The IBA The Interacting Boson Approximation Model Preliminary review of collective behavior in...
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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 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
Slide 2
J = 2, one phonon vibration Spherical nuclei
Slide 3
Slide 4
Even-even Deformed Nuclei Rotations and vibrations
Slide 5
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.
Slide 6
Slide 7
0+0+ 2+2+ 4+4+ 6+6+ 8+8+ Rotational states Vibrational
excitations Rotational states built on(superposed on) vibrational
modes Ground or equilibirum state
Slide 8
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
!
Slide 9
Slide 10
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.
Slide 11
Crucial for structure Microscopic origins of collectivity
correlations, configuration mixing and deformation: Residual
interactions
Slide 12
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.
Slide 13
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
Slide 14
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
Slide 15
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
Slide 16
Slide 17
Slide 18
0+0+ 2+2+ 6 +... 8 +... Vibrator (H.O.) E(I) = n ( 0 ) R 4/2 =
2.0 n = 0 n = 1 n = 2 V ~ C
Slide 19
0+0+ 2+2+ 4+4+ 6+6+ 8+8+ Rotational states Vibrational
excitations Rotational states built on (superposed on) vibrational
modes Ground or equilibrium state V ~ C 1 C 2
Slide 20
TINSTAASQ
Slide 21
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
Slide 22
Why do we need to simplify why not just calculate with the
Shell Model????
Slide 23
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
Slide 24
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 = # of val. protons + # 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
Slide 25
Modeling a Nucleus 154 Sm3 x 10 14 2 + 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: 26 2 + states Is it conceivable that
these 26 basis states are correctly chosen to account for the
properties of the low lying collective states?
Slide 26
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
Slide 27
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. Lets briefly review their
key properties.
Slide 28
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
Slide 29
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
Slide 30
Why s, d bosons?
Slide 31
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 +
Slide 32
Slide 33
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 ]
Slide 34
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
Slide 35
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 dont know
the structure of a phonon but, for many predictions, we dont need
to know its microscopic basis.
Slide 36
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. 0 2 4,2,0 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 100s 1000s 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 !!!!
Slide 37
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)
Slide 38
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
Slide 39
Brief, simple, trip into the Group Theory of the IBA DONT 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, dont 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
Slide 40
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:
Slide 41
Sub-groups: Subsets of generators that commute among
themselves. e.g: d d 25 generatorsspan 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 #
Slide 42
Simple example of dynamical symmetries, group chain,
degeneracies [H, J 2 ] = [H, J Z ] = 0 J, M constants of
motion
Slide 43
Lets 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).
Slide 44
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
Slide 45
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
Slide 46
Concept of a Dynamical Symmetry N OK, heres the key point :
Spectrum generating algebra !!
Slide 47
Group theory of the IBA U(6) 36 generators conserve N U(5) 25
generators conserve n d Suppose: H = 1 C U(6) + 2 C U(5) (1) All
states of a given nucleus have same N. So, if 2 = 0, i.e., H = 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)
Slide 48
OK, heres 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.
Slide 49
Group Structure of the IBA s boson : d boson : U(5) vibrator
SU(3) rotor O(6) -soft 1 5 U(6) Sph. Def. Magical group theory
stuff happens here Symmetry Triangle of the IBA
Slide 50
Classifying Structure -- The Symmetry Triangle Most nuclei do
not exhibit the idealized symmetries but rather lie in transitional
regions. Mapping the triangle. Sph. Deformed