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

46
Isospin-Symmetry interaction and many-body correlations M. Papa

description

Isospin-Symmetry interaction and many-body correlations M. Papa. Sez.-CT. Sez.-CT. Summary. Main ingredient of the model The Isovectorial Interaction Nuclear matter simulations The dynamical case and first applications Conclusive remarks. Sez.-CT. V loc = ∫ e int ( ρ 2 ,…)dr - PowerPoint PPT Presentation

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

Page 1: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

Isospin-Symmetry interaction

and many-body

correlations

M. Papa

Page 2: 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

Page 3: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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

Page 4: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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

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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

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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)

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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

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112Sn

The quasi-Fermi distributionIs maintained in time.

With no constraint the energydistribution rapidly changesin a Boltzman-like one

Au

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Pauli constrantOn Au+AU at low energy

Page 10: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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

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

related times

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PLF- fission

A1

A1

A2A1A1

A3

A2

A2 A3

A1A2

A3

A4time

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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

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PLF Fission at longer time

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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

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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

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Experimental Selection criteria to decouple the twodynamics

Isospin collaboration

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Hot source Multifr. Light partner Multifr.

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Heavy partner Multifr.

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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

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Degree of coherence for the γ-ray emission

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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

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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

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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

aU

ρρρ

ˆˆ

ˆˆ

ˆˆˆ

..LN

No Coulomb, we neglect eventual others highorder correlations

Usual symmetry term but modified ISOV. “bias term”

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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

aU

ˆ

ˆˆeff

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Nuclear Matter Simulations

Page 29: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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

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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

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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.

Page 32: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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.

Page 33: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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.

Page 34: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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

Page 35: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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.

Page 36: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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).

Page 37: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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.

Page 38: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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

Page 39: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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

Page 40: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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

Page 41: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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.

Page 42: Isospin-Symmetry interaction  and  many-body  correlations M. Papa

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

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[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).