Nuclear matter properties from self-consistent Green s function methods

Transcription

1 Nuclear matter properties from self-consistent Green s function methods V. Somà, P. Bożek 2 CEA, Centre de Saclay, Gif-sur-Yvette, France 2 Institute of Nuclear Physics of the Polish Academy of Sciences, Cracow, Poland University of Surrey 30 November 200

2 Outline! Introduction " what is nuclear matter definitions, limits of validity, EOS " why we study it applications and constraints " how we study it phenomenological vs. ab-initio approaches! Self-consistent Green s functions at zero & finite temperature! The need for three-body forces! Results I : equation of state! Results II : in-medium single-particle properties! Conclusions and current plans

3 Introduction: what is nuclear matter

4 Nuclear matter! strongly interacting nucleons (symmetric/pure neutron matter)! spin-unpolarized! homogeneous system! thermodynamic limit i.e. imagin N,V. oscopic quan ρ = N V " Weizsäcker semi-empirical mass formula E(N p,n n )=E B N + E surf N 2/3 + E Coul N 2 p N /3 + E Pauli (N n N p ) 2 /N energy per particle in symmetric nuclear matter at saturation density ρ 0

5 2 the proton are regarded The neutron and as thez projections same particle with vac 2F = E T St = isospin = 939 MeV, spin s = and isospin, corresponding to two p m = 939 MeV, spin s = and isospin t =, corresponding to two isospin z 2 T) = P (ρ, 2 system 2 2 = 2 and tzpp =. We assume the to be homogeneous in space and ssystem = 2 andtoisospin t = 2, corresponding to two isospin z m 2 MeV, pspin nwe P (ρ, T )==939 = 2 and tzp ==. assume the be homogeneous in space and t and t =. We assume the system to be homogeneous Ω z 2... in p z property high density! hyperons, etc n n-unpolarized. The first allows us to define a constant particle density Limits of validity tpz = = P2(ρ,and tz = 2. We assume the system to be homogeneous in T ) Ω P = 2 high density! hyperons, etc... spin-unpolarized. The first property allows us to define a constant partic n-unpolarized.p The property allows us to define a constant particle density P = P (ρ, T ) nuclei =study first V Ω N/V. We then only two situations: the case withallows an equal pro- partic spin-unpolarized. Thenuclei first property us tonumber define aofconstant low density! clusters V clusters N/V. then study two situations: thenumber case with an equal num P density = Ωρ! N/V. We then study only twowe situations: theonly case with an equal of prolow N/V. We matter, then study two situations: the case with only, an equal num s and neutrons, nuclear or only a system composed ofa neutrons P symmetric = Vρtons Ω and neutrons, symmetric nuclear matter, or system composed of neu s and neutrons, symmetric nuclear matter, or a system composed of neutrons only, V Ω P = and neutrons, symmetric nuclear matter, or a system composed of neu 0.2 ρ0tons utron matter. P = V neutron matter. density utron matter. 0.2 ρ0neutron matter. V density 0.2 ρ0 0.2 ρ0 ρ0 ρ0 ρ0 3 ρ ρ0 ρ0 3 ρ0 LG T = TcLG ρ0 ρ0 ρ0 3 ρ ρ0 ρ0 3 ρ0 T = TcLG The nucleons interact via strong forces, detail in the ne The nucleonstinteract strong forces, addressed in detail inaddressed the nextinsection. = TcLG via liquid-gas phase transition The nucleons interact via strong forces, addressed detail in the ne The nucleonstinteract via strong forces, addressed in detail in theapplications nextinsection. = Tc Weak liquid-gas phase transition interactions enter only indirectly in some (such as n ak interactions enter Weak only indirectly in some applications (such as neutron star interactionsinenter only indirectly in someasapplications (such as ne ak interactions enter only indirectly some applications (such neutron star superfluidity T = T and are nucleonic not presentmatter when pure nucleonic matter is considered. Th c matter) tter) and are not present when pure is considered. The effects of superfluidity T = T and are nucleonic not presentmatter when pure nucleonic matter is considered. Th c matter) tter) and are not present when pure is considered. The effects of T = T Coulomb interactions between protons are assumed to be negligible and th c ulomb interactions between protons are assumed to be negligible and thus electrot = T interactions betweentoprotons are assumed to be electronegligible and th c Coulomb ulomb interactions between protons are assumed be negligible and thus magnetic forces arebeginning. switched off from the beginning. gnetic forces are switched off from the magnetic forces arebeginning. switched off from the beginning. gnetic forces are switched off keep from the We ourselves in a non-relativistic framework. This assumption is We keep ourselves in a non-relativistic framework. This assumption is supported We keep ourselves in a non-relativistic framework. This assumption is Challenges We keep ourselves in by a non-relativistic This assumption the fact that theframework. nucleon mass is large enough; iswesupported shall discuss in th by themass fact is that the enough; nucleon mass is large enough; we shall discuss in th the fact that the nucleon large we shall discuss in the following (Section 3.) the possible inclusion of relativistic corrections and the fact that the nucleon mass is large enough; we shall discuss in the followingtheir conn 3.) the possible inclusion of relativistic corrections and their conn ction" 3.) the matter possible inclusion of relativistic corrections and their connection with nuclear as(section a thermodynamic ensemble (non-relativistic) three-body forces. and their connection with ction" 3.) the matter possible inclusion of relativistic corrections nuclear as(non-relativistic) a thermodynamic ensemble three-body forces. n-relativistic) three-body forces. equation offorces. state n-relativistic)!three-body! equation of state In this work only the case of nuclear matter in thermodynamic equilibri In this work only the case of nuclear matterequilibrium in thermodynamic equilibri this "work only the case of nuclear matter in thermodynamic will be considered. In the framework of statistical mechanics such a system may b nuclear matter asconsidered. a of system of interacting nucleons this "work only the case nuclear matter in thermodynamic equilibrium will be In interacting the framework of statistical mechanics such a system may b nuclear matter as a system of nucleons sidered. In the framework of statistical mechanics such a system may be regarded sidered. In the framework of statistical mechanics We use natural units, i.e. c = =such kb =a.system may be regarded! modified 2single-particle properties We use natural units, i.e. per c =unit = volume kb =.is sometimes denoted by n; however in t The number of particles! modified single-particle properties 2 = kb =. We use natural units, i.e. c = The number of particles per unit volume is sometimes denoted by n; however in t Limits of validity Challenges We use natural units, i.e.when c = speaking = kb =of nuclear. matter, one finds more often the notation ρ. The number of particles when per unit volume is sometimes n; often however in the literature, speaking of nuclear matter, denoted one findsby more the notation ρ. The number of particles per unit volume is sometimes denoted by n; however in the literature, n speaking of nuclear matter, one finds more often the notation ρ. n speaking of nuclear matter, one finds more often the notation ρ.

6 Introduction: why we study nuclear matter

7 Applications and constraints low-density neutron matter universal properties of Fermi systems nuclear matter near saturation density low-energy nuclear reactions benchmark development of EDF (e.g. extensions to neutron-rich systems) high-density/high-temperature nuclear matter astrophysical environment (SN, (proto-)neutron stars,...) high-density/high-temperature nuclear matter heavy-ion collisions

8 Cooling of neutron stars! direct Urca cooling n 3 p e v e, p 3 n e v e! modified Urca cooling n (n, p) 3 p (n, p) e v e. p n, p 3 n (n, p) e v e much less efficient ( < 0-4 ) [ Lattimer and Prakash, Science 304 (2004) ]! neutrino emissivities depend on the in-medium nucleon properties, in particular of the superfluid phase

9 repulsive than K 380 MeV at densities 3 0 could provide a stricter upper bound on the pressure. Our transport theory calculations provide a calibration for the transverse and elliptic flow Global constraints from flow observables [ Danielewicz et al., Science 298 (2002) ]! not excluded a phase transition above 4"0 Fig. 5. Zero-temperature EOS for neutron matter. The upper and lower shaded regions correspond to the pressure regions for neutron matter consistent with the experimental flow data after inclusion of the pressures from asymmetry terms with strong and weak density dependences, respectively. The various curves and lines show predictions for different neutron matter EOSs discussed in the text.! extrapolation to neutron matter model for the symmetry energy that de from th sons. A much p sion by terms i sure in with t The EO ening were a matter and the duce to EOS th tent wi EOS, a above of a d densiti phase quark-g the pre Our matter vae and such de volve a

10 Introduction: how we study nuclear matter

11 Ab-initio approaches model for the nuclear Hamiltonian H ( ) = T + V NN ( ) + V 3N ( ) + V 4N ( ) + fits two-body observables (NN phase shifts, deuteron) traditional/hard-core potentials (large cutoff) self-consistent Green s functions Brueckner-Hartree-Fock variational approaches AFDMC EFT interactions (intermediate cutoff) renormalized/soft potentials (small cutoff) perturbation theory

12 Phenomenological approaches effective interactions (Skyrme, Gogny,...) energy density functionals fit selected nuclei and nuclear matter properties phenomenological effective Lagrangian relativistic mean field fit nuclear matter properties lack predictive power

13 Self-consistent Green s functions at zero and finite temperature

14 Finite temperature Green s functions! single-particle propagator on the time-contour i G AαBβ (r,t; r,t )= T ψ Aα (r,t) ψ Bβ (r,t ) carry statistical mechanics information of the system tr e ˆρ β(h µn) Ô Ô = tr Ô = tr e β(h µn) iβ 0 t 2 t the chemical potential, and = consistency between macroscopic and microscopic observables other ab-initio approaches:! Bloch-De Dominicis ( BBG) " frozen correlations approximation! variational " work in progress

15 Equations of motion! equations of motion for the single-particle propagator i t + 2 2m G(, )=δ(, ) i... N- coupled equations dr 2 V (r r 2 ) G 2 (, r 2,t ;, r 2,t + ) decouple the hierarchy! self-energy C d3 Σ(, 3) G(3, )= i dr 2 V (r r 2 ) G 2 (, r 2,t ;, r 2,t + )! Dyson equation G(, ) = G 0 (, ) Σ(, )

16 In-medium T-matrix! T-matrix approximation for the two-particle propagator G 2 = T = ! energy per particle E N = Htot = Hkin ρ V ρ V + H pot V H pot = n = 2 2 [ [ T T ] ]! grand-canonical potential Ω = tr{ln[g ]} tr{σ G} + Φ Ω[G, Σ, Φ] = P V

17 !-derivability " there exists a class of approximations which automatically fulfills thermodynamic constraints [Baym & Kadanoff 6, 62] Σ(, )= δ Φ[G,V] δ G(, ) " Hartree-Fock, second order, T-matrix belong to this class [ Baym, Phys. Rev. 27 (962) ]

18 Spectral representation! matrix representation " ( i G(, G ) = i c (, ) G < (, ) ) G > (, ) G a (, ) ( T ψ() ψ = ( ) ψ ( ) ψ() ) ψ() ψ ( ) T a ψ() ψ ( )! equilibrium & homogeneous system: spectral GFs ig > (p, ω) = [ f(ω)] A(p, ω) ig < (p, ω) =f(ω) A(p, ω) " free particles dω A(p, ω) =. 2π A 0 (p, ω) =2πδ(ω p 2 /2m) A(p,! p ) (MeV - ) ! (MeV) n(p)! momentum distribution n(p) = dω 2π A(p, ω) f(ω) Z p = p (MeV)

19 Self-consistent procedure start with an ansatz for compute the T-matrix k T R (P, Ω) k = V (k, k )+ compute the self-energy A(p, ω) dp (2π) 3 dq R V (k, p) p Gnc (2π) 3 2 (P, Ω) qq T R (P, Ω) k Im Σ R dk dω p k (p, ω) = Im (2π) 3 T R (p + k, ω + ω ) p k 2π 2 2 p k Im T R (p + k, ω + ω ) k p b(ω + ω )+f(ω ) A(k, ω ) 2 2 dω Re Σ R (p, ω) =Σ HF (p, ω)+p π compute the spectral function A(p, ω) = Im Σ R (p, ω ) ω ω 2 Im Σ R (p, ω) ω p 2 /2m Re Σ R (p, ω) 2 + Im Σ R (p, ω) 2 repeat until convergence is achieved

20 Technical aspects numerical solution of coupled integro-differential equations iterative scheme use Fast Fourier Transform (FFT) and convolution theorem dω F (ω ω) F 2 (ω )= [ F T (t)f T 2 (t)] T discretization on a fixed-spacing grid cut-off dependence under control each point (T, ρ, δ, V) ~ 00 hours

21 The need for three-body forces! empirical values for saturation ρ sat ρ 0 =0.6 ± 0.0 fm 3 E sat /N B = 6 ± MeV C)D)0)EF4@G $! #! '# '$! ()*+',-..)%,/ () )%,/ 678)9:;<;=-.;>?)6@$A)%,/,B/)6@$A)%,/ (H! Coester band '$# '%!!"# $ $"# % %"# &!)D)!! [ Akmal et al., Phys. Rev. C 58 (998) ] [ Baldo and Maieron, J. Phys. G 34 (2007) ] New Coester band

22 Three-body forces

23 Three-body forces Realistic microscopic calculations cannot avoid the use of NNN forces Binding energies, saturation properties and radii Shell evolution Three-nucleon scattering Currently: microscopic NNN interactions only in light systems and INM Normal-ordered (average) part of NNN possibly sufficient Coupled-cluster in 4 He [Hagen et al. 2007] SCGF in INM [Somà, Bożek 2008] Perturbation theory in INM [Hebeler, Schwenk 2009] E / E CCSD body only 0-body 3NF -body 3NF estimated triples corrections 2-body 3NF residual 3NF () (2) (3) (4) (5)

24 Urbana three-body forces V Urbana ijk = V 2π ijk + V R ijk! modification of the internal structure of hadrons " #-excitation 2! exchange " and others:

25 Derivation of the effective potential! need to derive an effective two-body potential V eff 3 (q, q )= στ dk FT n(k) V (2π) 3 3 (k, q, q ) to be inserted in the T-matrix Fourier transform p V V = V + V eff 3 p 2 k k 2 spin-isospin average p 3 k 3 projection into partial waves q V S=0 J (P ) q = 4π 2 V eff 3 (q, q ) = Vs R (q, q )+Vs 2π (q, q ) + Vστ 2π (q, q ) σ σ τ τ + VSτ 2π (q, q ) S(q, q ) τ τ dω qq P J (Ω qq ) V R s V R s + Vs 2π + Vs 2π 3V 2π στ +9V 2π στ for J even for J odd

26 Results I: equation of state

27 Energy per particle in symmetric matter CD-Bonn E/N = 6.3 MeV ρ = 0.7 fm 3 Nijmegen E/N = 6.4 MeV ρ = 0.64 fm 3 E / N [ MeV ] 5 variational + TBF BHF + TBF CD-Bonn 0 CD-Bonn + TBF Nijmegen 5 Nijmegen + TBF / 0 [ Somà and Bożek, Phys. Rev. C 78 (2008) ]

28 Energy in neutron matter & symmetry energy E / N [MeV] CD-Bonn Nijmegen CD-Bonn + TBF Nijmegen + TBF BHF + TBF APR + TBF AFDMC + TBF E sym [ MeV ] variational + TBF BHF + TBF CD-Bonn + TBF Nijmegen + TBF / ! /! 0 [ Gandolfi et al., Phys. Rev. C 79 (2009) ] parabolic approximation E N (ρ, δ) = E N (ρ, δ = 0) + δ2 E sym (ρ)+o(δ 4 ) explicitly by Bomabaci and Lombardo [74] t δ ρ n ρ p ρ tot

29 Experimental constraints on the EoS incompressibility symmetry energy effective mass ω p p 2 = 2m K 0 =9 ρ 2 2 E/N ρ 2 ρ=ρ 0, δ=0 the asymmetry parameter. If w S 0 = 2 E/N 2 δ 2 = E sym (ρ 0 ) ρ=ρ 0, δ=0 ω p = p2 2m + Re Σ(p, ω p) ρ 0 [fm 3 ] E 0 [MeV] K 0 [MeV] S 0 [MeV] m /m Experiment 0.6 ± ± 20 ± ± CD-Bonn CD-Bonn + TBF Nijmegen Nijmegen + TBF

30 Pressure and entropy +,-,&,./0 +,-,'&,./0 direct (diagrammatic) calculation +,-,#!,./0 of P entropy from S/N Φ = 0 S N = T CD-Bonn ( ) CD-Bonn + TBF Nijmegen Nijmegen + TBF 3,4./0,56 () 7 dλ λ H pot(λv,g λ= ) ( E N µ + P ρ ) +,-,'!,./0 +,-,'#,./0 +,-,'$,./0 +,-,'%,./0 P [MeV fm -3 ],2,! T = 8 MeV T = 0 MeV T = 2 MeV T = 4 MeV T = 6 MeV T = 8 MeV T = 20 MeV spinodal region P < 0, ρ < 0 T T * $ ) # '! (' Ω[G, Σ, Φ] = P V 2BF 89(:;<< Nijmegen!!"#!"$!"%!"& ' '"# '"$ '"% '"&,2,! Figure 4.7: Pressure as a function of density for different temperatures with the CD Bonn potential (left panel) and the CD-Bonn potential plus three-body forces (right / 0 / panel). 0 chemical potential P ρ µ ρ 3BF coexistence region µ(ρ g )=µ(ρ l ) < 0. (4.5) T P (ρ g )=P (ρ l ) The T [MeV] border of this region is the spinodal line. Inside this region the system is unstable and tends to separate in two different phases, gas at a lower and liquid at a higher

31 Spinodal region %( %% %! )*+,-.. )*+,-../0-23"4 )*+,-../5/6,7 )*+,-../5/6,7/0-23"4 %( %% %! )*+,-.-/ )*+,-.-/023-4"5 )*+,-.-/ )*+,-.-/ "5 $' $' 6/9:2;< $& $( 70;<-=> $& $( $% $% $! $! ' ' &!!"# $ $"# % &!!"# $ $"# %!/8/!!!0:0!! potential T c (MeV) ρ c (fm 3 ) P c (MeV fm 3 ) P c ρ c T c CD-Bonn CD-Bonn + TBF Nijmegen Nijmegen + TBF [ Somà and Bo!ek, Phys. Rev. C 80 (2009) ]

32 Critical temperature in nuclear matter! Bloch-De Dominicis! Dirac-Brueckner-Hartree-Fock! Green s functions (NN only)! Green s functions (NN+NNN) " 9-20 MeV " 0-3 MeV " 6-9 MeV " ~2 MeV [Das et al. ; Baldo et al.] [Ter Haar et al ; Huber et al.] [Rios et al. ; VS, Bo!ek] [VS, Bo!ek] Limiting temperature in finite nuclei! Coulomb and surface effects 2 Natowitz et al. DB (Malfliet et al.) BHF+Bonn+TBF Chiral pert. (Kaiser et al.) BHF+Av4+TBF 0 the nucleus undergoes a mechanical instability before reaching Tc T lim [MeV] 8 6 #P = Pc + Ps (T) A [ Baldo et al., Phys. Rev. C 69 (2004) ]

33 Results II: single-particle properties

34 Single-particle properties! spectral representation ig < (p, ω) =f(ω) A(p, ω) ig > (p, ω) = [ f(ω)] A(p, ω) recall that the free spectral function is A 0 (p, ω) =2πδ(ω p 2 /2m) " quasiparticle approximation A qp (p, ω) =2πδ(ω p 2 /2m Re Σ(p, ω))! effective mass ω p p 2 = p=pf 2m where ω p = p2 2m + Re Σ(p, ω p)

35 Spectral function T = 0 T = 0 MeV A (p=0, ) [ MeV - ] A (p=0, ) [ MeV - ] CD-Bonn CD-Bonn + TBF [MeV] [MeV] [MeV]

36 Effective mass T = 0 m* / m T = 0 MeV m* / m CD-Bonn CD-Bonn + TBF p [MeV] p [MeV] p [MeV]

37 Spectral function - neutron matter T = 0 T = 0 MeV A (p=0, ) [ MeV - ] A (p=0, ) [ MeV - ] CD-Bonn CD-Bonn + TBF [MeV] [MeV] [MeV]

38 Effective mass - neutron matter T = 0 T = 0 MeV m* / m m* / m CD-Bonn CD-Bonn + TBF p [MeV] p [MeV] p [MeV]

39 Conclusions! first spectral calculations of the nuclear matter EOS with TBF! correct saturation properties! entropy not affected by nucleon correlations! study of the liquid-gas phase transition are included, T c 2 MeV ement with ot! single-particle properties " TBF effects above saturation density! spectral function " opposite effect in symmetric and in pure neutron matter Extensions of the technique:! asymmetric nuclear matter! explicit inclusion of superfluidity! application to nuclei

40 Gorkov approach to pairing Saclay-Surrey collaboration (V.S., C. Barbieri, T. Duguet) Gorkov-Green s function formalism at 2 nd order many-body long-range correlations (quenching of spectroscopic factors,...) pairing beyond HFB ab-initio no-core calculations in large spaces link between g.s. energy and normal/anomalous densities (EDF)

41 Spectral function f 7/ Z k/n / (2j+) S k ± S k ± Dyson st order (HF) 55 Ni p /2 E k ± [MeV] p Ni /2 55 Ni 57 Ni 0.0 p 57 /2 Ni p / p 3/2 0.0 p 0.0 p 3/2 3/ f 0.0 5/2 f 0.0 f 5/2 5/ f 7/ f 7/ S k ± p 3/2 f 5/2 E ± ε - k =E A A- k [MeV] + o - Ek [MeV] ε 57 Ni Dyson 2 nd order ( + FRPA) n =E n A+ - Eo A [MeV] S k ± Gorkov st order (HFB) 63 Ni p / E k ± [MeV] p 3/2 f 5/2 f 7/2 Gorkov 2 nd order in progress 65 Ni Gorkov 2 nd order + FRPA next