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

The ladder series for the three-particle scattering matrix

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 the three (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

Three hole-line contribution (MeV) (fm-1) M.B. & C. Maieron, PRC 77, (2008)

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 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 parametersin order to reproduce the scattering length and effective range for this channel (low energy data)

Explicit expression of the separable G-matrix

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

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

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

Conclusions for the “very low” density region of 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 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, (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 matter Two-body forces only.

density (fm-3) E/A (MeV) 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 and microscopic calculations (BBG) at sub-saturation densities. M.B. et al. Nucl. Phys. A736, 241 (2004)

Symmetry energy as a function of density. A comparison at low density. Microscopic results approximately fitted by

Symmetry energy

CAVEAT : EoS of symmetric matter at low density M. B. et al. PRC 65, (2001)

Problem : cluster formation at low density G. Roepke et al., PRL 80, 3177 (1998)

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. 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, (2008)

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)

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

A section (schematic) of a neutron star The astrophysical link

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.

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).

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/ , PRC 76, (2007).

Wigner – Seitz (WS) method Crystal matter is approximated with a set of independent spherical cells of the radius R c. The cell contains Z protons, N=A-Z neutrons, And Z electrons (to be electroneutral). β-stability condition:

Generalized energy density functional (GEDF) method Choice of F m : outside almost homogeneous neutron matter (LDA is valid for E mi ), inside, where the region of big ∂ρ/∂r exists, E ph dominates which KNOWS how to deal with it.

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.

Negele & Vautherin classical paper. Simple functional, and no pairing. Functional partly compatible with microscopic neutron matter EOS. The structure of nuclei and Z/N ratio are dictated by beta equilibrium

No drip regionDrip region Outer CrustInner Crust Position of the neutron chemical potential

Looking for the energy minimum at a fixed baryon density Density = 1/30 saturation density Wigner-Seitz approximation

The neutron matter EOS Solid line : Fayans functional ; Dashes : SLy4 Dotted line : microscopic (Av-18)

In search of the energy minimum as a function of the Z value inside the 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, (2007)

Comparing with ‘real’ nuclei. Proton density M. B., E.E. Saperstein, S.V. Tolokonnikov, PRC 76, (2007)

Kf Z A Acl Rc ___________________________________ ____________________________________ _____________________________________ 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)) 1 1

Making a comparison N & V Catania - Moskow

M. B., E.E. Saperstein, S.V. Tolokonnikov, PRC 76, (2007) The upper edge of the crust Comparison with N & V N & V

µ n for DF3 functional Two competing drip regions

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

EOS of the crust

Adiabatic index

EOS of the crust Comparing different Equations of State for low density

Pressure vs. density Drip

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 for the 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-body forces affects mainly “high” density EoS

Symmetry energy as a function of density The density dependence above saturation can be non-trivial (e.g. change of curvature)