OK.
So what am I gonna do here.
I'm gonna draw something-- I don't know-- like this.
Isn't it Look, it's shiny.
Now you know what it is.
It's a diamond.
OK.
So diamond is 100% carbon.
And we certainly know at room temperature and pressure
it seems to exist-- it actually is not thermodynamically
stable, but we'll get to that.
And we know it's carbon.
It's in fact sp3 hybridised.
So we've got four covalent bonds,
and they're all the same behaviour.
You know, in two dimensions we would sketch it like this.
We'd say well carbon's bonded to four other carbons,
and that continues.
But that is not-- that's two dimensions.
We want to know what this looks like in three dimensions.
And so we have this central carbon atom
with four other carbons coming out
of it and four covalent bonds.
And you know, we can create faces on that, if you will.
In fact, let me draw that for you a little bit bigger.
I'm trying to draw a tetrahedron.
So tetrahedron is this four-sided-- clearly
I'm not an artist-- four-sided.
That's-- yeah, they're supposed to be all exactly the same
shape.
But this is a tetrahedron.
A hedra, tetrahedra.
So it's got four sides.
And the positioning of atoms would be like one right
in the very centre of that thing.
You got one on each corner.
And you've got a covalent bond between each.
So that's the arrangement of bonds that we have.
But our human minds, our human brains,
sometimes like to make things a little bit simpler, if we can,
to visualise.
I mean we could just say that's the structure of a diamond,
and that repeats.
But you can actually go to a few--
when you introduce a few more carbon atoms,
you can actually-- you realize that there's actually
a cubic symmetry to it.
And that's nice, because we like simple geometry.
Oh, look at that, I broke my tetrahedron.
Now I glue it back together.
There we go.
All right.
So what I'm trying to do is I'm going to show you
how this cubic symmetry exists.
So here's a cube.
We're familiar with this.
We've seen this in face-centered cubic.
We've even seen it in the rock salt crystal structure.
And so what it involves is it involves
an atom at each corner.
And I'm going to draw here for you-- depict that there's
1/8 of an atom at each corner.
This is the back here now.
All right, so there's 1/8 of an atom at each corner,
and then there's atoms in face-centered positions.
Front face, top face, right face.
But there's also, of course, the left face hidden,
so I'll dash it in.
Back face, and the bottom face.
And so that gives us four atoms in total.
Four atoms from the FCC positions.
But we need more.
And we need to find spaces where we can fit an atom where
it will be touching four others with this kind
of a tetrahedral coordination, these four atoms nearest to it.
And I think the easiest way to do this is actually just
to show you, and then I'll try to go around and come back
to it and show you different ways to visualise it
to help make it more concrete.
If you look at this-- kind of this region here of this unit
cell.
And by this region, I mean towards
the bottom-- bottom half-- towards the left-hand side,
and towards the front.
Right in the middle of that bottom left front
area is a space.
And I could actually put an atom there, carbon atom there.
And then I could draw these bonds to that left side face,
the front-- sorry, the front bottom left,
the front face centre, and the bottom face centre position.
Now because of the perspective I've drawn this on,
it's a little hard to tell.
But that is, in fact, the same tetrahedral geometry
that I sketched before.
I could try-- I hope this doesn't go too bad,
but I could show you the faces here.
And the last one is in there.
There we go.
OK, so that is, in fact, a tetrahedral site.
So it may be a little hard to tell
that this is, in fact, the same tetrahedron as I had over here.
But you remember the rotate-- the angle that we're looking at
is different.
So this tetrahedron that I drew up here,
if you imagine you kind of rotated
that over so that this atom was in that corner,
this one then maybe became that bottom face.
You could have this atom taking that face position,
and this one here taking that left side face position.
It is, in fact, the same.
But let me show you some other ways to visualise that site,
and then we're going to go back to create finally
the entire diamond cubic crystal structure.
So what I'd like to do next to better show you
where that tetrahedral site exists,
because you might not have followed this, and admittedly
my artistic abilities are somewhat limited, as
hard as I try.
If I cut this thing in half, and front to back
I'm going to slice this cube in half, that's what I've done.
And then I'm going to slice it in the plane
of the page in half again.
And then finally, in this horizontal plane,
I'm going to slice it in half.
So what have I done?
I've now sliced it into eight little--
we'll call them sub-cubes.
Sub-cubes.
That is, they're these little cubes, but they're smaller.
They're within the original unit cell.
I could dot in this top front right one here for you.
I'll dot in the back so you can kind of see.
Well there's a cube, that's one of these sub-cubes.
And, in fact, in the very centre of that sub-cube is this
interstitial site that I'm talking about.
So right in the very centre of that front top right--
label that so we're clear here-- this sub-cube is towards
the front, towards the top, and towards the right.
It's opposite in the front plane from this bottom left front.
So that's where that is.
And then I'll draw the direction of contacting in here.
Maybe you'll see this tetrahedral a little bit more
clearly.
So that is, in fact, tetrahedral.
I could try to make it into a tetrahedral solid for you
by showing the triangular faces.
So that is-- there you go-- that's, again,
the tetrahedral interstitial site.
And I've actually drawn the alternating corners here
on purpose because, in fact, for every carbon
to be bonded to four others, we can only
fill half of these sub-cubes.
So there's eight sub-cubes, but only
half-- that is, four of them-- are filled.
A name for a space between other atoms is interstitial site.
It's a little bit unusual to refer
to an interstitial site in diamond,
because they're all-- every atom is carbon.
But if you start thinking about this as a space
that you could fit an atom between others,
it'll help you in other crystal structures that
have some similarities to this.
So let's continue on again.
We've spent a little time on this,
and I'm going to sketch for you another unit cell, a fresh one,
and we'll go ahead and identify which ones have carbon in them.
And then you'll know the diamond cubic crystal structure
like the back of your hand.
So here's a cube.
Dash in the back for you there.
And this time, I'm actually going to go ahead
and I'm going to cut the sub-cubes for you,
as we did before.
So I've got eight little identical cubes
making up the greater, the larger, unit cell.
And what I showed you on the previous slide
is I've got, in this front bottom left, a carbon.
In the front top right is a carbon.
And then what happens is these alternate in space
so that they're spaced out as far apart as they can.
That is, they wouldn't-- all four of them would not occupy
the bottom half of the unit cell or the right side.
They're going to alternate.
So they alternate top to bottom, left to right here.
Well in the back of the unit cell,
they're going to do the same thing.
So that means the back top left site will be filled,
and the back bottom right.
So this is-- sorry, the back top left and the back bottom right
will be filled.
So they alternate.
And a student a few years ago showed me a little trick
that was useful for him.
And if you just sort of picture those little sub-cubes--
and what I do for you is I shade in the exposed,
the free surfaces of the sub-cubes
that contain an occupied tetrahedral site--
you'll see the positioning of the tetrahedral sites perhaps
a little bit more clearly.
So there you go.
Those are the occupied tetrahedral sites,
or the positions of those carbon atoms
in amongst all the other carbon atoms in diamond cubic.
And so finally what I'll do is I'll just
position the FCC positioned ones for you.
And back in that corner there, top front right side.
And maybe for simplicity I won't dot
in the ones that are hidden for completeness.
So that is diamond cubic.
And this tetrahedral symmetry is a really important one.
It comes up in organic molecules,
it comes up in many semiconductor devices.
So understanding diamond cubic is a really good foundation
for proceeding to other crystal structures,
and other structures.