1. Introduction {#S1}
===============

The exponential-decay law is one of the most famous laws of physics, already carved in stone since the pioneering work of Ernest Rutherford \[[@R1]\], Maria Skłodowska-Curie \[[@R2]\] and others. It has withstood numerous tests \[[@R3]--[@R5]\] demonstrating that the decay of a radionuclide can be characterised solely by a single decay constant -- or equivalently by the half-life -- which is invariable in space and time. However, observations of periodic oscillations in measured decay rates of radioactive sources \[[@R6]--[@R13]\] have been heavily debated in the last decade \[[@R6]--[@R25]\]. Controversy arose at two levels: (i) at the observational level, with experimental data sets showing significant differences in stability of decay rates with time, and (ii) at the interpretational level, either ascribing the observed modulations to instabilities in the detection system, or advocating new physics to explain variability in the decay constants.

As much as the instability claims attract interest as inspiration for new physical theories and applications \[[@R14],[@R15]\], if true they would have major implications on traceability and equivalence in the common measurement system of radioactive substances. Variability of decay constants at permille level would limit the precision by which a half-life value could be assigned to a radionuclide, as well as the accuracy by which the SI-unit becquerel could be established through primary standardisation \[[@R26]\] and international equivalence demonstrated through key comparisons and the Système International de Référence (SIR) \[[@R27]\]. The implications at metrological level would eventually affect science built on the decay laws, from renormalisation of activity to a reference date for nuclear dosimetry to precise nuclear dating for geo- and cosmochronology.

At the heart of this controversy are the metrological difficulties inherent to the measurement of half-lives \[[@R28]--[@R30]\]. From a metrological point of view, it is obvious that instruments, electronics, geometry and background may vary due to external influences such as temperature, pressure, humidity and natural or man-made sources of radioactivity. Claims of variability of half-lives on the basis of deviations from an exponential decay curve can only be considered when the instrumental effects have been fully compensated and/or accounted for in the uncertainty budget. Jenkins et al. \[[@R9]\] claim to have done so before proposing their hypothesis that permille sized seasonal variations of decay rates of ^226^Ra and ^36^Cl are caused by solar influences on their decay constants \[[@R6]--[@R8]\]. Evidence has been collected to demonstrate instabilities in the decay of other radionuclides \[[@R10],[@R11]\] and by means of time-frequency analysis periodicity at shorter and longer term than 1 year have been claimed \[[@R11]--[@R13]\]. However, this interpretation is being challenged by the publication of data sets confirming a close adherence to exponential decay with residuals in the 10^−5^ range \[[@R16],[@R18],[@R20],[@R21],[@R23]\].

Authors of both convictions expressed the need for collecting evidence for different radionuclides measured with different detection techniques \[[@R7],[@R11],[@R13],[@R18],[@R23]\]. At national metrology institutes (NMIs) taking responsibility for establishing the unit becquerel, mono-radionuclide sources are kept and regularly measured for standardisation purposes as well as for determining half-lives. In addition, gamma-ray spectrometry laboratories keep records of quality control measurements on their spectrometers which provide useful information on long-term trends in activity measurements of a reference source. In this work, the hypothesis that decay constants vary through solar influence in phase with Earth--Sun orbital distance has been tested through the analysis of a unique collection of activity measurements repeated over periods of 200 days up to four decades at 14 laboratories distributed across the globe.

2. Measurements & analysis {#S2}
==========================

Precise activity measurement series were performed for alpha decay (^209^Po, ^226^Ra series, ^228^Th, ^230^U, ^241^Am), beta minus decay (^3^H, ^14^C, ^60^Co, ^85^Kr, ^90^Sr, ^124^Sb, ^134^Cs, ^137^Cs), electron capture (^54^Mn, ^55^Fe, ^57^Co, ^82,85^Sr, ^109^Cd, ^133^Ba), a mixture of electron capture and positron decay (^22^Na, ^65^Zn, ^207^Bi), and a mixture of electron capture and beta minus/plus decay (^152^Eu). More than 60 data sets were collected, some of which were performed over several decades. Some data sets excel in precision, others reveal vulnerability of different measurement techniques to external conditions. Characteristics of the data sets are summarised in [Table 1](#T1){ref-type="table"}.

The measurement techniques employed are as follows: ionisation current measurements in a re-entrant ionisation chamber (IC) or a hospital calibrator (HIC) \[[@R31],[@R32]\], net area analysis of full-energy γ-ray peaks (and integral spectrum counting) by γ-ray spectrometry with a HPGe detector (HPGe) \[[@R33]\], particle counting in a planar silicon detector in quasi-2πconfiguration (PIPS) \[[@R34]\], X-ray counting at a small defined solid angle with a gas-filled proportional counter (PC) \[[@R35],[@R36]\], live-timed β--γ anti-coincidence counting (LTAC) \[[@R37]\], triple-to-double coincidence counting with a liquid scintillation vial and three photodetectors (TDCR) \[[@R38]\], liquid scintillation counting (LSC) \[[@R38]\], particle and photon counting in a sandwich CsI (Tl) spectrometer (CsI) \[[@R39]\], internal gas counting (IGC) \[[@R40]\], and α-particle counting at a small defined solid angle with a large planar silicon detector (αDSA) \[[@R35],[@R36]\]. An overview of standardisation techniques and their sources of error can be found in the special issues 44(4) and 52(3) of Metrologia \[[@R41],[@R42]\] and references in \[[@R25],[@R28]\].

Exponential decay curves were fitted to the data and the residuals were inspected for annual modulations. The data sets were first compensated for (1) the presence of occasional outlier values, (2) abrupt systematic changes in the detector response, e.g. due to replacement of the electronics or recalibrations of the instrument, and (3) systematic drift extending over periods of more than 1 year, e.g. due to gas loss from an ionisation chamber, uncompensated count loss through pulse pileup in a spectrometer, activity build-up from decay products in a source, etc. The residuals were binned into 8-day periods of the year and averaged to obtain a reduced set of (maximum) 46 residuals evenly distributed over the calendar year. To the averaged residuals, a sinusoidal shape *A* sin(2π(*t*+*a*)/365) has been fitted in which *A* is the amplitude, *t* is the elapsed number of days since New Year, and *a* is the phase shift expressed in days. The fitted amplitude values can be considered insignificant if they are of comparable magnitude as their estimated standard uncertainty (see [Table 1](#T1){ref-type="table"}).

3. Discussion {#S3}
=============

The controversy started with the interpretation \[[@R7],[@R8]\] of *A* ≈ 0.15% modulations in the decay rate measurements of a sealed ^226^Ra reference source in an IC at the PTB between 1983 and 1998. The averaged residuals, shown in [Fig. 1A](#F1){ref-type="fig"}, have a sinusoidal shape with amplitude *A* = 0.083 (2)% and phase *a* = 59 days. An explanation through solar influence on the alpha or beta decay constants of nuclides in the ^226^Ra decay series seems unlikely, since the residuals are out of phase with the annual variation of the inverse square of the Sun--Earth distance, 1/*R*^2^ (renormalised to 0.15% amplitude in the [Figs. 1](#F1){ref-type="fig"}--[@R2] of this work). The real cause is of instrumental nature, since the modulations were significantly reduced after changing the electrometer of the IC \[[@R22],[@R25]\]. There is a remarkable correlation with average seasonal changes of radon concentration in air (*A* = 16 (2)%, *a* = 57 days) measured inside the laboratory from 2010 to 2016, but causality has not been proven.

At other institutes, annual modulations of smaller amplitude and different phase have been observed, which demonstrates the local character of the non-exponential behaviour. The data sets for ^226^Ra show a different level of instrumental instability, but the most stable ^226^Ra measurements prove invariability of its decay constant against annual modulations within 0.0025% to 0.005%. An example is shown in [Fig. 1B](#F1){ref-type="fig"}, comprising 4000 ^226^Ra ionisation current measurements over a period of 22 years at the NPL.

Stability is best achieved where the detector efficiency is least influenced by geometrical and environmental variations and where the signal of the radiation is easily separated from interfering signals and electronic noise. For example, measuring ^241^Am decay through alpha-particle detection with close to 100% detection efficiency would typically be more stable than through fractional detection of its low-energy photon emissions in a gas-filled pro-portional counter. For the alpha emitters, ^209^Po, ^226^Ra, ^230^U, and ^241^Am, the invariability of the decay constants was confirmed within the 10^−5^ level.

Comparably lower stability could be anticipated for beta-minus decay. Parkhomov \[[@R10]\] found 7 data sets of beta-decaying radionuclides exhibiting periodic variations of 0.1% to 0.3% amplitude with a period of 1 year. Fischbach et al. \[[@R8],[@R14],[@R15]\] suggested new theories in which the variable flux of anti-neutrinos from the Sun would significantly modulate the probability for β^−^ emission. From metrological point of view, instability in the detection efficiency for a pure beta emitter can be expected due to the continuous energy distribution of the beta particle which makes the count rate subject to threshold variations at the low-energy side and possibly incomplete detection probability at the high-energy side. However, measurements based on γ-ray emission subsequent to the β^−^ emission -- possibly through the decay of a short-lived daughter nuclide -- can be made more robust.

High-quality measurement data were collected for β^−^ emitters in [Table 1](#T1){ref-type="table"}, mostly obtained by IC but also with primary activity measurement techniques such as the triple-to-double coincidence ratio (TDCR) method and live-timed 4π β--γ anti-coincidence counting (LTAC). It was demonstrated for ^36^Cl \[[@R20]\], ^60^Co ([Table 1](#T1){ref-type="table"}) and ^90^Sr/^90^Y \[[@R23]\] that primary standardisation techniques like TDCR and LTAC are more stable than routine counting techniques, because each measurement provides information about the detection efficiency and automatically corrects for its fluctuations. Some IC measurements show remarkable stability, too, and refute the conclusions made about variability of the decay constants as well as the hypothesis of a significant solar influence on the decay rate. In [Fig. 2A](#F2){ref-type="fig"}, averaged residuals for ^134^Cs in an IC demonstrate stability within the 10^−5^ range. Evidence of stability down to the 10^−5^ level was found for the beta minus emitters ^60^Co, ^90^Sr, ^124^Sb, ^134^Cs and ^137^Cs, and down to the 10^−4^ level for ^3^H, ^14^C, and ^85^Kr. These results are in direct contradiction with the permille level oscillations for ^3^H, ^60^Co, ^90^Sr, and ^137^Cs reported by Parkhomov \[[@R10]\] and Jenkins et al. \[[@R11]\].

Radionuclides disintegrating by electron capture (EC) and β^+^ decay -- ^22^Na, ^54^Mn, ^55^Fe, ^57^Co, ^65^Zn, ^82^Sr/^82^Rb+^85^Sr, ^109^Cd, ^133^Ba, ^152^Eu, and ^207^Bi -- were investigated by the same techniques as α and β^−^ emission and, also here, stability within the 10^−5^ to 10^−4^ range was observed in most cases. An example is shown in [Fig. 2B](#F2){ref-type="fig"} for ^22^Na measured in the same period with the same IC as ^134^Cs in [Fig. 2A](#F2){ref-type="fig"}. The tiny modulations in the residuals for both nuclides are highly correlated, which is most likely a seasonal effect of instrumental origin. Clear evidence of annual modulations being of instrumental origin has been found in thousands of γ-ray spectrometry measurements with 8 HPGe detectors at the SCK, as shown in [Fig. 3](#F3){ref-type="fig"}: the modulations in measured decay rates for the alpha decay of ^241^Am and mixed EC, β^−^, and β^+^ decay of ^152^Eu are highly correlated but the amplitude differs from one detector to another. In other words, the modulations are linked to the instrument, not to the type of decay.

4. Conclusions {#S4}
==============

The experimental data in this work are typically 50 times more stable than the measurements on which recent claims for solar influence on the decay constants were based. The observed seasonal modulations can be ascribed to instrumental instability, since they vary from one instrument to another and show no communality in amplitude or phase among -- or even within -- the laboratories. The exponential decay law is immune to changes in Earth--Sun distance within 0.008% for most of the investigated α, β^−^, β^+^ and EC decaying nuclides alike.

Owing to the invariability of decay constants, there is no impediment to the establishment of the becquerel through primary standardisation at 0.1% range accuracy nor to the demonstration of equivalence of activity at international level over a time span of decades. It is normal for repeated activity measurements to show varying degrees of instability of instrumental and environ-mental origin and such auto-correlated variability should be taken into account next to statistical variations when setting alarm levels in quality control charts. Taking into account such instabilities and adhering to proper uncertainty propagation, no fundamental objections need to be made against half-life measurement with sub-permille uncertainties, nor against applying exponential decay formulas to calculate activity at a future or past reference time or to perform accurate nuclear dating.

Funded by SCOAP^3^.

The authors thank all past and present colleagues who contributed directly or indirectly to the vast data collection over different periods spanning six decades.

![**A.** Annual average residuals from exponential decay for ^226^Ra activity measurements with an IC at PTB from 1983 to 1998. The line represents relative changes in the inverse square 1/*R*^2^ of the Earth--Sun distance, normalised to an amplitude of 0.15%.\
**B.** Same for ^226^Ra activity measurements with the Vinten IC of NPL from 1993 to 2016, after renormalisation per calendar year.](nihms835317f1){#F1}

![**A.** Annual average residuals from exponential decay for ^134^Cs activity measurements with the IG12 IC at the JRC from 2010 to 2016.\
**B.** Same for ^22^Na.](nihms835317f2){#F2}

![Amplitude of average annual oscillations in the decay rates of ^241^Am and ^152^Eu measured by γ-ray spectrometry with 8 HPGe detectors at SCK between 2008 and 2016. The index refers to the detector number. A mixed ^241^Am-- ^152^Eu point source was measured 166--466 times in a fixed geometry at about 11 cm from the endcap using the 59 keV line of ^241^Am and the 122 keV, 779 keV and 1408 keV lines of ^152^Eu.](nihms835317f3){#F3}

###### 

Characteristics of the decay rate measurement sets analysed. The method acronyms are explained in the text. The period indicates the first and last year in which data were collected. The standard deviation is an indication of the uncertainty on the annual averaged data (maximum 46 data, covering 8-day periods), derived from the spread of the input data and the inverse square root of the number of values in each data group. The amplitude and phase are the result of the fit of a sinusoidal function to the averaged data. In bold are the amplitudes at 10^−6^--10^−5^ level. The estimated standard uncertainty on the amplitude is indicated between parentheses, its order of magnitude corresponding to that of the last digit of the value of *A*.

  --------------------------------------------------------------------------------------------------------------------------------------------------------
  Decay\           Nuclide                  Laboratory   Method                       Period\      \#Data   Rel. std\   Amplitude *A*\    Phase shift α\
  mode(s)                                                                             (year)                dev in %    in %              in days
  ---------------- ------------------------ ------------ ---------------------------- ------------ -------- ----------- ----------------- ----------------
  α                ^209^Po                  JRC          PIPS                         2013--2016   1539     0.024       **0.006** (5)     6

  α + β^−^         ^226^Ra                  PTB          IC                           1983--1998   1973     0.011       0.083 (2)         59

  α + β^−^         ^226^Ra                  PTB          IC                           1999--2016   2184     0.005       0.016 (1)         194

  α + β^−^         ^226^Ra                  ENEA         IC                           1992--2015   161      0.025       0.043 (5)         324

  α + β^−^         ^226^Ra                  NIST         IC \#1                       2008--2016   99       0.016       0.015 (3)         255

  α + β^−^         ^226^Ra                  NIST         IC \#2                       2012--2016   272      0.036       **0.002** (8)     8

  α + β^−^         ^226^Ra                  BIPM         IC                           2001--2015   136      0.015       **0.004** (3)     4

  α + β^−^         ^226^Ra                  JRC          IC                           2005--2015   1737     0.005       **0.003** (2)     363

  α + β^−^         ^226^Ra                  NPL          IC \#1                       1993--2016   4055     0.014       **0.0025** (18)   60

  α + β^−^         ^226^Ra                  NPL          IC \#2                       1993--2016   3996     0.005       **0.005** (1)     73

  α + β^−^         ^226^Ra                  NMISA        IC                           1992--2015   276      0.343       0.106 (60)        67

  α + β^−^         ^226^Ra                  ANSTO        IC                           2012--2015   700      0.015       **0.005** (3)     256

  α + β^−^         ^226^Ra                  ANSTO        HIC                          2008--2014   1749     0.077       0.009 (18)        82

  α + β^−^         ^226^Ra                  LNHB         IC \#1                       1998--2016   455      0.026       0.026 (6)         328

  α + β^−^         ^226^Ra                  LNHB         IC \#2                       1998--2016   498      0.028       0.042 (7)         294

  α                ^228^Th                  NIST         IC                           1968--1978   70       0.107       0.031 (22)        327

  α                ^230^U                   JRC          αDSA, PIPS, CsI, LSC, HPGe   2010--2011   5451     0.083       **0.007** (7)     173

  α                ^241^Am                  JRC          PC                           2004--2008   245      0.022       0.101 (16)        104

  α                ^241^Am                  SCK          HPGe \#8                     2008--2016   430      0.13        0.024 (28)        55

  α                ^241^Am                  SCK          HPGe \#26                    2013--2016   166      0.12        **0.002** (23)    304

  α                ^241^Am                  SCK          HPGe \#11                    2008--2016   402      0.12        0.055 (22)        242

  α                ^241^Am                  SCK          HPGe \#16                    2008--2016   382      0.13        0.079 (26)        290

  α                ^241^Am                  SCK          HPGe \#25                    2011--2016   245      0.12        0.079 (22)        236

  α                ^241^Am                  SCK          HPGe \#10                    2008--2016   466      0.14        0.115 (27)        259

  α                ^241^Am                  SCK          HPGe \#27                    2011--2015   238      0.45        0.095 (91)        280

  α                ^241^Am                  SCK          HPGe \#13                    2008--2016   434      0.12        0.167 (26)        235

  α                ^241^Am                  PTB          LSC                          2014--2016   574      0.004       **0.0006** (7)    260

  β^−^             ^3^H                     JRC          LSC                          2002--2014   706      0.112       0.048 (24)        197

  β^−^             ^3^H                     NIST         IGC                          1961--1999   21       0.75        0.18 (20)         149

  β^−^             ^14^C                    JRC          LSC                          2002--2014   706      0.075       0.013 (16)        92

  β^−^             ^14^C                    NMISA        TDCR                         1994--2014   32       0.250       0.067 (80)        59

  β^−^             ^60^Co                   NIST         IC                           1968--2007   250      0.050       **0.007** (7)     0

  β^−^             ^60^Co                   NIST         LTAC + IC                    2006--2014   26 + 7   0.036       **0.007** (9)     18

  β^−^             ^60^Co                   JSI          HPGe \#1--6                  1998--2016   15254    0.079       0.041 (14)        161

  β^−^             ^85^Kr                   NIST         IC                           1980--2007   98       0.035       0.036 (15)        153

  β^−^             ^90^Sr                   PTB          TDCR                         2013--2014   4493     0.009       **0.004** (2)     362

  β^−^             ^90^Sr                   PTB          IC                           1989--2016   2207     0.009       0.018 (2)         26

  β^−^             ^124^Sb                  JRC          IC                           2007         59       0.005       **0.003** (2)     241

  β^−^             ^134^Cs                  JRC          IC                           2010--2015   1065     0.002       **0.0051** (5)    48

  β^−^             ^137^Cs                  IRA          IC                           1984--2012   276      0.043       0.018 (9)         342

  β^−^             ^137^Cs                  NRC          IC \#1--3                    1995--2009   62       0.074       **0.006** (22)    147

  β^−^             ^137^Cs                  PTB          IC                           1997--2016   2149     0.005       0.014 (1)         29

  β^−^             ^137^Cs                  NIST         IC                           1968--2011   254      0.034       **0.004** (6)     33

  β^+^, EC         ^22^Na                   JRC          IC                           2010--2016   443      0.003       **0.0047** (6)    53

  EC               ^54^Mn                   JRC          IC                           2006--2009   156      0.007       **0.005** (1)     28

  EC               ^54^Mn                   PTB          IC                           2010--2016   716      0.011       0.014 (2)         78

  EC               ^55^Fe                   JRC          IC                           2004--2005   595      0.007       **0.004** (3)     187

  EC               ^57^Co                   NIST         IC                           1962--1966   97       0.089       0.055 (22)        324

  EC, β^+^         ^65^Zn                   JRC          IC                           2002--2003   140      0.026       **0.008** (4)     163

  EC(, β^+^)       ^82^Sr/^82^Rb + ^85^Sr   NIST         IC                           2007--2008   158      0.011       **0.0006** (27)   240

  EC(, β^+^)       ^82^Sr/^82^Rb            NIST         HPGe                         2007--2008   23       0.46        0.073 (75)        255

  EC               ^109^Cd                  JRC          IC                           2006--2010   125      0.017       0.015 (4)         18

  EC               ^109^Cd                  JSI          HPGe \#3, 4                  1998--2016   5414     0.139       0.035 (24)        346

  EC               ^109^Cd                  NIST         IC                           1976--1981   167      0.058       0.013 (15)        220

  EC               ^133^Ba                  NIST         IC                           1979--2012   131      0.042       0.028 (8)         74

  EC, β^−^, β^+^   ^152^Eu                  IAEA         HPGe \#1, 2                  2010--2016   143      0.113       0.020 (24)        162

  EC, β^−^, β^+^   ^152^Eu                  SCK          HPGe \#8                     2008--2016   1228     0.10        **0.006** (19)    242

  EC, β^−^, β^+^   ^152^Eu                  SCK          HPGe \#26                    2013--2016   499      0.10        0.027 (23)        178

  EC, β^−^, β^+^   ^152^Eu                  SCK          HPGe \#11                    2008--2016   1168     0.10        0.048 (21)        280

  EC, β^−^, β^+^   ^152^Eu                  SCK          HPGe \#16                    2008--2016   1260     0.08        0.062 (18)        285

  EC, β^−^, β^+^   ^152^Eu                  SCK          HPGe \#25                    2011--2016   723      0.10        0.080 (23)        213

  EC, β^−^, β^+^   ^152^Eu                  SCK          HPGe \#10                    2008--2016   1374     0.08        0.094 (16)        206

  EC, β^−^, β^+^   ^152^Eu                  SCK          HPGe \#27                    2011--2015   698      0.16        0.155 (34)        228

  EC, β^−^, β^+^   ^152^Eu                  SCK          HPGe \#13                    2008--2016   1249     0.11        0.161 (24)        214

  EC, β^−^, β^+^   ^152^Eu                  NIST         IC                           1976--2011   96       0.040       0.021 (9)         214

  EC, β^−^, β^+^   ^152^Eu                  PTB          IC                           1989--2016   2199     0.007       0.018 (1)         11

  EC(, β^+^)       ^207^Bi                  NIST         IC                           1971--2011   152      0.05        **0.004** (11)    23
  --------------------------------------------------------------------------------------------------------------------------------------------------------

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