Unusual Symmetries in Nuclei

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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Unusual Symmetries in Unusual Symmetries in NucleiNuclei

ASHOK KUMAR JAINASHOK KUMAR JAINIndian Institute of TechnologyIndian Institute of Technology

Department of PhysicsDepartment of PhysicsRoorkee, IndiaRoorkee, India


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OutlineOutline• Role of Intrinsic Wavefunction – Role of Intrinsic Wavefunction –

SignatureSignature

• Triaxial EllipsoidTriaxial Ellipsoid

• Odd Multipole Shapes – SimplexOdd Multipole Shapes – Simplex

• Only two planes of symmetry Only two planes of symmetry

• Only one plane of symmetryOnly one plane of symmetry

• Tetrahedral and Triangular shapesTetrahedral and Triangular shapes

• Tilted Axis Rotation – Planar and AplanarTilted Axis Rotation – Planar and Aplanar

• Magnetic Top and Chiral RotationMagnetic Top and Chiral Rotation

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

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Intrinsic Wavefunction and its consequences

The total wavefunction

IMK

IMK D

where is the intrinsic part. Axial symmetry is assumed so that

The total wavefunction must remain invariant under and

acting on internal and collective coordinates with the condition that

K

)(xR )(eR

)()( ex RR Since j is not a good q.n.,

J

JKJK C


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For K=0, the following holds

,)( 00 KKx rR

,)( 02

02

KKx rR so that and 12 r 1rOne may also write these expressions as

000)(

Ki

KJi

Kx eeR x

which leads to values =0 and =1 corresponding to r = +1 and r = -1.

Both and r are termed as the signature quantum number.


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Signature and angular momentum get connected by the relation

IM

IIM

IiIMKe YYeDR )1()( 0

Therefore,,)1( Ir

and the K=0 band can be classified as

=0, ,1r .,.........4,2,0I

=1, ,1r ....,.........5,3,1I

The GSB of even-even nuclei is the best example of r=+1 signature.

A K=0 band in an odd-odd nucleus has both r=+1 and r=-1 signature.


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For K ≠ 0, the intrinsic states are two fold degenerate because of

symmetry. Note that and time-reversal T the same effect on the wave-function. The time-reversed state has the negative value

of , so that

)(xR

)(xR

K

zjKxK

R 1

and

jKj

KjK

ji

Kxe .)1(.

The rotational wave function changes as

IKM

KIIMK

IiIMKe DDeDR

)1(

so that a rotationally invariant wave function may be constructed as

IKMK

KIIMKK

IMK DD

I

)1(16

12 2

1

2


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,22 )1( K

jKxR For odd-A nuclei, 2j odd.

ijix eeR x Now,

Iie eR and,

11 ex RRtogether with, lead us to the classification

.,,.........2

9,

2

5,

2

1I ,

2

1 ,ir

..,,.........2

11,

2

7,

2

3I ,

2

1 ir

In general, I = (α + even number).


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PARITY: Parity operator acts upon the intrinsic coordinates only.

,kkP 1Since parity commutes with jz, all the states in a given band having a quantum number K, have the same parity. It is possible to have K=0 bands with π and α quantum numbers independent of each other as the two are distinct from each other. Thus, K=0 bands may have

0,.....4,2,0 I

1,........5,3,1 I

0..,.........4,2,0 I

and,

1,......5,3,1 I

GSB


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Ellipsoid with D2 –symmetry – Asymmetric Top

• Full D2-symmetry :invariance with respect to the three rotations by 180 about each of the three principal axes.

• Finite gamma deformation – K not a good quantum number

•

• K=0 not allowed. Only even integers K=2,4,…etc. allowed.

• A typical rotational band may have I=2,4,6,….etc. as signature is still a good quantum number.

K

IKM

IIMK

IK

IM DDg .)1(),,,( 321


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Figure 10


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Odd-Multipole Shapes: Simplex quantum number

• Axial Symmetry – Y30 shapes only.

• violates the and symmetry, but preserves .

• Two minima in octupole def energy- two degenerate states arise

• Denote which is conserved

• K=0 band satisfy:

)(xR

P

PR x

1

xRPS

Is )1(1,........,3,2,1,0 sI

1....,,.........3,2,1,0 sI


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OO

V V

ε3< 0 ε3 >0

3-

2+

1-

0+

11/2±

9/2±

7/2±

5/2±

4+

2+

0+

3-

1-

9/2+

7/2+

5/2+

9/2-

7/2-

5/2-

ε3< 0 ε3 >0

E-E Odd-A E-E Odd-A


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For K≠ 0, the intrinsic states have a 2-fold Kramer’s degeneracy and

We obtain,

isI

...,.........2

5,

2

3,

2

1

isI

.......,,.........2

5,

2

3,

2

1

with the levels having I ≥ K allowed.


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Regions of Octupole Deformation

For example, Δl=3 orbitals for both neutrons and protons come close together just beyond Pb-208

d3/2 s1/2

g7/2

d5/2

j15/2

i11/2

g9/2

Y3

184

126p1/2

f5/2

p3/2

i13/2

h9/2

f7/2

Y3

82d3/2

h11/2

s1/2

g7/2

d5/2

Y3

50g9/2

p1/2

f5/2

p3/2

Y3

28

f7/2

20


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odd-A

3/2+

r = i

7/2+

1/2+

r = -i

5/2+

S = -i

1/2-

3/2+

5/2-

7/2+

1/2+ 3/2-

5/2+

7/2-

S = i

odd-A

5/2+

7/2+

9/2+

11/2+

even-even

O+

5/2±

7/2±

9/2±

11/2±

0+

2+

4+

2+

0+

1-

3-

4+

0+

1-

3-

5/2-

7/2-

9/2-

11/2-

5/2+

7/2+

9/2+

11/2+

1/2- 3/2-

5/2- 7/2-

ΔE

1/2+ 3/2+

5/2+

7/2+

ε3=0 ε3≠0 ε3≠0 +tunneling

I (I+1)

K=1/2

(a>0)

K=5/2

K=0

2+

4+


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FIGURE 14


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Density Distribution has two planes of symmetry

• Axial symmetry is lost for μ≠ 0

• Two independent planes of symmetry for μ even.

• Rotation about the long axis is possible.

•For

3Y ,0 ,1)( xR PTRy )(

Parity doublets of odd- or, even-angular momenta arise

,....4,2 I ,.........5,3,1 I


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FIGURE 15


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EXOTIC NUCLEAR SHAPES

SD Prolate Y20 + Y30 Y20 + Y31

Y20 + Y32 Y20 + Y33 Pure Y40


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Density distribution has one plane of symmetry

• Odd-μ components allowed

• Rotation is possible along the long axis as well as any one of the short axes

• Signature is not a good quantum number. Both even and odd-spins will occur

• Parity is not conserved; both parities will occur.

• Rotation about long axis

• Rotation about short axis

• Chiral partners obtained

,.........6,5,4 I

,...)10(,)9(,)8( 222 I


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FIGURE 16


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Tetrahedral and Triangle Symmetries

• Tetrahedral symmetry is related to Y-32 term

• Large shell gaps at N, Z=16, 20, 32, 40, 56,58,70,90-94

• and at N=136/142

• Tetrahedral Equilibrium shapes of the order of 0.13 for

,,, 401607068

1084040

8040 YbZrZr

142242100 Fm

• Obey simplex symmetry, parity doublets formed


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FIGURE 17


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Dipole Mom. ≠0 Dipole Mom. =0

4+

3-

2+

1-

0+

4- 4+

3+ 3-

2- 2+

1-1+

0- 0+

4- 4+

3+ 3-

2- 2+

1+ 1-

0- 0+

Pear-shape-like Octupole

Tetraheda

4-

3+

2-

1+

0-

Pure Tetrahedra

Dipole Mom.=0 Quadrupole Mom.=0


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Triangular Shape corresponds to Y-33 def.

• Has only one plane of symmetry

• Possible in proton rich N=Z nuclei

,,,,, 8076726864 ZrSrKrSeGe Mo84


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Tilted Axis Rotation

• Riemann - classical rotation about an axis different than principal axes possible in ellipsoids

• Planar Tilt – Axis of rotation in a principal plane

• P and Ry()T are conserve .

• Signature not conserved

• A band like is observed

• Observed in Magnetic Rotation Bands

,......6,5,4 I


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Figure 10


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Aplanar Tilt:

• Only parity is conserved

• Four distinct situations, related by Rx() and Ry()T exist

• Band obtained:

,......)6(,)5(,)4( 222 I

• Observed in Chiral Bands


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Figure 10


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Planar Tilt in Octupole shapes

• Ry()T=P

• Four distinct situations obtained by Rx() and P.

• Two degenerate bands emerge having

,.....6,5,4 I

• Shown in bottom panel of Fig 15


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Magnetic Top

• Mean Field remains nearly isotropic

• Anisotropy of currents – net magnetic dipole moment

• Coupling of high-j neutron and proton, one hole and the other particle

• Resultant angular momentum makes an angle with the principal axes

• Magnetic effects can dominate when deformation is small

• Some deformation necessary to close the blades of neutron and proton – Shears mechanism


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36 Lakshmi et al,2003


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Magnetic Rotation

Z=17-20, N=20-23: Cl,Ar,K,CaZ=21-23, N=17-20 & 25-28: Sc,Ti,VZ=25-28, N=21-24: Mn,Fe,Co,NiZ=28-40,N=42-50: Ni,Cu,Zn,Ga,Ge,As,

Se,Br,Kr,Rb,Sr,Y,ZrZ=44-51,N=50-64: Ru,Rh,Pd,Ag,Cd,In,Sn,Sb

N=82-88 (n-rich)Z=50-64,N=74-82: Sn,Sb,Te,I,Xe,Cs,

Ba,La,Ce,Pr,Nd,Sm,GdZ=76-82,N=126-140: Os,Ir, Pt,Au,Hg,Tl,Pb (n-rich)Z=80-86,N=110-126: Hg,Pb,Bi,Po,At,Rn


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Chiral Bands• Aplanar Tilt in Tri-axial shape

• First visualised in Odd-Odd nuclei

• Odd-proton aligned along the short axis

• Odd-neutron aligned along the long axis

• Rotational contribution along the intermediate axis

• Resultant of the three ang mom is out of the three planes

• Parity is conserved but Ry()T is broken

• Two pairs of identical ΔI=1 bands having the same parity

• Tunneling between the right- and the left-handed system gives rise to a splitting between the levels.


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Zhu et al, 2003

2(h11/2),

1ν(h11/2)


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• Observation of Chiral pair of bands in an odd-A nucleus confirms that the phenomenon is purely geometrical in nature.

• One can visualise similar possibilities in even-even nuclei also.


4242

Dedicated to my son

NaMaN

The beautiful soul


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Bibiliography

General References:

• A. Bohr and B.R. Mottelson, Nuclear Structure, Vol.1, Single Particle Motion, and Vol.2, Nuclear Deformations (Benjamin, 1969,1975).

• M.K. Pal, Theory of Nuclear Structure, (East-West Press,1982).

• P. Ring and P. Schuck, The Nuclear Many Body Problem, (Springer, 1980).

• K. Heyde, Basic Ideas and Concepts in Nuclear Physics, 2nd Edition (IOP Publishing, 1999).


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• L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Vol.3 of Course of Theoretical Physics (Pergamon Press, 1956).

• Table of Isotopes, Eighth Edition, Ed. Firestone et al. (Wiley, 1996).

Articles on Symmetries:

• W. Greiner and J.A. Maruhn, Nuclear Models (Springer, 1989).

• J. Dobaczewski, J. Dudek, S.G. Rohozinski, and T.R. Werner, Phys. Rev. C62, 014310, and 014311 (2000).

• P. Van Isacker, Rep. Prog. Phys. 62, 1661 (1999).

• S. Frauendorf, Rev. Mod. Phys. 73, 463 (2001).


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Refelction Asymmetric Shapes:

• P.A. Butler and W. Nazarewicz, Rev. Mod. Phys. 68, 349 (1996).

• A.K. Jain, R.K. Sheline, P.C. Sood, and K. Jain, Rev. Mod. Phys. 62, 393 (1990).

• R.K. Sheline, A.K. Jain, K. Jain, I. Ragnarsson, Phys. Lett. B219, 47 (1989).

• J. Gasparo, G. Ardisson, V. Barci and R.K. Sheilne, Phys. Rev. C62, 064305 (2000).

Tetrahedral and Triangular Shapes:

• X. Li and J. Dudek, Phys. Rev. C49, R1250 (1994).


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• S. Takami, K. Yabana, and M. Matsuo, Phys. Lett. B431, 242 (1998).

• M. Yamagami, K. Matsuyanagi, and M. Matsuo, Nucl. Phys. A672, 123 (2000).

Magnetic Rotation:

• S. Frauendorf, Nucl. Phys. A557, 259c (1993).

• R.M. Clark and A.O. Macchiavelli, Annu. Rev. Nucl. Part. Sci. 50, 1 (2000).

• Amita, A.K. Jain, and B. Singh, At. Data & Nucl. Data Tables 74, 283 (2000); revised version under publication (2003).

• A.K. Jain and Amita, Pramana-J. Phys. 57, 611 (2001).

• S. Lakshmi, H.C. Jain, P.K. Joshi, A.K. Jain and S.S. Malik, under publication.


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Chiral Bands:

• V.I. Dimitrov, S. Frauendorf, and F. Donau, Phys. Rev. Lett. 84, 5732 (2000).

• D.J. Hartley et al., Phys. Rev. C64, 031304(R) (2001).

• S. Zhu et al., Phys. Rev. Lett. 91, 132501 (2003).

Critical Point Symmetries:

• R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 87, 052503 (2001).

• F. Iachello, Phys. Rev. Lett. 91, 132502 (2003).

Unusual Symmetries in Nuclei

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