Production of 40 Ar by an overlooked mode of 40 K decay with implications for KAr geochronology

The decay of 40K to the stable isotopes 40Ca and 40Ar is used as a measure of time for both the K-Ca and KAr geochronometers, the latter of which is most generally utilized by the variant 40Ar/39Ar system. The increasing precision of geochronology has forced practitioners to deal with the systematic uncertainties rooted in all radioisotope dating methods. A major component of these systematic uncertainties for the K-Ar and 40Ar/39Ar techniques is imprecisely determined decay constants and an incomplete knowledge of the decay scheme of 40K. Recent geochronology studies question whether 40K can decay to 40Ar via an electron capture directly to ground state (ECground), citing the lack of experimental verification as reasoning for its omission. In this study, we (1) provide a theoretical argument in favor of the presence of this decay mode and (2) evaluate the magnitude of this decay mode by calculating the electron capture to positron ratio (ECground/β) and comparing calculated ratios to previously published calculations, which yield ECground/β between 150–212. We provide support for this calculation through comparison of the experimentally verified ECground/β ratio of 22Na with our calculation using the theory of β decay. When combined with measured values of β and β decay rates, the best estimate for the calculated ECground/β for 40K yields a partial decay constant for 40K direct to ground-state 40Ar of 11.6±1.5×10−13 a−1 (2σ ). We calculate a partial decay constant of 40K to 40Ar of 0.592± 0.014× 10−10 a−1 and a total decay constant of 5.475± 0.107× 10−10 a−1 (2σ ), and we conclude that although omission of this decay mode can be significant for K-Ar dating, it is minor for 40Ar/39Ar geochronology and is therefore unlikely to have significantly biased published measurements.


Introduction
40 K is a naturally occurring radioisotope of K with atomic abundance of 0.0117 % (Garner et al., 1975). 40 K undergoes a branched decay to 40 Ar and 40 Ca with a total halflife of ca. 1.3 Ga and is the basis of the K-Ca and the K-Ar geochronometers (Aldrich and Nier, 1948;Wasserburg and Hayden, 1955;Marshall and DePaolo, 1982). The K-Ar system is most often exploited using the variant 40 Ar/ 39 Ar method, wherein some of the 39 K in the sample is transmuted to 39 Ar by irradiation with fast neutrons, thereby allowing both the parent and the daughter nuclides to be measured as isotopes of Ar (Merrihue and Turner, 1966). The latter is widely used to date geological events that span Earth's history, from volcanic eruptions recorded in historical texts (e.g., Preece et al., 2018;Renne et al., 1997) to the earliest events in the solar system (e.g., Renne, 2000).
Advances in analytical precision have forced practitioners in geochronology to address systematic uncertainties that are inherent in all radioisotope dating methods, such as uncertainties in the measurement apparatus, prior assumptions made by the observer, or interference from environmental factors. For the K-Ar system, these uncertainties also include those that arise from imprecisely known decay rates of 40 K. In the geological literature, there have been two influential reviews of measurements of the 40 K decay rate. Beckinsale and Gale (1969) provided the first comprehensive review of measured and predicted decay rates, which became the basis of the convention adopted by Steiger and Jäger (1977) used by the geochronological community for the next 20 years. Subsequently, Min et al. (2000) provided a more lengthy, critical review of available specific activity data determined by direct measurements of decay and updated the derived decay rates for newer physical constants. More recently, the 40 K decay parameters were estimated by Renne et al. (2010Renne et al. ( , 2011, and although direct measurements of the 40 K decay were incorporated into the estimate, it was heavily weighted to an intercomparison with 238 U decay. The decay rate determined by Renne et al. (2010Renne et al. ( , 2011, and the Min et al. (2000) decay rates are the most frequently used in 40 Ar/ 39 Ar geochronology. These evaluations, along with those from the nuclear physics community, have been summarized recently by Cresswell et al. (2018Cresswell et al. ( , 2019. Despite decades of work and long-standing interest in 40 K decay, there remains uncertainty over the nature of the decay scheme. There is consensus that most 40 K decays by β − to 40 Ca or by electron capture to 40 Ar via an excited state and that a small amount (∼ 0.001 %) of 40 K decays to 40 Ar via β + . The early but influential review of 40 K decay by Beckinsale and Gale (1969) included these decay modes and also included a prediction of a second electron capture decay directly to the ground state of 40 Ar that would add an additional ∼ 2 % to the rate of decay from 40 K to 40 Ar. Many workers in geochronology (e.g., those who use the Min et al., 2000, decay constants) have ignored this prediction, and some nuclear physics tabulations do not clearly include it (Endt, 1990;Audi et al., 2003). The influential review by Min et al. (2000) described this decay mode as "unverified" and having a "questionable" existence.
However, the putative electron capture to ground-state decay-mode decay constant is of the same order of magnitude as the uncertainties in the decay rate of 40 K to 40 Ar and therefore may be a non-negligible and potentially important part of the geochronological system. Here, we describe the theoretical basis of this predicted decay mode and demonstrate the robust nature of the prediction via an analogous calculation of 22 Na decay. We describe experiments that could be made to measure this decay mode and also identify observations from nuclear physics experiments that offer evidence for its existence. We conclude that the evidence for this decay mode is strong and, despite the large uncertainty, should be considered in evaluations of the 40 K decay rate.

Historical overview
At present, 40 K has three experimentally verified decay modes (Fig. 1).
1. β − decay to 40 Ca, this mode can be verified by direct measurement of the β − emission.
2. Electron capture to an excited isomer of 40 Ar, followed by decay to the ground state of 40 Ar accompanied by emission of a 1.46 MeV γ -ray. Hereafter, we denote this decay mode as EC*. This mode can be verified by direct measurement of the γ emission.
3. β + decay from the ground state of 40 K to the ground state of 40 Ar (Engelkemeir et al., 1962), this is a very small component of the total decay rate and has been verified by direct measurement of the β + emission.
In their paper reporting the measurement of β + /β − , Engelkemeir et al. (1962), through private correspondence with Brosi and Kettle, proposed that an electron capture mode that goes directly to ground-state 40 Ar also exists, with an electron capture to positron ratio of 155. This decay mode is hereafter denoted EC ground . This decay mode has not been experimentally detected, in part because the measurement is much more difficult to make than the others. If it exists, it would contribute about 0.2 % to the total decay rate of 40 K or about 2 % to the 40 Ar branch. The EC ground decay mode was included in the review by Beckinsale and Gale (1969) and then subsequently in Steiger and Jäger (1977). This decay mode is also included in the widely used Evaluated Nuclear Structure Data File (ENSDF) and Decay Data Evaluation Project (DDEP) evaluations (Chen, 2017;Mougeot and Helmer, 2009, respectively). However, evaluations by Endt andVan der Leun (1973, 1978), Endt (1990), and Audi et al. (2003) do not explicitly include this decay mode, with Audi et al. (2003) giving a transition intensity that is the combined EC* and β + intensities. Min et al. (2000) have questioned its validity because there is no experimental verification and therefore do not include EC ground in their estimates.

Why there must be an EC ground decay mode
In nuclides that are too proton-rich and therefore radioactive, protons decay to correct this imbalance via two mechanisms. Either, (1) the nucleus undergoes electron capture wherein an orbital electron is captured by the nucleus, or (2) the nucleus emits a positron (β + ). Both processes are types of β decay and result in the transformation of a proton to a neutron to conserve charge, and they both also emit a neutrino in order to conserve lepton number and energy. These two processes are typically paired: coupled electron capture-β + is the second most abundant decay type on the chart of the nuclides, after β − decay (Audi et al., 2003). The electron capture (EC) and positron (β + ) decay modes are linked because both processes have the same initial and final nuclear states.
β + decay is always accompanied by EC, but the converse is not always true (Bambynek et al., 1977). This is because β + decay, unlike EC, requires a minimum amount of energy (∼ 1022 keV, equivalent to the combined rest masses of both a positron and an electron) in order to produce the emitted  son (1999) and Leutz et al. (1965), where 1 is the electron capture branch to the excited state of 40 Ar with y-ray emission (EC*), 2 is the electron capture direct to the ground state of 40 Ar (EC ground ), 3 is the positron decay to ground state of 40 Ar, and 4 is the β decay to the ground state of 40 Ca. The disputed decay mode, EC ground , is highlighted in red.
positron and an electron (the latter to satisfy charge conservation). The energy between initial and final states is the Q value, which is a function of the difference between the initial and final masses. This energy is shared between the outgoing neutrino, atomic excitation of the daughter system, recoil energy, and nuclear excitation of the daughter system. The energy that dictates if positron emission is possible is denoted Q EC . In the decay of 40 K, the EC* branch has an energy difference (Q EC ) between the initial and excited isomer state of only 44 keV. In contrast, the energy difference between 40 K and the ground state of 40 Ar is 1504.4 keV (Wang et al., 2017), an energy greater than the combined rest masses of the positron and electron. Therefore, the EC* branch, with an energy difference of only 44 keV, cannot be the complement to the β + decay, and the EC ground must exist to provide the β + complement. The experimental observation of the β + decay mode comes from a single measurement by Engelkemeir et al. (1962). We rely on this measurement to make our argument for the existence the EC ground . It is possible that the positrons observed arise from the pair production of the ∼ 1460 keV gamma; Engelkemier et al. (1962) discuss this possibility in their experiment, calculating this as 55 %-60 % of the total positron detection rate. However, positrons produced by this mode of pair production would be monoenergetic at 440 keV, whereas the observed positron energy spectra exceed this value, with a maximum of 491 keV. A reasonable fit is also observed between the measured β + energy spectrum and the theoretical third forbidden unique energy spectrum, supporting the argument that these β + are from a decay mode rather than arising from pair production.

Theory and calculation of EC ground /β +
In the decay of 40 K, the nuclide can reach a more stable state ( 40 Ca or 40 Ar) only by violating quantum selection rules. Quantum selection rules place formal constraints of the possible transitions of a system from one quantum state to another. In this case it places constraints on the possible set of transitions from the parent 40 K state to the daughter 40 Ar. Decays which violate these selection rules undergo slow, so-called "forbidden" unique transitions, which give 40 K its long ∼ 1.3 Ga half-life. The 40 K decay scheme itself is unusual because the coupled EC ground -β + and β − branches are the only third-order unique forbidden transitions known in nature. All 40 K decays undergo a parity reversal (where parity reversal is the change of sign in one of the spatial coordinates (x, y, z)) between the initial parent state and final daughter state. Therefore, we can define the selection rules as follows: is the change in spin from initial to final state following Krane and Halliday (1987). We can characterize each decay mode of 40 K by its degree of forbiddenness from the above selection rule. The EC* mode undergoes a spin change of J = 4 − 2 = 2 and is classified as a first-order unique forbidden decay. The three other decay modes of 40 K, including EC ground , all undergo a spin change of J = 4 − 0 = 4 and are classified as third-order unique forbidden decays.
The EC process occurs because the atomic electrons have a finite probability to be in the nucleus with the likelihood of being captured highest for those closest to the nucleus. A theoretical description of β emission was first given by Fermi (1934), while the possibility of electron capture was first recognized by Yukawa and Sakata (1935) and later developed by Bethe and Bacher (1936). Here we use Fermi theory of β − decay to calculate the EC ground /β + in the decay of 40 K.
We can use the ratio of orbital electron capture and positron emission to infer the existence of EC ground . The ratio br is defined as follows: where λ ec and λ β+ are the probability per unit time of electron capture or β + emission. In electron capture, orbital electrons can be captured from any orbital shell of the atom. The EC/β + is therefore the summation of the individual capture ratios from each shell. Following Bambynek et al. (1977), the total electron capture-to-positron ratio is as follows: where x is the shell, n x is the relative occupation number, C x contains the dependence of electron capture rates on nu-clear structure giving the forbiddenness classification, similar to the shape factor in β decay (Emery, 1975), f x is the integrated fermi function in β decay, f β + is the integrated positron spectrum, and C(W ) is the theoretical shape factor for allowed or forbidden transitions. A review of shape factors for 40 K transitions is provided by Cresswell et al. (2018Cresswell et al. ( , 2019. We initially simplify this equation to only consider the innermost K shell, the shell containing the electron with the highest probability to be captured by the nucleus: where λ K is the probability of K shell capture. For this capture, f K is defined as follows: where q K is the momentum of the neutrino particle, β K is the Coulomb amplitude of the wave function, and B K is the term for overlap and exchange corrections. Similarly, f β+ is defined as follows: where W is the total energy of the positron given by its kinetic energy E T and rest mass m e , defined above, and the momentum of the positron is given by p (Eq. 8); W 0 is the total normalized energy defined above; E max is the upper limit of the positron energy (equal to the Q value of the decay); and F (−Z, W ) is the Fermi function. We follow Bambynek et al. (1977) in the formula for C K C(W) which is given by where L = J and L = 1 for J = 0. The parameter λ n cannot be calculated in a straightforward manner and therefore we follow a typical assumption that λ n = 1 (Huber, 2011). This reduces the above expression to In a given decay, the change in charge from the initial to final state can lead to an imperfect overlap of the wave functions of these states. Furthermore, given the indistinguishability of electrons, there is the possibility of an exchange effect wherein an electron does not necessarily come from the orbital where the vacancy appears. For instance, it is possible that a vacancy may appear in the K-shell, but the captured electron from an outer shell is then subsequently filled by the inner shell electron (Bahcall, 1962;Bambynek et al., 1977). We follow Bahcall (1962) in implementing corrections for these effects, resulting in B K = 0.979. Following this, using nuclear data given in Bambynek et al. (1977), we estimate an EC ground /β + of 148. We first note that this value is in approximate concordance with the private correspondence value in Engelkemeir et al. (1962). However, this is only the capture ratio from the K shell, so we extend our model to a total electron capture ratio from all orbitals following Bosch et al. (1977). The total electron capture to positron ratio, an extension of Eq. (1), is given by We can simplify this equation by neglecting shells that make a negligible contribution. In 40 K the probability of capture is dominated by the two inner shells K and L1, with approximate probability of ca. ∼ 90 % and ∼ 10 % and a negligible contribution from the shells further out. We can therefore omit all shell captures except K and L1 to arrive at the total EC ground /β + ratio: The ratio of each shell capture can be solved with the following equation: where x = L1 and the other symbols have the same definition as above. Using this equation we calculate a total EC ground /β + of 164.
To further estimate the magnitude of the electron capture decay mode, we can perform another calculation of EC ground /β + following Fireman (1949). This simplified form of the calculating EC ground /β + is dependent only on the Q value (the difference between the initial and final state energies). This is given by λ EC λ β + = (η + 2) 8 0.450η 6.5 · 1 0.0676 + 1.25η + 8.48η 2 + 12.5η 3 + 1.74η 4 + 0.079η 6 , where η = Q m e − 2. We calculate an EC ground /β + of 272 using this method and the updated Q value of Wang et al. (2017). We note that despite discrepancies in these values for each method of evaluation, they are of the same order of magnitude. The differences in the values in these evaluations highlight the need for experimental measurement of EC ground /β + .

Comparison with other evaluations
Other theoretical evaluations of EC ground /β + for 40 K exist in the literature (Fig. 2). Pradler et al. (2013) and Mougeot (2018) report ratios of 150 and 212 ± 0.15, respectively (uncertainties are reported where they have been estimated). These workers use broadly similar methods to us. Mougeot (2018) uses higher-order corrections for both exchange and overlap and accounts for the dependence of λ K , that we set equal to 1 in Eq. (9) using the energy of the decay. Pradler et al. (2013) use the Fermi method and data from Bambynek et al. (1977) but only perform the calculation for K-shell electrons, resulting in a slightly differently calculated value than we report. Notably, all estimated values are of the same order of magnitude, similar to the ratio 155 reported in Engelkemeir et al. (1962) and our calculated value of 164. Currently, the most commonly used EC ground /β + value is calculated via the LogFT program, a program used in nuclear data evaluations (ENSDF collaboration, LogFT). However, the program is capable of only calculating first and second unique forbidden decay ratios, so the EC ground /β + value from LogFT of 200 ± 100 is an extrapolation, with the assumption that the increase in the ratio from second to third order is by the same factor as the increase from first to second order. Finally, Chen (2017) evaluates the 40 K decay data and reports a EC ground /β + value of 45.2 ± 1.4 without elaboration.
The variability between the modern estimates are driven primarily by choices when making the approximations necessary for these calculations to be tractable. Uncertainties on individual estimates that could be derived by propagating the uncertainties in the underlying experimental data are small, and where uncertainties are estimated, they are generally not explicated.
The value calculated by Mougeot (2018) of 212 ± 0.15 is currently the best estimate of the 40 K EC ground /β + . It is slightly higher than two other recent estimates, our value of 164 or that of Pradler et al. (2013) of 150. Given a broad consensus in calculated EC ground /β + over several decades and via a variety of methods, it appears highly likely that it falls in the range 150-212 (Fig. 2).

Comparison with 22 Na
To test the validity of our 40 K EC ground /β + estimate, we use the same calculations to estimate the experimentally constrained (EC/β + )* value for 22 Na decay. 22 Na is radionu- Figure 2. Comparison of theoretically calculated EC ground /β + of 40 K in this study using three methods: (1) Bambynek method (Bambynek et al., 1977), (2) Fireman method (Fireman, 1949), and (3) LogFT (ENSDF collaboration, 2020, LogFT). The value of Chen (2017) is not included in the figure as it is an extreme outlier, and the authors do not explain the method they use to reach this value. Our calculated ratios are compared to previous evaluations in the literature (Engelkemeir et al., 1962;Pradler et al., 2013;Mougeot, 2018). Uncertainties in these values are either intractable or, in the case of Mougeot (2018), too small to plot. Note the consistency in the estimated ratio from all of the methods. Uncertainties are unknown in all cases except Mougeot (2018), where the uncertainty is too small to plot. clide with a half-life of ∼ 2.6 years, it occurs in nature as a low-abundance cosmogenic nuclide produced by spallation of 40 Ar and is also produced synthetically by proton irradiation for use in positron emission tomography. Like 40 K, it decays by electron capture and positron emission. The main EC-β + pair for 22 Na decays initially to the excited state of 22 Ne, followed by a 1.27 MeV γ emission ( Fig. 3; Bé et al., 2010). This pair has a (EC/β + )* of approximately 0.1 and accounts for > 99.9 % of the total decay. A second ECβ + pair decays directly to the ground state of 22 Na with an (EC/β + ) ground of ∼ 0.02, but it is a minor component. Here, we calculate the (EC/β + )* for the main branch. Unlike 40 K, the dominant decay of 22 Na is the β + decay mode. This is due to the greater difference in energy between the initial and final states, as positron decay will have a greater possibility of occurring in decays with a greater mass differences between initial and final states (Emery, 1975). 22 Na is not a perfect analogue; however, it is probably the best choice that  Bé et al. (2010) and Leutz et al. (1965). An additional EC and β + decay pair that corresponds to approximately 0.056 % of the total decay of 22 Na has been omitted for clarity.
has both a tractable theoretical calculation and a wealth of experimental data that can be used readily for verification.
Unlike 40 K, there are numerous measurements of the electron capture to positron ratio for decay to the excited state of 22 Ne ( Fig. 4; Kreger, 1954;Vatai et al., 1968;Williams, 1964;McCann and Smith, 1969;MacMahon and Baerg, 1976;Bosch et al., 1977;Baerg, 1983;Schmidt-Ott et al., 1984;Sýkora and Povinec, 1986;Kunze et al., 1990;Nähle et al., 2008). Measurement of (EC/β + )* for 22 Na is accomplished by measurement of both of the gammas (which come from both the EC* and the β + * ) and the x-rays (which only come from the EC branch). Relative to the 40 K EC ground /β + , the 22 Na (EC/β + )* is a more straightforward measurement because of the higher activity (meaning higher count rate) and the higher energy of the x-ray emitted from the Auger electron, which an electron from the same atom that is emitted as a vacancy of an inner shell is filled. In a decay to the excited state of 22 Ne, the de-excitation 1.28 MeV γ will be associated with both electron capture and positron decay. However, those measured 1.28 MeV γ that are not accompanied by two 0.511 MeV annihilation photons can be used to distinguish between both processes. We use the experimental measurements to verify our calculations described above for 40 K.
Following a similar calculation using the Fermi method, our preferred method, to that used for our proposed estimate of the 40 K EC ground /β + , we estimate an (EC/β + )* of approximately 0.11. This is within the range of measured values of 0.105-0.115 (Fig. 4), suggesting that our calculation strategy of the 40 K EC ground /β + is accurate, and lends fur- Figure 4. Comparison of experimentally measured (EC/β + )* ratios of 22 Na (grey circles) adapted from Kunze et al. (1990) with our calculated value (dashed black line). Note the concordance of the theoretical and experimental determinations. The uncertainty in the Baerg (1983) determination is smaller than the symbol. ther confidence to the existence of the current unmeasured 40 K electron capture to ground-state decay.

Experimental verification of EC ground decay mode
In both β − and β + decay, an electron or positron is emitted that allows for direct detection and verification of the decay process. In contrast, electron capture cannot be detected directly. Methods to experimentally verify electron capture rely on indirect processes associated with the rearrangement of the atom following the capture of the orbital electron. Once the electron is captured the atom will rearrange itself to fill the vacancy, resulting in the emission of a characteristic xray or Auger electron with an energy defined by the binding energy of the shell vacancy of the daughter nucleus.
In the case of 40 K, verification of the EC ground decay can be achieved by measuring the characteristic x-rays (Di Stefano et al., 2017). The orbital electron with the highest probability of capture is from the K-shell; if this electron is captured, the resulting vacancy in the K-shell may be filled by an electron from any of the other shells (e.g., L, M), and a characteristic x-ray is emitted with an energy dependent on the particular shell that fills the vacancy. It is not necessary, however, that all K-capture processes result in the emission of an x-ray. By the Auger effect, a radiationless transfer may occur wherein the K-shell vacancy is replaced by two vacancies in the next outer shell, L, or one in the next two shells; L and M. The energies of the Auger electrons emitted in these transitions depend upon the 40 Ar product resulting from K-capture. Both electron capture decays to the ground and excited state of 40 Ar ( 40 Ar 2+ ) result in the same electron configuration and x-ray emissions. Di Stefano et al. (2017) suggested tagging x-rays with the de-excitation γ associated with electron capture to 40 Ar 2+ , which has a lifetime on the order of ∼ 10 −12 s (Di Stefano et al., 2017). Measuring these tagged x-rays experimentally will be challenging since it requires identifying a low-probability decay mode with x-ray signals present against a high background from the 40 Ar 2+ state. Further, as illustrated in Di Stefano et al. (2020), it is expected that 50 EC* decays occur for every 1 EC ground decay; therefore, a detector efficiency of ≥ 98 % is required to make sure that there is fewer than one mis-tagged EC* decay for each true EC ground decay. The experiment therefore requires an x-ray spectrometer able to resolve the Ar-K x-ray from other x-rays in the background and accurately account for the x-ray-γ -ray coincidence efficiency (≥ 98 %) to quantify x-ray emission rates in excess of those from the 40 Ar 2+ state. Given the complexity involved in this experiment, a pilot study was conducted at SUERC to measure characteristic x-rays from a KCl source. The experiment was not successful because the detector was not able to resolve the Ar-K x-ray sufficiently but demonstrates the potential of this method to detect the x-rays, given a sufficiently high-resolution detector. Full details are provided in the Supplement.
Ongoing attempts are being made to verify this decay mode by careful detection of the characteristic x-rays by the Potassium decay (KDK) experiment (Di Stefano et al., 2017;Stukel, 2018). Experimental verification has implications for (1) rare event physics, as it is a vital component in constraining the irreducible background and verifying results in the DArk MAtter (DAMA) experiment (Pradler et al., 2013); (2) the theory of βdecay (Fermi, 1934), as it is the only thirdorder unique forbidden electron capture decay known (Audi et al., 2003); and (3) K-Ar and 40 Ar/ 39 Ar geochronology, for which it is currently overlooked due to lack of experimental evidence. We further expand on the implications for geochronology below.

Relevance for geochronology
Geochronology with the K-Ar system requires either both the branching ratio and the total decay constant, or in the case of an 40 Ar/ 39 Ar age wherein the fluence monitor age is constrained independently of its K-Ar systematics (Merrihue and Turner, 1966), only the total decay constant. Using lower and upper bound values of EC ground /β + corresponding to 150 and 212 as described above, the decay constants calculated by Min et al. (2000) (λ EC * = 0.580±0.014×10 −10 a −1 and λ T = 5.463 ± 0.107 × 10 −10 a −1 ), and the β + /β − from Engelkemeir et al. (1962) (1.12±0.14×10 −5 ), we calculate a β + decay constant of 5.47 ± 0.69 × 10 −15 a −1 and a range of EC ground decay constants of 8.2-11.6 × 10 −13 a −1 . Combining these values with the Min et al. (2000) quantities yields a new partial decay constant for 40 K to 40 Ar (λ40 Ar ) that ranges from 0.588-0.592 × 10 −10 a −1 and total decay constant (λ T ) that ranges from 5.471-5.475 × 10 −10 a −1 . These ranges are within the uncertainties calculated by Min et al. (2000) for decay constants that do not include the EC ground decay mode. Existing and modified constraints on the decay modes are given in Table 1. Our preferred decay constants are those calculated with the EC ground /β + = 212 from Mougeot (2018). Consequently, K-Ar (and 40 Ar/ 39 Ar) ages calculated with these new decay constants will be younger than those calculated using the Min et al. (2000) decay constants. K-Ar dates are most sensitive to shifts in the decay constant because they incorporate the branching ratio, which is more strongly affected than the total 40 K decay constant. K-Ar ages will decrease by ∼ 1.4 %-2.0 % at 1 Ma, ∼ 1.1 %-1.5 % at 1 Ga, and ∼ 0.6 %-0.8 % at 4.5 Ga (Fig. 5). Ages determined using the 40 Ar/ 39 Ar method, for which the fluence monitor age is independently constrained (e.g., Kuiper et al., 2008;Rivera et al., 2011), are much less sensitive to the change in decay constant. Using Eq. (5) from Renne et al. (1998), and assuming calibration to a monitor with an age of 28.2 Ma, ages < 28 Ma increase only slightly, by < 0.002 %. There is no age difference at 28.2 Ma, the fluence monitor age. Ages then decrease for ages > 28.2 Ma, with ages decreased by 0.07 %-0.10 % at 2.5 Ga, and by 0.09 %-0.13 % at 4.5 Ga (Fig. 5).
The age of fluence monitors such as the Fish Canyon tuff sanidine (e.g., Morgan et al., 2014) determined by intercomparison with astronomically tuned ages of ash beds (Kuiper et al., 2008;Rivera et al., 2011) is also sensitive to revision of decay constants. Using the data published by Kuiper et al. (2008), and incorporating an EC ground decay mode, we calculate a new age for Fish Canyon sanidine of 28.200 ± 0.044 Ma, nominally lower but indistinguishable from the published value of 28.201 ± 0.044 Ma. Overall, the effects of an EC ground decay mode are unlikely to be significant for most current applications of 40 Ar/ 39 Ar geochronology. However, given the levels of analytical precision attainable by the K-Ar dating approach when dating geologically recent materials by K-Ar (e.g., Altherr et al., 2019), the EC ground decay mode will impact the accuracy of this chronometer.

Conclusion
The Fermi theory of β decay has decades of experimental support and is well established. We demonstrate this here by using these theories to accurately calculate the decay rate of a 22 Na, a nuclide with an experimentally verified decay rate. We have used this information to demonstrate the high likelihood that the suspected second electron capture decay mode of 40 K exists. Based on the cal- λ EC ground 11.6 ± 1.5 × 10 −13 a −1 13 This work λ40 Ar 0.592 ± 0.014 × 10 −10 a −1 2.4 This work λ T 5.475 ± 0.107 × 10 −10 a −1 2.0 This work Figure 5. Change in age, age, is the age of a given sample with the decay mode to ground state included, subtracted from the age with the decay mode to ground state omitted. Panel (a) shows the change in age using the 40 Ar/ 39 Ar equation with independently calibrated standards using both the lower bound (EC/β + = 150; grey) and upper bound (EC/β + = 212; black). Panel (b) shows the change in age using the K-Ar equation using both the lower bound (EC/β + = 150; grey) and upper bound (EC/β + = 212; black). Inset figures in each panel show the fractional differences in age by the inclusion of both the upper-and lower-bound EC/β + value. The larger difference in ages for the K-Ar system is due to the dependence on both the total decay constant and branching ratio. culations of Mougeot (2018), the best estimate of the partial decay constant for 40 K direct to ground-state 40 Ar is 11.6 ± 1.5 × 10 −13 a −1 (2σ ), and other calculations are no lower than about 8.2 × 10 −13 a −1 . Combining this with the decay constants published by Min et al. (2000) results in revised values of λ40 Ar = 0.592 ± 0.014 × 10 −10 a −1 and λ T = 5.475 ± 0.107 × 10 −10 a −1 . This addresses a long-standing question in K-Ar and 40 Ar/ 39 Ar geochronology and provides future workers with confidence that the 40 K EC ground decay mode exists. Just as important as providing support for its ex-istence, we also demonstrate that the magnitude of this decay mode is small enough that neglecting it has not yet resulted in significantly biased geochronological 40 Ar/ 39 Ar data. The same cannot be stated for the K-Ar dating approach, especially for geologically young materials.
Despite the strong grounding in theory, the EC ground decay mode has yet to be detected. The next step is experimental verification to determine the branching ratio. This will allow for a more complete evaluation of uncertainties associated with the decay mode and the branching ratio. This experiment is difficult but not intractable. Data availability. All data used are available in the Supplement and can be accessed publicly via https://doi.org/10.6084/m9.figshare.13280795 (Carter et al., 2020).
Author contributions. The study was conceived by JC and RBI. JC, AJC, and DCWS calculated the ratio of electron capture to beta activities and measured x-rays. JC, RBI, DFM and MMT calculated the effects for geochronology. JC wrote the manuscript with contributions from all authors.