rebecca granot and roi baer- can primordial helium survive in diamonds on geologic time scales?

8/3/2019 Rebecca Granot and Roi Baer- Can primordial helium survive in diamonds on geologic time scales?

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Can primordial helium survive in diamonds on geologic time scales?

Rebecca Granot and Roi Baer*

Institute of Chemistry and the Fritz Haber Center for Molecular Dynamics, the Hebrew University of Jerusalem,

Jerusalem 91904 Israel.

The 3He/4He isotopic ratio of helium inclusions in diamonds

have been used in the past to provide bounds and clues on the

formation of the solar system, the planets and Earth. Yet, it is

unclear whether primordial helium can actually survive in di-

amonds on geologic time scales (billions of years). In this paper

we provide a detailed theoretical study based on atomistic ab-

initio and tight-binding models. Various results of these models

compare well to known experimental data. We find helium re-

sides in cavities, where it has a chemical potential similar to that

in the surrounding. Large cavities are formed in diamonds spon-

taneously by diffusive coalescence of smaller carbon lattice va-

cancies from the time of its formation. Helium in cavities equili-

brates with the surrounding environment of diamond by diffu-

sion as lattice interstitials. Under high mantle temperatures(T>1000oC) the helium and vacancy equilibration processes are

fast on geologic timescales while in lower temperatures equili-

bration is slow. Thus helium isotopic ratios in diamonds may not

represent the primordial state of the deep mantle. Isotopic ratios

reflect those existing in the mantle surroundings of the last high-

temperature state of the diamond.

I. INTRODUCTION

Diamonds are unique in their imperviousness and their antiq-uity surpassing 3 billion years.1 They can be used to samplevarying depths of earths mantle, ranging from 150 km to 600km.2 As diamonds crystallized in the mantle, materials intheir surroundings were captured by them.3 Helium is an im-

portant such inclusion as the isotopic3

He/4

He abundanceratio is much higher than found in Earths atmosphere, oftenclose to the solar ratio.4 This lead to the speculation that he-lium inclusions are almost as old as diamonds themselves andreflect mantle composition 1-2 billion years ago5-9 Knowingthe history of helium isotopic ratios deep inside Earth mayhelp resolve some contradicting theories regarding the struc-ture and the dynamics of convective processes within themantle.10

Several groups investigated the stability of helium isotopicratios in diamonds by directly measuring its diffusivity.These resulted in a variety of estimates spanning more than13 orders of magnitude. The first measurements were made

using alpha particles injected into synthetic diamond, estimat-ing an activation energy of 1.04 eV for diffusion and a diffu-sion constant of ~10-7 cm2/sec at 1200oC.11 Subsequent mea-surements used naturally occurring helium in diamonds yield-ing much lower diffusivities D = 10-n cm2/sec: with n = 11 8,14 12, 17,13 and 19 7 at around 1200oC. Estimates of diffusivi-ty as low as 10-20cm2/sec.14 are based on the existence of hete-rogeneous regions in a single diamond crystal.15

The huge range of diffusivity estimates warrants complemen-tary theoretical work. It is the purpose of this article to offer adetailed account of the helium-diamond system, with theprimary goal of identifying the most stable type of heliuminclusion and determining whether it is stable enough to sur-vive the high temperatures and geologic time scales in di-

amond. Based on ab initio calculations and an atomistic tight-binding model of helium-carbon interactions, we study sever-al types of helium inclusion in diamonds. These include he-lium as lattice interstitials, in single carbon vacancies andhelium in many-fold vacancies, which we call cavities. Theresults are then used to explain the very low and very highmeasured diffusivity values. The broader meaning of theseresults is also discussed.

II. STRUCTURE OF HELIUM RELATED DEFECTS INDIAMOND

Our study is based on DFT/B3LYP calculations and on a re-

cently developed tight-binding model for helium-carbon16and helium-helium interactions. The calculations were per-formed on spherical-like diamond clusters with the surfacepassivated by hydrogen atoms. Two crystal sizes were used,for checking convergence with respect to size and for treatingsome of the problems which required placement of heliumatoms far from each other. The smaller cluster (denoted S0)contained 56 carbon atoms and 78 surface hydrogen atomswhile the larger cluster (denoted S1) consisted of 184 carbonatoms and 182 surface hydrogen atoms.

Figure 1: Several helium inclusions in diamond. From left to right:the tetrahedral interstitial site (showing nearest neighbouring car-bons in orange and next shell in green), followed by the most stableposition of 2,3 and 4 helium atoms in a single C-atom vacancy. Dueto their repulsive interactions helium atoms (marked red) try to max-imize their mutual distance within the available space in the cavity.The helium atoms are connected by bars as a visual aid for illustrat-ing the shape of the inclusion.

Pressure effects were determined by scaling distances in thecluster. This method gave bulk modulii of 390 and 480 GPafor the S0 and S1 clusters respectively and compared wellwith the experimental bulk modulus of diamond 443 GPa.17

a. Interstitial Helium

We first consider the helium atom as a diamond lattice inters-titial which was determined as the middle position of 4 tetra-gons as shown in Figure 1. The tetrahedral site has 4 nearestneighbor carbon atoms, at a distance of ~1.9, followed by asecond shell of 6 atoms at ~2.0 . Cluster distortion beyondthese shells was negligible. TB overestimated the interstitialhelium affinity18 energy by ~3eV compared to DFT/B3LYP.Hartree Fock calculations also exaggerated this energy by~1eV. Clearly there is a large contribution of correlation

energy in these large energy differences. TB will thus be used


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mainly for determining diffusion barriers. The DFT calcula-tion found the affinity energy to be -5.8eV. This result,coupled with the high mobility of this species shows that nat-ural diamonds have negligible amounts of interstitial helium.

b. Carbon vacancies in diamondA vacancy is a missing carbon atom in the diamond lattice.This defect may host helium atoms in place of the missingcarbon. We found the carbon vacancies benefit energeticallyfrom coalescing into multiple vacancies, which we term cavi-ties. Let ( )vE M denote the energy of a cluster with a fullyrelaxed multiple vacancy having M missing carbon atoms.The coalescing energy of a single vacancy with a ( )1M -fold cavity,

( ) ( ) ( )( ) ( ) ( )( )0 1 1v v v v v E M E M E E M E = + + (1)

was estimated in a series of DFT and TB calculations, giving( ) 4.8vE M eV = nearly independent of M . The forma-tion of a single carbon vacancy involves a larger energy,namely19 7.2 eV (in type I diamonds the vacancies will as-semble around nitrogen defects and this estimate is lower 20).Because of coalescence we subtract 4.8 eV, and conclude thatthe formation energy of a missing atom in diamond is

2.4eVC

E = (2)

Figure 2: Helium in a vacancy cavity. Left Panel: The lowest-energyconfiguration of a 20 helium atom containment inside a 5-fold cavity

in diamond. Right Panel: helium affinity vs. helium number density in the 2-C and 5-C atom cavity calculated using TB andDFT/B3LYP. 0 is 4 times the carbon number density in diamond.The data was fitted to an analytic expression as discussed in the text.

c. Helium in a single carbon vacancy

We calculated helium affinity for this position and found it tobe -2.4 eV using DFT/ B3LYP. We found that a single car-bon vacancy can accommodate up to 4 helium atoms in anenergetically favourable position compared to the interstitialdefect. The average helium affinity for the second, third andfourth atom as found using DFT is -3.2 eV. The arrangementsof the atoms in the vacancy are shown in Figure 1. It is seenthat the helium atoms strive to maximize the inter-atomic

distance while confined in the vacancy space.

d. Helium in a cavity (multiple carbon vacancy)

Next, we considered larger cavities. The helium affinity insuch cavities for 2 and 5 missing carbon atoms is shown inFigure 2. As the cavity grows, the single helium inside feels

little of the surrounding carbons, hence the helium is essen-tially in gas phase and its affinity energy comes close tozero. We now consider what happens as such cavities arefilled by several helium atoms. In Figure 2 we show the ar-rangement of 20 helium atoms in a 5-fold vacancy. The affin-ity energy drops as the helium density grows, due to heliumrepulsion. For high densities the helium affinity becomescomparable to the helium affinity of the interstitial site. Thiscan be seen by the form of the analytic expression we madeto fit the data:

( ) 2.2 0.455.8 (4.9 1.6 )repE eV e e eV

= + + (3)

Table 1: Activation energies for diffusion processes (eV)

Process Pressure (GPa)0 5 20

1 He-interstitial diffusion 1.41 1.38 1.202 He-vacancy diffusion via concerted exchange 6.5 6.5 6.33 Carbon vacancy diffusion 3.1 3.0 2.24 Single vacancy detachment from a multi-fold

cavity7.9 7.8 7.0

5 Conversion of single vacancy helium to intersti-tial helium

4.9 4.8 4.6

6 Conversion of single helium in cavity to intersti-tial helium

7.2 7.2 7.0

III. MOBILITY OF HELIUM AND RELATED DEFECTS

In order to estimate diffusion coefficients we calculated therate of transfer of the defect from a given site to the next lo-

cation using transition state theory.21

The saddle-point for thereaction path on the TB potential hypersurface was located tohigh precision using the recently developed spline method.22All carbon and helium atoms were allowed to move duringsuch a reaction while hydrogen atoms were held fixed. Theresulting activation energies are presented in Table 1 formantle pressures. The partition functions required for thediffusion calculations (rows 1,2 and 4 in Table 2) were com-puted using the harmonic approximation at the reactants po-tential valley and at the saddle-point configuration. We foundthe effecting attempt frequencies for these reactions were allon the order of 10-4 Eh/. We used this value in place of thepartition functions for the more involved rates (rows 3 and 5-

7 in Table 2) described below.The rate calculations were carried out for 3He and 4He iso-topes. We found quantum effects, such as zero-point energyand tunneling, negligible. Isotopic mass differences in thepartition-functions affected rates mildly because helium vi-brational modes mixed strongly with those of the crystal.

a. Mobility of interstitial helium in diamond

The interstitial helium at the tetrahedral site can hop to anadjacent tetrahedral site, 1.5 away. The saddle-point barrierfor this reaction is given in Table 1 row 2. The activationenergy lowers when pressure on the crystal is increased. Thecalculated diffusivity of interstitial helium, based on this bar-

rier, is given in Table 2 (row 1). The Our results compare

5

4

3

2

1

0

1

0 0.2 0.4 0.6 0.8 1

Helium

Affinity(eV

)

/0

Analytic

TB (2V)

DFT (2V)

TB (5V)


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well with experimentally measured interstitial diffusivity inartificial diamonds (Table 3).11 The conclusion of these expe-riments and our calculations is that the mobility of interstitialhelium in diamond under mantle conditions (T > 900oC) isvery high (~1 year to traverse the crystal). In the low temper-atures of the crust interstitial helium becomes immobile onlong time scales. The high diffusivity measurements reportedin ref. 11 were not observed in natural diamonds and werethus attributed to radiation damage.9, 23 Our results of highdiffusivity for the interstitial indicate that radiation damageis an unnecessary assumption since as noted above, naturaldiamonds contain negligible traces of interstitial helium.

Table 2: Calculated diffusivities and rates of various processes indiamond. At crust STP conditions, these were all close to zero.

Process P/T (GPa/oC)5/900 5/1200

1 Diffusivity of interstitial He in diamond(cm2/sec)

410-8 510-7

2 Diffusivity of a carbon vacancy in diamond.(cm2/sec)

110-13 310-11

3 Rate of single vacancy detachment from acavity (Ga-1)

510-5 3103

4 Diffusivity of vacancy He by concerted ex-change. (cm2/sec)

610-30 210-24

5 Rate of He escape from a single vacancyoccupied by 1 He atom to the interstitial site(Ga-1)

3108 51012

6 Rate of He escape from an empty cavity (Ga-1) 210-2 31047 Rate of He injection from gas phase into an

interstitial position (Ga-1)210-2 3104

b. Mobility of vacancies in diamond

The diffusivity of a single carbon vacancy is important for

assessing the rate of vacancy annealing. We found the activa-tion energy for vacancy diffusion in diamond is 2.2 at highmantle pressures (20 GPa) and 3.1 eV at zero pressure. Thisrange compares well with existing experimental data.24 Basedon this we estimate the diffusivity of a single vacancy givenin Table 2, row 2. For temperatures exceeding 1000oC thediffusivity is high and the equilibration rate of cavities is de-termined by the slower event of single vacancy detachmentfrom a cavity. The activation barrier for detachment is 7.8eV(Table 1, row 4) leading to a rate of evaporation given in Ta-ble 2 row 3. The rate is low but for temperatures exceeding1000oC it is sufficient for vacancy equilibration on geologictimescales.

Table 3: The diffusivity of interstitial helium in diamond for varioustemperatures at atmospheric pressure - comparison between theoryand experiment.

TemperatureoC

Diffusivity(10-7 cm2/sec)

Calculation Experiment111200 0.5 21450 3 71800 15 25

c. Mobility of vacancy helium in diamond

One plausible mobility mechanism for vacancy helium is themechanism of He-C concerted exchange i.e. the helium and

a nearby carbon atom swapping positions. We calculated the

activation energy for this exchange process as a function ofexternal pressure (Table 1, row 2) and found that diffusivitythrough this channel is very low (Table 2, row 4). Anothermechanism involves conversion of vacancy to mobile heliuminterstitial. This escape process is activated by a barrier of4.6-4.9 eV (Table 1, row 5), leading to relatively high escaperates at high temperatures (Table 2, row 5). Helium affinity incavities is higher and the rate of escape from them is consi-derably lower (Table 1, row 6 and Table 2, row 6). At lowmantle temperatures (T1000oC) the fraction of missing carbon

atoms is expC B

E k T where

CE is the formation

energy of a vacancy (Eq. (2)). This leads to the following

ratios of missing carbon atoms in the crystals: 1110 , 910

and 710 for temperatures of 1000, 1200 and 1400oC respec-tively. Assuming helium is abundant in the environment, andsince helium affinity of low density cavities is 0eV, the max-imal helium concentration in diamond can be readily esti-

mated. This is summarized in Table 4 and the values are inagreement with measured helium concentration in naturaldiamonds which range from 10-4 to 10-8 cc(STP)/g.6, 7, 14, 23.

Table 4: Estimated helium concentration in diamond for variousmantle temperatures assuming helium abundance in the mantle.

Temperature (oC) Helium concentration (cc(STP)/g)1000 310-71200 610-61400 610-5

IV. SUMMARY AND DISCUSSION

We presented a detailed atomistic model for helium in di-

amond. Our basic question was whether primordial helium


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entrapped during the formation of diamonds billions of yearsago can subsist to present times. Our model is based on thefollowing assumptions: (a) Mantle conditions, of 5-20 GPawith temperatures in the range of 900oC- 1400oC. (b) Di-amond remaining in these conditions for ~109 years and thenmoved to cooler environments; (c) Diamonds do not containsignificant concentrations of other inclusions. Our main find-ings can be summarized as follows:

1) The lowest energy helium interstitial is highly mobile attemperatures above 900oC, in quantitative agreement withexperiments.

2) Cavities form spontaneously in diamond by coalescenceof single carbon vacancies existing in it perhaps since itscrystallization phase. The rate of vacancy detachment andvacancy diffusivity allows cavity equilibration in tem-peratures exceeding ~1000C.

3) The most stable site of helium in diamond is in the cavi-ties. This result enabled us to estimate helium content indiamonds as a function of mantle temperatures, compar-ing well with measurements. Furthermore, the double-temperature helium release seen in stepped heating expe-riments were explained as resulting from the difference inhelium affinities of single vacancies and cavities.

4) At T > 1000oC full equilibrium of helium between variouscavities and between the cavities and the mantle surround-ings is attained on geologic time scales, with interstitialhelium acting as the quick channel for helium mobility.

5) At T < 1000oC helium encapsulated diamond is not ex-pected to equilibrate with the surrounding mantle on geo-logic times.

One of the open questions regarding helium in diamonds ishow diamonds initially trap helium.25 Our findings show thatat high temperatures (T>1000 oC) this issue is of no conse-quence due to thermal equilibration: helium can move in andout of the diamond at reasonably high rates and thus heliumin cavities equilibrates with the surrounding. At lower tem-peratures the question of entrapment remains. Only diamondsformed and retained at T

rebecca granot and roi baer- can primordial helium survive in diamonds on geologic time scales?

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