Wormhole Model for Neon-20

Nicholas S. Manton111email<EMAIL_ADDRESS>and Maciej
Dunajski222email<EMAIL_ADDRESS>

Department of Applied Mathematics and Theoretical Physics,
University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, U.K.

###### Abstract

A quantum mechanical model for the Neon-20 nucleus is developed that allows
for the splitting of a bipyramidal structure of five alpha-partices into an
alpha-particle and an Oxygen-16 nucleus. The geometry of the configuration
space is assumed to be a 3-dimensional spatial wormhole, and on the wormhole
background there is an attractive short-range potential. This leads to a
radial Schrödinger equation of the Heun form, which simplifies for threshold
bound states to an associated Legendre equation that has explicit solutions.
The energies of the true bound states for all spin/parities are numerically
calculated, and match those of the well-established $K^{\pi}=0^{+}$,
$K^{\pi}=0^{-}$, and certain higher rotational bands of Neon-20.

Keywords: Spatial Wormhole, Neon-20 Nucleus, Rotational Bands

##  1 Motivation

We develop a model for quantum states of the Neon-20 nucleus, where the
underlying configuration space is a spatial wormhole. Spacetime wormholes have
generated much interest, but are usually regarded as a science-fiction
fantasy. Here we find a wormhole that is physically realised – not as a
spacetime, but as a model configuration space for a spatially-extended,
deformable nucleus. The characteristic length scale of the wormhole is about 5
fm, comparable to the size of the Neon-20 nucleus. This wormhole model was
first outlined in [1], but not analysed there in any detail.

We start from the old idea that the ground-state configuration of Neon-20 is a
bipyramid of five alpha-particles [2, 3, 4], which splits relatively easily
into a single alpha-particle and a bound cluster of four alpha-particles
representing Oxygen-16. The bipyramid has an equilateral triangle of alphas in
the middle, but we ignore rotations in the plane of this triangle and treat
the bipyramid as axially symmetric. The angular momentum projection about the
symmetry axis is therefore $K=0$. We also neglect centre of mass motion.

Conventionally, one would have a Euclidean configuration space for the
separation vector of the two clusters. A central ball can then be cut out to
enforce a non-zero minimal separation, or a potential can be introduced that
strongly disfavours the clusters from overlapping. Such models have rather
singular behaviour near the bipyramid.

In contrast, our model has a smooth, curved configuration space near the
bipyramid, avoiding any singularity. It has a single radial coordinate $r$,
which accurately corresponds to the separation of the clusters asymptotically,
when they are well-separated, and additionally has the usual spherical polar
coordinates $(\theta,\phi)$ representing the direction of the separation axis.
Unusually, however, $r$ runs from $-\infty$ to $\infty$. The metric is that of
an Ellis–Bronnikov wormhole [5, 6]

$ds^{2}=dr^{2}+(r^{2}+a^{2})(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\,.$ (1)

On this background, there is a smooth, attractive potential energy depending
symmetrically on $r$, with its minimum at $r=0$ where the clusters merge into
the bipyramid. The radius of the wormhole’s throat, $a$, is related to the
linear size of the bipyramid.

To justify this geometry, consider a simple classical motion with $r$
monotonically decreasing through zero, and $(\theta,\phi)$ fixed. This models
an incoming alpha-particle coalescing instantaneously with the four-alpha
cluster to form the bipyramid, and a different alpha-particle from the
opposite side of the bipyramid getting ejected. The motion can be thought of
as analogous to that of a Newton cradle, with one ball coming in and a
different ball going out. If the incoming and outgoing alpha-particles are
distinguishable, it makes sense for $r$ to have the range $(-\infty,\infty)$.
However, note that the outgoing particle’s angular coordinates are antipodal
to those of the incoming particle, and because alpha-particles are
indistinguishable bosons, one should identify points in configuration space
with coordinates $(r,\theta,\phi)$ and $(-r,\pi-\theta,\phi+\pi)$. The true
configuration space is therefore not the complete wormhole, but its quotient
by $\mathbb{Z}_{2}$ under this identification.

The $\mathbb{Z}_{2}$ action has no fixed points, so the quotient space is
smooth. The quotient is the half-wormhole $r\geq 0$, where antipodal points
$(\theta,\phi)$ and $(\pi-\theta,\phi+\pi)$ on the 2-sphere at $r=0$ are
identified. This identification is required, because the bipyramid is
symmetric under the antipodal map. We will quantize the dynamics on the half-
wormhole, but to do this it is convenient to perform the quantization on the
complete wormhole, separating radial and angular variables as usual; the
$\mathbb{Z}_{2}$ symmetry then constrains physical states to be either
symmetric in $r$ with even angular momentum, or antisymmetric in $r$ with odd
angular momentum. A state’s parity is even/odd if the angular momentum is
even/odd.

This wormhole model is an anharmonic extension of a rovibrational model for
Neon-20, incorporating only the lowest-frequency vibrational mode of the
bipyramid, the singly-degenerate mode transforming under the
$A_{2}^{\prime\prime}$ representation of ${\cal D}_{3h}$ that tends to produce
a 1+4 cluster split. The mode’s oscillating amplitude corresponds to an
oscillation of $r$ around zero. The potential on the wormhole is minimal at
the bipyramid, just like the harmonic oscillator potential. Another similarity
is that the quantized rovibrational model has a bosonic constraint. Physical
states are those with either an even number of vibrational phonons and even
angular momenta, or an odd number of phonons and odd angular momenta.

Apart from the wormhole radius $a$, our model has just one adjustable,
dimensionless parameter $m$. It can take any positive value, and we found
initially that a good fit to experimental data was with $m=9.01$.
Mathematically, it is very convenient for $m$ to be a positive integer, so we
have fixed $m=9$. A consequence is that all the threshold bound states, just
below the continuum scattering states, have radial wavefunctions given by
analytic solutions of an associated Legendre equation.

Having fixed $m=9$, together with an energy scale, we obtain a good match to
the lowest observed rotational bands of Neon-20 with quantum numbers
$K^{\pi}=0^{+}$ and $K^{\pi}=0^{-}$, interpreted here as zero-phonon and one-
phonon bands. Bijker and Iachello have presented a rovibrational analysis of
Neon-20 states, incorporating all modes of small oscillation of the bipyramid
[7]. Our interpretation of some of the higher $K=0$ bands differs somewhat
from theirs. We agree on the two-phonon $0^{+}$ band, but interpret the second
$0^{-}$ band as a three-phonon band, rather than a band combining two distinct
phonons with opposite $\mathbb{Z}_{2}$ symmetries. Quantitatively, our results
are comparable, but as we have fewer adjustable parameters, we can predict
some of the parameters that they have fitted – in particular, the ratios of
the rotational constants $B$ associated to the various $K=0$ bands.

##  2 Quantum States on the Wormhole

We start with the classical Lagrangian dynamics for the relative motion of an
alpha-particle and a four-alpha cluster, and then quantize. The Lagrangian is

$L=\frac{1}{2}\mu\left({\dot{r}}^{2}+(r^{2}+a^{2})({\dot{\theta}}^{2}+\sin^{2}\theta\,{\dot{\phi}}^{2})\right)+\frac{V_{0}}{(r^{2}+a^{2})^{2}}\,,$
(2)

where the kinetic term is based on the wormhole metric, $\mu$ is the reduced
mass of the two clusters at large separation, and $V_{0}$ is a positive
constant so the potential is negative and attractive. The physical
interpretation of the wormhole geometry is that the inertial mass tensor is
separation-dependent, and different in the radial and angular directions. A
separation-dependent inertial mass in the radial direction has been previously
considered in the context of Neon-20 by Wen and Nakatsukasa [8].

The quantum Hamiltonian derived from $L$ is

$\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2\mu}\nabla^{2}-\frac{V_{0}}{(r^{2}+a^{2})^{2}}$
(3) $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2\mu}\left(\partial_{rr}+\frac{2r}{r^{2}+a^{2}}\,\partial_{r}+\frac{1}{r^{2}+a^{2}}\nabla_{\rm
ang}^{2}\right)-\frac{V_{0}}{(r^{2}+a^{2})^{2}}\,,$

where $\nabla^{2}$ is the Laplace–Beltrami operator for the wormhole metric
(1), and $\nabla_{\rm ang}^{2}$ is the usual angular part of the Laplacian in
spherical polars. $\nabla^{2}$ reduces to the standard Laplacian in spherical
polars when $a=0$. We separate variables, and write stationary state
wavefunctions as

$\Phi(r,\theta,\phi)=\chi(r)Y_{l,\widetilde{m}}(\theta,\phi)\,.$ (4)

$l$ is the usual angular momentum quantum number, and the index
$\widetilde{m}$ takes its usual $2l+1$ values. We are reserving $m$ to be a
parameter in the potential. The reduced, radial equation for stationary states
of energy $E$ is

$-\frac{\hbar^{2}}{2\mu}\left(\frac{d^{2}\chi}{dr^{2}}+\frac{2r}{r^{2}+a^{2}}\frac{d\chi}{dr}-\frac{l(l+1)}{r^{2}+a^{2}}\,\chi\right)-\frac{V_{0}}{(r^{2}+a^{2})^{2}}\,\chi=E\chi\,.$
(5)

We recall that bound states of the potential
$V=\mbox{const}\cdot(r^{2}+a^{2})^{-2}$ with zero angular momentum have, in
the context of kinks on the wormhole background, been considered in [9] and
[10].

We now change variable by setting $r=ax$, and write
$V_{0}=\frac{\hbar^{2}a^{2}}{2\mu}m^{2}$ and $E=\frac{\hbar^{2}}{2\mu
a^{2}}\varepsilon$. The radial equation then has the dimensionless form

$-\frac{d^{2}\chi}{dx^{2}}-\frac{2x}{x^{2}+1}\frac{d\chi}{dx}+\frac{l(l+1)}{x^{2}+1}\,\chi-\frac{m^{2}}{(x^{2}+1)^{2}}\,\chi=\varepsilon\chi\,.$
(6)

Equation (6) for general $\varepsilon$ is a confluent Heun equation, with two
regular singular points at $x=\pm i$, and a confluent singularity at infinity.
Its general solution is given by

$\displaystyle\chi(x)=c_{1}\left({x}^{2}+1\right)^{{\frac{m}{2}}}{\mbox{HeunC}}\left(0,-{\frac{1}{2}},m,-{\frac{\varepsilon}{4}},{\frac{m^{2}+1-l(l+1)+\varepsilon}{4}},-{x}^{2}\right)$
$\displaystyle+\,c_{2}\,x\left({x}^{2}+1\right)^{{\frac{m}{2}}}{\mbox{HeunC}}\left(0,{\frac{1}{2}},m,-{\frac{\varepsilon}{4}},{\frac{m^{2}+1-l(l+1)+\varepsilon}{4}},-{x}^{2}\right).$
(7)

There is more than one motivation for our choice of potential. It is short-
ranged compared to the centrifugal repulsion proportional to $l(l+1)$
(although not exponentially so). It is a multiple of the Ricci scalar
curvature of the wormhole, something that could arise from an alternative
treatment of canonical quantization. Finally, the potential’s algebraic form
means that for $\varepsilon=0$, eq.(6) usefully simplifies to the associated
Legendre equation, which can be explicitly solved.

##  3 The Energy Spectrum

Here we derive some general properties of the bound state spectrum of the
radial equation (6). True bound states are solutions with negative
$\varepsilon$, but there are also normalisable, threshold bound states with
$\varepsilon=0$.

If we write $\chi(x)=(x^{2}+1)^{-\frac{1}{2}}\,\eta(x)$, eq.(6) becomes

$-\frac{d^{2}\eta}{dx^{2}}+\frac{l(l+1)}{x^{2}+1}\,\eta-\frac{m^{2}-1}{(x^{2}+1)^{2}}\,\eta=\varepsilon\eta\,,$
(8)

which is easier to treat analytically. The transformed equation (8) is a 1-d
Schrödinger equation with effective potential (Fig.1)

$v_{\rm eff}(x)=\frac{l(l+1)}{x^{2}+1}-\frac{m^{2}-1}{(x^{2}+1)^{2}}\,.$ (9)

Figure 1. Effective potential with $m=9$ and $l=7$.

For bound states to exist, $v_{\rm eff}$ has to be negative somewhere, which
requires $m^{2}-1>l(l+1)$. On the other hand, if $m^{2}-1>2l(l+1)$ then the
integral of $v_{\rm eff}$ over IR is negative, and a bound state definitely
exists (by a variational argument). For general positive $m$ we therefore
expect bound states, but only for $l<m$.

Rather remarkably, there are threshold bound states for all $0<l<m$ when $m$
is a positive integer. To see this, we rewrite eq.(6) for $\varepsilon=0$ as

$(x^{2}+1)\frac{d^{2}\chi}{dx^{2}}+2x\frac{d\chi}{dx}-l(l+1)\chi+\frac{m^{2}}{x^{2}+1}\,\chi=0\,.$
(10)

Setting $z=ix$, this becomes the standard associated Legendre equation. We are
interested in solutions for real $x$ that are normalisable, and these are
associated Legendre functions evaluated on the imaginary $z$-axis. As this
axis does not pass through the regular singular points at $z=\pm 1$, the
solutions can be singular at these points, and we are not constrained to
impose $|m|\leq l$ as in the construction of spherical harmonics.

We can therefore assume that $\chi$ has the form

$\chi(x)=\frac{S(x)}{(x^{2}+1)^{\frac{m}{2}}}$ (11)

where

$S(x)=\sum_{k=0}a_{k}\,x^{k}\,.$ (12)

From (10) we then derive the recurrence relation

$a_{k+2}=-\frac{m(m-1)-l(l+1)-(2m-1)k+k^{2}}{(k+2)(k+1)}\,a_{k}\,.$ (13)

$\chi(x)$ is the radial wavefunction of a normalisable, threshold bound state
provided the series for $S$ truncates to a finite polynomial of degree less
than $m-1$. This occurs for all angular momenta in the range $0<l<m$, and we
denote the resulting polynomial $S_{l}^{m}(x)$. $S_{l}^{m}$ has degree
$m-l-1$. It is even if $m-l-1$ is even and odd if $m-l-1$ is odd, and has
$m-l-1$ nodes. A further threshold state exists for $l=0$, but this is not
normalisable because $S$ has degree $m-1$. Note that all the threshold states
are physically allowed provided $m$ is odd, because $m-l-1$ has the same
parity as $l$ in this case.

As explained in Sect.1, we have made the choice $m=9$. Usefully, for fitting
the Neon-20 spectrum, there are then threshold states for all angular momenta
up to $l=8$. For $m=9$, and $l$ decreasing from $8$ to $1$ the polynomials
$S_{l}^{9}(x)$ are respectively

$\displaystyle
1,\;x,\;1-15x^{2},\;3x-13x^{3},\;3-78x^{2}+143x^{4},\;15x-110x^{3}+99x^{5},$
$\displaystyle 5-165x^{2}+495x^{4}-231x^{6},\;5x-45x^{3}+63x^{5}-15x^{7},$
$\displaystyle 1-36x^{2}+126x^{4}-84x^{6}+9x^{8},$ (14)

and all their roots are real.

$\chi(x)$ is in fact, up to a normalisation constant, the associated Legendre
function of the second kind $Q_{l}^{m}(z)$, evaluated on the imaginary axis
$z=ix$. It has the explicit form, for $m$ an integer greater than $l$,

$Q_{l}^{m}(z)=(1-z^{2})^{\frac{m}{2}}\,\frac{d^{m}}{dz^{m}}\left[\ln\left(\frac{1+z}{1-z}\right)\frac{d^{l}}{dz^{l}}(1-z^{2})^{l}\right]\,.$
(15)

One might anticipate, from the expression for $Q_{l}(z)$ in ref.[11], an
additional polynomial inside the square brackets, but its $m$th derivative
vanishes. Also, because $m>l$, at least one derivative acts on the logarithmic
term, so there is no such term in the result.

Let us now consider the negative energy bound states. Because the threshold
bound state with angular momentum $l$ has $m-l-1$ nodes, it follows from the
Sturm oscillation theorem that there are $m-l-1$ negative energy states with
angular momentum $l$. These are confluent Heun functions, but as there are no
simple formulae for their energies, we have solved eq.(8) numerically to find
them. We use a shooting and bisection method on the interval $[-20,20]$, using
as initial condition at $x=0$ either that $\eta$ has zero slope (for the even
solutions) or zero value and a finite slope (for the odd solutions).

For each $l$, the lowest-energy wavefunction $\eta(x)$ is symmetric in $x$ and
has no nodes, but as the energy increases, the parity of $\eta$ alternates and
the number of nodes increases by 1. The physically allowed states are those
where the parity of $\eta$ matches the parity of $l$. The spectrum of
physically allowed states for $m=9$, including all the threshold states, is
shown in Fig.2.

Figure 2. Numerical bound state spectrum for $m=9$ and integer $l<9$ (dots),
compared with the anharmonic oscillator approximation (curves).

Although the true energies of the bound states are not known analytically,
they can be evaluated using an anharmonic oscillator approximation. The
expansion of $v_{\rm eff}$ up to $O(x^{4})$ is

$v_{\rm
eff}\simeq-(m^{2}-1)+l(l+1)+[2(m^{2}-1)-l(l+1)]x^{2}-[3(m^{2}-1)-l(l+1)]x^{4}\,.$
(16)

Using the harmonic oscillator wavefunctions for $v_{\rm eff}$ truncated at
order $x^{2}$, together with first-order perturbation theory to take account
of the $x^{4}$ term [12], we find the approximate energy levels

$\displaystyle\varepsilon_{n,l}$ $\displaystyle=$
$\displaystyle-(m^{2}-1)+l(l+1)+(2n+1)\sqrt{2(m^{2}-1)-l(l+1)}$ (17)
$\displaystyle\qquad-\frac{3}{2}\,\frac{3(m^{2}-1)-l(l+1)}{2(m^{2}-1)-l(l+1)}\left(n^{2}+n+\frac{1}{2}\right)\,,$

where $n\geq 0$ is the harmonic oscillator level. These approximate energies
are also shown in Fig.2 for $m=9$, as functions of continuous $l$. The lowest-
lying states for each even $l$ comprise the ground state rotational band, with
$n=0$, and above this we see the higher rotational bands. The $n$th rotational
band can be interpreted as rotational excitations of an $n$-phonon, purely
vibrational state.

For $m=9$, the anharmonic oscillator energies are accurate only up to about
$n=4$, but the phonon number $n$ is still a good label for all the bound
states. The rotational bands get shorter as $n$ increases, because the
threshold state is reached when $l=8-n$.

In each rotational band, we can express the energy for small $l$ as

$\varepsilon_{n,l}\simeq\varepsilon_{n}+b_{n}\,l(l+1)\,,$ (18)

where $b_{n}$ is the rotational constant (inversely proportional to the
effective moment of inertia of the band). The anharmonic oscillator
approximation gives, for $m=9$,

$b_{n}\simeq
1-\frac{1}{\sqrt{640}}(2n+1)-\frac{3}{640}\left(n^{2}+n+\frac{1}{2}\right)\,.$
(19)

The values for the lowest few bands are
$b_{0}=0.96\,,b_{1}=0.86\,,b_{2}=0.77\,,b_{3}=0.66$ and $b_{4}=0.54$. We see
from Fig.2 that these are quite accurate, even though the actual rotational
bands are significantly shifted up for $n=3$ and $n=4$.

A selection of bound state wavefunctions $\eta(x)$ is shown in Figs.3a, 3b,
3c.

Figure 3a. Bound states with $n=0$.

Figure 3b. Bound states with $n=1$.

Figure 3c. Bound states with $n=2$.

For given $n$, and independently of $l$, they have $n$ nodes in the full range
$-\infty<x<\infty$ and their widths hardly change with $l$ because they are
well approximated by harmonic oscillator wavefunctions in the simplified,
$l$-independent effective potential derived from (9),

$\widetilde{v_{\rm eff}}=-m^{2}+2m^{2}x^{2}\,.$ (20)

However, the threshold states, with maximal $l$, are broader and decay more
slowly for large $|x|$.

These wavefunctions may be compared to those in refs.[13] and [14], whose node
number varies with $l$. However, most of the latter nodes are in the cluster-
overlap region, and the wormhole wavefunctions are similar in shape if the
comparison is made outside this region.

##  4 Comparison with Neon-20 States

For the experimental spectrum of Neon-20, we use the TUNL tables [15]; see
also the review [16] and the ENSDF tables [17]. The identification of
rotational bands in Neon-20 goes back many decades. See, for example,
refs.[18, 19, 14, 20, 21] among many others, also Table 20.20 in [16], and the
recent discussion by Bijker and Iachello [7]. The ground state $0_{1}^{+}$
rotational band is well established up to $l=6$. There is also a well-
established, lowest $0^{-}$ rotational band with states from $l=1$ up to
$l=7$. We identify these bands with the 0-phonon and 1-phonon bands in our
model. There are a few more recognised $0^{+}$ bands. Following Bijker and
Iachello, we regard the $0_{2}^{+}$ and $0_{3}^{+}$ bands as arising from
other, symmetric vibrational excitations of the bipyramid, and identify the
experimental $0^{+}_{4}$ band as the 2-phonon band in our model. The states of
the $0^{+}_{4}$ band have been recognised as “higher nodal” states by Fujiwara
et al. [14], and our model gives them the expected wavefunction structure,
with a single node in the half-wormhole range $x>0$. We have also tentatively
identified one higher, observed rotational band with the 3-phonon band of our
model. A few further observed states can be assigned to the shorter bands with
4-, 5-, 6- and 7-phonons. $0^{+}$ bands have positive-parity states of even
angular momenta, and $0^{-}$ bands have negative-parity states of odd angular
momenta.

To calibrate our model, we need to shift the model ground state upwards to
match the experimental energy $E=0$, and then find the optimal physical energy
scale $\hbar^{2}/2\mu a^{2}$ by a least squares fit. The model’s dimensionless
ground state energy is $\varepsilon=-68.5$ (both numerically and in the
anharmonic oscillator approximation) so the shift upwards is by $68.5$.

In a preliminary fit we matched the states that are most confidently assigned
to rotational bands – those in the ground state $0^{+}_{1}$ band up to
$6^{+}$, the lowest $0^{-}$ band up to $7^{-}$, and the $0^{+}_{4}$ band up to
$4^{+}$. Following refs.[14, 21] and others, we identified the $7^{-}$ state
at 15.37 MeV to be in the $0^{-}$ band; it is a threshold state in our model.
Consistently with this, the preliminary fit gave a threshold energy just above
15 MeV. We then noted that the states in the TUNL table between 15.1 MeV and
15.9 MeV are unique candidates for all the threshold states in our model, with
spin/parities between $1^{-}$ and $8^{+}$. All the required spin/parities
occur, and there are almost no other observed states in this energy range, one
exception being the $8^{-}$ state assigned to the $2^{-}$ band of Neon-20.

For our final calibration, we therefore extended our ground state rotational
band to include the 15.87 MeV $8^{+}$ state as the threshold state. Other
models also predict that the $8^{+}$ state in the ground state band has a
similar energy [13]. There is one lower $8^{+}$ state at 11.95 MeV that is
often assigned to the ground state band, but this creates a sharp kink in the
band slope, and the state has other properties that makes this assignment
controversial, for example, its rather low E2 transition rate to the $6^{+}$
state in the ground state band [16]. We also followed Michel et al. [22] in
assigning the 15.16 MeV $6^{+}$ state to the observed $0^{+}_{4}$ band of
broad, higher nodal states. They argued that the energy of the $6^{+}$ state
in this band is in the range 14-15 MeV, rather than at 12.58 MeV. It becomes
the threshold state in our 2-phonon band. The broad $5^{-}$ state at 15.17 MeV
becomes the threshold state in the 3-phonon band. We have also identified
various lower-spin states in the TUNL table that are close in energy to the
states required for our model, both at and below the threshold, but these
states are more numerous, so we have some choice, and they have less
significance.

In the best fit of our model to all these 24 identified states, the conversion
factor from the shifted $\varepsilon$ to the physical energy $E$ is
$\hbar^{2}/2\mu a^{2}=0.222$ MeV. The threshold energy is then at $E_{\rm
thresh}=68.5\times 0.222\,{\rm MeV}=15.19$ MeV. Setting $\hbar=197.3$ MeV fm,
and $\mu=2982$ MeV (the reduced mass $\mu$ being almost exactly four-fifths of
the alpha-particle mass $m_{\alpha}$), we find $a=5.42$ fm. This is the radius
of the 2-sphere throat of the wormhole, and has order of magnitude the linear
size of the bipyramid. In Fig.4 we show the best fit of these observed states
to our model.

Figure 4. Experimental energies (boxes), numerical bound states (dots), and
the anharmonic oscillator approximation (curves).

The dimensionless rotational constant $b_{n}$, given in (19), converts to the
physical value $B_{n}=0.222\,b_{n}$ MeV for the $n$-phonon band. The
rotational constants of the rotational bands in our model are therefore
$B_{0}=0.213(0.212)$, $B_{1}=0.190(0.137)$, $B_{2}=0.170(0.105)$,
$B_{3}=0.146(0.101)$ and $B_{4}=0.120$, all in MeV. The bracketed quantities
are the rotational constants calculated in ref.[7] for the corresponding
bands. Although there are quantitative differences, there is a similar
decrease of $B_{n}$ as $n$ increases.

The detailed comparision between the experimental and model energies in MeV,
after the calibration, is as follows:

Experiment Model $0$-phonon band $\displaystyle[0,1.63,4.25,8.78,15.87]$
$\displaystyle[0,1.28,4.25,8.90,15.19]$ $1$-phonon band
$\displaystyle[5.79,7.16,10.26,15.37]$ $\displaystyle[5.04,6.97,10.39,15.19]$
$2$-phonon band $\displaystyle[8.7,9.0,10.8,15.2]$
$\displaystyle[8.36,9.38,11.71,15.19]$ $3$-phonon band
$\displaystyle[11.25,12.4,15.2]$ $\displaystyle[11.45,12.86,15.19]$ $4$-phonon
band $\displaystyle[13.2,13.9,15.3]$ $\displaystyle[13.11,13.80,15.19]$
$5$-phonon band $\displaystyle[14.6,15.4]$ $\displaystyle[14.49,15.19]$
$6$-phonon band $\displaystyle[14.5,15.1]$ $\displaystyle[14.95,15.19]$
$7$-phonon band $\displaystyle[15.3]$ $\displaystyle[15.19].$ (21)

The TUNL table has no state very close to the model prediction of 14.95 MeV
for the 6-phonon $0^{+}$ state. However, a recent review of data from
proton/Fluorine-19 scattering experiments [23] identifies some more $0^{+}$
states, including one at 14.9 MeV. This would better fit the 6-phonon band of
our model. There are also further $2^{+}$ states identified above 15 MeV, so
there is a 15.3 MeV alternative to our choice of the 15.1 MeV state as the
threshold 6-phonon state. If we had used these two states in our final
calibration, the threshold energy would have been slightly higher.

Identifying the 3-phonon ($n=3$) band of our model with observed states is
rather controversial. We find a good fit to the energies and spin/parities of
the observed $0^{-}$ band that Bijker and Iachello describe using a
combination of antisymmetric and symmetric one-phonon oscillations of alpha-
particles along the bipyramid axis. In favour of our interpretation, we note
that in earlier work on rovibrational models for Oxygen-16 and Calcium-40 [24,
25], it was found that 3-phonon states of a low-frequency vibrational mode
were needed to fit the experimental data. In anharmonic models with a
potential that flattens out, 3-phonon states are not of very high energy, and
cannot be ignored.

To confirm these band identifications, it would help to have predictions for
the frequencies of all vibrational modes of the bipyramid. It would also help
to clarify the status of the clear rotational band described as a $1^{-}$ band
in [16], and interpreted this way in [7]. The problem is that the only
experimentally confirmed states in this band have spin/parities
$1^{-},3^{-},5^{-},7^{-}$, so it looks like a $0^{-}$ band. We also note that
some of the states assigned to the two double-vibration $0^{-}$ bands in [7]
are indicated in [16] and in the TUNL table [15] to have isospin 1, rather
than the expected isospin 0. Greater clarity concerning the odd spin, negative
parity states of Neon-20 would be desirable.

It is of interest to relate the wormhole radius to the bipyramid and two-
cluster geometry more concretely using a simplified model. Suppose that
pointlike alpha-particles of mass $m_{\alpha}$ are located at

$\displaystyle\left(1,0,\frac{1}{4}z\right)\,,\left(-\frac{1}{2},\frac{\sqrt{3}}{2},\frac{1}{4}z\right)\,,\left(-\frac{1}{2},-\frac{\sqrt{3}}{2},\frac{1}{4}z\right)\,,\left(0,0,\sqrt{2}+\frac{1}{4}z\right)\,,$
$\displaystyle\left(0,0,-\sqrt{2}-z\right)\,.$ (22)

These form a tetrahedral four-alpha cluster centred at
$\left(0,0,\frac{1}{4}(\sqrt{2}+z)\right)$, accompanied by a single alpha-
particle at $\left(0,0,-(\sqrt{2}+z)\right)$. The cluster separation is
$s=\frac{5}{4}(\sqrt{2}+z)$. When $z=0$ these five alphas form a bipyramid of
double-tetrahedron shape.

Let us now introduce a spatial scale factor $c$ for this structure. The moment
of inertia about any axis through the origin, and orthogonal to the
$x_{3}$-axis is found to be

${\cal I}=\left(\frac{4}{5}s^{2}+3\right)m_{\alpha}c^{2}\,.$ (23)

This can be identified with the moment of inertia defined by the angular
kinetic energy in (2), ${\cal
I}=\frac{4}{5}m_{\alpha}(r^{2}+a^{2})=\frac{4}{5}(x^{2}+1)m_{\alpha}a^{2}$,
where we have approximated $\mu$ as $\frac{4}{5}m_{\alpha}$. Therefore

$\left(\frac{4}{5}s^{2}+3\right)c^{2}=\frac{4}{5}(x^{2}+1)a^{2}\,.$ (24)

When $z=0$ and $s=\frac{5}{4}\sqrt{2}$, then $x=0$, so the scale factor is
$c=\sqrt{\frac{8}{55}}\,a=2.07$ fm. The physical cluster separation is $R=sc$,
and a little further algebra gives

$R^{2}=\left(x^{2}+\frac{5}{11}\right)a^{2}\,.$ (25)

This relates the wormhole coordinate $x$ and wormhole radius $a$ to the
physical geometry of the clusters. Note that $R$ is not linearly related to
$x$.

The separation of the clusters when they merge into the bipyramid is
$R=\sqrt{\frac{5}{11}}\,a=3.65$ fm. This is the same as the separation of the
alpha and Oxygen-16 clusters calculated by Zhou et al. [26]. The ground state
wavefunctions extend from $x=0$ to approximately $x=0.7$. This corresponds to
a range of $R$-values, between $3.65$ fm and approximately $5.3$ fm. The
1-phonon wavefunctions have a peak at about $x=0.32$, corresponding to
$R=4.04$ fm, so the cluster separation hardly exceeds that of the ground state
wavefunctions. In the 2-phonon wavefunctions, the node is at $x=0.23$. The
cluster separation here is $R=3.86$ fm, so the bipyramid is only slightly
split. On the other hand, the peak of the 2-phonon wavefunctions is around
$x=0.54$. Here $R=4.68$ fm, which is a substantially larger cluster
separation, larger than the sum of the root mean square radii of an alpha-
particle ($1.63$ fm) and an Oxygen-16 nucleus (2.72 fm). This large cluster
separation in the 2-phonon, higher nodal states is in agreement with what is
found using a variety of microscopic cluster models, as illustrated in Fig.4.5
of ref.[14]. However, recall that our wavefunctions are only defined outside
the minimal cluster separation, $3.65$ fm.

##  5 Conclusions

We have proposed a model for the states of Neon-20 interpolating between the
rotational excitations of the ground-state bipyramid, and bound states of a
separated alpha-particle and a four-alpha cluster, i.e. an Oxygen-16 nucleus.
The model combines a radial coordinate $r$, related to the cluster separation
via a moment of inertia, and angular coordinates $(\theta,\phi)$ for the
spatial orientation of the axis joining the two clusters. $r=0$ at the
bipyramid.

Our significant novel idea is that the geometry of the configuration space is
a 3-d spatial wormhole – an Ellis–Bronnikov wormhole – with $SO(3)$ rotational
symmetry. The $SO(3)$ orbit at the wormhole throat at $r=0$ is not a point, as
in Euclidean space, but a 2-sphere of finite radius $a$, parametrising the
orientation of the bipyramid. On the wormhole configuration space, we have
added an attractive rotationally-symmetric, short-range potential, with rather
simple mathematical form. The quantum states of the model are interpreted as
rotational bands of the $n$-phonon excitations of the lowest-frequency
vibrational mode of the bipyramid, which tends to produce a $1+4$ cluster
split.

Our model differs from those based on Euclidean geometry, in that the cluster
separation is never less than what it is at the bipyramid. Also, the
centrifugal repulsive potential for states with non-zero angular momentum $l$
is not singular at the bipyramid, even though $r=0$. Another property, arising
from the choice of geometry and potential, is that each rotational band has an
angular momentum cut-off, and therefore only a finite number of bound states.
The threshold states at the top of the bands all have the same energy, and
form a sequence of increasing angular momentum $l$, up to $l=8$ for our choice
of dimensionless parameter, $m=9$. The observed Neon-20 spectrum gives support
for this picture. The radial wavefunctions of the threshold states have simple
analytic forms, obtained by solving an associated Legendre equation. This
allows us to establish, by the Sturm oscillation theorem, exactly how many
lower-lying, true bound states there are, but we have needed to find their
energies numerically.

We have found that the model’s energy spectrum matches the energies of
observed states of Neon-20 quite well, if we calibrate the energy threshold to
be at 15.19 MeV. The rotational bands include the well-established $0_{1}^{+}$
ground state band, the lowest $0^{-}$ band, and the $0^{+}_{4}$ band of broad,
“higher nodal” states, and it supports the assignment of observed $8^{+}$,
$7^{-}$ and $6^{+}$ states between 15 and 16 MeV to these bands as threshold
states. The calibration implies that the wormhole’s throat has radius $a=5.42$
fm, corresponding to the separation of the alpha-particle and Oxygen-16
cluster in the ground-state bipyramid being 3.65 fm.

Unlike the harmonic oscillator potential, our potential (9), shown in Fig.1,
is a finite well, and it would be interesting to analyse scattering states in
this potential, to determine the widths of resonant states along the lines of
ref.[13], and to consider alpha/Oxygen-16 fusion [8].

A challenge is to extend the model to include further vibrational excitations
of the bipyramid, and to understand theoretically all the vibrational
frequencies. The geometry of such an extended model will be higher-dimensional
and more complicated than a 3-d wormhole. It would also be interesting to
investigate if spatial wormholes can model the excitations of other nuclei
that split asymmetrically into a pair of clusters whose internal excitations
can be neglected.

## Acknowledgements

NSM thanks David Jenkins for helpful discussions.

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