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The Symmetry Energy from the Neutron-Rich Nucleus Produced in the Intermediate-Energy 40,48Ca and 58,64Ni Projectile Fragmentation *

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The Symmetry Energy from the Neutron-Rich Nucleus Produced in the Intermediate-Energy 40,48Ca and 58,64Ni Projectile Fragmentation * ISSN: 0256-307X 中国物理快报 Chinese Physics Letters Volume 29 Number 6 June 2012 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/cpl http://cpl.iphy.ac.cn C HINESE P HYSICAL S OCIETY CHIN. PHYS. LETT. Vol. 29, No. 6 (2012) 062101 The Symmetry Energy from the Neutron-Rich Nucleus Produced in the Intermediate-Energy 40;48Ca and 58;64Ni Projectile Fragmentation * MA Chun-Wang(êS!)**, PU Jie(Ê'), WANG Shan-Shan(王闪闪), WEI Hui-Ling(魏¦ ) Department of Physics, Henan Normal University, Xinxiang 453007 (Received 9 January 2012) In the framework of a modified Fisher model, using the isobaric yield ratio method, we investigate the fragments produced in the 140 A MeV 40;48Ca+9Be and 58;64Ni+9Be projectile fragmentation reactions. Using different approximation methods, asym=T (the ratio of symmetry-energy coefficient to temperature) of symmetric and neutron-rich fragments are extracted. It is found that asym=T of fragments depend on the reference nucleus and the neutron excess of fragments. The asym=T of the isobar decreases when the neutron-excess of the isobar increases, while for a fragment with the same neutron-excess, asym=T increases as the mass of the fragment increases but saturate when the mass of the fragment becomes larger. PACS: 21.65.Cd, 21.65.Ef, 21.65.Mn DOI: 10.1088/0256-307X/29/6/062101 The symmetry energy, which is an important pa- 140 A MeV 40;48Ca and 58;64Ni projectile fragmenta- rameter in the equation of the state of nuclear matter, tion data well.[25] In this Letter, the IYR for fragments is very important in nuclear physics and astrophysics. in the 40;48Ca and 58;64Ni projectile fragmentation[26] The symmetry energy of nuclear matter both of sub- will be revisited, and asym=T of neutron-rich frag- saturation and supra-saturation density, and at high ments will be extracted using the IYR methods. temperature are still unclear due to its complex de- Following the MFM theory,[23;24] the yield of a pendence on both density and temperature. Large fragment with mass number A and neutron excess difference of theoretical results of symmetry energy I(I ≡ N − Z), Y (A; I) is given by for nuclear matter has been demonstrated from differ- −휏 ent models, and even for the same model but using Y (A; I) = CA expf[W (A; I) + 휇nN + 휇pZ]=T different interactions.[1−4] Many experimental works + N ln(N=A) + Z ln(Z=A)g; (1) have concentrated on studying the nuclear equation of state and the liquid-gas phase transition in nu- where C is a constant. The A−휏 term originates from clear matter[5−11] in heavy-ion collisions (HIC). The the entropy of the fragment, 휏’s for all fragments are [23] symmetry energy and temperature of hot emitting identical; 휇n and 휇p are the neutron and proton sources at different densities and temperatures were chemical potentials, respectively; and W (A; I) is the also investigated using the isotopic yields in different free energy of the cluster, which is supposed to equal models.[9;5;12−21] its binding energy at a given T and density 휌. Using In the early 1980s, a study of the isotopic-yield dis- the semiclassical mass formula,[27;28] W (A; I) can be tributions of intermediate mass fragments produced in written as high-energy proton-induced multifragmentation reac- 2 1=3 tions showed that the distributions can be well de- W (A; I) = − asym(휌, T )I =A−ac(휌, T )Z(Z−1)=A scribed by a modified Fisher model (MFM),[22;23] in 2=3 + av(휌, T )A − as(휌, T )A − 훿(N; Z): which the isotope production is governed by the avail- (2) able free energy. Instead of attempting to determine a unique set of the parameters globally, parameters are For simplification, the T and 휌 dependences of ai(휌, T ) related to the isobaric yields in MFM. In the ratios in Eq. (2) are written as ai (i = v; s; c; sym), where between isobars, many terms contributing to the free the indexes v; s; c, and sym represent the coefficients energy cancel out and one can study the specific terms of volume-, surface-, Coulomb-, and symmetry-energy individually and discuss the meaning of the extracted terms, respectively; 훿(N; Z) is the pairing energy. It parameters more clearly. Huang et al.[24] addressed should be kept in mind that these coefficients still the advantage of extracting the coefficient of symme- depend on density and temperature and they actu- try energy to temperature (in the form of asym=T , ally include both the binding energy and the entropy where T is the temperature) of measured fragments contributions.[23] In the MFM model and other models by the isobaric yield ratio (IYR) methods. Correla- based on free energy, it is difficult to separate ai and tions between asym=T and IYRs were found to fit the T . Only ai=T can be extracted according to Eq. (2). *Supported by the National Natural Science Foundation of China under Grant No 10905017, the Program for Innovative Research Team (in Science and Technology) under Grant No 2010IRTSTHN002 in the Universities of Henan Province, and the Young Teacher Project in Henan Normal University. **Corresponding author. Email: [email protected] © 2012 Chinese Physical Society and IOP Publishing Ltd 062101-1 CHIN. PHYS. LETT. Vol. 29, No. 6 (2012) 062101 1=3 It should also be noted that for a neutron-rich nu- If replacing the [∆휇 + 2ac(Z − 1)=A ]=T term in cleus, the symmetry energy should be separated to Eq. (6) by ln[R(1; −1;A)], i.e., taking the IYR of the the surface-symmetry energy asurf and the volume- mirror nuclei as references, we have [29;30] symmetry energy avol. To compare the results asym A with Huang’s, the separation of asym is not included = fln[R(1; −1;A)] − ln[R(I + 2;I;A)] in this work. T 4(I + 1) 1=3 The yield ratio between isobars differing by 2 units − ac(I + 1)=(A T ) + ΔI g: (7) in I is defined as Taking the IYR of the I − 2 fragments as the ref- R(I + 2;I;A) = Y (A; I + 2)=Y (A; I) erences, we have = expf[W (I + 2;A) − W (I;A) a A + (휇 − 휇 )]=T sym = fln[R(I;I − 2;A)] − ln[R(I + 2;I;A)] n p T 8 + Smix(I + 2;A) − Smix(I;A)g; 1=3 − ΔI−2 + ΔI − 2ac=(A T )g: (8) (3) Taking the fragments produced in the 140 A MeV where Smix(I;A) = N ln(N=A) + Z ln(Z=A). Assum- 40;48Ca+9Be and 58;64Ni+9Be reactions[26] as exam- ing that as, ac, 휇n, and 휇p for the I and I + 2 isobars are the same, inserting Eq. (2) to Eq. (3), and taking ples, the asym=T of neutron-rich fragments will be ex- the logarithm of the resultant equation, one gets IYR tracted using Eqs. (6)–(8). for isobars with odd I, In Fig. 1, the IYRs for the fragments with I from −1 to 7 are plotted. The IYR for mirror nuclei of 64 9 ln[R(I + 2;I;A)] − ΔI = [∆휇 − 4asym(I + 1)=A the Ni+ Be reactions is absent due to the lack of 1=3 measured data. The IYRs for these isobars increase + 2ac(Z − 1)=A ]=T; (4) almost “linearly” as A of the fragments increases, but the “slope” decreases when I increases. Equation (5) where ΔI = Smix(I +2;A)−Smix(I;A), ∆휇 = 휇n −휇p, can well fit the IYRs for the mirror nuclei as shown A and Z are for the reference nuclei with I. The pair- by the lines. The fitted ac=T (∆휇/푇 ) for the mirror ing energy of the odd-I fragments is zero in Eq. (2) nuclei are: 0:3728 ± 0:0776 (−0:6427 ± 0:6237) of 40Ca according to Ref. [31]. reaction, 0:5053 ± 0:0791 (−0:8655 ± 0:7089) of 58Ni reaction, and 0:7842 ± 0:0622 (−1:6805 ± 0:5573) of ∆ 6 40Ca+9Be 48Ca+9Be 48Ca reaction, respectively. The n/p values of 40Ca, )- 4 I=-1 58 64 48 2 I=1 Ni, Ni, and Ca are 1.0, 1.07, 1.21, and 1.4, re- I=3 0 spectively. The extracted ac=T increases when n/p of I=5 I⇁֒I֒A -2 projectile increases, while ∆휇/푇 decreases when n/p -4 I=7 lnR( -6 of projectile increases. The extracted ac=T will be 6 ∆ used in the extraction of asym=T for other neutron- 4 58Ni+9Be 64Ni+9Be )- rich isobars using Eqs. (6)–(8). Due to the lack of data 2 64 0 in the Ni reaction, the average values of ac=T and -2 ∆휇/푇 of the 48Ca and 58Ni reactions are used instead. I⇁֒I֒A -4 In Fig. 2, the extracted asym=T ’s of fragments with -6 lnR( different I are plotted. The methods using Eqs. (6), 20 30 40 50 20 30 40 50 60 A A (7), and (8) to extract asym=T of fragments are labeled as (a), (b), and (c), respectively. Fig. 1. (Color online) The isobaric yield ratios {in the Firstly, we discuss the difference between the re- form of ln[R(I + 2;I;A)] − Δ} for fragments produced in the 140 A MeV 40;48Ca + 9Be and 58;64Ni + 9Be reac- sults of methods (a), (b), and (c).
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