Create Presentation
Download Presentation

Download Presentation
## Materia nucleare e dinamica nucleare

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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