# Nuclear polarisation and relativistic effects contributions to King plot
non-linearity

P. Munro-Laylim, V. A. Dzuba, and V. V. Flambaum  School of Physics,
University of New South Wales, Sydney 2052, Australia

###### Abstract

The effect of nuclear polarisation on the non-linearities of King plot for
isotope shift in Ca+, Sr+ and Yb+ ions is studied. These ions are currently
used in search for the manifestation of new interaction via non-linearities of
King plot. The role of relativistic effects is also studied. The best way to
achieve separation of electronic and nuclear variables (which is necessary for
the linearity) is to present the field shift as $F^{\prime\prime}\delta\langle
r^{\lambda}\rangle$, where the value of $\lambda$ is the result of numerical
relativistic calculations ($2\gamma<\lambda<2$,
$\gamma=\sqrt{1-(Z/137.036)^{2}}$). Using this ansatz we demonstrated that the
contribution of the relativistic effects to the non-linearity in spherical
nuclei is very small.

##  I Introduction

The Standard Model (SM) has been one of the most successful theories in
physics, proving to be accurate in its numerous predictions. However,
significant flaws still arise in the SM, including the lack of an explanation
for dark matter and matter-antimatter asymmetry, among other problems. The
apparent incompleteness leads us to explore for physics beyond the SM, with a
variety of avenues being tested to find new physics. One particular area of
exploration is with isotope shift (IS), where significant improvements in
measurements have been made in recent years - see e.g. Counts et al. (2020);
Knollmann et al. (2019); Miyake et al. (2019); Manovitz et al. (2019);
Figueroa et al. (2021). Within SM, the King plot for isotope shift King (1963)
is approximately linear, up to some small contributions. However, in Refs.
Counts et al. (2020); Miyake et al. (2019); Figueroa et al. (2021) small
deviations from linearity in the King plot have been observed. Multiple
explanations have been offered in Refs. Delaunay et al. (2017); Berengut et
al. (2018); Flambaum et al. (2018) to account for the observed non-linearity,
including higher order correction terms to field isotope shift (FIS) within
SM, but a possibility of a new interaction, mediated by a boson beyond SM, is
also a valid explanation for the observations.

It was proposed in in Refs. Delaunay et al. (2017); Berengut et al. (2018);
Flambaum et al. (2018) that this boson $\phi$ acts as a mediator between
neutrons and electrons, which may contribute a Yukawa potential to the nuclear
potential. Due to this interaction the boson affects FIS, however such a small
contribution would be impossible to detect among SM contributions in direct IS
measurements. This Yukawa potential would be more apparent in the King plots
due to its contribution to the non-linearity. This effect on the King plot
non-linearity has been examined in Ref. Counts et al. (2020); Frugiuele et al.
(2017); Rehbehn et al. (2021), however these explorations neglected SM
contributions to the non-linearities; while the work is notable, findings
cannot be conclusive as true constraints on the boson. Therefore, the
importance of accurately determining SM contributions to the King plot non-
linearity is apparent, as this will allow future works to apply more accurate
bounds on the new boson.

The non-linearities of King plot arise from small SM contributions to FIS,
which include quadratic field shift, relativistic effects and effects of
nuclear deformation.

In several previous papers Flambaum and Dzuba (2019); Allehabi et al. (2020,
2021), we examined the effect of the nuclear deformation on isotope shift in
heavy elements. We calculated various contributions to the non-linearity of
the King plots for Yb+ in Ref. Allehabi et al. (2021). An important conclusion
was that the non-linearity from the variation of the nuclear deformation in
different isotopes is comparable to the observed non-linearity in Ref. Counts
et al. (2020). We also found variation in the non-linearity between the
results of the calculations, based on different nuclear models, which is
caused by the variation of the nuclear shape between these models. The
sensitivity of the non-linearity calculations is also emphasised in Ref.
Müller et al. (2021), which found that experimental uncertainties in the
nuclear charge radii can lead to large variations in the non-linearity.

Nuclear polarisation has been suggested as a non-negligible contribution to
the non-linearity in Ref. Flambaum et al. (2018). In a recent paper Flambaum
et al. (2021), the effect of the nuclear polarisation on spectra of atoms and
ions was examined. To incorporate the effect into the standard atomic many-
body calculations, the results of the direct calculations of the corresponding
energy shifts were accurately fitted by the effective potentials, which may be
added to the Coulomb nuclear potential. In particular, the contributions to
the nuclear polarisation effects from the giant electric-dipole (E1) nuclear
resonance and rotational nuclear (E2) transitions have been calculated, with
the effective potentials derived for each contribution.

The giant resonance contribution is enhanced by the large collective E1
nuclear matrix elements. There is a simple explanation why E2 effect is
important in heavy atoms with a deformed nucleus. In light atoms nuclear
rotation is faster than electron motion, and electron sees averaged spherical
Coulomb nuclear potential. In heavy atoms nuclear rotation is slower but
electron moves faster. It sees a non-spherical nucleus, and this affects
electron energy levels and wave functions. If we use perturbation theory to
calculate this effect, it is described as a second order perturbation theory
sum with the non-diagonal matrix elements of the quadruple component (E2) of
the electron-nucleus electrostatic interaction, where in intermediate state
both electron and nucleus are excited. In heavy atoms typical virtual electron
excitation energy is bigger than the minimal nuclear excitation energy, which
is equal to the rotational interval. Small nuclear excitation energy (the
rotational interval) and collective nuclear matrix elements between the
rotational states are the reasons for the enhancement of this E2 nuclear
polarisation effect. Both the nuclear E2 matrix elements and the rotational
intervals depend on the isotope. An isotopic dependence of this effect
indicates an important contribution to the King plot non-linearity.

In this paper, we examine and calculate nuclear polarisation contributions to
the King plot non-linearity. We also study the role of the relativistic
effects and demonstrate that their contribution to the non-linearity is small.
We present a way of separation of the relativistic effects from other possible
contributions to the non-linearities. We consider Ca+, Sr+ and Yb+ ions since
they have already been used in the search for new interactions via non-
linearities of King plot Shi et al. (2016); Gorges et al. (2015); Solaro et
al. (2020); Knollmann et al. (2019); Manovitz et al. (2019); Dubost et al.
(2014); Counts et al. (2020).

##  II King plot and contributions to its non-linearity

In this work we study a contribution to the non-linearity of King plot caused
by nuclear polarizability and compare it to other possible sources of the non-
linearities, which exist within the standard model (SM). This includes
quadratic field shift (QFS), relativistic and nuclear deformation effects. An
expression for the IS, which contains all these effects, can be written as

$\displaystyle\nu_{a}=F_{a}\delta\langle r^{2}\rangle+K_{a}\mu^{AA^{\prime}}+$
(1) $\displaystyle G_{a}^{(2)}\delta\langle
r^{2}\rangle^{2}+G_{a}^{(4)}\delta\langle r^{4}\rangle+\nu_{a}^{NP}.$

Here $\nu_{a}$ is the IS for an atomic transition $a$, $K_{a}$, $F_{a}$,
$G_{a}^{(2)}$, and $G_{a}^{(4)}$ are electronic structure factors, which
depend on the transition but do not depend on the nuclear isotope;
$\delta\langle r^{2}\rangle$, and $\delta\langle r^{4}\rangle$ are changes of
the nuclear moments between isotopes $A$ and $A^{\prime}$,
$\mu^{AA^{\prime}}=1/A-1/A^{\prime}$. First line in (1) is the standard
expression for the IS contaning leading contribution to the FIS and the mass
shift. Second line contains higher-order contributions to FIS which we study
in present work. The term with $G_{a}^{(2)}$ is the QFS, the term with
$G_{a}^{(4)}$ is due to the relativistic effects and effects of changing
nuclear shape. The last term is the main focus of present work. It presents
the change of the frequency of an atomic transition between two isotopes due
to the difference in nuclear polarisation potentials for the two isotopes.
Note that in this term we do not separate nuclear and electronic variables.
This is because the nuclear polarisation potential depends on several nuclear
parameters (see below) and separation of the variables leads to unwanted
complications.

If we have two atomic transitions $a$ and $b$ and two corresponding equations
(1), then finding $\delta\langle r^{2}\rangle$ from one equation and
substituting it to another leads to

$\displaystyle\frac{\nu_{b}^{AA^{\prime}}}{\mu^{AA^{\prime}}}=\frac{G_{b}}{F_{a}}\frac{\nu_{a}^{AA^{\prime}}}{\mu^{AA^{\prime}}}+\bigg{(}K_{b}-\frac{F_{b}}{F_{a}}K_{a}\bigg{)}+$
(2)
$\displaystyle\bigg{(}G_{b}^{(2)}-\frac{F_{b}}{F_{a}}G_{a}^{(2)}\bigg{)}\frac{\delta\langle
r^{2}\rangle^{2}}{\mu^{AA^{\prime}}}+\bigg{(}G_{b}^{(4)}-\frac{F_{b}}{F_{a}}G_{a}^{(4)}\bigg{)}\frac{\delta\langle
r^{4}\rangle}{\mu^{AA^{\prime}}}$
$\displaystyle+\bigg{(}\nu_{b}^{NP}-\frac{F_{b}}{F_{a}}\nu_{a}^{NP}\bigg{)}\frac{1}{\mu^{AA^{\prime}}}.$

Here first line is the equation for the standard King plot. If we keep these
terms only, points, corresponding to pairs of isotopes, are all on the same
line on the $(\nu/\mu)_{a}$, $(\nu/\mu)_{b}$ plane. The terms in the second
and third lines can cause the non-linearities of the King plot since they
depend on isotopes. Note that each of these terms may vanish if
$G_{a}^{(2)}/G_{b}^{(2)}=F_{a}/F_{b}$, or
$G_{a}^{(4)}/G_{b}^{(4)}=F_{a}/F_{b}$, or
$\nu_{a}^{NP}/\nu_{b}^{NP}=F_{a}/F_{b}$.

##  III Calculation of the electronic structure factors

.

In this work we calculate electronic structure factors $F$, $G^{(2)}$,
$G^{(4)}$ and frequency shift $\nu^{NP}$ (see Eq. (1)). We consider Ca+, Sr+,
and Yb+ ions, which all have similar electronic structure, having one external
electron above closed-shell core. Therefore, we use the same approach for all
three systems, the approach which we used for Yb+ calculations in our previous
work Allehabi et al. (2021). In short, we perform the calculations in the
$V^{N-1}$ approximation, starting from the relativistic Hartree-Fock
calculations for the closed-shell core. The states of external electron are
calculated in the field of the frozen core. Core-valence correlations are
included by means of the correlation potential method Dzuba et al. (1987). The
correlation potential $\hat{\Sigma}$ (which is a non-local integration
operator) is calculated in the second order of the many-body perturbation
theory and added to the relativistic Hartre-Fock potential. Solving Hartree-
Fock equations with added $\hat{\Sigma}$ redefines the valence orbitals
leading to the so-called Bruekner orbitals (BO) for the valence states. FIS is
calculated as a perturbation caused by the change of the nuclear potentials
between two isotopes. Random phase approximation (RPA) is used to calculate
the energy shift. When the shift is caused by the change of nuclear volume and
shape between two isotopes, the fitting of the RPA results leads to the
determination of the $F$ and $G^{(4)}$ electronic structure factors. The
$G^{(2)}$ factor , responsible for the quadratic isotope shift, is calculated
as a second-order correction to the energy (see Allehabi et al. (2021) for
details). When the shift is caused by the difference in the nuclear
polarisation potentials between two isotopes, the RPA calculations produce the
values of $\nu^{NP}$. Calculated values of the $F$, $G^{(2)}$, $G^{(4)}$
factors for several states or transitions in Ca+, Sr+, and Yb+ are presented
in Table 1. Note that when the factors are presented for the states, the
factors for transitions should be calculated as a difference between
corresponding factors for the states. The accuracy of the calculations is
$\sim$ 1%. However, we present many digits in the calculated values of the
factors because the non-linearities are very sensitive to the ratio of the
factors (see last sentence of the previous section). Since calculations for
different states are done exactly the same way, we believe that the accuracy
for the ratios of the electronic factors $F$, $G$ and $\nu^{NP}$ is higher
than 1%.

Table 1:  Calculated electronic structure factors. Numbers in square brackets mean powers of 10. Ion | State or | $F$ | $G^{(2)}$ | $G^{(4)}$
---|---|---|---|---
| transition | GHz/fm2 | GHz/fm4 | GHz/fm4
Ca+ | $4s_{1/2}$ | 0.26026 | -0.8358[-4] | -0.3710[-4]
| $4p_{1/2}$ | -0.02240 | -0.1525[-6] | 0.3580[-5]
| $4p_{3/2}$ | -0.02262 | -0.3996[-10] | 0.2965[-5]
| $3d_{3/2}$ | -0.10735 | -0.6488[-8] | 0.1569[-4]
| $3d_{5/2}$ | -0.10673 | -0.6455[-8] | 0.1615[-4]
Sr+ | $5s_{1/2}$ | 1.25903 | -0.9380[-3] | -0.4268[-3]
| $5p_{1/2}$ | -0.07221 | -0.7237[-5] | 0.2511[-4]
| $5p_{3/2}$ | -0.08055 | 0.2619[-8] | 0.2767[-4]
| $4d_{3/2}$ | -0.35372 | -0.7741[-7] | 0.1185[-3]
| $4d_{5/2}$ | -0.34346 | -0.7529[-7] | 0.1151[-3]
Yb+ | $6s-5d_{5/2}$ | -17.6035 | 0.02853 | 0.013083
| $6s-5d_{3/2}$ | -18.0028 | 0.02853 | 0.013371

###  III.1 Nuclear Polarisation

The general effective potential for the nuclear polarisation contribution to
the atomic energy shift was introduced in Ref. Flambaum et al. (2018) to allow
for calculation in multielectron atoms and ions. It may be schematically
presented as

$V_{L}(r)=-\frac{e^{2}}{2}\frac{\alpha^{EL}_{0}}{r^{2L+2}+b^{2L+2}}\,,$ (3)

with the scalar nuclear polarizability due to the $EL$ nuclear transition from
state of the total angular momentum $L$ to the ground state

$\alpha^{EL}_{0}=\frac{8\pi}{2L+1}\frac{B(EL;L\to 0)}{E_{L}},$ (4)

$b$ in (3) is the cut-off parameter which depends on the nuclear properties,
notably nuclear charge $Z$, mass number $A$, nuclear radius $R_{0}$, and
nuclear deformation $\beta_{2}$. It was calculated and tabulated in Flambaum
et al. (2018). At large distances $r>>b$ this potential becomes an ordinary
polarisation potential. Parameters of this effective potential have been
fitted to reproduce directly calculated energy shifts to the accuracy $\sim$ 1
%.

The electric dipole component of this potential is dominated by the
contribution of the nuclear giant electric dipole resonance. The potential (3)
becomes

$V_{1}=-\frac{e^{2}}{2}\frac{\alpha_{0}^{E1}}{r^{4}+b^{4}}\,,\quad\alpha_{0}^{E1}=\frac{8\pi
B(E1)}{3E_{\rm GR}}\,.$ (5)

Here $\alpha_{0}^{E1}$ is the scalar electric dipole nuclear polarizability,
$B(E1)$ is the giant resonance transition probability,

$B(E1)=\frac{3}{8\pi}\frac{Z(A-Z)e^{2}}{AE_{GR}m_{p}},$ (6)

$A$ is atomic number, $E_{GR}$ is the energy of the giant dipole resonance,
$m_{p}$ is proton mass.

The electric quadrupole component of the potential (3) in deformed nuclei is
dominated by the nuclear rotational transition due to the small excitation
energy $E_{rot}$ and large collective matrix element of such transition.

$V_{2}=-\frac{e^{2}}{2}\frac{\bar{\alpha}_{0}^{E2}}{r^{6}+\tilde{b}^{6}}\,.$
(7)

Here coefficient $\bar{\alpha}_{0}^{E2}$ is equal to

$\bar{\alpha}_{0}^{E2}=\frac{8\pi}{5}\frac{B(E2)}{\bar{E}_{\rm rot}}\,.$ (8)

Note that $\bar{\alpha}_{0}^{E2}$ looks similar but not exactly equal to the
electric quadrupole scalar nuclear polarisability $\alpha_{0}^{E2}$. The
reason is that $\bar{E}_{\rm rot}=50$ keV is a parameter, which is numerically
close to the average nuclear rotational interval, but the actual dependence on
the rotational energy $E_{rot}$ in a specific isotope is weak due to the
domination of the electron excitation energy $E_{\rm electron}$ in the energy
denominators of the perturbation theory ($E_{\rm electron}+E_{\rm rot}$);
$B(E2)$ is the nuclear rotational transition probability

$B(E2)=\frac{1}{5}\frac{3}{4\pi}Z^{2}e^{2}R_{0}^{4}\beta_{2}^{2},$ (9)

$R_{0}$ is the nuclear radius, $\beta_{2}$ is the parameter of the nuclear
quadrupole deformation. Note that the potential (7) vanishes for spherical
nuclei. Strong dependence on the isotope appears here due to variations of the
quadrupole deformation parameter $\beta_{2}$ between two isotopes, similar to
the deformation effect in the field isotope shift resulting in the non-
linearity of the King plot Allehabi et al. (2021).

Numerical values for all parameters needed for the calculations can be found
in Ref. Flambaum et al. (2018).

##  IV Results

###  IV.1 Non-linearities of King plot

To calculate non-linearities of King plot we calculate IS for a pair of
transition using equation (1) and calculate function (2) which corresponds to
the King plot. Mass shift is not included in the calculations since it does
not contribute to the non-linearities. It just moves the King plot on the
vertical scale without affecting its shape. On the next step we fit the
function (2) by a straight line on the $\nu_{a}/\mu$, $\nu_{b}/\mu$ plane. The
non-linearities are defined as relative deviation $r$ of the calculated FIS
from the straight line,

$r^{AA^{\prime}}=\frac{\tilde{\nu}_{b}^{AA^{\prime}}-\nu_{sl}^{AA^{\prime}}}{\tilde{\nu}_{b}^{AA^{\prime}}},$
(10)

where $\tilde{\nu}\equiv\nu/\mu$ is given by formula (2), $\nu_{sl}$ is the
result of fitting by straight line.

To calculate FIS using (1) we need to know electronic structure factors $F$,
$G^{(2)}$, $G^{(4)}$, nuclear factors $\delta\langle r^{2}\rangle$,
$\delta\langle r^{4}\rangle$, and nuclear parameters for the nuclear
polarisation potentials $V_{1}$ and $V_{2}$ (see Eqs. (5,7)). Calculation of
the electronic structure factors was considered in previous section and
described in more details in our earlier paper on Yb+ Allehabi et al. (2021).
For the nuclear structure factors $\delta\langle r^{2}\rangle$, $\delta\langle
r^{4}\rangle$ we use the results of nuclear calculations performed by means of
Hartree-Fock-Bogolubov method with the covariant density functional called DD-
PC1 Agbemava et al. (2014). For Yb+ we also use the FIT1 nuclear model
Allehabi et al. (2021) in which nuclear parameters were chosen to fit the
observed non-linearities of the King plot. Note that nuclear calculations do
not give the value of the $\delta\langle r^{4}\rangle$ directly. Instead, they
give the value of the nuclear quadrupole deformation parameter $\beta_{2}$,
which can be used to calculate the nuclear density averaged over nuclear
rotation. Using this density we calculate corresponding $\delta\langle
r^{4}\rangle$ (see Ref. Allehabi et al. (2021) for more details). For the
nuclear polarisation potentials $V_{1}$ and $V_{2}$ we use nuclear parameters
presented in Ref. Flambaum et al. (2018).

To separate relative contributions to the non-linearities from different
effects, such as QFS, nuclear shape and nuclear polarisation, we remove in
turn all but one term from the second and third line of Eq. (2), do the
fitting by the straight line and calculate the non-linearities using Eq. (10).
The results are presented in Table 2. We study the non-linearities for several
pairs of transitions, following experimental publications. E.g. the
$4s-4p_{1/2,3/2}$ pair of transitions in Ca+ ion was studied in Refs. Shi et
al. (2016); Gorges et al. (2015), the $4s-3d_{3/2,5/2}$ pair of transitions in
Ca+ ion was studied in Ref. Solaro et al. (2020), the $5s-5p_{1/2}$,
$4d_{3/2}-5p_{1/2}$ pair of transitions in Sr+ ion was studied in Ref. Dubost
et al. (2014), and the $6s-5d_{3/2,5/2}$ pair of transitions in Yb+ ion was
studied in Ref. Counts et al. (2020).

It turns out that the results are sensitive to the details of the
calculations. In particular, they are sensitive to the values of the nuclear
parameters. Therefore, the uncertainties are close to the values of the the
calculated non-linearities. However, the order of magnitude for each source of
the non-linearities is established sufficiently reliably. We see that for all
considered ions the dominating contribution to the non-linearity of King plot
comes from the QFS and nuclear shape effects (terms associated with the
$G^{(2)}$ and $G^{(4)}$ coefficients in (1) and (2)), while the contribution
from nuclear polarisation is significantly smaller. In Yb+ the dominating
source of the non-linearity is nuclear deformation (associated with the
$G^{(4)}$ coefficient) in agreement with our earlier study Allehabi et al.
(2021). All considered isotopes of Ca and Sr have spherical nuclei. Therefore,
the $V_{2}$ nuclear polarisation potential is zero for them and there is no
corresponding contribution to the non-linearities. The calculated non-
linearities in the FIT1 nuclear model for Yb+ are in good agreement with
experiment, which is the result of the fitting by the nuclear deformation
parameters (see Ref. Allehabi et al. (2021) for details). The calculated non-
linearities in Ca+ and Sr+ are consistent with the limits found in experiments
Shi et al. (2016); Gorges et al. (2015); Dubost et al. (2014).

It might be useful to know the values of absolute contributions of the nuclear
polarisation to the IS. According to the calculations, the contribution from
the $V_{1}$ potential is $\sim$ 100 to 400 Hz for Ca+, $\sim$ 1.0 to 1.6 kHz
in Sr+, and $\sim$ 4 kHz in Yb+. The contribution from $V_{2}$ to IS in Yb+ is
$\sim\pm$ 100 Hz.

We see that nuclear polarization is not the dominating contribution to the
non-linearity of King plot. However, it is sufficiently large to affect the
analysis of the non-linearities in terms of the manifestation of new
interactions.

The accuracy of the calculation of all considered sources of the non-
linearities (QFS, nuclear deformation and nuclear polarisation) is
insufficient to subtract them from the observed non-linearities and use the
residual non-linearities to put the limits on a new interaction. Therefore,
the limits on new boson may be determined by the uncertainty of SM
contributions to the non-linearity rather than experimental errors. Order of
magnitude estimates for this situation have been presented in the Table V of
Ref. Flambaum et al. (2018). SM contributions to the non-linearity increase
with the nuclear charge $Z$. If new boson Compton wave length is comparable or
bigger than Bohr radius $a_{B}$, the new boson contribution to the non-
linearity does not increase with $Z$ and one could conclude that the best
limits on the new boson may be obtained in light atoms with spherical nucleus
like Ca, where the field shift contributions to the non-linearities are small.
However, a proper estimate of the non-linearities from the quadratic mass
shift is required here, and this may become another theoretical problem. Atoms
with medium spherical nuclei like Sr have much smaller quadratic mass shift
than Ca. Also, if the Compton wave length of new boson is smaller than Bohr
radius, the contribution of the new boson to the non-linearities increases
with $Z$. This gives another advantage to Sr in comparison with Ca.

Table 2:  Non-linearities of King plot coming from different sources. Numbers in square brackets mean powers of 10. Pair of | $G^{(2)}$ | $G^{(4)}$ | $V_{1}$ | $V_{2}$ | Expt.
---|---|---|---|---|---
isotopes |  |  |  |  | Counts et al. (2020)
Ca+, DD-PC1, $4s-4p_{1/2}$ and $4s-4p_{3/2}$
40 - 42 | -1.7[-9] | 2.2[-8] | -3.2[-9] |  |
42 - 44 | 3.3[-9] | -4.6[-8] | 6.4[-9] |  |
44 - 46 | 2.6[-10] | 4.6[-9] | 5.1[-10] |  |
46 - 48 | -2.5[-9] | 2.7[-8] | -4.9[-9] |  |
Ca+, DD-PC1, $4s-3d_{5/2}$ and $4s-3d_{3/2}$
40 - 42 | 2.1[-9] | 1.4[-8] | -4.3[-10] |  |
42 - 44 | -4.1[-9] | -3.0[-8] | 8.6[-10] |  |
44 - 46 | -3.1[-10] | 3.0[-9] | 6.7[-11] |  |
46 - 48 | 3.2[-9] | 1.7[-8] | -6.5[-10] |  |
Sr+, DD-PC1, $5s-5p_{1/2}$ and $5p_{1/2}-4d_{3/2}$
84 - 86 | 3.1[-5] | 2.2[-6] | 3.6[-9] |  |
86 - 88 | 3.6[-5] | 7.7[-7] | 1.8[-8] |  |
88 - 90 | -1.2[-5] | -5.6[-7] | -3.8[-9] |  |
90 - 92 | -3.7[-4] | -1.7[-5] | -1.2[-7] |  |
Yb+, DD-PC1, $6s-5d_{3/2}$ and $6s-5d_{5/2}$
168 - 170 | -4.5[-8] | 1.2[-8] | -1.6[-10] | -3.1[-9] | -1.9[-7]
170 - 172 | -2.9[-7] | -2.4[-7] | -1.3[-9] | -3.3[-8] | 2.7[-7]
172 - 174 | -8.8[-8] | 4.0[-7] | 1.0[-11] | 7.8[-9] | -4.9[-7]
174 - 176 | 4.0[-7] | -1.6[-7] | 1.5[-9] | 2.6[-8] | 4.1[-7]
Yb+, FIT1, $6s-5d_{3/2}$ and $6s-5d_{5/2}$
168 - 170 | -6.4[-8] | -1.6[-7] | -4.2[-11] | -2.5[-9] | -1.9[-7]
170 - 172 | 7.9[-8] | 2.2[-7] | 1.0[-10] | 3.5[-9] | 2.7[-7]
172 - 174 | -2.3[-8] | -4.3[-7] | -6.2[-10] | -5.9[-9] | -4.9[-7]
174 - 176 | 6.8[-9] | 3.6[-7] | 5.6[-10] | 4.8[-9] | 4.1[-7]

###  IV.2 The Role of Relativistic Effects

In non-relativistic case FIS is given by

$\nu^{\rm FIS}=F\delta\langle r^{2}\rangle.$ (11)

This formula is widely used to extract the change of nuclear charge radius
from the IS measurements. It is even used for this purpose for heavy elements
up to nobelium ($Z$=102) Raeder et al. (2018). However, in heavy atoms the use
of formula (11) should be reconsidered due to the role of relativistic
effects. They manifest themselves, in particular, in the dependence of the
electronic structure factor $F$ in (11) on the nuclear radius. This dependence
can be neglected for neighbouring isotopes for the extraction of
$\delta\langle r^{2}\rangle$ (which was the case for nobelium). But it is not
the case when the non-linearities of the King plot are considered. The
separation of the nuclear and electronic variables is a necessary condition
for the King plot to be linear. Given that the non-linearities can be very
small and at least four isotopes are needed to study them, the drift of the
coefficient $F$ before $\delta\langle r^{2}\rangle$ can no longer be
neglected. It was suggested in Ref. Flambaum et al. (2018) to use a different
formula

$\nu^{\rm FIS}=F^{\prime}\delta\langle r^{2\gamma}\rangle,$ (12)

where $\gamma=\sqrt{1-(Z\alpha)^{2}}$, $\alpha=1/137.036$ is the fine
structure constant. In this formula the electronic structure factor
$F^{\prime}$ practically does not depend on the nuclear isotope. It was
obtained by the relativistic consideration of a model problem in which the
nucleus was presented as a uniformly charged ball and approximate analytical
expressions for the wave functions for the $s$ and $p_{1/2}$ states inside the
nucleus were used. A more sophisticated formula was suggested in Ref. Müller
et al. (2021) using a similar approach.

In this section we study the problem numerically, using a more realistic
nuclear model with the Fermi distribution of nuclear charge and numerical wave
functions obtained in the relativistic Hartree-Fock calculations or in the
numerical calculations for the hydrogen -like ions. FIS is calculated as an
expectation value of the change of the nuclear potential between two nuclear
isotopes. The results are presented in the form, similar to (12),

$\nu^{\rm FIS}_{a}=\langle a|\delta
V_{N}|a\rangle=F_{a}^{\prime\prime}\delta\langle r^{\lambda}\rangle.$ (13)

The values of $\lambda$ are found from the condition that the electronic
structure factor $F^{\prime\prime}$ does not depend on the nuclear isotope.
The results are presented in Table 3 for $s_{1/2}$ and $p_{1/2}$ wave
functions of Ca+, Sr+, Yb+, Fr and E120+. It turns out that the values of
$\lambda$ do not depend on which kind of single-electron orbitals are used,
relativistic Hartree-Fock orbitals, Bruekner orbitals (the orbitals which
include core-valence correlations Dzuba et al. (1987)), or the orbitals
obtained for the hydrogen - like systems. Note that the values of $\lambda$
for $s_{1/2}$ and $p_{1/2}$ states are very close. This means that in the
approximation, when this difference is neglected, there is no contribution to
the non-linearities of King plot. This corresponds to the case
$F_{a}/F_{b}=G^{(4)}_{a}/G^{(4)}_{b}$ in (2).

The values of the power $\lambda$ are close to $2\gamma$ for light atoms only.
For heavy atoms the difference is significant. This is due to the difference
in the analytical and numerical considerations. Only leading in
$(Z\alpha)^{2}$ term was included in the expansion of the electronic density
inside the nucleus in Ref. Flambaum et al. (2018). This works well for small
$Z$ (accurate to $\leq$ 1% for $Z\leq 50$), but becomes increasingly
inaccurate for large $Z$. Numerical results show that even for superheavy
atoms the values of $\lambda$ are closer to 2 than to $2\gamma$. This
justifies the use of (11) for the extraction of the change of nuclear radius
between neighbouring isotopes of heavy atoms. The value of $\lambda$ for any
nuclear charge $Z$ can be interpolated with the accuracy $\sim$ 1% by the
formula

$\lambda(Z)=2-3.1968\times 10^{-4}Z-2.0632\times 10^{-5}Z^{2}.$ (14)

This formula is the result of fitting of the data from Table 3.

Table 3 shows results assuming the spherical nuclear charge distribution
described by the standard Fermi-type formula. When nuclear deformation is
taken into account, formula (13) becomes less accurate. The way of dealing
with the nuclear deformation was considered in our earlier paper Allehabi et
al. (2021).

Table 3:  Parameters $\lambda$ (formula (13)) for the $s_{1/2}$ and $p_{1/2}$ functions of different atomic systems. $R_{N}$ is nuclear radius (parameter of Fermi charge distribution) used in the calculations. $Z$ |  $R_{N}$(fm) | $\lambda_{s_{1/2}}$ | $\lambda_{p_{1/2}}$ | $2\gamma$
---|---|---|---|---
20 | 3.719 | 1.967 | 1.968 | 1.9786
38 | 4.867 | 1.963 | 1.966 | 1.9216
70 | 6.277 | 1.881 | 1.887 | 1.7194
87 | 6.834 | 1.818 | 1.826 | 1.5452
120 | 7.360 | 1.662 | 1.673 | 0.9658

## Conclusion

In this paper we calculated the contribution of the nuclear polarisation into
isotope shift and non-linearities of the King plot for Ca+, Sr+ and Yb+ ions.
We also studied the role of the relativistic effects and found that their
contribution to the non-linearities is small if we assume spherical
distribution of the nuclear charge. A formula was found which allows us to
isolate relativistic effects from other sources of the non-linearities. We
argue that the accuracy of the calculations of the non-linearities of the King
plot, which come from the field shift, is insufficient to isolate them from
possible contribution of a new interaction. Therefore, the limits on new boson
may be determined by the uncertainty of SM contributions to the non-linearity
rather than experimental errors. An important source of SM non-linearities is
variation of the nuclear deformation from isotope to isotope. Therefore, the
best choices for the new boson search are among atoms with medium mass
spherical nuclei, like Sr and Ca, where both the field shift and quadratic
mass shift contributions to the non-linearities are suppressed in comparison
with deformed heavy nuclei and very light nuclei.

## Acknowledgements

The work work supported in part by the Australian Research Council.

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