Transcript Three-body Force Effects on the Properties of Neutron-rich Nuclear Matter Zuo Wei Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China I.

Three-body Force Effects on the Properties
of Neutron-rich Nuclear Matter
Zuo Wei
Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou, China
I. Bombaci, G.F.Burgio, U. Lombardo, H.-J. Schulze,
A. Lejeune, J. F. Mathiot, B. A. Li , A. Li, Z. H. Li,
L.G.Cao, C.W.Shen, J.M.Dong, W. Scheid
NN2012, Hyatt Regency, San Antonio, Texas, USA
May 27 – June 2, 2012
Outline:
1. Introduction (Motivation)
2. Theoretical approaches
3. Results
4. Summary and conclusion
Properties of Asymmetric
Nuclear Matter
Review Paper: BA Li, LW Chen, CM Ko, Phys. Rep. 464 (2008)113
Theoretical Approaches
• Skyrme-Hartree-Fock
• Relativistic Mean Field Theory
• Relativistic Hartree-Fock
•
•
•
•
•
Variational Approach
Green’s Function Theory
Brueckner Theory
Dirac-Brueckner Approach
Effective Field Theory
Symmetry energy predicted by various many-body
theories ---- Extremely Large uncertainty at high densities!
?
C. Fuchs and H. H. Wolter,
EPJA30(2006)5
BHF
Greens function
Variational
Dieperink et al., PRC67(2003)064307.
Most recent results from BHF
Z.H. Li, U. Lombardo, H.-J. Schulze, Zuo et al., PRC74(2006)047304
Bethe-Goldstone Theory
•
Bethe-Goldstone equation and effective G-matrix
G(  ,  ; )  vNN  vNN 
k1k 2
k1k2 Q(k1 , k2 ) k1k2
   (k1 )   (k2 )  i
→ Nucleon-nucleon interaction:
G(  ,  ;  )
vNN  v2  V3eff
★ Two-body interaction :v2AV18 (isospin dependent)
★ Effective three-body force
V eff
3
→ Pauli operator :
Q(k1 , k2 )  1  nk1 1  nk2 
→ Single particle energy :
 (k )   2k 2 /(2m)  U (k )
→ “Auxiliary” potential : continuous choice
U (k )   n(k ' ) Re kk' G[ (k )   (k ' )] kk'
A
k'
Confirmation of the hole-line expansion of the EOS under
the contineous chioce (Song,Baldo,Lombardo,et al,PRL(1998))
Nuclear Matter Saturation Problem
The model of rigid nucleons interacting via realistic two-body forces fitting invacuum nucleon-nucleon scattering data can not reproduce the empirical
saturation properties of nuclear matter (Coestor band, Coestor et al.,
PRC1(1970)765)
Microscopic Three-body Forces
• Based on meson exchange approach
• Be constructed in a consistent way with the adopted two-body
force---------microscopic TBF !
• Grange et.al PRC40(1989)1040
Z-diagram
,
N

(b)
(c)
N
N
R
,

, 
N
N
, 
N
N
 ,  ,
N
N
 ,  ,
N
(a )
, 

, 

, 
, R
Effective Microscopic Three-body Force
• Effective three-body force V eff
3
eff
3
V
r ', r ' r , r 
1
2
1
2
 
1
 Tr   d r3d r3 ' n* r3 ' 1   r13 '1   r23 '
4
n

  
 W3 r1 ' , r2 ' , r3 ' r1 , r2 , r3  n r3 1   r13 1   r23 
→ Defect function: (r12)= (r12) – (r12)
★Short-range nucleon correlations (Ladder correlations)
★Evaluated self-consistently at each iteration
 Effective TBF ---- Density dependent
 Effective TBF ---- Isospin dependent for asymmetric
nuclear matter
EOS of SNM & saturation properties
TBF is necessary for reproducing
the empirical saturation property of
nuclear matter in a non-relativistic
microscopic framework.
Saturation properties:
 (fm-3)
EA (MeV) K (MeV)
0.19
–15.0
210
0.26
–18.0
230
W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot, NPA706(2002)418
TBF based on Bonn B interaction
Li ZengHua，H. J. Schulze, U. Lombardo, Wei Zuo， PRC 77, 034316 (2008)
Isospin dependence of the EOS of
asymmetric nuclear matter
Parabolic law
EA (, T ,  )  EA (, T ,0)  Esym (, T ) 2
W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot,
EPJA 14 (2002) 469
W. Zuo, Z.H.Li,A. Li, G.C.Lu,
PRC 69(2004)064001
Density dependence of symmetry energy
TBF effect
Thermal effect
Critical temperature for liquid-gas
phase transition in warm nuclear matter
SHF : 14-20 MeV
RMT : 14 MeV
DBHF: 10 MeV
BHF(2BF): 16 MeV
BHF(TBF): 13 MeV
BHF(Z-d): 11 MeV
Z-diagram
Full TBF
A possible explanation of the
discrepancy between the DBHF
and BHF predictions
W. Zuo, Z.H.Li,A. Li, U.lombardo,
NPA745(2004)34.
At the lowest mean field approximation, two problems of BHF
approach for predicting nuclear s.p. properties:
1. At densities around the saturation density, the predicted optical
potential depth is too deep as compared to the empirical value,
and it destroy the Hugenholtz-Van Hove (HVH) theorem.
Solution: to include the effect of ground state correlations
J. P. Jeukenne et al., Phys. Rep. 25 (1976) 83
M. Baldo et al., Phys. Lett. 209 (1988) 135; 215 (1988) 19
2. At high densities, the predicted potential is too attractive and
its momentum dependence turns out to be too weak for
describing the experimental elliptic flow data.
P. Danielewicz, Nucl. Phys. A673 (2000) 375
Improvement in two aspects:
1. Extend the calculation of the effect of ground state correlations
to asymmetric nuclear
W. Zuo et al., PRC 60 (1999) 024605
2. Include a microscopic three-body force (TBF) and the TBFinduced rearrangement contribution in calculating the s.p.
properties
W. Zuo et al., NPA706 (2002) 418; PRC 74 (2006) 014317
Single Particle Potential beyond the mean field approximation:
1. Single particle potential at lowest BHF level
G(  ,  ; )  vNN  vNN 
k1k 2
k1k2 Q(k1 , k2 ) k1k2
   (k1 )   (k2 )  i
G(  ,  ;  )
U BHF (k )   n(k ') Re kk ' G[ (k )   (k ')] kk '
A
k'
2. Ground state correlations
3. TBF rearrangement
 V3eff
1
TBF (k )   ij
ij
2 ij
 nk
Full s.p. potential：
ni n j
A
U (k )  U BHF (k )  U2 (k )  UTBF (k )
Single particle potential at the BHF level
In neutron rich matter :
Up<Un at low momenta
Up>Un at high enough momenta
W. Zuo, I. Bombaci and U. Lombardo, PRC
W. Zuo, L.G. Gao, B.A. Li et al., Phys. Rev. C72 (2005)014005 .
Pauli rearrangement contribution: Ground state correlations
1. The Pauli rearrangement is repulsive
2. It affects maily the s.p. potential at low momenta and
vanishes repaidly above Fermi momentum
3. It distories the linear beta-dependence of the s.p. potential
W. Zuo, I. Bombaci, U. Lombardo, PRC 60 (1999) 024605
Neutron-proton effective mass splitting in neutron-rich matter
1
1
 m dU 
m
k  d 


1



m m  dk  kF 
p dk  k
F
*
M*n > M*p
neutrons
protons
Comparison to other predictions:
DBHF: mn* > mp*
Dalen et al., PRL95(2005)022302
Z. Y. Ma et al., PLB 604 (2004)170
F. Sammarruca et al., nucl-th/0411053
Skyrme-like interactions:
mp* < mn* or mn* < mp*
B. A. Li et al., PRC69(2004)064602
W. Zuo, L.G. Gao, B.A. Li et al., Phys. Rev. C72 (2005)014005 .
TBF rearrangment contribution to s.p. potential in symmetric
nuclear matter
 V3eff
1
U TBF (k )   ij
ij
2 ij
 nk
ni n j
A
Effective mass
1. The TBF induces a strongly repulsive
rearrangement modification of the s. p.
potential at high densities and momenta.
2. The TBF rearrangement contribution is
strongly momentum dependent at high
densities and momenta.
Zuo, Lombardo, Schulze, Li, Phys. Rev. C74 (2006) 017304
S.p. potential including the TBF rearrangment contribution
W. Zuo, L.G. Gao, B.A. Li et al., Phys. Rev. C72 (2005)014005 .
Isospin dependence of the TBF rearrangment effect
Symmetry potential
1. Negligible at low densities around and
below the Fermi momentum.
2. Enhancement of the repulsion for
neutrons and the attraction for protons
at high densities
Effective masses
1. Remarkable reduction of the neutron
and proton effective masses.
2. Suppression of the isospin splitting
in neutron-rich matter at high
densities.
TBF effect on the 1S0 proton gap in
neutron star matter
TBF suppresses strongly the 1S0 proton
superfluidity in neutron stars
1. It reduces the energy gap from ~1 to ~0.5
2. It suppresses largely the density region
of the superfluidity
W. Zuo et al., PLB 595(2004)44
Proton superfludity:
Going from SNM to PNM,
the maximum gap value decreases
and the density domain enlarges
remarkably
W. Zuo et al., PRC75(2007)045806
3PF
2
proton and neutron superfluidity
in asymmetric nuclear matter
Neutron
W. Zuo et al., EurPhys. Lett. 84(2008)32001
Proton
TBF effect on the 3PF2 neutron gap in
neutron star matter and neutron stars
TBF enhances remarkably the 3PF2
neutron superfluidity in neutron star
matter and in neutron stars
3PF2 gap in Neutron star matter
3PF2 gap in Neutron stars
W. Zuo et al., Phys. Rev. C78(2008)015805
Summary
• The TBF provides a repulsive contribution to the EOS of nuclear matter and
improves remarkably the predicted saturation properties.
• The empirical parabolic law for the EOS of ANM can be extended
to the highest asymmetry and to the finite-temperature case.
• The TBF leads to a strong enhancement of the stiffness of symmetry energy at
high densities.
• The neutron-proton effective mass splitting is m*n > m*p
• The TBF induces a strongly repulsive and momentum-dependent
rearrangement contribution to the s.p. potential at high densities.
Thank you !