OK.
So I'd like to take a look at interstitial sites.
What are these?
These are spaces between other atoms, OK?
So for example, carbon is a small little atom.
And amongst iron, it can occupy a little space between them.
It goes into solid solution and it can dissolve.
You can't dissolve very much of it,
but that's an interstitial site.
But we've already seen interstitial sites
in another-- we see interstitial sites in sodium chloride,
as well.
Sodium chloride, NaCl.
So there we had these chlorine anions and FCC type
positions with sodium nestled in between them.
So that's a space between other atoms, isn't it?
We could say that the sodium is in one of those spaces.
It's in an interstitial site.
But what kind of interstitial site is it?
And to look at that, we need to actually go out
into three dimensions.
I mean, it looks like, in this case here,
this chlorine-- I'll label these for you here.
Chlorine and sodium.
It looks like the chlorine is touching four sodiums.
Or in fact, actually the way we often do it is we
would look at the sodium touching four chlorines.
But that's only in two dimensions, right?
That's two dimensions.
And we got to go to three dimensions.
So to do that, we need to go to the unit
cell for sodium chloride.
And you'll remember that we've got chlorines in corners here--
that's conventionally how it's illustrated--
and in the face centred positions.
Back in there, those hidden faces in there.
And then we've got sodium occupying these edge positions.
Edge positions like that.
And so there's 12 edges.
And what fraction of a circle-- of a sphere,
rather, is within the unit cell?
Only a quarter.
We've got a sphere that's cut into wedges, basically.
So we've got 12 positions on the edges times one quarter each.
That's three.
We know there's four chlorines.
So we're missing one.
And the one that we're missing is the one that I'm actually
interested in, the one right in the very centre of the cube.
And that one's not cut up at all.
That's in the very centre of the cube.
And that is probably the easiest position to picture for this.
When we're trying to picture it in three dimensions,
this interstitial site.
So that's the guy we're interested in.
And I drew in with the yellow colour the direction of nearest
neighbour contact, which chlorines is touching it.
And so I'd like to do that now.
So coming out of the page, let me draw the axes for you.
This would be x coming out of the page, y to the right,
and then up.
In the x direction, out of the page and into the page,
this sodium is making contact with those chlorines.
Along the y-axis, to the left and to the right,
it's contacting those two face centred atoms.
And then top to bottom, and it's the same distance,
even on my sketches, not quite right.
In the z direction, plus and minus,
this sodium is touching chlorines.
So how many is it touching?
It's touching one, two, three, four, five, six.
So we find that we have what we call a coordination number.
That is the number of atoms touching a given atom,
a coordination number of six.
And so that would be one way you could describe it.
But we often like to describe them another way.
And that's based on the geometric solid that's created.
Just like we have the tetrahedron in diamond,
or tetrahedral symmetry for methane.
We had tetrahedron, this four sided triangular faced solid.
That's a tetrahedron.
This is just a side, right?
Tetrahedral was four faces.
So if we make something this is on this thing here.
I'm going to connect as if we put
no triangular faces on this.
We've got one face there, and then slicing along
in the horizontal plane, we could then
create another triangular face on the front.
There's another one on the right here.
And there's another one on the back.
So I've created four triangular faces up in the top.
And then there's four more down to the bottom, as well.
So in total, there's eight faces.
And the structure, that geometric solid,
is called a octahedra.
Octahedra.
So the coordination number of six
corresponds to what we call the octahedral interstitial site.
Octahedral interstitial site.
And let me show you one more.
Look at this.
If we want to figure out how big that was,
what size cation would fit into that octahedral site?
What we could do is we could take a slice
right through that cube there.
This is beautiful, yet messy.
If you're just looking at it now, it looks like a mess.
Let's redraw that.
Let's start fresh for a moment just to show you
where I'm going with this.
I'm always doing this.
Didn't give myself enough room.
Let's move that down a little bit.
There we go.
All right, move that down.
That should be enough room.
And good.
I'm going to draw a beautiful cube here, one of my best.
And dash that, and we can see the back.
Excellent.
And what I want to do is I want to say, OK,
for the purpose of our discussion here,
let's just look at that central, interstitial site,
right in the very centre of the cube.
And as we know, it's making contact
with the top face, the bottom face, the front face,
the right side face, the left side face, and the back face.
And then if I slice this cube this way
in the plane of the board, the plane of the page,
straight up and down.
Let me just redraw that one.
I'm going to slice it like this.
Here we go.
OK.
I slice it like this.
Then this guy is cutting on the right spot, pesky little anion.
There he is.
OK, so I've sliced this thing in half.
And so I've made this little plane, this green plane.
And so that green plane, if I draw it,
would have some anions here.
It would an anion like that.
It would have an anion over here, one here,
another one here, and in the very centre would be this
cation.
OK And so that's kind of the interstitial-- that's
the site that we're looking at.
And make this just a little bigger
so it's actually making contact.
One thing that always happens.
Remember solids make-- they reduce their energy by forming
bonds, by increasing their nearest neighbour atoms.
So we always observe, in any structure,
that cations will be touching their nearest neighbour,
ations.
Now, the anions don't necessarily
make contact with other-- let me make it clear in the picture
here.
They don't necessarily make contact with each other.
So I'm going to just doctor that up a little.
There we go.
So in this case, these have been pushed apart.
We know they're in FCC type positions,
so they should be making contact with each other,
but they've been pushed apart by the presence of this cation.
And that's fine.
That can happen.
What we don't find happening is what
I'll draw in just a second for you here.
What we observe that does not occur in nature is this.
I'm drawing the same interstitial site,
just a little bit smaller.
And I'm just drawing it rotated by 45 degrees.
We never find this happening, OK?
Where the cation is not touching the nearest neighbour anions.
Now, that here doesn't happen.
That's high energy.
It's unstable.
It won't happen.
But let's consider the case right
at the threshold between this, where they're not touching,
and this where the cation has pushed the anions apart.
Let's see if we can consider that special case there.
I'm going to do that for your right now.
So here's an anion, another anion, another anion,
another anion, OK?
And now we're going to position that cation right there
in the middle of them, OK?
Beautiful.
And what we'd like to do is we have
to say, well, how big is the cation that
just perfectly fits?
That perfectly fits, geometrically ideal,
if you will.
Now, how could we express that?
Well, we could express it as if we said,
let's just define the size of the anion as one,
then you could define the size the cation
as a fraction of that.
That is, if I draw that same anion over here, OK?
Now, line up a cation in there, how big is it?
Is this dimension here, how does that dimension
compare to this dimension?
And you can see by my sketch here, which
is actually not a bad sketch.
It's perhaps not my best, but it's not too bad.
It looks like maybe it's on the order of a half, probably
a little bit less.
So we expect that our answer, our value,
should give us that the radius of the cation divided
by the radius of the anion is roughly 50%.
We think it should be.
It should, in affect, if my sketch is accurate, which
it is, although it's not great.
But we think it should be a little bit less than 50%.
So we can have a sense for it going
in what the answer should be.
And so now there's some nice little geometry
that works out here.
If I connect from the centre of this anion to the center
of this anion and again to the center of this one and then
across on the diagonal, I get some very interesting geometry,
because it goes right through the centre of the cation.
And we know that this distance here is twice
the radius of the anion.
So that's the radius of the anion times 2,
as is this dimension.
And this hypotenuse here, it's twice the radius of the anion
plus twice the radius of the cation, isn't it?
So now we've got some nice little fairly straightforward
maths that we can work out here.
This angle here.
What's that angle?
That angle's got to be 45 degrees.
So we could say OK, great, so sine
45 is going to be equal to opposite,
so that's 2 times the radius of the anion,
over the hypotenuse, 2 times the radius of the anion
plus the radius of the cation-- correction, times 2 of that.
So that's great.
The 2s cancel out.
And if we multiply across, we could say Ra sine 45
plus Rc sine 45 equals Ra.
Not too bad.
And what are we after?
This is the term we're after.
So we want to divide by Ra.
We can't just do that willy nilly.
We have to follow the rules of maths here.
So we're just going to divide both sides by Ra
and then everyone's happy.
And so that's great.
This is 1 on the right hand side there.
This is 1 over here.
So this is looking good.
And we're going to have, then, that Rc over Ra
is equal to 1 minus sine 45 over sine 45,
which everyone else is 0.414.
I'm just joking.
Everyone doesn't know that.
But you can calculate it.
And it's 0.414.
And hey, that's great, because that's a little bit less
than 50%, isn't it?
So that tells us the ideal size of a cation
that'll fit into the octahedral interstitial site.
And in fact, this would provide us
a basis for predicting some crystal structures.
There's some limitations, of course,
but a cation that's anything less than 0.414
times the anion radius would not occupy the octahedral site.
If it's less than that, well, mean
it would occupy a tetrahedral site.
Because tetrahedral is smaller.
But it could certainly occupy the octahedral site
if it was a little bit larger than this.
And so that gives us a little bit
of an understanding for the size of these interstitial sites.
And the relative sizes of these sites,
at least the ones that we're probably
most immediately interested in, would be tetrahedral.
This is one of our smaller ones.
Then octahedral, which is the one we just worked out.
And then there's a larger one, actually,
which is just right in the very centre of a cube.
It's the body centred position, that you may imagine right
in the very centre of the cube.
And so that's the octahedral.
That's the simple cubic interstitial site, or just
cubic interstitial site, actually
cubic interstitial site.
We'll just delete that.
Cubic interstitial site.
Perfect.
So that's a little look at interstitial sites.