Outline

2. Outline

3. Study the structure of neutron star crusts • MOTIVATION

4. MOTIVATION } What is the composition of the crust?

5. MOTIVATION Use molecular dynamics to study neutron star crust

6. MOTIVATION Known: Neutron star matter composed of Protons, neutrons and electrons Expected: Structure of neutron star crust is “Pasta” Relevant questions: • Does the structure of the pasta evolves as r, T and isospin asymmetry vary? • Are there phase transitions in NSM? • Does the symmetry energy depend on these structures and phase changes?

8. CLASSICAL MOLECULAR DYNAMICS • - Inter-particle potential • - Uses protons & neutrons • Proper dynamics and geometry • - Does not use “test particles” • - Does not use gaussian density distributions • - Produces fragmentswithoutexternalaid • - Deexcitesfragmentsnaturally • Uses a unique set of parameters • But . . . isclassic and not quantum Classical molecular dynamics

9. Classical Molecular Dynamics • CLASSICAL MOLECULAR DYNAMICS • Potential • Solve equation of motion (Verlet) • Recognize clusters (MSE) • Track evolutions in space-time

10. CMD can determine • Mass distributions • Critical phenomena • Caloric curves • Isoscaling • Nuclear “Pasta” • CLASSICAL MOLECULAR DYNAMICS

11. CLASSICAL MOLECULAR DYNAMICS

12. INFinite matter Procedure to study infinite nuclear matter • Create an infinite system • Select density r • Select temperature • Select isospin content • Equilibrate • Measure • Binding energy E(r,T) • Pressure p(r,T) • Compressibility K(r,T) • Obtain equation of state • Obtain phase diagram • Obtain structure at various r, T, isospin content • CLASSICAL MOLECULAR DYNAMICS

13. Pasta ! CLASSICAL MOLECULAR DYNAMICS Gnocchi Spaghetti Lasagna

14. “Pasta” shapes CLASSICAL MOLECULAR DYNAMICS X=0.5, T=0.1 MeV

15. Outline

16. How to characterize the “pasta” • Tools

17. Lindemann coefficient • Tools Kolmogorov statistic

18. Euler characteristic • Tools How to use the Euler number with the “pasta”

19. Example of use of topology Look alike, are they equal? • Tools

20. No, they have different Euler numbers and curvatures • Tools

21. Pasta can be characterized with topology • Tools

22. Tools

24. The structure of the nuclear star matter will be studied using the following tools: • Radial distribution function • Minkowskifunctionals • Volume, Surface, Curvature • Euler characteristic • Lindemann coefficient • Kolmogorov statistic • Structure of NSM

25. Pasta structures 4000 nucleons, x = 0.5, T = 0.2 MeV • Structure of NSM Pasta structures for neutron star matter (with screened Coulomb potential) systems with ρ = 0.05, 0.06, 0.07 and 0.085 fm−3. Protons are represented in orange, and neutrons in blue.

26. Effect of proton fraction • on proton structure • Structure of NSM Surface representation of protons at T = 0.2 MeV, ρ = 0.04 fm-3 for a system of 4000 nucleons.

27. Caloric curve Symmetric case (x=0.5) • 4000 nucleons • x = 0.5 • ρ= 0.05 fm-3 • Structure of NSM Phase transition! Internal energy per nucleon for symmetric NSM as a function of T, for ρ= 0.05 fm-3.

28. Significance of phase transition? • Structure of NSM New phases in the nuclear phase diagram!

29. Significance of phase transition? Liquid-Gas - Pasta, a glass transition? • Structure of NSM Liquid-Gas - Crystal, freezing? Pasta – Crystal: Latent heat needed 1st order phase transition?

30. Significance of phase transition? • Structure of NSM Here we will study the phase transition from crystalline pasta to liquid-like pasta

31. Structure of nuclear star matter (i.e. p + n + embedding electrons) Structure of NSM • Now will study phase changes within the pasta formed in NSM • Radial distribution function • Lindemann coefficient • Kolmogorov statistic

32. Radial distribution function Symmetric case (x=0.5) • Radial distribution function below and above the transition temperature. • Below the transition temperature the peaks are larger  long-range order typical of solids. Structure of NSM • Crystal-like below transition • Liquid-like above transition

33. Radial distribution function Symmetric case (x=0.5) • Pattern is maintained for all densities Structure of NSM • Crystal-like below transition • Liquid-like above transition

34. Lindemann coefficient Symmetric case (x=0.5) Structure of NSM • Lindemanncoefficient measures average displacement as a function of temperature. • The sudden change signals of a solid–liquid phase transition. high displacement liquid-like Low displacement crystal-like

35. Minkowskifunctionals Symmetric case (x=0.5) • Euler number χ and mean curvature as a function of temperature • χ < 0  more tunnels than voids, more connected matter. • χ > 0  more voids than connected matter, more cavity-like. Structure of NSM • There is a sharp transition for both Minkowskifunctionals at the same temperature.

36. Minkowskifunctionals Symmetric case (x=0.5) • The Euler characteristic χ as a function of temperature. • N = 4000 nucleons Structure of NSM • χ < 0  more tunnels than voids and isolated regions. • At lower density the structure is more cavity-like. • At higher density tunnels fill up yielding more compact structures.

37. Lindeman - Minkowski Symmetric case (x=0.5) Structure of NSM • Temperatures at which the phase transitions are observed according to Minkowski functional and Lindemann coefficient. • Both measures signal same transition temperatures.

38. Effect of isospin content Structure of NSM Structure of NSM • Now will study phase changes within the pasta formed in NSM with x = 0.1, 0.2, 0.3, 0.4, 0.5 • Radial distribution function • Euler characteristic • Kolmogorov statistic

39. Radial Distribution function Structure of NSM Structure of NSM • Radial distribution function • N=4000 nucleons • T = 0.2 MeV • At x = 0.4 peaks are more pronounced • At x = 0.2 system is more amorphous

40. Radial Distribution function • PP, PN, and NN correlation functions • T=0.2MeV, ρ=0.01fm−3 • x = 0.2 and x = 0.4 • N = 4000 • N-N are not too correlated Structure of NSM Structure of NSM

41. Euler characteristic Structure of NSM Structure of NSM • Euler characteristic χ as a function of temperature • The total number of nucleons is N=4000. • Change of χ indicates phase transition. • At low density χ < 0 •  more cavity-like • At higher density χ < 0 •  more compact structures

43. - Pasta structures are formed at low • temperatures and subcritical densities • Topology helps to study shapes of pasta

44. - Pastas show liquid-solid and solid-crystal phase transitions within the pastas. - A morphological re-arrangement of the pasta structures occurs as the proton fraction diminishes.

45. - The lowest explored proton fraction x = 0.1 attains a gnocchi-like structure. - Gnocchis themselves experience (inner) topological changes during its formation process.

46. Thanks ! • Current work: • Isospin-dependent phase diagram of nuclear matter • Momentum-dependent potentials • introduction of Pauli exclusion principle

47. Extra slides

48. In progress Extend what we know into the new dimension: Isospin ?

49. Quantum caveats I • Take nucleons as particles in a box • Compare to number of particles N • There are more states available than nucleons • Pauli blocking is not restrictive at • T > 1 MeV

50. Quantum caveats II • In the liquid-gas phase the inter-particle distance is larger than the de Broglie wavelength for all cluster sizes for T > 1 MeV