PROFESSOR CIMA: So Bravais started thinking about--
OK, that's fine. This is what the unit cell has to look like.
Where did the atoms go? Am I free to put the atoms anywhere?
So he was concerned with what's called a lattice. And what does that mean?
Well, Bravais took a look at these simple shapes in two dimensions and
realized, well, that's pretty easy, at least for some of them.
If I put-- Look at the square one here.
If I put an atom, or I put a lattice point at each of the corners, I will--
Each lattice point is exactly the same as the other.
So here's my space filling unit cell. And I can arrange atoms at each of the corners
and it will fill space. And the same is true of the rectangle here.
Same is true there. Now here's an interesting one.
He said-- But it gets a little bit more complicated,
because I can't put a lattice point in the middle.
Why do these two differ? Well, this one, the four nearest neighbors
with different distances, right?
But here, one in the center, the four nearest neighbors of every lattice
point, are equal distant from each other. So in other words, from this one rectangular
space group-- I mean not space groups--
Lattice -- Oh, I'm going to be a careful.
OK. So from this one rectangular space filling
unit cell, I can put lattices on it, two different lattices on it, that
have the same symmetry. In other words, they're both rectangular space
filling unit cells, but one has got an atom centered in the center
and the other one doesn't. That's--
OK. So now let's go to three dimensions.
Oh, no. Sorry, here's another one.
So what do we put at those-- So what's so cool about these points?
What are at these points? Are the atoms?
Sometimes, they're atoms but other times they're not.
So before I go on to three dimensions, I want to look at this.
Here's another tile. And you want to look at this very carefully
and see. OK.
What is the unit cell? Well, the unit cell is just that one.
You can see this is repeated here, is repeated here, and is repeated here.
So this is our unit cell in this pattern. What's at this lattice point, though?
What's at that lattice point is that. See?
These eight tiles are all arranged associated with that lattice point.
And then when I go to this lattice point, which eight tiles is it?
Those. And what about this point?
Those. So we call this the basis.
It's the thing that's at the lattice point. And it doesn't necessarily need to be just
an atom. It could be a molecule.
So in this case the basis is actually these eight tiles, all associated with
that lattice point. So it's just easier to see it.
Before we get into three dimensions, you should understand the two
dimensional one. That's the basis.
Oh, this one's really wild. What's the basis here?
So there's the unit cell. It turns out the basis is that.
