Isospin-Symmetry interaction and many-body correlations M. Papa

Summary Main ingredient of the model The Isovectorial Interaction Nuclear matter simulations The dynamical case and first applications Conclusive remarks

V loc =∫e int (ρ 2,…)dr e int from EOS =∫ρ 2 dr∞∑ k,l ρ k,l But one needs to exclude the self- energies k=l =∫ρ 2 dr → ∑ k≠l ρ k,l

Cuts: if σ>55 mb→ σ=55 mb Functional form N-N cross-section (from BNV) Search for a close particle K’ with a distance D<2σr N:Y Cycle on the generic particle k go to another particle k Ceck mean free path P=(v r ∙dt)/(ρσ) r:P < > go to another particle k do the scattering, evaluate blocking factor: f k f k’ r:P b >1 reset p k,p k’ go to another particle k Double collision excluded

Constraining the phase space Initialization local cooling-warming-procedure stabilization For each particle i, we define an ensemble K i of nearest identical particles with distances less than 3σ r,3σ p in phase space Distribute Z and N particles in phase according to Fermi gas model f i :1 > < γ m =1.02 m«K i γ m =0.98 m«K i p i (t)=γ i (p i (t-dt)+F i dt) x i (t)=x i (t-dt)+v i dt Cycle on particles Cycle on time t coolinig R;BE No good T stab R,B good Write good configuration Conf. In memory

Global minimizzation of the total energy under “Pauli constraint ” Stable configurations with internal kinetic energy around MeV/A (No solid) Stable configurations with good Binding Energy and nuclear radii in large mass interval- (σ r σ p ) Au results Monopole Zero-point motion At variance with others Approaches (AMD)

Constraint in the normal dynamics MF calculation Numerical integration Cycle on particles f i :1 > Multi-scattering between particles m«K i > Cycle on particles Collision routine Cycle on particles ≤

112 Sn The quasi-Fermi distribution Is maintained in time. With no constraint the energy distribution rapidly changes in a Boltzman-like one Au

Pauli constrant On Au+AU at low energy

CoMD-II and Impulsive Forces J.Comp.Phys. N2 403 (2005) L i 1,2 =(r 1 -r 2 )^p 1 i L f 1,2 =(r 1 -r 2 )^p 1 f p 1 i ≠p 1 f p 1 i = p 1 f L i 1,2 ≠L f 1,2 ∑L i k,m ≠∑L f k,m k ∊C Constraint C≡ ensemble of colliding particles In the generic time step belonging to a compact configuration -Rigid Rotation -Radial momentum scaling -Momentum Translation-Energy conservation Trivial solution

ComD-II results on relative velocities Multi-break-up processes and related times

PLF- fission A1A1 A1A1 A2A2 A1A1 A1A1 A3A3 A2A2 A2A2 A3A3 A1A1 A2A2 A3A3 A4A4 time

Average Time Scale At least 1 IMFAt least 3 target’s IMF tdtd tftf tctc t c >>t f +t d Note; at E * <3.5 MeV/A T ev >800 fm/c for one nucleon

PLF Fission at longer time

What about dynamical fission ? Alignment can be explained without supposing a vanishing collective angular velocity in some time interval. However a different behavior with respect the rigid rotation model it seen to occur for the biggest fragments in the time interval fm/c Φ plane A1A1 A3A3 A2A2

Average Dynamics of the angular momentum transfer Fluctuating Dynamics Non linear behaviour Partial overlap between Fission time and transfer of J Fluctuation of the relative orientation of the Fission plane and entrance channel plane

Experimental Selection criteria to decouple the two dynamics Isospin collaboration

Hot source Multifr.Light partner Multifr.

Heavy partner Multifr.

At long time our semi-classical N- body approach CoMD should well mimics the statistical model prediction. In this case we can also reasonably believe to the dynamical prediction at short time. Formal definition of coherence and incoherence for dynamical variables And references there in

Degree of coherence for the γ-ray emission

Restoring Pauli Principle through a multi-scattering procedure (branching). N-N scattering processes : Restoring of the total angular momentum conservation with a suitable algorithm which further constrains the equations of motion CoMD-II Model and Isospin interaction:    e a ρF(   G.S. configuration obtained with a cooling-warming procedure producing an effective Fermi motion Main ingredient of the model: T=0 i-singlet states NZ/2 couples V0V0 V1V1 T=1 i-triplet states (N 2 +Z 2 +NZ)/2 couples V 1 >V 0 a sym =(V1-V0)/4>0 Isovectorial interaction at density less than ≈1.5ρ 0 : From deuteron binding energy, low energy nucleon-nucleon scattering experiment Bao-an Li PRL Hp 0) : The European Physical Journal A - Hadrons and Nuclei ISSN: (Print) X (Online) Category: Regular Article - Theoretical Physics DOI: /epja/i EOS  =(ρ n -ρ p )/ρ

U  depends on the normalized Gaussian overlap integrals ρ K,J related to the nucleonic wave packets. For simplicity we consider a compact system with A>>1 Isovectorial interactions (major role), the Coulomb interactions and the Pauli Principle (minor role) produce: Correlation coefficient for the Neutron-Proton dynamics: it is a function of N,Z and of the average overlap integral (self- consistent dynamics) Starting from the same Hp 0) but implemented in a many-body framework we get: Leading factor Source of correlations: For moderate asymmetries β M <0 Effect mainly generated by the most simple cluster the Deuton { U τ Pauli principle U c

Isospin Forces with correlations The last condition asks for a further constraint I.M.F.A No Coulomb, we neglect eventual others high order correlations Usual symmetry term but modified ISOV. “bias term”

Inappropriate for time dependent problems and, most important it masks ISOVECTORIAL bias term Non local approximation for large compact systems Interac. Energy density I.M.F.A. α =0 e a =27MeV a sym =72 MeV >0

Nuclear Matter Simulations

Χ=N-Z However, we obtain always a parabolic dependence as a function of the asymmetry parameter which is able to fit the binding energies of Isobars nuclei α ≈0.15 Even if α≈ we have non negligible effects These effects can be independent from model details

Small correlation effect:  N,Z,  0.1÷0.15; low asymmetry:  Microscopic calculations for the 40 Cl+ 28 Si system at 40 MeV/A b=4 fm Average Isovectorial potential energy per nucleon: Stiff1-2 options produce an attractive behaviour as a function of the density; the soft option has a repulsive behaviour. These effects enhance the sensitivity of several observables from the different options describing the isospin potential. Rather relevant effects in the strength of the effective Isovectorial interaction

Limiting asymmetries  M positive  U  changes its behavior, from attractive to repulsive, for the Stiff options for the system with relevant asymmetry (dot- dashed line). It could be a “fingerprint” of a finite value of  for the stiff potentials.  =(N-Z)/A A=68   black) A=68  =0.31 (red)  0.1 ÷ 0.15 Extrapolated results I.M.F.A.

Different aspects of many-body correlations Many body correlations can describe in hot systems fluctuations around average value. Correlations between fluctuating variable modify the average dynamics For cold systems (static calculations) I.M.F.A.

LIMITING experiment: Isospin effects in the incomplete fusion of 40 Ca+ 40,48 Ca, 46 Ti systems at 25 MeV/A Data from CHIMERA multi-detector Higher probability of Heavy Residues (HR) formation for the system with the higher N/Z ratio ( 40 Ca+ 48 Ca) CoMD-II+GEMINI (statistical decay stage) calculations with the Stiff2 potential confirm the experimental trend CoMD-II calculations enlighten the dynamical nature of the Isospin effect (no statistical decay stage) Many body correlations effect Stiff1 (attractive): Higher HR yield Soft (repulsive): Multifragmentation mechanism Degree of Stiffness of the symmetry interaction Quadratic dependence of the probability distributions from  Average value of the results obtained for the three Different targets: = 1±0.10 Quantify the distance of the Stiff2 parametrization from The best one Sub. for publications in P.R.L.

I) CoMD-II calculations suggest the existence of correlations generated by the Isovectorial U τ interaction (DEUTERON effect ?!). - Because of the microscopic structure (alternate signs ), the Isospin Interaction Term is quite sensitive to A-body correlations and strong affects also simple observables. II) these correlations affect U τ both in magnitude and in sign. – In particular U τ produces, apart from a modified term proportional to  2, an Isovectorial “bias term” (not proportional to  2 ). This last term is responsible for the large differences with respect to I.M.F.A. III) Accordingly, the quantitative results on the investigation about the behavior of U τ could be strongly affected by the main features characterizing the model calculations (M.F. or Molec. Dynamics) Conclusive remarks

iv) Without any other suggestion we have used Form Factor taken fro EOS static calculations, which commonly gives informations only on the symmetry energy (β 2 dependence). v) However, information on the real strength of these correlations should be obtained through well suited experiments (not simple, already performed and approved experiments in Catania with, the CHIMERA detector; R.I.B. can help) vi) What about EOS ab initio calculations…? What about the α parameter? (correlations should be also here) In the presented scenario it is quite important to get information from such kind of theoretical approaches. These information can play a key role in dynamical models.

The Soft option you criticize has been used in a lot of BUUcalculations. These calculations are subjects also of Letters in this review. You quote AMD calculations, in particular the paper Progress. Of Theoretical Phys. 84 No. 5. May 1992, to confute our previous affirmation about the difficulties to reproduce the binding energies of light particles, with phenomenological interactions which are normally used to describe Heavy ion Collisions. This is surprising, in fact in that paper the authors clearly declare that they use an external parameter (external parameter with respect their approach) to reproduce the binding energies of light clusters. This parameter is the so called To, which regulate the way in which the zero point kinetic energy is subtracted. In our opinion, by careful reading the paper, one could also find others introduced parameters, that are related to the way in which the clusters are defined, which could be considered like free (se for example the so called ả). On the other hand it is well known the improvement of the prediction power that models can obtain if free parameters are introduced. Moreover, the authors use an interaction (Volkov) which does not corresponds to the interaction used in AMD calculations to study the Heavy Ion dynamics at Fermi energies (Gogny.Gogny-As).

The nice AMD calculations are very powerful also in predicts cluster structures and their first excited levels, but in these cases other interactions are used and the width of the wave packets is treated as a free parameter in the minimization procedure. However, if you let us to use the degree of freedom which AMD peoples use and declare in an explicit way, we obtain the following results from CoMD-II calculation concerning the binding energies of light particles: Eb=H+E0*X0 E0=zero point kinetic energy fixed to 11.8 MeV X0=reducing fraction T0=E0*X0 H= total energy per nucleon without zero point energy Eb=binding energy per nucleon Soft case With X0=1. we can get: Eb (d)=-1.1 Eb(α)=-4.5 Eb(12 C)=-7.5 Stiff 2 With X0=0.67 we can get: Eb (d)=-1.1 Eb(α)=-7. Eb(12 C)=-7.5 This results are not so bad, also by taking into account that our parameters are regulated to reproduce average binding energies, nuclear radii, GDR frequencies for nuclear masses larger that about 35. Conversely the parameter of the Volkov interaction (five parameters) are chosen to reproduce just the binding energies of the light particles.

Repulsive behaviour as a function of the density for all of the three options. Sensitivity to the Isospin interaction is reduced CoMD-II calculation with no correlation in the symmetry interaction Having obtained the limiting expression of β M in M.F. approximation, we can compare for a compact system with A=68 at density ρ g.s the contribution in the correlated case with the one associated to the M.F case. i.e. β M =-0.43 fm -3 if α=0.1 β M =0.065 fm -3 if α=0 It is enough a quite small value of dynamical correlation to change the sign of β M (and its behavior) and to heavly affect the strength

Isospin equilibration and Dipolar degree of freedom As shown from CoMD-II calculations, a useful observable to globally study the Isospin equilibration processes of the system also in multi-fragmentation processes, is the time derivative of the average total dipole [4]. -it is invariant with respect to statistical processes; -it depends only on velocities and multiplicities of charged particles; -it has a vector character allowing to generalize the equilibration process along the beam and the impact parameter directions; General condition for Isospin equilibration. V1,N1/Z1 V2,N2/Z2 V3,N3/Z3 Light particles THINKING TO THE SYSTEM IN A GLOBAL WAY γ γ γ [4] M. Papa and G. Giuliani, arxiv: v1 [nucl-th]. [5] M. Papa and G. Giuliani, submitted to Phys. Rev C

Neutron-proton relative motion DIFF.- FLOW “GAS”- ”LIQUID” relative Motion. Fragments relative motion.-REACTION DYNAMICS -A simple case as an example: neutrons and protons emission from a hot source. Relative changes of the average total dipolar signals along the beam direction Y G Y L Charge/mass asymmetries for the different “phases” N.L. Approximation

Isospin equilibration and dipolar degree of freedom: local form factors calculations During the first fm/c, when the system is in a compact configuration, the dipolar collective mode can be described by the non-local approximation (see the upper figure). For the system under study, the local form factors calculations produce also remarkable differences in isospin equilibration along the x component (impact parameter). In this case the Soft potential shows a higher degree of isospin equilibration with respect to the Stiff1 potential. In the impact parameter region of mid- peripheral collisions the system shows a higher sensitivity to the different parameterizations of the ISOVECTORIAL potentials. A more detailed analysys on results based on local form factors calculations is still working.

CoMD-II Collective rotation for the biggest fragment J=Average total spin (Collective and non) Error bars represent Dynamical fluctuations

[4] M. Papa and G. Giuliani, arxiv: v1 [nucl-th]. [5] M. Papa and G. Giuliani, submitted to Eur. Phys. Journ. [6] F. Amorini et al ; to be submitted for publication. [1] M. Papa, A. Bonasera and T. Maruyama, Phys. Rev. C (2001). [2] M. Papa G. Giuliani and A. Bonasera, Journ. Of Comp. Phys (2005). [3] G. Giuliani and M. Papa, Phys. Rev. C (R) (2005).