American Mineralogist, Volume 69, pages 104-111, 1984
Enumeration of 4-connected3-dimensionalnets and classificationof framework silicates:
linkagesfrom the two (52.8)2(5.82)1
2D nets
JosBpH V. Smith
Department of the Geophysical Sciences
The University of Chicago, Chicago, Illinois 60637
aNo J. Mrcnepr
Br,NNerr
Union Carbide Corporation
Tarrytown, New York 10591
Abstract
Two topologically distinct 3-connected2D nets can be developed from pentagonsand
octagons,and the centeredone can be distorted into an oblique net which is polytypic with
the zigzagnet. Many infinite seriesof polytypic 4-connected3D nets were enumeratedby
addition of updown linkagesto the 3-connectednodes,and by conversionof horizontal
branches of the two 2D nets into crankshaft and zigzagchains. Forty of the simpler nets
were selectedfor detailed description, and four are representedby bikitaite, dachiardite,
epistilbite and the CsAlSi5Ol2structure. Someof the nets can also be describedusing other
2D nets as a basis: particularly interesting theoretically is a polytypic net containing
pentagons and heptagons. All 3D nets obtained from the pentagon-octagon2D nets by
ionversion of horizontal branches into saw chains are too complex and distorted to be
described. Some of the polytypic series of nets are polytypic between series as well as
within a series.
Introduction
bend around nodes in responseto physical and chemical
forces.
Simple 2D nets were used as the basis of 4-connected
3D nets enumeratedin the first four papers of this series
(Smith, 1977,1978,1979;Smith and Bennett, l98l). The
present paper provides a selectedlist of 3D frameworks
derived from the centered and zigzag 2D nets with
(52.8)2(5.82)r
nodes. In the first subgroup, a vertical
branch is added to each 3-connectednode and the choice
of upward or downward corresponds to a two-color
enumeration.Distortion of a 2D net occurs when a model
is made from tetrahedral stars and plastic tubing; in
particular, a horizontal branch between adjacent up and
down branches turns into a crankshaft chain. Other 3D
nets are obtained by conversion of a horizontal branch
into a zigzagor a saw chain. Becauseall known natural
and synthetic materials with framework structures are
structurally elegant, complex inelegant nets with low
symmetry and many types of nodes are deliberately
omitted. The choice between selection and rejection is
arbitrary, and a few complex structures are listed for
safety. Cell dimensions were measured directly from
models made of plastic tetrahedral stars linked 3.1 cm
apart by Tygon tube segments; considerable variations
can be expectedin real crystal structures as the branches
0003-004x84/0I 02-0I 04$02.
00
Polytypic relation between the centered and zigzag
nets
The obvious geometrical arrangement of the (52.8),
(5.82)1nodes produces a centered net with space group
cmm (Fig. l) in which octagon D lies at the center of a
rectangularunit cell defined by ABA,B,. Each of the
pentagons obeys a mirror line of symmetry. The ideal
rectangular net can be homogeneouslydeformed into an
oblique net (space groupp2) in which A --+G, B --+ H, D
--+H', E -+ L, C -+ I, F --+ J. The unit cell becomesa
parallelogram defined by GHG'H', and the mirror lines
are lost. Although K and L, and I and J, are in a mirror
relation, the octagonsare not. The zigzagnet is topologically distinct from the centeredioblique nets; thus octagons N and O match G and H, but pentagonsT and U
replace M. The zigzag net has a rectangular unit cell
outlined by NPN'P' and the zigzag arrangementof octagons and pentagonsgives the spacegroup p gm. polytypism can occur between the oblique and zigzag nets
becausethey can be attached along lines of type dd to
give an infinite number of combinaiions. The 3D nets will
be arrangedin the figures to show this polytypic relation;
104
105
SMITH AND BENNETT: CLASSIFICATION OF FRAMEWORK SILICATES
CENTERED
Z'GZAG
Fig. l. Topologicaland geometricalrelationsbetweenthe
centered/obliqueand zigzag nets based on octagonsand
pentagons.
however, Table I gives cell dimensions only for the
centered arrangementand not for the oblique one.
Addition of up-down linkages
Figure 2 shows plans of the simpler 3D nets formed by
addition of up-down linkagesto the 2D nets. Fourteen 3D
Table l.
Arbl trary
nwnber
zt
220
12
nets (Table 1; 103, 22W232) were selected from the
infinite number of two-color combinations from the centered net. and nine were selected for the two-color
combinationsof the zigzagnet (Table l;223-241). Seven
pairs (2201233; 1031234; 2221235; 2231236; 2271237;
are related polytypically via the oblique
2291238;2311239)
(o) distortion of the centered (c) net, as shown by the
adjacent triplets in Figure 2.
It must be emphasizedthat these 23 nets were chosen
empirically out of an infinite number of possiblecombinations. To illustrate the complexity, Table 2 provides a
systematic enumeration of 3D nets obtained from the
centered2D net with the restriction that the color (f.e., up
or down) is constant in the horizontal direction. There are
eight rows to consider (l-E, extreme upper left, Fig. 2). If
all have the same color with a run sequence of 8, a
double-sheetstructure (Table 2, first line) is obtained in
which a 2D net with UUUUUUUU linkages is cross-
Propertiesof nets
o;
113szo;
r 1+2sazt
1+2szat)
r1t2sa3),
.'
1+sze2ay
r 1se5;
Hlghest
space group
a
2,
24
Iman
9
c
b
Angstrorn
tv
7.5
l9
24
Amnm
24
Arnnn
9
l9
24
Imam
9
l9
24
Cnrm
9
l+2szaz)
,
rl+zsaez)
) .,
1+szsza1
,1soae
24
Im2m
9
l9
'19
225
1 2 1+2szoe
1r1so3a2;,
24
Imam
9
19
226
12
Illlrm
9
l9
tz
P112/n
7.5
l0
24
Inmb
9
19
24
Al I 2/m
]0
15
9
48
Bbcm
]6
19
9
24
Al I 2/m
l0
15
48
Bbnrn
15
19
9
48
Anam
14
20
9
9
I03*
12
221
12
222
223
{ sr3a2
1.,
1+2s2se),
,,
+sea;
12 1a2s2oay
r1
224
12
12
227
1*2e2a1
r1+sa2s2),
.,
.,1+2soza
1 2 1+s2oe;
) .,(+s2sza)
228
t2
229
230
231
).,
).'{+s2oz2
1t2s2oa1,1t2 saza
.,
,,1szoa;
1 2 1oszs3;
3 ., szoa1{
.,
so5)
24 t +s2e
) t
12 1+2s2at1,1+2szaz).,{+2soaz
).,
232
24
233
24
1+2szat)
r1+2ssaz)
,
1+3s2e1r1+2se2to).'
7.5
t.J
9
?4
Pnnn
l0
235
),
1+2s2at
1,1+2sa3
24 1+3saz1
rl+2szdt'1
,
48
Anam
14
20
9
236
24
1+3s2ey
,1+2szaa'1
,
.'{+soa3
),
{+s203)
24
Pbm
l0
l4
9
237
24
Pbmm
l0
14
9
238
24
., 24
{+s2sz2)
1+2szae1
,1t2ss2a).,
''1s2e41
.,1so5;.,
24
1+s203;
Pnmm
I0
14
9
14
9
24
239
24'
240
24
szat1,1+2szta).,1+2ssza).,
1+2
+sa2
a2)''{s2o4
a31.r1
) .,
1+s2
241
24
).,{s2oa
) .,{ sss),
{+s203
48
Anam
1saea2yr1sos;''
12
Cmcrn
986
24
Pbm
l0
24
Pbm
t0
l4
9
i4
20
9
15
5
7.5
I 20'
I 12"
14
234
'tr0"
106
SMITH AND BENNETT: CLASSIFICATION OF FRAMEWORK SILICATES
linked to aZD net with DDDDDDDD linkages.When the
first four rows of branches point up and the next four
rows point down (UUUUDDDD), net 220 (Fig. 2, top
left) is obtained; its sequencesaround the 8- and 5-rings
are UUDDDDUU and UUUUU. The run sequenceof
2U,2D,2U,2D yields net 103alreadygiven by Smith
(1978),and run sequenceUDUDUDUD gives net 221.
Out of the combinationswith run lengthsof 4, 2 and I
only the nets 222-226are simple.
Nets 227-232 are shown becauseof special structural
features. Net 227 has chains of edge-sharing 4-rings
(unmarked nodes) crosslinked by parallel 4-rings (two
dots)in a patternwith monoclinicsymmetry.Its polytypic relative, net 228, has undulating chains of 4-rings
crosslinkedby a zigzagpattern of 4-rings.Nets 231 and
232 are composedentirely of chains of edge-sharing4-
rings at two levels. Finally nets 229 and 230 are interesting becauseU and D alternatearound the 8-rings;alternation is not possible around a 5-ring, and a 4-ring is
inserted between adjacent 5-rings.
Nine nets were chosenwith up-down linkagesfrom the
2D zigzag net. Nets 233-239 are related polytypically to
centerednets 220, 103,222-231via an oblique distortion
ofeach centerednet. Net 241is characterizedby alternation of nodes around each 8-ring, and net 240 by a 4-ring
bridge between each pair of adjacent 8-rings.
The following details appearedwhen models were built
from tetrahedral stars and plastic tubes. After the 3D net
has adjustedits shape(but not its connectivity), each pair
of adjacent dissimilar nodes in Figure 2 represents a
crankshaft chain (c) while adjacent similar nodes represent 4-rings arrangedlike stepping stones (s4s). Because
Table l. (cont.)
Arbi tra ry
number
Z,
242
12
243
o
Z"
lsaoa),{s303),{s2o3a).,
.,
1s4oa;.,
1s303).,
{ s2o3a)
Highest
space group
Bbmm
12
9
l3
5
1 s s a 1 , 1 s 4 o a 1 , 1 s s s ; . ,1 2
tz
l s a o a ) , 1 s 3 0 3 1 ' , 1 s 2 o 3 a1y2. , Pbnm
246
rz
1+2s321011
s4
ae
22
1 r, o ; . , 24
s3
247
251
aaz
a2J
l6
'14
Pnrm
tz
azv
'14
c
8
244
248
b
Angstrom
Al12/n
zq5
249
a
5
5
9
l3
A112/n
l0
i8
7.5
24 1+2s3t)
24
o1s4az).,{53a2r0,.,
1 2 1 + s 3 212s; .s,z1y s, ,a o z 1 . ,
24
Pncm
l0
'10
17
7.5
Pnmm
t8
'18
7.5
2 4 1 + s 3 212s; .5, zl;s, a e z ; . ,
1 2 1 + s 3 2 2 ; , 1 s 5 2 1 . , 1 s 4 o 24
z).,
2 4 1 + s 3 2 2 1 , 1 s s z 1 . , 1 s a o z96
1,
2 4 1 s 5 2 1 ' , 1 s 4 e 2 1 . , 1 s 3 2 324
1.,
1 2 1 s 5 z ; , 1 s a o r 1 . , 1 s 3 2 3 124
.,
4112/n
81l2/n
( a l s o p s e u d o - o r t h o r h o m bci e
c ] l a '14b ' 1 8c l 0 )
l0
108"
r07'
107"
t.)
9
l0
l8
Fddd
t5
Jb
l0
Pnam
l0
18
/.f
A1l2/n
l0
18
7.5
125"
Il0.
( a l s o p s e u d o - o r t h o r h o m bci e
c ] l a '10b 3 6 c 7 . 5 )
t2
1 s 5 z 1 ' , 1 s 4 o z ; ' , 1 s 3 2 324
1,
255
24
( s 4 7 )(ts 4 6 7 )( l5 3 7 3 ) t
zqA
12
1 + s 3 z 2 1 o 1 s a z z ; , , 1 s 3 2 3 11 +, ,
'10
4112/a
'10
'
1
5
(also pseudo-orthorhombic
cell a
b
c l8)
, 22)r(s323)'t
1 2 ( + s 3 2 2()s, 4
OR
24
Fddd
15
35
Pnnm
]0
l8
A1l2/n
]0
I8
Cell Dlmenslons
abc
7.60 8,6l 4,96
yl l4o
I 3.83 16.78 5.02
7. 5 1 0 . 3
18.7
8108.
l0
7.5
1060
* 0 r i g i n a 1 1 yd e s c r l b e dw i t h r c n o c l l n l c h a l f - c e l l .
Type
Name
Reference
SpaceGroup
98
bl kl tai te
Pl l 2,
'
242
248
CsAlSlS0tz
dachlardlte
Kocmanet al .
( 1974)
Arakl ( 1980)
250
e p ls t l l b i t e
8bm2
Gottardl and C'12/n1
l'leler (1953)
) l2lml
P e r r o t t a ( 1 9 6 7C
8.9
17.7
10.2
Bl24'
SMITH AND BENNETT: CLASSIFICATION OF FRAMEWORK SILICATES
(
107
1
l2
3a
J-a
:?
)>
220b)
233
229k\
229b\
238
lO3(c)
lO3(o)
234
231(c)
231(o)
?39
222lc)
222b)
220k)
224
225
228
2 2 3 b ) 236
dt.--z'-zt-z
227k)
2?7b)
237
232
240
241
Fig. 2. Selected3D nets obtained by addition ofup-down linkages to the centered(c)/oblique(o) andzigzag nets. op octagonal
prism, pp pentagonalprism.
6-rings in molecular sieves are permeable only by small
molecules,emphasiswill be placed on larger rings, especially in multiply-connected channel and cage systems.
Net 220consistsof zigzagchains offace-sharing pentagonal prisms (pp) along the c-axis crosslinked by crankshaft
chainsalong a. One-dimensionalchannelsare boundedby
elongated l2-rings. Net 221 contains crankshaft chains
parallel to the a-axis (Fig. 2), and the c-axis projection
(Fig. 3) shows saw(s) and c chains projecting onto a 63
net. The l-dimensional channel svstem is limited bv 8-
rings. Net 222 contains chains of face-sharingoctagonal
prisms along the c-axis crosslinked by c chains along the
a-axis. The 2D channel system is limited by elongated12rings perpendicular to c and S-rings perpendicular to a.
Net 233 projects down the c-axis onto a 4.82net (Fig. 3),
and its 3D channelsystemis limited by S-rings.Nets 224227 project down the c-axis onto complex 2D nets with 4-,
6-, and 8-rings, not shown in Figure 3, but the zigzag
feature of the polytypic variant 228 of net 227 eliminates
this property. Net 229 contains zigzagchains along the c-
l0t
SMITH AND BENNETT: CLASSIFICATI)N
Table 2. Enumeration of up-down 3D nets from (52.8)2.
(5.82)rnet. Subgroup with constant color in 8.8 direction
8
41
2222
'|lllllll
422
242
4lI l t
l4lll
II4II
22211
22121
21221
12221
22112
21212
22I'I1I
2I2III
UUUUUUUU
3hmt
UUUUDDDD
220
UUDDWDD1 0 3
u D t 4 | D U D221
uwuou,u unllstld
UUODDDUU
222
uuuuDutx, unlI Btad
uDoDDtu, unllstld
UDUUUUDU
unllstrd
UUDIIJUDUunlI st€d
UUD&'DU' unlI sted
UUOUUDDU
unllsted
UDDUUDU'223
UUDIXJDI'I'unlisted
UUd'UqJU un'llsted
UUDDUIT'D
unllst€d
UUUJUUJD
unllsted
2l l2l l
2I I I2I
2nll2
I 22II'I
I2I2II
I2II21
II22II
2IIIIII
I2TIIII
ll2lllr
l llzill
42l t
4121
4I I 2
24II
I42I
I4I2
uuDtDDtD
224
UUDUDTJUD
unll3trd
uuDuouDo unl I strd
UDDUUU'D unllstrd
UDO{IDDUDunll3tad
UDDIJDUUD225
UN'UDDUD unllstld
UU|)I'IUDU unI I stcd
UDDUfiIDU unllsted
unuouuj
unlI sted
utxJooulxj 226
ut uuDDuD unllsted
UUUUDUUDunllst€d
UUUUDTJDDunllsted
UUDDDIXJDunll sted
UODDDI'UD unllsted
UDDDDTJDDunl isted
axis, and projects onto a 63 net (not shown in Fig. 3); its
unconnectedchannels are walled by 6-rings and limited
by 8-rings. The channels in net 230 are similar. Net 231
projects down its b-axis onto a 4.82 net, and its 3D
channel system is limited by non-planar 8-rings. Net 232
has an even more complex 3D channel system also
limited by non-planar S-rings.
Net 233 contains horizontal zigzag chains of facesharing pentagonal prisms at two levels crosslinked by
vertical crankshaft chains to give elliptical channels
spannedby l2-rings and interconnected by 8-rings. The
zigzag chains of 4-rings in net 234 are crosslinked by
vertical crankshaftchains to give a 2D comrgated channel
system limited by 8-rings. Net 239 has a diferent type of
4-ring chain. Net 235 contains elliptical channelslimited
by l2-rings and interconnectedvia the face-sharingoctagonal prisms.
Net 236 contains 4-rings (pairs of dots) as bridges
Fg. 3. Alternative projections of two nets from Fig. 2,
Compare with Figs. 9 and 12 of Smith (1979). c crankshaft, h
horizontal branch, s saw, s4s 4-rings in stepping-stonepattern.
oF FRAMEW1RK SILICATES
'.,i^].,.
C't
.2....1..!....a.
zz
.:....1...!...1..
L2J
Z
2
98(c)
L
242
P . . . . .z. .
......j
z...,r,.i ) 2,Jl
zJ)rq
98(o)
244
Fig. 4. Selected3D netsobtainedby additionof zigzag(z)
linkagesto the centered(c)ioblique(o) and zigzag2D nets.
Continuousanddottedlinesshowchainsof horizontallinkages
at two levelsrelatedby the verticalzigzagchains.Net 98(c)can
be distortedto 98(o)whichis polytypicalongpq with zigzagnet
2214.
Nets242and243are polytypicacrossee andff.
between horizontal zigzag chains of 4-rings (unmarked
nodes),and its 3D channel system is limited by 8-rings.
Net 237 has a 4-ring bridge between each pair of 5-rings,
while net 238has 4-rings alternatingwith crankshafts.Net
238also has alternation ofU and D around each S-ring,as
also does net 241. Nets 238 and 240 have 4-ring bridges
between adjacent 8-rings. All these nets have complex
channel systems limited by 8-rings. Net 238 projects
down its a-axis onto a 63 net.
Addition of zigzag chains
Figure 4 shows how vertical zigzag chains (z) can be
combined with the centered/obliqueand zigzag2D nets.
Becausea horizontal branch of a 3-connected2D net is
converted into two inclined branches (zig and zag) of a
zigzag chain, the two adjacent branches of the 2D net
cannot be converted into zigzagbranches; otherwise the
3D net would have more than 4 connectionsat each node.
Each 4-connectednode ofthe nets in Figure 4 lies at the
intersection of one z chain and two horizontal branches.
Continuous and dotted lines distinguish between the two
Ievels of the horizontal branches. Net 98 (bikitaite) was
already listed in Smith (1979) because it can also be
described with a 63 2D net modified by crankshaft and
saw chains. The centered/oblique2D net yields two more
3D nets based on the choices for placement of three z
chains in edge-sharedpentagons.Net 242 is found in the
structure of CsAlSi5Olz (Araki, l9E0), and can be obtained from net 232 by replacement of each c chain
(branch between dissimilar nodes; Fig. 2, middle bottom)
by a z chain. Net 243 is similarly related to net 231; note
that the chains are depicted in diferent diagonal directions. Nets 242 and 243 are polytypic across the boundaries ee and ff.
All these nets have isolated channels spannedby nonplanar S-rings.Net 243 nearly projects down its Il t0] axis
onto a 63 net.
SMITH AND BENNETT: CLASSIFICATION OF FRAMEWORK SILICATES
D
246/247
Fig. 5. Two projections of nets 246 and 247. See text for
explanation.
Addition of zigzag chains and other structural
features
Centered net
Nets 246-257 are selected from the infinity of nets
produced by addition of zigzagchains and other structural features to the centered 2D net of pentagons and
octagons.
Nets 246 and 247 are the simplest polytypes of an
infinite series obtained by placing a zigzag chain at each
bridge between adjacent octagons and by converting
edges of the pentagons into a horizontal double-crankshaftchain(Fig. 5, left: heavylinesjoiningcircled nodes).
The c-axisprojection (Fig. 5, right) shows how the 3D net
can be r€presentedby conversion of some branchesfrom
a 2D net (4.5.8)2(5.5.8)r(5.8.8)r
into crankshaft (c) and .
saw (arrow) chains. Some horizontal branches are retained either as singleones (h) or as componentsof zigzag
chains (heavy lines and dots). There are two choices for
attachment of the double-crankshaftchains to the zigzag
chains, as is most easily seenby looking at the elongated
250/251
octagons at the right. Continuation of the oblique sequenceABC gives the monoclinic net246, while continuation of the zigzag sequenceBCD gives orthorhombic net
247.The 3D channelsystemsare limited by elliptical 8rings in all directions.
The combination of zigzag chains between the pentagons and stepping-stone4-rings (s4s) between the octagonsproducesa multiply-infinite seriesof 3D nets, two of
which are represented by dachiardite (Gottardi and
Meier, 1963)and epistilbite (Perotta, 1967).Although the
schematic projections of Figure 6 are convenient for
enumeration,the stereoplotsof dachiarditeand epistilbite
(Meier and Olson, 1978)are helpful. The four simplest
nets (Merlino, 1978)are projected in Figure 6. There are
two choices ofheight for each s4s in the centered 2D net
at the left. The first infinite series, which contains dachiardite(net 248),is basedon the sameheightfor d, d',
d", etc., and similarlyfor e, e', e", for f, f', f', andfor g, g',
g"; howeverd, e, fand g may have differentheights.The
two simplestchoicesof relative height are shown in the caxis projection (next diagram). This reveals the same 2D
net as in Figure 5, but with a diferent choice of crankshaftand saw chains.The obliquechoiceof4-rings d, e, f
yields the monoclinic cell of net 248, and positions p and q
would extend that net. The zigzagchoicee, f, g yields the
orthorhombic cell of net 249 which would extend to
positions r and s. Algebraically, this polytypic seriescan
be expressedby giving fractional coordinatesalong the aaxis (bottom) and the c-axis (left-hand side).
The second infinite series, which contains epistilbite
(net 250),hasd and d", etc. at the sameheightalongthe aaxis (e.g., ll2) and alternatepositions(e.9., d') displaced
by one-half. In the c-axis projection, the 4-rings of
horizontal linkages are now represented by continuous
and dashed lines (Fig. 5, two right-hand diagrams)'
Whereasall the s4slabeledd, d', d'project onto d for the
dachiardite net (248), they project alternately onto two
positions for the epistilbite net (250). Again there is an
infinite set of choicesfor s4sof types d, e, f, g, etc., and
@
ili@
(O) (rzz) (O)
sq
irtt:E
244/249
3;ir'=-r l;
v-/4274v
I(D
25o
Fig. 6. Projections for nets 248-251.See text for explanation.
SMITH AND BENNETT: CLASSIFICATION OF FRAMEWORK SILICATES
| - - - - . . . . . . . . , - _ __ ( / 2 1
-1..'... (o)
o..i..i--f/2 .../--\.1
..:v.
3t2J:
.rr$ral
ltt4l
o -....--,--,......(o)
ccc
| ----".....''-
0/2)
-./.---\..(t/4'l
3/2 \...,\Bz+t
ttz. -'\L/,
\.
'252/?s3
c- c
1..:
c
J-1.:,.,,.!:';;ii:_ij...::,.-'7.:..i'ji,
254
255
Fig. 7. Projections for nets 252-255.See text for explanation.
the two simplestonesare nets 250and 251.The unbrack- rings are representedby linked crankshaft chains (horieted heightsin the two right-hand diagramsrefer to the a- zontal rows of c, Fig. 7).
axis of the extreme left-hand drawing. Net 251 has an
Finally, for the centered 2D net of pentagons and
orthorhombic cell outlined by ABCD, and doubling of the octagons, Fig. 8 shows projections for the polytypic
d-repeat results in the bracketed fractions. The zigzag series represented by nets 256 and 257. Each branch
sequence0, ll4, ll2, 314produces the diamond glide betweentwo pentagonsis replacedby a crankshaft in the
planes in the space group Fddd (Table 1). Dachiardite left-hand drawing, and each branch between two octa(250)has a monocliniccell with cornersat height0 (t, t,,
gons by a zigzag chain. Adjacent crankshafts lie at the
v, v') and I (u, u', w, w').
same height along the d-axis, and becomejoined by s4s.
AII nets of these two polytypic series contain crinkled
Staggerifig the crankshafts by a a./2 produces highlysheets of 6-rings composed of linked crankshaft chains strainednets which are not given here. The c-axis projec(horizontal rows of c, Fig. 6), as described by Meier
tion revealsthe same2D nets found for nets252and253,
(1978)and Merlino (1975). Stacking faulrs in dachiardite but with a different set of 3D linkages.
and epistilbiteare describedby Gellenset al. (1982)and
Slaughter and Kane (1969). Various related structures Zigzag net
describedby Kerr (1963),Merlino (1975)and Sherman
The zigzagfeatureproducesan asymmetry in the 8-ring
and Bennett (1973) will be described in a later paper (Fig. l) which leads to nets of greater complexity than
which includes the mordenite net.
those developedfrom the centered2D net. All the nets in
Nets 252-255 are the simplest representatives of the
Figure 9 belong to polytypic series analogousto those of
infinite polytypic series formed by insertion of zigzag nets246-257.The zigzagfeature precludessimple projecchainsbetween adjacentpentagonsand crankshaft chains tion down the c-axis. Becausethe nets are complex and
between adjacent octagons (Fig. 7, left). The c-axis inelegant,they are not given numbersand are not listed in
projection shows how the zigzagchains along the a-axis Table l.
are replacedby crankshaft chains along the c-axis. These
c chains are connected by saw chains (arrows) to the
horizontal crankshaft chains at two levels (heavy line vs.
dots). Just as for the dachiardite and epistilbite series,
there are two choices. Superposition of the horizontal
crankshaftsat c equals ll2 and 312or at 0 and 1 gives the
polytypic series representedby nets 252 and 253 which
are analogousto the 2481249nets. Displacement of the
horizontal crankshafts by alT (Fig. 7, right-hand diagrams)gives the polytypic seriesrepresentedby nets 254
and 255which are analogoustotheZ50l25l nets. Bracketed fractions are obtained as in Figure 6. Net 254 is
monoclinicwith cell cornersat heightO (u, u', w, w,) and
I (r, r', v, v'). The c-axisprojectionsof nets2521253
are
based on 62T.61f)3 2D nets, of which the rectangular
one for 3D net 252 is found in the B net of yCrBr
Fig. 8. Projections for nets 256 and 257. See text for
(O'Keefe and Hyde, 1980,Fig. 30a).Crinkled sheetsof 6- explanation. Compare with Fig. 7.
SMITH AND BENNETT: CLASSIFICATION OF FRAMEWORK SILICATES
ill
Acknowledgments
We thank Nancy Weber for typing, and Union Carbide Corporation and the National science Foundation (gxant cHE 8G
23444)fot support.
b
References
Araki, T. (19E0)Crystal structure of a cesium aluminosilicate,
CslAlSisOrzl. Zeitschrift fiir Kristallographie, I 52, 207Jl3.
Gellens, L. R., Price, G. D. and Smith, J. V. (1982) The
structural relation between svetlozarite and dachiardite. Mineralogical Magazine, 45, 157-161.
Gottardi, G. and Meier, W. M. (1963)The crystal structure of
dachiardite. Zeitschrift fiir Kristallographie, I I 9, 53-64.
Kerr, I. S. (1963) Possible structures related to mordenite'
Fig. 9. Projectionsof nets obtained by addition of zigzag and
Nature, 197, 1194-1195.
other linkagesto the zigzag2D net. Heavy lines in a and b show
Kocman, V., Gait, R. I. and Rucklidge J. (1974) The crystal
projectionsof vertical 4-rings. Rows of dots in a show vertical 4structure of bikitaite, L(AlSirO6)' H2O. American Mineralrings displacedwith respect to ones shown in heavy line.
ogist,59,71-78.
Meier, W. M. (197S)Constituent sheets in the zeolite frameworks of the mordenitegroup. In L. B. Sandand F. Mumpton,
Eds., Natural Zeolites, p. 99-103. Pergamon,New York'
Addition of saw chains
Meier, W. M. and Olson, D. H. (1978)Atlas of zeolitestructure
types. PolycrystalBook Service,Pittsburgh,PA.
All 3D nets obtained by replacing branchesofboth the
(1975) Le strutture dei tettosilicati. Rendiconti
centered and zigzag 2D nets of linked pentagons and Merlino, S.
di Mineralogia e Petrologia, 31, 513-540.
Italiana
Societa
octagonsare complex with some severely distorted polyO'Keefe, M. and Hyde, B. G. (19E0)Plane nets in crystal
gons. All models were highly strained, and would not
chemistry. PhilosophicalTransactionsofthe Royal Society of
adopt a symmetrical shape even after several unit cells
London. 295. 553-618.
had been built. A systematic enumeration was not com- Perrotta, A. J. (1967)The crystal structure of epistilbite. Minerpleted becauseit became obvious that the models were
alogical Magazine, 36, 480-490.
inelegant and complex. This was disappointing because Sherman,J. D. and Bennett,J. M. (1973)Frameworkstructures
related to the zeolite mordenite. Advances in Chemistry
the models contain channelsbounded by l2-rings.
Series. l2l, 52-65. American Chemical Society, Washington,
D. C.
Conclusion
Slaughter, M. and Kane, W. T. (l!)69) The crystal structure of
disorderedepistilbite. Zeitschrift fiir Kristallographie, 130,68The present enumerationof4-connected 3D nets based
87.
on 2D nets containing pentagons and octagons.has reSmith, J. V. (1977) Enumeration of 4-connected3-dimensional
vealed many infinite series of nets which are not only
nets and classificationofframework silicates. I' Perpendicular
polytypic within a seriesbut also between some series.
linkagefrom simplehexagonalnet. American Mineralogist,62,
Forty of the simpler nets were selected for detailed
703JW.
description, and four are represented by bikitaite, da- Smith, J. V. (1973)Enumeration of 4-connected3-dimensional
chiardite, epistilbite and the CsAlSisOrz structure. Benets and classificationofframework silicates, II. Perpendicular and near-perpendicularlinkagesfrom 4.82,3'122and4.6-12
causethe nets can be distorted considerably, it would be
nets. American Mineralogist, 63, 960-969.
wise to allow a toleranceofat least 5%oin cell edgesand 5'
in cell angle when attempting to obtain a match with cell Smith, J. V. (1979) Enumeration of 4-connected3-dimensional
nets and classification of framework silicates, III. Combinadimensions of an unknown structure. Although the four
tion of helix, and zigzag, crankshaft and saw chains with
not
they
are
elegant,
are
structurally
observed structures
simple2D nets. American Mineralogist,g,551-562'
really simple; thus there are three types of circuit symbols Smith, J. V. and Bennett, J. M. (1981) Enumeration of 4for epistilbite and dachiardite. Furthermore, some of the
connected3-dimensionalnets and classificationof framework
simpler theoretical nets do not have observed countersilicates: the infinite set of ABC-6 nets; the Archimedeanand
parts. It is obvious that crystal-chemical factors are
o-related nets. American Mineralogist 66, 777-:188.
important in selection of the theoretical nets for actual
crystal structures, and that there is no simple route to
prediction of tetrahedral f4ameworks.
Manuscript received, December 29, 1982;
acceptedfor publication, July 29, 1983.