Nuclear and neutron matter EOS

Trento, 3-7 August 2009

How relevant is for PREX ?

1. Microsopic theory of Nuclear matter EOS. Comparison with phenomenological models

2. Symmetry energy

3. From homogeneous matter to nuclei

4. The Astrophysical link. Neutron Star crust structure and EOS. 5. Some conclusions and prospects

OUTLOOOK

Ladder diagrams for the scattering G-matrix

;Ge

QVVG A

kkkh kkGkkE

F

'||'2

1

',2

The ladder series for the three-particlescattering matrix

A

Akkk kk

h

kkGkke

kkkXXTkkke

kkGkk

E

Te

QGXGT

212''1

3213321

2]'''[

121

3

33

3

||'''

1''''||''

1

''||2

1

321

F

F

kkkkk

kkkk

'','',','

,,

2121

321

Three hole-line contribution

Two and three hole-line diagrams in terms of the Brueckner G-matrixs

Two hole-line (Brueckner)contributions. They take care of the repulsive short range correlations

Long range correlations(cluster formation and condensate………..) are included in thethree (or more) hole line diagrams

kf (fm-1)

B/A(MeV)

“Low” 1. The s-wave dominates density 2. The thre hole-lines are small (< 0.2 MeV) region 3. Three-body forces are negligible (< 0.01 MeV) 4. Effect of self-consistent U is small (see later)

s

p, d

f

M.B. & C. Maieron

Neutron matter EoS at low density

0.4 0.5 0.6 0.7 0.8

0.023 0.091 0.107 0.153 0.148

-0.630 -0.416 -0.526 -0.611 -0.592

0.485 0.389 0.515 0.648 0.651

0.156 0.122 0.123 0.121 0.095

0.012 -0.004 -0.005 -0.005 -0.006

Fk

3D

B

BU

RH

Three hole-line contribution

(MeV)

(fm-1)

M.B. & C. Maieron, PRC 77, 015801 (2008)

M.B. & C. Maieron, PRC 77, 015801 (2008)

A simple exercise in nuclear matter Calculate the neutron matter EOS at low density

)'()()',( kkkkV

]|)(|1[/)'()()',( 0 EGkkkkT

]|)(|1[/)'()()',( 0 EGQkkkkG

122 ][)( kk

Take a separable representation for the 1S0 channel

with e.g.

for which the free scattering matrix reads

)(0 EGwhere is the free two-body Green’s function. Then fix

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.

the parameters , in order to reproduce the scattering lengthand effective range for this channel (low energy data)

Explicit expression of the separable G-matrix

M.B. & C. Maieron, PRC 77, 015801 (2008)

M.B. & C. Maieron, PRC 77, 015801 (2008)

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

M.B. & C. Maieron, PRC 77, 015801 (2008)

QMC

Conclusions for the “very low” density regionof pure neutron matter

1. 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).

2. The dominant correlation comes from the Pauli operator

3. Both three hole-line and single particle potential effects are small and essentially negligible

4. Three-body forces negligible

5. The rank-1 potential is extremely accurate : scattering length and effective range determine completely the G-matrix.

6. Variational calculations are slightly above BBG. Good agreement with QMC.

In this density range one can get an accurate neutron matter EOS

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)

density (fm-3)

E/A(MeV)

Comparison between BBG (solid line) Phys. Lett. B 473,1(2000)and variational calculations (diamonds) Phys. Rev. C58,1804(1998)

Pure neutron matterTwo-body forces only.

density (fm-3)

E/A

(M

eV)

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 ?

B. Alex Brown PRL 85 (2000) 5296

Spread in the neutron matter EOS

Comparison between phenomenological forces andmicroscopic calculations (BBG) at sub-saturationdensities.

M.B. et al. Nucl. Phys. A736, 241 (2004)

Symmetry energy as a function of density. A comparisonat low density.Microscopic results approximately fitted by 6.0

0 )/(3.31

Symmetry energySymmetry energy

CAVEAT : EoS of symmetric matter at low density

M. B. et al. PRC 65, 017303 (2001)

Problem : cluster formation at low density

G. Roepke et al. , PRL 80, 3177 (1998)

termorbitSpinL

termSurfaceL

energyCoulombL

mndensityenergycmicroscopiL

partfreeL

LLLLLL

SO

grad

Coul

bulk

sogradCoulbulk

.).(0

0

Semi-microscopic approach

The last two terms are phenomenological, adjusted to reproducebinding, radius and single particle levels in finite nuclei.Fine tuning is definitely needed.

Going to finite nuclei

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)

Using microscopic EoS for Energy Density Functionals in nuclei Since the inclusion of the clusters in the lowdensity region of nuclei ground state would beunrealistic, 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)

Trying connection with phenomenology : the case.Density functional from microscopic calculations

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.

rel. mean field

Skyrme and Gogny

Pb208

A section (schematic)

of a neutron star

The astrophysical link

In the outermost part of the solid crust a latticeof is present, since it is the most stable nucleus.Going down at increasing density, the electron chemicalpotential starts to play a role, and beta-equilibrium impliesthe appearence of more and more neutron-rich nuclei.Theoretically, at a given average baryon density, one has to impose a) Charge neutrality b) Beta-equilibriumand 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.

Fe56

There has been a lot of work on trying to correlatethe finite nuclei properties (e.g. neutron skin) and Neutron Star structure. A possibility is to considera large set of possible EoS and to see numericallyif correlations are present among different quantities,like skin thikness vs. pressure or onset of the Urcaprocess (see. e.g. Steiner et al., Phys. Rep. 2005).

Here we take a different attitude : we try to predict bothNS structure and finite nuclei properties on the basis of microscopic calculations (estimating the theoretical uncertainity).

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).

Wigner – Seitz (WS) method

Crystal matter is approximated with a set of independent spherical cells of the radius Rc.

The cell contains Z protons, N=A-Z neutrons,

3)3/4( cRAAnd Z electrons (to be electroneutral).

β-stability condition:

.0)( epn

Generalized energy density functional (GEDF) method

))(1())(),(( rFrrE mmi

)())(),(())(),(( rFrrErrE mph

),0(1.0)( pmp R

fm.3.0md

,))/)exp((1()( 1 mmm dRrrF

Choice of Fm : outside almost homogeneous neutron matter (LDA is valid for Emi), inside, where the region of big ∂ρ/∂r exists,Eph dominates which KNOWS how to deal with it.

parts.cmicroscopiandogicalphenomenolfor

),,()(),( annorm EEE

al.etS.FayansbyfunctionalDF3theusedweFor phE

functionalcmicroscopitheusedweFor minormE

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 asmall addendum of 3-body forces.

Negele & Vautherin classical paper. Simple functional,and no pairing. Functional partly compatible with microscopicneutron matter EOS.

The structure of nuclei and Z/N ratio are dictated by betaequilibrium epn

No drip region Drip region

Outer Crust Inner Crust

Position of the neutron chemical potential

Looking for the energyminimum at a fixedbaryon density

Density = 1/30 saturationdensity

Wigner-Seitzapproximation

In search of theenergy minimum as a function ofthe Z value insidethe WS cell

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)

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)

Comparing with ‘real’ nuclei. Neutron density

M. B., E.E. Saperstein, S.V. Tolokonnikov, PRC 76, 025803(2007)

Comparing with ‘real’ nuclei. Proton density

M. B., E.E. Saperstein, S.V. Tolokonnikov, PRC 76, 025803(2007)

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

Dependence on the funcional . Black : pure Fayans Red : Fayans + micr.

1 2

1 Negele & Vautherin

2 Uniform nuclear matter (M.B.,Maieron,Schuck,Vinas NPA 736, 241 (2004))

11

Making a comparison

N & V Catania - Moskow

M. B., E.E. Saperstein, S.V. Tolokonnikov, PRC 76, 025803(2007)

The upper edge of the crustComparison with N & V

N & V

µn for DF3 functional

Two competing drip regions

Indications from the comparisons :

1. The functional must be compatible withlow density nuclear matter EoS

2. Different functionals give close cruststructures if they fulfill this condition .

To be checked with a wider set of functionals

-2

-1

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

CT-Moskow

E/A

(M

eV)

K_F (fm-1)

EOS of the crust

,

,,

3

1

3

4

)(

e

ek

d

Pd

P

keA

E

F

F

Adiabatic index

-4

-2

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

CT-MoskowShenN & VBPSLS300

E/A

(M

eV)

K_F

EOS of the crust

Comparing different Equations of State for low density

Pressure vs. density

Drip

0.0001

0.001

0.01

0.1

1

0.0001 0.001 0.01 0.1

CT-MoskowShenBPSLS370

Pre

ssur

e (

MeV

fm-3

)

density ( fm-3 )

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.

The shear modulus

The extension to finite temperature is needed forthe study of a) Supernovae b) Protostars c) Binary mergers

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.

3. Functionals that are compatible with microscopic nuclear matter EoS seem to give comparable results for the NS crust structure.

4. Other microscopically based functionals must be tested before firm conclusions can be reached.

Nuclear matter EOS. Uncertainity of the three-bodyforces affects mainly “high” density EoS

Symmetry energyas a function of density

The density dependenceabove saturation can benon-trivial (e.g. changeof curvature)