Temperature and symmetry energy of neutron-rich fragments in the 1A GeV ^124,136Xe+Pb reactions

Temperature and symmetry energy of neutron-rich fragments in the 1 A GeV 124,136 Xe+Pb reactions 1 corresp first-author Corresponding author. E-mail address: machunwang@126.com Corresponding author. E-mail address: machunwang@126.com machunwang@126.com 1 1 1 1 1 Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China In this work we study the symmetry-energy coefficient of neutron-rich nuclei, and the temperature dependence of nuclear symmetry energy at low temperatures. An isobaric method is used to extract the symmetry-energy coefficients of neutron-rich nucleus ( a sym ) at zero temperature ( T ) and a sym / T at nonzero temperature in the measured 1 A GeV 124,136 Xe+Pb reactions. T of fragment is obtained from the ratio of its a sym to a sym / T . The results show that, for fragment with the same neutron-excess ( I = N ‒Z), the heavier the fragment is, the higher T it has, and T tends to saturate around 1 MeV for the large mass fragments. It is also shown that the more neutron-rich the isobar is, the higher temperature it has. The T 2 dependence of symmetry energy of finite nucleus at low temperatures is verified by the extracted results. Temperature Symmetry energy Isobaric yield ratio Neutron-rich nucleus 1 10Introduction 1 Depending on both the density and temperature of nuclear matter, the nuclear symmetry energy (NSE) is very important not only in nuclear physics but also astrophysics. On the density dependence of NSE, several probes were used to study NSE theoretically and experimentally [ 1 1 , 2 2 , 3 3 , 4 4 , 5 5 , 6 6 , 7 7 , 8 8 , 9 9 , 10 10 , 11 11 , 12 12 , 13 13 , 14 14 ] . Many works have been performed on NSE of the supra-saturate nuclear matter in the hot emitting source of heavy-ion collisions (HIC) [ 1 1 , 2 2 ] . However, the NSE results show a large difference ranging from soft to stiff one, especially for high density nuclear matter. This necessitates further theoretical efforts on constraining the symmetry energy. On the temperature dependence of NSE, some theoretical work indicates a T 2 -dependence of symmetry energy of finite nucleus [ 15 15 ] . Comparing with the various thermometers of colliding source in HIC constructed by the emitted light particles [ 16 16 , 17 17 , 18 18 , 19 19 , 20 20 , 21 21 ] , the thermometer of heavy fragments in HIC is not fully investigated. In HIC, the measured fragment should not be of high temperature. Albergo et al . [ 16 16 ] used the isotopic thermometer to extract the temperature of heavy fragments, and the T of heavy fragment they obtained is lower than that of the light particles [ 22 22 , 23 23 ] . The recent IYR (isobaric yield ratio) thermometer of heavy fragments also indicates the low T of heavy fragment [ 24 24 , 25 25 ] . In this article, we present a new thermometer of heavy fragments using IYR, and use the thermometer to investigate the temperature of fragments produced in the 1 A GeV 124,136 Xe+Pb reactions. The non-zero temperature of symmetry energy coefficient to temperature ( a sym / T ) of neutron-rich fragment in HIC has been investigated using the IYR method in the framework of a modified Fisher model [ 9 9 , 10 10 , 11 11 , 12 12 ] . Previously, we reported the symmetry energy coefficient of nucleus at T =0, the surface-symmetry energy coefficient ( a sym ) and the volume-symmetry energy coefficient [ 13 13 ] . Based on the results in Refs.[11,13], we will extract the temperature of the measured fragments in the 1 A GeV 124, 136 Xe+Pb reactions [ 26 26 ] . 2 Theoretical method 1 The theoretical deduction is performed in the framework of the modified Fisher model, which relates the yield of a fragment to its free energy at temperature T [ 27 27 ] . The formula used to determine the a sym / T of the neutron-rich fragment in Refs.[11,12] is slightly changed to Eq.(1): <math display="block" id="M1"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msubsup><mi>a</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup></mrow><mrow><mi>A</mi><mi>T</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>{</mo><mi>ln</mi><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mi>I</mi><mo>−</mo><mn>2</mn><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>−</mo><mi>ln</mi><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>2</mn><mo>,</mo><mi>I</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable><mo>+</mo><mi>Δ</mi><msub><mi>E</mi><mi>c</mi></msub><mo>−</mo><msub><mi>Δ</mi><mrow><mi>I</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>Δ</mi><mi>I</mi></msub><mo>}</mo><mo>,</mo></mtd></mtr></mtable></math> where I = N ‒ Z , <math id="M2"><mrow><msubsup><mi>a</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup><mo>=</mo><mtext>4</mtext><msubsup><mi>a</mi><mrow><mtext>sym</mtext></mrow><mo>′</mo></msubsup></mrow></math> , and the prime means that the results are for the fragment; R is the yield ratio between the isobars differing 2 in I ; Δ E c = E c ( I )– E c ( I –2) is the difference between the Coulomb energy of the isobars; Δ I -2 and Δ I are the mixing term of N and Z of the isobars related [ Δ I =( N I+ 2 / A )+ Z I+ 2 ln( N I+ 2 / A )- N I ln( N I / A )- Z I ln( N I / A )]. In Eq.(1), T can hardly be known directly, and the term of <math id="M3"><mrow><msubsup><mi>α</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup><mo>/</mo><mi>A</mi><mi>T</mi></mrow></math> should be viewed as one parameter. For the zero-temperature finite nucleus, the symmetry energy coefficient can be determined using the isobaric method [ 13 13 ] as Eq.(2): <math display="block" id="M4"><mrow><mfrac><mrow><msubsup><mi>a</mi><mrow><mtext>sym</mtext></mrow><mo>*</mo></msubsup></mrow><mi>A</mi></mfrac><mo>=</mo><mo stretchy="false">[</mo><mi>B</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>−</mo><mi>B</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mi>Δ</mi><msub><mi>E</mi><mi>c</mi></msub><mo stretchy="false">]</mo><mo>/</mo><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo></mrow></math> where B ( I ) and B ( I +2) are the binding energy of the isobars [ 28 28 ] . In Eqs.(1) and (2), we take the same form of Coulomb energy of the nucleus as in Ref.[ 13 13 ]. Using Eqs.(1) and (2), the temperature of fragments in HIC can be determined as: <math display="block" id="M5"><mrow><mi>T</mi><mo>=</mo><mfrac><mrow><msubsup><mi>a</mi><mrow><mtext>sym</mtext></mrow><mo>*</mo></msubsup></mrow><mi>A</mi></mfrac><mo>/</mo><mfrac><mrow><msubsup><mi>a</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup></mrow><mrow><mi>A</mi><mi>T</mi></mrow></mfrac><mo>.</mo></mrow></math> Therefore, the a sym ~ T correlation can be investigated. 3 Results and discussion 1 In Fig.1 Fig.1 , the <math id="M6"><mrow><msubsup><mi>α</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup><mo>/</mo><mi>A</mi><mi>T</mi></mrow></math> of the fragments in the 1 A GeV 124,136 Xe+Pb reactions and <math id="M7"><mrow><msubsup><mi>α</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup><mo>/</mo><mi>A</mi><mi>T</mi></mrow></math> of the corresponding T =0 nuclei are plotted. Due to the effect of the nonzero temperature, there is large difference between the values of <math id="M8"><mrow><msubsup><mi>α</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup><mo>/</mo><mi>A</mi><mi>T</mi></mrow></math> and <math id="M9"><mrow><msubsup><mi>α</mi><mrow><mtext>sym</mtext></mrow><mo>*</mo></msubsup><mo>/</mo><mi>A</mi></mrow></math> , especially in fragments which have relative small mass. Taking the ratio of <math id="M12"><mrow><msubsup><mi>α</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup><mo>/</mo><mi>A</mi><mi>T</mi></mrow></math> and <math id="M13"><mrow><msubsup><mi>α</mi><mrow><mtext>sym</mtext></mrow><mo>*</mo></msubsup><mo>/</mo><mi>T</mi></mrow></math> of the fragments according to Eq.(3), the values of T related to the fragments can be determined. The obtained T of fragments are plotted in Fig.2 Fig.2 . For the nuclei having same I , T decreases with increasing mass, especially for fragments of which less nucleons being abraded from the projectile and are mostly produced in peripheral reactions [ 24 24 , 29 29 , 30 30 ] . The results show that, in the two Xe reactions, T of the same fragment has very little difference, except the fragment of small A in the same I nucleus-chain. It should be noted that, in models which predict the yield of fragment, the temperature is usually set as a constant, which will result in the disagreement between the theoretical and experimental yield. The variation of T of fragments shown in this work indicates that T changes with the fragment in theoretical calculation. After the fragment temperature is separated from the <math id="M14"><mrow><msubsup><mi>α</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup><mo>/</mo><mi>A</mi><mi>T</mi></mrow></math> results, by using the ( <math id="M15"><mrow><msubsup><mi>α</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup><mo>/</mo><mi>A</mi><mi>T</mi></mrow></math> )/ ( <math id="M16"><mrow><msubsup><mi>α</mi><mrow><mtext>sym</mtext></mrow><mrow><mo>*</mo><mo>′</mo></mrow></msubsup><mo>/</mo><mi>A</mi></mrow></math> ) ratio of the fragments, it is interesting to see how a sym / T depends on T as shown in HIC [ 9 9 , 10 10 , 11 11 , 12 12 , 31 31 ] . The correlation between a sym / T and the obtained T of the fragment is plotted in Fig.3 Fig.3 . Relative similar distributions of a sym / T~T correlations of fragments with different I are observed. For the measured fragments, a sym / T decreases with increasing T . Generally, the symbols of larger a sym / T are for the larger mass fragments, and the symbols of larger T values represent the smaller mass fragments in the same I chains. The temperature dependence of the symmetry energy of the fragments can be discussed as follows. According to theoretical calculation using the density-functional based on the Skyrme interaction (SKM), the relationship between the symmetry energy and the low temperature of the finite nucleus ( A , I ) was proposed as Eq.(4) [ 15 15 ] : <math display="block" id="M17"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>b</mi><mi>v</mi></msub><mo>+</mo><mn>0.42</mn><msup><mi>T</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msup><mi>I</mi><mn>2</mn></msup><mo>/</mo><mi>A</mi><mo>−</mo><mo stretchy="false">(</mo><msub><mi>b</mi><mi>s</mi></msub><mo>+</mo><mn>2.06</mn><msup><mi>T</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msup><mi>I</mi><mn>2</mn></msup><mo>/</mo><msup><mi>A</mi><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>,</mo></mrow></math> where b v and b s are the volume- and surface-symmetry energy coefficients, respectively. The first and second terms in RHS of Eq.(4) correspond to the volume symmetry energy and the surface symmetry energy of the ( A, I ) nucleus, respectively. The symmetry energy of the fragment at T is calculated, and the value of a sym /T of the fragment is deduced according to <math display="block" id="M18"><mrow><msub><mi>a</mi><mrow><mtext>sym</mtext></mrow></msub><mo>/</mo><mi>T</mi><mo>=</mo><msub><mi>E</mi><mrow><msup><mrow/><mrow><mtext>sym</mtext></mrow></msup></mrow></msub><mi>A</mi><mo>/</mo><mo stretchy="false">(</mo><msup><mi>I</mi><mn>2</mn></msup><mi>T</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></math> Figure 3 Figure 3 shows the correlation between a sym / T and T of the fragments in the 1 A GeV 124 Xe+Pb (crossed open symbols) and the 136 Xe+Pb (solid symbols) reactions. The symmetry energy of the 117 Cs nucleus, of which I =15, is calculated using different sets of b v and b s (for the S1, S2 and S3 lines in Fig.3 Fig.3 , b s / b v =1.68). Using such sets of b v and b s values, the a sym / T of nucleus depends very little on the b v values (result not shown). The T was varied from 0.04 MeV to 5 MeV in the calculation. In S1 (dash line), b v =19.69 MeV was suggested in Ref.[ 15 15 ]. In S2 (dot line), b v =32 MeV accords to the finite-range liquid-drop (FRLD) model [ 13 13 ] . In S3 (dash-dot line), the b v is simply changed to 29 MeV to see the effect of b v and b s parameters. All the S1‒S3 results can well reproduce the trend of the experimental a sym / T~T correlation. The result of S1 shows that the suggested b v in Ref.[ 15 15 ] agrees with the a sym / T of 117 Cs, while the S2 according to FRLD and S3 overestimate the experimental data. The calculated a sym / T verifies the T 2 dependence of the symmetry energy of finite nucleus, however, for a specific nucleus, proper values of b v and b s should be used. A simple fitting using the y =a+b x +c x 2 function was performed to see how a sym / T depends on the temperature, and from the fitting we have a=47.99, b=–27.34 and c=4.43. The b x term in the fitting function term cannot be omitted due to the complicated dependence of nuclear symmetry energy on A and I according to Eq.(4). In Fig.3 Fig.3 , when T is larger than ~3.5 MeV, the fitting result deviates from the experimental results, and the trend of the fitting result goes to the opposite direction of the experimental data, which indicates that T 2 dependence of the symmetry energy is only valid in the relative low temperature (for example, lower than 5 MeV). Actually, if the system temperature is larger than 5 MeV, the liquid-gas transition happens and the finite nucleus becomes unstable. 4 Conclusion 1 In conclusion, an IYR method is proposed to separate the energy part and temperature part of parameters obtained in the framework of a modified Fisher model. It is also instructive for other models based on the free energy. By analyzing the yield of the fragment produced in the 1 A GeV 124 Xe+Pb and 136 Xe+Pb reactions measured at GSI, the symmetry energy coefficient of fragment is found to be modified in different degrees by the temperature in HIC. T of fragment is determined by the ratio of its symmetry energy coefficient at T =0 and nonzero T in HIC. The dependence of a sym / T on T is shown. All these verify the T 2 -dependence of the symmetry energy of finite nucleus at low temperature, with a proper setting of b v and b s . References Li B A , Chen L W , Ko C M , et al . Phys Rep , 2008 , 464 : 113 - 181 . Phys Rep Chen L W , Ko C M , Li B A , et al . Front Phys China , 2007 , 2 : 327 - 357 . Front Phys China Tsang M B , Liu T X , Shi L , et al . Phys Rev Lett , 2004 , 92 : 062701 . Phys Rev Lett Ma Y G , Wang K , Wei Y B , et al . Phys Rev C , 2004 , 69 : 064610 . Phys Rev C Famiano M A , Liu T , Lynch W G , et al . Phys Rev Lett , 2006 , 97 : 052701 . Phys Rev Lett Zhou P , Tian W D , Ma Y G , et al . Phys Rev C , 2011 , 84 : 037605 . Phys Rev C Kuma S , Ma Y G , Zhang G Q , et al . Phys Rev C , 2012 , 85 : 024620 . Phys Rev C Pu J , Chen J H , Kumar S , et al . Phys Rev C , 2013 , 87 : 047603 . Phys Rev C Huang M R , Chen Z , Kowalski S , et al . Phys Rev C , 2010 , 81 : 044620 . Phys Rev C Ma C W , Wang F , Ma Y G , et al . Phys Rev C , 2011 , 83 : 064620 . Phys Rev C Ma C W , Pu J , Wei H L , et al . Eur Phys J A , 2012 , 48 : 78 . Eur Phys J A Ma C W , Pu J , Wang S S , et al . Chin Phys Lett , 2012 , 29 : 062101 . Chin Phys Lett Ma C W , Yang J B , Yu M , et al . Chin Phys Lett , 2012 , 29 : 092101 . Chin Phys Lett Gautam S , Sood A D , Puri R K , et al . Phys Rev C , 2011 , 83 : 034606 . Phys Rev C Lee S J and Mekjian A Z . Phys Rev C , 2010 , 82 : 064319 . Phys Rev C Albergo S , Costa S , Costanzo E , et al . Nuovo Cimento A , 1985 , 89 : 1 - 28 . Nuovo Cimento A Wang J , Wada R , Keutgen T , et al . Phys Rev C , 2005 , 82 : 024603 . Phys Rev C Su J and Zhang F S . Phys Rev C , 2011 , 84 : 037601 . Phys Rev C Westfall G D . Phys Lett B , 1982 , 116 : 118 - 122 . Phys Lett B Ma Y G , Natowitz J B , Wada R , et al . Phys Rev C , 2005 , 71 : 054606 . Phys Rev C Su J , Zhu L , Xie W J , et al . Phys Rev C , 2012 , 85 : 017604 . Phys Rev C Trautmann W , Bassini R , Begemann-Blaich M , et al . Phys Rev C , 2007 , 76 : 064606 . Phys Rev C Ma C W , Wang S S , Pu J , et al . Commun Theor Phys , 2013 , 59 : 95 - 98 . Commun Theor Phys Ma C W , Pu J , Ma Y G , et al . Phys Rev C , 2012 , 86 : 054611 . Phys Rev C Ma C W , Zhao X L , Pu J , et al . Phys Rev C , 2013 , 88 : 014609 . Phys Rev C Henzlova D , Schmidt K H , Ricciardi M V , et al . Phys Rev C , 2008 , 78 : 044616 . Phys Rev C Hirsch A S , Bujak A , Finn J E , et al . Phys Rev C , 1984 , 29 : 508 - 525 . Phys Rev C Audi G , Wapstra A H , Thibault C . Nucl Phys A , 2003 , 729 : 337 - 676 . Nucl Phys A Ma C W , Wei H L , Liu G J , et al . J Phys G Nucl Partic , 2010 , 37 : 015104 . J Phys G Nucl Partic Ma C W , Wei H L , Yu M . Phys Rev C , 2010 , 82 : 057602 . Phys Rev C Ma C W , Song H L , Pu J , et al . Chin Phys C , 2013 , 37 : 024102 . Chin Phys C We thank Prof. Roy Wada in Institute of Modern Physics, and Prof. Yu-Gang Ma in Shanghai Institute of Applied Physics, Chinese Academy of Sciences for their useful discussions. 10.13538/j.1001-8042/nst.2013.05.010 Supported by National Natural Science Foundation of China (NSFC) Projects (No.10905017) , the Program for Science & Technology Innovation Talents in Universities of Henan Province (No.2013HASTIT046) , and the Young Teacher Project in Henan Normal University. 2013-06-27 2013-10-01 2021-04-11 2013 Nuclear Science and Techniques 5 24