Nuclear matter, 2- and 3-body forces and Exotic nuclei in Brueckner Theory

Wasi HaiderDepartment of Physics, AMU, Aligarh.

Dedicated to Dr J R ROOK and Prof. M Z R Khan

Students: S. M. Saliem, B. Sharma, Manjari Sharma, Dipti Pachouri and Syed Rafi.

Collaborators: J. R. Rook, P. E. Hodgson, A. M. Kobos, E.D Cooper, K.F Pal, A.M. Street, S. Kailas, Y.K. Gambhir, A. Bhagwat, Hemalatha, J. Blomgren, Zafar A. Khan

1. Introduction

(a) Brief sketch of the theory of Nuclear Matter (effective Interaction) (b) Self consistency (BHF)

2. Binding Energy (symmetric ) (a) Two body force (Coester Band) (b) Three-body force (TBF) (c) Results 3. Nucleon Optical potential (a) Results (Recent)

4. EXOTIC Nuclei.

5. Summary

Introduction (a) Brief sketch of the theory of Nuclear Matter (effective interaction/G-matrix)

Relationship of Nuclear Matter with Nuclear Physics (NP): Main Aim of NP

To understand Nuclear Structure in terms of n/p and the strong force among the constituents.

One should start from some fundamental Theory- derive the existence and Properties of real nuclei NO SUCH THEORY… Non-Relativistic Schrödinger Eqn. for n/p interacting via the Realistic TWO-Body force (approx.) +3-body force.

THIS MANY-BODY PROBLEM IS TOO HARD TO SOLVE

Nuclear matter (NM) enters as simple FIRST STEPNM is a HYPTHETICAL SYSTEM : No Coulomb force

Equal no. of n/p. INFINITE in Coordinate space.

Translational Invariance… SPWF = Plane Waves

ONLY problem to solve… E/A as f (ρ) and the effective Interaction

Saturation Property of Nuclear Force.. E/A(ρ) minimum E0 at ρ0 .

Empirical Estimates of NM Prop = -16 ± 1 (MeV) , = 0.17 ± 0.01 Nucl./fm-3

K= 210 ± 30, S= 30.0 ± 3 (MeV)

Nuclear Matter theory with TWO-Body force should predict the above properties

0E 0

Nuclear EOS

Attempt to obtain EOS & OMP from basic Theory (NM) (a) BHF (b) Variational (c) DBHF

(Bethe, Brueckner, Gammel, Rajaraman, B. D. Day) Rev. Mod. Phys. 39(1967)719, Rev. Mod. Phys. 39(1967)745. Rajaraman & Bethe(Three Nucleon Correlations)

Only input is: NN-interaction + Nucleon Density in Target Nuclei

Φ0 = 1/√A! A [ Φ1(r1)Φ2(r2)……..ΦA(rA) ] H0 Φ0 = E0 Φ0, where E0 =∑En

H Ψ = E Ψ Goldstone expansion for E

E = E0 + <Φ0 ׀H1׀ Φ0 > +< Φ0 ׀H11/ (E0-H0) ΡH1 ׀Φ0> + ….

where P = 1 - Φ0> <Φ0

0.0 0.5 1.0 1.5 2.0 2.5

0

500

1000

1500

2000

1S0

V(r

) (M

eV)

r(fm)

Av-18

FIRST ORDER TERMS:

This would diverge as v is highly repulsive at short distances.

This is like first Born term: Full Schrodinger equation

mn v mn

c d m n a b m n

mn v cd cd v ab ab v mn

E E E E E E E E

+ + + …..

mmba EEEE

mnvababvmn

Ψrs(r1,r2) = Φrs(r1,r2) - (Q/e) G(W) Φrs(r1,r2).

vΨrs(r1,r2) = (v - v (Q/e) G(W) ) Φrs(r1,r2) = G(W) Φrs(r1,r2).

Ψrs(r1,r2) = Φrs(r1,r2) - (Q/e) v Ψrs(r1,r2)

This is the famous Bethe-Goldstone integral equation.

Summary

The sets of equations suggest that the single particle potential has to be calculated in a self consistent manner.

The above choice is called as the Brueckner-Hartree-Fock approximation (BHF).

The BINDING ENERGY of NUCLEAR MATTER is then

The figure shows the level of self- consistancy achieved in about 4-5 cycles (Av-18)

Results: No TWO-BODY force gives the correct Saturation property of the Symmetric Nuclear Matter. The Goldstone expansion converges rapidly. Hence there is no hope that higher order terms would improve this situation.

• THREE-Body forces are introduced to remedy this situation.

URBANA MODEL

NPA 401, 59 (1983) NPA 449, 219 (1986)

N N N N N N = +

N N N

,,

N

N*N N N

+

A. Lejeune, U. Lombardo, and W. Zuo, Phys. Lett. B 477, 45(2000);

We need to calculate VS(r), VT(r) and VR(r) and the corresponding defect functions.

0.0 0.5 1.0 1.5 2.0 2.5

-8

-6

-4

-2

0

2

4

6 kF= 1.4 fm-1

A =- 0.0058U = 0.0016

VR(r)

VT(r)

VS(r)/5

V (

MeV

)

r(fm)

0 1 2 3 4 5

-0.2

0.0

0.2

0.4

0.6

0.8

1S0

g(r

)

r(fm-1)

KF=1.1 fm-1

KF=1.33 fm-1

KF=1.4 fm-1

KF=1.5 fm-1

KF=2.0 fm-1

Pure neutron Matter:Results:

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

100

200

300

400

500

600

700

800

NEUTRON MATTER(TBF)

density(fm-3)

E(

)(M

eV

)

UV14+TNI BHF(OUR) UV14+TBF(OUR) UV14+TBF(VARITAIONAL) UV14+TNI(VARITAIONAL) AV14+TBF BHF(BALDO)

0.0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

density(fm-3)

ES(M

eV

)

UV14 UV14+UVII UV14+TNI

Symmetry Energy at normal density from

different NN-interactions are nearly same and close to

the expected result of about 30 MeV.

0 . 0 0 . 5 1 . 0 1 . 5 2 . 0

1 0 0

1 0 1

1 0 2

1 0 3

1 0 4

c

c / 1 0

d e n s i t y ( f m - 3 )

Me

V-f

m3

s o u n d v e l o c i t y

p r e s s u r e

m a s s d e n s i t y

U V 1 4 + T N I U V 1 4 + T B F U V 1 4

Nuclear optical Potential

Nucleon Scattering has provided a huge wealth of information about nuclear interaction

This Interaction is represented as a single Particle Potential (OPTICAL POTENTIAL):

U(E,r)=-V(E,r)-iW(E,r)+Vc(r) +(Vso(E,r) + iWso(E,r))

Empirically different components are represented in terms of a large no of parameters ( normally 12)

It has helped in organizing huge data set, however, there are ambiguities and very small predictive power of this model:

DATA: (p,n) Elastic, Reaction & Total cross-section, Polarisation, Spin-Rotation Non Relativistic Mod works upto 200 MeV (A=12-208)

Hence the quest to determine it Microscopically starting from the basic NN- interaction using some theory (BHF).

'dr'r)'r(U)'r(v)'r,r(G4)kr(j)kr(U 2JS

''LL

JS

''L'L'L''L 0

'LLL

JS

'LL

BHF: 1. AMOS-Group (Non-Local: Bonn)

2. Our-Group (Local: HJ, UV14, Av-14, Av18, Reid93, Nijm II)

We solve the radial Bethe Goldstone equation

Use BR prescription to define radial G-matrices such that the NM-potential

is reproduced. < Φrs g Φrs > = < Φrs v Ψrs >

2212121

)()),(),(),( drrrkjERrrgrr onpEXp

2212121

)()),(,(),( drrrkjERrrgrr onnEXn

1( , )nU r Ec

The G-matrices are folded over the nucleon densities to obtain

the central and spin-orbit components of the OMP.

= 22122212

)),(,()()),(,()( drERrrgrdrERrrgr npDp

nnDn

0 2 4 6 8-60

-50

-40

-30

-20

-10

0

10

20

30

Av 18

r (fm)

40Ca

Real C

en

tral P

ote

nti

al (M

eV

)

21MeV 26 30 40 45 50 65 75 8595107127155185225400

0 2 4 6 8

-30

-20

-10

0

21MeV 26 30 40 45 50 65 75 8595107127155185225400

Imag

inar

y C

entr

al P

ote

nti

al (

MeV

)

40Ca

r (fm)

The real and imaginary central parts for p-40Ca (21-400 MeV)

92 96 100 104 108 112 116 120 124 128 132 136 140

0.00

0.05

0.10

0.15

0.20

Ep=200 MeV

Pea

k V

alu

e (R

eal S

pin

-orb

it)

(MeV

)

A

p-Sn (96-136) Isotopes

Decrease of spin-orbit potential as more and more neutrons are added to a nucleus.

Predicted weakening of the Spin-Orbit interaction with the addition of Neutrons; M.Hemalatha,Y.K.Gambhir,W.Haider and S.Kailas.

Phys. Rev. C79(2009)057602

0 2 4 6 8 10 12-0.5

0.0

0.5

1.0

132Sn

96Sn

Sn isotopes at 50MeV using Uv-14+UVII

r (fm)

96Sn

98Sn

100Sn

102Sn

104Sn

106Sn

108Sn

110Sn

112Sn

114Sn

116Sn

118Sn

120Sn

122Sn

124Sn

126Sn

128Sn

130Sn

132Sn

Rea

l Spi

n O

rbit

Pot

enti

al (

Mev

)

Proton scattering from Sn-Isotopes at 295 MeV

Microscopic description of 295 MeV polarized protons incident on Sn isotopes. W. Haider, Manjari Sharma, Y. K. Gambhir, and S. Kailas, Phys. Rev. C 81, 034601 (2010).

Proton scattering from Pb-isotopes at 295 MeV

PHYSICAL REVIEW C 84, 037604 (2011)Microscopic description of proton scattering at 295 MeV from Pb isotopes

Syed Rafi, Dipti Pachouri, Manjari Sharma, A. Bhagwat, W. Haider, and Y. K. Gambhir

The first maxima in the spin-orbit force for p-Ni isotopes (52-114) at 65 MeV. The inset shows the neutron skin for the same isotopes.

J. Phys. G: Nucl. Part. Phys. 40 (2013) 065101

Syed Rafi, A Bhagwat, W Haider and Y K Gambhir

0 1 2 3 4 5 6 7 8 90.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

84Ca

36Ca

Ca Isotopes using Av18+3BF at 65MeV

Rea

l Spi

n O

rbit

Pote

ntia

l(M

eV)

r(fm)

36Ca38Ca40Ca42Ca44Ca46Ca48Ca50Ca52Ca54Ca56Ca58Ca60Ca62Ca64Ca66Ca68Ca70Ca72Ca74Ca76Ca78Ca80Ca82Ca84Ca

0 1 2 3 4 5 6 7 810-3

10-2

10-1

0 1 2 3 4 5 6 7 80.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07 P(r)

N(r)

r (fm)

De

nsity(N

ucle

on

/fm

3 )

22C

P(r)

N(r)

r (fm)

22C

Exotic Nucleus: 22C

Recent Reaction Cross-Section. Results for p- 22C at 40 MeV. K. Tanaka et al. PRL 104

(2010)062701. 19C………..754(22) mb 20C………..791(34) mb 22C………..1338(274) mbOur Brueckner Theory + Glauber

Theory results: 22C……1334 mb

Only extended density for the last two neutrons give results in excellent agreement with data.

Indicating a Halo structure for 22C

0 2 4 6 810-4

10-3

10-2

10-1

6He

r (fm)

P(r)

N(r)

The nucleus: 6He

The recent data on polarisation of protons from 6He at 71 MeV analysed in BHF.

The extended neutron density distribution suggests a HALO structure.

The nucleus: 9C

Li Isotopes

Syed Rafi, A. Bhagwat,W. Haider and Y. K. Gambhir

PHYSICAL REVIEW C 86, 034612 (2012)

Nucleon Optical potential with Three-Body forces

p-40Ca at 65 MeV

p-40Ca at 200 MeV

PHYSICAL REVIEW C 87, 014003 (2013)

Syed Rafi,Manjari Sharma,Dipti Pachouri,W. Haider,and Y. K. Gambhir

List of recently published research papers in refereed journals : 1. Microscopic Optical Model Potentials for p-Nucleus Scattering at Intermediate Energies, M.Hemalatha, Y.K.Gambhir, S.Kailas and W.Haider Phys.Rev.C75(2007)037602

2. Elastic scattering of 96 MeV neutrons from iron, yttrium and lead; A.¨Ohrn, J. Blomgren, P. Andersson, A. Atac, C. Johansson…+ W.Haider; Phys. Rev. C77(2008)024605

3. Predicted weakning of the Spin-Orbit interaction with the addition of Neutrons; M.Hemalatha,Y.K.Gambhir,W.Haider and S.Kailas. Phys. Rev. C79(2009)057602

4. Microscopic Local Optical Potentials and the Nucleon Nucleus Scattering at 65 MeV. W. Haider, Manjari Sharma, IJMPE Vol.19, No 3 465-482 (2010).

5. Microscopic description of 295 MeV polarized protons incident on Sn isotopes. W. Haider, Manjari Sharma, Y. K. Gambhir, and S. Kailas, Phys. Rev. C 81, 034601 (2010).

6. Neutron density distribution and the halo structure of 22C. Manjari Sharma, A. Bhagwat, Z. A. Khan, W. Haider, and Y. K. Gambhir Phys. Rev C 83, 031601(R) (2011).

7. Microscopic description of protons scattering at 295 MeV from Pb isotopes. Syed Rafi, Dipti Pachouri, Manjari Sharma, Ameeya Bhagwat, W. Haider and Y. K. Gambhir, Phys. Rev. C 84, 037604 (2011).

8. Microscopic Neutron optical potential in the energy region 65-225MeV. Syed Rafi and W.Haider International Journal of Modern Physics E Vol. 20, No. 9 (2011) 2017–2026.

10. Exact calculation of the Direct part of the nucleon-nucleus spin-orbit potential in Brueckner theory; Dipti Pachouri, Syed Rafi, Manjari Sharma and W.Haider; International Journal of Modern Physics E Vol. 21, No. 2 (2012) 1250010.

11. Microscopic optical potentials for nucleon - nucleus scattering at 65 MeV. Dipti Pachouri, Syed Rafi, W Haider Journal of Physics G: Nuclear and Particle Physics J. Phys. G: Nucl. Part. Phys. 39 (2012) 055101 (18pp)

12.Brueckner-Hartree-Fock based optical potential for proton- 4,6,8He and proton- 6,7,9,11Li scattering    Syed Rafi, A. Bhagwat, W. Haider, Y.K.Gambhir Phys.Rev. C 86, 034612 (2012)

14. Equation of state and the nucleon optical potential with three-body forces Syed Rafi, Manjari Sharma, Dipti Pachouri, W. Haider and Y. K. Gambhir

Phys.Rev. C 87, 014003 (2013).

15. A systematic analysis of microscopic nucleon–nucleus optical potential for p–Ni scattering Syed Rafi, A Bhagwat2, W Haider and Y K Gambhir J. Phys. G: Nucl. Part. Phys. 40 (2013) 065101

9. Microscopic Optical Potential from Argonne inter-nucleon potentials. Dipti Pachouri, Manjari Sharma, Syed Rafi, W. Haider International Journal of Modern Physics E; Vol.20, No.11 (2011)2317-2327.

Thank You