1. IBA – An Introduction and Overview Basic Ideas, underpinnings, Group Theory, basic predictions

2. Dynamical Symmetries Shell Model - (Microscopic) Geometric – (Macroscopic) Third approach — “ Algebraic ” Phonon-like model with microscopic basis explicit from the start. Group Theoretical Shell Mod. Geom. Mod. IBA Collectivity Microscopic

3. IBA – A Review and Practical Tutorial Drastic simplification of shell model  Valence nucleons  Only certain configurations  Simple Hamiltonian – interactions “Boson” model because it treats nucleons in pairs 2 fermions boson F. Iachello and A. Arima

4. Why do we need to bother with such a model? Remember 3 x 10 14 ? We simply MUST simplify the problem. As it turns out, the IBA is: a)The most successful macroscopic model b)The only collective model in which it is even possible in practice to calculate many observables

5. Shell Model Configurations Fermion configurations Boson configurations (by considering only configurations of pairs of fermions with J = 0 or 2.)

6. 0 + s-boson 2 + d-boson Valence nucleons only s, d bosons – creation and destruction operators H = H s + H d + H interactions Number of bosons fixed in a given nucleus: N = n s + n d = ½ # of val. protons + ½ # val. neutrons valence IBM Assume fermions couple in pairs to bosons of spins 0+ and 2+ s boson is like a Cooper pair d boson is like a generalized pair. Create ang. mom. with d bosons

7. Why s, d bosons? Lowest state of all e-e First excited state in non-magic s nuclei is 0 + d e-e nuclei almost always 2 +  - fct gives 0 + ground state  - fct gives 2 + next above 0 +

8. The IBA – an audacious, awesome leap 154 Sm3 x 10 14 2 + states Or, why the IBA is the best thing since tortellini Magnus Shell model Need to truncate IBA assumptions 1. Only valence nucleons 2. Fermions → bosons J = 0 (s bosons) J = 2 (d bosons) IBA: 26 2 + states Is it conceivable that these 26 basis states could possibly be correctly chosen to account for the properties of the low lying collective states?

9. Why the IBA ????? Why a model with such a drastic simplification – Oversimplification ??? Answer: Because it works !!!!! By far the most successful general nuclear collective model for nuclei ever developed Extremely parameter-economic Deep relation with Group Theory !!! Dynamical symmetries, group chains, quantum numbers

10. IBA Models IBA – 1No distinction of p, n IBA – 2Explicitly write p, n parts IBA – 3, 4 Take isospin into account p-n pairs IBFM Int. Bos. Fermion Model for Odd A nuclei H = H e – e(core) + H s.p. + H int IBFFMOdd – odd nuclei [ (f, p) bosons for  = - states ] Parameters !!!: IBA-1: ~2 Others: 4 to ~ 20 !!!

12. Note key point: Bosons in IBA are pairs of fermions in valence shell Number of bosons for a given nucleus is a fixed number N  = 6 5 = N  N B = 11 

13. Review of phonon creation and destruction operators is a b-phonon number operator. For the IBA a boson is the same as a phonon – think of it as a collective excitation with ang. mom. zero (s) or 2 (d). What is a creation operator? Why useful? A)Bookkeeping – makes calculations very simple. B) “Ignorance operator”: We don’t know the structure of a phonon but, for many predictions, we don’t need to know its microscopic basis.

14. Most general IBA Hamiltonian in terms with up to four boson operators (given, fixed N) IBA Hamiltonian AARRGGHHH !!! We will greatly simplify this soon but it is useful to look at its structure

15. What J’s? M-scheme Look familiar? Same as quadrupole vibrator. 6 +, 4 +, 3 +, 2 +, 0 + 4 +, 2 +, 0 + 2 + 0 + 3 2 1 0 n d Simplest Possible IBA Hamiltonian Excitation energies so, set  s = 0, and drop subscript d on  d What is spectrum? Equally spaced levels defined by number of d bosons

16. Most general IBA Hamiltonian in terms with up to four boson operators (given N) IBA Hamiltonian These terms CHANGE the numbers of s and d bosons: MIX basis states of the model Crucial for structureCrucial for masses

17. Most general IBA Hamiltonian in terms with up to four boson operators (given N) IBA Hamiltonian Complicated and not really necessary to use all these terms and all 6 parameters Simpler form with just two parameters – RE-GROUP TERMS ABOVE H = ε n d -  Q  Q Q = e[s † + d † s + χ (d † ) (2) ] Competition: ε n d term gives vibrator.  Q  Q term gives deformed nuclei. Note: 3 parameters. BUT: H’ = aH have identical wave functions, q.#s, sel. rules, trans. rates. Only the energy SCALE differs. STRUCTURE – 2 parameters. MASSES – need scale

18. Brief, simple, trip into the Group Theory of the IBA DON’T BE SCARED You do not need to understand all the details but try to get the idea of the relation of groups to degeneracies of levels and quantum numbers A more intuitive (we will see soon) name for this application of Group Theory is “Spectrum Generating Algebras”

19. To understand the relation, consider operators that create, destroy s and d bosons s †, s, d †, d operators Ang. Mom. 2 d † , d   = 2, 1, 0, -1, -2 Hamiltonian is written in terms of s, d operators Since boson number is conserved for a given nucleus, H can only contain “bilinear” terms: 36 of them. s † s, s † d, d † s, d † d Gr. Theor. classification of Hamiltonian IBA IBA has a deep relation to Group theory Note on ” ~ “ ‘s: I often forget them

20. Concepts of group theory First, some fancy words with simple meanings: Generators, Casimirs, Representations, conserved quantum numbers, degeneracy splitting Generators of a group: Set of operators, O i that close on commutation. [ O i, O j ] = O i O j - O j O i = O k i.e., their commutator gives back 0 or a member of the set For IBA, the 36 operators s † s, d † s, s † d, d † d are generators of the group U(6). Generators : define and conserve some quantum number. Ex.: 36 Ops of IBA all conserve total boson number = n s + n d N = s † s + d † Casimir: Operator that commutes with all the generators of a group. Therefore, its eigenstates have a specific value of the q.# of that group. The energies are defined solely in terms of that q. #. N is Casimir of U(6). Representations of a group: The set of degenerate states with that value of the q. #. A Hamiltonian written solely in terms of Casimirs can be solved analytically ex: or: e.g:

21. Sub-groups: Subsets of generators that commute among themselves. e.g: d † d 25 generators—span U(5) They conserve n d (# d bosons) Set of states with same n d are the representations of the group [ U(5)] Summary to here: Generators: commute, define a q. #, conserve that q. # Casimir Ops: commute with a set of generators  Conserve that quantum #  A Hamiltonian that can be written in terms of Casimir Operators is then diagonal for states with that quantum # Eigenvalues can then be written ANALYTICALLY as a function of that quantum #

22. Simple example of dynamical symmetries, group chain, degeneracies [H, J 2 ] = [H, J Z ] = 0 J, M constants of motion

23. Let’s ilustrate group chains and degeneracy-breaking. Consider a Hamiltonian that is a function ONLY of: s † s + d † d That is: H = a(s † s + d † d) = a (n s + n d ) = aN In H, the energies depend ONLY on the total number of bosons, that is, on the total number of valence nucleons. ALL the states with a given N are degenerate. That is, since a given nucleus has a given number of bosons, if H were the total Hamiltonian, then all the levels of the nucleus would be degenerate. This is not very realistic (!!!) and suggests that we should add more terms to the Hamiltonian. I use this example though to illustrate the idea of successive steps of degeneracy breaking being related to different groups and the quantum numbers they conserve. The states with given N are a “representation” of the group U(6) with the quantum number N. U(6) has OTHER representations, corresponding to OTHER values of N, but THOSE states are in DIFFERENT NUCLEI (numbers of valence nucleons).

24. H’ = H + b d † d = aN + b n d Now, add a term to this Hamiltonian: Now the energies depend not only on N but also on n d States of a given n d are now degenerate. They are “representations” of the group U(5). States with different n d are not degenerate

25. N N + 1 N + 2 ndnd 1 2 0 a 2a E 00 b 2b H’ = aN + b d † d = a N + b n d U(6) U(5) H’ = aN + b d † d Etc. with further terms

26. Concept of a Dynamical Symmetry N OK, here’s the key point -- get this if nothing else: Spectrum generating algebra !!

27. OK, here’s what you need to remember from the Group Theory Group Chain: U(6)  U(5)  O(5)  O(3) A dynamical symmetry corresponds to a certain structure/shape of a nucleus and its characteristic excitations. The IBA has three dynamical symmetries: U(5), SU(3), and O(6). Each term in a group chain representing a dynamical symmetry gives the next level of degeneracy breaking. Each term introduces a new quantum number that describes what is different about the levels. These quantum numbers then appear in the expression for the energies, in selection rules for transitions, and in the magnitudes of transition rates.

28. Group Structure of the IBA s boson : d boson : U(5) vibrator SU(3) rotor O(6) γ-soft 1 5 U(6) Sph. Def. Magical group theory stuff happens here Symmetry Triangle of the IBA (everything we do from here on will be discussed in the context of this triangle. Stop me now if you do not understand up to here )

29. Dynamical Symmetries – The structural benchmarks U(5) Vibrator – spherical nucleus that can oscillate in shape SU(3) Axial Rotor – can rotate and vibrate O(6) Axially asymmetric rotor ( “gamma-soft”) – squashed deformed rotor

30. Dynamical Symmetries I. U(6)  U(5)  O(5)  O(3) U(5) N n d n  J II. U(6)  SU(3)  O(3) SU(3) N (,  ) K J III. U(6)  O(6)  O(5)  O(3) O(6) N σ   J Vibrator Rotor Gamma-soft rotor

31. IBA Hamiltonian Complicated and not really necessary to use all these terms and all 6 parameters Simpler form with just two parameters – RE-GROUP TERMS ABOVE H = ε n d -  Q  Q Q = e[s † + d † s + χ (d † ) (2) ] Competition: ε n d term gives vibrator.  Q  Q term gives deformed nuclei.

32. Relation of IBA Hamiltonian to Group Structure We will see later that this same Hamiltonian allows us to calculate the properties of a nucleus ANYWHERE in the triangle simply by choosing appropriate values of the parameters

33. U(5) Spherical, vibrational nuclei

34. Most general IBA Hamiltonian in terms with up to four boson operators (given N) IBA Hamiltonian Mixes d and s components of the wave functions d+dd+d Counts the number of d bosons out of N bosons, total. The rest are s-bosons: with E s = 0 since we deal only with excitation energies. Excitation energies depend ONLY on the number of d- bosons. E(0) = 0, E(1) = ε, E(2) = 2 ε. Conserves the number of d bosons. Gives terms in the Hamiltonian where the energies of configurations of 2 d bosons depend on their total combined angular momentum. Allows for anharmonicities in the phonon multiplets. d

35. What J’s? M-scheme Look familiar? Same as quadrupole vibrator. 6 +, 4 +, 3 +, 2 +, 0 + 4 +, 2 +, 0 + 2 + 0 + 3 2 1 0 n d Simplest Possible IBA Hamiltonian – given by energies of the bosons with NO interactions Excitation energies so, set  s = 0, and drop subscript d on  d What is spectrum? Equally spaced levels defined by number of d bosons = E of d bosons + E of s bosons

36. Important as a benchmark of structure, but also since the U(5) states serve as a convenient set of basis states for the IBA U(5) Multiplets

37. Which nuclei are U(5)? No way to tell a priori (until better microscopic understanding of IBA is available). More generally, phenomenological models like the IBA predict nothing on their own. They can predict relations among observables for a given choice of Hamiltonian parameters but they don’t tell us which parameter values apply to a given nucleus. They don’t tell us which nuclei have which symmetry, or perhaps none at all. They need to be “fed”. The nuclei provide their own food: but the IBA is not gluttonous – a couple of observables allow us to pinpoint structure. Let the nuclei tell us what they are doing !!!! Don’t force an interpretation on them

38. E2 Transitions in the IBA Key to most tests Very sensitive to structure E2 Operator: Creates or destroys an s or d boson or recouples two d bosons. Must conserve N

39. T = e Q = e[s † + d † s + χ (d † ) (2) ] Specifies relative strength of this term χ is generally fit as a parameter but has characteristic values in each dynamical symmetry E2 electromagnetic transition rates in the IBA Finite, fixed number of bosons has a huge effect compared ot the geometrical model

40. Note: TWO factors in B(E2). In geometrical model B(E2) values are proportional to the number of phonons in the initial state. In IBA, operator needs to conserve total boson number so gamma ray transitions proceed by operators of the form: s † d. Gives TWO square roots that compete.

42. B(E2: J  J-2) Yrast (gsb) states J 2 4 6 8 10 12 654321654321 Geom. Vibrator IBA, U(5), N=6 Finite Boson Number Effects: B(E2) Values Slope = 1.51 2+ 0+2+ 0+

43. Classifying Structure -- The Symmetry Triangle Have considered vibrators (spherical nuclei). What about deformed nuclei ?? Sph. Def.

44. SU(3) Deformed nuclei (but only a special subset)

45.   M ( or M, which is not exactly the same as K)

47. Typical SU(3) Scheme SU(3)  O(3) K bands in (,  ) : K = 0, 2, 4, - - - - 

48. Totally typical example Similar in many ways to SU(3). But note that the two excited excitations are not degenerate as they should be in SU(3). While SU(3) describes an axially symmetric rotor, not all rotors are described by SU(3) – see later discussion

50. Another example of finite boson number effects in the IBA B(E2: 2  0): U(5) ~ N; SU(3) ~ N(2N + 3) ~ N 2 B(E2) ~N N2N2 N Mid-shell H = ε n d -  Q  Q and keep the parameters constant. What do you predict for this B(E2) value?? !!!

51. Signatures of SU(3)

52. E  = E  B (   g )  0 Z   0 B (   g ) B (   g ) E (  -vib )  (2N - 1)  1/6

53. O(6) Axially asymmetric nuclei (gamma-soft)

56. Note: Uses χ = o

58. 196 Pt: Best (first) O(6) nucleus  -soft

59. Xe – Ba O(6) - like

60. Classifying Structure -- The Symmetry Triangle Most nuclei do not exhibit the idealized symmetries but rather lie in transitional regions. Trajectories of structural evolution. Mapping the triangle.