Geometrical Symmetries in Nuclei-An Introduction

I.A.E.A. Workshop - I.C.T.P. Trieste, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003Nov.2003

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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


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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


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


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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.


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.


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


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

.


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.

'


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


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


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.


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P = 1 P ≠ 1 5 π ___________ 5± ___________ 4 π ___________ 4± ___________

→ 3 π ___________ 3± ___________ 2π ___________ 2± ___________ π = +1 or, -1

RULE NO. 1


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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


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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


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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


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P = 1, Rx (π) = 1 P = Rx (π) 8+ _______________ 6+ _______________ 6+ _______________ 5- _______________

→ 4+ _______________ 4+ _______________ 2+ _______________ 3- _______________ 2+ _______________

RULE NO. 4


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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


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


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


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.


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


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.


2424

Here,

,cos20 a Sinaa2

12222

01221 aa

,3

24 22

kSinBk

5002

1RB


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


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


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


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ND Prolate Prolate + Hexadeca

Prolate - Hexadeca

ND Oblate

SOME COMMON NUCLEAR SHAPES with Axial Symmetry

NOT SO COMMON SHAPES

Non-axial Quadru Y22


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


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


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


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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)


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


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.


35

K

M RI

Z

zJ


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.


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


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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


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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

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Geometrical Symmetries in Nuclei-An Introduction