Subterranean production of neutrons, 39 Ar and 21 Ne: Rates and uncertainties
Ondřej Šrámek, Lauren Stevens, William F. McDonough, Sujoy Mukhopadhyay, R. J. Peterson
SSubterranean production of neutrons, Ar and Ne: Rates and uncertainties
Ondˇrej ˇSr´amek a, ∗ , Lauren Stevens b , William F. McDonough b,c, ∗∗ , Sujoy Mukhopadhyay d , R. J. Peterson e a Department of Geophysics, Faculty of Mathematics and Physics, Charles University, V Holeˇsoviˇck´ach 2, 18000 Praha 8, Czech Republic b Department of Chemistry and Biochemistry, University of Maryland, College Park, MD 20742, United States c Department of Geology, University of Maryland, College Park, MD 20742, United States d Department of Earth and Planetary Sciences, University of California Davis, Davis, CA 95616, United States e Department of Physics, University of Colorado Boulder, Boulder, CO 80309-0390, United States
Abstract
Accurate understanding of the subsurface production rate of the radionuclide Ar is necessary for argon dating tech-niques and noble gas geochemistry of the shallow and the deep Earth, and is also of interest to the WIMP dark matterexperimental particle physics community. Our new calculations of subsurface production of neutrons, Ne, and Artake advantage of the state-of-the-art reliable tools of nuclear physics to obtain reaction cross sections and spectra(
TALYS ) and to evaluate neutron propagation in rock (
MCNP6 ). We discuss our method and results in relation to pre-vious studies and show the relative importance of various neutron, Ne, and Ar nucleogenic production channels.Uncertainty in nuclear reaction cross sections, which is the major contributor to overall calculation uncertainty, isestimated from variability in existing experimental and library data. Depending on selected rock composition, on theorder of 10 –10 α particles are produced in one kilogram of rock per year (order of 1–10 kg − s − ); the number ofproduced neutrons is lower by ∼ Ne production rate drops by an additional factor of 15–20,and another one order of magnitude or more is dropped in production of Ar. Our calculation yields a nucleogenic Ne / He production ratio of (4 . ± . × − in Continental Crust and (4 . ± . × − in Oceanic Crust andDepleted Mantle. Calculated Ar production rates span a great range from 29 ± − yr − in the K–Th–U-enriched Upper Continental Crust to (2 . ± . × − atoms kg-rock − yr − in Depleted Upper Mantle. Nucleogenic Ar production exceeds the cosmogenic production below ∼
700 meters depth and thus, a ff ects radiometric ages ofgroundwater. The Ar chronometer, which fills in a gap between H and C, is particularly important given the needto tap deep reservoirs of ancient drinking water.
Keywords: ( α, n ) neutrons, noble gases, Ar production rate, Ne production rate, fluid residence time
Contents1 Introduction 22 Overview 23 Alpha-particle production 34 Neutron production 4 α, n ) reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Neutrons from spontaneous fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 ( n , p ) and ( n , α ) ∗ Principal corresponding author. ∗∗ Corresponding author.
Email addresses: [email protected] (Ondˇrej ˇSr´amek), [email protected] (William F. McDonough)
Preprint submitted to Geochimica et Cosmochimica Acta September 13, 2016 a r X i v : . [ phy s i c s . g e o - ph ] S e p Results 6 Ar and Ne yields and production rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66.3 Coe ffi cients for plug-in formulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66.4 Uncertainty in the calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Argon-39 is a noble gas radionuclide with half-life of 269 ± Ar, Ar and Ne, in rocks are necessary for dating groundwaters and in proper interpretation of isotopicsignatures of gases originating in Earth’s interior (e.g., Graham, 2002; Tucker and Mukhopadhyay, 2014).Production of Ar in the Earth’s atmosphere is dominated by the cosmogenic production channel, where Ar isproduced from the abundant stable Ar ( ∼ Ar( n , n ) Ar(e.g., Loosli and Oeschger, 1968). The notation is the standard shorthand for a transfer reaction Ar + n → n + Arand n stands for a neutron. The Ar cosmogenic production keeps the atmospheric Ar / Ar mass ratio at a present-daysteady-state value of (8 . ± . × − (Benetti et al., 2007).Below the Earth’s surface Ar can also be produced cosmogenically, by cosmic ray negative muon interactions,in particular by negative muon capture on K (93.3 % of all potassium; µ − + K → ν µ + Ar). As the secondarycosmic ray muon flux drops o ff with depth, at depths greater than 2000 meters water equivalent (m.w.e.; ∼
700 m inthe rock) the nucleogenic production by K( n , p ) Ar dominates (Mei et al., 2010).Argon extracted from several deep natural gas reservoirs shows Ar / Ar ratios below the atmospheric value(Acosta-Kane et al., 2008; Xu et al., 2012). For this reason, underground gas sources have attracted the attentionof astroparticle physics projects that require a source of Ar with minimal radioactive Ar for low-energy rare eventdetection of WIMP Dark Matter interactions and neutrinoless double beta decay using liquid argon detectors.This paper provides an evaluation of the Ar nucleogenic production rate for specified rock compositions bycalculating the rate of Ar atoms produced by naturally occurring fast neutrons. The calculation also yields resultsfor other nuclides, in particular the rare stable Ne (0.27 % of neon). We describe in detail the methods of calculationof neutron, Ne and Ar yields and production rates, including the use of nuclear physics software tools (
SRIM , TALYS , MCNP6 ), and the input data, with the goal to present reproducible results. In section 2 we give the overview ofthe calculation. Section 3 deals with α -particle production. Section 4 discusses production of neutrons, both by ( α, n )reactions and by spontaneous fission. Neutron-induced reactions, in particular K( n , p ) Ar, are described in section5. In section 6 we present the results, including coe ffi cients for production rate evaluation (6.3) and an estimate ofthe uncertainty in the calculated results (6.4). A discussion follows in section 7 where we compare our calculationswith recent results (Yatsevich and Honda, 1997; Mei et al., 2010; Yokochi et al., 2012, 2013, 2014, sec. 7.1), calculatethe nucleogenic neutron flux out of a rock (7.2), and discuss the geochemical implications (7.3). Concluding remarksfollow in section 8.
2. Overview
Decays of natural radionuclides of U and Th in Earth’s interior produce α particles with specific known energies.The α particles slow down (i.e., lose kinetic energy) due to collisions with other atoms that form the silicate rock andeventually stop and form He atoms by ionizing nearby matter. A small fraction ( (cid:46) − ) of the α particles, beforelosing all their energy, participate in ( α, n ) reactions with light (atomic number Z (cid:46)
25) target nuclides to produce2ast ( ∼ U (mean SF neutron energy 1.7 MeV). Collectively all of these neutrons can thenparticipate in a number of interactions, including scattering and various reactions including ( n , p ) reactions. We areinterested in the ( n , p ) reaction on K to produce Ar (Figure 1).The number of Ar atoms produced per unit time per unit mass of rock is denoted S ( S for source). Tocalculate S it is necessary to know the neutron production rate as well as the neutron energy distribution. Theneutron spectra are required because reaction cross sections, including that of K( n , p ) Ar, strongly depend on theenergy of the incident particles. To calculate ( α, n ) neutron production and spectra, the production rate of α particlesand their energy distribution must be known. In sections 3 to 5 we discuss the production of α particles, neutrons, and Ar. A useful byproduct of the Ar calculation is the production rate of another noble gas nuclide, Ne, which isproduced in the ( α, n ) reaction on O with a limited contribution from Mg( n , α ) Ne reactions.We evaluate the results for a number of representative rock compositions. Our selection includes the three layersof Continental Crust (CC) of Rudnick and Gao (2014), the Bulk Oceanic Crust (OC) of White and Klein (2014), andthe Depleted Upper Mantle (DM) composition of Salters and Stracke (2004). Table 1 lists the elemental abundances.We also provide plug-in formulæ allowing inputs of an arbitrary elemental composition to obtain production rates.
3. Alpha-particle production
Alpha particles are produced in radioactive decays of naturally occurring
Th,
U, and
U and their daughternuclides along the decay chains. Other natural α emitters are not included as their contribution is negligible (the nextmost potent α emitter, Sm, gives 20 times fewer α particles per unit mass of rock per unit time than U). Eachindividual α -decay emits one α particle with specific energies, given by the decay scheme which is characterized bythe energy levels and intensities. We use decay data—half-lives, branching ratios, α energy levels, α intensities—from“Chart of Nuclides” available at the National Nuclear Data Center website (NNDC, 2016). The resulting α energyspectra are shown in Figure 2. Tabulated α energies and intensities as well as charts of decay networks can be foundin Supplementary Materials. The maximum α energy is 8.78 MeV from decay of Po in the thorium decay chain.The mean α energy in Th,
U, and
U chains is 6.0, 6.0, and 5.4 MeV, respectively.Decays of
Th,
U, and
U produce n α =
6, 7, and 8 α ’s per chain. The α production rate S α —number of α particles produced per unit time in unit mass of rock—of each decay chain is proportional to the elemental abundance(mass fraction) A of the parent (Th or U) and is obtained by simple multiplication, S α = n α A λ XN A M = n α A C , (1)where we have introduced the per-decay-to-rate conversion factor C , C ≡ λ XN A M , (2)in which λ is the decay constant and the fraction XN A / M is the number of parent nuclide atoms per unit mass ofelement (e.g., number of U atoms per 1 kg of natural U) where X is natural isotopic composition (mole fraction), M is standard atomic weight, and N A is Avogadro’s number (see Table 2 for values). The three equations (1), one foreach decay chain, can be recast into one plug-in formula to calculate α production rate from uranium concentration inppm and Th / U ratio, S α (cid:34) (cid:35) = (cid:32) + . A Th A U (cid:33) A U [ppm] . (3)Alpha production rates S α are evaluated in Table 3 for the representative rock compositions.Each α particle is emitted with a specific initial kinetic energy E α (Figure 2). It progressively loses its energy,mostly by inelastic scattering on atomic electrons and elastic scattering on nuclei. The range of an α particle, i.e., howfar it travels before stopping, is obtained by integration along its path,Range( E α ) = (cid:90) E α d x = E α (cid:90) d E α (cid:16) − d E α d x (cid:17) , (4)3nd is a function of the initial energy of the α particle in a given material. The material property − d E α / d x is thelinear stopping power. We use the SRIM software (srim.org; Ziegler et al., 2008), version
SRIM-2013.00 , to obtainthe mass stopping power − ρ − d E α / d x , where ρ is rock density (listed in Table 1). Mass stopping power is essentiallyidentical for all rock compositions we consider and its energy dependence is plotted in Figure 3. The range of anaverage-energy natural α ( ∼ µ m (Figure 3). We assume that α particle propagation isisotropic with no channeling in the crystal lattice on the grain scale.
4. Neutron production
The overall neutron production rate S n —number of neutrons produced per unit time in unit mass of rock—in eachdecay chain is calculated from S n = A C Y , (5)where the neutron yield Y , i.e., the number of neutrons produced from decay or fission of one atom of parent nuclide,is the sum of contributions from ( α, n ) reactions on various target nuclides and from spontaneous fission, Y = (cid:88) i Y i α, n + Y SF . (6)We now calculate the individual neutron yields. Energy spectra of the neutrons produced via various productionpathways are also discussed in the following sections. ( α, n ) reactions Neutron production by ( α, n ) was comprehensively described and quantified by Feige et al. (1968). Before an α particle loses all its kinetic energy, it can enter another atom’s nucleus to form a compound nucleus. The α has tohave enough energy to overcome the Coulomb barrier, i.e., the electromagnetic repulsion between the target nucleusand the α particle. The Coulomb barrier height V C is estimated from V C = π(cid:15) q q r = . Z Z R int [fm] MeV = . Z Z . A / + A / ) MeV , (7)where indices 1 and 2 refer to projectile and target, (cid:15) is vacuum permittivity, q is electric change, r is distance, Z isatomic number, R int (in femtometers) is the interaction radius taken as the sum of the two atomic radii, which in turnare approximated by a common empirical formula assuming nuclear volume to be proportional to the mass number A .As the Coulomb barrier height is proportional to the charge of the target nucleus, only relatively low Z target nuclidesallow for compound nucleus formation from natural α particles. Even though the compound nucleus can form with α energies below the Coulomb barrier due to quantum tunneling e ff ect, the interaction cross section drops rapidly.The compound nucleus is considered a short-lived intermediate state and from this compound nucleus there aremany possible channels that form various products nuclides. At energies of natural α particles, the most commonoutcome is that the compound nucleus then sheds a neutron to form a product nucleus. This constitutes an ( α, n )reaction. For example, α + O form a compound Ne ∗ , which then emits a neutron to produce the Ne product or, inshorthand, O( α, n ) Ne. The majority of the relevant ( α, n ) reactions are endothermic, i.e., their Q value is negative( Q value being reactant minus product rest masses), and the incoming α particle has to have energy above a threshold E th for the reaction to proceed. Specifically, the kinetic energy in the center-of-mass system before interaction has tobe larger than | Q | if Q <
0. In the laboratory reference frame, this translates into a relation for the threshold energy ofan endothermic reaction E th = − m + m m Q , (8)where m is the rest mass of projectile and m that of target nuclide.The Coulomb barrier and threshold energy result in a strong energy dependence of the ( α, n ) reaction cross section.Based on the magnitude of cross sections and the abundance of a particular nuclide in natural rocks, we identify 14target nuclides that are important for ( α, n ) neutron production in natural rocks. They are listed in Table 4 togetherwith the ( α, n ) reaction Q values, threshold energies E th (8), and Coulomb barrier heights V C (7).4he chance that an α particle, emitted with initial energy E α , participates in an ( α, n ) reaction on a light targetnuclide i is quantified by the thick target—meaning that the α stops within the medium—neutron production function P i ( E α ) = N i (cid:90) E α σ i α, n ( E α ) d E α d x d E α , (9)where N i is the atomic density of nuclide i ( i per unit volume) and σ i α, n is the ( α, n ) cross section for nuclide i . To get from neutron production function to ( α, n ) neutron yield consists of, for each decay chain and target nuclide i , accounting for all α decays and α energy levels within each decay, giving Y i α, n = decays (cid:88) k = b k levels (cid:88) l = f kl P i ( E kl ) , (10)where b k accounts for decay chain branching and f kl is the α intensity of level l in decay k with α energy E kl .To calculate the ( α, n ) neutron energy spectra, we evaluate the di ff erential neutron production function d P i / d E n ,d P i d E n ( E α , E n ) = N i (cid:90) E α d σ i α, n d E n ( E α , E n ) d E α d x d E α , (11)which is an equation analogous to (9) except that it calls for the neutron spectrum (or di ff erential cross section)d σ i α, n / d E n , instead of the integrated cross section σ i α, n ( E n denotes neutron energy). To obtain the yield spectrum (ordi ff erential yield) d Y i α, n / d E n , we follow the analogy through equation (10). Similar thick target methods for α particleshave been used to study nuclear reaction rates for astrophysics (Roughton et al., 1983), and these methods have beenused for thermonuclear reactions in energetic plasmas (Intrator et al., 1981).We use the nuclear physics code TALYS α, n ) neutron production cross sections σ i α, n as well as the emitted neutron energy spectrad σ i α, n / d E n . We use the default TALYS input parameters except for allowing the width fluctuation corrections calculatedusing the Moldauer model at all energies (see the
TALYS documentation for details). An example of our
TALYS inputfile is provided in Supplementary Materials. The
TALYS -calculated ( α, n ) cross sections for all considered targetnuclides are plotted in Figure 4.The calculated neutron production function P i (9) of various target nuclides is presented in Figure 5. The figureshows the relative importance of various target nuclides, which depends both on the cross section magnitude and thenuclide abundance in the rock. Figure 6 shows the neutron yield spectra d Y i α, n / d E n for each decay chain and targetnuclide from each of the three actinides. The mean energy of neutrons generated by ( α, n ) reactions is 1.8 MeV in Th decay chain, 1.6 MeV in
U chain, and 1.7 MeV in
U chain.
Spontaneous fission of
U produces 2.07 neutrons per fission (Shultis and Faw, 2002) and the SF branching ratiois 5 . × − (NNDC, 2016). The fission neutron spectrum is approximated by the Watt fission spectrum with neu-tron energy distribution following exp( − E n / a ) sinh √ bE n with parameters a = . b = .
811 MeV(SOURCES-4C, 2002), resulting in a mean SF neutron energy of 1.7 MeV. The fission neutron yield spectrumd Y SF / d E n is included in Figure 6. The number of neutrons produced in spontaneous fission of Th and
U, aswell as other nuclides with non-zero SF branching fraction in the decay chains ( U, Th,
Pa), is negligiblecompared to neutrons from ( α, n ) reactions.
5. Neutron-induced reactions: ( n , p ) and ( n , α ) To quantify the last step in the nuclear reaction sequence of Ar production, the K( n , p ) Ar exothermic reaction(Table 4), we use
MCNP6 , version
MCNP6 Beta3 , a general-purpose Monte Carlo N-Particle transport code developedat Los Alamos National Laboratory (mcnp.lanl.gov).
MCNP6 allows us to calculate the Ar yield per one neutron5here we specify the neutron energy spectrum for each of the ( α, n ) and SF neutron production channels. An exampleof our MCNP6 input file is provided in Supplementary Materials. In addition to Ar yields, we also use
MCNP6 tocalculate the Ne yield from the Mg( n , α ) Ne endothermic reaction (Table 4). From these nuclide yields andneutron production rates discussed in previous section, we can evaluate the Ar and Ne production rates S and S .
6. Results
The calculated yields Y and production rates S n of neutrons produced by both ( α, n ) reactions and by spontaneousfission are reported in Table 5. For the Upper CC compositions we report detailed results for each ( α, n ) target nuclideand each natural decay chain to show the relative importance of various neutron production channels. For the remain-ing representative compositions, only the O( α, n ) Ne, SF, and total neutron yields are presented. The maximumneutron yields are of the order of 3 × − neutrons per decay of 1 atom of long-lived radionuclide. With Upper CCcomposition, spontaneous fission is responsible for 11 % of total neutrons produced; with other compositions the SFcontribution ranges from 8 to 12 %. Of ( α, n ) target nuclides in Upper CC rocks, Al produces the most neutrons at32 % of total, followed by Na at 23 %, Si at 9.2 %, Si at 7.8 %, O at 7.0 %, Mg at 4.0 % and Mg at 2.4 %.These proportions obviously vary with composition, however, these seven target nuclides + SF neutrons account forat least 96 % of neutrons produced for all representative compositions we use. With the magnesium-rich DM com-position, Mg nuclides alone account for 69 % of neutrons. In terms of contributions of decay chains, with Upper CCcomposition
Th accounts for 57 % and
U for 41 % of neutrons produced. Again, exact proportions vary withcomposition but the trend, Th > U (cid:29) U, holds. Ar and Ne yields and production rates
The largest ( n , p ) and ( n , α ) nuclide yields (per neutron per elemental weight fraction of reacting nuclide, i.e., K orMg) are obtained with highest energy neutron spectra, coming from exothermic ( α, n ) reactions with positive Q values.That is, from ( α, n ) target nuclides C, O, Mg, and in particular Mg, which provides the highest energy neutrons.The maximum nuclide yields are of the order of 0.4 per neutron per weight fraction of K for Ar and 0.05 per neutronper weight fraction of Mg for Ne. The production rates, per year per kilogram of rock, also factor in the relativeimportance of neutron production channels. Therefore, the seven most neutron producing nuclides ( Al, Na, Si, Si, O, Mg, Mg) and SF are also responsible for most of Ar production. Neon-21 production by ( n , α ) isdominated by ( α, n ) neutrons from Mg (55–81 % of Ne produced depending on chosen composition). The resultsfor Ar and Ne production rates are reported in Table 5.Table 6 summarizes the subsurface production of α particles, neutrons, Ne, and Ar. Depending on selectedrock composition, on the order of 10 –10 α particles are produced in one kilogram of rock per year. The number ofneutrons produced is lower by ∼ Ne production rate drops by an additional factor of 15–20,and production rate of Ar decreases by at least another order of magnitude. In Table 6 we include production ratesof He, Ne, and Ar in units of cm STP per year per gram of rock, in order to facilitate easier comparison of ourresults to previous work. ffi cients for plug-in formulæ In Table 7 we provide coe ffi cients χ that one can use to calculate subsurface neutron production, Ne productionby ( α, n ) and nucleonic Ar production in a rock of arbitrary composition. The word “arbitrary” should be qualifiedwith “while close enough to a natural rock composition”. To indicate what “close enough” means, let us state thatwhile the provided coe ffi cients are based on a calculation with Upper CC composition, the empirical formulas yield aresult within 1 % of the actual value for all rock compositions we use here (Upper, Middle, and Lower CC, Bulk OC,Depleted Upper Mantle). The one exception is Ar production evaluation for DM composition, where the Upper CCcoe ffi cients overestimate Ar by 8 %.As an example, we use these coe ffi cients to evaluate the neutron and Ar production rate in the O ( α, n ) targetnuclide channel with α particles from U decay chain in a rock with Upper CC composition. The appropriate6eutron production coe ffi cient χ n ( O , U) in Table 7 is 2 . × and the neutron production rate is evaluated,using elemental abundances from Table 1, as S n = χ n ( O , U) × A O × A U = . × × . × . × − =
294 neutrons / (year kg-rock) , (12)which, as a consistency check, agrees with the rate reported in Table 5. With O target, the neutron production rate isequal to the Ne production rate by ( α, n ). The Ar production coe ffi cient χ ( O , U) in Table 7 is 4 . × and the Ar production rate is evaluated as S = χ ( O , U) × A O × A U × A K = . × × . × . × − × . = .
41 atoms / (year kg-rock) , (13)which again agrees with Table 5 result. A Python script which performs the evaluations (12) and (13) is provided viatinyurl.com / argon39. Estimates of rock composition generally carry large uncertainty, especially as one aims to infer the chemistry ofthe deep Earth (for example, the amount of heat-producing elements K, Th, and U in the Earth is only agreed uponup to a factor of a few; e.g., McDonough and ˇSr´amek, 2014). Still, it is worthwhile to estimate the uncertainty ofneutron, Ne and Ar production, assuming source rock composition is known. The additional uncertainty arisesfrom uncertainties in the nuclear physics models we employ and their underlying data sets, i.e., the evaluation ofstopping power (
SRIM ), neutron production cross sections (
TALYS ), and neutron interactions (
MCNP6 ).Half-lives of α -decaying nuclides, branching ratios, α energy levels, and α intensities have small uncertainties(NNDC, 2016); some uncertainties are listed in Table 2. We estimate the overall uncertainty of calculating α particleproduction to be below 1 %.Ziegler et al. (2010) report the accuracy of SRIM stopping power calculations when compared to experimentaldata, which is 3.5 % for stopping of α particles. Since these stopping powers are vital to implementation technologiesin the semiconductor industry, these values and uncertainty can be used with confidence.Estimation of ( α, n ) and ( n , p ) cross section uncertainties are challenging. Neutron reactions on common elementsin the few MeV energy range are important to many applications for fission reactors. Therefore one might expect data,models, and codes to have quite small uncertainties. Various nuclear data libraries exist, such as the ENDF / / endf) used by MCNP6 , which provide nuclear cross section data. The “evaluated nucleardata” come from assessment of experimental data combined with nuclear theory modeling. The evaluated librarydatasets come with no uncertainty estimate, however. Often large di ff / exfor.We integrate each cross section over energy up to a certain upper energy bound and calculate the relative di ff erencebetween the integrated data sets. We use this di ff erence as a measure of the cross section uncertainty (1 σ ).Clearly, this method of uncertainty estimation is not satisfying. However, given the lack of rigorous uncertaintyestimates provided by existing data libraries, the variability in the cross section data gives us at least some estimateof uncertainty. Koning and Rochman (2012) describe the method of rigorous uncertainty estimation in TALYS nucleardata evaluations which consists of a Monte Carlo approach to propagation of uncertainty in various nuclear modelparameters. Unfortunately, the available versions of
TENDL ( TALYS -based evaluated nuclear data library) do notinclude the uncertainty in ( α, n ) cross section and angular distribution.In the case of ( α, n ) reactions, we integrate TALYS -calculated cross sections, used in this study, and the availableexperimental dataset (using linear interpolation between the data points) up to 6 MeV (mean α energy in the Th7hain). The relative di ff erence between the integrals is taken as 1 σ uncertainty estimate. In the case of the K( n , p )reaction, the cross section from ENDF / B-VII.0 library used by
MCNP6 and an experimental dataset are integrated upto 3 MeV (above which the neutron energy spectrum is negligible). Again, the resulting relative di ff erence is adoptedas 1 σ uncertainty.Table 8 lists the various contributors to the calculation uncertainty. We considered the seven most important( α, n ) target nuclides, together with spontaneous fission accounting for >
96 % of neutron production. Experimentaldata from Flynn et al. (1978) (EXFOR entires A0509005, A0509007, A0509008) were used for Al, Si and Si;Norman et al. (1982) (C0731001) data for Na; combined Bair and Willard (1962) (P0120002) and Hansen et al.(1967) (P0116003) data for O. In the case of Mg and Mg where no experimental data are available in theEXFOR database, we arbitrarily set the uncertainty at 10 %. The experimental K( n , p ) cross section dataset isconstructed from Johnson et al. (1967) and Bass et al. (1964) experimental data. Nolte et al. (2006) report a non-zero K( n , p ) Ar cross section in the epithermal energy range. However, the cross section is smaller by at least 4 orders ofmagnitude compared to values for fast ( ∼ Ar production.Plots of cross sections for visual inspection can be found in Supplementary Materials.The uncertainty of the neutron propagation calculation is represented solely by the K( n , p ) cross section uncer-tainty, which is an oversimplification. Strictly speaking, one should consider the uncertainty in cross sections of allreactions that the neutron can participate in. Such complete treatment is beyond this study’s scope. However, K( n , p )has the largest of all ( n , p ) cross sections (Khuukhenkhuu et al., 2011), which justifies our simple approach. The N-particle simulations performed by MCNP6 also introduce a statistical uncertainty. Argon-39 yield tallies are repeateduntil the statistical uncertainty is below 0.5 %, therefore negligible compared to systematic uncertainty. Statisticaluncertainty of Ne tallies are as high as several tens of percent in some cases. However, as we show, the contributionof Mg( n , α ) Ne production is negligible relative to O( α, n ) Ne in rocks.The overall uncertainty of Ar production is calculated using standard error propagation rules following thissymbolic formula for Ar production rate evaluation,[ Ar production] = [ α decay] × [stopping power] − (cid:88) target [( α, n ) reaction] × [ K( n , p ) reaction] , (14)where each term in [brackets] carries its own uncertainty and these uncertainties are considered independent. Eval-uation of neutron and Ne production uncertainty is modified accordingly. We evaluate the overall uncertainty ofboth the neutron production calculation and Ne production calculation at <
20 %; the uncertainty estimate of Arproduction is 30 % (Table 8). We used the composition of Upper Continental Crust in the uncertainty estimation.The choice of elemental composition is reflected in the relative importance of various neutron production channelswhich somewhat changes the overall uncertainty with composition. These uncertainty estimates do not include theuncertainty in rock composition. To our knowledge, this is the first attempt at consistent uncertainty estimation of the Ar nucleonic production rate. Yokochi et al. (2014) state a 50 % uncertainty of their Ar production calculations,however, no reasoning is o ff ered to justify their estimate.Our calculation assumes a homogeneous elemental distribution in the rock. This is a good assumption for theEarth’s mantle. In the crust, however, the presence of accessory mineral phases introduces a heterogeneity in dis-tribution of some elements on the grain size scale, typically a few hundred µ m and particularly so for K, Th & U.In the Continental Crust the distribution of Th & U is heavily controlled by accessory phases (Bea, 1996). Giventhe mean free path of a MeV neutron in the rock, a few centimeters, the geometric e ff ect of this heterogeneity onneutron propagation is insignificant. However, as the range of a natural α particle is only few tens of microns, thisheterogeneity may change the neutron production both in terms of rate and energy content, depending on how various( α, n ) target nuclides are spatially distributed within the mineral phases (see Figures 5 and 6). This would in turna ff ect the neutron induced reaction yields of Ar and Ne. Martel et al. (1990) developed a simple geometric modelof spherical accessory phase inclusion in host rock and quantified their e ff ect on neutron yields relative to uniformdistribution of elements. In the limit of grain size exceeding the most energetic α particle range (25 µ m), they findthat presence of uraninite and monazite inclusions (deficient in the important light element ( α, n ) targets Al, Na, Si,O) in biotite decreases the neutron production by a factor of 5. The neutron and subsequently Ar production clearlydepends strongly on petrography. 8 . Discussion
We provide a comparison of nucleogenic production rates of neutrons, Ne, and Ar calculated in this studyto select recent evaluations (Yatsevich and Honda, 1997; Leya and Wieler, 1999; Mei et al., 2010; Yokochi et al.,2012, 2014). Similar calculations were previously performed by Fabryka-Martin (1988); Andrews et al. (1989, 1991);Lehmann et al. (1993); Lehmann and Purtschert (1997).Mei et al. (2010) calculated rates of Ar production for one representative granitic rock composition. We assumethat their reported Th and U abundance values were erroneously interchanged in their manuscript, as we are not awareof any granites where Th to U ratio is <
1. Using their corrected elemental composition, we calculate a neutronproduction rate at 5400 ±
700 neutrons / (kg yr), which is identical to the result of 5500 neutrons / (kg yr) calculated byMei et al. (2009). The agreement is expected as both Mei et al. (2009) and our calculation use ( α, n ) cross sectioninputs from TALYS . Mei et al. (2010) then estimate the Ar production rate to be 7 atoms per kg of rock per year.Our calculation gives 16 ± / (kg yr). The factor of two discrepancy stems from di ff erent methods of calculatingthe K( n , p ) Ar reaction yields. Mei et al. (2010) estimate the Ar production rate as proportional to the ratio of K( n , p ) Ar cross section to total neutron absorption cross section. It is not obvious whether—and how—Mei et al.(2010) account for the energy spectrum on the neutrons generated by ( α, n ) reactions and spontaneous fission. In thiswork, we use MCNP6 which calculates large numbers of histories of neutrons distributed along a calculated energyspectrum and accounts for all the possible interactions in the material of given composition. Of course, all thesemodels assume, perhaps incorrectly, a uniform distribution of elements.Yokochi et al. (2012) calculated nucleogenic Ar production for a few representative compositions using a “mod-ified version of SLOWN2 code”. With their Continental Crust composition with 2.0 % K, 5.0 ppm Th and 1.5 ppm U,their calculated production rate is 0.065 Ar atoms / (cm yr) which, assuming rock density 2.7 g cm − , translates to24 atoms / (kg yr). When we rescale our result for Middle CC composition which is the closest in K, Th, U abundanceto their K, Th, U, assuming other elemental abundances unchanged, we arrive at 13 ± / (kg yr), another factorof two di ff erence in results, in the other direction.Yokochi et al. (2013) calculated the Ar production rate for two specific rock compositions. Their rates were 4 to6 times above our calculations, in addition to showing some obvious inconsistencies (a lower Ar production rate fora composition richer in K, Th, U). Following ˇSr´amek et al. (2013) and subsequent discussion, Yokochi et al. (2014)updated their result for “Lava Creek Tu ff ” rock composition to 120 ±
60 atoms / (kg yr). Our calculation with theirK, Th, U input abundances yields 140 ±
40 atoms / (kg yr). Their SLOWN2 code calculation assumes mono-energeticneutrons at 2 MeV while our calculation accounts for the actual neutron energy spectrum. When we assume all ( α, n )and SF neutrons are produced with an initial energy of 2 MeV (while keeping the neutron yields unchanged), weobserve a 35% decrease in the Ar production rate. Yokochi et al. (2014)’s result falls in between our calculationwith the full neutron spectra (Figure 6) and our calculation with 2 MeV neutrons, and within the large uncertaintyagrees with our result.Table 9 breaks down Ne production into the O( α, n ) and Mg( n , α ) channels. The neutron induced Neproduction is negligible for crustal compositions and only contributes 3.4 % to Ne produced in a Depleted UpperMantle rock. We calculate the nucleogenic Ne / He production ratio at (4 . ± . × − in Continental Crust andat (4 . ± . × − in Oceanic Crust and Depleted Mantle. Within uncertainty, our result agrees with Yatsevich andHonda (1997) who calculated the Ne / He production ratio at 4 . × − . Their optimistic uncertainty of < α, n ) yield in their analysis. Using chemical abundances for the“crust” and “mantle” compositions of Mason and Moore (1982), which were the input concentrations in both Yatsevichand Honda (1997) and Leya and Wieler (1999) studies, we calculate Ne production rates of 469 ±
78 atoms / (kg yr)((1 . ± . × − cm STP / (g yr)) for the crust and 4 . ± . / (kg yr) ((1 . ± . × − cm STP / (g yr))for the mantle. Our results lie in between the lower rates of Leya and Wieler (1999) and the higher values of Yatsevichand Honda (1997), while both fall within our 1 σ uncertainty bounds; see also Ballentine and Burnard (2002). Near the ground, nucleogenic and fissiogenic fast neutrons produced in the rock can be ejected into the atmosphereand enhance the near-surface neutron flux derived from cosmic rays. Most of these neutrons that might be ejectedfrom the rock would originate within the first few tens of centimeters within the surface and from ∼ . × − n cm − s − for granite and 5 . × − n cm − s − for limestone. The factor of 15 di ff erence reflectsthe disparate Th and U content of the “hot” granite and the “cold” limestone. These fluxes only account for the ( α, n )and SF neutrons. It turns out that their contribution is minor compared to the cosmic ray neutron production, wherethe flux of 0.4 eV–0.1 MeV neutrons was measured by Yamashita et al. (1966) to be 2 . × − n cm − s − at sea level.Cosmic ray neutrons therefore constitute the dominant component ( >
99 %) of neutron flux at Earth’s surface.
Given the consequences of global climate change and the world population increase, a need is arising to explorepumping of groundwater from previously unused deep reservoirs. Interestingly, Gleeson et al. (2015) demonstrate thatonly ∼ H) dating (half-life 12 . ± .
02 years). The nextavailable radiometric tool on the age scale is Ar (half-life 269 ± H and also Kr (10 . ± .
014 years), and the longer-lived C (5700 ±
30 years). The number of Ar dating studiesand measurements so far were limited due to the large sample size and analytical requirement of the well establishedLow-Level Counting (LLC) method (Loosli and Purtschert, 2005). However, ongoing technological developmente ff orts, in particular the Atom Trap Trace Analysis (ATTA; e.g., Lu et al., 2014), now allow for a relatively fast Arage analysis of small sample volumes (Ritterbusch et al., 2014), and are expected to increase the number of analysesin the future. Understanding the relationship between the atmosphere- and subsurface-generated Ar is thus essentialin deep groundwater radiometric dating, in studies of fluid circulation in the Earth’s crust, circulation pathways andresidence times of ocean water masses, and in dating of deep ice cores.Hydrological and glaciological dating using Ar relies on counting Ar atoms in water / ice samples as Ar ac-tivity predictably decays exponentially, it is assumed, after sequestration of the sample from the atmosphere. Cosmo-genic neutron flux sharply decreases with depth in the soil / rock, and therefore, the atmospheric production mechanism Ar( n , n ) Ar becomes insignificant. However, any other mechanism of ambient Ar production at depth must befactored in the dating analysis in order to obtain correct ages. In principle, several noble gas nuclide productionmechanisms are possible underground. Cosmogenic production includes neutron spallation, thermal neutron capture,negative muon ( µ − ) capture, and fast muon induced reactions with nuclides abundant in the rock (Niedermann, 2002).Mei et al. (2010) showed that µ − capture dominates Ar production at depths down to 1800 m.w.e. ( ∼
700 m depth).At greater depths, the nucleogenic K( n , p ) Ar production dominates.In most old groundwater, Ar is below detection limit. This is due to the combination of relatively low productionrate of Ar in rocks, slow release from rock matrix to groundwater, and short half-life. Therefore, Ar studies forwater resource assessment are typically based on simple radioactive decay where the subsurface production does notcontribute significantly to the Ar budget and Ar is simply a decay tracer (e.g., Delbart et al., 2014; Edmunds et al.,2014; Mayer et al., 2014; Visser et al., 2013; S¨ultenfuß et al., 2011; Corcho Alvarado et al., 2007). Geochemicalsites where Ar subsurface production becomes important are high temperature environments such as geothermalsystem (Purtschert et al., 2009; Yokochi et al., 2013, 2014), highly radioactive environments (Andrews et al., 1989),or settings with intense water–rock interactions. Yokochi et al. (2012) presented a case study for such high Arsettings.To account for the nucleogenic Ar in groundwaters, Yokochi et al. (2012) proposed a novel method using mea-surements of argon isotopic composition in the fluid. The method is based on comparison of a measurement to amodeled evolution of Ar / Ar ∗ ratio in an argon-producing rock ( Ar ∗ denotes radiogenic Ar). Its age range ex-ceeds the usual limit of several half-lives of Ar — after which Ar concentration assumes a steady-state value inan Ar-producing rock while Ar continues to accumulate, thus Ar / Ar keeps evolving. The input in their modelare the Ar and Ar production rates in the rock, which then loses argon to fluid. Yokochi et al. (2012) present casestudies for two sites with available groundwater argon isotopic composition data, Milk River Sandstone and StripaGranite. In our e ff ort to reproduce their results, we have identified a mistake in their groundwater age calculation:We are only able to recover Yokochi et al. (2012)’s results if we intentionally divide (instead of multiply) with rockdensity when recalculating from wt.% K to number of K atoms per cm rock. In Table 10 we show both Yokochi10t al. (2012)’s calculation results and our “original recalculation” using their input values and rock density 2.7 g / cm for both Milk River Sandstone and Stripa Granite, chosen to best match their output. We also show the “correctedrecalculation,” where we appropriately multiply with rock density when recalculating from wt.% K to number of Katoms per cm rock. Fixing the calculation error results in a downward revision of both the lower and upper limitson groundwater age by a factor of ∼ Ar production rate of 0.015 atoms / cm / yr. Using arock composition of 2.4 ppm U, 6.3 ppm Th, 1.07 % K, and 3.91 % Ca (Andrews et al., 1991) we calculate a Arproduction rate of 9.45 atoms / kg / yr or, using a rock density of 2.6 g / cm (Andrews et al., 1991), 0.023 atoms / cm / yr.For Stripa Granite, Yokochi et al. (2012) use Ar production rate of 1.3 atoms / cm / yr. Using the Stripa Granitecomposition, including 3.84 %K, 33.0 ppm Th, 44.1 ppm U (Andrews et al., 1989), we calculate a Ar productionrate of 367 atoms / kg / yr or 0.95 atoms / cm / yr, using a rock density of 2.6 g / cm (Andrews et al., 1989). Using thesenew Ar production rates, the calculated groundwater ages lie within a factor of 2 of the corrected results of Yokochiet al. (2012). The lower limits on groundwater age for the two Milk River Sandstone samples, Well 8 and Well 9, arecalculated to be 6.1 kyr and 17 kyr. The upper limits on age are 22 Myr and 27 Myr. Both Stripa Granite samples, NIand VI, have Ar / Ar ∗ above the revised value for the reservoir rock assuming closed system. The upper limits ongroundwater age of the Stripa Granite samples are calculated to be 0.056 Myr and 0.35 Myr (Table 10).
8. Conclusions
Calculations of subsurface nucleogenic Ar production are extremely useful in hydrology applications and guidestrategies for obtaining argon for particle physics detectors, while Ne production is an integral component of geo-chemical interpretations of noble gas isotopic composition. We describe and calculate the production of neutrons in( α, n ) interactions of naturally emitted α particles and the nucleogenic production of noble gas nuclides Ar and Ne.We use new nuclear physics codes to evaluate ( α, n ) reaction cross sections and spectra ( TALYS ) and to track neutronpropagation in the rock (
MCNP6 ).Nucleogenic Ar production rate in a rock previously calculated by Mei et al. (2010) is a factor of two belowour result, the calculation of Yokochi et al. (2012) is a factor of two above our calculation, and the Yokochi et al.(2014) result lies within our uncertainty margin. We consider the sources of uncertainty in the calculation alongwith identifying areas that require better constraints. The largest uncertainty comes from uncertainties in the nuclearreaction cross sections, which we estimate from the variability among available experimental data and cross sectionsfrom nuclear data libraries. The overall calculation uncertainty is estimated to be ∼
30 % for Ar production, andwithin 20 % for Ne and neutron production. We expect this uncertainty could be decreased if uncertainty estimateswere available for the ( α, n ) cross sections within TALYS and for the ENDF / B-VII library data used by
MCNP6 . Ouranalysis yields a tighter uncertainty estimate compared to the recent study by Yokochi et al. (2014) who estimated alarge error bar (50 %) in the calculated Ar production rate.The Ar chronometer fills in a gap between H and C. Conventional radiometric Ar dating of hydrologicalreservoirs assumes a simple radioactive decay of initial Ar content in the groundwater. Non-negligible subsurface Ar production and / or groundwater–rock interaction in specific environments requires a more refined Ar / Ar ∗ chronometer proposed by Yokochi et al. (2012). Their original calculation for the fluid samples from the Milk RiverSandstone and Stripa Granite reservoirs overestimated the groundwater ages by a uniform factor of 7. An update usingour new Ar productions rates results in groundwater age limits which are a factor of 4–11 times lower compared tothe original analysis.The Supplementary Materials document the uncertainty estimate of ( α, n ) and ( n , p ) cross sections, and includedetails of natural decay chains, examples of our TALYS and
MCNP6 input files, and a link to argon39 , a Fortran 90code we wrote, which handles the overall calculation. Supplementary Materials are provided as a separate PDF document. cknowledgements We are grateful to Alice Mignerey and Bill Walters for invaluable discussions on broad topics in nuclear chemistry.Jutta Escher brought our attention to
TALYS . Mike Fensin provided helpful advice on
MCNP usage via the mcnp-forum discussion list. Milan Krtiˇcka advised on uncertainty estimate. C´ecile Gautheron shared her O( α, n ) Ne crosssection. Two reviewers provided thorough assessment and thoughtful comments on the manuscript. We gratefullyacknowledge support for this research from NSF EAR 0855791 CSEDI Collaborative Research: Neutrino Geophysics:Collaboration Between Geology & Particle Physics.
Author contributions
O ˇS and WFM proposed and conceived the study with SM. O ˇS conducted all simulations and incorporated modelsand constraints from WFM and SM. Graduate student LS participated early on in data modeling and interpretation ofthe results. RJP provided insights into the modeling of uncertainties of the nuclear physics data. The manuscript waswritten by O ˇS and all the authors contributed to it. All authors read and approved of the final manuscript.
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Non-zero amount of carbon was included in the CC and OC composition (0.5 %, 0.1 %, and 0.02 % of CO by weight for Upper,Middle, and Lower CC, and 0.05 wt% CO for OC). Rock density is taken from CRUST2.0 (Laske et al., 2001) for the crustal layers and fromPREM (Dziewonski and Anderson, 1981) for the depleted mantle (average from MOHO to 400 km depth). Elemental abundances not provided ina particular compositional estimate entered as n / a. Z symbol Upper CC Middle CC Lower CC Bulk OC DM Granite Limestone3 Li – – – – – 3.40E-05 n / a6 C 1.36E-03 2.73E-04 5.46E-05 1.36E-04 1.37E-05 2.30E-04 0.1107 N 8.30E-05 n / a – n / a – – n / a8 O 0.480 0.469 0.452 0.446 0.440 0.494 0.4889 F 5.57E-04 5.24E-04 5.70E-04 n / a 1.10E-05 1.28E-03 n / a11 Na 0.0243 0.0251 0.0197 0.0164 2.15E-03 0.0300 1.88E-0412 Mg 0.0150 0.0216 0.0437 0.0621 0.230 4.72E-03 2.50E-0313 Al 0.0815 0.0794 0.0894 0.0831 0.0227 0.0815 3.40E-0314 Si 0.311 0.297 0.250 0.234 0.210 0.318 0.032015 P 6.55E-04 6.55E-04 4.36E-04 4.36E-04 – 6.11E-04 1.75E-0416 S – – 3.45E-04 n / a – – n / a17 Cl 3.70E-04 1.82E-04 2.50E-04 n / a – – n / a19 K 0.0232 0.0191 5.06E-03 6.51E-04 6.00E-05 0.0370 2.40E-0320 Ca 0.0257 0.0375 0.0685 0.0843 0.0250 0.0130 0.35922 Ti 3.84E-03 4.14E-03 4.91E-03 6.59E-03 7.98E-04 2.84E-03 4.20E-0424 Cr – – – 3.17E-04 2.50E-03 – –25 Mn 7.74E-04 7.74E-04 7.74E-04 8.52E-04 1.05E-03 – –26 Fe 0.0385 0.0460 0.0655 0.0635 0.0627 0.0185 2.88E-0328 Ni – – – – 1.96E-03 – –38 Sr – – – – – 4.75E-04 –56 Ba – – – – – 1.88E-03 –90 Th 1.05E-05 6.5E-06 1.2E-06 2.10E-07 1.37E-08 2.47E-05 9.49E-0792 U 2.7E-06 1.3E-06 2E-07 7.00E-08 4.70E-09 1.95E-06 1.42E-06K / U 8609 14687 25320 9300 12766 18974 1690Th / U 3.9 5.0 6.0 3.0 2.9 13 0.67 ρ in g cm − Table 2: Atomic and decay quantities for natural long-lived parent nuclides. Natural isotopic composition X and relative atomic mass M from NIST(2016), half-lives t / from NNDC (2016), λ = ln 2 / t / . Atomic mass truncated to six digits. Numbers in parentheses give uncertainty in lastdigits, otherwise uncertainty beyond shown digits. The conversion factor C (eqn. 2) is the number of decays per unit time per unit mass of parentelement. Quantity Symbol Unit Th U UElemental abundance A kg-elem / kg-rockNatural isotopic composition X mol-nucl / mol-elem 1.0000 0.007204(6) 0.992742(10)Standard atomic weight M g mol − t / yr 14.0(1) 0.704(1) 4.468(3)Decay constant λ − s − α ’s per decay chain n α C kg-elem − s − able 3: Calculated production rates S α of α particles, as number of α ’s produced per second in 1 kilogram of rock, in a particular decay chain(columns 2–4) and the total (column 5). Composition Th U U TotalUpper Continental Crust 256 10.9 267 533Middle Continental Crust 158 5.24 128 292Lower Continental Crust 29.2 0.806 19.8 49.8Bulk Oceanic Crust 5.11 0.282 6.91 12.3Depleted Mantle 0.334 0.0190 0.464 0.817
Table 4: Target and product nuclides, Q values, threshold energies E th (eqn. 8), and Coulomb barriers V C (eqn. 7) of ( α, n ) reactions. Also Q and E th of neutron induced reactions. In cases where the product nuclide is unstable ∗ , we show the final stable nuclide as well. Energies in MeV.Nuclide rest masses for calculation of Q and E th taken from NIST (2016). Target Reaction Product
Q E th V C Al ( α, n ) P ∗ → Si − . Na ( α, n ) Al ∗ → Mg − . Si ( α, n ) S − . Si ( α, n ) S − . O ( α, n ) Ne − . Mg ( α, n ) Si 0 . Mg ( α, n ) Si 2 . F ( α, n ) Na ∗ → Ne − . O ( α, n ) Ne 0 . Fe ( α, n ) Ni ∗ → Co − . K ( α, n ) Sc ∗ → Ca − . Ti ( α, n ) Cr ∗ → V − . C ( α, n ) O 2 . Ca ( α, n ) Ti − . K ( n , p ) Ar 0 . / a Mg ( n , α ) Ne − . / a16 a b l e : C a l c u l a t e d r e s u lt s f o r n e u t r ony i e l d Y , n e u t r onp r odu c ti on r a t e S n , A r p r odu c ti on r a t e S A r , a nd N e p r odu c ti on r a t e by ( n , α )r eac ti on S n , α N e . N e p r odu c ti on r a t e by ( α , n )r eac ti on i s e qu a lt on e u t r onp r odu c ti on r a t e w it h O t a r g e t . D e t a il e d r e s u lt s — y i e l d s o rr a t e s fr o m eac hn e u t r onp r odu c ti on c h a nn e li n eac hd eca y c h a i n — a r e r e po r t e d f o r U pp e r CC r o c k c o m po s iti on . O n l y s e l ec t e dou t pu ti s li s t e d f o r o t h e r c o m po s iti on s . N e u t r ony i e l d s a r e g i v e np e r d eca yo f a t o m o f l ong - li v e d r a d i onu c li d e . P r odu c ti on r a t e s g i v e np e r y ea r p e r k il og r a m o fr o c k . N e u t r ony i e l d ( Y ) N e u t r onp r odu c ti on r a t e ( S n ) A r p r odu c ti on r a t e ( S A r ) N e p r od . r a t e by ( n , α )( S n , α N e ) t a r g e t T h U U T h U U S u m T h U U S u m T h U U S u m U pp er C o n t i n e n t a l C r u s t A l . e - . e - . e - . . . . . . . . . . . . N a . e - . e - . e - . . . . . . . . . . . . S i . e - . e - . e - . . . . . . . . . . . . S i . e - . e - . e - . . . . . . . . . . . . O . e - . e - . e - . . . . . . . . . . . . M g2 . e - . e - . e - . . . . . . . . . . . . M g1 . e - . e - . e - . . . . . . . . . . . . F . e - . e - . e - . . . . . . . . . . . . O . e - . e - . e - . . . . . . . . . . . . F e . e - . e - . e - . . . . . . . . . . . . K . e - . e - . e - . . . . . . . . . . . . T i . e - . e - . e - . . . . . . . . . . . . C . e - . e - . e - . . . . . . . . . . . . C a . e - . e - . e - . . . . . . . . . . . . SF . e - . e - . e - . . . . . . . . . . . . T o t a l . . . . . . . . M i dd l e C o n t i n e n t a l C r u s t O . . . . SF . . . . T o t a l . . . . . . . . L o w er C o n t i n e n t a l C r u s t O . . . . SF . . . . T o t a l . . . . . . . . B u l k O ce a n i c C r u s t O . . . . SF . . . . T o t a l . . . . . . . . . D e p l e t e d U pp er M a n t l e O . . . . SF . . . . T o t a l . . . . . . . . . . . . able 6: Summary of production rates of He, neutrons, Ne, and Ar in units of both the number of atoms or neutrons per year per kilogram ofrock and cm STP per year per gram of rock. Compositional estimates are described in Table 1. − kg − cm STP yr − g − Composition He neutrons Ne Ar He Ne ArUpper Continental Crust 1 . × . × − . × − . × − Middle Continental Crust 8 . × . × − . × − . × − Lower Continental Crust 1 . × . × − . × − . × − Bulk Continental Crust 9 . × . × − . × − . × − Bulk Oceanic Crust 3 . ×
260 15.8 0.0235 1 . × − . × − . × − Depleted Upper Mantle 2 . × . × − . × − . × − Table 7: Coe ffi cients for evaluation of neutron production (columns 3–5) and Ar production (columns 6–8) for arbitrary composition. Secondcolumn and third row indicate the elements whose abundances ( A as weight fractions) cross-multiply a particular coe ffi cient. Neutron productioncoe ffi cients in units of “neutrons per year per kg-rock per wt-frac-target-elem per wt-frac-chain-parent-elem”. Argon-39 production coe ffi cients inunits of “ Ar atoms per year per kg-rock per wt-frac-target-elem per wt-frac-chain-parent-elem per K-wt-frac”. See section 6.3 for examples.
Neutron production Ar productionChain Th U U Th U U( α, n ) target A Th U U Th U U Al Al 2.65e + + + + + + Na Na 6.07e + + +
10 5.04e + + + Si Si 1.95e + + + + + + Si Si 1.68e + + + + + + O O 8.77e + + + + + + Mg Mg 1.72e + + + + + + Mg Mg 1.01e + + + + + + F F 1.60e +
10 2.34e + +
10 1.78e + + + O O 9.50e + + + + + + Fe Fe 1.28e + + + + + + K K 1.10e + + + + + + Ti Ti 4.35e + + + + + + C C 3.61e + + + + + + Ca Ca 2.98e + + + + + + + + + + + + able 8: Estimate of calculation uncertainty and its various contributions, assuming chemical composition is known precisely. Contribution ofthe specific ( α, n ) channel and spontaneous fission to neutron and Ar production is shown in columns 3 and 4. Calculated with Upper CCcomposition. † No experimental data available, uncertainty was arbitrarily set at 10 %.
Uncert. est. Neutron Ar% % contrib. % contrib.Decay data, α production < Al( α, n ) cross section 36 32 25 Na( α, n ) cross section 7.7 23 15 Si( α, n ) cross section 7.3 9.2 13 Si( α, n ) cross section 20 7.8 5.4 O( α, n ) cross section 17 7.0 13 Mg( α, n ) cross section 10 † Mg( α, n ) cross section 10 † α, n ), neutron production 12Overall ( α, n ), Ar production 10 K( n , p ) cross section 28Neutron production calculation 13 Ne production calculation 17 Ar production calculation 30
Table 9: Calculated production rates of Ne by ( α, n ) and ( n , α ) and Ne / He ratio. Compositional estimates are described in Table 1.
Composition Ne prod. in atoms / kg-yr % contrib.( α, n ) ( n , α ) Total ( α, n ) ( n , α ) Ne / HeUpper Continental Crust 753 0.159 753 99.98 0.02 4 . × − Middle Continental Crust 415 0.165 416 99.96 0.04 4 . × − Lower Continental Crust 70.1 0.104 70.2 99.85 0.15 4 . × − Bulk Oceanic Crust 15.8 0.0452 15.8 99.71 0.29 4 . × − Depleted Upper Mantle 1.03 0.0366 1.06 96.55 3.45 4 . × − able 10: Update of Yokochi et al. (2012)’s Table 2, using their notation. Original recalculation:
We are only able to recover Yokochi et al.(2012)’s results if we intentionally divide (instead of multiply) with rock density when recalculating from wt.% K to number of K atoms percm rock. We use Yokochi et al. (2012)’s input values and rock density 2.7 g / cm for both Milk River Sandstone and Stripa Granite. Correctedrecalculation:
Appropriately, we multiply with rock density when recalculating from wt.% K to number of K atoms per cm rock. Recalculationwith new input:
We use our new values of Ar production rate P and rock density of 2.6 g / cm for both Milk River Sandstone (Andrews et al.,1991) and Stripa Granite (Andrews et al., 1989). The ratio R R (defined as R R ≡ Ar / Ar ∗ in the rock; R F is the ratio in the fluid) is expressed inunits of atmospheric R a = . × − ; α is the argon loss constant; φ R is porosity of the rock; Age low and Age up are the lower and upper limits onthe groudwater age. INPUT CALCULATIONRock age K P R R closed Age low α / φ R Age up Myr wt.% at. / cm / yr R a kyr year − Myr
Milk River Sandstone Well 8
Yokochi et al. (2012) values 83.5 1.1 0.015 193 27.1 7.3e-4 118Original recalculation ” ” ” 192 25.5 7.2e-4 107Corrected recalculation ” ” ” 26.4 3.5 7.2e-4 14.7Recalculation with new input ” ” 0.023 45.7 6.1 4.3e-4 21.9
Milk River Sandstone Well 9
Yokochi et al. (2012) values ” 1.1 0.015 193 75.2 3.2e-4 328Original recalculation ” ” ” 192 71.3 3.1e-4 299Corrected recalculation ” ” ” 26.4 9.8 3.1e-4 41.0Recalculation with new input ” ” 0.023 45.7 16.9 1.9e-4 26.7
Stripa Granite NI
Yokochi et al. (2012) values 1650 3.8 1.3 155 R F > R R R F > R R R F > R R R F > R R Stripa Granite VI
Yokochi et al. (2012) values ” 3.8 1.3 155 3.3 2.9e-3 2.5Original recalculation ” ” ” 153 3.1 2.9e-4 2.5Corrected recalculation ” ” ” 20.9 0.43 2.9e-4 0.34Recalculation with new input ” ” 0.95 15.9 R F > R R adioactive decay Th U U 678 Pb Pb Pb 446+ + α ¯ ν e e − +( )+ + α + + n Al Na Si Si O … P Al S S Ne … n p K Ar radiogenic α particlesnucleogenic neutronsnucleogenic Ar U spontaneous fi ssion + + n ~2 fi ssiogenic neutrons ~90% ~10% ( α ,n)(n,p) Figure 1: Overview of nucleogenic Ar production. ean α energy Th U α i n t en s i t y U Energy in MeV
Figure 2: Natural α emission energy spectra. Blue spikes represent intensity of each alpha emitted. Spike intensities sum up to number of alphasemitted in particular decay chain (6, 7, 8 for Th, U, U, respectively). Red dashed line shows the mean energy of emitted α particles. topping power Range Upper Cont. CrustDepl. Mantle S t opp i ng po w e r i n M e V c m g − α energy in MeV α r ange i n µ m Figure 3: Mass stopping power − ρ − d E / d x for an α particle in rock (dashed line, left vertical axis) and range of α particle (eqn. 4, solid lines,right vertical axis) as a function of α energy. Range plotted up to maximum energy of a natural α particle (8.78 MeV from decay of Po). Thedi ff erence between range in Upper Cont. Crust (red) and Depleted Mantle (green) simply reflects the di ff erence in rock density (Table 1). Curvesfor Middle and Lower CC and OC fall between the two shown. Al Na Si Si O Mg Mg F O Fe K Ti C Ca O Mg0100200300400500600 C r o ss s e c t i on i n m ba r n α energy in MeV Figure 4: Energy dependent ( α, n ) cross sections σ α, n for target nuclides considered in this study as calculated by TALYS version 1.6. One mbarn = − cm . Al Na Si Si O Mg Uses Upper Continental Crust rock composition P r obab ili t y o f ( α , n ) r ea c t i on Initial α energy in MeV Figure 5: Thick target ( α, n ) neutron production function P (i.e., neutron yield per one α particle; eqn. 9) plotted against the initial α energy forvarious target nuclides. Curves for the six most neutron producing targets are labeled and color-coded. Remaining targets plotted as thin black lines.The Upper CC rock composition is used. Di ff erent amplitudes for various target nuclides stem from the combined e ff ect of elemental abundance,natural isotopic composition, and energy dependence of ( α, n ) cross section. Al Na Si Si O Mg Th U D i ff e r en t i a l y i e l d s pe c t r u m i n M e V − spontaneous fission U Neutron energy in MeV
Figure 6: ( α, n ) neutron yield spectra d Y α, n / d E n plotted against neutron energy for each decay chain and six most neutron producing targets. Thefission neutron yield spectrum d Y SF / d E n is also plotted (non-negligible only in U decay). The Upper CC rock composition is used. Area beloweach curve integrates to the neutron yield Y α, n or Y SF (i.e., neutrons produced per decay of one atom of parent nuclide).(i.e., neutrons produced per decay of one atom of parent nuclide).