Phys Chem Minerals (1994) 21:489-500 PlfllSICS ]CHEMISTRY ]I]MIHERAIS �9 Springer-Verlag 1994

Mg Tracer Diffusion in Synthetic Forsterite and as a Function of P, T and fOz Sumit Chakrahorty ~'*, John R. Farver 2, Richard A. Yund 2, David C. Rubie s

1 Bayerisches Geoinstitut, Universitfit Bayreuth, D-95440 Bayreuth, Germany 2 Department of Geological Sciences, Brown University, Providence, RI 02912, USA

Received June 10, 1994/Revised, accepted July 20, 1994

San Carlos Olivine

Abstract. We present new experimental data on Mg trac- er diffusion in oriented single crystals of forsterite (FOl0o) and San Carlos olivine (Foga) between 1000-1300 ~ C. The activation energies of diffusion are found to be 400 (_+ 60) kJ/mol (~ 96 kcal/mol) and 27 5 (_+ 25) kJ/mol (~ 65 kcal/ mol) in forsterite and San Carlos olivine, respectively, along [001] at a fO2 of 10 -~2 bars. There is no change in activation energy of Mg tracer diffusion within this tem- perature range. Mg tracer diffusion in a nominally pure forsterite is found to be anisotropic (DjI o > DIj a > D jib) and a function of fO2. This fO2 dependence is different from that in olivine containing Fe as a major element, which suggests that the diffusion mechanism of Mg in forsterite is different from that in Fe-bearing olivine at least over some range of fO2. The diffusion mechanism in nominally pure forsterites may involve impurities present below the limits of detection or alternately, Si or Fe 3+ interstitial defects, Fe being present as impurity (ppm level) in forsterite. Pressure dependence of Mg tracer diffusivity in forsterite measured to 10 GPa in a multianvil apparatus yields an activation volume of approximately 1-3.5 cm3/ tool. It is found that presence of small amounts of hydro- gen bearing species in the atmosphere during diffusion anneal (fH2,~0.2 bars, fm0~0.24 bars) do not affect Mg tracer diffusion in forsterite within the resolution of our measurement at a total pressure of 1 bar. The observed diffusion process is shown to be extrinsic; hence extrapo- lation of the diffusion data to lower temperatures should not be plagued by uncertainties related to change of dif- fusion mechanism from intrinsic to extrinsic.

Present address: Mineralogisch-Petrographisches Institut, Uni- versit/it zu K61n, Ztilpicher StraBe 49b, D-50674 K61n, Germany Correspondence to: S. Chakraborty

Introduction

Knowledge of cation diffusion rates in olivine are neces- sary for a quantitative understanding and modeling of a variety of processes in earth and planetary sciences including creep in the earth's upper mantle, closure of ion-exchange geothermometers involving olivine and modeling observed compositional zoning profiles in oliv- ine to understand cooling rates and thermal histories of terrestrial rocks and meteorites. Although a substan- tial body of diffusion data exist already for olivines of several compositions (see, for example, Morioka and Na- gasawa 1991; Hermeling and Schmalzried 1984), there are important unresolved questions which this study ad- dresses through the determination of Mg tracer diffusion coefficients in forsterite (Fol00) and San Carlos olivine (Fo92). We first present a set of Mg tracer diffusion coeffi- cients as a function of temperature in a Fe-bearing oliv- ine (San Carlos) and as a function of pressure in forsterite (Fo~oo) which were not previously available, Secondly, we use these data to address the question of whether there is a compositional dependence of Mg tracer diffusi- vity in the olivine solid solution series. Thirdly, in view of some inconsistency among various available data sets for Mg tracer diffusion in forsterite (Morioka 1980; Sockel and Hallwig 1977; Sockel et al. 1980; Andersson 1987) we have obtained a set of tracer diffusion data in forsterite under carefully characterized conditions which allows us to constrain the influence of various factors on diffusion rates. These data will also be useful for comparisons with results of computer calculations of energetics of diffusion (activation energy, defect forma- tion energies etc.) in order to assess the possible mecha- nisms that control defect formation and migration in olivines.

Finally, we discuss whether the observed diffusion oc- curs by an intrinsic or extrinsic mechanism. This issue is of practical importance because the temperature of transition from extrinsic to intrinsic mechanism deter- mines the reliability of extrapolations of experimentally measured diffusion data to low temperatures.

490

Table 1. Analyses of olivine crystals Sample Forsterite Forsterite Ideal San Carlos olivine

(Fo 1 - Czochralski (Fo 2 - Verneuil Forsterite (Smithsonian grown)" grown) b 4t: 136718) c

Oxide wt % wt % wt % wt %

SiO 2 42.5-42.7 42.82 42.69 40.5 MgO 57.5-57.8 57.16 57.31 48.7 FeO 9.4 MnO 0.2 NiO 0.5 CaO < 0.01

Trace Fe (i) 122 ppm d Fe 11 ppm, elements Fe (ii) 180 ppm d A1 5 ppm,

Zn 4-10 ppm a Cu 3 ppm, V 10-15 ppm d Mn 3 ppm Ir 16-18 ppm Ca < 1 ppm Na < 10 ppm

Total 99.99 99.3

Analyses: a Takei and Kobayashi (1974). b Graham and Barsch (1969); Jaoul et al. (1980). c Smithsonian Institution. d ICP-AES at Bayerisches Geoinstitut; 0)'(il)-analyses of different sections of the same crystal

Experimental Method

Starting Material

Single crystals of synthetic forsterite (Fol) prepared using the Czochralski method by H. Takei at Tohoku University and natural Fo92 olivine from San Carlos, Arizona obtained from the Smith- sonian Museum (4t: 136718) were primarily used for the diffusion experiments. Two additional measurements were made on a second sample of synthetic forsterite (Fo2) prepared by the Verneuil tech- nique at Union Carbide Corporation. Chemical composition of these crystals, as provided to us, are given in Table 1. Additional trace element analyses for the Czochralski grown forsterite (Fo 1) used predominantly in this study were performed using an ICP- AES at Bayerisches Geoinstitut.

The crystals were oriented using Laue X-ray diffraction pat- terns. The oriented crystals were sawn into pieces 2 mm on a side and a surface perpendicular to one of the crystal axes was ground flat, polished and prepared for diffusion anneal. In order to test for the possibility of enhanced diffusion along surface defects intro- duced during mechanical polishing we measured diffusion rates perpendicular to (001) in a set of identical samples run in the same charge but prepared using three different polishing techniques: i) fine polishing using a diamond based polishing compound (0.25 gin), ii) brief etching with HF of a surface prepared as in (i), or iii) final polishing using a chemical polishing technique where the polishing compound was a highly alkaline colloidal silica solu- tion. No difference in retrieved diffusivities were observed from these differently polished samples (see later) and most runs were done using samples prepared following method (i).

Diffusion Anneal 1 Atmosphere

A 1000 ppm solution of 26MgO (~98 atom% enriched) was pre- pared by dissolving the appropriate amount of 26MgO powder in an aqueous solution containing enough HC1 to provide a stoi- chiometric equivalent of chloride ions to Mg ions (2 C1 per 1 Mg). Several drops of this solution were evaporated onto the polished smface of each crystal. The sample was then heated over several

hours to 350-400 ~ C to decompose the magnesium chlorides, leav- ing a 26MgO layer on the surface. The MgO layer consists of micro- granular particles that cover the surface of the olivine in homoge- neous patches. There is some recrystallization and surface migra- tion during the diffusion anneal, as evidenced by the presence of more isolated islands of coating on the surface after the anneals compared to the starting samples. This polycrystalline coating of 26MgO provides the tracer source for the diffusion experiments (see Cygan and Lasaga 1985). Additional experiments using solu- tions of 26MgO in HNO3 and oxalic acid were attempted. The nitric acid solution was not used because it reacted with olivine and corroded the surface. The precipitate from the oxalate was too dilute and did not provide a sufficiently enriched tracer source.

In order to induce Mg tracer diffusion, the 26MgO coated sam- ples were annealed at temperatures between 1000-1300 ~ C for vary- ing lengths of time. The oxygen fugacity was buffered for most runs at 10 - l z bars using a flowing gas mixture of C O - C Q hut in one case a mixture of H z + C O z was used. Some runs were carried out in air or other oxygen fugacities. The activity of silica (as~o~) relative to an arbitrary standard state would normally be buffered by the MgO used as the tracer source coexisting with the forsterite. However, attempts were made to test for the effects of perturbing asio2 by surrounding the crystal with powdered MgO, enstatite or none of these potentially buffering compounds. When nothing is added to the charge, the SiOz activity should be buffered by the coexisting MgO - often dissolved as excess in the crystal and also present as the tracer source for the diffusion anneals.

At the end of the diffusion anneal, the isotopic concentration of 26Mg on the surface of the crystal was often found to be lower than the initial concentration in the 26MgO. This could result from either incomplete coverage of the crystal surface by the precipitate or from a steady depletion of the tracer source during the diffusion anneal (see Cygan and Lasaga 1985). Non-uniform coverage of the surface by the tracer layer should not affect retrieved diffusivities (Tannhauser 1956), and, as discussed below, appropriate models for diffusion are available and have been tested for both constant and steadily depleting source conditions. However, reproducibilities of calculated diffusion coefficients and independence of the diffusion coefficient on the duration of diffusional anneal (see later) suggest that the variability of 26Mg concentration on the surface did not significantly affect the measurements. Given that the amount of

491

26Mg evaporated onto the surface greatly exceeds the amount nec- essary to maintain a constant source, we believe that the lower surface concentrations are the result of poor bonding of the polyc- rystalline tracer (as proposed by Cygan and Lasaga 1985 as well). The observations that surface concentration of the tracer shows no obvious correlation with duration of anneal and the concentra- tion of a6Mg on the surface of the same crystal may vary depending on the location of the analysis further suggest that non-uniform coverage rather than steady depletion is the reason for low Z6Mg on the surface.

Diffusion Anneal - High Pressure

Diffusion anneals at high pressures were carried out in a 1200 tonne uniaxial split-sphere multianvil apparatus. An MgO octahe- dron with an 18 mm edge length was used as the pressure medium and a stepped LaCrO3 heater was used to heat the sample and minimize thermal gradient. An axially placed W3%Re-W25%Re thermocouple was used to monitor temperature during the diffu: sion anneal. Details of the sample assembly as well as pressure and temperature calibrations may be found in Rubie et al. (1993a; 1993 b).

The sample size for the high pressure diffusion experiments were smaller than the 1 atmosphere anneals, i m m x l mm x 1 mm sized cubes of forsterite (Fol) were prepared as for the low pressure anneals and the edges of the cubes were bevelled to prevent a concentration of stress at sharp corners during pressurization, A polished surface on each of these bevelled cubes was then coated with a 26MgO layer following the procedure outlined above. The coated crystal was completely wrapped in a 25 pm thick Au foil and placed inside a Pt capsule surrounded by NaC1. Care was taken at each step to keep the sample assembly as dry as possible and details of the procedure used are provided in Rubie et al. (1993a).

Most of the compression and decompression were carried out at 600 ~ C to minimize the strength of the NaC1 surrounding the sample, following a P-T-time path as outlined in RuNe et al. (1993a). This procedure enables single crystals of olivine to be pres- surized to 10 GPa and subsequently recovered without ductile de- formation or fracturing (RuNe et al. 1993 a). Pressure and tempera- ture were maintained by a computer controlled system. Nominal annealing conditions ranged between 1100-1200 ~ C, 6-10 GPa, 18- 40 hours. Single crystals were recovered after the high pressure runs by dissolving the surrounding NaCI pressure medium and unwrapping the Au foil taking care not to disturb the tracer layer on the surface of the crystal.

Measurement of Diffusion Profiles

After the diffusion anneal, excess coating is removed by washing in ethanol in an ultrasonic bath which does not affect the crystal. Profiles of 26Mg concentration versus depth into the crystal were measured using a Cameca IMS 3 f ion microprobe at the Woods Hole Oceanographic Institute. Although some residual mounds of MgO are often left on the surface after washing, analyses were carried out in clean areas with a planar surface to ensure that a flat bottomed crater was obtained after sputtering. Details of the technique have been reported previously (e.g. Giletti et al. 1978). The sputtering employed a ~ 50 gm diameter primary beam of O- ions with an accelerating voltage of 13.1 KeV. The rastered area was a square ~ 150 to 175 gm on a side, and Mg + ions were analyzed. A mechanical aperture, carefully centered on the sput- tered area, was introduced into the ion optics, limiting the area from which data were collected to a circle 8 ~tm in diameter. Masses 26, 24, 25.5 (background), and 28 (Si) were measured while sputter- ing the sample. Monitoring Si helps to check for stoichiometry of analyses as well as to locate the position of the interface between oxide/silicate. Bore hole depths were determined using an optical interferometer and monochromatic green light (540 nm).

90

80

70

8o

I 40 30

20

10 0

a,

f r o

'7

0 0

b

[ ]

[ ]

Forsterite 1150 ~ | O2= 10"12bars 20.5 hours

[] []

[] []

[] IZl,ff] [] E ]

. . . . . . . . . . . . . . . . . .

0,5 1.0 1.5 2.0 2.5 3.0 Depth (gm)

Forsterite 1150 ~ f 02= 10"12bars

r . ~ i �9 , I = i = = I = = i i ! I i �9 ' I . . . . [ _ _ �9 ' = i = = i i i . . . . . . I i I I

0,5 1.0 1.5 2.0 2.5 3.0 Depth (gin)

Fig. 1. a A representative profile of 26Mg in forsterite single crystal (Fo 1) along [001] and (b) Linearized fit to inverse error function. The data are for run ~ O I-15 b, see Table 2 for details of run condi- tions

Fitting of Diffusion Profiles and Calculation of Diffusion Coefficients

The diffusion profiles can be fit to either a constant or a steadily depleting source solution. The diffusion equation for transport nor- mal to the surface of a semi-infinite volume with constant surface concentration is (Crank 1975):

Cx-C1 erf x (1) Co-C1 2 ~ t

where C x = the concentration at some depth x (x = 0 is the interface between oxide and silicate), Co=the initial concentration in the olivine, C1 =the concentration at the crystal surface, D =diffusion coefficient and ~ = duration of the diffusion anneal. Err is the error function. For a steadily depleting source according to Cx=o= C1--kt (k=a constant), solutions have been provided in Crank (e.g. eqn.3.32, 1975):

C=Co+4kti2erfc x

The diffusion coefficients from fitting these two equations were found to be very similar (within a factor of two) and within the scatter of the data (e.g. see run # 12a, Table 2). Thus, the D values presented below were calculated using a constant tracer source (Eq.(1)). The evidence presented above for the existence of an infi- nite reservoir and the likely operation of significant surface diffusion

492

Table 2. Summary of diffusion anneal conditions at 1 atm and data

Synthetic forsterite (Fo I)

O1- i b 1 I00 1 e- 121 None [001J 90000 2.70 e- 19 - 18.57 4.03e-19 - 18.39

Average 3.37 e- 19 - 18.47

O l - l a z 1100 le-12 None [001] 90000 2.32e-19 - 18.64 2.11e-19 -18 .68

Average 2.21 e-19 - 18 .66

O1-2 r 1100 air None [001] 346320 3.34 e- 18 - 17.48 3.28 e-18 - 17.48 4.42e-18 -1 7 .3 6

Average 3.68 e-i 8 - 17.43

O1-4 1100 1 e- 12 Enstatite [001] 84 000 6.91 e- 19 - 18.16 7.46e-19 -18 .13 6.91 e-19 - 18.16

Average 7.09 e-19 - 18 .15

O1-5 1100 1 e-12 Enstatite [001] 341400 3.32e-19 - 18.48 3.50e-19 -18 .46 3.66e-19 - 18.44

Aver age 3,49 e- 19 - 18 .46

O I-6 r 1100 air Enstatite [001 ] 94 788 2.88 e- 18 - 17.54 2,58e-18 -17.59

Average 2.73 e- 18 - 17 .56

O1-7 1125 1 e-6 Enstatite [001] 97200 4.70e-19 - 18.33 3.80e-19 - 18.42 7.46e-19 -18 .13

Average 5.32e-19 - 18 .27

O1-8 a 1100 I e-12 None [001] 86400 5.92e-19 - 18.23 2.57e-19 -18 .59 2.19e-19 -18 .66

Average 3.56 e- 19 - 18.45

O1-9 b 1100 1 e- 12 MgO [001] 87300 5.68 e- 19 - 18.25 5.26e-19 - 18.28 6.21e-19 -18 .21

Average 5.72 e- 19 - 18.24

O1-I0 1 I00 air MgO [001] 77400 6.48 e-19 - 18.19 9.44e-19 - 18.03

Average 7.96 e- 19 - 18 .10

O1-13 a 1300 1 e-12 None [001] 15300 2.42e-17 - 16.62 4.65e-17 -16 .33 2.40e-17 -16 .62

Average 3.15 e- 17 - 16.50

O1-12 a 1200 1 e-12 None [001] 44388 8.31 e-18 - 17.08 5.64e-183 -17 .253 9.7e-18 4 -17 .014 7.38e-18" - 1 7 . 1 3 " 5.62e-18 - 17.25

Average 6.52 e- 18 - 17.19

Ol- 14 a 1000 1 e- 12 None [001 ] 1 176 300 5.46 e-20 -- 19.26 6.73 e-20 - 19.17 6.00e-20 - 19.22

Average 6.06 e-20 - 19.22

Ol-15a 1150 le-12 None [00l] 73800 5.73e-18 -1 7 .2 4 4.89e-18 - 1 7 . 3 l

Average 5.31 e- 18 - 17.27

O1-15b 1150 le-12 None [001] 73800 7.85e-18 - 1 7 . 1 0 (Run at (H2/CO2 6.85e-18 -- 17.16

Brown mix) Univ.)

Run Temp. f02, Solid compd. Transpor t Dura t ion D log D number (~ bars a round sample direction (sec) (m2/sec)

Table 2 (continued)

Run Temp. fO2, Solid compd. Transpor t Dura t ion D log D number (~ bars a round sample direction (see) (mZ/see)

493

Average 7.35 e- 18 - 17.13

O1-24 r 1100 air None [001] 90 900 2.68 e- 18 - 17.57 5.75e-19 - 18.24 1.54e-18 - 17.81 7.84e-19 -18 .11

Average 1.39e-18 - 17.86

O1-25 1100 air None 1-001] 86400 2.84e-18 -- 17.55 2.13e-18 --17.67

Average 2.49 e- 18 - 17.60

O1-26 1100 le-12 None [010] 86400 5.59 e-20 -- 19.25 7.59e-20 - 19.12 6.78 e-20 -- 19.17

Average 6.65 e-20 - 19.18

O1-28 1100 1 e- 12 None [100] 93 600 1.48 e-19 - 18.83 1.16e-19 - 18.94 4.56e-20 - 19.34

Average 1.03 e-19 - 18.99

O1-31 1054 1 e-12 None [001] 664 560 1.40e-19 -- 18.85 9.09e-20 -- 19.04

Average 1.16 e- 19 - 18.94

Synthetic forsterite (Fo 2) O1-32 1100 le-12 None [001]

Average

O1-33 1100 Air None [001]

Average

82800

86400

1.03e-18 -17 .99 1.18e-18 -17 .93 8.81e-19 -18 .05 1.03 e- 18 - 17.99

1.63e-18 - 17.79 1.06e-18 - 17.97 6.96e-19 -18 .16 1.13e-18 -1 7 .9 5

San Carlos Olivine

O1-3 b 1100 le-12

Average

O1-8b 1100 le-12 Average

O1-13b 1300 le-12

Average

Ol-12b 1200

Average

Ol-14b 1000

Average

O1-34 1100

le-12

le-12

le-9

Enstatite [001 ] 346 320 1.04 e- 18 - 17.98 2.24e-18 - 17.65 1.64e-18 - 17.79

None [001] 86400 1.52e-18 - 17.82 - 17.82

None [001] 15 300 5.69 e-17 - 16.24 5.37e-17 -16 .27 4.53e-17 -1 6 .3 4 5.20e-17 - 16.28

None [-001] 44388 1.22e-17 -- 16.91 1.31e-17 --16.88 1.12e-17 --16.95 1.22e-17 - 16.91

None [001] 1 176300 4.29e-19 -- 18.37 4.77e-19 -- 18.32 4.53e-19 - 18.34

None [001] 86400 6.1e-18 - 17.21

* D calculated from profile of 26Mg/28Si 1 Read le-12 as 1 x 10 -12 etc. 2 H F etched after d iamond polished 3 D retrieved using eqn. (1), 4 D retrieved from same profile using steadily depleting reservoir model (see text for details) r Rapid heat up to run temperature, all other unmarked runs were heated up slowly

-t6

T (~

1300 1200 1100 1000 I I I t

Run Temp P Time D log D number (~ (GPa) (sec) (mZ/s)

-q7 OLP-1 1200 10 64620 1.19e-18 - 17.92 "~

7.66e-19 -18.11 E Average 9.78e-19 - 18.00 ~ -18

OLP-3 1100 10 136800 1.90e-19 - 18.72 1.46e-19 -18.83

Average 1.68e-19 -18.77 -19

OLP-4 1100 6 117000 2.13e-19 -18.67 1.79e-19 -18.74

Average 1.96e-19 -18.71 -206.1

OLP-5 1100 10 126000 9.91e-20 - 19.00 1.12e-19 -18.95 a 1.06e-19 - 18.98

during diffusion anneal in the MgO layer suggest that the solution of the diffusion equation for an infinite, well-stirred reservoir is reasonable for modelling the data.

According to Eq.(1), the inverse error function of the data from a depth profile should yield a linear array (e.g. Fig. 1 b). A least squares linear regression fit to the data then has a slope equal to 2(Dr) 1/2. Knowing the duration of anneal, t, the diffusion coeffi- cient D can be calculated from the slope of the regression line.

R e s u l t s

The experimental results are presented in Tables 2 (1 atmosphere) and 3 (high pressure) along with the details of the run conditions. Results are grouped into sets of bores made on different places on the surface of an indi- vidual crystal, and the average of these bores is the value of D assigned to that experiment. There are two forms of uncertainties - variation in D values obtained from different bores on the same crystal and variation in aver- age D values obtained from different crystals annealed at the same conditions. The reproducibility of the data, which gives an useful estimate of the uncertainty in D, is within a factor of three.

Within the uncertainty of the average D values for forsterite Fo 1, there is no significant effect of (i) the var- ious surface preparat ion techniques (run # 1 a, lb), (ii) the presence or absence of the SiO2 activity "buffering" compounds around the single crystal during the diffusion anneal (run ~: l b, 1 a, 4, 5, 8a, 9b), (iii) the duration of the diffusion anneal (run ~: 1 a, 1 b, 4, 5) and (iv) slow or rapid heating up for the diffusion anneal (run #2 , 6, 24, 25). Experiments carried out in two different labs (Bayreuth and Brown) at the same conditions yield com- parable results (run =~ 15 a, 15 b).

Runs numbered 15a and 15b were buffered at a fO2 of 10 -12 bars using gas mixes of C O - C O 2 and H2 - C O 2 , respectively. Minimization of Gibbs free energy in the system C - O - H at 1 bar, 1150~ and the known ratio of H2 and CO2 used in the experiment yield fugacities of other species as follows: fH2=0.203, fco = 0.244, fco~ = 0.306 and fH2o = 0.244 bars. This calcula- tion and the similarity of the diffusivities obtained from

494

Table 3. Summary of diffusion anneal conditions at high pressures and data

I I I 6.6 7.1 7.6

104/T (k) 8.1

T (~ 1300 1200 1100 1000

- 1 6 i j I

c~ -18

. J

-19

-20 I I I 6.1 6.6 7.1 7.6 8.1

b lOZ~/T (k)

Fig. 2. Temperature dependence of diffusivity in (a) forsterite (Fo 1) and (b) San Carlos olivine for runs at fOz=10 -lz bars, with no extra solid "buffering" compound added, illustrated in an Arrhen- ius plot. For activation energy and pre-exponential factors obtained by fitting these data (9 points for Fo 1, 5 for San Carlos), see text

runs 15 a and 15b shows that presence of small amounts of hydrogen bearing species in the atmosphere has no significant effect on Mg tracer diffusion in single crystal forsterite at 1 atmosphere total pressure. Temperature, pressure, crystallographic orientation, composit ion and oxygen fugacity were found to have significant, or at least measurable, effects on diffusivity. These are dis- cussed in detail below.

The results of all 1 bar experiments on forsterite and San Carlos olivine at an fOz of 10-12 bars, with no addi- tional solid compound or MgO around the crystal, are shown in an Arrhenius plot of the logarithm of D against reciprocal of absolute temperature (Fig. 2). The solid line is a linear regression fit to the data with an r2=0.93 for forsterite and 0.99 for San Carlos olivine. The fits yield the following diffusion parameters for the two com- positions over the range 100(~1300 ~ C for t ransport par- allel to the c-axis under a oxygen fugacity of 10-12 bars:

Forsterite: Do=9 . 6 x 10 .4 m2/sec, Q = 400 __ 60 k J/tool (96 __ 13 kcal/mol)

San Carlos: Do = 5.6 x 10 .8 m2/sec, Q = 275 4- 25 kJ/mol (65 4- 6 kcal/mol)

495

The results show that at any temperature below 1300 ~ C, Mg tracer diffusivity in forstrite is slower than in San Carlos olivine. At 1100 ~ C the difference is about a factor of four and this difference increases with decreasing tem- perature as a consequence of the higher activation energy for Mg tracer diffusion in forsterite compared to San Carlos olivine.

At 1100 ~ C and a fO2 of 10 -12 bars, Mg tracer diffu- sion is anisotropic in forsterite. Diffusion is fastest along the c-axis and decreases in the order DIIo>Dlla>DIIb. Using mean values from run ~ s 1 a, 1 b, 4, 5, 26 and 28 we find diffusivity along c~-4 x DII~-~6 x Diib, similar to the anisotropy obtained from interdiffusion studies in olivine (Clark and Long 1971; Buening and Buseck 1973; Misener 1974; Morioka 1980) and tracer diffusion studies in forsterite [Andersson (1987) found Dll b -~DI1~-~0.16 D[I ~ at 1100 ~ C]. However, considering the factor of three uncertainty in the measurement of diffu- sion coefficients, the magnitude of this anisotropy may be smaller.

The Mg tracer diffusion coefficient in the nominally pure single crystal forsterite Fo 1 is found to be a weak function offO2 at 1100 ~ C. A number of runs at 1100 ~ C were done in air (run ~# 2, 6, 24, 25) and the results from these runs were compared with the data at 1100 ~ C col- lected at a fO2 of 10-~z bars (run #~la, lb , 4, 5, 8a, 9b). The diffusion coefficients determined for Fo 1 in samples annealed in air are systematically greater (Fig. 3) than those determined from samples annealed at a fO2 of 10-12 bars. This fO2 dependence is found to be much smaller than the approximately pO 1/6 dependence of ca- tion diffusivity found in Fe-bearing olivine (Buening and Buseck 1973; Nakamura and Schmalzried 1984; Hermel- ing and Schmalzried 1984). However, on the other no- minally pure forsterite, Fo 2, there is no detectable differ- ence (Fig. 3) in diffusivities measured in air or at a p O 2 of 10-12 bars (run ~ 32, 33).

Mg tracer diffusion coefficients measured L (001) for Fo 1 at high pressures are given in Table 3 and are shown in Fig. 4. The data suggest that Mg tracer diffusion coef- ficients for forsterite decrease slightly with pressure. However, a few aspects of the high pressure experiments merit discussion here before the results may be interpret- ed. No attempts were made to control fO 2 or fI-I20 dur- ing the diffusion anneals. Using a similar setup as that used in this study, RuNe et al. (1993 a) found that initially dry Fe-bearing olivines picked up some hydrogen at high pressures during anneals. Thus, addition of some H to the samples during the high pressure diffusion anneals cannot be precluded. However, the low pressure data suggest that the presence of small amounts of hydrogen and water in the atmosphere do not significantly affect diffusion rates of Mg in Fo 1 and the effect of fO2 is measurable but small over a large range of fO2 (see dis- cussion above). This gives us some confidence in compar- ing the high pressure data with the low pressure results.

A further uncertainty arises from the measurement of temperatures at high pressures because of the effect of pressure on thermocouple EMF. This effect for a W- Re thermocouple is believed to be small (Williams and Kennedy 1969; Ohtani 1979; Tsuzaki and Takahashi

& c~

-17.0

-17.5

-18.0

-18.5

0 Foi

�9 Fo2

�9 ../ . - ../"

. . ~ ' " ....,"

................................................... . .

0

O

.O .-i." .-"""" i"""

O

- 1 9 . 0 I , I I I I I

-14.0-12.0 -10.0 - 8 . 0 - 6 . 0 - 4 . 0 - 2 . 0 0.0 2.0 Log fO 2

Fig. 3. Mg tracer diffusion data in different synthetic forsterites at 1100 ~ C as a function of fO2 (in bars) with no extra solid "buffer- ing" compound added. Symbols: Squares: Fol Circle: Fo2. The proposed alternative point defect models for explaining the fO2 dependence are illustrated by dotted (Model I) and dashed (Model II) lines

-17.0

-17.5

5 -18.0

Ch

b/] -18.5 O

-19.0

+ Air 0 f02=lO -'2 bar

+

..... 3.5 cma/mol

O 1 cm/mol 'E) ...... "'-. E) "@

-19.5 I I I I I I -2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0

P [CPa] Fig. 4. Mg tracer diffusion coefficient as a function of pressure at 1100 ~ C in synthetic forsterite Fo 1. Circles: 1 atm. data at fO2 =10 -~2 bars, (run 4#s la, lb, 4, 5) Cross: 1 atm. data in air (run 4#s 2, 6, 24, 25) and Cross in circle: High pressure data

1992; Luth 1993). Extrapolating the trend of the Williams and Kennedy (1969) relative correction one finds that the pressure correction should be around 15 ~ C for the conditions of our experiment. This correction is not very significant at the temperatures concerned for moderate values of activation volumes ( ~ 10 cm3/mol) but would be critical for lower activation volumes. However, the estimate of pressure effect on EMF is very tentative since there is no determination of absolute pressure correc- tions of any thermocouple at the P - T conditions of this study. Thus, although available data suggest that the temperature corrections are likely to be small, we refrain from making any quantitative corrections to the apparent temperatures read from the thermocouples at this time.

496

Discussion

Temperature Dependence of Diffusivities

A comparison of the Arrhenius relations reported in dif- ferent studies on Mg tracer diffusion in forsterite is shown in Fig. 5. The diffusion data obtained in this study for forsterite are slower and the activation energy is greater, than those of Morioka (1981) and Sockel and Hallwig (1977). The extrapolated combination of these latter two data sets (dotted lines in Fig. 5) have been used in various discussions to suggest a change in activa- tion energy, and hence diffusion mechanism, of Mg tracer diffusion in forsterite at around 1100 ~ C. Our measure- ments cover the entire temperature range of these pre- vious studies and there is no indication of any kink in the Arrhenius plot due to a change in activation energy (Fig. 5). Sockel et al. (1980) obtained a much greater acti- vation energy (as read from their graphical representa- tion and therefore not shown in Fig. 5) than that ob- tained in this study. Consequently, their diffusion rate is faster than those obtained by us at 1200 ~ C and slower at 1000 ~ C.

Our data are in agreement with the measurements of Andersson et al. (1989) and Andersson (1987) within the relatively large scatter (up to __ 1 order of magnitude at a given temperature) of their data. The Arrhenius fits to the temperature dependence of the data from Anders- son (1987) and this study are in excellent agreement and practically overlap (Fig. 5). However, the smaller scatter in our data leads to smaller calculated errors on the Arrhenius parameters. This agreement is particularly sig-, nificant because the experimental methods employed to determine the diffusion coefficients were different in the

1450"C 900 ~

-14.5 Morioko

-15.5

"~ XXxx \ \ ~'E -16.5 123

o - - 1 7 . 5

&H

-18.5 il

-19.5 Lw~ 5.5 6.0 6.5 7.0 7.5 8.0 8.5

104/T (k) Fig. 5. Comparison of Arrhenius relations obtained from various studies on Mg tracer diffusion in forsterite. The different lines are indexed by authors, the extent of the line show the temperature range of study. Dotted line is extrapolation. The references are: Morioka: Morioka (1980), S & H: Sockel and Hallwig (1977), An- dersson: Andersson (1987)

two studies - Andersson (1987) sputtered 26Mg enriched MgO or enstatite onto polished surfaces of forsterite to obtain the tracer source whereas we used a chemical precipitation technique as described above.

Mg diffusion data in Fe-bearing olivine of similar composition to the San Carlos olivine used in this study have been obtained by Jurewicz and Watson (1988). It is not clear whether their experiments yielded tracer or interdiffusion rates. Additionally, as they point out in their study, their data were not collected for determining an activation energy. Nevertheless, when normalized to the s a m e f O 2 conditions, their data at 1300 ~ C are consis- tent with our results within the limits of error of the two studies.

Pressure Dependence of Diffusivities

Keeping in mind the reservations regarding the high pressure data mentioned above, we can combine the high pressure and low pressure results to obtain an estimate of the activation volume of Mg tracer diffusion in forster- ite. A plot of the pressure dependence of Mg tracer diffu- sivity is shown in Fig. 4. Because of the uncertainty in the fO2 conditions of the high pressure anneals, we have calculated activation volumes by combining the high pressure data with two alternate data sets at 1 atmo- sphere; data collected in air and at a fO2 of 10 .22 bars. These two alternative comparisons yield activation vol- umes of 3.4+_0.5 and 1.1_+0.3 cm3/mol at 1100 ~ C, re- spectively. The error limit is that obtained from a statisti- cal fit of the data; the total limit could be much higher because of the unquantifiable sources of uncertainties discussed above. The value of activation volume found here is in very good agreement with the activation vol- ume of ~ 2 cm3/mol for F e - M g interdiffusion reported by Bertran-Alvarez et al. (1992) in their study using a multianvit apparatus. The activation volume found in this study is somewhat smaller than 5.5 cm3/mol re- ported by Misener (1974) for F e - M g interdiffusion ob- tained at much lower pressures using the piston cylinder apparatus. Considering the various sources of errors in the different studies, it is premature to discuss whether the various determinations are in disagreement and if so, the reasons for such disagreement. In particular, the uncertainty arising from the lack of knowledge of the pressure effect on the EMF of the thermocouple is spe- cially critical for the determination of small activation volumes, as are being found here. This should be the topic of a future detailed study. The present dataset only allows us to conclude that activation volumes for Mg tracer diffusion are small (on the order of a few cm3/mol), unless the pressure effect on the thermocouple EMF have been greatly underestimated. Nevertheless, the relatively low activation volumes found in this study as well as all other experimentally determined values indicate that activation volumes on the order of 21 cm3/mol, as found for example for Fe diffusion in fayalite in the computer simulation studies of Takeda (1990), are unlikely to be correct, unless there is a large compositional dependence of activation volume in the olivine solid solution series.

497

Dependence of Diffusivity on f02 and Diffusion Mechanism

In this study it was found that the tracer diffusivity of Mg in nominally pure forsterite Fo i as well as Fe-bear- ing olivine is a function of fO2. The fO2 dependence of Mg tracer diffusivity in San Carlos is consistent with an approximately fO 1/6 dependence (run # O13 b, O18 b, O134) as found in other diffusion studies on Fe-bearing olivines (e.g. Buening and Buseck 1973; Jurewicz and Watson 1988; Nakamura and Schmalzried 1984; Her- meling and Schmalzried 1984). The fOz dependence of Mg tracer diffusivity in nominally pure forsterite Fo 1 is a new result which we discuss in more detail below. The difference in fO2 dependence of Mg tracer diffusivity between San Carlos olivine and forsterites (Fo 1 and Fo 2) suggest that the diffusion mechanism of Mg is dif- ferent in the Fe-bearing San Carlos olivine from that in the nominally pure (very low Fe contents) forsterites, at least over certain ranges of fO2.

The impurity in forsterite Fo 1 most likely to be af- fected by a change in fO 2 is a transition metal such as Fe, which is the dominant impurity (120-180 ppm) in Fo l . The change in fO 2 would affect the Fe2+/Fe 3+ ratio in the forsterite which in turn affects diffusivity. In order to understand how a change of valence state of Fe may affect diffusivity of Mg, it is necessary to relate this change of valence state to the point defect chemistry of olivine. Fe may be in the M-site (as in olivines), in the Si site as Fe~ or occur interstitially. There are two alternative approaches to explain the observed pO2 de- pendence of diffusivity in the nominally pure forsterite:

Model I. If it is assumed that Fe always occupies its preferred M-site in the olivine structure, then the fO2 dependence of diffusivity (as well as vacancy concentra- tion) controlled by Fe 2 § - Fe 3 § equilibria would be pro- portional to approximately fO~/6 for a vacancy mecha- nism. As fO2 is decreased from an initially high value (~air), the amount of Fe 3+, the number of vacancies and diffusivity of Mg would all decrease. The equilibrium relations fix the ratio of Fe3+/Fe 2+ at any P - T - f O 2 condition, which is very small for the olivine structure (e.g. Nakamura and Schmalzried 1983). For a mineral with very low (few ppm) total Fe concentration (e.g. Fo 1), this implies a low Fe 3§ and vacancy concentra- tion. Thus, it is possible that at relatively reducing condi- tions the concentration of vacancies created by Fe 2§ - F e 3 + equilibria are overwhelmed by other vacancies (which may be intrinsic defects or vacancies related to the presence of other impurities) so that the diffusivity becomes independent of fOz. Thus, the observed weak fO2 dependence of diffusivity in Fo 1 could be the result of a combination of two kinds of behavior - a domain of fO21/6 dependence at high fO2 and a domain of fO 2 independence at low fO2. This is shown by the dashed line labelled I on Fig. 3.

If one accepts this scenario to explain the fO 2 depen- dence, then the transition between the two mechanisms can be calculated to occur at a fO 2 o f ~ 10 - 6 bars using the mean values of diffusivities measured at high and

low fO2. Further, extrapolation of relationships given in Nakamura and Schmalzried (1983) for Fe-bearing ol- ivines (e.g. Eq.(18)) allow one to estimate the Fe 3 + con- tent at which the transition occurs to be about 60 ppm (out of the total average Fe content of ~150 ppm in Fo 1). This corresponds to a vacancy concentration of

120 ppm in Fo i at 1100 ~ C and fOz = 10 -6 bars. Since these calculations are based on the extrapolation of the Nakamura and Schmalzried (1983) data and defect mod- el to practically Fe-free systems, they should be treated only as crude first order estimates of vacancy concentra- tions in forsterite.

The forsterite Fo2 on the other hand contains very little Fe (11 ppm) which means the concentration of Fe 3 § is inherently low in this sample at all fOz values so that the vacancy concentration and diffusivity are controlled by some other defects which are independent of fO2; this is analogous to the proposed low-fO2 case for Fo 1.

Modell II. A single stage scenario can also explain the weak fO2 dependence. For this purpose, we note that Fe 3-- in the Si site has been shown to exist in forsterite from EPR studies (Niebuhr 1975, 1976), but this defect usually requires Fe 3 + in the M-site as part of the charge balance condition (e.g. Nakamura and Schmalzried 1983). As discussed above, if these are the majority de- fects, then the pO2 dependence of diffusivity would have to be much stronger (pO2 ~/6) than that observed in this study. This suggests that defects involving Fe in an inter- stitial site is the most likely explanation. An inspection of possible defects (Stocker and Smyth 1978, Table 3) that fulfill these requirements indicates two possibilities for charge balance conditions that are consistent witti the above requirements:

[e'] = 4 [Sii"]

and

3 [ F e i ] = 2 [V~g],

where the Kr6ger-Vink notation has been used for the defects. Both of these indicate a pO 1/1~ dependence of V~g (and therefore, diffusivity). The observed dependence of diffusivity with pOa is for a change in pO2 of 12 orders of magnitude, the diffusivity changes by about an order of magnitude. This dependence is somewhat weaker than the pO 1/1~ dependence predicted by this model and the difference may be simply due to the lack of resolution of our data or due to more complex charge balance schemes involving some interaction among defects.

It might be possible to distinguish between the two charge balance conditions using forsterite crystals of dif- ferent Fe contents e.g. Fo 1 and Fo 2. The Mg diffusivities obtained in Fo2 for experiments performed in air and at an fO= of 10 -12 bars are very similar, in contrast to those from Fo l . This suggests that the pO2 depen- dence of diffusivity observed in the forsterite crystals con- taining small amounts of Fe (i.e. Fo 1) is more likely due to the presence of interstitial Fe, if a single stage model is preferred. At this stage, the resolution of our data is not adequate to choose between model I and

498

model II. However, model I has the advantage of ex- plaining the observations without invoking any new scheme of defect formation in silicate olivines. In any case, the above discussion serves to narrow down the possibilities for the mechanism of Mg diffusion in forster- ite by eliminating such simple scenarios as Schottky or Frenkel defect controlled diffusion.

Intrinsic or Extrinsic Diffusion?

A transition from an intrinsic to an extrinsic diffusion mechanism is marked by a kink in the Arrhenius plot (change in activation energy) of a diffusion coefficient. A kink in the Arrhenius plot was not observed within the experimental range in this study although it is ex- pected to occur at some temperature, implying that all measurements in this study were in a single regime. If the measured data are in the intrinsic regime, then lower temperature extrapolations are to be treated with cau- tion since at some temperature corresponding to the in- trinsic to extrinsic transition the Arrhenius relation used for extrapolations would change. On the other hand, if the measured data are on the low temperature side of the transition i.e. in an extrinsic regime, then extrapo- lations to lower temperatures are more reliable. The acti- vation energy may change nevertheless, due to factors such as changes in the defect speciation/association in the minerals with temperature. However, the uncertainty of the extrinsic to intrinsic transition would be elimi- nated.

It is useful to find a suitable operational definition for intrinsic and extrinsic diffusion in minerals. In simple substances such as pure metals, the definitions are straightforward. If the vacancies dominantly responsible for the transport process are thermally generated, the process is termed intrinsic. In this case, the vacancy con- centrations are a function of P and T only and indepen- dent of other factors. On the other hand, if the vacancies controlling the transport process are generated by the presence of impurities then the transport process is ex- trinsic. In complex substances such as minerals, however, this definition becomes ambiguous. For example, incor- poration of a cation B into a mineral AX (A = cation, X = anion) may be considered to be an impurity or alter- nately, formation of a solid solution (A, B) X. Diffusion coefficients in the resulting substance may be treated as diffusion in pure AX 'doped ' with B or diffusion in pure (A, B)X. However, in the extrinsic case, the vacancy con- centrations (and hence the transport rates) are propor- tional to the concentration of these impurities. In a ther- modynamic sense, then, in this regime the transport pro- cesses at a fixed major element composition are functions of chemical potentials of different components, in addi- tion to P and T. Therefore, it is operationally convenient to define the intrinsic regime as that where tracer- or self-diffusivities (vacancy concentrations) are indepen- dent of component activities. In contrast, in the extrinsic regime the diffusivities (vacancy concentrations) at a giv- en major element composition are functions of some component activities, in addition to P and T [see

Schmalzried (1983) for more detail]. Thus, the pO2 de-' pendence of diffusivity in San Carlos olivine as well as the forsterite Fo 1 imply that the measured diffusivity in these minerals are in the extrinsic regime. Additional evidence for extrinsic diffusion comes from the energetics of defect formation in olivines:

I) Formation Energy of Vacancies. In the intrinsic regime, the activation energy is made up of two parts an energy of formation of the thermally generated vacancies and an energy of migration of these vacancies to the neigh- bouring sites. In the extrinsic regime, the activation ener- gy consists only of the migration energy term and is hence smaller than in the intrinsic regime. Thus indepen- dent estimates of the energy required to form vacancies may be compared with the measured activation energies to provide information about whether diffusion is occur- ing in the extrinsic or in the intrinsic regime. The mini- mum defect formation energies obtained from model cal- culations (e.g. Lasaga 1980; Catlow 1987; Ottonello et al. 1990) are about 800kJ/mol, which is substantially greater than the observed activation energy for Mg tracer diffusion in forsterite obtained in this study. This suggests that the observed activation energies cannot be a sum of intrinsic vacancy formation and migration ener- gies, which in turn implies that the observed diffusion occurs by an extrinsic process.

II) Level of Purity Required for Intrinsic Diffusion. Giv- en a reasonable estimate of the vacancy formation ener- gy, one can calculate the concentration of impurities that is required to overwhelm the thermally generated vacan- cies at a given P and T and cause diffusion to occur by an extrinsic mechanism. One may then compare this with the level of purity likely to be observed in natural and synthetic materials to decide whether observed diffu- sivities are by an extrinsic or intrinsic mechanism. A very conservative estimate of intrinsic vacancy formation energy of about 550 kJ yields the concentration of vacan- cies from

Xv = e - AHv/2RT.

At 1300 ~ C, this yields Xv of about 2 x 10-1~ at 1000 ~ C X v is even lower at about 1 x 10-12 mol fraction. If addi- tion of each impurity ion creates a vacancy, then even impurities on the level of ppb would overwhelm the in- trinsic vacancy concentration. Thus, it is unlikely that even the synthetic crystals used in this study show intrin- sic diffusion behavior, as already argued from other lines of evidence.

The calculations above are of significance for a number of reasons: i) Unless intrinsic vacancy formation energies are un- usually low (a few kJ), temperatures are high (> 1300 ~ C), and crystals are abnormally pure, intrinsic diffusion be- havior is unlikely to ever be observed in silicates. A simi- lar conclusion has been arrived at for refractory oxides (Wuensch 1982). In particular, the requirement of purity makes it unlikely that diffusion in any natural silicate would have occurred by an intrinsic mechanism.

499

ii) Since observed diffusion behavior is extrinsic, low temperature extrapolations are unlikely to be affected by a kink in the temperature dependence of diffusivity due to the intrinsic to extrinsic transition. On the con- trary, high temperature extrapolation may be dangerous. However, we reiterate that changes in slopes on Arrhen- ius plots may occur nevertheless due to factors such as defect association, change in the identity of the majority defect, structural changes related to phase transitions etc. iii) The absolute values of diffusivites may depend on concentrations of impurities, at least for cases where dif- fusion is by a simple vacancy mechanism. Thus, unless the trace element characteristics are completely charac- terized, it is difficult to predict the absolute value of diffu- sivity in a mineral. The effect of trace elements on diffu- sion rates may be seen in this study using the two forster- ites, Fo 1 and Fo 2. The absolute values of diffusion rates under identical conditions are different (Table 2) as is the effect of a change of pO2 on diffusion in the two crystals. The situation may be somewhat better for tran- sition metal viz. Fe-bearing minerals. In these, at fixed P, T and fO2 the vacancy concentration, and hence diffu- sion rates, is largely controlled by the fixed Fei+/Fe 3+ ratio (factors such as asi02 or chemical potential of other components will have a finite, but likely small, effect).

In this connection, it is interesting to compare the dataset obtained in this study on the Fe-bearing San Carlos olivine with that obtained on Fo 1 as a function of temperature. We find that the scatter in the data for Fo 1 is much larger than that for the San Carlos olivine. Although the number of measurements on Fo 1 are much larger which contributes to this scatter and the fewer data points on San Carlos may fortuitously cluster, it is interesting to speculate on an alternate possibility. The defect characteristics in the Fe bearing (constant Fe con- tent) San Carlos olivine were completely defined by fix- ing the P, T, fO 2 and asi02. On the other hand, the Fe impurity in the synthetic forsterite Fo 1 has a heteroge- neous distribution (see Table 1) as is commonly expected in Czochralski grown crystals. At the low levels of Fe content, these differences may have been significant enough to affect diffusion rates, leading to the scatter of the data. indeed the effect of small differences in Fe contents on diffusivity are well illustrated by samples Fo 1 and Fo 2.

Conclusions

We have obtained tracer diffusion coefficients for volume diffusion of Mg in forsterite and San Carlos olivine. The diffusion coefficients as well as the activation energies of diffusion are found to be significantly different in these two compositions. It is found that Mg tracer diffusion in forsterite with some Fe impurity as well as in the Fe bearing San Carlos olivine are a function of pO2. However, the pO2 dependence is different for these two compositions, indicating that the mechanism of diffusion in forsterite with Fe impurity is different from that in San Carlos olivinc, at least over certain fO2 ranges. The pressure dependence of Mg tracer diffusion measured

to 10 GPa in forsterite is found to be weak yielding an activation volume of ~ 1-3.5 cm3/mol. Mg tracer diffu- sion in forsterite is found to be anisotropic with D II c >Dlla>Dii b. At a total pressure of 1 atmosphere, pres- ence of small amounts of hydrogen bearing species in the atmosphere are found to have no perceptible effects on Mg diffusion rates in forsterite.

The observed diffusion of Mg in these olivines is shown to be due to an extrinsic process so that low temperature extrapolation of these data should be more reliable than high temperature extrapolation.

Acknowledgments. We thank Prof. S. Karato (Univ. Minnesota), Prof. George Rossman (CalTech) and the Smithsonian Institution for providing the single crystals used in this study. SC thanks Sharon Webb for instructions on orienting single crystals using Laue photographs. We thank Hubert Schulze and Anna Dietel for technical help. Prof. D.L. Kohlstedt critically read the manu- script and model I explaining the fO2 dependence of D*g in Fo 1 is based on his suggestions; the responsibility for considering it viable and including it in the paper is, of course, ours. We thank Prof. Olivier Jaoul and an anonymous reviewer for their critical reviews.

References

Andersson K (1987) Materialtransport und Defektstrukturen in kri- stallinem Magnesiumorthosilikat bei h6heren Temperaturen, Ph.D Thesis, Teehnisehe Universitfit Clausthal

Andersson K, Borchardt G, Scherrer S, Weber S (1989) Self diffu- sion in Mg2SiO 4 (forsterite) at high temperature: A model case study for SIMS analyses on ceramic surfaces, Fresenius Z Anal Chem 333:383 385

Barrer RM, Bartholomew RF, Rees LVC (1963) Ion exchange in porous crystals Part II. The relationship between self- and ex- change-diffusion coefficients. J Phys Chem Solids 24:309 317

Bertran-Alvarez Y, Jaoul O, Liebermann RC (1992) Fe-Mg inter- diffusion in single crystal olivine at very high pressure and con- trolled oxygen fngacity: technological advances and initial data at 7 GPa, Phys. Earth Planetary Interiors 70:102-118

Brady JB (1975) Reference frames and diffusion coefficients. Amer Jour Sci b 275:954~983

Buening DK, Buseck PR (1973) Fe-Mg lattice diffusion in olivine. J Geophys Res 78:6852-6862

Catlow CRA, Cormack AN (1987) Computer modelling of silicates. Int Rev Phys Chem 6:227-250

Clark AM, Long JVP (1971) Anisotropic diffusion of nickel in oliv- ine. In: Thomas Graham memorial symposium on Diffusion Processes, 511-521, Gordon and Breach, New York

Crank J (1975) The mathematics of diffusion. Oxford Univ. Press, Oxford, UK

Cygan RT, Lasaga AC (1985) Self-diffusion of magnesium in garnet at 750 ~ C to 900 ~ C. Amer Jour Sci 285:328 350

Giletti BJ, Semet MP, Yund RA (1978) Studies in diffusion III. Oxygen in feldspars: an ion microprobe determination. Geo- chim Cosmochim Acta 42:45 57

Graham EK, Barsch GR (1969) Elastic constants of single-crystal forsterite as a function of temperature and pressure. J Geophys Res 74:5949

Hermeling J, Schmalzried H (1984) Tracer diffusion of the Fe-ca- tions in olivine (FexMgl_~)2SiO 4 (III). Phys Chem Minerals 11:161-166

Jaoul O, Froidevaux C, Durham WB, Michaut M (1980) Oxygen self-diffusion in forsterite: implications for the high temperature creep mechanism. Earth and Planetary Science Letters 47:391 397

Jurewicz AJG, Watson EB (1988) Cations in olivine, part 2: Diffu- sion in olivine xenocrysts, with applications to petrology and mineral physics. 99:186-201

500

Lasaga AC (1980) Defect calculations in silicates: olivine. Amer Mineral 65:1237-1248

Luth RW (1993) Measurement and control of intensive parameters in experiments at high pressures in solid media apparatus. In: Luth RW (ed) Experiments at high pressures and applications to the Earth's mantle, Short Course Handbook. Mineralogical Association of Canada 21:15 38

Manning JR (1968) Diffusion kinetics for atoms in crystals, Van Nostrand, Princeton. N J, 257 pp.

Misener DJ (1974) Cationic diffusion in olivine to 1400~ C and 35 kbar. In: Hofmann AW, Giletti BJ, Yoder HS Jr, Yund RA (eds) Geochemical Transport and Kinetics. Carnegie Institution of Washington

Morioka M (1980) Cation diffusion in olivine I. Cobalt and mag- nesium. Geochim Cosmochim Acta 44:759 762

Morioka M, Nagasawa H (1991) Ionic diffusion in olivine. In: Gan- guly J (ed) Diffusion, Atomic ordering and mass transport. Ad- vances in Physical Geochemistry 8. Springer, New York

Nakamura A, Schmalzried H (1983) On the nonstoichiometry and point defects of olivine. Phys Chem Minerals 10:27 37

Nakamura A, Schmalzried H (1984) On the Fe z + - M g interdiffu- sion in olivine (II). Ber Bunsenges Phys Chem 88:140-145

Niebuhr HH (1975) Electron spin resonance of ferric iron in forster- ite, Mg2SiO4. Acta Cryst A31, suppl $274

Niebuhr H (1976) Elektron-Spin Resonance yon dreiwertigen Eisen in Forsterit. Habilitationsschrift, FB Geowissenschaften, Mar- burg

Ohtani E (1979) Melting relations of FezSiO 4 upto about 200 kbar. J Phys Earth 27:189-208

Ottonello G, Princivalle F, Della Giusta A (1990) Temperature, composition and fO2 effects on intersite distribution of Mg and Fe 2 + in olivines: experimental evidence and theoretical inter- pretation. Phys Chem. Mineral 17:301 3i2

Rubie DC, Karato S, Yan H, O'Neill HStC (1993 a) Low differential stress and controlled chemical environment in multianvil high- pressure experiments. Phys Chem Miner 20:315-322

Rubie DC, Ross CR II, Carroll MR, Elphick SC (1993b) Oxygen self-diffusion in Na2Si40 9 liquid up to 10 GPa and estimation of high-pressure melt viscosities. Am Mineral 78:574 582

Schmalzried H (1983) Thermodynamics of compounds with narrow range of stoichimetry. Ber Bunsenges Phys Chem 87:726-733

Sockel HG, Hallwig D (1977) Ermittlung kleiner Diffusionkoeffi- zienten mittels SIMS in oxydisehen Verbindungen. Mikrochim Acta (Wien) 7:95-107

Sockel HG, Hallwig D, Schachtner R (1980) Investigations of slow exchange processes at metal and oxide surfaces and interfaces using secondary ion mass spectrometry. Materials Science and Engineering 42:59-64

Stocker RL, Smyth DM (1978) Effect of enstatite activity and oxy- gen partial pressure on the point-defect chemistry of olivine. Phys Earth and Planetary Interiors 16:145 156

Takeda H (1990) Diffusion of ions in mantle minerals at high pres- sure and high temperature. In: Marumo F (ed) Dynamic Pro- cesses of Material Transport and Transformtion in the Earth's Interior. Terra Scientific Publishing, Tokyo, pp 217-238

Takei H, Kobayashi T (1974) Growth and properties of MgzSiO single crystals. Jour Crystal Growth 23:121-124

Tannhauser DS (1956) Concerning a systematic error in measuring diffusion constants. J Appl Phys 27:662

Tsuzaki Y, Takahashi E (1992) Pressure corrections on thermocou- ple EMFs with a Multianvil apparatus. 29 th International Geo- logical Congress, Abstracts Vol. 1, 58

Williams DW, Kennedy GC (1969) Melting curve of diopside to 50 kilobars. J Geophys Res 74:4359-4366

Wuensch BJ (1982) Diffusion in stoichiometric close-packed oxides. In: Beniere F, Catlow CRA (eds) Mass Transport in Oxides. NATO-ASI series, 353-376