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Strong interaction in the nuclearmedium: new trends
Effective interactions and energyfunctionals: applications to
nuclear systems – II
Marcella Grasso
École Joliot-Curie
27 September - 3 October 2009
Lacanau - France
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Overview of the second
lecture• 1. Mean-field based models for the ground state. From HF to HFB (pairing included)
• 2. Mean-field based models to describe excited states. RPA derived as smallamplitude approximation from time-dependent HF equations. Pairing -> QRPA
• 3. The interaction (and then the functional) in the pairing channel
• 4. Isospin effects. Exotic nuclei. New phenomena… stronger and/or differentcorrelations?
• 5. Correlations. Going beyond mean field
• 6. Nuclear matter and its properties. The EoS (case of Skyrme)
• 7. Asymmetric matter, neutron matter. Isospin effects
• 8. Spin instabilities of nuclear matter and the Skyrme interaction
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1. Mean-field based modelsfor the ground state
- From HF to HFB (pairingincluded)
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• In HF mean field the ground state is aSlater determinant (completelyuncorrelated single-particle wavefunctions)
• Pairing correlations may be included atthe mean field level by introducing theconcept of quasiparticles (the number ofparticles is not conserved: symmetrybreaking)
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Ground state. Individual degrees offreedom. Uncorrelated state.
Mean field HF (EDF)
Ground state. Individual degreesof freedom. Uncorrelated state,but correlations explicitely in the
functional Kohn-Sham (DFT)
Mean field level but withquasiparticles. Pairingcorrelations are added.
HFB ot HF +BCS
in progress
Dynamical properties.
TDHF
Dynamical properties.
TDDFT
Small amplitude limit.
(Q)RPA (collectivestates)
Dynamical properties.
TDHFB
Beyond mean field. More generalground state (not a simple Slaterdeterminant) GCM multiparticle-multihole ext. RPA
Beyond mean field. Coupling single-particle and collective coordinates
Particle-vibrationcoupling
Second RPA
(also from small
amplitude limit of TDDM)
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No isotopes (Z=102)
Duguet et al. arXiv:nucl-th/0005040v1
Sn = E (Z,N)-E(Z,N-1)
Odd-even effect
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The Hartree-Fock-Bogoliubov (HFB) orBogoliubov-de Gennes equations
We can derive them in a very elegant wayin the simple case of:
- spherical system - two spin states ! and " - zero-range interaction acting only
between states with opposite spin
P. de Gennes, Superconductivity of metals and alloys
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The Hamiltonian is composed by a 1-body term and a 2-body term (zero-range
interaction of intensity V)
The mean field approximation leads to a 1-bodyeffective term that approximates H2
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Since Heff is a quadratic form in # and #+, it can be diagonalized byunitary transformations like the Bogoliubov transformations
The Bogoliubov transformations diagonalize Heff. It means:
One calculates:
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To derive the expressions for the mean field W and the pairing field $,we impose that the free energy F is stationary
where the mean value of H is defined as:
Temperaturedependence
Using the Bogoliubov transformations one gets the HFB equations.The solutions are the two-component wave functions (u,v) and theassociated quasiparticle energies %:
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It can be shown that F is stationary if:
Density
Pairingdensity
The temperature dependence iscontained in a Fermi factor
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2. Mean-field based modelsto describe excited states.
RPA derived as smallamplitude approximationfrom time-dependent HF
equations. Pairing -> QRPA
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Ground state. Individual degrees offreedom. Uncorrelated state.
Mean field HF (EDF)
Ground state. Individual degreesof freedom. Uncorrelated state,but correlations explicitely in the
functional Kohn-Sham (DFT)
Mean field level but withquasiparticles. Pairingcorrelations are added.
HFB ot HF +BCS
in progress
Dynamical properties.
TDHF
Dynamical properties.
TDDFT
Small amplitude limit.
(Q)RPA (collectivestates)
Dynamical properties.
TDHFB
Beyond mean field. More generalground state (not a simple Slaterdeterminant) GCM multiparticle-multihole ext. RPA
Beyond mean field. Coupling single-particle and collective coordinates
Particle-vibrationcoupling
Second RPA
(also from small
amplitude limit of TDDM)
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Two-neutron 0+ addition mode
QRPA in particle-particlechannelResponse function for 124Sn
Khan, Grasso, and Margueron
Excitation modes (small amplitudes)Quadrupole mode QRPA in
particle-hole channelResponse function for 22O
Khan, Sandulescu, Grasso,and Van Giai, PRC 66, 024309(2002)
Pair transfermode
Khan, Grasso, Margueron, submitted PRC
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The RPA equations can beintroduced in different ways
• 1) equations of motion method (Rowe. 1968) (starting fromthe stationary Schroedinger equation)
• 2) Green’s function techniques
• 3) From the small amplitude limit of time-dependentHartree-Fock (TDHF) equations
• Both 2) and 3) are based on the linear response theory
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Linear Response Theory
• Action of a time-dependent external
field: response of the system
• Limit of a weak perturbing field: linear
response
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RPA from TDHF equations
External perturbation:
We impose that &(t) corresponds to a Slater determinant. It
means: &2=&. The density satisfies the following equations of
motion:
We introduce a variation of the density (linear with the external
perturbation)
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Inserting in the equations of
motion and keeping only linear terms one finds theRPA equations (solving for f(t)=0). The matrices A
and B are written as:
One recognizes the residual interaction (different from V ifV is density dependent rearrangement term):
E is the energydensity functional
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Pairing included: linearapproximation from TDHFB
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Khan et al. PRC 66, 024309 (2002)
From time-depedent Hartree-Fock-Bogoliubov
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Variations
Variation of theparticle density
Variation of thepairing tensor
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After manipulations the Bethe-Salpeter equationcan be found
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3. The interaction (and thenthe functional) in the pairing
channel
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In mean field calculations withGogny
(Almost) the same interaction as inthe particle-hole (mean field) channel
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Pairing in Skyrme-HFB meanfield framework
• Non empirical pairing energy density functional derived at lowest order inthe two-nucleon vacuum interaction including Coulomb (many-bodyperturbation theory) (Lesinski, et al., arXiv:0809.2895 [nucl-th]
• Derived from a microscopic nucleon-nucleon interaction. If LDA is valid: fitto reproduce the gap in symmetric and neutron matter (Margueron, et al.Phys. Rev. C 77, 054309 (2008))
• Also dependence on the isovector density (Margueron)
• Empirical pairing energy density functional with constraints from nuclei:odd-even mass staggering, separation energies
• Also dependence on the isovector density (Yamagami, et al.,arXiv:0812.3197 [nucl-th])
• In the context of empirical pairing functional: New constraints? Pairingvibrations?
Matsuo, Proceedings of COMEX3, to be published Grasso, et al. Proceedings of NSD09, to be published Khan et al., 2009
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Model: Skyrme-HFB with zero-range pairinginteraction
and dependence on the isoscalar density
( )( )
( )21
0
0211 rr
rxVrrV
rrrr!
""
#
$
%%
&
'
(()
*++,
-!=! .
/
/0
!0 is currently chosen equal to the saturation density 0.16 fm-3
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Ultraviolet divergence. Renormalization ofthe HFB equations
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Thomas-Fermi approximation for thecalculation of the regular part of the
Green’s function
Increasingenergycutoff
50 (MeV)
45
35
30
20
Bulgac and Yu,2002
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4. Isospin effects. Exoticnuclei. New phenomena…stronger and /or different
correlations?
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Coulombcross section
Deduced photo-neutroncross section Adrich et al., PRL 95, 132501 (2005)
Coulomb excitation, GSI
PYGMY DIPOLERESONANCE
Evolution of excitation modesin exotic nuclei. One example.
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Change of magic numbers. Importance of abetter treatment of single-particle states
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Bastin, et al. 2002
Low energy.Collapse ofthe N=28 shellclosure atZ=14
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Tensor force• Shell model : T. Otsuka, et al., PRL 95, 232502
(2005)
• Relativistic HFB (one needs exchangecontribution so that the pion can be active) :W. Long, et al., PLB 640, 150 (2006)
• Non relativistic mean field:• Skyrme : G. Colò, et al., PLB 646, 227 (2007)
• Fit: Lesinski, et al., PRC 76, 014312 (2007)• Gogny : T. Otsuka, et al., PRL 97, 162501
(2006) aggiornare riferimenti
Not beyond mean field, but towards a more complete formalism
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Variation of the energy density (contributionsdepending on J)
( ) pnpn JJJJH !" ++=# 22
2
1
( ) ( ) ( ) )(4
31112
4
1)( 2
3rvlljjj
rrJ iiiii
i
iq !"
#$%
&'+'++= (
)
( )'
'02
2qq
qqq
SO JJdr
d
dr
dWU !"
##++$$
%
&''(
)+=
TC!!! +=
TC!!! +=
( ) ( )221121
8
1
8
1xtxttt
C+!!=" ( )
2211
8
1xtxt
C+!="
J -> spin-orbit density
Spin–orbit potential is modified by these contributions
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Effect due to the tensor
contribution with SLy5. Energy difference
between proton states 2s1/2 and 1d3/2 in Ca isotopes
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5. Correlations.Going beyond mean field
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Ground state. Individual degrees offreedom. Uncorrelated state.
Mean field HF (EDF)
Ground state. Individual degreesof freedom. Uncorrelated state,but correlations explicitely in the
functional Kohn-Sham (DFT)
Mean field level but withquasiparticles. Pairingcorrelations are added.
HFB ot HF +BCS
in progress
Dynamical properties.
TDHF
Dynamical properties.
TDDFT
Small amplitude limit.
(Q)RPA (collectivestates)
Dynamical properties.
TDHFB
Beyond mean field. More generalground state (not a simple Slaterdeterminant) GCM multiparticle-multihole ext. RPA
Beyond mean field. Coupling single-particle and collective coordinates
Particle-vibrationcoupling
Second RPA
(also from small
amplitude limit of TDDM)
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Beyond mean field. Adding correlations byusing more general wave functions than Slater
determinants
Generator Coordinate Method (GCM)
For the formalism: Bonche, et al., Nucl. Phys. A 510 (1990), 466
-Put together collective and single-particle degrees offreedom in a single quantum formulation (collectivephenomena, dynamics of large amplitude deformations)
-Variational method that extends configuration mixing tothe case of a continuous collective variable
-Projection methods (used to restore broken symmetries)are special forms of GCM (where the wave functions areknown a priori)
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• Given a family of N-body wave functions|'(q)( depending on a collective variableq the GCM determines approximatestates of the Hamiltonian of the form:
• The coefficients fk are found by imposingthat
is stationary with respect to variations )fk
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Some technical problems are known and studied: existenceof jumps and divergences (related to the functionals)
Solutions proposed in Bender et al., PRC 79 (2009), Duguetet al., PRC 79 (2009), Lacroix et al., PRC 79 (2009)
From D. Lacroix
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Ground state. Individual degrees offreedom. Uncorrelated state.
Mean field HF (EDF)
Ground state. Individual degreesof freedom. Uncorrelated state,but correlations explicitely in the
functional Kohn-Sham (DFT)
Mean field level but withquasiparticles. Pairingcorrelations are added.
HFB ot HF +BCS
in progress
Dynamical properties.
TDHF
Dynamical properties.
TDDFT
Small amplitude limit.
(Q)RPA (collectivestates)
Dynamical properties.
TDHFB
Beyond mean field. More generalground state (not a simple Slaterdeterminant) GCM multiparticle-multihole ext. RPA
Beyond mean field. Coupling single-particle and collective coordinates
Particle-vibrationcoupling
Second RPA
(also from small
amplitude limit of TDDM)
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Correlations. Multiparticle-multihole configurations. The
ground state is a superposition of Slater determinants
• Higher Tamm Dancoff approximation
• More refined method:
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Ground state. Individual degrees offreedom. Uncorrelated state.
Mean field HF (EDF)
Ground state. Individual degreesof freedom. Uncorrelated state,but correlations explicitely in the
functional Kohn-Sham (DFT)
Mean field level but withquasiparticles. Pairingcorrelations are added.
HFB ot HF +BCS
in progress
Dynamical properties.
TDHF
Dynamical properties.
TDDFT
Small amplitude limit.
(Q)RPA (collectivestates)
Dynamical properties.
TDHFB
Beyond mean field. More generalground state (not a simple Slaterdeterminant) GCM multiparticle-multihole ext. RPA
Beyond mean field. Coupling single-particle and collective coordinates
Particle-vibrationcoupling
Second RPA
(also from small
amplitude limit of TDDM)
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Beyond mean field. Particle-vibration coupling:coupling between single-particle and collective
degrees of freedom
These correlations modify the single particle spectrum (occupation numbersdifferent from 1 and 0 -> we can evaluate the spectroscopic factors; the single-
particle energies are shifted) and the low-lying excited states
In RPA framework (taking intoaccount ph excitations) the HFmass operator is modified by:
with:
Polarizationdiagram
Correlationdiagram
These diagrams have to be subtractedto reduce the double counting relatedto the use of RPA in the description ofparticle-vibration coupling
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These correlations also affect the excited states …
Effects of particle-vibration coupling onthe single-particlespectrum
Neutron states in 208Pb
Bernard, Van Giai, Nucl. Phys. A 348(1980), 75
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Ground state. Individual degrees offreedom. Uncorrelated state.
Mean field HF (EDF)
Ground state. Individual degreesof freedom. Uncorrelated state,but correlations explicitely in the
functional Kohn-Sham (DFT)
Mean field level but withquasiparticles. Pairingcorrelations are added.
HFB ot HF +BCS
in progress
Dynamical properties.
TDHF
Dynamical properties.
TDDFT
Small amplitude limit.
(Q)RPA (collectivestates)
Dynamical properties.
TDHFB
Beyond mean field. More generalground state (not a simple Slaterdeterminant) GCM multiparticle-multihole ext. RPA
Beyond mean field. Coupling single-particle and collective coordinates
Particle-vibrationcoupling
Second RPA
(also from small
amplitude limit of TDDM)
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Extensions of RPA. Problem of theQuasiboson Approximation (violation of the
Pauli principle)
Gambacurta, et al. PRC 80, 014303 (2009) and references therein
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Ground state. Individual degrees offreedom. Uncorrelated state.
Mean field HF (EDF)
Ground state. Individual degreesof freedom. Uncorrelated state,but correlations explicitely in the
functional Kohn-Sham (DFT)
Mean field level but withquasiparticles. Pairingcorrelations are added.
HFB ot HF +BCS
in progress
Dynamical properties.
TDHF
Dynamical properties.
TDDFT
Small amplitude limit.
(Q)RPA (collectivestates)
Dynamical properties.
TDHFB
Beyond mean field. More generalground state (not a simple Slaterdeterminant) GCM multiparticle-multihole ext. RPA
Beyond mean field. Coupling single-particle and collective coordinates
Particle-vibrationcoupling
Second RPA
(also from small
amplitude limit of TDDM)
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Second RPA
Excitationoperator:
1p1h +
2p2h
It can be derived by a linearization of Time DependentDensity Matrix (TDDM)
Tohyama and Gong, Z. Phys. A - Atomic Nuclei 332, 269 (1989)
Lacroix, Ayik, Chomaz, Prog. Part. Nucl. Phys. 52, 497 (2004)
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where:
Matrix elementsAphp’p’’h’h’’ couple1p1h with 2p2h
(particle-phononcoupling is thusincluded in acomplete way)
Diagonalization of:
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6. Nuclear matter and itsproperties. The EoS (case of
Skyrme)
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SymmetricFrom symmetric to neutron matter. Yp=Z/A
Isospin effects
Differences between symmetric nuclear matter EoSs and neutron-richmatter EoSs (density dependence of symmetry energy)
With an EDF model (for instance from theSkyrme density functional) we can calculate the
Equation of State (EoS) of nuclear matter
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Equation of State
Symmetric nuclear matter Pure neutron matter
From Sagawa, Nuclear Bulk Properties, MSU 2008
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Saturation density
Compressibility
Symmetry energy
WITHSOMESKYRMEFORCES
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How these quantities are known?
• Saturation point. Extracted from electronscattering experiments: the central density ofheavy nuclei is always the same, independentlyof the nucleus.
Empirical values
• From the EoS with Skyrme (eliminating surfaceand spin-orbit and with &n = &p = &/2) the volumeenergy is:
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• Incompressibility (curvature at saturation point).From studies on monopole giant resonance(theory and experiment)
Accepted value (systematic of Blaizot (RPA))
From the Skyrme EoS
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• Isoscalar effective mass (very importantfor level densities around Fermi energy). Itcan be related to the energy of theisoscalar quadrupole giant resonance
Empirical value:
From Skyrme EoS:
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7. Asymmetric matter,neutron matter. Isospin
effects
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The EoS (Yp=Z/A, I=(N-Z)/A))
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Isospin effects. Importance ofsymmetry energy.
• Empirical value (from masses using models).
Accepted value
• From the equation of state:
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From Skyrme EoS
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About the symmetry energy. Dependenceon &
• We have mentioned the correlations betweensome quantities characterizing the EoS. A studyof the correlations {K,&0, aI} would allow a bettercontrol on some terms of the interaction (or theenergy density functional). Density-dependentterm?
• Heavy ion collisions, fragmentation: one of theobjectives is to get a better knowledge of thedensity dependence of the symmetry energy.This scenario is richer if exotic ions areconsidered (dependence on neutron/protonasymmetry). Marie-France Rivet Lecture
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• Collective modes in asymmetric nuclei?
• Neutron star masses and radii. URCA process andcooling (role of the symmetry energy)
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8. Spin instabilities ofnuclear matter and the
Skyrme interaction
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Landau-Migdal parameters
In Landau theory of Fermi liquids nuclear matter properties can bewritten in terms of matrix elements of the interaction at Fermi surface
The coefficients F, F’, G, G’ are functions of the anglebetween k1 and k2. A multipole expansion provides theLandau-Migdal parameters Fl, F’l, Gl, G’l (only l=0 and 1 fora zero-range interaction like Skyrme).
The stability of HF solutions constrains the values ofthese parameters (sum rules)
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A detailed analysis in:
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Ferromagnetic phase diagram
Spin asymmetry density :
Susceptibility :
Instability for if :
Asymmetric nuclear matter :
Symmetric nuclear matter :
Diag.
!"# ññ=ñ
s
unstable
stable
F /V
sñ
!!
"
#
$$
%
&
pppn
npnn
÷÷
xx=x
/1/1
/1/11 21
ë,ë
[ ] 21/1 ëë=÷Det !
[ ] 21/1 ë+ë=÷Tr
or
fñ [ ] 0/1 =÷Det [ ] 0/1 >÷Trand
0
/1
2
2
=
VF=
sñs
!" #
#
From Sagawa, Nuclear Bulk Properties, MSU 2008
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Microscopic calculations of theferromagnetic instabilty
Results:No ferromagnetic instability
On the other hand, Skyrme functional is unstable at high density or largeasymmetries
Magneticsusceptibility
S. Fantoni, et al., PRL 87, 018110 (2001)!
I. Vidana et al., PRC 65, 035804 (2002), 66, 045801 (2002)!
I. Bombaci et al., PLB 632, 638 (2006), ...
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J. Meyer 2003
Margueron 2002
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Proposed in Margueron and Sagawa, to be published in J. Phys. G(work on nuclear matter). New terms:
where the spin and spin-isospin densities are:
Applied to ground state of nuclei in:
To be published J. Phys. G
How to cure? An extended Skyrme interaction
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Conclusions: necessity ofimproving predicting power
(exotic nuclei)
• Fitting criteria ? Changing observables ?
• Functional and/or interaction ?
• Models ?