Symmetry energy extraction from primary fragments in intermediate heavy-ion collisions

Symmetry energy extraction from primary fragments in intermediate heavy-ion collisions 1 2 first-author 1 corresp Corresponding author, huangmeirong@impcas.ac.cn Corresponding author, huangmeirong@impcas.ac.cn huangmeirong@impcas.ac.cn 1 1 2 1 2 1 1 1 1 2 1 1 2 1 1 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China College of Information and Software Engineering, University of Electronic Science and Technology, Chengdu 610054, China College of Information and Software Engineering, University of Electronic Science and Technology, Chengdu 610054, China An improved method is proposed for the extraction of the symmetry energy coefficient relative to the temperature, a sym / T , for the heavy-ion reactions near the Fermi energy region, based on the modified Fisher Model (MFM). This method is applied to the primary fragments of the Anti-symmetrized Molecular Dynamics (AMD) simulations for the reactions of 40 Ca + 40 Ca at 35 MeV/nucleon. The density and the temperature at the fragment formation stage are extracted using a self-consistent method. Intermediate energy heavy ion reactions Symmetry energy Density Temperature Modified Fisher model Self-consistent method I. INTRODUCTION 1 The symmetry energy term in the nuclear equation of state (EOS) intimately relates to a wealth of the dynamical process of nuclear reactions, the structure character of nuclei and astrophysical phenomena [ 1 1 ]. The investigations of the symmetry energy, especially focusing on its density dependence, have been conducted through many observables such as isotopic ratio [ 2 2 ], isospin diffusion [ 3 3 ], neutron-proton emission ratio [ 4 4 , 5 5 ], giant monopole resonance [ 6 6 ], pygmy dipole resonance [ 7 7 ], giant dipole resonance [ 8 8 ], collective flow [ 9 9 ] and isoscaling [ 10 10 - 12 12 10-12 ]. In our recent works, we utilized the isotopic yields to extract the symmetry energy coefficient relative to the temperature, a sym / T , in the framework of the nuclear phase transition theory [ 13 13 - 17 17 13-17 ]. Along this scenario, a sym / T was extracted, using m -scaling [ 18 18 ] and isobaric yield ratios [ 19 19 ]. Ono et al. also independently introduced a generalized free energy, K ( N , Z ), and extracted the a sym / T values from their quadratic distributions [ 20 20 ]. In this method, for a given Z , all contributions from the volume, surface, Coulomb and pairing terms in the free energy are squeezed into an N proportional term and a constant, ξ ( Z ) N + η ( Z ), as a course approximation. While temperature is one of the key variables in characterizing nuclear reactions, it is very difficult to determine the temperature of hot nuclear matter in a dynamical process. Several nuclear thermometers have been proposed [ 21 21 ]. These include the slope of energy spectra [ 22 22 , 23 23 ], momentum fluctuations [ 24 24 , 25 25 ], double isotope yield ratios [ 26 26 ] and excited state distributions [ 27 27 ]. However they may not be generally applicable in all circumstances and even for a given system the extracted temperature values from these thermometers may be quite different from each other [ 21 21 ]. In our recent work, temperature is evaluated from a ( μ n -μ p )/ T analysis, but no density determination was possible [ 28 28 ]. In our another recent work, the isotopic yield ratio method was applied to extract a sym / T values from the experimentally reconstructed primary fragment yields. These ratios were compared to those calculated from AMD primary generated fragment yields obtained using Gogny interactions with different density dependencies of the symmetry energy [ 29 29 , 30 30 ]. In the analysis, we found that the extracted a sym / T values change according to the interactions used. From the dependence on interaction, the density, symmetry energy and temperature at the time of fragment formation were determined in a self-consistent manner. In this article, an improved method is used to extract the a sym / T values. In the improved method, all available isotope yields are employed. The improvement is made possible by taking into account the mass dependent temperature [ 31 31 ] in the free energy in an iterative manner. Using the self-consistent manner, the density and the temperature at the time of fragment formation are carefully determined from the obtained a sym / T values from different interactions with different density dependencies of the symmetry energy term, i.e., the standard Gogny interaction which has an asymptotic soft symmetry energy (g0), the Gogny interaction with an asymptotic stiff symmetry energy (g0AS) and the Gogny interaction with an asymptotic super-stiff symmetry energy (g0ASS) [ 12 12 , 32 32 ]. Our present analyses are conducted in the framework of AMD. There are three major reasons to use AMD as the event generator for this work. One is its capability to reproduce the experimental isotope yields. AMD results, such as multiplicity, angular distribution and energy spectra, have often been compared with those from the experimental data for intermediate energy heavy ion collisions and reproduce them reasonably well [ 32 32 - 39 39 32-39 ]. In one of our recent works in Ref.[ 30 30 ], the yields of the experimentally reconstructed primary hot isotopes are well reproduced by those of the AMD simulations. Second is to eliminate the secondary cooling effect. As shown in Ref.[ 19 19 , 30 30 ], the sequential decay of the primary hot isotopes significantly alters the yield distribution and distorts the information inherent in the primary hot fragment yields. Third is to simplify the initial conditions using zero impact parameter to eliminate the effects of transverse flow and neck emission among others. Thus the AMD events are generated for central collisions ( b =0 fm) of 40 Ca + 40 Ca at 35 MeV/nucleon. II. IMPROVED MFM MODEL AND EXTRACTION OF A SYM / T 1 In the framework of MFM, the yield of an isotope with mass A and I = N - Z ( N neutrons and Z protons) produced in a multifragmentation reaction, can be given as [ 14 14 , 16 16 - 19 19 16-19 , 40 40 ] <math display="block" id="M1"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi>I</mi><mn>,</mn><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><msub><mi>Y</mi><mn>0</mn></msub><mo>⋅</mo><msup><mi>A</mi><mrow><mo>−</mo><mi>τ</mi></mrow></msup><mi>exp</mi><mo stretchy="false">[</mo><mfrac><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>I</mi><mn>,</mn><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>μ</mi><mi>n</mi></msub><mi>N</mi><mo>+</mo><msub><mi>μ</mi><mi>p</mi></msub><mi>Z</mi></mrow><mi>T</mi></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mi>N</mi><mi>ln</mi><mo stretchy="false">(</mo><mfrac><mi>N</mi><mi>A</mi></mfrac><mo stretchy="false">)</mo><mo>+</mo><mi>Z</mi><mi>ln</mi><mo stretchy="false">(</mo><mfrac><mi>Z</mi><mi>A</mi></mfrac><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Using the generalized Weisz ä cker-Bethe semiclassical mass formula [ 41 41 , 42 42 ], W ( I , A ) can be approximated as <math display="block" id="M2"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>I</mi><mn>,</mn><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mi>v</mi></msub><mi>A</mi><mo>−</mo><msub><mi>a</mi><mi>s</mi></msub><msup><mi>A</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><msub><mi>a</mi><mi>c</mi></msub><mfrac><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>−</mo><msub><mi>a</mi><mrow><mtext>sym</mtext></mrow></msub><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mi>A</mi></mfrac><mo>−</mo><msub><mi>a</mi><mi>p</mi></msub><mfrac><mi>δ</mi><mrow><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>δ</mi><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>Z</mi></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>N</mi></msup></mrow><mn>2</mn></mfrac><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> In Eq.(1), A - τ and Nln ( N/A )+ Zln ( Z/A ) originate from the increases of the entropy and the mixing entropy at the time of the fragment formation, respectively. μ n ( μ p ) is the neutron (proton) chemical potential. τ is the critical exponent. In this work, the value of τ =2.3 is adopted from the previous studies [ 40 40 ]. In general coefficients, a v , a s , a sym , a p and the chemical potentials are temperature and density dependent. In this formulation a constant volume process at equilibrium is assumed in the free energy and therefore the term "symmetry energy" is used throughout this work along Ref. [ 43 43 ]. If one assumes a constant pressure at the equilibrium process [ 44 44 ], the term "symmetry enthalpy" should be used. Experimentally, whether the equilibrium process takes place at constant pressure or volume can not be determined, and thus we use "symmetry energy" throughout the paper, keeping in mind the ambiguity [ 43 43 ]. In the previous analyses [ 19 19 , 30 30 , 45 45 , 46 46 ], the temperature in Eq.(1) was assumed to be identical to the temperature of the fragmenting source and treated as a constant for all isotopes. However as seen below, this temperature turns out to be (fragment) mass dependent. This mass dependence on the temperature was not recognized in the previous analyses, just because the mass dependence was masked by the larger error bars. In this improved method, the error bars become small and the mass dependence becomes evident. In order to take into account the mass dependence of the temperature in Eq.(1), the temperature T is replaced by an apparent temperature T ( A ). We attribute this mass dependence to the system size effect as discussed in Section IV. In the improved MFM formulation, therefore, this system size effect is empirically realized by reducing the apparent temperature as A increases as T ( A )= T 0 (1- kA ). T 0 is the temperature of the fragmenting source and k is a constant quantifying the mass dependence. In order to study the density, temperature and symmetry energy in the fragmenting source, the improved MFM of Eq.(1) is utilized to extract the a sym / T 0 value from the available isotope yields. Since the a sym / T 0 value in Eqs.(1) and (2) depends on 5 parameters, a v , a s , a c , a p and Δ μ (defined by Δ μ = μ n - μ p ), the optimization process of these parameters is divided into the following three steps to minimize the ambiguity for each parameter. For a given k value 1. Optimize Δ μ / T 0 and a c / T 0 values from mirror isobars and fix these parameter values. 2. Optimize a v / T 0 , a s / T 0 and a p / T 0 values from N=Z isotopes. 3. Using extracted parameters in step (1) and step (2), a sym / T 0 values are extracted from all available isotopes. Comparing the extracted a sym / T 0 values with three different interactions, the density of the fragmenting source is extracted. Using this density, the value of the symmetry energy coefficient, a sym , for each interaction is determined. The temperature is then calculated as the ratio of a sym to a sym / T 0 . It is expectable that if the k value is properly selected which means the mass dependence is well considered, a constant T 0 is obtained. Since the k value is small as seen below, we perform the optimization of the parameter k in an iterative manner in the following analysis, that is, in the first round k = k 1 =0 is set in T ( A )= T 0 (1- kA ) and calculate the temperature as a function of A , using steps (1)–(3). From this plot a new k value, <math id="M3"><mrow><mi>k</mi><mo>=</mo><msub><mi>k</mi><mn>1</mn></msub><mo>'</mo></mrow></math> , is extracted from the slope. In the second round, <math id="M4"><mrow><mi>k</mi><mo>=</mo><msub><mi>k</mi><mn>2</mn></msub><mo>=</mo><msub><mi>k</mi><mn>1</mn></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>k</mi><mn>1</mn></msub><mo>'</mo></mrow></math> is used for the steps (1)–(3) and a new k value, <math id="M5"><mrow><mi>k</mi><mo>=</mo><msub><mi>k</mi><mn>2</mn></msub><mo>'</mo></mrow></math> , is extracted. If <math id="M6"><mrow><msub><mi>k</mi><mn>2</mn></msub><mo>'</mo></mrow></math> is 0 within a given error range, the iteration stops and the k 2 value is fixed as the mass dependent parameter of the apparent temperature and the T 0 value is determined. Otherwise the iteration continues. The details of steps (1)–(3) are first described below for a given k value. In the step (1), following Ref. [ 19 19 ], the isotope yield ratio between isobars with I +2 and I , R ( I +2, I , A ), is utilized, which is <math display="block" id="M7"><mtable columnalign="left"><mtr><mtd><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>2,</mn><mi>I</mi><mn>,</mn><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>2,</mn><mi>A</mi><mo stretchy="false">)</mo><mo>/</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>I</mi><mn>,</mn><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mi>exp</mi><mo>{</mo><mo stretchy="false">[</mo><msub><mi>μ</mi><mi>n</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>p</mi></msub><mo>+</mo><mn>2</mn><msub><mi>a</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mi>Z</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>/</mo><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mtd></mtr><mtr><mtd><mo>−</mo><mn>4</mn><msub><mi>a</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>/</mo><mi>A</mi><mo>−</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>1,</mn><mi>Z</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>/</mo><mo stretchy="false">[</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>k</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>2,</mn><mi>I</mi><mn>,</mn><mi>A</mi><mo stretchy="false">)</mo><mo>}</mo><mn>,</mn></mtd></mtr></mtable></math> where Y ( I , A ) is the yield of isotopes with I and A , and Δ( I +2, I , A )= S mix ( I +2, A ) - S mix ( I , A ). When the above equation is applied for a pair of mirror nuclei of odd mass isotopes with I = -I and I , the symmetry energy term, pairing term and mixing entropy terms drop out and the following equation is obtained. <math display="block" id="M8"><mrow><mi>ln</mi><mo stretchy="false">[</mo><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mn>,</mn><mo>−</mo><mi>I</mi><mn>,</mn><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>/</mo><mi>I</mi><mo>=</mo><mo stretchy="false">[</mo><mi>Δ</mi><mi>μ</mi><mo>+</mo><msub><mi>a</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>/</mo><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo stretchy="false">]</mo><mo>/</mo><mo stretchy="false">[</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>k</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>.</mn></mrow></math> For all available mirror isobars, Δ μ / T 0 and a c / T 0 are optimized in Eq.(4). The ln[ R ( I , - I , A )]/ I values and the fit result for k =0 is shown in Fig.1 Fig.1 . In the step (2) we apply Eq.(1) to the isotopes with N=Z with the extracted Δ μ / T 0 and a c / T 0 values in the step (1). For the N = Z = A /2 isotopes, the ratio of the free energy relative to the temperature can be calculated from Eq.(1) and Eq.(2) without the symmetry energy term as <math display="block" id="M9"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo>/</mo><mn>2,</mn><mi>A</mi><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo>/</mo><mn>2,</mn><mi>A</mi><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>k</mi><mi>A</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>ln</mi><mo stretchy="false">[</mo><mfrac><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi>A</mi><mo>/</mo><mn>2,</mn><mi>A</mi><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><msup><mi>A</mi><mi>τ</mi></msup></mrow><mrow><msub><mi>Y</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">]</mo><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>k</mi><mi>A</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><mover accent="true"><mrow><msub><mi>a</mi><mi>v</mi></msub></mrow><mo stretchy="true">˜</mo></mover></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mi>A</mi><mo>−</mo><mfrac><mrow><msub><mi>a</mi><mi>s</mi></msub></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><msup><mi>A</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mfrac><mrow><msub><mi>a</mi><mi>c</mi></msub></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mfrac><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mrow><msub><mi>a</mi><mi>p</mi></msub></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mfrac><mi>δ</mi><mrow><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>A</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>k</mi><mi>A</mi><mo stretchy="false">)</mo><mi>ln</mi><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where <math id="M10"><mrow><mover accent="true"><mrow><msub><mi>a</mi><mi>v</mi></msub></mrow><mo stretchy="true">˜</mo></mover><mo>=</mo><msub><mi>a</mi><mi>v</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msub><mi>μ</mi><mi>n</mi></msub><mo>+</mo><msub><mi>μ</mi><mi>p</mi></msub><mo stretchy="false">)</mo></mrow></math> . The value of <math id="M11"><mrow><mi>l</mi><mi>n</mi><mo stretchy="false">[</mo><mfrac><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi>A</mi><mo>/</mo><mn>2,</mn><mi>A</mi><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><msup><mi>A</mi><mi>τ</mi></msup></mrow><mrow><msub><mi>Y</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">]</mo></mrow></math> on the right of the second equation can be calculated from the simulated or experimental values when the τ value is fixed. Non-zero values show the deviation of the mass distribution of N=Z isotopes from the power law distribution determined by the critical exponent [ 40 40 ]. When other τ values are used, the parameter values change accordingly. In order to eliminate the constant Y 0 , all isotope yields are normalized to the yield of 12 C [ 40 40 , 18 18 , 19 19 ]. For the first round with k =0, the renormalized values of <math id="M12"><mrow><mo>−</mo><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo>/</mo><mn>2,</mn><mi>A</mi><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac></mrow></math> from the AMD events with the g0 interaction are plotted as a function of the isotope mass A using solid points in Fig. 2(a) Fig. 2(a) . The values of <math id="M13"><mrow><mover accent="true"><mrow><msub><mi>a</mi><mi>v</mi></msub></mrow><mo stretchy="true">˜</mo></mover><mo>/</mo><msub><mi>T</mi><mn>0</mn></msub></mrow></math> , a s / T 0 and a p / T 0 are used as free parameters to fit the given <math id="M14"><mrow><mo>−</mo><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo>/</mo><mn>2,</mn><mi>A</mi><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac></mrow></math> values, employing Eq.(5). A typical search result is shown by open circles in Fig. 2(a) Fig. 2(a) for the case of the g0 interaction. Similar quality results are obtained for the events generated using the g0AS and g0ASS interactions. One should note that the value of a p / T 0 makes a small contribution and that the contribution is evident as staggering in the - F ( A /2, A /2)/ T 0 vs. A plot. Therefore the essential free parameters in this step are <math id="M15"><mrow><mover accent="true"><mrow><msub><mi>a</mi><mi>v</mi></msub></mrow><mo stretchy="true">˜</mo></mover><mo>/</mo><msub><mi>T</mi><mn>0</mn></msub></mrow></math> and a s / T 0 . The extracted parameter values are summarized in Table 1 Table 1 for the first round ( k =0) and the final round ( k =0.007). a Δ μ / T 0 values are taken from the step (1). 1 1 k 1 1 interaction 1 1 <math id="M17"><mrow><mover accent="true"><mrow><msub><mi>a</mi><mi>v</mi></msub></mrow><mo stretchy="true">˜</mo></mover><mo>/</mo><msub><mi>T</mi><mn>0</mn></msub></mrow></math> 1 1 a s / T 0 1 1 a c / T 0 1 1 a p / T 0 1 1 Δ μ / T 0 a a 1 1 0.0 1 1 g0 1 1 1.77 1 1 2.74 1 1 0.1040 1 1 0.427 1 1 -0.254 1 1 1 1 g0AS 1 1 1.76 1 1 2.66 1 1 0.1080 1 1 0.661 1 1 -0.272 1 1 1 1 g0ASS 1 1 1.77 1 1 2.72 1 1 0.1030 1 1 0.778 1 1 -0.252 1 1 0.007 1 1 g0 1 1 1.55 1 1 2.29 1 1 0.0805 1 1 0.416 1 1 -0.189 1 1 1 1 g0AS 1 1 1.53 1 1 2.20 1 1 0.0830 1 1 0.651 1 1 -0.202 1 1 1 1 g0ASS 1 1 1.55 1 1 2.29 1 1 0.0800 1 1 0.766 1 1 -0.186 In step (3), Eq.(1) is applied to the yields of all isotopes with N=Z and N Z . From Eq.(1), a sym / T 0 and Δ μ / T 0 values can be related to the modified free energy, <math id="M18"><mrow><mfrac><mrow><mi>Δ</mi><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac></mrow></math> as <math display="block" id="M19"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><mi>Δ</mi><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mfrac><mrow><msub><mi>a</mi><mrow><mi>s</mi><mi>y</mi><mi>m</mi></mrow></msub></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mi>A</mi></mfrac><mo>−</mo><mfrac><mrow><mi>Δ</mi><mi>μ</mi></mrow><mrow><mn>2</mn><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mi>Z</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> where <math id="M20"><mrow><mfrac><mrow><mi>Δ</mi><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac></mrow></math> is the free energy relative to the temperature, <math id="M21"><mrow><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac></mrow></math> , subtracted by the calculated contributions of the volume, surface, Coulomb and pairing terms, using the parameters in Table 1 Table 1 . Resultant <math id="M22"><mrow><mfrac><mrow><mi>Δ</mi><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac></mrow></math> values are shown by symbols in Fig. 2(b) Fig. 2(b) . They exhibit quadratic relationships with minimum values close to zero. The minimum values are at or near N=Z isotopes and therefore reflect approximately the difference between the data and fitting points in Fig. 2(a) Fig. 2(a) . In this step, the a sym / T 0 and the Δ μ / T 0 values are optimized. Since the Δ μ / T 0 values are extracted from the step(1), the optimization is made around the values in the fifth column of Table 1 Table 1 in a small margin. The a sym / T 0 values are extracted from the quadratic curvature of the isotope distribution for each given Z and plotted in Fig. 2(c) Fig. 2(c) separately for the g0, g0AS and g0ASS interactions. As one can see for the first round with k =0, the extracted a sym / T 0 values increase as Z increases in all cases, and they more or less parallel each other. III. SELF-CONSISTENT DETERMINATION OF DENSITY AND TEMPERATURE 1 In order to determine the density and temperature at the time of the fragment formation, the parallel behavior of the observed a sym / T 0 values in Fig. 2(c) Fig. 2(c) is utilized. As suggested in Ref. [ 12 12 ], the observed differences are attributed to the difference of the symmetry energy at the density at the time of the fragment formation. The ratios between g0/g0AS and g0/g0ASS of a sym / T 0 for the first round are shown in Fig. 3(a) Fig. 3(a) . The ratios show flat distributions as a function of Z for both cases. The extracted average ratio values are shown by lines in the figure and the values are given in the first column of Table 2 Table 2 . In Fig. 3(b) Fig. 3(b) the symmetry energy coefficient is plotted as a function of the density for the three interactions used in the calculations and in Fig. 3(c) Fig. 3(c) their ratios, R sym = a sym ( g 0)/ a sym ( g 0AS) and R sym = a sym ( g 0)/ a sym ( g 0ASS), are plotted. Using the ratio values determined from Fig. 3(a) Fig. 3(a) and the density dependence of the R sym values in Fig. 3(c) Fig. 3(c) , the implied densities of the fragmenting sources are indicated by the shaded vertical areas shown in Fig. 3(c) Fig. 3(c) . The extracted density values for each case are given in the second column of Table 2 Table 2 . Assuming that the nucleon density should be same for the three different interactions used, the nucleon density of the fragmenting source is determined from the overlap of the extracted values. This assumption is reasonable for the central collisions because the nucleon density is mainly determined by the stiffness of the EOS and not by the density dependence of the symmetry energy term. From the overlapped density area in Figs. 3(c) Figs. 3(c) , ρ / ρ 0 = 0.67 ± 0.02 is extracted as the density at the time of the fragment formation. Using this density value, the corresponding symmetry energy values at that density are extracted for the three different interactions from Fig. 3(b) Fig. 3(b) . They are given in the third column of Table 2 Table 2 . 1 1 k 1 1 interaction 1 1 R sym 1 1 ρ / ρ 0 1 1 a sym (MeV) 1 1 0.0 1 1 g0 1 1 1 1 1 1 26.6±0.3 1 1 1 1 g0/g0AS 1 1 1.14±0.02 1 1 0.71±0.05 1 1 1 1 1 1 g0AS 1 1 1 1 1 1 23.7±1.3 1 1 1 1 g0/g0ASS 1 1 1.41±0.03 1 1 0.66±0.02 1 1 1 1 1 1 g0ASS 1 1 1 1 1 1 18.7±0.7 1 1 0.007 1 1 g0 1 1 1 1 1 1 26.8±0.2 1 1 1 1 g0/g0AS 1 1 1.14±0.02 1 1 0.72±0.05 1 1 1 1 1 1 g0AS 1 1 1 1 1 1 24.0±1.3 1 1 1 1 g0/g0ASS 1 1 1.40±0.03 1 1 0.66±0.02 1 1 1 1 1 1 g0ASS 1 1 1 1 1 1 18.7±0.7 Once the symmetry energy value is determined for a given interaction, the temperature, T 0 , can be calculated as T 0 = a sym / ( a sym / T 0 ). The extracted T 0 values are shown as a function of Z by open symbols for the first round in Fig. 4 Fig. 4 . The larger errors of T 0 , comparing to those in Fig. 2(c) Fig. 2(c) , originate from the errors of a sym and a sym / T 0 extracted for each interaction which are shown in the third column of Table 2 Table 2 and Fig. 2(c) Fig. 2(c) , respectively. The temperature values extracted from the three different interactions agree with each other very well and show a monotonic decrease as Z increases from 5 MeV at Z = 4 to 3 MeV at Z = 18. From this slope the extracted temperature as a function of A , T 0 = 5.5 (1 - 0.012A), is determined for the first round, assuming A 2 Z . The iteration is repeated four times in this work. The same plots as Fig.2 Fig.2 , but with the k value for the fourth (final) round, k =0.007, are shown in Fig.5 Fig.5 and the extracted parameters are also given in Table 1 Table 1 . A very similar quality of results to those of the first round with k =0 were obtained, even though the optimized parameter values are quite different between those of the first round ( k =0) and of the fourth round ( k =0.007). The extracted a sym / T 0 values parallel each other and show a rather flat distribution as a function of Z for Z up to 15 in Fig. 5(c) Fig. 5(c) . As seen in Fig. 4 Fig. 4 , in which the extracted T 0 values are shown by closed symbols as a function of Z for the fourth round with k =0.007, the extracted T 0 values are consistent with 5.5 MeV within the error bars. Since T 0 values show a flat distribution as a function of Z , the iteration is stopped at this round and T 0 =(5.5±0.2) MeV is taken as the temperature of the emitting source. The extracted parameter values of R sym , density and symmetry energy for the fourth round are very similar to those of the first round as shown in Table 2 Table 2 . The values and errors of these parameters are essentially determined by the ratios and their errors of the a sym / T 0 values between different interactions as discussed in Fig.3(a) Fig.3(a) . These ratio values are stable between the first and fourth rounds, even though the optimized parameter values in Table 1 Table 1 are quite different between these rounds. The extracted value of the mass dependent factor k has some errors, but the above fact ensures that the extracted density, temperature and symmetry values and their errors in Table 2 Table 2 are rather stable independent of the choice of the k values within its error bar, when the parameters in Table 1 Table 1 are optimized for the given k value. The decreasing trend of the temperature as A increases is often observed in heavy ion collisions and normally attributed to variations in impact parameter [ 47 47 ]. The heavier fragments tend to be produced in more peripheral collisions and therefore show a lower temperature. However in this study all events analyzed are generated in the same class of events, central collisions with b =0 fm. Therefore we attribute the decreasing trend may originate to different fragment formation processes rather than to the centrality of the events. IV. SUMMARY 1 An improved method is proposed for the extraction of the ratio of the symmetry energy coefficient relative to the temperature, a sym / T ( A ), taking into account the mass dependence of the apparent temperature, based on the MFM model. This method is applied for the central collisions of the AMD events generated for 40 Ca + 40 Ca at 35 MeV/nucleon. The Gogny interactions, g0, g0AS and g0ASS, with three different density dependencies of the symmetry energy are employed. As a function of IMF charge Z , the ratios of the extracted a sym / T 0 values from different interactions are essentially constant and reflect the differences of the symmetry energy at that density at the time of the fragment formation. Using this correlation, ρ / ρ 0 = 0.67 ± 0.02 is evaluated as the density at the time of fragmenting source and the symmetry energy value at that density are extracted for each interaction. The temperature values are then determined as T 0 =5.5 MeV. The apparent temperatures show a monotonic decrease as the fragment Z increases, changing from 5 MeV to 3 MeV when Z increases from 4 to 18 due to the system size effect. References B A Li , L W Chen and C M Ko . Recent progress and new challenges in isospin physics with heavy-ion reactions . Phys Rep , 2008 , 464 : 113 - 281 . DOI: 10.1016/j.physrep.2008.04.005 http://doi.org/10.1016/j.physrep.2008.04.005 Recent progress and new challenges in isospin physics with heavy-ion reactions M B Tsang , W A Friedman , C K Gelbke , et al . Isotopic scaling in nuclear reactions . Phys Rev Lett , 2001 , 86 : 5023 - 5026 . DOI: 10.1103/PhysRevLett.86.5023 http://doi.org/10.1103/PhysRevLett.86.5023 Isotopic scaling in nuclear reactions M B Tsang , T X Liu , L Shi , et al . Isospin diffusion and the nuclear symmetry energy in heavy ion reactions . Phys Rev Lett , 2004 , 92 : 062701 . DOI: 10.1103/PhysRevLett.92.062701 http://doi.org/10.1103/PhysRevLett.92.062701 Isospin diffusion and the nuclear symmetry energy in heavy ion reactions M Famiano , T Liu , W G Lynch , et al . Neutron and proton pransverse emission ratio measurements and the density dependence of the asymmetry term of the nuclear equation of state . Phys Rev Lett , 2006 , 97 : 052701 . DOI: 10.1103/PhysRevLett.97.052701 http://doi.org/10.1103/PhysRevLett.97.052701 Neutron and proton pransverse emission ratio measurements and the density dependence of the asymmetry term of the nuclear equation of state W J Xie and F S Zhang . Probing the density dependence of the symmetry energy with central heavy ion collisions . Nuc Sci Tech , 2013 , 24 : 050502 . Probing the density dependence of the symmetry energy with central heavy ion collisions T Li , U Garg , Y Liu , et al . Isotopic dependence of the giant monopole resonance in the even-A 112-124Sn isotopes and the asymmetry term in nuclear incompressibility . Phys Rev Lett , 2007 , 99 : 162503 . DOI: 10.1103/PhysRevLett.99.162503 http://doi.org/10.1103/PhysRevLett.99.162503 Isotopic dependence of the giant monopole resonance in the even-A 112-124Sn isotopes and the asymmetry term in nuclear incompressibility A Klimkiewicz , N Paar , P Adrich , et al . Nuclear symmetry energy and neutron skins derived from pygmy dipole resonances . Phys Rev C , 2007 , 76 : 051603 . DOI: 10.1103/PhysRevC.76.051603 http://doi.org/10.1103/PhysRevC.76.051603 Nuclear symmetry energy and neutron skins derived from pygmy dipole resonances L Trippa , G Colo and E Vigezzi . Giant dipole resonance as a quantitative constraint on the symmetry energy . Phys Rev C , 2008 , 77 : 061304 . DOI: 10.1103/PhysRevC.77.061304 http://doi.org/10.1103/PhysRevC.77.061304 Giant dipole resonance as a quantitative constraint on the symmetry energy Z Kohley , M Colonna , A Bonasera , et al . Sensitivity of intermediate mass fragment flows to the symmetry energy . Phys Rev C , 2012 , 85 : 064605 . DOI: 10.1103/PhysRevC.85.064605 http://doi.org/10.1103/PhysRevC.85.064605 Sensitivity of intermediate mass fragment flows to the symmetry energy H S Xu , M B Tsang , T X Liu , et al . Isospin fractionation in nuclear multifragmentation . Phys Rev Lett , 2000 , 85 : 716 - 719 . DOI: 10.1103/PhysRevLett.85.716 http://doi.org/10.1103/PhysRevLett.85.716 Isospin fractionation in nuclear multifragmentation M B Tsang , C K Gelbke , X D Liu , et al . Isoscaling in statistical models . Phys Rev C , 2001 , 64 : 054615 . DOI: 10.1103/PhysRevC.64.054615 http://doi.org/10.1103/PhysRevC.64.054615 Isoscaling in statistical models A Ono , P Danielewicz , W A Friedman , et al . Isospin fractionation and isoscaling in dynamical simulations of nuclear collisions . Phys Rev C , 2003 68 : 051601(R) . For g0AS, x=-1/2 and for g0ASS x=-2 are used in Eq.(2) in Ref. [12] . DOI: 10.1103/PhysRevC.68.051601 http://doi.org/10.1103/PhysRevC.68.051601 Isospin fractionation and isoscaling in dynamical simulations of nuclear collisions A Bonasera , F Gulminelli and J Molitoris . The Boltzmann equation at the borderline. A decade of Monte Carlo simulations of a quantum kinetic equation . Phys Rep , 1994 , 243 : 1 - 124 . DOI: 10.1016/0370-1573(94)90108-2 http://doi.org/10.1016/0370-1573(94)90108-2 The Boltzmann equation at the borderline. A decade of Monte Carlo simulations of a quantum kinetic equation Bonasera A , Bruno M , Dorso C O , et al . Critical phenomena in nuclear fragmentation . Riv. Nuovo Cimento , 2000 , 23 : 1 - 102 . Critical phenomena in nuclear fragmentation M E Fisher . The theory of equilibrium critical phenomena . Rep Prog Phys , 1967 , 30 : 615 - 730 . DOI: 10.1088/0034-4885/30/2/306 http://doi.org/10.1088/0034-4885/30/2/306 The theory of equilibrium critical phenomena R W Minich , S Agarwal , A Bujak , et al . Critical phenomena in hadronic matter and experimental isotopic yields in high energy proton-nucleus collisions . Phys Lett B , 1982 , 118 : 458 - 460 . DOI: 10.1016/0370-2693(82)90224-6 http://doi.org/10.1016/0370-2693(82)90224-6 Critical phenomena in hadronic matter and experimental isotopic yields in high energy proton-nucleus collisions A S Hirsch , A Bujak , J E Finn , et al . The description of fragmentation as a critical phenomenon in high energy P-nucleus collisions . Nucl Phys A , 1984 , 418 : 267 - 287 . DOI: 10.1016/0375-9474(84)90553-0 http://doi.org/10.1016/0375-9474(84)90553-0 The description of fragmentation as a critical phenomenon in high energy P-nucleus collisions M Huang , Z Chen , S Kowalski , et al . A novel approach to isoscaling: The role of the order parameter <math id="M23"><mrow><mi>m</mi><mo>=</mo><mfrac><mrow><msub><mi>N</mi><mi>f</mi></msub><mo>−</mo><msub><mi>Z</mi><mi>f</mi></msub></mrow><mrow><msub><mi>A</mi><mi>f</mi></msub></mrow></mfrac></mrow></math> . Nucl Phys A , 2010 , 847 : 233 - 242 . DOI: 10.1016/j.nuclphysa.2010.07.004 http://doi.org/10.1016/j.nuclphysa.2010.07.004 A novel approach to isoscaling: The role of the order parameter m=Nf−ZfAf M Huang , Z Chen , S Kowalski , et al . Isobaric yield ratios and the symmetry energy in heavy-ion reactions near the Fermi energy . Phys Rev C , 2010 , 81 : 044620 . DOI: 10.1103/PhysRevC.81.044620 http://doi.org/10.1103/PhysRevC.81.044620 Isobaric yield ratios and the symmetry energy in heavy-ion reactions near the Fermi energy A Ono , P Danielewicz , W A Friedman , et al . Symmetry energy for fragmentation in dynamical nuclear collisions . Phys Rev C , 2004 , 70 : 041604(R) . DOI: 10.1103/PhysRevC.70.041604 http://doi.org/10.1103/PhysRevC.70.041604 Symmetry energy for fragmentation in dynamical nuclear collisions C C Guo , J Su and F S Zhang . Comparison between nuclear thermometers in central Xe+Sn collision . Nucl Sci Tech , 2013 , 24 : 050513 . Comparison between nuclear thermometers in central Xe+Sn collision G D Westfall , B V Jacak , N Anantaraman , et al . Energy dependence of nuclear matter disassembly in heavy ion collisions . Phys Lett B , 1992 , 116 : 118 - 122 . DOI: 10.1016/0370-2693(82)90988-1 http://doi.org/10.1016/0370-2693(82)90988-1 Energy dependence of nuclear matter disassembly in heavy ion collisions B V Jacak , G D Westfal , C K Gelbke , et al . Measurement of complex fragments and clues to the entropy production from 42-137-MeV/nucleon Ar + Au . Phys Rev Lett , 1983 , 51 : 1846 - 1849 . DOI: 10.1103/PhysRevLett.51.1846 http://doi.org/10.1103/PhysRevLett.51.1846 Measurement of complex fragments and clues to the entropy production from 42-137-MeV/nucleon Ar + Au S Wuenschel , A Bonasera , L W May , et al . Measuring the temperature of hot nuclear fragments . Nucl Phys A , 2010 , 843 : 1 - 13 . DOI: 10.1016/j.nuclphysa.2010.04.013 http://doi.org/10.1016/j.nuclphysa.2010.04.013 Measuring the temperature of hot nuclear fragments H Zheng , G Giuliani and A Bonasera . Density and temperature of fermions and bosons from quantum fluctuations . Nucl Sci Tech , 2013 , 24 : 050512 . Density and temperature of fermions and bosons from quantum fluctuations S Albergo , S Costa , E Costanzo , et al . Temperature and free-nucleon densities of nuclear matter exploding into light clusters in heavy-ion collisions . Nuovo Cimento A , 1985 , 89 : 1 - 28 . DOI: 10.1007/BF02773614 http://doi.org/10.1007/BF02773614 Temperature and free-nucleon densities of nuclear matter exploding into light clusters in heavy-ion collisions D J Morrissey . Excited state production and temperature measurement in a heavy ion reaction . Phys Lett B , 1984 , 148 : 423 - 427 . DOI: 10.1016/0370-2693(84)90730-5 http://doi.org/10.1016/0370-2693(84)90730-5 Excited state production and temperature measurement in a heavy ion reaction R Wada , M R Huang , W P Lin , et al . IMF production and symmetry energy in heavy ion collisions near Fermi energy . Nucl Sci Tech , 2013 , 24 : 050501 . IMF production and symmetry energy in heavy ion collisions near Fermi energy W P Lin , R Wada , M R Huang , et al . Experimental reconstruction of primary fragments with kinematical focusing method . Nucl Sci Tech , 2013 , 24 : 050511 . Experimental reconstruction of primary fragments with kinematical focusing method W Lin , X Liu , M R D Rodrigues , et al . Novel determination of density, temperature, and symmetry energy for nuclear multifragmentation through primary fragment-yield reconstruction . Phys Rev C , 2014 , 89 : 021601(R) . DOI: 10.1103/PhysRevC.89.021601 http://doi.org/10.1103/PhysRevC.89.021601 Novel determination of density, temperature, and symmetry energy for nuclear multifragmentation through primary fragment-yield reconstruction J Łukasik , G Augerb , M L Begemann-Blaich , et al . Fragmentation in peripheral heavy-ion collisions: from neck emission to spectator decays . Phys Lett B , 2003 , 566 : 76 - 83 . DOI: 10.1016/S0370-2693(03)00811-6 http://doi.org/10.1016/S0370-2693(03)00811-6 Fragmentation in peripheral heavy-ion collisions: from neck emission to spectator decays A Ono . Antisymmetrized molecular dynamics with quantum branching processes for collisions of heavy nuclei . Phys Rev C , 1999 , 59 : 853 - 864 . DOI: 10.1103/PhysRevC.59.853 http://doi.org/10.1103/PhysRevC.59.853 Antisymmetrized molecular dynamics with quantum branching processes for collisions of heavy nuclei A Ono and H Horiuchi . Antisymmetrized molecular dynamics of wave packets with stochastic incorporation of the Vlasov equation . Phys Rev C , 1996 , 53 : 2958 - 2972 . DOI: 10.1103/PhysRevC.53.2958 http://doi.org/10.1103/PhysRevC.53.2958 Antisymmetrized molecular dynamics of wave packets with stochastic incorporation of the Vlasov equation A Ono , S Hudan , A Chbihi , et al . Compatibility of localized wave packets and unrestricted single particle dynamics for cluster formation in nuclear collisions . Phys Rev C , 2002 , 66 : 014603 . DOI: 10.1103/PhysRevC.66.014603 http://doi.org/10.1103/PhysRevC.66.014603 Compatibility of localized wave packets and unrestricted single particle dynamics for cluster formation in nuclear collisions A Ono and H Horiuchi . Antisymmetrized molecular dynamics for heavy ion collisions . Prog Part Nucl Phys , 2004 , 53 : 501 - 581 . DOI: 10.1016/j.ppnp.2004.05.002 http://doi.org/10.1016/j.ppnp.2004.05.002 Antisymmetrized molecular dynamics for heavy ion collisions R Wada , K Hagel , J Cibo , et al . Entrance channel dynamics in 40Ca + 40Ca at 35A MeV . Phys Lett B , 1998 , 422 : 6 - 12 . DOI: 10.1016/S0370-2693(98)00033-1 http://doi.org/10.1016/S0370-2693(98)00033-1 Entrance channel dynamics in 40Ca + 40Ca at 35A MeV R Wada , K Hagel , J Cibor , et al . Reaction mechanisms and multifragmentation processes in 64Zn + 58Ni at 35A-79A MeV . Phys Rev C , 2000 , 62 : 034601 . DOI: 10.1103/PhysRevC.62.034601 http://doi.org/10.1103/PhysRevC.62.034601 Reaction mechanisms and multifragmentation processes in 64Zn + 58Ni at 35A-79A MeV R Wada , T Keutgen , K Hagel , et al . Reaction dynamics and multifragmentation in Fermi energy heavy ion reactions . Phys Rev C , 2004 , 69 : 044610 . DOI: 10.1103/PhysRevC.69.044610 http://doi.org/10.1103/PhysRevC.69.044610 Reaction dynamics and multifragmentation in Fermi energy heavy ion reactions S Hudan , R T De Souza and A Ono . Short timescale behavior of colliding heavy nuclei at intermediate energies . Phys Rev C , 2006 , 73 : 054602 . DOI: 10.1103/PhysRevC.73.054602 http://doi.org/10.1103/PhysRevC.73.054602 Short timescale behavior of colliding heavy nuclei at intermediate energies A Bonasera , Z Chen , R Wada , et al . Quantum nature of a nuclear phase transition . Phys Rev Lett , 2008 , 101 : 122702 . DOI: 10.1103/PhysRevLett.101.122702 http://doi.org/10.1103/PhysRevLett.101.122702 Quantum nature of a nuclear phase transition C F Von Weizsäcker . Zur theorie der Kernmassen . Z Phys , 1935 , 96 : 431 - 458 . DOI: 10.1007/BF01337700 http://doi.org/10.1007/BF01337700 Zur theorie der Kernmassen H A Bethe and R F Bacher . Stationary states of nuclei . Rev mod Phys , 1936 , 8 : 82 - 229 . DOI: 10.1103/RevModPhys.8.82 http://doi.org/10.1103/RevModPhys.8.82 Stationary states of nuclei P Marini , A Bonasera , A McIntosh , et al . Constraining the symmetry term in the nuclear equation of state at subsaturation densities and finite temperatures . Phys Rev C , 2012 , 85 : 034617 . DOI: 10.1103/PhysRevC.85.034617 http://doi.org/10.1103/PhysRevC.85.034617 Constraining the symmetry term in the nuclear equation of state at subsaturation densities and finite temperatures L G Sobotka . Nuclear asymmetry enthalpy . Phys Rev C , 2011 , 84 : 017601 . DOI: 10.1103/PhysRevC.84.017601 http://doi.org/10.1103/PhysRevC.84.017601 Nuclear asymmetry enthalpy Z Chen , S Kowalski , M Huang , et al . Isocaling and the symmetry energy in the multifragmentation regime of heavy-ion collisions . Phys Rev C , 2010 , 81 , 064613 . DOI: 10.1103/PhysRevC.81.064613 http://doi.org/10.1103/PhysRevC.81.064613 Isocaling and the symmetry energy in the multifragmentation regime of heavy-ion collisions W Lin , X Liu , M R D Rodrigues , et al . Experimental reconstruction of primary hot isotopes and characteristic properties of the fragmenting source in heavy-ion reactions near the Fermi energy . Phys Rev C , 2014 , 90 , 044603 . DOI: 10.1103/PhysRevC.90.044603 http://doi.org/10.1103/PhysRevC.90.044603 Experimental reconstruction of primary hot isotopes and characteristic properties of the fragmenting source in heavy-ion reactions near the Fermi energy W Trautmann , R Bassini , M Begemann-Blaich , et al . Thermal and chemical freeze-out in spectator fragmentation . Phys Rev C , 2007 , 76 , 064606 . DOI: 10.1103/PhysRevC.76.064606 http://doi.org/10.1103/PhysRevC.76.064606 Thermal and chemical freeze-out in spectator fragmentation This work was partially supported by the visiting senior international scientists programs (2012T1JY3-2010T2J22) of CAS. The authors thank Prof. J. B. Natowitz and Prof. A. Bonasera for their financial support of their work in the US. 10.13538/j.1001-8042/nst.26.S20508 Supported by the National Natural Science Foundation of China (Nos. 11075189 and 11205209), 100 Persons Project (Nos. 0910020BR0 and Y010110BR0), ADS project 302 (No. Y103010ADS) of the Chinese Academy of Sciences, the U.S. Department of Energy (No. DE-FG03-93ER40773) and the Robert A. Welch Foundation (No. A0330) 2015-01-09 2015-01-28 2015-04-20 2015-04-20 2021-04-11 2015 Nuclear Science and Techniques 2 26