Dynamics and Correlations in Exotic Nuclei
YITP, Kyoto 2011
NN effective interaction from Brueckner Theory
and Applications to Nuclear Systems
U. Lombardo
Dipartimento di Fisica e Astronomia and INFN-LNS,Catania (Italy)
outline:
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Overview of the Brueckner theory of Nuclear Matter
3BF within the meson-exchange model of the interaction
Skyrme-type parametrization of the BHF energy functional
In medium Nucleon-Nucleon cross section
Transport Parameters and dissipation of collective modes in NS
Optical – Model Potential for nucleon-Nucleus collisions at E<200 MeV.
Collaboration
M. Baldo, HJ Schulze, INFN Catania
I. Bombaci, Pisa University
U. Lombardo, Catania University and INFN-LNS
A. Lejeune, Liege University
J.F. Mathiot, Clermont-Ferrand,CEA
W. Zuo, ZH Li . IMP-CAS, Lanzhou
H.Q. Song, Fudan University
The non-relativistic nuclear many-body problem
Nuclear matter is an homogeneous system made of pointlike protons and neutrons
interacting through the Hamiltonian
A
A
H   Ti   vij
i 1
ji
(no Coulomb)
i j
v strongly constrained by the most recent data (few thousands, up to 350
MeV) on NN phase shifts, energy scattering parameters, and deuteron
binding energies.
As an example, Argonne v18 potential :
Nucleon-Nucleon phase shifts
Wiringa, Stoks, Schiavilla, PRC51, (1995) 38
Due to the short range repulsive core of the NN interaction,
standard perturbation theory is not applicable.
Bonn B and Exper phase shifts
Many Body Approaches
Non-relativistic:
•
•
•
Brueckner-Bethe-Goldstone expansion
Variational method
Chirl perturbation theory - > Schwenk
Relativistic :
•
DBHF method
Ab initio approaches, realistic NN potentials, no free parameters
The Brueckner-Bethe-Goldstone theory of Nuclear Matter
auxiliary single - particle potential :
A
A
i 1
i j
H   Ti   vij  (T  U )  (V  U )  H 0  H 1
healing
1S
In the BBG expansion:
• H1 expansion
0
is recast in terms of G-matrix
G ;    V NN 
V
k a kb
k a kb Q k a kb
NN
  e( k a ,  )  e( k b ,  )
G ;  
• and then rearranged
V Y12 = GF 12
according to the order of correlations : twobody, threebody ,……(hole line expansion)
2
E
3k
1
(   F 
kk ' G  ;   kk ' a    

A
5 2 m 2  k , k ' k
F
The BBG expansion
 BHF approximation
Two and three hole-line diagrams in terms
of the Brueckner G-matrixs
Bethe-Faddeev
The variational method in its practical form
Pandharipande & Wiringa, 1979; Lagaris & Pandharipande, 1981
Method used to calculate the upper bound to the ground state energy:
 H
E
 E0

Trial w.f. is the antisymmetrized product of two-body correlation
. functions acting on an unperturbed ground state Φ:
 (r1 , r2 ,...)   i  j f (rij ) (r1 , r2 ,....)
The correlation function f represents the correlations induced by the
two-body potential. It is expanded in the same spin-isospin, spin-orbit
and tensor operators appearing in the NN interaction.
The correlation function is obtained from
  H
0
f  
Dependence on the Many-Body Scheme
Nucleon-Nucleon Interaction:
Argonne v18
(Wiringa, Stoks & Schiavilla, Phys. Rev. C51, 38 (1995))
same NN interaction
BBG:
Catania group
Neutron Matter
?
Symmetric Matter
Variational
(Akmal, Pandharipande
& Ravenhall, PRC 58, 1804 (1998))
It makes us confident that these approximations do work!
Convergence of hole line expansion
At the level of 3 hole line approximation the EoS is independen of auxiliary potential
2+3 hole
UG =gap choice
UC=continuous choice
E0= <K> + Σi<V>i = - 16 .0 MeV
Song,Baldo,Giansiracusa and Lombardo. PRL 81 (1998):
3 hole lines: Bethe-Faddeev equation
<V>2= - 40 MeV
<V>3= 3 MeV
<V>4= 0.5 MeV
B.Day,1987
saturation point
3BF in Nuclei
puzzle of analyzing power
Ishikawa Phys.Rev. C59 (1999)
N-d el.scatt. At E=3MeV
10B
Caurier, P. Navrdtil, W.E. Ormand, and J.P. Vary,
Phys. Rev. C64, 051301 (2001)
Meson-exchange Model of 2- and 3- body interaction
P. Grangé et al., PRC 40, 1040 (1989):
, ,σ,ω exchange
+
two-body
,N*
Fujita-Miyazawa
model
N
+
Dirac sea polarization
Z-diagrams are introduced as the relativistic correction to n.r. BHF approach
r 8/3
D E A = 4.2 ( r )
MeV
0
Brown,Weise,Baym,Speth, Nucl.Part.Phys.1987
Nucleon Excitations
Δ - resonance
spin-flip of one quark
S=1/2 Lπ =1- Jπ=1/2+
2s
a radially excited three-quark state
1s
P11 (1440)
coupling constants from N* and Δ decay withs + quark structure
N* - resonance
Input: meson parameters fitting
NN experiment phase shifts
Output: 2BF and 3BF
meson parameters
based on the
N
No adjustable parameters !
Double-selfconsistent approach of BHF-3BF
saturating one particle
VNNN (  ) 
*
*


 13 23 123 V 1' 2 '3' 1'2 '1'3 '
33 '
Ф12 two-body correlation function : V Ф12 = G Фo12
V
(0)
V
(1)
V
(2)
= V Ю G - matrix Ю V
NN
=V +V
NN
= ЧЧЧ
(0)
NNN
(r ) ЮЧЧЧ
(0)
NNN
(r ) Ю
Esym (ρ) = BA (ρ)|SNM - BA (ρ)|PNM
asystiff
asysoft
•
•
•
saturation point:
=0.17 fm-3 , E= -16 MeV
symmetry energy at saturation
Sv≈ 30 MeV
incompressibility at saturation
K ≈ 244 MeV
Danielewicz Plot
HIC collective flows
ZH Li, U Lombardo and H-J Schulze, W,Zuo
PRC(2008)
sub-saturation densities
K∞(ρo)
BHF
Esym (ρo)
244
29.4
Empirical
240±20
28-32
source
MGR
B-W mass table
Esym (0.1)
23.
24.1±0.8
DGR
L
74.4
65.1 ± 15.5
PGR
NUSYM10, Riken 2010
Esym (ρ)=Esym (ρo)+ L/(3 ρo) (ρ-ρo)+Ksym /(18ρo2)(ρ-ρo)2
From HIC (isoscaling, multigragmentation, n-p products,..) still ambiguous
EoS from BHF : individual contributions
and comparison with DBHF
NN excitations (Z-diagrams)
nucleon resonances (F-M) :
Δ(1232),R*(1440)
strong compensation !
BHF with 2+3 body force
The G-matrix expansion (full Brueckner theory) is converging
The non relativistic BHF theory with two body force is consistent
with other n.r. many body approaches (e.g., variational method)
Including relativistic effects (Z-diagrams) the BHF EoS fully overlaps
the DBHF
The three-body force pushes the saturation density to the empirical
value, but its effect is small on the saturation energy (1-2 MeV) and
compression modulus (220->240 MeV) is small
reconciling nuclei and nuclear matter
Ambitious task is to build up a unified EDF for nuclei and nuclear matter:
The question is:
To what extent a Skyrme-type forces can reproduce the BHF (numerical) EDF
as well as experimental observables?
Our hope in such study is to learn what is missing in the theory of nuclear
matter and which Skyrme parametrizations cannot match nuclear matter
we performed a ‘weighted fit’ of nuclear matter (theory) and nuclei
(experiment) via the Skyrme functional:
Gambacurta,Li,Colo’,Lombardo, Van Giai,Zuo, PRC 2011
Skyrme-like Energy Functionals (LNSx)
splitting U into (S,T)
Despite large deviations in the individual spin-isospin components of energy potential U(S,T)
large compensation gives rise to similar EoS specially around saturation
Since U(S,T) probe different observables (compressibility,symmetry energy, GR’s,…) it looks
convenient to fit U(S,T) instead of total U
Simultaneous Skyrme-like fit of nuclear matter (theory) and nuclei (exp) changing the
relative weights:
е
i
sky
th
wi (U ST
- U ST
)+
i
е
j ,a
w j (Oath - Oasky )
j
th=BHF Oα = nuclear binding energies and radii.
Changing the relative weights we produce different parametrizations: LNSn (n=1,2,3,…)
HF calculations from the Skyrme-like parametrizations of BHF
theor <- ( LNS1 – LNS5 ) -> exp
closed-shell nuclei b. energy and radii
Sn Isotopic chain
Giant Resonances
open problems: tensor and spin-orbit
Energy Density Functional on Microscopic Basis
Baldo,Robledo,Schuck,Vinas
finite range term
(surface)
Microscopic Transport Equation:
p2
ep =
+ Up
2m
I i (n) =
е
Ii j =
j
n= 1- n
е
жd s ij ч
ц
зз
чЧ(ni ' n j 'n%%
%%
i n j - ni n j ni ' n j ')
зи d Wч
ч
ш
In BHF nuclear potential and cross sections are calculated from G-matrix
j
Applications:
 simulations of heavy ion collisions (test-particle method)
from linearization : n = n0 + δn -> collective modes
 nuclear fluidodynamics: transport parameters in dense matter
(high-energy HIC, neutron stars)
Phenomelogical Transport Eq.s
gradT
Heat Conduction Eq
heat flux
2 

   u    T    (  u )  K T
3  t

A
(K thermal conductivity)
Navier-Stokes Eq.





 u   u  f    p  V   u   T u
3
 t




bulk viscosity
¶ ux
¶x
shear viscosity
¶ ux
¶y
Transport Parameters
Linearizing the Boltzmann equation the connection with phenomenological
transport equations is established as well as the transport parameters
shear viscosity
1 2
hT =
r vF C (l )W (r )
20
2
4 eF
W
-1
=
т
0
thermal conductivity
s ( E, J )
dE
2eF
2p
т
0
1
dJ
[1- E / 4eF ] 2 s ( E, J )
2p
1
kT =
pF vF2 H (m)W (r )
12
is the in-medium NN cross section
In-medium NN cross sections
To calculate the cross sections at high density the free scattering amplitude
1
T (E) = V + V
T (E)
E- K
must be replaced by in-medium scattering amplitude (G-matrix)
G( E ) = V + V
Q
G(E )
E - Ho
medium effects in the cross section:
Pauli blocking (Q): nucleons scatter into unoccupied states
Strong mean field (H0) between two collisions
Compression of the level densities in entry and exit chennels
theoretical framework (G-matrix)
quasiparticle
spectrum
in-medium dσ/dΩ
p2
e p=
+
2m
t
d s np (J )
dW
е е
t'
< pp ' | G
p '< p f
m*2
:
4p 2h 4
е
SS z S z'
tt '
p2
| pp ' > a =
+ Ut
*
2mt
| бp | G S S S ' (J ) | p 'с|2
z z
mean free path, viscosity, heat conductivity,…
In medium effects on the scattering cross section
dσ/dΩ(θ) =N2 |<p|G (θ)|p’ >|2
A) -------------Finite Range Interaction---------------------------
}N
G
}N
level density:
жd e p ч
ц
з
ч
N=з
зи dp ч
ч
ш
F
- 1
m*
=
pF
effective mass vs β
empirical OMP data support the
prediction m*n>m*p in ANM
effective mass vs ρ
Mean field and Effective Mass vs β
empirical OMP data support the
prediction m*n>m*p in ANM
We expect that in neutron-rich matter. σnn is less suppressed
than σpp
B)--------------------Pauli blocking p>pF----------------------------------------pF
momentum transfer
p ≈ p’ ≈ pF
θ
Δp = 2pF sin (θ/2) =
Fermi sphere
In CM frame
{
=0
>0
pf=0 free space
pf >0 in medium
backward and forward scatterings sizably suppressed
by Pauli blocking
transport parameters in NS
structure of NS:
chemical composition,superfluidity, phase
transitions,…
transport phenomena
rotation, glitches, cooling,collective modes,…
NS internal structure
nucleons,hyperons, kaons, quarks
in beta-equilibrium with leptons
 beta-equilibrium with electrons and muons : p + e¯ n + 
 hyperonized matter:
n + n  n +  ( p + ¯)
at
 > 2o
 kaon condensation
n  p + K¯
at
 > 2-3o
 transition to quark matter
HP  QP (u,d,s)
at
 ~ 6o
restrict on only the first item (p,n. e- )
transition to QGP (in progress)
Viscosity controls the rotational dynamics
and the damping of collective modes
NS is a viscous fluid rapidly rotating
Glitches and post-glitches relaxation time
(superfluidity ?)
coupling between rotation and collective modes
( r-modes)
Chandrasekhar Instability
Thermal conductivity controls thermal evolution
of Neutron Stars (Tsurf vs. Tcore)
Z
 0.11  thr  0.3 fm3
A
nn  (nn) + 
(nn)  nn
minimal cooling:
ANM with β = β (ρ)
p + e¯  n + e
n  p + e¯ + e
p + ¯  n + 
n  p + ¯ + 
charge conservation :  p = e  k pF  keF
chemical equilibrium : n   p  e
n   p  4 Esym 
energy loss replacing a proton with a neutron,
namely the symmetry energy ( aA in B - W mass formula, N  Z )
3
1  4 Esym 
p  

2  ckF 
Cross Sections in β-stable matter
ANM with β = β (ρ)
nn collisions
p + e¯  n + e
n  p + e¯ + e
np collisions
non linear behaviour of proton mean field and effective mass
shear viscosity
1 2
hT 2 =
r vF C (l
20
Beta-stable
й4eF dE
) ккт
кл 0 2eF
2p
т
0
- 1
щ
1
dJ
2
(1 - E / 4eF ) s ( E, J )ъъ
2p
ъ
ы
SNM vs.PNM
electrons
Shternin & Yakovlev PRD(2008), APR
Benhar & Carbone, arXiv09112.0129,CBF
σo(Ω) → σ(Ω) : η~10·η0 Flowers & Itoh, ApJ (1979)
isospin effect: η(proton) → η(neutron) at higher density
m → m* : ηn >> ηe Shternin & Yakovlev PRD(2008)
no r-mode damping
thermal conductivity
4 eF
1
dE
kT =
pF vF2 H (m) т
12
2eF
0
2p
т
0
1
dJ
(1- E / 4eF ) 2 s ( E, J )
2p
NS Cooling
Yakovlev et al, Phys.Reports 354 (2001) 1
Dissipation of r-modes
A non radial collective excitation of a NS is described by a
velocity field
l
2
r
r
ж
ц
r
R i wt
r
dv = iwx = iwe
e R С (зз ч
Ylm )
ч
ч
з
иR ш
l
collective energy:
r 2
1
E = т dV r | dv |
2
dissipation time scale (viscosity only):
1
1 dE
==t
2 E dt
R
т
dV h (r ) | С i v j |2
0
η(r )= η(ρ(r)) density profile of NS is required !
Tolman-Oppenheimer-Volkov (TOV) and nuclear EoS
Input  Equation of State P=P(, p)
Output  Mass-Radius plot
Time scale of nonradial mode damping
from shear viscosity
Li,Lombardo,Peng, PRC (2008)
integration over the star (0≤r≤R):
r → ρ(r) → η(ρ)
→ TOV
constant mass approximation
1
h
= (l - 1)(2l + 1) 2
tV
rR
r=
Me
4p
= const
R3
3
l=2 r-mode
tk (s)
0.54 1016
1.5 1016
3.0 1015
6.0 1015
viscosity time scale
thermal cond.
Transition to deconfined phase
in neutron stars
Despite the contribution of quarks to viscosity is smaller than hadrons the phase transition
is pushing the hadron phase to higher density allowing an extra contribution to viscosity
Chandrasekhar Instability (1970)
Y22 - nonradial mode:  ~ ω0 (Coriolis force)
Inertial frame
J0 >> J22 which decreases, being Y22 a sourse
of gravitational radiation (General Relativity)
Corotating frame
J’22 =
(J-J22 )
|J’22| increases more and more
accompanied by larger frequency and amplitude
and then more angular momentum loss for radiation
Since rotating NS exist, GR instability must be stopped by
Some damping of Y22 the best candidate is viscosity
interplay between GR driving instability and viscosity damping
critical velocity:
1
1
+
= 0
t GR (wc ) t V (wc )
expected to be detected in terrestrial labs ( LIGO,VIRGO,…)
interplay between GR driving instability and viscosity damping
critical velocity:
1
1
+
= 0
t GR (wc ) t V (wc )
Ω~1000 Hz → τGR ~ 100 sec(depending only on rotation)
constant density approximation underestimates the effect of viscosity
Nuclear Potential
Optical Potential vs. nucleon-Nucleus Scattering
mass – shell relation
E
2k 2
2m
 ( k , E ) 
2k 2
2m
 1 (k , E )  2 (k , E )  
BBG hole-line expansion in G-matrix
Hughenholtz-Van Hove theorem:
EF
=
BA
(r = r 0 )
on-shell self.energy
U opt ( E )   R (k ( E ), E )  i I (k ( E ), E )
procedure: for a given approximation of ∑ one solve k=k(E)
and determines selfconsistently the on-shell selfenergy
Nuclear Potential
VNN = Vbare +
е
V123 = Vbare + V2 (r )
3
dU
Up =
A = d
dn p
dn p
й1
щ
1
dG
к е n p n p ' < pp ' | G | pp ' > A ъ= е n p ' < pp ' | G | pp ' > A + е n p n p ' < pp ' | dn | pp ' > A
p
к
ъ
2 pp ''
л2 pp ''
ы p'
dV3 (r )- 1
dG
dG- 1
1- n
=-G
G= - G
G- G
G
dnp
dnp
E
dnp
core polarization Dirac sea pol.
+
+
+
…
n and p self-energies
(k , E )  bhf (k , E )  cpol (k , E )  tbf ( k , E )
p,E
EF
p,E
EF
p,E
EF
Dirac Sea
Evidence for “large” mass ?
Nice et al. ApJ 634, 1242 (2005)
PSR J0751+1807
M = 2.1 +/- 0.2
Demorest et al. Nature 467, (2010)
J1614-2230
M =1.97 +/- 0.04
Ozel, astro-ph /0605106
EXO 0748 – 676
M > 1.8
Quaintrell et al. A&A 401, 313 (2003)
NS in VelaX-1
1.8 < M < 2
ρ ≈ 10 ρ0
OMP Imaginary part:absorption from BHF
from a virtual collisions with the Fermi sea
ε > εF
Im ∑bhf
ε < εF
Im ∑cpol
Im ∑3bf = 0
At E>0 Im ∑bhf ≠ 0 only
Which information can be extracted from N-ion collisions? We
must first single out the volume contribution from others
(surface,spin-orbit,Coulomb)
from NM only the strengths
Vv(E,ρn,ρp),Wv(E,)ρn,ρp)
on energy (or momentum ) ,density and isospin
From Koning & Delaroche (NP 2003) parametrization of OMP fitting nucleon-A experimental
cross sections one can extract the strength g(E) of the volume component
n-A total cross sections
p-A react.cross sections
g (E)
V vol (r , E ) =
1+ e
( r - R )/ a
56Fe
+n
The strength is energy ,density,
and isospin dependent
--Volume term: real part-- vs. energy
no Coulomb correction
with Coulomb correction
Comparison between BHF and empirical OMP
--Volume term: real part-- vs. asymmetry
208Pb
28Si
absorption
Volume term (absorption)
Vs. asymmetry
Volume term (absorption)
vs. energy
theory underestimates empirical data
especially at high energy
Conclusions
The Brueckner theory with 3bf makes predictions is consistent with the empirical
properties of nuclear matter around the saturation density
The meson-exchange model of the interaction allows to build up the two and three
body force with the same meson parameters adjusted on the experimantal phase
shifts on NN scattering
The NN virtual excitations (Z-diagrams) embody most relativistic effects , making
non relativistic BHF comparable with DBHF
Preliminary Skyrme parametrization of the BHF potential energy
The estimate of the time-scale of viscosity dissipation of collective r-modes
(with and without deconfined quark phase) is comparable with time-scale of
gravitational waves emission, confirming that the viscosity is the mechanism
that guarantees the stability of rotating NS
The BHF self-energy gives a microscopic basis to the optical-model potential,
(volume part) describing nucleon-Nucleus collisions