The Nuclear Equation of State

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1 The Nuclear Equation of State Abhishek Mukherjee University of Illinois at Urbana-Champaign Work done with : Vijay Pandharipande, Gordon Baym, Geoff Ravenhall, Jaime Morales and Bob Wiringa National Nuclear Physics Summer School 2007 Abhishek Mukherjee: Nuclear EOS, 1

2 Outline EOS of Hot Dense Matter Variational Theory for Finite T EOS The Hamiltonian Results Conclusions Abhishek Mukherjee: Nuclear EOS, 2

3 EOS of Hot Dense Matter Nuclear Matter Infinite Matter with Neutrons + Protons only Symmetric Nuclear Matter Equal Number of Neutrons and Protons Pure Neutron Matter Neutrons only Asymmetric Nuclear Matter Unequal number of neutrons and protons Abhishek Mukherjee: Nuclear EOS, 3

4 EOS of Hot Dense Matter Why is the EOS interesting/useful? Supernova evolution Cooling of Neutron Stars Heavy Ion Collisions Novel Many Body Systems Abhishek Mukherjee: Nuclear EOS, 4

5 EOS of Hot Dense Matter Abhishek Mukherjee: Nuclear EOS, 5

6 Variational Theory for Finite T EOS Challenges for a microscopic theory: 1. Include correlations beyond mean field 2. Good description of the excited states (for T 0) 3. Calculate the energy, E(ρ, T ) accurately 4. Calculate the entropy, S(ρ, T ) accurately Abhishek Mukherjee: Nuclear EOS, 6

7 Variational Theory for Finite T EOS Correlated Basis States Landau Fermi Liquid Theory Eigenstates of Interacting System one - one Fermi Gas States (FGS) Correlated Basis States (CBS) Ψ I ) = CBS) = G FGS] [FGS G G FGS] Abhishek Mukherjee: Nuclear EOS, 7

8 Variational Theory for Finite T EOS Correlation Operator G = S i<j F(r ij ) Generalization of Jastrow method of pair correlation functions Correlations are included in the functions F(r ij ) (Ψ I H Ψ I ) VCS (FHNC/SOC) (Ψ I H Ψ J ) No such method Abhishek Mukherjee: Nuclear EOS, 8

9 Variational Theory for Finite T EOS Gibbs-Bogoliubov variational principle for T 0 F V is an approximation for F F F V Tr(ρ V H) + T Trρ V lnρ V. Analogy : Rayleigh-Ritz variational principle for T = 0 E 0 (Φ 0 H Φ 0 ) e.g. the Akmal-Pandharipande-Ravenhall EOS Phys. Rev. C 58, 1804(1998) Abhishek Mukherjee: Nuclear EOS, 9

10 Variational Theory for Finite T EOS Variational Hamiltonian What we want to do H v Ψ I ) = E I v Ψ I ) = k ɛ T (k)n I (k) Ψ I ) Variational density matrix ρ v (T ) e Hv /T = e E I v /T Ψ I )(Ψ I S v (T ) = k ( n(k) ln n(k) + (1 n(k)) ln(1 n(k))) Tr (ρ v H) = 1 I e E I v /T e E v I /T (Ψ I H Ψ I ) I Abhishek Mukherjee: Nuclear EOS, 10

11 Variational Theory for Finite T EOS The Orthogonality Problem The Correlated basis states are not Orthonormal to each other Ψ I )(CBS) Θ I Orthonormal Correlated Basis States (OCBS) H v Θ I = E I v Θ I = k ɛ T (k)n I (k) Θ I Tr (ρ v H) = 1 I e E I v /T e E v I /T [(Ψ i H Ψ i ) + E I ] E I ( Orthogonality Corrections) (Ψ I H Ψ J ), non diagonal matrix elements i Abhishek Mukherjee: Nuclear EOS, 11

12 Variational Theory for Finite T EOS Our Solution Microcanonical ensemble instead of a canonical ensemble ρ MC v = 1 Θ I Θ I N M M(T ) the microcanonical ensemble at temperature T N M = # of elements in M Mixed orthonormalization procedure (Lówdin + Gram-Schmidt). E I (Orthogonality Corrections) 0 I Details: A. Mukherjee and V.R. Pandharipande, Phys. Rev. C 75, (2007) Abhishek Mukherjee: Nuclear EOS, 12

13 The Hamiltonian The Hamiltonian H = i 2 2 2m + i<j v ij + i<j<k V ijk + i<j δv ij. v ij = NN potential (Argonne v18) V ijk = NNN potential (Urbana IX) δv ij = Relativistic Boost Corrections Abhishek Mukherjee: Nuclear EOS, 13

14 The Hamiltonian Argonne v18 Av18 = Long Range Part + Intermediate Range Part + Short Range Part Long Range Part One Pion Exchange Potential Intermediate Range Part : Motivated by Meson Exchange Theory Short Range Part : Phenomenological Abhishek Mukherjee: Nuclear EOS, 14

15 The Hamiltonian Urbana IX and Relativistic Boost Corrections Urbana IX UIX = Two Pion Exchange + Phenomelogical Repulsion Fits Binding Energy of Light nuclei + Binding of nuclear matter at saturation Abhishek Mukherjee: Nuclear EOS, 15

16 The Hamiltonian Urbana IX and Relativistic Boost Corrections Urbana IX UIX = Two Pion Exchange + Phenomelogical Repulsion Fits Binding Energy of Light nuclei + Binding of nuclear matter at saturation Relativistic Boost Corrections Leading order corrections due to Special Relativity Abhishek Mukherjee: Nuclear EOS, 15

17 Results The (temperature and density dependent) eigenvalue spectrum of H V is an additional parameter to be varied. E I v = k ɛ T (k) n(k) Effective mass approximation ɛ T (k) = k 2 2m (ρ, T ) + U(ρ, T ) Abhishek Mukherjee: Nuclear EOS, 16

18 Results Minimize the energy of the two body cluster at finite temperature Long range part of the wavefunctions constrained by the model (uncorrelated) wavefunction Euler-Lagrange equations Correlation functions(f) F has variational parameters : 2 correlation lengths (d c, d t ), spin-isospin quenching parameter (α) and the effective mass (m ) Abhishek Mukherjee: Nuclear EOS, 17

19 Results 100 Symmetric Nuclear Matter Free Energy (MeV) Density (fm -3 ) Abhishek Mukherjee: Nuclear EOS, 18

20 Results Pure Neutron Matter Free Energy (MeV) Density (fm -3 ) Abhishek Mukherjee: Nuclear EOS, 19

21 Results Symmetric Nuclear Matter 1 5 HDP m*/m 5 LDP Density (fm -3 ) Abhishek Mukherjee: Nuclear EOS, 20

22 Results 1 Pure Neutron Matter 0.9 m*/m Density (fm -3 ) Abhishek Mukherjee: Nuclear EOS, 21

23 Results Pion Condensation π 0 s condensation = Softening of the spin-isospin sound mode (same quantum number as a pion) Pions are absorbed in the NN interaction Pionic modes Tensor interactions Pion condensation = enhancement of the tensor correlations d t increases dramatically in the HDP Abhishek Mukherjee: Nuclear EOS, 22

24 Results 6 10 PNM HDP Tensor Correlation Length (d t / r 0 ) SNM HDP 10 PNM LDP 10 SNM 20 SNM 5 SNM LDP Density (fm -3 ) Abhishek Mukherjee: Nuclear EOS, 23

25 Conclusions The method of correlated basis functions which has in the past been successfully applied to study the ground state of dense strongly interacting systems has been extended to study the thermodynamics and response at finite temperature. An EOS for nuclear matter at low and moderate temperatures is being developed. In future the influence of thermal pions at very high temperature and pairing at low densities will be examined. Abhishek Mukherjee: Nuclear EOS, 24

26 Appendix Correlated Basis States Correlation operator G = S i<j F ij F ij contains the same operators as the NN interaction. Generalization of the Jastrow correlation functions. Correlated Basis States (CBS) Ψ I ) = G Φ I ΦI G G Φ I Abhishek Mukherjee: Nuclear EOS, 25

27 Appendix Orthogonalization The CBS are not orthogonal to each other Φ I ) (CBS) orthonormalization Θ I (OCBS) OCBS = Orthonormalized Correlated Basis States Θ I H Θ I = (Φ I H Φ I ) + Orthogonality Corrections Abhishek Mukherjee: Nuclear EOS, 26

28 Appendix Variational Hamiltonian H V Θ I {n I (k, σ z )} = n I (k, σ z )ɛ V (k, σ z ) Θ I {n I (k, σ z )}, k,σz For present Calculations: = E V I Θ I {n I (k, σ z )} ɛ V = 2 k 2 2m (ρ, T ) + constant Abhishek Mukherjee: Nuclear EOS, 27

29 Appendix We define ρ V = 1 N M I M Θ I Θ I M = Microcaninical ensemble of the Variational Hamiltonian H V F V = 1 N M I M (Ψ I H Ψ I ) + Orthogonality Corrections TS V (T ) Abhishek Mukherjee: Nuclear EOS, 28

30 Appendix Vanishing Orthogonality corrections We showed that Orthogonality Corrections, in the thermodynamic limit. F V = 1 N M I M (Ψ I H Ψ I ) TS V (T ) First Term : Variational Chain Summation = Fermi Hypernetted Chain summation + Single Operator Chain summation Second Term : Trivial Abhishek Mukherjee: Nuclear EOS, 29