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\title{Microscopic Interpretation of $\boldsymbol{K^{\pi}=0_{2}^{+},2_{\gamma}^{+}}$
Bands in Strongly Deformed Nuclei\thanks{Support from the U.S.
National Science Foundation, PHY-0140300, and the Southeastern
Universities Research Association is gratefully acknowledged.}}

\runningheads{MICROSCOPIC INTERPRETATION OF
$K^{\pi}=0_{2}^{+},2_{\gamma}^{+}$ BANDS IN...}{A.~GEORGIEVA,
G.~POPA,  J.~P.~DRAAYER}


\begin{start}
\author{A.~Georgieva}{1,2}, \coauthor{G.~Popa}{3},
\coauthor{J.~P.~Draayer}{2}

\address{Institute of Nuclear Research and Nuclear Energy,\\
Bulgarian Academy of Sciences, Sofia 1784, Bulgaria}{1}

\address{Department of Physics and Astronomy, Louisiana State
University}{2}

\address{Department of Physics, Rochester Institute of Technology}{3}

\begin{Abstract}
An overarching $sp(4,R)$ structure is used to identify and link
trends in properties of the low-lying, non-yrast states of
strongly deformed nuclei. Properties of nuclei within the linked
sets are reproduced by using a proton-neutron version of the
pseudo-$SU(3)$ shell model.
\end{Abstract}
\end{start}

\section[]{Introduction}

An interpretation of the properties of the low-lying spectra of
deformed even-even atomic nuclei is usually based first on
understanding the structure of states that belong to the ground
state band (gsb) and then on properties of the excited bands,
including especially the excitation energies of the band-head
configurations. While some light nuclei display well-developed
rotational characteristics, far and away the majority of nuclei
with rotational yrast bands are observed in the heavier shells,
that is, among nuclei from the lantanide and actinide regions. An
investigation of the properties of low-lying, non-yrast bands
reveals large differences in their behavior. Non-yrast bands of
particular interest are those built on the lowest excited
$J^\pi=0^{+}$ and $K^\pi=2^{+}, ~J^\pi=2^+$ states \cite
{Garetrev} and are traditionally interpreted as band-heads of the
so-called $\beta$ and $\gamma$ bands, a labelling that follows
from an interpretation of these modes as quadrupole surface
vibrations of a deformed liquid drop model \cite{BMb}. The
underlying structures of these bands are described differently
depending upon the model one chooses to apply \cite{RevSh}. With
the development of new experimental equipment and the accumulation
of more data, new model approaches \cite{IBA} have been introduced
in order to interpret and describe a variety of observed low-lying
spectra. There are now many examples against which to test model
interpretations of such states. In this report we focus on the
first two excited bands which we identify by their band-head
quantum numbers $K^{\pi }=0_{2}^{+}$ and $K^{\pi
}=2_{\gamma}^{+}$.

We give a brief review of the existing data on these bands along
with an interpretation of their structure in terms of an $sp(4,R)$
classification scheme, which has been tested empirically and shown
to be convenient for a unified description of the low-lying yrast
energies of even-even nuclei \cite{dggrr,DGM}. As with the
previous study, this investigation is aimed at gaining an
empirical understanding of the trends in the behavior of these
excited bands. Since these low-lying excited non-yrast bands are
considerably more complex than the gsb, a deeper investigation
into their microscopic structure is required.

To interpret and reproduce properties of the low-lying spectra of
deformed even-even nuclei, we apply a proton-neutron version of
the pseudo-$SU(3)$ shell model \cite{DraM}. This scheme is
particularly useful since it combines a consideration of the
microscopic structure of nuclei with simple but general symmetry
principles. Specifically, the pseudo-$SU(3)$ model has been shown
to be appropriate for a description of the low-lying spectra of
the strongly deformed nuclei \cite{aplDraM}. Another advantage of
this approach is that it gives a geometrical interpretation of
many-nucleon states through an established relationship between
the $SU(3)$ invariants and the shape variables $\beta$ and
$\gamma$ of the Geometrical Collective Model \cite{GCMre}.

\section{Empirical Investigation and the Classification Scheme}

In an empirical investigation of the yrast state energies of
deformed nuclei, we achieved an accurate and unified theoretical
description by superimposing a classification scheme that linked
nuclear species within major valence shell sets \cite{dggrr}. This
classification scheme is very convenient because it depends only
on two numbers, namely the total number of valence bosons $N=N_\pi
+N_\nu $ and the third projection $F_0=\dfrac 12(N_\pi -N_\nu )$
of the $F$-spin. Equivalently, this is a classification of nuclei
in terms of the operators $N_\pi =\dfrac 12(N_p-N_p^1)$ and $N_\nu
=\dfrac 12(N_n-N_n^1)$. These two operators give the number of
proton and neutron valence pairs within a given shell beyond their
respective closed cores (the usual magic numbers denoted here by
$N_p^1$ and $N_n^1$) which serve as the vacuum state of the
system.

The nuclei belonging to a major nuclear shell, defined by its
bordering magic numbers -- $(N_{p}^{(1)},N_{n}^{(1)}|N_{p}^{(2)}$,
$N_{n}^{(2)})$, where $N_{p}^{(2)}>N_{p}^{(1)}$ and
$N_{n}^{(2)}>N_{n}^{(1)}$, are subdivided into two $sp(4,R)$
multiplets determined by whether $N$ is even or odd. The even
$H_{+}$ ($N$ even) and odd $H_{-}$ ($N$ odd) spaces of the
symplectic algebra~\cite{brsp} are further reduced by an
application of the operators $N$ and $F_{0}$ to a definite vector
within one of these subspaces. Each of the nuclei considered in
this study has a unique $N$ and $F_{0}$ value.

As for the case of the empirical investigation of gsb phenomena
\cite{dggrr}, in the present study we focused on the behavior of
the $J^\pi=0_{2}^{+}$ and $J^\pi=2_{\gamma }^{+}$ levels in some
$F_{0}$ multiplets belonging to different shells. In order to
observe some common trends in the variety of structures
encountered, we focused on well-deformed nuclei from the
lanthanide region \cite{BS}, specifically considering the behavior
of the energies of the $J^\pi = 2^{+}$ state of the gsb, the first
excited $K^\pi = 0_{2}^{+}$ state and $K^\pi = 2_{\gamma }^{+}$
band-head, in the longest $F_{0}=0$ multiplet of the
$(50,82|82,126)_{+}$ shell.

\begin{figure}[b]
\centerline{\epsfig{file=files/popa_fig1.eps,width=70mm}}
\caption{The experimental and theoretical energies of the ground
band $J^{\pi}=0^{+}$ and $J^{\pi}=2^{+}$ states and the non-yrast
$K^{\pi}=0_2^{+}$ and $ K^{\pi}=2_{\gamma}^{+}$ states of deformed
nuclei in the $F_0=0$ multiplet of Sp(4,R). The experimental
values are indicated with bars and the calculated numbers with
shapes. \label{f1-ani}}
\end{figure}

As we can see from Figure~1, the behavior of the $0_2^{+}$ and
$2_\gamma ^{+}$ band-head states is strikingly irregular, in
contrast with the smooth behavior of the $2^{+}$ energies of the
yrast band. This is particularly pronounced in the middle of the
shell ($10<N<22$), which is a region of well-deformed nuclei, as
can be appreciated from the typical $L(L+1)$ rotational spacing of
the yrast band states. The gsb $J^\pi =2^{+}$ energies for these
nuclei lie on an almost straight line at $\sim 0.07$~MeV. In
contrast, the energies of $0_2^{+}$ and $2_\gamma ^{+}$ states
oscillate out of phase as a function of $N$. The trends in the
positions of the energies of these states form a pattern that is
almost symmetric with respect to the middle of the rotational
region at $N=16$ for $^{164}Dy$. At this point, the energy of the
first excited $K^\pi =0^{+}$ state has its highest value and the
band-head of the $\gamma$ band has its minimum value. To either
side of $^{164}$Dy for the $^{160}$Gd and $^{168}$Er we have
$E(2_\gamma ^{+})<E(0_2^{+})$. However, away from $^{164}$Dy, to
the left ($^{152}$Nd and $^{156}$Sm) and to the right ($^{172}$Yb,
$^{176}$Hf), the two non-yrast $J^\pi =0_2^{+}$ and
$K^\pi=2_\gamma ^{+}$ states change their ordering in energy,
$E(2_\gamma ^{+})>E(0_2^{+})$. Three loops are formed by the lines
connecting the energies of these states. The first and third loops
are quite similar. Our aim is to understand and reproduce this
behavior, which has many different model interpretations
\cite{Garetrev}.

\section{Outline of the Proton-Neutron Version of the Pseudo-$\boldsymbol{SU(3)}$ Model}

The complicated type of behavior noted above is usually strongly
dependent upon the microscopic structure of the nucleus \cite{BS}.
It is well known that the pseudo-$SU(3)$ model is successful in
giving a description of the low-energy spectra and electromagnetic
transition strengths in heavy deformed nuclei \cite{PoHiDr}. In
contrast to a classification scheme where protons and neutrons
pairs are counted as bosons, the proton-neutron version of the
pseudo-$SU(3)$ shell model is a fully microscopic theory that
respects the Pauli principle.

The model is an extension of the $SU(3)$ shell model \cite{El} for
heavy nuclei, and in particular for the lanthanides and actinides
where one finds most of the strongly deformed nuclei. In the
pseudo-$SU(3)$ version of the model, the pseudo-shell
$\widetilde{\eta }=\eta -1$ is defined as the original `parent'
shell $\eta$ without its highest intruder level $j=\eta
+\dfrac{1}{2}$. In the pseudo-shell, containing only the normal
parity states, the corresponding pseudo spin-orbit interaction is
negligible and hence the $ SU(3)$ symmetry is restored. This
mapping, from the $\eta$ to the $ \widetilde{\eta }=\eta -1$
shell, yields a symmetry governed reduction of the model space to
a subset of $SU(3)$ irreducible representations (irreps) that
correspond to the largest intrinsic deformation \cite{Vargas}.

The proton and neutron occupancies $n_{\sigma }$ ($\sigma =\pi$
and $\nu$, respectively) are determined by filling the Nilsson
single-particle levels from below \cite{Nils} with pairs of
particles in each level at a fixed value for the deformation
($\beta \sim 0.3$) for all nuclei. (The changes in predicted
occupancies as a function of deformation are rather rare over the
normal range of deformation, $\beta \sim 0.25$ to $\beta \sim
0.35$.) Further, we consider only nucleons in normal parity orbits
to be spectroscopically active with those in the unique parity
orbitals relegated to a renormalization role. This is an
assumption that is consistent with what has been done in the past
and one that is known to work well for low-lying configurations
\cite{JEJPDAF}. Since the nuclei in an $F_{0}=0$ multiplet have an
equal number of valence protons and neutrons, the classification
number $N$ is equal to the number of valence particles of each
kind.

The $n_\sigma ^{+}$ nucleons in a normal parity pseudo-shell
$\widetilde{ \eta }_\sigma $ are considered in a `strong-coupled'
basis, for which the proton ($\pi$) and neutron ($\nu$)
pseudo-$SU(3)$ irreps $(\lambda _\sigma ,\mu _\sigma)$ are first
coupled to a total $(\lambda,\mu)$ with $\rho$ labelling multiple
occurrences of the same irrep: $(\lambda _\pi ,\mu _\pi)\otimes
(\lambda _\nu,\mu_\nu)\rightarrow \rho(\lambda,\mu)$. The
quadrupole-quadrupole ($Q\cdot Q$) interaction is a part of the
$SU(3)$ second order operator $C_2=\dfrac 14(Q\cdot Q+3L^2)$ and
gives a dominant weight to the `stretched' coupled representations
$(\lambda,\mu )=(\lambda _\pi+\lambda _\nu,\mu _\pi+\mu _\nu)$.
However, in order to describe the rich and complex structure of
the nuclear spectra, we need to include at least $5$ or $6$
additional proton and neutron pseudo-$SU(3)$ irreps, which are
selected according to their largest $C_2$ values. A combination of
all of these irreps gives a large space of product
representations, so they are further truncated in the same way to
about $12$ or so. The angular momentum quantum number $L$, with
multiplicity label $K$, is determined through the reduction
$SU(3)\supset SO(3)$. The partition of the valence protons and
neutrons, into normal ($^{+}$) and abnormal ($^{-}$) parity
orbits, and the leading pseudo-$SU(3)$ irreps, obtained for the
nucleons in the normal parity orbits, are given in
Table~{\ref{irreps}}.

\begin{table} [hb]
\caption{\label{irreps} Occupation numbers for members of the
$F_{0}=0$ multiplet.}\smallskip
\begin{small}\centering
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}ccccccccccc}
\hline  \noalign {\smallskip}
nucleus & $N$ & $n_{\nu }$ & $n_{\nu }^{+}$ & $n_{\nu }^{-}$ & $n_{\pi }$ & $%
n_{\pi }^{+}$ & $n_{\pi }^{-}$ & \multicolumn{3}{c}{leading
$SU(3)$ irreps}
\\
&  &  &  &  &  &  &  & $(\lambda ,\mu )$ & $(\lambda _{\pi },\mu
_{\pi })$ & $(\lambda _{\nu },\mu _{\nu })$ \\ \hline\noalign
{\smallskip}
${}^{152}$Nd & 10 & 10 & 6 & 4 & 10 & 6 & 4 & (30,0) & (12,0) & (18,0) \\
${}^{156}$Sm & 12 & 12 & 6 & 6 & 12 & 6 & 6 & (30,0) & (12,0) & (18,0) \\
\hline\noalign {\smallskip}
${}^{160}$Gd & 14 & 14 & 8 & 6 & 14 & 8 & 6 & (28,8) & (10,4) & (18,4) \\
${}^{164}$Dy & 16 & 16 & 10 & 6 & 16 & 10 & 6 & (30,8) & (10,4) & (20,4) \\
${}^{168}$Er & 18 & 18 & 10 & 8 & 18 & 10 & 8 & (30,8) & (10,4) & (20,4) \\
\hline\noalign {\smallskip}
$^{172}${Yb} & 20 & 20 & 12 & 8 & 20 & 12 & 8 & (36,0) & (12,0) & (24,0) \\
$^{176}${Hf} & 22 & 22 & 12 & 10 & 22 & 12 & 10 & (8,30) & (0,12)
& (8,18)
\\ \hline
\end{tabular*}
\end{small}
\end{table}

The development of a computer code that enables one to calculate
the reduced matrix elements of the physical operators between
different $SU(3)$ irreps \cite{BaDra}, makes it possible to
include interactions that break the $SU(3)$ symmetry. The
importance of pairing modes, in the middle of the deformed region,
has been pointed out in some studies of $K^{\pi }=0^{+}$ states
\cite{Miko}, so these terms are included in our model Hamiltonian.
The Hamiltonian, that is appropriate for the description of the
nuclei considered, includes spherical single-particle terms for
both protons and neutrons, $H_{sp}^{\sigma }$, proton and neutron
pairing terms, $ H_{P}^{\sigma }$, an isoscalar
quadrupole-quadrupole interaction, $Q\cdot Q$ , and four smaller
`rotor-like' terms that preserve the pseudo-$SU(3)$ symmetry:
\begin{multline}
H=H_{sp}^{\pi}+H_{sp}^{\nu}-G_{\pi}H_{P}^{\pi}-G_{\nu}H_{P}^{\nu}-
\frac{1}{2}\chi Q\cdot Q   \\
+aJ^{2}+bK_{J}^{2}+a_{3}C_{3}+a_{sym}C_{2}\ .  \label{eq:ham}
\end{multline}
In the above formula, $C_{2}$ and $C_{3}$ are the second and third
order invariants of $SU(3)$, which are related to the axial and
triaxial deformation of the nucleus. In the calculations we
assumed fixed values for the proton and neutron single-particle
and pairing interaction strengths, as well as for the
quadrupole-quadrupole interaction strength. The strength factor
multiplying $Q\cdot Q$ was taken as $35A^{-5/3}$, and the proton
and neutron interaction strengths were chosen to be $G_{\pi
}=21/A$ and $G_{\nu }=17/A$ \cite{PoHiDr}. The pairing force is
known to induce $K$-band mixing and hence deviations from axial
symmetry, which in general is favored by the quadrupole-quadrupole
interaction \cite{BaEsDr}.

The other interaction strengths were varied to give a best fit for
the second $J^\pi = 0^{+}$, first $J^\pi = 2^{+}$ and $K^\pi =
2_{\gamma }^{+}$ states. The fits were done in the following way:
the $C_{3}$ interaction strength, $a_{3}$, was varied to fit the
energy of the second $0^{+}$ state. The interaction strength $b$
of $K_{J}^{2}$ was varied to fit the energy of the gamma $2^{+}$
band-head, which is not necessary the second $2^{+}$ state. The
interaction strength $a$ of $J^{2}$ was varied -- but only
slightly -- to give a best fit to the moment of inertia of the
ground state band.

\section{Results}

A microscopic interpretation of the relative position of a
collective band, as well as that of the levels within the band,
follows from an evaluation of the primary $SU(3)$ content of the
collective state. The latter are closely linked to nuclear
deformation. This results from a connection between the
microscopic quantum numbers $(\lambda,\mu)$ and the collective
shape variables ($\beta $,$\gamma$) \cite{GCMre}.

For $^{160}$Gd, $^{164}$Dy, and $^{168}$Er, in the middle of the
shell, the $ \gamma $ band is below the respective $K^\pi
=0_2^{+}$ band. The ground and $ \gamma $ bands belong to the same
$SU(3)$ irrep, when the leading configuration has triaxial
character (nonzero $\mu \approx \lambda $). The leading $SU(3)$
irreps, for the three nuclei, have the quantum numbers $\mu>0$ and
$\lambda>\mu$. The gsb and the $\gamma $ band have similar $SU(3)$
content, and the percentage content of the dominant irrep is
always higher in the $\gamma$ band. There is low content of these
irreps in the other bands. The $K^\pi =0_2^{+}$ band lies mainly
in another $SU(3)$ product configuration. For the $^{164}$Dy case,
where the band-head of the $\gamma$ band reaches its highest
energy, there is no mixing, that is, the band-head is $100\%$ in
the $(10,4)\otimes (14,10)$ $\rightarrow$ $(24,14)$ irrep. Note
that this happens also when $\mu$ has its largest value.

If the leading $SU(3)$ configuration is prolate ($\mu =0$), the
$K^{\pi
}=0_{2}^{+}$ and $K^{\pi }=2^{+}$ of the $\gamma $ band come from the same $%
SU(3)$ irrep. This is the case for both $^{152}$Nd and $^{156}$Sm
at the beginning of the sequence of nuclei that we considered for
which $E(2_{\gamma }^{+})>E(0_{2}^{+})$. The ground states are
spread over almost all the $SU(3)$ irreps considered in each
calculation, with a maximum that is less than $40\%$ in the most
symmetric $(12,0)\otimes (18,0)$ $ \rightarrow $ $(30,0)$
configuration for $^{156}$Sm. The $K^{\pi }=0_{2}^{+}$ and $\gamma
$ band-heads are also mixed but with about $75\%$ in the coupled
$(12,0)\otimes (12,6)$ $\rightarrow $ $(24,6)$ configuration. For
$^{152}$Nd, a state of angular momentum $J^{\pi} = 2^{+}$, $K^{\pi
}=2^{+}$ is not known experimentally. As a result of our
calculation we predict this state to be at $1.31$~MeV within a
range of uncertainty that is consistent with others for known
states.

At the end of the series, the experimental situation is very
similar to the one at the beginning. The ground band for the
$^{172}$Yb nucleus lies almost $100\%$ in the fully symmetric
representation $(36,0)$ with a very small admixture of
$(12,0)\otimes (16,10)$ $\rightarrow $ $(28,10)$, a configuration
that plays an important but not dominant role in the $\gamma $ and
$K^\pi =0_2^{+}$ band-head configurations. The $\gamma$ band shows
the greatest amount of mixing with the $K^\pi =0_2^{+}$ band, with
the largest percentage (about $75\%$) in the triaxial irrep
$(4,10)\otimes (16,10)$ $\rightarrow $ $(20,20)$. The $L-$even
states of the $K^\pi =0_2^{+}$ and $ \gamma $ bands are almost
degenerate. In the case of $^{176}$Hf, the protons and neutrons in
the normal parity states fill more than half of the shell. Hence
the $SU(3)$ quantum numbers for the leading proton $(0,12)$ and
neutron $(8,18)$ irreps have $\mu >\lambda $, which corresponds to
oblate shapes and as a result the pseudo-$SU(3)$ quantum numbers
for the leading coupled irrep $(8,30)$ have also $\lambda <\mu $.
In this case, the energy of the $J^\pi =0_2^{+}$ state is less
than the energy of the $K^\pi =2^{+}$ state, as for $^{172}$Yb and
in the first region, but the $SU(3)$ content in the wave functions
is similar to the one in the middle region. The $SU(3)$ content is
almost the same in the gsb and $K^\pi =2_\gamma ^{+}$ state, and
this agrees with the picture described in the middle region. The
difference is that in this case the energy of $K^\pi =2_\gamma
^{+}$ state is higher than the one of the first excited $0^{+}$
state, which comes from the respective eigenvalues of the $C_3$
invariant for these states. This type of nuclei requires further
investigation.

\begin{table} [hb]
\caption{\label{coefs} Interaction strengths of the
Hamiltonian.}\smallskip
\begin{small}\centering
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}crrrrrrrrr}
\hline  \noalign {\smallskip}
coef./nucleus & $^{152}$Nd & $^{156}$Sm & $^{160}$Gd & $^{164}$Dy & $^{168}$%
Er & $^{172}$Yb & $^{176}$Hf \\ \hline\noalign {\smallskip}
$a_{3}\times 10^{-4}$ & 2.57 & 2.59 & 1.93 & 0.65 & 0.75 & 0.31 & 0.43 \\
$a$ & 0.000 & 0.000 & 0.001 & -0.001 & -0.002 & -0.001 & -0.007 \\
$b$ & 0.00 & 0.55 & 0.153 & 0.042 & 0.022 & 0.12 & 0.3 \\
$a_{s}$ & 0.0000 & 0.0000 & 0.0035 & 0.0008 & 0.0008 & 0.001 & 0.006 \\
\hline
\end{tabular*}
\end{small}
\end{table}

The parameters of the Hamiltonian (\ref{eq:ham}) that were
obtained though a fitting procedure applied to all of the nuclei
considered in this study, are given in Table~{\ref{coefs}.} It is
important to note that the pairing interactions split and mix the
different $SU(3)$\ irreps, and that this mixing introduces
triaxiality in the system. The quadrupole-quadrupole interaction
drives the proton and neutron systems towards prolate shapes if
the oscillator shell is less than half full, towards oblate shapes
if the respective shell is more than half full, and to large
$\beta $ values at maximum asymmetry for shells which are roughly
half full. In addition to the quadrupole-quadrupole ($\chi$) and
the pairing strengths ($G_\pi$ and $G_\nu$ ) which change very
smoothly as a function of mass, the `fine tuning' of the energies
of the non-yrast band states required the use of the other four
parameters $a_{3},$ $a$, $b$ and $a_{sym}$, which were sufficient
determining the correct behavior of the states under consideration
and differences in energies of the nuclei with equivalent
configurations. The $ SU(3)$ configurations that enter into the
anaylysis result from the fact that the current version of the
model focuses only on the particles in the normal parity orbits.
The nuclei with the same $SU(3)$ leading irreps (see
Table~\ref{irreps}) differ by the number of particles in the
unique parity states of the Nilsson scheme. With the interaction
strengths given in Table~\ref{coefs}, the theoretical spectra of
the nuclei considered are in good agreement with one another
(systematic changes in interaction strengths as a function of
mass) and with the experimental data, not only for band-head
configurations that we focused on here, but also for the excited
states within those bands.

\section{Conclusions}

In the present application, we achieved a good description of the
complex properties of a series of deformed nuclei using a small
configuration space. This is a result of the microscopic basis of
the model, that correctly takes into account the distribution of
particles among the single-particle levels of the valence shell.
The combination of proton and neutron valence shells, which are
different from one another, are very important in obtaining these
results. The coupled pseudo-$SU(3)$ representations that emerge
from this analysis, yield information about the deformation of
each system. The truncation scheme that is used is also governed
by symmetry principles, and tracks the onset of a deformation
trough in the coupled representation space. The Hamiltonian of the
model includes some terms that are not invariants of $SU(3)$ and
therefore split and mix the resultant eigenvectors. In particular,
the pairing interactions play a very important role in determining
the distribution of the eigenstates across the allowed $SU(3)$
configurations. The four parameters that are used for finely
tuning the spectra not only permit a very good reproduction of the
experimental data, but also predict the position of states that
have not yet been experimentally identified.

\section*{Acknowledgments}

Support from the U.S. National Science Foundation, PHY-0140300,
and the Southeastern Universities Research Association is
gratefully acknowledged.

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