The IBA The Interacting Boson Approximation Model Preliminary review of collective behavior in nuclei Collective models, and why the IBA Basic ideas about the IBA, including a primer on its Group Theory basis The Dynamical Symmetries of the IBA Practical calculations with the IBA T rust me, while it may sound scary, it is really simple

J = 2, one phonon vibration Spherical nuclei

Even-even Deformed Nuclei Rotations and vibrations

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.

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

Systematics and collectivity of the lowest vibrational modes in deformed nuclei Notice that the the  mode is at higher energies (~ 1.5 times the  vibration near mid-shell)* and fluctuates more. This points to lower collectivity of the  vibration. * Remember for later !

Development of collective behavior in nuclei Results primarily from correlations among valence nucleons. Instead of pure “shell model” configurations, the wave functions are mixed – linear combinations of many components. Leads to a lowering of the collective states and to enhanced transition rates as characteristic signatures. How does this happen? Consider mixing of states.

Crucial for structure Microscopic origins of collectivity correlations, configuration mixing and deformation: Residual interactions

How can we understand collective behavior Do microscopic calculations, in the Shell Model or its modern versions, such as with density functional theory or Monte Carlo methods. These approaches are making amazing progress in the last few years. Nevertheless, they often do not give an intuitive feeling for the structure calculated. Collective models, which focus on the structure and symmetries of the many-body, macroscopic system itself. Two classes: Geometric and Algebraic We will illustrate collective models with the algebraic model, the IBA, by far the most successful and parameter-efficient collective model. First, for background, we discuss how to treat them with a geometric approach.

Can describe collective oscillations of the nuclear shape in terms of collective coordinates, in the context of the “Bohr” Hamiltonian  is elongation of nuclear ellipsoidal shape – related to ratio of major and minor axes.  is the deviation from axial symmetry along the major axis. It is measured as an angle from 0 to 30 degrees

H = T + T rot + V  Not the full story –deformed nuclei can rotate – need to include the Euler angles. Distinction between body-fixed and lab coordinates. Rotational spectra

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

Vibrator (H.O.) E(I) = n (   0 ) R 4/2 = 2.0 n = 0 n = 1 n = 2 V ~ C  

Rotational states Vibrational excitations Rotational states built on (superposed on) vibrational modes Ground or equilibrium state V ~ C 1     C 2  

TINSTAASQ

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

Why do we need to simplify – why not just calculate with the Shell Model????

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

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: 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

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 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 Talmi – “flying elephants” Beware of multi-parameter models that appear to do well. There are not so many truly independent observables

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  Basically the IBA is a Hamiltonian written in terms of s and d bosons and their interactions. It is written in terms of boson creation and destruction operators. Let’s briefly review their key properties.

XX X X X X X X +++… Phonon number not constant Compare “phonon” models. Excitations are particle-hole excitations relative to Fermi surface 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  g.s 1-phonon 2 + state

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

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 +

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

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

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. 0 (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.

Why are creation and destruction operators useful? Example: Consider a vibrational nucleus with quadrupole phonon excitations, with angular momentum 2+. These are vibrational excitations, with energy, say, E. Therefore the lowest energy state will have N=0 of these vibrations (because each one costs energy E). The lowest excited state will have one of them, and excitation energy E. The next excited states will have angular momena 4,2,and 0 formed by vector coupling two vibrations, each with angular momentum 2. These 3 states will be degenerate at an energy 2E ,2,0 0 E 2E 1 2 Why is the B(E2: 4 – 2) = 2 x B(E2: 2– 0) ?? The wave functions in the shell model are each linear combinations of 100’s – 1000’s of components and it is not obvious why such a simple result emerges. However, it is trivial to see using destruction operators WITHOUT EVER KNOWING ANYTHING ABOUT THE DETAILED STRUCTURE OF THESE VIBRATIONS !!!!

That relation is based on the 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 Group is called U(6)

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

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 name for this application of Group Theory is “Spectrum Generating Algebras” Next 8 slides give an introduction to the Group Theory relevant to the IBA. If the discussion of these is too difficult or too fast, don’t worry, you will be able to understand the rest anyway. Throughout these lectures, the symbol on a slide means that the slide contains some technical stuff that can be skipped

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 : Spectrum generating algebra !!

Group theory of the IBA U(6) 36 generators conserve N U(5) 25 generators conserve n d Suppose: H = α 1 C U(6) + α 2 C U(5) (1) All states of a given nucleus have same N. So, if α 2 = 0, i.e., H = α 1 C U(6) only, then all states would be degenerate. But these states have different n d. Thus, if we consider the full eq. 1, then the degeneracy is broken because C U(5) gives E = f (n d ). In group notation Dyn. Symm. U(6)  U(5)  … Recall: O(3)  O(2)

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

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