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PROFESSOR CIMA: So now we know what the d is, though.
The d is the distance between these planes. So let me rewrite Bragg's Law.
And just so I don't have to carry around the n all the time, we're just
going to assume the n equals 1 for here. But just keep it in mind.
Every time you see a lambda, you should be able to put an n in there.
2 * d * sine theta. So the first diffraction peak we're looking
at here. Now, you know from our past lectures that
--hkl-- the spacing between planes is related to their
Miller indices. And that's why we spent all that time worrying
about Miller indices. Because you can see right now-- as soon as
I can plug in here-- if I know what x-ray radiation I'm using,
I can calculate the angle by which it diffracts.
STUDENT: Professor. PROFESSOR CIMA: Yo.
STUDENT: Yes, I had a question. On the drawing in the middle board, if those
planes of atoms reflect the x-rays, what stops the ray that reflects at
B prime from reflecting back into the crystal at C?
PROFESSOR CIMA: Well, I think that's a very legitimate question.
But let me tell you a little secret about this.
What's actually radiating are the atoms, not these planes.
It's just a mathematical convenience to call them planes.
So there's another way to treat the math, which unfortunately doesn't
allow you to make a nice little drawing like this.
But it imagines that each one of these atoms radiates, spherically, a ray
when it'***. And it turns out you can write those as sums
of exponentials-- exponentials that look a little funny.
They have a square root of minus 1 in their exponent.
Remember I said, what's the square root of minus 1?
It's just a phase. It moves the sine to cosine.
If you multiply it by square root of minus 1, that's all it's doing.
And if you sum up all those radiators-- it really is just adding up all those exponentials--
you end up with exactly the same relationship. So in that formalism, it's doing exactly what
you said. It doesn't care where the radiation's going.
But in only certain directions do they all add up.
So I know that's a long-winded explanation and not entirely
intellectually satisfying. But that's really what's happening.
So let's apply this. So the next step is, we're going to plug this
in here. And we're going to put everything that's a
constant in our experiment on the left and everything that's sort of a variable
on the right. So in other words, when I put this in here
and I want to get rid of that square root, you end up with (sine squared)
theta_hkl. So, in other words, there's an angle for every
specific plane that we're talking about.
And then I'll do this. So for any particular experiment, I'm going
to use a specific wavelength of radiation of x-rays and I'm going to use a
specific crystal. That should be an a.
That's the lattice constant. And so this over here, for any given experiment
now, is a constant. And so what that says is, as I change h squared
plus k squared plus l squared-- when I go from one plane to the
next-- I'm going to find the diffraction from those
planes at different angles. In fact, you can see, in general, as h squared
plus k squared plus l squared increases, this increases.
The angle increases.