# Nuclear structure study of 20-23Na isotopes with ab initio no-core shell-
model

Chandan Sarma  Department of Physics, Indian Institute of Technology Roorkee,
Roorkee 247667, India  Praveen C. Srivastava  Corresponding author:
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Roorkee, Roorkee 247667, India

###### Abstract

We have done a systematic no-core shell-model study of 20-23Na isotopes. The
low-energy spectra of these sodium isotopes consisting of natural and un-
natural parity states were reported, considering three realistic interactions:
inside nonlocal outside Yukawa (INOY), charge-dependent Bonn 2000 (CDB2K), and
the chiral next-to-next-to-next-to-leading order (N3LO). We also present the
mirror energy differences in the low-energy spectra of $|T_{z}|$ = 1/2 mirror
pair (21Na - 21Ne). Apart from the energy spectra, we have also reported the
electromagnetic transition strengths and moments. Finally, considering all
three realistic interactions, we report the point-proton radii and neutron
skin thicknesses.

###### pacs:

21.60.Cs, 21.30.Fe, 21.10.Dr, 27.20.+n

##  I Introduction

For the last two decades, the no-core shell-model (NCSM) NCSM_r2 ; NCSM_r1 ;
MVS2009 ; A=10 ; C.Forssen ; stetcu1 ; stetcu2  became a major ab initio
approach to solve nuclear many-body problems starting from realistic $NN$ and
$3N$ interactions. The NCSM approach is very successful in explaining nuclear
structural properties of $p$-shell nuclei Phys.Rev.C573119(1998) ;
Phys.Rev.C502841(1994) ; PRL84 ; PRC62 ; Choudhary1 ; Choudhary2 ; Choudhary3
; arch1 ; Phys.Rev.C54001(2021) ; reviewJPG ; NCSM_p8 , although it is less
explored for the $sd$-shell nuclei mainly due to the significant increase in
computational resources. Some approximations become necessary, in terms of
small model spaces considered, to tackle lower $sd$-shell nuclei within the
NCSM formalism. Like any other many-body methods, approximations employed will
cause deficiencies in reproducing the correct results of some of the nuclear
observables. However, understanding the nature of such deficiencies is
important for improving the nuclear Hamiltonians taken into account to study
nuclei of a particular region of the nuclear chart.

The information of low-lying energy states together with electromagnetic
properties can reveal the internal structures of atomic nuclei. And, the
nuclei around the $N=Z$ work as perfect testing grounds for different many-
body methods of nuclear structure. Earlier, we explored neon chain from $A$ =
18 to $A$ = 24 within the NCSM formalism chandan . The calculated results for
energy-spectra, $B(M1)$ and magnetic moment are in good agreement with the
experimental data, however, operator that depends on the long-range part of
nuclear wavefunctions such as $B(E2)$ and quadrupole-moment are far away from
the experimental data. Recently, the $E2$ strengths between the ground and the
first excited state of 23Na and its mirror nucleus 23Mg are measured using
Coulomb excitation measurement vs_imsrg1 . The experimental $E2$ strengths of
both nuclei are compared to the $ab$ $initio$ valence space in-medium
similarity group (VS-IMSRG) results and standard shell model results within
the $sd$ space. A similar kind of theoretical investigation is reported in
Ref. vs_imsrg2  for a range of mirror nuclei pairs across the $sd$-space,
including $|T_{z}|$ = 1/2 mirror pairs (21Na-21Ne) and (23Na-23Mg). This study
shows that missing $E2$ strengths for VS-IMSRG are mostly isoscalar in nature.
In Ref. arch3 , the electric quadrupole and magnetic dipole moments of 20-31Na
isotopes are studied along with other nuclei across the $sd$-space using
valence-space Hamiltonian constructed from ab initio VS-IMSRG and coupled
cluster effective interaction (CCEI). Recently, the isospin-symmetry breaking
in the mirror-energy difference (MED) of $sd$\- and $pf$-shell nuclei are
investigated in Ref. MED  within the ab initio  VS-IMSRG. This study shows
that while the calculated MEDs of 20Na ($1^{+}_{2}$) and 21Na
(1/2${}^{+}_{1}$) are close to the experimental MEDs, the calculated MED of
19Na (1/2${}^{+}_{1}$) is small compared to the experimental one.

The charge radius is another fundamental property of atomic nuclei apart from
low-energy spectra and electromagnetic observables, that can reveal the
structure of nuclei to a great extent. In recent years, several experiments
have been performed to measure the charge radii of different isotopic chains.
On the other hand, the calculated charge radii with different many-body
theories having error quantification procedures facilitate the comparison of
theoretical and experimental results directly. Recently, the charge radii of
Na isotopes have been extracted from accurate atomic calculations in Ref.
Na_rc . Suzuki et. al.  in Ref. Na_skin  studied the development of neutron
skin along the Na isotopic chain. Using shell-model, the variation of charge
and matter radius along the sodium isotopic chain is reported in Ref.
subhrajit .

In this work, we have performed nuclear structure study of 20-23Na isotopes
within the formalism of ab initio no-core shell model using three realistic
interactions, namely inside nonlocal outside Yukawa (INOY) INOY ; INOY2 ;
INOY3 , charge-dependent Bonn 2000 (CDB2K) CDB2K1 ; CDB2K2 , and the chiral
next-to-next-to-next-to-leading order (N3LO) QCD ; N3LO . We have calculated
the g. s. energies, low-energy spectra comprising of both natural and un-
natural parity states, electromagnetic transition strengths and moments, and
also the point-proton radii of these sodium isotopes. The maximum basis space
we have reached in this work is $N_{max}$ = 4 for all four sodium isotopes.
This work is important to check the consistency of our previous works on the
neon chain chandan  as well as other works on the lower mass $sd$-shell nuclei
arch1 ; arch2  using the NCSM formalism.

This paper is organized as follows: In section II, the basic formalism of the
NCSM approach is given. The $NN$ interactions used in this work are briefly
described in section III. In section IV, the NCSM results for energy spectra,
electromagnetic properties, and point-proton radii are presented. Finally, we
summarize our work in section V.

##  II Formalism of no-core shell-model

In this approach, nucleons are considered to be non-relativistic point
particles interacting via realistic $NN$ and $NN$ \+ $3N$ interactions. In the
present work, we are using only the $NN$ interactions for which the
Hamiltonian can be written as follows NCSM_r2 :

$\displaystyle
H_{A}=T_{rel}+V=\frac{1}{A}\sum_{i<j}^{A}\frac{({\vec{p}_{i}-\vec{p}_{j}})^{2}}{2m}+\sum_{i<j}^{A}V^{NN}_{ij},$
(1)

where, $T_{rel}$ is the relative kinetic energy and $V^{NN}_{ij}$ is the $NN$
interaction containing nuclear part as well as Coulomb part.

In the NCSM approach, the solution of a non-relativistic Schrödinger equation
for an A-nucleon system is found by performing a large-scale matrix
diagonalization in a many-body Harmonic oscillator basis. The many-body basis
is constructed by taking the Slater determinant (M-scheme) of the single-
particle Harmonic oscillator orbitals having a truncation parameter $N_{max}$.
The $N_{max}$ shows the number of major shells taken above the minimum
configuration allowed by the Pauli exclusion principle. The use of the
M-scheme many-body basis along with $N_{max}$-truncation allows the smooth
separation of the center of mass (CM) coordinates from the relative
coordinates. Large $N_{max}$ calculations are necessary to obtain converged
results while dealing with hard-core potentials having strong short-range
correlations. However, large $N_{max}$ calculations are computationally
challenging, thus we need some renormalization technique to soften such
potentials. Two such renormalization procedures that are popularly used in
NCSM calculations are the Okubu-Lee-Suzuki (OLS) OLS1 ; OLS2 ; OLS3 ; OLS4
transformation and similarity renormalization group (SRG) SRG1 ; SRG2 . Both
these procedures soften the hard-core part of nuclear interactions eventually
to obtain converged results within a computationally tractable model space
controlled by the $N_{max}$ parameter. In this work, we are using the OLS
technique to obtain an effective Hamiltonian.

To facilitate the derivation of the OLS effective Hamiltonian, the center-of-
mass (c.m.) Hamiltonian, $H_{c.m.}$ is added to the original Hamiltonian
Equation 1:

$H^{\Omega}_{A}=H_{A}+H_{c.m.}=\sum_{i=1}^{A}[\frac{p_{i}^{2}}{2m}+\frac{1}{2}m\Omega^{2}r_{i}^{2}]\\\
+\sum_{i<j}^{A}[V_{NN,ij}-\frac{m\Omega^{2}}{2A}(r_{i}-r_{j})^{2}].$ (2)

Here, $H_{c.m.}$ = $T_{c.m.}$ \+ $U_{c.m.}$ and $T_{c.m.}$ and $U_{c.m.}$ are
the kinetic and potential terms for the center of mass co-ordinate. $H_{A}$,
being translationally invariant which does not change the intrinsic properties
once $H_{c.m.}$ is added to it.

In order to develop effective interactions, first the infinite HO basis is
separated into $P$\- and $Q(=1-P)$ spaces by using projection operators. The
$P$-space contains all the HO orbitals allowed by the truncation parameter
$N_{max}$ and the $Q$-space contains all the excluded HO orbitals. The
effective Hamiltonian is then constructed by using the OLS unitary
transformation on $H^{\Omega}_{A}$ given in Equation 2. The OLS transformation
induces up to A-body terms even if we start with the $NN$ Hamiltonian given in
Equation 1. In this work, we are considering only up to 2-body terms as
contributions from those terms would be dominant over other many-body terms.
In the final step, the $H_{c.m.}$ is subtracted, and the Lawson projection
term, $\beta(H_{\mbox{c.m.}}-\frac{3}{2}\hbar\Omega)$ Lawson  is added to the
Hamiltonian. In this work, the value of $\beta$ is taken to be 10 and the
addition of the Lawson projection terms shifts energy levels arising due to
the excitation of the CM. So, the final form the Hamiltonian can be written
as:

$H_{A,eff}^{\Omega}=P\left\\{\sum_{i<j}^{A}\left[\frac{{(\vec{p}_{i}-\vec{p}_{j})}^{2}}{2mA}+\frac{m{\Omega}^{2}}{2A}{(\vec{r}_{i}-\vec{r}_{j})}^{2}\right]\right.\\\
\left.+\sum_{i<j}^{A}\left[V^{NN}_{ij}-\frac{m{\Omega}^{2}}{2A}{(\vec{r}_{i}-\vec{r}_{j})}^{2}\right]_{\rm
eff}+\beta\Bigg{(}H_{\mbox{c.m.}}-\frac{3}{2}\hbar\Omega\Bigg{)}\right\\}P.\\\
\quad$ (3)

The effective Hamiltonian of Equation 3 is dependent on the number of nucleon
(A), HO frequency ($\Omega$), and number of HO orbitals considered in the
$P$-space controlled by the truncation parameter, $N_{max}$. In order to
minimize the effect of many-body terms in developing an effective Hamiltonian,
a large model space is needed for NCSM. For the extreme limit of
$N_{max}\rightarrow\infty$, the effective Hamiltonian of Equation 3 reduces to
the bare Hamiltonian of Equation 1 and the NCSM results with effective
Hamiltonian reduce to the exact solution.

##  III Realistic $NN$ interactions

In this work, we have performed NCSM calculations using three realistic $NN$
interactions namely, inside nonlocal outside Yukawa (INOY) INOY ; INOY2 ;
INOY3 , charge dependent Bonn $NN$ interaction (CDB2K) CDB2K1 ; CDB2K2 , and
the chiral N3LO QCD ; N3LO .

The inside nonlocal outside Yukawa (INOY) INOY ; INOY2 ; INOY3  is a
phenomenological potential having a nonlocal part at a short distance ($\leq$
1 fm) and a Yukawa tail similar to local Argonne $v18$ av18  potential at a
large distance ($\geq$ 3 fm). A smooth cutoff in the transition region (1-3
fm) connects the local to the nonlocal part, and the range of locality and
nonlocality can be controlled explicitly by adjusting the parameters. It can
represented as follows:

V${}^{full}_{ll^{\prime}}(r,r^{\prime})$ = W${}_{ll^{\prime}}(r,r^{\prime})$
\+ $\delta(r-r^{\prime})$ F${}^{cut}_{ll^{\prime}}(r)$
V${}^{Yukawa}_{ll^{\prime}}(r)$

where the cut-off function is defined as

$F^{cut}_{ll^{\prime}}(r)=\begin{cases}1-e^{-[\alpha_{ll^{\prime}}(r-R_{ll^{\prime}})]^{2}}&for~{}r\geq
R_{ll^{\prime}},\\\ 0&for~{}r\leq R_{ll^{\prime}},\\\ \end{cases}$

and $W_{ll^{{}^{\prime}}}(r,r^{{}^{\prime}})$ and
$V_{ll^{{}^{\prime}}}^{Yukawa}(r)$ are the phenomenological nonlocal part and
the local part of the interaction, respectively. The parameters
$\alpha_{ll^{\prime}}$ and $R_{ll^{\prime}}$ are considered to be independent
of the angular momenta having values 1.0 fm-1 and 1.0 fm, respectively. The
other parameters of W${}_{ll^{\prime}}(r,r^{\prime})$ are determined by
fitting $NN$ data and the binding energy of 3He. This interaction model breaks
charge independence and charge symmetry as it is essential to reproduce all
low-energy experimental parameters, including $np$, $pp$, and $nn$ scattering
lengths, to high precision. To describe a few-nucleon system reasonably well,
additional $3N$ forces are necessary along with local $NN$ potentials, but
INOY $NN$ interaction does not need $3N$ force to provide the binding energies
of $3N$ systems. The effects of three-nucleon forces are partly included in
the non-local part of this interaction. The non-local part of the $NN$
interaction is mainly due to the internal structure of nucleons.

The second interaction we employed in this work is the charge-dependent Bonn
2000 (CDB2K)CDB2K1 . It is a one-boson exchange $NN$ potential that includes
all the mesons $\pi$, $\eta$, $\rho$, and $\omega$, the masses of which are
less than the nucleon mass. However, the $\eta$-meson is dropped from the
potential as it has a vanishing coupling with the nucleon. Additionally, two
partial-wave dependent scalar-isoscalar $\sigma$ bosons are also introduced.
This charge-dependent potential reproduces accurately charge symmetry breaking
(CSB) and charge independence breaking (CIB) in all partial waves with
J$\leq$4 as predicted by the Bonn full model CDB2K2 . In this model, three
$NN$ interactions are constructed: a proton-proton, a neutron-neutron, and a
neutron-proton potential, and the differences between them are measured by CSB
and CIB. These potentials fit the world proton-proton data available in the
year 2000 with $\chi^{2}$/datum = 1.01 and the corresponding neutron-proton
data with $\chi^{2}$/datum = 1.02 below 350 MeV. The CDB2K potential uses the
original form of the covariant Feynman amplitudes of meson exchange without
local approximation. So, the off-shell behavior of this $NN$ potential is
different from other local $NN$ potentials.

The third interaction used in our current study is a chiral $NN$ interaction
at next-to-next-to-next-to-leading order (N3LO) N3LO  derived from the quantum
chromodynamics using chiral perturbation theory ($\chi$PT). The $\chi$PT is a
systematic expansion of the nuclear potential in terms of
(Q/$\Lambda_{\chi}$)ν, where Q is a low-momentum scale or pion mass,
$\Lambda_{\chi}$ $\approx$ 1 GeV is the chiral symmetry breaking scale and
$\nu$ $\geq$ 0 QCD . For a definite order $\nu$, there is a finite number of
unique and calculable contributing terms. The long-ranged part of the nuclear
interaction is associated with the exchange of pions, whereas the short-ranged
part is defined in terms of contact terms. The charge dependence of this
interaction is important for a good fit of the low-energy $pp$ and $np$ data.
There are, in total, 29 parameters of the N3LO potential. Three of them are
low-energy constants (LECs), $c_{2}$, $c_{3}$, and $c_{4}$ that appear in the
$\pi N$ Lagrangians. The most important fit parameters are 24 contact terms
that dominate the partial waves with L $\leq$ 2, and the remaining two
parameters are two charge-dependent contact terms. The $NN$ interaction at
this order can be compared to high-precision phenomenological potential AV18
av18  in terms of the accuracy in reproducing the $NN$ data below 290 MeV lab
energy.

The NCSM calculations have been performed with the pAntoine code pAntoine1 ;
pAntoine2 ; pAntoine3 .

##  IV Results and Discussions

In this section, we have reported NCSM results of different nuclear
observables. The NCSM calculations are computationally challenging for medium
mass nuclei mainly due to the huge dimensions of the Hamiltonian matrix. In
Table 1, the $M$-dimensions of the Hamiltonian matrices are shown for 20-23Na
isotopes corresponding to the model space from $N_{max}$ = 0 to 6. We are able
to reach basis size up to $N_{max}$ = 4 for all four sodium isotopes (shown in
blue color in Table 1) with the available computational resources. The maximum
dimension we are able to reach is 1.1 $\times$ 109 for 23Na corresponding to
$N_{max}$ = 4 model space.

The first step of the NCSM calculation is to decide an optimum frequency for
which different observables are calculated later on. The optimum frequency for
a particular interaction is decided by plotting the ground state energies for
a particular interaction corresponding to different $N_{max}$. Subsequently,
the frequency corresponding to the minimum g.s. energy calculated at the
highest $N_{max}$ is taken as the optimum frequency for a particular
interaction. In Figure 1, the g.s. energies of 21,22Na isotopes are shown with
realistic $NN$ interactions: INOY, CDB2K and N3LO for $N_{max}$ = 2 and 4
model spaces. The optimum frequencies for INOY, CDB2K, and N3LO for both
isotopes are 20-, 16-, and 14-MeV, respectively. Similarly, the optimum
frequencies for other Na-isotopes are calculated.

Table 1:  The m-scheme dimensions of the ground states of Na isotopes corresponding to different $N_{max}$ are shown. The dimensions up to which we have reached are shown in blue. $N_{max}$ |  20Na (2+) |  21Na (3/2+) |  22Na (3+) |  23Na (3/2+)
---|---|---|---|---
0 | 394 |  1935 | 6116 | 1.3 $\times 10^{4}$
2 | 4.0 $\times 10^{5}$ | 1.5 $\times 10^{6}$ | 4.6 $\times 10^{5}$ | 8.4 $\times 10^{6}$
4 | 5.9 $\times 10^{7}$ | 1.9 $\times 10^{8}$ | 5.2 $\times 10^{8}$ | 1.1 $\times 10^{9}$
6 | 3.6 $\times 10^{9}$ | 1.1 $\times 10^{10}$ | 3.0 $\times 10^{10}$ | 6.5 $\times 10^{10}$

Figure 1: The variation of calculated g.s. energies of 21,22Na with HO
frequencies for different $N_{max}$ are shown corresponding to three different
$NN$ interactions. The horizontal lines show the experimental binding energies
of the g. s.

After deciding the optimum frequency for a particular interaction, the low-
energy spectra and other nuclear observables are calculated corresponding to
the optimal frequency. In order to compare the results of different
interactions for the low-energy spectra of a particular nucleus, we use root
mean square deviation (rms) of energy spectra defined as:

$\displaystyle
E_{rms}=\sqrt{\frac{1}{N}\sum_{i}^{N}(E_{exp}^{i}-E_{th}^{i})^{2}}$ (4)

Here, $E_{exp}^{i}$ and $E_{th}^{i}$ are, respectively, the experimental and
the calculated energy for the $i^{th}$ state of a particular nucleus. The $N$
is the number of states considered for calculating the rms value, and the
$E_{rms}$ provides the quality of theoretical results compared to the
experimental data.

###  IV.1 Natural-parity low-lying states of 20-23Na:

Figure 2: Natural parity low-energy spectra of 20,21,22Na. Figure 3: Natural
parity low-energy spectra of 23Na.

The top panel of Figure 2 shows the positive-parity low-lying spectra of 20Na.
As shown in the figure, 2+ is the experimental ground state, which is well
reproduced by all three interactions, namely INOY, CDB2K, and N3LO. However,
the calculated ground state energies are only a few keV lower than the
energies of the first excited states for all three interactions. The INOY and
N3LO interactions correctly reproduce the first excited state of 20Na though
the excitation energies are 58- and 7- keV, respectively, compared to the
experimental excitation energy of 596 keV. None of the three realistic
interactions are able to reproduce the excitation spectra above the first
excited state. However, the excitation energies of $4_{1}^{+}$ state obtained
for INOY and $3_{3}^{+}$ state obtained for CDB2K are only 150 keV away from
the experimental data. The rms deviation in energies for the eight lowest
states are 1.146-, 0.720-, and 0.941- MeV, for INOY, CDB2K, and N3LO $NN$
interactions, respectively.

The low-lying spectra of 21Na for INOY, CDB2K, and N3LO interactions are shown
along with the experimental data in the second panel of Figure 2. The figure
shows that the CDB2K and N3LO reproduce the correct ordering of low energy
states up to the fifth excited state ($5/2^{+}$). On the other hand, the INOY
interaction is able to reproduce correct ordering only up to the second
excited state (7/2+). A significant deviation of 2.907 MeV is observed between
the calculated excitation energy with INOY and the experimental data for the
$1/2_{1}^{+}$ state. The root-mean-square deviations corresponding to nine
low-lying states between the experimental and calculated results for the INOY,
CDB2K, and N3LO are 1.645, 0.375, and 0.439 MeV. However, the rms deviations
are 0.336, 0.388, and 0.396 MeV corresponding to INOY, CDB2K, and N3LO for the
yrast band states (3/2+-5/2+-7/2+-9/2+-11/2+).

For the case, 22Na, the ground state (3+) is well reproduced by all the three
interactions considered in this work as shown in the third panel of Figure 2.
While the CDB2K interaction is able to reproduce low-energy spectra up to the
second excited state (0+), the N3LO interaction is able to reproduce only the
ground state and the first excited state in the correct order. The calculated
excitation energies of the 4+ state with CDB2K and N3LO interactions are only
56 and 73 keV away from the experimental data.

In Figure 3, we have shown the low-energy spectra of 23Na. While the INOY
interaction is able to reproduce the correct ordering of low-energy spectra up
to the second excited state (7/2+), the other two interactions, namely CDB2K
and N3LO, are able to reproduce only the g. s. (3/2+) and first excited state
(5/2+) correctly.

Figure 4: Low-energy un-natural parity states of 20,21,22Na. Figure 5: Low-
energy un-natural parity states of 23Na.

###  IV.2 Un-natural parity low-lying states of 20-23Na:

In this section, we are discussing the low-lying unnatural parity states of
20-23Na. The ground states of all four isotopes have positive parity, and the
unnatural negative parity states are shown in Figure 4 and Figure 5. The first
panel of Figure 4 shows the negative parity states of 20Na. As can be seen
from the figure, all three interactions are able to reproduce 2- as the
correct lowest negative parity state. While CDB2K and N3LO are able to
reproduce the correct sequence of states up to 3${}^{-}_{1}$, the ordering of
2${}^{-}_{2}$ and 3${}^{-}_{1}$ are seen to be reversed for INOY interaction.
The 2${}^{-}_{4}$ state obtained for N3LO interaction is in good agreement
with the un-confirmed (2)${}^{-}_{4}$ state, so the 4.150 MeV state of 20Na
negative parity spectra could be 2- as predicted from our NCSM calculation.

The second panel of Figure 4 shows the un-natural parity states of 21Na. From
the figure, we can see that the correct sequence of states is correctly
reproduced by INOY interaction. While the calculated excitation energies of
3/2${}^{-}_{1}$ (5/2${}^{-}_{1}$) states are less by 240 keV (337 keV)
compared to the experimental states, they are higher by 576 keV (1008 keV) for
3/2${}^{-}_{2}$ (1/2${}^{-}_{1}$) states compared to the experimental data.
The other two interactions fail to reproduce the correct ordering of low-
energy spectra including the g.s.

The negative parity spectra of 22Na are shown in the third panel of Figure 4.
While the INOY interaction is able to reproduce the correct ordering up to
3${}^{-}_{1}$ state, the other two interactions fail to reproduce the correct
spectra including the g. s. The calculated 2${}^{-}_{1}$ state with INOY is in
good agreement with the experimental data.

The low-energy negative parity states of 23Na are shown in Figure 5. The CDB2K
and N3LO interactions are able to reproduce the g.s. correctly, while the
calculated results with INOY show 7/2${}^{-}_{1}$ as the lowest negative
parity state. The calculated spectra using CDB2K match the experimental
ordering of states. The excitation energy of the 5/2${}^{-}_{1}$ state for
CDB2K is only 50 keV higher than the experimental data; however, the
calculated excitation energies of 3/2${}^{-}_{1}$, 3/2${}^{-}_{2}$ and
7/2${}^{-}_{1}$ states for the same interaction are lower than the
experimental data.

In Figure 6, the lowest positive and negative parity states of 20-23Na
isotopes are compared for the model space going from 0$\hbar\Omega$ to
4$\hbar\Omega$ for the case of N3LO interactions. Calculated results
corresponding to CDB2K and INOY are shown only for 3$\hbar\Omega$ and
4$\hbar\Omega$ model spaces. We observed that the excitation energies of the
unnatural states improve by increasing the model space size for all four Na
isotopes. Further improvements in excitation energies of unnatural parity
states can be expected for even larger model space. Of the three interactions
considered in this work, INOY shows more excitation energies of the
corresponding negative parity states, even for the highest model spaces
considered. The deviation with INOY interaction is very large corresponding to
0$\hbar\Omega$ to 1$\hbar\Omega$ and 0$\hbar\Omega$ to 1$\hbar\Omega$ model
space. Because of this reason, we have shown N3LO results corresponding to the
above two model spaces in Figure 6.

Figure 6: The excitation energies of the lowest unnatural parity states of
20-23Na isotopes are shown with respect to the g. s. For N3LO, results are
shown from model space from 0$\hbar\Omega$ to 4$\hbar\Omega$ and for the other
two interactions, 3$\hbar\Omega$ and 4$\hbar\Omega$ results are shown.

###  IV.3 Mirror energy difference in the low-energy spectra of $A$ = 21
mirror pairs:

The isospin symmetry breaking can be studied by considering pairs of mirror
nuclei where the number of protons and neutrons are interchanged. It is
responsible for the difference in excited spectra of mirror pairs. Though the
Coulomb interaction among the protons is the major source of isospin symmetry
breaking, some contributions also come from the nuclear part. The mirror
energy difference (MED) between the analogous states of mirror pairs are
defined as:

$\displaystyle MED_{J}=E_{J}(T_{z}=-T)-E_{J}(T_{z}=+T).$ (5)

Here, $E_{J}$ are the excitation energies of analogous states with angular
momentum $J$ in a mirror pair with $T_{z}$ = $\pm T$. The MED provides an
estimation of the isospin symmetry breaking of a particular interaction.

In this work, we have shown a comparison between the low-energy states of 21Na
and 21Ne which form a $|T_{z}|$ = 1/2 mirror doublet in Figure 7. The spectra
of 21Ne are taken from Ref. chandan . All results are for $N_{max}$ = 4 at the
corresponding optimum frequencies of each interaction. The figure shows that
the yrast band states:
3/2${}^{+}_{1}$-5/2${}^{+}_{1}$-7/2${}^{+}_{1}$-9/2${}^{+}_{1}$-11/2${}^{+}_{1}$
are slightly different in energy (less than 50 keV) between the analogous
states of two mirror pairs. However, some other low-energy states show a large
MED.

In Figure 8, we show the comparison between the experimental and calculated
MEDs of some of the low-lying states, which show a large MED. For comparison,
we also include MED of 5/2${}^{+}_{1}$, which is a yrast band state.
Experimentally, a small MED (18 keV) is seen for 5/2${}^{+}_{1}$, while the
1/2${}^{+}_{1}$ shows a large MED of 370 keV. The calculated MEDs for this
state (1/2${}^{+}_{1}$) are 122, 100, and 78 keV with INOY, CDB2K and N3LO,
respectively. The other three states, 3/2${}^{+}_{2}$, 5/2${}^{+}_{2}$, and
5/2${}^{+}_{3}$ show MEDs $\sim$ 200 keV. The calculated MEDs with all three
realistic interactions underestimate corresponding to the experimental data.
It is observed that out of the three realistic interactions, INOY produces the
highest MEDs between analogous states of the mirror pairs considered in this
work.

A comparison of the neutron and proton occupancies of different HO orbitals is
shown in Figure 9 for the 1/2${}^{+}_{1}$ state of (21Na - 21Ne) mirror pair.
As shown in the figure, a significant contribution comes from the HO orbitals
up to 0d3/2. Among the HO orbitals that contribute most to the 1/2${}^{+}_{1}$
wavefunction, the occupancies of proton and neutron 1s1/2 orbitals are
significantly different for all three interactions. While the proton (neutron)
occupancies of 21Na(1/2${}^{+}_{1}$) state are comparable to the neutron
(proton) occupancies of 21Ne(1/2${}^{+}_{1}$), the calculated MEDs for INOY,
CDB2K, and N3LO are 122, 100 and 78 keV, respectively. This might arise from
the isospin non-conserving two-body matrix elements involving 1s1/2 orbital
along with the contributions from the Coulomb part of the Hamiltonian.

Figure 7: Comparison of the low-energy states of 21Na and 21Ne. Figure 8:
Mirror energy difference between the low-energy states of 21Na and 21Ne.
Figure 9: Comparison of the neutron and proton occupancies of different HO
orbitals for the 1/2${}^{+}_{1}$ state of (21Na - 21Ne) mirror pair.  Figure
10: Probability distributions over 0$\hbar\Omega$ (in blue), 2$\hbar\Omega$
(in green), and 4$\hbar\Omega$ (in red) subspaces are shown for the $N_{max}$
= 4 NCSM ground state of 23Na considering three interactions.

Figure 11: Neutron and proton occupancies of 20-23Na isotopes are shown for
INOY, CDB2K and N3LO interactions.

###  IV.4 Ground state energies and electromagnetic properties

In this section, we are going to discuss the binding energies of the g. s. and
the electromagnetic properties of all four sodium isotopes considered in this
work. All the results presented in Table 2 are with respect to the optimal
frequencies of each interaction corresponding to the highest $N_{max}$
calculation. The experimental data included in the table are taken from Refs.
NNDC ; Qandmag .

We first discuss the binding energies of the g.s. As shown in Table 2, the
binding energy of 20Na g. s. is -145.960 MeV and the calculated result with
INOY interaction overbind the g. s. by only 281 keV. While the other two
interactions significantly underbind the g. s of 20Na. In Figure 12, we have
plotted the g. s. binding energies of sodium isotopes obtained from all three
interactions considered in this work. From the figure, it is seen that the
INOY interaction overbinds the g. s. of all four Na-isotopes, while the other
two interactions underbind corresponding g. s. significantly. On average,
while the INOY interaction overbinds the g. s. by 5.140 MeV, the CDB2K and
N3LO underbind by 13 and 11 MeV, respectively. In Figure 10, we have shown the
probability distributions over different model spaces from $N_{max}$ = 0 to 4
for the NCSM calculation of 23Na g. s. for $N_{max}$ = 4 calculation. From the
figure, we see that a maximum contribution of 18.7 % comes from $N_{max}$ = 4
space for the N3LO interaction among the three interactions. The neutron and
proton occupancies up to 0f7/2 orbital are shown in Figure 11 for the g. s. of
20-23Na isotopes corresponding to three different interactions. From the
figure, a significant difference is seen for the occupancies of
$\nu(1s_{1/2}$) and $\pi(1s_{1/2}$) orbitals for INOY and N3LO interactions.
These differences in occupancies might have caused the differences in the
binding energies of the g. s. of 20-23Na isotopes for INOY and N3LO. Less
occupancies of 1s1/2 orbitals for INOY can be correlated to more binding of
the g. s. compared to CDB2K and N3LO interactions.

Apart from the binding energies of the g. s., Table 2 shows the calculated
quadrupole and magnetic moments of the g. s. obtained from all three
interactions. In Figure 13 and Figure 14, we have plotted the calculated
quadrupole and magnetic moments of the g. s. of sodium isotopes, respectively,
and compared them with the experimental data. The overall trends of g. s.
moments are followed by all three interactions, though there are some
deviations between the calculated and the experimental values. The
electromagnetic transition strengths, more specifically calculated $B(E2)$
values, are significantly less than the experimental values, as can be seen
from Table 2. The prime reason for this is that $B(E2)$ is a long-range
operator. While the OLS transformation renormalizes the short-range part of
nuclear interaction and short-range operators, it weakly renormalizes the
long-range operators like $B(E2)$. Thus better results for $B(E2)$ can be
obtained once we perform calculations with higher $N_{max}$. For comparison,
we also included the valence-space in-medium similarity renormalization group
(VS-IMSRG) results of $B(E2;5/2^{+}_{1}\to 3/2^{+}_{1}$) transitions for 21Na
and 23Na isotopes in Table 2. For these two cases, the $B(E2)$ strengths
obtained for VS-IMSRG are better than the NCSM results for all three
interactions considered in this work. However, the calculated values of $E2$
transition strengths from VS-IMSRG and NCSM are still far from the
experimental data. Out of the three interactions considered in our work, N3LO
reproduces better results for $B(E2)$ transition strengths. On the other hand,
the $B(M1)$ transition strengths depend on the spin and isospin coordinates
only for which converged results can be achieved easily unlike $B(E2)$
transitions.

Figure 12: Comparison of the calculated ground state energies of Na-isotopes with the experimental data. Figure 13: Comparison of the calculated ground state quadrupole moments of Na-isotopes with the experimental data. Figure 14: Comparison of the calculated ground state magnetic moments of Na-isotopes with the experimental data. Table 2:  Electromagnetic observables of 20-23Na corresponding to the largest $N_{\mbox{max}}$ at their optimal HO frequencies. Quadrupole moments, magnetic moments, $E2$ and $M1$ transitions are in barn (b), nuclear magneton ($\mu_{N}$), $e^{2}$ fm4 and $\mu_{N}^{2}$ respectively. Experimental values are taken from Refs. NNDC ; Qandmag . The VS-IMSRG results for $B(E2)$ transitions are taken from Ref. vs_imsrg2 .  20Na |  EXPT |  INOY |  CDB2K |  N3LO |  VS-IMSRG
---|---|---|---|---|---
$E_{g.s.}(2^{+})$ |  -145.960 |  -146.24 |  -131.22 |  -134.57 |  -
$Q(2^{+}$) |  0.101(8) |  0.04 |  0.07 |  0.09 |  -
$\mu$($2^{+}$) |  0.3694(2) |  0.34 |  0.68 |  0.86 |  -
$B(E2;4_{1}^{+}$ $\rightarrow$ $2_{1}^{+}$) |  NA |  6.33 |  10.50 |  13.08 |  -
$B(M1;2_{1}^{+}$ $\rightarrow$ $3_{1}^{+}$) |  NA |  0.68 |  0.12 |  0.06 |  -
21Na |  EXPT |  INOY |  CDB2K |  N3LO |  VS-IMSRG
$E_{g.s.}(3/2^{+})$ |  -163.09 |  -165.39 |  -148.92 |  -152.20 |  -
Q($3/2^{+}$) |  0.138(11) |  0.05 |  0.07 |  0.07 |  -
$\mu$($3/2^{+}$) |  2.38610(4) |  2.21 |  2.53 |  2.68 |  -
$B(E2;5/2_{1}^{+}$ $\rightarrow$ $3/2_{1}^{+}$) |  134(10) |  19.58 |  27.96 |  33.12 |  56.1
$B(E2;7/2_{1}^{+}$ $\rightarrow$ $5/2_{1}^{+}$) |  55(27) |  12.53 |  22.93 |  30.27 |  -
$B(E2;7/2_{1}^{+}$ $\rightarrow$ $3/2_{1}^{+}$) |  72(27) |  7.90 |  12.80 |  16.68 |  -
$B(M1;5/2_{1}^{+}$ $\rightarrow$ $3/2_{1}^{+}$) |  0.1513(18) |  0.24 |  0.19 |  0.22 |  -
$B(M1;7/2_{1}^{+}$ $\rightarrow$ $5/2_{1}^{+}$) |  0.36(9) |  0.32 |  0.20 |  0.13 |  -
22Na |  EXPT |  INOY |  CDB2K |  N3LO |  VS-IMSRG
$E_{g.s.}(3^{+})$ |  -174.152 |  -179.51 |  -159.93 |  -162.82 |  -
$Q(3^{+})$ |  0.180(11) |  0.10 |  0.12 |  0.13 |  -
$\mu$($3^{+}$) |  1.746(3) |  1.77 |  1.81 |  1.83 |  -
$B(E2;1_{1}^{+}$ $\rightarrow$ $3_{1}^{+}$) |  0.03457(29) |  $<$ 0.01 |  0.22 |  0.60 |  -
$B(E2;5_{1}^{+}$ $\rightarrow$ $3_{1}^{+}$) |  19.0(15) |  3.87 |  6.12 |  7.15 |  -
$B(M1;0_{1}^{+}$ $\rightarrow$ $1_{1}^{+}$) |  4.96(18) |  4.37 |  7.90 |  12.60 |  -
23Na |  EXPT |  INOY |  CDB2K |  N3LO |  VS-IMSRG
$E_{g.s.}(3/2^{+})$ |  -186.55 |  -193.90 |  -172.82 |  -175.79 |  -
$Q(3/2^{+})$ |  0.104(1) |  0.05 |  0.06 |  0.07 |  -
$\mu$($3/2^{+}$) |  2.21750(3) |  2.03 |  2.20 |  2.30 |  -
$B(E2;5/2_{1}^{+}$ $\rightarrow$ $3/2_{1}^{+}$) |  124(23) |  18.38 |  26.38 |  30.53 |  56.9
$B(E2;7/2_{1}^{+}$ $\rightarrow$ $3/2_{1}^{+}$) |  47.4(58) |  6.95 |  11.55 |  14.67 |  -
$B(M1;5/2_{1}^{+}$ $\rightarrow$ $3/2_{1}^{+}$) |  0.403(25) |  0.35 |  0.30 |  0.31 |  -

###  IV.5 Point-proton radius and neutron skin

We have also investigated the point-proton radii of sodium isotopes in
addition to the low-energy spectra and electromagnetic properties. The point-
proton radius is a long-range operator like the $B(E2)$ operator. Both of
these two operators are sensitive to the long-range part of the nuclear
wavefunctions. So, to obtain converged results for these observables, NCSM
calculations with large $N_{max}$ model space are required. The point-proton
radius is extracted from the experimental charge radius by using the following
formula:

$\langle r^{2}\rangle_{p}=\langle r^{2}\rangle_{c}-\langle
R_{p}^{2}\rangle-\frac{N}{Z}\langle R_{n}^{2}\rangle-\frac{3}{4m_{p}^{2}}\\\ $
(6)

Here, $\langle R_{p}^{2}\rangle$ and $\langle R_{n}^{2}\rangle$ are,
respectively the squared charge radius of proton and neutron. The last term of
the equation is the Darwin-Foldy term related to relativistic correction in
natural units. These values are taken to be $\langle R_{p}^{2}\rangle^{1/2}$ =
0.8783(86) fm, $\langle R_{n}^{2}\rangle$ = -0.1149(27) fm2 charge_radii  and
3/(4$m_{p}^{2}$) = 0.033 fm2 C.Forseen . The squared point-proton radius
relative to the center of mass of all nucleons is evaluated with

$r_{p}^{2}=\frac{1}{Z}\sum_{i=1}^{Z}\lvert\vec{r_{i}}-\vec{R}_{CM}\rvert^{2}.$
(7)

The operator in the above equation is a two-body operator. It is reduced to a
more suitable form involving one-body and two-body operators to evaluate two-
body matrix elements for this operator. After finding the two-body matrix
elements, the expectation value of the $r_{p}$ operator is similar to the
calculations of the ground-state energies. A similar method is applied to
calculate atomic nuclei’s point-neutron radii ($r_{n}$); however, experimental
measurement of $r_{n}$ is challenging, unlike $r_{p}$ measurement.

In Figure 15, the calculated $r_{p}$ and $r_{n}$ of 21,22Na are shown as
function of $\hbar\Omega$ for $N_{max}$ = 2 and 4 model spaces. The figure
shows that at a lower HO frequency, the calculated $r_{p}$ and $r_{n}$
decrease with increasing $N_{max}$, while at a high HO frequency, they
increase with increasing $N_{max}$. So, a region of calculated $r_{p}$ and
$r_{n}$ around the crossing points is observed, which is independent of
$N_{max}$. That crossover can be taken as a true converged point-proton radius
rp_1 ; rp_2  and point-neutron radius. This work takes the intersection points
of two $r_{p}$ and $r_{n}$ curves corresponding to $N_{max}$ = 2 and 4 as the
converged radii. For example, the first panel of Figure 15 shows that the
$r_{p}$ curves of 21Na for INOY interaction crosses each other almost at 2.30
fm. Similarly, the crossing point for $r_{n}$ curves is at 2.68 fm. So, the
point-proton and point-neutron radii of 21Na corresponding to INOY interaction
are 2.30 and 2.68 fm, respectively. Similarly, the converged $r_{p}$ and
$r_{n}$ are calculated for all four sodium isotopes corresponding to the three
realistic interactions considered. The calculated $r_{p}$ from the NCSM method
are reported in Table 3 and compared with the experimental data which are
taken from charge_radii . From Figure 16, we see that the converged $r_{p}$
are underpredicted by NCSM results with all three interactions. While the N3LO
interaction shows the correct trend of experimental $r_{p}$, the INOY and
CDB2K show odd-odd sodium isotopes 20,22Na to have large $r_{p}$ compared to
odd-even sodium isotopes 21,23Na. Out of the three realistic interactions, the
N3LO reproduces slightly better results compared to the other two. This is due
to large occupancies of the $\pi(1s_{1/2}$) orbital for N3LO interaction
compared to INOY and CDB2K interactions as can be seen from the right side
figure of Figure 11.

Figure 15: Root mean square radii of point-proton (solid line) and point-
neutron (dotted line) distribution as a function of HO frequency for different
$N_{max}$. Table 3:  The point-proton radii (rp) of 20-23Na isotopes
calculated with INOY, CDB2K and N3LO are shown. The experimental data are
taken from charge_radii . All the units are given in fm.

rp Expt. INOY CDB2K N3LO 20Na 2.8494(445) 2.30 2.43 2.48 21Na 2.8951(308) 2.10
2.21 2.46 22Na 2.8674(20) 2.44 2.37 2.45 23Na 2.8779(104) 2.22 2.36 2.41

From the knowledge of both $r_{p}$ and $r_{n}$, the neutron skin thickness,
$r_{np}$ can be calculated using $r_{np}$ = $r_{n}$ \- $r_{p}$. In Figure 17,
the neutron skin thicknesses of the sodium isotope ground states are shown.
The experimental data for the figure is taken from Na_skin . From the figure,
we see that the calculated $r_{np}$ of 20,23Na g. s. are within the
experimental error for all three interactions. While $r_{np}$ of 22Na is in
good agreement with the experimental data for INOY, for the other two
interactions, they are slightly away from the experimental data. A significant
mismatch is observed for 21Na $r_{np}$ corresponding to INOY and CDB2K
interactions. However, the N3LO result for the same is close to the
experimental value.

Figure 16: Comparison of the calculated ground state point-proton radii
($r_{p}$) of Na-isotopes with the experimental data. Figure 17: Comparison of
the calculated ground state neutron skin thickness ($r_{np}$) of Na-isotopes
with the experimental data.

##  V Conclusions

In this work, we have investigated the low-lying nuclear structure properties
of sodium isotopes 20-23Na within the ab initio NCSM formalism using three
realistic interactions, namely INOY, CDB2K, and chiral N3LO. We studied the
low-energy spectra of Na-isotopes, including both natural and unnatural parity
states, electromagnetic properties, point-proton radii, and neutron skin
thicknesses. We observed a good agreement of the g.s. binding energy of 20Na
for INOY interaction with the experimental data. However, for other Na-
isotopes, INOY interaction overbinds the corresponding g.s. The CDB2K and N3LO
underbind the g.s. of all four sodium isotopes. The INOY interaction results
for the g.s. energies are better than CDB2K and N3LO interactions, this is
because three-body force effects are absorbed in the nonlocal part of the INOY
interaction.

Among the electromagnetic properties, the quadrupole and magnetic moments of
the g.s. follow the same trend as in the experimental data. The $B(M1)$
transition being independent of spatial coordinates, converged results close
to experimental data can be achieved at a smaller basis space. However, the
situation is different for the $B(E2)$ transition that depends on the long-
range part of nuclear wavefunctions. Among the three realistic interactions,
the N3LO reproduces better results for $E2$ transition strength for all four
Na-isotopes. However, only one-third of the experimental transition strengths
are obtained for N3LO interactions. Comparing with $B(E2)$ results of another
ab initio method VS-IMSRG, the $B(E2;5/2_{1}^{+}\to 3/2_{1}^{+})$ for 21Na and
23Na are 56.1 and 56.9 $e^{2}fm^{4}$ for VS-IMSRG, the NCSM calculation with
N3LO provide 33.12 and 30.56 $e^{2}fm^{4}$. Compared to the experimental
results, the $ab$ $initio$ results including VS-IMSRG are significantly less.

The point-proton radii ($r_{p}$) is also a long-range observable, just like
the $B(E2)$. In order to obtain the converged $r_{p}$, we employed the
“crossing-point” method, and the converged $r_{p}$ for three different
interactions are compared to the available experimental data. We observed that
the g.s. $r_{p}$ obtained from NCSM calculation are less than the experimental
data. However, among the three interactions, N3LO reproduces slightly better
results compared to the other two interactions and follows the same
experimental trend. Similarly, we also calculated the converged root mean
square radius of the neutron distribution ($r_{n}$) and the neutron skin
thickness ($r_{np}$) of the g. s. of Na-isotopes. Except for the case of 21Na,
the calculated $r_{np}$ are in good agreement with the experimental data.

## ACKNOWLEDGMENTS

We acknowledge financial support from SERB (India), CRG/2019/000556. We would
like to thank Petr Navrátil for providing us his NN effective interaction code
and Christian Forssén for making available the pAntoine.

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