#  The shapes of nuclei

G.F<EMAIL_ADDRESS>Department of Physics and Institute for Nuclear
Theory,
University of Washington, Seattle, Washington 98915, USA

###### Abstract

Gerry Brown initiated some early studies on the coexistence of different
nuclear shapes. The subject has continued to be of interest and is crucial for
understanding nuclear fission. We now have a very good picture of the
potential energy surface with respect to shape degrees of freedom in heavy
nuclei, but the dynamics remain problematic. In contrast, the early studies on
light nuclei were quite successful in describing the mixing between shapes.
Perhaps a new approach in the spirit of the old calculations could better
elucidate the character of the fission dynamics and explain phenomena that
current theory does not model well.

##  I Introduction

In the early 1960’s, Bohr and Mottelson pointed out some puzzling experimental
data: some light nuclei thought to be spherical in shape had excited energy
levels exhibiting characteristics of deformed nuclei. This was taken up first
by Engeland en65  and then Gerry Brown, who saw an opportunity to test the
realistic nuclear interactions that were being developed at the time. The
nuclei 16O and 18O were the first subjects of study br66 . As his graduate
student, I worked on a parallel study of Ca isotopes as part of my thesis
project. Later, a more definitive study of the Ca nuclei was carried out by
Gerace and Green ge67 . As a general conclusion, one saw that the mixing
between shapes could be understood with the realistic interactions derived
from nucleon-nucleon scattering data.

Since those early days of nuclear structure physics, the subject of nuclear
deformation has matured. First of all, we now know that the shape coexistence
is ubiquitous in the low-energy spectra of nuclei across the periodic table,
affecting even the fission properties of the heaviest nuclei. Also, we now
have computational tools to describe and predict the static features of the
landscape of nuclear shapes. However, the dynamics of shape change, ie. how
different shapes mix together, has been a challenging problem in the theory of
heavy nuclei and is still not well understood. I shall describe some work I
have been engaged in recently, to develop a new approach to fission dynamics
in the spirit of the old studies on light nuclei.

##  II Theory of static deformations

Nuclei are highly deformable, and the first task is to construct reliable
models of the nuclear potential energy surface, ie. the energy of
configurations as a function of deformation coordinates such as the
expectation values of quadrupole and higher moments. An immediate question is
how to define both energy and shape of a configuration: the operators for
these two quantities do not commute. The resolution of this conundrum is that
we are only dealing with approximate wave functions, not the true eigenstates
of the Hamiltonian. In practice, theory relies on the mean-field
representation of wave functions as products of single-particle orbitals. The
present-day calculational framework is very similar to the Density Functional
Theory (DFT) of condensed matter physics. One defines an effective
interaction, which may depend on the local density. The energies and shapes
are determined by minimizing the energy expectation value just as in Hartree-
Fock (HF) theory.

Several families of DFT in use today, and all have considerable predictive
power on static properties of nuclei. In the examples discussed below, I will
show results obtained with the Gogny D1S functional. de10 . It has 14
parameters, 3 of them fixed and 11 adjusted many years ago D1S  to reproduce
general nuclear properties. An example showing its considerable predictive
power is recent systematic study of the low-energy spectroscopy of even-even
nuclei de10 . I will come back to findings from this study later. Another
success of the DFT approach is verification of additional shape minima at very
high deformation in heavy nuclei. The potential energy surface that a
fissioning nucleus traverses has at least two minima and perhaps more. Fig. 1
shows the energy versus deformation in a typical fissile nucleus.

Figure 1: Potential energy surface for 236U covering the range of deformations
up to the scission point. The curve was calculated by minimizing the D1S
density functional with the mass quadrupole moment constrained to the value on
the abscissa. The point marked “I” is the global minimum. The points marked
“A” and “B” are saddle points on the fission path, with a second minimum “II”
in between.

##  III How nuclei change shape

Up to now, the only practical approach for treating shape dynamics in heavy
nuclei is the Generator Coordinate Method (GCM) proposed by Hill and Wheeler
hi53 . Formally, one generates a continuum of configurations by minimizing the
Hamiltonian in the presence of an external field. Thus, the configuration
$|\Psi_{\lambda}\rangle$ is defined by minimizing

$\langle\Psi_{\lambda}|\hat{H}-\lambda\hat{Q}|\Psi_{\lambda}\rangle$ (1)

where $\hat{H}$ is the Hamiltonian or energy-density functional and $\hat{Q}$
is a one-body operator such as the axial quadrupole field

$\hat{Q}_{20}=\hat{z}^{2}-(\hat{x}^{2}+\hat{y}^{2})/2.$ (2)

Then one constructs an effective Hamiltonian from information about the
diagonal and off-diagonal matrix elements
$\langle\Psi_{\lambda}|\Psi_{\lambda}^{\prime}\rangle$ and
$\langle\Psi_{\lambda}|\hat{H}|\Psi_{\lambda}^{\prime}\rangle$.

Computationally, the GCM can be implemented in two ways. In the first way, the
moment $q=\langle\Psi_{\lambda}|\hat{Q}|\Psi_{\lambda}\rangle$ is treated as
the position variable in a one-dimensional Schrödinger equation. It is
straightforward to derive formulas for potential energy function $V(q)$ and a
kinetic energy operator

$-{1\over 2}{\partial\over\partial q}{1\over{\cal I}(q)}{\partial\over\partial
q}$ (3)

in approximations such as the Gaussian Overlap Approximation gi83 . This one-
dimensional collective approach works well for treating the ground state
tunneling under the barriers, but generalizing it to include excited states is
cumbersome be11 . A more technical problem is necessity to define at least
four or five deformation operators to completely specify the shape of the
fissioning nucleus. It is possible, but not easy, to set up and solve the
corresponding higher-dimension Schrödinger equation; a two-dimensional
approximation was carried out in Ref. go05 .

The second computational method is to discretize the GCM configurations with a
mesh of $\lambda$ values and minimize the Hamiltonian in that finite basis.
There are obstacles to this approach as well. On a purely technical level, the
fact that the configurations are not orthogonal leads to numerical
instabilities. More serious on a fundamental level, it is not clear how to
calculate interaction matrix elements between configurations when they are
constructed via an energy density functional. Several plausible prescriptions
are possible, but the results can be unphysical if the wrong prescription used
ro10 . Such problems never occurred in the early work on light nuclei—we used
orthogonal bases and we had a pretty good idea of the interactions.

To build an alternative to the GCM approach, we should start by constructing
an orthonormal basis within the mean-field framework. The dynamics can be
developed later from the off-diagonal interactions in this basis. In the old
work, the basis was constructed from the harmonic oscillator Hamiltonian by
using the associated SU(3) group structure to organize the many-body
configurations. This is obviously too crude for treating heavy nuclei, and it
is better to use the DFT to build the orbitals. In the nuclear context, one
could use DFT orbitals and still preserve orthogonality by to constraining
them by their quantum numbers rather than by their deformations. For most
nuclei, the mean-field potential is axially symmetric and has good parity at
the DFT minima. This allows one to assign orbital quantum numbers $K^{\pi}$,
where $K$ is the angular momentum about the symmetry axis and $\pi$ is the
parity of the orbital. The DFT minimization can be carried out taking as a
constraint the number of nucleons having given values of $K$, $\pi$, and
isospin projection $\tau_{z}$. Wave functions that differ in filling numbers
are automatically orthogonal, so this should be very helpful for constructing
an orthogonal basis. In the remaining sections of this article, I will go
through some examples illustrating the use of a $K^{\pi}$ partitioning to
define the configurations.

Before going on to the examples, it is instructive to see how the partitioning
works in a semiclassical limit. Assume that the nucleus is spherical and the
phase space density is uniform in spheres of radius $R$ in coordinate space
and $k_{F}$ in momentum space. Then the number of nucleons having orbital
quantum number $M$ is given by

$n_{M}=g\int^{R}d^{3}r\int^{k_{F}}{d^{3}k\over(2\pi)^{3}}\delta(M-(\vec{r}\times\vec{k})_{z})$
(4)

where $g=2$ is the spin degeneracy of the nucleon and $M$ is the orbital
angular momentum about the $z$-axis. The integrals can be carried out
analytically; the result is

$n_{M}={9N\over 4k_{F}R}\left((1/2+x^{2}){\rm
arccos}\,x-3x(1-x^{2})^{1/2}\right)$ (5)

where $x=M/(k_{F}R)$. The same formula applies to ellipsoidally deformed
nuclei with the radius $R$ replace by the transverse radius of the ellipsoid.
Fig. 2 shows the distribution for the neutrons in the spherical nucleus 208Pb,
comparing the mean-field fillings with the formula (5).

Figure 2: $M$ distribution of occupied neutron orbitals in the nucleus 208Pb.
Histogram: spherical shell model. Dashed line Eq. (5 with $R=7.1$ fm,
$k_{F}=1.35$ fm-1.

##  IV Examples

The shells of the three-dimensional harmonic oscillator model are completely
filled or empty at nucleon numbers 8, 20, 40,… The first two examples here are
nuclei at those possibly magic numbers: 40Ca and 80Zr. The third example deals
with the early shape changes in a fissioning 236U nucleus.

###  IV.1 40Ca

The spectroscopy of 40Ca has been known since the early 1960’s. Fig. 3 shows
the states in the spectrum that are relevant to the discussion.

Figure 3: The spectrum of 40Ca, showing the first three levels of the deformed
band and the gamma transitions that establish the deformation of the band ma71

The first excited state in the spectrum has angular moment zero, which is rare
among the 600 or so known even-even nuclei. The excited $0^{+}$ together with
the $2^{+}$ and $4^{+}$ above it form the lowest members of a rotational band.
The evidence that these states are part of a band comes from the energy
spacing and from the electromagnetic transition rates between states,
indicated by arrows in the Figure. The measured quadrupole transition strength
between two lowest states in the band is
$B(E2,2^{+}_{1}\\!\rightarrow\\!{0^{+}_{2}})=250\pm 35$ e2 fm4 ma71 . On the
scale of a single-particle quadrupole moment this is very large and would
require an axis ratio of $a_{z}/a_{\perp}\approx 1.55$ to model the band as an
ellipsoidal rotor. The corresponding intrinsic mass quadrupole moment in the
band is

$\langle\,\hat{Q}_{20}\,\rangle=117~{}~{}{\rm fm}^{2}.$ (6)

Now let us compare the GCM and the $K^{\pi}$-partition approaches to determine
the structure of the band. We start with the GCM, applying a field
$\lambda\hat{Q}_{20}$ to the spherical ground state configuration. The
minimization is performed small increments of $\lambda$ up to the point where
the configuration has the deformation (6) extracted from experiment. The
results are shown in Fig. 4 as the open circles.

Figure 4: Energies of 40Ca configurations calculated by the GCM (open circles)
and by $K^{\pi}$-constrained minimization (black circle).

The energy increases monotonically as the deformation increases. At the
deformation corresponding to the one extracted from experiment, the excitation
energy is 13 MeV. This is way off from the observed band head at 3.35 MeV.
There will be a small gain of energy when one takes into account that the
calculated energy is that of the band as a whole, but the energy gain is
probably too small to approach the experimental value. Anyway, a big surprise
comes when we look at the shape of the nucleus, shown as in the middle panel
of Fig. 5. One sees that the GCM-generated configuration no longer has good
parity; there is a strong octupole component as well as the quadrupole
deformation. This would imply that the band should have odd-parity members
interleaved between the even-parity members. Since that is not the case, so we
can concluded that GCM carried out this way has failed.

Figure 5: Density distribution of the 40Ca configurations. Left: ground state;
center: GCM configuration at $q=108$ fm2; right: 4p-4h $K^{\pi}$-constrained
configuration.

Now let’s try the $K^{\pi}$-constrained approach using the $K^{\pi}$
partitions from Ref. ge67 . The ground state configuration has the fillings of
the spherical shell model. The deformed configuration was built by taking the
lowest 4-particle 4-hole excited state in a deformed harmonic oscillator
potential. This is the Nilsson model; its diagram of orbital energies is shown
in Fig. 6. We carry out the DFT minimization again, but now use the $K^{\pi}$
quantum numbers of the occupied orbitals to constrain the minimization. The
results for the spherical and deformed configurations are shown in Fig. 4 as
the black square and black circle, respectively. The predicted quadrupole
moment of the deformed configuration, 105 fm2, agrees well with (6). The
energy is still too high, but it is lower than the GCM energy and so is a
better candidate for understanding the band structure. Of course, one can
obtain the configuration by GCM minimization, but doing this would require a
different starting point or additional shape-dependent constraints. One final
point: as may be seen in the right-hand panel of Fig. 5, the configuration
found by starting from the 4p-4h hole configuration preserves the even parity
of the band.

Figure 6: Single-particle energies of orbitals around the Fermi level of 40Ca
as a function of deformation. The deformed band is attributed to 4-particle
4-hole excitation shown with solid and open circles (after Fig. 1 of Ref. ge67
.

Ignoring the energy problem, one can try to calculate the mixing of deformed
and spherical states, as was done by the early researchers. Unfortunately, as
I mentioned earlier, there is no consistent way to extract configuration-
interaction elements from a DFT. But DFT should still be useful to construct
the configurations. The Hamiltonian matrix elements might be evaluated using
these configurations as in the condensed-matter hybrid procedure.

###  IV.2 80Zr region

At the next harmonic oscillator shell closure ($N=20$ at 80Zr), the
competition between spherical and deformed configurations plays out
differently. The coexistence question here was addressed in the DFT by Zheng
and Zamick zh91 . They tried several energy functionals in the Skyrme family
and found that a deformed configuration came out lower than the spherical.
While in 40Ca there were 4 particle jumps needed to connect the
configurations, 12 jumps were needed in 80Zr. The Gogny D1S functional yields
rather similar results. The corresponding potential energy surface is shown in
Fig. 7. The filled square and circle show the minima for the spherical and
deformed configurations, obtained by constraining the $K^{\pi}$ partitions
according to the spherical shell model and the 12p-12h Nilsson configuration.
The latter has a very large deformation, $Q_{20}\approx 400$ fm2, at an energy
$~{}3.5$ MeV above the global minimum. The deformation is very robust with
respect to the choice of energy functional. In terms of the dimensionless
deformation parameter $\beta$, the Skyrme functionals and the Gogny D1S all
give $\beta=0.44\pm 0.01$. The authors of Ref. zh91  also note that a simple
formula derived earlier by myself is quite accurate.

The energetics are more complicated. First of all, the shell model
configuration is not the lowest energy minimum at $q=0$. The actual minimum
has broken parity with a large octupole moment and a very small quadrupole
moment. The entire potential energy surface up to high deformation can be
generated by GCM procedure, provided one starts with a configuration that
already has an octupole deformation. It is shown as the solid line in the
Figure.

Figure 7: The 80Zr potential energy surface. The solid line is generated by
the GCM procedure, starting from a octupole-deformed ground state minimum. The
$K^{\pi}$-constrained minimization starting from the partition of the
spherical shell model configuration is shown at the black square. Starting
with the 12p-12h partitions gives the filled circle, coinciding with the
second minimum of the GCM potential energy curve.

While the 12p-12h is not the lowest configuration in the Gogny DFT,
correlation effects can change the ordering. Recently de10  a systematic study
of low-energy spectroscopy was carried out including all five quadrupole shape
degrees of freedom in the GCM framework. In that way, rotational energies and
some shape mixing effects are taken into account. As an example of the
predictive power of the Gogny functional and the method, it was found that
calculated quadrupole transition moments of deformed nuclei agree with
experiment to $~{}10\%$ accuracy. For the 80Zr nucleus, the authors found that
the rotational energy was enough to bring the band head of the highly deformed
configuration down to the ground state. Fig. 8 compares their calculated
$2^{+}$ excitation energies across the chain of the heavy $N=Z$ even-even
nuclei. One sees that $2^{+}$ excitation energy in 80Zr agrees very well with
experiment. It is also the most highly deformed in the chain, judging by the
excitation energy of the lowest $2^{+}$ state. The heaviest measured nucleus
in the Figure, 96Pd, is a real prediction, as the experimental measurement
ce11  was reported the following year. So, despite the mixed performance of
DFT in 40Ca, we find that the energy functionals becomes quite successful in a
heavier region of nuclei.

Figure 8: Systematics of $2^{+}$ excitations in the heaviest known $N=Z$
nuclei. Solid black lines: older experiments; red dashed line: Gogny/GCM
theory de10 . Theory and experiment are visually indistinguishable for the
nuclei 76Sr and 80Zr. The blue dotted line shows the new measurement at 96Pd
ce11 .

##  V Fission dynamics

My ultimate goal is to gain a better understanding of the dynamics of
fissioning nuclei. As mentioned at the beginning, the GCM approach works well
for describing spontaneous fission as tunneling through a barrier in the
potential energy surface. But when the excitation energy is above the barrier,
it is far from clear what theoretical approach is justified. In any case, the
spontaneous fission show that the part of the interaction responsible for
pairing is very important ro14 . This suggests that the time-dependent
Hartree-Fock-Bogoliubov approximation (TD-HFB) might be appropriate for above-
barrier dynamics. Recently, computer power has become available to test the
TD-HFB for fission using current nuclear energy functionals. Such a study was
carried out by Bulgac and collaborators bu16 . Their starting point was a very
deformed configuration past the second barrier and at a small excitation
energy over the potential energy surface. They found that fission would take
place, but the duration before the fragment formation could be very long.

A weak point of the TD-HFB approximation is that it takes into account only a
very restricted set of interaction matrix elements to propagate the system
from one configuration to another. Namely, configurations that can be
generated as two-quasiparticle excitations of the starting configuration are
included in some way, but all four-quasiparticle transitions are neglected. In
the example of configuration mixing in 40Ca, the HFB approximation would be
useless. There is no pairing condensate, and even if there were one, the
intermediate configurations are only partially represented by the allowed pair
jumps.

While it would still be an enormous challenge to make a realistic treatment of
fission in the discrete-configuration approach, we can at least see how the
landscape could be traversed. We concentrate on the two minima and the barrier
between them. Fig. 9 shows the potential energy surface of 236U again,
expanding the horizontal scale in the region of the first barrier.

Figure 9: 236U HFB potential energy surface between the first and second
minima and the energies of the partition-constrained HF wave functions. Solid
and dashed lines are the HF and HFB potential energy surfaces respective.
Diamonds show the energy and quadrupole moment of the partition-constrained
minima.

Figure 10: Path from the first to second minimum in the $(q_{20},q_{40})$
plane. The left-hand panel shows the HFB energies constrained by $q2$. The
minima and intermediate maximum are marked by filled circles. The right-hand
panel shows the same path in the HF approximation. The filled diamonds are the
partition-constrained minima. In principle, either method should produce the
same global minimum.

The lower curve was obtained by the GCM in the HFB approximation, which
includes pairing effects. The more jagged curve above that is the GCM in the
HF, ie. ignoring pairing effects. The HF configurations have definite
$K^{\pi}$ partitions that change abruptly along the curve. The black dots are
three configurations generated by the $K^{\pi}$-constrained minimization, with
the partitions taken from the GCM results. It is interesting to compare the
$K^{\pi}$ partitions found with the Gogny D1S energy function with other
simpler ways for building a many-particle wave function. The simplest is the
Nilsson model mentioned earlier. One finds that it gives exactly the same
ground state $K^{\pi}$ partition as the DFT. Another model simpler than the
DFT is the Finite-Range Liquid Drop Model mo09 . Here the orbital energies are
computed in a deformed potential well, but the shape of the well is determined
by an energy functional based in part on the liquid drop model. It also gives
the same $K^{\pi}$ partition as the other models. This suggests that the
$K^{\pi}$ filling of the ground state is a quite robust property of the mean-
field wave function. And it is certainly easier to specify, compared with the
many shape parameters needed to describe unambiguously configurations obtained
by the GCM. However, at the A and the II points along the fission path the
partition obtained with the Gogny functional differs by one pair jump from the
FRLD partition. It is likely that they are nearly degenerate, and would be
mixed by the pairing interaction.

For dynamics, an important consideration is the number of steps it takes to
get from one configuration to another via the two-particle interaction in the
Hamiltonian. This suggests a distance measure between configurations as the
number of pairing jumps needed to get from one to the other. Fig. 10 shows a
plot of the path from configuration I to II via the barrier top at A, showing
both the quadrupole and hexadecapole coordinates of the intermediate
configurations. The left-hand plot is the GCM in the HFB approximation, which
is seen to be smooth with very little structure. The right-hand plot shows the
same path in the HF approximation. It is broken into a sequence of segments,
each segment having a specific $K^{\pi}$ partition. Analyzing these segments,
one can determine that it takes 6 pair jumps to get from I to A and 8 pair
jumps to get from A to II. But it would wrong to concluded that it takes 14
pair jumps to get from I and II. In fact, examination of end-points reveals
that the configuration II can be reached from I by changing the orbits of only
6 pairs. That of course requires surmounting a higher potential barrier. Which
path is more important in the dynamics is not obvious; one needs to know
interaction matrix elements between configurations as well as their energies
to begin to address this question.

##  VI Conclusion

The configuration-interaction method has been very successful in light nuclei,
and it should be possible, at least in a statistical way, to extend it to
heavy nuclei. I have concentrated exclusively on the basis of wave functions,
advocating the use of DFT constrained by $K\pi$ partitioning. This still
leaves the interaction matrix elements to be determined. The same problem
exists in condensed-matter theory, and there a hybrid approach has been quite
successful sa16 .

It would take a large effort to carry out this program in nuclear physics.
However, there are good reasons, rooted in experimental findings, to undertake
that effort. Empirically, there is strong evidence that the fissioning nucleus
is close to a statistical equilibrium near the scission point ra11 . The big
open theoretical question is whether we can explain that in terms of
Hamiltonian dynamics with realistic interactions.

Fluctuation phenomena also remain unexplained. Here are two examples. In 1978
Keyworth and collaborators measured the fission cross sections for neutron-
induced fission of 235U separating out the individual angular momentum
channels. Besides the fluctuations associated with the compound nucleus states
near the waypoints I and II, they saw fluctuations on much larger energy scale
mo78 . The data for one of the angular channels is shown in Fig. 11.

Figure 11: Average fission width of the $J=4$ compound-nucleus states in 236U
as a function of excitation energy. The dashed line is the predicted width
from the Bohr-Wheeler theory with one open channel. The solid curve is a fit
to the experimental data in Ref. mo78 .

It may be that the discrete states near the barrier tops are responsible. A
discrete basis would be very helpful here.

Another old experiment exhibiting unexplained fluctuations is the measurement
of angular distributions of fission fragments by Huizenga, Loveland, and
collaborators be69 . The distributions very close to threshold could be fitted
with the usual theory, but on examination of a more extended range they found:
“Further attempts to fit the data… were unsuccessful.”

In summary, I think there is a good case for trying a discrete configuration
approach to shape dynamics in heavy nuclei as an alternative to the GCM.

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