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PROFESSOR: OK.

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We're still on Exam 2 of
the fall 2009 class.

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We're going to be doing
problem number 2 now.

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This is the x-ray problem,
as I like to call it.

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It's kind of exciting.

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Let's talk about what we need
to know before we actually

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reasonably try to attempt
this problem.

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So I would review these
things again.

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We want to know emission line
nomenclature, how to name

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emission lines.

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We want to know Moseley's Law.

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We want to know Bragg's Law,
and I would also emphasize,

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I'll get onto it later, you
want to know how to derive

00:00:50.250 --> 00:00:52.420
Bragg's Law as well.

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We want to know, this is
pronounced Bremsstrahlung,

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it's a German word which means
breaking radiation.

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So Bremsstrahlung radiation.

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And we also want to know some
reflection rules, and I'll

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sort of elucidate
that a bit more.

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But let me start you off
in that direction.

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So getting on to part A, we
are told that somebody has

00:01:10.390 --> 00:01:14.080
been horsing around with
our x-ray machine.

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We've all had this
problem before.

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And they've changed
the target.

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So let me first tell you what a
target is, and how the x-ray

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machine looks.

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Then we'll be able to understand
what exactly it is

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we're trying to figure out.

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An x-ray machine, basically
the x-ray source involves

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electrons being accelerated
into a

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target material, a metal.

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And then that gives off a whole
bunch of electromagnetic

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radiation, so photons.

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Some of those photons we'll talk
a little about later, in

00:01:40.610 --> 00:01:42.530
the Bremsstrahlung spectrum,
are very

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characteristic of the material.

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So I'm going to call, you know,
we have long wavelength

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photons, short wavelength
photons.

00:01:49.150 --> 00:01:52.980
Some of the photons have a very
characteristic wavelength

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or frequency that corresponds
to a particular electron

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transition.

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We're going to talk more about
this in a little bit, but I'm

00:02:00.320 --> 00:02:03.030
going to call this the
K-alpha photon, OK?

00:02:03.030 --> 00:02:07.680
So that just happens to
be in metals, x-ray.

00:02:07.680 --> 00:02:13.310
We know that for particular
materials, we have particular

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K-alpha wavelengths.

00:02:15.070 --> 00:02:17.710
And what's basically happened
in this problem is that, you

00:02:17.710 --> 00:02:20.940
know, maybe we knew what this
material was before, and then

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all of a sudden we weren't
around, somebody switched it

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on us, and now we have to
figure out what it is.

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Don't you hate when
that happens?

00:02:29.480 --> 00:02:35.180
So the best way to approach this
problem is to understand

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Moseley's Law, and then have
it on your equation sheet.

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We allow all of our students to
have an equation sheet, and

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also a periodic table.

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So don't panic if you can't
memorize this equation.

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Our students are expected to,
not to memorize it, but to

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understand it.

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So this is Moseley's Law.

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And Moseley's Law basically
tells us, let me go through

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these variables.

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We have the wave number.

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We have a Rydberg constant.

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This is a bunch of physical
constants, sort of

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agglomerated into one big one,
to save us some time.

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We have the final energy level,
where the electron

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ends, and we have the initial
energy level where it begins.

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And then we have z, which
designates our element.

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So it's the number of protons
we have in the

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nucleus, for example.

00:03:20.540 --> 00:03:22.890
And then we have this sigma.

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OK?

00:03:23.710 --> 00:03:26.320
So, you know, in order to solve
this problem, we want to

00:03:26.320 --> 00:03:27.645
know what element.

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So obviously, we're
looking for z.

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Now the question is, do we
know everything else?

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So yeah, we definitely do.

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Let's just go through each
individual thing that we know.

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So we know Rydberg constant.

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That's just, you know, have that
on your equation sheet.

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It's a bunch of constants
agglomerated together.

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We know that, for K-alpha
we're told,

00:04:04.380 --> 00:04:06.700
this is K-alpha radiation.

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Let me draw a cartoon
for you here.

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OK?

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This is our nucleus, OK?

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This is n equals 1, this
is n equals 2.

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I'm going to go kind
off the board.

00:04:20.340 --> 00:04:21.420
This is 3.

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OK?

00:04:23.480 --> 00:04:29.120
And K-alpha corresponds to an
electron dropping down from n2

00:04:29.120 --> 00:04:33.290
to n1 to here, and then
giving off a photon.

00:04:33.290 --> 00:04:37.260
So I'm going to draw a photon as
a squiggly line like this.

00:04:37.260 --> 00:04:38.055
OK?

00:04:38.055 --> 00:04:39.700
So this is going to be h nu.

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This will give you our
K-alpha radiation.

00:04:44.570 --> 00:04:45.640
So we know all of
those things.

00:04:45.640 --> 00:04:47.760
We know our wave number here.

00:04:47.760 --> 00:04:49.230
So we know pretty
much everything.

00:04:49.230 --> 00:04:50.520
But what is sigma?

00:04:50.520 --> 00:04:54.590
Well, sigma is the correction
factor that Moseley found.

00:04:54.590 --> 00:04:56.530
Now, if you're looking at this
and you're thinking, wait,

00:04:56.530 --> 00:04:59.610
this looks a lot like the
Rydberg equation, it's because

00:04:59.610 --> 00:05:00.640
it's sort of is.

00:05:00.640 --> 00:05:02.500
The only difference is that the
Rydberg equation is only

00:05:02.500 --> 00:05:05.040
for hydrogenic-type atoms. OK?

00:05:05.040 --> 00:05:07.170
So we're talking about hydrogen,
we're talking about

00:05:07.170 --> 00:05:09.670
helium plus 1, lithium plus 2.

00:05:09.670 --> 00:05:12.490
We can't assume that our target
is hydrogenic, so we

00:05:12.490 --> 00:05:14.870
can't use the Rydberg
equation.

00:05:14.870 --> 00:05:17.260
So we're using Moseley's
equation.

00:05:17.260 --> 00:05:20.650
And what Moseley found, just
to show you where the sigma

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sort of comes from, and this
is actually reviewed in, I

00:05:23.930 --> 00:05:27.630
think, lecture 17 of the
online postings.

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What Moseley found was that if
you plot your wave number

00:05:32.990 --> 00:05:38.110
versus your z, you've got
points like this.

00:05:40.680 --> 00:05:41.930
You've got some sort of line.

00:05:46.120 --> 00:05:46.946
OK?

00:05:46.946 --> 00:05:49.220
So this is what Moseley found.

00:05:49.220 --> 00:05:50.790
And what Moseley basically
did, was he fit

00:05:50.790 --> 00:05:51.553
an equation to it.

00:05:51.553 --> 00:05:54.140
And the equation looks a lot
like the Rydberg equation.

00:05:54.140 --> 00:05:56.480
And because these don't
intersect the origin, we've

00:05:56.480 --> 00:05:59.910
got this fitting factor
right here.

00:05:59.910 --> 00:06:03.920
We know that for a K-alpha
transition, we're looking at a

00:06:03.920 --> 00:06:05.230
sigma equals 1.

00:06:09.430 --> 00:06:12.270
OK?

00:06:12.270 --> 00:06:14.250
So we're good.

00:06:14.250 --> 00:06:15.060
We know our sigma.

00:06:15.060 --> 00:06:15.820
We know our n's.

00:06:15.820 --> 00:06:18.150
We know n final is
going to be 1.

00:06:18.150 --> 00:06:20.840
We know n initial is
going to be 2.

00:06:20.840 --> 00:06:23.000
We know the Rydberg constant,
and we know our wave number,

00:06:23.000 --> 00:06:25.030
because we're given our
wavelength here.

00:06:25.030 --> 00:06:27.390
So we can go ahead and actually
just calculate z.

00:06:27.390 --> 00:06:28.630
And let's do that right now.

00:06:28.630 --> 00:06:31.185
So let me rewrite this
equation so that

00:06:31.185 --> 00:06:32.435
it makes more sense.

00:06:39.120 --> 00:06:41.790
Rydberg constant.

00:06:41.790 --> 00:06:42.540
1 over.

00:06:42.540 --> 00:06:43.610
our n final.

00:06:43.610 --> 00:06:48.090
That's 1 squared minus
1 over, our

00:06:48.090 --> 00:06:49.920
initial point is 2 squared.

00:06:52.880 --> 00:06:56.920
We're looking for our z, and
we're going to subtract off 1,

00:06:56.920 --> 00:06:59.010
because we're dealing with
a K-alpha transition.

00:06:59.010 --> 00:07:01.760
So you're expected to either
remember that, or have it on

00:07:01.760 --> 00:07:02.760
your question sheet.

00:07:02.760 --> 00:07:05.210
There's no way to derive that
unless you've got all this

00:07:05.210 --> 00:07:07.240
data, which you won't have.

00:07:07.240 --> 00:07:10.160
So we've got this squared.

00:07:10.160 --> 00:07:14.160
We know everything, so we can
basically reduce this.

00:07:14.160 --> 00:07:16.090
This is why on the answer
sheet, you see

00:07:16.090 --> 00:07:17.900
something like this.

00:07:17.900 --> 00:07:19.440
You kind of skip all
these steps, and we

00:07:19.440 --> 00:07:21.570
went right to this.

00:07:21.570 --> 00:07:24.990
3/4 z minus 1 squared.

00:07:24.990 --> 00:07:26.490
And now you can just
solve for z.

00:07:26.490 --> 00:07:27.740
No problem.

00:07:43.660 --> 00:07:44.070
OK?

00:07:44.070 --> 00:07:48.600
So that's 4 over 3 lambda
Rydberg constant to

00:07:48.600 --> 00:07:50.125
the half plus 1.

00:07:50.125 --> 00:07:53.090
And that's going to
give you about 23.

00:07:53.090 --> 00:07:54.290
You might get 22.9.

00:07:54.290 --> 00:07:55.530
You might get 23.1.

00:07:55.530 --> 00:07:57.010
We're talking about 23.

00:07:57.010 --> 00:07:59.260
That's going to be
the vanadium.

00:07:59.260 --> 00:08:02.950
That's the answer to the first
part of this problem.

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OK.

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We're going to take a second and
clean up, and come right

00:08:05.440 --> 00:08:06.690
back and finish the problem.

00:08:09.610 --> 00:08:12.130
We're going to start part
B now of problem 2

00:08:12.130 --> 00:08:14.340
on the second exam.

00:08:14.340 --> 00:08:18.780
So we basically have used the
Moseley equation to find out

00:08:18.780 --> 00:08:19.470
an element.

00:08:19.470 --> 00:08:21.200
And now we're going to switch
the problem up a little bit

00:08:21.200 --> 00:08:25.230
more, and we're going to ask
you to find the smallest

00:08:25.230 --> 00:08:27.590
defraction angle for a
particular situation.

00:08:27.590 --> 00:08:29.210
So what's the situation?

00:08:29.210 --> 00:08:31.630
Let's go back to our
x-ray machine.

00:08:31.630 --> 00:08:34.690
The way you actually do x-ray
defraction, you know, it's

00:08:34.690 --> 00:08:37.210
acronymized XRD.

00:08:37.210 --> 00:08:40.130
The way you do it, is you
have to generate x-rays.

00:08:40.130 --> 00:08:42.790
So we accelerate an electron
into some material.

00:08:42.790 --> 00:08:46.130
That material gives off a
whole bunch of photons.

00:08:46.130 --> 00:08:49.320
You then filter all the photons,
so you only pick up a

00:08:49.320 --> 00:08:50.980
certain wavelength.

00:08:50.980 --> 00:08:53.510
And we're going to use the
K-alpha wavelength.

00:08:53.510 --> 00:08:57.820
And then you get these K-alpha
radiation, which then probes

00:08:57.820 --> 00:08:58.710
your sample.

00:08:58.710 --> 00:08:59.720
So here we go.

00:08:59.720 --> 00:09:01.990
This is what we're looking
at for this part.

00:09:01.990 --> 00:09:06.480
We've got K-alpha radiation
coming in, and it's hitting

00:09:06.480 --> 00:09:09.460
our structure, which has
a crystal, hopefully.

00:09:09.460 --> 00:09:11.040
You're going to do XRD.

00:09:11.040 --> 00:09:13.660
And then you're getting off
some defraction, anything

00:09:13.660 --> 00:09:16.480
that's reflecting off
of the crystal in

00:09:16.480 --> 00:09:17.730
some way, like this.

00:09:17.730 --> 00:09:20.900
So the thing you need to know
to do part B, is you need to

00:09:20.900 --> 00:09:23.570
know Bragg's Law, Bragg's
equation.

00:09:23.570 --> 00:09:28.280
And that's number 3 on the win
list. So I drew this picture

00:09:28.280 --> 00:09:31.170
here just to define the
variables, but one thing I

00:09:31.170 --> 00:09:35.040
would stress for you at home, is
to please, go home and, you

00:09:35.040 --> 00:09:37.570
know, search online, Google
image search even.

00:09:37.570 --> 00:09:38.690
Look for a picture like this.

00:09:38.690 --> 00:09:41.020
Search Bragg's Law and look
for a picture like this.

00:09:41.020 --> 00:09:42.840
And it's actually from this.

00:09:42.840 --> 00:09:45.110
It's very easy, just
using geometry, to

00:09:45.110 --> 00:09:46.130
derive Bragg's Law.

00:09:46.130 --> 00:09:48.470
Bragg's Law is just the
geometric interpretation of

00:09:48.470 --> 00:09:49.400
this picture.

00:09:49.400 --> 00:09:50.250
OK?

00:09:50.250 --> 00:09:54.570
So Bragg's Law, in its
traditional format, is n

00:09:54.570 --> 00:09:56.650
lambda equals 2d sine theta.

00:09:56.650 --> 00:09:58.510
And I've defined, I drew
the picture to show

00:09:58.510 --> 00:10:00.910
the variables here.

00:10:00.910 --> 00:10:02.310
We have n, which is--

00:10:02.310 --> 00:10:03.430
we'll talk about
n in a second.

00:10:03.430 --> 00:10:06.900
We have lambda, which is
our incoming radiation.

00:10:06.900 --> 00:10:11.030
We have d, which is the spacing
between planes.

00:10:11.030 --> 00:10:14.560
Whatever plane orientation
you're looking at.

00:10:14.560 --> 00:10:18.070
2d is the spacing, is there
because you need to come in,

00:10:18.070 --> 00:10:18.785
and then come out again.

00:10:18.785 --> 00:10:22.510
So you're doing twice the
distance between the planes.

00:10:22.510 --> 00:10:25.830
Theta is the angle
at which you're

00:10:25.830 --> 00:10:27.920
incident on the material.

00:10:27.920 --> 00:10:30.180
And n is going to be your
defraction index.

00:10:30.180 --> 00:10:32.030
We're not going to worry about
that in this class.

00:10:32.030 --> 00:10:35.120
We're going to assume it's
one, for this class

00:10:35.120 --> 00:10:36.460
and for this exam.

00:10:36.460 --> 00:10:39.670
So the question is asking us to
find the smallest possible

00:10:39.670 --> 00:10:41.010
angle of defraction.

00:10:41.010 --> 00:10:43.590
What that means, is that we're
getting all this radiation,

00:10:43.590 --> 00:10:44.970
it's hitting our material,
we're moving

00:10:44.970 --> 00:10:45.930
our material around.

00:10:45.930 --> 00:10:48.680
The question is, what's the
smallest angle at which we're

00:10:48.680 --> 00:10:52.710
going to see a peak
on our XRD graph?

00:10:52.710 --> 00:10:56.120
I rearranged the equation here,
so theta equals the

00:10:56.120 --> 00:10:58.550
inverse sign of lambda
over 2d.

00:10:58.550 --> 00:11:02.340
And we're looking for as small
a theta as possible.

00:11:02.340 --> 00:11:05.070
This question is really just
mathematical manipulation.

00:11:05.070 --> 00:11:07.340
I need to know a couple
important details that

00:11:07.340 --> 00:11:11.510
hopefully you've read
about in advance.

00:11:11.510 --> 00:11:14.160
So we want to find the smallest
theta possible.

00:11:14.160 --> 00:11:16.660
That's the objective
for this problem.

00:11:16.660 --> 00:11:20.360
I've drawn the arc sine
function here.

00:11:20.360 --> 00:11:24.600
To minimize theta, we want to
have the argument, which I

00:11:24.600 --> 00:11:26.900
call x here, we want
the argument to

00:11:26.900 --> 00:11:28.450
be as small as possible.

00:11:28.450 --> 00:11:32.700
The smaller it is, the closer
to 0 you are for y, which is

00:11:32.700 --> 00:11:34.910
theta, in our case.

00:11:34.910 --> 00:11:38.010
So we want to minimize the
term in parentheses.

00:11:38.010 --> 00:11:42.710
And the way you do that, small
theta means small lambda over

00:11:42.710 --> 00:11:46.410
2d, which means you want
to have a large d.

00:11:46.410 --> 00:11:49.820
And the reason I'm not using
lambda, the wavelength, as the

00:11:49.820 --> 00:11:52.620
knob, is because
lambda is set.

00:11:52.620 --> 00:11:56.460
Lambda is predetermined by the
target material you've chosen.

00:11:56.460 --> 00:12:00.060
So for our problem,
we have titanium

00:12:00.060 --> 00:12:01.360
as our target material.

00:12:01.360 --> 00:12:03.900
Which is, we can't change
the lambda K-alpha.

00:12:03.900 --> 00:12:05.740
It's set by that material.

00:12:05.740 --> 00:12:07.730
So we have lambda K-alpha--

00:12:07.730 --> 00:12:08.960
we have k-Alpha coming
off, which is

00:12:08.960 --> 00:12:10.980
characteristic of titanium.

00:12:10.980 --> 00:12:14.500
So we can't use lambda
as a knob.

00:12:14.500 --> 00:12:17.430
The only thing we can mess
around with here, to figure

00:12:17.430 --> 00:12:19.660
out this problem, is we
have to look at d.

00:12:19.660 --> 00:12:22.800
So we want to have a large d
to minimize this, and to

00:12:22.800 --> 00:12:25.680
minimize your theta.

00:12:25.680 --> 00:12:27.490
To get a large d, we
need to know the

00:12:27.490 --> 00:12:29.770
equation for planar spacing.

00:12:29.770 --> 00:12:31.100
OK?

00:12:31.100 --> 00:12:32.620
This is your equation
for planar spacing.

00:12:32.620 --> 00:12:35.890
And as we sort of talked about
in problem 1, in the prior

00:12:35.890 --> 00:12:41.205
video, d planar spacing is the
distance between one plane of

00:12:41.205 --> 00:12:43.120
a particular orientation, and
another plane of the same

00:12:43.120 --> 00:12:44.670
exact orientation.

00:12:44.670 --> 00:12:44.980
OK?

00:12:44.980 --> 00:12:48.470
So we're looking at maybe a 011
plane in this unit cell,

00:12:48.470 --> 00:12:50.920
and a 011 plane in
this unit cell.

00:12:50.920 --> 00:12:52.590
What's the distance
between the two?

00:12:52.590 --> 00:12:54.740
That's what this d is.

00:12:54.740 --> 00:12:55.170
OK?

00:12:55.170 --> 00:12:58.230
So we want to have a large d.

00:12:58.230 --> 00:13:02.050
And let's look at what we have
in the equation for d.

00:13:02.050 --> 00:13:03.380
We have these hkl.

00:13:03.380 --> 00:13:07.350
These are the coefficients the
Miller indices of the plane.

00:13:07.350 --> 00:13:09.540
And we have a, which
is the lattice

00:13:09.540 --> 00:13:11.330
constant of the material.

00:13:11.330 --> 00:13:12.500
We're looking at--

00:13:12.500 --> 00:13:13.470
what are we looking at here?

00:13:13.470 --> 00:13:15.750
We're looking at tantalum.

00:13:15.750 --> 00:13:17.900
We can't change the lattice
constant for constant

00:13:17.900 --> 00:13:19.100
temperature and pressure.

00:13:19.100 --> 00:13:20.720
So this is nonnegotiable.

00:13:20.720 --> 00:13:22.670
We can't change what a is.

00:13:22.670 --> 00:13:26.120
The only things that we can
toggle, the only switch we can

00:13:26.120 --> 00:13:28.540
toggle now, is h, k, and l.

00:13:28.540 --> 00:13:31.580
So the only thing we can do,
is look at different plane

00:13:31.580 --> 00:13:32.750
orientations in the crystal.

00:13:32.750 --> 00:13:35.350
So this is the logic I used
to go through the problem.

00:13:35.350 --> 00:13:37.850
So you want to have as small
an h, k, and l as possible,

00:13:37.850 --> 00:13:42.340
because the smaller these are,
the larger d is, et cetera.

00:13:42.340 --> 00:13:45.020
So the final piece of
information where we have to

00:13:45.020 --> 00:13:50.610
be clever is knowing how to get
the smallest h, k, and l.

00:13:50.610 --> 00:13:52.630
We're told that we're
looking at tantalum.

00:13:52.630 --> 00:13:56.020
And the students in this class
have access to a periodic

00:13:56.020 --> 00:13:58.130
table, which has a vast quantity
of information about

00:13:58.130 --> 00:13:59.520
all the elements in it.

00:13:59.520 --> 00:14:01.140
One of the things it
has is the crystal

00:14:01.140 --> 00:14:02.780
structure of pure elements.

00:14:02.780 --> 00:14:07.810
So for tantalum, we're looking
at a bcc structure.

00:14:10.350 --> 00:14:13.020
So we're looking at bcc.

00:14:13.020 --> 00:14:17.450
bcc, from your reflection rules,
which is the number 5

00:14:17.450 --> 00:14:19.170
on the things we need to know
to do this problem--

00:14:21.930 --> 00:14:27.430
for bcc, you have to have your
h plus your k plus your l--

00:14:27.430 --> 00:14:28.680
let me write it over here--

00:14:31.220 --> 00:14:34.130
h plus k plus l--

00:14:34.130 --> 00:14:35.430
must be even.

00:14:38.130 --> 00:14:40.200
That's a rule, OK?

00:14:40.200 --> 00:14:41.960
Proving that is a little bit
beyond the scope of this

00:14:41.960 --> 00:14:44.300
class, but that's a rule that
we went over in class, and

00:14:44.300 --> 00:14:46.270
it's in the lectures as well.

00:14:46.270 --> 00:14:47.860
So we have a couple
things now.

00:14:47.860 --> 00:14:51.000
We want to minimize h, k, and
l, and they have to be even.

00:14:51.000 --> 00:14:53.650
So basically, what that means
is, we have to go through a

00:14:53.650 --> 00:14:54.830
couple permutations.

00:14:54.830 --> 00:14:55.540
So let's think about it.

00:14:55.540 --> 00:14:56.520
0 0 0.

00:14:56.520 --> 00:14:58.940
That's even, but that doesn't
really correspond to a plane.

00:14:58.940 --> 00:15:02.750
That doesn't make, you know,
that's not going to help us.

00:15:02.750 --> 00:15:05.190
1 0 0, or 0 1 0.

00:15:05.190 --> 00:15:06.590
So let me write some
of those down.

00:15:06.590 --> 00:15:08.800
Let's look at, you
know, 1 0 0.

00:15:08.800 --> 00:15:11.410
Well, that's not even, so
we can't look at that.

00:15:11.410 --> 00:15:14.780
So what's the next smallest
thing we could go?

00:15:14.780 --> 00:15:16.030
How about 1 1 0?

00:15:21.050 --> 00:15:23.540
Those are the smallest
coefficients we can have for a

00:15:23.540 --> 00:15:25.440
plane which has the rule
that the h plus k

00:15:25.440 --> 00:15:27.770
plus l must be even.

00:15:27.770 --> 00:15:31.200
So this could be 1 1 0, this
could be 1 0 1, this

00:15:31.200 --> 00:15:32.335
could be 0 1 1.

00:15:32.335 --> 00:15:34.790
But what we're basically talking
about, if you remember

00:15:34.790 --> 00:15:43.300
from the first problem, is the
family of 1 1 0 planes.

00:15:43.300 --> 00:15:46.130
So I plugged in just 1
1 0 for h, k, and l.

00:15:46.130 --> 00:15:52.480
You get a d of 3.31 over
the square root of 2.

00:15:52.480 --> 00:15:55.670
And then with that d, you know
that you've just maximized the

00:15:55.670 --> 00:15:59.270
size of your d, which means
we've minimize the size of

00:15:59.270 --> 00:16:04.640
lambda over 2d, and that means
we've minimized our theta.

00:16:04.640 --> 00:16:07.200
And now for theta, we can easily
calculate that we're

00:16:07.200 --> 00:16:11.650
looking at an angle, if we plug
in for d, which is 3.31

00:16:11.650 --> 00:16:14.180
over the square root
of 2, here.

00:16:14.180 --> 00:16:21.250
And if we plug in for lambda,
which is given to us as 2.75

00:16:21.250 --> 00:16:30.160
angstroms, we get an answer
of 36 degrees.

00:16:30.160 --> 00:16:32.735
Put that right in the middle.

00:16:32.735 --> 00:16:35.370
And I put in the middle to
emphasize that this is sort of

00:16:35.370 --> 00:16:37.990
the logic you have to go
through, sort of a, you know,

00:16:37.990 --> 00:16:41.470
circular logic, to get
to that answer.

00:16:41.470 --> 00:16:43.370
So that's the answer
to part B.

00:16:43.370 --> 00:16:45.060
And remember, this is
pretty much math.

00:16:45.060 --> 00:16:47.360
Just manipulation until the very
end, where we discussed

00:16:47.360 --> 00:16:50.380
the reflection rules for bcc.

00:16:50.380 --> 00:16:53.470
I want to go on to the last part
now, and I'm just going

00:16:53.470 --> 00:16:55.690
to erase this.

00:16:55.690 --> 00:16:56.940
We only need a little
bit of room.

00:17:00.180 --> 00:17:02.510
This last question was actually
my favorite question

00:17:02.510 --> 00:17:04.970
in the entire class.

00:17:04.970 --> 00:17:07.340
And actually, we only had
3 students get it

00:17:07.340 --> 00:17:09.840
right, out of 480.

00:17:09.840 --> 00:17:12.980
So this was a great question,
and if you get it right, then

00:17:12.980 --> 00:17:14.610
you're doing really
well in the class.

00:17:14.610 --> 00:17:17.940
So let's look at C.

00:17:17.940 --> 00:17:20.400
It's really a chatty question.

00:17:20.400 --> 00:17:22.240
We're not looking
for a number.

00:17:22.240 --> 00:17:27.180
The question basically asks us,
we want you to talk about,

00:17:27.180 --> 00:17:30.620
to sketch the emission spectrum
of an x-ray target

00:17:30.620 --> 00:17:33.780
that's bombarded by photons
instead of electrons.

00:17:33.780 --> 00:17:37.820
So let's go back over to what
we said in the beginning.

00:17:37.820 --> 00:17:41.920
When we generate x-rays,
generally the way we do it, is

00:17:41.920 --> 00:17:45.970
we accelerate an electron
into a target material.

00:17:45.970 --> 00:17:47.750
In part B, the target material
was titanium,

00:17:47.750 --> 00:17:49.010
but it could be anything.

00:17:49.010 --> 00:17:51.350
A very common target
material is copper.

00:17:51.350 --> 00:17:55.380
copper K-alpha is a very
standard target material, or

00:17:55.380 --> 00:17:57.490
wavelength to use.

00:17:57.490 --> 00:18:00.945
You accelerate your electron,
and you generate the spectrum

00:18:00.945 --> 00:18:03.080
of electromagnetic radiation.

00:18:03.080 --> 00:18:03.490
OK?

00:18:03.490 --> 00:18:04.400
So this is our spectrum.

00:18:04.400 --> 00:18:08.080
We have large wavelength, we
have small wavelength, we have

00:18:08.080 --> 00:18:09.580
some wavelengths
in the middle.

00:18:09.580 --> 00:18:14.010
And generally, what we would
see, coming back over here,

00:18:14.010 --> 00:18:15.570
I'm going to draw
it like this.

00:18:15.570 --> 00:18:18.270
This is what you'd
normally see, if

00:18:18.270 --> 00:18:19.520
you're using electrons.

00:18:33.440 --> 00:18:35.630
Professor Sadoway refers to
this as the whale-shaped

00:18:35.630 --> 00:18:38.530
curve, probably because
we're in New England.

00:18:38.530 --> 00:18:40.410
But this is what you
would normally see.

00:18:40.410 --> 00:18:43.110
Now, this y-axis is
the intensity.

00:18:43.110 --> 00:18:47.040
That's basically the number of
photons you count at some

00:18:47.040 --> 00:18:49.430
specific wavelength.

00:18:49.430 --> 00:18:50.790
And you see these spikes.

00:18:50.790 --> 00:18:54.410
And these spikes correspond
to K-alpha.

00:18:54.410 --> 00:18:54.990
This is K-beta.

00:18:54.990 --> 00:18:56.850
We'll talk about what they
are in a second.

00:18:56.850 --> 00:18:59.110
L-alpha and L-beta.

00:18:59.110 --> 00:18:59.590
And et cetera.

00:18:59.590 --> 00:19:02.240
You'd have M-alpha, N-beta.

00:19:02.240 --> 00:19:03.550
You go all the way down.

00:19:03.550 --> 00:19:06.090
But they have very
low intensity.

00:19:06.090 --> 00:19:08.360
So this is what we would
normally see if we use

00:19:08.360 --> 00:19:11.270
electrons to hit our sample.

00:19:11.270 --> 00:19:13.110
But we're not using electrons.

00:19:13.110 --> 00:19:15.960
And this is actually the answer
we got for probably 90%

00:19:15.960 --> 00:19:18.360
of the solutions
from students.

00:19:18.360 --> 00:19:21.730
They just drew the
Bremsstrahlung radiation,

00:19:21.730 --> 00:19:23.890
Bremsstrahlung, breaking
radiation.

00:19:23.890 --> 00:19:26.770
And they walked away, and
thought they had full credit.

00:19:26.770 --> 00:19:29.000
But in reality, this is not the
correct answer, because

00:19:29.000 --> 00:19:30.590
you think about what's actually

00:19:30.590 --> 00:19:32.400
causing this whale shape.

00:19:32.400 --> 00:19:35.100
Once you understand where the
whale shape comes from, then

00:19:35.100 --> 00:19:36.350
you understand what happens
if we're using

00:19:36.350 --> 00:19:39.550
photons instead of electrons.

00:19:39.550 --> 00:19:42.090
So let's actually understand
where these

00:19:42.090 --> 00:19:43.010
characteristics come from.

00:19:43.010 --> 00:19:46.260
There's two things to pick
up from this plot.

00:19:46.260 --> 00:19:47.640
We have a whale shape.

00:19:47.640 --> 00:19:50.670
We've also got these spikes.

00:19:50.670 --> 00:19:52.750
So first, let's talk
about where the

00:19:52.750 --> 00:19:54.470
whale shape comes from.

00:19:54.470 --> 00:19:55.720
Let me just do it with blue.

00:19:58.640 --> 00:20:00.440
We'll talk about the spikes in
a second, because they're

00:20:00.440 --> 00:20:03.030
actually a different
phenomenon.

00:20:03.030 --> 00:20:05.960
The whale shape comes from--

00:20:05.960 --> 00:20:07.530
let me draw it for you.

00:20:07.530 --> 00:20:09.250
Let's zoom in.

00:20:09.250 --> 00:20:11.880
Let's zoom in on our
target here.

00:20:11.880 --> 00:20:14.460
So here's our target material.

00:20:14.460 --> 00:20:15.890
We have an electron
hitting it.

00:20:15.890 --> 00:20:17.160
It's just a metal.

00:20:17.160 --> 00:20:18.530
And we've got photons
coming up.

00:20:18.530 --> 00:20:20.300
Let's zoom in really close
to the surface

00:20:20.300 --> 00:20:21.566
of that target material.

00:20:21.566 --> 00:20:24.740
Let's go back over here.

00:20:24.740 --> 00:20:25.990
So we've got--

00:20:33.430 --> 00:20:36.100
looks something like this--

00:20:36.100 --> 00:20:37.860
we've got an electron,
which I'll draw in

00:20:37.860 --> 00:20:42.220
yellow, coming in.

00:20:45.110 --> 00:20:45.825
So here's our electron.

00:20:45.825 --> 00:20:46.970
It's about to hit
our material.

00:20:46.970 --> 00:20:49.160
Notice I've drawn it with some
crystal structure, because

00:20:49.160 --> 00:20:51.540
that's what we're going to talk
about, really zoomed in,

00:20:51.540 --> 00:20:52.670
the atomic level.

00:20:52.670 --> 00:20:56.240
Now this electron, we've talked
about this in class.

00:20:56.240 --> 00:20:58.540
What can happen is that this
electron could come in--

00:20:58.540 --> 00:21:00.750
here's its path, normally,
if it wasn't going to get

00:21:00.750 --> 00:21:01.560
deflected--

00:21:01.560 --> 00:21:03.820
it can come in, and it
can be deflected.

00:21:03.820 --> 00:21:05.260
OK?

00:21:05.260 --> 00:21:07.310
So it can get deflected off--

00:21:07.310 --> 00:21:10.070
let me draw the following
path in blue.

00:21:10.070 --> 00:21:12.750
So it can go off like
this, like this.

00:21:12.750 --> 00:21:14.400
It could actually go straight
through, without being

00:21:14.400 --> 00:21:16.230
deflected at all.

00:21:16.230 --> 00:21:22.680
It could actually be reflected
back, like this.

00:21:22.680 --> 00:21:25.250
And these different reflections
correspond to the

00:21:25.250 --> 00:21:27.790
electron being accelerated.

00:21:27.790 --> 00:21:28.690
It's getting accelerated.

00:21:28.690 --> 00:21:33.410
If we define our system like
this, so here's our x and our

00:21:33.410 --> 00:21:36.210
y, the electron's getting
accelerated in the

00:21:36.210 --> 00:21:37.410
y-direction.

00:21:37.410 --> 00:21:39.800
This is just a simple location,
but what happens, is

00:21:39.800 --> 00:21:42.880
when you accelerate a charge,
you generate radiation.

00:21:42.880 --> 00:21:45.150
You generate electromagnetic
radiation.

00:21:45.150 --> 00:21:46.800
Accelerating charge.

00:21:46.800 --> 00:21:49.830
So, you know, here in the first
path, where it comes

00:21:49.830 --> 00:21:52.630
through, and it gets deflected
only by a little bit, you

00:21:52.630 --> 00:21:55.970
generate low energy radiation,
low energy photons.

00:21:55.970 --> 00:21:59.340
So what we're talking about
here is a very large

00:21:59.340 --> 00:22:02.960
wavelength photons coming out.

00:22:02.960 --> 00:22:05.210
So we have an electron.

00:22:05.210 --> 00:22:06.170
It gets deflected.

00:22:06.170 --> 00:22:07.190
It's accelerated.

00:22:07.190 --> 00:22:10.150
And from that point, we're
also generating a photon.

00:22:10.150 --> 00:22:12.520
Same thing for all these other
paths, but for the larger

00:22:12.520 --> 00:22:13.490
angle deflection.

00:22:13.490 --> 00:22:14.580
So this one here.

00:22:14.580 --> 00:22:16.875
And as you actually start
deflecting back, you're

00:22:16.875 --> 00:22:21.400
generating very high energy
electromagnetic radiation.

00:22:21.400 --> 00:22:26.300
And this is all because the
electron is basically, you

00:22:26.300 --> 00:22:27.570
know, is getting accelerated.

00:22:27.570 --> 00:22:30.360
You can actually have, you
know, the maximum energy

00:22:30.360 --> 00:22:33.700
photon you can give off here is
a photon that corresponds

00:22:33.700 --> 00:22:36.180
to this electron coming
in and getting stopped

00:22:36.180 --> 00:22:37.870
completely by the atom.

00:22:37.870 --> 00:22:40.210
Think electrostatic repulsion.

00:22:40.210 --> 00:22:43.580
So it comes in, and it gets
stopped dead in it's tracks.

00:22:43.580 --> 00:22:46.320
So you have some energy, it
was moving some kinetic

00:22:46.320 --> 00:22:48.300
energy, and now has
0 kinetic energy.

00:22:48.300 --> 00:22:51.420
The energy of your photon that
gets given off is basically

00:22:51.420 --> 00:22:52.700
the difference in the two.

00:22:52.700 --> 00:22:55.290
And that's what we call here,
on this plot, the short

00:22:55.290 --> 00:22:56.190
wavelength limit.

00:22:56.190 --> 00:22:59.480
We're going to write SWL.

00:22:59.480 --> 00:23:05.340
Because this is lambda, which
means that energy moves in the

00:23:05.340 --> 00:23:06.140
other direction.

00:23:06.140 --> 00:23:08.420
Lambda goes up, energy
goes down.

00:23:08.420 --> 00:23:11.830
Lambda goes down,
energy goes up.

00:23:11.830 --> 00:23:13.680
So this is our short wavelength
limit, which means

00:23:13.680 --> 00:23:17.050
that is the maximum, the highest
energy we can generate

00:23:17.050 --> 00:23:20.850
from this setup here.

00:23:20.850 --> 00:23:25.370
So all I'm saying is that you
can get deflections in any

00:23:25.370 --> 00:23:28.520
angle across this way, you can
get deflections in angle and

00:23:28.520 --> 00:23:32.650
generate any number of
wavelengths of photons with

00:23:32.650 --> 00:23:35.440
the minimum wavelength being
this one, that's generated

00:23:35.440 --> 00:23:37.620
from this situation.

00:23:37.620 --> 00:23:40.300
So I'll just make it very,
very small for you.

00:23:43.590 --> 00:23:46.030
So you generate all these
wavelengths, you create all

00:23:46.030 --> 00:23:48.860
this energy of photons, and
that's what basically

00:23:48.860 --> 00:23:52.680
corresponds to the whale-shaped
curve.

00:23:52.680 --> 00:23:55.510
You're more likely to get
deflections that correspond to

00:23:55.510 --> 00:23:57.390
something around here, because
you have higher intensity, and

00:23:57.390 --> 00:24:01.670
over here, you're less likely
to see those happening.

00:24:01.670 --> 00:24:03.230
So that's our whale shape.

00:24:03.230 --> 00:24:05.140
And now, let's return
to the problem.

00:24:05.140 --> 00:24:11.420
Let's ask, when we use photons,
what's the situation?

00:24:11.420 --> 00:24:15.460
Well, photon means we're no
longer looking at an electron.

00:24:27.670 --> 00:24:29.330
Our photon is coming in.

00:24:29.330 --> 00:24:31.710
And the thing about photons, is
that they're not going to

00:24:31.710 --> 00:24:33.400
get deflected like that.

00:24:33.400 --> 00:24:34.260
They don't have a charge.

00:24:34.260 --> 00:24:36.800
There's no electrostatic
propulsion, for example.

00:24:36.800 --> 00:24:39.583
So a photon will either only be
absorbed and re-emitted, or

00:24:39.583 --> 00:24:41.910
it will go through
the material.

00:24:41.910 --> 00:24:45.490
So what that means is that
there's no process now that

00:24:45.490 --> 00:24:48.110
will accelerate a
charge to create

00:24:48.110 --> 00:24:49.910
Bremsstrahlung radiation.

00:24:49.910 --> 00:24:52.830
And if there's no process to
accelerate a charge, then

00:24:52.830 --> 00:24:55.430
there's no way to get
Bremsstrahlung radiation.

00:24:55.430 --> 00:24:58.425
So the first thing you need to
do to get most the points on

00:24:58.425 --> 00:25:00.450
this problem is to say, look!

00:25:00.450 --> 00:25:02.720
There's no Bremsstrahlung
radiation anymore.

00:25:02.720 --> 00:25:03.970
There's no charge being
accelerated.

00:25:13.140 --> 00:25:15.120
So that's sort of the
first answer.

00:25:15.120 --> 00:25:18.230
The second thing to realize is
that these spikes still exist.

00:25:18.230 --> 00:25:20.090
Because the reason for these
spikes is a different

00:25:20.090 --> 00:25:22.970
mechanism than it is for
the Bremsstrahlung.

00:25:22.970 --> 00:25:27.545
These characteristic peaks
correspond to the movement of

00:25:27.545 --> 00:25:29.670
an electron to a different
energy level, and then a

00:25:29.670 --> 00:25:30.680
dropping back down.

00:25:30.680 --> 00:25:33.670
So it's very characteristic
of the material itself.

00:25:33.670 --> 00:25:36.190
So a photon can still come in.

00:25:36.190 --> 00:25:39.120
It could still liberate
or move an electron.

00:25:39.120 --> 00:25:40.990
It could just knock an electron
out of its shell.

00:25:40.990 --> 00:25:43.000
Perhaps n equals 1.

00:25:43.000 --> 00:25:45.580
And then the n equals 2 electron
will drop down and

00:25:45.580 --> 00:25:46.870
fill in the n equals 1 shell

00:25:46.870 --> 00:25:49.500
corresponding to K-alpha radiation.

00:25:49.500 --> 00:25:54.110
If you had n equals 3 electron
dropping down to n equals 1,

00:25:54.110 --> 00:25:57.190
you'd have your K-beta
radiation.

00:25:57.190 --> 00:26:02.180
And likewise from, you know, 3
to 2, 4 to 2, L-alpha, L-beta.

00:26:02.180 --> 00:26:05.150
So that's the key take-home
message here.

00:26:05.150 --> 00:26:08.280
The message is that your
Bremsstrahlung, the thing that

00:26:08.280 --> 00:26:10.350
you see in the notes in the
class all the time, the whale

00:26:10.350 --> 00:26:13.670
shape with the spikes, that's
two specific mechanisms. The

00:26:13.670 --> 00:26:17.380
whale, the Bremsstrahlung,
corresponds to the breaking of

00:26:17.380 --> 00:26:18.840
an electron.

00:26:18.840 --> 00:26:21.180
These peaks correspond to the
ejection of an electron, and

00:26:21.180 --> 00:26:24.560
an electron's moving around
energy levels within the atom.

00:26:24.560 --> 00:26:24.910
OK?

00:26:24.910 --> 00:26:27.830
So our learning objectives for
this problem, the things that

00:26:27.830 --> 00:26:30.960
we've taken home from the
problem overall, is that we

00:26:30.960 --> 00:26:34.650
know, for example, Rydberg
equation is only for

00:26:34.650 --> 00:26:37.530
hydrogenic-type atoms, and we
know how to use the Moseley's

00:26:37.530 --> 00:26:38.580
equation now.

00:26:38.580 --> 00:26:40.040
That was part 1.

00:26:40.040 --> 00:26:43.730
Part 2, we know Bragg's Law,
and how to use it, and what

00:26:43.730 --> 00:26:45.440
all the variables mean.

00:26:45.440 --> 00:26:47.080
That's something you should
learn and take home.

00:26:47.080 --> 00:26:50.390
And I really highly recommend
that you derive Bragg's Law

00:26:50.390 --> 00:26:52.540
from the geometric
interpretation.

00:26:52.540 --> 00:26:59.030
And part C, we really wanted
to probe and understand who

00:26:59.030 --> 00:27:01.370
conceptually understood
what was happening.

00:27:01.370 --> 00:27:03.720
And part C tells us, the thing
that we learned from it is

00:27:03.720 --> 00:27:07.260
that this Bremsstrahlung
radiation with the peaks that

00:27:07.260 --> 00:27:11.860
we saw in class, that's two
separate mechanisms occurring.

00:27:11.860 --> 00:27:14.820
So that's problem number 2,
and I hope you did well.