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Nuclear and neutron matter EOS
How relevant is for PREX ?
Trento, 3-7 August 2009
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OUTLOOOK

• Microsopic theory of Nuclear matter EOS.
• Comparison with phenomenological models
• Symmetry energy
• From homogeneous matter to nuclei
• 4. The Astrophysical link.
• Neutron Star crust structure and EOS.
• 5. Some conclusions and prospects

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Ladder diagrams for the scattering G-matrix
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The ladder series for the three-particle scatterin
g matrix
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Three hole-line contribution
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Two hole-line (Brueckner) contributions. They
take care of the repulsive short range
correlations
Long range correlations (cluster formation and
condensate ..) are included in the three
(or more) hole line diagrams
Two and three hole-line diagrams in terms of
the Brueckner G-matrixs
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Neutron matter EoS at low density
f
p, d
B/A (MeV)
s
kf (fm-1)
Low 1. The s-wave dominates
density 2. The thre hole-lines are
small (lt 0.2 MeV) region 3.
Three-body forces are negligible (lt 0.01 MeV)
4. Effect of
self-consistent U is small (see later)
M.B. C. Maieron
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Three hole-line contribution
(fm-1)
(MeV)
M.B. C. Maieron, PRC 77, 015801 (2008)
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M.B. C. Maieron, PRC 77, 015801 (2008)
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A simple exercise in nuclear matter
Calculate the neutron matter EOS at
low density
Take a separable representation for the 1S0
channel
with e.g.
for which the free scattering matrix reads
where
is the free two-body Greens function. Then fix
the parameters
in order to reproduce the scattering length
and effective range for this channel (low energy
data)
The in-medium G-matrix reads
where Q is the Pauli operator. Compare
G-matrix and T-matrix. Everything is
analytical. The neutron matter energy can be
calculated by simple integration.


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Explicit expression of the separable G-matrix
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M.B. C. Maieron, PRC 77, 015801 (2008)
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M.B. C. Maieron, PRC 77, 015801 (2008)

• Gezerlis and J. Carlson, Pnys. Rev. C 77,032801
  (2008)
• Quantum Monte Carlo calculation

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QMC
M.B. C. Maieron, PRC 77, 015801 (2008)
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Conclusions for the very low density
region of pure neutron matter

• Only s-wave matters, but the unitary limit is
  actually
• never reached. Despite that the energy is ½
  the kinetic energy
• in a wide range of density (for unitary
  0.4-0.42 from QMC).
• The dominant correlation comes from the Pauli
  operator
• Both three hole-line and single particle
  potential effects are small
• and essentially negligible
• Three-body forces negligible
• The rank-1 potential is extremely accurate
  scattering length
• and effective range determine completely the
  G-matrix.
• Variational calculations are slightly above BBG.
• Good agreement with QMC.

In this density range one can get an accurate
neutron matter EOS
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Confronting with exact GFMC for v6 and v8
at higher dednsity
Variational and GMFC Carlson et al. Phys. Rev.
C68, 025802(2003) BBG M.B. and C. Maieron,
Phys. Rev. C69,014301(2004)
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Pure neutron matter Two-body forces only.
E/A (MeV)
density (fm-3)
Comparison between BBG (solid line)
Phys. Lett. B
473,1(2000) and variational calculations
(diamonds)
Phys. Rev. C58,1804(1998)
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E/A (MeV)
density (fm-3)
Including TBF and extending the comparison to
very high density. CAVEAT TBF are not exactly
the same.
In any case, is it relevant for PREX ?
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Spread in the neutron matter EOS
B. Alex Brown PRL 85 (2000) 5296
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Comparison between phenomenological forces
and microscopic calculations (BBG) at
sub-saturation densities.
M.B. et al. Nucl. Phys. A736, 241 (2004)
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Symmetry energy as a function of density. A
comparison at low density.
Microscopic results approximately fitted by
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Symmetry energy
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CAVEAT EoS of symmetric matter at low density
M. B. et al. PRC 65, 017303 (2001)
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Problem cluster formation at low density
G. Roepke et al. , PRL 80, 3177 (1998)
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Going to finite nuclei
Semi-microscopic approach
The last two terms are phenomenological,
adjusted to reproduce binding, radius and single
particle levels in finite nuclei. Fine tuning
is definitely needed.
M.B., C. Maieron, P. Schuck and X. Vinas, NPA
736, 241 (2004) M.B. , P. Schuck and X. Vinas,
PLB 663, 390 (2008) L.M. Robledo, M.B., P. Schuck
and X. Vinas, PRC 75, 051301 (2008)
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Using microscopic EoS for Energy
Density Functionals in nuclei Since
the inclusion of the clusters in the
low density region of nuclei ground state
would be unrealistic, we need the nuclear
matter EoS where they are suppressed. The
simplest way to do that is to consider only
short range correlations (i.e. Brueckner
level)
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Trying connection with phenomenology the
case. Density functional from microscopic
calculations
rel. mean field
Skyrme and Gogny
microscopic functional
The value of r_n - r_p from mic. fun. is
consistent with data, which are centered around
0.15 but with a large uncertainity.
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The astrophysical link
A section (schematic)
of a neutron star
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In the outermost part of the solid crust a
lattice of is present, since it is the
most stable nucleus. Going down at increasing
density, the electron chemical potential starts
to play a role, and beta-equilibrium implies the
appearence of more and more neutron-rich
nuclei. Theoretically, at a given average baryon
density, one has to impose a) Charge
neutrality b)
Beta-equilibrium and then mimimize the energy.
This fixes A, Z and cell size. At higher density
nuclei start to drip. Highly exotic nuclei are
then present in the NS crust.
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There has been a lot of work on trying to
correlate the finite nuclei properties (e.g.
neutron skin) and Neutron Star structure. A
possibility is to consider a large set of
possible EoS and to see numerically if
correlations are present among different
quantities, like skin thikness vs. pressure or
onset of the Urca process (see. e.g. Steiner et
al., Phys. Rep. 2005). Here we take a different
attitude we try to predict both NS structure
and finite nuclei properties on the basis of
microscopic calculations (estimating the
theoretical uncertainity).
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• A semi-microscopic self-consistent method to
  describe the inner crust of a neutron star
• WITHIN the Wigner-Seitz (WS) metod
• With PAIRING effects included.

M. Baldo, U. Lombardo, E.E. Saperstein, S.V.
Tolokonnikov, JETP Lett. 80, 523 (2004). Nuc.
Phys. A 750, 409 (2005). Phys. At. Nucl.,
68, 1812 (2005). M. Baldo, E.E. Saperstein,
S.V. Tolokonnikov, Nuc. Phys. A 775, 235
(2006). - Eur. Phys. J. A 32, 97 (2007)
M. Baldo, E.E. Saperstein, S.V. Tolokonnikov,
arxiv preprint nucl-th/0703099 , PRC 76,
025803 (2007).

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Wigner Seitz (WS) method

• Crystal matter is approximated with a set of
  independent spherical cells of the radius Rc.
• The cell contains Z protons, NA-Z neutrons,

And Z electrons (to be electroneutral). ß-stabili
ty condition
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Generalized energy density functional (GEDF)
method
Choice of Fm outside almost homogeneous neutron
matter (LDA is valid for Emi), inside, where the
region of big ??/?r exists, Eph dominates which
KNOWS how to deal with it.
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S.A. Fayans, S.V. Tolokonnikov, E.L. Trykov, and
D. Zawisha, Nucl. Phys. A 676, 49
(2000). Describes a set of long isotopic chains
(with odd-even effects) with high accuracy.

from the Brueckner theory with the Argonne force
v18 and a small addendum of 3-body forces.
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The structure of nuclei and Z/N ratio are
dictated by beta equilibrium
Negele Vautherin classical paper. Simple
functional, and no pairing. Functional partly
compatible with microscopic neutron matter EOS.
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Outer Crust
Inner Crust
No drip region
Drip region
Position of the neutron chemical potential
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Looking for the energy minimum at a fixed baryon
density
Density 1/30 saturation density
Wigner-Seitz approximation
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The neutron matter EOS
Solid line Fayans functional Dashes
SLy4 Dotted line microscopic (Av-18)
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In search of the energy minimum as a function
of the Z value inside the WS cell
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Neutron density profile at different Fermi momenta
.
.
.
.
.
.
.
.
.
.
.
M.B. , U. Lombardo, E.E. Saperstein and S.V.
Tolokonnikov, Phys. of Atomic Nuclei 68,
1874 (2005)
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Proton density profile at different Fermi momenta
M.B. , U. Lombardo, E.E. Saperstein and S.V.
Tolokonnikov, Phys. of Atomic Nuclei 68,
1874 (2005)
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Comparing with real nuclei. Neutron density
M. B., E.E. Saperstein, S.V. Tolokonnikov, PRC
76, 025803(2007)
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Comparing with real nuclei. Proton density
M. B., E.E. Saperstein, S.V. Tolokonnikov, PRC
76, 025803(2007)
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Dependence on the funcional . Black pure
Fayans
Red Fayans micr.
Kf Z A Acl
Rc ___________________________________
0.7 68 1398 343
30.65 51 1574 225
31.89 ________________________________
____ 0.9 56 1324 386
23.41 24 857
132 20.25 __________________________
___________ 1.1 20 601
181 14.73 20
635 172 14.99
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2
1
1
1
1 Negele Vautherin
2 Uniform nuclear matter (M.B.,Maieron,Schuck,Vi
nas NPA 736, 241 (2004))
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Making a comparison
N V
Catania - Moskow
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The upper edge of the crust Comparison with N
V
N V
M. B., E.E. Saperstein, S.V. Tolokonnikov, PRC
76, 025803(2007)
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µn for DF3 functional
Two competing drip regions
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Indications from the comparisons 1. The
functional must be compatible with low density
nuclear matter EoS 2. Different functionals
give close crust structures if they fulfill
this condition . To be checked with a wider set
of functionals
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EOS of the crust
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Adiabatic index
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EOS of the crust
Comparing different Equations of State for low
density
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Pressure vs. density
Drip
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The shear modulus
In many applications also the shear modulus is
needed. In the WS approximation the shear
modulus is determined only by the
coulomb energy of the lattice. However the
coulomb energy depends indirectly on the
functional through the values of Z. How large
is the baryonic contribution ?
Beyond the WS approximation.

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The extension to finite temperature is needed
for the study of a)
Supernovae b) Protostars
c) Binary mergers
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SOME CONCLUSIONS AND PROSPECTS

• 1. Through a semi-microscopic approach one can
  establish
• a connection between exotic and non-exotic
  nuclei, studied
• in laboratory and e.g. the structure of
  neutron star crust.
• 2 In EDF based on microscopic EoS one
  has to take care
• of the problem of cluster formation at
  low density.
• Functionals that are compatible with
  microscopic nuclear
• matter EoS seem to give comparable results
  for the NS
• crust structure.
• Other microscopically based functionals must be
  tested
• before firm conclusions can be reached.

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Nuclear matter EOS. Uncertainity of the
three-body forces affects mainly high density
EoS
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Symmetry energy as a function of density The
density dependence above saturation can
be non-trivial (e.g. change of curvature)