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Rayleigh fractionation of stable isotopes from a multicomponent source
Geochimica et Cosmochimica Acta, Vol. 64, No. 2, pp. 299 –306, 2000 Copyright © 2000 Elsevier Science Ltd Printed in the USA. All rights reserved 0016-7037/00 $20.00 ⫹ .00
Pergamon
PII S0016-7037(99)00181-7
Rayleigh fractionation of stable isotopes from a multicomponent source JYOTIRANJAN S. RAY and R. RAMESH* Physical Research Laboratory, Ahmedabad 380 009, India (Received December 15, 1998; accepted in revised form May 12, 1999)
Abstract—A formulation of the Rayleigh equation for the stable isotopic evolution of a multicomponent source reservoir is presented. Its applicability to the carbon and oxygen isotopic evolution of a fluid-rich carbonate magma and the crystallizing calcite carbonatite is demonstrated using data from the Amba Dongar carbonatite complex, Deccan province, India. The initial ␦13C of the parent magma for this complex has been estimated to be ⫺5.3 ⫾ 0.2‰ relative to V-PDB. Copyright © 2000 Elsevier Science Ltd bination of the fractionation factors between the other components and the former component, see Appendix); (2) the compound that forms is in instantaneous isotopic equilibrium with the source before being removed; (3) the process is isothermal; and (4) the abundance of the heavier isotope is much smaller than that of the lighter isotope of the element under consideration (e.g., [13C] ⬍⬍ [12C]). Assumption (3) may not be required for high temperature processes, where the temperature dependence of the isotopic fractionation factor is small. The equation that describes the stable isotopic evolution of the multicomponent reservoir, in ␦ notation, is:
1. INTRODUCTION
The Rayleigh equation, which depicts the stable isotopic evolution of a homogeneous reservoir from which a phase is continously extracted, is well known (Rayleigh, 1896; Broecker and Oversby, 1971):
␦ ⫺ ␦ o ⬇ 10 3共 ␣ ⫺ 1兲ln f
(1)
where ␦ and ␦o, respectively, are the stable isotopic compositions of an element in the reservoir to start with and when a fraction f of the original amount of the element is left, ␣ the isotopic fractionation factor between the phase separating out and the reservoir. There are instances where a reservoir has many discrete components, with different isotopic ratios, each contributing isotopes of an element to a compound/phase forming and separating out. For example, in the formation of hydrothermal graphite, CO2 ⫹ CH4 3 2C ⫹ 2H2O (Deines, 1980), CO2 and CH4 each contribute one atom of carbon to the graphite. Similarly, the oxygen isotopic composition of the rain water condensing from a cloud mass will be controlled by that of the relative amounts of liquid and vapor phases present in the cloud (Craig and Gordon, 1965). Other important natural examples are the calcite precipitation at the expense of CO2 and H2O from fluid/magma, metamorphic decarbonation, and serpentinization processes. An approximate treatment for a twocomponent source has already been reported by Pineau et al. (1973). Here we treat the general problem of Rayleigh isotopic fractionation from a multicomponent source and derive a precise equation for the stable isotopic evolution of the parent reservoir. We illustrate this model by the example of calcite precipitation from a magma, using the stable carbon and oxygen isotope data from Amba Dongar carbonatite complex, Deccan flood basalt province, India.
共 ␦ ⫺ ␦ o兲 ⬇ 10 3
冉 冊 冉
冊
␣ c⫺1 af ⫺ b ln ⫺ ln f a a⫺b
(2)
where ␦ and ␦o denote the isotopic composition of the source at any time t ⬎ 0, and at t ⫽ 0, respectively and f is the fraction of remaining atoms of the element under consideration left in the source. ␣c⫺1 is the temperature-dependent equilibrium fractionation factor of the element, between the product c and the first (major) source component 1. The constants a and b are weighted mean values of the equilibrium fractionation factors between the different source components and the major component. In the case of a, the weights are the numbers of atoms of the element under consideration contributed by various source components. In the case of b, besides this, the weight factor includes the critical values of f (i.e., the values of f corresponding to the steps at which different source components exhaust). The detailed derivation of the above equation and the definitions of a and b are given in the Appendix. The stable isotope ratios of the product c and the multicomponent source s can be related by a fractionation factor A: A⫽
2. RAYLEIGH ISOTOPIC FRACTIONATION FROM A MULTICOMPONENT SOURCE (RIFMS)
Rc ␣ c⫺1 R c/R 1 ⫽ ⫽ R s 共a ⫺ b/f 兲 共a ⫺ b/f 兲
(3)
where R c, R s, and R 1 denote stable isotope ratios of the product, the multicomponent source and the first source component, respectively. Therefore, the stable isotopic composition of the product as a function of f will be:
We assume that: (1) the different components of the source are always in stable isotopic equilibrium with each other (i.e., the stable isotopic composition of the source reservoir at any time can be expressed as a product of isotopic composition of the most abundant component of the source and a linear com-
(␦ ⫺ ␦o)product ⬇
冉
冊
⫹ 103
* Author to whom correspondence should be addressed (ramesh@ prl.ernet.in). 299
冉 冊 冋冉 冊 冉 冊册
␣c⫺1 ␣c⫺1 ␦ ⫺ (␦ ) a ⫺ b/f source a ⫺ b o source
␣c⫺1 ␣c⫺1 ⫺ a ⫺ b/f a⫺b
(4)
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Carbon and oxygen isotopic studies on carbonatites from many complexes world-wide have revealed that the ␦13C and ␦18O of carbonatite carbonates often show systematic correlated variations, in the ␦18O range of 5 to 15‰ (Deines, 1989). Such a commonly observed relationship is believed to be a result of some magmatic processes and not due to secondary alteration (Deines, 1989). Pineau et al. (1973) observed a correlation with a slope of ⬃0.4 in some carbonatites and suggested that this could be a result of precipitation of calcite (a two component Rayleigh) from a CO2-H2O magmatic fluid. However, to obtain a theoretical slope of 0.4, they had to assume that H2O and the associated silicate magma do not contribute more than 30% of the total oxygen required for the evolution of the carbonatites. This is clearly not valid, because H2O is known to be one of the major fluid components associated with the carbonate magmas (Gittins, 1989). Moreover, the fractionation factors used by them (Bottinga, 1968) are not applicable to higher temperatures. These have been modified recently (Chacko et al., 1991). In a subsequent study, Deines (1989) showed that these correlation trends have an average slope of 0.4, which he attributed to isotopic fractionation during simultaneous crystallization of carbonate and silicate phases or during carbonate-silicate melt unmixing. However, he used rather unrealistic values for some parameters (viz. molar ratio of carbon between carbonatite and silicate rock, carbonatesilicate oxygen, and carbon fractionation factors) and neglected the role of associated fluids (CO2 ⫹ H2O). Determining the slope of the ␦13C-␦18O plot does not constrain the isotopic composition of the initial magma. The majority of carbonatites worldwide that shows the above correlation is calcite carbonatite. Therefore, we will restrict our discussion to this variety. Because calcite is the major and sometimes the only carbonate phase in calcite carbonatites, the isotopic composition of calcite and carbonatite may be considered equivalent. The C (and O) isotopic fractionation behaviors of a carbonate melt and calcite are probably very similar (Deines, 1989). Hence, it is reasonable to assume the fractionation factors of C and O between calcite and the carbonate melt as unity, implying that the crystallization of calcite from a pure carbonate melt may not cause measurable isotopic variations. By contrast, if the calcite crystallizes in equilibrium with the associated magmatic fluids, then significant fractionation will occur. Carbonate magmas carry a large amount of CO2-H2O fluids during their evolution (Gittins, 1989). If calcite fractionally crystallizes from such a melt–fluid system, then its formation reaction will probably be: Ca2⫹ ⫹ CO2 ⫹ H2O 3 CaCO3 ⫹ 2H⫹, as proposed by Pineau et al. (1973). It is not obvious that a pure Rayleigh-type process controls the differentiation of carbonate magmas. Intermediate models involving larger degrees of global equilibration will introduce less important isotopic variations. In assuming a Rayleigh-type process, we follow the suggestion of Pineau et al. (1973). Isotopically, CO2 is the only source of carbon for calcite, whereas oxygen is contributed by both CO2 and H2O. So the carbon isotopic evolution will follow a single-component Rayleigh fractionation process, whereas that of oxygen, a two-component Rayleigh process. Carbon isotopic evolution of the source and carbonatite precipitate then will be given respectively by:
␦13Cs ⫽ (1000 ⫹ ␦13Csi)[f(c)](␣ ⫺1) ⫺ 1000
(5)
␦13Ccal ⫽ ␣c (1000 ⫹ ␦13Csi)[f(c)](␣ ⫺1) ⫺ 1000
(6)
c
3. AN APPLICATION
and c
where ␣c is the fractionation factor of carbon between calcite and CO2, f(c) is the fraction of remaining carbon in the source, and ␦13Csi is the initial carbon isotopic composition of the source. The oxygen isotope evolution of the carbonatite precipitate is derived from Eqn. 4 as follows: From the definition of A and a value of f ⬇ 1 (i.e., the first precipitate), o 1000 ⫹ ␦18Oocal ␣c⫺1 ⫽ A(f ⬇ 1) ⫽ o a⫺b 1000 ⫹ ␦18Os
from Eqn. 3. Rearranging, we get:
␦18Oocal ⫽
冉 冊
冉
冊
冊 冉
冊
o o ␣c⫺1 ␣c⫺1 ␦18Oso ⫹ 103 ⫺1 a⫺b a⫺b
then using this in Eqn. 4),
␦18Ocal ⫽ 103
冉
o o ␣c⫺1 ␣c⫺1 ⫺1 ⫹ ␦18Os a ⫺ b/f(o) a ⫺ b/f(o)
(7)
where ␣ is the fractionation factor of oxygen between calcite and the largest source component. f(o) is the fraction of remaining oxygen in the reservoir; a and b are to be determined from Eqns. A-9 and A-10, respectively, of the appendix and ␦18Os from Eqn. 2. To determine these parameters we need the values of pj’s (number of atoms contributed by various source components to the product), P (total number of atoms contributed), ␣j⫺1 (fractionation factor between jth source component and the first/largest component), rj⫺1 (ratio of the initial number of moles of the jth and the first components) and f j (jth critical value of f, when the jth reservoir gets exhausted), (see Appendix). In this calcite crystallization process, p1 ⫽ 2, p2 ⫽ 1, P ⫽ 3, ␣j⫺1 is either ␣H2O⫺CO2 or ␣CO2⫺␣H2O (oxygen isotope fractionation factor) depending on which one of the two source components is the largest. f j value, the jth critical value of f(o), will depend on the abundance of the smallest component. Here we discuss the isotopic evolution of the source and the carbonatite precipitate in two different cases: when (I) CO2 is the largest source component, (II) H2O is the largest source component. o c⫺1
3.1. Case I First, we consider a case when CO2 is the largest source component. Here rj⫺1 is rH2O⫺CO2 and ␣j⫺1 is ␣H2O⫺CO2. The value of f(o) at which H2O will be completely used up in the calcite formation is given by: f H 2O ⫽ 1 ⫺
3rH2O⫺CO2 (from Eqn. A-8 of Appendix) 2 ⫹ rH2O⫺CO2
(8)
The other parameters are: a ⫽ 31 (2 ⫹ ␣H2O⫺CO2) (from Eqn. A-9 of Appendix)
(9)
Rayleigh fractionation of stable isotopes
301
Fig. 1. Covariation of oxygen and carbon isotopic compositions of a two component source (CO2 ⫹ H2O), relative to the initial source compositions during fractional crystallization of calcite. r H2O⫺CO2 is the initial molar ratio of H2O to CO2. End of each curve marks the exhaustion of the H2O reservoir (Case I of the RIFMS model). See text for discussion.
冉
b ⫽ 32 (␣H2O⫺CO2 ⫺ 1)
1 ⫺ rH2O⫺CO2 2 ⫹ rH2O⫺CO2
冊
(from Eqns. A-10 and A-8 of Appendix)
(10)
To calculate ␦13C and ␦18O of calcite at the same time of formation, we relate f(c) and f(o) by: f(c) ⫽
冉
冊
冉
1 ⫺ rH2O⫺CO2 2 ⫹ rH2O⫺CO2 ⫹ f(o) 3 3
冊
(11)
The carbon and oxygen isotope composition of the source can then be calculated putting the values of the above parameters for appropriate r2⫺1 values in Eqns. 5 and 2, respectively. Figure 1 shows the covariation of (␦13C-␦13Co) and (␦18O␦18Oo) of the source for different rH2O⫺CO2 values. The end of each evolution curve marks the exhaustion of the smallest source component (i.e., H2O). Similarly the carbon (oxygen) isotope composition of the calcite precipitated can be found from Eqns. 6 and 7, respectively, as a function of the fraction of remaining carbon (oxygen) in the source. Figure 2a shows the isotopic evolution curves of calcite in a ␦13C vs. ␦18O space, for different rH2O⫺CO2 values. In this example, the temperature of calcite formation is taken to be 700°C and the initial source ␦13C and ␦18O values are ⫺5.5‰ and 8.5‰, respectively. The fractionation factors used are taken from Richet et al. (1977) and Chacko et al. (1991). Model calculations also showed that the slope of the isotopic evolution curves is not very sensitive to temperature (Fig. 2b); however, it is sensitive to rH2O⫺CO2 values (Fig. 2a). In fact, a particular slope of ␦13C vs. ␦18O of calcite does not correspond to a unique temperature.
Fig. 2. (a) Plot of ␦13C vs. ␦18O showing the isotopic evolution curves for calcite (crystallized at 700°C) generated by the RIFMS model for carbonatites in Case I (CO2 is the largest source component). ⽧ represents the initial composition of the carbonate magma. (b) Isotopic evolution curves for calcite at r H2O⫺CO2 (initial molar ratio of H2O to CO2) ⫽ 0.95 at three different temperatures. ⽧, initial magma. The arrows show the direction of isotopic evolution during fractional crystallization.
3.2. Case II In this case, H2O is the largest source component; here rj⫺1, is rCO2⫺H2O and ␣j⫺1 is ␣CO2⫺H2O. The value of f(o) at which CO2 is exhausted is: f CO2 ⫽ (1 ⫺ rCO2⫺H2O)/(1 ⫹ 2 rCO2⫺H2O) from Eqn A-8 of Appendix)
(12)
(from Eqn. A-9 of Appendix)
(13)
and other parameters are: a ⫽ 1 ⫹ 32 (␣CO2⫺H2O ⫺ 1)
冉
b ⫽ 32 (␣CO2⫺H2O ⫺ 1)
1 ⫺ rCO2⫺H2O 1 ⫹ 2 rCO2⫺H2O
冊
(from Eqns. A-10 and A-8 of Appendix) f(o) and f(c) are related by
(14)
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Fig. 3. Plot of ␦13C vs. ␦18O showing isotopic evolution curves (at 700°C) generated by the RIFMS model for calcite carbonatites in the Case II (H2O is the largest source component). Different curves are for different r CO2⫺H2O (initial molar CO2/H2O ratio). ⽧, initial magma. Here the calculations are done up to f (c) (fraction of remaining carbon in the source) of 0.001. The arrow shows the direction of isotopic evolution.
冉
f(c) ⫽ 31 1 ⫺
冊 冉
冊
1 1 ⫹ 31 2 ⫹ f(o) rCO2⫺H2O rCO2⫺H2O
(15)
Using these parameters, ␦13C and ␦18O values at different f(o) were calculated from Eqns. 6 and 7, respectively, for source compositions similar to those used in Case I. Figure 3 shows the isotopic evolution curves at 700°C for different rCO2⫺H2O values. The fractionation factors used are taken from Richet et al. (1977), Friedman and O’Neil (1977), and Chacko et al. (1991). Calculations were done up to f(c) ⫽ 0.001 and at these points f(o) ⬎ fcritical, (⫽fCO2). That is why the end points of all evolution curves have the same ␦13C value. In this example, the effect of changing rCO2⫺H2O values is much larger on the slope of the evolution curves than in Case I, whereas the variation in ␦18O generated is much smaller. The slopes of the correlated ␦13C and ␦18O variations (⬍1.0) observed in calcite carbonatites complexes can be explained by fractional crystallization only by Case I of the RIFMS model (slopes ⬍ 1; Fig. 2a). This is consistent with the suggestion that CO2 is probably the major fluid associated with primary carbonate magmas (Kjarsgaard and Hamilton, 1989). However, in the cases where water happens to be the dominant fluid component, there also the Case I of the model may apply if water rich fluids are expelled from the magma as fenitizing fluids producing a residual CO2-rich fluid before the onset of crystallization. The rH2O⫺CO2 values not only control the slopes of the evolutionary trends but also the maximum ␦13C and ␦18O values that can be reached by the calcite. Calcite carbonatites also contain other minerals like magnetite, apatite, phlogopite, fluorite, and other silicates and oxides.
Fig. 4. Plot of ␦13C vs. ␦18O showing RIFMS model curves for four carbonatites (A, B, C, and D), which are precipitating from four different CO2-rich carbonate magmas (r H2O⫺CO2 ⫽ 0.9) at 700°C (case I). ⽧ represent the initial isotopic compositions of the magmas. Also shown are the fields for upper mantle (Chazot et al., 1997; Cartigny et al., 1998) marked M and the primary igneous carbonatites, PIC (Keller and Hoefs, 1995).
Although abundances of these minerals are very small compared to calcite, their crystallization may affect the oxygen isotope composition of the source magma. However, the lack of fractionation factors between these minerals and melt makes it difficult to evaluate the exact isotopic effects. Using oxygen isotopic fractionation factors for magnetite-water (Bottinga and Javoy, 1973), biotite–water (assumed to be the same as muscovite–water; Bottinga and Javoy, 1973), and calcite–water (Friedman and O’Neil, 1977), we find that fractional crystallization (Rayleigh type) of magnetite and biotite (ⱕ10% by weight) from a carbonate melt at 800°C enhances the ␦18O (by ⬃1‰) of the melt. The crystallization of apatite probably depletes the ␦18O of the melt at magmatic temperatures [inferred from apatite-water fractionation factor of Shemesh et al. (1988). In this calculation, it was assumed that the carbonate melt and calcite share similar oxygen fractionation properties. Hence, in our model curves a range of ⫾1‰ in the initial ␦18O composition of the source can be considered to take into account the possible variations caused by the crystallization of minerals other than calcite. 3.3. Isotopic Evolution of Calcite Carbonatites In one of the earliest stable isotopic studies on carbonatites, Taylor et al. (1967) suggested that the carbon and oxygen isotopic compositions of carbonatites were homogeneous and defined a field for carbonatites in a ␦13C versus ␦18O space. This field was subsequently modified with more data coming in and termed as primary igneous carbonatite field (PIC) (Fig. 4). However, in most of the later studies it was observed that the
Rayleigh fractionation of stable isotopes
C and O isotopic compositions of unaltered magmatic carbonatites do not necessarily fall inside the so-called PIC field (Pineau et al., 1973; Deines and Gold, 1973; Nelson et al., 1988; Deines, 1989) and in many complexes they show some kind of correlated variation (Deines, 1989). It seems that there are two kinds of unaltered carbonatites: the first does not show significant variation in ␦13C and ␦18O, suggesting no significant isotopic fractionation during their crystallization [e.g., natrocarbonatites of Oldoinyo Lengai (Keller and Hoefs, 1995); calcite carbonatites of Shillong plateau (Ray et al., 1999)]. The second shows correlated variations of ␦13C and ␦18O. The latter, which largely consists of calcite carbonatites, probably acquires its isotopic composition during its fractional crystallization from a CO2-rich carbonate magma (case - I of the model). To elucidate this, we consider the fractional crystallization of four different carbonate magmas with different initial ␦13C and ␦18O values at 700°C. Figure 4 shows the model evolution curves at an initial molar H2O/CO2 ratio of 0.9. The four different initial compositions are extreme compositions for carbonate magmas that are chosen considering mantle like ␦13C value (⫺6.0 ⫾ 1.0‰; Cartigny et al., 1998, and references therein) and ␦18O value in the range of 6.0 to 9.0‰ (Rosenbaum et al., 1994). The model curves have average slopes of ⬃0.9 and generate ␦13C and ␦18O values that extend beyond the PIC field. The RIFMS model contains four unknowns : temperature, rH2O⫺CO2, initial ␦13C, and ␦18O of the source. In the present example of calcite carbonatite crystallization/precipitation, the temperature of crystallization/precipitation and the rH2O⫺CO2 (initial molar H2O/CO2 ratio) can be estimated from fluid inclusion studies. As the ␦18O of the mantle rocks does not vary much, the ␦18O of carbonate magma derived therefrom can be assumed in the suggested range of 6 to 9‰ (Rosenbaum et al., 1994) and should be affected by a ⫾1.0‰ variation to account for the fractionation caused by the crystallization of other minerals. With the help of the above parameters, one can generate series of RIFMS model curves for various initial ␦13C values to fit the observed data in a given complex and estimate the best initial ␦13C for the magma. As the ␦13C of a carbonate magma is believed to reflect that of its source region (Deines, 1989), the model can therefore provide estimates of the ␦13C of the mantle source regions of carbonatites. It may also help in the search of recycled carbon in carbonatite source regions. To exemplify application of the model to a natural case, we use the ⬃65.0 Ma old Amba Dongar carbonatite complex of Deccan flood basalt province, India (Ray, 1997). For this purpose, ␦13C and ␦18O data in calcite carbonatites (unaltered, determined by petrographic studies) were measured following the procedures described in Ray and Ramesh (1998). Some data from earlier work were also included (Sarkar et al., 1985; Gwalani et al., 1993; Simonetti et al., 1995; Srivastava and Taylor, 1996; Viladkar, 1996; Schleicher et al., 1998). The data, when plotted in a ␦13C-␦18O space, show a positive trend and some of them fall in the PIC field (Fig. 5). In the absence of fluid inclusion data in early-crystallized calcites and apatites in Amba Dongar, we must assume the temperature of crystallization but the slope of the model curves is not very sensitive to temperature (Fig. 2b). The average crystallization temperature is assumed to be 800°C [considering the earlier estimates of temperature of crystallization of these rocks in the range of
303
Fig. 5. Plot of ␦13C vs. ␦18O of calcite carbonatites from Amba Dongar carbonatite complex, Deccan province, India. The dashed line represents the RIFMS model curve fitted to the data for initial compositions of ␦13C ⫽ ⫺5.3‰ and ␦18O ⫽ 8.5‰ ⽧ CM ⫽ initial carbonate parent magma). The solid curves represent ⫾1.0‰ variation in the ␦18O of the magma. Model calculations are done for a temperature of 800°C and a r H2O⫺CO2 (initial molar ratio of H2O to CO2) value of 0.9. Also shown are the fields of mantle (M) and primary carbonatites (PIC) as in Fig. 4. Data source: F (21 points), this work; 䡺 (4 points) Simonetti et al. (1995); ‚ (2 points), Gwalani et al. (1993); 〫 (2 points) Sarkar et al. (1985); E (3 points), Srivastava and Taylor (1996); Œ (17 points), Viladkar (1996) and Schleicher et al. (1998). Linear regression analysis (Williamson, 1968) gave a slope of 0.49 with a correlation coefficient of 0.67, significant at 0.01 level (Student’s t-test, Bevington, 1969).
600 –900°C; Barker (1989) and references therein]. As the rH2O⫺CO2 value decides the critical value of f and thus the maximum values of ␦13C and ␦18O that can be reached by carbonatites (see Fig. 2a), it is possible to constrain the value of rH2O⫺CO2 from experimental data. We have shown earlier that for slopes ⬍ 1, a CO2 dominant fluid is required; therefore, rH2O⫺CO2 has to be less than one. Further considering the range of observed ␦18O values, rH2O⫺CO2 has to be close to 0.9. Initial ␦18O of the carbonate magma for Amba Dongar is assumed to be 8.5. This was estimated from the ␦18O of the mantle source region for Deccan tholeiites [inferred from the ␦18O ⫽ ⬃6.3‰ value for the least contaminated Deccan tholeiites; Matsuhisa et al. (1986)] and the ⬃2.2‰ fractionation between the carbonate magma and the mantle olivine at 1000°C (Rosenbaum et al., 1994) to be 8.5‰. A ⫾1.0‰ variation in the initial ␦18O is also considered. The model curves were generated for various initial ␦13C values so as to fit the observed data, and it was found that the curves for ␦13C ⫽ ⫺5.3 ⫾ 0.2‰ best explained the data (Fig. 5). Therefore, the forward model yields the following isotopic compositions of the parent carbonate magma for Amba Dongar: ␦13C ⬇ ⫺5.3‰ and ␦18O ⬇ 8.5‰. Thus the ␦13C of the mantle source region of Amba Dongar probably had ␦13C value of ⫺5.3 ⫾ 0.2‰, within the upper mantle range (Fig. 5).
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J. S. Ray and R. Ramesh 4. SUMMARY AND CONCLUSIONS
We have presented a general model describing the Rayleigh isotopic fractionation from a multicomponent source, and shown that it is consistent with the standard single component case and to the approximate equation reported earlier for a specific two component case. With accurately determined temperature-dependent fractionation factors, this model can be applied to various natural processes to understand the isotopic evolution of the sources and the products. We have illustrated the applicability of this model to the fractional crystallization of calcite carbonatites of the Amba Dongar complex in the Deccan province, and suggest that the isotopic compositions of these rocks are consistent with their fractional crystallization from a CO2-rich carbonate magma. The mantle source region of this complex probably had a ␦13C of ⫺5.3 ⫾ 0.2‰, which fits the ␦13C range for the upper mantle. Acknowledgments—K. Pande, P. P. Patel, N. Sharma, and J. R. Trivedi participated in the field work. We also thank S. Krishnaswami and two anonymous reviewers for critical comments. REFERENCES Barker D. S. (1989) Field relations of carbonatites. In Carbonatites: Genesis and Evolution (ed. K. Bell), pp. 38 – 69, Unwin Hyman. Bevington P. R. (1969) Data reduction and error analysis for the Physical Sciences, McGraw Hill. Bottinga Y. (1968) Calculation of fractionation factors for carbon and oxygen isotopic exchange. J. Phys. Chem. 72, 800 – 808. Bottinga Y. and Javoy M. (1973) Comments on oxygen isotope geothermometry. Earth Planet. Sci. Lett. 20, 250 –265. Broecker W. S. and Oversby V. (1971) Chemical Equilibria in the Earth. McGraw-Hill. Cartigny P., Harris J. W., Phillips D., Girad M., Javoy M. (1998) Subduction-related diamonds?—The evidence for a mantle-derived origin from coupled ␦13C–␦15N determinations. Chem. Geol. 147, 147–159. Chacko T., Mayeda T. K., Clayton R. N., and Goldsmith J. R. (1991) Oxygen and carbon isotope fractionation between CO2 and calcite. Geochim. Cosmochim. Acta 55, 2867–2882. Chazot G., Lowrey D., Menzies M., and Mattey D. (1997) Oxygen isotopic composition of hydrous and anhydrous peridotites. Geochim. Cosmochim. Acta 61, 161–169. Craig H. and Gordon L. I. (1965) Deuterium and oxygen-18 variations in the ocean and marine atmosphere. In Stable Isotopes in Oceanographic studies and Paleotemperatures (ed. E. Tongiori), pp. 9 –130, Consiglio Nazionale delle Ricerche, Laboratorio di Geologia Nucleare, Pisa. Deines P. and Gold D. P. (1973) The isotopic composition of carbonatites and kimberlite carbonates and their bearing on the isotopic composition of the deep-seated carbon. Geochim. Cosmochim. Acta 37, 1709 –1733. Deines P. (1980) The isotopic composition of reduced organic carbon. In Handbook of Environmental Isotope Geology (eds. P. Fritz and J. C. Fontes), pp. 329 – 406, Springer-Verlag. Deines P. (1989) Stable isotope variations in carbonatites. In Carbonatites: Genesis and Evolution (ed. K. Bell), pp. 301–359, Unwin Hyman. Friedman I. and O’Neil J. L. (1977) Compilation of stable isotope fractionation factors of geochemical interest. In Data of Geochemistry, 6th edition, 440-KK, U.S. Geol. Surv. Paper. Gittins J. (1989) The origin and evolution of carbonate magmas, In Carbonatites: Genesis and Evolution (ed. K. Bell), pp. 580 –599, Unwin Hyman. Gwalani L. G., Rock N. M. S., Chang W. J., Fernandez S., Alle`gre C. J., and Prinzhofer A. (1993) Alkaline rocks and carbonatites of Amba Dongar and adjacent areas, Deccan Igneous Province, Gujarat,
India: 1. Geology, Petrography and Petrochemistry. Mineral. Petrol. 47, 219 –253. Keller J. and Hoefs J. (1995) Stable isotope characteristics of recent natrocarbonatite from Oldoinyo Lengai. In Carbonatite Volcanism: Oldoinyo Lengai and the Petrogenesis of Natrocarbonatites (ed. K. Bell and J. Keller). IAVCE I, Proc. Volcanol. 4, pp. 113–123. Kjarsgaard B. and Hamilton D. L. (1989) The genesis of carbonatites by immiscibility. In Carbonatites: Genesis and Evolution (ed. K. Bell), pp. 388 – 404, Unwin Hyman. Matsuhisa Y., Bhattacharya S. K., Gopalan K., Mahoney J. J., and MacDougall J. D. (1986) Oxygen isotope evidence for crustal contamination in Deccan basalts. Terra Cognita 6, 181. Nelson D. R., Chivas A. R., Chappell B. W., and McCulloch M. T. (1988) Geochemical and isotopic systematics in carbonatites and implications for the evolution of oceanic-island sources. Geochim. Cosmochim. Acta 52, 1–17. Pineau F., Javoy M., and Alle´gre C. J. (1973) E´tude syste´matique des isotopes de l’oxyge´ne, du carbone et du strontium dans les carbonatites. Geochim. Cosmochim. Acta 37, 2363–2377. Ray J. S. (1997) Stable and radioisotopic constraints on the evolution of Mesozoic carbonatite–alkaline complexes of India, Ph. D. thesis, M. S. Univ. Baroda. Ray J. S. and Ramesh R. (1998) Stable carbon and oxygen isotope analysis of natural calcite and dolomite mixtures using selective extraction procedure. J. Geol. Soc. India 52, 323–332. Ray J. S., Ramesh R., and K. Pande (1999) Carbon isotopes in Kerguelen plume derived carbonatites: Evidence for recycled inorganic carbon. Earth Planet. Sci. Lett. 170, 205–214. Rayleigh J. W. S. (1896) Theoretical considerations respecting the separation of gases by diffusion and similar processes. Philos. Mag. 42, 493–593. Richet P., Bottinga Y., and Javoy M. (1977) A review of hydrogen, carbon, nitrogen, oxygen, sulfur, and chlorine stable isotope fractionation among gaseous molecules. Ann. Rev. Earth Planet. Sci. 5, 65–101. Rosenbaum J. M., Walker D., and Kyser T. K. (1994) Oxygen isotope fractionation in the mantle. Geochim. Cosmochim. Acta 58, 4767– 4777. Sarkar A., Bhattacharya S. K., Srivastava R. K., and Chandrasekaran V. (1985) Stable isotope studies in Indian carbonatites. In 3rd National symposium on Mass-spectrometry, India, E 10, 1–3. Schleicher H., Kramm U., Pernicka E., Schidlowski M., Schmidt F., Subramaniam V., Todt W., and Viladkar S. G. (1998) Enriched subcontinental upper mantle beneath southern India: Evidence from Pb, Nd, Sr and C-O isotopic studies on Tamil Nadu carbonatites. J. Petrol 39, 1765–1785. Shemesh A., Kolodny Y., and Luz B. (1988) Isotope geochemistry of oxygen and carbon in phosphate and carbonate of phosphorite francolite. Geochim. Cosmochim. Acta 52, 2565–2572. Simonetti A., Bell K., and Viladkar S. G. (1995) Isotopic data from the Amba Dongar carbonatite complex, west-central India: Evidence for an enriched mantle source. Chem. Geol. 122, 185–198. Srivastava R. K., and Taylor L. A. (1996) Carbon- and oxygen-isotope variations in Indian carbonatites. Int. Geol. Rev. 38, 419 – 429. Taylor H. P., Jr., Frechen J. Jr., and Degens F. T. (1967) Oxygen and carbon isotope studies of carbonatites from Laacher See District, West Germany and Alno district, Sweden. Geochim. Cosmochim. Acta 31, 407– 430. Viladkar S. G. (1996) Geology of the carbonatite-alkalic diatreme of Amba Dongar, Gujarat. A monograph published by Gujarat Mineral Development Corporation, Ahmedabad, India. Williamson J. H. (1968) Least square fitting of a straight line. Can J. Phys. 46, 1845–1847. APPENDIX 1. Derivation of the RIFMS Model Equation Notations n, no
Number of atoms of the element under consideration in the whole source reservoir at the time t ⬎ o and at t ⫽ 0, respectively.
Rayleigh fractionation of stable isotopes N i , N oi Number of molecules of the ith component of the source at t ⬎ 0 and at t ⫽ 0, respectively. r j⫺1 Ratio of the initial number of moles of the jth component to the 1st component (N oj/N o1). The latter is chosen to be the biggest component so that rj⫺1 ⱕ 1 for all j. L Total number of components in the reservoir at t ⫽ 0. pi Number of atoms contributed by the ith source component to the compound formed (e.g., in the case of calcite precipitation from CO2 and H2O mixture, CO2 contributes 2 atoms of oxygen and H2O, one). P Total number of atoms of the element under consideration in the compound formed; P ⫽ ¥L1 p i R Isotopic ratio of the element under consideration; subscripts s, c, i denote, respectively this element in the source reservoir, in the compound formed and in the ith source component; o denotes the value at t ⫽ 0. ␣i⫺j Fractionation factor between the ith and jth component (⫽R i/ R j). Note that ␣i⫺i ⫽ 1 for all the values of i. A Overall fractionation factor between the compound formed and the multicomponent source reservoir (⫽R c/R s). f Fraction of atoms of the element under consideration left in the source (⫽n/n o). fj jth critical value of f, when the jth reservoir gets exhausted. ␦ Isotopic composition expressed in permil [(R/R std ⫺ 1) 䡠 103].
In addition to these variables g, a, and b are defined below.
305
Defining
b ⫽
n ⫽ and
冘 冘 1
冘 冘 L
Rs ⫽
1
Rs ⫽
L
A⫽
(A-3) p jN j
hence, p jN j ⫽
(A-4)
no f
pj n 共1 ⫺ f 兲 P o
冉
pj PN oj n f ⫺ no 1 ⫺ P o no
冋 冉
pj n f ⫺ no 1 ⫺ P o
(A-5)
冊册
Pr j⫺1
冘
L 1
p ir i⫺1
冊册
(A-6)
using Eqn. A-11. Hence, 共 ␣ c⫺1 ⫺ a ⫹ b/f 兲 共a ⫺ b/f 兲
冉
冉
L 1
冊 冊 冉
g ⫹ b/f a ⫺ b/f ⫹b
(A-14)
df f
(A-15)
df af 2 ⫺ bf
冊
(A-16)
dR s ⫽ Rs
冉 冊冋 g a
d共af ⫺ b兲 共af ⫺ b兲
册 冋 ⫹
d共af ⫺ b兲 共af ⫺ b兲
册 冋册 ⫺
df f
(A-17)
p ir i⫺1
冉 冊 冉 冊 冋
1n
Rs ⫽ R os
g 共af ⫺ b兲 ⫹ 1 1n a 共a ⫺ b兲
册
⫺ 1n f
(A-18)
1n(R/R os) can be approximated to (␦-␦o)10⫺3 for small values of ␦-␦o. Therefore, (␦ ⫺ ␦ o) ⬇ 103
冉 冊 冉
␣c⫺1 af ⫺ b 1n a a ⫺ b
冊
⫺ 1n f
(A-19)
Eqn. A-19 describes the isotopic evolution of the multicomponent source reservoir. When the different source components in the source are not in stoichiometric proportions, the smallest of them will get exhausted first (this is why we chose index 1 for the largest source component). Each time a source component gets exhausted, the parameters ␦o, P, a, b, and f are to be reinitialized for further evolution and this will generate L-1 discontinuities in the ␦ vs. f plot. 2. Simpler Forms of the Equation
Pr j⫺1
冘
(A-13)
(A-7)
By defining f j, the jth critical value of f, when the jth source component exhausts (i.e., N j ⫽ 0), given by (from Eqn. A-7): fj ⫽ 1 ⫺
(A-12)
Integrating and applying the initial condition that at f ⫽ 1, R s ⫽ R os, we get
p jN j ␣j⫺1
冋
(A-11)
Expanding the second term into partial fractions we obtain,
p jN jR j
Therefore, p jN j ⫽
b f
R c/R 1 ␣ c⫺1 Rc ⫽ ⫽ R s 共a ⫺ b/f 兲 共a ⫺ b/f 兲
dR s df ⫽g Rs af ⫺ b
Also, the number of atoms removed from the jth component and the total number of atoms removed from the reservoir are related by p jN oj ⫺ p jN j ⫽
(A-10)
where
dR s ⫽ Rs
Using ␣j⫺1 ⫽ R j/R 1 and f ⫽ n/n o we get l
p j␣j⫺1 f j
df dR s ⫽ 共 A ⫺ 1兲 Rs f
(A-2)
L
冘
1
Treating the total reservoir as a single unit, the Rayleigh fractionation equation (Broecker and Oversby, 1971) in differential form can be written as:
L 1
L
冉 冊
Defining g ⫽ ␣c⫺1-a, we have
p jN j
冘
Rs ⫽ R1 a ⫺
(A-1)
1
Rl
1 P
p jN oj
L
(A-9)
1
and using Eqns. A-6 to A-10 in A-4 we obtain
A⫺1⫽ no ⫽
p j␣j⫺1
L
and
Derivations We define:
冘
1 P
a ⫽
(A-8)
We now show how Eqn. A-19 can be reduced to a single component Rayleigh equation (L ⫽ 1, P ⫽ p 1 and ␣1⫺1 ⫽ 1; r1⫺1 ⫽ 1 and r2⫺1 ⫽ 0; f1 ⫽ 0). Now, a ⫽ 1 and b ⫽ 0; therefore, ␦-␦o ⬇ 103 (␣c⫺1 ⫺ 1)1n( f ). In the case of initial molar quantities of different source components being equal (i.e., N o1 ⫽ N o2 ⫽ N o3 . . . N oL), r j-1 is unity for all values of j (⫽1 to L) and f j ⫽ 0 for all values of j (see Eqn. A-8). From Eqn. A-10 we see that b ⫽ 0 and Eqn. A-19 reduces to (␦-␦o) ⫽
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103 [(␣c⫺1/a) ⫺ 1]1n(f), similar to the equation for a single component case, but with an effective fractionation factor given by (␣c⫺1/a). For a two-component case (calcite precipitation: Ca2⫹ ⫹ CO2 ⫹ H2O 3 CaCO3 ⫹ 2H⫹), Pineau et al. (1973) derived a simplified equation:
再冋
␦ ⫺ ␦ o ⬇ 10 3
共 ␣ c⫺1 ⫺ 1兲 ⫺
⫻ 1n f ⫹
2 3
冋冉
共 ␣ 2⫺1 ⫺ 1兲 3
冊
册
冉 冊册冎
1 ⫺ r 2⫺1 1 共␣2⫺1⫺1兲 1⫺ 2 ⫹ r 2⫺1 f
dR s ⫽ Rs
冉
冊
冉冊
␣ c⫺1 df df ⫺1 ⫹b 2 a f f
(A-21)
and the solution yields as for Eqns. A-18 and A-19:
冋冉
␦ ⫺ ␦ o ⬇ 10 3
冊
冉 冊册
1 ␣ c⫺1 ⫺ 1 1n f ⫹ b 1 ⫺ a f
(A-22)
(A-20)
where the subscripts c, 1, and 2 denote CaCO3, CO2 and H2O, respectively. Under some approximations, the above described multicomponent Rayleigh Eqn. A-19 reduces to this equation. This can be readily seen by doing the integration of Eqn. A-17 with an approximated value of A in Eqn. A-13 A ⬇ ( ␣ c⫺1/a)(1 ⫹ b/f ) ⬃ ( ␣ c⫺1/a) ⫹ (b/f ), as both ␣c⫺1 and a are close to unity. Therefore, Eqn. A-12 becomes:
For the two component case, a ⫽ (1/3)(2 ⫹ ␣ 2⫺1 ) from Eqn. A-9, b ⫽ (1/3)(2f 1 ⫹ ␣ 2⫺1 f 2 ) from Eqn. A-10 and f 1 ⫽ 1 ⫺ [3/(2 ⫹ r 2⫺1 )]; f 2 ⫽ 1 ⫺ [3r 2⫺1 /(2 ⫹ r 2⫺1 )] from Eqn. A-8. Hence, b ⫽ (2/3)( ␣ 2⫺1 ⫺ 1)[(1 ⫺ r 2⫺1 )/(2 ⫹ r 2⫺1 )]. Again, [(␣c⫺1/a) ⫺ 1] can be approximated to be (␣c⫺1 ⫺ 1) as a ⬃ 1. Using this, and the relation for b in (A-22), we obtain the Pineau et al. (1973)’s approximate Eqn. A-20.
Pergamon
PII S0016-7037(99)00181-7
Rayleigh fractionation of stable isotopes from a multicomponent source JYOTIRANJAN S. RAY and R. RAMESH* Physical Research Laboratory, Ahmedabad 380 009, India (Received December 15, 1998; accepted in revised form May 12, 1999)
Abstract—A formulation of the Rayleigh equation for the stable isotopic evolution of a multicomponent source reservoir is presented. Its applicability to the carbon and oxygen isotopic evolution of a fluid-rich carbonate magma and the crystallizing calcite carbonatite is demonstrated using data from the Amba Dongar carbonatite complex, Deccan province, India. The initial ␦13C of the parent magma for this complex has been estimated to be ⫺5.3 ⫾ 0.2‰ relative to V-PDB. Copyright © 2000 Elsevier Science Ltd bination of the fractionation factors between the other components and the former component, see Appendix); (2) the compound that forms is in instantaneous isotopic equilibrium with the source before being removed; (3) the process is isothermal; and (4) the abundance of the heavier isotope is much smaller than that of the lighter isotope of the element under consideration (e.g., [13C] ⬍⬍ [12C]). Assumption (3) may not be required for high temperature processes, where the temperature dependence of the isotopic fractionation factor is small. The equation that describes the stable isotopic evolution of the multicomponent reservoir, in ␦ notation, is:
1. INTRODUCTION
The Rayleigh equation, which depicts the stable isotopic evolution of a homogeneous reservoir from which a phase is continously extracted, is well known (Rayleigh, 1896; Broecker and Oversby, 1971):
␦ ⫺ ␦ o ⬇ 10 3共 ␣ ⫺ 1兲ln f
(1)
where ␦ and ␦o, respectively, are the stable isotopic compositions of an element in the reservoir to start with and when a fraction f of the original amount of the element is left, ␣ the isotopic fractionation factor between the phase separating out and the reservoir. There are instances where a reservoir has many discrete components, with different isotopic ratios, each contributing isotopes of an element to a compound/phase forming and separating out. For example, in the formation of hydrothermal graphite, CO2 ⫹ CH4 3 2C ⫹ 2H2O (Deines, 1980), CO2 and CH4 each contribute one atom of carbon to the graphite. Similarly, the oxygen isotopic composition of the rain water condensing from a cloud mass will be controlled by that of the relative amounts of liquid and vapor phases present in the cloud (Craig and Gordon, 1965). Other important natural examples are the calcite precipitation at the expense of CO2 and H2O from fluid/magma, metamorphic decarbonation, and serpentinization processes. An approximate treatment for a twocomponent source has already been reported by Pineau et al. (1973). Here we treat the general problem of Rayleigh isotopic fractionation from a multicomponent source and derive a precise equation for the stable isotopic evolution of the parent reservoir. We illustrate this model by the example of calcite precipitation from a magma, using the stable carbon and oxygen isotope data from Amba Dongar carbonatite complex, Deccan flood basalt province, India.
共 ␦ ⫺ ␦ o兲 ⬇ 10 3
冉 冊 冉
冊
␣ c⫺1 af ⫺ b ln ⫺ ln f a a⫺b
(2)
where ␦ and ␦o denote the isotopic composition of the source at any time t ⬎ 0, and at t ⫽ 0, respectively and f is the fraction of remaining atoms of the element under consideration left in the source. ␣c⫺1 is the temperature-dependent equilibrium fractionation factor of the element, between the product c and the first (major) source component 1. The constants a and b are weighted mean values of the equilibrium fractionation factors between the different source components and the major component. In the case of a, the weights are the numbers of atoms of the element under consideration contributed by various source components. In the case of b, besides this, the weight factor includes the critical values of f (i.e., the values of f corresponding to the steps at which different source components exhaust). The detailed derivation of the above equation and the definitions of a and b are given in the Appendix. The stable isotope ratios of the product c and the multicomponent source s can be related by a fractionation factor A: A⫽
2. RAYLEIGH ISOTOPIC FRACTIONATION FROM A MULTICOMPONENT SOURCE (RIFMS)
Rc ␣ c⫺1 R c/R 1 ⫽ ⫽ R s 共a ⫺ b/f 兲 共a ⫺ b/f 兲
(3)
where R c, R s, and R 1 denote stable isotope ratios of the product, the multicomponent source and the first source component, respectively. Therefore, the stable isotopic composition of the product as a function of f will be:
We assume that: (1) the different components of the source are always in stable isotopic equilibrium with each other (i.e., the stable isotopic composition of the source reservoir at any time can be expressed as a product of isotopic composition of the most abundant component of the source and a linear com-
(␦ ⫺ ␦o)product ⬇
冉
冊
⫹ 103
* Author to whom correspondence should be addressed (ramesh@ prl.ernet.in). 299
冉 冊 冋冉 冊 冉 冊册
␣c⫺1 ␣c⫺1 ␦ ⫺ (␦ ) a ⫺ b/f source a ⫺ b o source
␣c⫺1 ␣c⫺1 ⫺ a ⫺ b/f a⫺b
(4)
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J. S. Ray and R. Ramesh
Carbon and oxygen isotopic studies on carbonatites from many complexes world-wide have revealed that the ␦13C and ␦18O of carbonatite carbonates often show systematic correlated variations, in the ␦18O range of 5 to 15‰ (Deines, 1989). Such a commonly observed relationship is believed to be a result of some magmatic processes and not due to secondary alteration (Deines, 1989). Pineau et al. (1973) observed a correlation with a slope of ⬃0.4 in some carbonatites and suggested that this could be a result of precipitation of calcite (a two component Rayleigh) from a CO2-H2O magmatic fluid. However, to obtain a theoretical slope of 0.4, they had to assume that H2O and the associated silicate magma do not contribute more than 30% of the total oxygen required for the evolution of the carbonatites. This is clearly not valid, because H2O is known to be one of the major fluid components associated with the carbonate magmas (Gittins, 1989). Moreover, the fractionation factors used by them (Bottinga, 1968) are not applicable to higher temperatures. These have been modified recently (Chacko et al., 1991). In a subsequent study, Deines (1989) showed that these correlation trends have an average slope of 0.4, which he attributed to isotopic fractionation during simultaneous crystallization of carbonate and silicate phases or during carbonate-silicate melt unmixing. However, he used rather unrealistic values for some parameters (viz. molar ratio of carbon between carbonatite and silicate rock, carbonatesilicate oxygen, and carbon fractionation factors) and neglected the role of associated fluids (CO2 ⫹ H2O). Determining the slope of the ␦13C-␦18O plot does not constrain the isotopic composition of the initial magma. The majority of carbonatites worldwide that shows the above correlation is calcite carbonatite. Therefore, we will restrict our discussion to this variety. Because calcite is the major and sometimes the only carbonate phase in calcite carbonatites, the isotopic composition of calcite and carbonatite may be considered equivalent. The C (and O) isotopic fractionation behaviors of a carbonate melt and calcite are probably very similar (Deines, 1989). Hence, it is reasonable to assume the fractionation factors of C and O between calcite and the carbonate melt as unity, implying that the crystallization of calcite from a pure carbonate melt may not cause measurable isotopic variations. By contrast, if the calcite crystallizes in equilibrium with the associated magmatic fluids, then significant fractionation will occur. Carbonate magmas carry a large amount of CO2-H2O fluids during their evolution (Gittins, 1989). If calcite fractionally crystallizes from such a melt–fluid system, then its formation reaction will probably be: Ca2⫹ ⫹ CO2 ⫹ H2O 3 CaCO3 ⫹ 2H⫹, as proposed by Pineau et al. (1973). It is not obvious that a pure Rayleigh-type process controls the differentiation of carbonate magmas. Intermediate models involving larger degrees of global equilibration will introduce less important isotopic variations. In assuming a Rayleigh-type process, we follow the suggestion of Pineau et al. (1973). Isotopically, CO2 is the only source of carbon for calcite, whereas oxygen is contributed by both CO2 and H2O. So the carbon isotopic evolution will follow a single-component Rayleigh fractionation process, whereas that of oxygen, a two-component Rayleigh process. Carbon isotopic evolution of the source and carbonatite precipitate then will be given respectively by:
␦13Cs ⫽ (1000 ⫹ ␦13Csi)[f(c)](␣ ⫺1) ⫺ 1000
(5)
␦13Ccal ⫽ ␣c (1000 ⫹ ␦13Csi)[f(c)](␣ ⫺1) ⫺ 1000
(6)
c
3. AN APPLICATION
and c
where ␣c is the fractionation factor of carbon between calcite and CO2, f(c) is the fraction of remaining carbon in the source, and ␦13Csi is the initial carbon isotopic composition of the source. The oxygen isotope evolution of the carbonatite precipitate is derived from Eqn. 4 as follows: From the definition of A and a value of f ⬇ 1 (i.e., the first precipitate), o 1000 ⫹ ␦18Oocal ␣c⫺1 ⫽ A(f ⬇ 1) ⫽ o a⫺b 1000 ⫹ ␦18Os
from Eqn. 3. Rearranging, we get:
␦18Oocal ⫽
冉 冊
冉
冊
冊 冉
冊
o o ␣c⫺1 ␣c⫺1 ␦18Oso ⫹ 103 ⫺1 a⫺b a⫺b
then using this in Eqn. 4),
␦18Ocal ⫽ 103
冉
o o ␣c⫺1 ␣c⫺1 ⫺1 ⫹ ␦18Os a ⫺ b/f(o) a ⫺ b/f(o)
(7)
where ␣ is the fractionation factor of oxygen between calcite and the largest source component. f(o) is the fraction of remaining oxygen in the reservoir; a and b are to be determined from Eqns. A-9 and A-10, respectively, of the appendix and ␦18Os from Eqn. 2. To determine these parameters we need the values of pj’s (number of atoms contributed by various source components to the product), P (total number of atoms contributed), ␣j⫺1 (fractionation factor between jth source component and the first/largest component), rj⫺1 (ratio of the initial number of moles of the jth and the first components) and f j (jth critical value of f, when the jth reservoir gets exhausted), (see Appendix). In this calcite crystallization process, p1 ⫽ 2, p2 ⫽ 1, P ⫽ 3, ␣j⫺1 is either ␣H2O⫺CO2 or ␣CO2⫺␣H2O (oxygen isotope fractionation factor) depending on which one of the two source components is the largest. f j value, the jth critical value of f(o), will depend on the abundance of the smallest component. Here we discuss the isotopic evolution of the source and the carbonatite precipitate in two different cases: when (I) CO2 is the largest source component, (II) H2O is the largest source component. o c⫺1
3.1. Case I First, we consider a case when CO2 is the largest source component. Here rj⫺1 is rH2O⫺CO2 and ␣j⫺1 is ␣H2O⫺CO2. The value of f(o) at which H2O will be completely used up in the calcite formation is given by: f H 2O ⫽ 1 ⫺
3rH2O⫺CO2 (from Eqn. A-8 of Appendix) 2 ⫹ rH2O⫺CO2
(8)
The other parameters are: a ⫽ 31 (2 ⫹ ␣H2O⫺CO2) (from Eqn. A-9 of Appendix)
(9)
Rayleigh fractionation of stable isotopes
301
Fig. 1. Covariation of oxygen and carbon isotopic compositions of a two component source (CO2 ⫹ H2O), relative to the initial source compositions during fractional crystallization of calcite. r H2O⫺CO2 is the initial molar ratio of H2O to CO2. End of each curve marks the exhaustion of the H2O reservoir (Case I of the RIFMS model). See text for discussion.
冉
b ⫽ 32 (␣H2O⫺CO2 ⫺ 1)
1 ⫺ rH2O⫺CO2 2 ⫹ rH2O⫺CO2
冊
(from Eqns. A-10 and A-8 of Appendix)
(10)
To calculate ␦13C and ␦18O of calcite at the same time of formation, we relate f(c) and f(o) by: f(c) ⫽
冉
冊
冉
1 ⫺ rH2O⫺CO2 2 ⫹ rH2O⫺CO2 ⫹ f(o) 3 3
冊
(11)
The carbon and oxygen isotope composition of the source can then be calculated putting the values of the above parameters for appropriate r2⫺1 values in Eqns. 5 and 2, respectively. Figure 1 shows the covariation of (␦13C-␦13Co) and (␦18O␦18Oo) of the source for different rH2O⫺CO2 values. The end of each evolution curve marks the exhaustion of the smallest source component (i.e., H2O). Similarly the carbon (oxygen) isotope composition of the calcite precipitated can be found from Eqns. 6 and 7, respectively, as a function of the fraction of remaining carbon (oxygen) in the source. Figure 2a shows the isotopic evolution curves of calcite in a ␦13C vs. ␦18O space, for different rH2O⫺CO2 values. In this example, the temperature of calcite formation is taken to be 700°C and the initial source ␦13C and ␦18O values are ⫺5.5‰ and 8.5‰, respectively. The fractionation factors used are taken from Richet et al. (1977) and Chacko et al. (1991). Model calculations also showed that the slope of the isotopic evolution curves is not very sensitive to temperature (Fig. 2b); however, it is sensitive to rH2O⫺CO2 values (Fig. 2a). In fact, a particular slope of ␦13C vs. ␦18O of calcite does not correspond to a unique temperature.
Fig. 2. (a) Plot of ␦13C vs. ␦18O showing the isotopic evolution curves for calcite (crystallized at 700°C) generated by the RIFMS model for carbonatites in Case I (CO2 is the largest source component). ⽧ represents the initial composition of the carbonate magma. (b) Isotopic evolution curves for calcite at r H2O⫺CO2 (initial molar ratio of H2O to CO2) ⫽ 0.95 at three different temperatures. ⽧, initial magma. The arrows show the direction of isotopic evolution during fractional crystallization.
3.2. Case II In this case, H2O is the largest source component; here rj⫺1, is rCO2⫺H2O and ␣j⫺1 is ␣CO2⫺H2O. The value of f(o) at which CO2 is exhausted is: f CO2 ⫽ (1 ⫺ rCO2⫺H2O)/(1 ⫹ 2 rCO2⫺H2O) from Eqn A-8 of Appendix)
(12)
(from Eqn. A-9 of Appendix)
(13)
and other parameters are: a ⫽ 1 ⫹ 32 (␣CO2⫺H2O ⫺ 1)
冉
b ⫽ 32 (␣CO2⫺H2O ⫺ 1)
1 ⫺ rCO2⫺H2O 1 ⫹ 2 rCO2⫺H2O
冊
(from Eqns. A-10 and A-8 of Appendix) f(o) and f(c) are related by
(14)
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J. S. Ray and R. Ramesh
Fig. 3. Plot of ␦13C vs. ␦18O showing isotopic evolution curves (at 700°C) generated by the RIFMS model for calcite carbonatites in the Case II (H2O is the largest source component). Different curves are for different r CO2⫺H2O (initial molar CO2/H2O ratio). ⽧, initial magma. Here the calculations are done up to f (c) (fraction of remaining carbon in the source) of 0.001. The arrow shows the direction of isotopic evolution.
冉
f(c) ⫽ 31 1 ⫺
冊 冉
冊
1 1 ⫹ 31 2 ⫹ f(o) rCO2⫺H2O rCO2⫺H2O
(15)
Using these parameters, ␦13C and ␦18O values at different f(o) were calculated from Eqns. 6 and 7, respectively, for source compositions similar to those used in Case I. Figure 3 shows the isotopic evolution curves at 700°C for different rCO2⫺H2O values. The fractionation factors used are taken from Richet et al. (1977), Friedman and O’Neil (1977), and Chacko et al. (1991). Calculations were done up to f(c) ⫽ 0.001 and at these points f(o) ⬎ fcritical, (⫽fCO2). That is why the end points of all evolution curves have the same ␦13C value. In this example, the effect of changing rCO2⫺H2O values is much larger on the slope of the evolution curves than in Case I, whereas the variation in ␦18O generated is much smaller. The slopes of the correlated ␦13C and ␦18O variations (⬍1.0) observed in calcite carbonatites complexes can be explained by fractional crystallization only by Case I of the RIFMS model (slopes ⬍ 1; Fig. 2a). This is consistent with the suggestion that CO2 is probably the major fluid associated with primary carbonate magmas (Kjarsgaard and Hamilton, 1989). However, in the cases where water happens to be the dominant fluid component, there also the Case I of the model may apply if water rich fluids are expelled from the magma as fenitizing fluids producing a residual CO2-rich fluid before the onset of crystallization. The rH2O⫺CO2 values not only control the slopes of the evolutionary trends but also the maximum ␦13C and ␦18O values that can be reached by the calcite. Calcite carbonatites also contain other minerals like magnetite, apatite, phlogopite, fluorite, and other silicates and oxides.
Fig. 4. Plot of ␦13C vs. ␦18O showing RIFMS model curves for four carbonatites (A, B, C, and D), which are precipitating from four different CO2-rich carbonate magmas (r H2O⫺CO2 ⫽ 0.9) at 700°C (case I). ⽧ represent the initial isotopic compositions of the magmas. Also shown are the fields for upper mantle (Chazot et al., 1997; Cartigny et al., 1998) marked M and the primary igneous carbonatites, PIC (Keller and Hoefs, 1995).
Although abundances of these minerals are very small compared to calcite, their crystallization may affect the oxygen isotope composition of the source magma. However, the lack of fractionation factors between these minerals and melt makes it difficult to evaluate the exact isotopic effects. Using oxygen isotopic fractionation factors for magnetite-water (Bottinga and Javoy, 1973), biotite–water (assumed to be the same as muscovite–water; Bottinga and Javoy, 1973), and calcite–water (Friedman and O’Neil, 1977), we find that fractional crystallization (Rayleigh type) of magnetite and biotite (ⱕ10% by weight) from a carbonate melt at 800°C enhances the ␦18O (by ⬃1‰) of the melt. The crystallization of apatite probably depletes the ␦18O of the melt at magmatic temperatures [inferred from apatite-water fractionation factor of Shemesh et al. (1988). In this calculation, it was assumed that the carbonate melt and calcite share similar oxygen fractionation properties. Hence, in our model curves a range of ⫾1‰ in the initial ␦18O composition of the source can be considered to take into account the possible variations caused by the crystallization of minerals other than calcite. 3.3. Isotopic Evolution of Calcite Carbonatites In one of the earliest stable isotopic studies on carbonatites, Taylor et al. (1967) suggested that the carbon and oxygen isotopic compositions of carbonatites were homogeneous and defined a field for carbonatites in a ␦13C versus ␦18O space. This field was subsequently modified with more data coming in and termed as primary igneous carbonatite field (PIC) (Fig. 4). However, in most of the later studies it was observed that the
Rayleigh fractionation of stable isotopes
C and O isotopic compositions of unaltered magmatic carbonatites do not necessarily fall inside the so-called PIC field (Pineau et al., 1973; Deines and Gold, 1973; Nelson et al., 1988; Deines, 1989) and in many complexes they show some kind of correlated variation (Deines, 1989). It seems that there are two kinds of unaltered carbonatites: the first does not show significant variation in ␦13C and ␦18O, suggesting no significant isotopic fractionation during their crystallization [e.g., natrocarbonatites of Oldoinyo Lengai (Keller and Hoefs, 1995); calcite carbonatites of Shillong plateau (Ray et al., 1999)]. The second shows correlated variations of ␦13C and ␦18O. The latter, which largely consists of calcite carbonatites, probably acquires its isotopic composition during its fractional crystallization from a CO2-rich carbonate magma (case - I of the model). To elucidate this, we consider the fractional crystallization of four different carbonate magmas with different initial ␦13C and ␦18O values at 700°C. Figure 4 shows the model evolution curves at an initial molar H2O/CO2 ratio of 0.9. The four different initial compositions are extreme compositions for carbonate magmas that are chosen considering mantle like ␦13C value (⫺6.0 ⫾ 1.0‰; Cartigny et al., 1998, and references therein) and ␦18O value in the range of 6.0 to 9.0‰ (Rosenbaum et al., 1994). The model curves have average slopes of ⬃0.9 and generate ␦13C and ␦18O values that extend beyond the PIC field. The RIFMS model contains four unknowns : temperature, rH2O⫺CO2, initial ␦13C, and ␦18O of the source. In the present example of calcite carbonatite crystallization/precipitation, the temperature of crystallization/precipitation and the rH2O⫺CO2 (initial molar H2O/CO2 ratio) can be estimated from fluid inclusion studies. As the ␦18O of the mantle rocks does not vary much, the ␦18O of carbonate magma derived therefrom can be assumed in the suggested range of 6 to 9‰ (Rosenbaum et al., 1994) and should be affected by a ⫾1.0‰ variation to account for the fractionation caused by the crystallization of other minerals. With the help of the above parameters, one can generate series of RIFMS model curves for various initial ␦13C values to fit the observed data in a given complex and estimate the best initial ␦13C for the magma. As the ␦13C of a carbonate magma is believed to reflect that of its source region (Deines, 1989), the model can therefore provide estimates of the ␦13C of the mantle source regions of carbonatites. It may also help in the search of recycled carbon in carbonatite source regions. To exemplify application of the model to a natural case, we use the ⬃65.0 Ma old Amba Dongar carbonatite complex of Deccan flood basalt province, India (Ray, 1997). For this purpose, ␦13C and ␦18O data in calcite carbonatites (unaltered, determined by petrographic studies) were measured following the procedures described in Ray and Ramesh (1998). Some data from earlier work were also included (Sarkar et al., 1985; Gwalani et al., 1993; Simonetti et al., 1995; Srivastava and Taylor, 1996; Viladkar, 1996; Schleicher et al., 1998). The data, when plotted in a ␦13C-␦18O space, show a positive trend and some of them fall in the PIC field (Fig. 5). In the absence of fluid inclusion data in early-crystallized calcites and apatites in Amba Dongar, we must assume the temperature of crystallization but the slope of the model curves is not very sensitive to temperature (Fig. 2b). The average crystallization temperature is assumed to be 800°C [considering the earlier estimates of temperature of crystallization of these rocks in the range of
303
Fig. 5. Plot of ␦13C vs. ␦18O of calcite carbonatites from Amba Dongar carbonatite complex, Deccan province, India. The dashed line represents the RIFMS model curve fitted to the data for initial compositions of ␦13C ⫽ ⫺5.3‰ and ␦18O ⫽ 8.5‰ ⽧ CM ⫽ initial carbonate parent magma). The solid curves represent ⫾1.0‰ variation in the ␦18O of the magma. Model calculations are done for a temperature of 800°C and a r H2O⫺CO2 (initial molar ratio of H2O to CO2) value of 0.9. Also shown are the fields of mantle (M) and primary carbonatites (PIC) as in Fig. 4. Data source: F (21 points), this work; 䡺 (4 points) Simonetti et al. (1995); ‚ (2 points), Gwalani et al. (1993); 〫 (2 points) Sarkar et al. (1985); E (3 points), Srivastava and Taylor (1996); Œ (17 points), Viladkar (1996) and Schleicher et al. (1998). Linear regression analysis (Williamson, 1968) gave a slope of 0.49 with a correlation coefficient of 0.67, significant at 0.01 level (Student’s t-test, Bevington, 1969).
600 –900°C; Barker (1989) and references therein]. As the rH2O⫺CO2 value decides the critical value of f and thus the maximum values of ␦13C and ␦18O that can be reached by carbonatites (see Fig. 2a), it is possible to constrain the value of rH2O⫺CO2 from experimental data. We have shown earlier that for slopes ⬍ 1, a CO2 dominant fluid is required; therefore, rH2O⫺CO2 has to be less than one. Further considering the range of observed ␦18O values, rH2O⫺CO2 has to be close to 0.9. Initial ␦18O of the carbonate magma for Amba Dongar is assumed to be 8.5. This was estimated from the ␦18O of the mantle source region for Deccan tholeiites [inferred from the ␦18O ⫽ ⬃6.3‰ value for the least contaminated Deccan tholeiites; Matsuhisa et al. (1986)] and the ⬃2.2‰ fractionation between the carbonate magma and the mantle olivine at 1000°C (Rosenbaum et al., 1994) to be 8.5‰. A ⫾1.0‰ variation in the initial ␦18O is also considered. The model curves were generated for various initial ␦13C values so as to fit the observed data, and it was found that the curves for ␦13C ⫽ ⫺5.3 ⫾ 0.2‰ best explained the data (Fig. 5). Therefore, the forward model yields the following isotopic compositions of the parent carbonate magma for Amba Dongar: ␦13C ⬇ ⫺5.3‰ and ␦18O ⬇ 8.5‰. Thus the ␦13C of the mantle source region of Amba Dongar probably had ␦13C value of ⫺5.3 ⫾ 0.2‰, within the upper mantle range (Fig. 5).
304
J. S. Ray and R. Ramesh 4. SUMMARY AND CONCLUSIONS
We have presented a general model describing the Rayleigh isotopic fractionation from a multicomponent source, and shown that it is consistent with the standard single component case and to the approximate equation reported earlier for a specific two component case. With accurately determined temperature-dependent fractionation factors, this model can be applied to various natural processes to understand the isotopic evolution of the sources and the products. We have illustrated the applicability of this model to the fractional crystallization of calcite carbonatites of the Amba Dongar complex in the Deccan province, and suggest that the isotopic compositions of these rocks are consistent with their fractional crystallization from a CO2-rich carbonate magma. The mantle source region of this complex probably had a ␦13C of ⫺5.3 ⫾ 0.2‰, which fits the ␦13C range for the upper mantle. Acknowledgments—K. Pande, P. P. Patel, N. Sharma, and J. R. Trivedi participated in the field work. We also thank S. Krishnaswami and two anonymous reviewers for critical comments. REFERENCES Barker D. S. (1989) Field relations of carbonatites. In Carbonatites: Genesis and Evolution (ed. K. Bell), pp. 38 – 69, Unwin Hyman. Bevington P. R. (1969) Data reduction and error analysis for the Physical Sciences, McGraw Hill. Bottinga Y. (1968) Calculation of fractionation factors for carbon and oxygen isotopic exchange. J. Phys. Chem. 72, 800 – 808. Bottinga Y. and Javoy M. (1973) Comments on oxygen isotope geothermometry. Earth Planet. Sci. Lett. 20, 250 –265. Broecker W. S. and Oversby V. (1971) Chemical Equilibria in the Earth. McGraw-Hill. Cartigny P., Harris J. W., Phillips D., Girad M., Javoy M. (1998) Subduction-related diamonds?—The evidence for a mantle-derived origin from coupled ␦13C–␦15N determinations. Chem. Geol. 147, 147–159. Chacko T., Mayeda T. K., Clayton R. N., and Goldsmith J. R. (1991) Oxygen and carbon isotope fractionation between CO2 and calcite. Geochim. Cosmochim. Acta 55, 2867–2882. Chazot G., Lowrey D., Menzies M., and Mattey D. (1997) Oxygen isotopic composition of hydrous and anhydrous peridotites. Geochim. Cosmochim. Acta 61, 161–169. Craig H. and Gordon L. I. (1965) Deuterium and oxygen-18 variations in the ocean and marine atmosphere. In Stable Isotopes in Oceanographic studies and Paleotemperatures (ed. E. Tongiori), pp. 9 –130, Consiglio Nazionale delle Ricerche, Laboratorio di Geologia Nucleare, Pisa. Deines P. and Gold D. P. (1973) The isotopic composition of carbonatites and kimberlite carbonates and their bearing on the isotopic composition of the deep-seated carbon. Geochim. Cosmochim. Acta 37, 1709 –1733. Deines P. (1980) The isotopic composition of reduced organic carbon. In Handbook of Environmental Isotope Geology (eds. P. Fritz and J. C. Fontes), pp. 329 – 406, Springer-Verlag. Deines P. (1989) Stable isotope variations in carbonatites. In Carbonatites: Genesis and Evolution (ed. K. Bell), pp. 301–359, Unwin Hyman. Friedman I. and O’Neil J. L. (1977) Compilation of stable isotope fractionation factors of geochemical interest. In Data of Geochemistry, 6th edition, 440-KK, U.S. Geol. Surv. Paper. Gittins J. (1989) The origin and evolution of carbonate magmas, In Carbonatites: Genesis and Evolution (ed. K. Bell), pp. 580 –599, Unwin Hyman. Gwalani L. G., Rock N. M. S., Chang W. J., Fernandez S., Alle`gre C. J., and Prinzhofer A. (1993) Alkaline rocks and carbonatites of Amba Dongar and adjacent areas, Deccan Igneous Province, Gujarat,
India: 1. Geology, Petrography and Petrochemistry. Mineral. Petrol. 47, 219 –253. Keller J. and Hoefs J. (1995) Stable isotope characteristics of recent natrocarbonatite from Oldoinyo Lengai. In Carbonatite Volcanism: Oldoinyo Lengai and the Petrogenesis of Natrocarbonatites (ed. K. Bell and J. Keller). IAVCE I, Proc. Volcanol. 4, pp. 113–123. Kjarsgaard B. and Hamilton D. L. (1989) The genesis of carbonatites by immiscibility. In Carbonatites: Genesis and Evolution (ed. K. Bell), pp. 388 – 404, Unwin Hyman. Matsuhisa Y., Bhattacharya S. K., Gopalan K., Mahoney J. J., and MacDougall J. D. (1986) Oxygen isotope evidence for crustal contamination in Deccan basalts. Terra Cognita 6, 181. Nelson D. R., Chivas A. R., Chappell B. W., and McCulloch M. T. (1988) Geochemical and isotopic systematics in carbonatites and implications for the evolution of oceanic-island sources. Geochim. Cosmochim. Acta 52, 1–17. Pineau F., Javoy M., and Alle´gre C. J. (1973) E´tude syste´matique des isotopes de l’oxyge´ne, du carbone et du strontium dans les carbonatites. Geochim. Cosmochim. Acta 37, 2363–2377. Ray J. S. (1997) Stable and radioisotopic constraints on the evolution of Mesozoic carbonatite–alkaline complexes of India, Ph. D. thesis, M. S. Univ. Baroda. Ray J. S. and Ramesh R. (1998) Stable carbon and oxygen isotope analysis of natural calcite and dolomite mixtures using selective extraction procedure. J. Geol. Soc. India 52, 323–332. Ray J. S., Ramesh R., and K. Pande (1999) Carbon isotopes in Kerguelen plume derived carbonatites: Evidence for recycled inorganic carbon. Earth Planet. Sci. Lett. 170, 205–214. Rayleigh J. W. S. (1896) Theoretical considerations respecting the separation of gases by diffusion and similar processes. Philos. Mag. 42, 493–593. Richet P., Bottinga Y., and Javoy M. (1977) A review of hydrogen, carbon, nitrogen, oxygen, sulfur, and chlorine stable isotope fractionation among gaseous molecules. Ann. Rev. Earth Planet. Sci. 5, 65–101. Rosenbaum J. M., Walker D., and Kyser T. K. (1994) Oxygen isotope fractionation in the mantle. Geochim. Cosmochim. Acta 58, 4767– 4777. Sarkar A., Bhattacharya S. K., Srivastava R. K., and Chandrasekaran V. (1985) Stable isotope studies in Indian carbonatites. In 3rd National symposium on Mass-spectrometry, India, E 10, 1–3. Schleicher H., Kramm U., Pernicka E., Schidlowski M., Schmidt F., Subramaniam V., Todt W., and Viladkar S. G. (1998) Enriched subcontinental upper mantle beneath southern India: Evidence from Pb, Nd, Sr and C-O isotopic studies on Tamil Nadu carbonatites. J. Petrol 39, 1765–1785. Shemesh A., Kolodny Y., and Luz B. (1988) Isotope geochemistry of oxygen and carbon in phosphate and carbonate of phosphorite francolite. Geochim. Cosmochim. Acta 52, 2565–2572. Simonetti A., Bell K., and Viladkar S. G. (1995) Isotopic data from the Amba Dongar carbonatite complex, west-central India: Evidence for an enriched mantle source. Chem. Geol. 122, 185–198. Srivastava R. K., and Taylor L. A. (1996) Carbon- and oxygen-isotope variations in Indian carbonatites. Int. Geol. Rev. 38, 419 – 429. Taylor H. P., Jr., Frechen J. Jr., and Degens F. T. (1967) Oxygen and carbon isotope studies of carbonatites from Laacher See District, West Germany and Alno district, Sweden. Geochim. Cosmochim. Acta 31, 407– 430. Viladkar S. G. (1996) Geology of the carbonatite-alkalic diatreme of Amba Dongar, Gujarat. A monograph published by Gujarat Mineral Development Corporation, Ahmedabad, India. Williamson J. H. (1968) Least square fitting of a straight line. Can J. Phys. 46, 1845–1847. APPENDIX 1. Derivation of the RIFMS Model Equation Notations n, no
Number of atoms of the element under consideration in the whole source reservoir at the time t ⬎ o and at t ⫽ 0, respectively.
Rayleigh fractionation of stable isotopes N i , N oi Number of molecules of the ith component of the source at t ⬎ 0 and at t ⫽ 0, respectively. r j⫺1 Ratio of the initial number of moles of the jth component to the 1st component (N oj/N o1). The latter is chosen to be the biggest component so that rj⫺1 ⱕ 1 for all j. L Total number of components in the reservoir at t ⫽ 0. pi Number of atoms contributed by the ith source component to the compound formed (e.g., in the case of calcite precipitation from CO2 and H2O mixture, CO2 contributes 2 atoms of oxygen and H2O, one). P Total number of atoms of the element under consideration in the compound formed; P ⫽ ¥L1 p i R Isotopic ratio of the element under consideration; subscripts s, c, i denote, respectively this element in the source reservoir, in the compound formed and in the ith source component; o denotes the value at t ⫽ 0. ␣i⫺j Fractionation factor between the ith and jth component (⫽R i/ R j). Note that ␣i⫺i ⫽ 1 for all the values of i. A Overall fractionation factor between the compound formed and the multicomponent source reservoir (⫽R c/R s). f Fraction of atoms of the element under consideration left in the source (⫽n/n o). fj jth critical value of f, when the jth reservoir gets exhausted. ␦ Isotopic composition expressed in permil [(R/R std ⫺ 1) 䡠 103].
In addition to these variables g, a, and b are defined below.
305
Defining
b ⫽
n ⫽ and
冘 冘 1
冘 冘 L
Rs ⫽
1
Rs ⫽
L
A⫽
(A-3) p jN j
hence, p jN j ⫽
(A-4)
no f
pj n 共1 ⫺ f 兲 P o
冉
pj PN oj n f ⫺ no 1 ⫺ P o no
冋 冉
pj n f ⫺ no 1 ⫺ P o
(A-5)
冊册
Pr j⫺1
冘
L 1
p ir i⫺1
冊册
(A-6)
using Eqn. A-11. Hence, 共 ␣ c⫺1 ⫺ a ⫹ b/f 兲 共a ⫺ b/f 兲
冉
冉
L 1
冊 冊 冉
g ⫹ b/f a ⫺ b/f ⫹b
(A-14)
df f
(A-15)
df af 2 ⫺ bf
冊
(A-16)
dR s ⫽ Rs
冉 冊冋 g a
d共af ⫺ b兲 共af ⫺ b兲
册 冋 ⫹
d共af ⫺ b兲 共af ⫺ b兲
册 冋册 ⫺
df f
(A-17)
p ir i⫺1
冉 冊 冉 冊 冋
1n
Rs ⫽ R os
g 共af ⫺ b兲 ⫹ 1 1n a 共a ⫺ b兲
册
⫺ 1n f
(A-18)
1n(R/R os) can be approximated to (␦-␦o)10⫺3 for small values of ␦-␦o. Therefore, (␦ ⫺ ␦ o) ⬇ 103
冉 冊 冉
␣c⫺1 af ⫺ b 1n a a ⫺ b
冊
⫺ 1n f
(A-19)
Eqn. A-19 describes the isotopic evolution of the multicomponent source reservoir. When the different source components in the source are not in stoichiometric proportions, the smallest of them will get exhausted first (this is why we chose index 1 for the largest source component). Each time a source component gets exhausted, the parameters ␦o, P, a, b, and f are to be reinitialized for further evolution and this will generate L-1 discontinuities in the ␦ vs. f plot. 2. Simpler Forms of the Equation
Pr j⫺1
冘
(A-13)
(A-7)
By defining f j, the jth critical value of f, when the jth source component exhausts (i.e., N j ⫽ 0), given by (from Eqn. A-7): fj ⫽ 1 ⫺
(A-12)
Integrating and applying the initial condition that at f ⫽ 1, R s ⫽ R os, we get
p jN j ␣j⫺1
冋
(A-11)
Expanding the second term into partial fractions we obtain,
p jN jR j
Therefore, p jN j ⫽
b f
R c/R 1 ␣ c⫺1 Rc ⫽ ⫽ R s 共a ⫺ b/f 兲 共a ⫺ b/f 兲
dR s df ⫽g Rs af ⫺ b
Also, the number of atoms removed from the jth component and the total number of atoms removed from the reservoir are related by p jN oj ⫺ p jN j ⫽
(A-10)
where
dR s ⫽ Rs
Using ␣j⫺1 ⫽ R j/R 1 and f ⫽ n/n o we get l
p j␣j⫺1 f j
df dR s ⫽ 共 A ⫺ 1兲 Rs f
(A-2)
L
冘
1
Treating the total reservoir as a single unit, the Rayleigh fractionation equation (Broecker and Oversby, 1971) in differential form can be written as:
L 1
L
冉 冊
Defining g ⫽ ␣c⫺1-a, we have
p jN j
冘
Rs ⫽ R1 a ⫺
(A-1)
1
Rl
1 P
p jN oj
L
(A-9)
1
and using Eqns. A-6 to A-10 in A-4 we obtain
A⫺1⫽ no ⫽
p j␣j⫺1
L
and
Derivations We define:
冘
1 P
a ⫽
(A-8)
We now show how Eqn. A-19 can be reduced to a single component Rayleigh equation (L ⫽ 1, P ⫽ p 1 and ␣1⫺1 ⫽ 1; r1⫺1 ⫽ 1 and r2⫺1 ⫽ 0; f1 ⫽ 0). Now, a ⫽ 1 and b ⫽ 0; therefore, ␦-␦o ⬇ 103 (␣c⫺1 ⫺ 1)1n( f ). In the case of initial molar quantities of different source components being equal (i.e., N o1 ⫽ N o2 ⫽ N o3 . . . N oL), r j-1 is unity for all values of j (⫽1 to L) and f j ⫽ 0 for all values of j (see Eqn. A-8). From Eqn. A-10 we see that b ⫽ 0 and Eqn. A-19 reduces to (␦-␦o) ⫽
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103 [(␣c⫺1/a) ⫺ 1]1n(f), similar to the equation for a single component case, but with an effective fractionation factor given by (␣c⫺1/a). For a two-component case (calcite precipitation: Ca2⫹ ⫹ CO2 ⫹ H2O 3 CaCO3 ⫹ 2H⫹), Pineau et al. (1973) derived a simplified equation:
再冋
␦ ⫺ ␦ o ⬇ 10 3
共 ␣ c⫺1 ⫺ 1兲 ⫺
⫻ 1n f ⫹
2 3
冋冉
共 ␣ 2⫺1 ⫺ 1兲 3
冊
册
冉 冊册冎
1 ⫺ r 2⫺1 1 共␣2⫺1⫺1兲 1⫺ 2 ⫹ r 2⫺1 f
dR s ⫽ Rs
冉
冊
冉冊
␣ c⫺1 df df ⫺1 ⫹b 2 a f f
(A-21)
and the solution yields as for Eqns. A-18 and A-19:
冋冉
␦ ⫺ ␦ o ⬇ 10 3
冊
冉 冊册
1 ␣ c⫺1 ⫺ 1 1n f ⫹ b 1 ⫺ a f
(A-22)
(A-20)
where the subscripts c, 1, and 2 denote CaCO3, CO2 and H2O, respectively. Under some approximations, the above described multicomponent Rayleigh Eqn. A-19 reduces to this equation. This can be readily seen by doing the integration of Eqn. A-17 with an approximated value of A in Eqn. A-13 A ⬇ ( ␣ c⫺1/a)(1 ⫹ b/f ) ⬃ ( ␣ c⫺1/a) ⫹ (b/f ), as both ␣c⫺1 and a are close to unity. Therefore, Eqn. A-12 becomes:
For the two component case, a ⫽ (1/3)(2 ⫹ ␣ 2⫺1 ) from Eqn. A-9, b ⫽ (1/3)(2f 1 ⫹ ␣ 2⫺1 f 2 ) from Eqn. A-10 and f 1 ⫽ 1 ⫺ [3/(2 ⫹ r 2⫺1 )]; f 2 ⫽ 1 ⫺ [3r 2⫺1 /(2 ⫹ r 2⫺1 )] from Eqn. A-8. Hence, b ⫽ (2/3)( ␣ 2⫺1 ⫺ 1)[(1 ⫺ r 2⫺1 )/(2 ⫹ r 2⫺1 )]. Again, [(␣c⫺1/a) ⫺ 1] can be approximated to be (␣c⫺1 ⫺ 1) as a ⬃ 1. Using this, and the relation for b in (A-22), we obtain the Pineau et al. (1973)’s approximate Eqn. A-20.