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 E 0 at ρ 0.  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

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

Φ 0 = 1/√A! A [ Φ 1 (r 1 )Φ 2 (r 2 )……..Φ A (r A ) ] H 0 Φ 0 = E 0 Φ 0, where E 0 =∑E n H Ψ = E Ψ Goldstone expansion for E E = E …. where P = 1 - ‌Φ 0 > <Φ 0 ‌

FIRST ORDER TERMS: This would diverge as v is highly repulsive at short distances. This is like first Born term: Full Schrodinger equation

+ + + …..

Ψ rs (r 1,r 2 ) = Φ rs (r 1,r 2 ) - (Q/e) G(W) Φ rs (r 1,r 2 ). vΨ rs (r 1,r 2 ) = (v - v (Q/e) G(W) ) Φ rs (r 1, r 2 ) = G(W) Φ rs (r 1,r 2 ). Ψ rs (r 1,r 2 ) = Φ rs (r 1,r 2 ) - (Q/e) v Ψ rs (r 1,r 2 ) 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* + A. Lejeune, U. Lombardo, and W. Zuo, Phys. Lett. B 477, 45(2000);

We need to calculate V S (r), V T (r) and V R (r) and the corresponding defect functions.

Pure neutron Matter: Results: Symmetry Energy at normal density from different NN-interactions are nearly same and close to the expected result of about 30 MeV.

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)+V c (r) +(V so (E,r) + iW so (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).

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. =

The G-matrices are folded over the nucleon densities to obtain the central and spin-orbit components of the OMP. =

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

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

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, (2010).

Proton scattering from Pb-isotopes at 295 MeV PHYSICAL REVIEW C 84, (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) Syed Rafi, A Bhagwat, W Haider and Y K Gambhir

Exotic Nucleus: 22 C Recent Reaction Cross-Section. Results for p- 22 C at 40 MeV. K. Tanaka et al. PRL 104 (2010) C………..754(22) mb 20 C………..791(34) mb 22 C………..1338(274) mb Our Brueckner Theory + Glauber Theory results: 22 C……1334 mb Only extended density for the last two neutrons give results in excellent agreement with data. Indicating a Halo structure for 22 C

The nucleus: 6 He  The recent data on polarisation of protons from 6 He at 71 MeV analysed in BHF.  The extended neutron density distribution suggests a HALO structure.

The nucleus: 9 C

Li Isotopes Syed Rafi, A. Bhagwat,W. Haider and Y. K. Gambhir PHYSICAL REVIEW C 86, (2012)

Nucleon Optical potential with Three-Body forces p- 40 Ca at 65 MeV

p- 40 Ca at 200 MeV

PHYSICAL REVIEW C 87, (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) 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) 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) Microscopic Local Optical Potentials and the Nucleon Nucleus Scattering at 65 MeV. W. Haider, Manjari Sharma, IJMPE Vol.19, No (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, (2010). 6. Neutron density distribution and the halo structure of 22 C. Manjari Sharma, A. Bhagwat, Z. A. Khan, W. Haider, and Y. K. Gambhir Phys. Rev C 83, (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, (2011). 8. Microscopic Neutron optical potential in the energy region MeV. 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) 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) (18pp) 12.Brueckner-Hartree-Fock based optical potential for proton- 4,6,8 He and proton- 6,7,9,11 Li scattering Syed Rafi, A. Bhagwat, W. Haider, Y.K.Gambhir Phys.Rev. C 86, (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, (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) 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)

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