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A Review of The Nuclear Shell Model

A Review of The Nuclear Shell Model. By Febdian Rusydi. Why We Need the Model?. To describe and predict nuclear properties associated with the structure. This Review will focus on: Angular Momentum & parity, J  Ground and excited state configuration Magnetic moment, .

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A Review of The Nuclear Shell Model

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  1. A Review ofThe Nuclear Shell Model By Febdian Rusydi

  2. Why We Need the Model? • To describe and predict nuclear properties associated with the structure. • This Review will focus on: • Angular Momentum & parity, J • Ground and excited state configuration • Magnetic moment, 

  3. Presentation Overview • Historical development • Why Shell Model: The Evidences • How to develop the model • How to explain the ground and excited state configuration of an nucleus • How to determine the magnetic moment of the nucleus

  4. Historical Development • 1927-28: Statistical Law of Fermions developed by Fermi • 1932-33: Magic Number2, 8, 20, 28, 50, 82, 126 pointed out by Barlett & Elsasser • 1934: The nuclear structure model begun to discuss. Fermi Gas Model developed, then applied to nuclear structure. • 1935: Liquid Drop Model by Weizsäcker • 1936: Bohr applied LDM to nuclear structure The magic number remained mystery…

  5. Binding Energy per Nuclear Particle 4He and 12C  -cluster Solid Red  Experimental Dash Black  Semi-empirical

  6. Why Shell Model? • Old-fashioned thought: • nucleons collide with each other. No way for shell model. Nuclear scattering result: • that thought doesn’t fit the data! • Magic number even doesn’t look to support shell model! BUT Indication that nuclear potential can be approached by a Potential-Well  Experiment evidence Atomic physics  electron orbits around the core But, how is inside the core??? ?

  7. The Evidence #1:Excitation Energy of First 2+State N/Z=20/20 N/Z=126/82 Z=30 N/Z=50/40 N/Z=82/60 Z=50 Z=70 Review Physics Letter 28 (1950) page 432

  8. The Evidence #2:Neutron Absorption X-section (Logarithmic) E. B. Paul, “Nuclear & Particle Physics”, North Holland Pub. Comp., 1969, page 259

  9. The Evidence #3:Neutron Separation Energy Frauenfelder & Henley, “Subatomic Physics”, Prentice Hall, 1991, page 488

  10. Conclusion so far… • Nuclear structure BEHAVES alike electron structure • Magic number  a Closed Shell • Properties: • Spherical symmetric • Total angular momentum = 0 • Specially stable

  11. Presentation Overview • Historical development • Why Shell Model: The Evidences • How to develop the model • How to explain the ground and excited state configuration of an nuclei • How the determine the magnetic moment of the nuclei to

  12. Let’s Develop the Theory! • Keyword: • Explain the magic number • Steps: • Find the potential well that resembles the nuclear density • Consider the spin-orbit coupling

  13. Shell Model Theory:The Fundamental Assumption • The Single Particle Model • Interactions between nucleons are neglected • Each nucleon can move independently in the nuclear potential

  14. Various forms of the Potential Well Central potential Residual potential Cent. Pot >> Resd. Pot, then we can set   0. Finally we have 3 well potential candidates! Full math. Treatment: Kris L. G. Heyde, Basic Ideas and Concepts in Nuclear Physics, IoP, 1994, Chapter 9

  15. The Closed Shell:Square Well Potential The closed shell  magic number

  16. The Closed Shell:Harmonic Potential The closed shell  magic number

  17. The Closed Shell:Woods - Saxon Potential The closed shell  magic number But… This potential resembles with nuclear density from nuclear scattering

  18. The Closed Shell:Spin-Orbit Coupling Contribution • Maria Mayer (Physical Review 78 (1950), p16) suggested:, • There should be a non-central potential component • And it should have a magnitude which depends on the S & L • Hazel, Jensen, and Suess also came to the same conclusion.

  19. The Closed Shell:Spin-Orbit Coupling Calculation The non-central Pot. Energy splitting Experiment: Vls = negative  Energy for spin up < spin down Full math. Treatment: Kris L. G. Heyde, Basic Ideas and Concepts in Nuclear Physics, IoP, 1994, Chapter 9

  20. SMT: The Closed Shell Povh, Particle & Nuclei (3rd edition), Springer 1995, pg 255

  21. SMT: The Ground State • How to determine the Quantum Number J ?[1] • J (Double Magic number or double closed shell) = 0+. If only 1 magic number, then J determined by the non-magic number configuration. • J determined from the nucleon in outermost shell (i.e., the highest energy) or hole if exist. •  determined by (-1)l, where l(s,p,d,f,g,…) = (0, 1, 2, 3, 4, …). To choose l: consider hole first, then if no hole  nucleon in outermost shell.

  22. SMT: The Ground State (example)

  23. SMT: Excited State • Some conditions must be known: energy available, gap, the magic number exists, the outermost shell (pair, hole, single nucleon). • Otherwise, it is quite difficult to predict precisely what is the next state.

  24. SMT: Excited State (example) • Let’s take an example 18O with ground state configuration: • Z= 8 – the magic number • N=10 – (1s1/2)2 (1p3/2)4 (1p1/2)2 (1d5/2)2 or (d5/2)2 • If with E ~ 2 [MeV], one can excite neutron to (d5/2) (d3/2), then with E ~ 4 [MeV], some possible excite states are: • Bring 2 neutron from 1p1/2 to 2d5/2 (d5/2)4 0  J  5 • Bring 2 neutron from 2d5/2 to 2d3/2 (d3/2)2 0  J  3 • Bring 1 neutron from 2d5/2 to 1f7/2 (f7/2)1 1  J  6 • Some other probabilities still also exist

  25. Mirror Nuclei 15NZ=715OZ=8 If we swap protons & neutrons, the strong force essentially does not notice it Discrepancy The prediction of SMT fail when dealing with deformed nuclei. Example: 167Er Theory 7/2 - Exprm  7/2 + Collective Model!  SMT: Mirror & Discrepancy

  26. SMT: Mirror Nuclei (Example) Povh, Particle & Nuclei (3rd edition), Springer 1995, pg 256

  27. SMT: The Magnetic Moment • Since L-S Coupling  associated to each individual nucleon • SO   sum over the nucleonic magnetic moment Full math. Treatment: A. Shalit & I. Talmi, Nuclear Shell Model, page 53-59

  28. Conclusions • How to develop the model • Explain the magic number • Single particle model • Woods – Saxon Potential • LS Coupling Contribution • Theory for Ground & Excited State • Treat like in electron configuration • J can be determined by using the guide • Theory for Magnetic Moment •  is sum over the nucleonic magnetic moment

  29. Some More Left… • Some aspects in shell Model Theory that are not treated in this discussion are: • Quadruple Moment – the bridge of Shell Model Theory and Collective Model Theory. • Generalization of the Shell Model Theory – what happen when we remove the fundamental assumption “the nucleons move in a spherical fixed potential, interactions among the particles are negligible, and only the last odd particle contributes to the level properties”.

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