# Global systematics of octupole excitations in even-even nuclei

L.M. Robledo<EMAIL_ADDRESS>http://gamma.ft.uam.es/robledo Departamento
de Física Teórica, Módulo 15, Universidad Autónoma de Madrid, E-28049 Madrid,
Spain  G.F. Bertsch  Institute for Nuclear Theory and Dept. of Physics, Box
351560, University of Washington, Seattle, Washington 98915, USA

###### Abstract

We present a computational methodology for a theory of the lowest octupole
excitations applicable to all even-even nuclei beyond the lightest. The theory
is the well-known generator-coordinate extension (GCM) of the Hartree-Fock-
Bogoliubov self-consistent mean field theory (HFB). We use the discrete-basis
Hill-Wheeler method (HW) to compute the wave functions with an interaction
from the Gogny family of Hamiltonians. Comparing to the compiled experimental
data on octupole excitations, we find that the performance of the theory
depends on the deformation characteristics of the nucleus. For nondeformed
nuclei, the theory reproduces the energies to about $\pm 20$ % apart from an
overall scale factor of $\approx 1.6$. The performance is somewhat poorer for
(quadrupole) deformed nuclei, and for both together the dispersion of the
scaled energies about the experimental values is about $\pm 25$ %. This
compares favorably with the performance of similar theories of the quadrupole
excitations. Nuclei having static octupole deformations in HFB form a special
category. These nuclei have the smallest measured octupole excitation energies
as well as the smallest predicted energies. However, in these cases the
energies are seriously underpredicted by the theory. We find that a simple
two-configuration approximation, the Minimization After Projection method,
(MAP) is almost as accurate as the full HW treatment, provided that the
octupole-deformed nuclei are omitted from the comparison. This article is
accompanied by a tabulation of the predicted octupole excitations for 818
nuclei extending from dripline to dripline, computed with several variants of
the Gogny interaction.

##  I Introduction

The octupole excitations of nuclei have been well-studied theoretically on a
case-by-case basis but there has never been a global study for a fixed
Hamiltonian and well-defined computational methodology. Such studies are
important for several reasons. Seeing the systematic trends, one can better
assess the deficiencies in the Hamiltonian or the underlying theory, which
could hopefully lead to improvements on both sides. Also, the predictive power
of the theory with the given Hamiltonians can be measured by the comparison to
a large body of nuclear data. In this work we carry out a study of this kind
using the Hartree-Fock-Bogoliubov (HFB) approximation extended by Generator
Coordinate Method (GCM). Earlier studies of the octupole degree of freedom
using this and similar methods are in Refs. ma83 ; he01 ; eg91 ; sk93b ; he94
; ga98 . A competing methodology is based on the quasiparticle random phase
approximation; recent application to octupole modes may be found in Refs. se02
; co03 ; an09 . For a general review of the theory of octupole deformations
and collective excitations, see Ref. bu96 .

A global theory not only needs to treat the consequences of static octupole
deformations in HFB ground states but also to treat the more ordinary
situation where the degree of freedom appears more as a collective vibration
of a symmetric HFB ground state. The latter is typically treated by RPA or
QRPAse02 ; co03 ; an09 , but the most of the studies consider a small body of
nuclei chosen by considerations emphasizing one characteristic or another, for
example semi-magic isotope chains. Our study is the first to encompass not
only magic and semimagic ordinary nuclei, but the quadrupole- and octupole-
deformed nuclei as well. This follows in spirit the studies of the nuclear
quadrupole degrees of freedom in Refs. sa07 ; de10 . We mention that our GCM
coordinate is a one-dimensional variable labeled by the mass octupole moment.
A two-dimensional treatment of the octupole deformations treating the
quadrupole deformation as a separate degree of freedom is important in theory
of fissiongo05 , and is likely to play a role in spectroscopy as wellme95 .

The HFB fields and quasiparticle wave functions are assumed to have the
following symmetries: time reversal, axial symmetry, and the $z$-component of
isospin. We can only consider even-even nuclei under these restrictions. The
restriction to axial symmetry is harmless in spherical nuclei, but for
deformed nuclei it causes two problems. The first is that theory only treats
the $K=0$ excitations of deformed nuclei. As we will see, some of the
identified octupole excitations very likely have nonzero $K$ quantum number.
The second difficulty that arises with deformed nuclei is that angular
momentum is not a good quantum number of the HFB/GCM wave function. On a
practical level, we shall compare the calculated excitation energies with the
spectroscopic $0^{+}\rightarrow 3^{-}$ transitions, assuming that the
rotational inertias can be neglected.

The calculations are carried Gogny’s form of the interaction in the
Hamiltonian. In particular, the D1S Gogny interaction has been well-tested in
many HFB calculations and also gives good results in (Q)RPA pe08  and GCM
extensions of HFB ro02b . Specific results for that interaction will be
presented in the text, and results for other Gogny interactions are provided
in the supplementary material accompanying this article.

##  II Implementing the GCM

###  II.1 GCM

In the GCM, an external field is added to the Hamiltonian to generate a set of
mean-field configurations to be taken as a basis for the HW minimization. We
take for the generating field the mass octupole operator,
$\hat{Q}_{3}=\sqrt{\frac{4\pi}{7}}r^{3}Y^{3}_{0}(\hat{r})=z^{3}-\frac{3}{2}z(x^{2}+y^{2}).$.
We label the solutions of the HFB equations in the presence of the field
$\lambda\hat{Q}_{3}$ by the expectation value of $\hat{Q}_{3}$,

$\langle q|\hat{Q}_{3}|q\rangle=q.$ (1)

For convenience, we will use the nominal value of $\beta_{3}$ instead of $q$
in discussing the wave functions. These are related by the formula
$q=\sqrt{9/28\pi}(1.2)^{3}A^{2}\beta_{3}$. We also fix the (average) center-
of-mass of the nucleus at the origin with the constraint
$\langle|\hat{z}|\rangle=0$ to avoid a spurious octupole moment associated
with the position of the nucleus.

The GCM wave function is constructed by combining the configurations
$|q\rangle$ to build a correlated wave function $|\sigma\rangle$. This is
expressed formally in the GCM as an integral over configurations

$|\sigma\rangle=\int dq\,f_{\sigma}(q)|q\rangle.$ (2)

The function $f$ in Eq. (2) is to be determined by applying the variational
principle to the expression

$E=\frac{\langle\sigma|H|\sigma\rangle}{\langle\sigma|\sigma\rangle}.$ (3)

While Eq. (3) and (4) define the GCM formally, further approximations are
required to arrive at a well-defined computational methodology. One way common
in the literature is to keep the formal integral Eq. (3) and use the Gaussian
overlap approximation to calculate the matrix elements in Eq. (4), as was done
in Ref. de10  to map the quadrupole deformation onto a collective Hamiltonian,
and in Ref. ro10b  for the octupole degree of freedom. A quite different way
is the discrete basis Hill-Wheeler method, first carried out for the octupole
excitations in Ref. ma83 . This method, which we will follow here,
approximates the integral using a discrete set of configurations. The
minimization is equivalent to solving the matrix eigenvalue equation

$\sum_{j}\langle q_{i}|H|q_{j}\rangle c_{j}=E\langle q_{i}|q_{j}\rangle
c_{j}.$ (4)

The states will have good parity if the basis is reflection symmetric, i.e. if
$|-q_{i}\rangle$ is in the basis if it contains $|q_{i}\rangle$.

For either method one needs the overlap integrals between configurations
$\langle q|q^{\prime}\rangle$, the matrix elements of Hamiltonian $\langle
q|H|q^{\prime}\rangle$ and the matrix elements of one-body operators such as
$\langle q|\hat{Q}_{3}|q^{\prime}\rangle$. The basic overlap integral is
computed with the Onishi formulaonishi . The matrix elements of one-body
operators and products of one-body operators are then evaluated using the
generalized Wick’s theorembalian . Unfortunately, the Gogny interaction cannot
be expressed in this way due to its $\rho^{1/3}(\vec{r})$ density dependence.
This gives rise to well-known ambiguities in treating the interaction as a
Hamiltonian in a multiconfiguration space. Of the various prescriptions
available, we use the "mixed density" method. Here the $\rho$ in the
$\rho^{1/3}$ factor is replaced by $\rho_{BB}(\vec{r})$ given by

$\rho_{BB}(\vec{r})=\frac{\langle
q|\hat{\rho}(\vec{r})|q^{\prime}\rangle}{\langle q|q^{\prime}\rangle}$ (5)

and the resulting $\vec{r}$-dependent interaction is evaluated in the usual
way. The mixed-density prescription was introduced in Ref bo90  and first
applied to parity-projected HFB as "Prescription 2" in Ref eg91 . It is
consistent with the mean field limit and is a scalar under symmetry
transformations ro07 . Another prescription which seems plausible at first
sight is to use the projected density for $\rho^{1/3}$. However, this gives
unphysical results for octupole deformationsro10a .

While the configurations $|q\rangle$ constructed with the octupole constraint
have mixed parity, the HW solutions restore the parity quantum number,
provided that we use a basis that contains the both signs of $q$ in the
included configurations. In effect, the parity projection needed to calculate
spectroscopic properties can be obtained from the HW minimization without any
extra effort. However, as a practical matter, it is easier to define the
parity operator in the harmonic oscillator basis and use it to construct
$|-q\rangle$ from $|q\rangle$ thus avoiding a separate HFB minimization for
the $-q$ configuration.

The HW states of interest are the lowest lying even- and odd-parity states of
spectrum, which we call $|e\rangle$ and $|o\rangle$. Taking them to be
normalized, the energies of ground state $E_{e}$, the odd parity state
$E_{o}$, and the excitation energy difference $E_{3}$ are given by

$E_{e}=\langle e|H|e\rangle;\,\,\,\,E_{o}=\langle
o|H|o\rangle;\,\,\,\,E_{3}=E_{o}-E_{e}$ (6)

We follow the usual procedure to solve the matrix equation Eq. (4), using if
necessary the singular value decomposition to avoid difficulties with an
overcomplete space.

One first diagonalizes the overlap matrix and transforms all of the matrices
to the diagonalized basis. Often there will be vectors which very small norms
and the basis is truncated to exclude vectors whose norms are less than a
certain value $n_{min}$. The Hamiltonian is diagonalized in this basis, called
the collective space, to give the HW energies. The eigenvectors are used to
calculate matrix elements of other operators between energy eigenstates.

The main problem with the discrete Hill-Wheeler method is that the calculated
values cannot be considered reliable unless both the range of deformations has
been fully covered and that the singular value decomposition has been set to a
robust truncation. For most of the nuclei, we shall take as a basis the set of
$\beta_{3}$ from -0.5 to +0.5 in steps of 0.025. For lighter nuclei, the range
is extended from -1.2 to +1.2. The calculations are carried out as a function
of the dimension $N_{basis}$ of the singular-value truncation. There is
generally a broad range of $N_{basis}$ for which the excitation energies have
converged to some value; we take the value on this plateau as the HW result.
An example is shown in detail in the next section.

The computation of the HW starting matrices is not trivial, requiring $N^{2}$
Hamiltonian matrix elements for a basis size $N$. While this is not an
important issue here, if one were to attempt GCM calculations in more than one
variable, the number of states $N_{basis}$ could be large. It is therefore of
interest to investigate the accuracy of simpler approximations using fewer
configurations. One of the simplest treatments is to take two configurations,
$|q_{e}\rangle$ and $|q_{o}\rangle$, for the even-parity and odd-parity state,
respectively. The values of $q$ are chosen to minimize the projected energies
of the configuration. We follow Ref. sa07  calling this the Minimization After
Projection (MAP) procedure. The deformations and energies at the minima
denoted $\beta_{3p},E_{p}$ and $\beta_{3m},E_{m}$ for the two projected
states. The MAP excitation energy is defined as

$E_{3}^{MAP}=E_{m}-E_{p}$ (7)

One last general point of the computational procedure needs to be mentioned.
While the individual HFB configurations are constructed with the desired
proton and neutron particle numbers, the mixed configurations in the HW wave
function may have slightly different expectation values of $N$ and $Z$. The
energy depends strongly on $\langle N\rangle$ and $\langle Z\rangle$, and
changes must be corrected for. We do this by adding to the HW Hamiltonian the
term $\lambda_{p}(\hat{Z}-Z)+\lambda_{n}(\hat{N}-N)$, where $\lambda_{p,n}$
are the nucleon chemical potentials at $\beta_{3p}$ bo90 .

###  II.2 HFB

The constrained HFB calculations were carried out using the code HFBaxial
written by one of us (L.M.R.). It uses a harmonic oscillator basis specified
by the length parameters $b_{z}$ and $b_{t}$ of the oscillator potential and
the number of shells $N_{osc}$ in the basis. For the calculations reported
here we have taken a fixed spherical basis for all nuclei with oscillator
length parameters $b_{z}=b_{t}=2.1$ fm. The number of oscillator shells
included in the basis is 10,12, and 14 for nuclei in the ranges
$Z=[8,50],[52,82]$, and $[84,100]$ respectively. This is more than enough to
provide converged results for energy differences. We report on the results for
the D1S Gogny interaction in sections below. More detailed results for the D1S
as well as for other interactions of the Gogny form are given in the
supplemental material epaps .

##  III Examples

In this section we will go through the details for four examples illustrating
the application to a spherical nucleus, 208Pb, a well-deformed nucleus, 158Gd,
the nucleus 226Ra whose HFB ground state has a static octupole deformation,
and a light nucleus having a very large transitional octupole moment, 20Ne. A
summary of the results for these nuclei is given in Table 2 at the end of this
Section.

###  III.1 208Pb

The nucleus 208Pb is a paradigm for a doubly magic nucleus. It is one of the
very few nuclei whose first excited state has $J^{\pi}=3^{-}$ quantum numbers.
The excitation energy is 2.62 MeV and the transition rate is strongly
collective with strength of $B(E3,\uparrow)=0.611$ e2b3 or 34 Weisskopf
unitswu . For the theory, we first shown HFB and projected energies of the GCM
configurations in Fig. 1.

Figure 1:  Energy of 208Pb as a function of octupole deformation $\beta_{3}$.
Open circles: HFB energy of constrained configurations ; Solid squares: energy
$E_{e}$ of the even-parity projected wave function; Solid circles: the odd-
parity projected energy $E_{o}$. See the Appendix for explanation of the
fitted lines.

The minimum energy projected configurations, ie. the MAP states, are at
$\beta_{3p}\approx 0.0375$ and $\beta_{3m}\approx 0.075$. One sees that the
energy of the ground state is lower by projecting from a nonzero $\beta_{3}$;
the associated correlation energy has the order of magnitude of one MeV. The
MAP approximation to the excitation energy $E_{3}$ is given by the difference
of the minima of the plus- and minus-projected energy curves, which is about
4.2 MeV.

To see how the calculated $E_{3}$ depends on the basis, we show it in Fig. 2
as a function of $N_{basis}$. The difference of MAP energies is the open
square, and solid circles show the results with various truncations. The full
basis set is comprised of the 41 configurations between $\beta_{3}=-0.5$ to
$\beta_{3}=+0.5$ in steps of 0.025. The truncation is carried out by the
singular-value decomposition.

Figure 2:  Excitation energy $E_{3}$ in 208Pb as a function of the
configuration space choice. Solid circles: HW using the singular value
decomposition to keep $N_{basis}$ states; solid square: HW with the two MAP
states; open square: Energy difference of the two MAP states.

One sees that the energy has converged at about $N_{basis}\approx 14$ and the
numerics remain stable up to much larger values. The converged energy, 4.0
MeV, is fairly close to the difference of MAP energies. In fact, one can do
even better in the 4-dimensional space allowing the MAP configurations to mix.
This is shown as the solid square in the figure. We note that our excitation
energy of 4.0 MeV is close to the value found in Ref. he01  using the GCM/HW
method but with the Skyrme SLy4 interaction.

We see here that the MAP could be a very useful simplification, but its
validity depends on the circumstances. It is instructive to examine the GCM/HW
wave function and compare it with MAP. These are shown in Fig. 3, for both the
ground state and the odd-parity excited state. The wave function amplitudes
are formally defined by the integral

$g_{\sigma}(\beta_{3})=\int d\beta_{3}^{\prime}{\cal
N}^{1/2}(\beta_{3},\beta_{3}^{\prime})f_{\sigma}(\beta_{3}^{\prime})$ (8)

where $f$ is normalized $1=\int d\beta_{3}\,d\beta_{3}^{\prime}{\cal
N}(\beta_{3},\beta_{3}^{\prime})f_{\sigma}(\beta_{3}^{\prime})f_{\sigma}(\beta_{3})$.
The above relation establishes the connection between the standard GCM
amplitudes $f$ with the amplitudes $g$ entering the expansion of the GCM wave
functions in terms of orthogonal states $|q\rangle_{\textrm{orth}}=\int
dq^{\prime}{\cal N}^{-1/2}(q,q^{\prime})|q^{\prime}\rangle$. The square root
of the norm overlap has to be understood in terms of the relation $\int
dq^{\prime\prime}{\cal N}^{1/2}(q,q^{\prime\prime}){\cal
N}^{1/2}(q^{\prime\prime},q^{\prime})={\cal N}(q,q^{\prime})$. The ground and
excited state wave functions can be distinguished by the amplitude at
$\beta_{3}=0$, which is finite for the even-parity ground state and zero for
the odd-parity excited state. The HW wave function and the MAP approximation
are shown as solid and dashed lines, respectively. It is clear that the MAP
configuration is a good approximation to the full wave function of both the
ground and excited states, for this particular nucleus.

Figure 3:  Wave function amplitudes. See text for explanation.

More insight into the collective physics of the octupole degree of freedom can
be obtained comparing with simple models of the excitation (See Appendix). If
the configuration energies and interactions can be treated as quadratic
functions of the deformation coordinate, and the matrix elements between
different configurations can be treated by the GOA, the GCM/HW reduces to the
RPA and is exact. The line through the HFB energy curve in Fig. 1 is a
quadratic fit. It appears to be well satisfied. Also, the energy of the even-
parity projected configuration follows well the predicted dependence according
to the GOA, Eq. (16). This shown as the line through the even-parity projected
energies in the figure. Thus two of the conditions are met to reduce the
GCM/HW theory to an RPA of a single collective state.

###  III.2 158Gd

Our example of a strongly deformed nucleus is 158Gd. It has a $3^{-}$
excitation at 1.04 MeV with a transition strength $B(E3\uparrow)=0.12$ e2b3.
The energies from the GCM calculation are shown in Fig. 4. Overall, the energy
curves look quite similar to those for 208Pb. The HFB curve is also well fit
by a quadratic dependence on $\beta_{3}$ but the curvature here is much
shallower. The projected energy function $E_{e}(\beta_{3})$ also has a similar
shape to the curve for 208Pb, and can be fitted by the same functional form,
Eq. (14). The ratio of MAP minimum points is found to be
$\beta_{3p}/\beta_{3m}\approx 2$, similar to the situation for 208Pb. The
excitation energy $E_{3}$ comes out to about 1.7 MeV, much smaller than the
208Pb value. This is to be expected in view of the softer HFB curve. The
correlation energy of the ground state, $E_{0}-E_{e}$, is similar to the 208Pb
value, about one MeV.

Figure 4:  Energy of 158Gd as a function of octupole deformation $\beta_{3}$.
Open circles: HFB energy of constrained configurations ; Solid squares: energy
$E_{e}$ of the even-parity projected wave function; Solid circles: the odd-
parity projected energy $E_{o}$. The line along the HFB values is the function
$E_{q}=E_{0}+K_{1}\beta_{3}^{2}$ with $K_{1}=48.8$ MeV fitted to the values
$\beta_{3}\leq 0.05$. The line along the $E_{e}$ values is the fit motivated
by the Gaussian overlap approximation,
$E_{e}=E_{q}-K_{2}\beta^{2}/(1.0+\exp(\alpha\beta^{2}))$, with $K_{2}$ and
$\alpha$ fitted.

Experimentally, the situation is complicated by the deformation and the
splitting of the octupole strength into different $K$-bands. There are three
negative parity bands known experimentally at low energy. There is a $K=1^{-}$
with an $1^{-}$ state at 977 keV, a $K=0^{-}$ with the $1^{-}$ state at 1263
keV and finally a $K=2^{-}$ with a $2^{-}$ state at 1793 keV. Our excitation
energy of 1.7 MeV should be compared with the 1263 keV of the $1^{-}$ state of
the $K=0^{-}$ band. The theoretical value is stretched by a factor 1.4 with
respect to the experimental value (see discussion below). Note that the
measured octupole transition at 1.04 MeV is not relevant for the comparison
because it corresponds to a different $K$ value.

###  III.3 226Ra

226Ra has the lowest $3^{-}$ excitation energy of any in the compilation ki02
, $E_{3}=320$ keV. It also has the highest transition strength in the
compilation, $W(E3)=54$ Weisskopf unitswu . On the theory side, the nucleus is
predicted to deformed both in the quadrupole ($\beta_{2}\approx 0.3$) and the
octupole degrees of freedom. The HFB/GCM energy curve, shown in Fig. 5, has a
minimum at $\beta_{3}\approx 0.13$.

This nucleus is very interesting for our survey, not only because of the
static octupole deformation, but because the theory is seen to fail badly if
the large amplitude fluctuations are not properly accounted for. The predicted
excitation energies for different treatments of the GCM configurations are
shown in Table 1. The most naive theory (top line) would ignore the GCM
construction and simply take the HFB minimum and project from that. The
overlap $\langle-q|q\rangle$ at the HFB minimum is essentially zero and the
$E_{3}$ comes out less than 1 keV. In the next approximation we consider
(second line), we take the single configuration that gives the MAP ground
state. Here the deformation is much closer to zero. However, the $E_{3}$
calculated as the difference between the even and odd projected states is now
far too large, 1.7 MeV. Of course in the full MAP approximation we should take
the configurations at different $\beta_{3}$ for odd and even projections. This
is done in line 3 of the Table, and now the $E_{3}$ has the correct order of
magnitude. Adding more configurations, the valued do not change much on an
absolute MeV scale, but on a relative scale there is a considerable change.
The most complete HW treatment, on the bottom line, underpredicts the energy
by a factor of $\approx 2$.

Figure 5:  Energy of 226Ra as a function of octupole deformation $\beta_{3}$
as in Figs. 1,4g. A very similar plot is shown in Fig. 3 of Ref. eg91 .

We also show the HW and MAP wave functions in Fig. 3. It is clear that the
full wave functions are far from harmonic and that the MAP approximation fails
badly.

$N_{q}$ | $\beta_{3}$ | $E_{e}$ | $E_{o}$ | $E_{3}$
---|---|---|---|---
1 | 0.15 | -1722.63 | -1722.63 | 0.00
1 | 0.05 | -1722.71 MeV | -1721.01 | 1.7 MeV
2 | 0.05,0.15 | -1723.43 |  | 0.37
3 | 0.05,0.1,0.15 | -1723.45 |  | 0.31
4 | 0.025,0.075,0.125,0.175 | -1723.53 |  | 0.22
12 | [-0.5,0.5] |  |  | 0.16
Table 1:  Calculated energies of 226Ra with various choices of the
configuration set.  Figure 6:  Energy of 20Ne as a function of octupole
deformation $\beta_{3}$ as in Figs. 1,4,5.  Figure 7:  Nucleon density
distribution in 20Ne at $\beta_{3p}$ (left) and $\beta_{3m}$ (right).

###  III.4 20Ne

20Ne illustrates some differences that one sees in treating light nuclei by
the GCM/HW, first studied by this method in Ref. ma83 . Due to the incipient
alpha clustering, the equilibrium octupole deformation of the projected
configurations can be very large. The HFB and projected energies are shown in
Fig. 6. Note that the HFB energy deviates from a quadratic dependence on the
deformation, and looks almost linear at large $\beta_{3}$. Fig. 7 shows the
density distribution at the two projected minima. One sees a compact localized
density, suggestive of an alpha particle, outside a nearly spherical core.
Since the alpha emission threshold is rather low in this nucleus, one should
expect a softness in with respect to the generator coordinate corresponding to
alpha cluster separation. In a multipole representation, this requires
changing both the quadrupole and the octupole deformation. This is in fact
what occurs in our GCM wave functions. Fig. 8 shows their deformations in the
two multipolarities. The coupling of the multipolarities can cause problems,
however. We will come back to this in the Appendix, referring to the coupling
in 16O, also shown on the figure.

Figure 8:  Deformation of the octupole-constrained HFB configurations for 16O and 20Ne.  Nucleus |  $E_{3}$ (MeV) | $W(E3)$
---|---|---
| Exp. | Present | Other | Theory | Eq. | Exp.
20Ne | 5.6 | 6.7 | 5.2a | 12. | (11) | 13.
208Pb | 2.6 | 4.0 | 4.0b | 53. | (12) | 34.
158Gd | 1.04 | 1.93 |  | 11.6 | (11) | 12.
226Ra | 0.32 | 0.16 |  | 43. | (11) | 54.
Table 2:  Summary of results for the four examples discussed in the text.
References for column 4, other theory: a) ma83 ; b) he01 .

##  IV Systematics

We have applied the HFB/GCM/HW theory across the chart of nuclides including
818 nuclei between $8\leq Z\leq 110$. About 6% of them are octupole deformed
in the HFB ground state. The nuclei are shown in Fig. 9. Favorable conditions
for static octupole deformation occur when a high-$j$ intruder orbital is
close to an opposite-parity orbital with three units less of orbital angular
momentum near the Fermi energybu96 , which happens for $Z$ and $N$ values
around 36, 56, 88, and 134. The regions around Ba and Ra are well-known in
earlier studies. We also find static deformations near 80Zr and near $Z\approx
N\approx 56$ (for this region, see also Ref. he94 . There are also
calculations in the literature reporting static octupole deformations in other
regions as wellmo08 ; zh10 . In any case, the HFB deformation is not an
observable. Physically, one can only measure excitation energies and
transitions strength. These are compared with experiment in the two
subsections following.

Figure 9:  Chart of the nuclides shown those calculated in the present study.
Those in black have static octupole deformations in HFB. Except for the nuclei
near $N\sim Z\sim 40$, the nucleon numbers correspond well to the numbers 56,
88, and 136 listed in Ref. bu96  as especially favorable for octupole
deformation.

###  IV.1 Excitation energies

We now compare theory with the experimental data from the review by Kibédi and
Spear ki02 . The excitation energies of the 284 tabulated nuclei with $Z\geq
8$ are shown in Fig. 10, plotted as a function of $A$. The data show a strong
overall $A$-dependence as well as shell-related fluctuations. The line shows a
fit to the smooth trend in $A$ with the phenomenological parameterization
$E(A)=103/A^{0.85}$ MeV. The most pronounced fluctuation about the trend is
the rise and sudden drop near $A=208$; the drop is to low values is due to the
extreme softness in the octupole mode. The theoretical energies, shown as
triangles, replicate the overall trend with $A$ and the dramatic fluctuation
at $A\sim 208$. However, overall the theoretical energies are too high,
particularly in the light nuclei.

Figure 10:  Octupole excitation energies as a function of mass number $A$.
Circles: experiment; triangles: theory.

A more detailed comparison of theory and experiment may be seen on the scatter
plot Fig. 11. For excitation energies above 1 MeV, the theoretical values
track the experimental but scaled by a factor. Around 1 MeV and below the
theoretical values become closer to experiment. The lowest energy measured
excitations are in the Ra isotopes, where the theoretical HFB wave functions
have static octupole deformations. The theory reproduces the low energies to
several hundred keV on an absolute energy scale, but does not do well on the
logarithmic energy scale shown in the figure.

Figure 11:  Octupole excitation energies, comparing the theory with
experiment. Filled circles are excitations with measured $B(E3)$ strengths;
open circles are other identified octupole transitionski02 .

We also make some quantitative assessment of the performance of the theory,
which should be useful in the future for comparing with other theories. We use
the same performance measures as was used to assess theories of quadrupole
excitationssa07 ; de10 , namely to compare ratios of theoretical to
experimental quantities on a logarithmic scale. In terms of
$R_{E}=\log(E(th)/E(exp))$ we determine the average value

$\bar{R}_{E}=\langle R_{E}\rangle$ (9)

and the dispersion about the average,

$\sigma_{E}=\langle(R_{E}-\bar{R}_{E})^{2}\rangle^{1/2}.$ (10)

The results are shown in Table 3. The first line shows the comparison taking
the full HW treatment on the theoretical side and the full data set on the
experimental side. One sees that the predicted energy is systematically too
high, by a factor of $e^{0.44}\approx 1.6$. This is similar to the situation
with the quadrupole excitations. There the understanding is that the wave
function is missing components that would be included in collective theories
using Thouless-Valatin inertial parameters. There may be other reasons for the
systematic overprediction here that we will come back to in Sect. V. The
dispersion in the values is $\sigma_{E}\approx 0.4$, corresponding to errors
in the ratio of theory to experiment of $-30\%$ to $+50\%$. This is larger
than the global dispersion found for the GCM-based theories of quadrupole
excitations. However, we saw in Fig. 11 that there are differences in the
nuclear structure that are responsible for the variable performance of the
theory. Most importantly, the nuclei with calculated static octupole
deformations should be treated separately. Taking out these nuclei, the
dispersion decreases dramatically, as shown on the second line of the Table. A
further distinction can be made between well-deformed and other nuclei,
spherical and soft, respect to ordinary quadrupole deformations. A good
theoretical indicator for deformed nuclei is the ratio of $4^{+}$ to $2^{+}$
excitation energies, called $R_{42}$. The values are available for the Gogny
D1S interaction from the global study de10 , and we use them to set the
condition $R_{42}>2.9$ to define the set of well-deformed nuclei. The results
are shown in the third and fourth rows of the table. One sees that the
dispersion becomes even narrower for the nuclei in the nondeformed set. Thus,
we can claim that the HFB/GCM/HW methodology is quite successful for
nondeformed nuclei, when allowing for the overall scale factor. On the other
hand, the deformed set is significantly poorer, with the average predicted
energies higher and a larger dispersion. A possible cause of this poorer
performance could be the misidentification of transitions in deformed nuclei.
We have assumed here that all transitions are associated with the axially
symmetric octupole operator ($K=0$). As discussed in the next section, it is
clear that some of the measured energies are for transitions with $K\neq 0$
(see also the 158Gd example). Since all the $K$ values in spherical nuclei are
degenerate, this would explain the better overall agreement there.

|  | HW | MAP
---|---|---|---
Selection | Number | $\bar{R}_{e}$ | $\sigma_{e}$ | $\bar{R}_{e}$ | $\sigma_{e}$
all | 284 | 0.45 | 0.40 |  |
$\beta_{3}=0$ | 277 | 0.55 | 0.23 | 0.59 | 0.22
$\beta_{3}=0$, def. | 59 | 0.62 | 0.32 | 0.75 | 0.26
$\beta_{3}=0$, sph. | 196 | 0.52 | 0.19 | 0.53 | 0.17
Table 3:  Performance of the HW theory for excitation energies compared to the
experimental data tabulated in Ref. ki02 . The performance measures $r_{E}$
and $\sigma_{E}$ are given in Eq. (9) and (10) of the text. The performance of
MAP is shown as well on lines 2-4 for subsets of nuclei selected by
deformation criteria.

###  IV.2 Transition strengths

The octupole transition strength is computed from the proton octupole
transition matrix element $\langle o|\hat{Q}_{3}\frac{1+t_{z}}{2}|e\rangle$.
In a strongly deformed nucleus, the excitation is in a $K=0$ odd-parity band
and the spectroscopic matrix element from the $3^{-}$ state in the band is
given by

$B(E3,3^{-}\rightarrow 0^{+})=\frac{e^{2}}{4\pi}\langle
o|\hat{Q}_{3}\frac{1+t_{z}}{2}|e\rangle^{2}.$ (11)

This formula was used in Ref. eg91  to estimate the octupole transition
strengths in Ra isotopes and other possible octupole-deformed nuclei. On the
other hand, if the state $|e\rangle$ is spherical, then the excitation induced
by $Q_{3}$ gives a state $|o\rangle$ that has good angular momentum and the
transition strength can be calculated directly as

$B(E3,3^{-}\rightarrow 0^{+})=\frac{7e^{2}}{4\pi}\langle
o|\hat{Q}_{3}\frac{1+t_{z}}{2}|e\rangle^{2}.$ (12)

Notice that this is a factor of 7 larger than Eq. (11). The reason for the
difference is that Eq. (12) gives a total octupole transition strength, while
Eq.(11) only gives the transition strength for the $K=0$ components.

Besides these limiting cases, there are soft nuclei which should fall in
between. Thus, it is imperative to restore good angular for the theory to have
a global applicability. While angular momentum projection has been carried out
in the pasteg93 ; su94 ; eg96 , it is beyond the scope of this article.
Instead, we examine here the range of predicted values using a theoretical
marker of the deformation to distinguish nuclei falling in the different
categories. Fig. 12 shows the ratios of theoretical to experimental $B(E3)$
values, using the experimental data set from Ref. ki02  and Eq. (11) for the
theory. The data is plotted as a function of the quantity $R_{42}$, the ratio
of the lowest $4^{+}$ to $2^{+}$ excitation energies. Values around 2 or less
are characteristic of spherical nuclei, while strongly deformed nuclei have
$R_{42}\geq 3$. We take the values for $R_{42}$ from the spectroscopic
calculations of Ref. de10 , based on HFB/GCM with the same Gogny D1S
interaction used for the theory here. The plot show a lot of scatter, but one
can see two groups of nuclei, the lefthand representing deformed nuclei. There
is a trend visible in the $B(E3)$ ratios consistent with the above discussion.

Figure 12:  Ratio of theoretical octupole transition strength to experimental,
with the theoretical strength obtained using Eq. (11). The horizontal axis is
the ratio $R_{42}$ from the theory of Ref. de10 . Experimental $B(E3)$ values
are from Ref. ki02 .

To make the analysis more quantitative, we examine the logarithmic averages
$\bar{R}$ dividing the nuclei into two group according to $R_{42}$. The
results are shown in Table 4. Since we use Eq. (11) to determine $R$, we
should find $\bar{R}=0$ for the first row of the Table. In fact, the average
is about 40 % high. For the second row, if all the nuclei were spherical, the
strength should be a factor of 7 larger. This implies that the $\bar{R}$
calculated with the deformed formula should give a value $0.33-\log(7)=-1.6$.
The value found, -0.99, shows that there is an important effect of the
deformation but that it to simplistic to assume that these nuclei are all
spherical.

Selection | Number | $\bar{R}$ | $\sigma$
---|---|---|---
Deformed, $R_{42}>2.9$ | 41 | 0.34 | 0.5
Other, $R_{42}<2.9$ | 112 | -0.99 | 0.7
Table 4:  Ratio of theoretical to experimental $B(E3)$ strengths. The second
column is the number of nuclei in the data set.

We note that the enhancement of the $B(E3)$ for the less deformed nuclei is
evident in the projected calculations for 16O (eg93 ) and Pb isotopes near
$A=208$ (eg96 ). Also, in Ref. sk93a  the authors remark on a strong
disagreement between theory and experiment for 96Zr. This is the case if one
uses Eq. (11), but that nucleus is spherical according to the $R_{42}$
criterion and Eq. (12) gives a satisfactory agreement. agreement is satisfying

It is of interest to examine the nuclei that deviate most strongly from the
theory. In Fig. 12 there is a group of three outlier nuclei in the upper
right-hand corner. The nuclei are 170Er and its neighbors. In these cases, the
experimental transitions are likely to be to excited states with $K\neq 0$.
The lowest $1^{-}$ excitation in 170Er at 1.26 MeV has a $K=1^{-}$ character,
and the first $K=0^{-}$ is higher by 0.6 MeV. There are some studies in the
literature in which the $K$-dependence of the octupole excitation is
examinedsk93b ; ya01 ; zb06 . In Ref. ya01 ; zb06  the $K=0^{-}$ bands were
found to be higher in energy than other $K$ values.

The other glaring anomaly is the nucleus 64Zn at $R_{42}\approx 2.4$, which
has a grossly underpredicted $B(E_{3})$. It turns out that the quadrupole
deformation of this nucleus changes sign as $\beta_{3}$ is increased. The
ground state at $\beta_{3}=0$ is oblate, but it switches to another minimum
with a prolate shape at moderate values of $\beta_{3}$. The very small
predicted $B(E3)$ is due to the small overlap between the oblate and prolate
configurations. Clearly, the GCM must include explicitly both quadrupole and
octupole degrees of freedom to properly treat this nucleus. A few other nuclei
with similar Z values show the same behavior. We note that the $B(E3)$ comes
out much closer to experiment if both even and odd states are taken from
configurations having the same sign of quadrupole moment.

##  V Discussion

We have demonstrated that a global theory of the octupole degree of freedom
can be constructed using the HFB/GCM/HW methodology. The theory reproduces the
secular trend of the excitations, the effects of an incipient static octupole
deformation, and the most visible shell effects. However, the theory has
obvious deficiencies. Most notably, we require a overall scaling factor of 1.6
to make quantitative comparison with experiment. It is urgent to understand
what physics is needed to make predictions on an absolute energy scale. There
are several possible reasons for the absolute errors. One is the Hamiltonian
itself. Besides the Gogny interaction, there have been calculations with the
BCP interaction, interactions from the Skyrme family and from relativistic
mean-field theory. Ref. ro10b  found that the D1S Gogny interaction and the
BCP interaction gave significant differences in the odd-parity excitations of
Ra isotopes. The Gogny interaction is guided by nuclear Hartree-Fock theory,
and one of the characteristics is a nucleon effective mass less than the
physical mass. This implies that single-particle excitation energies will be
higher than for a non-interacting system, and these effects could carry over
to the collective excitations as well. We note that the calculation of the
208Pb in Ref. he01  using a Skyrme interaction with a similar effective mass
to D1S agrees with our results. However, the Relativistic Mean Field
Hamiltonian also has a small effective mass, but excellent agreement was
obtained for $E_{3}$ in an isotone chain by (Q)RPA an09 .

This brings up another source of systematic error in the GCM/HW, the
restriction of the degrees of freedom in the excitation to a single variable.
It is well-known in the theory of quadrupole excitations that time-odd
components must be included in the wave function to obtain good moments of
inertia ro00 . For large amplitude deformations, this can be achieved by self-
consistent cranking. When no time-odd components are allowed in the angular
momentum projected (AMP) GCM calculation the excitation energy is stretched
with respect to standard cranking calculations by a factor of around 1.4. This
correction factor is compatible with the discrepancies observed between our
results and the experiment in the case of 158Gd as well as with the overall
1.6 factor for the negative parity excitation energies discussed previously.

More generally, one can introduce methods that would reduce to (Q)RPA in the
small amplitude limit. The raises the question of how well (Q)RPA would
perform in a global context. As shown in the Appendix, for a large fraction of
nuclei the GCM/HW methodology is essentially equivalent to (Q)RPA in a single
collective variable. For these nuclei, the (Q)RPA is justified and is very
likely to give lower excitation energies.

The interaction of the octupole with the quadrupole degree of freedom is an
interesting problem that appears in several contexts in our study. First, the
HFB static quadrupole deformation of many nuclei invalidates a spectroscopic
interpretation of the observables for the physical angular momentum
eigenstates of the system. We saw this most directly in the discussion of the
$B(E3)$ transition strengths. The solution is to carry out angular momentum
projection. Another aspect missing from our study is the inclusion of $K\neq
0$ excitations in deformed nuclei. This has been done in HFB-BCS in Ref. sk93b
; da99 ; zb06  and in HFB in Ref. ya01 . Since $K\neq 0$ bands can fall below
the $K=0$ octupole excitation band, it is essential for a complete theory of
the octupole excitations in deformed nuclei.

Some aspects of the quadrupole-octupole mixing may require a two-dimensional
GCM to describe properly. It was clear in the light nuclei that octupole and
quadrupole deformations are strongly coupled in forming alpha-clusters. Also
we found that the severe problem describing the $B(E3)$ in 64Zn could be
traced to the coupling. We note that the two-dimensional GCM has been
implemented in the past. In Refs. me95  the coupled GCM was applied to the
complex spectroscopy of the nucleus 194Pb. Also, the microscopic theory of
asymmetric fissiongo05  requires at least a two-dimensional GCM.

One last aspect of the theory should be mentioned. We have seen in the
examples that the correlation energy of the ground state associated with the
$K=0$ octupole excitation is of the order of one MeV. This can have an
important influence on the theory of the nuclear masses. We plan to
investigate the systematics of the correlation energy in a future publication.

The approximation of a single degree of freedom can break down in different
ways. One is if the coupling between different multipoles is important in
determining the configurations. This is the case for light alpha-particle
nuclei. Fig. 8 shows the $\beta_{2}$ and $\beta_{3}$ deformations of the GCM
configurations for 16O and 20Ne. It may be seen that the $\beta_{3}$
deformation carries a $\beta_{2}$ deformation along with it for all but the
smallest values of $\beta_{3}$. Whether this is physical or not depends on the
matrix elements of the interaction connecting the different configurations.
For 16O, the admixtures are perturbative and thus should not change
$\beta_{2}$. However, if the configurations are sampled on coarse mesh, there
will be significant admixture of quadrupole excitations and the Gaussian
overlap approximation will fail. As mentioned in the Introduction, a two-
dimensional treatment of the GCM taking the quadrupole and octupole
deformations as independent variables is important in fissiongo05 . It has
also been carried out for the 194Pb nucleusme95  which is very soft with
respect to quadrupole deformations.

The single-operator approximation is also problematic due to the fragmentation
of octupole strength in the full spectrum. Roughly speaking, the octupole
strength has two important branches: the low collective excitation that is
under study here, and the high-lying excitation characterized as
$3\hbar\omega$ in the harmonic oscillator model. Our generating field
introduces amplitudes of both into the constrained wave function.

##  VI Acknowledgments

We thank H. Goutte, P.-H. Heenen and W. Nazarewicz for reading and comments on
the manuscript. This work was supported in part by the U.S. Department of
Energy under Grant DE-FG02-00ER41132, and by the National Science Foundation
under Grant PHY-0835543. The work of LMR was supported by MICINN (Spain) under
grants Nos. FPA2009-08958, and FIS2009-07277, as well as by Consolider-Ingenio
2010 Programs CPAN CSD2007-00042 and MULTIDARK CSD2009-00064.

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## Appendix: Simplified approximations and limits

It is important to understand the limiting behavior of any computationally
demanding theory, both to check the reliability of the calculations as well as
to see whether approximations are justified that would simplify the
calculations. For the GCM/HW methodology, the theory becomes analytic or
nearly so if a few conditions are met. One requirement is that there be only a
single degree of freedom necessary to describe the excitation of the system.
There are simple Hamiltonians that satisfy this condition. Examples are the
Lipkin modelRS , ro92 , where the degree of freedom is the number of particles
in the excited orbital, and the two-particle problem treated in Ref. ha00
where the degree of freedom is the center-of-mass displacement. In the last
model and other like it the theory becomes analytic and reduces to RPA the if
overlap integrals satisfy the Gaussian Overlap Approximation and the matrix
elements of the Hamiltonian reduce to a quadratic functions times the overlap.
In fact the relation to RPA remains even if there are many degrees of freedom
in the GCM br68 ; ja64 .

To make the discussion concrete, let us assume that there is a single
continuous degree of freedom $q$ and we can write the overlap integral and the
Hamiltonian matrix element as

$\langle q^{\prime}|q\rangle=e^{-(q-q^{\prime})^{2}/q_{s}^{2}}$ (13)
$\frac{\langle q^{\prime}|H|q\rangle}{\langle
q^{\prime}|q\rangle}=E_{0}+\frac{1}{2}v(q+q^{\prime})^{2}-\frac{1}{2}w(q-q^{\prime})^{2}$
(14)

The solution obtained by the Hill-Wheeler construction is identical to the
solution of the RPA equation for the operator $\hat{Q}$ that generates the GCM
states $|q\rangle$. The HW wave functions have the form of Gaussians in the
variable $q$ and the excitation energy is given by

$\hbar\omega_{RPA}=q_{s}^{2}\sqrt{vw}.$ (15)

Let us now compare with the MAP approximation. Here one first calculates
projected energies as a function of $q$,

$\frac{\langle e|H|e\rangle}{\langle
e|e\rangle}=2vq^{2}\frac{v-we^{-4(q/q_{s})^{2}}}{1+e^{-4(q/q_{s})^{2}}}$ (16)

and

$\frac{\langle o|H|o\rangle}{\langle
o|o\rangle}=2vq^{2}\frac{v+we^{-4(q/q_{s})^{2}}}{1-e^{-4(q/q_{s})^{2}}}$ (17)

The energies are then minimized with respect to $q$. The results for a range
of values of the ratio $w/v$ are given in Table 5. The ratios $q_{0}/q_{e}$
are close to $\sqrt{3}$, which may reflect the harmonic oscillator character
of the exact HW wave functions. In the last columns we compare the MAP
excitations energy with the RPA values. They are remarkably close.

$w/v$ | $q_{e}$ | $E_{e}$ | $q_{o}$ | $E_{o}$ | $E_{o}-E_{e}$ | $\hbar\omega_{RPA}$
---|---|---|---|---|---|---
1.5 | 0.226 | -0.0125 | 0.390 | 1.212 | 1.225 | 1.225
2. | 0.292 | -0.0421 | 0.509 | 1.373 | 1.415 | 1.414
4. | 0.400 | -0.232 | 0.716 | 1.782 | 2.01 | 2.00
8. | 0.469 | -0.721 | 0.870 | 2.207 | 2.93 | 2.83
Table 5:  The MAP solution in the harmonic limit. Deformations are in units of
$q_{s}$ and energies are in units of $vq_{s}^{2}$. The last column shows the
(Q)RPA excitation energy, Eq (15).

As a general conclusion, we find that if the MAP conditions are satisfied, the
energies are close to the RPA performed with a single collective variable. For
those nuclei, it would better to extend the space for the calculation using
more RPA degrees of freedom than by going to large amplitudes in a single
collective variable.

It would be nice to find a criterion to test for validity of the simplified
treatment. The first condition we can check is the ratio $q_{o}/q_{e}$. This
is graphed in Fig. 13 for the 284 nuclei tabulated in Ref. ki02 .

Figure 13:  Ratio of MAP deformations $\beta_{3p}/\beta_{3m}$ for nuclei with
measured $E_{3}$ ki02 .

There is a strong peak at $\beta_{3m}/\beta_{3p}\approx 1.9$. This is slightly
higher than the single-mode (Q)RPA, but still close enough to make a further
investigation of the quadratic Hamiltonian approximation. There are also wings
on the distribution extending from 0.9 (16O) to 3.2 (230U). Excluding the
wings below 1.7 and above 2.2, the peak contains 80 % of the measured nuclei.

To examine the validity of the quadratic approximation, we compared the
extracted coefficients $vq_{s}^{2}$ and $wq_{s}^{2}$ at the two deformations
$\beta_{3p}$ and $\beta_{3m}$. If the quadratic approximation is valid, they
should be equal. For example, the values of $\beta_{3p}$ and $\beta_{3m}$ at
the closest mesh points are 0.0375 and 0.075, respectively. The values of
$v\beta_{3p}^{2}$ and $w\beta_{3p}^{2}$ extracted at that mesh point are 0.23
MeV and 1.72 MeV, respectively. The corresponding numbers for $\beta_{3m}$ are
0.94 MeV and 7.20 MeV, very close to 4 times the values at $\beta_{3p}$. This
is just what is expected given $\beta_{3m}/\beta_{3p}=2$, showing that 208Pb
satisfies the conditions for the quadratic Hamiltonian. With these values for
$v$ and $w$, the RPA energy formula Eq. (15) gives 3.9 MeV, close to the
GCM/HW value of $4.0$ MeV. The results for the nuclei within the peak of Fig.
14 is shown as a scatter plot of the ratios. In general, the $w$ term follows
a quadratic dependence very well. The $v$ term can have large deviations,
particularly for nuclei that are soft to octupole deformations. However, for
most of the nuclei, the quadratic approximation is valid to an accuracy far
better than needed, given the overall performance of the theory in non-
octupole deformed nuclei at the 25% level in the scaled energies.

Figure 14:  Ratio of MAP deformations $\beta_{3m}/\beta_{3p}$ for nuclei with
measured $E_{3}$ ki02 .