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ROFESSOR CIMA: OK, so today how do we find our way around a crystal?
Well, there's a couple of conventions. The first is direction.
And we start with direction, making sure that we understand
the principal axes. So if we have a cubic system, we always start
with a right-hand set of axes.
So what I mean by right-hand again is, if that's my x direction, I put my
right hand in the direction of x, fingers pointing in the direction of
positive x. And then I rotate my right hand into y, and
my thumb will be pointed in the z direction.
So that's a right-handed set. Now if I draw a cubic lattice on this roughly here, we define a direction as
being a vector. So let's do this vector here, going out one
unit on the x direction. And we will describe this vector by its endpoint.
So in other words, since this goes to the point 1-0-0, we call that
direction [100]. You notice I have these square-looking brackets.
And likewise, if I want to do-- maybe I should do a different color just to--
if I want to do this, you can see that that goes to [010].
Oh, let's not confuse everybody. That goes to the point 0-1-0.
So that guy is the vector [010]. So pretty simple.
The one that will go across the diagonal then is like this.
And that goes to 1-1-1. So its direction is the [111]
vector. That's pretty trivial.
The part that gets a little confusing where we have another convention is
when I pick a vector that goes like this, across that
back face, for example. So what's its direction?
Well, what we do-- or what I like to do--
there's two things you can do. I've seen students do it two different ways,
and both give you the same result.
The first is just to pick up this vector and not let it change direction
and move it so that the starting point is at your origin.
So if that's the case, you kind of have to think in your mind the
extension of the z and the y-axis. And I'm going to put this down here-- and
unfortunately it's kind of superimposed--
but you can see what I'm doing. It's going to go down to a point that's 0
in x, -1 in y, right? Back.
And then -1 in z. And this will be what we call our direction.
So that'll be the zero. Now, because we don't put the minus sign here
in our convention, what we do is we put it on top.
This is what we'll do throughout the crystallographic notation.
It's called a macron. It's called a macron.
I got to do the z here. I didn't do that one.
It's got to go-- that's got to have a -1 also.
The other way to do this, and it's equivalent, is to pick up the axes and
move the axes up here so that I have, well, I guess I'll call it x prime, y
prime, and z prime. Same thing.
I just picked up the axes. And then, of course, in terms of this origin,
put the origin at the base here, I get the same direction.
Now, there was a question right up here. Did I answer it?
STUDENT: Yeah. PROFESSOR CIMA: Yeah, OK, got it.
Oh, I got another example here. Let's--
this has gotten a little too crowded. Let's do another one here.
I've done sort of unit. So, here's my x, y, and z.
So what about this one here? It doesn't end up conveniently on a lattice
point. In fact, it ends up--
well, it's 1-- comes out 1, 0 on the y, and 1/2 on the z.
So here what you do is, you do not do 1- 0 -1/2.
That's a no. What you do is, you multiply this by the smallest
integer to get rid of all the fractions.
So, obviously, what you're going to do is multiply by two.
So that direction is the [201]. The [201].
Another way to think of this, if I were to extend this out.
And I'm afraid I didn't do a very good job. I meant to be 1/2, but let's say--
there it is. That's closer to 1/2.
If I draw the unit cell in front here and extend this out just by
multiplying by a scalar, it does intersect a unit cell point at x of 2,
y of 0, and z of 1. So there's two ways to think of this.
One is, you can write this down and then clear the fractions by
multiplying by a constant. Or you can just, in your mind, extend this
out to a point where it does hit integer values of the lattice.
Either way works.