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

If we have more than one quadrupole phonon (boson) what total angular momenta do we have? Use m-scheme for BOSONS (no Pauli Principle)

Deformed Nuclei What is different about non-spherical nuclei? They can ROTATE !!! They can also vibrate – For axially symmetric deformed nuclei there are two low lying vibrational modes called  and  So, levels of deformed nuclei consist of the ground state, and vibrational states, with rotational sequences of states (rotational bands) built on top of them.

E(I)  ( ħ 2 /2 I )J(J+1) R 4/2 = 3.33 Rotational Motion in nuclei

Rotational states Vibrational excitations Rotational states built on (superposed on) vibrational modes Ground or equilibrium state

Systematics and collectivity of the lowest vibrational modes in deformed nuclei Notice the smooth systematics and low energies for the  mode near mid-shell, compared with the erratic behavior and higher energies of the  mode. This points to lower collectivity of the  vibration.

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

Shell Model Configurations Fermion configurations Boson configurations (by considering only configurations of pairs of fermions with J = 0 or 2.) The IBA Roughly, gazillions !! Need to simplify

Modeling a Nucleus 154 Sm3 x states Why the IBA is the best thing since jackets Shell model Need to truncate IBA assumptions 1. Only valence nucleons 2. Fermions → bosons J = 0 (s bosons) J = 2 (d bosons) IBA: states Is it conceivable that these 26 basis states are correctly chosen to account for the properties of the low lying collective states?

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 +

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

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

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 !!!

F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, England, 1987). F. Iachello and P. Van Isacker, The Interacting Boson-Fermion Model (Cambridge University Press, Cambridge, England, 2005) R.F. Casten and D.D. Warner, Rev. Mod. Phys. 60 (1988) 389. R.F. Casten, Nuclear Structure from a Simple Perspective, 2 nd Edition (Oxford Univ. Press, Oxford, UK, 2000), Chapter 6 (the basis for most of these lectures). D. Bonatsos, Interacting boson models of nuclear structure, (Clarendon Press, Oxford, England, 1989) Many articles in the literature Background, References

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 

Me et al.

An early conference on the IBA (Featuring young Alison)

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.

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”

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

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:

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 #

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

Let’s illustrate 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).

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

N N + 1 N + 2 ndnd 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

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

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.

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 )

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

U(5) Spherical, vibrational nuclei

What J’s? M-scheme Look familiar? Same as quadrupole vibrator. 6 +, 4 +, 3 +, 2 +, , 2 +, 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

E (n d,, J) = α n d + β n d (n d + 4) + 2γ ( +4) + 2δJ (J + 1) Degeneracy breaking of multiplets [C j terms] But Ψ’s remain unchanged since H [ U(5)]) cannot change (mix) n d But, U(5) is a rich symmetry that allows anharmonicities Harmonic U(5): β, γ, δ = 0; E ~ n d U(5)

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 T = e Q = e[s † + d † s + χ (d † ) (2) ] Specifies relative strength of this term

E2 transitions in U(5) χ = 0 so T = e[s† + d†s] Can create or destroy a single d boson, that is a single phonon. 6 +, 4 +, 3 +, 2 +, , 2 +, n d

Vibrator (H.O.) E(I) = n (   0 ) R 4/2 = 2.0 Vibrator

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.

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

SU(3) Deformed nuclei

  M

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

Signatures of SU(3) E  = E  B (   g )  0 Z   0 B (   g ) B (   g ) E (  -vib )  (2N - 1)  1/6

Signatures of SU(3)

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

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?? !!!

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

Note: Uses χ = o

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

Xe – Ba O(6) - like

Classifying Structure -- The Symmetry Triangle Most nuclei do not exhibit the idealized symmetries but rather lie in transitional regions. Mapping the triangle. Sph. Deformed

The IBA: convenient model that spans the entire triangle of colllective structures The IBA: convenient model that spans the entire triangle of colllective structures  H = ε n d -  Q  Q Parameters: ,  (within Q)  /ε  Sph. DrivingDef. Driving H = c [ ζ ( 1 – ζ ) n d 4N B Q χ ·Q χ - ] Competition: : 0 to infinity  /ε Span triangle with  and  Parameters already known for many nuclei c is an overall scale factor giving the overall energy scale. Normally, it is fit to the first 2 + state.