AmericanMineralogist, Volume75, pages 1159-1 169, 1990

Ringso topology and the density of tectosilicates

L.tns Srrxnuon, M.S.T. BurowrusrrDepartment of Geology and Geophysics, University of California at Berkeley, Berkeley, California 94720, U.S.A.

Ansrucr

A review of density data for tectosilicates shows that variations in density within a singleframework type are typically an order of magnitude smaller than the twofold variationsamong different framework types. To help explain this large range of densities, we haveanalyzed the geometric and topologic properties of different frameworks. We used thegeometry and statistics of clusters to construct a predictive model for the frameworkdensity of tectosilicates. Ring statistics and geometries, together with a simple theory ofthe effects of ring formation on framework density, lead to the definition of a framework-specific characteristic ring size. We show that this characteristic ring size increases withincreasing framework density.

INrnooucrroN

Tectosilicates, including the ubiquitous silica poly-morphs and feldspars as well as feldspathoids, scapolites,and the technologically important zeolites, are the mostabundant minerals in the Earth's crust (see, e.g., Hurlbutand Klein, 1977). Although the few hundred distinct spe-cies vary widely in color, habit, composition, and otherphysical properties, we focus here on how variations indensity are controlled by differences in the underlyingframework structure, the continuous three-dimensionalnetwork of corner-sharing SiOo and AlOo tetrahedra com-mon to all tectosilicates. In addition to its fundamentalcrystallographic significance, the relationship betweendensity and framework structure is central to the designand synthesis of new low-density zeolites important toindustry (Smith and Dytrych, 1984; Davis et al., 1988;Brunner and Meier, 1989) and may also elucidate therelation between the highly variable framework struc-tures and compression of silicate liquids (Stixrude andBukowinski, 1989).

Although the total number of tectosilicate species isIarge, many have topologically identical frameworks, andonly about 80 distinct underlying framework structureshave been observed in naturally occurring and syntheticspecimens (Meier and Olson, 1988; Smith, 1977, 1978,1979; Smith and Bennett, 1981, 1984). The variation indensity among different types of frameworks is muchlarger than variations within a single framework type. Forexample, the differences in molar volume among themyriad feldspar species are more than an order of mag-nitude smaller than the difference between the molar vol-ume of the densest known framework, coesite, and thatofthe sparsest, faujasite, which differ in density by a factorof two, whereas other theoretically proposed structuresare less dense by a factor of three than coesite (Meier,1986). The goal ofthis paper is a description ofthe ge-

ometry and topology of framework structures that canexplain this remarkable variation in density.

The fact that local geometries [as measured by (Si,Al)-Oand (Si,Al)-(Si,Al) distances and coordination numberslin different framework structures are nearly identicalmakes the wide range of observed densities even moreremarkable. This has led several authors to examine therelationship between density and topological elements offrameworks such as clusters and rings. For instance, acorrelation between cluster populations and density hasbeen observed @runner, I 979; Akporiaye and Price, I 989)and a relationship between the smallest ring in a frame-work and its density has recently been noted (Liebau,1988; Brunner and Meier, 1989).

Here we combine topologic and geometric measures ofclusters and rings to examine more closely their relation-ship to density. We first review the concept of frameworkdensity and describe in some detail its wide variabilityamong tectosilicates. We then describe the notion of acluster and present a simple predictive model that relatesthe topology and geometry of clusters to framework den-sity. Finally, we critically examine the definition of a ringand introduce a theory for the effect of ring formation oncluster size and, thus, framework density.

Fru,unwonxs AND FRAMEwoRK DENsrrY

Framework density (FD), used by many previous au-thors in studies of tectosilicates (e.9., Brunner and Meier,1989), is defined as the number of tetrahedrally coordi-nated atoms, or T atoms (usually Si or Al), per unit vol-ume. This is a convenient measure, since it allows us todirectly compare tectosilicates with different chemicalcompositions. The wide range of FD for the frameworktypes considered in this study is shown in Table l. Al-though many species may share a single framework type,the variability of FD within a single framework type ismuch less than the total variability in FD. For example,

0003-{04x/90/09 l0-t I 59$02.00 I 1 5 9

Framework

I 160

Tlale 1. Tectosilicate framework densities

STIXRUDE AND BUKOWINSKI: TECTOSILICATES

Reterence

Lawton and Rohrbaugh(1seo)

Gramlich and Meier (1970)Robson et al. (1973)Fischer (1966)calligaris er al. (1982)Meier and Kokotailo (1965)Gard and Tait (1972)Stades and Gard (1959)Merlino et al. (1975)Barrer and Villiger (1969)Rinaldi et al. (1974)Galli (1971)Fischer (196i1)Alberti and Vezzalini

(1 983)Bartl and Fischer (1967)Meier (1961)Sieber and Meier (1974)LOns and Schulz (1967)Jarchow (1965)Pechar et al. (1983)Gottardi and Meier (1963)Perotta (1967)Gerke and Gies (1984)Gies (1983)Kokotailo et a]. (1985)Rohrman et al. (1985)Schlenker et al. (1985)Kocman 6t al. (1974)wyckoff (1982)Wyckoff (1982)Colville and Ribbe (1968)Deiseroth and Muller-

Buschbaum (1973)Peacor (1973)Tak6uchi et al. (1973)Levien ard Papike (1976)Haga (1973)cohen et al. (1977)Levien et al. (1980)Geisinger et al. (1987)

aRbFELDSPARS

BolI

a'

t

t a 't a

Al=50% r rr

_ t a) r l r

l a

t t '

a t '

r or l

l t

, atolio

aa

a

a

.l*

41.10 1 2 3 ^ - , + s

(cATroN RADTUS (A)) 'Fig. l. Summary of density data for the feldspar framework

type, plotted as the volume per T atom vs. the average radiuscubed ofthe interstitial cations, taken from Shannon (1976). Thealkali feldspar series is plotted as circles and the alkaline-earthseries as squares. End-member compositions are indicated byenlarged symbols. The data from the Ca-Sr-Ba series are fromBambauer and Nager (1981), the Na-K series from Kroll et al.(1986), and the K-Rb series from McMillan et al. (1980). Thedifference between high (Al-Si disordered) and low (Al-Si or-dered) structural states (not shown) is comparable to the size ofthe larger symbols. The total variation in framework volume (orframework density) shown is less than 120,6, small comparedwith the variation among different framework types (Fig. 3).

cations from the structure. Then, since in the actual struc-ture each T atom is bonded to four O atoms, each ofwhich in turn is bonded to one other T atom, we thinkof T atoms that share a common O as being linked toone another in the simplified structure. The relationshipbetween simplified and actual structures is illustratedschematically in Figure 2. Since all of the properties con-sidered here, including framework density and measuresof cluster and ring size, are identical in both the simplifiedand actual structures, the two structural representationsare completely equivalent for the purposes of this paper.

This conceptual picture of tectosilicate structure can b€used to examine the relationship between frameworkdensity and measures of local geometry, such as bondlength and coordination number. In many crystal struc-tures, variations in these quantities can be related directlyto variations in density. For example, the densities ofmany materials with simple structures, such as metals,can be characterized entirely by a packing fraction ofat-oms that depends mostly on interatomic distance andcoordination number (e.g., Kittel, 1976). These conceptscan sometimes be fruitfully applied to silicates: the dif-ference in density between quartz and stishovite can beattributed to the increase from tetrahedral to octahedralcoordination (Stishov and Popova, 196l) and the in-

17.O

o<16.2

ol!

- 45.4l l

! o..e=Jo> /ts.EYEQ +s.oUJ

! +z.zEla

zsM-18

Linde-ARhoGmeliniteChabazitezK-'OffretiteErioniteLevyneLinde-LPhillipsiteStilbiteGismondineHeulandite

LaumontiteMordeniteLosodSodaliteCancriniteNatroliteDachiarditeEpistilbiteDodecasilMelanophlogitezsM-22zsM-23zsM-48BikitaiteParac€lsianTridymiteFeldsparCaGarOo

CristobaliteCaAl,SirO"MarialiteBanalsiteCordieriteQuartzCoesite

14.27 3.090 14.13

12.88 3.201 14.1714.17 3 .101 14 .1914.59 3.131 15.0414.47 3.141 15.0614.68 3.128 15.0915.52 3.131 16.0015.64 3.136 16.1915.23 3.177 16.3916.37 3.104 16.4315.82 3.144 16.5016.29 3.125 16.6815.29 3.197 16.7717.12 3.115 17.36

17.78 3.084 17.5117.03 3.130 17.5315.78 3.221 17.7017.20 3.136 17.8116.58 3.179 17.8717.73 3.111 17.9217.34 3.148 18.1617.65 3.13€ 18.2218.47 3 .111 18 .6818.96 3.124 19.4119.73 3.092 19.5720.00 3.081 19.6419.92 3.091 19.7520.29 3.100 20.292'1.45 3.090 21.2422.21 3.081 21.81n.zs 3.096 22j621.49 3.131 22.14

23.28 3.074 22.7023.36 3.075 22.8021.75 3.141 22.6322.52 3.128 23.1423.15 3.146 24.2026.52 3.057 25.4329.28 3.086 28.87

Note.'The mineral chosen to represent each framework type is listedalong with the reference to its structure. For alternative species see com-pilations by Meier and Olson (1988), Smith (1977, 1978, 1979), and Smithand Bennett (1981, 1984). L is the average T-T distane. Units ot FD andFD* ar€ nm-3 and L is in angstroms. FD. is defined by fD' : FD(UL)",where the standard T-T distance L is chosen to be 3.1 A.

it has long been recognized that the FD offeldspars variesinversely with the ionic radius of the interstitial cation(see the review by Smith and Brown, 1988). However,Figure I shows that this variation, along with changes inFD due to differences in Al-Si ordering and relative con-centration, is less than l2o/o compared with the twofoldtotal variability among different framework types. Thus,a description of the geometric and topological variationsamong different framework types on which FD dependsmost strongly is central to an understanding of the den-sities of tectosilicates.

The measure of framework density suggests a simpli-fied conceptual picture oftectosilicate structure that willbe useful in characterizing different framework types (seealso Smith, 1982, p. I 6 I ff). Since framework density de-pends only on the number of T atoms, we form the sim-plified structure by removing all O atoms and interstitial

Fig.2. Relationship between actual tectosilicate structure andthe simplified conceptlral structure used in this paper. On theleft is the bitetrahedron, the largest structural unit shared by alltectosilicates, consisting of two T atoms (solid circles) surround-ed by their coordinating O atoms (open circles). A portion ofthesodalite structure, together with its simplified version (far right),derived by eliminating O atoms and interstitial cations and draw-ing links between T atoms that share a common O.

crease with pressure in the density of quartz can be at-tributed to a decrease in T-T distance (Stixrude and Bu-kowinski, 1988, 1989). However, the variability of localgeometries among different framework types is insuffi-cient to explain the observed variability of FD amongtectosilicates. Applying our simplified conceptual pictureof tectosilicate structure, we see that each T atom is al-ways linked to four other T atoms, and thus all frame-work structures are fourfold coordinated. Further, thevariations in T-T distance, although inversely related toFD, are insufficient to explain the total variability of FD.This is illustrated in Figure 3, which shows that the vari-ability of FD* (FD scaled to a standard T-T distance) is9090 of the variability of FD itself. The inability of localgeometries to explain framework densities provides theprimary motivation for the examination of larger scaleframework elements such as clusters and rings.

Cr,usrnnsClusters in tectosilicate framework structures are the

physical expression of the coordination sequences firstapplied extensively to tectosilicates by Meier and Moeck(1979) and Brunner (1979). A cluster consists of a cen-tral T atom, the four T atoms linked to it, all the Tatoms linked to these four and so on. The four T atomslinked to the central T atom are referred to as the firstlinked neighbors. All the T atoms linked to the first linkedneighbors (except of course for the central T atom) arereferred to as second linked neighbors. All the T atomslinked to the second linked neighbors (except the firstlinked neighbors) are referred to as third linked neighborsand so on. More specifically, for a cluster of size Q withnumbers of first through Qth linked neighbors i{,, N., Nr,. . . , Na, the total number of T atoms in the cluster Mnis given by:

oMo:r*Pry ( l )

1 l 6 l

TECTOSIL ICATESFRAMEWORK DENSITY

aa ,

FD'=FD(L/1")5

1 2 1 6 2 0 _ 2 1 2 AFD (nm- ' )

Fig. 3. The framework density of all the frameworks listedin Table I plotted against their framework density scaled to astandard T-T distance (I,. : 3.1 A). fnis shows the wide vari-ability of framework density among tectosilicates and the in-ability of variations in T-T distance (Z) to account for thisvariability.

As an illustrative example, the number of Oth linkedneighbors for a cluster in a Bethe lattice (Bethe, 1935;Domb, 1960) with a coordination number of four (Fig.4) is given by

Nn:4 x 3o- t (2)

and the total number of T atoms in such a cluster of sizeQ is given by:

a - rM a : l + 4 x ) 3 i : 2 x 3 0 - l . ( 3 )

t:0

This particularly simple framework, sometimes referredto as a tree, will prove usefirl when we discuss rings be-low. In addition to Nn and Mn we have computed D,the average distance to the Oth linked neighbor shell foreach of the framework structures considered in this studyup to Q: 6. Some of the resulls of these computations,discussed in detail in Appendix 1, are listed in Tables 2and 3.

In order to relate clusters to tectosilicate density, weconstruct a simple model that allows the calculation offramework density from the number of,T atoms in a clus-ter (Nn and Mn) and the spatial dimensions of the cluster(Dn). In this model, we consider a framework to be com-posed of many clusters of a given size Q. Although onemay divide the framework into clusters in such a waythat the overlap among clusters is minimize4 in general,adjacent clusters will share some number of T atoms. Weassume that only the outermost T atoms, -those in theQth linked neigfubor shell, are shared and that each of

STIXRUDE AND BUKOWINSKI: TECTOSILICATES

2 1

I

E320

o[!

t 5

t 2

t t 6 2

Fig. 4. Bethe lattice complete through the third linked neigh-bor shell.

these Nn atoms is shared on average between two clusters.If we also approximate the shape of each cluster as asphere, the calculated framework density for our model is

FD: (Mo-, + YzN)/V(D) (4)

where

V(Dd: 4zrD3n/3' (5)

In order to assess the validity of this simple model, wecompare FD calculated from Equation 4, using Q: 6, tothe actual FD for all the framework structures listed inTable l. Figure 5 shows the excellent agreement betweenEquation 4 and actual framework densities: the root-mean-square deviation between model and data is 3010.We chose O : 6 since this is the smallest cluster size thatcontains all the fundamental rings of the frameworkstructures examined here (see Table 4 and discussion ofrings below). For much larger Q, the approximation thatonly atoms in the Qth shell are shared will become lessaccurate, whereas for a much smaller @ different frame-works become indistinct: in the limit of Q: 1, all frame-works have Nn:4. Nevertheless, the calculated FD val-ues are insensitive to small changes in Q: for Q: 5, theroot-mean-square deviation between calculated and ac-tual FD is 4ol0. Thus, although the assumptions uponwhich the model is based can only be approximately cor-

the model is remarkably accurate in predicting ac-framework densities from the properties of micro-

scopic clusters ofT atoms, suggesting a close associationbetween the structure of clusters and that of the entireframework.

Further examination of the properties of clusters showsthat the variability of topological measures (Ma,Na),ratherthan geometric measures (Dn), of cluster size are primar-

STIXRUDE AND BUKOWINSKI: TECTOSILICATES

TABLE 2. Cluster statistics

Framework Ar, N2 ,V" M ,v, ^1.

zsM-18Linde-ARhoGmeliniteChabazitezK-sOffretiteErioniteLevyneLind+LPhillipsiteStilbiteGismondineHeulanditeLaumontiteMordeniteLosodSodaliteCancriniteNatroliteDachiarditeEpistilbiteDodecasilMelano-

phlogitezsM-22zsM-23zsM-48BikitaiteParacelsianTridymiteFeldsparCaGa,OoCristobaliteCaAlrSi20sMarialiteBanalsiteCordieriteQuartzCoesite

4.000 9.5294.000 9.0004.000 9.0004.000 9.0004.000 9.0004.000 9.0004.000 9.3334.000 9.3334.000 9.33i]4.000 9.3334.000 9.0004.000 10.2224.000 9.0004.000 10.6674.000 10.0004.000 11.6674.000 10.0004.000 10.0004.000 10.0004.000 8.8004.000 1 1.6674.000 1 1.6674.000 11.647

4.000 12.0004.000 12.0004.000 12.0004.000 11 .6674.000 12.0004.000 10.0004.000 12.0004.000 10.0004.000 11 .0004.000 12.0004.000 11 .0004.000 1 1.0004.000 1 1.0004.000 10.6674.000 12.0004.000 10.000

17.294 29.294 45.176 65.41217.000 28.000 42.000 60.00017.000 28.000 42.000 60.00017.000 29.000 45.000 65.00017.000 29.000 45.000 64.00017.000 29.000 45.000 64.00018.000 30.667 48.667 72.00018.000 30.667 48.667 71.33318.000 30.667 48.000 68.66718.333 31.000 47.000 68.00018.000 32.000 49.000 69.50019.111 34.667 57.111 80.00018.000 32.000 48.000 67.00020.444 36.000 59.383 85.33319.393 32.667 52.000 74.00022.OOO 38.000 60.333 88.00020.000 34.000 53.000 76.00020.000 34.000 52.000 74.00020.000 34.000 54.000 78.00018.800 35.200 52.800 75.60022.000 38.667 63.000 93.00022.000 39.333 64.333 92.33323.824 40.588 64.000 92.353

24.783 42.261 67.652 98.00023.000 41.000 64.667 94.33923.000 41.167 65.000 95.16723.833 40.667 64.3&1 93.66724.OOO 42.667 69.333 98.66721.000 37.000 57.000 81.00025.000 44.000 67.000 96.00021.000 38.000 57.000 81.00024.000 41.000 63.000 91.00024.000 42.000 64.000 92.00024.000 41.000 62.000 90.00022.000 41.000 64.667 92.00023.000 42.000 66.000 97.00023.393 M.667 66.000 102.00030.000 52.000 80.000 116.00022.500 47.000 83.000 126.000

rect,tual

Note.' No is the mean number of neighbors in the Oth linked neighborshell of a single T atom averaged over all T atoms in the structure'

ily responsible for the observed variation in frameworkdensity. There is, for example, a good correlation betweencluster population (M) and framework density (Fig. 6).Similar correlations between Nn and FD have been ob-served by several previous authors using slightly differentdata sets (Brunner, 1979; Akporiaye and Price, 1989). Incontrast, Figure 7 shows that the geometry of clusters, asmeasured by Dn, is nearly independent of framework type.The standard deviation of Du for all framework structuresis only 100/0. An explanation of variations in the topolog-ical size of clusters, and thus framework density, is there-fore of primary importance. In the next section we showthat variations in ring statistics provide this explanation.

RrNcs

Although a lucid discussion of the possibility for widestructural variation in tetrahedral frameworks appearedas early as 1932 it Tachaiasen's pioneeiing work onglasses, the systematic description of the topology of dif-ferent framework types is more recent, apparently origi-nating with Bernal's (1964) suggestion that rings, rather

A/ote.' Do is the mean distanc€ from a single T atom to neighbors in itsOth linked neighbor shell averaged over all T atoms in the structure.

than the coordination numbers so profitably employed indescribing ionic and metallic materials, should be the pri-mary topological measure of covalently bonded struc-tures. Thus, the characterization of physical models ofsilica glass built shortly thereafter (Evans and King, 1966;Bell and Dean, 1966) included the first complete mea-surements of the ring statistics of a tectosilicate structure(King, 1967). However, the definition of a ring used inthese early papers, very similar to that used by Smith(1977,1978,1979) and Smith and Bennett (1981, 1984)in their more recent work on the derivation of crystallinetectosilicate structures, differs from the one used in thepresent study; a detailed discussion of the definition of aring is therefore required.

In the most general sense, a ring may be taken as anyreturning path in the framework. However, this is not avery useful definition since there is an infinite number ofsuch rings, and the number of rings of a given size in-creases without limit with the size of the ring. Thus forpractical and aesthetic reasons, identification of a smallset of rings as being somehow fundamental is desirable.The older definition, first given by King (1967), counted

1 163

CLUSTER MODELFOR FD

t 2 1 5 2 0 - 2 1 2 8FD (nm- ' )

Fig. 5. Illustration of the excellent agreement between ob-served framework density and that predicted by the simple mod-el relating cluster density to macroscopic density, which is de-scribed in the text (Eq. 4).

the set of six smallest rings, one through each of the sixpairs of T-T links emanating from a T atom, as funda-mental. This definition, however, leads to ambiguities inring counting (Smith, 1978) and igrrores the common oc-currence of more than one ring of the same size passingthrough a single pair of T-T links (Belch and Rice, 1987).On a practical level, many different framework structures

250

TECTOSILICATES .CLUSTER TOPOLOGY

a a

o . '

aa

aa a

7.tl'.

ba a

t 2 2E

Fig. 6. Plot showing the good correlation (r: 0.88) betweenthe cluster population (M) and framework density, indicatingthat most of the variation in cluster density is accounted for bychanges in the number of T atoms in a cluster rather than bychanges in geometric size.

STIXRUDE AND BUKOWINSKI: TECTOSILICATES

TABLE 3. Cluster geometry

Framework D, D64D3D2

zsM-18Linde-ARhoGmeliniteChabazitezK-5OffretiteErioniteLevyneLinde-LPhillipsiteStilbiteGismondineHeulanditeLaumontiteMordeniteLosodSodaliteCancriniteNatroliteDachiarditeEpistilbiteDodecasilMelanophlogitezsM-22zsM-23zsM-48BikitaiteParacelsianTridymiteFeldsparCaGar04CristobaliteCaA12Si,OoMarialiteBanalsiteCordieriteQuartzCoesite

3.090 5.0823.201 5.3073.101 5.2153.131 5.2103.141 5.2433.128 5.1953.131 5.1763.136 5.1763.177 5.2693.104 5.0893.144 5.2373.125 5.0593.197 5.3303.115 5.0583.084 5.0963.130 5.0673.221 5.3503.136 5.2323.179 5.2403.1 1 1 5.1693.148 5.0953.133 5.0713.111 5.0873.124 5.0903.092 4.9993.081 4.9783.091 5.0263.100 5.0283.090 5.0313.081 5.0313.096 5.0303.131 5 .1103.074 4.9783.075 4.9843.141 5.0433.128 5.1003.146 5.0613.057 4.9533.086 5.037

7.210 9.3507.504 9.5047.286 9.2177.251 9.2317.332 9.3107.229 9.2357 .173 9.1877.168 9.1637.274 9.2316.936 8.8107.279 9.1217.236 9.3597.377 9.2387.231 9.3246.993 9.0107.325 9.4037.355 9.3567.132 9.0847.222 9.2097.109 8.7267.353 9.4417.322 9.3967.192 5.2487.222 9.2847.O40 9.0977.007 9.0616.950 9.0987.091 9.1266.834 8.5946.845 8.9086.841 8.5956.893 8.8446.732 8.7406.723 8.5937.003 8.7486.926 8.8296.805 8.5856.710 8.6896.822 8.402

11.398 13.47011.524 13.6091 1 .216 13 .17911.222 13.2281't.267 13.2231 1 .195 13 .1 1411.240 13.25411.200 13 .187't1.221 13.22310.893 13.0241 1 .062 13.1731 1 .332 13.38811.213 13.37311.272 13.39910.946 12.86011.507 13.682'11.420 13.50811.050 13.08911.285 13.30610.828 12.99611.506 13.68511 .419 13.64611 .390 13.54911.382 13.62311.225 13.3351 1 .184 't3.27411.151 13 .30111.188 't3.42310.496 12.50310.921 13.02710.504 12.57710.889 12.92210.666 12.72810.569 12.55710.776 12.92610.807 12.98010.810 12.77010.867 12.93110.176 12.405

2 1

I

E3zo

o4

1 6

t 2

t 5 2 0F D ( n m - ' )

I r64 STIXRUDE AND BUKOWINSKI: TECTOSILICATES

l 4

t 2

TECTOSI L I CATESCLUSTER GEOMETRY I

II

NAR R INGS

o < l 0

o- 8

il(JZ 6

a6 +

0 1 2 3 , + 5 5S H E L L N U M B E R : Q

Fig. 7 . Summary of cluster geometry in tectosilicates, show-ing the average distance to each ofthe first six linked neighborshells for all frameworks listed in Table l. The horizontal widthof the symbols shown for each Do (pairs of Gaussian curvesdrawn out to + twice the standard deviation) is proportional tothe probability of finding a framework with a particular value ofDn. The observed Dn values are compared with two limitingcases: (1) Dn expected ifevery atom in the Qth linted neighborshell closed a planar ring of slze 2Q, and (2) Dn expected if everyatom in the Qth shell lay at the end of a chain of length Q withT-T-T angles equal to the ideal tetrahedral angle of 109.47". Thetotal variation in Dn values is small compared with variationsin cluster density (Fig. 6) and compared with the two limitingcases shown, indicating that cluster geometry is approximatelyindependent of framework type.

have identical ring statistics by this definition (Smith,1977), suggesting that a new definition would be useful.We find the simplest and most appealing definition of afundamental ring, introduced by Marians and Hobbs(1989b), to be any ring that cannot be divided into twosmaller ones. It is free of the ambiguities of the olderdefinition and results in a much greater variability of ringstatistics among different framework structures. Some ofthe differences between this and the older definition offundamental rings are illustrated schematically in Fig-ure 8.

We used this new definition of a fundamental ring andan algorithm described in Appendix I to determine thering statistics for all the framework structures examinedin this study (see Table 4). These are the first measure-ments of ring statistics of tectosilicates, oth€r than thesilica polymorphs (Marians and Hobbs, 1989b), that arebased on the new definition.

To analyze the ring statistics of tectosilicates, we intro-duce a simple theory of the effects of ring formation onframework density. We shall argue that, in general, ringformation tends to decrease framework density, and thatforming a small ring affects the density much more than

Fig. 8. On the left is a schematic representation of a portionof the cristobalite structure showing the 12 fundamental six-membered rings that pass through each T atom (after Mariansand Hobbs, 1989a). The older definition of a fundamental ring(see text) allows only six rings to pass through any T atom. Onthe right is a schematic representation of a portion of a hypo-thetical framework. The older definition of fundamental ringswould consider as fundamental the two four-membered ringsand the six-membered ring. The definition used in this paperalso counts the two four-membered rings but counts the seven-membered ring instead of the six-membered ring, since the lattercan be divided into two smaller rings (see dso Smith, 1978).Note that each structure shown is complete through the secondlinked neighbor shell of rhe enlarged central atom and, for clar-ity, includes only those third linked neighbors that close rings;in the case ofcristobalite, this is half of the 24 tolal third linkedneighbors.

forming a larger one. To illustrate this we will consideronly clusters, rather than entire frameworks, and assumethat ring formation affects only cluster topology (Ma,Nn)and not geometry (Dn). These limitations are justified,

since we have shown that cluster topology is closely re-lated to framework density (Fig. 5) and that cluster ge-ometries of frameworks with widely varying ring statis-tics are very similar (Fig. 7). Further, we make thesimpli&ing assumption that the relative efect of differentsizes of rings is the same for all frameworks. This allowsus to base our discussion on a simple framework, theBethe lattice described earlier, which will yield an ana-lytic expression for the effect of ring formation on frame-work density.

Figure 9 shows the effect of ring formation for differentsizes ofrings on the cluster topology ofthe Bethe lattice.For example, the formation of a three-membered ringbelonging to the central T atom means that two firstneighbors ofthe central T atom become linked to eachother. However, because of the restriction that every Tatom can be linked to only four others, these two firstneighbors must now break their links with T atoms inthe second neighbor shell of the central T atom, thuspruning from the tree two branches radiating outwardfrom two T atoms in the second neighbor shell. Similarly,forming a four-membered ring belonging to the central Tatom means that two T atoms in the second neighborshell must coalesce into one. Again, the restriction of four-coordination then requires that four branches be prunedbeginning in the third neighbor shell. More generally, for

STIXRUDE AND BUKOWINSKI: TECTOSILICATES I 165

Taau 4. Ring statistics

\n fo t K :

Framework 1 21 1

zsM-18Linde-ARhoGmeliniteChabazitezK-sOffretiteErioniteLevyneLinde-LPhillipsiteStilbiteGismondineHeulanditeLaumontiteMordeniteLosodSodaliteCancriniteNatrolileDachiarditeEpistilbiteDodecasilMelanophlogitezsM-22zsM-23zsM-48BikitaiteParacelsianTridymiteFeldsparCaGa.OoCristobaliteCaAlrSi106MarialiteBanalsiteCordieriteQuartzCoesite

0.180.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00

2.123.003.003.003.003.002.672.672.672.673.00'1.78

3.001.332.000.332.OO2.002.004.000.330.330.350.000.000.000.330.002.O00.002.001.000.001.001.001.001.330.002.O0

1.760.000.000.000.000.000.000.000.000.000.002.220.003.330.005.000.000.000.000.005.005.004.415.223.333.331.673.330.000.000.000.000.000.003.330.000.000.000.00

o.715.001.001.001.001.002.002.002.001.670.004.000.001.386.000.004.004.004.000.000.000.00'1.24

0.785.005.007.504.003.00

12.004.506.00

12.006.002.00

10.504.676.001.50

1.240.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00

0.001.002.006.006.006.004.004.004.006.674.002.224.001.332.672.000.000.000.00

12.801.331.330.000.000.000.001.332.678.000.00s.00

16.000.00

20.005.332.000.00

40.002.00

0.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.007.200.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00

40.000.009.00

0.000.00

10.000.000.000.003.333.3!!0.006.675.001 . 1 10.001 . 1 13.333.335.000.00

10.000.006.673.335.292.611.671.670.830.000.000.00

22.500.000,000.003.33

10.006.670.002.50

0.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00

11.00

0.71 4.488.00 4.940.00 4.967.00 4.981.00 4.860.00 4.839.33 5.195.33 5.101.38 4.850.33 s.fft0.00 4.800.00 5.110.00 4.522.O0 4.98

20.00 5.691.q) 5.3!l

14.00 5.4082.00 5.5620.00 5.740.00 5.323.67 5.510.00 5.230.00 5.330.00 5.250.00 5.580.00 5.580.00 5.720.00 5.670.00 5.430.00 6.000.00 6.090.00 6.330.00 6.000.00 6.470.00 5.510.00 5.990.00 6.930.00 7.389.00 6.00

/vofe.'The average number ot rings ot size Kpassing through a single T atom averaged over all T atoms in the structure is given by f(rg.' Characteristic ring size defined by Equation 1 1.

a tree of size Q, the number of T atoms pruned from thetree, P(Q,K) by the formatl:l,o, u ring of size K is:

P ( Q , K ) : 2 x > 3 i : 3 @ - s - l ( 6 )

for odd-membered rings and

@ - s I

P ( Q , K ) : l + 4 x ) 3 i : ) x 3 o - s - I ( 7 )

for even-membered rings, *]rere ,S is the linked neighborshell in which a ring closes: S is l/2(K - l) for odd ringsand l/2K for even rings. We define the relative pruningefficiency, P*(K) as the large Q limit of P(Q,K) norrnal-izedlo P(Q,3):

P*(K) : nm P(Q,K)/P(Q,3). (8)

Thus, for odd-membered rings:

P*(K) : l/3vdK 3) (9)

and for even-membered rings:

P*(K): 2/3h(K-2) (10)

A plot of P*(K) (Fig. 9) shows it to be rapidly decreasingfunction of ring size, emphasizing the greater effect ofsmall rings on framework density.

Our theory predicts that a measure of ring size thattakes into account the greater role played by small ringswill increase with increasing framework density. Thus, wedefine a characteristic ring size:

K*: > Kf(K)w(K)/> f(K)w(/O, (ll)

where the sum are over K f(K) is the number of K ringsin a structure (Table 4) and W(K) is a weighting functionwhich is a decreasing function of K. If, as the theory sug-gests, we identify W(K) with P*(K), we find a significantpositive correlation between K* and framework density(Fig. l0). We attribute this result to the greater pruningefficiency of small rings compared with large ones. Thus,despite the simplifications built into our theory, it servesto elucidate the close relationship between ring.statisticsand framework density.

I 166 STIXRUDE AND BUKOWINSKI: TECTOSILICATES

t n 7N

an(9zE

( ) 5Fau.UJF

c)o a 5

-()

1 . 0

:z

o-. . 0 . E()zlrJ

o o .oqg

lrJ

(Jz o.tz.=u(L

|jJ 0.2

F

Jlrls 0 .0

TREE PRUNINGBY

RING FORMATION

3 4 5 6 7 I 9 1 0 1 1 1 2R I N G S I Z E : K

Fig. 9. The etrect of ring formation on framework density.The inset shows a cluster of size 3 from the Bethe lattice, withT atoms indicated by symbols and T-T links indicated by liehtlines. The bold lines indicate the effect ofring formation. In theupper right portion of the cluster, a three-membered ring isformed by linking two T atoms in the first linked neighbor shell,as shown by the solid bold line. The dashed bold line indicatesthe pruning required by the constraint of four-coordination.Similarly, in the bottom left portion of the cluster, a four-mem-bered ring is formed by equating two T atoms in the secondlinked neighbor shell, indicated by the bold line encircling thetwo T atoms. Again, the bold dashed line indicates the portionsof the cluster that must be pruned to maintain four-coordina-tion. The large circles plotted are the relative pruning efrcienciesfor different-sized rings from Equations 9 and 10. The line is anexponential fit to guide the eye. The plot shows that small ringshave a much greater effect on framework density than large ones.

Our observation that characteristic ring size increaseswith framework density is fully consistent with previousobservations, including the recent work ofliebau (1988)and Brunner and Meier (1989), noting a systematic trendtoward lower observed framework density with decreas-ing size of the smallest ring in the framework. Connec-tions between the existence of small rings in a networkand small values of Nn have also been noted (Brunner,1979; Akporiaye and Price, 1989), again fully consistentwith our results.

The perhaps counterintuitive relationship between ringsize and framework density is topological in nature, viathe pruning mechanism, and is not attributable to anygreater propensity for large rings to crumple or fold upcompared with small rings, as has been suggested (Mar-ians and Hobbs, 1989b). In fact, rings with larger topo-logical size (K) are also geometrically larger. By measur-ing the geometry of rings in tectosilicates, we show thatplausible measures of the effective density of a ring de-crease with increasing ring size K. We define an effective

TECTOSILICATTSI

a

a

a

a

a

at4a

Ia

a a

a

a

o l

a

F D ( n m - r )

Fig. 10. Plot showing the positive correlation (r: 0.83) be-tween characteristic ring size, K* (Eq. I l), and framework den-sity (FD) as predicted by the theory of the effects of ring for-mation on framework density described in the text andsummarized in Figure 9.

3 4 5 6 7 I 9 l 0 l l 1 2R I N G S I Z E : K

Fig. I l. Geometry of rings in tectosilicates as measured bythe radius ofgyration (Eq. 13) and illustrated schematically inthe upper left portion ofthe figure. Each symbol represents theaverage radius ofgyration for all rings ofsize Kin one structure.The single example of three-membered rings (in ZSM-18; R" :

1.7419) is not shown for convenience. Observed values are com-pared with two limiting cases: (1) planar rings, for which Ro isa maximum and (2) the value of Ro expected if the etrectivedensity of rings (Eq. 12) remained constant with increasing ringsize starting at K: 3. The two lower curves correspond to effectivering dimensionalities of 3 (lowermost curve) and 2 (see Eq. 15).All Ro values are scaled to a standard T-T distance of 3. I A tofacilitate comparison.

21t 51 2

o<

I( r 5

2o

!+ol!

oa3 5

E

TECTOSI LI CATESRING GTOMETRY

rist

density ofa ring:

p(K) u. K/R(, (r2)where Ro is the radius of gyration (root-mean-square dis-tance from the center of mass):

RL: 0/2K') 2 r",

where ru is the distance between T atoms i and j and d isthe effective dimensionality of a ring, which may bethought ofas lying between two and three, since rings arenearly planar structures. We have determined Ro for allthe rings of this study and compared these values to thevalue of Ro for planar rings:

Ro (Planar) : l/2Lcsc(tr/K), (14)

where Z is the T-T distance, and to variations in Ro forconstant effective ring density:

R"[p(K) : constant] d Kt/d. (15)

Figure I I shows that, although rings deviate systemati-cally from planarity with increasing ring size, the fact thatRo increases much more rapidly than Equation 15 for d: 2 ot d:3 means that any plausible measure of effectivering density will decrease with increasing ring size. Thus,rings crumple only slightly, and one should not think ofincreasing characteristic ring size with increasing densityas being accommodated by the crumpling of rings. Thisindicatos that the density of rings, which are two-coor-dinated, nearly two dimensional structures, cannot besimply related to the density of three-dimensional four-coordinated frameworks.

Suvrtvrnnv

We have examined in detail the density of tectosili-cates, minerals based on frameworks with identical co-ordination numbers and nearly identical bond lengths yetvastly different densities. We have shown how topologi-cal elements of frameworks, such as clusters, can be usedto form predictive models of framework density. We de-scribed the advantages of a new definition of rings andpresented the first measurements of ring statistics of tec-tosilicates, other than the silica polymorphs, based onthis definition. Finally, we have described a simple theorythat relates ring formation to framework density and usedthis to define a characteristic ring size that shows a strongpositive correlation with framework density. The coun-terintuitive increase in ring size with increasing densityis attributed to a topological pruning mechanism and isnot related to deformation of rings.

Although the picture relating ring statistics to frame-work density presented here is simple and can be refined,it provides a basis for rationalizing the densities of tec-tosilicates. It provides further support for the importanceof structures containing four- and three-membered ringsin the search for new low-density zeolites. Finally, it issufficiently general to provide a first-order description ofthe complex changes in silicateJiquid framework struc-

ture caused by increasing pressure (Stixrude and Bukow-inski, 1990).

AcxNowr,nocMENTs

we thank F. Liebau and J.V. Smith for their thorough reviews and C.

Lithgow, D. Snyder, and H.-R. Wenk for helpful comments that im-proved the manuscript. L.S. also thanks C.S. Marians for many enjoyable

discussions. This work was supported by NSF grant EAR-8816819 and

the Institute of Geophysics and Planetary Physics at the Iawrence Liv-

ermore National l-aboratory. Figure 9 is reprinted from Science,250,

541ff. with the kind permission of AAAS

RnrnnBNcns crrED

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Mnxr;scnrp'r RrcErvED DpcsMnER 18, 1989MeNuscnrp'r AccEprED Aucusr 10, 1990

AppnNorx 1. Mn,c.suREMENT oF cLUsrER ANDRING STATISTICS

The description ofthe algorithm for finding cluster parame-ters, Nn anld Mn, and ring statistics, (1Q, is presented in a waythat allows the results to be easily reproduced by hand, at leaslfor a small Q. For Q Larger than three or four, the logical oper-ations are too numerous. We used a computer program to gen-erate all the results of this paper. Throughout, the quartz struc-ture is used as an example, since it contains the smallest numberof T atoms per unit cell.

The first step is to identiff each atom in the structure with afour-part label: (iS,k,l), where I uniquely identifies all the atomsin the unit cell and j,k,l define a lattice translation vector in thebasis defined by the lattice vectors a,b,c (i.e., t: ia + fi + kc,where t is a translation vector). T-T links are then identified bysearching for all T atoms that lie within a certain distance ofeach T atom in the unit oell (3.5 A is an appropriate cutoffforall the structures in this study). The result is a linkage table thatcontains all linkage information for the entire structur€. That is,if we know that (tI,0,0,0) is linked to (iz,l,O,l), then (i 1,0 +.1',0+ H,0 + /,) is linked to (i2,1 + 1,0 + l{,1 + l') for all j"lc,l'.As an example, if we identifo the atoms in the unit cell of quartzin the following way:

T atom Atomic coordinates fu: 0.4697\

(1,0,0,0) (u,0,0)(2,0,0,0) (0,u,2/3)(3,0,0,0) (-u,-u,t/3),

then the resulting linkage table is

T atom Linked to

(1,0,0,0) (2,0,-1,-1), (2,r,0,-r),(3,0,-1,0), (3,0,0,0)(2,0,0,0) (1, - l ,0,1), (1,0,1,1), (3, - 1,0,0), (3,0,0,0)(3,0,0,0) (1,0,0,0), (1,0,1,0), (2,0,0,0), (2,1,0,0).

Because the linkage table establishes the linkage for the entirestructure, all subsequent operations involved in finding clusterand ring statistics involve only the linkage table, and no furtherreference to atomic coordinates is made.

The cluster up to O : 3 of atom (1,0,0,0) in quartz is shownin Figure lA. The first neighbor shell is obtained directly fromthe linkage table, whereas the second neighbor shell is construct-ed by finding the linked neighbors ofthe atoms in the first linkedneighbor shell and so on for larger Q. For instance, the linkagetable tells us that the first linked neighbors of atom (2,0,0-1,0-l ) a re (1 , -1 ,0 -1 ,1 - l ) , (1 ,0 ,1 - l , l - l ) , (3 , -1 ,0 -1 ,0 - l ) , and (3 ,0 ,0 -1,0-1). Once the cluster is constructed, the number of distinctneighbors in each neighbor shell are counted to give Nnall,d Mn

The fact that several atoms appear more than once in the Q: 3 shell indicates that six-membered rings exist (although theyare not necessarily fundamental). Similarly, if two atoms in the

0 : 3 shell were linked to each other (this does not occur in ourexample), this would indicate five-membered rings (again notnecessarily fundamental). To determine whether a ring is fun-damental, we compzue minimal path distances to path distances

tt69

Fig. lA. The cluster of atom (1,0,0,0) in quartz up to O: 3.The bold lines indicate a fundamental six-membered ring.

along the ring (Marians and Hobbs, 1989b). The path distancebetween two T atoms is simply the number of T-T links tra-versed along any framework path connecting the two. Althoughthere are an infinite number of such paths in an infinite frame-work, we are primarily interested in the minimal path distance,which can be found directly from the clusters. The minimal path

distance between one T atom and a T atom in its Oth neighborshell is equal to p. Thus, the minimal path distances betweenall pairs of T atoms in the entire structure are found from theclusters of the T atoms in the unit cell. Marians and Hobbs(1989b) have shown that a ring is fundamental if the shortestdistance between any pair ofT atorns along the ring is equal toits minimal path distance. Thus, the steps involved in determin-ing a fundamental ring are (l) construct a potential ring by in-spection of the cluster; (2) determine the shortest path distancesalong the ring, again by inspection ofthe ring; and (3) comparethese path distances with minimd path distances. For the ringoutlined in the cluster above:

(1,0,0,0) - (2,0,- l ,- l ) - (1,- l ,- 1,0)

I

(3,0,- 1,0) - (2,0,- 1,0) - (3,- l ,- 1,0),

we see that the shortest path distance along the ring between, forexample, (2,0,-1,-l) and (2,0,- 1,0) is three. From the clusterof atom (2,0,0,0), we see that the minimal path distance between(2,0,0,0) and (2,0,0,1) is three, and thus the minimal path dis-tance between (2,0,- l,- l) and (2,0,- 1,0) is also three, the sameas the shortest path distance along the ring. By checking all Tatom pairs in the ring in this way, we would find that this par-

ticular six-membered ring is fundamental.

STIXRUDE AND BUKOWINSKI: TECTOSILICATES