a model for projectile fragmentation collaborators: s. mallik, vecc, india s. das gupta, mcgill...
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A MODEL FOR PROJECTILE FRAGMENTATION
Collaborators: S. Mallik, VECC, India S. Das Gupta, McGill University, Canada
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Gargi Chaudhuri
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Different Stages of Projectile Fragmentation Our Model Results Comparison with different experimental data Summary
CONTENTS
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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)
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Projectile Projectile
Multi-fragmentationMulti-fragmentation EvaporationEvaporation Target Target
AbrasionAbrasion
Pictorial Scenario
PLFProjectile fragmentation
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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
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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)
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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
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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
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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
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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 )
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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
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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
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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
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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
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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
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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
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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…….
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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
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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)……
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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