# Nuclear charge radii of germanium isotopes around $N$ = 40

S. J. Wang  A. Kanellakopoulos111Present address: HEPIA, HES-SO University of
Applied Scienes and Arts Western Switzerland, Geneva, Switzerland X.F. Yang
<EMAIL_ADDRESS>S. W. Bai  J. Billowes  M. L. Bissell  K. Blaum  B.
Cheal C. S. Devlin  R. F. Garcia Ruiz J. Z. Han  H. Heylen  S. Kaufmann  K.
König  Á. Koszorús S. Lechner  S. Malbrunot-Ettenauer222Present address:
TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada W. Nazarewicz  R.
Neugart  G. Neyens  W. Nörtershäuser  T. Ratajczyk P.-G. Reinhard  L. V.
Rodríguez  S. Sels  L. Xie  Z. Y. Xu333Present address: Department of Physics
and Astronomy, University of Tennessee, 37996 Knoxville, TN, USA D. T.
Yordanov  Y. M. Yu  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  School of
Physics and Astronomy, The University of Manchester, Manchester M13 9PL,
United Kingdom  Max-Planck-Institut für Kernphysik, D-69117 Heidelberg,
Germany  Oliver Lodge Laboratory, Oxford Street, University of Liverpool,
Liverpool, L69 7ZE, United Kingdom  Massachusetts Institute of Technology,
Cambridge, MA, USA  Experimental Physics Department, CERN, CH-1211 Geneva 23,
Switzerland  State Key Laboratory of Precision Measurement Technology and
Instruments, Key Laboratory of Photon Measurement and Control Technology of
Ministry of Education, Department of Precision Instrument, Tsinghua
University, Beijing 100084, China  Institut für Kernphysik, TU Darmstadt,
D-64289 Darmstadt, Germany  Institut für Kernchemie, Universität Mainz,
D-55128 Mainz, Germany  Technische Universität Wien, Karlsplatz 13, AT-1040
Wien, Austria  Department of Physics and Astronomy and FRIB Laboratory,
Michigan State University, East Lansing, Michigan 48824, USA  Institut für
Theoretische Physik, Universität Erlangen, Erlangen, Germany  Institute de
Physique Nucléaire, CNRS-IN2P3, Université Paris-Sud,Université Paris-Saclay,
91406 Orsay, France  Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

Collinear laser spectroscopy measurements were performed on 68-74Ge isotopes
($Z=32$) at ISOLDE-CERN, by probing the
$4s^{2}4p^{2}\,{}^{3}\\!P_{1}\rightarrow 4s^{2}4p5s\,^{3}\\!P_{1}^{o}$ atomic
transition (269 nm) of germanium. Nuclear charge radii are determined via the
measured isotope shifts, revealing a larger local variation than the
neighboring isotopic chains. Nuclear density functional theory with the Fayans
functionals Fy($\Delta r$,HFB) and Fy(IVP), and the SV-min Skyrme describes
the experimental data for the differential charge radii $\delta\langle
r^{2}\rangle$ and charge radii $R_{\rm c}$ within the theoretical
uncertainties. The observed large variation in the charge radii of germanium
isotopes is better accounted for by theoretical models incorporating ground
state quadrupole correlations. This suggests that the polarization effects due
to pairing and deformation contribute to the observed large odd-even
staggering in the charge radii of the Ge isotopic chain.

###### keywords:

Collinear laser spectroscopy , Nuclear charge radii , Nuclear density
functional theory , Ground state correlation

††journal: Physics Letters B

##  1 Introduction

The charge radius is one of the basic properties of atomic nuclei. Based on
general considerations, the nuclear charge radii are expected to globally
scale with the nuclear mass number $A$ as $A^{1/3}$. It is known, however,
that the local deviations from this bulk behavior are large; they are due to
various effects, including shell structure variations, shape deformations,
nuclear superconductivity, and continuum coupling [1]. This sensitivity makes
nuclear charge radii excellent laboratories of structural phenomena such as
halo structures [2, 3, 4], nuclear magicity and nucleonic pairing [5, 6, 7, 8,
9], nuclear correlations [10, 11, 12], and static shape deformations and shape
coexistence [13, 14, 15, 16]. The nuclear charge radius has thus become one of
the key observables to explore the structure of atomic nuclei in different
regions of the chart of nuclides [2, 6, 9, 17, 18] and to provide benchmarks
for the tests of nuclear many-body methods and interactions [19, 20, 21].

The region of nuclei around the semi-magic 68Ni and doubly magic 78Ni has been
a topic of intensive experimental and theoretical studies [22, 23, 24, 25, 26,
27, 28, 29, 30]. In the last decade, a series of experiments has been
conducted in order to gain a deeper understanding of the nuclear structure in
this region. These experimental investigations include high-precision laser
spectroscopy and mass measurements, resulting in rich data on the nuclear
properties, such as nuclear masses [29, 31, 32], spins [13, 33],
electromagnetic moments [34, 35, 36], and charge radii [8, 11, 12]. Insightful
information on shell evolution [33, 34, 35], emergence of collectivity around
$N=40$ [11, 32, 35, 37], and shape coexistence [13, 31] near 78Ni has been
obtained from these measurements.

More specifically, based on earlier experimental measurements of the nuclear
charge radii in this mass region including nickel ($Z=28$) [17], copper
($Z=29$) [10, 12], zinc ($Z=30$) [11], and gallium ($Z=31$) [38, 39], a change
of nuclear structure has been observed around $N=40$ with protons being added
above the $Z=28$ shell closure [1]. The ground states of germanium and
selenium isotopes around $N=40$ are expected to have a fairly complex
structure due to large quadrupole correlations and shape coexistence effects
[24] resulting in configuration mixing. According to the multiple Coulomb
excitation analysis [40], the ground state configurations of 70,72,74,76Ge are
associated with modest deformations. The excited $0^{+}_{2}$ state is well
deformed in 70Ge and it is nearly spherical in 74,76Ge. In the nucleus 72Ge,
the two lowest 0+ states are strongly mixed [40, 41], as evidenced by their
similar quadrupole invariants $\langle Q^{2}\rangle$ [42, 43] (For the shape
mixing analysis of 70,72,74,76Ge, see Fig. 37 of Ref. [24] and Ref. [44].).

Theoretical potential energy surfaces for the Ge isotopic chain [45, 46, 47,
48, 49, 50] are predicted to be fairly soft, with shallow coexisting
spherical, prolate, and oblate minima. In some cases, $\gamma$-soft or
triaxial shapes are expected (For a comprehensive analysis of quadrupole
invariants in the Ge chain, see Ref. [41].). Due to the transitional character
of the nuclei considered, there is a considerable spread between predictions
of different theoretical models. The question that will be addressed in this
paper is: “In which way does the complex pattern of quadrupole collectivity of
the Ge isotopes impact their charge radii?”

Prior to the present measurements, laser spectroscopic information on the
charge radii of germanium isotopes, even for the stable ones 70,72-74,76Ge, is
scarce [51, 52]. This is due to the numerous experimental challenges, such as
the difficulty to produce radioactive germanium in ionic form at ISOL
facilities, and the optical transition which is not easily accessible by
commonly used laser devices, as detailed in Ref. [36]. Here, we report the
first nuclear charge radii measurement of radioactive germanium isotopes using
high-resolution collinear laser spectroscopy, benefiting from the combination
of laser frequency-mixing and second harmonic generation to produce 269-nm
light [36]. The measured root mean squared (rms) charge radii of 68-74Ge
isotopes are compared with the adjacent isotopic chains revealing an enhanced
odd-even staggering (OES) effect around $N=40$. Our results are interpreted on
the basis of nuclear density functional theory (DFT) using three energy
density functionals, highlighting the impact of pairing, deformation, and
zero-point ground-state quadrupole correlations (GSC) on nuclear charge radii.

Figure 1: Hyperfine structure spectra of 68-74Ge isotopes measured at COLLAPS
for the atomic germanium transition of
$4s^{2}4p^{2}\,{}^{3}\\!P_{1}-4s^{2}4p5s\,^{3}\\!P_{1}^{o}$. The red solid
lines and vertical dashed lines show the fits to the data and the center of
gravity of each spectrum, respectively. Stable isotope 72Ge was used as the
reference for the isotope shifts measurements to 68-74Ge.

##  2 Experimental method

The experiment was performed at the COLLAPS setup [53] at ISOLDE-CERN. Details
of the general setup and experiment can be found in Ref. [36]. In brief, the
germanium isotopes were produced by impinging 1.4-GeV protons on a
$\textrm{ZrO}_{2}$ target and the released atoms were ionized by a plasma ion
source. The extracted germanium ions were accelerated to 50 keV and then mass
separated. Prior to being delivered to the COLLAPS beamline, the ions were
first cooled in a gas-filled linear Paul trap (ISCOOL) [54] for 5 ms and
released as bunches with a temporal width of 5 $\mu$s. At COLLAPS, the
germanium ions were neutralized in-flight by passing through a charge-exchange
cell (CEC) filled with sodium vapor and the ground state
$4s^{2}4p^{2}\,{{}^{3}\\!}P_{1}$ of Ge I was populated [55]. The atomic
germanium beam was then collinearly overlapped with a frequency-mixed and
frequency-doubled continuous-wave (cw) laser beam at a wavelength of 268.9 nm
to match the Doppler shifted atomic transition
$4s^{2}4p^{2}\,{{}^{3}\\!}P_{1}-4s^{2}4p5s\,{{}^{3}\\!}P_{1}^{o}$. The laser
frequency was stabilized by a wavelength meter, which was regularly calibrated
with a stabilized diode laser locked to one of the hyperfine components of
87Rb atom. The kinetic energy of the germanium beam was tuned by applying a
varying voltage to the CEC, in order to resonantly excite the germanium atoms
into its $4s^{2}4p5s\,{{}^{3}\\!}P_{1}^{o}$ state through Doppler tuning [1].
Four photomultiplier tubes were used to detect the subsequently emitted
fluorescence photons as a function of the tuning voltage [36]. In this manner,
hyperfine structure (hfs) spectra of germanium isotopes were recorded. To
compensate for any possible long-term drifts in the ion velocity or in the
laser frequency, hfs spectra of all studied germanium isotopes were measured
alternating with the stable reference isotope 72Ge.

##  3 Experimental results

Typical hfs spectra of 68-74Ge isotopes shown in Fig. 1 were analyzed using a
$\chi^{2}$-minimization approach implemented in the SATLAS package [56]. The
extracted magnetic and quadrupole hfs parameters $A$ and $B$ of the odd-$A$
isotopes have already been reported in Ref. [36]. Based on the extracted
center of gravity (COG) of each spectrum, $\nu^{A}$, isotope shifts
$\delta\nu^{72,A}=\nu^{A}-\nu^{72}$ of 68-74Ge isotopes were calculated with
respect to the COG ($\nu^{72}$) of the reference isotope 72Ge, as presented in
Table 1. The differential mean-square charge radii $\delta\langle
r^{2}\rangle^{72,A}\equiv\langle r^{2}\rangle^{A}-\langle r^{2}\rangle^{72}$
can then be calculated from the isotopes shifts ($\delta\nu^{72,A}$) according
to:

$\delta\nu^{72,A}=K_{\rm MS}\frac{m_{A}-m_{72}}{m_{A}m_{72}}+F\delta\langle
r^{2}\rangle^{72,A}$ (1)

where $K_{\rm{MS}}$ and $F$ are the atomic mass-shift and field-shift factors,
respectively, of the probed atomic transition, and $m_{A}$ are the atomic
masses taken from Ref. [57].

Table 1: Isotope shifts $\delta\nu^{72,A}$, changes in mean-square charge
radii $\delta\langle r^{2}\rangle^{72,A}$ and absolute charge radii $R_{\rm
ch}$ of 68-74Ge isotopes. Statistical errors are shown in curved brackets.
Systematic errors in square brackets arise from the uncertainty on the atomic
mass-shift and field-shift factors. The experimentally calibrated atomic
factors of $F=+317(58)$ MHz/fm2 and $K_{\rm{MS}}=-9(22)$ GHz u are used for
the calculation of the charge radii. See text for more details.

$A$ | $I^{\pi}$ | $T_{1/2}$ | $\delta\nu^{72,A}$ | $\delta\langle r^{2}\rangle^{72,A}$ | $R_{\rm ch}$
---|---|---|---|---|---
|  |  | (MHz) | (fm2) | (fm)
68 | 0+ | 270.9 d | -51(4) | -0.184(13)[67] | 4.036(3)[8]
69 | 5/2- | 39.05 h | -72(2) | -0.244(6)[62] | 4.029(2)[8]
70 | 0+ | Stable | -35(2) | -0.121(6)[36] | 4.044(2)[4]
71 | 1/2- | 11.43 d | -69(3) | -0.223(9)[43] | 4.032(2)[5]
72 | 0- | Stable | 0 | 0 | 4.059(2)[0]a
73 | 9/2+ | Stable | 8(4) | 0.031(13)[15] | 4.063(3)[2]
74 | 0- | Stable | 52(2) | 0.174(6)[41] | 4.080(2)[5]

  * a

a: $\langle r^{2}\rangle_{\rm{\mu e}}^{1/2}$ calculated from the electron
scattering and muonic atom spectroscopy data [51].

In general, the atomic mass-shift and field-shift factors can be calibrated
from the experimentally known charge radii of at least three stable isotopes
using a King-plot analysis [51] or calculated by using advanced atomic theory
[1]. For germanium isotopes, nuclear radius parameters $\varLambda_{\rm{\mu
e}}^{72,70}$, $\varLambda_{\rm{\mu e}}^{74,72}$, which in a first
approximation is equal to differences in the mean square charge radii
$\delta\langle r^{2}\rangle$, are known experimentally from a combined
analysis of electron scattering and muonic atom spectroscopy data [51].
Therefore, a King plot analysis [51] has been performed using the measured
$\delta\nu^{72,A}$, leading to the atomic factors of $F$ = +317(58) MHz/fm2
and $K_{\rm{MS}}=-9(22)$ GHz u. To further validate these calibrated atomic
factors, we performed atomic calculations using the configuration interaction
approach with many-body perturbation theory (CI$+$MBPT) implemented in the
AMBiT software [58], resulting in $F=453(100)$ MHz/fm2 and
$K_{\rm{MS}}=-14(5)$ GHz u, where theoretical uncertainties are estimated by
considering the possible impact due to the higher-order electronic
correlations. Within the uncertainty, these calculated atomic factors are in
agreement with those obtained from the King-plot analysis.

Figure 2: (a) Changes in mean-square charge radii of the 68-74Ge isotopes
compared with the neighbouring isotopic chains of nickel, copper, zinc and
gallium. Note that the data are arbitrarily offset for better visualization.
The brown empty stars present the radii extracted using the $F$ and $M$
factors from CI+MBPT calculations. (b) $\langle r^{2}\rangle_{\rm{gs}}-\langle
r^{2}\rangle_{0}$ of ground state of the nickel, copper, zinc, gallium and
germanium isotopes obtained by subtracting the spherical droplet model
reference $\langle r^{2}\rangle_{0}$ [59]. See text for more details.

Table 1 and Fig. 2(a) (solid black stars) show the differential charge radii
$\delta\langle r^{2}\rangle^{72,A}$ of 68-74Ge calculated by using the
calibrated atomic factors. The systematic uncertainties for $\delta\langle
r^{2}\rangle^{72,A}$ in Table 1 are dominated by the uncertainties of the
atomic factors. For comparison, the values of $\delta\langle
r^{2}\rangle^{72,A}$ based on the theoretically calculated atomic factors are
also presented in Fig. 2(a) (open orange stars); they are within the
systematic uncertainties of those obtained with calibrated atomic factors. The
absolute charge radii of 68-74Ge listed in Table 1 are calculated using the
charge radius of 72Ge from the electron scattering and muonic atom
spectroscopy data [51].

##  4 Discussion

Figure 2(a) presents a systematic comparison of the differential charge radii
$\delta\langle r^{2}\rangle$ of several isotopic chains in this mass region.
To better show nuclear structure effects we have subtracted the spherical
droplet model reference $\langle r^{2}\rangle_{0}$ [59]. Such obtained charge
radii $\langle r^{2}\rangle-\langle r^{2}\rangle_{0}$ of nickel [17], copper
[12], zinc [11], gallium [39] and germanium are shown in Fig. 2(b) where odd
and even-$N$ isotopes are represented by open and filled symbols,
respectively. Although the charge radii of these five isotopic chains show a
similar trend, appreciably larger variations in the charge radii and a
significantly enhanced OES effect are seen for the germanium isotopes.

As discussed above, the light germanium isotopes around $N=40$ are soft
systems in which shape coexistence phenomena are present. Deformation effects,
whether related to static deformations or dynamic fluctuations, are expected
to impact charge radii. To have a better understanding of the charge radii of
68-74Ge, we performed DFT calculations using three energy density functionals,
namely two Fayans functional parametrizations Fy($\Delta r$, HFB) [6, 19] and
Fy(IVP) [60], as well as the Skyrme parametrization SV-min [17, 61]. All three
functionals were optimized to the same large set of experimental data as
described in Ref. [61]. The functional Fy($\Delta r$, HFB) has been calibrated
using the data on differential charge radii of calcium isotopes [19]; this
additional information substantially impacts the pairing functional and is
crucial for reproducing the observed OES of charge radii. The trend of charge
radii along the Cd isotopic chain was well reproduced [62] with this
functional. But in the mid-shell region of the semi-magic Sn and Ni isotopes,
the slope of differential radii turned out to be overestimated [9, 17] whereas
the behavior at the shell closures was in good agreement with the experimental
data [9, 8]. This was attributed to missing isovector pairing correlations. In
order to capture the underlying physics, the new functional Fy(IVP) was
developed in Ref. [60] by using the additional data on charge radii of calcium
and lead isotopes.

Figure 3:  Potential energy computed with SV-min as function of quadrupole
deformation $\beta_{2}$ for 68,70,72,74Ge (full lines) and 69,71,73Ge (dashed
lines). The energy is normalized to zero at the spherical shape. Figure 4:
(a-c) Differential charge radii $\delta\langle r^{2}\rangle$, (d-e) charge
radii $R_{c}$ and (g-i) $\Delta^{3}(\langle r^{2}\rangle,N)$ of germanium
isotopes compared to DFT results using Fy($\Delta r$, HFB), Fy(IVP), and SV-
min energy density functionals. The statistical errors on the theoretical
values come from the calibration uncertainties of the parametrizations. One
can also consider the difference between Spherical and GSC variants as a rough
estimate of systematic errors. See text for more details. Figure 5: (a-c)
Contribution of the static and dynamic deformation to the nuclear charge radii
of $68-74$Ge isotopes calculated by Fy($\Delta r$, HFB), Fy(IVP), and SV-min
functionals. See text for more details.

As a representative example of DFT calculations for the germanium isotopes,
Fig. 3 shows the potential-energy curves for 68-74Ge calculated with SV-min
along the dimensionless quadrupole deformation $\beta_{2}$ deduced from the
calculated intrinsic proton quadrupole moments:

$\beta_{2}=4\pi\frac{\langle
r^{2}Y_{20}\rangle}{3ZR^{2}}\;,\;R=1.2A^{1/3}\,\mathrm{fm}\;.$ (2)

The energy curves for even-even isotopes are extremely soft. As a consequence,
their ground-state deformations are not well defined; the corresponding
collective wave functions are in fact coherent superpositions of nearly-
degenerate states having different quadrupole deformations. For the odd-$A$
isotopes, deformations are better defined due to a strong quadrupole
polarization exerted by an odd neutron occupying a deformed single-particle
level. In our calculations, we assumed that the quantum numbers $K^{\pi}$ of
the blocked one-quasiparticle state correspond to the observed ground-state
quantum numbers $I^{\pi}$ given in Table 1. As seen in Fig. 3, the quadrupole
polarization effects are substantial, and they strongly depend on the
geometric character of the blocked orbital [63, 64].

It is thus not straightforward to unambiguously determine the ground state
configuration from the calculated mean-field minima alone. In fact, the
situation shown in Fig. 3 calls for a multi-reference theory such as the
Generator Coordinate Method (GCM). Here, we handle GCM at the level of the
Gaussian overlap approximation [65]. The actual procedure is explained in
Refs. [66, 67]. We compute the collective deformation path along axially
symmetric shapes by quadrupole constrained DFT calculations on a cylindrical
grid in coordinate space [68]. This yields the collective potential energy
curve as shown in Fig. 3 together with the corresponding collective mass and
moment of inertia. The potential landscape is soft in triaxial $\gamma$
direction which allows to interpolate potential energy, collective mass and
inertia into the full triaxial $\beta$-$\gamma$-plane. The ground-state wave
function of the resulting collective Bohr Hamiltonian determines the GSC
corrected radii. As a rule of thumb, the ground-state fluctuation of shapes
cover typically a range of $\Delta\beta_{2}\approx 0.2$. The final DFT results
for the charge radii of germanium isotopes are presented in Fig. 4. Results
labeled “Spherical”, “Deformed” and “GSC”, represent calculations at spherical
shape, minimum-energy configuration, and with GSC added, respectively. The
statistical errors are computed based on the calibration uncertainties of the
energy density functionals [69, 70]. As a rough estimate of systematic errors
one can consider the difference between “Spherical” and “GSC” variants.

As seen in Fig. 4 (a-f), considering the theoretical and experimental
uncertainties, all three functionals provide a fair agreement with
experimental data for $\delta\langle r^{2}\rangle$ and $R_{\rm c}$. It is to
be noted that, in general, $R_{\rm c}^{\rm Spherical}<R_{\rm c}^{\rm
Deformed}<R_{\rm c}^{\rm GSC}$. Indeed, since the rms charge radius depends on
quadrupole correlations as $\langle
Q^{2}\rangle\propto\langle\beta_{2}^{2}\rangle$, the difference $\Delta R_{\rm
c}^{\rm s}\equiv R_{\rm c}^{\rm Deformed}-R_{\rm c}^{\rm Spherical}$ measures
the contribution from the static deformation, while the $\Delta R_{\rm c}^{\rm
s+d}\equiv R_{\rm c}^{\rm GSC}-R_{\rm c}^{\rm Spherical}$ also measures the
impact of dynamic fluctuations. As shown in Fig. 5, it is interesting to see
that $\Delta R_{\rm c}^{\rm s+d}$ changes rather gradually for even-even
isotopes, which is consistent with the data on quadrupole invariants $\langle
Q^{2}\rangle$ [40, 41]. The relation between the three radii, $R_{\rm c}^{\rm
Spherical}$, $R_{\rm c}^{\rm Deformed}$, and $R_{\rm c}^{\rm GSC}$ reflects
the character of underlying potential energies. For Fy(IVP) and SV-min $\Delta
R_{\rm c}^{\rm s}$ is smaller and $\Delta R_{\rm c}^{\rm s+d}$ is larger as
68-74Ge isotopes are predicted to be very soft in these models. On the other
hand, for Fy($\Delta r$, HFB), $\Delta R_{\rm c}^{\rm s}$ and $\Delta R_{\rm
c}^{\rm s+d}$ is similar as this model predicts smaller dynamic deformations.

While the performance of the three functionals for the general trend of
$\delta\langle r^{2}\rangle$ and $R_{\rm c}$ is rather similar, the local
variation of the nuclear charge radii, namely the OES defined as
$\Delta^{3}(\langle r^{2}\rangle,N)=\langle
r^{2}\rangle^{N}-\frac{1}{2}(\langle r^{2}\rangle^{N-1}+\langle
r^{2}\rangle^{N+1})$, shows more model dependence (Fig. 4(g-i)). Specifically,
the relatively large OES observed in the germanium chain is captured by the
Fy($\Delta r$, HFB) functional in all three calculation variants, and is very
well described by the “GSC” variant of the SV-min model. Overall, the
calculations that consider zero-point correlations describe the OES of the
germanium radii fairly well. This indicates that the observed large OES in
68-74Ge can be attributed to both pairing and shape polarization effects,
which – as shown in Fig. 3 – are appreciable in the transitional germanium
isotopes.

##  5 Summary and conclusion

The charge radii of 68-74Ge isotopes were measured employing the collinear
laser spectroscopy technique. By comparing with the neighbouring isotopic
chains, we found that charge radii of germanium isotopes around $N=40$ exhibit
large OES. Quantified DFT calculations using three different energy density
functionals were augmented by considering ground-state quadrupole
correlations. Within the uncertainties, all calculations are consistent with
the experimental data on differential and absolute charge radii of 68-74Ge.
The OES of the charge radii of germanium isotopes is better described by the
calculation including the GSC. One can thus conclude that the behavior of
charge radii along the Ge isotopic chain is strongly influenced by the
polarization effects due to both pairing and shape deformation. This work also
highlights the recent progress in the quantified DFT calculations for the
deformed open shell nuclei.

## Acknowledgments

We acknowledge the support of the ISOLDE collaboration and technical teams.
This work was supported by the National Key R&D Program of China (Contract No.
2022YFA1604800, 2023YFA1606403), the National Natural Science Foundation of
China (Nos:12027809,12350007); the BriX Research Program No. P7/12, FWO-
Vlaanderen (Belgium), GOA 15/010 from KU Leuven; the UK Science and Technology
Facilities Council grants ST/L005794/1 and ST/P004598/1; the BMBF Contracts No
05P18RDCIA, 05P21RDCI1, the Max-Planck Society, the Helmholtz International
Center for FAIR (HIC for FAIR); the EU Horizon2020 research and innovation
programme through ENSAR2 (No. 654002); the computing center RRZE of the
Friedich-Alexander university Erlangen; and by the U.S. Department of Energy
under Award Number DE-SC0013365 (Office of Science) and DE-SC0023175 (Office
of Science, NUCLEI SciDAC-5 collaboration).

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