A MODEL FOR PROJECTILE FRAGMENTATION

Collaborators: S. Mallik, VECC, India S. Das Gupta, McGill University, Canada

1

Gargi Chaudhuri

Different Stages of Projectile Fragmentation Our Model Results Comparison with different experimental data Summary

CONTENTS

2

PROJECTILE FRAGMENTATION (Different Stages)

Collision of the projectile & target nuclei above certain energy (> 100 MeV/n)

(COLLISION)

Part of the projectile goes into the participant & remaining part (projectile spectator or PLF) gets sheared off (ABRASION)

Hot, abraded PLF (As, Zs) expands to about 3V0 – 4V0.(V0-normal nuclear

volume) & breaks up into many fragments (MULTIFRAGMENTATION)

The excited fragments de-excite by sequential decay (EVAPORATION)

3

Projectile Projectile

Multi-fragmentationMulti-fragmentation EvaporationEvaporation Target Target

AbrasionAbrasion

Pictorial Scenario

PLFProjectile fragmentation

4

Abrasion Stage Calculation

Abrasion Cross section

)(2 ,,, iZNiZNa bbPbSSSS

We use an impact parameter dependent temperature profile T(b) for the PLF

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Ni58+Be9

Ni58+Ta181

Sn124+Sn119

AS/A

0

(b-bmin

)/(bmax

-bmin

)

PLF size for different reactions

Overlapping volume V(b) (participant region) of projectile & target using straight-line geometry

( for different impact parameter b )

0

00

0

)( )( NV

bVbNZ

V

bVbZ s

Ss

S

PLF Size : average number of proton (<ZS>) and neutron (<NS> )

Probability of formation of PLF (Ns ,Zs)

)()()(, bPbPbPSSSS ZNZN

using minimal distribution

Vs(b)=V0-V(b)

5

Ref: S. Mallik, G.Chaudhuri & S. Das Gupta Phys. Rev. C 83 (2011) 044612

T is independent of projectile beam energy T depends on impact parameter (b).

Temperature of PLF

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Ni58+Be9

Ni58+Ta181

Sn124+Sn119

AS/A

0

(b-bmin

)/(bmax

-bmin

)0.0 0.2 0.4 0.6 0.8 1.02

3

4

5

6

7

8

T

emp

erat

ure

(M

eV)

(b-bmin

)/(bmax

-bmin

)

)(

)(5.45.7)(

minmax

min

bb

bbbT

T independent of As/Ao

For all reactions

As(b)/A0

Simplest parametrization

From many sets of experimental data

]/)([ 5.45.7)( 0AbAbT S

]/)([)( 0AbATbT S

0.0 0.2 0.4 0.6 0.8 1.02

3

4

5

6

7

8

Ni58+Be9

Ni58+Ta181

Sn124+Sn119

Tem

per

atu

re (

MeV

)

(b-bmin

)/(bmax

-bmin

)

It depends upon the wound of the original projectile which is (1.0 – As/A0)

6

Multi-fragmentation StageHigh

excitation energy

Expansion

Density fluctuation

Breaking into composites and

nucleons

Thermodynamic Equilibrium

@ freeze-outHot primary fragments production

PLF(As,Zs)

Canonical Thermodynamical Model (CTM)

Evaporation Stage :-(based on Monte-Carlo Simulation)

Weisskopf’s evaporation theory Decay Channels:- p, n, α, d, t, 3He, γ

Hot primary fragments

Evaporation Model

Cold Secondary fragments

7

Ref: G.Chaudhuri & S.Mallik Nucl. Phys. A 849 (2011) 190

Ref: C. B. Das , S. Das Gupta et al. , Phys . Rep. 406 (2005) 1

Canonical Thermodynamical Model (CTM)

) ()(! ,,

, ,

,,

,

Sjij

Sjiiji ji

nji

ZN ZnjNnin

Qji

SS

Baryon & charge conservation constraints

ni,j=No of fragments with i neutrons & j protons

Canonical Partition function of PLF AS (ZS ,NS)

ωi,j=Partition function of the fragment ni , j

Computationally difficult !

Recursion relation An exact computational method which

avoids Monte Carlo by exploiting some

properties of the partition function

Most important feature of our model

Possible to calculate partition function of very large nuclei within seconds

Crux of the model

8

CTM contd…Partition function of the fragment ni ,j

})(

31

)({1

exp 0

2223

2

0, aT

a

jis

a

ikaTaW

Tq ji

Intrinsic part of the partition function

jiji qTjimh

V,

2/33, } )( 2{

translational intrinsic

Liquid drop model Fermi-gas model

Average no. of composites {i,j} or Multiplicity

NZ

jNiZjiji Q

Qn

,

,,,

SS

iSS

iSS

iZN

TZNaTZN

ZNTZNm n,

,,,,,

,,,, Cross-section after multi-fragmentation stage:-

abrasionmultifragmentation 9

Model summary………..

Results………

Different

Target-projectile combinations

Incident energy

Observables

Comparison with experimental data

10

observablesAs(b) Zs(b)

CTM +evaporatio

n

Abrasion Model

PLF size

Freeze-out volume=3V0

]/)([5.45.7)( 0AbAbT SProjectile size (A0,Z0 )& target size (At, Zt )

Comparison of theoretical and experimental temperature profile

Experimental Temperature Profile

By isotope thermometry method

Good agreement

0 10 20 30 400

2

4

6

8

10

12

0 10 20 30 40 500

2

4

6

8

10

12107Sn+119Sn

Te

mp

era

ture

(M

eV

)

Zbound

124Sn+119Sn

solid lines modelSquares with error bars data

= ZS - No. of Z=1 fragments

Zbound

Experiment:- 600 MeV/nucleon (ALADIN @GSI) 107Sn and 124Sn on natural Sn

11

0 10 20 30 400.0

0.5

1.0

1.5

2.0

2.5

<M

IMF>

107Sn+119Sn

0 10 20 30 40 500.0

0.5

1.0

1.5

2.0

2.5124Sn+119Sn

Zbound

dashed lines modelsolid lines data

Variation of IMF multiplicity with Zbound

IMF size: 3 ≤ Z ≤ 20

Nice agreement with data

Experiment :- 600 MeV/nucleon (ALADIN @GSI) 107Sn and 124Sn on natural Sn

12

Differential Charge Distribution in Projectile Fragmentation

Lower Zbound range

higher T of PLF breaks into many fragments of very small charge.

Steeper Charge distribution

Higher Zbound range

Lower T of PLF fragmentation is less, both low & high Z fragments

“U” shaped Charge distributiondashed lines modelsolid lines data

Experiment:- 600 MeV/nucleon (ALADIN)

(At different Zbound intervals)

13

0 10 20 30 4010-6

10-3

100

103

106

109

1012

0 10 20 30 40 5010-6

10-3

100

103

106

109

1012

x105

x102

x100

x10-2

x10-4

zbound

/z0=0.0-0.2

zbound

/z0=0.2-0.4

zbound

/z0=0.4-0.6

zbound

/z0=0.6-0.8

zbound

/z0=0.8-1.0

Proton Number(Z)

Cro

ss

-se

cti

on

(m

b)

107Sn+119Sn124Sn+119Sn

zbound

/z0=0.0-0.2

zbound

/z0=0.2-0.4

zbound

/z0=0.4-0.6

zbound

/z0=0.6-0.8

zbound

/z0=0.8-1.0

x105

x102

x100

x10-2

x10-4

0.0 0.2 0.4 0.6 0.80.0

0.2

0.4

0.6

0.8

1.0107Sn+119Sn

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0124Sn+119Sn

<Z

max

>/Z

0

Zbound

/Z0

Largest Cluster in Projectile Fragmentation

)(Pr ,1

max , mNZ

Zz

zm zzZ

ss

sm

m

sNsZ

Average size of largest cluster

dashed lines modelsolid lines data

Experiment :- 600 MeV/nucleon (ALADIN @GSI) 107Sn and 124Sn on natural Sn

Nice agreement

with experiment

Probability that zm is the largest cluster

14

Charge Distribution in Projectile Fragmentation58Ni+9Be

140 MeV/nucleon (MSU)

136Xe+208Pb 1 GeV/nucleon (GSI)Experimentally

Different Beam EnergyTheoretically

Same Temperature Profile

5 10 15 20 25 3010-1

100

101

102

103

104

Cro

ss

-se

cti

on

(m

b)

Proton Number (Z)

10 20 30 40 50 6010-1

100

101

102

103

104

Proton Number (Z)

Cro

ss

-se

cti

on

(m

b)

5 10 15 20 25 3010-1

100

101

102

103

104

Cro

ss

-se

cti

on

(m

b)

Proton Number (Z)

40 44 48 52 5610-1

100

101

102

103

104

Proton Number (Z)C

ros

s-s

ec

tio

n (

mb

)

58Ni+181Ta 140 MeV/nucleon (MSU)

129Xe+27Al 790 MeV/nucleon (GSI)

15

Ref: S. Mallik, G.Chaudhuri & S. Das Gupta Phys. Rev. C 84 (2011) 054612

The trend is nicely reproduced for all the reactions

dashed lines modelsolid lines data

Isotopic Distribution in Projectile Fragmentation

10-7

10-5

10-3

10-1

101

103

Z=6 Z=9

Z=12

10-7

10-5

10-3

10-1

101

103

Z=15

0 4 8 1210-7

10-5

10-3

10-1

101 Z=18

Cro

ss

-se

cti

on

(m

b)

0 4 8 12

Z=21

0 4 8 12

Z=24

Neutron Excess (N-Z)

0 4 8 1210-7

10-5

10-3

10-1

101

Z=27

58Ni+9Be 140 MeV/nucleon (MSU Experiment)

Circles joined by dotted lines modelSquares with error bars data

Nice agreement with data

16

SUMMARY The model for projectile fragmentation is grounded in traditional concepts of

heavy-ion reaction (abrasion) plus the well known model of multifragmentation (Canonical Thermodynamical Model).

The model is in general applicable and implementable above a certain beam energy.

Universal temperature profile (depending on impact parameter) is introduced as input for different target-projectile combinations & widely varying energy of the projectile.

The model is able to successfully reproduce a wide variety of experimental observables like charge & mass distribution, isotopic distributions, IMF multiplicity, size of largest cluster .

Microscopic BUU calculations is being done in order to estimate the size & excitation of the initial PLF at different impact parameters.

The work is in progress…….

17

18

Fluctuation in number of IMFs for small Projectile like fragments:-

Black solid lines dataRed dotted lines direct calculation

Zbound=ZS- No. of Z=1 fragment

Zbound=Non-integer

0 10 20 30 400.0

0.5

1.0

1.5

2.0

2.5

<M IM

F>

0 10 20 30 40 500.0

0.5

1.0

1.5

2.0

2.5

Zbound

Experiment :-

Zbound=Integer

(Due to event by event measurement)

Theoretical Calculation :-

(Due to average no. of fragment calculation)

1 2 3 4 5-0.25

0.00

0.25

0.50

0.75

1.00

1.25

<MIM

F>

1 2 3 4 5 6-0.25

0.00

0.25

0.50

0.75

1.00

1.25

Zbound

Sn107+Sn119 Sn124+Sn119

Zbound=3

<MIMF>=1

Zbound=5

<MIMF>=1

Zbound=4

MIMF=1MIMF=0

<MIMF> is calculated by modifying the CTM with experimental decay scheme of different energy levels.

Fluctuation contd…

Black solid lines dataRed dotted lines direct calculationBlue triangles modified calculation

Sn107+Sn119 Sn124+Sn119

1 2 3 4 5-0.25

0.00

0.25

0.50

0.75

1.00

1.25

<MIM

F>

1 2 3 4 5 6-0.25

0.00

0.25

0.50

0.75

1.00

1.25

Zbound

AX3 BX3 +neutron(s)AX4 BX3 +neutron(s)+protonAX5 BX3 +neutron(s)+2 protons……

AX5 BX5 +neutron(s)AX5 BX3 +CHe2 +neutron(s)……

AX4 BX4 +neutron(s)AX4 CHe2 +DHe2 +neutron(s)……

24

0 4 80.8

1.0

1.2

1.4

Zbound

/Z0

=0.2-0.4

0 4 8

Zbound

/Z0

=0.4-0.6

Proton Number(Z)

<N

>/Z

0 4 8 120.8

1.0

1.2

1.4Z

bound/Z

0

=0.6-0.8