Gargi Chaudhuri

Variable Energy Cyclotron Centre India

NuSYM11

Examining the different functional relationships between the isoscaling and

isotopic observables and symmetry

energy coefficient csym in the framework

of the canonical thermodynamical model.

Study the effects of secondary decay on the observables sensitive to symmetry energy

Formation of highly excited nuclear system (high E* & internal pressure)

expansion break-up into hot de-excitation cold secondary (density fluctuations) primary fragments fragments

Thermodynamic equilibrium

prior to break up at < 0

Intensiveexchange of mass, charge, energy during expansion

STATISTICAL MODELS

Statistical Multifragmentation Models

Canonical Thermodynamical Model (CTM)(Subal Das Gupta et al.)(A grand canonical version is also there)

No Monte Carlo(much simpler)

Monte Carlo simulations

probability of a break-up channel (final state)

statistical weight in the available phase-space

Canonical partition function of a nucleus A0 (N0,Z0)

all partitions

) ()(! 0,0,

}{ , ,

,,

,

00ZnjNni

nQ ji

jji

ip ji ji

nji

ZN

ji

Baryon & charge conservation

Computationally difficult

An exact computational method

which

avoids Monte Carlo by exploiting

some

properties of the partition functionMost important feature of our model

Possible to calculate partition function of very

large nuclei within seconds

ji

jZiNjiZN QiN

Q,

,,, 0000

1

Crux of the model

No of composites with i neutrons and j protons

Partition function of the composite {i,j}

ni,jwi,j

Recursion relation

For A > 4 Liquid drop formula

Interaction

})(

)({1

exp0

22

3/1

23

2

0, aT

a

jis

a

ikaTaW

Tq ji

translational Intrinsic

jiji qTjimh

V,

2/33, } )( 2{

Partition function of the fragment {i,j}

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

The available volume is Vf = V(freeze-out volume) – V0 (volume of

A0)

No interaction between composites except for Coulomb and

excluded-volume correction.

Coulomb interaction between composites through Wigner Seitz

approximation

CTM contd...…

Csym=23.5 MeV Constant Csym as input Fragments at normal density

A0, Z0, ,T, /0

Average no. of primary fragments

Q N , Z for all N,Z

The evaporation model is based on Monte-Carlo Simulation.

Weisskopf’s evaporation theory used. Decay Channels:- p, n, α, d, t, He3, γ, fission Outputs:-Secondary fragments

Evaporation Model

00

00

,

,,,

ZN

jZiNjiji Q

Qn Secondary

fragments

CanonicalThermodynamical

Model

Ref: G.Chaudhuri et. al. Nucl. Phys. A 849 (2011) 190

CTM contd…Inputs

Outputs

hot cold

2

2

2

2

1

14AZ

AZ

TCsym

Tn

Isoscaling parameter

Grand canonical model

)exp(),(

),(),(

1

221 ZNc

ZNY

ZNYZNR

])()[(1

exp),(21 ZNT

cZNR pn

Two reactions 1 & 2 have different isospin asymmetry but nearly same T

Formula from EES model

(Formula 1)

(related to n)

Isoscaling (source) Formula

M.B. Tsang et. al., PRC 64, 054615 (2001)

3 6 9 12 150.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Cin

sym=15.0 MeV

n (

MeV

)

T (MeV)

A1=168 A

2=186

Z1=75 Z

2=75

Cin

sym=23.5 MeV

Eq. 1

Eq. 1

The curves with different values of Cs approach each other as T rises

Extracting Cs from n using Eq. 1 difficult because of T dependence

n increases with T from our model ; it is a constant in formula 1.

Formula 1 cannot be exact at finite T

112/124Sn on 112/124 Sn central collisions

2

2

2

2

1

14AZ

AZ

TCsym

• T 0

• At very high T

n must be an evolving f(n) of T from T =0 to very high T

leads to formula 1

n= T ln[N0 (2)/N0

(1)]

• No simple formula at any general T

(not a function of Csym)

Formula 1 does not have any explicit T dependence

Similar result from Percolation Model

-1 0 1 2 3 4 5 6 7 8 9 1010-1

100

101

R21

(N,Z

)

Neutron Number (N)

Z=5Z=4

Z=3

Z=2Z=1

-1 0 1 2 3 4 5 6 7 8 9 1010-1

100

101

R21

(N,Z

)

Neutron Number (N)

Z=5Z=4

Z=3

Z=2Z=1

3 6 9 12 15 18 21 24 2710-3

10-2

10-1

100

101

102

103

104

R21

(N,Z

)

Neutron Number (N)

Z=14

Z=12Z=10

Z=8

Z=6

-1 0 1 2 3 4 5 6 7 8 9 1010-1

100

101

Neutron Number (N)

R21

(N,Z

)

Z=1

Z=2 Z=3Z=4

Z=5

3 6 9 12 15 18 21 24 2710-3

10-2

10-1

100

101

102

103

104

R21

(N,Z

)

Neutron Number (N)

Z=14

Z=12Z=10

Z=8Z=6

Primary

3 6 9 12 15 18 21 24 2710-3

10-2

10-1

100

101

102

103

104

R21

(N,Z

)

Neutron Number (N)

Z=14

Z=12Z=10

Z=8

Z=6

Secondary

112/124Sn on 112/124 Sn central collisions

Isoscaling approximately valid

for secondary fragments; improves for higher Z

values

Isoscaling nicely valid for primary fragments linear slope, can be

extracted easily

A1=168 , A2=186 Z1=Z2=75

Isoscaling in CTM

3 4 5 6 7 80.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 Primary Secondary

Iso

sc

ali

ng

Pa

ram

ete

r(

)Temperature (MeV)

0 3 6 9 12 150.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Iso

sc

ali

ng

Pa

ram

ete

r(

)

Temperature (MeV)

From Slope of the ratios From

n

calculated from the slope of the ratios

Variation of Isoscaling Parameter With Temperature

α from slope of the ratios & from n match very well

Isoscaling parameter decreases marginally after evaporation from T=4 MeV

A1=168 , A2=186 Z1=Z2=75

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

Is

os

ca

lin

g P

ara

me

ter(

)

Input Symmetry Energy (Cin

sym)

A1=168 , A2=186 Z1=Z2=75

Variation of Isoscaling Parameter With Input Symmetry energy for primary and secondary fragments

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

Is

os

ca

lin

g P

ara

me

ter(

)

Input Symmetry Energy (Cin

sym)

T=5 MeV black lineT=7 MeV red line

Secondary solid linePrimary dotted line

7 MeV

5 MeV

vs cinsym almost linear

for primary fragments. Temperature independent for very low Cin

sym. becomes less sensitive to Csym

after secondary decay, especially for higher T

Similar results from SMM

Extraction of symmetry energy from

isoscaling analysis should be done

cautiously.

2

2

2

1

4

)()(

AZ

AZ

TZZCsym

A. Ono. et al., PRC68, 051601(R) (2003)

α(z) isoscaling slope parameter of a fragment of charge z

<Ai> average mass number of

a fragment of charge z produced by source i(=1,2)Assumptions

1. Isotopic distributions are essentially Gaussian.

2. Free energies contain only bulk terms.

Approximate grand canonical expression connecting csym with Z/<A> of fragments

This formula tested with the canonical thermodynamical model coupled with an evaporation code. Results presented for both the primary(at beak up stage) fragments and for the secondary fragments (after evaporation).

Isoscaling fragment formula (Formula 2)

(Formula 2)

]

),((exp[),( 1,,,,

ZN

ZNFZZNY

pn

pnpn

)(2

)(exp)(),(

2

20

,, Z

IIZKZNY

Ipn

)(2)(2

ZC

TAZ

symI

Yield from grand canonical model

Gaussian approximation on grand canonical expression

Isospin variance related to Csym

Formula 3

Fluctuation formula (Formula 3)

Csym is the symmetry energy of fragment Zσ(Z) is width of isotopic distribution

Ad. R. Raduta. et al., PRC 75,044605 (2007)

& 4fromformula

2) both decrease after evaporation.

For primary fragments, Csym/T from formulae 1, 2 & 3 close to each other.

Variation with Z (Input Csym=23.5 MeV)Csym/T=/4

4 8 12 16 20 24 28 320.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Proton Number (Z)

/

4 8 12 16 20 24 28 320.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Proton Number (Z)

/

4 vs Z

4

4 8 12 16 20 24 28 320

4

8

12

16

20

Co

ut sy

m/T

Proton Number (Z)

4 8 12 16 20 24 28 320

4

8

12

16

20

Co

ut sy

m/T

Proton Number (Z)

Csym/T vs Z

Large increases after evaporation for formulae 2 & 3; Small decrease after evaporation for formula 1

Primary dotted lineSecondary solid line

2

2

2

2

1

1

A

Z

A

Z

Source Formula (1) black lineFragment Formula (2) red lineFluctuation Formula (3) blue line

T =5.0 MeV

increases while fromformula 2) decreases after evaporation

Similar trends of results as in the case of input Csym=23.5 MeV for formula (2) & (3)

For formula (1), Csym/T increases after evaporation in contrary to Csym=23.5 MeV

4 8 12 16 20 24 28 320.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Proton Number (Z)

/

4 8 12 16 20 24 28 320

4

8

12

Co

ut sy

m/T

Proton Number (Z)

4 8 12 16 20 24 28 320.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Proton Number (Z)

/

& 4vs Z

4

4 8 12 16 20 24 28 320

4

8

12

Co

ut sy

m/T

Proton Number (Z)

Csym/T vs Z

Secondary solid line

Primary dotted line

Variation with Z (Input Csym=15.0 MeV)Csym/T=/4

2

2

2

2

1

1

A

Z

A

Z

Source Formula (1) black lineFragment Formula (2) red lineFluctuation Formula (3) blue line

T =5.0 MeV

5 6 7 8 9 10 11 12 13 14 150

4

8

12

16

20

24

Csy

m/T

Proton Number (Z)

5 6 7 8 9 10 11 12 13 14 150

4

8

12

16

20

24

Csy

m/T

Proton Number (Z)

Comparison with experimental data (Solid line) M.Mocko , P.hD Thesis MSU

Before evaporationThe results from primary from all the three formulae are close to each other

Csym / T from projectile fragmentation reactions

Formulae 2 & 3 after evaporation

Model Abrasion + CTM + evaporation

5 6 7 8 9 10 11 12 13 14 150

4

8

12

16

20

24

Csy

m/T

Proton Number (Z)

Source Formula (1) black lineFragment Formula (2) red lineFluctuation Formula (3) blue line

Similar results from AMD + Gemini codes from isobaric yield ratios.

58/64Ni on Be at 140 MeV/n

cold

hot

cold

Csym/T increases from primary. Results close to experimental values.

Csym deduced is about 2-3 times input Csym

Csym/T decreases from primary.

and is away from experimental values.

Formula 1 after evaporation

T =4.25 MeV

PRC 81,044620(2010)

Variation of Output Csym with Input Csym

0 5 10 15 20 25 300

5

10

15

20

25

30

Ou

tpu

t S

ym

me

try

En

erg

y (

CO

ut s

ym)

Input Symmetry Energy (Cin

sym)

0 5 10 15 20 25 300

5

10

15

20

25

30

Ou

tpu

t S

ym

me

try

En

erg

y (

CO

ut s

ym)

Input Symmetry Energy (Cin

sym)

7 MeV

5 MeV

0 5 10 15 20 25 300

5

10

15

20

25

30

O

utp

ut

Sy

mm

etr

y E

ne

rgy

(C

Ou

t sy

m)

Input Symmetry Energy (Cin

sym)

Upper = 7 MeVLower = 5 MeV

0 5 10 15 20 25 300

10

20

30

40

50

60

70

80

O

utp

ut

Sy

mm

etr

y E

ne

rgy

(C

Ou

t sy

m)

Input Symmetry Energy (Cin

sym)

Secondary solid linePrimary dotted line

Linear correlation between input and output Csym for hot fragments. For hot fragments, output Csym is almost equal to that of input Csym. For formula 1, Csym

out becomes less sensitive to input Csym after secondary decay. For formula 2 & 3, For the cold fragments, output Csym is 2-3 times of the input Csym .

As temperature increases, disagreement between input and output Csym

increases .

Formula (2) & (3)

Formula (1)

Source Formula (1) Fragment Formula (2) Fluctuation Formula (3)

7 MeV red line5 MeV black line

Formula 1

Formula 2T=5 MeV

Source sizes used 168, 177, 186; Z=75

T from different pairs of sources

Secondary Red solid line

Primary Black dotted line

-0.06 -0.03 0.00 0.03 0.06-6

-4

-2

0

2

4

6-0.06 -0.03 0.00 0.03 0.06

-6

-4

-2

0

2

4

6

T

(N/A)2

T

(Z/A)2

-0.06 -0.03 0.00 0.03 0.06-6

-4

-2

0

2

4

6-0.06 -0.03 0.00 0.03 0.06

-6

-4

-2

0

2

4

6

T

(N/A)2

T

(Z/A)2

Primary Csym=22.61 MeV

Secondary Csym=21.74 MeV

Csym from hot & coldfragments very close

-0.06 -0.03 0.00 0.03 0.06-6

-4

-2

0

2

4

6-0.06 -0.03 0.00 0.03 0.06

-6

-4

-2

0

2

4

6

TT

(N/A)2

(Z/A)2

-0.06 -0.03 0.00 0.03 0.06-6

-4

-2

0

2

4

6-0.06 -0.03 0.00 0.03 0.06

-6

-4

-2

0

2

4

6

TT

(N/A)2

(Z/A)2

Primary Csym=29.57 MeV

Secondary Csym=62.06 MeV

Csym from primary close to input Csym Secondary results almost twice of primary

Isoscaling nicely valid for the hot fragments, only approximately valid for the cold. After evaporation, the isoscaling parameter increases or decreases depending on value of input Csym.

becomes less sensitive to input Csym after secondary decay.

Isoscaling parameter

Isoscaling (source) formula (1)

n analysis reveal that Formula 1 is not good for finite temperature.

Linear correlation between input and output Csym for the hot

fragments.

Csymout becomes less sensitive to Csym

in after secondary decay.

Need to decrease Csymin in the model in order to match

experimental data.

These formulae not good for extraction of Csym from cold fragments.

Results from both the formulae close to each other for hot and cold fragments.

For the hot fragments, Csym deduced close to that of the input Csym.

For the cold fragments, Csym/T values agree with that from the experiment.

Csym deduced is more than twice that of the input value for the cold fragments .

Isoscaling (fragment ) formula (2) & the fluctuation formula (3)

Prof. Francesca Gulminelli, LPC Caen, France.2. Prof. Subal Das Gupta, McGill University, Montreal, Canada.3. Swagata Mallik , VECC, Kolkata, INDIA.

At high temperature, in the multifragmentation regime, no formula gives a

satisfactory reproduction of the input csym

In multifragmentation models, fragments formed at normal density....hence not advisable to extract density dependence from such models..

The different models have to be carefully compared to a large number of independent observables before one can safely draw any conclusion.

CollaboratorsCollaborators

ni (Ai, Zi ) & Ei* (Ai, Zi ) t=0

(Excited Fragments from CTM)

ni (Ai, Zi ) & Ei* (Ai, Zi ) t=0

(Excited Fragments from CTM)

1st Monte-Carlo Simulation(Evaporation/fission or not)

1st Monte-Carlo Simulation(Evaporation/fission or not)

2nd Monte-Carlo Simulation (which type of evaporation or fission)

2nd Monte-Carlo Simulation (which type of evaporation or fission)

3rd Monte-Carlo Simulation(Ek of evaporated particle)

3rd Monte-Carlo Simulation(Ek of evaporated particle)

Adjustment of A, Z & E* Adjustment of A, Z & E*

nf(Af, Zf )(Secondary Fragments)

nf(Af, Zf )(Secondary Fragments)

NONO

YESYES

Calculation of different decay widths (Weisskopf Formalism)Calculation of different decay widths (Weisskopf Formalism)

Energetically further evaporation/fission or

not

Energetically further evaporation/fission or

not

NONO

t=t+Δt≤ttot

t=t+Δt≤ttot

YESYES

Block diagram of the evaporation model

Sn112 +Sn112 central collisions

Dissociating system N0= 93, Z0 = 75, A0 = 168

Reaction 1

N0= 111, Z0 = 75, A0 = 186

Sn124 +Sn124 central collisions

Dissociating system

Reaction 2

For A=5 & 6, include Z= 2 & 3 ; A=7, include Z= 2, 3 & 4 For higher masses drip lines computed using liquid-drop formula and include all isotopes within these boundaries