# Pre-Equilibrium Clustering in Production of Spectator Fragments in Collisions of Relativistic Nuclei

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## Abstract

**:**

## 1. Introduction

## 2. Description of AAMCC-MST Model

- Removal of participating nucleons from the initial nuclei (abrasion) resulting in the formation of an excited prefragments;
- De-excitation of prefragments (ablation) by their decay into spectator nucleons and nuclear fragments.

## 3. Fragmentation of ${}^{197}$Au in Nuclear Emulsion

## 4. Multiplicity of Spectator Nucleons in ${}^{208}$Pb–${}^{208}$Pb Collisions

## 5. Yields of Various Elements from Fragmentation of ${}^{\mathbf{16}}$O

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MST | Minimum Spaning Tree; |

SMM | Statistical model of multifragmentation; |

AAMCC | Abrasion-Ablation Monte Carlo for Colliders; |

AAMCC-MST | AAMCC model with MST-clustering algorithm. |

## References

- Puddu, G.; Arnaldi, R.; Chiavassa, E.; Cicaló, C.; Cortese, P.; De Falco, A.; Dellacasa, G.; Ferretti, A.; Floris, M.; Gagliardi, M.; et al. The zero degree calorimeters for the ALICE experiment. Nucl. Instrum. Methods A
**2007**, 581, 397–401. [Google Scholar] [CrossRef] - Svetlichnyi, A.; Nepeyvoda, R.; Pshenichnov, I. Using Spectator Matter for Centrality Determination in Nucleus-Nucleus Collisions. Particles
**2021**, 4, 227–235. [Google Scholar] [CrossRef] - Svetlichnyi, A.O.; Pshenichnov, I.A. Formation of Free and Bound Spectator Nucleons in Hadronic Interactions between Relativistic Nuclei. Bull. Russ. Acad. Sci. Phys.
**2020**, 84, 911–916. [Google Scholar] [CrossRef] - Pshenichnov, I.A.; Dmitrieva, U.A.; Svetlichnyi, A.O. Secondary Nuclei from Peripheral and Ultraperipheral Collisions of Relativistic Heavy Ions. Bull. Russ. Acad. Sci. Phys.
**2020**, 84, 1007–1011. [Google Scholar] [CrossRef] - Pshenichnov, I.A.; Kozyrev, N.A.; Nepeivoda, R.S.; Svetlichnyi, A.O.; Dmitrieva, U.A. Properties of Spectator Matter in Nuclear Collisions at NICA. Phys. Part. Nucl.
**2021**, 52, 591–597. [Google Scholar] [CrossRef] - Botvina, A.S.; Mishustin, I.N.; Begemann-Blaich, M.; Hubele, J.; Imme, G.; Iori, I.; Kreutz, P.; Kunde, G.J.; Kunze, W.D.; Lindenstruth, V.; et al. Multifragmentation of spectators in relativistic heavy-ion reactions. Nucl. Phys. A
**1995**, 584, 737–756. [Google Scholar] [CrossRef][Green Version] - Bondorf, J.; Botvina, A.; Iljinov, A.; Mishustin, I.; Sneppen, K. Statistical multifragmentation of nuclei. Phys. Rep.
**1995**, 257, 133–221. [Google Scholar] [CrossRef] - Bairathi, V.; Haque, M.R.; Mohanty, B. Selecting specific initial configurations using spectator neutrons in U+ U collisions. Phys. Rev. C
**2015**, 91, 054903. [Google Scholar] [CrossRef][Green Version] - Bairathi, V.; Chatterjee, S.; Haque, M.R.; Mohanty, B. Probing Pb+Pb collisions at $\sqrt{{s}_{NN}}=2760$ GeV with spectators. Phys. Lett. B
**2016**, 754, 144–150. [Google Scholar] [CrossRef][Green Version] - Adamovich, M.I.; Aggarwal, M.M.; Alexandrov, Y.A.; Amirikas, R.; Andreeva, N.P.; Anzon, Z.V.; Avetyan, F.A.; Badyal, S.K.; Bakich, A.M.; Baklitskaya, E.; et al. Multifragmentation of Gold nuclei in the interactions with photoemulsion nuclei at 10.7 GeV/nucleon. Z. Phys. A
**1997**, 359, 277–290. [Google Scholar] [CrossRef] - Acharya, S.; Adamová, D.; Adler, A.; Adolfsson, J.; Aggarwal, M.M.; Aglieri Rinella, G.; Agnello, M.; Agrawal, N.; Ahammed, Z.; Ahmad, S.; et al. Data-Driven Model for the Emission of Spectator Nucleons as a Function of Centrality in Pb-Pb Collisions at Lhc Energies. Available online: https://cds.cern.ch/record/2712412?ln=bg (accessed on 28 December 2021).
- El-Nagdy, M.; Abdelsalam, A.; Badawy, B.; Zarubin, P.; Abdalla, A.; Yasin, M.N.; Saber, A.; Mohamed, M.; Ahmed, M. Channels of projectile fragmentation of 16O nucleus in nuclear emulsion. J. Phys. Commun.
**2018**, 2, 035010. [Google Scholar] [CrossRef][Green Version] - Nepeivoda, R.S.; Svetlichnyi, A.O. Dependence of n/p-ratio in spectator matter on the energy and mass of colliding nuclei. Mem. Fac. Phys.
**2021**, 1, 2110302. [Google Scholar] - Loizides, C.; Kamin, J.; d’Enterria, D. Improved Monte Carlo Glauber predictions at present and future nuclear colliders. Phys. Rev. C
**2018**, 97, 054910. [Google Scholar] [CrossRef][Green Version] - Ericson, T. The statistical model and nuclear level densities. Adv. Phys.
**1960**, 9, 425–511. [Google Scholar] [CrossRef] - Scheidenberger, C.; Pshenichnov, I.A.; Sümmerer, K.; Ventura, A.; Bondorf, J.P.; Botvina, A.S.; Mishustin, I.N.; Boutin, D.; Datz, S.; Geissel, H.; et al. Charge-changing interactions of ultrarelativistic Pb nuclei. Phys. Rev. C
**2004**, 70, 014902. [Google Scholar] [CrossRef][Green Version] - Gaimard, J.J.; Schmidt, K.H. A reexamination of the abrasion-ablation model for the description of the nuclear fragmentation reaction. Nucl. Phys. A
**1991**, 531, 709–745. [Google Scholar] [CrossRef] - Prim, R.C. Shortest connection networks and some generalizations. Bell Syst. Tech. J.
**1957**, 36, 1389–1401. [Google Scholar] [CrossRef] - Cormen, T.H.; Leiserson, C.E.; Rivest, R.L.; Stein, C. Introduction to Algorithms; MIT Press: Cambridge, MA, USA, 2009; pp. 624–642. [Google Scholar]
- Hauger, J.A.; Warren, P.; Albergo, S.; Bieser, F.; Brady, F.P.; Caccia, Z.; Cebra, D.A.; Chacon, A.D.; Chance, J.L.; Choi, Y.; et al. Multifragmentation of the remnant produced in the reaction of 1A GeV gold with carbon. Phys. Rev. C
**1998**, 57, 764–783. [Google Scholar] [CrossRef] - Viola, V.E.; Kwiatkowski, K.; Natowitz, J.B.; Yennello, S.J. Breakup densities of hot nuclei. Phys. Rev. Lett.
**2004**, 93, 132701. [Google Scholar] [CrossRef] - Natowitz, J.B.; Hagel, K.; Ma, Y.; Murray, M.; Qin, L.; Shlomo, S.; Wada, R.; Wang, J. Caloric curves and nuclear expansion. Phys. Rev. C
**2002**, 66, 031601. [Google Scholar] [CrossRef][Green Version] - De, J.N.; Samaddar, S.K.; Viñas, X.; Centelles, M. Nuclear expansion with excitation. Phys. Lett. B
**2006**, 638, 160–165. [Google Scholar] [CrossRef][Green Version] - Allison, J.; Amako, K.; Apostolakis, J.; Arce, P.; Asai, M.; Aso, T.; Bagli, E.; Bagulya, A.; Banerjee, S.; Barrand, G.; et al. Recent developments in Geant4. Nucl. Instrum. Methods A
**2016**, 835, 186–225. [Google Scholar] [CrossRef] - Weisskopf, V. Statistics and Nuclear Reactions. Phys. Rev.
**1937**, 52, 295–303. [Google Scholar] [CrossRef] - Fermi, E. High Energy Nuclear Events. Prog. Theor. Phys.
**1950**, 5, 570–583. [Google Scholar] [CrossRef] - Trautmann, W.; Adloff, J.C.; Begemann-Blaich, M.; Bouissou, P.; Hubele, J.; Imme, G.; Iori, I.; Kreutz, P.; Kunde, G.J.; Leray, S.; et al. The rise and fall of multifragment production in
^{197}Au + C, Al, and Cu reactions at E/A = 600 MeV. Nucl. Phys. A**1992**, 538, 473–481. [Google Scholar] [CrossRef] - Cherry, M.; Dabrowska, A.; Deines-Jones, P.; Holynski, R.; Jones, W.; Kolganova, E.; Olszewski, A.; Sengupta, K.; Skorodko, T.Y.; Szarska, M.; et al. Fragmentation and multifragmentation of 10.6 A GeV gold nuclei. Phys. Rev. C
**1995**, 52, 2652. [Google Scholar] [CrossRef] - Abelev, B.; Adam, J.; Adamova, D.; Adare, A.M.; Aggarwal, M.M.; Aglieri Rinella, G.; Agnello, M.; Agocs, A.G.; Agostinelli, A.; Ahammed, Z.; et al. Centrality determination of Pb-Pb collisions at$\sqrt{{s}_{NN}}=2.76$ TeV with ALICE. Phys. Rev. C
**2013**, 88, 044909. [Google Scholar] [CrossRef][Green Version] - Barrette, J.; Bellwied, R.; Bennett, S.; Bersch, R.; Chang, W.C.; Cleland, W.E.; Clemen, M.; Cole, J.D.; Cormier, T.M.; Dai, Y.; et al. Light fragment yields from central Au+Au collisions at 11.5A GeV/c. Phys. Rev. C
**2000**, 61, 044906. [Google Scholar] [CrossRef][Green Version] - Bleicher, M.; Hillmann, P.; Reichert, T.; Steinheimer, J.; Sombun, S.; Herold, C.; Limphirat, A.; Yan, Y. Transport model calculations of deuteron production in relativistic hadron and heavy-ion collisions. Astron. Nachr.
**2019**, 340, 977–982. [Google Scholar] [CrossRef] - Gaebel, V.; Bonne, M.; Reichert, T.; Burnic, A.; Hillmann, P.; Bleicher, M. Understanding the energy dependence of B
_{2}in heavy ion collisions: Interplay of volume and space-momentum correlations. Eur. Phys. J. A**2021**, 57, 1–7. [Google Scholar] [CrossRef] - Sosin, Z.; Błocki, J.; Kallunkathariyil, J.; Łukasik, J.; Pawłowski, P. Alpha-cluster model of atomic nuclei. Eur. Phys. J. A
**2016**, 52, 1–12. [Google Scholar] [CrossRef][Green Version] - Wang, X.; Dong, G.; Gao, Z.; Chen, Y.; Shen, C. Tetrahedral symmetry in the ground state of
^{16}O. Phys. Lett. B**2019**, 790, 498–501. [Google Scholar] [CrossRef] - Brewer, J.; Mazeliauskas, A.; van der Schee, W. Opportunities of OO and pO collisions at the LHC. arXiv
**2021**, arXiv:2103.01939. [Google Scholar]

**Figure 1.**Peripheral (

**left**) and central (

**right**) events of ${}^{208}$Pb${-}^{208}$Pb interactions at $\sqrt{{s}_{NN}}=2.76$ TeV simulated by the Glauber Monte Carlo model v3.2 [14]. The directions of motion of the initial nuclei are perpendicular to the figure plane. The spectator nucleons of the nucleus A are indicated by turquoise circles, nucleus B is indicated by yellow ones, while all participant nucleons are indicated by dark red ones.

**Figure 2.**Schematic representation of a result of the MST-clustering algorithm. Solid turquoise circles denote nucleons (vertices) and red circles denote the resulting clusters of nucleons.

**Figure 4.**Average maximum charge of the fragment $\langle {Z}_{\mathrm{max}}\rangle $, the average multiplicity of intermediate mass fragments $\langle {M}_{\mathrm{IMF}}\rangle $, and the average multiplicities of hydrogen and helium fragments, $\langle {N}_{\mathrm{Z}=1}\rangle $ and $\langle {N}_{\mathrm{Z}=2}\rangle $, in collisions of 10.7 GeV/nucleon ${}^{197}$Au with ${}^{109}$Ag nuclei calculated using the AAMCC model with (solid points) or without (open points) the MST algorithm as functions of ${Z}_{\mathrm{b}3}$ or ${Z}_{\mathrm{bound}}$, see the definitions in the text. Experimental data [10] on projectile fragmentation in NIKFI BR-2 nuclear emulsion are represented by triangles. The data obtained for fragmentation of 600A MeV/nucleon ${}^{197}$Au on a copper target using the ALADIN [6] detector are represented by stars.

**Figure 5.**Average multiplicities of neutrons (

**left**) and protons (

**right**) in ${}^{208}$Pb–${}^{208}$Pb collisions at $\sqrt{{s}_{\mathrm{NN}}}=5.02$ TeV as functions of the collision impact parameter calculated using the AAMCC model with and without MST-clustering (solid and dotted histograms, respectively). ALICE data [11] are represented by circles.

**Figure 6.**Probabilities of production of a certain number of $\alpha $-particles (

**left**) and certain elements (

**right**) as spectators in collisions of ${}^{16}$O with energies of 3.7 GeV/nucleon and 200 GeV/nucleon, respectively, with ${}^{16}$O nuclei calculated by the AAMCC model with and without the MST-clustering (solid and dotted histograms). Points represent experimental data on the fragmentation of ${}^{16}$O on light nuclei (CNO) of photoemulsion [12].

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**MDPI and ACS Style**

Nepeivoda, R.; Svetlichnyi, A.; Kozyrev, N.; Pshenichnov, I.
Pre-Equilibrium Clustering in Production of Spectator Fragments in Collisions of Relativistic Nuclei. *Particles* **2022**, *5*, 40-51.
https://doi.org/10.3390/particles5010004

**AMA Style**

Nepeivoda R, Svetlichnyi A, Kozyrev N, Pshenichnov I.
Pre-Equilibrium Clustering in Production of Spectator Fragments in Collisions of Relativistic Nuclei. *Particles*. 2022; 5(1):40-51.
https://doi.org/10.3390/particles5010004

**Chicago/Turabian Style**

Nepeivoda, Roman, Aleksandr Svetlichnyi, Nikita Kozyrev, and Igor Pshenichnov.
2022. "Pre-Equilibrium Clustering in Production of Spectator Fragments in Collisions of Relativistic Nuclei" *Particles* 5, no. 1: 40-51.
https://doi.org/10.3390/particles5010004