Single Particle and Collective Modes in Nuclei

1. Single Particle and Collective Modes in Nuclei Lecture Series R. F. Casten WNSL, Yale Sept., 2008

2. TINSTAASQ

3. You disagree? So, an example of a really really stupid question that leads to a useful discussion:Are nuclei blue? nucleus

4. Sizes and forces • Uncertainty Principle: DE Dt > h  Dm Dx/c > h • Nuclear force mediated by pion exchange: m ~ 140 MeV • Range of nuclear force / nuclear sizes ~ fermis --------------------------------------------------------------------------------- • Uncertainty Principle: Dx D p > h •  Characteristic nuclear energies are 105 times atomic energies: 10 ev  1 MeV

6. Probes and “probees” E = h /  Energy of probe correlated with sizes of probee and production devices Atoms – lasers – table top Nuclei – tandems, cyclotrons, etc – room size Quarks, gluons – LHC – city size

9. Overview of nuclear structurealsoSome preliminariesIndependent particle modeland clustering in simple potentialsConcept of collectivity(Note: many slides are VG images – and contain typos I can’t easily correct)

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

11. Evolution of structure – First, the data • Magic numbers, shell gaps, and shell structure • 2-particle spectra • Emergence of collective features, deformation and rotation

12. The magic numbers:” special benchmark numbers of nucleons

14. 2+ 0+ B(E2: 0+1  2+1)   2+1 E20+12 Be astonished by this: Nuclei with 100’s of nucleons orbiting 1021 times/s, not colliding, and acting in concert !!!

15. The empirical magic numbers near stability • 2, 8, 20, 28, (40), 50, (64), 82, 126 • This is the only thing I ask you to memorize.

16. “Magic plus 2”: Characteristic spectra < 2.0

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

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

19. (Z = 52) Neutron number6870 72 74 76 78 80 82 Val. Neutr. number1412 10 8 6 4 2 0

20. Lots of valence nucleons of both types R4/2 ~3.33

21. Deformed nuclei – rotational spectra 8+ 6+ 4+ 2+ 0+ Rotor E(I)  (ħ2/2I )I(I+1) R4/2= 3.33 BTW, note value of paradigm in spotting physics (otherwise invisible) from deviations

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

23. Sudden changes in R4/2 signify changes in structure, usually from spherical to deformed structure Def. Def. Sph. Sph. Onset of deformation Onset of deformation as a phase transition

24. Another, simpler observable 1/E2 – Note similarity to R4/2 E2, or 1/E2, is among the first pieces of data obtainable in nuclei far from stability. Can we use just this quantity alone? R4/2 Observable E2 Nucleon number, Z or N

25. B(E2; 2+ 0+ )

27. Basic Models • (Ab initio calculations using free nucleon forces, up to A ~ 12) • (Microscopic approaches, such as Density Functional Theory) • Independent Particle Model  Shell Model and its extensions to weakly bound nuclei • Collective Models – vibrator, transitional, rotor • Algebraic Models – IBA

28. One on-going success story

29. Ui Vij r = |ri - rj|  r Independent particle model: magic numbers, shell structure, valence nucleons.Three key ingredients First: 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.

30. 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

32. - =

35. But nuclei are 3- dimensional. What’s new in 3-dimensions?Angular momentum, hence centrifugal effects. 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. Hence raising one and lowering the other can lead to similar energies and to “level clustering”: H.O: E = ħ (2n+l) E (n,l) = E (n-1, l+2) e.g., E (2s) = E (1d) Add spin-orbit force

36. nlj: Pauli Prin. 2j + 1 nucleons

37. Too low by 14 Too low by 12 Too low by 10

38. 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?

39. 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 !!!

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

42. So, modify Harm. Osc. By squaring off the outer edge. Then, add in a spin-orbit force that lowers the energies of the j = l + ½ orbits and raises those with j = l – ½

43. 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

44. 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

45. a) Hence J = 0