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fermions cj
N bosons
A nucleons
valence nucleons
N nucleon pairs
L = 0 and 2 pairss,d
even-even nuclei
2.2 The Interacting Boson Approximation
A. Arima, F. Iachello, T. Otsuka, I Talmi
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Schrödinger equation in second quantisation
N s,d boson system
with
N=cte
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The hamiltonian is written in terms of the 36 generators of U(6):
])~
(,)~
[( '''"
†"'
† kll
kll bbbb
(2k 1)(2k' 1)( 1)k k' (kk '' k"")k ", "
[( 1)k k ' k "k k' k"
l"' l l'
l' l"(bl
† ˜ b l"' ) "k "
k k' k"
l" l' l
ll"' (bl "
† ˜ b l' ) "k " ]
.
kll bb )
~( '
†
Construction of a dynamical symmetry as an example U(5) limit:
[(d† ˜ d )k ,(d† ˜ d ) '
k ' ]
kk ddbb )~
()~
( †2
†2
(2k 1)(2k' 1)( 1)k k' (kk '' k"")k ", "
k k' k"
2 2 2
(d† ˜ d )"k" [( 1)k k ' k " 1]
Now, consider the 10 odd k generators only and then the angular momentum ones.
LnN
SUU
d
)3(0)5(0)5()6(
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U(5)
SU(3)
SO(6)196Pt
156Gd
110Cd
U(5): Vibrational nuclei
SO(6): -unstable nuclei
SU(3): Rotational nuclei (prolate)
Dynamical symmetries of a N s,d boson system
U(5) SO(5) SO(3) SO(2) {nd} () L M
U(6) SO(6) SO(5) SO(3) SO(2)[N] <> () L M SU(3) SO(3) SO(2)
() L M
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Analytic solutions are associated with each dynamical symmetry:
U(5): H = 1C1[U(5)] + ’1C2[U(5)] + 4C2[O(5) ] +5C2[ SO(3)] E(nd,L) = 1nd+ ’1nd( nd+ 4) + 4(+3) +5L(L+1)
H = 1C1[U(5)] + ’1C2[U(5)] + 2C2[SU(3) ] +3C2[SO(6) ] +4C2[SO(5) ] + 5C2[ SO(3)]
Again using first and second order Casimir operators a six parameter hamiltonian results::
which needs to be solved numerically.
SU(3): H = 2C2[SU(3)] +5C2[ SO(3)] E(,L) = 2+5L(L+1)
SO(6): H = 3C2[SO(6)] + 4C2[O(5) ] +5C2[ SO(3)] E(,L) = 3(+4) + 4(+3) +5L(L+1)
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A dynamical symmetry leads to very strict selection rules that can be used to test it.
If an operator is a generator of a subalgebra G then due to the property:
[Gi,Gj]=kcijkGk.and
E = a f() =>Ek = a f() k withk {Gk}
it cannot connect states having a different quantum number with respect to G.
Example: 196Pt and its E2 properties.
)2(2 )~
(ˆ sddsT E 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 ||=1
Experimentally: Q(2+1) = +0.66(12) eb
B(E2)= 0?
H.G. Borner, J. Jolie, S.J. Robinson, R.F. Casten, J.A. Cizewski, Phys. Rev. C42(1990) R2271
195Pt(n,)196Pt + GRID method
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Nuclear shapes associated with the four dynamical symmetries
The shapes can be studied using the coherent state formalism.
01
;,,
N
dsN
N
using the intrinsic state (Bohr) variables:
0)(2
sincos
1,;,, 220
N
dddsN
N
Then the energy functional:
,;,,,;,,
,;,,ˆ,;,,),;,,(
NN
NHNNE
can be evaluated for each value of and
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U(5) limit: irrelevant: spherical vibrator
SO(6) limit: flat: -unstable rotor
SU(3) limit: prolate rotor
SU(3) limit: oblate rotor
SU(3)
E U(5)
SO(6)
00 ;0 00 ;0
0;0 00 60;0 00
0°
60°
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Shape phases and critical point solutions.
QQN
naH d ˆ.ˆ)1(ˆˆ
Most nuclei are very well described by a very simple IBA hamiltonianof Ising form:
with two structural parameters and and a scaling factor a
)2()2( )~
()~
(ˆ ddsddsQ with
generatessphericalshape
generatesdeformedshape
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U(5)
The rich structure of this simplehamiltonian are illustrated by the extended Castentriangle
SU(3)
2/7
1 U(5) limit U(6) U(5) SO(5) SO(3) 0,0 SO(6) limit U(6) SO(6) SO(5) SO(3)
27,0 SU(3) limit U(6) SU(3) SO(3)
The simple hamiltonian has four dynamical symmetries
QQN
naH d ˆ.ˆ)1(ˆˆ
SO(6)
2/7SU(3)
27,0 SU(3) limit U(6) SU(3) SO(3)
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60000:)3cos( 003
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 observeshape phase transitions of the groundstate configuration.
They are analogue to phase transitions in crystals
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Thermodynamic potential: );,( TP
Externalparameters
Orderparameter
Energy functional: ),;,,( NE
L. Landau
Landau theory of continuous phase transitions (1937) describes theseshape phase transitions.
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
PT
),( 11min TP ),( 21min TP
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T
c
c
c
min
Tc
P0,T,min
Tc
P0,T,
First order phase transition with P = P0= const
c
c
c
min
c
P0,T,min
Tc
P0,T,Second order phase transition
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...),(),(),();,( 4320 TPCTPBTPATP
);,( 0TP
);,( TP
should be continuous everywhere.
if discontinuous at 0 : first order phase transition.
2
2 );,(
TP if discontinuous at 0: second order phase transition.
with TPTPB ,;0),(
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Extremum are at:
0]),(4),(3),(2[ 2 TPCTPBTPA0);,(
TP
C
ACBB
8
3293
0
2
Our case: TPTPCTPB ,0),(,0),(
0);,(
2
2
TP
0),(12),(6),(2 2 TPCTPBTPA
0),(0 TPAalways:
cATPATPB ),(,0),(0
cATPATPB ),(,0),(0
0),( TPC
and
),(4
),( 2
TPC
TPBAc
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...),(),(),();,( 4320 TPCTPBTPATP
0)),(),(),(( 22 TPCTPBTPA
Both minima become degenerated at:
),(),(4),( 2 TPCTPATPB
or at
),(2
),(
TPC
TPBc
c 0),( TPA
),(4),(
),(2
TPCTPB
TPA
To fullfill this equation and the one for 0 ),(32
),(9),(
2
TPC
TPBTPA
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00
0),( TPB 0),( TPB
B
-A
0
C
ACBB
8
3293 2
0
C
ACBB
8
3293 2
0
Solution:
First order phase transitions at: Second order at:
),(4
),(),(
2
TPC
TPBTPA
),(4),(
),(2
TPCTPB
TPA
0),(),( TPBTPA
0
),( TPA
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...4321)1(
1 64222
So we can absorb it by allowing negative values !
60000:)3cos( 003
and
06000 00 ßß
Energy functional in coherent state formalism
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One obtains then:
...),,(),,(
),,(),(),;,,(
43
20
NCNB
NANENE
when we fix N: ...),(),(),();,( 4320 TPCTPBTPATP
72
)1)(1(4),,( NNB
)}84)(1({),,( 2 NNNA
54
85
840),,(
2
2
0
N
NNA
prolate-oblate00),,( 0 NB
N
The first order phase transitions should occur when
spherical-deformed
The isolated second order transition at:
8584
0,0),,(),,( 00
NN
andNBNA
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spherical = 0
prolatedeformed > 0
oblatedeformed < 0
Triple point ofnuclear deformation
: first order transition
: isolated second order transition
00
00
00 I
III
II
P
T
J.Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linnemann, V. Werner, Phys. Rev. Lett. 89 (2002)182502
Landau theory and nuclear shapes.
Thermodynamic potential:
External parameters
Order parameter
Energy functional:
(P,T;) E(N,)
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The shape phase transitions can be seen by the groundstate energies.
E
SU(3)
O(6)
SU(3)
U(5) (N=40)
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The quadrupole moment corresponds to the control parameter 0:
N=10 N=40
N=10 N=40
A sensitive signature is in particular the B(E2;22+-> 21
+)
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J. Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linnemann, V. Werner, Phys.Rev.Lett. 89(2002)182502P. Cejnar and J. Jolie, Rep. on Progress in Part. and Nucl. Phys. 62 (2009) 210.
dynamical symmetry
First order phasetransition
Second order phasetransition (isolated)
sphericalprolatedeformed
oblatedeformed
Conclusion: the following shapes phase transitions are obtained:
J. Jolie, R.F. Casten, P. von Brentano, V. Werner, Phys.Rev.Lett. 87(2001)162501
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Examples for the prolate-oblate and sherical-prolate phase transition
PbHgPtOsWHfYb
104 106 108 110 112 114 116 118 120 122 124 126
200Hg198Hg
196Pt194Pt
192Os190Os188Os
186W184W182W
180Hf
Samarium isotopes
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N=10N=20
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50
82
Normal states Intruder states
110Cd
2p-2hpair + Q+ ...
K. Heyde, et al. Nucl. Phys. A466 (1987) 189.
inmix
mixn
HH
HHˆˆ
ˆˆThis can be described in the IBM by a N (normal) plus N+2 (intruder) systemwhich might mix. :
with
and)0(||)0(|| )
~~()( ddddssssHmix
2.2.4 Core excitations
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Iz = + 1/2
Iz = - 1/2
Up(6)
Uh(6)
U(6)
Also new kinds of symmetries are possible: Intruder or I-spin
always fulfilled[H,Iz] =0
[H,I2] = 0
[H,I+] = [H,I-] = 0 intruder-analog state
good intruder Spin
K. Heyde, C. De Coster, J. Jolie, J.L. Wood, Phys. Rev. C 46 (1992), 541
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I=0
I=1/2
I=1
I=2
I=3/2
+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
Intruder analog states
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The cadmium isotopes are unique in several respects.
-) protons nearly fill the Z=50 (Sn) shell;
-) neutrons are near mid-shell (N=66)
-) there are 8 stable isotopes of Cd.
This is clearly the mass region where we can learn about nuclear structure.
2.3 A case study: 112Cd
M. Délèze, S. Drissi, J. Jolie, J. Kern, J.P. Vorlet, Nucl. Phys, A554 (1993) 1
Pd (,2n) Cd reaction allowedthe establishment of multiphononstates up to nd=6.
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5.39ps
0.68ps0.73ps
But, above 1.2 MeV additional states exist and build asecond collective structure.
< 2.1ps
< 2.8ps
0.7ps
and
The additional states are intruder states presenting 2 particle- 2 hole excitations across Z=50. They lead to shape coexistence and can be described using the SO(6) limit.
E[N,nd,,L] = nd + (+3) +L(L+1)
intrL] = (+3) + (+3) +L(L+1)
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U(6)
U
U(5) O(6)
U
U
O(5)
O(3)
J. Jolie and H. Lehmann, Phys. Lett B342 (1995)
normal states
112Cd
Intruder states
)0(||)0(|| )~~
()( ddddssssHmix is a O(5) scalar.
Symmetries can play a dominant role in shape coexistence.
Wavefunctions with O(5) symmetry have fixed seniority of d-bosons.
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)0(
)0(
5
1
5
1
6
555
6
1
ddssddss
)0(||)0(|| )~~
()( ddddssssHmix
)0)(0(0]2[ T
0)0(2]2[0)0(0]2[
5
1
6
55
6
1
TT
Cannot connect intruder withnormal states
=N+2
max=N0)0(2]2[
5
1
6
5
TH effmix
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Moreover one can rewrite:
)0(||)0(|| )~~
()( ddddssssHmix
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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’ ) Elevel J and placements of E from excitation function varying En
from angular distributions
n = 1.25ps1.200.42
= 1.16ps0.490.27
= 0.67ps0.210.13
= 1.20ps0.830.35
= 0.42ps0.100.07
= 0.51ps0.170.10
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(n,n´) with 3.4 and 4 MeV neutrons for lifetimes and coincidences allowed theextension 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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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
Absolute B(E2) values for the decay of three phonon states in 110Cd
![Page 42: fermions c j N bosons A nucleons valence nucleonsN nucleon pairs L = 0 and 2 pairs s,d even-even nuclei 2.2 The Interacting Boson Approximation A](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e965503460f94b99fed/html5/thumbnails/42.jpg)
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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Fribourg-Kentucky-Köln Data B(E2) Values in Six Valence Proton Configurations
Neutron Cd Intruder Ru Ba
Number B(E2;230A) B(E2;2101) B(E2;2101)
27182362
558175664
1415457086166
61167748668 2430
169870
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).