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):

iFM

icum

iFM

iPM

fc ccccf

−−

= (7.1)

Now we can write a simple mass balance equation:

)1( fciFMfc

icum

iPM 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)

PP

cum

P=cum

cum

Q Q

R

PM

PM

PM

FM

FM

differentiatedmagma

parentalmagma

cumulate

FM

% o

xide

A

% o

xide

A

% o

xide

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)

4/3

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

ik

ik

k

icum 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

oliol

iPMi

FM ffcc

c−−

=

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

> x<-read.table("MaunaLoa.data",sep="\t") > x<-as.matrix(x) > WR<-x[,1] # composition of the parental magma > min<-x[,2] # composition of olivine > f<-c(0.05,0.1,0.15) # degrees of fractional crystallization > for (i in 1:length(f)){ > y<-(WR-min*f[i])/(1-f[i]) > x<-cbind(x,y) > } > colnames(x)<-c("WR","ol",f) > 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

4/4

• 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<-read.table("basalt.data",sep="\t") > x<-as.matrix(x) > WR<-x[,1] # composition of parental magma > mins<-x[,2:ncol(x)] # comp. of fractionating minerals > f<-c(0.2,0.3,0.5) # mineral proportions in cumulate > fc<-0.2 # degree of fractionation > ccum<-mins%*%f # composition of the cumulate > crl<-(WR-ccum*fc)/(1-fc) # and of residual liquid > x<-cbind(x,ccum,crl) > colnames(x)<-c("basalt",colnames(mins),"cum.","dif.magma") > 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 should not greatly exceed one.

[reversed Exercise 7.2]

• Given the compositions of the parental magma (MORB basalt), differentiated melt and crystallizing minerals (file basalt2.data, Table 7.3), estimate by the least-squares method the degree of fractional crystallization and likely mineral proportions in the cumulate

1 Note that the parameter intercept needs to be set to FALSE in order for the solution to pass through the origin.

Exercise 7.3

4/5

Table 7.3. Compositions of a primitive MORB basalt, a differentiated melt (DM) and crystallizing phases

basalt DM olivine diopside anorthite

SiO2 49.79 50.13 40.01 54.69 48.07 Al2O3 16.95 17.02 0 0 33.37 FeO 8.52 9.69 14.35 3.27 0 MgO 8.59 7.22 45.64 16.51 0 CaO 12.17 11.26 0 25.52 16.31 Na2O 2.61 2.98 0 0 2.25

> x<-read.table("basalt2.data",sep="\t") > x<-data.matrix(x) > A<-x[,-1] > y<-x[,1] # parental magma composition > ee<-lsfit(A,y,intercept=FALSE) > fc<-1-ee$coeff[1] # degree of fractional crystallization > f<-ee$coeff[-1] # if these are normalized to sum up to > f<-f/sum(f) # 100 % we get the mineral proportions > cat(round(100*fc,3),"% fc ","\n") > print(f*100,4)

20.003 % fc olivine diopside anorthite 19.97 30.05 49.98

> cat("\nRsquared: ",sum(ee$residuals^2),"\n")

Rsquared: 8.11834e-32

To check the solution, we can invoke the direct modelling as in Exercise 7.2:

> mins<-x[,-(1:2)] > parent<-x[,1] > cum<-mins%*%f > estimated<-(parent-fc*cum)/(1-fc) > print(round(estimated,2))

[,1] SiO2 50.13 Al2O3 17.02 FeO 9.69 MgO 7.22 CaO 11.26 Na2O 2.98

7.1.3 Trace elements — direct modelling

The concentration of a trace element in course of fractional crystallization obeys the Rayleigh law (Rollinson 1993):

cc

FL D

0

1= −( )

(7.5)

Where: c0 = initial concentration of the trace element in the

unfractionated melt cL = concentration of the trace element in the fractionated

melt F = fraction of the melt remaining (1→0); (1 – F) =

degree of fractionation D = bulk distribution coefficient for crystallizing phases:

10

100

1.0

F

0.10 0.2 0.4 0.6 0.8 1.0

0.10.01

12

5

D =

10

cL

c0

Fig. 7.3. Dependence of melt composition on fraction of the melt left (F) for various bulk distribution coefficients (D) during Rayleigh fractional crystallization. The shaded region denotes the “forbidden zone” limited by possible maximum enrichment of a strongly incompatible element (D = 0: see Eq. 7.9)

4/6

D Kdi Xii

= ∑

(7.6)

Instantaneous solid:

c Dc Dc Fsi

LD= = −

01( )

(7.7)

Bulk cumulate:

c c FFs

D

=−−0

11

(7.8)

Fractional crystallization depletes quickly compatible elements from the melt. This fact can be used to confirm that fractional crystallization and not partial melting was responsible for the observed variations (Fig. 7.3). The minimum degree of fractionation can be constrained using a strongly incompatible element. As D approaches zero, the Rayleigh equation (Eq. 7.5) changes into:

cc F

L

0

1→

(7.9)

Data file basalt3.data (Table 7.4) contains selected trace-element data for a basalt together with distribution coefficients of its main constituents (Albarède 1995 — p. 494)

Table 7.4. Trace-element data for a basalt and distribution coefficients for its main mineral constituents

basalt ol cpx plg

Ni 150 15 1 0 Sr 100 0 0.1 2.0 Yb 3 0.05 0.35 0.25 Rb 10 0 0 0

• Calculate the melt composition following 20% fractional crystallization of 30 % olivine, 20 % diopside and 50 % plagioclase

• What is the instantaneous and bulk composition of the cumulate?

> x<-read.table("basalt3.data",sep="\t") > x<-data.matrix(x) > c0<-x[,1] # parental magma composition > Kd<-x[,-1] # table of distribution coefficients > f<-c(0.3,0.2,0.5) # mineral proportions in the cumulate > F<-0.8 # fraction of the melt left > D<-Kd%*%f # bulk distribution coefficients > cL<-c0*F^(D-1) # melt composition > cS<-D*cL # instantaneous solid > cSavg<-c0*(1-F^D)/(1-F) # average solid > result<-cbind(c0,D,cL,cS,cSavg) > colnames(result)<-c("c0","D","cL","cS","cSavg") > print(round(result,1))

Exercise 7.4

4/7

c0 D cL cS cSavg Ni 150 4.7 65.7 308.8 487.2 Sr 100 1.0 99.6 101.5 101.8 Yb 3 0.2 3.6 0.8 0.7 Rb 10 0.0 12.5 0.0 0.0

Exercise 7.5

• Plot a graph illustrating the evolution of normalized trace-element concentrations log(cL/c0) with the

decreasing fraction of melt left (F). Do so for different values of the bulk distribution coefficient (D = 0.01, 0.1, 1, 2, 5, 10; Eq. 7.5).

> F<-seq(1,0,by=-0.05);D<-1 > plot(F,F^(D-1),xlab="F",ylab=expression(c[L]/c[0]),type="l",

ylim=c(0.1,10),log="y") > D<-c(0.01,0.1,2,5,10) > for (i in 1:length(D)){ > points(F,

F^(D[i]-1),type="l") > }

7.1.4 Trace elements — reverse modelling

The identification of fractionating phases is facilitated by log–log plots of whole-rock trace-element concentrations (e.g. Fig. 7.4), in which the originally exponential trends are converted to linear ones:

log( ) log( ) ( ) log( )cL c D F= + −0 1 (7.10)

Of obvious choice are compatible elements, because their contents decrease sharply in course of fractionation. For granitoids are commonly used Rb, Sr, Ba whose distribution coefficients (Kd) are relatively well known (Hanson 1978; Table 7.5). Somewhat trickier are REE (e.g. Hanson 1980) as their distribution is, to a large extent, controlled by accessory phases, whose role may be difficult to assess (Gromet & Silver 1983; Sawka 1988; Evans & Hanson 1993). For instance zircon has large Kd for HREE, allanite for LREE, whereas both titanite and apatite prefer middle REE (Fig. 7.5). Among the main rock-forming minerals, feldspars have low Kd for all REE with exception of Eu. The magnitude of the Eu anomaly in plagioclase decreases with increasing fO2 and temperature (Hanson 1980). Clinopyroxene strongly prefers middle and heavy REE; similar patterns – even though at higher total contents – has also amphibole. On the other hand, biotite is characterized by low contents of all REE (Fig. 7.5).

1000

bi

Kfplg

hb

100

1000

500 2000

Ba (ppm)

A: 42% hb + 32% plg + 12% KF + 13% bi

10

10 10

1010

10

20

20

30

50

40

30

B: 26% hb + 47% plg + 27% bi

ABKozárovice

intrusionBlatná

intrusion

Fig. 7.4. Ba vs Sr patterns for the Kozárovice (diamonds) and Blatná (squares) intrusions of the Central Bohemian Plutonic Complex. Labelled vectors correspond to 10% fractionation of the rock-forming minerals. Shown are also effects of up to 60% fractional crystallization of Hbl, Pl, Bt ± Kfs (assemblages A, B) (Janoušek et al. 2000a)

Table 7.5. Typical mineral/melt distribution coefficients for Rb, Sr and Ba of rock-forming minerals from dacitic and rhyolitic rocks (Hanson 1978)

Mineral Rb Sr Ba garnet 0.0085 0.015 0.017 hypersthene 0.0027 0.0085 0.0029 clinopyroxene 0.032 0.516 0.131 amphibole 0.014 0.22 0.044 biotite 3.26 0.12 6.36 K-feldspar 0.659 3.87 6.12 plagioclase 0.041 4.4 0.31

4/8

7.2 Binary mixing In fact, this model approximates several petrogenetic processes, the most important being magma mixing and contamination. Let’s have a binary mixture of components A and B. If fraction of A is denoted as f:

f AA B

=+

(7.11)

concentration of any given element in the mixture M is:

BBABAM cccffcfcc +−=−+= )()1( (7.12)

for two elements, X and Y, we can write analogously (Faure 1986):

Y XY Y

X XY X Y X

X XM MA B

A B

B A A B

A B

=−−

+−−

( )( )

(7.13)

7.2.1 Major–element based mixing test

The Eq. 7.12 defines a straight line in the diagram cA–cB vs. cM–cB with the slope corresponding to the fraction f of the end member A. Fitting a least-squares regression line to major-element data in this plot is also the principle of the mixing test as employed by Fourcade & Allégre (1981) and Castro et al. (1990) (Fig. 7.6a).

7.2.2 Trace-element based mixing test

Castro et al. (1990) used the theoretical proportions obtained from the major-element based mixing test for calculation of theoretical contents of trace elements in the putative hybrid. Then they compared the calculated and observed concentrations in a manner similar to Fig. 7.6b.

In the Central Bohemian Plutonic Complex, associated with the Kozárovice granodiorite are K-rich pyroxene- and amphibole-bearing monzonitic rocks (e.g. Lučkovice melamonzonite–monzogabbro). Within granodiorite occur bodies of

biotite–amphibole quartz monzonite, whose hybrid origin is strongly supported by both the field evidence and presence of disequilibrium textures on a mineral scale (Janoušek et al. 2000b). In data file koza.data are stored compositions of all three granitoid types: granodiorite, monzonite and quartz monzonite (Table 7.6).

ZIRCON

GARNET

APATITE

ANORTHOCLASE

PLAGIOCLASE

K-FELDSPAR

HORNBLENDE

CLINOPYROXENE

HYPERSTHENE

BIOTITE

MIN

ER

AL/

MAT

RIX

MIN

ER

AL/

MAT

RIX

1

1

4

10

50

100

400

0.1

0.01

0.05

0.1

Ce CeNd NdSm SmEu EuGd GdDy DyEr ErYb Yb

Fig. 7.5. Mineral/melt distribution coefficients for REE in dacites and rhyolites (Hanson 1978)

Exercise 7.6

4/9

Table 7.6. Major-element compositions of the Kozárovice granodiorite, Lučkovice monzonite–monzogabbro and quartz monzonite, their presumed hybrid (Janoušek et al. 2000b)

A: Kozárovice

M: quartz monzonite

B: Lučkovice

SiO2 64.60 59.58 49.21 TiO2 0.57 0.72 1.02 Al2O3 14.99 14.8 13.69 FeO 2.79 4.08 6.96 Fe2O3 1.27 1.69 2.47 MnO 0.08 0.14 0.15 MgO 2.37 4.11 8.53 CaO 3.44 5.33 9.74 Na2O 3.12 2.84 1.89 K2O 4.34 4.19 3.61

• Test on major elements, whether the quartz monzonite could correspond to a hybrid between Kozárovice

granodiorite and Lučkovice monzogabbro • If so, determine the proportion of granodiorite in the mixture • Given that granodiorite contains 1154 ppm and monzogabbro 2329 ppm Ba, calculate its expected

concentration in the quartz monzonite

a

c-c

MB

c -cA B-10 100 20

-10

10

0

20

Si

TiAl

Fe2+ Fe3+Mn

MgCa

NaK

HY

BR

ID/ B

ASI

C

0

1

2

3

Ba Rb Sr Zr Hf Ce Y Ni CoLa Cr

b

Fig. 7.6. a Major-element based mixing test Fourcade & Allégre (1981) for the Kozárovice quartz monzonite; ca, cb and ch correspond to wt % oxide in the acid end-member (Kozárovice granodiorite), basic end-member (Lučkovice monzonite), and suspected hybrid (quartz monzonite); the slope of the regression line gives the proportion of the acid component f;

b Trace-element mixing test Castro et al. (1990) that compares the actual trace-element contents in the putative hybrid (solid) with theoretical composition of a mixture with 68 % of the acid end member (dashed). (Janoušek et al. (2000b)

4/10

> x<-read.table("koza.data",sep="\t") > x<-as.matrix(x) > mix1<-x[,1]-x[,3] > mix2<-x[,2]-x[,3] > plot(mix1,mix2,xlim=c(-10,15),

ylim=c(-5,10),pch=16, xlab=expression(c[a]-c[b]), ylab=expression(c[h]-c[b]),cex=1.3)

> abline(h=0) > abline(v=0) > text(mix1,mix2+0.5,rownames(x),cex=0.75) > lq<-lsfit(mix1,mix2,intercept=F) > abline(lq,lty="dashed") > # Fig. 7.6 > print(lq$coeff)

Intercept X

0.0004768997 0.6841033439

> lq$coeff["X"]*1154+(1-lq$coeff["X"])*2329

X 1525.178

7.2.3 Radiogenic isotopes — direct modelling

a) Single isotopic ratio

The mixing equation for two end-members A and B that have concentrations of given element cA, cB and corresponding isotopic ratios IA, IB is:

−+

=

M

BB

M

AAM c

fcI

cfc

II)1(

(7.14)

Eqs 7.12 and 7.14 can be, after eliminating f, developed into:

( )( ) BA

BBAA

BAM

ABBAM cc

IcIccccIIcc

I−−

+−−

=

(7.15)

and this is an equation of a hyperbola in the c–I (e.g. Sr–87Sr/86Sr) space.

In the isotope-based modelling of the binary mixing are frequently used plots such as 1/Sr–(87Sr/86Sr)i (i.e. age-corrected Sr isotopic ratios), where the mixing hyperbolae change into straight lines. For suite of co-genetic rocks, a non-zero slope of this line implies that some open process has played role, such as magma mixing or wall-rock contamination Briquet & Lancelot (1979). On the other hand, samples that originated from the same magma by various degrees of closed-system fractional crystallization preserve identical initial isotopic ratios (forming horizontal trends). The parameter f can be calculated if the isotopic compositions and element concentrations for both end-members as well as the isotopic composition of the presumed hybrid are known:

( )BBAABAM

MBB

cIcIccIIIc

f+−−

−=

)( (7.16)

-10 -5 0 5 10 15

-50

510

ca − cb

c h−

c b

TiO2

Al2O3

FeO

Fe2O3

MnO

MgOCaO

Na2OK2O

Fig. 7.7. Major-element based mixing test for the Kozárovice quartz monzonite (Exercise 7.6)

4/11

During the ascent, basaltic magma is contaminated by the surrounding paragneiss. Their compositions are given in the Table 7.7 (Albarède 1995 – p. 5)

• Plot a theoretical mixing hyperbola between basalt and gneiss in the Sr–87Sr/86Sr and 1/Sr–87Sr/86Sr spaces • Calculate the 87Sr/86Sr ratio in a mixture containing 50 % of the gneiss • Determine the proportion of gneiss in the mixture that has 87Sr/86Sr = 0.710

> ca<-100 > ia<-0.712 > cb<-400 > ib<-0.704 > f<-seq(0,1,by=0.05) > cm<-ca*f+(1-f)*cb > names(cm)<-f > im<-ia*ca*f/cm+ib*cb*(1-f)/cm > names(im)<-f > par(mfrow=c(1,2)) > srlab<-expression(""^87*Sr/""^86*Sr) > plot(cm,im,xlab="Sr(ppm)",ylab=srlab,type="b") > plot(1/cm,im,xlab="1/Sr(ppm)",ylab=srlab,type="b") > # Fig. 7.8 > f<-0.5 > cm<-ca*f+(1-f)*cb > im<-ia*ca*f/cm+ib*cb*(1-f)/cm > im

[1] 0.7056 > im<-0.710 > f<-cb*(ib-im)/(im*(ca-cb)-ia*ca+ib*cb) > f

[1] 0.923077

Exercise 7.7

100 200 300 400

0.70

40.

706

0.70

80.

710

0.71

2

0.004 0.008

0.70

40.

706

0.70

80.

710

0.71

2

1/Sr (ppm)Sr (ppm)

87S

r 86

Sr

87S

r 86

Sr

Fig. 7.8. Theoretical mixing hyperbola resulting from contamination of basalt by the surrounding gneiss (Exercise 7.7)

Table 7.7.

A: gneiss B: basalt

Sr 100 ppm 400 ppm 87Sr/86Sr 0.712 0.704

4/12

b) Pair of different isotopic ratios

From Eqs. 7.12 and 7.14:

)1()1(

fcfcfcIfcI

IBA

BBAAM −+

−+=

(7.17)

Using this formula, two sets of isotopic ratios (typically Sr and Nd) can be calculated for pre-set proportions of the end member A (f) and the mixing hyperbola plotted in the IA–IB (e.g., 87Sr/86Sr – 143Nd/144Nd) space.

The asymptotes are defined as (Albarède 1995):

qSrSrq

SrSr

x AB

−

−

=1

86

87

86

87

0 q

NdNdq

NdNd

y BA

−

−

=1

144

143

144

143

0 (7.18)

and curvature:

B

A

NdSrNdSr

q)/()/(

=

(7.19)

(straight line is obtained only if q = 1).

Basaltic magma is contaminated by the surrounding gneiss. Their compositions are given in the Table 7.8 (Albarède 1995 — p. 22)

Table 7.8.

A: gneiss B: basalt

Sr 200 ppm 100 ppm 87Sr/86Sr 0.710 0.703

Nd 20 ppm 2 ppm 143Nd/144Nd 0.511 0.513

• Calculate the Sr concentrations, 87Sr/86Sr isotopic ratios, Nd concentrations and 143Nd/144Nd isotopic ratios of

mixtures containing 0, 5, 10, … 100 % of the gneiss; print the result in a tabular form • Plot a theoretical mixing hyperbola between basaltic magma and gneiss in the 87Sr/86Sr– 143Nd/144Nd space • Define the asymptotes

> ca1<-200 > cb1<-100 > ia1<-0.71 > ib1<-0.703 > f<-seq(0,1,by=0.05) > cm1<-ca1*f+(1-f)*cb1 > im1<-ia1*ca1*f/cm1+ib1*cb1*(1-f)/cm1 > ca2<-20;cb2<-2 > ia2<-0.511;ib2<-0.513 > cm2<-ca2*f+(1-f)*cb2 > im2<-ia2*ca2*f/cm2+ib2*cb2*(1-f)/cm2 > x<-cbind(cm1,im1,cm2,im2) > rownames(x)<-f > colnames(x)<-c("Sr","87Sr/86Sr","Nd","143Nd/144Nd") > print(x)

Exercise 7.8

4/13

Sr 87Sr/86Sr Nd 143Nd/144Nd 0 100 0.7030000 2.0 0.5130000 0.05 105 0.7036667 2.9 0.5123103 0.1 110 0.7042727 3.8 0.5119474 0.15 115 0.7048261 4.7 0.5117234 0.2 120 0.7053333 5.6 0.5115714 0.25 125 0.7058000 6.5 0.5114615 0.3 130 0.7062308 7.4 0.5113784 0.35 135 0.7066296 8.3 0.5113133 0.4 140 0.7070000 9.2 0.5112609 0.45 145 0.7073448 10.1 0.5112178 0.5 150 0.7076667 11.0 0.5111818 0.55 155 0.7079677 11.9 0.5111513 0.6 160 0.7082500 12.8 0.5111250 0.65 165 0.7085152 13.7 0.5111022 0.7 170 0.7087647 14.6 0.5110822 0.75 175 0.7090000 15.5 0.5110645 0.8 180 0.7092222 16.4 0.5110488 0.85 185 0.7094324 17.3 0.5110347 0.9 190 0.7096316 18.2 0.5110220 0.95 195 0.7098205 19.1 0.5110105 1 200 0.7100000 20.0 0.5110000 > plot(im1,im2,xlab=expression

(""^87*Sr/""^86*Sr), ylab=expression(" "^143*Nd/" "^144*Nd),type="b")

> # Fig. 7.9 > q<-(ca1/ca2)/(cb1/cb2) > x0<-(ib1-q*ia1)/(1-q) > y0<-(ia2-q*ib2)/(1-q) > x0

[1] 0.70125

> y0

[1] 0.5105

0.703 0.705 0.707 0.709

0.51

100.

5115

0.51

200.

5125

0.51

30

87Sr 86Sr

143 N

d 14

4 Nd

Fig. 7.9. Theoretical mixing hyperbola between basalt and gneiss (Exercise 7.8)

H.L. Mencken's Law:

Those who can — do. Those who can't — teach.

Martin's Extension: Those who cannot teach — administrate.

Unnamed Law

If you can't learn to do it well, learn to enjoy doing it badly.