1. Isospin dependence of the nuclear phase transition near the critical point Zhiqiang Chen Institute of Modern Physics (Lanzhou) Chinese Academy of Sciences IWND09 Shanghai, China, August 23-25, 2009

2. collaboration R. Wada 1, A. Bonasera 1,3,4, T. Keutgen 5, K. hagel 1, M. Huang 1,2, J. Wang 2, L. Qin 1, J. B. Natowitz 1, T. Materna 1, S. Kowalski 6, P. K. Sahu 1 and T. akagawa 7 1 Cyclotron Institute, Texas A&M University, College Station, Texas 77843, USA 2 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China 3 Laboratori Nazionali del Sud, INFN, via Santa Sofia, 62, 95123 Catania, Italy 4 Libera Universita Kore di Enna, 94100 Enna, Italy 5 Institut de Physique Nuclea´ire and FNRS, Universite Catholique de Louvain, B- 1348 Louvain-la-Neuve, Belgium 6 Institute of Physics, Silesia University, Katowice, Poland 7 Riken, 2-1 Hirosawa, Wako-shi, Saitama, Japan 351-0198

3. Outline Introduction background: Liquid-gas phase transition in nuclear matter Isospin dependence of nuclear phase transition Experimental evidence AMD model calculation Summary Phase transitions and critical phenomena The Landau free energy description

4. Introduction Phase transitions and critical phenomena Phase Diagram Coexistence curves for a typical pure PVT system. Point A is the tripe point and point C is the critical point. The dashed line is an example of fusion curve with negative slope. Textbook: L. E. REICHL > 2 nd Edition

5. Liquid He 4 The coexistence curves for He 4 Liquid He 3 The coexiste nce curves for superfluid phases of He 3 when no magnetic field is applied

6. Liquid He 3 -He 4 Mixtures The phase diagram for a liquid He 3 -He 4 mixture plotted as a function of temperature T and mole fraction x3 of He 3 λ-line extended from T=2.19K for x3=0 to about T=1.56K for x3=0.282. Later experiments extended the λ-line down to T=0.87K for x3=0.67.

7. Landau free energy In the late 1930s, Ginzburg and Landau proposed a mean field theory of continuous phase transitions which relates the order parameter to underlying symmetries of the system. Ψ is Landau free energy, m is order parameter,H is its conjugate variable, a-c are variable parametes. Textbook: Kerson Huang, >

8. Phase diagram of the m6 model For simplicity we take c to be a positive fixed constant; but a and b are variable parameters. This is a line of first-order transitions, there are three minima.

9. Phase transition in nuclear matter Recent time a large body of experimental evidence has been interpreted as demonstrating the occurrence of a phase transition in finite nuclei at temperatures(T) of the order of 10MeV and at densities, ρ,less than half of the normal ground state nuclear density. Even though strong signals for a first and a second-order phase transition have been found. Remain questions: The Equation of State of Nuclear matter (NEOS) near the critical point. The roles of Coulomb, symmetry, pairing and shell effects.

10. Isospin dependence of nuclear phase transition Our experiment The experiment was performed at the K-500 superconducting cyclotron facility at Texas A&M University. Quadrant Si+CsI detector telescope, θ = 17.5 o ± 2.5 o and θ = 22.5 o ± 2.5 o. IMFs (3 ≤Z ≤18) typically 6-8 isotopes I projectileN/ZtargetN/Z 64 Ni1.28 58 Ni1.07 64 Zn1.13 64 Ni1.28 70 Zn1.33 112 Sn1.24 124 Sn1.48 197 Au1.49 232 Th1.58 A. Bonasera, et al., Phys. Rev. Lett. 101, 122702 (2008)

11. Z=6Z=12 Z Real Typical linearized isotope spectra are shown for Z=6 and 12 cases. The number at the top of each peak is the mass number assigned. Linear back ground is assumed from valley to valley for a given Z. Each Gaussian indicates the yield of the isotope above the back ground.

12. Mass distribution for the 64 Ni + 124 Sn system at 40 MeV/nucleon for I=0. The line are power law fits with exponents 2.32±0.02 (odd-odd nuclei, dashed line), 3.86±0.04 (even-even nuclei, full line) respectively. Data analysis Odd-odd (open symbols) and even-even(closed symbols) different which suggests that pairing is playing a role in the dynamics.

13. The observance of this power law suggests that the mass distributions may be disscussed in terms of a modified Fisher model. A. Bonasera et al., Rivista Nuovo Cimento, 23(2000) 1. Where y 0 is a normalization constant,τ=2.3 is one critical exponent, β is the inverse temperature, and Δμ=F(I/A) is the difference in chemical potential between neutron and proton, i.e., the Gibbs free energy per particle, F, near the critical point.

14. Ratio versus symmetry energy for the 64 Ni + 64 Ni case at 40 MeV/nucleon. The dashed lines are fits using a ground state symmetry energy, and a ‘‘temperature’’ of 6 MeV. The I 0 isotopes are indicated bythe open circles and the full circles, respectively. The I =0isotopes are given by the full square Ratio vs symmetry energy

15. (1) (2) (3) (4) New order parameter for IMF isotope distributions Free energy versus symmetry term 64 Ni + 58 Ni 64 Ni + 232 Th

17. Free energy versus symmetry term for the case 64 Ni+ 232 Th. The full line is a free fit based on Landau O(m6) free energy. The dashed-dotted line is obtained imposing in the fit b = −sqrt(16/3ac) and it is located on a line of first order phase transition. The dotted line corresponds to b = −sqrt(4ac), i.e. superheating.

18. Isospin dependence of the nuclear phase transition ‘‘apparent temperature’’ versus I/A of the compound nuleus. H/T versus (I/A) of the compound nucleus obtained from the data fit to the Landau free energy,

19. AMD Calculation AMD (Antisymmetrized Molecular Dynamics) Akira Ono (Tohoku University,Japan) AMD-3.4 version,Gogny force, Gogny-AS force A. Ono, H. Horiuchi, T. Maruyama, and A. Ohnishi, Phys. Rev.Lett. 68, 2898 (1992). A.Ono and H. Horiuchi, Prog. Part. Nucl. Phys. 53, 501 (2004). Modified GEMINI model for secondary decay: Discrete levels of the excited states of light fragments with Z≤15 are taken into account. The Hauser-Feshbach formalism is extended to the particle decay of these fragments when the excitation energy is below 50 MeV. R. J. Charity et al., Nucl. Phys. A483, 371 (1988). R. Wada et al., Phys. Rev. C 69, 044610 (2004).

20. Density dependence of the symmetry energy of nuclear matter for the Gogny force (solid line) and for the Gogny-AS force (dashed line).

21. F/T vs I/A for 40 AMeV 64 Ni + 64 Ni before decay after decay

22. Summary Experimental evidence for isospin dependence of the nuclear phase transition was observed. The properties of the critical point depend on asymmetry of the system. The nuclear phase transition is analogous to the well known superfluid λ transition in He 3 -He 4 mixtures. AMD calculation shows that secondary decay effects and a corresponding loss of information on the primary fragmenting system are very important.