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Diapositiva 1

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Title: Diapositiva 1


1
Isospin-Symmetry interaction and many-body
correlations M. Papa
2
Summary
  • Main ingredient of the model
  • The Isovectorial Interaction
  • Nuclear matter simulations
  • The dynamical case and first applications
  • Conclusive remarks

3
Vloc?eint(?2,)dr eint from EOS ??2dr8?k,l
?k,l
But one needs to exclude the self- energies
kl ??2dr ? ?k?l ?k,l
4
N-N cross-section (from BNV)
Cycle on the generic particle k
Functional form
Double collision excluded
Search for a close particle K with a
distance Dlt2sr
NY
go to another particle k
Ceck mean free path
go to another particle k
go to another particle k
gt
lt
rP
P(vrdt)/(?s)
do the scattering, evaluate blocking factor
fk fk
reset pk,pk
gt1
gt1
rPb
Cuts if sgt55 mb? s55 mb
5
Constraining the phase space
For each particle i, we define an ensemble Ki of
nearest identical particles with distances less
than 3sr ,3sp in phase space
Initialization local cooling-warming-procedure
stabilization
Cycle on particles
No good
?m1.02 mKi
gt
Distribute Z and N particles in phase according
to Fermi gas model
R,B
pi(t)?i(pi(t-dt)Fidt) xi(t)xi(t-dt)vidt
f?i1
RBE
good
?m0.98 mKi
lt
Conf. In memory
Write good configuration
Tstab
Cycle on time tcoolinig
6
Global minimizzation of the total energy under
Pauli constraint
Stable configurations with internal kinetic
energy around 16-18 MeV/A (No solid)
At variance with others Approaches (AMD)
Stable configurations with good Binding
Energy and nuclear radii in large mass interval-
(sr sp)
Au results
Monopole Zero-point motion
7
Constraint in the normal dynamics
Multi-scattering between particles mKi
Cycle on particles
gt
f?i1
MF calculation Numerical integration
gt
Collision routine

Cycle on particles
Cycle on particles
8
The quasi-Fermi distribution Is maintained in
time. With no constraint the energy distribution
rapidly changes in a Boltzman-like one
112Sn
Au
9
Pauli constrant On AuAU at low energy
10
CoMD-II and Impulsive Forces
J.Comp.Phys. N2 403 (2005)
Li1,2(r1-r2)p1i
Lf1,2(r1-r2)p1f
Li1,2?Lf1,2
p1i?p1f p1i p1f
?Lik,m??Lfk,m
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
11
ComD-II results on relative velocities
Multi-break-up processes and related times
12
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13
PLF- fission
A2
A1
A3
A1
A2
A1
A2
A3
A2
A1
A4
A3
time
14
Average Time Scale
At least 1 IMF
At least 3 targets IMF
td
tf
tc
Note at Elt3.5 MeV/A Tevgt800 fm/c for one
nucleon
tcgtgttftd
15
PLF Fission at longer time
16
What about dynamical fission ?
A3
A2
Fplane
A1
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
17
Average Dynamics of the angular momentum
transfer
Fluctuating Dynamics
Partial overlap between Fission time and transfer
of J
Fluctuation of the relative orientation of
the Fission plane and entrance channel plane
Non linear behaviour
18
Experimental Selection criteria to decouple the
two dynamics
Isospin collaboration
19
Hot source Multifr.
Light partner Multifr.
20
Heavy partner Multifr.
21
Formal definition of coherence and incoherence
for dynamical variables
And references there in
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.
22
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23
Degree of coherence for the ?-ray emission
24
CoMD-II Model and Isospin interaction
Main ingredient of the model
G.S. configuration obtained with a
cooling-warming procedure producing an effective
Fermi motion
Restoring Pauli Principle through a
multi-scattering procedure (branching). N-N
scattering processes
The European Physical Journal A - Hadrons and
Nuclei ISSN 1434-6001 (Print) 1434-601X
(Online) Category Regular Article - Theoretical
Physics DOI10.1140/epja/i2008-10694-2
Restoring of the total angular momentum
conservation with a suitable algorithm which
further constrains the equations of motion

Isovectorial interaction at density less than
1.5?0
From deuteron binding energy, low energy
nucleon-nucleon scattering experiment
EOS
Hp 0)
V1
T1 i-triplet states (N2Z2NZ)/2 couples
et?(?)-?(0)ea?F(r/r0)b2
?(?n-?p)/?
T0 i-singlet states NZ/2 couples
V0
asym(V1-V0)/4gt0
V1gtV0
Bao-an Li PRL 78 1997
25
Starting from the same Hp 0) but implemented in a
many-body framework we get
Ut depends on the normalized Gaussian overlap
integrals ?K,J related to the nucleonic wave
packets.
For simplicity we consider a compact system with
Agtgt1
Leading factor

Ut Pauli principle Uc

Isovectorial interactions (major role) , the
Coulomb interactions and the Pauli Principle
(minor role) produce
Source of correlations
Effect mainly generated by the most simple
cluster the Deuton
For moderate asymmetries ßMlt0
Correlation coefficient for the Neutron-Proton
dynamics it is a function of N,Z and of the
average overlap integral (self-consistent
dynamics)
26
Isospin Forces with correlations
No Coulomb, we neglect eventual others high order
correlations
ISOV. bias term
Usual symmetry term but modified
I.M.F.A
The last condition asks for a further constraint
27
Inappropriate for time dependent problems and,
most important it masks ISOVECTORIAL bias term
gt0
Non local approximation for large compact systems
Interac. Energy density
ea27MeV asym72 MeV
I.M.F.A.
a0
28
Nuclear Matter Simulations
29
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
?N-Z
a0.15
Even if a0.03-0.02 we have non negligible
effects
These effects can be independent from model
details
30
Microscopic calculations for the 40Cl28Si system
at 40 MeV/A b4 fm
Small correlation effect a(N,Z,r)0.10.15 low
asymmetry b0.088
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
31
Limiting asymmetries
b(N-Z)/A A68 b0.088 (black) A68 b0.31
(red) a0.10.15 Extrapolated results
I.M.F.A.
bM positive Ut 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 a for the stiff potentials.
32
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.
33
LIMITING experiment Isospin effects in the
incomplete fusion of 40Ca40,48Ca, 46Ti systems
at 25 MeV/A
Data from CHIMERA multi-detector
Sub. for publications in P.R.L.
Higher probability of Heavy Residues (HR)
formation for the system with the higher N/Z
ratio (40Ca48Ca)
CoMD-IIGEMINI (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)
Degree of Stiffness of the symmetry interaction
Many body correlations effect
Stiff1 (attractive) Higher HR yield
Quadratic dependence of the probability
distributions from d. Average value of the
results obtained for the three Different
targets
ltgexpgt 10.10 Quantify the distance of
the Stiff2 parametrization from The best one
Soft (repulsive) Multifragmentation mechanism
34
Conclusive remarks
I) CoMD-II calculations suggest the
existence of correlations generated by the
Isovectorial Ut 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 Ut both in magnitude
and in sign. In particular Ut 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 Ut could
be strongly affected by the main features
characterizing the model calculations (M.F. or
Molec. Dynamics)
35
iv) Without any other suggestion we have used
Form Factor taken fro EOS static calculations,
which commonly gives informations only on the
symmetry energy (ß2 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 a 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.
36
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 ?). 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).
37
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 EbHE0X0 E0zero point
kinetic energy fixed to 11.8 MeV
X0reducing fraction
T0E0X0 H total
energy per nucleon without zero point energy
Ebbinding energy
per nucleon Soft case With X01. we can get Eb
(d)-1.1
Eb(a)-4.5 Eb(12 C)-7.5 Stiff 2 With
X00.67 we can get Eb (d)-1.1 Eb(a)-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.
38
CoMD-II calculation with no correlation in the
symmetry interaction
Repulsive behaviour as a function of the density
for all of the three options. Sensitivity to the
Isospin interaction is reduced
Having obtained the limiting expression of ßM in
M.F. approximation, we can compare for a compact
system with A68 at density ?g.s the contribution
in the correlated case with the one associated to
the M.F case. i.e. ßM-0.43 fm-3 if
a0.1 ßM0.065 fm-3 if a0 It is enough a quite
small value of dynamical correlation to change
the sign of ßM (and its behavior) and to heavly
affect the strength
39
Isospin equilibration and Dipolar degree of
freedom
THINKING TO THE SYSTEM IN A GLOBAL WAY
?
?
?
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
4 M. Papa and G. Giuliani, arxiv0801.4227v1
nucl-th. 5 M. Papa and G. Giuliani,
submitted to Phys. Rev C
General condition for Isospin equilibration.
40
  • A simple case as an example neutrons and protons
    emission from
  • a hot source.

YG YL Charge/mass asymmetries for the different
phases
N.L. Approximation
Neutron-proton relative motion DIFF.-FLOW
GAS-LIQUID relative Motion.
Fragments relative motion.-REACTION DYNAMICS
Relative changes of the average total dipolar
signals along the beam direction
41
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.
Preliminary results
42
CoMD-II Collective rotation for the biggest
fragment
JAverage total spin (Collective and non)
Error bars represent Dynamical fluctuations
43
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44
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45
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46
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).
4 M. Papa and G. Giuliani, arxiv0801.4227v1
nucl-th. 5 M. Papa and G. Giuliani,
submitted to Eur. Phys. Journ. 6 F. Amorini et
al to be submitted for publication.
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