Few-particle correlations in nuclear systems

Transcription

1 Trento, Few-particle correlations in nuclear systems Gerd Röpke, Rostock

2 Outline Quantum statistical approach to nuclear systems at subsaturation densities, spectral function Correlations and cluster formation in warm nuclear matter and the nuclear matter equation of state, cluster virial expansion, quasiparticle concept Symmetry energy, quantum condensates, and further signatures of correlations in warm nuclear matter Few-particle correlations in finite nuclear systems (nuclei)

3 Supernova explosion T.Janka

4 Core-collapse supernovae Density. electron fraction, and temperature profile of a 15 solar mass supernova at 150 ms after core bounce as function of the radius. Influence of cluster formation on neutrino emission in the cooling region and on neutrino absorption in the heating region? K.Sumiyoshi et al., Astrophys.J. 629, 922 (2005)

5 Composition of supernova core Mass fraction X of light clusters for a post-bounce supernova core K.Sumiyoshi, G. R., PRC 77, (2008)

6 Nuclear matter phase diagram Core collapse supernovae T. Fischer et al., ApJS 194, 39 (2011)

7 Nuclear matter phase diagram Exploding supernova T. Fischer et al., arxiv

8 Many-particle theory

9 Different approximations

10 Different approximations

11 Cluster decomposition of the self-energy T-matrices: bound states, scattering states Including clusters like new components chemical picture, mass action law, nuclear statistical equilibrium (NSE)

12 Different approximations Ideal Fermi gas: protons, neutrons, (electrons, neutrinos, ) bound state formation Nuclear statistical equilibrium: ideal mixture of all bound states (clusters:) chemical equilibrium continuum contribution Second virial coefficient: account of continuum contribution, scattering phase shifts, Beth-Uhl.E. chemical & physical picture Cluster virial approach: all bound states (clusters) scattering phase shifts of all pairs medium effects Quasiparticle quantum liquid: mean-field approximation Skyrme, Gogny, RMF Chemical equilibrium with quasiparticle clusters: self-energy and Pauli blocking Generalized Beth-Uhlenbeck formula: medium modified binding energies, medium modified scattering phase shifts Correlated medium

13 Quasiparticle picture: RMF and DBHF But: cluster formation Incorrect low-density limit C. Fuchs et al.; J.Margueron et al., Phys.Rev.C 76, (2007)

14 Quasiparticle approximation for nuclear matter But: cluster formation Incorrect low-density limit Klaehn et al., PRC 2006

15 Different approximations Ideal Fermi gas: protons, neutrons, (electrons, neutrinos, ) bound state formation Nuclear statistical equilibrium: ideal mixture of all bound states (clusters:) chemical equilibrium continuum contribution Second virial coefficient: account of continuum contribution, scattering phase shifts, Beth-Uhl.E. chemical & physical picture Cluster virial approach: all bound states (clusters) scattering phase shifts of all pairs medium effects Quasiparticle quantum liquid: mean-field approximation Skyrme, Gogny, RMF Chemical equilibrium with quasiparticle clusters: self-energy and Pauli blocking Generalized Beth-Uhlenbeck formula: medium modified binding energies, medium modified scattering phase shifts Correlated medium

16 Ideal mixture of reacting nuclides mass number A, charge Z A, energy E A,ν,K, ν internal quantum number, K: center of mass momentum Nuclear Statistical Equilibrium (NSE)

17 Effective wave equation for the deuteron in matter In-medium two-particle wave equation in mean-field approximation # 2 p 1 + Δ 1 + p 2 & 2 % + Δ 2 ( Ψ d,p ( p 1, p 2 ) + (1 f p1 f p2 )V ( p 1, p 2 ; p 1 +, p 2 +)Ψ d,p ( p 1 +, p 2 +) $ 2m 1 2m 2 ' p 1 +,p 2 + Add self-energy Pauli-blocking = E d,p Ψ d,p ( p 1, p 2 ) Fermi distribution function f p = [ e ( p 2 / 2m µ)/ k T B +1] 1 Thouless criterion E d (T,µ) = 2µ BEC-BCS crossover: Alm et al.,1993

18 Pauli blocking phase space occupation p z cluster wave function (deuteron, alpha, ) in momentum space P P - center of mass momentum p y p x Fermi sphere The Fermi sphere is forbidden, deformation of the cluster wave function in dependence on the c.o.m. momentum P momentum space The deformation is maximal at P = 0. It leads to the weakening of the interaction (disintegration of the bound state).

19 Shift of the deuteron binding energy Dependence on nucleon density, various temperatures, zero center of mass momentum thin lines: fit formula G.R., NP A 867, 66 (2011)

20 Shift of the deuteron binding energy Dependence on center of mass momentum, various densities, T=10 MeV thin lines: fit formula G.R., NP A 867, 66 (2011)

21 Few-particle Schrödinger equation in a dense medium 4-particle Schrödinger equation with medium effects ([ E HF ( p 1 ) + E HF ( p 2 ) + E HF (p 3 ) + E HF (p 4 )])Ψ n,p ( p 1, p 2, p 3, p 4 ) + (1 f p1 f p2 )V ( p 1, p 2 ; p $ 1, p2 $ )Ψn,P ( p $ 1, p2 $, p3, p 4 ) p 1 $,p 2 $ + { permutations} = E n,p Ψ n,p ( p 1, p 2, p 3, p 4 )

22 Shift of Binding Energies of Light Clusters Symmetric matter G.R., PRC 79, (2009) S. Typel et al., PRC 81, (2010)

23 Parametrization Single-nucleon quasiparticle energies E τ (p, T, n B, Y e ) (DBHF, Skyrme, RMF, ) Bound state energies E A,Z, ν (p, T, n B, Y e ) G.R., NP A 867, 66 (2011)

24 Composition of dense nuclear matter mass number A, charge Z A, energy E A,ν,K, ν: internal quantum number, Inclusion of excited states and continuum correlations Medium effects: self-energy and Pauli blocking shifts of binding energies, Coulomb corrections due to screening (Wigner-Seitz,Debye)

25 Different approximations Ideal Fermi gas: protons, neutrons, (electrons, neutrinos, ) bound state formation Nuclear statistical equilibrium: ideal mixture of all bound states (clusters:) chemical equilibrium continuum contribution Second virial coefficient: account of continuum contribution, scattering phase shifts, Beth-Uhl.E. chemical & physical picture Cluster virial approach: all bound states (clusters) scattering phase shifts of all pairs medium effects Quasiparticle quantum liquid: mean-field approximation Skyrme, Gogny, RMF Chemical equilibrium with quasiparticle clusters: self-energy and Pauli blocking Generalized Beth-Uhlenbeck formula: medium modified binding energies, medium modified scattering phase shifts Correlated medium

26 Two-particle correlations Generalized Beth-Uhlenbeck Approach for Hot Nuclear Matter M. Schmidt, G.R., H. Schulz Ann. Phys. 202, 57 (1990)

27 Cluster virial expansion for nuclear matter within a quasiparticle statistical approach Generalized Beth-Uhlenbeck approach Avoid double counting Generating functional G.R., N. Bastian, D. Blaschke, T. Klaehn, S. Typel, H. Wolter, NPA 897, 70 (2013)

28 α-α scattering phase shifts C.J.Horowitz, A.Schwenk, Nucl. Phys. A 776, 55 (2006)

29 α-n scattering phase shifts C.J.Horowitz, A.Schwenk, Nucl. Phys. A 776, 55 (2006)

30 Correlations in the medium cluster mean-field approximation

31 Intermediate nuclei Quantum statistical calculation of cluster abundances in hot dense matter G.R., J. Phys.: Conf. Series 436, (2013)

32 Pauli blocking in symmetric matter 0.5 free proton fraction X p T =11 MeV T =10 MeV T = 9 MeV T = 8 MeV T = 7 MeV T = 6 MeV T = 5 MeV T = 4 MeV baryon density n B [fm -3 ] Free proton fraction as function of density and temperature in symmetric matter. QS calculations (solid lines) are compared with the NSE results (dotted lines). Mott effect in the region n saturation /5.

33 EOS at low densities from HIC Yields of clusters from HIC: p, n, d, t, h, α chemical constants

34 Comparison: excluded volume approach Light element mass fractions from different approaches M. Hempel, J. Schaffner-B., S. Typel, G.R. Phys. Rev. C 84, (2011)

35 Determination of thermodynamic parameters from light cluster yields in HIC G.R. et al., PRC 2013 T [MeV] [3], NSE [3], QS [5], coalescence [5], QS [5], NSE [8], fluctuations n B [fm -3 ] Analysis of different measured light cluster yields ([3] Kowalski et al., [5] Natowitz et al., [8] Yennello et al.) to infer the freeze-out density and temperature values. In-medium quantum statistical (QS) agrees with coalescence and fluctuation analysis, NSE gives too low densities.

36 Internal energy per nucleon Isotherms T[MeV] thin lines: NSE S. Typel et al., PRC 81, (2010)

37 Internal energy per nucleon Quantum statistical approach: Cluster? Condensate? EOS for symmetric matter - low density region?

38 Clustering phenomena in nuclear matter below the saturation density FIG. 8. Energy curves of DFSs due to a and 16O clustering in" the symmetric nuclear matter by the use of the BB sb4d force. The" density of matter is normalized by the saturation density of the" uniform matter with the Fermi sphere, r0=0.206 fm 3. The presentation" of the curves is similar to that in Fig. 4." Hiroki Takemoto et al., PR C 69, (2004)

39 Light clusters and symmetry energy symmetry energy per baryon U sym [MeV] T =10, NSE T = 5, NSE T = 2, NSE T =10, QS T = 5, QS T = 2, QS T =10, RMF T = 5, RMF T = 2, RMF baryon density n [fm -3 ] K. Hagel et al. arxiv:

40 Symmetry energy: low density limit K. Hagel et al. arxiv: Danielewicz and Lee (2013) Kowalski et al. (2007) Natowitz et al. (2010), T=4 MeV Natowitz et al. (2010), T=8 MeV x= E sym [MeV] E sym [MeV] Danielewicz and Lee (2013) Kowalski et al. (2007) x=0 x=1 x= n [fm -3 ] n [fm -3 ]

41 Influence of cluster formation on β equilibrium T = 5 MeV µ ν = 0 (no neutrinos) ideal mixture RMF shifts light clusters Ideal mixture RMF shifts Including light elements proton fraction Y p baryon density n B [fm -3 ]

42 Finite systems: Self-conjugate 4n nuclei

43 Alpha cluster structure of Be 8 Contours of constant density, plotted in cylindrical coordinates, for 8Be(0+). The left side is in the laboratory frame while the right side is in the intrinsic frame. R.B. Wiringa et al., PRC 63, (01)

44 α cluster in astrophysics Crust of neutron stars Protons in droplets (heavy nuclei) α-cluster outside, at the surface, condensate? S. Typel, Proc. Int. Workshop XII Hadron Physics

45 Four-nucleon energies at finite density Solution of the in-medium wave equation, T = 0 bound state (α particle) with Pauli blocking 4 free nucleons at the Fermi energy (continuum)

46 212 Po: α on top of 208 Pb Woods-Saxon potential (Delion 2013) Thomas-Fermi nucleon density Pauli-blocking of the α particle energy [MeV], baryon density x 1000 [fm -3 ] nucleon single-particle potential baryon density n B (Shlomo), x 1000 TF baryon density n B, x 1000 α intrinsic energy, blocking: TF density critical density n Mott,α = 0.03 fm -3 tunneling energy: MeV α particle dissolved distance r [fm] Cluster: center of mass motion as collective degree of freedom, Separation of the c.o.m. motion from the internal motion. Exact wave equations?

47 Summary Correlations (cluster formation, quantum condensates) are essential in low-density matter, occurring in astrophysical processes. Correlations are suppressed with increasing density (Pauli blocking). The low-density limit of the nuclear matter EoS can be rigorously treated. The Beth-Uhlenbeck virial expansion is a benchmark. An extended quasiparticle approach can be given for single nucleon states and nuclei. In a first approximation, self- energy and Pauli blocking is included. An interpolation between low and high densities is possible. Compared with the standard quasiparticle approach, significant changes arise in the low-density limit due to clustering. Examples are Bose-Einstein condensation (quartetting), and the behavior of the symmetry energy and β equilibrium. Finite Systems (nuclei): Clusters in low-density regions, quantum condensates. Possibly preformed clusters at surface, α decay.

48 Thanks to D. Blaschke, C. Fuchs, Y. Funaki, H. Horiuchi, J. Natowitz, T. Klaehn, Z. Ren, S. Shlomo, P. Schuck, A. Sedrakian, K. Sumiyoshi, A. Tohsaki, S. Typel, H. Wolter, Z. Xu, T. Yamada, B. Zhou for collaboration to you for attention D.G.

49 Influence of cluster formation on β equilibrium Dependence on T µ ν = 0 (no neutrinos) Ideal mixture RMF shifts Including light elements proton fraction Y p T=5, ideal mixture T=5, RMF shifts T=5, light clusters T=10 T=10 T=10 T=20 T=20 T= baryon density n B [fm -3 ]

50 Free energy per nucleon preliminary correlated medium

51 Symmetric nuclear matter: Phase diagram

52 Liquid-vapor phase transition blue: no light cluster, green: with light clusters, QS, red: cluster-rmf S. Typel et al., PRC 81, (2010)

53 Pauli blocking and Mott effect Two different fermions (a,b: proton,neutron) form a bound state (c: deuteron). c q = p F(q, p)a p b q p Is the bound state a boson? Commutator relation + [ c q,c q " ] = F(q, p)f * + + ( q ", p ") a p b q p,b q " p " a p " p, p " + a p b q p b # + +b # q p # q p # [ ] a # p a p b q p a p # a p b # b # = a p a + + p # δ q p, q # p # b # q p # q p # b q p a + + # a p b # a p a # p p + + b q p b # + [ c q,c q " ] = F(q, p)f * ( q ", p ") ( δ p, p " a + a p " )δ p q p, " averaging p, p " q p # q p # b q p a + + # b # a p a # p p q p # + + b q p b # b q p δ p, p # = ( δ p, p # a + p # a p )δ q p, # q p # q p # q # + [ q p b " q " p " b q p δ p, p " ] = F(q, p)f * (q, p)δ q, q # F(q, p)f * ( q #, p #) ( a + a p # )δ p q p, # p p, p # & + [ c q,c q " ] = δ q, q " 1 F(q, p)f * (q, p) a + + ( p a p + b q p '( p + a p b q p a p # a + p # a p b q p p b # + q p # b q p δ p, # + [ q p + ( b # q # p # b q p )δ p, p # ] ( b q p ) Fermionic substructure: phase space occupation, excluded volume ) + * + p

54 Density determination from light cluster yields Y a /Y p 4 * Yh 2 /Yt 2 * (Yp +Y n +2 Y d +3 Y t +3 Y h +4 Y a ) Data [4] T = 4 MeV T = 5 MeV T = 6 MeV T = 7 MeV T = 8 MeV T = 9 MeV T =10 MeV T =11 MeV n B [fm 3 ] Cluster yield ratios to infer the freeze-out density: Experimental data (stars) for T = 5; 6; 7; 8; 9; 10; 11 MeV (increasing density) in comparison with the NSE values (thin dotted lines) and QS calculations (bold straight lines).

55 Scattering phase shifts in matter

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