# Analytic Response Relativistic Coupled-Cluster Theory: The first application
to indium isotope shifts

B.K. Sahoo  Atomic and Molecular Physics Division, Physical Research
Laboratory, Navrangpura, Ahmedabad 380009, India<EMAIL_ADDRESS>A.R.
Vernon  School of Physics and Astronomy, The University of Manchester,
Manchester M13 9PL, United Kingdom<EMAIL_ADDRESS><EMAIL_ADDRESS>,  R.F. Garcia Ruiz  School of Physics
and Astronomy, The University of Manchester, Manchester M13 9PL, United
Kingdom  EP Department, CERN, CH-1211 Geneva 23, Switzerland
<EMAIL_ADDRESS>C.L. Binnersley  School of Physics and
Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
J. Billowes  School of Physics and Astronomy, The University of Manchester,
Manchester M13 9PL, United Kingdom  M.L. Bissell  School of Physics and
Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
T.E. Cocolios  KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001
Leuven, Belgium  G.J. Farooq-Smith  KU Leuven, Instituut voor Kern- en
Stralingsfysica, B-3001 Leuven, Belgium  K.T. Flanagan  School of Physics and
Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
Photon Science Institute Alan Turing Building, University of Manchester,
Manchester M13 9PY, United Kingdom  W. Gins  KU Leuven, Instituut voor Kern-
en Stralingsfysica, B-3001 Leuven, Belgium  R.P. de Groote  KU Leuven,
Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium  Department of
Physics, University of Jyväskylä, Survontie 9, Jyväskylä, FI-40014, Finland
Á. Koszorús  KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001
Leuven, Belgium  G. Neyens  EP Department, CERN, CH-1211 Geneva 23,
Switzerland  KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001
Leuven, Belgium  K.M. Lynch  EP Department, CERN, CH-1211 Geneva 23,
Switzerland  F. Parnefjord-Gustafsson  KU Leuven, Instituut voor Kern- en
Stralingsfysica, B-3001 Leuven, Belgium  C.M. Ricketts  School of Physics and
Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
K.D.A Wendt  Institut für Physik, Johannes Gutenberg-Universität Mainz,
D-55128 Mainz, Germany  S.G. Wilkins  EN Department, CERN, CH-1211 Geneva 23,
Switzerland  X.F. Yang  School of Physics and State Key Laboratory of Nuclear
Physics and Technology, Peking University, Beijing 100871, China  KU Leuven,
Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium

###### Abstract

With increasing demand for accurate calculation of isotope shifts of atomic
systems for fundamental and nuclear structure research, an analytic energy
derivative approach is presented in the relativistic coupled-cluster theory
framework to determine the atomic field shift and mass shift factors. This
approach allows the determination of expectation values of atomic operators,
overcoming fundamental problems that are present in existing atomic physics
methods, i.e. it satisfies the Hellmann-Feynman theorem, does not involve any
non-terminating series, and is free from choice of any perturbative parameter.
As a proof of concept, the developed analytic response relativistic coupled-
cluster theory has been applied to determine mass shift and field shift
factors for different atomic states of indium. High-precision isotope-shift
measurements of 104-127In were performed in the 246.8-nm (5p 2P3/2
$\rightarrow$ 9s 2S1/2) and 246.0-nm (5p 2P1/2 $\rightarrow$ 8s 2S1/2)
transitions to test our theoretical results. An excellent agreement between
the theoretical and measured values is found, which is known to be challenging
in multi-electron atoms. The calculated atomic factors allowed an accurate
determination of the nuclear charge radii of the ground and isomeric states of
the 104-127In isotopes, providing an isotone-independent comparison of the
absolute charge radii.

###### pacs:

00.00, 20.00, 42.10

††: New J. Phys.

Keywords: Article preparation, IOP journals

##  1 Introduction

The removal or addition of neutrons to the nucleus produces changes in the
energy of atomic transitions, known as the isotope shift (IS). These small
variations, typically less than 10-6 with respect to the atomic energy levels,
can probe fundamental aspects of the electron-nucleus interaction, e.g., the
size of the nucleus [1], the existence of new bosons [2, 3], new spin-
independent interactions [4, 5] and long-range neutrino-mediated forces [6].
Currently, extensive experimental efforts worldwide have been focused on the
development of complementary techniques to perform high-precision measurements
of IS in atomic transitions, across different isotopic chains [7, 8, 9, 10].
Alongside the experimental progress, the development of many-body methods
plays a central role in these studies as it provides the means to extract
nuclear-structure and fundamental-physics parameters from experimental
observations [11]. Reliable atomic calculations are critical to establish firm
conclusions from high-precision experiments in nuclear [12] and fundamental-
physics research [13].

Most of our present knowledge on the nuclear charge radius of unstable nuclei
is derived from IS measurements in atomic transitions performed by laser
spectroscopy experiments [12]. The calculation of atomic physics factors which
are needed to decouple many-body electron correlations from nuclear-structure
variations present the main challenges in the interpretation of IS
measurements. The coupled-cluster (CC) method is considered as the gold
standard for treating electron-correlation effects [14]. However, the current
methods used to calculate atomic physics operators present serious drawbacks
that can generate uncontrolled theoretical uncertainties. The commonly used
expectation-value-evaluation (EVE) approach [15, 16], for example, involves
non-terminating series, and the finite-field (FF) approach [17] depends on the
choice of a perturbation parameter. To overcome these problems in this work we
implement and demonstrate, for the first time in atomic systems, the analytic-
response (AR) theory within the CC framework [18] to determine IS shift
parameters of atomic systems.

The atomic factors involved in the IS measurements can be empirically obtained
for even-proton elements [19], where independent charge radii measurements
from electron scattering and muonic atoms exists for three or more stable
isotopes. However, this is not the case for elements with odd-proton number,
where only up to two stable isotopes exists and the accuracy of all charge-
radii values obtained from isotope shifts measurements relies on atomic
physics calculations. Accurate determination of the charge radii of
radioactive isotopes is not only relevant for nuclear structure research, but
can provide a deeper insight into nuclear matter [20, 21]. Motivated by the
current nuclear structure interest in the study of ISs around proton number
$Z$ = 50 [22, 23, 24, 25], our theoretical developments were used to perform,
for the first time, ab-initio calculations of atomic factors for indium (In)
atom ($Z$ = 49). The In isotope chain offers a comprehensive laboratory to
test these theoretical developments. The long chain of isotopes increases the
precision in cancelling out the nuclear contribution to the IS, while the
presence of at least one isomeric nuclear state at each mass allows for an
mass-independent measure of the field-shift (FS) contribution to the IS. This
provides a stringent constraint to test our theoretical calculations by
increasing the precision on the experimentally determined atomic factors.
Moreover, several atomic transitions are experimentally accessible, and
precise data on transitions to high-lying states [26] can be combined with our
new measurements and calculations to evaluate the individual atomic level-IS
(LIS), allowing a direct study of the IS factors for each level.

##  2 Theory

The IS of an energy level, $i$, between an isotope, $A$, with mass, $m_{A}$,
and an isotope, $A^{\prime}$, with mass, $m_{A^{\prime}}$, is given [27] by a
product of nuclear and atomic factors as 111The factor of $h$ is dropped in
the notation of this work unless relevant i.e. IS$=\delta E_{i}$. However
where values are presented for comparison to experiment the factor is
included.

$\displaystyle\delta E_{i}$ $\displaystyle=$ $\displaystyle F_{i}\delta\langle
r^{2}\rangle+K_{i}^{\text{MS}}\frac{m_{A}-m_{A^{\prime}}}{m_{A}m_{A^{\prime}}},$
(1)

where $\delta\langle r^{2}\rangle=\langle r^{2}\rangle_{A}-\langle
r^{2}\rangle_{A^{\prime}}$ is the difference between the nuclear mean-square
charge radii of the two isotopes [28, 29]. Higher-order effects and non-linear
corrections to expression 1 are expected to be lower than 1$\%$ [30], and are
thus neglected in our present study. The atomic part is factorized in the
constants $F_{i}$ and $K_{i}^{\text{MS}}$, which are the FS and mass shift
(MS) contributions to the LIS, respectively.  The FS factor,
$F_{i}=\frac{\langle\Psi_{i}|\sum_{e}F(r_{e})|\Psi_{i}\rangle}{\langle\Psi_{i}|\Psi_{i}\rangle}$,
for atomic level, $i$, described by the wave function, $|\Psi_{i}\rangle$, is
calculated using the operator defined by

$\displaystyle F(r_{e})$ $\displaystyle=$ $\displaystyle-\frac{\delta
V_{\text{nuc}}(r_{N},r_{e})}{\delta\langle r_{N}^{2}\rangle}\;,$ (2)

where $r_{N}$ is the nuclear radius ($\langle r_{N}^{2}\rangle$ is the mean)
and $r_{e}$ is the electronic coordinate. The electrostatic potential due to
the nuclear charge, $V_{\text{nuc}}(r_{N},r_{e})$, is evaluated by assuming a
spherically-symmetric Fermi nuclear charge distribution defined by

$\rho_{\text{nuc}}(r_{N})=\frac{\rho_{0}}{1+e^{(r_{N}-c)/a}}\;\;,$ (3)

for the normalization factor, $\rho_{0}$. $c$ is the half-charge radius and
$a$ is related to the skin thickness [31]. The total MS constant is expressed
as the sum of the normal MS (NMS),
$K_{i}^{\text{NMS}}=\frac{\langle\Psi_{i}|\sum_{e}H_{NMS}(r_{e})|\Psi_{i}\rangle}{\langle\Psi_{i}|\Psi_{i}\rangle}$,
and specific MS (SMS), $K_{i}^{\text{SMS}}=\frac{\langle\Psi_{i}|\sum_{k,l\geq
k}H_{SMS}(r_{kl})|\Psi_{i}\rangle}{\langle\Psi_{i}|\Psi_{i}\rangle}$ for the
inter-electronic distance, $r_{kl}=|\vec{r}_{k}-\vec{r}_{l}|$, between the
electrons located at $r_{k}$ and $r_{l}$. These constants are obtained using
the relativistic expressions of the operators given by [32]

$\displaystyle
H_{NMS}(r_{i})=\vec{p}_{i}^{\;2}-\frac{\alpha_{e}Z}{r_{i}}\vec{\alpha}_{i}^{D}\cdot\vec{p}_{i}-\frac{\alpha_{e}Z}{r_{i}}\left\\{(\vec{\alpha}_{i}^{D}\cdot\vec{C}_{i}^{(1)})\vec{C}_{i}^{(1)}\right\\}\cdot\vec{p}_{i}\;\;,$
(4)

and

$\displaystyle
H_{SMS}(r_{ij})=\vec{p}_{i}\cdot\vec{p}_{j}-\frac{\alpha_{e}Z}{r_{i}}\vec{\alpha}_{i}^{D}\cdot\vec{p}_{j}-\frac{\alpha_{e}Z}{r_{i}}\left\\{(\vec{\alpha}_{i}^{D}\cdot\vec{C}_{i}^{(1)})\vec{C}_{i}^{(1)}\right\\}\cdot\vec{p}_{j}\;\;.$
(5)

In the above expressions, $\vec{p}$ is the momentum operator, $\alpha_{e}$ is
the fine structure constant, $Z$ is the atomic number, $\vec{\alpha}^{D}$ is
the Dirac matrix and $\vec{C}^{(1)}$ is the Racah operator of rank one. It is
worth noting here is that these expressions in the non-relativistic limit
become $H_{NMS}(r_{i})=\vec{p}_{i}^{\;2}$ and
$H_{SMS}(r_{ij})=\vec{p}_{i}\cdot\vec{p}_{j}$. Since $H_{SMS}$ is a two-body
operator, evaluation of $K_{i}^{SMS}$ using the expectation value expression
is computationally cumbersome.

##  3 The relativistic coupled cluster theory and the analytic-response
approach

Traditionally, the finite-field (FF) approach is adopted through a suitable
many-body method for the determination of IS factors, like the configuration-
interaction (CI) approach, as they involve both the one-body and the two-body
operators. It is also observed that evaluation of expectation value of
$\vec{p}^{\;2}$ exhibits strong electron-correlation effects. This introduces
difficulties in calculating using either the FF and expectation-value-
evaluation (EVE) approaches, as the calculations do not converge with the
inclusion of higher-order effects in the atomic wave functions [16]. In fact,
this is also one of the reasons $\langle\vec{p}^{\;2}\rangle$ is often
approximated from the experimental energy in the heavy atomic systems
following the Virial theorem [33]. As pointed out in Refs. [34, 35], it is
imperative to include both pair-correlation and core-polarization effects
rigorously for accurate calculations of the IS. The coupled cluster (CC)
method incorporates both these effects to all orders. Moreover, a truncated CC
method, unlike a truncated CI method, is free from the size-extensivity and
size-consistency problems appearing in many-body methods (e.g. see Ref. [14]).
In this work, we apply relativistic CC (RCC) theory to account for the
relativistic effects in our calculations.

###  3.1 Basic aspects

The atomic wave function of a state in an atomic system with a closed-shell
configuration and with a valence orbital ($v$) can be expressed in the RCC
theory as (e.g. see Refs. [16, 35, 36] and therein)

$\displaystyle|\Psi_{v}\rangle\equiv
e^{\\{T+S_{v}\\}}|\Phi_{v}\rangle=e^{T}\\{1+S_{v}\\}|\Phi_{v}\rangle,$ (6)

where $|\Phi_{v}\rangle=a_{v}^{+}|\Phi_{0}\rangle$ with the Dirac-Hartree-Fock
(DHF) wave function, $|\Phi_{0}\rangle$, of the closed-core (in this work
$[4d^{10}5s^{2}]$). Here $T$ is the RCC excitation operator embodying
electron-correlation effects from $|\Phi_{0}\rangle$ and the $S_{v}$ operator
incorporates correlation of the electron from the valence orbital along with
the core-valence interactions. Amplitudes of the RCC operators and energies
are obtained using the following equations

$\displaystyle\langle\Phi_{0}^{L}|(He^{T})_{c}|\Phi_{0}\rangle=0\;,$ (7)

and

$\displaystyle\langle\Phi_{v}^{L}|(He^{T})_{c}S_{v}|\Phi_{v}\rangle=E_{v}\langle\Phi_{v}^{L}|S_{v}|\Phi_{v}\rangle-\langle\Phi_{v}^{L}|(He^{T})_{c}|\Phi_{v}\rangle,\
\ \ \ $ (8)

where $H$ is the atomic Hamiltonian and the subscript $c$ indicates connected
terms. The superscript, $L$, over the reference states indicates
$L^{\text{th}}$-excited determinants with respect to the reference
determinants appearing in the ket states. $E_{0}$ and $E_{v}$ are the exact
energies of the states containing the closed-core (i.e. for the In+ ion) and
the closed-core with valence orbital, $v$, (i.e. for the In atom),
respectively. Both the $T$ and $S_{v}$ RCC operators are normal ordered with
respect to $|\Phi_{0}\rangle$. For convenience we carry out all the
calculations using normal-ordered operators, designated by subscript $N$. The
normal-ordered Hamiltonian is defined as
$H_{N}=H-\langle\Phi_{0}|H|\Phi_{0}\rangle$, for the DHF energy,
$E_{DHF}=\langle\Phi_{0}|H|\Phi_{0}\rangle$, using which the above amplitude
solving equations for the RCC operator are given by

$\displaystyle\langle\Phi_{0}^{L}|\bar{H}_{N}|\Phi_{0}\rangle=0\;,$ (9)

and

$\displaystyle\langle\Phi_{v}^{L}|\bar{H}_{N}S_{v}|\Phi_{v}\rangle=\Delta
E_{v}\langle\Phi_{v}^{L}|S_{v}|\Phi_{v}\rangle-\langle\Phi_{v}^{L}|\bar{H}_{N}|\Phi_{v}\rangle\;.\
\ \ \ $ (10)

Here $\bar{H}_{N}=(H_{N}e^{T})_{c}$, $\Delta E_{0}=E_{0}-E_{DHF}$ is the
correlation energy of the closed core and $\Delta E_{v}=E_{v}-E_{0}$ is the
electron affinity (EA) of the electron in the valence orbital, $v$. We are
interested in the EA values in this work, which are evaluated by

$\displaystyle\Delta
E_{v}=\langle\Phi_{v}|\bar{H}_{N}\\{1+S_{v}\\}|\Phi_{v}\rangle\;.$ (11)

It is clear from the above that both Eqs. (10) and (11) are correlated. In our
calculations we have considered Dirac-Coulomb-Breit (DCB) interactions in the
atomic Hamiltonian, $H^{a}$. Further, we only consider all possible single-
and double-excitation configurations in our RCC theory (RCCSD method).
Excitation energy between two states are estimated from the difference of
their EA values.

###  3.2 The finite-field approach to isotope shifts

Since all the relevant FS, NMS and SMS operators are scalar, they can be
included with the atomic Hamiltonian to estimate their contributions to the
energies. On the other hand, by expressing the total Hamiltonian as
$H=H^{a}+\lambda_{v}^{O}O$ with the atomic DCB Hamiltonian, $H^{a}$, and $O$,
representing one of the FS, NMS or SMS operators for an arbitrary parameter,
$\lambda_{v}^{O}$, it is possible to express the energy (here EA) in the FF
approach as

$\displaystyle\Delta E_{v}=\Delta E_{v}^{(0)}+\lambda_{v}^{O}\Delta
E_{v}^{(1)}+{\cal O}(\lambda_{v}^{O})^{2}\;.$ (12)

The superscripts (0), (1), and ${\cal O}(\lambda_{v}^{O})^{2}$ denote the
zeroth, first and higher-order contributions respectively. It can be noted
that the ${\cal O}(\lambda^{O})^{2}$ contributions are not of our interest. It
clearly follows that

$\displaystyle<O>\equiv\Delta E_{v}^{(1)}\simeq\left.\frac{\partial\Delta
E_{v}}{\partial\lambda_{v}^{O}}\right|_{\lambda_{v}^{O}=0}.$ (13)

This obviously follows the Hellmann-Feynman (H-F) theorem [37, 38], but it has
two major problems. First, the behaviors of FS, NMS and SMS operators are very
different, the choice of $\lambda_{v}^{O}$ has to be distinct for estimating
the FS, NMS and SMS constants reliably, and they can also be atomic state
dependent. Secondly, we assume ${\cal O}(\lambda^{O})^{2}$ contributions are
neglected in the FF approach based on the choice of the $\lambda_{v}^{O}$
value without removing them. Usually the electron correlation effects
contribute significantly to these quantities. Therefore, the IS constants
inferred from the FF approach are subjected to large numerical uncertainty.
Nevertheless, we use $\lambda_{v}^{O}=1\times 10^{-6}$ to determine all the IS
constants to perform the calculations in different states only for making
comparative analysis of the results in our study.

###  3.3 The expectation-value-evaluation approach

One can find several recent works that present high-precision results of many
properties in atomic systems, e.g. hyperfine structure constants [36, 39], by
employing the RCC theory. These calculations are carried out using the EVE
approach. Since the IS constants are the expectation values of the respective
operators, we can evaluate them in the EVE approach using the RCC theory
expression

$\displaystyle<O>\equiv\frac{\langle\Psi_{v}|O|\Psi_{v}\rangle}{\langle\Psi_{v}|\Psi_{v}\rangle}=\frac{\langle\Phi_{v}|\\{1+S_{v}^{\dagger}\\}e^{T\dagger}O_{N}e^{T}\\{1+S_{v}\\}|\Phi_{v}\rangle}{\langle\Phi_{v}|\\{1+S_{v}^{\dagger}\\}e^{T\dagger}e^{T}\\{1+S_{v}\\}|\Phi_{v}\rangle}$
(14)

by determining the wave functions using the Hamiltonian $H\equiv H^{a}$. The
advantage of using this approach is that it is possible to analyse and observe
the roles of various physical effects to the determination of the properties,
whereas one can obtain only the final results in the FF approach without
actually understanding the behavior of electron-correlation effects
explicitly. Evidently, this approach too has many shortcomings. First, both
the numerator and denominator of the above expression have non-terminating
series. Secondly, the SMS operator is a two-body operator, so its normal-
ordered form will have two components in the calculations as (e.g. refer to
[35])

$\displaystyle O_{N}\equiv O_{N}^{1}+O_{N}^{2},$ (15)

where superscipts 1 and 2 correspond to the effective one-body and two-body
parts. For the properties that are described by one-body operators, such as
hyperfine structure constants, we have adopted an iterative procedure to
account for contributions from the aforementioned non-terminating series in
the numerator and denominator [36]. However, it is impractical to apply a
similar technique for the effective two-body terms, as it becomes unmanageable
to compute contributions from the two-body components of the SMS operator
using a diagrammatic procedure even in the RCCSD method approximation. Thus,
we estimate contributions by selecting only important diagrams representing
the two-body components of the SMS operator based on the knowledge gained from
our earlier studies (see discussions in Ref. [35]). This may lead to large
errors in the results. The third notable drawback of the EVE approach is, it
does not satisfy the H-F theorem [14]. This can be understood from the simple
argument of Thouless [40], that the form of Eq. (14) does not follow the
energy-evaluating expression given by Eq. (11).

###  3.4 The analytic-response approach

The aforementioned problems of (i) unwanted contributions from ${\cal
O}(\lambda_{v}^{O})^{2}$ in the FF approach, (ii) the appearance of non-
terminating series in the EVE approach, (iii) the analysability of the roles
of various physical effects in the determination of properties, and (iv)
satisfying H-F theorem in the determination of the IS constants using the RCC
theory, can all be circumvented by adopting the AR procedure as suggested by
[18].  The uniqueness of this approach is it uses features from both the FF
and EVE procedures, in which Eq. (13) is directly obtained by perturbing the
RCC operators due to $O$ as

$\displaystyle T=T^{(0)}+\lambda_{v}^{O}T^{(1)}+{\cal
O}(\lambda_{v}^{O})^{2},$ (16)

and

$\displaystyle S_{v}=S_{v}^{(0)}+\lambda_{v}^{O}S_{v}^{(1)}+{\cal
O}(\lambda_{v}^{O})^{2},$ (17)

where $T$ and $S_{v}$ are the RCC operators for the total Hamiltonian,
$H=H^{a}+\lambda_{v}^{O}O$, and superscripts $(0)$ and $(1)$ indicate the
unperturbed and the first-order perturbed corrections due to $O$,
respectively. Substituting the above expanded form of the operators into Eqs.
(9) and (10), and then equating the zeroth-order and first-order terms in
$\lambda^{O}$ gives the equations for the unperturbed and perturbed RCC
operators, respectively.  Similarly, the first-order terms from the expansion
in Eq. (11) will correspond to the expectation values of the operator $O$.
Thus, using the normal-ordered form of the operators, we can get

$\displaystyle\langle\Phi_{0}^{L}|\bar{H}^{a}_{N}T^{(1)}|\Phi_{0}\rangle$
$\displaystyle=$
$\displaystyle-\langle\Phi_{0}^{L}|\bar{O}_{N}|\Phi_{0}\rangle,$ (18)
$\displaystyle\langle\Phi_{v}^{L}|(\bar{H}^{a}_{N}-\Delta
E_{v}^{(0)})S_{v}^{(1)}|\Phi_{v}\rangle$ $\displaystyle=$ $\displaystyle\Delta
E_{v}^{(1)}\langle\Phi_{v}^{L}|S_{v}^{(0)}|\Phi_{v}\rangle,$ (19)
$\displaystyle-\langle\Phi_{v}^{L}|(\bar{H}^{a}_{N}T^{(1)}+\bar{O}_{N})\\{1+S_{v}^{(0)}\\}|\Phi_{v}\rangle,$

and

$\displaystyle\Delta
E_{v}^{(1)}=\langle\Phi_{v}|\bar{H}_{N}S_{v}^{(1)}+(\bar{H}_{N}T^{(1)}+\bar{O}_{N})\\{1+S_{v}^{(0)}\\}|\Phi_{v}\rangle.$
(20)

Here, $\bar{O}_{N}=(O_{N}e^{T^{(0)}})_{c}$, and the superscripts $(0)$ and
$(1)$ in the energies indicate the zeroth and first-order contributions,
respectively. The AR equations have the advantages that were mentioned above.
It can be noted that the lowest-order contributions (DHF results) in the EVE
and AR approaches are the same, while they are different in the FF procedure.
Again, the above equations are modified appropriately for the evaluation of
the SMS constants as

$\displaystyle\langle\Phi_{0}^{L}|\bar{H}^{a}_{N}T^{(1)}|\Phi_{0}\rangle$
$\displaystyle=$
$\displaystyle-\langle\Phi_{0}^{L}|\bar{O}_{N}^{1}+\bar{O}_{N}^{2}|\Phi_{0}\rangle,$
(21) $\displaystyle\langle\Phi_{v}^{L}|(\bar{H}^{a}_{N}-\Delta
E_{v}^{(0)})S_{v}^{(1)}|\Phi_{v}\rangle$ $\displaystyle=$ $\displaystyle\Delta
E_{v}^{(1)}\langle\Phi_{v}^{L}|S_{v}^{(0)}|\Phi_{v}\rangle$ (22)
$\displaystyle-$
$\displaystyle\langle\Phi_{v}^{L}|(\bar{H}^{a}_{N}T^{(1)}+\bar{O}_{N}^{1}+\bar{O}_{N}^{2})\\{1+S_{v}^{(0)}\\}|\Phi_{v}\rangle,$

and

$\displaystyle\Delta
E_{v}^{(1)}=\langle\Phi_{v}|\bar{H}_{N}S_{v}^{(1)}+(\bar{H}_{N}T^{(1)}+\bar{O}_{N}^{1}+\bar{O}_{N}^{2})\\{1+S_{v}^{(0)}\\}|\Phi_{v}\rangle,$
(23)

due to the two-body nature of the SMS operator. The AR approach also involves
a slight computational challenge compared with the FF and EVE approaches as it
requires storing matrix elements of the additional one-body and two-body
operators than the atomic Hamiltonian.

Ideally, if results from all the three, FF, EVE and AR, approaches agree with
each other then the results can be assumed as very reliable. However, it is
difficult to achieve good agreement between the results from all these
procedures in heavy atomic systems using approximated many-body methods and
due to large numerical uncertainties associated with the implementation of the
EVE and FF approaches. Nonetheless, the results obtained using the AR approach
at the given level of approximation in the many-body theory should be treated
as more valid owing to the aforementioned merits of this procedure.

Table 1:  Comparison of FS, NMS and SMS factors of the six states in indium
from the FF, EVE and AR approaches obtained using the RCCSD method.

Method | 5P1/2 | 5P3/2 | 6S1/2 | 7S1/2 | 8S1/2 | 9S1/2
---|---|---|---|---|---|---
$F$ (GHz/fm2)
FF | 1.544 | 1.491 | -0.437 | -0.155 | -0.069 | -0.033
EVE | 1.275 | 1.299 | -0.408 | -0.135 | -0.061 | -0.033
AR | 1.435(6) | 1.442(6) | -0.383(1) | -0.1281(5) | -0.0559(25) | -0.0307(5)
$K_{\text{NMS}}$ (GHz.u)
FF | 749 | 711 | 364 | 170 | 98 | 63
EVE | 1340 | 375 | 458 | 201 | 113 | 71
AR | 774(41) | 734(37) | 340(5) | 163(2) | 96(1) | 61.7(5)
Experiment† | 768 | 731 | 367 | 171 | 99 | 65
$K_{\text{SMS}}$ (GHz.u)
FF | -470 | -403 | 119 | 38 | 17 | 9
EVE | -1048 | -899 | 136 | 42 | 18 | 10
AR | -638(71) | -533(69) | 94(26) | 29(8) | 13(4) | 8.6(5)
Experiment∗ | -536(122) | -507(111) | 169(51) | 55(42) | 24(80) | -13(66)
$\text{LIS}^{113,115}$ (MHz)
Experiment | 277(10) | 272(6)[26] | 17(6) | 12(6) | 9(12) | 2(10)

$\dagger$ Level energies from [41] were used.

* To determine $K_{\text{SMS}}$ from Eq. 1, the measured differential ISs, $\delta E^{113,115}$, were combined with FS factors from the AR approach and $\delta\left\langle r^{2}\right\rangle_{\mu}^{113,115}$ = 0.157(11) fm2 [42]

##  4 Isotope shift measurements

The results of the calculations have been combined with complementary
measurements to perform a comprehensive theoretical and experimental study of
the FS and SMS constants of the indium atom. Further, they are used to provide
accurate nuclear charge-radii of 104-127In. As indium has only two naturally
occurring isotopes (113,115In), exotic isotopes were produced at the on-line
isotope-separator facility ISOLDE at CERN. To produce the neutron-rich indium
isotopes, 115-127In, a beam of 1.4-GeV protons impinged onto the neutron
converter of a thick UCx target. The converter suppressed nearby caesium mass
contamination and increased utilizable neutron-rich indium yields [43]. The
neutron-deficient indium isotopes, 104-115In, were produced by impinging the
protons directly onto a thick LaC2 target [44]. The indium isotopes diffused
through the target material and their ionization was enhanced by the use of
the resonant ionization ion source RILIS [45]. The produced [45] indium ions
were then accelerated to 40 keV, mass separated, and injected into a gas-
filled linear Paul trap (ISCOOL) [46, 47]. Ion bunches of 2 $\mu$s temporal
width, were then re-accelerated to 40 keV and deflected into the CRIS beamline
[48, 49]. The indium ions were then neutralised with a sodium-filled vapor
cell, with an efficiency of up to 60% and predicted relative atomic
populations of 57% and 37% respectively for the 5p 2P3/2 metastable state and
5p 2P1/2 ground state [50]. The remaining ion fraction was removed by
electrostatic deflectors, and the neutralized atom bunch was collinearly
overlapped with two pulsed lasers, one for excitation and another for non-
resonant ionization. The atoms were resonantly excited using two different UV
transitions in separate measurements. The first using 246.8-nm laser light for
the 5p 2P3/2 $\rightarrow$ 9s 2S1/2 atomic transition. The second using
246.0-nm laser light for the 5p 2P1/2 $\rightarrow$ 8s 2S1/2 atomic
transition. The resonant laser light was produced by frequency tripling the
light from an injection-locked Ti:Sapphire laser system [51]. This laser was
seeded using a narrow-band M Squared SolsTiS continuous-wave Ti:Sapphire
laser, and pumped using a LEE LDP-100MQ Nd:YAG laser, producing pulsed narrow-
band 740(738)-nm laser light at 1 kHz. This light was then frequency tripled
to 246.8(246.0)-nm light by the use of two non-linear BiB3O6 crystals [52], 3
mW of laser light was used to saturate both transitions. The excited atoms
were then ionized by a non-resonant 532-nm step, The frequency of the resonant
first step was scanned and the resulting ions were deflected onto a detector,
producing the hyperfine spectra from which the IS were obtained. The
determined IS values are displayed in Table 2.

##  5 Comparison with experiment and evaluation of nuclear mean-squared charge
radii

###  5.1 King plot analysis

Since the changes in the mean-square charge radii are independent of the
atomic transitions, the nuclear dependence can be removed by comparing the IS
of two atomic transitions. A combination of the IS using Eq. (1), for two
atomic transitions, $i$ and $j$, can be expressed as

$\mu_{A,A^{\prime}}\delta
E^{A,A^{\prime}}_{j}=\frac{F_{j}}{F_{i}}\mu_{A,A^{\prime}}\delta
E^{A,A^{\prime}}_{i}+M_{j}-\frac{F_{j}}{F_{i}}M_{i},$ (24)

with $\mu_{A,A^{\prime}}=\frac{m_{A}m_{A^{\prime}}}{m_{A}-m_{A^{\prime}}}$.
Hence, in a ‘King’ plot [28] of $\mu_{A,A^{\prime}}\delta
E^{A,A^{\prime}}_{j}$ versus $\mu_{A,A^{\prime}}\delta E^{A,A^{\prime}}_{i}$,
the gradient provides the FS ratio, $F_{j}/F_{i}$, between two transitions,
and the MS differences can be extracted from its intercept.

Figure 1:  King plots of the 246.0-nm and 246.8-nm and the 410.2-nm and
451.1-nm transitions. Inset: the ratio of isomer shift values allowed mass-
shift-independent determination of $\frac{F_{246.0}}{F_{246.8}}=1.04(9)$.
Theoretical values are indicated by $\frac{F_{246.0}}{F_{246.8}}_{AR}$. The
shaded area indicates the uncertainty of the fits. Error bars include
statistical and systematic uncertainties (indicated by the black part of the
error bar)

The King plot obtained for the transitions measured in this work (246.8-nm (5p
2P3/2 $\rightarrow$ 9s 2S1/2) and 246.0-nm (5p 2P1/2 $\rightarrow$ 8s 2S1/2)),
and previous measurements in the 410.2-nm (5p 2P1/2 $\rightarrow$ 6s 2S1/2)
and 451.1-nm (5p 2P3/2 $\rightarrow$ 6s 2S1/2) transitions [53] are shown in
Fig. 1. The calculations and experimental data agree within 1$\sigma$, using
the AR or FF approaches.

###  5.2 Isomer shifts

The availability of several isomeric nuclear states in the indium isotope
chain allows a further test of the theoretical calculations. For isomeric
states, the factor $\frac{m_{A}-m_{A^{\prime}}}{m_{A}m_{A^{\prime}}}$ tends to
0, and the Eq. (24) can be approximated as $\frac{\delta E^{m}_{i}}{\delta
E^{m}_{j}}=\frac{F_{i}}{F_{j}}$. This assumption corresponds to an uncertainty
of up to 0.02 MHz for the excitation energies of the isomers in this work
(<400 keV). Therefore, isomer-shift measurements provide a test of the FS
factors and are less sensitive to systematic uncertainties present in the King
plot analysis. Previous measurements have not reported values for isomer
shifts in the indium atom as they are relatively small and require
particularly high precision [53]. The new measurements reported here allowed
the extraction of isomer shifts for the 246.8-nm (5p 2P3/2 $\rightarrow$ 9s
2S1/2) and 246.0-nm (5p 2P1/2 $\rightarrow$ 8s 2S1/2) transitions. The
extracted FS ratios from the measured isomer shifts are shown in the inset of
Fig. 1. This ratio agrees with the value obtained from the King plots, and is
within 1$\sigma$ of the presented theoretical calculations.

###  5.3 Experimental level specific mass shift

Calculations of SMS are notably challenging. To the authors’ knowledge, they
have not yet been reported for the indium atom. Moreover, a reliable
experimental test is also difficult as optical measurements provide the
difference of SMS between two states and their individual contribution cannot
be separated. Yet calculations of the atomic FS and MS factors are typically
performed for individual atomic-energy levels, with the difference between two
states used to determine the atomic factors for a transition used to measure
an isotope shift. In this work the individual atomic-level isotope shift (LIS)
values were determined by combining the IS measurements with measurement of
transitions to high-lying atomic states in indium [26]. As the contribution to
the IS of a transition from an atomic state decreases with the principle
quantum number of the state, in measurements to high-lying Rydberg states the
IS contribution from the upper state becomes negligible [54]. This allowed the
LIS to be determined for each state, and then the specific-mass-shift
contribution to the individual state, $l$, could be evaluated for comparison
to the calculations. The new measurements of this work provide access to the
8S1/2 and 9S1/2 states. For example, using the 5p 2P3/2 $\rightarrow$ 5s2 np
2P1/2,3/2 transition (27$\leq$n$\leq$35) IS measured for 113,115In [26], a LIS
of the 5p 2P3/2 state of $\text{LIS}_{\text{P3/2}}^{113,115}$=272(6) MHz was
reported. Using the IS value measured with the 5p 2P3/2 $\rightarrow$ 6s 2S1/2
transition of $\text{LIS}_{\text{P3/2-6s}}^{113,115}$=255.4(5) MHz [55], gives
a LIS of $\text{LIS}_{\text{6s}}^{113,115}$ = 17(6) MHz. This LIS value can in
turn be used to determine the LIS of the 5p 2P1/2 state from the 5p 2P1/2
$\rightarrow$ 6s 2S1/2 transition [39], giving
$\text{LIS}_{\text{5P1/2}}^{113,115}$ = 277(10) MHz. All of the LIS values
determined from the new measurements of this work and from literature (6S1/2
and 7S1/2 states [39, 55, 56]) are presented in Table 1. The LIS value,
$\text{LIS}^{113,115}_{l}$, of a state, $l$, is the sum of the field-shift
(volume isotope shift) and mass-shift contributions given by

$\displaystyle\begin{aligned} \text{LIS}^{113,115}_{l}&=F_{l}\delta\langle
r^{2}\rangle^{113,115}\\\
&+(K_{l}^{\text{NMS}}+K_{l}^{\text{SMS}})\frac{m_{113}-m_{115}}{m_{113}m_{115}}\end{aligned}$
(25)

Using the calculated state FS atomic factors and relativistic normal mass
shift factors, $K_{l}^{\text{NMS}}$, given in Table 1, and the literature
value of $\delta\left\langle r^{2}\right\rangle_{\mu}^{113,115}$ = 0.157(11)
fm2 [42] allowed evaluation of the SMS factors for individual states,
$K_{\text{SMS}}^{\text{Exp}}$. The experimental results and theoretical
calculations are shown in Table 1. The new calculations presented here,
adopting the AR approach, agree within 1$\sigma$ of the experimental values,
in addition to the values from the FF approach. In contrast, the EVE results
present large discrepancies.

###  5.4 Comparison with nuclear mean-squared charge radii

Combining the IS measurements and the calculated FS and MS constants in Eq.
(1), a value of $\delta\left\langle r^{2}\right\rangle^{113,115}$ = 0.163(4)
fm2 is obtained for the root-mean-square charge radii difference between the
stable isotopes 113,115In, in good agreement with the muonic atom result of
$\delta\left\langle r^{2}\right\rangle_{\mu}^{113,115}$ = 0.157(11) fm2 [42].
The nuclear charge radii of the exotic indium isotopes were extracted from the
measured IS and the calculated FS and SMS constants from the AR approach. The
extracted $\delta\left\langle r^{2}\right\rangle^{115,A}$ values are given in
Table 2 and are plotted in Fig. 2. The reported uncertainties of the
calculated atomic factors using the AR approach were evaluated from a
perturbative estimation of the neglected triples contribution. The atomic
masses used were taken from [57]. The values obtained from the FF and EVE
approaches are also shown in Fig. 2 for comparison.

Table 2:  IS measured with the 246.0-nm (5p 2P1/2 $\rightarrow$ 8s 2S1/2) and 246.8-nm ( 5p 2P3/2 $\rightarrow$ 9s 2S1/2 ) transitions, and $\delta\left\langle r^{2}\right\rangle^{115,A}$ values extracted using the AR approach.  A | I |  $IS^{115,A}$ (MHz) |  $\delta\left\langle r^{2}\right\rangle^{115,A}$ (fm2)
---|---|---|---
|  | 246.0 nm | 246.8 nm | 246.0 nm | 246.8 nm
104 | ($5^{+}$) | -1805(10) | -1753(20) | -1.19(5) | -1.11(5)
105 | $\frac{9}{2}^{+}$ | -1510(10) | -1540(20) | -1.00(5) | -0.97(5)
106 | $7^{+}$ | -1381(10) | -1362(20) | -0.91(4) | -0.86(4)
107 | $\frac{9}{2}^{+}$ | -1166(10) | -1178(20) | -0.77(4) | -0.74(4)
108 | $2^{+}$ | -1033(10) | -978(20) | -0.68(3) | -0.61(3)
108 | $7^{+}$ | -1046(10) | -1011(20) | -0.69(3) | -0.64(3)
109 | $\frac{9}{2}^{+}$ | -835(10) | -855(20) | -0.55(3) | -0.54(3)
110 | $7^{+}$ |  | -729(20) |  | -0.46(2)
111 | $\frac{9}{2}^{+}$ | -555(30) | -542(20) | -0.37(3) | -0.34(2)
113 | $\frac{9}{2}^{+}$ | -265(5) | -278(5) | -0.175(9) | -0.175(9)
114 | $5^{+}$ | -175(5) | -171(10) | -0.116(5) | -0.109(8)
115 | $\frac{9}{2}^{+}$ | 0 | 0 | 0 | 0
115 | $\frac{1}{2}^{-}$ | 26(8) | 33(5) | 0.018(5) | 0.022(3)
116 | $5^{+}$ | 89(5) | 99(20) | 0.058(5) | 0.06(1)
116 | $8^{-}$ | 86(8) | 99(2) | 0.056(7) | 0.061(4)
117 | $\frac{9}{2}^{+}$ | 243(5) | 265(3) | 0.160(9) | 0.167(8)
117 | $\frac{1}{2}^{-}$ | 261(6) | 282(4) | 0.173(9) | 0.179(8)
118 | $5^{+}$ | 330(5) | 329(2) | 0.22(1) | 0.20(1)
118 | $8^{-}$ | 324(5) | 324(3) | 0.21(1) | 0.20(1)
119 | $\frac{9}{2}^{+}$ |  | 475(3) |  | 0.30(2)
119 | $\frac{1}{2}^{-}$ |  | 488(4) |  | 0.30(2)
120 | $(5)^{+}$ | 531(5) | 556(5) | 0.35(2) | 0.35(2)
120 | $(8^{-})$ | 500(5) | 530(2) | 0.33(2) | 0.33(2)
121 | $\frac{9}{2}^{+}$ |  | 654(2) |  | 0.41(2)
121 | $\frac{1}{2}^{-}$ |  | 661(3) |  | 0.41(2)
122 | $5^{+}$ | 704(5) | 674(5) | 0.46(3) | 0.41(3)
122 | $8^{-}$ | 687(5) | 658(8) | 0.45(3) | 0.40(3)
123 | $\frac{9}{2}^{+}$ |  | 756(3) |  | 0.46(3)
123 | $\frac{1}{2}^{-}$ |  | 751(2) |  | 0.46(3)
124 | $(3)^{+}$ |  | 809(10) |  | 0.49(3)
124 | $(8^{-})$ |  | 810(3) |  | 0.49(3)
125 | $\frac{9}{2}^{+}$ |  | 941(4) |  | 0.58(4)
125 | $\frac{1}{2}^{-}$ |  | 926(5) |  | 0.57(4)
126 | $3^{+}$ |  | 1026(3) |  | 0.63(4)
126 | $(8^{-})$ |  | 1019(5) |  | 0.62(4)
127 | $\frac{9}{2}^{+}$ | 1115(5) | 1129(4) | 0.73(5) | 0.69(4)
Figure 2: a) $\delta\left\langle r^{2}\right\rangle^{115,A}$ values for the
104-127In isotopes extracted from the IS measurements of four optical
transitions and using the calculated FS and MS factors. The spread in values
from each approach is indicated by the colored areas. The shaded area
‘Literature’ indicates the uncertainty from literature FS and MS factors [42,
53]. b) $\sqrt{\left\langle r^{2}\right\rangle^{A}}$ compared to Sn ($Z$ = 50)
[58] and Cd ($Z$ = 48) [9] isotopes.

Remarkably, the extracted $\delta\left\langle r^{2}\right\rangle$ values agree
for all four optical transitions, which gives confidence in the accuracy of
the calculations. The absolute charge radii, $\sqrt{\left\langle
r^{2}\right\rangle^{A}}$, using the reference isotope 115In (4.615 fm [42]),
are compared to its isobaric neighbours Sn ($Z$ = 50) [58] and Cd ($Z$ = 48)
[9] in Fig. 2. The effect of inaccurate calculation of the MS factors using
the EVE approach is seen to be significant, causing a large discrepancy
between the values extracted from the four transitions. Previously, literature
values [53, 42] were normalized to the neighboring tin and cadmium isotopes
and the $\delta\left\langle r^{2}\right\rangle_{\mu}^{113,115}$ value. This
introduces large uncertainties (yellow area in Fig. 2), and prevents an
independent comparison of the nuclear charge radii with neighbouring elements.
Our theoretical calculations have therefore enabled the first independent
comparison of absolute charge radii for an odd-proton system around the $Z$ =
50 nuclear closed shell to be made.

##  6 Conclusion

In conclusion, we present in this work a new theoretical method to perform
accurate calculations of FS and MS constants in atomic systems. These
constants are critical to separate electronic and nuclear structure effects in
the interpretation of IS measurements for fundamental and nuclear-physics
research. This new theoretical method uses an analytic-energy-derivative
approach in the RCC framework, and solves fundamental problems related to the
evaluation of operators, which have been present in previous atomic physics
calculations. Precise IS measurements in the indium atom were used as an
exhaustive experimental test for these theoretical developments. A good
agreement was found with all available experimental data. The existence of
several isomers and the access to high-lying states in the indium atom allow
the separation of FS from MS, providing a stringent test for the calculations.
Our calculations of the atomic physics factors are essential to extract
nuclear charge radii values from isotope shifts measurements of exotic indium
isotopes [59]. These results can be extended to different elements across the
nuclear chart. This is especially important for odd-proton nuclei, which rely
on atomic theory to extract charge radii from laser-spectroscopy measurements.
Our theoretical developments will help to provide a deeper insight in the
evolution of the nuclear charge radius for different numbers of protons and
neutrons, which is of great importance for our understanding of nuclear
structure [9, 60, 1, 61] and nuclear matter [20].

##  7 Acknowledgments

This work was supported by ERC Consolidator Grant No.648381 (FNPMLS); STFC
grants ST/L005794/1, ST/L005786/1, ST/P004423/1 and Ernest Rutherford Grant
No. ST/L002868/1; GOA 15/010 from KU Leuven, BriX Research Program No. P7/12;
the FWO-Vlaanderen (Belgium); the European Unions Grant Agreement 654002
(ENSAR2); National Key R&D Program of China (Contract No:2018YFA0404403), the
National Natural Science Foundation of China (No:11875073) and we acknowledge
the financial aid of the Ed Schneiderman Fund at New York University. B. K. S.
acknowledges use of the Vikram-100 HPC cluster of Physical Research
Laboratory, Ahmedabad. We would also like to thank the ISOLDE technical group
for their support and assistance, the University of Jyväskylä for the use of
the injection-locked cavity, and the Physikalisch-Technische Bundesanstalt
(PTB) for the use of their voltage divider.

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