## Introduction

The continental lithospheric mantle (CLM) has been chemically altered by infiltrating carbon- and water-rich (C-O-H) fluids throughout its history1,2,3. This process, generally referred to as “mantle metasomatism,” is prevalent and involves enrichment of the CLM in volatiles and highly incompatible trace elements, as reflected by the mineralogy and chemical composition of CLM samples (xenoliths and xenocrysts) and their host alkaline magmas4,5,6,7,8,9. Constraining the timing of these events and the nature of the fluids involved is an ongoing challenge for our understanding of the CLM evolution and history, including its long-term chemical and physical properties10,11,12,13.

Diamonds are a primary target for studies of deep C-O-H fluids since metasomatic interaction between such fluids and CLM rocks often leads to their formation14,15,16. Diamonds grow from carbon supplied by the fluid14,17,18 and thus their crystallization ages represent the timing of CLM metasomatic events. However, there is no technique available to date diamonds directly because the diamond lattice does not contain sufficient quantities of any radioisotope. As a result, current approaches use radiogenic isotope dating of silicates and sulfide inclusions to obtain the age of their host diamonds and the metasomatic events in which they grew19,20,21,22,23. The logic is based on the key assumption that diamonds and the inclusions they encapsulate are syngenetic (i.e., formed simultaneously). This idea is supported by observations that the morphology of included minerals are commonly imposed by the diamond24,25,26,27. On the other hand, increasing evidence points to the possible protogenetic origin of inclusions (pre-existing minerals incorporated during diamond crystallization) that has generated a debate about the significance of the information recorded by inclusions for the age of their host diamonds28,29,30,31. Nevertheless, it is also possible that inclusions are protogenetic but still record the age of diamond formation, provided that the inclusion and the diamond-forming fluid reached equilibrium during the metasomatic event32.

Some diamonds encapsulate direct samples of deep C-O-H mantle fluids in the form of high-density supercritical fluid (HDF) microinclusions, which vary between saline, silicic, and carbonatitic compositions33,34,35. The potential of dating these HDF-bearing diamonds would provide a means to circumvent the assumptions concerning diamond age determination from mineral inclusions. In addition, HDFs offer a unique record on the compositions of the fluid involved in the metasomatic event35,36, whereas mineral inclusions only allow the composition to be indirectly inferred from geochemical proxies or modeling based on mineral/melt partition coefficients. To date, however, the limited available Sr-Nd-Pb isotopic data on HDF-bearing diamonds do not offer a straightforward age significance35,36,37. Thus, many studies have used the temperature and time dependency of nitrogen atom aggregation in the diamond lattice to place temporal constraints on the formation of HDF-bearing diamonds. But this method has limitations as well, since nitrogen aggregation is more sensitive to temperature than time, resulting in variable diamond formation ages between a few kiloyears and billions of years as a function of possible temperature histories between the time of diamond formation and its ascent to the surface by kimberlite eruption38,39. Moreover, along a cratonic geotherm, variable nitrogen aggregation states in diamonds from the same locality do not necessarily denote different ages, rather the nitrogen aggregation variability could indicate derivation from different depths and therefore reflect differences in temperature.

Recently, a different approach has been suggested for constraining the timing of HDF-bearing diamond formation and C-O-H metasomatism by combining He isotope analyses with U-Th-He abundance measurements40,41. Like (U-Th)/He chronometry of other minerals42,43, this approach is based on the accumulation of 4He atoms produced by α-decay of U and Th, while at the same time it considers all possible sources contributing to the budget of He in a diamond. For example, it accounts for the presence of significant initial 4He in many C-O-H fluid-bearing diamonds. This concept has been investigated in a study of alluvial HDF-bearing diamonds41, which showed a positive correlation between 4He and U-Th concentrations, reflecting the low diffusivity of He in diamond at mantle temperatures44,45,46,47. However, that study did not take into account the possibility of some diffusive loss of He, which is critical for a meaningful interpretation of (U-Th)/He data and for constraining crystallization ages43.

This study reveals strong relationships between helium concentrations, 4He/3He ratios, (U-Th)/He ratios, and the nitrogen aggregation characteristics of ten HDF-bearing diamonds from the De Beers Pool and Finsch mines in the southwest Kaapvaal craton, South Africa. Taking into account the impact of possible He diffusion on the diamonds 3He/4He ratios with time (previously estimated to fall within a large range between D = 10−16–10−21 cm2 s−1 44,45,46,47), and considering the thermal and tectonic history of the Kaapvaal craton, we suggest an upper limit of ~1.8 ± 0.2 × 10−19 cm2 s−1 for the diffusivity of He in HDF-bearing diamonds. This diffusion limit provides a major step forward for constraining diamond crystallization ages based on U-Th-He geochronology, and for unraveling the timing of C-O-H metasomatic events in the context of regional tectonics and volcanism. For the diamonds analyzed in the present study, it constrains the timing of crystallization during three alteration events, each by a different fluid agent. The youngest episode, by highly saline fluids, was coeval to the Kaapvaal late-Mesozoic kimberlite eruptions, and silicic and carbonatitic fluids were responsible for preceding metasomatic events that took place during the Paleozoic and Proterozoic.

## Results

### Helium in HDF-bearing diamonds from the southwest Kaapvaal craton

We report helium contents and isotopic compositions of ten HDF-bearing diamonds from De Beers Pool and Finsch mines (Supplementary Data 148) that were previously analyzed for nitrogen aggregation and HDFs’ major- and trace-element compositions49,50. The South African HDF-bearing diamonds have low 3He/4He ratios (Fig. 1a) compared to both mid-ocean-ridge basalts (MORB, with 3He/4He = 8 ± 1 Ra; where Ra is the atmospheric ratio of 1.39 × 10−6) and ocean island basalts (OIBs, 3He/4He = 5–50 Ra), including the so-called “low-3He/4He” OIBs (3He/4He < 7 Ra)51,52. Rather, their 3He/4He ratios are characterized by Ra values that range between MORB and continental crust (CC, 3He/4He < 1 Ra)53 (Fig. 1a, b). A strong connection is observed between the measured 3He/4He ratios, the HDFs’ major-element compositional type (saline, silicic, or high-Mg carbonatitic), and the diamond nitrogen aggregation characteristics (Figs. 1 and 2a). The saline HDF-bearing diamonds are characterized by 3He/4He ratios of 2.7–4.4 Ra (4He/3He = 163–272 × 103), whereas the diamonds containing silicic and high-Mg carbonatitic HDFs have much lower 3He/4He ratios between 0.07 and 0.69 Ra (4He/3He = 1000–11,000 × 103). At the same time, the saline HDF-bearing diamonds analyzed here contain nitrogen only in A centers (nitrogen atom pairs replacing two adjacent carbon atoms), whereas diamonds of silicic and high-Mg carbonatitic HDF compositions contain nitrogen in both A and B centers (with 25–35% in B centers, where four nitrogen atoms and an atomic vacancy substitute for five carbon atoms). Available literature data for seven additional HDF-bearing diamonds from De Beers Pool and two from the neighboring Koffiefontein mine54 are overall consistent with this observation; all are characterized by 3He/4He = 3.4–5.1 Ra (4He/3He = 143–214 × 103), nitrogen is solely in A centers, and nearly all (eight of nine) have saline HDF compositions (Figs. 1 and 2a). The new data also show a strong positive relationship between 4He/3He and (U + Th)/He ratios (Fig. 2b), which is expressed on a 4He/3He vs. (U + Th)*/3He diagram (where (U + Th)* = (1.291 × 238U + 0.2969 × 232Th) Ga−1 and is the 4He production rate), with the lowest values in the saline HDF-bearing diamonds and higher values in those carrying silicic and carbonatitic fluids. This relationship, which also correlates with increasing nitrogen aggregation in the host diamonds, reflects the ingrowth of radiogenic 4He with time since diamond formation at depth. Moreover, excluding one diamond with an unusually high U/Th ratio (i.e., with Th/U < 1)49, the saline HDF-bearing diamonds’ 4He/3He vs. (U + Th)*/3He defines a positive trend, which, if interpreted as an age, corresponds to 90 ± 42 Ma and an initial 4He/3He ratio of 162 ± 25 × 103 (3He/4He = 3.9–5.3 Ra; Fig. 2c; explained in “Methods—‘U-Th-He isochron ages’”). This apparent age is in agreement with the low-aggregated nitrogen of these HDF-bearing diamonds and the eruption ages of their host kimberlites (De Beers Pool—85 ± 5 Ma, Finsch—118 ± 3 Ma, e.g., refs. 55,56,57; the Finsch host volcanic is an olivine lamproite, formerly Type 2 kimberlite or orangeite, but we are using the generalized term kimberlite throughout the manuscript).

### Diffusion of helium in HDF-bearing diamonds

The prime source of He in HDF-bearing diamonds is the microinclusions41,58,59. Additional possible sources of He from the diamond lattice are negligible, or can be avoided by crushing release of He from an inner fragment of deep mined diamonds (as discussed in detail by Timmerman et al. 41, and here in the Supplementary information—“The budget of He in HDF-bearing diamonds”). Timmerman et al. 41 also assumed that He diffusion is slow and insignificant; but considering the large range of estimated He diffusivities of between D = 10−16–10−21 cm2 s−1 44,45,46,47, this process may impact the diamond’s potential to retain He over geological time scales of hundreds of millions to billions of years, influencing the estimated age. To determine the possible effects of He loss by diffusion on the 3He/4He ratios and (U-Th)/He age determinations of HDF-bearing diamonds, we modeled the change in the total He budget in a diamond due to radioactive production of 4He and removal by diffusion of 4He and 3He (details are in Methods—‘Diffusion model of He in diamonds’).

In order to understand the role of diffusion of He in diamonds, the important terms that control the contents of He are r2/D for both 3He and 4He (where r[L] is the diamond radius and D[L2/T] is the diffusion coefficient) and 1/λ for 4He (where λ [1/T] is the decay constants of U and Th isotopes). At D values applicable to monocrystalline diamonds (10−20–10−21 cm2 s−146), 4He is almost immobile over hundreds of millions to billions of years (Fig. 3 and Fig. S1). In these cases, diffusion is primarily confined to the diamond’s edge (i.e., for a diamond with r = 2.5 mm, there is <250 µm of diffusion in 1 Ga if D < 10−21 and <750 µm of diffusion in 1 Ga if D < 10−20) and thus it has little effect on the ingrowth of 4He and the evolving 3He/4He ratio within the diamond. For example, the total loss of 4He and 3He after 500, 1000, and 3000 Ma is 4%, 5%, and 9%, and 5%, 7%, and 13%, respectively, and the change in 3He/4He ratio is negligible (1.5%, 2.5%, and 4%, respectively). In such cases, (U-Th)/He approximates closed-system evolution. Even at D = 1 × 10−20 cm2 s−1 most of the diamond shows only small effects from diffusion over billions of years, with <37% of 3He and <27% of 4He lost over time periods of ~3000 Ma, which lowers the 3He/4He ratio by 14%. At D > 10−19 cm2 s−1, on the other hand, He loss is significant, with the effect that substantial 3He is permanently lost from the diamond by diffusion, while substantial 4He is lost by diffusion and added by radioactive decay. At D ≥ 1 × 10−18 cm2 s−1, He loss by diffusion is fast, and the content of 3He and 4He in the diamond drops quickly after formation, having a major effect on the diamond’s 3He/4He ratio. Thus, within a portion of the range of available experimental diffusivities of D = 10−16–10−21 cm2 s−144,45,46,47, our model results indicate that the budget of He in a diamond is affected by diffusion over geological time scales and its effects must be taken into account.

When considering 3He/4He model ages of HDF-bearing diamonds (Fig. 4), the diffusion effects translate to older ages, which increases with increasing diffusivities and diamond age. This can be explained as follows: in the case of no diffusion, 3He remains constant and 4He is added over time due to production by radioactive decay of U and Th. With diffusion, 4He is added by radioactive decay, while 3He and 4He are lost by diffusion. As a result of the loss by diffusion, it requires more time, and higher initial 3He and 4He concentrations, to reach the present-day 3He/4He ratio from any initial Ra value (Fig. 4); we assume an initial 3He/4He in the range 5–9 Ra, representing common values for MORB, the CLM, and subducted components (Fig. 1). This means that the assumption of no diffusion (closed system, effectively for D < 10−21) yields a minimum age of the diamond, while if diffusion is important, the true crystallization age is older, and depends on the diffusion history (Fig. 4c).

### The significance of microinclusion homogeneity

Another important factor for determining the (U-Th)/He ages of natural HDF-bearing diamonds is the homogeneity of the microinclusion composition and density through the diamond. This is because U and Th compositions are obtained by laser ablation inductively coupled plasma mass spectrometry (ICP-MS), whereas the He isotope ratio and content are measured by bulk crushing and static MS analyses, two destructive methods that sample different diamond volumes, with the data then combined for (U-Th)/He age calculations.

As a rule, the major and trace-element composition of HDF microinclusions in an individual diamond is homogeneous, and to date, out of ~300 HDF-bearing diamonds, only a few show significant radial (core-to-rim) compositional changes, for example, refs. 18,34,60. The microinclusion compositions are homogeneous in the De Beers Pool and Finsch diamonds analyzed here49,50, as demonstrated by the close Th/U ratios of duplicate analyses of the same diamond (Fig. S2). Variations in trace-element concentrations in the diamond reflect variations in the spatial abundance of microinclusions. Such variations indicate that for some of the De Beers Pool and Finsch HDF-bearing diamonds the inclusion-density scatter is small, whereas for others it is somewhat larger. The bulk trace-element composition of a diamond (including its U and Th content), however, can be closely represented by averaging a few laser ablation ICP-MS analyses of different parts in the diamond. This was shown by Rege et al.,61 who compared the laser ablation averages of two HDF-bearing diamonds to their composition determined by INAA analyses62 (Fig. S2), which average a much larger diamond volume, similar to He analyses41,54.

The ability to approximate the bulk U and Th of an HDF-bearing diamond does not always allow for (U-Th)/He age determination. For example, in two carbonatitic HDF-bearing diamonds from Finsch, the internal microinclusion-bearing and overgrown microinclusion-free zones could not be entirely separated for He analyses. In these cases, a shortfall of He compared to U and Th (measured solely in the microinclusion-bearing zone) results in an apparent high (U-Th)/He ratios (Fig. 2b). In contrast, the 3He/4He ratios of these diamonds are not affected by microinclusion-density variations. All other De Beers Pool and Finsch HDF-bearing diamonds analyzed in the present study had no such issues, and their (U-Th)/He are used for calculating ages (Fig. 2c and 5; excluding a single diamond mentioned above, with an unusually high U/Th ratio).

## Discussion

Helium isotope ratios of mantle samples provide fundamental geochemical information on their sources and can distinguish the origin of volatiles in the Earth interior51,63,64. For example, the measured 3He/4He ratios of the South African saline HDF-bearing diamonds (2.7–5.2 Ra; Fig. 1a) and their “errorchron” initial value (4.6 ± 0.7 Ra; Fig. 2c) strengthen the connection between saline HDF and recycled subducted surface material36, similar to “low-3He/4He” OIBs52,63,65. Extreme radiogenic (3He/4He « 1 Ra) signatures, on the other hand, cannot be directly related to any mantle component as sampled by oceanic or continental basalts51,52,66 nor to surficial helium because helium is not recycled by subduction67. Therefore, the low 3He/4He « 1 Ra in xenoliths from the Kaapvaal craton has been attributed to their Proterozoic emplacement age and post-emplacement production of radiogenic 4He in the host crystals (i.e., olivine, clinopyroxene), compared to xenoliths with Mesozoic emplacement ages in which 3He/4He ≈ 5 Ra68 (Fig. 1b and Supplementary Data 269, excluding three xenoliths of recycled oceanic-crustal protoliths from Roberts Victor). In diamonds, the low diffusivity of He (discussed below) allows radiogenic 3He/4He signatures to develop from the time of diamond formation in the mantle, irrespective of their emplacement age by kimberlites. The South African silicic and carbonatitic HDF-bearing diamonds are such examples; they display radiogenic 3He/4He ratios and correlated nitrogen aggregation characteristics (Fig. 2), which support their formation during earlier metasomatic events than those recorded by the saline HDF-bearing diamonds from the same kimberlites. A key issue, however, is whether the He retention characteristics in HDF-bearing diamonds allow meaningful geochronology by (U-Th)/He age determination, which can be used to unravel the timing of these metasomatic events.

The diffusion calculations have important implications for constraining the ages implied by the diamonds. While diffusivities lower than D ~ 10−19 cm2 s−1 have only small impacts on model ages, a major difference occurs over small changes above that value (Fig. 4c). Additional constraints on the diffusion rates in HDF-bearing diamonds and therefore their ages can be achieved by considering independent geological evidence (Fig. 5). For example, diamond ON-DBP-332 yields an age of 0.4 Ga at D = 1 × 10−19 cm2 s−1 that increases to 1.2 Ga at D = 4 × 10−19 cm2 s−1, whereas ON-FCH-349 yields an age of 1.4 Ga at D = 1 × 10−19 cm2 s−1 that increases to 4 Ga at D = 2 × 10−19 cm2 s−1. The latter model age result is unreasonably old based on the known geological history of the Kaapvaal craton, as indicated by Re-depletion ages (TRD) of peridotite xenoliths70 (Fig. 5b). Moreover, when considering the oldest documented ages for monocrystalline diamonds from the Kaapvaal (~3.3 Ga)19,71, the range of possible diffusivities is limited to D ≤ 1.93 × 10−19 cm2 s−1 (for 3He/4He0 ≈ 8 Ra). Compared to monocrystalline diamonds, HDF-bearing diamonds are less stable for long periods of time under mantle conditions and more likely to be resorbed. Such diamonds are unlikely to survive the thermal and tectonic history prior to craton stabilization at ~2.6 Ga72,73,74. This translates to a maximum potential 3He/4He model age of 2.6 Ga and thus an upper limit of D ≈ 1.8 × 10−19 cm2 s−1, which we propose here as the best current estimate for the maximum diffusivity of He in HDF-bearing diamonds, until further constrained by experimental studies. The uncertainty on the average U content impacts this estimation by ±0.2 × 10−19 cm2 s−1 (i.e., DHe in diamond = 1.8 ± 0.2 × 10−19 cm2 s−1). Therefore, diamond ON-FCH-349 formed from carbonatitic HDFs between 750 and 2600 Myr (Figs. 5 and 6) and based on the upper limit of D ≈ 1.8 × 10−19 cm2 s−1, the timing of diamond formation of ON-DBP-332 during silicic metasomatism can be constrained between a minimum age of 300 Ma and a maximum of 540 Ma.

The Kaapvaal CLM in South Africa is one of the most studied Archean provinces and is commonly used as an exemplar for understanding the evolution of cratonic CLM11,70,73. Our new data on HDF-bearing diamonds reveal three episodes of chemical enrichment events in the southwest Kaapvaal CLM, each by a different C-O-H metasomatic agent, following the intensive chemical depletion by melting during its accretion in the Archean. The oldest episode by carbonatite fluids could have taken place sometime in the Proterozoic, after craton stabilization at 2600 Ma and until 750 Ma (Fig. 6). Within this large time frame, this event could possibly be tied with mantle enrichment in the southwest Kaapvaal CLM between 1720 and 940 Myr, as evidenced by subcalcic garnets with sinusoidal rare earth element patterns from Finsch74, or the Namaqua–Natal Orogeny between 900 and 1250 Myr75. However, a connection to older mantle metasomatic events, which were likely coeval to large surface tectonics and volcanism in the Kaapvaal craton, could be inferred as well (Fig. 6).

The time frame for the metasomatic event by Paleozoic C-O-H silicic fluids is more limited, between 300 and 540 Myr (Fig. 6). This time interval coincides with a second garnet enrichment event in Finsch between 300 and 500 Myr and overlaps with the Damara orogeny (ca. 500 Ma)76, but is distinct from younger and older records of silicic metasomatism associated with the Karoo and Bushveld volcanism77,78. The youngest episode, by highly saline fluids, controlled the metasomatism that also led to diamond formation in the Kaapvaal CLM during the Cretaceous, as indicated by the majority of saline compositions in HDF-bearing diamonds with low-aggregated nitrogen (25 of 26) from De Beers Pool, Koffiefontein, and Finsch kimberlites49,50,54,79, and their 90 ± 42 Ma age indicated by the positive trend on the (U-Th)/He isochron diagram (Fig. 2c). This metasomatism could reflect a single event before the Finsch eruption at 118 Ma, or a few episodes of saline fluid infiltration events that started before 118 Ma and continued until the De Beers Pool kimberlites eruptions at 85 ± 5 Ma (Fig. 6). Considering the incompatible chemical behavior of helium80 and the susceptibility of the CLM to metasomatic enrichment processes, we propose that the similarity between 3He/4He values of saline HDFs (2.7–5.2 Ra; Fig. 1b) and xenoliths with late-Mesozoic emplacement ages (5.1 ± 2.4 Ra) reflects the impact of the last metasomatic episode on the 3He/4He isotopic composition of the CLM. This probably erased the record of previous C-O-H fluid episodes unless encapsulated in diamonds. Low 3He/4He ratios are also documented for the De Beers Pool Wesselton kimberlite (1.6–3.7 Ra; Fig. 1b)81, suggesting a possible link between saline HDFs and kimberlite through recycled subducted surface material36,81. Further studies are needed to clarify possible genetic implications of this relationship, but it is emphasized here that the coeval timing of saline HDFs with late-Mesozoic kimberlite eruptions in the southwest Kaapvaal Craton strongly suggest such a link.

HDF-bearing diamonds are documented in many localities from all Archean cratons. Our diffusion model and constraints on He diffusivity in diamonds indicates an upper limit of D ≈ 1.8 × 10−19 cm2 s−1, which provides a means to directly place an upper and lower limit on the timing of diamond crystallization and C-O-H metasomatism in different CLM provinces, by U-Th-He geochronology of HDF-bearing diamonds. For the southwest Kaapvaal CLM, the new findings thus delineate a sequence of different types of invading diamond-forming C-O-H fluid types over its history, carbonatitic during the Proterozoic, silicic during the Paleozoic, and saline during the Cretaceous, that led to fertilization, oxidation, and hydration of its deepest parts through space and time. The apparent link between metasomatism, HDF-bearing diamond formation, and tectonics and volcanism at Kaapvaal craton (Fig. 6), elucidates possible relationships between events occurring on Earth’s surface and in the shallow CC and chemical events involving different C-O-H fluid types in the mantle below.

## Methods

### Samples, preparation, and analytical methods

HDF-bearing diamonds from the De Beers Pool and Finsch kimberlites were selected from two suites that were previously analyzed for their nitrogen content and aggregation, and HDF microinclusions major and trace-element compositions49,50.

Helium isotope analyses were performed at the Noble Gas Lab at the Lamont–Doherty Earth Observatory of Columbia University. After coarse crushing of the diamond, inner microinclusion-bearing fragments were selected and cleaned ultrasonically in a mixture of concentrated HF and concentrated HNO3 for >2 h, washed with ethanol and milli-Q water, dried at 130 °C, weighed, and loaded in a stainless-steel electromagnetic crusher. Gas extraction was performed under vacuum by sequential crushing with an automated piston (180-piston cycles in each crushing sequence) to obtain the He gas preferentially trapped in the microinclusions58,82. The amount of gas released over time from the different crushing sequences follows a power law until the diamond is no longer releasing He (>95% extraction); total helium contents and isotopic composition for each diamond are reported in Supplementary Data 148.

The extracted gas was purified by passage through a liquid nitrogen cooled charcoal trap at 77 K and by exposing the gas to a SAES getter at room temperature, before it was collected at 14 K on a chryogenically cooled trap. Helium was released from the charcoal at 45 K and abundance and isotopic analysis was performed with a MAP 215-50 (Mass Analyzer Products, Manchester, UK) noble gas mass spectrometer by peak jumping83,84. 4He is detected with a Faraday cup, and 3He with a channeltron multiplier operated in ion counting mode. The mass spectrometric system is calibrated with known aliquots of helium from gas collected in Yellowstone National Park (the MM Standard with 3He/4He = 16.45 Ra85,86). Standard reproducibility during the period of the measurements was ~1% (1σ) and ~1.5% (1σ) for 4He and 3He, respectively. At room temperature, the crusher plus mass spectrometer blank is typically 1 × 10−10 ccSTP (cubic-centimeters-at-standard-temperature-and-pressure) for 4He, and its 3He/4He ratio is indistinguishable from atmospheric ratio within error84. The reported uncertainties include standard reproducibility, internal measurement precision, uncertainty associated with the nonlinearity correction, and the uncertainty in the blank subtraction.

### U-Th-He isochron ages

An isochron approach is used for saline HDF inclusion-rich diamonds, based on a simplification of the general equation for closed-system evolution of 4He/3He ratios87. Given that the helium in a sample under closed-system conditions reflects the initial amount plus the amount produced by radioactive decay,

$${}^{\,4}{\rm{H}}{\rm{e}}{/}^{3}{\rm{He}}={}^{\,4}{\rm{H}}{\rm{e}}/{}^{\,3}{\rm{H}}{{\rm{e}}}_{{\rm{i}}}+{}^{\,4}{\rm{H}}{\rm{e}}\ast /3{\rm{He}}$$
(1)

where 4He/3Hei is the initial ratio and 4He* is the amount of radiogenic 4He produced by the decay of U and Th. The amount of radiogenic 4He* produced over time in turn is equal to

$${}^{\,4}{\rm{He}}\ast ={8}\cdot {}^{238}{\rm{U}} \cdot ({{e}}^{\lambda 238t}\cdot-1)+{7}\cdot{}^{235}{\rm{U}} \cdot ({{e}}^{\lambda 235t}\cdot-1)+6\cdot {}^{232}{\rm{Th}} \cdot ({{e}}^{\lambda 232t}\cdot-1),$$
(2)

where t is the age of formation and 238U, 235U, and 232Th are the molar concentrations. Using the approximation ex ≈ (1 + x), Eq. (2) can be simplified to:

$${{}^{4}}{\rm{He}}{\ast} =t \cdot \{{8\cdot{\lambda }_{238}}\cdot{{}^{238}{\rm{U}}}+{7\cdot {\lambda }_{235}}\cdot {{}^{235}{\rm{U}}}+{6\cdot {\lambda }_{232}}\cdot {{}^{232}{\rm{Th}}}\}=t \cdot({\rm{U}}+{\rm{Th}})\ast$$
(3)

where (U + Th)* is the present-day production rate of 4He. Therefore, Eq. (1) can be expressed as

$${}^{4}{\rm{H}}{\rm{e}}/{}^{3}{\rm{H}}{\rm{e}}={}^{4}{\rm{H}}{\rm{e}}/{}^{3}{\rm{H}}{{\rm{e}}}_{{\rm{i}}}+t \cdot \{{8 \cdot {\lambda }_{238}}\cdot {{}^{238}{\rm{U}}}+{7\cdot {\lambda }_{235}}\cdot {{}^{235}{\rm{U}}}+{6\cdot {\lambda }_{232}}\cdot {{}^{232}{\rm{T}}}{\rm{h}}\}/{{}^{3}{\rm{He}}}$$
(4)

and

$${}^{4}{\rm{H}}{\rm{e}}/{}^{3}{\rm{H}}{\rm{e}}={}^{4}{\rm{H}}{\rm{e}}/{}^{3}{\rm{H}}{{\rm{e}}}_{{\rm{i}}}+t\cdot ({\rm{U}}+{\rm{Th}})\ast /{}^{3}{\rm{H}}{\rm{e}}$$
(5)

(U + Th)* can be further simplified to (U + Th)* = (1.291. 238U + 0.2969. 232Th) Ga−1, 238U/235U = 137.88, λ238 = 0.155125 Ga−1, λ235 = 0.98485 Ga−1, and λ232 = 0.04948 Ga−1.

Equation (5) shows that under closed-system conditions, a series of samples formed at the same time with the same initial 4He/3He will fall on a line on an “isochron” plot of 4He/3He vs. (U + Th)*/3He (Fig. 2c), and the slope will correspond to the age, as long as approximation ex ≈ (1 + x) is valid. At 100 Ma, the approximate age for our saline HDF diamonds, the offset is only ~0.6% (600 kyr); thus, the approximation and the “isochron” approach works well for these samples. We note, however, that with older ages, the true age increasingly diverges from the production rate age estimate such that, for example, at 1.0 Ga the offset is 5.9% (59 Ma), at 2.0 Ga it is 11.6% (132 Ma), and at 3.0 Ga it is 17.1% (515 Ma). We calculate our ages and errors using IsoplotR87, which argues that because the parent and daughter nuclides are analyzed separately, their uncertainties are uncorrelated and the isochron regression can be done using the simple least-squares fitting method of York88.

### Diffusion model of He in diamonds

We assume that during the metasomatic event, the crystallizing diamond and its microinclusions are in equilibrium with respect to He content. This assumption is valid based on the short time it takes to reach equilibrium after diamond formation. According to the large amount of inclusions, their micrometer size and the distance between neighboring inclusions (a few micrometers on average33,89), equilibrium between the microinclusions and the diamond will be reached shortly (<1 Ma) after formation by diffusion. This conclusion is based on the time scale of diffusion (r2/D < 0.3 Ma) for a microinclusion size of ~1 µm and diffusion coefficient (D) of 10−21 cm2 s−1. Continuous diffusion only affects the He budget in the diamond and microinclusions if He actually diffuses out of the diamond. We also assume that He is lost forever (He = 0) once outside the HDF-bearing diamond (which explains the boundary conditions in Eq. (6), below). This is based on the fact that diffusion coefficients for mantle minerals surrounding the diamond are many orders of magnitude higher (e.g., Ol 10−3–10−4 cm2 s−1; Opx 10−2 cm2 s−1; Cpx 10−2 cm2 s−1) and He contents are much lower than in HDF-bearing diamonds90,91,92,93,94,95. The change in the 4He budget in the diamond, therefore, equals the production of 4He by radioactive decay of U and Th in the microinclusions minus the removal of 4He from the diamond due to diffusion (assuming a constant diffusion coefficient) according to Fick’s second law. We assume a sphere and radial symmetry in the distribution of He in the diamond, as well as homogeneous microinclusion composition and density, for example, refs. 33,96. Moreover, in the absence of experimental constraints about possible differences between 3He and 4He diffusivities in diamonds, we assume that D(3He) ≈ D(4He). The mass balance equation of He, as a function of the distance from the diamond’s center (r) and time, taking into account both diffusion and radioactive decay is:

$$\frac{\partial C}{\partial t}=f\left(t\right)+\frac{1}{{r}^{2}}\frac{\partial }{\partial r}\left({r}^{2}D\frac{\partial C}{\partial r}\right)\qquad C\left(R,t\right)=0\qquad {C}\left(r,0\right)={C}_{0}$$
(6)

where C [M/L3] is the 4He content, t [T] is the time since the diamond formed, r [L] is the distance from the center of the diamond, R [L] is the diamond radius, D [L2/T] is the diffusion coefficient and C0 [M/L3] is the initial concentration of 4He. f(t) is the production rate of 4He from radioactive decay of 238U, 235U, and 232Th described as:

$$f(t)={}^{238}{\rm{U}}\{8\cdot {\lambda }_{238}\cdot {e}^{{\lambda }_{238}(T-t)}+7/137.88\cdot {\lambda }_{235}\cdot{e}^{{\lambda }_{235}(T-t)}+6\cdot K\cdot {\lambda }_{232}\cdot {e}^{{\lambda }_{232}(T-t)}\}$$
(7)

where 238U [M/L3] is the 238U concentration today, λ238, 235, and 232 [1/T] are the decay constants of 238U, 235U, and 232Th respectively, T [T] is the diamond formation age, t is an age between T and today, 238U/235U is the atomic ratio of 137.88 today, and Κ = 232Th/238U today.

The concentration of 3He with time is determined only by diffusion of helium according to Fick’s second law (i.e., f(t) = 0 in Eq. (6)). Equation (6) does not have an analytical solution, and therefore we solved97 it by finite difference using the Crank–Nicolson method98 (the equation derivation is given below). To evaluate the change in the total He budget in the diamond as a function of time for different He diffusivities, between D = 10−21 and 10−18 cm2 s−1, we solve Eq. (6) and integrate it across the diamond using a hypothetical example. Figure 3 and Fig. S1 shows forward modeling results for the change in 4He and 3He content and 3He/4He ratio with time after diamond formation, in a diamond with r = 2.5 mm, 3He0 = 1.1 × 10−15 mole g−1, and 4He0 = 1 × 10−10 mole g−1 (i.e., 8 Ra), U = 5 × 10−10 mole g−1, and Th/U = 4, for different He diffusivities. These parameters are approximately average values of the studied De Beers Pool and Finsch diamonds and close to those of available HDF-bearing diamonds54,59. The model results indicate that within the range of available experimental He diffusivities, the budget of He in a diamond is affected by diffusion over geological time scales (further details are in the main text).

3He/4He model ages of HDF-bearing diamonds are calculated back in time, based on the present-day (measured) concentrations of U, Th, and 4He and 3He (e.g., Figure 4; an example is shown for diamond ON-FCH-349; Supplementary Data 148). This is done by iterations and the best fit of the numerical solution of Eq. (6) to the measured concentrations of 3He and 4He. The model runs for a range of ages, in which for each age the 3He0 is calculated according to the analytical solution of 3He vs. time:

$${}^{\,3}{\rm{H}}{{\rm{e}}}_{0}={}^{\,3}{\rm{H}}{{\rm{e}}}_{({\rm{measured}})}/\left(\frac{6}{{\pi }_{\,}^{2}}\mathop{\sum}\limits_{n=1}^{\infty }\frac{1}{{n}^{2}}\exp\{-D{n}^{2}{\pi }^{2}t/{R}^{2}\}\right)\,({\rm{Eq.}}\ 6.20\ {\rm{in}}\ {\rm{ref.}}{\,}^{98}).$$
(8)

Thus, for a measured 3He/4He and U/Th/He today, the model determines the (3He/4He)0 at any point in time under the determined diffusion conditions set by the value of D. The age of the diamond and the included HDF (i.e., the 3He/4He model age) is constrained for a given D by the apparent asymptotic form of the curve, assuming an initial (3He/4He)0 in the range 5–9 Ra, which represents common values for MORB, the CLM, and subducted components (e.g., Fig. 1).

The effects of diffusion on 3He, 4He, and 3He/4He through time are illustrated in Fig. 4. In each of the calculations, the model starts in the past, with an initial Ra value, and calculates how long it takes to reach the measured (present) concentrations of 4He and 3He. Since the formation of the diamond in the past, 4He is added by radioactive production of U and Th and at the same time, 3He and 4He are removed by diffusion. In the case of no diffusion, 3He is constant with time, and 4He is only added by production. As a result of the removal by diffusion, at any point in time the 3He/4He will be lower with diffusion than with no diffusion (see also Fig. 3). Therefore, if diffusion is important it requires higher initial 3He and 4He concentrations and more time to reach the 3He/4He present ratio (starting with the initial Ra ratio; Fig. 4). That is, the condition of “no diffusion” yields the minimum estimate of the diamond formation age, and depending on the diffusion rates, the true ages may be older (further details are in the main text).

Crank–Nicolson development for Eq. (6):

$$\frac{\partial C}{\partial t}=\frac{2D}{r}\frac{\partial C}{\partial r}+D\frac{{\partial }^{2}C}{\partial {r}^{2}}+f(t)$$
$$\frac{\partial C}{\partial t}\Rightarrow \frac{{C}_{i}^{j+1}-{C}_{i}^{j}}{\Delta t}$$
$$\frac{{\partial }^{2}C}{\partial {r}^{2}}\Rightarrow \frac{1}{2{\left(\triangle r\right)}^{2}}\left(\left({C}_{i+1}^{j+1}-2{C}_{i}^{j+1}+{C}_{i-1}^{j+1}\right)+\left({C}_{i+1}^{j}-2{C}_{i}^{j}+{C}_{i-1}^{j}\right)\right)$$
$$\frac{\partial C}{\partial r}\Rightarrow \frac{1}{2}\left(\frac{\left({C}_{i+1}^{j+1}-{C}_{i-1}^{j+1}\right)}{2\left(\triangle r\right)}+\frac{\left({C}_{i+1}^{j}-{C}_{i-1}^{j}\right)}{2\left(\triangle r\right)}\right)$$
$$C\Rightarrow \frac{1}{2}\left({C}_{i}^{j+1}+{C}_{i}^{j}\right)$$
$$A1=\frac{1}{2}\left({r}_{i}-\frac{1}{\triangle r}\right)$$
$$A2=\frac{\triangle r}{D\triangle t}+\frac{1}{\triangle r}$$
$$A3=-\frac{1}{2}\left({r}_{i}+\frac{1}{\triangle r}\right)$$
$$B1=-A1$$
$$B2=\frac{\triangle r}{D\triangle t}-\frac{1}{\triangle r}$$
$$B3=-A3$$
$${AA}=\left[\begin{array}{ccccccc}A1+A2 & A3 & 0 & \ldots & \ldots & \ldots & 0\\ 0 & A1 & A2 & A3 & 0 & \ldots & 0\\ 0 & 0 & A1 & A2 & A3 & \ldots & 0\\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & A1 & A2 & A3 & 0\\ 0 & \ldots & \ddots & 0 & A1 & A2 & A3\\ 0 & \ldots & \ldots & 0 & 0 & A1 & A2\end{array}\right]$$
$${BB}=\left[\begin{array}{ccccccc}B1+B2 & B3 & 0 & \ldots & \ldots & \ldots & 0\\ 0 & B1 & B2 & B3 & 0 & \ldots & 0\\ 0 & 0 & B1 & B2 & B3 & \ldots & 0\\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & B1 & B2 & B3 & 0\\ 0 & \ldots & \ddots & 0 & B1 & B2 & B3\\ 0 & \ldots & \ldots & 0 & 0 & B1 & B2\end{array}\right]$$
$${C}_{N}=\left[\begin{array}{c}0\\ {C}_{0}\\ \vdots \\ {C}_{0}\end{array}\right]$$
$$\left[{C}^{j+1}\right]=\left[{{AA}}^{-1}\right]\left(\left[{BB}\right]{C}^{j}+{f}^{j+1}+\left[{C}_{N}\right]\right)$$
$${C}_{{\mathrm{tot}}}^{j+1}=\frac{3\mathop{\sum }\nolimits_{i=1}^{n-1}{C}_{i}^{j}{{r}_{i}}^{2}}{{R}^{3}\triangle r}+\frac{3{C}_{n}\triangle r}{2R}$$

where j is the time index, i the spatial index, n the number of time steps, Ctot the integrated concentration in the diamond, f is a vector with the number of cells according to the spatial discretization, with the production term of He (f(t)), and R the diamond radius.