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, 

Presentation Overview 1.Historical development 2.Why Shell Model: The Evidences 3.How to develop the model 4.How to explain the ground and excited state configuration of an nucleus 5.How to determine the magnetic moment of the nucleus

Historical Development : Statistical Law of Fermions developed by Fermi : Magic Number 2, 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…

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

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???

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

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

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

Conclusion so far… Nuclear structure BEHAVES alike electron structure Magic number  a Closed Shell Properties: 1.Spherical symmetric 2.Total angular momentum = 0 3.Specially stable

Presentation Overview 1.Historical development 2.Why Shell Model: The Evidences 3.How to develop the model 4.How to explain the ground and excited state configuration of an nuclei 5.How the determine the magnetic moment of the nuclei to

Let’s Develop the Theory! Keyword: Explain the magic number Steps: 1.Find the potential well that resembles the nuclear density 2.Consider the spin-orbit coupling

Shell Model Theory: The Fundamental Assumption The Single Particle Model 1.Interactions between nucleons are neglected 2.Each nucleon can move independently in the nuclear potential

Various forms of the Potential Well Residual potential Central 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

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

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

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

The Closed Shell: Spin-Orbit Coupling Contribution Maria Mayer (Physical Review 78 (1950), p16) suggested:, 1.There should be a non-central potential component 2.And it should have a magnitude which depends on the S & L Hazel, Jensen, and Suess also came to the same conclusion.

The Closed Shell: Spin-Orbit Coupling Calculation The non-central Pot. Energy splitting Experiment: V ls = 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

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

SMT: The Ground State How to determine the Quantum Number J  ?[1][1] 1.J  (Double Magic number or double closed shell) = 0 +. If only 1 magic number, then J  determined by the non-magic number configuration. 2.J determined from the nucleon in outermost shell (i.e., the highest energy) or hole if exist. 3.  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.

SMT: The Ground State (example) How to configure ground state of nucleus NuclideZ and N number Orbit assignmentShell Model JJ Note 6 HeZ= 2 N= 2 (1s 1/2 ) 2 s 1/ Double magic number 11 BZ= 5 N= 6 (1s 1/2 ) 2 (1p 3/2 ) -1 (1s 1/2 ) 2 (1p 3/2 ) 4 p 3/2 3/ p 3/2 Closed shell 12 CZ= 6 N= 6 (1s 1/2 ) 2 (1p 3/2 ) 4 p 3/ Double Closed shell 15 NZ= 7 N= 8 (1s 1/2 ) 2 (1p 3/2 ) 4 (1p 1/2 ) -1 (2 nd mg.#) p 1/2 1/ p 1/2 16 OZ= 8 N= 8 (2 nd mg.#) p 1/ Double magic number 17 FZ= 9 N= 8 (1s 1/2 ) 2 (1p 3/2 ) 4 (1p 1/2 ) 2 (1d 2 ) 1 (2 nd mg.#) d 5/2 5/2 + 1 proton in outer shell 27 MgZ= 12 N= 15 (2 nd mg.#) (1d 5/2 ) 4 (2 nd mg.#) (1d 5/2 ) 6 (2s 1/2 ) -1 s 1/2 1/2 + 4 proton 1d 5/2 1 2s 1/2 37 SrZ= 38 N= 49 (3 rd mg.#) (2p 3/2 ) 4 (1f 5/2 ) 6 (3 rd mg.#) (2p 3/2 ) 4 (1f 5/2 ) 6 (2p 3/2 ) 4 (1g 9/2 ) -1 g 9/2 9/2 + Closed f 5/2 1 g 9/2

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.

SMT: Excited State (example) Let’s take an example 18 O with ground state configuration: –Z= 8 – the magic number –N=10 – (1s 1/2 )2 (1p 3/2 )4 (1p 1/2 )2 (1d 5/2 )2 or (  d 5/2 )2 If with E ~ 2 [MeV], one can excite neutron to (  d 5/2 ) (  d 3/2 ), then with E ~ 4 [MeV], some possible excite states are: –Bring 2 neutron from 1p 1/2 to 2d 5/2  (  d 5/2 ) 4 0  J  5 –Bring 2 neutron from 2d 5/2 to 2d 3/2  (  d 3/2 ) 2 0  J  3 –Bring 1 neutron from 2d 5/2 to 1f 7/2  (  f 7/2 ) 1 1  J  6 –Some other probabilities still also exist

SMT: Mirror & Discrepancy Mirror Nuclei 15 N Z=7  15 O Z=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: 167 Er Theory  7/2 - Exprm  7/2 + Collective Model! 

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

SMT: The Magnetic Moment Since L-S Coupling  associated to each individual nucleon SO   sum over the nucleonic magnetic moment values of g l and g s protonNeutron glgl gsgs 10 Full math. Treatment: A. Shalit & I. Talmi, Nuclear Shell Model, page 53-59

Conclusions 1.How to develop the model -Explain the magic number -Single particle model -Woods – Saxon Potential -LS Coupling Contribution 2.Theory for Ground & Excited State -Treat like in electron configuration -J  can be determined by using the guide 3.Theory for Magnetic Moment -  is sum over the nucleonic magnetic moment

Some More Left… Some aspects in shell Model Theory that are not treated in this discussion are: 1.Quadruple Moment – the bridge of Shell Model Theory and Collective Model Theory. 2.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”.