Comparison between nuclear thermometers in central Xe+Sn collision

Comparison between nuclear thermometers in central Xe+Sn collision 1 2 first-author 1 2 1 2 3 corresp Corresponding author. E-mail address: fszhang@bnu.edu.cn Corresponding author. E-mail address: fszhang@bnu.edu.cn fszhang@bnu.edu.cn The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China Beijing Radiation Center, Beijing 100875, China Beijing Radiation Center, Beijing 100875, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou 730000, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou 730000, China The temperature of fragmenting source in central heavy-ion collisions at Fermi energy is investigated by the isospin-dependent quantum molecular dynamics model in combination with the statistical decay model GEMINI. Five different nuclear thermometers are used to extract nuclear temperature. We find that the He and Li isotope temperature reaches a plateau at about 70–100 MeV/nucleon of beam energy. The slope temperature and the quadrupole fluctuation temperature give high values. The quantum slope temperature and the quantum quadrupole fluctuation temperature are more close to the He and Li isotope temperatures. Heavy-ion collision Nuclear temperature Phase transitions 1 Introduction 1 The nuclear matter may undergo a phase transition from the liquid ground-state to a gas of nucleons when the nuclei was heated. The nature of the interaction between nucleons is similar to the Van der Waals forces between molecules. The uncertainties of the nuclear equation of state (EOS) make the study of nuclear liquid-gas phase transition be important and meaningful [ 1 1 , 2 2 ] . Experimental signals of phase transitions have been observed by many experimental groups [ 3 3 , 4 4 , 5 5 , 6 6 ] . Many theoretical efforts to find signatures of phase transitions in heavy-ion collisions (HICs) have been performed a few years ago [ 7 7 , 8 8 , 9 9 ] . HICs at intermediate energies offer an opportunity to heat nuclei, and the nuclear multifragmentation is long assimilated to the nuclear liquid-gas phase transition. Temperature is one of the most important degrees of freedom in describing phase transition. However, the nuclear matter created in HICs is a non-equilibrium, finite, and open system. It is more difficult to determine the temperature in nuclear matter than that in ordinary matter. Different from ordinary thermometer, nuclear thermometers are more complicated. Several nuclear thermometers have been proposed in the past days. These approaches can be divided into three families [ 2 2 , 10 10 ] : (1) Kinetic approach. Kinetic approach is based on the assumption that energy spectra of particles obey the Maxwell- Boltzmann distribution. The temperature can be derived from the slopes of the kinetic energy spectra [ 11 11 , 12 12 ] or the momentum fluctuations [ 13 13 ] . It has been suggested that the slopes of light product spectra in nuclear reactions lead to very high "temperatures" . The temperatures extracted from Maxwell-Boltzmann kinetic approach probably reflect not only the thermal properties of the system, but also the collective energies coming from the dynamics of the nuclear collision. Recently, the Fermi-Dirac kinetic approach is considered to improve this type of thermometer [ 14 14 , 15 15 ] . (2) Population approaches. The underlying idea for this method is that the relative populations of the produced clusters or its excited states are assumed to obey a Boltzmann distribution. Population of excited states [ 16 16 ] and double ratios of isotopic yields [ 17 17 ] are two of the most often used methods. Employing the isotopic thermometer, Pochodzalla et al . had given the caloric curve which is taken as the evidence of the occurrence of a liquid-gas type phase transition [ 4 4 ] . (3) Thermal-energy approaches. The temperatures at freeze-out are obtained from the excitation energy which is extracted by measuring evaporation cascade from a thermalized source by varying neutron-to-proton ratio N / Z . An example is the isospin thermometer [ 18 18 ] . In this paper, we attempt to compare five different nuclear thermometers. The central collisions of 129 Xe+ 120 Sn at Fermi energy are simulated by the isospin-dependent quantum molecular dynamics (IQMD) model together with the statistical decay code GEMINI. The double ratio temperature, the slope temperature, the quantum slope temperature, the quadrupole fluctuation temperature, and the quantum quadrupole temperature are compared. 2 IQMD transport model 1 The IQMD model [ 19 19 , 20 20 ] is based on the same principles as the quantum molecular dynamics model. With consideration of the different mean field potentials of proton and neutron, and the production of pion and kaon, the IQMD model has been widely and successfully used for the analysis of collective flows, stopping, pion and kaon multiplicities in HICs at incident energy below 2GeV/nucleon [ 20 20 ] . The statistical code GEMINI was proposed by R. J. Charity in 1980s [ 21 21 ] . It can be utilized to treat the decay of a compound nucleus in fusion reaction and excited fragments in HICs. GEMINI code was always applied to the transport model to statistically deexcite the hot fragments [ 22 22 ] . Recently, within the framework of the IQMD+GEMINI model, the odd-even effect in the yields of the final fragments has been well reproduced [ 23 23 ] . Using the same model we have investigated the multiplicities and the kinetic energy spectra in central HICs, the simulations are in very good agreement with the experimental data. The slope temperature and the isotope temperature were also studied [ 14 14 , 24 24 ] . The transport with modified (the medium factor of <math id="M1"><mrow><mn>1</mn><mo>−</mo><mn>0.2</mn><mi>ρ</mi><mo>/</mo><msub><mi>ρ</mi><mn>0</mn></msub></mrow></math> ) nucleon-nucleon elastic cross section [ 25 25 ] and soft EOS [ 26 26 ] are used to study the nuclear temperature. To compensate the fermionic feature the method of the phase-space density constraint in the constrained molecular dynamics model [ 27 27 ] is applied. 3 Results and discussion 1 3.1 He and Li isotope temperature 2 Figure 1 Figure 1 displays charge distribution of the central collision ( b < 1.6fm) of 129 Xe+ 120 Sn at 50 MeV/nucleon. Both the simulation by IQMD with and without GEMINI code are displayed in the figure. The experimental data are shown by the symbol. Firstly, it is clearly seen that IQMD+GEMINI can reproduce the experimental data quite well. Secondly, with the help of GEMINI code, the yield of heavier fragments will decrease and that of light particles will increase, due to deexcitation of the hot fragments. The good agreement with experimental data, makes the study of the double ratio temperature be possible and believable. We introduce one of double ratio temperature in which He and Li isotopes are used, named <math id="M2"><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">HeLi</mi></mrow></msub></mrow></math> . It is defined by <math display="block" id="M3"><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">HeLi</mi></mrow></msub><mo>=</mo><mn>13.3</mn><mi mathvariant="normal">MeV</mi><mo>/</mo><mi>ln</mi><mrow><mo>(</mo><mrow><mn>2.2</mn><mfrac><mrow><mrow><mrow><msub><mi>Y</mi><mrow><mi>L</mi><mn>6</mn><mi>i</mi></mrow></msub></mrow><mo>/</mo><mrow><msub><mi>Y</mi><mrow><mi>L</mi><mn>7</mn><mi>i</mi></mrow></msub></mrow></mrow></mrow><mrow><mrow><mrow><msub><mi>Y</mi><mrow><mi>H</mi><mn>3</mn><mi>e</mi></mrow></msub></mrow><mo>/</mo><mrow><msub><mi>Y</mi><mrow><mi>H</mi><mn>4</mn><mi>e</mi></mrow></msub></mrow></mrow></mrow></mfrac></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> where Y is the yield of the isotope. The <math id="M4"><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">HeLi</mi></mrow></msub></mrow></math> can be calculated using this formula. 3.2 The slope and the quantum slope temperature 2 Figure 2 Figure 2 illustrates the calculated (open circle) and experimental (open star) kinetic energy spectra of free protons in central collisions of 129 Xe+ 120 Sn at E lab =50 MeV/nucleon. The solid lines in each panel denote Maxwell-Boltzmann fit. The fit formula is expressed as <math display="block" id="M5"><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>∝</mo><mfrac><mrow><mrow><mo>(</mo><mrow><mi>E</mi><mo>−</mo><msub><mi>E</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mi>T</mi><mrow><mi mathvariant="normal">slope</mi></mrow><mn>2</mn></msubsup></mrow></mfrac><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><mrow><mo>(</mo><mrow><mi>E</mi><mo>−</mo><msub><mi>E</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">slope</mi></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> where E 0 reflects the repulsive Coulomb forces [ 27 27 ] . Taking into account the Fermi nature of nucleon, the slope temperature was rewritten by Bauer [ 29 29 ] <math display="block" id="M6"><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">slope</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>A</mi><mo>−</mo><msub><mi>A</mi><mi>f</mi></msub></mrow><mrow><mi>A</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mfrac><mn>2</mn><mn>5</mn></mfrac><msub><mi>E</mi><mi>F</mi></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>5</mn><msup><mi>π</mi><mn>2</mn></msup><msubsup><msup><mi>T</mi><mo>′</mo></msup><mrow><mi mathvariant="normal">slope</mi></mrow><mn>2</mn></msubsup></mrow><mrow><mn>12</mn><msubsup><mi>E</mi><mi>F</mi><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> where E F is the Fermi energy. <math id="M7"><mrow><msub><msup><mi>T</mi><mo>′</mo></msup><mrow><mi mathvariant="normal">slope</mi></mrow></msub></mrow></math> names the quantum slope temperature. 3.3 Quadrupole fluctuation and the quantum quadrupole fluctuation temperature 2 One of new nuclear temperature which is based on momentum fluctuations of detected particles was proposed in Ref.[ 17 17 ]. The momenta of particles were assumed to obey the Maxwell-Boltzmann distribution <math display="block" id="M8"><mrow><mi>f</mi><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>π</mi><mi>m</mi><msub><mi>T</mi><mrow><mi mathvariant="normal">fluct</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></mfrac><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><msubsup><mi>p</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>p</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>p</mi><mi>z</mi><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><mi>m</mi><msub><mi>T</mi><mrow><mi mathvariant="normal">fluct</mi></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> where m is the mass of the particle, p x , p y and p z are three momentum components of the particle. The variance <math id="M9"><mrow><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>y</mi></mrow><mn>2</mn></msubsup></mrow></math> can be obtained from the quadrupole <math id="M10"><mrow><msub><mi>Q</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>=</mo><msub><mi>p</mi><mi>x</mi></msub><mo>−</mo><msub><mi>p</mi><mi>y</mi></msub></mrow></math> distribution through <math display="block" id="M11"><mrow><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>y</mi></mrow><mn>2</mn></msubsup><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><msup><mi>d</mi><mn>3</mn></msup><mi>p</mi></mrow></mrow></mstyle><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>Q</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mi>f</mi><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>.</mo></mrow></math> Figure 3 Figure 3 shows the distribution of <math id="M12"><mrow><msub><mi>Q</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></mrow></math> for free protons. The form is similar to the experimental results [ 13 13 ] . Then, we can get the <math id="M13"><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">fluct</mi></mrow></msub></mrow></math> from these formulae, the results are shown in Fig.4 Fig.4 . More details will be discussed in the following section. The fluctuation temperature was improved as follows [ 18 18 ] : <math display="block" id="M14"><mrow><msub><mi>T</mi><mrow><mtext>fluct</mtext></mrow></msub><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>4</mn><msubsup><mi>E</mi><mi>F</mi><mn>2</mn></msubsup></mrow><mrow><mn>35</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow><mrow><mn>15</mn></mrow></mfrac><msubsup><msup><mi>T</mi><mo>′</mo></msup><mrow><mtext>fluct</mtext></mrow><mn>2</mn></msubsup></mrow></msqrt><mo>,</mo></mrow></math> where E F is the Fermi energy of nuclear matter and <math id="M15"><mrow><msubsup><msup><mi>T</mi><mo>′</mo></msup><mrow><mi mathvariant="normal">fluct</mi></mrow><mrow/></msubsup></mrow></math> is named as the quantum quadrupole temperature. 3.4 Comparison between nuclear thermometers 2 Within the IQMD+GEMINI model, we calculated the nuclear temperatures from 129 Xe+ 120 Sn central collision at incident energies from 30 MeV/nucleon to 100 MeV/nucleon. Five nuclear thermometers are investigated in this work: the double ratio temperature T HeLi , the slope temperature T slope , the quantum slope temperature <math id="M17"><mrow><msub><msup><mi>T</mi><mo>′</mo></msup><mrow><mi mathvariant="normal">slope</mi></mrow></msub><mo>,</mo></mrow></math> the quadrupole fluctuation temperature T fluct , and the quantum quadrupole temperature <math id="M18"><mrow><msub><msup><mi>T</mi><mo>′</mo></msup><mrow><mi mathvariant="normal">fluct</mi></mrow></msub></mrow></math> . The results are shown in Fig.4 Fig.4 . Firstly, it can be seen that different thermometers give very different values of nuclear temperature. The discrepancy between various nuclear temperature steadily grows with increasing incident energy. All temperatures exhibit an increase of value with increasing incident energy. <math id="M19"><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">HeLi</mi></mrow></msub></mrow></math> reaches a plateau at about 70–100MeV/nucleon of beam energy. Secondly, <math id="M20"><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">slope</mi></mrow></msub></mrow></math> and <math id="M21"><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">fluct</mi></mrow></msub></mrow></math> are very close to each other, because both of these two approaches assumed energy distribution (Eq.(2)) and the momentum distribution (Eq.(4)) are the classical Maxwell-Boltzmann type. These two temperatures are the highest of all. It is because besides the thermal motion, they also contain the Fermi motion. Thirdly, <math id="M22"><mrow><msub><msup><mi>T</mi><mo>′</mo></msup><mrow><mi mathvariant="normal">slope</mi></mrow></msub></mrow></math> and <math id="M23"><mrow><msub><msup><mi>T</mi><mo>′</mo></msup><mrow><mi mathvariant="normal">fluct</mi></mrow></msub></mrow></math> are both smaller than <math id="M24"><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">slope</mi></mrow></msub></mrow></math> and <math id="M25"><mrow><msub><mi>T</mi><mrow><mi mathvariant="normal">fluct</mi></mrow></msub></mrow></math> . This is because the Fermi motion of nucleon is removed. But due to the different methods of elimination of the Fermi motion, the results are different. 4 Conclusion 1 Within the IQMD+GEMINI model, various nuclear thermometers have been investigated for the central collisions of 129 Xe+ 120 Sn at incident energy range from 30 to 100 MeV/nucleon. These nuclear thermometers are not exactly coincident with each other. 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