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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. - PowerPoint PPT Presentation
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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)
(MeV
)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
,,
NN*
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 (M
eV)
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(
MeV
)
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
30Av 18
r (fm)
40Ca
Rea
l Cen
tral
Pot
entia
l (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
oten
tial (
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
Peak
Val
ue (R
eal S
pin-
orbi
t) (M
eV)
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
96SnSn 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
Pote
ntia
l (M
ev)
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)
Den
sity
(Nuc
leon
/fm3 )
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