles are concerned, the agreement is instead already quite satisfactory. We emphasize that the absolute results of our simulations shown in Fig. 1 have been obtained without making use of any normalization factor. The spectra at intermediate and backward angles have been attributed by Iwata et al. (2001) to the partial superposition of two components. The equilibrium component corresponds to the lower kinetic energy (En < 20 MeV) part of the spectra, where an increasing number of emitted neutrons with decreasing energy can be observed, while the cascade/pre-equilibrium component is given by a wide peak extending up to a few hundreds MeV/A. The theoretical simulations made by QMD + FLUKA appear to agree in predicting the presence of both these distributions. From the computational point of view, typical CPU times required to perform these simulations on a commercial PC amount to tens of seconds for an event of intermediate mass ion–ion collision. As an example, a Ne + Ar collision is completed in 10–20 s on a PC equipped by a Pentium 4 Processor, with 3.00 GHz CPU. The slowness is due to the QMD part of the simulation, in particular, to the evolution of the colliding nuclei before and after their overlapping. Up to now most of the efforts were devoted to physics: significant speed gains should be possible by a careful optimization of the code. 3.2. Fragment isotopic distributions When two ions collide at energies of a few hundreds MeV/A, highly excited fragments can be formed, which in turn decay to lower mass and charge ones by de-excitation processes. Isotopic distributions for projectile-like fragments have been measured by Reinhold et al. (1998) at GSI in the case of 129Xe ions at a 790 MeV/A bombarding energy, impinging on an aluminium target. We have simulated about 10,000 129Xe + 27Al reaction events at the same bombarding energy using QMD + FLUKA. The results of our simulations are presented in Fig. 2. In each panel the production cross-section is shown for the isotopes of a different element, ranging from Z = 54 to Z = 40 (projectile-like isotopes). Together with the results

of our theoretical simulations, also the experimental data taken from Reinhold et al. (1998) are plotted. We can appreciate a better agreement between the theoretical and experimental distributions for the elements the atomic number of which is very close to the projectile Z value (Z  54–50), than for the lighter fragments (Z  44–40). While the first ones are more copiously produced, the last ones are more rarely emitted, as follows by comparing their production cross-sections. The results of our simulations refer to fragment emitted over the whole solid angle. At the considered energy, higher mass and charge fragments are produced in more peripheral collisions, and propagate in forward direction, whereas the lower mass and charge ones can be emitted also at larger angles. Furthermore, at fixed Z, the simulations seem to overestimate the N richer tails, while underestimating the higher Z ones. Indeed, an asymmetry between the two tails is evident also from the experimental data, but to a lesser extent than obtained by the theoretical simulations. Reinhold et al. (1998) explain their results suggesting that neutron-deficient fragments are formed from prefragments with high excitation energies, and consequently a crucial contribution to their formation could be played by evaporation processes. Since neutrons can evaporate more easily than protons, due to Coulomb barrier effects on the last ones, the evaporation of neutrons would give rise to these fragments. According to this scenario, the theoretical excess of N richer fragments could be ascribed to the fact that the excitation energies of the fragments at the end of the fast stage of the collision process are not high enough. We observe that repeating the same simulations by using a completely different model to describe the fast stage of ion–ion collisions, i.e. the relativistic QMD code RQMD2.4 developed at the University of Frankfurt a few years ago by Sorge et al. (1989a,b), and already coupled to FLUKA by our Collaboration (Aiginger et al., 2005), leads to very similar results, also plotted in Fig. 2. A comparison between the results of the two models shows that at a 790 MeV/A bombarding energy the isotopic distributions are rather insensitive to the fact that fully relativistic potentials are used, or non-relativistic ones, as are the cases of RQMD and QMD, respectively. Including a pre-equilibrium phase after the fast stage of the collisions, with a smooth transition to the equilibrium stage described by statistical models, could help in gaining further insight on the reasons of the discrepancies with experimental data. The total theoretical charge fragment yield as a function of Z is presented in Fig. 3 for completeness, including also fragments with Z < 40, the distributions of which have not been measured in the experiment by Reinhold et al. (1998). 4. Conclusions We have performed simulations of ion–ion collision events, in the approximation of neglecting target geometry effects such as beam energy loss and reinteraction during propagation, by using the QMD we have developed in

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Fig. 2. Isotopic distributions of projectile-like fragment production cross-sections (barn) for a 790 MeV/A 129Xe beam hitting an Al target. Each panel contains the results for the isotopes of a different element. The results of the theoretical simulations of about 10,000 129Xe + 27Al reaction events made by QMD + FLUKA are shown by open squares, while the experimental data taken from Reinhold et al. (1998) are shown by crosses. The results obtained by a modified version of the fully relativistic code RQMD2.4 + FLUKA are also superimposed on each plot (dotted lines).

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Fig. 3. Charge fragment yield for the same projectile–target combination considered in Fig. 2. The theoretical results predicted by QMD + FLUKA (filled squares) are compared to those predicted by RQMD2.4 + FLUKA (asterisks).

the last few years, interfaced to the statistical de-excitation module taken from the general purpose Monte Carlo FLUKA code. Results of simulations at bombarding energies around and below 100 MeV/A have been shown in other papers, whereas here results were shown concerning different projectile–target combinations at energies of a few hundreds MeV/A. Specific observables, such as neutron double-differential production cross-sections at different angles and fragment isotopic distributions, have been calculated. Comparisons of the results with experimental data suggest an overall reasonable agreement, though some discrepancies also appear, which need further investigation. In particular, the inclusion in the model of the angular dependence of nucleon–nucleon cross-sections and of further pre-equilibrium emissions during the thermalization of the hot nuclei left after the fast reaction stage described by QMD, can in our opinion be crucial to further improve the agreement with the experimental data. References Aichelin, J. ‘‘Quantum’’ molecular dynamics – a dynamical microscopic nbody approach to investigate fragment formation and the nuclear equation of state in heavy-ion collisions. Phys. Rep. 202, 233–360, 1991. Aiginger, H., Andersen, V., Ballarini, F., et al. The FLUKA code: new developments and application to 1 GeV/n iron beams. Adv. Space Res. 35, 214–222, 2005. Ballarini, F., Battistoni, G., Cerutti, F., et al. Nuclear models in FLUKA: present capabilities, open problems and future improvements, in: Haight, R.C., Chadwick, M.B., Kawano, T., Talou, P. (Eds.), Proceedings of ND2004, ‘‘International Conference on Nuclear Data for Science and Technology’’, Santa Fe, NM, September 26–October 1 2004. AIP Conference Proceedings 769, 1197–1202, 2005. Boal, D.H., Glosli, J. Computational model for nuclear reaction studies: quasiparticle dynamics. Phys. Rev. C 38, 2621–2629, 1988. Fasso`, A., Ferrari, A., Ranft, J., Sala, P.R. FLUKA: a multi-particle transport code, CERN Yellow Report 2005-10, INFN/TC_05/11, 1– 387, 2005.

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