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					Eutactic Structures

(a)   Octahedral and Tetrahedral Holes

Ionic structures have traditionally been described by starting
with a close packing of anions and inserting cations in the voids
present in the close packed anionic system. Although this is
useful for visualization purposes a literal acceptance of this
doctrine has several flaws, two of which I point out below:

      (1) The Coulombic interaction between anions is repulsive, so
      they would prefer to maximize their separation rather than
      minimize it.

      (2) This assumes that anions are much larger than cations,
      but modern techniques have shown that in crystals this is
      not necessarily true (i.e. O2- has a crystal radius of 126 pm,
      while Na+ has a crystal radius of 116 pm).

The arrangements of ions in crystals have been investigated by
Mike O'Keeffe (Acta Cryst. A 33, 924-927 (1977)) who concludes
that the ions in ionic structures approximate close packing
arrangements in order to maximize the volume of the structure,
while keeping the cation-anion distance at its optimal value. He
calls these structures Eutactic.

Keeping these considerations in mind it is still a useful
visualization tool to consider ionic structures beginning from the
closest packing concept. The figure below shows the location of
the octahedral (O) and tetrahedral holes (T and T') in relation to
the close packed layers.
When you look at the image perpendicular to the layers you see
the octahedron looking down a threefold axis (a triangular face of
the octahedron is in the plane of the screen), the tetrahedra are
also seen looking down a threefold axis (looking down one of the
T-O bonds). The T-O bond perpendicular to the plane of the
screen is pointing up for the T holes and down for the T' holes.
The lower figure shows the location of the octahedral holes in
the FCC cell (which results from cubic close packing). There is an
octahedral hole at the midpoint of each edge, and in the center
of the unit cell, where the octahedron can clearly be seen. The
tetrahedral holes are located in the center of each octant of the
cubic cell.

Remembering that atoms located on a corner, edge or face are
only 1/8, 1/4 and 1/2 contained in the unit cell, we can count the
number of close packing ions and holes per FCC unit cell.

       4 Close packing ions
       4 Octahedral holes
       8 Tetrahedral holes

For an HCP unit cell the ratios are the same, but there are only
half as many ions per unit cell.

(b)   Structures Based on Filling of Octahedral Holes

There are several structures that can be formed by filling some
or all of the octahedral holes in a close paced (eutactic) anion
lattice. The exact structure type depends upon the fraction of
octahedral holes that are filled and whether the anion lattice is
hcp or ccp.
                    Fraction of
                                    Close Packed
 Structure Type     Octahedral                       Space Group
                    Holes Filled
Rock Salt (NaCl)       100%          Cubic cp           Fm3m
      NiAs             100%        Hexagonal cp        P63/mmc
      CdCl2             50%          Cubic cp
      CdI2              50%        Hexagonal cp         P3m1
  TiO2 (rutile)         50%        Hexagonal cp*       P42/mnm
Al2O3 (corundum)      66.7%        Hexagonal cp          R3c

*The hcp layers in rutile are buckled, so it is an approximation to
call them close packed.

Pictures and crystallographic descriptions of each of these
structures can be found by clicking on the name of the structure
in the table above. A few comparisons between these structures
are given below.

1. All of the octahedral holes are filled in both the rock salt and
NiAs structures, but the contrast in anion lattice packing (ccp vs.
hcp) has some important consequences. If we view each
structure as being constructed from cation centered polyhedra,
each octahedron in rock salt shares edges with its neighbors,
while in NiAs the Ni-centered octahedron shares both edges and
faces with neighboring octahedra. This results in a fairly short
cation-cation distance in NiAs, which makes this structure type
unstable when the cation-anion bond becomes too ionic.

2. Another difference between rock salt and NiAs is the anion
coordination. In rock salt it is octahedral, which means that both
cation and anion have the exact same coordination environment.
In NiAs both ions are six-coordinate but the anion has trigonal
prismatic coordination. The higher symmetry of the rock salt
structure is very favorable for ionic compounds (though the rock
salt structure is not limited to ionic compounds), provided there
is a 1:1 stoichiometry.

3. Each octahedron in corundum shares both edges and faces, as
in the NiAs structure. However, the vacancies on the octahedral
holes are distributed so that each octahedron shares only one of
its faces with a neighboring octahedron, as opposed to sharing
two opposite faces as in NiAs. This modification allows the
cations in corundum to shift away from each other, thereby
reducing the cation-cation repulsion. This helps to explain why
ionic compounds, such as Al2O3 and Ga2O3, are found with the
corundum structure.

4. From the table above it’s not clear what distinguishes the
rutile structure from the CdI2 structure. Both can be described
as hcp arrays of anions (noting that the anion array is distorted in
rutile) with 50% of the octahedral holes filled. However, they
differ in the way the octahedral holes are filled. In rutile the
octahedral holes are filled in a manner that produces chains of
edge-sharing octahedra running parallel to thee c-axis. These
chains are linked with other chains corners to form a 3D
structure. In CdI2 layers of filled octahedral holes alternate
with layers of vacant octahedral holes to form sheets of edge-
sharing octahedral which are held together by relatively weak van
der Waals interactions between iodine layers. This type of
interaction is most stable when the anion is relatively large and
has a low formal charge (soft anions such as I- and Br-). There
are several hydroxides that take the CdI2 structure. In these
compounds the hydrogen ions reside between the oxide layers and
hydrogen bonding is responsible for holding the layers together.
5. The CdCl2 structure is very similar to CdI2, with van der Waals
interactions holding layers of edge shared octahedra together.
They differ primarily in their anion packing arrangement: ccp in
CdCl2 and hcp in CdI2. Both structure types can be considered
layered 2D structures in terms of their bonding and should be
amenable to intercalation of cations in between the layers.

(c)   Structures Based on Filling of Tetrahedral Holes

The following structure types can be formed by filling
tetrahedral holes of a close packed array of anions (usually)

                     Fraction of
                                      Close Packed        Space
 Structure Type      Tetrahedral
                                      Arrangement         Group
                     Holes Filled
 (Anti)fluorite         100%            Cubic cp         Fm3m
Wurtzite (zincite)       50%          Hexagonal cp       P63mc
                         50%             Cubic cp         P43m
  (zinc blende)
    Diamond              50%             Cubic cp         Fd3m

A few notes on structures based on filling tetrahedral holes are
given below:

1. Just as filling all of the octahedral holes in a ccp anion array
leads to the rock salt structure where each octahedron shares all
of its edges with neighboring octahedra, filling all of the
tetrahedral holes in a ccp array leads to the antifluorite
structure where each tetrahedron shares each of its edges with
neighboring tetrahedra. A similar situation exists for an hcp
anion array where face sharing polyhedra results (NiAs  face
sharing octahedra). However, as far as I know, face sharing
tetrahedra do not exist (you can geometrically show that the M-
M distance is shorter than the M-X distance!). Consequently, it is
not possible to fill all of the tetrahedral holes in an hcp anion

2. The fluorite (e.g. CaF2) and antifluorite (Na2O) differ from
each other in that the anion and cation positions are reversed. In
the fluorite structure the cation is surrounded by 8 anions in a
cubic coordination, while the anion is tetrahedrally coordinated
(and vice versa in antifluorite).

3. The sphalerite and wurtzite structures have the common
feature that both anions and cations are tetrahedrally
coordinated. Furthermore, in each structure the tetrahedra are
linked to each other by corner sharing. To understand the
difference between the two structures we must look at the
stacking of tetrahedra, ABAB... in wurtzite and ABCABC... in
sphalerite (click here for illustration). The strong structural
similarity suggests that the two structures should have similar
stabilities. This is borne out by the observation that many
compounds are known to exist in both structure types (ZnS,
ZnSe, CdS, ZnO, CuI, CuCl, AgI, SiC, ...). The general wisdom is
that sphalerite becomes increasingly favored as the covalent
character of thee bonding increases. This may arise from the
fact that strictly speaking the site symmetry is tetrahedral
(43m) only in sphalerite.

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