Hi, let's take a look at determining
crystallographic planes.
And we're going to use a set of numbers
similar to what's done in directions, except we
call them Miller indices.
And they're determined differently, obviously,
because they're describing a plane.
So the process is going to start by remembering
that you can translate a plane if necessary,
and it doesn't change the plane as long as it's parallel
to the original.
That's what translation implies.
We're translating it parallel to the original.
But we have to do this step if the plane cross
through our origin.
Translate plane-- so origin is not on the plane.
Of course, another way of stating that same fact is you
could define a new origin.
You can define a new origin that's not on the plane.
The origin is your tool to work with
to use to make your life easier.
So once we've done that, then we can determine distance
to intercept plane, by travelling along each axis from
the origin.
In this step, you have to realise that you may have
infinity introduced.
If the plane is parallel to one of the axes,
you could travel infinitely far along
that axis and never intersect the plane.
So when we've got infinity in there,
we're kind of funny people in material science.
We like to keep a notation a very specific way.
And with fractions, we don't like fractions,
you know that from directions.
And infinity signs are pesky.
They use up a lot of space.
We'd rather express it as 0.
How do you make infinity into 0?
Take the reciprocal.
So we take the reciprocal each and every time.
And even if there's no infinity in the particular plane
that you're working with, you still
take the reciprocal to be consistent.
So you take the reciprocal.
And then we can close in parentheses.
And often, we just use the placeholder,
letters H, K, and L to refer to general Miller indices
for a plane.
If there's any negative signs we move them over the top,
like what we do for directions.
And I'll write that down, just to be complete.
Negatives go above, and no commas.
No commas.
All right.
Good.
So let's take a look at a plane.
Let's take a look at my all time favourite plane.
This plane here cuts off the back
on the left corner of the cube.
So we can actually in this case start
with this conventional origin as our origin.
So what we're going to do is I'm going
to proceed through in this same order, here.
I'm going to determine the intercept.
I'm going to take the reciprocal,
and then I'm going to enclose.
OK, so we're going to do that.
We're going to determine the intercept, take the reciprocal,
and then we're going to enclose for x, y, and z.
So from this origin here, we travel 1 in the x direction.
We would travel from that origin, 1 in the y direction
you hit the plane.
And from this origin here, we would
travel 1 in the z direction and hit the plane.
There's no reductions necessary, and the enclosure then
becomes 1, 1, 1.
This is the 1,1,1 plane.
What about if we take a look at another plane over here?
Perhaps this plane, right here.
All right, so this plane crosses through the y-axis
at 1/2 of the lattice parameter in the y direction.
So again, we're going to go systematically.
I mean, there are shortcuts you can take,
but let's do it systematically at first.
Intercept reciprocal enclosure.
I actually missed a step, and that would be reduction.
And it's going to be important on this step here, reduction
and then enclosure.
So again, we can start with this conventional origin.
We would travel 1 in the x direction, y and z.
We would travel 1/2 lattice parameter in the y, 1/2.
And how far do we travel to z?
Well, it's actually parallel to z.
So we would travel infinitely far.
So our reciprocal gives us 1, 2, 0.
The reciprocal has taken care of the reduction, actually,
so I should have thought of that.
So we didn't need to do that extra reduction step.
If we had needed to, we would have cleared the fraction just
by multiplying across by the lowest common denominator.
And so the enclosure then becomes 1, 2, 0.
That's the 1, 2, 0 plane.
Finally, we could look at families of planes.
And these are planes that look the same two dimensions.
They've got the same atoms in the same positions.
They would have the same linear packing fraction.
So for example, the faces of a cube, well,
the faces of the cube are this front face
is the 0-- sorry, correction.
It's the 1, 0, 0.
You can see the zeros in the y and z positions
tell you it's parallel to y and z.
It's actually in the y-z plane.
It crosses through only the x-axis.
So that specific plane is 1, 0, 0.
This side over here would look the same.
Pick a crystal structure, FCC, DCC.
You draw it in two dimensions, simple cubic.
Draw it in two dimensions.
For that specific crystal system that you've picked,
the atoms will look the same-- the plane will look
the same in two dimensions.
And so that makes it part of the same family of planes.
This plane over here specifically
is 1, 0, 0, and of course the top is 0, 0, 1.
Together, we would call all of those the 0, 0,
1 family of planes.
And for this cube, there's actually
only three specific planes in the 0, 0, 1 family of planes.
We've moved the zeros to the left hand
side, the non-zero values to the right,
and reduced the number of negative signs.
In fact, there's no significance here to negative signs.
You multiply a plane by negative 1, it's the same plane.
So if you can multiply by negative 1
and reduce the number of negative signs for the plane,
you do it.
All right.
That's it.
Thank you very much.