American Mineralogist, Volume 73, pages 105-118, 1988

Polytypism in : A polyhedral approach to energy calculations

Rrcrr,lno N. Annorr, Jn. Department of Geology, Appalachian State University, Boone, North Carolina 28608, U.S.A' Cnl.RLrs W. BunNruvr Department of Earth and Planetary Sciences,Harvard University, Cambridge,Massachusetts 02138, U.S'A.

Ansrucr There has beenwide speculationabout the structural factors responsiblefor the observed frequency of the different polytypes. The purpose of this investigation is to id€ntit' the important factors and to test the relevant ones by using calculated cohesive energies to discriminate between alternative model structures.Sets of structureswere constructed, basedon , pyrophyllite, , , and anandite. Each model set was specifi- cally designedso that all structuresin a given set would have essentiallythe same short- range repulsive forces and essentiallythe same van der Waals forces. The diference be- tween the cohesiveenergies of two structurescould then be ascribedessentially entirely to Iong-rangeCoulomb electrostaticforces. The method is a natural extension of the poly- hedral approach to crystal chemistry. Of the various parameters designed to characteize deviations from structural ideality, only Az and a are specificallyrelated to the articulation ofthe various coordination poly- hedra. Other parametersmeasure the departure from ideal geometry of individual coor- dination polyhedra.We identify Az as the best parameterfor predicting the stablepolytype. Most micas with Az lessthan 0.1 A are lM polytypes,regardless of composition and other structwal parameters. Most micas with Az greaterthan 0. I A are 2M, polytypes regardless of composition and other structural parameters.The conspicuousexceptions include the lithian micas and the brittte mica anandite, [Ba(Mg,Fe)r(SirFe)O'o(OH)S].Though signif- icantly variable, a bearsno apparentrelationship to the stability of the I M vetsts the 2M ' polytype. The calculations support both theoretical and empirical observations regarding (a) the predominance of lM and 2M, polltypes and (b) the scarcity of 2M' and 20 polytypes. Al-Si "disordering" in micas is subject to certain rules: (a) the principle of aluminum avoidance and (b) the same ratio of Al to Si in all tetrahedral sheets.The relative posi- tioning oftetrahedral Al cations on opposite sidesofthe octahedralsheet seems not to be important. For muscovite-2M,, there are at least 48 crystal-chemically reasonableAl-Si orderings with very nearly the same low energy.The different orderings should be more or less equally representedin actual structures.Calculated eneryiesare most sensitive to interlayer Al-Al separations.In lM and 2Mt polytpes, Al-Si orderings consistenturith a 2, axis parallel to b are especiallyfavorable.

INrnonucrroN scarcityof 2Mr,2O, and6tlpolytypes (Radoslovich,,l959, There are fundamental differencesin the unitJayer 1960;Giiven, 1971;Thompson,l98l),thereislittlecon- structures of dioctahedral and trioctahedral micas (Bai- sensusand much speculation(Giiven, l97l; Bailey, 1984) ley, 1984). The differencesare in some combination re- regarding the structural factors responsiblefor the over- sponsible for the general observation that most diocta- whelming predominance of 2M,-dioctahedral and lM- hedral micas are 2M, polytypes, using the notation of trioctahedral micas. Ramsdell (1947), whereas most trioctahedral micas are The purpose of this researchis to identify the relevant ]M (Batley,1984).Other simple polltypes-2Mr, 20, 37:, structural factors controlling mica polytypism and to test and6fl(SmithandYoder, 1956)-arecomparativelyrare the important ones by comparing calculatedcohesive or largely restricted to specialcompositions; for instance, energiesfor appropriate model structures.Although the anandire [Ba(Mg,Fe)r(SirFe)O,o(OH)S],which is a 20 primary goal is a better understandingof polytypism, this brittle mica, and the lithian micas, many of which are3T end is not achievedwithout paying considerableattention or 2M, polytypes (Bailey, 1984). Although there are ex- to short-range ordering of tetrahedral cations (Abbott, cellent crystal-chemical grounds for understanding the 1984;Abbott et al., 1986).The calculationshelp to iden- 0003-004x/88/0102-0 105$02.00 105 106 ABBOTT AND BURHNAM: POLYTYPISM IN MICAS

TABLE1. Structuralparameters of trioctahedraland dioctahe- dral micas

Trioctahedral Dioctahedral ldeal (24 structures) (22 structures) nvg T-o 1Ay 1 66(0.01) 1.64(0.02) Avg r (') 109.5 110 5(0.5) 1110(0 8) nvs Ml-O 1A1 2.08(0.02) 2.21(0.06) Avs.t(M1) (") 54.75 s8.9(0.6) 61(1) Avg.M2-o- (A) 2.08(0.02) 1.95(0.04) Avg '/(M2)(') 54.7s 58.8(0.6) 57.0(0.4) x(lH) Avg a (') 0 6(3) 11(4) Avg Az (A) 0 0.02(0.02) 0 18(0.06) Avg.s -0.333 0.333(0004) -0.37(0,01) oburul* Note: The table includesonly nongermanian,nonlithian, and nonbritile -.---* trioctahedraland dioctahedralmicas from Bailey(1984). Values in paren- oapical v theses representone standard deviation. H ------> z(1 11)

tify the direct and indirect influences on polytypism of the following factors: (l) OH = F exchange,(2) Al-Si x(1 f1) y(111) ordering, and (3) Ml + vacancy exchange(trioctahedral vs. dioctahedral structures). bc Fig. 1. parameters.(a) Projectionof tetrahedral Rnr,rvaNr STRUCTURALpARAMETERS Structural sheetonto (001).The shadedtetrahedra are in the idealconfig- The differencesin unit-layer structuresofdioctahedral uration(a : 0). The symmetryof the sheetis 6 mm. Solidout- and trioctahedral micas can be quantified, for the purpose linedtetrahedra are rotated through the anglea. The symmetry of comparison, in a number of ways. Severalof the stan- of the tetrahedralsheet is reducedto 3m. (b) Projectionof part dard parametersfor 69 structural determinations of di- ofa unit layeronto /M setting(0 1 0), looking down the unit layer : : octahedral and trioctahedral micas have been compiled diad.The following sites are indicated: T tetrahedralsite, M : : by Bailey (1984). The parametersfor 14 brittle micas octahedralsite, Oo.*, basaloxygen, O.o." apicaloxygen. The followingstructural parameters are indicated: the 06""1-T-0"'1""1 have beencompiled by Guggenheim(198a). The param- angle,r; the octahedralflattening angle, ,!; and the tetrahedral eters measure the departure given of a structure from shift,s. (c)View ofportion ofunitJayerstructure, looking down (Pauling, ideality 1930;Jackson and West, 1930;Gruner, lM settinga axis.The parameterAz is indicated. 1934;Pabst, 1955). Table I presentsthe statisticson the structural parameters for 46 nonlithian and nongerma- nian micas from Bailey (1984).The parametersare ex- plained in Figure I and include (l) the tetrahedral rota- ger, 1982; Hazen, 1985) of Si and Al tetrahedra.The av- tion, a (0" < o < 30.), (2) the obuor-T-ouoi*,bond angle, eragesof the M-O- distancesare different for dioctahedral a (aia.ur: 109"28'),(3) an angularmeasure of octahedral and trioctahedral micas, but this is merely a reflection of flattening, ,1,({,o*r: 5444'), (4) the interlayer tetrahedral the size of the common M substituents, M2 : Al and shift, s (approximately0.333a), and (5) a measureof de- Ml : vacancy in the former versus Ml : M2 : Mg or parture from coplanarity of the basal oxygens, Az : Fe2* in the latter. The averagevalues for r/ (Ml or M2) - [z(Oo^u,)."* z(Oo^J--][c sin B], (0 A < az < 0.35 A). are directly proportional to the averagesof the M-O dis- Essentially,Az measuresthe amplitude of corrugationsin tances(M : Ml or M2). The tetrahedralshift, s, is greater the surfacedefined by the basal oxygens.In Table I we in the dioctahedral micas than in the trioctahedral micas have also included statisticson the important mean bond becausethe disparity in the sizes of the Ml (: vacant) lengths.T-O-, MJO, and !fl-Q-. and M2 coordination polyhedra is greater in the former Most of the parameters-FO, r, M-Q- (M : Ml or than in the latter. M2),'lr, and s-are specifically related to the geometries Unlike the other parametersin Table l, a and Az are of individual coordination polyhedra. These parameters direct measuresof the manner of articulation of the var- are strongly dependent on cation substitution, hence on ious coordination polyhedra. Both parametersare sensi- bulk composition. The mean TlO distance,for instance, tive to variations in the other parameters.Hence, they is a measureof the ratio of Al to Si (Smith, 1954;Baur, happen to be distinctly different in the dioctahedral and 1970; Hazen and Burnham, 1973).The similarity in the trioctahedral micas, but only becauseof the differences averagesof the T-O- distancesfor the dioctahedral and in the common Ml and M2 substituents.Because they trioctahedral micas simply reflects the common ratio of are not specificallyrelated to the geometriesof individual Al to Si (approx. %).The value of z is essentiallythe same polyhedra, but rather to the way the polyhedra are con- for dioctahedral and trioctahedral micas by virtue of the nected, a and Az share certain advantagesin identifying inherent rigidity and incompressibility (Hazen and Fin- causesunderlying the obvious correlation between the ABBOTT AND BURNHAM: POLYTYPISM IN MICAS r07 polytype and the occupancyofthe octahedralsites (Table \Y 2Y1211220 3r trioctahedral.r-v l): a 0) dioctahedralooo-a l. The value ofa is independentofstructural variations o L due to homogeneoussize-scaling. Two structuresthat are c') 0) distinguishableon the basis of T-O and M-O values may o.^ .,;,':tiii:!:S:ii:t{+9,,.ii1g:.1i}i.,:i:iF,' be the samein other respectsif the ratios of T-O to Ml- D- O and M2-O are the same in the two structures. One structure is simply homogeneouslybigger than the other. Problems associatedwith size-scalingof this sort may be more insidiously embedded in the complex chemistries ". :o and structuresof micas than might otherwisebe indicated Buoo by a simple inspection of T-O and M-O distances.For t,i this reason,we have avoided the size-dependentparam- eters, such as TE and NFO, in order to avoid false or ambiguous conclusions based on distinctions between az (A) parts structures,or thereof, that differ only in size-scaling. Fig. 2. Plot of d vs. Lz for nonlithianmicas culled from 2. Though Az does vary in responseto homogeneous Bailey(1984) and Guggenheim (198a). R&R : Richardsonand size-scalingbecause it is a function ofc sin B, the depen- Richardson(1982), T&R - Takedaand Ross (1975), talc : Rav- dency is small. This can be seen when it is considered ner and Brown (1973),pyrophyllite : Lee and Guggenheim that the percantvariation in Az as measuredfor instance (198r). by l))o/Az-.u., where o : the standard deviation of the z(Oo.*,)values, is very much greater than the anticipated : - percent variation in (c sin il/n (: l0 A), where n is the refined (flnal R 19.50/o)and may be discredited. Ma- number of unit layersin one c translation,i.e., n: I for terial from the same locality has since been shown to be Ihe IM polytpe, n : 2 for 2M,, etc. Consequently,vari- disordered with only a statistical tendency toward 2M' ations in A,z arelargely independent ofvariations due to stacking(Bailey, pers. comm., 1987).(7) Except for an- homogeneoussize-scaling. A better definition for Az might andite, the tetrahedral rotation (a) bears no obvious re- be r[z(Oo".",)** - z(Oun.ur)-,"],which would be truly in- lationship to the polytype. (8) The combination of high dependent of homogeneous size-scaling.However, we A.z and low a may be crucial in stabilizing the 2O poly- have elected to retain the original formulation of Az for type. (9) The brittle micas have extremevalues for a. The the sakeof comparison with other investigations. margaritesand two of the clintonites have the highest a 3. Finally, it should be noted ttrat Lz and a are signif- values, and the polytype, 2M, and 1M, respectively,de- icantly more variable than any of the other parameters, pends on Az in a manner that is consistentwith the po- both within and between the major dioctahedral and tassiun micas. At the other extreme, ananditeshave very trioctahedral groupings. low a values. (10) There are lM-tioctahedral Ba-rich Figure 2 is a plot of a and Az for the nonlithian micas micas with low Az values.Thus, the Ba in anandite is not from Bailey (1984) and Guggenheim(1984). The trioc- necessarilyresponsible for the 20 slacking' tahedral micas are plotted as solid symbols; the diocta- With regard to the relative stability of IM and 2M, hedral micas, as open symbols. The shapeof the symbol polytypes, Az is the single most important parameter of indicates the polytype. Specificchemical traits are keyed those listed in Table 1 and in Bailey's (1984) and Gug- by number to individual symbols wherever a coherent genheim's (1984) more detailed tabulations. We recog- group, such as the paragonites,could not be encircled nize three kinds of comrgations: conveniently. The group marked Ge includes synthetic 1. Pure conugations. This type of distortion comple- germanian micas in which Ge has been substituted for ments a vacancy in the Ml site and, hence,is prominent Si. The data points marked T&R (Takedaand Ross, 1975), in dioctahedralmicas. Pure distortions may also occur in R&R (Richardson and Richardson, 1982), talc, pyro- trioctahedral micas in conjunction with the ordering of phyllite, and 4 (anandite),are the basesfor model struc- very diferent cations on the Ml and M2 sites,such as in tures used in our energy calculations.The following fea- the trioctahedral lithian micas (Bailey, 1984). Pure cor- tures should be noted: (l) Most of the trioctahedral micas rugations,as defined here, have nothing to do with cation are lM polytypes.(2) Most of the dioctahedral micas are substitution on tetrahedral sites. Effectsdue to pure cor- 2M, polytypes. (3) With two exceptions (phengites),the rugations may, however, be difficult to separatefrom ef- lM micas are restricted to low values of Az (less than fects due to cation substitutions on the tetrahedral sites. approximately0.10 A). (4) With three exceptions,rhe2M, Certainty that comrgationsare pure, therefore,exists only micas are restricted to high values of Az (greater than in talc, pyrophyllite, celadonite, or other micas with no approximatelv0.10 A). (5) The dioctahedralmica cela- substitution on the tetrahedral sites. donite, with a low Az, is a IM polytype. (6) The triocta- 2. Ordered substitutional corrugations.Individual and hedral mica clintonite, with a high Az, is a 2M, polytype. mean bond lengths for tetrahedrally coordinated Al are But the structure (Akhundov et al., 1961)was very poorly substantiallylonger than for tetrahedrally coordinated Si r08 ABBOTT AND BURHNAM: POLYTYPISM IN MICAS

(Smith, 1954; Baur, 1970;Hazen and Burnham, 1973). structural determinations. The unit-layer structures for Thus, among other changesin a mica structure, the sub- both talc and pyrophyllite were idealized slightly to con- stitution of Al for Si produces differencesin the z coor- form with |M-A/m symmetry. The actual structuresare dinatesofthe basaloxygens. Ifthere is short-rangeAl-Si triclinic in one-layer (10-A) polytypes that defy descrip- ordering, the resulting distortions may be modulated so tion according to conventional stacking theories (Smith as to produce coherent corrugations. Substitutional and and Yoder, 1956;Thompson, l98l). Unlike true micas, pure corrugationsmay be involved in simple constructive in talc and pyrophyllite, the basal oxygensin the top of or destructive interference, or some complex combina- one layer arejuxtaposedwith thosein the bottom ofthe tion of these. The possible effects of such interferences next layer in such a way that the cations ofthe respective are an interesting subject for speculation,but lie beyond tetrahedral sheets do not superimpose when projected the immediate concernsof this investigation. down c*. Nonetheless,the observedunit-layer structures representsa special case.In margarite there for both talc and pyrophyllite are very closeto the slightly is a strict alternation of Al and Si tetrahedra(Guggenheim modified C2/m stuctures for the parametersreported in and Bailey, 1975)in accordancewith Loewenstein's(1954) Table 2. In the hypothetical model structures, adjacent principle of aluminum avoidance. Each basal oxygen is sheetsof basal oxygens were positioned as they are in shared equally between one Al and one Si such that the mrcas. position ofeach basal oxygen is influenced equally by the None of the structure determinations provided coor- two cations. Consequently, the corrugations, which are dinates for H. Initially, this problem was avoided by con- significant (Fig. 2), must be regardedas pure. sidering the OH to be replacedby F, thus neglectingin- 3. Random (disordered)substitutional distortions. The fluencesof OH dipoles. Most of the models were tested observed spacegroup of muscovite-2M,, C2/c, permits as F-micas. Later on, further work was deemed unwise at least partial ordering of Al and Si on the tetrahedral without taking into account the influencesof OH bonds, sites.Yet there is no evidencefor long-rangeordering of whose orientations are so very different in trioctahedral any kind (Bailey,1975,1984). We will addressthis prob- and dioctahedralmicas (Bassett,1960; Giese, 1984).A lem in more detail in subsequentsections. On the aver- seriesof OH model structures was manufactured in ac- age,the two distinct tetrahedral sitesin C2lc cordancewith the findings of Giese (1984) regarding the are populated by the same Yt ratio of AVSi. The reported Iength and orientation of the OH bond. For both trioc- Az values measurea pure distortion, but for an average tahedral and dioctahedral model structures,the OH bond structure,possibly giving a false impression of the actual length was fixed at 0.9 A. In the former, the bond was shapeofthe surfacedefined by the basal oxygens.Ifthere oriented perpendicular to (001) and directed away from is short-rangeordering, the basal oxygensmay be mod- the plane ofthe octahedralcations. In the latter, the OH ulated in small domains, perhaps in a manner that is bond was oriented in the unit-layer mirror plane [i.e., contrary to that suggestedby the measured Az for the (010) in lM and 20, (ll0) in 2M,, and (1,1,0) iyr2M,l averagestructure. On the other hand, ifthere is not even and inclined 12"from (001). short-range ordering, Az for the average structure may The structure determinations by Takeda and Ross measure the only systematic disturbance of the surface (1975) are unique in providing the only direct compari- defined by the basal oxygens-a systematic disturbance son between naturally coexisting lM and 2M, micas. superimposedon random substitutional effects. Among other observations, they showed that the two In this study, agreal deal ofeffort was devoted to sort- polytypes have virtually the samecomposition and unit- ing out the relative importance of pure corrugationsver- layer structure. In this casealone, there was no need to sus substitutional effects.This end was achievedby vary- construct one or more hypothetical polytypes from the ing the relevant parameters in the following model unitJayer structure ofone real polytype. Energy calcula- structures and discriminating between the alternative tions were performed on the actual structures. structureson the basis of calculatedcohesive enereies. For muscovite, a hypothelical IM polytype was de- rived by transformation of the unit-cell geometry and TBsr srnucruREs atomic coordinatesfrom the initial 2M, structure of Ta- Five unit-layer structures, and modifications thereof, ble 2. The initial 2M, atomic coordinateswere very stghtly were tested in two or more of the four polytypic config- modified to conform with A/munit-layer symmetry.This urations: lM, 2M,, 2Mr, and,20. The unitJayer models was necessaryin order to insure a unique mapping of the were derived from well-refined structure determinations atomic coordinatesfrom the 2M, unit-cell orientation to on natural specimens:talc (Rayner and Brown, 1973), Ihe lM unit-cell orientation. pyrophyllite (ke and Guggenheim,I 9 8 I ), biotite (Takeda Hypothetical 20, 2M,, and 2M, polytypes were fash- and Ross, 1975), muscovite (Richardson and Rich- ioned using the initial C2/ m unit-laver structuresof both ardson, 1982),and anandite(Filut et al., 1985).Table 2 talc and pyrophyllite (Table 2). Only the hypothetical lM lists unit-cell dimensions, mean bond lengths, and rele- polytype was constructed from the initial 2O anandite vant structural parametersfor the model structures.With structure (Table 2). the exception of pyrophyllite and talc, the parameters Table 3 reports the real and hypothetical models tested correspond essentially to those for the experimental in the investigation. Each row in Table 3 representsa ABBOTT AND BURNHAM: POLYTYPISM IN MICAS 109

TeeLe2. Structural parameters and lattice geometries of model structures

Biotite Pyro- Talc phyllite Muscovite Anandite C2lm C2lc C2lm C2lc Pnmn a (A) 5.293 5.331 5.329 5.160 5.199 5.439 b(A) N3 9.179 9.231 9.234 8.966 9.027 9.509 c (A) 9.499 10.173 20.098 9.334 20.106 19.878 Bf) 100 10016 95.09 100 95.78 90 r1-o-(A) 1 622 1.659 1.657 1.617 1.643 1.620 z(r1)(.) 109.5 10916 110.4 110.3 109.4 11 1.0 112.5 T2:O(A) 1 662 1.643 1.799 r[r2)f) 109.s 110.2 111 .0 112.6 q(") 0 1 76 7.7 10.2 11.0 0.9 az (A) 0 0.01 0.01 0.02 0.24 0.22 0.30 s +0.333 -0.324 -0.335 -0 334 -0.383 +0.378 10.337 Ml-o- (A) 2.067 2.086 2.086 (2.253)' 2.097 l/(M1)f) 54.75 57.97 59.2 59.2 (61.7). 59.8 M2-o-(A) 2.076 2.086 2.068 1.912 1.940 2.236 ,r(M2)f) 5475 s8.09 58 9 58.9 57.1 56.6 55.3 r./3-o-(A) 2.228 {(M3)f) 55.2 I\/4-O-(A) 2.120 ,/(M4) f) 60.2 - M1-O and P(M1)relative to centerof M1 (vacancy). model set,all of whosemembers have essentiallythe same anandite-2o (Pnmn) permit partial (C2/c, Pnmn) or no unit-layer structure, at least within the limitations of (l) (C2/m) ordering of tetrahedralcations. In muscovite-2M, polytype-imposed symmetry differencesand (2), where and biotite-2M,, where limited ordering is possible, the relevant, Al-Si ordering differences.Each entry in Table structural determinations (Richardson and Richardson, 3 under the different polytype headingsis the number of I 982; Takeda and Ross, 1975) indicate that Al and Si are distinct Al-Si (Fe-Si, in the case of anandite) ordered essentially completely disordered. The structure of an- i'Fe structures that were constructed for the corresponding andite (Filut et al., 1985)indicates that and one-third polytype. The rationale for the limited choice of Al(Fe)- presentedin Si orderings (in contrast to Giese, 1984) is TneLe3. Modelsets on whichenergy calculations were per- the next section. For pyrophyllite and talc models, the formed tetrahedral sheetsare pure SirOr; henceone structure per polytype in each model set is sufficient. For the model End Model char- member acteristics lM 20 2M, sets labeled prs in Table 3. Si-O and Al-O (ort"Fe-O) Biotite model sets- bond lengths were adjusted by distance-least-squares 8 prs F4 analysis (Baerlocheret al., 1977). The details of the oH4- 8 calculationsare describedin a later section.The Al(i"Fe)- OHDLS4_ I Si orderings for the polytypes of the non-ots model sets Muscovite model sets. are distinguishablefrom one another only on the basis of F4 I oH4- I chargedistribution. For instance,in each 1M structure of OH Az:O 4 - I the non-prs OH-biotite model set, the atomic coordi- OHDLS4- 12 nates conform to C2/ m, whereasthe chargedistribution Pyrophyllitemodel sets (owing to the ordering of Al and Si) is in a lower-order F 1- F Az: O spacegroup (P2b P2, PI, and P;T7;see Abbott, 1984). OH 1 11 For each structure in the ors-adjusted OH-biotite model OH Lz: 0 1 11 set, both the chargedistribution and the atomic coordi- OH nates have the same reduced symmetry. Talc model sets F 1 1- The comrgations were removed from some model sets F Lz: 0 1 1- by setting the z coordinate of each basal oxygen to the OH 1 11',I 1- averageof the z values in the actual comrgated strucure. OH A,z: 0 1 One model set for OH-pyrophyllite (a : 0) was con- Anandite model sets OH 24 structed with the tetrahedral rotation (a) set to zero. OH DLS Ordering of tetrahedral cations Note: Numberof Al-Si (Fe-Siin the case of anandite)orderings. Boldface values indicate lowest-energypolytype. The important observed spacegroups for bioti1'r-(A/c * Resultsreported earlier (Abbott et al., 1986) and only summarized in 2M, and C2/m in 1tr4), muscovire-2M, (C2/c), and nere 110 ABBOTT AND BURHNAM: POLYTYPISM IN MICAS

of the Si are disordered on one of the two distinct sites proach is justified becausesymmetrically equivalent or- (T2 : FenSin),with the remainderof the Si on the other derings have the same energy,hence are redundant, and site (Tl : Si). As we alreadynoted, Bailey (1975,1984) becausecrystal-chemically unreasonablestructures will has argued that there is no evidence for long-range or- have high energies,hence correspondinglylow represen- dering in either the 2M, or the lM polytypesof muscov- tations according to Boltzmann's law. ite or biotite, even when the structures are reconsidered We have adopted the third approach here, selecting in reduced-symmetryspace groups. Recent nuclear mag- only those Al-Si ordered structuresthat fulfill the follow- netic resonancestudies (Sanz and Serratosa, 1984; Sanz ing criteria: et al., 1986;Herrero et al., 1985;Herrero, 1987)indicate L For the muscovite and biotite model sets, each2M, a somewhat ill-defined short-range ordering that does, and 1M Al-Si ordered structure is symmetrically distinct however,comply with Loewenstein's(1954) principle of under the space-groupoperations of C2/c and C2/m, re- aluminum avoidance. Other workers (Gatineau, 1964; spectively.For the anandite model sets, each2O and lM Gatineauand Mering, 1966;Abbott, 1984)have postu- Fe-Si-ordered structure is symmetrically distinct under lated models for short-rangeordering, basedon X-ray or the operations of Pnmn and P2/m, respectively. electrondiftaction observations.Abbott et al. (1986)have 2. All tetrahedral sheetsin a given structure have the consideredthe consequencesand energeticsof short-range sameratio of Al to Si (Abbou, 1984;Abbott et al., 1986). ordering oftetrahedral cations. The problem ofdisorder- This eliminatesfrom considerationany subgroupof C2/m ing versus short- or long-rangeordering is crucial to the containing the mirror plane or the a-glide plane and any calculation of realistic cohesiveenergies. subgroupof Pnmn containing the mirror plane.A11 model The cohesiveenergy ofa structure dependson nearest- structuresobey Loewenstein's(1954) principle of alumi- neighborforces (Burnham, 1985).In the ionic modeling num avoidance(Sanz and Serratosa,1984; Sanz et a1., approach,these forces make senseonly in the context of 1986;Herrero et al., 1985;Herrero, 1987). individual sites in a stmcture being occupied by discrete After screeningthe possibilities (28 for lM; 1820 for atoms (i.e.,T : Si or Al) and not by hybrid atoms (e.g., 2M,; 1820 for 2O), there are only four symmetrically dis- T : Al,Si,,) (Giese,1984; Burnham, 1985).Presumably, tinct and crystal-chemicallyreasonable lM orderedstruc- the real structure consistsof a Boltzmann distribution of tures (Abbott, 1984), twelve 2M, structures,and twelve differently orderedunit cells.At present,the problem can 20 structures.Only one of the 2M, model subsetsin Ta- be accommodatedonly crudely by three approaches: ble 3 (or"s-adjustedOH-muscovite) includes all l2 of the l. Consider all possible ordering schemes. Even in Al-Si orderings.Each of the other muscovite and biotite seeminglysimple cases,this can lead to an overwhelming model sets-created for an earlier study (Abbott et al., number of structures.The problem becomesintractable, 1986)-includes only 8 of the l2 possibilities,though we and for this reasonalone, the approachhas enjoyed little, are confident, on the basis of the 2M, structure calcula- ifany, popularity. tions on all 12,that our conclusionsare unaffectedby this 2. Select,at random, a manageablesubset of structures earlier omission. Two of the 4 possible anandite-lM from the total number of possibilities. This approachhas structuresand 8 of the 12 possible anandite-2o structures beenused with somemeasure of successby Giese(1984, were eliminated becausethey were not consistent with 1986)who selected100 Al-Si orderedmuscovite struc- the observed(Filut et al., I 985) partial ordering ofFe and tures from 1820 possibilities and 100 margarite struc- si. tures from I 2 870 possibilities.The procedurehas at least Each ofthe ordered structurescan be characterizedby one serious flaw in caseswhere only one or two particu- a spacegroup that is subordinate to the spacegroup of larly favorable structuresexist out ofhundreds ofpossi- the disordered or partially ordered structure. The space bilities, as in the caseof margarite (Giese, 1984).In such groups are reported in subsequenttables. cases,the one or more exceptional casesmay be over- We have included tetrahedral-site orderings that vio- looked, ifthey cannot be anticipated a priori. In the case late what one of us (Abbott, 1984) has referred to as Gii- of margarite (Giese, 1984), the one favorable structure ven's rule. Gi.iven(1971) argued that two apical oxygens could be anticipated and was included in the set ofstruc- of Al tetrahedraforming the sameshared octahedral edge tures tested.The approachdoes not discriminate between should be especiallyunfavorable with respectto the local symmetrically equivalent structures,nor does it discrim- balancing of electrostatic charge. The rule refers to the inate between crystal-chemically reasonableand unrea- situation where both of the tetrahedra in Figure lb are sonablestructures. occupiedby Al. The sum of the Pauling (1960) bond 3. Use a set of ordered structures consisting of only strengthsreaching each apical oxygen is 1.75. Thus, two symmetrically distinct arrangementsthat obey reason- electrostaticallyundersaturated oxygens are juxtaposed. able crystal-chemicalprinciples. In their energy calcula- tions concerningthe relationship ofAl-Si ordering to the nr,s analysis position of Na in albite, by using symmetrically distinct The tetrahedralbond lengthsfor the hypothetically Al- Al-Si orderings that obey Loewenstein's(1954) principle Si ordered muscovite and biotite models (and Fe-Si or- of aluminum avoidance,Post and Burnham (1987)lim- dered anandite models) were adjusted by distance-least- ited the number of orderingsfrom 1820 to 56. The ap- squaresanalysis, using the program ols-zo (Baerlocheret ABBOTT AND BURNHAM: POLYTYPISM IN MICAS

I 640 I 780 TABLE4. T-O bond lengths (A) prescribedfor ols refinement l 6ff 1771 Muscovite and biotite si-o6"r 1,612 Al-o*""r 1.754 .S si-o"p,*, 1.633 Al-Oai""r 1.777 I ozJ o-.n-oo*, 2.648 O"pi*r-Oo"*r 2.882 "3 | 760 oo"*,-o*"d 2.628 O*"-O*"" 2.860 to | 617 l 756 l? Anandite ll ra Si-Oe"d 1.623 Fe-Oo"s 1.881 ti; 1612 | 752 Si-O"p,- 1.613 Fe-O"d""r 1'857 r 606 [ 2.689 O"pio-Oo"*r 3.107 o-b"r-ooa"ar1 2.645 t 2.711 O*"-Oo"." 2.996 I 6Cr0 174o | 2.594 X o*.d-oo"g I 2.646 AI [ 2.568 I Hazen&Burnham(1983) Notei There are three distinct 0"@-06d distances Eaur(1981) and three distinct Ob""d-Obsddistances for the Si tet- basedon pyrophyllite rahedra. Fig. 3. T-O bond lengthsas a functionof Xa' : Al/(Si + Al). Theheavy lines (marked 3) arebased on the Si-O distances in pyrophylliteand the averageofthe slopesofthe lines I and mulated by extrapolationfrom the dimensionsof the pure 2,which are due to Hazenand Burnham (1 973) and Baur (198 1), Si-O tetrahedron, through the dimensions of the Feu,Siu, respectively.The lower and upperheavy lines correspond re- disordered tetrahedron, to the dimensions for the hypo- spectivelyto the variationin T-Oo.",,and T-O,o,*,.The middle thetically pure Fe tetrahedron.In the extrapolation, a dis- heavyline is for the weightedmean T-O distance: 0.75(T- tinction was made among bonds involving different kinds or*,) * 0.25(T-O"pt",r). of bridging oxygens,i.e., Fe-Oo"".,-Sivs. Si-O0.."'-Si.By this means, four kinds of Oou"u1-O6u"'and four kinds of al., 1977 Villager, 1969). Only the atomic coordinates Ou,.ur4"pi."rbonds could be prescribed. The prescribed (x, y, z) for the basal oxygens and z coordinates for tet- bond lengths are reported in Table 4. Note that, unlike rahedral cations were allowed to vary. In the refinements, the Al-Si micas, in anandite the T-Ouo,"'bonds are short- the O-O interactions were assigned'lo of the weighting er than the T-Oo",.' bonds (Filut et al., 1985). assignedto the T-O interactions.This weightinggave rea- The unit-cell dimensions of each polytype were not sonable results and is consistent with weightings used permitted to change during the course of or,s analysis. elsewhere,as discussedby Burnham (1985). Changesin unit-cell geometry can be exaggeratedin oH The mean catiorcXygen bond length for a given tet- analysis, leading to structures for which the calculated rahedral site (T:O) in any Al-Si mica can be used as a eneryiescannot be compared realistically. Unconstrained measureof the proportion of Al and Si on that site. Two or,s analysesoften lead to unrealistic changesin the ar- algorithms (Fig. 3) are in common use for evaluating the ticulation of the tetrahedral and octahedralcoordination fraction of Al, X^r : Al/(Si * Al), on a given tetrahedral polyhedra. Even so, the conservative approach adopted site:T-O- : I .60 8 + 0. I 63XA, (Hu en and Burnham, I 97 3) here can lead to anomalous structures that are generally and Ts : 1.623+ 0.129X ^t(Baur, I 98 I ). In the present characterizedby unrealistic A-site coordination polyhe- analysiswe have formulated two new algorithms in order dra. Fortunately, the anomalous structures have corre- to treat the long T-O,o'*' bond and the short T-O**, bonds spondingly anomalous calculatedenergies and are there- separately. fore easilyrecognized. The oIs-adjusted /Mmodifications The mean tetrahedralbond length in pyrophyllite (Lee of OH-muscovite afford a good example. The unusually and Guggenheim,1981) is SrE : 1.617 A; it includes low calculated energiesfor the P2b Pl, and Pl' struc- 3 x Si-Ou"".,of 1.612A, and Si-O.oi"",of 1.633A. Using tures-when compared with the energiesfor the various thesebond lengths and the averageofthe slopes(Fig. 3) 2M, strvctures-are due to unrealistically short K-O dis- determined by Hazen and Burnham (1973) and Baur tances.The K polyhedra are not comparable in Ihe IM (1981),we get and 2M, structures.By way of contrast, the K polyhedra are essentiallythe same in the IM and 2Mr structuresof T-O*""':1.612 + 0.142X^l each biotite model set becauseAz is close to zero. and ENBncv cALcULATIoNS T4"pi*r: 1.633+ 0.144XN. If meaningful results are taken as an indication, there The mean O*orOo.*, and mean Ouor-4up*r distanceswere is reasonablej ustification for treating many silicate struc- scaled accordingly, relative to these distances in pyro- tures as ionic (Burnham, 1985).The proof of this rests phyllite. The relevant bond lengthsprescribed for the ors essentially on the generally favorable comparison be- calculationsare given in Table 4. tween propertiescalculated on the basisof the ionic mod- For anandite, the tetrahedral bond lengths were for- el and measuredproperties. In the ionic model, there are lt2 ABBOTT AND BURHNAM: POLYTYPISM IN MICAS three important contributions to the cohesiveenergy, us- (a) The differencesare anticipatedto be small. This makes ing the terminologyofBurnham (1985):a long-range,or the method especiallyeffective in identifying an unusual Coulomb, electrostaticterm; a short-rangerepulsive term, structure for which the energydeviates significantly from and a van der Waals term. Of the severalcomputer pro- the mean. ft) The formulation for the Coulomb electro- grams now available for calculating the cohesiveenergy, static energy is exact, and in principle the energy can be we used wvrrN (Busing, l98l). In our calculations,we computed to any desired degreeof precision. (c) The sig- have included neither the short-range repulsive energy nificanceofthe resultsis directly proportional to the care nor the van der Waals energy.We report only the Cou- taken in the construction of the model structures.In the lomb electrostaticenergy. This simplification is justified end, we point to the consistencyof our resultswhen com- in this study for the following reasons: pared with the natural statesof micas and our expecta- l. The models are based on the refined structures of tions basedon crystal-chemicalprinciples. naturally occurring minerals. We have assumed,a priori, that the structures representequilibrium or, at the very Summary of previousresults least, near-equilibrium configurations. Under these cir- Some of our resultspertaining to biotite and muscovite cumstances,the absolute magnitude of the short-range have been reportedelsewhere (Abbott et al., 1986).For repulsive energy is approximately 100/oof the magnitude the sake of comparison with the latest calculations, the of the long-range Coulomb electrostatic energy (Giese, findings of this earlier work are briefly summarized here: 1984),and the van der Waalsenergy is small. (l) For trioctahedral OH-micas (OH-biotite), the lM 2. Our structureswere fashionedin such a way that the polytype is roughly 1.75 kJ/anion more stable than the nearest-neighborbonding relationships are, as nearly as 2M, polytype. (2) For trioctahedral F-micas (F-biotite), possible, identical in all members of a given model set. the energy difference between the lM and 2M, polytypes This applies even to the ors-adjusted structures.The es- is about 9 I kJlanion in favor of |M. (3) For dioctahedral sential differencesbetween any two members of a given OH- and F-micas (OH- and F-muscovite), rhe 2M, poly- model set involve some combination of (a) the relative type is 1.2 to 2.1 kJlanion more stable than IM. (4) For position of the A(Fe)-Si ordered tetrahedral sheets on a given polytype and octahedral sheetstructure, the var- opposite sides ofthe octahedral sheet,(b) the manner of ious Al-Si orderings (with or without prs refinement) have stacking successivelayers (the polytype), and (c) the ge- slightly different energies.(a) ln 2M, and lM polytypes ometry of the interlayer-cation coordination polyhedron of biotite or muscovite, the lowest-energyAl-Si orderings (K, Ba, vacancy).Under these circumstances,the short- are consistentlythe ones with the most even distribution range energy is essentiallyidentical for all members of a of tetrahedral Al atoms. That is, the most favorable or- model set, at least according to the currently available deringsmaximize the closestAl-Al separations.(b) In 1M methods for evaluating the short-rangeenergy (Burnham, and 2M, polytypes, Al-Si orderings consistent with a 2, 1985).It is therefore assumedthat the differencesin cal- axis parallel to b are especially favorable. The lowest- culated energiescorrespond to structural differencesfor energy 2M, ordeings are in subgroupsP2,/n, P2r, and which the differencesin short-rangeenergy terms and van P2,/c;the lowest-energy1M ordering is in subgroupP2,. der Waals terms are negligible. On the other end of the spectrum,Al-Si orderingswith a Becauseof the way we have constructedour model sets, 2 axis parallel to b have by far the highest eneryies.(5) the energydifference between two structuresin the same With the corrugations removed from the OH-muscovite model set is directly related to the manner of articulation (A,z:0), there is essentiallyno energydifference between of the diferent polyhedral units in the structures. The the lowest-energyAl-Si ordered 2M, and lM polytypes. structuresof a given model set may be thought of as being constructedfrom a fixed set ofcation coordination poly- New results and discussion hedra. In this sense,our calculations are a natural exten- Pyrophyllite and talc. Table 5 reports the results of our sion of Hazen's(1985; Hazen and Finger, 1982)"poly- calculations for talc and pyrophyllite. These model sets hedral approach to comparative crystal-chemistry." were prepared in order to examine the influence of pure Two notes of caution are in order: (l) We emphasize distortions on stacking,without complications due to Al- that any attempt to discriminate betweendifferent struc- Si substitutions. tures on the basis of calculated Coulomb electrostatic The calculations on the various model sets for pyro- energiesalone is meaningfulonly for membersof the same phyllite indicate that 2M, is the most stable poll4ype when model set. Structuresfrom different model sets,differing Az + 0 and a * 0. The range of calculated energiesin in nearest-neighbor bonding relationships, cannot be the unmodified OH model set spansa differenceof 1.74 compared realistically without taking into account both kJ/anion. Only the energiesfor the 2M, and 20 polytypes short-rangeand van der Waals interactions. (2) In some differ by more than one standard deviation (o : 0.7 kI/ cases,the rangeof calculatedenergies for the members of anion) from the mean energy(-7001.9 kJ/anion), such a model set is exceedinglysmall in comparison with the Ihat Err, I E,, 1 Err, 1 E2o. It should be noted that absolutevalues. This raisesthe important question of the the energyof the 2M, polytype is only 0. I I kJlanion less significanceof differencesthat may amount to lessthan I than that of the lM polytype. For F-pyrophyllite, the in 7000 kJ. The readershould bear the following in mind: differencein the energiesof the lM and 2M, polytypesis ABBOTT AND BURNHAM: POLYTYPISM IN MICAS 113 greater, approximately 0.56 kJ/anion, and still in favor polytypes. When a and Az are both close to zero, as in of the latter polytype. When the structure is modified so talc, the coordination polyhedron for the A site (vacant) that Az is zero, the relative stability of the lM and 2M, is essentially hexagonal prismatic-the coordination polytypes reverses,and the difference in the energiesof number is 12 and essentiallythe same in all four poly- the two polytypes increasesto 1.36 kJ/anion for OH-py- types. It follows that if the geometry of the A polyhedron rophyllite and 0.9 kJlanion for F-pyrophyllite. This re- alone determinesthe polytype, two possibilities may sub- versal shows the influence ofpure corrugations in stabi- tend: (l) The different polytypes should have the same lizing the 2M , polytype relative to the I M polytype. When frequency, or (2) the stacking sequencein a given struc- the structure is modified so that a is zero, the relative ture should be random. In fact, neither possibility gains stability ofthe various polytypes changes:Err, 1 E,, I much support from natural examples (Fig. 2). Even among Err, 1 Ero, favoing the 2M, polytype. In the a : 0 the low-a trioctahedral micas, the lM polytype seemsto model set, only the 2M, and 20 polytypes have eneryies dominate. differing by more than I o (0.7 kJ/anion) from the mean That 20 and 2Mt micas are indeed scarcein nature can energy (-7052.0 kJlanion). The high calculated energy surely be ascribedto the fact that most trioctahedral and for the 20 polfiype in the pyrophyllite model setsis con- dioctahedral micas have a values that differ significantly sistent with the extreme scarcity (Bailey, 1984) of this from zero (Fig. 2). Under thesecircumstances, the lowest- polytype in nature. energypoll'type is 2M, or 1M dependingon the presence Calculations on talc show that the four hypothetical or absence,respectively, of corrugations in the surface polytypes have nearly the same energy,regardless of the definedby the basaloxygens. It is interestingto note how- differencesbetween the model sets. For the unmodified ever that the three 2M'lithian micas reported by Bailey OH-talc model set, the difference between the highest (1984) all have a values around 5", and the one phengite- (2M,) and,lowest(1,41) eneryies is only 0.1 kJ/anion, and 2M, in Figure 2 has a value of a of approximately I l"! all of the calculatedenergies lie within I o (0.04 kJ/anion) At present, we can only speculateabout the causesfor of the mean (-6680.76 kJlanion). The calculated ener- theseanomalies. giesfor the IM, 2M,, and 20 polytypes differ by lessthan The difference in the calculated energiesfor 2M' and 0.03 kJ/anion! The very small differencesare obviously 2O pyrophyllites (Table 5) deservesadditional comment. insignificant; the four polytlpes are equally stable, hence In both of the model sets that include these polytypes, equally likely. Alternatively, the layer-stackingscheme, if the 2M, polytype has a significantly lower energy than it obeys a Boltzmann distribution, would be random. It the 20 polytype. In the unmodified model set, the energy is interesting to note that when the structure is modified differenceis 1.04 kJ/anion, whereasin the modified mod- by setting Az lo zero (from the already low initial value el set (a : 0), the difference is approximately twice as of 0.005 A1, the differencein the energiesof the 2M, arrd great, 2.06 kJlanion. This discrepancymight not other- /M polytypes increasesslightly, giving a correspondingly wise command much attention if both polytypes had slight preference(0.244.28 kJ/anion) Io lhe lM poly- higher energiesthan either 2M' or lM under all circum- type. This supportsthe observation (Fig. 2) thal Az is the stances.But when a : 0 and A,z> 0, the very low energy important parameter with regard to the relative stability of the 2M, polytype hints at a specialstructural problem. of the lM and 2M, polytypes. Figure 4a shows the A-site coordination polyhedron in For both the OH-pyrophyllite structures and OH-talc the 20 polytype under the conditions just stipulated, i.e., structures,the relative order ofthe four polytypes is con- a : 0 and L,z > 0. The polyhedron is an orthorhombic sistentwith Thompson's (1981) suggestionthat the ge- parallelepipedwith edgesparallel to the 20 a, b, and c ometry of the A-site coordination polyhedron (vacant in axes. Basal oxygens occupy the corners of the parallel- pyrophyllite and talc) is important in explaining the scar- epiped, and the A site is at the center. We shall refer to city of the 2M, and 20 polytypes. When the tetrahedra the O-O edgelengths as d. (short), d' (long), and d". The are rotated (o > 0) as in pyrophyllite, the coordination short edgein the (001) face,d", has a length equal to the polyhedron for the A site is trigonal antiprismatic (octa- shortest Oruor-Oru.urdistance. The edge length dt equals hedral) in the IM and 2M, polytypes and trigonal pris- t/3d,; and d., which is the shortest interlayer O-O sepa- matic in 2M, and 20 polytypes.Thompson (1981; Ra- ration, is approximately equal to d" (d" = d,). The lr:ace doslovich, 1959, 1960; Giiven, l97l) argued that the of the unit-layer mirror plane is shown on the (001) faces antiprismatic arrangement should be more stable, be- of the polyhedron. The four shortest interlayer O-O dis- causeinterlayer O-O repulsionsare less.Our calculations tanceshave the same length, d.. The next shortest inter- for the OH-pyrophyllites lend support on two accounts: layer distances are four in number, end-face diagonals (1) When o I 0, the 20 and 2M, polytypes have signif- eachhaving length\/@ +@. Other interlayer O-O dis- icantly higher energiesthan either the IM or the 2M, tancesare significantly longer. Figure 4b showsthe A-site polytype. (2) When q : 0, the order of stability of the polyhedron for the 2M, polytype under the same condi- polytypeschanges such that the hypothetical 2M,and 2O tions, a : 0 and A,z > 0. The polyhedron can be derived polytypeshave, respectively,the lowest and highestener- from the 20 A-site polyhedron by rotating one of the gies,thus emphasizingthe importance of a rotation with (00 I ) faces60' about the normal to (00 I ). The (00 1) faces regard to the stability (or instability) of the 2M, and 20 have the samedimensions as in the 20 polytype, d" : (O- tt4 ABBOTT AND BURHNAM: POLYTYPISM IN MICAS

TneLe6. Coulomb electrostaticenergies for anandite, Ba(Mg,Fe)"(Si.Fe3*)O'o(OH)S

Atomic Charge Energy Polytype coordinates distributions (kJ/anion) Non-DLSadiusted structure 1M Pzln n -6038.87 1M P2ln P1 -6044.17 Pnmn Pn2n -6038.04 ab 20 Pnmn P2,22, -6039.29 20 Pnmn P2jl n11 -6035.00 Fig.4. A-sitecoordination polyhedron in (a)the 20 polytype 20 Pnmn P11211n -6039.05 : and(b) the 2M, polytype,when Az > 0, and a 0. DLS-adjustedStruCtures 1M n- -6132.77 1M P1- -6124.71 |n 2M, polytype, the 20 Pn2n" -6139.34 O)o"*,distance, ard dt: \,3d,. lhe -6132.73 interlayer are of \ength d". 20 P2122; two shortest O-O separations 20 P2,ln11- -6122.59 The next shortest interlayer O-O separationsare six in 20 P112,1n" -6139.17 +@. interlayer number, each having lengthr/@ Other Note.'Boldtace values indicate low-energystructures. O-O distances are substantially longer. The eight A-O . Atomic coordinatesand charge distributionare consistentwith bond lengths are the same (: \/5d"/2) in both 2M, and samespace group. 20 polytypes! Recalling that a Coulomb electrostaticre- pulsive force varies inversely with the square of the in- teratomic distance,the sum of the repulsions due to the averagestructure are not responsiblefor the observed20 eight shortestinterlayer O-O distanceswould be approx- polytype. imately 200/omore for the 20 polytype than it would be In the second model set, the Si-O and Fe-O bond for the 2M, polytype. The 2M, polytype should thus be lengths were adjusted by ols. For thesecalculations, two more stablethan the 2O polytype. The argument is anal- of the 20 structures (Pn2n and Pll2'/n) gave signifi- ogousto the one offeredby Thompson (1981; Radoslo- cantly lower energiesthan the other structures.This sug- vich, 1959, 1960,Giiven , 197l) for the preferenceof tri- geststhat the actual structure may have short-rangeor- gonal antipismatic (octahedral) coordination over trigonal dering, possibly consisting of domains of the prismatic coordination. crystallographically equivalent complexions (Abbott, Anandite. Table 6 reports the resultsof our calculations 1984) of one or both of the low-energy configurations. on anandite. The first model set that we constructedwas The observed 20 polytype appears to be stabilized by predicted on the basis that Fe and Si were disordered on high Az and low a in conjunction with distortions caused one of the two crystallographically distinct tetrahedral by Fe-Si ordering. sites,as indicated by the structure refinement (Filut et al., Muscovite. Table 7 and Figure 5 present our most re- 1985). The averagestructure was treated as a fair repre- cent calculations involving ors-adjusted OH-muscovite sentationofwhat would be, in this case,a specialkind of structures.Ofinterest here are the very different charac- substitutional distortion. The energieswere calculatedfor teristics of the resultsfor the two polytypes, lM and 2M,. different Fe3+-Si charge-orderingsin which the bond The anomalously low energiesfor three of the ,lM struc- lengthswere left unmodified by ors. One of the lM struc- tures are due to unrealistically short prs-modeled K-O tures has a significantlylower energythan any other struc- distances.The K polyhedra are simply not comparable ture. This suggeststhat the distortions representedby the in the lM and 2M' ors-adjusted structures; hence the

Teele5. Coulombelectrostatic energies for pyrophylliteand talc model structures

Polytypeenergies (kJ/anion) End Modelchar- member acteristics 1M 20 2M, 2M, Pyrophyllitemodel sets T -6391.72 -6392.28 F Az: 0 -6461.80 -6460.90 OH -7002J8 -7000.74 -7002.59 -7001.78 OH Az: 0 -7069.36 -7068.00 OH a:0 -7052.24 -7050.98 -7051.89 -70s3.04 Talc model sets -6094.69 f -6094.76 -6095.43 T Az: O -6095.67 OH -6680.80 -6680.77 -6680.70 6680.77 OH Lz: 0 -6681.67 -6681.39 Note: Boldfacevalues indicatelowest-energy polytype. ABBOTT AND BURNHAM: POLYTYPISM IN MICAS 115

TneLe7. Coulombelectrostatic eneroies for muscovite

T-T': Energy <-4 Structure Al-Al'- (kJ/anion) Mean +_3' Muscovite-tM P2,I -999? P1l oo4o,Jct9 I ( _aann-oouu. rr r2(126.91) Pi+ 66g2.20 | fz+ 6382.20 | a(2Yr) Muscovite-2M, PA 31', -oo | /.cY I Pttl 31', 6613.47| -6615^^.- 60(2^, 67) P2T 31', ;6i3:i3 | P2.f 31', -6618.15) 4 P2lcl 32' 6612.85 q9l -#i;.fi ) 661s^^.- 64(2^. 7e) o y'- P21ltt 32', i L lv 42, -6613.72 o! Pt+ ) o!1 . Pnf -:::l El. -ooto l! Pt+ -oor/.cc9: ! az('a') ca 33', I t P2,+ 33', 6618.32 -::139:J P2,14 34', -oorsrqzsoy o o Pzln+ 34', oo rz.ro Ij (\ 6 \o \o Intralayerrelationships: muscovite-2M, \o \o rO I Structures that obey Gtiven's rule -6615.64\2 24]. -6616.26(2.41 Structuresthat violateGUven's rule ) Energy(kJ/anion) Interlayerrelationships: muscovite-2M1 -661 Structureswith c alide 5 57(230) Structureswith n glide -6615 43(2.68) = structuresviolate 6ilven's rule Structureswith 2 axis -6613 96(2.13) Structureswith 21axis -6618.24(0.14) Fig. 5. (a)Labeling oftetrahedral sites in referenceto Table (b) for 12Al-Si orderingsin muscovite- Note: Values in parenthesesrepresent one standard deviation 7. Histogramof energies . See Figure5a. 2M,, t Structuresthat obey Gtjven'srule. f Structuresthat violateGiiven's rule. to be somewhat less than the averagefor the lM struc- tures. That there are so many (12 plus symmetrically equivalent complexions, making 48 altogether) more or structures do not meet the criteria we have established Iess equally accessiblelow-energy states suggeststhat it for a valid comparison. The calculated energiescannot shouldbe disordered(Giese, 1984). As for the lM poly- therefore be used as a reliable guide to the relative sta- type, a hypothetical, disorderedmtscovite-2M' may con- bility of the 1M versus Ihe 2M, polytype. However, the sist of small domains of the different Al-Si orderings(Ab- K-O distancesare similar in the four 1M structures, so bott, 1984),or it may be a mosaicof individual, differently these structuresconstitute a valid set for comparing the ordered unit cells. efects of different Al-Si orderings.Likewise, the K-O dis- On closer examination of Table 7, it becomesevident tances are similar in the twelve 2M, structures; hence (Fig. 5) that there is a bimodal distribution of 2M' ener- they too constitute a valid set for comparison. Problems gies.For this reason,it is instructive to sort the 12 struc- with the modeling of the A-site geometry were not en- tures into subsets,each being characterizedby one or counteredin the other model sets. more common features.There are four distinct unitJayer The energiesfor the four Al-Si ordered 1M structures structures,identified as 3l',32',33', and 34'. The desig- are very different (o : 126.91kJ/anion), with three struc- nations refer to the Al-occupied sites as they are labeled tures(P2,, Pl, P7')having the very lowestenergies of all in Figure 5a. Note that, within the limits of the standard of the 2M, and lMstructures and the remainingP2 struc- deviation, the averageenergy is the same for each ofthe ture having the very highest energy.The Boltzmann dis- unitJayer groups. Evidently, the intralayer distribution tribution law would suggestthat a hypothetical 1M mus- of Al and Si (within the constraints set up earlier) has no covite should be dominated by the Pzr, PL, and PT? bearing on the bimodal distribution of energiesand, be- arrangements.The hypothetical structurewould probably causethe four unit-layer structuresare very different (in consist of small domains of the different Al-Si orderings terms of the relative positioning of the Al atoms), essen- and symmetrically equivalent orderings (Abbott, 1984), tially no bearing on the preferencefor the 2M' polytype. or it may consist of a mosaic of individual, differently The 12 structurescan be sortedin another way, according ordered unit cells. to elementsof the spacegroups. In combinations or in- In contrast, the twelve 2M, structureshave very nearly dividually, the symmetry elements2,2,, c, and n relate the sameenergy (o : 2.48 kJ/anion). The Boltzmann dis- one unit layer to another and, hence,relate to the struc- tribution law suggeststhat the different Al-Si orderings ture of the interlayer region. When sorted in this way would be more or less equally represented.The actual (Table 7), the averageenergy and standard deviation for energywould be closeto the average,whichjust happens the goup characterizedby the 2' axis-space groupsP2r/n, 116 ABBOTT AND BURHNAM: POLYTYPISM IN MICAS

P2, (31' or 33'), and P2r/c-are substantiallylower than to characterizedeviations from structural ideality, Azhas the averagesand standard deviations for the other groups. the most influence over the relative stability of lM and This indicates that layers related by 2, are particularly 2M, polytypes. Most micas with Az less than approxi- favorable. The group characterizedby a 2-fold rotation mately 0.1 A form lM polytypes, regardlessof compo- has the highest averageenergy. Evidently, regardlessof sition and other structural parameters.The lMpolytype the unitJayer structure,relating adjacentlayers by a 2-fold representsthe default condition, for the absenceof dis- rotation is not a particularly good solution to the polytype tortions except a rotation. Most micas with Az greater problem. The calculationssuggest (see Abbott et al., 1986) than approximately 0. 1 A form 2M, polytypesregardless that a complex interplay betweenthe unit-layer structure of composition and other structural parameters. Con- (i.e., Al-Si ordering) and structure of the interlayer region spicuousexceptions include the lithian micas and anan- is responsiblefor stabilizing the 2M, polytype. dite. Six of the 2M, structtres, correspondingto the 33' and 3. Energycalculations for pyrophyllite, talc, muscovite, 34' unit-layer structures-with double daggersin Table and biotite support the generalobservations regarding the 7-violate Giiven's (1971)rule. The averageenergy and stability of I M and 2M , polytypes,and identify the most standarddeviation for this group are essentiallythe same important kind of structural distortion to be pure corru- as for the group that obeys Giiven's rule (Table 7). Evi- gations in the surfacedefined by the basal oxygens-that dently, the influence of intralayer interactions involving is, corrugations directly related to the structure of the tetrahedral Al polyhedra across the intervening octahe- octahedral sheet,rather than corrugations due to substi- dral sheetis minimal. tutions within the tetrahedral sheets.Of course, comr- Structuraldeterminations indicate that there is no long- gations ofany kind in pyrophyllite and talc are pure. In range ordering of Al and Si in muscovite-2M, (Balley, muscovite and biotite, the distortions, as measured by 1975, 1984). We suggestfurther that the disorder is not Az, canbe thought of as pure comrgations, superimposed some manifestation of short-rangeordering in small but on disorderedsubstitutional distortions. For anandite,the discretedomains. As our calculationsshow, and as point- high Az associatedwith the 2O polytype seemsto be a ed out by Giese(1984, 1986),there are simply too many manifestation of ordered substitutional distortions in the Al-Si orderings with nearly the same energy. However, tetrahedral sheets. we suggestthat the disorder, real though it is on a unit- 4. At high Az, alow a in conjunction with pure distor- cell scale,is subject to certain rules: the principle ofalu- tions (hypothetical OH-pyrophyllite, a : 0) stabilizesthe minum avoidance(Sanz and Serratosa,1984; Sanz et al., 2M, polytype, whereasa low a (lessthan approximately 1986;Herrero et al., 1985;Herrero, 1987)and the same 3") in conjunction with substitutional distortions (anan- ratio of Al to Si in each sheet (Abbott, 1984). If this dite) may stabilize the 20 polytype. interpretation is correct, then by analogy with the pyro- 5. The calculations on pyrophyllite and talc support phyllite calculations,the preferenceof muscovitefor 2M, Thompson's (1981; Radoslovich, 1959, 19601,Giiven, is probably a manifestation of pure distortion of the av- l97l) proposal regarding the scarcity of 2M' and 20 erage structure, superimposed on disordered (but not polytypes for micas with rotated tetrahedra (o > 0). This completely random) substitutional distortions. This is in accounts for the vast majority of trioctahedral and di- contrastto anandite,for which ordered substitutional dis- octahedralmicas (Fig. 2). tortions are implicated in the stabilization of the 2O poly- 6. We suggestthat the disordering of Al and Si in mus- type. The 3Z polytype of muscovite, which is well-or- covite and biotite is subjectto certain rules (Abbott, 1984; dered with respect to Al and Si (Giiven and Burnham, Abbott et a1., 1986), chief among which are (a) the prin- 1967), may be stabilized by the effectsof ordered substi- ciple of aluminum avoidance (Sanzand Serratosa,1984; tutional distortions. Sanzet a1.,1986; Herrero et al., 1985;Herrero, 1987) and (b) a constantratio ofAl to Si in all tetrahedralsheets (Abbott, 1984). On this basis, there are four crystal- CoNcr-usroxs chemically reasonableordered unit-layer structures for 1. F has a grealinfluence in stabilizing the 1M polytype IM and 20 polytypes and four different ordered unit- in biotite, though the lM polytype is also favored for layer structures for 2M, and 2M, polytypes. In a given OH-biotite. F = OH exchangehas little or no influence biotite or muscovite model set, the lowest-energyAl-Si in stabilizing the 2M, polytype in dioctahedral micas orderings are consistently the ones with the most evenly (muscovite and pyrophyllite). Substitution of F for OH distributed tetrahedralAl. That is, the most favorable Al- is thus much more important for than it is for Si orderingsmaximize the closestA1-Al separations.This dioctahedral micas. Given the difference in the position is consistentwith the findingsofSanz and Serratosa(l 984), ofthe H in dioctahedral versus trioctahedral layers, this Sanz et al. (1986), Herrero et al. (1985), and Herrero makes sense because K+-H+ repulsion will be much (1987).In lM and 2M, polytypes,Al-Si orderingscon- greaterin the latter. As would be expected,F has no spe- sistentwith a 2, axis parallel to b areespecially favorable. cial influence in stabilizing talc-lMbecause the A site is The relationshipsare not as clearcutfor hypothetical lM- vacant. dioctahedral structures,though the tendency appearsto 2. Of the various parameters(Bailey, 1984) designed be the same. ABBOTT AND BURNHAM: POLYTYPISM IN MICAS lt'7

7. We supportGiese's (1984, 1986)explanation for the Akhundov, YA., Mamedov, K.S., and Belov, N.Y (1961)The crystal Nauk SSSR,Earth Sciences, disordering of Al and Si in muscovite. There are many structureofbrandisite. Doklady Akademii 137,438440 Al-Si ordering schemeswith very nearly the same low Baerlocher,C., Hepp,A., and Meir, W.M. (1977)ors-ze: Program for the energy.According to the Boltzmann distribution law, the simulation of crystal structures.Institute of Crystallographyand Pe- different Al-Si arrangementswould be more or lessequal- trography, ETH, Zurich, Swirzerland. ly representedin the real structure. Bailey, S.W. (1975) Cation ordering and pseudosymmetryin layer sili- cates.American Mineralogist, 60, 175-187. 8. Giiven's (1971) rule concerningthe relative posi- -. (19S4) Crystal chemisrry of the micas Mineralogical Society of tioning of tetrahedral Al atoms on opposite sides of the America Reviews in Mineralogy, 13, 13-60. octahedral sheet does not seem to be important in the Bassett,WA (1960) Role of hydroxyl orientation in mica alteration. energeticsof micas. By all indications the effect is mini- GeologicalSociety of America Bulletin, 7 l, 449 poly- mal. Baur, W H. ( I 970) Bond length variation and distorted coordination hedra in inorganic crystals. American Crystallographic Association 9. Energy calculations for the 2M, ors-adjusted OH- Transactions,6, 129-155. muscovite structuresshow that the ordering of Al and Si, -(1981) Interatomicdistance predictions for computersimulation subject to the constraints cited above, has little apparent of crystal structures.In M. O'Keeffe and A. Navotsky, Eds., Structure influence on the energyattributable to the internal struc- and bonding in crystals,vol. II, p. 3l-52. Academic Press,New York. Burnham, C.W. (1985) Mineral structure energeticsand modelling using ture of the unit layer, but may have considerableinflu- the ionic approach.Mineralogical Societyof America Reviewsin Min- ence on the energy attributable to the interlayer region. eralogy,14,347-388. Hence, for 2M, structures, the interlayer Al-Al separa- Busing,W R ( I 98 l) wr'aIN:A computer program to model moleculesand tions are more important than the intralayer Al-Al sep- crystals in terms of potential energy functions. Oak Ridge National p. arations. By analogy, the interlayer relationships in lM Laboratory,Special Paper ORNL-5747, 169 Filut, M A, Rule,A.L., and Bailey,S.W. (1985) Refinement of anandite- polytypesare probably more important than the intralay- 2Or structtre. American Mineralogist, 70, 1298-1308. er relationships. Gatineau, P L (1964) Structure reelle de la muscovite. Repartition des 10. For the purpose of discriminating among the struc- substitutions isomorphiques. Bulletin de la Soci6t6Frangaise de Mi- tures ofa model set, as defined here, the use ofthe Cou- n6ralogieet de Cristallographie,87, 321-355. les justifiable. Gatineau, P.L., and Mering, J. (1966) Relations ordre-desordredans lomb electrostatic energy alone is If a set of substitutions isomorphiques des micas. Centre National de la Re- structuresto be compared can be manufactured in such chercheScientifique, Groupe Frangaisdes Argiles, Bulletin, 18,67-74. a way that the nearest-neighborinteratomic distancesand Giese,R.F., Jr. (1984)Electrostatic energy models of micas. Mineralogical bond anglesare the samein all of the structuresand hence Societyof America Reviewsin Mineralogy, 13, 105-144. -(1986) Detection of short-rangeorder in substitutionally disor- the structuresdiffer only in the articulation of a fixed set dered cation sites of phyllosilicates.International Mineralogical Asso- ofpolyhedral units, then the short-rangerepulsive energy ciation Abstractswith Program, I I L and van der Waals energywill not vary significantly from Gruner, J W. (1934) The crystal structuresof talc and pyrophyllite. Zeit- one structure to another. Under thesecircumstances, dif- schrift fiir Knstallographie, 88, 412419. ferencesin the cohesive energieswill be due almost en- Guggenheim,S. (1984) The brittle micas. Mineralogical Societyof Amer- ica Reviewsin Mineralogy,13, 61-104. tirely to differencesin the long-range Coulomb electro- Guggenheim,S., and Bailey, S.W. (1975) Refinement of the margarite static interactions. The method is potentially very structure in subgroup symmetry American Mineralogist, 60, 1023- powerful in discriminating between structuresthat differ 1029 only in the ordering and articulation of a fixed set of Giiven, N. (1971) Structural factors controlling stacking sequencesin di- micas.Clays and Clay Minerals,19, 159-165. polyhedral units. The effectivenessof the method de- octahedral Giiven, N., and Burnham, C.W. (1967) The of 3T mus- pends on the strategyused in the design of the compari- covite Zeitschrift fiir Kristallographie, 125, l-6. son and the skill in manufacturing the model sets. The Hazen, R.M. (1985) Comparative crystal chemistry and the polyhedral approachis a natural extensionof Hazen's(1985; Hazen approach. Mineralogical Society of America Reviews in Mineralogy, and Finger, 1982) polyhedral approach to comparative 14,317-346. Hazen, R.M , and Bumham, C W. (1973) The crystal structure of one- crystal-chemistry. layer and annite. American Mineralogist, 58, 889-900. Hazen,RM, and Finger,LW. (1982)Comparative crystal chemistry. Wiley, New York AcrNowr,nncMENTS Herrero, C.P (1987) Monte Carlo simulation and calculation of electro- static energiesin the analysesof Si-Al distribution in micas. In L.G. Most of the calculationswere performed on the geophysicscomputing Schultz, H. van Olphen, and F.A. Mumpton, Eds., Proceedingsof the facilities at Harvard University, during R.N.A.'s tenure as a postdoctoral International Clay Conference,Denver, 1985, p.24-30. Clay Mineral fellow The researchwas supported by NSF Grant EAR 79-20095 to Society,Bloomington, Indiana C W B We greatly appreciatethe helpful comments of the reviewers,J. Herrero, C.P, Sanz,J., and Serratosa,J M (1985) Tetrahedralcation E Postand S. W. Bailey. ordering in layer silicatesby'?eSi mun spectroscopy.Solid State Com- munications.53. 151-154. Jackson,W.W., and West, J. (1930) The crystal structure of muscovite, 76, 21 1-227. RBrpnnxcns cltet KAl,(AlSi3)Oro(OH)r.Zeitschrift fiir Kristallographie, Lee, J.H, and Guggenheim,S. (1981) Singlecrystal X-ray refinement of Abbott, R.N., Jr (198a) Al-Si ordering in lM trioctahedral micas. Ca- pyrophyllite-1?"c American Mineralogist, 66, 350-357. nadian Mineralogist, 22, 659-667. Loewenstein,W. (1954) The distribution of aluminum in the tetrahedra Abbott, R.N., Jr., Burnham,C.W., and Post,J E. (1986)Energetics of of silicatesand aluminates-American Mineralogist, 58, 92-96. polytypism in di- and trioctahedral micas. Intemational Mineralogical Pabst,A (1955) Redescriptionofthe singlelayer structure ofthe micas AssociationAbstracts with Program, 40 American Mineralogist, 40, 967-97 4. 118 ABBOTT AND BURHNAM: POLYTYPISM IN MICAS

Pauling,L (1930)Thestructureofthemicasandrelatedminerals.Pro- Sanz,J., Herrero, C.P, and Senatosa,JM (1986) Tetrahedralcation ceedingsofthe National Academy ofScience, 16, 123-129 distribution in phyllosilicates 2:l by "Si NMR spectroscopy.Interna- - (l 960) The nature ofthe chemicalbond (3rd edition). Cornell Uni- tional Mineralogical AssociationAbstracts with Program, 220. versityPress,Ithaca,NewYork. Smith,J.V.(1954)AreviewoftheAl-OandSi-OdistancesActaCrys- Post,J.E, and Bumham, C.W. (1987)Structure-energy calculations on tallographica,13,479. low and high albite. American Mineralogist, 72,507-514. Smith, J.V., and Yoder, H.S (1956) Experimentaland theoreticalstudies Radoslovich,E.w (1959)Structural control of polymorphismin micas. of mica polymorphs.Mineralogical Magazine, 31,209-235. Nature, 183,253. Takeda,H., and Ross,M. (1975)Mica polytypism:Dissimilarities in the -(1960) The structure of muscovite, KAlr(Si3Al)Oro(OH)r.Acta crystal structuresof coexisting lM and 2M, biotite. American Miner- Crystallographica,13, 919-932 alogist, 60, I 030-1040. Ramsdell,L.S.(1947)StudiesonsiliconcarbideAmericanMineralogist, Thompson,J.B., Jr. (1981) Polytypismin complex crystals:Contrasts 32,64-82 betweenmica and classicalpolytypes. In M. O'Keeffe and A. Navrot- Rayner, J.H., and Brown, G (1973) The crystal structure oftalc. Clays sky, Eds, Structureand bonding in crystals, vol II, p. 167-196. Aca- and Clay Minerals, 21, 103-114. demic Press,New York. Richardson,S.M.,andRichardson,J.W,Jr.(1982)CrystalstructureofaVillager,H (1969)orsmanual.InstituteforCrystallographyandPetrog- pink muscovite from Archer's Post, Kenya: Implications for reverse raphy, ETH, Zunch, Switzeiland. pleochroismin dioctahedralmicas American Mineralogist, 67, 69-75. Sanz,J., and Serratosa,J.M. (1984),,Siand,?Al high-resolution MAs-NMR spectraof phyllosilicates Journal of the American Chemical Society, MnNuscrrpr RECETVEDJvw 2, 1987 106.4790-4793 MaNuscnrsr AccEprEDOcrosen 9. 1987