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On the Description of Complex Inorganic Crystal Structures.

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Volume 22
-
Number 2
February 1983
Pages 69-170
International Edition in English
On the Description of Complex Inorganic Crystal Structures
By Sten Andersson*
Dedicated to Professor Harald Schafer on the occasion of his 70th birthday
To describe a structure is to describe chemistry. A description of structure should be simple
yet also open up ways for an understanding of complex structures and their properties.
Such interrelationships are gained on inspection and discussion of structures in models of
interpenetrating partial structures and on application of symmetry operations to whole
structures. Transition regions between interpenetrating partial structures can be approximated to periodic minimal surfaces. Chemical reactions, e. g. in ion-exchange, in heterogeneous catalysis, or in molecular separations in complicated zeolites will take place on these
minimal surfaces.
1. Introduction
To describe a structure is to describe chemistry-and
what is chemistry? Whether a substance is a solid, a liquid
or a gas, we split it or decompose it into its component
parts, i. e. building blocks, phases, polyions, molecules,
atoms, electrons etc. to find out what it is made of. The
analysis may be carried out by means of simple chemical
methods, diffraction methods or quantum physical methods, and when it is complete we try to put the components
of the substance together again to see if we can synthesize
what we have just analyzed. When we think we have found
out something about a structure and about the effective
forces that hold the atoms of a substance together, we try
to predict what happens with different combinations of
these substances under various conditions of, e. g., temperature and pressure.
Solid substances are almost always crystalline and have
structures which are periodic, but the larger the repeat unit
[*I Prof. Dr. S. Andersson
Oorganisk Kemi 2, Kemicentrum
Box 740, S-22007 Lund (Sweden)
Angew. Chem. lnr. Ed. Engl. 22 (1983) 69-81
the more difficult it becomes to describe the structure. Really large complex structures are beyond comprehension
and we cannot explain why they exist. With an appropriate
description, however, we may sometimes be led to think
that we understand a structure, understand how it is
formed, and how to explain its properties.
Recently, on the occasion of Wilhelm Klemm’s 85th
birthday, a series of review articles appeared in this journal
in which von Schnerzng, Simon, Hoppe and Bronger outline
their methods for the description of crystal structures“-41.
It is immediately apparent that these descriptions stem
from the same school-each of these authors were at
Munster at the same time for a number of years-and that
they are also strongly influenced by the chemistry of cluster compounds, in its infancy at that time and pioneered by
Harald Schiifer. Another concept for the description of
structures, the concept of “nested polyhedra” was developed by Parthe to simplify the description of complex
cubic structures[51. There are, of course, many ways in
which to describe a structure, and I doubt whether we can
say at this stage that any one is better than another.
Nevertheless, what we expect from a structure description is undisputed:
0 Verlag Chemie GmbH, 6940 Weinheim. 1983
0570-0833/83/0202-0069 $ 02.50/0
69
1. It should be simple, and should show as many relationships as possible to other structures.
2. It should facilitate the explanation of diffusion and
reaction mechanisms.
3. It should be useful in explaining physical properties
like conductivity, hardness, etc.
4. It should help us to interpret high-resolution electron
microscopic lattice images -this often means trying to
understand the defect structure of a crystal.
The first statement means, among other things, that a
structure description should provide us with many ideas
and associations. The description should thus allow us to
apply Birkhoff s equation for the aesthetic measure[61:
where 0 may contain elements of the associations (of order) and ideas, and C is the complexity, which in a polygon, for example, is measured by the number of sides. In
simple terms this means a structure description should be
“beautiful”. (When solving the large block structures of
various niobium oxides, Wadsley and I used to have two
criteria for the proposed models: The chemical formula,
and the beauty17].) We are reminded of an old definition for
beauty-as “that which gives us the greatest number of
ideas in the shortest space ot time” (Hernsterhuis).
2. Simple Structures
The principal idea is that a complex structure is built up
of much simpler structures[281.What are much simpler
structures and how can they be combined to give a complex structure? This is what we shall deal with in this article.
Before we outline our methods for the description of
complex structures, we shall first describe simple structures. Such structures are, e. g., hexagonal close packing
(hcp) of the atoms, cubic close packing (ccp), body centered cubic (bcc), hexagonal primitive (hp), cubic primitive
(cp), or just simply primitive packing of lower symmetry
(Fig. 1).
Two simple and identical structures can interpenetrate
to give a new structure, as shown in Figure 1. Examples are
NaCl, diamond, and Mg3Cr,Al18.Three simple and identical structures can also interpenetrate to give a new structure.
This can be vividly demonstrated using the concept of
rod packing”]: The alloy Cr,Si has the so-called A15 structure, in which infinite Cr-chains with short metal-metal
distances run in three perpendicular directions. One explains via this peculiarity the existence of many superconductors with this type of structure, e. g. Nb3Sn and Nb3Ge.
These chains with short metal-metal distances are represented by rods (arranged with quadratic mesh) which interpenetrate to form the simple, cubic A15 structure (Fig. 1).
Four simple and identical structures can also interpenetrate-to form the garnet structure [Ca3A12(Si04)3]shown
in Figure 2. In this case, we have hexagonal meshes of rods
70
0
0
0
.
0
.
.
......
......
0
.
0
.....
.....
.....
.....
.....
0
0
.
.
0
0
m
0
.....
.....
.....
.....
.....
. . . . _
+
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
~
0
0
0
0
0
0
0
0
0
-
O
O
O
D
0
0
0
0
0
0
0
0
0
~.
Fig. 1. Top: A simple structure.-Center: Two simple structures interpenetrate.-Bottom: Three simple structures interpenetrate to form the Cr,Si
structure (A15-type).
in each simple structure, where each rod consists of facesharing octahedra (filled) and trigonal prisms (empty) and
all the oxygens are used to form the rods. Tetrahedra and
twisted cubes (for the larger cations) are created between
the rods by the oxygen atoms of adjacent rods. This structure is also taken up by alloys like Th,Bi, in which all the
Th atoms are in the oxygen positions of the garnet structure, and the Bi atoms are in the interstices formed by the
Th rods (the large cation position in the garnet structure).
Two simple and different structures can interpenetrate
to form a structure. One example is Re03, shown in Figure
3, another is NiAs, in which the hcp structure of As atoms
interpenetrate a hp structure of Ni atoms. In Cr3Si the silicon structure (bcc) interpenetrates the chromium rod struc-
Fig. 2. Four simple structures interpenetrate to form the garnet structure.
Angew. Chem. Int. Ed. Engl. 22 (1983) 69-81
ture, which itself consists of three identical interpenetrating structures. In W3Fe3C the octahedral structure of
tungsten atoms interpenetrates the tetrahedral structure of
the iron atoms (Fig. 5 ) and the carbon atoms in the tungsten octahedra form a third structure, which is the same as
that of the zinc atoms in MgZnz, and the oxygen atoms in
cristobalite (idealized).
-0-0
0-0-0
I I
o-oI I
I
I
o-o-o
.. .. .. ..
.... .. ..
.
D
D
D
O
O
0
0
+
a
*
.
.. .. .. .. .. ..
.. .. .. .. .. ..
......
0
0
D
0
I 1 I
I 1 I
0-0-0
0-0-
I I I
0-0-0
o-o-o
0-0-
o-o-o
--0
lI
0--
I
p
o-o-o
I
o-o-o
Fig. 4. Shear by parallel shifts.
O
0
0
Fig. 3. Two different structures interpenetrate.
Of course, many more examples could be given. Between these simple structures there exist many relationships which we will leave undefined at this stage. Suffice it
to say that from one simple structure many other simple
structures are easily obtained by small shifts of atoms in
various directions. Good examples of these topological operations are found, for example, in Burnighausen’s
work[’’.
3. Complex Structures
“Most isometries are familiar in everyday life. When
you walk straight forward you are undergoing a translation. When you turn a corner it is a rotation: when you as:
cend a spiral staircase, a twist (screw). The transformation
that interchanges yourself and your image in an ordinary
mirror is a reflection, and it is easy to see how you could
combine this with a rotary reflection or a glide reflection,
respectively” (from Coexter’s Introduction to Geometry).
By using a point instead of “you”, we describe the symmetry operations that give rise to the 230 space groups. By using building blocks of atoms, which may be parts of simple
structures, we analogously describe the symmetry operations that give the complex structures. The symmetry operations that describe the space groups leave all distances
and angles unchanged, and similarily the building blocks
are left unchanged during a transformation.
We thus assume that complex structures can be described as parts of simple, or simpler structures. We shall
demonstrate this with a number of examples.
4. Parallel Translation
Figure 4 shows how a simple structure (Fig. 1) may be
repeated by a single parallel translation to build up a more
complex structure. This operation, called crystallographic
shear, or slip, produces two interesting new features in the
structure: In the shear planes there are triangles of atoms,
in which each of the three participating atoms is now pentacoordinated, and the sheared complex structure is more
dense than the simple structure. Two examples of formal
shear structures are those of compounds Nb307F and
Angew. Chem. Int. Ed. Engl. 22 (1983) 69-81
o-o-o
o-o-o
0
0
0
l
? I1 l1 l1 !-U Ii Il
b-l-% b 1 I I 1 7-1-p
0-0-0
O
l
R-Nb205 (Fig. 5). In both cases the simple parent structure type is ReO,.
The structure of TiNb,O, (Fig. 5 ) contains columns of
the ReO, structure (3 x 3 octahedra in size) stretching infinitely in the direction perpendicular to the plane of the
picture. The structure is formally derived from double
translations of the parent Re03 structure, or equally well
from a single translation of the Nb3O7Fstructure. Figure 5
shows the structure of NaNbI3O3,, which also is derived
from a double translation of the parent ReO, structure or a
single translation of the Nb307Fstructure.
Two different column sizes of the same Re03 parent
structure type, occurring in the same crystal in an ordered
way, are found in the structure of W4Nbz6077171.
This important compound was first prepared by Gruehn in Schufer’s laboratory in Miinster. W4Nb2,077~could
easily be described as composed of alternate slabs of the oxides
WNbl2O3,and W3Nbl4OU,and is called an intergrowth or
hybrid compound, the first example of its kind (Fig. 6).
A variation in composition of a compound due to random variations of column or slab size were predicted earlier by David Wadsley in Melbourne. He, together with
Sanders and Allpress, were also the first to use the now so
well known lattice imaging technique in a classical study
on various niobium oxides[Io1.I shall always remember my
feelings of joy upon receiving his enthusiastic letters, always accompanied by many lattice images of these oxides
with random shear planes. After his untimely death we
named these faults, now also known to be so common in
minerals, metals and alloys, Wadsley defects, in honour of
the man who proposed and predicted their occurrence so
many years before they were actually discovered. Schafer
realized at an early date the importance of Wadsley’s pioneering work on oxides, which he wholeheartedly supported.
Nowadays, microscopes are far more better and it is
possible to achieve atomic resolution almost routinely. A
study by Bovin et al. on crystals of NaNb,3033clearly demonstrates this[”]. Columns, sized 3 x 5 octahedra and
similar in structure to those of TiNb207(cf. Fig. 5, bottom
left), are intergrown with the normal column of the
NaNb13033structure (Fig. 7). Thus, it is vividly clear how
the crystal is able to accomodate variations in the sodium
content.
It is obvious that for these transition metal oxides structure description is now a well established procedure, with
full theoretical and experimental support from computer
calculations and high resolution electron microscopy, The
structures of many, many more oxides can be developed
71
CuA12. If the translation operation is continued as shown
in Figure 8, larger columns of Cr3Si can build into the
structure of Mo3CoSi, which can also be derived from the
CuA12 structure.
Complete experimental support for this structure description is obtained from Stenberg’s lattice images of
Mo3CoSi (Fig. 9)[l3].Furthermore, we note that the short
metal-metal distances, typical for Cr3Si in three dimensions, are restricted to two dimensions for the single translation structures, and to one dimension for double shear
structures.
Fig. 5 . Top: Structure of W,Fe,C. The structure of the tungsten atoms (red spheres), with carbon in the center of the (yeilow) octahedra interpenetrate the tetrahedral iron structure (yellow spheres).-Center: The structures of Nb,O,F (left) and R-Nb205(right) shown in two different projections.-Bottom: The structures of
TiNb20, (left) and NaNb,,03, (right).
by simple parallel translation. For further detailed information o n this subject the reader is referred to the literature[’’].
In Section 2 we described the Cr3Si structure as a rod
structure, but here we can describe it in a slightly different
way, as shown in Figure 8. The shaded regions consist of
five irregular tetrahedra-an arrangement which Schubert
calls a “Tetraederstern”. In Figure 8 we have applied single and double translation (shear) to the structure and obtained two very important structure types, Zr,Al, and
72
5. Parallel Reflection
Figure 10 illustrates twinning of a hcp structure. This
simple parallel reflection is called “chemical twinning”, or
also “unit cell t ~ i n n i n g ’ ” ’ ~New
~ ] . polyhedra are created in
the twin plane by this operation, in this case trigonal
prisms. The first example is cementite, Fe3C (see Fig. 10
and 11). The hcp slabs are four iron atoms wide, and the
trigonal prisms in the twin planes contain carbon atoms. If
the slabs are alternately 4 and 3 iron atoms wide, we obtain
Angew. Chem.
Int.
Ed. Engl. 22 (1983) 69-81
Fig. 6. Structure of the intergrowth or hybrid compound of W4Nb260n
formed by the components W3NbI4Ou and WNb12033.
the structure of Fe5C2, which is of the Pd5B2type. Variation of stoichiometry can easily be achieved by varying the
block size (Fig. 1l), and this has recently been verified experimentally by electron
There are many
Fig. 8. Top: Structure of Cr3Si (A15-type) (left) and single steps of the shear
operation (right) (cf. text).-Center left: Zr4A13 structure (idealized) is
formed from the Cr3Si structure by a vertical shear of each second slab followed by a connection of the tetrahedra-stars. Center right: Structure of
CuA1, formed by an additional shear operation of the same type along the
broken line in Zr,A13. (In both cases the central atoms are omitted.)-Bottom: The structure of Mo3CoSi generated by the same shear operation but
only of each second slab of Cr3Si.
Fig. 7. Lattice image of a defect in a crystal of NaNb1303,. The intergrowth
of the NaNbi303, structure with a TiNb20,-type structure is easily discernible (region between the arrows) [ll].
compounds with the Fe,C structure, of which we shall
mention only three here, namely SbCI3, BiCls, and, as a
special case, Xe03. Likewise, ccp nets may be twinned,
and this operation also produces trigonal prisms in the
twin planes. The structures of Re3B, CaTi,O, and lillianite
(Pb,Bi&) are typical examples (Fig. 11). P ~ r t h k [ ’has
~]
pointed out that unit-cell twinning is also observed in
mixed hcp and ccP structures, for
in sc3C0 and
one modification of TbNi.
Angew. Chem. hi.Ed. Engl. 22 (1983) 69-81
Fig. 9. High resolution lattice image of Mo,CoSi. The light dots are the center of mass of the hexagonal antiprisms, the weaker dots correspond to the
square antiprisms. The intersection of the arrows is the Cr3Si structure.
73
t
0
0
0
0
0
.O
xt
L
-
Y
Fig. 1 I . Top: Structures of Fe3C and FesC2. Carbon atoms fill the yellow triganal prisms, between the green and red hcp slabs of the Fe atoms (yellow
spheres). Note that the green slabs in FesCL(right) are not as wide as those in
Fe3C (left).-Bottom: The structure of Re,B, CaTilO, and lillianite (trigonal
prisms = blue polyhedra).
Fig. 10. Top: Twinning in hcp. Heights are in units of 1/12th. Trigonal
prisms in the twin planes are shown, as are also some octahedra.-Bottom:
The structure of Fe,C formed by four atom layer slabs (as indicated on the
bottom left).
Unit-cell twinning of a primitive cubic structure is also
possible. One example is the structure of Ru4Si3shown in
Figure 12 as twinned RuSi (CsC1 type), but here the reflection is combined with a translation. Rh5Ge3(Ru5Si3is isostructural) follows the same scheme, but the slab size is only
five atoms wide (Fig. 12), whereas in Ru4Si3 it is eight
atoms wide. Naturally, one expects a crystal to have faults
which correspond to various distances between the twin
planes. The presence of faults was confirmed in some early
electron microscopic studies on Ru4Si3 crystals, but since
no really high resolution was available, we had to make a
histogram analysis of the fringes in the one-dimensional
lattice images"61. Figure 13 shows the various c-axis
lengths found, corresponding to the different RuSi block
sizes in the Ru,Si3 crystal.
h -
6. Complex Parallel Intergrowth
The extremely important alloy structure type W,Fe, (pphase) is best described as an intergrowth of slabs of two
structure types, namely those of Zr4A13 and MgCu2 (Fig.
14). This suggests that disorder may exist, since the two
different slabs can vary in size and orientation; this has in
74
Fig. 12. Top: Structure of Ru& formed by RuSi (eight atomic layers).-Bottom: The structure of Ru5Si3(RhSGe,) generated by RuSi-slabs (five atomic
layers).
Angew. Chem. l n f . Ed. Engl. 22 (1983) 69-81
t
Ru5Si3
20
15
10
t
t
t
t
t
t
Ru&
t
t
t
[A1
t’
Calculated
I
periodrities
Another spectacular example of intergrowth is found in
the crystal structure of Ta6S (Fig. 14). Slabs of the Zr4A13
type are intergrown with slabs of the FeSC2type and thus
form the Ta,S structure. The tantalum atoms occupy the
Fe, Zr and A1 positions, and the sulfur atoms lie in the trigonal prisms (at the positions of the carbon atoms in
FeSC2).
7. Parallel Translation and Reflection in
Combination
Fig. 13. Frequency distribution of the c-axis lengths found in five different
crystals of Ru&.
fact been confirmed by electron diffractionr”]. One of the
reasons we include the so-called @-phasestructure here is
that it can be sheared and twinned, and thus lends itself for
the description of at least three more structures.
We have already given one example of this very common combined operation for describing structures in Section 5. Another example is the M-phase (Nb48Ni39A1,3)
structure, which is accurately derived by twinning and
shear of the @-phase structure (Fig. 15). If the p-phase
slabs are extended in size a hypothetical structure is obtained, and with thinner slabs we get the FeSiW, structure
shown in Figure 15. If the operation is continued by an additional shear, the melilith (Ca,MgSi,O,) structure is obtained. Excellent experimental support for this description
is again obtained from Stenberg’s electron microscopic investigations. Figure 16 shows part of a crystal of the Mphase containing large parts of the p-phase, sheared and
twinned as in the M-phase structure1’*].
8. Cyclic Operations: Translation and Reflection
Fig. 14. Top: Structure of WsFe, composed of Z t A l , and MgCu2 slabs.-Bottom: Structure of TadS composed of Zr4A13 and FesC, slabs.
Angew. Chern. Int. Ed. Engl. 22 (1983) 69-81
A simple example of this is shown in Figure 17, where
the tetragonal bronze structure is generated by a cyclic
twin operation from the Re03 structure. A large part of the
bronze structure is recognizable in the center; the central
octahedron is formed partly by the cyclic operation and
then merely completed. In addition, the BaNbz06structure
is formed in the twin planes. Naturally, this offers a geometrical model for the formation of the compound by nucleation. Repetition to a larger crystal may then occur if it
is energetically possible, e.g. if the right ions are present
under the right conditions of temperature and pressure.
The structure of W,Si, is a further example for this operation: The simple construction of a fourling of the Cr,Si
structure (Fig. 17) enables the demarcation of a column of
WsSi3, whose structure is commonly described with square
antiprisms, octahedra, and hexagonal antiprisms. Here
also the structure in the center was completed by the insertion of an individual atom.
The Zr4Al, structure is obtained by a shear (or translation) operation in the Cr3Si structure. If a Zr4A13column,
similar to the Cr3Si column above, is used again for a
fourling construction we obtain the Ta2S structure, as
shown in Figure 17. Bursill and Hyde have shown that the
tetragonal bronze structure can also be derived by a rotation of one-dimensional columns of the ReO, stru~ture~’~].
A similar transformation mechanism has also been proposed by us for the formation of the Ni3P and w&, structures from that of Cr3Si[201.
75
L
4-k
p-Phase
6
0
p-Phase
pPhase
Fig. 16. High resolution lattice image of an M-phase crystal, with large regions
of the @-phase.(One glide plane is indicated by an arrow.)
i
p-Phase
9. Cyclic Intergrowth
Cyclic intergrowth has recently been used by us[z11to describe several structures; here we shall only mention a few
cases. Figure 18 shows a simple model for this operation
and clearly demonstrates how the important o-phase
(FeCr) structure is very accurately derived by intergrowing
Cr,Si and Zr4A13. This provides an alternative but simple
and attractive description of the o-phase, previously described with Kagome nets.
Simon has described the suboxides of the alkali metals
discovered by him as cluster compounds with the structural units Rb902or CS,,03,
respectively[2z1.
An alternative
description is possible for the suboxide Cs703by a cyclic
intergrowth of an hcp structure and the Zr,Al, structure
(Fig. 18). In the center of this cyclic intergrowth we note a
cluster of three face sharing octahedra, namely the Cs, ,03
unit, which represents the basis of the Cs703 structure
(shown inside the heavy line).
w
Mu-Phase
10. Complex Applications
Fig. 15. Top: The M-phase structure. The glide planes (g) transform the intergrowing slabs of the p-phase into each other.-Center: A hypothetical structure with thicker slabs of the p-phase.-Bottom: The FeSiW, structure composed of thinner pphase slabs.
76
The validity and accuracy of this general method for the
description of structures has been demonstrated experimentally with high resolution electron microscopy for
some groups of compounds. We have shown earlier[231
that
some complex alloy structures and even some so-called
giant cubic structures could be very accurately described
with these methods. Examples are y-brass, Rh7Mg44, and
Cu4Cd3.
Another unusually complex structure is found in the silicate paulingite [K,(Ca, Ba),.,(Si, Al),z024.14 H,O], which
has a cubic unit cell containing about 2500 atomsf241.The
methods discussed here led to an alternative description of
this structure'251,which differs considerably from the one
given by Samson et al. A reflection operation as initial step
Angew. Chern. Int. Ed. Engl. 22 (1953) 69-51
Ba Nb1O6
Ba Nb2 O6
Fig. 17. Top left: A fourling of perovskite (or ReOz) generated by two perpendicular reflections, to give the tetragonal bronze structure in the center (framed). The
main structural units of BaNb206(partially isolated in the upper part) are formed along the twin planes.-Top right: Fourling of the CrlSi structure (formed by
fourfold rotation) generates the W,Si, structure.-Bottom right: Fourling of the CnSi structure with the region of the W5Si3structure in the center (framed).- Bottom left: Fourling construction of the ZraAI, structure to give the Ta2S structure (framed with bold lines).
converted the very simple structure of cristobalite (SiO,)
into the gismondine structure (CaAl2Si2Os.4 H20)f261.A
fourling of this structure (Fig. 19) formed by two orthogonal reflections generates the structure of merlionite
(K5Ca2A19Si23064.24
HzO) in the center. This mode of
operation also leads to the structure of phillipsite
(KCaAI3Si5oI6.6HzO) from two pieces of gismondine in
Angew. Cheni. h i . Ed. Engl. 22 (1983) 69-81
the mirror plane region. If a whole packet of atoms is now
taken from the center of this fourling construction (only Si
atoms are shown in Fig. 20) and put together in three-dimensional space by a translation operation, the paulingite
structure is obtained (Fig. 20). The coordinates of all atoms
of the paulingite structure, as calculated from the gismondine structure, agree exactly with the data of a crystal
77
Fig. 18. Top left: Cyclic intergrowth of two simple structures.-Top right: Cyclic intergrowth ofthe ZrdA13 (view along the hexagonal c-axis) and Cr$i StrUctureS to
give the 0-phase structure (framed in the center).-Bottom left: The complete u-phase. The same region as in upper right has been framed in the center.-Bottom
3
is formed (only Csright: Cyclic intergrowth of the hcp and ZraA13structures to give the Cs,O, structure (indicated by a bold line). In the center the C s l 1 0 cluster
atoms). The unit cell is indicated by broken lines.
structure analysis by Samson et al., which again confirms
the validity and accuracy of our description[”].
Our method for the description of these large cubic
structures can, of course, be used to solve unknown complex structures as well. The systematic tracing of relationships to much simpler structures in the reciprocal lattice of
a complex structure provides a direct method for structure
determination. It was with this in mind that Fulth and I undertook the structure determination of a very large cubic
structure, that of zeolite N with a lattice constant a=36.9
Crystals of this
A, and about 3500 atoms in the unit
zeolite were grown by FZth using a method developed by
him for silicates. In the Patterson function we quickly
found a simple relationship to the sodalite structure
78
[Na4(AISi04)3CI],and by taking out exactly half of all the
atoms from this structure, the remainder could be fitted to
the space group of zeolite N, namely Fd3. This accounts
for half the number of tetrahedra in the structure shown in
Figure 20, and it was clear from this development that a
second partial structure still had to be inserted in the sodalite skeleton in order to construct the complete structure
of zeolite N. By trial and error we found that this missing
part is a fragment of the zeolite K5 structure (Fig. 20). This
part was taken in a similar way from the zeolite K5 structure as described for the sodalite fragment, namely with
the criterion to fulfill the symmetry of the space group of
zeolite N. Both structures, sodalite and K5 thus interpenetrate into the complete skeleton of zeolite N (Fig. 20)[271-a
formidable and elegant example of a structure, generated
Angew. Chem. Inr.
Ed. Engl. 22 (1983) 69-81
Fig. 19. Fourling of gismondine G (hatched region) and merlionite M in the
center. In the region of the mirror planes (indicated by arrows) phillipsite P
is formed by two parts of G.
by different interpenetrating complex point systems, which
are operated as a whole by symmetry[28J.
11. Periodic Minima1 Surfaces
Some of these interpenetrating structures are hard to describe and understand but if we persist then some new and
exciting developments emerge. Consider first:
a) Cu4Cd3, a pyrochlore-like structure with mainly Cd
atoms which interpenetrates a complex Laves phase
structure, of mainly Cu atoms.
b) Zeolite N, a sodalite structure which interpenetrates a
zeolite K5 structure.
c) W3Fe3C, a tungsten octahedral structure which interpenetrates a tetrahedral structure of iron atoms (Fig.
5).
d) Cr,Mg,Al,8, where two identical structures interpenetrate.
If we study especially that part of the Cu4Cd3structure
which is built up of Friauf polyhedra intergrown in a manner typical for Laves phases, then we can recognize that a
very smooth surface, close to the interface between the two
interpenetrating structures, covers as well as separates the
two parts (pyrochlore and Laves phase) in three dimensions. We propose that this interface is an infinite periodic
minimal surface.
The Schwarz periodic minimal surfaces have been systematically described by S ~ h o e n [ ’ ~One
’ . of the most spectacular is the so called Neovius surface[30’.According to
Hilbert, it has the diamond symmetry and also contains the
famous “monkey addle"'^'' (cf. Fig. 21).
Angew. Chem. Int. Ed. Engl. 22 (1983) 69-81
S c r i ~ e n [has
~ ~ ’suggested that a minimal surface might
play a role in fluid or liquid crystals and that the structures
differ on each side. Other crystallographical applications
for these surfaces have been suggested by M ~ c k a y [ ~
Ac~’.
tual use of the concept has been found for a cubic liquid~ ~ ’Fon,
crystalline phase by Larsson, Fontell and K r ~ g [ and
tefl[351.
Truncated octahedra are one type of polyhedra that
fill space and if half these are taken out of space-filling
packing in a primitive way, the cubic liquid structure is obtained. In this case the truncated octahedra share only
square faces, and the structure, which corresponds to
Schwarz’s primitive periodic minimal surface, is the one
found by Larsson. Fontell, and Krog. On the other hand, if
half of the polyhedra are removed from the space-filling
packing of truncated octahedra in such a way that the remaining polyhedra share hexagonal faces, a cubic face
centered structure is obtained (diamond symmetry), which
represents half of the Si,AI structure of zeolite N (Fig.
20).
It is clear that the interface between, e. g., the interpenetrating W and Fe structures in W3Fe3C, the interpenetrating Laves and pyrochlore-like structures in Cu4Cd3, and
the interpenetrating s o d a h and ZK 5 structures in zeolite
N can all be approximated to a periodic minimal surface, as
shown in Figure 21 for the tetrahedral part of Fe3W3C.
This is the first periodic minimal surface of Schwarz; it has
diamond symmetry. We can now continue this description
by saying that the skeleton atoms in the Linde A and K 5
zeolites also form a minimal surface[331,like the one found
by Fontelf et al. in their liquid crystals. Faujasite (zeolite X
and y), which belongs to the same family as the zeolite N,
as well as the pyrochlore structures may also be treated
similarly. Finally, the hexagonal tunnel structures and the
hexagonal bronze structures are also topologically related
to Schoen’s H, or H-T structures[36J.
A useful concept in the description of structures has
been the close packing of spheres in a plane. Intuitively
however, we feel that atoms of different sizes, like Fe and
W, Cd and Cu, will pack more efficiently on a curved surface than on a plane. It is obvious therefore, that on these
periodic minimal surfaces, atoms will pack with non-Euclidian geometry, and that reactions like ion-exchange,
heterogeneous catalysis or molecular separations occur on
these minimal surfaces.
A logical step leads us to three or even four interpenetrating structures, as in Cr3Si or garnet. Do such minimal
surfaces exist in these cases? Can a minimal surface divide
space into three or even more interpenetrating subvolumes? This I do not know.
12. Concluding Remarks
In this article a number of structures have been described by geometrical methods, which we consider are exact and general, and from which we can obtain some understanding of defect structures and of how transformations and diffusion occur.
Formerly it used to be almost impossible for us to imagine how a complex crystal is formed, but now there is a
glimmer of hope. If we assume that Nature treads well
79
rig. LU. l o p left: A model of a packet of atoms (only silicon atoms are shown) taken out from the fourling center in Fig. 19.-Center left: The paulingite structure.-Top right: The sodalite part of the zeolite N structure. Blue spheres mark the unit cell.-Center right: The ZKS part of the zeolite N structure. The blue units
(identical with red) mark the unit cell. Bottom: The complete structure of zeolite N. A model of quartz in the left corner has been included for the purpose of comparison.
80
Angew. Chem. Inl. Ed. Engl. 22 (1983) 69-81
known paths, e . g . she builds parts of simple structures
first, and then uses the classical symmetry operations, o r
periodic minimal surfaces, to build complex structures, we
begin to feel a sense of excitement at understanding something of the phenomenon.
Having worked on the methods for the description of
structures, and having had a very straightforward application in the solution of the giant structure of zeolite N, we
are unable to resist making a more human and exultant
conclusion: We feel we can at least challenge Nature and
solve whatever she is able to repeat.
This work was supported by the Swedish Natural Science
Research Council. I thank Pro$ H . G. von Schnering, Dr.
W . Honle [Stuttgart) and Dr. F. J . Lincoln (Western Australia) for a critical reading of the manuscript.
Received: September 27, 1982 [A 443 IEI
German version: Angew. Chem. 95 (1982) 67
~~
Fig. 21. Top: The Neouius minimal surface. The saddle lines of Hilbert’s
monkey saddle a r e indicated.-Bottom right: A monkey saddle after Hilbert
which builds the surface on which we believe ion exchange, catalysis and
molecular sieve reactions occur in structures of, e.9.. the faujasite type.Bottom left: The D minimal surface as formed in the region between the interpenetrating structures of the W2Fe3C structure (cf. Fig. 5).
Can we explain why complex structures exist? The
achievement of an energy minimum is the plausible but
not very helpful answer. The energy minimum is necessary,
but is it enough?
It is clear that the transformations discussed above seem
to be invariant, or nearly so, and since Nature aspires to
attain maximum symmetry we suggest that she prefers
complex structures to coexisting phases. We notice that
complex structures like Cu,Cd3 are better packed than the
metals, which means that the minimal surface in the interface region contains atoms with very efficient packing. It is
here, in the interface region that the five-membered ring o r
icosahedral packing occurs in the structure.
Angew. Chem. Inr. Ed. Engl. 22 (1983) 69-81
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81
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