American Mineralogist, Volume 68, pages 287-306, 1983

Quantitative characterization of site-occupancies in minerals

F. C. HawrHoRNE

Department of Earth Sciences, University of ManitobaWinnipeg, Manitoba, Canada R3T 2N2

Abstract

Site-occupancies in minerals may be derived by a variety of experimental methods. Theuse of these methods is briefly reviewed here and particular attention is paid to theproblems and limitations associated with the derivation of quantitative site-occupanciesfrom experimental data. Diffraction methods are the most highly developed of thesetechniques, and for two species disordered over two or more sites can yield results theprecision of which is dependent on the difference in the scattering powers of the atomsinvolved. Mdssbauer spectroscopy is used primarily for Fe2* and Fe3* site-occupancydeterminations. The precision of the results depends on the resolution of the spectrum;with decreasing resolution, parameter correlation becomes a problem in the spectrumrefinement and precision is strongly decreased. Hydroxyl-band infrared spectroscopy hasbeen used for site-occupancy determinations in OH-bearing minerals. However, thismethod is affected by clustering, and if it is to be used for quantitative site-occupancydeterminations, either the absence of clustering has to be demonstrated or the method hasto be further developed to allow for clustering. There has recently been considerabledevelopment in the use ofelectronic absorption spectroscopy for qualitative site-occupan-cy studies. Some quantitative work is now appearing, and there is considerable potentialhere for further development. Other methods used for qualitative or semi-quantitativework-'pray diffraction, X-ray photoelectron diffraction, X-ray and neutron diffractionpowder profile-refinement, nuclear magnetic resonance spectroscopy, electron paramag-netic resonance spectroscopy-have potential for fully quantitative work, but moredevelopment is required. Each method has its own area of particular usefulness, and thesemethods should be viewed as complementary rather than mutually exclusive.

Introduction

In the past twenty years, much information has becomeavailable concerning long-range atomic ordering in thecrystal structures of minerals. This has been accompa-nied by investigations into the relationships betweenmineral composition, structural state, physical propertiesand ambient conditions of crystallization and/or equilibra-tion. The experimental techniques by which such order-ing is characterized have also undergone considerabledevelopment during this period. Conventional diffractiontechniques have been refined, and are now capable ofproducing results of high precision. A number of spec-troscopic techniques have been applied to the character-ization of ordering in the past fifteen years, and some ofthese methods have been developed until they are nowcapable of giving quantitative results of high precision.

All of these methods have inherent limitations thatpertain to the experimental method; most diffractiontechniques require reasonably large, good quality crys-tals, many spectroscopic methods are not applicable tospecific atomic species of interest, e/c.; these limitationsare quite well-known. When such techniques are used to

0003-004x/83/0304-0287$02.00 287

quantitatively determine ordering, there are other inher-ent problems associated with derivation of site-popula-tions from the experimental results. It is these problemsand the techniques that have been developed to deal withthem that are the subject of this review. For the sake ofcompleteness, most of the common methods of site-occupancy characteization will be briefly discussed.However, I will concentrate on those techniques that arein general use, because it is for these techniques that theproblems associated with quantitative site-populationderivation have been examined in detail. Experiencesuggests that similar problems will be encountered inother techniques when they are used extensively for site-population characterization.

Diffraction methods

Dffiaction

The unit cell of a crystal contains atoms in variouspositions, and electro-magnetic radiation (with a wave-length comparable to lattice spacings) scattered fromdifferent atoms will have phase differences with respect to

288 HAWTHORNE: OUANTITATIVE CHARACTERIZATION OF SITE.OCCUPANCIES

each other. During Bragg diffraction, the amplitude F1,11of the scattered wave is given by

Fnn : 51 exp 2zri(lrx1 + lcyl + lz)

exp (-Bj sin2 0/tr2) (l)

where S.; is the scattering factor at the jth site in the cell, x.;are the positional coordinates ofthejth site, and Bi is theisotropic temperature factor, a function ofthe vibrationaldisplacement of the atom(s) at the jth site.

In a standard diffraction experiment, a large number ofstructure factors (lF1,pl values) are measured; this numberis generally well in excess of the number of variableparameters of a crystal structure, and consequently theseparameters may be determined from the structure factors.Examination of Equation (l) shows these parameters tobe: (1) scattering factors at the n sites in the unit cell, (2)positional parameters for the n sites in the unit cell, (3)vibrational parameters at the n sites in the unit cell,together with a scale factor that relates the observed datato the calculations of Equation (1). In most structures, thepresence of symmetry significantly reduces the number ofvariable parameters. More realistic models for the ther-mal vibration of atoms may be used, but this does notaffect the subject of this paper. In general, the chemicalcomposition of the crystal used in the diffraction experi-ment is known; thus the scattering factors are not general-ly considered as variable once the atoms have beencorrectly identified. The exception to this occurs whenatoms are disordered over more than one crystallographi-cally unique site, when the scattering factors at these sitesare composite averages of all the atoms occupying thesesites in the crystal. With the assumption that all atoms ata site have the same positional coordinates and thermalparameters, equation (l) may be modified to

exp (-Bj sin2d/L2; (2)

where n is the number of occupied sites in the cell, a1i arethe site-occupancies of the jth site by the ith atom, anO I isthe scattering factor of the ith atom. It is instructive to re-write Equation (1) in this way as it emphasizes that asingle diffraction experiment can only determine thecomposite scattering from each site in the structure.

A vacancy may be considered as a scattering specieswith a scattering power of zero. Thus a site in a crystalcan always be considered to be fully occupied by scatter-ing species, even if it is not fully occupied by atoms. Asingle diffraction experiment gives the total scatteringpower S; at each occupied site in a crystal structure:

with a1.; subject to the constrairrt that the site be complete-ly occupied by scattering species:

?uu: tAs we are primarily concerned with the derivation of

site-occupancies, the form of the scattering factorf is ofconsiderable interest. Most site-occupancy studies aregenerally carried out with X-rays or thermal neutrons.For X-rays, the scattering factor/is a complex functionof sin d/tr (Fie. l). For neutrons, the scattering is indepen-dent of sin 0/I. For neutron scattering, equation (3) hassolut ionsfor i :2and j : | - -+ monly; wheni > 2 ( threeor more scattering species distributed over two or moresites), there is no general solution for a single neutrondiffraction experiment. For X-ray diffraction, the relativedecrease of scattering power with increasing sin 0/)rdiffers slightly between scattering species. Thus equation(3) becomes a set of N equations, where N is the numberof unique reflections recorded in the X-ray diffractionexperiment. In principle, these equations can be solvedfor any number of scattering species (n) distributed overany number of sites (m). Hawthorne and Grundy (1977)used this idea to refine the occupancy of three scatteringspecies over three sites in an amphibole; they obtainedextremely large standard deviations using a full-matrixmethod, and concluded that it is impractical to derivesite-occupancies of more than two scattering speciesdistributed over two or more unique sites in a crystal.

Site-occ upancy derivation

In the earliest studies concerned with site-occupancies,an overall temperature factor was applied to all atoms inthe structure. On Fourier projections, site-occupancieswere evaluated by electron counts or peak height valuesat the sites involved in the disorder (e.g., Whittaker,1949). An alternative method involved the calculation ofR-factors for several possible alternative distributions ofcations, generally for a subset of the structure factor datathat was particularly sensitive to the site-occupanciesadopted (Steinfink, 1962;FanelI et al.,1963). Later, site-occupancies were determined by manual adjustment ofthe scattering factors such that the isotropic temperaturefactors at the sites involved were approximately equal(Ghose and Hellner, 1959; Ghose, 1961). There were twomain drawbacks to this method. The solution was not aleast-squares solution and least-squares standard devi-ations could not be assigned to the derived site-popula-tions. Second, the assumption that the isotropic tempera-ture factors at the sites involved in the atom disorder areequal is not necessarily valid. Burnham (1962) reviewedseveral factors upon which the magnitude of the isotropictemperature factors depend; such aspects as coordinationnumber and degree of cation (or anion) substitutionaldisorder can significantly affect isotropic temperaturefactors.

(4)n

s2j : r

rn*, : j (i t.r) exp 2m(hx; + kyj + tz,J)

s: : ) f ial (3)

HAWTHORNE: QUANTITATIVE CHARACTERIZATION OF SITE-OCCUPANCIES 289

occupancies to be equal to the total site-group chemistryindicated by a chemical analysis (Finger, 1969b):

b.;ar; : C1 (6)J

o' ' 'o" i '^ '"^ i '

ot '

Fig. l. X-ray scattering factors as a function of sin d/L forsome mineralogically important cations.

A more satisfactory solution would involve the directdetermination of site-populations from least-squares re-finement of diffraction data. This technique was proposedby Fischer (1963, 1968), who suggested two methods bywhich this could be done: (l) multiplying the scatteringcurve of an atom by a variable that is a refinable parame-ter in the least-squares procedure; (2) expressing thescattering power of the atoms occupying a site by theweighted sum of the individual scatterers; thus

f-: ar^fr * az^fz (5a)

f^: at^fr + (l - a1j/2 (5b)

Fischer (1966) used equation (5a) to determine the Fe-Mg site-occupancies in a cummingtonite originally refinedby Ghose (1961). There were considerable correlationproblems and the final solution had to be "normalized" tothe known cell content. Burnham and Radoslovich (1963)used equation (5b) to successfully determine K-Na occu-pancies of the interlayer cation position in coexistingmuscovite and paragonite.

In a refinement of the structure of grunerite, Finger(1969a) found that when the site-occupancies of the fourM sites were refined, the sum ofthe refined site-occupan-cies did not agree with the bulk chemistry of the crystalcalculated from a chemical analysis. This situation wasalso observed in other cases (e.g., Burnham et al., l97l).In Equation (1), Si and exp(-B1sin20/)r,2) are of similarform. As F6p1 is the sum of products of these functions,they are highly correlated and the presence of sin 0/\dependent systematic error in the diffraction data inducessystematic error in both S1 and 81. This general problemwas circumvented by constraining the sum of the site-

where bi is the multiplicity of the jth site and C; is the cellcontent of the itn scattering species.

Finger (1969a) applied this method to site-occupancyrefinement of grunerite, obtaining considerably reducedvariable correlation and rapid convergence of the refine-ment. This constraint procedure was incorporated in theprograms nnne (Finger, 1969c) and r.nrNe4 (Finger andPrince, l97O and has since been used in the refinement ofsite-occupancies in numerous minerals.

Constraints of the type indicated in Equations (4) and(6) may be thought of as rigid constraints, as the con-strained parameters are forced to exactly obey the con-straint equation. In the case ofEquation (4), this is a validassumption. For Equation (6), it is not as the cell content(CJ of the i'n scattering species is generally an experimen-tally derived quantity and its exact value is not known. lnaddition, the diffraction data also contains information onthe bulk composition of the crystal. Both these factorscan be given weight in the refinement if the method ofWaser (1963) is used to apply the constraints (Rollet,1970). In this procedure, the constraints are treated inexactly the same way as the observational equations, andthe suitably weighted squares of the residuals resultingfrom the constraint equations are incorporated in theminimization function

w i A 4 +

where AF1 is the difference ofthe observed and calculatedstructure factor for the i'n reflection, and App is thedi-fference between the calculated constraint parameter cn(calc.) and its observed value c. (obs.). The weightingparameter wk reflects the confidence with which thevalue of cn (obs.) is known; thus wp should be someinverse function of the standard deviation of c., withexact constraints having very high weight. However, therelationship between wi and wp (their relative magni-tudes) is not all that clear. Where a bulk chemicalconstraint is only approximate, an overall charge-balanceequation may be required to ensure that the overallformula remains electrostaticallv neutral.

Precision

The precision with which site-occupancies can be mea-sured depends on the difference in scattering factors ofthe atoms involved. X-ray scattering factors are a func-tion of atomic number. Thus it is easy to distinguishscatterers of disparate atomic number (such as Mg and Feor Al and Fe) but ordering of atoms that are adjacent ornear-neighbors in the periodic table (Al and Si, Fe and

oI Uo

o

oL

X

) *u Ap? (7)M:>I

o

o

N

N

2% HAWTHORNE: OUANTITATIVE CHARACTERIZATION OF SITE-OCCUPANCIES

X - r o y s c o t t e r i n o f o c t o r +, N

o o o

3' -Neul ron sco t te r inn . rTr . - r " . r ,on i lO

- t6c .>

Fig. 2. A comparison of X-ray (o) and Neutron (O)scattering factors as a function of atomic number for somemineralogically important cations; X-ray scattering factors arecalculated for sin 0/)r : 0.30.

Mn) cannot be reliably characterized in this fashionwithout resorting to special techniques. Using bulk chem-ical constraints of the form of Equation (6), Mg/Fe site-occupancies are routinely determined with assigned stan-dard deviations of -0.005 atoms. For AVSi ordering,assigned standard deviations are generally >0.1 anddirect determination of site-occupancies by normal X-raystructure refinement is not practical. At high values of sin0/\, the relative difference in X-ray scattering factorsbetween Al and Si increases, reaching -157o at sin 0/\ -0.8-0.9. Fischer (1965) first proposed the determinationAl/Si ordering by the refinement of high-angle X-ray data;Fischer and Zehme (1967) used this method for micro-cline, and Kosoi e/ al. 0974\ used it for aluminousorthopyroxenes, quoting a (root?) mean square error of0.04; both studies gave results that were in accord withthe observed mean bond lengths. Conversely, Brown e/al. (1974) were unsuccessful in attempts to refine Al/Sisite-populations in sanidine using high-angle X-ray data(Phillips and Ribbe, 1972); they also noted large correla-tions (-0.87) between the occupancy parameters and theanisotropic temperature factor coefficients, suggestingthat refinement of high-angle X-ray data is not an ade-quate method for characterization of AVSi ordering infeldspars. Direct determination of Al/Si site-occupanciesby refinement of high-angle data is probably now feasibleusing current state-of-the-art crystallographic techniques;however, the precision of such a determination is notknown.

For neutron scattering, the scattering length of theelements increases gradually with increasing mass num-ber, but is also subject to irregular fluctuations (Fig. 2).For most elements, the scattering length is positive, but

for a few (e.9.,H, Ti, Mn) it is negative. The significantdifferences in scattering lengths between some elementsadjacent in the periodic table allow site-occupancy char-acterization that is not feasible by direct X-ray methods.Although the number of instances where this is possible islimited (Fig. 2), the fact the Ti and Mn have negativescattering lengths means that relative ordering of Fe-Tiand Fe-Mn can be very precisely characterized by neu-tron diffraction. This has not yet been exploited to anygreat extent, and thus the precision attainable on aroutine basis is not yet known. The neutron scatteringlengths of Al and Si differ by -20%. This difference hasbeen used successfully by Brown et al. (1974), Pince etal. (1973) and other workers to successfully refine AVSisite-occupancies to an adequate degree ofprecision (o -

0.02-0.03) using constrained site-occupancy refinementtechniques.

The standard deviations quoted above are routinelyobtained in current site-occupancy refinements. Howev-er, these values may be somewhat misleading, as they arecalculated with the assumption that the crystal composi-tion is known exactly. In many cases, the precision of thebulk constraint calculated from the chemical analysis ismuch less than the precision of the site-occupanciesdetermined. More reasonable standard deviations shouldbe obtained using "soft" constraints, as in equation (7),but this has not yet been used extensively.

Disorder of three or more species

In order to obtain an exact solution to equation (3) for i) 2, more than one experiment is necessary. In general,for i species there must be (i-1) values of S;; that is, (i-l)diffraction experiments or a total of (i-1) diffraction ex-periments plus externally derived site-occupancies. Thevalue of such combined studies has been emphasizedpreviously (Burnham et al.,l97l; Hawthorne and Grun-dy, 1977).

In a well-refined crystal structure, the mean bondlengths of coordination polyhedra are generally deter-mined to quite high precision. Such mean bond lengthsare related to the size ofthe cations and anions constitut-ing the coordination polyhedron, and thus information onsite-occupancy may be obtained. This situation is compli-cated by the fact that the factors controlling the variationof mean bond lengths in crystals are not completelyunderstood despite considerable effort in this area (Shan-non, 1975, 1976; Brown and Shannon,1973;Baur,1974,1978). Thus simple comparison of ionic radii sums cannotbe expected to yield accurate answers except in the moreextreme and/or fortuitous cases. However, many studieshave shown that in groups ofisomorphous structures, themean bond lengths of individual coordination polyhedramay be related fairly precisely to the mean ionic radii ofthe constituent cations (Novak and Gibbs, 1970; Brown,1978; Cameron and Papike, 1980; Hawthorne, 1981a).This is presumably the result of the fact that the ligancy

HAWTHORNE: QUANTITATIVE CHARACTERIZATION OF SITE-OCCUPANCIES 291

and local environment of the coordination polyhedronremain reasonably constant across the series of struc-tures, and the polyhedron responds linearly to the varia-tions in cation size without any other factors affecting thestereochemistry. Where such relationships are well-char-acterized, they can be used to derive site-occupancies orcontribute to site-assignments in structure refinements.

For i > 2, equation (3) may have ranges of parametervalues that lead to physically impossible solutions (i.e.,negative site-occupancies), thereby placing limits on pos-sible site-occupancies. In most cases, these limits will beso wide as to be of little value. However, an advanta-geous combination of positive and negative scatteringspecies in neutron diffraction can lead to a limited rangeof real solutions that can provide useful information. Asan example of this, consider the site-occupancy refine-ment of potassian oxy-kaersutite (Kitamura et al., 1975).When the observed neutron scattering densities at thethree octahedrally coordinated M sites are consideredtogether with the unit cell contents from the chemicalanalysis, allowable solutions to the site-occupancy equa-tions occur for M(1) Ti site-occupancies between -0.25and -0.29; as this accounts for nearly 90Vo of the Ti in theunit cell, an adequate answer to the question of Tidistribution in oxy-kaersutite was derived from the ex-periment even though an exact solution was not obtained.

Specific radiation dffi action

The essential feature of these methods, as developedby Duncan et al. (1973), is to perturb normal Braggscattering by a competing mechanism that is related to thepresence ofthe cation (or anion) ofinterest in the mineral.The difference between the scattering in the normal andperturbed conditions consists predominantly of informa-tion concerning the site-distribution of the atom of inter-est. Methods ofthis type have not been used to any greatextent, but they have the potential to characterize atomsthat cannot easily be examined by other methods due tointerference from other atoms in the structure.

Duncan et al. (1975) and Johnston and Duncan (1975)used anomalous X-ray scattering to derive the site-distri-butions of Fe2+ and Mn3* in cordierite and tourmalinerespectively. Here, a specific type of atom is induced toscatter anomalously by using X-rays of a wavelengthclose to its absorption edge. The results are comparedwith those of a normal scattering experiment, in principleisolating the anomalous scattering contribution and mea-suring the distribution of the anomalous scatterer. Themethod was demonstrated using both powder difractionof FeKa and CuKa radiation, and single-crystal (10 x l0x I mm plate) diffraction of white X-radiation usingenergy dispersive detection. Only three or four reflec-tions were measured in each case; a much larger data setis preferable, low counting times notwithstanding. Assuggested by Whittaker (1975), the use of synchrotonradiation would speed up measurement time and allow

the use of smaller specimens with the energy dispersivemethod.

Duncan et al. (1973) suggest using yray diffraction witha moving rray source. Under these conditions, anyM<issbauer sensitive isotopes present will also shownuclear scattering when the source is moving at theresonance velocity. Comparison of the diffraction on andoff resonance gives information concerning the phase ofthe contribution from the Mossbauer sensitive atom; thusordering may be characterized. No details of such adetermination are yet available.

X-ray photoelectron spectroscopy is a technique ofconsiderable potential with regard to the characterizationof order/disorder in minerals. At present, the low resolu-tion of the technique precludes its use on a quantitativelevel, but future improvements in both intensity andresolution should promote its use in this area. A review ofthe application of this technique to geology is given byBancroft et al. (1979). Photoelectrons generated belowthe surface of a single-crystal may be diffracted beforethey reach the surface of the crystal. The resultantejected electron flux shows directional anisotropy that isa function of the structure of the sample crystal. Adams etal. (1978) have used this technique of X-ray photoelectrondffiaction to characterize cation ordering in micas. Thepotential of this technique is considerable, but muchdevelopment is required before the results are fullyquantitative.

Powder dffiaction methods

The discussion given above has centered on single-crystal difraction. Many minerals, both natural and syn-thetic, do not occur in crystals suitable for single-crystaldiffraction experiments. Quantitative information on or-dering can be derived by powder diffraction provided thestructure is not too complex. In a powder diffractionpattern, reflections generally overlap and thus individualintensities cannot be measured directly. However, muchof the information in these intensities is still present in thepowder pattern. This may be extracted and used forstructure refinement using the profile-fitting refinementprocedure of Rietveld (1969). Although primarily used forpowder neutron diffraction, the method has also beenused for site-occupancy characteization in powder X-raydiffraction (Nord, 1977; Nord and Stefanidis, 1980).

Mdssbauer spectroscoPy

Mdssbauer resonance

The Mossbauer effect is the recoil-free emission andabsorption of yrays by a specific atomic nucleus. Theemission of a yray during a nuclear transition normallycauses a recoil of the emitting atom; this recoil energydissipates by transfer to the phonon spectrum of thestructure. As the phonon spectrum is quantized, thistransfer must occur in integral multiples of the phonon

HAWTHORNE: QUANTITATIVE CHARACTERIZATION OF SITE-OCCUPANCIES

energy and the probability exists that no energy is trans-ferred. The effective line-width of the zero-phonon (re-coil-free) process is that of the yray; this is extremelysmall (Wertheim, 1964, Chap. 4, Fie.l) in relation to thecharacteristic energies of interaction between the nucleusand its surrounding electrons. If the zero-phonon yrayencounters another nucleus, its energy may be absorbedby raising that nucleus to an excited state, provided thetransition energies of the emission and absorption eventsare equal to within the line-width of the 7ray. As the line-width of the yray is much smaller than the characteristicinteraction energies between nuclei and electrons, achange in structural environment is generally sufficient tobring the two nuclei out of resonance. However, theenergy of the Tray may be modulated by applying adoppler shift to bring the system into resonance. In thisway, nuclear transition energies may be compared indiferent environments.

A change in the s-electron density at the nucleus of anatom will result in a shift in the nuclear energy levels.Where such a variation occurs between emitter andabsorber, the processes are separated in the energyspectrum by an amount known asthe Isomer Sftrl GS) orChemical Sltrlt (CS). This quantity is thus a measure ofthe relative s-electron density at the nucleus. Two factorsare principally responsible for variations in isomer shift.Screening of s-electrons from the nucleus by valenceelectrons is strongly affected by valence state and degreeof covalent bonding. Thus isomer shift may be used tocharacterize valence state and coordination number. Ifthe nucleus does not have a uniform charge density, aquadrupole moment arises which can interact with theelectric field gradient (EFG) at the nucleus to lift thedegeneracy of the nuclear states. The splitting of thenuclear energy levels gives rise to a series of possibletransitions between the ground and excited states of thenucleus, the number ofwhich depends on the nuclear spinquantum number. For 57Fe (I : ll2) in paramagneticmaterials, the ground state is not split and the first excitedstate (1 : 3/2) is split into two levels; the two allowabletransitions give rise to a doublet in the energy spectrum;the separation of the two components is known as theQuadrupole Splitting (QS) and is thus a measure of theEFG at the nucleus. Principal factors affecting the EFG atthe nucleus are the non-spherical electron distribution inthe atom itself, a function of valence state. and the non-spherical component of the crystal field. Thus quadrupolesplitting may be used to characterize valence state andvariations in structural environment.

Sit e-oc c up ancy de rivatio n

There are thirty or so isotopes that are sensitive to theMtissbauer effect. There have been some studies of Sband Sn in minerals (Stevens and Maclntosh, 1977; Bakerand Stevens, 1977) but virtually all quantitative site-population studies have involved 57Fe. Early studies ofiron in minerals (Pollak et al.,1962; de Coster er al.,1963)

were reconnaissance in nature. Gibb and Greenwood(1965) correctly assigned the spectrum of cummingtoniteby a qualitative consideration of site-populations. Theintroduction of least-squares refinement techniques tospectrum fitting led to quantitative and more extensivestudies of iron site-populations in silicate minerals (Ban-croft et al., 1966, l967a,b; Evans et al., 1967, Ghose andHafner, 1967; Virgo and Hafner, 1968, 1969).

Bancroft (1967, 1970) and Bancroft et al. (1967a) haveconsidered the assumptions inherent in using the M0ss-bauer effect for site-population characterization ofiron insilicates when two distinct quadrupole split doubletsoccur in the spectrum. We may generalize this to rquadrupole split doublets in the spectrum. The area undera peak is given by (Bancroft et al.,1967a)

t ,

il = T

fifaoriG(ni, f;, oo)ni (8)

where f1 is the recoil-free fraction of the absorber at thesite giving rise to the peak i, f" is the recoil-free fraction ofthe emitter, o6 is the maximum resonant absorptioncross-section 1: peak height per absorbing atom), 4 isthe half-width of the peak (peak width at half-height),G(ni,fi,oo) is the saturation correction, and n; is thenumber of atoms per formula unit at the site giving rise tothe peak i. Expressing the area of the j'h peak as afunction of the total absorption

/ r / rA t / > A i = n j f 2 C , n t ( 9 )

/ i = t / i = l

where

G;(n;,fi,oe)

Gi(n;,fi,os)

In most studies, Ci is assumed to be unity. Consideringthe individual terms in the above expression for Ci, thisgenerally seems to be a reasonable assumption, but hasnever been proven to be correct in a solid solution series.Saturation corrections approach unity for thin absorbers,and small deviations from this value should be self-cancelling providing that the ni values are not radicallydifferent from each other (Bancroft, 1973). This is not thecase when measureable saturation does occur; this situa-tion can be corrected by running at lower absorberconcentrations. Extensive experimental evidence hasconfirmed that half-widths of single-peaks are approxi-mately equal. The recoil-free fractions at energeticallysimilar sites should be equal. In principle, the recoil-freefractions can be determined, but this has rarely beendone. Some studies have derived C values different fromunity but such results have often been somewhat contro-versial.

An experimental Mcissbauer spectrum is shown inFigure 3. Each data point represents the number ofcounts recorded over that particular source velocity

c,=+1lj Tj

zo

ceo

G

z

e

o

HAWTHORNE: QUANTITATM CHARACTERIZATION OF SITE-OCCUPANCIES 293

ai is the ith peak haH-width (full width at half peak height),and b is the background intensity. The area of the ith peakmay be written as

Ai = 24(! - y(0)J/2 (ll)

Slight sinusoidal and linear deviations occur in the back-ground intensity due to source movement and instrumen-tal drift; b is modified in the above intensity equation toaccount for these features. Least-squares refinement pro-cedure minimizes Ro, the weighted sum of the squares ofthe residuals for each data point (spectrum channel):

0 1 ?

V E L O C I T Y ( I 4 M . / S E C . ] R o = wJvP - vil'z

where w. is the weight of the rfr observations (usuallyassigned from counting statistics), yP is tne observedbeam intensity at the r'h channel, and yi is the calculatedbeam intensity at the rft channel (calculated using theleast-squares parameter estimates). Law (1973) has con-sidered the case when there is no systematic error in thedata or the model. In this case, an ideal residual definedas

R r = w,ty9 - y1l2

(r2)m

r = l

(13)m

r = l

-? -t ur.l.-rt i"r.rr '... ,

t 4

Fig. 3. The M6ssbauer spectrum of potassian ferri-taramite.

interval, and the length of the vertical dash representstwo standard deviations based on counting statistics. Thecounts at the margins of the spectrum represent thebackground counts and are approximately constant; theaverage background value is termed the off-resonancecount. Towards the center of the spectrum, the countsdecrease; this is due to the resonant absorption of Traysof this specific energy by the sample. The ideal shape of asingle absorption peak (line) in a Mdssbauer spectrum isLorentzian. Thus the observed spectrum consists of aseries of Lorentzian peaks, the number and characteris-tics of which are a function of the crystal structure of thesample. Unfolding the observed envelope into its compo-nent Lorentzian peaks is done using least-squares refine-ment techniques. The intensity of the transmitted beam yas a function of its energy x for a spectrum containing Ilines may be expressed as

b - v(o)il + 4 ( x - x ( O ) J ' z l a ?

where x(0)1 is the resonant velocity of the ith peak, y(0)i isthe beam intensity at the resonant velocity ofthe ith peak,

where y[ is the calculated beam intensity at the rschannel (calculated using the correct parameter values),would be non-zero on account of random error in thedata. Ry follows the chi-squared distribution; if a set offitted parameters is a valid approximation to the correctset of values, then Ro is a value from this distribution(Law, 1973). To test the hypothesis that the least-squaresparameter estimates are a valid approximation for thecorrect values, the percentage points of the chi-squareddistribution are used to assess the probability that R1would exceed Ro. When refining spectrum parameters, adecrease in Ro is generally associated with an improve-ment in the model, until an acceptable fit occurs when Rolies between the 99% and lVo points of the chi-squareddistribution. Within these limits, statistical tests cannotdiscriminate between alternate models (Law, 1973). Giv-en two statistically acceptable solutions to the spectrum-fitting procedure, the simpler solution is to be preferredon statistical grounds alone. However, if additional infor-mation is available from another technique that indicatesthat the more complex solution is correct, then the morecomplex solution is to be preferred. In this case, weconstrain our solution on the basis of external informa-tion, and the simpler solution is rejected not on statisticalgrounds, but because it does not conform with theseexternal constraints.

Statistical tests are used as a measure of comparisonbetween alternate models. with the residual as a measureof the agreement between the observed data and a

(10)I

v=b-2i = I

HAWTHORNE: QUANTITATIVE CHARACTERIZATION OF SITE-OCCUPANCIES

corresponding set of calculated values derived from theleast-squares solution. Statistical acceptability and a lowresidual are no guarantee that the derived model iscorrect; they merely indicate that the derived modeladequately (but not exclusively) explains the observeddata. The most obvious examples of this are the M<iss-bauer spectra of the amphiboles of the magnesiocum-mingtonite-grunerite series (Bancroft et al., 1967a:Hafner and Ghose, 1971). The C2lm amphibole structureindicates that the correct (ideal) spectrum model shouldconsist of four quadrupole-split Fe2+ doublets. However,statistically acceptable fits are obtained for two quadru-pole-split Fe2+ doublets. Doublet intensity (Bancroft eral.,1967a; Hafner and Ghose, l97l; Ghose and Weidner,1972) and local stereochemistry (Hawthorne, l98lb)show that three of the four doublets overlap almostperfectly in the spectra and have very similar IS and eSvalues. The correct (ideal) four-doublet model cannot beresolved, as the spectrum does not contain sufficientinformation for this to be done. This is not to say thatuseful information cannot be derived from the two-peaksolution in this particular case; however, it does illustratethat statistical acceptability and correctness are not nec-essarily concomitant.

For most problems of interest in mineralogy, Mdss-bauer spectra consist of a complex overlap of peaks thatreflect diferential occupancy of more than one crystallo-graphic site in the mineral structure. In such cases, theprecision with which the parameters of interest can bedetermined is strongly a function of the degree of overlapof the constituent peaks. This has been investigated byDollase (1975) for the case oftwo overlapped peaks usingsimulated spectra. Several interesting features emergedfrom this work. First, for equal area and equal widthLorentzian peaks, separate maxima in the envelope occurfor peak separations greater than llt/T?0.6) peak width.Second, the uncertainly associated with the derived peakarea is a function of the peak separation and the peak-to-background ratio. Because Fe3+ shows much less varia-tion than Fe2* in quadrupole splitting values for mostminerals, complex Fe3* spectra generally show far great-er overlap than comparably complex Fe2+ spectra. Con-sequently, Fe3+ site-occupancies are usually far less-precisely determined than Fe2+ site-occupancies.

The problems associated with peak overlap and thecorresponding correlation problems in the least-squaresfitting procedure may be alleviated somewhat by incorpo-rating into the model additional information that is exter-nal to the spectral data itself. There are two types ofconstraints that are widely used:

(l) Width constraints: Mrissbauer spectral examina-tion of many end-member compounds suggests that thepeak widths corresponding to Fe2+ in various sites areequal in a specified sample. Thus in the spectrum refine-ment procedure, the width of all the peaks can beconstrained to be equal, with a considerable improvementin both the convergence rate and standard deviations.

This is a commonly used constraint; however, it may notalways be valid. In a solid solution, variation in the localenvironment ofa site can cause peak broadening (substi-tutional broadening); if the amount of variation differsamong the different unique sites in a structure, ditrerentialsubstitutional broadening may result, leading to differenthalf-widths for different Fe2+ doublets. For Fe3* in verydilute concentrations, relaxation broadening (Blume,1966; Wignall, 1966) may produce peak-width asymme-try, as appears to be the case in clinozoisites (Dollase,1973). Despite these cases cited here, for complex mixed-valence minerals, a general equal peak-width constraintmay be advisable unless there is strong evidence tosuggest otherwise; there is a danger that a single Fe3+doublet with an unconstrained peak-width may pick up allsorts of minor contributions that arise from minor errorsin several Fe2+ doublets and end up considerably distort-ed from its correct configuration.

(2) Area constrarnts: Experimental evidence suggeststhat the individual peaks of a quadrupole-split doubletshould have approximately equal intensity (Bancroft eral., 1967b). This is another very powerful constraint thatis frequently used in spectral refinement. This constraintmay not be strictly correct as preferred orientation in asample can give rise to intensity asymmetry in quadru-pole-split doublets (see, for example, the single-crystalspectra given by Duncan and Johnston, 1974, and Gold-man et al., 1977). Fortunately, mixing and grinding thesample with an inert equidimensional filler such as sugarsatisfactorily removes preferred orientation. Generally anasymmetry of a few percent is present in most spectra,but is generally not sufficient to interfere with the spec-trum fitting and useful operation of equal-intensity con-straints. Such residual asymmetry could be crudely han-dled by including an overall asymmetry parameter in therefinement process, but such a procedure is not generallyused.

In complex, poorly-resolved spectra, such as thatshown in Figure 3, the final solution is only as good as theconstraints which were used in the fitting procedure. Ifthe constraints are true, then the fitted parameters arevalid; however, if the constraints Ere wrong (or inappro-priate) then the solution will be wrong. As emphasizedabove, the selection of constraints requires careful con-sideration of the problem at hand, and the constraintsflnally used may represent a compromise between the"ideal" constraints and the constraints that give the mostrealistic solution. These width and area constraints arerigid constraints as applied in normal spectrum fittingprocedures. These constraints are not exact (width andarea constraints, even if ideally exact, are perturbed bysubstitutional broadening and residual orientation and, inaddition, the spectrum contains information on relativepeak widths and relative areas of QS doublet compo-nents). Consequently they should be used as "soft"constraints rather than hard constraints, as discussed inthe section on difraction.

HAWTHORNE: QIIANTITATIVE CHARACTERIZATION OF SITE OCCUPANCIES 295

Precision

Least-squares refinement is a widely used method inscience. The reason for this is that once convergence hasbeen attained, some kind of precision (a standard devi-ation) can easily be assigned to the variable parameters ofthe model. However, some Mdssbauer studies in theliterature do not report standard deviations for theirassigned site-population. These values are not usefulwithout some estimation (a minimum estimation in thecase of least-squares modelling) of their precision. Thesite-occupancy (xj) of the M<issbauer-sensitive atom atthe jth site may be written as

can be derived from the variation ofthe upper and lowervelocity area ratios with Tray incidence angle, withouthaving to resolve the individual quadrupole-split peaks.Duncan and Johnston (1973) have used this method toderive site-occupancies in olivine, where great difrcultyis encountered in resolving and assigning individual quad-rupole-split doublets in the powder Mcissbauer spectrum.Duncan and Johnston (1974) used this method to assignsite-occupancies in cordierite, but their results are ques-tioned by Goldman et al. (1977). This method seems tohave potential for poorly resolved but fairly simple spec-tra; however, for more complex spectra, the solution isprobably not very well defined.

Vibrational spectroscoPY

Vibrational spectroscopy involves the interaction be-tween electro-magnetic radiation and the vibrationalmodes of a crystal. A vibrational mode in a crystal willabsorb electromagnetic radiation if the frequencies of thevibration and the radiation are coincident and if theexcited vibration results in a change in the dipole momentof the crystal; this gives rise to infrared absorptionspectroscopy. Electromagnetic radiation may be elasti-cally scattered by a crystal (Rayleigh scattering). Scatter-ing may also occur inelastically. In this case, the scatter-ing episode is accompanied by a vibrational transition inthe crystal, where energy is absorbed from or imparted tothe scattered radiation; this is the Raman effect and gives

rise to Raman spectroscoPY.Both methods have found considerable application for

order-disorder phenomena in solids, and comprehensivereviews are given by White (1967, 1974) and White andKeramidas (1972). These authors also give a classificationof the different types of ordering that can occur, deriva-tive from a parent structure. Most types of orderingconsidered involve a change in cell size and/or symmetry.The principal applications of vibrational spectroscopyinvolve the characterization of such ordering by factorgroup analysis of the vibrational spectra (White andDeAngelis, 1967; DeAngelis et al., 1971). Such an ap-proach is not applicable to many problems of mineralogi-cal interest where relative cation ordering at non-equiva-lent sites does not affect the factor group symmetry of thecrystal. However, where a particular vibrational band canbe associated with the vibration at a specific site in acrystal, variation in the cation type occupying that sitecan be correlated with a shift in frequency ofthe associat-ed spectral band. Most work in this area has involvedinfrared spectroscopy. The Raman effect requires exacttranslational symmetry over several unit cells in order toproduce a sharp spectrum (White, 1975). Perturbation ofthis symmetry by order-disorder causes rapid peakbroadening and loss of detail in the Raman spectrum,suggesting that Raman spectroscopy will be of limitedapplication to this type of order-disorder. Experimentalevidence to this efect can be seen by comparing the

X.x : : & -m.;

(14)

where \ is the relative area of the doublet assigned to x.;,Xt is the total amount of Mcissbauer sensitive species perunit cell, and m; is the rank of the equipoint of the j'h site(the number of jth sites in the unit cell).

Thus the standard deviation of the site-population ismade up of contributions from \ and Xt. Generally thestandard deviation associated with Xt is neglected (a1-though there is no reason why this should be done), andthe standard deviation ofthe site-population is the scaledstandard deviation of the area ratio \. In order that thecalculated standard deviations be correctly calculated, allvariable parameters must have been varied in the lastIeast-squares cycle and the full variance-covariance ma-trix must be used in the error propagation calculation forthe area ratio.

Standard deviations derived from nonlinear least-squares refinement must be treated with caution. If theestimated standard deviations are small enough that thefunctions are linear over a range of several standarddeviations, then the normal methods of testing linearhypotheses can be applied; this is hopefully the generalcase in most uses of least-squares refinement. However,ifthere are departures from linearity in this range, as is tobe expected ifhigh correlations give rise to large standarddeviations, then variance-ratio tests are no longer exact.This is presumably the origin of the "imprecisely deter-mined" standard deviations in the study of Dollase(1975), and this effect should be borne in mind whenassessing the significance of site-populations derivedfrom spectra in which high correlations are encountered.

Single-crystal Mdssbauer method

For a single-crystal absorber, the area ratio of thepeaks of a quadrupole-split doublet is a function of theangle the incoming yray beam makes with the electricfield gradient at the Mrissbauer nucleus. If a M<issbauer-sensitive species occupies more than one crystallographi-cally unique site in a structure, the response of atoms ateach site to changing incidence angle of the Tray beamwill be different, provided significant differences in theEFG occur at each of these sites. Thus site-occupancies

296 HAWTHORNE: QUANTITATIvE CHARACTERIZATIqN oF SITE.ICCI]PANCIES

Raman spectra for actinolite (White, 1975) and tremolite(Blaha and Rosasco, 1978).

A random solid solution should exhibit the vibrationalbands characteristic of its end-members, with a linearshift in frequency with composition. This type of behav-ior has often been observed for mineral series, and isgenerally accompanied by peak-broadening in the centralregions of the series. Such peak-broadening in the infra-red spectra is a useful qualitative indicator ofdisorder inseries that show order-disorder over more than oneunique site in a crystal (Martin, 1970; Estep et al., l97l;Farmer and Velde, 1973). Burns and Huggins (1972) andKovach et al. (1975) noted non-linear variation withcomposition for some bands in the infrared spectra ofolivines and pyroxenes. They suggested that this was dueto significant cation ordering over the octahedral sites inthe structures of these minerals. Huggins (1973) furtherdeveloped this idea to quantitatively derive site-popula-tions in manganiferous olivines; however, no compara-tive studies of this method have yet been made. perhapsthe most precise method of characterization involves thecorrelation of band position with order/disorder parame-ter for a fixed mineral composition.

The hydroxyl stretching region

A fragment in many common hydroxyl-bearing miner-als is the edge-sharing octahedral trimer [M3@12(OH)](@ : unspecified anion); this is embedded in the struc-tures of the humites, the amphiboles, the micas, talcs andchlorites.

The fundamental band of the O-H stretching vibrationoccurs from 1500-3800 cm-r. The exact position of thisband in the infrared region is a function ofthe strengths ofthe hydrogen--oxygen bond; strong bonds are associatedwith higher frequencies, the lower end of the range beingcharacteristic of symmetrical hydrogen bonds (Hamiltonand Ibers, 1968). In the spectra ofminerals containing thiscluster, the fundamental band generally occurs from3600-3700 cm-r, indicative ofa strong hydroxyl bond andlittle or no hydrogen bonding. Minerals containing acompletely ordered arrangement of cations in this clustershow a single sharp hydroxyl stretching band in thisregion. However, the principal band in minerals of inter-mediate composition shows considerable fine structure.Bassett (1960) was first to make the connection betweenfine-structure in the OH stretching region and the cationarrangement in the coordinating sites when he interpretedsuch structure in the spectra of biotites as being due tovacancies in the octahedral layer. Vedder (1964) con-firmed this association between OH fine-structure andcation configuration in the octahedral layer ofphlogopitesin a very detailed study. Strens (1966) and Burns andStrens (1966) interpreted the fine-structure bands in am-phiboles in terms of cation disorder over the octahedrally-coordinated sites in the structure, and used the relativeintensity ofthe various bands to derive the frequency ofoccurrence of the different possible combinations of

cations and the cation ordering over the non-equivalentsites in the structure. Similar quantitative interpretationswere subsequently made for the micas (Wilkins, 1968)and the talcs (Wilkins and Ito, 1967), and rhe method hasbeen used in systematic site-occupancy studies (Burnsand Prentice, 1968; Burns and Greaves, 1971).

Frequency shift

Strens (1974) has shown that the frequency shift oftheindividual bands in the fine-structure is a function of theelectronegativity of the bonded cations. This suggeststhat the frequency shift is related to the relative covalen-cy of the cation-hydroxyl bond. The resolution of thesebands in the hydroxyl spectrum is a function ofboth bandwidth and frequency shift. The intrinsic width of thebands in end-member ordered structures is -5 cm-r,which is usually broadened to -6 cm-l by instrumentaland minor substitutional effects. The frequency shift isgenerally considerably greater than the band width; thusthe principal resolution problems arise from accidentaloverlap of configurations having similar frequency shifts.One complicating factor is the presence of cation substi-tutions at sites not coordinated to the hydroxyl and theirpossible effect on the frequency ofthe bands. This is bestdocumented for the amphiboles, in which alkali cationsentering the A-site increase the principal OH stretchingfrequency considerably (-t24 cm-1 in riebeckite (Strens,1974); 30-40 cm-r for hastingsite and pargasite (Semet,1973);56cm I and 62cm-t respectively forrichterite andpotassium-richterite (Rowbotham and Farmer, 1973)).This suggests that the chemically more complex mineralswill give so many bands that the spectra will become toocomplex to be resolved. Where cation substitutions insites not coordinated to OH do not have such a greateffect on the stretching frequency, the peaks may merelybe broadened rather than split into different peaks. Again,this will cause problems as such substitutional broadeningwill decrease the resolution of the spectrum and lead toIess well-defined band intensities.

The derivation of band intensities

The derivation of band intensities from a measuredspectrum is a difficult problem even in the simplest ofcases, and some of the difficulties involved have beendiscussed by Strens (1974). Three main methods havebeen used: (1) graphical resolution, (2) use of a curveresolver, and (3) least-squares refinement.

Several problems are common to all three methods.First, it is usually necessary to apply substantial baselinecorrections to the observed spectrum (see Burns andGreaves, 1971, Figs. I and 2). Most studies do notmention how this is done, but Strens (1974) indicates thatthe baseline is usually drawn in freehand by the experi-menter. Obviously such a subjective procedure cannot beconsidered satisfactory. Similar problems in the fitting ofMcissbauer spectra are handled by a baseline refinementtechnique. Such an approach is also necessary for the

HAWTHORNE: OUANTITATIVE CHARACTERIZATION OF SITE-OCCUPANCIES 297

infrared technique if the method is to give objectiveresults. The second factor involved in each ofthe resolu-tion methods is the form of the band shape. In principle,vibrational bands are Lorentzian but line-broadening usu-ally results in a band shape that approximates a skewedGaussian (Strens, 1974). If a normal Gaussian is used, thismay give rise to spurious weak bands in the fitted spectra.

The first two methods outlined above are highly subjec-tive; however, both are useful for deriving starting param-eters for numerical fitting procedures. The latter willusually involve the least-squares technique, possibly withthe use of linear constraints, with the major advantagethat standard deviations can be derived.

Other considerations

The derivation of site-occupancies from band intensi-ties assumes that the band intensity for a specific configu-ration is related to the frequency of occulTence of thatconfiguration in the same way that all other bands arerelated to their corresponding configurations. Thus thetransition moment of the OH vibration should be indepen-dent of the type of configuration. Work on micas (Roux-het, 1970) suggests that this is not so, and the limitedamount of work done on amphiboles in this regard is alsoapparently in accord with this conclusion (Strens, 1974).The occurrence of weak hydrogen bonding may signifi-cantly affect the transition moment of the OH band,whether or not the strength of the hydrogen bonding isrelated to the cation configuration at the coordinatingoctahedral sites.

In many structures, the hydroxyl position may beoccupied by F, Cl (and O2-; in addition to OH. In order toapply this method, it is necessary to assume that there israndom mixing of OH, F, Cl anions with no segregation atspecific cation configurations. Unfortunately, this doesnot appear to be the case, as Fe2* tends to avoid F-coordinated sites (Rosenburg and Foit, 1977).

Hydroxyl infrared spectra have found most use in site-occupancy studies of Mg/Fe2* and Mg/Fe2+/Fe3*/Al or-dering over the M(1), M(2) and M(3) sites in amphiboles.It is instructive to consider the specific case of Mg/Fe2*ordering in C2lm amphiboles, following the work of Law(1976). When Fe2+ and Mg only are present at theoctahedrally coordinated sites coordinating a single hy-droxyl-2M(l) + M(3)-, the OH absorption band consistsof four component peaks:

A 2M(1) + M(3) : 3Ms

B 2M(l) + M(3) : 2Mg + Fe2* O (=2B' + B")

C 2M(l) + M(3) : Mg .r 2Fe2+ O (=2C' + C")

D 2M(1) + M(3) : 3Fe2*

In principle, both B and C should be resolvable intotwo components (B', B"; C' , C") that depend on whetheror not the unique atom is at an M(l) or M(3) site. No suchfitting procedure has been used. The total iron content of

the M(l) and M(3) positions in a binary (Me,Fe) amphi-bole solid solution is given by

T : B + 2 C + 3 D (15)

If the total Fe content of the amphibole is known fromthe chemical analysis, the Fe content of the M(2) + M(4)sites may be derived by difference.Consider the equations relating peak intensity to molefraction:

A= p t2 t t t : ( l - 0 , ) r (1 - 03 )B : p r 2 h + Z r q Q t L t t

: ( r - 6)26t + 2l - 6)6Jt - 6) (16)

c : Qr r t " + 2p th th : Qr2( t - h ) + 2 ( l - Q)Qf t tD : Qr2h: Qrzb t

where pi and @1 are the Mg and Fe2* occupancies of theM(i) site. As @r = ll2(T - @3), where T is known from thebulk composition, each of the peak intensities can berelated to the single variable f3. Thus four values of @3are obtained, fairly precise estimates ofthe site-occupan-cies. An alternate method of obtaining site-occupanciesusing the intensity ratios of peaks B and C was developedby Strens (1966,1974), but this has been disputed by Lawfl976\.

Clustering is the tendency for arrangements of likecations (e.g., FeFeFe, MgMgMg) to occur more oftenthan expected for random mixing. Strens (1966) notedthat the infrared technique was sensitive to this effect,and suggested that clustering of Fe2* or Mg in amphibolesis indicated if observed intensities of the D or A bands (Do

and Ao) exceed the values (D" or A") calculated assumingrandom mixing. At this stage, it seemed that the presenceor absence of clustering was an additional piece ofinformation that could be derived from the band intensi-ties. However, Law (1976) questioned the validity of thiscriterion for clustering, and Whittaker (1979) showed thatthe presence of clustering affects the relative intensity ofall bands in the OH spectrum. This means that theprocedure of Law (1976) that was outlined above for thederivation of M(1) and M(3) site-occupancies in amphi-boles is only valid in the absence of cation clustering (or

anticlustering: the mutual avoidance of like cations). Tosummarize the results of Whittaker (1979), clustering ofFe increases the intensity of both the A and D bands(although not equally), and also affects the intensities ofthe B and C bands in a complex manner that is a functionof the bulk composition of the mineral. These resultswere for one particular model of cation interaction; asindicated by Whittaker (1979), other models are possible,leading to different but equally complex band intensitybehavior. Considering the large number of factors thatcan a.ffect the band intensities, it is questionable whetheror not the spectrum contains sufficient information toderive a unique model. Certainly more work is required in

298 HAWTHORNE: QT]ANTITATIVE CHARACTERIZATIqN oF SITE-)CCI]PANCIES

this vein if the OH band infrared method is going toproduce quantitative site-occupancies for minerals inwhich clustering has not been disproved by other means(e.9., sreu with analytical capability).

The precision of the OH infrared method is not well-documented. The two examples given by Law (1976)have standard deviations of 0.02 and -0.06 atoms/siterespectively (ignoring the uncertainty in the bulk compo-sition of the minerals). The precision of site-occupancydeterminations will be a function of the bulk compositionof and ordering pattern exhibited by the mineral, and ageneral statement of precision is not possible.

Although the arguments outlined above were devel-oped for amphiboles, they are applicable in principle toother similar structure types. If clustering proves to be acommon effect, it will almost certainly preclude anywidespread use of the OH band infrared method forquantitative characteization of cation ordering withoutconsiderable development in the method of spectrumanalysis. On the other hand, if the absence of clusteringcan be demonstrated (or assumed for qualitative work),this is an extremely useful technique, particularly for fine-grained material with ordering of non-Mossbauer sensi-tive species.

Electronic absorption spectroscopy

The 3d-orbitals of a transition metal ion are energetical-ly degenerate in a spherically symmetric electric potentialfield. Application of an electric potential field of lowersymmetry removes this degeneracy and leads to a split-ting of the orbital energy levels. When light passesthrough a crystal containing a transition metal ion, certainwave lengths are absorbed by transition of electronsbetween the ground and excited states of the ion. As thecharacter and amount of separation of the orbital energylevels is a function of the potential field applied by thecrystal, occupancy of different sites in a crystal willproduce absorption of different wavelengths of light.Hence optical absorption spectroscopy is ofpotential useto characterize site-occupancy in crystals both qualita-tively (by observation and assignment of spectral bandsto specific cations in specific sites-Burns, 1970a) andquantitatively (by comparison of spectral band intensi-ties-Burns, 1970b). However, this method is notstraightforward because the interpretation of the spectrais not simple, as indicated for example by the manyconflicting studies on the orthopyroxenes (White andKeester, 1966, 1967; Bancroft and Burns, 1967; Burns,1968, 1970a; Runciman et al.. 1973l. Goldman and Ross-man, 1976, 1977b).

Excitation occurs through interaction of the electricfield of the incident radiation with a component of thedipole moment of the absorbing species; consequently theabsorption is dependent both on orientation and thesymmetry properties of the mineral. Thus for completecharacteization of absorption, polarized absorptionspectra are required. This is of particular importance in

minerals with polyvalent transition metal cations, inwhich the spectra may also be complicated by inter-element electron transitions, that is, charge transferbands where the electron transitions take place fromcation to anion, anion to cation or cation to cation. Thesetransitions are particularly common when adjacent cat-ions have variable valence states (e.g., Fe2* and Fe3+,Ti3+ and Tia*) and are a major cause of color andpleochroism in silicates (Manning and Nickel, 1969;Fayeand Nickel, 1970; Burns, l98l). The intensity of this typeof band is one to two orders of magnitude larger than theintensity of spin-allowed d-d bands. Thus overlap ofthese two types of band in an absorption spectrum canobscure the d-d transition bands. The occurrence ofmorethan one type of transition-metal cation in the structure,as is common in many minerals, will further complicatethe interpretation of the absorption spectra (c/. Mooreand White, 1972). For the derivation of quantitative site-populations by this method, additional problems arise.The intensity of an absorption is a function of theprobability of the transition. The transition probability isgoverned by the Laporte selection rule (King, 1964)which forbids transitions between 3d orbitals. When atransition-metal cation occupies a site without a center ofsymmetry, mixing the 3d and 4p orbitals allows a transi-tion, the intensity of which is a function of the degree of3d-4p mixing that is, in turn related to the deviation of thecation environment from centrosymmetry. Thus when atransition-metal cation occurs in more than one site in acrystal, the intensities of the d-d bands are a function ofboth the cation occupancies and the deviation of thecation environment from centrosymmetry. In centrosym-metric sites, some 3d-4p orbital mixing may occur byvibronic coupling and hence very weak bands can resultfrom cations in these sites. Local disorder can also giverise to intensification of d-d bands in solid solutions;Robbins and Strens (1972) have noted this affect in micasand termed it "substitutional intensification". This couldlead to non-linear relationship in Beer's law type plots(band intensity ys. concentration). Rossman (1975,1976\has demonstrated tremendous enhancement of spin-for-bidden d-d bands by antiferromagnetic exchange in hy-droxy- and oxy-bridged Fe3* clusters as compared withisolated Fe3+ polyhedra. These considerations suggestthat if optical absorption spectroscopy is to be used forquantitative derivation of site-populations, it will be nec-essary to empirically calibrate the method using samplesof known site-occupancy.

Extensive electronic absorption studies began in themid-1960's (Farrel and Newnham, 1965; Burns, 1965;White and Keester, 1966) when there was a tendency tointerpret the spectra purely in terms of d-d bands.Recognition of the importance of charge-transfer process-es in minerals (Littler and Williams, 1965; Faye et aI.1968; Robbins and Strens, 1968) led to improvements inspectrum interpretation. Manning and Nickel (1969) andFaye and Nickel (1970) resolved the wide absorption

N u c l e a r M a g n s t r c

Fig. 4. Nuclear energy levels for a nucleus with spin 3/2,showing the effect of the Zeeman and quadrupole interactions(left) and the resultant nuclear magletic resonance (right).

bands in the spectra of pyroxenes and amphiboles intocomponent peaks of narrower half-width, using a skewedGaussian peak shape and a Dupont curve resolver. In asimilar study on the olivines, Burns (1970b) used theresultant band intensities to assign Fe2+ site-occupanciesto the M(l) and M(2) sites, evaluating the relationshipbetween the peak area ratios and the Fe2* site-occupancyratios in two fayalites and assuming it to be constantacross the series.

Goldman and Rossman (1977a,b,1978) and Goldman etal. (1978) have shown that electronic absorption spectros-copy can be a powerful method for qualitatively assigningFe2+ site-populations, and is particularly useful in assign-ing quadrupole split doublets in Mossbauer spectra toFe2* at specific sites in the structure under examination.Goldman and Rossman (1979) have also used this methodto derive quantitative cation distributions in orthopyrox-enes, correlating the intensity of bands due to Fe2+ atM(2) with the molar concentration of Fe2+ at this site.Their method was calibrated using orthopyroxenes ofknown site-occupancy.

This method has not had extensive use in quantitativesite-population studies as yet but, as indicated by Ross-man (1979), has considerable potential in this area. Whenthe method requires unfolding of overlapping peaks as inthe case of the olivines (Burns, 1970b), the computationalproblems are similar to those of the infrared method forhydorxyl-bearing silicates. When overlap problems canbe avoided (Goldman and Rossman, 1979), the principaldfficulty would appear to be adequately describing theshape of the background under the absorption band ofinterest, particularly when affected by the tail of a closelyadjacent peak. In more complex solid solutions, substitu-tional broadening and "intensity borrowing" (Manning,1973) may introduce considerable complexity into thecalibration of this method.

Nuclear magnetic resonance spectroscopy

Nuclear magnetic resonance (NMR) is the resonanceabsorption and emission of electromagnetic radiation byan atomic nucleus in a magnetic field. An atomic nucleushas angular momentum that can be expressed as Ih,

where I is called the nuclear spin quantum number andcan take integer or half-integer values, and h is Planck'sconstant. As the nucleus is charged and has spin, it has amagnetic multipole moment that can interact with anapplied magnetic field to produce a series of preferredorientations of the nuclear magnetic moment. Thesecorrespond to the number ofquantum states indicated byI, the nuclear spin quantum number. Resonant absorptionof energy corresponding to the energy separations of thequantum states can cause excitation between thesestates. Measurement of the energy absorbed by thesetransitions allows the characteization of the energy levelseparations. This is done by placing the specimen in anoscillating magnetic field and measuring the absorption ofradio-frequency energy transmitted through the sample;thus the radio-frequency field is held constant and themagnetic field is varied.

There are two principal types of interaction betweenthe nucleus and a magnetic field:

(1) interaction between the field and the nuclear mag-netic dipole moment (Zeeman interaction) that results in2l + | equally spaced energy levels and one resonanceabsorption line.

(2) interaction between the field and the nuclear electricquadrupole moment. In general, this is a much smallereffect and can be treated as a second order perturbationon the Zeeman energy levels (Volkoff, 1953).

The effects of each interaction on the energy levels of anucleus with spin 3/2 is shown in Figure 4. The splitting ofthe energy levels is a function of the strength of themagnetic field and, in the case of the quadrupole interac-tion, the orientation of the magnetic field with regard tothe environment of the nucleus. The energy separationsbetween the perturbed Zeeman sublevels are no longerequal, and the single resonance line from the Zeemaninteraction is split into 2I components. For half-integralspins, there is a central component and 2l - I satellitessymmetrically located about the central component; forintegral spins, there are 2I shifted components.

Nuclear quadrupole resonance has been used qualita-tively to measure order/disorder of atoms with nuclearspin I > ll2. The central component (-ll2 e +ll2transition) depends only on second order terms of theperturbing electric quadrupole interaction, whereas thesatellite lines (11/2 <-> +312 and t3l2 <> -+512 etc.transitions) depend on first order perturbation terms.Atomic disorder at surrounding sites results in differentelectric field gradients at crystallographically-equivalentpositions throughout the crystal. This leads to broadeningof the satellite lines, which may not be observed incrystals with considerable disorder. This effect has beenused qualitatively to measure AVSi order/disorder infeldspars (see Smith, 1974), andreferences therein), AUSidisorder and Na positional disorder in nepheline (Brink-mann et al., 1972), AUSi order in cordierite (Tsang andGhose. 1972) and considerable disorder in tourmaline(Tsang and Ghose, 1973). Spin-echo spectra (using a

+ 3n -- '

Z e e m o n Z e e m o n o n d

leve l c quodruPo lei n l e r o c l i o n i

HAWTHORNE: QUANTITATM CHARACTERIZATION OF SITE OCCUPANCIES 299

3m HAWTHORNE: QUANTITATIVE CHARACTERIZATION OF SITE-OCCUPAN1IES

+8_8,

zer.o-ffeldenerg5r levelr

pulsed rather than a continuous radiofrequency field)have been used to characterize order/disorder in spinels(Rotter et al.,1977; Shirakashi and Kubo, 1979).

Nuclear magnetic resonance can also be used to char-actenze order/disorder between diamagnetic and para-magnetic cations in crystals. The presence of paramag-netic ions in the immediate vicinity of the nucleus ofinterest considerably modifies the local magnetic fieldwhen compared with the field from an environment ofdiamagnetic ions. This change in the magnetic field inturn produces a change in the energy level separation(s)which causes a "paramagnetic shift" in the resonancelines observed in the NMR spectrum. In a crystal withsubstitutional disorder at next-nearest-neighbor posi-tions, the NMR spectrum consists of the diamagneticspectrum plus one or more paramagnetically shifted spec-tra corresponding to the number of configurations of thenext-nearest neighbors. The relative intensities of thevarious spectral lines are related to the frequency ofoccurrence of the next-nearest-neighbor configurations.Such paramagnetically shifted lines were recognized in

, l? t

--- -== +t- ,t t

tt! t

\_

high-field nuclearlevele splitttng

the 27Al spectra of almandine (Ghose, 1964; Brinkmanand Kaeser, 1972), and the relative intensities of thepossible configurations correlated well with the chemicalcomposition of the garnet. Sanz and Stone (1977) andKalinichenko et al. (1977) used this technique withH*NMR spectra in micas and amphiboles to characterizeMg/Fe order/disorder at the octahedrally coordinatedsites h these minerals. The problems associated withderivation of site-populations should be much the same asthose encountered in the infrared method. Sanz andStone (1977, 1979) have also shown the value of NMRspectroscopy applied to short-range order in minerals.

Electron paramagnetic resonance spectroscopy

Electron paramagnetic resonance (EPR) is the reso-nance absorption and emission of electromagnetic radia-tion by electrons of a paramagnetic ion. A paramagneticion with n unpaired electrons acts as a system with a spinS : n/2. The crystal field lifts the degeneracy of theelectronic states and an applied magnetic field lifts thedegeneracy of the spin states. Consequently the system

- ,

+g-r,

+,

Fig. 5. Schematic energy-level diagram for Mn2* showing the electronic levels in zero and in strong magnetic field together withthe splitting due to nuclear spin; some typical transitions are shown by arrows (from Bleany and Ingram, l95l).

HAWTHORNE: QIIANTITATM CHARACTERIZATION OF SITE OCCUPANCIES 301

Fig. 6. Mn2* electron paramagnetic resonance spectrum ofdiopside; note the two overlapping groups of hyperfine lines inthe central region of the spectrum, due to Mn2+ at the M2(strong) and Ml (weak) sites (from Ghose and Schindler, 1969).

has S + 1 energetically distinct levels; transitions be-tween these levels are known as fine-structure lines. Thepresence of nuclear moments in the paramagnetic ion (or

in the surrounding ligands) can also contribute to the localmagnetic field at the electron. The number of possibleorientations of a nuclear moment are 2l + l, and thuseach fine-structure absorption line will be split into 2I + Ihyperfine lines. The selection rules allow transitionsbetween levels of the same nuclear spin quantum numberand differing by one in their electronic spin quantumnumber. A schematic energy level diagram for Mn2+ andthe spectrum of Mn2* in diopside (Ghose and Schindler,1969) are shown in Figures 5 and 6. Fine-structure linesvary in intensity, whereas the hyperfine lines from asingle fine-structure absorption are of equal intensity.General discussions of EPR and its application to miner-alogy are given by Schindler (1968) and Ghose (1968).Most applications to mineralogy have involved Mn2* orFe3+, although Ti3* (Weeks, 1973), Eu2* and Gd3+(Morris, 1975) have been examined. This technique dif-fers from the others outlined in this paper in that it is usedto examine trace paramagnetic components in a diamag-netic structure. If the paramagnetic ions are not magneti-cally dilute, their magnetic moments interact and a broadstructureless resonance results. The EPR is complemen-tary to the other major component techniques describedhere.

Electron paramagnetic resonance has been used tosemi-quantitatively measure order/disorder of atoms bycomparing the intensities of absorption of atoms at difer-ent sites in a mineral. This is illustrated by the Mn2+absorption in diopside shown in Figure 6. Superimposedon the strong central hyperfine sextet is a weaker hyper-fine sextet with a slightly different spacing. Ghose andSchindler (1969) interpreted these as due to Mn2+ at theM2 and Ml sites respectively in diopside; from therelative intensity of the lines, site-occupancies were de-rived, assuming an equal transition probability at eachsite. There has not been sufficient work in this area torealistically evaluate the possible limits of accuracy andprecision. However, this may change as there is consider-able potential for application of EPR to magneticallydilute minerals, particularly gemstones.

Accuracy

An experimental result consists of the true value to-gether with random and systematic errors characteristicof the experimental method. Precision is an estimate ofthe random elTor component. The systematic elTors arefar more difficult to examine. This can be done byanalysis of standard samples in which the measuredquantity is already known (i.e., from the preparationmethod). This is feasible for diffraction methods by cationsite-occupancy refinement in proven stoichiometric com-pounds in which cation disorder is not feasible (e.9.'

disorder of Ca and S in CaSOa), and for spectroscopicmethods by examining ordered end-member composi-tions where all the relevant sites are completely occupiedby the spectroscopically active species (e.8., orthoferro-silite). There have been no systematic studies of this sort.The alternative method is to examine the same samplewith a variety of techniques of approximately equalprecision. The difference in the results is then a measureof the differences in their systematic errors, and thesatisfactory agreement of several results suggests thatsystematic errors in each of the methods are not signifi-cant at the attained level of precision.

Table I shows selected results for X-ray diffraction andMcissbauer spectroscopy on the same samples, where theMdssbauer spectra were well-resolved and each determi-nation is of comparable precision. The X-ray results forcummingtonite and anthophyllite have been modifiedslightly to account for small amounts of Ca and Mn;details of this procedure are given by Hawthorne (1983).

The results are identical within the quoted precision,

suggesting systematic errors well below the assignedprecision, and hence accurate results. For more complexminerals, there are no suitable data for a comparison ofthis sort. Some available data are given in Table 2, butstandard deviations are often not assigned, a Mn compo-nent is present in some of the X-ray site-occupancydeterminations, and in the case ofglaucophane the crystal

Table l. Comparison of site-occupancy determinations of highprecision in minerals with well-resolved Mossbauer spectra

Mloeral Slte X-ray Irossbsuer Reference

l{1Orthopyroxene ,,p

u(1)C@ldgtonLte U(1,2,3)

u(4)Gruuer i te u(1,2,3)

u4Anthophylllte vl r 1

0.743(3) 0 .745(4)

0 .9s7(3) 0 .955(4)

0 .76( - ) 0 .78(L)

0 .21( - ) o .22O(4)

0 .985(8) 0 .96s(8)

0 .826(10) 0 .839(8)

0 .5r r54) 0 ,s29( - )

0 .028(3) 0 .022( - )

(1 )

( 2 , 3 )

(4 ,5 )

( 6 , 7 )

Referqcea: (1) Bwl@ et aL. (19?1); (2) Glpse (1961) and'iiechet Qg66), nodified bg uthor; (3) Ghoae ed' weifu@r (1972);

tn) fingio os:6gd; -

(5) iarner and @roee Q971); (6) ri.ngq (1970)'

ae rcdifi.ed. bv etl@li (7) seifert (1978).

302 HAWTHORNE: QUANTITATIVE CHARACTERIZATI1N oF SITE-)CCUPANCIES

l r (1)Iteotanoan ll(2)ferro-ecttnol t te M(3)

f e t .

M1

nohquistire il3Ref.

H(1)

craucopbene il[3]Ref.

o. ze(-)rel]0 . 2 9 ( - ) F e ; r0 . 4 0 ( - ) P e -

'

(2)

o. ro 1syr"2"l0.05(2)Fe; r0 .59(9)Fe- '

( ) j

o. rs {-) rell0 . 1 5 ( - ) F e ; r0 . 3 2 ( - ) F e - '

( 7 )

o. so{o)rell0.47 (6 )Fe; r0 . 2 3 ( 5 ) F e - '

(3 )

0 .51 ( - ) Fe*0.46 (-)Fe*+0.15FeF0. 58 (-) Fe*

(1)

0 .39( - )Fe*u0,055 (-) Fe-0.55 (-)Fe*

(4)

o. rs(z)re"2l0.09(2) Fe i .0 .29(2)Fe- '

(6 )

Ee*.=_7"' ' = I,tn. Refe"erces: (1) tt itchell et aL. (1,9?1); (i l Bm@d Gze@es (197J); (i) I@ (1926); G) fru8tutd @d. Hhittakq(1975); (5) Ian vhittqkex (1981); (6) papike atd, ctuk.(1,oss); 0)B@@oft ffid Bw@ (1969).

Table 2. Comparison of site-occupancy determinations incomplex minerals

Mlnersl X-ray lto6sbauer Infrared

and gloss over some of the inherent difrculties. Experi-ence should tell us that "methods of considerable poten-tial" are also methods with considerable unrecognizedproblems.

Of the techniques surveyed here, single-crystal X-raydiffraction is the most widely applicable. Any binarycombination of atoms ditrering significantly in atomicnumber can be handled, with site-occupancies deter-mined fairly precisely (-0.005 for Mg/Fe site-occupan-cies, -0.01 for Na/Ca site-occupancies). In addition,stereochemical information is derived and this can lead tofurther definition of site-occupancies and/or an internalcheck on the accuracy of the result. Single-crystal neu-tron diffraction is less widely used, but has similarcharacteristics and is particularly useful for site-occupan-cies involving AUSi (precision -0.03), Fe/Mn and Fe/Tiordering. Mrissbauer spectroscopy is highly specific inapplication,.and most work has focussed on ordering ofFe2* and Fe3*. However, the ubiquity of Mg/Fe2+ andAUFe3* diadochy in minerals makes this an importanttechnique in site-occupancy determination. The ability toaccurately distinguish between Fe2* and Fe3* is ofpartic-ular utility. In well-resolved spectra, Fe2+ site-occupan-cies can be determined with a precision of -0.005. Withdecreasing spectral resolution, parameter correlation be-comes a problem and the precision of site-occupancydetermination decreases strongly. Hydroxyl-band infra-red spectroscopy seems to be capable of adequate preci-sion (-0.02 on Mg/Fe2+ site-occupancies). However, themethod is perturbed by clustering and further develop-ment of this method to allow for this efect is necessary.Electronic absorption spectroscopy is now being used forquantitative site-occupancy determination, and there ispotential here for further development. X-ray and neu-tron diffraction powder profile-refinement, y-ray difrac-tion, X-ray photoelectron diffraction, nuclear magneticresonance spectroscopy and electron paramagnetic reso-nance spectroscopy have potential for quantitative char-acterization of site-occupancies, but more development isrequired.

The determination of site-occupancies in minerals isnot a trivial problem, particularly in the more chemicallycomplex species. Any single method can only resolvebinary site-occupancies sensitive to that particular tech-nique. Complete experimental characterization of site-occupancies frequently requires a combination of severaltechniques that should be regarded as complementaryrather than competetive. The very specific sensitivity ofmany of the spectroscopic techniques make them anatural counterpart to both diffraction techniques andother spectroscopic methods. The difrculties and ambi-guities involved in an experimental method can often besubstantially reduced by using a combination of experi-mental techniques pertinent to the specific problem athand. The utility ofthis approach is evident from severalstudies (Bancroft et al.,1967a; Bancroft and Burns, 1969;Hawthorne and Grundy, 1977; Goldman and Rossman,

used in the X-ray study apparently deviated from theaverage chemistry of the bulk sample and so the resultsare not strictly comparable. The results for holmquistiteagree fairly well. There is considerable variation amongthe different results for the manganoan ferro-actinolite,but without standard deviations it is not apparent whetheror not these differences are significant; one might expectthe standard deviations in the Mrissbauer study to be atleast as large as those of the holmquistite study, sopossibly the differences are not significant. Certainly acomparative study of this sort on some of the morecomplex minerals by current techniques is desirable.

In well-characterized mineral groups, the accuracy ofsite-occupancy determination by structure refinementcan be checked from mean bond-length constituent ca-tion/anion radius relationships, where it is known thatsuch relationships are well-behaved (i.e., not significantlyperturbed by substitutions at other sites in the structure).The sensitivity of this method depends on the differencein ionic radius of the cations/anions involved, and on thecoordination number of the site. In some cases. ionic radiidifferences are sufficient to assess the accuracy of themethod at the level of assigned precision (e.g., AVSirefinement using neutron difraction data), but this is notgenerally the case. For such cases as Mg1Fe2+ or AVFe3+ordering, where precisions of -0.005 atoms per site canbe expected from X-ray diffraction data, assessment ofaccuracy is an order of magnitude larger than the preci-sion except perhaps in optimum cases where the struc-tures are very well characterized.

Conclusions

The methods for site-population determination all havetheir specific advantages and disadvantages; there is nophilosopher's stone for site-occupancy characterization.When introducing new techniques to this field, there hasbeen a tendency in the past to overemphasize their utility

HAWTHORNE: QUANTITATIVE CHARACTERIZATION OF SITE.OCCUPANCIES 303

1977a: Goldm an et al. , 1977), the combination of methodsleading to a much better characterization of site-popula-tions than would otherwise have occurred.

AcknowledgmentsThe author is particularly grateful to Dr. Wayne Dollase whose

comments greatly improved an earlier version of this paper. Iwould like to thank George R. Rossman and Associate EditorSubrata Ghose for their comments. Financial support during thiswork was provided by a Research Fellowship and grant to theauthor from the Natural Sciences and Engineering ResearchCouncil of Canada.

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Manuscript received, February 25, 1982;acceptedfor publication, October 8, 1982.