# Study of proton-unbound states in [24]Al relevant for the [23]Mg$(p,\gamma)$
reaction in novae

E. C. Vyfers  Department of Physics and Astronomy, University of the Western
Cape, P/B X17, Bellville 7535, South Africa  V. Pesudo  Department of Physics
and Astronomy, University of the Western Cape, P/B X17, Bellville 7535, South
Africa  iThemba LABS, P.O. Box 722, Somerset West 7129, South Africa  S.
Triambak<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of the Western Cape, P/B X17, Bellville 7535, South Africa  P. Adsley  iThemba
LABS, P.O. Box 722, Somerset West 7129, South Africa  Cyclotron Institute and
Department of Physics & Astronomy, Texas A&M University, College Station,
Texas 77843, USA  B. A. Brown  Department of Physics and Astronomy and the
Facility for Rare Isotope Beams, Michigan State University, East Lansing,
Michigan 48824-1321, USA  H. Jivan  School of Physics, University of the
Witwatersrand, Johannesburg 2050, South Africa  M. Kamil  Department of
Physics and Astronomy, University of the Western Cape, P/B X17, Bellville
7535, South Africa  D. J. Marin-Lambarri  Department of Physics and Astronomy,
University of the Western Cape, P/B X17, Bellville 7535, South Africa  iThemba
LABS, P.O. Box 722, Somerset West 7129, South Africa  Instituto de Física,
Universidad Nacional Autónoma de México, Av. Universidad 3000, Mexico City
04510, Mexico  R. Neveling  iThemba LABS, P.O. Box 722, Somerset West 7129,
South Africa  J. C. Nzobadila Ondze  Department of Physics and Astronomy,
University of the Western Cape, P/B X17, Bellville 7535, South Africa  P.
Papka  Department of Physics, University of Stellenbosch, Private Bag X1, 7602
Matieland, Stellenbosch, South Africa  L. Pellegri  iThemba LABS, P.O. Box
722, Somerset West 7129, South Africa  School of Physics, University of the
Witwatersrand, Johannesburg 2050, South Africa  B. M. Rebeiro  Department of
Physics and Astronomy, University of the Western Cape, P/B X17, Bellville
7535, South Africa  B. Singh  Department of Physics and Astronomy, University
of the Western Cape, P/B X17, Bellville 7535, South Africa  F. D. Smit
iThemba LABS, P.O. Box 722, Somerset West 7129, South Africa  G. F. Steyn
iThemba LABS, P.O. Box 722, Somerset West 7129, South Africa

###### Abstract

Background: The nucleosynthesis of several proton-rich nuclei is determined by
radiative proton-capture reactions on unstable nuclei in nova explosions. One
such reaction is [23]Mg$(p,\gamma)$[24]Al, which links the NeNa and MgAl
cycles in oxygen-neon (ONe) novae.

Purpose: To extract [23]Mg$(p,\gamma)$[24]Al resonance strengths from a study
of proton-unbound states in [24]Al, produced via the 24Mg(3He,$t$) reaction.

Methods: A beam of [3]He2+ ions at 50.7 MeV was used to produce the states of
interest in [24]Al. Proton-triton angular correlations were measured with a
$K=600$ QDD magnetic spectrometer and a silicon detector array, located at
iThemba LABS, South Africa.

Results: We measured the excitation energies of the four lowest proton-unbound
states in 24Al and place lower-limits on $\Gamma_{p}/\Gamma$ values for these
four states. Together with USD-C shell-model calculations of partial gamma
widths, the experimental data are also used to determine resonance strengths
for the three lowest [23]Mg$(p,\gamma)$[24]Al resonances.

Conclusions: The energy of the dominant first [23]Mg$(p,\gamma)$ resonance is
determined to be $E_{r}=481.4\pm 1.1$ keV, with a resonance strength
$\omega\gamma=18\pm 6$ meV.

††preprint: APS/123-QED

##  I Introduction

The [23]Mg$(p,\gamma)$[24]Al reaction is a crucial link between the NeNa and
MgAl cycles in oxygen-neon (ONe) novae. Consequently, an accurate
determination of this stellar reaction rate is critical for an improved
understanding of elemental abundances up to Ca Kelly  _et al._ (2013); José
_et al._ (2004). At peak nova temperatures $(T_{9}=0.2-0.4)$, the dominant
contribution to this process is through the first resonance above the proton
threshold in 24Al (c.f. Fig. 1), at $E_{\mathrm{x}}=2345.1\pm 1.4$ keV Basunia
and Chakraborty (2022); Lotay et al. (2008).

Direct measurements of the 23Mg($p,\gamma)$ reaction rate are challenging,
primarily because of small cross sections at astrophysically relevant energies
and the difficulty in producing intense 23Mg radioactive ion beams with
sufficient purity. As a result, there have been several attempts to determine
the 23Mg($p,\gamma)$ reaction rate indirectly Wallace and Woosley (1981);
Herndl et al. (1998); Wiescher  _et al._ (1986); Kubono et al. (1995); Visser
et al. (2007), using theoretical estimates of the partial proton and gamma
widths of relevant states in [24]Al. For isolated narrow resonances, the
resonant contribution for each level can be obtained from its resonance
strength

$\omega\gamma=\frac{(2J_{r}+1)}{8}\frac{\Gamma_{p}\Gamma_{\gamma}}{\Gamma},$
(1)

where $J_{r}$ is the spin of the resonance, $\Gamma_{p}$ and $\Gamma_{\gamma}$
are the partial proton and gamma widths of the resonant state, and
$\Gamma=\Gamma_{p}+\Gamma_{\gamma}$ is its total width.

Figure 1: [23]Mg$(p,\gamma)$[24]Al resonances discussed in this work. Nominal
energies (in keV) and spin-parity assignments are from Ref. Basunia and
Chakraborty (2022).

To date, the only direct [23]Mg$(p,\gamma)$ measurement was performed by
Erikson et al. Erikson et al. (2010) at the TRIUMF-ISAC facility, where a
radioactive 23Mg ion beam was incident on a hydrogen gas target at the DRAGON
recoil spectrometer Hutcheon  _et al._ (2003). Prompt $\gamma$ rays were
detected with an array of bismuth germanate (BGO) scintillators surrounding
the target, while the recoils were identified using a combination of
ionization chamber and microchannel plate (MCP) detectors located at the focal
plane of the spectrometer. However, the experiment was severely affected by a
dominant time-varying [23]Na contamination in the beam Erikson et al. (2010).
This made it difficult to position the resonance at the center of the gas-
target and obtain its $\omega\gamma$ precisely. Nonetheless, the energy of the
lowest resonance and its corresponding resonance strength were determined to
be $E_{r}=485.7_{-1.8}^{+1.3}$ keV and $\omega\gamma=37.8_{-15.4}^{+20.5}$
meV, respectively Erikson et al. (2010).

Considering the above, in this work we report complementary measurements of
proton-unbound states in [24]Al, produced via the [24]Mg$(^{3}{\rm
He},t)$[24]Al reaction.

##  II APPARATUS

The experiment was performed at the iThemba LABS cyclotron facility, where a
10 pnA, 50.7-MeV dispersion-matched 3He2+ beam was bombarded on an $\approx
300~{}\mu\text{g}/\text{cm}^{2}$-thick MgF2 target on a carbon backing. The
target was located in the scattering chamber of the $K=600$ QDD magnetic
spectrometer, which was configured in $0^{\circ}$ mode, with the beam stop
located inside the first dipole magnet Neveling  _et al._ (2011). Momentum-
analyzed reaction ejectiles were detected in a $1/4^{\prime\prime}$-thick
plastic scintillator at the spectrometer focal plane, after passing through
two vertical drift chambers (VDCs). The VDCs determined the horizontal and
vertical positions of the ejectiles crossing the focal plane, while the
plastic scintillator was used for particle identification (PID) purposes, and
to generate the data acquisition (DAQ) trigger. The PID plots were generated
using the relative time difference between the cyclotron RF and DAQ trigger
signals, together with the energy deposited in the scintillator. Fig. 2 shows
the PID plot obtained for this experiment, with the triton group explicitly
highlighted.

Figure 2: PID spectrum from energy deposited in the scintillator vs relative
time of flight (tof), as discussed in the text.

An array of five 400-$\mu$m-thick MMM-type Double-sided Silicon Strip
Detectors (DSSDs) called CAKE Adsley  _et al._ (2017) was used to detect
protons from unbound states in [24]Al, in coincidence with triton focal-plane
events. These detectors were placed upstream of the target location in a
backward-facing lampshade configuration and covered $\approx 25\%$ of the
total solid-angle. Each DSSD comprised 16 ring channels and 8 sector channels.
The rings had an angular range of $\theta_{lab}=115\degree-165\degree$, for
the array.

##  III DATA ANALYSIS

###  III.1 Triton singles

The [24]Mg$(^{3}{{\rm He}},t)$ singles spectrum, obtained with the appropriate
triton PID gate, is shown in Fig. 3. Because the [24]Mg$(^{3}{{\rm He}},t)$
reaction $Q$-value is $\sim 10$ MeV lower than [25,26]Mg$(^{3}{{\rm He}},t)$
reactions, one can safely assume negligible contributions from other Mg
isotopes in this spectrum. However other contaminant peaks from carbon and
oxygen in the target foil cannot be ruled out. To determine such
contributions, we took additional $(\isotope[3]{He},t)$ data with a Li2CO3
target, whose spectrum is also shown in Fig. 3. Three contaminant peaks are
evident in the region of interest (ROI). We associate these peaks to states in
[16]F, produced via the [16]O$(^{3}{{\rm He}},t)$ reaction.

All triton peaks were fit using a lineshape function defined by a Gaussian
distribution convolved with a low-energy exponential tail Triambak  _et al._
(2006); Kamil  _et al._ (2022). Peak centroids $(\mu_{t})$ and areas $(A_{t})$
were recorded for later analysis. We next used known excitation energies in
24Al Basunia and Chakraborty (2022) to perform an in-situ energy calibration
of the focal-plane spectrum. This was done using a quadratic fit of the form

$E_{\rm fit}(i)=a_{0}+a_{1}\mu_{t}(i)+a_{2}\mu_{t}(i)^{2}.$ (2)

For this procedure, we used the Nuclear Data Sheets Basunia and Chakraborty
(2022) to identify calibration peaks within the range $425\leq
E_{\mathrm{x}}\leq 3328$ keV. The 1.5 and 1.6 MeV states were not included in
the analysis, because of possible unresolved doublets in the region Basunia
and Chakraborty (2022); Visser et al. (2007) and the presence of the
contaminant [16]F ground state peak.

In the first trial, the above calibration yielded a poor fit, with reduced
chi-squared value, $\chi^{2}/\nu=5.1$. Further analysis showed that more
reasonable agreement ($\chi^{2}/\nu=1.2$), with a $p$ value of 31%, was _only_
obtained if we excluded two other points (the 500 and 1261 keV states) from
the calibration. The residuals from these two fits are shown in Fig. 4. We use
the results of this regression analysis to independently determine the
excitation energies of observed states in [24]Al that are relevant for the
[23]Mg$(p,\gamma)$ reaction rate. These are listed in Table 1. For the lowest
[23]Mg$(p,\gamma)$ resonance, our calibration yields $E_{\mathrm{x}}=2345.5\pm
1.1$ keV, which is in excellent agreement with a previous $\gamma$-ray
measurement Lotay et al. (2008); Basunia and Chakraborty (2022). Our data do
not show explicit evidence of the recently reported state at 2605 keV Zegers
et al. (2008).

The contaminant 721 keV peak from [16]F does not significantly affect our
energy determination for 2345 keV state. However it is important that this
contamination is corrected for, when the area of the 2345 keV peak is used
determine its corresponding proton branching-ratio. Such a correction is
trivial for this case, thanks to the additional high statistics triton peak
associated with the 424 keV state in [16]F. Since this peak appears in both
spectra (c.f. Fig. 3 around channel number 390), the correction factor was
determined from the relative areas of both the contaminant peaks.

Figure 3: Triton focal-plane spectra obtained for the MgF2 and Li2CO3 targets. Nominal 24Al energies are labeled (in keV), with the state corresponding to the first [23]Mg$(p,\gamma)$ resonance marked in blue.  Figure 4: Residuals for the quadratic fit in Eq. (2), obtained after including (red circles) and excluding (green squares) the 500 and 1261 keV states in the regression analysis. Table 1: Extracted energies for the four lowest [23]Mg$(p,\gamma)$ resonances. Ref. Basunia and Chakraborty (2022) | This work
---|---
$E_{\mathrm{x}}$ (keV) |  $E_{\mathrm{x}}$ (keV) |  $E_{r}$ (keV)
$2345.1\pm 1.4$ | $2345.5\pm 1.1$ | $481.4\pm 1.1$
$2523\pm 6$ | $2517\pm 2$ | $653\pm 2$
$2805\pm 10$ | $2790\pm 2$ | $925\pm 2$
$2878\pm 6$ | $2871\pm 2$ | $1007\pm 2$

###  III.2 Triton-proton coincidences

As mentioned previously, charged-particle-like events from
[24]Mg$(\isotope[3]{He},t)$[24]Al${}^{*}(p)$[23]Mg were detected using the
DSSD array described in Ref. Adsley  _et al._ (2017). The energy and timing
signals for the array were digitized using CAEN V785 ADCs and V1190A TDCs,
respectively. Triton-proton ($t$-$p$) coincidences were selected by gating on
the prompt TDC timing peak, which was around 40 ns wide and further imposing
an energy condition on the sector and ring channels (so that they are within
300 keV). The DSSDs were energy calibrated using a [226]Ra $\alpha$-source at
a later time, when the beam was off.

Figure 5: $t$-$p$ coincidence data, with proton groups from states in 24Al
highlighted by diagonal bands. The upper ($p_{0}$) and lower ($p_{1}$) bands
are from transitions to the ground and first excited states in 23Mg,
respectively. The ellipse around the $p_{0}$ proton group is for the lowest
23Mg($p$,$\gamma$)24Al resonance.

Fig. 5 shows the coincident $t$-$p$ energy loci obtained from this experiment.
Since each ring of the DSSD array corresponds to a polar angle $(\theta)$ with
respect to the beam-axis, the proton branching ratios $B_{p}(\theta)$ can be
determined by gating on the relevant proton groups in Fig. 5, and making use
of the formula

$B_{p}(\theta)=\frac{N_{tp}(\theta)}{N_{t}}\frac{1}{\epsilon(\theta)}.$ (3)

Here $N_{tp}(\theta)$ are the registered $t$-$p$ coincidences for a particular
ring, $N_{t}$ are the corresponding triton singles and $\epsilon(\theta)$ is
the proton detection efficiency of the ring. For states with definite spin and
parity, these yields are expected to follow the simple angular distribution
Noé  _et al._ (1974)

$B_{p}(\theta_{\mathrm{c.m.}})=\sum_{k=\text{even}}A_{k}P_{k}(\cos\theta_{\mathrm{c.m.}}),$
(4)

whose integrated yield gives the total proton branching ratio for each state

$\frac{\Gamma_{p}}{\Gamma}=\int_{-1}^{1}B_{p}(\theta_{\mathrm{c.m.}})d(\cos\theta_{\mathrm{c.m.}})=A_{0}.$
(5)

To perform the above analysis, we first determined $\epsilon(\theta)$ for each
active ring of the DSSD array using GEANT4 Monte Carlo simulations. These
values were used to obtain an initial set of $B_{p}(\theta)$ for the first
four [23]Mg$(p,\gamma)$[24]Al resonances. Next, a similar analysis was also
performed on the [16]O$(^{3}{{\rm He}},t)\isotope[16]{F}^{*}(p)$ data,
obtained with the Li2CO3 target. As [16]F is unbound and $\Gamma_{p}/\Gamma=1$
for its observed states NND ; Lee  _et al._ (2007), this important procedure
determined an effective normalization111A weighted mean of the results from
the 424 and 721 keV-state data yielded a normalization factor $\kappa=1.51\pm
0.04$. to correct the [24]Al results, which had previously only relied on
simulated efficiencies. The renormalized $t$-$p$ distributions for the first
four proton-unbound states in [24]Al are shown in Fig. 6.222As the coincidence
data for the 2345 and 2523 keV states were statistics limited, for these cases
the angular yield at each bin was obtained by combining data from 4 adjacent
strips in the DSSDs., with the converged fit results for $A_{0}$ and their
associated $\pm 1\sigma$ uncertainties shown in the figure insets. If one
assumes Gaussian distributions for these values, it is evident that their
coverage probabilities also include disallowed regions of parameter space,
with $\Gamma_{p}/\Gamma>1$. Consequently, we use the fit-results in Fig. 6 to
present lower-limits on $\Gamma_{p}/\Gamma$ values for all four states. These
are listed in Table 2.

Figure 6: Legendre polynomial fits to the measured proton-triton angular
distributions. The states in 24Al are labeled by their nominal energies a:
$E_{\rm x}=2345$ keV, b: $E_{\rm x}=2523$ keV, c: $E_{\rm x}=2805$ keV and d:
$E_{\rm x}=2878$ keV.

##  IV RESULTS

Table 2: Parameters for key 23Mg($p$,$\gamma$)24Al resonance reactions.

Assigned $J^{\pi}$ | $E_{r}$ | $\Gamma_{p}/\Gamma~{}^{\text{a}}$ | ${\Gamma_{\gamma}}^{\text{b}}$ | $\omega\gamma$
---|---|---|---|---
(from Ref. Herndl et al. (1998)) | (keV) |  | (meV) | (meV)
$3^{+}$ | $481.4\pm 1.1$ | $\geq 0.64$ | 23 | $18\pm 6$
$4^{+}$ | $653\pm 2$ | $\geq 0.55$ | 39 | $34\pm 11$
$2^{+}$ | $925\pm 2$ | $\geq 0.87$ | 71 | $62\pm 18$
$3^{+}$ | $1007\pm 2$ | $\geq 0.99^{\text{c}}$ | 8 | …

${}^{\text{a}}$Unless specified otherwise, lower limits are quoted at the 90%
confidence level (CL).

${}^{\text{b}}$Evaluated using a USD-C Magilligan and Brown (2020) shell model
calculation.

${}^{\text{c}}$ Quoted at the 99.9% CL.

Although we are unable to quote $\Gamma_{p}/\Gamma$ values with sufficient
precision, it is still possible to extract meaningful resonance strengths from
our data, particularly for the first three states.333Since
$\Gamma_{p}/\Gamma\approx 1$ for the 1007 keV resonance, its $\omega\gamma$
value is given by its partial gamma width. For this part of the analysis we
relied on shell-model-calculated partial gamma widths ($\Gamma_{\gamma}$),
obtained using a recently developed USD-C isospin non-conserving Hamiltonian
Magilligan and Brown (2020). Next, we used Monte Carlo simulations to generate
Gaussianly distributed random deviates around the measured central $A_{0}$
values. Each simulated variable $A_{0}^{\text{sim}}$ was used to calculate a
corresponding $\chi^{2}$ with respect to the measured data points in Fig. 6.
Simultaneously, the simulations also determined associated $\omega\gamma$
values, making use of the relation

$\left(\frac{\Gamma_{p}}{\Gamma_{\gamma}}\right)_{\text{sim}}=\frac{A_{0}^{\text{sim}}}{1-A_{0}^{\text{sim}}}.$
(6)

For each state, the simulated $\omega\gamma$ that yielded the minimum
$\chi^{2}$ with respect to our data444Events that yielded
$\Gamma_{p}/\Gamma_{\gamma}<0$ were rejected from the analysis. was accepted
as the true value, with its $\pm 68\%$ CL statistical uncertainties obtained
from the values that yield $\chi^{2}=\chi^{2}_{\text{min}}+1$. Clearly, since
$\omega\gamma$ is proportional to $\Gamma_{\gamma}$, a more realistic
uncertainty on the former should include a contribution from the shell model
calculation. In order to estimate these uncertainties, we performed similar
calculations of level lifetimes in the mirror [24]Na nucleus, which were then
compared with experimentally measured values Basunia and Chakraborty (2022).
This comparison is shown in Table. 3, based on which we conservatively assign
a 30% relative uncertainty to each calculated $\Gamma_{\gamma}$ value for
[24]Al. These systematic uncertainties from theory are the dominant
contributions to the final uncertainties in our extracted $\omega\gamma$
values, which are listed in Table 2.

Table 3: Half-lives of states in the mirror [24]Na. A comparison between
theory and experiment

Energy${}^{\text{a}}$ | $J_{n}^{\pi}$ |  $T_{1/2}$ (USD-C)${}^{\text{b}}$ |  $T_{1/2}$ (expt)${}^{\text{a}}$
---|---|---|---
(keV) |  | (fs) | (fs)
$563$ | $2_{1}^{+}$ | $37000$ | $35000\pm 500$
$1341$ | $2_{2}^{+}$ | $350^{\text{c}}$ | $50\pm 13$
$1345$ | $3_{1}^{+}$ | $30$ | $26\pm 8$
$1347$ | $1_{2}^{+}$ | $2200$ | $4400\pm 300$
$1514$ | $5_{1}^{+}$ | $28$ | $27\pm 6$
$1846$ | $2_{3}^{+}$ | $310$ | $171\pm 35$
$1886$ | $3_{2}^{+}$ | $18$ | $26\pm 6$
$2513$ | $3_{3}^{+}$ | $23$ | $10\pm 5$
$2563$ | $4_{2}^{+}$ | $10$ | $<17$
$2904$ | $3_{4}^{+}$ | 38 | $35\pm 8$
$2978$ | $2_{4}^{+}$ | 6.5 | $<17$
$3216$ | $4_{3}^{+}$ | 12 | $16\pm 6$

${}^{\text{a}}$ From Ref. Basunia and Chakraborty (2022).

${}^{\text{b}}$ Effective charges and $g$-factors for the $E2$ and $M1$
operators were taken from the global sd-shell fit results in Ref. Richter  _et
al._ (2008).

${}^{\text{c}}$ This state makes a transition to the $1_{1}^{+}$ level at 472
keV Basunia and Chakraborty (2022) with a 95% branch, and is predicted to be
$M1$-dominated. When corrected for the difference between the theory-predicted
(0.53 MeV) and measured (0.87 MeV) transition energy, its calculated half-life
is 79 fs.

The final values from Table 2 were used to evaluate [23]Mg$(p,\gamma)$
reaction rates with the RatesMC Monte Carlo code described in Refs. Longland
_et al._ (2010); Iliadis  _et al._ (2010). The non-resonant (direct capture)
contribution was determined from a polynomial expansion of the $S$-factor

$S(E)\approx
S(0)+S^{\prime}(0)E+\frac{1}{2}S^{\prime\prime}(0)E^{2}~{}\mathrm{keV~{}b},$
(7)

whose coefficients were taken from Ref. Herndl et al. (1998). The results for
individual resonances identified in this work are shown in Fig. 7, with the
fractional contribution of each component shown in Fig. 8.

Figure 7: Median 23Mg($p$,$\gamma$)24Al reaction rates obtained from this
work. Direct capture (DC) and resonant contributions are shown separately,
with the latter evaluated using the results listed in Table 2. Figure 8:
Fractional contributions of direct capture (DC) and low-lying resonances to
the total 23Mg($p$,$\gamma$)24Al reaction rate. The uncertainty bands are
evaluated using the Monte Carlo prescription described in Ref. Longland  _et
al._ (2010); Iliadis  _et al._ (2010).

##  V Summary

To summarize, this work reports a study of states in [24]Al, which are
important to determine the [23]Mg$(p,\gamma)$[24]Al nuclear reaction rate in
ONe novae. Together with shell-model calculations of partial gamma widths, our
experimental data are used to determine resonance strengths for the first
three [23]Mg$(p,\gamma)$ resonances. Our measured excitation energy for the
dominant lowest resonance is in excellent agreement with a previous
$\gamma$-ray measurement Lotay et al. (2008). Making use of the most recent
atomic mass data compilation Wang  _et al._ (2021), this translates to a
resonance energy $E_{r}=481.4\pm 1.1$ keV. Our extracted $\omega\gamma$ value
for this resonance is $18\pm 6$ meV. These values may be compared with the
lower-precision direct-measurement results of Erikson et al. Erikson et al.
(2010), $E_{r}=485.7_{-1.8}^{+1.3}$ keV and
$\omega\gamma=37.8_{-15.4}^{+20.5}$ meV, which relied on extensive Monte Carlo
simulations and a joint likelihood analysis to determine their median and $\pm
68\%$ CL values. The quoted uncertainties in their work were the result of a
highly asymmetric joint probability density function (PDF), which showed
significant tailing and a double-humped structure for its projected
$\omega\gamma$ probability distribution. Although our extracted $\omega\gamma$
value is in reasonable agreement with the result reported by Erikson it et
al., it appears to be more consistent with the left peaked-structure of their
PDF for $\omega\gamma$ (c.f. Fig. 16 in Ref. Erikson et al. (2010)), which is
centered around $\sim 23$ meV instead.

###### Acknowledgements.

We are thankful to Alejandro García for insightful discussions. This work was
partially funded by the National Research Foundation (NRF), South Africa under
Grant No. 85100. B.A.B acknowledges funding support from the National Science
Foundation, under Grant. No. PHY-2110365. E.C.V thanks the NRF-funded
MaNuS/MatSci program at the University of the Western Cape for financial
support during the course of his M.Sc.

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