American Mineralogist, Volume 7 5, pages 17Ul82, 1990

Error propagation in calculations of structural formulas

Manro J. Gr,rnalrrra, flowano W. DavDepartment of Geology, University of California, Davis, California 95616, U.S.A.

AssrRAcr

We have derived the algorithms for propagating analytical uncertainty through calcu-lations of structural formulas including relative formula proportions of Fe2* and Fe3*, thedistribution of cations among structural sites, and relevant cation sums. We analyzedapproximately 50 spots each on a standard augite, bytownite, and hornblende and prop-agated the uncertainties in the measured oxide concentrations through the calculation ofstructural formulas of each mineral.

The analytical uncertainty of the oxide weight percent of an element severely underes-timates the true uncertainty in the formula proportions of multivalent cations and cationsthat are assigned to more than one structural site. Ifthe contributions ofcovariance to thetotal uncertainty are neglected, however, the uncertainty in the formula proportions of Siand of multivalent or multisite cations may be overestimated by as much as 25o/o. Co-variance makes a relatively small contribution to the observed standard deviations ofother cations except for data sets in which a few "bad" analyses are present. Whetheruncertainties in formula proportions are magnified or reduced, compared to the uncer-tainties in the corresponding oxide weight percent, depends not only on the compositionof the mineral, the structural formula to which the cations are assigned, and the type ofnormalization used in the calculation, but also on the relative sizes of the uncertainties inthe oxide weight percent and the magnitude of the covariance terms.

For the bytownite, augite, and hornblende we studied, the propagated uncertainties offormula proportions of cations of major elements generally are less than l-3o/o. However,they range as high as 360/o for calculated Fe3* and 25o/o for cations that lie in more thanone structural site.

INrnooucrroN

The electron microprobe yields chemical data in theform of weight percents of the oxides from which struc-tural formulas of minerals are calculated. Anall.tical un-certainties in the oxide concentrations can be determinedby standard statistical methods, but determining the pre-cision associated with cation proportions is not as simple.

Understanding the effect of analytical uncertainty oncalculations of structural formulas is especially importantfor two reasons. First, we commonly want to knowwhether a microprobe analysis conforms to stoichiomet-ric constraints. Unless uncertainties in calculated num-bers of cations can be determined, it is not clear whethercation sums are significantly different than those requiredby stoichiometry. The smallest uncertainties that can beexpected are derived by propagating counting errorsthrough the calculation of the structural formula.

Second, there are certain petrologically important pa-rameters derived from a calculated structural formula forwhich a corresponding oxide cannot be analyzed directly.Examples of these include the distribution of cationsamong sites in a mineral structure and the relative abun-dances of multivalent cations such as Fe. The uncertain-ties associated with these parameters cannot be estimateddirectly by using the uncertainty of a measured oxide con-

0003404x/90/0 I 024 l 70$02.00

centration; they must be determined by propagating theerrors in all the oxide concentrations through the calcu-lation.

In this paper, we evaluate several methods of deter-mining the uncertainties in structural formulas and theprincipal controls on error magnification. Our resultsdemonstrate that complete elror propagation includingcovariance terms is advisable for precise work. For manyroutine applications, however, there is little difference be-tween measured uncertainties of oxide concentrations andthe propagated uncertainties ofthe corresponding calcu-lated cation proportions, except for multivalent cationsand cations assigned to more than one structural site.

The basic equation

If some calculated value Yis a function of several mea-sured variables, Y: flxr, xz, '" xn), then the uncertaintyof I is given by the basic equation for error propagation(Bevington, 1969):

o,r: o,.r(0Y/6xt)2 + o2,2(dY/dx)2* 2o,r-r(0Y/6xr) (0Y/0xr) + "'

or (Hahn and Shapiro, 1967)

o'": f f, o*,,,(dY/6x,)(0Y/0x,) (l)i : r j : l

170

GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS t7l

Table 1 contains the definition of all variables. Two as-sumptions underlie the derivation of these expressions:(l) The measured variables and calculated parameters arederived from approximately normally distributed popu-lations, and (2) the error-propagation equation is an ap-proximation based on a Taylor series expansion aboutthe point where each of the variables takes on its meanvalue. The contribution of terms of higher order than thefirst partial derivatives is assumed to be negligible (Rod-dick, 1987).

Both of these assumptions are satisfied by calculationsof structural formulas based on microprobe analyses.Measured oxide concentrations are likely to be normallydistributed because, if a series of replicate measurementsare subject only to random errors, the frequency distri-bution of those values will be Gaussian or normal (Bertin,1980). In addition, instrumental error often is normallydistributed (Hahn and Shapiro, 1967), and a major sourceof uncertainty in microprobe measurements is the count-ing error, which also has a normal distribution (Bertin,1980). Finally, the mean of a parameter dependent onfour or more variables from different populations tendsto be normally distributed even for small values of n(Roddick, 1987, who credited Eisenhart, 1963, and Hahnand Shapiro, 1967).

Roddick (1987) also pointed out that the omission ofthe higher-order terms of the Taylor expansion producesexact errors for linear functions and is usually valid forother functions. It is necessary to verifu that the functionsemployed in calculations of structural formulas are linearwithin the limits of analytical uncertainty of the mea-sured variables. We examined this problem by calculat-ing the contribution ofthe next-higher-order term in theerror-propagation equation to the uncertainties associ-ated with the calculated structural formulas of the threeexamples below. The worst case we found was a higher-order contribution equal to 0.530/o ofthe value calculatedby the basic equation. Therefore, we concluded that thefunctions for calculating structural formulas from oxidedata are linear within the regions of interest.

Previous work

Several approaches to error propagation appear in thegeological literature (Dalrymple and Lanphere , 197l1' An-derson, 1976; Ludwig, 1980; Rees, 1984; Hodges andCrowley, 1985; Hodges and McKenna, 1987; see Rod-dick, 1987, for a complete summary), but, to our knowl-edge, analysis of uncertainties in calculated structural for-mulas in minerals has not been reported. The Monte Carlomethod (e.g., Anderson, 1976) simulates an error distri-bution in the calculated value by large numbers of re-peated calculations. The input parameters for each cycleare derived from random variations in the measured val-ues within the limits of their measured uncertainties.Roddick (1987) presented a standard numerical method,derived from the basic error-propagation equation, thattakes correlation oferrors among variables into accountand incorporates a test for linearity.

TABLE 1. Definition of variables

oi,

cation basis for structural-formula calculationoxygen basis for structural-formula calculationnumber of cations of the fth element in a structural for-

mulanumber of cations of the ith element in cation-basis strue

tural formulanumber of cations of the ith element in oxygen-basis

structural formulanumber of cations in the oxide of the ith elementcounting error: (4)'2formula weight of the oxide of the fth elementtotal Fe: elt': qlt'number of cations of the last element (D to fill a structural

sitenumber of measurementsnumber of counts

2 *'no'= t:-, total number of oxygens on a cation basis

2*'nnumber of oxygen anions in the oxide of the fth elementnumber of cations of element i remaining after element i is

last to fill a sitenumber of cations that can be accommodated in a struc-

tural site

:2 *,tC, sum of mole numbers of cations-".

: Z

**, sum of mole numbers of oxygen anions

weight percent of the oxide of the ,th element: w,lf,lf: fraction of total oxygen anions that reside in

the fth oxide: wlAl€: fraction of total cations that reside in the fth ox-

idekth measurement of the fth variable

: ) x,/n: mean value of the ith variable

partial derivative of xi with respect to v/, evaluated at x4,,all (v&)i+i constant

: o,*lo,g,,i coefficient of correlation between x' and xt

:tll(n - 1)l > (xr - x,,-)(\r - xrJ: covariance of nt l

measurements of x and x: orr: variance, a special case of covariance: (4)'/2: standard deviation of n measurements x,

eF

Fe'lanlt

n

oc

r!

SF

wt

nnXu

Xi,^

AxtlAvt

Appr,rclrroN To cArrcuLATIoNs oFSTRUCTURAL FORMUI,AS

The method of propagating errors proposed herein isto derive analytical formulations that make use of thebasic error-propagation equation (Eq. l). The methodproduces errors that are "exact" provided the assump-tions implicit in the use of the basic equation are valid.

Analytical solutions are specific to individual types ofcalculations because they require partial derivatives ofthe functions by which desired values are obtained. Suchspecific solutions may be desirable for routine proceduressuch as the calculation of mineral formulas from chemi-cal data. Calculations of structural formulas are based onsimple functions having simple derivatives. Consequent-

r72 GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS

ly, the analytical formulation for propagating uncertain-ties of cation proportions is widely applicable and easilyprogrammed.

The formula proportion of a cation is a function of themeasured concentrations of all the oxides, or c,: f(w,,wr, .-. w,).Hence, by Equation I,

n n

o?,:2)o*,*o(0c,/0w,)(Ec,/0w). (2)j : t k=l

Equation 2 reveals that determining the uncertainty as-sociated with a calculated formula proportion of cationsrequires only the covariance matrix for the measured ox-ide concentrations (o,,,o, Table l) and partial deriva-tives of the functions foi the calculated numbers of cat-ions (6c,/0w).If the errors in the oxide concentrations areuncorrelated, the covariance terms can be dropped fromthe equation, and the resulting expression is

. {o;,: D o2r(0c,/6y)2. (3)j : l

The function for calculating the formula proportion ofa particular cation depends upon whether it is normalizedon an oxygen basis (superscript O) or on a cation basis(superscript c). For an oxygen-basis calculation,

In order to propagate the uncertainties in the measuredoxides through the calculation of structural formulas, thederivatives and the elements of the covariance matrixmust be substituted into Equation 2.

In addition to the formula proportions of cations ofeach element, the relative abundances of Fe2* and Fe3*,the distribution of cations among sites, and cation sumsare routinely included in calculations of structural for-mulas. Below, we analyze the functions by which theseparameters are derived and calculate their partial deriv-atives. In all cases, the functions are simply linear com-binations of the functions for calculating formula pro-portions, derived above, and, hence, should also behavein a linear fashion within the ranges of their uncertainties.

The valence of Fe

The electron microprobe cannot discriminate betweenFe2* and Fe3* and the analyst has the option ofreportingall the Fe in a sample as either FeO or FerOr. If total Fe(Fe*) is reported as FeO, the abundances ofFe2* and Fe3*cations can be calculated from the oxygen deficiency thatarises from the attempt to normalize the structural for-mula to a fixed number of oxygens and fixed number ofcations. The oxygen deficiency is the difference betweenbo, the number of oxygens to which the structural for-mula is normalized, and o., the number of oxygens cal-culated on a cation basis. For each missing oxygen, 2FeO's are converted to FerOr, and the extra oxygen "cre-ated" balances the deficiency.

The function for calculating the number of Fe3* cationsis c."r* : 2(bo - o

") (Table I ). Expanding o", this expression

can be written as

The partial derivatives of this function with respect tothe oxide weight percents have the form

.

dcp,,, / 6 w i: za"{t"c#"e)

The number of Fe2* cations is calculated by the difference

CF;2+ : CFe, - Cp&+1 Q)

and the partial derivatives by

6cr.*/0w,: )cr.Jdw, - 6cp"t*/0w,.

The terms on the right-hand side of the derivatives ofEquations 6 and 7 have the form of the derivatives ofEquation 5. The uncertainties associated with the for-mula proportions of Fe2* and Fe3* can be calculated bysubstituting the partial derivatives of Equations 6 and 7and the elements of the covariance matrix of oxide con-centrations into Equation 2.

w,(c7*/f,)bo w,lCbocp : _ - " : : _ * (4 )

The partial derivative of cf with respect to the weightpercent of the oxide of the same cation is given by

)cf/0w,solebo - w,lel€b"

(so)' '

(6)whereas the partial derivative of cf with respect to theweight percent of an oxide of a different cation is givenby

(0cl/0w,),*,: -W

The function for the formula proportion of a particularcation normalized to a cation basis is

w,(cy/f,)b' w,hb.c r : s= : ; .

( 5 )

The two forms of the partial derivatives of this functionare

1ci/0w,:s.lcb" - w,(lt),b"

and

(6q/0w1),*1: -W

GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS t73

Site assignments

The uncertainties associated with formula proportionsbefore assignment of the total Fe to Fe2* and Fe3+ cationsand cations to structural sites are the same as for thegeneral structural-formula calculations described previ-ously. However, major differences arise as a result of dis-tributing cations of a single element between two struc-tural sites. For example, Al is distributed betweentetrahedral and octahedral sites in many minerals.

Cations are assigned to sites in three possible ways. Ifall of the cations of a particular element are assigned toa single site, for example, c5i : c1a1s;, then the derivativesrequired are simply those of Equations 4 or 5.

Ifa cation is the last to fill a site, but the site is filledwithout using all the cations of that element, the functionfor calculating the number of cations of the last speciesassigned to the site is

l , : s - ) c ; ( 8 )

where { is the number of cations of the last species to fillthe site, s is the total number of cations the site can ac-commodate, the cre vzlus5 are numbers of cations calcu-lated on a cation basis, andT includes only species in thesite besides {. The partial derivatives ofthis function aregiven by

01,/0wo

where theT again refers only to cations in the site otherthan 1,. Note that similar functions would also apply tostructural formulas normalized to an oxygen basis, inwhich case c! would be the appropriate term in the pre-vious two equations. These derivatives commonly implythat o1+1a1 : o' for minerals in which all Si is tetrahedrallycoordinated and that durpa: oM4c"in those normalizationschemes that exclude transition-metal cations from theM4 site of amphiboles.

The cations of an element remaining after previous as-signment to another site are usually allocated to only oneother site. For example, the Al cations remaining afterassignment to the tetrahedral site are usually allocated toa single, octahedral site. The function for calculating thenumber of cations assigned to a second site in this man-ner is

r,: c1 - 1,, (9)

where r, is the remaining number of cations of element i.The panial derivatives necessary for error propagationare then

0r,/0wn: )ci/dwr - 01,/6w0.

The terms on the right-hand side of the derivatives ofEquations 8 and t have the form ofone ofthe derivativesof Equations 4-9. Substitution of the appropriate deriv-atives and the covariance matrix of the oxide analvses

into Equation 2 permits the calculation of uncertaintiesin { or r,.

Cation sums

Cation sums commonly are used to judge the qualityof a microprobe analysis and to examine petrological be-havior in discriminant diagrams (e.g., Laird and Albee,l98l). In order to calculate the uncertainty associatedwith a cation sum, it is necessary to calculate the partialderivatives of the function by which the sum is calculat-ed. The partial derivative of the sum is given by dc",-/0w,: 6r"1U*, * 0cr/0w, I 0c"/0w,"' . The terms on theright are the derivatives of one or more of the Equations4-9, and the uncertainty of the cation sum is calculatedby inserting this expression into Equation 1.

MnrHoos

In order to test the eftciency and applicability of theseuncertainty calculations, we analyzed three natural min-erals using the automated wns Cameca sxso electron mi-croprobe at the University of California, Davis. Analysesof 30 s for between 50 and 60 spots were performed oneach ofthree standard natural minerals using other ana-lyzed natural minerals as standards. The operating con-ditions were 15 keV accelerating potential, a spot size ofapproximately 3 pm, and a beam current of 20 nA forboth hornblende and augite analyses and l0 nA for by-townite. Matrix conections were done using a ZAFscheme.

Three standards were analyzed as unknowns: Kakanuihornblende and augite (Jarosewich and others, 1980) anda bytownite collected from a gabbro sill in Crystal Bay,Minnesota. Structural formulas for bytownite were nor-malized to eight oxygens assuming all Fe to be Fe3*. Au-gite structural formulas were normalized to four total cat-ions and six oxygen anions. Hornblende formulas werecalculated on a 23-oxygen basis and by three cation nor-malization schemes as discussed below. Analyses in whichany of the oxides differed by three or more standard de-viations from the mean were discarded as "bad" analv-ses. Average analyses are presented in Table 2.

Rrsur-rsComparison of propagated and rneasured uncertainties

Tables 3. 4. and 5 contain the structural formulas ofbytownite, augite, and hornblende calculated from themean oxide analyses in Table 2 and the uncertainties inthe cation proportions calculated in four ways. ColumnI in each table contains the standard deviations of thecation proportions calculated by a complete error prop-agation (Eq. 2) ofthe uncertainties in the weight percentoxide analyses (Table 2). We tested the validity of thecomplete error propagation by calculating structural for-mulas for each individual spot analysis and determiningthe standard deviations ofthe -50 such cation propor-tions. These estimates (column 4, Tables 3, 4, and 5) arevirtually identical to uncertainties obtained by completepropagation ofthe uncertainties in the weight percent ox-

174 GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS

TrEle 2. Average microprobe analyses of bytownite, augite, and hornblende

Bytownite'n : 5 9

Augite-'n : 5 2

Hornblende.-n : 5 4

E ("/4 1%) 1(Wsio, 48.94TiO, n.d.Alro3 32.13FerO3 0.43CrrO. n.d.FeO n.d.MnO n.d.MgO n.d.CaO 15.22NarO 2.76K,O 0.07

99.29

0.26

0.220.09

o.230.090.02

0.53

0.6820.14

0.16

0.180.35

0.400.834.56

50.160.848.04

0 .156 .180 .15

16.5915.971 .20n.o.

99.12

0.455.920.98

36.282.57

27.490.830.962.96

0.08o.870.18

2.980.542.400.140.190.59

40.204.93

15.02

n.o.10.540.10

12.969.742.402.03

97.92

0.300.150.18

1 .493 . 1 1

32.83

0.450 . 1 10.320.250.09o.o7

0.230.050.08

0.0s0.160.040.140.150.04

0.76 0.153.07 0.591.21 0.25

4.28 0.42102.92 3.50

2.49 0.172.54 0.233.83 0.603.47 0.38

Note: E"(%) : percent counting error: 100(y'n")/n", where 4: counts; n: number of analyses; n.d. : not determined; ry: weight percent otthe oxide; o : standard deviation; 6o/e: pa(@nl standard deviation.

- Total Fe as FerO..-- Total Fe as FeO.

ides (column l, Tables 3, 4, and 5). We conclude that ouranalytical formulations of the error propagation yield acorrect estimate of the true dispersion of the structuralformulas due to analytical uncertainty.

It might be proposed that the uncertainty in the oxideweight percent (Table 2) is an adequate measure of theuncertainty in a cation proportion. Our results show thatthis can be a dangerous assumption in cases for whichthe proportion of multivalent cations must be estimatedor cations must be distributed among structural sites, butotherwise the estimate is very good. The relative uncer-tainties in many cations (column l, Tables 3,4, and 5)are very similar to the relative uncertainties in the cor-responding oxides (Table 2; column 3, Tables 3,4, and5). However, the relative uncertainties of the oxide con-centrations of Al2O3, FeO, and NarO (Table 2) severelyunderestimate the relative uncertainties of tolAl,, tolf, p"z+,

and Fe3* in augite and hornblende and MaNa and ̂ Na inhornblende. Thus, our calculations show that uncertaintyin multisite and multivalent cations cannot be estimated

directly from the uncertainty in the corresponding oxideconcentrations.

Correlation effects

Complete error propagation (Eq. 2) is time consuming,and it would be useful to know if comparable results couldbe obtained by partial error propagation (Eq. 3). Com-plete error propagation (colurnn l, Tables 3, 4, and 5)requires a covariance matrix and takes correlation ofan-alytical errors into account, whereas partial error propa-gation (column 2; Tables 3, 4, and 5) requires only stan-dard deviations of the measured variables and is validonly ifthe analytical errors are uncorrelated.

The uncertainties estimated by partial error propaga-tion are, generally, somewhat higher than those estimatedby including covariance terms (Tables 3, 4, and 5), but afew values are actually slightly lower. If covariance isneglected, the standard deviation of Si is overestimatedby l0-l5o/o (relative), the standard deviations of the Fe3*and Fe2" contents of augite and hornblende are overes-

TABLE 3. Average structural formula* and various measures of analytical uncertainty for bytownite

cations(e) oYo

0.530.68

20.14

SiAIF€F*-.

NaKCa

2.24761.73920.01474.0015

o.24580.00400.74900.99885.0003

0.00690.00910.00300.0064

0.00730.00130.00980.01240.0070

0.310.52

20.410.17

2.9732.501.311.240.14

0.00800.00990.00300.0067

0.00760.00130.01050.01280.0075

0.360.57

20.41o.17

3.0932.501.401.280.15

0.01190.01180.0030

0.00760.00130.01 12

3.1132.831.49

0.0069 0.310.0091 0.520.0030 20.410.0060 0.1s

0.0073 2.790.0013 32.500.0098 1.310.023 1.230.0070 0.14

Notej (1) complete error propagation (Eq.2); (2) partial error propagation (Eq.3); (3) product of fractional standard deviation of oxide and ci; (4)standard deviations of cation proportions calculated from individual spot analyses.

'Cations normalized to 8 oxygens.'t Fe reported as Fe.O".

GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS

TABLE 4. Average structural formula. and various measures of analytical uncertainty for augite

t75

cations(e) o"/o

0.384.04

Si{arAl

16rAlTiFe3+i*

CrMnMgFe2*CaNa

1.82640.1 7362.0000

0.17160.02310.03630.00430.00480.90020.1 5200.62300.08482.0001

0.00610.0061

0.00700.00140.01300.00150.00130.00610.01 170.00520.0024

0.333.50

4.076.01

35.9036.2327.530.67/ .oY0.842.84

0.00700.0070

0.00740.00140.01450.00150.00130.00640.01390.00540.0025

4.295.90

40.1136.2527.46o.719 . 1 2o.872.92

0.00830.0017

0.00170.00140.00090.001s0.00130.00750.00390.00600.0025

0.450.98

0.985.922.57

36.2827.490.832.570.962.96

0.0061 0.330.0061 3.50

0.0070 4.070.0014 6.020.0130 35.940.0010 36.290.0013 27.520.0061 0.670.01 17 7 .690.0052 0.840.0024 2.84

Note: (1) complete error propagation (Eq. 2); (2) partial error propagation (Eq. 3); (3) product of fractional standard deviation of oxide and c/; (4)standard deviations of cation proportions calculated from individual spot analyses.- Cations normalized to 4 total cattons.

*'Fe3* calculated as described in text.

timated by l2-25o/o, and the uncertainty in ANa of horn-blende is overestimated by l7ol0. With the exception ofSi, cations assigned only to one site and having only onevalence state show little difference between the completeand partial error propagations.

If covariance terms are neglected, a small number of"bad" or "outlying" data may lead to either overesti-mates or underestimates of the uncertainty. The bytown-ite data in Table 3 are based on 59 analyses. We repeatedthe error analysis including a single "bad" analysis thatoriginally was excluded because it had a low oxide totaland because the concentrations of many of the oxideswere greater than three standard deviations from themean. The resulting differences between the complete andpartial propagation oferrors are larger when the bad anal-ysis is included than when it is not (Table 3). The singlebad analysis imparts a greater correlation among errorsin the oxides that results in the covariance terms havinga greater contribution to the uncertainty.

The differences between complete and partial errorpropagation are not large, but the relative differences maybe as high as 25o/o. Partial error propagation (Eq. 3) is abetter estimate of the complete uncertainty than the per-cent uncertainty in the oxide concentration for all cationsexcept Si in hornblende (Table 5). Clearly, it is advisableto include the covariance terms (Eq. 2) where high pre-cision is necessary.

Evaluating the quality of microprobe analysesBytownite. A problem facing any microprobe user is

determining what constitutes a good analysis. Obviously,oxide totals for an anhydrous mineral should be nearl00o/0. In addition, stoichiometric constraints on someminerals are well understood. For example, a feldsparanalysis calculated with 8 oxygens should have 5 cations.Moreover, the sum of the trivalent and tetravalent cat-ions in the tetrahedral sites should be 4, the sum of the

divalent and univalent cations should be l, and the num-ber of trivalent cations in the tetrahedral site should beequal to one more than the total number of divalent cat-10ns.

It is possible to decide whether the cation sums areconsistent with stoichiometric constraints by propagatingthe errors in the oxide concentrations through the cal-culation of the structural formula. These uncertainties areshown in Table 3. Within analytical uncertainty, by anyof the methods of propagating it, the sum of the tetra-hedrally coordinated cations is 4, the sum of univalentand divalent cations is l, and the total number of cationsis 5. Furthermore, the diference between total trivalentcations and I plus the number of Ca is only 0.0049, wellwithin the uncertainties of both the sum and the differ-ence either by partial or complete error propagation.Therefore, the plagioclase analysis is consistent with stoi-chiometric constraints within analytical precision.

Commonly, it is necessary to evaluate the quality ofpublished microprobe analyses for which standard devia-tions ofcation proportions are given, but covariances arenot. Partial error propagation (Eq. 2) does not require thecovariance among the oxide concentrations and for cat-ion sums in bytownite yields very good approximationsof the uncertainty. For example, Equation 2 yields anuncertainty of 0.0064 for the sum of Al + Si + Fe3+cations, and Equation 3, 0.0067 (Table 3). Partial errorpropagation also provides good approximations of theuncertainty for the sums Na + K + Ca and the totalnumber of cations in bytownite (Table 3). Consequently,in the absence of covariance information, we recommendusing partial error propagation to obtain the uncertaintiesassociated with cation sums in plagioclase.

In order for an analysis to be considered acceptable,each critical cation sum must lie within uncertainty ofthe value required by stoichiometry. If there are strongcorrelations between two cation sums being used to judge

176 GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS

TABLE 5, Average structural formula' and various measures of analytical uncertainty for hornblende

cations(4)

Sir.tAl

> tet

rqAlFe3+'r

TiMgte.'Mn> M1 ,2 ,3

CaNa> M 4

5.81852 . 1 8 1 5n

0.38000.65910.53662.79520.61620.0128J

1 .51010.4899

0 .18340.37570.5591

0.04510.0451

0.06060.1 2810.01610.06020.12020.0132

0.03730.0373

0.04220 01360.0453

15.9519.453.002.15

19.51103.13

2.47/ . o I

23.01J-OZ

8 . 1 0

0.04940.0494

0.06170.1 5880.01640.05730.14520.0132

0.04020.0402

0.04950.01340.0523

0.852.26

16.2424.093.062.05

23.561 03.13

2.668.21

0.04400.0264

0.00460.02790.01650.06970.02640.0132

0.03840.0188

0.00700.0130

0.761 .21

1 .214.283.072.494.28

102.92

2.543.83

0.782.O7

0.0452 0.780.0452 2.07

0.0608 16.000.1281 19.440.0161 3.000.0605 2.160j204 19.540.0132 103.13

0.0374 2.480.0374 7.63

0.0421 22.960.0137 3.650.0452 8.08

3.833.47

26.993.569.3s

NaK> A

Notej (1) complete error propagation (Eq.2); (2) partial error propagation (Eq.3); (3) product of fractional standard deviation of oxide and c/; (4)

standard deviations of cation proportions calculated from individual spot analyses.. Calculated on a basis of 23 oxygens with cations normalized to 13 excluding Na, K, and Ca'

't Fe3* calculated as described in text.

the analysis, the two criteria are not independent and theformer statement may be a necessary, but not sumcient,criterion for acceptability at a given level ofconfidence.For example, a strong negative correlation between thesum of Na + K + Ca and the sum of Al + Si + Fe inbytownite must exist because the two sums contain allcations and the total cations are approximately constant.For the bytownite we studied,

D - - 0 . 9 1 ." A l + S i + F e ' N a l K f C a

where p is the correlation coefficient. Because of the strongcorrelation, calculation of the probability that both cat-ion-sum criteria are satisfied simultaneously requires notonly the variance of the two sums, but the covariance aswell. The derivation of the such probability functions isbeyond the scope ofthis paper, but the necessary covari-ance terms for the minerals we studied can be calculatedby using the covariance matrices ofthe oxide concentra-tions given in Appendix I, the derivatives of Equation4-9, and Equation 8 of Roddick (1987).

Augite. The stoichiometry of the augite (Table 4) can-not be judged in the same way as that of the bytownitebecause we have chosen to normalize both the oxygenand the cations in the formula unit and all sites are filledwith a specified number of cations. Because the formulaunit was normalized to 4 cations, after allocating enoughSi and Al to fill 2 tetrahedral sites, the remaining 2 oc-tahedral sites must necessarily contain 2 cations.

The calculated number of Fe3* cations (Table 4) is theonly available stoichiometric test of the quality of theaugite analysis. Notice that the calculated number of Fe3'cations is significantly different from zero, despite beingvery small. The calculated value is about 200lo higher than

the value obtained by wet-chemical analysis (Jarosewichand others. 1980; P. Schiffman, 1986, personal commu-nication), but the uncertainty associated with the calcu-lation is about 360/0. The uncertainty is large, in part,because the amount of Fe3* is small, but the method ofestimating Fe3* content is accurate within the limits ofanalytical precision.

Charge-balance criteria that might be used to test thequality of the augite analysis are not independent of theFe3* calculation. For example, the criterion

t't12Tia* + Al3* + Fe3+ + Cr3*l - ['Na + t41Al] : 0

requires that the charge excess created by the substitutionof trivalent and tetravalent cations in the M sites mustbe balanced by substitutions of univalent cations in theM sites and trivalent cations in the tetrahedral sites. Thiscriterion is satisfied exactly for the mean augite compo-sition because either positive or negative Fe3* was cal-culated from each spot analysis in order to achieve chargebalance. Unless the amount of Fe in an analysis is verysmall, it generally will be possible to balance the chargeexactly for all spots, and both the mean and standarddeviation of the charge-balance criterion will be zero.Consequently, the criterion is not a very stringent test ofthe quality ofthe analysis because such charge balance isrequired by the calculation of the structural formula.

For many Fe-rich minerals, failure to meet such charge-balance criteria may imply that negative Fe3* was notpermitted by the calculation procedure. Failure to allownegative Fe3* contents in structural formulas may pro-

duce a systematically biased mean and standard devia-tion of the apparent Fe3*. Consider, for example, an au-gite that contains no Fe3*. Analytical error in the weight

percent of each oxide should lead to a dispersion of thecalculated Fe3* to values slightly above and below zero.Failure to include negative values will bias the mean val-ue of calculated Fe3* to a small positive value and willbias the standard deviation to lower values in such a waythat the small positive mean may appear to be signifi-cantly different than zero. If the mean of several analysesleads to a negative Fe3+ content that is significantly dif-ferent from zero, there is serious, systematic error eitherin the analyses or the assumptions on which the calcu-lation is based. Consequently, it is important to permitnegative Fe3* in the calculations in order to evaluateproperly the significance ofthe Fe3+ calculation.

Hornblende. The hornblende analyses also cannot betreated in the same way as the bytownite analysis becauseall sites except the A site are filled with a specified num-ber of cations. Many normalization schemes have beendevised to calculate structural formulas in amphiboles(see Robinson et al., 1982, for a detailed review), but weexamined only three schemes. In the 13eNI(C scheme,the sum of all cations excluding Na, K, and Ca are nor-malized to 13. This scheme excludes Fe and Mg fromand maximizes Na in the M4 site (Stout, 1972). In the15eNK scheme, the sum of all cations excluding Na andK is normalized to 15. Of the three schemes considered,this one provides the minimum number of Fe3* cationsand forces all the Na into the A site (Stout,1972).

For each normalization scheme, the cations were as-signed to structural sites in the order Si, Al, Fe3*, Ti4*,Mg, Fe2*, Mn, Ca, and Na (Robinson et al., 1982). Whena site was filled, the remaining cations were placed in thenext available site. The eight tetrahedral sites were filledfirst with Si, Al, Fe3*, and Tia* if necessary. The remain-ing cations were placed first into the five Ml, M2, andM3 sites and then into the two M4 sites. All the remain-ing cations were assigned to the A site.

A poor analysis or poor choice of calculation schemeis indicated by more than eight Si cations, fewer thaneight tetrahedral cations or two M4 cations, or by morethan one cation in the A site. A negative number of Fe3+cations is unacceptable, and Robinson et al. (1982) sug-gested that a normalization resulting in Ca in the Ml,M2, and M3 sites is also unacceptable.

The l3eNKC normalization was the only scheme thatyielded an acceptable structural formula for the horn-blende analysis. Notice that the calculated Fe3* (Table 5)is significantly different than zero and that other stoichio-metric constraints mentioned above are met. Moreover,the calculated proportion of Fe3* exceeds the reportedvalue (Jarosewich and others, I 980) by less than one stan-dard deviation of the calculated value. The I 5eNrK schemeyielded -0.88 t 0. 16 Fe3* cations, demonstrating thatthe negative value for Fe3* was not entirely the result ofanalytical uncertainty. Likewise, the I 5eK normalizationscheme required 0.16 + 0.04 Ca cations in the Ml, M2,and M3 sites, and the propagated uncertainty demon-strates that the Ca in these sites is not merely an artifactofanalytical error.

t77

The site charge-balance criterion (Papike et aI., 1974),

["(Na + K) +_r5r(Al + Fe3+ + 2Ti + Cr)(tujAl + *oNa;1 :6,

also is not independent ofour treatment ofthe calculatedFer*. The mean value of the site charge-balance criterionand its uncertainty are zero because all spots analyzedachieved charge balance by means of calculated Fe3*.Consequently, the site charge-balance criterion providesno information that is independent of the Fe3* calcula-tion.

Application to discrirninant diagrams

The structural formulas of minerals are commonly usedas discriminators of petrologic behavior. For example,the covariation of MaNa and t6lAl + Fe3* + Ti + Cr inamphibole was used to discriminate among metamorphicgrades in mafic schist from Vermont (Laird and Albee,I 98 1). Although the data upon which such plots are basedseldom contain rigorous estimates ofthe errors associatedwith the cation sums, the information is critical for eval-ualing their signifi cance.

The microprobe data and uncertainties reported in Ap-pendix I ofLaird and Albee (1981) are an unusual ex-ample of data that permit a detailed analysis. We prop-agated the errors (Eq. 3) in the oxide concentrationsthrough the calculation of the number of MaNa cationsand the sum of t6lAl + Fe3* * Ti + Cr cations for onesample each from the biotite zone, the garnet zone, andthe kyanite-staurolite zone of Laird and Albee (1981).Although the uncertainties generally are larger than thesymbols published in Figure 2 of Laftd and Albee (1981),our analysis demonstrates that analytical error is not asignificant source of scatter on this graph (Fig. l) andsupports the conclusion that the observed variations ofamphibole chemistry reflect real variations in metamor-phic conditions.

Laird and Albee (1981) reported no information aboutcorrelation among the oxides analyzed. Using the datafrom the hornblende analyzed in this study, we attemptedto evaluate the validity of error propagation in the ab-sence of this information by examining uncertainty in thesum. The estimated uncertainty of the sum Al + Fe +Ti + Cr is 0.1120 by Equation 3, without covarianceterms, and 0.0899 by Equation 2, with all covarianceterms. Considering the purposes for which these diagramsare commonly used, the differences in the uncertaintiescalculated by the two methods are very small.

For some applications, such as regression analysis (e.g.,Ludwig, 1980; Roddick, 1987), it is advisable to evaluatecorrelation between calculated parameters. We used thecovariance matrix for our hornblende analyses (App. l)to examine correlation between the number of MaNa cat-ions and the cation sum t6lAl * Fe3+ + Ti + Cr (Eq. 8 inRoddick, 1987). The strong positive correlation betweenthe calculated parameters in this single sample (p : 0.69)might pose a problem for curve fitting or discriminationbetween very similar samples. However, the analytical

GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS

178 GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS

Trele 6, Structural formula, counting errors, and error magnifications for bytownite

Uncertainties based on counting errors Magnifications.

7E (%l(hvp.)

51%)(hvp.)

4

(hyp)

2E.

(meas.)

6flo

(meas.)

3E (%)(meas.)

'|

(cations)

SiAIFe3*

NaK

2.24761.73920.01474.0015

0.24580.00400.47900.9988

5.0003

0.00220.00270.00010.0018

0.00200.00020.00280.0034

0.0020

0.100 . 1 60.370.04

0.814.880.370.34

0.04

0.00400.00490.00010.0018

0.00090.0000o.oo270.0030

0.0021

0.180.280.390.04

0.580.761 . 0

0.s60.871 . 2

1 . 21 . 11 . 1

Total

0.950.990.88

0.370.350.360.30

0.04

Notei (1) 8-oxygen basis structural formula; (2) absolute uncertainty obtained by propagating measured counting error (Table 2; Eq. 3); (3) column2 as percent of column 1; (4) absolute uncertainty obtained by propagating a hypothetical counting enor ol 0.32oh (Eq. 3); (5) column 4 as percent ofcolumn 1 ; (61 o"h trom column 1 , Table 3, based on complete error propagation (Eq. 2), oxide uncertainty from Table 2l (ll E" fA from column 5, 4(o/d in oxide = 0.32.

. Error magnifications calculated as (uncertainty% cation)/(uncertainty% oxide).

uncertainties attached to a single sample are clearly muchsmaller than the size of the fields in which they are en-closed (Fig. 1), and the differences between fields must bereal. Assuming our correlation data (App. l) apply ingeneral, they might be used to estimate correlation amongparameters derived from microprobe analyses of otherplagioclase, augite, or hornblende samples.

Error magnification

The uncertainties obtained by propagating countingerror through the calculations of structural formulas mightbe considered a measure ofthe unattainable "best" struc-tural formula for a perfectly homogeneous mineral grainanalyzed under perfectly stable analltical conditions. Evenif the only source of uncertainty in an oxide analysis iscounting statistics, however, uncertainties in the struc-

tural formula are commonly magnified by factors of upto 10 or 15. For example, counting errors of about 0.50/oofthe concentration ofiron oxide (Table 2) are magnifiedto 2.25o/o and7.260/o of calculated Fe3t in hornblende andaugite, respectively (column 3, Tables 7 and 8).

The way in which uncertainties are magnified dependsprimarily on the composition of the mineral and the nor-malization scheme by which the structural formula is cal-culated. In order to isolate the effects of mineral com-position from other contributions to error magnification,we considered counting errors for a hypothetical analysisin which each element was analyzed for 105 counts. Thisvalue is intermediate between the number of counts com-monly accumulated for major and minor elements andyields a counting error of 0.32010. The hypothetical count-ing errors in the oxide analyses were propagated using

TABLE 7. Structural formula, counting errors, and error magnifications for augite

Unc€rtainties based on countino errors Magnifications'

7E"(1")(hyp.)

66-/0

(meas.)

c

1%)(hypJ

3E"(/"1(meas.)

2

(meas.)

4

(hvp.)

1

(cations)

0.596.4

0.733.6

0 . 1 92.05

1.920.36

18.760.380.370.294.370.310.35

SirltAl

rotAlTiFe3*

MnMgFe"

Na

1.82640.17362.0000

0.1 7160.02310.03630.00430.00480.90020.1 5200.62300.08482.0001

0.00120.0012

0.00130.00020.00260.00010.00010.00110.00250.00110.0005

0.070.72

0.770.867.262.972.400.121.62o.170.58

0.00360.0036

0.00330.00010.00680.00000.00000.00260.00660.00190.0003

4.21 . 0

1 41 . 01 . 00.813.00.880.96

6.01 . 1

581 . 21 . 20.91

1 40.971 . 1

Notei (1) 4-cation basis structural formula; (2) absolute uncertainty obtained by propagating measured counting error (Table 2; Eq.3); (3) column 2as percent of column 1; (4) absolute uncertainty obtained by propagating a hypothetical counting enor ol O.32o/" (Eq. 3); (5) column 4 as percent ofcolumn 1; (6) o% from column 1, Table 4, based on complete error propagation (Eq. 2), oxide uncertainty from Table 2; (7) 4 (%) from column 5, 4 (%)in oxide : 0.32.

'Error magnifications calculated as (uncertainty% cation)/(uncertainty% oxide).

GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS

Tlele 8, Structural formula, counting errors, and error magnifications for hornblende

t79

Uncertainties based on counting errors Magnifications.

1

(cations)

2

(meas.)

3E (/4(meas.)

71V"\(hvp)

5E (%)(hvp.)

4E

(hvp.)

6ooh

(meas.)

5 lrrrAl

t tet

r6tAl

Fe$t lMgF€F+M n

> M1 ,2 ,3

CaNa> M 4

5.81852.1 8158

0.38000.65910.s3662.79520.61620.01284.9999

1 .51010.48992.0000

0.18340.37570.5591

0.00650.0065

0.00630.01450.00310.00450.01260.0005

0.00380.0038

0.00570.00150.0057

0.01 170.01 17

0.00960.02030.00190.00820.01950.0000

0.00540.0054

0.00640.00140.0069

1 34.50.980.864.61 . 0

7.89.81 . 10.919.61 . 0

1 . 12 R

1 11 . 1

0.110.30

1.642.250.580.162.003.49

0.250.77

3.090.401.08

1 . 01 . 7

0.200.53

o.621 . 7

0.972.0

6.01 . 0

NaK> A

2.483.120.35o.293.060.32

0.361 . 1 2

3.470.361.23

/Vote: (1) 13e/VKabasis structural formula; (2) absolute uncertainty obtained by propagating measured counting error (Table 2; Eq. 3); (3) column 2as percent of column 1; (4) absolute uncertainty obtained by propagating a hypothetical counting enor ol 0.32o/" (Eq. 3); (5) column 4 as p€rcent ofcolumn 1; (6) o% from column 1, Table 5, based on complete error propagation (Eq. 2), oxide uncertainty from Table 2; (7) 4 (%) from column 5, 4 (7din oxide : 0.32.' Error magnifications calculated as (uncertainty% cation)/(uncertainty% oxide).

Equation 3, which does not include covariance terms andassumes that errors are uncorrelated.

Columns 4 and 5 (Tables 6,7, and 8) contain the ab-solute and relative uncertainties obtained by partial errorpropagation (Eq. 3) ofthe "hypothetical" 0.320lo countingerrors in the oxide concentrations. Column 7 lists thecorresponding error magnification, measured by the ratioof the percent standard deviations in the cation and thecorresponding oxide (cation/oxide). The uncertainties inmost single-site or single-valence cations (e.g., column 5,Table 6) are comparable to the assumed counting errorof 0.32o/o, and the errors in cation sums are much lessthan in the individual cations. On the other hand, theuncertainty in cations of major elements such as Si com-monly is smaller than 0.320/0, and the uncertainties ofmultivalent and multisite cations are much larger (col-umn 5, Tables 6, 7, and 8).

For the hypothetical analyses in columns 4 and 5 (Ta-bles 6, 7, and 8), the magnification of uncertainty in asingle-site, single-valence cation compared to the corre-sponding oxide (column 9, Tables 6, 7 , and 8) is entirelythe effect of the mineral composition and normalizationscheme. Although covariation and variable uncertaintiesin the oxide concentrations affect such magnifications ina real analysis, the uncertainties reported in columns 4and 5 and the magnifications in column 7 were calculatedusing Equation 3, which contains no covariance terms,and the same relative uncertainty for all oxides. Conse-quently, columns 5 and 7 illustrate only the efects of therelative abundances of the oxides on the error propaga-tion for the chosen normalization scheme.

Equation 3 can be rewritten to show the dependence of

error magnification on composition for hypotheticalanalyses in which the relative uncertainties are the samein all oxides (See App. 2 for the derivation). For the spe-cial case in which o./w, : o.,/w, for all i and, j,

,-*@,:lu-x?),*>tnf". (ro)

The first term on the right side of Eq.uation l0 is themain control on the error magnification. That term is theproportion of total oxygen that resides in all the oxides 7not equal to i. The effect ofthis term is best understood

0 . 5 l 0 t 5 2 0 2 . 5

HAI * F"3'* Ti * c,

Fig. l. Propagated uncertainties in graph of MoNa versus I6rAl+ Fe3+ + Ti + Cr in amphiboles from mafic schist in Vermont.The dot-dash line encloses samples from the biotite zone; thesolid line, samples from the garnet zone; and the dashed line,samples from the staurolite-kyanite zone. One-sigma uncertain-ties obtained by partial error propagation (Eq. 3) of reporteduncertainties in oxide concentrations are shown for a represen-tative sample from each zone. Modified after Fig. 2b in Lairdand Albee (1981).

180 GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS

by considering an example. The bytownite (Table 2) con-tains about 49 wto/o SiOr, which contains about 560/o ofthe total oxygen anions in the analysis. IfSi is consideredto be the lth cation, the first term in Equation 10 is about(0.44)'z: 0. 19. The summation term in Equation l0 isalways a small positive number less than one because theaverage value of the five terms (Xre) is about 0.09, whichmust be squared, and the sum is about 0.04. Consequent-ly, the error magnification is commonly less than one forSi and other cations of major elements.

The uncertainty and error magnification in the formulaproportion ofSi are significantly larger for a fixed-cationnormalization than for a fixed-oxygen normalization. Forformula proportions normalized to a fixed number of cat-lons,

( l l )

where the X; terms are the fraction of all cations that arecontributed by the oxide i (Table l). The derivation ofthis equation follows from Equations 3 and 5 is parallelto that presented in Appendix 2.

The larger error magnification for Si arises primarilybecause SiO, contains more anions than cations. For ex-ample, the augite (Table 2) contains about 50 wto/o SiOr,accounting for about 610/o of the anions and 460lo of thecations. If Si is the fth cation, the first terms on the rightside of Equations l0 and I I are, respectively, (0.39F :0.15 and (0.54y :0.29. Because the second terms on theright tend to be small in both cases, the error magnifica-tion is larger for the fixed-cation normalization than forthe fixed-oxygen scheme. The formula proportion of Siin the augite for the fixed-cation normalization scheme is1.8264 + 0.0061 (Table 4, column l) whereas for a six-oxygen basis, we calculated 1.8320 + 0.0044, or a differ-ence in the standard deviations of 30-500/0. For cationsother than Si, the differences in the standard deviationsand error magnifications obtained from fixed-cation andfixed-anion normalizations are less pronounced and, forthe most part, insignificant.

The error magnification is normally greater than onefor minor elements in the hypothetical analyses (columns5 and 7, Tables 6, 7, and 8). If the fth oxide is only Iwtolo of the total sample, the proportion of oxygen thatresides in other oxides is commonly more than 990/o sothat the first term on the right side of Equation I I isabout (0.99)'? : 0.98. The summation term is small butlarger than the case for which Si is the fth cation becausethe average value of Xr9 now must include contributionsfrom SiO, that are larger than those from other oxides.Consequently, the right-hand side of Equation 10 is com-monly larger than one for minor elements, and the un-certainty in the oxide is magnified in the formula pro-portion ofcations.

The error magnification for minor amounts of calcu-lated Fe3* is very large. FeO is a major oxide in both

augite and hornblende, yet the propagated uncertaintiesof Fe3* (columns 5 and 7, Tables 7 and 8) are about 58and 10 times larger, respectively, than the assumed 0.320loerror in the weight percent of FeO. The small, calculatedamount of Fe3* and the error magniflcation depend onthe analyses ofall oxides (Eq. 6). From Equations 3 and6, we find that the error magnification for Fe3* is givenby

o "ru-/

cr,,,

o nrn/ won

The first term on the right side of Equation l2 dominatesfor all silicates poor in Fe3*. From Equation 6, it can beshown that the ratio so/s" approaches bo/b" as the amountofcalculated Fe3* approaches zero and the first term mustbecome very large. The second term on the right is alwaysa small number because the mole fractions are between0 and I and their differences are squared. For example,the first term for the augite is about 322 and the secondterm is about 0.18.

It is widely appreciated that the uncertainty in calcu-lated Fe3' depends heavily on the silica analysis,but allmajor elements and some minor elements may contrib-ute in a major way depending upon the relative sizes ofthe derivatives and variances in Equation 3. For exam-ple, the observed uncertainty of wto/o SiO' in the augitestudied here (Table 2) contributes

lo'^,o, (lcr.,-/ }lt ,o,)'l-only about 330/o of the total variance in Fe3*-whereasMgO, CaO, and FeO contribute 260/o, l7o/o, and I lol0,respectively, and NarO contributes about l2o/o despite itslow abundance. This large contribution arises, in part,because the partial derivative ofFe3t cations with respectto wto/o NarO is about four times larger than the deriva-tives with respect to other oxides.

These calculations show that whether errors are mag-nified or reduced is primarily a function of the compo-sition of the mineral for a given normalization scheme.With few exceptions, the complete analysis (column 6),including the measured and variable uncertainties of theoxides and covariance terms, produced the same direc-tion of magnification or reduction of errors as column 7,which included no covariance and constant hypotheticaluncertainties in the oxides. Error magnification for cal-culated amounts of multivalent cations may be extremeand may depend in a significant way on oxides other thansior.

CoNcr,usroNs

The analytical method of propagating uncertainties isa useful method for calculations of structural formulasthat correctly reproduces the true dispersion of structuralformulas due to analytical uncertainty. For many roulineapplications, our results show that the uncertainties ofmeasured oxide concentrations can be reasonable esti-mates of the complete uncertainties for single-valence,

:lf,:- '] '[?

6, - x?Y) (,2)

ffi:[,'-"r' *i,rn'),

GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS 1 8 1

single-site cations. However, partial error propagation (Eq.3), which omits the covariance terms, is not a difficultcalculation and is a better approximation of the completeuncertainty for virtually all cations. It is advisable to in-clude the covariance terms in the calculated uncertaintiesof Fe2*, Fe3*, multisite cations, or in any case for whichhigh precision is needed.

Uncertainties in structural formulas are commonlymagnified or reduced compared to the uncertainties inthe corresponding oxide concentrations. The error mag-nification depends not only on the composition of themineral, the normalization scheme used, and the struc-tural formula to which the cations are assigned, but alsoon the relative sizes ofuncertainties in the oxides and thecontribution of covariance. Error magnifications com-monly depend on the analyses of all elements. For ex-ample, the uncertainty of Fe3* in the augite studied herecontains major contributions not only from SiO, but alsofrom the other major oxides and NarO. Consequently, ananalyst wanting to minimize uncertainties in a particularapplication may determine the dependence of the errormagnification on the various elements in order to find anoptimum analytical scheme.

Microprobe analysts commonly use stoichiometric cri-teria as indicators ofthe quality ofan analysis. The error-propagation technique outlined here permits a rigorousevaluation of how well a structural formula meets suchcriteria for cases where information concerning correla-tion is available. The method also establishes a means ofevaluating the quality of microprobe analyses in the lit-erature for which laboratory reproducibilities are the onlyreported uncertainties. Finally, calculated uncertainties ofcation sums can also be used to evaluate the effect ofanalytical error on discriminant diagrams. The chemicalvariations of calcic amphiboles on the graph of MaNa ver-sus t6lAl * Fe3+ + Ti + Cr, illustrated by Laird and Albee(1981), are clearly larger than any effects of analy'ticaluncertainty, supporting their conclusion that the varia-tions represent real differences in metamorphic condi-tions. Covariance among parameters derived from a sin-gle microprobe analysis can be obtained by inserting thederivatives ofEquations 4-9 and covariance matrices ofoxide concentrations into Equation 8 ofRoddick (1987)for applications where such correlation is ofinterest.

AcxNowlnncMENTS

This paper is based on work submitted as part ofthe Ph.D. dissertationof M.J.G. Jo Laird kindly provided us with detailed analytical data onamphibole analyses. We are grateful for critical comments by A. A. Fin-nerty and P. Schiffman on early drafts of the manuscript and a helpfulreview by J. C. Roddick. This work was supported by NSF grants EAR85-1 5890 and EAR87-20368.

RnrpnrNcns crrnoAnderson, G.M. (1976) Enor propagation by the Monte Carlo method in

geochemical calculations. Geochimica et Cosmochimica Acta, 40, 1533-l 538 .

Bertin, E.P. (1980) Introduction to x-ray spectrometric analysis, 485 p.Plenum Press, New York.

Bevington, P.R. (1969) Data reduction and error analysis for the physical

sciences, 336 p. McGraw-Hill, San Francisco.Dalrymple, G.B., andLanphere, M.A. (1971)4fu/'Artechnique of K-Ar

dating: A comparison with the conventional technique. Earth and Plane-tary Science Letters, 12, 300-308.

Eisenhart, C. (1963) Realistic evaluation ofthe precision and accuracy ofinstrument calibration systems. Journal ofResearch ofthe Natural Bu-reau ofStandards C. Engineering and Instrumentation, 67C, 16I-1E8(not seen: extracted from Geochimica et Cosmochimica Acta, 5 l, 1987).

Hahn, G.J., and Shapiro, S.S. (1967) Statistical models in engineering,355 p. Wiley, New York.

Hodges, K Y., and Crowley, P.D. (1985) Error estimation and empiricalgeothermobarometry for pelitic systems. American Mineralogist, 70,702-709.

Hodges, K.V., and McKenna (1987) Realistic propagatiou of uncertaintiesin geologic thermobarometry American Mineralogist, 72, 67 l-680.

Jarosewich, E., Nelen, J.A., and Norberg, J.A. (1980) Reference samplesfor electron microprobe analyses. Geostandard Newsletter, 4, 4347.

Laird, Jo, and Albee, A.L. (1981) Pressure, temperature, and time indi-cators in mafic schist: Their application to reconstructing the polyme-

tamorphic history of Vermont. American Journal of Science, 281, 127-l 7 5 .

Ludwig, K.R. (1980) Calculation of uncertainties of U-Pb isotope data.Earth and Planetary Science I€tters, 46,212-220.

Papike, J.J., Cameron, K.L., and Baldwin, K. (1974) Amphiboles andpyroxenes: Characterization of other than quadrilateral components.Geological Society of America Abstracts with Programs, 6, 1053.

Rees, C.E. (1984) Error propagation calculations. Geochimica et Cos-mochimica Acta. 48. 2309-2311.

Robinson, Peter, Spear, F S., Schumacher, J.C., Laird, Jo, Klein, Cornelis,Evans, B.W., and Doolan, B.L. (1982) Phase relations of metamorphicamphiboles: Natural occurrence and theory. In Mineralogical Societyof America Reviews in Mineralogy, 9B,l-227.

Roddick, J.C. (198?) Generalized numerical error analysis with applica-tions to geochronology and thermodynamics. Geochimica et Cosmo-chimica Acra, 51, 2129-2135.

Stout, J.H. (1972) Phase petrology an mineral chemistry of coexistingamphiboles from Telemark, Norway. Journal ofPetrology, 13, 99-146.

Mexuscrrsr RrcErvED JeNuanv 17, 1989Mexuscnrsr AccEprED Sesrevuen 28, 1989

Apppunrx

APPENDTX Trar-e A1.

1. CovlnrlNcE MATRTcEs

Correlation matrix for bytownite analysis

Naro Alros sio, Ito CaO FerOo

NaroAl20osio,KrOCaOFerO.

1.000.11 1 .000.26 0.210.02 -0.280.14 0 .180.00 -0.22

1.00-0.12

o.24- 0 . 1 0

1.000.000.0s

1.00-0.2'l 1 .00

t82 GIARAMITA AND DAY: CALCULATIONS OF STRUCTURAL FORMULAS

Apperorx TABLE A2. Correlation matrix for augite analysis

Na.O Mgo Al203 sio, CaO Tio, C12O3 MnO FeO

NaroMgoAlrossio,CaOTio,

MnOFeO

1.000 0 1

-0.070.280.160.16

-0.080 . 1 50 1 3

1.000.120.170.07

-o.2'l0 . 1 5

-0.36o.24

1.000.29

-0.050 . 1 80 . 1 80.230.07

1.000 . 1 5

-o.24-0.12-0.05

0.21

1.00-0.04

0.60-0 .11

0 . 1 1

1.00-0.09-0.01-0.30

1.000 0 6

-0.041.00

-0.09

Apperorx TABLE A3. Correlation matrix for hornblende analvsis

Tio,Naro Mgo Al203 sio, KrO

1.00-0.06 1.00

NaroMgoAl,o3sio,KrOCaOTio,MnOFeO

1.00-0.47

0.150.180.16

-0.24-0 .13-0 03

0 . 1 0

1.00-0.12

0.01-0.20

0.400.02

,0.05- 0 . 1 8

1.000.310.02

-0.12-0.02- 0 1 6

0.04

1.000.04

-0 04-0.09-0.30-0.03

1.00- 0 .14-0.21

0.220.o2

1.00-0.01-0.'t7

0.05

1.00-0.04

0.12

AppnNnrx 2. Ennon MAGNTFTcATToN

The calculated abundance ofa cation, ci, has a variancethat is given by Equation 3, in the absence ofsignificantcovanance terms:

o?,: 2 o2n@c10w,)2. (3)

We are interested to know how the ratio o,/c, compareslo o*,/w, for an oxygen-basis calculation. Dividing Equa-tion 3 by cf, we have

Substituting for c,, 6c,/dw,, and 0c,/0w, from Equations 4and canceling terms,

i:e{':__y4)' . 2"r($)'

For the case in which o.,/w,: o*/wj for all i and j, andtaking the square root,

( l0 )

Equation 10 clearly shows that the error magnification ofa cation i depends on the proportion oftotal oxygen thatresides in the oxide ofthat cation.

#fr :f' =u**,' . yr 2, (A,, $)'

ffi:[,r - "r, * >, rnf"'*:'*(#)' . +2"r(#)'