Symmetry Energy within the Brueckner-Hartree-Fock approximation

Transcription

1 Symmetry Energy within the Brueckner-Hartree-Fock approximation Isaac Vidaña CFC, University of Coimbra International Symposium on Nuclear Symmetry Energy Smith College, Northampton ( Massachusetts) June 17 th -20 th 2011

2 In collaboration with: U. Coimbra: C. Providência, C. Ducoin U. Barcelona: A. Polls U. Surrey: A. Rios IPN, Orsay: J. Margueron

3 Motivation Isospin asymmetric nuclear matter is present in: Nuclei, especially those far away from the stability line in astrophysical systems (neutron stars) A well-grounded understanding of the properties of isospin-rich nuclear matter is necessary for both nuclear physics astrophysics However, some of these properties are not well constrained. In particular the density dependence of the symmetry energy is still an important source of uncertainties.

4 Some properties of asymmetric nuclear matter can be obtained from: the analysis of experimental data in heavy ion collisions (e.g., ID, double n/p ratios, GDR, ) the analysis of existing correlations between different quantities in bulk matter finite nuclei (e.g. δr versus L) PREX JLAB A major effort is being carried out to study experimentally the properties of asymmetric nuclear systems. Experiments at CSR, GSI (FAIR), RIKEN, GANIL, FRIB can probe the behavior of the symmetry energy close and above saturation density. Astrophysical observations of compact objects window into nuclear matter at extreme isospin asymmetries

5 In this talk Study of density dependence of the symmetry energy within the BHF approximation and comparison with effective models (Skyrme RMF). Analysis of correlations between L and K sym. Special attention to correlations of L with neutron skin thickness and crust-core transition point in neutron stars. based on: Phys. Rev. C 80, (2009) Phys. Rev. C 83, (2011)

6 Equation of State of Asymmetric Nuclear Matter Charge symmetry expansion of (E/A) ANM on even powers of isospin asymmetry β=(ρ n -ρ p )/(ρ n +ρ p ) E A (",#) = E SNM (") + S 2 (")# 2 + S 4 (")# 4 + O(6) E SNM (") = E (",# = 0), A S 2 (") = 1 2 # 2 E / A, S #$ 2 4 (") = 1 # 4 E / A 24 #$ 4 $ = 0 $ = 0 In good approximation: S 2 (") ~ E A (",# =1) $ E (",# = 0) A

7 E SNM (ρ) commonly expanded around saturation density ρ 0 E SNM (") = E 0 + K 0 2 $ % " # " 0 3" 0 ' ) ( 2 + Q 0 6 $ % " # " 0 3" 0 ' ) ( 3 + O(4) E 0 = E SNM (" = " 0 ) # $16 MeV K 0 = 9" 0 2# 2 E SNM (") #" 2 "= " 0 $ 240 ± 20 MeV Q 0 = 27" 0 3# 3 E SNM (") #" 3 "= " 0 $ % MeV Similarly S 2 (ρ) can be also characterized with few bulk parameters around ρ 0 $ S 2 (") = E sym + L " # " ' 0 ) + K $ sym " # " 0 % 3" 0 ( 2 % 3" 0 ' ) ( 2 + Q sym 6 $ " # " 0 % 3" 0 ' ) ( 3 + O(4) #S L = 3" 2 (") 0 K sym = 9" 2# 2 S 2 (") 0 Q #" "= " #" 2 sym = 27" 3# 3 S 2 (") 0 0 "= " 0 #" 3 "= " 0 Less certain predictions of different models vary largely

8 Combining the expansions of E SNM (ρ) and S 2 (ρ) one arrives at E A (",#) = E 0(#) + K 0(#) % " $ " 0 (#)( ' * 2 3" 0 (#) ) 2 + Q 0(#) 6 % ' " $ " 0 (#) 3" 0 (#) ( * ) 3 + O(4) where " 0 (#) = " 0 $ 3" 0 L K 0 # 2 + O(4) E 0 (") = E 0 + E sym " 2 + O(4) $ K 0 (") = K 0 + K sym # 6L # Q ' $ 0 L) " 2 + O(4) Q % K 0 (") = Q 0 + Q sym # 9L Q ' 0 )" 2 + O(4) 0 ( % ( K τ K 0

9 BHF approximation of ANM Bethe-Goldstone Equation Partial sumation of pp ladder diagrams G (") = V + V E " (k) = h2 k 2 U " (k) = % " ' 2m " % k'$k F " ' Energy per particle Q " # E # E ' + i$ G " ( ) + Re[ U " (k)] r k k r ' G # = E " (k) + E "' (k') ( ) r k k r ' A Pauli blocking Nucleon dressing E A (",#) = 1 A, $, k+k F $ % ' h 2 k 2 2m $ [ ] Re U $ ( r k ) ( * ) Infinite sumation of two-hole line diagrams

10 Few words on the NN and NNN forces used Argonne V18 (Av18) NN potential V ij = " p V p (r ij )O O ij ij p=1,18 p=1,14 $ r = 1, ( " # r " i j ),S ij, L r # S r r,l 2,L 2 " i # r r ( " j ),( L # S r ) 2 ' % () * 1, r + i # r r O p=15,18 ij = T ij,(" i # r " j )T ij,s ij T ij,( $ zi + $ zj ) [ ] [ ( + j )] Urbana IX (UIX) NNN potential V UIX ijk = V 2" R ijk + V ijk V ijk 2" : Attractive Fujita-Miyazawa force V ijk 2" = A + cyclic X ij = Y m " r ij Y(x) = e"x x % '{ X ij,x jk } r # i $ r # j, r # j $ r ( ) r { # k } X ij, X jk # i $ r # j + T( m " r ij )S ij " 2 $ # [ ][ r # i $ r # j, r # j $ r # k ] ( 1" ) T(x) = 1+ 3 ex x + 3 ' e(x x 1( 2 2 ex % x ( ) 2 ( * ) π π eff V NN Reduced to an effective density-dependent 2BF r ( ij ) = V UIX ( r i, r, r j k )n r i, r, r " ( r j k )d 3 r k V ijk R : Repulsine Phenomenological V R ijk = B " T 2 ( r ij )T 2 r jk cyclic ( ) A, B fit to reproduce the saturation point

11 BHF nucleon mean field in ANM Isospin splitting of mean field in ANM U n ~ U 0 + U sym " U p ~ U 0 "U sym # Symmetry potential U sym = U n "U p 2#

12 G-matrix gives access to in-medium NN cross sections " ## ' = m * * # m #' 2J +1 16$ 2 h 4 LL'SJ 4$ G LL'SJ ## ' %## ' 2, ##'= nn, pp,np

13 Skyrme Phenomenological approaches Lyon group SLy SkI family Relativistic mean field models Non-linear Walecka models (NLWM) with constant coupling constants: NL3, TM1, GM1, GM3 FSU Density dependent hadronic models (DDH) with density dependent coupling constants: TW, DD-ME1, DD-ME2, DDHδ Quark meson coupling model QMC Nuclear matter: system of non-overlaping MIT bags interacting through exchange of scalar and vector mean fields

14 Bulk parameters of E SNM (ρ) S 2 (ρ) Model r 0 E 0 K 0 Q 0 E sym L K sym Q sym K t γ BHF (3BFa) BHF (3BFb) BHF (2BF) SLy SLy230a SkI NL TM FSU TW QMC HIC at intermediate energies consistent with # " S 2 (") = E sym % ( $ ' " 0 ) * ) = L 3E sym

15 Symmetry Energy versus L Recent extracted values of L BHF (Adapted from M. B. Tsang et al, Phys. Rev. Lett. 102, (2009)) (Adapted from D. V. Shetty S. J. Yennello, Pramana 75, 259 (2010))

16 Density dependence of S 2 and L Spin-Isospin contributions to E sym and L (S,I) E sym L (0,0) (0,1) (1,0) (1,1) F.G L(") # 3" 0 $S 2 (") $" Total Larger contribution to E sym and L from S=1 channel (tensor)

17 Effect of three-body forces 3BF increase the slope of S 2 (ρ) astrophysical consequences: larger Y p earlier onset of direct URCA stiffer EoS larger M max of neutron star

18 Correlation of K sym K τ with L S 2 (MeV) Skyrme BHF S 2 (MeV) RMF BHF S 2 : crossing at ρ ~ 0.11 fm -3, S 2 (0.11)~24±4 MeV (expected from finite nuclei constraints at ρ<ρ 0 ) L : tendency to cross at ρ ~ ρ 0 /3 K sym : no crossing observed

19 Neutron Skin Thickness Symmetry Energy Neutron skin thickness Density ρ (fm -3 ) "R = r n 2 # r p 2 r (fm) Typel Brown showed that δr calculated in mean field models is very sensitive to the slope of the symmetry energy. Typel-Brown correlation

20 Fully self-consistent finite nuclei calculation based on BHF approach too difficult δr estimated to lowest order in the diffuseness corrections (Steiner et al., Phys. Rep. 411, 325 (2005)) "R # 3 5 t t = " c # 0 (" c )(1$ " c 2 ) E s 4%r o 2 # 0 (" c ) o ( ) E SNM (#) $ E 0 d# # E sym /S 2 (#) $1 # 0 (" c ) o d# # E SNM (#) $ E 0 thickness of semi-infinite asymmetric matter ( ) $1/ 2 ( ) 1/ 2

21 Correlation of the Neutron Skin Thickness δr with L K sym Linear increase of δr with L K sym Not surprising: Pressure δr In n-rich matter pressure increases with L at fixed ρ P (",#) = " 2 % ' L# 2 + K 0 +K sym # 2 3" 0 ( ) " $ " 0 ( +L* 3" 0 )

22 Neutron Stars Symmetry Energy: Crust-Core transition density The crust of a neutron star is very important for a number of observable properties : thermal evolution glitches X-ray burst. It is very important to understand well the crust-core transition region Which constraints are set by the isospin dependence of the nuclear EoS on the transition region? (Picture from Nicolas Chamel ) How sensitive is to the symmetry energy?

23 Crust-core transition estimated from crossing of β-equilibrium EoS and spinodal instability line Curvature Matrix: $ "µ n /"# n "µ n /"# p 0 ' ) C = "µ p /"# n "µ p /"# p 0 ) % 0 0 "µ e /"# e ( " D nn D np 0% +k 2 $ ' D pn D pp 0 $ ' + 4(e k # positive definite: Tr(C ) > 0, Det(C ) > 0 2 " % $ ' 0 1 )1 $ ' # 0 )1 1 ρ td ρ tt : we consider first this point Thermodynamical spinodal upper bound of the the real ρ t ( ~ 15% larger than TF calculation of nuclear pasta ) (Figures courtesy of C. Ducoin C.Providência)

24 Correlation of Y pt and ρ t with L spinodal Clear decreasing correlations Proton fraction Y pt Density ρ t [fm -3 ] β stability ( Higher L { Lower Y pt Lower ρ t Dispersion of Y pt due to dispersion of E sym µ n " µ p = 4#S 2 ($) = µ e " µ % e Y 1/ 3 * p (1" 2Y p ) = 4S 2 ($) * ) hc 3' 2 $ ( ) 1/ , Proton fraction Y pt Density ρ t [fm -3 ] Proton fraction Y pt L [MeV] L [MeV] Density ρ t [fm -3 ]

25 L- P t correlation: an open issue Xu et al., PRC, 2009 Decrease of P t Increase of P with L in n-rich matter at fixed density P (",#) = " 2 BUT % ' L# 2 + K 0 +K sym # 2 3" 0 ( ) " $ " 0 ( +L* 3" 0 ) Moustakidis et al., PRC, 2010 Increase of P t ρ t (fm -3 ) Y pt L (MeV)

26 Role of other bulk parameters Dynamical vs Themodynamical spinodal Fits on pairs of parameters P t = ax 1 + bx 2 + c 0.5 P dt (MeV fm -3 ) P dt (MeV fm -3 ) L (MeV) [E sym L] (MeV) Transition point well correlated P dt (MeV fm -3 ) Skyrme Relativistic [L K sym01 ] (MeV) Unclear L-P t correlation Inclusion of E sym more reliable corelation Good correlation with L and K sym at ρ=0.1 fm -3

27 Neutron Skin Thickness The Crust-Core Transition Density Neutron Star Crust Neutron Skin are made out of neutron rich matter at similar densities Neutron Star Heavy nucleus Both are governed by EoS at subnuclear densities in particular by S 2 (ρ) its derivatives Inverse correlation between δr and ρ t (Horowiz Piekarewicz) an accurate measurement of neutron skin in neutron rich nuclei can provide considerable valuable information on the crust-core transition density. (PREX JLAB)

28 Summary Conclusions Study of S 2 (ρ) within the BHF approximation comparison with effective models (Skyrme RMF). L=66.5 MeV compatible with values deduced from different observables. Larger contribution to E sym L from S=1 channels (tensor). 3BF increase the slope of S 2 (ρ). Correlation of ρ t Y pt with L: Robust correlation of ρ t with L. Correlation L-Y pt more disperse due to dispersion on E sym. Correlation of P t with L: Opposite contributions difficult prediction. Improvement with combination of L K sym at ρ ~ 2/3ρ 0.

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