Here we'll describe
where the tetrahedral
sites and the octahedral
sites are located on a face centered cubic
or FCC crystal structure and a question first would be why would be interested in
this?
So we could look at structure of something like
aluminum oxide. It turns out this structure consists of
the oxygen in the face centered cubic structure
and the aluminum then
in these interstitial sites and depending on the
type of aluminum oxide crystal structure,
determines exactly where they're located
but for gamma alumina which just one type
of aluminum structure they are both in tetrahedral
and octahedral sites
And the reason this happens is aluminum if we look at the
iconic radius, it's much smaller than the ionic radius
of oxygen. So we have the oxygen's in this
crystal structure and then the spaces between those ions we can
fit the much smaller aluminum. That's a reason we are interested so let's look and
see exactly where the sites are
So what I have drawn here is just the start of
the first couple layers of a FCC structure so
FCC structure and what we're looking at is
the Miller Indices plane
labeled 111 so that has a hexagonal symmetry
and if you remember an FCC structure can be envisioned as an
ABCABC...
type packing where we have a hexagonal layer. We put now
the second layer above the openings
not above the atoms themselves, and we put the third layer
above the openings that
are not
directly above the first layer. We do not put the third layer here because
that would be directly above the "A" layer but we put it here
here ect. and so
if we look at just a few atoms in the second structure, again these
would extend out in the same type of packing as the first layer,
then we can look at if we put just one atom
in that third layer, so let's say we put one atom
this orange atom, what we've done is say
we're going to put an atom here. Well this space then
underneath that atom that's
under here so let's go back and say
this then that space that we're going to create
is a tetrahedral site. It's smaller
certainly then the atoms of the FCC structure
As we create a tetrahedral site and here it's just shown with the
third atom much lighter. So notice the
third atom is not above an atom in the first layer
and we can see the tetrahedral structure
These four atoms form a tetrahedron
and we've identified the location then
of this tetrahedral site. So
turns out we can identify the locations now in a unit cell. So what I've done
is drawn just the four corners
of the unit cell FCC structuring
Of course there is going to be an atom here. Atom on this face, this face
this face, and the one in the back. The
green locations are tetrahedral sites. One quarter in, one quarter up
one-quarter over and
if we look at the structure we can see there are eight
sites. These are interstitial sites
And the unit cell of the FCC structure
has four atoms in the unit cell so
8 tetrahedral sites unit cell, 4 atoms. The 4 atoms correspond to
we have 6 faces. Atoms on each face. It's shared with the
unit cell next to it so it's six times 1/2
so we have three atoms from the faces
And then the corner we have 8 atoms but they are shared
each of them with 7 others so there's only 1/8
on the corners. You can see now we have a total then of four atoms
in the unit cell 8 interstitial sites we have
two interstitial tetrahedral sites
two interstitial tetrahedral sites for every atom of the FCC structure
what we want to now look at is octahedral sites
again I'm showing the FCC
the 111 plane. Miller indices 111 the hexagonal structure
here's the "B" layer. Now the "C" layer remember
is not located above the "A" layer so I've indicated now three atoms
in the "C" layer so we can visualize the
octahedral sites, so these three atoms here shown
lighter. See that this atom in layer "C" is not above the atom
Right? This is layer "A" this is layer "B" and this atom is layer "C"
and since "C" are not above the atoms in layer
"A" this site here then is the octahedral site and it has
six-fold coordination. It's surrounded by six atoms of the FCC
structure
but we call it an octahedral site because
these six atoms form, if we connect
the centers of the atoms, a structure that has eight sides
an let's try and visualize that. And the easiest way is to
look at this atom draw a line to the center of this atom
in the"C" layer, this atom in the "C" layer
and then this atom in the
"B" layer
and then connect them so we have a square
And we have this atom above, this atom below, and maybe it's easy
so what we're looking at and why they call octahedral site
so what we have is four atoms
in a
square atom above so
if I connect a similar atom
below again connecting, what we show here is that
eight sided structure. Four triangles on top, four triangles on bottom. 6
coordination
so we call an octahedral site because of this
structure. This eight sided structure that
forms by connecting atoms
and then the octahedral sites
reproduce the FCC structure and now I've drawn the octahedral sites in
The octahedral sites happen to be located along
each side. So there is one octahedral site on each side
and there's one in the center so if we wanted to add up and determine the number
of octahedral sites
So one site is in the center
and then we have 4, 8
12 sites
but these are each shared with three other
so only 1/4 three other unit cells
only 1/4 are there so, so this is three
from the edges. So
a total then of four octahedral sites
So four octahedral sites per unit cell. Remember
we showed earlier we had four atoms per unit cell so we have one octahedral site
for each atom and as we showed previously
two tetrahedral sites. So a total of three interstitial sites for
each atom in the FCC structure