# Accuracy of the mean-field theory in describing ground-state properties of
light nuclei

Yu-Ting Rong<EMAIL_ADDRESS>Guangxi Key Laboratory of Nuclear Physics
and Technology, Guangxi Normal University, Guilin, 541004, China

###### Abstract

The ground-state properties of helium, beryllium, and carbon isotopes are
studied using the relativistic mean-field model and its corrections, including
three types of center-of-mass corrections and two types of rotational
corrections. The accuracy of the mean-field approach in describing the binding
energies is investigated, and a mass-dependent behavior of the mean-field
approximation is obtained by averaging the results calculated from six
selected effective interactions. Quadrupole deformations, root-mean-square
charge radii, root-mean-square matter radii, and neutron skins are also
discussed for each isotopic chain. Our conclusion is that, unlike heavy
nuclei, the exchange terms in the center-of-mass correction for describing
radius cannot be neglected for light nuclei. The mean-field approximation with
center-of-mass and rotational corrections effectively describes the energies
and radii of light nuclei, but it fails to describe the deformations and shell
evolutions of the helium and beryllium isotopic chains due to the absence of
many-body corrections between valence neutrons.

###### pacs:

##  I INTRODUCTION

Light nuclei have attracted considerable attention due to their unique
properties. One of these properties is the cluster phenomenon [1, 2, 3], such
as the presence of two-alpha cluster structure in 8Be, the Hoyle state in 12C
[4, 5, 6], and the tetrahedral shape in 16O [7, 8, 9]. Additionally, with the
development of rare-isotope beam facilities, the exploration of the drip line
[10, 11] associated with the isospin limitation and the halo phenomena in
extremely neutron- or proton-rich nuclei [12] has become a hot topic. Another
important aspect is the shell evolution, where both experimental and
theoretical investigations have suggested the possible existence of the magic
number 6 in some semi-magic unstable light nuclei. Notably, there have been
proposals for subshell closures in 8He [13, 14] and 14C [15, 16]. Conversely,
the traditional magic number 8 is significantly compromised in 12Be [17, 18,
19].

Since a light nucleus consists of few nucleons, a real Hamiltonian for it can
be constructed with bare nucleon-nucleon interactions obtained from
scattering. Together with the improvements in computer performance, ab initio
methods are being utilized to study exotic nuclear properties in light nuclei
from first principle, and these methods are now being extended to heavier
regions as well [1] . While density functional theories (DFTs) [20] using
universal effective interactions have been successful in describing many
nuclear phenomena for nuclei with mass numbers $A\geq 16$, it is commonly
thought that DFTs are not suitable for light nuclei. This is due to the mean-
field approximation used in DFTs, which erases the few-body correlations
between nucleons and poses significant difficulties for accurately describing
properties such as binding energy, charge radius, neutron skin, and surface
thickness in the light mass region.

Covariant density functional theories (CDFTs), which take Lorentz symmetry
into account, provide microscopic frameworks for a global description of
atomic nuclei [21, 22]. These approaches have been extended to study light
nuclei [23, 24, 25, 26, 27, 27, 28, 29, 30], with 11Li serving as a typical
example [23, 24]. In Ref. [23], pairing and continuum effects were considered,
and both the binding energies and radii of the isotopic chain from 6Li to
11Li, as well as the halo structure of 11Li, were successfully reproduced.
Additionally, the $\alpha$-clustering and halo structures in beryllium and
boron nuclei, along with several prominent cluster structures in both the
ground and intrinsic excited states of $\alpha$ nuclei from 12C to 32S, were
well-described by relativistic mean-field (RMF) calculations [25]. Recently,
halo structures in 22C and 17B were also well described by the deformed
relativistic Hartree-Bogoliubov model in continuum (DRHBc model) [28, 29]. The
triangular shape in 12C was studied through parity and angular momentum
projections based on the multidimensionally constrained relativistic Hartree-
Bogoliubov (p-MDCRHB) model [30]. These works suggest that results for light
nuclei calculated with CDFTs and their corrections appear to be reliable, at
least for bulk properties such as binding energies, densities, and single-
particle levels. If this is indeed the case, comparing ab initio calculations
with CDFTs will help us understand the connections between the two methods and
even build new-generation DFTs with realistic nuclear forces, such as the
relativistic chiral nucleon-nucleon interactions [31, 32].

Therefore, in this work, our aim is to systematically study the bulk
properties of nuclei with proton number $Z<8$ using various CDFTs. Our primary
objective is to evaluate the ability of these approaches to accurately
describe the ground-state properties of light nuclei. Specifically, we will
investigate whether the effective interactions, which are constrained by the
properties of heavy nuclei, can provide reliable predictions for the ground-
state properties of light nuclei within the CDFT frameworks. Moreover, we will
examine the ability of CDFTs to accurately describe the properties of 4He,
which serves as the starting nucleus for $\alpha$ cluster structures. We will
also investigate whether the CDFTs can reproduce the observed shell evolutions
in these isotopes.

To do this, we will investigate even-even nuclei in the light mass region,
utilizing three distinct effective interactions: nonlinear meson exchange (NL-
ME), density-dependent meson exchange (DD-ME), and density-dependent point
coupling (DD-PC). Our calculations are based on the multidimensional
constrained relativistic Hartree-Bogoliubov (MDCRHB) model [33, 34],
accounting for both center-of-mass and rotational corrections. For the first
time, we compute the root-mean-square (rms) radii including full microscopic
center-of-mass correction, with exchange terms previously deemed negligible in
the $Z>8$ mass region [35]. In Sec. II, we provide a brief overview of the
MDCRHB model and the corrections applied in this study. In Sec. III, we
present the calculated binding energies, radii, deformations, and potential
energy curves for helium, beryllium, and carbon isotopes, along with detailed
discussions of the results. Finally, in Sec. IV, we summarize our findings.

##  II THEORETICAL FRAMEWORK

###  II.1 Mean-field description

In the present work, the RHB theory is employed to provide a unified
description of the relativistic mean-field and the pairing correlations via
the Bogoliubov transformation [36]. The RHB equation reads

$\int d^{3}\bm{r}^{\prime}\left(\begin{array}[]{cc}h-\lambda&\Delta\\\
-\Delta^{*}&-h+\lambda\end{array}\right)\left(\begin{array}[]{c}U_{k}\\\
V_{k}\end{array}\right)=E_{k}\left(\begin{array}[]{c}U_{k}\\\
V_{k}\end{array}\right),$ (1)

where $h$ is the single-particle Hamiltonian, $\Delta$ is the pairing field,
$\lambda$ is the Fermi energy, $E_{k}$ is the quasiparticle energy, and
$(U_{k},V_{k})^{T}$ is the wave function. The single-particle Hamiltonian

$h=\alpha\cdot p+\beta[m+S(r)]+V(r)+\Sigma_{R}(r),$ (2)

consists of the kinetic energy term, the scalar potential $S(r)$, the vector
potential $V(r)$ and the rearrangement potential $\Sigma_{R}(r)$. $m$ denotes
the mass of nucleon. For meson-exchange interactions,

$\displaystyle S(r)=g_{\sigma}\sigma,$ (3) $\displaystyle
V(r)=g_{\omega}\omega_{0}+g_{\rho}\rho_{0}\cdot\tau_{3}+e\dfrac{1-\tau_{3}}{2}A_{0},$
$\displaystyle\Sigma_{R}(r)=\dfrac{\partial
g_{\sigma}}{\partial\rho_{V}}\rho_{S}\sigma+\dfrac{\partial
g_{\omega}}{\partial\rho_{V}}\rho_{V}\omega_{0}+\dfrac{\partial
g_{\rho}}{\partial\rho_{V}}\rho_{V}\tau_{3}\rho_{0},$

where $g_{\sigma},g_{\omega}$, and $g_{\rho}$ are coupling constants of
$\sigma,\omega_{0}$ and $\rho_{0}$ meson fields, $A_{0}$ is the time-like
component of the Coulomb field, $e$ is the charge unit for protons, $\rho_{S}$
and $\rho_{V}$ are isoscalar and isovector densities. For point-coupling
interactions,

$\displaystyle S(r)=$
$\displaystyle\alpha_{S}\rho_{S}+\alpha_{TS}\rho_{TS}\tau_{3}+\beta_{S}\rho_{S}^{2}+\gamma_{S}\rho_{S}^{3}$
(4)
$\displaystyle+\delta_{S}\Delta\rho_{S}+\delta_{TS}\Delta\rho_{TS}\tau_{3},$
$\displaystyle V(r)=$
$\displaystyle\alpha_{V}\rho_{V}+\alpha_{TV}\rho_{V}\tau_{3}+\gamma_{V}\rho_{V}^{3}$
$\displaystyle+\delta_{V}\Delta\rho_{V}+\delta_{TV}\Delta\rho_{TV}\tau_{3}+e\dfrac{1-\tau_{3}}{2}A_{0},$
$\displaystyle\Sigma_{R}(r)=$
$\displaystyle\dfrac{1}{2}\dfrac{\partial\alpha_{S}}{\partial\rho_{V}}\rho_{S}^{2}+\dfrac{1}{2}\dfrac{\partial\alpha_{V}}{\partial\rho_{V}}\rho_{V}^{2}+\dfrac{1}{2}\dfrac{\partial\alpha_{TV}}{\partial\rho_{V}}\rho_{TV}^{2},$

where
$\alpha_{S},~{}\alpha_{V},~{}\alpha_{TS},~{}\alpha_{TV},~{}\beta_{S},~{}\gamma_{S},~{}\gamma_{V},~{}\delta_{S},~{}\delta_{V},~{}\delta_{TS}$,
and $\delta_{TV}$ are coupling constants for different channels, $\rho_{TS}$
and $\rho_{TV}$ are time-like components of isoscalar current and time-like
components of isovector current, respectively.

The pairing field reads

$\displaystyle\Delta(\bm{r}_{1}\sigma_{1},\bm{r}_{2}\sigma_{2})=\int
d^{3}\bm{r}_{1}^{\prime}d^{3}\bm{r}_{2}^{\prime}\sum_{\sigma_{1}^{\prime}\sigma_{2}^{\prime}}$
(5) $\displaystyle
V(\bm{r}_{1}\sigma_{1},\bm{r}_{2}\sigma_{2},\bm{r}_{1}^{\prime}\sigma_{1}^{\prime},\bm{r}_{2}^{\prime}\sigma_{2}^{\prime})\kappa(\bm{r}_{1}^{\prime}\sigma_{1}^{\prime},\bm{r}_{2}^{\prime}\sigma_{2}^{\prime}),$

where $V$ is the effective pairing interaction and $\kappa$ is the pairing
tensor. In this work, we use a separable pairing force of finite range with
pairing strength $G=728$ MeV$\cdot$fm3 and effective range of the pairing
force $a=0.644$ fm [37].

The deformations with $V_{4}$ symmetry are allowed in the MDCRHB model. The
deformation parameter $\beta_{\lambda\mu}$ is determined by

$\beta_{\lambda\mu}=\dfrac{4\pi}{3AR^{\lambda}}Q_{\lambda\mu},$ (6)

where $R$ is the radius of the nucleus, $A$ is the number of nucleons, and
$Q_{\lambda\mu}$ is the intrinsic multipole moment calculated from the vector
density by

$Q_{\lambda\mu}=\int d^{3}r\rho_{V}(\bm{r})r^{\lambda}Y_{\lambda\mu}(\Omega),$
(7)

where $Y_{\lambda\mu}$ is the spherical harmonic.

###  II.2 Corrections

We consider both the center-of-mass and rotational corrections for the
calculated binding energies. The resulting binding energy is

$E_{\rm B}=-E_{\rm MF}-E_{\rm c.m.}-E_{\rm rot.}.$ (8)

where $E_{\rm MF},~{}E_{\rm c.m.}$, and $E_{\rm rot.}$ denote the energies of
mean field, center-of-mass correction and rotational correction, respectively.
The center-of-mass energy can be evaluated analytically from the harmonic
oscillator (HO) states. Using the usual parameterisation of the oscillator
constant from the Nilsson model one obtains an estimate as [38]

$E_{\rm c.m.}^{\rm HO}=-\dfrac{3}{4}\cdot 41A^{-1/3}~{}\rm MeV,$ (9)

where $\hbar\omega=41A^{-1/3}$ is adopted. The other way is the so call
microscopic method, in which the center-of-mass correction can be given by
calculating the change in binding energy from projection-after-variation in
first-order approximation as follow

$E_{\rm c.m.}^{\rm mic}=-\dfrac{\langle P_{\rm c.m.}^{2}\rangle}{2mA},$ (10)

with

$\displaystyle\langle P_{\rm c.m.}^{2}\rangle=$
$\displaystyle\sum_{i}\upsilon_{i}^{2}p_{ii}^{2}-\sum_{i,j}\upsilon_{i}^{2}\upsilon_{j}^{2}p_{ij}p_{ij}^{*}$
(11)
$\displaystyle+\sum_{i,j}\upsilon_{i}u_{i}\upsilon_{j}u_{j}p_{ij}p_{\bar{i}\bar{j}},$

where $i$ and $j$ denote the actual quasiparticle states. $\upsilon_{i}$ is
the occupation probability. $p_{ii}^{2}$ is the expectation value of the
square of the quasiparticle momentum operator, and $p_{ij}$ is the off-
diagonal matrix element of the quasiparticle momentum operator. We call them
direct and exchange terms latter in this work. The rotational energy
correction is calculated by

$E_{\rm rot.}=-\dfrac{1}{2}\sum_{k=1}^{3}\dfrac{\langle
J_{k}^{2}\rangle}{I_{k}},$ (12)

where $k$ denotes the axis of rotation, $J_{k}$ denotes the component of the
angular momentum in the body-fixed frame of a nucleus. The moment of inertia
$I_{k}$ is a linear combination of Inglis-Belyaev formula and the moment of
inertia of rigid rotor, i.e., $I_{k}=0.8I_{k}^{\rm IB}+0.2I_{k}^{\rm rigid}$,
with the Inglis-Belyaev formula [39, 40]

$I_{k}^{\rm
IB}=\sum_{i,j}\dfrac{(u_{i}\upsilon_{j}-\upsilon_{i}u_{j})^{2}}{E_{i}+E_{j}}|\langle
i|J_{k}|j\rangle|^{2}.$ (13)

Similar to the energy, the radius calculated from RHB model needs corrections,
too. With center-of-mass correction, the square of the radius is estimated
after RHB calculation by

$R^{2}=R_{\rm MF}^{2}-R_{\rm c.m.}^{2}.$ (14)

where $R_{\rm MF}$ and $R$ are rms radius before and after center-of-mass
correction, and $R_{\rm c.m.}$ is the rms radius of the center of mass. We
compare the radius calculated from HO approximation by

$(R_{\rm c.m.}^{\rm HO})^{2}=\dfrac{3\hbar^{2}}{2mA\cdot 41A^{-1/3}},$ (15)

and from mean-field expectation values by

$\displaystyle(R_{\rm c.m.}^{\rm mic})^{2}=$
$\displaystyle\sum_{i}\upsilon_{i}^{2}(r^{2})_{ii}-\sum_{i,j}\upsilon_{i}^{2}\upsilon_{j}^{2}r_{ij}r_{ij}^{*}$
(16)
$\displaystyle+\sum_{i,j}\upsilon_{i}u_{i}\upsilon_{j}u_{j}r_{ij}r_{\bar{i}\bar{j}}.$

Note that the second and third terms in Eq. (16) are called the exchange
terms, and usually omitted because of their fairly small effects in the mean-
field models [35]. We take these terms into account in this work to
investigate the information given by them in light nuclei. The above formulas
are employed for both proton radii ($R_{p}$) and neutron radii ($R_{n}$). The
charge radius ($R_{c}$) is obtained from the proton radius by [41]

$R_{c}^{2}=R_{p}^{2}+(0.862~{}{\rm fm})^{2}-(0.336~{}{\rm fm})^{2}N/Z,$ (17)

in which the proton and neutron spin-orbit contributions to the charge radius
[42, 43] are neglected.

In the above calculations, the ground states are obtained by applying the
variational principle with a Bogoliubov vacuum, and the effects of fundamental
translational invariance and rotational symmetry are estimated approximately.
However, wave functions and observables with certain symmetries cannot be
achieved. The standard way to restore the broken symmetries and calculate
observables with good quantum numbers is through the projection-after-
variation (PAV) technique [44, 45, 46, 47, 48, 49]. Recently, a projected
multidimensionally-constrained relativistic Hartree-Bogoliubov (p-MDCRHB)
model has been developed by incorporating the parity and angular momentum
projections into the MDCRHB model. In this model, both the triaxial and
octupole shapes are allowed. The wave function with a certain angular momentum
$J$ and parity $\pi$ is obtained by

$|\Psi_{\alpha,q}^{JM\pi}\rangle=\sum_{K}f_{\alpha}^{JK\pi}\hat{P}^{J}_{MK}\hat{P}^{\pi}|\Phi(q)\rangle,$
(18)

where $K$ represents angular momentum projection onto $z$-axis in the
intrinsic frame, $f_{\alpha}^{JK\pi}$ is the weight function, and $q$
represents a collection of the deformation parameters. The operator
$\hat{P}_{MK}^{J}$ projects out the component with angular momentum $J$ and
its projection $M$ from the deformed mean-field wave function
$|\Phi(q)\rangle$, and $\hat{P}^{\pi}$ is the parity projection operator. To
approximately restore the average proton and neutron numbers, two correction
terms are added to the Hamiltonian kernel ${\cal H}$ as in Ref. [50]. Finally,
the weight function $f_{\alpha}^{JK\pi}$ and the eigenvalue
$E_{\alpha}^{J\pi}$ are obtained by solving the generalized eigenvalue
equation [36, 51]

$\sum_{K^{\prime}}\left\\{{\cal
H^{\prime}}_{KK^{\prime}}^{J\pi}(q;q)-E_{\alpha}^{J\pi}{\cal
N}_{KK^{\prime}}^{J\pi}(q;q)\right\\}f_{\alpha}^{JK^{\prime}\pi}=0,$ (19)

where ${\cal N}$ is the norm kernel, and ${\cal H^{\prime}}$ is the
Hamiltonian kernel with particle number correction.

This model has been used to study the low-lying states related to exotic
nuclear shapes, such as the triangular shape associated with three-$\alpha$
configuration in 12C [30] and the octupole correlations in 96Zr [52]. In this
work, we restrict the calculations to axial and reflection symmetry and
perform angular momentum projection after variation to discuss the cluster
structure and shell evolution in the light mass region. For simplicity, the
configuration mixing associated with shape fluctuation is beyond the scope of
this work.

##  III RESULTS AND DISCUSSIONS

The DFTs are well-established for studying the properties of heavy nuclei,
while they are thought to be difficult to describe light nuclei due to the
deficiency of many-body correlations. However, as mentioned above, previous
studies also found that proper treatments of the corrections based on the DFTs
can help us reproduce the structures of light nuclei successfully. In this
work, the ground-state properties are systematically calculated in the CDFT
framework, with the center-of-mass and rotational corrections considered. The
accuracy of the mean-field theory in describing the ground-state properties of
light nuclei is investigated by comparing the calculated bulk properties with
the corresponding experimental data and results from other models. To estimate
the uncertainty from input parameter sets, we consider eight representative of
them, which are classified into three types: (1) the nonlinear meson-exchange
(NL-ME) interactions, including NLSH [53], TM1 [41], NL5(A) [54], and PK1
[35]; (2) the density-dependent meson-exchange (DD-ME) interactions, including
PKDD [35], DD-ME2 [55], and DD-LZ1 [56]; and (3) the density-dependent point-
coupling (DD-PC) interaction DD-PC1 [57]. These interactions have been
demonstrated to accurately describe the ground-state properties of heavy
nuclei.

###  III.1 Binding Energies

We first test the validity of those parameter sets by calculating the binding
energies of 4He, 8Be, and 12C using the eight parameter sets mentioned above.
For each one, the corresponding center-of-mass energy correction $E_{\rm
c.m.}$ is determined by the one used to fit the parameter set, i.e., Eq. (9)
is adopted for NLSH and TM1, but Eq. (10) is adopted for others. For deformed
nuclei, the rotational energy correction $E_{\rm rot.}$ is non-zero and
calculated by Eq. (12), in which the mean-field wave functions are used to
calculate the expectation values. The calculated energies per nucleon $E_{\rm
B}/A$’s are shown in Fig. 1. The numbers in the bars are the ratios of the
mean-field and center-of-mass energies to the total energy calculated by the
MDCRHB model with each parameter set, respectively. The black dashed lines are
experimental $E_{\rm B}/A$’s taken from AME2020 [58].

Figure 1: (Color online) Energies per nucleon $E_{\rm B}/A$’s for (a) 4He, (b)
8Be, and (c) 12C calculated by the MDCRHB model with eight selected parameter
sets. The black dashed lines denote the experimental $E_{\rm B}/A$’s taken
from AME2020 [58]. The values in the columns are the ratios of the mean-field
and center-of-mass energies to the total calculated energies, i.e., $-E_{\rm
MF}/E_{\rm B}$ and $-E_{\rm c.m.}/E_{\rm B}$.

In Fig. 1(a), we find the mean-field (center-of-mass) energy contributes about
53% (47%) to the total energy of 4He, and the rotational energy correction is
zero. The $E_{\rm B}/A$’s calculated by the NLSH and TM1 parameter sets
overestimate the experimental value for about 1 MeV, while those calculated by
the other parameter sets compare well with the experimental value. This
indicates that the phenomenological mass-dependent formula for center-of-mass
correction determined from the properties of heavy nuclei is not suitable for
extending to light nuclei. However, with microscopic center-of-mass
correction, this overestimation problem can be prevented, as suggested by Long
et al. [35].

For 8Be in Fig. 1(b), the mean-field energies contribute to $59.9\%-71.1\%$ of
the $E_{\rm B}/A$ values, which are larger than those in 4He and become the
dominant part of the binding energy. The center-of-mass energy correction
accounts for more than 20% of the $E_{\rm B}/A$, and that for the rotational
correction is about 10%. This result infers that for deformed light nuclei,
not only is the center-of-mass energy correction essential for determinating
the calculated binding energy, but the rotational correction is also
indispensable. For the heavier nucleus, 12C, the mean-field, center-of-mass,
and rotational motions take up about 82%, 13%, and 5% of the $E_{\rm B}/A$’s,
respectively, except for the results calculated with DD-LZ1 and DD-PC1, where
the nucleus is spherical and there is no rotational correction. Comparing the
results obtained from different parameter sets, we find that the $E_{\rm
B}/A$’s obtained by using PK1 and PKDD are very close to the experimental
values for all these three nuclei, while NL5(A), DD-ME2, DD-LZ1, and DD-PC1
systematically underestimate the experimental values by a few hundred keV. In
conclusion, the binding erergies of these alpha nuclei are reproduced with
CDFTs when both microscopic center-of-mass and rotational corrections are
taken into consideration.

Next, we extend these calculations to even-even nuclei in helium, beryllium
and carbon isotopes in order to evaluate the effectiveness of mean-field
models in accurately describing the binding energies of light nuclei
carefully. According to the above discussions, only the parameter sets fitted
with microscopic center-of-mass correction are used in the following
discussions. The axial and reflection symmetry shapes are allowed. The
obtained $E_{\rm B}/A$’s of 4,6,8He, 6,8,10,12,14Be and 10,12,14,16,18,20C are
listed in table 1. From table 1, we find that all the calculated binding
energies reproduce the experimental values with high accuracy. The largest
deviation of the $E_{\rm B}/A$ values is 0.416 MeV, calculated by DD-PC1 in
8Be.

As an example, the energies in the mean-field approximation and with
corrections calculated with PK1 parameter set are clearly visible in Fig.
2(b1-b3), together with the corresponding quadrupole deformations in Fig.
2(a1-a3). we find that the $E_{\rm B}/A$’s are all well-reproduced with
center-of-mass and rotational energy corrections, especially for the two-alpha
cluster structure nucleus 8Be. Although this nucleus does not have the largest
$E_{\rm B}/A$ in this isotopic chain in the mean field, it is repaired after
corrections, indicating that the correction terms bring cluster effect to the
mean-field description to some extent.

Table 1: The calculated energy per nucleon $E_{\rm B}/A$, root-mean-square (rms) matter radius $R_{m}$, rms charge radius $R_{c}$, and quadrupole deformation parameter $\beta_{20}$ with relativistic Hartree-Bogoliubov model. For energies, the microscopic center-of-mass correction with full exchange terms are used. For radii, results with and without exchange terms ($R_{f}$ and $R_{d}$) are listed for comparison. The rotational energy correction calculated by Eq. (12) is included in $E_{\rm B}/A$ when the nucleus is deformed. Experimental (Expt.) values of $E_{\rm B}/A$ and $\beta_{20}$ are taken from Refs. [58] and [59], respectively, except otherwise noted. The experimental radii and the corresponding references are also listed. Nucleus | Interaction |  $E_{\rm B}/A$ (MeV) |  $R_{c,f}$ (fm) |  $R_{c,d}$ (fm) |  $R_{m,f}$ (fm) |  $R_{m,d}$ (fm) | $\beta_{20}$
---|---|---|---|---|---|---|---
4He | Expt. | 7.074 | 1.681(4) [60] |  |  |
| NL5(A) | 7.008 | 1.846 | 1.846 | 1.663 | 1.663 | 0.000
| PK1 | 7.076 | 1.838 | 1.838 | 1.654 | 1.654 | 0.000
| PKDD | 7.017 | 1.848 | 1.848 | 1.665 | 1.665 | 0.000
| DD-ME2 | 6.951 | 1.894 | 1.894 | 1.716 | 1.716 | 0.000
| DD-LZ1 | 6.842 | 1.982 | 1.982 | 1.812 | 1.812 | 0.000
| DD-PC1 | 6.723 | 1.925 | 1.925 | 1.750 | 1.750 | 0.000
6He | Expt. | 4.879 | 2.068(11) [61] | 2.30(7) [62] | 1.024(66)
| NL5(A) | 4.941 | 1.935 | 1.947 | 2.129 | 2.118 | 0.000
| PK1 | 4.973 | 1.929 | 1.940 | 2.130 | 2.120 | 0.000
| PKDD | 4.873 | 1.946 | 1.958 | 2.154 | 2.144 | 0.000
| DD-ME2 | 4.835 | 2.005 | 2.014 | 2.211 | 2.202 | 0.000
| DD-LZ1 | 4.858 | 2.100 | 2.109 | 2.295 | 2.286 | 0.000
| DD-PC1 | 4.892 | 1.997 | 2.013 | 2.171 | 2.157 | 0.000
8He | Expt. | 3.925 | 1.929(26) [61] | 2.45(7) [62] | 0.40(3) [63]
| NL5(A) | 4.022 | 1.953 | 1.977 | 2.429 | 2.429 | 0.000
| PK1 | 3.983 | 1.951 | 1.974 | 2.449 | 2.449 | 0.000
| PKDD | 3.833 | 1.968 | 1.992 | 2.475 | 2.475 | 0.000
| DD-ME2 | 3.715 | 2.036 | 1.974 | 2.541 | 2.541 | 0.000
| DD-LZ1 | 3.695 | 2.130 | 2.150 | 2.610 | 2.610 | 0.000
| DD-PC1 | 3.950 | 2.010 | 2.039 | 2.448 | 2.448 | 0.000
6Be | Expt. | 4.487 |  |  |  |  |
| NL5(A) | 4.388 | 2.467 | 2.452 | 2.166 | 2.157 | 0.000
| PK1 | 4.413 | 2.474 | 2.460 | 2.169 | 2.161 | 0.000
| PKDD | 4.319 | 2.500 | 2.486 | 2.194 | 2.186 | 0.000
| DD-ME2 | 4.297 | 2.553 | 2.541 | 2.252 | 2.246 | 0.000
| DD-LZ1 | 4.331 | 2.630 | 2.618 | 2.341 | 2.334 | 0.000
| DD-PC1 | 4.345 | 2.494 | 2.471 | 2.204 | 2.191 | 0.000
8Be | Expt. | 7.062 |  |  |  |  |
| NL5(A) | 6.931 | 2.481 | 2.431 | 2.342 | 2.290 | 1.175
| PK1 | 7.040 | 2.459 | 2.410 | 2.319 | 2.267 | 1.145
| PKDD | 6.981 | 2.475 | 2.426 | 2.335 | 2.284 | 1.158
| DD-ME2 | 6.928 | 2.527 | 2.477 | 2.390 | 2.338 | 1.213
| DD-LZ1 | 6.852 | 2.615 | 2.563 | 2.482 | 2.428 | 1.307
| DD-PC1 | 6.646 | 2.581 | 2.531 | 2.448 | 2.396 | 1.263
10Be | Expt. | 6.498 | 2.361(17) [18] | 2.30(2) [64] | 1.071(${}^{+26}_{-20}$)
| NL5(A) | 6.472 | 2.292 | 2.279 | 2.297 | 2.268 | 0.353
| PK1 | 6.574 | 2.273 | 2.259 | 2.282 | 2.252 | 0.356
| PKDD | 6.510 | 2.306 | 2.289 | 2.312 | 2.282 | 0.385
| DD-ME2 | 6.431 | 2.325 | 2.313 | 2.331 | 2.303 | 0.316
| DD-LZ1 | 6.374 | 2.268 | 2.261 | 2.270 | 2.246 | 0.001
| DD-PC1 | 6.230 | 2.319 | 2.311 | 2.321 | 2.294 | 0.033
12Be | Expt. | 5.721 | 2.503(15) [18] | 2.59(6) [64] | 0.88(${}^{+24}_{-12}$)
| NL5(A) | 5.844 | 2.321 | 2.319 | 2.490 | 2.464 | 0.000
| PK1 | 5.834 | 2.312 | 2.310 | 2.495 | 2.470 | 0.000
| PKDD | 5.724 | 2.331 | 2.327 | 2.508 | 2.484 | 0.000
| DD-ME2 | 5.746 | 2.371 | 2.368 | 2.534 | 2.510 | 0.000
| DD-LZ1 | 5.821 | 2.392 | 2.389 | 2.554 | 2.531 | 0.000
| DD-PC1 | 5.932 | 2.368 | 2.365 | 2.511 | 2.485 | 0.000
14Be | Expt. | 4.994 |  | 3.16(38) [64] |
| NL5(A) | 5.284 | 2.488 | 2.475 | 2.853 | 2.280 | 0.789
| PK1 | 5.255 | 2.472 | 2.460 | 2.867 | 2.835 | 0.785
| PKDD | 5.112 | 2.490 | 2.478 | 2.889 | 2.856 | 0.797
| DD-ME2 | 5.039 | 2.509 | 2.503 | 2.917 | 2.886 | 0.758
| DD-LZ1 | 4.911 | 2.413 | 2.419 | 2.896 | 2.871 | 0.123
| DD-PC1 | 5.195 | 2.503 | 2.459 | 2.853 | 2.821 | 0.730
Nucleus | Interaction |  $E_{\rm B}/A$ (MeV) |  $R_{c,f}$ (fm) |  $R_{c,d}$ (fm) |  $R_{m,f}$ (fm) |  $R_{m,d}$ (fm) | $\beta_{20}$
---|---|---|---|---|---|---|---
10C | Expt. | 6.032 |  |  |  |  | 0.701(${}^{+32}_{-34}$)
| NL5(A) | 5.958 | 2.567 | 2.532 | 2.333 | 2.304 | 0.397
| PK1 | 6.056 | 2.557 | 2.521 | 2.318 | 2.288 | 0.398
| PKDD | 5.997 | 2.583 | 2.546 | 2.348 | 2.317 | 0.429
| DD-ME2 | 5.914 | 2.608 | 2.573 | 2.374 | 2.346 | 0.361
| DD-LZ1 | 5.814 | 2.558 | 2.525 | 2.319 | 2.294 | 0.019
| DD-PC1 | 5.713 | 2.577 | 2.541 | 2.345 | 2.318 | 0.066
12C | Expt. | 7.680 | 2.4702(22) [65] | 2.35(2) [66] |  $-0.40(2)$ [67]
| NL5(A) | 7.608 | 2.469 | 2.440 | 2.327 | 2.297 | $-0.350$
| PK1 | 7.744 | 2.436 | 2.408 | 2.292 | 2.263 | $-0.320$
| PKDD | 7.700 | 2.467 | 2.439 | 2.325 | 2.296 | $-0.346$
| DD-ME2 | 7.660 | 2.491 | 2.463 | 2.349 | 2.320 | $-0.347$
| DD-LZ1 | 7.633 | 2.376 | 2.354 | 2.228 | 2.204 | 0.002
| DD-PC1 | 7.269 | 2.481 | 2.453 | 2.340 | 2.312 | $0.000$
14C | Expt. | 7.520 | 2.5025(87) [65] | 2.33(7) [66] |
| NL5(A) | 7.438 | 2.466 | 2.443 | 2.431 | 2.405 | 0.000
| PK1 | 7.517 | 2.446 | 2.423 | 2.417 | 2.391 | 0.000
| PKDD | 7.484 | 2.463 | 2.440 | 2.427 | 2.402 | 0.000
| DD-ME2 | 7.495 | 2.493 | 2.470 | 2.449 | 2.424 | 0.000
| DD-LZ1 | 7.623 | 2.474 | 2.452 | 2.436 | 2.414 | 0.000
| DD-PC1 | 7.562 | 2.521 | 2.497 | 2.474 | 2.448 | 0.000
16C | Expt. | 6.922 |  | 2.74(3) [66] | 0.323(18)
|  |  |  |  |  |  |  $0.356^{+0.25}_{-0.23}$ [68]
| NL5(A) | 6.950 | 2.512 | 2.494 | 2.644 | 2.619 | 0.327
| PK1 | 6.994 | 2.492 | 2.475 | 2.639 | 2.614 | 0.320
| PKDD | 6.938 | 2.512 | 2.494 | 2.653 | 2.628 | 0.326
| DD-ME2 | 6.935 | 2.547 | 2.529 | 2.679 | 2.654 | 0.316
| DD-LZ1 | 7.015 | 2.522 | 2.506 | 2.674 | 2.652 | 0.192
| DD-PC1 | 7.049 | 2.566 | 2.548 | 2.671 | 2.645 | 0.324
18C | Expt. | 6.426 |  | 2.86(4) [66] | 0.289(${}^{+20}_{-13}$)
| NL5(A) | 6.553 | 2.569 | 2.555 | 2.828 | 2.802 | $-0.381$
| PK1 | 6.567 | 2.550 | 2.536 | 2.832 | 2.807 | $-0.380$
| PKDD | 6.479 | 2.566 | 2.552 | 2.842 | 2.816 | $-0.374$
| DD-ME2 | 6.456 | 2.602 | 2.587 | 2.864 | 2.839 | $-0.372$
| DD-LZ1 | 6.526 | 2.571 | 2.559 | 2.841 | 2.817 | $-0.318$
| DD-PC1 | 6.580 | 2.615 | 2.601 | 2.837 | 2.810 | $-0.362$
20C | Expt. | 5.961 |  | 2.98(5) [69] | 0.405(${}^{+89}_{-45}$)
| NL5(A) | 6.171 | 2.615 | 2.607 | 2.987 | 2.957 | $-0.468$
| PK1 | 6.146 | 2.596 | 2.588 | 2.999 | 2.970 | $-0.468$
| PKDD | 6.045 | 2.612 | 2.604 | 3.001 | 2.972 | $-0.457$
| DD-ME2 | 6.006 | 2.649 | 2.640 | 3.011 | 2.983 | $-0.457$
| DD-LZ1 | 6.026 | 2.609 | 2.602 | 2.953 | 2.926 | $-0.413$
| DD-PC1 | 6.177 | 2.664 | 2.656 | 2.985 | 2.956 | $-0.453$
Figure 2: (Color online) Ground-state properties for helium ($Z=2$), beryllium
($Z=4$), and carbon ($Z=6$) isotopes. The evolutions of (a) the quadrupole
deformation parameter $\beta_{20}$, (b) the energy per nucleon $E_{\rm B}/A$,
(c) the root-mean-square (rms) charge radius $R_{c}$, (d) the rms mass radius
$R_{m}$, and (e) the neutron skin ratio between pure mean-field and mean-field
with corrections, $R_{np}/R_{np,{\rm MF}}$, calculated with the PK1 effective
interaction are shown. For center-of-mass (c.m.) correction, "c.m.[A]" is
corrected by Eq. (9) for energies and Eq. (15) for radii; "c.m.[B]" is
corrected by Eq. (10) for energies and Eq. (14) for radii with only direct
term in Eq. (16); "c.m.[C]" is the same as "c.m.[B]" but with exchange terms.
The green dashed line in (b) represents the result with rotational energy
correction. The measured $\beta_{20}$, $E_{\rm B}/A$, $R_{c}$, and $R_{m}$
listed in table 1 are denoted by open squares. The $R_{c}$’s with open
diamonds are derived from proton radii obtained from measurement of charge-
changing cross sections [70, 71, 16] with Eq. (17).

The above results naturally brings a question: How does the percentage of the
mean-field energy to the total energy in a nucleus depends on the mass number?
To do this, we study the particle-number dependence by averaging the $E_{\rm
MF}/E_{\rm B}$ ratios calculated from the six selected effective interactions
NL5(A), PK1, PKDD, DD-ME2, DD-LZ1, and DD-PC1 for each nucleus. Meanwhile, the
standard deviations of the ratios are also given as model uncertainties. Fig.
3 displays the average $-E_{\rm MF}/E_{\rm B}$ values of the even-even helium,
beryllium and carbon isotopes. Note that 16O, 20Ne, and 40Ca are added to
constrain the particle-number dependence in the medium-mass region. It can be
observed that the expected behavior, i.e., a favorable relation between the
$-E_{\rm MF}/E_{\rm B}$ and the mass number in the light mass region until
$A=40$ is obtained. For heavier nuclei, this dependence can be neglected.
Given that the $-E_{\rm MF}/E_{\rm B}$ ratios for nuclei with the same mass
number are slightly different, e.g., 0.795$\pm$0.057 for 10Be and
0.778$\pm$0.065 for 10C, we find the isospin effect exists but can be
neglected in the mass-number dependent behavior. To give a specific form of
the mass-number dependent behavior, we fit all the energy ratios shown in Fig.
3 and obtain the following relation

$-E_{\rm MF}/E_{\rm B}=1.06e^{-2.90/A},$ (20)

where the mean-field energy ratios accumulate exponentially as the number of
particles increases. Predicting by Eq. (20), when $A=42$, the mean-field
energy is 99% equals to the total energy.

Figure 3: (Color online) The ratios of the mean-field energy to the binding
energy, $-E_{\rm MF}/E_{B}$, as a function of the mass number $A$. The ratios
are average values of the results calculated by using NL5(A), PK1, PKDD, DD-
ME2, DD-LZ1, and DD-PC1. The corresponding errorbars are their root-mean-
square deviations. The solid curve is fitted by all the average values in this
figure.

###  III.2 Radii

Next, we discuss the radii. The radius is a quantity of great interest in
describing a nucleus, providing information on deformations [72], exotic
structures such as halos [61, 66] and neutron/proton skins [73], short-range
correlations of nucleon-nucleon interactions [74], and shell evolutions [18],
etc. Previous studies have shown that the inclusion of relativistic and
center-of-mass corrections impacts the quality of energy density functionals
optimized for charge radii data [75]. In this work, we focus on the treatment
of the center-of-mass correction. The rms radius of the center of mass is
calculated with HO approximation [Eq. (15)] or expectation values from RHB
states with or without exchange terms [Eq. (16)].

Table 1 lists the radii calculated by utilizing the microscopic center-of-mass
correction with the six selected parameter sets. The radii corrected by Eq.
(15) are not listed because they are simple to calculate and independent of
RHB wave functions. As can be seen from table 1, the exchange terms change
both $R_{c}$’s and $R_{m}$’s from those with only the direct term, except for
4He. To observe this effect clearly, we calculate the differences between the
rms matter (charge) radii with and without exchange terms, i.e.,
$R_{m,f}-R_{m,d}$ ($R_{c,f}-R_{c,d}$), with PK1, NL5(A), and DD-LZ1 and show
them in Fig. 4. It is evident that all the $R_{m,f}-R_{m,d}$ values are
greater than or equal to zero, manifesting that the inclusion of the exchange
terms increases $R_{m}$. While $R_{c,f}-R_{c,d}<0$ is obtained for 6He and 8He
with these three parameter sets and 14Be with DD-LZ1, meaning that $R_{c}$ may
be reduced or increased depending on the choice of effective interactions for
an individual nucleus. Overall, the differences range from 0.0 fm to 0.05 fm.
Especially for 8Be, this effect introduces a two percent deviation to the
calculated radius. Considering the unprecedented level of precision offered by
new experimental techniques, which allows for the exploration of new physics
and the elucidation of unclear observables [76, 77], the exchange terms cannot
be ignored for these nuclei in the mean-field calculations.

Figure 4: (Color online) The difference between the (a) rms matter radius
$R_{m}$ and (b) rms charge radius $R_{c}$ calculated with microscopic center-
of-mass correction with and without exchange terms (labeled by $R_{m,f}$ and
$R_{m,d}$ for matter radius, respectively) for helium, beryllium, and carbon
isotopes. The results are calculated with PK1 (red), NL5(A) (blue), and DD-LZ1
(green) parameter sets.

According to the data in table 1, we plot the radii calculated with PK1 and
different center-of-mass corrections, and compare them to the mean-field and
measured values in Fig. 2(c-d). In Fig. 2(c1), the mean-field $R_{c}$’s
calculated with PK1 are around 2.0 fm and decrease as the neutron number
increases. The curve including the center-of-mass correction with HO
approximation shows a similar tendency in the isotopic chain as the mean-field
result, but the corresponding $R_{c}$’s are smaller. This can be easily
understood because this approximation only considers the mass-number
dependence. However, with microscopic center-of-mass corrections, i.e., by
using Eq. (15), the isospin effect is considered, whether with or without
exchange terms. For 4He, where the nucleons occupy only the $s$ single-
particle level, the exchange terms are zero. For 8He, the reduction of the
$R_{c}$ caused by the exchange terms cannot be neglected. Comparing these
results with the experimental charge radii, the calculated value for 4He is
larger, while the values for 6He and 8He are reasonably reproduced. Overall,
the calculations with full microscopic correction perform better than the
other three methods in describing the experimental curve, particularly for the
$R_{c}$’s of 4He and 8He. However, a remaining issue is that the calculated
relation $R_{c}(^{6}\rm{He})$< $R_{c}(^{8}\rm{He})$ contradicts the
experimental observation [61]. This inconsistency is associated with many-body
correlations and will be discussed later from the perspective of deformation.

Regarding $R_{m}$ in Fig. 2(d1), It is clear that the treatment of the
correction significantly affects the calculated radii in these nuclei. When
compared with the data in Ref. [62], which reveal 6He and 8He as halo nuclei,
it is surprising that the calculated $R_{m}$’s for 8He are large enough, and
the one with full microscopic center-of-mass correction is very close to the
experimental value. We can further highlight the effects of different
corrections by calculating the neutron skin ratio, i.e., $R_{np}/R_{np,{\rm
MF}}$, where $R_{np}=R_{n}-R_{p}$ is the neutron skin thickness, and $R_{n}$
($R_{p}$) is the rms neutron (proton) radius. In Fig. 2(e1), one can observe
that the ratios calculated by Eq. (16) are lower than 1.0, while those
calculated by Eq. (15) are larger than 1.0. This indicates that the
microscopic methods reduce the neutron skins, but the HO approximation with
energy fixed by the heavy nuclei increases them.

Combining both binding energy and radius, in Fig. 5, the $E_{B}/A$’s for 4He
are shown as a function of it’s proton radii $R_{p}$’s calculated with the six
selected effective interactions, and compared with other models and the
experimental point. It is noteworthy that all the calculated points are on the
right side of the measured point. Calculated with effective interactions, the
mean-field models, including MDCRHB model and DRHF model [78, 79], provide
proper description for the $E_{\rm B}/A$ but predict larger $R_{p}$ compared
to the experimental value. In contrast, ab initio methods employing realistic
nuclear forces yield smaller deviations between the calculated and
experimental $R_{p}$’s, but larger deviations for $E_{\rm B}/A$’s. Our
conjecture is that incorporating radii or densities into the fitting procedure
may result in mean-field models that exhibit greater consistency with
experimental measurements.

Figure 5: (Color online) Energies per nucleon $E_{\rm B}/A$’s for 4He as a
function of the proton radius $R_{p}$’s calculated by the MDCRHB model with
full microscopic center-of-mass (c.m.) correction, in comparison with RBHF
model with Bonn A interaction [80], DRHF model with PKO1 [78] and PKA1 [79]
interactions, Faddeev-Yakubovsky (FY) equations with N4LO [81], and no-core
shell model (NCSM) with N3LO NN potentials [82].

Then, we turn our attention to the beryllium isotopes. The $R_{c}$’s
calculated with PK1 for beryllium isotopes are shown in Fig. 2(c2). Obviously,
those with center-of-mass corrections are smaller than the corresponding
$R_{c}$ values in the mean-field calculation, consistent with the increase in
binding energy due to this correction. Furthermore, we find that the
microscopic type of center-of-mass correction with only a direct term
drastically shrinks the $R_{c}$’s of 6Be and 8Be, while the inclusion of the
exchange terms enlarges the value for 8Be notably. For heavier nuclei, the
results with different center-of-mass corrections closely converge. When
compared to the experimental $R_{c}$’s, our results show a systematic
shrinkage for 10Be and 12Be. This situation can be somewhat ameliorated by
taking the beyond mean-field effects into consideration. The symmetry
restoration, in particular, helps to achieve a larger $\beta_{20}$ from near
spherical shape [83, 84] and increase the $R_{c}$’s of the studied nuclei.
Detailed insights into this will be provided in the subsequent section. For
14Be, the $R_{c}$ obtained from PK1 closely aligns with the value derived from
the charge-changing cross section measurement [70]. This could be associated
with the large $\beta_{20}$ calculated with PK1. In Fig. 2(d2), the correction
effect for the $R_{m}$ is similar to that for $R_{c}$ in Fig. 2(c2), and the
specifics are not revisited. The calculated $R_{m}$’s for 10Be and 12Be
accurately reproduce the experimental values, and that for 14Be falls within
the limits of the experimental error. We examine the neutron skin ratios in
Fig. 2(e2) and obtain the same conclusion as with helium isotopes: The
microscopic center-of-mass correction diminishes the neutron skin when
compared to the mean-field result, while the simple HO approximation yields an
opposite tendency.

For carbon isotopes, the reduction of the charge and matter radii by employing
center-of-mass correction from the mean-field values is still visible. As
shown in Fig. 2(c3), the $R_{c}$’s calculated with full microscopic center-of-
mass correction are very close to those calculated with HO approximation in
this isotopic chain, and the same situation holds for $R_{m}$’s in Fig. 2(d3).
This suggests that, from the perspective of radii, it is a valid approximation
for these nuclei that the center-of-mass motion behaves as a harmonic
oscillator vibration. When compared with the experimental radii, the $R_{c}$’s
in Fig. 2(c3) with corrections reproduce the experimental data well for 12C
and 18C. Notably, the $R_{c}$ for 12C with PK1 is comparable to the measured
values, in contrast to the value calculated with TMA in a previous study [16].
However, those with pure mean-field calculations are closer to the results
derived from measurements of charge-changing reactions [71], rather than the
corrected values. As for $R_{m}$, the corrected values closely match the
experimental data for 18C and 20C, but they surpass the experimental data for
12C and 16C. Experimental findings [66] indicate that the smallest $R_{m}$ in
this isotopic chain is observed in 14C. Nevertheless, when calculated with
PK1, the $R_{c}$ value for 14C is overestimated with all types of center-of-
mass corrections, and the reduction of the radius in this nucleus throughout
the evolution of the isotopic chain cannot be achieved. In comparison with the
results from DRHBc model [85], which adopts a spherical Dirac Woods-Saxon (WS)
basis and considers continuum effects, our results align with theirs, except
for 16C. The difference arises from the softness of the potential energy
surface (PES) in the $\gamma=\arctan(\sqrt{2}\beta_{22}/\beta_{20})$ direction
with triaxial degree of freedom, as demonstrated in Ref. [26]. Additionally,
Fig. 2(e3) underscores that the neutron skin can serve as a probe to
differentiate between types of center-of-mass correction.

###  III.3 Deformation and Shell Evolution

The quadrupole deformations calculated with the six selected parameter sets
for the studied light nuclei are illustrated in the last column of table 1.
Apparently, for most of the studied nuclei, the deformation parameters
calculated with different interactions are similar, except for some of those
calculated with the point coupling interactions DD-PC1 and DD-LZ1, which tend
to have spherical ground states rather than deformed ground states.

We then focus on the results calculated with PK1 as an example. 4,6,8He are
spherical according to the MDCRHB model, implying that the valence neutrons
are uniformly distributed around the surface of the alpha particle. This
situation naturally leads to an increase in the charge and matter radii as the
number of neutrons increases. However, charge radii extracted from the
measured isotope shifts reveal a significant reduction in the charge radius
from 6He to 8He [61]. The authors interpreted it as a change in the
correlations of the excess neutrons: in 6He, the two neutrons are correlated
so that on average they spend more time together on one side of the core
rather than on opposite sides; While for 8He, the four excess neutrons are
distributed in a more spherically symmetric fashion in the halo, resulting in
less smearing of the charge in the core. The quadrupole deformation obtained
from proton inelastic scattering also supports this picture [61]. In other
words, both the charge radii and deformations from experiments reveal that the
mean-field approximation misses the correction among the valence neutrons,
which is essential in helium isotopes.

Superficially, it is the zero quadrupole deformation that causes the
inconsistent results with the experiment. Suppose the experimental
deformations can be reproduced by some corrections, then the calculated radius
is modified by [86]

$R^{2}=\left(1+\dfrac{5}{4\pi}\beta_{2}^{2}\right)R_{\rm sph}^{2},$ (21)

where $R_{\rm sph}$ is the rms radius for a spherical nucleus. By
incorporating the experimental $\beta_{2}$ into Eq. (21), the rms $R_{c}$ for
6He is calculated to be 2.296 fm, and that for 8He is 2.012 fm, based on the
full microscopic center-of-mass correction. These values are larger than the
measured values of 2.068 fm and 1.929 fm, respectively, and the trend of
$R_{c}(^{6}\rm{He})$> $R_{c}(^{8}\rm{He})$ > $R_{c}(^{4}\rm{He})$ is
reproduced.

Since angular momentum projection usually changes the quadrupole deformation
of the mean-field ground state from near zero to a larger value [83, 84], we
attempt to reproduce the experimental deformations for 6He and 8He by applying
angular momentum projection after variation. In Fig. 6(a-c), the mean-field
and projected potential energy surfaces (PES’s) for 4,6,8He calculated with
DD-PC1 effective interaction are shown. Note that here DD-PC1 is used instead
of PK1 since symmetry restoration calculation with the latter effective
interaction has not been realized. From Fig. 6(a-c), we find that the
quadrupole deformations for these three nuclei are zero in the mean field,
consistent with the results in Fig. 2(a1). With angular momentum projection,
the PES for 4He is very soft, while those for 6He and 8He evidently reach
deformed energy minima. Precisely, the locations of the energy minima for
4,6,8He are $\beta_{20}=0.20,0.90$ and $-0.50$. The corresponding energy
differences between the projected and mean-field energy minima are 0.01 MeV,
0.22 MeV, and 0.17 MeV, respectively. We conclude that the experimental
deformations for 6He and 8He can be partly explained by doing angular momentum
projection, and the influence of symmetry restoration on 4He can be neglected.
However, the issue might persist after configuration mixing calculation due to
the softness of the PES’s, similar to the case of 32Mg calculated with PC-F1
[84]. Detailed discussions of configuration mixing are beyond the scope of
this article.

For beryllium isotopes, in Fig. 2(a2), we find the studied nuclei are prolate
except for 6Be and 12Be. As expected, the deformation of 8Be is
$\beta_{2}=1.145$, forming a typical two-alpha cluster structure. With more or
fewer neutrons, the alpha cluster structure diminishes or even disappears.
12Be, which consists of four extra neutrons compared to 8Be, is a well-known
nucleus in discussions about shell evolution. The disappearance of the magic
number $N=8$ has been suggested in this nucleus through various observables
measured in experiments, such as lifetime [17], charge radius [18], and
single-neutron removal cross sections [19]. However, as listed in table 1, it
is predicted to be spherical by the RHB model with all the selected effective
interactions. This indicates that the magic number $N=8$ naturally arises in
every mean-field description for this nucleus.  A natural understanding of
this result is that the correlations between nucleons are lost with the mean-
field approximation, whereas it is crucial for determining the shell closure
in this mass region. For example, in the molecular-orbital models, by coupling
with the spin-triplet states, the energy of the $(3/2^{-})^{2}(1/2^{+})^{2}$
configuration for the four valence neutrons is almost the same as, or even
lower than, the $(3/2^{-})^{2}(1/2^{-})^{2}$ corresponding to the closed
$p$-shell configuration. This results in the breaking of the neutron magic
number $N=8$ [87, 88]. However, the inclusion of this effect is difficult
within a single-reference configuration.

Similar to the discussion in helium isotopes, the radius for 12Be calculated
using Eq. (21) with the spherical mean-field radius of 2.312 fm and
experimental quadrupole deformation $\beta_{2}=0.88$ is 2.644 fm, which is
slightly larger than the measured charge radius of 2.503(15) fm [18]. We also
present the projected PES’s calculated with the DD-PC1 effective interaction
in Fig. 6(d-f). In this figure, we observe that the distance between the two
alpha particles in 8Be enlarges after projection, as the corresponding
$\beta_{20}$ in the projected energy minimum is larger than that in the mean
field. The same phenomenon occurs for 10Be. However, it is hard to conclude
that 12Be is well-deformed after projection, because the projected PES appears
very soft in the figure. We increase the pairing strength for neutrons by
$20\%$ and obtain a stiffer projected PES with a prolate energy minimum. This
suggests that an enhancement of pairing interactions may contribute to
breaking up the shell closure in this nucleus.

Figure 6: (Color online) The mean-field and the projected potential energy
surfaces for 4,6,8He and 8,10,12Be calculated with the DD-PC1 effective
interaction. The red squares mark the mean-field ground states, and the blue
crosses represent the energy mimima after angular momentum projection.

Finally, we discuss the carbon isotopes. Calculated with PK1, in Fig. 2(a3),
we find that 10,16C are prolate, 14C is spherical, and 12,18,20C are oblate.
These shapes align with those calculated with the MDC-RMF model, except for
18C [26], which is predicted to be triaxially deformed its ground state. In
comparison with the DRHBc model, our results are consistent with theirs,
except for 16C. This discrepancy arises from the softness of the PES in the
$\gamma$ direction with triaxial degree of freedom, as shown in Ref. [26].
It’s worth noting that in Ref. [59], the deformation parameter is obtained
from electric quadrupole transition of the nucleus, and distinguishing between
prolate ($\beta_{20}>0$) and oblate ($\beta_{20}<0$) shapes is not feasible.
Consequently, when compared to the results from Ref. [59], the calculated
deformations for 16,18,20C are quite accurate. For 10C, prediction from the
MDCRHB model indicates a smaller quadrupole deformation parameter compared to
the experimental value of 0.701 [59]. As for 12C, our result supports an
oblate shape with $\beta_{20}=-0.40(2)$ [67].

14C, with six protons and eight neutrons, provides an important platform to
study the possible existence of the magic number 6 in certain semimagic
unstable nuclei. The fact that the systematics of proton radii, $B(E2)$ values
and the empirical proton-subshell gaps for most carbon isotopes are comparable
to those for proton-closed shell Oxygen isotopes manifests 14C is a doubly-
magic nucleus [16]. In the MDCRHB model, 14C is predicted to be spherical in
its ground state using all the effective interactions we employed. When
compare with the cases of 8He and 12Be, we conclude that mean-field
calculations consistently yield strong spin-orbit coupling for the $1p$ state,
leading to the magic number 6. As a result, shell closures are achieved in 8He
and 14C. However, the poor description of deformations in helium and beryllium
isotopes shows the absence of the many-body correlations in the mean-field
approximation.

##  IV SUMMARY

We investigate the accuracy of the mean-field approach in describing light
nuclei in this work. To achieve this goal, the ground-state properties,
including binding energy, quadrupole deformation, rms charge radius, rms
matter radius, and neutron skin, of helium, beryllium and carbon isotopes with
the MDCRHB model and its corrections are studied. Eight effective interactions
are used in order to estimate the theoretical uncertainty from interactions.
The full microscopic center-of-mass correction for radius is considered for
the first time, which has been neglected in the descriptions of medium and
heavy nuclei. Furthermore, the angular momentum projection on the potential
energy surface after mean-field calculations is performed.

We find the binding energies of the studied nuclei are well described by the
MDCRHB model with microscopic center-of-mass correction and rotational
correction. The average mean-field energies calculated with these corrections
exhibit an exponential relation to the calculated binding energies, with a
coefficient of mass correlation. Regarding the radius, the exchange terms in
the center-of-mass correction cannot be neglected, which contrasts with the
situation in heavier mass region. With the PK1 effective interaction, most of
the charge and matter radii closely match the experimental values. The neutron
skin ratio can be used to distinguish the type of center-of-mass correction,
and the neutron skin becomes smaller with microscopic center-of-mass
correction. Shell closures are achieved in 8He and 14C due to the consistently
strong spin-orbit coupling for the $1p$ state predicted by mean-field
calculations. However, the poor description of deformations in helium and
beryllium isotopes indicates the absence of many-body correlations in the
mean-field approximation. We find the deformation is a key property to test
the ability of mean-field models in describing light nuclei, and angular
momentum projection after variation can partially aid in reproducing the
deformations in helium and beryllium isotopes. Further utilization of
realistic nucleon-nucleon interactions to calculate ground states within this
framework would be of interest.

###### Acknowledgements.

Helpful discussions with Xiang-Xiang Sun, Shan-Gui Zhou and Bing-Nan Lu are
gratefully acknowledged. This work has been supported by the National Natural
Science Foundation of China (Grants No. 12205057), the Central Government
Guides Local Scientific and Technological Development Fund Projects (Grants
No. Guike ZY22096024), the Science and Technology Plan Project of Guangxi
(Grants No. Guike AD23026250), the promotion project of scientific research
ability of young teachers in universities of Guangxi (Grant No. 2022KY0055),
and the the Natural Science Foundation of Guangxi (Grant No.
2023GXNSFAA026016). The results in this paper are obtained on the High-
performance Computing Cluster of ITP-CAS.

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#  AI Ethics: Software Practitioners’ and Lawmakers’ Points of View

Arif Ali Khan  University of Oulu, Finland  Muhammad Azeem Akbar
Lappeenranta-Lahti University of Technology, Finland  Muhammad Waseem  Wuhan
University, China  Mahdi Fahmideh  University of Southern Queensland,
Australia  Aakash Ahmad  University of Ha’il, Saudi Arabia  Peng Liang  Wuhan
University, China  Mahmood Niazi  King Fahd University of Petroleum and
Minerals, Saudi Arabia  Pekka Abrahamsson  University of Jyvaskyla, Finland

###### Abstract

Despite their commonly accepted usefulness, Artificial Intelligence (AI)
technologies are concerned with ethical unreliability. Various guidelines,
principles, and regulatory frameworks are designed to ensure that AI
technologies bring ethical well-being. However, the implications of AI ethics
principles and guidelines are still being debated. To further explore the
significance of AI ethics principles and relevant challenges, we conducted an
empirical survey of 99 AI practitioners and lawmakers from twenty countries
across five continents. Study findings confirm that transparency,
accountability, and privacy are the most critical AI ethics principles. On the
other hand, lack of ethical knowledge, no legal frameworks, and lacking
monitoring bodies are found the most common AI ethics challenges. The impact
analysis of the challenges across AI ethics principles reveals that conflict
in practice is a highly severe challenge. Our findings stimulate further
research, epically empowering existing capability maturity models to support
the quality assessment of ethics-aware AI systems.

##  1 Introduction

Artificial Intelligence (AI) becomes necessary across a vast array of
industries including health, manufacturing, agriculture, and banking [1]. AI
technologies have the potential to substantially transform society and offer
various societal benefits, which are expected to happen from high-level
productivity and efficiency. In line with this, the ethical guidelines
presented by the European commission expert group on AI highlights that (AI
HLEG) [2]:

“AI is not an end in itself, but rather a promising means to increase human
flourishing, thereby enhancing individual and societal well-being and the
common good, as well as bringing progress and innovation.”

However, the promising benefits of AI technologies have been considered with
worries that the complex and opaque systems might bring more social harm than
good [1]. People start thinking beyond the operational capabilities of AI
technologies and investigate the ethical aspects of developing strong and
potentially life consequential technologies. US government and private
companies do not consider the virtual implications of decision-making systems
in health, criminal justice, employment, and creditworthiness without ensuring
that these systems are not coded intentionally or unintentionally with
structural biases [1].

Concomitant with advances in AI systems, we witness the ethical failure of
those systems. For example, a high rate of unsuccessful job applications that
was decided by the Amazon recruitment system was later found biased against
women applicants and triggered interest in AI ethics [3]. The need for
developing policies and principles that address the ethical aspects of AI
systems is timely and critical. The ethical harm in AI systems might may
jeopardize human control and these potential threats set a stage for AI ethics
discussion. AI systems are not only concerned with technical efforts, but also
need to consider the social, political, legal, and intellectual aspects.
However, AI’s current state of ethics is broadly unknown to the public,
practitioners, policy, and lawmakers [4].

Extensively, the ethically aligned AI system should meet the following three
components through the entire life cycle [2]: 1) the proposed AI system should
comply with all the applicable laws and regulations, 2) adhere to ethical
principles and values, and 3) should be technically and socially robust. To
the best of our knowledge, no empirical study is conducted that cover and
explore the above three core components based on the data collected from
industrial practitioners and lawmakers. For instance, Vakkuri et al. [4]
conducted a survey study to determine industrial perceptions based only on
four AI ethics principles, Lu et al.[5] conducted interviews with researchers
and practitioners to understand the AI ethics principles implications and the
motivation for rooting these principles in the design practices. Similarly,
Leikas et al. [6] mainly focused on AI ethics guidelines.

In this study, insights are provided by encapsulating the views and opinions
of practitioners and lawmakers regarding the AI ethics principles and
challenges by collecting survey data from 99 respondents across 20 different
countries.

##  2 Background

Generally, the AI ethics is classified under the umbrella of applied ethics,
which mainly consider the ethical issues that happen because of developing,
deploying, and using the AI systems. It focuses on linking how an AI system
could raise worries related to human autonomy, freedom in a democratic
society, and quality of life. Ethical reflection across AI technologies could
serve in achieving multiple societal purposes [2]. For instance, it can
stimulate focusing on innovations that aim to foster ethical values and bring
collective well-being. Ethically aligned or trustworthy AI technologies can
flourish sustainable well-being in society by bringing prosperity, wealth
maximization, and value creation [2].

It is vital to understand the development, deployment, and use of AI
technologies to ensure that everyone can build a better future and live a
thriving life in the AI-based world. However, the increasing popularity of AI
systems has raised concerns such as reliability and impartiality of decision-
making scenarios [2]. We need to make sure that decision-making support of AI
technologies must have an accountable process to ensure that their actions are
ethically aligned with human values that should not be compromised [2].

In this regard, different organizations and technology giants developed
committees to draft the AI ethics guidelines. Google and SAP presented the
guidelines and policies to develop ethically aligned AI systems [7].
Similarly, the Association of Computing Machinery (ACM), Access Now, and
Amnesty International jointly proposed the principles and guidelines to
develop an ethically mature AI system [7]. In Europe, the independent high-
level expert group on artificial intelligence (AI HLEG) developed the
guidelines for promoting trustworthy AI [2]. The Ethically Aligned Design
(EAD) guidelines are presented by IEEE, consisting of a set of principles and
recommendations that focus on the technical and ethical values of AI systems
[8]. In addition, the joint ISO/IEC international standard committee proposed
the ISO/IEC JTC 1/SC 42 standard which covers the entire AI ecosystem,
including trustworthiness, computational approach, governance,
standardization, and social concerns [9].

However, various researchers claim that the extant AI ethics guidelines and
principles are not effectively adopted in industrial settings. McNamara et al.
[10] conducted an empirical study to understand the influence of the ACM code
of ethics in the software engineering decision-making process. Surprisingly,
the study findings reveal that no evidence has been found that the ACM code of
ethics regulate decision-making activities. Vakkuri et al. [11] conducted
multiple interviews to know the status of ethical practices in the domain of
the AI industry. The study findings uncover the fact that various guidelines
are available; however, their deployment in industrial domains are far from
being mature. The gap between AI ethics research and practice remains an
ongoing challenge. To bridge this gap, we conducted an empirical study to know
the significance of AI ethics principles, challenges, and their impact by
encapsulating the views of AI practitioners and lawmakers.

##  3 Setting the Stage

Previously, we conducted a systematic literature review (SLR) to provide an
in-depth overview of AI ethics challenges and principles [12]. This study aims
to empirically validate the SLR findings based on the insights provided by the
AI practitioners and lawmakers. AI practitioners have higher ethical
responsibilities compared to others. Practitioners often make the design
decisions of complex autonomous systems with less ethical knowledge. The
magnitude of risks in AI systems makes practitioners responsible for
understanding ethical attributes. To achieve reliable outcomes, it is
essential to know the practitioners understanding of AI ethics principles and
challenges. Law resolves everyday conflicts and sustains order in social life.
People consider law an information source as it impacts social norms and
values [13]. The aim of considering this type of population (lawmakers) is to
understand the application of the law to AI ethics. The data collected from
legislation personnel will uncover the question, of whether standing AI ethics
principles are sufficient, or is there a need for innovative standards [13]?

The primary goal of this study is to understand the significance of the AI
ethics principles, challenges, and the severity of the challenges across the
principles. We used industrial collaboration contacts to search the AI
practitioners and send a formal invitation to participate in this study.
Moreover, various law forums across the world were contacted and requested to
participate in this survey. The targeted populations were approached using
social media networks including LinkedIn, WeChat, ResearchGate, Facebook, and
personal email addresses. The survey instrument consists of four core
sections: 1) demographics 2) AI ethics principles 3) challenges 4) challenges
impact on principles. The survey questionnaire also includes open-ended
questions to know the novel principles and challenges that were not identified
during the SLR [12]. The Likert scale is used to evaluate the significance of
each principle and challenge and assess the severity level of the challenging
factors. The survey instrument is structured both in English and Chinese. The
software industry in China is flourishing like never before, where AI is
taking the front seat and is home to some of the leading technology giants in
the world, such as Huawei, Alibaba, Baidu, Tencent, and Xiaomi. However, it
would be challenging to collect the data from the Chinese industry because of
the language barriers. Mandarin is the national and official language in
China, unlike India, where English is commonly used for official purposes.
Therefore, the Chinese version of the survey instrument is developed to cover
the major portion of the targeted population. Both English and Chinese
versions of the survey instrument are available online for replication [14].

The piloting of the questionnaire is performed by inviting three external
qualitative software engineering research experts. The expert’s suggestions
were mainly related to the overall design, and understandability of the survey
questions. The suggested changes were incorporated, and the survey instrument
was online deployed using Google forms (English version) and Tencent
questionnaire (Chinese version). The data were collected from September 2021
to April 2022 and we received 107 responses. The manual review revealed that
eight responses were incomplete and we only considered 99 responses for the
final data analysis.

##  4 Survey Results

Frequency analysis is performed to organize the descriptive data and it is
more suitable for analyzing a group of variables both for numeric and ordinal
data. We noticed that 99 respondents from 20 countries across 5 continents
with 9 roles and 10 different backgrounds participated in the survey study
(see Figure 1(a-c)). The organizational size (number of employees) of survey
participants mostly ranges from 50 to 249, which is 28% of the total responses
(see Figure 1(d)). Of all the responses, majority (48%) have 3-5 years of
experience working with AI focused projects as practitioners or lawmakers (see
Figure 1(e)).

Participants were asked to explain their opinions about the perceived
importance of AI systems in their organization. The majority of the
participants positively agreed. For instance, 77% mentioned that their
respective organizations consider ethical aspects in AI processes or develop
policies for AI projects, 12% answered negatively, and 10% were not sure about
it (see Figure 1(f)). We mapped the respondents’ roles across nine different
categories using thematic mapping (see Figure 1(b)). The final results show
that the most of the respondents (29%) are classified across the law
practitioner category. Similarly, the domains of the participants’
organizations are conceptually framed in 10 core categories and the results
revealed that most (19%) of the organizations are working on smart systems
(see Figure 1(c)).

The survey responses are classified as average agree, neutral and average
disagree (see Figure 2(a-b)). We observed that average $\geq$ 60% of the
respondents positively confirmed the AI ethics principles and challenges
identified in the SLR study [12]. For instance, one survey participant
mentioned that:

“The listed AI ethics principles are comprehensive and extensive to cover
various aspects of ethics in AI.”

Moreover, we selected the most frequently reported seven challenging factors
and six principles discussed in our SLR study [12]. The aim is to investigate
the severity impact of the seven challenges (i.e., lack of ethical knowledge,
vague principles, highly general principles, conflict in practice, interpret
principles differently, lack of technical understanding, and extra
constraints) across the six AI ethics principles (i.e. transparency, privacy,
accountability, fairness, autonomy, and explainability). The survey
participants were asked to rate the severity impact using the Likert scale:
short-term (insignificant, minor, moderate) and long-term (major, and
catastrophic) (see Figure 2(c)). The results reveal that most challenges have
long-term impacts on the principles (major, and catastrophic). For example,
interpret principles differently challenge is likely to have a long-term
impact (i.e., 50% major, and 27% catastrophic) on the transparency principle.

Further, the non-parametric statistical analysis is performed to understand
the significant differences between both types of populations (AI
practitioners and lawmakers). Because of the allowed word limits, the
statistical discussion and dataset are provided online [14].

Figure 1: Demographic details

##  5 Data Interpretation

The results illustrate that the majority of the survey participants positively
agreed to consider the identified list of AI ethics principles (see Figure
2(a)). We noticed that 77.8% of survey respondents thought transparency as the
most significant AI ethics principle. This is an interesting observation as
transparency is equally confirmed as one of the core seven essential
requirements by AI HLEG [2] for realizing the ‘trustworthy AI’. Transparency
provides detailed explanations of logical AI models and decision-making
structures understandable to the system stakeholders. Moreover, it deals with
the public perceptions and understanding of how AI systems work. Broadly, it
is a societal and normative ideal of “openness”.

The second most significant principle to the survey participants was
accountability (71.7%). It refers to the expectations or requirements that the
organizations or individuals need to ensure throughout the lifecycle of an AI
system. They should be accountable according to their roles and applicable
regulatory frameworks for the system design, development, deployment, and
operation by providing documentation on the decision-making process or
conducting regular auditing with proper justification. Privacy is the third
most frequently occurred principle, supported by 69.7% of the survey
participants. It refers to preventing harm, a fundamental right specifically
affected by the decision-making system. Privacy compels data governance
throughout the system lifecycle, covering data quality, integrity, application
domain, access protocols, and capability to process the data in a way that
safeguards privacy. It must be ensured that the data collected and manipulated
by the AI system shall not be used unlawfully or unfairly discriminate against
human beings. For example, one of the respondents mentioned that

“The privacy of hosted data used in AI applications and the risk of data
breaches must be considered.”

In general, the survey findings of AI ethics principles are in line with the
widely adopted accountability, responsibility, and transparency (ART)
framework [15] and the findings of an industrial empirical study conducted by
Ville et al. [2]. Both studies [2] [15] jointly considered transparency and
accountability as the core AI ethics principles, which is consistent with the
findings in this survey. However, we noticed that privacy has been ignored in
both mentioned studies [2] [15], but is placed as the third most significant
principle in this survey. The reason might be that, as more and more AI
systems have been placed online, the significance of privacy and data
protection is increasingly recognized. Presently, various countries embarked
on legislation to ensure the protection of data and privacy.

Further, the results reveal that the majority of the survey respondents
($>$60%) confirmed the identified challenging factors [12] (see Figure 2(b)).
Lack of ethical knowledge is considered as the most significant challenge by
(81.8%) the survey participants. It exhibits that knowledge of ethical aspects
across AI systems is largely ignored in industrial settings. There is a
significant gap between research and practice in AI ethics. Various guidelines
and policies devised by researchers and regulatory bodies discussed different
ethical goals for AI systems. However, these goals have not been widely
adopted in the industrial domain because of limited knowledge of scaling them
in practice. The findings are in agreement with the industrial study conducted
by Ville et al. [11], which summarises the overall findings by stating that
ethical aspects of AI systems are not particularly considered, and it mainly
happened because of a lack of knowledge, awareness, and personal commitment.
We noticed that no legal frameworks (69.7%) is ranked as the second most
common challenge for considering ethics in the AI domain. The proliferation of
AI technologies in high-risk areas starts mounting the pressure of designing
ethical and legal standards and frameworks to govern them. It highlights the
nuances of the debate on AI law and lays the groundwork for a more inclusive
AI governance framework. The framework shall focus on most pertinent ethical
issues raised by the AI systems, the use of AI across industry and government
organisations, and economic displacement (i.e. the ethical reply to the loss
of jobs as a result of AI-based automation). The third most common challenging
factor is lacking monitoring bodies, and it was highlighted by (68.7%) of the
survey participants. Lacking monitoring bodies refers to the lack of
regulatory oversight to assess ethics in AI systems. It raises the issue of
public bodies’ empowerment to monitor, audit, and oversee the enforcement of
ethical concerns in AI technologies by the domain (e.g., health, transport,
education). One survey respondent mentioned that

“I believe it shall be mandatory for the industry to get standard approval
from monitoring bodies to consider ethics in the development process of AI
systems.”

Monitoring bodies extensively promote and observe the ethical values in
society and evaluate technology development associated with ethical aspects of
AI [2]. They would be tasked to advocate and define responsibilities and
develop rules, regulations, and practices in a situation where the system
takes a decision autonomously. The monitoring group should ensure “ethics by,
in and for design” as mentioned in AI HLEG [2] guidelines.

Additionally, the survey participants elaborated on new challenging factors.
For instance, one of the participants mentioned that

“Implicit biases in AI algorithms such as data discrimination and cognitive
biases could impact system transparency.”

Similarly, the other respondent reported that

“Biases in the AI system’s design might bring distress to a group of people or
individuals.”

Moreover, a survey respondent explicitly considered the lack of tools for
ethical transparency and AI biases as significant challenges to AI ethics. We
noticed that AI biases is reported as the most common additional challenge. It
will be interesting to further explore (i) the type of biases that might be
embedded with the AI algorithms, (ii) the causes of these biases, and (iii)
corresponding countermeasures to minimize the negative impact on AI ethics.

Finally, the severity impact of the seven challenging factors is measured with
respect to the six AI ethics principles (see Figure 2 (c)). For the
transparency principle, we noticed that the challenging factor interpret
principles differently has significant long-term impacts, and 77% (i.e., 50%
major, and 27% catastrophic) of the survey participants agreed to it. The
interpretation of ethical concepts can change for a group of people and
individuals. For instance, the practitioners might perceive transparency
differently (more focused on technical aspects) than law and policymakers, who
have broad social concerns. Furthermore, lack of ethical knowledge has a
short-term impact on the transparency principle, and it is evident from the
survey findings supported by 52% (7% insignificant, 25% minor, and 20%
moderate) responses. Lack of knowledge could be instantly covered by attaining
knowledge, understanding, and awareness of transparency concepts.

Conflict in practice is deemed the most significant challenge to the privacy
principle. Hence, 74% (i.e., 53% major, and 21% catastrophic) survey
respondents considered it a long-term severe challenge. Various groups,
organizations, and individuals might have opinion conflicts associated with
privacy in AI ethics. It is critical to interpret and understand privacy
conflicts in practice. We noticed that (82%) of survey participants considered
the challenging factor extra constraints as the most severe (long-term)
challenge for both accountability and fairness principles. Situational
constraints, including organizational politics, lack of information, and
management interruption, could possibly interfere with the accountability and
fairness measures. It could negatively impact the employee’s motivation and
interest to explicitly consider ethical aspects across the AI activities.
Interestingly, (82%) of the survey respondents considered conflict in practice
as the most common (long-term) challenge for autonomy and explainability
principles.

Overall, we could interpret that conflict in practice is the most severe
challenge, and its average occurrence is $>$77% for all the principles. It
gives a general understanding to propose specific solutions that focus on
tackling the opinion conflict regarding the real-world implication of AI
ethics principles. The results further reveal lack of ethical knowledge  has
an average (38%) short-term impact across selected AI ethics principles. The
lack of knowledge gap could be covered by conducting training sessions,
workshops, certification, and encouraging social awareness of AI ethics.
Knowledge increases the possibility of AI ethics success and acceptance in the
best practice of the domain.

Figure 2: AI Ethics Principles and Challenges

##  6 Lessons Learned

The study findings result in the following lessons learned:

  1. 1.

Emerging Roles: Besides practitioners, the role of policy and lawmakers is
also important in defining the ethical solutions for AI-based systems. Based
on our knowledge, this study is the first effort made to encapsulate the views
and opinions of both types of populations.

  2. 2.

Confirmatory Findings: This study empirically confirms the AI ethics
principles and challenging factors discussed in our published SLR study [12].

  3. 3.

Adherence to AI principles: The most common principles (e.g., transparency,
privacy, accountability) and challenges (e.g., lack of ethical knowledge, no
legal frameworks, lacking monitoring bodies) must be carefully realized in AI
ethics.

  4. 4.

Risk-aware AI ethics principle design: The challenging factors have mainly
long-term severity impacts across the AI ethics principles. It opens a new
research call to propose solutions for minimizing or mitigating the impacts of
the challenging factors.

The given catalogue (see Figure 2) of principles and challenging factors could
be used as a guideline for defining ethics in the AI domain. Moreover, the
catalogue is a starting point for further research on AI ethics. It is
essential to mention that the identified principles and challenging factors
only reflect the perceptions of 99 practitioners and lawmakers in 20
countries. More deep and comprehensive empirical investigation with wider
groups of practitioners would be useful to generalize the study findings
globally. This, together with proposing a robust solution for integrating
ethical aspects in AI design and process flow, will be part of our future
work.

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