1131
The Canadint Minerala gi st Vol.33, pp. I 131-1l5l (1995)
BORATEMINERALS. I. POLYHEDMLCLUSTERS AND FUNDAMENTALBUILDING BLOCKS
PETERC. BURNSE Departnuntof EanhSciences, tlniversity of Cambridge,Downing Street, Catnbridge CB2 iEQ, U.K.
JOELD. GRICE ResearchDivisiaa Carudian.Mweum of Nature,P.O. Box 3443, Station D, Ottawa,Ontario KIP 6P4
FRANKC. HAWTHORNE Departmcntof GeologicalSciences, tlniversity of Manitoba,Wittnipeg, Manitoba R3T 2M
ABSIRA T
In general,the structuresof the borateminerals are ba$edon BQ3and B$a polyhedra,which occur as discreteoxyanions or polymerizeto form finite clusters,chains, sheets and frameworls. The B-Q bonds (Q: unspecifiedanion) are of much higber bond-valencethan the interstitial bonds,and thus borateninerals readily lend themselvesto hierarchicalclassification based on the topological characterof ttre FBB (fundamentalbuilding block) and the structuralunit. Here, we derive topologically and metrically possiblefinite clust€rsof the form [BnQ.], where 3 3 n 3 6, and identify those clustersthat occur as FBBs of the structues ofborate minerals.In addition,we havedeveloped graphical and algebraicdescriptors ofthe topologicaland chemical aspectsof tle clustersand their modeof polymerization.ln fhe structuresof the borafeminerals based on FBBs with 3 3 z 5 6, all FBBs arepolyhedral rings or decoratedpolyhedral rings. Moreover,three-membered polyhedral rings are almostcompletely dorninant;the only exceptionis the four-memberedpolyhedral ring in borcarite.Three-membered polyhedral rings occur in the following order of preference:
Keywords:borate minerals, hierarchy of structures,fundamental building block, crystalstructure, boron, structure classification.
Sol,IIr,IaIRs
En g6n6ral,les structuresdes min6rauxborat6s ont commeunit6s de basedes polybdresB03 et BO4,presents sous forme d'anionsdistincts ou en agencementspolym6ris6s, qui sont soit des regroupementslimit6s de polybdres,des chalnes,des feuillets ou des trames.I,es liaisons B-$ ($: anion non sp6cifi6) possOdentune valencede liaison beaucoupplus 6levdeque les liaisons interstitielles, de lelle sort€ que les min6rauxboratds se pret€nt tout natuellement tr un sch6made classification hidrarchiquefond6 sur le caractbretopologique du bloc structuralfondamental et de I'unite structuralede base.Nous d6rivons ici tous les agencementsfinis possiblesselon les critdrestopologiques et m6triquesapproprids pour les agencemetrtsde tylre ;BrQ.l, danslesquels 3 3 z 3 6, et nousidentifions les agencementsqui serventde bloc structuralfondamental dans les structures de mindraux borat6s. En plus, nous ddvelopponsles atnibuts graphiqueset alg6briquesrequis pow d6crire les aspects topologiqueset chimiquesdes agencemsnts et leur modede polymdrisation.Dans touts structured'un mindralborat6 impliquant un bloc structuralfondamental avec 3 3 n 3 6, ce bloc constitueun armeaude polybdres,ddcor6 ou non. De plus, les an:reauxi trois membressont fortement pr6dominants.La seule exception, en fai! est la borcarite, qui contient un anneaul quatre polybdres.Les anneauxi trois polybdresse rencontrentavec une fr6quencedans I'ordre
Mots-cl€s: min6raux borat6s,hidrmchie des strucfires, bloc structural fondamental,strucn[e cristalline, bore, classification structurale.
* Presentaddress: Department of Earth andPlanetary Sciences, Univenity of New Mexico, Albuquerque,New Mexico 87131- 1116.U.S.A. Ll32 TT{E CANADIAN MINERALOGIST
IrrrnoousnoN structuresand, ultimately, the paragenesisof borate minerals. Boronhas an ionic radiusof 0.1I A (Shann611976;, and hencecan occur in both niangular and tetrahedral The schemzof Christ & Clark ( 1977) coordination where bonded to oxygen. BQ3 (Q: O2-, OH-) groups have an average B-Q bond-valence In their crystal-chemical classification of borate approximatelyequal to 1 valenceunit (y.u.), and BQa structures, Christ & Cluk (1977) emphasizedthe groups have an averageB-{ bond-valenceapproxi- importanceof polymerizationof BQ3tiangles and B$, mately equal to t/e,v.u. Hence, both BQ3 and BQo tetahedra to form clusters that are compact,insular groups can polymerize by sharing comers without gtoups, referred to as fundamental building bloclcs violating the valence-sumrule (Brown 1981). Such (FBB). The FBBs forrn the basis of the classification polymerizationis very common in both minerals and scheme of Christ & Clark (1977). Their structural syntheticinorganic compounds,and gives rise to great classificationis basedon threeprincipal criteria: (1) the structural diversify. In general, a borate structure numberof boron atomsin the FBB, (2) the numberof contains clusters of corner-sharingBQ3 and BQa BQ3niangles and BQotetrahedra in the FBB, and (3) polyhedra, which occur as discrete polyanions the mode of polymerizationbetween the FBBs, giving or polymerizeto form larger clusters,chains, sheets or isolated, moffied isolate4 chains, moffied chains, frameworks.The excesscharge of the array of borate sheets and modified sheets.Cbrist & Clark (1977) polyhedrais balancedby the presenceof low-valence proposeda notation (n:iA,+ jT) for eachFBB, which interstitial cations. In the structures of most borate gives the total number of boron atomsin the FBB, as minerals, the B-$ bonds are of much higher bond- well as the number of B$3 triangles (A) and BQa valence(> 0.7 v.u.) ftan ths ls6nining cation-$bonds tetrahedra(Q. For example,the FBB 5:2A+37hasfive (< 0.3 v.u.). Thus, the borate structuresreadily lend boron atomsin the FBB, of which two occur as B0" themselvesto classification on tJre basis of the triangles,and three as BO4tetrahedra. geometryofthe clustersofborate polyhedra. The utility of organizing crystal structures into The occurrenceoflarge FBBs hierarchical sequenceshas long been recognized. Bragg (1930) first classified the silicate structures In developing their classification, Cbrist & Clark accordingto the geometryof the polymerizationof the (1977) consideredthe structuresofthe hydrousborate (Si,Al)O4 tetahedr4 and this schemewas generalized mineralsavailable at that time, as well asthe structures to include sfructuresbased on polymerizedtetrahedra of some anhydrous minerals and hydrous and by Znltai (1960) and Liebau (1985). Suchhierarchical anhydroussynthetic inorganic compounds. They noted classificationsserve to order our knowledge and to that the FBBs of borale structuresare generallysmall, facilitate comparison of crystal structuresowhich is as neruly all sfuctures known at that time were based infrinsically quite diffrcult. Howeveromuch additional upon FBBs with six or less boron atoms. The one insight can be derived from such sftuctural schemeso exceptionwas preobrazhenskite, the structureofwhich particularly regarding the underlying controls on is basedon FBBs containingnine boron atoms.Since bond topology (Hawthorne 1983, 1994) and mineral the work of Cbrist & Clark (1977), however, several paragenesis(Moore L965, L973, Hawthorne 1984). borate structuresthat contain even larger FBBs have There have been several classifications proposed beenreported. Cttice et al. (1994) solvedthe structures specifically for borate structures @dwards & Ross of pringleite{Cae[B2sO2s(oH)ra][BoOo(OID6].13H2O] 1960,Christ 1960,Tennyson L963, Ross & Edwards and ruitenbergite { Car[B2eO2s(OH)rs][B6Oo(OH)o] 1967, Heller 1970, Christ & Clark 1977). Previous .13H2O),both of which containFBBs with 12 boron classifications were reviewed by Christ & Clark atoms.Several synthetic compoundshave 12 or more (1977).Their classificationhas proven very usefirl over boronatoms in the FBB [i.e., Ag6fBpOls(OII)6].3H2O the past fifteen years,and has been widely used.The (Skakibaie-Moehadamet al. 7990),Nax[B,2Oro(OII)n] basisof the Christ & Clark (1977) schemeis the degree (vlenchetti & Sabelli 1979), Na6[Cu2{ B 16024(OtD 10} I .l2H2O of polymerization within the simplest sffuctural unit. @ehm 1983), K6[UO2 { B r6Oz(OID8} ]. I 2H2O They porfayed this with the descriptor n.iA + JT, @ehm I 985), and K5H{CuaO[820O32(OtD8] ].33H2O wheren is the total numberof boron atomswithin this (Heller & Pickardt 1985)1.These more complex unit, which contains i BQ3 and j BQ, polyhedra. structurespresent a problem for the Christ & Clark Although this is usefirl information, it does not give (1977) schemeof classification,which cannotuniquely any indication about the topology of the cluster of distinguishFBBs with its nomenclature.For example, polyhedranor about the degreeto which the cluster is the FBBs v/ith twelve boron atoms in pringleite, translatedthroughout the crystal structure.These two ruitenbergite, Na3[B,rOro(OH)a]and A96[8,rO,, stuctural featuresare the ones we incorporatein the (OI{)61.3H2Oall contain six BQ3triangles and six BQ, presentscheme of classification,as they are essential tetrahedra.and the notation for these FBBs is thus to the understandingof the hierarchy of the borate l2:6L+67. However"examination of theFBBs of these FUNDAMENTAL BTJILDINGBLOCKS IN BORATE MINERALS LL33
b)
Frc. l. Examplesof the FBB 12:6L+4T:a) pringleite, b) Nar[Bt2O2e(OH)a].
structures(Frg. 1) shows that the FBBs in pringleite structures,a descriptorfor borate FBBs that includes and ruitenbergiteare twelve-memberedrings of poly- information on connectivity of the polyhedra is hedrawith alternatingBQ3 triangles and BQo tenahedr4 required. In developing such a descriptor, it is whereas the FBBs in Na3[B12O26(OH)a]and necessaryto sFike a balancebetweetr the amount of Ag6[B12O13(OH)6].3H2Oeach contain six three- information conveyed and the complexity of the membered rings of polyhedra containing BQo descriptor.dthough our methoddoes not alwaysresult tetrahedraand one BQ3triangle, and theserings form a in a unique descriptorfor the FBB, considerablymore larger ring by sharing 804 tetrahedra(Fig. 1). In the information is included than in previousschemes. classification of Christ & Clark (LW7), structures basedon FBBs with identical numbersof BQ3triangles B-B graphs and BQatetrahedra always have identical descriptors, even where the structural arrangementsare very The representationof large polyhedral clusters is different; their notation does not indicate the topo- considerably simplified by omitting the anions, as logical characteristicsof their linkage. This is of is common in topological considerationsof crystal generalsignificance; Burdett (1986)has shown that the structures(Smith 1977, 1988, Hawthorne 1983, 1990). energy difference between two structures can be Such graphs are used to show B-B connectivity expressedin terms of the first few disparatemoments relationships;B-$-B bondsare shown as a single line of the elecffonic energy density-of-statesof the two connectingthe boron atoms, and nonbridging anions structures.In stuctural terms (Hawthorne 1994), the me omitted completely.Inforrration on the coordina- important terms in the formulation of energy involve tion number of the boron atoms is retainedby using differences in coordination number lwhich are different symbols(A and [) for the nodesof the graph. describedin the notation schemeof Christ & Clark B-B graphs are used extensively in this paper; (1977)l and differencesin local linkage of the poly- examplesare shown in Figure 2o where they may be hedra [which arc notf. Thus it is desirableto incor- compared with the conventional representations porate information on local linkage of polyhedrainto involving all the atomsof the cluster. a descriptionof FBBs. FBB descriptor A PnoposnnDrscnrpron pon Fuluawrrret Butr-Dtr{cBLocKs IN BoRATFS Borate FBBs must contain either BQ3triangles or BQ4 tetraledra, or both. Examination of borate In this series of papers, we intend to develop a structuresshows that many FBBs also contain rings of hierarchyofborate structuresbased on a FBB that we BQ, polyhedra. The importance of three-membered define as a stongly bondedclusier ofborate polyhedra rings is sriking; almost all large FBBs contain these that is repeatedby the tanslational symmetryoperators rings. Thus, it is necessarythat the descriptor also to give the sfructuralunit (Hawthorne1983, 1985).As denotesthe occurrenceof rings in the FBBs. a part of the developmentof the hierarchy of borate The FBB descriptorproposed here is basedon: (1) Lt34 THE CANADIAN MINERALOGIST
o) \, o IAIA ".-r*l.y ? l:14 A-I-^ \.1./ "^?tr:[o]<42tr>i
242tr:
i-tl-A 4:2A+2!l
2 2!:
3n:[0]trltrltrl Flc. 2. Examplesof borate clusters,including B-B connec- I 3:37 tivity diagramsand descriptor.For eachcluster, t'wo sets o4-o of symbols are given; the first is as proposedhere, the secondis asproposed by Christ & Clark (1977). FTNDAMENTAL BIIILDING BLOCKS IN BORAIE MINERAI,S 1135 the number of borate polyhedra in the FBB, (2) the
thatis [3]-connectedis [<3L]l.A list of thepolyhedra o) or clustersthat areconnected to the centralunit follows the [ ] delimiters; each cluster that is separately 4A:<3A>=<34> connectedto the central unit is terminated by the symbol l; note that the order of the listing of these is clustersis not importanl. b) ^l\i Consideran oxygenatom that is sharedamong three BQotetrahedra @i9.4). The descriptorfor this cluster 3Atr:4Atr)=(2Atr> is 3n:[0]n I nlnl, indicatingthat it containsthree BQa tetrahedra,each of which is connectedto a central anion. The cluster given in Figure 2k also containsan c) oxygenatom that is sharedamong three BQ, tetrahedra. In this case,the clusteralso containstwo i\i Inrge FBBs h) The new descriptor effectively distinguishes 43tr:<3o>= o\ /o ? ortfo A / l\ o-B-o-l{ T\T o-{-oir-o o\ lJl /o o.J'o xxi I /\ o 4tr:<3tr>=4tr> 4tr:{ Fig.4b o\ /,o\ /.o o'P 1\ '-J--i-o\-zo ,/t\. ,/1\. :j.". ,), \-/ *i-. \-y o'\ rlo,'.o 5tr:<5tr> 5u:<4tr>=<3I1> '-1..o-.--.o1 -o .iIl'sv' 'A\.,l-' .z\. o'\ :\r{ 5tr:<3tr>=<39>-z3n> 5u:.4tr>-<3tr> Fig.4c o o\? ::fi' ?..-o-P-o l]'-t.l lor,9 | *i .J-o-r\?"''"j (> X II 5tr:[0]<3tr>| <3tr> I 5tr:<4tr>=<4tr> o\? B-o-B-o l\,o I oB-o * lro'\l r-l B--o-8. to o/ 5tr:{ Flo. 4. Thepossible 2- or higherconnected clusters for 3 < n 3 5 containingonly !. suitablymodified for the presenceof A anddecorations The graphsof the clustersfor 3 3 n < 5 ate grven in wherenecessary. Figure 6. In a later paper, we vTill sxamine possible Other possibleclusters were obtainedby permuting factorsthat control the existence(or otherwise)ofthese A for n in the thirry-nine clustersofFigures 4 and 5. clustersas FBBs in boratestructures. FT]NDAMENTAL BI]ILDING BLOCKS IN BORATE MINERALS IL39 Fig.5a of / I_I :"a')io-' Fig.5b O Ao ? ,o-fio o-f -o-P'-o-;-o r-i-"-?.o..1 ^T\r-r '{;,i o\_/o d> '-i-'-i-o/to \ \./ JJ r-r \{./ .)t; 6tr:<4try=sJtr)=<3tr> 6t1:[<3tr>]={tr> I =<3n> | =<3tr> | oo tl o-B-o-B-o )a"-',f ll 0(' tt i-\ ,{A-l-"# o-B-o-B-o tsI tl oo #-){ tl \-/ o-B-o-B-o lt 6tr:€ tr>=<3tr>=<3tr>=<3 tr> oo 6u.<4tD=<4tr> 1.t40 TTIE CANADIAN MINERALOGIST Fig'5c o\ zo ,o ,B-o-3' 1 ,ro o-a;o?.o/o' d| ? )-:: +DI_I ff fbr_I ort!"-t.o ,Bi-o-Br^ ooo 6o:{<4s>=<4tr>}=<3tr> 6tr:{<3o>=a3o>}=<5n> xw#.s. o-l 'vi/T-\ t-I /t-o-tro 6tr:4u>={<4tr><4u>) 6tr:{ <3r:>=<4U>=<3tr>}=<3tr> Fig.5d \-o..,P o)r-o-r'o '-ll"-;g-" /l_AI-l '-'$r{{ \t/ "t..xDI_r r-I ort-o-tao 6u:{ <3n>=<4tr>=<3tr> }=<3tr> 6u:<4tr>E{<3 tr>=<4u>=<3tr> }= <4tr> o.rjo_, I-I r-r .e' /i\ .-t'o-rZ7^':a\ /JJ\. /\ \/ t/ I-I v' o/.-o-tao rr 6tr:<4tr>= <3tr)=<3tr>=<4tr> = <4tr> { ) 6u:{411>=altr>}=<4u> FUNDAMENTAL BI'ILDING BLOCKS IN BORATE MINERAI-S tl4l Fig.5e o\ zo o\ /o f r_t I-I "-'ti4lj/-7 'W/ \ "fKh l=\ ---oto r-r if \/ i l-r g>=<3tr>=<4tr> o{"lLo 6n: { < }=<3tr> 6u: { <3s>=<3l1>=<4t]1I*3ot o\ o\ /o 'n:AX" A r_! a E Dat )z:i',--l I -r 9t9,/9Yrll-af ?'\ r \\ -a|,/rl\Lfu o-B-oJ-o -N,Z -1.r7' e.! rz _a_r \L/ l/ Dv-\ I_I or,/ i-I o ,8.. 6tr:<3tr>={<3E)=(4tr>=<3tr> } =<3tr> ?? >t-"-t<: 6tr:<3EI>-<3tr)=<3tr> Fig.5f Oarro o ./o' \ ro :t?'"J'"-',1 {,,& lt';" ',4. $ o\.ro r'CI{ a'\ 6tr:[$]<3o> 6tr:<4tr>-<3tr> l<3tr>lc3o> | o o I-l l]'-f ,/l ffir,, '{) l<>. t$ii*'I-r ;i"i': I-I ;>i._")'(" 6n:[Q]<3o> | <3o>=a3s> | 6o:[$]<3u>| <3o>=<3n>| rt42 TIIE CANADIAN MINERALOGIST Fig.59 oo tt or,2 o\.ro o-l-o-fro r-I '"rJ.-.-JfJ.-""ajlto o'\ AT\ :T*T' fi>t-I \lr_I ti J'.l-" I 6tr: Ttn Occunnmcceor FBBs u.rnm Srnuctup;es Within each set of clusters (Figs. 4, 5), tlere are oF BoRATEMnunars severaldifferent possibleFBBs with different numbers of A and n. The frequency of occurrenceof each In the second paper of this series, we intend to specific FBB within the three sets correspondingto arrange all borate minerals within a hierarchy of 3B:<3B>, 4B:<3B>=<38>and 5B:<3B>-<3B>is structures.In this paper, we restrict our discussion given in Figure 8. For n = 6, FBBs in the set to the FBBs tlat occur in the stuctures of borate 6B:[0]<3B>l<38>l<38>lare of two kinds: ten are minerals, and ignore the overall connectivity of the 3A3n:[0] 38:<38> 4Bz<48> 49;aggy=<38> - - ^ ^'. 'fli'i.. l-l I- H l-t ?r_t fiJ +' i i.. i i ^.^.,2\/\ T-Ji- t ltll I l t\t t\t t\t t\t t\t-l t\ \l t\l-l^ t\l-a t-I l---f l-rrq H H I__I I_A l_l a_ Hr-{H l-a I ^- { l-a 48:{<38>=<$l{E{<3l1=<3B>} 5B:<58> t___{ ____.| l_a A- l a .A . IX,L14.l)!,XIIXJ,LXI (](jo(](-}{-}il{-i 5B:<4B>=<38> (} {1}fl} fli (}{i il} (} il} $} {}i} fl}{\} il} a'aaaaa5B:<3B>=<3B>=<3B> 5B:(0)<38>l 58:<48>=48> *f?*Q****QQf?Q*A 58:{<3B>=<3B>}=<48> --r-- .-^\ .-^\ w^ww Q***'**g-QQQ *QQ*Q**Q*}Q Ftc. 6. All possible2- or higher connectedclusters for n = 3, 4,5, not contain a 38:<3B> 48:<4B> 48:<3B>=<38> 48: {<38>=<38>} = {=<3B> } 58:<5B> 5B:<48>=<38> 58:<3B>=<3B)=<3B> 58:<38>_€B> 5B:[0]<3B>l<38> | 58:<48>=<48> 58:[<38>=<38>J=<4B> 24681012't416t820 Frequency Ftc. 7. The frequencyof occurrenceof eachof the elevenclasses of clusten (3 3 r 3 5) in boratemineral structures. with the exceptionof the FBB 4[:<4n> in borcarite, Post-DoctoralFellowship to PCB and an Operating noneof tlese havebeen observed in mineral structures. Grant to FCH. Clare Hall, Cambridge,supported this However, there are also several sets of clusters with workvia a researchfellowship to PCB. Reviewsof an 3 3 n 3 6 that contain only three-memberedrings earlier version of this manuscript by Dn. Paul B. (Figs. 4, 5) that have not yet beenfound in any borate Moore and John M. Hughes, and editorial work by mineral structure. R.F. Martin, were invaluable in preparingthis manu- script. Sumuanv RSFERB{CFJ (1) We have developed a new representationand descriptorfor borateclusters that includesinformation Bmna, H. (1983): Hexasodium(cyclo-decahydroxotetra- on the total numberof boron atoms.the coordinationof cosaoxohexadeca-borato)dicuprate(Il)dodecahydrate, the boron atoms (A: triangular, n: tenahedral),the Na6[Cu2{ B 160z+(0II)10} ]' 12H2o . Aca Cry stallogr. C39, connectivity of A and n, the presenceof rings, and 20-22. the connectivitywithin and betweenthe rings. (1985): Hexapotassium(cyclo-octahydroxotetra- (2) We have derived all clusters that are at leasr cosaoxohexadeca-borato)dioxouraaate(Vl)dodecahy- [2]-connected,but not more than which [4]-connected, drate, K6ruO2{ B r6o24(OfD8} l' 12HrO. Acta Cry stallo gr. havetbree to six polyhedra. c41,642-645. (3) In the structures of borate minerals containing borateFBBs with 3 S z S 6, FBBs incorporatingonly BRAGc,W.L. (1930):The structureof silicates. Z Kristallogr. three-memberedrings of polyhedra are almost domi- 74,237-305. nant. One FBB that is a four-memberedring of poly- hedrais known (in borcarite). BRowN,I.D. (1981):The bond-valencemethod: an empirical (4) Three-memberedrings of polyhedraoccur in borate approach[o chemicalstructure and bonding./n Structure mineral sfuctures in the following order of preference: and Bonding in Crystalstr (M. O'Keeffe & A. Navrotsky, This work was supportedby the Natural Sciences BuRDErr,J.K. (1986): Structural-elecnonicrelationships in and Engineering ResearchCouncil of Canadavia a the solid state.Mol. Struct.Energ. 1,209-275. FUNDAMENTAL BTIILDING BLOCKS IN BORATE MINERALS lt45 3tr:<3u> 2LlrJ:<2Ltr> IL2tr: 4U:<3tr>=<3tr> lA3tr: 5E:<3rl>-<3E> 1A4tr:<3tr>- 2 5 9 ]0 Frequency Flc. 8. The frequency of occurrenceof selectedclusters as FBBs in borate mineral strucu.rres. BIRNS,P.C. & HewrnonNs,F.c. (1993a):Hydrogen bonding & - (L99M): Kaliborite: an example of a in colemanite:an X-ray and structure-energystttdy. Can. crystallographically symmetrical hydrogen botd. Can. Mineral.31.297-3M. Minzral.32, 885-894. & - (1993b):Hydrogen bonding in meyer- - & - (1995):Hydrogen bonding in borcarite, hofferite: an X-ray and structure energy study. Caz. an unusual carbo-borate mineral. Mineral. Mag. 59, Mineral 3l.305-312. 297-3M. & - (1994a): Hydrogen bonding in Crnrsr, C.L. (1960): Crystal chemistryand systematicclassi- tunellite. Can.Mineral. 32 895-902. fication of hydrated borate minerals. Am Mineral. 45, 334-3N. & - (1994b): Structure and hydrogen bonding in preobrazhenskite,a conplex heteropolyhedral & CI-ARK,J.R. (1977): A crystal-chemicalclassifi- borate.Can- Mineral. 32. 387-396. cation of borate structureswith emphasison hydrated borates.P&ys. Chem. Mincrals 2, 59-87. & - (L99k): Structure and hydrogen bondingin inderborite,a heteropolyhedralsheet structue. Conw, D.I.A. (1978):Ba.sic Techniqucs of tombtuntorfutl Can Mineral. 32, 533-539. Theory.Wt7ey,New York, N.Y. T146 THE CANADIAN MINERALOGIST EDwARDS,J.O. & Ross, V.F. (1960): Srrucruralprinciples (1973): Pegmatitephosphates: descriptive mineral- of the hyrlrated polyborates.J. Inorg. Nurl. Chem. 15, ogy and crystal chemistry.Mineral. Rec.4, 103-130. 329-337. Ross,V.F. & EDwARDS,J.O. (1967):The structuralchemistry Gnrcp, J.D., BuRNs,P.C. & Hawrronnp, F.C. (1994): of the borates. /n The Chemistry of Boron and is Determination of the megastructures of the borate Compounds (E.L. Muetterties, ed.). John Wiley, New polymorphs pringleite and ruitenbergite. Can Mineral. York, N.Y. (155-207), 32,1,-1,4. SuaNNoN,R.D. (1976): Revised effective ionic radii and HewruonNe, F.C. (1983): Graphical enumerationof poly- systematic studies of interatomic distaaces in halides hedralclusters. Acta Crystallogr. A39, 724-736. andchalcogenides. Acta Crystallogr.432, 75l-7 67. (1984): The crystal structureof stenoniteand the SKAKTBATE-MoGHADAM,M., I{srm., G. & Tnnm., U. (1990): classification of the aluminofluoride minerals. Caz. Die Kristallstrukturvon A96[B'2O1s(OII)6].3H2O,einem Mineral.22,245-251. neuenDodekaborat. Z. Kristallogr. 190, 85-96. (1985): Towards a structural classificationof SMrrq J.V. (1977): Enumeration of 4-connected3-dimen- minerals: the riWaT2qnminerals. Am. Mineral. 70, sional nets and classification of framework silicates. I. 4554',73. Perpendicularlinkage from simple hexagonal ne!..Arn. Mineral.4,703-709. (1990): Structural hierarchy in ,7t0l17ia$") minerals.Z Kristallngr. 192, 1-52. (1988): Topochemistryof zeolites and related materials. 1. Topology and geometry. Chem. Rev. 8, (1994): Structural aspectsof oxide and oxysalt 149-182. crystals.Acta Crystallogr.850, 481-510. SurNo,S., Cum, J.R.,PAptrG, J.J. & KoNNRr, J.A. (L973): I{uln, G. (1970): Darstellung und Systemarisierungvon Crystal-structurerefinement of cubic boracite. Arn. Boraten und Polyboraten. Fortschr. Chem. Forschun.p Mineral. 58. 691-697. 15.206-280. TtsNNysoN,C. (1963):Eine SystematikderBorate aufkristall- & Ptcxanor, J. (1985): Uber ein Ikosaborat-ionin chemischerGrundlage. Fortschr. Mineral. 41, 6+91. hydratisiertenKalium- und Natriumkupferpolyboratan.Z. Natu rfo rsch. 40b, 462466. ZoL"rlu, T. (1960): Classificationof silicates and other minerals with tetrahedral structures.Am. Mineral. 45. Lmsau, F. (1985):Strucnral Chemistryofsilicates. Spinger- 960-973. Verlag, Berlin, Germany. Mmtcrrnrn, S. & Sanru:, C. (L979):A new boratepolyanion in the structure of NqfB12O2e(OH)A.Acta Crysnllngr. 835,2488-2493. Moons, P.B. (1965): A structural classification of Fe-Mn Received. December 8, 1994, revised m.anuscript accepted orthophosphatehydrates. Arn Mineral. 50, 2052-2062. April 6, 1995. FUNDAMENTAL BTJILDINGBLOCKS IN BORATE MINERALS 1.L47 AppnrDx I TABLE AI. DISJOINTCYCLE DECOMPOSITIONAND CYCLE STRUCTURE Considerthe B-B graph shownin Figure Al; let us OF THE AUTOMORPHISMGROUP P OF call this graph G. This is a labeled graph in which THE CRAPHG IN FICUREA1 verticesrepresent borate polyhedra ofunspecified type, and edgesrepresent linkage betweenthese polyhedra Disjoirucycle Cyclestructurc* suchthat the polyhedralcluster represented consists of decomposition two three-memberedrings with two polyhedra in comrnon. The problem is to determine how many distinct clustersof borate triangles and tetrahedraare (1) (2) (3) (4) si possible; this reduces to determining how many (2) (4) (1 3) !l J2 distinct black (tetrahedron,E) and white (triangle, A) coloringsof the graphin Figure A1 are distinct. (1) (3) (2 4) s?rl The automorphismgroup, P, of a graph is the (r 3) (24) t collection of all permutationsof the vertex labelings that preserve isomorphism. The collection of all x possible pennutations of the vertex labelings is the s aredummy variables &aI carrythe cycle symmetric group .So(where n = 4 in Fig. A1). P is a structureof thedisjoint cycle decomposition. subgroup of So, and the complementary disjoint subgroupofSn definesall labelingsofthe graph G that are distinct. The disjoint cycle decomposition and cycle structure of P for G of Figure A1 is given in N =1 124+2.22.2+221=g (M) Table A1. The correspoadingcycle index of G, 4' denotedZ(P) is given by There are thus nine distinct possible arangementsof ,N four borate polyhedra that consist of two three- Z@) =L 2 ilrXrcvt=\15+x|+pS - ' 1l^t1 memberedrings with two polyhedrain common. ll,l DEI, lp,l" I following standard nomenclature in combinatorial theory (Brualdi 1991, Cohen 1978). From the unweightedversion of P6lya'stheorem, the numberof AppnNox tr distinct schemes,lSl, is given by Z(P:rn), that is, by substitutionof thenumber of differentcolors (i.e.,Zor We wish to deriveall geomefically possibleclusters L: m = 2) for the dummy variables/ez); of z tetrahedra(specifically for n = 4 to n = 6) subject to the condition that tetrahedralink by sharingcorners only and that eachoxygen is not bondedto more than two boron atoms. We will begin by deriving all topologically possible clustersoand then will discard those clusters in which the tetrahedra are inter- penetrant,deformed beyond chemical feasibility, are not at least two-connected,are greater than four- connected,or which form disconnectedrings. The linkagebetween polyhedra is definedby the edgeset of the correspondinggaph, and hencewe are concemed with the combinatorialcharacteristics of the edgeset of the graph. First, it is necessaryto emphasizethat this is a topological problem rather than a geometrical problem (as was the casein Appendix I). Hence the relevant automorphismgroup for the vertex set is the symmetric group ,Sr. This has a corresponding permutationgroup, Pry W = n(n-lY2l, that permutes the edgeset, and it is the cycle structureof P" that can be used,together with the weightedversion of P6lyas theorem, to calculate the inventory of topologically distinct ,urangements.A convenientway to deal with the structureof the edgeset is to allow the operations of S, to permute the elementsof the vertex set, and Ftc. A1. A labeled B-B graph, G, in which the vertices representpolyhedra and the edgesdenote linkage between to derive the correspondingoperations of P1yby exam- polyhedra- ining the resulting structureof the adjacencymatrix. 1148 4 The elementsof the adjacencymatrix, l4 b ...fl, caa I I be chosenfrom the set {0,1}, indicatingno linkageor linkage, respectively,between tetrahedra. T\us m = 2 2 2 and 3 3 4 1 $ =+126+9.2a+14,*)=LL (A5) Ftc. A2. A graphwith z = 4 (left), the adjacencymatrix for tle Thus there are 1l topologically distinct clusters for graph (middle), and the conespondinggeneral adjacency matrix (right). n = 4.We may derive information on their structureby using the weighted version of P6lya's theorem. The matrix elements [a, b ....J] are chosen from the set {0,1}; let us assignweights {u,v} to the set{0,1}. We can then derive the inventory of arrangements,invlSl, by substitutingthe weight functions Figwe A2 shows a specific graph with n = 4, the k adjacencymahix ofthat graph, and the corresponding wr,=-=l Z,w(r)k = vkarV lel,n (.{6) generaladjacency manix that we usehere. Considerfirst the casefor n = 4. T\e disjoint cycle for the dummy variabless{til n the cycle index. The decomposition and elements of the cycle structure resulting inventory of arrangementsis for So are shown in Table A2, together with the = correspondingelements for Pry. The resulting cycle invls J "l@+v)6+9(u+v)2(u2+f)2+8,la3 +f)2 structurefor Pu is thus 24 +6(u2l+tP)(ua+va)i = (A7) 1- u6+usv+2u4tP+3u3f+2u2va+mf+v6 z(p)=h tsf+eslsl+8,s1+6sls|l (A3) Each term in the inventory correspondsto a certain The number of distinct rurangements,lSl, can now be number of edges, and the coefficient of each terrn derived from tle unweighted version of P6lya's denoteshow many topologically distinct arrangements theoremby substitutingthe numberof possiblevalues, occur with that uumber of edges;thus the term 2ua$ moof the matrix elementsfor s, in the cycle structure: indicatesthat there are two distinct arrangementswith two (v = 2) edges.Now we require that all verticesbe lSt= Z(p:m)=fi1m6+Sm!+tt#l (,{4) at least two-connected;a necessary-but-not-sufficient condition for this is that there be at leastfour edgesin the graphs,and this meansthat we are only concemed with inventory terms for which the exponentof v is TABLE 42. DTSTOINT CYCLE DECOMPOSITIONS AND CYCLE greaterthan or equalto 4 (1e.,r, the numberofvertices STRUCTURES OF 51 AND P6 in the vertex set of the graph):2*v4, a,f andv6. These graphs Disjoli! cycle Cycle Disjoint cyclc Cyclc are shown in Figure .A.3.Note that one of the de@mposldonofS. suucturc dcomposition of P6 stru9lurc (l) (2)(3) (4) ri (a) (b) (c) (O @)A rf (l 2) (3)(4) (a) (b c) (d e) (f) o 3) (2)(4) J"r \a c) (b) (di k) t4 o 4) (2)(3) 3-l (a e) (bt @ (A o) (2 3) (4) (s b) (c) (d) (e!) o) (24) (3) @aQ)Gn k) EN (r) (2)(3 (b (c @ 4) @) A e) (1, (1 2) (3 4',, (a)(be)cOA (t3)Q4) (qt (b) (c d) (e) o4)(23) (4 f) (b e) (c) (a (1 2 3) (4t (a b c) (df e) N (1 3 2) (4) (acb)(de!, r (r24)Q' (ade)(bfc) (r 4 2) (3) J-r QeA@cf) (r34)Q) (dec)(bdJ) 2 (r 4 3)(2) (oce)(bfd) LT tr -t o)(234) ": (adb)(ceJ) J FIc. ,{3. Graphsfor possibleborate clusterswith n = 4 that (t) (24 3) (abd)(cfe) contain at least four vertices. Boxes are drawn around (1214) -l -t r,l (qdfc)(be) the graphs of interest, i.e., those that are at least two- connectedand are not gfeaterthan four-connected. FIINDAMENTAL BI]ILDING BLOCKS IN BORATE MINERAT.S tL49 arrangementshas a one-connectedvertex, andhence is not needed;the otherthree graphs are relevant, and the resultingclusters are shownin Figure 4. @ ED\-VDtzN Having illustrated the method, we can now take a short cut in the calculationsfor z = 5 and n = 6. T\e different permutationsof ,Socan be divided up into conjugacyclasses, the structureof which is denotedby @ @DQKlow the svmbol n .lI- 4!Lu) (A8) @ lGEl'ai1A tc--l @ ic-@ TABLE A3. CYCLESTRUCTURES AND NUMBER OF ELEMENTSIN S" AND P"o,+t)aFORa=5AND6 t4 L@i Cycle structure Cyclcstrucrurc s5 N,* Pto r ffi FIc. A4. Graphsfor possibleborat€ clusterswith z = 5 that contain at least five vertices. Boxes are drawn around ri I rlo tle graphs of interes! i.e., those that are at least two- ri ri 10 sl si connectedand are not gfeaterthan four-connected.Where the graphs meet these criteria but are deformedbeyond rl ri 15 r?ri chemical feasibility, or contain interpenetranttetrahedxa or disjoint rings, the boxesare drawn using brokenlines. r?rl 20 slsi sl sl 20 sl sl sl slsl 30 ri s? sl 24 J5 in which s{kd ars the dummy variablesused above, s6 N{. Pts and.l(k,g) denotesj cycles of length ft in the permuta- tion g e ,S,.The number of conjugacyclasses in ,Snis p(n), the numberof partitionsof the integern. Thus we s! 1 Dl can derive the conjugacy-classstructure of S, from rl sl 15 sl sl p(n); the correspondingcycle index of S, is thus the sum of the elementsof the conjugacyclass of ,S, each .t?'rt 45 ri rl multiplied by an appropriatecoefficient that denotes sl s?sl the numberof disjoint cycle decompositionsof ^lnwith that specific cycle-stucture. The conjugacyclasses of .? 4A "l ri ,i ,!5and 56 areshown in Table A3. How do we derivethe sl s| s] 120 sl sl s?.rl appropriatecoefficients? For a specific cycle sfucture [e.g.,s s for 55:a specificexample is (L 2 3)(4 5)], we t3 4A s3 calculatethe number of distinct anangementsffor s s the numberof distinct r?rl 90 sl sl si {e.5.,Q 2 !@ 5)}: in this case, anangementsis 5 x 4 x 3/3 for s and 2 x l/2 for s, and sl sl 90 sl sl si thetotal numberis therefore5 x 4 x 3 x2l3 x2=201. The resultant values for 55 and '56 are given in sl rl 144 si Table .{3. Now althoughthe conjugacyclasses of P1y sl na sl,f aredifferent from thoseof,So, corresponding conjugacy classesin the cycle indicesof eachgloup musthave the same coefficient, and each operation in So has a *N = numberof elemenlsof S" (andP,,".rl correspondingoperation in Pp. Thus, to derive the with the cycle structureindicated completecycle index for P1y,we need only to derive I 150 TIIE CANADIAN MINERALOGIST the cycle sffucture for one operation within each For n = 6, the cycle structurefor P15is conjugacyclass. The correspondingconjugacy classes and associatedcoefficients for P1e(associated with 55) z@ = 5slsj+60sfs!++osf sf and P15 (associatedwith S) are gtven in Table A.3. *tsls+l From TableA.3, we canwrite down the cycle structures +1 s0slslsl+ I 2os{slsls}++os! for P1e and P15, and proceed with calculating the +Mafi+I20,s\;ff (Ar2) patterninventory as was doneabove for n = 4. For z = 5, the resulting cycle structurefor P16is The total numberof distinct arangementsis given by r -^ Z(P:2): = z(P) * [slO+l0srasl+15($+20sls! +z0sls\s[+30sls]+zaQr1 (A9) Z(P:2)= J 12rs+l5.2tt+60{2s +40.27 +120.2s The total numberof distinct arrangementsis given by +40'2s+144'2j+120'231=156 (A13) Z(P:2): The conespondingpattern inventory is given by Z(P:D = J l2to+10.2a.23+15.22.24 +20.2.23 120'- 1 -. invls= [(u+v)ts+15 (u+v)1 (u2+f1a +20'2'2'2+30.2.?]+2a.?]l= 34 (A10) fr +60(u+v)3 (u2+*)6 +4o (u+v)3 Qlsf)A The correspondingpattern inventory is given by +180(z+v) 1*+*11#a/)3 +120(u+v) (u6+v6) | .. .r. Q.3+*)Q.f+f)2 invlSl= l(u+v)to+ t0(u+D4 n?)3 +40 (u3+f)s +144(us+f)3 fi @2 +L20(u3+tf) (u6w6)21 (A14) +15(u+v)2QluP)a+20(u+v)Qlaf)3 = ul 5+ u\ 4y +2ur3 f .+5 ut21)3 +9 ar I v4 +20(u+v)(u3+f1@6+l'f) + 1,5 ulo tf +21,ue ra *24rt rl a2avf$ +30(u2+*)Qf+f)2+24(usuf)21 (A11) +21u6ve + l,5 a5 v10 +9 da i | +5i,3 v12 = uro+ ue v +ZuB tP +h/ f +6u6va +6us f q27]rt3q*taa:fs +6t/v6+4# i7 +Zuzv8 +we +vL0 Requiring that all vertices are two-connectedis a Requiring that all vertices be two-connected is a necessary-but-not-sufficientcondition that restrictsour necessary-but-not-sufficientcondition that restrictsour attentionto arrangementswith the exponentof v equal interest to arrangementswith the exponentof v equal to or greaterthan 6. Exponentsof v greater than 12 to or greater than 5; these 20 graphs are shown in must involve arangements with at least one five- Figure A4. Nine ofthe 20 graphshave a zero- or one- connectedvertex and can be discounted.As can be connected vertex; five graphs require very highly read from the above pattern inventory, there are distorted or interpenetranttetrahedra. The remaining 119 relevant arrangements;these are shown in six graphs are geometricallypossible, and are repre- Figure A'5. Geometricallypossible clusters are shown sentedin Figure 4. in Fieure 5. FUNDAMEIIIAL BTIILDING BLOCKS IN BORATE MINERAIS 1151 @ Eooue survevsGa UU$OD'E4LAfz @ |oelooel6Nlia@e@$ e>Eauaess€s4b& @ oo0 €HOHolsels e€e€@ess cisslai @ @e0@ E€L€Jaao@ s s ssffiN @ i-€ies@€ffi@ffis@s@ s:Q:@.i @ €@@ig__as€oi-ej @ Ftc. A5. Graphsfor possibleborate clusterswith n = 6 that contain at least six vertices. Boxes are drawn around the graphsof interest, i.e., those that axeat least two-con- nectedand are not greaterthan four-connected.Where the graphsmeet thesecriteria but are deformedbeyond chemicalfeasibility, or contain interpenetranttetrahedra or disjoint rings, the boxesare drawn using broken lines.=4tr>}E {4tr>=4tr>} 1138 TTIE CANADIAN MINERALOGIST =<3tr>}=<4tr> <3u> o 6tr:<4tbE{<4s1=14tr>} oo g o..l I "i\:.i{ | 'B-o-B-o o-F-1 I /-to o\l o-!-"-?-"-?- :".".1ii.: U>t-I $ jod I o 6n:[$]<3n>l<4tr> I 6tr:<4tr>=<5tr]> Ftc.5. Thepossible 2- or higherconnected clusters for n = 6 conainingonly fJ.