1
00:00:00,480 --> 00:00:04,060
All right. I guess we can we can get started now. Welcome back.
2
00:00:04,080 --> 00:00:07,650
It's the 13th lecture of the condensed matter course.
3
00:00:07,980 --> 00:00:12,120
When last we left off, we were talking about scattering the basic experiment.
4
00:00:12,630 --> 00:00:20,340
If you have some sample, you have a wave at some way vector K that you send into the sample and then it's scattered off to some wave vector.
5
00:00:20,970 --> 00:00:29,130
K prime. And by measuring how much and in what directions you have scattering, you're able to do something about what's actually in the sample.
6
00:00:29,160 --> 00:00:32,910
A couple of things that we know about these type of scattering experiments.
7
00:00:32,940 --> 00:00:40,080
First of all, in order to get scattering from chaotic prime, you must have K minus K prime equals reciprocal lattice vector g.
8
00:00:40,410 --> 00:00:49,340
And we derived in the last lecture that's equivalent to the Bragg condition two d sine theta equals and lambda.
9
00:00:49,350 --> 00:00:54,299
In other words, since a reciprocal lattice vector is orthogonal to a family of lattice planes,
10
00:00:54,300 --> 00:01:03,800
what you're actually getting is diffraction off a diffraction grating with lattice plane spacing D orthogonal to that vector g.
11
00:01:04,170 --> 00:01:12,989
The intensity of scattering at any reciprocal lattice vector is proportional to the structure factor of g squared,
12
00:01:12,990 --> 00:01:17,190
where the structure factors the 48 transform of the scattering potential in the unit cell.
13
00:01:17,340 --> 00:01:22,410
So this intensity gives you some amount of information about what's actually in the unit cell.
14
00:01:22,740 --> 00:01:26,220
Now, how you actually do this experiment is a little more complicated.
15
00:01:26,370 --> 00:01:28,200
There are several methods one can go about.
16
00:01:28,620 --> 00:01:38,250
There is the first method, which is the best method, but also the hardest method, best bit hardest, which I'll discuss only very briefly.
17
00:01:38,920 --> 00:01:43,230
The best method, but also the hardest method is to use a single crystal.
18
00:01:44,010 --> 00:01:48,750
Single crystal. So a single crystals look like this.
19
00:01:49,110 --> 00:01:55,980
This is a gorgeous single crystal of sodium chloride, a big single piece of the material without any defects.
20
00:01:56,400 --> 00:02:05,250
Now what you'll find so you take a single crystal, you put it in the path of your of your wave, and you look for four waves being scattered off of it.
21
00:02:05,340 --> 00:02:11,340
What you find is if you take your single crystal, you put it in some arbitrary orientation in front of your your beam.
22
00:02:11,340 --> 00:02:15,100
Here, for most orientations, you won't get any scattering at all.
23
00:02:15,120 --> 00:02:20,430
In other words, you will not manage to align the families of lattice planes just at the right angle,
24
00:02:20,580 --> 00:02:27,900
such that the Bragg condition is actually satisfied. So in order to actually get any scattering, what you need to do is to rotate the crystal.
25
00:02:29,130 --> 00:02:38,310
Rotate the crystal. Another way to get scattering is to vary the wavelength very lambda, do one or the other,
26
00:02:38,310 --> 00:02:43,620
and then at some angle of the crystal or at some value of the wavelength, you get scattering.
27
00:02:43,740 --> 00:02:47,400
You write down in which direction you get the scattering, you measure the intensity of the scattering,
28
00:02:47,520 --> 00:02:54,090
and you can deduce a lot of information about the structure factor and therefore about the structure of the crystal.
29
00:02:54,330 --> 00:03:00,569
Now, we're not actually going to go into how you analyse such data in any detail in this course.
30
00:03:00,570 --> 00:03:03,870
And the reason we're not going to do it is because this method is very rarely used,
31
00:03:04,140 --> 00:03:08,070
and the reason it's very rarely used is because it's actually the hardest thing to do.
32
00:03:08,460 --> 00:03:14,730
The reason it's hard is because, almost without exception, materials don't like to form nice big crystals.
33
00:03:14,730 --> 00:03:18,360
Sodium chloride is a suggestion. You have these beautiful big crystals of sodium chloride.
34
00:03:18,660 --> 00:03:23,280
Quartz makes nice big crystals. You can probably make a nice big crystal of sugar something.
35
00:03:23,280 --> 00:03:27,540
I mean, so there are a couple of materials that you can get to form nice big crystals, but most materials don't.
36
00:03:27,750 --> 00:03:34,020
If you cook up some new material in your laboratory or you find some new material in nature, usually it's not crystalline.
37
00:03:34,020 --> 00:03:40,200
It doesn't form big crystals. Frequently it's powder. Or even worse than that, you can have what's known as Polycrystalline.
38
00:03:40,560 --> 00:03:48,000
So this is a picture of a Polycrystalline material where you have tiny little crystal lights, which are about on the size of a micron.
39
00:03:48,000 --> 00:03:49,980
So there's a little crystal here, a little crystal here,
40
00:03:50,190 --> 00:03:54,900
and those crystals have their axes miss oriented with respect to each other, and they're also smushed together.
41
00:03:55,140 --> 00:03:59,400
You can think of it as like a powder of tiny little crystals all stuck together.
42
00:03:59,400 --> 00:04:03,719
And typically materials look like this when you make them. It's very, very difficult.
43
00:04:03,720 --> 00:04:09,750
It takes a very good chemist and very good material scientist to turn something that looks like this into something that looks like this.
44
00:04:10,080 --> 00:04:16,620
So most of the time, when one is doing experiment, one is working with something that looks like this, it's probably crystal.
45
00:04:17,190 --> 00:04:25,920
So to take data on a polycrystalline sample, you use what's known as the device Scherrer method to buy Scherrer.
46
00:04:28,800 --> 00:04:35,430
The buy is the same guy from the Buy Theory of Vibrations and Solids share someone else obviously.
47
00:04:36,810 --> 00:04:40,050
And the device share. This method is also known as powder diffraction.
48
00:04:43,690 --> 00:04:52,800
Diffraction. Because you can use a powder of the material, lots of tiny, tiny little crystal lights in a powder.
49
00:04:53,010 --> 00:04:57,210
And you can put that in your sample and get data out of it and actually make sense of it.
50
00:04:57,480 --> 00:05:01,770
Now, the general principle of the idea of powder diffraction or doing diffraction on
51
00:05:01,770 --> 00:05:08,130
polycrystalline samples is if you're sending a wave into your into your crystal here,
52
00:05:08,250 --> 00:05:11,550
if you imagine for a second you got a big a big single crystal,
53
00:05:11,700 --> 00:05:16,409
and you could rotate that single crystal around if there was any direction in which
54
00:05:16,410 --> 00:05:20,010
you rotated that single crystal and you could get diffraction out at this angle.
55
00:05:20,250 --> 00:05:26,280
Then somewhere in your powder, somewhere in your polycrystalline sample, the axes will be correctly oriented.
56
00:05:26,460 --> 00:05:32,790
In other words, the poly crystal material or the powder represents every possible orientation of the material.
57
00:05:32,970 --> 00:05:37,379
So somewhere in your material you'll have a situation where your family of lattice
58
00:05:37,380 --> 00:05:43,110
planes is aligned just in the right way in order to get diffraction from K2 K Prime.
59
00:05:43,110 --> 00:05:49,800
If there's any orientation that will do it, you'll find that crystal light with the right orientation somewhere in your sample.
60
00:05:50,700 --> 00:05:53,010
That's the general principle we're going to live by.
61
00:05:53,220 --> 00:06:00,000
Now, if if you can get diffraction in this direction with some lattice planes that are sort of lined up parallel this way,
62
00:06:00,240 --> 00:06:06,420
you can also imagine that if you lined up your lattice planes this way, you could get diffraction out this way.
63
00:06:07,170 --> 00:06:07,770
And in fact,
64
00:06:08,010 --> 00:06:17,640
you can imagine rotating those crystal lights around the incoming axis and you realise that you can get a whole cone of diffraction outwards.
65
00:06:17,820 --> 00:06:25,860
If there's any direction that you can scatter, you can rotate that crystal light around the incoming access and get a get a cone of outgoing waves.
66
00:06:26,140 --> 00:06:30,360
So the way the experiment actually looks at in a setup is kind of like this.
67
00:06:30,660 --> 00:06:35,010
There's an incident x ray beam, there's a polycrystalline or powder powder specimen here.
68
00:06:35,340 --> 00:06:39,870
And then you get these cones of outgoing waves, which you then try to measure.
69
00:06:40,050 --> 00:06:46,110
Remember that the it's marked here, so you don't forget the total deflection angle is two theta very important.
70
00:06:46,770 --> 00:06:51,239
A common source of source of error and to theta can go anywhere from zero,
71
00:06:51,240 --> 00:06:56,040
meaning it's not scattered at all to 180 degrees, meaning it's directly back scattered.
72
00:06:56,230 --> 00:07:04,620
Now, the way one used to measure it is you would take this sort of round environment and you would line the inside with photographic paper,
73
00:07:04,770 --> 00:07:07,770
and you would just expose the photographic paper to the x rays,
74
00:07:07,770 --> 00:07:17,070
and you would see these nice conical signals coming out for these various different cones coming out like this from your polycrystalline sample.
75
00:07:17,400 --> 00:07:21,000
But in modern days, photographic paper is not used.
76
00:07:21,000 --> 00:07:27,059
It's much more it's much more convenient and actually much more accurate and much more sensitive to use semiconductor detectors.
77
00:07:27,060 --> 00:07:32,490
So it's usually what you actually get out of your sample, out of your experiment now is a plot that looks like this,
78
00:07:32,790 --> 00:07:37,859
where what's plotted across the bottom is the scattering angle and the intensity vertically.
79
00:07:37,860 --> 00:07:42,450
So you'll see there's a peak here at 20 something degrees, another peak at 30 something degrees,
80
00:07:42,450 --> 00:07:45,509
which might correspond to this ring and then that ring and so forth.
81
00:07:45,510 --> 00:07:49,410
And this is the kind of data that we're going to spend our time analysing.
82
00:07:49,620 --> 00:07:57,809
Okay. So this is sort of sort of a classic type of question for exams, analyse a powder diffraction pattern with the device sharing method.
83
00:07:57,810 --> 00:08:01,200
So there's a couple of rules for following this method, so I'll write them down.
84
00:08:01,920 --> 00:08:06,210
It's fairly difficult to find a book that explains this well, I think as explained well in my book,
85
00:08:06,420 --> 00:08:10,860
but if you don't like the explanation of my book, you're kind of out of luck because there aren't a lot of other books to explain it.
86
00:08:11,100 --> 00:08:20,129
So sorry about that, but that's just how it is. So rule number zero is no the wavelength, no your wavelength.
87
00:08:20,130 --> 00:08:23,850
Lambda Why do you know you're wavelength lambda? Well, you're the experimentalist.
88
00:08:24,090 --> 00:08:27,750
You put in the X-ray beam, you had the X-ray apparatus.
89
00:08:27,900 --> 00:08:31,620
So, you know, presumably what the wavelength of the X-ray is. So it's a good thing to start with.
90
00:08:31,860 --> 00:08:36,150
Know your wavelength, probably know it pretty accurately, actually, if you're a good experimentalist,
91
00:08:37,200 --> 00:08:47,160
step one measure the angles, measure angles, angle of scattering, which is two theta, not theta.
92
00:08:47,610 --> 00:08:52,110
Remember, that is going to be important. So here's some data we're actually going to analyse where they've done.
93
00:08:52,110 --> 00:08:54,540
The first two steps for this is aluminium.
94
00:08:54,540 --> 00:09:03,240
They're doing scattering off of aluminium, I guess, in this country and the wavelength is known extremely precisely in this setup.
95
00:09:03,390 --> 00:09:11,030
This is done with a typical X-ray tube and you only have six digits of precision as to how how precisely they know the wavelength and
96
00:09:11,040 --> 00:09:18,419
remember the reason they know their wavelength so precisely is because this is an electronic transition between two iron states in an atom.
97
00:09:18,420 --> 00:09:21,899
It's an electron falling down from one eigen state to another eigen state.
98
00:09:21,900 --> 00:09:25,230
So the the energies of the eigen states are very, very precisely defined.
99
00:09:26,370 --> 00:09:27,179
So so you know,
100
00:09:27,180 --> 00:09:34,740
the wavelength with very high precision then it you know you get this data all these peaks as a function of of scattering angle to theta.
101
00:09:35,040 --> 00:09:43,200
And fortunately in this particular picture they've actually measured the angles for us 38.43 degrees, 44.6, seven degrees and so forth.
102
00:09:43,410 --> 00:09:48,630
And I've also labelled the peaks with letters so we can keep track of them peak A, peak B peaks and so forth.
103
00:09:49,740 --> 00:09:55,800
Just one hint that if you're ever given a powder diffraction pattern and you have to measure the angles yourself,
104
00:09:55,980 --> 00:10:02,280
it will really help a lot if you get the angles very precise. If you get the angles off by a little bit, you can get very confused.
105
00:10:02,280 --> 00:10:07,230
It won't be obvious what the result is supposed to be, so try to measure it as precisely as you can.
106
00:10:08,070 --> 00:10:13,200
For each scattering angle you measure step two angle.
107
00:10:13,320 --> 00:10:20,520
Sorry. Angle. So step two is calculate the lattice plane spacing.
108
00:10:20,520 --> 00:10:34,800
So lattice plane spacing, plane spacing using Bragg's law and that is D equals lambda over two sine theta.
109
00:10:36,060 --> 00:10:43,050
So for each scattering angle, you see that's representing a family of lattice planes with the spacing d equals lambda over two sine theta.
110
00:10:43,320 --> 00:10:46,020
Now you might think it should be and lambda over to sine theta,
111
00:10:46,230 --> 00:10:52,350
but in fact you don't need the end up here because the higher values of end will correspond to different theta is that you will also measure,
112
00:10:52,350 --> 00:10:57,690
so you'll just be over counting if you put an end upstairs. So you just need to keep lambda over to sine theta.
113
00:10:57,690 --> 00:11:02,820
Each angle corresponds to one one lattice plane spacing.
114
00:11:03,090 --> 00:11:06,240
Okay. All right, so step three.
115
00:11:06,780 --> 00:11:13,350
So this is for us, this is sort of not in the real world. We can assume cubic of some sort assume cubic.
116
00:11:13,920 --> 00:11:22,560
It's actually not a bad assumption in the real world too. Meaning simple cubic, simple boxy or FCC.
117
00:11:23,040 --> 00:11:26,189
I have never seen in 25 years of exams.
118
00:11:26,190 --> 00:11:27,870
I have never seen a 30 year exam.
119
00:11:28,050 --> 00:11:34,470
There's asked you to figure out a powder diffraction pattern for anything more complicated than simple cubic beaker FCC.
120
00:11:34,770 --> 00:11:38,999
Those are the only lattices you'll run into on your 30 year exam. That's almost a promise.
121
00:11:39,000 --> 00:11:42,540
I mean, they could do something for the first time, but. But you're pretty.
122
00:11:42,540 --> 00:11:46,290
It's pretty good assumption that this is all you're ever going to run into in the real world.
123
00:11:46,290 --> 00:11:49,859
It's you'll run into things that are more complicated, but it's not a bad thing to start with.
124
00:11:49,860 --> 00:11:54,450
This is an assumption. And if you can't get the data to fit these, then you go on to more complicated possibilities.
125
00:11:54,900 --> 00:12:00,900
All right. Assuming cubic we also have will remember that the lattice plane spacing this
126
00:12:00,900 --> 00:12:06,780
is something we derived corresponding to Miller indices h k l is the lattice
127
00:12:06,780 --> 00:12:11,790
constant a divided by the square root of h squared plus k squared plus l squared
128
00:12:13,140 --> 00:12:18,959
is something we derive I think in the last lecture or maybe the one before. So what does that give us.
129
00:12:18,960 --> 00:12:27,540
That gives us that a squared over D squared is h squared plus K squared plus L squared.
130
00:12:28,080 --> 00:12:33,690
Now we don't know what A is yet. We're going to try to figure it out, but we know a bunch of D's.
131
00:12:33,690 --> 00:12:34,860
So what is this going to tell us?
132
00:12:34,860 --> 00:12:45,149
The ratio of D's are going to be these k's and L's are integers, so the ratio of one over D squared are going to be an integer ratios.
133
00:12:45,150 --> 00:12:51,840
Okay. So we're going to declare this thing, this quantity h group plus case, but elsewhere we're going to call it N and step.
134
00:12:52,170 --> 00:12:56,730
Let's see what up we step up to step four. We're going to look for integer ratios.
135
00:12:58,210 --> 00:13:03,450
Four integer ratios. Ratios of N.
136
00:13:07,400 --> 00:13:15,470
And then once we have the integer ratios of and we will look for selection roles, look for selection roles.
137
00:13:20,880 --> 00:13:32,940
Rules. Rules. Okay, so let's, let's recall from the last lecture what the selection rules were given h can l if we had a simple cubic lattice.
138
00:13:33,660 --> 00:13:39,960
Simple cubic lattice. All h can l are allowed.
139
00:13:40,560 --> 00:13:47,250
All H can now correspond to reciprocal lattice vectors for the simple cubic layout lattice.
140
00:13:47,250 --> 00:13:51,870
So these are all allowed for a basic lattice.
141
00:13:52,230 --> 00:14:04,530
We derive the selection rule that h plus k plus l l must be even must be even for an FCC lattice.
142
00:14:04,950 --> 00:14:21,150
We derive the selection rule that hk l all even or all odd, so we can actually start making a table of the possibilities.
143
00:14:23,130 --> 00:14:33,150
Actually, maybe I'll put the table over here. So over first column we'll put h can l this is the family possible family's last place.
144
00:14:33,150 --> 00:14:39,060
We'll write the number n which is h squared plus k squared plus l squared.
145
00:14:39,450 --> 00:14:44,970
And then we'll write simple cubic boxy and FCC and ask whether they're allowed.
146
00:14:45,450 --> 00:14:49,499
So what's the simple as possible family of last place we can be talking about?
147
00:14:49,500 --> 00:14:59,390
We could be talking about 100. So n a square plus case proposal squared is one simple cubic yes that's allowed for simple cubic all a
148
00:14:59,400 --> 00:15:05,219
squared plus k all h can l are allowed for simple cubic bc it's not allowed because one +00 is odd.
149
00:15:05,220 --> 00:15:12,120
So this is no, the FCC is not allowed because they're not all odd or all even so it's not allowed for FCC.
150
00:15:12,600 --> 00:15:16,680
How about the next simplest 1110?
151
00:15:17,130 --> 00:15:20,610
A square plus l squared is two simple cubic well.
152
00:15:20,610 --> 00:15:27,749
Yes, that's also allowed bc well one plus one plus zero is even one plus one plus zero is two.
153
00:15:27,750 --> 00:15:38,970
So that's allowed. Yes, FCC is not allowed because they're not all odd or all even 111 what l squared is three.
154
00:15:39,270 --> 00:15:46,770
It's allowed for simple cubic for BC. It's not allowed because the sum is odd, but it is allowed for FCC because they're all odd.
155
00:15:47,310 --> 00:15:50,580
Then 200 keep going.
156
00:15:50,580 --> 00:15:54,780
So two squared +00 squared is four. It's allowed for simple cubic.
157
00:15:55,110 --> 00:15:59,820
The sum is even so it's allowed here and there all even so it's allowed here.
158
00:16:00,120 --> 00:16:10,140
And you can make up a big table of these things. It's worth doing to keep track of which indices you should observe for which possible lattice type.
159
00:16:10,410 --> 00:16:18,790
Now, from this type of table, you can make a list of the possible integers n that you would see for the different lattice types.
160
00:16:19,010 --> 00:16:30,750
So for simple cubic cubic you get and the possible ends you get are one, two, three, four, five, six, eight.
161
00:16:31,470 --> 00:16:41,290
So what happened to seven. So all the possible a's case ls are allowed, but you'll discover if you try there's no h can l that if you square them,
162
00:16:41,310 --> 00:16:45,950
take eight squared plus k squared plus l squared you'll get seven. Just can't find integers.
163
00:16:45,960 --> 00:16:51,870
It's sort of a number theory type of result that you just can't find three integers that when squared and added together give you seven.
164
00:16:52,020 --> 00:17:05,219
So seven is missing and so forth. And I think the next one that's missing is 15 for BC, for BC and is two, four, six, eight, so forth and so on.
165
00:17:05,220 --> 00:17:10,080
I think the first one that's missing is 28. Actually, FCC is the most interesting one.
166
00:17:10,500 --> 00:17:20,740
FCC, the series is an equals three, three, four, eight, 11, 12, 16, 16 and so forth.
167
00:17:20,810 --> 00:17:23,160
Actually, I think I wrote these all down already. Yeah.
168
00:17:23,160 --> 00:17:30,180
So on this on this slide, so here's exactly building that whole table again, we wrote down the first four elements of this table.
169
00:17:30,360 --> 00:17:34,530
Here's the ends. A square was case growth was L squared. One, two, three, four, five, six.
170
00:17:34,770 --> 00:17:38,820
When you get to seven, there's no set of Miller indices that when squared together gives you seven.
171
00:17:40,500 --> 00:17:46,200
And you look at FCC and the possible the possible indices that give you that are allowed for FCC.
172
00:17:46,200 --> 00:17:51,570
The first the first one is 111. Those are all odd that that gives you a total of three.
173
00:17:51,840 --> 00:17:57,840
The next one that's allowed is 200. Those are all even that gives you two squared +00 squared is four.
174
00:17:57,870 --> 00:18:04,260
That's allowed. The next one that's allowed is 220. All even two squared plus two squared plus zero squared is eight.
175
00:18:04,500 --> 00:18:09,690
The next one that's allowed is 311. All odd that gives you 11 and so forth.
176
00:18:09,690 --> 00:18:13,590
So these are the series that we want to want to look for.
177
00:18:14,550 --> 00:18:18,330
So actually let's do that for this aluminium data. Oh, and here's, here's.
178
00:18:18,420 --> 00:18:22,290
A series written out. One, two, three, four, five, six, eight, nine.
179
00:18:22,530 --> 00:18:29,520
Missing seven, 15 and 23. I I'll even integers excluding 28, 60 and so forth.
180
00:18:29,640 --> 00:18:34,620
Actually, it's not a coincidence. The 28 happens to be four times seven and 60 happens to be four times 15.
181
00:18:34,630 --> 00:18:42,960
You can convince yourself why that's true. And then FCC is three, four, eight, 11, 12 and and and so forth.
182
00:18:42,990 --> 00:18:48,930
Okay, so here's the data again. Let's go through this procedure, all the steps, one by one.
183
00:18:49,110 --> 00:18:53,250
So here we measured all of the the angles are someone measured it for us here.
184
00:18:53,300 --> 00:18:56,670
It's very nice of them. So we make a table of all these angles.
185
00:18:56,670 --> 00:19:00,060
We calculate D equals lambda over to sine theta.
186
00:19:00,600 --> 00:19:04,770
So here is the lambda over to sine theta for all of these scattering peaks.
187
00:19:05,310 --> 00:19:12,210
Then we have this statement that one or a over eight squared over squared is squared, plus K squared plus L squared.
188
00:19:12,420 --> 00:19:16,460
And we want to find integer ratios of one over dx squared.
189
00:19:16,470 --> 00:19:17,310
So how do we do that?
190
00:19:17,580 --> 00:19:28,250
Well, let's declare this number here 2.3405 Angstroms we'll call it this of a and we'll make a table of this of a squared over dx squared where d is,
191
00:19:28,250 --> 00:19:37,050
is all of these possible values of D. So the first slot is by definition 12.3 squared over 2.3 squared is one.
192
00:19:37,380 --> 00:19:42,540
But the second 12.3 squared over 2.02 squared is 1.33 and so forth and so on.
193
00:19:42,540 --> 00:19:46,739
We get this in this table here. Those aren't integers, but they're pretty close to integers.
194
00:19:46,740 --> 00:19:51,060
If you multiply it by three, you get a set of things that are really, really close to integers,
195
00:19:51,360 --> 00:20:01,710
and you'll see that that pattern is really three, four, eight, 11, 12, 16, 1920, which is the FCC series three, four, eight, 11, 12, 16, 1920.
196
00:20:02,040 --> 00:20:12,870
So, so we've concluded from this that the the aluminium, the aluminium scattering pattern we found is corresponds to an FCC lattice.
197
00:20:13,110 --> 00:20:17,309
We can actually go on a little bit further to calculate lattice constants.
198
00:20:17,310 --> 00:20:29,760
This might be step five, calculate lattice constant the size of the units L constant a which would be d times the square root of k squared
199
00:20:30,150 --> 00:20:35,309
plus l squared plus h squared because I did that the wrong order e squared plus case proposal squared usually.
200
00:20:35,310 --> 00:20:39,629
All right, same thing. And if we do that there,
201
00:20:39,630 --> 00:20:43,590
the Miller indices have written down for their for the peaks and here's the
202
00:20:43,590 --> 00:20:47,669
A's and they're all pretty close to the same to about three or four digits.
203
00:20:47,670 --> 00:20:53,910
They're not exactly the same. So there's probably some error in the measurement of the angles or something like that has gone wrong.
204
00:20:54,150 --> 00:20:59,970
But within four digits of accuracy, we've measured the lattice constant, a four for aluminium.
205
00:21:00,240 --> 00:21:08,280
Okay, so that's generally how it works. Now when you get good at this, you don't even have to really do this whole calculation.
206
00:21:08,280 --> 00:21:12,120
You can sort of look at the pattern and have a pretty good idea what it is.
207
00:21:12,450 --> 00:21:15,599
So this oh, there's aluminium there.
208
00:21:15,600 --> 00:21:19,020
There's the Miller Indices labelled appropriately.
209
00:21:19,650 --> 00:21:28,290
So I'm going to show you a pattern. And without actually doing any calculation, we can tell what the lattice type is.
210
00:21:28,290 --> 00:21:32,220
Now, there's only three possibilities. If you give me a reason, just give me a reason.
211
00:21:32,460 --> 00:21:36,780
If it's simple cubic, what's the reason? Sound very good.
212
00:21:36,780 --> 00:21:40,830
All right, so that's a so No. Seven. That's a so that one's years.
213
00:21:41,550 --> 00:21:50,370
So you can see here that the first peak so that the spacing of the peaks is not completely uniform and that's because of the sine theta factor.
214
00:21:50,670 --> 00:21:56,460
That sine theta so expresses things out more when you're at low angles than it is that at at high angles.
215
00:21:56,790 --> 00:22:00,239
But, but you can kind of see that one.
216
00:22:00,240 --> 00:22:04,380
The spacing from 1 to 2 isn't so different from 2 to 3, 3 to 4, 4 to 5, 5 to 6.
217
00:22:04,500 --> 00:22:08,700
But then there's a hole where seven is supposed to be, and then 8 to 9, ten, 11, 12.
218
00:22:08,700 --> 00:22:10,829
And you see if they actually there's a hole where 15 is supposed to be.
219
00:22:10,830 --> 00:22:14,400
And if you go all the way up to 23, you'll notice there's a hole where 23 is supposed to be.
220
00:22:14,550 --> 00:22:17,310
And that's the series that you expect for a simple cubic.
221
00:22:17,310 --> 00:22:24,240
So you can just look at this at the scattering pattern for barium Titan eight and know that barium type Nate is a simple cubic lattice.
222
00:22:25,410 --> 00:22:32,549
That's how it works. Pretty cool. All right. It's only very tight. And it is a fairly important material for the Optoelectronics industry.
223
00:22:32,550 --> 00:22:37,230
It's used for various things. Now, there's more information in this.
224
00:22:38,040 --> 00:22:42,599
There is it. There's more information in this scattering pattern than we've used so far.
225
00:22:42,600 --> 00:22:47,880
All we've used is the angles of scattering. We haven't used anything about the intensity of scattering.
226
00:22:48,090 --> 00:22:51,990
And there's a lot of information in the intensity of scattering that we can make use of.
227
00:22:52,350 --> 00:23:00,500
So let's see if we can understand what how much scattering we should get for Miller Indices H Candle.
228
00:23:00,870 --> 00:23:08,130
Well, first of all, there's the structure factor, acid H candle, which gets squared now.
229
00:23:08,550 --> 00:23:14,700
All right, let's even write it out. So the structure factor, remember this s equals s lattice times s basis.
230
00:23:15,690 --> 00:23:24,030
Times s basis. And as Gladys does nothing more than enforce selection rules.
231
00:23:24,240 --> 00:23:32,229
Enforce selection rules. So we've done that.
232
00:23:32,230 --> 00:23:35,680
So that's not going to be interesting for us as a basis.
233
00:23:35,980 --> 00:23:39,879
Well, there's only one atom in this basis because it's just one type of atom.
234
00:23:39,880 --> 00:23:45,310
A single, single atom in the basis is just a pure FCC lattice with no, no interesting basis.
235
00:23:45,550 --> 00:23:50,890
So all you get is the the form factor for aluminium and this form factor for aluminium
236
00:23:51,070 --> 00:24:00,460
decays decays with increasing with increasing increasing g as we mentioned last time,
237
00:24:00,670 --> 00:24:06,190
or increasing angle or increased increase theta.
238
00:24:09,010 --> 00:24:14,700
So we expect that the total scattering is going to drop slowly as a function of increased scattering angle.
239
00:24:15,070 --> 00:24:18,970
But there's actually that's not all that contributes to the the intensity.
240
00:24:19,180 --> 00:24:27,100
There's two other factors which contribute to the intensity of first is the probability, probability of alignment.
241
00:24:30,480 --> 00:24:36,480
So this is the probability that a crystal light will align in just the right way in order to get scattering.
242
00:24:36,660 --> 00:24:42,030
It will come back to that in a moment. And the last factor is known as the Lorenz factor.
243
00:24:42,540 --> 00:24:45,690
LORENZ Or sometimes. LORENZ Polarisation factor.
244
00:24:46,260 --> 00:24:54,540
Polarisation factor, which is really a geometric factor.
245
00:24:54,540 --> 00:24:56,880
We're not going to say too much about it, but I'll show you what it looks like.
246
00:24:57,150 --> 00:25:05,219
So this function is a function which varies fairly rapidly as a function of angle, but then when you add in intermediate angles,
247
00:25:05,220 --> 00:25:11,880
it's fairly flat between around 60 and 140, and then it goes back up eventually at around 140, it starts going back up.
248
00:25:12,180 --> 00:25:17,490
It's the Lorenz polarisation factor. It's really a feature of how you set up your experiment.
249
00:25:17,700 --> 00:25:24,010
And so if you're an experimentalist, you will know what this factor is in advance before you do any sort of wave,
250
00:25:24,240 --> 00:25:25,440
before you do any sort of measurement.
251
00:25:25,680 --> 00:25:33,600
Now, there's a there's sort of a glitch here, which is that sometimes when people present you with x ray patterns like this one,
252
00:25:33,840 --> 00:25:38,040
they'll have already divided by the Lorenz polarisation factor here.
253
00:25:38,040 --> 00:25:41,250
And other times they will not have divided by the range polarisation factor.
254
00:25:41,430 --> 00:25:45,120
So really, if someone shows you data like this, you should ask Is that raw data?
255
00:25:45,120 --> 00:25:48,480
Or have you divide it out by this Lorenz polarisation factor already?
256
00:25:48,690 --> 00:25:55,800
And embarrassingly enough, on many exams they'll ask you a question and they won't tell you whether they've divided by this or not.
257
00:25:55,830 --> 00:26:01,980
So if you want to be safe, you should say you should have told me whether you divide it by this factor or not, I'm assuming whatever.
258
00:26:02,310 --> 00:26:06,150
So anyway, so that's that's this factor.
259
00:26:06,360 --> 00:26:11,460
And it also for most angles. So here we're only scattering up to about 120 degrees here.
260
00:26:11,760 --> 00:26:18,150
So this so this factor here is decaying as a function of angle up to 120 degrees.
261
00:26:18,330 --> 00:26:24,750
This factor here is also decaying as a function of angle. So we'd sort of naively guess that we should always decay as a function of angle.
262
00:26:24,750 --> 00:26:29,790
As you increase the angle and and more or less the scattering amplitude is decaying is a function of angle.
263
00:26:30,030 --> 00:26:33,600
But we haven't looked at this piece here. Probability of alignment.
264
00:26:33,930 --> 00:26:40,229
Well, you might think to yourself, you know, all the crystal lights are aligned completely randomly, so that that factor should be trivial.
265
00:26:40,230 --> 00:26:43,260
It should be just one. But think about this for a second.
266
00:26:44,220 --> 00:26:47,400
You remember when we talked about families of lattice planes.
267
00:26:47,640 --> 00:26:53,100
If we're thinking about a lattice plane spacing, which is associated with the one one, one family of lattice planes,
268
00:26:53,310 --> 00:26:59,430
it could have been the 111 bit family of large planes or the one one bar one or the one bar one one or the one by one bar one.
269
00:27:00,030 --> 00:27:07,470
I mean, there's eight possibilities, which would all give you exactly the same lattice plane spacing that you can scatter off of.
270
00:27:07,830 --> 00:27:17,520
So in fact, this probability of alignment is actually the multiplicity of a scale.
271
00:27:19,350 --> 00:27:24,179
So if we go back to this table here to see where yeah.
272
00:27:24,180 --> 00:27:26,010
Add the multiplicity on to this table.
273
00:27:26,310 --> 00:27:33,480
If we're thinking about the 100 and the Miller indices, there's six possible directions which would present the same lattice.
274
00:27:33,480 --> 00:27:37,290
Plane spacing is 100, the six possible axes of the cube.
275
00:27:37,620 --> 00:27:45,689
But if we're looking at the 110, there would be 12 different phases for 12 different faith ways you could face the cube,
276
00:27:45,690 --> 00:27:54,570
that you would present the same lattice plane spacing. And someone wanted a chocolate bar for realising that 3 to 1 has a multiplicity of 48.
277
00:27:54,960 --> 00:28:01,320
So these are the factors that we want to keep track of if we want to understand the intensity of the scattering.
278
00:28:01,320 --> 00:28:04,560
Now let's look at this.
279
00:28:05,490 --> 00:28:08,370
Let's look at two peaks that are actually fairly close to each other, an angle.
280
00:28:08,370 --> 00:28:11,580
Why do I want to look at two things that are fairly close to each other an angle?
281
00:28:11,850 --> 00:28:19,200
Well, if you both F and L are decaying kind of slowly as a function of angle.
282
00:28:19,410 --> 00:28:28,230
So here, if we're take if we take angles somewhere in the middle, you know, around 100, 120, then then this thing is pretty flat.
283
00:28:28,380 --> 00:28:32,130
This thing is decaying only slowly, and we can only worry about the multiplicity.
284
00:28:32,400 --> 00:28:37,110
So let's take these two peaks here. Three, one, one and two, two, two, three, one, one and two, two, two.
285
00:28:37,500 --> 00:28:40,530
They're they they have these F's and L's in them.
286
00:28:40,530 --> 00:28:45,870
But an L are pretty much the same because we're scattering at about the same angle for three, one, one and two, two, two.
287
00:28:45,870 --> 00:28:51,900
Just a slight difference in the angle. But the multiplicity is let's back up to the multiplicity here.
288
00:28:52,860 --> 00:28:59,700
311 has a where is it. 311 has a multiplicity of 24 where a two, two, two has a multiplicity of eight.
289
00:29:00,060 --> 00:29:06,930
So we would guess at the peak, four, three, one, one should be three times higher than the peak for eight and indeed the peak four,
290
00:29:06,930 --> 00:29:11,610
three, one one is pretty close to three times higher than the peak for 2 to 2.
291
00:29:12,090 --> 00:29:15,750
Okay. So that's how the multiplicity is get in the game as well.
292
00:29:16,320 --> 00:29:23,520
Okay. So these intensities of scattering can actually be very useful for analysing data as well.
293
00:29:23,520 --> 00:29:28,470
I'm going to show you a sort of artificial example of where this might be useful.
294
00:29:28,790 --> 00:29:34,339
So here is some fairly bad data for a taken on iron.
295
00:29:34,340 --> 00:29:38,960
That was bad for a couple of reasons. The first reason is because the peaks are pretty broad.
296
00:29:39,200 --> 00:29:42,319
The second reason is because you only see three peaks with three peaks.
297
00:29:42,320 --> 00:29:45,170
It's pretty hard to tell which series you're talking about here.
298
00:29:45,170 --> 00:29:49,370
You can't count out to the seventh and see if the seventh is missing because you only have three peaks.
299
00:29:49,760 --> 00:29:56,239
So this is going to be a little bit difficult, but we have the amplitudes of the scattering and that actually might be might be helpful.
300
00:29:56,240 --> 00:30:05,690
So we'll see if we can we can make use of that. So we go through the same exercise of measuring the the angles, calculate the latest plane spacing,
301
00:30:05,990 --> 00:30:11,870
take the ratio of the first lattice plane spacing to the various last plane spacing da2
302
00:30:11,870 --> 00:30:18,770
.03 squared over 1.4 squared is two is 22.13 squared over 1.17 squared is about three.
303
00:30:19,100 --> 00:30:25,819
We realise those are in the integer ratios of one, two and three. So those could be two possibilities.
304
00:30:25,820 --> 00:30:32,540
Either it's simple cubic and then these scattering peaks correspond to 100110 and 111,
305
00:30:32,750 --> 00:30:41,120
meaning this integer and a squared plus k squared plus l squared is one, two or three or it could be and is two, four and six.
306
00:30:41,120 --> 00:30:48,050
Those would still be in the ratio of one, two and three, but they would correspond to Miller Indices of 110200 and 211,
307
00:30:48,410 --> 00:30:51,770
which when added squared and added up, give you two, four and six.
308
00:30:51,980 --> 00:30:55,160
So we don't know which one it is just by looking at this data.
309
00:30:56,450 --> 00:31:06,469
So what do we do? Well, there's a couple of approaches we could take. One approach is that we could try calculating the the lattice constant.
310
00:31:06,470 --> 00:31:13,400
A So if we calculated that we could assume simple cubic calculate the lattice constant and we
311
00:31:13,400 --> 00:31:18,260
would get d times square root of a square plus k squared plus l squared gives us 2.03 angstroms.
312
00:31:18,480 --> 00:31:24,260
We did the same thing for BC. We'd get something that square root of two bigger because we're multiplying all the ends by two.
313
00:31:24,290 --> 00:31:28,459
So this thing inside the square root is multiplied by two overall.
314
00:31:28,460 --> 00:31:31,510
So instead of getting 2.3, we get 2.86 OC.
315
00:31:31,670 --> 00:31:36,950
So we have a different lattice plane, a different size of the, of the unit.
316
00:31:36,960 --> 00:31:40,010
So conventional units l for a BC case.
317
00:31:40,370 --> 00:31:42,409
Now why does that help us?
318
00:31:42,410 --> 00:31:48,590
Because we don't know what the lattice constant is for iron, but it does help us because we could try calculating the atomic density.
319
00:31:49,660 --> 00:31:58,059
For the simple Cuba case, we would have one atom per 2.03 angstroms cubed, whereas for a BBC it's two atoms.
320
00:31:58,060 --> 00:32:03,010
Two atoms in the conventional unit cell per 2.86 angstroms cubed.
321
00:32:03,010 --> 00:32:06,160
And those densities differ from each other by a factor of square root of two.
322
00:32:06,460 --> 00:32:10,930
So if we knew the atomic density of ion, we could tell which one it is, so we'd be done.
323
00:32:11,650 --> 00:32:15,160
But suppose we don't happen to know the atomic density of ion. So what do we do then?
324
00:32:15,250 --> 00:32:23,610
Okay. There are other. Other things we could invoke. We can look at the scattering amplitudes, which was what I was leading up to now, remember.
325
00:32:23,670 --> 00:32:28,410
So at this at this scattering angle, the Lorentz factor is decreasing.
326
00:32:28,770 --> 00:32:32,310
The form factor is always decreasing as a function of increased angle.
327
00:32:32,520 --> 00:32:38,400
So you would sort of naively expect that A should be bigger than B should be bigger than C, but it's not C is bigger.
328
00:32:38,640 --> 00:32:45,660
The only thing that could possibly explain that is if the multiplicity of the peak C is bigger than the multiplicity of B.
329
00:32:46,260 --> 00:32:50,740
So if you look at the simple cubic, the simple cubic, we identify the first peak is 100.
330
00:32:50,760 --> 00:32:54,020
The second peak is 110 and the third peak as 111.
331
00:32:54,030 --> 00:33:00,660
The multiplicity of B is bigger than multiplicity of C. So there's no reason you would ever get the C peak to be bigger than B peak.
332
00:33:01,020 --> 00:33:05,579
However, for BC, the multiplicity of the C peak to one one is 24 ways.
333
00:33:05,580 --> 00:33:10,620
The multiplicity of peak is 200, so c should actually be bigger than B.
334
00:33:11,040 --> 00:33:14,820
Now if you just look at the multiplicity as you would expect a factor of four,
335
00:33:14,940 --> 00:33:22,470
that C should be four times bigger than B and it's not four times bigger than B, it's only about two, maybe two and a bit bigger times bigger than B.
336
00:33:22,740 --> 00:33:26,969
But the reason it's not a four, four times bigger is because F is dropping,
337
00:33:26,970 --> 00:33:30,360
the form factor is dropping and Lawrence Factor is dropping also at the same time.
338
00:33:30,570 --> 00:33:34,710
So it's competing with this decreasing effect from these other two terms.
339
00:33:34,980 --> 00:33:41,040
But at any rate, just by looking at this data, we can we can tell just from the amplitudes that this must be BC.
340
00:33:41,880 --> 00:33:48,360
Of course, we also know that iron is not simple cubic because there's only one element that takes a simple cubic lattice,
341
00:33:48,360 --> 00:33:53,880
and that's polonium and iron isn't polonium. And so that does it also in sort of a cheating way to go about it.
342
00:33:54,990 --> 00:33:58,830
All right. So let's do one more example.
343
00:33:59,970 --> 00:34:05,940
This is an example of neutron scattering. Neutron scattering, very similar to X-ray scattering.
344
00:34:06,120 --> 00:34:09,540
But it's a little simpler because the form factor is just a constant.
345
00:34:09,630 --> 00:34:16,440
It's just a nuclear scattering length. It's not a function of g. So what's the latest type?
346
00:34:18,190 --> 00:34:22,350
Oh, my gosh. Oh, yes, yes.
347
00:34:22,780 --> 00:34:25,620
Not so good at this. So. Yes.
348
00:34:25,630 --> 00:34:35,260
So it's so you can tell it's it's this is FCC without doing any calculation because of the pattern that you can sort of imagine if this is three,
349
00:34:35,260 --> 00:34:41,650
this is four, then there's a big spacing to eight, then a little spacing to 11, 12, close together.
350
00:34:41,830 --> 00:34:45,970
So you can kind of see that the spacing is the right spacing for an FCC lattice.
351
00:34:47,170 --> 00:34:54,700
Okay. So but if you want to go through the whole the whole the whole story, we can go through the whole story carefully, measure the angles,
352
00:34:55,180 --> 00:35:05,260
calculate the lattice plane spacings, take the ratios of the one over the one over the square of the last plane spacings you get one, 1.3, 2.6.
353
00:35:05,410 --> 00:35:12,790
We really multiply these by three. We realise these are integer ratios, three, four, pretty close to eight, pretty close to 11 actually.
354
00:35:12,790 --> 00:35:17,740
The data isn't so great and if you back up well you'll realise that these peaks are pretty broad.
355
00:35:17,980 --> 00:35:21,130
They're not really that accurately resolved.
356
00:35:21,340 --> 00:35:24,610
It's not the greatest data. This was data taken in 1959.
357
00:35:24,610 --> 00:35:28,950
I mean, modern, modern data is much better. But but okay,
358
00:35:28,980 --> 00:35:40,270
it's you can definitely see that it's going to be FCC and you can also label the the Miller Indices and get an estimate of the lattice constant.
359
00:35:40,270 --> 00:35:45,910
Okay. So the lattice constant, maybe you don't know it within a few percent, but you have a pretty good estimate of what it is.
360
00:35:46,330 --> 00:35:51,940
But in fact, what we have here and we can write out the multiplicity is we can label the peaks as well.
361
00:35:52,030 --> 00:35:57,879
Actually, I noticed that by doing this you'll notice that there's you would have predicted a slight peak here,
362
00:35:57,880 --> 00:36:00,910
a peak of 40 zero peak here where the shoulder is.
363
00:36:00,910 --> 00:36:06,640
It's not there you. But if you kind of look at it carefully, you see that there's kind of a weak shoulder.
364
00:36:06,940 --> 00:36:11,380
You can imagine that it's just a very small peak that's sitting there now.
365
00:36:12,480 --> 00:36:17,100
So this tells us where where the peaks are. But we there's more information that we would like to get.
366
00:36:17,310 --> 00:36:24,780
We would like to know what the basis is. So we know that titanium carbide titanium carbide is F.C.C.
367
00:36:25,020 --> 00:36:32,130
And it has to have a basis with the basis that it has two elements in it, with the basis of two atoms.
368
00:36:32,640 --> 00:36:38,000
Of two atoms. And we'd like to know what is the form of the basis.
369
00:36:38,010 --> 00:36:42,040
In other words, where do you put the titanium? Where do you put the carbon in the unit cell?
370
00:36:42,510 --> 00:36:51,299
So can we figure that out? Well, okay, so arbitrarily we can we can always assign titanium to the position at position 000.
371
00:36:51,300 --> 00:36:55,230
And the unit cell will just define the titanium position to be 000.
372
00:36:55,530 --> 00:37:01,080
But then carbon we'll put at position u, v w which we don't know.
373
00:37:01,440 --> 00:37:05,580
Incidentally, I should have mentioned Titanium Carbide is actually a a really important material.
374
00:37:05,590 --> 00:37:09,420
Does anyone know? Here's another one. Anyone know why? Titanium Carbide is really important?
375
00:37:12,070 --> 00:37:16,480
Use it for what? Yeah, yeah. Anything that needs to be really hard then.
376
00:37:16,630 --> 00:37:21,670
Yeah. Okay. So. So Titanium Carbide is one of the hardest materials out there.
377
00:37:21,670 --> 00:37:25,360
Short of diamond, I think there's only one or two others that are as hard as titanium carbide.
378
00:37:25,360 --> 00:37:29,319
It's easy to, um. And it's easier to make titanium carbide.
379
00:37:29,320 --> 00:37:36,940
It's not as expensive a diamond. It's so whenever you need anything that needs to be very tough, it's very frequently used for things like tools.
380
00:37:36,940 --> 00:37:40,300
If you need a saw or a drill bit, that's going to cut through something very hard.
381
00:37:40,510 --> 00:37:46,239
It's very frequently titanium carbide. So that's why they did this experiment way back in in 1959,
382
00:37:46,240 --> 00:37:50,770
because it was an interesting material even back in 1959, and they didn't even know its structure.
383
00:37:51,220 --> 00:37:57,790
So. Okay, so I think they did maybe they did know this structure from something else.
384
00:37:57,790 --> 00:38:04,779
I don't I'm not sure, but I don't I don't think they knew it. So anyway, we don't know where the carbon is in the, in itself.
385
00:38:04,780 --> 00:38:10,750
So we're going to see if we can use this intensity data from these peaks to figure out where in the units all it is.
386
00:38:11,590 --> 00:38:20,050
So generally the piece we're going to be interested in is the basis structure factor, which is which ends up getting squared.
387
00:38:20,800 --> 00:38:28,630
And that is, well, maybe I'll write this out here first as basis equals sum over alpha in the units l
388
00:38:28,750 --> 00:38:38,980
and units l e to the i g dot are alpha times b alpha for the two atoms in the cell.
389
00:38:39,310 --> 00:38:50,500
So space is squared here will be v titanium plus b carbon e to that will write it as either the two pi i h k l
390
00:38:51,010 --> 00:38:59,590
dotted into u v w and then you take that thing and you square it and that will give you the basis structure factor.
391
00:38:59,860 --> 00:39:08,470
Now if you knew what these nuclear scattering lengths were, b titanium in b b carbon, you might have a fighting chance of figuring out what you were.
392
00:39:09,610 --> 00:39:14,620
But suppose we didn't know that. Can we still make any progress? Well, we can make a little bit of progress.
393
00:39:14,620 --> 00:39:24,130
If I give you a hint. This is a kind of hint. You get an exam. The kind of hit you get on an exam is that F.C.C. with two atom basis.
394
00:39:25,210 --> 00:39:28,960
Atom basis. There's only two common ones.
395
00:39:28,960 --> 00:39:33,130
Only two common ones. Common possibilities.
396
00:39:38,400 --> 00:39:42,450
In fact, I don't even know of any cases that are not one of these two.
397
00:39:42,480 --> 00:39:45,480
There may be even a theorem that you can't have anything else. I'm not sure about that.
398
00:39:45,480 --> 00:39:49,590
So don't quote me. But anyway, the two possibilities we've both we've seen them both before.
399
00:39:49,890 --> 00:39:54,719
There's a so-called zinc blend structure, which is the same as the gallium arsenide structure,
400
00:39:54,720 --> 00:40:02,220
which we talked about in class, which puts you BW at the position of one quarter, one quarter, one quarter.
401
00:40:04,500 --> 00:40:14,850
And the other possibility we also discussed is the sodium chloride structure which has you've w at one half, one half, one half.
402
00:40:17,520 --> 00:40:21,630
So can we figure out which one of these two possibilities it is.
403
00:40:23,160 --> 00:40:31,260
Okay. Well, let's see if we can do it. So let's try first try one, try zinc blend structure.
404
00:40:32,130 --> 00:40:35,400
Okay. So in this case s basis.
405
00:40:37,480 --> 00:40:48,730
Squared equals well, we have b titanium plus B carbon either the two pi i h k l and then one quarter of one quarter,
406
00:40:48,730 --> 00:40:55,209
one quarter, which we can also write this factor here.
407
00:40:55,210 --> 00:41:01,600
This exponential factor here is also equivalent to E2 the pi over two times I
408
00:41:01,900 --> 00:41:08,920
times h plus k plus l or equivalently this would be I to the h plus k puzzle.
409
00:41:10,720 --> 00:41:31,450
So I'm out of room here. We're here so we have s basis squared equals b titanium plus v carbon times I to the h plus k plus l squared.
410
00:41:32,020 --> 00:41:36,520
Okay. And what is that as a function of H canal? Well, there's three possibilities.
411
00:41:37,180 --> 00:41:45,970
Case one, I mean, in case A is that H plus k plus l is a multiple of four equals four times an integer n.
412
00:41:47,140 --> 00:41:57,130
So in that case we get SBC squared equals v titanium plus v carbon absolute value squared.
413
00:41:57,940 --> 00:42:08,290
The second possibility case b e is h plus plus l is still even, but it's not a multiple of four.
414
00:42:08,290 --> 00:42:20,170
So we'll call for n plus two, in which case we get spaces squared equals B titanium minus B carbon squared.
415
00:42:21,430 --> 00:42:27,720
And the third case, Casey. Is H plus k plus l is odd.
416
00:42:29,070 --> 00:42:45,450
Odd in which case s basis squared is B titanium plus or minus i times b carbon squared which is b titanium squared plus b carbon squared.
417
00:42:48,140 --> 00:42:53,510
Good. All right. So with the stare at that for a second.
418
00:42:53,690 --> 00:42:59,149
And what can you conclude from this? Well, no matter what B, titanium and B carbon are, as long as they're real,
419
00:42:59,150 --> 00:43:04,580
which they are the largest of the three possibilities is always either case or case.
420
00:43:04,580 --> 00:43:10,550
B, if they be titanium, be carbon to have the same sine in case A is the largest of the three possibilities
421
00:43:10,730 --> 00:43:14,300
if they have opposite sign in case B is the largest of three possibilities.
422
00:43:14,570 --> 00:43:19,070
So the largest peaks should definitely have even H plus K plus l.
423
00:43:19,370 --> 00:43:26,990
Now let's look at this data and you'll see H plus K, plus L and you'll notice that.
424
00:43:28,280 --> 00:43:35,740
The highest peaks always have od plus k plus l systematically always there.
425
00:43:35,750 --> 00:43:40,220
So if 111 is high, 311 is high 331 is high, 333 is high.
426
00:43:40,400 --> 00:43:47,900
All the other ones are small are even so this does not agree with the data does not agree but agree.
427
00:43:51,470 --> 00:44:02,000
Well, we can try something else. We can try the sodium chloride structure, so try sodium chloride, in which case we have.
428
00:44:02,840 --> 00:44:07,160
Well, I wrote it over here. Somewhere over there is one half, one half, one half.
429
00:44:07,460 --> 00:44:15,860
So s basis squared is then b titanium plus.
430
00:44:16,400 --> 00:44:27,350
B carbon times e to the to pi i h k now dotted into one half, one half, one half, and then all squared.
431
00:44:28,620 --> 00:44:40,920
And that we can be right as v titanium plus v carbon times minus one to the H plus k plus l all squared.
432
00:44:41,490 --> 00:44:50,430
So there's two cases we can have. Case one case or case a is h plus k plus l is odd.
433
00:44:51,030 --> 00:44:56,220
In which case we have B titanium minus v carbon squared is the answer.
434
00:44:57,300 --> 00:45:12,150
Whereas case B case v is h plus k plus l equals, even in which case we have b titanium plus b carbon squared.
435
00:45:14,660 --> 00:45:18,140
Okay. So now what's which one is largest here?
436
00:45:18,470 --> 00:45:25,440
Well, you can see that if vitamin B carbon have the same sign, then the even will always be larger.
437
00:45:25,460 --> 00:45:29,550
Whereas if they have the opposite sign, then the odd will have larger.
438
00:45:29,570 --> 00:45:33,770
Well, if you look at the data, the H plus Capossela Ott is clearly much larger.
439
00:45:33,950 --> 00:45:46,790
And so that tells us it's it's consistent with this case that if B titanium be titanium has opposite sine it sine of be carbon.
440
00:45:49,510 --> 00:45:54,219
So the large peaks are be titanium minus be carbon squared with be titanium and be carbon having
441
00:45:54,220 --> 00:46:00,550
opposite side and the small peaks are the sum of the two with them having opposite sine as well.
442
00:46:01,070 --> 00:46:04,600
Okay. Everyone happy with that? More or less.
443
00:46:04,850 --> 00:46:07,960
Little bit. Yes, please. Yes. All right. Thank you.
444
00:46:08,350 --> 00:46:11,950
All right. Good. So let's see.
445
00:46:11,950 --> 00:46:16,690
So here's the summary. Summary slide. So we conclude it has to be sodium chloride structure.
446
00:46:18,010 --> 00:46:21,639
Let's see. So we can actually go on and discuss.
447
00:46:21,640 --> 00:46:25,150
Oh, here. Let's discuss how you make neutrons for a second. So.
448
00:46:26,420 --> 00:46:32,780
We discussed how you make how you make ashtrays in the last in the last last lecture, I think.
449
00:46:33,080 --> 00:46:38,180
But we haven't discussed how you make neutrons. This is a neutron facility escalation.
450
00:46:38,180 --> 00:46:44,000
Neutron source, also right down the road next to the synchrotron source, rather large and expensive apparatus.
451
00:46:44,690 --> 00:46:48,140
What we need is we need neutrons with a wavelength on the order of one angstrom.
452
00:46:49,310 --> 00:46:56,540
How do we get that? Well, if the wavelength is one angstrom, the momentum is two pi h bar over lambda.
453
00:46:56,900 --> 00:47:09,050
And from that we can calculate that the energy we need is p squared over two m equals approximately 80 million electron volts or about 800 kelvin.
454
00:47:09,150 --> 00:47:12,770
You put in the appropriate factor Botswana's constant to make energy a temperature.
455
00:47:13,100 --> 00:47:18,530
And this might sound to you like it's hot, but in fact, 800 Kelvin is really, really cold for neutrons.
456
00:47:18,740 --> 00:47:21,980
Why is that? Well, how do you get the neutrons in the first place?
457
00:47:23,720 --> 00:47:27,710
Making neutrons make neutrons.
458
00:47:28,250 --> 00:47:31,790
Well, back in the old days, they didn't have neutrons.
459
00:47:32,060 --> 00:47:36,260
And then along came the the Second World War and the Atomic Bomb Project.
460
00:47:36,410 --> 00:47:42,830
And after the atomic bomb project, they had nuclear reactors. And nuclear reactors have neutrons as a by-product.
461
00:47:42,890 --> 00:47:48,170
So possibly one is a by-product of vision.
462
00:47:51,470 --> 00:47:56,120
And you can imagine that when a neutron comes off of a of a nucleus and some fission process,
463
00:47:56,300 --> 00:48:01,040
it's coming out with millions of electron volts rather than miller electron volts.
464
00:48:01,040 --> 00:48:07,910
So it's extremely, extremely energetic. So this 800 Kelvin is extremely low energy.
465
00:48:08,150 --> 00:48:14,090
There's a more favoured method for making because people don't like to have atomic reactors in their backyard these days.
466
00:48:14,840 --> 00:48:21,020
There's a more favourite method for making neutrons and in fact it's more efficient method as well, which is known as spoliation.
467
00:48:22,640 --> 00:48:26,480
And that's what this source is. This neutron source is a separation source.
468
00:48:26,780 --> 00:48:33,409
The idea of a separation source is not unlike the idea of an X-ray tube, where you take usually a proton,
469
00:48:33,410 --> 00:48:41,690
a proton, and you accelerate it to about a giga electron volt and you hit it onto a target target.
470
00:48:42,950 --> 00:48:47,600
And the idea is to smash some nucleus, some nuclei, and kick off some neutrons.
471
00:48:48,170 --> 00:48:57,110
Now, again, since you're hitting it with such a high a high energy, you're going to kick off neutrons with some very, very high energy as well.
472
00:48:57,410 --> 00:49:04,160
So whichever method you use to create neutrons, they're coming off with an incredibly high energy and you have to make them cold.
473
00:49:04,430 --> 00:49:13,220
So to make them cold, what you use is you use a moderator and a moderator is basically just a big tank of some substance
474
00:49:13,490 --> 00:49:18,770
where the neutrons can go in the substance and bounce around giving off their energy to the substance.
475
00:49:18,770 --> 00:49:21,920
So frequently people use carbon graphite.
476
00:49:22,250 --> 00:49:27,200
Graphite is carbon graphite, carbon water, heavy water, lots of materials,
477
00:49:27,200 --> 00:49:32,870
anything that the neutrons can bump into and give off their energy to and eventually out.
478
00:49:32,870 --> 00:49:40,430
The other end of this thing you get you get neutrons and you want to kill them down until they have about 800 kelvin worth of energy.
479
00:49:40,700 --> 00:49:43,280
Now, once you have the the neutrons coming out,
480
00:49:43,490 --> 00:49:53,330
you need to somehow figure out how to make them a monochromatic monochrome, meaning you just want to get one wavelength.
481
00:49:53,330 --> 00:49:59,190
Remember, that was the first rule. Know your wavelength. So you have to monochrome eight.
482
00:49:59,210 --> 00:50:02,980
So how do you monochrome one way method one is to distract.
483
00:50:04,040 --> 00:50:08,240
So you can use a crystal. It's known we mentioned this in the case of X-rays as well.
484
00:50:08,450 --> 00:50:11,600
Different from a known crystal. From known crystal.
485
00:50:15,380 --> 00:50:21,140
And that sometimes works. But there's another method which is also used, which is very nice for neutrons,
486
00:50:21,380 --> 00:50:29,390
which is a time of flight method, time of flight, which works in the following way.
487
00:50:29,930 --> 00:50:35,240
So you have a beam of neutrons coming in over here, and then you have sort of a window which you can open and close,
488
00:50:35,660 --> 00:50:39,200
open and close whenever you want, and then you have a long spacing.
489
00:50:39,610 --> 00:50:43,730
L From here to here and another window which you can open and close.
490
00:50:44,210 --> 00:50:47,960
So you open up and close the window very quickly to let in a little pulse of neutrons.
491
00:50:48,260 --> 00:50:52,220
And then at some time Delta T later you open up the other window.
492
00:50:52,700 --> 00:51:00,230
So then you know l you know T So if it made it from the first window to the second window in Delta T You know, it's velocity.
493
00:51:00,590 --> 00:51:03,740
So L divided by delta t is its velocity.
494
00:51:03,830 --> 00:51:07,250
If it's no, it's velocity, you know, it's momentum. If you know it's momentum, you know it's wavelength.
495
00:51:07,610 --> 00:51:12,110
So you can select by knowing how by giving it a certain amount of time to get from
496
00:51:12,110 --> 00:51:15,530
the first window to the second window you can select for a particular velocity,
497
00:51:15,530 --> 00:51:19,700
therefore predict particular momentum, therefore a particular wavelength.
498
00:51:20,720 --> 00:51:25,670
One more thing that I do want to mention about neutrons will say maybe just one or two more things about them in the.
499
00:51:25,750 --> 00:51:29,560
Next lecture. But I do want to mention now is one of the reasons people like neutrons.
500
00:51:30,400 --> 00:51:38,460
Is the neutrons actually have a spin. And that enables you to see things that have important where the spin is important.
501
00:51:38,470 --> 00:51:42,730
If you have more spins in one region than another region that neutrons are sensitive to,
502
00:51:42,730 --> 00:51:48,500
that they feel to the local magnetic environment as a potential, whereas photons do not.
503
00:51:48,520 --> 00:51:52,840
They don't care about the local magnetic environment. Neutrons do because the neutrons have some spin.
504
00:51:53,140 --> 00:51:57,250
Okay, we'll stop there and we'll pick up. Finish this off next next lecture.