Nuclear and neutron matter EOS How relevant is for PREX ? Trento, 3-7 August 2009
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
The ladder series for the three-particle scattering matrix
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
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 (< 0.2 MeV) region 3. Three-body forces are negligible (< 0.01 MeV) 4. Effect of self-consistent U is small (see later) M.B. & C. Maieron
Three hole-line contribution (fm-1) (MeV) M.B. & C. Maieron, PRC 77, 015801 (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 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.
M.B. & C. Maieron, PRC 77, 015801 (2008) • Gezerlis and J. Carlson, Pnys. Rev. C 77,032801 (2008) • Quantum Monte Carlo calculation
QMC M.B. & C. Maieron, PRC 77, 015801 (2008)
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
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)
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)
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 ?
Spread in the neutron matter EOS B. Alex Brown PRL 85 (2000) 5296
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
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)
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)
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 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.
The astrophysical link Asection(schematic) of a neutron star
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 preprintnucl-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, And Z electrons (to be electroneutral). β-stability condition:
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 ∂ρ/∂r exists, Eph 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.
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.
Outer Crust Inner Crust No drip region Drip region Position of the neutron chemical potential
Looking for the energy minimum at a fixed baryon density Density = 1/30 saturation density Wigner-Seitz approximation
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, 025803(2007)
Comparing with ‘real’ nuclei. Proton density M. B., E.E. Saperstein,S.V. Tolokonnikov, PRC 76, 025803(2007)
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
2 1 1 1 1 Negele & Vautherin 2Uniformnuclearmatter (M.B.,Maieron,Schuck,Vinas NPA 736, 241 (2004))
Making a comparison N & V Catania - Moskow
The upper edge of the crust Comparison with N & V N & V M. B., E.E. Saperstein,S.V. Tolokonnikov, PRC 76, 025803(2007)
µn 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