Nuclear matter, 2- and 3-body forces and Exotic nuclei in
Brueckner Theory
Wasi Haider
Department 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 STEP
NM 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
E0 = -16 ± 1 (MeV) ,
K= 210 ± 30,
0
= 0.17 ± 0.01 Nucl./fm-3
S= 30.0 ± 3 (MeV)
Nuclear Matter theory with TWO-Body force should predict the above properties
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
HΨ=EΨ
Goldstone expansion for E
E = E0 + <Φ0 ‫׀‬H1‫ ׀‬Φ0 > +< Φ0 ‫ ׀‬H11/ (E0-H0) ΡH1 ‫׀‬Φ0> + ….
where P = 1 - Φ0> <Φ0
FIRST ORDER TERMS:
2000
Av-18
1500
V(r) (MeV)
1
S0
1000
500
This would diverge as v is highly repulsive at
short distances.
0
0.0
This is like first Born term: Full
Schrodinger equation
0.5
1.0
1.5
r(fm)
2.0
2.5
 mn v mn  +
m n v ab ab v m n
E a  Eb  E m  E m
+
mn v cd cd v ab ab v mn
+ …..
 Ec  Ed  Em  En  Ea  Eb  Em  En 
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.
=
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);
URBANA MODEL
NPA 401, 59 (1983)
NPA 449, 219 (1986)
We need to calculate VS(r), VT(r) and VR(r) and the corresponding
defect functions.
-1
R
kF= 1.4 fm
A =- 0.0058
U = 0.0016
V (r)
6
4
V (MeV)
2
VT(r)
0
-2
VS(r)/5
-4
-6
-8
0.0
0.5
1.0
1.5
2.0
2.5
r(fm)
0.8
-1
KF=1.1 fm
1
S0
0.6
-1
KF=1.33 fm
-1
KF=1.4 fm
-1
0.4
KF=1.5 fm
g(r)
-1
KF=2.0 fm
0.2
0.0
-0.2
0
1
2
3
-1
r(fm )
4
5
Pure neutron Matter:
Symmetry Energy at normal density from
Results:
different NN-interactions are nearly same and close to
the expected result of about 30
NEUTRON MATTER(TBF)
MeV.
70
800
UV14+TNI BHF(OUR)
UV14+TBF(OUR)
UV14+TBF(VARITAIONAL)
UV14+TNI(VARITAIONAL)
AV14+TBF BHF(BALDO)
700
50
500
ES(MeV)
E()(MeV)
600
UV14
UV14+UVII
UV14+TNI
60
400
40
30
300
200
20
100
10
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-3
density(fm )
1.4
1.6
1.8
2.0
0
0.0
0.1
0.2
0.3
-3
density(fm )
0.4
0.5
104
mass density
103
3
MeV-fm
pressure
UV14+TNI
UV14+TBF
UV14
102
c
101
sound velocity
c/10
100
0.0
0.5
1.0
density(fm-3)
1.5
2.0
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 NNinteraction using some theory (BHF).
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
U
JS
LL'
(kr )  j (kr )
L
LL'
 4
L' '


0
G (r, r ' ) v
L'
JS
L' L' '
(r ' ) U
JS
LL' '
Use BR prescription to define radial G-matrices such that the NM-potential
is reproduced.
< Φrs g Φrs >
=
< Φrs v Ψrs >
2
(r ' )r ' dr'
The G-matrices are folded over the nucleon densities to obtain
the central and spin-orbit components of the OMP.
n
Uc (r1 , E )
=


 nn  
 np  
  n (r2 ) g D ( r1  r2 ,  ( R), E )dr2    p (r2 ) g D ( r1  r2 ,  ( R), E )dr2 

  nn  
 

(
r
,
r
)
g
(
r

r
,

(
R
),
E
)
j
(
k
r
 r2 )dr2 
o
 n 1 2 EX 1 2
1

  np  
 
  p (r1 , r2 ) g EX ( r1  r2 ),  ( R), E ) jo (k r1  r2 )dr2
30
Av 18
40
Ca
Ca
0
Imaginary Central Potential (MeV)
20
40
Real Central Potential (MeV)
10
0
-10
21MeV
26
30
40
45
50
65
75
85
95
107
127
155
185
225
400
-20
-30
-40
-50
-10
21MeV
26
30
40
45
50
65
75
85
95
107
127
155
185
225
400
-20
-30
-60
0
2
4
6
8
0
2
r (fm)
The real and imaginary central parts for p-40Ca (21-400 MeV)
4
r (fm)
6
8
1.0
Sn isotopes at 50MeV using Uv-14+UVII
0.20
96
0.15
0.10
Ep=200 MeV
0.05
0.00
Real Spin Orbit Potential (Mev)
Peak Value (Real Spin-orbit) (MeV)
Sn
0.5
96
Sn
Sn
100
Sn
102
Sn
104
Sn
106
Sn
108
Sn
110
Sn
112
Sn
114
Sn
116
Sn
118
Sn
120
Sn
122
Sn
124
Sn
126
Sn
128
Sn
130
Sn
132
Sn
98
132
Sn
0.0
p-Sn (96-136) Isotopes
92 96 100 104 108 112 116 120 124 128 132 136 140
A
-0.5
0
2
4
6
8
10
r (fm)
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
12
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
2.2
Ca Isotopes using Av18+3BF at 65MeV
36
Ca
Ca
40
Ca
42
Ca
44
Ca
46
Ca
48
Ca
50
Ca
52
Ca
54
Ca
56
Ca
58
Ca
60
Ca
62
Ca
64
Ca
66
Ca
68
Ca
70
Ca
72
Ca
74
Ca
76
Ca
78
Ca
80
Ca
82
Ca
84
Ca
38
2.0
Real Spin Orbit Potential(MeV)
1.8
1.6
36
1.4
Ca
1.2
1.0
84
0.8
Ca
0.6
0.4
0.2
0.0
0
1
2
3
4
5
r(fm)
6
7
8
9
Exotic Nucleus: 22C
-1
10
P(r)
N(r)
P(r)
N(r)
0.07
0.06
22
22
C
C
3
Density(Nucleon/fm )
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) mb
Our Brueckner Theory + Glauber
Theory results:
22C……1334 mb
Only extended density for the last
two neutrons give results in
excellent agreement with data.
0.05
0.04
-2
10
0.03
0.02
Indicating a Halo structure for 22C
0.01
-3
0.00
10
0
1
2
3
4
5
r (fm)
6
7
8
0
1
2
3
4
5
r (fm)
6
7
8
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.
-1
10
6
He
P(r)
N(r)
-2
10
-3
10
-4
10
0
2
4
r (fm)
6
8
The nucleus: 9C
Li Isotopes
PHYSICAL REVIEW C 86, 034612 (2012)
Syed Rafi, A. Bhagwat,W. Haider and Y. K. Gambhir
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…+
Phys. Rev. C77(2008)024605
W.Haider;
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.
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.
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
Thank You