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INSTITUTE OF PHYSICS PUBLISHING
PHYSICA SCRIPTA
Phys. Scr. T125 (2006) 1–7
doi:10.1088/0031-8949/2006/T125/001
Orbits, shapes and currents
Stefan Frauendorf
Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA
and
IKH, FZ-Rossendorf, PF 510119, 01314 Dresden, Germany
E-mail: [email protected]
Received 14 July 2005
Accepted for publication 1 September 2005
Published 28 June 2006
Online at stacks.iop.org/PhysScr/T125/1
Abstract
Shapes and current distributions of nuclei and alkali clusters are discussed in terms of
the single particle motion of fermions in the average potential. For small particle number,
an interpretation in terms of the lowest spherical harmonics is presented. For large
particle number, an interpretation in terms of classical periodic orbits is presented.
PACS numbers: 21.10.Re, 21.10.Dr, 23.20.Lv, 24.10.Pa, 36.40.Cg, 73.22.−f, 73.23.Ra
1. Introduction
Sven Gösta Nilsson’s work [1] that is commemorated at this
meeting treated for the first time the motion of nucleons in a
realistic spheroidal potential generated by all other nucleons.
The famous Nilsson diagrams, which show the single particle
energies as functions of the deformation parameter, gave us a
first quantitative estimate of the nuclear shape. This seminal
paper comprises in an ingeniously simple way, the concept of
the mean field approach:
Z
δ e[ρ(Er )] dτ → {t + V [ρ(Er )] − ωl z }ψi = i ψi → ρ(Er ).
(1)
The average potential V [ρ(Er )] is generated by variation
of some energy density functional with respect to the nucleon
density ρ(Er ). Finding the energies i and wavefunctions ψi of
the nucleons in the potential V [ρ(Er )] permits us to calculate
the density ρ(Er ). Making the potential self-consistent with
density determines the shape of the density distribution and
the potential, which is called the nuclear shape. Inclusion of
the cranking term −ωl z permits us to calculate the current
E r ) of a rotating nucleus that carries a finite
distribution j(E
amount of angular momentum.
Since Nilsson’s paper, nuclear shapes have been studied
in great detail using a variety of mean field approaches. They
all gave very similar shapes. Moreover, the shapes of alkali
atom clusters were calculated in a mean field approximation
similar to that of nuclei. Here, one calculates the energies
and wavefunctions of one electron in the average potential
generated by all other delocalized conduction electrons and
the positive ions, which are treated as a ‘jellium’ of positive
charge. As discussed later, the cluster shapes are very similar
0031-8949/06/125001+07$30.00
to nuclear shapes. These examples illustrate the general
observation that the shapes of all finite fermion systems that
have near constant density in the interior which goes to
zero within a thin surface layer, are essentially the same.
These shapes reflect the quantized motion of the fermions
in a leptodermic potential. The relation between this quantal
motion and the shape that the system takes will be discussed.
There are two simple situations. For small particle number the
shape of the system reflects the shape of the lowest quantal
orbitals. Since the shape of the molecules reflects the shape of
the s- and p-valence orbitals, it is called the chemical regime.
For large particle number, the combined geometry of many
single particle orbitals determines the shape of the system. If
one is interested only in the gross structure, the shapes can be
discussed in terms of periodic classical orbits of the fermions
in the average potential. There is a certain analogy to the
construction of a concert hall. In order to avoid echoes, one
studies how sound impulses bounce from the walls and travel
through the interior (periodic orbits). It is called the acoustic
regime.
Clusters are simpler than nuclei, because only the
delocalized conduction electrons determine the shape. In
nuclei, both the protons and the neutrons do this. If both
prefer the same shape, the resulting shape is similar to those
of clusters. If the protons and neutrons prefer different shapes,
the result will be a compromise. A comparisons of systematic
calculations for clusters, as shown in figure 1, with those in
e.g. [3] clearly reveals this simple interplay of protons and
neutrons. Hence, only one kind of fermion as in the case of
clusters may be considered.
The current distribution of rotating nuclei and alkali
clusters exposed to a magnetic field is also largely determined
by the quantization of the fermionic motion. The two
© 2006 The Royal Swedish Academy of Sciences
Printed in the UK
1
Stefan Frauendorf
2. The chemical regime
Figure 1. The deformation parameters of heavy alkali clusters as
calculated in [2] using the shell correction method. The deformation
parameters αl approximately correspond to the standard deformation
parameters βl introduced by a multipole expansion of the surface.
phenomena are related as follows. The external magnetic
field induces electric currents in a cluster that are carried by
the conduction electrons. These currents generate a magnetic
moment. A nucleus does not rotate like a rigid body. The
currents that generate the total angular momentum consist of
two parts
E r ) = jErig (E
j(E
r ) + jEsh (E
r ),
jErig (E
r ) = ρ(E
r )ωE × rE.
(2)
The term jErig (E
r ) is the current of a rigid body that rotates
with the angular velocity ω.
E The deviation jEsh (E
r ) is caused by
quantized motion of the fermions. It represents the residual
current in the rotating frame of reference that is fixed to the
body. This residual currents correspond to the electric currents
in a cluster, where the magnetic field should be expressed by
the Larmor frequency.
2
In the case of molecules the direction of the chemical bonds is
determined by the density distribution of the valence s-orbital
(isotropic spherical harmonic Y00 ) and the valence p-orbitals
(the anisotropic spherical harmonics Y1m ). The interaction
with the partners in the bonds may cause a strong mixing of
the valence s- and p-orbitals, which is called hybridization.
The hybrid orbitals have new shapes, like the three bonds of
the sp2 hybrid, which lie in a plane with a mutual angle of
120◦ . In the lightest fermion systems only the s-, p- and dorbitals are occupied. The shape of the system reflects the
shape of these orbitals. Figure 2 shows the shapes of the
lightest alkali clusters and nuclei. In most cases they closely
resemble the shapes of the density distributions obtained
by filling the s-, p- and d-orbitals in a spherical potential.
The nodal structure of the first eigenmodes in a compact
potential has to be similar to these orbitals, which means a
similar density distribution. The admixture of higher shells
modifies the radial profile of the orbitals such that the density
changes from the interior value to zero within a thin surface
layer. Such a profile minimizes the total energy of the energy
density functional, which has a minimum at the saturation
density. As a consequence, the
P shapes (equidensity surfaces)
resemble the surfaces R = l,m n lm |Ylm |2 , which are also
shown. Moreover, the energy density functionals favour a
small surface area because of the positive surface energy.
Since the surface tension of nuclei is larger than that of
clusters, the nuclear surface follows less well the shapes
of the valence orbitals, because a more rounded shape has
a lower surface energy. If the state l,
√m 6= 0 is only half
filled, the hybrid orbital (Ylm + Yl−m )/ 2 is formed, which
has a lower surface energy. The hybridization is also driven
by the tendency to clusterization. The N = 10 system is an
example. As seen in figure 3, the calculated shape is somewhat
reflection asymmetric. The asymmetry can be generated by
some admixture of the harmonics Y2±1 to the Y1±1 and of
Y30 to Y20 (see figure caption). The mixing coefficients of this
hybridization are determined by the tendency to form the two
particularly stable subsystems, N = 8 and N = 2. The energy
gain in forming two closed shell clusters has to compete with
the loss due to the larger surface of the two fragments. The
shape is a compromise, showing partial clusterization. As seen
in figure 2, the shapes of the N = 12, 16 and 18 systems
also differ somewhat from the reflection-symmetric shapes
generated by filling the spherical orbitals. The asymmetry is
caused by a similar hybridization mechanism as discussed for
N = 10 (see [4]).
The rotational and magnetic response reflects the current
distribution generated by combining the lowest orbitals
such that the cranking term −ωl z is taken into account.
Figure 4 shows the velocity field in the rotating frame of
E r )/ρ(E
reference vEsh (E
r ) = j(E
r ) − ωE × rE for the N = 4 system.
The presence of strong vortices demonstrates the dramatic
deviation of the current from rigid rotation. The pattern is
dominated by the transition current between the l = 1, m = 0like and l = 1, m = 1-like orbitals without radial nodes. It is
combined with some irrotational current. The example is a
harmonic oscillator potential at equilibrium deformation as
discussed in [5]. In this case, the moment of inertia takes
Orbits, shapes and currents
(l = 0, m = 0) n = 2 = 00 2
008 20 2
00 2101
00 210 2
008 20 2 (21 + (2 − 1)) 2
008 20 2 212
00 210 2 (11 + (1 − 1))1
00810 −2
00810 −1
008
008 201
00 20 20 −2 (21 + (2 − 1)) −2
00 20 20 −1 (21 + (2 − 1)) −1
00 20
00830 2
Figure 2. The shapes of the lightest alkali clusters and N = Z nuclei as calculated
P in [4] using the ultimate jellium energy density
functional. Shown are the surfaces of half density. In addition, the surfaces R = l,m n lm |Ylm |2 are shown, where the occupation number n lm
of each orbital is given as (lm)n .
Figure 3. The shape of the N = 10 cluster as in figure 2. The upper
panel shows the surfaces generated from spherical harmonics,
where the configuration 002 102 112 1(−1)2 202 is left and
the hybridized configuration 002 102 (11 + 0.5 × 21)2
(1(−1) +0.5 × 2(−1))2 (20 + 0.23 × 30)2 is right.
The lower panel shows the shape calculated in [4].
the value for rigid rotation Jrig , i.e. the contributions of the
vortices to the total angular momentum cancel. In the case
of a more realistic leptodermic potential, the contributions
Figure 4. The velocity field in the rotating frame of four fermions
in a spheroidal harmonic oscillator potential, the axis ratio of which
is chosen to satisfy the condition for equilibrium deformation
(see [5]).
from the currents in the body fixed frame do not cancel in
general. The deviations of the moment of inertia from the
rigid body value will be discussed for large systems. They are
stronger than for small systems, because the potential deviates
3
Stefan Frauendorf
more from the oscillator one. However, it is stressed that the
flow pattern deviates strongly from rigid rotation, even if the
moment of inertia is close to the rigid body value.
3. The acoustic regime
Now the case of large particle number is discussed, which
corresponds to a mesoscopic scenario. The level density,
binding energy, and other quantities may be considered as
a sum of a smooth part that represents the properties of a
small but macroscopic droplet and a shell part that describes
the consequences of the quantal motion of the fermions near
the Fermi surface. The shapes are the result of a compromise
between the liquid drop energy, which prefers a spherical
shape to reduce the surface energy, and the shell energy,
which seeks a shape for which the level density near the
Fermi surface is low. The gross dependence of shapes on
the particle number is discussed, which corresponds to some
average of adjacent clusters or nuclei. The gross shell structure
is described by means of the periodic orbit theory (POT), a
detailed presentation of which was given in [6]. The energy is
decomposed into a smooth part Ẽ, which is the energy of the
droplet, and an oscillating part E sh , which describes the shell
energy,
X
E = Ẽ + E sh ,
E sh =
Eβ ,
(3)
Figure 5. Classical periodic orbits in a spheroidal cavity at
moderate deformation.
β
where β labels the periodic orbits that generate the shell
energy. The gross shell structure is given by the few shortest
orbits. For simplicity, the case of a cavity, i.e. an infinite
potential step at the surface is discussed. Then the classical
orbits are composed of straight lines and specular reflection
on the surface. The energy
e is conveniently expressed by
√
the wavenumber k = 2me/h̄. For the spherical cavity, the
periodic orbits are equilateral polygons. The upper panel of
figure 5 shows the triangle and the square, which are the
simplest. Each polygon belongs to a threefold degenerate
family, which corresponds to the possible orientations of
the polygon in space that are generated by the three Euler
angles. All orbits of a family have the same length L, the
circumference of the polygon. As seen in figure 5, there are
two types of orbits in the case of a cavity with moderate
spheroidal deformation. The equator orbits are the regular
polygons in the equatorial plane. They belong to a onefold
degenerate family generated by the possible rotations of the
polygon in the plane. The meridian orbits are the polygons
in a plane that goes through the symmetry axis. They belong
to a twofold degenerate family, which is generated by the
rotation of the meridian plane about the symmetry axis and a
shift of the reflection points on the surface within the meridian
plane.
The contribution from each family to the shell energy is
given by
2
h̄
kF L β
,
Eβ =
Aβ (kF ) sin(L β k + νβ )D
(4)
τβ
γ Ro
where Ro = ro N 1/3 is the radius of a sphere with the same
volume as the spheroidal cavity. The period of revolution
√ of
a particle moving with the Fermi momentum h̄kF = 2meF
on the orbit is denoted by τβ . Each term in the sum oscillates
4
Figure 6. Shell energy for a cavity-like spheroidal potential. For
α = 0, the shape√is spherical, for α = 0.5, the shape is prolate with
an axis ratio of 3,
√ and for α = −0.5, the shape is oblate with
an axis ratio of 1/ 3. The dashed and full lines correspond to
sin(L 4 kF + ν4 ) = 1, −1, respectively. The figure is relevant for alkali
clusters. Taken from [7].
with the frequency kF as a function of the length L β of the
orbital. The Maslov index νβ is a constant phase (see [6]). The
amplitude Aβ depends on the degeneracy of the periodic orbit:
the more symmetries a system has, the greater the degeneracy,
and the more pronounced are the oscillations of the shell
energy. The damping factor D(k L β /γ Ro ) is a decreasing
function of its argument, which filters out the shortest orbits
that are responsible for the gross shell structure.
3.1. Shapes
For a qualitative discussion of the shapes, it is sufficient
to consider only the family of tetragons. Figure 6 shows
the shell energy of N fermions in a spheroidal cavity with
deformation α. The volume of the cavity is constant, V =
4
πro3 N . The figure includes the lines of constant length of
3
the tetragons L 4 kF + ν4 = π(2n + 1/2) (dashed) and L 4 kF +
ν4 = 2π(2n − 1/2) (full), which correspond to the maxima
and minima of the sin-function in equation (4), respectively.
At spherical shape, the shell energy oscillates as function
of the particle number N , because L 4 ∝ N 1/3 . The minima
at N = 58, 92, 136 are the spherical shell closures. When
the cavity takes a prolate deformation (α > 0), the meridian
orbits become longer. In order to keep the length constant,
the size of the cavity must decrease, which corresponds to
a smaller particle number N . Hence, the lines of constant
Orbits, shapes and currents
length of the meridian orbits are down sloping. The equator
orbits become shorter at prolate deformation. In order to
keep the length constant, N must increase, and the lines of
constant length are up sloping. Figure 6 shows the interference
pattern between the meridian and equator orbits. There is
a system of down sloping valleys and ridges emaneting
from the minima and maxima at spherical shape, which
follow the lines of constant length of the meridian orbits.
Superimposed is a system of up sloping valleys and ridges
that follows the lines of constant length of the equator orbits.
The contribution from the meridian orbits is stronger because
they are twofold degenerate, whereas the equator orbits are
only onefold degenerate. On the oblate side, the meridian
orbits become shorter and the equator orbits longer, and the
particle number N must increase and decrease respectively,
in order to keep the length constant. The landscape for the
shell energy of the realistic nuclear potential is similar to
the cavity case, except that the spin–orbit term changes the
magic numbers for spherical shell closure to N = 50, 82, 126.
Hence, the following interpretation applies to nuclei as well if
the appropriate degree of shell filling is considered.
The shape is spherical (α = 0) for closed shells (N =
58, 92, 136). Taking particles away, the equilibrium shape is
located in the valley on the prolate side that is generated by
the meridian orbits. The deformation gradually increases in
the valley. If N decreases further, the equilibrium shape moves
over the saddle on the ridge that is generated by the meridian
orbits. The deformation decreases abruptly after the saddle.
Figure 1 shows the smooth increase of the deformation α with
decreasing N and its sudden decrease. This N -dependence
of the quadrupole deformation is also experimentally well
known for nuclei. Reference [8] first pointed out that it reflects
the down sloping meridian valleys.
The meridian ridges and valleys are less down sloping on
the oblate side than on the prolate side. The different slope
can be explained in terms of geometry. In the lower panel of
figure 5, consider the meridian square orbit that has two of its
sides perpendicular and two parallel to the symmetry axis of
the cavity. If a is the length of the symmetry semi-axis and b
the length of the other two axes, then the length of the orbit
L 4 ∝ a + b. The volume of the cavity is V = 4πab2 /3. It has
to be the same as for spherical shape V = 4π Ro3 /3. This
means L 4 ∝ b/Ro + 1/(b/Ro )2 . This function has a larger
negative slope for b/Ro < 1 (prolate) than for b/Ro > 1
(oblate). In fact, the slope becomes positive for b/Ro >
21/3 . The system tries to avoid the mid-shell mountain at
spherical shape by taking a deformed shape. It is energetically
favourable to go to the prolate side, because the argument of
the sin-function changes more rapidly. This is the explanation
of [9] for the preponderance of prolate over oblate shapes,
which can be seen in figure 1 and which is well known for
nuclei.
The weaker valley-ridge structure generated by the equator orbits is less important for the quadrupole deformation.
However, it generates the hexadecapole deformation. As seen
in figure 6, the meridian and equator valleys cross in the middle of the shell at N ≈ 76, α ≈ 0.3 and N ≈ 114, α ≈ 0.25.
The constructive interference generates the local minimum,
which corresponds to the favoured shape. Figure 1 shows
that these are the quadrupole deformation parameters α of
Figure 7. Shapes of heavy alkali clusters as calculated in [2] using
the shell correction method. The arrows have the same length.
mid-shell clusters, and that the hexadecapole deformation
α4 ≈ 0 for them. Moving along the meridian valley to larger
or smaller N , brings the system out of the equator valley,
which makes the shell energy less negative. It is energetically
favourable to change the shape such that the equator diameter remains the same as in the middle of the shell. Then the
length of the equator orbits does not change, and their contribution to the shell energy remains at the most negative value.
The hexadecapole deformations α4 in figure 1 are positive
below and negative above the middle of the shell. Figure 7
shows that the equator diameter of the equilibrium shapes is
approximately constant indeed. The measured nuclear shapes
follow the same sequence through a shell: lemon-like in the
lower part of the shell, spheroidal in the middle, and barrellike in the upper part of the shell.
3.2. Currents and moments of inertia
Here, the case of slow rotation is considered, which means
that the periodic orbits are close to the discussed polygons (for
details concerning this simplification see [7]). The important
quantity is the rotational flux through the orbit. It is defined in
the same way as the magnetic flux, which takes its role in the
case of clusters in a magnetic field,
h̄8(θ ) = 2m Aβ ω cos θ.
(5)
Here, Aβ is the area enclosed by the orbit, and θ is the angle
between the normal of its plane and the axis of rotation. The
rotation manifests itself in the appearance of an additional
modulation factor M in the expression for the shell energy,
X
X
Eβ →
Mβ E β .
(6)
β
β
The modulation factor is an oscillating function of 8β , which
is the flux through the orbit perpendicular to the rotational
axis. These oscillations determine the shell energy at high
spin, which is discussed in [7]. Only the first term (quadratic)
of the expansion into powers of 8β is considered, which gives
an expression for the moment of inertia,
X
J = Jrig + Jsh ,
Jsh =
Jβ ,
(7)
β
5
Stefan Frauendorf
Figure 8. The experimental ground state shell energy (upper panel)
and the shell moment of inertia of unpaired nuclei (lower panel) as
functions of the neutron number N . The different symbols give the
proton number Z . Taken from [7].
where Jrig is the rigid body value. For the discussion, only
one term in the sum over the periodic orbits is kept. Then the
moments of inertia are given by
Jshk =
h̄ 2 2
(k Ak )2 E shk ,
eF2 F
Jsh⊥ =
h̄ 2 2
(k A⊥ )2 E sh⊥ ,
2eF2 F
(8)
where E(kF )k is the contribution of the equator orbits to the
ground-state shell energy (4), and E(kF )⊥ the corresponding
contribution of the meridian orbits. The areas of the respective
orbits are Ak and A⊥ .
Figure 8 shows the experimental ground state shell
energies of heavy nuclei as function of the neutron number,
which are obtained in the usual manner by subtracting
the energy of a spherical droplet from the experimental
ground state energies. In the lower panel the experimental
shell contribution to the moment of inertia (‘shell m. o.
i.’) of unpaired nuclei is shown. It is obtained by fitting
the expression I 2 /2Jexp to the experimental yrast energies
above spin I = 20h̄, where the pair correlations are essentially
quenched. The experimental shell moment is the difference
Jsh = Jexp − Jrigid (for details see [7]). The deviations of the
moments of inertia from the rigid body value are strong. Their
6
N -dependence is similar to the one of the ground state shell
energy, which is expected from the relation (8). Equation
(8) gives a relative scale of Jsh ≈ (h̄ 2 /1000 MeV2 )A4/3 E sh ,
which is correct as the comparison of the two panels
shows. However, concerning POT, there is one important
difference between ground state energies and moments of
inertia. Whereas all orbits contribute to the ground state shell
energy, only the orbits that carry rotational flux contribute
to the shell moment of inertia. That is, only the meridian
orbits contribute if the rotational axis is perpendicular to
the rotational axis, and only the equator orbits contribute
if the rotational axis is parallel to the rotational axis. This
is indicated by the subscripts k and ⊥ for the parallel and
perpendicular orientation of the rotational axis. On the other
hand, both the meridian and equator orbits contribute to the
ground state shell energy, i.e. E sh = E shk + E sh⊥ . The two
orientations of the rotational axis can be easily distinguished.
If it is perpendicular to the symmetry axis, one observes
a regularly spaced rotational band. If it is parallel to the
symmetry axis, the yrast line becomes an irregular sequence
of states that contains many high-K isomers.
The strongly negative value of Jsh at the magic number
N = 126 reflects the shell closure. The POT interpretation is
the same as discussed above for E sh . The dip at N = 82 is less
deep, because Z is mid shell. As for E sh , the positive value of
Jsh around N = 90 is caused by the meridian ridge. Nuclei in
this region rotate about an axis perpendicular to the symmetry
axis and show rapid alignment of nucleon angular momentum
with this axis (back bending). The bump in Jsh demonstrates
that the meridian orbits are responsible, because only they are
perpendicular to the axis of rotation and thus carry rotational
flux. The bump of Jsh around N = 110 is caused by the
equator ridge. Nuclei in this region rotate about the symmetry
axis, which is reflected by the appearance of many highK isomers. This demonstrates that the equator orbits must
be responsible, because only they are perpendicular to the
rotational axis and carry rotational flux.1 The equator orbits
generate only a shoulder in E sh , because the meridian orbits
make a negative contribution that increases with N .
Figure 9 shows the velocity field of a nucleus within the
group around N = 90, which have a positive shell moment
of inertia. As expected, there is a substantial current in the
body fixed frame, which has the same direction as the rotation.
It causes the positive shell moment of inertia. The velocity
field resembles the one generated by nucleons moving on the
meridian orbits shown in the insert. As discussed in [10],
nuclei outside this region have a similar current distribution
in the rotating frame. However, the velocities are opposite to
the rotation, which is consistent with a negative shell moment
of inertia. A more quantitative relation between the currents
and the motion of particles on classical orbits remains to be
established.2
The negative shell energy that causes super deformation
of rotating nuclei is substantial. At a first glance one might
expect that this would correspond to a negative shell moment
of inertia. However, superdeformed nuclei are known to
1
For most of the yrast states the rotational axis is not completely parallel to
the symmetry axis. That is why Jsh does not become positive. It does become
positive in microscopic calculations that assume parallel rotation [7].
2 Rotation about the symmetry axis was not studied in [10].
Orbits, shapes and currents
4. Conclusion
Figure 9. The velocity field of 162 Yb in the laboratory system (left)
and the body fixed system (right) at angular momentum of about
15h̄. Taken from [10]. The inset shows the tetragonal orbits that are
responsible for the current in the rotating frame. The arrows indicate
that more particles run anti-clockwise than clockwise.
The shapes of nuclei and alkali clusters reflect the quantized
motion of the fermions at the Fermi surface in the average
potential. They are not sensitive to the interaction that
generates the potential, i.e. their properties are universal to
all leptodermic potentials. The relation between the quantal
single particle motion and the shapes and currents is relatively
transparent for small and large particle number. For small
particle number, the pattern originates from the density
and current distributions of the first (hybridized) spherical
harmonics. For large particle number, the pattern is generated
by the shortest classical orbits in the potential. The orbit length
controls the deformation. Constant length of the orbits in the
meridian plane determines the quadrupole deformation, and
constant length of the orbits in the equator plane determines
the hexadecapole deformation. The current pattern is universal
as well. In the absence of pair correlations, the currents of
rotating nuclei strongly deviate from rigid flow. In heavy
nuclei, this is reflected by the moments of inertia at high spin
differing from the rigid body value. The difference can be
related to the classical periodic orbits. It suggests a strong
magnetic response of alkali clusters.
Acknowledgments
This work was supported by the US Department of Energy
under contract DE-FG02-95ER40934.
References
Figure 10. Classical orbits in a superdeformed cavity.
have moments of inertia that are close to the rigid body
value. The reason is that super deformation is caused by
other types of orbits [11], examples of which are shown in
figure 10. Superdeformed nuclei rotate about an axis
perpendicular to the symmetry axis. Therefore the equator
orbits do not contribute to the shell moment of inertia. The
meridian orbits are of the butterfly type. They do not carry flux
because the contributions of the two wings compensate each
other. (The motion is clockwise in one wing and anticlockwise
in the other.) Since none of the orbits carries rotational flux,
the moment of inertia takes the rigid body value.
[1] Nilsson S G 1955 Mat.-Fys. Medd. K. Dan. Vidensk. Selsk.
29 1
[2] Frauendorf S and Pashkevich V V 1996 Ann. Phys., Lpz. 5 34
[3] Möller P, Nix J R, Myers W D and Swiatecki W J 1995
At. Data Nucl. Data Tables 59 185
[4] Koskinen M, Lipas P O and Manninien M 1995 Z. Phys. D
35 285
[5] Bor A and Mottelson B 1975 Nuclear Structure II
(New York: Benjamin)
[6] Brack M and Bhaduri R K 1997 Semiclassical Physics
(Reading: MA: Addison-Wesley)
[7] Deleplanque M A, Frauendorf S, Pashkevich V V and Chu S Y
Z. Phys. A 283 269
[8] Strutinsky V M, Magner A G, Ofengenden S R and Dossing T
1977 Z. Phys. A 283 269
[9] Frisk H 1990 Nucl. Phys. A 511 309
[10] Fleckner J, Kunz J, Mosel U and Wüst E 1980 Nucl. Phys. A
339 227
[11] Yamagami M and Matsuyanagi K 2000 Nucl. Phys. A 672 123
7