4. Structures Based on Linked Polyhedra
(a)
General Considerations
As we have seen the octahedron and tetrahedron are the two
dominant cation coordination polyhedra in solid state chemistry.
In this section we are going to look at structures as though they
were constructed from these polyhedral building blocks, which
share corners, edges and faces from extended structures.
Before we go further, consider the energetic implications of
sharing corners, edges and faces. In each case the cation-cation
distance, M-M, can be expressed in terms of the cation-anion
distances, M-X (see page 32 in West).
Polyhedron
Tetrahedra
Octahedra
Corner Sharing
2.00 M-X
2.00 M-X
Edge Sharing
1.16 M-X
1.41 M-X
Face Sharing
0.67 M-X
1.16 M-X
You can see that from a Coulombic point of view, sharing edges
and faces is less favorable then sharing corners, particularly for
tetrahedra. The situation is so sever for face sharing tetrahedra
(M-M distance is shorter than M-X) that this structural
arrangement is not observed. As we saw in the structure of
corundum, in cases of edge and face sharing, it is common for
cations to shift away from each other if possible. This will be
illustrated in some of the structures discussed in this section.
Another point which we have not yet discussed is the relationship
between stoichiometry and coordination number. For example if a
structure contains isolated octahedra (cation centered) then its
stoichiometry must be MX6, and the anion coordination number is
one. Now if we want to maintain octahedral coordination about
the cation, but change the stoichiometry to MX2 (as in rutile or
CdI2) the coordination number of the anion must increase. The
relationship between stoichiometry and coordination number for a
binary MaXb compound is:
bCNx = aCNM
So for the rutile form of TiO2 the coordination number of
oxygen is:
2CNO = 1(6)
CNO = 3
For CdI2 the numbers are identical and the iodine CN is also
three. Of course it is not possible to increase the anion CN
without linking the octahedra together in some manner. If we
deal with extended structures (excluding structures with isolated
clusters of polyhedra) then we can infer some information
regarding the polyhedral connectivity from thee anion
coordination number (or vice versa for anion centered pollyhedra,
as in fluorite).
Anion CN = 1  Isolated octahedra/tetrahedra
Anion CN = 2  Corner sharing octahedra/tetrahedra
Anion CN  3  Some edge or face sharing + corner sharing
This is a very simple, yet powerful tool for predicting and/or
rationalizing crystal structures. For example once we accept the
fact that Ti4+ will take octahedral coordination (for which it has a
strong preference), charge balance tells us the composition must
be TiO2 and the CN analysis tells us to expect a structure where
the octahedra share both edges (or faces) and corners. However,
keep in mind the limitations of this analysis.
(1)
We can only determine the coordination number in this
manner, not the coordination geometry. For example, the
anion coordination is (nearly) trigonal planar in rutile, while
it is trigonal pyramidal in CdI2.
(2)
This approach can be extended to ternary and more complex
stoichiometries (see for example perovskite, spinel or
garnet) but it is not generally applicable if the anions occupy
multiple crystallographic (Wyckoff) sites (even for binary
compounds).
(b)
Structures Based on Tetrahedra
Scheelite (CaWO4)
Let’s start our discussion of structure types based on tetrahedra
with the scheelite (CaWO4) structure. If we treat the larger,
lower valent cation (Ca2+ ion CaWO4) as a charge compensating
ion, then the scheelite structure can be described as consisting
of isolated MX4n- tetrahedra. This structure type is favored for
AMO4 compounds where A is a large electropositive ion (generally
an alkali metal, alkaline earth or lanthanide ion) and M is a smaller
cation in a high oxidation state (e.g., W6+, Mo6+, Re7+, Ru7+, I7+).
The highly positive oxidation state of the M cation favors the
scheelite structure in two ways. First of all, the very strong
bonds to oxygen discourage formation of more than one such
bond to a given oxygen. This destabilizes competing structures
which might have some corner sharing tetrahedra. Secondly, the
highly positive oxidation state allows the relatively large anion to
cation ratio associated with the scheelite structure.
-Cristobalite (SiO2) [Idealized]
The crystal chemistry of SiO2 is extremely complex for a
compound that is very simple from a stoichiometric point of view.
Three distinct structure types are known: cristobalite, tridymite
and quartz. Furthermore, both cristobalite and quartz form high
and low temperature polymorphs. All of these structures contain
three dimensional networks of corner sharing SiO4 tetrahedra.
The tetrahedral coordination of silicon combined with the
stoichiometry tell us that oxygen is two coordinate. The
structures differ from each other in both the long range linkages
of tetrahedra and the Si-O-Si bond angles. We are only going to
consider the idealized structure of -cristobalite (high
temperature form). This structure can easily be visualized by
starting with the structure of elemental Si (diamond structure)
and placing an oxygen midway between every pair of Si atoms.
This gives a highly symmetric structure with linear Si-O-Si
bonds. However, I call this idealized structure because it is now
known that the tetrahedra rotate which leads to a partial
collapse of the framework. This reduces the Si-O-Si bond angle
to ~147 and lowers the symmetry from Fd3m to F4d2
(standard setting is I42d). Interestingly, the unit cell
dimensions are still cubic within experimental error (this is a good
illustration of the principle that symmetry elements and not unit
cell dimensions determine the crystal system).
The structure of tridymite can be derived from the wurtzite
structure in the same way that the structure of cristobalite can
be derived from sphalerite (placing Si on all of the sites in
wurtzite then inserting oxygen ions between the silicon atoms)
Cuprite (Cu2O)
Just as we generated the ideal -cristobalite structure from the
diamond structure, we can generate the cuprite (Cu2O) structure
starting from a body centered cubic array of oxygens and placing
copper ions midway between oxygen ions. This exercise is slightly
different in cuprite though, because each oxygen ion in a bcc
lattice has eight near neighbors and copper ions are only inserted
into half of the midpoint sites. The resulting coordination
environments are tetrahedral about oxygen and linear about Cu.
Ag2O and Pb2O are isostructural; Zn(CN)2 and Cd(CN)2 adopt anticuprite structures.
Zeolites
The term zeolite refers to a rather large class of aluminosilicate
framework structures built from corner sharing TO4 tetrahedra
(T = Si4+, Al3+). Zeolites have a Ti:O ratio of 2:1, just like the
various forms of SiO2. However, unlike SiO2, the zeolite
framework contains large voids or pores, which are generally
filled with water, cations and/or other molecular species. For
this reason zeolites are often called microporous materials
(though the term microporous is broader than zeolite,
encompassing compounds which do not contain aluminum or
silicon). This rather unique structural feature is responsible for
the widespread use of zeolites in a variety of commercially
important applications (e.g., catalysis, including cracking of crude
oil, molecular sieves, ion exchange, gas separations, sorbents, gas
sensors, etc.).
The replacement of Si4+ ion with an Al3+ ion necessitates the
presence of a counter ion (such as Na+) to balance charge. The
counter ion usually sits somewhere in one of the pores or tunnels
in the structure. The need to accommodate this ion helps to
stabilize the porous framework structure over a more condensed
structure, such as formed by SiO2. A detailed treatment of
zeolites is beyond the scope of this course, but to give a feel for
the structural aspects of zeolites three highly symmetric (and
commercially useful) zeolites are described in the table below.
All three structures are constructed from building blocks called
-cages (link here) or sodalite cages. The -cage can be
described as a truncated octahedron containing 24 tetrahedral
cations (Al3+ or Si4+). The outside surface of a -cage contains
square faces and hexagonal faces (much like a C60 molecule
contains hexagonal and pentagonal faces).
Framework
Space Group
# of T ions per
Unit Cell
Aperature Size
Si:Al Ratio
Description
Sodalite
Zeolite A
Faujisite
Im3m
Pm3m
Fd3m
12
24
192
--
4Å
1:1
1:1
7.4Å
1-1.5 Zeolite Y
1.5-3 Zeolite X
-cages linked
-cages fused together (with
together thee an oxygen ion)
square faces
at the square
faces
-cages linked
together at
the hexagonal
faces
Applications of
Zeolite A:
Molecular sieving (i.e., separating small molecules like H2O,
CO2, NH3 from hydrocarbon mixtures), catalysis –
selectively converting primary alcohols to olefins.
Faujisite:
Catalysis – cracking of crude oil (converting long chain
hydrocarbons to shorter chain molecules)