American Mineralogist, Volume 58,pages32-42, 1973
Braceletsand Pinwheels:
A Topological-Geometrical
Approach
to the CalciumOrthosilicate
and Alkali SulfateStructures
Peur Bnre,NMoonr, Department of the GeophysicalSciences,
The Untuersityof Chicago,Chicago,Illtnois 60637
Abstract
Glaserite, IGNaISO*], and its high temperature isotype "silico-glaserite," c-Ca"[SiOn]; merwinite, Ca"Mg[SiOnJr;larnite, p-Ca,[SiOd; room temperature p-trG[SOn];bredigite, ca. CazMg
[SiOn]-;K[LiSO+J; and palmierite, K,Pb[SO4Jzare the basesof the atornic arrangements for over
100 compounds, Some of these compounds are of considerable interest to the cement, blast
furnace, brick and fertilizer industries.
The glaserite structure type consists of one large alkali cation which is ideally l2-coordinated
by oxygen atoms, six of which define the vertices of an elongate trigonal antiprism and six of
which reside in an hexagonal ring in the plane of the large alkali. The tetrahedral grouping
around tho antiprism defines a "pinwheel" where the apical oxygens point either up (z) or
down (d).
Tho bracelet is a mathematical object, a loop with z nodes involving rz symbols, where,m (
n. For the pinwheel, n - 6 (an hexagonal ring) and m :2
(u or d). Combinatorially, the
total number of distinct bracelets is 13. For any bracelet there is a pinwheel which, idealized,
defines the maximum coordination number of the central large alkali. The maximum coordination number is 12-p whero 0 < p S 5 and where p are the number of tetrahedral apical oxygens coordinating to the alkali. The bracelets can be used to construct, by condensation, ideals
of real and hypothetical atomic arrangements found in the CadSiOd polymorphs and many of
the alkali sulfates.
The coordination polyhedra ofinterest include 7(tetrahedron); M(octahedron, : p : 6); Xtlr-el
(which, for p : 0, has ideal symmetry 121m1;Yt ot (point symmetry 3m) and f'trzl (suboctahedron).
Condensation of the bracelets and their associatedpinwheels defines the maximum coordination
numbers for all the polyhedra in the ideal model. Thesemodels can be used to classify known structures and to retrieve hypothetical ones, one of which may correspond to bredigite.
Introduction
eral problem of the structural aspectsof calcium
orthosilicatepolymorphismin detail. The attemptsto
show structural interrelationshipsamong key compoundshave beenmeagerindeed,augmentedby the
fact that only a few reasonablyprecisestructurerefinements of the calcium orthosilicates have been
reported. But few papers have attempted to afford
the generalrelationshipsamong thesestructuresand
none to my knowledge utilizes the polyhedral approach which shall be the themeof this contribution.
The structuretypes included in this discussionare
glaserite,KaNa[SOr]2,and
merwinite,CasMg[SiO+]2;
its high temperatureisotype "silico-glaserite", o-Ca2
[SiO+J;larnite, F-Caz[SiO+];room temperatureB-K2
[SOr] and its myriad isotypes;bredigite, o,'-(Ca,
Mg)zlSiOrl; KtLiSO4l; and palmierite,KzPblSO+lz.
In a recent detailed atomic arrangementanalysis
of merwinite, CagMg[SiO+]z(Moore and Araki,
1972), it becameabundantlyclear that a review of
the calciumorthosilicatestructureswas necessary
in
order to advance our knowledge of these related
structures.As this review progressed,it becameevident that extensive confusion and uncertainty
aboundsin the literature. This is not surprising in
light of the fact that the literature on the subject of
calcium orthosilicatesis voluminous, largely on account of their crucial importance in the cement,
clinker, slag, fertilizer and brick industries.
This paper is not intended to be an exhaustive
treatiseon the subject,but shall draw from pertinent
literature only where that knowledge is imperative
in advancing some general geometrical models of
The Glaserite Modek Its Geometry
these structures.Despite the fact that no less than
100 compoundshave been characteized as isotypes
All compoundsdiscussedevincea profound hexaof the structure types which have bearing on this gonal substructurewhich, in many instances,has
discussion,only a handful of papers treat the gen- led to obfuscationin the literature regardingidenti12
33
BRACELETS AND PINWHEELS
fi.cationby powder diffractometry. Glaserite is the
simplest arrangement,the one whose geometry defines the substructureobserved in the other compounds. Glaserite, KaNa[SO+]zand the structurally
havebeeninvestisimilar aphthitalite,KeNaNa[SO+],2,
gated by severalworkers but highly accuratestructure refinementsappearto be wanting.
The structureofaphthitalite appearsas a polyhedral
diagram in Figure l. It derives from the study of
Bellanca (1943). The large cation polyhedra, when
idealized,consistofthree kinds, and illustration ofthe
first two in Figure 2 shall be the key theme in this
subject. The glaseritestructure was investigatedby
Gossner (1928) and re-examined by Fischmeister
(1962) from the standpoint of cation disorder. One
largecation coordinationpolyhedron(Fig. ?-a)consists
of a l2-coordinated trigonal antiprism with six
additional meridional anions in hexagonal outline,
with ideal point symmetryZZ/m.It is identicalto the
idealizedpolyhedron discussedby Moore and Araki
(1972)aboutthe Ca(l) cation in merwinite.The second
large cation coordination polyhedron involves,when
idealized,l0 nearestanionswhich possess3z trigonal
symmetry(Fig. 2b). Again, six anionsare situated on
the verticesof a meridionalhexagon,threeaboveshare
the tetrahedral base and one below the tetrahedral
apex.This polyhedroncorrespondsto the idealsfound
for the Ca(2) and Ca(3) coordination polyhedra in
merwinite. The remaining polyhedron, of point
symmetry32/m,is a trigonal antiprism,whoseideal is
the octahedron. This correspondsto the Mg coordination polyhedron in merwinite. In glaserite,
Gossner(1928)ascribesK to this polyhedron,which
hasbeendocumentedby Wyckoff (1965),but it is more
likely that Na belongsto this sitewith the K cationsin
the l2-coordinated site.
Severalgeometricalgamescan be played with the
glaseritestructuretype. If the trigonal antiprismof the
l2-coordinated polyhedron is idealized to an octahedron and definedwith an edgeequal to the tetrahedral edge,the hexagonalclose-packednetwork in
Figure 3a results. Geometricall!, the apices of the
tetrahedra are at the same heights as those of the
octahedra,so becomesecondnearestneighbors,and
the true coordination number of the central large
cation is reducedto 6. In actuality, the ionic radii of
the K* and Su* cations are propitiously related to
assurea close approximation to l2-coordination by
oxygens. This is demonstratedby construction in
Figure 4. From the ionic radii tables of Shannonand
Prewitt (1969),an ideal KOu octahedronis constructed
o?
o,l-
LJ
FIo. l. Polyhedral diagram of the glaserite atomic arrangement. The central trigonal antiprism and six circumjacent tetrahedra are stippled' The KOr" arrangement is
shown as a spoke diagram.
using16* : 1.38A and ts.- : 1.40A. This resultsin
an ideal octahedraledge distance,I : 3.93A. For a
regularoctahedron,the altitude,ft, normal to opposing
faceswould be (l/2) tan 60" sin 70o 32' : 0.825 l.
Thus, ft : 3.24A. Now, definethe tetrahedraledge :
X[rzl
-SZtn
o
Sztn
c
Fro. 2. Three large cation coordination polyhedra. a.
Xr"r, consisting of trigonal antiprism (triangles) and six
meridional anions. b. Yt'or, which occurs above and below
tetrahedra. c. F*t, the cuboctahedron.
34
PAUL BRIAN MOORE
Fro. 3a. Condensation of trigonal antiprisms and circumjacent tetrahedra to form the sheet of Xt,1 polyhedra.
that this ideal model closely approximatesthe 12coordinated polyhedron, where the meridional distancesare only slightly longer than the apical octahedral distances.
In the actual glaseritestructure,the KOr2 polyhedron is significantlydistorted,such that the "octahedral" component is elongatedto a trigonal antiprism. Such an arrangementcan be appreciatedin
Figure 1, where the approximateO(2) coordinates
would lead to an estimatedaltitude h - 0.600 x
7.3 A = 4.4 A, a very significantdilation from the
3.2 A altitude compr.rtedon the basis of ionic radii
and the ideal octahedron.This accommodates
the
geometrical problem encountered in Figure 4. A
tetrahedronwhoseedgeis only half that of the ideal
octahedron would be too small for the O-O'
tetrahedralinteratomicdistancewhich would be 3.93
/2 : L96 A. The actual tetrahedraledge is approximately 20 percent larger than the geometrical
construction and contributes to further dilation of
the ideal "octahedron"into an elongatetrigonal antiprism.
The pinwheel. The key to the relationship of
glaseritewith the other structuresdiscussedherein is
the dispositionof the tetrahedraabout the octahedral
Fro. 3b. Condensation of trigonal antiprisms (-octahedra) and tetrahedra to form the sheet of MgOu octahedra
and SiO+ tetrahedra in merwinite. The merwinite cell is
outlined.
l/2; this results in the six circumjacenttetrahedral
apicesaround the r.entraloctahedronsituatedin the
plane of the large cation at the octahedral center.
Define the distancefrom the octahedralcenterto its
vertex : Z. The distancefrom the octahedralcenter
to the tetrahedralapex in the plane would be L' :
6/r/ \/rL + 6/r/\/dL
: r.063z.rt is ctear
Frc. 4. Construction of MOe trigonal antiprism (-octahedron) and one of the circumjacent tetrahedra where the tetrahedral edge is
half the octahedral edge. Q are the octahedral
vertices, P the tetrahedral apex. The dashed line
approximates the [SO.] tetrahedron when M K*.
BRACELETS AND PINWHEELS
35
component of the KO', polyhedron. This grouping,
featured in Figure 5a, definesa cluster with point
symmetry 3Z/m and I shall refer to it as the "pinwheel". Condensationof such pinwheelsleadsto the
diagram in Figure 3a. We have encounteredthe
complementof this pinwheelin the close-packedslab
of the merwinite structure,where six SiOntetrahedra
are groupedaround the MgOooctahedronas depicted
in Figure 5b. Condensationof this pinwheelleadsto
the hexagonalclose-packedsheetin Figure 3b. Such
0b
an arrangementis complementaryto the glaserite
arrangementsincethe six tetrahedralverticesare asfar FIc. 5. a. Pinwheel for the Xt'r cation in glaserite. b. Pinwheel for the Mgrtr octahedron in merwinite.
removed as possible from the octahedral core and
cannotbe associatedin the inner coordinationsphere.
Now, the geometricalrestrictionis imposedon the
To derive and describethe other structuresin this
paper, we shall engage in a topological exercise pinwheels thus derived. We consider only those
tetrahedral altitudes which conform to the glaserite
utilizing bracelettheory.
model, that is, the height of the tetrahedron from
The Pinwheel Bracelets
base to oposing vertex is exactly half the height of
There doesnot appearto be a strict mathematical the trigonal antiprism. Thus, associatedwith each
definition for the term "bracelet," but Gardner bracelet configuration in Figure 6 are tetrahedral
(1969) appliesthe term to a cyclic chain involving apicalpositionswhich may potentiallybe coordinated
symbols which shall be manipulatedin a combina- to the central large cation. For convenienceof vistorial way. Thus, a braceletis definedas a cycliccol- ualization, potentially coordinating apical positions
lection of n-beads or nodes involving rn symbols are drawn as bold disks.
wherem 1 n. It is alsoto be notedthat the bracelets
It is necessaryto discusscomplementarityof the
may possessthe property of cornplementarity,that bracelets in terms of the glaserite structure. As
is, where the interchangeof symbols for some ru Figure 1 reveals,the repeat distancenormal to the
leadsto further solutions.l
hexagonaloutline is the repeat of the pinwheel and
The pinwheelgrouping is teducedto a bracelet its complementabove.In the ideal construction,the
a braceproblem in the following manner. In any pinwheel, central large cation at level z = 0 possesses
tetrahedralapicespoint either up (r.r) or down (d). let whosecomplementcorrespondsto the ideal coorThus, the symbolsare distributedat the nodal points dination of the central large cation at z = l/2.
of an octahedronand the possiblecombinations, Thus, for any specifiedbracelet and its associated
including complements,are derived. We set n = 6 pinwheel,the complementarypinwheel occurs above
and m = 2. The complementsin this case are de- and specification of any one leads to the other.
rived by a 60o rotationof the symbolsin their proper It is alsoworthy to note that (4 + 2)" and (4 t 2)u
a self-complement,where the central
order. The solutions are 6u = l; 5u * ld : 2; each possesses
=
=
possible
13
6,
with
3u
3d
at
4;
4u * 2d
I
large cations both levels would possessthe same
are
depicted
arrangements
arrangementsin all. These
ideal configuration. The remaining large cations
combination
by
the
in Figure 6. They are symbolized
situated above and below the tetrahedronare auto(u * d),, where (u * d.),,indicatesits complement. matically defined: in the ideal arrangement' they
coordinateto six oxygensassociatedwith the trigonal
antiprism, to the three oxygens of the tetrahedral
l Some definitions are in order. Depending on their
base. and to the tetrahedral vertex, resulting in an
utility, nodes may represent vertices of a polyhedron, the
center of a polyhedron, etc. Connections between nodes ideal coordination number of 10.
are branches (for example, edges of a polyhedron). If
nodes and branches are connected such that tracing a path
without retracing one's steps leads to the starting point then
the chain of nodes and branches is cyclic and constitutes
a loop. In this study, each node has a symbol ascribed
to it (in this case. either "u" or "d").
Discussion of the Structures
The pinwheelsderived from the braceletscan now
be used as modules for constructing hypothetical
structures.Such an approach is reminiscent of the
PAUL BRIAN MOORE
U
d
u
(3'3)b
d
( 5 * l )o
u
u
(5*l)o'
u
(4t2) c'
u
d
(3*3)b'
u
d
d
d
Fto. 7. Condensation of the (4 * 2)a - ( 4 +
bracelets to form a periodic array.
u
(4*2lo=(4*2lo'
(3*3)o
(3*3)c
u
u
d
d
=(4,2)ti
(4+215
d
(3+3) o,
u
l?+?\a/
2)d
initially used when the first pinwheel is constructed,
which limit to some extent the symbols permissible
for the adjacent bracelets.Some modules, such as
(3 + 3)" and (4 * 2)o,leadto simplearrangements
with small cells, others do not. For example, (3 +
3), must alwaysbe usedin combinationwith some
other braceletat the samelevel to generatea repeating pattern.
It is only necessaryto specify the connectionof
braceletscorrespondingto one level along the pinwheel repeat axis. This is clear since the complementary solution at z * l/2 is automatically defined and that the remaininglarge cation sites above
(6+910
Fta. 6. The thirteen distinct bracelets involving six nodes
and two symbols distributed over trigonal antiprismatic
vertices. Solid disks represent tetrahedral apices coordinating
to the central large cation,
study of Smith and Rinaldi (1962) who generated
cells based on parallel 4- and S-memberedtetrahedral loops, and used these solutionsto clarify the
geometricalideals of the feldspars,harmotome and
paracelsianstructures.
Despitethe fact that 13 distinct pinwheelsexist,
the possiblecombinationsin periodic arrays appear
to be quite limited. This derives from the symbols
FIc. 8. Condensation of pinwheels corresponding to Figure /.
BRACELETS AND PINWHEELS
37
and below the tetrahedraare dictatedby the orientationsof thosetetraliedra.
Considerthe bracelet(4 + 2), which has a selfcomplement.Figure 7 revealsthat the condensation b
to the
of the braceletsof the samekind corresponds
The symnodal points of the hexagonaltessellation.
bols around any hexagonare identical to any other
hexagonwithin the repeatunit. This solutionappears
in Figure 8. It has formula unit 4M2[TO+] and
possesses
the spacegraupPnma for the geometrical
ideal. In the inorganicworld, it correspondsto the
atomic arrangementof room temperatureK2[SO4].
Glaserite. The glaseritestructureis derivedfrom
ideal l2-coordithe bracelet(3 + 3)" and possesses
nation. Its complementabove,(3 + 3),,, possesses
a coordination polyhedron which is a trigonal antiFrc. 9. Polyhedral diagram of the room temperature
prism. It is evidentwhy the K atom was placedin IGtSO,l structure. Compare with Figure 8. The KOro
the (3 + 3), location and Na in (3 * 3)"', re- polyhedron (=Yrtot; is shown as a spoke diagram.
versedfrom the inferred ordering schemesuggested
by Gossner(1928).
complete
The crystalcell criteriafor glaserite,KNaKz[SO+]g Unfortunately, Robinsondid not present
can
information
such
but
data,
distance
interatomic
are a = 5.65 and c = 7.29 A. This is compared
(NH4)r[WS4]
the
of
the
structure
for
found
be
the high temperawith "silico-glaserite,"
"-Caz[SiOn],
(NHa)*
ture calcium orthosilicate polymorph: Douglas isotypereportedby Sasv6ri(1963)' The
nearest
possesses
9
pinwheel
grouped
in
the
cation
(1952) reportsa = 5.46 and c = 6.76 A.
to
3.10
from
ranging
N-S
with
atoms
and larnite neighbor S
Room temperatureK,ISO 4l(B-KrlSO,,D
which
(NHa)-S
polyhedron,
remaining
(B-CarlSiOal. The structureof room temperature 3.58 A. The
as
possesses'
Kr[SO4] (also called B-KrtSO+l) was determined is situatedabovethe tetrahedralbase,
from
ranging
N(1)-S
with
by Robinson ( 1958) ; in addition, the isotypes expected,l0-coordination
is also noteCszlCrOrl (Miller, 1938) and KzlCrO+l (Zachafia- 3.47 to 3.96 A. The Sasv6riarticle
which an atin
few
the
sen and Ziegler (1931)) were investigatedfrom a worthy in that it is one of
on
structures
these
structuralstandpoint.Wyckoff (1965) lists 47 com- tempt has been made to discuss
a more generalbasis.
poundsbelongingto this structuretype.
structurewas reported
The larnite, B-Ca2[SiO+],
:
For B-Kr[SOn],the crystalcell criteria area 7.46,
a
(1952)
as
distorted
Midgley
by
F-Kz[SO+]struc:
:
:
4.The
b 5.78,c 10.08A, spacegrotp Pnma,Z
are a = 5'48,
parameters
cell
crystal
The
ture.
:
5.78
a-direction is the pinwheel repeat and b
= 94o33',P2r/n' The
c/6 : 5.82 A emphasizesthe pseudo-hexagonal b = 6.76, c = 9.28 A, B
= bp,
with
comparison
F-Kz[SOn] is evidently a1
characterof the x-axis projection.Thesevaluesare to
the
be comparedwith a : 5.65and c : 7.29L reported bt = 48,cr = cfl.The spacegroup of larnite is
Larnite
proper
orientation.
the
in
Pnm:q
of
subgroup
by Gossner(1928)for glaserite.
The point symmetryof the (4 + 2)" pinwheelis is topologically equivalentto B-K2'[SO+]but is geo'a
m andthe structurereportedby Robinson(1958) is metrically distorted in such manner to violate the
shown as a polyhedral diagram in Figure 9. The K mirror plane of the ideal arrangementin Figure 8'
atoms at the two levels are split away from the ideal The explanationappearsto be a matter of compropinwheel center, with K at z - 0 oriented in the mise between ideal geometry (see Figure 4) and
direction towwd the three oxygen atoms at tetra- actu:alcootdinationnumber.The evidenceappearsin
hedral apices.I also note that the pseudo-trigonal Midgley'sinteratomicdistancetables.Ca(2), which
antiprism oxygens in that direction arc dilated is in the pinwheel, possessesonly 8-coordination
whereas those at the opposing vertices ate com- with Ca-O 2.36-2.80A, and Ca(1) above the
pressed.This suggestsan approximationof 6 * 3 :
tetrahedral base possesses6 inner coordination
spheresranging ftom 2.30'2.75 A and 6 outer co9-coordination.
38
PAUL BRIAN MOORE
Ftc. 10. Condensation of bracelets in Figure 6 which may
correspond to the bredigite arrangement.
ordinationspheresfrom 2.98-3.56A. It is clearfrom
this discussion that the ideal arransements are
approximated in crystals only when In approach
to the geometrical ideal in Figure 4 is evident.
Briefly, controlling factors are the ionic radius ratio
of the large cation to the small tetrahedrally coordinatedcation, a point which shall be discussed
in greater detail further on.
Bredigite @'-Ca2 [SiO+]): An Attempt At
Postulating Its Structure
Evidently, no three-dimensionalsolution of the
bredigite atomic arrangement has been reported.
This is not surprising in light of its large cell with
fairly low symmetry.Douglas (1952) presenreda
detailed single crystal investigation on the compound, which is also calledoo,-Ca,2[SiO*].
Shereports
a = L0.93,b = 6.75, c = 18.41A, spacegroup
Pmnn, Z = 16. Thesevalues are essentiallythose
of the "ideal" B-K2,[SO4]
equivalentof BJarnite,but
with doubleda and c axes.It is clear that the pinwheel repeatdirectionis the b-axis.Douglas'study
appearsto be a carefulinvestigationand she points
out the cell relationshipwith B-K'[SO4]. In addition, she notes that the intensity distribution indicates a centrosymmetric
crystal.
A structurefor bredigiteis postulatedwhich is
consistentwith the cell criteria of Douglasand which
is derived directly from the bracelet combination
which allows Pmnn symmetry.Figure 10 is a condensationdiagramof the braceletswhichleadto ideal
symmetryPmnn, and Figure 11 is the corresponding condensationof pinwheels which effectively is
an idealizedmodel of the proposedstructure.It is
made up of 2Ca in the bracelet (3 -f 3),, 2 Ca
i n ( 3 * 3 ) " , , 2 C ai n ( 3 + 3 ) , , 2 C a i n ( 3 + 3 ) , ,
and & Ca in (4 + 2),. Figure 6 revealsthat the
ideal coordinationnumberswould be 8, lO, 12, 6,
and 9 respectively.This would suggesta bredigite
compositionCa2BM2Mz'ITO+]ro
whereM andM' are
smaller"impurity" cationsof maximum coordination
numbers 6 and 8, respectively.This would suggest
a solid solution range of Mg'* for Ca2* between
(Car.zaMgo.z5)
and (Ca1.s6Mgb.r2),
wherethe upper
bound is Mg'* in the (3 + 3),, environment.This
range is close to compositionsof synthetic bredigite
reviewedby Biggar (1971). The remaining 16 Ca
are in the positions above and below the tetrahedra
and are automatically defined from the bracelets
specified beforehand. Their coordination numbers
average10.
No assertionis made that this is the correct structure for bredigite; the actual structure can only be
revealed through careful three-dimensionalcrystal
structure analysis.I only wish to emphasizethat an
TneLE 1. AI. ell
Sur,r'lrs
SrnucrunBs: For,var.
COtvtpOSnONS
ft4l : tetrahedron
MlBl : octahedron
ltal : n6ds5 of 3 2/m polyhedron
fttol:3zpolyhedron
f'tnl : nedss of cuboctahedron
Fro. 11. Condensation of pinwheels correspondingto Figure 10.
glaserite
merwinite (distortion)
0-Kz SOr
larnite (distortion)
palmierite
kalsilite
x|r2l Y 2t1oI M t6)[ Ttrt g 01,
xtsl Y Isl Y I8lM t6lI ?"Ia]0d z
xtelrtr0l[?Ft410{]
xI6tYIslTl4lO4
ylrzlf ,trotlTta)o elz
Ftelrl4l[?"t4104]
39
BRACELETS AND PINWHEELS
arrangementwith Pmnn symmetrycan be retrieved
in the same manner that the ideal Pnma arrangement for B-K2[SO4] was retrieved with no prior
knowledge of its crystal structure. Becauseof obvious structural relationships among these compounds,particularlyevidentin projectiondown the
axis normal to the pseudo-hexagonal
outline where
electron densitieswould be similar. careful threedimensionalanalysisis the only good tool for ascertainingthe true arrangement.
\
\
.t
.l
-l
\
Bo.j3
I
I
A l + 0 . 3 , 3+ ' 4
Additional StructureTypes With Pinwheels
0il1+1.84
The idealizedcoordinationpolyhedraencountered
in this study includesthe pinwheelpolyhedronwhose
coordinationnumber is at least 6 (the octahedron)
and whose possible maximum is 12, when all
meridional oxygens are present. It is symbolized
as Xt"t where6 1 n 3 12; for n = 6, the resulting
octahedronis symbolizedas M. The secondpolyFrc. 12. Coordination polyhedron associated with Ba in
hedron with ideal point symmety 3m is ideally
Ba[Al,O,]. Solid disks represent meridional oxygens which
lO-coordinatedand shall be symbolizedas Yt101. coordinate to the central Ba atom. Note that three meridional
The third polyhedronis the tetrahedron,symbolized positions which would complete the cuboctahedron are missas T. Table 1 summarizesthe compositionsand struc- lng'
tures consideredin this study; I emphasizethat the
coordination numbers refer to the ideal model and
TheKILiSO,'Jstructuretype. Someconfusionexists
actual crystalsmay show true coordination numbers in the literature regardingthis structuretype. Bradley
which are lower on accountof distortion.
(1925) approximately solved the structure and the
Many compounds in the literature have com- polar character of the crystals was well-established.
positions suggestingordered derivativesof the sim- The structurerevealsthe tetrahedralpinwheel(6 + O)"
pler structure typ€s. For example, among the 47 and its self-complement.Such an arrangementposcompoundslisted by Wyckoff (1965) representing sessesideal 9-coordination. In this arrangement,the
the B-Kr[SOr] structure type, are compositionslike meridionaloxygensare rotated awayfrom the trigonal
KBaIPO+1,KCa[POa],NaCa[POn](low),e/c. It is antiprism. The Li atoms occur in tetrahedral coinstructive to inquire how the alkali and alkali-earth ordination,defininga [LiSOl'- tetrahedralframework
cations are distributed. According to Table 1, the with spacegroup P63.When the tetrahedralatomsare
generalformulacanbe writtenXt'glytlol[TO4]where identical, the Ba[AlrOn] structure results,with space
9-coordination for X is the ideal number for the grotp P6"22.Figure 12 preserttsthe disposition of
(4 + 2)" pinwheel.Struck and White (1962), in circumjacentoxygenatomsaround the centralbarium
their structure analysisof KBa[POa], have estab- atom. In this arrangement,the meridionaloxygensare
lished X = Ba and Y = K. Although a structure rotated exactly 30oaway from the trigonal antiprism.
analysisof KCa[POn]has not beendone,it is reason- This definesnine of the twelveverticesof a polyhedron,
able to assumethat X = Ca. We shall note through- symbolizedFrt'r, featuredin Figure 2c. It is ideally
out this discussion that the smaller cations tend the cuboctahedron,an Archimedean solid with 12
toward the X positionsand the larger cationstoward vertices, 24 edgesand 14 faces, and is a frequent
the Y positions.
coordinationpolyhedronin alloy and metal structures
particularly
inSeveral other structure types are
whenthe centralatom and its coordinatingatomshave
triguing and are now discussed.Although they do similar crystal radii. In addition, it is the coordination
not fall within the restrictionsof the structuresbased polyhedron of an anion with respectto its nearest
on ,the condensationof pinwheels, similar if not neighbor anions in ideal cubic close-packing.Moveidentical coordination polyhedra prevail in their -"nt fro- the Xt"r polyhedronto the Ft"r polyhedron
is geometricallycontinuouseventhough the polydehra
arrangements.
LJ
40
PAUL BRIAN MOORE
PALM
IERITE
(3 + 3)" pinwheel found in glaserite.The K atoms
reside in the Y position with actual point symmetry
3m. This is yet another examplewhere ordering of
different cationsleadsto site preferenceof the larger
cation in the YO,opolyhedron.
The merwinite structure: its topological identity
with glaserite. Moore and Araki (1972) havepresented an extensivediscussionon the geometry of
the merwinite(Ca3Mg[SiO+]2,
a = 13.25,b : 5.29,
c = 9.33 A, I = 91.90o,P21/a,Z = 4) structure
and pointed out its profound pseudo-hexagonal
characterwhen idealized(i.e., freed from geometrical distortions). Despite its complicatedstructure,
the presentstudy revealsthat merwiniteis a distorted
equivalentof the glaseritestructure, comparedwith
the B-larnite structure as the distorted equivalent
of the B-K2[SOa]structure.The authorshave shown
that the geometrical ideal of merwinite can be
written XIa2lY2tlotM[TOn],which is identical to
the generalglaseriteformula. In the actual structure, where X, Y = Ca"*, M - Mg',* and Z =
Sin*, the coordination numbers are closer to
xt8)YtetYrs'tMlToi2.
Figure 14 illustratesthe relationshipin the stacking of the cations for merwinite and glaseritealong
the (pseudo-) hexagonalaxis. The correspondence
Frc. 13. Stacking sequencein the palmierite
in actual heights along the a-axis of merwinite and
arrangement. Heights are given as Angstriim
the c-axisof glaseriteis apparent.
units.
Although there is no evidence that a higher
temperature polymorph of merwinite exists, it is
are not topologically identical since the former is logical to assume that if such a polymorph oc3--connectedand the latter is 4-connected. This curred, it would possessthe glaserite arrangement,
movementis facilitated by rotation of the tetrahedra just as "silico-glaserite"is the high temperature
in the pinwheel about their trigonal axes. Kalsilite, polymorph of BJarnite. Although Wyckoft (1965)
K[A1SiO4] possessesthe K[LiSO1] structure but classifieshigh-KzlSO+l and high-NazlSOrl with the
oxygendisorder occursin the structure(Perrotta and KtLiSO4l structure type, there is no evidencesupSmith, 1965).Wyckoff (1965,p. il3) lists 15 com- porting this classification.More logical would be
poundswhich may possess
the K(LiSOn]structurebut their classificationwith the aphthitalite (glaserite)
it appearsmore likely that they actuallybelongto the arrangement.A self-consis'tent
relationship between
aphthitalitestructuretype (P3zl) sincethe ,.?', atoms the alkali sulfate and calcium orthosilicatestructures
are large cations such as Ca'* , Ba"*, K* and Na*. is evident: at low temperatures,the olivine structure
The palmierite, KzPbISOnf"
structure type. About type exists; at intermediate temperatures,the B10 compunds crystallize in the palmierite structure Kz[SO+] structure and its geometrical distortions;
type, most notably the alkaline earth ortho- and at high temperature the glaserite arrangement
phosphates.This arrangementcrystallizesin space prevails. It is noted that the proposed structure of
group R3z and differsfrom the other structuressince 'bredigite sharesaspectsof its structure with both
the pinwheeland its complementno longer make up p-Kr[SO4] and glaserite arrangements and may
the c-axis repeat.As Figure 13 reveals,the sequence representa fleld of structural transition where propalongthe cu-axis
involves'.. XIt')-Y"ot-T-Terties of both low and high temperaturepolymorphs
frtor ... . The Pb : X cationcorresponds
to the are present.
-20.9-
Pbx[2]
<-- K Y[oJ
ci
- 8.4-
r [cl
, [cl
K ylol
-0.0-
<F- to 1[tzl
BRACELETS AND PINWHEELS
MERWINITE
o
4l
Y polyhedronsharesa face with the small highly
GLASERITE the
charged tetrahedral group but the X polyhedron
M
y[roJ
T
x[4
T
yIoJ
M
y[oJ
T
x[4
T
y[oJ
M
Fro. 14. Comparison of stacking sequencesbetween (a)
merwinite and (b) glaserite.
Coordination Number and Structure
For an ideal arrangement, if a pinwheel has
3 6,
coordination number Xrr2-at, where O 1p
then its complement has coordination number
Xr6+pt. Pinwheel and complement average to rnean
coordination number Xt'gl. This is the maximum
average coordination number for the X polyhedra
in glaserite-derivative structures; the actual structures on account of geometrical distortion may
possess average coordination number less than this
value.
For mixed cations where the differences in ionic
radii are considerable (such as Ca'* and Mg2* or
K* and Na*), p will tend toward 0 for the larger
cation oxygen coordination and the smaller cation
will be accommodated in its complementary polyhedron. But it is more difficult to propose an ordering scheme between the Xt^) and ytrot polyhedra.
Geometrically, the two polyhedra are similar and
the coordination number of Y will tend toward a
higher value than n for most structures discussed in
this study. A further controlling factor appears to
be the cation-cation repulsion effect. It is noted that
shares faces only with similar large polyhedra of
relatively low central positive charge. Even in
palmierite, the X polyhedra are shielded from the
tetrahedra by the intervening Y polyhedra. Thus,
cation-cationrepulsionsbetweenthe Z and Y polyhedra are particularly violent and can be minimized
in two ways: in the caseof cationsof samecharge
but different ionic radius, the larger cation will tend
toward the Y polyhedron;and for cationsof different
chargebut similar ionic radius, that cation of lower
charge will tend toward the Y polyhedron. Thus,
for KBa[POa] and KzPb[SO+]2,the K atoms are
located in the Y polyhedra. Likewise, in the
aphthitalite struc,ture,the larger K* cations are locatedin the Y polyhedra.
Severaltrends are advancedon the basisof foregoing discussions:
1. For distributions over the X polyhedra for
structuresinvolving cations of different crystal radii
in XtL2-pt,p will tend toward 0 for the large cation
and will tend toward 6 for the small cation in the
complementaryPinwheel.
2. For distributionbetweenX and Y, the lower
charged cation among cations of similar crystal
radius will tend toward the Y polyhedron.
3. For distribution betweenX and Y, the cation
of larger radius among cations of samecharge will
tend toward the Y polyhedron.
It is worthwhile to note that one of the problems
in the study of high temperaturepolymorphs,especially the Q32[SiOe1structures,is the inversion of
such structuresupon quenching.Investigators,especially in the cement and blast furnace industries,
have empirically observed that the presence of
"impurity atoms" can stabilize high temperature
Caz[SiOe] structures. Especially good cations for
this purpose are Na*, Mg'*, Sr'* and Ba2*- The
explanation for the stabilizing influence of these
cationsappearsstraightforward.In structures,where
p differs considerablybstween a pinwheel and its
complement,larger cations will order over the sites
of low p and the smallercationsover sitesof high p.
Likewise, in structures where a pinwheel and its
complement are identical, ordering will occur between the X and Y polyhedra. Arguing from the
oppositedirection, it shouldbe possibleto selectjust
the right cations to assurethe stability of that structure. This emphasizesthe pressing need for more
accuraterefinementsof the crystal structuresof the
42
PAUL BRIAN MOORE
various polymorphs with especialemphasison the
site preferencesof the "impurity" cations.
Addendum
During construction of this manuscript while in Australia, I corresponded with Dr Walter Eysel concerning
crystals of calcium orthosilicates. He informed me of his
studies on A"[BX.] and &[B&]
compounds. Upon my
return, Dr. Eysel kindly presented me with his doctoral
dissertation (Eysel, 1968), which is an extensive study
on glaserite and other alkali sulfate compounds, and many
of his tabulations on refined cell parameters constitute new
information. His study appears to be the most exhaustive
review on the A"[Bx"] and lr[B&] compounds to date. My
approach to this problem appears unique and with little
duplication of the Eysel study. Accordingly, I submitted
this manuscript with but minor alteration and urge interested
readers to the outstanding study of Dr. Eysel as well.
Acknowledgments
Opportunity to think about general problems in structur€ type generation and classification was afforded by professor A. E. Ringwood. I thank professor Ringwood and
the staff at the Institute of Geophysics and Geochemistry,
Australian National University, for much cooperation and
hospitality. This work was supported by a Dreyfus Foundation award and funds provided by the Institute of Advanced Studies, Australian National University.
References
Barr.eNce, A. (1943) Sulla struttura dell'aftitalite. period.
Mineral 14, l-29.
Brcc,ln, G. M. (1971) Phase relationships of bredigite
(Ca"MgSLO-) and of the quaternary compound (CaoMeAl'sio.)
in the system cao-Mgo-AlzorSiog.
cement Concrete Res. l, 493-513.
BReoLEy,A. J. (1925) The crystal structute of lithium potassium sulfate, Phil. Mag. 49, 1225-1237.
DouGLAs, A. M. B. (1952) X-ray investigation of bredigite.
Mineral. Mag. 29, 875-884.
Evser, W. (1968) Kristallchemie polymorphie und Mischkristallbildung uon Silikaten und Germanaten der Ver-
bindungstypur A"BO, und A,BO". Ph.D. Thesis. Technische Hochschule Aachen.
FrscuuersrBn, H. F. ( 1952) Riintgenkristallographische
Ausdehnungsmessungenan einigen Alkalisulfaten. Monatsh. Chem. 93, 420434.
GenoNrn, M. (1969) Mathematical games: Boolean algebra, Venn diagrams and the propositional calculus. ^Jci.
Amer. 2?4, 113-114.
GossNnn,B. (1928) Ueber die Kristallstruktur von Glaserit
und Kaliumsulfat. Neues lahrb. Mineral. Abt. A BeilageBand, 57,89-116.
Mrocrrv, C. M. ( 1952) The crystal structure of p-dicalcium
silicate. Acta Crystallogr. 5, 3O7-312.
Mrrren, J. J. (1938) The crystal structure of caesium
chromate, CsCrOn. Z. Kristallogr. 99, 32-37.
Moone, P. B., lNo T. Anexr (1972) Atomic arrangement
of merwinite, Ca"Mg[SiO]2, an unusal dense-packedstructure of geophysical interest. Amer. Mineral. 57, 13551374.
Pennorre, A. J., eNo J. V. Surrn (1965) The crystal
structure of kalsilite, KAlSiO.. Mineral. Mag, 35, 588595.
RorINsoN, M. T. (1958) The crystal structures of p-KrSO.
and of B-IGPOaF. I. Phys. Chem. 62, 925-928.
Slsv,(nr, K. (1963) The crystal structure of ammonium
thiotungstate (NlL)'\ryS,. Acta Crystallogr. 16, 719-724.
SueNNoN, R. D., eNp C. T. Pnswmr (1969) Effective
ionic radii in oxides and fluorides. Acta Crystallogr.825,
925-946.
Surnr, J. V., euo F. RrNlrtr (1962) Framework structures
formed from parallel four- and eight-membered rings.
Mineral. Mag. 31, 202-212.
SrRUcK, C. W., eNp J. G. WHrre (1962) The unit cell dimensions and crystal structure of KBaPOr. Acta Crystallogr. 15,290-291.
Wvcrorr, R. W. G. (1965) Crystal Structures,Vol. 3,2nd
ed. John Wiley and Sons, New York. pp. 9l-116.
Zecxlnnsrx, W. H., eNo G. E. Zrscren (1931) The
crystal structure of potassium chromate, KzCrOn. Z,
Kristallogr. tO, 164-173.
Manuscript receiued May 26, 1972; accepted for
publication, tuly 18, 1972.