1. Structured Mixed Phase of Nuclear Matter Toshiki Maruyama (JAEA) In collaboration with S. Chiba, T. Tatsumi, D.N. Voskresensky, T. Tanigawa, T. Endo, H.-J. Schulze, and N. Yasutake 1 Mixed phase at first-order phase transitions. Its non uniform structures. Its equation of state.

2. 2 Phase transitions in nuclear matter Liquid-gas, neutron drip, meson condensation, hyperon mixture, quark deconfinement, color super-conductivity, etc. Some of them are the first-order  mixed phase EOS of mixed phase Single component congruent (e.g. water) Maxwell construction satisfies the Gibbs cond. T I =T II, P I =P II,  I =  II. Many components noncongruent (e.g. water+ethanol) Gibbs cond. T I =T II, P i I =P i II,  i I =  i II. No Maxwell construction ! Many charged components (nuclear matter) Gibbs cond. T I =T II,  i I =  i II. No Maxwell construction ! No constant pressure !

3. 3 In the mixed phase with charged particles, non- uniform “Pasta” structures are expected. [Ravenhall et al, PRL 50(1983)2066, Hashimoto et al, PTP 71(1984)320] Depending on the density, geometrical structure of mixed phase changes from droplet, rod, slab, tube and to bubble configuration. differentbulk picture Quite different from a bulk picture of mixed phase. Need to take into account the effects of pasta structures when we calculate the EOS. Surface tension & Coulomb

4. 4 (I) Low density nuclear matter Collapsing stage of supernova explosion (non  equil)  Liquid-gas phase transition T=0 (not realistic) electron gas + nuclear liquid T>0 (1~several MeV) nuclear gas + nuclear liquid Neutron star crust (  equil)  Neutron drip T=0 neutron liquid + nuclear liquid

5. Field equations to be solved Relativistic Mean Field (RMF) model ： Lorentz-covariant Lagrangian L  with baryon densities, meson fields , ,  electron density and the Coulomb potential is determined. Local density approx for Fermions ： Thomas Fermi model for baryons and electron Consistent treatment for potentials and densities ：  Coulomb screening by charged particles [T.M. et al,PRC72(2005)015802; Rec.Res.Dev.Phys,7(2006)1] 5

6. 6 Choice of parameters -- Properties of uniform matter and nuclei-- Symmetric matter: Energy minimum at  B =0.16 fm  and E/A=  MeV

7. 7 Chemical equilibrium fully consistent with all the density distributions and fields. EOM for fields

8. 8 Numerical calculation of mixed-phase Assume regularity in structure: divide whole space into equivalent and neutral cells with a geometrical symmetry (3D: sphere, 2D : cylinder, 1D: plate).  Wigner-Seitz cell approx. Give a geometry (Unif/Dropl/Rod/...) and a baryon density  B. Solve the field equations numerically. Optimize the cell size (choose the energy-minimum). Choose an energy-minimum geometry among 7 cases (Unif (I), droplet, rod, slab, tube, bubble, Unif (II)). WS-cell

9. Density profiles in WS cell Pasta structures in matter (case of fixed Y p ) Y p =0.5 T=0 Y p =0.1 T=0 9

10. 10 In the case of symmetric (Y p =0.5) nuclear matter,  p (r) and  n (r) are almost equal.  e (r) is approximately independent. p & n are Congruent.  Maxwell construction may be possible for pn matter with uniform e. But baryon and electron are not congruent.  Maxwell constr. of pne matter is impossible. Y p =0.5 T=0

11. 11 Clustering mechanism of symmetric nuclear matter (Note that the total pressure is positive due to electron pressure.) For symmetric matter at T=0, Maxwell constr leads: negative pressure region (  B  0 ) is unstable.  mixed phase (clustering) But at some density just below  0, uniform matter is favored due to finite size effects (surface and Coulomb). Y p =0.5 Maxwell & bulk Gibbs

12. 12 Y p =0.1 T=0 In the case of asymmetric (Y p <0.5) nuclear matter,  p (r) and  n (r) are different.  e (r) is approximately independent.  Maxwell constr does not satisfy Gibbs cond. Non congruent

13. 13 EOS and the structure size T=0 Pasta structures (droplet, rod,…) appear with the change of density. Pasta  energy gain

14. 14 mechanical instability (dP/d  <0) formation of “pasta” (?) phase coexistence (bulk Gibbs) Instability of uniform matter at finite T Due to the surface tension and the Coulomb interaction, the region of inhomogeneous matter should be limited ( “pasta” < coexistence ). Its relation with Mechanical instability is under investigation. For Y p <0.5, chemical instability should be taken into account.  future work

15. 15 EOS with pasta structures in nuclear matter at T>0 Pasta structure at T 10 MeV coexistence region (Maxwell for Y p =0.5 and bulk Gibbs for Y p <0.5) is meta- stable. Uniform matter is allowed in some coexistence region due to finite-size effects. spinodal region (dP/d  <0) is unstable.  clusterize Spinodal region is included in the pasta region (?). Symmetric matter Y p =0.5 Asymmetric matter Y p =0.3

16. 16 Strong surface tension and weak Coulomb  large R Extreme case  no minimum. (pasta unstable) [Voskresensky et al, PLB541(2002)93; NPA723(2003)291]  Uniform matter appear at some part of coexistence region. size of structure R Structure size (Charge screening) Dependence of E/A on R.

17. 17 Coulomb screening effects Compare different treatments of Coulomb int. Smaller structure size for “no Coulomb screening” calc.  Coulomb screening enlarges the structure size. Narrower pasta region for “no Coulomb screening” calc.  Coulomb screening makes pasta unstable. (uniform electron)

18. 18 Surface tension effects Compare different nuclear surface tension Weak surface Normal surface tension Smaller structure size for weak surface tension.  Stronger surface tension enlarges structure size. Wider pasta region for weak surface tension.  Weaker surface tension makes EOS close to bulk Gibbs.

19. 19 Beta equil. Beta-equilibrium case Only “droplet” structure appears. The change of EOS due to the non-uniform structure is small. The proton fraction Y p is drastically influenced by the non-uniform structure. T=0

20. 20 Y p -P B phase diagram of matter Symmetric matter has constant value of Y p =0.5. It corresponds to the Maxwell construction. In general cases, liquid and gas phases have different values of Y p. In the case of small Y p, the retrograde transition (gas-mix-gas) may occur. But by surface tension and the Coulomb interaction, it might be suppressed.

21. 21 Mixed phase at higher densities Kaon condensation [PRC 73(2006)035802, RecentResDevPhys7(2006)1] Hadron-quark transition [PRD 76(2007)123015, PLB 659(2008)192]

22. 22 From a Lagrangian with chiral symmetry K single particle energy (model-independent form) Threshold condition of condensation ( II ) Kaon condensation

23. Kaon condensation in uniform matter With increase of density, fraction of proton and electron increases so as to decrease neutron energy. At around 3  0 there appears K  and neutron fraction further decreases. At higher density,  p  n since K  attracts p more than n. Due to the strong attraction between K  and p, pressure decreases. (density gradient becomes negative)  First-order phase transition  Phase separation  Non uniform matter 23 Particle fraction

24. 24 EOM for fields (RMF model) Kaon field K(r) added.

25. Kaonic pasta structure 25

26. 26 At 2-3    hyperons are expected to appear.  Softening of EOS  Maximum mass of neutron star becomes less than 1.4 solar mass.  Contradicts the obs >1.5 M sol Possibility to resolve this problem by introducing quark phase in neutron stars. ( III ) Hadron-quark phase transition 1.4 ( only pn) } hyperons [Schulze et al, PRC73 (2006) 058801]

27. 27 Coupled equations to get density profile, energy, pressure, etc of the system

28. 28 Hadron-quark droplet Rearrangement of charge Quark phase is negatively charged.  u quarks are attracted and ds quarks repelled. Same happens to p in the hadron phase. Localization of e into hadron phase.

29. 29 EOS of matter Full calculation is close to the Maxwell construction (local charge neutral). Far from the bulk Gibbs calculation (neglects the surface and Coulomb).

30. 30 Density at r Mass inside r Total mass and Radius Pressure ( input of TOV eq.) Solve TOV eq. Structure of compact stars Bulk Gbbs Full calc  surf =40 MeV/fm 2 Maxwell const. Density profile of a compact star (M=1.4 solar mass)

31. 31 Mass-Radius relation of compact stars Full calc yields the neutron star mass very close to that of the Maxwell constr. The maximum mass are not very different for three cases.  surf =40

32. 32 Maxwell construction : Assumes local charge neutrality (violates the Gibbs cond.) and neglects surface tension. Bulk Gibbs calculation : Respects the balances of  i between  phases but neglects surface tension and the Coulomb interaction. Full calculation: includes everything. Strong surface  Large R, charge-screening  effectively local charge neutral.  close to the Maxwell construction. Weak surface  Small R  Coulomb ineffective  close to the bulk Gibbs calc. Equation of state of mixed phase ?

33. 33 Summary We have studied ``Pasta’’ structures of nuclear matter at the first-order phase transitions. Pasta structures appear in liquid-gas transition at low- density, meson condensation at high-density, and hadron- quark transition. Coulomb screening and stronger surface tension enlarges the structure size. If surface tension is strong, Maxwell construction is effectively valid.