vol. 72 pp. 757-825, 2010 17 copyright © mineralogical

17Reviews in Mineralogy & GeochemistryVol. 72 pp. 757-825, 2010 Copyright © Mineralogical Society of America

1529-6466/10/0072-0017$10.00 DOI: 10.2138/rmg.2010.72.17

Diffusion in Oxides

James A. Van Orman1,2 and Katherine L. Crispin1

1Department of Geological SciencesCase Western Reserve University

10900 Euclid AvenueCleveland, Ohio 44106 U.S.A.

[email protected]

2Department of Chemical EngineeringCase Western Reserve University

10900 Euclid AvenueCleveland, Ohio 44106 U.S.A.

INTRODUCTION

Non-silicate oxide minerals are minor but important constituents of many igneous and metamorphic rocks, a major component of Earth’s lower mantle, and are well represented in planetary and meteoritic materials. Oxide minerals also have important roles in technology, for example as semiconductors, thermal and electrical insulators, fuel cell components, substrates for thin films, photovoltaic materials, and as products of metal oxidation. Because of their technological applications, and their fundamental interest to geosciences, materials science, physics and chemistry, the diffusion properties of many oxide minerals have been studied intensively, using a wide range of experimental, analytical and computational approaches. In many cases, particular attention has been devoted to deciphering the atomic-level mechanisms involved in diffusion. With the possible exceptions of metals and halides, oxides have probably been studied in more detail with regard to their point defect and diffusion properties than any other group of minerals.

The oxide minerals considered in this chapter are relatively simple in terms of their structure and chemistry, but nonetheless exhibit quite complicated diffusion behavior in many cases. Due to the simplicity of the minerals, and to the amount and quality of data available, the origin of many of these complicated behaviors is fairly well understood. In magnetite, for example, cation diffusion rates have a complex dependence on oxygen fugacity. This dependence is due to internal redox reactions, and to a transition from an interstitial diffusion mechanism at low oxygen fugacities to a vacancy mechanism under more oxidizing conditions. Oxygen and titanium diffusion rates in rutile also vary strongly with fO2

, due to internal reduction of titanium and the associated production of oxygen vacancies and titanium interstitials. Some cations in rutile exhibit strong diffusional anisotropy, which is thought to result from rapid diffusion along interstitial “channels” that extend along the c direction in the rutile structure. In periclase, trivalent cations diffuse rapidly compared to divalent cations, opposite to the trend observed in most silicate minerals. This behavior arises from the Coulombic attraction between trivalent cations and cation vacancies. Similar interactions between oxygen vacancies and cation vacancies appear to be responsible for lattice diffusion of oxygen in periclase.

In this chapter, we focus on four oxide minerals—periclase, spinel, magnetite and rutile—that are important in high-temperature geochemistry and geophysics, and for which a significant volume of diffusion data are available. A large body of experimental data also exists

758 Van Orman & Crispin

on diffusion in corundum, and this has been reviewed recently by Doremus (2006). Wüstite is addressed here as a solid solution with periclase; for information on cation self-diffusion in the pure phase see McKee (1983) and references therein. Diffusion in silica minerals is reviewed by Cherniak (2010).

PERICLASE

Periclase is an important constituent of Earth’s lower mantle, thought to comprise roughly 15 to 20% of its mass, and is also an important technological material. Its mass transport properties are important for many applications, and a large body of experimental diffusion data has been acquired over the past several decades. Periclase has also been the focus of many theoretical studies that have helped establish the diffusion mechanisms and have provided quantitative predictions of point defect formation and migration energies, as well as absolute diffusion rates. These predictions are, in general, in good agreement with the experimental results. The convergence of theory and experiment in the study of diffusion in periclase illustrates the power of a combined approach for understanding the mechanisms and controls on diffusion in minerals, and provides a benchmark for multi-disciplinary studies of diffusion in more complicated minerals.

General considerations

Periclase has the same simple structure as halite (NaCl) and is stable over the entire range of temperature and pressure relevant to Earth’s interior. Its broad range of stability has made it possible to study diffusion over a large range of temperatures (to 2500 °C) and pressures (to 35 GPa). The simple structure and stoichiometry of periclase make it somewhat easier, compared to most minerals, to infer the point defect mechanisms involved in diffusion. Periclase has a single cation site and a single anion site, each with octahedral coordination, as well as interstitial sites with tetrahedral coordination. Theoretical calculations indicate that interstitial magnesium and oxygen in periclase are energetically unfavorable. The enthalpy of formation for a Frenkel defect, which involves either Mg or O moving from its regular lattice site to an interstitial site, leaving behind a vacancy, is predicted to be very large, on the order of 1200 kJ/mol for Mg Frenkel defects and perhaps even greater for O Frenkel defects (Hirsch and Shankland 1991, and references therein). The concentrations of Mg and O interstitials are thus expected to be very low. A Schottky defect, formed when both an Mg atom and an O atom leave their lattice sites to create vacancies, is predicted to have a formation enthalpy on the order of 650 kJ/mol (Alfè and Gillan 2005; Karki and Khanduja 2006), only about half as large as that of a Frenkel defect. Cation and anion vacancies are thus predicted to be much more favorable than interstitials, and vacancy mechanisms for diffusion are therefore be expected to predominate in periclase. The experimental data for oxygen and most cations are indeed consistent with diffusion by a vacancy mechanism. However, some small cations like Be2+ may diffuse by an interstitial mechanism.

Although the Schottky defect formation enthalpy is smaller than the formation enthalpy for Frenkel defects, it is still quite large and the concentrations of intrinsic cation and anion vacancies are therefore expected to be small. At equilibrium, the concentration of oxygen and magnesium vacancies in pure MgO is given by:

′′ = = −

••V V

G

RTSf

Mg O exp ( )2

1

where GSf is the free energy of formation of the Schottky pair, R is the gas constant, T is the

absolute temperature, and the brackets refer to concentrations. In this equation, the identity of the point defect is expressed using Kröger-Vink notation; ′′VMg represents a vacant Mg site with

Diffusion in Oxides 759

an effective charge (relative to that of the occupied site) of −2, and VO•• represents a vacant O site

with an effective charge of +2. For a free energy of formation of 650 kJ/mol, the equilibrium concentrations of intrinsic vacancies are less than a few ppm at any temperature up to the melting point (~2850 °C). Even for a formation energy as low as 500 kJ/mol, intrinsic vacancy concentrations are <70 ppm at the melting point, and much smaller at lower temperatures.

In addition to intrinsic vacancies, extrinsic vacancies are present in periclase to balance the charge of aliovalent solutes. Even synthetic MgO crystals of ultra-high purity contain extrinsic cation vacancies to compensate trivalent cation impurities. Extrinsic anion vacancies are much less abundant in most samples than cation vacancies, because positively charged cation solutes (mainly trivalent cations such as Fe3+, Al3+ and Cr3+) tend to be present at higher concentrations than solutes with effective negative charges (monovalent cations such as Li+ and Na+). The commercial “nominally pure” MgO crystals used in most experimental diffusion studies are calculated to contain cation vacancies at the level of ~15-1000 ppm (e.g., Wuensch 1975; Oishi et al. 1983), which would likely predominate over intrinsic cation vacancies at all temperatures up to the melting point.

As in other highly ionic crystals, oppositely charged point defects in periclase may bind to form pairs and other associates. These defect associates can have a strong influence on diffusion rates. Cation and anion vacancies may bind to form electrically neutral pairs, which have been inferred by some authors to be the defect primarily responsible for oxygen diffusion (Ando et al. 1983; Yang and Flynn 1994). Cation vacancies also may bind to positively charged trivalent cations to form either negatively charged pairs, neutral dimers consisting of two trivalent ions and a vacancy on adjacent cation sites, or larger clusters. In general, the larger defect associates (dimers and clusters) are expected to be significant only at relatively high concentrations and low temperatures (e.g., Carroll et al. 1988), but cation-vacancy pairs may be present at significant concentrations over a broad range of conditions. The formation of these pairs impedes the motion of cation vacancies, and thus reduces the diffusivity of unbound cations that diffuse by a vacancy mechanism. On the other hand, diffusion of the trivalent cation is enhanced by up to several orders of magnitude when a vacancy is bound to it on an adjacent cation site (Perkins and Rapp 1973; Van Orman et al. 2009).

Oxygen

More than a dozen experimental studies on oxygen self-diffusion in periclase have been performed over the last five decades (See Appendix – Table 1). Most of the early studies determined diffusion coefficients by measuring the rate of exchange of 18O with a gas phase, and relied on bulk measurements of the isotopic composition. Later studies, beginning in the 1980s, employed secondary ion mass spectrometry (SIMS) and proton activation techniques to measure 18O diffusion profiles in periclase samples that had undergone diffusive exchange with an isotopically enriched gas or solid reservoir. One study included in Table 1 (see Appendix), by Narayan and Washburn (1973), determined oxygen diffusion coefficients indirectly, based on measurements of the shrinkage rate of dislocation loops, where oxygen self-diffusion was interpreted to be the rate-limiting step in dislocation climb.

The diffusion coefficients determined from these studies span three orders of magnitude at a particular temperature (Fig. 1a), neglecting two studies that used MgO doped with lithium (Shirasaki et al. 1973; Oishi et al. 1987). Diffusion coefficients determined in the studies on Li-doped samples are more than an order of magnitude larger than from any of the other studies, consistent with an enhancement of oxygen diffusivity by extrinsic vacancies created to compensate Li+ on cation sites. The activation enthalpies determined from these studies, 186 kJ/mol (Shirasaki et al. 1973) and 279 kJ/mol (Oishi et al. 1987) have relatively large uncertainties, due to the small temperature range covered in each study and to experimental problems including Li volatilization, but agree reasonably well with recent theoretical calculations of the migration energy for oxygen vacancies, 261 kJ/mol (Ita and Cohen 1997)


Figure 1. Summary of experimental diffusion data for oxygen in periclase at atmospheric pressure. In (a), dashed lines refer to data from studies using Li+-doped periclase; solid curves show data from studies on nominally pure MgO, or MgO doped with trivalent cations. In (b), the experimental data (solid curves) are compared with calculated bulk diffusion coefficients for crystals with different dislocation densities (dashed curves, calculated using Eqn. 2). Abbreviations: N73 – Narayan and Washburn (1973); O60 – Oishi and Kingery (1960); O83 – Oishi et al. (1983); O87 – Oishi et al. (1987); R66n – Rovner (1966) Norton crystals; R66se – Rovner (1966) Semi-Elements crystals; R83 – Reddy and Cooper (1983); S73 – Shirasaki et al. (1973) 3.5 at.% Li-doped; S73ls – Shirasaki and Hama (1973) “loosely sintered”; S73ws – Shirasaki and Hama (1973) “well sintered”; Y – Yoo et al. (1984, 2002); Y94 – Yang and Flynn (1994); Y96 – Yang and Flynn (1996).

a.

4 5 6 7 8 9 10

-28

-26

-24

-22

-20

-18

-16

-14

O in periclase

10 16m -2

10 14m -2

10 12m -2

log

D (m

2 /s)

104/T (K)

10 10m -2

Calculated bulk diffusioncoefficients, for differentdislocation densities

4 5 6 7 8 9

-24

-23

-22

-21

-20

-19

-18

-17

-16

O in periclase

R83

O83

Y84,02

N73

Y96Y94

R66nR66se

S73ws

S73ls

O87

O60

log

D (m

2 /s)

104/T (K)

S73

H72

a. b.

a.

4 5 6 7 8 9 10

-28

-26

-24

-22

-20

-18

-16

-14

O in periclase

10 16m -2

10 14m -2

10 12m -2

log

D (m

2 /s)

104/T (K)

10 10m -2

Calculated bulk diffusioncoefficients, for differentdislocation densities

4 5 6 7 8 9

-24

-23

-22

-21

-20

-19

-18

-17

-16

O in periclase

R83

O83

Y84,02

N73

Y96Y94

R66nR66se

S73ws

S73ls

O87

O60

log

D (m

2 /s)

104/T (K)

S73

H72

a. b.

(a)

(b)


and 234 kJ/mol (Karki and Khanduja 2006). However, as pointed out by Oishi et al. (1987), the interpretation of the activation enthalpy in these experiments is subject to considerable uncertainty. The measured activation enthalpy would be equivalent to the migration enthalpy for oxygen if Li resided solely on cation sites and did not bind significantly with oxygen vacancies to form defect associates, but it is not clear whether these conditions were met in the experiments. Further measurements of oxygen diffusivity in samples with a broad range of Li concentration, over a greater temperature interval, would undoubtedly shed more light on this issue. In general, the use of monovalent cation dopants in diffusion studies on periclase appears to be a promising avenue for further exploration of oxygen diffusion mechanisms.

Oxygen diffusion coefficients in periclase crystals that contain mainly trivalent rather than monovalent cation impurities—the type used in all studies except the two discussed above that used Li-doped samples—do not appear to vary significantly with the concentration of trivalent cations. Ando et al. (1983) studied both “pure” MgO and samples doped with between 310 and 12,900 ppm Fe, in experiments performed under oxidizing conditions where a substantial proportion of the iron must have been present as Fe3+, and found no significant variation in oxygen diffusivity with the level of doping. Similarly, Henriksen et al. (1983) found no significant variation in oxygen diffusion coefficients among samples that were nominally pure, or that were doped with Sc3+ at 550 ppm and 1400 ppm. These experimental observations indicate that free oxygen vacancies are not the species responsible for oxygen diffusion in periclase containing predominantly trivalent impurities, because while trivalent solutes enhance the concentration of cation vacancies, they decrease the concentration of free oxygen vacancies according to the law of mass action. Hence, if oxygen diffusion were to occur by means of free vacancies, a negative correlation would be expected between the trivalent cation concentration in the periclase sample and the oxygen diffusivity. Ando et al. (1983) proposed that in samples doped with trivalent cations oxygen diffuses by means of a neutral species whose concentration is independent of the level of doping, and suggested that this neutral species was a bound pair consisting of a cation vacancy and an anion vacancy. Cation-anion vacancy pairs have similarly been inferred to contribute to anion diffusion in alkali halides (e.g., Fuller 1966) and later studies reinforced the inference that vacancy pairs were largely responsible for oxygen diffusion in MgO. Yang and Flynn (1994) found quantitative agreement between their experimental results on oxygen diffusion in the high-temperature intrinsic regime and theoretical calculations on the diffusivity of ( ′′VMg − VO

••) pairs, and in later experiments (Yang and Flynn 1996) observed no significant variation in oxygen diffusivity between samples with ~30 and ~3 ppm trivalent impurities. Van Orman et al. (2003), in a high-pressure experimental study, found no variation in oxygen diffusivity along an Al3+ gradient in either single crystals or grain boundaries, suggesting that vacancy pairs (or another neutral species) are responsible for oxygen diffusion through periclase grain boundaries as well as single crystals, and at high as well as low pressures.

Yang and Flynn (1994, 1996) employed thin-film diffusion couples of exceptionally high quality grown by molecular beam epitaxy, and determined oxygen diffusion coefficients that were smaller than in any other study. The samples were grown on a polished MgO substrate and consisted of a 200-500 nm layer of epitaxial single-crystal Mg16O deposited at the base, followed by a 5 nm Mg18O tracer layer and 30-250 nm top layer of Mg16O. Two of these samples were clamped face to face and packed in MgO powder to prevent evaporation during the high-temperature diffusion anneals. Oxygen isotope profiles across the sample were measured before and after each diffusion experiment using SIMS depth profiling. The activation enthalpy for oxygen diffusion at high temperatures was found to be 667 kJ/mol, much higher than calculated values for the migration enthalpy of oxygen but consistent with the activation enthalpy for intrinsic diffusion, which includes half the formation enthalpy of the Schottky pair. Further, Yang and Flynn (1994) showed that their experimental oxygen diffusion coefficients in the high-temperature regime fell between the narrow band of values for intrinsic diffusion with no pairing


and with complete pairing of oxygen vacancies to cation vacancies. A later theoretical study of defect formation and migration energies in MgO by Ita and Cohen (1997) also yields intrinsic oxygen diffusion coefficients in close agreement with the Yang and Flynn (1994) values.

At lower temperatures, Yang and Flynn (1994) found a deviation from intrinsic behavior, with lower activation enthalpy (257 kJ/mol). The lower temperature results have an Arrhenius slope similar to those found in many other experimental studies of oxygen diffusion, but with lower diffusivity values. A similar low-temperature regime, with diffusion coefficients an order of magnitude higher, was found in a study of epitaxially-grown MgO with even higher chemical purity but a higher density of unidentified structural defects (Yang and Flynn 1994, 1996). Evidently, diffusion in this low temperature regime is enhanced by these structural defects. The activation enthalpy for oxygen diffusion in the low-temperature regime observed by Yang and Flynn (1994, 1996) and in many studies of oxygen diffusion using bulk gas-exchange methods, is similar to the measured activation enthalpy of 252 kJ/mol for oxygen diffusion along dislocation cores (Narayan and Washburn 1973). This suggests that dislocations may have enhanced oxygen diffusion in these studies, and that variations in dislocation density among the samples used might explain at least some of the wide scatter in oxygen diffusion coefficients determined in different experimental studies. Other experimental observations support this interpretation. Oishi et al. (1983) found that oxygen diffusion coefficients decreased by one to two orders of magnitude when samples were chemically polished to remove surface damage produced by crushing or cleaving the samples in preparation for gas-exchange experiments. Also, Henriksen et al. (1983) found that oxygen diffusivity increased by a factor of ~5 in a single crystal that had been deformed prior to the diffusion anneal. In general, the influence of dislocations and other extended defects becomes more important when lattice diffusion is slow, as is the case for intrinsic oxygen diffusion in periclase, especially at lower temperatures.

Figure 1b shows calculated values for the effective diffusivity of oxygen in periclase single crystals with four different dislocation densities, along with the experimental data for periclase single crystals (excluding those that were doped with Li). The effective diffusivity is calculated as:

Deff = Dlat + ρaDdisl (2)

where Dlat is the lattice diffusion coefficient, assumed to follow the Arrhenian dependence determined by Yang and Flynn (1994) at high temperatures, Ddisl is the diffusion coefficient in a dislocation core assumed to follow the Arrhenian dependence determined by Narayan and Washburn (1973), ρ is the dislocation density, and a is the cross-sectional area of a dislocation. In writing Equation (2), diffusion along subgrain boundaries is not considered, although subgrains would be expected to contribute to oxygen transport in deformed periclase crystals. Thus the calculated curves can be taken to represent the lower limit on effective diffusivity at a given dislocation density, for the lattice and dislocation diffusion coefficients used in the calculations. The lowest dislocation density shown in the figure, 1010 m−2, is typical of well-annealed crystals, yet even in this case dislocations are predicted to contribute significantly to diffusion at temperatures below 1400 °C. The highest dislocation density shown, 1016 m−2, is relevant to a highly deformed crystal, subjected to shear stress on the order of 1 GPa according to the shear stress/dislocation-density relationship reported for MgO by Takeuchi and Argon (1976). In this case the intrinsic regime is not seen; diffusion along dislocations dominates oxygen transport at all temperatures.

The influence of pressure on oxygen diffusion in periclase has been addressed in one experimental (Van Orman et al. 2003) and several theoretical studies (Ita and Cohen 1997, 1998; Karki and Khanduja 2006; Ito and Toriumi 2007). Van Orman et al. (2003) determined oxygen diffusion coefficients in single crystals and grain boundaries at 2273 K and pressures of 15-25 GPa. Oxygen diffusion coefficients in the single crystals were found to decrease significantly with increasing pressure between 15 and 25 GPa, with an activation volume of ~3.3 cm3/mol.


However, the diffusion coefficients obtained from these high-pressure experiments are larger than those obtained from most studies at atmospheric pressure. It was suggested (Van Orman et al. 2003) that oxygen transport in the high-pressure experiments may have been enhanced by dislocations generated during non-hydrostatic compression of the samples. In this case, the activation volume determined in this study would be that for diffusion along dislocations—which, as discussed above, may be the relevant activation volume for oxygen diffusion in periclase in many cases. The oxygen grain boundary diffusion coefficients determined by Van Orman et al. (2003) were seven orders of magnitude higher than the volume diffusion coefficients measured in the same study, and had a similar pressure dependence.

Because the high-pressure experiments on oxygen diffusion in MgO appear to have been conducted in an extrinsic regime, the influence of pressure on the intrinsic diffusion of oxygen has been addressed only by theoretical calculations. Ita and Cohen (1997) used a Gordon-Kim method to determine the Gibbs free energy of formation of Schottky pairs in MgO, and the Gibbs free energy of oxygen migration at pressures up to 140 GPa and temperatures of 1000 to 5000 K. The activation volume for intrinsic oxygen diffusion derived from these results at 0 to 20 GPa is 9 cm3/mol, and decreases at higher pressures. Karki and Khanduja (2006) studied defect formation and migration energies in MgO using calculations based on density functional theory. The activation volume for oxygen intrinsic diffusion derived from their results between 0 and 20 GPa is 7.6 cm3/mol, and also decreases at higher pressures. The Schottky pair formation energy found in these two studies is in good agreement with the value determined by Alfè and Gillan (2005) using a quantum Monte Carlo method. Ito and Toriumi (2007) studied diffusion in MgO over a wide range of temperatures and pressures using molecular dynamics simulations with empirically fitted force fields. The activation volume for intrinsic diffusion of oxygen derived from their results at 0 to 20 GPa is 13.5 cm3/mol. In general, the Schottky formation energy found by Ito and Toriumi (2007) is higher than those found by Ita and Cohen (1997) or Karki and Khanduja (2006), especially at high pressures. The migration energies determined in all three studies are similar at low pressures, but at very high pressures the migration energies derived by Ito and Toriumi (2007) are significantly smaller than those found in the other two studies.

Magnesium

Magnesium self-diffusion in periclase has been studied extensively using both radioactive and stable isotope tracers (See Appendix – Table 2). Important information on Mg self-diffusion is also available from studies of ionic conductivity in periclase samples doped with trivalent cation impurities. Sempolinski and Kingery (1980) studied periclase doped with various levels of trivalent cations and found that the data were consistent with a simple model wherein cation vacancies are the charge-carrying species and their concentration is controlled by the trivalent cation substituents through charge balance. Conductivity data for samples doped with 65 to 1500 ppm trivalent cations yield consistent data on the diffusivity of the cation vacancies, with an Arrhenius relation (Sempolinski and Kingery 1980):

DH

RTVm

′′−= × −

Mg

3 8 10 35. exp ( )

where is the diffusion coefficient for cation vacancies in m2/s, Hm is the migration enthalpy for cation vacancies (220 kJ mol−1), R is the gas constant (8.3145 J mol−1 K−1), and T is the temperature in Kelvin. The migration enthalpy determined by Sempolinski and Kingery (1980) is in reasonable agreement with recent theoretical calculations of 241 kJ/mol (Ita and Cohen 1997) and 218 kJ/mol (Karki and Khanduja 2006). Information on the rate of diffusion of cation vacancies derived from these ionic conductivity measurements is extremely useful because it can be used to calculate the coefficient for Mg self-diffusion as a function of the concentration of cation vacancies and the temperature, according to the relation:


D x D fMg V VMg Mg= ′′ ′′ ( )4

where xVMg′′ is the fraction of cation sites that are vacant and f is a correlation factor with numerical value of 0.78145 for the face-centered cubic cation sublattice in periclase (Shewmon 1989, p. 111).

Magnesium diffusion coefficients calculated by combining Equations (3) and (4) are compared with experimental data on Mg self-diffusion in Figure 2. The data are reasonably consistent with the calculations, for cation vacancy concentrations between ~25 and 2000 ppm. Early experimental studies, published prior to 1973, are consistent with high vacancy concentrations, while those published more recently are generally consistent with a vacancy concentration of ~50 ppm, within the range expected for high-purity MgO single crystals that are presently available from commercial suppliers based on the trace impurity contents reported by the manufacturers. When the Mg diffusion coefficients determined in the various experimental studies are corrected to a common cation vacancy concentration, they are in good agreement (Fig. 2).

Although it was suggested in some early studies (Harding et al. 1971; Harding and Price 1972) that intrinsic Mg self-diffusion was measured at high temperatures, this interpretation of the data now seems unlikely. The inference of intrinsic diffusion was made on the basis of data covering a short high-temperature interval on the Arrhenius plots, with an apparent activation enthalpy somewhat higher than at lower temperatures. It is not clear, however, that a distinct high temperature segment is resolvable; re-fitting each of the Harding et al. (1971) data sets, obtained using Ventron and Monocrystal MgO crystals, with a single Arrhenius line yields activation energies of 242 and 257 kJ/mol, respectively, with r2 values of 0.98 and 0.97. Similarly, fitting the Harding and Price (1972) data set with a single Arrhenius line yields an activation energy of 245 kJ/mol with an r2 value of 0.99. On the whole, the Mg diffusion data for periclase appear to be consistent with an extrinsic vacancy mechanism, with an activation energy similar to values for the migration energy determined in theoretical and ionic conductivity studies. This interpretation is consistent with recent first-principles calculations of the formation energy for Schottky defects in periclase (Alfè and Gillan 2005; Karki and Khanduja 2006), which show (a) that the concentrations of intrinsic cation vacancies will be far below the extrinsic concentrations in real crystals at all temperatures up to the melting point, as noted above, and (b) that the activation energy in the intrinsic regime should be ~600 kJ/mol, far higher than observed in any experimental study.

The influence of pressure on Mg self-diffusion has been determined in one experimental study (Van Orman et al. 2003) and has also been addressed by several theoretical studies. Van Orman et al. (2003) measured Mg self-diffusion coefficients in single crystals and fine-grained polycrystals at 2273 K and pressures of 15 to 25 GPa. The single crystal data were found to be consistent with data from studies at atmospheric pressure using samples with comparable purity. Magnesium diffusion in the high-pressure experiments appears not to have been affected significantly by the presence of dislocations or other extended defects, in contrast to oxygen diffusion in the same experiments (Van Orman et al. 2003). Oxygen is much more prone to the influence of extended defects because its intrinsic diffusivity in the lattice is so slow; lattice diffusion of Mg is much faster due to the relatively high concentration of extrinsic cation vacancies. In combination with the atmospheric pressure data, the high-pressure data for Mg self-diffusion yield an activation volume of 3.0 cm3/mol, which is in reasonable agreement with the migration volumes determined over a similar pressure range using various theoretical methods (Ita and Cohen 1997; Karki and Khanduja 2006; Ito and Toriumi 2007). At higher pressures the theoretical calculations diverge significantly, although all predict a decreasing activation volume at higher pressures (Fig. 3). Clarifying the pressure dependence of cation diffusion in MgO is an important goal for the future, as it has important implications for rates of


34

56

78

910

11-2

2

-21

-20

-19

-18

-17

-16

-15

-14

-13

-12

100

ppm

25 p

pm

500

ppm

log D (m2/s)

104 /T

(K)

2000

ppm

M85

S92

W73H71

L57

H72M

g in

per

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se

Cal

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s, fo

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t ca

tion

vaca

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conc

entra

tions

45

67

89

1011

-22

-21

-20

-19

-18

-17

-16

-15

-14

-13

-12

Mg

in p

eric

lase

X v = 5

0 pp

m

Wue

nsch

& V

asilo

s, 19

73

Saka

guch

i et a

l., 19

92

log D (m2/s)

104 /T

(K)M

artin

elli e

t al.,

1985

Har

ding

et a

l., 1

971

Har

ding

& P

rice,

197

2 Lind

ner &

Par

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1957

Fig

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2. S

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of e

xper

imen

tal d

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fusi

on d

ata

for

mag

nesi

um i

n pe

ricl

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at a

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are

ba

sed

on

the

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ta,

and

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(i

ndic

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by

th

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from

mea

sure

men

ts o

f th

e ca

tion

vaca

ncy

diff

usiv

ity

(Sem

polin

ski

and

Kin

gery

198

0),

usin

g E

quat

ions

(3)

and

(4

). I

n th

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set,

the

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of 5

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= D

exp(

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; H

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al.

(198

5) –

50

ppm

; Sak

aguc

hi e

t al.

(199

2) –

50

ppm

.


diffusion in Earth’s deep mantle. At the pressure of Earth’s core-mantle boundary (~140 GPa) the migration enthalpy extrapolated from experimental data assuming a constant activation volume of 3.0 cm3/mol is 640 kJ/mol, whereas theoretical values are 445 kJ/mol (Ita and Cohen 1997), 390 kJ/mol (Karki and Khanduja 2006) and 205 kJ/mol (Ito and Toriumi 2007). These differences in migration enthalpy correspond to differences in the Mg self diffusion coefficient of six orders of magnitude at 4000 K (and even larger differences at lower temperatures).

Other group IIA divalent cations

Diffusion data have been acquired for all five stable elements in group IIA of the periodic table. A compilation of the experimental results is presented in Table 3 (see Appendix), and a summary is shown on an Arrhenius plot in Figure 4.

Be2+. Diffusion of divalent beryllium in nominally pure MgO has been investigated by Harding and Mortlock (1966) at temperatures of 1000 to 1700 °C and by Harding (1973a) from 636 to 2341 °C. Both studies used 7Be as a radioactive tracer, and employed serial sectioning techniques to measure the diffusion profiles. The two studies yielded consistent results, despite using MgO crystals from two different sources (Monocrystals and Ventron), and no difference was observed between experiments performed in air vs. an argon atmosphere (Harding 1973a). In each case the diffusion coefficients were well described by a single Arrhenius equation over the entire temperature range studied, with activation energy of 154 kJ/mol (Harding and Mortlock 1966) and 162 kJ/mol (Harding 1973a). The small activation energy, rapid diffusion rates relative to other divalent cations, and low degree of scatter in the diffusion coefficients (perhaps implying that variations in purity from sample to sample are not important) led Harding (1973a) to tentatively suggest an interstitial diffusion mechanism.

0 50 100 150

200

250

300

350

400

450

Ito & Toriumi, 2007

Karki & Khanduja, 2006

Van Orman et al., 2003

Hm =

Em +

PV

m (k

J/m

ol)

Pressure (GPa)

Mg pressuredependence

Ita & Cohen, 1997

Figure 3. Experimental and theoretical results on the pressure dependence of Mg diffusion in MgO. Theoretical values of the migration enthalpy from Karki and Khanduja (2006) and Ito and Toriumi (2007) are those reported in the original reference. The theoretical values of Ita and Cohen (1997) were calculated from the reported values of the Gibbs free energy of migration (Gm) at different pressures according to the relation Hm = Em + P(dGm/dP)T, with the reference value of Em (220 kJ/mol) taken to be the migration energy determined by Sempolinski and Kingery (1980). The experimental values of the migration enthalpy, shown as squares, were similarly calculated as Hm = Em + PVm, where Em is the migration energy determined by Sempolinski and Kingery (1980) and Vm (3.0 cm3/mol) is the activation volume determined by Van Orman et al. (2003).


Ca2+. Calcium diffusion in nominally pure MgO has been studied by Rungis and Mortlock (1966) and Harding (1973b) using a thin surface deposit of 45Ca radiotracer with diffusion profiles determined by serial sectioning; by Wuensch and Vasilos (1968) using thin film and vapor exchange techniques with electron microprobe analysis of the diffusion profiles; and by Yang and Flynn (1994, 1996) using high-purity MgO single crystals grown by molecular beam epitaxy with an embedded Ca-doped tracer layer, with diffusion profiles characterized by SIMS depth profiling. As with Mg there is considerable scatter among the different studies in the diffusion coefficients determined at a particular temperature, but much of this scatter is removed by correcting the data to a common cation vacancy concentration. Yang and Flynn (1994) found quantitative agreement between their experimental results and theoretical predictions for diffusion by a vacancy mechanism, at the impurity levels relevant to their samples (10-50 ppm cation vacancies). Yang and Flynn (1996) used samples of exceptionally high purity, with ~3 ppm extrinsic cation vacancies, and found diffusion coefficients an order of magnitude smaller than in their 1994 study, consistent with diffusion by an extrinsic vacancy mechanism.

Sr2+. Strontium diffusion in nominally pure MgO was studied between 1000 and 1600 °C by Mortlock and Price (1973), using 85Sr radiotracer thin films and serial sectioning. Two distinct zones were identified in the diffusion profiles—near the surface where the radiotracer

4 5 6 7 8 9 10 11-22

-21

-20

-19

-18

-17

-16

-15

-14

-13

-12Mg 2+; W73

Mg 2+; S92

log

D (m

2 /s)

104/T (K)

Mg2+; M85

Mg 2+; H71

Mg2+; HP72

Mg 2+; L57

Ca 2+; H73b

Ca 2+; W68

Ca2+; R66

Ca2+; Y94

Ca 2+; Y96

Sr 2+; M73

Ba 2+; H67

Ba2+; H72

Be 2+; H66

Be 2+; H73a

Group IIA cationsin periclasexv

= 50 ppm

Figure 4. Summary of experimental diffusion data for group IIa cations in periclase at atmospheric pressure. Data for each cation have been corrected to a common cation vacancy concentration of 50 ppm (as in Fig. 2b), with the exception of Be which may diffuse by an interstitial mechanism. Abbreviations and estimated experimental cation concentrations are as follows: H66 – Harding and Mortlock (1966); H73a – Harding (1973a); L57 – Lindner and Parfitt (1957), 500 ppm; H71 – Harding et al (1971), 500 ppm; HP72 – Harding and Price (1972), 500 ppm; W73 – Wuensch et al. (1973), 100 ppm; M85 – Martinelli et al. (1985), 50 ppm; S92 – Sakaguchi et al. (1992), 50 ppm; R66 – Rungis and Mortlock (1966), 500 ppm; W68 – Wuensch and Vasilos (1968), 70 ppm; H73b – Harding (1973b), 500 ppm; Y94 – Yang and Flynn (1994), 30 ppm; Y96 – Yang and Flynn (1996), 3 ppm; M73 – Mortlock and Price (1973), 500 ppm; H67 – Harding (1967), 500 ppm; H72 – Harding (1972), 500 ppm.


was deposited the concentration gradient was relatively steep, and a segment with much shallower gradient extended much deeper into the crystal. This second, deeper zone, which yielded diffusion coefficients two orders of magnitude higher than in the near-surface region, was found to disappear if the samples were pre-annealed at 1700 °C, and was attributed to dislocation-enhanced transport. Enhancement of Sr mobility by dislocations is not surprising given the low solubility of the large Sr2+ cation in periclase; cations with low solubility in the crystal lattice are often found to have high relative concentrations in extended crystal defects (e.g., Hiraga et al. 2004).

Ba2+. Diffusion of Ba2+ in nominally pure MgO was studied by Harding (1967) at 1000-1700 °C and by Harding (1972) over an extended temperature range from 1100 to 2500 °C. Both studies utilized 133Ba radiotracer thin films and serial sectioning. As with Sr2+, the diffusion profiles from these studies have relatively steep gradients near the surface and “tails” extending much deeper into the crystal. The atmosphere in which the experiments were conducted (air vs. argon) was found not to have a significant influence on the diffusion rates. Harding (1972) suggested that three regimes could be identified on an Arrhenius plot, corresponding to different diffusion mechanisms with different activation energies. These three regimes are not obviously distinguishable, and the data are actually represented quite well by a single line. A non-weighted least squares fit of the entire data set to a single Arrhenius equation yields an activation energy of 292 kJ/mol with r2 of 0.98, which is in reasonable agreement with the value of 326 kJ/mol determined by Harding (1967) over a smaller temperature interval.

Influence of ionic radius. Among the elements of group IIA the activation energy increases and the diffusion coefficient decreases with increasing cation radius (Fig. 5). The small activation energy and large diffusivity for Be2+, the smallest of the group IIA cations, may result from diffusion by an interstitial rather than a vacancy mechanism, but for the larger cations an extrinsic vacancy mechanism appears to be well established. The observed correlations are broadly consistent with expectations based on elastic strain considerations (e.g., Mullen 1966); a larger cation is expected to encounter a larger energy barrier as it passes through a constriction during a jump from one site to the next, and thus would have a larger activation energy and, all else being equal, slower diffusivity. However, this general expectation is not observed for diffusion of the transition metals, where there is no obvious correlation between ionic radius and diffusivity. For the group IIa cations, which share similar valence electron shell configurations, the size of the ion appears to play a primary role in governing its

0.4 0.6 0.8 1.0 1.2 1.4-18

-17

-16

-15

-14

Be Mg Ca Sr Ba

Group IIA cationsin periclase at 1573K

log

D (m

2 /s)

Ionic Radius (�)0.5 1.0 1.5

150

200

250

300

350

Mg Be Ca Sr Ba

Q (k

J/m

ol)

Ionic Radius (�)

Group IIA cationsin periclasea. b.

Figure 5. Variation with ionic radius of the activation energy (a) and diffusion coefficient at 1573 K (b) for group IIa cations in periclase. In (b), the diffusion coefficients for all cations except Be have been corrected to a cation vacancy concentration of 50 ppm. Ionic radii are from Shannon (1976).


diffusivity, while for the transition metals other factors, including the crystal field effect and the dipole polarizability, may come into play.

Group IIIA and IIIB trivalent cations

Diffusion of trivalent cations differs fundamentally from diffusion of divalent cations in that their substitution into periclase must be coupled with that of a charge-balancing species. In most cases the charge-balancing species is a cation vacancy, which is the point defect that controls diffusion of the trivalent cations, as well as most other cations. The coupled substitution of cation vacancies and trivalent cations leads in general to a strong concentration dependence of the diffusivity; the concentration of cation vacancies increases with the concentration of the trivalent cations, and thus the diffusivity increases. In the simplest case the diffusion coefficient is a linear function of the concentration. However, as has been observed in the alkali halides and other strongly ionic crystals, positively charged solutes and negatively charged cation vacancies are attracted to each other in periclase, and tend to form strongly bound M3+-vacancy pairs (and larger defect complexes, at low temperatures and high concentrations). The formation of bound pairs enhances the mobility of the trivalent cation, because it increases the fraction of trivalent cations that have vacant nearest-neighbor cation sites. On the other hand, the formation of bound pairs “steals” vacancies from other cations, and thus reduces their mobility. These effects are most important at high trivalent cation concentrations, and at relatively low temperatures.

A summary of diffusion data for Group IIIA and IIIB cations, at low concentration, is shown in Figure 6, and a compilation of experimental results is given in Table 4 (see Appendix).

Al3+. Diffusion of Al3+ in periclase has been studied using MgO-Al2O3 (Whitney and Stubican 1971a) and MgO-MgAl2O4 diffusion couples (Whitney and Stubican 1971b; Van Orman et al. 2009), with electron microprobe analyses of the experimental diffusion profiles. Whitney and Stubican (1971a,b) calculated Al diffusion coefficients in MgO as a function of concentration by graphical analysis of the diffusion profiles using an equation presented by Wagner (1969). Interdiffusion between MgO and Al2O3 (Whitney and Stubican 1971a) is complicated by the formation of MgAl2O4 spinel between them, resulting in a continuously moving interface between the periclase and spinel; despite this complication, the results of these experiments are in reasonable agreement with those using spinel as the source of diffusant (Fig. 7). In both studies the diffusion coefficient was found to be a strong function of concentration.

Van Orman et al. (2009) performed experiments similar in design to those of Whitney and Stubican (1971b) but instead of performing a graphical analysis to determine diffusion coef-ficients, the Al diffusion coefficient as a function of concentration was derived by numerically fitting the diffusion profiles to a theoretical model for diffusion in the presence of bound Al3+-vacancy pairs. The theoretical model used was based on the model presented by Lidiard (1955) for diffusion of a divalent cation in an alkali halide crystal, and extended to a trivalent cation in a simple metal oxide by Perkins and Rapp (1973). The equation derived by Van Orman et al. (2009) for the Al diffusion coefficient as a function of the Al cation fraction, xAl, is:

D Dx x

G RT G RTAl

Al Al=

− +

−+

−

−

2

3

4 16 16

1

576 2

2

2 2

1

2

exp( / ) exp( / )

xx

G RTAl

16

1

322

5+−

exp( / )

( )

where G2 is the Gibbs free energy of the point defect association reaction:

′′ + ↔ ( )′•V VMg Mg Mg MgAl Al ( )6

and D2 is the diffusion coefficient of the bound (VMgAlMg)′ pair. Synthetic diffusion profiles


based on Equation (5) with the appropriate boundary conditions were found to provide a good description of the experimental diffusion profiles at all conditions, from 1577-2273 K and 1 atm to 25 GPa. The average binding energy of the Al-vacancy pair derived from fitting the experimental diffusion profiles, −50 kJ/mol, is in good agreement with the temperature-corrected theoretical values derived from shell model calculations (Gourdin and Kingery 1979; Carroll et al. 1988). The activation energy and activation volume for diffusion of the (VMgAlMg)′ pair were found to be similar to those for Mg self-diffusion. Analysis of the frequencies of Al-vacancy and Mg-vacancy exchange—both of which are involved in the motion of the Al-vacancy pair—shows that the Al jump frequency is only about one third the Mg jump frequency. Despite this, diffusion of Al is about an order of magnitude faster than Mg at the same conditions, due to the attraction of cation vacancies to Al3+.

Ga3+. The concentration-dependent diffusion of Ga3+ in MgO was studied by Crispin and Van Orman (2010) using an experimental design similar to that used by Van Orman et al.

5.0 5.5 6.0 6.5

-13.5

-13.0

-12.5

-12.0

MgO-MgAl2 O

4

MgO-MgAl2O4;Whitney & Stubican, 1971b

log

D (m

2 /s)

104/T (K)

Al-Mg interdiffusionin periclase; xAl = 0.02

MgO-Al2O3;Whitney & Stubican, 1971a


Figure 7. Summary of ex-perimental data for Al-Mg interdiffusion in periclase. All data are shown at an Al cation fraction of 2%.

3.5 4.0 4.5 5.0 5.5 6.0 6.5

-15.0

-14.0

-13.0

-12.0Groups IIIA, IIIB, IVAcations in periclase

Ge 4+; Harding, 1973b

Y 3+; Berard, 1971


Al 3+ (100 ppm)

log

D (m

2 /s)

104/T (K)

Ga 3+ (100 ppm); Crispin & Van Orman, 2010

Sc3+; Solaga & Mortlock, 1970

Figure 6. Summary of experi-mental diffusion data for group IIIa, IIIb and IVa cations in periclase. Data for Y, Sc and Ge are from radiotracer stud-ies, and data for Ga and Al are from interdiffusion studies that characterized the concentration dependence of the diffusion coefficient. Diffusion coeffi-cients for Ga and Al are shown at cation concentrations of 100 ppm, similar to the aliovalent impurity concentrations in the periclase samples used in the tracer diffusion studies.


(2009) for Al3+. These authors found that cation vacancies bind more tightly to Ga3+ than they do to Al3+, with a binding energy of −83 kJ/mol, which appears to be the primary reason Ga diffuses more rapidly than Al.

Sc3+. A series of measurements of Sc3+ diffusion was made by Solaga and Mortlock (1970) using radiotracer (46Sc) and sectioning methods. Each experiment was performed at 1773 K for a duration of 50 hours, and the amount of radiotracer applied to the surface of the MgO crystal was varied between 0.5 and 105 ppm. For experiments performed with surface tracer concentrations less than about 50 ppm the diffusion coefficients were found to be constant. Beyond 50 ppm the diffusive penetration distance increased with the tracer concentration, and the diffusion profiles showed clear evidence for concentration-dependent diffusivity.

Y3+. Yttrium diffusion in nominally pure MgO was studied by Berard (1971) using a radiotracer (91Y) and sectioning method. No mention is made of the concentration of radiotracer used, and the diffusion profiles were fit to a thin film solution with the diffusion coefficient independent of concentration.

Tetravalent cations

Tetravalent cations, like trivalent cations, are expected in most cases to be charge-balanced by cation vacancies in periclase. Cation vacancies should also bind more strongly to tetravalent cations than to trivalent cations, due to the greater Coulombic attraction. To the authors’ knowledge, only one study of tetravalent cation diffusion in periclase has been published, on diffusion of Ge4+ (Harding 1973b). A small concentration of tracer was used in these experiments, which the author suggested would not increase the concentration of cation vacancies in the MgO crystal significantly. Unfortunately no information on the binding energy between Ge4+ and cation vacancies can be extracted from the Harding (1973b) experiments. However, the diffusion coefficients for Ge4+ were found to be higher than the Ca2+ diffusion coefficients reported in the same paper, despite the higher activation energy of Ge4+. Ordinarily an ion with larger activation energy is expected to diffuse more slowly. It seems likely that the enhanced diffusivity of Ge4+ results from strong binding to cation vacancies.

Transition metals

An extensive body of experimental data exists on the diffusion of 10 transition metals in periclase, including Sc and Y which were discussed above in the section “Group IIIA and IIIB trivalent cations”. A compilation of the published experimental data is presented in Tables 4-7 (see Appendix). The transition metals differ in several ways from the elements considered above, in that many of them (a) are stable in multiple valence states, (b) have extensive solid solution with MgO, and (c) have partially filled d-orbitals resulting in non-spherical valence electron shells.

Cr. Diffusion of chromium has been studied experimentally by several groups, under rela-tively oxidizing conditions where it is thought to have been present primarily as Cr3+. As with the trivalent cations discussed in section “Group IIIA and IIIB trivalent cations”, diffusion of Cr3+ is strongly concentration-dependent and its attraction to cation vacancies appears to have a significant influence on its diffusion. Tagai et al. (1965) studied diffusion of Cr3+ in air by depositing a thin oxide film and analyzing diffusion profiles with an electron microprobe. In contrast to later studies, no concentration dependence of the diffusion coefficient was noted. Greskovich and Stubican (1969) examined interdiffusion in the MgO-Cr2O3 system (which, as in the MgO-Al2O3 system discussed above, is complicated by the formation of a spinel phase, MgCr2O4) and found that diffusion of Cr3+ in MgO was a strong and quasi-linear function of concentration. Osenbach et al. (1981) obtained similar results using MgO crystals doped with 0.05 and 0.56 cation% Cr3+ as the diffusion source, rather than Cr2O3. Crispin and Van Orman (2010) examined interdiffusion between nominally pure MgO single crystals and polycrystal-line (Mg0.99Cr0.01)O over a large range of temperature at 2 GPa, and numerically fitted the dif-


fusion profiles to an equation analogous to Equation (5). The results of this study were found to be in broad agreement with the previous interdiffusion studies, and the binding energy between Cr3+ and cation vacancies was found to be −22 kJ/mol, much smaller in absolute value than for Ga3+ or Al3+. Weber et al. (1977, 1980) performed a completely different type of experiment to determine the concentration dependence of Cr3+ diffusion in periclase, examining 51Cr radio-tracer diffusion in samples doped with Cr3+ at different levels up to 0.6 cation%. The results are in broad agreement with the interdiffusion studies. The diffusion coefficients determined in each of the studies are in reasonably good agreement at low Cr3+ concentrations (Fig. 8).

Compared to the other trivalent cations that have been investigated Cr3+ is the slowest, despite having an ionic radius and polarizability nearly identical to Ga3+, the fastest of the trivalent cations (Fig. 9). Crispin and Van Orman (2010) suggested that the much slower diffusivity, larger migration energy and smaller binding energy of Cr3+ to vacancies could be explained in terms of the crystal field effect, which deepens the energy well for Cr3+ on cation sites in periclase.

Mn. Three studies of Mn diffusion in periclase have been published. Tagai et al. (1965) deposited a thin layer of manganese oxide onto the polished surface of MgO crystals and annealed the samples in air. A later study by Weeks and Chatelain (1978) showed that under these conditions at least two different oxide layers are formed near the surface, including an MnO2 layer that extends dendritically downward into the periclase. Thus, chemical exchange in the Tagai et al. (1965) experiments probably involved processes considerably more complicated than simple diffusion of Mn2+ within a single phase. Weeks and Chatelain (1978) performed experiments in evacuated quartz tubes where the source of diffusant was either a vapor in equilibrium with Mn metal, as fine-grained particles or a ~10 µm thin film deposited onto the MgO single crystal. These experiments were presumably buffered by the presence of Mn and MnO, and manganese is expected to have been predominantly divalent under these conditions. Diffusion of Mn2+ was characterized in the quenched samples by a combination of electron paramagnetic resonance spectroscopy and electron microprobe analysis of the diffusion profiles. No evidence for concentration dependence of the diffusion coefficient was mentioned, and the equations used to fit the diffusion profiles are based on the assumption that the diffusion coefficient during the experiment was constant.

Jones and Cutler (1971) studied Mn(1−x)O-MgO interdiffusion over a wide range of composition, both in air and under reducing conditions. For the experiments in air, an (Mn,Mg)O solid solution was packed around an MgO single crystal, in order to avoid the formation of intermediate phases during the interdiffusion process, and diffusion profiles were measured both by serial sectioning with wet chemical analysis and by electron microprobe. The cation vacancy concentration along each profile was inferred based on the relationship between Mn3+ content and total Mn concentration determined by Jones and Cutler (1966) under the same conditions, making the reasonable assumption that the cation vacancy concentration was equal to half the Mn3+ concentration. In air, the interdiffusion coefficient was found to be a linear function of the cation fraction of vacancies, up to ~2% vacancies, and the activation enthalpy for diffusion, 171 kJ/mol, was found to be independent of the composition. Jones and Cutler (1966) found that the cation fraction of Mn3+ (and associated cation vacancies) increased exponentially with the concentration of Mn. As a result, the Mn-Mg interdiffusion coefficient also increases exponentially with Mn concentration. Based on the data reported by Jones and Cutler (1966, 1971) we find the following relationship for Mn-Mg interdiffusion in air: DMn-Mg ∝ exp(16.9xMn). Jones and Cutler (1971) also performed an MnO-MgO interdiffusion experiment at 1500 °C under a reducing atmosphere, with fO2

= 10−4 Pa. In this case the interdiffusion coefficient was also found to be an exponential function of the cation fraction of Mn, but with a weaker dependence on composition: DMn-Mg ∝ exp(4.6xMn). These results imply that the fraction of Mn3+ in MgO increases less steeply with Mn concentration under reducing conditions


than under highly oxidizing conditions. The diffusion coefficients determined under reducing conditions by Jones and Cutler (1971) and Weeks and Chatelain (1978) are consistent with those determined in air by Jones and Cutler (1971), when the Mn concentration is very small (consistent with cation vacancy concentrations of ~10-100 ppm).

Co. Cobalt diffusion in periclase at low concentrations was investigated by Wuensch and Vasilos (1962) at 1000-1810 °C, using diffusion couples consisting of a thin layer of oxide sand-wiched between two nominally pure MgO single crystals. Diffusion profiles in the quenched

1E-3 0.01 0.110-16

10-15

10-14

10-13

Cr; Weber et al., 1980Cr; Osenbach et al., 1981 Cr; G

reskovich &

Stubican, 1969

D (m

2 /s)

xM3+

Cr; Crispin & Van Orman, 2010

Ga; Crispin & Van Orman, 2010

Al; Whitney & Stubican, 1971b

Al; Van Orman et al., 2009

Cr3+, Al3+ & Ga3+

in periclase at 1773 K

Figure 9. Experimental data on the concentration dependence of the diffusion coefficient for trivalent cations in periclase.

4.5 5.0 5.5 6.0 6.5-17

-16

-15

-14

Weber et al., 1977

Weber et al., 1980

Crispin & Van Orman, 2010 (100 ppm)

log

D (m

2 /s)

104/T (K)

Tagai et al., 1965

Cr3+ diffusion in periclaseat low concentrations

Figure 8. Summary of experimental diffusion data for Cr3+ in nearly pure MgO.


samples were measured using an electron microprobe. The experiments at 1000-1200 °C were performed in air, and at higher temperatures (1410-1810 °C) under more reducing conditions, in a gas-air mixture of unspecified composition. Despite the different atmospheres used, all data are consistent with a single Arrhenius curve, and no compositional dependence of the diffusion coefficient was noted.

Zaplatynsky (1962) studied MgO-CoO interdiffusion in experiments with a single crystal of MgO surrounded by either CoO or (Co0.3Mg0.7)O powder. A series of experiments was per-formed in air for different durations, and the “average” interdiffusion coefficient at a particular temperature was determined by measuring the rate of motion of a contour of fixed concentra-tion, identified as a color boundary observed under transmitted light. Because the technique relies on the motion of a single concentration contour, it does not allow determination of the interdiffusion coefficient as a function of composition across the diffusion couple. However, the experiments using CoO on one side of the diffusion couple yielded diffusion coefficients an order of magnitude larger than the experiments using Co0.3Mg0.7O, indicating a substantial concentration dependence of Co diffusion in air.

Yurek and Schmalzried (1974) explicitly studied the compositional dependence of CoO-MgO interdiffusion in air, at 1300 °C. Diffusion couples consisted of single crystals of CoO and MgO, and diffusion profiles were measured using an electron microprobe. These authors also report data on Co tracer diffusion, cation vacancy concentrations, and cation vacancy diffusion coefficients in (Co,Mg)O solid solutions in air at 1300 °C, from the Ph.D. dissertations of Dieckmann (1975) and Schwier (1973). Both the Co-Mg interdiffusion coefficients and Co tracer diffusion coefficients were found to depend exponentially on composition, with DCo-Mg ∝ exp(7.4xCo) and DCo

* ∝ exp(6.8xCo). The cation fraction of vacancies was found to have a similar exponential dependence on composition, with xV ∝ exp(5.4xCo). The somewhat greater dependence of the interdiffusion and tracer diffusion coefficients on composition, compared to the cation vacancy concentrations, appears to result from the greater mobility of cation vacancies in CoO-rich solid solutions; cation vacancies reportedly diffuse an order of magnitude more rapidly in pure CoO than in pure MgO (Yurek and Schmalzried 1974).

Ni. Diffusion of nickel has been studied both in tracer diffusion and NiO-MgO interdiffusion experiments, and in both oxidizing and reducing atmospheres. Mimkes and Wuttig (1971) performed 63Ni radiotracer experiments using two different nominally-pure MgO single crystals, one of which was deformed to increase the dislocation density by a factor of ~100. These experiments demonstrated that dislocations have a negligible influence on Ni diffusion in MgO. Harding (1972) also performed 63Ni radiotracer experiments on nominally pure MgO crystals, using both air and Ar atmospheres during the diffusion anneals. The results from the experiments performed in air were indistinguishable from those performed in Ar. On the other hand, when a small amount of Al was added to the tracer solution deposited on the crystals, the diffusion coefficients for Ni increased by a factor of ~4, indicating that diffusion was enhanced by cation vacancies introduced to charge-balance Al3+ that diffused into the crystal during the experiment. Wuensch and Vasilos (1962, 1971) also performed experiments using nominally pure MgO single crystals, with both vapor sources and NiO thin films used as the diffusant, and electron microprobe analysis of the diffusion profiles. Although these experiments were, strictly speaking, interdiffusion experiments, the NiO concentrations were small and the results similar to those from the Harding (1972) and Mimkes and Wuttig (1971) tracer studies. Wuensch and Vasilos (1962, 1971) found that Ni diffusion coefficients were similar in MgO crystals with 270 and 820 ppm cation impurities, respectively, and like Harding (1972) found that that Ni diffusion coefficients were insensitive to the atmosphere used, at the low Ni concentrations investigated.

Blank and Pask (1969) studied interdiffusion in the NiO-MgO system both in air and under vacuum, using Boltzmann-Matano analysis to numerically evaluate the interdiffusion coefficients as functions of composition. In contrast to the studies discussed above, where


the Ni concentration along the diffusion profiles was low, these authors found a significant influence of the atmosphere on the interdiffusion coefficients. Under vacuum, interdiffusion coefficients were found to be insensitive to the concentration of NiO over the entire range of composition, and similar to Ni tracer diffusion coefficients in nominally pure MgO. In air, at low Ni concentration, the Ni-Mg interdiffusion coefficients are similar to those determined under vacuum, but the Ni-Mg interdiffusion coefficient increases exponentially with the Ni concentration (Fig. 10a). These results, combined with the tracer studies discussed above, suggest that nickel is present almost exclusively as Ni2+ at all concentrations under vacuum, and at low concentrations in air. At higher concentrations in air, the proportion of Ni3+ (and thus cation vacancies) increases exponentially with Ni concentration. Appel and Pask (1971) found similar results using similar techniques to analyze Ni-Mg interdiffusion profiles in air. They found DNi-Mg ∝ exp(βxNi), with β = 2.16, compared to β =2.68 from the results of Blank and Pask (1969). Wei and Wuensch (1973) found a similar exponential dependence of the Ni

Figure 10. Summary of ex-perimental data for diffusion of Ni in periclase. (a) Ni-Mg interdiffusion coefficients as a function of Ni cation frac-tion and temperature (all data from Blank and Pask 1969). In air, the interdif-fusion coefficients increase strongly with Ni concentra-tion. Under vacuum the in-terdiffusion coefficients are independent of composition and similar to the values in air at xNi = 0) (b) Ni tracer diffusion coefficients and interdiffusion coefficients at xNi ~ 0)

4 5 6 7 8

-17

-16

-15

-14

-13

-12

Appel & Pask, 1971

Wei & Wuensch, 1973

W & V, 1971

Wuensch & Vasilos, 1962

Harding, 1972

Blank & Pask, 1969

log

D (m

2 /s)

104/T (K)

Mimkes & Wuttig, 1971

Ni2+ in periclase

5.8 6.0 6.2 6.4 6.6 6.8-16.0

-15.5

-15.0

-14.5

-14.0

air vacuum

0 ≤ xNi ≤ 1

xNi = 0

xNi = 0.8

xNi = 0.6

xNi = 0.4

log

D (m

2 /s)

104/T (K)

Ni-Mg interdiffusionin periclase

xNi = 0.2

4 5 6 7 8

-17

-16

-15

-14

-13

-12

Appel & Pask, 1971

Wei & Wuensch, 1973

W & V, 1971

Wuensch & Vasilos, 1962

Harding, 1972

Blank & Pask, 1969

log

D (m

2 /s)

104/T (K)

Mimkes & Wuttig, 1971

Ni2+ in periclase

5.8 6.0 6.2 6.4 6.6 6.8-16.0

-15.5

-15.0

-14.5

-14.0

air vacuum

0 ≤ xNi ≤ 1

xNi = 0

xNi = 0.8

xNi = 0.6

xNi = 0.4

log

D (m

2 /s)

104/T (K)

Ni-Mg interdiffusionin periclase

xNi = 0.2

(a)

(b)


tracer diffusion coefficient on Ni concentration, in experiments performed on 12 different crystals with 0 to 67.9 at.% NiO with 63Ni as a radiotracer. This approach is laborious but avoids problems inherent in the Boltzmann-Matano analysis of interdiffusion profiles, where the concentration dependence of the tracer diffusion coefficient depends on an analysis of the detailed shape of the diffusion profile. Wei and Wuensch (1973) found β =3.16 for Ni tracer diffusion, and β =1.88 for calculations of the Ni-Mg interdiffusion coefficient based on their tracer diffusion results.

A summary of Ni2+ diffusion coefficients in MgO is shown in Figure 10b. Included on this Arrhenius plot are data from tracer diffusion experiments as well as Ni-Mg interdiffusion experiments in air at low Ni concentrations, and under vacuum. Data from the various studies are in reasonably good agreement, despite differences in sample purity and experimental design.

Fe. Iron diffusion has been studied extensively, in part because periclase in Earth’s lower mantle is thought to be primarily a solid solution of (Mg,Fe)O. A number of studies have addressed the dependence of Fe-Mg interdiffusion rates on iron concentration, oxygen fugacity and temperature, and recent studies have also investigated the influence of pressure (Holzapfel et al. 2003; Yamazaki and Irifune 2003) and dissolved water (Demouchy et al. 2007; Kohlstedt and Mackwell 2008). A compilation of expressions for the Fe-Mg interdiffusion coefficient in ferropericlase is given in Table 7 (see Appendix).

Iron is similar to Mn, Co and Ni in that it may exist in both the divalent and trivalent states in periclase. However, it is somewhat more complicated because Fe3+ is present even under highly reducing conditions, especially at high total Fe concentrations; the end-member wüstite does not exist in the stoichiometric form within its regime of stability (e.g., Darken and Gurry 1945). Also, Fe3+ may occupy both octahedral cation sites and tetrahedral interstitial sites (Waychunas 1983; Hilbrandt and Martin 1998; Jacobsen et al. 2002; Otsuka et al. 2010), while other trivalent transition metals appear to have a strong preference for cation sites (e.g., Blank and Pask 1969). Early studies of Fe diffusion in periclase were performed by Wuensch and Vasilos (1962) and Tagai et al. (1965), at relatively low concentrations and with diffusion profiles measured by electron microprobe. Most of the experiments were performed in air, and the concentration of Fe was not reported. Interpretation of the results is uncertain because the Fe concentration is now known to have a strong influence on the diffusion rate, which varies with the fO2

(e.g., Blank and Pask 1969; Sata and Goto 1982). It is likely that diffusion in the early experiments by Wuensch and Vasilos (1962) and Tagai et al. (1965) was enhanced by cation vacancies formed due to oxidation of Fe2+.

The Fe-Mg interdiffusion coefficient in ferropericlase is a strong function of Fe concentration even under highly reducing conditions, similar to Mn-Mg but in contrast to Ni-Mg. Rigby and Cutler (1965) performed interdiffusion experiments in the Fe1−xO-MgO system under a reducing H2-H2O atmosphere (although neither the composition of the gas mixture nor the oxygen fugacity were reported), and found that the proportion of iron present as Fe3+ and the Fe-Mg interdiffusion coefficient both increased strongly with the total concentration of iron. As with Co in air, much but not all of the increase in diffusivity with Fe concentration could be attributed to the greater proportion of Fe3+ and attendant increase in cation vacancy concentrations; the mobility of cation vacancies was also found to increase significantly with the fraction of iron. We found that the data of Rigby and Cutler (1965) could be cast as exponential functions of the Fe cation fraction, for xFe up to 0.3, with DFe-Mg ∝ exp(11.4xFe) and xV ∝ exp(9.4xFe). Several subsequent studies found a similar dependence of the Fe-Mg interdiffusion coefficient on composition. Blank and Pask (1969) studied Fe1−xO-MgO interdiffusion under vacuum, using metallic Fe as a diffusion source, and found DFe-Mg ∝ exp(βxFe), with values of β 7.5 to 9.3 for temperatures between 1150 and 1350 °C. Sata and Goto (1982) found a similar exponential dependence of the interdiffusion coefficient on Fe concentration in experiments conducted in a similar temperature range, and over a range of fO2

. They found that β varied


with fO2, from 9.7 at 10−6 Pa to 7.5 at 3×10−3 Pa. The interdiffusion coefficient was found to

be proportional to f mO2, with the exponent m between 0.2 and 0.3. Bygdén et al. (1997) also

found an exponential dependence of the interdiffusion coefficient on iron concentration, with a slightly larger value of β = 12.4. Their experiments were performed under purified Ar, and were probably at an fO2

near the iron-wüstite buffer. Bygdén et al. (1997) used polycrystalline rather than single-crystal diffusion couples, yet obtained diffusion coefficients somewhat lower than in any other study. Grain boundaries apparently played little role in enhancing bulk diffusion in their experiments, and the relatively low diffusivities found in this study may be explained by the highly reducing conditions of the experiments. Mackwell et al. (2005) performed Fe-Mg interdiffusion experiments over a range of fO2

, at 1320 to 1400 °C, and fit their data with an exponent m = 0.19, at the low end of the range found by Sata and Gato (1982) (Fig. 11). They expressed the interdiffusion coefficient as a function of x pFe, with p = 0.73, rather than as an exponential function of iron concentration as in the studies above. Despite the difference in functional form, each of these studies found a similar dependence of the interdiffusion coefficient on Fe concentration, as shown in Figure 12.

Chen and Peterson (1980) studied 59Fe radiotracer diffusion in two different MgO-FeO solid solutions, with xFe of 0.5 and 0.76, respectively, at temperatures of 900-1100 °C and oxygen fugacities of 10−11 to 10−6 Pa. They found that the iron tracer diffusion coefficient was dependent on the oxygen fugacity and iron concentration, similar to the results for Fe-Mg interdiffusion discussed above, being proportional to

fO

1/6

2 and to exp(7.7xFe). The activation

enthalpy for Fe tracer diffusion in the sample with xFe = 0.76 was found to be quite small, 121 kJ/mol in an atmosphere with a constant ratio of CO2/CO. This value represents the sum of the migration enthalpy and the enthalpy of the reaction for incorporating a cation vacancy, an oxygen atom and two electron holes into (Mg,Fe)O. Correcting to constant oxygen fugacity using the fO2

1 6/ dependence determined by the authors yields a still smaller value of ~30 kJ/mol for the migration enthalpy, which is very small.

Blank and Pask (1969) suggested that the activation enthalpy for Fe-Mg interdiffusion increased with Fe concentration, while Mackwell et al. (2005) suggested the opposite. The

-6 -5 -4 -3 -2 -1-14.0

-13.5

-13.0

-12.5

Mackwell et al., 2005 (1523 K)


Sata & Goto, 1982

log

D (m

2 /s)

log fO2


Fe-Mg interdiffusionin Mg

0.8Fe

0.2O

Figure 11. Dependence on oxygen fugacity (in Pa) of the Fe-Mg interdiffusion coefficient in periclase (Fe0.2Mg0.8O).


apparent discrepancy between these two studies may result in part from the difference in experimental conditions—the Blank and Pask (1969) experiments, performed in vacuum with metallic iron as the diffusion source, likely followed the iron-wüstite oxygen buffer, along which the fO2

increases with temperature, while the temperature dependence in the Mackwell et al. (2005) experiments was determined from experiments at constant oxygen fugacity. Part of the discrepancy may also arise from the small temperature interval that was studied in each case, 1150-1350 °C for Blank and Pask (1969) and 1320-1400 °C for Mackwell et al. (2005); this results in a large uncertainty in determining the activation enthalpy in each case. It seems reasonable to conclude that variations in activation enthalpy with composition in the Fe1−xO-MgO system are not yet well constrained.

An Arrhenius plot comparing Fe-Mg interdiffusion results from different studies, at the composition (Mg0.8Fe0.2)O, is shown in Figure 13. Much of the scatter among the various studies may result from differences in the oxygen fugacity of the experiments. The Blank and Pask (1969) and Bygdén et al. (1997) experiments were performed under vacuum and purified Ar, respectively—i.e., under highly reducing conditions, probably near the stability limit of wüstite. Based on the fO2

dependence documented by Sata and Gato (1982) and Mackwell et al. (2005), it is not surprising that these two studies yielded the lowest interdiffusion coefficients. The Rigby and Cutler (1965) experiments were performed using an H2-H2O gas, but neither the mixing proportion nor the oxygen fugacity was specified.

Mackwell et al. (2005) presented a global expression for the Fe-Mg interdiffusion coefficient in periclase as a function of xFe, fO2

, and temperature:

D f xx

RTFe-Mg O FeFe

2= × − −

−2 9 10209 000 96 0006 0 19 0 73. exp

, ,. . m /s2 ( )7

Sata and Goto (1982) presented a similar equation for the Fe-Mg interdiffusion coefficient as a function of fO2

and temperature, but without explicitly considering the temperature dependence. These equations provide a good description of the data over the range of conditions studied in the experiments. Extrapolating the equations far from the conditions studied, however, may

0.0 0.1 0.2 0.3-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

Rigby & Cutler, 1965 (1573 K) H 2-H 2

O

Bygden et al., 1997 (1573 K) argon

Blank & Pask, 1969 (1523 K) vacuum

Blank & Pask, 1969 (1623 K) vacuum

Sata & Goto, 1982 (1573 K) fO 2=1e-4 Pa

log

D (m

2 /s)

xFe

Mackwell et al., 2005 (1573 K) fO 2

=1e-4 Pa

Concentration dependenceof D

Fe-Mg in ferropericlase

Figure 12. Dependence on Fe cation fraction of the Fe-Mg interdiffusion coefficient in ferropericlase.


result in large errors, due to changes in the defect structure and/or diffusion mechanisms. For example, Fe-Mg interdiffusion coefficients in air (Blank and Pask 1969) are far slower than would be predicted by an extrapolation of Equation (7), have a larger activation enthalpy, and are independent of iron concentration. Under these conditions a large proportion of the iron is trivalent, and the defect structure of ferropericlase may be quite different than under reducing conditions, perhaps more similar to wüstite in which defect clusters such as interstitial Fe3+-vacancy combinations are important (e.g., McKee 1983).

Demouchy et al. (2007) studied the influence of dissolved hydrogen on Fe-Mg interdiffusion in periclase. Experiments were performed along the Ni-NiO buffer under a water pressure of 300 MPa and temperatures of 1000-1250 °C. Compared to anhydrous conditions, the Fe-Mg interdiffusion coefficients were enhanced by a factor of ~3 to 4 (Fig. 14); dissolved hydrogen thus has a significant influence on interdiffusion rates, but the effect is rather small compared to variations in oxygen fugacity or iron concentration. Demouchy et al. (2007) attributed the increased interdiffusion coefficients to the incorporation of defect associates formed between a cation vacancy and a proton, with an overall charge neutrality condition of [ ] [{( ) } ]FeMg O Mg

• •= − ′′ ′OH V . Although the activation enthalpy under hydrous conditions reported by Demouchy et al. (2007) is rather high compared to the values determined in experiments under anhydrous conditions, 270 kJ/mol for the MgO end-member, it refers to the apparent activation enthalpy with fO2

buffered by the Ni-NiO equilibrium. Along this buffer the oxygen fugacity increases with temperature, and this leads to an enhanced activation enthalpy. As pointed out by Kohlstedt and Mackwell (2008), the calculated value at constant oxygen fugacity is 208 kJ/mol, in line with anhydrous studies. Thus, dissolved hydrogen does not appear to have a significant influence on the temperature dependence of Fe-Mg interdiffusion.

The influence of pressure on Fe-Mg interdiffusion in periclase was studied by Holzapfel et al. (2003) at 8-23 GPa and 1653-2073 K, and by Yamazaki and Irifune (2003) at 7-35 GPa and 1753-1973 K. Holzapfel et al. (2003) determined an activation volume of 3.3 cm3/mol, similar to the value of 3.0 cm3/mol determined for Mg self-diffusion in periclase (Van Orman et al. 2003), while Yamazaki and Irifune (2003) found a significantly smaller value, 1.8 cm3/

6.0 6.5 7.0 7.5

-14

-13

-12

-4.3

-3

-2

Rigby & Cutler, 1965; H2 -H

2 O

Bygden et al., 1997; argon

log

D (m

2 /s)

104/T (K)

Blank & Pask, 1969; vacuum

Mackwell et al., 2005

log fO2

-1Temperature dependenceof D

Fe-Mg in Mg

0.8Fe

0.2O

Figure 13. Temperature dependence of Fe-Mg interdiffusion in ferropericlase (fO2 in Pa).


mol. The activation volumes probably do not reflect simply the migration volumes of Fe and Mg, but also include variations in diffusivity due to changes in oxygen fugacity with pressure. Holzapfel et al. (2003) used a Ni-NiO assembly to buffer the oxygen fugacity. Yamazaki and Irifune (2003) did not include an oxygen buffering assemblage; their assemblies included Re and, at high pressure, graphite/diamond, but it is not clear that they were buffered by either Re-ReO2 or C-CO-CO2. It is possible that the difference in activation volume determined in the two studies results from the different fO2

-P curves followed by the experimental sample assemblies. The pressure dependence of Fe-Mg interdiffusivity in ferropericlase with 20 mol% FeO is shown in Figure 15, in comparison with results for Mg2+ (Van Orman et al. 2003) and Al3+ in MgO (Van Orman et al. 2009).

Chen et al. (2008) recently performed 59Fe radiotracer diffusion experiments on MgO doped with 5000 ppm Al3+, over a range of fO2

. Because a small concentration of radiotracer was used, the vacancy concentration was controlled by the Al3+ dopant; thus, while the ratio Fe3+/(Fe3++Fe2+) increased with increasing fO2

, variations in the cation vacancy concentration were extremely small. The diffusivity of iron was found to increase with increasing oxygen fugacity, indicating that Fe3+ diffuses more rapidly than Fe2+ in periclase. Possible reasons for the more rapid diffusivity of Fe3+ are its Coulombic attraction to cation vacancies (e.g., Gourdin and Kingery 1979; Van Orman et al. 2009), smaller size and lesser crystal field stabilization (Crispin and Van Orman 2010). The activation enthalpy for diffusion of Fe2+ determined by Chen et al. (2008) is 259 kJ/mol. This value is significantly higher than other values determined for Fe2+ diffusion in periclase, but includes a contribution due to the temperature dependence of the free vacancy concentration, which is significant for the doping level of 5000 ppm Al3+ used. Correcting the data to a common free vacancy concentration using a binding energy between Al3+ and cation vacancies of −50 kJ/mol (Van Orman et al. 2009), the migration enthalpy is 228 kJ/mol, which is within the range of values determined in other studies.

Zn. Diffusion of Zn2+ was studied at 1000-1645 °C by Wuensch and Vasilos (1965) using a vapor source of ZnO, with electron microprobe analysis of the quenched-in diffusion profiles. The surface concentration of Zn2+ following the diffusion anneals was 5-10 at.%. No

0.0 0.1 0.2 0.310-15

10-14

10-13

10-12

"dry"

"wet"

Demouchy et al., 2007

Influence of H2O on Fe-Mg

interdiffusion in ferropericlasefO2

= 10-2 Pa; 1523K; xFe

=0.2

DFe

-Mg (m

2 /s)

xFe

Mackwell et al., 2005

Figure 14. Influence of H2O on Fe-Mg interdiffusion in ferropericlase. The “wet” experiments of Demouchy et al. (2007) were performed at PH2O = Ptot = 300 MPa.


concentration dependence of the diffusion coefficient was evident in the diffusion profiles.

Pd. Diffusion of Pd in MgO was investigated by de Bruin and Tangtreeratana (1981) using a 103Pd radiotracer and serial sectioning. The diffusion anneals were performed in air at 700-1100 °C. A discontinuity in the diffusion coefficients was observed at a temperature between 800 and 850 °C, with the diffusion coefficient dropping abruptly by an order of magnitude above the transition temperature, and having a larger activation energy in the high temperature regime. This behavior was attributed to a change in valence from Pd2+ at low temperatures to Pd0 at higher temperatures. It is interesting to speculate that the slower diffusivity of Pd0 results in part from a Coulombic repulsion between it and a cation vacancy (each of which has an effective charge of −2 in periclase). The larger size of atomic Pd also may be important.

Cd. Diffusion of cadmium in MgO, thought to be present as Cd2+, was studied by Harding and Bhalla (1971) at high temperatures (1800-2300 °C) using a 115Cd radiotracer and serial sectioning techniques. Because Cd and its oxides are quite volatile at high temperatures, the diffusion anneals were performed in double-walled MgO/graphite containers, in an argon atmosphere. The diffusion coefficients are well represented by a single Arrhenius curve.

Divalent transition metals—summary. Diffusion data for the divalent transition metals in nearly pure MgO are summarized in Figure 16 on an Arrhenius plot. The diffusion coefficients for all elements fall within a relatively narrow band, and there are no obvious correlations with ionic radius, dipole polarizability or crystal field stabilization energy; to determine whether these ionic parameters have a significant influence on diffusion rates would require careful control of sample purity and experimental environment to ensure that the diffusion coefficients for each element were determined under identical conditions.

The dependence of the interdiffusion coefficient on concentration is summarized in Figure 17 and Table 7 (see Appendix), for transition metal cations that are predominantly divalent. The concentration dependence is generally larger in air than under reducing conditions (with the exception of iron, which forms defect clusters at high concentrations in air), indicating

0 5 10 15 20 25 30 35

1E-15

1E-14

1E-13

1E-12

xFe =0.2; 1973K

xFe =0.2; 1973K

Mg; 2273K; Wuensch &Vasilos, 1973

Mg; 2273K; Van Orman et al., 2003

Al; xAl =100 ppm; 2273K; Van Orman et al., 2009

Al; xAl =0.02; 2273K; Van Orman et al., 2009

Fe-Mg; Holzapfel et al., 2003

D (m

2 /s)

Pressure (GPa)

Fe-Mg; Yamazaki & Irifune, 2003

Pressure dependenceof cation diffusion inpericlase

Figure 15. Summary of experimental data on the pressure dependence of cation diffusion rates in periclase and ferropericlase. The reported activation volumes are: Mg – 3.0 cm3/mol (Van Orman et al. 2003); Al – 3.2 cm3/mol (Van Orman et al. 2009); Fe-Mg – 3.3 cm3/mol (Holzapfel et al. 2003), 1.8 cm3/mol (Yamazaki and Irifune 2003).


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Ni

Co

log

(D/D

xMe=

0) (m

2 /s)

xMe

Air

Mn

Fe

Fe

Mn

Reduced

xMe

Figure 17. Concentration dependence of the metal-Mg interdiffusion coefficient for divalent transition metals in periclase, in air and under reducing conditions.

4 5 6 7 8 9 10-18

-17

-16

-15

-14

-13

-12lo

g D

(m2 /s

)

104/T (K)

Zn; W65

Divalent transitionmetals in MgO

Fe; C08

Mn; W78

Co; Y74

Mn; J71 Fe; B97

Fe; B69

Ni; W71

Ni; W62

Ni; H72

Ni; B69Ni; M71

Cd; H71

Co; W62Pd; d81

Figure 16. Summary of experimental data for the diffusion of divalent transition metals in nearly pure MgO. Abbreviations: J71 – Jones and Cutler (1971); W78 – Weeks and Chatelain (1978); C08 – Chen et al. (2008); B69 – Blank and Pask (1969); B97 – Bygdén et al. (1997); Y74 – Yurek and Schmalzried (1974); W62 – Wuensch and Vasilos (1962); M71 – Mimkes and Wuttig (1971); H72 – Harding (1972); W71 – Wuensch and Vasilos (1971); W65 – Wuensch and Vasilos (1965); d81 – de Bruin and Tangtreeratana (1981); H71 – Harding and Bhala (1971). Iron tracer diffusion data from Chen et al. (2008) were measured in a sample doped with 5000 ppm Al3+; in this figure the values are corrected to an Al3+ concentration of 100 ppm using the dependence of the free vacancy concentration on Al concentration determined by Van Orman et al. (2009).


that the proportion of trivalent cations, and charge-balancing cation vacancies, increases more steeply with concentration under oxidizing conditions. Taken together, the data obtained in air and under reducing conditions indicate that interdiffusion of Fe has the strongest dependence on concentration, followed by Mn, Co and Ni in that order.

Hydrogen

Diffusion of deuterium and hydrogen in MgO has been studied by Gonzalez et al. (1982) by analyzing the change with time in the infrared absorbance of the OD- and OH- bands in MgO crystals held in D2O, and in a few cases H2O, atmospheres at high temperature. These authors examined both undoped MgO crystals and crystals doped with one of fourteen different cations: H, Li, Na, Al, V, Cr, Fe, Co, Ni, Cu, Ga, Ag, Au and Yb. Similar results were obtained for each of the crystals except for those doped with Li, in which diffusion was much faster. In the Li-doped samples, deuterium diffusion at 1073-1173 K was described by an Arrhenius equation with activation enthalpy of 183 kJ/mol and D0 of 0.015 m2/s. In the other samples, diffusion of deuterium could only be detected using the infrared absorbance method at temperatures above ~1750 K, near the limit of the furnaces used. At 1873 K, the diffusion coefficient for deuterium in these samples was found to be ~10−10 m2/s, three orders of magnitude slower than in the Li-doped samples, extrapolated to the same temperature. An experiment in which hydrogen and deuterium diffused simultaneously into MgO under the same conditions provided an estimate of the isotope effect. The ratio DH/DD was found to be 1.1 ± 0.1.

Gonzalez et al. (1982) found that diffusion of hydrogen in MgO was enhanced significantly by the application of an electric field, but not to the same degree as in quartz or other materials with open channels. Under a field of 2000 V/cm, hydrogen diffusion became significant at 1100 K or higher, in contrast to ~1750 K when no field was applied. The diffusion coefficients under the applied electric field were not reported, but based on their Figure 6, which compares the hydrogen loss for samples with and without an electric field, an enhancement of approximately two orders of magnitude in the hydrogen diffusivity can be inferred.

Gonzalez and Chen (2002) presented a further review of hydrogen mobility in MgO and other oxide minerals, including a discussion of radiation effects and information derived from studies of the internal redistribution of hydrogen-related defects. These authors also discuss diffusion of the hydride species, in addition to the hydroxyl ions discussed above. The hydride ion consists of a proton with two electrons occupying an oxygen vacancy, rather than a proton bound between an oxygen atom and magnesium vacancy (the hydroxyl defect), and may be important when the concentration of oxygen vacancies is large.

SPINEL

Spinel, with end-member formula MgAl2O4, is one of the major minerals in Earth’s upper mantle, a common mineral in metamorphic rocks, and an important constituent of calcium-aluminum inclusions (CAIs), condensates from the solar nebula during the earliest history of the solar system. Spinel has cubic symmetry, and diffusion is therefore isotropic. Oxygen atoms are in a nearly closest-packed arrangement, similar to their packing in periclase. Cations, on the other hand, are distributed on distinct octahedral and tetrahedral sites, occupied primarily by Al3+ and Mg2+, respectively. The existence of more than one cation sublattice makes the situa-tion with regard to point defects and diffusion much more complicated than for periclase. Spinel also contains octahedral interstitial sites, and two different types of tetrahedral interstitial sites.

At high temperatures a wide range of solid solution exists between MgAl2O4 and Al2O3, extending to compositions with up to ~90 mol% Al2O3 at temperatures approaching the melting point. The excess Al2O3 occupies tetrahedral sites and is accommodated primarily by the creation of cation vacancies on the octahedral (Al3+) sites rather than on the tetrahedral sites (Jagodzinski


and Saalfeld 1958). There is relatively little solid solution between MgAl2O4 and MgO; spinel in equilibrium with periclase at temperatures up to nearly 2000 °C is nearly stoichiometric with only up to ~1.016 Mg atoms per formula unit (Watson and Price 2002). Excess MgO in spinel would likely be accommodated by oxygen vacancies; however, to the authors’ knowledge no experimental studies have addressed point defects or diffusion in spinel with excess MgO.

Many different diffusion mechanisms are conceivable in spinel. For example, Mg could diffuse by means of vacancies on either the Mg or Al sites, by hopping through one or more types of interstitial sites, or by means of defect pairs or clusters of various types. Calculations on point defect formation and migration energies have been performed, based on a classical ionic description of the crystal lattice (Chiang and Kingery 1990; Chen et al. 1996; Ball et al. 2008; Murphy et al. 2009), but the large number of possible point defect reactions and lack of knowledge regarding the detailed distributions of point defects make it impossible to clearly identify the diffusion mechanism that operates in any particular case.

Oxygen

A compilation of experimental diffusion data for oxygen (and cations) in magnesium aluminate spinels is provided in Table 8 (see Appendix), and the results are shown on an Arrhenius plot in Figure 18. In each set of experiments single-crystal or polycrystalline samples were annealed in 18O-enriched gas, and diffusive exchange was characterized either by bulk analysis of the quenched samples (Ando and Oishi 1972, 1974; Oishi and Ando 1975) or by measurement of the 18O concentration profile in quenched samples by proton activation analysis (Reddy and Cooper 1981) or SIMS depth profiling (Ryerson and McKeegan 1994). There is good agreement among the studies on the temperature dependence of O volume diffusion, but even among the stoichiometric MgAl2O4 samples there is variation of more than an order of magnitude in the diffusion coefficient at a particular temperature. The diffusion coefficients determined by analysis of the 18O diffusion profiles in stoichiometric spinel single crystals (Reddy and Cooper 1981; Ryerson and McKeegan 1994) are smaller than in the other studies,

5 6 7 8

-22

-21

-20

-19

-18

-17

-16

-15

-14

-13

O; R81; xAl2O3=1.8

O; R81;xAl2O3=3.5

O; A74

O; A74; xAl2O3=1.2

O; O75O; R81

O; A72

O; R94

Mg; L02

Mg; S92

log

D (m

2 /s)

104/T (K)

Mg; L58

O and Mgin spinel

Figure 18. Summary of experimental data for oxygen and magnesium self-diffusion in spinel. Abbreviations: A72 – Ando and Oishi (1972); A74 – Ando and Oishi (1974); O75 – Oishi and Ando (1975); R81 – Reddy and Cooper (1981); R94 – Ryerson and McKeegan (1994); L58 – Lindner and Åkerström (1958); S92 – Sheng et al. (1992); L02 – Liermann and Ganguly (2002).


and in good agreement with each other both in activation enthalpy and in absolute diffusivity. The higher diffusivities obtained using bulk exchange methods might be due to inadequate characterization of the sample geometry, enhanced transport along near-surface defects formed during cleavage or polishing, or by inappropriate assumptions regarding the surface exchange kinetics (Reddy and Cooper 1981).

Oishi and Ando (1975) found that oxygen diffusion in polycrystalline spinel with ~1 µm grain size was four orders of magnitude faster than in single-crystal spinel. When diffusion coefficients were calculated on the basis of the grain size rather than the polycrystalline particle size, good agreement was found with the single crystal data, indicating that diffusion along grain boundaries was extremely rapid.

There is some disagreement between the two studies that examined oxygen diffusion in non-stoichiometric spinel single crystals with excess alumina. Ando and Oishi (1974), in bulk exchange experiments, found little difference in the diffusion coefficients for stoichiometric crystals and those with compositions MgO⋅1.2Al2O3 and MgO⋅2.2Al2O3. Reddy and Cooper (1981), on the other hand, found that oxygen diffusion was significantly faster in spinel with excess alumina. Oxygen diffusion rates in MgO⋅1.8Al2O3 and MgO⋅3.5Al2O3 were found to be a factor of ~4 and ~50 higher, respectively, than in stoichiometric spinel. Excess alumina in spinel is accommodated by the production of vacancies on octahedral Al sites, but it must also suppress the concentration of oxygen vacancies. Hence, if oxygen diffused by a vacancy mechanism one would expect oxygen diffusion in excess-alumina spinel to be slower rather than faster than in stoichiometric spinel. Reddy and Cooper (1981) suggested that oxygen diffusion might instead occur by an interstitial mechanism, or by a mechanism involving cation vacancies, since the concentration of these defects increases with increasing solid solution of alumina. The formation energy for O interstitial defects in spinel has been calculated to be 513 kJ/mol (Ball et al. 2008), although it could be reduced significantly if interstitial oxygen were bound to any of a variety of oppositely charged point defects. For example, binding to AlMg

• , which would be present at high concentration in alumina-rich spinel, is calculated to lower the energy relative to the isolated defects by 167 kJ/mol (Ball et al. 2008). To the authors’ knowledge no information exists on the migration energy for isolated O interstitial defects, or for defect associates involving oxygen interstitials.

Magnesium

Self-diffusion of magnesium in spinel is at least three orders of magnitude faster than self-diffusion of oxygen (Fig. 18). The earliest data on Mg self-diffusion in MgAl2O4 spinel were cited as unpublished results of Lindner by Lindner and Åkerström (1958), and were based on experiments using a thin film of 28Mg deposited onto a polycrystalline aggregate. Sheng et al. (1992) studied Mg self-diffusion in MgAl2O4 spinel single crystals, by examining the exchange of Mg stable isotopes with 25Mg-doped, spinel-saturated melts in the CaO-MgO-Al2O3-SiO2 system. Diffusion profiles were measured using SIMS in spot mode. The Mg self-diffusion coefficients reported by Sheng et al. (1992) are an order of magnitude smaller than those cited by Lindner and Åkerström (1958). It is possible that the faster diffusion rates reported by Lindner and Åkerström (1958) are due to rapid transport along grain boundaries in the polycrystalline samples used, but differences in point defect concentrations due to deviations from stoichiometry or aliovalent impurities might also explain the discrepancy. The spinel crystals used by Sheng et al. (1992) have a reported composition consistent with a slight (~1%) alumina excess.

The temperature dependence of the Mg self-diffusion coefficient determined by Sheng et al. (1992) is quite large – the reported activation enthalpy is 384 kJ/mol. This is much larger than the calculated energy for Mg migration via interstitials or vacancies on the Mg or Al sublattices, ~50, ~50 and ~100 kJ/mol, respectively (Murphy et al. 2009). It is also nearly a


factor of two larger than the activation enthalpy for Mg self-diffusion determined from Fe-Mg interdiffusion experiments (discussed below) by Liermann and Ganguly (2002). Information on cation diffusion mechanisms in spinel is insufficient to draw any firm conclusions, but a possible explanation for the large activation enthalpy determined by Sheng et al. (1992) is that diffusion in their experiments was not in a purely extrinsic regime.

Fe-Mg interdiffusion

Two experimental studies of Fe-Mg interdiffusion in spinel have been performed (Freer and O’Reilly 1980; Liermann and Ganguly 2002), which produced quite different results. Interdiffusion coefficients determined by Freer and O’Reilly (1980) are about two orders of magnitude higher than those determined by Liermann and Ganguly (2002) (Fig. 19). Also, Freer and O’Reilly (1980) found complicated diffusion profiles that appeared to indicate strong compositional dependence of the interdiffusion coefficient, while Liermann and Ganguly (2002) found simple error function diffusion profiles that indicated a negligible influence of iron concentration. The discrepancies between the two studies might be related to differences in oxygen fugacity. Liermann and Ganguly (2002) performed experiments at 2-3 GPa with fO2

controlled by the graphite capsule, while Freer and O’Reilly (1980) performed experiments in evacuated silica glass tubes with uncertain fO2

. The compositional dependence observed by Freer and O’Reilly (1980) might be explained by more oxidizing conditions, similar to Ni-Mg and Mn-Mg interdiffusion in periclase, which are much more strongly concentration-dependent in air than under reducing conditions, as discussed above. Diffusion in the Freer and O’Reilly (1980) experiments may also have been enhanced by diffusion along extended defects formed during coarse polishing with 600-grit carborundum, or by mineral surface or vapor phase transport within the incompletely sintered polycrystalline pellets used.

The self-diffusion coefficients for Fe2+ and Mg2+ determined by Liermann and Ganguly (2002) from numerical modeling of their diffusion profiles are close to the Mg self-diffusion coefficients determined by Sheng et al. (1992) in MgAl2O4 spinel, within the temperature range

4 5 6 7 8 9-18

-17

-16

-15

-14

-13

-12

-11

Cr0.8 -Al; S08

Cr0 -Al; S08

Cr; S85

H; F02

Al-Mg; Z96

Mg; S92

Al-Mg; W02

H; O09

Fe-Mg 25 wt%

Fe-Mg 5 wt%; F80

Mg; L58Mg; L02

log

D (m

2 /s)

104/T (K)

Fe; L02

Cations in spinel

Figure 19. Summary of experimental data for cation diffusion in spinel. Abbreviations: L58 – Lindner and Åkerström (1958); S92 – Sheng et al. (1992); L02 – Liermann and Ganguly (2002); S85 – Stubican et al. (1985); O09 – Okuyama et al. (2009); F02 – Fukatsu et al. (2002); Z96 – Zhang et al. (1996); W02 – Watson and Price (2002); S08 – Suzuki et al. (2008); F80 – Freer and O’Reilly (1980).


where the two studies overlap, but the temperature dependence is nearly a factor of two lower than that reported by Sheng et al. (1992), and similar to the activation energies determined for divalent cation diffusion in periclase.

Suzuki et al. (2008) reported Fe-Mg interdiffusion coefficients from two experiments using MgAl2O4/chromite spinel diffusion couples, in which Cr-Al interdiffusion was negligible. Like Liermann and Ganguly (2002), they found that the Fe-Mg interdiffusion coefficient was independent of the Mg# (where Mg# = Mg/(Mg+Fe2+)). However, the Fe-Mg interdiffusion coefficient was found to depend strongly on the Cr# (where Cr# = Cr/(Cr+Al)), increasing by an order of magnitude as Cr# increased from 0 to 0.9. The interdiffusion coefficients determined by Liermann and Ganguly (2002) are similar to those determined by Suzuki et al. (2008) when extrapolated to the same experimental conditions.

Mg-Al interdiffusion

The kinetics of spinel growth due to the interdiffusion between MgO and Al2O3 has been examined experimentally by several authors (Whitney and Stubican 1971a; Zhang et al. 1996; Watson and Price 2002; van Westrenen et al. 2003). Zhang et al. (1996) and Watson and Price (2002) inferred Mg-Al interdiffusion coefficients by applying simple models to the Al and Mg concentration profiles measured across the spinel layer grown between MgO and Al2O3 dur-ing each experiment. A critical assumption of these models is that the spinel layer grows by counterdiffusion of Mg2+ and Al3+, with oxygen being effectively immobile. Although oxygen diffusion in spinel single crystals is quite slow, as discussed above, van Westrenen et al. (2003) demonstrated substantial 18O mobility across a polycrystalline spinel layer grown at 15 GPa and 2273 K. This indicates that counterdiffusion of oxygen and cations is likely to contribute to spinel growth, at least at high pressure, and suggests caution in interpreting Al and Mg concen-tration profiles in terms of Al-Mg interdiffusion alone.

The Al-Mg interdiffusion coefficients inferred from spinel growth experiments are larger than Mg self-diffusion coefficients (Fig. 19), and also larger than Al self-diffusion coefficients inferred from Cr-Al interdiffusion experiments, which are discussed below. The spinel grown in the Al2O3-MgO interdiffusion experiments is fine-grained and deviates significantly from stoi-chiometry near the boundary with corundum, especially at high temperatures. It is thus possible that Al-Mg interdiffusion in these experiments was enhanced by grain boundary transport or by cation vacancies generated to compensate excess alumina. However, Watson and Price (2002) noted no dependence of Al-Mg interdiffusion rates on time, despite significant coarsening of the spinel, and Zhang et al. (1996) found only a slight dependence of the interdiffusion coef-ficient on composition in their experiments, with the diffusion coefficient appearing to decrease slightly with excess alumina.

Redfern et al. (1999) studied the kinetics of Mg-Al cation ordering in MgAl2O4 spinel as a function of temperature. The order-disorder process involves short-range diffusion of Mg and Al as these two cations are exchanged between tetrahedral and octahedral sites, but it is not clear whether the mechanism is the same as that involved in Mg-Al interdiffusion. Nonetheless, the activation enthalpy for order-disorder determined by Redfern et al. (1999), 230 kJ/mol, is similar to the values determined for Mg-Al interdiffusion.

Cr-Al interdiffusion

Suzuki et al. (2008) studied interdiffusion between natural MgAl2O4 spinel and chromite spinel crystals at pressures of 3 to 7 GPa and temperatures between 1400 and 1700 °C. The inter-diffusion coefficients were found to be a strong function of composition, increasing by an order of magnitude as the atomic fraction Cr/(Cr+Al) = xCr increased from 0.1 to 0.8. This composi-tional dependence was represented as DCr-Al ∝ exp(10.3xCr

0.54). Modeling of the interdiffusion coefficients in terms of the Cr and Al self-diffusion coefficients indicated that diffusion of Al is more than an order of magnitude faster than Cr. The temperature dependence was found to be


fairly large, with an activation enthalpy of 520 kJ/mol at 3 GPa, and independent of composition. The pressure dependence was found to be small, with an activation volume of 1.36 cm3/mol.

An Arrhenius plot for tracer diffusion of 51Cr in single-crystal MgAl2O4 spinel was reported by Stubican et al. (1985), cited as the unpublished data of Osenbach and Stubican. These values can be compared to the Cr-Al interdiffusion coefficients determined by Suzuki et al. (2008), extrapolated to pure MgAl2O4 spinel (the Cr-Al interdiffusion coefficient approaches the Cr self-diffusion coefficient as the Cr concentration approaches zero). At the highest temperature reported by Stubican et al. (1985), 1600 °C, the 51Cr diffusion coefficient is similar to that de-termined by Suzuki et al. (2008), but at lower temperatures the Stubican et al. (1985) values are significantly higher. Stubican et al. (1985) report an activation enthalpy of 337 kJ/mol, which is only ~60% of the value determined by Suzuki et al. (2008).

Hydrogen

The diffusion coefficients of hydrogen and/or deuterium in magnesium aluminate spinels have been determined from vapor-exchange experiments, based on changes with time in the integrated infrared absorbance of a single-crystal spinel plate held within a given atmosphere for a specified time. The results of these studies are in good agreement, and demonstrate that hydrogen diffusion is at least an order of magnitude faster than any other species in spinel (Fig. 19). González et al. (1987) presented a single measurement of the deuterium diffusivity in a non-stoichiometric spinel crystal with Al/Mg ratio of ~7, at 1600 K, giving a value of 6×10−12 m2/s. Fukatsu et al. (2002) determined hydrogen diffusion coefficients in Mg1−xAl2O4−x spinel, with x = 0, 0.1, 0.2 and 0.3. The diffusion coefficients determined in experiments in which ini-tially “dry” samples were annealed in a “wet” atmosphere were found to be indistinguishable from those determined from outgassing experiments in which samples pre-saturated with water were annealed in a completely dry atmosphere. A slight dependence of the hydrogen diffusivity on composition was noted, with the diffusivity in stoichiometric samples ~50% higher than in samples with x = 0.3. Fukatsu et al. (2002) also measured the electrical conductivity of their samples and found that it was too high to be explained in terms of proton conduction. Cation vacancies (either Mg or Al) were instead inferred to be the predominant charge carriers. The activation enthalpy for electrical conduction was found to be 248 kJ/mol, within the range of values for cation diffusion discussed above.

Okuyama et al. (2009) examined the exchange of deuterons and protons in experiments in which a plate of spinel was pre-equilibrated in either an H2O or D2O atmosphere, and then annealed in an atmosphere containing the other isotope, at the same water vapor pressure. Their results are similar to those of Fukatsu et al. (2002) for bulk chemical diffusion of water, and in addition demonstrate that the H/D interdiffusion coefficient is independent of the partial pres-sure of water vapor.

MAGNETITE

Magnetite is a member of the spinel group and has the ideal formula Fe3O4. It is a common mineral in igneous and metamorphic rocks, and has been studied extensively in connection with oxygen isotope thermometry. Magnetite is also an important industrial mineral, as one of the primary phases formed during the oxidation of iron. Much of the early work on diffusion in magnetite was motivated by the desire to understand the oxidation kinetics in detail. In contrast to spinel proper, in which there is a high degree of ordering of Mg2+ on the tetrahedral sites and Al3+ on the octahedral sites, magnetite is an “inverse” spinel at low temperatures, with Fe2+ and Fe3+ occupying the octahedral sites in equal proportions, and Fe3+ being the sole occupant of the tetrahedral sites (Verwey et al. 1947; Shull et al. 1951). At higher temperatures electron interchange leads to an apparently random distribution of Fe2+ and Fe3+ among the octahedral and tetrahedral sites (Dieckmann and Schmalzried 1977b).


Magnetite is stable relative to wüstite (Fe1−xO) and hematite (Fe2O3) only within a restricted range of oxygen fugacity. Under most conditions magnetite has a slight cation deficit, accommodated by vacancies on the normally occupied octahedral and/or tetrahedral cation sites. However, at low oxygen fugacities, where magnetite is at or near equilibrium with wüstite, magnetite has a slight excess of cations as interstitial defects (Dieckmann 1982). The dependence of point defect concentrations on fO2

provides a convenient route to obtain information on diffusion mechanisms, which has been put to good use in studies of cation diffusion and, to a lesser extent, oxygen diffusion. As a result, cation diffusion mechanisms in magnetite are much better understood than they are in spinel.

Magnetite and ulvöspinel (Fe2TiO4) form a continuous solid solution series, and a large number of studies have been performed on point defects and cation diffusion in titanomagnetite. Although titanium may exist as both Ti4+ and Ti3+, it appears that it is primarily tetravalent in titanomagnetite and that the redox equilibria in the solid solution are controlled primarily by iron (Aggarwal and Dieckmann 2002a).

A compilation of experimental data on diffusion in magnetite is provided in Tables 9-12 (see Appendix).

Oxygen

Diffusion of oxygen in magnetite has been studied by various groups over a wide temperature range, from 251 °C to 1150 °C. Measurements of diffusivity extending to such low temperatures are possible because oxygen diffusion in magnetite is quite fast—several orders of magnitude faster than in spinel, which has the same basic structure and also has nearly closest-packed oxygen atoms. Castle and Surman (1967) examined the diffusive exchange of 18O between water vapor and magnetite crystals at low temperatures (251-550 °C) by continuously sampling and analyzing the isotopic composition of the vapor from the diffusion chamber. By using small magnetite particles (<0.2 µm) and keeping the gas volume small relative to the volume of magnetite, they were able to measure diffusion coefficients as low as 10−24 m2/s. Castle and Surman (1969) used the same technique to determine oxygen diffusion coefficients as a function of oxygen fugacity, using gas mixtures with different H2/H2O ratios. They found that the oxygen diffusion coefficient increased under reducing conditions (i.e., at higher H2/H2O), where there is an excess of cations in magnetite and a corresponding increase in oxygen vacancies; this suggests that oxygen diffuses by a vacancy mechanism. They found that the diffusion coefficient was proportional to fO2

0 27− . , and that the activation enthalpy (~71 kJ/mol) was insensitive to the oxygen fugacity.

Giletti and Hess (1988) performed experiments between 500 and 800 °C in which 18O was exchanged between water and magnetite at different water pressures and oxygen fugacities. The oxygen diffusion coefficients were determined from fits to the oxygen isotopic profiles measured in the quenched magnetite samples by SIMS depth profiling. The Giletti and Hess (1988) results are in good agreement with those of Castle and Surman (1967, 1969) where the studies overlap in temperature, but have a much stronger dependence on temperature (Fig. 20), with an activation enthalpy more than twice as large (188 kJ/mol). Giletti and Hess (1988) found that oxygen diffusivity was insensitive to water pressure or to fO2

within the precision of their measurements, and suggested a change in oxygen diffusion mechanism at ~500 °C, between the temperatures of their experiments and those of Castle and Surman (1967, 1969).

Crouch and Robertson (1990) inferred oxygen diffusion coefficients from steady-state creep experiments in both the power law (dislocation) creep regime and Nabarro-Herring (volume diffusion) creep regime, in “dry” CO2/CO atmospheres. The oxygen diffusivities they determined are similar to those measured by Giletti and Hess (1988) and Castle and Surman (1967, 1969), but have a stronger temperature dependence, with an activation enthalpy of 264 kJ/mol. They found that the oxygen diffusion coefficient varied as fO2

0 56− . , i.e., in the same direction


found by Castle and Surman (1969) under hydrothermal conditions at lower temperatures, but with an exponent about twice as large.

Millot and Niu (1997) and Millot et al. (1997) studied oxygen diffusion under a broad range of fO2 within the stability field of magnetite at 1150 and 800 °C, respectively. Similarly to Castle and Surman (1969) and Crouch and Robertson (1990) they found that oxygen diffusivity decreased with increasing oxygen fugacity, with DO ∝ fO2

−1 2/ , under relatively reducing conditions. However, at both temperatures they found a transition to a different regime, with DO ∝ fO2

1 6/ , under more oxidizing conditions (Fig. 21). Oxygen diffusion in these two regimes was interpreted as being due to oxygen vacancies under reducing conditions, and to bound oxygen-cation vacancy pairs under more oxidizing conditions (similar to the diffusion mechanism inferred for oxygen in MgO).

Sharp (1991) inferred temperature-dependent oxygen diffusion coefficients in magnetite based on the variation in oxygen isotopic abundances preserved in natural magnetite crystals of different size in a calcite marble from the Bancroft terrane of the Ontario Grenville province. The activation enthalpy of 211 kJ/mol obtained from these data lies between the values deter-mined by Giletti and Hess (1988) and Crouch and Robertson (1990), but the inferred diffusion coefficients are about two orders of magnitude smaller. The absolute diffusivities inferred from the natural samples using Sharp’s method depend on knowledge of the cooling rate at tem-peratures near the closure temperature for magnetite-calcite exchange. It is possible that the assumed cooling rate was underestimated; however, to bring the results of Sharp (1991) into agreement with those of Giletti and Hess (1988) would require an increase in assumed cooling rate of two orders of magnitude. Sharp (1991) suggested that the slower diffusivities inferred from the Bancroft marbles, relative to those found by Giletti and Hess (1988), could be attrib-uted to the absence of fluid during metamorphism of the natural samples. However, this would not explain why the oxygen diffusivity values determined by Sharp (1991) are also slower than those inferred from the “dry” creep experiments of Crouch and Robertson (Crouch and Rob-ertson 1990). The experimental studies on oxygen diffusion in magnetite have focused on the nominally pure iron oxide end-member, while the composition of the natural magnetite crystals studied by Sharp (1991) was not reported. It is possible that aliovalent impurities in the natural crystals suppressed oxygen vacancy concentrations, thus lowering the diffusivity.

8 10 12 14 16-25

-24

-23

-22

-21

-20

-19

-18

-17

-16

-15

C & S, 1969; H2 /H

2 O=10 -3

C & S, 1969; H2 /H

2 O=1.2Sharp, 1991

Giletti & Hess, 1988

Crouch & Robertson, 1990

lo

g D

(m2 /s

)

104/T (K)

O in magnetite

Castle & Surman, 1967

Figure 20. Summary of experimental data on the temperature dependence oxygen diffusion in mag-netite.


Iron

Tracer diffusion of iron has been studied extensively both in pure magnetite and in tit-anomagnetites. Himmel et al. (1953) studied 59Fe tracer diffusion in fine-grained magnetite polycrystals prepared by oxidizing pure iron. The polycrystals used were nearly stoichiometric, with a slight cation deficit, and the diffusion anneals were performed in an Ar/water vapor at-mosphere. Schmalzried (1962) also studied 59Fe tracer diffusion in fine-grained polycrystalline magnetite, with varying degrees of sintering, as a function of oxygen fugacity. Schmalzried (1962) found that the Fe tracer diffusion coefficient increased with oxygen fugacity, suggesting a vacancy mechanism. Dieckmann and Schmalzried (1977b) later pointed out the importance of studying single-crystal or fully dense coarse-grained polycrystalline samples—they showed that cation vacancy concentrations increase significantly in fine-grained, poorly sintered magnetite. Nonetheless, the study of Schmalzried (1962) clearly showed the importance of oxygen fugac-ity as a control on point defect populations and cation self-diffusion in magnetite. Another early study of 59Fe tracer diffusion by Ogawa et al. (1968) used polycrystalline magnetite sintered at 1450 °C in pure O2, and performed diffusion anneals in a sealed quartz tube. The oxygen fugacities in these experiments are not known, and it is therefore not possible to place these results in the appropriate context for rigorous comparison to more recent results performed in controlled atmospheres. Results on Fe self-diffusion in magnetite as a function of temperature are compared in Figure 22, which includes results from studies in uncontrolled atmospheres, and later results along the magnetite-wüstite and magnetite-hematite oxygen buffers.

A large number of later studies on dense, large-grained magnetite utilizing different techniques have shown conclusively that diffusion of iron occurs by two different mechanisms in magnetite. At low oxygen fugacities, where magnetite is stoichiometric or has a cation excess, the predominant point defects on the cation sublattices are Frenkel pairs, in which an iron cation moves from its normally occupied octahedral or tetrahedral lattice site into an interstitial site. In this regime iron diffuses either by a direct interstitial mechanism, in which cations jump only along interstitial sites, or by a more complicated interstitialcy mechanism in which atoms travel by jumping between cation and interstitial sites. As oxygen fugacity increases, iron interstitials decrease in abundance, leading to a decrease in the diffusivity of iron by an interstitial mechanism. Meanwhile vacancies form on the cation sublattices and, under

-22 -20 -18 -16 -14 -12 -10 -8 -6 -4-23

-22

-21

-20

-19

-18

-17

-16

-15

-14

773 K

1073 K

Castle & Surman, 1969

Millot et al., 1997

lo

g D

(m2 /s

)

log fO2

O in magnetitefO2 dependence

Millot & Niu, 1997

1423 K

Figure 21. Summary of experimental data on the fO2

dependence of oxygen diffusion in magnetite.


sufficiently oxidizing conditions, diffusion by a vacancy mechanism becomes predominant. The change in point defect populations with fO2

leads to a concave-upward relationship between DFe and fO2

, with a minimum in DFe at intermediate oxygen fugacity (Fig. 23). This relationship between oxygen fugacity, point defects and iron diffusivity was first elucidated by Dieckmann and Schmalzried (1977a,b), who measured Fe tracer diffusion coefficients over a broad range of fO2

and temperature within the stability field of magnetite. The finding was later supported by thermogravimetric studies of point defects in magnetite (Dieckmann 1982); studies of the diffusion mechanism inferred from the isotope effect for Fe diffusion (Peterson et al. 1980) and the kinetics of point defect relaxation (Dieckmann and Schmalzried 1986); and Mössbauer spectroscopy studies of the atomic jumps involved in Fe diffusion (Becker et al. 1990, 1993). Iron tracer diffusion experiments at a much lower temperature (500 °C) are also consistent with the picture described above, wherein diffusion occurs by a mechanism involving interstitials at low oxygen fugacities, and by vacancies at higher oxygen fugacities (Atkinson et al. 1983).

Point defect model. The dependence of iron diffusivity on oxygen fugacity can be rationalized in terms of a simple point defect model. As pointed out by Dieckmann (1998), the point defect equilibria can be written in a simple form without considering the two distinct cation sites in the magnetite structure. The formation of cation vacancies (on octahedral and/or tetrahedral sites) is described by:

32

32

1

382

23

3 4Fe O Fe Fe OFe+ ++ ↔ + +V ( )

and the formation of Fe2+ or Fe3+ interstitials is described by:

Fe Fen

I InV Fe V+ ++ ↔ + ( )9

where VI represents a vacant interstitial site, and VFe a vacant cation site.

6 7 8 9 10-16

-15

-14

-13

-12

-11

Dieckmann & Schmalzried, 1977a; minima

Ogawa et al., 1968Dieckmann & Schmalzried, 1977a; MW buffer

Dieckmann & Schmalzried, 1977a; MH buffer

Himmel, 1953

log

D (m

2 /s)

104/T (K)

Izvekov, 1958Fe in magnetite

Figure 22. Summary of experimental data on the temperature dependence of iron diffusion in magnetite. The experiments of Himmel (1953), Izbekov and Gorbunova (1959), and Ogawa et al. (1968) were conducted in unknown atmospheres. Dieckmann and Schmalzried (1977a) studied Fe diffusion under a wide range of fO2

; the results are shown at conditions along the wüstite-magnetite and magnetite-hematite buffers, and at intermediate conditions where the Fe diffusivity is at a minimum (see Fig. 23).


Considering the law of mass action for reaction 8 yields the following expression for the concentration of iron vacancies:

[ ][ ]

[ ]( )

/

//V K

a

a

KaV

VFe

O

Fe OO

Fe

Fe= ≈

+

+

2 3

3 2

2 3

1 32 32

3 4

2410

where KV is the equilibrium constant for Equation (8) and the oxygen activity with fO2

=1 atm. The last equality in Equation (10) arises because in pure magnetite aFe3O4

≈ 1, and [Fe2+] ≈ 1; it is only approximate because under oxidizing conditions, near equilibrium with hematite, deviations from stoichiometry become detectable and a small correction factor, also involving the oxygen activity, is needed (Dieckmann 1982). A similar expression can be written for the concentration of iron interstitials, based on the mass action law for Equation (9):

[ ] ( )/Fe O2In

IK a+ −= 4 112 3

where KI is the equilibrium constant for Equation (9). Based on Equations (10) and (11) one expects the concentration of iron vacancies to increase and the concentration of iron interstitials to decrease with increasing oxygen fugacity, with exponents of 2/3 and −2/3, respectively. This relationship has been demonstrated experimentally in thermogravimetric studies (Dieckmann 1982; Dieckmann and Schmalzried 1986). Within the simple point defect framework described above the tracer diffusion coefficient for iron can be expressed as a sum of the diffusivities by interstitial and vacancy mechanisms. Dieckmann and Schmalzried (1977a) fit their tracer diffusion data to the following equation:

D DH

RTa D

H

RTaV

VI

IFe O O2 2

∗ −= −

+ −

[ ]/

[ ]/exp exp (0 2 3 0 2 3 112)

with an activation enthalpy of −140 kJ/mol for diffusion by a vacancy mechanism and 613 kJ/mol for diffusion by an interstitial (or interstitialcy) mechanism. Note that the activation enthalpy

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1-16

-15

-14

-13

-12

-11

-10

1673 K1573 K1473 K1373 K1273 K

log

D (m

2 /s)

log fO2

(atm)

1173 K

Fe in magnetite

Figure 23. Variation with fO2 of the Fe diffusion coefficient in magnetite, at different temperatures. Based

on experimental data and equations given by Dieckmann and Schmalzried (Dieckmann and Schmalzried 1977a). Iron is thought to diffuse by a mechanism involving interstitials (likely an interstitialcy mechanism) under highly reducing conditions, and by a vacancy mechanism under highly oxidizing conditions.


for iron diffusion by a vacancy mechanism is negative, which indicates that the diffusivity by this mechanism decreases as the temperature increases (Fig. 23). This counterintuitive behavior arises because the formation enthalpy for cation vacancies in magnetite is strongly negative (Dieckmann 1982), and has a larger absolute value than the positive migration enthalpy (Dieckmann et al. 1987). With increasing temperature, the concentration of cation vacancies decreases enough to inhibit diffusion by a vacancy mechanism, even though each vacancy is of course more mobile at higher temperature. For interstitials, on the other hand, both the formation and migration enthalpies are strongly positive (Dieckmann et al. 1987); both the concentration and mobility of interstitials increase with temperature, and diffusion by an interstitial(cy) mechanism therefore increases steeply with increasing temperature.

Dieckmann and Schmalzried (1986) inferred, based on point defect relaxation experiments, that in the low-fO2

regime where Frenkel defects predominate there are actually two cation diffusion mechanisms involving interstitials, each with a different temperature dependence, and each having a correlation factor consistent with an interstitialcy mechanism. The reader is referred to that paper for a refinement to Equation (12) that provides a slightly more accurate description of the tracer diffusion of Fe. For a more detailed description of diffusion mechanisms in magnetite see Peterson et al. (1980), Dieckmann et al. (1987), and Becker et al. (1990, 1993).

The influence of titanium. Iron tracer diffusion and point defect equilibria have also been examined extensively in magnetite-ulvöspinel solid solutions, (TixFe1−x)3−δO4 (Aragón and McCallister 1982; Aggarwal and Dieckmann 2002a,b). The variation of DFe with fO2

is similar to that in pure magnetite. Adding Ti to the solid solution increases the concentration of cation vacancies, and decreases the concentration of cation interstitials (Aggarwal and Dieckmann 2002a), thus enhancing the diffusion of Fe by a vacancy mechanism while suppressing diffusion by an interstitial mechanism (Aggarwal and Dieckmann 2002b). The addition of 30 at.% Ti on cation sites increases the diffusivity of Fe by a vacancy mechanism by a factor of ~1000 relative to pure magnetite, and decreases the diffusivity of Fe by an interstitial mechanism by a factor of ~20. On a plot of DFe vs. fO2

, like that shown in Figure 23, the minimum in the curve shifts to lower fO2

and to higher DFe with increasing Ti concentration (Aggarwal and Dieckmann 2002b).

Cation vacancy diffusivity. The rates of diffusion of iron and other cations by a vacancy mechanism depend on the concentration and diffusivity of cation vacancies. Cation vacancy diffusion coefficients in magnetite have been determined by Nakamura et al. (1978), Yamauchi et al. (1983) and Dieckmann and Schmalzried (1986), from experiments on the kinetics of relaxation of point defects following a step-wise change in the oxygen fugacity. The vacancy diffusion coefficients determined in these studies agree within about a factor of two. Based on a detailed study of the Mössbauer spectra in magnetite at different temperatures and oxygen fugacities, Becker et al. (1990, 1993) inferred that vacancy diffusion in magnetite involves the octahedral cation sublattice almost exclusively. The experiments of Aggarwal and Dieckmann (2002a,b) demonstrate that the addition of Ti in solid solution has only a small influence on the mobility of cation vacancies—the enhancement of cation diffusivity by a vacancy mechanism when Ti is present is due almost entirely to the increased concentration of cation vacancies.

Other cations

In pure magnetite, tracer diffusion coefficients have been determined experimentally for Co (Dieckmann et al. 1978), Cr (Dieckmann et al. 1978; Hodge 1978), Mn and Ti (Aggarwal and Dieckmann 2002b). Dieckmann et al. (1987) also re-analyzed data from a Ph.D. dissertation (Petuskey 1977) on interdiffusion of Al3+, extrapolated to pure magnetite. Tracer diffusion coefficients have also been measured for Co, Mn and Ti in titanomagnetites (Aggarwal and Dieckmann 2002b). Expressions for the diffusion of these cations as functions of temperature and oxygen fugacity are given in Table 11 (see Appendix). All of these cations


behave similarly to Fe in their response to oxygen fugacity, with diffusivity varying as under reducing conditions and as under more oxidizing conditions, consistent with interstitial(cy) and vacancy diffusion mechanisms, respectively (Fig. 24). The diffusivities of Co and Mn are similar to Fe under all conditions, while Ti4+, Al3+ and Cr3+ are substantially slower. Figure 25 shows Arrhenius plots of cation diffusion coefficients in magnetite at oxygen fugacities along the wüstite-magnetite (Fig. 25a) and magnetite-hematite buffers (Fig. 25b). Along the wüstite-magnetite buffer, which is relatively reducing, diffusion by interstitial(cy) mechanisms tends be predominant. Cation diffusion is generally slower in titanomagnetite than in magnetite under these conditions (Fig. 25a), because interstitials are suppressed in the presence of Ti4+. Along the more oxidizing magnetite-hematite buffer, where vacancy mechanisms predominate, cation diffusion in titanomagnetite is generally much faster than in pure magnetite (Fig. 25b).

Dieckmann et al. (1987) re-analyzed data from Grman and Jesenak (1978) and Eveno and Paulus (1974) on Ni diffusion in magnetite with substantial Ni in solid solution, in regimes interpreted to be dominated by interstitials and vacancies, respectively. The dependence of Ni diffusivity on fO2

is broadly similar to that of the other cations. However, Ni diffuses signifi-cantly faster than Fe, Mn and Co under reducing conditions where the interstitialcy mechanism dominates, and significantly slower than Fe, Mn and Co under oxidizing conditions where a vacancy mechanism dominates.

Fe-Ti interdiffusion. Several studies have been conducted on interdiffusion in magnetite-ulvöspinel (Fe3O4-Fe2TiO4) solid solutions. Petersen (1970) determined average interdiffusion coefficients by examining the rate of development of exsolution lamellae in natural titanomagnetite crystals, after heating in air. Price (1981) performed experiments that were based on a similar concept, but in this case determining average interdiffusion coefficients by examining the rate of homogenization of microstructures in natural titanomagnetite crystals. The homogenization experiments were conducted under vacuum in sealed silica glass tubes, so that the oxygen fugacity was constrained only to be within the regime of stability of the crystals used. Freer and Hauptman (1978) and Aragon et al. (1984) performed interdiffusion experiments using synthetic magnetite and titanomagnetite, with the diffusion profiles measured using an electron microprobe and diffusion coefficients determined as functions of composition using Boltzmann-Matano analysis. Aragon et al. (1984) employed various binary solid-state

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��

��

��

��

�

log

D (m

2 /s)

�� f��

�

�

��

Figure 24. Variation with fO2

of cation diffusion coef-ficients in magnetite. Fe, Co, Cr, Al, and Ni were reported by Dieckmann and Schmalz-ried (1987). Mn and Ti were reported by Aggarwal and Di-eckmann (2002b).


buffering assemblages and CO-CO2 gas mixtures to control the oxygen fugacity, while most of the Freer and Hauptman (1978) experiments were performed under vacuum in sealed silica glass tubes. The results of these studies are in order of magnitude agreement with each other, and with Fe and Ti tracer diffusion coefficients in magnetite and titanomagnetite (Fig. 26). Both Freer and Hauptman (1978) and Aragon et al. (1984) found the Fe-Ti interdiffusion coefficient to be proportional to exp(βxTi), with broadly similar values of β: 13.3 and 15.1, respectively. Aragon et al. (1984) found a positive relationship between DTi and fO2

with concave downward curvature, and inferred that Ti diffusion occurred by a vacancy mechanism over the full range of fO2

in which magnetite is stable. This result is inconsistent with the direct tracer diffusion measurements of Aggarwal and Dieckmann (2002a,b) who found clear evidence for Ti diffusion by both interstitial and vacancy mechanisms, in both pure magnetite and in titanomagnetites with various concentrations of Ti. Along the quartz-fayalite-magnetite oxygen buffer, the Fe-Ti interdiffusion coefficients determined by Aragon et al. (1984) are an order of magnitude smaller than Ti tracer diffusion coefficients determined by Aggarwal and Dieckmann (2002b), at the same temperature and composition, and two orders of magnitude smaller than Fe tracer diffusion coefficients.

RUTILE

Rutile is a common accessory mineral in igneous and metamorphic rocks. It is also a semi-conductor at high temperatures, a photocatalyst, and a potential photovoltaic material, and these applications have provided additional motivation for studies of its diffusion properties. Rutile has the ideal formula TiO2, but at high temperatures and/or low oxygen fugacities it exhibits significant non-stoichiometry, being represented by the formula TiO2−x, with x taking values up to ~0.008 depending on temperature and fO2

. The oxygen deficiency in non-stoichiometric rutile

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Figure 25. Cation diffusion in magnetite and titanomagnetite at the (a) wüstite-magnetite and (b) magnetite-hematite buffer. Diffusion rates in titanomagnetite are broadly similar to those in pure magnetite under reducing conditions, while under oxidizing conditions—where the vacancy mechanism dominates, and vacancy concentrations are enhanced by the presence of Ti4+—diffusion rates in titanomagnetite are orders of magnitude faster.


appears to be accommodated both by oxygen vacancies and by Ti interstitials, with oxygen va-cancies dominating at high temperatures and relatively oxidizing conditions, and Ti interstitials at lower temperatures and more reducing conditions (Kofstad 1967; Marucco et al. 1981). As discussed below, non-stoichiometry in rutile has an important influence on diffusion of both oxygen and cations, with oxygen vacancies promoting oxygen diffusion, and Ti interstitials promoting the diffusion of titanium and several other cations. Natural rutile also may contain significant divalent and trivalent cation impurities, which may be charge-balanced by oxygen vacancies and/or cation interstitials, and pentavalent cation impurities (Nb5+ and Ta5+) that may be charge-balanced by cation vacancies.

Unlike the other minerals considered in this chapter, which are isometric, rutile is tetragonal and hence has anisotropic diffusion properties. In particular, some small monovalent, divalent and trivalent cations exhibit strong anisotropy, with diffusion parallel to the c-axis several orders of magnitude faster than perpendicular to the c-axis. These cations appear to diffuse as interstitials along channels with low electron density that run parallel to the c-axis in rutile (e.g., Sasaki et al. 1985). A compilation of experimental diffusion data for rutile is provided in Tables 13-16 (see Appendix).

Oxygen

Oxygen diffusion in rutile has been studied by several groups, under a wide range of conditions and using a variety of experimental and analytical techniques. A summary of the data is shown as an Arrhenius plot in Figure 27. There is a general consensus that oxygen diffusion in rutile occurs by a vacancy mechanism. Haul and Dümbgen (1965) presented an extensive set of measurements of oxygen diffusivity at 710-1300 °C using bulk exchange techniques with a gaseous 18O tracer, finding that the results for diffusion parallel to the c-axis could be described by a single Arrhenius equation. No dependence of oxygen diffusivity on fO2

was found, between 10−6 and 1 bar, and thus oxygen diffusion in these experiments was inferred to be in an extrinsic regime, with oxygen vacancies present mainly to charge-compensate Al3+ impurities. Arita et

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xTi =0.2

Fe-Ti interdiffusion cation tracer

xTi=0.2

TixTi =0

Fe-Ti; P81

Fe-Ti; F78; xTi =0.15

log

D (m

2 /s)

104/T (K)

Fe-Ti; A84

Fe-Ti interdiffusionin magnetite

xTi =0

Fe

xTi =0.15

Figure 26. Arrhenius plot showing experimental results on iron-titanium interdiffusion in magnetite. The results from Freer and Hauptman (1978) and Aragon et al. (1984) are extrapolated to xTi = 0.15, similar to the bulk composition of the crystals used for homogenization experiments by Price (1981). Iron and tita-nium tracer diffusion coefficients (Aggarwal and Dieckmann 2002b) are also shown, at xTi = 0 and xTi = 0.2.


al. (1979) later performed experiments using both “pure” and Cr3+-doped rutile, under relatively oxidizing conditions, finding significantly faster oxygen diffusion in the Cr3+-doped crystal. These results are also consistent with diffusion by a vacancy mechanism in the extrinsic regime. In a later complementary study, Millot and Picard (1988) performed experiments on nominally pure crystals under conditions where rutile has significant non-stoichiometry (high temperature, reducing conditions). These authors found oxygen diffusivity to be a linear function of the oxygen deficit, indicating that under these conditions the predominant oxygen vacancies are formed to compensate Ti3+ (Fig. 28).

Considering the wide scatter that often exists in experimental studies of oxygen diffusion in minerals, there is surprisingly good agreement among studies of oxygen diffusion in rutile. The

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O; M88; 1373 K

Ti; A78; 1373 K

Ti; A78; 1331 KTi; A78; 1273 K

Ti; H85; 1773 K

Ti; H85; 1673 K

Ti; H85; 1573 KTi; H85; 1473 K

Ti; H85; 1373 K

log

D (m

2 /s)

log fO2

(atm)

Ti; H85; 1273 K

Influence of fO2

on diffusionof Ti and O in rutile

Figure 28. Variation with fO2

of Ti and O diffusion coefficients in rutile. As the oxygen fugacity decreases, rutile becomes increasingly deficient in oxygen, with the deficiency compensated both by oxygen vacancies and by Ti interstitials. The increase in diffusivity with increasing degree of non-stoichiometry indicates that Ti diffuses by an interstitial mechanism, and that oxy-gen diffuses by a vacancy mechanism. Abbreviations: H85 – Hoshino et al. (1985); A78 – Akse and Whitehurst (1978); M88 – Millot and Picard (1988).

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M98; "slower"

O in rutile hydrothermal, || C-axis hydrothermal, ⊥ C-axis dry, || C-axis

A79; doped 0.08 mol% Cr2O3

A79

M98; "slower"

H65

D93; "synthetic"

log

D (m

2 /s)

104/T (K)

D93; "natural"

D81

M98; "faster"

Figure 27. Summary of ex-perimental results on the tem-perature dependence of oxy-gen diffusion in rutile, under dry conditions in air, and un-der hydrothermal conditions. Although the presence of wa-ter is generally considered to enhance the rates of diffusion-controlled geological process-es, it appears to inhibit diffu-sion of oxygen in rutile. Ab-breviations: A79 – Arita et al. (1979); H65 – Haul and Düm-bgen (1965); D81 – Derry et al. (1981); D93 – Dennis and Freer (1993); M98 – Moore et al. (1998).


three experimental studies that used nearly pure, stoichiometric crystals under “dry” conditions (Haul and Dümbgen 1965; Arita et al. 1979; Derry et al. 1981) found diffusion coefficients for oxygen parallel to the c-axis that differed by less than a factor of three at the same temperatures (Fig. 27). Much wider scatter in diffusion coefficients may be expected in natural rutile samples, which may have much higher concentrations of impurities than the synthetic crystals used in these studies. Samples that contain predominantly trivalent impurities like Cr3+, Al3+ and Fe3+ would be expected to have higher diffusivities, and those that contain high abundances of Nb5+ and/or Ta5+ may have lower diffusivities because these positively charged impurities suppress oxygen vacancy concentrations, The measured temperature dependence of oxygen diffusivity is also similar among the three studies, with reported activation enthalpies of 251, 251 and 283 kJ/mol, respectively. The diffusion coefficients of Bagshaw and Hyde (1976) at 1000 °C are an order of magnitude higher than in the other three studies, and have relatively large errors, attributed by the authors to an inadequate representation of the sample geometry in the bulk exchange experiments employed. These data are closer to those obtained using Cr3+-doped (Arita et al. 1979) and highly non-stoichiometric (Millot and Picard 1988) rutile samples. It is not clear whether the Bagshaw and Hyde (1976) diffusion coefficients are higher due to lesser sample purity or to experimental errors.

Haul and Dümbgen (1965) found oxygen diffusion in rutile to be slightly anisotropic at 1347 °C, with diffusion perpendicular to the c-axis being ~60% faster than parallel to it. At a lower temperature, 806 °C, Gruenwald and Gordon (1971) found somewhat larger anisotropy (a factor of ~5), with diffusion perpendicular to c again being faster than parallel to c. Moore et al. (1998) found the opposite sense of anisotropy under hydrothermal conditions, with diffusion parallel to the c-axis a factor of ~5 faster than diffusion perpendicular to c, over the temperature range 800-1000 °C.

Influence of H2O. The influence of dissolved water on diffusion in rutile has also been studied, first by Haul and Dümbgen (1965), who examined diffusion in water-saturated air at atmospheric pressure. Interestingly, these authors found oxygen diffusivity to be about a fac-tor of two slower in the presence of water. In many cases, for example in olivine (Costa and Chakraborty 2008), small amounts of dissolved water enhance rather than inhibit oxygen diffu-sion rates. Later, Dennis and Freer (1993) and Moore et al. (1998) studied oxygen diffusion un-der hydrothermal conditions at high pressures (and thus high water fugacities), and reinforced the early result that dissolved water inhibits oxygen diffusion in rutile. Dennis and Freer (1993) found slower diffusivities compared to “dry” experiments, by one to two orders of magnitude depending on the temperature. To explain this they suggested that water may dissolve in rutile as hydroxyl ions on oxygen lattice sites, consuming oxygen vacancies in the process. Moore et al. (1998) found complex diffusion profiles in their hydrothermal experiments, indicating dif-fusion by two mechanisms differing in rate by about an order of magnitude. Both mechanisms were found to operate under hydrous conditions, with the slow mechanism dominant. Under anhydrous conditions the fast mechanism alone was found to operate, with oxygen diffusivity consistent with data from the other studies on pure stoichiometric rutile (Haul and Dümbgen 1965; Arita et al. 1979; Derry et al. 1981).

Tetravalent and pentavalent cations

Titanium. Titanium self-diffusion in rutile has been studied as a function of temperature, oxygen fugacity and crystallographic orientation. Several different groups (Venkatu and Poteat 1970; Lundy and Coghlan 1973; Akse and Whitehurst 1978; Hoshino et al. 1985) have studied Ti self-diffusion along the c direction in air, with results that are in good agreement, particularly at high temperature (Fig. 29). Lundy and Coghlan (1973) and Hoshino et al. (1985) also studied Ti self-diffusion perpendicular to the c-axis, with each group finding diffusion to be slightly faster than in the c direction (by ~20 to 300%, depending on temperature) and to have a somewhat lower activation energy. Akse and Whitehurst (1978) and Hoshino et al.


(1985) examined the influence of oxygen fugacity on Ti self-diffusion coefficients, with Akse and Whitehurst (1978) focusing on low temperatures and low fO2

, and Hoshino et al. (1985) examining a much broader range of conditions. The results of these two studies are quite consistent, both finding that the Ti self-diffusion coefficient increases under more reducing conditions (Fig. 28). The fO2

dependence of Ti diffusivity is similar to that for O diffusivity, but Ti diffusion is several orders of magnitude faster. For oxygen the enhancement of diffusion under reducing conditions is due to the production of oxygen vacancies, while for Ti it is due to the production of Ti interstitials. Hoshino et al. (1985) suggested an interstitialcy rather than a pure interstitial mechanism for Ti, wherein an interstitial Ti atom jumps to a neighboring occupied Ti site, displacing the lattice Ti atom into a second neighboring interstitial site. These authors found that the Ti self-diffusion coefficient varied with oxygen fugacity as D f n

Ti O2∝ −1/ ,

with n = 4.16 for diffusion parallel to c and n = 4.28 for diffusion perpendicular to c.

Akse and Whitehurst (1978) found a sharp drop in Ti diffusivity at low fO2, below 10−16

atm. The drop in diffusivity occurs at the rutile stability limit, where a more oxygen deficient phase (a so-called Magnéli phase) becomes stable. The drop in diffusivity was attributed to obstruction by shear planes formed during the phase transition.

Zirconium and hafnium. Sasaki et al. (1985) studied diffusion of Zr4+ (among several other impurity cations, discussed below) as a function of temperature, oxygen fugacity and crystallographic direction, using radiotracer techniques with serial sectioning. The dependence of Zr4+ diffusivity on fO2

and crystallographic orientation were found to be similar to those for Ti self-diffusion, and an interstitialcy mechanism was inferred. Diffusion of Zr in rutile was found to be about two orders of magnitude slower than Ti, at 1773 K, and to have a somewhat larger activation energy (leading to a greater difference in diffusivity between Zr and Ti at lower temperatures). These observations may be accounted for by the greater size of Zr4+ (0.072 nm in octahedral coordination) compared to Ti4+ (0.0605 nm). Cherniak et al. (2007) studied Zr4+ and Hf4+ diffusion in rutile, using Rutherford Backscattering Spectroscopy (RBS) to measure the diffusion profiles. Their results for Zr are quite different than those obtained by Sasaki et al. (1985); in particular, Cherniak et al. (2007) found much smaller diffusion coefficients, a smaller activation energy, and no significant dependence of the diffusivity on oxygen fugacity. Cherniak et al. (2007) suggested that the diffusion mechanism was different in the two studies,

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Ti; Hoshino et al., 1985

Ti; Lundy & Coghlan, 1973

Ti; Venkatu & Poteat, 1969

log

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Ti; Akse et al., 1978Zr; Sasaki et al., 1985

Hf; Cherniak et al., 2007

Zr; Cherniak et al., 2007

Tetravalent cationsin rutile (in air)

Figure 29. Summary of experimental data on diffusion of tetravalent cations in stoichiomet-ric rutile (in air).


due to the very different Zr concentrations used in the experiments. Cherniak et al. (2007) suggested that only a small amount of Zr could be accommodated in rutile as interstitials, and that the interstitialcy mechanism operating at the low Zr concentrations used by Sasaki et al. (1985) may be masked by a slower vacancy mechanism that dominates the diffusive flux at the much higher concentrations studied by Cherniak et al. (2007). Cherniak et al. (2007) found that Zr and Hf had similar diffusivities, as might be expected because of their similarity in size, and that diffusion of Hf was only slightly anisotropic.

Niobium. Diffusion of Nb5+ in air, in an unspecified orientation, has recently been studied by Sheppard et al. (2007, 2009) in both nominally pure rutile and in crystals doped with 4.3 at.% Nb. Diffusion profiles in quenched samples were measured using secondary ion mass spectrometry (SIMS) depth profiling. The measured diffusion profiles were quite complicated, particularly in the undoped crystals (Sheppard et al. 2007), with extremely high Nb concentrations near the surface possibly reflecting the presence of residual tracer, or surface reaction to a Nb-Ti oxide. Such complicated profiles may result from analytical artifacts (due, for example, to the presence of residual surface tracer) and/or from complicated diffusion behavior, due for example to concentration-dependent diffusivity, as would be expected for an aliovalent cation like Nb5+ at high concentrations. Sheppard et al. (2007, 2009) calculated the diffusion coefficient using a small segment of the concentration profile that was quasi-linear in the logarithm of concentration vs. the square of distance, which was interpreted to represent simple concentration-independent diffusion from a plane source. It is not clear that this interpretation of the profiles is valid, and we consider these results with skepticism. Further work on Nb diffusion, and on diffusion in Nb-doped samples, would be useful, and reliable results would no doubt provide important information on point defects and diffusion mechanisms in rutile.

Divalent and trivalent cations

A summary of diffusion data for divalent and trivalent cations in rutile is shown in Figure 30 as an Arrhenius plot. Sasaki et al. (1985) performed radiotracer sectioning experiments to determine the diffusion coefficients of Sc, Cr, Mn, Fe, Co, and Ni in rutile single crystals, as functions of temperature, fO2

and crystallographic orientation. Under the conditions of the experiments, Sc and Cr were trivalent, Co and Ni were thought to be predominantly divalent, and Fe and Mn had mixed valence. Diffusion of each cation was found to have a dependence on fO2

similar to that found for Ti self-diffusion (Hoshino et al. 1985), both parallel to and perpendicular to the c-axis. In terms of the directional dependence of diffusivity, the trivalent cations Sc3+ and Cr3+ were found to have little anisotropy, but with diffusion parallel to c slightly faster than perpendicular to c, opposite to Ti and Zr. Like the tetravalent cations, these were interpreted to diffuse by an interstitialcy mechanism, which involves cooperative motion of the impurity cation and titanium. The divalent cations, in contrast, exhibit strong anisotropy, with rapid diffusion along the c axis; in air at 800 °C, the diffusion coefficient for Co is 2800 times higher parallel to c than perpendicular to c. The rapid diffusion of these divalent cations parallel to the c-axis was suggested to result from interstitial migration along “channels” oriented parallel to c in the rutile structure. Cobalt was inferred to dissolve substitutionally, but to diffuse rapidly along an interstitial channel when pushed there by an interstitialcy-type motion, returning to a substitutional site in the lattice only after traveling many atomic distances along the channel. The mixed-valence cations Fe and Mn were found to have intermediate behavior, diffusing ~20-80 times faster in the c direction than in the plane perpendicular to it. Sasaki et al. (1985) suggested that diffusion of these mixed-valence cations represented a combination of divalent ions diffusing by an interstitial mechanism (with large anisotropy) and trivalent ions diffusing by an interstitialcy mechanism (with little anisotropy).

Diffusion of iron in rutile has also been studied by several other groups. Izbekov and Gorbunova (1959) performed an early study of Fe diffusion using radiotracer techniques, using


both total absorption and layer removal techniques. The diffusion coefficients are much smaller, by 2 to 3 orders of magnitude, than those obtained in later studies. Sasaki et al. (1984) studied Fe diffusion in both pure rutile and in Fe-doped samples. A steep drop in the diffusivity was found for Fe contents exceeding 0.35 at.%; at 800 °C, the diffusion coefficient for Fe parallel to the c-axis drops by nearly three orders of magnitude as the Fe doping level increases from 0.35 to 2.0 at.%. Following the explanation suggested by Akse and Whitehurst (1978) for the sharp decline in Ti self-diffusivity under highly reducing conditions, Sasaki et al. (1984) suggested that rutile becomes unstable at Fe doping levels beyond ~0.35 at.%, resulting in a phase transformation and formation of shear planes that obstruct ion movement. One might speculate that Izbekov and Gorbunova (1959) used a high Fe tracer concentration in their experiments, and that this explains the small diffusion coefficients obtained, but it is not possible to confirm this based on the information provided in the paper. Two studies (de Biasi and Grillo 1996; Egerton et al. 2001) used electron paramagnetic resonance (EPR) techniques to study Fe3+ diffusion in powdered rutile samples, utilizing the change with time in the EPR signal for substitutional Fe3+ as iron was absorbed from a source mixed with the rutile powder. Neither of these studies reported Fe3+ diffusion coefficients, but did report the activation energy for diffusion from measurements of the absorption kinetics at different temperatures. de Biasi and Grillo (1996) determined an activation energy of 215 kJ/mol from measurements at 1273-1373 K, while Egerton et al. (2001) determined an activation energy of 110 kJ/mol from measurements at 883-973 K. This value is similar to those obtained by Sasaki et al. (1985) for Fe diffusion in air, parallel to (125 kJ/mol) and perpendicular to (135 kJ/mol) the c-axis.

Two EPR studies were also performed to study the diffusion of Cr3+ in rutile (de Biasi and Fernandes 1994; Egerton et al. 2000). These studies found activation energies of 130 and 150 kJ/mol, respectively, similar to that found by Sasaki et al. (1985) using radiotracer/sectioning measurements. The studies also confirm the finding by Sasaki et al. (1985) that chromium diffusion is significantly slower than iron diffusion in rutile.

Experimental studies on diffusion of the larger divalent cations Ba2+ (Nakayama and Sasaki 1963) and Pb2+ (Cherniak 2000) indicate that these cations behave more similarly to the trivalent and tetravalent cations, with relatively small diffusion coefficients and little anisotropy. These cations appear to be too large to diffuse by a fast interstitial mechanism along the c-oriented channels in rutile, as Co2+, Fe2+ and Mn2+ appear to do. The diffusion

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Fe; S85

Co; S85

Mn; S85

Fe; Sasaki et al., 1985Ni; Sasaki et al., 1985

Co; Sasaki et al., 1985

Mn, perp.;S85

lo

g D

(m2 /s

)

104/T (K)

Pb; Cherniak, 2000Divalent and trivalentcations in rutile (in air)

Figure 30. Summary of ex-perimental data for the diffu-sion of divalent and trivalent cations in stoichiometric rutile (in air, with the excep-tion of the Pb2+ experiments which were performed under moderately reducing condi-tions). Note that for some of the divalent cations (Mn, Co, Fe) diffusion is strongly anisotropic, with much faster diffusion rates parallel to the c crystallographic axis. This anisotropy is attributed to rap-id diffusion along interstitial channels parallel to the c axis, pathways that cations with small radius and/or charge are able to exploit.


coefficients for Ba2+ are about two orders of magnitude larger than those for Pb2+, despite the larger size of Ba2+ (0.135 nm vs. 0.119 nm for Pb2+; Shannon 1976). Although this seems counterintuitive, it is actually consistent with the elastic model discussed by Van Orman et al. (2001)—diffusivity is predicted to increase with radius for cations much larger than the one that normally occupies the site. On the other hand, it is possible that transport in the Ba2+ experiments was enhanced by greater impurity concentrations or by non-diffusive transport related to reactions with the BaCO3 tracer layer that was used.

Monovalent cations

Diffusion of Li+ in rutile was studied at temperatures of 80-360 °C by Johnson (1964) in out-diffusion experiments wherein a crystal uniformly doped with Li was placed in a water or molten salt bath and annealed at constant temperature. The bath was designed to approximate a perfect sink for Li, maintaining a zero-concentration boundary condition at the surface of the rutile crystal during the experiment. The concentration distribution within the rutile crystal following each experiment was characterized by optical absorption measurements. Diffusion of Li was found to be extremely rapid parallel to the c-axis, with very small activation energy, 32 kJ/mol. The degree of anisotropy was reported to be extremely large, with diffusion perpendicular to the c-axis slower by a factor of at least 108, although no diffusion data were actually reported for this direction and the basis for inferring this large factor is not obvious. Johnson and Krouse (1966) studied the mass dependence of DLi in rutile by measuring the 6Li/7Li ratio along a lithium diffusion profile. The ratio of the diffusion coefficients for these two isotopes was found to vary as the inverse square root of the mass, within experimental error, as expected for an interstitial mechanism (the mass dependence is smaller for other diffusion mechanisms, including the vacancy mechanism, where correlation of successive jumps becomes important and reduces the mass dependence of the diffusion coefficient).

Diffusion of hydrogen and its isotopes deuterium and tritium has been studied by Caskey (1974), Johnson et al. (1975) and Cathcart et al. (1979). Each of these studies was conducted under relatively oxidizing conditions, where rutile is nearly stoichiometric. All three studies found diffusion parallel to c to be much faster than perpendicular to c, and the absolute values of the diffusion coefficients for each direction are in reasonably good agreement at the temperatures where the studies overlap (Fig. 31). Diffusion of hydrogen in the c direction is

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3H; C79 "fast"

1H/ 2H; J75

3H; Cathcart et al., 1979

3H; Caskey, 1974

1H/ 2H; Johnson et al., 1975

log

D (m

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Li; Johnson, 1964

Monovalent cationsin rutile

Figure 31. Summary of dif-fusion data for monovalent cations in rutile. Hydrogen and lithium diffuse very rapidly in the c direction, apparently diffusing inter-stitially along channels with low electron density.


quite fast, but several orders of magnitude slower than Li. Cathcart et al. (1979) suggested that hydrogen was present in rutile primarily as OH−, and that hydrogen diffuses as H+ hopping from one oxygen site to a neighboring one. The bonding of hydrogen to oxygen atoms in the lattice provides a plausible explanation for the slower diffusivity of H+ relative to Li+ in rutile; apparently Li+ is able to diffuse freely along channels parallel to c in the rutile structure, while H+ must spend time trapped at oxygen lattice sites.

Cathcart et al. (1979) found evidence for two diffusion mechanisms in the direction perpendicular to c. The dominant mechanism is inferred to be due to the transport of H+. The other mechanism, which is evidenced only in the near-surface portion of the diffusion profiles, is a factor of 103-104 slower and has an activation energy nearly twice as large. This “slow” mechanism was attributed to the diffusion of H2 molecules, based on the presence of an H2 stretching mode in the Raman spectrum near the surface of the sample.

ACKNOWLEDGMENTS

We are grateful for the reviews of the manuscript by Daniele Cherniak, Youxue Zhang and an anonymous reviewer. This work was supported in part by the National Science Foundation under Grants No. 0337125 and 0838141.

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APPENDIX

Tables 1-16. See text for discussion.

Tabl

e 1.

Oxy

gen

self

-dif

fusi

on in

per

icla

se.

Tem

pera

ture

(K

)D

o (m

2 /s)

Q

(kJ/

mol

)D

T

(m2 /

s)E

xp.

Met

hod†

Ana

lysi

s†R

efer

ence

Not

es

1773

-202

36.

76×1

0−4

536

—G

SXG

SMS

Ois

hi e

t al.

(198

3)N

orto

n cr

ysta

ls15

73-1

773

2.20

×10−

1321

3—

GSX

GSM

SO

ishi

et a

l. (1

983)

Nor

ton

crys

tals

1121

-157

37.

80×1

0−8

279

—G

SXG

SMS

Ois

hi e

t al.

(198

7)40

0 pp

m L

i-do

ped

1573

-202

32.

50×1

0−10

261

—G

SXG

SMS

Ois

hi a

nd K

inge

ry (

1960

)15

80-1

820

1.90

×10−

837

0—

GSX

GSM

SR

eddy

and

Coo

per

(198

3)13

73-1

700

1.37

×10−

646

0—

(see

not

es)

Nar

ayan

and

Was

hbur

n (1

973)

disl

ocat

ion

loop

shr

inka

ge r

ates

1293

-172

32.

40×1

0−11

233

—G

SXG

SMS

Shir

asak

i and

Ham

a (1

973)

“wel

l-si

nter

ed”

1293

-152

31.

60×1

0−11

252

—G

SXG

SMS

Shir

asak

i abd

Ham

a (1

973)

“loo

sely

-sin

tere

d”15

23-1

723

1.85

×10−

5 *43

0—

GSX

GSM

SSh

iras

aki a

bd H

ama

(197

3)“l

oose

ly-s

inte

red”

1323

-171

14.

50×1

0−11

252

—G

SXG

SMS

Has

him

oto

et a

l. (1

972)

1363

-169

35.

20×1

0−11

186

—G

SXG

SMS

Shir

asak

i et a

l. (1

973)

3.5

atom

ic %

Li-

dope

d12

73-1

923

1.80

×10−

1031

3—

TH

INSI

MS-

DP

Yoo

et a

l. (1

984,

200

2)12

48-1

423

1.53

×10−

553

7—

GSX

GSM

SR

ovne

r (1

966)

Nor

ton

crys

tals

1023

-124

82.

12×1

0−11

327

—G

SXG

SMS

Rov

ner

(196

6)N

orto

n cr

ysta

ls11

48-1

423

2.40

×10−

934

3—

GSX

GSM

SR

ovne

r (1

966)

Sem

i-E

lem

ents

cry

stal

s13

00-2

400

(see

not

es)

—T

HIN

SIM

S-D

PY

ang

and

Flyn

n (1

994)

D =

7.6

×10−

3 exp(

−80

,221

/T)

+

1×10

−13

exp(

−30

,910

/T)

m2 /

s18

73-2

273

1.53

×10−

5 *53

7*—

TH

INSI

MS-

DP

Yan

g an

d Fl

ynn

(199

6)15

73-1

873

2.12

×10−

11*

327*

—T

HIN

SIM

S-D

PY

ang

and

Flyn

n (1

996)

1673

——

1.3×

10−

19T

HIN

SIM

S-D

PH

enri

ksen

et a

l. (1

983)

“pur

e” c

ryst

als

1673

——

1.9×

10−

19T

HIN

SIM

S-D

PH

enri

ksen

et a

l. (1

983)

550

ppm

Sc-

dope

d16

73—

—1.

6×10

−19

TH

INSI

MS-

DP

Hen

riks

en e

t al.

(198

3)14

00 p

pm S

c-do

ped

1673

——

5.8×

10−

19T

HIN

SIM

S-D

PH

enri

ksen

et a

l. (1

983)

defo

rmed

cry

stal

2273

——

4.3×

10−

16SS

XSI

MS-

SMV

an O

rman

et a

l. (2

003)

P =

15

GPa

V =

3.3

cm

3 /m

ol22

73—

—2.

1×10

−16

SSX

SIM

S-SM

Van

Orm

an e

t al.

(200

3)P

= 1

6 G

Pa V

= 3

.3 c

m3 /

mol

2273

——

5.7×

10−

17SS

XSI

MS-

SMV

an O

rman

et a

l. (2

003)

P =

25

GPa

V =

3.3

cm

3 /m

ol

* R

e-fit

fro

m d

ata

in th

e or

igin

al p

aper

† Abb

revi

atio

ns: G

SX =

gas

-sol

id e

xcha

nge;

TH

IN =

thin

film

sou

rce;

SSX

= s

olid

-sol

id e

xcha

nge;

GSM

S =

gas

sou

rce

mas

s sp

ectr

omet

ry; S

IMS-

DP

= s

econ

dary

ion

mas

s sp

ectr

omet

ry d

epth

pro

filin

g; S

IMS-

SM =

sec

onda

ry io

n m

ass

spec

trom

etry

spo

t mea

sure

men

ts


Tabl

e 2.

Mag

nesi

um s

elf-

diff

usio

n in

per

icla

se.

Tem

pera

ture

(K

)D

o

(m2 /

s)Q

(kJ/

mol

)D

T

(m2 /

s)E

xp. M

etho

d†A

naly

sis†

Ref

eren

ceN

otes

2178

-262

37.

43×1

0−6

334

—R

AD

SSH

ardi

ng a

nd P

rice

(19

72)

Ven

tron

Ele

ctro

nics

cry

stal

s

1723

-217

37.

48×1

0−10

151

—R

AD

SSH

ardi

ng a

nd P

rice

(19

72)

Ven

tron

Ele

ctro

nics

cry

stal

s

1538

-172

35.

40×1

0−5

309

—R

AD

SS

H

ardi

ng a

nd P

rice

(19

72)

Ven

tron

Ele

ctro

nics

cry

stal

s

1593

-202

51.

20×1

0−9

154

—R

AD

SSH

ardi

ng e

t al.

(197

1)M

onoc

ryst

als

Co.

cry

stal

s

1406

-159

35.

40×1

0−5

309

—R

AD

SS

H

ardi

ng e

t al.

(197

1)M

onoc

ryst

als

Co.

cry

stal

s

1593

-201

51.

60×1

0−9

174

—R

AD

SSH

ardi

ng e

t al.

(197

1)M

onoc

ryst

als

Co.

cry

stal

s

1406

-159

35.

40×1

0−5

309

—R

AD

SS

H

ardi

ng e

t al.

(197

1)V

entr

on E

lect

roni

cs c

ryst

als

1673

-187

32.

49×1

0−5

331

—R

AD

SSL

indn

er a

nd P

arfit

t (19

57)

1100

-125

08.

84×1

0−10

215

— (

see

note

s)N

RA

Mar

tinel

li et

al.

(198

5)gr

owth

of

epita

xial

Mg18

O

from

gas

in a

n el

ectr

ic fi

eld

973-

1573

4.80

×10−

923

1—

RA

DSI

MS-

DP

Saka

guch

i et a

l. (1

992)

1374

-267

34.

19×1

0−8

266

—G

SXT

OF

MS

Wue

nsch

et a

l. (1

973)

2273

——

2.9×

10−

15SS

XSI

MS-

SMV

an O

rman

et a

l. (2

003)

P =

15

GPa

V =

3.0

cm

3 /m

ol

2273

——

2.6×

10−

15SS

XSI

MS-

SMV

an O

rman

et a

l. (2

003)

P =

16

GPa

V =

3.0

cm

3 /m

ol

2273

——

6.1×

10−

16SS

XSI

MS-

SMV

an O

rman

et a

l. (2

003)

P =

25

GPa

V =

3.0

cm

3 /m

ol

† Abb

revi

atio

ns: R

AD

= r

adio

trac

er; G

SX =

gas

-sol

id e

xcha

nge;

SSX

= s

olid

-sol

id e

xcha

nge;

SS

= s

eria

l sec

tioni

ng; N

RA

= n

ucle

ar r

eact

ion

anal

ysis

; SIM

S-D

P =

sec

onda

ry io

n m

ass

spec

trom

etry

dep

th p

rofil

ing;

TO

F M

S =

tim

e-of

-flig

ht m

ass

spec

tros

copy

; SIM

S-SM

= s

econ

dary

ion

mas

s sp

ectr

omet

ry s

pot m

easu

rem

ents

.


Tabl

e 3.

Gro

up I

A a

nd I

IA c

atio

n di

ffus

ion

in p

eric

lase

.

Dif

fusi

ng

Spec

ies

Tem

pera

ture

R

ange

(K

)D

o

(m2 /

s)Q

(k

J/m

ol)

Exp

. M

etho

d†A

naly

sis†

Ref

eren

ce

2 H10

73-1

173

1.5×

10−

218

3G

SXIR

Gon

zale

z et

al.

(198

2)

Be

1273

-197

31.

41×1

0−9

154

RA

DSS

Har

ding

and

Mor

tlock

(19

66)

Be

909-

2613

1.99

×10−

916

2R

AD

SSH

ardi

ng (

1973

a)

Ca

1183

-197

32.

95×1

0−9

206

RA

DSS

Run

gis

and

Mor

tlock

(19

66)

Ca

1063

-212

38.

90×1

0−8

266

SSX

EPM

AW

uens

ch a

nd V

asilo

s (1

968)

Ca

2123

-267

33.

43×1

0−7

309

RA

DSS

Har

ding

(19

73b)

Ca

1000

-130

01.

04×1

0−8

243

TH

INSI

MS-

DP

Yan

g an

d Fl

ynn

(199

4)

Ca

1000

-130

06.

49×1

0−11

*21

5*T

HIN

SIM

S-D

PY

ang

and

Flyn

n (1

996)

Sr12

73-1

873

6.00

×10−

828

1R

AD

SSM

ortlo

ck a

nd P

rice

(19

73)

Ba

1273

-197

37.

00×1

0−6

326

RA

DSS

Har

ding

(19

67)

Ba

1773

-272

32.

80×1

0−6

338

RA

DSS

Har

ding

(19

72)

Ba

1373

-169

54.

20×1

0−6

318

RA

DSS

Har

ding

(19

72)

* R

e-fit

fro

m d

ata

in th

e or

igin

al p

aper

† Abb

revi

atio

ns:

GSX

= g

as-s

olid

exc

hang

e; R

AD

= r

adio

trac

er;

SSX

= s

olid

-sol

id e

xcha

nge;

TH

IN =

thi

n fil

m s

ourc

e; I

R =

inf

rare

d sp

ectr

osco

py;

SS =

ser

ial

sect

ioni

ng; E

PMA

= e

lect

ron

prob

e m

icro

anal

ysis

; SIM

S-D

P =

sec

onda

ry io

n m

ass

spec

trom

etry

dep

th p

rofil

ing


Tabl

e 4.

Tri

vale

nt a

nd te

trav

alen

t cat

ion

diff

usio

n in

per

icla

se.

Dif

fusi

ng

Spec

ies

Tem

pera

ture

R

ange

(K

)D

o

(m2 /

s)Q

(kJ/

mol

)D

T

(m2 /

s)E

xp.

Met

hod†

Ana

lysi

s†R

efer

ence

Not

es

Al

1577

-227

36.

76×1

0−7

213

—SS

XE

PMA

Van

Orm

an e

t al.

(200

9)pa

ir d

iffu

sivi

ty;

P =

0-2

5 G

Pa; V

= 3

.5 c

m3 /

mol

Al

1577

-227

34.

50×1

0−9

184

—SS

XE

PMA

Van

Orm

an e

t al.

(200

9)lis

ted

para

met

ers

are

extr

apol

ated

to

100

ppm

triv

alen

t cat

ions

; P

= 0

-25

GPa

Sc17

73—

—7.

40×1

0−16

RA

DSS

Sola

ga a

nd M

ortlo

ck (

1970

)

Cr

1573

-197

38.

80×1

0−8

285

—T

HIN

EPM

ATa

gai e

t al.

(196

5)

Cr

1626

-182

61.

02×1

0−7

294

—R

AD

SSW

eber

et a

l. (1

977)

Cr

1623

-182

34.

12×1

0−7 *

315*

—R

AD

SSW

eber

et a

l. (1

980)

“pur

e” M

gO

Cr

1656

-176

84.

23×1

0−13

*84

*—

RA

DSS

Web

er e

t al.

(198

0)0.

3 m

ol %

Cr-

dope

d

Cr

1573

-227

35.

25×1

0−7

237

—SS

XE

PMA

Cri

spin

and

Van

Orm

an (

2010

)pa

ir d

iffu

sivi

ty; P

= 2

GPa

Cr

1573

-227

37.

81×1

0−10

220

—SS

XE

PMA

Cri

spin

and

Van

Orm

an (

2010

)lis

ted

para

met

ers

are

extr

apol

ated

to

100

ppm

triv

alen

t cat

ions

; P =

2 G

Pa

Ga

1563

-227

31.

38×1

0−7

190

—SS

XE

PMA

Cri

spin

and

Van

Orm

an (

2010

)pa

ir d

iffu

sivi

ty; P

= 0

-2 G

Pa

Ga

1563

-227

32.

53×1

0−10

123

—SS

XE

PMA

Cri

spin

and

Van

Orm

an (

2010

)lis

ted

para

met

ers

are

extr

apol

ated

to

100

ppm

triv

alen

t cat

ions

;P

= 0

-2 G

Pa

Y16

73-2

033

2.11

×10−

629

9—

RA

DSS

Ber

ard

(197

1)

Ge

2123

-267

33.

40×1

0−5

386

—R

AD

SSH

ardi

ng (

1973

b)

* R

e-fit

fro

m d

ata

in th

e or

igin

al p

aper

† Abb

revi

atio

ns: S

SX =

sol

id-s

olid

exc

hang

e; R

AD

= r

adio

trac

er; T

HIN

= th

in fi

lm s

ourc

e; S

SX =

sol

id-s

olid

exc

hang

e; E

PMA

= e

lect

ron

prob

e m

icro

anal

ysis

; SS

= s

eria

l sec

tioni

ng

814 Van Orman & CrispinTa

ble

5. D

ival

ent t

rans

ition

met

al d

iffu

sion

in p

eric

lase

.

Dif

fusi

ng

Spec

ies

Tem

pera

ture

R

ange

(K

)D

o

(m2 /

s)Q

(kJ/

mol

)D

T

(m2 /

s)E

xp.

Met

hod†

Ana

lysi

s†R

efer

ence

Not

es†

Mn

1573

-197

34.

10×1

0−11

117

—T

HIN

EPM

ATa

gai e

t al.

(196

5)ai

rM

n12

23-1

511

9.40

×10−

1020

3—

(see

not

es)

Wee

ks a

nd C

hate

lain

(19

78)

Mn-

MnO

buf

fer;

vap

or e

xcha

nge

with

E

PR

Mn

1653

-183

84.

70×1

0−10

164

—SS

XSS

/WC

A, E

PMA

Jone

s an

d C

utle

r (1

971)

liste

d pa

ram

eter

s ar

e ex

trap

olat

ed to

10

0 pp

m c

atio

n va

canc

ies;

air

Mn

1773

——

8.00

×10−

15SS

XSS

/WC

A, E

PMA

Jone

s an

d C

utle

r (1

971)

f O2=

10-4

Pa

Fe12

73-2

083

8.83

×10−

917

5—

TH

INE

PMA

, XR

AW

uens

ch a

nd V

asilo

s (1

962)

air

Fe15

73-1

963

1.30

×10−

720

5—

TH

INE

PMA

Taga

i et a

l. (1

965)

iron

met

al s

ourc

e; a

irFe

1573

-196

33.

20×1

0−8

176

—T

HIN

EPM

ATa

gai e

t al.

(196

5)fe

rric

chl

orid

e so

urce

; air

Fe13

73-1

573

6.46

×10−

625

9—

RA

DSS

Che

n et

al.

(200

8)M

gO d

oped

with

500

0 pp

m A

l3+

Fe13

73-1

573

6.50

×10−

822

7—

RA

DSS

Che

n et

al.

(200

8)lis

ted

para

met

ers

extr

apol

ated

to 1

00

ppm

Al3+

‡

Fe14

23-1

623

5.82

×10−

1210

5—

SSX

EPM

AB

lank

and

Pas

k (1

969)

vacu

umFe

1473

-160

35.

30×1

0−11

152

—SS

XE

PMA

Byg

dén

et a

l. (1

997)

argo

n at

mos

pher

eC

o12

73-2

083

5.78

×10−

919

9—

TH

INE

PMA

, XR

AW

uens

ch a

nd V

asilo

s (1

962)

air

Co

1573

——

1.62

×10−

15SS

XE

PMA

Yur

ek a

nd S

chm

alzr

ied

(197

4)ai

rN

i12

73-2

083

1.80

×10−

920

3—

TH

INE

PMA

, XR

AW

uens

ch a

nd V

asilo

s (1

962)

air

Ni

2173

-273

31.

80×1

0−9

203

—T

HIN

EPM

A, X

RA

Wue

nsch

and

Vas

ilos

(196

2)ar

gon

atm

osph

ere

Ni

1473

-157

36.

00×1

0−10

174

—R

AD

Mim

kes

and

Wut

tig (

1971

)ai

rN

i21

73-2

773

1.40

×10−

631

8—

RA

DSS

Har

ding

(19

72)

air

and

argo

n at

mos

pher

esN

i18

23-2

123

1.30

×10−

1015

4—

RA

DSS

Har

ding

(19

72)

air

and

argo

n at

mos

pher

esN

i14

73-1

673

9.83

×10−

1019

0—

SSX

EPM

AB

lank

and

Pas

k (1

969)

vacu

umN

i14

73-1

673

4.44

×10−

1018

1—

SSX

EPM

AB

lank

and

Pas

k (1

969)

air

Ni

1619

——

7.19

×10−

16SS

XE

PMA

App

el a

nd P

ask

(197

1)ai

rN

i15

76—

—1.

55×1

0−15

SSX

SSW

ei a

nd W

uens

ch (

1973

)ai

rZ

n12

73-1

918

1.48

×10−

917

9—

TH

INE

PMA

Wue

nsch

and

Vas

ilos

(196

5)Pd

973-

1073

3.00

×10−

916

8—

RA

DSS

de B

ruin

and

Tan

gtre

erat

ana

(198

1)Pd

1123

-137

32.

00×1

0−5

274

—R

AD

SSde

Bru

in a

nd T

angt

reer

atan

a (1

981)

Cd

2051

-254

01.

70×1

0−6

328

—R

AD

SSH

ardi

ng a

nd B

hala

(19

71)

‡ The

val

ues

for

the

Arr

heni

us p

aram

eter

s ar

e de

rive

d fr

om th

e ex

trap

olat

ion

of th

eir

data

to 1

00 p

pm A

l3+, b

ased

on

free

vac

ancy

con

cent

ratio

ns f

or A

l-do

ped

MgO

cal

cula

ted

usin

g th

e da

ta o

f Van

Orm

an e

t al.

(200

9)† A

bbre

viat

ions

: TH

IN =

thin

film

sou

rce;

SSX

= s

olid

-sol

id e

xcha

nge;

RA

D =

rad

iotr

acer

; EPM

A =

ele

ctro

n pr

obe

mic

roan

alys

is; S

S =

ser

ial s

ectio

ning

; WC

A =

wet

che

mic

al a

naly

sis;

XR

A =

x-

ray

abso

rptio

n; E

PR =

ele

ctro

n pa

ram

agne

tic r

eson

ance


Tabl

e 6.

Sum

mar

y ex

pres

sion

s fo

r Fe

-Mg

inte

rdif

fusi

on in

fer

rope

ricl

ase

(xFe

is th

e ca

tion

frac

tion

of F

e).

Ref

eren

ceSu

mm

ary

Exp

ress

ion

(D

is in

m2 /

s)T

(K

)f O

2 (P

a)x F

eN

otes

Mac

kwel

l et a

l. (2

005)

Df

xx

T=

×−

−−

20

1025

136

1154

66

019

073

.ex

p((

,,

)/)

..

OF

eF

e2

1593

-167

310

−1 -1

0−4.

30.

1-0.

27f O

2 in

Pa

Rig

by a

nd C

utle

r (1

965)

Dx

T=

×−

628

1011

3625

425

8.

exp(

.)e

xp(

,/

)F

e13

78-1

588

0.1-

0.3

H2-

H2O

Sata

and

Got

o (1

982)

DD

x Df

f

=

=−

−−

0

02

00296

0194

2188

22

exp(

)

ln.

(ln

).

(ln

).

βFe

OO

where

annd

OO

β=

++

00175

00977

777

22

2.

(ln

).

(ln

).

ff

1324

-142

310

−2.

5 -10−

60.

3-0.

8f O

2 in

Pa

Byg

dén

et a

l. (1

997)

Dx

T=

×−

−5

310

124

1828

111

.ex

p(.

)exp

(,

/)

Fe

1473

-160

3ar

gon

0.1-

0.7

Bla

nk a

nd P

ask

(196

9)D

xx

T=

×−

+−

46

1024

012

232

2340

412

.ex

p(.

)exp

((

,,

)/)

Fe

Fe

1423

-162

3va

cuum

0.05

-0.5

Hol

zapf

el e

t al.

(200

3)D

xT

PT

=×

−+

×−

−9

810

1587

630

669

397

106

7.

exp(

,/

)exp

((

,.

)/)

Fe

whe

re P

is in

Pa

1653

-207

3N

i-N

iO0.

07-0

.37

8-23

GPa

Yam

azak

i and

Iri

fune

(20

03)

Dx

PT

=×

−−

×+

×−

−4

110

2718

113

590

17

9510

712

.ex

p[(

,,

)(

.)/

]F

e

whe

re P

is in

Pa

1573

-197

30.

19-0

.41

7-35

GPa

Dem

ouch

y et

al.

(200

7)D

xx

T=

×−

−−

510

3247

39

622

40

8F

eF

e.

exp(

,,

)/)

1273

-152

310

−2.

3 -10−

40.

01-0

.25

f H2O

= 3

00 M

Pa

† Equ

atio

n fit

to d

ata

give

n in

sou

rce


Table 7. Concentration dependence of transition metal interdiffusion coefficient, tracer diffusion coefficient, and cation vacancy concentration in periclase. Each is proportional to exp(βxMe), where xMe is the cation fraction of the transition metal.

Element Property Atmosphere β Reference*Mn D Air 16.9 Jones and Cutler (1971)

*Mn D fO2 = 10−4 Pa 4.6 Jones and Cutler (1971)

Co D Air 7.4 Yurek and Schmalzried (1974)

Co D* Air 6.8 Yurek and Schmalzried (1974)

Co xV Air 5.4 Yurek and Schmalzried (1974)

Ni D Vacuum 0 Blank and Pask (1969)

Ni D Air 2.7 Blank and Pask (1969)

Ni D Air 2.2 Appel and Pask (1971)

Ni D Air 1.9 Wei and Wuensch (1973)

Ni D* Air 3.2 Wei and Wuensch (1973)

Fe D H2-H2O 11.4 Rigby and Cutler (1965)

Fe D Vacuum 7.5-9.3 Blank and Pask (1969)

Fe D fO2 = 10−6 Pa 9.7 Sata and Goto (1982)

Fe D fO2 = 3×10−3 Pa 7.5 Sata and Goto (1982)

Fe D Argon 12.4 Bygdén et al. (1997)

Fe D Air 0 Blank and Pask (1969)

Fe D* fO2 = 10−11 − 10−6 Pa 7.7 Chen and Peterson (1980)

Fe xV H2-H2O 9.4 Rigby and Cutler (1965)* Parameters fit to data given in source


Tabl

e 8.

Dif

fusi

on in

spi

nel.

Dif

fusi

ng

Spec

ies

Tem

pera

ture

R

ange

(K

)D

o

(m2 /

s)Q

(kJ/

mol

)D

T

(m2 /

s)E

xp.

Met

hod†

Ana

lysi

s†R

efer

ence

Not

es

O16

93-1

903

1.50

×10−

156

9—

GSX

GSM

SA

ndo

and

Ois

hi (

1972

)

O15

83-1

967

1.60

×10−

638

5—

GSX

GSM

SO

ishi

and

And

o (1

975)

O17

05-2

012

8.90

×10−

543

9—

GSX

GSM

SA

ndo

and

Ois

hi (

1974

)

O17

74-1

989

2.20

×10−

444

4—

GSX

GSM

SA

ndo

and

Ois

hi (

1974

)x A

l2O

3 = 1

.2

O16

25-1

925

1.06

×10−

641

5—

GSX

NR

AR

eddy

and

Coo

per

(198

1)

O18

31—

—2.

71×1

0−17

GSX

NR

AR

eddy

and

Coo

per

(198

1)x A

l2O

3 = 1

.8

O18

00—

—3.

98×1

0−16

GSX

NR

AR

eddy

and

Coo

per

(198

1)x A

l2O

3 = 3

.5

O17

81—

—6.

46×1

0−16

GSX

NR

AR

eddy

and

Coo

per

(198

1)x A

l2O

3 = 3

.5

O17

47—

—3.

29×1

0−16

GSX

NR

AR

eddy

and

Coo

per

(198

1)x A

l2O

3 = 3

.5

O13

78-1

673

2.20

×10−

740

4—

GSX

SIM

S-D

PR

yers

on a

nd M

cKee

gan

(199

4)

Mg

1534

-182

67.

46×1

0−3

384

—(s

ee n

otes

)SI

MS-

SMSh

eng

et a

l. (1

992)

mel

t-cr

ysta

l exc

hang

e

Mg

1173

-157

32.

00×1

0−2

360

—R

AD

Lin

dner

and

Åke

rstr

öm (

1958

)

Mg

1246

-159

81.

90×1

0−9

202

—SS

XE

PMA

Lie

rman

n an

d G

angu

ly (

2002

)

Fe12

46-1

598

1.80

×10−

919

8—

SSX

EPM

AL

ierm

ann

and

Gan

guly

(20

02)

Cr

1573

-187

32.

45×1

0−6

337

—R

AD

SSSt

ubic

an e

t al.

(198

5)

H14

73-1

673

1.60

×10−

521

0—

GSX

IRO

kuya

ma

et a

l. (2

009)

H13

73-1

673

1.40

×10−

521

2—

GSX

IRFu

kats

u et

al.

(200

2)

Al-

Mg

1473

-187

31.

17×1

0−2

374

—SS

XE

PMA

Zha

ng e

t al.

(199

6)

Al-

Mg

1473

-227

32.

50×1

0−6

235

—SS

XE

PMA

Wat

son

and

Pric

e (2

002)

Cr-

Al

1673

-197

3(s

ee n

otes

)51

6 (a

t 1 a

tm)

—SS

XE

PMA

Suzu

ki e

t al.

(200

8)D

x0

40

5410

103

011

7=

+−

exp(

.)

..

Cr

V =

1.3

6 cm

3 /m

ol

Fe-M

g10

73-1

307

9.26

×10−

133

4—

SSX

EPM

AFr

eer

and

O’R

eilly

(19

80)

Fe =

5 w

t%

† Abb

revi

atio

ns: G

SX =

gas

-sol

id e

xcha

nge;

RA

D =

rad

iotr

acer

; SSX

= s

olid

-sol

id e

xcha

nge;

GSM

S =

gas

sou

rce

mas

s sp

ectr

omet

ry; N

RA

= n

ucle

ar r

eact

ion

anal

ysis

; SIM

S-D

P =

sec

onda

ry io

n m

ass

spec

trom

etry

dep

th p

rofil

ing;

SIM

S-SM

= s

econ

dary

ion

mas

s sp

ectr

omet

ry s

pot m

easu

rem

ents

; EPM

A =

ele

ctro

n pr

obe

mic

roan

alys

is; S

S =

ser

ial s

ectio

ning

; IR

= in

frar

ed s

pect

rosc

opy


Tabl

e 9.

Oxy

gen

self

-dif

fusi

on in

mag

netit

e.

Tem

pera

ture

R

ange

(K

)D

o

(m2 /

s)Q

(kJ/

mol

)E

xp.

Met

hod†

Ana

lysi

s†R

efer

ence

Not

es

575-

823

3.20

×10−

1871

GSX

GSM

SC

astle

and

Sur

man

(19

67)

753-

1373

1.20

×10−

526

4(s

ee n

otes

)C

rouc

h an

d R

ober

tson

(19

90)

infe

rred

fro

m d

ata

on p

ower

law

and

di

ffus

ion

cree

p

773-

1073

3.50

×10−

1018

8W

SXSI

MS-

DP

Gile

tti a

nd H

ess

(198

8)

773-

1123

4.30

×10−

1121

1(s

ee n

otes

)Sh

arp

(199

1)ba

sed

on n

atur

al is

otop

ic v

aria

tions

in

mag

netit

e gr

ains

fro

m a

slo

wly

-co

oled

mar

ble

823-

1123

1.80

×10−

1771

GSX

GSM

SC

astle

and

Sur

man

(19

69)

H2/

H2O

= 1

.2

823-

1123

1.80

×10−

1771

GSX

GSM

SC

astle

and

Sur

man

(19

69)

H2/

H2O

= 0

.001

† Abb

revi

atio

ns: G

SX =

gas

-sol

id e

xcha

nge;

WSX

= w

ater

-sol

id e

xcha

nge;

GSM

S =

gas

sou

rce

mas

s sp

ectr

omet

ry; S

IMS-

DP

= s

econ

dary

ion

mas

s sp

ectr

omet

ry d

epth

pro

filin

g


Tabl

e 10

. Cat

ion

diff

usio

n in

mag

netit

e.

Dif

fusi

ng

Spec

ies

Tem

pera

ture

R

ange

(K

)D

o (m

2 /s)

Q(k

J/m

ol)

Exp

. M

etho

d†A

naly

sis†

Ref

eren

ceN

otes

Fe10

43-1

473

2.50

×10−

522

6R

AD

SSIz

beko

v (1

958)

unkn

own

buff

er

Fe10

23-1

273

5.20

×10−

423

0R

AD

(see

not

es)

Him

mel

(19

53)

unkn

own

buff

er; s

urfa

ce a

ctiv

ity

Fe13

73-1

473

4.56

×10−

714

8R

AD

SSO

gaw

a et

al.

(196

8)un

know

n bu

ffer

Fe‡

1173

-167

31.

42×1

0−5

195

RA

DSS

Die

ckm

ann

and

Schm

alzr

ied

(197

7a)

WM

buf

fer

Fe‡

1173

-167

38.

14×1

0−6

175

RA

DSS

Die

ckm

ann

and

Schm

alzr

ied

(197

7a)

MH

buf

fer

Fe‡

1173

-167

31.

38×1

0−5

197

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)*W

M b

uffe

r

Fe‡

1173

-167

39.

10×1

0−6

175

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)*M

H b

uffe

r

Co‡

1179

-148

31.

44×1

0−5

194

RA

DSS

Die

ckm

ann

et a

l. (1

978)

WM

buf

fer

Co‡

1179

-148

34.

58×1

0−6

170

RA

DSS

Die

ckm

ann

et a

l. (1

978)

MH

buf

fer

Mn‡

1373

-157

35.

48×1

0−6

188

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)*W

M b

uffe

r

Mn‡

1373

-157

35.

50×1

0−7

140

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)*M

H b

uffe

r

Ni

1173

-137

34.

27×1

0−3

260

SSX

EPM

AG

rman

and

Jes

enak

(19

78)

Fe3O

4-N

i 0.2Fe

2.8O

4; 0

.3-0

.4 lo

g un

its a

bove

the

QFM

buf

fer

Ni

1433

-159

34.

32×1

0−6

196

RA

DSS

Eve

no a

nd P

aulu

s (1

974)

Ni 0

.37F

e 2.6

3O4;

0.7

-0.8

log

units

ab

ove

the

MH

buf

fer

Cr‡

1483

-168

37.

17×1

0−4

335

RA

DSS

Die

ckm

ann

et a

l. (1

978)

WM

buf

fer

Cr‡

1483

-168

31.

04×1

0−3

307

RA

DSS

Die

ckm

ann

et a

l. (1

978)

MH

buf

fer

Al‡

1553

-177

31.

46×1

0−5

265

RA

DSS

Die

ckm

ann

et a

l. (1

978)

¥W

M b

uffe

r

Al‡

1553

-177

36.

89×1

0−5

250

RA

DSS

Die

ckm

ann

et a

l. (1

978)

¥M

H b

uffe

r

Ti‡

1373

-157

32.

77×1

0−5

267

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)W

M b

uffe

r

Ti‡

1373

-157

33.

29×1

0−5

208

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)M

H b

uffe

r

vaca

ncy

1573

-172

31.

4×10

−5

136

Nak

amur

a et

al.

(197

8)f O

2=10

1 -10

5 Pa

‡ The

se A

rrhe

nius

par

amet

ers

wer

e ca

lcul

ated

alo

ng th

e w

üstit

e-m

agne

tite

(WM

) or

mag

netit

e-he

mat

ite (

MH

) bu

ffer

(as

not

ed)

usin

g th

e eq

uatio

ns g

iven

in th

e so

urce

. The

f O2 w

as c

alcu

late

d al

ong

the

buff

er f

rom

the

equa

tions

in H

uebn

er (

1971

).

¥ Bas

ed o

n a

refit

of

Petu

sky

1977

(T

hesi

s).

*Fro

m d

ata

com

pila

tion

in T

able

5 o

f Agg

arw

al a

nd D

ieck

man

n (2

002b

).† A

bbre

viat

ions

: RA

D =

rad

iotr

acer

; SSX

= s

olid

-sol

id e

xcha

nge;

SS

= s

eria

l sec

tioni

ng; E

PMA

= e

lect

ron

prob

e m

icro

anal

ysis


Tabl

e 11

. Cat

ion

diff

usio

n in

tita

nom

agne

tite.

Dif

fusi

ng

Spec

ies

Tem

pera

ture

R

ange

(K

)D

o (m

2 /s)

Q

(kJ/

mol

)E

xp.

Met

hod†

Ana

lysi

s†R

efer

ence

Not

es

Fe‡

1373

-157

33.

38×1

0−7

165

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)W

M b

uffe

r

Fe‡

1373

-157

32.

33×1

0−4

147

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)M

H b

uffe

r

Co‡

1373

-157

31.

59×1

0−6

185

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)W

M b

uffe

r

Co‡

1373

-157

31.

30×1

0−3

171

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)M

H b

uffe

r

Mn‡

1373

-157

38.

81×1

0−7

185

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)W

M b

uffe

r

Mn‡

1373

-157

37.

95×1

0−4

163

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)M

H b

uffe

r

Ti‡

1373

-157

31.

33×1

0−2

332

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)W

M b

uffe

r

Ti‡

1373

-157

38.

55×1

0−4

184

RA

DSS

Agg

arw

al a

nd D

ieck

man

n (2

002b

)M

H b

uffe

r

Fe-T

i76

3-10

032.

38×1

0−7

208

(see

not

es)

Pric

e (1

981)

mic

rost

ruct

ure

hom

ogen

izat

ion;

va

cuum

; “se

lf-b

uffe

red”

; na

tura

l mag

netit

e x T

i~0.

15

Fe-T

i87

3-12

735.

4×10

−16

40(s

ee n

otes

)Pe

ters

en (

1970

)ex

solu

tion

lam

ella

e fo

rmat

ion;

ai

r; n

atur

al m

agne

tite

x Ti~

0.15

Fe-T

iln

D =

−22

.71

+ 1

5.09

x Ti –

196

30/T

Ara

gon

et a

l. (1

984)

QFM

buf

fer

Fe-T

iln

D =

−15

.17

+ 1

3.3x

Ti –

258

70/T

Fr

eer

and

Hau

ptm

an (

1978

)“s

elf-

buff

ered

”

‡ The

se A

rrhe

nius

par

amet

ers

wer

e ca

lcul

ated

alo

ng th

e w

üstit

e-m

agne

tite

(WM

) or

mag

netit

e-he

mat

ite (

MH

) bu

ffer

(as

not

ed)

usin

g th

e eq

uatio

ns g

iven

in th

e so

urce

. The

f O2 w

as c

alcu

late

d al

ong

the

buff

er f

rom

the

equa

tions

in H

uebn

er (

1971

). A

ll pa

ram

eter

s re

fer

to th

e co

mpo

sitio

n x T

i = 0

.2, w

here

xT

i is

the

frac

tion

of a

ll ca

tion

site

s oc

cupi

ed b

y T

i.† A

bbre

viat

ions

: RA

D =

rad

iotr

acer

; SS

= s

eria

l sec

tioni

ng


Tabl

e 12

. Sum

mar

y ex

pres

sion

s fo

r ca

tion

diff

usio

n in

mag

netit

e an

d

titan

omag

netit

e as

fun

ctio

ns o

f te

mpe

ratu

re a

nd f O

2.

Dif

fusi

ng

Spec

ies

DV

,o

(m2 /

s)Q

V

(kJ/

mol

)D

I,o

(m2 /

s)Q

I (k

J/m

ol)

Ref

eren

ce

Cr

5.12

×10−

13−

7.3

3.84

×105

752.

5D

ieck

man

n et

al.

(198

7)

Al

3.24

×10−

13−

64.9

6.92

×10

681.

0D

ieck

man

n et

al.

(198

7)

Fe*

1.68

×10−

14−

123.

19.

79×1

0361

8.2

Agg

arw

al a

nd D

ieck

man

n (2

002b

)

Co*

2.20

×10−

15−

144.

88.

22×1

0361

2.0

Agg

arw

al a

nd D

ieck

man

n (2

002b

)

Mn*

2.02

×10−

16−

176.

72.

48×1

0360

4.6

Agg

arw

al a

nd D

ieck

man

n (2

002b

)

Ti*

1.37

×10−

14−

107.

62.

96×1

0468

8.7

Agg

arw

al a

nd D

ieck

man

n (2

002b

)

Fe (

x Ti =

0.2

) *1.

36×1

0−13

−16

7.5

2.33

×101

562.

3A

ggar

wal

and

Die

ckm

ann

(200

2b)

Co

(xT

i = 0

.2) *

5.55

×10−

13−

148.

01.

47×1

0258

4.8

Agg

arw

al a

nd D

ieck

man

n (2

002b

)

Mn

(xT

i = 0

.2) *

5.64

×10−

13−

149.

92.

47×1

0156

8.3

Agg

arw

al a

nd D

ieck

man

n (2

002b

)

Ti (

x Ti =

0.2

) *4.

07×1

0−13

−13

0.4

3.20

×109

840.

4A

ggar

wal

and

Die

ckm

ann

(200

2b)

The

dif

fusi

on c

oeffi

cien

t for

eac

h ca

tion

is e

xpre

ssed

as:

D

DQ

RTa

DQ

RTa

VV

II

*

,

/

,

/ex

p(/

)ex

p(/

)=

−+

−−

0

23

0

23

OO

22

whe

re a

O2 i

s f O

2/ fO

20 , w

ith f O

20 =

1 a

tm

*Cal

cula

ted

from

all

data

rep

orte

d in

Tab

le 5

of A

ggar

wal

and

Die

ckm

ann

(200

2b).

822 Van Orman & CrispinTa

ble

13. O

xyge

n se

lf-d

iffu

sion

in r

utile

.

Tem

pera

ture

R

ange

(K

)D

o (m

2 /s)

Q

(kJ/

mol

)D

T

(m2 /

s)E

xp.

Met

hod†

Ana

lysi

s†R

efer

ence

Not

es

873-

1373

1.14

×10−

1116

9—

WSX

SIM

S-D

PD

enni

s an

d Fr

eer

(199

3)na

tura

l, 10

0 M

Pa h

ydro

ther

mal

873-

1373

2.41

×10−

1217

3—

WSX

SIM

S-D

PD

enni

s an

d Fr

eer

(199

3)sy

nthe

tic, 1

00 M

Pa h

ydro

ther

mal

1173

-147

32.

40×1

0−6

283

—G

SXN

RA

Der

ry e

t al.

(198

1)||

c-ax

is

1079

——

1.70

×10−

19G

SXG

SMS

Gru

enw

ald

and

Gor

don

(197

1)⊥

c-a

xis

1079

——

3.20

×10−

20G

SXG

SMS

Gru

enw

ald

and

Gor

don

(197

1)||

c-ax

is

1373

——

1.45

×10−

16G

SXG

RA

VM

illot

and

Pic

ard

(198

8)||

c-ax

is;

log

D =

−17

.02

− 0

.123

log

fO

2 w

here

−16

< lo

g f O

2 < −

9 (a

tm)

1273

——

2.15

×10−

16G

SXG

SMS

Bag

shaw

and

Hyd

e (1

976)

smal

l par

ticle

s

1269

——

6.78

×10−

17G

SXG

SMS

Bag

shaw

and

Hyd

e (1

976)

larg

er p

artic

les

983-

1573

2.00

×10−

725

1—

GSX

GSM

SH

aul a

nd D

ümbg

en (

1965

)

1023

-127

34.

70×1

0−7

258

—W

SXN

RA

Moo

re e

t al.

(199

8)“f

aste

r” ||

c-a

xis

1023

-127

35.

90×1

0−5

330

—W

SXN

RA

Moo

re e

t al.

(199

8)“s

low

er”

|| c-

axis

1053

-127

31.

50×1

0−7

258

—W

SXN

RA

Moo

re e

t al.

(199

8)“f

aste

r” ⊥

c-a

xis

1173

-127

35.

08×1

0−6

316

—W

SXN

RA

Moo

re e

t al.

(199

8)“s

low

er”

⊥ c

-axi

s

1150

-145

03.

40×1

0−7

251

—G

SXSI

MS-

DP

Ari

ta e

t al.

(197

9)||

c-ax

is

1150

-145

02.

00×1

0−8

204

—G

SXSI

MS-

DP

Ari

ta e

t al.

(197

9)||

c-ax

is, 0

.08

mol

% C

r 2O

3 do

ped

† Abb

revi

atio

ns: W

SX =

wat

er-s

olid

exc

hang

e; G

SX =

gas

-sol

id e

xcha

nge;

SIM

S-D

P =

sec

onda

ry io

n m

ass

spec

trom

etry

dep

th p

rofil

ing;

NR

A =

nuc

lear

rea

ctio

n an

alys

is; G

SMS

= g

as s

ourc

e m

ass

spec

trom

etry

; GR

AV

= g

ravi

met

ric

anal

ysis


Tabl

e 14

. Tet

rava

lent

cat

ion

diff

usio

n in

rut

ile.

Dif

fusi

ng

Spec

ies

Tem

pera

ture

R

ange

(K

)D

o

(m2 /

s)Q

(k

J/m

ol)

Exp

. M

etho

d†A

naly

sis†

Ref

eren

ceN

otes

Ti

1173

-157

36.

40×1

0−6

257

RA

DSS

Ven

katu

and

Pot

eat (

1970

)||

c-ax

is; a

ir

Ti

1470

-178

34.

60×1

0−6

251

RA

DSS

Lun

dy a

nd C

oghl

an (

1973

)||

c-ax

is; a

ir

Ti

1470

-178

32.

40×1

0−7

203

RA

DSS

Lun

dy a

nd C

oghl

an (

1973

)⊥

c-a

xis;

air

Ti

1373

-167

36.

50×1

0−4

277

RA

DSS

Hos

hino

et a

l. (1

985)

|| c-

axis

; fO

2 = 1

.5×1

0−2

Pa

Ti

1373

-167

34.

60×1

0−4

268

RA

DSS

Hos

hino

et a

l. (1

985)

⊥ c

-axi

s; f O

2 = 1

.5×1

0−2

Pa

Ti

1373

-167

33.

68×1

0−6

253

RA

DSS

Hos

hino

et a

l. (1

985)

|| c-

axis

; air

Ti

1373

-167

31.

75×1

0−6

233

RA

DSS

Hos

hino

et a

l. (1

985)

⊥ c

-axi

s; a

ir

Ti

1273

-137

32.

18×1

0−4

210

RA

DSS

Aks

e an

d W

hite

hurs

t (19

78)

|| c-

axis

; air

Ti

1273

-137

32.

93×1

0646

5R

AD

SSA

kse

and

Whi

tehu

rst (

1978

)||

c-ax

is; f

O2 =

1.7

×10−

11 P

a

Hf

1023

-132

32.

50×1

0−12

227

SSX

RB

SC

hern

iak

et a

l. (2

007)

⊥ c

-axi

s; a

ir, Q

FM, N

NO

Hf

1073

-127

39.

10×1

0−15

169

SSX

RB

SC

hern

iak

et a

l. (2

007)

|| c-

axis

; air

, QFM

, NN

O

Zr

1023

-137

39.

80×1

0−15

170

SSX

RB

SC

hern

iak

et a

l. (2

007)

|| c-

axis

; air

, QFM

, NN

O

Zr

1373

-177

42.

31×1

0−7

291

RA

DSS

Sasa

ki e

t al.

(198

5)||

c-ax

is; a

ir

Zr

1373

-177

48.

40×1

0−7

288

RA

DSS

Sasa

ki e

t al.

(198

5)⊥

c-a

xis;

air

† Abb

revi

atio

ns: R

AD

= r

adio

trac

er; S

SX =

sol

id-s

olid

exc

hang

e; S

S =

ser

ial s

ectio

ning

; RB

S =

Rut

herf

ord

back

scat

teri

ng s

pect

rosc

opy


Tabl

e 15

. Tri

vale

nt a

nd d

ival

ent c

atio

n di

ffus

ion

in r

utile

.

Dif

fusi

ng

Spec

ies

Tem

pera

ture

R

ange

(K

)D

o (m

2 /s)

Q

(kJ/

mol

)E

xp.

Met

hod†

Ana

lysi

s†R

efer

ence

Not

es

Cr

973-

1673

1.36

×10−

913

8R

AD

SSSa

saki

et a

l. (1

985)

|| c-

axis

; air

Cr

973-

1673

1.08

×10−

816

1R

AD

SSSa

saki

et a

l. (1

985)

⊥ c

-axi

s; a

ir

Sc11

76-1

773

1.98

×10−

818

8R

AD

SSSa

saki

et a

l. (1

985)

|| c-

axis

; air

Sc11

76-1

772

1.57

×10−

916

9R

AD

SSSa

saki

et a

l. (1

985)

⊥ c

-axi

s; a

ir

Ba

1373

-147

31.

24×1

0−11

172

RA

DSS

Nak

ayam

a an

d Sa

saki

(19

63)

|| c-

axis

; air

Ba

1373

-147

35.

42×1

0−8

264

RA

DSS

Nak

ayam

a an

d Sa

saki

(19

63)

⊥ c

-axi

s; a

ir

Pb97

3-13

733.

92×1

0−10

250

SSX

RB

SC

hern

iak

(200

0)⊥

c-a

xis,

syn

thet

ic r

utile

; QFM

, NN

O

Pb10

73-1

775

1.55

×10−

1024

2SS

XR

BS

Che

rnia

k (2

000)

both

dir

ectio

ns, n

atur

al r

utile

; QFM

, NN

O

Fe10

73-1

273

1.98

×10−

623

0R

AD

SSIz

beko

v an

d G

orbu

nova

(19

59)

pres

sed

pow

der;

air

Fe10

43-1

273

1.92

×10−

523

2R

AD

SSIz

beko

v an

d G

orbu

nova

(19

59)

pres

sed

pow

der;

vac

uum

Fe87

4-16

732.

73×1

0−7

124

RA

DSS

Sasa

ki e

t al.

(198

5)||

c-ax

is; a

ir

Fe97

2-12

733.

08×1

0−8

135

RA

DSS

Sasa

ki e

t al.

(198

5)⊥

c-a

xis;

air

Co

796-

1173

2.24

×10−

413

2R

AD

SSSa

saki

et a

l. (1

985)

|| c-

axis

; air

Co

1073

-130

51.

86×1

0−5

180

RA

DSS

Sasa

ki e

t al.

(198

5)⊥

c-a

xis;

air

Ni

975-

1179

1.34

×10−

512

6R

AD

SSSa

saki

et a

l. (1

985)

|| c-

axis

, air

Mn

865-

1664

5.82

×10−

713

7R

AD

SSSa

saki

et a

l. (1

985)

|| c-

axis

; air

† Abb

revi

atio

ns: R

AD

= r

adio

trac

er; T

HIN

= th

in fi

lm s

ourc

e; S

SX =

sol

id-s

olid

exc

hang

e; S

S =

ser

ial s

ectio

ning

; SIM

S-D

P =

sec

onda

ry io

n m

ass

spec

trom

etry

dep

th p

rofil

ing;

RB

S =

Rut

herf

ord

back

scat

teri

ng s

pect

rosc

opy


Tabl

e 16

. Mon

oval

ent c

atio

n di

ffus

ion

in r

utile

.

Dif

fusi

ng

Spec

ies

Tem

pera

ture

R

ange

(K

)D

o (m

2 /s)

Q

(kJ/

mol

)E

xp. M

etho

d†A

naly

sis†

Ref

eren

ceN

otes

Li+

353-

633

2.95

×10−

532

(see

not

es)

OPT

AJo

hnso

n (1

964)

|| c-

axis

; out

-dif

fusi

on in

to

wat

er o

r al

kali

nitr

ate

bath

s

1 H/2 H

623-

973

1.80

×10−

757

GSX

IRJo

hnso

n et

al.

(197

5)||

c-ax

is

1 H/2 H

887-

994

3.80

×10−

512

4G

SXIR

John

son

et a

l. (1

975)

⊥ c

-axi

s

3 H42

8-57

37.

50×1

0−10

38G

SXA

RA

DC

aske

y (1

974)

|| c-

axis

3 H42

8-57

32.

70×1

0−10

55G

SXA

RA

DC

aske

y (1

974)

⊥ c

-axi

s

3 H77

3-11

831.

77×1

0−6

107

GSX

SSC

athc

art e

t al.

(197

9)⊥

c-a

xis,

“fa

st”

3 H92

4-11

758.

90×1

0−6

187

GSX

SSC

athc

art e

t al.

(197

9)⊥

c-a

xis,

“sl

ow”

3 H52

7-97

38.

50×1

0−7

72G

SXSS

Cat

hcar

t et a

l. (1

979)

|| c-

axis

† Abb

revi

atio

ns: G

SX =

gas

-sol

id e

xcha

nge;

OPT

A =

opt

ical

abs

orpt

ion;

IR

= in

frar

ed s

pect

rosc

opy;

AR

AD

= a

utor

adio

grap

hy; S

S =

ser

ial s

ectio

ning

vol. 72 pp. 757-825, 2010 17 copyright © mineralogical

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