I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
11
Geometrical Symmetries in Geometrical Symmetries in Nuclei-An IntroductionNuclei-An Introduction
ASHOK KUMAR JAINASHOK KUMAR JAINIndian Institute of TechnologyIndian Institute of Technology
Department of PhysicsDepartment of PhysicsRoorkee, IndiaRoorkee, India
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
22
OUTLINEOUTLINE
Brief IntroductionBrief Introduction Mean Field and Spontaneous Symmetry BreakingMean Field and Spontaneous Symmetry Breaking Symmetries, Unitary Transformation, & MultipletsSymmetries, Unitary Transformation, & Multiplets Discrete SymmetriesDiscrete Symmetries Nuclear ShapesNuclear Shapes Collective Hamiltonian, Wave-function, etc.Collective Hamiltonian, Wave-function, etc. Spheroidal Shapes – constraints on KSpheroidal Shapes – constraints on K Symmetric Top – Even-Even NucleiSymmetric Top – Even-Even Nuclei
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
33
Basic Idea - Mean FieldBasic Idea - Mean Field
Basic Nucleon-Nucleon interaction remains invariant Basic Nucleon-Nucleon interaction remains invariant under all the basic operations like Translational under all the basic operations like Translational invariance in space and time, Inversion in space invariance in space and time, Inversion in space and time, rotational invariance in space etc.and time, rotational invariance in space etc.
When many nucleons collect to make a nucleus, it When many nucleons collect to make a nucleus, it generates a generates a mean fieldmean field, which may break one or, , which may break one or, more of these symmetries. If the mean field were more of these symmetries. If the mean field were also to conserve all the basic operations, we will also to conserve all the basic operations, we will see very little structure in the nucleus.see very little structure in the nucleus.
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
4
6+
4+
2+
0+
8+
10+
2+
3+
4+
5+
6+
7+
8+
8-
7-
6-
5-
4-
8+
6+
4+
2+
0+
3-
2-
1-
5-
4-
6-
6+
4+
2+
0+
7-
6-
5-
4-
3-
6-
5-
4-
3-
2-
6+
5+
4+
3+
168Er
2.0
1.5
1.0
0.5
0
MeV
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
55
Spontaneous Breaking of the Spontaneous Breaking of the SymmetrySymmetry
This leads to the concept of spontaneous breaking of This leads to the concept of spontaneous breaking of the symmetries by the mean field of the nucleus. the symmetries by the mean field of the nucleus. Basic N-N interaction preserves all the symmetries Basic N-N interaction preserves all the symmetries yet the mean field may break them.yet the mean field may break them.
An example:An example:Transition from Shell Model to Nilsson ModelTransition from Shell Model to Nilsson Model
As the mean field changes, so does the spectrum. As the mean field changes, so does the spectrum. Further, the mean field changes at every pretext. Further, the mean field changes at every pretext. Change the angular mom, the excitation, the N/Z Change the angular mom, the excitation, the N/Z ratio, or the particle number, the mean field changes.ratio, or the particle number, the mean field changes.
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
66
How to obtain the Mean Field?How to obtain the Mean Field?Theorists are always concerned about this big Theorists are always concerned about this big
question. Fortunately, the gross properties, or the question. Fortunately, the gross properties, or the major phenomena come out OK for a variety of major phenomena come out OK for a variety of mean fields, which differ from each other only mean fields, which differ from each other only slightly.slightly.
As we shall see, the main features can be As we shall see, the main features can be understood by the application of symmetry understood by the application of symmetry arguments ro the mean field which in turn arguments ro the mean field which in turn depends on the nuclear shapes. depends on the nuclear shapes.
As most nuclei are now known to be deformed As most nuclei are now known to be deformed rather than being spherical, these arguments rather than being spherical, these arguments become widelybecome widely applicable.applicable.
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
77
Symmetry operation Symmetry operation Group of Unitary Transformation Group of Unitary Transformation Û Û An observable An observable
Unitary Transformation, Degeneracy Unitary Transformation, Degeneracy and Multipletsand Multiplets
QuuQ †1† uu
Since
Invariance under U implies that QuuQ 1 and
0],[ Qu
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
88
An useful Unitary transformation arising from H is exp(-i t H ), which gives
,QQee itHitH for all t
It implies a commutation of Q with H, and Q becomes invariant as H is a generator of time,
)1(' itHe itH
)1('z
ji Jie z Similarly, rotation about the z-axis can be generated by
For Jz to be invariant,
0],[ zJH
.
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
99
Concept of Multiplets and DegeneracyConcept of Multiplets and Degeneracy
, EHIf
'' )1( EJiHH z
.
and we also have
• either is an eigenstate of both H and Jz or,
• the eigenvalue E has a degeneracy.
Thus, and, both are eigenstates of H with the same energy eigenvalue E. An energy eigenstate can have n-fold degeneracy if n-fold rotation of about the z-axis leaves invariant. This degeneracy will be lifted if an interaction or, deformation violates this symmetry and multiplet arises.
'
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
10
An Example
J is a good quantum number for spherical
symmetry. A small deformation breaks the symmetry. If the deformed potential has axial symmetry
about the z-axis, Jz is the only conserved quantity. The (2j+1) fold degeneracy is lifted and multiplet
arises. Therefore, the quantum number is
used to label the states.
β
g 9/2
deformation
Ω=1/2
3/2
5/2
7/2
9/2
0.0
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
1111
Discrete Symmetries in NucleiDiscrete Symmetries in Nuclei
Most commonly encountered discrete symmetries in rotating nuclei are
1. parity P
2. rotation by π about the body fixed x, y, z axes, Rx (π), Ry (π,), Rz (π)
3. time reversal T, and
4. TRx (π), TRy (π), TRz (π).
These are all two fold discrete symmetries, and their breaking causes a doubling of states.
See Dobaczewski et al(2000) for complete classification
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
1212
Simple rules to work out the consequences of these symmetries:
1. When P is broken, we observe a parity doubling of states. A sequence like I+, I+1+, I+2+, … turns into I, I+1, I+2, … .
2. When Rx (π) is broken, states of both the signatures occur. The two sequences like I, I+2,…etc. and I+1, I+3, …etc. having different signatures, merge into one sequence like I, I+1, I+2, I+3 … etc.
3. When Ry (π) T is broken, a doubling of states of the allowed angular momentum occurs. A sequence like I, I+2, I+4, … etc. becomes 2(I), 2(I+2), 2(I+4), …, each state now occurring twice (chiral doubling).
4. When P=Rx (π), the two signature partners will have different parity. Thus states of alternate parity occur. We obtain a sequence like I+, I+1-, I+2+, … etc.
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
13
P = 1 P ≠ 1 5 π ___________ 5± ___________ 4 π ___________ 4± ___________
→ 3 π ___________ 3± ___________ 2π ___________ 2± ___________ π = +1 or, -1
RULE NO. 1
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
14
Rx (π) = 1 Rx (π) ≠ 1 8 π ___________ 6 π ___________ 6 π ___________ 5 π___________
→ 4 π ___________ 4 π___________ 2 π ___________ 3 π___________ π = +1 or, -1 2 π ___________
RULE NO.2
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
15
Ry (π) T = 1, Rx (π) = 1 Ry (π) T ≠ 1, R x (π) = 1 8+ _______________ _______________ 6+_______________
→ 6+ _______________ _______________ 4+ _______________ 4+ _______________ _______________ 2+ _______________ 2+ _______________
RULE NO. 3
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
16
Ry (π) T = 1, Rx (π) ≠ 1 Ry (π) T ≠ 1, R x (π) ≠ 1 5+ _______________ _______________ 5+ _______________ 4+ _______________ _______________ 4+ _______________
→ 3+ _______________ ______________ 3+ ______________ 2+ _______________ ______________ 2+ ______________
RULE NO. 3
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
17
P = 1, Rx (π) = 1 P = Rx (π) 8+ _______________ 6+ _______________ 6+ _______________ 5- _______________
→ 4+ _______________ 4+ _______________ 2+ _______________ 3- _______________ 2+ _______________
RULE NO. 4
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
18
Part of the TABLE from Frauendorf (2000)
S.No. P Rx () Ry ()T Level sequence
1. S S S ......,.........)4(,)2(, III
2. S D S .....,.........)2(,)1(, III
3. S D D .,.........)2(2,)1(2,2 III
4. S S D ,......)4(2,)2(2,2 III
5. S D )(xR ,.......)2(,)1(, III
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
1919
NUCLEAR SHAPESNUCLEAR SHAPES
Radius vector of the surface of an arbitrary deformed bodyRadius vector of the surface of an arbitrary deformed body
)],(1[),( *
,,0
YRR
Different spherical harmonics have definite but different geometric symmetries and may occur in the mean field.
Most common is the λ = 2 quadrupole term. Higher order terms occur in specific regions of nuclei.
Permanent non-spherical shape ↔ Rotational motion
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
2020
Define surface in body-fixed frame rather than space-fixed
)],(1[),( *,0 YaRR
Here time-independent parameters are used. Empirical evidence exists for the quadrupole, quad + hexadeca, quad + octupole. While axial shapes are most common,evidence exists for non-axial shapes also.
Following possibilities give rise to these many shapes
• High Spin Configurations
• Non-yrast configurations
• Unusual N/Z ratios
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
2121
Tetrahedral and Triangular shapes: Besides the usual octupole shape like Y30, predictions of Y31 and Y32 shapes also exist.
Tilted Axis Rotation: A New Dimension has been provided to the whole scene by the possibility of having rotation about an axis other than one of the principal axes. This has given rise to the new types of rotational bands like the Magnetic Rotation bands and the Chiral bands.
Additional types of symmetry breakings and ensuing rotational structures are expected.
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
2222
The Collective Hamiltonian (Bohr and Mottleson,1975; MK Pal,1982)
][2
1 22.
,
CBVTH
20
150
1
4
3MARRB o
where
.12
1.
)(
2
3)2)(1(
0
220
R
ZeSRCCC cs
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
2323
Transform to body-fixed principal axes frame
.2
1
2
1
2
1 223
1
2.
aCaBVTTH kK
krotvib
Now the equation has a rotor term also. The three moments of inertia about the three principal axes appear explicitly.
Quadrupole (λ=2) motion
,2
1
2
1)(
2
1 222.
22.
CBHk
kk
where (β,γ) parameters have been used.
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
2424
Here,
,cos20 a Sinaa2
12222
01221 aa
,3
24 22
kSinBk
5002
1RB
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
2525
Quantization of the classical H
).,,,,(),,,,(2
1
2
33
11
2
32132122
2
24
4
2
ECR
SinSinB
kk k
Separable in β and γ coordinates by wriiting
),,,,()(),,,,( 321321 f
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
2626
The β-equation is
),()(1
22
11
2 2
224
4
2
EffB
Cd
d
d
d
B
The rotor plus γ-motion equation is
),,,(),,,(
32
sin4
13
3sin
1
321321
2
2
k
k
k
RSin
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
2727
For rigidity against γ-motion, only rotational motion is left,
).(),,(
3
24
1321321
2
2
kSin
Rk
It is known that
IMK
IMK DIIDR )1(
2
IMK
IMKZ MDDR
IMK
IMKz KDDR
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
28
ND Prolate Prolate + Hexadeca
Prolate - Hexadeca
ND Oblate
SOME COMMON NUCLEAR SHAPES with Axial Symmetry
NOT SO COMMON SHAPES
Non-axial Quadru Y22
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
2929
Spheroidal shape – Axial symmetry about say the z-axis.
0, zyx
.13
1
)(3
1)(
3
1
3212
3212
2
32122
IMK
yx
DKII
RRRR
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
3030
For a general ellipsoid and the coefficients of are not equal. We have
Terms like R+ R+ and R- R-, and (R+ R- + R- R+) arise. The last operator leaves unchanged.
However, the first two operators change to
and respectively. Thus, we have a mixing of K with K+2 and K-2 in the eigenfunction.
zyx 22 , yx RR
22 )(4
1 RRRx
22 )(4
1 RRRy yx iRRR
IMKD
IMKD
IMKD 2
IMKD 2
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
3131
The eigenfunctions look like
., 321321 IMK
k
IKM
I Dg
where K differ by 2.
Constraints on K-values: When we go from space-fixed to body-fixed frame, an arbitrariness arises in assigning the set of parameters (β,γ) and the three Euler angles for a given set of
Parameters. This is met by requiring that the wavefunction remain invariant under a set of rotation operators R1, R2, and R3:
2
.,2
,2
,2
,0,0,0,,0 321
RRR
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
32
y
x
z
x
y
z
x
y
x'
y'
z'
R3 (π/2,π/2,π)
z'
z'
y'
x'
R2 (0,0,π/2)
x'
y'
R1 (0,π,0)
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
3333
It puts the following conditions on the wavefunction:
(i)
zyz JiJiJi eeeR 321),,( 321
where,
KIIK
IK gg
)1)(()( (ii) )()1()( I
KKI
K gg It restricts the K to even integers only, and we obtain
),()1()( IK
IIK gg
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
3434
The wavefunction remains invariant only if written as
K
IKM
IIMK
IKM
I DDg )1()(),,,( 321
With K as even integers only.
Axial Symmetry – Symmetric Top: Gamma motion is frozen,
IKM
IIMK
IK
IMK DDg )1(),,( 321
IKM
IIMK DD
I
)1(
16
122
With K as even integer only.
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
35
K
M RI
Z
zJ
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
3636
For K=0, only even-I values are allowed else the wavefunction vanishes,
IMK
IMK D
I020 8
12
It can be further proved that only K=0 is the allowed K-value for the case of axial symmetry.
I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003
3737
KnNB
C
2
12,
GSB: K=0, Nβ=0, Nγ=0, nγ=0, I=0, 2, 4, ……etc.
Gamma band: Nβ=0, Nγ=1 (one γ-phonon), K=2, I=2,3,4,…etc.
Beta band: Nβ=1, Nγ=0, K=0, I=0,2,4,……etc.
12, InNB
C
22
)1(2
12
12
2
5
KII
KnNE
Even-Even Nuclei: K=0 GSB, β-bands, γ-bands
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
38
Energy Spectrum of Bohr Hamiltonian Even-Even Nuclei
6 __________ 6 __________ 5 __________ 6__________ 4 __________ 4 __________ 4 __________ 5__________ 2 __________ 2 __________ 7- __________ 0 __________ 0 __________ 2 __________ 4__________ 5- __________ K=0 6+__________ K=2 0 __________ K=4 3- __________ n β=2, n γ=0 n β=1, n γ=1 5+__________ K=0 n β =0, n γ =2 1- __________ n β=0, n γ=2 K=0
n λ =3 =1
4+__________ 4+ __________ 2+__________
3+ __________ 0+__________ 2+ __________ K=0 6+__________ K=2 n β=1, n γ=0 n β=0, n γ=1 4+__________ 2+__________ 0+__________ K=0 n β=0, n γ=0 Based on λ=2 vibration λ=3 vibration
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
39
20+_______________4463 18+_______________3845 16+_______________3292 12+___________2910 14+ _______________2745 13-____________ 2817 11+___________2656 10+___________2346 11-____________ 2368 12+ _______________2165 9+___________2133 8+___________1872 9-____________ 1985 7+___________1669 7-____________ 1682 10+ _______________1602 6+___________1459 5-____________ 1469
5+___________1286 4+____________~1369
3
1═════════ 1356
1352
4+___________1128 2+_____________1171 K=0 8+ _______________1096 3+___________1002 0+_____________1087 Octupole band 2+___________900 K=0 K=2 β band 6+ _______________666 γ band 4+ _______________329 2+ _______________102 0+ _______________0 K=0
GS band Er16268