Comprehensive treatment of correlations at different. energy scales in nuclei using Green s functions

Transcription

1 Comprehensive treatment of correlations at different energy scales in nuclei using Green s functions CISS07 8/30/2007 Lecture 1: 8/28/07 Propagator description of single-particle motion and the lin with experimental data Lecture 2: 8/29/07 From Hartree-Foc to spectroscopic factors < 1: inclusion of long-range correlations Lecture 3: 8/29/07 Role of short-range and tensor correlations associated with realistic interactions Lecture 4: 8/30/07 Dispersive optical model and predictions for nuclei towards the dripline Adv. Lecture 1: 8/30/07 Adv. Lecture 2: 8/31/07 Saturation problem of nuclear matter & pairing in nuclear and neutron matter Quasi-particle density functional theory Wim Dichoff Washington University in St. Louis Green s function V 1

2 The two most elusive numbers in nuclear physics What are these numbers? In what sense are they elusive? What is the history? Three-body forces? Relativity? Give up? What has been learned from (e,e p)? What really decides the saturation density? Nuclear Matter with SRC? No LRC? Conclusions Green s function V 2

3 Empirical Mass Formula Global representation of nuclear masses (Bohr & Mottelson) B = b vol A b surf A 2 / b sym (N Z) 2 A 3 5 Z 2 e 2 R c Volume term b vol = MeV Surface term b surf = MeV Symmetry energy b sym = MeV Coulomb energy R c = 1.24 A 1/3 fm Pairing term must also be considered Green s function V 3

4 Empirical Mass Formula Plotted: most stable nucleus for a given A Green s function V 4

5 Central density of nuclei Multiply charge density at the origin by A/Z Empirical density = 0.16 nucleons / fm 3 Equivalent to F = 1.33 fm -1 Nuclear Matter N = Z No Coulomb A, V but A/V = ρ fixed Two most important numbers b vol = MeV and F = 1.33 fm -1 Green s function V 5

6 Historical Perspective First attempt using scattering in the medium Bruecner 1954 Formal development (lined cluster expansion) Goldstone 1956 Low-density expansion Galitsii 1958 Reorganized perturbation expansion (60s) Bethe & students involving ordering in the number of hole lines BBG-expansion Variational Theory vs. Lowest Order BBG (70s) Clar, Pandharipande Variational results & next hole-line terms (80s) Day, Wiringa Three-body forces? Relativity? (80s) Urbana, CUNY Confirmation of three hole-line results (90s) Baldo et al. New insights from experiment NIKHEF Amsterdam about what nucleons are up to in the nucleus (90s & 00s) JLab Green s function V 6

7 Old pain and suffering! Figure adapted from Marcello Baldo (Catania) Green s function V 7

8 Lowest-order Bruecner theory (two hole lines) l G JST ( K, E) 'l' = l V JST 'l' Spectrum εbhf ( ) = h2 2 Self-energy Energy E A = 4 ρ + 1 2ρ m,m' Σ BHF dq q 2 l V JST f ql" G BG 2π l" 0 ( ) 3 2m + Σ BHF ;ε BHF ( ;E) = 1 ν d 3 θ ( 2π) 3 F m,m' ( ) h2 2 d 3 ( 2π) θ 3 F ( ) 2m G f BG angle-average of f G BG ( 1, 2 ;E) = θ ( 1 F )θ 2 F E ε( 1 ) ε 2 ( ) ( ) + iη ( q;k, E) ql" G JST K, E ( ( )) < F standard choice d 3 ' ( 2π) θ ' 3 F d 3 ' ( 2π) θ ' 3 F ( ) r ( ) r ( ) 'l' all continuous choice r 'mm' G r + r ';E + ε ( ' ) r 'mm' BHF ( ) r r 'mm' G( r + r ';ε BHF ( ) + ε BHF ( ' )) r r 'mm' Green s function V 8

9 M. van Batenburg (thesis, 2001) & L. Lapiás from 208 Pb (e,e p) 207 Tl Occupation of deeply-bound proton levels from EXPERIMENT SRC Not Not a Fermi Fermi gas! gas! LRC Up to 100 MeV missing energy and 270 MeV/c missing momentum Covers the whole mean-field domain for the FIRST time!! Confirmation of theory Green s function V 9

10 Where are the last protons? Answer is coming! Jlab data PRL93, (2004) Rohe et al. 12 C Location of highmomentum components integrated strength OK! There There are are high-momentum components in in the the nuclear ground ground state! state! Green s function V 10

11 Energy Sum Rule (Migdal( Migdal, Galitsii, Koltun...) Finite nuclei E A A 0 = Ψ ˆ A 0 H Ψ 0 = 1 2 αβ α T β n αβ ε F α de E S h (α;e) n αβ = Ψ 0 A a α + a β Ψ 0 A = 1 π ε F de Im G(β,α;E) Nuclear matter S h (α;e) = E A = ρ n Ψ n A 1 a α Ψ 0 A 2 δ E E A A 1 ( ( 0 E n )) = 1 Im G(α,α;E) π d 3 ε F h 2 2 de ( 2π) 3 2m + E S h (;E) Presumes Presumes only only two-body two-body interactions! interactions! Correct Correct description description of of experimental experimental spectral spectral function function should should yield yield good good E/A!! E/A!! Green s function V 11

12 Where does binding come from (really)? 16 O PRC51,3040(1995) Quasiholes contribute 37% to the total energy High-momentum nucleons (continuum) contribute 63% but represent only about 10% of the particles!! Green s function V 12

13 Saturation density and SRC Saturation density related to nuclear charge density at the origin. Data for 208 Pb A/Z *ρ ch (0) = 0.16 fm -3 Charge at the origin determined by protons in s states Occupation of 0s and 1s totally dominated by SRC as can be concluded from recent analysis of 208 Pb(e,e p) data and theoretical calculations of occupation numbers in nuclei and nuclear matter. Depletion of 2s proton also dominated by SRC: 15% of the total depletion of 25% (n 2s = 0.75) Conclusion: Nuclear saturation dominated by SRC and therefore high-momentum components Green s function V 13

14 Elastic electron scattering from 208 Pb Mean field Experiment B. Frois et al. Phys. Rev. Lett. 38, 152 (1977) Green s function V 14

15 Self-consistent treatment of SRC in nuclear matter Interaction Self-energy Dyson equation Green s function V 15

16 Results from Nuclear Matter 2nd generation (2000) Spectral functions for = 0, 1.36, & 2.1 fm -1 Common tails on both sides of ε F Momentum distribution : only minor changes occupation in nuclei depleted similarly!?! Green s function V 16

17 Self-consistent spectral functions S(,E) (MeV -1 ) = 2.1 fm -1 E - ε F (MeV) Green s function V 17

18 Saturation with self-consistent spectral functions in nuclear matter reasonable saturation properties Contribution to the energy per particle before integration over the single-particle momentum at high momentum for two densities Green s function V 18

19 Saturation of Nuclear Matter Ladders and self-consistency for Nuclear Matter Ghent group Dewulf,, Van Nec & Waroquier St. Louis Stoddard, WD Phys. Rev. Lett. 90,, (2003) Green s function V 19

20 Self-consistent spectral functions Distribution below ε F broadens for high momenta and develops a common tail at high missing energy Slight increase in occupation < F to 85% at F = 1.36 fm -1 compared to Phys. Rev. C44, R1265 (1991) & Nucl. Phys. A555, 1 (1993) Self-consistent treatment of Pauli principle Interaction between dressed particles weaer (reduced cross sections for both pn and nn) Pairing instabilities disappear in all channels Saturation with lower density than before and reasonable binding Contribution of long-range correlations excluded Green s function V 20

21 Self-consistent Green s functions and the energy of the ground state of the electron gas GW approximation G self-consistent sp propagator W screened Coulomb interaction RPA with dressed sp propagators Electron gas : -XC energies (Hartrees) Method r s = 1 r s =2 r s =4 r s =5 r s =10 r s =20 Reference QMC CA OB94;OHB99 GW GG Green s function HB98V 21 RPA

22 What about long-range correlations in nuclear matter? Collective excitations in nuclei very different from those in nuclear matter Long-range correlations normally associated with small q Contribution to the energy lie dq q 2 very small (except for e-gas) Contributions of collective excitations to the binding energy of nuclear matter dominated by pion-exchange induced excitations?!? Green s function V 22

23 Inclusion of Δ-isobars as 3N- and 4N-force Nucl. Nucl. Phys. Phys. A389, A389, (1982) F [fm -1 ] third order a) b) c) d) e) f) Sum Green s function V 23

24 Inclusion of Δ-isobars as 3N- and 4N-force 2N,3N, and and 4N 4N from from B.D.Day, PRC24,1203(81) Rings with Δ-isobars : Nucl. Nucl. Phys. Phys. A389, A389, (1982) PPNPhys 12, 12, (1983) No sensible convergence with Δ-isobars Green s function V 24

25 Nuclear Saturation without π-collectivity Variational calculations treat LRC (on average) and SRC simultaneously (Parquet equivalence) so difficult to separate LRC and SRC Remove 3-body ring diagram from Catania hole-line expansion calculation about the correct saturation density Hole-line expansion doesn t t treat Pauli principle very well Present results improve treatment of Pauli principle by self- consistency of spectral functions => more reasonable saturation density and binding energy acceptable Neutron matter: pionic contributions must be included (Δ)( Green s function V 25

26 Pion collectivity: nuclei vs. nuclear matter Pion collectivity leads to pion condensation at higher density in nuclear matter (including Δ-isobars) => Migdal... No such collectivity observed in nuclei LAMPF / Osaa data Momentum conservation in nuclear matter dramatically favors amplification of π-exhange interaction at fixed q V π (q) = f 2 π 2 m π q 2 m π 2 + q 2 In nuclei the same interaction is sampled over all momenta Phys. Lett. B146,, 1(1984) Needs Needs further further study study Exclude collective pionic pionic contributions to to nuclear matter matter binding binding energy energy Green s function V 26

27 Two Nuclear Matter Problems The The usual one one With π-collectivity and only nucleons Variational + CBF and three hole-line results presumed OK (for E/A) but not directly relevant for comparison with nuclei! NOT OK if Δ-isobars are included Relevant for neutron matter The The relevant one?! Without π-collectivity Treat only SRC But at a sophisticated level by using self-consistency Confirmation from Ghent results Phys. Rev. Lett. 90, (2003) 3N-forces difficult π... Relativity? Green s function V 27

28 Comments Relativity Saturation depends on NNσcoupling in medium but underlying correlated twopion exchange behaves differently in medium m * 0 with increasing ρ opposite in liquid 3 He appears unphysical Dirac sea under control? sp strength overestimated too many nucleons for < F Three-body forces Microscopic models yield only attraction in matter and more so with increasing ρ Microscopic bacground of phenomenological repulsion in 3N- force (if it exists)? 4N-, etc. forces yield increasing attraction with ρ Needed in light nuclei and attractive! Mediated by π-exchange Argonne group can t t get nuclear matter right with new 3N-force Green s function V 28

29 Conclusions Good understanding of role of short-range correlations Depletion of Fermi sea: nuclear matter OK for nuclei Confirmed by experiment High-momentum components # of protons experimentally confirmed Long-range correlations crucial for distribution of sp strength Energy per particle from self-consistent Green s functions Better understanding of nuclear matter saturation SRC dominate (don t treat LRC from pions) We now what protons are up to in nuclei!! Green s function V 29

30 Some pairing issues in infinite matter Gap size in nuclear matter & neutron matter Density & temperature range of superfluidity Resolution of 3 S 1-3 D 1 puzzle (size of pn pairing gap) Influence of short-range correlations (SRC) Influence of polarization contributions Relation of infinite matter results & finite nuclei Review: e.g. Dean & Hjorth-Jensen, RMP75, 607 (2003) Green s function V 30

31 Puzzle related to gap size in 3 S 1-3 D 1 channel Mean-field particles Early nineties: BCS gaps ~ 10 MeV Alm et al. Z.Phys.A337,355 (1990) Vonderfecht et al. PLB253,1 (1991) Baldo et al. PLB283, 8 (1992) Dressing nucleons is expected to reduce pairing strength as suggested by in-medium scattering Green s function V 31

32 Results from Nuclear Matter (N=Z) 2nd generation (2000) Momentum distribution: only minor changes when self-consistency is included Occupation in nuclei: Depleted similarly! Thesis Libby Roth Stoddard (2000) Green s function V 32

33 Green s function and Γ-matrix approach (ladders) Single-particle Green s function G(,t 1,t 2 ) = i T c (t 1 )c + (t 2 ) Dyson equation: = + Σ G(,ω) = G(,ω) = G (0) (,ω) + G (0) (,ω)σ(,ω)g(,ω) 1 ω 2 /2m Σ(,ω) S(,ω) = 2ImG(,ω) Self-energy Σ, Γ-matrix = Γ Γ = + Γ Pairing instability possible Finite temperature calculation can avoid this Green s function V 33

34 Self-energy G(,ω) = 1 ω 2 /2m Σ(,ω) S(,ω) = 2ImG(,ω) Real and imaginary part of the retarded self-energy F = 1.35 fm -1 T = 5 MeV = 1.14 fm -1 Note differences due to NN interaction Green s function V 34

35 Spectral functions Strength above and below the Fermi energy as in BCS But broad distribution in energy BCS not just a cartoon of SCGF but both features must be considered in a consistent way CDBonn interaction at T=0 Green s function V 35

36 BCS: a reminder NN correlations on top of Hartree-Foc: + c + + Bogoliubov transformation a = uc + vc 2 with u = 1 2 v 2 1± ε µ, E() = (ε (ε µ) 2 + Δ() 2 µ) 2 + Δ() 2 ε, Gap equation Spectral function S(,ω) Δ( ) = 2 d <, V ', ' > Δ( ') 2E( ) Ε ε µ Ε ω Green s function V 36

37 Green s function V 37 Solution of the gap equation ) ( 2 ) ( ) ( E Δ = ω δ E() = (ε µ) 2 + Δ() 2 ) ( 2 ') ( ' ',, ) ( ' E V Δ > < = Δ ω with and ω=0 Define: = > + < > < > < > + < ') ( ) ( ') ( ) ( ' ' ') (,2, ',, ',, ) ( 2 V E V V V E δ δ ω δ δ M M L M O M K Eigenvalue problem for a pair of nucleons at ω=0 Steps of the calculation: Assume Δ() and determine E() Solve eigenvalue equation and evaluate new Δ() If lowest eigenvalue ω<0 enhance Δ() (resp. δ()) If lowest eigenvalue ω>0 reduce Δ() Repeat until convergence

38 Gaps from BCS for realistic interactions Nuclear matter 3 S 1-3 D 1 momentum dependence Δ() different NN interactions very similar to pairing gaps in finite nuclei for lie particles...!? Neutron matter for np pairing no strong empirical evidence...?! 1 S 0 Early nineties: BCS gaps ~ 10 MeV Alm et al. Z.Phys.A337,355 (1990) Vonderfecht et al. PLB253,1 (1991) Baldo et al. PLB283, 8 (1992) T = 0 Mean-field particles Green s function V 38

39 Beyond BCS in the framewor of SCGF Generalized Green s functions: Extend G(,t 1,t 2 ) = i T c (t 1 )c + (t 2 ) Anomalous propagators Generalized Dyson equation: ω t Σ(,ω) Δ + (,ω) + i T cc i Tcc G(,t 1,t 2 ) = i Tc + c + i Tc + c = G F F + G Gorov equations Δ(,ω) ω + t + Σ(,ω) G (,ω) pair F + (,ω) F(,ω) G pair(,ω) = Leads to e.g. = - Δ G pair = G - G Δ F = Δ G includes all normal self-energy terms Green s function V 39

40 Anomalous self-energy: Δ & generalized Gap equation Δ = = Δ Δ() = 2 d V f (ω) = ( ) f ( ) dω d ω 1 f ω ω S( ω ω,ω) S (, ω ) Δ( ) pair 1 e βω +1 Fermi function If we replace S(,ω) by HF approx. and S pair (,ω) by BCS: Usual Gap equation If we tae S pair (,ω) =S(,ω) : With S pair (,ω) : Corresponds to the homogeneous solution of Γ-matrix eq. The above and self-consistency Green s function V 40

41 Consistency of Gap equation (anomalous self-energy) and Ladder diagrams Iteration of Gorov equations for anomalous propagator generates + and all other ladder diagrams at total momentum and energy zero (w.r.t. 2µ) plus anomalous self-energy terms in normal part of propagator So truly consistent with inclusion of ladder diagrams at other total momenta and energies Green s function V 41

42 Features of generalized gap equation Δ() = 2 d V ( ) f ( ) dω d ω 1 f ω ω S( ω ω,ω) S (, ω ) Δ( ) pair 1 2 χ ' χ Dashed: Spectral strength only at 1 energy Dashed-dot: Effect of temperature (5 MeV) Solid: Includes complete strength distribution due to SRC Related studies by Baldo, Lombardo, Schuc et al. use BHF self-energy Green s function V 42

43 CDBonn yields stronger pairing than ArV18 Green s function V 43

44 Proton-neutron pairing in symmetric nuclear matter Using CDBonn Dashed lines: quasiparticle poles Solid lines: dressed nucleons No pairing at saturation density! Green s function V 44

45 Pairing and spectral functions Spectral functions S(,ω) dashed = A(,ω) S pair (,ω) solid = A S (, ω) ρ = 0.08 fm -3 T = 0.5 MeV = 193 MeV/c 0.9 F Expected effect Green s function V 45

46 Pairing in neutron matter Dressing effects weaer, but non-negligible CDBonn Green s function V 46

47 Comparison for neutron matter with CBF & Monte Carlo PRL95,192501(2005) SCGF