## N3Nifrac N3Nio x m3miF101

In practice, stable-isotope ratio measurements are compared with standard measurements to cancel out analytical bias. International standards are used, e.g. the standard mean ocean water (SMOW) value for oxygen and hydrogen. The fractionated isotope composition of a given element in a sample (SAM) is reported in terms of its deviation from a standard (STD) composition: it is expressed using S with a mass (or mass ratio) as a superscripted index, and the standard reference as a subscripted index:

The factor 1000 in this expression "amplifies" the generally small differences found in stable-isotope compositions. For an element with three isotopes it is useful to plot S values against each other, for instance S29/28Si versus S30/28Si as in Fig. 10.2. In mass-dependent fractionation, the S values are predicted to correlate. Substitution of Eqn (10.1) into Eqn (10.2) with (N2/NOfrac = (N2/Ni)sam and (N2/NO0 = (N2/N1)std etc. yields the relation

For example, substituting F = 0.01 and precise values of O-isotope masses into Eqn (10.3a) gives S17O = 0.516 x S16O (Section 10.5). In most cases F is indeed low and mi/m1 is close to unity; then Eqn (10.3a) may be approximated by

which gives S17O « 0.5 x S16O. Relations (10.3a, b) enable one to test the hypothesis of mass-dependent fractionation for isotope ratios in a three-isotope plot.

Equilibrium stable-isotope fractionation envisages the redistribution of species among coexisting phases (gaseous, liquid or solid) governed by chemical bonds in a steady-state equilibrium. This redistribution is mass dependent and Eqns (10.1—

10.3) apply to it. The steady-state and equilibrium conditions imply that, in contrast with kinetic fractionation, the effect is independent of pathways. Examples are plentiful and embrace different molecules in a gas or liquid mixture containing the same element and also coexisting minerals in metamorphic or magmatic rocks.

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o Barred e Porphyritic • Dark inclusions ® Bulk

Chondrules o Barred e Porphyritic • Dark inclusions ® Bulk

Fig. 10.2 Mass-dependent fractionation of Si isotopes in CAIs and chondrules. The isotopic 29Si /28Si and 30Si/28Si ratios are presented relative to the terrestrial standard NBS-28 as S(30Si) and S(29Si) (see Eqn 10.2 for the S notation). Silicon is a slightly volatile element. Evaporation (leaving behind heavy Si) and recondensation affected its isotope compositions. On the basis of the analysis of a large number of stony meteorites, the average solar system S(30Si) = -0.5%. All samples fall on the normal mass-dependent fractionation line, which passes through this value. After Clayton etal. (1985, 1988).

Fig. 10.2 Mass-dependent fractionation of Si isotopes in CAIs and chondrules. The isotopic 29Si /28Si and 30Si/28Si ratios are presented relative to the terrestrial standard NBS-28 as S(30Si) and S(29Si) (see Eqn 10.2 for the S notation). Silicon is a slightly volatile element. Evaporation (leaving behind heavy Si) and recondensation affected its isotope compositions. On the basis of the analysis of a large number of stony meteorites, the average solar system S(30Si) = -0.5%. All samples fall on the normal mass-dependent fractionation line, which passes through this value. After Clayton etal. (1985, 1988).

Phenomenologically, the equilibrium fractionation for isotopes 1 and 2 between two phases a and b is described by a fractionation factor a:

The factor a generally deviates further from unity with decreasing temperature, which leads to the important use of stable-isotope systematics as thermometers. The relation between a (Eqn 10.4) and S (Eqn 10.2) is useful in practice. From Eqn (10.4),

From the definition of S, and bearing in mind that lnx ~ 1 + x if x is small, it follows that

Equilibrium isotope fractionation has its basis in quantum mechanics, which only allows specific energy levels for particles in atoms or molecules. Here we merely give a broad outline of the principle; the subject is treated in detail in a number of excellent reviews, e.g. O'Neil (1986) and Hoefs (2005).

In the case of the translational (straight) or rotational motion of a molecule or atom, the quantum energy levels are so close together that they effectively form a continuous range, and classical mechanics applies. However, for the vibrational motion relating to an elastic bond within a molecule or a crystal, the spacing between the energy levels is significant. Further, the allowed energy levels depend on the masses of the atoms or ions forming the bond. Heavier isotopes of the same element have a higher probability of lower energy levels of vibration. In other words, heavier isotopes form stronger bonds in the same crystal or molecule than lighter ones. Differences in this energy behaviour between different molecule or crystal species lead to the fractionation of stable-isotope ratios between them. At temperatures approaching absolute zero this effect is strongest. At high temperatures approaching 1000 K the relative difference between the energy levels is so small that there is generally no isotope fractionation. The temperature dependence is not a simple function. At temperatures higher than ~ 300 K, ln a varies approximately proportionally to 1/ T2 for most substances, and at lower temperatures this changes gradually towards 1/T. The differences in S18O values between silicate minerals reach 5%c at 1000 K and 24%c at 300 K, and although these differences are small they can be used to determine the temperatures of the respective chemical reactions, including metamorphism (Section 25.4). In three-isotope plots such as S17O versus S18O or S29Si versus S30Si, the calculated slope of a correlation line for equilibrium fractionation is exactly the same as that for a kinetic process (Eqn 10.3a).

In some cases the relationships (10.3) appear to be invalid, and it can be shown that this problem does not result from mixing. Then mass-independent fractionation (MIF) could be involved. Mass-independent fractionation can result from different processes. Thus, quantum-mechanical symmetry effects in molecules can be a reason (Thiemens, 1996). For instance, in reactions involving ozone, molecular symmetry kinetically favours the yield of 17O16O16O and 18O16O16O relative to 16O16O16O; therefore such reactions as 2O3 ^3O2 leave behind 16O-depleted ozone O3 and 16O-enriched residual oxygen O2. Also the non-symmetric molecules, e.g. 16O-C-17O or 16O-Si-18O, were found to be more stable than the corresponding symmetric ones, 16O-C-16O or 16O-Si-16O. In the S17O versus S18O plot, this type of mechanism predicts a correlation with slope ~ 1, in contrast with the value ~ 0.5 that follows from Eqns (10.3a, b).

Another MIF mechanism envisages the isotope-dependent photodissociation of a chemical compound (e.g. an O-bearing or S-bearing compound, Sections 10.5 and 27.9 respectively; Farquhar et al., 2001) and may be observed if the products of the dissociation have been preserved. The "pure" effect of MIF can be expressed as a difference between the observed fractionation and that predicted for the mass-dependent effect. For example, the MIF of the sulphur isotope 33 S is defined as A33S « ¿33S - 0.515 x ¿34S (Fig. 27.21).

Silica-isotope fractionation in CAI: a process of partial evaporation and recondensation

Silicon is on the verge between moderately volatile and refractory behaviour at the surmised high temperatures of CAI formation. Silicon-isotope ratios define a good correlation (Fig. 10.2) with a slope of 0.5, conforming to the relation of Eqn (10.3a, b) for the masses of Si isotopes. Compared with the average solar system ratios, heavy-isotope enrichment seems to dominate, but light-isotope-enriched Si is also seen. The fractionation line passes through the mean solar system Si ratios, which proves that the CAIs showing these effects are in fact fractionated products of the solar nebula (compare with Fig. 3.4). The inclusions enriched in heavy isotopes are generally coarse-grained and are probably residues of partial evaporation. The lighter isotopes would be preferentially lost during sublimation of the solid or evaporation from a liquid, possibly combined with diffusion. Those depleted in heavy isotopes are fine-grained and may represent condensates from a gas phase containing previously evaporated light Si.

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