Eur. Phys. J. A (2012) 48: 78 DOI 10.1140/epja/i2012-12078-5
THE EUROPEAN PHYSICAL JOURNAL A
Regular Article – Theoretical Physics
Symmetry energy extracted from fragments in relativistic energy heavy-ion collisions induced by 124,136Xe Chun-Wang Maa , Jie Pu, Hui-Ling Wei, Shan-Shan Wang, Heng-Li Song, Sha Zhang, and Li Chen Department of Physics, Henan Normal University, Xinxiang 453007 China Received: 31 January 2012 / Revised: 2 May 2012 c Societ` Published online: 4 June 2012 – a Italiana di Fisica / Springer-Verlag 2012 Communicated by B. Ananthanarayan Abstract. In the framework of a modified Fisher model, the ratio of the symmetry-energy coefficient to temperature (asym /T ) is extracted from the fragment produced in the 124,136 Xe induced reactions using the isobaric yield ratio methods based on different approximations. It is found that for nuclei with the same neutron excess (I ≡ N − Z), asym /T increases when the mass of the fragment increases, while for isobar asym /T decreases when I increases. It is also found that the extracted asym /T of the nucleus has very little dependence on the n/p ratio of the projectile, target, and the incident energies in the reaction.
1 Introduction Depending on both density (ρ) and temperature (T ), the symmetry energy of nuclear matter is important for understanding not only many aspects of nuclear physics but also important issues in astrophysics, like the liquid-gas phase transition of asymmetric nuclear matter, the dynamical evolution of massive stars, and the supernova explosion mechanisms [1–9]. Heavy-ion collisions (HIC) can produce nuclear matter of sub-saturation and suprasaturation density with high temperatures, which make it a unique method for studying the symmetry energy of low- and high-density nuclear matters. At the same time, a very neutron-rich nucleus, which has low neutron density distribution in its skirt, can been produced in HIC. But the properties of them, especially their symmetry energy, are not well known. Studies of multiple fragment emission have been performed to use fragment yield distributions to explore the symmetry energy in the emitting source at different ρ and T [10–22]. But no consistent results are obtained for the symmetry energy between the experimental and the theoretical results of different models [1]. The symmetry energy of fragments emitted from a union source is expected to be the same, thus should have no dependence on the mass of fragment. But the measured fragments, which are in ground states, firstly undergo a sequential decay after they are created in the early reaction time. It is not well known to what degree the sequential decay modifies the symmetry energy of fragments which are inherited from the emitting source. In approaches to extract symmetry energy that are based on free energy, all terms that can be determined from exa
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periment appear in the form of ai /T times some function of mass number A, charge number Z, or neutron number N , where i represents the coefficients of the different terms contributing to free energy [23–25]. In a modified Fisher model (MFM), Huang et al. extracted the ratio of the symmetry-energy coefficient to temperature (asym /T ) of the n/p symmetric nuclei in multifragmentation reactions by an isobaric yield ratio (IYR) method [26], but no result for more neutron-rich nuclei is obtained. Since the reactions involved in refs. [25,26] are from near the Fermi energy to very high energy, it is supposed that the MFM is proper in this energy range. In this article, following the ideas in refs. [26,27], we shall analyze the yields of fragments in the 124,136 Xe projectile fragmentation [28] and the 136 Xe spallation [29] at an incident energy of 1 A GeV performed at the fragment Separator (GSI, Darmstadt), in which the measured fragments have mass range from 5 to above 130, and neutron excess I (I ≡ N −Z) range from –1 to 21. These high-quality data provide an opportunity to investigate the symmetry energy of very neutron-rich nuclei by the IYR method.
2 Model descriptions The MFM is a natural choice for reproducing the power law distribution of isotopic yield [30]. The experimental isotopic-yield distribution is well reproduced by the MFM in the works of the Purdue Group [23–25] using nine free parameters, and one of the parameters (the volume coefficient, typically) is arbitrarily fixed to the nominal groundstate value. In the yield ratio of isobars, many terms contributing to free energy which only depend on the mass cancel out, thus making it easy to study the specific terms
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individually and discuss the meaning of the extracted parameters more clearly [26]. The correlations between IYR and the coefficients of different energy terms have been deduced in ref. [26]. Here some of the equations are rewritten to clearly introduce the IYR methods. In the MFM, the yield of a fragment with A and I, Y (A, I) is given by [25] Y (A, I) = CA−τ exp [W (A, I) + μn N + μp Z]/T +N ln(N/A) + Z ln(Z/A) , (1) where C is a constant; A−τ originates from the entropy of the fragment and τ is independent of the fragment size [25]. μn and μp are the chemical potentials of neutron and proton, respectively, and W (A, I) is the free energy of the cluster at T , which is equal to the binding energy of the cluster. At given ρ and T , the binding energy can be parameterized as the Weisz¨acker-Beth formula [32,33] W (A, I) = −asym (ρ, T )I 2 /A − ac (ρ, T )Z(Z − 1)/A1/3 +av (ρ, T )A − as (ρ, T )A2/3 − δ(N, Z),
(2)
where the indices v, s, c, and sym represent the volume, surface, Coulomb, and symmetry energies, respectively. Actually, the coefficients contain contributions from both the binding energy and the entropy of the cluster since they depend on T [25]. For simplification, ai (ρ, T ) is written as ai (i represents the different indices). For neutronrich nuclei, the symmetry energy should include a surfacesymmetry-energy term [34–36]. To compare the results with Huang’s, the surface-symmetry energy is not introduced since it makes the extraction very complicated. Defining the yield ratio between isobars differing 2 in I, R(I + 2, I, A) = Y (A, I + 2)/Y (A, I) = exp [W (I + 2, A) − W (I, A) +(μn − μp )]/T + Smix (I + 2, A) −Smix (I, A) ,
(3)
where Smix (I, A) ≡ N ln(N/A) + Z ln(Z/A). For the oddI isobars, the pairing energies are zero [37]. Assuming that as , ac , μn , and μp for the I and I + 2 isobars are the same, inserting eq. (2) into eq. (3), and taking the logarithm of the resultant equation, one gets IYR for isobar with odd I, ln[R(I + 2, I, A)] − ΔI = Δμ − 4asym (I + 1)/A +2ac (Z − 1)/A1/3 T, (4) where ΔI ≡ Smix (I + 2, A) − Smix (I, A), Δμ = μn − μp , Z is the charge of the reference nucleus with I. For the mirror nuclei isobars, Δ−1 = 0, from eq. (4) one gets ln[R(1, −1, A)] = (Δμ + ac A2/3 )/T.
(5)
One can obtain Δμ/T and ac /T of mirror nuclei using eq. (5). Assuming that Δμ/T and ac /T of the fragments are the same as those of the mirror nuclei, from eq. (4) one gets A asym = Δμ + 2ac (Z − 1)/A1/3 /T T 4(I + 1) − ln[R(I + 2, I, A)] + ΔI .
(6)
If replacing the [Δμ + 2ac (Z − 1)/A1/3 ]/T term in eq. (6) by ln[R(1, −1, A)], i.e., taking the IYR for mirror nuclei as reference, one gets A asym = ln[R(1, −1, A)] − ln[R(I + 2, I, A)] T 4(I + 1) −ac (I + 1)/(A1/3 T ) + ΔI . (7) Taking IYR for the I − 2 fragments as references, i.e., replacing the [Δμ + 2ac (Z − 1)/A1/3 ]/T term in IYR for the I fragments by IYR for the I − 2 fragments, one gets A asym = ln[R(I, I − 2, A)] − ln[R(I + 2, I, A)] T 8 −ΔI−2 + ΔI − 2ac /(A1/3 T ) .
(8)
In eqs. (7) and (8), an extra ac /T term appears due to the difference between Z of isobars. In eqs. (6)–(8), different approximations are adopted in the extracting of asym /T of fragments. In eq. (6), a simple replacement of Δμ/T and ac /T of all fragment by those of the mirror nuclei taken. In eq. (7), the Δμ/T and ac /T terms in IYR for the I fragments is replaced by those of the mirror nuclei, while the Δμ/T and ac /T terms in IYR for the I fragments are replaced by those of the I − 2 fragments in eq. (8). As to the extra ac /T term in eqs. (7) and (8), the ac /T extracted from the mirror nuclei can be used.
3 Result and discussion In fig. 1, the yield of fragment in the 1 A GeV 124,136 Xe + Pb [28] and 136 Xe + H [29] reactions are plotted. The dependence of the yield of fragment on the n/p ratio of projectile, i.e., the shift of the centroid of the isobaric distribution is shown, especially in the large I fragments, which is like the isospin effect in the fragment production in HIC [38–42]. In fig. 2, the IYRs for fragments are plotted. The IYRs for fragments increase monotonic as the mass become larger. For fragments with the same I, the IYRs for the fragments in the 124,136 Xe induced fragmentation have a large difference, especially when I is large. The IYRs for the fragments in the 136 Xe + Pb/H reactions almost overlap. It is easy to conclude that IYRs are greatly influenced by the n/p ratio of the projectiles, but have dependence on the targets used. The trends of IYRs for the mirror nuclei
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Fig. 1. (Color online) The yield of fragment (in mb) measured in the 1 A GeV 124,136 Xe + Pb projectile fragmentation [28] and 136 Xe + H spallation [29]. I = N − Z is the neutron excess of fragment.
Fig. 2. (Color online) The isobaric yield ratios (in the form of ln R(I + 2, I, A) − Δ) for the fragments in the 124,136 Xe reactions. The lines in the inserted figure of the first panel are the fitting results by eq. (5).
in the three reactions can be well fit by eq. (5) except for the odd-even staggering, as shown by the inserted figure in the I = −1 panel of fig. 2. Using eqs. (6)–(8), asym /T of the fragments in the three 124,136 Xe induced reactions are extracted. The results are plotted in fig. 3. To describe the results conveniently, the results of eq. (6) are labeled as (a), eq. (7) as (b), and eq. (8) as (c). Generally, the extracted asym /T of the fragments are found to be nonuniform, but depends on the mass. In (a), which uses the ac /T and (μn − μp )/T of the mirror nuclei directly, the asym /T is generally larger than those in (b) and (c). For the fragments with the same I, the asym /T has little difference among the three reactions when A is small, but the difference increases when A becomes larger. Due to the mass of the mirror, the nuclei measured are less than 30, the ac /T and (μn −μp )/T of the
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Fig. 3. (Color online) The asym /T of the fragments extracted in the 1 A GeV 124,136 Xe projectile fragmentation and 136 Xe spallation using eqs. (6)–(8). The squares, circles, and triangles denote the results of the 1 A GeV 136 Xe + Pb, 124 Xe + Pb, and 136 Xe + H reaction, respectively. (a), (b) and (c) denote the different methods used. See the text for explanation.
mirror nuclei may not be suitable for the more neutronrich and larger mass fragments (for the I ≥ 5 fragments, their masses are all much larger than 30). In (a) and (c), the asym /T increases as the A of the fragment becomes larger when I ≤ 9; when I > 9, in a certain range of A, the asym /T of fragments increases as the A of the fragments increases and then plateauing. The large difference among the results in (a) in the three reactions is due to the different Δμ/T fitted from the ln R(1, −1, A), while the results in (c) overlap and show no systematic dependence on the reaction. To compare the asym /T of isobars with different I, the results are replotted according to (a), (b), and (c) in fig. 4, which are labeled as panel (a), (b), and (c), respectively. In panel (a), the asym /T of the isobar decreases when I increases, but the difference between the neighbor isobars becomes smaller when I increases. In panel (b), similar results of the I = 1 fragments as in ref. [26] (which are plotted as stars) are found, in which the asym /T increases from 10 to 24 when A increases from 10 to 35. The asym /T of the I = 3 fragments (open symbols) also increases as the A of the fragment increases, but is smaller than those of the I = 1 isobars. In panel (c), the asym /T of isobars also decreases as their I increases and plateauing after a certain A, and the values of the plateaus for the different I chains are very close. For a specific I, the asym /T increases from 1 to 37 as the A of the fragment increases. The difference between the asym /T of the I = 13 and I = 15 is very small. Very little dependence of the asym /T on the n/p ratio of the projectile and the target used in the collisions is found in (c), which is also similar to the results in ref. [26].
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Fig. 5. (Color online) The asym /T of the fragments with specific mass as a function of I in the 1 A GeV 136 Xe + Pb reaction. Results of methods (a) and (c) are denoted by the full and open symbols, respectively. Fig. 4. (Color online) The asym /T of fragments in the 1 A GeV 124,136 Xe induced reactions. Panels (a), (b), and (c) are for the results using eqs. (6), (7) and (8), respectively. The different filling of squares, circles, and triangles denote the results for fragments with different I in the 1 A GeV 136 Xe + Pb, 124 Xe + Pb, and 136 Xe + H reactions, respectively. The stars in panel (b) are the results from the 64 Zn + 112 Sn reaction in ref. [26].
For the sake of specification, the asym /T of isobars with specific mass (of A range from 41 to 121) in the 136 Xe + Pb reaction are plotted as a function of I in fig. 5. It is clearly shown that for isobars, the asym /T decreases both in (a) and (c). But the asym /T decreases faster in (c), which can be easily seen in fig. 3. Taking the A = 91 isobars as example, from I = 7 to 13, the per nucleon binding energy for these nuclei are 8.61358, 8.67084, 8.69326, and 8.68487 MeV, respectively. And for the A = 101 isobars, from I = 7 to 15 the per nucleon binding energy 8.5115, 8.56082, 8.58817, 8.60128, and 8.59305 MeV [43], respectively. The difference among the isobars are very small. As I increases, the density distributions in the surface region of isobars will decrease. The decreasing asym /T in isobars can be explained by the decrease of density in the surface region of nuclei. The results in (a) and (c) show large difference. Since the difference between the Coulomb-energy term is very small in isobars, the main difference between eqs. (6) and (8) is the approximation methods of Δμ/T . Δμ depends on ρ and T , which has a great influence on the symmetry energy of nuclear matter [9]. Considering the large difference between the mass of the mirror nuclei and the very neutron-rich fragment in the 124,136 Xe induced reactions, there is no strong reason to directly use the Δμ/T of mirror nuclei in extracting the asym /T of the very neutron-rich fragments by eq. (6), while replacing
the Δμ/T term by the neighbor IYR avoids the shortage of eq. (6) and is more proper. In summary, in the MFM, the asym /T of the very neutron-rich fragment produced in the 1 A GeV 124,136 Xe projectile fragmentation and 1 A GeV 136 Xe spallation are extracted by the IYR methods based on different approximations. For the very neutron-rich fragments, the directly use of the ac /T and the (μn − μp )/T of the mirror nuclei is found to be not a proper approximation. It is found that for fragments with same neutron excess, the asym /T increases when A becomes larger but plateauing when the mass is large enough, while for the isobar, the asym /T decreases when I increases. The decrease of the asym /T of isobars is on account of the decrease of density in the surface region of the neutron-rich nuclei. It is also found that the asym /T shows very little dependence on the n/p ratio of the projectiles and the targets. The measured yields are modified by sequential decay [20,21,26], which has been noticed but not well known. A large difference between the experimental and theoretical results is found but hard to explain [26]. Since the sequential decays for fragments are quite different, the asym /T of fragments is not constant. At the same time, the temperatures in the discussion are at the moment when the fragments are finally formed after the sequential decay with different excitation energy. Thus the temperature has a wide distribution. The similar asym /T for the I = 1 fragments in this work and ref. [26] suggest that though the incident energies differ largely, the asym /T of fragments tend to be coincident after the sequential decay. The asym /T obtained is an apparent one since it is not known to what degree it is distorted by the sequential decay. But the IYR method still provides a way to understand the symmetry energy of the neutron-rich matter at nonzero temperature.
Eur. Phys. J. A (2012) 48: 78 We thank Dr. Mei-Rong Huang at the Institute of Modern Physics and Prof. Yu-Gang Ma at the Institute of Applied Physics (Chinese Academy of Science) for useful discussion. This work is supported by the National Natural Science Foundation of China under grant No. 10905017, and the Program for Innovative Research Team (in Science and Technology) under Grant No. 2010IRTSTHN002 in the Universities of Henan Province, China.
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