# Nuclear charge radii of Na isotopes: A tale of two theories

B. Ohayon  Corresponding author<EMAIL_ADDRESS>Institute for Particle
Physics and Astrophysics, ETH Zürich, CH-8093 Zürich, Switzerland  R.F. Garcia
Ruiz  Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,
USA  Z. H. Sun  Physics Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  Department of Physics and Astronomy, University of
Tennessee, Knoxville, Tennessee 37996, USA  G. Hagen  Physics Division, Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  Department of
Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996,
USA  T. Papenbrock  Department of Physics and Astronomy, University of
Tennessee, Knoxville, Tennessee 37996, USA  Physics Division, Oak Ridge
National Laboratory, Oak Ridge, Tennessee 37831, USA  B. K. Sahoo
<EMAIL_ADDRESS>Atomic, Molecular and Optical Physics Division, Physical
Research Laboratory, Navrangpura, Ahmedabad 380058, Gujarat, India

###### Abstract

The accuracy of atomic theory calculations limits the extraction of nuclear
charge radii from isotope shift measurements of odd-proton nuclei. For Na
isotopes, though precise spectroscopic measurements have existed since more
than half a century, calculations by different methods offer a wide range of
values. Here, we present accurate atomic calculations to reliably extract the
Na charge radii. By combining experimental matter radii with nuclear coupled-
cluster calculations based on nucleon-nucleon and three-nucleon forces, we
constrain the parameters obtained from the atomic calculations. Therefore,
this study guides atomic theory and highlights the importance of using
accurate atomic and nuclear computations in our understanding of the size of
light nuclei.

Introduction.— The understanding of the evolution of the size of a nucleus
with extreme numbers of protons and neutrons is a challenge for microscopic
nuclear theory Caurier  _et al._ (2001); Lapoux  _et al._ (2016); Lonardoni
_et al._ (2018); Ekström  _et al._ (2015); Somà  _et al._ (2020); Novario  _et
al._ (2020); Somà  _et al._ (2021). In recent years, simultaneous developments
in experimental techniques, as well as the atomic and nuclear theory, have
provided great advances in our understanding of the nuclear size away from
stability Miller  _et al._ (2019); Garcia Ruiz  _et al._ (2016); de Groote
_et al._ (2020); Koszorús  _et al._ (2021). Accurate nuclear charge radii
calculations with quantifiable uncertainties are now becoming available for
the light and medium mass nuclei, enabling detailed comparisons with
experimental data Lapoux  _et al._ (2016); Lonardoni  _et al._ (2018); Novario
_et al._ (2020); Heylen  _et al._ (2021); Koszorús  _et al._ (2021). Light
nuclear systems with mass number $A<40$, whose properties can now be assessed
by different ab initio many-body methods Lu  _et al._ (2013); Lonardoni  _et
al._ (2018); Lapoux  _et al._ (2016); Maaß  _et al._ (2019); Novario  _et al._
(2020); Somà  _et al._ (2020), are a testing ground for nuclear theory.
However, nuclear charge radii measurements of these systems are scarce, with
most of the available data obtained from spectroscopic measurements of atomic
isotope shifts (ISs) Yordanov  _et al._ (2012); Heylen  _et al._ (2021);
Koszorús  _et al._ (2021), whose uncertainties are dominated by calculated
atomic parameters. The reason for this is that in light elements, the nuclear
volume contributions to the ISs, which allows a measure of the charge radius,
decrease exponentially with atomic number, leaving the mass shifts (MSs),
strongly affected by electronic correlations Heylen  _et al._ (2021), to
dominate. Hence, the accuracy of atomic theory is one of the current main
limitations for extending our knowledge of the nuclear size in this frontier
region of the nuclear chart.

IS measurements cannot be compared directly with atomic calculations. Where at
least three stable isotopes exist, independent charge radii measurements
determined by muonic x-ray transition energies or elastic electron scattering
Fricke  _et al._ (1995) can be used in combination with IS measurements to
benchmark atomic calculations. However, this procedure cannot be applied to
most odd-proton elements as they have only one stable isotope. The isotopes of
Na ($Z=11$) provide a distinct example of how joint developments in both
atomic and nuclear theory are critical to guide our understanding of the
evolution of the nuclear size. The IS measurements for these isotopes have
existed since more than four decades Huber  _et al._ (1978). However, it has
been a major challenge to perform accurate atomic calculations required to
extract the nuclear charge radii values from the experimental data. For this
reason, Na isotopes are some of the rare cases where matter radii are known
with higher precision than charge radii Suzuki  _et al._ (1998, 2014).

In this Letter, we report on new accurate relativistic atomic calculations,
with quantifiable uncertainties, that enable the extraction of charge radii
values for the Na isotopic chain. The available experimental data on matter
radii alongside the recent progress of nuclear theory are used to establish
constraints on atomic parameters, providing guidance to the developments of
atomic many-body theory.

Isotope shifts.— Changes in the root-mean-squared nuclear charge radii,
$\delta r_{c}^{2}$, can be inferred from measurements of ISs, $\delta\nu$,
using the linear expression King (2013)

$\delta\nu^{A^{\prime},A}=K(1/M_{A^{\prime}}-1/M_{A})+F(\delta
r_{c}^{2})^{A^{\prime},A},$ (1)

where $K$ and $F$ are the transition-dependent MS and field-shift (FS)
constants, respectively, which are to a good approximation isotope-
independent, and $M_{A}$ is the nuclear mass for atomic number $A$. Higher
order corrections are smaller than the current experimental and theoretical
uncertainties for the cases studied in this work. The MS constant is usually
separated into $K=K_{\text{NMS}}+K_{\text{SMS}}$, with $K_{\text{NMS}}$ and
$K_{\text{SMS}}$ the normal mass shift (NMS) and the specific mass shift (SMS)
constants, respectively. For atomic transitions in light systems,
$K_{\text{NMS}}$ can be estimated with a few per-mil precision by scaling the
experimental excitation energy ($E^{ex}$) as $K_{\text{NMS}}=E^{ex}m_{e}$,
where $m_{e}$ is the electron mass Bethe and Salpeter (1977); Tupitsyn  _et
al._ (2003); Kozlov and Korol (2007); Baugh  _et al._ (1998); Kaufman and
Martin (1991); Li  _et al._ (2012); Kozhedub  _et al._ (2010). However,
accurate calculations of $K_{\text{SMS}}$ are extremely challenging for
systems with more than six electrons Bubin  _et al._ (2013); Puchalski  _et
al._ (2015); Zubova  _et al._ (2016); Maaß  _et al._ (2019); Yerokhin  _et
al._ (2020); Hornyák  _et al._ (2020); Puchalski  _et al._ (2021). When given,
the reported $K_{\text{SMS}}$ uncertainties in many-electron systems are
typically larger than $10\%$ Cheal  _et al._ (2012); Ferrer  _et al._ (2014);
Heylen  _et al._ (2016); Zuhrianda (2017); Sahoo  _et al._ (2020); Koszorús
_et al._ (2021); Heylen  _et al._ (2021), thus limiting the effectiveness of
comparisons between experimental charge radii with those calculated using
nuclear theory.

Atomic theory.— We employ the relativistic coupled-cluster (RCC) theory, which
is well suited for the accurate evaluation of correlations in many-electron
atomic systems Lindgren and Morrison (1982). Traditionally, two procedures
have been used to carry out these calculations; finite-field (FF) Berengut
_et al._ (2003); Tupitsyn  _et al._ (2003); Korol and Kozlov (2007); Roy and
Majumder (2015) and expectation-value-evaluation (EVE) Safronova and Johnson
(2001); Sahoo (2010), using many-body methods including the RCC theory Sahoo
(2010); Roy and Majumder (2015). These proved to have several limitations as
detailed in Refs. Vernon  _et al._ (2020); Sahoo and Ohayon (2021). The
analytic response-based RCC (AR-RCC) theory was developed to circumvent these
problems.

Recently, we have used the AR-RCC theory with singles and doubles
approximation (AR-RCCSD) to estimate the IS constants for transitions in the
indium atom Vernon  _et al._ (2020); Sahoo  _et al._ (2020), but calculations
for Na and Mg+ require further development to precisely estimate these
constants by including higher-order electron correlations. Thus we extend our
AR-RCC theory to account for full triples excitations (AR-RCCSDT method). This
method was recently bench-marked by performing extensive calculations on
lithium-like systems, for which more accurate methods are available, and was
found to be the most reliable over the calculations carried out adopting the
FF and EVE approaches Sahoo and Ohayon (2021). To validate the method for
many-electron systems, we compare the calculated and measured values in Mg+,
which has a similar electronic structure to Na. For Na, we develop a hybrid
method based on a comparison of the neutron skin thickness deduced from the
matter-radii measurements and those calculated by applying nuclear many-body
theory.

Table 1: Comparison of IS constants, $F$ in (MHz/fm${}^{2})$, and $K_{\text{SMS}}$ in (GHz amu), for the D1 and D2 lines from the AR-RCCSD/T methods with literature values. The experimental $F$ in Mg+ is taken from a semi-empirical method Yordanov  _et al._ (2012). The experimental $K_{\text{SMS}}$ in Mg+ is derived from the results of Batteiger  _et al._ (2009) by subtracting the NMS and FS contributions.  | D1 | D2
---|---|---
$F$ | Na | Mg+ | Na | Mg+
AR-RCCSD | $-38.9$ | $-126.1$ | $-38.9$ | $-126.1$
AR-RCCSDT | $-39.2(3)$ | $-126.3(7)$ | $-39.2(3)$ | $-126.3(7)$
Ref. Tupitsyn  _et al._ (2003) | $-36.45$ | $-123.2$ |  |
Ref. Berengut  _et al._ (2003) | $-39$ |  | $-39$ | $-127$
Ref. Korol and Kozlov (2007) | $-33$ | $-127$ | $-33$ | $-127$
Ref. Safronova and Johnson (2001) | $-38.42$ | $-125.81$ | $-38.43$ | $-125.82$
Ref. Sahoo (2010) | $-38.76$ | $-126.22$ | $-38.80$ | $-126.32$
Exp. Yordanov  _et al._ (2012) |  | $-127(12)$ |  |
K${}_{\text{SMS}}$ | Na | Mg+ | Na | Mg+
AR-RCCSD | $132$ | $404$ | $132$ | $404$
AR-RCCSDT | $109(3)$ | $374(7)$ | $109(3)$ | $374(7)$
$\Delta$NNLOGO+$r_{m}$ | $105.3(1.3)$ |  |  |
Ref. Tupitsyn  _et al._ (2003) | $~{}98.5$ | $406.1$ |  |
Ref. Berengut  _et al._ (2003) | $109(24)$ | $379(12)$ | $108(24)$ | $373(6)$
Ref. Korol and Kozlov (2007) | $116$ | 378 | $116$ | 378
Ref. Roy and Majumder (2015) |  | $365$ |  | $366$
Ref. Safronova and Johnson (2001) | $97$ | 362 | $97$ | 361
Ref. Sahoo (2010) | $114.4$ | 398.8 | $112.3$ | 389.9
Exp. Batteiger  _et al._ (2009) |  | $369.3(3)$ |  | $367.7(3)$

Nuclear theory.— Accurate ab initio calculations of charge radii are
particularly challenging for open-shell nuclei. Shell-model calculations based
on non-perturbative effective interactions derived from methods like valence-
space in-medium similarity renormalization group Hergert  _et al._ (2016);
Stroberg  _et al._ (2017); Miyagi  _et al._ (2020) and shell-model based
coupled-cluster Sun  _et al._ (2018), are complicated for nuclei where valence
spaces consist of more than one major shell. Alternatively, methods based on
single reference states that explicitly break symmetries may provide a
conceptually simpler approach Somà  _et al._ (2020); Novario  _et al._ (2020);
Somà  _et al._ (2021). However, such approaches carry uncertainties from the
lack of symmetry restoration that are somewhat difficult to quantify. In this
work, we follow Ref. Novario  _et al._ (2020) and employ single-reference
coupled-cluster (CC) theory.

Table 2:  Ground states properties of the Na isotopic chain. The first two columns give the mass and neutron numbers. Columns 3 and 4 give the experimental charge radii differences, and absolute values respectively, for $K_{D1}=388(3)$ and $F_{D1}=-39.2(3)$, with uncorrelated uncertainties in parenthesis, and correlated ones from our estimation of $K$ are in square brackets. Columns 5-7 give the charge, proton, neutron radii and skin, respectively, while the neutron skin is given in the last column, as calculated via $\Delta$NNLO(450).  $A$ | $N$ |  $(\delta r_{c}^{2})^{23,A}\,$fm2 |  $r_{c}\,$fm |  $r_{c}^{Th}\,$fm |  $r_{p}^{Th}\,$fm |  $r_{n}^{Th}\,$fm |  $r_{np}^{Th}\,$fm
---|---|---|---|---|---|---|---
19 |  8 |  |  | 2.99(3) | 2.87(3) | 2.58(3) | -0.29(3)
20 |  9 | -0.60(9)[52] | 2.891(16)[92] |  |  |  |
21 | 10 | -0.13(5)[33] | 2.972 (8)[57] | 3.03(3) | 2.92(3) | 2.82(3) | -0.09(3)
22 | 11 | -0.16(2)[16] | 2.967 (3)[27] |  |  |  |
23 | 12 |  | 2.9935(38) | 3.01(3) | 2.90(3) | 2.92(3) | 0.02(3)
24 | 13 | -0.02(4)[15] | 2.990 (7)[25] |  |  |  |
25 | 14 | 0.12(5)[28] | 3.013 (8)[47] | 2.97(3) | 2.87(3) | 2.98(3) | 0.11(3)
26 | 15 | 0.34(2)[41] | 3.049 (4)[68] |  |  |  |
27 | 16 | 0.60(5)[52] | 3.091 (8)[86] | 3.01(3) | 2.92(3) | 3.11(3) | 0.19(3)
28 | 17 | 0.89(7)[63] |  3.139(11)[101] |  |  |  |
29 | 18 | 1.39(10)[73] |  3.217(15)[114] | 3.03(3) | 2.94(3) | 3.25(3) | 0.31(3)
30 | 19 | 1.68(15)[82] |  3.262(23)[128] |  |  |  |
31 | 20 | 2.19(9)[91] |  3.339(13)[137] | 3.06(3) | 2.96(3) | 3.33(3) | 0.37(3)
33 | 21 |  |  | 3.13(3) | 3.04(3) | 3.49(3) | 0.45(3)

For the CC calculations we employ the recently developed $\Delta$NNLOGO
interaction Jiang  _et al._ (2020) with a momentum cutoff of 450 MeV. These
calculations are performed in the singles and doubles (CCSD) approximation
Kümmel  _et al._ (1978); Bartlett and Musiał (2007); Hagen  _et al._ (2014)
starting from an axially symmetric Hartree-Fock (HF) reference state. Parity,
particle number, and the projection of total angular momentum onto the
symmetry axis are conserved quantities. The HF calculations are performed in a
harmonic-oscillator basis consisting of 15 major oscillator shells
($N_{\rm{max}}=14$), with a spacing of $\hbar\omega=16$ MeV. The three-body
interaction has an additional energy cut given by $E_{\rm{3max}}=16$ MeV,
which is sufficiently large for the nuclei we compute. Once the HF solution is
converged, a more accurate density matrix is computed using second-order many-
body perturbation theory Tichai  _et al._ (2019). Diagonalization of this
density matrix then yields the natural orbital basis Tichai  _et al._ (2019);
Novario  _et al._ (2020); Hoppe  _et al._ (2021). Following Refs. Hoppe  _et
al._ (2021); Heinz  _et al._ (2021) the normal-ordered Hamiltonian in the two-
body approximation Hagen  _et al._ (2007); Roth  _et al._ (2012) is then
truncated to a smaller model space ($N_{\rm{max}}^{\rm{nat}}=12$) according to
the occupation numbers of the natural orbits. The proton and neutron radii are
calculated as ground-state expectation values, and charge radii include
corrections from the Darwin-Foldy term and spin-orbit contributions (see
Supplemental Material for details). So far, we can only estimate the effects
of symmetry restoration from projected HF calculations. The uncertainties of
nuclear radii from model-space truncation and uncertainties from the
interaction are estimated to be 2$-$3% following Refs. Novario  _et al._
(2020); Koszorús  _et al._ (2021).

Odd-mass nuclei, such as the Na isotopes considered in this work, are more
complicated than even-even nuclei because of the unpaired last nucleon. We
performed quadrupole constrained HF calculations for a range of oblate and
prolate deformations and found that in all cases (except for 25Na) the prolate
HF minimum provides the optimal reference state for the CC calculations. For
25Na, starting from an oblate reference state yield the largest binding
energy.

Results and discussion.— Table 1 shows our results of $F$ and
$K_{\text{SMS}}$, computed with the AR-RCCSD and AR-RCCSDT methods, for the D1
and D2 atomic transition lines of Na and Mg+. The calculated energies are
compared with the experimental values in the Supplemental Material. We find
that by including triples excitations, the energy calculation accuracy is
improved by an order of magnitude. The differences between the AR-RCCSD and
AR-RCCSDT values for $F$ in all the states are found to be negligible. This
finding is in line with our calculations for Li-like systems Sahoo and Ohayon
(2021), where it was also found that both the EVE and AR methods produce
reliable results for $F$, with only the FF method showing some spurious
deviations. These two facts explain the agreement between our calculation and
the previously reported FF Berengut  _et al._ (2003); Roy and Majumder (2015)
and EVE Sahoo (2010); Safronova and Johnson (2001) results from the
literature. The FF results of Tupitsyn  _et al._ (2003); Korol and Kozlov
(2007) show a more significant deviation. Our uncertainty for $F$ is given by
estimating the magnitude of neglected QED effects as detailed in Ref. Sahoo
and Ohayon (2021).

In contrast to the results for calculations of $F$, we find triples
excitations to be significant for the $K_{\text{SMS}}$ constants in both
systems, with their magnitude much larger than in Li-like systems Sahoo and
Ohayon (2021). An indication to this behaviour comes from the fact that in
both Na and Mg+, the value of $K_{\text{SMS}}$ for the ground state changes
sign between the results from the mean-field Dirac-Hartree-Fock (DHF) and AR-
RCCSD/T methods (see Supplemental Material). This highlights the critical role
of electron correlations in the determination of the above constants. On the
other hand, higher-order relativistic effects are found to be small, thus
strengthening our assumption that $K_{\text{NMS}}$ may be taken from the
scaling-law with sufficient accuracy. Both the limited reliability of the FF
and EVE approaches for estimating $K_{\text{SMS}}$ Sahoo and Ohayon (2021);
Dorne  _et al._ (2021), combined with the major role of triples excitations,
which are implemented fully here for the first time, account for the major
differences found between the earlier reported values of $K_{\text{SMS}}$ in
these systems. Only the values of Berengut  _et al._ (2003), who utilized the
FF approach in a non-relativistic calculation, agrees with our calculations
for both systems. However their values for the individual levels differ
considerably, as shown in the Supplemental Material.

Even though higher-level, such as quadruple excitations, do not contribute
directly in the AR-RCC theory owing to the one-body and two-body forms of the
FS and SMS operators, their inclusion can change the amplitudes of the
unperturbed wave operators, causing an indirect modification to our results.
An uncertainty of $2-3\%$ was estimated by analyzing such contributions in a
perturbative approach. For Na, this uncertainty is an order of magnitude
smaller than that reported in the literature Berengut  _et al._ (2003). To
validate the reliability of our calculated $K_{\mathrm{SMS}}$, and their
uncertainty, we carry out two benchmark tests; one from the atomic physics
side, and the other from the side of nuclear physics.

Figure 1:  Semi-empirical estimation of $K_{\mathrm{SMS}}$. We compare the
calculated $r_{np}$ from Table 2 (filled band), with $r_{np}$ deduced from:
matter radii (taken from Suzuki  _et al._ (1998, 2014)), the field shift given
in this work, and calculated spin orbit corrections. The comparison is done
for different values of $K_{\mathrm{SMS}}$, and returns:
$K_{\mathrm{SMS}}=105.3(1.3)\,$GHz amu.

Benchmark with Mg+.— We take advantage of the precise IS measurements for Mg+
Batteiger  _et al._ (2009). Applying Eq. (1), with $K_{\mathrm{NMS}}$ taken
from the scaling law, and $F=-126.3(7)$ MHz/fm2 from this work, as well as
$(\delta r_{c})^{2}_{24,26}=-0.158(9)$ fm2 (see Supplemental Material), we
obtain $K_{\text{SMS}}=369.3(3)$ GHz amu and $K_{\text{SMS}}=367.7(3)$ GHz amu
for the D1 and D2 transitions, respectively. These values agree within one
standard deviation with our calculated values for $K_{\text{SMS}}=374(7)$ GHz
amu for both transitions.

Benchmark with matter radii.— In Table 2 we give the result for the ground
state properties for Na isotopes calculated by
$\Delta$NNLO${}_{\mathrm{GO}}(450)$. Our uncertainty estimation is based on
similar calculations in Ne, Mg, K, and Ca for the neutron skin Hagen  _et al._
(2015) and radii Novario  _et al._ (2020); Koszorús  _et al._ (2021).

Instead of comparing directly with the matter radii, the neutron skin, defined
as the difference between its proton and neutron radii, $r_{np}=r_{n}-r_{p}$,
can be used to establish physical limits for the atomic parameters. Figure 1
shows the nuclear theory results for $r_{np}$ alongside the semi-empirical
results obtained using available matter radii Suzuki  _et al._ (1998, 2014),
our calculated $F=-39.2(3)$ MHz/fm2 and spin-orbit corrections, and charge
radii values extracted from the IS data using different values of the atomic
parameter $K_{\mathrm{SMS}}$. As $r_{c}$ is extremely sensitive to
$K_{\mathrm{SMS}}$, we vary this parameter, while keeping $F$ constant, and
compare the extracted neutron skin with the accurate theoretical prediction.
This semi-empirical fit returns $K_{\mathrm{SMS}}=105.3(1.3)\,$GHz amu, which
agrees with the direct calculation using AR-CCSDT of
$K_{\mathrm{SMS}}=109(3)\,$GHz amu. A comparison with previous calculations is
given in Fig. 2.

Figure 2:  Comparison of $K_{\mathrm{SMS}}$ in the D1 line of Na as calculated
by: a. Sahoo (2010), b. Safronova and Johnson (2001), c. Korol and Kozlov
(2007), d. Berengut  _et al._ (2003), e. Tupitsyn  _et al._ (2003), and in
this work. The filled area is from the semi-empirical estimation illustrated
in Fig. 1.  Figure 3:  Absolute charge radii of Ne, Na and Mg as a function of
neutron number. The data-points from Marinova  _et al._ (2011); Huber  _et
al._ (1975); Pescht  _et al._ (1977); Huber  _et al._ (1978); Touchard  _et
al._ (1982); Yordanov  _et al._ (2012) give the central values with errorbars
from statistical uncertainty including that of the reference isotope. The full
and broken lines give the spread resulting from the atomic parameters
calculated in this work and in Ohayon  _et al._ (2019). The filled bands
represent the spread of values calculated by the nuclear theory. For Na,
another band gives the values returned from the semi-empirical neutron-skin
fit of Fig. 1.

These two benchmarks validate the reliability of the central values and
uncertainties calculated by the atomic theory, and serve as a striking test of
the AR-CCSDT method and its applicability in light many-electron systems.

Extracted charge radii.— With the Na atomic constants calculated in this work
(Table 1), and the measured IS by Huber  _et al._ (1975); Pescht  _et al._
(1977); Huber  _et al._ (1978); Touchard  _et al._ (1982), we deduce $(\delta
r_{c}^{2})^{23,A}$ directly from Eq. (1). Our results are given in Table 2,
and shown in Fig. 3 along with predictions from nuclear theory. For
completeness, we also include the charge radii of Ne Marinova  _et al._
(2011); Ohayon  _et al._ (2019) and Mg, with the latter extracted form IS
measurements Yordanov  _et al._ (2012), and using our calculated
$F=-126.3(7)\,$MHz/fm2 (see Supplemental Material for more details). Overall,
nuclear theory predictions for $r_{c}$ agree very well with experiment for
both Ne and Mg, from $N=10$ up to $N=18$. This agreement suggests that the
$K_{\mathrm{SMS}}$ value expected for Na isotopes sits at the lower edge of
the confidence interval given by our atomic calculations. This conclusion is
very much in line with the semi-empirical prediction from the matter radii
(neutron skins), as well as the difference between experimental and calculated
$K_{\mathrm{SMS}}$ in Mg+. Further developments in atomic theory are needed to
increase the precision in the calculated atomic parameters, e.g. by including
a relativistic calculation of $K_{\mathrm{NMS}}$, accounting for QED effects,
and including quadruple excitations.

Summary and outlook.— We report new developments of atomic theory that enable
the extraction of nuclear charge radii of Na isotopes from existing isotope
shift measurements, as well as a state-of-the-art ab initio nuclear
calculation of the ground-state properties of odd-mass Na isotopes. Our
results were combined with the experimental data of matter radii to extract
the neutron skin values of these isotopes and establish semi-empirical
constraints to the atomic physics parameters. These results provide an
important benchmark to guide the development of atomic theory as currently
both charge radii and neutron skin uncertainties of Na isotopes are dominated
by the calculated atomic parameters. Nuclear theory predicts a clear “kink” in
the increase of charge radii at $N=20$ and a monotonous increase of the
neutron skin. As there is limited experimental data in this region, these
predictions provide a further motivation to perform new experiments in these
neutron-rich systems. Similar calculations are needed for other odd-$Z$
elements, for which the atomic theory input is critical to extract nuclear
charge radii values from isotope shift data Heylen  _et al._ (2021). These
developments are timely with the progress of new radioactive beam facilities,
such as FRIB in the US, where exotic light nuclear systems will be produced up
to the proton and neutron drip lines.

###### Acknowledgements.

The atomic physics calculations were carried out using the Vikram-100 HPC
cluster of Physical Research Laboratory, Ahmedabad, India. This work is
supported by Department of Energy, Office of Science, Office of Nuclear
Physics, under Award numbers DE-SC0021176, DE-SC0021179, DE-FG02-96ER40963 and
DE-SC0018223. Computer time was provided by the Innovative and Novel
Computational Impact on Theory and Experiment (INCITE) programme. This
research used resources of the Oak Ridge Leadership Computing Facility located
at Oak Ridge National Laboratory, which is supported by the Office of Science
of the Department of Energy under contract No. DE-AC05-00OR22725.

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Nuclear charge radii of Na isotopes: A tale of two theories

Supplemental Material

##  .1 Calculations of energies

To validate the calculations using the employed methods, and check the
accuracy of the wave functions, we present the electron affinities (opposite
of ionization energies) of the $3s~{}^{2}S_{1/2}$, $3p~{}^{2}P_{1/2}$ and
$3p~{}^{2}P_{3/2}$ states of Na and Mg+ in Table 3. The calculations are done
assuming infinite nuclear mass, and then corrected using the IS constants. We
present the calculations from the DHF method, and approximating the RCC theory
at the singles and doubles excitations (RCCSD method) and at the singles,
doubles and triples excitations (RCCSDT method). The Breit and QED
contributions from the RCCSDT method are also listed separately. The final
results are compared with the experimental values Baugh  _et al._ (1998);
Kaufman and Martin (1991). Compared with experiment, our calculated energies
are accurate to few $10^{-4}$, with triples excitations crucial to reach this
accuracy. We find higher order relativistic and recoil corrections to be
negligible in this level of accuracy.

Table 3:  Calculated electron affinities (in cm-1) of Na and Mg+ using the DHF, RCCSD and RCCSDT methods. Corrections due to the Breit and QED interactions are given separately, as well as the finite mass recoil correction (NMS+SMS) determined in this work. The final results are compared with the experimental values. Method | $3s~{}^{2}S_{1/2}$ | $3p~{}^{2}P_{1/2}$ | $3p~{}^{2}P_{3/2}$
---|---|---|---
23Na
DHF | 39951.56 | 24030.34 | 24014.12
RCCSD | 41355.63 | 24464.16 | 24445.78
RCCSDT | 41449.84 | 24498.53 | 24479.31
$+$Breit |  $-1.86$ |  $-1.42$ |  $-0.32$
$+$QED |  $-3.56$ |  $0.21$ |  $0.12$
$+$Recoil |  $-1.09$ |  $-0.53$ |  $-0.53$
Final | 41443(10) | 24496(7) | 24478(7)
Exp. Baugh  _et al._ (1998) | 41449.451(2) | 24493.281(2) | 24476.085(2)
24Mg+
DHF | 118824.0 | 84293.8 | 84203.5
RCCSD | 121179.9 | 85545.6 | 85448.0
RCCSDT | 121267.6 | 85600.7 | 85502.1
$+$Breit |  $-8.2$ |  $-8.7$ |  $-3.3$
$+$QED |  $-8.1$ |  0.8 |  0.6
$+$Recoil |  $-3.0$ |  -1.6 |  -1.6
Final | 121248(20) | 85591(10) | 85497(10)
Exp.Kaufman and Martin (1991) | 121267.64(5) | 85598.33(5) | 85506.76(5)

##  .2 IS constant for individual states

In the main text we give the IS constants for relevant transitions. These are
taken from the differences between the constants for the individual levels,
given here.

In Tables 4 and 5, we present the $F$ and K${}_{\text{SMS}}$ values of the
$3s~{}^{2}S_{1/2}$, $3p~{}^{2}P_{1/2}$ and $3p~{}^{2}P_{3/2}$ states of Na and
Mg+, respectively, using the DHF, AR-RCCSD and AR-RCCSDT methods. The final
quoted value is calculated using the Breit Hamiltonian. Its uncertainty is
given by estimating numerical uncertainty, partial quadruple correlations, and
unaccounted-for QED effects.

Our results for the individual states are compared with the literature where
available, with a comparison for the IS parameters for the D1 and D2 lines
given in the main text. As can be seen from the tables, previous calculations
do not quote uncertainties and vary considerably.

Table 4: IS constants of the $3s~{}^{2}S_{1/2}$, $3p~{}^{2}P_{1/2}$ and $3p~{}^{2}P_{3/2}$ states in Na using the DHF, AR-RCCSD and AR-RCCSDT methods. Previously reported calculated values are also mentioned. Method | $3s~{}^{2}S_{1/2}$ | $3p~{}^{2}P_{1/2}$ | $3p~{}^{2}P_{3/2}$
---|---|---|---
F (MHz/fm2)
DHF | $-29.717$ | $-0.008$ | $-0.000$
AR-RCCSD | $-37.317$ | $1.591$ | $1.603$
AR-RCCSDT | $-37.762$ | 1.495 | 1.480
$+\Delta$Breit | 0.022 | 0.003 | 0.001
Final | $-37.7(3)$ | 1.50(3) | 1.48(3)
Ref. Safronova and Johnson (2001) | $-36.825$ | $1.597$ | $1.603$
Ref. Sahoo (2010) | $-37.035$ | $1.725$ | $1.765$
K${}_{\text{SMS}}$ (GHz amu)
DHF | $-221.967$ | $-115.735$ | $-115.537$
AR-RCCSD | $101.658$ | $-29.765$ | $-29.810$
AR-RCCSDT | 72.121 | $-37.074$ | $-36.960$
$+\Delta$Breit | 0.235 | 0.141 | 0.028
Final | $72(3)$ | $-37(1)$ | $-37(1)$
Ref. Berengut  _et al._ (2003) | 69 | $-40$ | $-39$
Ref. Safronova and Johnson (2001) | 53.94 | $-43.36$ | $-43.39$
Ref. Sahoo (2010) | 73.2 | $-41.2$ | $-39.1$
Table 5: Same as Table 4, but for Mg+ Method | $3s~{}^{2}S_{1/2}$ | $3p~{}^{2}P_{1/2}$ | $3p~{}^{2}P_{3/2}$
---|---|---|---
F (MHz/fm2)
DHF | $-104.568$ | $-0.059$ | $-0.000$
AR-RCCSD | $-116.388$ | $9.836$ | $9.860$
AR-RCCSDT | $-116.686$ | $9.724$ | 9.746
$+\Delta$Breit | $0.096$ | $-0.006$ | $-0.009$
Final | $-116.6(7)$ | $9.72(8)$ | $9.74(8)$
Ref. Safronova and Johnson (2001) | $-116.01$ | $9.800$ | $9.811$
Ref. Sahoo (2010) | $-116.102$ | $10.119$ | $10.222$
K${}_{\text{SMS}}$ (GHz amu)
DHF | $-563.278$ | $-600.518$ | $-598.181$
AR-RCCSD | 125.451 | $-278.034$ | $-277.721$
AR-RCCSDT | 69.965 | $-303.543$ | $-303.232$
$+\Delta$Breit | 1.051 | 0.471 | 0.026
Final | 71(3) | $-303(5)$ | $-303(5)$
Ref. Berengut  _et al._ (2003) | 83.0 | $-296.0$ | $-290.0$
Ref. Safronova and Johnson (2001) | 38.0 | $-324.0$ | $-323.0$
Ref. Sahoo (2010) | $78.9$ | $-319.9$ | $-311.0$
Ref. Roy and Majumder (2015) | $-206.5$ | $-571.6$ | $-572.1$

##  .3 Charge radii and their differences

Neglecting contributions from higher moments, IS calculations and measurements
pertain to the RMS charge radii differences between the isotopes. To connect
these with absolute charge radii, one has in the very least use one absolute
$r_{c}$. In principle, elastic electron scattering experiments offer a model-
independent access to radii and other moments of the nuclear charge
distribution, however, for the S-D shell nuclei, their results have large
uncertainties and in many cases disagree with one another Fricke  _et al._
(1992). Muonic X-ray measurements are more precise, and agree better with one
another. However, they give access to the model-independent Barrett-equivalent
radius $R_{k}^{\alpha}$ of the charge distribution Barrett (1970). Direct
inference of $r_{c}$ from $R_{k}^{\alpha}$ must rely on a model-dependent
assumption on the shape of the nuclear distribution. If this dependency is
taken into account, it results in too large an uncertainty, an example for Mg
is given in Kowalska (2006).

To obtain precise and model-independent $r_{c}$, the gold standard is to
combine electron scattering and muonic X-ray measurements Fricke and Heilig .
This is done by determining the ratio between $R_{k}^{\alpha}$ and $r_{c}$,
which reads

$\nu^{2}=\frac{r_{c}^{2}}{(3/5)R_{k}^{\alpha 2}}.$ (2)

The proportionality factor $\nu$ tends to $1$ for a hard sphere distribution,
and increases slowly up to $Z=60$ Angeli (2002). However, for the isotopes
considered in this work, no such $\nu$ factors are available in the
literature.

When their results are tabulated in a model-independent way, $\nu$ may be
determined with high accuracy of order $10^{-4}$ from scattering measurements
Mazanek (1992), as the main uncertainties tend to cancel in the ratio Fricke
_et al._ (1995). Such information is available for 26Mg Soundranayagam  _et
al._ (1988), and 24Mg Li  _et al._ (1974); De Vries  _et al._ (1987). The
resulting $r_{c}^{\mu e}$ is given in Table 6, and its uncertainty is
dominated by that of nuclear polarization corrections given in Fricke and
Heilig .

Table 6: Summary of electromagnetic moments of stable isotopes determined by muonic X-ray and electron scattering experiments. The structure of the given uncertainties is: (statistical)[Nuclear Polarization]{proportionality factor}.  |  $R^{\alpha}_{k}$ (fm) | $\nu$ |  $r_{c}^{\mu e}$ (fm)
---|---|---|---
23Na | 3.8492(7)[30] | 1.004{1} |  2.9935(5)[23]{30}
24Mg | 3.9291(5)[30] | 1.0040{3} | 3.0556(4)[23]{9}
26Mg | 3.8992(8)[26] | 1.0031{1} | 3.0297(6)[20]{3}
|  $\delta R^{\alpha}_{k}$ (fm) |  $(\delta r_{c}^{\mu e})^{2}$ (fm2) |
24,26Mg | -0.0299(9)[10] | -0.158(4)[5]{6} |

To estimate $r_{c}^{\mu e}$ for 23Na, for which model-independent analysis of
electron scattering measurements is not available, we interpolate between
$\nu$ of $Z=6-16$ elements extracted from De Vries  _et al._ (1987); Fricke
_et al._ (1995). To account for this interpolation, a conservative uncertainty
in $\nu$ is taken. Our result is given in Table 6.

Table 7: Updated charge radii of Mg isotopes. Neutron deficient (ND) and
neutron rich (NR) datasets are analyzed separately. For radii differences
statistical uncertainties are in parenthesis and systematics related to
$K^{\prime}$ are in square brackets. For the absolute radii only the total
uncertainty is tabulated.

$A$ | Yordanov  _et al._ (2012) |  $(\delta r_{c}^{2})^{26,A}$ fm2 |  $r_{c}$ fm
---|---|---|---
21 |  0.173(24)[ 63] |  0.223(25)[29] | 3.0663(60)
22 |  0.214( 5)[ 51] |  0.251( 5)[23] | 3.0709(37)
23 |  0.053( 6)[ 34] |  0.080( 6)[16] | 3.0428(31)
24 |  0.140( 5)[ 25] | 0.158( 9) | 3.0556(25)
25 | -0.030( 4)[ 11] | -0.022( 5)[ 5] | 3.0261(24)
26 | 0 | 0 | 3.0297(21)
27 | -0.008( 4)[ 10] | -0.016( 4)[ 5] | 3.0270(29)
28 |  0.216( 9)[ 27] |  0.202( 9)[10] | 3.0628(36)
29 |  0.256( 6)[ 36] |  0.234( 6)[14] | 3.0681(39)
30 |  0.473( 5)[ 56] |  0.446( 5)[18] | 3.1023(44)
31 |  0.710(13)[ 79] |  0.676(13)[22] | 3.1393(52)
32 |  0.948( 6)[101] |  0.909( 6)[26] | 3.1761(53)

$K^{\prime}_{\text{ND, D1}}=951.3(4)$ GHz amu, $K^{\prime}_{\text{NR,
D1}}=952.1(4)$ GHz amu and $F_{\text{D1}}=-126.3(7)$ Mhz/fm2.

Table 8: Ground state nuclear radii of Na. All values are in fm. $A$ | $r_{c}$ | $r_{p}$ | $r_{m}$ | $r_{n}$
---|---|---|---|---
| Main Text | Eq. (4) | Suzuki  _et al._ (1998, 2014) | Eq. (5)
20 | 2.90(9) |  | 2.73(3) |
21 | 2.98(6) | 2.86(6) | 2.75(3) | 2.63(10)
22 | 2.97(3) |  | 2.88(5) |
23 |  2.993(4) |  2.89(4) | 2.90(3) | 2.91(7)
24 | 2.99(3) |  | 2.84(4) |
25 | 3.01(5) | 2.92(5) | 2.89(3) | 2.86(7)
26 | 3.04(7) |  | 2.93(4) |
27 | 3.08(8) | 3.00(9) | 3.01(2) | 3.01(7)
28 |  3.13(10) |  | 3.11(2) |
29 |  3.21(11) |  3.13(12) | 3.15(4) | 3.16(10)
30 |  3.25(13) |  | 3.25(2) |
31 |  3.33(14) |  3.24(14) | 3.31(2) | 3.34(8)

In the main text, $(\delta r_{c}^{\mu e})^{2}$ for 24,26Mg, along with our
calculated F, is used to determine mass shift constants from Eq. (1). To
extract $(\delta r_{c}^{\mu e})^{2}$, we first calculate the Barrett radii
differences $\delta R$ assuming a correlation between the nuclear polarization
corrections that reproduces their uncertainties as given in Fricke and Heilig
, which are considered an upper limit. $\delta(r_{c}^{\mu e})^{2}$ are then
determined from $\delta R_{k}^{\alpha}$ and $\nu$ with Eq. (2), accounting for
all correlations. The uncertainty contribution stems from statistics, nuclear
polarization and $\nu$ on similar footing. The total uncertainty in
$\delta(r_{c}^{\mu e})^{2}$ given here is smaller then that used by Yordanov
_et al._ (2012) as they utilized a different procedure for determining them.

The ISs in the D1 transition of 21-32Mg+ were measured at ISOLDE Yordanov  _et
al._ (2012). Combining their results with our $F=-126.3(7)$ MHz/fm2, allows us
to extract $(\delta r_{c}^{2})^{26,A}$ more accurately than semi-empirically.
The results are given in Table 7 and shown in Fig. 3 of the main text.

##  .4 Extraction of neutron skin thickness

We detail the relevant ground state properties of Na needed to extract the
neutron skin thickness, which is the difference between the point neutron and
point proton RMS radii

$r_{np}=r_{n}-r_{p}.$ (3)

The point proton radius $r_{p}$ is given by Yamaguchi  _et al._ (2011)

$r_{p}^{2}=r_{c}^{2}-R_{p}^{2}-\frac{N}{Z}R_{n}^{2}-\frac{3\hbar^{2}}{4m_{p}^{2}c^{2}}+\delta_{\text{SO}}^{2},$
(4)

with $R_{p}=0.8414(19)$ fm the latest CODATA value for the RMS charge radius
of the proton Tiesinga  _et al._ (2019), and $R_{n}^{2}=-0.116(2)$ fm2 that of
the neutron Tanabashi  _et al._ (2018). The last term is the Darwin-Foldy
correction Friar  _et al._ (1997).

Utilizing $r_{p}$ and $r_{m}$, $r_{n}$ may now be approximated as
Kamuntavičius (1997)

$r_{m}^{2}=\frac{Z}{A}r_{p}^{2}+\frac{N}{A}r_{n}^{2}.$ (5)

$r_{np}$ is now given by Eq. (3). Its uncertainty is deduced taking into
account the correlation between $r_{n}$ and $r_{p}$ which are both dependant
on $r_{c}$. Our results for the ground-state moments in the Na chain are
summarized in Table 8.