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2.2 The Interacting Boson Approximation
even-even nuclei
A nucleons
N bosons
L 0 and 2 pairs
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Schrödinger equation in second quantisation
Ncte
N s,d boson system
with
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The hamiltonian is written in terms of the 36
generators of U(6)

.

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Dynamical symmetries of a N s,d boson system
? U(5) ? SO(5) ? SO(3) ? SO(2) nd
(t) L M U(6) ? SO(6) ? SO(5)
? SO(3) ? SO(2) N ltsgt (t)
L M ? SU(3) ? SO(3) ?
SO(2) (l,m) L M
U(5) Vibrational nuclei
SO(6) g-unstable nuclei
SU(3) Rotational nuclei (prolate)
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Again using first and second order Casimir
operators a six parameter hamiltonian
results which needs to be solved
numerically.
H k1C1U(5) k1C2U(5) k2C2SU(3)
k3C2SO(6) k4C2SO(5) k5C2 SO(3)
Analytic solutions are associated with each
dynamical symmetry
U(5) H k1C1U(5) k1C2U(5) k4C2O(5)
k5C2 SO(3) E(nd,t,L) k1nd k1
nd( nd 4) k4t( t3) k5L(L1)
SU(3) H k2C2SU(3) k5C2 SO(3)
E((l,m),L) k2(l2m2 lm 3(lm))k5L(L1)
SO(6) H k3C2SO(6) k4C2O(5) k5C2
SO(3) E(s,t,L) k3 s( s4) k4t(
t3) k5L(L1)
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A dynamical symmetry leads to very strict
selection rules that can be used to test it.
Example 196Pt and its E2 properties.
If an operator is a generator of a subalgebra G
then due to the property
Gi,GjSkcijkGk.
and
EY a f(a) Y gtEYk a f(a) Yk
with Yk ? GkY
it cannot connect states having a different
quantum number with respect to G.
is an SO(6) generator
E2 transitions between different SO(6)
representations are forbidden.
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Also quadrupole moments are equal to zero
because of SO(5) (seniority) and the d-boson
number changing E2 operator Dt1
Experimentally Q(21) 0.66(12) eb
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Nuclear shapes associated with the four dynamical
symmetries
The shapes can be studied using the coherent
state formalism.
using the intrinsic state (Bohr) variables
Then the energy functional
can be evaluated for each value of b and g.
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E
U(5)
SU(3)
g
SO(6)
0
b
b
U(5) limit irrelevant spherical vibrator
SO(6) limit flat g-unstable rotor
SU(3) limit prolate rotor
SU(3) limit oblate rotor
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Shape phases and critical point solutions.
Most nuclei are very well described by a very
simple IBA hamiltonian of Ising form
with
with two structural parameters h and c and a
scaling factor a.
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The simple hamiltonian has four dynamical
symmetries
The rich structure of this simple hamiltonian are
illustrated by the extended Casten triangle
h 1
h 0
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Energy functional in coherent state formalism
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Shape phase transitions in the atomic nucleus.
When studying the changes of the nuclear shape
one might observe shape phase transitions of the
groundstate configuration.
They are analogue to phase transitions in crystals
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Landau theory of continuous phase transitions
(1937) describes these shape phase transitions.
L. Landau
J.Jolie, P. Cejnar, R.F. Casten, S. Heinze, A.
Linnemann, V. Werner, Phys. Rev. Lett. 89 (2002)
182502. P. Cejnar, S. Heinze, J.Jolie, Phys.
Rev. C 68 (2003) 034326
Thermodynamic potential
Order parameter
External parameters
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T
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with
should be continuous everywhere.
if discontinuous at x0 first order phase
transition.
if discontinuous at x0 second order phase
transition.
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Extremum are at
Our case
and
always
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Both minima become degenerated at
To fullfill this equation and the one for x0
F
Fc
or at
x
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Solution
x0
B
-A
First order phase transitions at
Second order at
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Energy functional in coherent state formalism
and
So we can absorb it by allowing negative b values
!
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One obtains then
when we fix N
The first order phase transitions should occur
when
spherical-deformed
prolate-oblate
The isolated second order transition at
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Landau theory and nuclear shapes.
first order transition
isolated second order transition
Triple point of nuclear deformation
oblate deformed b lt 0
prolate deformed b gt 0
spherical b 0
Thermodynamic potential
Energy functional
E(N,h,cb,g)
F(P,Tx)
Order parameter
External parameters
J.Jolie, P. Cejnar, R.F. Casten, S. Heinze, A.
Linnemann, V. Werner, Phys. Rev. Lett. 89
(2002)182502
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The shape phase transitions can be seen by the
groundstate energies.
U(5)
(N40)
h
E
O(6)
SU(3)
SU(3)
c
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The quadrupole moment corresponds to the control
parameter b0
N10
N40
A sensitive signature is in particular the
B(E222-gt 21)
N10
N40
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Conclusion the following shapes phase
transitions are obtained
oblate deformed
dynamical symmetry
First order phase transition
Second order phase transition (isolated)
spherical
prolate deformed
J. Jolie, R.F. Casten, P. von Brentano, V.
Werner, Phys.Rev.Lett. 87(2001)162501
J. Jolie, P. Cejnar, R.F. Casten, S. Heinze, A.
Linnemann, V. Werner, Phys.Rev.Lett.
89(2002)182502 P. Cejnar and J. Jolie, Rep. on
Progress in Part. and Nucl. Phys. 62 (2009) 210.
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Examples for the prolate-oblate and
sherical-prolate phase transition
Samarium isotopes
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N10 N20
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Normal states
Intruder states
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2.2.4 Core excitations
50
110
Cd
This can be described in the IBM by a N (normal)
plus N2 (intruder) system which might mix.
D E(2p-2h) - Dpair DQnp ...
with
and
K. Heyde, et al. Nucl. Phys. A466 (1987) 189.
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Also new kinds of symmetries are possible
Intruder or I-spin
IBM
U(6)
Iz 1/2
Up(6)
EIBM
Iz - 1/2
Uh(6)
H,Iz 0
always fulfilled
H,I2 0
good intruder Spin
H,I H,I- 0
intruder-analog state
K. Heyde, C. De Coster, J. Jolie, J.L. Wood,
Phys. Rev. C 46 (1992), 541
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Intruder analog states
I2
I3/2
I1
I1/2
I0
1/2
3/2
-1/2
-3/2
1
2
-1
-2
0
Iz
H. Lehmann, J. Jolie, C. De Coster, B. Decroix,
K. Heyde, J.L. Wood, Nucl. Phys. A 621 (1997) 767
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2.3 A case study 112Cd
The cadmium isotopes are unique in several
respects. -) protons nearly fill the Z50 (Sn)
shell -) neutrons are near mid-shell (N66) -)
there are 8 stable isotopes of Cd. This is
clearly the mass region where we can learn about
nuclear structure.
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EN,nd,t,L k1 nd k4 t(t3) k5 L(L1)
But, above 1.2 MeV additional states exist and
build a second collective structure.
0.7ps
lt 2.8ps
lt 2.1ps
0.73ps
and
0.68ps
5.39ps
The additional states are intruder states
presenting 2 particle- 2 hole excitations across
Z50. They lead to shape coexistence and can be
described using the SO(6) limit.
EintrN,S,t,L k3 S(S3) k4 t(t3) k5
L(L1)
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Symmetries can play a dominant role in shape
coexistence.
is a O(5) scalar.
Wavefunctions with O(5) symmetry have fixed
seniority of d-bosons.
normal states
Intruder states
D
112
Cd
J. Jolie and H. Lehmann, Phys. Lett B342 (1995)
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Cannot connect intruder with normal states
SN2
SmaxN
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Moreover one can rewrite
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Inelastic Neutron Scattering (INS) experiment at
the Van de Graaff Accelerator of the University
of Kentucky (Prof S.W. Yates, Lexington USA).
(n,n g)
Elevel Jp and placements of Eg from excitation
function varying En d,t from angular
distributions
n
t 1.25ps1.200.42
t 1.20ps0.830.35
t 1.16ps0.490.27
t 0.42ps0.100.07
q
t 0.67ps0.210.13
t 0.51ps0.170.10
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(n,n) with 3.4 and 4 MeV neutrons for lifetimes
and coincidences allowed the extension to higher
low-spin states (P.E. Garrett et al. Phys. Rev.
C75 (2007)054310
There it becomes difficult to describe the
details.
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Absolute B(E2) values for the decay of three
phonon states in 110Cd
Harmonic vibrator (collective model)
finite N effects (IBM in the U(5)-limit)
intruder states within a U(5)-O(6) model
neutron-proton degree of freedom and symmetry
breaking
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Confirmation of the U(5)-O(6) picture.
F. Corminboeuf, T.B. Brown, L. Genilloud, C.D.
Hannant, J. Jolie, J. Kern, N. Warr and S.W.
Yates, Phys. Rev. C 63 (2001) 014305.
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B(E2) Values in Six Valence Proton Configurations

Fribourg-Kentucky-Köln Data
It works well only in a given shell.
M. Kadi, N. Warr, P.E. Garrett, J. Jolie, S.W.
Yates, Phys. Rev. C68 (2003) 031306(R).