coexisting pyroxenes — a multivariate statistical approach

11
Coexisting pyroxenes " a multivariate statistical approach* ANDERS LINDH _~(~ Lindh,A. 1975: Coexisth~ pyroxgues--a multivariate s t a t ~ approach. L/dos 8, 151--161. A population of 117 coexisting nontlkaline pyroxene pain has been studied stagy to .d e,,mt is by the Fe/Mg-ratio, by the Ca ¢ o n t e n t ~ y of clinopyro~ by the content of tetrahcKlral AI. Pc and tetrahedral AI are found to be negativelycorrelated. A principal compo- nent analysis based on the variation of Si, AI Iv, A! vl, Fe, MS, Mn, Ca is perfmmed. Dropping of highly correlated variables does not affect the result significantly. The first prim/pal compo- nent reflectsthe major chemical variation in Fe and Mg. When using ferrous and ferrk iron as separate entries of the analysis,either the second or the third component demonstrates a tempe- rature dependence. It is, however, not possible to obtain pure tmmpm~ture and chemical com- ponents due to the composition not being uncomdatnd to temperature of formation. From these components a graph reflecting temperature of formation has been ¢omm'uctnd. Anders L/ndh, Departnumt of Mineralogy and Petrolofy, O~verslty of LJmd, ~Ol,~,gatan 13, S-223 62 Lumt, Swedm. The thermal history of pyroxenes has been ap- proached by studying the compositional variation of both single and coexisting pyroxene phases. From tie-line intersections in the pyroxene quad- rilateral, Hess (1941) stated implicitly that when two pyroxenes coexist, the Ca-poor phase is the more iron-rich one. Studies of these intersections continued and efforts were made to distinsuish between metamorphic and magmatic pyroxenes from the position of the intersection point (Muir & Tilley 1957, 1958, Wilson 1960). After the pioneering work of Ramberg & de Vore (1951) on the physico-chemical backsround of element distribution, it was possible to base the investiga- tions on the distribution coeffi~ent which, con- trary to the intersection points, has a thermo- dynamic meaning. At the besitming both pyrox- ene series appeared to be more or less ideal solid solutions with respect to magnesium and iron (Kretz 1961, Mueller 1961). The distribution coefficient, ir opx_epx__XCP: (1 - vepxx "~s' Where opx stands -XMg for orthopyroxene, cpx for clinopyroxene, and Xas for the mole fraction Mg Mg+ Fe' was found to vary with temperature (Kretz 1961), and in 1963 Kretz suggested a tentative diagram relating K~lg ©px to temperature. Already in 1962, Binns had tried to subdivide the 8ranulite facies by using the iron- magnesium distribution between coexisting pyrox- enm, Unfortunately he adopted a distribution model of no physko-ch~'.ical relevance. Binns' work provoked Davidsson (1968) to investigate the relations between the two pyroxene phases. He also took into account the ordering of Fez+and Mg s+on nonequivalent cation positions in orthopyroxene, which was reported by Ghose (1965). As a result, from this orderinfl it was possible Ibr him to show that the iron-magnesium ratio of the pyroxenes and the content of Ca in orthopyroxene influence the distribution. Later Grover & Orville (1969) from a theoretica! one site-two site distribution model arrived at virtually the same result. The importance of Ca was still later stressed by Blan- der (1972), showing the great influence from the content of Ca in ¢linopyroxene on the distribu- tion of Fe and Mg. Besides the Fe/Mg-ratio and the content of Ca, there are other elements which may influence K~ epx, e.g. Si, AI, Na, Mn, and Ti. In order to evaluate the effect of the other elemevts and the temperature, a multivariate statistical method is applied. The present analysis is based on iitera- * This is a corrected version of an article by Antlers Lindh which was published in L/tAos 7 (1974), pp. 1 9 ~ 2042. We apolosize to Mr. Lindh for this inconvenience.

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Coexisting pyroxenes " a multivariate statistical approach*

ANDERS LINDH

_ ~ ( ~ Lindh, A. 1975: Coexisth~ pyroxgues--a multivariate s t a t ~ approach. L/dos 8, 151--161.

A population of 117 coexisting nontlkaline pyroxene pain has been studied s t a g y to . d e , , m t is

by the Fe/Mg-ratio, by the Ca ¢ o n t e n t ~ y of c l i n o p y r o ~ by the content of tetrahcKlral AI. Pc and tetrahedral AI are found to be negatively correlated. A principal compo- nent analysis based on the variation of Si, AI Iv, A! vl, Fe, MS, Mn, Ca is perfmmed. Dropping of highly correlated variables does not affect the result significantly. The first prim/pal compo- nent reflects the major chemical variation in Fe and Mg. When using ferrous and ferrk iron as separate entries of the analysis, either the second or the third component demonstrates a tempe- rature dependence. It is, however, not possible to obtain pure tmmpm~ture and chemical com- ponents due to the composition not being uncomdatnd to temperature of formation. From these components a graph reflecting temperature of formation has been ¢omm'uctnd.

Anders L/ndh, Departnumt of Mineralogy and Petrolofy, O~verslty of LJmd, ~Ol,~,gatan 13, S-223 62 Lumt, Swedm.

The thermal history of pyroxenes has been ap- proached by studying the compositional variation of both single and coexisting pyroxene phases. From tie-line intersections in the pyroxene quad- rilateral, Hess (1941) stated implicitly that when two pyroxenes coexist, the Ca-poor phase is the more iron-rich one. Studies of these intersections continued and efforts were made to distinsuish between metamorphic and magmatic pyroxenes from the position of the intersection point (Muir & Tilley 1957, 1958, Wilson 1960). After the pioneering work of Ramberg & de Vore (1951) on the physico-chemical backsround of element distribution, it was possible to base the investiga- tions on the distribution coeffi~ent which, con- trary to the intersection points, has a thermo- dynamic meaning. At the besitming both pyrox- ene series appeared to be more or less ideal solid solutions with respect to magnesium and iron (Kretz 1961, Mueller 1961).

The distribution coefficient,

i r opx_epx__XCP: (1 - vepxx " ~ s ' Where opx stands

-XMg

for orthopyroxene, cpx for clinopyroxene, and Xas

for the mole fraction Mg Mg+ Fe' was found to vary

with temperature (Kretz 1961), and in 1963 Kretz suggested a tentative diagram relating K~lg ©px

to temperature. Already in 1962, Binns had tried to subdivide the 8ranulite facies by using the iron- magnesium distribution between coexisting pyrox- enm, Unfortunately he adopted a distribution model of no physko-ch~'.ical relevance. Binns' work provoked Davidsson (1968) to investigate the relations between the two pyroxene phases. He also took into account the ordering of Fe z+ and Mg s+ on nonequivalent cation positions in orthopyroxene, which was reported by Ghose (1965). As a result, from this orderinfl it was possible Ibr him to show that the iron-magnesium ratio of the pyroxenes and the content of Ca in orthopyroxene influence the distribution. Later Grover & Orville (1969) from a theoretica! one site-two site distribution model arrived at virtually the same result. The importance of Ca was still later stressed by Blan- der (1972), showing the great influence from the content of Ca in ¢linopyroxene on the distribu- tion of Fe and Mg.

Besides the Fe/Mg-ratio and the content of Ca, there are other elements which may influence K ~ epx, e.g. Si, AI, Na, Mn, and Ti. In order to evaluate the effect of the other elemevts and the temperature, a multivariate statistical method is applied. The present analysis is based on iitera-

* This is a corrected version of an article by Antlers Lindh which was published in L/tAos 7 (1974), pp. 1 9 ~ 2042. We apolosize to Mr. Lindh for this inconvenience.

152 Acders Lindh

ture data representing most published complete analyses of coexisting nonalkaline pyroxenes. The analytical accuracy may be of different quality which introduces scattering of the plotted pairs. "H~e studied pyroxene pairs are assumed to represent equilibria. Contrary to Blander (1972) we furthermore assume that the equilibrium -.'emperature is s.hailar for each element of the ~ame sample. Deviations from this situation will result in further scattering of the plotted pairs.

The principal component analysis, PCA The principal component method is a convenient way to handle a large number of data including many variables. From the observations, the linear correlation coefficients among the variables are computed and put together into a correlation matrix. This raatrix gives information about the correlations among all the variables. Then the eigenvalues and eige~vectors of the matrix are calculated. The eigenvectors are the principal components along which the variance of the population is channelled, the corresponding eigen- value gives the fraction of the total variance ,:on- nected with the respective eigenvector. The magni- tude and the sign of the vectorial elements give the importance and the direction of the contribution from one particular variable to the vector. Fur- thermore, the eigenvectors may be used as new orthogonal co-ordinate axes, spanning a multi- dimensional space. The observation points are transformed from the original co-ordinate system defined by the original variables to the new co- ordinate system. The computer programmC~which has been used gives projections of this space onto a two-dimensional subspace defined by two or the three eigenvectors connected with the largest eigenvalues. The programme is a slightly modified version of a PCA programme made by Blackith & Reyment ( 1971).

PCA of coexisting pyroxenes All the chemical analyses of the present I I 7 pyrox- ene pairs have been published earlier. The inves- tigation is based on the following variables in the two pyroxenes: Si, AI Iv, AI vx, Fe s+, Ire s+, Fe tot, Mg, Mn, and Ca calculated as ions on the basis

x,r o p x - c p x of 6 oxygens and, in addition, on ~'..DMg (calculated both from Fe t°t and from Fe s+) and

Kt~c~ - ~ ' . The last distribution coefficient is cal- culated according to the formula

• -

'

where Xcs stands for the mole fraction Ca

Ca+ Mg+ Fet°t+ Mn+ AI vP and IV and VI as

superscripts at AI denote the coordination num- ber. Na and Ti have been excluded from the statistical analysis. Ti occurs only sporadically in the published analyses, the low content in most pyroxenes suggests that Ti does not markedly in- fluence the distribution of other elements. The number of Na-analyses is not large enough and because of the low Na-content in nonalkaline pyroxenes, the relative analytical accuracy is probably not very good.

Most of the data come from wet-chemical analyses, but some are microprobe determina- tions. In the latter case there is no possibility to discriminate between ferric and ferrous iron. If one element (e.g. Fe *+ or Fe s+) is lacking in any pair, and this element is one of the entries of the analysis, this pair must he eliminated from the statistical calculations. Mn is missing in nine of the presented analyses, thus conclusions including Mn are not based on the whole population. Some of the listed variables are highly correlated, e.g. S i - AI w, F e - M g - Mn. One element in such a combination might preferentially he dropped, because introducing a new element which is highly correlated with one or more of those already present gives minor new information for the statistical treatment. Reducing the number of variables has to be done by steps, because they may share an important fraction of the discrimi- nating 1,3wer, the contribution of only one of them not being significant (Blackith & Reyment 1971). The results and conclusions presented in the following pages are results obtained from many statistical analyses, where the number of variables and the number of coexisting pyroxene pairs are varied. Multivariate statistical treatment was first applied to mineral analyses by Middleton 0964), but it is not very much used. Saxena (1969) with this technique first studied the distribution of elements between two coexisting phases.

Results and discussion Before the results from the statistical analysis are given, two of the current methods of subdividing

Coexistit~ l ~ o x e ~ s 153

metamorphic metamorphic

2& pyroxenes u~ pyroxenes

29

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G00 0.08 QI6 0,2& KOpX-©px

Dco rocks into the magmatic and the metamorphic groups are demonstrated. In the form of histo- grams, Fig. 1 illustrates the ranges of K~'~s cpx for magmatic and metamorphic rocks and Fig. 2 the partitioning of Ca for the same groups of rocks. Obviously it is impossible to draw definite conclusions of rock provenience from distribu- tion coefficients despite a certain grouping of K~ePX-values and it is even more impossib!e to subdivide a certain metamorphic facies from those parameters only.

The correlalion matrix

There is no substantial difference in the correla- tion matrix when the number of observations, i.e. the number of pyroxene pairs, or the number of variables, is changed. Consequently the quantity of data may be regarded sufficient. The correla- tion matrix found in Table 1 is obtained from 93 pyroxene pairs, using all variables except KOpX-epx z)c, . Some correlations found are obvious either from earlier studies or f rom known substi- tutions in the pyroxene lattices. Within the pyrox- ene groups such correlations are found betwo=n Si and AI tv (opx-0..~7, c p x - 1.00) and between Mg and Fe '+ (opx- 0.97, cpx- 0.87). The correla- tion between Mg and Fe s+ is significantly lower

Fhr. 2. Distribution coefficient K~cx-ePx for the metamorphic and ntafgmatic pyroxene pairs.

in the clinopyroxenes than in the orthopyroxenes. Probably this difference reflects the higher and more variable content of octahedral AI and the more varying content of Ca in clinopyroxene compared to orthopyroxene. Between the groups we have correlations like Fe *+- Fe *+ (0.94), Mg- Mg (0.91), and Mn-Mn (0.86). Besides these obvious correlations there is a rather ~gh correla- tion between Mn and F~ + (opx 0.71 and cpx 0.62). It is difficult to ascribe this correlation to a crystal- chemical cause, since the distribution of Mn and Fe s+ between the pyroxenc phases differs (Lindh, in press). Conseq, Jently it is more plausible that the correlation reflects a normal geochemical differentiation trend. There is also a correlation between A! rv and Fe t+ (opx-0.54, cpx-0.53). Correlations between AI tv and Fe *+ are expected (see for example Ramberg 19.¢2) to be positive. Ramberg (1952) ascribed the normal positive correlation between Fe t÷ and AI TM to an increasing covalent character of the metal-~.~,,gen bond whe:l part of Si is substituted by AI. The magmat- ic pyroxenes as a whole are more magnesium-rich than the metamorphic ones (Fig. 3), but there is no significant difference in the content of Ai tv (Fig.

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4) between these groups. Consequently the expect- ed positive correlation between AI xv and Fe is blurred. However, blurring does not explain the negative correlation. Therefore most of an AI- surplus is suggested to be taken up by the other coexisting ferro-maguesiim silicates, e.g. biotites and amphiboles. Hereby the change in bonding character, in whatever terms it is described, is bound to lower the free energy for the iron end- member. A corresponding ~.%'rement of the free energy should take place ,n the pyroxenes. In- strumental in this interpretation is the assumption that the decrement is more pronounced in those minerals which accept most AI. One observation suppo~ling this suggestion is the weak but positive correlation between the content of tetrahedral AI in both pyroxenes and K ~ cpx (opx 035, cpx 0.20) in combination with the fact that clinopyrox- ene takes more AI than orthopyroxene.

The correlation of Ca-content between ortho- pyroxene and clinopyroxene is significant but rather low (-0.58). The low correlation coeffi- cient is explained by the following facts: If the Fe/Mg-ratio is kept constant the Ca-distribution would vary with temperature along a smooth non- linear curve, reflecting the trunc,~ted hoep-shaped solvii. However, there is a considerable variability

Coexistin8 pjc,oxenea 153

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F'qr. 4. Composition runlles (number of Airy per 6 oxyllens, cpx) of metamorphic and mslpmmtic pyroxenes.

in the Fe/Mg-ratio, which causes a scattering along the curve (DavicL~on 1968). In addition there is pr,,tmbly a magnesimn-enrichrr~:nt of the pyroxenes with increasing temperature causing a deflection o r the theoretical smooth distribution curve. Furthermore, the Cat-determinations of orthopyroxene are impaired by large errors, partly due to low Ca-content and partly to a consider- able risk of impurities from clinopyroxene and plagioclase. Davidsson's (1968) and Blander's (1972) deduced correlations between Ca-content and Mg-distribution are confirmed. Especially Ca-content of clinopyroxene affects the distribu- tion coefficient (Caopx_ vopx-c:, - n '~a and g~-DMlige|+) - - v.~. •

Cacpx_ ~px-cpx - - 0.34). uMgi ]ge| +) - -

Davidsson (1968) pointed out that when the iron content of coexisting pyroxcnes is increased the distribution coefficients for Mg and Fe approach unity. With the present notation this is equivalent to an increase in vopx-epg with increasing Wt.DMx

content of Fe .+. However, the correlation betwec n v o P z - C p z the iron content of the pyroxenes and Z~D31g

is rather weak and negative. No fundamental opposition can be raised against Daviclsson's argu- ments and consequently the discrepancy must be

t,,.og, x - cpz i n - explained in some other way. ~DNj creases with increasing temperature (Kretz 1961) and at the same time the present pyroxenes be-

156 Ant~sLtMIt

come more Mg-rich. Obviously the temperature effect dominates over the compositional effect.

If K ~ s - e p x is introduc~t into the analysis, it is found to be highly correlated to X ~ x (correla- tion coefficient 0.97). A high positive correlation is expected since ~c~ x and X ~ x vary antithet- ically. The observed correlation between ~ n x

vopx-epx does not call for any special and Z~DMg explanation. It is only due to the fact that there is a certain correlation between the Fe/Mg-ratio of the pyroxenes and K ~ epx and that Fe z+ and Mn 2+ covary.,

The principal components There is no apriori theoretical argument that con- clusions of petrological interest should arise from the principal components. All other parameters being constant, the distribution of elements be- tween two phases in most cases responds to a change in temperature. In natural samples there is no possibility to keep all variables constant. If a sufficiently large number of data is attended to, there should be a certain grouping ot' the pyroxene analyses, where the population variance should be channelled along certain paths due to crystal- chemical constraints laid upon the rock bulk com- position. One of these paths intuitively is due to changing temperature conditions.

There is no straightforward method to ascribe the different eigenvectors of the correlation ma- trix to certain composition or physical parameters. When i~terpreting the result we first have to con- sider the known greatest variations in the popula- tion and then obvious correlations known from the correlation matrix. Normally there is no possibility to ascribe all eigenvectors to a special physical or chemical parameter. The three eigen- vectors representing the greatest variance for different numbers of variables and observa- tions are given in Tables 2--5. The greatest variation of the population is found in the Fe/ Mg-ratio; consequently the first principal com- ponent should reflect this variation, which also is confirmed. The first eigenvector is principally made up of a variation in Fe and, when present as a vectorial element, Mg. This variation is correlat- ed to Mn and to AI and Si reflecting the antithetic relations between AI xv and Fe previously discuss- ed. Thus, this vecto," is supposed to represent the major chemical variation of the population. The negative correlation found between the Ai n' and FO + vectorial components supports the interpre- tation of the first principal component as a major

Tab/e 2. Based on 101 pyroxene pairs.

Vectorial Eigenvectora Elements 1 2

% o f total variance 39 2 2

Si °px 0.39 0 . 1 7

AI v'x°px - 0 .37 - 0 . I 0 Fctot opx 0.43 - 0 . 2 0

Ca °px 0.01 - 0 . 5 9

Si cpx 0.44 0 . 0 5

AI v lcpx - 0 .26 - 0 .08

FetOt cpx 0.38 - 0.39

CaCP x 0.06 0 . 6 2 KOpX-cpx

DMg(FetOt ) - - 0.34 - 0 .17

Table 3. Based on 100 pyroxene pairs.

Vectorial Eigenvectors Elements 1 2

13

- 0 . 0 4

- O . M ,

- 0 . 3 5

0.28

0 .19

- 0 . 6 6

- 0 . 2 2

- 0.01

0 .40

Table 4. B a s e d o n 107 pyroxene pairs.

Vectorial Eigenvectors Elements 1 2

% o f total variance 45 17 13

Si °px 0.27 - 0.01 - 0 .30

AI I v ° p x - 0.29 0 .02 0 .29

A I v I ° p ~ - 0.22 0 .02 0 .22

FetOt opx 0.34 - 0 .02 0 .22

M g °px - 0.33 - 0 .08 - 0 .23

Mn °px 0.26 - 0 .07 0.31

Ca °px 0.06 0 .57 - 0 .10

Si epx 0.29 0.01 - 0.31

AI I v e p x - 0 .30 - 0.01 0 .32

Aiv lep x - 0 .17 0 .03 0 .25

FetOt epx 0.32 0 .12 0 .25

Mgep x - 0 .30 0 .06 - 0.33

MnCp ~ 0.24 0 . 0 0 0 .30

CaeP x 0.04 - 0.51 - 0.13

,px-cpx - 0 .20 0 .19 0 .14 " DMg(FetOt) 0.03 0 .59 - 0.09 K ~ X a - c p x

% o f total variance 31 17 12

Si °px 0.39 0.11 0 .06

AI T M - 0.36 - 0 . 06 0.03

Fe s+°px - 0.02 - 0 .07 - 0.64

F e 9+°px 0.43 - 0 .05 - 0 .25

Ca °px 0.18 - 0 . 5 2 0 .27

Si epz 0.44 0 .05 - 0 .00

AI T M - 0.29 - 0 .15 - 0 .20

Fe e+opx - 0 . 1 1 - 0 . 2 1 - 0 . 5 9

Fe a+Opx 0.43 - 0 .22 - 0 .19

CaOp x - 0.03 0 .63 - 0 .14 KOpX-op x DMg! Fea+) -- 0.16 - 0 .43 0.07.

Coextsti~pyroxems 157

Tab/e 5. B a s e d o n 93 p y r o x e n e pa i rs .

Vec to r i a l Ei l~vec to rs Elemen t s I 2 3

% o f t o t a l va r i ance 39 I $ 13

gi °px 0.26 0.05 - 0 . 2 8

A! t vopx - 0 .29 - 0.05 0.27

AI vx°px - 0.21 - 0 .04 0 .20

Fo a+°px 0.01 - 0.05 0.14

FeS+oP x 0 .34 - 0.09 0 .20

M g °px - 0 .34 O.Ol -- 0 .24

MnOp x 0.28 - 0 .20 0.18

C a °px 0.10 O.M 0.05

Si epx 0.29 0.07 - 0.27

AIIVep x - 0.29 - 0 .06 0.27

AI T M -- 0 .16 - 0.03 0.27

Fea+oP x - 0.04 - 0.02 0.41

F e t + e P x 0.34 0.05 0.23

Mgep z - 0 . 2 9 0.17 - 0 . 3 1

Mnep x 0.26 - O. I I 0.20

c g e P x - 0.02 - 0.47 - 0.23

K ~ x - e p x 0.07 0.56 0.07 KOpX-epx uMsCFeS+) -- 0.12 0.26 0.14

chemical component. If this negative correlation has been blurred or even reversed in sigr:, bonding properties would ha~e outweighed the chemical contribution. The contribution to the f ~ t princi- pal component from Ca is minor and probably erratic, the variation not even always reflecting the negative correlation between X ~ ~ and Xc epx However, if the contribution from • •

K ~ Opx to the first principal component was a pure chemical one it should have been positive (Davidsson 1968), but it is always negative. Con- sequently the first principal component does not reflect a pure chemical variation but probably there are interfering temperature effects.

The Ca-variation is found to be only weakly correlated to the Fe/Mg-ratio (Table 1) and the variation is rather large. Hence, the Ca-variation ought to appear in one of the eigenvectors asso- ciated with the high eigenvalues; it is also found to make the largest contribution to either the second (Tables 2m5) or to the third eigenvector. If all the highly correlated compositional variables are used, but K~q~ -cpx is dropped in the statis- tical analysis, the Ca-variation comprises the third eigenvector--in the other cases the second vector. This is easily explained by the fact that by drop- ping v o p x - c p x • ~DC~ we decrease the joint Ca-variance of the population, by which its part of the total

variance diminishes. Further dropping of varia- bles correlated with iron and silicon ~ the relative amount of Ca-variation again, hence Ca- variation appears as the second eigenvector. Be- sides the Ca-variation there is also a contribution from Fe and M8, especially in clinopyroxene. This again confirtns the interrelations between Ca and the Fe/Ms-ratio. The last of the three eigenvectors associated with the three largest eigenvalues can- not he easily interpreted. It is made up of contri- butions from all the variables, Ca and K ~ -cpx normally making small contributions. In one case, where the number of variables is reduced (Table 3), the third eigenvector reflects the ferric/ferrous ratio, that is the oxidation state of the mineral. Evidently dropping va~ables as Mg, Mn, or Al;V will not affect the principal components radically.

When using ferrous and ferric iron separately, a large part of the variation of K~ '~ epg is channelled parallel to the Ca-variation (Tables 3 and 5). If total iron is used (Tables 2 and 4), the major part of tl~e K~uX~epX-variation is chan- nelled parallel to the first, fourth, fifth, or sixth eigenvector. The significance of these last three eigenvectors is no: understood. The differe,ce in channelling cannot be due to any difference in the population, because the same observations have been used both with total iron and with ferric and ferrous iron separgtely, and the difference remains. However, at present it cannot be given a reason- able mineralogical interpretation, but probably it depends on a mathematical or statistical cause. When interpreting the results from the analysis the difference is very important.

After the observations have been transformed to a coordinate system spanned by the eigen- vectors, they are pr~m:ted onto the subspace de- fined by the first eigenvector and the 'Ca-variation eigenvector'. One of the axes now takes up the main part of the Fe/Mg-variation and the asscgiat- ed variation in Si. AI, Mn, and Ca, and the other the main part 3f the Ca-variation and the K~ePX-variation-N.B. if ferrous and ferric iron are used as separate entries. Temperature is known to affect the Mg-Fe-distribution (Kretz 1961, 1963) and of necessity also affects the Ca- distribution, the immiscibility gap being narrowed with increasing temperature. However, both Mg- and Ca-distributi,3n also depend upon a number of cow lx~sitional parameters, so the variation of these properties cannot be taken directly as tem- perature indicators. It is evident from the eigen- vector of 'ca-variation' that K ~ cpx varies negatively with Ca in clinopyroxene (Tables 2--5).

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Ccwx/st/~ pj~oxenes . 159

It is possible to interpret this covadance as a temperature effect. From this discussion it is in- tuitively felt that graphs obtaita~ from projec. tions onto the subspace mentioned above give in- formation of the tempcwature of formation or rather the te .mpmatme of quelgh_ in a of the inter- pyroxene element distribution. Fig. 5 is one example of such a 8raph. Pymxenes obtained from intrusive and effusive roc~ are found to plot in the lower left part and those from metamorphic rocks towards the upper right part of the graph. A decreasing metamorphic grade should be indi- cated as a further tramlation towards the upper right part Of the graph. To test the relevance of the graph the pyroxenes analysed by Davids,son (1968) are attended to. Nine ofDavidsson's pyrox- erie pairs are from two different areas in the Quaraiding district, Aithorpe Peak and North Danging. The additional three are from various places in the same district. The pyroxenes from Althorpe Peak were thought to he isothermic and so were those from North Danging. Davidsson's using of all his data when showing the dependence of K~P~t~ epx on the Fe/Mg-ratio, implicitly suggests that he considers all his pyroxene paics to have equilibrated at the same or very near the same temperature. From the graph (Fig. 5) they are seen to plot around a line. Bitms' six pyroxenes (1962), which according to him would reflect different metamorphic grades, appear at first sight to be unsystematically distributed. However, an analysis of the order of the points compared to those of Davidsson's suggests a metamorphic gradient among pyroxenes with different Fe/Mg- ratios. This gradient is in harmony with the gra- dient found by Binns (op. cit.) from geological evider.~e. The four magmatic rocks investigated by Best & Mercy (1967) have crystalfized at a rather low temperature, they consequently plot in the low part of the magmatic and the highest part of the metamorphic areas. From normal magmatic differentiation it is possible to conclude the order in which the pyroxenes crystallized at falling temperature, an order which is reflected in this graph as a decrease in temperature. Analysing Davidsson's (1968) pyroxenes more in detail it is found that those from Althorpe Peak represent slightly lower temperature than those from North Danging. As Davidsson has already pointed out, his pyroxene pair number 9 is anomalous.

To get further confirmation of the graph, the pyroxene pairs from the Bushveld intrusion re- ported by Atkins (1969) have been plotted. They have been omitted from the statistical analysis

to provide an independent test of the graph. Fig. 5 reveals than to plot in the magmatic part of the mwph as they are expected to.

If the distribution coefficient K ~ , epx is ex- cluded from the analysis the three first elgenvec- togs remain virtually unaffected. When plotted the discriminatory pow~" of the dropped coeffi- cient is evident, because the temperature separat- ing in the graph becomes very small.

The graph in Fig. 5 is calculated from the eigen- vectors found in Table 3. The new coordinates are obtained through a variable transformation defined by the matrix multiplication of the appro- priate eigenvector col~nn and the raw data ma- trix.

If we make a least square linear regression of Davidsson's pyroxene Imirs, minimizing the de- viations in the ordinata we achieve equation (a).

y - - - 0.40x + 1.29 (a)

Furthermore, if we suggest that the directional coefficient for isothermic conditions remains con- stant equal to --0.40 all over the diagram, the constant term in equation (a) becomes u tempera- ture indicator. A higher temperature is indicated by a decrement of the constant. However, there is nothing to support this suggestion in those parts of the graph which are remote from the meta- morphic part. When applying equation (a) to pyroxenes formed at very high temperatures or of extreme composition, caution is needed because the empirical nature of the diagram must be borne in mind.

According to the previous discussion, Ca and Fe vary along a smooth curve if temperature is kept constant. To test the graph (Fig. 5) we chose pyroxene pairs which, according to the graph, represent near isothermic conditions. A plot of the pyroxenes in the pyroxene quadrilateral reveaL~ . ~ m e scattering along the ~u.,~s (Fig, 6). This scattering can be ascribed to analytical errors and to small variations in temperature. Even if the minerals were strictly isothermic, some scattering would remain due to the quadrilateral not proper- ly allowing for components like AI and Mn.

This type of diagram (Fig. 5) is the result of a statistical analysis which has been given a physico- chemical interpretation. However, at present it is not possible to give any theoretically founded evidence for this graph. From studies of crystalli- zation paths of pyroxenes it is known that these paths are more complicated in detail than this i-ather straightforward type of analysis. Conse- quently, this type of diagram cannot be claimed

160 AndersL indh

MgsiO 3

/ I I

/ i I I . . . \

FeSiO,j

Fig. 6. Compositional variation of isothermic pyroxene pairs. The three dashed lines show typical tie-lines between coexisting minerals.

to be the final answer to the question of how the composition of coexisting pyroxenes is influenced by temperature. However, founded on this type of graph it should be possible to subdivide the granulite facies rocks with a higher degree of secu- rity than has hitherto been possible.

Conclusior t

Multivariate statistical analysis has proved to be a possible tool to discriminate between pyroxenes formed at different temperatures. However, the paths along which the variance is channelled are not possible to ascribe to pure temperature and bulk chemistry effects. Interactions occur partly due to temperature and bulk chemistry not being uncorrelated quantities. Consequently, this statis- tical method cannot be expected to give the final answer. The graph (Fig. 5) might be improved if it was possible to find more pyroxene pairs which are isothermic, in addition to those reported by Davidsson (1968) to confirm or adjust equation (a). These pairs should rather be magmatic than metamorphic and should have a wide composi- tional variation.

By varying the entries of the analyses small changes in eigenvalues and sometimes reversals of eigenvectors occur. The changes are easy to explain except for the change introduced by splittiag total iron into ferrous and ferric i ron

The correlations found among the different elements confirm the results of earlier studies,

e ~cept for the negative correlation between tetra- hedral aluminium and iron, Earlier studies have not succeeded in discriminating between third element influences and temperature influences on K ~ ' ~ cpx. The present statistical method to some extent evades this mtliculty by constructing tl, e graph (Fig. 5).

Acknowledgements. - I am indebted to Professor H. Ram ~rg, head of the institute. Dr. T. Ekstr6m read the manuscript and his constructive criticism is gratefully ackn ~wledged. Financial support was given by the Swedish Nat1 rat Science Research Council.

References

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C o e x t s t l q pyroxenea 161

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A~epted for publication April 1974 Printed April 1975

11 ~ Lithos 2/75