Theoretical Study of Isotope Production in The Peripheral Heavy-ion Collision 136Xe + Pb at 1 GeV/nucleon
1 Theoretical Study of Isotope Production in The Peripheral Heavy-ion Collision Xe + Pb at 1 GeV/nucleon H. Imal * , R. Ogul Department of Physics, Faculty of Science, University of Selçuk, 42079, Konya, Turkey
Abstract
We have studied the fragment yields emitted from the fragmentation of excited projectile nuclei in peripheral collisions of Xe + Pb at 1 GeV/nucleon, and measured with the high-resolution magnetic spectrometer, the Fragment Separator (FRS) of GSI. The mass, charge and isotope distributions of nuclear fragments formed in the reactions were calculated within a statistical ensemble approach and compared to the experimental data. The ensemble of excited projectilelike source nuclei were created in the framework of a previous analysis of similar reactions performed at 600 MeV/nucleon (ALADIN-experiments, GSI). The overall agreement between theory and experiment was very satisfactory in reproducing the experimental data of isotope yields measured in the heavy-ion collisions. It is seen that a broad range of isotopes of the elements between Z=3 and Z=54 (near-projectile isotopes) can be reproduced from fragmentation of Xe projectiles. Keywords; isotope production, radioisotopes, nuclear reactions Introduction
Radioactive isotopes are produced mainly in research reactors, accelerators, and separation facilities. They are widely used in various fields including medicine, research, industry, and radiation processing of foods. This paper aims to search for the isotope yields of nuclear fragments produced in high energy heavy-ion reactions. When a large amount of energy is deposited in a nucleus, the excited nuclear matter disintegrates into several fragments. It is called multifragmentation if there are at least 3 intermediate mass fragments (IMFs) with
Z ≥
3 in one fragmentation event. It has been observed in nearly all types of high-energy nuclear reactions induced by hadrons, photons, and heavy ions (see, for example, Refs. [1-5]). At low excitation energies (
Ex ≤ Xe + Pb , at an incident beam energy of 1 GeV/nucleon. Production cross sections of projectilelike isotopes were measured with the high-resolution magnetic spectrometer, the Fragment Separator (FRS) of GSI, Darmstadt [21]. The heavy-ion beams were delivered from the universal linear accelerator (UNILAC) to the SIS18 heavy-ion synchrotron, where they were extracted and guided through the target area to the FRS. The FRS was used for the separation and analysis of the isotopes emitted from the excited projectiles. For the interpretation of experimental data, we have carried out the calculations in the framework of the statistical multifragmentation model (SMM) for nuclear multifragmentation process. Statistical simulation of the formation of radioactive isotopes
There are several models in the literature developed to analyse high energy nuclear collisions at various energies. At low excitation energies up to Ex = 1 MeV/nucleon, disintegration of heavy nuclei is very well described by the compound nucleus model introduced by Niels Bohr in 1936 [6]. At higher energies, besides the dynamical and kinetic models, the statistical models are found to be very suitable to reproduce the experimental data, especially in describing the fragment production [15, 22]. In this study, we consider the statistical ensemble version of SMM. The same parameters that were used for a previous analysis of ALADIN experimental data obtained for similar reactions at 600 MeV/nucleon [15], were considered. The general properties of the considered ensembles of residual nuclei are given in Ref. [23]. Description of the statistical model
The SMM assumes statistical equilibrium of the excited nuclear system with mass number A charge Z , and excitation energy E x (above the ground state) within a low-density freeze-out volume. In the SMM, all breakup channels (partitions j) composed of nucleons and excited fragments are considered and the conservation of baryon number, electric charge number, and energy are considered. Besides the breakup channels, also the compound-nucleus channels are included and the competition between all channels is permitted. Thus, the SMM covers the conventional evaporation and fission processes occurring at low excitation energy as well as the transition region between the low- and high-energy de-excitation regimes. In the thermodynamic limit, as demonstrated in Ref. [24], the SMM is consistent with the nuclear liquid-gas phase transition when the liquid phase is represented by an infinite nuclear cluster. In the model, light nuclei with mass number A ≤
Z ≤
A > F AZ are parametrized as a sum of the bulk, surface, Coulomb and symmetry energy contributions, respectively, as follows: 𝐹𝐹 𝐴𝐴𝐴𝐴 = 𝐹𝐹 𝐴𝐴 , 𝐴𝐴𝐵𝐵 + 𝐹𝐹 𝐴𝐴 , 𝐴𝐴𝑆𝑆 + 𝐸𝐸 𝐴𝐴 , 𝐴𝐴𝐶𝐶 + 𝐸𝐸 𝐴𝐴 , 𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠 (1) The standard expressions for these terms are 𝐹𝐹 𝐴𝐴 , 𝐴𝐴𝐵𝐵 = ( −𝑊𝑊 − 𝑇𝑇 / 𝜀𝜀 ) 𝐴𝐴 (2) where, T is the temperature, 𝜀𝜀 is related to the level density, and W = 16 𝑀𝑀𝑀𝑀𝑀𝑀 is the binding energy of infinite nuclear matter, F A , ZS = B A / (( T C2 − T )/(T C2 + T ) ) / (3) where B = 18 𝑀𝑀𝑀𝑀𝑀𝑀 is the surface energy coefficient, and Tc = 18 MeV is the critical temperature of infinite nuclear matter; 𝐸𝐸 𝐴𝐴 , 𝐴𝐴𝐶𝐶 = 𝑐𝑐 𝑍𝑍 / 𝐴𝐴 / (4) where 𝑐𝑐 = (3/5)( 𝑀𝑀 / 𝑟𝑟 )(1 − ( 𝜌𝜌 / 𝜌𝜌 ) / ) (5) is the Coulomb parameter (obtained in the Wigner-Seitz approximation) with the charge unit e and 𝑟𝑟 = 1.17 fm; 𝐸𝐸 𝐴𝐴 , 𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠 = 𝛾𝛾 ( 𝐴𝐴 − 𝑍𝑍 ) / 𝐴𝐴 (6) where γ = 25 MeV is the symmetry energy parameter. These parameters are those of the Bethe-Weizsäcker 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. However, these parameters, especially the symmetry coefficient γ , can be different in hot nuclei at multifragmentation conditions, and they should be determined from corresponding experimental data as shown throughout the recent studies by various groups [15, 16, 25, 19]. According to the microcanonical treatment [1], total number of nucleons, total charge and energy are fixed, and the statistical weight of a partition j is calculated by W j = 𝑀𝑀𝑒𝑒𝑒𝑒 (S j (E ∗ , A, Z)) (7) where, ξ is the normalization constant (the partition sum), given by 𝜉𝜉 = ∑ 𝑀𝑀𝑒𝑒𝑒𝑒 ( 𝑆𝑆 𝑗𝑗 ( 𝐸𝐸 ∗ , 𝐴𝐴 , 𝑍𝑍 )) 𝑗𝑗 (8) and S j is the entropy of the channel j, depending on the fragments in this partition as well as on the parameters of the system. In the canonical ensemble the temperature, the total nucleon number and charge are supposed to be fixed for all partitions. The probabilities of different break-up channels are determined in terms of Gibbs free energy and temperature, instead of the entropy of the channel in the microcanonical treatment [1, 2]. At low energy excitations, the decay channels of compound nucleus are also included and the competition between all channels is permitted. Thus, the conventional evaporation and fission processes in the transition region between the low and high energy deexcitation regimes are also included in this model. Further details of the model can be found in Ref. [2]. Here we particularly stress two main achievements of statistical models in theory of nuclear reactions: first, a clear understanding has been reached that sequential decay via compound nucleus must give a way to nearly simultaneous break-up of nuclei at high excitation energies; and second, the character of this change can be interpreted as a liquid-gas type phase transition in finite nuclear systems. The results obtained in the nuclear multifragmentation studies can be applied in several other fields. First, the mathematical methods of the statistical multifragmentation can be used for developing thermodynamics of finite systems [26]. These studies were stimulated by recent observation of extremely large fluctuations of energy of produced fragments, which can be interpreted as the negative heat capacity [27, 28]. A very important advantage of the statistical approach to the cluster production is that the statistical equilibration is generally achieved in the astrophysical conditions [29]. We can demonstrate the links for the neutron-rich isotope production in both cases in Refs. [30, 31]. De-excitation of hot fragments
For excitation energies
Ex ≤
A ≤
16 can be described by the Fermi break-up model. In the microcanonical approximation we consider all possible breakup channels satisfying the conservation of energy, momentum, and particle numbers A and Z. We assume that the probability of each event channel is proportional to the occupied states in the phase space. The weight of the channels containing n particles with masses m i ( i = 1 , · · · , n ) is given by 𝑊𝑊 𝑗𝑗𝑠𝑠𝑚𝑚𝑚𝑚 = 𝑆𝑆𝐺𝐺 � 𝑉𝑉 𝑓𝑓 ( ) � 𝑛𝑛−1 � 𝛱𝛱 𝑖𝑖=1𝑛𝑛 𝑠𝑠 𝑖𝑖 𝑠𝑠 � / ∙ ( ) ( 𝑛𝑛−1 ) 𝛤𝛤� ( 𝑛𝑛−1 ) � ∙ ( 𝐸𝐸 𝑘𝑘𝑚𝑚𝑛𝑛 − 𝑈𝑈 𝑗𝑗𝐶𝐶 ) 𝑛𝑛− (9) where 𝑚𝑚 = ∑ 𝑚𝑚 𝑛𝑛𝑚𝑚=1 𝑚𝑚 is the total mass, 𝑆𝑆 = Π 𝑚𝑚=1𝑛𝑛 (2 𝑆𝑆 𝑚𝑚 + 1) is the spin degeneracy factor (S i is the i-th particle spin), G = Π 𝑗𝑗=1𝑘𝑘 𝑛𝑛 𝑗𝑗 ! is the particle identity factor, n j is the number of particles of kind j, E kin is the kinetic energy of nuclei and 𝑈𝑈 𝑗𝑗𝐶𝐶 is the Coulomb interaction energy between nuclei, which are related to the energy balance as described in Ref.[33]. The total excitation energy Ex can be expressed using the conservation of total energy. Sequential decay modes of primary hot fragments with mass number A >
16 were studied nearly 60 years ago as excited modes of compound nuclei [32]. This mechanism has been investigated extensively, and it was shown that compound nucleus models successfully reproduce the experimental data [2]. The emission width of a particle j emitted from the compound nucleus (A, Z) is given by 𝛤𝛤 𝑗𝑗 = ∑ ∫ 𝜇𝜇 𝑗𝑗 𝑔𝑔 𝑗𝑗 ( 𝑖𝑖 ) 𝜋𝜋 ℏ 𝜎𝜎 𝑗𝑗 ( 𝐸𝐸 ) 𝜌𝜌 𝐴𝐴′𝑍𝑍′ ( 𝐸𝐸 𝐴𝐴𝑍𝑍∗ −𝐵𝐵 𝑗𝑗 −𝐸𝐸 ) 𝜌𝜌 𝐴𝐴𝑍𝑍 ( 𝐸𝐸 𝐴𝐴𝑍𝑍∗ ) 𝐸𝐸 𝐴𝐴𝑍𝑍∗ −𝐵𝐵 𝑗𝑗 −𝜖𝜖 𝑗𝑗 ( 𝑖𝑖 ) 𝐸𝐸𝐸𝐸𝐸𝐸 (10) Here the sum is taken over the ground and all particle-stable excited states 𝜖𝜖 j ( i ) (i=0,1,…,n) of the fragment j, 𝑔𝑔 j ( i ) =(2 𝑆𝑆 𝑗𝑗 ( 𝑚𝑚 ) + 1) is the spin degeneracy factor of the i-th excited state, μ 𝑗𝑗 and B 𝑗𝑗 are corresponding reduced mass and separation energy, 𝐸𝐸 𝐴𝐴𝐴𝐴∗ is the excitation energy of the initial (mother) nucleus, and E is the kinetic energy of an emitted particle in the centre-of-mass frame. In Eq. (10), 𝜌𝜌 𝐴𝐴𝑍𝑍 and 𝜌𝜌 𝐴𝐴′𝑍𝑍′ are the level densities of the initial (
A, Z ) and final (daughter) (A ′ , Z ′ ) compound nuclei in the evaporation chain. The cross section σ 𝑗𝑗 ( 𝐸𝐸 ) of the inverse reactions (A ′ , Z ′ ) + j = (A, Z) was calculated using the optical model with nucleus-nucleus potential [1]. This evaporation process was simulated by the Monte Carlo method and the conservation of energy and momentum was strictly controlled in each emission step. After the analysis of experimental data, we come to conclusion that at sufficient large excitation energies (more than 1 MeV per nucleon) it is reasonable to include the decreasing the symmetry energy coefficient in mass formulae, that leads to adequate description of isotope distributions [16, 34, 19]. An important process of de-excitation of heavy nuclei (approximately, A ≥ 𝜌𝜌 𝑠𝑠𝑒𝑒 = ( 𝐸𝐸 ) , as follows: Γ f = AZ ( E AZ ∗ ) ∫ ρ sp ( E AZ ∗ − B f − E ) dE E AZ ∗ − B f (11) where 𝐵𝐵 𝑓𝑓 is the height of the fission barrier which is determined by the Myers-Swiatecki prescription. For approximation of 𝜌𝜌 𝑠𝑠𝑒𝑒 we have used the results of the extensive analysis of nuclear fissility and branching ratios Γ 𝑛𝑛 / Γ 𝑓𝑓 (see Ref. [2]). Calculations and comparisons with experimental data
In this section, we compare the results of our calculations for producing the fragment mass and charge yields in the reactions Xe + Pb at 1 GeV/nucleon, and the experimental data obtained for the same reaction system in Ref. [21]. In the calculations, the statistical ensemble version of SMM was performed to describe the properties of isospin asymmetric reaction system (projectile and target nuclei have different neutron to proton ratio N/Z ). The primary hot fragments emitted from excited quasiprojectile sources were assumed to be formed during the preequilibrium process long before the statistical equilibrium stage. The deexcitation process leading to the formation of final cold fragments starts after the disintegration of the system into primary hot fragments. The present analysis is based on Ref. [23], where the distribution of excited projectile sources were characterized by a correlation of decreasing source mass number A corresponding to increasing excitation energy per nucleon E x /A and by a saturation of excitation energy per nucleon at around E x /A =8 MeV/nucleon. Mass and charge distributions
For a quantitative comparison of theory and experiment, normalizations have been carried out with respect to the measured cross sections in the interval 20 ≤ Z ≤
25, as shown in Fig.1. We obtained the factor 0.0258 mb per theoretical event for fragmentation of Xe projectiles. In Fig.1, the upper panel shows the theoretical results obtained from ensemble calculations performed for 100 000 reaction events for mass distribution of final cold fragments, and the experimental data given in Ref. [21]. The lower panel shows the charge distribution as function of charge number Z, for the region of A ≥
Z ≥
10. In the experiments, the production cross sections were restricted to the final cold residues in the charge number range
Z ≥
10 since the correction for the limited angular transmissions through the FRS was performed for
Z ≥
10 (see, Ref. [21]). This is because, the determination of the transmission corrections for lighter isotopes is rather difficult due to the overlap of contributions from different reaction mechanisms producing these light isotopes. As a result, the U-shape of nuclear mass and charge distributions shown in Fig.1 evolves in the region between
Z ≥
10 and the heaviest residues close to the projectile. Theoretical mass and charge distributions follow a similar trend to experimental ones measured in reactions with considerably high excitation energies. Theoretical results of these distributions (for
Z <
10) are not shown in Fig.1, since the experimental data are not available in this region of lighter particles. In short, we have shown that the mass and charge numbers of the particles formed in the reactions decrease as the excitation energy accumulated in projectile sources increases. Consequently, a qualitative agreement between theoretical and experimental results with considerable differences has been noticed. The observed differences in the distribution plots are mostly originated from the fact that the isotopes are not fully covered in the experiments. An extensive analysis of these differences was given in Refs. [15, 19] for the relativistic heavy-ion collisions measured at FRS. In the Fermi energy region, where the projectile velocity is comparable with the Fermi velocity, nucleon exchange between the reaction partners during the collision become important. In the dynamical stage of the collision, N/Z values of quasiprojectile and quasitarget sources move towards the N/Z of composite system as a result of isospin diffusion, that produces significant changes in fragmentation picture (see, e.g., Refs. [35, 36]).
Fig. 1.
Mass and charge distributions of isotopes produced in the projectile fragmentation of the reaction system Xe + Pb . Upper panel shows the production cross sections of the produced isotopes as a function of the isotope mass number A, and lower panel as a function of the isotope charge number Z. In both panels, the red solid circles show the experimental data, and the empty blue circles the theoretically predicted results. Isotope distributions
Isotope distribution observables may provide information in isospin asymmetry dynamics of the reaction systems. In Fig.2, we show the experimental data of the isotope distributions of some selected isotopes given in Ref. [21] and compared to the presently predicted results in the framework of the statistical ensemble approach. The full isotope distribution inside the angular acceptance of the FRS was applied for the experimental measurements [21]. It is seen from this figure that there is a slight discrepancy between predictions and experimental data. This is because, our recent analysis of ALADIN data shows that the isotopic cross section values (isotope distributions) are more sensitive to the variation of symmetry energy parameter at low density freeze-out region. For this reason, the best agreement with experimental data was achieved by taking the reduced symmetry term coefficient into account instead of standard value of 25 MeV. In the present calculations, we have taken the modified values of symmetry energy parameter obtained for a similar reaction system in Ref. [19]. Since this paper aims to reflect an overview of isotope production rather than providing a detailed discussion of the modifications of these model parameters, we will refer the reader to Refs. [15, 16, 35] for detailed information on such optimization calculations. One may also see the production cross sections of some selected radionuclides in Fig.2, whose names are highlighted in red, such as F (for use as a tracer in PET scans), K (for use of dating purposes, and as the largest source of natural radioactivity), Ca (for use of dating of carbonate rocks), and Sr (as one of the most dangerous Fig. 2. Production cross sections of some selected isotopes measured in the reactions
Xe + Pb at 1 GeV/nucleon of relativistic projectile energy. The red solid circles show the experimental data, and the solid lines the theoretically predicted results. Isotope name in red refers to see the selected examples of some radionuclides including F, K, Ca, and Sr (see the text). fission product of nuclear fallout). As can be seen from the figure, all these isotopes have lower cross-section values, in other words, lower lifetimes than those of the most stable isotopes located around the maximum point of the isotope distribution curves. For example, one may see from the figure that positron emitter F , which is used for medical PET imaging, has a cross-section of 1.94 mb, while that of the most stable one F located at maximum, is 7.04 mb. Moreover, isotopes with a cross section of less than about 1 mb probably have lifetimes in the order of milliseconds and therefore cannot be directly observed, but their decay channels can be measured [20]. Conclusions
In conclusion, apart from the producing isotopes in research reactors for medical and industrial use, it is also made for research purposes at low and high energy accelerators by various facilities. The chemical isotopes are mainly produced in stellar evolution including supernova explosions and formation of stars. There may be several elements occur in universe even though they cannot be found on earth, and some of them may be detectable in cosmic rays. Various research groups at the several accelerator facilities are trying to produce new elements/isotopes. In some laboratories including GSI, JINR, and RIKEN new elements in between Z=107 (Bohrium-Bh) and Z=118 (Oganesson-Og) were discovered in the period of the years 1981 and 2006 [37]. They are mostly unstable and decay after a short time in milliseconds, therefore they are not directly observed, but their decay chains can be measured. However, theoretical calculations have predicted the island of stability, whereby the isotopes of superheavy elements might have considerably longer lifetime [38]. So far, thousands of radioactive isotopes are predicted to lie within the particle stability limits in nuclear chart, and around fifty per cent of them are identified. Although, the Radioactive Ion Beams (RIB) facilities contribute substantially to work in this area, more work would be worthwhile to produce exotic isotopes contained in the chart. As a result of many years of research in large ion-beam accelerator systems it was also succeeded to develop a new form of radiotherapy the so called heavy-ion and/or hadron therapy. The advantage of this new treatment modality is that the ion beam selectively damages tumour tissues while sparing the surrounding healthy tissues. Consequently, we show that neutron-rich radioactive isotopes can be produced in nuclear reactions in a wide range of energy regime. The main examples of such reactions are fission, fusion, nuclear fragmentation, and nucleon transfer reactions [10,39]. At high energy heavy-ion collisions starting from the Fermi energy regime (20-50 MeV/nucleon) up to relativistic energies (1 GeV/nucleon), nuclear multifragmentation reactions become superior to produce neutron-rich and proton-rich isotopes [40, 41]. Further experiments are needed to extract information for the properties of neutron-rich exotic nuclei towards neutron dripline in nuclear chart. This kind of studies will also provide us with very useful tools for investigating the properties of stellar matter at extreme conditions, because of the similarities of nuclear and stellar matter.
Acknowledgements
The authors gratefully acknowledge enlightening discussions with A. Botvina and W. Trautmann on FRS and ALADIN experiments and their interpretation.
References [1]
A.S. Botvina, A.S. Iljinov, I.N. Mishustin, J.P. Bondorf, R. Donangelo, K. Sneppen, Statistical simulation of the break-up of highly excited nuclei, Nucl. Phys. A 475(4) (1987) 663-686. [2]
J.P. Bondorf, A.S. Botvina, A.S. Iljinov, I.N. Mishustin, and K. Sneppen, Statistical multifragmentation of nuclei, Phys. Rep. 257 (1995) 133-221. [3]
D.H.E. Gross, Statistical decay of very hot nuclei-The production of large clusters, Rep. Prog. Phys. 53 (1990) 605-658. [4]
V.E. Viola, et al. Light-ion-induced multifragmentation; The ISIS project, Phys. Rep. 434 (2006) 1-46. [5]
G. Bertsch and P.J. Siemens, Nuclear fragmentation, Phys. Lett. B 126 (1983) 9-12. [6]
N. Bohr, Neutron capture and Nuclear Constitution, Nature (London) 137 (1936) 344-348. [7]
A.L. Goodman, J.I. Kapusta and A.Z. Mekjian, Liquid-Gas Phase Instabilities and Droplet Formation in Nuclear Reactions, Phys Rev C 30 (1984) 851-865. [8]
J. Desbois, R. Boisgard, C. Ngô, and J. Nemeth, Multifragmentation of Excited Nuclei Within a Schematic Hydrodynamical and Percolation Picture, Z. Phys. A 328 (1987) 101-113. [9]
W.A. Friedman, Rapid massive cluster formation, Phys. Rev. C 42 (1990) 667-673. [10]
H. Geissel, P. Armbruster, et al., The GSI projectile fragment separator (FRS): A versatile magnetic system for relativistic heavy ions, Nucl. Instrum. Meth B 70 (1992a) 286-297. [11]
H. Geissel, K. Beckert, et al. First storage and cooling of secondary heavy-ion beams at relativistic energies. Phys. Rev. Lett. 68 (1992b) 3412-3415. [12]
D.J. Morrissey and B.M. Sherill, In-flight Separation of Projectile Fragments, Lect. Notes Phys. 651 (2004) 113–135. [13]
M. Huyse, The Why and How of Radioactive-Beam Research, Lect. Notes Phys. 651 (2004) 1–32.
P. Van Duppen, Isotope Separation On Line and Post Acceleration, Lect. Notes Phys. 700 (2006) 37–77 (Springer-Verlag Berlin Heidelberg). [15]
R. Ogul, et al., Isospin-dependent multifragmentation of relativistic projectiles, Phys. Rev. C 83 (2011)024608. [16]
N. Buyukcizmeci, R. Ogul and A.S. Botvina, Isospin and symmetry energy effects on nuclear fragment production in liquid-gas phase transition region, Eur. Phys. J. A 25 (2005) 57-64. [17]
R. Ogul, et al., Surface and symmetry energies in isoscaling for multifragmentation reactions, J. Phys. G: Nucl. Part. Phys. 36 (2009) 115106. [18]
N. Buyukcizmeci, et al., Isotopic yields and symmetry energy in nuclear multifragmentation reactions, J. Phys. G: Nucl. Part. Phys. 39 (2012)115102. [19]
H. Imal, et al., Theoretical study of projectile fragmentation in the 112Sn+112Sn and 124Sn +124Sn reactions at 1 GeV/nucleon, Phys. Rev. C 91 (2015) 034605. [20]
R. Ogul, N. Buyukcizmeci, A. Ergun and A.S. Botvina, Production of neutron-rich exotic nuclei in projectile fragmentation at Fermi energies, Nucl. Sci. Tech. 28 (2017) 28:18. [21]
D. Henzlova, K.-H. Schmidt, M.V. Ricciardi, et al., Experimental investigation of the residues produced in the 136Xe+Pb and 124Xe+Pb fragmentation reactions at 1 A GeV, Phys. Rev. 78 (2008) 044616. [22]
A.S. Botvina, N. Buyukcizmeci, M. Erdogan, et al., Modification of surface energy in nuclear multifragmentation, Phys. Rev. C 74 (2006) 044609. [23]
Botvina AS, et al. Multifragmentation of spectators in relativistic heavy-ion reactions, Nucl. Phys. A 584 (1995) 737-756. [24]
S. Das Gupta and A.Z. Mekjian, Phase transition in statistical model for nuclear multifragmentation, Phys. Rev. C 57 (1998) 1361-1365. [25]
A. Ono, et al., Isospin fractionation and isoscaling in dynamical simulations of nuclear collisions, Phys. Rev. C68 (2003) 051601(R). [26]
B.A. Li , A.R. de Angelis and D.H.E. Gross, Statistical model analysis of ALADIN multifragmentation data, Phys. Lett. B 303 (1993) 225-229. [27]
M. D’Agostino, F. Gulminelli, et al., Negative heat capacity in the critical region of nuclear fragmentation: an experimental evidence of the liquid-gas phase transition, Phys. Lett. B 473 (2000) 219-225. [28]
R. Ogul and A.S. Botvina, Critical temperature of nuclear matter and fragment distributions in multifragmentation of finite nuclei, Phys. Rev. C 66 (2002) 051601(R). [29]
N. Buyukcizmeci, A.S. Botvina, I.N. Mishustin, et al., A comparative study of statistical models for nuclear equation of state of stellar matter, Nucl. Phys. A 907 (2013) 13–54. [30]
P.N. Fountas, G.A. Souliotis, M. Veselsky and A. Bonasera, Systematic study of neutron-rich rare isotope production in peripheral heavy-ion collisions below the Fermi energy, Phys. Rev. C 90 (2014) 064613. [31]
A. Kaya, N. Buyukcizmeci and R. Ogul, Isotopic distribution in projectile fragments above the Coulomb barrier, Turk. J. Phys. 42 (2018) 659-667. [32]
W. Weisskopf, Statistics and nuclear reactions, Phys. Rev. 52 (1937) 295. [33]
A.S. Lorente, A.S. Botvina, and J. Pochodzalla, Production of excited double hypernuclei via Fermi breakup of excited strange systems, Phys. Lett. B 697(3) (2011) 222-228 . [34]
D. Henzlova, A.S. Botvina, K-H. Schmidt et al., Symmetry energy of fragments produced in multifragmentation, J. Phys. G 37 (2010) 085010. [35]
H. Imal, N. Buyukcizmeci, R. Ogul and A.S. Botvina, Isospin dependence of fragment yields in peripheral heavy ion collisions, Eur. Phys. Journ. A 56 (2020) 110. [36]
M.B. Tsang, et al., Isospin diffusion and the nuclear symmetry energy in heavy ion reactions, Phys. Rev. Lett. 92 (2004) 062701. [37]
P.J. Karol, R.C. Barber, B.M. Sherrill, E. Vardaci and T. Yamazaki, Discovery of the element with atomic number Z=118 completing the 7th row of the periodic table (IUPAC Technical Report), Pure. Appl. Chem. 88(1-2) (2016) 155-160. [38]
V. Zagrebaev, A. Karpov and W. Greiner, Future of super heavy element research: Which nuclei could be synthesized within the next few years? Journal of Physics: Conference Series, 420 (2013) 012001. [39]
A. Kelic, k.H. Schmidt, T. Enqvist, et al., Isotopic and velocity distributions of Bi-83 produced in charge-pickup reactions of Pb-208(82) at 1 AGeV, Phys. Rev. C 70 (2004) 064608. [40]
S. Barlini, S. Piantelli, G. Casini, et al., Isospin transport in Kr-84+Sn-112, Sn-124Sn collisions at fermi energies, Phys. Rev. C 87 (2013) 054607. [41]