Estimation of the slope of nuclear symmetry energy via charge radii of mirror
nuclei
Sakshi Gautam1,2, Anagh Venneti2, Sarmistha Banik 2

and B. K. Agrawal3

1 Department of Physics, Panjab University, Chandigarh -160014, India.
2 Department of Physics, Birla Institute of Technology and Science Pilani,
Hyderabad Campus, Hyderabad- 500078, India.
3 Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata-700064, India.
Email<EMAIL_ADDRESS>
###### Abstract

Charge radii of mirror nuclei are calculated by implementing pairing effects
with the Hartree-Fock Bogoliubov approximation. Correlations between the
difference of charge radii ($\Delta R_{ch}$) and slope of nuclear symmetry
energy (L) are examined for different mirror nuclei pairs of varying masses
using 40 different Skyrme energy density functionals. $\Delta R_{ch}-L$
correlations are found to be robust for the binding constraints imposed on
density functionals. We observe that $\Delta R_{ch}$ and $L$ show better
correlations in relatively heavier pairs than those obtained in the lighter
pairs. Our calculations impose a constraint on the slope of nuclear symmetry
energy as -20 MeV $\leq L\leq$ 55 MeV with 68% confidence band using available
measurements on charge radii. This is a moderately soft symmetry energy, in
contrast to stiff and soft symmetry energy indicated by PREX-II and CREX
measurements of neutron skin thickness in ${}^{208}Pb$ and ${}^{48}Ca$,
respectively. Our result is also in agreement with celestial constraints
obtained from observational data for neutron stars.

##  1 Introduction

Energy per particle for asymmetric nuclear matter can be expressed
approximately as [1];

$E(\rho,\delta)=E(\rho,0)+E_{sym}(\rho)\delta^{2},$ (1)

where $E(\rho,0)$ is the energy of symmetric nuclear matter,
$\delta=\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}}$ is the isospin asymmetry
parameter and $\rho_{n/p}$ is the density of neutrons/protons, respectively.
$E_{sym}$ is the nuclear symmetry energy, which represents the energy
difference between pure neutron matter ($\delta=1$) and symmetric nuclear
matter ($\delta=0$). It measures the change in the nuclear binding of the
system as the neutron-to-proton ratio is varied at a fixed baryonic number.
The nuclear symmetry energy around saturation density $\rho_{0}$ is expanded
up to second order as;

$\displaystyle E_{sym}(\rho)$ $\displaystyle=$ $\displaystyle
J+L\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)+\frac{1}{2}K_{sym}\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)^{2},$
(2) where $\displaystyle J$ $\displaystyle=$ $\displaystyle
E_{sym}(\rho_{0}),$ (3) $\displaystyle L$ $\displaystyle=$ $\displaystyle
3\rho_{0}\left(\frac{\partial E_{sym}}{\partial\rho}\right)_{\rho=\rho_{0}},$
(4) $\displaystyle K_{sym}$ $\displaystyle=$ $\displaystyle
9\rho_{0}^{2}\left(\frac{\partial^{2}E_{sym}}{\partial\rho^{2}}\right)_{\rho=\rho_{0}}.$
(5)

where $J$ is the nuclear symmetry energy at saturation density, L is the slope
of nuclear symmetry energy and $K_{sym}$ is the curvature of nuclear symmetry
energy. The slope of nuclear symmetry energy (L) is estimated from the
pressure of pure neutron matter at saturation density and is related as;

$P_{N}(\rho_{0})\approx\frac{\rho_{0}L}{3}.$ (6)

The value of $J$ is found to be close to 32 MeV from various studies on
nuclear structure [1]. However, it is the slope of nuclear symmetry energy
(L), which is crucial to understand the extrapolation of symmetry energy to
lower and higher densities [2, 3]. A correct estimation of L is important to
study the properties of proton- or neutron-rich heavy nuclei as well as to
understand the physics of neutron stars.

Strong interactions play a significant role for both neutron star matter and
the finite nuclei. Neutron degeneracy pressure acts against inward
gravitational pull in the former, unlike in the finite nucleus, wherein it
acts against surface tension. In finite nuclei, a higher value of L, due to an
increase in neutron pressure, will push the neutron surface (see eqn. 6), thus
leading to an enhanced neutron distribution in neutron-rich nuclei. Therefore,
the slope of symmetry energy determines neutron-skin thickness, _viz., the
difference in the root mean square (r.m.s.) radii of neutrons and protons
distributions_ , in finite nuclei as well as the radius of neutron stars. Note
that curvature of symmetry energy, $K_{sym}$, also plays a significant role in
astrophysical processes, where high densities are involved and its value is
poorly constrained at present [4, 5]. Thus, multi-messenger observations from
the merging of neutron stars provide excellent probes of high-density nuclear
matter to complement the knowledge gained from terrestrial measurements

Since, $L$ is not a directly measurable quantity, various probes of L have
been proposed in recent times from the studies involving neutron skins in
finite nuclei [2, 6, 7, 8, 9], collective flows, and fragmentation observables
in heavy-ion collisions [10, 11, 12, 13]. In addition, astrophysical
observations of Gravitational waves relating to tidal deformability and radius
of neutron stars [14, 15, 16, 17] have been employed to constrain the values
of $L$. The values of $L$ estimated from different observables are usually at
variance, thus, the density dependence of symmetry energy remains
inconclusive. Neutron skin thickness for $Pb^{208}$ and $Ca^{48}$ nuclei have
been measured by PREX-I/II [18, 19, 20] and CREX [21] experiments,
respectively, to obtain constraints on the values of $L$. For example,
improved neutron skin thickness of $Pb^{208}$ by PREX-II hints towards a
stiffer symmetry energy. The strong correlation between neutron skin thickness
in $Pb^{208}$ and $L$ yields $L=106\pm$37 within the relativistic energy
density functional [22]. On the other hand, neutron skin thickness for
$Ca^{48}$ nucleus from CREX results in much softer symmetry energy with L =
30.61 $\pm$ 6.74 MeV [23] with relativistic energy density functionals. This
disagreement in the simultaneous reproduction of both measurements reflects an
incomplete knowledge of the equation of the state of neutron-rich matter.
Similar conclusions of marginal overlap of L values are reported by Tagami et
al. [24] with around 200 equations of state. They proposed L values for CREX
and PREX-II results to be between 0-51 and 76-165 MeV, respectively. Likewise,
a tension between CREX and PREX results is recently reported in Ref. [25],
where Bayesian inference of nuclear symmetry energy and neutron skins in
$Pb^{208}$ and $Ca^{48}$ is performed. The values of symmetry energy are
inferred separately from CREX and PREX-II, compatible with each other at
90$\%$ confidence interval and inconsistent with each other at 68.3$\%$
confidence interval. Similar inadequacy of relativistic energy density
functionals in the simultaneous understanding of CREX and PREX-II results is
reported in Ref. [26] as well. Therefore, there is a large stimulus in the
theoretical nuclear physics community to further investigate the slope of
nuclear symmetry energy.

The advancements in laser beam technology result in the precise measurements
of nuclear charge radii [27] and thus open new opportunities in estimating
neutron-skin thicknesses, under isospin symmetry of nucleon-nucleon
interaction. At the same time, measuring neutron radii is a bit challenging
and involves sensitive electro-weak interactions. An alternative clean
electromagnetic probe, _viz_., charge radii in mirror nuclei, is proposed to
estimate the slope of nuclear symmetry energy [28]. It is based on the
assumption that under perfect charge symmetry, the neutron radius in a given
nucleus is the same as the proton radius in its mirror counterpart. Therefore,
neutron skin thickness ($\Delta R_{np}$) in a nucleus is equivalent to the
difference in proton radii ($\Delta R_{ch}$) of mirror nuclei pairs (under no
Coulomb interactions) [28, 29]; i.e.,

$R_{n}(X^{N}_{Z})-R_{p}(X^{N}_{Z})=R_{p}(Y^{Z}_{N})-R_{p}(X^{N}_{Z}),$ (7)

where $X^{N}_{Z}$ and $Y^{Z}_{N}$ are mirror nuclei pairs. As mentioned above,
the foremost attempt was carried out by Brown for the mirror nuclei pairs of
${}^{34}\textrm{Ar}$-34S, ${}^{52}\textrm{Ni}$-52Cr and
${}^{54}\textrm{Ni}$-54Fe using 48 Skyrme energy density functionals [28]. It
was shown that the difference in charge radii is proportional to $(N-Z)\times
L$. Later, this linear correlation between $\Delta R_{ch}$ and L was also
confirmed by Yang and Piekarewicz [29], using relativistic density
functionals. Following these studies, constraints on L are established using
recent precise measurements on radii of proton-rich nuclei of 36Ca/38Ca and
54Ni. The deduced values of L are 5-70 MeV [30] and 21-88 MeV [31],
respectively. Another very recent attempt in this direction was carried out by
An _et al._ [32], where $\Delta R_{ch}$ and L correlations are observed for
different pairs using six relativistic and six non-relativistic energy density
functionals. Their deduced slope values from comparison with measured radii
are reported to be between 22.5 and 51.55 MeV. Recently, correlations between
charge radii differences of mirror nuclei and neutron star observables are
also investigated [33]. Therefore, we see that a lot of studies on charge
radii calculations are still in progress for obtaining a stringent constraint
on L. However, this claim of establishing a clean probe of L was challenged in
a study by Reinhard and Naraewicz [34], wherein it is observed that $\Delta
R_{ch}$ is an inferior indicator of L as it is influenced by pairing
correlations in nuclei. Lately, the above finding was verified by Huang _et
al_. [35] for different mirror nuclei pairs for around 20 Skyrme functionals.
These investigations thus pointed out that even precise measurements of
$\Delta R_{ch}$ cannot put any stringent constraints on the slope of the
nuclear symmetry energy.

In the present work, we shall re-visit the $\Delta R_{ch}$ $vs$ L correlation
in light of pairing effects using 40 Skyrme energy density functionals. We
shall also examine the robustness of these correlations by further considering
only those density functionals that predict nuclear binding energies to be
within $\pm 10$ MeV of the measured nuclear bindings. Further, relatively
better correlations of $\Delta R_{ch}$ $vs$ L are seen in heavier nuclei, and
possible bounds on L are reported. A brief outline of the theoretical
framework is presented in Section II and results are discussed in Section III,
with a summary in Section IV.

##  2 Theoretical approach

The present investigation of the correlation between the difference in charge
radii of mirror nuclei and the slope of nuclear symmetry energy is carried out
for 40 Skyrme density functionals, for which L varies between approximately
-30 MeV to 160 MeV. This wide range of L covers most of the values of L
employed in previous studies. Various Skyrme forces used in this work are
listed in Table LABEL:table:1. The values of saturation density, binding
energy per nucleon at saturation, incompressibility coefficient, symmetry
energy coefficient, slope of symmetry energy, and curvature of symmetry energy
for various functionals are also listed in the Table.

The nuclei involved in the present study are open-shell nuclei, therefore,
pairing effects will be significant and are treated using Hartree-Bogoliubov
transformations [55]. The Hartree-Fock-Bogoliubov equation for nucleons is:

$\begin{pmatrix}h&\Delta\\\
-\Delta^{*}&-h^{*}\end{pmatrix}\begin{pmatrix}U_{i}\\\
V_{i}\end{pmatrix}=E_{i}\begin{pmatrix}U_{i}\\\ V_{i}\end{pmatrix}$ (8)

in which $E_{i}$ is the quasi-particle energy, $(U_{i},V_{i})$ is the quasi-
particle wave function and h is the single-particle Hamiltonian of the
nucleon, which comprises contributions from the kinetic term, mean-field term,
and Fermi energy. Here $\Delta$ represents the pairing potential:

$\Delta_{jk}=\frac{1}{2}\sum_{lm}\langle jk|V^{n,p}|lm\rangle\lambda_{lm}$ (9)

where $\lambda$ is the pairing tensor and $V^{n,p}$ is the pairing force. The
pairing channel is parameterized by a density-dependent delta-pairing force
with mixed volume and surface features, as:

$V^{n,p}_{pair}(r)=V^{n,p}_{0}(r)\left(1-\frac{1}{2}\frac{\rho(r)}{\rho_{0}}\right)\delta(r-r^{{}^{\prime}}),$
(10)

with $V^{n,p}_{0}(\bf r)$ the pairing strength for neutrons (n) and protons
(p) (specific for each density functional), $\rho(\bf r)$ the isoscalar local
density. As the pairing force introduced is of zero range, a cut-off in the
quasi-particle space is introduced, which we take as 60 MeV in the present
calculations.

The charge radii ($R_{ch}$) of different nuclei are calculated using the
relation:

$R_{ch}=\sqrt{R_{p}^{2}+0.8^{2}}.$ (11)

Here, $R_{p}$ is the point proton radius, and the factor of 0.8 accounts for
finite-size correction [56, 57]. The correlation between the difference of
charge radius in mirror nuclei and the slope of symmetry energy is analyzed
through least-square analysis and is quantified using the Pearson correlation
coefficient.

Table 1: Values of saturation density ($\rho_{0}$), binding energy per nucleon at saturation density ($E_{0}$), incompressibility coefficient (K), symmetry energy coefficient (J), slope of symmetry energy (L) and curvature of symmetry energy (Ksym) for various Skyrme energy density functionals. All the quantities are in units of MeV, except for $\rho_{0}$ which is in fm-3. . |  |  |  |  |  |  |
---|---|---|---|---|---|---|---
Force | $\rho_{0}$ | $E_{0}$ | K | J | L | $K_{sym}$ | Ref.
Zs | 0.16 | -15.86 | 233.12 | 26.71 | -29.27 | -401.43 | [45]
MSK3 | 0.157 | -15.79 | 233.24 | 28.00 | 7.04 | -283.52 | [51]
BSK1 | 0.157 | -15.80 | 231.29 | 27.81 | 7.19 | -281.83 | [46]
MSK4 | 0.157 | -15.79 | 231.16 | 28.00 | 7.20 | -284.05 | [51]
MSK8 | 0.157 | -15.79 | 229.30 | 27.93 | 8.26 | -280.01 | [52]
MSK7 | 0.157 | -15.79 | 231.21 | 27.95 | 9.41 | -274.63 | [52]
MSK6 | 0.157 | -15.79 | 231.16 | 28.00 | 9.63 | -274.33 | [51]
SIII | 0.145 | -15.85 | 355.35 | 28.16 | 9.91 | -393.73 | [36]
MSK9 | 0.157 | -15.80 | 233.31 | 27.99 | 10.37 | -270.23 | [52]
SKP | 0.162 | -15.95 | 201.02 | 30.00 | 19.67 | -266.60 | [38]
SKX | 0.155 | -16.06 | 271.11 | 31.09 | 33.17 | -252.12 | [39]
MSK2 | 0.157 | -15.83 | 231.63 | 30.00 | 33.35 | -203.44 | [51]
SKXC | 0.155 | -15.86 | 268.16 | 30.20 | 33.6 | -238.39 | [39]
MSK1 | 0.157 | -15.83 | 233.72 | 30.00 | 33.9 | -200.02 | [51]
SGII | 0.158 | -15.59 | 214.63 | 26.83 | 37.6 | -145.90 | [37]
HFB9 | 0.159 | -15.92 | 231.42 | 30.00 | 39.89 | -150.03 | [44]
UNE1 | 0.159 | -15.80 | 220.00 | 28.99 | 40.00 | -176.51 | [48]
KDE | 0.164 | -15.97 | 223.69 | 31.97 | 41.4 | -141.83 | [49]
UNE0 | 0.160 | -16.05 | 229.99 | 30.54 | 45.08 | -185.74 | [47]
KDE0 | 0.161 | -16.08 | 228.49 | 32.99 | 45.23 | -144.78 | [49]
SKM* | 0.160 | -15.78 | 216.66 | 30.03 | 45.77 | -155.94 | [42]
SLY4 | 0.159 | -15.97 | 229.90 | 32.00 | 45.96 | -119.73 | [50]
SLY7 | 0.158 | -15.90 | 229.74 | 31.99 | 46.94 | -114.34 | [50]
SLY6 | 0.159 | -15.92 | 229.84 | 31.96 | 47.45 | -112.71 | [50]
SKb | 0.155 | -15.99 | 263.14 | 23.88 | 47.5 | -78.46 | [40]
SLY5 | 0.161 | -15.98 | 229.91 | 32.01 | 48.15 | -112.76 | [50]
KDE1 | 0.164 | -16.21 | 227.33 | 34.58 | 54.7 | -127.12 | [49]
SKT1 | 0.161 | -15.98 | 236.11 | 32.01 | 56.2 | -134.83 | [53]
SKI4 | 0.159 | -15.93 | 235.48 | 35.45 | 60.43 | -40.56 | [54]
SIV | 0.151 | -15.96 | 324.53 | 31.22 | 63.49 | -136.72 | [36]
SKOP | 0.160 | -15.75 | 222.31 | 31.94 | 68.9 | -78.83 | [41]
SKa | 0.155 | -15.99 | 263.14 | 32.91 | 74.6 | -78.46 | [40]
SKO | 0.160 | -15.83 | 223.32 | 31.97 | 79.14 | -43.17 | [41]
S272 | 0.155 | -16.28 | 271.49 | 37.39 | 91.67 | -67.79 | [43]
SKT4 | 0.159 | -15.94 | 235.48 | 35.45 | 94.1 | -24.46 | [53]
S255 | 0.157 | -16.33 | 254.92 | 37.39 | 95.05 | -58.34 | [43]
SKI3 | 0.157 | -15.96 | 257.97 | 34.83 | 100.49 | 73.04 | [54]
SKI2 | 0.157 | -15.73 | 240.37 | 33.33 | 104.16 | 70.69 | [54]
SKI5 | 0.155 | -15.83 | 255.57 | 36.61 | 129.17 | 159.97 | [54]
SKI1 | 0.161 | -15.98 | 242.53 | 37.51 | 160.96 | 234.67 | [53]

##  3 Results and discussion

The present study is carried out for five different pairs of mirror nuclei:
14O$-$14C, 18Ne$-$18O, 44Cr$-$44Ca, 58Zn$-$58Ni and 60Ge$-$60Ni. The mass of
these pairs ranges from 14 to 60 units, and this range covers the mass units
for which experimental measurements of charge radii are also available. As
mentioned previously, the present investigation is done for 40 different
Skyrme energy density functionals taking into account the pairing correlations
as discussed in Section 2.

Figure 1: $\Delta R_{ch}$ as a function for L for the pairs of 14O $-$18C,[a &
f] 18Ne $-$18O [b & g], 44Cr $-$ 44Ca [c & h], 58Zn $-$ 58Ni [d & i] and 60Ge
$-$ 60Ni [e & j]. Left and right panels display the results with and without
pairing effects. The dashed lines represent least-square linear fits and the
corresponding Pearson correlation coefficients are mentioned in the panels.

We have performed the calculations for several nuclei considered with and
without the inclusion of the pairing term. In Fig. 1, the difference in the
charge radius of mirror nuclei as a function of the slope of symmetry energy
are plotted as solid circles obtained with pairing (left panels) and without
pairing term (right panels) for five pairs of mirror nuclei. The dashed lines
represent the least-square linear fits. From the figure, we notice that there
is almost no correlation for the 14O $-$ 14C pair. However, correlations
significantly improve for the rest of the pairs of mirror nuclei considered,
as reflected in the values of Pearson correlation coefficients which are 0.66,
0.83, 0.8 and 0.8 for 18Ne $-$18O, 44Cr $-$ 44Ca, 58Zn $-$ 58Ni and 60Ge $-$
60Ni, respectively.

Recent studies in Refs. [34, 35] have shown that the inclusion of pairing
effects weakens the correlation between the difference in charge radii of
mirror nuclei and L. To verify this aspect here, we have performed the
calculations without taking into account the pairing effects. This is achieved
by setting the pairing strength to zero and the corresponding results are
shown in the right panels of Fig. 1. Once again a poor correlation is found in
14O $-$ 14C pair. The value of Pearson correlation coefficients now enhance
and are 0.76, 0.9, 0.89, and 0.96 for 18Ne $-$18O, 44Cr $-$ 44Ca, 58Zn $-$
58Ni and 60Ge $-$ 60Ni, respectively. Our findings thus establish the results
reported in Refs. [34, 35]. It is worth mentioning that correlations reported
in the findings of Ref. [35] are different than those obtained in the present
study. This could be due to different Skyrme energy density functionals used
in both studies and different treatments of pairing force (only surface
contribution taken into account in Ref. [35]). Moreover, $\Delta R_{ch}$ and L
correlations are also affected by the correlation between the curvature of
symmetry energy $K_{sym}$ and L [35], which is different in both studies. This
aspect will be discussed later in Fig 3. It is worth mentioning that pairing
force also plays a significant role in the static and dynamics aspects of
fission phenomena experienced by heavy atomic nuclei as well [58, 59]. For
example, pairing correlations have a strong impact on the lifetime of
spontaneous fission [58] as well as on fission yields of $Pu^{240}$ [59].

Figure 2: Same as Fig. 1, but for density functionals with binding energy
constraint of $\pm 10$ MeV (for details, see text).

Next, we assess the role of pairing effects on the robustness of these
correlations. For this, we re-examine $\Delta R_{ch}$ $vs$ L correlation for
Skyrme forces that reproduce the nuclear bindings of corresponding nuclei to
be within $\pm 10$ MeV of experimental values. The data on nuclear bindings is
taken from AME2020 atomic mass table [60]. The imposition of binding energy
constraint filters out various Skyrme forces. The results with binding energy
constraints are displayed in Fig. 2. We observe that $\Delta R_{ch}$ $vs$ L
correlations do not change significantly with this imposed constraint of
binding energy. A poor correlation in 14O $-$ 14C pair still exists and
relatively stronger correlations without pairing effects are also observed.
Moreover, relatively heavier mirror nuclei pairs show relatively good
correlations between $\Delta R_{ch}$ and L. This analysis thus proves the
robustness of these correlations and establishes the fact that pairing effects
do weaken the correlation between the difference in charge radii of mirror
nuclei and the slope of nuclear symmetry energy. Note that we do not see any
clear systematics in mass effects on the Pearson Correlation coefficient as
differences in mirror charge radii are also influenced by the neutron-proton
asymmetry of the pair [28].

Figure 3: (a) The Curvature of symmetry energy $K_{sym}$ as a function of the
slope of symmetry energy L. Regression line is shown and a strong correlation
is seen. (b) Pearson correlation coefficient for $\Delta R_{ch}$ and L
correlation in the mirror nuclei pair of ${}^{60}\textrm{Ge}-^{60}$Ni for
calculations with different pairing strengths.

To understand the difference between the correlations obtained in the present
study and Ref. [35], we have shown the correlation between the curvature of
symmetry energy and slope of symmetry energy in Fig. 3 (upper panel). We see
that a relatively stronger correlation (of 0.97) is seen in the present study,
which can change the correlations between $\Delta R_{ch}$ and L. Therefore,
types of density functionals (Skyrme or relativistic) change the correlation
between $\Delta R_{ch}$ and L as the curvature of symmetry energy influences
this correlation, and thus different values of Pearson correlation coefficient
are reported in the literature even for same mirror nuclei pair.

Next, we also examine whether the inclusion of pairing effects systematically
decreases the correlation or not. We performed the calculations for the
heaviest pair of 60Ge $-$ 60Ni, where the best correlation is seen, with
pairing effects implemented to their $80\%$ of the full strength. This is
achieved by scaling the pairing strengths of neutrons and protons in different
Skyrme forces accordingly and results are shown in Fig. 3 (bottom panel). From
the figure, we see that the Pearson correlation coefficient obtained with
$80\%$ pairing strength lies in between those obtained without pairing effects
and with full pairing strength. Note that when pairing strengths are reduced
below $80\%$, then pairing effects become too feeble for most of the Skyrme
forces, to have any observable effect and results are close to those obtained
without pairing force taken into account.

Figure 4: $\Delta R_{ch}$ as a function of L for pairs of 36Ca $-$ 36S [a], 34Ar $-$ 34S [b], 38Ca $-$ 38Ar [c], and 54Ni $-$ 54Fe [d]. Experimental measurements are shown by horizontal bands and data is taken from Brown _et al_. 2020 [a,c] [30], Brown 2017 [28] [b], and Pineda _et al._ 2021 [d] [31]. Dashed lines represent the least square linear fits and dark (light) shaded regions represent 68% (95%) confidence bands. Table 2: Experimental values of charge radii Rch and difference in charge radii $\Delta R_{ch}$ for mirror nuclei pairs of 34Ar $-$ 34S, 36Ca $-$ 36S, 38Ca $-$ 38Ar and 54Ni $-$ 54Fe. Values in the parentheses indicate systematic uncertainties. Nucleus | $R_{ch}$ | $\Delta R_{ch}$ | Ref.
---|---|---|---
Ar34 | 3.3657(21) |  |
S34 | 3.284(2) | 0.082(9) | [28]
Ca36 | 3.4484(27) |  |
S36 | 3.2982(12) | 0.150(4) | [30]
Ca38 | 3.4652(17) |  |
Ar38 | 3.4022(15) | 0.063(3) | [30]
Ni54 | 3.7370(30) |  |
Fe54 | 3.6880(17) | 0.049(4) | [31]

Lastly, we attempt to obtain a constraint on the slope of nuclear symmetry
energy from the experimental findings available on charge radius. Calculated
values of $\Delta R_{ch}$ _vs_ L are plotted as solid circles in Fig. 4. The
measured values of charge radii are available only for a few nuclei and these
pairs are 34Ar $-$ 34S [28], 36Ca $-$ 36S [30], 38Ca $-$ 38Ar [30] and 54Ni
$-$ 54Fe [31]. The values of measured charge radii for different nuclei and
the difference in charge radii are shown in Table 2. Experimental $\Delta
R_{ch}$ values are displayed by horizontal bands in the figure. We observe
that there is a very weak correlation seen in 38Ca $-$ 38Ar, 36Ca $-$ 36S and
34Ar $-$ 34S pairs, with Pearson correlation coefficients lie between 0.3 and
0.5. Therefore, these pairs do not impose any noteworthy constraint on the
slope of the nuclear symmetry energy. On the other hand, a slightly better
correlation is seen in and 54Ni $-$ 54Fe pair, with a Pearson correlation
coefficient of 0.75. Note that almost the same correlation is seen in Ref.
[34], where the Coefficient of Determination is found to be 0.69 (which
corresponds to a correlation coefficient close to 0.8) for 54Ni $-$ 54Fe pair,
with HFB pairing case. It is worth mentioning that correlation is also
significantly affected by the type of pairing scheme [34]. Dark (light) shaded
regions display 68% (95%) confidence bands. Therefore, 54Ni $-$ 54Fe pair
imposes a relatively better constraint on the slope of the nuclear symmetry
energy. The predicted values of L are $-20\leq L\leq 55$ for 68% confidence
interval for 54Ni $-$ 54Fe pair. The maximum value of L becomes 62 MeV in 95%
confidence interval. Incidentally, this constraint is close to the L value
reported recently by Lattimer _et al._ [61] from combined analysis of neutron
skin measurements and neutron star properties. Note that this value of L hints
towards moderately soft symmetry energy and lies in between those estimated by
CREX and PREX measurements, where soft and stiff forms of symmetry energy are
inferred, respectively. The intermediate values of L are also proposed by Hu
_et al._ [62], where calculations using ab-inito approach for nuclear forces
from chiral EFT are performed. Their predicted slope parameter $L=37-66$ MeV
is found to be consistent with nuclear-scattering data as well. The above
constraint is also in agreement with the one reported by S$\pi$RIT
collaboration [63], where spectral pion ratio at high transverse momenta in
Sn+Sn collisions is utilized to deduce the slope of the nuclear symmetry
energy. Further, this moderately softer form of nuclear symmetry energy is
also in consensus with a recent study [64], where EoS is constrained using
observational data on neutron stars and low-mass X-ray binaries. An optimal
range of L is found to be $50.79^{+15.16}_{-9.24}$ MeV in 68% confidence
interval when Bayesian analysis is done using priors chosen from PREX-II
measurements [22, 63, 65, 66]. Therefore, from our analysis, we conclude that
the more accurate difference in the charge radii in heavier mirror nuclei
pairs provides an alternative and independent probe to constrain the slope of
the symmetry energy. Thus, new measurements on the charge radii of relatively
heavier nuclei, along with neutron-skin measurements will be crucial for
obtaining a much more stringent constraint on the slope of nuclear symmetry
energy.

##  4 SUMMARY

Correlations between the difference in charge radii of mirror nuclei ($\Delta
R_{ch}$) and the slope of nuclear symmetry energy (L) are revisited by
invoking pairing effects in the calculations. The said correlation is
calculated for mirror nuclei pairs of 14O $-$14C, 18Ne $-$18O, 44Cr $-$ 44Ca,
58Zn $-$ 58Ni and 60Ge $-$ 60Ni using 40 Skyrme energy density functionals. It
is seen that $\Delta R_{ch}$ is better correlated with L in heavier nuclei
relative to those in lighter counterparts, with much stronger correlations
obtained when pairing effects are not taken into consideration. These features
remain preserved when density functionals are constrained for nuclear binding
energy which reflects the robustness of these correlations. An attempt to
derive a feasible value for slope of symmetry energy is made by comparing the
calculations on charge radii differences with measurements of 34Ar $-$ 34S,
36Ca $-$ 36S, 38Ca $-$ 38Ar, and 54Ni $-$ 54Fe pairs. Our constraint from
measured $\Delta R_{ch}$ hints at moderately soft symmetry energy, with
predicted values of L are -20 MeV $\leq L\leq$ 55 MeV with 68% confidence
band, which is in harmony with recent studies done on simultaneous
compatibility of neutron skins reported in CREX and PREX-II data and
astrophysical observations on neutron stars. Further, our study predicts that
new measurements on the charge radii of relatively heavier nuclei will be
fruitful in achieving independent constraints on the slope of the nuclear
symmetry energy. Such constraints can then be complemented with measurements
of neutron skin to impose much better information about the behavior of
nuclear symmetry energy.

##  5 Acknowledgements

AV would like to acknowledge the CSIR for the support through the CSIR-JRF
09/1026(16303)/2023-EMR-I. SG partially acknowledges CSIR for the support
vide. grant no. 03/1513/23/EMR-II.

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