On decay constants and orbital distance to the Sun—part III: beta plus and electron capture decay

The hypothesis that seasonal changes in proximity to the Sun cause variation of decay constants at permille level has been tested for radionuclides disintegrating through electron capture and beta plus decay. Activity measurements of 22Na, 54Mn, 55Fe, 57Co, 65Zn, 82+85Sr, 90Sr, 109Cd, 124Sb, 133Ba, 152Eu, and 207Bi sources were repeated over periods from 200 d up to more than four decades at 14 laboratories across the globe. Residuals from the exponential nuclear decay curves were inspected for annual oscillations. Systematic deviations from a purely exponential decay curve differ from one data set to another and appear attributable to instabilities in the instrumentation and measurement conditions. Oscillations in phase with Earth’s orbital distance to the sun could not be observed within 10−4–10−5 range precision. The most stable activity measurements of β+ and EC decaying sources set an upper limit of 0.006% or less to the amplitude of annual oscillations in the decay rate. There are no apparent indications for systematic oscillations at a level of weeks or months.


Introduction
This is part III of a series of three papers investigating annual modulations in measured radioactive decay rates and in particular the claim that decay constants change at permille level in phase with the seasonal variations in Earth-Sun distance. In part I [1], long-term measurements of alpha decay were collected from metrology laboratories across the globe. The decay rates of 209 Po, 226 Ra, 228 Th, 230 U, and 241 Am sources showed no oscillations in phase with Earth's orbital distance to the sun within 10 −5 -10 −6 range precision. The most stable activity measurements of α decaying sources set an upper limit of 0.0006% to 0.006% to the amplitude of annual modulations in the decay rate. In part II [2], evidence was collected and analysed for β − decaying nuclides. The amplitudes of annual sinusoidal modulations in the most stable measurements were below 0.007% for 60 Co, 134,137 Cs, 90 Sr, and 124 Sb and below 0.05% for 3 H, 14 C, and 85 Kr. In part III, the focus is on radionuclides disintegrating through electron capture (EC) and beta plus (β + ) decay.
The rationale behind the project has been discussed in parts I [1] and II [2] and in a summary paper [3]. The issue is similar as for α and β − decay. Claims have been made in the literature that there are violations of the exponential decay law in the shape of seasonal modulations of permille level amplitude. Theories have been proposed that predict variability of the decay constants. One of the prevailing ideas is that radioactive decay is stimulated by interaction with neutrinos-either solar neutrinos or relic neutrinos from dark matter-and seasonal changes in neutrino flux reaching Earth would cause non-exponential decay. The metrological community has an interest in investigating this issue, because the exponential decay law and the invariability of the decay constants constitute the cornerstone of the common measurement system for radioactivity and all its applications. The aim is to anticipate the metrological consequences if new insights necessitate a different view on radioactivity.
Whereas the experimental evidence of permille-sized seasonal modulations in radioactive decay has been mostly focussed on β − decay (see e.g [4][5][6].), similar cases have been reported involving EC and β + decay. O'Keefe et al [7] reanalysed 22 Na/ 44 Ti decay rate ratio data measured by Norman et al [8] and reported a weak annual variation at sub-permille level. Ionisation chamber measurements of 152 Eu at PTB have been known to show seasonal effects [5,10], but PTB metrologists related them to varying laboratory conditions affecting the instrumentation [9,10]. Jenkins and Fischbach [11] claimed that the decay of 54 Mn is influenced by solar flares and Mohsinally et al [12] reported correlations with solar storms. Jenkins et al [5] included 54 Mn in the list of nuclides with permille level annual modulations, whereas this was refuted by Silverman [13] on the basis of a mathematical analysis of half-life measurement data by Van Ammel et al [14] at the JRC (see section 3.2). From a metrological point of view, the mere observation of seasonal modulations is insufficient proof of the variability of decay constants as long as instrumental instability cannot be ruled out as a plausible cause of the observed effects [15][16][17].
In this work, activity measurements of 22 [18], detection of gamma rays in a high-purity germanium spectrometer (HPGe) [19], and counting of x-rays at a defined low solid angle with a gas wire proportional counter (PC) [20].
Exponential decay curves were fitted to the measured decay rates and the residuals were inspected for annual modulations, using the same methodology as in [1][2][3]. The residuals from the fitted decay curve were binned into 8 d 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) was 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 standard uncertainty on the fitted amplitude was determined as the value which increases the variable χ 2 / (χ 2 /υ) 0 by a value of one; the chi square χ 2 was divided by the reduced chi square (χ 2 /υ) 0 of the fit to protect against unrealistic uncertainty evaluations, e.g. due to correlations between measurements. A summary table of the sinusoid parameter fit values for most of the data sets has been published in [3].
In this paper, graphs are shown of residuals of integrated count rates or ionisation currents (for convenience all types of signals will be represented by the same symbol I) over the measured period as well as multi-annual averages taken over fixed 8 d periods of the year. The uncertainty bars are indicative only: for the individual data they often refer to a short-range repeatability, and for the annual averaged data (maximum 46 data, covering 8 d periods) they were derived from the spread of the input data and the inverse square root of the number of values in each data group. As a reference measure for the expected solar influence, a functional curve is included representing the annual variation of the inverse square of the Sun-Earth distance, 1/R 2 , renormalized to an amplitude of 0.15% (which is typical for the magnitude of the effect claimed by Jenkins et al [5]).

Decay characteristics
The decay of 22 Na (2.6029 (8) a) proceeds through β + emission (90.36%) and electron capture (9.64%), both predominantly to the 1275 keV state of 22 Ne [21]. The 1275 keV γ-ray and the 511 keV annihilation quanta are easily detectable in an IC or a γ-ray spectrometer. It is an important radionuclide for calibration of γ-ray spectrometers.
According to O'Keefe et al [4], there is a weak annual modulation at sub-permille level (0.034%) in the 22 Na/ 44 Ti decay rate ratio.

22 Na @JRC
At the JRC (Geel, Belgium), the decay of a 22 Na source in aqueous solution inside a sealed glass vial was followed in the IG12 IC between 2010 and 2016. The fitted half-life is in excellent agreement with the evaluated value in literature [21]. The residuals to an exponential decay curve presented in figure 1 do not exceed 0.02%. Consequently, the presence of permille-level modulations at frequencies of days, weeks, months or even a few years can be excluded. The annual averaged residuals in figure 2 show hints of a very small residual annual oscillation with an amplitude of only A = 0.0047 (6)% and a phase of a = 53 d. The amplitude and phase are almost identical as for 134 Cs measured in the same conditions [2,3], which points to a common origin of physical or instrumental nature. These results set a new upper limit to the solar effect on 22 Na decay-if there is any at allwhich is an order of magnitude lower than in [4].

22 Na @NIST
From 1968 to 1985, a 22 Na source was measured 90 times in the NIST ionisation chamber 'A' [22], from which 87 data were selected for analysis. Linear corrections were applied to the decay rates to compensate for gradual slippage of the source holder as a function of time [23]. The residuals to an exponential decay curve, shown in figure 3, are generally smaller than 0.2% in magnitude, which precludes the presence of permille level modulations with frequencies between a day and a few years. The best fitting sinusoidal modulation to the annual averaged residuals in figure 4 has a negligible amplitude comparable to its standard uncertainty and is out of phase with the JRC data (A = 0.019 (12)%, a = 204 d).

Decay characteristics
Manganese-54 (312.19 (3) d) decays almost uniquely by electron capture to the 834.855 keV excited level of 54 Cr, followed by a gamma transition to the ground state [21]. It is one of the important mono-energetic γ-ray emitters used for calibrations of γ-ray spectrometers.   There are claims that the decay of 54 Mn is influenced by solar neutrinos, visible in seasonal variations of the decay rates [5], and correlations with solar flares [11] and solar storms [12].

54 Mn @JRC
Two 54 Mn sources in aqueous solution inside a sealed ampoule were measured between 2006 and 2009 in the IG12 IC at the JRC, at initial ionisation currents of 685 pA (#1) and 69 pA (#2) [14]. The data sets of 102 (#1) and 54 (#2) measurements have been combined in one residuals plot (figure 5). The standard deviation is less than 0.01%, which precludes the presence of permille-level modulations at a time scale between a day and a few years. The remaining cyclic instability (A = 0.005 (1)% and a = 28 d) in the annually averaged residuals in figure 6 has the same low amplitude as the 22 Na data in section 2.2. The 54 Mn data set of JRC has also been analysed by Silverman [13], who came to similar conclusions.

54 Mn @PTB
Between 2010 and 2016, a 54 Mn source was measured 724 times in the IG12/A20 IC at the PTB, and 716 data were selected for analysis. Since the data set showed some trending behaviour (<0.25%) over time, it was subdivided in three multi-annual time regions and realigned through a linear transformation in each region. Annual modulations in the residuals remain unaffected by this detrending procedure, while the interfering effects of long-term instabilities are suppressed. The resulting residuals are presented in figure 7 and the annually averaged residuals in figure 8. There is a distinct annual oscillation (A = 0.014 (2)%, a = 78 d) which has resemblance with effects seen for 85 Kr, 90 Sr, 137 Cs [2], 133 Ba (section 9.3), and 152 Eu (section 10.5) in the same IC. The effect may be related with the normalisation through a 226 Ra check source, which has modulations of the same amplitude

Decay characteristics
Iron-55 (2.75 (1) a) decays by EC almost uniquely to the ground state of 55 Mn; therefore it produces practically no gamma emission [21]. It can be detected through the x-rays and Auger electrons emitted in the course of atomic rearrangements. The main x-ray has an energy of 5.9 keV, which can be difficult to separate well from interfering signals and to measure in stable conditions.

55 Fe @JRC
From August 2004 to May 2005 at the JRC, Van Ammel et al [24] measured the decay rate of an electrodeposited 55 Fe source on a copper backing and covered with an aluminium foil in a fixed defined low solid angle counter [20] with beryllium window and argon(90%)-methane(10%) filled wire proportional counter (PC). The residuals in figure 9 are small but show autocorrelations which complicate the uncertainty assessment of the half-life measurement [15,24]. Some of these autocorrelations reappear in the annual averages in figure 10 (since the experiment covered less than one year) but in spite of the metrological difficulties it has been demonstrated that physically induced annual cycles, if they occur, must be extremely small (A = 0.004 (3)%, a = 187 d).

Decay characteristics
Cobalt-57 (271.80 (5) d) decays by 100% EC to the excited levels of 136.47 keV (99.82%) and 706.42 keV (0.18%) in 57 Fe [21]. Some characteristic energies of γ-rays emitted in the decay are 14 keV, 122 keV and 136 keV, which are very useful for efficiency calibration of γ-ray spectrometers at the low-energy side.

57 Co @NIST
From 1962 to 1966, a 57 Co source was measured 99 times in the NIST IC 'A' [22], from which 2 data were excluded from analysis. The residuals to an exponential decay curve, shown in figure 11, are generally smaller than 0.2% in magnitude. The fitted sinusoidal function to the annual averaged residuals in figure 12 is well below permille level (A = 0.055 (22)%, a = 187 d).

65 Zn @JRC
The decay of a 65 Zn source in aqueous solution was measured by Van Ammel et al [25] in the IG12 and ISOCAL III ICs of the JRC between March 2002 and May 2003, in the frame of a half-life determination. The IG12 (20th Century Electronics, UK) ionisation chamber is a well-type ionisation chamber filled with argon to 2 MPa pressure, shielded with 5-cm-thick lead bricks and connected to a current integrating electrometer which incorporates an external feedback air-spaced capacitor. When the output voltage reaches a lower or upper discriminator level, a precision digital voltmeter is sampled. The ISOCAL III ionisation chamber has no shielding, and the resulting current is measured directly with a Keithley 617 electrometer. Each measurement consisted of typically 3000 samples of the current, at regular time intervals of 1 s. Outliers were removed and one average value was taken per measurement. In total, 140 data points from both ICs were combined and annually averaged data are presented in figure 13. The annual oscillations do not rise above the 10 −5 level (A = 0.008 (4)%, a = 163 d).  [26] with the ionisation chamber 'AUTOIC' [27], which is different than IC 'A' used for most half-life measurements in the last half century [22]. Whereas the daughter nuclide 82 Rb was in equilibrium with the 82 Sr, the 85 Sr contribution-mainly significant towards the end of the experiment-required the fit of an additional exponential function to the decay curve, the 85 Sr half-life parameter being fixed at the literature value. The ratio of the AUTOIC responses to the 82 Sr/ 85 Sr components was 2.5 at the reference time.

Strontium
Additional activity measurements (Source No.1952) were performed by γ-ray spectrometry using a HPGe detector. The ratio of the net gamma-ray emission rates of the 776.517 keV 82 Sr γ-ray line to the 661.657 keV 137 Cs γ-ray line was followed for 222 d. The residuals to the fit of the decay curves have been published by Pibida et al [26].

Decay characteristics
Cadmium-109 (462.29 (30) d) decays by EC to the isomeric state of 109 Ag, followed by emission of an 88 keV γ-ray [21]. This nuclide is often used for efficiency calibrations of γ-ray spectrometers in the low-energy region.

109 Cd @JRC
At the JRC between 2006 and 2010, measurements with the IG12 IC of a 109 Cd source in aqueous solution inside a glass ampoule were used for an accurate half-life determination [28]. In parallel, an additional series of measurements was performed by γ-ray spectrometry, relative to an 241 Am reference source. The spectrometry residuals were within 0.1%-0.3% and the IC data ( figure 15) within 0.03%-0.08%, which is less precise than for other nuclides due to the lower initial source activity and IC current (11.7 pA). The residual annual instability of the IC data ( figure 16) has an amplitude of A = 0.015 (4)% and a phase of a = 18 d.

109 Cd @JSI
Besides the 88 keV γ ray, 22 keV x-rays are emitted after the EC decay of 109 Cd or as a consequence of internal conversion of 109m Ag (T 1/2 = 39.7 s). Both emissions are regularly measured in HPGe detectors at the JSI (Slovenia). Due to the low energy of the radiation, their detection is very sensitive to absorption effects and small changes in geometry.     to residuals in the flat regions, one obtains annual averages (figure 21) with a mild seasonal effect (A = 0.035 (24)%, a = 346 d). More importantly, these data sets, in comparison with the 60 Co data [2] on the same detectors, demonstrate how detection instability strongly depends on the energy of the radiation.

109 Cd @NIST
Between 1976 and 1981, the decay of a 109 Cd source was measured in the IC 'A' at the NIST [22]. Linear detrending corrections (<0.4%) have been applied over three time zones to suppress the influence of long-term drift on the residuals and to focus the search on systematic annual effects. The residuals to an exponential decay curve, presented in figure 22, are typically a few permille. The annual averages in figure 23 have magnitudes around a permille, but appear to be randomly
According to Jenkins et al [5] the decay of 133 Ba exhibits 'strong' annual modulations as well as a 2-yearly oscillation tentatively associated with a 'Rieger' r-mode oscillation in the solar neutrino flux [29]. It was repeated by Sturrock et al [30] that 'these results are compatible with a solar influence' and that 'it is possible that 133 Ba measurements are also subject to a non-solar (possibly cosmic) influence'.

133 Ba @NIST
A 133 Ba source was measured 138 times from 1979 to 2012 with the IC 'A' [22] at the NIST. A linear adjustment was made for the time period after 1991 and 7 among the most recent measurements were excluded from the analysis because of a growing bias. In figure 24, the residuals from exponential decay do not exceed a magnitude of 0.15%. Since the data were not evenly distributed, several averaged residuals in figure 25 show a similar precision and a relatively mild seasonal modulation (A = 0.028 (8)%, a = 74 d). The data do not confirm the claim of 'strong' annual modulations [5].

133 Ba @PTB
From 1996 to 2016, the ionisation current from a 133 Ba source was measured 2158 times with an IC at the PTB and the data were analysed relative to the 226 Ra check source. The decaycorrected IC output in figure 26 shows a 2-permille-level    long-term drift, very similar to instabilities observed with other nuclides measured over the same period (see e.g. 85 Kr [2]). This was compensated for in this work by linear adjustments over 4 time regions and exclusion of 61 extreme data (in value or uncertainty). The resulting residuals in figure 27 have a standard deviation of 0.09% and show some local non-random structures. It is suspected that this particular solution is chemically not stable, which might-at least partly-explain why the residuals are larger than expected considering the rather high ionisation current. The averaged residuals in figure 28 do not adhere smoothly to a sinusoidal function, but nevertheless the fit exhibits an annual modulation of comparable amplitude as for other nuclides in the same IC (A = 0.015 (4)%, a = 46 d).
Sturrock et al [30] analysed the 152 Eu decay rates measured from 1990 to 1995 with the IG12/A20 IC at the PTB and observed annual oscillations in the 0.1% range. They concluded that 'these results are compatible with a solar influence, and do not appear to be compatible with an experimental or environmental influence'. In section 10.5, it will be shown that these oscillations reduced significantly since the replacement of the electrometer in October 1998, which invalidates this interpretation.

152 Eu @IAEA
At the Terrestrial Environment Laboratory of the IAEA (Seibersdorf, Austria), a 152 Eu point source was measured on two HPGe detectors for quality control between 2010 and 2016. Four data sets were derived from the backgroundcorrected count integral of the γ-ray spectra and from the sum of net peak areas of the 40, 122, 344, 778, 964 and 1408 keV emissions. After elimination of extreme data, 143 residuals of typically 0.1%-0.3% were combined in figure 29. The average deviations in figure 30 are free of annual oscillations below permille level (A = 0.020 (24)%, a = 162 d).

152 Eu @SCK
A wealth of 152 Eu γ-ray spectrometry data was collected at the SCK•CEN (Belgium) on 8 HPGe detectors, in parallel with 241 Am mixed in the same source [1]. The net peak areas of three transitions at 122 keV, 779 keV and 1408 keV have been followed between 2008 and 2016. The decay-corrected peak areas (see e.g. figure 31) show a quasi-linearly increasing trend-probably caused by uncompensated count loss through pulse pileup-and a jump in 2011 due to a change in data acquisition system [31]. The 24 data sets were linearised and connected by means of the fit of two slopes and a scaling factor. Three groups of results were averaged: two detectors   There is clearly no common annual effect in the activity of the 152 Eu source, therefore amplitudes above 0.01% cannot be ascribed to variability of the decay constant in correlation with Earth-Sun distance.
Several authors have speculated that beta decay would be susceptible to 'external' influences and alpha decay not, which would explain why less seasonal effects have been observed in alpha decay (see e.g. [5-7, 29, 32]). From a metrological point of view, an explanation could be that it is easier to establish measurement stability for alpha particles with high, discrete energies than for beta particles with lower, variable energies. When not measuring the particles but the subsequent γ-ray emission, the metrological difficulty is similar for both types of decay. It turns out that the annual oscillations of the 241 Am (α decay) γ-ray data are almost identical to those of the 152 Eu (EC, β − , β + decay) data

152 Eu @NIST
A data set of 96 decay rate measurements obtained between 1976 and 2011 with the IC 'A' [22] of the NIST have been analysed, after applying a linear detrending correction for data obtained after 1986. The residuals in figure 36 are less than 0.15% in amplitude and the annual averages in figure 37 show no explicit seasonality (A = 0.021 (9)%, a = 214 d).

152 Eu @PTB
At the PTB (Germany), IC measurements of a 152 Eu source were repeated 2515 times between 1989 and 2016. The raw ionisation currents obtained before 1999 show 0.1% modulations [10], which can be significantly reduced by analysing the data relative to the 226 Ra check source. Most of the resulting residuals are well within 0.1% (figure 38), which leaves little room for potential claims about permille-sized modulations in the 152 Eu decay rates at any frequency between a day and several decades. There is a distinct but small annual sinusoidal effect (A = 0.018 (1)%, a = 11 d) in the averaged residuals (figure 39) which appears to be instrument-specific (see e.g. 54 Mn in section 3.3, 90 Sr, 137 Cs in [2] and 226 Ra in [1]).
A reanalysis was done of the subset of the raw-i.e. not normalised to the check source-152 Eu data published by Schrader [10]

207 Bi @NIST
The decay of a 207 Bi source was followed over four decades with the IC 'A' [22] at the NIST. Linear adjustments were made for slow geometrical changes in the source holder [23], and additionally for long-term detrending purposes (over three multi-annual time zones) in this work. The residuals from exponential decay in figure

Conclusions
The decay through β + and EC processes shows as little variability with respect to annual oscillations as α or β − decay. No systematic oscillations in phase with Earth-Sun distance were found. The observed stability of 22 Na, 54 Mn, 65 Zn, 82,85 Sr and 207 Bi decay rates-within A = 0.008% or better-is bound by the same instrumental limitations as similar IC measurements for α or β − emitting nuclides. The experimental evidence is equally convincing for 55 Fe (A = 0.004%) in spite of its low-energy radiation and measurement in a gas proportional counter. Good stability was demonstrated for 109 Cd (A = 0.015%) and 152 Eu (A = 0.01%) measured through IC and γ-ray spectrometry. The data sets presented in parts I-III of this work are typically 50 times more stable than the ones in literature which inspired theories of non-exponential decay. The best experimental evidence confirms the validity of the exponential decay law within the 10 −5 -10 −6 level and no proof could be found of violations of the invariability of decay constants.
It has to be recognised that all long-term measurements are vulnerable to instrumental instabilities, which have to be taken into account in the interpretation of observations of autocorrelations in residuals to the exponential decay curve. The measurements with the PTB ionisation chamber, for example, show a small recurrent modulation for several radionuclides which is incompatible with the theory that seasonal changes in the solar neutrino flux changed the decay rate, because it was not reproduced by any of the other laboratories. The fact that non-exponential behaviour in γ-ray spectrometry differs from one spectrometer to another within the same laboratory (see SCK, JSI) strongly suggests that instrumental instability is the root cause rather than a global physical phenomenon.
Owing to the invariability of decay constants, there is no impediment to the establishment of the SI unit becquerel through primary standardisation at 0.1% range accuracy nor to the demonstration of equivalence of activity at the international level over a time span of decades. It is normal for repeated activity measurements to show varying degrees of instability of instrumental and environmental 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.