unusual symmetries in nuclei
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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
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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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.
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
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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
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
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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
I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003
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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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I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.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.
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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).
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