Liquid Drop Model

Fission Process

The National Superconducting

Cyclotron Laboratory@Michigan State University

Bet ty Tsang, Tohoku University , Sendai, Japan

Dual Nature of Nuclei – from Shell model to Isospin Diffusion

Shell Model

Magic numbers

N=2

N=10

N=20

Magic number

N=2

N=10

N=20

Maria Goeppert-Mayer & Hans

Jensen, 1963 Noble Prize

winners for the Nuclear Shell

Model.

Spectroscopic Factors:

measure the single

structure of the

valence nucleons.

Spectroscopic factors from transfer reactions

RM

EX

jl

d

dd

d

S

)(

)(

,

Pros:

We know the exact state of the

nucleon transferred.

Good understanding of the

experimental technique and

reaction theory (DWBA).

Lots of data from past 40 years.

Cons:

Do we measure the “absolute”

spectroscopic factors?

Data appear to give inconsistent

results

14.3 MeV Ca40(d,p)Ca41

0

1

10

0 20 40 60 80 100 120 140 160 180

angle (deg)

d.c

.s (

mb

/sr)

CH

DWBA

Hjorth

One of the important technique

to understand the structure of

the rare nuclei.

• Published spectroscopic factors show large fluctuations from analysis to analysis

• Consequence of using different optical model potentials and parameters for the

DWBA reaction model.

Spectroscopic Factors from literaturesExample: 1p1/2 neutron SF in 13C = 12C+n

Basic assumptions

of DWBA

The reaction is dominated by 1-step direct transfer.

Elastic Scattering is the main process in the entrance and

exact channels.

A(d,p)B B(p,d)A

TDWBA = <Apf|V|Bdi>

DWBA

EX

jl

d

dd

d

S

)(

)(

,

Extraction of

Spectroscopic Factor

For each angular distribution:

1. Fit first peak only (emphasize on maximum and shape)

2. Require more than 1 data point

3. Use global proton optical potential and standardized parameters.

4. Construct d potential from p & n potential using the Adiabatic

Approximation (Soper-Johnson).

14.3 MeV Ca40(d,p)Ca41

0

1

10

0 20 40 60 80 100 120 140 160 180

angle (deg)d

.c.s

(m

b/s

r)

CH

DWBA

Hjorth

RM

EX

jl

d

dd

d

S

)(

)(

,

Procedure

1. Digitize (p,d)

and (d,p)

angular

distribution

data from

literature.

2. Run DWBA

calc’s with

“standard”

parameter set.

3. Extract SF

12C(d,p)13Cgs

The spectroscopic factors deduced in a systematic and consistent way show that we can extract spectroscopic factors within the measurement uncertainties.

Apply the technique to a large data set

Liu et al, PRC 69, 064313 (2004)

Systematic extraction of SF’s

We studied 79

nuclie by digitizing

~ 430 angular

distributions from

literature

for (p,d) & (d,p)

reactions on target

from Z=3-24

Z=3 Li 6, 7, 8Z=4 Be 9, 10, 11Z=5 B 10, 11, 12Z=6 C 12, 13, 14, 15Z=7 N 14, 15, 16Z=8 O 16, 17, 18, 19Z=9 F 19, 20Z=10 Ne 21, 22, 23Z=11 Na 24Z=12 Mg 24, 25, 26, 27Z=13 Al 27, 28Z=14 Si 28, 29, 30, 31Z=15 P 32Z=16 S 32, 33, 34, 35, 36, 37Z=17 Cl 35, 36, 37, 38Z=18 Ar 36, 37, 38, 39, 40Z=19 K 39, 40, 41, 42Z=20 Ca 40, 41, 42, 43, 44, 45, 47, 48, 49Z=21 Sc 45, 46Z=22 Ti 46, 47, 48, 49, 50, 51Z=23 V 51Z=24 Cr 50, 51, 52, 53, 55

Digitization of ~430 angular distributions from literature

for (p,d) & (d,p) reactions on target from Z=3-24

Data come from many groups over 40 years.

-- Require quality control

How to assess the uncertainties of the procedure?

A+pB+d S+

B+dA+p S-

Equivalent processes S+ = S-

Self Consistency Checks

Sn 79 nuclei from Li to Cr

(p,d) : S+ 47 nuclei

(d,p) : S- 55 nuclei

(p,d) & (d,p) 18 nuclei

Comparison of (p,d) and (d,p) reactions

0.1

1

10

0.1 1 10

SF(p,d)

SF

(d,p

)

pd vs. dp

line

By requiring the chi-square per degree of freedom is 1, we

obtain nominal uncertainty of 20% for each measurement.

Comparison with Endt’s resultsEndt in 1977 compiled SF’s of the s-d shell nuclei from

(p,d), (d,p) – 50% uncertainty

(p,d), (d,p), (d,t), (3He, a) – 25% uncertainty

0.1

1

10

0.1 1 10

Endt's best SF

Data

line

Endt vs. Data

There are some scattering of the values but there is a strong

correlation between present analysis and Endt values

Textbook Example: Spectroscopic factors of Ca isotopesDirect Nuclear Reaction Theories by Austern; pg 291

l=7/2, S=1, 2, 0.75, 4, 0.5, 6, 0.25, 8ACa = 40Ca +(A-40)n Assume 40Ca is a good inert core.

IsotopeS_n s valence-nIPM Endt shell Expt

Ca40 15.51 1d3/2 4 4 4 4.31

Ca41 9.367 1f7/2 1 1 0.85 1 1.03

Ca42 11.12 1f7/2 2 2 1.6 1.81 1.82

Ca43 8.04 1f7/2 3 0.75 0.58 0.75 0.63

Ca44 11.27 1f7/2 4 4 3.1 3.64 3.82

Ca45 7.761 1f7/2 5 0.5 0.5 0.41

Ca47 6.546 1f7/2 7 0.25 0.256 0.25

Ca48 8.846 1f7/2 8 8 7.38 7.06

Ca49 4.43 2p3/2 1 1 0.918 0.66

Sc45 12.27 1f7/2 4 4 0.6 0.35 0.32

Sc46 8.687 1f7/2 5 0.5 0.34 0.36 0.57

Ti46 13.14 1f7/2 4 4 2.58 2.6

Ti47 9.578 1f7/2 5 0.5 0.025

Ti48 11.89 1f7/2 6 6 0.09

Ti49 8.438 1f7/2 7 0.25 0.24

Ti50 10.76 1f7/2 8 8 6.4

Ti51 5.721 2p3/2 1 1 1.21

V51 0 1f7/2 8 8 1.06

Cr51 10.21 1f7/2 7 0.25 0.35

Cr52 12.49 1f7/2 8 8 6.3

Cr53 7.115 2p3/2 1 1 0.36

Cr54 9.421 2p3/2 2 2

Cr55 2p3/2 3 0.75 0.85

Ca 0.1

1

10

39 41 43 45 47 49

A

sp

ec

tro

sc

op

ic f

ac

tor

Data

Austern IPM model

Shell model

nS 12

11

j

nS

IPM (Austern, pg 291)For n even

For n odd

40-48Ca isotopes have good single particle states with spherical cores

SF for 49Ca is lower than IPM and shell model predictions.

Comparison with Austern’s IP Model

Most experimental SF values are less than predictions.

There are no constant quenching even for close shell nuclei.

Discrepancies may be explained by including interaction

between nucleons and core

0.1

1

10

0.1 1 10

IPM SF

Ex

pt

SF

Be

line

B

C

N

O

Na

Mg

Al

Si

P

S

Cl

Ar

K

Ca

Sc

Ti

V

Cr

20% line

-20% line

Li

F, Ne

Compare with Modern Shell Model (Oxbash)

Good agreement with most isotopes

Outliners: deformed nuclei and isotopes with small SF’s

(Ne)

0.1

1

10

0.1 1 10

Shell Model SF

Ex

pt

SF

Be

line

B

C

N

O

Na

Mg

Al

Si

P

S

Cl

Ar

K

Ca

Sc

Ti

20% line

-20% line

Li

F

(e,e’p) – sensitive to interior of the wave-functions

Quenched by 35% compared to IP(S)M

Other measurements of Spectroscopic Factors

Knockout – sensitive to the tail of the wave-functions

Depends on Separation energy compared to SM

Conclusions

1. We have extracted ground state neutron

spectroscopic factors for 79 (Z=3-24) nuclei.

2. 40Ca to 48Ca isotopes follow the simple IPM

predictions Good valence nucleons around spherical cores

No quenching for gs n-orbital for the closed shell

nuclei of 40Ca?

3. Most SF’s fall short of IPM predictions but

agree with modern day shell model

calculations – SM interactions take care of

most of the long-range interactions.

4. There are puzzling differences of these values

compared with the SF’s obtained from (e,e’p)

and knockout reactions.

Discrepancies larger than the quoted experimenttal errors.

Shape of 1st peak is not the same

0.1

1

10

0 50 100 150

Problem : Disagreement between measurements

11.8 MeV PRC64(2001)034312

12 MeV PR164(1967)1274

11B(d,p)12B

qcm

Data from Lee (PR136(1964)B971) are consistently high

Cross-comparisons weed out “bad” data

40Ca(d,p)41Ca

1

10

0 10 20 30 40 50 60

Ed=12 MeV

1

10

0 10 20 30 40 50 60

Ed=11 MeV

qcm

3/2AaAaB SV 3/1

)1(

A

ZZaC

A

ZAasym

2)2(

Neutron Number N

Pro

ton

Nu

mb

er Z 3/2

AaAaB SV 3/1

)1(

A

ZZaC

A

ZAasym

2)2(

asym=30-42 MeV for infinite NM

Inclusion of surface

terms in symmetry

2

23/2 )2()(

A

ZAAaAa SV

symsym

5o

EOS ? ?

0 o

exotics10 km

Relevance to dilute and dense n-rich objects

Sizes of nuclei with

n-halo and n-skin

Stability of Neutron

Star and its structure

• Prospects are good

for improving

constraints further.

• Relevant for

supernovae - what

about neutron stars?

What is known about the EOS of symmetric matterE(, ) E(, 0) Ssym() 2

Danielewicz, Lacey, Lynch (2002)

Experimental setup

MSU, IUCF, WU collaboration

Sn+Sn collisions involving 124Sn, 112Sn at E/A=50 MeV

Miniball + Miniwall

4 multiplicity array

Z identification, A<4

LASSA

Si strip +CsI array

Good E, position,

isotope resolutions

Ring Counter

Annular Si+CsI array

Z of projectile-like residue

HiRA group picture, 1999

Isoscaling constructed from Measured Isotopic yieldsT.X Liu et al. PRC 69,014603

P T

Isoscaling from Relative Isotope Ratios

R21=Y2/ Y1

TZTN pne//

MB Tsang et al. PRC 64,054615

Simple derivation of the isoscaling law

• Basic trends from Grand Canonical ensemble:

– Yields term with exponential dependence on the chemical potentials.

• Ratios to reduce sensitivity to secondary decays:

• Scaling parameters C,

( )

( )

),(),(),(

)/exp(12

),(/),(exp),(

*

int

int

ZNfZNYZNY

TEJZwhere

ZNZTZNBZNZNY

HOTCOLD

i

ii

pnHOT

feeding correction

( ) TZTNC

ZNY

ZNYZNR

//

1

221

pne),(

),(,

TT pn /,/ a

Isoscaling in statistical models

Primary distributions show good isoscaling

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

WCI statistical model working group (2004)

Isoscaling in

Antisymmetrized

Molecular Dynamical

model

A. Ono et al. PRC 68,051601 (2003)

Isoscaling observed in many reactions

Y2/ Y1

TZN pne/)(

PRL, 86, 5023 (2001)

86Kr+116Sn,124Sn86Kr+58Ni,64NiE/A=35 MeV

More Data

58Ni+58Ni58Fe+58FeE/A=30,40,47

Souliotis et al(2003)

Shetty et al (2003)

p,4He+116Snp,4He+124SnE/A>1 GeV

Botvina,Trautmann (2002)

b

P T

P T

P T

Q Value, Sep. E

ECoul Esym

Separation Energy

ECoul Esym

Chemical Potentials

ECoul Esym p n

R21exp[(-Sn·N- Sp·Z)/T]

R21exp[((-Sn+ fn*)·N+(-Sp +fp

*+ )·Z)/T]

R21exp[(-n·N- p·Z)/T]

Symmetry energy from AMD

a depends on symmetry term interactions

A. Ono et al. PRC 68,051601 (2003)

40Ca+40Ca

48Ca+48Ca

60Ca+ 60Ca

Isospin diffusion in the projectile-like region

Basic ideas:

• Peripheral reactions

• Asymmetric collisions 124Sn+112Sn, 112Sn+124Sn

-- diffusion

Lijun Shi

Projectile

Target

)/()( ZNZN

Isospin diffusion in the projectile-like region

Basic ideas:

• Peripheral reactions

• Asymmetric collisions 124Sn+112Sn, 112Sn+124Sn

-- diffusion

• Symmetric Collisions 124Sn+124Sn, 112Sn+112Sn

-- no diffusion

• Relative change between

target and projectile is the

diffusion effect

Target

)/()( ZNZN

Isoscaling of mixed systems

Y21 exp(aN+Z)

Experimental: isoscaling;Y21 exp(aN+Z)

Theoretical : = (N-Z)/(N+Z)

x=experimental or theoretical isospin

observable

x=x124+124 Ri = 1.

x=x112+112 Ri = -1.

112112124124

1121121241242

xx

xxxR

i

Rami et al., PRL, 84, 1120 (2000)

Isospin Transport Ratio

a 1 – 2 (EES,SMM,AMD)

BUU predictionsLijun Shi

Experimental

results are in

better

agreement with

predictions

using hard

symmetry

terms

E(, ) E(, 0)Ssym() 2

Ssym()

Summary

A lot of work has been done on isoscaling.

Robust observable

Seen in many different reactions

Promising tool to study symmetry energy with

heavy ion collisions – Isospin Diffusion

Acknowledgements

P. Danielew icz , C.K. Gelbke, T.X. Liu, X.D. Liu, W.G. Lynch,

L.J. Shi, R. Shomin, M.B. Tsang, W.P. Tan, M.J. Van

Goethem, G. Verde, A. Wagner, H.F. Xi, H.S. Xu, Akira Ono,

Bao-An Li, B. Davin, Y. Larochelle, R.T. de Souza, R.J.

Charity, L.G. Sobotka , S.R. Souza, R. Donangelo

Bill Friedman

Spectroscopic factors:

Jenny Lee, Xiaodong Liu

Isospin Diffusion

BUU predictions E(, ) E(, 0)Ssym() 2

Ssym() ()

Including the

momentum

dependence in

the mean-field

in BUU changes

the agreement

Need more

experimental

constraints

B.-A. Li, C. B. Das,

S. Das Gupta, and C.

Gale

Phys. Rev. C 69,

011603 (2004)

Isospin Diffusion from BUU

Symmetry Energy

Esym() = ee sym

kin

sym

int

)( 0

32

25.12

MeVekin

sym

0 = 0.16 fm-3

14MeV (/0)2

14MeV (/0)

14MeV (/0)1/3

38.5(/0)21(/0)2

esym

int

/0

esym

int

The density dependence

of asymmetry term is

largely unconstrained.

E(, ) E(, 0)Ssym() 2

Pressure and collective flow dynamics

• Both the elliptical and transverse flow reflect the pressure created in the collisions

density

contours

pressure

contours

Danielewicaz, Lacey, Lynch Science, Dec 2002

Discrepancies in the data is larger than that quoted

by the authors

1

10

0 5 10 15 20 25 30 35 40

11B(d,p)12B

11.8 MeV Liu at al, PRC64(2001)034312

12 MeV Schiffer et al, PRev164(1967)1274

Discrepancies in the data is larger than the

uncertainties quoted by the authors

33S(d,p)34S

12 MeV Van Der Baan, NPA173(1971)456

12 MeV Crozier et al, NPA198(1972)209

0.01

0.1

1

0 10 20 30 40 50 60

Problems with small SF determinations

19F(d,p)20F SF<0.1

16 MeV Fortune et al, PRC6(1972)21

Cross-sections are small and data fluctuate

0.01

0.1

1

0 10 20 30 40 50 60

qcm

Compare with Shell Model (Oxbash)

Good agreement with most isotopes within +20%

0

0.5

1

1.5

2

5 15 25 35 45

A

Ex

pt

SF

/Sh

ell

Be

B

C

N

O

Na

Mg

Al

Si

P

S

Cl

Ar

K

Ca

Sc

Ti

Li

F,Ne

Compare with Shell Model (Oxbash)

No n-separation energy dependence quenching

0

0.5

1

1.5

2

-5 0 5 10 15

S_n

Ex

pt

SF

/Sh

ell

Be

B

C

N

O

Na

Mg

Al

Si

P

S

Cl

Ar

K

Ca

Sc

Ti

Li

F,Ne

Take A(d,p)A+1 stripping reaction as an example:

can be expressed in terms of summation

over the complete set of :

is the overlap function defined as :

The theoretical spectroscopic factor is given by

A

i1

A

f

f

A

f

i

f

A

i r 1)(

)(ri

f

A

i

A

f

i

f r 1)(

i

fS

2))(( drS i

f

i

f

Calculation

The theoretical differential cross sections for a particular reaction were

calculated by the modified version of code TWOFNR based on the

DWBA model. Global Optical Model Potential and JLM Optical

Model Potential were used.

DWBA Theory

For the reaction of A(a,b)B, the transition amplitude (T) is :

For (d,p) reaction in zero-range approximation

baaiABbf rdrdrVrT

),(),( )(*)(

)()()(*

0 nA

m

lpnBBABAA rYrrDMJMMMjJ

342

0 105.1 fmMeVD

))()((21

ddpdppnAjljlAB msmmsmsjmlsmrRSV

Optical-Model potential

where and

Thus 13 parameters are needed to be adjusted to reproduce the observed

elastic scattering experiment.

• Different sets of parameters were used for the same reaction at different

energies.( Parameters are quite sensitive to the fitting procedure)

• Global optical model potential is used to avoid such sensitivity

)(4)()(

1)1.(0.2)()( 0 D

D

WVSOSOc xfdx

dWxfWixf

dr

d

rVxfVVrU

1)1()( xi

i exfiii aArrx /)( 3

1

DWBA Adiabatic CH JLM

Proton potential Chapel-Hill [43] Chapel-Hill [43] JLM [47,48]

Deuteron potential Daehnick [45] Adiabatic [53] from CH Adiabatic [53] from JLM

Target r.m.s radius /density Shell model

n-binding potential Woods-Saxon

r 0 =1.25, a=0.65

Woods-Saxon

r 0 =1.25, a=0.65

Woods-Saxon

r 0 =1.25, a=0.65

Hulthen finite range factor 0.7457 0.7457 0.7457

Vertex constant D02 15006.25 15006.25 15006.25

JLM potential scaling λ N/A N/A λ v =1.0 and λ w =0.8 [54]

Non-Locality potentials p 0.85; n N/A; d 0.54 p 0.85; n N/A; d 0.54 p 0.85; n N/A; d 0.54

Summary of the input parameters used in DWBA code

TWOFNR (Surrey version)Source : PRC 69 (2004) 064313

No adjustment of parameters for the entire range of isotopes

Digitization of ~430 angular distributions from literature

for (p,d) & (d,p) reactions on target from Z=3-24

Strength lies in the numbers.

To test the method in the quality control:

1. Compare to Endt’s “Best” values when

available.

2. Compare SF’s derived from (p,d) and (d,p)

reactions separately to estimate the

uncertainties in our method.

Nuclear

physics

can be

fun