Strain model

The observation of the departure from cubic symmetry above Tm co-incident with the appearance of the central peak scattering serves to resolve the conflict between dynamic and lattice strain models. The departure from cubic symmetry may be attributed to a shift in the atomic equilibrium position associated with the soft-mode anharmonicity. In such a picture, the central peak then becomes the precusor to a Bragg reflection for the new structure. 

Figure 3. Cartoon illustrating the lattice strain model of trace element partitioning. For an isovalent series of ions with charge n+ and radius entering crystal lattice site M, the partition coefficient,, can be...

Figure 4. Fits of lattice strain model to experimental mineral-melt partition coefficients for (a) plagioclase (run 90-6 of Blundy and Wood 1994) and (b) elinopyroxene (ran DC23 of Blundy and Dalton 2000). Different valence cations, entering the large cation site of each mineral, are denoted by different symbols. The curves are non-linear least squares fits of Equation (1) to the data for each valence. Errors bars, when larger than symbol, are 1 s.d. Ionic radii in Vlll-fold coordination are taken from Shannon (1976).

Wood and Blundy (2001) developed an electrostatic model to describe this process. In essence this is a continuum approach, analogous to the lattice strain model, wherein the crystal lattice is viewed as an isotropic dielectric medium. For a series of ions with the optimum ionic radius at site M, (A(m))> partitioning is then controlled by the charge on the substituent (Z ) relative to the optimum charge at the site of interest, (Fig. 10) ... 

The approach taken here is to use the lattice strain model to derive the partition coefficient of a U-series element (such as Ra) from the partition coefficient of its proxy (such as Ba) under the appropriate conditions. Clearly the proxy needs to be an element that forms ions of the same charge and similar ionic radius to the U-series element of interest, so that the pair are not significantly fractionated from each other by changes in phase composition, pressure or temperature. Also the partitioning behavior of the proxy must be reasonably well constrained under the conditions of interest. Having established a suitable partition coefficient for the proxy, the partition coefficient for the U-series element can then be obtained via rearrangement of Equation (2) (Blundy and Wood 1994) ... 

Wood and Blundy (1997) adapted the lattice strain model to describe lanthanide partitioning between clinopyroxene and melt as a function of crystal composition, pressure and temperature. In developing the model, they arrived at relationships between and respectively, crystal composition, and pressure and temperature ... 

Figure 12. Application of the lattice strain model to Nb-Ta fractionation by clinopyroxene. The calculated is based on the...

There is only one determination of Z)pb in orthopyroxene, that of Salters et al. (2002) at the mantle solidus at 2.8 GPa. This value (0.009 0.006) is within error of that calculated from the Dsr value of McDade et al. (2003a) under similar conditions, using the lattice strain model, i.e., 0.0024 0.0012. However, the uncertainties on both measurements should not be taken as strong support for the potential of Sr as a proxy for Pb. Still, there is no evidence for the anomalously low Z)pb values observed in clinopyroxene. 

Figure 13. Electrostatic model fitted to partition coefficients for cations entering the M2-site in orthopyroxene, based on the experiments of McDade et al. (2003a,b). The curves are fits to Equation (7) and can be used to estimate and Do(m2) > from which D-p ui) can be calculated via the lattice strain model. The fit parameters are given in the legend.

Van Westrenen et al. (2001a) present a model of lanthanide and Sc partitioning between the garnet X-site and melt. The model is a variant of the lattice strain model of clinopyroxene-melt partitioning of Wood and Blundy (1997), and is based on 160 experimental garnet-melt pairs in the pressure-temperature range 2.5-7.5 GPa and 1450-1930°C. The model includes composition-sensitive expressions for and accounts for the non-linear variation in with composition, as follows ... 

Figure 14. U-Th fractionation by garnet as a function of X-site dimension ) for 33 experimental garnet-melt pairs from the sonrees listed in the legend. r Jx) is assnmed eqnal to r (x) as given by Eqnation (22a). Note the mnch larger U-Th fractionation prodneed by garnet relative to clinopyroxene (Fig. 1). The emved line shows the prediction of the lattice strain model (at fixed temperature of 1500°C) using the Vlll-fold ionic radii in Table 2 and, as given by Equation (22b). Errors...

Figure 17. Electrostatic model fitted to partition coefficients for cations entering the M4-site in amphibole, based on the experiments of Brenan et al. (1995) and La Tourrette et al. (1995). A single M4-site is assnmed, rather than M4 and M4, as proposed by Botlazzi et al. (1999). The curves are fits to Eqnation (7) and can be nsed to estimate Do(M4), from which Z)pa(M4) can be calculated via the lattice strain model. Becanse of the mnltiplicity of sites in amphibole, it is unlikely that extrapolation of the curves to zero charge gives a reliable estimate for Dr . The fit parameters are Zo(M2> = 1-87 and ps = 38.1 A (La Tonrrette et al. 1995), and 2.31, 36.1 A (Brenan et al. 1995).

This expression is relatively imprecise because of the scarcity of data. Also, the oxidation state of Pb in these experiments is not known. However, it is interesting that for those experiments in which both Dpb and Dsr have been determined, the Dpb/Dsr ratio is consistently less than would be predicted from the 2+ lattice strain model using parameters presented above. As in the case of clinopyroxene, increasing the effective Vlll-fold ionic radius of Pb in plagioclase, to 1.38 A, does retrieve the observed ratios. Thus one can... 

Spinels. There are limited experimental data on uranium and thorium partitioning between magnetite and melt (Nielsen et al. 1994 Blundy and Brooker 2003). Both studies find U and Th to be moderately incompatible. Blundy and Brooker s results for a hydrous dacitic melt at 1 GPa and 1025°C give Du and D h. of approximately 0.004. The accuracy of these values is compromised by the very low concentrations in the crystals and the lack of suitable SIMS secondary standards for these elements in oxide minerals. Nonetheless, these values are within the range of Djh of magnetites at atmospheric pressure 0.003-0.025 (Nielsen et al. 1994). It is difficult to place these values within the context of the lattice strain model, firstly because there are so few systematic experimental studies of trace element partitioning into oxides and secondly because of the compositional diversity of the spinels and their complex intersite cation ordering. 

Figure 24. Lattice strain model applied to zircon-melt partition coefficients from Hinton et al. (written comm.) for a zircon phenocryst in peralkaline rhyolite SMN59 from Kenya. Ionic radii are for Vlll-fold coordination (Shannon 1976). The curves are fits to Equation (1) at an estimated eraption temperature of 700°C (Scaillet and Macdonald 2001). Note the excellent fit of the trivalent lanAanides, with the exception of Ce, whose elevated partition coefficient is due to the presence of both Ce and Ce" in the melt, with the latter having a much higher partition coefficient into zircon. The 4+ parabola cradely fits the data from Dj, and Dy, through Dzi to Dih, but does not reproduce the observed DuIDjh ratio. We speculate that this is due to melt compositional effects on Dzt and (Linnen and Keppler 2002), and possibly other 4+ cations, in very silicic melts. Because of its Vlll-fold ionic radius of 0.91 A (vertical line), Dpa is likely to be at least as high as Dwh, and probably considerably higher.

Figure 2.3 Schematic representation of the induced strain model of transition state stabilization. Source Redrawn from Copeland (2000).

FIGURE 12.4 S = 7/2 EPR of the [8Fe-7S] P-cluster in Azotobacter vinelandii nitrogenase. The experimental spectrum (trace A) has been simulated in the absence (trace B) and the presence (trace C) of D-strain modeled as a correlated distribution in the zero-field parameters D and E. 

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