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Lectures on
Geometrical Symmetries in Nuclei
Ashok Kumar Jain
Department of Physics
Indian Institute of Technology Roorkee
INDIA
I.A.E.A. Workshop on
Nuclear Structure and Decay Data:Theory and Evaluation
November 17-28, 2003
held at the A. Salam International Centre for Theoretical Physics,
Trieste, Italy
1
Geometrical Symmetries in Nuclei*
Introduction
Symmetries in nature, art, and architecture fascinate us. We are charmed by objects
which are symmetric and, therefore, beautiful. Most of these symmetries are geometric in
nature, and are related to the external appearance. We, however, come across many other
types of symmetries in physics, which are quite different from the purely geometric or,
spatial symmetries.
The theorem of Emmy Noether enunciates that each continuous symmetry is related
to a conserved quantity, or, a constant of motion. We, accordingly, have constants related
to symmetries of translation, rotation, and reflection in space and time. Thus, invariance
under the time translation leads to the conservation of the total energy of a closed system.
Likewise, invariance under space translation, and rotation leads to the conservation of
linear momentum, and angular momentum respectively. Besides the continuous
symmetries, we also come across discrete symmetries like reflection or, inversion of space
which leads to the conservation of parity. Time reversal invariance can also be added to
this list, which manifests itself as Kramer’s degeneracy in single nucleon orbits. Most
common among the discrete symmetries are the point-group symmetries widely used in the
classification of crystal structure. These symmetries have also found very useful
application in molecules and nuclei.
Besides these, we have the dynamical symmetries and the fundamental gauge
symmetries in nature. However, complex systems like atoms, molecules, and nuclei have
their own set of symmetries which can be geometrical as well as dynamical, and emerge
from the complexity of the system. Certain algebraic symmetries related to the various
group structures such as U(5), SU(3), and SO(6) have also been identified in complex
systems such as nuclei. These result in a characteristic set of patterns of energy levels, and
transition patterns between them. A more recent development along the same line is the
observation of simple behavior in systems at the critical point of quantum phase
transitions. This simple behavior has been interpreted as the occurrence of a dynamical
*
These lectures are dedicated to my son NamaN, a beautiful soul.
2
symmetry such as X (5) in
152
Sm. Systems lying at the critical point of first and second
order phase transitions are being closely scrutinized for similar behaviour. We shall,
however, not discuss this kind of symmetries.
Mean Field and Spontaneous Symmetry Breaking
The concepts of mean field and spontaneous breaking of symmetries of the mean
field play an important role in explaining the observed band structures. The fundamental
nucleon-nucleon interaction, taken to be a two body force, should be invariant under all the
basic transformations like translation, rotation, and inversion in space and time. However a
collection of nucleons such as a nucleus giving rise to a mean field, may break one or more
of these symmetries even though the fundamental N-N interaction does not. Such a
symmetry breaking is termed as the spontaneous breaking of symmetry. The concept of
spontaneous symmetry breaking is crucial to understand a large variety of characteristic
pattern of levels observed in experiments. However, as we shall see, additional varieties of
patterns are predicted and waiting to be observed.
If the nuclei also obeyed the basic symmetries of the N-N interaction, we would
miss much of the richness in the band structure of nuclei. It is the spontaneous breaking of
one or, more of these symmetries, which leads to a rich band structure. It enables us to
classify and label the levels into various bands and infer information about the nature of
the mean field. For example, the levels energies of 168Er shown in the column on the left of
Fig. 1 begin to look meaningful and also beautiful, when classified into bands as on the
right hand side.
Symmetry, Unitary transformation, Degeneracy, and Multiplets
A symmetry in quantum mechanics can be represented by a group of unitary
transformations Û in the Hilbert space. An operator Q, representing an observable,
transforms as
Q  u †Qu
under the unitary transformation u. Since u †  u 1 for unitary transformations, invariance
of Q under u implies that
Q  u 1Qu
3
i.e., [u , Q ]  0 .
This is a well known result from quantum mechanics. If the unitary operator happens to
arise from the Hamiltonian of the quantum system, the operator Q leads to a conserved
quantity. In this situation, the unitary operator defined by H is e-i t H, and
e itH Qe itH  Q, for all t.
Thus, a commutation of Q with u  e itH also implies a commutation of Q with H, and Q is
conserved.
Note that H is the generator of time translation because
 '  e itH   (1  itH )
represents a new state obtained by translation in time. Likewise,
 '  e ijz   (1  iJ z )
represents a new state obtained by rotation by θ about the z–axis. Here, Jz is the z–
component of the angular momentum operator and is the generator of rotation about the zaxis. If Jz is an invariant operator, we have
[H , J z ]  0 .
Also if, H  E , we have
H '  H (1  iJ z )  E ' .
This means that 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.
This leads us to the concept of degeneracy and multiplets. An energy eigenstate can have
n-fold degeneracy if n-fold rotation of  about the z–axis leaves  invariant. An
interaction or, deformation, which violates this symmetry, will lift the degeneracy and a
multiplet will emerge.
As a simple example, consider a single particle moving in a spherically symmetric

central potential carrying an angular momentum j ; it has an energy which does not depend
on jz. This state has a (2j+1) fold degeneracy, where j is the angular momentum quantum
number. However, a slight deformation of the potential splits the degeneracy of the j-
4
multiplet, and we get a characteristic level pattern. Such a symmetry breaking is witnessed
when go from the solutions of the spherical shell model to the deformed shell model, or,
the Nilsson model, as shown in Figure 2. If the potential has an axial symmetry about the
Z-axis, Jz is the only conserved quantity and the corresponding quantum number Ω can be
used to label the state.
Discrete Symmetries in Nuclei
Most commonly encountered discrete symmetries in rotating nuclei correspond to
the parity P, rotation by π about the body fixed x, y, z axes, Rx(π), Ry(π,), Rz(π), time
reversal T, and TRx(π), TRy(π), TRz(π). These are all two fold discrete symmetries, and their
breaking causes a doubling of states. Dobaczewski etal. (2000) have done a detailed
classification of the mean field solutions according to the discrete symmetries of a double
point group denoted by D2h (Landau and Lifshitz, 1956) ; this includes all the symmetries
listed above. We can enunciate the following simple rules to work out the consequences of
these symmetries on a rotational band comprising of levels with angular momentum
quantum numbers I, I+1, I+2, …etc.:
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, … [Fig. 3(a)].
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 and shifted in energy
with respect to each other, merge into one sequence like I, I+1, I+2, I+3 … etc.
[Fig. 3(b)].
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) [Fig. 3(c)].
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. [Fig. 3(d)].
Since all these symmetries have a two-fold degeneracy, a breaking of each of them
individually doubles the number of states. Frauendorf (2001) has listed the consequences
of breaking up of those symmetries which are relevant for the two-body rotating
5
Table I: Consequences of spontaneous breaking of one or more of the discrete symmetries
of the rotating mean field. Here x is chosen as the axis of rotation. A symbol D means the
mean field becomes different under the corresponding operation and S means it remains
same. When another operation is shown as an entry, it means the two are identical. The last
column shows the spectrum arising for a given set of conserved/broken symmetries.
Although only positive parity is shown in rows 1-5, parity can be negative also (taken from
Frauendorf, 2001).
S.No.
P
Rx ()
Ry ()T
1.
S
S
S
I  , ( I  2)  , ( I  4)  ,...............
2.
S
D
S
I  , ( I  1)  , ( I  2)  ,..............
3.
S
D
D
2I  ,2( I  1)  ,2( I  2)  ,..........
4.
S
S
D
2I  ,2( I  2)  ,2( I  4)  ,......
5.
S
D
Rx ( )
I  , ( I  1)  , ( I  2)  ,.......
6.
D
S
S
I  , ( I  2)  , ( I  4)  ,.........
7.
D
D
S
I  , ( I  1)  , ( I  2)  ,......
8.
D
S
D
2I  ,2( I  2)  ,2( I  4)  ,..........
9.
D
D
Rx ( )
I  , ( I  1)  , ( I  2)  ,........
10.
Rx ( )
D
S
I  , ( I  1)  , ( I  2)  ,........
11.
Rx ( )
D
D
2I  ,2( I  1)  ,2( I  2)  ,........
12.
R y ( )T
S
D
I  , ( I  2)  , ( I  4)  ,............
13.
R y ( )T
D
D
I  , ( I  1)  , ( I  2)  ,..........
14.
Rx ( )
D
Rx ( )
I  , ( I  1)  , ( I  2)  ,.........
15.
D
D
D
2I  ,2( I  1)  ,2( I  2)  ,...........
Level sequence
6

Hamiltonian H=T+V- j x , which we reproduce here as Table I. All the possibilities
presented in the table can be worked out by using these rules either alone or, in
combination.
Nuclear Shapes
In order to proceed further, it is necessary to recollect some basic ideas of nuclear
shapes. It is well known that the surface of an arbitrarily deformed body can be expressed
by the radius vector along the polar angles  and  as
R( ,  )  R0 [1     ,  Y *  ( ,  )] .
 ,
Here R0 is the radius of an equivalent volume sphere. The terms λ=0, 1, 2, 3, 4 etc.
correspond to the monopole, dipole, quadrupole, octupole, and hexadecapole shapes. In
general, we get 2λ-pole deformation for a given λ. These spherical harmonics have definite
geometrical symmetries and may occur in the mean field of the nucleus. In nuclei, the
monopole shape oscillation may occur only at very high excitations due to the
incompressible nature of the nuclear matter. The dipole term simply corresponds to a
translation of the nucleus and does not have a physical significance. Therefore, the lowest
order term of utmost importance is the λ = 2 quadrupole term. Higher order terms play a
role in specific mass regions of nuclei, but λ = 2 term is the most widespread and globally
occurring shape in nuclei.
A permanent non-spherical shape gives rise to the possibility of observing
rotational motion. It is, then, more convenient to consider the nuclear surface in the body
fixed frame rather than the space fixed frame. The nuclear surface in the body fixed frame
can also be described by a similar relation,
R( ,  )  R0 [1   a , Y *  ( ,  )]
where a  have been introduced as the new time-independent parameters in the body fixed
frame, which coincides with the principal axes. The parameters a  are related to   as,
a    D'  ()  ' .
'
7
The Y2μ term corresponding to λ=2 has five components labeled by μ =  2,  1, 0.
The μ = 0 component corresponds to the situation where full rotational symmetry is
maintained about one of the three principle axes say the z-axis. In this situation, the other
two axes (x- and y-) are equal. We call such a shape as a spheroid. For x = y < z situation,
we get a prolate spheroid, and for x = y > z, we get an oblate spheroid. It is now known
that the prolate spheroid is the most common shape in nuclei, although oblate shape is also
known to occur near the magic numbers.
The next most commonly observed shape is the λ = 4 hexadecapole shape, which
is generally superposed on the quadrupole shape, and is found with small amplitude only.
A small λ = 3 octupole shape is now believed to occur in certain pockets of nuclei and is
also superimposed on the quadrupole shape. Further, much of the experimental evidence
favours the occurrence of μ = 0 component of the various multipoles. However, attention
has now been focused on μ ≠ 0 components of the various multipoles and their
consequences. This corresponds to the introduction of non-axial or, axially-asymmetric
degrees of freedom. Some common nuclear shapes corresponding to the various multipoles
are shown in Figure 4. We also show in Figure 5 some extraordinary or, exotic shapes.
Observation of one or, more of these varied shapes in nuclei has become a distinct
possibility with enhancement in our experimental capabilities. While the ground state
configurations of nuclei may not support all of these shapes, we now have the possibility
of observing high spin configurations, non-yrast configurations, and configurations with
abnormal N/Z ratio (the nuclei away from the line of stability) which may support one or,
more of these novel shapes.
Each of these shapes is obtained by a distinct symmetry breaking of the mean field
and, therefore, leaves a characteristic impression on the level pattern due to the lifting of
degeneracy. Those operations which leave these geometrical shapes invariant, when
coupled with the time-reversal and space-inversion (parity) operators, provide a fertile
ground for observing fascinating patterns of levels in nuclei.
An additional new dimension to the whole scenario has been provided by the
realization that rotation is also possible about an axis other than one of the principal axes.
8
This is particularly true for the tri-axial shapes where rotation about a tilted axis has
successfully explained the observed features and phenomena such as the Magnetic
Rotation and Chiral Rotation. This leads us to additional types of symmetry breakings and
ensuing consequences.
The Collective Hamiltonian
The Collective Hamiltonian for an irrotational flow of fluid can be written as (Bohr
and Mottelson, 1975; M.K. Pal, 1982)
.
1
H  T  V   [ B  
2  ,
2
2
 C    ] ,
where
B  1  0 Ro5 
3 1
 MAR02
4
C  C5  Cc  R02 S (  1)(  2) 
3 ( Ze) 2   1
.
.
2 R0 2  1
Here 0 is the equilibrium density of nuclear matter. Note that the space-fixed frame and
parameters have been used. This is a classical Hamiltonian of a vibrator for each
1
2
C 
( ,  ) mode having a classical frequency of vibration given by       .
 B 
Transformation of this Hamiltonian to body-fixed principal axes frame assumes a
particularly simple form given by
H  Tvib  Trot  V 
.
1
B
a
  
2 
2

2
1 3
1
  k2   C  a  .

2 K 1 k
2 
This equation is written in term of the parameters a  defined in the body fixed frame. The
first and the last term represent the energies of a vibrator, and the second term corresponds
to a rotator with k (k  x, y, z ) as the three components of the moment of inertia in the
body-fixed frame. The pure vibrator Hamiltonian in the space fixed frame becomes a
vibrator plus a rotator Hamiltonian in the body-fixed frame.
9
Quadrupole (  2) motion
If we consider only   2 terms, it can be shown that
H
1
1
1
B(  2   2 2 )   k  k2  C 2 ,
2
2 k
2
where  ,  parameters have been used. Here,
a 20   cos  , a 22  a 2  2 
1
2
Sin , a21  a21  0
2 

 k  4 B 2 Sin 2    k
,
3 

B
1
 0 R05 .
2
Quantization of this Hamiltonian leads to the Schrodinger equation
 2  1  4 
1

 
 4


 2
Sin3

  Sin3 
 
 2 B   

k

2 2 1
Rk  C 2  (  ,  ,1 , 2 , 3 )  E (  ,  ,1 , 2 , 3 ).
2 k
2

This equation is separable in β- and γ- coordinates, so that
( ,  ,1 , 2 , 3 )  f ( )( ,1 , 2 , 3 ),
where f(β) satisfies the β-equation,
2
 2 1 d 4 d
1
 1 
2

 f (  )  Ef (  ),


C


 2 B  4 d
d 2
2 B  2 

and Φ(γ,θ1, θ2, θ3) satisfies the rotor plus γ-motion equation,


Rk2
  1  Sin 3   1
 2
 sin 3 
2
 4 k
sin    k

3


 ( , 1 ,  2 ,  3 )   ( , 1 ,  2 , 3 )



.


If the nucleus is rigid for γ-vibration, only rotational part is left in the rotor plus γ motion equation, and we obtain
10
1

4
Rk2
2 

Sin    k

3 

 (1 ,  2 ,  3 )   (1 2 3 ).
2
The operators Rk , (k  x, y, z ) are the components of the rotational angular


momentum operator R along the body-fixed axes x, y, and z. The components of R along
the space-fixed axes are donated by X, Y, Z. It is known that
RX , RY   iRZ ,......etc
but
R , R   iR ,......etc
x
y
z
I
( 1 2 3 ) . It
Also, it can be shown that (1 2 3 ) are nothing but the function DMK
is also known that these satisfy the eigen-value equations,
2
I
I
,
R DMK
 I ( I  1) DMK
I
I
,
RZ DMK
 MDMK
I
I
.
Rz DMK
 KDMK
Spheroidal Shapes
When the ellipsoidal body has an axis of symmetry along one of the principal axes,
we get a spheroid. Let z-axis be the symmetry axis, which is perpendicular to the x- and yaxis. In this situation γ = 0, and
x   y , z  0 .
It is a consequence of the general rule that there cannot be any rotation about an axis of

symmetry. The equation in R reduces to

1 2
1  2
R x  R y2  (1 2 3 )   R  R 2  (1 2 3 )
3
3






1
I
1 2 3 .
I I  1  K 2 DMK
3
11
For a general ellipsoid,  x   y   z , and the coefficients of Rx2 , R y2 are not equal. We can
write R x2 
1
1
( R  R ) 2 , and R y2   ( R  R ) 2 , where R  R x  iR y . This leads us to
4
4
terms of the type R+ R+ and R- R-, and (R+ R- + R- R+). The last operator leaves
I
I
I
I
unchanged. However, R+ R+ and R- R- change DMK
to DMK
DMK
 2 and DMK  2 respectively.
I
The eigen functions, therefore, become a mixture of DMK
with K differing by ± 2.
I
The equation for rotor plus γ-motion can also be solved by using the DMK
functions.
Its eigen functions are of the type
I
1 2 3 .
 I M  ,1 2 3    g KI  DMK
k
where K differ by ± 2. This equation corresponding to rotor plus γ-motion is difficult to
solve as it leads to a chain of coupled differential equations.
Constraints on K-values
We recall that  2  are the shape parameters in space-fixed frame and define the
shape uniquely. When we transform to the body-fixed axes and bring in the parameter
a 2  or, (β,γ) (θ1 θ2 θ3), an arbitrariness creeps into the labeling of the body-fixed frame.
The body-fixed axes (which coincide with the principal axes of the body) can be chosen in
many ways. Restricting to right-handed frames only, it can be shown that there are 24
different ways to choose the body-frame (Pal, 1982). For each such choice, we get
different (β,γ,θ1 θ2 θ3) values. However, any change of body-frame which does not change
  ' s should leave the wave function invariant. This is ensured by considering the effect
of the rotation operators R1, R2, and R3 on the wave function (Fig. 6).
It is possible to obtain all the 24 frames by an application of one or, more of the
three rotation operators,
    

R1 0,  ,0, R2  0,0, , R3  , ,  .
2 2 2 

12
Here, R( 1 ,  2 ,  3 ) denotes an operator consisting of
R(1, 2 ,3 )  ei1 J z e
i 2 J y
ei 3 J z .
A combination of these operators can give the 24 different sets of body-fixed axes, which
give different (β,γ, θ1 θ2 θ3) for same a  ' s . We, therefore, demand that the wave function
should remain invariant under these three operations. The three operators affect the
functions as follows. The β remains unaffected in all the cases.
(i) R1y (0,  ,0) :    , 1  1 ,  2   2   ,  3   3 .
Hence,
g
I
K
I
1 , 2 , 3 
( ) DMK
K
I
  g KI ' ( ) DMK
' (1 ,  2   , 3 )
K'
  g KI ' ( ) DMI  K ' (1 , 2 , 3 )( 1) I  K '
K'
I
Equating the coefficients of DMK
on both sides,
g KI ( )  g I K ( )(1) I  K

(ii) R2z  0,0,  :   
2

,
1  1 ,  2   2 ,  3   3 

2
Hence,
g
I
K
I
( ) DMK
(1 , 2 , 3 )
K


I
  g KI ' ( )DMK

' 1 ,  2 ,  3 
2

K'
I
K'
  g KI ' ( ) DMK
' ( 1 , 2 , 3 )i
K'
Again equating the coefficients,
g KI ( )  i K g KI ( ).
Using this relation again to replace g KI ( ), we get
g KI ( )  (1) K g KI ( ) .
13
It restricts K to even-integer values only. Combining the two relations from (i) and (ii), we
obtain
g KI ( )  (1) I g I K ( ),
with K as even integers only.
4


  
(iii) R3  , ,   :    
, 1  1  ,  2   2 
, 3  3  
3
2
2
2 2 
Therefore,
g
I
K
I
( ) DMK
(1 , 2 , 3 )
K
 I
I  
  g KI ' (  120 0 ) DKK
, ,   DMK
(1 , 2 ,  3 )
'
2 2 
KK '
Equating the coefficients, we get

I  
g KI ( )   g KI ' (  120 0 ) DKK
, , 
'
2 2 
K'
Incorporating the property from (ii), the wave function remains invariant if written as

I
 I M ( , 1 , 2 , 3 )   g KI ( ) DMK
 (1) I DMI  K

K
where K = even integers only.
Axial Symmetry – Symmetric Top
If γ-motion is frozen, g KI ( ) becomes independent of γ. Further, K (=Ω) is a good
quantum number for a spheroidal shape (Figure 7). The summation on K disappears. We,
therefore, have

I
I
 MK
(1 , 2 ,  3 )  g KI DMK
 (1) I DMI  K


2I  1 I
DMK  (1) I DMI  K
2
16


where normalization condition has been used. Remember that K is allowed to have eveninteger values only.
14
For K = 0, even integer values alone are allowed for I else the wave-function
vanishes. Therefore,
I
 MK
0 
2I  1 I
DMK 0
8 2
It can be shown that only K=0 is allowed in the case of spheroidal symmetry. To
show, consider the action of R2 (0,0,  ) for rotation by an arbitrary angle  about the zaxis, which is also the symmetry axis. Therefore,
g KI ( )  e iK g KI ( )  e

iK (  ')
2
g KI ( )
 e iK ' g KI ( )
Applying this again, we get
g KI ( )  ei 2 K g KI ( )
'
Since this relation must be valid for any value of  , K=0 only.
Even-even nuclei: K=0 ground state band, β-bands and γ- bands
When the axially symmetric deformed nucleus acquires small oscillations in β, and
γ, it is possible to write the total energy E of the nucleus as
5

E     N     
2

,
2
1

 
2
I ( I  1)  K
 2n  K  1 
2

 2


where,
 
C
, N   2n  I  1 ,
B
 
C
B
, N   2n 
1
K,
2
The lowest lying band corresponds to no β-phonon (Nβ=0), and no γ-phonon (Nγ=0, nγ=0,
K=0) excitation. Since K=0 allows only even angular momentum states, we obtain K=0,
I=0, 2, 4,…… all of even parity for the ground rotational band.
15
Another rotational band arises for Nβ=0, Nγ=1 (one γ-phonon) i.e. n  =0, K=2. This
K=2 γ-band can have any integer I = 2, 3, 4, ……. etc.
A K=0 β-band arises for Nβ=1, Nγ=0. Since K=0, we again have I = 0, 2, 4, ……,
and parity positive. Higher phonon excitations can be constructed by taking more than one
β-, or/and γ-phonons.A schematic diagram showing the various possible bands based on
λ=2 phonon excitations is shown in Fig. 8. An example of these bands is shown in Fig 9.
On the side, we also show the example of an octupole phonon excitation and a band built
on it.
Intrinsic Wave function and its consequences
We now introduce the intrinsic wave function also. It is necessary as the total
wave-function of a nucleus is most conveniently written as a product of an intrinsic part
and a rotational part. This is particularly needed in odd-A or, odd-odd nuclei. Also, the
intrinsic wave-function alone contains the information of parity.
Signature Quantum Number
An important consequence of introducing the intrinsic wave-function is the
emergence of signature quantum number for a spheroidal shape. Let z be the symmetry
axis and the quantization axis. The nucleus, as a consequence of the spheroidal shape, has a
reflection symmetry in the x-y plane. The total wave-function
I
I
MK
   DMK
must remain invariant under a transformation Rx ( ) acting on the intrinsic coordinates,
and Re ( ) acting on the collective coordinates such that
Rx ( )  Re ( ) .
For axial symmetry, K   , and   becomes  K , where  K   C J  JK .
J
For K =0 state, the intrinsic state must return to itself when operated twice by
Rx ( ) . Therefore,
Rx ( )  K 0  r K 0 ,
Rx2 ( )  K 0  r 2  K 0 ,
16
so that,
r2 1
r  1 .
and
One may also write these expressions as
Rx ( )  K 0  e  iJ x  K 0  e i  K 0
which leads to the values =0 and =1 corresponding to r = +1 and r = -1 respectively.
Both  and r are termed as the signature quantum number.
Also,
I
 iI
Re ( ) DMK
YMI  (1) I YMI .
0  e
Therefore,
r  (1) I ,
and the K  0 rotational band gets divided into two domains as
  0 , r  1,
I  0,2,4,..........
  1, r  1,
I  1,3,5,.............
An example of α=0, K=0 band has already been shown in Fig. 9. A K=0 band in odd-odd
nucleus has both α = 0 and α = 1 signatures.
For K  0 , the intrinsic states are two-fold degenerate as a consequence of the
invariance with respect to rotation by 1800 about the x (or, y) axis. It is interesting to note
that this operation has the same effect as the time reversal operator. The time reversed state
is denoted by K and has the negative eigenvalue of j z , so that
 K  R x1  K .
Since  K   C j  jK , we have
j
 K  e ij . K   (1) j  K . j  K
x
j
I
The effect of Re ( ) on DMK
is given by
I
I
Re DMK
 e iI DMK
 (1) I  K DMI  K .
A rotationally invariant wave-function can be constructed as,
17
1
I
MK
1  R 1 x Re  2I 21  2  K DMKI

2
 8 
1
1

 2I  1  2
I

 K DMK
 (1) I  K  K DMI  K
2 
16




For odd-A nuclei,
Rx2  K  (1) 2 j  K ,
where 2j is odd. Now,
R x  e  ij x  e  i
implies that  
1
1
for r  i and    for r  i . Similarly
2
2
Re  e iI
implies that I 
1
3
for re  i and I  for re  i . The condition
2
2
R x1 Re  1 requires that
I
1 5 9
1
, , ,.........., for   , r  i,
2 2 2
2
and
I
3 7 11
1
, , ,..........., for    , r  i .
2 2 2
2
In general,
I  (  even number).
The favoured signature levels come down in energy whereas the unfavoured
signature levels are pushed up. This corresponds to the situation shown in the first row of
Table I. An example of bands with α=1/2 and α=-1/2 signatures are shown in Fig.10. We
have chosen to show a band based on a i13/2 orbital with K=1/2; the
1
2
j   1 2 matrix
element (which decides the decoupling parameter) plays an important role in lowering the
energies of the favoured signature levels. This is the decoupling effect. In this case, the
decoupling effect is so strong that the levels 13/2,17/2, … lie lowest although K is very
18
small. This leads to the well known observation of decoupled bands. When the signature is
no longer a good quantum number i.e. Rx () is not a conserving operation, we get only one
sequence of levels such as I 
1 3 5 7
, , , ,......... etc. This situation corresponds to the
2 2 2 2
second row of Table I.
It may be noted that at higher rotational frequencies, the Coriolis force becomes
important and leads to significant K-mixing. Therefore, the time reversal as well as the full
D2-symmetry are broken. The only good quantum numbers that survive at high spins are
signature α and parity π.
Parity
If the intrinsic Hamiltonian preserves the parity, the corresponding wave-function
has fixed parity. Since parity operator P commutes with jz,
P k  k ,   1 ,
and all states in a given band have the same parity π. 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
I  0  ,2  ,4  ,........., α = 0,
or, I  0  ,2  ,4  ,............ , α = 0,
and
I  1 ,3 ,5  ,.............., α = 1,
or, I  1 ,3 ,5  ,................ , α =1 .
Ground rotational bands of even-even nuclei are known to exhibit
I  0  ,2  ,4  ,..........  0
band, and octupole vibrational bands of even-even nuclei display
I  1 ,3 ,5  ,........   1
band.
19
Parity and Time-reversal violating terms




Under the parity operation P , r  r , and p   p, but spin s and time t remain

unchanged. A Hamiltonian containing terms like r .s or, s. p , violates parity. Similarly,





under the time-reversal operation T , t  t , p   p, and s  s but r remain


unchanged. Terms like r . p and r .s , when present in the Hamiltonian, violate time-reversal

invariance. This violation leads to a doublet structure in the spectrum as both P and

T correspond to two-fold discrete symmetry. A connection between rotational motion and



P and T - violating Hamiltonian occurs if the system, while violating R x ( ) symmetry,
 
 
preserves the R x P or, R x T symmetry (see Table 1).
Ellipsoid with D2 –symmetry – Asymmetric Top
A general ellipsoid does not have axial symmetry, but has full D2-symmetry. In
other words, the system is invariant with respect to the three rotations by 1800 about each
of the three principal axes (tri-axial shape). The nucleus has a finite γ-deformation
different from 00 or, multiples of 2π/3. Even if the γ-motion is frozen, K is not a good
quantum number, and the wave function may look like


I
 MI ( , 1 , 2 , 3 )   g KI DMK
 (1) I DMI  K . .
K
Since K is allowed to have only even-integer values, it can take values K=2, 4, …etc. K=0
is not allowed as axial symmetry has been lost. Parity and signature are still good quantum
numbers as P=1 and Rx(π) = 1. Besides these two operations, Ry(π) T is also conserved.
(we are assuming rotation about the x-axis, which is also the long axis of the ellipsoid). A
typical rotational band may have I=2, 4, 6, ……etc. This situation also corresponds to the
first row of Table I. This situation is shown in the upper panel of Fig. 11.
20
Odd-Multipole Shapes: Simplex quantum number
An odd-multipole shape such as Y30 (octupole deformation) has an axial symmetry,
say about the long axis. It violates the Rˆ x ( ) , and P̂ symmetry, but preserves Rˆ x Pˆ . The
reflection symmetry is broken and two degenerate states with identical shapes arise. These
correspond to the two minima in the octupole deformation energy (Fig.12). This situation
corresponds to the 9th row of Table I. The operation Rˆ x Pˆ corresponds to a reflection in a
plane containing the symmetry axis. We denote this combined operation by
Sˆ  Pˆ Rˆ x1 ,
where Ŝ acts on the intrinsic variables.

K= 0 band: The intrinsic states with K= 0 are eigenstates of Ŝ as well as T . We
have
Sˆ K 0  s K 0  ei  K 0 .
I
I
I
Since Pˆ  SˆRˆ x , and Rˆ x DMK
 0  (1) DMK  0 , we get
  s(1) I ,
where π is the eigenvalue of P̂ . Hence, the K= 0 band can be classified as
I   0  ,1 ,2  ,3 ,........, s  1 ,
or, I   0  ,1 ,2  ,3 ,............., s  1 .
For K  0, the intrinsic states have a two-fold degeneracy with respect to Tˆ (Kramer’s
degeneracy). The band gets classified as






1 3 5
I
, , ,............s  i ,
2 2 2
1 3 5
, , ,................, s  i ,
or, I 
2 2 2
where only levels with I  K occur. Since Sˆ K   K , the positive and negative parity
states having the same spin are degenerate, giving rise to the phenomenon of parity
doublets.
21
Examples of Octupole Deformed Nuclei
It is now understood that the shell effects play a very major role in stabilizing a
given configuration towards a particular nuclear shape. Nuclei lying in a narrow range
beyond 208Pb, and to a lesser extent the nuclei in the neutron-excess light rare-earths, have
been found to be prone to octupole deformation of the Y30 type. As shown in Figure 13,
appropriate orbitals with l  3 are observed to be very close together and near the Fermi
energy for nuclei just beyond
1.42 MeV apart in
209
209
208
Pb. For example, the 1j15/2 and 2g9/2 neutron orbitals are
Pb, while the 1i13/2 and 2f7/2 proton orbitals are 1.70 MeV apart in
Bi. The corresponding orbitals in the rare-earth are the 1i13/2 and 2f7/2 neutron orbitals
and the 1h11/2 and 2d5/2 proton orbitals. In the lighter nuclei, the 1g9/2 and 2p3/2 orbitals
come close together near particle number 34. An early review of the experimental
systematics which support the octupole deformation appears in Jain et al. (1990), and a
detailed account of theory and experiment related to the octupole shapes appears in Butler
and Nazarewicz (1996).
As pointed out earlier, parity doublets arise of the type I   0  ,1 ,2  ,....... in eveneven, and I  



1 3 5
, , ,.... in odd-A nuclei. These parity doublet (PD) bands split into
2 2 2
two, if the barrier separating the two octupole minima has a finite height. Due to the
tunneling between the two mirror octupole shapes, the two bands of opposite parity are
displaced in energy with respect to each other with the even spins energetically favoured.
(Figure 14). A similar situation exists for odd-A nuclei, also shown in Figure 14. For
K=1/2 bands, the rotational band is further modified by the octupole decoupling parameter
a   a. p , so that
1
I

1 

E  E 0  A I ( I  1)  a (1) 2 ( I  ) .
2 

It is obvious that a  ( p  1)  a  ( p  1) . Thus, the decoupling parameters for K=1/2
bands have opposite sign but nearly same absolute value. The possibility of tunneling
22
along with octupole decoupling further complicates the energies of the K=1/2 band as
shown in Figure 14.
An example of the spectrum where PD bands have seen is presented in Figure 15.
Here we show the level scheme of
interpretation of
225
Ra taken from Gasparo et al. (2000). The first
225
Ra in terms of octupole deformation was provided by Sheline et al.
(1989). Experimental work of Gasparo et al. (2000) further confirms this interpretation.
We find that 5 PD bands can be identified in the observed spectrum. Each pair of PD band
has been assigned a labelling K  s z ,   and for K=1/2 bands,
 j  , the octupole

decoupling parameter. The value of  indicates the degree of parity mixing in the single
particle states.
Density distribution has two planes of symmetry
The axial-symmetry is lost when we consider shapes like Y3  ,   0 . The density
distribution has only two independent planes of symmetry for μ even, and it is possible to
have a rotation about the long axis. Besides Rx ( )  1 , now, R y ( )T  P . Parity doublets
of even or, odd-angular momenta arise. We, therefore, expect a level pattern like
I  2  ,4  ,.... etc., or, I  1 ,3 ,5  ,......... etc. This corresponds to the situation of 12th row
in Table I. This situation is shown in the top panel of Fig. 16.
If the axis of rotation is perpendicular to one of the symmetry planes, and the




rotation axis is denoted as the x-axis, we have R x ( )  P and R y ( )T  P . Signature is
not a good quantum number. One obtains a pair of parity doublet bands such as the one in
case of axial symmetry given by I  4  ,5  ,6  ,........ and I  4  ,5  ,6  ,........... etc. This
corresponds to the situation shown at row no. 13 of Table I. This situation is shown in the
middle panel of Fig. 16 and is discussed separately under the tetrahedral symmetry in the
following section.
23
Density distribution has only one plane of symmetry
Further reduction in the symmetry occurs if only one plane of symmetry is
supported by the nuclear shape; now one may also have odd μ components. Such a shape is
shown in Figure 16. Rotation is possible along the long axis as well as any one of the short
axes. Signature is not a good quantum number in either case; therefore, both even and odd
spins will be found in the rotational sequence. Parity is also not conserved; therefore, both


the parities will occur. When the rotation is about the long axis, R y ( )T  P and four


distinct situations can be obtained by the application of R x ( ) and P . We obtain a
sequence such as I   4  ,5  ,6  ,......... This represents the situation shown at row no. 12 of
Table I. This situation is depicted in the top panel of Fig. 17. However, when the rotation is


about one of the short axes, R x ( )  P , four distinct situations are obtained by the


application of R x ( ) and R y ( )T . We obtain two nearly degenerate sequence such as
I   (8  ) 2 , (9  ) 2 , (10  ) 2 ,........ . The two I  1 degenerate sequences with alternating
parity represent bands which are chiral partners (one is left-handed and the other is right
handed). This represents the situation shown at row no. 11 of Table I, and in the bottom
panel of Fig. 17.
Tetrahedral and Triangle Symmetries in Nuclei
It has been suggested that there is a possibility of observing a four fold
degeneracy in the level patterns of a number of N ~ 136 isotones [Li and Dudek, 1994].
This symmetry arises due to the λ = 3, μ ≠ 0 components in the nuclear shape discussed in
the proceeding sections. In particular, a tetrahedral symmetry is expected to break both the
spherical symmetry and the symmetry by inversion. More specifically, a deformation of
Y32 ( ,  ) is related to the TdD symmetry group having two 2-dimensional and one 4dimensional irreducible representations. This means that there exist three families of
multiplets: two are doubly degenerate and one is quadruply degenerate.
24
Theoretical spectrum of single particle states as a function of the
deformation parameter a 32 (the coefficient of the Y32 term) reveals strongly increasing gaps
at Z=32, E  2MeV , at Z=40 with E  3MeV , and a huge gap at Z=56, 58 with
E  4MeV (Figure 18). Calculations reveal strong tetrahedral-symmetry effects at
N,Z=16, 20, 32, 40, 56-58, 70, 90-94, for both neutrons / protons and 136/142 for neutrons
only. These minima in tetrahedral shapes coincide with oblate and/or prolate minima in
energy. A ten dimensional minimization in energy for  ,  , a3 (   0,1,2,3) , and
a 4  (   0,1,2,3,4) shapes leads to tetrahedral equilibrium shapes of a32  0.13,0.13,0.15 ,
and 0.11 respectively for
80
40
160
Zr40 ,108
40 Zr68 , 70 Yb 40 , and
242
100
Fm142 . The tetrahedral nuclei also
obey the simplex symmetry and lead to parity-doublet bands but with one important
difference. Since these nuclei will not have any significant dipole moment, the E1
transitions, which are a typical feature of axial octupole nuclei will be absent. In Figure
19, we compare the rotational spectrum of an axial-octupole nucleus with a tetrahedral
rotor. A pear shaped octupole nucleus has considerable dipole moment and hence strong
E1 and E2 transitions. On the other hand, a tetrahedral “pyramid” shape rotor having some
quadrupole shape has zero dipole moment and the lowest multipole transitions will be of
pure E2 type. However, in the ideal case of a pure tetrahedral rotor, only E3 transitions will
be seen.
Recent calculations of Yamagami et al (2000) suggest the possibility of
exotic shapes which break the reflection and axial symmetries in proton rich N=Z nuclei:
64
Ge, 68Se, 72Kr, 76Sr , 80Zr , and
84
Mo . In particular, the oblate ground state of
68
Se is very
soft against Y33 triangular deformation, and the low-lying spherical minimum coexisting
with the prolate ground state in
80
Zr is extremely soft against the Y32 tetrahedral
deformation. The Y33 triangular deformation has only one plane of symmetry and a
rotational spectrum very different from the Y32 tetrahedral shape. There are no known
examples of Y32 and Y33 symmetries so far and their experimental discovery is a distinct
possibility.
25
Rotation about an axis other than the principal axis - Tilted Axis Cranking
So far we have considered situations where rotation is always about one of the
principal axes of the body. Riemann had pointed out the possibility of having ellipsoidal
shapes of equilibrium when the vorticity of internal motion of a non-rigid system leads to
uniform rotation about an axis different from the principal axes of the density distribution.
It is possible to have nuclear configurations which support rotation about an axis lying in
one of the principal planes (planar tilted axis cranking) or, rotation about an axis lying out
of the three principal planes (aplanar tilted axis cranking).
We consider the effect of planar and aplanar tilted axis of rotation for a tri-axial
shape. First, we consider the possibility of rotation about an axis lying in one of the

principal planes. Here parity P is conserved and so is R y ( )T . Signature is not a good
quantum number; therefore, all the spins will be seen. They will have the same parity. We
observe a sequence like I   4  ,5  ,6  ,...... etc. This corresponds to the row no. 2 of Table
I. The situation is shown in the middle panel of Fig. 11.
A further doubling of states occurs if the rotation axis is out of all the principal
planes (aplanar TAC). In this case only parity is conserved. We obtain four distinct
situations by the operation of Rx ( ) and R y ( )T . This gives rise to a rotational sequence
of the type I   (4  ) 2 , (5  ) 2 , (6  ) 2 ,...... , where each spin occurs twice and is nearly
degenerate. This represents the situation at row no 3 of Table I, and is depicted in the
bottom panel of Fig 11.
In the case of odd-multipole shapes such as octupole shape, planar TAC gives rise
to a situation where R y ( )T  P . Parity is no longer an invariant operation. We obtain four
distinct situations by the application of Rx ( ) and P. Two nearly degenerate rotational
sequences emerge, which have I   4  ,5  ,6  ,..... etc. This is identical to the situation at
row no. 13 of Table I and is shown in Fig. 16 (bottom panel).
26
Magnetic Rotation – A Magnetic Top
In a recent development, it has been shown that the isotropy of the mean field can
be broken in a way other than having an anisotropic charge density distribution. The new
kind of anisotropy arises from the currents and, therefore, net magnetic dipole moments
instead of net electric quadrupole moment. Such a situation arises in nuclei when higher
lying high-j neutron particle (hole) combines with a high-j proton hole (neutron) in such a
way that they make a right angle with each other. The resultant angular momentum about
which the nucleus appears to rotate, therefore, makes an angle with the principal axes
(Figure 20). In order that the magnetic effects of current anisotropy dominate, the
deformation should be small. As shown in Figure 20, a net magnetic dipole moment is
generated implying an anisotropy in currents. Higher angular momentum states are
generated by the closing of the neutron and the proton blades as in a pair of shears, hence
the name shears mechanism (Frauendorf, 1993). As a consequence, we obtain a ‘rotation’
like band. The Rx(π) symmetry is broken and signature is no longer a good quantum
number. We obtain a  I=1 band as shown in the example of
consecutive levels of band B5 in
134
Ce in Figure 21. The
134
Ce are connected by strong M1 transitions, with the
M1 intensity decreasing as the shears closes (Fig.22), which is the result of a decrease in
the dipole moment with increasing spin. The very first example of MR band, as recently
reported (Malik et al., 2004), appears to be that of
83
Kr. However, a large number of such
cases have been discovered which are spread in the A=80, 110, 130, 190 mass regions
(Amita et al., 2000).
Chiral Bands
Besides a planar tilt, it is also possible to have an aplanar tilt; the axis of rotation
does not coincide with any of the three principal axes, and also does not lie in any of the
principal planes. Such a situation is best visualized in a triaxial odd-odd nucleus. If the
configuration is such that the odd-proton alignment is along the short axis, the odd-neutron
alignment is along the long axis, and the rotational contribution is along the intermediate
axis, we obtain three angular momenta perpendicular to each other and the resultant
angular momentum acquires an aplanar tilt. Note that the rotation has been taken along the
27
intermediate axes as the moment of inertia about this axis is maximum and the rotational
energy is minimum. While parity is still conserved, such an arrangement breaks the Ry(π)T
symmetry. The two situations shown in the upper part of this panel have a right handed
sense of rotation. On the other hand, the two situations shown in the lower part have a lefthanded sense of rotation. This symmetry breaking doubles the number of levels and we
should observe two pairs of identical  I=1 bands having the same parity. These have been
termed as chiral bands.
In real nuclei, the bands will be shifted in energy because of the tunneling between
the right-handed and the left-handed states. It may be noted that existence of triaxiality and
an optimum quadrupole deformation play an important role in breaking the chiral
symmetry. Dimitrov et al (2000) presented the first results of an aplanar TAC calculation
which support the existence of chiral bands in 134Pr. However, the more recent observation
of a chiral pair of bands in an odd-A nucleus
135
Nd (Zhu et al. 2003) has confirmed that
chiral rotation is a purely geometric phenomenon, and not confined to odd-odd systems
alone. The level scheme of
135
Nd is presented in Figure 23, where the bands A and B
become chiral partners at higher rotational frequencies. The triaxial shapes shown in the
upper part of the figure are labeled by l, s, and i-axes which stand for long, short, and
intermediate axes, respectively. In the order s-i-l, these axes form a “right-handed” system
in the ellipsoid on the left and a “left-handed” system in the ellipsoid on the right - forming
a chiral doublet. The corresponding bands A and B are shown in the level scheme. The two
bands come very close to each other at a rotational frequency of about 0.45 MeV, and
become interlaced with each other. Since the two are based on the same configuration
h
2
11 / 2

, vh111/ 2 , there are good number of linking transitions between the two bands. In
Figure 24, we show the example of a pair of such bands in
135
Ce, an odd-A nucleus
(Lakshmi et al, unpublished). It is highly interesting to note that what we observe here is
two-way connection between the pair of bands, which is seen for the first time. This
confirms that the pair of bands have same configuration. This also confirms that the chiral
bands are purely a geometrical phenomenon arising out of the special situation of the three
vectors.
28
Conclusions
We have discussed the basics of the geometrical symmetries and their
consequences in nuclei. Connection between the various shapes and band structures was
emphasized. Unusual shapes were also considered. Recent discoveries like the magnetic
rotation and chiral rotation were discussed, which involve rotation about a tilted axis rather
than the usual principal axis. An attempt was made to present a simple guide, which will
be useful to the experimentalists.
Acknowledgements
Financial support from the I.A.E.A.(Vienna), D.S.T. (Govt. of India), and D.A.E.
(Govt. of India) in the form of research projects is gratefully acknowledged.
29
Bibiliography
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30
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31