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Nuclear Structure

Y. K. Gambhir. Department of Physics, Indian Institute of Technology, Bombay. Nuclear Structure. THEORY. Radius = r 0 A 1/3 , r 0 =1.2 fm. A = 8, R=2.4 fm. A = 216, R=7.2 fm. 9 Be  2.50 fm. 208 Pb  5.45 fm. 9. Ground State Properties. ..

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Nuclear Structure

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  1. Y. K. Gambhir Department of Physics, Indian Institute of Technology, Bombay Nuclear Structure THEORY

  2. Radius = r0 A 1/3, r0=1.2 fm A = 8, R=2.4 fm A = 216, R=7.2 fm 9Be  2.50 fm 208Pb  5.45 fm 9

  3. Ground State Properties

  4. .. = = ≈ 1014 .. VERY VERY DENSE MATTER Atom. Vol Nucl. Den ( 10-8) 3 Atom. Den Nucl. Vol (10-13) 3 A- nu. Vol. A.( 4π/3) (0.8) 38 . Most of Nucl. Vol. is Empty t = ≈ t = 30% (4π/3) (1.2 A1/3) 3 Nucl. Vol. 27

  5. N-N int.: V. Strong, Net Attractive Short range, State Dep. Non - Central

  6. Nucleus: A (N+Z) – Body Problem

  7. Can Not be Solved: Difficulties: ● Mathematical ● Two-Body Interaction (in the Nucleus) Approximate Methods: Models Developed: Many Models Exists

  8. Mean Field Concept:

  9. Advantage is Freedom to Choose ChooseZero (Minimum) p M Mean Field Helps to reduce A-Body Problem One Body Problem

  10. Phenomenological Phenomenological (Shell Model,…) Microscopic (BBHF) • H.O. + l.s

  11. Plan ●Mean Field Concept ●Shell Model ● Magic Nuclei : TDA - RPA ● Open Shell Nuclei a. Configuration Mixing b. Truncations: Seniority, BPA …. c. BCS – Quasiparticle Method d. HF, HFB, PHF, PHFB

  12. Plan NO CORE ● ab intio Shell Model ● DDHF – Skyrme Type Interaction ● RMF – Rel. Mean Field

  13. B Schrodinger Equation Bound State Problem Basis Expansion Method

  14. B Basis Expansion Method ,

  15. Step I: Choice of Basis (Mean Field) Step II: Construction of ΦI - A Nucleons Unperturbed Energies εI Step III: Setting of Ham. Matrix Step IV: Diagonalization of

  16. Step I: Choose Core, Valence Level, s.p. Energy εI εI: Expt., Calculated or Parameters ћω =41A-1/3 Step II: Orthonormal Basis Set ΦI Group Theoretical Method Step II: Orthonormal Basis Set ΦI Group Theoretical Method Step III: Setting up Hamiltonian Matrix Requires Two –Body Matrix Elements Realistic, Phenomenological, Empirical

  17. Step IV: Diagonalization of H-Matrix Repeat for each Jπ Hamiltonian:

  18. : Particle Creation (destruction) Operator Vacuum Obeys

  19. Application to Closed Shell Nuclei

  20. Hole Levels : h1, h2, h3, …. Particle Levels : p1, p2, p3, … • 1p - 1h : Lowest Energy – Excitation • Higher Order (Energy) Excitations 2p - 2h, 3p – 3h, …..

  21. Equation of Motion Method Operator Obeys: acts as step up down Operator Vacuum (g.s.) is defined by

  22. if Set of Operators Obey Step up operator:

  23. We require and its HC It Contains Two Terms:

  24. Use and Where

  25. Notice Define

  26. The Matrices

  27. where In Coupled Representation: JΠ T

  28. × Hole Particle Matrix Element W is Racah Coefficient

  29. X , Y are Eigen Vectors of the Matrix Step Up Operator With Norm Both X and Y can be large

  30. Illustration 16O ● Step I:

  31. ● Step II: Construction of h-p Basis For Jπ = 0– For

  32. For State

  33. Step III: Two – Body Interaction M=W=0.15, H=0.4 and B=0.3Gaussian Shape, Strength = - 40 MeV Step IV: Diagonalization Results for Both for and Similar Results for Other States

  34. Open Shell Nuclei Illustration 58Ni Step I: Mean Field Core: Valence Levels: s. p. Energies : 0.0, 0.78 and 1.08 MeV

  35. Step II: Orthonormal Basis Set 2 Valence Neutrons ; No of Basis States are:

  36. Step III: Kuo – Brown Inte. M.E. Step IV: Diagonalization. Results for 58Ni, 60Ni, 62Ni and 64Ni.

  37. Is Nuclear problem solved? NOReason : Huge number of Basis Ф:For 112Sn: 12 neutrons in five s.p. statesThe Number of States Ф are for

  38. Solution: Truncation Schemes:Seniority Truncation SchemeSeniority (): No. of Nucleons Left After all Pairs Coupled to J = 0 are Removed.Even – Even:  = 0, 2, 4 are OK Odd - Even :  = 1, 3, 5 are OK

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