Equation of state for the dense matter of neutron stars and supernovae

Transcription

1 Equation of state for the dense matter of neutron stars and supernovae SFB/Tr7 Summer School Albert Einstein Institut Herbert Müther Equation of state of Nuclear Matter, Golm 004 Folie 1

2 Normal matter versus nuclear matter Distance between atoms: m 10 m Radius of nucleus: 3*10-15 m 0.3 m Compress matter to to nuclear matter: Some densities: Earth g/cm g/cm 3 3 White dwarf ~10 ~ g/cm g/cm 3 3 Nuclear matter g/cm g/cm 3 3 Equation of state of Nuclear Matter, Golm 004 Folie

3 Gravitation to compress matter: r F F F Balance of forces: grav grav press F press 0 G m( r) m ρ( r) f r f P( r) P( r r) r [ P( r) P( r r) ] G m( r) r r dp dρ G m( r) ρ( r) dρ dr r Mass inside the shell of radius r: m(r) m( r r) m( r) 4π r d m( r) dr 4π r ρ( r) r ρ( r) So we need the equation of state: P(ρ) Simple EoS: ideal gas PV N T P ρ T Equation of state of Nuclear Matter, Golm 004 Folie 3

4 Equation of state of Nuclear Matter, Golm 004 Folie 4 Non-relativistic degenerate Fermi gas single particle energies m i i ε Energy and density for infinite system of Fermions: ( ) ( ) d V d x d F i π π π Density: Energy: Pressure ρ π π π π ρ m dv de P m d m V E d V N F F F F Note T0 Note T0 Fermi momentum: F

5 Density g/cm Phase transition to quar gluon plasma? ( filling factor of nuclear matter: ~1/3 1/4) Homog. neutron matter Nuclear Pasta its cheaper to buy a neutron: p E p e n ν ( p F ) E e ( e F ) > E ( if n F E including rest masses e n ) 10 6 White dwarf stabilized by degenerate e-gas 10 1 Normal Matter stabilized by valence e- Equation of state of Nuclear Matter, Golm 004 Folie 5

6 Its not neutrons only: decomposition of neutron star matter Σ n Equation of state of Nuclear Matter, Golm 004 Folie 6

7 Outline of this lecture strong interaction realistic NN interaction meson exchange many-body forces many-body theories hole-line expansion selfconsistent Green function relativsitic aspects superfluidity interaction with neutrinos phase transitions pion condensation aon condensation quar matter Equation of state of Nuclear Matter, Golm 004 Folie 7

8 Few references: R. Machleidt, Adv. Nucl. Phys.19 (1989) 185 H. Heiselberg, M. Hjorth-Jensen, Phys. Rep. 38 (000) 37 H. Müther, A.Polls, Prog. Part. & Nucl. Phys. 45 (000) 43 A. Amal, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C58 (1998) 1804 M. Praash et al. Phys. Rep. 80 (1997) 1 H. Riffert, H. Müther, H. Herold, H. Ruder: Matter at High Densities in Astrophysics Springer Tracts in M.P. 133(1996) D.Blasche, N.K.Glendenning, A.Sedraian:Physics of Neutron Stars Interiors Physics of Neutron Stars Interiors Springer (001) Equation of state of Nuclear Matter, Golm 004 Folie 8

9 Basic Concept of microscopic nuclear structure study: Basic Concept of microscopic nuclear structure study: Realistic NN interaction: model of the interaction adjust parameter to describe NN data Solve Many-Body problem at normal densities: describe saturation of nuclear matter E/A -16 MeV, ρ fm -3, F 1.36 fm -1 and finite nuclei without any adjustable parameter Predict properties of matter at densities above ρ 0 Equation of state of Nuclear Matter, Golm 004 Folie 9

10 Equation of state of Nuclear Matter, Golm 004 Folie 10 Relativistic Meson-Exchange Model π,ρ,σ,... qq quar- versus meson-exch. - - q - Ingredients: 1) Dirac spinor of nucleons ) meson nucleon vertices λ λ λ m E m m E u 1 ), ( vector 4 pseudoscalar 4 scalar meson 4 5 Γ Γ Γ υ µυ µ σ γ π γ π π q i m f g g i g v v v ps ps s s 3) meson propagator mesons for vector for scalar mesons µ µ µ υ µ µυ q q q q g P q q P s s [ ] [ ] ) ( ') ( ) ( ') ( ) ', ( u u P u u V α α α α Γ Γ

11 Solve NN Problem relat: Bethe-Salpeter eq.: nonrel.: Lippmann-Schwinger eq.: Ψ Φ E T Φ V Ψ T V V 1 H 0 1 E H V Ψ iη 0 T iη propagate spinors only only fix fix 0 0 define: Blanb. Sugar explicitly: Equation of state of Nuclear Matter, Golm 004 Folie 11

12 Typical One-Boson-Exchange potential: good fit of NN data with only a few parameter π : dominates long range, tensor force σ : medium range attraction ω : short range repulsion ρ : short range tensor force other mesons for fine tuning however: there is no σ in the particle data boolet!?! σ π Equation of state of Nuclear Matter, Golm 004 Folie 1

13 Local versus nonlocal Interaction Interaction depends on relative coordinate r but not on center of mass R: <r V r> Transfomation to momentum space: < V > dr dr < r ><r V r><r > dr dr e ir e -i r <r V r> If the interaction is local (<r V r> V(r) δ(r-r )) < V > dr e i(- )r V(r) the representation in momentum space depends on q- only; Otherwise it also depends on Nonlocality corrsponds to momentum dependence: OBE is non-local: CD Bonn, Nijm1 Equation of state of Nuclear Matter, Golm 004 Folie 13

14 Local and non-local NN interactions Ψ Deuteron 3 3 Ψ( S1) Ψ( D1 ) Energy of the deuteron: one number originating from different sources -. TS TD VSS VDD VSD TS TD VSS VDD VSD CDBo V Nijm Nijm Reid Equation of state of Nuclear Matter, Golm 004 Folie 14

15 Wavefunctions of deuteron from various interactions r D-state probability: a measure of the tensor force Local interactions stiff V18, Nijm Nonlocal interactions CdBonn, Nijm1 soft Equation of state of Nuclear Matter, Golm 004 Folie 15

16 Hartree-Foc: first step in many-body theories Basis of all EoS Variational approach: find Slaterdeterminant which yields minimal E Φ A i h< f i< F i h j i t j ε δ i U j ih V h< F 1 E h ij i ( t ε ) i U j jh ih V hj h i h U j nucleons move independently in the mean field U determine single-particle wave functions i self-consistent trivial for nuclear matter Equation of state of Nuclear Matter, Golm 004 Folie 16

17 Energy of nuclear matter in HF approx. E HF T CDBo V Nijm Reid Correlations are necessary to obtain binding energy Local interactions are stiffer than non-local V,Φ,Ψ Potential uncorrel wave correl wave r Correlations: <V> more attractive <T> larger Equation of state of Nuclear Matter, Golm 004 Folie 17

18 Various ways to account for correlation effects hole line expansion, Bruecner theory determine effective operator, such that Coupled cluster method, (Exp S) determine exact wave function in form self-consistent evaluation of Green function e.g. G(,, t) T Ψ a ( t) a (0) Ψ Variational approach (Monte Carlo techniques) Ψ H Ψ δ 0 Ψ Ψ Φ H eff P Ψ Φ E Φ Ψ exp ( S ) Φ Equation of state of Nuclear Matter, Golm 004 Folie 18

19 Equation of state of Nuclear Matter, Golm 004 Folie 19 Hole line expansion, Bruecner th: Our aim: solve Ψ HΨ E 1) define appropriate model wave function Φ i i i i H V H U V H U V U T V T H ϕ ϕ ε ϕ :Slaterdet. of ) ( ) ( ) ( Φ ) Determine H eff mit Φ Φ eff H E Lined eff eff eff eff eff V H E Q V V V H E Q V V V V H H 0 0 mod 0 mod 0 Lined Cluster theorem Note: - order expansion according to number of hole lines - V eff is a many-body operator - choice of U 1 influences convergence

20 Equation of state of Nuclear Matter, Golm 004 Folie 0 Bruecner Hartree Foc for nuclear matter C m ij w G ij t w G K h w K Q V d V w G i j F j i i i > < < * 3 ) ( ) ( ), ( ), ( ) ( ε ε ε Bethe Goldstone equation corresponds to LS eq. in medium Pauli quenching - dispersive qu. G less attractive than T T h t V V T Problems: self-consistent solution how to choose particle spectrum (conventional vs continous) convergence (3 hole lines contr.?)

21 Particle spectrum and convergence: Solve 3-body problem in medium: Bethe Fadeev eq. H.Q.Song et al: Phys.Rev.Lett. 81 (1998) 1584 using V14 Hartree-Foc: 30 MeV BHF: -10 or -17 MeV BHF 3: -15 MeV Equation of state of Nuclear Matter, Golm 004 Folie 1

22 Dependence on Interaction: Energy of nuclear matter Soft nonlocal interactions yield more binding and softer EoS EHF T Vπ E CDBo V Nijm Reid Correlations are necessary to obtain binding energy Pionic (tensor) correlations are very important Correlations a fingerprint of the interaction Employ Hellmann-Feynman theorem to determine expectation values: H ( λ) H H ( λ) Ψ λ < Ψ V Ψ > 0 λ V > E λ Ψ λ Eλ λ > λ 1 Equation of state of Nuclear Matter, Golm 004 Folie

23 Saturation point of nuclear matter: the Coester band We dont get energy and density V18 CD Bonn Equation of state of Nuclear Matter, Golm 004 Folie 3

24 Saturation in finite nuclei: 16 O Include effects of long range correlations: generalized ring Equation of state of Nuclear Matter, Golm 004 Folie 4

25 Relativistic effects: Dirac BHF Ingredients of meson exchange: 1) Dirac spinor of nucleons u(, λ) E m 1 λ m E m λ - π,ρ,σ,... - q - ) meson nucleon vertices Γ s 4π g s scalar meson σ attraction Γ v 4π µ [ g γ ] vector v ω repulsion V V scalar 1 1 V vect γ Σ Hartree 1 Σ s µ γ µ µ γ Σµ Should we not consider the relativistic structure of Σ change of Dirac spinors in medium Equation of state of Nuclear Matter, Golm 004 Folie 5

26 Note: ε( m* 0) m Σ m Σ s s Σ 0 effective mass small no binding in DHF correlation effects in G reduce Σ s and Σ 0 binding in DBHF large cancellation Equation of state of Nuclear Matter, Golm 004 Folie 6

27 Effective masses decrease with density enhancement of small Dirac component Equation of state of Nuclear Matter, Golm 004 Folie 7

28 DBHF reproduces saturation point of nuclear matter... and improves for finite nuclei Equation of state of Nuclear Matter, Golm 004 Folie 8

29 How reliable is the EoS???? with respect to the many-body approach with respect to the NN interaction with respect to relativistic dynamic and/or 3N forces Equation of state of Nuclear Matter, Golm 004 Folie 9

30 with respect to the many body approach Both calculations use V18 interaction Equation of state of Nuclear Matter, Golm 004 Folie 30

31 with respect to the NN interaction Energy density for β-stable nucleon matter Equation of state of Nuclear Matter, Golm 004 Folie 31

32 with respect to relativistic effects or 3 N forces Energy density pure neutron matter Equation of state of Nuclear Matter, Golm 004 Folie 3

33 Equation of state of Nuclear Matter, Golm 004 Folie 33 Single-particle Greens function: Def: Ψ Ψ Θ Ψ Ψ Θ ) ( ) ( ) ( ) ( ) ( ) ( ), ( t a t a t t t a t a t t t t ig particle propagation hole propagation Fourier Transformation: ( ) ( ) Ψ Ψ Ψ Ψ F h F p A A i S d i S d i E E a i E E a g η ω ω ω ω η ω ω ω ω η ω η ω ω δ δ δ γ γ γ,, ) ( ) ( ), ( 1 1 Spectral function S h (, ω ): probability for the removal of particle with momentum and energy ω

34 Calculate Greens function Solve Dyson equation g( αβ, ω) g ( αβ, ω) g0( αγ, ω) Σ( γδ, ω) g g(, ω) ω ( t ( δβ, 0 ω γδ infinite matter 1 Σ(, ω)) ± Evaluate self-energy Σ(,ω) iη ) self-consistency required Equation of state of Nuclear Matter, Golm 004 Folie 34

35 The nucleon self-energy Σ(, ω) Σ p1h (, ω) Σh 1p (, ω) hole-line expansion: Ignore Σ Σ h1p i.e.imaginary part at ω<µ Equation of state of Nuclear Matter, Golm 004 Folie 35

36 Self-energy on-shell: Σ(,ωt Σ) PhD thesis of Tobias Fric Equation of state of Nuclear Matter, Golm 004 Folie 36

37 Examples for Spectral functions S(,ω) The effect of self-consistency: QPGF: quasiparticle approx. in evaluating Σ SCGF: self-consistent Spectral function and momentum distribution: µ n( ) S(, ω) dω - Equation of state of Nuclear Matter, Golm 004 Folie 37

38 Spectral function and momentum distribution Correlations lead to partial occupancies for momenta below F : Compare with effects of finite T Correlations lead to nucleons with high momenta How can we observe these momenta? Equation of state of Nuclear Matter, Golm 004 Folie 38

39 Explore momentum distribution in (e,e )p? High momenta but small nissing enegies: to ground-state of A-1 Nuleons with large momenta only at large missing energies ω << ε F large excitation energy in residual nucleus Exp: Blomqvist et al. Mainz Equation of state of Nuclear Matter, Golm 004 Folie 39

40 Spectral function at high momenta and missing energies: Experimental data: D. Rohe and I. Sic Theory: Local density approx. We need: Better description of spectral function at low energies Equation of state of Nuclear Matter, Golm 004 Folie 40

41 with respect to strange matter e n Σ ν we have to calculate selfenergies for Σ and Λ G Σ,Λ we have to solve generalized Bethe Goldstone eq. (M. Hjorth-Jensen) Σ n Equation of state of Nuclear Matter, Golm 004 Folie 41

42 But correlations lead to abundancies of excited baryons already at normal densities: π Mutual polarizations of baryons in the medium Equation of state of Nuclear Matter, Golm 004 Folie 4

43 Propagation, creation and absorption of Neutrinos ν ν q,ω n Neutral current: mean free path of Neutrino e ν q,ω n p Charged current: pe nν n pe ν URCA Cross section proportional to Lindhard function: density of excitations with momentum q and energy ω p Equation of state of Nuclear Matter, Golm 004 Folie 43

44 Example: mean free path of Neutrinos: Importance of RPA: Results of mean field (BHF) (blac, Jerome Margueron) Equation of state of Nuclear Matter, Golm 004 Folie 44

45 Effects of partial occupations: Effects are very important for charged currents: Pauli effect supresses URCA Equation of state of Nuclear Matter, Golm 004 Folie 45

46 Between Nuclei and Nuclear Matter: Pasta n density p x Thomas Fermi Approximation versus BHF calculation in a Wigner Seitz Box Equation of state of Nuclear Matter, Golm 004 Folie 46

47 Typical density profile Energy versus density (in various WS cells) Equation of state of Nuclear Matter, Golm 004 Folie 47

48 Shell Effects: Enhance proton abundance Reduce pairing gap Equation of state of Nuclear Matter, Golm 004 Folie 48

49 Conclusions nuclear part of the EoS rather well established interaction model? many-body forces? relativistic effects? up to around 5 ρ 0 inhomogenous matter nuclear pasta shell effects problems for EoS at larger densities strangeness excited baryons quar gluon plasma more information: neutrino production etc. pairing properties Equation of state of Nuclear Matter, Golm 004 Folie 49