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Cation ordering in orthopyroxenes from two stony-iron meteorites: implications for cooling rates and metal-silicate mixing
Geochimica et Cosmochimica Acta, Vol. 64, No. 7, pp. 1291–1297, 2000 Copyright © 2000 Elsevier Science Ltd Printed in the USA. All rights reserved 0016-7037/00 $20.00 ⫹ .00
Pergamon
PII S0016-7037(00)00405-6
Cation ordering in orthopyroxenes from two stony-iron meteorites: Implications for cooling rates and metal-silicate mixing JIBAMITRA GANGULY* and MARILENA STIMPFL Department of Geosciences, University of Arizona, Tucson, AZ 85721, USA (Received July 13, 1999; accepted in revised form November 3, 1999)
Abstract—We have determined the cooling rates of orthopyroxene crystals from two group IVA stony iron meteorites—Steinbach (ST) and Sa˜o Joa˜o Nepomuceno (SJN)— on the basis of their Fe–Mg ordering states. The rate constant was calibrated as a function of temperature by controlled cooling experiments using orthopyroxene crystals separated from ST. These data were used along with earlier calibrations of the equilibrium intracrystalline fractionation of Fe and Mg as a function of temperature for crystals separated from both meteorites to calculate their cooling rates. The site occupancies of the orthopyroxene crystals were determined by single-crystal X-ray diffraction subject to the bulk compositional constraints. The closure temperatures (T c) of cation ordering for the untreated crystals from SJN are ⬃400°C, whereas those from ST vary between ⬃430 and 470°C. Reconciliation of the metallographic and orthopyroxene cooling rate data, within the framework of the metal–silicate mixing model of Haack et al. (1995), suggests that these two stony irons had cooled at a similar rate of ⬃400°C/Ma through the closure temperatures for cation ordering in the orthopyroxenes. This was followed by slow cooling for ST at ⬃50°C/Ma at T ⬍ 425°C. Similar slow cooling was not recorded by the metals in SJN, which implies that if this stony iron were subjected to slow cooling, it must have been below 350°C. The similar cooling rates above 425°C for both ST and SJN, as required to reconcile the metal and orthopyroxene cooling rate data, is at variance with the earlier notion (Rasmussen et al., 1995) of distinctly different cooling rates for the high and low Ni IVA irons and stony irons. The cation ordering and metallographic cooling rate data are also amenable to an alternative interpretation, which requires two different parent bodies for the two stony irons, and mixing of the metal and silicate components of ST after the metals had cooled below the closure temperature of Fe–Ni interdiffusion. However, the available textural data for ST seems to argue against such metal-silicate mixing model. Copyright © 2000 Elsevier Science Ltd with a model of origin in a common asteroidal core. Also, the metal–silicate mixing of the IVA stony irons remains a puzzling issue. Moren and Goldstein (1978, 1979) argued that the IVA irons meteorites did not originate in a core, but crystallized from metallic melts dispersed throughout the mantle of the parent body. Ulff–Møller et al. (1995) argued that the silica saturated mineralogy of the IVA stony irons could not have crystallized from magmas that evolved in an olivine rich mantle. They suggested a model of metal-silicate mixing near the core-mantle boundary in which the magmas were separated from the refractory residue at different stages of partial melting and injected into the core through fractures that presumably developed by solidification shrinkage of the core. According to them, this model also helps explain the low Fe/Mn ratio of pyroxenes in the IVA stony irons, as Fe would be reduced by equilibration with the metal. Scott et al. (1996) showed that even though the IVA irons have extremely diverse cooling rates which are difficult to reconcile with a model of their core origin in a parent body, the chemical characteristics of the IVA metal phases are consistent with fractional crystallization of an asteroidal core, which contradicts the “raisin bread” model of Moren and Goldstein (1978, 1979). Haack et al. (1996) carried out transmission electron microscopic (TEM) studies of the disordered clinobronzite in the ST meteorite by using electron diffraction and high-resolution imaging techniques, and concluded from these data that it had cooled at ⬃100°C/hr through 1200°C, whereas the IVA irons had cooled much slower cooling, ⬍300°C/y, in
1. INTRODUCTION
Steinbach (ST) and Sa˜o Joa˜o Nepomuceno (SJN) are stony iron meteorites consisting of a unique mixture of metals and ⬃60 vol% of silicates made of coarse-grained mixture of tridymite, orthopyroxene and clinopyroxene. Scott et al. (1996) showed that the chemical properties of the metal phases of these meteorites are closely related to those of IVA irons, which suggest a common origin for both types. Rasmussen et al. (1995) determined the metallographic cooling rates (MCR) of the IVA irons and stony irons by using the updated data for the Fe–Ni phase diagram and Ni diffusion coefficients (Saikumar and Goldstein, 1988). They found a wide range of values from 19 to 3400 K/Ma, which could be separated into two groups according to the Ni content of the meteorites: one group with Ni ⬍ 8.4 wt% with a very large range of cooling rate between ⬃250 K/Ma and 3400 K/Ma, apparently correlating inversely with the Ni content; the other a relatively high-Ni group (8.6 –10.5 wt%) with much slower and similar cooling rates between 20 and 60 K/Ma. The mean MCR for ST is ⬃20 K/Ma, whereas that of SJN is ⬃900 K/Ma. These cooling rates are compatible with the MCR versus Ni content systematics observed for the IVA iron meteorites. The large span of MCR for IVA irons poses severe problems
This research was supported by a NASA Grant NAGW 3638. * Author to whom correspondence should be addressed (ganguly@ geo.arizona.edu). 1291
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the range 1200 –1000°C, as evidenced by the absence of dendrites in large troilite nodules. To explain the metal–silicate mixing and to reconcile the suggestion of Scott et al. (1996) about the core origin of the metals with their diverse metallographic cooling rates, Haack et al. (1996) developed a four-stage evolutionary model for the IVA irons and stony-irons: (1) high-velocity impact of the asteroidal parent body after ⬃95% crystallization of the core; (2) gravitational reassembly of the silicate mantle and the metal core fragments within a dust cloud; followed by (3) cooling through temperature range (⬃750 – 400°C) required for the development of exsolution textures in Fe–Ni alloys; and (4) catastrophic disruption 450 My ago, which represents the cosmic ray exposure age of most IVA irons (Voshage and Feldman, 1979). In this model, the metals are expected to record diverse cooling rates as these were buried at random depths after the gravitational reassembly. However, as discussed by Rasmussen et al. (1995), the problem with the breakup–reassembly model is the apparent exclusive yield of the high Ni meteorites from a single slowly cooled fragment requiring a meteorite yield factor equivalent to that of a randomly sampled single core. The quenched Fe2⫹–Mg ordering state between the M1 and M2 structural sites of an orthopyroxene crystal is an important recorder of its cooling rate (Ganguly, 1982). The closure temperature of cation ordering falls in the range of ⬃300 – 400°C for cooling rates in the range of 1–104°C/Ma (Ganguly et al., 1997). Thus, the Fe–Mg ordering state of orthopyroxene provides an important method of determining the cooling rate of the silicate component somewhat below the temperature range at which the metallographic cooling rates are valid (e.g., Ganguly et al., 1994). If the break-up and reassembly model of Haack et al. (1996) were valid, then one would expect to retrieve similar cooling rates from both metal textures and Fe–Mg ordering data in orthopyroxenes because these would have cooled in the same environment through their respective closure temperatures. In view of the complexity of the problems of metal–silicate mixing and thermal history of the IVA irons and stony irons, and the lack of any quantitative constraints on the low temperature cooling rates of the silicate components, we undertook an investigation of the cooling rates of the orthopyroxene crystals in ST and SJN. 2. RETRIEVAL OF COOLING RATES FROM CATION ORDERING IN ORTHOPYROXENES: METHOD AND DATA
2.1. Outline of Theory and Numerical Method The method of retrieval of cooling rate from the Fe2⫹–Mg ordering state of orthopyroxene has been discussed in a number of earlier publications (e.g., Ganguly, 1982; Ganguly et al., 1994; Ganguly and Domeneghetti, 1996). Thus, we provide here only a brief summary of the method and the underlying principles. In orthopyroxenes, Fe2⫹ and Mg fractionate between two nonequivalent octahedral sites, M1 and M2, with a preference of Fe2⫹ for M2 and that of Mg for M1 sites. Mueller (1967, 1969) developed a kinetic theory of order-disorder by treating the fractionation of cations between the nonequivalent structural sites in terms of a homogeneous exchange reaction, viz.:
disordering 3 Fe2⫹(M2) ⫹ Mg(M1) 7 Mg(M2) ⫹ Fe2⫹(M1), 4 ordering and developed an expression for the change of site occupancy as a function of time under isothermal condition. In this formulation, in which the homogeneous exchange reaction has been assumed to be a second-order elementary kinetic process, the time scale of ordering is related to the initial and final ordering states through the rate constant for the disordering ¡ process, ( K) and the equilibrium Fe2⫹–Mg distribution coefficient, k D, between the two sites ((Fe2⫹/Mg)M1/(Fe2⫹/Mg)M2). The rate of change of site occupancies or ordering state is ¡ ¢ governed by both disordering ( K) and ordering ( K) rate con¢ ¡¢ stants, but K is eliminated by using the relation that k D ⫽ K/ K. The k D is defined as the ratio of (Fe2⫹/Mg) in the M1 to that in the M2 site. The choice of which rate constant is to be retained is arbitrary. Using Mueller’s theory, Ganguly (1982) developed a numerical method of determining the cooling rate of orthopyroxene on the basis of its quenched or observed ordering state. The method involves calculation of the ordering state of a crystal from an initial temperature in a series of isothermal steps approximating an assumed T-t relation following a smooth functional form (e.g., an asymptotic relation described by 1/T ⫽ 1/T o ⫹ t, where T 0 is the initial temperature and is a constant with the dimension of K ⫺1 t ⫺1 ) and varying the latter (i.e., in the asymptotic model) until a satisfactory match is obtained between the quenched ordering state in the numerical simulation and that observed in a natural crystal. (This method can also be used to retrieve the thermodynamic and kinetic parameters governing the ordering process by controlled cooling experiments, as discussed below.) The retrieved cooling rate is quite insensitive to errors in the determination of T 0 as long as it is within the domain at which thermodynamic equilibrium is maintained. However, it is very sensitive to errors in the determination of site occupancies of ¡ ¢ the natural crystals and of k D and K (or K) as a function of temperature and composition. In order to avoid the problem of ¡ the compositional effects, we determined both k D and K by using crystals separated from ST and SJN meteorite fragments. 2.2. Site Occupancies of Orthopyroxenes from the IVA Stony Iron Meteorites The site occupancies of the orthopyroxene crystals were determined by single crystal structure refinements, subject to the bulk compositional constraints. Stimpfl et al. (1999) carried out a detail comparative study of the procedures of site occupancy determination by using different strategies but the same X-ray data set, and concluded that the two widely used programs SHELXL-93 (Sheldrick, 1993) and RFINE90 (which is an updated version of RFINE4, Finger and Prince, 1975) essentially yield the same results that are not significantly affected by whether one uses ionic or atomic scattering factors. However, the use of ionic scattering factors yield better goodness of fit and weighted agreement index (R w), and smaller standard deviation on the site occupancy data. The results obtained from either program were also in excellent agreement
Cooling rates of stony-iron meteorites
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Table 1. Initial (I) and adjusted (adj) bulk compositions of the orthopyroxene crystals from Sao Joao Nepomuceno (SJN) and Steinbach (ST) meteorites used for the determination of cooling rates. An initial composition represents the mean of a large number of spot analyses. The numbers within parentheses are the estimated SDs from the mean in the right justified form. Sample
SJN23 (l)
SJN23 (adj)
SJN24 (l)
SJN24 (adj)
SJN25 (l)
SJN25 (adj)
ST71 (l)
ST71 (adj)
ST76 (l)
ST76 (adj)
ST77 (l)
ST77 (adj)
Si 1.986 (3) 1.984 1.985 (3) 1.983 1.985 (5) 1.981 1.986 (4) 1.9833 1.988 (1) 1.9889 1.990 (3) 1.9849 Al 0.014 (5) 0.016 0.015 (3) 0.017 0.015 (4) 0.019 0.014 (2) 0.0167 0.009 (1) 0.0111 0.009 (3) 0.0151 Ti 0.001 (0) 0.000 0.001 (0) 0.000 0.001 (1) 0.001 0.001 (1) 0.0005 0.001 (1) 0.0003 0.000 (0) 0.0000 Cr 0.018 (1) 0.017 0.019 (1) 0.018 0.012 (1) 0.019 0.016 (1) 0.0158 0.011 (1) 0.0105 0.016 (1) 0.0151 Mg 1.668 (4) 1.667 1.664 (6) 1.662 1.669 (10) 1.665 1.641 (7) 1.6424 1.665 (3) 1.6625 1.648 (7) 1.6503 Fe 0.275 (3) 0.274 0.281 (5) 0.279 0.274 (3) 0.274 0.305 (3) 0.3053 0.304 (3) 0.3015 0.304 (3) 0.3047 Mn 0.018 (1) 0.018 0.019 (1) 0.019 0.018 (1) 0.018 0.016 (1) 0.0160 0.015 (1) 0.0146 0.016 (1) 0.0160 Ni 0.000 (0) 0.000 0.000 (0) 0.000 0.000 (0) 0.000 0.000 (0) 0.0000 0.000 (0) 0.0000 0.000 (0) 0.0000 Ca 0.023 (1) 0.023 0.021 (1) 0.021 0.023 (1) 0.023 0.020 (1) 0.0200 0.011 (1) 0.0106 0.014 (1) 0.0140 Na 0.001 (1) 0.002 0.001 (1) 0.002 0.001 (1) 0.001 0.000 (0) 0.0000 0.000 (0) 0.0000 0.000 (0) 0.0000 Fe3⫹ 0.000 (0) 0.000 0.000 (0) 0.000 0.000 (0) 0.000 0.000 (0) 0.0000 0.000 (0) 0.0000 0.000 (0) 0.0000 Total 4.004 4.000 4.004 4.000 4.004 4.000 3.999 4.000 4.004 4.000 3.997 4.000 Charge 12.012 12.000 12.013 12.000 12.012 12.000 12.002 12.000 12.006 12.000 11.999 12.000 Al(T)a 0.014 0.016 0.015 0.017 0.015 0.019 0.014 0.017 0.009 0.011 0.009 0.015 Al(M)a 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 d(chrg(M)b 0.018 0.016 0.019 0.017 0.020 0.019 0.018 0.017 0.013 0.011 0.016 0.015 d(M-T)c 0.004 0.000 0.004 0.000 0.005 0.000 0.004 0.000 0.004 0.000 0.007 0.000 M-cations 2.004 2.000 2.004 2.000 2.004 2.000 1.999 2.000 2.007 2.000 1.998 2.000 a
Al(T): Al in T site; Al (M): Al in M site. d(chr(M)): exess positive charge in M site due to the substitution for 2⫹ cations. c d(M-T): difference between the charge excess in M site and the charge deficiency in T site. b
with those obtained from the new refinement strategy developed by Kroll et al. (1997, and personal communication). We separated a number of orthopyroxene crystals from ST and SJN and from these selected three crystals from each meteorite on the basis of their optical, dimensional, and X-ray properties for single-crystal structure refinements. The site occupancies were determined by using the X-ray data, ionic scattering factors, and the bulk compositional constraints in the program RFINE 90. The details of X-ray data collection and the procedure of site occupancy determination have been discussed in detail by Stimpfl et al. (1999) and are, thus, not repeated here for the sake of brevity. The observed X-ray intensities were weighted according to the scheme built into the program, which results in a normal distribution with mean 0 and variance 1 of the weighted difference between the observed and calculated intensities. Following Ganguly and Domeneghetti (1996), Fe2⫹ and Mn (⬃0.005 atomic fraction) were treated as one component, as these cations fractionate almost equally between the M1 and M2 sites (Hawthorne and Ito, 1978), and are designated as Fe*. After the X-ray data collection, the composition of each crystals was determined by a large number of spot analyses in an electron microprobe against mostly synthetic standards, as described in Stimpfl et al. (1999). Each crystal was found to be homogeneous within the resolution of the microprobe spot analyses, which were carried out by using 15 kV accelerating voltage, 20 –25 nA beam current, and counting times of 20 s at the peak and 20 s at the background. The average compositions along with their respective standard deviations (SDs) were used to determine the statistically most probable bulk composition of each crystal by a computer aided mathematical projection of the average composition of each element, weighted by their respective SDs, on to the n-dimensional surface that satisfies the crystal chemical constraints for orthopyroxene. This
method is based on the work of Dollase and Newman (1984), and is described in detail by Stimpfl et al. (1999). The average and adjusted compositions of each crystal used for the cooling rate calculations are summarized in Table 1. For all crystals, the adjusted composition of each element was found to be within the ⫾1 error of its average composition. The site occupancy data of each crystal are summarized in Table 2. 2.3. Determination of the Thermodynamic and Kinetic Parameters Stimpfl et al. (1999) determined the equilibrium k *D, defined as (Fe*/Mg)M1/(Fe*/Mg)M2, as a function of temperature for two orthopyroxene crystals, one separated from the ST meteorite and the other from SJN. The equilibrium k D-s for the ST crystals were determined at 700, 800, 900, and 1000°C by both ordering and disordering experiments at each temperature, which constrained the equilibrium values very tightly, whereas those of the SJN crystals were determined by disordering experiments at 700 and 800°C and tightly constrained by both ordering and disordering experiments at 750°C. The fitted linear relations between lnk *D versus 1/T for the two crystals are essentially the same, which is expected as these crystals (Fs14 and Fs15) have similar compositions. The site occupancy of another low Fe crystal (Fs11) from an Antarctic meteorite was also determined very precisely by using single crystal X-ray diffraction data but different structure refinement strategy (Kroll et al., 1997) by Lueder (1995; cited and discussed in Stimpfl et al., 1999). The k *D values retrieved from these data are also very similar to those from the ST and SJN crystals. A least-squares regression of the three low Fe data sets yields (Stimpfl et al., 1999): ln k *D ⫽ ⫺
2854共⫾86兲 ⫹ 0.603共⫾0.095兲 T
(1)
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Table 2. Summary of the site occupancy data, k *D, closure temperatures (T c) and cooling time constants () of the orthopyroxene crystals from Sa˜o Joao Nepomuceno (SJN) and Steinbach (ST). The SD stands for the estimated standard deviation of the site occupancies of Fe* (i.e., Fe ⫹ Mn) and Mg arising from those of the structural data, as given by the refinement program RFINE90. The uncertainty of the structural and bulk compositional data leads to uncertainties of factors of ⬃10 and 5 in the values (and hence cooling rates) for the SJN and ST samples, respectively. Sa˜o Jao Nepomuceno Sample no.
Steinbach
SJN23
SJN24
SJN25
ST71
ST76
ST77
M2 site Mg Fe* Na Ca
0.6945 0.2815 0.0015 0.0225
0.6905 0.2875 0.0015 0.0205
0.6961 0.2803 0.0010 0.0226
0.6744 0.3057 0.0000 0.0200
0.6900 0.2994 0.0000 0.0110
0.6788 0.3072 0.0000 0.0140
M1 site Mg Fe* Al Cr Ti
0.9719 0.0107 0.0000 0.0173 0.0001
0.9712 0.0104 0.0000 0.0180 0.0004
0.9692 0.0109 0.0000 0.0194 0.0005
0.9680 0.0156 0.0000 0.0160 0.0010
0.9725 0.0167 0.0000 0.0110 0.0000
0.9712 0.0138 0.0000 0.0150 0.0000
T site Si Al SD k *D T c (°C) , K⫺1Y⫺1 (dT/dt)Tc (°C/Ma)
1.9840 0.0160 0.0010 0.0270 405 8.5E-10 390
1.9830 0.0170 0.0008 0.0260 396 4.5E-10 200
1.9810 0.0190 0.0009 0.0280 409 1.2E-09 558
1.9830 0.0170 0.0009 0.0340 442 2.0E-08 10225
1.9890 0.0110 0.0009 0.0400 473 7.25E-08 40347
1.9850 0.0150 0.0009 0.0310 429 4.60E-09 2267
where the uncertainties represent ⫾1 (SD). ¡ ¢ In principle, each rate constant, K and K, can be retrieved by modeling the data on the change of either ordering or disordering state as a function of time and temperature if the k D is known as a function of temperature (e.g., Besancon, 1981; Ganguly, 1982; Anovitz et al., 1988; Saxena et al., 1987). However, contrary to the expectation from the kinetic model of Mueller (1967, 1969), Stimpfl et al. (1997) found that the value of a rate constant at a fixed temperature depends on whether it is derived from ordering or disordering experiments. In partic¡ ular, K derived from disordering experiment using Mueller’s kinetic theory (1967; 1969) was found to be significantly larger than that derived from ordering experiments. Similar observation was made by O’Neill (1994) for Fe2⫹-Mg order-disorder in spinels. A theoretical explanation of this difference in the retrieved values of a specific rate constant from the ordering and disordering experiments of orthopyroxene is still lacking. However, even though it cannot reconcile the data from the two types of experiments, Mueller’s kinetic model provides a satisfactory phenomenological representation of the data from either type of experiment (Besancon, 1981; Saxena et al., 1987; Anovitz et al., 1988; Stimpfl et al., 1997). Since a rate constant is not only a function of temperature (and pressure) but also of composition (Besancon, 1981; Ganguly, 1982; Ganguly and Tazzoli, 1994), we decided to deter¡ mine K as a function of temperature using an orthopyroxene crystal from the ST meteorite, which obviates the need for any compositional correction on the rate constant. Also, because the value of a rate constant seems to depend on the nature of the ¡ experiment (ordering versus disordering), we determined K as a function of temperature by ordering the orthopyroxene crystal so it is compatible with that in the natural process of cooling. In this work, we deviated from the conventional approach (e.g., Besancon, 1981; Ganguly, 1982; Anovitz et al., 1988;
¡ Saxena et al., 1987) of determining K by modeling the change of ordering state or site occupancies as a function of time at different temperatures. Instead, we carried out controlled cooling experiments which permit retrieval of a rate constant as a function of temperature by a much smaller number of experiments and crystal structure refinements than in the conventional method. A crystal from the ST meteorite was first annealed at a desired temperature (T 0 ) and then cooled at a fixed cooling rate, which was controlled by a computer. The cooling rates, in °C/min, were as follows: 11.3 for the sample ST7, 2.4 for ST6, 0.48 for ST5/1, and 0.0115 for ST 5/2 (Fig. 1). The duration of annealing for equilibration at T 0 was guided by earlier experience (Stimpfl et al., 1999). After cooling to a temperature of 650°C, which is much below the closure temperature (T C) expected from the available kinetic and thermodynamic data and the imposed cooling rates, the sample was quickly removed from the furnace and quenched. Both annealing at T 0 and the cooling experiments were carried out in sealed silica tubes in the presence of a wu¨stite-iron buffer. The buffer was tested for its reactivity by sealing it with a relatively small amount of physically separated NiO inside a silica tube and holding the assembly inside a furnace for half an hour at 550°C. The NiO was found to completely convert to Ni. The results of controlled cooling experiments and the modeling of the data are illustrated in Fig. 1. The quenched site occupancy in each experiment is shown by a filled circle along with its ⫾1 uncertainty arising from that in the structural data. The bulk compositions and site occupancies of the crystals (ST5, ST6, and ST7) used for the controlled cooling experiments were determined in the same way as those of the untreated crystals (the complete data are not shown for the sake of brevity, but are available from the authors). The solid lines are the calculated evolution of the ordering states of the orthopyroxene crystals according to the imposed cooling rates and a
Cooling rates of stony-iron meteorites
Fig. 1. Controlled cooling rate experiments using orthopyroxene crystals from the Steinbach meteorite to derive the disordering rate constant ( ¡ K) during the ordering process of Fe* and Mg between the M1 and M2 sites. Fe* ⫽ Fe2⫹ ⫹ Mn (the total atomic fraction of Mn is ⬃0.005). The slanted solid line represents the equilibrium ordering path. ST5, ST6, and ST7 are crystal numbers. The quenched occupancy of Fe* in the M2 site for each cooling rate is shown by filled circles with ⫾1 error bars. The corresponding initial annealing temperatures are shown on the equilibrium path. The solid and dashed lines represent the calculated evolution of the ordering states for the imposed cooling rates and rate constants given by Eqns. 2 and 3, respectively.
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Fig. 2. Arrhenian plot illustrating the dependence of the disordering ¡ rate constant, ¡ K, on temperature ( ¡ K⫽¡ K 0 e⫺Q/RT). (a) Both K 0 and Q are derived from the controlled cooling rate experiments (Fig. 1); (b) ¡ only K 0 was derived from the controlled cooling rate experiment, whereas Q was constrained to 71409 cal/mol; (c) the result from the optimized expression of Ganguly and Tazzoli (1994).
sition, but if we constrain it to the latter value, then the ¡ optimization analysis through MINUIT yields K 0 ⫽ 3.10 ⫻ 1011/min. The corresponding Arrhenian relation is then: ¡ K ⫽ 3.1 ⫻ 10 11 e⫺35938/T/min
¡ single temperature dependent disordering rate constant, K, that leads to the best fit of the four quenched ordering states. The ¡ retrieved K can be expressed in an Arrhenian form as: ¡ K ⫽ 7.6 ⫻ 10 12e⫺39510/T/min
(2)
which implies an activation energy (Q) of 78506 cal/mol. ¡ ¢ In principle, both k D and K (or K) can be retrieved from the controlled cooling experiments, but since there are four unknown constants in the expressions of these parameters, it is desirable to have data from more than four controlled cooling experiments. In modeling the present set of data, we used the k D values from Eqn. 1, which incorporates the data for a Steinbach orthopyroxene, instead of treating it as an adjustable ¡ parameter. The pre-exponential parameter ( K 0 ) and Q in Eqn. 2 were derived by interfacing the calculation of the change of ordering state as a function of temperature with an optimization program, MINUIT (James and Roos, 1975), and treating both parameters as adjustable variables. The program was made to ¡ find the best combination of K 0 and Q that provides the best overall match, according to the least-squares method, between the experimental and calculated values of the quenched ordering states. The retrieved activation energy seems somewhat too ¡ high compared to that derived for K from the ordering experiments by using orthopyroxene crystals of ⬃Fs50 composition, which is 71409 cal/mol (Stimpfl et al., 1997, and work in progress). Of course, Q could change as a function of compo-
(3)
The Arrhenian relations given by Eqns. 2 and 3 are illustrated in Fig. 2. ¡ Using the available data on the disordering rate constant, K, derived primarily from disordering experiments, Ganguly and Tazzoli (1994) obtained an optimized Arrhenian relation, in ¡ which K 0 was expressed as a function of composition, and Q was assumed to be constant. Using an orthopyroxene composition of Fs15, their relation is compared with those obtained in this work in Fig. 2. The difference between the Arrhenian relations derived above and that obtained by Ganguly and Tazzoli (1994) partly lies in the difference between the nature of change of Fe–Mg site occupancy (ordering versus disordering) in the two data sets (also see Stimpfl et al., 1997), and also probably to a larger compositional dependence of the rate constant than assumed by Ganguly and Tazzoli (1994) from meager data of this nature. 3. COOLING RATES FROM THE CATION ORDERING DATA
¡ Using the above data on k D and K, we calculated the cooling rates of the natural crystals so that the quenched ordering states in the calculated evolution of ordering state as a function of temperature match those determined for the natural crystals. The cooling was assumed to follow an asymptotic (1/T ⫽ 1/T 0 ⫹ t) model. The T C for each sample along with the retrieved values of the constant , corresponding to the Arrhenian relation given by Eqn. 2, are summarized in Table 2. Use of the
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J. Ganguly and M. Stimpfl 4. METAL-SILICATE MIXING AND IMPLICATIONS OF THE ORTHOPYROXENE COOLING RATES
Fig. 3. Comparison of the cooling rates at ⬃400 ⫺ 450°C derived from the Fe-Mg ordering data in orthopyroxenes (open diamonds), and kinetic parameter given by Eqn. 2, with those derived from the metallographic data (filled circles) in Sa˜o Joa˜o Nepomuceno and Steinbach meteorites. The kinetic data is given by Eqn. 2. The vertical bars represent ⫾1 errors in the cooling rate values.
constrained Arrhenian relation, Eqn. 3, yields values that are larger by factors of ⬃6 and ⬃8 for the ST and SJN samples, respectively, than those given in Table 2. It is generally recognized that the cooling rate derived from a particular temperature sensitive property of a mineral is strictly applicable only near the closure temperature of that property. At the latter condition, however, the calculated cooling rates are not very sensitive to the choice of the cooling model even though these may differ significantly at other temperatures depending on the cooling model. The retrieved cooling rates at approximately the mean closure temperatures for the two groups of crystals, i.e., 400°C for SJN and 450°C for Steinbach, using the values summarized in Table 2 are illustrated in Fig. 3 along with the respective metallographic cooling rates. The cooling rates obtained from ¡ the constrained K (Eqn. 3) are larger by factors of ⬃6 and ⬃8 for ST and SJN crystals, respectively, since dT/dt ⫽ ⫺ T2. Although they suggested a cooling rate of ⬃20°C/Ma for ST, Rasmussen et al. (1995) were able to fit the entire spectrum of metallographic data only by considering a two-step cooling model, namely a high temperature step above ⬃440 ⫺ 460°C characterized by a cooling rate of ⬃150°C/Ma, followed by slower cooling of ⬃20°C/Ma. Both of these cooling rates have been illustrated in Fig. 3. Considering the errors in the determination of ordering states of orthopyroxene arising from those in the structural (Table 2) and compositional data (Stimpfl et al., 1999), we estimate uncertainties of around factors of 10 and 5 in the calculated cooling rates for SJN and ST orthopyroxene crystals, respectively. Rasmussen et al. (1995) tentatively estimated the uncertainty of the metallographic cooling rates to be ‘better than a factor of 3⬘. We have assigned an uncertainty of this magnitude to the MCR data.
The thermal history and metal-silicate mixing model of Haack et al. (1996) requires the metal and silicate components of IVA stony irons to experience similar thermal history below the temperature of development of the metal textures. The data illustrated in Fig. 3 are compatible with this model only if the cooling rate data for both metals and orthopyroxenes are pushed to their statistical error limits in two opposite directions. In that case, one has to accept similar cooling rate, ⬃400°C/ Ma, for both ST and SJN in the high temperature regime, i.e., above 400 – 450°C. Calculation of the evolution of Fe2⫹-Mg ordering state, using the unconstrained kinetic data (Eqn. 2), shows that no significant ordering would take place if the ST samples cooled by ⬃50°C/Ma at T ⬍ 425°C. Because the SJN metals do not record slow cooling at low temperature, one has to conclude that slow cooling in this stony iron ensued at a temperature, ⬍350°C, at which the metals were unable to record it because of extremely sluggish Fe–Ni interdiffusion rate which has an exponential dependence on temperature. The reconciliation of the orthopyroxene and metal cooling rates becomes extremely difficult within the framework of the breakup-reassembly model of Haack et al. (1996) if one accepts ¡ the constrained kinetic data for the rate constant, K, for the Fe–Mg ordering in orthopyroxene (Eqn. 3). Although this would improve the agreement between the metal and orthopyroxene cooling rates for the SJN samples by pushing the latter upward by a factor of 8 –9, the cooling rates for the ST orthopyroxene crystals would be faster by factors of 5 to 7 than those shown in Fig. 3, making them too fast compared to the metal cooling rates. Explanation of these data would then necessitate cooling of the silicates and metals in ST in different environments, shallow and deep-seated, respectively, and mixing after the metals had cooled through the closure temperatures for the development of the metal textures. On the other hand, the metals and silicates of SJN would have to cool in the same environment and thus had to be assembled prior to the development of metal textures. Consequently, these two stony irons would have to be derived from two different parent bodies, which broke up at different stages relative to the temperature of development of the metal textures, and reassembled gravitationally to produce metal-silicate mixtures. However, a strong argument against this model comes from the textural data of ST, presented by Scott et al. (1996, Fig. 3b), which shows silicates, including orthopyroxenes, embedded in what seems to be a large (⬃3.4 cm) single crystal of metal with exsolution textures oriented in the same direction throughout the section. This type of embedding would seem very unlikely if the metals and silicates were assembled after the metals cooled below 350°C. This is a stronger argument against the two parent body model than the notion (Rasmussen et al., 1995) that Scott et al. (1996) were able to relate the interelement properties of metals in these meteorites to a fractional crystallization model of a common metallic core. This is because they assumed a complex nonequilibrium partitioning behavior of S between solid and liquid metal as a function of S concentration, and an admittedly unrealistic assimilation model in order to match the observed interelement data to a fractional crystallization trend. Even then, the fractional crystallization-assimila-
Cooling rates of stony-iron meteorites
tion model was not completely successful (e.g., Co versus Ni trend). In summary, we favor a single parent body model for the high and low Ni groups, primarily on the strength of the textural data, as discussed above. The reconciliation of the metal and orthopyroxene cooling rate data within the framework of a single parent body model suggests that both ST and the SJN cooled fairly rapidly and at a similar rate of ⬃400°C/Ma at high temperature, after the gravitational reassembly of the disrupted parent body created a random mixture of the metal and silicate components, as envisioned by Haack et al., 1996. This was followed for ST by slow cooling at ⬃50°C/Ma at T ⬍ 425°C. The slow cooling did not ensue for SJN until the temperature dropped below ⬃350°C where the diffusion kinetics in the metals were too slow to record the change of cooling rate. The similar cooling rates for both ST and SJN above 425°C, as required to reconcile the metal and orthopyroxene cooling rate data, is at variance with the earlier notion (Rasmussen et al., 1995) of distinctly different cooling rates for the high and low Ni iron and stony irons. The question, however, remains as to why the most probable estimate of cooling rate of each orthopyroxene crystal from Steinbach is much faster than those from Sa˜o Joa˜o Nepomuceno, although the cation ordering states of both groups of orthopyroxenes were determined by the same method. Acknowledgments—Constructive reviews by Drs. Herbert Kroll, Ed Scott, and an anonymous reviewer are gratefully acknowledged. Ed Scott is especially thanked for his patience and willingness to discuss several issues through many E-mail correspondences which have substantially influenced the final interpretation of our data. We are grateful to Dr. Ed Olsen for the donation of the Steinbach sample from the collection of the Natural History Museum of Chicago and to the Smithsonian Institution for the donation of the Sa˜o Joa˜o Nepomuceno sample. REFERENCES Anovitz L. M., Essene E. J., and Dunham W. R. (1988) Order-disorder experiments on orthopyroxenes: Implications for the orthopyroxene geospeedometer. Am. Mineral. 73, 1060 –1073. Besancon J. R. (1981) Rate of cation ordering in orthopyroxenes. Am. Mineral. 66, 965–973. Dollase W. A. and Newman W. I. (1984) Statistically most probable stoichiometric formulae. Am. Mineralogist 69, 553–556. Finger L. W. and Prince E. (1975) A system of FORTRAN IV computer programs for crystal structural computations. NBS Tech Note, 854. Ganguly J. (1982) Mg–Fe order-disorder in ferromagnesian silicates: II. Thermodynamics, kinetics and geological applications. In: (ed.) S. K. Saxena, Advances in Physical Geochemistry (pp. 58 –99) 2 Springer-Verlag, New York. Ganguly J. and Domeneghetti M. C. (1996) Cation ordering of orthopyroxenes form the Skaergaard intrusion: Implications for the subsolidus cooling rates and permeabilities. Contrib. Mineral. Petrol. 122, 359 –367. Ganguly J. and Tazzoli V. (1994) Fe2⫹-Mg interdiffusion in orthopyroxene: Retrieval from the data on intracrystalline exchange reaction. Am. Mineral. 79, 930 –937. Ganguly J., Stimpfl M., and Molin G. (1997) Orthopyroxene chronom-
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etry of meteorites: II. Towards a general relation between cooling rate and closure temperature of cation ordering and application to the Steinbach meteorite. Proc. 28th Lunar Planet. Sci. Conf. 391–392. Ganguly J., Yang H., and Ghose S. (1994) Thermal history of mesosiderites: Quantitative constraints from compositional zoning and Fe–Mg ordering in orthopyroxenes. Geochim. Cosmochim. Acta 58, 2711–2723. Haack H., Scott E. R. D., Love S. G., Brearley A. J., and McCoy T. J. (1996) Thermal histories of IVA stony-iron and iron meteorites: Evidence for asteroid fragmentation and reaccretion. Geochim. Cosmochim. Acta 60, 3103–3113. Hawthorne F. C. and Ito J. (1978) Refinement of the crystal structures of Mg0.776Co0.224)SiO3 and (Mg0.925Mn0.075)SiO3. Acta Crystallographica B34, 891– 893. James F. and Ross M. (1975) MINUIT, a system for function minimisation and analysis of the parameter error and correlations. Comput. Phys. Commun. 10, 343–347, CERN/DD, Internal Report 75/20. Kroll H., Lueder T., Schlenz H., Kirfel A., and Vad T. (1997) The Fe2⫹-Mg distribution in orthopyroxene: A critical assessment of its potential as a geospeedometer. Eur. J. Mineral. 9, 733–750. Moren A. E. and Goldstein J. L. (1978) Cooling rate variations of group IVA iron meteorites. Earth Planet. Sci. Lett. 40, 151–161. Moren A. E. and Goldstein J. L. (1979) Cooling rates of group IVA iron meteorites determined from a ternary Fe-Ni-P model. Earth Planet. Sci. Lett. 43, 182–196. Mueller R. F. (1967) Model for order-disorder kinetics in certain quasibinary crystals of continuously variable composition. J. Phys. Chem. Solids 28, 2239 –2243. Mueller R. F. (1969) Kinetics and thermodynamics of intracrystalline distribution. Min. Soc. Am. Spec. Papers 2, 83–93. O’Neill H. StC. (1994) An experimental study of the factors influencing kinetics of cation ordering in oxide spinels. European Science Foundation Programme on Kinetic Processes in Minerals and Ceramics. Rasmussen K. L., Ulff-Møller F., and Haack H. (1995) The thermal evolution of the IVA iron meteorites: Evidence from metallographic cooling rates. Geochim. Cosmochim. Acta 59, 3049 –3059. Saikumar V. and Goldstein J. I. (1988) An evaluation of the methods to determine the cooling rates of iron meteorites. Geochim. Cosmochim. Acta 52, 715–726. Saxena S. K., Tazzoli V., and Domeneghetti M. C. (1987) Kinetics of Fe2⫹-Mg distribution in aluminous orthopyroxenes. Phys. Chem. Minerals 15, 140 –147. Scott E. R. D., Haack H., and McCoy T. J. (1996) Core crystallization and silicate-metal mixing in the parent body of the IVA iron and stony-iron meteorites. Geochim. Cosmochim. Acta 60, 1615–1631. Sheldrick G. M. (1993) SHELXL93. Program for crystal structure refinement. University of Go¨ttingen, Germany. Stimpfl M., Ganguly J., and Molin G. (1997) Orthopyroxene chronometry of meteorites: I. Experimental determination of thermodynamic and kinetic parameters. Proc. 28th Lunar Planet. Sci. Conf. 1379 – 1380. Stimpfl M., Ganguly J., and Molin G. (1999) Fe2⫹-Mg order-disorder in orthopyroxene: Equilibrium fractionation between the octahedral sites and thermodynamic analysis. Contrib. Mineral. Petrol. 136, 297–309. Ulff-Møller F., Rasmussen K. L., Prinz M., Palme H., Spettel B., and Kallemeyn G. W. (1995) Magmatic activity on the IVA parent body: Evidence from silicate-bearing meteorites. Geochim. Cosmochim. Acta 59, 4713– 4728. Vorshage H. and Feldmann H. (1979) Investigations on cosmic ray produced nuclides in iron meteorites, 3. Exposure ages, meteoroid sizes and sample depths determined by mass spectrometric analyses of potassium and rare gases. Earth Planet. Sci. Lett. 45, 293–308.
Pergamon
PII S0016-7037(00)00405-6
Cation ordering in orthopyroxenes from two stony-iron meteorites: Implications for cooling rates and metal-silicate mixing JIBAMITRA GANGULY* and MARILENA STIMPFL Department of Geosciences, University of Arizona, Tucson, AZ 85721, USA (Received July 13, 1999; accepted in revised form November 3, 1999)
Abstract—We have determined the cooling rates of orthopyroxene crystals from two group IVA stony iron meteorites—Steinbach (ST) and Sa˜o Joa˜o Nepomuceno (SJN)— on the basis of their Fe–Mg ordering states. The rate constant was calibrated as a function of temperature by controlled cooling experiments using orthopyroxene crystals separated from ST. These data were used along with earlier calibrations of the equilibrium intracrystalline fractionation of Fe and Mg as a function of temperature for crystals separated from both meteorites to calculate their cooling rates. The site occupancies of the orthopyroxene crystals were determined by single-crystal X-ray diffraction subject to the bulk compositional constraints. The closure temperatures (T c) of cation ordering for the untreated crystals from SJN are ⬃400°C, whereas those from ST vary between ⬃430 and 470°C. Reconciliation of the metallographic and orthopyroxene cooling rate data, within the framework of the metal–silicate mixing model of Haack et al. (1995), suggests that these two stony irons had cooled at a similar rate of ⬃400°C/Ma through the closure temperatures for cation ordering in the orthopyroxenes. This was followed by slow cooling for ST at ⬃50°C/Ma at T ⬍ 425°C. Similar slow cooling was not recorded by the metals in SJN, which implies that if this stony iron were subjected to slow cooling, it must have been below 350°C. The similar cooling rates above 425°C for both ST and SJN, as required to reconcile the metal and orthopyroxene cooling rate data, is at variance with the earlier notion (Rasmussen et al., 1995) of distinctly different cooling rates for the high and low Ni IVA irons and stony irons. The cation ordering and metallographic cooling rate data are also amenable to an alternative interpretation, which requires two different parent bodies for the two stony irons, and mixing of the metal and silicate components of ST after the metals had cooled below the closure temperature of Fe–Ni interdiffusion. However, the available textural data for ST seems to argue against such metal-silicate mixing model. Copyright © 2000 Elsevier Science Ltd with a model of origin in a common asteroidal core. Also, the metal–silicate mixing of the IVA stony irons remains a puzzling issue. Moren and Goldstein (1978, 1979) argued that the IVA irons meteorites did not originate in a core, but crystallized from metallic melts dispersed throughout the mantle of the parent body. Ulff–Møller et al. (1995) argued that the silica saturated mineralogy of the IVA stony irons could not have crystallized from magmas that evolved in an olivine rich mantle. They suggested a model of metal-silicate mixing near the core-mantle boundary in which the magmas were separated from the refractory residue at different stages of partial melting and injected into the core through fractures that presumably developed by solidification shrinkage of the core. According to them, this model also helps explain the low Fe/Mn ratio of pyroxenes in the IVA stony irons, as Fe would be reduced by equilibration with the metal. Scott et al. (1996) showed that even though the IVA irons have extremely diverse cooling rates which are difficult to reconcile with a model of their core origin in a parent body, the chemical characteristics of the IVA metal phases are consistent with fractional crystallization of an asteroidal core, which contradicts the “raisin bread” model of Moren and Goldstein (1978, 1979). Haack et al. (1996) carried out transmission electron microscopic (TEM) studies of the disordered clinobronzite in the ST meteorite by using electron diffraction and high-resolution imaging techniques, and concluded from these data that it had cooled at ⬃100°C/hr through 1200°C, whereas the IVA irons had cooled much slower cooling, ⬍300°C/y, in
1. INTRODUCTION
Steinbach (ST) and Sa˜o Joa˜o Nepomuceno (SJN) are stony iron meteorites consisting of a unique mixture of metals and ⬃60 vol% of silicates made of coarse-grained mixture of tridymite, orthopyroxene and clinopyroxene. Scott et al. (1996) showed that the chemical properties of the metal phases of these meteorites are closely related to those of IVA irons, which suggest a common origin for both types. Rasmussen et al. (1995) determined the metallographic cooling rates (MCR) of the IVA irons and stony irons by using the updated data for the Fe–Ni phase diagram and Ni diffusion coefficients (Saikumar and Goldstein, 1988). They found a wide range of values from 19 to 3400 K/Ma, which could be separated into two groups according to the Ni content of the meteorites: one group with Ni ⬍ 8.4 wt% with a very large range of cooling rate between ⬃250 K/Ma and 3400 K/Ma, apparently correlating inversely with the Ni content; the other a relatively high-Ni group (8.6 –10.5 wt%) with much slower and similar cooling rates between 20 and 60 K/Ma. The mean MCR for ST is ⬃20 K/Ma, whereas that of SJN is ⬃900 K/Ma. These cooling rates are compatible with the MCR versus Ni content systematics observed for the IVA iron meteorites. The large span of MCR for IVA irons poses severe problems
This research was supported by a NASA Grant NAGW 3638. * Author to whom correspondence should be addressed (ganguly@ geo.arizona.edu). 1291
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the range 1200 –1000°C, as evidenced by the absence of dendrites in large troilite nodules. To explain the metal–silicate mixing and to reconcile the suggestion of Scott et al. (1996) about the core origin of the metals with their diverse metallographic cooling rates, Haack et al. (1996) developed a four-stage evolutionary model for the IVA irons and stony-irons: (1) high-velocity impact of the asteroidal parent body after ⬃95% crystallization of the core; (2) gravitational reassembly of the silicate mantle and the metal core fragments within a dust cloud; followed by (3) cooling through temperature range (⬃750 – 400°C) required for the development of exsolution textures in Fe–Ni alloys; and (4) catastrophic disruption 450 My ago, which represents the cosmic ray exposure age of most IVA irons (Voshage and Feldman, 1979). In this model, the metals are expected to record diverse cooling rates as these were buried at random depths after the gravitational reassembly. However, as discussed by Rasmussen et al. (1995), the problem with the breakup–reassembly model is the apparent exclusive yield of the high Ni meteorites from a single slowly cooled fragment requiring a meteorite yield factor equivalent to that of a randomly sampled single core. The quenched Fe2⫹–Mg ordering state between the M1 and M2 structural sites of an orthopyroxene crystal is an important recorder of its cooling rate (Ganguly, 1982). The closure temperature of cation ordering falls in the range of ⬃300 – 400°C for cooling rates in the range of 1–104°C/Ma (Ganguly et al., 1997). Thus, the Fe–Mg ordering state of orthopyroxene provides an important method of determining the cooling rate of the silicate component somewhat below the temperature range at which the metallographic cooling rates are valid (e.g., Ganguly et al., 1994). If the break-up and reassembly model of Haack et al. (1996) were valid, then one would expect to retrieve similar cooling rates from both metal textures and Fe–Mg ordering data in orthopyroxenes because these would have cooled in the same environment through their respective closure temperatures. In view of the complexity of the problems of metal–silicate mixing and thermal history of the IVA irons and stony irons, and the lack of any quantitative constraints on the low temperature cooling rates of the silicate components, we undertook an investigation of the cooling rates of the orthopyroxene crystals in ST and SJN. 2. RETRIEVAL OF COOLING RATES FROM CATION ORDERING IN ORTHOPYROXENES: METHOD AND DATA
2.1. Outline of Theory and Numerical Method The method of retrieval of cooling rate from the Fe2⫹–Mg ordering state of orthopyroxene has been discussed in a number of earlier publications (e.g., Ganguly, 1982; Ganguly et al., 1994; Ganguly and Domeneghetti, 1996). Thus, we provide here only a brief summary of the method and the underlying principles. In orthopyroxenes, Fe2⫹ and Mg fractionate between two nonequivalent octahedral sites, M1 and M2, with a preference of Fe2⫹ for M2 and that of Mg for M1 sites. Mueller (1967, 1969) developed a kinetic theory of order-disorder by treating the fractionation of cations between the nonequivalent structural sites in terms of a homogeneous exchange reaction, viz.:
disordering 3 Fe2⫹(M2) ⫹ Mg(M1) 7 Mg(M2) ⫹ Fe2⫹(M1), 4 ordering and developed an expression for the change of site occupancy as a function of time under isothermal condition. In this formulation, in which the homogeneous exchange reaction has been assumed to be a second-order elementary kinetic process, the time scale of ordering is related to the initial and final ordering states through the rate constant for the disordering ¡ process, ( K) and the equilibrium Fe2⫹–Mg distribution coefficient, k D, between the two sites ((Fe2⫹/Mg)M1/(Fe2⫹/Mg)M2). The rate of change of site occupancies or ordering state is ¡ ¢ governed by both disordering ( K) and ordering ( K) rate con¢ ¡¢ stants, but K is eliminated by using the relation that k D ⫽ K/ K. The k D is defined as the ratio of (Fe2⫹/Mg) in the M1 to that in the M2 site. The choice of which rate constant is to be retained is arbitrary. Using Mueller’s theory, Ganguly (1982) developed a numerical method of determining the cooling rate of orthopyroxene on the basis of its quenched or observed ordering state. The method involves calculation of the ordering state of a crystal from an initial temperature in a series of isothermal steps approximating an assumed T-t relation following a smooth functional form (e.g., an asymptotic relation described by 1/T ⫽ 1/T o ⫹ t, where T 0 is the initial temperature and is a constant with the dimension of K ⫺1 t ⫺1 ) and varying the latter (i.e., in the asymptotic model) until a satisfactory match is obtained between the quenched ordering state in the numerical simulation and that observed in a natural crystal. (This method can also be used to retrieve the thermodynamic and kinetic parameters governing the ordering process by controlled cooling experiments, as discussed below.) The retrieved cooling rate is quite insensitive to errors in the determination of T 0 as long as it is within the domain at which thermodynamic equilibrium is maintained. However, it is very sensitive to errors in the determination of site occupancies of ¡ ¢ the natural crystals and of k D and K (or K) as a function of temperature and composition. In order to avoid the problem of ¡ the compositional effects, we determined both k D and K by using crystals separated from ST and SJN meteorite fragments. 2.2. Site Occupancies of Orthopyroxenes from the IVA Stony Iron Meteorites The site occupancies of the orthopyroxene crystals were determined by single crystal structure refinements, subject to the bulk compositional constraints. Stimpfl et al. (1999) carried out a detail comparative study of the procedures of site occupancy determination by using different strategies but the same X-ray data set, and concluded that the two widely used programs SHELXL-93 (Sheldrick, 1993) and RFINE90 (which is an updated version of RFINE4, Finger and Prince, 1975) essentially yield the same results that are not significantly affected by whether one uses ionic or atomic scattering factors. However, the use of ionic scattering factors yield better goodness of fit and weighted agreement index (R w), and smaller standard deviation on the site occupancy data. The results obtained from either program were also in excellent agreement
Cooling rates of stony-iron meteorites
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Table 1. Initial (I) and adjusted (adj) bulk compositions of the orthopyroxene crystals from Sao Joao Nepomuceno (SJN) and Steinbach (ST) meteorites used for the determination of cooling rates. An initial composition represents the mean of a large number of spot analyses. The numbers within parentheses are the estimated SDs from the mean in the right justified form. Sample
SJN23 (l)
SJN23 (adj)
SJN24 (l)
SJN24 (adj)
SJN25 (l)
SJN25 (adj)
ST71 (l)
ST71 (adj)
ST76 (l)
ST76 (adj)
ST77 (l)
ST77 (adj)
Si 1.986 (3) 1.984 1.985 (3) 1.983 1.985 (5) 1.981 1.986 (4) 1.9833 1.988 (1) 1.9889 1.990 (3) 1.9849 Al 0.014 (5) 0.016 0.015 (3) 0.017 0.015 (4) 0.019 0.014 (2) 0.0167 0.009 (1) 0.0111 0.009 (3) 0.0151 Ti 0.001 (0) 0.000 0.001 (0) 0.000 0.001 (1) 0.001 0.001 (1) 0.0005 0.001 (1) 0.0003 0.000 (0) 0.0000 Cr 0.018 (1) 0.017 0.019 (1) 0.018 0.012 (1) 0.019 0.016 (1) 0.0158 0.011 (1) 0.0105 0.016 (1) 0.0151 Mg 1.668 (4) 1.667 1.664 (6) 1.662 1.669 (10) 1.665 1.641 (7) 1.6424 1.665 (3) 1.6625 1.648 (7) 1.6503 Fe 0.275 (3) 0.274 0.281 (5) 0.279 0.274 (3) 0.274 0.305 (3) 0.3053 0.304 (3) 0.3015 0.304 (3) 0.3047 Mn 0.018 (1) 0.018 0.019 (1) 0.019 0.018 (1) 0.018 0.016 (1) 0.0160 0.015 (1) 0.0146 0.016 (1) 0.0160 Ni 0.000 (0) 0.000 0.000 (0) 0.000 0.000 (0) 0.000 0.000 (0) 0.0000 0.000 (0) 0.0000 0.000 (0) 0.0000 Ca 0.023 (1) 0.023 0.021 (1) 0.021 0.023 (1) 0.023 0.020 (1) 0.0200 0.011 (1) 0.0106 0.014 (1) 0.0140 Na 0.001 (1) 0.002 0.001 (1) 0.002 0.001 (1) 0.001 0.000 (0) 0.0000 0.000 (0) 0.0000 0.000 (0) 0.0000 Fe3⫹ 0.000 (0) 0.000 0.000 (0) 0.000 0.000 (0) 0.000 0.000 (0) 0.0000 0.000 (0) 0.0000 0.000 (0) 0.0000 Total 4.004 4.000 4.004 4.000 4.004 4.000 3.999 4.000 4.004 4.000 3.997 4.000 Charge 12.012 12.000 12.013 12.000 12.012 12.000 12.002 12.000 12.006 12.000 11.999 12.000 Al(T)a 0.014 0.016 0.015 0.017 0.015 0.019 0.014 0.017 0.009 0.011 0.009 0.015 Al(M)a 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 d(chrg(M)b 0.018 0.016 0.019 0.017 0.020 0.019 0.018 0.017 0.013 0.011 0.016 0.015 d(M-T)c 0.004 0.000 0.004 0.000 0.005 0.000 0.004 0.000 0.004 0.000 0.007 0.000 M-cations 2.004 2.000 2.004 2.000 2.004 2.000 1.999 2.000 2.007 2.000 1.998 2.000 a
Al(T): Al in T site; Al (M): Al in M site. d(chr(M)): exess positive charge in M site due to the substitution for 2⫹ cations. c d(M-T): difference between the charge excess in M site and the charge deficiency in T site. b
with those obtained from the new refinement strategy developed by Kroll et al. (1997, and personal communication). We separated a number of orthopyroxene crystals from ST and SJN and from these selected three crystals from each meteorite on the basis of their optical, dimensional, and X-ray properties for single-crystal structure refinements. The site occupancies were determined by using the X-ray data, ionic scattering factors, and the bulk compositional constraints in the program RFINE 90. The details of X-ray data collection and the procedure of site occupancy determination have been discussed in detail by Stimpfl et al. (1999) and are, thus, not repeated here for the sake of brevity. The observed X-ray intensities were weighted according to the scheme built into the program, which results in a normal distribution with mean 0 and variance 1 of the weighted difference between the observed and calculated intensities. Following Ganguly and Domeneghetti (1996), Fe2⫹ and Mn (⬃0.005 atomic fraction) were treated as one component, as these cations fractionate almost equally between the M1 and M2 sites (Hawthorne and Ito, 1978), and are designated as Fe*. After the X-ray data collection, the composition of each crystals was determined by a large number of spot analyses in an electron microprobe against mostly synthetic standards, as described in Stimpfl et al. (1999). Each crystal was found to be homogeneous within the resolution of the microprobe spot analyses, which were carried out by using 15 kV accelerating voltage, 20 –25 nA beam current, and counting times of 20 s at the peak and 20 s at the background. The average compositions along with their respective standard deviations (SDs) were used to determine the statistically most probable bulk composition of each crystal by a computer aided mathematical projection of the average composition of each element, weighted by their respective SDs, on to the n-dimensional surface that satisfies the crystal chemical constraints for orthopyroxene. This
method is based on the work of Dollase and Newman (1984), and is described in detail by Stimpfl et al. (1999). The average and adjusted compositions of each crystal used for the cooling rate calculations are summarized in Table 1. For all crystals, the adjusted composition of each element was found to be within the ⫾1 error of its average composition. The site occupancy data of each crystal are summarized in Table 2. 2.3. Determination of the Thermodynamic and Kinetic Parameters Stimpfl et al. (1999) determined the equilibrium k *D, defined as (Fe*/Mg)M1/(Fe*/Mg)M2, as a function of temperature for two orthopyroxene crystals, one separated from the ST meteorite and the other from SJN. The equilibrium k D-s for the ST crystals were determined at 700, 800, 900, and 1000°C by both ordering and disordering experiments at each temperature, which constrained the equilibrium values very tightly, whereas those of the SJN crystals were determined by disordering experiments at 700 and 800°C and tightly constrained by both ordering and disordering experiments at 750°C. The fitted linear relations between lnk *D versus 1/T for the two crystals are essentially the same, which is expected as these crystals (Fs14 and Fs15) have similar compositions. The site occupancy of another low Fe crystal (Fs11) from an Antarctic meteorite was also determined very precisely by using single crystal X-ray diffraction data but different structure refinement strategy (Kroll et al., 1997) by Lueder (1995; cited and discussed in Stimpfl et al., 1999). The k *D values retrieved from these data are also very similar to those from the ST and SJN crystals. A least-squares regression of the three low Fe data sets yields (Stimpfl et al., 1999): ln k *D ⫽ ⫺
2854共⫾86兲 ⫹ 0.603共⫾0.095兲 T
(1)
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J. Ganguly and M. Stimpfl
Table 2. Summary of the site occupancy data, k *D, closure temperatures (T c) and cooling time constants () of the orthopyroxene crystals from Sa˜o Joao Nepomuceno (SJN) and Steinbach (ST). The SD stands for the estimated standard deviation of the site occupancies of Fe* (i.e., Fe ⫹ Mn) and Mg arising from those of the structural data, as given by the refinement program RFINE90. The uncertainty of the structural and bulk compositional data leads to uncertainties of factors of ⬃10 and 5 in the values (and hence cooling rates) for the SJN and ST samples, respectively. Sa˜o Jao Nepomuceno Sample no.
Steinbach
SJN23
SJN24
SJN25
ST71
ST76
ST77
M2 site Mg Fe* Na Ca
0.6945 0.2815 0.0015 0.0225
0.6905 0.2875 0.0015 0.0205
0.6961 0.2803 0.0010 0.0226
0.6744 0.3057 0.0000 0.0200
0.6900 0.2994 0.0000 0.0110
0.6788 0.3072 0.0000 0.0140
M1 site Mg Fe* Al Cr Ti
0.9719 0.0107 0.0000 0.0173 0.0001
0.9712 0.0104 0.0000 0.0180 0.0004
0.9692 0.0109 0.0000 0.0194 0.0005
0.9680 0.0156 0.0000 0.0160 0.0010
0.9725 0.0167 0.0000 0.0110 0.0000
0.9712 0.0138 0.0000 0.0150 0.0000
T site Si Al SD k *D T c (°C) , K⫺1Y⫺1 (dT/dt)Tc (°C/Ma)
1.9840 0.0160 0.0010 0.0270 405 8.5E-10 390
1.9830 0.0170 0.0008 0.0260 396 4.5E-10 200
1.9810 0.0190 0.0009 0.0280 409 1.2E-09 558
1.9830 0.0170 0.0009 0.0340 442 2.0E-08 10225
1.9890 0.0110 0.0009 0.0400 473 7.25E-08 40347
1.9850 0.0150 0.0009 0.0310 429 4.60E-09 2267
where the uncertainties represent ⫾1 (SD). ¡ ¢ In principle, each rate constant, K and K, can be retrieved by modeling the data on the change of either ordering or disordering state as a function of time and temperature if the k D is known as a function of temperature (e.g., Besancon, 1981; Ganguly, 1982; Anovitz et al., 1988; Saxena et al., 1987). However, contrary to the expectation from the kinetic model of Mueller (1967, 1969), Stimpfl et al. (1997) found that the value of a rate constant at a fixed temperature depends on whether it is derived from ordering or disordering experiments. In partic¡ ular, K derived from disordering experiment using Mueller’s kinetic theory (1967; 1969) was found to be significantly larger than that derived from ordering experiments. Similar observation was made by O’Neill (1994) for Fe2⫹-Mg order-disorder in spinels. A theoretical explanation of this difference in the retrieved values of a specific rate constant from the ordering and disordering experiments of orthopyroxene is still lacking. However, even though it cannot reconcile the data from the two types of experiments, Mueller’s kinetic model provides a satisfactory phenomenological representation of the data from either type of experiment (Besancon, 1981; Saxena et al., 1987; Anovitz et al., 1988; Stimpfl et al., 1997). Since a rate constant is not only a function of temperature (and pressure) but also of composition (Besancon, 1981; Ganguly, 1982; Ganguly and Tazzoli, 1994), we decided to deter¡ mine K as a function of temperature using an orthopyroxene crystal from the ST meteorite, which obviates the need for any compositional correction on the rate constant. Also, because the value of a rate constant seems to depend on the nature of the ¡ experiment (ordering versus disordering), we determined K as a function of temperature by ordering the orthopyroxene crystal so it is compatible with that in the natural process of cooling. In this work, we deviated from the conventional approach (e.g., Besancon, 1981; Ganguly, 1982; Anovitz et al., 1988;
¡ Saxena et al., 1987) of determining K by modeling the change of ordering state or site occupancies as a function of time at different temperatures. Instead, we carried out controlled cooling experiments which permit retrieval of a rate constant as a function of temperature by a much smaller number of experiments and crystal structure refinements than in the conventional method. A crystal from the ST meteorite was first annealed at a desired temperature (T 0 ) and then cooled at a fixed cooling rate, which was controlled by a computer. The cooling rates, in °C/min, were as follows: 11.3 for the sample ST7, 2.4 for ST6, 0.48 for ST5/1, and 0.0115 for ST 5/2 (Fig. 1). The duration of annealing for equilibration at T 0 was guided by earlier experience (Stimpfl et al., 1999). After cooling to a temperature of 650°C, which is much below the closure temperature (T C) expected from the available kinetic and thermodynamic data and the imposed cooling rates, the sample was quickly removed from the furnace and quenched. Both annealing at T 0 and the cooling experiments were carried out in sealed silica tubes in the presence of a wu¨stite-iron buffer. The buffer was tested for its reactivity by sealing it with a relatively small amount of physically separated NiO inside a silica tube and holding the assembly inside a furnace for half an hour at 550°C. The NiO was found to completely convert to Ni. The results of controlled cooling experiments and the modeling of the data are illustrated in Fig. 1. The quenched site occupancy in each experiment is shown by a filled circle along with its ⫾1 uncertainty arising from that in the structural data. The bulk compositions and site occupancies of the crystals (ST5, ST6, and ST7) used for the controlled cooling experiments were determined in the same way as those of the untreated crystals (the complete data are not shown for the sake of brevity, but are available from the authors). The solid lines are the calculated evolution of the ordering states of the orthopyroxene crystals according to the imposed cooling rates and a
Cooling rates of stony-iron meteorites
Fig. 1. Controlled cooling rate experiments using orthopyroxene crystals from the Steinbach meteorite to derive the disordering rate constant ( ¡ K) during the ordering process of Fe* and Mg between the M1 and M2 sites. Fe* ⫽ Fe2⫹ ⫹ Mn (the total atomic fraction of Mn is ⬃0.005). The slanted solid line represents the equilibrium ordering path. ST5, ST6, and ST7 are crystal numbers. The quenched occupancy of Fe* in the M2 site for each cooling rate is shown by filled circles with ⫾1 error bars. The corresponding initial annealing temperatures are shown on the equilibrium path. The solid and dashed lines represent the calculated evolution of the ordering states for the imposed cooling rates and rate constants given by Eqns. 2 and 3, respectively.
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Fig. 2. Arrhenian plot illustrating the dependence of the disordering ¡ rate constant, ¡ K, on temperature ( ¡ K⫽¡ K 0 e⫺Q/RT). (a) Both K 0 and Q are derived from the controlled cooling rate experiments (Fig. 1); (b) ¡ only K 0 was derived from the controlled cooling rate experiment, whereas Q was constrained to 71409 cal/mol; (c) the result from the optimized expression of Ganguly and Tazzoli (1994).
sition, but if we constrain it to the latter value, then the ¡ optimization analysis through MINUIT yields K 0 ⫽ 3.10 ⫻ 1011/min. The corresponding Arrhenian relation is then: ¡ K ⫽ 3.1 ⫻ 10 11 e⫺35938/T/min
¡ single temperature dependent disordering rate constant, K, that leads to the best fit of the four quenched ordering states. The ¡ retrieved K can be expressed in an Arrhenian form as: ¡ K ⫽ 7.6 ⫻ 10 12e⫺39510/T/min
(2)
which implies an activation energy (Q) of 78506 cal/mol. ¡ ¢ In principle, both k D and K (or K) can be retrieved from the controlled cooling experiments, but since there are four unknown constants in the expressions of these parameters, it is desirable to have data from more than four controlled cooling experiments. In modeling the present set of data, we used the k D values from Eqn. 1, which incorporates the data for a Steinbach orthopyroxene, instead of treating it as an adjustable ¡ parameter. The pre-exponential parameter ( K 0 ) and Q in Eqn. 2 were derived by interfacing the calculation of the change of ordering state as a function of temperature with an optimization program, MINUIT (James and Roos, 1975), and treating both parameters as adjustable variables. The program was made to ¡ find the best combination of K 0 and Q that provides the best overall match, according to the least-squares method, between the experimental and calculated values of the quenched ordering states. The retrieved activation energy seems somewhat too ¡ high compared to that derived for K from the ordering experiments by using orthopyroxene crystals of ⬃Fs50 composition, which is 71409 cal/mol (Stimpfl et al., 1997, and work in progress). Of course, Q could change as a function of compo-
(3)
The Arrhenian relations given by Eqns. 2 and 3 are illustrated in Fig. 2. ¡ Using the available data on the disordering rate constant, K, derived primarily from disordering experiments, Ganguly and Tazzoli (1994) obtained an optimized Arrhenian relation, in ¡ which K 0 was expressed as a function of composition, and Q was assumed to be constant. Using an orthopyroxene composition of Fs15, their relation is compared with those obtained in this work in Fig. 2. The difference between the Arrhenian relations derived above and that obtained by Ganguly and Tazzoli (1994) partly lies in the difference between the nature of change of Fe–Mg site occupancy (ordering versus disordering) in the two data sets (also see Stimpfl et al., 1997), and also probably to a larger compositional dependence of the rate constant than assumed by Ganguly and Tazzoli (1994) from meager data of this nature. 3. COOLING RATES FROM THE CATION ORDERING DATA
¡ Using the above data on k D and K, we calculated the cooling rates of the natural crystals so that the quenched ordering states in the calculated evolution of ordering state as a function of temperature match those determined for the natural crystals. The cooling was assumed to follow an asymptotic (1/T ⫽ 1/T 0 ⫹ t) model. The T C for each sample along with the retrieved values of the constant , corresponding to the Arrhenian relation given by Eqn. 2, are summarized in Table 2. Use of the
1296
J. Ganguly and M. Stimpfl 4. METAL-SILICATE MIXING AND IMPLICATIONS OF THE ORTHOPYROXENE COOLING RATES
Fig. 3. Comparison of the cooling rates at ⬃400 ⫺ 450°C derived from the Fe-Mg ordering data in orthopyroxenes (open diamonds), and kinetic parameter given by Eqn. 2, with those derived from the metallographic data (filled circles) in Sa˜o Joa˜o Nepomuceno and Steinbach meteorites. The kinetic data is given by Eqn. 2. The vertical bars represent ⫾1 errors in the cooling rate values.
constrained Arrhenian relation, Eqn. 3, yields values that are larger by factors of ⬃6 and ⬃8 for the ST and SJN samples, respectively, than those given in Table 2. It is generally recognized that the cooling rate derived from a particular temperature sensitive property of a mineral is strictly applicable only near the closure temperature of that property. At the latter condition, however, the calculated cooling rates are not very sensitive to the choice of the cooling model even though these may differ significantly at other temperatures depending on the cooling model. The retrieved cooling rates at approximately the mean closure temperatures for the two groups of crystals, i.e., 400°C for SJN and 450°C for Steinbach, using the values summarized in Table 2 are illustrated in Fig. 3 along with the respective metallographic cooling rates. The cooling rates obtained from ¡ the constrained K (Eqn. 3) are larger by factors of ⬃6 and ⬃8 for ST and SJN crystals, respectively, since dT/dt ⫽ ⫺ T2. Although they suggested a cooling rate of ⬃20°C/Ma for ST, Rasmussen et al. (1995) were able to fit the entire spectrum of metallographic data only by considering a two-step cooling model, namely a high temperature step above ⬃440 ⫺ 460°C characterized by a cooling rate of ⬃150°C/Ma, followed by slower cooling of ⬃20°C/Ma. Both of these cooling rates have been illustrated in Fig. 3. Considering the errors in the determination of ordering states of orthopyroxene arising from those in the structural (Table 2) and compositional data (Stimpfl et al., 1999), we estimate uncertainties of around factors of 10 and 5 in the calculated cooling rates for SJN and ST orthopyroxene crystals, respectively. Rasmussen et al. (1995) tentatively estimated the uncertainty of the metallographic cooling rates to be ‘better than a factor of 3⬘. We have assigned an uncertainty of this magnitude to the MCR data.
The thermal history and metal-silicate mixing model of Haack et al. (1996) requires the metal and silicate components of IVA stony irons to experience similar thermal history below the temperature of development of the metal textures. The data illustrated in Fig. 3 are compatible with this model only if the cooling rate data for both metals and orthopyroxenes are pushed to their statistical error limits in two opposite directions. In that case, one has to accept similar cooling rate, ⬃400°C/ Ma, for both ST and SJN in the high temperature regime, i.e., above 400 – 450°C. Calculation of the evolution of Fe2⫹-Mg ordering state, using the unconstrained kinetic data (Eqn. 2), shows that no significant ordering would take place if the ST samples cooled by ⬃50°C/Ma at T ⬍ 425°C. Because the SJN metals do not record slow cooling at low temperature, one has to conclude that slow cooling in this stony iron ensued at a temperature, ⬍350°C, at which the metals were unable to record it because of extremely sluggish Fe–Ni interdiffusion rate which has an exponential dependence on temperature. The reconciliation of the orthopyroxene and metal cooling rates becomes extremely difficult within the framework of the breakup-reassembly model of Haack et al. (1996) if one accepts ¡ the constrained kinetic data for the rate constant, K, for the Fe–Mg ordering in orthopyroxene (Eqn. 3). Although this would improve the agreement between the metal and orthopyroxene cooling rates for the SJN samples by pushing the latter upward by a factor of 8 –9, the cooling rates for the ST orthopyroxene crystals would be faster by factors of 5 to 7 than those shown in Fig. 3, making them too fast compared to the metal cooling rates. Explanation of these data would then necessitate cooling of the silicates and metals in ST in different environments, shallow and deep-seated, respectively, and mixing after the metals had cooled through the closure temperatures for the development of the metal textures. On the other hand, the metals and silicates of SJN would have to cool in the same environment and thus had to be assembled prior to the development of metal textures. Consequently, these two stony irons would have to be derived from two different parent bodies, which broke up at different stages relative to the temperature of development of the metal textures, and reassembled gravitationally to produce metal-silicate mixtures. However, a strong argument against this model comes from the textural data of ST, presented by Scott et al. (1996, Fig. 3b), which shows silicates, including orthopyroxenes, embedded in what seems to be a large (⬃3.4 cm) single crystal of metal with exsolution textures oriented in the same direction throughout the section. This type of embedding would seem very unlikely if the metals and silicates were assembled after the metals cooled below 350°C. This is a stronger argument against the two parent body model than the notion (Rasmussen et al., 1995) that Scott et al. (1996) were able to relate the interelement properties of metals in these meteorites to a fractional crystallization model of a common metallic core. This is because they assumed a complex nonequilibrium partitioning behavior of S between solid and liquid metal as a function of S concentration, and an admittedly unrealistic assimilation model in order to match the observed interelement data to a fractional crystallization trend. Even then, the fractional crystallization-assimila-
Cooling rates of stony-iron meteorites
tion model was not completely successful (e.g., Co versus Ni trend). In summary, we favor a single parent body model for the high and low Ni groups, primarily on the strength of the textural data, as discussed above. The reconciliation of the metal and orthopyroxene cooling rate data within the framework of a single parent body model suggests that both ST and the SJN cooled fairly rapidly and at a similar rate of ⬃400°C/Ma at high temperature, after the gravitational reassembly of the disrupted parent body created a random mixture of the metal and silicate components, as envisioned by Haack et al., 1996. This was followed for ST by slow cooling at ⬃50°C/Ma at T ⬍ 425°C. The slow cooling did not ensue for SJN until the temperature dropped below ⬃350°C where the diffusion kinetics in the metals were too slow to record the change of cooling rate. The similar cooling rates for both ST and SJN above 425°C, as required to reconcile the metal and orthopyroxene cooling rate data, is at variance with the earlier notion (Rasmussen et al., 1995) of distinctly different cooling rates for the high and low Ni iron and stony irons. The question, however, remains as to why the most probable estimate of cooling rate of each orthopyroxene crystal from Steinbach is much faster than those from Sa˜o Joa˜o Nepomuceno, although the cation ordering states of both groups of orthopyroxenes were determined by the same method. Acknowledgments—Constructive reviews by Drs. Herbert Kroll, Ed Scott, and an anonymous reviewer are gratefully acknowledged. Ed Scott is especially thanked for his patience and willingness to discuss several issues through many E-mail correspondences which have substantially influenced the final interpretation of our data. We are grateful to Dr. Ed Olsen for the donation of the Steinbach sample from the collection of the Natural History Museum of Chicago and to the Smithsonian Institution for the donation of the Sa˜o Joa˜o Nepomuceno sample. REFERENCES Anovitz L. M., Essene E. J., and Dunham W. R. (1988) Order-disorder experiments on orthopyroxenes: Implications for the orthopyroxene geospeedometer. Am. Mineral. 73, 1060 –1073. Besancon J. R. (1981) Rate of cation ordering in orthopyroxenes. Am. Mineral. 66, 965–973. Dollase W. A. and Newman W. I. (1984) Statistically most probable stoichiometric formulae. Am. Mineralogist 69, 553–556. Finger L. W. and Prince E. (1975) A system of FORTRAN IV computer programs for crystal structural computations. NBS Tech Note, 854. Ganguly J. (1982) Mg–Fe order-disorder in ferromagnesian silicates: II. Thermodynamics, kinetics and geological applications. In: (ed.) S. K. Saxena, Advances in Physical Geochemistry (pp. 58 –99) 2 Springer-Verlag, New York. Ganguly J. and Domeneghetti M. C. (1996) Cation ordering of orthopyroxenes form the Skaergaard intrusion: Implications for the subsolidus cooling rates and permeabilities. Contrib. Mineral. Petrol. 122, 359 –367. Ganguly J. and Tazzoli V. (1994) Fe2⫹-Mg interdiffusion in orthopyroxene: Retrieval from the data on intracrystalline exchange reaction. Am. Mineral. 79, 930 –937. Ganguly J., Stimpfl M., and Molin G. (1997) Orthopyroxene chronom-
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