# Laser spectroscopy of indium Rydberg atom bunches by electric field
ionization

A. R. Vernon  KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001
Leuven, Belgium  Corresponding author<EMAIL_ADDRESS>C. M. Ricketts
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  T. E. Cocolios
KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium  B.
S. Cooper  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  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  R. F.
Garcia Ruiz  EP Department, CERN, CH-1211 Geneva 23, Switzerland
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  F. P.
Gustafsson  KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001 Leuven,
Belgium  G. Neyens  KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001
Leuven, Belgium  Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  H. A. Perrett  School of Physics and Astronomy, The University of
Manchester, Manchester M13 9PL, United Kingdom  B. K. Sahoo  Atomic, Molecular
and Optical Physics Division, Physical Research Laboratory, Navrangpura,
Ahmedabad 380009, India  Q. Wang  School of Nuclear Science and Technology,
Lanzhou University, Lanzhou 730000, China  F. J. Waso  Stellenbosch
University, Merensky Building, Merriman Street, Stellenbosch, South Africa  X.
F. Yang  School of Physics and State Key Laboratory of Nuclear Physics and
Technology, Peking University, Beijing 100871, China

###### Abstract

This work reports on the application of a novel electric field-ionization
setup for high-resolution laser spectroscopy measurements on bunched fast
atomic beams in a collinear geometry. In combination with multi-step resonant
excitation to Rydberg states using pulsed lasers, the field ionization
technique demonstrates increased sensitivity for isotope separation and
measurement of atomic parameters over non-resonant laser ionization methods.
The setup was tested at the Collinear Resonance Ionization Spectroscopy
experiment at ISOLDE-CERN to perform high-resolution measurements of
transitions in the indium atom from the 5s25d 2D5/2 and 5s25d 2D3/2 states to
5s2($n$)p 2P and 5s2($n$)f 2F Rydberg states, up to a principal quantum number
of $n$ = 72. The extracted Rydberg level energies were used to re-evaluate the
ionization potential of the indium atom to be
$46\,670.1055(21)\text{\,}{\mathrm{cm}}^{-1}$. The nuclear magnetic dipole and
nuclear electric quadrupole hyperfine structure constants and level isotope
shifts of the 5s25d 2D5/2 and 5s25d 2D3/2 states were determined for
113,115In. The results are compared to calculations using relativistic
coupled-cluster theory. A good agreement is found with the ionization
potential and isotope shifts, while disagreement of hyperfine structure
constants indicates an increased importance of electron correlations in these
excited atomic states. With the aim of further increasing the detection
sensitivity for measurements on exotic isotopes, a systematic study of the
field-ionization arrangement implemented in the work was performed and an
improved design was simulated and is presented. The improved design offers
increased background suppression independent of the distance from field
ionization to ion detection.

## Introduction

The ability to separate and study small quantities of isotopes from a large
ensemble without losses is the limiting factor of many experimental studies in
modern nuclear physics [1, 2, 3, 4], as exotic isotopes of interest can often
only be produced at low rates (fewer than 100s of ions per second) and their
accumulation into substantial quantities is prevented by their short half-
lives. Furthermore, sensitive detection or separation of small quantities of
isotopes also has numerous technological applications [5, 6, 7, 8, 9, 10, 11].
Fast beam collinear laser spectroscopy techniques have allowed high-precision
measurements on short-lived isotopes, down to rates of fewer than 100 ions per
second [3, 12, 13]. These approaches use the Doppler compression of an
accelerated atomic beam to enable high-precision laser spectroscopy
measurements to be performed in a collinear geometry [14]. This technique is
now being implemented at radioactive ion beam facilities worldwide, giving a
resolution of a few 10s of MHz, which is sufficient to resolve the hyperfine
structure for nuclear physics studies [15, 16, 17, 18].

Motivated by a need for higher sensitivity to access exotic isotopes produced
at rates lower than a few ions per second, a variation of the technique, the
Collinear Resonance Ionization Spectroscopy (CRIS) [19, 20] experiment at
CERN-ISOLDE [21] has been developed. The technique is based on resonant laser
excitation of atom bunches [22, 23] using a high-resolution pulsed laser,
followed by resonant or non-resonant ionization of the excited atoms. The ions
are then deflected away from the atoms which were not resonantly excited,
allowing ion detection measurements with significantly reduced background. The
experiment has so far reached a background suppression factor of
$2.5\text{\times}{10}^{8}\text{\,}$ providing a detection sensitivity down to
yields of around 20 atoms per second [24]. The main source of ion background
for the technique is due to collisional re-ionization of the atom beam (often
also containing a substantial amount of isobaric contamination) with residual
gas atoms along the collinear laser overlap volume. For this reason,
considerable effort is given to reach ultra-high vacuum pressures
($<$$1\text{\times}{10}^{-9}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$)
in this overlap region. The work reported here demonstrates that the
incorporation of field ionization, previously tested on continuous atom beams
[25, 26], can further increase the sensitivity of measurements on bunched
atomic beams by also compressing the measurements into a narrow ionization
volume. In addition, we show the approach has advantages for measurements of
atomic parameters when combined with the multi-step pulsed narrow-band laser
excitation. The sensitivity of the approach is further improved by removing
the need for a powerful non-resonant laser ionization step, which often
contributes substantially to the re-ionization background. This is due to non-
resonant ionization of contaminant atomic species, which are often neutralised
into excited atomic levels[27], that are easily ionized with powerful laser
light. Using the field-ionization technique, high-resolution measurements were
performed of the energies of the 5s2($n$)p 2P and 5s2($n$)f 2F Rydberg series
(intermittently over principal quantum numbers $n$ = 12 to $n$ = 72), which
additionally enabled the re-evaluation of the ionization potential of the
indium atom. The production of atom bunches was implemented using an ablation
ion source [28] with naturally abundant 113In and 115In isotopes.

In typical measurements using atomic transitions to extract nuclear structure
parameters, both the upper and lower atomic states of the transition have a
nuclear-structure dependence. However due to the vanishing nuclear structure
dependence of the Rydberg states [29, 30], the transition measurements made
here allowed extraction of the nuclear structure dependent parameters of the
lower atomic states alone. These measurements therefore allowed extraction of
the hyperfine structure constants of the 5s25d 2D5/2 and 5s25d 2D3/2 states
and their level isotope shifts (LISs), the shift in the level energies between
the 113,115In isotopes. This allows direct comparison to atomic calculations
of the isotope shift contribution from a single atomic state, in contrast to
typical atomic transitions measurements where upper and lower state
contributions are combined. The experimentally determined LISs, hyperfine
structure constants and indium atom ionization potential are compared to
calculations from relativistic coupled-cluster theory [31]. Comparisons with
direct measurements of atomic states of differing symmetry, such as 2D5/2 and
2D3/2, can give a valuable benchmark to assist the extraction of nuclear
structure observables with higher accuracy [32, 28]. In addition, the
development of accurate calculations for states of different symmetry has the
potential to probe new aspects of the atomic nucleus and its interactions [33,
34, 35, 36, 37].

## Methods

### Experimental setup

The measurements reported here were performed following a modification of the
CRIS experimental setup [38, 39]. A schematic layout of the modified setup is
shown in Figure 1a). In+ ions were produced using an ablation ion source
(detailed in Refs. [28, 38]), with a pulsed 532-nm Litron LPY 601 50-100 PIV
Nd:YAG laser focused to produce a fluence of $>$0.5 J/cm2 on a solid indium
target (99% purity).

Figure 1: a) A schematic diagram of the layout of the modified CRIS
experiment, to use field ionization. Electrostatic deflectors guide the ions
created by ablation to be neutralised. Atoms follow multi-step resonant laser
excitation to Rydberg states, and are subsequently field ionized and counted
by a particle detector. b) The time-of-flight distribution of the bunch
created by the ablation ion source.

This produced high-intensity bunches of indium ions at the 100 Hz repetition
rate of the laser with a typical bunch width of
$2.0(3)\text{\,}\mathrm{\SIUnitSymbolMicro s}$ (see Figure 1b). This ablation
laser was used as the start trigger to synchronise the atomic bunches in the
interaction region with the pulsed lasers subsequently used for spectroscopy.
A set of ion optics were used to focus and accelerate the In+ ions to 25 keV
and deflect them by 34∘ to overlap with two laser beams in a collinear
geometry. The acceleration creates a kinematic separation in the transition
frequency of the naturally abundant isotopes 115In (95.72%) and 113In (4.28%),
which greatly enhances the isotope selectivity of the approach compared to in-
source laser spectroscopy separation or measurement techniques [5] such as
laser induced breakdown spectroscopy (LIBS) [40], resonant ionization mass
spectroscopy (RIMS) [41] or in-gas cell laser ionization spectroscopy (IGLIS)
[42].

The ions were subsequently neutralised, using a sodium-filled charge-exchange
cell heated to 300(10)∘C, with an efficiency of 60(10)%, where 64% of the
atomic population is simulated to be in the 5s25p 2P3/2 metastable state [27].
The ions which were not neutralised were deflected electrostatically following
the cell. After approximately 80 cm of flight (the maximum
$4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ bunch width corresponds to a spatial
spread of 81 cm at 25 keV), the indium atoms were then excited using either
the 5s25p 2P3/2 $\rightarrow$ 5s25d 2D5/2 (325.6 nm) or 5s25p 2P3/2
$\rightarrow$ 5s25d 2D3/2 (325.9 nm) transition, depending on the Rydberg
series to be studied.

Figure 2: The multi-step laser excitation schemes used in this work on the
indium atom. Field ionization was used following the two laser steps to a
Rydberg state. The lifetime of the intermediate 5s25d 2D5/2, 5s25d 2D3/2
states is 7.0(4) ns [43].

The multi-step ionization schemes used in this work are shown in Figure 2. The
first step light was produced using a Spectron Spectrolase 4000 pulsed dye
laser with DCM dye dissolved in ethanol, this produced fundamental light at
650 nm which was frequency doubled using a BaB2O4 crystal to 325 nm. The
linewidth of the laser was $\sim$14 GHz, allowing excitation of all of the
hyperfine structure of the 5s25p 2P${}_{3/2}^{o}$ state, while the 699 GHz
separation between the 5s25d 2D5/2 and 5s25d 2D3/2 states required tuning the
laser to each fine structure transition. The dye laser was pumped with 532-nm
light from the second output head of the Litron LPY 601 50-100 PIV Nd:YAG
laser used for ablation. The second excitation step was scanned in laser
frequency to perform high-resolution laser spectroscopy, from either the 5s25d
2D5/2 or 5s25d 2D3/2 state to a Rydberg state of the 5s2($n$)f or 5s2($n$)p
series, ranging over $n$ = 14-72 (763-723 nm). The high-resolution infrared
laser light was produced using an injection-locked Ti:Sapphire laser [44, 45]
pumped by a LEE LDP-100MQ Nd:YAG laser and seeded using a narrowband
continuous-wave Matisse Ti:Sapphire laser by Spectra-Physics. This provided
the pulsed narrowband (20(5) MHz [45]) laser light to be used for
spectroscopy. The resonantly excited indium Rydberg states were then field
ionized in a longitudinal geometry by thin wire grids with a field gradient of
$7.5\text{\,}\mathrm{kV}\text{\,}{\mathrm{cm}}^{-1}$. Technical details of
this setup are given in the Systematics of the field-ionization setup section.
Spatial alignment of the atom and ion paths was performed using irises and
Faraday cups. The ion beam waist was measured to be around 3(1) mm using an
iris [38], below this a reduction in beam current begin to be observed. This
was measured $\sim$30 cm from the neutralisation cell. Following ionization
the ions were deflected by 20∘ onto a ETP DM291 MagneTOFTM detector and the
recorded count rate was used to produce the hyperfine spectra as a function of
the infared laser frequency.

### Coupled-cluster calculations

In order to compare to our experimental results, the indium atom ionization
potential, hyperfine structure constants A${}_{\text{hf}}$ and
B${}_{\text{hf}}$, and atomic isotope shift factors were calculated using
relativistic coupled-cluster (RCC) theory as outlined below. The
A${}_{\text{hf}}$ and B${}_{\text{hf}}$ constants were calculated using an
expectation-value evaluation approach as described in Ref. [46]. While the
atomic parameters for the isotope shift were calculated using an analytic
response approach (AR-RCC), an approach developed for increased accuracy for
the evaluation of isotope shift contributions, as described in Ref. [32].

In RCC theory, the wave function ($|\Psi_{v}\rangle$) of an atomic state with
a closed-core and a valence orbital $v$ can be expressed as

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

where $|\Phi_{v}\rangle$ is the mean-field wave function, defined as
$|\Phi_{v}=a_{v}^{\dagger}|\Phi_{0}\rangle$, with the Dirac-Hartree-Fock (DHF)
wave function of the closed-core, $|\Phi_{0}\rangle$. Here, $T$ and $S_{v}$
are the RCC excitation operators which incorporate electron correlation
effects by exciting electrons in $|\Phi_{0}\rangle$ and $|\Phi_{v}\rangle$,
respectively, to the virtual space. The amplitudes of the RCC operators and
energies were obtained by solving the following equations

$\displaystyle\langle\Phi_{K}|\left(He^{T}\right)_{c}|\Phi_{0}\rangle$
$\displaystyle=$ $\displaystyle\delta_{K,0}E_{0}\;,$ (2)

and

$\displaystyle\langle\Phi_{L}|\left(He^{T}\right)_{c}S_{v}+\left(He^{T}\right)_{c}|\Phi_{v}\rangle$
$\displaystyle=$
$\displaystyle\left(\delta_{L,0}+\langle\Phi_{L}|S_{v}|\Phi_{v}\rangle\right)E_{v}\;,$
(3)

where $H$ is the atomic Hamiltonian, and $|\Phi_{K}\rangle$ and
$|\Phi_{L}\rangle$ denote the excited determinants with respect to
$|\Phi_{0}\rangle$ and $|\Phi_{v}\rangle$. Here, $E_{0}$ and $E_{v}$
correspond to the energies of the closed-core and the closed-core with the
valence orbital respectively. Thus, the difference between $E_{v}$ and $E_{0}$
gives the binding energy or the negative of the ionization potential (IP) of
the electron from the valence orbital, $v$. The hyperfine structure constants
of the unperturbed state were evaluated by

$\displaystyle\frac{\langle\Psi_{v}|O|\Psi_{v}\rangle}{\langle\Psi_{v}|\Psi_{v}\rangle}$
$\displaystyle=$
$\displaystyle\frac{\langle\Phi_{v}|[1+S_{v}^{\dagger}]e^{T^{\dagger}}Oe^{T}[1+S_{v}]|\Phi_{v}\rangle}{\langle\Phi_{v}|[1+S_{v}{\dagger}]e^{T^{\dagger}}e^{T}[1+S_{v}]|\Phi_{v}\rangle},$
(4)

where $O$ is the hyperfine interaction operator. In the above expression, the
non-terminating series of $e^{T^{\dagger}}Oe^{T}$ and $e^{T^{\dagger}}e^{T}$
in the numerator and denominator, respectively, were calculated by adopting a
self-consistent iterative procedure as described in Ref. [46]. All-possible
singles and doubles excitations were included in the RCC calculations (RCCSD)
for determining the energies and hyperfine structure constants. The
calculations were performed by first considering the Dirac-Coulomb (DC)
Hamiltonian, then including the Breit and lower-order quantum electrodynamics
(QED) interactions as described in Ref. [47]. Corrections due to the Bohr-
Weisskopf (BW) effect to the hyperfine structure constants were estimated by
considering a Fermi-charge distribution of nucleus.

The AR-RCC approach adopted to determine the field shift (FS), normal mass
shift (NMS) and the specific mass shift (SMS) constants was implemented by
evaluating the first order perturbed energies due to the respective operators,
as discussed in Ref. [32]. The AR-RCC theory calculations were also truncated
using the singles and doubles excitation approximation (AR-RCCSD) when used to
calculate the FS, NMS and SMS constants in this work. Contributions from the
DC Hamiltonian and corrections from the Breit and QED interactions were
evaluated explicitly and are shown in Table 2.

## Analysis and results

A summary of the high-resolution Rydberg series measurements is presented in
Figure 3. A range of wavelengths from 720 to 770 nm (14000-12900 cm-1) were
used to cover the transitions to Rydberg states in this work ($n$ = 12-72), as
shown in Figure 3a). The energies of the states in the Rydberg series
(vertical black dashes in Figure 3a)) were estimated using the Rydberg formula
[48] extrapolating from the energies of the five lowest principal quantum
number atomic states ($n$ = 4-9 for ($n$)f, $n$ = 5-10 for ($n$)p), taken from
literature [49]. See the Section Evaluation of the ionization potential of the
indium atom for details. Figures 3b) and 3c) show hyperfine spectra obtained
for the 5s25d 2D5/2 $\rightarrow$ 5s2($n$)f 2F5/2,7/2 and 5s25d 2D3/2
$\rightarrow$ 5s2($n$)f 2F5/2 transitions respectively. The hyperfine
structure resulting from the 5s25d 2D5/2 and 5s25d 2D3/2 states is visible in
these spectra, while the contribution from the Rydberg state in both cases is
vanishingly small due to the reduced overlap of the electronic wavefunctions
at the nucleus [29, 30]. The fine structure splitting between 5s2($n$)f 2F5/2
and 5s2($n$)f 2F7/2 Rydberg states has been measured to be $<$1 MHz [50] and
smaller than the linewidth of the laser used in this work. The upper states of
transitions from 5s25d 2D5/2 are therefore denoted as 5s2($n$)f 2F5/2,7/2 to
indicate that excitation to both the 5s2($n$)f 2F5/2 and 5s2($n$)f 2F7/2
states are included.

Figure 3: Summary of the high-resolution measurements of the transitions to
Rydberg states from the 2P1/2 5s2($n$)p, 2P3/2 5s2($n$)p and 2F5/2 5s2($n$)f
series, showing a) the spectrum of resonances measured in the 770-720 nm
range, with an inset in black indicating energies for the members of the
Rydberg series using Expression 5. Example hyperfine spectra of 115In from the
b) 5s25d 2D5/2 and c) 5s25d 2D3/2 lower states, resolvable from $n$ = 15-34.
Fits to the spectra are indicated in red.

### Hyperfine structure constants and isotope shifts

The extracted magnetic dipole and electric quadrupole hyperfine constants,
A${}_{\text{hf}}$ and B${}_{\text{hf}}$, of the 5s25d 2D5/2 and 5s25d 2D3/2
states for 113,115In are displayed in Table 1. The constants were determined
by least-square minimisation fitting [51] of the obtained hyperfine spectra to
the well known hyperfine structure relations [52] with A${}_{\text{hf}}$ and
B${}_{\text{hf}}$ as free parameters. A Voigt line profile [53] was used in
the fitting with the Gaussian and Lorentzian components and transition
intensities as free parameters.

The presented A${}_{\text{hf}}$ and B${}_{\text{hf}}$ values are an average of
the results from well-resolved hyperfine spectra obtained in this work, over
principal quantum numbers $n$ = 15-34. For Rydberg states below $n\approx$20,
the applied laser power saturated the transition and this led to power
broadening which reduced the resolution of those hyperfine spectra. The
transitions above $n\approx$30 were not saturated (see Section Systematics of
the field-ionization setup), because the transition probability scales
approximately[54] as $n^{-3}$. For these transitions a greater ablation laser
fluence and ion source extraction potential were required to obtain a similar
resonant signal level. This resulted in an increased energy spread of the ion
bunch, increasing the Gaussian contribution to the linewidth to greater than
100 MHz and obscuring the hyperfine spectra in those cases. No statistically
significant deviation was seen for contributions from the Rydberg states to
the hyperfine structure constants down to $n$ = 12. The larger uncertainty of
the extracted B${}_{\text{hf}}$ values was due to their small magnitudes
compared to the laser linewidth of 20(5) MHz. While the larger uncertainty on
the 113In values was due to the lower statistics, related to its lower natural
abundance of 4.28% [55], in combination with a reduction in excitation
efficiency due to the 14-GHz linewidth of the dye laser used for the first
step transition which was centered at a frequency for 115In. The isotope
shifts for the levels (LIS) were also extracted and are displayed in Table 1.
In order to correct for the effect of an ion velocity distribution created in
the ablation process on the measured LIS between 113,115In, the time of
flights of 113,115In gated by the on-resonance laser frequencies were used to
ensure LISs were measured from the same velocity component [38, 56] removing
the Doppler shift from the presence of any distribution of velocities.

|  |  115In |  113In |
---|---|---|---|---
State |  | A${}_{\text{hf}}$ (MHz) | B${}_{\text{hf}}$ (MHz) | A${}_{\text{hf}}$ (MHz) | B ${}_{\text{hf}}$ (MHz) | LIS ‡ (MHz)
5s25d 2D5/2 | Exp. | 151.2(9) | 33(20) | 151(6) | 28(200) |  $-$103(50)
| Theor. |  |  |  |  |
| DHF | 1.87 | 2.53 | 1.86 | 2.50 |
| RCCSD | 39.66 | 24.84 | 39.57 | 24.53 |
|  $+$Breit | 39.80 | 24.84 | 39.71 | 24.53 |
|  $+$QED | 40.00 | 24.99 | 39.91 | 24.68 |
|  $+$BW | 39.87 | 25.01 | 39.78 | 24.70 |  $-$67.3(3.2)†
5s25d 2D3/2 | Exp. |  $-$64(1) | 41(20) |  $-$68(5) | 20(60) |  $-$98(30)
| Theor. |  |  |  |  |
| DHF | 4.37 | 1.83 | 4.36 | 1.81 |
| RCCSD | $-9.89$ | 17.82 | $-9.87$ | 17.60 |
|  $+$Breit | $-9.89$ | 17.81 | $-9.87$ | 17.59 |
|  $+$QED | $-9.87$ | 17.92 | $-9.85$ | 17.70 |
|  $+$BW |  $-$9.74 | 17.98 |  $-$9.72 | 17.72 |  $-$67.7(3.2)†
Table 1:  Hyperfine structure constants and LISs of the 5s25d 2D5/2 and 5s25d
2D3/2 states measured for 113,115In.
$\dagger$ \- Theoretical LISs were calculated using the F, K${}_{\text{SMS}}$
and K${}_{\text{NMS}}$ constants from Table 2 obtained from the AR-RCCSD
approach, combined with the experimentally measured change in root-mean-square
charge radius, $\delta\left\langle r^{2}\right\rangle_{\mu}^{113,115}$ =
-0.157(11) fm2, taken from Ref. [57], which gave the LIS uncertainty for the
‘calculated’ values in the table shown in brackets.
$\ddagger$ \- Here the sign convention of the 113In frequency centroid minus
the 115In frequency centroid was used in the isotope shift formulae [32].

The calculated A${}_{\text{hf}}$ and B${}_{\text{hf}}$ values of the 5s25d
2D3/2 and 5s25d 2D5/2 states using the RCCSD method are presented in Table 1.
Literature nuclear magnetic dipole moment values of $\mu$ = 5.5289 $\mu_{N}$
(Ref. [58]) and nuclear electric quadrupole moment values of $Q$ = 0.80 b
(Ref. [28]) for 113In, and of $\mu$ = 5.5408 $\mu_{N}$ (Ref. [59]) and $Q$ =
0.81 b (Ref. [28]) for 115In were used to evaluate the A${}_{\text{hf}}$ and
B${}_{\text{hf}}$ values in Table 1, from the calculated quantities of
A${}_{\text{hf}}I$/$\mu$ and B${}_{\text{hf}}$/$Q$. The DHF values of
A${}_{\text{hf}}$ were calculated to be 4.37 MHz and 1.87 MHz, whereas the
RCCSD calculations gave $-$9.74 MHz and 39.87 MHz compared to the experimental
values $-$64(2) MHz and 151.2(9) MHz for the 5s25d 2D3/2 and 5s25d 2D5/2
states of 115In, respectively. The B${}_{\text{hf}}$ values are within the
1$\sigma$ uncertainty of the experimental results, although the experimental
uncertainty was large. The difference in the BW correction to the
A${}_{\text{hf}}$ values between 113In and 115In were found to be negligibly
small ($>$0.01 MHz). Contributions from Breit and QED interactions were also
found to be small. This indicates that electron correlations due to core-
polarization effects play the principal role in bringing the results close to
the experimental values. Thus, the experimental A${}_{\text{hf}}$ values
deviate in contrast to the LIS calculations using the AR-RCCSD calculations at
the same level of truncation to singles and doubles excitation. An explanation
demands including triples excitations or employing a more rigorous theoretical
approach for the evaluation of A${}_{\text{hf}}$ factors.

In Table 1, a comparison is also made between calculated LIS values with the
measurements for the 5s25d 2D3/2 and 5s25d 2D5/2 states. The calculated FS
(F), NMS (K${}_{\text{NMS}}$) and SMS (K${}_{\text{SMS}}$) constants, used to
determine the calculated LIS values, are reported in Table 2 along with the
included corrections. For comparison to our relativistic ab-initio
calculations of K${}_{\text{NMS}}$, the K${}_{\text{NMS}}$ constant values
from the non-relativistic approximation are also shown, estimated by the
relation K${}_{\text{NMS}}=E_{i}m_{e}$ and experimental energies [49],
$E_{i}$. Unlike the A${}_{\text{hf}}$ hyperfine structure constants, we find a
good agreement between the measured and theoretical values for the LISs of the
5s25d 2D3/2 and 5s25d 2D5/2 states by substituting the calculated IS
constants.

State | Method | F (MHz/fm2) | K${}_{\text{NMS}}$ (GHz amu) | K${}_{\text{SMS}}$ (GHz amu)
---|---|---|---|---
5s25d 2D5/2 | DHF | $\sim 0.0$ | 277.95 | $-38.48$
| AR-RCCSD | 293.33 | 198.76 | $-55.92$
|  $+$Breit | 293.56 | 198.72 | $-55.41$
|  $+$QED | 288.39 | 198.97 | $-56.41$
| Exp. |  | 226.20777(13) |
5s25d 2D3/2 | DHF | $\sim 0.0$ | 277.78 | $-37.39$
| AR-RCCSD | 295.99 | 198.59 | $-55.69$
|  $+$Breit | 296.19 | 198.54 | $-55.22$
|  $+$QED | 291.00 | 198.88 | $-56.28$
| Exp. |  | 226.59111(13) |
Table 2:  Calculated F, K${}_{\text{SMS}}$ and K${}_{\text{NMS}}$ constants of
the 5s25d 2D5/2 and 5s25d 2D3/2 states for the In atom using DHF and AR-RCCSD
approaches. Experimental level energies for the non-relativistic
K${}_{\text{NMS}}=E_{i}m_{e}$ approximation were taken from Ref. [49]. The
factors were used to calculate the LIS values given in Table 1, using the
expression LIS = F$\delta\left\langle r^{2}\right\rangle_{\mu}^{113,115}$ \+
$\mu$(K${}_{\text{NMS}}$ \+ K${}_{\text{SMS}}$), where $\mu$ =
(m113+m115)/(m113m115) is the mass modification factor using atomic masses
from Ref. [60]

### Rydberg state energies

In order to reduce the systematic error of the measured transition
frequencies, reference scans were performed every few hours using transitions
to the 5s221f 2F5/2 state. When the 5s25p 2P3/2 $\rightarrow$ 5s25d 2D3/2
first step transition was used, this was performed using the 5s25d 2D3/2
$\rightarrow$ 5s221f 2F5/2 transition. While for the 5s25p 2P3/2 $\rightarrow$
5s25d 2D5/2 first step transition, the 5s25d 2D5/2 $\rightarrow$ 5s221f 2F5/2
transition was used. This allowed measurements of the Rydberg series to be
referenced to the same 5s221f 2F5/2 state. The absolute energy of the 5s221f
2F5/2 state was determined for the first time in this work, using an average
of measurements of the 5s25d 2D5/2 $\rightarrow$ 5s221f 2F5/2,7/2 and 5s25d
2D3/2 $\rightarrow$ 5s221f 2F5/2 transition energies, combined with literature
values for the 5s25d 2D5/2 and 5s25d 2D3/2 states, taken from Ref. [61]. This
gave an averaged value of 46420.309(5) cm-1 (1391645845(138) MHz) for the
energy of the 5s221f 2F5/2 state, as presented in Table 3. This was the
largest contribution to the final energy level uncertainty. Other sources of
systematic uncertainty to the absolute energy measured for the 5s221f 2F5/2
state are also presented in Table 3. Where $\sigma(\text{T}_{\text{B}})$ is
the systematic ion source voltage uncertainty, which determines the
uncertainty in the energy of the atom beam, $\text{T}_{\text{B}}$, and
therefore the transition energy through the Doppler shift. And
$\sigma(\lambda)$ is the manufacture quoted absolute accuracy of the
HighFinesse WSU2 wavemeter used. The wavemeter was drift stabilized by
simultaneous measurement of a Toptica DLC DL PRO 780 diode laser locked to the
5s2S1/2 $\rightarrow$ 5p 2P3/2 F = 2 - 3 transition of 87Rb using a saturated
absorption spectroscopy unit [62]. Transitions to the other principal numbers
of the 5s2($n$)f and 5s2($n$)p series were then correlated with the closest
5s221f 2F5/2 reference scans in time to determine the relative centroid shift
of their hyperfine structure. These centroid shifts are presented in Tables 4
and 5 for the series measured using the 5s25p 2P3/2 $\rightarrow$ 5s25d 2D5/2
(325.6 nm) or 5s25p 2P3/2 $\rightarrow$ 5s25d 2D3/2 (325.9 nm) as first step
transitions respectively. The centroid shifts for the 5s2($n$)f and 5s2($n$)p
series were then used to determine their absolute energy levels using the
absolute value for the 5s221f 2F5/2 state, as reported in Table 3.

Transition | 5s221f 2F5/2 (MHz) |  $\sigma$ Lit. [61] (MHz) |  $\sigma(\text{T}_{\text{B}})$ (MHz) |  $\sigma(\lambda)$ (MHz)
---|---|---|---|---
5s25d 2D5/2 $\rightarrow$ 5s221f 2F5/2,7/2 | 1391645920(166) | 150 | 70 | 2
5s25d 2D3/2 $\rightarrow$ 5s221f 2F5/2 | 1391645782(151) | 150 | 19 | 2
Average | 1391645845(138) |  |  |
Table 3:  Determination of the absolute energy level of the 5s221f 2F5/2
reference state. The final value is the weighted mean from two sets of
reference transition measurements, 5s25d 2D5/2 $\rightarrow$ 5s221f 2F5/2,7/2
and 5s25d 2D3/2 $\rightarrow$ 5s221f 2F5/2.
‘Lit.’ refers to the uncertainty on the lower state energy taken from
literature [61].

The energy levels for the members of the 5s2($n$)f 2F5/2 and 5s2($n$)f
2F5/2,7/2 series shown in Tables 5 and 4, and Figure 2 have agreement between
them, using the evaluated energy of the 5s221f2F5/2 reference from Table 3.
The few principal quantum numbers with available values in literature [63],
for the 5s2($n$)f 2F5/2,7/2, 5s2($n$)p 2P1/2 and 5s2($n$)p 2P3/2 states, have
agreement well within uncertainty.

|  |  |  | Literature |
---|---|---|---|---|---
|  | Centroid shift | Energy level |  energy level[49] |
Series | $n$ | (MHz) |  (cm-1) |  (cm-1) | $\delta_{n}$
2F5/2,7/2 | 12 | -15510479(20) | 45902.9328(6)[46] | 45902.92(22) | 0.04010(4)
5s2($n$)f | 13 | -12098644(10) | 46016.7393(5)[46] |  | 0.04028(5)
| 14 | -9393432(60) | 46106.976(2)[5] |  | 0.04052(8)
| 15 | -7212037(60) | 46179.739(2)[5] |  | 0.0407(1)
| 16 | -5427634(80) | 46239.260(3)[5] |  | 0.0408(1)
| 18 | -2710813(20) | 46329.8837(7)[46] |  | 0.0406(1)
| 20 | -769186(6) | 46394.6494(2)[46] |  | 0.0407(2)
| 21 | 0 | 46420.3070(5)[46] |  | 0.0408(2)
| 22 | 666548(10) | 46442.5403(3)[46] |  | 0.0408(2)
| 23 | 1247917(400) | 46461.93(1) |  | 0.0408(9)
| 24 | 1758074(100) | 46478.950(5)[5] |  | 0.0407(6)
| 25 | 2208136(60) | 46493.962(2)[5] |  | 0.0406(5)
| 26 | 2607211(20) | 46507.2739(7)[46] |  | 0.0404(4)
| 28 | 3280668(20) | 46529.7380(7)[46] |  | 0.0403(5)
Table 4:  Energy levels of the 2F5/2,7/2 5s2($n$)f Rydberg series states determined from energy level shifts relative to the 2F5/2,7/2 5s221f reference state, alongside the quantum defects, $\delta_{n}$, of the levels. Statistical uncertainty is indicated in parenthesis. Systematic uncertainty from the reference state is indicated in braces. |  |  |  | Literature |
---|---|---|---|---|---
|  | Centroid shift | Energy level |  energy level [48] |
Series | $n$ | (MHz) |  (cm-1) |  (cm-1) | $\delta_{n}$
2P1/2 | 22 | -1842830(20) | 46358.8365(7)[46] |  | 3.2238(2)
5s2($n$)p | 23 | -922829(20) | 46389.5244(6)[46] |  | 3.2236(2)
| 24 | -132449(20) | 46415.8886(6)[46] | 46415.871(26) | 3.2235(2)
| 25 | 551531(5) | 46438.7037(2)[46] |  | 3.2233(2)
| 26 | 1147394(6) | 46458.5796(2)[46] |  | 3.2231(2)
| 28 | 2129967(60) | 46491.355(2)[5] | 46491.338(26) | 3.2229(2)
| 30 | 2900676(60) | 46517.063(2)[5] | 46517.043(26) | 3.2226(2)
2P3/2 | 22 | -1816214(20) | 46359.7243(6)[46] |  | 3.1969(2)
5s2($n$)p | 24 | -112789(7) | 46416.5444(2)[46] | 46416.526(26) | 3.1966(2)
2F5/2 | 16 | -5427114(30) | 46239.278(1)[46] |  | 0.0404(1)
5s2($n$)f | 18 | -2711018(10) | 46329.8768(4)[46] |  | 0.0408(1)
| 19 | -1663438(5) | 46364.8203(2)[46] |  | 0.0409(1)
| 20 | -769307(10) | 46394.6453(4)[46] |  | 0.0409(2)
| 21 | 0 | 46420.3070(5)[46] |  | 0.0408(2)
| 22 | 666346(5) | 46442.5336(2)[46] |  | 0.0411(2)
| 23 | 1247803(10) | 46461.9289(5)[46] |  | 0.0410(3)
| 24 | 1757899(8) | 46478.9439(3)[46] |  | 0.0410(3)
| 25 | 2208075(100) | 46493.960(5)[5] |  | 0.0407(7)
| 26 | 2607218(100) | 46507.274(5)[5] |  | 0.0404(7)
| 27 | 2962711(20) | 46519.1321(7)[46] |  | 0.0402(5)
| 28 | 3280581(4000) | 46529.7(1) |  | 0.04(1)
| 30 | 3823828(9) | 46547.8558(3)[46] |  | 0.0401(6)
| 32 | 4268023(10) | 46562.6726(4)[46] |  | 0.0410(7)
| 33 | 4460517(30) | 46569.093(1)[5] |  | 0.0409(9)
| 34 | 4636375(60) | 46574.959(2)[5] |  | 0.040(1)
| 38 | 5205941(60) | 46593.958(2)[5] |  | 0.040(2)
| 40 | 5428856(6) | 46601.3938(2)[46] |  | 0.039(1)
| 45 | 5861534(20) | 46615.8264(8)[46] |  | 0.039(2)
| 50 | 6170831(20) | 46626.1434(6)[46] |  | 0.042(3)
| 53 | 6316157(200) | 46630.991(5)[5] |  | 0.037(7)
| 54 | 6359037(200) | 46632.421(7)[5] |  | 0.042(8)
| 55 | 6399878(300) | 46633.78(1) |  | 0.04(1)
| 57 | 6475198(50) | 46636.296(2)[5] |  | 0.034(5)
| 60 | 6574308(600) | 46639.60(2) |  | 0.03(2)
| 61 | 6603688(70) | 46640.582(2)[5] |  | 0.040(7)
| 62 | 6631893(20) | 46641.5228(7)[46] |  | 0.046(6)
| 65 | 6709179(30) | 46644.101(1)[5] |  | 0.048(7)
| 68 | 6776687(200) | 46646.353(5)[5] |  | 0.04(1)
| 69 | 6796960(60) | 46647.029(2)[5] |  | 0.05(1)
| 70 | 6816857(100) | 46647.693(4)[5] |  | 0.04(1)
| 72 | 6853899(300) | 46648.928(9)[5] |  | 0.03(2)
Table 5:  Energy levels of the 2P1/2 5s2($n$)p, 2P3/2 5s2($n$)p and 2F5/2
5s2($n$)f Rydberg series states determined from energy level shifts relative
to the 2F5/2 5s221f reference state, alongside the quantum defects,
$\delta_{n}$, of the levels. Statistical uncertainty is indicated in
parenthesis. Systematic uncertainty from the reference state is indicated in
braces.

### Evaluation of the ionization potential of the indium atom

The energy levels of the 2P1/2 5s2($n$)p, 2P3/2 5s2($n$)p, 2F5/2 5s2($n$)f and
2F5/2,7/2 5s2($n$)f Rydberg series states determined in this work are shown in
Figure 2 in comparison to the accepted literature ionization potential (IP) of
the indium atom [55]. The energies of the the $n^{\text{th}}$ Rydberg series
states, En, can be determined using the Rydberg expression [64, 65]

$E_{n}=\text{IP}+\frac{R_{\text{115In}}}{(n-\delta_{n})^{2}}\;,$ (5)

where $\delta_{n}$ is the quantum defect [66], a measure of the difference in
electronic structure for the Rydberg series of a multi-electron atom compared
to hydrogen, included as the effective principal quantum number
$n^{*}=$n$-\delta_{n}$. The effect due to the finite mass of 115In compared to
the electron, is given by the Rydberg constant $R_{\text{115In}}$ [65] of
$109\,736.79\text{\,}{\mathrm{cm}}^{-1}$, which was derived from Penning trap
atomic nuclei mass measurements [67, 68]. Expression 5 can be fitted to the
experimental energy levels, leaving the IP and $\delta_{n}$ as free
parameters. The result of this is shown by the black lines in Figure 4a).
Expression 5 was fitted to lower-lying $n$ states of the series [49] to
predict laser frequency scan ranges and give $n$ assignments.

Figure 4:  a) Extracted energies of the 5s2($n$)p 2P1/2, 5s2($n$)p 2P3/2,
5s2($n$)f 2F5/2 and 5s2($n$)f 2F5/2,7/2 Rydberg series measurements.
Literature values, labeled as ‘lit.’, were taken from Ref. [48]. The black
lines indicates the Rydberg expression values for the 2F5/2,7/2 5s2($n$)f
(solid line) and 2P1/2,3/2 5s2($n$)p series (dashed line). The ionization
potential is indicated by the dashed grey line. b) Determination of the
ionization potential of the indium atom using a global simultaneous fit using
Expressions 5 and 6 to $\delta_{n}$ values of the 5s2($n$)f 2F5/2, 2P1/2,3/2
5s2($n$)p, 2P1/2,3/2 5s2($n$)p and 2P1/2,3/2 5s2($n$)p of this work, along
with literate (‘lit’) values for the 2P1/2,3/2 5s2($n$)p series [48] and 2S1/2
5s2($n$)s [69] series. The red markers indicate series $\delta_{n}$ values
after the global fitting (IP = 46670.1055(21) cm-1). The IP value determined
from the fit to each series independently are shown in their corresponding
subplots. The black lines indicate series $\delta_{n}$ values determined by
Expression 6 from lowest-lying states [49] ($n$ = 4-10 for ($n$)f, $n$ = 5-10
for ($n$)p and $n$ = 6-10 for ($n$)s)

The $\delta_{n}$ values evaluated from our experimental energies are shown in
Figure 4b) alongside the Ritz expansion values (black lines) from [70]

$\delta_{n}=a_{0}+\frac{a_{1}}{(n-\delta_{0})^{2}}+\frac{a_{2}}{(n-\delta_{0})^{4}}+\frac{a_{3}}{(n-\delta_{0})^{6}}\;\;,$
(6)

where $a_{0,1,2,3}$ and $\delta_{0}$ are parameters fitted to measurements of
energies of lower-lying states for each series from literature [71, 72, 73].
This gave the behaviour of the $\delta_{n}$ values for increasing $n$. The
$a_{0,1,2,3}$ and $\delta_{0}$ values obtained for these series are given in
Table 6. The value for the IP can be determined by using it as a free
parameter to fit the $\delta_{n}$ values to this expression. The importance of
measuring Rydberg states over a wide range of $n$, not just for high-lying $n$
states, for fitting the IP with the $\delta_{n}$ values is clearly seen in
Figure 4 b), as the uncertainty in $\delta_{n}$ scales as ${n^{*}}^{-3}$.
Furthermore, deviations in the experimental $\delta_{n}$ from that expected by
Expression 6 can be used to identify deficiencies in the energy level
measurements being used to determine the IP, due to a susceptibility to stray
electric fields (in principle avoided by separation of the field ionization
from the laser excitation step in a collinear setup) or perturbing
configurations lying above the IP. Large perturbations were found in the
5s2($n$)d 2D series from the 2D term of the 5s5p2 configuration [69]. No
statistically significant deviation outside of the values of Expression 6 was
observed within the accuracy of the 5s2($n$)f 2F5/2, 5s2($n$)f 2F5/2,7/2,
5s2($n$)p 2P1/2, 5s2($n$)p 2P3/2 series measurements performed in this work.

The $\delta_{n}$ values were fitted to Expression 6 for each Rydberg series to
obtain the value for the IP independently for each. The resulting IP values
are presented in Table 6 and in the sub plots of Figure 4b). The IP values
extracted from the series measurements and from literature are in good
agreement. The $\delta_{n}$ obtained from the 5s2($n$)p 2P1/2,3/2 and
5s2($n$)s 2S1/2 series taken from literature [48, 69] are also shown in Figure
4b) and the resulting IP values in Table 6.

As the value of the IP is a common parameter for all of the series, a global
simultaneous fit with the IP as a free parameter was performed using the
$\delta_{n}$ values from the series measurements in this work in addition to
literature, taking into the account the error of the individual $\delta_{n}$
values and reducing possible sources of systematic error. This yielded a
combined value for the IP of $46\,670.1055(21)\text{\,}{\mathrm{cm}}^{-1}$, an
improvement over the previous highest precision literature value for the IP of
$46\,670.106(6)\text{\,}{\mathrm{cm}}^{-1}$ (Ref. [48] from the 5s2($n$)p
2P1/2,3/2 series.

The difference of 0.2% of the theoretical IP from the experimental value is
well within that expected under the RCSSD approximation [74], in contrast to
difference by a factor of 5 observed in the $A_{\text{hf}}$ constants. This
further highlights the difference in electron correlation trends for the
calculation of hyperfine structure constants, in contrast to energies, for the
same RCSSD level of approximation.

Series | IP (cm-1) | $\delta_{0}$ | $a_{0}$ | $a_{1}$ | $a_{2}$ | $a_{3}$
---|---|---|---|---|---|---
Theory |  |  |  |  |  |
DHF | 41507.11 |  |  |  |  |
RCCSD | 46762.85 |  |  |  |  |
$+$Breit | 46725.95 |  |  |  |  |
$+$QED | 46763.57 |  |  |  |  |
Literature |  |  |  |  |  |
($n$)p 2P1/2,3/2, Ref.[48] | 46670.106(6) |  |  |  |  |
($n$)s 2S1/2, Ref.[69] | 46670.107(10) |  |  |  |  |
($n$)f 2F5/2,7/2, Ref.[72], 7$\leq n\leq$10 | 46670.110(50) |  |  |  |  |
This work |  |  |  |  |  |
($n$)f 2F5/2 | 46670.110(4) | 0.04 | 0.041 | -0.153 | 0.714 | -2.296
($n$)f 2F5/2,7/2 | 46670.102(6) | 0.04 | 0.041 | -0.153 | 0.714 | -2.296
($n$)p 2P3/2 | 46670.105(4) | 3.22 | 3.196 | 0.382 | 0.125 | 3.1454
($n$)p 2P1/2 | 46670.106(7) | 3.25 | 3.223 | 0.380 | 0.112 | 3.1258
Global fit |  |  |  |  |  |
(This work & | 46670.1055(21) |  |  |  |  |
Literature) |  |  |  |  |  |
Table 6: Values for the ionization potential of the indium atom evaluated by
fitting the the quantum defects of the 5s2($n$)f 2F5/2 and 5s2($n$)p 2P3/2,1/2
Rydberg series, fitting to the series separately and simultaneously (‘sim.
fit.’). The determined $a_{0},1,2,3$ and $\delta_{0}$ parameters for
Expression 6 are presented.

## Systematics of the field-ionization setup

The field ionization of the Rydberg states in this work was performed using
the electrode arrangement shown in Figure 5, located in the position indicated
in Figure 1. Three consecutive grids of parallel gold wires, of
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness and with
$1\text{\,}\mathrm{mm}$ separation between wires were used as electrodes to
create the field for ionization, as shown in Figures 5a), and 5b). The outmost
grids were used to provide ground shielding. The wire grids were mounted on a
printed circuit board and spaced $4\text{\,}\mathrm{mm}$ apart, resulting in
an average electric field gradient of
$7.5\text{\,}\mathrm{kV}\text{\,}{\mathrm{cm}}^{-1}$ for the 3 kV potential
applied to the innermost grids. The arrangement included two parallel
electrostatic deflector plates with opposing polarity before the grids, in
order to deflect background ions. These background ions originate from non-
resonant processes in the preceding $120\text{\,}\mathrm{cm}$ flight path
between the charge exchange cell and the field-ionization grids, which is
referred to as the ‘laser-atom interaction region’. The region surrounding the
grids is further called the ‘field-ionization region’, and the region between
the last field-ionization grid and the ion detector will be referred to as the
‘post-ionization region’. The regions are also indicated in Figure 1.

Figure 5: a) A schematic of the electrode arrangement used in this work b) The
electric potential of the field-ionization arrangement used for the
measurements of this work, and c) the accompanying electric field gradient,
simulated from measurements of voltages applied during this work.

This longitudinal electric field-ionization arrangement was chosen so the
closely spaced grids could provide a small ionization volume of
$0.4\text{\,}{\mathrm{cm}}^{3}$ with a very well defined electric field
gradient. This small volume has to be compared to the
$120\text{\,}{\mathrm{cm}}^{3}$ volume where non-resonant laser ionization
would normally take place. This is a reduction by a factor of 300 in volume in
which collisional ionization of the atom beam can occur with residual gases.
This is a substantial source of background which can be removed when field
ionization is used. An additional advantage of this setup is that the grids
approximate a plane geometry for the electric field, removing the dependence
on transverse displacement of the atom beam on the field gradient experienced
by the atoms. This can be seen in Figure 5c) where very difference in electric
field gradient is observed on a $1\text{\,}\mathrm{mm}$ level, outside of the
grid regions, for an offset of $5\text{\,}\mathrm{mm}$ from the beam axis (to
simulate an incident atom beam of this width), compared to traditional tube
field-ionization geometries where large differences in electric field gradient
can be found transverse to the beam axis [26, 75]. The electrostatic
simulations were performed using COMSOL Multiphysics ® [76].

The spread in the position where the Rydberg atoms are ionized is determined
by the ionization probability for the Rydberg atom in the electric field
gradient created by the electric potential. Therefore the ionization
probability in a given electric field gradient ultimately determines the
spread in electric potential the ions are produced and the energy spread of
the ion beam as it exits the ‘field-ionization region’. The situation of the
Rydberg atom bunch encountering a step increase in electric field as they
travel into ‘field-ionization region’ is equivalent to the application of a
pulsed electric field to the Rydberg atoms at rest, which has been studied
more extensively [77, 78, 79]. In the adiabatic limit where the classical
electron motion is fast compared to the electric field pulse, the field
necessary to reach saturation of ionization for the ensemble of Rydberg atoms
(to ionize all Rydberg states above a given $n$ within the pulse duration) is
calculated [78] to be $E_{n}-\text{IP}=-5.97F^{1/2}$ V/cm, corresponding to a
field gradient of

$F_{sat}=3.38\times\frac{10^{8}}{n^{*4}}\;\text{V/cm},$ (7)

using the parameters for an indium Rydberg atom. This is similar to the
commonly used estimate for the critical field ionization strength in the case
of a static electric field [75, 80] of $F_{crit}\approx 3\times
10^{8}/n^{*4}\;\text{V/cm}$. The classical Kepler period [79] of

$\tau_{K}=2\pi\frac{m_{e}a_{0}^{2}}{\bar{h}}n^{*3}\;\;,$ (8)

for an electron in the 5s270f 2F5/2 state is
$\tau_{K}$=$52\text{\,}\mathrm{ps}$, where $a_{0}$ is the Bohr radius and
$m_{e}$ the electron mass. This can be used to estimate the cutoff for the
adiabatic limit. The distance within which Rydberg atoms can be assumed to be
ionized applying the electric field gradients according to Expression 7, is
then given by $l_{sat}=\tau_{K}\nu_{B}$, for an atom beam of velocity
$\nu_{B}$. In the case of atomic 115In at $T_{\text{B}}$ =
$25\text{\,}\mathrm{keV}$ this corresponds to
$10.65\text{\,}\mathrm{\SIUnitSymbolMicro m}$. This results in a minimum
energy spread of $\Delta E$ = $8\text{\,}\mathrm{eV}$ for the electric
gradient of $F$=$7.5\text{\,}\mathrm{kV}\text{\,}{\mathrm{cm}}^{-1}$ used in
this work ($\Delta E=l_{sat}F$). The corresponding time-of-flight broadening
for this minimum energy spread is well below the
$4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ ion bunch width from the ablation
ion source and was not resolvable in this work. In order to go below this
intrinsic energy spread, higher electric field gradients would be required to
ensure ionization in a short distance, although scaling as $F\propto n^{*-2}$
appears in the sub-ps regime [79].

Figure 6: a) The count rate on resonance for a typical non-resonant
photoionization last step versus the field-ionization scheme corresponding to
their total ionization efficiencies, b) fitting the saturation curve [81] of
the 5s25d 2D3/2 $\rightarrow$ 5s226f 2F5/2,7/2 transition, giving a saturation
fluence of 293(140) $\mu$J/cm2 and c) background rates with and without
electrostatic deflectors to estimate background in the ‘interaction’ and
‘post-ionization’ region.

Figure 6a) shows the detected ion-count rate of a measurement performed using
the 5s25p 2P3/2 $\rightarrow$ 5s25d 2D5/2 (325.6 nm) first step followed by
non-resonant ionization using 532-nm light produced by a Litron TRLi HR
120-200 Nd:YAG laser. This was recorded in order to make a comparison with the
total ionization efficiency of the field-ionization setup, using the same
first step followed by the 5s25d 2D3/2 $\rightarrow$ 5s226f 2F5/2,7/2
transition and field ionization. The measurements were performed less than a
minute apart following optimization of the overlap of the 735-nm, 532-nm and
325-nm light with the neutral atoms, aligned using two irises before and after
the interaction region of the beamline (as indicated in Figure 1). The laser
pulses were overlapped in time using a photodiode at the laser exit window of
the beamline. A maximum output pulse fluence of 55 mJ/cm2 was used for the
532-nm step, with no discernible decrease in count rate observed down to 44
mJ/cm2. Meanwhile the 5s25d 2D3/2 $\rightarrow$ 5s226f 2F5/2,7/2 transition
appeared to not be saturated as indicated in Figure 6b), with a maximum of
250(20) $\mu$J/cm2 available and the estimated saturation fluence of 293(140)
$\mu$J/cm2. A scatter of around 10% in beam intensity was due to shot-to-shot
variations intrinsic to the ablation ion source setup used [82, 28]. While
this makes an exhaustive study difficult, this underlines two issues under
these typical measurement conditions: i) The efficiency of a non-resonant
laser ionization scheme is typically lower than a field-ionization setup [83].
This can be due to the non-linear nature of non-resonant photoabsorption into
the continuum [83], reduced efficiency of collecting and detecting ions from a
larger volume or re-neutralisation of the ions (at pressures of around
$1\text{\times}{10}^{-9}\text{\,}\mathrm{mbar}$ this can still give an
appreciable contribution [27]). ii) A larger laser fluence is required in
order to saturate transitions to high $n$ Rydberg states such as the 5s25d
2D3/2 $\rightarrow$ 5s226f 2F5/2,7/2 transition, as the transition strength
decreases with $n$ [84]. An appropriate $n$ Rydberg state has to be chosen to
ensure saturation of the transition in addition to a electric field gradient
to ensure ionization according to Expression 7. This is an additional
validation of the fact that the technique lends itself well to use on bunched
atomic beams, where high laser fluence pulsed lasers can be used.

In order to study the factor of reduction in collisional background using the
field-ionization setup shown in Figure 5, measurements were performed at
pressures raised by a factor of 10 compared to the nominal operating level,
increasing the signal for the collisional background rate. The pressure in the
‘interaction’ region of length $l_{1}=$$120\text{\,}\mathrm{cm}$ was raised to
$5\text{\times}{10}^{-9}\text{\,}\mathrm{mbar}$
($\rho_{1}=$$1.2\text{\times}{10}^{14}\text{\,}{\mathrm{m}}^{-3}$), and the
‘post-ionization’ region of length $l_{2}=$$30\text{\,}\mathrm{cm}$ was raised
to $5\text{\times}{10}^{-8}\text{\,}\mathrm{mbar}$
($\rho_{2}=$$1.2\text{\times}{10}^{15}\text{\,}{\mathrm{m}}^{-3}$).

For a measured neutral beam current of
$I_{B}=$$6.0(1)\text{\times}{10}^{6}\text{\,}\mathrm{a}$toms/s, and a
collisional ion beam current, IC, the cross section for collisional ionization
can be defined as

$\sigma=\frac{I_{C}}{I_{B}\rho l}\;\;\text{cm}^{2}\;.$ (9)

As both regions will have the same value of $\sigma$, measurements of the
collisional ion currents can be used as a consistency check for the reduction
in ionization volume using the known atom path lengths, $l$, and residual gas
densities, $\rho$. The remaining background ion current from applying $\pm$2
kV electrostatic deflectors in the ‘field-ionization’ region gave the
collisional ion current for the ‘post-ionization’ region, while applying the
ground potential gave the background ion current from both ‘interaction’ plus
‘post-ionization’ regions. The measured ion currents were I${}_{C}^{l2}$ =
35(5) ions/s and I${}_{C}^{l1+l2}$ = 55(5) ions/s, respectively, as shown in
Figure 6c). From these measurements the collisional ionization cross sections
for the indium atom incident upon residual gas atoms at 25 keV were determined
to be $\sigma_{1}$ = $2.3(8)\text{\times}{10}^{-16}\text{\,}{\mathrm{cm}}^{2}$
and $\sigma_{2}=$$1.62(23)\text{\times}{10}^{-16}\text{\,}{\mathrm{cm}}^{2}$
for the ‘interaction’ and ‘post-ionization’ regions, respectively. The larger
error of $\sigma_{1}$ results from taking the difference between
I${}_{C}^{l2}$ and I${}_{C}^{l1+l2}$ to determine I${}_{C}^{l1}$. The cross
sections are in agreement and are of the expected order of magnitude at a beam
energy of 25 keV [85]. This demonstrates a consistency for a factor of five in
length (and volume assuming a homogeneous beam diameter) reduction (from 150
cm to 30 cm, as indicated in Figure 1) for the source of collisional
background ions. In addition this shows that the largest source of remaining
atom-beam related background is due to ions created by collisional ionization
with residual gases in the ‘post-ionization’ region, which are not able to be
removed by the electrostatic deflectors in the ‘field-ionization’ region. The
background suppression of the design in this work is therefore limited by the
length of the ‘post-ionization region’ and the vacuum pressure in that region.

The simulated electric field gradient of Figure 5c) highlights an additional
consideration when using parallel wires for field ionization. The
approximation of a planar electric potential breaks down as the wires are
approached and inhomogenities in the penetrating field create a large spike in
the experienced electric field gradient. This property is in fact useful for
defining the point of ionization and reducing the ion energy spread, however
the potential geometry of Figure 5b) creates three positions where the
electric field gradient is greatest and approximately equal in magnitude. It
is therefore crucial for the critical field for ionization saturation to be
applied to avoid ionization across more than one position which would result
in a maximum energy spread of the magnitude of the potential applied.

### An improved field-ionization setup

Although the design used in this work effectively removes background from
collisional ions created in the ‘interaction’ region before the ‘field-
ionization’ region, the remaining background from ions created in the ‘post-
ionization region’ can still be substantial.

With this consideration, an improved design has been developed to detect only
those ions created inside the small volume of the field-ionization grids and
is presented in Figure 7. The principle of the design is to create an energy
shift for the ions created in the ‘field-ionization’ region. This introduces
energy selectivity for the Rydberg states ionized in the ‘field-ionization’
region, distinguishing them from other background sources of ions which will
remain at the initial beam energy. Compared to the design used for the
measurements in this work, this removes the demand for a short ‘post-
ionization region’ with the best possible vacuum conditions.

In this improved design, the opposite polarity deflector plates (Figure 5a,b)
) are replaced by segmented flat electrodes [75] of the same polarity, but
with a potential difference of around 500 V between them to provide the
equivalent electrostatic deflection of background ions created before the
field-ionization grids (Figure 8a,b) ). These electrodes are labelled as
“segmeneted electrostatic deflectors” in Figure 7 a). Electrostatic lenses are
also included following the field-ionization grids, labelled as “acceleration
lenses” in Figure 7 a). These additional ion optic elements were designed with
simple planar geometries to be compatible with fabrication using metal traces
on printed circuit boards [86].

Figure 7: a) A schematic of the improved electrode arrangement simulated from
this work b) The electric potential of an optimised field-ionization setup,
and b) the accompanying electric field gradient, simulated using voltages
compatible with a 25 keV beam.

This design allows the outer grids to be held at a higher and adjustable
electric potential without compromising the advantages of the previous field-
ionization arrangement. The segmented electrodes allow the Rydberg atoms to
enter a high potential without an abrupt increase in electric field gradient
causing ionization. The removal of the opposite polarity deflector plates
allows the potential of the first grid to be raised without introducing a
large asymmetry in the electric potential or a large electric field gradient
transverse to the atom beam axis. In addition, this creates a well defined
electric field gradient without the need for outer grounding grids. The
principle of this arrangement is to reduce the electric field gradient between
the first and second grids, moving the step to high electric field gradient to
the middle grid instead. The step can be made greater by applying an opposite
polarity potential to the last grid, as shown in Figure 7. This localizes the
field-ionization region to a raised potential, resulting in an increase in
beam energy of Rydberg states which are ionized in this potential. The
‘acceleration lenses’ following the field-ionization grids, are then used for
extraction from the raised potential.

The resulting ion trajectories from this ionization arrangement are shown in
Figure 8a), where the increase in beam energy of ions created inside the
field-ionization region is indicated. The ions with the beam energy of
interest can be selectively detected following electrostatic deflection,
because the deflection introduces an angular separation of ions with different
energies, as seen in Figure 8a). Using slits (or a position sensitive
detector) to select ions of a given beam energy, allows the ions created by
field ionization to be distinguished from any other background source of ions
created in the ‘interaction’ or ‘post-ionization’ regions. This not only
includes collisional ions, but ions created by field ionization of
collisionally excited or re-neutralised atoms in the field of the
$20\text{\,}\mathrm{\SIUnitSymbolDegree}$ bend to the ion detector,
photoionization or molecular breakup, as all of these sources of background
ions will remain at the lower beam energy. Alternatively the beam energy could
be measured directly [87], or the difference in detected time of flight of the
ions could be used as a gate if the bunch width was sufficiently narrow. For
example, the time of flight separation introduced in the ‘post-ionization’
region for the ions travelling at 25 keV is around 15 ns (Figure 8b), so a
bunch window narrower than this would be needed. The incident temporal atom
bunch width of $4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ in this work would
prevent this. The beam energy difference for ions created by field ionization
could be enhanced by using lower incident beam energies or higher potential
for the ionization apparatus, however the design of the ion optics then
becomes more critical to avoid ion transmission loses.

The improved field-ionization design outlined here therefore offers improved
background suppression over the design used for measurements in this work, by
providing selectivity of ions created by field ionization independent of the
length and vacuum quality of the ‘post-ionization’ region. In general, the
background suppression factor for the improved field-ionization design
compared to non-resonant laser ionization can be expressed as

$S_{BG}=\frac{L}{l_{ion}}\frac{P_{L}}{P_{l}}\;\;.$ (10)

where $L$ is the path length of the ‘interaction’ plus ’post-ionization’
regions, $l_{ion}$ is the path length in which ionization can take place
inside the ‘field-ionization’ region, and $P_{L}/P_{l}$ is the ratio vacuum
pressure in the two regions. This is under the approximation of a homogeneous
gas composition in the regions and a uniform atomic beam diameter. The energy
selectivity offers the prospect of a reduction in ionization volume by a
factor of $1.6\text{\times}{10}^{5}\text{\,}\mathrm{,}$ down from a region of
length $L$ = $150\text{\,}\mathrm{cm}$ to the $l_{ion}$ =
$10.65\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for the adiabatic cut-off
assumed in the field-ionization model of Expression 7, where $l_{ion}$ =
$l_{sat}$ = $\tau_{K}\nu_{B}$. However the electrostatic bend used in the CRIS
experimental setup, combined with adjustable slits to select an ion path
incident on the detector has an energy resolution of around $\sigma_{E}$ =
$1.5\text{\,}\mathrm{keV}$, which can only guarantee a selectivity of the
ionization volume down to $l_{ion}$ = $F$/$\sigma_{E}$. For the value of $F$ =
$7.5\text{\,}\mathrm{kV}\text{\,}{\mathrm{cm}}^{-1}$ used in this work, this
corresponds to a volume reduction by a factor of
$1.25\text{\times}{10}^{3}\text{\,}$. Below this limit, direct beam energy
measurement, or ion time-of-flight measurement using ion bunches narrower than
15 ns would be necessary to determine the actual energy spread and confirm the
precise background suppression factor.

When combined with extreme-high vacuum technologies [88] to improve the vacuum
quality in the field-ionization region (increase the $P_{L}/P_{l}$ ratio),
this technique has the potential to reduce the dominating collisional
background ion contribution to a vanishingly low level when compared to other
sources of background, such as non-resonant ionization from the lower pulse
energy resonant step laser light, the dark count rate of the detector
($\sim$0.08 cps for an ETP DM291 MagneTOF), or residual radioactivity in the
setup.

Figure 8: a) The simulated ion trajectories using the field-ionization
arrangement shown in Figure 7 and b) the corresponding simulated time-of-
flight difference for ions from field ionization versus background.

## Conclusion

The use of ion cooling and bunching has allowed the most sensitive
measurements of exotic atoms and molecules containing short-lived isotopes to
date [3, 1], by concentrating measurements on ion bunches into a narrow time
window, in order to improve background suppression and additionally allowing a
high duty cycle for high-resolution and high-detection efficiency pulsed laser
ionization spectroscopy [39, 38, 19].

In this work we have implemented field ionization with the Collinear Resonance
Ionization Spectroscopy (CRIS) technique, to further increase the selectivity
(and thus sensitivity) of high-resolution measurements of hyperfine spectra of
isotopes in atom bunches. This allows the ionization to take place in a narrow
spatial window in addition to the narrow time window, substantially reducing
background due to collisional ions created alongside the atoms of interest in
larger ionization volumes. Here we have demonstrated a factor of five in
ionization volume reduction and corresponding background suppression, when
accounting for vacuum pressure. In principle this will allow measurements of
exotic isotopes with yields down to 4 atoms per second at the CRIS experiment.
However, a further factor of $>$400 improvement in background suppression of
collisional ionization shown to be possible with an improved design, which
also makes background suppression independent of distance from field
ionization to ion detection by incorporating an increase in beam energy of the
field-ionized Rydberg atoms. Furthermore, as a non-resonant pulsed laser step
is no longer necessary to ionize the atom bunches, this removes a significant
source of photo-ionization background, in addition to removing a source of AC
Stark shifts in measurements from short-lived metastable states [62].

By using bunched atomic beams the technique is well suited to the use of
narrow-band pulsed lasers, taking advantage of the high spectral density to
saturate transitions to high-lying Rydberg states required for field
ionization. The 5s2($n$)p 2P and 5s2($n$)f 2F Rydberg series states in the
indium atom up to $n$ = 72 were studied and used to evaluate the ionization
potential of the indium atom to be
$46\,670.1055(21)\text{\,}{\mathrm{cm}}^{-1}$, in agreement with, and
improving upon the precision of previous measurements. Furthermore, the
technique allows high resolution measurements of the hyperfine structure
constants and isotope shifts of individual atomic states directly.

The nuclear magnetic dipole, nuclear electric quadrupole hyperfine structure
parameters and isotope shifts of the 113In and 115In isotopes, for the 5s25d
2D5/2 and 5s25d 2D3/2 states were measured. The experimental results were
compared to DHF, RCCSD and AR-RCCSD calculations, where a good level of
agreement was found with experimental isotope shifts and the ionization
potential of the indium atom. While the RCCSD calculations showed an
improvement over DHF calculations for the A${}_{\text{hf}}$ constants, giving
the correct signs, the magnitudes were underestimated, indicating that
electron correlations play a crucial role in the hyperfine stucture constants
of these 5s25d 2D5/2 and 5s25d 2D3/2 states, demanding further theoretical
study.

Improvements in highly sensitive detection techniques compatible with precise
laser spectroscopy are required to measure the nuclear structure of the most
exotic nuclei produced at radioactive beam facilities, important for
developing nuclear theories [89, 90, 91], in addition to paving the way for
measurement of new observables from atomic nuclei when combined with precise
calculations of the parameters of atomic states and their symmetries [35, 37,
33, 92]. Furthermore, high sensitivity ionization of isotopes has many
potential applications, such as the separation of nuclear waste [5],
enrichment of nuclear fuel [11], collection of nuclear isomers [9],
“ultra”-trace analysis [7], research of nuclear-spin-dependent effects [8, 10]
and highly-purified nuclear decay spectroscopy [6].

## Acknowledgements

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; the FWO-Vlaanderen (Belgium);
National Key R&D Program of China (Contract No: 2018YFA0404403), the National
Natural Science Foundation of China (No:11875073). We would also like to thank
A.J. Smith and The University of Manchester workshop for their support. B.K.S.
acknowledges use of the Vikram-100 HPC cluster of Physical Research
Laboratory, Ahmedabad for carrying out atomic calculations.

## Author contributions statement

A.R.V and K.T.F. conceived the experiment(s). A.R.V, C.M.R, B.S.C., F.P.G,
Q.W, R.F.G., H.A.P., F.J.W conducted the experiment(s). A.R.V and C.M.R
analysed the results. A.R.V and B.K.S. prepared the manuscript. B.K.S
performed the coupled cluster method atomic calculations. J.B., T.E.C., G.N.,
R.F.G., K.T.F and X.Y. provided advice, supervision and/or equipment. All
authors reviewed the manuscript.

## Competing interests

The authors declare no competing interests.

## Additional information

…

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