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  • ALBARÈDE F. 1995. Introduction to the Geochemical Modeling.– Cambridge University Press, pp. 1–543.

    BRYAN W.B., FINGER L.W. & CHAYES F. 1969. Estimating proportions in petrographic mixing equations by least-squares approximation.– Science 163: 926–927.

    CASTRO A., DE LA ROSA J.D. & STEPHENS W.E. 1990. Magma mixing in the subvolcanic environment: petrology of the Gerena interaction zone near Seville, Spain.– Contrib. Mineral. Petrol. 105: 9–26.

    COX K.G., BELL J.D. & PANKHURST R.J. 1979. The Interpretation of Igneous Rocks.– George Allen & Unwin, pp 1–450.

    EVANS O.C. & HANSON, G.N., 1993. Accessory-mineral fractionation of rare-earth element (REE) abundances in granitoid rocks.– Chem. Geol. 110: 69–93.

    FAURE G. 1986. Principles of Isotope Geology.– J. Wiley & Sons, Chichester, pp. 1–589.

    FOURCADE S. & ALLÈGRE C.J. 1981. Trace elements behavior in granite genesis: a case study. The calc-alkaline plutonic association from the Quérigut Complex (Pyrénées, France).– Contrib. Mineral. Petrol. 76: 177–195.

    GROMET L.P. & SILVER L.T. 1983. Rare earth element distribution among minerals in a granodiorite and their petrogenetic implications.– Geochim. Cosmochim. Acta 47: 925–939.

    HANSON G.N. 1978. The application of trace elements to the petrogenesis of igneous rocks of granitic composition.– Earth Planet. Sci. Lett. 38: 26–43.

    HANSON G.N. 1980. Rare earth elements in petrogenetic studies of igneous systems.– Ann. Rev. Earth Planet. Sci. 8: 371–406.

    JANOUŠEK V., BOWES D.R., ROGERS G., FARROW C.M. & JELÍNEK E. 2000a. Modelling diverse processes in the petrogenesis of a composite batholith: the Central Bohemian Pluton, Central European Hercynides. J. Petrol. 41: 511–543.

    JANOUŠEK V., BOWES D.R., BRAITHWAITE C.J.R. & ROGERS G. 2000b. Microstructural and mineralogical evidence for limited involvement of magma mixing in the petrogenesis of a Hercynian high-K calc-alkaline intrusion: the Kozárovice granodiorite, Central Bohemian Pluton, Czech Republic.– Trans. Royal Soc. Edinburgh: Earth Sci. 91: 15–26.

    ROLLINSON H.R. 1993. Using geochemical data: Evaluation, presentation, interpretation.– Longman, pp 1–352.

    SAWKA W.N. 1988. REE and trace element variations in accessory minerals and hornblende from the strongly zoned McMurry Meadows Pluton, California.– Trans. Royal Soc. Edinburgh: Earth Sci. 79: 157–168.

    WILSON M. 1989. Igneous Petrogenesis.– Unwin Hyman, pp 1–466.

    Modelling igneous petrogenesis

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    7.1 Fractional crystallization

    7.1.1 Major elements — direct modelling

    The effects of fractional crystallization are well observable in binary diagrams of major-element oxides, in which the individual data points can be viewed as variously fractionated liquids. As x axis is usually plotted the so-called fractionation index, most commonly an oxide, whose concentration changes drastically in course of the crystallization. For acid igneous rocks the most useful fractionation index is SiO2 (such binary graphs of silica versus major-element oxides are known as Harker plots). For more basic compositions MgO or mg number (mg#) are more appropriate. Whereas undisturbed crystallization of the same cumulate results in linear correlations not unlike those produced by a variety of alternative scenarios (such as magma mixing, assimilation, partial melting and restite unmixing (Wall et al. 1987), the abrupt changes in mineral composition or proportions of fractionating minerals may cause inflections betraying the operation of fractional crystallization (Fig. 7.1).

    Graphical approach to modelling of fractional crystallization in Harker plots is shown by Fig. 7.2 after Cox et al. (1979). Plotted are the compositions of the parental magma (PM) and crystallizing phases (cum); fractionated magma (FM) develops along vectors in extension of the cum–PM tight lines. The degree of fractional crystallization ffc, defined as a proportion of already crystallized magma (0→1), is given by the lever rule (Wilson 1989):

    i FM

    i cum

    i FM

    i PM

    fc cc ccf

    − −

    = (7.1)

    Now we can write a simple mass balance equation:

    )1( fc i FMfc

    i cum

    i PM fcfcc −+= (7.2)

    Where: ffc = degree of fractional crystallization, PM = parental (undifferentiated) magma, cum = cumulus, FM = fractionated magma

    % SiO2% SiO2% SiO2

    plagioclase in

    plagioclase in

    apatite in

    olivine - cpx

    Al O2 3 P O2 5MgO

    Fig. 7.1. Harker plots for a cogenetic volcanic suite whose members are linked by fractional crystallization of olivine, clinopyroxene, plagioclase and apatite. The inflections result from changes in proportions of crystallizing phases after Wilson (1989)

    P P

    cum

    P=cum

    cum

    Q Q

    R

    PM

    PM

    PM

    FM

    FM

    differentiated magma

    parental magma

    cumulate

    FM

    % o

    xi de

    A

    % o

    xi de

    A

    % o

    xi de

    A

    % SiO2% SiO2% SiO2

    Ú

    Õ

    Ý Ý

    Õ

    Fig. 7.2. Evolution of the residual magma composition due to (simultaneous) fractional crystallization of one (P), two (P–Q) and three (P–Q–R) minerals – after Cox et al. (1979)

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    Concentration of element i in cumulate is a sum of its concentrations in individual minerals (ck) multiplied by their modal proportions (fk):

    i k

    i k

    k

    i cum fcc ∑=

    (7.3)

    The data file MaunaLoa.data (Table 7.1) contains major-element compositions of a basalt and olivine (fo88) that has crystallized from basaltic magma (Albarède 1995 – p. 5).

    • Assuming that the chemistry of basalt corresponds to that of the parental melt, calculate the compositions of fractionated magma after 5, 10 and 15 % crystallization of pure olivine.

    From Eq. 7.2:

    )1( ol ol

    i ol

    i PMi

    FM f fcc

    c − −

    =

    Where: PM = parental magma composition (basalt), ol = olivine, FM = the unknown chemistry of the differentiated melt

    > x x WR min f for (i in 1:length(f)){ > y x } > colnames(x) print(round(x,2)) WR ol 0.05 0.1 0.15 SiO2 51.63 39.90 52.25 52.93 53.70 TiO2 1.94 0.00 2.04 2.16 2.28 Al2O3 13.12 0.00 13.81 14.58 15.44 FeO 10.80 11.70 10.75 10.70 10.64 MgO 8.53 47.80 6.46 4.17 1.60 CaO 9.97 0.28 10.48 11.05 11.68 Na2O 2.21 0.00 2.33 2.46 2.60

    File basalt.data (Table 7.2) contains analyses of typical MORB basalt and some of its rock-forming minerals (Albarède 1995 – p. 8)

    Table 7.2. Composition of a typical MORB basalt and its rock-forming minerals

    basalt olivine diopside anorthite

    SiO2 49.79 40.01 54.69 48.07 Al2O3 16.95 0 0 33.37 FeO 8.52 14.35 3.27 0 MgO 8.59 45.64 16.51 0 CaO 12.17 0 25.52 16.31 Na2O 2.61 0 0 2.25

    Exercise 7.1

    Exercise 7.2

    Table 7.1. Compositions of a Hawaiian basalt and its olivine

    basalt olivine

    SiO2 51.63 39.9 TiO2 1.94 0 Al2O3 13.12 0 FeO 10.8 11.7 MgO 8.53 47.8 CaO 9.97 0.28 Na2O 2.21 0

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    • Calculate the composition of residual melt following 20% fractional crystallization of a cumulate consisting of 20 % olivine, 30 % diopside and 50 % anorthite

    • What is the composition of cumulate? (see Eq. 7.3)

    > x x WR mins f fc ccum crl x colnames(x) print(round(x,2)) basalt olivine diopside anorthite cum. dif.magma SiO2 49.79 40.01 54.69 48.07 48.44 50.13 Al2O3 16.95 0.00 0.00 33.37 16.68 17.02 FeO 8.52 14.35 3.27 0.00 3.85 9.69 MgO 8.59 45.64 16.51 0.00 14.08 7.22 CaO 12.17 0.00 25.52 16.31 15.81 11.26 Na2O 2.61 0.00 0.00 2.25 1.12 2.98

    7.1.2 Major elements — reverse modelling by least-squares method

    As shown in the preceding section, in modelling fractional crystallization it is possible to view the parental magma composition as a mixture of differentiated melt and crystallized minerals (Eq. 7.2). Let’s build a matrix A in whose first column will be stored the differentiated magma composition and in the following ones mineral compositions; y should be a vector with parental melt composition. The vector x should contain, as its first element, fraction of the melt remaining (i.e. 1 – degree of fractional crystallization), followed by relative proportions of minerals in the cumulate (recast to sum of 1). Then the mass balance can be written in a matrix form (Bryan et al. 1969).

    Axy = (7.4)

    and solved using the least-squares method. Albarède (1995) discusses in a detail all the necessary mathematics that is behind the solution. For us really matters that in R the least-squares method is implemented as the function:

    lsfit (A, y, intercept = FALSE1)

    The outcome of this function is a list, of which the most interesting is the component $coefficients, corresponding the vector x as defined above, and a component $residuals with deviations between the calculated and observed magma compositions. The sum of squares of residuals R2 is a useful quantifier for the goodness of fit. Even though it shows a marked tendency to decrease with increasing number of phases involved in the calculation, as a rule of thumb it shou

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