1. Shell Model and Collective Models in Nuclei Experimental and theoretical perspectives
R. F. Casten
WNSL, Yale Univ.
RIKEN, January, 2010

2. Lecture 1 Introduction to Nuclear Structure and the Independent Particle Model (IPM)

3. The scope of Nuclear Structure Physics
The Four Frontiers
Proton Rich Nuclei
Neutron Rich Nuclei
Heaviest Nuclei
Evolution of structure within these boundaries
Terra incognita — huge gene pool of new nuclei
We can customize our system – fabricate “designer” nuclei to isolate and amplify specific physics or interactions

4. Themes and challenges of Modern Science
Complexity out of simplicity -- Microscopic
How the world, with all its apparent complexity and diversity can be constructed out of a few elementary building blocks and their interactions
Simplicity out of complexity – Macroscopic
How the world of complex systems can display such remarkable regularity and simplicity

5. Simple Observables - Even-Even (cift-cift) Nuclei. .
Simple Observables - Even-Even (cift-cift) Nuclei
1000 4+
400 2+
Masses 0+ Jπ
E (keV)

6. Survey: Empirical evolution of structure
Magic numbers, shell gaps, and shell structure
2-particle spectra
Emergence of collective features –Vibrations, deformation, and rotation

7. Energy required to remove two neutrons from nuclei
(2-neutron binding energies = 2-neutron “separation” energies) Sn Ba Sm Hf Pb 5 7 9 11 13 15 17 19 21 23 25 52 56 60 64 68 72 76 80 84 88 92 96
100
104
108
112
116
120
124
128
132
Neutron Number
S(2n) MeV
N = 82
N = 126
N = 84

8. 2+ 0+ 8

9. 2+ 0+
B(E2: 0+1  2+1)   2+1 E20+12 9

10. The empirical magic numbers near stability
2, 8, 20, 28, (40), 50, (64), 82, 126
These are the famous magic numbers that have been benchmarks of structure for 60 years. Recently, with studies of exotic nuclei we are now beginning to realize that they are not as robust as we have thought. Studies in this area are one of the major thrusts of nuclear structure research today.

11. Shell Structure
Mottelson (Nobel Prize for the Unified Model, 1975) – ANL, Sept
Shell gaps, magic numbers, and shell structure are not merely details but are fundamental to our understanding of one of the most basic features of nuclei – independent particle motion. If we don’t understand the basic quantum levels of nucleons in the nucleus, we don’t understand nuclei. Moreover, perhaps counter-intuitively, the emergence of nuclear collectivity itself depends on independent particle motion (and the Pauli Principle).

12. “Magic plus 2”: Characteristic spectra
~ 1.3 -ish 12

13. Lowering of energies, development of multiplets. R4/2  ~2-2.4
What happens with both valence neutrons and protons? Case of few valence nucleons:
Lowering of energies, development of multiplets. R4/2  ~2-2.4 13

14. Spherical vibrational nuclei
Vibrator (H.O.)
E(I) = n ( 0 )
R4/2= 2.0
n = 0,1,2,3,4,5 !!
n = phonon No.

15. Lots of valence nucleons of both types: emergence of deformation and therefore rotation (nuclei live in the lab frame)
R4/2  ~3.33

16. Deformed nuclei – rotational spectra
Rotor
E(I)  ( ħ2/2I )I(I+1)
R4/2= 3.33 0+ 2+ 4+ 6+ 8+

17. Broad perspective on structural evolution: R4/2
Note the characteristic, repeated patterns

19. Sudden changes in R4/2 signify changes in structure, usually from spherical to deformed structure
Onset of deformation 

20. Think about the striking regularities in these data
Think about the striking regularities in these data. Take a nucleus with A ~ The summed volume of all the nucleons is ~ 60 % the volume of the nucleus, and they orbit the nucleus ~ 1021 times per second! Instead of utter chaos, the result is very regular behavior, reflecting ordered, coherent, motions of these nucleons. This should astonish you. How can this happen??!!!! Much of understanding nuclei is understanding the relation between nucleonic motions and collective behavior

21. B(E2; 2+  0+ )

22. 22

23. Ab initio calculations: One on-going success story

24. But we won’t go that way – too complicated for any but the lightest nuclei. We will make some simple models – microscopic and macroscopic Let’s start with the former, the Independent particle model and its daughter, the shell model

25. Independent particle model: magic numbers, shell structure, valence nucleons. Three key ingredients
First: 
Vij r Ui
r = |ri - rj|
Nucleon-nucleon force – very complex
One-body potential – very simple: Particle in a box ~
This extreme approximation cannot be the full story. Will need “residual” interactions. But it works surprisingly well in special cases.

26. Second key ingredient: Quantum mechanics
Particles in a “box” or “potential” well
Confinement is origin of quantized energies levels 3 1 2
Energy ~ 1 / wave length
n = 1,2,3 is principal quantum number
E up with n because wave length is shorter

27. 27

28. - = 28

29. 29

30. Nuclei are 3-dimensional
What is new in 3 dimensions?
Angular momentum
Centrifugal effects

31. 31

32. Radial Schroedinger wave function
Higher Ang Mom: potential well is raised and squeezed. Wave functions have smaller wave lengths. Energies rise
Energies also rise with
principal quantum number, n.
Raising one, lowering the other can give similar energies – “level clustering”:
H.O: E = ħ (2n+l)
E (n,l) = E (n-1, l+2)
e.g., E (2s) = E (1d)

33. 33

34. Third key ingredient
Pauli Principle
Two fermions, like protons or neutrons, can NOT be in the same place at the same time: can NOT occupy the same orbit.
Orbit with total Ang Mom, j, has 2j + 1 substates, hence can only contain 2j + 1 neutrons or protons.
This, plus the clustering of levels in simple potentials, gives nuclear SHELL STRUCTURE

35. nlj: Pauli Prin. 2j + 1 nucleons35

36. The plot gives B.E.s PER nucleon.
We can see how to improve the potential by looking at nuclear Binding Energies.
The plot gives B.E.s PER nucleon.
Note that they saturate. What does this tell us? 36

37. Consider the simplest possible model of nuclear binding.
Assume that each nucleon interacts with n others. Assume all such interactions are equal.
Look at the resulting binding as a function of n and A. Compare this with the B.E./A plot.
Each nucleon interacts with 10 or so others. Nuclear force is short range – shorter range than the size of heavy nuclei !!! 37

38. ~ Compared to SHO, will mostly affect orbits
at large radii – higher angular momentum states 38

39. 39

40. orbits and raises those with
The nuclear potential: a rounded square well (Wood-Saxon shape) works quite well in reproducing the magic numbers provided we add in a spin-orbit force* that lowers the energies of the
j = l + ½
orbits and raises those with
j = l – ½
* Maria Goeppert Mayer, Haxel, Jensen, and Suess, 1948, Nobel Prize 1963

41. Clusters of levels + Pauli Principle  magic numbers, inert cores
Concept of valence nucleons – key to structure. Many-body  few-body: each body counts.
Addition of 2 neutrons in a nucleus with 150 can drastically alter structure

42. Independent Particle Model
Put nucleons (protons and neutrons separately) into orbits.
Key question – how do we figure out the total angular momentum of a nucleus with more than one particle? Need to do vector combinations of angular momenta subject to the Pauli Principal. More on that later. However, there is one trivial yet critical case.
Put 2j + 1 identical nucleons (fermions) in an orbit with angular momentum j. Each one MUST go into a different magnetic substate. Remember, angular momenta add vectorially but projections (m values) add algebraically.
So, total M is sum of m’s
M = j + (j – 1) + (j – 2) + …+ 1/2 + (-1/2) + … + [ - (j – 2)] + [ - (j – 1)] + (-j) = 0
M = So, if the only possible M is 0, then J= 0
Thus, a full shell of nucleons always has total angular momentum 0. This simplifies things enormously !!!

43. 43 43

44. a)
Hence J = 0 44 44

45. 45 45

46. Ignore protons (magic), consider 51 neutrons
Let’s do 91 40Zr51
Ignore protons (magic), consider 51 neutrons 46 46

47. 47 47

48. 48 48

49. 49 49

50. 50 50

51. 51 51

52. 52 52

53. 53 53

54. 54 54

55. 55 55

56. Independent Particle Model
Some great successes (for nuclei that are “doubly magic plus 1”).
Clearly inapplicable for nuclei with more than one particle outside a doubly magic “core”. In fact, in such nuclei, it is not even defined. Thus, as is, it is applicable to only a couple % of nuclei.
Residual interactions and angular momentum coupling to the rescue.

57. Homework
Note that only those circled can be done before considering residual interactions 57 57

58. Independent Particle Model – Uh –oh !!!
Trouble shows up