Materianucleare e dinamica nucleare CORTONA2006 Dedicato ad Adelchi
Schema della relazione • 1. Introduzione • 2. Teoria a molti corpi dell’ equazione di stato della • materia nucleare. • 3. Indicazioni sulla EoS dai dati osservativi sulle stelle • di neutroni. • 4. Indicazioni sulla EoS dalla fenomenologia sulle • collisioni fra ioni pesanti. • 5. Dinamica nucleare in collisioni a bassa energia. • 6. Conclusioni e considerazioni generali.
A section (schematic) of a neutron star (from D. Page web site)
SN 1987a Exploding Beforeexplosion
Microscopic nuclear matter EoS • Bethe-Brueckner-Goldstone expansion • Variational method • Monte-Carlo methods
The BBG expansion Two and three hole-line diagrams in terms of the Brueckner G-matrixs
Graphical representation of the Brueckner self-consisten potential
The ladder series for the three-particle scattering matrix
Neutronmatter PLB473, 1 (2000)
Neutronmatter PLB473, 1 (2000)
Evidence of convergence . The final EOS is independent on the choice of the single particle potential . The three hole-line contribution is small in the continuous choice Phys. Rev. C65, 017303 (2001).
The CCM scheme from the variational principle Problem of the hard core
The variational method in its practical form The problem of non-central correlations
Channel dependent correlation factors The pair distribution function
Classification of diagrams (terms of the expansion) for spinless bosons Nodal Simple Elementary Irreducible Composite Theorem : to the energy and to the distribution function g(r) only connected and irreducible diagrams contribute, the reducible or disconnected ones cancel out identically (similarity with the “linked-cluster” theorem) .
Closed equations for N ed X excluding elementary diagrams Generalizing the integral equation for N One gets then two coupled integral equations (non linear) which allow to obtain both N and X and then g(r). Along a similar procedure g(r) has the following expression Since the integral equations are convolutions, they can be easily solved with the method of the Fourier transform. What about elementary diagrams ? It is very easy, but they cannot be summed up. HNC/0 HNC/1 HNC/2 ………….
Energy minimization Once obtained the distribution function g(r) in terms of the correlation factors, the energy functional has to be minimized. In principie this should lead to Eulero-Lagrange equations for f . At HNC/0 level one finds • where is directly related to the static structure function, which in turn • can be written in terms of the distribution function g(r) . • Which is the physical meaning ? • includes the effect of the mean on the particle – particle interaction • and takes into account mainly the long range correlations (“screening”) • 2. The Eulero-Lagrange equations include mainly short range correlations. • In general elementary diagrams express short range correlations. A.D. Jackson et al., Phys. Rep. 86 (1982) 55
Fermions with only central interactions In this case the wave function of the unperturbed stae is a Slater determinant. The modulus square of a Slater determinant is also a Slater determinant.
Nuclear matter case The correlation factors must be in this case actually operators. This complicates the theory. In particular it is not possibile to sum up “hypernetted chains”, but only some sub-series, the ones which select a particular operator, the “single operator chains” ( SOC ). PRC66(2002)0543308
Summary of the formal comparison • The CCM and BBG are essentially equivalent, which indicates that the • w.f. is of the type , if one gets the Brueckner approximation Once the single particle potential is introduced, the methods are not variational at a given truncation. • The main differences in the variational method • a) The correlation factors are local and momentum independent • (eventually gradient terms). • b) No single particle mean field is introduced, so that the meaning • of “clusters” is quite different • c) Chain summations include long range correlations • Short range 3-body cluster calculated in PRC 66 (2002) 0543308
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.
Confronting with “exact” GFMC for v6 and v8 Variational and GMFC : Carlson et al. Phys. Rev. C68, 025802(2003) BBG : M.B. and C. Maieron, Phys. Rev. C69,014301(2004)
“Very low” density Puzzling “quadratic” behaviour of interaction energy Dotted lone : ½ E(free gas) Stars : Friedman & Pandharipande, Nucl. Phys. A361,501(1981) Urbana potential
These results are suggestive of the “unitary limit” behaviour of neutron matter : -a >> d >> r where a is the scattering length, r is the effective range and d is the average distance between particles. For a the only scale is given by d and the corresponding energy scale can be only the kinetic energy with a density independent factor
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, work in progress
Three hole-line contribution (fm-1) (MeV)
Conclusions for the “low” density region • Only s-wave matters, but the “unitary limit” is actually • never reached • The dominant correlation comes from the Pauli operator • Both three hole-line and single particle potential effects are small • but not completely negligible • Three-body forces negligible • The EFT can give only an estimate : one has to go beyond the • scatt. length and effective range approximation (full phase shift) • 6. Some discrepancy between variational and BBG (higher part) In this density range one can get the “exact” neutron matter EOS
Neutron and Nuclear matter EOS. Comparison between BBG and variational method. PRC 69, 018801 (2004), X.R. Zhou et al.
Symmetry energy as a function of density Proton fraction as a function of density in neutron stars AP becomes superluminal and DU process is at too high density
The baryonic Equations of State HHJ : Astrophys. J. 525, L45 (1999) BBG : PRC 69 , 018801 (2004) AP : PRC 58, 1804 (1998)
Summary of the comparison • Similarities and differences between variational and BBG • At v6-v8 level excellent agreement between var. and BBG • as well as with GFMC (at least up to 0.25 fm-3) • for neutron matter. • For the full interaction (Av18) good agreement between • var. and BBG up to 0.6 fm-3 (symmetric and neutron • matter). • 4. The many-body treatment of nuclear matter EOS can be • considered well understood. Main uncertainity is TBF at high • density (above 0.6 fm-3).
Possible tests in of EOS from H.I. collisions and from observations on astrophysical compact objects • 1. Compressibility : H.I. Flows in H.I. • NS masses • Symmetry energy : H.I. Particle production • Isotopic distribuitions • NS DU process and cooling • 3. Liquid-gas phase transition : H.I. Multifragmentation • Limiting temperature • NS Proto-neutron stars • Crust structure
Further comparison : DBHF and phen. EoS EoS selection by T. Klahn, 2006, nucl-th/0602038
The value of the compressibility at saturation does not fix the EoS behaviour at high density Figure from C. Fuchs et al., nucl-th/0511070