New limits on double-beta decay of 190Pt and 198Pt

F.A. Danevicha,111Corresponding author. E-mail address<EMAIL_ADDRESS>(F.A. Danevich)., M. Hultb, A. Junghansc, D.V. Kasperovycha, B.N.
Kropivyanskya, G. Lutterd, G. Marissensb, O.G. Polischuka, M.V. Romaniuka, H.
Strohb, S. Tessalinae, V.I. Tretyaka, B. Waree

aInstitute for Nuclear Research of NASU, 03028 Kyiv, Ukraine

bEuropean Commission, Joint Research Centre, Retieseweg 111, 2440 Geel,
Belgium

cHelmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstrasse 400 01328 Dresden,
Germany

dDepartment of Environmental Engineering, Technical University of Denmark, DTU
Risø Campus, 4000, Roskilde, Denmark

eJohn de Laeter Centre for Isotope Research, GPO Box U 1987, Curtin
University, Bentley, WA, Australia

###### Abstract

A search for double-beta decay of 190Pt and 198Pt with emission of
$\gamma$-ray quanta was realized at the HADES underground laboratory with a
148 g platinum sample measured by two ultralow-background HPGe detectors over
8946 h. The isotopic composition of the platinum sample has been measured with
high precision using inductively coupled plasma mass spectrometry. New lower
limits for the half-lives of 190Pt relative to different channels and modes of
the decays were set on the level of $\lim T_{1/2}\sim 10^{14}-10^{16}$ yr. A
possible exact resonant $0\nu KN$ transition to the 1,2 1326.9 keV level of
190Os is limited for the first time as $T_{1/2}\geq 2.5\times 10^{16}$ yr. A
new lower limit on the double-beta decay of 198Pt to the first excited level
of 198Hg was set as $T_{1/2}\geq 3.2\times 10^{19}$ yr, one order of magnitude
higher than the limit obtained in the previous experiment.

##  1 Introduction

Double-beta decay was considered for the first time by Goeppert-Mayer in 1935
[1]. The neutrino accompanied mode of the decay with emission of electrons
(two-neutrino double-beta decay, $2\nu 2\beta^{-}$) is allowed in the Standard
Model of particles and interactions (SM) and is already observed in eleven
nuclei with measured half-lives in the range $T_{1/2}\sim(10^{19}-10^{24})$ yr
[2]. Another possibility could be $2\beta^{-}$ decay without neutrino
emission, neutrinoless double-beta decay ($0\nu 2\beta^{-}$). However, this
process violates the lepton number conservation and is only possible if
neutrinos are massive Majorana particles [3]. Thus, the $0\nu 2\beta^{-}$
decay is the most sensitive test of the lepton number conservation law and one
of the most promising tools to study properties of the neutrino and the weak
interaction. In general, the neutrinoless process can be mediated by many
effects beyond the SM and is considered as one of the most powerful probes of
the SM [4, 5, 6, 7]. Despite the attempts, the $0\nu 2\beta^{-}$ decay is
still not observed: the most sensitive experiments give upper half-life limits
in the range $\lim T_{1/2}\sim(10^{24}-10^{26})$ yr. This allows to bound the
effective Majorana neutrino mass at the level of $\lim\langle
m_{\nu}\rangle\sim(0.1-0.5)$ eV (see, e.g., the reviews [6, 7, 8, 9] and the
recent original works [10, 11, 12, 13, 14, 15, 16]).

Other channels of double-beta decay are “double-beta plus” decay processes:
double-electron capture ($2\varepsilon$), electron capture with positron
emission ($\varepsilon\beta^{+}$), and double-positron decay ($2\beta^{+}$).
Although allowed in the SM, the two-neutrino mode of the $2\varepsilon$
process (theoretically the fastest decay channel due to the biggest values of
the phase space factor) has yet to be indisputably discovered. There are
indications of $2\nu 2K$ decay with $T_{1/2}\sim 10^{21}$ yr in 130Ba [17, 18,
19] and 78Kr [20, 21]. However, the indications for 130Ba $2\nu 2K$ decay was
obtained only in geochemical experiments by detection of an anomaly in the
traces of xenon isotopic concentration in barite minerals. The results of the
experiment [20, 21] with a proportional chamber need to be confirmed. Recently
an observation of the $2\nu 2K$ decay of 124Xe with $T_{1/2}=(1.8\pm
0.5)\times 10^{22}$ yr was claimed by the XENON collaboration by using the
XENON1T dark-matter detector [22]. As for the neutrinoless double-beta plus
decay processes, their mechanisms are the same as for the decay with electrons
emission. Moreover, there are some additional arguments to develop
experimental methods to search for the $0\nu 2\varepsilon$,
$0\nu\varepsilon\beta^{+}$ and $0\nu 2\beta^{+}$ decays, taking into account
the potential to investigate the possible contribution of the right-handed
currents to the $0\nu 2\beta^{-}$ decay rate if observed [23], and an
interesting possibility of a resonant $0\nu 2\varepsilon$ process (see [24]
and references therein).

Two platinum isotopes are potentially unstable relative to the double-beta
decay: 190Pt with the decay energy $Q_{2\beta}=1401.3(4)$ keV [25] and the
isotopic abundance $\delta=0.012(2)\%$ [26], and 198Pt with the decay energy
$Q_{2\beta}=1050.3(21)$ keV [25] and the isotopic abundance
$\delta=7.356(130)\%$ [26]. A simplified decay scheme of 190Pt is shown in
Fig. 1.

Figure 1: A simplified decay scheme of 190Pt [27]. The energies of the excited
levels and of the emitted $\gamma$ quanta are in keV (the relative intensities
of the $\gamma$ quanta are given in parentheses). $Q_{2\beta}$ and
$Q_{\alpha}$ are the double-beta decay and alpha decay energies of 190Pt,
respectively.

The nuclide remains one of few nuclides with a possibility of exact resonant
$0\nu 2\varepsilon$ decay. The resonant character of the process is expected
in the $0\nu KN$ transition to the 1,2 1326.9(10) keV excited level of the
daughter [24]. It should be stressed that the decay channel was not considered
in the previous work [28] due to a rather poor accuracy of the $Q_{2\beta}$
value at the time of the experiment (in 2011 the recommended $Q_{2\beta}$ was
1383(6) keV [29]). A status of the decay as a potential resonance can be
characterized by the resonance parameter $R_{f}$:

$R_{f}=\Gamma_{f}/(\Delta^{2}+\Gamma_{f}^{2}/4),$ (1)

where $\Gamma_{f}=\Gamma_{1}+\Gamma_{2}$ is the de-excitation width of the
electron shell of the daughter nuclide. The degeneracy parameter $\Delta$ is
equal to $Q_{2\beta}-E_{exc}-E_{b1}-E_{b2}$, where $E_{exc}$ is the energy of
the daughter nucleus excited level, $E_{bi}$ are the binding energies of the
captured electrons on the atomic shells of the daughter atom. Taking into
account the presently recommended $Q_{2\beta}$ value [25] the resonance
parameter for 190Pt (normalized on the value for the $0\nu 2\varepsilon$ decay
54Fe $\rightarrow$ 54Cr) can reach one of the biggest values $\approx
7.0\times 10^{8}$ among the possible resonant transitions
[24]222Unfortunately, the 1326.9-keV level characteristics are known with a
rather limited accuracy: the uncertainty of the level energy is $\pm 1.0$ keV,
spin is 1 or 2, while parity of the level is unknown.. The method of ultralow-
background HPGe $\gamma$-ray spectrometry was applied in the present study to
search for different channels and modes of $2\varepsilon$ and
$\varepsilon\beta^{+}$ processes in 190Pt, including the possible resonant
transitions in the nuclide.

A simplified decay scheme of another possible $2\beta$ unstable isotope of
platinum, 198Pt, is presented in Fig. 2. The $2\beta^{-}$ transition of 198Pt
to the first $2^{+}$ 411.8 keV excited level of 198Hg is expected to be
accompanied by emission of $\gamma$-ray quanta with energy 411.8 keV that can
be detected by $\gamma$-spectrometry methods too.

Figure 2: A simplified decay scheme of 198Pt [30]. The energies of the excited
level and of the emitted $\gamma$ quantum are in keV. $Q_{2\beta}$ is the
double-beta decay energy of 198Pt.

In the next Section we describe the high-purity platinum sample, precise
measurements of its isotopic composition, and the ultralow background HPGe
detector system used in the experiment. The data analysis and obtained limits
on the $2\varepsilon$ and $\varepsilon\beta^{+}$ processes in 190Pt and the
$2\beta^{-}$ decay of 198Pt to the first excited level of the daughter are
presented in Section 3. A summary of the experiment is given in the
Conclusions section.

##  2 Experiment

###  2.1 Platinum sample, isotopic composition of the material

A disk-shaped sample of metallic platinum with a diameter of 25.04(1) mm, a
thickness of 14.07(2) mm, and with a mass of 148.122(1) g was used in the
experiment. The purity grade of the platinum is 99.95%333In May 2016 the Pt
disk was used as a neutron transmission target in the Institute of Radiation
Physics of the Helmholtz-Zentrum (Dresden). It was $\approx 1$ m away from the
electron beam (30 MeV) and the neutron producing radiator. The target was
irradiated by a neutron flux of about $200-400$ neutrons per second for about
one week. The neutron spectrum starts around $10-100$ keV and ends at around
10 MeV with a maximum around $1-2$ MeV.. The representative isotopic abundance
of 190Pt in normal terrestrial materials has a rather big uncertainty
$\delta=0.012(2)\%$ [26]. Thus, special mass-spectrometry measurements of the
sample were realized.

The Pt isotopic measurements were acquired using a sector field ICP-MS ELEMENT
XR (ThermoScientific) at the John de Laeter Centre for Isotope Research,
Curtin University. Measurements of masses 190Pt,192Pt, 194Pt, 195Pt, 196Pt,
and 198Pt were performed in low resolution mode using electrostatic scanning
(e-scan, i.e., peak jumping) from a set magnet mass at 190Pt. Due to the
drastically different abundances of Pt isotopes, the ELEMENT XR’s triple
detection mode is advantageous for such analyses as isotopes such as 190Pt and
192Pt can be measured in a pulse-counting detection mode, while the remaining
masses can be analysed in analogue mode all within the same analytical
session. Prior to the analysis of each sample, a blank solution of 2% HNO3 was
measured to correct for background. A summary of the platinum isotopic
composition, as well as numbers of nuclei of the isotopes in the sample are
presented in Table 1.

Table 1: Isotopic composition ($\delta$) of the platinum sample measured in the present work and the numbers of nuclei of each isotope in the sample calculated by using the measured isotopic concentrations. The combined standard uncertainties of the isotopic abundances are given with a coverage factor $k=2$ (approximately 95% level of confidence). The representative isotopic abundances from [26] are given too. Isotope |  $\delta$ (%) | Number of nuclei
---|---|---
| IUPAC [26] | this work | in the sample
190Pt | 0.012(2) | 0.0127(1) | $5.81(5)\times 10^{19}$
192Pt | 0.782(24) | 0.7759(16) | $3.548(7)\times 10^{21}$
194Pt | 32.864(410) | 32.6511(522) | $1.4929(24)\times 10^{23}$
195Pt | 33.775(240) | 33.6884(526) | $1.5403(24)\times 10^{23}$
196Pt | 25.211(340) | 25.5376(419) | $1.1677(19)\times 10^{23}$
198Pt | 7.356(130) | 7.3343(115) | $3.353(5)\times 10^{22}$

###  2.2 Ultralow-level gamma-ray spectrometry measurements

The platinum sample was measured in an ultralow-background HPGe-detector
system located 225 m underground in the laboratory HADES (Belgium) [31, 32].
The detector system consists of two p-type Extended Range HPGe-detectors
facing each other (see a schematic of the set-up in Fig. 3). Both the
detectors were manufactured by Canberra semiconductor (Olen, Belgium). The
detectors were shielded by 35 mm electrolytic copper (innermost) then 40 mm
ultralow-level lead and on the outside 145 mm lead. The main characteristics
of the HPGe detectors are presented in Table 2, more details of the detector
system can be found in [33].

Figure 3: Schematic of the experimental ultralow-background set-up with the HPGe detectors and the platinum sample. Table 2: Properties of the HPGe-detectors used in the present experiment. FWHM denotes the full width at half of maximum of $\gamma$-ray peak measured with a 60Co gamma-ray source. | Ge7 | Ge15
---|---|---
Energy resolution (FWHM) at 1333 keV | 2.2 keV | 1.8 keV
Relative efficiency | 90% | 85%
Crystal mass | 1778 g | 1840 g
Endcap / Window material | HPAl / HPAl | HPAl / HPAl
Dead layer (front) | 0.3 $\mu$m | 0.3 $\mu$m
HPAl = High Purity Aluminum

The data with the Pt sample were collected starting from February 6th, 2018
over a period of 805 days in 6 runs covering a live time of 8946 h (373 d).
The measurements of background without sample were carried out for 674 h (28
d). The energy calibration and stability check was performed using reference
point sources containing 60Co, 137Cs and 241Am.

##  3 Results and discussion

###  3.1 Radionuclides detected in the platinum sample

The sum energy spectra measured by the Ge7 and Ge15 detectors with the Pt
sample and without sample (background) are shown in Fig. 4. The majority of
the peaks can be assigned to 40K and nuclides of the 232Th, 235U and 238U
decay families. There are also peaks of 22Na, 26Al, 54Mn, 60Co, 137Cs and
110mAg. The traces of 22Na, 26Al and 110mAg that were detected are presumingly
reminiscence from neutron activation of minor impurities of the Pt-sample
during the neutron experiment in Dresden and during air-transport. Radioactive
22Na and 26Al can be also cosmogenically generated in the aluminum details of
the HPGe detectors [34]. 54Mn and 60Co are typical cosmogenic radionuclides
that can be produced in Pt, copper and some other materials of the set-up.
Presence of 137Cs can be a result of the set-up (sample) pollution after the
Chernobyl or (and) Fukushima Daiichi nuclear disasters. In the Pt data there
is a clear $\gamma$ peak with energy 137.2 keV due to the $\alpha$ decay of
190Pt to the 137.2 keV excited level of 186Os [35]444The results of the
$\alpha$ decay investigation will be presented in a separate report.. Also
peaks due to neutron-gamma reactions on the materials of the set-up were
observed: in particular, a 139.7-keV peak of 75mGe produced by neutron-gamma
reaction on 74Ge, a 198.4-keV peak from 70Ge(n,$\gamma$)71mGe reaction, a
202.6-keV peak from 115In(n,$\gamma$)116In reaction. There is also a peak of
41Ar with energy 1293.6 keV due to operation of the BR-1 nuclear reactor of
the Belgian nuclear research centre. The peak is present on few specific days
(approximately 15 days in total) during the measurements when air blows from
the reactor towards the inlet of the ventilation for the HADES laboratory. We
have decided to do not exclude the data with the 41Ar peak taking into account
a rather mild effect of the radioactivity in the energy intervals of interest.

Figure 4: Energy spectra in the energy intervals $20-800$ keV (a) and
$800-2700$ keV (b) measured with the platinum sample over 8946 h (solid
histogram) and without sample over 674 h (normalized to 8946 h, dots) by the
ultralow-background HPGe-detector system. The energy of the $\gamma$ peaks is
in keV.

The energy dependence of the energy resolution in the sum energy spectrum
measured with the Pt sample by the detectors Ge7 and Ge15 was determined for
the low energy region ($65-352$ keV) by using intense X-ray and $\gamma$-ray
peaks with energies 65.1 keV and 66.8 keV ($K_{\alpha 2}$ and $K_{\alpha 1}$
X-ray of Pt), 137.2 keV (190Pt), 238.6 keV (212Pb), 295.2 keV and 351.9 keV
(214Pb) as:

$\mathrm{FWHM(keV)}=1.08(12)+0.020(8)\sqrt{E_{\gamma}},$ (2)

where $E_{\gamma}$ is in keV. For the energies above 352 keV we use an
approximation obtained by analysis of the $\gamma$ peaks 238.6 keV (212Pb),
295.2 keV and 351.9 keV (214Pb), 583.2 keV (208Tl), 609.3 keV (214Bi), 1173.2
keV and 1332.5 keV (60Co) and 1460.8 keV (40K):

$\mathrm{FWHM(keV)}=0.55(8)+0.049(4)\sqrt{E_{\gamma}}.$ (3)

The specific activity555In this paper “specific activity” is the activity per
unit mass of the sample and not to the mass of a certain isotope. of all
detected radionuclides was calculated using the following formula:

$A=(S_{sample}/t_{sample}-S_{bg}/t_{bg})/(\epsilon\cdot\eta\cdot m),$ (4)

where $S_{sample}$ ($S_{bg}$) is the area of a peak in the sample (background)
spectrum; $t_{sample}$ ($t_{bg}$) is the time of the sample (background)
measurement; $\epsilon$ is the $\gamma$-ray emission intensity of the
corresponding transition; $\eta$ is the full energy peak efficiency; $m$ is
the sample mass. The detection efficiencies were calculated with the EGSnrc
simulation package [36, 37], the events were generated homogeneously in the Pt
sample. The Monte Carlo models of the two detectors have being validated
through several participations in proficiency tests and an additional
validation measurement was done using a 57Co point $\gamma$ source. The
standard deviation of the relative difference between the simulations and the
experimental data is $3.3\%$ for $\gamma$-ray peaks with energies 122.1 keV
and 136.5 keV for the detectors Ge7 and Ge15. The estimated specific
activities of radioactive impurities in the platinum are presented in Table 3.

Table 3: The specific activities of the detected radionuclides. The upper limits are given at 90% confidence level (C.L.), the reported uncertainties are the combined standard uncertainties calculated by summing the systematic and statistical uncertainties in quadrature. The reference date is the start of the measurement (February 6th, 2018). Chain | Nuclide | Specific activity (mBq/kg)
---|---|---
|  22Na | $\leq 0.5$
|  26Al | $\leq 0.6$
|  40K | $\leq 13$
|  54Mn | $\leq 0.9$
|  60Co | $\leq 0.8$
|  137Cs | $\leq 0.4$
|  110mAg | $9.2\pm 0.4$
232Th |  228Ra | $\leq 5.7$
|  228Th | $10.8\pm 1.3$
235U |  235U | $\leq 16$
|  231Pa | $\leq 81$
238U |  234Th | $247\pm 134$
|  234mPa | $\leq 89$
|  226Ra | $6.8\pm 0.9$
|  210Pb | $2600\pm 570$

###  3.2 Limits on the double-beta decay processes in 190Pt

No peculiarity was observed in the experimental energy spectra that could be
ascribed to the $2\beta$ decay processes in 190Pt or 198Pt. Thus, we set
limits on different modes and channels of the decays by using the following
formula:

$\lim T_{1/2}=N\cdot\eta\cdot t\cdot\ln 2/\lim S,$ (5)

where $N$ is the number of nuclei of interest in the sample (see Table 1),
$\eta$ is the detection efficiency for the $\gamma$-ray (X-ray) quanta
searched for, $t$ is the measuring time, and $\lim S$ is the number of events
of the effect which can be excluded at a given C.L. In the present work all
the $\lim S$ values and the half-life limits are given with 90% C.L. The
detection efficiencies of the detector system to the $\gamma$ (X-ray) quanta
expected in different modes and channels of the double-beta processes in 190Pt
and 198Pt were simulated with the EGSnrc simulation package [36, 37], the
decay events were generated by the DECAY0 events generator [38].

A cascade of X-rays and Auger electrons due to deexcitation of the Os electron
shell with individual energies in the energy interval $\approx(61.5-73.4)$ keV
is expected in the $2\nu 2K$ and $2\nu KL$ capture in 190Pt. However, the
energies of the $L$ X-rays are very low, $(7.8-12.5)$ keV, and therefore
heavily attenuated by the sample and also by the aluminium-windows of the
HPGe-detectors. The Auger electrons avoid detection for the same reason. Thus,
the response of the detector system to the $2\nu 2K$ and $2\nu KL$ decays of
190Pt was built assuming the following energies and intensities of X-rays from
the $K$ shell of the Os atom (only the X-rays with the intensities higher than
0.5% were considered): 61.5 keV ($K_{\alpha 2},27.5\%$), 63.0 keV ($K_{\alpha
1},47.3\%$), 71.1 keV ($K_{\beta 3},5.45\%$), 71.4 keV ($K_{\beta
1},10.50\%$), 73.4 keV ($K_{\beta 2},3.69\%$) [39]. The detection efficiencies
to the X-ray quanta were simulated by the EGSnrc code. To estimate $\lim S$
values for the $2\nu 2K$ and $2\nu KL$ decays, the energy spectrum taken with
the Pt sample was fitted in the energy interval $49-86$ keV by the $2\nu 2K$
($2\nu KL$) distribution666The $2\nu 2K$ and $2\nu KL$ distributions are
almost similar in the region of fit. and a sum of several Gaussian functions
to describe the 63.3-keV $\gamma$-ray peak of 234Th, X-ray peaks of Pt, Tl,
Pb, Bi and Po present in the spectrum, plus a straight line to describe the
continuous distribution. The result of the fit is shown in Fig. 5. Despite an
empirical approach to build the background model, the quality of the fit is
good: $\chi^{2}$/n.d.f.$=39.1/50=0.78$ (where n.d.f. is a number of degrees of
freedom). The fit returned an area of the $2\nu 2K$ ($2\nu KL$) distribution
$S=164\pm 125$ counts that indicates no evidence of the effect searched for.
An excluded effect can be estimated by the simple Gaussian approach as $\lim
S=369$ counts at 90% C.L. Taking into account the Monte-Carlo simulated
detection efficiency for the whole $2\nu 2K$ ($2\nu KL$) effect $\eta=0.0092$
($\eta=0.0047$), the following half-life limits were obtained for the $2\nu
2K$ and $2\nu KL$ capture in 190Pt:

$T_{1/2}^{2\nu 2K}~{}(^{190}$Pt, g.s.$~{}\rightarrow~{}$g.s.)
$\geq~{}1.0\times 10^{15}$ yr,

$T_{1/2}^{2\nu KL}~{}(^{190}$Pt, g.s.$~{}\rightarrow~{}$g.s.)
$\geq~{}5.2\times 10^{14}$ yr.

The half-life limits, the energies of the X-ray quanta ($E_{\gamma}$), which
were used to set the $T_{1/2}$ limits, the detection efficiencies ($\eta$) and
values of $\lim S$ are presented in Table 4777Only statistical errors
originating from the data fluctuations were taken into account in the present
study in the estimations of the half-life limits, systematic effects have not
been included in the half-life limits evaluations., where also results of the
previous experiment [28] are given for comparison. The limit for the $2\nu 2K$
decay half-life slightly exceeds the one reported in the previous work [28],
while the limit for the $2\nu KL$ capture is obtained for the first time.

Figure 5: The energy spectrum collected with the Pt sample by the HPGe detector system in the energy region where $K$ X-ray quanta are expected for the $2\nu 2K$ and $2\nu KL$ decays of 190Pt. The fit of the data by the background model is shown by solid line, the peaks included in the model of the background are shown by dotted lines, while the excluded effect is presented by dashed line (the excluded distribution is multiplied by a factor 2 to improve visibility). Energy of $\gamma$ and X-ray quanta are in keV. Table 4: The half-life limits on $2\varepsilon$ and $\varepsilon\beta^{+}$ processes in 190Pt and $2\beta^{-}$ decay of 198Pt to the first excited level of 198Hg. The energies of the X-ray or $\gamma$ quanta ($E_{\gamma}$), which were used to set the $T_{1/2}$ limits, are listed with their corresponding detection efficiencies ($\eta$) and values of $\lim S$. The results of previous experiment [28] are given for comparison. Values of $\lim S$ and $T_{1/2}$ are given at 90% C.L. All the empirical fits from which the signal upper limits are derived describe the data well, with the goodness-of-fit values within $\chi^{2}$/n.d.f.$=(0.39-0.84)$. Transition | Level of | $E_{\gamma}$ | $\eta$ | $\lim S$ | Experimental limit
---|---|---|---|---|---
| daughter | (keV) |  | (counts) |  $T_{1/2}$ (yr)
| nucleus (keV) |  |  |  | Present work | [28]
190Pt$~{}\to$ 190Os |  |  |  |  |  |
$2\nu 2K$ | g.s. | $61.5-73.4$ | 0.0092 | 369 | $\geq 1.0\times 10^{15}$ | $\geq 8.4\times 10^{14}$
$2\nu KL$ | g.s. | $61.5-73.4$ | 0.0047 | 369 | $\geq 5.2\times 10^{14}$ |
$2\nu 2\varepsilon$ |  $2^{+}$ 186.7 | 186.7 | 0.0093 | 39 | $\geq 9.8\times 10^{15}$ |  $\geq 8.8\times 10^{14}$$~{}^{*}$
$2\nu 2\varepsilon$ |  $2^{+}$ 558.0 | 558.0 | 0.0285 | 50 | $\geq 2.3\times 10^{16}$ | $\geq 5.6\times 10^{15}$
$2\nu 2\varepsilon$ |  $0^{+}$ 911.8 | 725.1 | 0.0393 | 31 | $\geq 5.2\times 10^{16}$ | $\geq 4.5\times 10^{15}$
$2\nu 2\varepsilon$ |  $2^{+}$ 1114.7 | 1114.7 | 0.0145 | 31 | $\geq 1.9\times 10^{16}$ | $\geq 1.0\times 10^{16}$
$2\nu KN$ | 1,2 1326.9 | 1326.9 | 0.0491 | 81 | $\geq 2.5\times 10^{16}$ |
$0\nu 2K$ | g.s. | $1253.2-1254.0$ | 0.0500 | 44 | $\geq 4.7\times 10^{16}$ | $\geq 5.7\times 10^{15}$
$0\nu KL$ | g.s. | $1314.1-1317.0$ | 0.0493 | 44 | $\geq 4.6\times 10^{16}$ | $\geq 1.7\times 10^{16}$
$0\nu 2L$ | g.s. | $1375.0-1380.0$ | 0.0484 | 62 | $\geq 3.2\times 10^{16}$ | $\geq 3.1\times 10^{16}$
$0\nu 2\varepsilon$ |  $2^{+}$ 186.7 | 186.7 | 0.0072 | 39 | $\geq 7.6\times 10^{15}$ | $\geq 6.9\times 10^{14}$
$0\nu 2\varepsilon$ |  $2^{+}$ 558.0 | 558.0 | 0.0234 | 50 | $\geq 1.9\times 10^{16}$ | $\geq 4.5\times 10^{15}$
$0\nu 2\varepsilon$ |  $0^{+}$ 911.8 | 725.1 | 0.0361 | 31 | $\geq 4.8\times 10^{16}$ | $\geq 3.6\times 10^{15}$
$0\nu 2\varepsilon$ |  $2^{+}$ 1114.7 | 1114.7 | 0.0117 | 31 | $\geq 1.6\times 10^{16}$ | $\geq 9.8\times 10^{15}$
Res. $0\nu KN$ | 1,2 1326.9 | 1326.9 | 0.0488 | 81 | $\geq 2.5\times 10^{16}$ |
Res. $0\nu LM$ | (0,1,2)+ 1382.4 | 1195.7 | 0.0497 | 42 | $\geq 4.9\times 10^{16}$ | $\geq 2.9\times 10^{16}$
$2\nu\varepsilon\beta^{+}$ | g.s. | 511 | 0.0953 | 134 | $\geq 2.9\times 10^{16}$ | $\geq 9.2\times 10^{15}$
$0\nu\varepsilon\beta^{+}$ | g.s. | 511 | 0.0942 | 134 | $\geq 2.9\times 10^{16}$ | $\geq 9.0\times 10^{15}$
$2\nu\varepsilon\beta^{+}$ |  $2^{+}$ 186.7 | 511 | 0.0926 | 134 | $\geq 2.8\times 10^{16}$ | $\geq 8.4\times 10^{15}$
$0\nu\varepsilon\beta^{+}$ |  $2^{+}$ 186.7 | 511 | 0.0922 | 134 | $\geq 2.8\times 10^{16}$ | $\geq 8.4\times 10^{15}$
198Pt$~{}\to$ 198Hg |  |  |  |  |  |
$2\beta^{-}$ ($2\nu+0\nu$) |  $2^{+}$ 411.8 | 411.8 | 0.0414 | 31 | $\geq 3.2\times 10^{19}$ | $\geq 3.5\times 10^{18}$
∗ The limit in [28] was set for the $2\nu 2K$ transition to the 186.7 keV
level of 190Os.

The $2\nu$ double-electron capture in 190Pt can undergo to several excited
levels of 190Os (see Fig. 1). In this case, in addition to X-rays cascade,
$\gamma$-peaks at the energy of the excited level (in the transition to the
first $2^{+}$ 186.7 keV excited level) and also at the energies of the
transitions between the initial and final levels of the daughter are expected.
To estimate $\lim S$ value for a possible 186.7-keV peak area, the
experimental spectrum was fitted in the energy interval $176-194$ keV by a
model consisting of a straight line, peaks of 235U and 226Ra with energies
185.7 keV and 186.2 keV, respectively, and a Gaussian peak at 186.7 keV with
the width fixed according to the formula (2) to describe the effect searched
for. The fit returns a 186.7-keV peak area $S=-28(39)$ counts, that
corresponds to $\lim S=39$ counts following the recommendations [40]. A part
of the energy spectrum gathered with the Pt sample in the vicinity of $\gamma$
peak 186.7 keV expected in the $2\varepsilon$ decay of 190Pt to the $2^{+}$
186.7 keV excited level of 190Os is presented in Fig. 6. The half-life limits
for the transitions to higher excited levels of 190Os were obtained in a
similar way. The limits are presented in Table 4.

Figure 6: The energy spectrum gathered with the Pt sample in the vicinity of
$\gamma$ peak 186.7 keV expected in the $2\varepsilon$ decay of 190Pt to the
$2^{+}$ 186.7 keV excited level of 190Os. The fit of the data is shown by
solid line, while the excluded peak is presented by dashed line. The dotted
lines show the peaks of 235U and 226Ra included in the background model. The
energies of the background and expected $\gamma$ peaks are in keV.

In the $0\nu$ double-electron capture in 190Pt the energy excess is assumed to
be emitted by bremsstrahlung $\gamma$-ray quanta with an energy
$E_{\gamma}=Q_{2\beta}-E_{b1}-E_{b2}-E_{exc}$. To estimate values of $\lim S$
for the $0\nu$ double-electron captures from $K$ and $L$ shells, the
experimental spectrum was fitted in the regions of the expected peaks with
energies (for the g.s. to g.s. transition) $1253.6\pm 0.4$ keV, $1315.5\pm
1.4$ keV and $1377.5\pm 2.5$ keV for the $0\nu 2K$, $0\nu KL$, and $0\nu 2L$
captures, respectively. The variations of the expected peaks energy are due to
the $Q_{2\beta}$ value uncertainty and different binding energies of the $L$
atomic shells. Thus, the energy of a peak searched for was free parameter of
the fits within the variations. A straight line was taken to describe the
continuous background. The background model included background $\gamma$ peaks
present in the energy intervals of the fits: 1238.1 keV of 214Bi (to estimate
a $\lim S$ for the $0\nu 2K$ decay), and 1377.7 keV of 214Bi plus 1384.3 keV
of 110mAg (in the case of the $0\nu 2L$ capture). The results of the fits are
shown in Fig. 7. The biggest peak areas were taken to derive the $\lim S$
values for the peaks searched for (see Table 4).

Figure 7: The energy spectrum gathered with the Pt sample in the energy
intervals where the $\gamma$ peaks from the $0\nu 2K$ (a), $0\nu KL$, and
$0\nu 2L$ (c) captures in 190Pt to the ground state of 190Os are expected. The
fits of the data are shown by solid lines (see text for details of the
background models), while the excluded peaks are presented by dashed lines.
The horizontal lines (above the arrows labelling the energy of the peaks
searched for) show the energy intervals corresponding to the uncertainties of
the expected energies of the peaks. The energy of the $\gamma$ peaks are in
keV.

Limits on the neutrinoless $2\varepsilon$ transitions to the excited levels of
the daughter were set by using the $\lim S$ values obtained to estimate the
half-life limits for the two-neutrino processes. However, the detection
efficiencies for the $0\nu$ modes are slightly smaller due to the energy
transfer to the bremsstrahlung $\gamma$-ray quanta (absent in the $2\nu$ decay
mode, where the energy is carried out by neutrinos).

The $0\nu KN$ transition in 190Pt to the 1,2 1326.9 keV excited level of 190Os
is of special interest: the decay can be an exactly resonant, up to six orders
of magnitude faster, if the degeneracy parameter
$\Delta=Q_{2\beta}-E_{exc}-E_{b1}-E_{b2}\sim 10$ eV. However, with the
uncertainties of the decay energy ($\pm 0.4$ keV [25]) and of the 1326.9-keV
level energy ($\pm 1.0$ keV [27]) the degeneracy parameter lies in the
interval from $-0.13$ keV to $+0.26$ keV with a combined uncertainty $\pm 1.1$
keV, that is too big to make a clear conclusion about resonance enhancement of
the transition. Nevertheless, development of experimental techniques to study
the decay at, as much as possible, high sensitivity level is an important
task. The energy spectrum gathered with the Pt sample was fitted in the energy
interval $1310-1357$ keV to estimate a half-life limit on the decay. The
background was described by a straight line and by a peak at 1332.5 keV
($\gamma$-peak of 60Co). The position of the peak searched for was bounded
within $\pm 1$ keV, while the peak width was fixed as FWHM $=2.33$ keV
according to the formula (3). The fit (with $\chi^{2}/$n.d.f.$=41.4/88=0.47$)
returned a peak area $S=29\pm 32$ counts888If no bounds were set to the peak
parameters, the energy of the peak was $1328.1(4)$ keV, the peak width FWHM
$=2.28(40)$ keV, and the peak area $S=41(22)$ counts.. Considering the
peculiarity as a statistical fluctuation we have obtained $\lim S=81$ counts.
Taking into account the detection efficiency for $\gamma$-ray quanta with
energy 1326.9 keV ($\eta=0.0488$), the resonant $0\nu KN$ transition in 190Pt
to the 1,2 1326.9 keV excited level of 190Os is limited as:

$T_{1/2}^{0\nu KN}~{}(^{190}$Pt, g.s.$~{}\rightarrow~{}1326.9$
keV$)~{}\geq~{}2.5\times 10^{16}$ yr.

The energy spectrum in the region of interest, the result of the fit and
excluded peak for the possible resonant $0\nu KN$ transition in 190Pt to the
1,2 1326.9-keV excited level of 190Os is shown in Fig. 8 (a).

The bound on the transition to the 1326.9 keV level of 190Os is limited by
60Co background, that can be attributed entirely to the setup contamination
(see Table 3) and not to the Pt sample. One can try to estimate a limit on the
transition by analysis of the no-sample spectra subtracted from the data taken
with the Pt sample, eliminating in such a way the 60Co background. A fit of
the background-subtracted spectrum by a straight line (to describe the
continuous background) plus a Gaussian peak with the bounded position and
fixed width (the effect searched for) is shown in panel (b) of Fig. 8999A fit
of the data with no bounds on the peak position and width returned the peak
area $S=34(17)$ counts and the peak position 1327.8(5) keV, however, with an
abnormally small peak width FWHM $=0.4(3)$ keV.. However, statistical
fluctuations in the background-subtracted spectrum are rather big due to a
comparatively short time of the background measurement. As a result, the limit
from the background-subtracted spectrum ($\lim S=109$ counts) is worse than
the one from the sample-only spectrum. Nevertheless, the both approaches give
comparable results providing a useful cross-check of the analysis.

Figure 8: The energy spectra gathered with the Pt sample (with no sample) by
the detector system where the $\gamma$ peak from possible resonant $0\nu KN$
transition to the 1,2 1326.9 keV level of 190Os is expected (a). The
background-subtracted spectrum is shown in panel (b). The fits of the data are
shown by solid lines (see text for details of the background models), while
the excluded peaks are presented by dashed lines.

The $0\nu LM$ transition of 190Pt to the $(0,1,2)^{+}$ 1382.4(2) keV level of
190Os can be characterized as “near” resonant one since the degeneracy
parameter in this case is rather big $\Delta=(2.9-6.1)\pm 0.6$ keV. The
estimation of half-life limit for the decay was performed by analysis of the
experimental data in the vicinity of $\gamma$ peak with energy 1195.7(2) keV
emitted in the de-excitation of the 1382.4 keV level. The result of the data
fit in the energy interval $1177-1220$ keV by a simple model constructed from
a straight line and a Gaussian peak at 1195.7(2) keV with a fixed width is
presented in Fig. 9.

Figure 9: The energy spectrum in the vicinity of $\gamma$ peak 1195.7 keV
expected in the near resonant $0\nu LM$ transition of 190Pt to the
$(0,1,2)^{+}$ 1382.4 keV level of 190Os (b). The fit of the data is shown by
solid line, while the excluded peaks are presented by dashed line.

The annihilation peaks at 511 keV were analyzed in the data with (without) the
Pt sample to estimate half-life limits for the electron capture with positron
emission in 190Pt to the energetically allowed transitions to the ground state
and to the excited 186.7 keV level of the daughter. A fit of the data with
(without) the Pt sample gives the 511-keV peak area $S=3143\pm 71$ counts for
measurement time 8946 h ($S=266\pm 18$ counts for 674 h) that leads to the
difference: $-388\pm 249$ counts that results in $\lim S=134$ counts. The
energy spectrum taken with the Pt sample and the background spectrum in the
vicinity of the 511 keV annihilation peak with the results of fits are shown
in Fig. 10. The detection efficiencies are slightly different for the decays
to the ground state and to the first 186.7 keV excited level of the daughter,
as well as for the $2\nu$ and $0\nu$ modes of the decays. The obtained half-
life limits are presented in Table 4.

Figure 10: The energy spectrum measured with the Pt sample for 8946 h in the
vicinity of the 511 keV annihilation peak. The background data (measured over
674 h, normalized on 8946 h) are shown by dots. The fit of the data obtained
with the Pt sample is shown by solid line, the fit of the background spectrum
is presented by dashed line.

The obtained limits on the possible double-beta processes in 190Pt are higher
up to an order of magnitude than the limits set in the previous study [28].
Moreover, the expected peaks positions for the $0\nu 2\varepsilon$ decays were
far from the actual ones, taking into account a rather different $Q_{2\beta}$
value and much bigger uncertainty at the time when the experiment [28] was
carried out. For this reason, the half-life limits for the $0\nu 2K$, $0\nu
KL$ and $0\nu 2L$ captures can be considered as obtained for the first time.

However, the sensitivity of the present study is very far from theoretical
estimations of the 190Pt decay probability. For instance, the calculations of
the 190Pt half-life relative to the resonant $0\nu 2\varepsilon$ transitions
to the 1326.2 keV [1382.4 keV] level of 190Os are in the interval
$T_{1/2}\sim(3.3\times 10^{26}-1.6\times 10^{30})$ yr [$T_{1/2}\sim(1.0\times
10^{30}-6.5\times 10^{30})$ yr] (assuming the effective Majorana neutrino mass
$\langle m_{\nu}\rangle=0.1$ eV and the weak axial-vector coupling constant
$g_{A}=1.27$) (see Table IV in [24]). Nevertheless, achievement of the
sensitivity on the level of $T_{1/2}\sim(10^{26}-10^{27})$ yr looks realistic
taking into account the great progress in the ultralow-background HPGe [14]
and low-temperature bolometers [13, 15, 16, 41] detection techniques, the
experimental approaches that look the most promising for possible large-scale
searches for the resonant $0\nu 2\varepsilon$ decays.

###  3.3 Limit on the $2\beta^{-}$ decay of 198Pt to the first excited level
of 198Hg

A limit on the $2\beta^{-}$ transition of 198Pt to the $2^{+}$ 411.8 keV
excited level of 198Hg was set by analysis of the energy spectrum measured
with the Pt sample. The spectrum was fit in the energy interval $390-430$ keV
by a straight line (to describe the continuous background) and a Gaussian
function at 411.8 keV with the peak width determined by the formula (3)
returning an effect area $S=-13\pm 26$ counts, that corresponds to $\lim S=31$
counts (the energy spectrum in the region of the fit is presented in Fig. 11).
Taking into account the detection efficiency ($\eta=0.0414$ both for the
$2\nu$ and $0\nu$ modes of the decay), the following limit was obtained:

$T_{1/2}^{(2\nu+0\nu)2\beta^{-}}(^{190}$Pt, g.s.$~{}\rightarrow~{}411.8$
keV$)\geq~{}3.2\times 10^{19}$ yr.

Figure 11: Part of the energy spectrum measured with the Pt sample where a
peak with energy 411.8 keV after the $2\beta^{-}$ decay of 198Pt to the first
$2^{+}$ 411.8 keV excited level of 198Hg is expected. The fit of the data is
presented by solid line, the excluded $\gamma$ peak is shown by dashed line.

The limit is one order of magnitude higher than that set in the previous
experiment [28]. Theoretical calculations for the $2\nu 2\beta^{-}$ decay of
198Pt to the ground state of the daughter are in the interval $T_{1/2}\sim
5\times 10^{21}-5\times 10^{23}$ yr [42, 43]. Similar estimations give
semiempirical formulae: $T_{1/2}=(3.0\pm 1.5)\times 10^{23}$ yr [44] and
$T_{1/2}=3.3\times 10^{22}$ yr [45]. A probability of the $2\nu 2\beta^{-}$
decay to the $2^{+}$ 411.8 keV excited level is expected to be several orders
of magnitude lower due to a smaller value of the phase space factor and the
spin change. The theoretical estimations of the $0\nu 2\beta^{-}$ decay of
198Pt to the ground state of 198Hg are much higher: $T_{1/2}\sim 3\times
10^{26}-2\times 10^{28}$ yr [42, 46, 47, 48]101010The calculations were
performed for the effective Majorana neutrino mass $\langle
m_{\nu}\rangle=0.1$ eV and $g_{A}\approx 1.27$., while the $0\nu 2\beta^{-}$
transition to the $2^{+}$ 411.8 keV excited level of 198Hg is expected to be
further suppressed. Thus, the sensitivity of the present experiment is rather
far from the theoretical predictions for the two-neutrino mode of the decay,
not to say for the neutrinoless process.

All the limits obtained in the present experiment are summarized in Table 4
where also results of the previous study [28] are given for comparison.

##  4 Summary and conclusions

A high-purity disk-shaped platinum sample with mass 148 g was measured in an
ultralow background HPGe-detector $\gamma$-ray spectrometry system located 225
m underground at the HADES laboratory over 8946 hours aiming at searching for
double-beta decay of 190Pt and 198Pt with emission of $\gamma$-ray quanta. The
isotopic composition of the platinum sample has been measured with high
precision using inductively coupled plasma mass spectrometry. No effect was
observed but lower limits on the half-lives for the different channels and
modes of the decays of 190Pt were set on the level of $\lim T_{1/2}\sim
10^{14}-10^{16}$ yr. A possible exact resonant $0\nu KN$ transition to the 1,2
1326.9 keV level of 190Os is limited for the first time as $T_{1/2}\geq
2.5\times 10^{16}$ yr. A new improved limit is set for the $2\beta^{-}$
transition of 198Pt to the $2^{+}$ 411.8 keV excited level of 198Hg as
$T_{1/2}\geq 3.2\times 10^{19}$ yr. All the obtained limits exceed the
previously obtained values up to one order of magnitude mainly thanks to a
substantially bigger exposure ($55.2$ kg$\times$day in the present work and
$3.2$ kg$\times$day in [28]). However, the sample geometry was not optimised
since it was an existing sample foreseen for another project. Thus, a further
improvement of the present experiment sensitivity can be achieved by decrease
of the sample thickness by production of thin disk with a diameter comparable
to the detectors size.

The sensitivity of the experiment to the 190Pt decays, particularly to the
potentially resonant transition to the 1326.9 keV level of 190Os, could be
advanced by using platinum enriched in the isotope 190Pt, increasing the
exposure and detection efficiency by utilization of thin samples and multi-
crystal system of HPGe $\gamma$-ray detectors or low-temperature bolometers.
However, such an experiment could be considered after more accurate
determination of the 1326.9-keV level energy (presently known with a rather
big uncertainty $\pm 1.0$ keV). Nevertheless, we realize that implementation
of such an experiment is practically a rather difficult task, first of all due
to the inaccessibility of methods for enrichment of platinum isotopes in the
hundreds kilograms scale requested for a competitive experiment (in terms of
the Majorana neutrino mass) even in a case of an exact resonant transition.

##  5 Acknowledgments

This work received support from the EC-JRC open access scheme EUFRAT under
Horizon-2020, project number 2018-35375-4 (PLATOS). O.G. Polischuk and M.V.
Romaniuk were supported in part by the project “Investigation of double-beta
decay, rare alpha and beta decays” of the program of the National Academy of
Sciences of Ukraine “Laboratory of young scientists” (the grant number
0120U101838). F.A. Danevich, D.V. Kasperovych and V.I. Tretyak were supported
in part by the National Research Foundation of Ukraine Grant No. 2020.02/0011.

## References

  * [1] M. Goeppert-Mayer, Double Beta-Disintegration, Phys. Rev. 48 (1935) 512.
  * [2] A. Barabash, Precise Half-Life Values for Two-Neutrino Double-$\beta$ Decay: 2020 Review, Universe 6 (2020) 159.
  * [3] J. Schechter, J.W.F. Valle, Neutrinoless double-$\beta$ decay in SU(2)$\times$U(1) theories, Phys. Rev. D 25 (1982) 2951.
  * [4] F.F. Deppisch, M. Hirsch, H. Päs, Neutrinoless double-beta decay and physics beyond the standard model, J. Phys. G 39 (2012) 124007.
  * [5] S.M. Bilenky, C. Giunti, Neutrinoless double-beta decay: A probe of physics beyond the Standard Model, Int. J. Mod. Phys. A 30 (2015) 1530001.
  * [6] S. Dell’Oro, S. Marcocci, M. Viel, F. Vissani, Neutrinoless Double Beta Decay: 2015 Review, Advances in High Energy Physics 2016 (2016) 2162659.
  * [7] M.J. Dolinski, A.W.P. Poon, W. Rodejohann, Neutrinoless Double-Beta Decay: Status and Prospects, Annu. Rev. Nucl. Part. Sci. 69 (2019) 219.
  * [8] A. Giuliani, A. Poves, Neutrinoless Double-Beta Decay, Advances in High Energy Physics 2012 (2012) 857016.
  * [9] O. Cremonesi, M. Pavan, Challenges in Double Beta Decay, Advances in High Energy Physics 2014 (2014) 951432.
  * [10] A. Gando et al. (KamLAND-Zen Collaboration), Search for Majorana Neutrinos Near the Inverted Mass Hierarchy Region with KamLAND-Zen, Phys. Rev. Lett. 117 (2016) 082503.
  * [11] G. Anton et al. (EXO-200 Collaboration), Search for Neutrinoless Double-$\beta$ Decay with the Complete EXO-200 Dataset, Phys. Rev. Lett. 123 (2019) 161802.
  * [12] S.I. Alvis et al. (Majorana Collaboration), Search for neutrinoless double-$\beta$ decay in 76Ge with 26 kg yr of exposure from the Majorana Demonstrator, Phys. Rev. C 100 (2019) 025501.
  * [13] O. Azzolini et al., Final Result of CUPID-0 Phase-I in the Search for the 82Se Neutrinoless Double-$\beta$ Decay, Phys. Rev. Lett. 123 (2019) 032501.
  * [14] M. Agostini et al. (GERDA Collaboration), Final Results of GERDA on the Search for Neutrinoless Double-$\beta$ Decay, Phys. Rev. Lett. 125 (2020) 252502.
  * [15] D.Q. Adams et al. (CUORE Collaboration), Improved Limit on Neutrinoless Double-Beta Decay in 130Te with CUORE, Phys. Rev. Lett. 124 (2020) 122501.
  * [16] E. Armengaud et al. (CUPID-Mo Collaboration), New Limit for Neutrinoless Double-Beta Decay of 100Mo from the CUPID-Mo Experiment, Phys. Rev. Lett. 126 (2021) 181802.
  * [17] A.P. Meshik et al., Weak decay of 130Ba and 132Ba: Geochemical measurements, Phys. Rev. C 64 (2001) 035205.
  * [18] M. Pujol et al., Xenon in Archean barite: Weak decay of 130Ba, mass-dependent isotopic fractionation and implication for barite formation, Geochim. Cosmochim. Acta 73 (2009) 6834.
  * [19] A. Meshik et al., Weak Decay of Tellurium and Barium Isotopes in Geological Samples: Current Status, JPS Conf. Proc. 14 (2017) 020702.
  * [20] Yu.M. Gavrilyuk et al., Indications of $2\nu 2K$ capture in 78Kr, Phys. Rev. C 87 (2013) 035501.
  * [21] S.S. Ratkevich et al., Comparative study of the double-K-shell-vacancy production in single- and double-electron-capture decay, Phys. Rev. C 96 (2017) 065502.
  * [22] XENON Collaboration, Observation of two-neutrino double electron capture in 124Xe with XENON1T, Nature 568 (2019) 532.
  * [23] M. Hirsch et al., Nuclear structure calculation of $\beta^{+}\beta^{+}$, $\beta^{+}$EC, and EC/EC decay matrix elements, Z. Phys. A 347 (1994) 151.
  * [24] K. Blaum et al., Neutrinoless double-electron capture, Rev. Mod. Phys. 92 (2020) 045007.
  * [25] M. Wang et al., The AME 2020 atomic mass evaluation (II). Tables, graphs and references, Chinese Phys. C 45 (2021) 030003.
  * [26] J. Meija et al., Isotopic compositions of the elements 2013 (IUPAC Technical Report), Pure Appl. Chem. 88 (2016) 293.
  * [27] B. Singh, J. Chen, Nuclear Data Sheets for A=190, Nucl. Data Sheets 169 (2020) 1.
  * [28] P. Belli et al., First search for double-$\beta$ decay of platinum by ultra-low background HP Ge $\gamma$ spectrometry, Eur. Phys. J. A 47 (2011) 91.
  * [29] G. Audi, A.H. Wapstra, C. Thibault, The AME2003 atomic mass evaluation (II). Tables, graphs and references, Nucl. Phys. A 729 (2003) 337.
  * [30] H. Xiaolong, K. Mengxiao, Nuclear Data Sheets for A=198, Nucl. Data Sheets 133 (2016) 221.
  * [31] E. Andreotti et al., Status of underground radioactivity measurements in HADES (2011), Proceedings of 3rd International Conference on Current Problems in Nuclear Physics and Atomic Energy, Kyiv, 7-12 June 2010, Publishing Department of KINR, Ukraine. P. 601-605.
  * [32] M. Hult, G. Lutter, G. Marissens, H. Stroh, JRC Serving policy and science at the HADES underground research facility - a casebook, EUR 30207 EN, Publications Office of the European Union, Luxembourg (2020). https://doi.org/10.2760/056079, JRC120311
  * [33] J.S.E. Wieslander et al., The Sandwich spectrometer for ultralow-level $\gamma$-ray spectrometry, Appl. Radiat. Isot. 67 (2009) 731.
  * [34] B. Majorovits, I. Abt, M. Laubenstein, O. Volynets, Aluminum as a source of background in low background experiments, Nucl. Instr. Meth. A 647 (2011) 39.
  * [35] P. Belli et al., First observation of $\alpha$ decay of 190Pt to the first excited level ($E_{exc}=137.2$ keV) of 186Os, Phys. Rev. C 83 (2011) 034603.
  * [36] I. Kawrakow et al., The EGSnrc Code System: Monte Carlo simulation of electron and photon transport. Technical Report PIRS-701, National Research Council Canada (2017).
  * [37] G. Lutter, M. Hult, G. Marissens, H. Stroh, F. Tzika, A gamma-ray spectrometry analysis software environment, Appl. Rad. Isot. 134 (2018) 200.
  * [38] O.A. Ponkratenko et al., Event Generator DECAY4 for Simulating Double-Beta Processes and Decays of Radioactive Nuclei, Phys. At. Nucl. 63 (2000) 1282.
  * [39] S.Y.F. Chu, L.P. Ekström and R.B. Firestone, WWW Table of Radioactive Isotopes, database version 1999-02-28 from URL http://nucleardata.nuclear.lu.se/toi/
  * [40] G.J. Feldman, R.D. Cousins, Unified approach to the classical statistical analysis of small signals, Phys. Rev. D 57 (1998) 3873.
  * [41] V. Alenkov et al., First results from the AMoRE-Pilot neutrinoless double beta decay experiment, Eur. Phys. J. C 79 (2019) 791.
  * [42] A. Staudt et al., Calculation of $2\nu$ and $0\nu$ double-beta rates, Europhys. Lett. 13 (1990) 31.
  * [43] M. Hirsch et al., Operator expansion method and nuclear $\beta\beta$ decay, Phys. Rep. 242 (1994) 403.
  * [44] B. Pritychenko, On Double-Beta Decay Half-Life Time Systematics, arXiv:1004.3280 [nucl-th].
  * [45] X.J. Ren et al., Systematic law for half-lives of double-$\beta$ decay with two neutrinos, Phys. Rev. C 89 (2014) 064603.
  * [46] J. Barea et al., Limits on Neutrino Masses from Neutrinoless Double-$\beta$ Decay, Phys. Rev. Lett. 109 (2012) 042501.
  * [47] J. Barea et al., Nuclear matrix elements for double-$\beta$ decay, Phys. Rev. C 87 (2013) 014315.
  * [48] F. Iachello et al., Recent results in double beta decay, Nucl. Part. Phys. Proc. 265 (2015) 25.