Introduction

Ice cores are unique archives for past climate conditions1,2,3,4. A prerequisite for retrieving climate information from ice cores is reliable dating of the ice. Continuous timescales are commonly obtained by layer counting, stratigraphic matching or orbital tuning, but these methods can be hampered, especially by disturbed stratigraphy or layer thinning. This is particularly the case for the bottom sections of ice cores, where climate information up to one million years and beyond can be stored5,6,7. If stratigraphic continuity is not preserved, alternative methods may be employed, e.g. using combinations of CH4 and δ18Oatm8,9, the time-dependent growth of air-hydrates7,10 or the radioactive decay of 36Cl and 10Be11,12. In particular, based on the increase with time of crustal 40Ar in the atmosphere, ages going back to several million years have been obtained for discrete ice core samples6,13,14,15,16. However, all these methods have limitations and are not generally applicable. As a consequence, the ages of numerous basal ice core sections remain elusive, despite the significance of the paleoclimate information contained in the ice5,17,18,19.

81Kr (t1/2 = 229 ka) has long been identified as an ideal isotope for radiometric dating of water and ice in the age range from thirty thousand to over one million years20,21,22. It is produced by cosmic rays in the stratosphere and uniformly distributed throughout the atmosphere with an isotopic abundance 81Kr/Kr = (9.3 ± 0.3) × 10−13 23. Being chemically inert, the isotopic abundance of 81Kr in the environment is not altered by geo-chemical processes and thus preserves the undisturbed age information. Atom Trap Trace Analysis (ATTA)24,25, an analytical method that detects single atoms via their fluorescence in a magneto-optical trap, has overcome the difficulty of detecting 81Kr at the extremely low isotopic abundance levels of 10−14 − 10−12 in the environment, enabling a wide range of applications in the earth sciences26. The sample size required for 81Kr analysis was ~100 kg of ice a decade ago when 81Kr dating of Antarctic blue ice was demonstrated27. It has since been reduced to ~10 kg of polar ice in recent analyses28,29,30,31,32,33. However, this sample size requirement is still too large to employ the full potential of 81Kr dating, especially for recovering paleoclimate information from deep ice cores.

The main limitation for reducing the sample size as well as extending the age range of 81Kr dating has been the cross-sample contamination in the ATTA system. It is caused by a discharge needed for exciting the krypton atoms from the ground to a metastable level, in which laser cooling and trapping can be applied25,34. Recently, optical excitation has been realized as an alternative method to the discharge excitation for producing metastable krypton atoms. This progress was made possible by the development of an intense and resonant light source at 124 nm34.

In this work, we have developed an ATTA system based on optical excitation for 81Kr analysis of water and ice samples. By avoiding the cross-sample contamination caused by the discharge excitation, the 81Kr background is reduced by two orders of magnitude. As a result, the required sample size is lowered to 100 nL STP of krypton, which can be extracted from ~1 kg of polar ice. The reduced 81Kr background moreover leads to the age range of 81Kr dating being significantly extended. We validate the optical-excitation ATTA system by analyzing reference samples that were previously measured with a discharge-excitation ATTA system using ~30 times larger sample size. We confirm the low background with a sample depleted in both 81Kr and 85Kr, produced in the course of this work with an electromagnetic mass separator. We show 81Kr dating of 1-kg ice core samples from Taylor Glacier, Antarctica, whose gas ages are precisely known from stratigraphic alignment. The uncertainties of the measured 81Kr ages approach the fundamental limit imposed by the Poisson statistics of the small number of 81Kr atoms contained in 1 kg of ice.

Results

Optical-excitation ATTA for 81Kr analysis

Figure 1a illustrates the ATTA system based on optical excitation. The krypton atoms are cooled to  − 160 °C in the source tube using liquid nitrogen. At the tube outlet, metastable krypton atoms are generated using optical excitation, i.e. a two-photon transition (124 nm + 819 nm) to a higher lying intermediate level followed by a spontaneous decay to the metastable level (Fig. 1b). The intense and resonant 124 nm light needed for excitation is produced by a recently developed low-pressure krypton discharge lamp34. The resulting metastable krypton beam is collimated by transverse laser cooling in a tilted-mirror configuration and gently focused. After being longitudinally slowed down in a Zeeman slower, the krypton atoms are captured in a magneto-optical trap. The target isotope is chosen by tuning the laser frequencies to the isotope-specific resonances of the 811 nm atomic transition for laser cooling and trapping (Fig. 1b). Individual 81Kr (or 85Kr) atoms are detected in the trap by imaging their fluorescence onto an EM-CCD camera. For 100 nL STP of krypton derived from modern air, the typical loading rates for 81Kr and 85Kr are 40 atoms/h and 600 atoms/h, respectively. To account for drifts in the detection efficiency during 81Kr and 85Kr analysis, the loading rate of the stable 83Kr (isotopic abundance = 11.5%) is also measured and used for normalization. The 83Kr loading rate is measured by quenching the 83Kr atoms in the metastable level with 810 nm light and detecting the 878 nm fluorescence from the 83Kr atoms as they deexcite to the ground level (Fig. 1b)25. The 81Kr abundance of a sample relative to the modern atmospheric value is obtained as

$${R}_{81}=\frac{{\left({L}_{81}/{L}_{83}\right)}_{{{{\rm{sample}}}}}}{{\left({L}_{81}/{L}_{83}\right)}_{{{{\rm{modern}}}}}},$$
(1)

where L81 and L83 are the loading rates of 81Kr and 83Kr, respectively, measured both for the sample and for a krypton reference derived from modern air. The 85Kr abundance of a sample is determined by comparing to a krypton reference with known 85Kr abundance (see Method section).

Fig. 1: Optical-excitation ATTA for 81Kr and 85Kr analysis of 1-kg ice samples.
figure 1

a Schematic of the system. The resonant 124 nm light needed for optical excitation is generated by a krypton discharge lamp. After undergoing several laser cooling stages, the individual 81Kr and 85Kr atoms are captured and detected in a magneto-optical trap. The turbo-molecular pumps (TMP) are configured to recirculate the sample in the vacuum system. b Atomic transitions and energy levels of krypton relevant for optical excitation, laser cooling and detection. See Method section for a more detailed level scheme.

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The whole vacuum system is differentially pumped by turbo-molecular pumps in a recirculation configuration so that a sample can be measured for an extended time (Fig. 1a). A Zr-Al getter removes all contaminants except noble gases. Due to residual leaks of ambient air into the vacuum system (on the rear side of the turbo-molecular pumps), krypton increases with a rate of 22 ± 5 pL STP/h as measured with a residual gas analyzer. The resulting 81Kr background is two orders of magnitude lower than that in previous ATTA systems using discharge excitation. Due to the residual leaks argon increases with a rate of ~200 nL STP/h, leading to a ~50% drop in the loading rates (in addition to the drop due to the decreasing lamp intensity) for a 100 nL STP krypton sample after ~12 h of measurement. The loading rate decrease is likely due to quenching collisions between argon and metastable krypton. Helium and neon are not present in sufficient quantities to affect the measurement. To recover the loading rates, after a 12-h measurement the sample is recollected and the argon (2-3 μL STP) removed using gas chromatography35. The sample loss during the entire process (recovery+argon removal) is  <5%. Therefore, the sample can be measured several times and be preserved for future use. These are important advantages of the optical-excitation ATTA. For the samples in this work, two 12-h measurements were performed followed by a measurement of a krypton reference.

Validation measurements

In order to validate the optical-excitation ATTA system for ~100 nL STP sample size, we use krypton samples that have previously been measured on a discharge-excitation ATTA system using ~3 μL STP sample size. The measured 81Kr abundances are compared in Fig. 2. The results are in good agreement over the entire range from 0 to 100 pMKr (percent Modern Krypton). The 81Kr abundance measured in ~100 nL STP krypton derived from Hefei air (Table 1) is 1.7σ below the modern value. Given the good agreement of the validation measurements (Fig. 2 and Method section) this particular result is probably a statistical outlier. To check the level of krypton contamination, we produce a sample depleted in both 81Kr and 85Kr using an electromagnetic mass separator36 (see Method section). With the optical-excitation ATTA system, for the depleted sample we measure 81Kr and 85Kr abundances which are ~0.13% and ~0.20% of the modern air values, respectively (Table 1). This is consistent with the krypton increase rate measured in the vacuum setup, confirming that contamination with modern air is insignificant.

Fig. 2: Validation measurements for the optical-excitation ATTA system.
figure 2

81Kr abundances measured using optical-excitation ATTA on ~100 nL STP krypton samples in comparison with corresponding measurements using discharge-excitation ATTA on ~3 μL STP krypton samples. The shown error bars correspond to 1σ standard deviation. Reduced χ2 = 1.1 for the measurements with respect to the line-of-perfect-agreement going from (0 pMKr, 0 pMKr) to (100 pMKr, 100 pMKr), indicated by the solid line.

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Table 1 81Kr and 85Kr results for 1-kg ice-core samples from Taylor Glacier, Antarctica, measured using optical-excitation ATTA
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81Kr dating of 1-kg ice core samples from Taylor Glacier, Antarctica

We perform 81Kr analysis on 1-kg ice core samples from Taylor Glacier, Antarctica, which were stratigraphically dated by comparing the atmospheric methane, oxygen isotopic composition, and carbon dioxide signals to well-dated records from Antarctic ice cores37,38 (see Method section). The gas has been extracted from the ice samples at Princeton University by melting the ice and freezing the released gas into a stainless steel tube submersed in liquid helium after removing the non-noble gases by gettering (see Method section). At USTC, the krypton has been separated from the argon by gas chromatography so that the pure krypton can be admitted to the ATTA system35. From the 1-kg ice samples around 100 nL STP of krypton have been obtained as expected given the typical air content of 100 mL STP/kg for polar ice and the krypton fraction in air of 1.1 ppm39. The results for the Taylor Glacier samples are listed in Table 1. The 85Kr abundances are slightly higher than the background in the system as observed with the depleted sample (Table 1) suggesting that a minor contamination with modern air may have entered during sample processing or storage. The resulting corrections to the 81Kr abundances (Table 1 and Method section) are less than 0.3%. The measured 81Kr abundances are significantly below the modern value. Using the radioactive decay law and the atmospheric 81Kr history23, the corrected 81Kr abundances translate into 81Kr ages (see Method section). The fundamental age uncertainty given by the Poisson statistics of the small number of 81Kr atoms in the samples (see next section) is  ± 8 ka for both samples, about half of the measurement uncertainty. As fundamental and measurement uncertainty are mutually independent, they are added in quadrature yielding the 81Kr-age uncertainties listed in Table 1. In addition, the age uncertainty due to the uncertainty of the 81Kr half-life (t1/2 = 229 ± 11 ka40) is 5%. This uncertainty would shift all 81Kr ages up or down together and can be re-evaluated in the future when the 81Kr half-life is determined more precisely. Within these uncertainties, the 81Kr ages agree with the gas ages obtained for the Taylor Glacier samples based on the stratigraphic alignment.

Fundamental age uncertainty of 81Kr dating

The number of 81Kr atoms in 1 kg of modern polar ice is only ~2500. At this sample size, the 81Kr-age uncertainty due to the Poisson statistics of the small number of atoms becomes relevant. The initial number of 81Kr atoms in an ice sample N0 statistically fluctuates following a Poisson distribution with mean 〈N0〉. Given a fixed initial number of atoms in the ice sample, the fraction of 81Kr atoms that has not decayed after a certain time t follows a binomial distribution. The two distributions combine to a new Poisson distribution with mean 〈Nt〉 = 〈N0〉e−t/τ and standard deviation \(\sqrt{\langle {N}_{t}\rangle }\) for the number of 81Kr atoms Nt in the sample after time t (see Method section), where τ is the mean lifetime of 81Kr. If the age t of the ice sample is determined based on Nt, then the age uncertainty merely arising from the fundamental statistical processes described above is

$$\Delta {t}_{{{{\rm{fund}}}}}=\frac{\tau }{\sqrt{\langle {N}_{t}\rangle }}=\frac{\tau }{\sqrt{\langle {N}_{0}\rangle }}\sqrt{{e}^{t/\tau }}.$$
(2)

In Fig. 3 the fundamental 81Kr-age uncertainty is shown for different amounts of polar ice (assuming an air content of 100 mL STP/kg). For 10 kg of ice, the fundamental 81Kr-age uncertainty is less than 2% over most of the relevant age range, and therefore negligible in typical applications. For 1 kg of ice, between 200 and 1700 ka the fundamental 81Kr-age uncertainty is  <5%, which is smaller than the current measurement uncertainty (also indicated in Fig. 3) but has to be taken into account. For 0.1 kg of ice the fundamental age uncertainty exceeds 10% for ages <400 ka and >1100 ka, becoming a limiting age uncertainty for certain applications.

Fig. 3: Fundamental 81Kr-age uncertainty, given by the Poisson statistics of the number of 81Kr atoms in an ice sample.
figure 3

It is shown for different amounts of polar ice (assuming an air content of 100 mL STP/kg). The measurement uncertainty for 1 kg of ice is shown as a dashed line.

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Discussion

Owing to the crucial advance in the 81Kr detection enabled by optical-excitation ATTA, 81Kr can now be measured in 1 kg of ice and less, approaching fundamental age uncertainty limits of 81Kr dating. Moreover, the upper age limit of the 81Kr dating range is significantly extended. In Fig. 4 the relative age uncertainty as a function of 81Kr age is shown for different ice amounts assuming an air content of 100 mL/kg, typical for polar ice. If the acceptable age uncertainty is set to 20%, then the upper age limit of 81Kr dating is 1.5 million years for a 1 kg ice sample.

Fig. 4: 81Kr-age uncertainty for the optical-excitation ATTA system developed in this work.
figure 4

The combined measurement and fundamental uncertainty are shown as a function of age for different amounts of polar ice (assuming an air content of 100 mL STP/kg).

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The sample size reduction down to 1 kg of ice or less enables 81Kr dating of ice core samples with higher depth resolution and in cases hitherto not possible due to sample size limitations. In particular, basal ice core sections with unknown or ambiguous ages, and of limited sample sizes can now be dated with 81Kr. For example, it is now feasible to perform 81Kr dating on basal ice from the Guliya ice cap on the Tibetan Plateau, for which 36Cl measurements suggest ages of more than 500 ka, whereas the other Tibetan glaciers investigated so far seem to be of Holocene origin17,41,42. Another application concerns the stability of the Greenland ice sheet. 40Ar dating of the GRIP bottom ice indicates that the summit of Greenland has been continuously ice-covered for the past million years6 whereas exposure dating of the rocks below the neighboring GISP2 ice core permits ice-free conditions during multiple interglacials of the past million years18. The limited amount of bottom ice left from the historic GISP2 and GRIP ice cores has in the past precluded to perform 81Kr dating on them. Similarly, 81Kr dating of basal ice from the Camp Century ice core would provide an independent confirmation of the recent finding that northwestern Greenland deglaciated during Marine Isotope Stage 1143. In the same way as for Greenland, 81Kr dating of bottom ice at different locations on the West Antarctic ice sheet19,44 can provide constraints on the extent to which the West Antarctic ice sheet has melted during the last interglacial, with implications for predicting future sea-level rise due to climate change. A further application is to assist the quest for a continuous ice core in Antarctica that reaches beyond the Mid-Pleistocene Transition, so that its causes and the relation to atmospheric CO2 levels can be examined45,46. A required sample size of only 1 kg facilitates the use of 81Kr for dating the old bottom ice retrieved by ongoing coring initiatives in Antarctica47,48,49. 81Kr can also provide constraints in cases where ice masses with different ages have substantially mixed, e.g. as a consequence of shear close to the bedrock. 81Kr and 40Ar are unique tracers for assessing mixing as 81Kr changes exponentially with time whereas the 40Ar/38Ar ratio of atmospheric air changes linearly and is sensitive beyond the dating range of 81Kr13.

Besides dating ice cores, 81Kr analysis on small samples can contribute to other applications in earth sciences, such as the recently proposed 81Kr exposure dating of zircon on the Earth surface50. Moreover, the low krypton background in optical-excitation ATTA allows to analyze sub-ppt levels of krypton in ultra-pure xenon gas for the next generation of dark matter detectors51,52,53.

The measurement uncertainty for 81Kr analysis can be further decreased in the future, especially with improvements of the optical excitation, such as integrating multiple lamps or developing a lamp with a longer lifetime and a higher intensity of resonant 124 nm light. With a magnitude higher 81Kr-loading rate, the 81Kr-age uncertainty approaches the fundamental limit imposed by the Poisson statistics of the small number of 81Kr atoms in a sample.

Methods

Validation measurements

The 85Kr and 81Kr results for the validation measurements shown in Fig. 2 of the manuscript are listed in Table 2. The leak-corrected 81Kr abundances \(R^{\prime}\) (see below) are shown for the optical-excitation ATTA measurements.

Table 2 85Kr and 81Kr data for the validation measurements
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Stratigraphically constrained ages of ice samples from Taylor Glacier, Antarctica

1 kg samples are from an ice core collected near the terminus of the Taylor Glacier during the 2015/16 Antarctic field season. As ice flow in this region is complex, most traditional methods and constraints for ice dating are not possible for this core. Instead, its chronology was established using a technique of value matching well mixed trace atmospheric gases (CO2, CH4, and δ18O -O2) to the well dated EPICA Dome C (EDC) ice core record37,38. Samples from this core range from ~120 ka near the surface to ~150 ka at 15 meters depth. For this study, we targeted samples within the Termination II interval (~136–129 ka), as glacial terminations are associated with large variations in CO2, CH4, and δ18O -O2 (Fig. 5), which provide strong constraints for stratigraphic alignment to well dated records.

Fig. 5: Stratigraphic alignment of Taylor Glacier samples to EPICA Dome C (EDC).
figure 5

In gray, atmospheric CH4 (top), δ18O -O2 (middle), and CO2 (bottom) records from the EDC ice core records. Orange (Taylor Glacier 1) and blue (Taylor Glacier 2) points show stratigraphically determined age of these samples based on alignment using these trace gas measurements (Table 3). The measurement uncertainties are not visible as they are too small (see Table 3).

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The Taylor Glacier 2 sample comes from the top 10 centimeters of a ~30 cm segment of ice that contains the abrupt CH4 increase that marks the end of Termination II. Due to the abrupt nature of this feature, its alignment to well dated ice cores is unambiguous. In addition, this feature may be synchronized to absolutely dated speleothem records. This is based on evidence that - where precisely dated polar ice cores and speleothem records overlap - abrupt changes in calcite δ18O recorded in Chinese speleothems associated with millennial scale climate variability and glacial terminations are synchronous with abrupt atmospheric CH4 variations54. Based on 230Th dating of Sanbao cave records, the end of Termination II, has been precisely dated to 128.91 ± 0.06 ka55. Given the tight constraints on ice core stratigraphic alignment and absolute age, we consider 129 ± 1 ka to be a robust estimate of age and uncertainty.

Taylor Glacier 1 is 15 cm in length, with existing measurements of δ18O -O2 and CO2 directly above and below the sample (Table 3). Synchronization to the well dated EDC ice core record is well constrained by a maximum in δ18O -O2 and mid-Termination CO2 concentrations. While analytical uncertainties of these measurements in the Taylor Glacier and EDC records are relatively low, variations in these gas properties are less abrupt than CH4. In addition, alignment via CO2 concentrations may be complicated, as offsets between ice core CO2 records have been previously observed56. However, no evidence exists for a CO2 offset between Taylor Glacier samples and the reference EDC record. If we consider the possibility of a large (10 ppm) offset between cores, this would shift the alignment by less than 1 ka in either direction. Given the reported 2 ka age uncertainty of the reference EDC record for this interval57 and  <1 ka uncertainty in the alignment of the Taylor Glacier record to EDC, we consider the 130 ± 3 ka to be a robust estimate of age and somewhat conservative estimate of age uncertainty.

Table 3 Taylor glacier trace gas measurements
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Air extraction from the ice samples

Prior to gas extraction from ice core samples, the outer ~2 mm of the sample was shaved off using a bandsaw in a  −30 °C freezer room. Samples were then cut to fit into a stainless-steel extraction vessel in which a magnetic stir bar was placed. The vessel was pumped down on a vacuum line while chilled with ethanol to remove ambient air before gas extraction. Gases from the sample were then released via ice melting and simultaneously cryogenically trapped onto silica gel cooled to  −196 °C. Once the ice was entirely melted, the meltwater was stirred for an additional 30 minutes so any dissolved gases were transferred to the silica gel. After gas extraction and transfer, all reactive gases were removed by exposure to a Zr/Al getter heated to 900 °C. Finally, the noble gases were cryogenically transferred to a stainless-steel tube via submersion in liquid helium.

Krypton level scheme

A simplified energy level and transition scheme is given in Fig. 1. In Fig. 6 a more detailed scheme with the relevant quantum numbers of the different hyperfine levels and the repumping transitions is shown. For the cooling transition (red solid arrow), four repumpers (red dashed arrow) are used in the collimation stage to transfer and maintain all atoms in the F = 13/2 hyperfine state. In the magneto-optical trap as well as the focusing stage, two repumpers are sufficient to keep the atoms in F = 13/2. No repumpers are needed in the Zeeman slower.

Fig. 6: Detailed energy level and transition scheme for optical excitation and trapping of 85Kr and 83Kr.
figure 6

The two isotopes have a nuclear spin of I = 9/2, for 81Kr (I = 7/2) the scheme is similar. The energy levels are given in Paschen Notation. The angular momentum numbers  J and F are indicated for the different hyperfine levels. The dashed arrows indicate the repumping transitions.

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Determination of the 81Kr and 85Kr abundances

The 81Kr loading rate for a sample is obtained as

$${L}_{81}=\frac{{N}_{81}}{{t}_{81}},\Delta {L}_{81}=\frac{\sqrt{{N}_{81}}}{{t}_{81}},$$
(3)

where N81 is the number of 81Kr atoms counted during the measuring time t81. To account for changes in the detection efficiency, the 83Kr loading rate is also measured and used for normalization. During the 12-h measurement, it is measured every 5 min for 10 s, providing the 83Kr loading rate L83 averaged over a whole 81Kr measurement, with a typical uncertainty of ~2%. The 81Kr abundance R81 of the sample is then obtained according to Eq. (1). Its value is given relative to the modern atmospheric value in the units of pMKr (percent Modern Krypton). The term \({\left({L}_{81}/{L}_{83}\right)}_{{{{\rm{modern}}}}}\) in Eq. (1) is the average of several reference measurements around a sample and has an uncertainty of ~1%. Our reference krypton is derived from modern air (in the year 2018), hence \({R}_{81}^{\,{{\mathrm{ref}}}\,}=100\,{\mathrm{pMKr}}\).

The 85Kr abundance of a sample is determined as

$${R}_{85}=\frac{{\left({L}_{85}/{L}_{83}\right)}_{{{{\rm{sample}}}}}}{{\left({L}_{85}/{L}_{83}\right)}_{{{{\rm{ref}}}}}}\cdot {R}_{85}^{{{\mathrm{ref}}}\,}.$$
(4)

\({\left({L}_{85}/{L}_{83}\right)}_{{{{\rm{ref}}}}}\) is the average of several reference measurements around a sample and has an uncertainty of ~1%. The 85Kr abundance in the atmosphere is changing due to variations in the anthropogenic production. Therefore, the 85Kr abundance is given in the absolute units dpm/cc (decay per minute per cubic centimeter STP of krypton), with 100 dpm/cc corresponding to 85Kr/Kr = 3.03 ⋅ 10−11. \({R}_{85}^{\,{{\mathrm{ref}}}\,}\) is the 85Kr abundance of the reference krypton, which was extracted from air in the year 2018.

Correction for contamination during analysis

Residual leaks of atmospheric air into the ATTA system cause minor krypton contamination during analysis. Using a residual gas analyzer, the increase of krypton with time has been determined to 22 ± 5 pL STP/h. Based on the measurement time the average amount of contaminant krypton Krleak can be obtained. In this work, the samples have been measured twice with each measurement lasting 12 h. The average amount of contaminant krypton over the whole measurement time is thus 264 ± 60 pL STP, which is less than 0.3% for a 100 nL STP krypton sample. Correction is therefore only relevant for samples with low 81Kr and 85Kr abundances.

The fraction of contaminant krypton and its uncertainty are

$${\eta }_{{{{\rm{leak}}}}}=\frac{{{{\mathrm{Kr}}}}_{{{{\rm{leak}}}}}}{{{{\mathrm{Kr}}}}_{{{{\rm{sample}}}}}+{{{\mathrm{Kr}}}}_{{{{\rm{leak}}}}}}\approx \frac{{{{\mathrm{Kr}}}}_{{{{\rm{leak}}}}}}{{{{\mathrm{Kr}}}}_{{{{\rm{sample}}}}}},$$
(5)

where Krleak and Krsample are the krypton amount from the leak and the sample, respectively. The 81Kr abundance corrected for the atmospheric leak in the vacuum system follows as

$${R^{\prime}_{81}}=\frac{{R}_{81}-{\eta }_{{{{\rm{leak}}}}}{R}_{81}^{{{{\rm{leak}}}}}}{1-{\eta }_{{{{\rm{leak}}}}}},$$
(6)

where \({R}_{81}^{\,{{\mathrm{leak}}}\,}\)=100 pMKr is the 81Kr abundance in modern air.

The 85Kr abundance corrected for the atmospheric leak accordingly follows as

$${R^{\prime}_{85}}=\frac{{{R}_{85}}-{\eta }_{{{{\rm{leak}}}}}{R}_{85}^{{{{\rm{leak}}}}}}{1-{\eta }_{{{{\rm{leak}}}}}},$$
(7)

where \({R}_{85}^{\,{{\mathrm{leak}}}\,}\) is the 85Kr abundance in Hefei air at the time of measurement. The 85Kr abundance in Hefei air is measured weekly58.

Correction for contamination of the sample

The 85Kr abundance remaining after leak correction can be used to quantify and correct contamination of the sample with modern air. If the sample is supposed to be 85Kr free (i.e. older than 60 a), the fraction of contaminant krypton in the sample is

$${\eta }_{{{{\rm{cont}}}}}=\frac{R^{\prime}_{85}}{{R}_{85}^{{{\mathrm{cont}}}\,}},$$
(8)

where \(R^{\prime}\) is the leak-corrected 85Kr abundance of the sample and \({R}_{85}^{\,{{\mathrm{cont}}}\,}\) is the 85Kr abundance used for the contaminant air. The 81Kr abundance corrected for contamination with modern air then follows as

$${R}_{81}^{\,{{\mathrm{corr}}}}=\frac{{R^{\prime}_{81}}-{\eta }_{{{{\rm{cont}}}}}{R}_{81}^{{{{\rm{cont}}}}}}{1-{\eta }_{{{{\rm{cont}}}}}},$$
(9)

where \({R}_{81}^{\,{{\mathrm{cont}}}\,}\) = 100 pMKr is the 81Kr abundance of the contaminant krypton.

Determination of the 81Kr ages

The raw, measured 81Kr abundances R81 of the ice samples are shown in Table 1. For determining the 81Kr ages, the 81Kr abundances corrected for modern air contamination \({R}_{81}^{\,{{\mathrm{corr}}}\,}\) are calculated as described above. Based on the initial atmospheric 81Kr values \({R}_{81}^{0}(t)\) as reconstructed in23, the 81Kr age of the sample t is obtained by numerically solving the decay equation

$${R}_{81}^{\,{{\mathrm{corr}}}}={R}_{81}^{0}(t)\cdot {{\mathrm{exp}}}\,(-t/\tau ),$$
(10)

where τ = 330 ka is the mean lifetime of 81Kr. Since the changes of the 81Kr abundances in the atmosphere are slow compared to the exponential decay, the right side of Eq. (10) is decreasing monotonically with t leading to a unique solution. By repeating the procedure for the interval \({R}_{81}^{\,{{\mathrm{corr}}}}\pm \Delta {R}_{81}^{{{\mathrm{corr}}}\,}\) we obtain the uncertainties of the 81Kr ages. There is an additional systematic age-uncertainty of 5% due to the uncertainty of the 81Kr half-life, which would shift all 81Kr ages up or down together. This uncertainty can be corrected in the future with a more precise measurement of the 81Kr half-life.

Fundamental 81Kr dating uncertainty

The initial number of 81Kr atoms N0 in an ice sample statistically fluctuates following the Poisson distribution

$${p}_{0}({N}_{0})=\frac{{\langle {N}_{0}\rangle }^{{N}_{0}}}{{N}_{0}!}{e}^{-\langle {N}_{0}\rangle }$$
(11)

with mean 〈N0〉. Given a fixed initial number of atoms N0, the number of atoms Nt that has not decayed after time t follows a binomial distribution

$${p}_{{{{\rm{decay}}}}}({N}_{t}| {N}_{0})=\left(\begin{array}{c}{N}_{0}\\ {N}_{t}\end{array}\right){e}^{-t/\tau {N}_{t}}{(1-{e}^{-t/\tau })}^{{N}_{0}-{N}_{t}},$$
(12)

with N0 ≥ Nt. The probability that the number of 81Kr atoms in an ice sample after time t is Nt, irrespective of the initial number of atoms N0, is obtained from the two probability distributions as

$$p({N}_{t})= {\sum}_{{N}_{0}={N}_{t}}^{\infty }{p}_{0}({N}_{0})\, {p}_{{{{\rm{decay}}}}}({N}_{t}| {N}_{0})\\ = \frac{{\left(\langle {N}_{0}\rangle {e}^{-t/\tau }\right)}^{{N}_{t}}}{{N}_{t}!}{e}^{\langle {N}_{0}\rangle {e}^{-t/\tau }},$$
(13)

which is a Poisson distribution with mean 〈Nt〉 = 〈N0〉e−t/τ and standard deviation \(\sqrt{\langle {N}_{t}\rangle }\). Based on the number Nt of 81Kr atoms in an ice sample, the 81Kr age of the ice sample can be determined as

$$t=\tau \,\log \left(\frac{\langle {N}_{0}\rangle }{{N}_{t}}\right).$$
(14)

The age uncertainty only arising from the statistical uncertainty of the 81Kr atom number Nt in the sample is

$$\Delta {t}_{{{{\rm{fund}}}}}=\frac{\tau }{\sqrt{\langle {N}_{t}\rangle }}=\frac{\tau }{\sqrt{\langle {N}_{0}\rangle }}\sqrt{{e}^{t/\tau }}.$$
(15)

81Kr and 85Kr depleted sample

To check the 81Kr and 85Kr background in the system, we have produced a krypton sample depleted in 85Kr and 81Kr using an electromagnetic mass separator36. The system was originally designed for argon isotopes. Separation of the stable krypton isotopes is possible by using the doubly-ionized krypton fraction, which is in a similar mass-to-charge range as the singly-ionized argon isotopes. The targeted krypton ions are selectively captured by positioning aluminum-foil in the isotope-specific target area. The doubly-ionized target isotopes of krypton are thus implanted into the aluminum foil while the ions of the undesired krypton isotopes are omitted. For the 85Kr and 81Kr depleted sample the aluminum-foil is positioned, such that krypton isotopes with masses lower than 82 (78Kr,80Kr and 81Kr) and higher than 84 (85Kr and 86Kr) are omitted. 82Kr and 84Kr are partially collected while 83Kr is fully collected. The resulting isotopic abundances of the stable krypton isotopes in the depleted sample have been measured with a quadrupole mass spectrometer (Table 4).

Table 4 Isotopic abundances of the stable krypton isotopes for krypton derived from atmosphere and the krypton sample depleted in 81Kr and 85Kr
Full size table

As expected, 78Kr, 80Kr and 86Kr have almost vanished, the isotopic abundances of 82Kr and 84Kr are reduced, whereas 83Kr has an isotopic abundance ~4.5 times higher than in atmospheric krypton. Since with the ATTA system 81Kr and 85Kr are measured relative to 83Kr, the removal of the stable isotopes other than 83Kr does not affect the 81Kr/83Kr and 85Kr/83Kr ratios. For the measurement of the depleted sample shown in Table 1, we use ~22 nL STP of depleted krypton sample which corresponds to ~100 nL STP of krypton before depletion. With a ~5 times larger sample size, the 81Kr and 85Kr abundances in the sample were determined to be <0.05% of the modern air values. Therefore, the depleted sample serves to check the 81Kr and 85Kr background in the optical-excitation ATTA system.