# Reconciling differential radii in the silver chain through improved
measurement and ab initio calculations

B. Ohayon<EMAIL_ADDRESS>The Helen Diller Quantum Center, Department
of Physics, Technion-Israel Institute of Technology, Haifa, 3200003, Israel
J. E. Padilla-Castillo  Fritz-Haber-Institut der Max-Planck-Gesellschaft,
Faradayweg 4-6, 14195 Berlin, Germany  S. C. Wright  sidwright@fhi-
berlin.mpg.de Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg
4-6, 14195 Berlin, Germany  G. Meijer  Fritz-Haber-Institut der Max-Planck-
Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  B. K. Sahoo
<EMAIL_ADDRESS>Atomic, Molecular and Optical Physics Division, Physical
Research Laboratory, Navrangpura, Ahmedabad 380058, Gujarat, India

###### Abstract

Nuclear charge radius differences in the silver isotopic chain have been
reported through different combinations of experiment and theory, and clear
disagreement is observed beyond the assigned uncertainties. This study
investigates this issue by combining high-accuracy calculations for six low-
lying states of atomic silver with an improved measurement of the
$5s^{2}S_{1/2}-5p^{2}P_{3/2}$ transition optical isotope shift. Our
calculations predict measured electronic transition energies in Ag I at the
0.3% level, the highest accuracy achieved in this system so far. We calculate
electronic isotope shift factors by employing analytical response relativistic
coupled-cluster theory, and find that a consistent charge radius difference
between 107,109Ag is returned when combining our calculations with the
available optical isotope shift measurements. We therefore recommend an
improved value for the mean-squared charge radius difference between 107Ag and
109Ag as $0.207(3)[4]$ fm2, in marginal agreement with the value derived from
muonic Ag experiments, and an updated set of charge radii differences across
the isotopic chain.

##  I Introduction

The mean square nuclear charge radius difference, $\delta r^{2}$, is a unique
probe of structural changes in isotopic chains Yang  _et al._ (2023),
complementary to the binding energy per nucleon. In principle, $\delta r^{2}$
can be extracted from optical isotope shifts (ISs, denoted $\delta\nu$), via
the approximate relation

$\displaystyle\delta\nu_{A,A^{\prime}}$ $\displaystyle\simeq$ $\displaystyle
K\mu_{A,A^{\prime}}+F\delta r^{2}_{A,A^{\prime}},$ (1)

where $\mu_{A,A^{\prime}}=1/M_{A}-1/M_{A^{\prime}}$ is the difference between
inverse nuclear masses of isotopes $A$ and $A^{\prime}$. $K$ denotes the mass
shift (MS) factor, and $F$ the field shift (FS) factor of a transition with
frequency $\nu$.

When $\delta r^{2}_{A,A^{\prime}}$ of two or more isotopic pairs have been
measured, usually via muonic atom cascade X-ray spectroscopy Ford and Wills
(1969), the atomic factors $K$ and $F$ of Eq. (1) can be directly extracted
from a linear fit called a calibrated King Plot (CKP) King (1963), having two
or more data-points. Using this information, $\delta r^{2}_{A,A^{\prime}}$ can
then be extracted across a chain of isotopes via optical isotope shift
measurements, and without further muonic atom experiments. This is the case
for most elements with an even number of protons blue(Z) Fricke and Heilig
(a). For the odd-$Z$ elements, there aren’t three or more stable isotopes
available that are needed for carrying out traditional cascade spectroscopy
measurements (see, however, Ref. Adamczak  _et al._ (2023); Heines  _et al._
(2023)). So in order to apply Eq. (1) to measurements of ISs in a chain of
isotopes, one has to rely on the calculation of the IS factors.

For elements in which $\delta r^{2}_{A,A^{\prime}}$ for a pair of isotopes has
been measured, it is sufficient to calculate or estimate one of the two IS
factors, and extract the other factor via Eq. (1). Usually, it is $F$ that is
evaluated, as it is believed to be less susceptible to electron correlation
effects. This method is sometimes called a partial CKP. It is used in odd-$Z$
elements with at least two stable isotopes (see e.g. Bissell  _et al._ (2016);
Procter  _et al._ (2012, 2015)). It is also useful for light even-$Z$ elements
in which a CKP yields larger uncertainties (e.g. Gangrsky  _et al._ (2004);
Blaum  _et al._ (2008); Ohayon  _et al._ (2019, 2022a); König  _et al._
(2023)). The partial CKP method has been considered to give both precise and
accurate results, as it relies on a calculation of $F$, much easier than of
$K$, and the use of differential radii from muonic cascade measurements which
are considered reliable at the few am level. Nevertheless, with tremendous
advancement in the development of atomic many-body methods, as well as
availability of ever-increasing computational power, it is possible today to
calculate $K$ values precisely in some systems. This enables extracting all
optical $\delta r^{2}_{A,A^{\prime}}$ beyond the accuracy obtained in either a
partial or a full CKP approach Sahoo  _et al._ (2020); Koszorús  _et al._
(2021); Schelfhout and McFerran (2021); Ohayon  _et al._ (2022b); Schelfhout
and McFerran (2022); Sahoo and Ohayon (2023); Karthein  _et al._ (2023);
Penyazkov  _et al._ (2023), which are forever limited by complex nuclear
corrections to the muonic energy levels.

Figure 1: Differential radii of ground-state silver nuclei as given in Table
5. Error bars show $68\%$ confidence intervals taking into account both
statistical and systematic uncertainties. The results of a DFT calculation
from Ref. Reponen  _et al._ (2021) are also shown. The results of this work
are in disagreement with Dinger  _et al._ (1989) and Ferrer  _et al._ (2014),
and agree with Reponen  _et al._ (2021), though with significantly reduced
uncertainties.

A particularly interesting case is that of silver (Ag, $Z=47$), having two
stable isotopes with a similar natural abundance, 107Ag and 109Ag. ISs of
synthetic Ag nuclei have been measured for three transitions. In 1975, the
$5s-5p~{}^{2}P_{1/2}$ $338\,$nm and the $5s-5p~{}^{2}P_{3/2}$ $328\,$nm lines
were measured with a hollow-cathode Fabry–Pérot interferometer using neutron-
irradiated targets of 108mAg and 110mAg Fischer  _et al._ (1975). The first
online experiment was performed at the GSI accelerator facility, where ISs in
the $4d^{9}5s^{2}~{}^{2}D_{5/2}-6p~{}^{2}P_{3/2}$ $548\,$nm line were measured
for neutron-deficient isotopes and isomers Dinger  _et al._ (1989). Further
neutron-deficient isotopes were measured with the $328\,$nm line at the LISOL
facility Ferrer  _et al._ (2014). Recently, at the IGISOL facility, ISs of a
long chain of isotopes, extending from 96Ag to 121Ag were measured; again
using the $328\,$nm line Reponen  _et al._ (2021). The center-of-gravity ISs
resulting from these efforts have been interpreted via partial CKPs, with
different choices for $\delta r^{2}_{109,107}$ and $F$, yielding inconsistent
results for the differential radii of the chain $\delta r^{2}_{109,A}$, as
shown in Fig. 1.

In this work we largely resolve the inconsistencies in differential radii of
Ag reported thus far. To do this, we have performed state-of-the-art ab initio
calculations of IS factors in low-lying levels of Ag I. Our results indicate
that semi-empirical estimations of $K$ and $F$, used in other studies of Ag,
are unreliable, and that the uncertainties accompanying these estimates should
be significantly increased in other similar systems. We also present improved
optical isotope shift of the $328\,$nm transition in naturally abundant Ag,
and perform a global analysis of the result with the available literature
data. Our calculations of $K$ and $F$ for these transitions in Ag, combined
with the data, produce a consistent value for $\delta r^{2}_{109,107}$ within
a few percent. This checks the consistency of our calculations and put limits
on possible additional shifts due to ‘off-diagonal’ hyperfine interactions
which mix different electronic or fine structure states.

Having validated our calculation, we use the available optical isotope shift
data to provide differential radii for the silver isotope chain spanning
$A=96-105$, $A=114-121$, and six long-lived isomers. These results reduce the
uncertainty in $\delta r^{2}_{109,A}$ by up to a factor of seven. A Comparison
with prior works pinpoints the reasons for past inconsistencies. Finally, the
recommended differential radii are compared with state-of-the-art nuclear
theory calculations. Whilst the overall trend of the nuclear calculations
agrees with the atomic data, a few interesting deviations are noticed.

##  II Atomic theory

###  II.1 Method of calculation

Table 1:  Calculated electron affinities (EAs, where zero energy refers to the ground state of Ag II) of the considered states in Ag at different levels of approximation. The estimated excitation energies (EEs) are also quoted. Unless otherwise stated, all values are in cm-1. Our final results are compared with the experimental values (denoted Exp.). Differences between our calculated and experimental values are shown as $\Delta$ in percentage. The last column gives the results of prior calculations available in the literature which are closest to experiment.  State | DHF | MP2 | RCCSD | +T |  $+$Basis |  $+$Breit |  $+$QED | Total | Exp. Loock  _et al._ (1999); Pickering and Zilio (2001) |  $\Delta$% | Lit.
---|---|---|---|---|---|---|---|---|---|---|---
EAs |  |  |  |  |  |  |  |  |  |  |
$5s$ | $50376$ | $61014$ | $60408$ | $441(110)$ | $193(96)$ | $-59$ | $-22(22)$ | $60961(148)$ | $61106.5(2)$ | $0.2(2)$ |  $60823$ Nayak  _et al._ (2006)
$5p~{}^{2}P_{1/2}$ | $26730$ | $30771$ | $31007$ | $455(114)$ | $46(23)$ | $-36$ | $4(4)$ | $31477(116)$ | $31554.4(2)$ | $0.2(4)$ |  $31066$ Nayak  _et al._ (2006)
$5p~{}^{2}P_{3/2}$ | $26148$ | $29862$ | $30089$ | $442(111)$ | $39(20)$ | $-24$ | $-4(4)$ | $30543(112)$ | $30633.7(2)$ | $0.3(4)$ |  $30184$ Nayak  _et al._ (2006)
$6s$ | $17115$ | $18641$ | $18455$ | $45(11)$ | $21(11)$ | $-8$ | $-3(3)$ | $18510(16)$ | $18550.3(2)$ | $0.2(1)$ |  $18494$ Nayak  _et al._ (2006)
$6p~{}^{2}P_{1/2}$ | $11786$ | $12726$ | $12680$ | $89(22)$ | $9(5)$ | $-8$ | $1(1)$ | $12771(23)$ | $12809.0(2)$ | $0.3(2)$ |  $12656$ Chaudhuri and Chattopadhyay (2022)
$6p~{}^{2}P_{3/2}$ | $11618$ | $12502$ | $12467$ | $94(24)$ | $8(4)$ | $-5$ | $-1(1)$ | $12563(24)$ | $12606.6(2)$ | $0.3(2)$ |  $12452$ Chaudhuri and Chattopadhyay (2022)
EEs |  |  |  |  |  |  |  |  |  |  |
$5s-5p~{}^{2}P_{1/2}$ | $23646$ | $30243$ | $29400$ | $-14(4)$ | $147(73)$ | $-24$ | $-25(25)$ | $29484(78)$ | $29552.061(1)$ | $0.2(3)$ |  $29496$ Chaudhuri and Chattopadhyay (2022)
$5s-5p~{}^{2}P_{3/2}$ | $24228$ | $31152$ | $30319$ | $-2(0.)$ | $154(77)$ | $-36$ | $-18(18)$ | $30418(79)$ | $30472.703(1)$ | $0.2(3)$ |  $30451$ Dzuba  _et al._ (2021)
$5s-6s$ | $33261$ | $42373$ | $41953$ | $396(99)$ | $172(86)$ | $-51$ | $-19(19)$ | $42451(105)$ | $42556.152(2)$ | $0.2(3)$ |  $42329$ Nayak  _et al._ (2006)
$5s-6p~{}^{2}P_{1/2}$ | $38590$ | $48288$ | $47727$ | $352(88)$ | $185(92)$ | $-51$ | $-23(23)$ | $48190(108)$ | $48297.402(3)$ | $0.2(3)$ |  $47765$ Chaudhuri and Chattopadhyay (2022)
$5s-6p~{}^{2}P_{3/2}$ | $38758$ | $48512$ | $47940$ | $347(87)$ | $185(93)$ | $-54$ | $-21(21)$ | $48398(103)$ | $48500.805(2)$ | $0.2(3)$ |  $47969$ Chaudhuri and Chattopadhyay (2022)

The Ag atom has the ground state configuration $[4d^{10}]~{}5s$. We are
interested to probe isotope shifts of the $5s-5p~{}^{2}P_{1/2;3/2}$, $5s-6s$
and $5s-6p~{}^{2}P_{1/2;3/2}$ transitions in this work. Thus, it is necessary
to determine wave functions of the ground state as well as for the
$5p~{}^{2}P_{1/2;3/2}$, $6s$ and $6p~{}^{2}P_{1/2;3/2}$ states of Ag I. To
obtain all these states conveniently, we first calculate wave function
($|\Psi_{0}\rangle$) of the common closed-shell core configuration $[4d^{10}]$
of all these states. We employ the coupled-cluster theory in the relativistic
framework (RCC method) to determine $|\Psi_{0}\rangle$ by expressing it as
Cizek and Paldus (1980)

$\displaystyle|\Psi_{0}\rangle=e^{S_{0}}|\Phi_{0}\rangle,$ (2)

where $S_{0}$ is the RCC excitation operator carrying information on electron
correlation effects and the reference state $|\Phi_{0}\rangle$ is obtained in
the Dirac-Hartree-Fock (DHF) method. We use the Dirac-Coulomb (DC) Hamiltonian
($H$), which takes into account the lowest-order relativistic effects as well
as the finite-size of the nucleus by employing a two-parameter Fermi-charge
distribution given by

$\displaystyle\rho(r)=\frac{\rho_{0}}{1+e^{(r-b)/a}},$ (3)

where $\rho_{0}$ is the normalization constant, $b$ is the half-charge radius
and $a=2.3/4\ln(3)$ is an approximate skin thickness. The nuclear potential is
given by

$\displaystyle V(r)=-\frac{Z}{\mathcal{N}r}\times\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle\left\\{\begin{array}[]{rl}\frac{1}{b}(\frac{3}{2}+\frac{a^{2}\pi^{2}}{2b^{2}}-\frac{r^{2}}{2b^{2}}+\frac{3a^{2}}{b^{2}}P_{2}^{+}\frac{6a^{3}}{b^{2}r}(S_{3}-P_{3}^{+}))&\mbox{for
$r_{i}\leq b$}\\\
\frac{1}{r_{i}}(1+\frac{a62\pi^{2}}{b^{2}}-\frac{3a^{2}r}{c^{3}}P_{2}^{-}+\frac{6a^{3}}{b^{3}}(S_{3}-P_{3}^{-}))&\mbox{for
$r_{i}>b$},\end{array}\right.$ (6)

where the factors are

$\displaystyle\mathcal{N}$ $\displaystyle=$ $\displaystyle
1+\frac{a^{2}\pi^{2}}{b^{2}}+\frac{6a^{3}}{b^{3}}S_{3}$
$\displaystyle\text{with}\ \ \ \ S_{k}$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{\infty}\frac{(-1)^{l-1}}{l^{k}}e^{-lb/a}\ \ \ $
$\displaystyle\text{and}\ \ \ \ P_{k}^{\pm}$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{\infty}\frac{(-1)^{l-1}}{l^{k}}e^{\pm l(r-b)/a}.$
(7)

In the above expression, $b$ is obtained using the relation

$\displaystyle b=\sqrt{\frac{5}{3}R^{2}-\frac{7}{3}a^{2}\pi^{2}},$ (8)

where $R$ is the approximate root mean square radius and calculated as

$\displaystyle R=(0.836A^{1/3}+0.57)~{}\text{fm}.$ (9)

Amplitudes of the $S_{0}$ excitation operator are obtained by solving the
equation

$\displaystyle\langle\Phi_{0}^{*}|\left(He^{S_{0}}\right)_{l}|\Phi_{0}\rangle=0,$
(10)

where $|\Phi_{0}^{*}\rangle$ represents all possible excited state
determinants with respect to $|\Phi_{0}\rangle$ and subscript $l$ means linked
terms. Once $|\Psi_{0}\rangle$ is obtained, we determine wave function
($|\Psi_{v}\rangle$) of a state with the $[4d^{10}]$ closed-core and a valence
orbital $v$ by defining Lindgren  _et al._ (1983); Mukherjee and Pal (1989)

$\displaystyle|\Psi_{v}\rangle$ $\displaystyle=$ $\displaystyle
e^{S_{0}+S_{v}}|\Phi_{v}\rangle$ (11) $\displaystyle=$ $\displaystyle
e^{S_{0}}\left\\{1+S_{v}\right\\}|\Phi_{v}\rangle,$

where $|\Phi_{v}\rangle=a_{v}^{\dagger}|\Phi_{0}\rangle$ is the modified
reference state containing valence orbital, and $S_{v}$ includes excitation
configurations due to correlation effects by the valence electron. The
amplitude solving equation for $S_{v}$ is given by

$\displaystyle\langle\Phi_{v}^{*}|\left\\{\left(He^{S_{0}}\right)_{l}-E_{v})\right\\}S_{v}+\left(He^{S_{0}}\right)_{l}|\Phi_{v}\rangle=0,$
(12)

where $|\Phi_{v}^{*}\rangle$ denotes for all possible excited state
determinants with respect to $|\Phi_{v}\rangle$. The energy of the respective
state is given by

$\displaystyle
E_{v}=\langle\Phi_{v}|\left(He^{S_{0}}\right)_{l}\\{1+S_{v}\\}|\Phi_{v}\rangle.$
(13)

We use here a normal-ordered Hamiltonian with respect to $|\Phi_{0}\rangle$,
so that $E_{v}$ corresponds to the electron affinity (EA) rather than total
energy of the state. Excitation energies (EEs) can be obtained from the
differences between EAs of the calculated states.

We calculate both the MS and FS factors via analytical-response relativistic
coupled-cluster (AR-RCC) theory. For the evaluation of the MS factors, they
are divided into one-body part, known as normal mass shift (NMS) constant
($K^{\mathrm{NMS}}$) and two-body part, known as specific mass constant
($K^{\mathrm{SMS}}$) as defined in Refs. Palmer (1987); Shabaev (1988);
Shabaev and Artemyev (1994). Using the Fermi-charge distribution, the FS
operator is defined as

$\displaystyle{\hat{F}}=-\frac{\delta V(r)}{\delta R}=-\frac{\partial
V(r)}{\partial b}\frac{\delta b}{\delta
R}~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}r\leq b.$ (14)

Our intention is to calculate the above IS factors as the first-order energy
correction ($E_{v}^{(1)}$) by solving the following equation

$\displaystyle(H-E_{v}^{(0)})|\Psi_{v}^{(1)}\rangle$ $\displaystyle=$
$\displaystyle(E_{v}^{(1)}-H_{\mathrm{IS}})|\Psi_{v}^{(0)}\rangle,$ (15)

where $H_{\mathrm{IS}}$ is the respective IS Hamiltonian. The superscripts
$(0)$ and $(1)$ denote contributions from the DC Hamiltonian $H$ and first-
order correction due to $H_{\mathrm{IS}}$, respectively. In the AR-RCC method,
they can be obtained by expanding the RCC operators as Sahoo  _et al._ (2020)

$\displaystyle S_{0}=S_{0}^{(0)}+\lambda S_{0}^{(1)}\ \ \ \text{and}\ \ \
S_{v}=S_{v}^{(0)}+\lambda S_{v}^{(1)}.$ (16)

The first-order excitation operator amplitudes are obtained by

$\displaystyle\langle\Phi_{0}^{*}|\left(He^{S_{0}^{(0)}}S_{0}^{(1)}+H_{\mathrm{IS}}e^{S_{0}^{(0)}}\right)_{l}|\Phi_{0}\rangle=0$
(18) and
$\displaystyle\langle\Phi_{v}^{*}|\left\\{\left(He^{S_{0}^{(0)}}\right)_{l}-E_{v}^{(0)})\right\\}S_{v}^{(1)}+\left(He^{S_{0}^{(0)}}S_{0}^{(1)}\right)_{l}$
$\displaystyle\times\left\\{1+S_{v}^{(0)}\right\\}+\left(H_{\mathrm{IS}}e^{S_{0}^{(0)}}\right)_{l}\left\\{1+S_{v}^{(0)}\right\\}$
$\displaystyle+E_{v}^{(1)}S_{v}^{(0)}|\Phi_{v}\rangle=0.\ \ \ \ \ $

In the above equation, the expression for an IS factor is given by

$\displaystyle E_{v}^{(1)}$ $\displaystyle=$
$\displaystyle\langle\Phi_{v}|\left(He^{S_{0}^{(0)}}\right)_{l}S_{v}^{(1)}+\left(He^{S_{0}^{(0)}}S_{0}^{(1)}\right)_{l}\left\\{1+S_{v}^{(0)}\right\\}$
(19)
$\displaystyle+(H_{IS}e^{S_{0}^{(0)}})_{l}\left\\{1+S_{v}^{(0)}\right\\}|\Phi_{v}\rangle.$

In this work, we considered electron correlation effects first from the single
and double excitations in the RCC theory (RCCSD method) up to 20$s$, 20$p$,
19$d$, 18$f$, 16$g$, 14$h$, and 12$i$ orbitals. Since considering triple
excitations among all these orbitals was not feasible with the available
computational resources, we have allowed triple excitations up to 15$s$,
15$p$, 15$d$, 11$f$ and 10$g$ orbitals to account for the correlation effects
along with the singles and doubles excitations among the aforementioned
orbitals (RCCSDT method). Also, corrections due to the Breit and QED
corrections are included using the 20$s$, 20$p$, 19$d$, 18$f$ and 16$g$
orbitals.

To demonstrate contributions to the IS factors at different levels of
approximations in the atomic Hamiltonian, we give results using the DC
Hamiltonian and corrections due to the Breit (given as $+$Breit) and QED
(given as $+$QED) effects as the RCCSD values. Differences of the RCCSD values
from the larger basis set are given as ‘$+$Basis’ contribution. Differences in
the results between the RCCSD and RCCSDT methods are listed as $+$T. We also
present calculated energies at the second perturbation theory (MP2 method)
using the 20$s$, 20$p$, 19$d$, 18$f$ and 16$g$ orbitals to demonstrate
importance of considering an all-order method like RCC theory for accurate
calculations of properties in Ag.

Though the RCC and AR-RCC methods used here are in principle capable of
producing very precise results, real calculations are often less precise due
to practical effects such as the choice of basis functions. Most of the
uncertainty in our calculated energies and IS factors would stem from the
frozen orbitals in the estimations of the triples contributions. High-lying
orbitals can also contribute to some extent at the RCCSD method approximation.
The Breit interaction contributions are included self-consistently at the DHF
and RCC methods, so their contributions due to higher-order correlation
effects are anticipated to be negligibly small. Since we have used a model
potential to estimate QED effects Sahoo (2016), these estimations are not very
reliable but represent typical order of magnitudes to both energies and IS
factors. Thus, we assign 100% uncertainties to them.

###  II.2 Results and Discussion

####  II.2.1 Energies

In Table 1 we give results for the calculated EAs. They are given first at the
DHF approximation, which already captures the gross level structure. When
taking into account electron correlations through either MP2 or RCCSD, the EAs
of the $n=5$ manifold increase by $\mathcal{O}$($15\%$). The effect is
stronger than in the isoelectronic Cd II, where it is $8-9\%$, indicating that
electron correlations are more important in Ag I. For the $n=6$ manifold, the
increase is half the size, hinting that correlations in more weakly bound
single-valence states play smaller roles, as expected. Introducing correlation
effects through triple excitations increases the EAs of the $n=5$ manifold by
$0.7-1.5\%$, twice as much as in Cd II, and three times that for the $n=6$
manifold. The uncertainty tied to missing quadruple electron excitation
contributions to EAs is expected to be small. The basis set extrapolation
increases the EAs for all levels as well.

The Breit and QED contributions are found to be small but not negligible. We
find that for states with $ns$ valence orbitals, the Breit and QED corrections
are comparable in magnitude while for states with the $np$ subshells, the
former is much larger, as was pointed in Kozioł and Aucar (2018). Our
approximate QED correction to the 5s EA is found to be smaller then most other
literature values, as compiled in Table VIII of Kozioł and Aucar (2018). For
this reason, we ascribe a $100\%$ uncertainty to it and to the corresponding
corrections to the IS operators. Although these uncertainties are negligible
with respect to other contributions, they point that moving to heavier or
multiply-charged systems without losing accuracy would necessitate a
refinement of the QED treatment.

The total EAs are $0.2-0.3\%$ away from experiment, in reasonable agreement
with our uncertainty estimates. They are more accurate than the closest
results from the literature Nayak  _et al._ (2006); Chaudhuri and
Chattopadhyay (2022), which do not quote uncertainties, by up to a factor of
$8$. Due to the stronger electron correlations, the total uncertainty is a
factor $2-3$ higher than in our prior work on Cd II. All in all, we undershoot
the experimental energies, which indicates a more complete treatment of
electron correlations is necessary for accurate estimations of the results.
For Cd II, we overshot the experimental energies, indicating possibly
underestimated many-body QED effects.

The EEs are calculated from the differences of the other level EAs to that of
the ground level at different approximations. Similarly to the EAs, the EEs
are $0.2\%$ away from experimental measurements, within our uncertainty
estimation. The EEs of the $n=5$ doublet are less accurate than in the prior
works Chaudhuri and Chattopadhyay (2022); Dzuba  _et al._ (2021), while those
of the higher states are more so Chaudhuri and Chattopadhyay (2022); Nayak
_et al._ (2006). It is interesting to note that the contribution of triple
amplitudes is similar for each state in the manifold. This means that the
uncertainty is dominated by basis extrapolation for the first two EEs.

####  II.2.2 IS factors

Table 2:  Calculated IS factors $F$, $K^{\mathrm{SMS}}$ and $K^{\mathrm{NMS}}$ for selected levels in Ag I. For each of the calculated values, we first list factors relative to the ground state of the Ag II ion, followed by factors for optical transitions from the 5s ground state of Ag I. S.L. refers to the scaling law used for $K^{\mathrm{NMS}}$, as discussed in the text.  State | DHF | AR-RCCSD | +T |  $+$Basis |  $+$Breit |  $+$QED | Total |
---|---|---|---|---|---|---|---|---
$F$ MHz/fm2 |  |  |  |  |  |  |  |
$5s$ | $-2527$ | $-3852$ | $129(32)$ | $-30(15)$ | $11$ | $19(19)$ | $-3723(40)$ |
$5p~{}^{2}P_{1/2}$ | $-17$ | $-163$ | $-37(9)$ | 1(1) | 1 | 1(1) | $-197(9)$ |
$5p~{}^{2}P_{3/2}$ | $-0.$ | $-126$ | $-42(11)$ | 2(1) | 1 | 1(1) | $-165(11)$ |
$6s$ | $-412$ | $-517$ | $~{}15(4)$ | $-1(1)$ | 1 | 2(1) | $-499(5)$ |
$6p~{}^{2}P_{1/2}$ | $-5$ | $-32$ | $-11(3)$ | 1(0.) | 0. | 0.(0.) | $-42(3)$ |
$6p~{}^{2}P_{3/2}$ | $-0.$ | $-25$ | $-15(4)$ | 1(0.) | 0. | 0.(0.) | $-39(4)$ |
$5s-5p~{}^{2}P_{1/2}$ | $-2510$ | $-3523$ | $166(42)$ | $-31(15)$ | $10$ | $19(19)$ | $-3525(48)$ |
$5s-5p~{}^{2}P_{3/2}$ | $-2527$ | $-3555$ | $171(43)$ | $-31(16)$ | $10$ | $18(18)$ | $-3557(49)$ |
Ref. Dinger  _et al._ (1989) |  $-2625$, $-3146$ |  |  |  |  |  | $-4265(341)$ |
Ref. Ferrer  _et al._ (2014); Reponen  _et al._ (2021) |  |  |  |  |  |  | $-4300(300)$ |
$5s-6s$ | $-2115$ | $-3221$ | $114(29)$ | $-28(14)$ | $10$ | $17(17)$ | $-3223(36)$ |
$5s-6p~{}^{2}P_{1/2}$ | $-2522$ | $-3680$ | $141(35)$ | $-30(15)$ | $11$ | $19(19)$ | $-3680(43)$ |
$5s-6p~{}^{2}P_{3/2}$ | $-2527$ | $-3683$ | $144(36)$ | $-30(15)$ | $11$ | $19(19)$ | $-3683(43)$ |
$K^{\mathrm{SMS}}$ GHz u |  |  |  |  |  |  |  |
$5s$ | $-1611$ | $1346$ | $115(29)$ | $-4(2)$ | $5$ | $1(1)$ | $1463(29)$ |
$5p~{}^{2}P_{1/2}$ | $-553$ | $342$ | $68(17)$ | $-3(2)$ | $-1$ | $-0.(0.)$ | $405(17)$ |
$5p~{}^{2}P_{3/2}$ | $-464$ | $370$ | $64(16)$ | $-3(1)$ | $-0.$ | $-0.(0.)$ | $432(16)$ |
$6s$ | $-253$ | $176$ | $28(7)$ | $-2(1)$ | $1$ | $0.(0.)$ | $202(7)$ |
$6p~{}^{2}P_{1/2}$ | $-150$ | $89$ | $23(6)$ | $-1(1)$ | $0.$ | $-0.(0.)$ | $88(6)$ |
$6p~{}^{2}P_{3/2}$ | $-128$ | $75$ | $25(6)$ | $-1(0.)$ | $0.$ | $-0.(0.)$ | $99(6)$ |
$5s-5p~{}^{2}P_{1/2}$ | $-1058$ | $1005$ | $47(12)$ | $-1(0.)$ | $6$ | $1(1)$ | $1058(12)$ |
$5s-5p~{}^{2}P_{3/2}$ | $-1146$ | $976$ | $51(13)$ | $-1(1)$ | $5$ | $1(1)$ | $1031(13)$ |
Ref. Ferrer  _et al._ (2014) |  |  |  |  |  |  | $150(450)$ |
$5s-6s$ | $-1358$ | $1171$ | $87(22)$ | $-2(1)$ | $4$ | $0.(0.)$ | $1260(22)$ |
$5s-6p~{}^{2}P_{1/2}$ | $-1461$ | $1281$ | $92(23)$ | $-3(1)$ | $5$ | $1(1)$ | $1375(23)$ |
$5s-6p~{}^{2}P_{3/2}$ | $-1482$ | $1271$ | $90(23)$ | $-3(2)$ | $5$ | $1(1)$ | $1363(23)$ |
$K^{\mathrm{NMS}}$ GHz u |  |  |  |  |  |  |  | S.L.
$5s$ | 3393 | 808 | $36(9)$ | $11(5)$ | $-1$ | $-1(1)$ | $852(11)$ | 1005
$5p~{}^{2}P_{1/2}$ | 1200 | 367 | $51(13)$ | $5(3)$ | $-1$ | $0.(0.)$ | $422(13)$ | 519
$5p~{}^{2}P_{3/2}$ | 1111 | 351 | $49(12)$ | $5(3)$ | $-1$ | $-0.(0.)$ | $405(13)$ | 504
$6s$ | 667 | 274 | $2(1)$ | $1(1)$ | $-0.$ | $-0.(0.)$ | $277(1)$ | 305
$6p~{}^{2}P_{1/2}$ | 396 | 181 | $5(1)$ | $1(1)$ | $-0.$ | $0.(0.)$ | $187(1)$ | 211
$6p~{}^{2}P_{3/2}$ | 376 | 176 | $5(1)$ | $1(1)$ | $-0.$ | $-0.(0.)$ | $182(1)$ | 207
$5s-5p~{}^{2}P_{1/2}$ | $2193$ | $441$ | $-14(4)$ | $5(3)$ | $-0.$ | $-1(1)$ | $431(5)$ | 486
$5s-5p~{}^{2}P_{3/2}$ | $2282$ | $457$ | $-13(3)$ | $5(3)$ | $-1$ | $-1(1)$ | $448(4)$ | 501
$5s-6s$ | $2727$ | $534$ | $34(9)$ | $9(5)$ | $-1$ | $-1(1)$ | $575(10)$ | 700
$5s-6p~{}^{2}P_{1/2}$ | $2998$ | $627$ | $31(8)$ | $9(5)$ | $-1$ | $-1(1)$ | $665(9)$ | 764
$5s-6p~{}^{2}P_{3/2}$ | $3017$ | $632$ | $32(8)$ | $9(5)$ | $-1$ | $-1(1)$ | $671(9)$ | 798

In Table 2, we give the IS factors evaluated at the different levels of
approximation. We first discuss the FS factors from our calculations. At the
DHF level, $F_{ns}$ is large stemming from strong overlap with the nucleus,
$F_{nP1/2}$ is small but non-negligible, and $F_{nP3/2}$ is negligible. Our
result for $F_{326}=F_{nP3/2}-F_{ns}$ is close to a the Hartree-Fock
calculation in Dinger  _et al._ (1989) but differs from their Dirac-Fock
calculation. Introducing electron correlations at the AR-RCCSD approximation
increases $F_{5s}$ by $50\%$, and for $F_{6s}$ by $25\%$. This trend is
similar to that seen with the EAs. Correlations also increase $F$ of the $nP$
levels, thus reducing magnitudes for the fine-structure intervals. Triple
excitation contributions to $F_{5s}$ reduce its magnitude by $3\%$, twice as
much and of opposite sign as in Cd II. In Zn II, triple contributions to
$F_{4s}$ are of the same sign and five times smaller then in Ag I Sahoo and
Ohayon (2023). These observations demonstrate the non-trivial nature of
correlation effects from the high-level excitations, and motivate extending
these calculations to the homologous levels in Cu I in order to gain further
insight. As in Cd II and Zn II, the contribution of approximate QED correction
is much larger than that of the Breit interaction. Nevertheless, it is still
small compared with our uncertainty estimation.

All in all, it is clear that it is difficult to estimate $F$ in Ag I with sub-
percent accuracy. In any case, trying to push much below that, e.g. by
enlarging the basis set or implementing quadruple excitations, has little
gain, as the unknown contribution to $\delta r^{2}$ from the isotopic change
in the shape of the charge distribution is on the order of $2\%$ Fricke and
Heilig (b), which we add as a systematic uncertainty to all $\delta r^{2}$
estimations in this work. This limitation increases in heavier, more charged,
or more deformed, systems.

To our knowledge, ours is the first ab initio calculation of the FS factor in
Ag I levels. Nevertheless, it was semi-empirically extracted from the $5s$
hyperfine structure, yielding $F_{328,\mathrm{SE}}=-4265(341)$ MHz/fm2 as
given in Dinger  _et al._ (1989). A similar estimation was also used in recent
works Ferrer  _et al._ (2014); Reponen  _et al._ (2021). $F_{328,\mathrm{SE}}$
is larger than our recommended value by two of its standard deviations. A
quarter of the difference is directly related to the missing contribution from
the $5p$ level, and another quarter from their rough estimation of the higher-
moment contribution of $2.4\%$. These observations support other studies (see
Berengut  _et al._ (2003); Kowalska (2006); Ohayon  _et al._ (2019)) which
suggest that semi-empirically estimated values of $F$ are about 20% too large.
Whilst this effect was already reported by Torbohm et al in 1985 Torbohm  _et
al._ (1985), semi-empirical values for $F$ are still often assigned a much
smaller margin of error in the literature.

The SMS factor $K^{\mathrm{SMS}}$, tied with a two-body operator, is entirely
affected by electron correlations. So much so that its calculated value is
meaningless at the mean field approximation. At the AR-RCCSD approximation,
$K^{\mathrm{SMS}}_{5s}$ is of similar magnitude in Ag I and in Cd II. However,
$K^{\mathrm{SMS}}_{5p}$ for both levels of the doublet is $2.5$ times larger
in Ag. Triple amplitude contributions are twice larger for
$K^{\mathrm{SMS}}_{5s}$ and an order of magnitude larger for both
$K^{\mathrm{SMS}}_{5p}$ states. Somewhat surprisingly, $K^{\mathrm{SMS}}$
converges fast with increasing the basis size. This is fortunate as the
calculation of $K^{\mathrm{SMS}}$ at the AR-CCSDT approximation already
requires several months of computation time on a medium-sized high-performance
cluster. As in Cd II, Breit and QED corrections to $K^{\mathrm{SMS}}$ are
completely negligible. Thus the total uncertainty is dominated by our
estimation of contributions from the missing quadruple excitations.

To our knowledge, there aren’t any ab initio calculations of
$K^{\mathrm{SMS}}$ to be compared with in the literature. Nevertheless, we can
test the heuristic used in Ferrer  _et al._ (2014), that
$K^{\mathrm{SMS}}_{\mathrm{SE}}\approx
0.3(9)K^{\mathrm{NMS}}_{\mathrm{S.L.}}=150(450)\,$GHz u, where the NMS factor
was estimated through the scaling-law
$K^{\mathrm{NMS}}_{\mathrm{S.L.}}=-m_{e}\Delta E=501\,$GHz u. Even though a
large uncertainty is attached to this semi-empirical calculation, our result
lies two standard deviations away, putting some doubts on the reliability of
using this method. Thus, the interpretations of IS measurements in refs
Bernard  _et al._ (1985); Ahmad  _et al._ (1985); Alkhazov  _et al._ (1989,
1990); Zimmermann  _et al._ (1994); Georg  _et al._ (1998); Lee  _et al._
(2013); Studer  _et al._ (2020); Choi  _et al._ (2020) should likely be
reevaluated, since they used heuristic estimates of $K^{\mathrm{SMS}}$.

Although $K^{\mathrm{NMS}}$ is estimated from a one-body operator, it is
highly affected by electron correlations, with the AR-CCSD value only a third
of the DHF value. Triple excitations contribute around $10\%$ to the $n=5$
manifold states, as seen in Cd II. For the $n=6$ manifold, the effect of
correlations is smaller; half and a few percent in the AR-RCCSD method and
after including contributions from the triple excitations, respectively. Breit
contributions to $K^{\mathrm{NMS}}$ are found to be small but not negligible,
while QED contributions are negligible. A difference of $10$ to $20\%$ between
the calculated $K^{\mathrm{NMS}}$ and the one semi-empirically estimated via
scaling-law is observed for all levels. It is much larger than the difference
seen in Cd II, indicating that its origin is from electron correlations,
stronger in Ag I, than from the relativistic effects, stronger in Cd II. A
discussion on this phenomenon is found in Ref. Palmer (1987).

##  III Measurements on the $\mathbf{5s~{}^{2}S_{1/2}\rightarrow
5p~{}^{2}P_{3/2}}$ line at 328 nm

Figure 2: Laser induced fluorescence spectroscopy of a buffer gas cooled Ag
atomic beam. (a) Level scheme for the ${}^{2}S_{1/2}\rightarrow{}^{2}P_{3/2}$
transition in 107,109Ag. The relative intensities of the transitions are
indicated. (b) Example spectra taken using a single frequency probe beam with
$s=0.012$ (upper), a single frequency probe beam with $s=0.002$ (middle), and
a two-frequency probe beam whose frequencies are separated by
$2\nu_{\mathrm{AOM}}\approx 410$ MHz (lower). Resonance lines are labelled by
isotope and total angular momentum numbers ($F,F^{\prime}$) for the
transition. The red shaded box indicates the region where the (1,2)
transitions of 107,109Ag are almost overlapping. (c) Example time-of-flight
trace of the fluorescence when using the double-pass AOM as discussed in the
text, illustrating the observation windows used in the analysis. A sudden
change in fluorescence signal occurs when the AOM is switched on or off. The
inset shows a spectrum using observation window (ii), and the fitted frequency
offset $\Delta$ between the (1,2) lines of the Ag isotopes. The contributions
of the individual isotopes are shown in transparent blue, with the solid red
line their sum which is the combined fit function. (d) A plot of the fitted
value of $\Delta$ versus $2\nu_{\mathrm{AOM}}$ for observation windows (i) and
(ii), used to extract the isotope shift of the (1,2) lines.

To improve the available measurements for the naturally occurring isotopes of
Ag, we perform cw laser induced fluorescence spectroscopy with a buffer gas
cooled atomic beam. The spectrometer used for these measurements has been
described previously Doppelbauer  _et al._ (2022); Hofsäss  _et al._ (2023);
Röser  _et al._ (2024). Briefly, Ag atoms of natural isotopic abundance
($52\%$ 107Ag and $48\%$ 109Ag) are produced by laser ablation inside a
cryogenically-cooled copper cell, thermalise by colliding with He buffer gas
at a temperature of 3 K, and exit the cell as a slow, pulsed atomic beam. At a
distance of 70 cm from the exit of the buffer gas cell, the atoms interact
with a low intensity probe laser beam which excites the
$5s~{}^{2}S_{1/2}\rightarrow 5p~{}^{2}P_{3/2}$ transition near 328 nm. The
laser light is produced by frequency doubling a narrow-linewidth ring dye
laser (Sirah Matisse 2DX) at 656 nm. The frequency of the 656 nm light is
recorded using a commercial wavemeter (High Finesse WS8-10, calibrated with a
temperature stabilised HeNe laser) which provides an absolute accuracy of 20
MHz at 328 nm. Fluorescence is collected using a photomultiplier tube whose
photocurrent is delivered to a transimpedance amplifier to generate a time-of-
flight trace. Based on the range of arrival times at the detector, we estimate
that the range of velocities in the beam covers 90 to 130 m/s (full-width at
half-maximum). The transverse velocity width of the atomic beam is restricted
using a 2 mm square aperture mounted immediately in front of the detector.
Orthogonality with the atomic beam direction is ensured using a set of
alignment irises mounted on the detection vacuum chamber; we additionally
verify and limit residual Doppler shifts due to misalignment by comparing
spectra by arrival time at the detector, as discussed later.

Figure 2a shows the relevant energy levels for 107,109Ag, both of which have a
nuclear spin of $I=1/2$. Levels within the ground (excited) levels are
labelled by their total angular momentum quantum number $F$ ($F^{\prime}$),
and the optical transitions are labelled by their respective line intensities.
In the experiment, the hyperfine structure in the ${}^{2}S_{1/2}$ and
${}^{2}P_{3/2}$ states is considered. The large splitting ($\sim 1.8$ GHz) in
the $5s$ ground state means that atoms excited on the $F=0\rightarrow
F^{\prime}=1$ and $F=1\rightarrow F^{\prime}=1$ hyperfine transitions are
optically pumped to the $F=1$ ($F=0$) levels after an average of 2 (3) photon
scattering events, respectively. The process of optical pumping tends to
reduce the amplitudes and increase the width of these lines as compared to the
closed $F=1\rightarrow F^{\prime}=2$ transition, and necessitates minimizing
the number of photon scattering events per atom to obtain the best spectra.
Secondly, the hyperfine splitting in the $5p~{}^{2}P_{3/2}$ state is roughly
three times the natural linewidth of the transition, which modifies the
fluorescence lineshape by the interference between photon scattering pathways
Brown  _et al._ (2013). To minimize this effect, we use laser light linearly
polarized at an angle $\theta_{\mathrm{m}}=\cos^{-1}(1/\sqrt{3})$ to the
detector direction. This is the so-called “magic angle” at which the
anisotropic part of the fluorescence emission is zero, interference between
scattering paths disappears, and a symmetric lineshape is recovered Brown  _et
al._ (2013). The relatively large solid angle of our collection optics of
nearly $\pi/4$ steradians further suppresses the interference effect by
roughly a factor 2.

Figure 2b shows three spectra obtained with our spectrometer. The upper two
spectra are taken with a single-frequency probe laser, at two different probe
laser intensities $I_{0}$. We label the spectra by the two-level saturation
parameter $s=I_{0}/I_{sat}$, where $I_{\mathrm{sat}}=\frac{\pi
hc\Gamma}{3\lambda^{3}}=82.5$ mW/cm2 is the two-level saturation intensity for
the transition. Solid red lines show fits using a set of Lorentzian functions,
where the full-width at half maximum $\Gamma/(2\pi)$ is allowed to vary
between resonances in order to account for optical pumping. For the spectrum
at lower intensity, the relative line intensities agree well with the relative
line intensities given in panel (a), indicating that optical pumping has been
largely avoided. For the higher intensity spectrum, we use the relative peak
heights to estimate that when the laser is tuned to the $(F,F^{\prime})=(1,2)$
resonance, the atoms scatter on average six photons. This means that the
effect of photon recoil shifts and broadens the resonance lines by at most 0.1
MHz. We note that the number of photon scattering events derived from the line
intensities is a factor two (less) than derived using a simple two-level rate
equation model and the estimated intensity of the probe light. The Lorentzian
linewidth of the $(1,2)$ resonances, being unaffected by optical pumping, is
$\Gamma/(2\pi)=25.4(1.0)$ MHz. Fitting with a Voigt lineshape resulted in a
slightly reduced Lorentzian linewidth, $\Gamma/(2\pi)=24.4(6)$ MHz, with a
Gaussian linewidth of below 4 MHz (full width at half maximum), corresponding
to a transverse velocity spread of below 3.1 m/s. The radiative lifetime
extracted from the Voigt fits, $\tau=1/\Gamma=6.50(16)$ ns, is slightly below
that reported by Carlsson et al. Carlsson  _et al._ (1990). One might expect
that the magnetic field in our detector, measured to be $\sim 0.3$ G, would
lead to a broadening of the lines by the Zeeman effect and may account for
this difference.

The absolute frequencies of the line centres from six spectra, taken over two
days and using a range of probe laser intensities, varied by less than 2 MHz
(standard deviation), and intervals between resonance lines varied by less
than 1.5 MHz (standard deviation). The uncertainty here is dominated by the 10
MHz uncertainty of the wavemeter at 656 nm; in a previous experiment using the
same spectrometer we found agreement with the precisely measured
${}^{1}S_{0}\rightarrow{}^{1}P_{1}$ 399 nm line in Yb at the 5 MHz level
Doppelbauer  _et al._ (2022). Doppler shifts due to misalignment of the probe
laser and atomic beams is at most 2 MHz, which we estimate by plotting the
fitted line centres versus forward velocity in the atomic beam, and
extrapolating to zero velocity. We extract the magnetic dipole hyperfine
interaction constants $A_{107,109}$ in each state for the two Ag isotopes and
present these in Table 3. The ground state splittings agree with high accuracy
microwave measurements Wessel and Lew (1953); Dahmen and Penselin (1967) to
within 2.5 MHz (109Ag) and 0.4 MHz (107Ag), and for the excited state we agree
with quantum beat spectroscopy measurements of Carlsson et al. Carlsson  _et
al._ (1990) to better than 0.4 MHz. This suggests that the isotope shift
between 107,109Ag measured from the same spectra may have an uncertainty of
2-3 MHz due to the uncertainty of the wavemeter frequency measurements.

To improve the accuracy of the isotope shift measurements, we introduced a
single acousto-optical modulator (AOM) into the optical setup, operated in
double-pass configuration using a cat-eye lens. The AOM is driven at a radio
frequency $\nu_{\mathrm{AOM}}$, which can be varied via a voltage controlled
oscillator, and is monitored using a radio frequency counter. We deliberately
allow both the zero-th order (i.e. unshifted) and twice-shifted beams to be
present, such that the probe laser light is now composed of two frequency
components separated by $2\nu_{\mathrm{AOM}}$, and whose spatial overlap and
pointing varies by less than 0.4 mrad as $\nu_{\mathrm{AOM}}$ is varied. An
example spectrum is shown in the lower trace of figure 2b, where arrows show
the displacement in the frequency axis introduced by the AOM. We choose
$\nu_{\mathrm{AOM}}$ to be around 200 MHz such that the frequency-shifted
component of the probe beam excites the 107Ag, $F=1-F^{\prime}=2$ transition,
whilst the other component simultaneously excites this transition in 109Ag.
This enables detecting both isotopes whilst only scanning a small ($\sim$200
MHz) range with the laser, indicated by the red shaded box in the spectrum.
Moreover, we can rapidly switch between single and dual-frequency probe light
as atoms fly through the detector via the RF driving power to the AOM.

Figure 2c shows a time of flight fluorescence trace using the AOM method, and
with the laser tuned to the $(1,2)$ 107Ag resonance. The RF power to the AOM
is set to give equal optical power in the two frequency components, and is
switched off for roughly 200 $\mu$s as the atoms fly through the detector. We
use the fluorescence within this observation window for a “reference spectrum”
in order to fix the 107Ag isotope position in each measurement. This largely
removes contributions to the IS from any slow drift in the wavemeter frequency
axis over timescales longer than a minute. The shaded regions in the figure
labelled (i) and (ii) are observation windows for which both frequency
components are present in the probe light. We fit spectra derived from these
observation windows to a model containing the resonances of both isotopes,
where the hyperfine splittings are fixed to those given in Table 3, the 107Ag
(1,2) resonance is fixed to that measured in the reference spectrum, and the
natural abundance ratio determines the total contribution from each isotope to
the spectrum. This leaves only the offset frequency $\Delta$ between the (1,2)
resonances of 107,109Ag as a free parameter, beside a common amplitude term
and a $y$-axis offset. The inset to figure 2 displays an example spectrum with
$2\nu_{\mathrm{AOM}}=404.02$ MHz, showing the two isotope components of the
signal and the interval $\Delta$ in the underlying fit function.

Table 3:  Summary of the experimental results for the ${}^{2}S_{1/2}\rightarrow{}^{2}P_{3/2}$ transition in 107,109Ag. All values are given in MHz. Absolute frequencies of hyperfine lines for isotope $\alpha$ are labelled as $\nu_{\alpha}(F,F^{\prime})$, and $\bar{\nu}_{\alpha}$ denotes the gravity centre for this isotope. $\bar{\nu}_{\mathrm{nat.}}$ is the mean of the gravity centers for the two isotopes, weighted by the natural isotopic abundance, which approximates the observed line-center in a low-resolution measurement with a naturally abundant sample.  | This work | Literature
---|---|---
$\nu_{109}(1,2)$ | $913~{}548~{}760(20)$ |
$\nu_{107}(1,2)$ | $913~{}549~{}171(20)$ |
$A_{109}(^{2}P_{3/2})$ | $-36.9(3)$ |  $-36.7(7)$ Carlsson  _et al._ (1990)
$A_{109}(^{2}S_{1/2})$ | $-1979.4(1.1)$ |  $-1976.932~{}075(17)~{}$Dahmen and Penselin (1967)
$A_{107}(^{2}P_{3/2})$ | $-31.7(6)$ |  $-31.7(7)$ Carlsson  _et al._ (1990)
$A_{107}(^{2}S_{1/2})$ | $-1713.0(8)$ |  $-1712.512~{}111(18)$ Dahmen and Penselin (1967)
$\bar{\nu}_{109}$ | $913~{}548~{}293(20)$ |
$\bar{\nu}_{107}$ | $913~{}548~{}766(20)$ |
$\bar{\nu}_{\mathrm{nat.}}$ | $913~{}548~{}539(20)$ |  $913~{}548~{}593(60)$ Pickering and Zilio (2001)
$\delta\nu_{109,107}(1,2)$ | $410.9(6)$ |
$\delta\bar{\nu}_{109,107}$ | $473.2(7)$ |  $467(4)$ Reponen  _et al._ (2021)
|  |  $476(10)$ Uhlenberg  _et al._ (2000)

We repeat this procedure for different values of $2\nu_{\mathrm{AOM}}$ and
plot $\Delta$ versus $2\nu_{\mathrm{AOM}}$ in figure 2d. In the ideal case,
this should result in a linear relationship with a gradient 1, whose $x$-axis
intercept (i.e. where $\Delta=0$) returns the isotope shift between the
$(1,2)$ lines of the two Ag isotopes. The linear fits to the data in
observation windows (i) and (ii) have slightly different intercepts, which we
attribute to a small residual misalignment of the probe light to the atoms,
and the atoms in the two observation windows having slightly different forward
velocities. The slopes of the two curves are $0.96(2)$ and $1.03(2)$ in
regions (i) and (ii) respectively. We take the average of the two intercepts
as the $x$-axis intercept, and half their difference as the 67% confidence
interval, and apply small corrections to this value to arrive at our value for
the true isotope shift as follows. Firstly, we consider the residual alignment
difference between the two frequency components in the probe light, which for
a forward velocity of 120 m/s tends to reduce the line separation by 150 kHz.
Secondly, the spectral intensity of the probe light near the 107Ag resonance
is about a factor 4 larger (i.e. $s\approx 0.016$) in the reference spectrum,
which would shift the $(1,2)$ line centre of 107Ag in the reference spectrum
by +70 kHz. We both correct for these effects and increase the systematic
error bar accordingly. Thirdly, the background magnetic field in the detector
($\sim 0.3$ G) leads to Zeeman shift-induced broadening of the hyperfine lines
and potentially to a small differential shift. Since the nuclear spins of
107,109Ag are identical, and the nuclear magnetic moments are within about
$15\%$ of one another Stone (2005), this effect is negligible. AC Stark shifts
of the ${}^{2}P_{3/2}$ and ${}^{2}S_{1/2}$ states by the excitation light are
well below the kHz level for the intensities used in our measurements and can
be neglected. The result, $\nu_{107}(1,2)-\nu_{109}(1,2)=410.9(6)$MHz, is
consistent with that measured using the wide range, single frequency probe
data, $410.7(7)$ MHz, though the latter has a few MHz systematic uncertainty
associated with the linearity of the wavemeter.

In deriving an improved value for the isotope shift of the
$5s~{}^{2}S_{1/2}\rightarrow 5p$ ${}^{2}P_{3/2}$ gravity centre, we take
advantage of the ${}^{2}S_{1/2}$ hyperfine-structure measurements of Dahmen
and Penselin (1967), whose stated uncertainty is below 0.1 kHz. We use the
values in Table 3 for the hyperfine splitting of the ${}^{2}P_{3/2}$ states,
and use the result of figure 2d to fix the isotope shift of the $(1,2)$ lines.
This interval contributes most to the uncertainty of the gravity centre
isotope shift, and therefore dominates the error bar.

##  IV Differential radii

###  IV.1 Between stable isotope pairs

Table 4:  Center of gravity ISs in MHz relative to the ground state, estimated from the indicated data sources, including this work (TW). The corresponding $\delta r^{2}_{109,107}$ in fm2 are extracted employing the factors given in Table 2. The uncertainty of the charge radii in parentheses includes both experiment and atomic factor uncertainties. The uncertainty in square brackets is added to allow for unknown shape variations.  Interval | Source | $\delta\nu_{109,107}$ |  $\delta r^{2}_{109,107}$
---|---|---|---
$5s-5p~{}^{2}P_{1/2}$ | Fischer  _et al._ (1968); Badr  _et al._ (2004) | $473(4)$ | $-0.207(3)[4]$
$5s-5p~{}^{2}P_{3/2}$ | TW | $473.2(7)$ | $-0.204(3)[4]$
$5s-6s$ | TW, Elbel and Fischer (1962); Fischer  _et al._ (1968); Badr  _et al._ (2004) | $368(7)$ | $-0.212(4)[4]$
$5s-6p~{}^{2}P_{3/2}$ | Badr  _et al._ (2001, 2006) | $414.0(6)$ | $-0.207(3)[4]$
Recommended |  | $\mathbf{-0.207(3)[4]}$
Muonic Ag | Fricke and Heilig (b) |  | $-0.198(4)[5]$
Interpolated | Dinger  _et al._ (1989) |  | $-0.148(1)[31]$
Compilation | Angeli and Marinova (2013) |  | $-0.148(1)$

We now combine our calculations of $F$ and $K$ with optical isotope shifts to
estimate the charge radius difference of the stable isotope pair, $\delta
r^{2}_{109,107}=r_{107}^{2}-r_{109}^{2}$. The calculated excited state field
shift factors, $F_{i}-F_{5s}$, spans a range of $-3223(26)$ to $-3683(43)$
GHz/fm2; an order of magnitude larger than their individual uncertainties (see
Table 2). Thus, a useful consistency check of our $F$ and $K$ calculations, as
well as the experimental ISs, is satisfied when applying equation 1 to each
optical transition returns the same value of $\delta r^{2}_{109,107}$. The
relative ISs are calculated with a Ritz-type analysis (see e.g. Ohayon  _et
al._ (2019)) of our measurement and those given in Elbel and Fischer (1962);
Fischer  _et al._ (1968); Badr  _et al._ (2001, 2004, 2006). The results are
given in Table 4. As, to our knowledge, no electron scattering experiments
have been performed on these isotopes, their charge distribution profile
differences are not known. We are thus inclined to add an uncertainty of $2\%$
to the extracted radii, in accordance with the estimation of the magnitude of
this effect in prior work Fricke and Heilig (b), in which a nuclear model was
assumed.

It is readily seen that $\delta r^{2}_{109,107}$ in Table 4 reasonably agree
with each other within our uncertainty estimation. This important test
indicates that no large spurious deviations are present in our calculations.
Moreover, it serves as a strong indication that a hyperfine interaction which
shifts the center of gravity of each isotope with respect to the other, causes
a smaller deviation in the extracted $\delta r^{2}$ than our reported
uncertainty. Our final recommended value, which is the mean weighted by the
uncorrelated errors, is in reasonable agreement with $\delta
r^{2}_{109,107}=(R_{k}^{107}/V_{2}^{107})^{2}-(R_{k}^{109}/V_{2}^{109})^{2}$,
where the Barret-equivalent moments $R_{k}$ are measured in muonic atom X-ray
spectroscopy Fricke and Heilig (b), and the proportionality factor $V_{2}$ are
estimated with a Fermi distribution, to which we added a relative shape-
variation uncertainty. It is illuminating to note that in both muonic and
electronic silver, it is the (unknown) shape change which dominates the error
in the extracted radii. This motivates to perform an elastic electron
scattering experiment to determine the shape change. The marginal agreement
between radii differences extracted all-optically and from muonic atoms is
somewhat expected considering recent work in the medium-mass region Bissell
_et al._ (2016); Ohayon  _et al._ (2022b); Sahoo and Ohayon (2023), pointing
to the need to reanalyze the cascade energies with modern tools (e.g. Santos
_et al._ (2005); Indelicato (2013); Trassinelli  _et al._ (2016); Michel  _et
al._ (2017); Antognini  _et al._ (2022); Oreshkina (2022); Valuev  _et al._
(2022); Valuev and Oreshkina (2024)). It is also worth noting that in Ref.
Reponen  _et al._ (2021), a more conservative uncertainty estimate for the
muonic data was employed, which might be warranted considering the above.

Figure 3: Two-dimensional King Plot whose data-points are given in bold in
Table 5. The vertical axis is the reduced IS in the
$5s~{}^{2}S_{1/2}-5p~{}^{2}P_{3/2}$ transition. The horizontal axis is the
reduced IS in the $4d^{9}5s^{2}~{}^{2}D_{5/2}-6p~{}^{2}P_{3/2}$ transition.
The resulting slope (full line) lies two standard deviations away from that
calculated empirically (dotted line) in Dinger  _et al._ (1989).

In their partial CKP interpretation of ISs of radioisotopes for the $548\,$nm
line, the authors of Ref. Dinger  _et al._ (1989) used an interpolated value
for $\delta r^{2}_{109,107}$ which barely agrees with our recommended one (see
Table 4). Moreover, adopting this value as given without the dominating
uncertainty in the most recent compilation Angeli and Marinova (2013), would
result in a deviation of $60$ times the tabulated error.

We conclude this section by focusing on the IS of the fine-structure of the
$5p$ doublet. Calculating it from the factors of Table 2 and the recommended
$\delta r^{2}_{109,107}$ from Table 5 results in $-8.3(4)\,$MHz; in tension
with the experimental value of $0\pm 4\,$MHz, whose uncertainty is completely
decided by a single photoelectric measurement Fischer  _et al._ (1968).
Extending precision laser spectroscopy to the IS of the $5s-5p~{}^{2}P_{1/2}$
$338$ nm line could help to shed light on this issue. The experimental method
detailed in this article could be straightforwardly applied.

###  IV.2 Among isotope and isomer chains

We interpret the ISs measured in radioisotopes of Ag in terms of differential
radii. First, the nuclei are divided into three groups. The first consists of
those whose ISs were measured only for the cooling line (Refs. Fischer  _et
al._ (1975); Ferrer  _et al._ (2014); Reponen  _et al._ (2021)). Here, $\delta
r^{2}_{109,A}$ can be estimated directly using our calculated IS factors,
including the added $2\%$ uncertainty for the unknown differences in shape.
The results are given in Table 5. The second group includes the four nuclei
whose ISs were measured for both the $328\,$nm and the $548\,$nm lines (data
from this work and Refs. Badr  _et al._ (2001); Dinger  _et al._ (1989);
Ferrer  _et al._ (2014); Reponen  _et al._ (2021), given in bold in Table 5).
Because the $548\,$nm line involves a $4d^{9}5s^{2}$ configuration, i.e. a
4$d$-hole state, calculating $F_{548}$ and $K_{548}$ accurately is far beyond
the scope of this article. We interpret these data by making use of a two-
dimensional KP linear equation

$\displaystyle\delta\bar{\nu}_{A,A^{\prime}}^{328}$ $\displaystyle=$
$\displaystyle
K_{328,548}+F_{328,548}~{}\delta\bar{\nu}_{A,A^{\prime}}^{548},$ (20)

where
$\delta\bar{\nu}_{A,A^{\prime}}^{i}\equiv\delta\nu_{A,A^{\prime}}^{i}/\mu_{A,A^{\prime}}$,
$F_{328,548}\equiv F_{328}/F_{548}$ and $K_{328,548}\equiv
K_{328}-F_{328,548}K_{548}$. A Monte-Carlo linear regression, shown in Fig. 3,
returns a slope $F_{548,328}=1.0(3)$, and intercept $K_{548,328}=3.4(1.9)$ THz
u. It also results in posterior ISs for the cooling line,
$\delta\nu_{109,A}^{328,\mathrm{KP}}$, given in Table 5, along with their
corresponding radii. The third group consists of nuclei for which measurements
exist only for the $548\,$nm line. We use the joint distribution of the slope
and intercept to project their ISs from the $548\,$nm line to the $328\,$nm
line. The results are also given in Table 5, with the corresponding radii.

The results of the fit can be used to check for the reasons for
inconsistencies found in the literature. The fitted slope deviates by two of
its standard deviations from $F^{\mathrm{SE}}_{548,328}=0.33(3)$, estimated
from the semi-empirical FS factors given in Dinger  _et al._ (1989). Combining
our calculated factors for the cooling line with the fitted slope and
intercept, we find $F_{548}=-3^{+1}_{-2}~{}$GHz/fm2 and
$K_{548}=-2^{+2}_{-1}~{}$THz u. Although it is roughly estimated, our FS
factor is highly inconsistent with the semi-empirical estimation
$F_{548}^{\mathrm{SE}}=-12(1)~{}$GHz/fm2 Dinger  _et al._ (1989). Indeed, the
authors of Ref. Dinger  _et al._ (1989) observed that when
$F_{548}^{\mathrm{SE}}$ was combined with $\delta r^{2}_{109,107}$ from muonic
atoms, a surprising crossing of isotopic chains appeared around $Z=50$. To
remedy this issue, they elected to interpolate $\delta r^{2}_{109,107}$ from
that of neighbouring isotones. These epicycles resulted in
$K_{547}^{\mathrm{SE}}=4.4(2.7)~{}$THz u which was considered to agree with
the HF calculation by Bauche Bauche (1974). However, as seen in Table 2, a HF
calculation can not even predict the sign of the SMS, which value dominates
that of the total MS. Considerably reducing the errors of the data-points in
Fig. 3, or introducing new ones via measurements, would help shed light on
these issues while reducing uncertainties in the extracted radii.

Our recommended $\delta r^{2}_{109,A}$ are compared with prior extractions in
Table 5 and Fig. 1. Although significantly different IS factors are used, we
find agreement with the radii given in Ref. Reponen  _et al._ (2021) within
uncertainties. This is due to the mitigating effect of enforcing $\delta
r^{2}_{109,107}$ from muonic atoms. However, the all-optical radii have
significantly smaller uncertainties, by up to a factor of 5. This in turn
motivates more accurate IS measurements for the neutron deficient Ag, whose
radii uncertainties are now dominated by experiment.

A larger disagreement is observed when comparing our results with those of the
GSI group Dinger  _et al._ (1989), with the largest deviation ($4\sigma$) for
$\delta r^{2}_{109,106m}$. This is due to the reasons described above; namely
the different estimation of $\delta r^{2}_{109,107}$ (see Table 4), and the
highly different $F_{548}$. Whereas most of the nuclei whose ISs were measured
with the $548\,$nm line at GSI were also measured with the cooling line later,
105Ag, 105mAg, and 106mAg were only measured with the $548\,$nm line. This
work endows them with reliable and precise $\delta r^{2}_{109,A}$.

We conclude this section by comparing $\delta r^{2}_{109,A}$ from this work,
and that calculated by state of the art density functional theory as done in
Reponen  _et al._ (2021) and shown in Fig. 1. Focusing first on the neutron
rich isotopes, we see agreement for the odd-odd nuclei, and a disagreement for
the odd-even ones. On the proton-rich side, an agreement is seen for all
nuclei except for 96Ag, as discussed in Reponen  _et al._ (2021), and 102Ag,
which lies 6 standard errors from the DFT calculation. These discontinuities
are further emphasized when looking at the ladder-type differences, given in
the last column of Table 5.

Table 5:  Center of gravity isotope shifts (GHz) and differential radii (fm2),
of short-lived silver isotopes and isomers. When two references are given, the
value is their weighted average. Numbers in bold are an input to the King Plot
of Fig. 3. $\nu_{A,109}^{328,\mathrm{KP}}$ is the mean and standard deviation
of the posterior distribution of the $328\,$nm line IS which is output by the
fit. The differential radii are calculated from $\nu_{109,A}^{328}$ and, when
available, $\nu_{109,A}^{328,\mathrm{KP}}$, using the IS factors from Table 2
They are compared with prior works (see also Fig. 1). The uncertainties in
parenthesis are tied to the experimental IS measurements, and those in square
Brackets stem from our calculation of the IS factors, including an added $2\%$
uncertainty for the unknown shape changes. The absolute radius can be obtained
by adding $r$(109Ag)$=4.564(2)\,$fm Fricke and Heilig (b) in quadrature. The
last column includes the ladder-type difference for ground state nuclei and
the isomer shifts for nuclear isomers.

A | $\delta\nu_{109,A}^{548}$ | Ref. | $\delta\nu_{109,A}^{328}$ | Ref. | $\delta\nu_{109,A}^{328,\mathrm{KP}}$ | $\delta r_{109,A}^{2}$ | Ref. Reponen  _et al._ (2021) | Ref. Dinger  _et al._ (1989) |  $\delta r_{A,A+2}^{2}$
---|---|---|---|---|---|---|---|---|---
96 |  |  | $2.64^{+1.1}_{-0.8}$ | Reponen  _et al._ (2021) |  | $-1.26^{+22}_{-31}[4]$ | $-1.21^{+27}_{-19}[14]^{\dagger}$ |  | $-0.20(27)[1]$
97 |  |  | $4.36(28)$ | Ferrer  _et al._ (2014); Reponen  _et al._ (2021) |  | $-1.70(8)[5]$ | $-1.55(8)[15]$ |  | $0.41(8)[1]$
98 |  |  | $3.66(21)$ | Ferrer  _et al._ (2014); Reponen  _et al._ (2021) |  | $-1.46(6)[4]$ | $-1.36(6)[13]$ |  | $0.31(8)[1]$
99 |  |  | $3.20(9)$ | Ferrer  _et al._ (2014); Reponen  _et al._ (2021) |  | $-1.29^{+2}_{-3}[4]$ | $-1.20^{+3}_{-2}[12]^{\dagger}$ |  | $0.27(5)[1]$
100 |  |  | $2.85(20)$ | Ferrer  _et al._ (2014); Reponen  _et al._ (2021) |  | $-1.14(6)[3]$ | $-1.02(8)[11]$ |  | $0.43(7)[1]$
101 | $\mathbf{4.672(5)}$ | Dinger  _et al._ (1989) | $\mathbf{2.54(23)}$ | Ferrer  _et al._ (2014); Reponen  _et al._ (2021) | $2.55(17)$ | $-1.02(5)[3]$ | $-0.98(10)[10]$ | $-0.670(3)[135]$ | $0.31(5)[1]$
102 |  |  | $1.60(17)$ | Reponen  _et al._ (2021) |  | $-0.712(48)[20]$ | $-0.67^{+5}_{-4}[8]^{\dagger}$ |  | $0.06(6)[0]$
103 | $\mathbf{3.302(4)}$ | Dinger  _et al._ (1989) | $\mathbf{1.58(30)}$ | Reponen  _et al._ (2021) | $1.74(8)$ | $-0.711(23)[20]$ | $-0.63(7)[7]$ | $-0.482(2)[98]$ | $0.314(24)[9]$
104 | $\mathbf{2.939(3)}$ | Dinger  _et al._ (1989) | $\mathbf{1.71(22)}$ | Reponen  _et al._ (2021) | $1.65(14)$ | $-0.648(39)[18]$ | $-0.61(6)[6]$ | $-0.416(2)[83]$ |
105 | $1.926(6)$ | Dinger  _et al._ (1989) |  |  | $0.894(22)$ | $-0.397(6)[11]$ |  | $-0.296(2)[63]$ | $0.193(6)[5]$
107 | $\mathbf{0.9781(5)}$ | Badr  _et al._ (2001) | $\mathbf{0.4732(7)}$ | TW |  | $-0.207(0)[6]$ | $-0.198(2)[20]$ | $-0.148(1)[31]$ | $0.207(0)[6]$
114 |  |  | $-0.850(3)$ | Reponen  _et al._ (2021) |  | $0.408(1)[11]$ | $0.384(1)[50]$ |  | $0.116(3)[3]$
115 |  |  | $-0.995(5)$ | Reponen  _et al._ (2021) |  | $0.480(1)[13]$ | $0.454(3)[60]$ |  | $0.114(2)[3]$
116 |  |  | $-1.040(9)$ | Reponen  _et al._ (2021) |  | $0.524(3)[15]$ | $0.500(10)[60]$ |  | $0.107(3)[3]$
117 |  |  | $-1.181(6)$ | Reponen  _et al._ (2021) |  | $0.594(2)[17]$ | $0.568(3)[70]$ |  | $0.107(2)[3]$
118 |  |  | $-1.203(5)$ | Reponen  _et al._ (2021) |  | $0.631(1)[18]$ | $0.607(3)[80]$ |  | $0.108(2)[3]$
119 |  |  | $-1.348(5)$ | Reponen  _et al._ (2021) |  | $0.701(1)[20]$ | $0.675(3)[90]$ |  | $0.090(2)[3]$
120 |  |  | $-1.379(4)$ | Reponen  _et al._ (2021) |  | $0.739(1)[21]$ | $0.715(2)[90]$ |  |
121 |  |  | $-1.461(3)$ | Reponen  _et al._ (2021) |  | $0.791(1)[22]$ | $0.767(1)[100]$ |  |
|  |  |  |  |  |  |  |  |  $\delta r_{A,Am}^{2}$
$~{}99m$ |  |  | $3.58(68)$ | Ferrer  _et al._ (2014); Reponen  _et al._ (2021) |  | $-1.39(19)[4]$ | $-1.18^{+26}_{-22}[12]$ |  | $-0.11(19)[0]$
$101m$ |  |  | $2.40(24)$ | Ferrer  _et al._ (2014) |  | $-0.98(7)[3]$ | $-0.91(6)[10]$ |  | $0.04(8)[0]$
$105m$ | $2.229(10)$ | Dinger  _et al._ (1989) |  |  | $1.21(8)$ | $-0.485(21)[13]$ |  | $-0.321(2)[65]$ | $-0.088(22)[2]$
$106m$ | $2.049(30)$ | Dinger  _et al._ (1989) |  |  | $1.30(18)$ | $-0.474(52)[13]$ |  | $-0.271(4)[52]$ |
$108m$ |  |  | $0.443(9)$ | Fischer  _et al._ (1975) |  | $-0.159(3)[4]$ |  | $-0.120(13)[20]$ |
$110m$ |  |  | $-0.689(60)$ | Fischer  _et al._ (1975) |  | $0.0551(17)[16]$ |  | $0.036(5)[16]$ |

† Our analysis suggests that when adopting the tabulated asymmetric
statistical uncertainties in $\delta\nu_{109,A}^{328}$, the signs of the
corresponding uncertainties in $\delta r_{109,A}^{2}$ should be swapped as
compared with those given in Table 1 and Figure 2 of Ref. Reponen  _et al._
(2021).

##  V Summary and outlook

Nuclear charge radius differences in the silver isotopic chain deviate between
experiments at the few sigma level, as seen in Fig. 1. To find the origin of
these deviations, and reconcile them, we performed high accuracy calculations
of isotope shift factors in the low-lying states of atomic silver (Tables 1
and 2), as well as new precise spectroscopic measurements in the silver
$5s-5p~{}^{2}P_{3/2}$ line (Table 3), and combined this information in a
global analysis (Tables 4 and 5).

Our measurement of the center-of-gravity isotope shift is in slight tension
with the recent collinear laser spectroscopy measurement Reponen  _et al._
(2021), which also affects all of the radii extracted from that campaign.
Nevertheless, the tension is not enough to explain the disagreements in the
radii determinations reported by other studies, narrowing down the suspects to
the used IS factors and/or differential radius of the stable pair.

To test the latter, we extracted the differential radius of the stable pair
from the isotope shifts of four transitions to the ground-state, two of which
make use of our new measurement in the $328\,$nm line (see table 4). We find a
reasonable agreement between these radii, which tests the accuracy of the
calculated isotope shift factors, and confirms the absence of off-diagonal
hyperfine effects, Based on this, we recommended a radius difference for the
stable pair. It is found in marginal agreement with the one extracted from
muonic atom cascade X-ray spectroscopy Fricke and Heilig (b). It also
disagrees with the value used in prior works Dinger  _et al._ (1989); Ferrer
_et al._ (2014), which was interpolated from neighbouring nuclei. The three
values of the differential radii used by us and in prior works partially but
not fully explain the disagreements in the silver chain, which must thus
originate from the choice of field shift factor.

We show this by making a projection of our calculated factors from the
$5s-5p~{}^{2}P_{3/2}$ line to the
$4d^{9}5s^{2}~{}^{2}D_{5/2}-6p~{}^{2}P_{3/2}$ line using a King Plot (Fig.
King (1963)), which also benefits from our new precise IS measurement. The
projected factors of the $548\,$nm line highly disagree with the ones
evaluated semi-empirically, thus pointing that this is the main culprit of the
disagreements.

Having shed light on prior disagreements we provide in Table 5 transparent and
reliable differential radii in the silver isotopic chain. Their trend is found
to be generally consistent with that from a state-of-the-art nuclear theory
calculation. Nevertheless, the calculated shape-staggering effect is
overestimated on the neutron-rich side, and there are discontinuities around
$A=96$ and $A=102$ which call for further attention, as seen in Fig. 1. With
the much smaller systematic uncertainties afforded by this work, the radii of
proton-rich silver nuclei could now be greatly improved with more accurate
measurements, and the maximum information can be extracted from new
measurements extending even further towards the drip lines Ge  _et al._
(2024).

###### Acknowledgements.

We thank Mikael Reponen for useful comments. B.K.S. acknowledges the use of
ParamVikram-1000 HPC facility at Physical Research Laboratory (PRL), Ahmedabad
for carrying out the atomic calculations and his work at PRL is supported by
the Department of Space, Government of India. B.O. is thankful for the support
of the Council for Higher Education Program for Hiring Outstanding Faculty
Members in Quantum Science and Technology.

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