Braunite: Its Structure and Relationship to Bixbyite, and Some Insights On

American Mineralogist, Volume61, pages1226-1240, 1976

Braunite:its structureand relationship to bixbyite,and some insights on the genealogy of fluoritederivative structures

Pnul B. Moonn e,NnTereHhnu ARnxr Departmentof the GeophysicalSciences The Uniuersityof Chicago Chicago,Illinois 60637

Abstract

Braunite,Mn'+Mn!+SiOn, e:9.403(16), c : 18.668(32)A, tetragonal holosymmetric, spacegroup I4t/acd, Z = 8,is a derivativestructure offluorite with orderedanion vacancies. R : 0.036(Rw: 0.050)for 5ll non-zeroreflections. The structuresof cubic and orthorhombicbixbyite (a-MnrO3), braunite, and probably braunite-ll(Mn'z+MnllSiOrn), are related according to the spacegroup lbca. The regionsin commoninclude sheets of edge-and corner-linked octahedra (the A andl' sheetsof bixbyite). Braunitesdiffer in havingB sheetsin placeofthe A'sheetsin bixbyite.The B sheetsconsist ol Mn2+in cubiccoordination, Mn3+ in octahedralcoordination, and Sia+ in tetrahedralcoordination with respectto oxygen. Averagepolyhedral distances are (8)Mn(l)2+ -O 2.33,(6)Mn(2)e+-O 2.04,(6)Mn(3)3+-O 2.05, (6)Mn(4)3+-O 2.04,and(4)Si-O l.6lA. The Mns+Oegroups are distorted as a resultof Jahn- Teller effects,forming elongatetetragonal bipyramids which are further distortedby shared polyhedraledges. With help from an enumerationtheorem, a cooperativelattice gameis proposedwhich convenientlyclassifies all structuresderived from fluorite.

Introduction dinarybraunite. The questionthen aroseas regards Braunite has been a crystal-chemicalenigma the correctcrystal structure, the relationshipbetween since it was first characterizedby Haidinger in the two braunites,and their relationshipwith bix- 183l Variously interpreted as (Mn,Si)rOr, byite,a-Mnroa. With thesefacts at hand, we con- 3Mn2+Mn4+Or.MnSiO,or 3MnzOs.MnSiOs,the cludedthat the brauniteproblem required further problematicalcompound has beena centerof con- study. troversyconcerning the role of silicaand the oxida- We havealso explored more deeply the subtle relation statesof the transitionmetal. Dispute also cen- tion betweenbraunite and bixbyitewith the large tered around its crystal class and space group: family of anion-deficientfluorite derivativestructures.Perhaps the most intriguing part of thisstudy is Aminoff (1931)proposed the spaceEroup I4r/acd and Bystrdmand Mason(1943) proposed 14c2, the not the braunitestructure per se,but its relationship latterstudy including a crystalstructure analysis. In with other equallycomplex compounds. addition,they concluded that the correctformula was 3MnrO3'MnSiOr.Those authorsalso presentedan Experimental accountof earlierstudies on the species. A single crystal of braunite was selectedfrom a There the matter stood until de Villiers and Herb- specimenfrom St.Marcel, Piedmont, Italy, originally stein(1967) investigated braunites from severallocal- from the collectionof GeorgeL. English(University itiesand concludedfrom singlecrystal study that the of Chicagonumber 1663).Gorgeu (1893) reported spacegroup wasin fact I4r/acd, the group originally MnO 74.40;O 7.50;Fe2O3, Al,Og 3.80;CaO 0.50; proposedby Aminoff.A variantof braunite(braun- MgO, KrO, NarO 1.00;PbO, CuO 0.15;CoO 0.30; ite-Il) was also characterized,which required a BaO trace;HrO 0.20;SiOz 9.80; PrOu 0.05, gangue doublingof the c axis,a halvingof the amount of 2.60 percentfor Piedmontmaterial. Accepting the SiOz,but a spacegroup which was the sameas or- sum of cations: 8 and the sum of 02- : 12.the t226 BRAUNITE 1227

formula unit contains (Mn31rMgBl.Ca31.XMn3l,, Three-dimensionalPatterson synthesis, P(uuw), in- Fe31")Si1loOrz. dicateda structurebased on fluorite,with a' : a/2 Thecrystal measured 0. 140 mm alongar,0.l37mm andc' : c/4 asthe subcellperiod. This required that alonga2 and 0. 137 mm alongc andwas mounted on a all cationpositions be occupiedbut that vacancies Pailredsemi-automated diffractometer with an a axis occurredin the anion positions.Since we did not parallelto themachine rotation axis. Utilizing graph- detect any violations of the space group l{t/acd ite-monochromatizedMoKar radiation (I (which is uniquely determined),severe restrictions 0.709264),reflections with indiceshkl, hkl (h > k) wereplace on atom locationsand the structurewas weregathered to k : l0 with an upperlimit of sind/tr solvedwithout difficulty. The trial parametersfor all = 0.80 on eachlevel. Other details:scan speed lo independentcations were used toward a p synthesis min-', 20 secondstationary background measure- (Ramachandranand Srinivasan,1970) from which mentson eachside of the peak,scan width rangeof the anionpositions were recovered. 4.20. The trial structureconsisted of onecation in cubic Absorptioncorrection employed the Gaussianin- coordination,one in tetrahedralcoordination, and tegrationmethod (Burnham,1966) with ten faces three independentcations in octahedralcoordina- approximatingthe crystal shape.The maximum tion. Approximatebond distancesrevealed that Sia+ transmissionfactor computed to 0.414and the min- aloneappeared in the tetrahedralsite. From the unit imum, 0.305.Unobserved reflections with intensity formulawe assigned Mn(l) : 0.78Mn'?+* 0.l6Mg'+ lessthan 2o(l) wereset 1o : o(1). After the correc- * 0.06Ca'?+to the cubicsite, and the trivalent(Mnt* tions for absorption,Lorentz, and polarizationfac- * Fe3+)cations were distributedequally over the tors, equivalentreflection pairs l(hkl) andl(hkl) were octahedralsites. This leadsto 19.3electrons at the averaged;896 independent l4l wereavailable for the Mn(l) siteinstead of 23for Mn'*, yieldingan "occu- ensuingstudy. pancy" of 0.84if the Mn'+ scatteringcurve is used. The cell parametersfor braunite were more pre- For refinement,we employedMn2+ for Mn(l), ciselyascertained by a powderdiffractometric trace Mn(2),Mn(3), and Mn(4),Sio+ for Si,and O'- for on the samesample (CuKal radiation,graphite mon- oxygen.X-ray scatteringfactors were obtained from ochromator,l/2" min-'scan speed).The tracewas the tablesof Cromerand Mann (1968).Anomalous correctedfor absorptionby thesample and alignment dispersioncorrection ernployed the tablesof Cromer of the specimenslide. Twenty-nine indexed lines were and Liberman(1970). Full-matrix least-squares re- used in a least-squaresrefinement, the results of finementproceeded from FLMXLS, a program de- whichappear in Tablel. velopedby T. Araki with optionsfor bond distances and angles,derived from the familiarORFLS pro- Solutionand refinementof the structure gram of Businget al. (1962).The final cycleincluded Sincethe spacegroup we determinedfor braunite one scalefactor, one secondaryextinction coefficient did not matchthat of Bystrrimand Mason(1943), we (Zachariasen,1968), eight atomic coordinatepara- assumedthat the structureof our crystalwas un- meters,thirty-seven anisotropic thermal vibration pa- known.Further on, we will discussthe relationship rametersand one occupancyparameter for Mn(1). betweenthe two. At convergence,R : 0.076and R. : 0.072for all 896

TAsr-r L Braunite and relatedcompounds. Structure cell parameters

Braunite Braunite- I I Bixbyite

arA) 9.408(16) s.4l (l) 9.440(s) 9.4146(r) 9 .41s7(3) uifri 9,423s(s) - 9.4047(3) "tij 18.668(32) Le.64(2) 37.76(1) Space group r41/ acd ll4c2 141/acd Ias Pcab 7 8 I 15 16 2+MnE+sio!2 Forrnula (th,ca) 1Ca,t'tr12*UnlfsiO24 (lfrt6.9sFe9.s2)203 lfttzog

Reference This study Bystron and Mason de Villiers and Herbstein Geller (1971) ( 1s43) (1s67) t228 P.B MOORE AND T. ARAKI

Tnnlr 2. Braunite.Atom coordinates* byiteswith spacegroup Ia3. Metrically,both kindsof bixbyite possessapproximately the samecell parameters and differ only in distortion from cubic symmetry l'frr(1) 8 222 0 r/4 L/8 for the orthorhombic crystal. To the 48 equivalent Inh(2) 16 10 0 0 oxygen atoms in a cell of cubic bixbyite there corre- Itr (3) I6 2 r/4 0.21s7(i) 0 spond 6 sets of 8 equivalent oxygen atoms in the (4) I'tr 16 2 0.23r8(r) l/4+x r/8 orthorhombic bixbyite. Two questionsthen arise re- si 8 4o r/4 3/8 garding bixbyite and braunite: do similar group-subgroup relations exist between the two, and to what 0 (1) (4) (2) 32 1 0.1487 0.8s37 (4) 0.94s3 extentis the isomorphism of the oxygenpacking con- o(2) 32 I 0.14s7(3) 0.0734(3) 0.0s69 (2l 0 (3) 32 r 0.0787(4) 0.Ls47(3) 0.92s0(2)served? *Estinated It is convenientto inquire first about their space parentheses standard errors in refer to groups. Braunite crystallizes in space group I4r/acd the last digit. The atons designated are followed by the nunber of positions and point sFnnetry. and cubic bixbyite in Ia3. But I4r/acd is not a subgroup of 1c3,since the elementsof the former are not contained in the latter when either one or two unit reflections;and R : 0.036and R, = 0.050for the5l I cells of Ia3 arc considered.Rather, the nearestsub- = reflectionsabove background error where R, : group relation is I a3d(ar,a",as)-l 4 r/ acd(araz,c a t). -l We then ask what is the maximal subgroupcommon tE,(l f. | F.l)"/>-fll'l2, with w : oF|.The "good- to I4'/acd(abe2,c : 2at) and la3(ar,ar,as).This is lbca nessof fit," ^S= E,ll4l-l F"ll,/(n-m), wheren : its table is given in Table 7. The numberof independent.F's and m : numberof pa- and multiplication tameters,is 1.23. The secondaryextinction formal relation is S(14'/acdteye2,c))S(la3:at,a2,as) : coefficient,co, is 2.0(3)X l0-? andthe scalefactor, s, S(lbca:a,b,c),where the S is the set of symmetry is 0.810(5).Finally, the "occupancy"parameter for elements and a denotes intersection. The group-subgroup relationshipcan be shown diagram- Mn(l ) is 0.828(6),in goodagreement with the chem- ical analysisfor St. Marcelbraunite. Thus, the minor matically, uiz.'. Ca'+ and Mg2+substitute at this site. Ia3 (8) Table 2 liststhe atomic coordinates,Table 3 the I4'/acd (32) anisotropicthermal vibration parameters, and Table 4 the root-mean-squaredisplacements and the orien- tationsof the ellipsoidsof vibration.Table 5 liststhe \,,," structurefactorsl and Table 6 the polyhedralinter- ,,u,o, atomicdistances and angles. + Pbcs (8\ Brauniteand bixbyite: groupsubgroup relationships The structuresof bixbyiteand braunite are related, but theyare not isomorphic.We inquireas to what This states that for the equivalent points in I4'/acd extentthey are related,and proceedby examining and for those coordinatesin Ia3 wherez is in the as- group-subgrouprelationships between the two and axial position, a nearestgroup whoseelements are in seekout thosesubsets of the equivalentpoint sets both is lbca. The orthorhombic group contains half whichare isomorphicin the two structures. the elementsin 14t/acd and one-third the elementsin Bixbyiteand brauniteare metrically related (Table 1a3. Since its equivalent point set has sixteen ele- non-equiva- I ) anddiffer by the c-crystallographicaxis of braunite ments, it remains to inquire if the three being double that of bixbyite.Geller (1971),in a lent oxygens for braunite match with three coordi- detailedanalysis of structuresof syntheticbixbyites, natesfor the oxygensin bixbyite. Indeed,these can be : -0.146, -0.1l0) notedthat purea-Mn2O, crystallizes in spacegroup found, and they are O(l) (0.149, -0.147, -0.084); Pcabbut that small amountsof Fd+ resultin bix- (0.129, O(2) (0.146, 0.073, 0.114),(0.147,0.084, 0.128); and O(3) = (0.079, 0.135, -0.150), (0.083,0.129, -0.146) for braunite I To receivea copy of this material, order document AM-76-031 from the BusinessOffice, Mineralogical Society of America, 1909 and bixbyite respectively. Since the bixbyite cell K Street, N.W., Washington, D.C. 20006. Pleaseremit $100 in shapewas chosen,the z coordinatesin braunite were advance for the microfiche doubled. Finally, the coordinatefor oxygen reported BRAUNITE 1229

Tesln 3. Braunite.Anisotropic thermal vibration parameters (x ICF)*

Brr Bzz Bsg 9tz Brs Bzs

rfrt(1) 3s.0(27) = Btt 4.3(4) -24.I (1s)

-1.4(4) lhr(2) 14.e (8) 17.6(e) 3.8(2) - J.Jt/J 0.3(3) lfrt (3) r7.0(9) 13.7(s) 4.1(2) 0 1.2(4) 0 lftr (4) 1s. s (12) = Brr 3.s(z) - 1.7(6) -0.2(5) = -9zz

Si 13.o(24) = Brr 4.8(6) 0 00

0 (1) 23.8(3r) 18.8(30) s.s(8) r.7 (23) -o.s(r3) -0.2(13) 0 (2) ls.s (30) 14.6(30) s.8(8) - 2.3(2r) -0.8(13) 3.1(12) 0 (3) 22.6(s2) rs.s (31) 6.7(8) 0.7(2s) -2.3(r3) 0.6(12)

Coefficients in exp[-(Brrhz * 3zzk2 * gggl2 + 2312hk+ 2113hl'+ 2Bzsk[)]. by Geller (1971) was reorientedto conform with the (l/4+z), (l/2-tz), and(3/4+z). Therefore'we con- transformation Pcab - Pbca. cludethat bixbyiteand braunite contain sheets which Inspection of the space group lbca (Table 7) re- arelocally isomorphic to eachother and which occur veals that eight symmetry elements are generated by at (z+ml2) in bixbyite and (21-m/4)in braunite, the composition of symmetry elements wherem is an integer.Call thesethe I sheets.The I in braunite [],(l)] .11,a161,b61,2,",]andthe remaining eight by the distinctkind of sheetsat lz'*(2m-l )/81 composition of these with [1,2',",].For the real bix- we call the I sheets. phase, byite and braunite crystals,11,a61,b61,21",] generate a Light is caston the interesting braunite-Il, sheetof crystal structurewhich is isomorphic in both first describedby de Villiersand Herbstein(1967). crystals. The composition with i(l) leads to com- For this phase,the c axis of brauniteis doubled pleted octahedralsheets, and thesewith [,2u"1] leads (Table l), but the spacegroup remainsunchanged. to equivalentsheets which are translated(l/2+z) in The idealformula is Mn,uSiOrnor Mn2+MniISiOrn' bixbyite. In braunite, the composition with 4u"y in We now proposea structurefor thisphase. Consider place of 2,,", leads to equivalent sheets displaced bixbyite in Pbca orientation. Since Pbca is a sub-

Trgr-B 4. Braunite. Parameters for the ellipsoids of vibration*

oic B([2) Atorn ui oia 0ib oic B([2) Aton ui oia eib

0.s3(6) Mn(1) I 0.070(8) qJ 4S 90 1.03(7) 51 | 0.076(7) 2 0. 087(s) on 90 I80 2 0.076(7) 0 J 0. 163(s) +J 135 90 3 0.092(6) 90

(s) Mn(2) I 0.072(3) 39 69 s8 0.s6 (2) o(l) I o.090 (7) /5 163 90 0.76 2 0.083 (3) 76 53 140 z 0.0s8(7) 109 96 160 70 5 o. os6 (2) 1)A 45 68 J 0.10s (7) 154 106

I'&t(3) 1 o .078(3) 90 0 s0 0.ss(2) o(2) I 0.070(s) 108 150 66 0.63(s) 2 0 .07e(3) 51 90 141 z 0. 083 (8) 158 80 109 z1 0.0e2(2) 39 90 JI 3 o.109 (7) t02 62

(5) l'ftr(4) I 0.079(4) 4J 45 90 o.s4 (3) o (3) I 0.082(8) 78 L64 79 0.76 2 o.08 1 (3) 98 82 169 2 0.094(7) 145 105 l2]. 0.088(3) L34 46 79 3 0. lls (6) t23 87 33

= ith principal axis and the 1= ith principal axis; ui = rns arnplitude; eia, oib, eic angles between the paraneters aie also stated. ce11 axes gl, !z and c. Tfie equivalent isotropic thennal vibration t230 P. B. MOORE AND T. ARAKI

Tlnlr 6. Braunite.Polyhedral interatomic distances and angles*

Mn(r) Mn(3)

4l'h(r)-o(1Yt) 2.rs3 (s)i 2lh(3)-o(r)€l r .906(s) 4 -o(2) 2.s0r (6) 2 -o(2) r .e71 (6) 2 -o(3) 2.267(s) average 2,327 average 2.048 2 o(2) -o(2Y8) 2.s76(6) 62.0(1) 4 o(2) -o(r)(3) 2.638(6) 68.6(r) 2.50s(6) 7s.s(2) 2 o(rlsL6111ts)2.694 (8) 77.s(2) 2.638(6) 85.7(2) 4 O(2) -0(I)'u 2.8s3(7) 7s.2(r) 2.678(6) 78.0(2) average 2,709 71 t 2.7e2(8) s4.2(2) 2,Fsz(r) 94.4(3) 3.246(8) r0r.8(1) 2 o(lYtLo(lY6) 164.0 (2) 3.3r2(8) 104.7(1) 2 O(2) -O(2)\r' r70 .6 (2) aveTage 2,887 90.0 si o(3) -o(3Y2) 140.7(2) o(2) -0(u4l r7e.8(1) 4 Si -o3l2, I.612(4)

2 o(3lfts131tst 2.626(8) roe.l(2) 4 O(3)z'-O(3)b' 2.635(7) 109.6(r) aveTage 2.632 109.5

l"tn(2) Mn(4)

Mn(2) -0 (2) I .866(2) Mn(4)-o(rt6) l.s46(s) -O (2)(l) r .866(2) _o(2)4' I .92r(5) -0 (3) 2.028(s) -o(1Y3) 1.e46(s) -o (3)0) 2.028(5) -6 121{l 0} r.921(3) -o(r)... 2 .212(s) _oislll 2.240 (s) -o (l IU 2.212(S) -0 (3)o' 2.240 (s) average 2.u55 average 2.036

o(2) --o(5) 2.60s(6) 83.9 (2) 0 (2)(lL0(3)(l) 2 .60s(5) 83.s (2) o(r) -o(3) 2.7sr (7) 80.8(r) o (rllLo (3)(u ) a

o(1) -o(r)(1) 180.0(0) o(3)(3)-o(3)6) r79.9 (2) o(2) -o(2)(ll r80.0(0) r77.0(r) o(3) -o(3It) r80.0 (0) Bg}si:B[3]li; r77.0(r) *siril"" regions of lr{n(2) anal l,tr(4) are listed for conparison, e)elained in the text. Based on the list in Table 2, the equivalent points are (l) -x, -y, -zi Q) l/2-x, y, -zt (3) x, L/2+y, -zt (4) l/2-x, l/Z+y, zt (S) L/2+x, l/2-y, -z; (6) r/4+y, t/4+x, l/.4+z; (7) -x, l/2-y, z; (8) l/4-y, L/4-x, t/4-z; (9) 3/a+y, L/4+x. t/4-z; (tO) t/4+y, s/4-x, l/4-z; (lt) l/z-x, l/2-y, I/2-z; (r2) -x, y, t/2+z; (r3) x, -y, t/2-zt (14) s/4-y, l/4-x, ll4+2. groupof lbcaand conserv€s 211sy, it follows that there repeatfor orthorhombicbixbyite is . .. IAA'lr. . . are two kinds of sheetsnormal to the c axis.Call with the z translationbeing (2"-l2m/4) for A; and thesethe A and I' sheetsrespectively. For cubic lz",*(2m*l)/4lfor A'. For braunite,the cellrepeat bixbyite,these sheets would be equivalent.The cell is . . .lABlo.. . with the z translation being BRAUNITE t23l

T,lsls 7. Group multiplicationtable for the symmetryelements in lbca+

(b) 2(a) c (b) b (a) c (a) b (c) a (b) I r i (1) i(2) 2tG) 2r (b) 4@) 2(c) 2

(c) (b) lr i (1) i(2) 21(c) 2r(b) 21@) 2 (c) 2 (b) 2 (a) c (b) b (a) a (c) c (a) b a (b) (a) b (c) b (a) a (c) c (b) I I i (2) i (r) 2 (c) 2 (b) 2 (a) 2r (c) 2r G) 2t@) a c (c) (a) 2 (c) 2 (b) i (r) I I a (c) c (b) b (a) b (c) aO) c(a) 2i O) 21@) 21 2 (a') (c) 2tG) 2r (c) 2r (b) i (2) b (c) a (b) c (a) a (c) c (b) b (a) 2 b) 2 2 i (i) a (b) i (2) c (a) 2r (c) | 2t@) 2rO) I 2 (a) 2 (b) b(a) cO) i (r) a (c) b (a) b (c) c (a) i (2) 2r (b) I 2r(c) 2 (a) I 2(c) (b) i (2) a (b) b (c) 2r (a) | 2rbl 2(c) I a (c) i (I) c (2) (b) i (1) b (a) 2(c) I 2t@) 2r (b) c(a) aO) i c (a) a (c) b (a) i (1) 2(b) I zr(") i (2) b (c) c (b) (1) c (b) a (c) 2(a) I b (c) i (2) a i 2 (c) 2(a) I co) I zr(") 2t@) 2rG) I 2O) 2(c) b (a) I | 2(a) a(c) I z(tl 2r 2r (c) c (a) I G) zr(t) b (c) I ao) I

the inversion at the origin, i(2) the inversion The elenents nean the folluing: I is the identity, I the body-centering; i(l) parentheses. For glide the parentheses at (* * +). For screws and rotations, the axes to which they are parallel are in Planes, refer to the axes to which these planes are nornal '

-l/4, (za+2m/8) foi A, and lzs*(2m+1)/81for B. For the origin of their cell is shifted by (0, l/8), braunite-Il,the postulatedsequence is .--lA'A near-coincidenceof their atomic cordinate parame- BAln... withthe z translationbeing . . -[2ft(4m+l)/ ters can be found with ours (Table 8). The only ex- l6l and lzo*(4m*3/l6)l for A; (za,*4m/16)for A'; ception is their O(l) x-parameter [linked to their and lzpl(4m+2)/l6l for B. We note that eachB Si(l)1, which has a reversedsense of direction. Two is adjacentto two I's in the braunites.In order to noteworthy results obtain flrom this difference: first, conservethe symmetryelements of the B slab, an the reverseddirection of O(1) destroysthe 4'-screw I slabis placedat theorigin for bixbyiteand braunite; axis in I4r/acd, and second,the Mn3+ cationsin the A and the A' slabis at the origin for braunite-Il The sheet remain in octahedral coordination but with nextsequence would be . - .l,AA'AA'A874-. . withthe changed orientation. These observationsemphasize ,4 slabat the origin and the z translationof B being the strong homometric character inherent in the fzp+(6mi3)/24]. Suchlong sequencesare probably braunite structure.We suspectthat with their limited unstablefor theyrequire a long-rangeorder between data set,Bystr

Tlsre 8. Comparisonof atomiccoordinate parameters for braunite+

BystrUrnand Mason (1943)* This Study

l{n(1) o ,/o r/e Mn(1) 0 r/4 r/8 ,, t. I'frt(2) 0 s/4 7/8 ltr (l; " o 3/4 7/8 Mn(3) 0.236 3/4+x 7/8 ltu(4)(3) 0.232 3/4+x 7/8 I'kr(4) .236 l/4+x I/8 l"fn(4) ,232 !/4+x l/8 l'&r(5) .264 .236 .000 I,hr3) L/4 .216 0 lln(6) .264 .236 .250 nnizj ter L/4 r/4 1/4

si (1) 0 L/4 7/8 5i(12) 0 r/4 7/8 si (2) U 3/q r/8 si(r3) 0 slq L/8

0(1) .104 .340 .927 o(3)11l- .o7s .36s otc o(2) .129 .619 .942 o(2)'r, .146 .s73 .943 0 (3) .t32 .386 .058 oili(3) .149 .ss4 .055 o(4) . r04 .659 .073 oisjtsr .o7s .63s .075 0(s) .135 .884 .942 o(1) .L49 .8s4 .945 0(6) .t32 .LLz .058 o(2) .146 .073 .057

space group !492 lqJzg9

After shifting origin (0 -1l4 I/8). The equivalent positions, listed as superscripts, are provided in Table 6, and situatedat z : 0, l/4, l/2, and3/4, areisomor- octahedra form a sheet through corner- and edge- phicto theI andl' sheetsin bixbyite.Figures I a and sharing. If the octahedra were regular, a sheet of lb afford a comparisonbetween the I sheetsin cubic close-packingwould appear, since along one brauniteand bixbyite.In braunite,the Mn(2) and direction, edgeswould be shared,and orthogonal to Mn(3) atomsdefine a planarcheckerboard, and their it, corners would be shared. Along a row in the

FIc. I s. The,4 sheetin the bixbyitecrystal structure drawn from the coordinatesof Geller ( I 97I ). The origin of the cell in that study FIc. ll The,4 sheetin the braunitecrystal structure. Locations is shown, the sheetreoriented for convenientcomparison with Fig. of nonequivalentatoms conform to Table 2. la. BRA U NITE l 233

The "shuffiing" of anions disguisesa deeperrela- tion of the braunite and bixbyite structures.Braunite, bixbyite, pyrochlore, and several more complex structuretypes are derivativestructures of the fluorite structuretype and are obtained from an ordereddefi- cit of anions. Figure 3a featuresthe regular fluorite checkerboardas a systemof edge-linkedcubes. One route of ordering anion vacancies(!) leads to the important pyrochlore Xt")M;")O.,,a structure type (Fig. 3b). How many discreteordered holesare there on the verticesof the cube?Utilizing Polya's(1937) enumerationtheorem, there are twenty-threediscrete arrangements,discussed in Appendix I and shown in Figure 6. It is immediatelyseen that fluorite is based on the u'(1) arrangementalone; pyrochlore on u'(1) and 'd'(3); bixbyite on u'd'(2) and u6d2(3); and braunite on z'(1); uud"(2),uud2(3); and undn(4),that is, the cube,two kinds of distortedoctahedra, and the tetrahedron, respectively.Thus, braunite is in one sensea shuffiedbixbyite, but in another sensea novel Frc. 2,q The B sheetin braunite Note tetrahedral.octahedral, structure type based on a different combination of and cubic coordination for Si, Mn(4), and Mn(l) respectively. ordered anion vacanciesover the fluorite structure type. Enumeration of orderedholes in latticesderives braunite and bixbyite l-sheets, the edge-sharingoc- from Appendix I, and in Appendix II a cooperative curs in units of three octahedra and is then inter- lattice game is proposed to characterizethese ar- rupted by corner-sharing;this arises from a major rangementsin a quick and straightforward way. departureof the Mn(3)O. octahedronaway from the 2/m symmetry implicit in the cubic dense-packed sheet. The,Blayer (Fig. 2a)is not isomorphicto theA' 1: A) layer in bixbyite (Fig. 2b). Thesesheets, also ori- entedparallel to {001},are situatedat z : l/8,3/8, 5/8, and 7/8 in braunite, and consist of linked Mn(l)O, cubes, Mn(4)O. distorted octahedra, and SiOntetrahedra. Although the cationsdefine a checkerboard pattern and although the oxygens between the A- and B-layer cationsare isomorphicto thosein bixbyite, only eight of the twelve oxygensabove the planeof the B-layercations are isomorphic in the two structures.Fig. 2b showsthat four oxygens[two O(7) oxygensand two O(10) oxygensof Geller (1971)l must be "shuffied" to achieve isomorphism of the bixbyite A-layer with the braunite B-layer. The group-subgroup relations require that some equivalent points in Ia3 are not isomorphic to points in I4r/acd. Basedon the braunite cell, theseinclude the 4x2x2: l6 atoms requiring shuffiing.In Figure 2b, it is seenthat from shuffiing the C polyhedron re- ceivestwo additional coordinating oxygensto form a FIc 2s. The,4' sheetin bixbyite,oriented for direct comparison distorted cube, and the D polyhedron losestwo to with Fig 2a. The coordinatesare from Geller (1971) The arrows form a distorted tetrahedron.Finally, polyhedron E show the atoms which must be shulied to create the I sheel in achievesa more resular octahedralcoordination. braunite P. B. MOORE AND T. ARAKI

ture type, and the presenceof shared polyhedral edges.The bond distancesand angles are listed in Table 6; atom designationsand the sharededges are convenientlyshown as Schlegeldiagrams in Figure 4. These diagrams are the planar maps of the three- dimensionalpolyhedra and conserveorder of vertices and paths along edges.The Mn(2)Ou and Mn(4)Ou octahedra are similar in that their six shared edges define Hamiltonian paths which are isomorphic to each other with respectto the elongatedMn3+-O2- bonds. The opposingpair ofelongated bonds arising from the d high-spinelectronic distribution in a weak crystal field are called the apical bonds, and the four shortened planar bonds are called the meridional bonds. The Mn(2)06 octahedron of point symmetry I has a distribution of sharededges and apicalbonds which suggests 2/m pseudosymmetry, as does Ihe u6*(3) arrangement.Indeed, listing the bond distancesand FIc. 3n. The unit checkerboard representingthe fluorite arrange- angles in Table 6 according to the "equivalent" ment. distances for 2/m shows that deviations range at most about 0.lA for O-O' edges and about 3o for O-Mn(2)-O' angles. The apical bonds are Bond distances,angles and their distortions 2Mn(2)-O(l ) : 2.21A and the meridionalbonds are A major problem in describingthe bond distances, 2Mn(2)-o(2): 1.874and 2Mn(2)-o(3) : 2.034. angles,and their distortions for braunite is the com- We shalldemonstrate later that the splittingof the bined effectof the severeJahn-Teller distortion of the meridional bonds by 0.16A is a consequenceof Mn3+O. octahedra,the relation to the fluorite struc- deviations from local electrostatic neutrality. The Mn(a)-0(6) octahedronof point symmetry 2likewise possesses2/m pseudosymmetry,but this is more seri- ously degradedthan in Mn(2)O. by the unsymmet- rical distribution of nearest-neighboredge-sharing polyhedra, the Mn(l)O, cube distributed such that the 2/m pseudosymmetryis broken. As a result, "equivalent" O-O' distancesdeviate by as much as 0.3A and angles by 8o.The apical bonds are 2Mn(a)-O(3) : 2.244, and the meridional bonds are 2Mn(4)-o(l): l.9sAand 2Mn(4)-o(2) : t.92A; owingto localelectrostatic neutrality, the meridional bondsare split by only 0.03A.Bond distancesfor Mn(2) and Mn(4) arelisted in Table6 as increasing valuesfor Mn(2); thoselisted for Mn(4) areaccord- ing to the isomorphismof the Schlegeldiagrams in Figure4. It is apparentthat Mn(2) and Mn(4) are similarin theirgeometry. The Mn(3)O. octahedronof point symmetry2, which also sharessix of its edges,differs markedly from Mn(2) and Mn(4). Although the Mn(3)-O bond distancesare not unusual [2Mn(3)-O(l) : : : FIc. 3s. The unit sheetrepresenting the crystalstructure of 1.91,2Mn(3)-O(2) t.97,2Mn(3)-O(3) 2.27A), pyrochlore.Note octahedralcoordination about M and cubic the polyhedronis severelydistorted. This is partic- coordinationabout X. ularly evident if the O(3)-Mn(3)-O(3;t'r- l4lo BRAUNITE t235

Frc. 4 Schlegeldiagrams of the Mn-o polyhedra in braunite These diagrams show the order of the atom nomenclatureas usedin Table 6 Sharededges are dashed,and the adjacent cation is listed.The cation in the centerof the polyhedronis listedin the centerof the diagram. Apical oxygensshowing elongatebond distancesare drawn as disks apical angle is noted, and clearly results from the body diagonals,which deviate from 180oby 9o and Mn-Mn repulsionsacross shared edges. The Schlegel l6o as a result of the twist. The SiOotetrahedron of diagram in Figure 4 revealsthat the sharededges are point symmetry4 is nearlyregular as only cornersare along one side of the polyhedron so that Mn(3) sharedwith other polyhedra. moves away from the O(2)-O(2)('z)edge and toward Electrostatic bond strength sums the O(l)(3)-O(l)(4)edge. The result is a bending of O(l)-Mn(3)-O(3)t'r by 39' away from regular tetra- For sphericalions, the bond strength,s, is obtained gonal coordination. It is likely that this kind of dis- by dividing the charge of the cation by its coordina- tortion is a key featurein the stability of braunite and tion number. For minerals and other stable ionic bixbyite structure types, since the polyhedron more crystalsthe sum p, : 2si of the i cations bonded to closelyrepresents the shapederived from the fluorite the anion should deviate,Ap", only slightly from the cube, which as uod2(2)is a highly distorted octahe- chargeof the anion with reversedsign. The deviations dron. correlatewith deviationsin individual bond distances The Mn(l)O, cube of point symmetry 222 shares away from polyhedral averages.Baur (1970)has dis- all twelve of its edgeswith other polyhedra.It is best cussed such calculations and correlations in great describedas a twisted cube,and possessesfour short detail. 4Mn(1)-O1t;''' : 2.154 and four long 4Mn(l)-O(2) Owing to the non-sphericalelectron-density distri- 2.504 bonds. The average bond distance, bution about Mns+ (HS), Kampf and Moore (1976) = Mn(l)-O : 2334, is identical to the value obtained proposed, on empirical grounds, s 4/12 for elon- by Shannonand Prewitt (1969)on a garnet structure. gate apical bonds and s : 7/12 for the shortened : The O-Mn(l)-O anglesrange from 62o to 78o com- meridional bonds, distinct from s 6/12 for a regu- pared with the 70.5' angle for the regular cube. The lar octahedral anion coordination about a spherical most significantangular distortionsare found for the trivalent cation. Calculating Lp,, we observe that 1236 P. B. MOORE AND T. ARAKI

-0.25, O(2)has Ap, = 0.00,O(l ) has Lp" : and O(3) Thus, for G, D, and R as specified, there are 22 has Ap" : +0.25. The local cation undersaturation discretearrangements which are distinguishableun- about O( 1) doubtless correlates with the short der cubic holosymmetry, 23 discrete arrangements Mn(1)-O(l )(1)distances and oversaturatedO(3) with under the cubic enantiomorphic class, 34 arrange- the long Mn(2)-O(3) meridional distancesobserved ments under the tetragonal holosymmetric, and 43 for braunite. arrangementsunder the tetragonal enantiomorphic cfass.It follows thatlF\43211 - 2/mll : 1, Appendix I lF$/m3 the only arrangementhaving chirality being found in - : Enumeration of uacancies on the uertices of the cube undo.Also,lFl422ll lF$/m 2/m 2/mll 9, the chiraf arrangments being one in u6*, two in u't, We are interested in the permutation group, G, three in two in ugff, and one in u'f . whose elementsare basedon the point groups 4/m3 u4X, The configurationsare shown diagrammaticallyin 2/m (cubic holosymmetric), 432 (cubic enantiomor- Figure 6. Under each category, u'd", those grouped phic), 4/m 2/ m 2/ m (tetragonal holosymmetric), and together are equivalent for the cube but distinct for 422 (tetragonal enantiomorphic). The domain, D, the tetragonal prism. They are codedu'd"(m), where a consistsof eight elements,Idrdr,...,dr), which are is an occupancy,r the number of occupiedsites (: the eight vertices of the cube or tetragonal prism, coordination number), d, the vacancies of which and the range,R, consistsof two elements,lu,dl .The there are r, and m is an integer for codifying. The elementsof the range may be conceivedas two colors "*", meansthat the arrangementhas a chiral or as statementslike "yes" and "no." Here, d corre- symbol, mate. Interchanging solid (: above) and open (: spondsto a vacancyat the vertexofthe cube and a to disks leavesthe configuration unchanged,so an occupancy.Diagrams of the discretedistributions below) theseare shown by "equal" signs.Thus, unf(3) is an of vacanciescan be conveniently shown as squares arrangementon a cube and involves one of the six (Fig. 6). A vacancyon the vertex above the plane of discreteways of distributing four vacanciesover the projection shall be shown as a solid disk, and a va- cube's vertices.When the tetragonal prism is chosen cancy on a vertex below the plane as an open disk. as the domain, it is broken into two discretearrange- The cycle indices, PIG:DI, for the four crystal which over classesacting upon D are'. ments,uod'(3) and undn(3t) are isomorphic the cubic domain but distinguishableover the tetra- G\4/m3 2/mt Pl4/m3 2/m:Dl gonal prismatic domain. Since S14/m 2/m I : (x?f l3xi + 8x?x3-t 6xix| + SxlxA+ 12x,,) 2/mlCSl4/m3 2/ml, the tetragonal prismatic do- * main has a larger number of discreteconfigurations. Gl432l P1432:Dl In Appendix II, these modules shall be used as "players" in a lattice game. Sincea tetragonal board :*rr,+exlt8x?x?+6xz) is used, there are 34 kinds of players, those acting under the tetragonalholosymmetric class as the basis Gl4/m 2/m 2/ ml P14/m 2/ m 2/m:Dl for the permutationgroup. Nine of theseplayers have chiral mates. If handednessis counted as distinct, : + In the 34 kinds * of exl4-2xixl + 4xi) then there are 43 "players." turn, of players can be reducedto 22 possiblearrangements Gl422l P1422:Dl: +Sxi+2x,^)over the verticesof a cube,with only undn(6)having a *Ot chiral mate. The maximal point symmetriesof the The patterninventories, F{P:R}, are: arrangementsfor the two holosymmetric classesas basesfor the permutations are given in Table 9. Pl4/m3 2/m:Dl FIP:u,dl: u8+ u1d-f 3u,il -l 3usd * 6uad i- ... Appendix II P1432:Dl FIP:u,dj: uE* u1dI 3u6&-l 3u5fr* Tuada* ... Cooperatiuelqtlice games P\4/m 2/m 2/m:DlFlP:u,dl: The thirty-four discrete patterns in Appendix I can be utilized as "players" in a three-dimensionallattice u" * uId * 5u6fr-f 5uufr-t l\uadn* ... game.The "board" consistsofa checkerboardofany P\422:Dl FIP:u,dl: specified planar edge dimensions. The "black" u8* u1dl- 6u8il l- 7u5&* l3uadar ... squares of the checkerboard will be called the unit BRAUNITE t237

Trslr 9. Point groupsfor the vacanciesat the verticesof a cube*

u8 (l) 4/n 3 z/n u5d2(s) 3 z/n u4dq(t) 4nn ,rada14; ii srn 4ln 2/n 2/n 2/n 4nun 42n

uTal (r) 3m ,-rsa3(r) m u+d+(lt) rrada(s) 3n n n 2nrn m

,r6d2( t ) 2nn usd3(tt) uada(z) n ,raaa(e ) 2run I m 2

u6d2( tt) usd3(z) n uadq12t1 ,rada(6t) n n I

u6a2(z) 2nn usd31Zt; u4d4(3) 2/n 2/n 2/n 2nm I 2/n 2/n 2/n

u6d212t; usd3(g) 3n uaaa1 st; 2 m 2/n

*After each configuration symbol, the point symmetry of the schene is given for a cubic donain followed by the syllllnetry for a tetTagonal prisnatic domain. Synbols with a I'trr in thern inply the tetragonal prismatic donain. For uidr, where s

checkerboard; the "red" squares will be called the conceived.Any arrangementof a unit checkerboard complementarycheckerboard. Fout rules are required which is identicalto that of its complementarycheck- for play: erboard is called self-complementary.This means that Rule I. The unit checkerboard and com- the same kinds of u'd(m), u'' d"' (m')..' occur on plementary checkerboardhave translational proper- both boards and theseoccur in the same order. The ties, the linear translationscorresponding to the vec- complementary board may have to be rotated to tors of the two edgesof the board. achievean exact match with the unit checkerboard. Rule 2. Adjacent players can be linked at a com- This. in effect. admits the existenceof screw axes mon corner only if the symbols of each match. The normal to the boards. possible matching symbols at a corner are no disk; The unit checkerboardof pyrochlore (Fig. 5a) is solid disk; open disk; or both solid and open disk. one such example of a self-complementarystructure. Rule 3. The complementary checkerboardadopts It is generatedfrom rz8(l) and u"fr(3). The board's open disks in place of solid disks of the unit checker- edgesare each twice the length of the fluorite board. board. If open disks or no disks appear on the unit The stacking axis requiresfour boards. A more ele- checkerboard no open disk can be placed at that gant example of a self-complementarystructure is position on the complementary checkerboard.The that of calzirtite, CarZrZroTirO'. (Fig. 5b). It is gen- player is free to add solid disksproviding rules I and erated from z'(l), u'd(l), and u"t(3). This structure, 2 are satisfied. too, requiresfour repeats.The unit board has length Rule 4. The articulated complementary checker- three times that of fluorite. All self-complementary board becomesthe new unit checkerboardand the structures only require specificationsof the unit game is continued until a repeatedpattern along the checkerboard, since the structure is repeated by the stacking axis of the boards is achieved. appropriate choice and orientation of a screw axis. With respect to the fluorite structure, the unit Thus, in Figures5a and 5b, the 4r-screwaxis is added checkerboardcorresponds to a sheetof edge-linked to complete the sequence.Self-replicating patterns cubes, the players being u'(l). The complementary are the composition of a two-sidedplane group with checkerboard is also a sheet of edge-linkedcubes a screw axis normal to it. We are presentlyretrieving which share edgeswith the tops of the cubesof the in a systematicway just those arrangementswhich unit checkerboardbelow. are self-complementary. Several rather interesting kinds of games can be A curious, and probably rather limited version of l 238 P, B, MOORE AND T. ARAKI

Ftc. 5. Checkerboardsfound in brauniteand relatedstructures. Vacancies at the verticesofthe cubeare shown as solid disks(above) or as open disks (below). The occupied cubes are crossed Loci of screw axes are shown. (a) Unit board for pyrochlore. (b) Unit board for calzirtite. (c) Unit board for bixbyite. (d) Complementary board for bixbyite. (e) Complementaryboard for braunite (comparewith 5d). (f) Unit board for braunite (comparewith 5c). self-complementaryarrangements are thosewe call u"E(2) and u"t(3) are distorted so that they are pseudo-self-complementary.In such arrangemented,equivalent in the cubic structure.The easeof forma- the kindsof u'd"(m),u'' d"'(m'),. . . arethe samewith tion of the orthorhombic structure may reflect the respectto (r,s),(r',s'), . '. but the(m), (m'), ... arc presenceof two kinds of vacancy orderings on the not all the same.For sucharrangements, although the undeformed fluorite cubes for the cube, and three coordinationnumbers are of the samekind for the kinds for the tetragonal prism. Furthermore, the articulatedboards, the polyhedralshapes are not su- spacegroup Pbca for orthorhombic bixbyite allows perimposable.Bixbyite is one suchexample; in Fig- the existence of the three kinds of distortions, ure5c, it is seenthat theunit checkerboard consists of u6d(2t), u"fr(2), and u6*(3). u"*(2t) and u"d(3), but the complement(Fig. 5d) Structureswhose unit and complementdo not have consistsof uu*(2) and u6&(2t).The structureis re- the samekinds of (r,s),(r',s'). . . or in which theseare peatedby applyingthe 2,-screwto unit and com- not in the same order are c,allednon-self-replicating. plement. In the actual cubic bixbyite structure, For these,freedom in the choiceof solid diskson the BRAUNITE t239 complementaryboard is permissible.The braunite B- 7 6 4 u7d uGd2 u4d4 layer is one such example(Fig. 5e): the pattern, made up of aE(I and is ?- e- ?-r h? (H u6t(3), uada(4), not self-replicating -l ), I sinceinspection shows that each squareon the com- L1I: LJ L_L_l t! o---€ ?----r A- plementary checkerboard has one open disk. The I tr I IlI aJ oJ F complementto braunite (Fig. 5f) is recognizedas the unit checkerboardof bixbyite. The repeatpattern for uldt ?---- fr H ffi tzl -tl t2l -t I braunite is made completeby applying a 4,-screwto r----a 4 a---s o---{ the compositeof unit and complement. HH F--r r?---- ll l-l I *l2ll *l2l I-rt Beyond the mere enjoyment of playing these e--{ a---a oJ games,we are extendingour study to exploresystem- ?----r ?---r l3l atically and retrieve all patterns and seek out those L1J r----a structureswhich satisfy these patterns. We proceed H I3fl from linessimilar to the study of spacegroup geneal- H 2 3 5 ogies and their application to structure types (see u2d6 uJ d5 dd3 H Neubiiser and Wondratschek, 1966). Completed t4l rJ H o---a three-dimensional composites of boards with the ttl l-1 ril T_T H LLJ a!--J oJ same axial dimensions are called "Zellengleich," and H Fr H H I tr I -TI'J rt -dl-t I compositeboards whose arrangementsbelong to the o---a n lllj o-J i:l same point group are called "Klassengleich." One r< IH H desirableresult is the systematicretrieval of both *l -t fil=f-] t2l r-1 n 6 I I * a!-4 IJ tr lJ (LJ aJ spacegroup and structuretype associated with these (H - H H r? patterns. .l,3lJ *l? | t-l l2ll t-1 *l6tI -tl It is clear that the fluorite arrangementis .J aJ dEJ aJ o_-r particularly adaptiveand admits a rangeof coordination numbers, from eight to at least three, and we H ?----'1 t3l 1'-1 suspectthat a large number of as yet unrecognized d--J t1.l G-4n structure types exist. Frc. 6. The discrete vacancieson the vertices of a cube (numer- Since the checkerboardsclassify "holes," the task als) and a tetragonalprism (numerals,and numeralsplus letters). of describing the structures is much easier. In fact, A vacancyabove the plane of projection is shown as a solid disk bixbyite, braunite, pyrochlore, and calzirtite are tri- and a vacancy below the plane as an open disk. Coordination vial. We are presentlyexploring structuresof rather numbers head the patterns.Chiral arrangementsare starred.Ar- monstrous size,all of which upon preliminary exam- rangementswhich are equivalentby reorientationare shown. Under undn,for example, there are six kinds ofvacancy diStribu- ination prove to be anion-deficientfluorite derivative tions over the cube (1,2,3,4, 5, and 6) and ten kindsover the structures.These include magnussonite,32Mnu (OH) tetragonalprism (1, 2, 3,4, 5, 6, lt,2t,3t, and 6l) disregarding (AsO3)3, Ia3d; stenhuggarite, 8CarFerSbrO2(AsO3)r, chiral mates. I4r/a; cafarsite,4Ca.MnrFenTir(OH)n(AsOa)rr, Pr3; and, perhaps most complicated of all, parwelite, l6MnuSbAsSiO,",A2/a. trostatic effects,that is, from the local arrangement of nearest-neighborcoordination polyhedra.We believe Addendum that this is hardly the complete state of affairs, and Shortly after this study was submitted for pub- our discussionon bond distancesand strengthsin- lication, deVilliers (1975) announced solution and cludes both the effects of Jahn-Teller distortioh and refinement of the braunite crystal structure on a electrostaticrepulsion across shared edges;Table 6 sample from Ldngban, Sweden.The two studiesare and the attendant discussionshow that consistencyis not only completely independentbut proceed from therein obtained. One minor point concerns the quite different approachesand emphasizequite dif- statedM(l)-O distanceof 2.23A in the earlierstudy, ferent aspectsof the problem. Therefore, we decided which is really2.34 A as shown in the table on inter- not to alter our manuscriptexcept in responseto the atomic distancesand should be comparedwith Mn'z+ suggestionsby the referee,which materially improved in eight-fold oxygen coordination, not six, from the the text. Readers of both papers will note that de tablesof Shannon and Prewitt (1969). Villiers' study does not mention Jahn-Teller dis- Regarding the details of the structure refinement, tortion but attributes polyhedral distortion to elec- both studies are in good agreement,save for a con- t240 P. B. MOORE AND T ARAKI sistently higher average of about 0.01 A in bond BysrRdM,A. ANDB. MnsoN (1943)The crystalstructure of braun- distancesin de Villiers' study, doubtlessarising from ite, 3Mn,O, MnSiOr. Ark. Kemi Mineral Geologi, 16, No. 15, l -8. differencesin cell dimensions:a and c are0.024 A and Cnonrn, D T. AND D. LIstnMeN (1910) Los Alamos Scientifc 0.035 A larger than our values. These differences Laboratory, Uniu of Calif. Rep. LA-4403, UC-34. reflect the different techniques in cell refinement, and - ANDJ. B. MINN (1968)X-ray scatteringfactors computed it is not possiblehere to ascertainwhich is closer to from numerical Hartree-Fock wave-functions. Acta Crystallogr. the true values. Scientistswho tabulate bond dis- A24,32t-324. DpVrr-lrer.s,J.P. R. (1975)The crystal structure of braunitewith tances should be cautioned about this differencein reference to its solid solution behavior Am. Mineral. 60, the two studies. 1098-t 104. De Villiers emphasizesthe importance of silica De Vrr-lrens, P R nNo F. H. Hrnrsrrrr (1967) Distinction be- substitution, which increasesto at least 40 weight tween two members of the braunite group. Am Mineral. 52, percent at high temperature. In agreementwith de 20-30 Gellen, S. (1971) Structuresof a-MnrOs, (MnorrrFeoo,r)rO3and Villiers, we suspect a coupled relationship (Mno r" Feou3)2Og and relation to magnetic ordering. Acta Crys' Mn3+Mn3+=Mn'+Sin+. Indeed, utilizing cooperative tallogr. B.21, 82 I -828. lattice games,a large family of derivative structures GoRGEU,A. (1893) Sur les oxydes de mangandsenaturels. Bal/. can be obtained, all consistentwith tetragonalcells, Soc fr. Mineral 16, 133-148. but in many instances,suggesting the appearanceof HnrorNcsn, W. ( I 83I ) Mineralogical account of the ores of man- ganese. Trans. R Soc Edinburgh, ll, l19-142. superstructurelines. Kevpr, A. R. ,ruo P B. Moone (1976) The crystal structureof a hydrated manganesephosphate. Am Mineral. 61, Acknowledgments bermanite, 000-000 This study was supported by the National ScienceFoundation NEUBUsER,J eNp H. WoNonnrscssK (1966) Untergruppender grant NSF G,4-40543(Geochemistry) and through the Materials Raumgruppen Kristallogr Tech. l, 530-543 ResearchLaboratory grant (NSF) administeredto the University P6lv,q, G. (1937) Kombinatorische Anzahlbestimmungen fiir of Chicago. Gruppen, Graphen, und chemischeVerbindungen. Acta Math 6E, t46-254. References Re.necHnNon,rr,G. N. eup R. SntNlvnslu (1970)Fourier Meth- AMINoFF,G. (1931)Lattice dimensionsand spacegroup of braun- ods in Crystallography Wiley-lnterscience, New York, p. ite K Suenska Vetensk Akad, Handl.9, 14-22 96-t 19. Beun, W H. (1970)Bond lengthvariation and distortedcoordina- SsrNNoN, R. D lNo C T Pnewlrr (1969) Effectiveionic radii in tion polyhedra in inorganic crystals. Trans Am. Crystallogr oxides and fluorides. Acta Crystallogr. 825,925-946. Assoc.6, 129-155. ZncHnnr,csrN,W. H. (1968) Experimental tests of the general BunNnnvt,C. W. (1966) Computation of absorption corrections, formula for the integrated intensity of a real ctysta'I.Acta Crys- and the significanceof theend effectAm Mineral.5l, 159-167. tallogr. A24,212-214 BusrNc,W R , K O. Menrrr and H A. Lrvv (1962)ORFLS, a Fortran crystallographic least-squares program. U.S. Natl. Manuscript receiued, February 5, 1976; Tech. Idorm Seru. ORNL-TM-305. acceptedfor publication, July I 3, 1976.