The isotope 50 V is present in the natural mixture of vanadium with a very low abundance of 0.250(10)% [1] . Taking into account the mass difference between 50 V and 50 Ti (2207.6 ± 0.4 keV [2] ), and between 50 V and 50 Cr (1038.06 ± 0.30 keV [2] ), both electron capture (EC) of 50 V to 50 Ti and β − decay of 50 V to 50 Cr are possible (the decay scheme of 50 V is shown in Fig. 1 ). However, decays of 50 V to the ground states of 50 Ti and 50 Cr are strongly suppressed by the very large spin change ∆J = 6 in both the cases. The only excited levels on which decay of 50 V can undergo are the 2 + 1553.8 keV level of 50 Ti, and the 2 + 783.3 keV level of 50 Cr. Both the decay channels are fourfold forbidden non-unique (∆J ∆π = 4 + ). Since in both channels decay goes to the excited levels of daughter nuclei, de-excitation γ-ray quanta can be detected by γ spectrometry of a vanadium sample. While the 50 V electron-capture transition to the 2 + 1553.8 keV level of 50 Ti is observed in several experiments, the β − decay of 50 V to the 2 + 783.3 keV level of 50 Cr remains unobserved (despite two claims of detection that have been disproved in the subsequent more sensitive investigations). The history of 50 V decays investigations is summarized in Table 1 (see also recent review [15] ).
The decay of 50 V is of especial interest since the transitions involve several different nuclear matrix elements with the associated different phase-space factors multiplied by the axial-vector coupling constant g A [16] . This constant plays an important role in the neutrinoless double β decay probability calculations [17, 18, 19, 20, 21] . Recent calculations in nuclear shell model [16] result in the following (partial) half-lives for the two decay modes: T EC 1/2 = (5.13 ± 0.07)[(3.63 ± 0.05)] × 10 17 yr given for g A = 1.00 [1.25] ; for the β − -decay branch, T β 1/2 = (2.34 ± 0.02)[(2.00 ± 0.02)] × 10 19 yr. In this work we report measurement of the 50 V EC decay half-life and search for β − decay of the nuclide using HPGe γ spectrometry of a 955 g vanadium sample.
A disk-shaped sample of metallic vanadium with diameter of 100.1 mm and thickness of 19.9 mm with mass of 955.21 ± 0.02 g, provided by Goodfellow Cambridge Ltd was used in the experiment. The vanadium disk was stored underground as soon as it was received by JRC-Geel in 2008 so that cosmogenic activation would be minimized. It was measured using an ultra low-background HPGe-detector system located 225 m underground in the laboratory HADES (Belgium). The detector system, named Pacman, consists of two HPGe-detectors facing each other [22] . The experiment was realized in two stages with different amount of Perspex in the inner volume of the lead/copper shield. At the start not all Perspex was available but due to time constraints it was judged beneficial to start the measurements anyhow. A schematic view of the two setups with HPGe detectors and the vanadium sample is shown in Fig. 2 . The main characteristics of the HPGe detectors are presented in Table 2 , more details can be found in [22, 23] .
At the first stage of the experiment in setup I the vanadium sample was measured for 34.74 d, then the detectors were running for 38.16 d to measure background data without sample. The distance between the detectors Ge10 and Ge11 was 21 mm in setup I. The energy spectra accumulated with the vanadium sample and without sample in setup I are shown in Fig. 3 . Then the experiment was continued in setup II for 110.55 d with the vanadium sample and over 21.70 d to measure background without sample. The distance between the detectors Ge10 and Ge11 was 23 mm in setup II. Additional Perspex pieces were installed in setup II to minimize air inside so that to suppress background due to radon. The energy spectra gathered in setup II are shown in Fig. 4 . The insertion of the Perspex details decreased background caused by 222 Rn daughters. In particular the counting rates in the γ-ray peaks of 214 Bi with energies 609.3 keV and 1764.5 keV were decreased by 3-5 times. 
Energy (keV)
Counts / 5 keV The energy spectra measured in the two setups are rather similar. The majority of the peaks could be assigned to 40 K and nuclides of the 232 Th, 235 U, and 238 U decay chains. Besides, there are also clear peaks of 138 La and 176 Lu in the data taken with the vanadium sample that is evidence of the V-sample contamination by La and Lu. No unidentified peaks were observed.
The energy dependence of the energy resolution in the sum energy spectrum of the detectors Ge11 and Ge10 in setups I and II was estimated by using clear γ-ray peaks with energies E γ = 201.8 keV and 306.8 keV ( 176 Lu), 583.2 keV ( 208 Tl), 609.3 keV and 1120.3 keV ( 214 Bi), 788.7 keV ( 138 La), 911.2 keV ( 228 Ac) as (E γ is in keV):
(1)
Massic activities in the vanadium sample of 40 K, 138 La, 176 Lu, daughters of the 232 Th, 235 U, and 238 U decay chains were calculated with the following formula:
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; η is the γ-ray emission intensity of the corresponding transition; ε is the full energy peak efficiency; m is the sample mass. The detection efficiencies were calculated with EGSnrc simulation package [24, 25] , the events were generated homogeneously in the V sample. The calculations were validated using a liquid solution containing 133 Ba, 134 Cs, 137 Cs, 60 Co, and 152 Eu. The standard deviation of the relative difference between the simulations and the experimental data is 2.5% for γ-ray peaks in the energy interval 53 keV-1408 keV for Ge10 detector, and is 4% for γ-ray peaks in the energy interval 80 keV-1408 keV for Ge11 detector. The estimated massic activities of radioactive impurities in the vanadium sample are presented in Table 3 . There is a clear peak with energy 1553.8 keV in all the energy spectra accumulated with the vanadium sample that can be ascribed to the electron capture decay of 50 V to the 2 + 1553.8 keV level of 50 Ti. The peak is absent in the background data. In order to estimate the half-life of 50 V for the EC decay channel the sum energy spectrum of all the measurements with the vanadium sample was analyzed. A part of the spectrum in the energy region of interest is presented in Fig. 5 . The exposure for 50 V is (2.25 ± 0.09) × 10 22 nuclei of 50 V×yr. The spectrum was fitted in the energy interval (1520-1585) keV by a sum of a first order polynomial function (to describe the continuous distribution near the peak) and by Gaussian function (to describe the γ-ray peak). The fit with a very good value of χ 2 /n.d.f. = 100.2/126 = 0.795 (where n.d.f. is number of degrees of freedom) returns the following peak parameters: energy of the peak is 1553.90 (12) The detection efficiencies for different detectors in the two setups for γ-ray quanta with energy 1553.8 keV were simulated with the help of the EGSnrc package [24, 25] . The detection efficiencies are given in Table 4 .
The half-life of 50 V relative to the electron capture to the 2 + 1553.8 keV level of 50 Ti (T 1/2 ) was calculated by using the following formula: 
where N is number of 50 V nuclei in the sample [N = 2.823(113) × 10 22 ], η i and t i are detection efficiencies and times of measurement for the two detectors in the two setups (given in Table 4 ), S is area of the peak with energy 1553.8 keV obtained by the fit of the data of the sum energy spectrum shown in Fig. 5 (S = 654 ± 27 counts). By using these data the half-life of 50 V has been calculated as T EC 1/2 = [2.774 +0.119 −0.110 (stat)] × 10 17 yr. In addition to the ≈ 0.2% statistical uncertainty of the Monte Carlo simulated detection efficiency we conservatively assess a 4% 2 systematic uncertainty on the calculated detection efficiency of the detector system to the 1553.8 keV γ-ray quanta. An indirect confirmation of a rather small systematic of the detection efficiency can be seen in Table 4 and Fig. 6 where the T EC 1/2 values determined from the data of measurements with two different detectors in setups I and II are presented. The difference between the half-life values is well within the statistical errors, that does demonstrate stability of the half-life result and its independence neither on the detector nor the experimental setup.
Variation of the energy interval of fit from 1520-1540 keV (starting point) to 1570-1585 keV (final point), changes T EC 1/2 up to 1.1%. Finally, we account 4.0% for uncertainty in the number of 50 V nuclei in the sample due to the accuracy of the representative isotopic abundance of the isotope [1] . The summary of the systematic uncertainties is given in Table 5 . A historical perspective of half-life of 50 V is presented in Fig. 7 . It is interesting to note that early experiments claimed too short half-lives. That can be explained, first of all, by utilization of rather low energy resolution detectors like proportional counters and NaI(Tl) scintillation counters (see Table 1 ). Other possible reasons for obtaining a too short half-life can be using nonpure samples, high background with possible interferences of γ rays of different origin (including cosmogenic activation, since most of the earlier experiments were performed in laboratories on the ground level), less good electronics, stability problems of long measurements. The latter point is especially crucial in conditions of a poor energy resolution. 10 [14] ). The results are presented by dots, while the limits are shown by arrows. The early positive claims of EC decay in 50 V with too short half-lives were obtained with low resolution detectors: proportional and NaI(Tl) scintillation counters [4, 5] , NaI(Tl) scintillation counter [7] . The half-lives measured with the help of HPGe detectors in works [10, 11, 12, 13, 14] and in the present study are in a reasonable agreement.
3.3 Limit on β − decay of 50 V to the 2 + 783.3 keV excited level of 50 Cr
There is no peak with energy ≈ 783 keV in the sum energy spectrum that can be interpreted as β − decay of 50 V to the 2 + 783.3 keV excited level of 50 Cr. Thus, we have set a lower half-life limit on the decay with the following formula:
where N is the number of 50 V nuclei in the sample, η i and t i are detection efficiencies (for 783.3 keV γ-ray quanta) and times of measurement for the two detectors in the two setups, and lim S is the number of events of the effect searched for which can be excluded at a given confidence level. The detection efficiencies for different detectors were simulated with the help of the EGSnrc package [24, 25] .
To estimate the value of lim S the sum energy spectrum with exposure (2.25 ± 0.09) × 10 22 nuclei of 50 V×yr was fitted by a background model that includes the effect searched for (a peak , superimposed on nearby intensive peaks, were fixed taking into account their relative intensities in the sub-chains. All the peak widths were fixed taking into account the dependence of the energy resolution on energy of γ-ray quanta (1).
The best fit, achieved in the energy interval 761-818 keV with χ 2 /n.d.f.= 0.815, returned an area 3.3 ± 15.5 counts in an expected 783.3 keV peak that is no evidence of the effect searched for. 3 The fit and excluded peak are shown in Fig. 8 . According to [27] we took lim S = 28.7 counts and, taking into account the detection efficiencies to 783.3 keV γ-ray quanta (given in Table 6 ), obtain the following limit on the β − decay of 50 V to the 2 + 783.3 keV excited level of 50 Cr:
T β 1/2 ≥ 8.9 × 10 18 yr at 90% C.L. The limit is approximately two times weaker than the limit T β 1/2 ≥ 1.9 × 10 19 yr reported in [14] . The sensitivity of the present experiment is lower mainly due to a rather high radioactive contamination of the vanadium sample that produce background in the region of interest.
Therefore, an advanced experiment should utilize a radio-pure vanadium sample. A possibility of a deep purification of vanadium from radioactive impurities has been demonstrated in [14] . Thus, aiming to estimate requirements to experiments able to detect the decay, we assume a level of background already achieved in setup II without sample (see Fig. 4 ). We consider two vanadium containing samples: a metallic vanadium of the natural isotopic composition with the sizes and geometry the same as in the present experiment, and a second one in form of vanadium oxide (V 2 O 5 ), enriched in the isotope 50 V to 50%. We assume the bulk density of enriched vanadium oxide sample to be 0.5 of the solid V 2 O 5 density (3.36 g/cm 3 ). To get the same number of 50 V nuclei (2.82 × 10 22 ), the size of the enriched sample was chosen to be ⊘50 × 2.57 mm, with a distance between the detectors H = 3 mm. Expected background counting rates and the Monte Carlo simulated detection efficiencies of the Pacman setup with the samples are given in Table 7 . The background of the detectors dominates in the experimental conditions, with the contribution from the EC process in 50 V an order of magnitude smaller. While the assumed enriched source contains the same number of 50 V nuclei as the metallic one with the natural isotopic composition, the detection efficiency with the enriched source is about three times higher. As a result, an experiment with enriched source has a higher sensitivity [see Fig. 9 , (a)]. Moreover, utilization of enriched 50 V would allow to observe clearly the β − decay of 50 V (assuming the theoretically predicted half-life T β − 1/2 = 2 × 10 19 yr [16] ) with a 3σ accuracy over about 200 d of data taking, while an experiment utilizing a V-sample of natural isotopic composition needs more than three years to detect the process with a similar accuracy (see Fig. 9, (b) ). (1) in the geometry of the present experiment (with a V-sample ⊘100 × 20 mm and distance between the detectors Ge10 and Ge11 H = 21 mm); (2) with a V 2 O 5 -sample enriched in the isotope 50 V to 50% with sizes ⊘50 × 2.57 mm and H = 3 mm. Only background without sample together with contribution due to the EC decay of 50 V are assumed.
The half-life of 50 V relative to the EC to the 2 + 1553.8 keV level of 50 Ti is measured as T EC 1/2 = (2.77 +0.20 −0.19 ) × 10 17 yr. The value is in agreement with the result of the recent experiment [14] and the theoretical predictions [16] . The β − decay of 50 V to the 2 + 783.3 keV level of 50 Cr is limited as T β 1/2 ≥ 8.9 × 10 18 yr at 90% C.L. The limit is about 2 times weaker than that set in the work [14] . Further improvement of the experiment sensitivity could be achieved by utilization of highly purified vanadium samples. Moreover, using of a sample enriched in 50 V would allow detection of the β − decay. The accuracy of T EC 1/2 will also be improved with a source enriched in 50 V both thanks to improvement of statistics and reduction of the uncertainty in the 50 V isotopic abundance.
This work received support from the EC-JRC open access scheme EUFRAT under Horizon-2020, project No. 22-14. D.V.K. and O.G.P. 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.D. greatly acknowledges the Government of Ukraine for the quarantine measures that have been taken against the Coronavirus disease 2019 that substantially reduced much unnecessary bureaucratic work.