Phys Chem Minerals (2014) 41:181–188 DOI 10.1007/s00269-013-0636-y
ORIGINAL PAPER
Application of the cBX model to the calculation of diffusion parameters of He in olivine F. Vallianatos • V. Saltas
Received: 5 June 2013 / Accepted: 21 October 2013 / Published online: 2 November 2013 Ó Springer-Verlag Berlin Heidelberg 2013
Abstract The validity of the thermodynamic cBX model is tested in terms of the experimentally determined diffusion coefficients of He in a natural Fe-bearing olivine (Fo90) and a synthetic end-member forsterite (Mg2SiO4) over a broad temperature range (250–950 °C), as reported recently by Cherniak and Watson (Geochem Cosmochim Acta 84:269–279, 2012). The calculated activation enthalpies for each of the three crystallographic axes were found to be (134 ± 5), (137 ± 13) and (158 ± 4) kJ mol-1 for the [100], [010] and [001] directions in forsterite, and (141 ± 9) kJ mol-1 for the [010] direction in olivine, exhibiting a deviation of \1 % with the corresponding reported experimental values. Additional point defect parameters such as activation volume, activation entropy and activation Gibbs free energy were calculated as a function of temperature. The estimated activation volumes (3.2–3.9 ± 0.3 cm3 mol-1) of He diffusion in olivine are comparable with other reported results for hydrogen and tracer diffusion of Mg cations in olivine. The pressure dependence of He diffusion coefficients was also determined, based on single experimental diffusion measurements at 2.6 and 2.7 GPa along the [001] direction in forsterite at 400 and 650 °C. Keywords Diffusion Helium Olivine Forsterite cBX model
F. Vallianatos V. Saltas (&) Laboratory of Geophysics and Seismology, Department of Environmental and Natural Resources Engineering, Technological Educational Institute of Crete, Chania, Greece e-mail:
[email protected]
Introduction The diffusion of He in minerals plays a key role for the proper implementation of noble gas thermochronological method, which is of great importance to the history of tectonic and surficial processes, the thermal history of petroleum reservoirs and other applications in the Earth, and planetary sciences (Harrison and Zeitler 2005; Mahon et al. 1998; Cassata et al. 2009, 2011; Gourbet et al. 2012; Huber et al. 2011) provided important new constraints on the dynamics and geochemical evolution of Earth’s mantle. The production of He during the radioactive a-decay of 238 U, 235U and 232Th, which are found in accessory minerals (apatite, zircon, titanite), is compensated by its diffusion to the boundaries of the material, and the ideas of production–diffusion processes have been applied in order to determine the thermal history of the sample (Farley 2000; Reiners et al. 2004; Boyce et al. 2005; Herman et al. 2007; Wolfe and Stockli 2010). The behavior of He diffusion is rather complicated and unpredictable depending on the different kinds of its distribution (inclusions, aggregations, etc.) in the mineral lattice (Trull and Kurz 1993; Tolstikhin et al. 2010). Furthermore, the experimental difficulties and constraints under high pressure and temperature conditions limit the available diffusion data, and the existence of an alternative approach would be invaluable. In this sense, the cBX thermodynamical model proposed by Varotsos and Alexopoulos (1977, 1980, 1982, 1986; Varotsos et al. 1978; Alexopoulos and Varotsos 1981) predicts the diffusion coefficients from the elastic properties of solids and has found application to the calculation of defect parameters over a broad range or materials, such as metals, noble gas solids, lead and alkali halides, fluorides, diamond, semiconductors and superionic solids (Alexopoulos et al. 1986; Varotsos 2007a, b, c, d, 2008,
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2009; Varotsos and Alexopoulos 1984, 1986; Varotsos et al. 1985, 1991; Zhang and Wu 2012; Sakellis 2012). Its validity has been recently extended in self- or heterodiffusion in geomaterials, such as silicate minerals and periclase (Zhang et al. 2010, 2011; Zhang and Wu 2011, 2013; Zhang 2012; Dologlou 2011), supporting previous results on the application of the cBX model to the estimation of thermodynamical and rheological parameters of earth’s mantle (Vallianatos and Eftaxias 1992, 1994; Vallianatos et al. 1995). In addition, the model has been proposed to interpret the generation of seismic electric signals (SES) emitted prior to large earthquakes (Varotsos 2005) from crystalline geomaterials in earthquake focal area (see a discussion on Vallianatos et al. 2004). In the present work, we apply the thermodynamic cBX model to estimate, based on bulk and expansivity elastic data, the experimentally determined diffusion parameters of He parallel to the crystallographic axes of natural olivine and synthetic forsterite over a broad temperature range, as reported recently by Cherniak and Watson (2012).
Validity of cBX model in terms of He diffusion in olivine In the case of a single diffusion mechanism, the self- or tracer-diffusion coefficient is related to the activation Gibbs free energy gact through the following Arrhenious equation: h act i D ¼ f a2 m exp g =kB T ð1Þ where f is a correlation factor that depends on the structure and the diffusion mechanism, a is the jump distance that is actually equal to the lattice constant, and m is the corresponding jump frequency (Varotsos and Alexopoulos 1980). The latter is of the order of the Debye frequency vD, but it depends on the mass of the diffusant for a given host material and may be expressed by: rffiffiffiffiffiffiffi mm m ¼ mD ð2Þ mi where vD and mm refer to the host material and mi stands for the mass of the diffusant (Decker et al. 1977). The Gibbs free energy gact for the activation process (formation and migration of defects) may be described in terms of the bulk properties of the solid according to the socalled cBX thermodynamic model: gact ¼ cact BX
ð3Þ
where B is the isothermal bulk modulus, X is the mean volume per atom, and cact stands for a dimensionless factor
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that may be considered in a first approximation to be independent of temperature and pressure. Its value is usually found to be\1, as reported in many applications of the cBX model (Alexopoulos and Varotsos 1981; Varotsos and Alexopoulos 1986; Varotsos et al. 1991). The value of cact may be roughly estimated from Eq. (1), by the following equation: 2 kB T1 f a1 m cact ¼ ln ð4Þ D1 B1 X1 considering that the diffusion coefficient D1 and the other parameters are known at a given temperature T1. However, the above estimation has been applied when only a single measurement of diffusion is available but may lead to considerable variations of cact due to its small temperature dependence and the experimental uncertainties of diffusion coefficients at different temperatures (Zhang 2012). Alternatively, a mean value of cact may be also derived by substituting Eqs. (2) and (3) in (1) and taking the natural logarithm of both sides: rffiffiffiffiffiffiffi mm BX ln D ¼ ln f a2 mD : ð5Þ cact kB T mi A linear behavior of lnD versus BX/kBS implies the validity of the cBX model for the material under investigation, while the slope of the line gives a direct estimation of cact. This alternative methodology has been utilized in the present case, in order to reduce the experimental uncertainties arising either from a single diffusion measurement or from the other aforementioned parameters in Eq. (4). In this way, the uncertainties resulting from the appropriate choice of Debye frequency (optic or acoustic frequency) as well as from the correlation factor which value is not always known are incorporated to an independent term in Eq. (5), and the value of cact arises as a ‘‘mean value’’ from all the available experimental diffusion data. Linear approximations of the temperature dependences of the bulk modulus B(T) and the mean volume X(S) have been considered, according to the following equations (Varotsos and Alexopoulos 1986; Zhang et al. 2010): oB BðTÞ ¼ Bo þ ðT To Þ ð6Þ oT P where Bo is the bulk modulus at room temperature (300 K) and ðoB=oTÞP its temperature derivative that remains constant below the melting point temperature. XðTÞ ¼ Xo ½1 þ bo ðT To Þ=nZ
ð7Þ
where Xo is the volume of the unit cell, bo is the volume thermal expansion coefficient at room temperature, n is the number of atoms in the chemical formula of the mineral,
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and Z is the number of formula units per unit cell. However, we have to mention that these linear approximations of B(T) and X(T) are of limited validity over a broad temperature range since, at high temperatures which is the case of minerals in mantle conditions, deviations from linearity are common, introducing in this way uncertainties to the estimation of elastic parameters in the framework of the cBX model. In the case of forsterite, Bo = 127.4(5)GPa, ˚ 3 and ðoB=oTÞP = -0.028(1) GPa K-1, Xo = 290.79(9) A -5 -1 bo = 2.69(1) 9 10 K (Katsura et al. 2009). The corresponding values of elastic parameters for olivine were adopted by Liu and Li (2006), [Bo = 129 GPa, ˚3 ðoB=oTÞ = -0.019(2) GPa K-1, Xo = 292.13(10) A P
and bo = 2.73(34) 9 10-5 K-1]. By substituting the above values in Eqs. (6) and (7) (for nZ = 28), B(T) and X(S) may be estimated accurately for temperatures below the melting point.
Fig. 1 Experimental diffusion coefficients Dexp of He in forsterite for the three crystallographic directions and He in olivine at [010] direction, as a function of the quantity BX/kBS. The linear behavior indicates the validity of the cBX model while the parameter cact is estimated in each case directly from the slope of the line
Considering the experimental values of diffusion coefficients of He in olivine and forsterite for the three crystallographic axes as reported by Cherniak and Watson (2012), a linear variation of logD versus BX/kBS is observed in each case, as it is depicted in Fig. 1, indicating clearly the validity of the cBX model. From the slope of the lines, the values of cact were estimated for each of the three crystallographic directions and are shown in Table 1. These values are necessary for the calculation of point defect parameters in the next section. We have to point out that due to the different methods that have been used to determine the value of cact, a careful approach is essential for the correct implementation of the cBX model, which otherwise may lead to misleading results. For example, our estimate for cact in olivine (0.169 ± 0.011) is remarkably different with that (cact = 0.6814) reported by Zhang et al. (2010) for oxygen diffusion in olivine, which was calculated from the characteristic temperature (Tc = 2,358 K), according to the compensation effect. A source of error that is associated with the determination of cact is the proper choice of the value of the correlation factor f. In the lack of any available theoretical or experimental values, the value f = 0.745, which has been reported for diffusion of Fe-cation in olivine by Hermeling and Schmalzried (1984) and has been used in the cases of oxygen and hydrogen diffusion in olivine (Zhang et al. 2010; Zhang 2012), should not be the same for different kind of diffusion mechanisms, such as He diffusion. So, without the limitation of available experimental diffusion data at elevated temperatures, the method of cact estimation which we applied in the present case could be a better approach, in order to eliminate any related uncertainties in the application of the cBX model. This is verified in the next section where the predicted activation enthalpies according to the cBX model converge to the reported experimental values. The calculated diffusion coefficient values Dcalc according to the cBX model and the corresponding experimental values Dexp by Cherniak and Watson have been plotted separately as a function of inverse temperature in Fig. 2
act act Table 1 Calculated values of cact, activation enthalpy (hact calc ), activation entropy (s ), activation Gibbs free energy (g ) and activation volume act (t ) for the three crystallographic axes of forsterite sample and the [010] direction of olivine
Direction
cact
-1 hact calc (kJ mol )
-1 hact exp (kJ mol )
sact (kb units)
gact (kJ mol-1)
tact (cm3 mol-1)
Forsterite [100]
0.160 ± 0.006
(133.1–134.2) ± 5.1
135 ± 5
(2.87–2.91) ± 0.16
(106.4–121.4) ± 5
(3.21–3.26) ± 0.23
[010]
0.163 ± 0.015
(136.8–137.1) ± 12.6
137 ± 13
(2.95–2.97) ± 0.30
(116.6–121.8) ± 13
(3.29–3.31) ± 0.36
[001]
0.189 ± 0.005
(157.1–158.5) ± 4.3
159 ± 4
(3.38–3.44) ± 0.17
(124.0–143.6) ± 4
(3.79–3.83) ± 0.26
(140.2–140.7) ± 9.2
142 ± 9
(1.78–1.81) ± 0.17
(128.8–131.3) ± 9
(3.86–3.87) ± 0.31
Olivine [010]
0.169 ± 0.011
The experimental values of
hact exp
(Cherniak and Watson 2012) are also included for comparison
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Fig. 2 Arrhenius plots of the experimental (points) and calculated (lines) diffusion coefficients of He in forsterite and olivine at the three principal crystallographic axes. Note the different temperature scale in the [010] direction
(Arrhenius plots). Obviously, the calculated values are in very good agreement with the experimental ones, within the experimental uncertainties (correlation factor R [ 0.98), for each one of the crystallographic axes. We have to note that in the case of the [010] direction, the calculations gave comparable predicted values, regardless of the kind of mineral (olivine or forsterite) implying in this way the insensitive role of the small He atoms to the diffusion process along the same direction. However, the structural anisotropy of forsterite results in different estimates of He diffusion coefficients for each of the three principal directions, in agreement with the experimental values. In addition, the kind of the diffusant atom should be significant to the diffusion anisotropy since diffusion experiments of rare-earth elements in natural olivine have shown little dependence of diffusion on the crystallographic orientation (Cherniak 2010), whereas similarities of diffusion anisotropy have been observed in the case of hydrogen diffusion in single crystals of olivine (Demouchy and Mackwell 2006). Extrapolation of the predicted diffusion coefficients for the fast diffusion direction (Fig. 2) at the temperature of the earth’s upper mantle (1,500 °C) results at a value of the order of 10-10 m2 s-1. This value supports clearly the results reported by Hart et al. (2008) that length scale of He isotope heterogeneities smaller than a few tens of meters would not survive at 1.5 Gyr residence time in the upper mantle. Indeed, an estimation of mean diffusion distances of He at 1,500 °C according to the simplified equation x * (Dt)1/2 results in a time period of 0.3 Gyr for a distance of 1 km and a few days for 1 cm while at nearsurface conditions (50 °C), He is practically immobile.
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Calculation of point defect parameters based on the cBX model An important point defect parameter is the activation volume tact , which describes the volume change in the host material due to the formation of point defects and the lattice distortion caused by their migration and is expressed as (Varotsos and Alexopoulos 1986): ogact act t ¼ : ð8Þ oP T By differentiating Eq. (3), we finally take: oB tact ¼ cact X 1 : ð9Þ oP T i act Similarly, the activation entropy sact ¼ ogoT and P
the activation enthalpy (hact ¼ gact þ Tsact ) may be estimated as a function of temperature, from the following relations: oB act act s ¼ c X þbB ð10Þ oT P and h
act
oB ¼ c X B TbB T oT act
ð11Þ P
where b is the thermal (volume) expansion coefficient, which, in general, is a function of temperature and pressure. In the case of forsterite at ambient pressure, a linear relationship of b with temperature has been applied,
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b ¼ 2:69 105 þ 2:12 108 ðT 300Þ K1 ; according to Katsura et al. (2009), while for olivine, the corresponding relationship is: b ¼ 2:73ð34Þ 105 þ 2:22ð81Þ 108 T 0:538 T 2 ðK1 Þ: The values of the pressure derivative of the bulk modulus of forsterite and olivine are ðoB=oPÞT = 4.2(2) and 4.61, respectively (Duddy et al. 1995; Liu and Li 2006). By substituting the above values in Eqs. (9)–(11), as well as the estimated mean values of cact and X(S), B(T) for the experimental temperature values of He diffusion in forsterite and olivine, the values of tact , sact, hact and gact were calculated and plotted in Fig. 3 as a function of temperature, for each of the three crystallographic directions. The range of values of the above estimations and their corresponding errors that were calculated by taking into account the experimental uncertainties of each quantity in Eqs. (3) and (9)–(11) are summarized in Table 1. We note that our estimated values of sact are positive since, in the equivalent form of Eq. (10), sact = cactbBX(d - 1), the so-called Anderson–Gru¨neisen parameter d defined as d ¼ ð1=bBÞðoB=oT ÞP , usually exceeds unity [for forsterite d = 6.17 at RT (Kroll et al. 2012)] while b is positive. We should also note that from a thermodynamic point of i act view we have sact ¼ ogoT and hact ¼ gact þ Tsact ¼
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We note that the anisotropy of He diffusion in forsterite is expressed via different values of the parameter cact, which affect all the related thermodynamic quantities, according to Eqs. (9)–(11). As a consequence, our estiact act mated values of cact (cact 100 c010 \c001 ) are consistent with the reported values of activation energy by Cherniak and Watson (2012) where for He diffusion in forsterite, hact 100 act act hact \h : In addition, comparable values of c were also 010 001 found for the case of He diffusion along the same direction either in olivine or in forsterite, in accordance with the act previous experimental findings (hact 010;olivine h010;forsterite ). To the best of our knowledge, there are no theoretical or experimental corresponding values of activation volume available in the literature. However, reported values of activation volume of hydrogen diffusion in forsterite at 1 GPa and 1,000–2,000 K (Zhang 2012) are fairly close (4.02–8.50 cm3 mol-1, depending to the direction) with our estimates if we take into consideration that, except for the pressure dependence, the size of each diffusant atom may affect to some extent the activation volume. In accordance with this, calculated activation volumes based on the cBX model of oxygen self-diffusion in Mg2SiO4 at 8 GPa and 1,000–2,200 K differ
P
gact T ½ogact =oT P which after differentiation leads to the expression T ½osact =oT P ¼ ½ohact =oT P , obeyed by the activation enthalpy and entropy. As we can see in Fig. 3, the temperature dependence of gact is almost linear decreasing with T. Taking into account the aforementioned thermodynamic relations, we lead to the conclusion that the activation enthalpy and entropy are almost constant, as in the present case. Our estimated values of activation enthalpy according to the cBX model are in excellent agreement (error \1 %) with the experimental activation energies of He diffusion in olivine (Cherniak and Watson 2012). Comparable value of activation enthalpy has been also measured recently by Cherniak et al. (2012) for Ne diffusion in forsterite (155 ± 12 kJ mol-1) although the diffusion coefficient was 5 orders of magnitude lower than He diffusion. In a recent experimental investigation of hydrogen diffusivities in San Carlos olivine single crystals by Demouchy and Mackwell (2006), the highly diffusive H has activation energies as (204 ± 94) kJ mol-1 for diffusion along [100] and [010] direction, and (258 ± 11) kJ mol-1 along [001] direction. These values are also in good agreement, within the experimental errors, with our predicted values according to the cBX model. The above-reported activation energies refer to inert and/or small diffusant atoms such as Ne and H, while for diffusion of heavier cations, higher activation energies are expected (Cherniak 2010).
Fig. 3 Estimated values of activation volume (tact), activation entropy (sact), activation enthalpy (hact) and activation Gibbs free energy (gact), as a function of temperature for the three crystallographic axes in forsterite and the [010] direction in olivine, according to the cBX model
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substantially (17.11–17.17 cm3 mol-1) with our estimated values (Zhang et al. 2011). However, in contradiction with the previous reported values of activation volumes, experimental results on Mg tracer diffusion in single crystals of forsterite and San Carlos olivine between 1,000 and 1,300 °C yield an activation volume of approximately 1–3.5 cm3 mol-1 (Chakraborty et al. 1994). Helium diffusivity in forsterite may be also investigated as a function of pressure in terms of the cBX model. Differentiating Eq. (5) with respect to pressure, we may derive the following expression for the activation volume [see page 126 of Varotsos and Alexopoulos (1986)]: tact ðP; TÞ ¼ kB T
o ln D o lnðf a2 mÞ þ kB T oP T oP T
ð12Þ
where the second term of the right side is equal to jðc 2=3Þ, assuming that the volume variation of the jump frequency m is well described by the Gru¨neisen constant c. Since this term is only a few percent of the first one, we may ignore it in a good approximation, and thus, Eq. (12) provides a direct way to determine the activation volume at any pressure from the slope of the lnD versus P plot. A linear behavior of this plot corresponds to an activation volume independent of pressure, but in general, tact is pressure dependent according to the following expression: 0 P 1 Z tact ðP; TÞ ¼ tact ð0; TÞ exp@ jact dPA ð13Þ 0
ln
DðP; TÞ Dð0; TÞ
tact ð0; TÞ jo c o P kB T act j tact ð0; TÞ 2 þ P 2kB T
¼
ð17Þ
which describes the variation of diffusion coefficient as a function of pressure at a given temperature, if the parameters jo, co, jact and tact(0, T) are known. The compressibility of activation volume jact may be derived from (14) if we insert the value of tact given by Eq. (9), resulting in an expression for jact that depends only on the bulk properties:
2 o B=oP2 T act : ð18Þ j ¼ jo ðoB=oPÞT 1 The isothermal first and second derivative of the bulk modulus in Eq. (18) can be estimated from elastic data reported by Zhang (2012) for H diffusion in forsterite from ambient pressure up to 12 GPa. The least-squares method suggests a linear relationship for B(P), implying that ðo2 B=oP2 ÞT is negligible and thus, jact jo . A comparable value of jact = 2.7jo is also derived according to Eq. (17) from experimental data used by Zhang (2012) for H diffusion in olivine in order to calculate the pressure dependence of diffusion coefficients, in the framework of the cBX model. For the values jo = 1/127.4 GPa-1 of the bulk compressibility, co = 1.2 for the Gru¨neisen constant (Kroll et al. 2012) and our estimates of activation volumes for the [001] direction in forsterite (3.81–3.83 9 10-6 m3 mol-1, see Table 1), the plot of logD versus pressure according to Eq. (17) is depicted in Fig. 4, at two different temperatures
if we recall that the compressibility of activation volume jact is defined as: 1 otact act j ¼ act : ð14Þ t oP T If we assume to a first approximation that jact is independent of pressure, Eq. (13) may be expanded as: 1 act 2 act act act t ðP; TÞ ¼ t ð0; TÞ 1 j P þ ðj PÞ þ 2 ð15Þ where tact ð0; TÞ is the activation volume at zero pressure, o ln Dð0; TÞ 2 tact ð0; TÞ ¼ kB T þkB Tjð0; TÞ cð0; TÞ : oP 3 T ð16Þ By substituting Eq. (12) and (16) in Eq. (15), in the approximation of jo co jc [jo = j(0, S), co = c(0, S)] and for c 2/3, we finally obtain:
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Fig. 4 Pressure dependence of He diffusion coefficients in forsterite according to the cBX model at 400 and 650 °C for different possible values of jact. The data points correspond to the experimental values reported by Cherniak and Watson, for He diffusion in forsterite along the [001] direction
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(400 and 650 °C) where the corresponding experimental data of He diffusion coefficients along the [001] direction in forsterite at high pressures (2.6 and 2.7 GPa) have been reported by Cherniak and Watson (2012). The different curves in each temperature correspond to different possible values of the compressibility of activation volume, jact, that were calculated previously. Our estimates deviate slightly from the experimental data, within the experimental errors, but additional experimental data at higher pressures are necessary for a reliable inspection of the validity of the cBX model. We have to mention at this point that the pressure as compared to temperature variation has relatively little effect on diffusion of He in forsterite as reported by Cherniak and Watson (2012) from their supplementary experiments that were carried out at high pressures (2.6 and 2.7 GPa). Their experimental finding is consistent with the cBX model since the pressure effect on the activation volume is negligible in this case. Indeed, if we consider the ratio of the activation volumes at pressure P with respect to ambient pressure Po, we obtain from Eq. (9) that: oB tact ðP; TÞ X oP P;T 1 : ð19Þ ¼ tact ðPo ; TÞ Xo oB oP Po ;T 1 Taking into account that the ratio oB 1 = oP P;T oB oP Po ;T 1 is of the order of unity in our case and that X/Xo is close to 0.98 (as calculated from extrapolated data for olivine at 1,300 K, Zhang 2012), we conclude that the activation volume according to Eq. (19) has a small variation with pressure.
Conclusions In summary, the temperature variation of the diffusion coefficients of He in natural olivine and synthetic forsterite (Mg2SiO4) has been tested over a wide range of temperatures using the cBX model that interconnects point defect parameters with bulk properties, i.e., the isothermal bulk modulus B, the volume thermal expansion coefficient b and the atomic volume X. The values of cact in the cBX model have been determined for each of the three crystallographic directions following an alternative approach than those reported so far, and different thermodynamic quantities such as the activation enthalpy, activation entropy, activation Gibbs energy and the activation volume were predicted in each case. The estimates of activation enthalpy of He diffusion in forsterite in each of the crystallographic directions are in good agreement with the reported experimental data. The pressure dependence of He diffusion coefficients in forsterite was also investigated giving a
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promising prediction which we expect to be verified by additional He diffusion experiments over a broad pressure range. The calculation of He diffusion in olivine by using the proposed model is feasible if the necessary thermodynamic data of minerals are available, permitting the estimation of He diffusion properties in the lower mantle. Acknowledgments The authors thank B. Zhang and two other anonymous reviewers for their helpful and constructive comments on the manuscript. This work was supported by the THALES Program (MIS 380208) of the Ministry of Education of Greece and the European Union in the framework of the project entitled ‘‘Integrated understanding of Seismicity, using innovative Methodologies of Fracture mechanics along with Earthquake and non extensive statistical physics – Application to the geodynamic system of the Hellenic Arc. SEISMO FEAR HELLARC’’.
References Alexopoulos K, Varotsos P (1981) Calculation of diffusion coefficients at any temperature and pressure from a single measurement. II. Heterodiffusion. Phys Rev B 24:3606–3609 Alexopoulos K, Lazaridou M, Varotsos P (1986) Activation volumes in lead halides and other solids. Phys Rev B 33:2838–2841 Boyce JW, Hodges KV, Olszewski WJ (2005) He diffusion in monazite: implications for (U–Th)/He thermochronometry. Geochem Geophys Geosyst 6:Q12004–Q12012 Cassata WS, Renne PR, Shuster DL (2009) Argon diffusion in plagioclase and implications for thermochronometry: a case study from the Bushveld Complex, South Africa. Geochem Cosmochim Acta 73:6600–6612 Cassata WS, Renne PR, Shuster DL (2011) Argon diffusion in pyroxenes: implications for thermochronometry and mantle degassing. Earth Planet Sci Lett 304:407–416 Chakraborty S, Farver JR, Yund RA, Rubie DC (1994) Mg tracer diffusion in synthetic forsterite and San Carlos olivine as a function of P, T and fO2. Phys Chem Miner 21:489–500 Cherniak DJ (2010) REE diffusion in olivine. Am Mineral 95: 362–368 Cherniak DJ, Watson EB (2012) Diffusion of helium in olivine at 1 atm and 2.7 GPa. Geochem Cosmochim Acta 84:269–279 Cherniak D, Thomas J, Watson EB (2012) Diffusion of neon in olivine and quartz. Mineral Mag 76(6):1572 Decker DL, Ross RA, Evenson WE, Vanfleet HB (1977) Pressure effects on the diffusion and solubility of Zn in Pb. Phys Rev B 15:507–513 Demouchy S, Mackwell S (2006) Mechanisms of hydrogen incorporation and diffusion in iron-bearing olivine. Phys Chem Miner 33:347–355 Dologlou E (2011) Interrelation of the equation of state of MgO and self diffusion coefficients. J Appl Phys 110(3):036103 Duddy TS, Zha C, Downs RT, Mao H, Hemely RJ (1995) Elasticity of forsterite to 16 Gpa and the composition of the upper mantle. Nature 378:170–173 Farley KA (2000) Helium diffusion from apatite: general behavior as illustrated by Durango fluorapatite. J Geophys Res 105:2903– 2914 Gourbet L, Shuster DL, Balco G, Cassata WS, Renne PR, Rood D (2012) Neon diffusion kinetics in olivine, pyroxene and feldspar: retentivity of cosmogenic and nucleogenic neon. Geochem Cosmochim Acta 86:21–36
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188 Harrison TM, Zeitler PK (2005) Fundamentals of noble gas thermochronometry. Rev Mineral Geochem 58:123–149 Hart SR, Kurz MD, Wang Z (2008) Scale length of mantle heterogeneities; constraints form helium diffusion. Earth Planet Sci Lett 69:507–516 Herman F, Braun J, Senden TJ, Dunlap WJ (2007) (U–Th)/He thermochronometry: mapping 3D geometry using micro-X-ray tomography and solving the associated production–diffusion equation. Chem Geol 242:126–136 Hermeling J, Schmalzried H (1984) Trace diffusion of the Fe-cations in olivine (FexMg1-x)2SiO4 (III). Phys Chem Miner 11:161–166 Huber C, Cassata WS, Renne PR (2011) A lattice Boltzmann model for noble gas diffusion in solids: the importance of domain shape and diffusive anisotropy and implications for thermochronometry. Geochem Cosmochim Acta 75:2170–2186 Katsura T, Shatskiy A, Manthilake MA, Geeth M, Zhai S, Fukui H, Yamazaki D, Matsuzaki T, Yoneda A, Ito E, Kuwata A, Ueda A, Nozawa A, Funakoshi K (2009) Thermal expansion of forsterite at high pressures determined by in situ X-ray diffraction: the adiabatic geotherm in the upper mantle. Phys Earth Planet Int 174:86–92 Kroll H, Kirfel A, Heinemann R, Barbier B (2012) Volume thermal expansion and related thermophysical parameters in the Mg, Fe olivine solid-solution series. Eur J Mineral 24:935–956 Liu W, Li B (2006) Thermal equation of state of (Mg0.9Fe0.1)2SiO4 olivine. Phys Earth Planet Int 157:188–195 Mahon KI, Harrison TM, Grove M (1998) The thermal and cementation histories of a sandstone petroleum reservoir, Elk Hills, California: part 2: in situ oxygen and carbon isotopic results. Chem Geol 152:227–256 Reiners PW, Spell TL, Nicolescu S, Zanetti KA (2004) Zircon (U– Th)/He thermochronometry: He diffusion and comparisons with 40 Ar/39Ar dating. Geochem Cosmochim Acta 68:1857–1887 Sakellis I (2012) Determining the activation volumes in ZnO. J Appl Phys 112(1):013504 Tolstikhin I, Kamensky I, Tarakanov S, Kramers J, Pekala M, Skiba V, Gannibal M, Novikov D (2010) Noble gas isotope sites and mobility in mafic rocks and olivine. Geochem Cosmochim Acta 74:1436–1447 Trull TW, Kurz MD (1993) Experimental measurements of 3He and 4 He mobility in olivine and clinopyroxene at magmatic temperatures. Geochem Cosmochim Acta 57:1313–1324 Vallianatos F, Eftaxias K (1992) The application of cBX model for the calculation of the activation for creep with depth in the earth’s lower mantle. Phys Earth Planet Inter 71:141–146 Vallianatos F, Eftaxias K (1994) Some aspects on estimation methods for activation volume. Acta Geophys Pol XLII(1):13–22 Vallianatos F, Eftaxias K, Vasilikou-Dova A (1995) A material science approach for the evaluation of the rheological state into the earth’s lower mantle. Radiat Eff Defects Solids 137:217–221 Vallianatos F, Triantis D, Tzanis A, Anastasiadis C, Stavrakas I (2004) Electric earthquake precursors: from laboratory results to field observations. Phys Chem Earth 29:339–351 Varotsos P (2005) The physics of seismic electric signals. TERRABUB, Tokyo
123
Phys Chem Minerals (2014) 41:181–188 Varotsos PA (2007a) Calculation of point defect parameters in diamond. Phys Rev B 75(17):172107 Varotsos P (2007b) Self-diffusion in sodium under pressure revisited. J Phys Condens Matter 19:176231–176235 Varotsos P (2007c) Comparison of models that interconnect point defect parameters in solids with bulk properties. J Appl Phys 101:123503–123508 Varotsos P (2007d) Defect volumes and the equation of state in aPbF2. Phys Rev B 76(9):092106 Varotsos P (2008) Point defect parameters in b-PbF2 revisited. Solid State Ion 179:438–441 Varotsos P (2009) The importance of an harmonic effects in models that interconnect point defect parameters with bulk properties in solids. J Appl Phys 105(8):083524 Varotsos P, Alexopoulos K (1977) Calculation of the formation entropy of vacancies due to anharmonic effects. Phys Rev B 15:4111–4114 Varotsos P, Alexopoulos K (1980) Calculation of diffusion coefficients at any temperature and pressure from a single measurement. I. Self diffusion. Phys Rev B 22:3130–3134 Varotsos P, Alexopoulos K (1982) Current methods of lattice defect analysis using dilatometry and self-diffusion. Phys Status Solidi (b) 110:9–31 Varotsos P, Alexopoulos K (1984) Connection between the formation volume and formation Gibbs energy in noble-gas solids. Phys Rev B 30:7305–7306 Varotsos PA, Alexopoulos KD (1986) Thermodynamics of point defects and their relation with bulk properties. North-Holland, Amsterdam Varotsos P, Ludwig W, Alexopoulos K (1978) Calculation of the formation volume of vacancies in solids. Phys Rev B 18: 2683–2691 Varotsos P, Alexopoulos K, Lazaridou M (1985) Interconnection of isothermal elastic data with self-diffusion in sodium. Phys Rev B 31:8263–8264 Varotsos P, Eftaxias K, Alexopoulos K (1991) Comment on ‘‘Selfdiffusion in tungsten’’. Phys Rev B 43:5170–5171 Wolfe MR, Stockli DF (2010) Zircon (U–Th)/He thermochronometry in the KTB drill hole, Germany, and its implications for bulk He diffusion kinetics in zircon. Earth Planet Sci Lett 295:69–82 Zhang B (2012) Diffusion of hydrogen in (Mg, Fe)2SiO4 and high pressure polymorphs refined by the cBX model. J Asian Earth Sci 54–55:9–17 Zhang B, Wu X (2011) Prediction of self-diffusion and heterodiffusion coefficients in zircon. J Asian Earth Sci 42:134–141 Zhang BH, Wu XP (2012) Calculation of self-diffusion coefficients in diamond. Appl Phys Lett 100(5):051901–051903 Zhang BH, Wu XP (2013) Diffusion of aluminium in MgO: a thermodynamic approach. Chin Phys B 22(5):056601–056605 Zhang B, Wu X, Xu J, Zhou R (2010) Application of the cBX model for the calculation of oxygen self-diffusion coefficients in minerals. J Appl Phys 108:053505–053506 Zhang B, Wu X, Zhou R (2011) Calculation of oxygen self-diffusion coefficients in Mg2SiO4 polymorphs and MgSiO3 perovskite based on the compensation law. Solid State Ion 186:20–28
