All righty.
Let's take a look at perovskite.
And that has the chemical formula,
generally-- well, the general formula
that's often given to it is barium titanate,
which has the perovskite crystal structure.
So that's two cations.
Barium, titanium, and then oxygen. And you'll
notice that, well, that there's two types.
So that creates a little bit of an issue for me because you
know how I like to use these colours.
I like to make my anions blue.
So what am I going to do?
Because I got two types of cations.
So, well, I'll have to use two colours that are kind of red.
And, so, I'm going to call them, as well,
I'm going to say that this is the A cation, right?
This is a convention that's often used.
And then I'm going to use another colour.
Purple.
Isn't that pretty?
Anyway, there you go.
I'm going to call that the B cation.
OK.
And so we're going to proceed through with that.
And I drew a careful cube there before I started recording.
And what we're going to do is we're going to position
at the centre.
At the face centre of each of those cubes is where
we're going to position an anion.
And this is one way of looking at it.
And you'll come across different ones
where the unit cell is shifted over,
with different anions in the corner positions and such.
But this is one way to look at it.
So we're going to put the anions in the centre of each of these
faces.
So we got the face diagonal.
That's the one running in all directions, all crossing.
So right in the middle of it is where we should have the anion.
So go ahead, there you go.
I've put those in.
And I can get rid of the face diagonals to clean it up a bit.
And, actually, let's just do some quick maths here.
So we've got six faces on the cube,
and each one we're only concerned
with the half of the anion that's inside.
So that's three.
And that works out.
That hits our formula just right.
So now we have to find a way to put one barium and one titanium
into this unit cell.
And of course the titanium's going to be that purple colour.
So, OK, what about the A cations?
So the A cations are going to go into the corner
positions of the cube.
OK?
So there we go.
We've occupied each of the corner positions,
and, again, we've got eight corners on a cube times 1/8
of a sphere inside the cube.
So that's one, which is fantastic.
Our stoichiometry is working out.
And then what about the B cation?
That's the titanium.
Well, that fellow is going to go right in the very centre
of the cube.
And I've drawn in faintly here the direction of contact
between this central cation here-- the B cation,
if you will-- and the nearest neighbour oxygens.
And so the coordination number for the B cations
is very straight forward.
One, two, three, four, front, back, left, right, top, bottom.
Coordination number of six.
That's an octahedral site.
It's a little bit more difficult, however,
to see the coordination number for the A cation.
So I want to take a moment to look
at that in a little more detail with you,
and I think that the best way to do
that is going to be like this.
To duplicate this charge.
And you can just see I'm making another unit cell.
And now we were originally only concerned
with that one unit cell, but what
if I were to draw another unit cell that was just exactly
to the right of the first?
OK?
So there it is.
Now we can start to see the atoms that
exist beyond the one we were looking at originally.
OK?
So I'm going to turn that off for a second
and make it invisible.
And what I want to do is I'm going to use that yellow colour
here, as some of you may know I sometimes do,
and use that to indicate the direction of contact,
just like we did here for the central cation.
But now I'm going to do it for the barium, if you will.
The A cation, I mean.
OK?
So the A cation is going to be touching this face
centred-- that's too thick.
Let's making a thinner line here.
OK, it's going to be touching-- I didn't
get it to actually match up.
There we go.
OK, so it's touching that face centred one, right?
It's touching across that face tag diagonal direction.
So that means it's also touching this right side face
centred anion.
And then of course across the front face of the cube here
as well it's also touching.
OK?
So, so far we've got one, two, ah, whoops.
We've got one, two three.
I'll write that in for you.
One, two, three.
OK.
So that's great.
That's within the unit cell.
But that front top right corner barium cation is sharing atoms
also, or touching this coordinate by-- it's got
nearest neighbour atoms in this right hand unit cell.
So, again, what I can do is I can
draw the direction of contact to this right side face centred,
and then of course also to this right side
face centered up at the front.
OK?
And that means that we've got-- that was one, two, three,
there's four, five.
But then we also have unit cells--
we have a couple other unit cells, don't we?
We've got a unit cell that also exists right over here,
in front of our first unit cell.
Let me position that as carefully as I can,
I need a little tweak.
OK.
It's not cooperating for some reason.
Oh, I think I shifted it.
OK.
Well, that's not bad, right?
Rotated it slightly, but anyway.
So what we were doing is then we're seeing,
OK, well it's touching those ones.
Well, so it's going to touch this face centred
in this anion-- in this unit cell
that is now out in front of our original unit cell.
And then see this right side anion right here?
I'll draw a little yellow dot there.
In fact, what I could do is I could also turn off correction.
I'm going to turn off these other unit cells.
So see this guy?
That's the one I'm talking about.
This anion is also touching that.
OK?
So now we've got five-- and did I miss one?
Wait so that should be good.
So that makes it five, six, seven.
OK.
And I turn those other guys back on so
that it'll help you to see this final atom that we haven't yet
added to the tally.
And that's this unit cell.
The anion in the top face there of this unit cell, that
exists sort of towards the right and in front
of our original unit cell.
So then that's our last anion touching this cation.
So we'll draw that one in and give that number eight.
So we've got eight so far, but that geometry
is duplicated above the plane.
These one, two, three, four, and the plane ones I've labelled
one, four, eight, and seven-- those ones are shared,
so we're not going to count those again.
We're only going to count these four below.
Their counterparts in the plane above.
So we've got four above, four anions
in the plane, and four below the plane, for a total of 12.
So the coordination number for-- let me
make that font a little bit bigger here-- here we go.
Coordination number for the anions-- Oh, correction.
Look at this.
I got so excited, or confused I guess I should say.
What I meant to do is, this is the one we were looking at,
right?
We were looking at this cation here.
This coordination number is 12.
All right?
That's what we just showed.
Now, in fact, the anions are linearly coordinated.
So that is the anion-- clean this up a second here.
Go back to just our original one.
In fact, what I could do is I could turn off that.
The anions are only touching along this direction here.
So there would be another B cation above,
and the anions would have coordination number--
with their nearest neighbour anions in that direction--
of two.
So we have the B cations, coordination number of 12.
That's, sorry, A cations the coordination number of 12.
B cations, coordination number of six.
And that is the perovskite crystal structure.
Last thing that I thought I'd show you,
which is sometimes a little confusing,
is what does the actual arrangement of the anions
look like?
OK.
So what we can do now to do that is draw in our surrounding unit
cells.
And what I want to do is show you.
OK, there's a direction between these anions, the two
face centred ones.
And from the top face centred down
to the bottom is another-- well, actually, what I'm doing is
I'm just trying to draw in for you just the anions to show
their positioning.
OK.
And so then we have to draw this unit
cell which exists out here.
And then we can see ah, OK, so the anions
touch across this way.
Correction, they don't really touch that way.
It's just the geometry between them.
And then, finally, that back unit cell
allows us to complete this little bit of geometry
for the anions.
So all I've done is I've connected
the face centred anions here to show you
the arrangement of anions.
Because sometimes the arrangement of anions
is described as face centred cubic,
which is actually inaccurate.
Because they're in the face centred positions.
But if I draw now for you just the anions that
are touching this cube that I've drawn,
we've got these corner ones that are
contributed from those top face centred atoms.
And we've got these corner ones contributed
from the bottom face centred atoms.
OK, then we have these mid plane ones
or the ones on the-- just referring to this blue cube,
the left side face, the right side face, the front face
centred position, and the back face centred position.
But the top and the bottom are occupied by those A cations.
So if I turn off all those other unit cells,
you'll see the positioning of the anions are on their own,
with apparently an extra little bit of the line that I
didn't need to draw in.
It's not completely face centred cubic.
They are in the face centred positions
but not the face centred cubic positions.
They are in the face centres of the cube,
but not in face centred cubic positions.
All right?
So we looked at quite a bit of stuff there.
I hope that helped.
It is fairly complicated structure.