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Summary
• Main ingredient of the model
• The Isovectorial Interaction
• Nuclear matter simulations
• The dynamical case and first applications
• Conclusive remarks
Vloc=∫eint(ρ2,…)dr
eint from EOS
=∫ρ2dr∞∑k,l ρk,l
But one needs to exclude the self-energies k=l
=∫ρ2dr → ∑k≠l ρk,l
mb 34.5σ
mb E
887.4 E0.0533.54σ
MeV E310
mb E
239380
E
180227.5σ
mb E
93074
E
11.14822.5σ
MeV 310E40
mb E
5057-
E
9069.26.95σ
mb E
11748-
E
3088.55.3σ
MeV 40E25
mb 381 σ
mb 80.6 σσ
MeV 25E0
np
rrnn
r
2rr
np
2rr
nn
r
2rr
np
2rr
nn
r
np
ppnn
r
Cuts: if σ>55 mb→ σ=55 mb
Functional formN-N cross-section (from BNV)
Search for a close particleK’ with a distance
D<2σr
N:Y
Cycle on the generic particle k
go to anotherparticle k
Ceck mean free path
P=(vr∙dt)/(ρσ)r:P
<>
go to anotherparticle k
do the scattering, evaluateblocking factor: fk fk’
r:Pb
>1 >1
resetpk,pk’
go to anotherparticle k
Double collisionexcluded
Constraining the phase space
Initialization local cooling-warming-procedure stabilization
For each particle i, we define an ensemble Ki of nearest identical particles with distances less than 3σr ,3σp in phase space
Distribute Z and Nparticles in phaseaccording to Fermi gas model
fi:1
>
<
γm=1.02 m«Ki
γm=0.98 m«Ki
pi(t)=γi(pi(t-dt)+Fidt)
xi(t)=xi(t-dt)+vidt
Cycle on particles
Cycle on time tcoolinig
R;BE
No good
Tstab
R,B
good
Write good configuration
Conf.In memory
Global minimizzation of the total energy under “Pauli constraint”Stable configurations with internal kinetic energyaround 16-18 MeV/A (No solid)
Stable configurations with good Binding Energyand nuclear radii in large mass interval- (σr σp)
Au results
MonopoleZero-pointmotion
At variance with others Approaches(AMD)
Constraint in the normal dynamics
MF calculation Numerical integration
Cycle on particles
fi:1>
Multi-scattering between particles m«Ki>
Cycle on particles
Collisionroutine
Cycle on particles
≤
112Sn
The quasi-Fermi distributionIs maintained in time.
With no constraint the energydistribution rapidly changesin a Boltzman-like one
Au
Pauli constrantOn Au+AU at low energy
CoMD-II and Impulsive Forces
J.Comp.Phys. N2 403 (2005)
Li1,2=(r1-r2)^p1
i Lf1,2=(r1-r2)^p1
f
p1i≠p1
f
p1i = p1
f
Li1,2≠Lf
1,2
∑Lik,m≠∑Lf
k,m
ωL Ikkk m rωpp ^'
k
kkkkk r 2
'''' )( rrp
pp
k C
kkk N
''''''' p
pp
Ck
k Tm2
'''2p
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
|||| 21
21 ir
fr
ir
fr pp
rr
rrpp
2int4
1R
d
d
Trivial solution
ComD-II results on relative velocities Multi-break-up processes and
related times
PLF- fission
A1
A1
A2A1A1
A3
A2
A2 A3
A1A2
A3
A4time
Average Time Scale
At least 1 IMF At least 3 target’s IMF
td tf tc
tc>>tf+td
Note; at E*<3.5 MeV/A Tev>800 fm/c for one nucleon
PLF Fission at longer time
What about dynamical fission ?
A
vr ik
i
ik
k
^
Nn
n
220 i
iiyy zxmI
yy
yeffyy
JI
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 40-150 fm/c
ΦplaneA1
A3A2
Average Dynamics of the angular momentum transfer
Fluctuating Dynamics
Non linear behaviour
Partial overlap between Fission time andtransfer of J
Fluctuation of the relative orientation of theFission plane and entrance channel plane
Experimental Selection criteria to decouple the twodynamics
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 coherenceand incoherence for dynamical variables
Z
kkrtD
1
)(
N
tD
tD
N
s
s 1
)(
)(
IncoherentD
CoherentDf
dt
tdDtV
)()(
0t)(Df
)()()( tDtDtD fi
2
k
k2c
dt
)E(Vd
cE6
e4
dE
dP
2
k
fk
2i
dt
)E(dV
cE6
e4
dE
dP
dE
dP
dE
dP
dE
dP ictot
dte c
-iEttm
0 dt
dV
dt
dV(E) dEdE
dPE
dt
d
0
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:
)2)((8
1)
8
1
8
3( ,,,
01,
01pnppnnaa VVVVV
Restoring of the total angular momentum conservation with a suitable algorithm which further constrains the equations of motion
CoMD-II Model and Isospin interaction:
eaρF(
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 V0
V1T=1 i-triplet states (N2+Z2+NZ)/2
couples
V1>V0asym=(V1-V0)/4>0
Isovectorial interaction at density less than ≈1.5ρ0:
From deuteron binding energy, low energy nucleon-nucleon scattering experiment
Bao-an LiPRL 78 1997
Hp 0)
: The European Physical Journal A - Hadrons and Nuclei ISSN: 1434-6001 (Print) 1434-601X (Online)Category: Regular Article - Theoretical PhysicsDOI:10.1140/epja/i2008-10694-2
EOS
=(ρn-ρp)/ρ
A
1=kj kj,ρ3A
4=S (S)F'=F'
(N.L.) Factors Formfor ionApproximat Local Non
;A
1=jk kj,ρ=JS
Soft1/2
g.sSJS
=(J) F'Stiff2; 1=(J) F'Stiff1; JS+gsS
J2S=(J)F'
uF(u)F'
1)]-A
1=kj kτ,
jτ(2δkj,[ρF'
02ρsyma
=τN.L.U ; 1)]-
A
1=kj kτ,
jτ(2δkj,(J)[ρF'
02ρsyma
=τU
nucleons ofcouple for integral overlapAverage I'I,ρ
PN,ρ2NZPP,ρ2ZN,Nρ2NMβ ;Mβ F'
02syma
N.L.U
~
2~~~
U depends on the normalized Gaussian overlap integrals ρK,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:
ρ~α)(1)2
ρ~ ρ~α)((1ρ~ that so ρ~ ,ρ~ρ~
PP,NN,
PN,PP,NN,PN,
ρ~Z,N,αα 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 frameworkwe get:
Leading factor
Source of correlations:
For moderate asymmetries βM<0
Effect mainly generatedby the most simple cluster
the Deuton
{Uτ
Pauli principle
Uc
Isospin Forces with correlations
The last condition asks for a furtherconstraint
I.M.F.A
4202
22
0
symτ
PPNN
..)]ρα(0,2
1)β|
β
)ρα(β,
4
1)ρα(0,
2
1[(1F'Aρ
2ρ
aU
ρρρ
ˆˆ
ˆˆ
ˆˆˆ
..LN
No Coulomb, we neglect eventual others highorder correlations
Usual symmetry term but modified ISOV. “bias term”
Inappropriate for time dependent problems and, most importantit masks ISOVECTORIAL bias term
Non local approximation for large compact systems
Interac. Energydensity
I.M.F.A. α=0ea=27MeV asym=72 MeV
>0
202
22
0
symτ )β|β
)ρα(β,
4
1)ρα(0,
2
1[(1F'Aρ
2ρ
aU
ˆ
ˆˆeff
Nuclear Matter Simulations
Χ=N-Z
However, we obtain always a parabolic dependence
as a function of the asymmetry parameterwhich is able to fit the binding energies of
Isobars nuclei
α≈0.15
Even if α≈0.03-0.02 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 40Cl+28Si system at 40 MeV/A b=4 fm
ρs
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
Uchanges 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.
Msg
A
Msg
A
FS
aU
FS
aU
'2
'
'2
..
0
..
0
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
AA UU '
0,' MFC
For cold systems (static calculations)
I.M.F.A.
LIMITING experiment: Isospin effects in the incomplete fusion of 40Ca+40,48Ca, 46Ti 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 (40Ca+48Ca)
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
γ)g.s.(s/s)g.s.F(s/sStiff1 0.5δ
Stiff2 0δSoft 0.5δ
small is g.s.)/sg.s.s(s if ,g.s.)/sg.s.s(sδ1)g.s.(s/sF'
Quadratic dependence of the probability distributions from Average value of the results obtained for the three Different targets:
<exp>= 1±0.10 Quantify the distance of the Stiff2 parametrization from The best one
Sub. for publicationsin P.R.L.
I) CoMD-II calculations suggest the existence of correlations generated by the Isovectorial Uτ 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 Uτ both in magnitude and in sign. – In particular Uτ 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 thebehavior of Uτ 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 (β2 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 theα 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 thatthey use an external parameter (external parameter with respect theirapproach) 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 nucleonSoft caseWith X0=1. we can get:Eb (d)=-1.1 Eb(α)=-4.5Eb(12 C)=-7.5Stiff 2
With X0=0.67 we can get:Eb (d)=-1.1Eb(α)=-7.Eb(12 C)=-7.5This 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 βM in M.F.approximation, we can compare for a compact systemwith A=68 at density ρg.s the contribution in the correlatedcase with the one associated to the M.F case. i.e.βM=-0.43 fm-3 if α=0.1βM=0.065 fm-3 if α=0
It is enough a quite small value of dynamical correlationto change the sign of βM (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;
0V
General condition for Isospin equilibration.
V1,N1/Z1
V2,N2/Z2 V3,N3/Z3
Light particles
Light particles
THINKING TO THE SYSTEM IN A GLOBAL WAY
)(tV
γ
γ
γ
[4] M. Papa and G. Giuliani, arxiv:0801.4227v1 [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
YG YL Charge/mass asymmetries for the
different “phases”
N.L. Approximation
Isospin equilibration and dipolar degree of freedom: local form factors calculations
During the first 150-200 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)
A
vr ii
i
C
^
Nn
n
Error bars representDynamical fluctuations
[4] M. Papa and G. Giuliani, arxiv:0801.4227v1 [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. C64 026412 (2001).[2] M. Papa G. Giuliani and A. Bonasera, Journ. Of Comp. Phys. 208 406-415 (2005).[3] G. Giuliani and M. Papa, Phys. Rev. C73 03601(R) (2005).
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