Transcript Symmetries Galore

Symmetries Galore
“Not all is lost inside the triangle”
(A. Leviatan, Seville, March, 2014)
R. F. Casten
Yale
CERN, August, 2014
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
What is the force that binds nuclei?
Why do nuclei do what they do?
•Simplicity out of complexity – Macroscopic
How the world of complex systems can display such remarkable regularity
and simplicity
What are the simple patterns that nuclei
display and what is their origin ?
We will look at a model that lives in both camps
The macroscopic perspective (many-body quantal
system) often exploits the idea of symmetries
These describe the basic structure of the object. Geometrical
symmetries describe the shape.
Symmetry descriptions are usually analytic and parameter-free
( except for scale). (Scary group theory comment.)
One model, the Interacting Boson Approximation (IBA) model,
is expressed directly in terms of symmetries.
Unfortunately, very few physical systems, especially in atomic
nuclei, manifest a symmetry very well.
We will see that this statement is now being greatly modified
and symmetries may play a much larger role than heretofore.
Simple Observables - Even-Even Nuclei
2+
1000
Relative B(E2) values

1

1
E (4 )
R4 / 2 
E (2 )
4+
330
B ( E 2; 41  21 )
2+
100
B ( E 2; 21  01 )
0+
0
Jπ
E (keV)
B ( E 2; J i  J f ) 
1
2J i  1
 i E2 
2
f
Example of a geometrical symmetry – ellipsoidal,
axially symmetric nuclei
Nuclei that are non-spherical can rotate (1952)!
Rotational energies follow the quantum symmetric top:
8+
E(J)Rot.  (
E(0)Rot.  0,
ħ2/2I
)J(J+1)
E(2)Rot.  6,
R4/2= 3.33
6+
4+
2++
0
E(4)Rot.  20
6+
690
4+
330
2+
100
0+
0
J
E (keV)
Amplifies
structural
Benchmark differences
Paradigm
Value of paradigms
?
700
333
Centrifugal
stretching
100
Without
0
rotor
Rotor
paradigm
( ħ2/2I ) J(J + 1)
Deviations
Identify additional
degrees of freedom
Rotational states built on
So, identification of the
(superposed on)
vibrational modes
characteristic predictions of
Real life example
a
symmetry both tells us the basic
nature of the system and can
highlight specific deviations from
the perfect symmetry which can
point to additional degrees of
freedom. Vibrational
excitations
Rotor
Rotational
states
8+
Ground or
equilibrium
state
E(I)  ( ħ2/2I )I(I+1)
6+
4+
2++
0
R4/2= 3.33
The IBA – A collective model built on a
highly truncated shell model foundation
(only configurations with pairs of nucleons coupled to
states with angular momentum
0 (s bosons) or 2 (d bosons)
Embodies the finite number of valence nucleons. Like the
shell model but opposed to traditional collective models,
the predictions depend on N and Z
Shell Model Configurations
Fermion
configurations
The IBA
Roughly, gazillions !!
Need to simplify
Boson
configurations
(by considering only
configurations of
pairs of fermions
with J = 0 or 2.)
s bosons and d
bosons as the basic
building blocks of
the collective states
Huge truncation of the shell model
Symmetries of the IBA
U(5)
s and d bosons:
6-Dim. problem
vibrator
U(6)
SU(3)
rotor
Magical group
theory stuff
happens here
R4/2= 2.5
Def.
R4/2= 2.0
R4/2= 3.33
Sph.
IBA Symmetry Triangle
O(6)
γ-soft
Three Dynamic symmetries,
nuclear shapes
What are these symmetries?
Idealized structures whose
predictions follow analytic
formulas, with states labeled
by good quantum numbers.
Proliferation of Symmetries
O(6) PDS – SU(3) QDS
QDS
PDS
AoR
ENTER PDS,QDS:
Recent
work
(Leviatan,
Van Isacker,
Alhassid,
However,
most nuclei
do not
exhibit
the idealized
symmetries:
Bonatsos,
Pietralla,
Cakirli, rfc, ..):Most
symmetries
notinside
limitedthe
to
So,
is theirCejnar,
role just
as benchmarks?
nuclei lie
vertices:
triangle
permeated
by symmetry
elements:
AoR,
QDS, PDS
triangle
where
chaos and
disorder are
thought
to reign.
PDS, QDS: what are these things?
They are various situations in which some of
the features of a Dyn.Sym. persist even though
there is considerable symmetry-breaking.
PDS: Some of the levels have the pure
symmetry [such as SU(3)] and others are
severely mixed
QDS: Some of the degeneracies characteristic
of a symmetry persist and some of the wave
function correlations persist.
Typical SU(3) Scheme
(for N valence nucleons)
(b,g) vibrations
.
.
.
.
Characteristic signatures:
• Degenerate bands
within a group
(l,m)
• Vanishing B(E2) values
between groups
SU(3)
O(3)
What do real nuclei look like – what are the data??
(b,g) vibrations
Totally typical example
Similar in many ways to SU(3).
But note that the excited (b,g) vibrational
excitations are not degenerate as they should be in SU(3). Also there
are collective B(E2) values from the g band to the ground band.
Most deformed rotors are not SU(3).
Clearly, SU(3) is severely broken
Or so we thought
B(E2) values in deformed rotor nuclei (typical values)
b
12 Wu
2 Wu
300 Wu
So, “clearly”, this violates the predictions of SU(3). So, realistic
calculations have broken the symmetries (mixed their basis states).
The degeneracies, quantum numbers, selection rules of the
symmetry no longer apply. …. Or so we thought
What is an SU(3)-PDS?
A bit weird!!
It is based on the IBA SU(3) symmetry BUT breaks it while
retaining pure SU(3) symmetry ONLY the ground and g
bands which preserve SU(3) exactly. All other states are
severely mixed!
Why would we need such a thing? We have excellent fits
to the data with numerical IBA calculations that break
SU(3). Why would we think g band preserves SU(3)?
Partial Dynamical Symmetry (PDS)
SU(3)
(l,m)
(l,m)
(l,m)
PDS: ONLY g and ground bands are pure SU(3).
(l,m)
So, expect PDS to predict vanishing B(E2) values between these bands
as in SU(3). How can that possibly work since empirically these B(E2)
values are collective !? BUT, g to ground B(E2)s CAN be finite in the PDS
SU(3) PDS: B(E2) values
g, ground states pure SU(3), others mixed. Generalized T(E2)
T(E2)PDS = e [ A (s+d + d+s) + B (d+d)] = e [T(E2)SU(3) + C (s+d + d+s) ]
First term gives zero from SU(3) selection rules. But not second.
Hence, relative g to ground B(E2) values:
M(E2: Jg – Jgr)
eC < Jgr | (s+d + d+s) | Jg >
-------------------- = -----------------------------------M(E2: Jg – J’ gr)
eC < J’gr | (s+d + d+s) | Jg >
eC cancels
Relative INTERband g to gr B(E2)’s are parameter-free!
So, Leviatan: PDS works well for 168 Er (Leviatan, PRL, 1996).
Accidental or a new paradigm?
Tried extensive test for 47 Rare earth nuclei.
Key data will be relative g to ground B(E2) values.
Illustrative
results
Transitional nuclei
Lets look into these predictions and comparisons a little deeper.
Compare to “Alaga Rules” – what you would get for a pure rotor for
RATIOS of squares of transition matrix elements [B(E2) values] from
one rotational band to another.
5:100:70 Alaga
168-Er: The Alaga rules, valence space, and collectivity
PDS has pure g, gr band K,
so why differ from Alaga?
Ans: PDS (from IBA) is
valence space model:
predictions are Nval – dep.
PDS differs from Alaga
solely due to N-Dep effects
that arise from the fermion
underpinning (Pauli
Principle).
Valence collective models that agree with the PDS have minimal
mixing.
Predictions deviating from the PDS signal configuration mixing.
Using the PDS to better understand collective model
calculations (and collectivity in nuclei)
• PDS B(E2: g – gr): the sole reason they differ from the Alaga
rules is that they take into account the finite number of
valence nucleons.
• Therefore, IBA calculations (like WCD) that agree with the
PDS differ from the Alaga rules purely because of finite
valence nucleon number effects. Not heretofore recognized.
• IBA – CQF deviates further from the Alaga rules, agrees
better with the data (and has one fewer parameter).
• IBA – CQF: The differences from the PDS are due to mixing
• Can use the PDS to disentangle valence space from mixing!
Now a highly selective O(6) PDS
Again, as one enters the triangle expect vertex symmetries to be
highly broken. Consider structures descending from O(6)
(s,t)
Calculate wave functions throughout triangle, expand in O(6) basis
s
good along a line
descending from O(6)
Remnants of pure O(6) (for the ground state band only) persist along a line in the
triangle extending into the region of well-deformed rotational nuclei
Order, Chaos, and the Arc of
regularity
Order and chaos? What happens
generally inside the triangle
3
6+, 4+, 3+, 2+, 0+
2
4+, 2+, 0+
1
2+
0
0+
nd
Arc of Regularity is a narrow zone in the triangle
with high degree of order amidst regions of chaotic
behavior. What is going on along the AoR?
Whelan, Alhassid, ca 1989
Role of the arc as a “boundary” between two classes of structures
AoR – E(0+2) ~ E(2+2)
b<g
b<g
Not just a theoretical curiosity: 8 nuclei in the rare
earth region have been found to lie along the arc
All SU(3) degeneracies and all analytic ratios of 0+
bandhead energies persist along the Arc.
QDS
The symmetry underlying the Arc of Regularity
is an SU(3)-based Quasi Dynamical Symmetry
Summary
O(6) PDS – SU(3) QDS
PDS
QDS
AoR
Elements of structural symmetries abound
throughout the triangle
Principal Collaborators:
R. Burcu Cakirli
D. Bonatsos
Klaus Blaum
Aaron Couture
Thanks to Ami Leviatan, Piet Van Isacker, Michal Macek, Norbert
Pietralla, C. Kremer for discussions.
BACKUPS
SU(3)
(l,m)
(l,m)
(l,m)
• States labelled by quantum
numbers (l,m)
(l,m)
• Degenerate bands within irrep
T(E2) = eQ =
e[(s†
d +d
†s)
-
7
2
(d † d )(2)]
Creation and destruction operators as
“Ignorance operators”
Example: Consider the case we have just discussed – the spherical vibrator.
Why is the B(E2: 4 – 2) = 2 x B(E2: 2– 0) ??
Difficult to see with Shell Model wave functions with 1000’s of components
However, as we have seen, it is trivial using destruction operators WITHOUT
EVER KNOWING ANYTHING ABOUT THE DETAILED STRUCTURE OF THESE
VIBRATIONS !!!! These operators give the relationships between states.
4,2,0
2E
2
E
2
1
0
0
Finite Valence Space effects on Collectivity
PDS B(E2: g to Gr)
Alaga
Note: 5 N’s up, 5 N’s down:
1 with N
Most dramatic, direct evidence for explicit valence space size in emerging collectivity
R.F. Casten, D.D. Warner and A. Aprahamian , Phys. Rev. C 28, 894 (1983).
~ 0.03
~ 0.1 x ~0.03 ~ 0.003
Concepts of group theory
Generators, Casimirs, Representations, conserved quantum numbers,
degeneracy splitting
Generators of a group: Set of operators , Oi that close on commutation.
[ Oi , Oj ] = Oi Oj - Oj Oi = Ok 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†s + d†d = ns + nd
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
Let’s illustrate group chains and degeneracybreaking.
†
Consider a Hamiltonian that is a function ONLY of:
d†d
That is:
ss +
H = a(s†s + d†d) = a (ns + nd ) = 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
H’ = H + b d†d = aN + b nd
Now, add a term to this Hamiltonian:
Now the energies depend not only on N but
also on nd
States of a given nd are now degenerate. They
are “representations” of the group U(5).
States with different nd are not degenerate
2a N + 2
a
H’ = aN + b d†d = a N + b nd
N+1
b
2
1
0
0
2b
N
0
nd
E
U(6)
H’ = aN
U(5)
+
b d†d
Concept of a Dynamical Symmetry
Each successive
term:
• Introduces a new
sub-group
N
• A new quantum
number to label the
states described by
that group
• Adds an
eigenvalue term
that is a function of
the new quantum
number, hence
• Breaks a previous
degeneracy
Now, WHY are spin increasing transitions so small in Alaga rules
and so much larger in the data than in the PDS?
DR = 4 !!
0, 4
4
0, 2
0, 0
Jint , R
2
0
Jtot
2g
Jtot
2,
0
Jint , ~R
~ Forbidden
Deviations from Alaga: g – gr. band mixing. E2 matrix element :
ME(E2) = [ a < || >unpert
+ b < || > mix ]
Can be larger or smaller than the unperturbed.
BUT, if unperturbed is forbidden, then
ME(E2) = [b < | > mix] > unpert (which is zero).
So mixing always increases forbidden/weak transitions.
Review of phonon creation and destruction operators
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.
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).