Electronic Polarizabiliw of Oxygen and Various Cations in Selected Triclinig Minerals: Point.Dipole Theory

909

The Canadian M ineralo gi st Vol. 32,pp. 909-918(1994)

ELECTRONICPOLARIZABILIW OF OXYGEN AND VARIOUSCATIONS IN SELECTEDTRICLINIG MINERALS: POINT.DIPOLETHEORY

RICHARD N. ABBOTT, JN. Depamnentof Geology,Appalnchian State unirersity, Boone, Nonh Caroliru 28608, U.S.A.

ABSTRACT

Point-dipoletheory is usedto calculatethe indices of refraction and orientationof the optical indicatrix in selegtedtriclinic pyro' minerals: fy-i*, wollastonite, schizoliie @In-rich pectolite), microcline, albite, walstromite @aCa2SgO)' kaolinite' for the constituentspecies '(atomsphyllite-/A, talc-.IA, and triclinic forms of clinochlore and chlodtoid. Electronic polarizabilities and OH groups)are optimized for the best agreementbetween observed and calculatedoptical propertie-s.Asmany as six electronicpo'iarizatitities ian le determinedin this way for eachmineral. The samecations in symmetricallydistinct sites were assignedthe samepolarizability. Four kinds of oxygen atoms were distinguishedaccording to..coordination: bridging t4lAl, [I), oxygennot oxygen 1i-O-r, Z = Si oi I41Al),non6ridcing oxygen (IGxo, X representinga cation other than Si, or toii"O io Si, t4iAl, or H (i.e., O-XP, anOttyOtoig o*yg"o @:O-X).The calculationsshow that the electronicpolarizability of an atom (or OH group) dependson the local struitural setting, i.e., polanz,ablhnesare site-specific.Electronic polarizabilities arenot readily ftansferableand thereforenot strictly additive.

Keywords:optical properties,point-dipole theory, electronic polarizabilities, triclinic minerals.

Somuann

La th6orie des dipolesponctuels sefi A calculerles indices de r6fraction et l'orientation de I'indicatrice optique de certains mindrauxtricliniquei: fy-i6, wollastonite, schizolite (pectolite riche en Mn), microcline, albite, walstromite @aCa2Si3Oe). kaolinite, pyrophyllite-1,A,t:/lc:-:r/.,et les formesricliniques du clinochloreet de la chloritoid€.lrs polarisabilitCsd.lgctrontques OesespOc6s p*Gntgs (atomeset groupesOII) sont optimisdespour obtenir la meilleure concordanceentre propri6tds optiques observeesei calculdei. On peuidetirminer jusqu? six polarisabilit6s 6lectroniquesde cette fagon pour chaqrreespdce mindrale.lrs cationsd'un seul 6l6mentimpliqu6 donscles iites sym6triquementdistincs sont attribu6sla m€mepolarisabilitd. On distinguequaUe sortes d'atomes O'oxygani selonleur coordinence:atome participant i une liaison I-G-I (Irepr6sente Si ou tat4D,'31e6sparticipant h une liaison-{O-X" (X reprdsenteun cation autre que Si, t4lAt, ou II), atomed'oxygdne non li6 A 5i, rel61i,'euH (k, O-XJ, et aromed'oxylbnqassoci6 e un groupehyclroxyle (H-O-XJ.D'aprbs lescalculs'la polarisabilit6 6lectroniqued'un atome'(oud'un groupe-OF! tt6pendraitdeJ ag&cementsstructuralD( locaux. C'est donc dire que les polari- sabilitls'd6pendent des sites stru-cturaux.Les polarisabilit6s-dlectroniquesne sont donc pas facilement transfdrables,ni strictementadditives. (traduit par la R€daction)

Mots-cl6s:propri€t€s optiques, th6orie des dipolx ponctuels,polarisabifit6s 6lectroniques, min6raux ticliniques.

brrnoouc"noN more than 28 sites in the unit cell. Calculationswere done on large structuresby taking advantageof the The point-dipole theory has beenused successfirlly approximate additivity of electronic polarizabilities to calculatethe orientationof the optical indicatrix in (Lasaga& Cygan 1982) and by describingthe sfruc- low-symmetry (monoclinic and triclinic) minerals. tures in terms of groupsof atoms.Thus the structures The approachused by Abbott (1993) was somewhat used in the earlier calculations(Abbott 1993) were compromisedby the limited size of a crystal structure describedin tems of the cation sites only, with the (numberof atomsin one unit cell) on which the calcu- elecfronicpolarizability ofthe anionsdistributed over lations could be performed.For reasonsrelated to the the cation sites. Using electronic polarizabilities that allocation of memory by the computer-programcom- are linear combinationsof publishedvalues (Lasaga & piler, the theory could not be appliedto sftuctureswith Cygan 1982),the resultswere found to be surprisingly 910 THE CANADIAN MINERALOGIST

successfulin reproducingthe orientationof the opticat THEoRyarvo MilHoo indicatrix in kyanite, wollastonite,and vaiious feldspars. Unfortunately,the calculatedprincipal For a given frequencyof light, the wavelengthin an indices of refraction were in most casesnot altoE"th", anisotropiccrystal (henceindex of refraction) can be satisfactory. calculatedfrom a knowledgeof the structureand the This paperpresents results using I modified version electronicpolarizabilities of the speciesmaking up the of_the computer program, capable of performing structure(Abboft 1993,Pohl 1978,Pohl & Rath 1979" calculationson structuresrvith as many ai 40 sites in Lager et al. 1987). According to the point-dipole the unit cell. Calculationswere performedon selected theory, as extendedby Abbott (1993), the relationship triclinic minerals (Table l). The relativelv small among structure,electronic polarizability, and refrac- number of minerals in the study was dictated by the tion of light can be summarizedconveniently in four following constraints:(l) less than 40 sitesin the unit equations.The lrst equation(Cummins et al. 1976, cell, (2) simple chemistry,(3) known structure,and Poht 1978)gives the local electric field, F(k), ar site k (4) known optical properties.AI1 atomic sites, excepr as a function of the local electric fields at all sites.k,. thoseoccupied by hydrogen,were included in the cal- in the unit cell. culations.Triclinic minerals are of special interest becausethe orientationof the optical indicatix is not F(k) = E + (1/V)\, L(kt') %(k') F(k'). (1) constrainedby symmetry. Whereaslinear combinations.ofpublished elec- The vectorE is the macroscopic(externally applied) tronic polarizabilities were used (Lasaga& Cygan electric field, L(kk') is the Lorentz factor tensorfor 1982) and not allowed t9 uT.y in the previous'ial- the pair t-r', oro6'; is the electronicpotarizalility of (Abbott 'v culations 1993),in this study the electronic sp"ii". k', and ii tl" volume of the unit cell. The polarizabilities were optimizedso as to minimize the macroscopic electric field, E, is equated with discrepancybetween observedan^d calculated optical the vibration direction (hence plane of polarization) P.roPe{e1(grincipal indices of refractionand orienta- of the incident light. The Lorentz factor tensors, tion of the indicatrix). In this way, it is possibleto L(kk'), dependonf on the geometryof the structure, determineas many as six electronicpolarizabilities, and were catcutatedusing ihe metirodof Cummins the number.correspondjngto the degreesof freedomin et at. (1976).The polariza'bilities,oroft,), are usually fhe.lefragti-veproperties for a given wavelengthof determinedbmpu&lly (Lasagau Cit;; issz, rutr incident light. For a triclinic mineral, there arJthree 1988).Equation t fortor a syitem of fo.* equations degreesof freedom associatedwith the principal solvablefor theindividual local electric fielOs, h6;. indices of refraction, and aaother three degreesof The dielectric susceptibility tensor, 21,l, th"o freedom associatedwith the principal directions obtainedfrom the locatelectric n;lds (pohl itTg): of vibration (i.e., orientationof the indicatrix). The objectivesare (l) to calculate. the _optigalproperties yE = (ll\ & on(k)rG) e) using completestructures of minerals,(2) to calculate optimized electronicpolarizabilities, and (3) for each Referred to a Cartesianbaseo x (column matrix of tlpe 9f {om, blt mainly for oxygen,to compareopti- coordinalevariables a y, arrdz),the coefficientsof the mized electronicpolarizabilities in different structural dielectric susceptibility tensor describetle surfaceof settings. an ellipsoid,

xrlx=1. (3)

The principal axes of the ellipsoid are parallel to the principal axesof the optical indicatrix. The directions of the principal axes and their magnitudes(eigen- TABLE T. TRICUMC MINERAI,S USB) N CAI.CIJIATCINS vectorsand eigenvalues,respectively) are found by

OPIICAL PROPERTIES diagonalizingthe dielectric susceptibility tensor (e.g., Julian& Bloss 1987).The principalindices ofrefrac- Kyaail6 Wier & cn66 (rf49) hr d, al, (1985) Wollastoriae ohshi (1984) D3d d, ar. (t985) tion are then simply related Schlzolir6. ohaohi (1978) to the eigenvalues, d 4l Deor e, ar, O9E .. 6, Microclino Blsi e, at (19Y/) hlr e, ar. (1985), wlEcboll & wi!ftU (1951) Albile Ambmst6! e, al (1990) Dc.r et d (198t. nt= (y11+l)v, (4) Whcholl & wirch6u 0951) WrlilrcEib hrr Glsr & Ctae, (Iffi) Alfm e, d f1955) Pyrophyllirc - bo & coprhoio (1!r9) D6€r er al Gi8t &olirito Dh! & vor Drut6 (1S9) De6! a, aL (1985) wherevalues of z1(i = 1, 2, 3) correspondto indices of Igl. -- PordikaEir & Buditr (ie8l) kr er ar. (1985) Clltrochloro Jods & Fe$ (180) D6sr e, ar. (1985) refraction a, B, andy. chlodrold Ha|Ien (19&) Do6 er aJ. (1985), PtilIiF & cliffs! (x9El) The quantity Xti + 1 is the familiar dielectric

. M!"doh p*totito. constatt. All calculationswere referredto the D wave- . . proFri* Opd€l Ffod6d b Fdolit6. length(X,o = 5893A). ELECTROMC POLARZABILITY IN TRICLINIC MINERALS 9tL

The possibility of useful relationshipswith other tion (in terms of electronic polarizability) between optical constantsmay not have eluded the diligent atomsof a given elementoccupying symmetrically reader. The relationship of popular optical constants distinct sites. This contradicts stict additivity of the (1.e.,Gladstone-Dale, Drude, and specific refractivity) electronic polarizabilities, admittedly a very approxi- to equations2 and4 is discussedin the Appendix. mate relatibnship(Lasaga & Cygan 1982).Pohl and Calculationswere done using the computerpro- co-workers(Pohl 1978,Pohl et al. 1978,Pohl & Rath gram OPT, written in TURBO PASCAL by the author 1979)have shownthat the electronicpolarizability for for an lBM-compatible PC. The program uses the a given atom dependson the local structural setting. Ewald methodof lattice summations,as formalizedby This is partly reflectedin the different valuesreported Cummins et al. (1976), and various subroutinesand by various investigatorsfor elecfionic polarizabilities numericalmethods documented by Cooper(1981) oiselectedelements (Jaffe 1988, Pohl 1978'Pohlet al. andBoisen & Gibbs(1985). The programis available 1978,Pohl & Rath 1979).Additivity is supportedonly from the author. insofar as the range of electronicpolarizabilities Calculating Lorentz factor ten$orsconsumes most reportedfor a given elementis generallysmall. of the computingtime, and storing them, for rapid With the hopeful intention of preservingsome ves- retrieval, can requirelarge amountsof memory.In this tige of additivity, oxygen atomswere distinguishedin latestversion of the program,the maximumnumber of a generalway accordingto coordination.Four typesof sites in ttre unit cell is 40. This limitation is imposed oxygen atomswere recognized:(l)'Ob" oxygen (stack size) of the TURBO atoms bridging two terahedrally coordinatedcations by the memory allocation 'Onb" PASCAL compiler. (T-O-T), (2) nonbridging oxygen atomscoor- Electronic polarizabilities were optimized by vary- dinatedto one tetrahedrallycoordinated cation (Si, Al) the individual values so as to minimize the and non-tetrahedrallycoordinated cations (74-X")' ing 'Ox" discrepancybetween observedand calculatedoptical (3) oxygen atoms coordinatedto neither tetra- properties.Whereas optimization has been shown to hedrally coordinatedcations nor hydrogen(G-Xr;' and be amenableto the methodof least squaresif applied (4) "Oh" oxygen atoms coordinatedto hydrogen to structuresmore symmetricalthan monoclinic (H-G-xr).This strategy,or compromise,proled^to be to the identification of Q-ageret a\.1987),in thepresent calculations (applied satisfactoryand more amenable to triclinic structures),each cycle of calculationof general trends than adopting the extreme position refractive propertieswith a new set of polarizabilites of assigninga different electronicpolarizability to requires a large amount of computing time. each symmetrically independentoxygen atom. For a Exploratory efforts to understandthe behavior of the given structure,symmetrically distinct cations of the variables during optimization indicated that a simple iame elementwere assignedthe sameelectronic polar- manual searchwas adequate,both with regard to izability. expectedlevels of precisionand reasonableagreement The hydrogenion, having no electrons,is not with observedoptical properties.The strategystarts polarizable.However, the presenceof H+ no doubt has with nominal values for the polarizability of the i profound effect on the polarizability of hydroxyl cationsand the samepolarizability for eachoxygen oiygen, both with regard to the magnitude and atom. Starting values for electronicpolarizabilities anisotropyof the polarizability(Bunn 1961).The were generally taken from Lasaga & Cygan (1982). influence of hydrogen atoms is difficult to model in The one value for oro(O)was varied until the mean of point-dipole theory. Here, hydrogensites are ignored, the calculatedindiceis of refraction matchedthe mean and the hydroxyl group is treatedas a single polariz- of the observedindices of refraction. Then the bire- able speciescentered at the oxygensite. fringences(fF and $-cr) were adjustedby varying the individual polarizabilitiesuntil the best agteementwas Rlsut-rs obtained between calculated and observedoptical properties(indices of refraction and orientationof the Kyanite optical indicatrix). Resultsthus obtainedappear to be unique insofar as no further improvementwas The resultson kyanite are reportedin Figure l. The producedby any small departuresof the electronic structure determinationused in the calculations polarizabilitiesfrom the final values, i.e', the final (Winter & Ghose 1979) pettains to kyanite of electronic polarizabilities correspondto a true mini- unknown, but presumablynear-end-member, com- mum in the discrepancybetween observed and calcu- position. The calculatedand observeddirections of lated optical properties.In no casedid exploratory vibration are illustratedin the cyclographicp'rojection, efforts indicate a second (or third, e/c.) independent where X, Y, andZ correspond,respectively, to the cL minimum. p, and y indices of refraction.The observedindices of During the courseof such calculationson different refraction (Frg. 1) are for kyanite of unknown,but pre- minerals, it becameapparent that satisfactoryresults sumably near-end-member,composition (Deer et al, could not be obtained without making some distinc- 1985).The calculatedindicatrix is misorientedby 9L2 THE CANADIAN MINERALOGIST

(T00) for near-end-memberwollastonite, (Ca1.96rFeq.s* Mno.oo2Mgo.o3)Si0.ee4o3(Ohashi 1984). Calculated directionsof vibration plot very closeto (within 5. of) the correspondingobserved directions of vibration in the cyclographic projection; calculated and observed indices of refraction are in close agreement.The observedindices of refraction (Frg. 2) are for wollastonite of composition(Cq.e63Feo.oo,Mno.oorMgo.ooz) (Sie.ee6Feo.e6Al6.oo)O,@eer et al. 1985). *too,ro I t0t0l In wollastonite, all oxygen atoms are either bridging or nonbridging.The resultswere greatly improved by recognizingtwo types of nonbridgingoxygen atoms.These are distinguishableon the basis of tlte numberof nearest-neighborCa atoms,O-SiCa2 and O-SiCa3, with the latter configuration havin! the lower electronicpolarizability for the oxygen.The polarizability of the bridging oxygen atoms was (r00) o calculated significantly lower than the polarizabilities of either I observed type of the nonbridging oxygen atoms.The value for KYANITts op(Ca) is substantiallylower than that reportedby Lasaga& Cygan (1982). No improvementin the cn(si) = 0.00(0.08) A3 a = L.773(L.7L2) on(Al) = 0.25 (0.13) F = 1.719(7.127) crn(onb) = 1.35(1.31) y = L727 (L.7/7) on(Ox) = 0.95 (1.31)

Fto. 1. Kyanite:cyclographic projection of calculatedand x observeddirections of vibration.Optimized electronic polarizabilities(values in parentheses, from Lasaga& Y Cygan1982). Calculated and observed (in parentheses.l indicesof refraction.References for observeddirections t00rl (0r0) of vibrationand observed indices of refractionare listed in Table1.

about 25o with respectto rotation about the X vibration direction. By adjustingthe polarizabilities,the indicatrix could be made to reorient to wirhin 10o of calculated the observedorientation, but not without causing tt00l I observed unacceptablylarge errors in the calculatedindices of refraction. All three calculated indices of refraction WOLI,ASTOMTE agreewell with observedvalues. The optimized elec- = c = tronic polarizabilities (Fig. l) can be comparedwith cn(si) 0.11 (0.08) A 1.618(1.618) valuestabulated by Lasaga& Cygan (1952). No com- on(Ce) = 1.03(1.66) F = L.625 (L.62e) bination of polarizabilities with a nonzero value for on(ob) - 1.23(L.3r) r = 1.630(1.631) ao(Si) improvedthe results.The good agreement betweenobserved and calculatedrefractive properties oo(onb) = 1.55(1.31) was achievableonly by acknowledginga significant (onb) = t.46 (L.3I) distinction between the electronic polarizabilities of oxygen atoms coordinatedto Si (Onb, O_SiAl) and oxygenatoms coordinated only to Al (Ox, O-Alr). Ftc. 2. Wollastonite:cyclographic projection of calculated and observeddirections of vibration. Optimizedelectronic polarizabilities(values parenthesei, Wollastoniteand schizolite in from Lasaga& Cygan 1982). Calculatedand observed(in parentheses) indices of refraction. Referencesfor observeddirectious Figure 2 shows the results on wollastonite. The of vibration and observedindices of refraction are listed structure determination used in the calculations was in Table1. ELECTROMC POI-ARZABILITY IN TRICLINIC MINERAI,S 913 results could be producedwith any combinationof Il00l assignedpolarizabili-ties involving cro(Ca)efeater than approximately1.05 A3. The results on schizolite (an Mn-rich variety of pectolite) are glven in Figure 3. The structuredetermination usedin the calculationswas for schizoliteof compositionNa(Ca1.226Mne.63sFes. 136Mge.000)HSi3Oe (Ohashi & Finger 1978). Agreementbetween observedand calculatedrefractive properties is not quite as good as for wollastonite,although it shouldbe noted that the comparisonhere is with the observed indices of refraction reported for a non-end-member pectolite of unknowncomposition and observeddhec- tions of vibration for presumedend-member pectolite (Deer et al. L985). The calculated value for 1 is notably too low, and apparentlynot corectable by varying the polarizabilities.The optimizedpolarizabilities are comparableto values for comparableatomic sites in wollastonite.The role of the hydrogen atoms in pectolite is not entirely resolved (Ohashi & Finger IVIICROCI,INE 1978),but they are consideredto be associatedwith cp(K) = 0.ee (1.98)A3 cp(si) = 0.00 (0.08) op(Al) = 0.00 (0.13) cp(ob) = t.46 (L.3L) Frc.4. Microcline:cyclogrErhic projection ofcalculated and observeddirections of vibration.Optimized electronic polarizabilities(values in parentheses,from Lasaga& Cygan1982). Calculated and observed (in parentheses) indicesof refraction.References for observeddirections of vibrationand observed indices of refractionare listed inTable1.

the secondtype of bridging oxygen,with the lower electronicpolarizability.

Microcline and albite

In C-centeredfeldspars, there are 52 atomic sitesin SCHTZOLITE the unit cell, exceedingby 12the maximumnumber of 40 sites acceptablein the computerprognm. In order to reduce the number of sites per unit cell in the cn(si) = 0.11 (0.08)A o = 1.609(1.610) feldspars,cell parametersand atomic coordinateswere = crn(Ca) = t.lz (L.66) 0 1.615(1.61s) expressedin terms of a primitive unit-cell consistent cn(Na) = 0.64 (L.74) 1 = 1.620 (1.645) with the following transformation: an (ob) = 1.13 (1.31) It o o on(Onb) = 1.43 (1.31) = l+.s 0.5-1 (5) c,(onb) = 1.34 (1.31) l.:l0 0 I lt) This reducesthe number of speciesper unit cell by projection Frc. 3. Schizolite(Mn-rich pectolite):cyclographic one-half. The results were then referred back to the and observeddirections of vibration. of calculated C-centeredgeometry of the unit cell. electronicpolarizabilities (values in paren- standard Optirrized usedin the calculations theses,from Lasaga & Cygan 1982). Calculatedand The structuredetermination (K6.er5Nas.ert observed (in parentheses)indices of refraction. was for microcline of composition Referencesfor observeddirections of vibration and Rbg.ggtAlsi3os@lasi et al. 1987).The results given observedindices ofrefraction are listed in Table 1. n Fidure 4 for microcline were obtainedwith polariz- 914 TIIE CANADIAN MINERALOGIST

abilities determinedby Pohl (pers. comm.). In this optim2ed oco(Na)is significantly lower than the value case,the polarizabilities were not optimized. The cal- offered by Lasaga& Cygan (L952). Again, as for culated optical properties were comparedwith = those microcline,%(Si) %(Al) = 0.0workedbest. observed(Fig. a) for end-membermicrocline (Winchell& Winchell 1951).Note tharcL(K) usedin Walstrotnite the calculationsis half the value reported6y Lasaga& Cygan (1982), and this differenceis compensatedby a Walstromite(BaCarSirOe) is a cyclosilicatewith somewhathigher ao(Ob). The results could not be three-memberedrings of SiOa tetrahedra.Results on rmprovedby changfuigthe valuesfor o6(K) and a"(O). this unusualstructure are given in Figure 6, The struc- Similarly,no combinationof assigned'polarizabiiitiesture determinationused in the calculationswas for with nonzerovalues for oco(Si)or oro(Al)improved the walstromiteof unknown, but presumablynear-end- agreement.For practical purposes,ihe optical proper- membercomposition (Dent Glasser& Glasser1968t. ties of microcline (and albite, Fig. 5) are detJrmined The calculatedoptical propertieswere comparedwith by the oxygenand potassium (or sodium)atoms. those observed(Fig. 6) for near-end-memberwalstro- Resultson albite are given in Figure5. The struc- mite (Alfors et al. 1,965).Calculated and observed hre determinationused in the calculationswas for an directionsof vibration agreereasonably well, as do alhite of approximate composition (Nas.rreCas.rr1.,calculatedand observedvalues for 0 and y. The low Alr.00rsi2.eeeo,(Armbruster et al. L990').ftri, cAiii- calculatedvalue for o seemsnot to be correctableby ated optical propertieswere comparedwith those varying the polarizabilities. This problem and the observed(Fig. 5) for end-memberalbite (Winchell & unusually high value for oo(Si) may be related ro Winchell 1951).The optimizedcq(Ob) used in the the necessarilystrained ariiculation of the three- calculationsis similar to that usedin microcline.The memberring of SiOo tetrahedra.Alternatively, the

Ir00l

t0r0l

1001l

O calculated I observed WALSTROMITE

ALBITE cn (si) = 0.32 (0.08)A3 op(Na) = 0.64 (1.14) A cp (ca) = 1.90(1.66) op(si) = 0.00 (0.08) on (Ba) = 2.60 (3.93) cp(Al) = 0.00(0.13) crn (ob) = 1.47(1.3L) c p (ob) = t.42 (L.37) oo (Onb) = 0.93 (1.31)

Ftc. 5. Albite: cyclographicprojection of calculatedand Ftc. 6. Walstromite: cyclographic projection of calculated observeddirections of vibration. Optimizedelectronic and observeddirections of vibration. Optimizedelectronic polarizabilities (valuesin parentheses,from Lasaga& polarizabilities(values in parentheses,from Lasaga& Cygan 1982). Calculatedand observed(in parentheses) Cygan 1982). Calculated and observed(in parentheses.l indices of refraction. Referencesfor observeddirections indices of refraction. Referencesfor observeddirections of vibration and observedindices of refraction are listed of vibration and observedindices of refraction are listed in Table1. in Table1. ELFTTRONIC POI.ARZABILITY IN TRICLIMC MINERAI-S 91.5 anomalousvalue for o6(Si) may be an artifact of a TABLE 3. OFTIMIZED POTARIZABIIJTIES AND CALCUB*IB' OPI1CAL PROPERTIESIN TRICUNIC poorly determinedsffucture (R = 0.16, Dent Glasser& CHI,ORITAID Glasser1968). The value for cro(Ba)is compatible with high valuesreported by PohI (1978) and infened on(si), A3 0.0 (1982). from Lasaga& Cygan o.25 "n{[6lat) cn(Fo) 0.30 Kaolinite, pyroplryIlite, talc, clinochlore 1.40 5(onb) an(Ox) L.26 In order to reducethe number of sites per unit cell cn(oh) 1.54 in the phyllosilicates,cell parametersand atomic coordinateswere expressedin terrnsof a primitive unit-cell arglo,x^c(') e.2 (-0) Y ^ b 38.4 (-o) consistentwith the following transforrnation: "nglo, ^ angle, Z a 37.0 (-0) Ir o o llx 1.704 (7.713) = 0.5 0 (6) "p llY 1.71s (r.7r9) l-o.5 t llz r.723 (r.723) lt:lo 0 l l,) ' Obsnsd values in ptrenthosos. See Table 1 for The results were then referred back to the standard tg fgtoncss. C-centeredunit cell (Table 2). Except as noted in Table 2 for the clinochlore, the structure determinations used in the calculations (Table 1) were for near-end-membercompositions. Thus in phyllosilicates, except for chlorite-group Also, exceptfor the clinochlore,the calculatedoptical minerals, the polarizability of oxygen appearsto be propertieswere comparedwith those observed fiansferable. (Table 2) for end-membercompositions (Deer et al. 1985). The observedoptical properties (Table 2) for Chloritoid clinochlore pertain to a hypothetical mid-range composition of clinochlore (Ftg. 82 in Deer et al. L985). In order to reducethe numberof sites per unit cell Indices of refraction and the orientationof the optical in chloritoid, cell parametersand atomic coordinates indicatrix were reasonablywell reproducedin each were expressedin termsof a primitive unit-cell consis- case.The most significant departurefrom observed tent with the following transformation: relationships was with regard to pyrophyllite, for which the calculatedorientation ofthe optical indica- 0.5 -0.s 0 trix was rotatedapproximately 24" about cs relative to 010 a) the observedorientation. The problem could not be V:,001 ltl correctedby varying the polarizabilities in any way. Appropriatepolarizabilities for hydroxyl groups in The resultsof the calculationswere then referredback dioctahedralphyllosilicates seemto be lower than to the standardC-centered unit cell (Table 3). values for trioctahedralphyllosilicates. Except for The structuredetermination used in the calculations clinochlore,a,(Ob) and ao(Onb)are the sameand pertains to chloritoid of composition (Fel+rt fairly restricted(1.25 or 1.28A3) in the phyllosilicates. Mgo.reMn6.6)Al3.e3Feai1Si2.oro1e(oH)r.u (Hanscom 1980).The calculatedoptical propertieswere com- oared with fhose observed(Table 3) for chloritoid of unkoown composition (Door et aI. 1985).The calcu- TABIIA OYIIMZEDPOIAXIZABIIJIIESANDC4T'UNTED' OPNWPROPERTIFSTOR of the X vibration direction is approx- TRICUNC FHYII.O$IICTIES lated orientation imately correct,but, like in pyrophyllite, the calculated NAOM PiNOPI|]IIJTE re cuNmm (close vibra- apGD A3 o.o o.o 0.0 0.0 indicatrix was rotated about c* to the X gtt6ht) 0.26 0.65 tion direction) by 38' from the correct alignment. de(Ms) 0.53 of refraction are acceptablyclose to dD(ob) r.?t 1.2E t2 L.47 Calculatedindices ap(onb) 1.25 r.2A l2a 1.47 observedvalues. dp(oh), ld6r* 1,2t 1.1o 1.55 1.41 1.25 1.17 5(oh),3!]f., DIscussIoN ql6,X^.(') re3 (10) 1J.0 (10) e.6 (10) 24.9 (4.s-7) srgl6, Y ^ a s.? (1-35) a.l (0) 3.0 (0) 0.8 (0-2.5) a'db,z^b 03 (0) 24.6 (o) a8 (0) 67 (0) The point-dipole theory as applied here treats each' c llx Ls1 Ols3) rJ58 (1.552) 1-536 (1J39) 1.s73 (1.s81) p ll v 1J59 (1J59) rJ90 (1.t88) 1.5S (1J89) 1.591 (1J81) atom as though its polarizability were the samein all t llz 1.560 (1J60) 1,01 (1.61D) 1J91 (1.189) 1.59 (1J8O dLections,Le., individual polarizabilities are isotropic. ' Obi6ftd valo6 ia pMlh.@ b Tabl6 1 for 6f66!s. '. Sib @dlt (ryH by Mg + A. Lager et al. (1987) pointed out that discrepancies f Iffir . prphyUt@Ik OH in taolioito, OH of hlc-&6 dsr b clhffiE. Srr(r6t e gibbib-lit€ OS h leu!tu' OH ol hnd6l&o ddt in clineiloF. betweencalculated and observedindices of refraction 916 TIIE CANADIAN MINERALOGIST

TABII4. SUMMARYOFOI'TTMZEDELBMONICPOIARZABILITES tive valuesreponed by Lasaga& Cygan (1982). In A'BCDBFGEIJKI44 microcline and albite, calculationsusing the values aP(sl), o.o 0.11 0.11 0.0 0.0 0.32 0.o 0.0 0.0 0.0 o.o 0.08 of Lasaga& Cygan (1982) for cro(K)and ao(Na), "o{tllar) -- 0.0 0.0 -- -. 0,13 togetherwith their valuesfor oro(Si),oco(At) and'oto(O) 5{tclaD O.23 -. 0,26 O.6j -- -- 0.2j 0.13 (Table 4), produceincorrect-indice3 of refraction eP(Ms) 0.53 0.2..__ 0.48 o!G6) 0.30 1.66 (microcline:a = 1.520,F = L.524,y = 1.550;albite: '- sp(ca) 1.03 1.X2.. .. 1,9 -- 1.66 qe(K) a = L.527, = 1.540, = 1.546),and none of the -" 0.99 -- 1.98 0 \ -- 5(N!) O.U 0.64 -" -- 1.14 calculateddirections of vibration are closer than dp(B!) " 2.6 -- 3.93 approximately35o to the observeddirections. For ap(ob) "- 1,23 1.13 1.46 !.42 1,47 t,23 r2a r.2a L47 -" 1.31 q!(o!b) ,..3J 1.s5 1.43 -" -. O.93 t.2t r.28 r2a L47 1.& Llr other cations (Mg, Fe, Ba), polarizabilitiesare less x.46 1.33 -- 1.31 than values reported dp(or) O.95 L26 r37 by, or inferred from, Lasaga& 5(oh) lmor 1.23 r.10 1.35 1.4? -- Cygan (1982), but consistentwith other determina- ap(ot) a!'!. L.X -- ." 1.47 -- qP(oh) tions(Jaffe 1988). In Table 4, there are22 polaizabilities ' for oxygen A - hrul|€, B- mn@drq C E s<c, D E d@lh. E r - u,cr -E ffioqafbib. (excludinghydroxyl oxygen).The averagevalue is [""ffiJ: "d.if;lff'# :fiH:*"u" (standard !&s Vd@ tod Ileg! & crs! (1982), 6Fts doG6) ald a-(Ba), shich aD 1.32A5 deviation:0.16 A3), which is close ,- Ef"*d &* kes & cysu (1982, TaU6 l). " $6 @piod by Ma + d to the value of 1.31A3 reportedby Lasaga& Cygan (1982). However, the range of optimized polarizabilities indicatesclearly that the averagevalue will not work well for any one of the minerals.The polar- m1y be due in large part to the use of isotropic polariz- izability ofoxygen is site-specific,and thereis signifi- abilities. In the presentcalculations, discrepancies that cant variation from one site to another.Table 4 affords might otherwisearise from this effect rnayt" compen- few useful generalitiesregarding the variability of satedwholly or in part by exaggeratedvalues for one cto(O).Polarizabilities for bridging oxygen-atoms or more of the optimizedpolarizabilities. Given tle show a bimodal distribution"l.l3-1.28 A3 and generally good agreementbetween calculated and 1.42-L.47A3. The lower modeincludes oxygen atoms observedoptical properties,the problem would be in wollastonite, pectolite, and phyllosilicates (except difficult to recognize. clinochlore); the higher mode includes oxygen atoms Table 4 summarizesthe optimized elechonicpolar- in the feldspars,walstromite and clinochlore. izabilities. For a given element, the range of values Polarizabilitiesof nonbridgingoxygen atomsrange indicatesthat polarizability dependson the local struc- from 0.93A3 in walstromid:o-L, i X3 in clinochlore. tural setting, i, e., polarizabilities are site-specific. In each of the phyllosilicates, the polarizabilities for Even for an elementhaving similar coordinationin bridging and nonbridgingoxygen atomswere found to different minerals,polarizabilities can be very differ- be the same.In other minerals having both bridging ent. Transferabilityto other strucfurespresumes some and nonbridging oxygen atoms (wollastonite, schizo- systematicrelationship between polarizability and one lite, walstromite), the polarizabilities are different. In or more site-specificproperties, for instancesite wollastonite and schizolite,the oro(Ob)is less than potential. Unfortunately, the 1imi1e6data-base offers or(Onb), whereasin walstronite,'oro(Ob)is greater little more than vaguegeneralities. With regardto site than oco(Onb).Only two of the mineials, kyanite and potential(Smyth & Bish 1988,Smyth 1989),there is chloritbid, have oxygen atomsthat are sharedwitl no recognizablesystematic relationship. neitherSi nor H. In theseminerals, a-(Ox) is ouite dif- Tetrahedrallycoordinated Si or Al have, for practi- ferent,0.95 A: in tyanlte, and1.26 A5'in cltoritoiO. cal purposes,zero polarizability, except in wollas- Polarizabilities for hydroxyl groups range from tonite, schizolite, and walstromite.The exceptionsare l.l0 Ar in pyrophylliteto 1.55A3 in talc. Lower perhapsnoteworthy for having unusually distorted polarizabilitiesare associatedwith the dioctahedral SiOo tetrahedra.Octahedrally coordinared Al has a phyllosilicates,and there appearsto be little difference polarizability of 0.25 or 0.26 A3, exceptin pyro- between polarizabilities for inner (pyrophyllite-like) phyllite, for which good resultscould be obtai'.edonly and surface(gibbsite-like) hydroxyl groups.The high- by using a significantlyelevated value for c"(t61Al), er polarizabilities are associatedwith trioctahedral 0.65 A3. This may be a casewhere the efevated phyllosilicates,and againthere appearsto be little dif- ooll6rAt; compensatesfor anisotropicpolarizability of ferencebetween the polarizabilitiesof inner (talc-like) S9 OH group. The polarizabitiry for Ca is 1".03-1.22 and surface(brucile-like) groupsofhydroxyl. Ar, exceptin walsftomite,for which the best value for ocn(Ca)was found to be approximately 1.9 43. In aI CoNCLUSIoNS three mineralswith Ca, s.(Ca) is sienificantlv different from the value of 1.66 A3 reportia Uv Lasasa& l. Electronicpolarizabilities cari be determinedby Cygan(1982)._The polarizabilities for Wi tO.O+A) minimizing the discrepancybetween calculated and K (0.98 A3; are approximatelyhalf of the respec- and observedoptical properties.Using optimizedelec- FJ ECTRONIC POT,ARZABILITY IN TRICLIMC MINBRAI-S 917

(1987):A re- tronic polarizabilities,the optical propertiescalculated Br-asr,A., Dr Por,Bu.st, C. &7'xtzz', P.F. Pellotsalo microcline: mineralogical for triclinic minerals (indices and orientation of the examination of the and genetic considerations'Can, Mineral' are very close to observedoptical implications optical indicatrix) 25.527-537. properties. of a 2. Optimized electronicpolarizabilities for atoms Br.oss,F.D. (1971): Crystallographyand Crystd Chcntistry: given elementvary significantly dependingon the an Introduction. Holt, Rinehart& Winston, New York' local structuralsetting. Thus electronicpolarizabilities are not readily transferable.Electronic polarizabilities BoIsEN.M.B., Jn. & Gtsss, G.V. (1985):Mathematical may be transferablewithin a mineral group (e.9., Crystallography. Rev.Mineral. 15. feldspars,dioctahedral phyllosilicates, e/c'), but even then, only with considerablecaution. BuNu, C.W. (1961): Chemical Crystallography;4n 3. Earlier work (Abbott 1993) shows,and the present Introduction to Optical and X-ray Methods(2nd ed.). calculationsconfirm" that the essentialinformation for ClarendonPress, Odord, U.K. the orientationof the optical indicatrix is containedin (1981): to Pascal Scientists. the Lorentz factor tensors;hence the orientation CooPER,J.W. Introduction for JohnWiley & Sons,New York. dependsflrstly on the geometryof the structure.The of the optical indicatrix of effect on the orientation P.G., DtnnvIun,D.A., Muml, R.W' & Newueu' the CuMMD{s, varying the distribution of polarizability over R.J. (19?6): Applications of the Ewald method' atomic sites only becomesimportant wherethe calcu- I. Calculationof multiple lattice sums.Acn Crystallogr. lated birefringenceis small. This caveatto an earlier a32.847-853. conclusion(Abbott 1993) follows directly from the behavior of the orientationof the indicatrix during DEER,W.A., HowrB,R.A. & Zussuar'1,J. (1985):An the optimizationprocedure. Introdaction to the Rock Forming Minerals. Longoan' Note add.edin proof, Resultspresented in this paper London.U.K. have been testedwith a new computerprogram, (1968): The crystal OPTRFN, developedby (and availablefrom) the DrNr GI-esssn,L.S. & Gr,assen,F.P. Am Mineral,s3' 9-13. author.Using the methodof least squares,the prograrn structureof walstromite. searchesfor the set of electronic polarizabilities that R. (1980): The structureoftriclinic chloritoid and gives discrepancybetween observed and Harscou, the least chloritoid polymorphism.,4oaMineral. 65, 53+539. calculatedoptical properties.For each mineral, the polarizabilities prograrq confirms that the electronic JAFFE,H.W. (1988): Crystal Chemistryanl Reftactivity, (t0.02 A3) determinedhere do indeedcorrespond to a CambridgeUniversity Press,Cambridge, U.K. minimum discrepancybetween observedand calculatedoptical properties. Joswrc. W. & FuFss,H. (1990): Refinementof a one-layer triclinic chlorite. Clays Clay Minerals 38'216-218. ACIC.{Owr-EDcElvIEl.rTS JuuAN,M.M. & Bloss, F.D. (1987):Matrix calculationof I thankThomas Armbruster and two anonymous optical indicatrix pafimeters from central cross sections refereesfor theirhelpflrl comments. throughthe indexellipsoid. An. Mineral.72'612'6L6. LAcER.G.A., ARMBRUSTER,T. & PonL D. (1987):Prediction from crystallographic RrrrnrNces of refractive indices in minerals data: application and limitations of the point-dipole model.Pftys. Chem^ Minerak 14, L77-180. Annotr, R.N., JR. (1993):Calculation of the orientationof the optical indicatrix in monoclinic and triclinic crystals: LASAGA,A.C. & Cvcar, R.T. (1982):Electronic and ionic point-dipolemodel. An Mturcral78,952-956. the polarizabilities of silicate minerals.Am. Mineral.67, 328-334. Alrons, J.T., SrnrsoN,M.C., Mermsws, R.A. & PArsr, A. (1965): Sevennew barium minerals from eastemFresno Lnn. J,H. & GuccBNsBnra,S. (1979): Single crystal X-ray County,California. Am. Mineral.s0, 314-340. retinementof pyropylhte-lTc. Aru Mineral. ffi,350-357.

ARMBRUsTER,T., BURGI,H.B., KuNz, M., Gxos, E., Mar*oenwo, J.A. (1976): The Gladstone-Dalerelationship. BnOwNMaNN.S. & LEIERT,C. (1990):Variation of dis- I. Derivationof new constants.Can- Minerol, 14' 498'502. placement parametersin structure refinements of low albrte.Am. Mtueral.75, 135-140. (1978): The Gladstone-Dalerelationship. II. Trendsamong constane. Caa Mineral,16,169-174. BrsH.D.L. & VoN Dnssl-E,R.B. (1989): Rietveld refinement of non-hydrogenatomic positions in kaolinite. Clays CIay (1979)', The Gladstone-Dale relationship. Mircrals 37.289-296. III. Somegeneral applications. Can, Mineral. l7 '71'76. 918 T}IE CANADIAN MINERALOGIST

(1981): The Gladstone-Dale relationship.IV. The ------& Rern, R. (1979): point-dipole rheory and compatibility concept and its application. Can. Mineral. optical birefringenceof calcite-typecarbonates. Acla 19,441,450. Cryst all og r. A35.694-695. Onasrr, Y. (1984): Polysynthetically+winnedstructures Surnr, J.R. (1989): Electrostaticcharacterization of oxygen of enstatiteand phys. wollastonite. Chem.Minerals 10, sites in minerals. Geochirn.Cosmochim- Aaa 53, 1,1,b1.- 217-229. 1110.

& Fnlcrn, L.W. (1979): The role of octahedral & Brss, D.L. (1988): Crystal Structuresand cationsin pyroxenoidcrystal chemistry.I. Bustamite, Cation Sites of the Rock-FormingMinerals. Allen & wollastonite,and pectolite-schizolite-ieranditeseries. Unwin, Boston.Massachuseus. Am Mineral.$,n+288. WTNcHELL,A.N. & WrNcHnu-,H. (1951): Elements PERDrKArsrs, of B. & Bunzlarr, H. (l9gl): Strukrur_ Optical Mineralogy ^l/.John Wiley & Sons,New york. y:{"iryryC an Talk Mg3(OH)rSiaOrc). Z. Kristailogr. 156,1,77-L86. Wnnen, J.K. & GHosE,S. (1979):Thermal expansion and high- temperaturecrystal chemisrryof the AirSiO, poly- PHILLrps,W.R. & GnnreN,D.T. (19g1):Optical Mineralogy. morphs.Art. Mineral.g,573-586. W.H. Freeman,San Francisco. California.

Pout, D. (1978):Electronic polarizabilities of ionsin doublv refractingcrystals. Acta Crystailogr. A34, 574-579.

Ecr, J.C.& Kr_esre,K.-H. (197g):Determination ofelectronic polarizabilities ofions in orthosilicates.Acta ReceivedNovember 23, Igg3, revised manuscript accepted Crystallogr.A%,rcn-1,028. March28, 1994.

APPENDD( DRT]DECONSTANT, GLADSTONE_DALE CONSTANT. SPECIFICREFRACTIVTTY

Combiningequations 2 and4: K1 may both be expressedas tensors.Combining equation3A with 2A: (e-I)E = (r/v)>k%G) F(k), (lA) (e + 2)KpE = (rtm) & o6G) F(k) (4A) where e (= z2) is the dielectric constant in this casea tensor,and I is the identity matrix. Becausethe density, Howeverthis equationis of questionablevalue p, is the mass,rFt, per unit cell volume,% equationlA becauseit separatesotherwise dependent tensors, may be rewritten: e and K1. 'The popular Gladstone-Daleconstant (spe- (e-I)/p = (Ltm) E >k %G) F(k) (2A) cific refractivity), Kc, can be related to the Drude constant: The coefficient (e-I)/p is the tensorforrn of the Drude (or Newton-Drude)constant, Ko (e.g.,Bloss 1971). Ke= (er/z+ 1) Kc (5A) Thus the Drude constanthas a sound basis in point- dipole theory. In this case,it is not clear that the Gladstone-Dale The relationship of point-dipole theory to other constanthas a meaningfultensorial representation. optical constantsis lessusefirl. For instance,the spe_ WhereasGladstone-Dale constants have proven ciflc refractivity, K1, is given by the Lorenz-Lorentz valuable for predicting mean indices of refraction on equation(Bloss 1971),and can be relatedsimply to the basisof compositionand density (e.g., Mandarino the nontensorial Drude consuulr: 1976, 1978, 1979, 1981,),there appearsro be no extension by which the principal indices of Ko = (e + 2)K" (3A) refraction of an anisotropic crystal can be anti- cipated,nor car the orientationof the indicatrix be This equationextends naturally to tensors,i,e,, e and predicied.