1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,830 Commons License. 3 00:00:03,830 --> 00:00:06,840 Your support will help MIT OpenCourseWare continue to 4 00:00:06,840 --> 00:00:10,510 offer high-quality educational resources for free. 5 00:00:10,510 --> 00:00:13,390 To make a donation, or view additional materials from 6 00:00:13,390 --> 00:00:17,490 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,490 --> 00:00:18,740 ocw.mit.edu. 8 00:00:21,960 --> 00:00:24,080 PROFESSOR: OK, OK, Settle down. 9 00:00:24,080 --> 00:00:25,380 Settle down. 10 00:00:25,380 --> 00:00:26,430 It's 11:05. 11 00:00:26,430 --> 00:00:27,940 It's time to learn. 12 00:00:27,940 --> 00:00:29,190 All right. 13 00:00:32,580 --> 00:00:36,130 Last day we were looking at the emissions spectrum of the 14 00:00:36,130 --> 00:00:38,370 target in the x-ray tube. 15 00:00:38,370 --> 00:00:43,430 The target is the anode, and it's being a bombarded by 16 00:00:43,430 --> 00:00:46,970 electrons that have been accelerated across a potential 17 00:00:46,970 --> 00:00:50,370 difference of some tens of thousands of volts, and this 18 00:00:50,370 --> 00:00:54,720 was what we saw as the output, intensity versus wavelength. 19 00:00:54,720 --> 00:00:58,360 We see this whale shaped curve which we have attributed to 20 00:00:58,360 --> 00:01:02,060 bremsstrahlung, which is the breaking radiation. 21 00:01:02,060 --> 00:01:04,530 And then if the voltage is high enough, high 22 00:01:04,530 --> 00:01:05,550 enough to do what? 23 00:01:05,550 --> 00:01:09,650 High enough to eject inner shell electrons, we will get 24 00:01:09,650 --> 00:01:14,310 the cascade, and the cascade will give rise to specific 25 00:01:14,310 --> 00:01:17,780 values of wavelength associated with transitions 26 00:01:17,780 --> 00:01:19,570 within the target atom. 27 00:01:19,570 --> 00:01:21,240 There's too much talking. 28 00:01:21,240 --> 00:01:25,450 I want it absolutely silent or else somebody is going to 29 00:01:25,450 --> 00:01:27,530 leave. That's the only way it works here. 30 00:01:27,530 --> 00:01:28,440 I talk. 31 00:01:28,440 --> 00:01:29,170 You listen. 32 00:01:29,170 --> 00:01:32,320 And if you have a problem with that, there's a door here. 33 00:01:32,320 --> 00:01:33,200 There's a door there. 34 00:01:33,200 --> 00:01:35,180 And there's a door at the back. 35 00:01:35,180 --> 00:01:37,670 So let's get it straight right now. 36 00:01:37,670 --> 00:01:39,210 It's the only way it works here. 37 00:01:45,040 --> 00:01:49,760 Now what happens is that these characteristic lines are 38 00:01:49,760 --> 00:01:51,850 calculable in part. 39 00:01:51,850 --> 00:01:55,560 The K alpha line, the L alpha line we saw from Moseley's 40 00:01:55,560 --> 00:01:59,930 law, which is up here on the slide. 41 00:01:59,930 --> 00:02:04,020 and the leading edge of the whale shape curve, the 42 00:02:04,020 --> 00:02:07,950 bremsstrahlung is calculable by the Duane-Hunt Law. 43 00:02:07,950 --> 00:02:09,900 The rest of this is not calculable. 44 00:02:09,900 --> 00:02:12,510 So there it is. 45 00:02:15,690 --> 00:02:19,206 What I want to do today is harness this radiation. 46 00:02:19,206 --> 00:02:22,210 The reason we were studying x-rays in first place was that 47 00:02:22,210 --> 00:02:24,960 we said they had a length scale that was comparable to 48 00:02:24,960 --> 00:02:27,380 that of atomic dimensions. 49 00:02:27,380 --> 00:02:32,840 And so now we want to close the circle here and probe 50 00:02:32,840 --> 00:02:35,760 atomic arrangement by x-ray diffraction. 51 00:02:35,760 --> 00:02:37,320 And it goes by this three-letter 52 00:02:37,320 --> 00:02:40,060 initialization XRD. 53 00:02:40,060 --> 00:02:44,180 And in order to use x-ray diffraction to probe atomic 54 00:02:44,180 --> 00:02:47,520 structure, we're going to make another model. 55 00:02:47,520 --> 00:02:48,930 And this model is going to be simple. 56 00:02:48,930 --> 00:02:51,010 And it's going to be inaccurate, but it's going to 57 00:02:51,010 --> 00:02:54,080 be good enough to explain the phenomena that 58 00:02:54,080 --> 00:02:55,220 we're going to study. 59 00:02:55,220 --> 00:02:58,880 So the first thing I want to do in constructing a model 60 00:02:58,880 --> 00:03:03,410 that will allow to use x-ray diffraction, is to model the 61 00:03:03,410 --> 00:03:05,240 atoms as mirrors. 62 00:03:05,240 --> 00:03:07,250 That's the first assumption that we make. 63 00:03:07,250 --> 00:03:08,960 We model the atoms as mirrors. 64 00:03:14,110 --> 00:03:17,470 And when we model the atoms as mirrors, this means that we 65 00:03:17,470 --> 00:03:19,990 will invoke the laws of specular reflection. 66 00:03:19,990 --> 00:03:23,940 So if this table top is a plane of atoms, and I have a 67 00:03:23,940 --> 00:03:27,030 beam coming down at an angle, the laws of specular 68 00:03:27,030 --> 00:03:30,370 reflection say that the angle of incidence will equal the 69 00:03:30,370 --> 00:03:31,830 angle of reflection. 70 00:03:31,830 --> 00:03:35,830 So that's all we're doing by modeling the atoms as mirrors. 71 00:03:35,830 --> 00:03:42,425 Laws of speculative reflection apply. 72 00:03:50,570 --> 00:03:55,350 So that simply means that the angle of incidence, theta 73 00:03:55,350 --> 00:03:58,846 incidence equals theta reflection. 74 00:03:58,846 --> 00:04:03,250 And you'll see how that comes into play in a minute. 75 00:04:03,250 --> 00:04:05,560 And then the second thing we're going to do in order to 76 00:04:05,560 --> 00:04:10,720 get intensities is to use the concept of constructive and 77 00:04:10,720 --> 00:04:12,910 destructive interference. 78 00:04:12,910 --> 00:04:13,900 OK. 79 00:04:13,900 --> 00:04:16,169 So we apply interference criteria. 80 00:04:24,060 --> 00:04:29,150 And I will explain what that means in a 81 00:04:29,150 --> 00:04:30,930 second with a diagram. 82 00:04:30,930 --> 00:04:38,710 But in dense text, it simply means that in phase rays-- 83 00:04:38,710 --> 00:04:41,920 it sounds like this rain in Spain falls mainly in the 84 00:04:41,920 --> 00:04:44,845 plain-- in-phase rays amplify. 85 00:04:49,030 --> 00:04:55,510 And out-of-phase rays dampen. 86 00:04:59,880 --> 00:05:03,110 So at some point, you'll study this in physics, but we need 87 00:05:03,110 --> 00:05:06,630 it right now, so I'll give you a little bit of foreshadowing, 88 00:05:06,630 --> 00:05:09,120 and then you'll be ahead of the game when you 89 00:05:09,120 --> 00:05:10,310 meet this in physics. 90 00:05:10,310 --> 00:05:16,740 So what I'm going to do is I'm going to show you two rays, 91 00:05:16,740 --> 00:05:21,910 and they both have the same wavelength. 92 00:05:21,910 --> 00:05:25,160 And I want to show you in phase. 93 00:05:25,160 --> 00:05:27,730 And so I want to illustrate this concept here, in-phase 94 00:05:27,730 --> 00:05:28,760 rays amplify. 95 00:05:28,760 --> 00:05:30,930 So let's give one ray here. 96 00:05:30,930 --> 00:05:33,610 I'm going to depict it as a wave. All right, so 97 00:05:33,610 --> 00:05:34,890 that's ray number 1. 98 00:05:34,890 --> 00:05:37,000 And beneath it, is ray number 2. 99 00:05:37,000 --> 00:05:38,140 It's got the same wavelength. 100 00:05:38,140 --> 00:05:41,550 So that means crest-to-crest distance is the same, and if 101 00:05:41,550 --> 00:05:45,750 it's in phase, in phase means that crest lines up with 102 00:05:45,750 --> 00:05:47,470 crest, trough lines up with trough. 103 00:05:47,470 --> 00:05:50,960 So I'm going to make another one seemingly identical. 104 00:05:50,960 --> 00:05:55,440 It's supposed to be identical to the capacity of 105 00:05:55,440 --> 00:05:56,890 my ability to draw. 106 00:05:56,890 --> 00:06:01,670 And so what this means is that these will amplify. 107 00:06:01,670 --> 00:06:06,050 So 1 plus 2, 1 plus 2 will give me something 108 00:06:06,050 --> 00:06:07,570 that looks like this. 109 00:06:07,570 --> 00:06:08,920 I'm going to line it up again. 110 00:06:08,920 --> 00:06:14,190 Only now the amplitude is double the amplitude of a 111 00:06:14,190 --> 00:06:14,960 single ray. 112 00:06:14,960 --> 00:06:16,820 That's what I'm trying to depict here. 113 00:06:16,820 --> 00:06:17,430 OK. 114 00:06:17,430 --> 00:06:19,560 So this is amplification. 115 00:06:19,560 --> 00:06:21,680 Now over here, I want to do the same thing. 116 00:06:21,680 --> 00:06:24,710 Only I'm going to have two rays, same 117 00:06:24,710 --> 00:06:27,130 wavelength, but out of phase. 118 00:06:27,130 --> 00:06:35,290 OK, so in phase is here, out of phase is to the right. 119 00:06:35,290 --> 00:06:40,060 So we'll have ray number 3, which does this. 120 00:06:40,060 --> 00:06:43,320 And ray number 4, I'm going to line up with the crest of ray 121 00:06:43,320 --> 00:06:46,760 number 3, the trough of ray number 4, but it's going to be 122 00:06:46,760 --> 00:06:48,160 the same wavelength. 123 00:06:48,160 --> 00:06:52,430 So the distance between successive crests is the same. 124 00:06:52,430 --> 00:06:56,310 Only they're going to be offset by exactly half 125 00:06:56,310 --> 00:06:57,450 wavelength. 126 00:06:57,450 --> 00:07:00,800 So the crest here of wave 3 lines up with the 127 00:07:00,800 --> 00:07:01,880 trough of wave 4. 128 00:07:01,880 --> 00:07:05,400 The trough of wave 3 lines up with the crest of wave 4, and 129 00:07:05,400 --> 00:07:08,050 so on, and so on, and so on. 130 00:07:08,050 --> 00:07:10,340 And if I drew this thing accurately, it would be 131 00:07:10,340 --> 00:07:11,080 obvious to you. 132 00:07:11,080 --> 00:07:13,630 But since I can't draw very well, then we're going 133 00:07:13,630 --> 00:07:16,170 to take 3 plus 4. 134 00:07:16,170 --> 00:07:18,950 And the sum of the crest and the trough is 0. 135 00:07:18,950 --> 00:07:21,160 The sum of the trough and the crest is 0. 136 00:07:21,160 --> 00:07:23,470 And so the sum here is 0. 137 00:07:23,470 --> 00:07:25,150 All right this is total destruction. 138 00:07:25,150 --> 00:07:27,750 This is cancellation. 139 00:07:27,750 --> 00:07:31,370 So in the extreme, you can have destructive interference 140 00:07:31,370 --> 00:07:34,750 that dampens to the point of obliteration. 141 00:07:34,750 --> 00:07:37,960 OK, so that's the two extremes, full amplification 142 00:07:37,960 --> 00:07:40,460 and complete cancellation. 143 00:07:40,460 --> 00:07:46,060 So now, I want to take this concept of planes as mirrors 144 00:07:46,060 --> 00:07:55,330 and rules of interference criteria, and now what I want 145 00:07:55,330 --> 00:07:56,970 to do is illuminate. 146 00:07:56,970 --> 00:07:58,940 And I want to use the full board here. 147 00:07:58,940 --> 00:07:59,770 So here's what we're going to do. 148 00:07:59,770 --> 00:08:06,260 We're going to take in phase, coherent-- 149 00:08:06,260 --> 00:08:07,350 that's what coherent means. 150 00:08:07,350 --> 00:08:10,900 Coherent means that the radiation is in phase and 151 00:08:10,900 --> 00:08:12,150 monochromatic. 152 00:08:14,000 --> 00:08:14,860 What's that mean? 153 00:08:14,860 --> 00:08:20,180 Monochromatic means one color, which means single wavelength. 154 00:08:20,180 --> 00:08:24,100 So I have something of a single wavelength in phase. 155 00:08:24,100 --> 00:08:33,120 Coherent, monochromatic, incident radiation, so I've 156 00:08:33,120 --> 00:08:34,720 got a plurality of rays. 157 00:08:34,720 --> 00:08:37,930 I have this thing flooded by a beam. 158 00:08:37,930 --> 00:08:39,380 So what I'm going to do, is I'm going to 159 00:08:39,380 --> 00:08:41,750 draw some atoms here. 160 00:08:45,740 --> 00:08:50,210 And I'll put a second batch on top. 161 00:08:50,210 --> 00:08:52,530 And now I'm going to model these as mirrors. 162 00:08:52,530 --> 00:08:57,430 So I'm going to have the incident beam come in like so. 163 00:08:57,430 --> 00:09:01,400 The incident beam comes in like so at an angle theta. 164 00:09:01,400 --> 00:09:05,010 So this is theta incident. 165 00:09:05,010 --> 00:09:06,930 This is the angle of incidence. 166 00:09:06,930 --> 00:09:10,280 And according to our model, the beam is reflected at the 167 00:09:10,280 --> 00:09:11,220 same angle. 168 00:09:11,220 --> 00:09:13,710 So this is theta, reflection of theta. 169 00:09:13,710 --> 00:09:16,570 Reflection equals theta incidence. 170 00:09:16,570 --> 00:09:18,810 So that's the first thing that we do. 171 00:09:18,810 --> 00:09:23,470 And then the second thing we do, is we use the laws of 172 00:09:23,470 --> 00:09:24,260 interference. 173 00:09:24,260 --> 00:09:28,200 So I'm going to take a second beam, and it's going to go 174 00:09:28,200 --> 00:09:31,050 down to this atom here. 175 00:09:31,050 --> 00:09:34,360 So I'm going to bring it down like so. 176 00:09:34,360 --> 00:09:36,910 And it's going to come in at the same angle. 177 00:09:36,910 --> 00:09:39,540 And it's going to leave at the same angle. 178 00:09:39,540 --> 00:09:41,100 And if you'll excuse the drawing, these 179 00:09:41,100 --> 00:09:42,295 lines should be parallel. 180 00:09:42,295 --> 00:09:44,920 They should be coming in parallel, and they should be 181 00:09:44,920 --> 00:09:48,320 leaving parallel where these angles theta i and 182 00:09:48,320 --> 00:09:50,110 theta r are the same. 183 00:09:50,110 --> 00:09:53,450 Now how do we invoke the interference criteria? 184 00:09:53,450 --> 00:09:56,070 Well what I'm going to do is put a marker here. 185 00:09:56,070 --> 00:09:57,250 I'll put a marker. 186 00:09:57,250 --> 00:10:01,060 And I'll call the first one ray number 1, and the second 187 00:10:01,060 --> 00:10:03,670 one ray number 2. 188 00:10:03,670 --> 00:10:07,950 And what I'm going to show is if I start from ray number 1-- 189 00:10:07,950 --> 00:10:10,240 maybe it's time for colored chalk-- 190 00:10:10,240 --> 00:10:15,020 so if I start from ray number 1, to show that it's in phase 191 00:10:15,020 --> 00:10:19,470 with ray number 2, I'm going to draw this little waveform, 192 00:10:19,470 --> 00:10:21,080 and they're both lined up. 193 00:10:21,080 --> 00:10:24,780 So you can see crest lines up with crest. Trough lines up 194 00:10:24,780 --> 00:10:27,720 with trough, and they both have the same wavelength. 195 00:10:27,720 --> 00:10:31,280 So that makes the point coherent, monochromatic. 196 00:10:31,280 --> 00:10:33,820 But now things get interesting, because you can 197 00:10:33,820 --> 00:10:37,710 see that ray number 1 travels a shorter distance than ray 198 00:10:37,710 --> 00:10:42,590 number 2, and we can put on some coordinates here and mark 199 00:10:42,590 --> 00:10:47,190 the geometry, so we can say that the point at which the 200 00:10:47,190 --> 00:10:50,530 ray number 2 has to start traveling a longer distance, 201 00:10:50,530 --> 00:10:52,890 I'll drop a normal down, and I'm going to 202 00:10:52,890 --> 00:10:55,020 call this point A. 203 00:10:55,020 --> 00:10:56,740 This is point A. 204 00:10:56,740 --> 00:10:59,180 The bottom here is point B. 205 00:10:59,180 --> 00:11:02,620 And then up here where it catches up with ray number 1, 206 00:11:02,620 --> 00:11:04,490 the reflected ray number 1, I'm going to 207 00:11:04,490 --> 00:11:06,840 call this point C. 208 00:11:06,840 --> 00:11:12,120 So you can you can see that ray number 2 has to travel 209 00:11:12,120 --> 00:11:16,440 this extra distance, AB plus BC. 210 00:11:16,440 --> 00:11:22,930 And now, if I want to get out to here where I'm now in the 211 00:11:22,930 --> 00:11:27,510 reflected zone, and I want ray number 1 reflected and ray 212 00:11:27,510 --> 00:11:32,100 number 2 reflected to continue to be in phase, there's a 213 00:11:32,100 --> 00:11:35,670 geometric constraint on the dimension of AB 214 00:11:35,670 --> 00:11:37,840 plus BC, isn't there? 215 00:11:37,840 --> 00:11:42,090 That length, AB plus BC must be a whole number of 216 00:11:42,090 --> 00:11:43,416 wavelengths. 217 00:11:43,416 --> 00:11:47,870 It must be an integer number of wavelengths of whatever 218 00:11:47,870 --> 00:11:50,600 this thing is, otherwise, these two will 219 00:11:50,600 --> 00:11:52,210 not being in phase. 220 00:11:52,210 --> 00:11:54,240 So that's the interference criterion. 221 00:11:54,240 --> 00:12:00,540 And n can equal 1, 2, 3, and so on, so this is called the 222 00:12:00,540 --> 00:12:02,840 order of reflection. 223 00:12:07,530 --> 00:12:09,630 And then we can go through the trigonometry. 224 00:12:09,630 --> 00:12:10,840 I'm not going to do it in class. 225 00:12:10,840 --> 00:12:11,810 It's just a waste of time. 226 00:12:11,810 --> 00:12:13,490 It'll bore everybody to tears. 227 00:12:13,490 --> 00:12:16,720 But if you want to do that at home at some point when you've 228 00:12:16,720 --> 00:12:18,670 got nothing to do, you can go through. 229 00:12:18,670 --> 00:12:19,790 You've got all the numbers here. 230 00:12:19,790 --> 00:12:21,340 You know the value of lambda. 231 00:12:21,340 --> 00:12:24,150 You know the distance AB, and you know what 232 00:12:24,150 --> 00:12:25,270 this distance is. 233 00:12:25,270 --> 00:12:27,780 This distance between successive planes is your 234 00:12:27,780 --> 00:12:30,400 dhkl, isn't it? 235 00:12:30,400 --> 00:12:33,210 So if you go through all of this analysis, you will show 236 00:12:33,210 --> 00:12:41,090 that n lambda is equal to 2 times the d of the hkl spacing 237 00:12:41,090 --> 00:12:44,800 times the sign of theta. 238 00:12:44,800 --> 00:12:46,900 And this is called Bragg's Law. 239 00:12:50,230 --> 00:12:54,610 And Bragg's Law governs the reflection of incident 240 00:12:54,610 --> 00:12:56,590 radiation by a crystal. 241 00:12:56,590 --> 00:12:59,160 Now you'll notice I didn't put an apostrophe here. 242 00:12:59,160 --> 00:13:02,120 Some people put the apostrophe here, because the 243 00:13:02,120 --> 00:13:03,510 man's name was Bragg. 244 00:13:03,510 --> 00:13:06,670 Some people put the apostrophe here because, in fact, it was 245 00:13:06,670 --> 00:13:08,900 father and son. 246 00:13:08,900 --> 00:13:13,380 And they both together won the Nobel Prize in 1915. 247 00:13:13,380 --> 00:13:17,430 Now there have been people who lived to see, as Nobel Prize 248 00:13:17,430 --> 00:13:20,830 winners, one of their children win the Nobel Prize. but this 249 00:13:20,830 --> 00:13:24,820 is the only time in history a father and son team together 250 00:13:24,820 --> 00:13:27,430 won the Nobel Prize for the same work. 251 00:13:27,430 --> 00:13:31,070 And there are undoubtedly people sitting in this room 252 00:13:31,070 --> 00:13:33,510 who think the fact that the father and son could work 253 00:13:33,510 --> 00:13:37,240 together in physics for an extended period of time alone 254 00:13:37,240 --> 00:13:40,180 is deserving of the Nobel Prize. 255 00:13:40,180 --> 00:13:47,160 Now in 3.091, I'm going to keep it simple, always choose 256 00:13:47,160 --> 00:13:51,930 first order reflection, always n equals 1 in Bragg's Law. 257 00:13:54,920 --> 00:13:58,860 So therefore, we will write Bragg's Law as lambda equals 258 00:13:58,860 --> 00:14:01,810 2d sin theta. 259 00:14:01,810 --> 00:14:05,760 And to make the point, the d is specific to a particular 260 00:14:05,760 --> 00:14:06,930 set of planes. 261 00:14:06,930 --> 00:14:09,870 So it's a d spacing of the hkl planes. 262 00:14:09,870 --> 00:14:14,480 And it's the theta associated with the correct reflection of 263 00:14:14,480 --> 00:14:16,300 the hkl planes. 264 00:14:16,300 --> 00:14:20,050 Now how does destructive interference come into play? 265 00:14:20,050 --> 00:14:23,400 Destructive interference comes into play should there be a 266 00:14:23,400 --> 00:14:26,280 situation where I have a third ray. 267 00:14:26,280 --> 00:14:28,690 I'm going to bring a third ray in also at the 268 00:14:28,690 --> 00:14:30,360 same angle of incidence. 269 00:14:30,360 --> 00:14:34,810 And that ray is halfway between these two rays. 270 00:14:34,810 --> 00:14:38,300 And what situation could that be? 271 00:14:38,300 --> 00:14:40,720 That could be a situation in which the crystal structure 272 00:14:40,720 --> 00:14:44,350 has in a plane out from the board, an atom that sits 273 00:14:44,350 --> 00:14:46,890 halfway between these other two atoms. 274 00:14:46,890 --> 00:14:48,730 And if that happens, you can see. 275 00:14:48,730 --> 00:14:50,160 We can go through the derivation. 276 00:14:50,160 --> 00:14:52,890 But there's going to be a path length difference that's 277 00:14:52,890 --> 00:14:56,370 exactly 1/2 wavelength different. 278 00:14:56,370 --> 00:14:57,300 That's bad. 279 00:14:57,300 --> 00:15:00,420 It's not just there's going to be some measure of reduction. 280 00:15:00,420 --> 00:15:02,620 It's going to be total cancellation. 281 00:15:02,620 --> 00:15:08,240 And so by going through the set of crystal structures and 282 00:15:08,240 --> 00:15:11,580 recognizing that when you have 1/2 wavelength difference, you 283 00:15:11,580 --> 00:15:13,280 get destructive interference. 284 00:15:13,280 --> 00:15:15,100 And therefore, you will see no line. 285 00:15:15,100 --> 00:15:18,730 The incident radiation will come in at this angle, and at 286 00:15:18,730 --> 00:15:21,980 the reflection angle for that particular plane, there will 287 00:15:21,980 --> 00:15:24,030 be nothing detected. 288 00:15:24,030 --> 00:15:28,960 So you can say that you have a combination of selection rules 289 00:15:28,960 --> 00:15:35,190 that involve the integral of both the rules of interference 290 00:15:35,190 --> 00:15:38,180 and interaction with the crystal structure. 291 00:15:38,180 --> 00:15:43,290 So let's generalize this and say if we take interference 292 00:15:43,290 --> 00:15:54,530 criteria plus the crystal structure, that is to say, the 293 00:15:54,530 --> 00:15:59,870 instant relationship of atom positions for a particular 294 00:15:59,870 --> 00:16:06,190 specimen, the combination of that will give rise to the set 295 00:16:06,190 --> 00:16:11,456 of expected reflections. 296 00:16:11,456 --> 00:16:15,760 So only when you satisfy the Bragg criterion do you get 297 00:16:15,760 --> 00:16:17,200 reflection. 298 00:16:17,200 --> 00:16:18,840 So you move the specimen. 299 00:16:18,840 --> 00:16:23,900 And only at that special angle will you get the reflections, 300 00:16:23,900 --> 00:16:25,750 will you get constructive interference. 301 00:16:25,750 --> 00:16:28,480 So this, in fact, is a fingerprint. 302 00:16:28,480 --> 00:16:33,780 This is a fingerprint of the crystal. 303 00:16:33,780 --> 00:16:35,920 In this case, we're not getting the chemical identity. 304 00:16:35,920 --> 00:16:38,310 We're getting the structural identity. 305 00:16:38,310 --> 00:16:40,600 So we can determine if something is 306 00:16:40,600 --> 00:16:42,770 BCC, FCC, and so on. 307 00:16:42,770 --> 00:16:51,500 So for example in BCC, that's exactly the case that I just 308 00:16:51,500 --> 00:16:52,440 illustrated. 309 00:16:52,440 --> 00:16:55,020 So in BCC we have atoms at the eight corners. 310 00:16:57,650 --> 00:17:00,540 And we have an atom dead center. 311 00:17:00,540 --> 00:17:02,160 There's a central atom. 312 00:17:02,160 --> 00:17:12,410 And the central atom lies in 0 0 2 plane, doesn't it? 313 00:17:15,240 --> 00:17:18,080 The base of this is 0 0 1. 314 00:17:18,080 --> 00:17:21,140 And the top is an 0 0 1. 315 00:17:21,140 --> 00:17:23,680 But the central atom is an 0 0 2. 316 00:17:23,680 --> 00:17:27,580 And 0 0 2 is halfway between successive 0 0 1. 317 00:17:27,580 --> 00:17:30,170 So can you see that light that comes in? 318 00:17:30,170 --> 00:17:34,300 And it's going to be constructively reinforced. 319 00:17:34,300 --> 00:17:39,390 Off of 0 0 1 is going to be destructively reinforced off 320 00:17:39,390 --> 00:17:46,930 of 0 0 2, and the result is that in BCC, no 0 0 1 321 00:17:46,930 --> 00:17:48,205 reflection observed. 322 00:17:53,410 --> 00:17:56,050 The same thing happens in face centered cubic, right? 323 00:17:56,050 --> 00:17:58,350 What's the face of face centered cubic look like? 324 00:17:58,350 --> 00:18:01,520 It looks like this, doesn't it? 325 00:18:01,520 --> 00:18:04,750 So there's an 0 0 1. 326 00:18:04,750 --> 00:18:06,750 Here's an 0 0 1. 327 00:18:06,750 --> 00:18:08,270 What about this space centered atom? 328 00:18:08,270 --> 00:18:09,900 Where is it? 329 00:18:09,900 --> 00:18:12,020 That's an 0 0 2. 330 00:18:12,020 --> 00:18:13,750 It's halfway between 0 0 1. 331 00:18:13,750 --> 00:18:17,400 So at the angle where you would've expected, if you use 332 00:18:17,400 --> 00:18:21,990 the Bragg Law and calculate the angle at which for your 333 00:18:21,990 --> 00:18:25,400 particular crystal, because you know the d spacing. 334 00:18:25,400 --> 00:18:28,140 You fix the wavelength of your x-radiation 335 00:18:28,140 --> 00:18:29,700 coming out of the generator. 336 00:18:29,700 --> 00:18:32,790 At the angle where you would expect to see reflection off 337 00:18:32,790 --> 00:18:35,730 of 0 0 1, you will see nothing, because 0 0 2 is 338 00:18:35,730 --> 00:18:38,820 canceling 0 0 1. 339 00:18:38,820 --> 00:18:41,090 So you don't have to worry about all this. 340 00:18:41,090 --> 00:18:43,910 This has all been tabulated for you. 341 00:18:43,910 --> 00:18:48,940 Somebody has gone through and done all of this. 342 00:18:48,940 --> 00:18:50,300 Well, this is just making the point. 343 00:18:50,300 --> 00:18:52,140 See, there's a simple cubic. 344 00:18:52,140 --> 00:18:53,570 All the planes are reflecting. 345 00:18:53,570 --> 00:18:55,340 There's body-centered cubic. 346 00:18:55,340 --> 00:18:56,250 That's a/2. 347 00:18:56,250 --> 00:18:57,620 There's face-centered cubic a/2. 348 00:18:57,620 --> 00:18:59,960 0 0 2 is going to cancel 0 0 1. 349 00:18:59,960 --> 00:19:01,340 You're not going to see anything there. 350 00:19:01,340 --> 00:19:03,450 So people have gone through. 351 00:19:03,450 --> 00:19:07,840 And they've made this set of rules for reflection. 352 00:19:07,840 --> 00:19:09,620 So simple cubic, you get reflection 353 00:19:09,620 --> 00:19:11,000 from all the planes. 354 00:19:11,000 --> 00:19:14,260 Body-centered cubic, you just get these planes here. 355 00:19:14,260 --> 00:19:17,660 And it turns out that there's a simple rule that compactly 356 00:19:17,660 --> 00:19:21,790 represents which planes are going to reflect in 357 00:19:21,790 --> 00:19:23,270 body-centered cubic. 358 00:19:23,270 --> 00:19:32,480 And in BCC, the hkl, it's the planes for which h plus k plus 359 00:19:32,480 --> 00:19:34,240 l is an even number. 360 00:19:37,610 --> 00:19:39,850 h plus k plus l is even. 361 00:19:39,850 --> 00:19:41,730 And 0 was counted as even. 362 00:19:41,730 --> 00:19:47,370 So for example, 0 0 1 h plus k plus l is 1. 363 00:19:47,370 --> 00:19:48,550 It doesn't reflect. 364 00:19:48,550 --> 00:19:52,170 0 0 2, 0 plus 0 plus 2 is 2. 365 00:19:52,170 --> 00:19:53,540 It does reflect. 366 00:19:53,540 --> 00:19:54,180 And so on. 367 00:19:54,180 --> 00:19:56,010 You just go down the whole line. 368 00:19:56,010 --> 00:19:59,350 And so these are in ascending order of h squared plus k 369 00:19:59,350 --> 00:20:01,300 squared plus l squared, because that's a nice way of 370 00:20:01,300 --> 00:20:03,570 deciding how to add them up. 371 00:20:03,570 --> 00:20:06,860 And now in FCC the selection rule is 372 00:20:06,860 --> 00:20:08,160 a little bit different. 373 00:20:08,160 --> 00:20:13,790 In FCC you get reflection from planes when you write the HKL 374 00:20:13,790 --> 00:20:25,480 such that h, k, and l must be either all even numbers, or 375 00:20:25,480 --> 00:20:31,640 all odd numbers, or some people like to say unmixed. 376 00:20:31,640 --> 00:20:34,210 Some crystallographers say unmixed, meaning you can't 377 00:20:34,210 --> 00:20:36,120 have a combination of even numbers and odd number. 378 00:20:36,120 --> 00:20:40,890 So for example, 0 0 1 won't work because 0 is a zero. 379 00:20:40,890 --> 00:20:41,760 That's even. 380 00:20:41,760 --> 00:20:43,160 1 is odd. 381 00:20:43,160 --> 00:20:49,890 But 0 0 2, h plus k plus l all even or all odd, there you go 382 00:20:49,890 --> 00:20:51,730 all even or all odd unmixed. 383 00:20:51,730 --> 00:20:53,410 So this will work. 384 00:20:53,410 --> 00:20:54,120 And then so on. 385 00:20:54,120 --> 00:20:57,160 So you can go through and see which ones work. 386 00:20:57,160 --> 00:20:57,500 All right. 387 00:20:57,500 --> 00:21:01,140 So the next one here is in the sequence. 388 00:21:01,140 --> 00:21:02,950 This sums to 1. 389 00:21:02,950 --> 00:21:04,260 And what would sum to 2? 390 00:21:04,260 --> 00:21:06,500 It would be 0 1 1. 391 00:21:06,500 --> 00:21:10,310 And 0 1 1 doesn't work either because zero is even. 392 00:21:10,310 --> 00:21:11,450 And 1 is odd. 393 00:21:11,450 --> 00:21:13,500 The next one of the sequence is 1 1 1. 394 00:21:13,500 --> 00:21:14,580 These are all odd. 395 00:21:14,580 --> 00:21:16,060 So that one reflects. 396 00:21:16,060 --> 00:21:20,240 So in FCC, you don't see 0 0 1, 0 1 1. 397 00:21:20,240 --> 00:21:21,890 But 1 1 1 does reflect. 398 00:21:21,890 --> 00:21:24,760 And then 0 0 2, which is the next one, because 0 squared 399 00:21:24,760 --> 00:21:26,950 plus 0 squared plus 2 squared is 4. 400 00:21:26,950 --> 00:21:29,520 This is even even. 401 00:21:29,520 --> 00:21:31,190 So that one works. 402 00:21:31,190 --> 00:21:32,560 So there's the sequence. 403 00:21:32,560 --> 00:21:38,430 And so now what we can do is use this technique in order to 404 00:21:38,430 --> 00:21:39,520 make measurements. 405 00:21:39,520 --> 00:21:42,400 But before we do so, I want to show you the experimental 406 00:21:42,400 --> 00:21:44,310 measurement, one way. 407 00:21:44,310 --> 00:21:47,490 There's several ways of conducting the measurement. 408 00:21:47,490 --> 00:21:49,790 And so the first way I'm going to show you is called 409 00:21:49,790 --> 00:21:51,040 diffractometry. 410 00:21:54,670 --> 00:21:56,650 And you can do this over in Building 13. 411 00:21:56,650 --> 00:21:59,530 If you get yourself a UROP, you might be assigned to make 412 00:21:59,530 --> 00:22:00,210 some measurements. 413 00:22:00,210 --> 00:22:04,150 So diffractometry is a form of x-ray diffraction. 414 00:22:04,150 --> 00:22:05,680 It's one of the techniques. 415 00:22:05,680 --> 00:22:12,940 And the way it works is you fix the wavelength and vary 416 00:22:12,940 --> 00:22:15,590 the angle of incidence, you don't 417 00:22:15,590 --> 00:22:17,520 rotate the x-ray generator. 418 00:22:17,520 --> 00:22:21,480 You rotate specimen and present continually varying 419 00:22:21,480 --> 00:22:22,470 angles to the thing. 420 00:22:22,470 --> 00:22:26,950 So this shows the technique in operation. 421 00:22:26,950 --> 00:22:28,790 So you have a specimen sitting here. 422 00:22:28,790 --> 00:22:33,870 The specimen can either be a thin film, or in other 423 00:22:33,870 --> 00:22:36,840 instances, we have a finely ground powder. 424 00:22:36,840 --> 00:22:40,040 So each of the powder grains presents a different angle. 425 00:22:40,040 --> 00:22:42,990 And coming out of the collimator is the 426 00:22:42,990 --> 00:22:44,530 monochromatic. 427 00:22:44,530 --> 00:22:46,630 We want a single wavelength. 428 00:22:46,630 --> 00:22:50,880 So this collimator means it's monochromatic and coherent. 429 00:22:50,880 --> 00:22:53,440 So that beam comes and strikes the specimen. 430 00:22:53,440 --> 00:22:56,470 And for historical reasons, instead of calling this the 431 00:22:56,470 --> 00:22:59,860 angle of incidence, and this the angle of reflection, they 432 00:22:59,860 --> 00:23:02,560 call the angle that goes to the detector 2 theta. 433 00:23:02,560 --> 00:23:07,600 In other words, the projection of the beam, this is theta 434 00:23:07,600 --> 00:23:08,930 incident equals theta r. 435 00:23:08,930 --> 00:23:11,070 So this is really 2 theta incident, or 436 00:23:11,070 --> 00:23:12,280 it's 2 theta reflected. 437 00:23:12,280 --> 00:23:13,020 It's the same thing. 438 00:23:13,020 --> 00:23:15,840 So you'll see x-ray data usually reported 439 00:23:15,840 --> 00:23:17,410 in units of 2 theta. 440 00:23:17,410 --> 00:23:18,800 And here's the detector. 441 00:23:18,800 --> 00:23:23,220 And then all you do is you rotate the specimen. 442 00:23:23,220 --> 00:23:26,450 And by rotating the specimen with a detector, you're able 443 00:23:26,450 --> 00:23:29,600 to get the entire set of reflections. 444 00:23:29,600 --> 00:23:33,980 And whenever you move through an angle that satisfies the 445 00:23:33,980 --> 00:23:37,210 Bragg Law, you get a peak in intensity. 446 00:23:37,210 --> 00:23:39,420 And here's what the output would look like. 447 00:23:39,420 --> 00:23:42,550 So you're plotting intensity as a function of 2 theta. 448 00:23:42,550 --> 00:23:46,040 So somewhere along here at around, it looks like about 38 449 00:23:46,040 --> 00:23:49,610 degrees, you have satisfied the Bragg angle, and you get 450 00:23:49,610 --> 00:23:51,100 constructive interference. 451 00:23:51,100 --> 00:23:53,890 When you move off of 38 degrees, destructive 452 00:23:53,890 --> 00:23:56,470 interference reigns supreme, and you get almost no 453 00:23:56,470 --> 00:23:57,090 reflection. 454 00:23:57,090 --> 00:23:59,950 Then you get to the right angle for 2 0 0, you get 455 00:23:59,950 --> 00:24:00,600 reflection. 456 00:24:00,600 --> 00:24:02,550 Well, all you get is these lines. 457 00:24:02,550 --> 00:24:04,360 You don't get these numbers. 458 00:24:04,360 --> 00:24:06,030 These numbers you don't get for free. 459 00:24:06,030 --> 00:24:08,280 So here's the experiment that we're going to do. 460 00:24:08,280 --> 00:24:10,010 We're going to run our x-ray generator 461 00:24:10,010 --> 00:24:11,780 with a copper target. 462 00:24:11,780 --> 00:24:13,590 Why do I tell you the copper target? 463 00:24:13,590 --> 00:24:15,760 Because you're going to fix the wavelength. 464 00:24:15,760 --> 00:24:18,270 And you fix the wavelength by choosing the target. 465 00:24:18,270 --> 00:24:23,960 So we've got copper target in our x-ray generator, OK? 466 00:24:23,960 --> 00:24:26,970 So remember, the sample is not the target. 467 00:24:26,970 --> 00:24:29,570 The sample is what is being irradiated. 468 00:24:29,570 --> 00:24:34,540 This is being bombed by the electrons in the tube, copper 469 00:24:34,540 --> 00:24:42,000 target in x-ray tube, and that means that the lambda of the 470 00:24:42,000 --> 00:24:43,700 radiation is lambda copper. 471 00:24:43,700 --> 00:24:47,500 And I'm going to use lambda copper K alpha because I know 472 00:24:47,500 --> 00:24:52,570 it's wavelength to five significant figures. 473 00:24:52,570 --> 00:24:56,950 It's 1.5418 angstroms. 474 00:24:56,950 --> 00:24:58,100 And here's the data set. 475 00:24:58,100 --> 00:25:00,200 I looked this up in the literature. 476 00:25:00,200 --> 00:25:04,050 Here's the data set from the experiment. 477 00:25:04,050 --> 00:25:06,700 And that's all you get, a set of 2 thetas. 478 00:25:06,700 --> 00:25:08,770 So here's my challenge to you. 479 00:25:08,770 --> 00:25:11,710 I'm telling you you've got an unknown sample that's cubic, 480 00:25:11,710 --> 00:25:13,200 and there's your data set. 481 00:25:13,200 --> 00:25:17,940 The task for you is determine two parts to the question. 482 00:25:17,940 --> 00:25:19,890 And we're going to do it together. 483 00:25:19,890 --> 00:25:25,530 First part, determine the crystal structure. 484 00:25:25,530 --> 00:25:32,390 So it's either FCC, BCC, or simple cubic. 485 00:25:32,390 --> 00:25:36,970 And the second part is determine the lattice 486 00:25:36,970 --> 00:25:39,690 constant, the value of the lattice constant. 487 00:25:39,690 --> 00:25:41,620 So you can get quantitative measurements. 488 00:25:41,620 --> 00:25:43,420 So I'm going to show you how to do it. 489 00:25:43,420 --> 00:25:45,310 So how are we going to do it? 490 00:25:45,310 --> 00:25:47,680 We're going to use this technique. 491 00:25:47,680 --> 00:25:50,460 This is my self-help book for you. 492 00:25:50,460 --> 00:25:51,650 And here's the key. 493 00:25:51,650 --> 00:25:53,340 Here's how I came up with this. 494 00:25:53,340 --> 00:25:55,240 There's a way to unravel this. 495 00:25:55,240 --> 00:25:58,730 And the way you unravel this is to take these two 496 00:25:58,730 --> 00:26:00,000 relationships. 497 00:26:00,000 --> 00:26:05,530 You have lambda equals 2d sin theta is 1. 498 00:26:05,530 --> 00:26:08,000 And you also know that d-- 499 00:26:08,000 --> 00:26:10,900 in fact, I'm going to keep writing hkl subscripts here-- 500 00:26:10,900 --> 00:26:14,250 you know the dhkl is equal to a. 501 00:26:14,250 --> 00:26:16,760 That's this lattice constant. 502 00:26:16,760 --> 00:26:18,580 Be sure this is lattice constant. 503 00:26:18,580 --> 00:26:22,750 This is the cube edge measurement lattice constant 504 00:26:22,750 --> 00:26:27,430 divided by the square root of h squared plus k 505 00:26:27,430 --> 00:26:30,150 squared plus l squared. 506 00:26:30,150 --> 00:26:32,550 So what I do is I combine the two. 507 00:26:32,550 --> 00:26:36,140 If you combine these two, you can end up with this 508 00:26:36,140 --> 00:26:37,516 relationship. 509 00:26:37,516 --> 00:26:43,210 If you combine the two, you get lambda squared over 4a 510 00:26:43,210 --> 00:26:50,610 squared equals sin squared theta over h squared plus k 511 00:26:50,610 --> 00:26:52,400 squared plus l squared. 512 00:26:52,400 --> 00:26:53,560 And this is the key. 513 00:26:53,560 --> 00:26:55,060 Why is it the key? 514 00:26:55,060 --> 00:26:57,260 The value of lambda is set by you. 515 00:26:57,260 --> 00:26:59,150 You chose the target. 516 00:26:59,150 --> 00:27:01,180 The value of a is set by the sample. 517 00:27:01,180 --> 00:27:02,800 That's the lattice constant. 518 00:27:02,800 --> 00:27:05,510 So the ratio of two constants is a constant. 519 00:27:05,510 --> 00:27:06,330 Agreed? 520 00:27:06,330 --> 00:27:09,330 So this is a constant. 521 00:27:09,330 --> 00:27:12,180 So if the left side of the equation is a constant, the 522 00:27:12,180 --> 00:27:14,450 right side of the equation must be a constant. 523 00:27:14,450 --> 00:27:16,300 But h, k, and l vary. 524 00:27:16,300 --> 00:27:17,580 And theta varies. 525 00:27:17,580 --> 00:27:20,550 But the ratio of the variation, when mapped into 526 00:27:20,550 --> 00:27:24,890 sin square theta over h squared plus k squared plus l 527 00:27:24,890 --> 00:27:26,860 squared, this must be a constant. 528 00:27:26,860 --> 00:27:30,600 And that's going to be the way I work through 529 00:27:30,600 --> 00:27:33,640 this delightful problem. 530 00:27:33,640 --> 00:27:34,790 So let's see. 531 00:27:34,790 --> 00:27:38,110 Start with 2 theta values, and generate a set of sin squared 532 00:27:38,110 --> 00:27:39,580 theta values. 533 00:27:39,580 --> 00:27:42,320 That's what Sadoway says first. So I took 2 theta. 534 00:27:42,320 --> 00:27:44,860 And all I did was make 1 theta, and then took the sin 535 00:27:44,860 --> 00:27:46,140 of it, and then took the square the sin. 536 00:27:46,140 --> 00:27:50,640 So this is the data set transmogrified into sin 537 00:27:50,640 --> 00:27:51,780 squared theta. 538 00:27:51,780 --> 00:27:53,140 All right, what's the next thing he says? 539 00:27:53,140 --> 00:27:58,180 Normalize by dividing through with the first value. 540 00:27:58,180 --> 00:28:00,080 So I'm just going to take this whole series and 541 00:28:00,080 --> 00:28:02,320 divide it by 0.143. 542 00:28:02,320 --> 00:28:05,160 And now I end up, instead of 0.143, 0.191. 543 00:28:05,160 --> 00:28:06,800 I have 1, 1.34, and so on. 544 00:28:06,800 --> 00:28:07,940 That's what I mean by normalize. 545 00:28:07,940 --> 00:28:10,070 Normalize to the first entry. 546 00:28:10,070 --> 00:28:11,620 OK, What's next thing he says? 547 00:28:11,620 --> 00:28:13,800 Clear fractions. 548 00:28:13,800 --> 00:28:16,400 Clear fractions from the normalized column. 549 00:28:16,400 --> 00:28:20,750 So I'm going to multiply this by a common number. 550 00:28:20,750 --> 00:28:23,810 Because 1.34, remember this is experimental data. 551 00:28:23,810 --> 00:28:24,400 It's noisy. 552 00:28:24,400 --> 00:28:26,300 1.34 looks like 4/3. 553 00:28:26,300 --> 00:28:27,480 Doesn't it? 554 00:28:27,480 --> 00:28:29,680 It's fuzzy logic. 555 00:28:29,680 --> 00:28:33,990 And 2.67 looks like 2 and 2/3, 3 and 2/3. 556 00:28:33,990 --> 00:28:35,430 That's roughly 4. 557 00:28:35,430 --> 00:28:36,840 5 and 1/3. 558 00:28:36,840 --> 00:28:38,650 The 3 looks like a magic number. 559 00:28:38,650 --> 00:28:44,920 So if a multiply this column by 3, I get 3, 4, 8, 11, 12. 560 00:28:44,920 --> 00:28:46,120 What does it say next? 561 00:28:46,120 --> 00:28:50,040 Speculate on the h, k, l values, that if expressed as h 562 00:28:50,040 --> 00:28:52,950 squared plus k squared plus l squared would generate the 563 00:28:52,950 --> 00:28:56,440 sequence of the clear fractions column. 564 00:28:56,440 --> 00:28:58,630 And then that's going to take me to the selection rule. 565 00:28:58,630 --> 00:29:00,550 So I say, well how do I get 3? 566 00:29:00,550 --> 00:29:02,530 It's 1 plus 1 plus 1. 567 00:29:02,530 --> 00:29:05,990 4 is 2 squared plus 0 squared plus zero squared. 568 00:29:05,990 --> 00:29:09,130 8 is 2 squared plus 2 squared plus 0 squared. 569 00:29:09,130 --> 00:29:11,630 So I'm generating this thing. 570 00:29:11,630 --> 00:29:14,180 And now it's pretty obvious, right? 571 00:29:14,180 --> 00:29:16,240 Because now I've got to use these. 572 00:29:16,240 --> 00:29:17,440 And what do I see here? 573 00:29:17,440 --> 00:29:20,320 Well, 1 1 1 are all odd. 574 00:29:20,320 --> 00:29:23,030 2 0 0, all even. 575 00:29:23,030 --> 00:29:25,770 2 2 0, all even. 576 00:29:25,770 --> 00:29:28,680 Gee, this looks like it conforms to the selection 577 00:29:28,680 --> 00:29:36,040 rules for Bragg reflection from an FCC crystal 578 00:29:36,040 --> 00:29:38,270 And then to make sure that I'm on to something, 579 00:29:38,270 --> 00:29:39,890 what I do is this. 580 00:29:39,890 --> 00:29:43,640 Compute for each theta the value of sin square theta over 581 00:29:43,640 --> 00:29:47,050 h squared plus k squared plus l squared on the basis of 582 00:29:47,050 --> 00:29:48,070 those assumed values. 583 00:29:48,070 --> 00:29:51,430 What I'm doing is I'm saying if my hunch is right, whatever 584 00:29:51,430 --> 00:29:55,630 I choose for the theta and the assumed value h squared plus k 585 00:29:55,630 --> 00:29:58,110 squared plus l squared, that should be a constant that 586 00:29:58,110 --> 00:29:59,430 doesn't change. 587 00:29:59,430 --> 00:30:00,450 So let's test it. 588 00:30:00,450 --> 00:30:01,950 And sure enough, look. 589 00:30:01,950 --> 00:30:06,290 0.0477, 0.0478, so it looks like I'm on to something. 590 00:30:06,290 --> 00:30:08,630 And furthermore, I know what this is. 591 00:30:08,630 --> 00:30:13,630 That 0.0477 is this. 592 00:30:13,630 --> 00:30:17,090 This is 0.0477. 593 00:30:17,090 --> 00:30:18,200 And I know my lambda. 594 00:30:18,200 --> 00:30:19,900 That's 1.5418. 595 00:30:19,900 --> 00:30:21,710 So now I can calculate my a. 596 00:30:21,710 --> 00:30:24,080 So now I've determined that it's FCC. 597 00:30:24,080 --> 00:30:27,770 Plus if I go ahead and I calculate the a value, I get 598 00:30:27,770 --> 00:30:32,980 3.53 angstroms. And if it's FCC and it's 3.53 angstroms, I 599 00:30:32,980 --> 00:30:35,970 bet it's nickel. 600 00:30:35,970 --> 00:30:37,640 So that's how you index this stuff. 601 00:30:40,630 --> 00:30:41,420 So there we are. 602 00:30:41,420 --> 00:30:43,890 There are the selection rules. 603 00:30:43,890 --> 00:30:45,420 Now here's a little trick. 604 00:30:45,420 --> 00:30:46,970 Let me show you. 605 00:30:46,970 --> 00:30:49,940 Because when I go to index this, the first thing I do is 606 00:30:49,940 --> 00:30:52,310 I go for low hanging fruit. 607 00:30:52,310 --> 00:30:56,170 If you start looking for whether it's simple cubic, 608 00:30:56,170 --> 00:30:58,880 face-centered cubic, or body-centered cubic, let's 609 00:30:58,880 --> 00:31:00,550 look at the sum. 610 00:31:00,550 --> 00:31:04,480 So I've got simple cubic, body-centered cubic, and 611 00:31:04,480 --> 00:31:05,790 face-centered cubic. 612 00:31:05,790 --> 00:31:09,570 So the first plane on simple cubic, it's going to give me 613 00:31:09,570 --> 00:31:10,270 everything. 614 00:31:10,270 --> 00:31:15,570 It's going to give me 1, 2, 3, 4, 5, 6. 615 00:31:15,570 --> 00:31:18,030 And FCC, what does it give us? 616 00:31:18,030 --> 00:31:19,760 If you start looking down there, it gives you 617 00:31:19,760 --> 00:31:24,920 3, 4, 8, 11, 12. 618 00:31:24,920 --> 00:31:26,260 Well, this is so different. 619 00:31:26,260 --> 00:31:29,580 3, 4, 8, 11 is so different from 1, 2, 3, 4. 620 00:31:29,580 --> 00:31:31,550 What does BCC give you? 621 00:31:31,550 --> 00:31:36,930 It gives you 2, 4, 6, 8, 10, 12. 622 00:31:36,930 --> 00:31:38,560 Can you see you have a problem here? 623 00:31:38,560 --> 00:31:41,090 It's pretty easy to pull out FCC. 624 00:31:41,090 --> 00:31:42,460 But look at these two. 625 00:31:42,460 --> 00:31:44,530 Since you don't know, you're just normalizing. 626 00:31:44,530 --> 00:31:45,330 You can't tell. 627 00:31:45,330 --> 00:31:48,940 If I give you a number sequence that is in the order 628 00:31:48,940 --> 00:31:52,630 2, 4, 6, 8, 10, how do you know that that couldn't be 1, 629 00:31:52,630 --> 00:31:53,880 2, 3, 4, 5? 630 00:31:57,700 --> 00:31:59,060 There's a hook here, though. 631 00:31:59,060 --> 00:32:01,960 Look at the sequence of h plus h squared plus k 632 00:32:01,960 --> 00:32:03,200 squared plus l squared. 633 00:32:03,200 --> 00:32:06,830 Do you notice that there's no combination of h squared plus 634 00:32:06,830 --> 00:32:10,230 k squared plus l squared that gives you 7? 635 00:32:10,230 --> 00:32:11,170 You get 6. 636 00:32:11,170 --> 00:32:12,760 And then the next one is 8. 637 00:32:12,760 --> 00:32:17,170 But there is a sequence that gives you 14. 638 00:32:17,170 --> 00:32:22,800 So if you have a seventh line, if the seventh line to the 639 00:32:22,800 --> 00:32:28,040 first line, the ratio of h squared plus k squared plus l 640 00:32:28,040 --> 00:32:35,530 squared for angle number 7 to h squared plus k 641 00:32:35,530 --> 00:32:38,180 squared plus l squared. 642 00:32:38,180 --> 00:32:43,230 For angle number 1, if it's equal to 7-- 643 00:32:43,230 --> 00:32:48,120 and point of fact it's not 7:1, it's 14:2-- 644 00:32:48,120 --> 00:32:51,820 and if it's equal to 8 for the seventh angle-- 645 00:32:51,820 --> 00:32:53,600 then it must be simple cubic. 646 00:32:53,600 --> 00:32:56,300 And now you've sorted it all out. 647 00:32:56,300 --> 00:32:57,650 So you're an expert now. 648 00:32:57,650 --> 00:32:59,530 Now you can do it. 649 00:32:59,530 --> 00:33:01,970 OK, so you're going to get some practice on homework. 650 00:33:01,970 --> 00:33:04,470 So you can index crystals. 651 00:33:04,470 --> 00:33:06,940 You can determine a crystal structure. 652 00:33:06,940 --> 00:33:08,360 All right, a couple of other things we can do. 653 00:33:08,360 --> 00:33:11,650 There is a second technique. 654 00:33:11,650 --> 00:33:18,680 And it's called Laue diffraction after von Laue, 655 00:33:18,680 --> 00:33:21,120 who got the Nobel Prize in 1914. 656 00:33:21,120 --> 00:33:24,440 He beat the Braggs by one year for this technique. 657 00:33:24,440 --> 00:33:25,750 So he shines. 658 00:33:25,750 --> 00:33:29,580 In this case for Laue, it's a slightly different technique. 659 00:33:29,580 --> 00:33:33,630 What Laue does, is he uses light x-rays. 660 00:33:36,860 --> 00:33:42,070 That means variable lambda. 661 00:33:42,070 --> 00:33:44,980 So how would you get a fixed value of lambda? 662 00:33:44,980 --> 00:33:47,910 Well, you would pick off one of those lines, like maybe the 663 00:33:47,910 --> 00:33:51,000 K alpha line because it's a nice, singular line. 664 00:33:51,000 --> 00:33:53,660 And repress the bremsstrahlung and so on. 665 00:33:53,660 --> 00:33:56,370 But if I want a variable lambda, where do I go for 666 00:33:56,370 --> 00:33:57,940 variable lambda? 667 00:33:57,940 --> 00:33:59,080 I repress the lines. 668 00:33:59,080 --> 00:34:00,490 And I go for the bremsstrahlung. 669 00:34:00,490 --> 00:34:04,870 So now I've got lines varying all across the x 670 00:34:04,870 --> 00:34:06,030 region of the spectrum. 671 00:34:06,030 --> 00:34:08,080 So I use variable lambda. 672 00:34:08,080 --> 00:34:13,670 And I fixed the angle of incidence, whereas with 673 00:34:13,670 --> 00:34:17,700 diffractometry, I use a single value of lambda and vary. 674 00:34:17,700 --> 00:34:20,830 So here's the cartoon which shows what's going on. 675 00:34:20,830 --> 00:34:23,960 I'll do it one more time. 676 00:34:23,960 --> 00:34:30,580 So this is called camera obscura, 677 00:34:30,580 --> 00:34:32,530 which is darkened room. 678 00:34:32,530 --> 00:34:34,600 That's all this means is dark room. 679 00:34:34,600 --> 00:34:38,110 So let's say I've got the specimen on the back wall of 680 00:34:38,110 --> 00:34:42,860 this thing, and what I've got is white x-rays coming in. 681 00:34:42,860 --> 00:34:46,920 So the white x-ray enters through the face. 682 00:34:46,920 --> 00:34:50,210 I've got the plate here, and I've got the specimen sitting 683 00:34:50,210 --> 00:34:51,800 somewhere in between. 684 00:34:51,800 --> 00:34:55,110 And what happens is I get a spot pattern. 685 00:34:55,110 --> 00:34:56,390 I get a spot pattern. 686 00:34:56,390 --> 00:35:13,640 And the spot pattern is imitative of the symmetry of 687 00:35:13,640 --> 00:35:14,890 atomic arrangement. 688 00:35:20,790 --> 00:35:25,030 So I have to say a little bit about symmetry so we know what 689 00:35:25,030 --> 00:35:26,590 that means. 690 00:35:26,590 --> 00:35:29,250 So let's take a look at symmetry. 691 00:35:29,250 --> 00:35:32,430 So I'm going to talk a little bit about rotational 692 00:35:32,430 --> 00:35:33,680 symmetries. 693 00:35:35,910 --> 00:35:37,740 So let's look at that. 694 00:35:37,740 --> 00:35:43,150 So first one I'm going to look at is an 0 0 1 plane. 695 00:35:43,150 --> 00:35:46,270 So if I just take a projection of an 0 0 1 plane, 696 00:35:46,270 --> 00:35:47,300 it looks like this. 697 00:35:47,300 --> 00:35:47,810 Doesn't it? 698 00:35:47,810 --> 00:35:50,650 It's just the cube edge, or the cube bottom. 699 00:35:50,650 --> 00:35:55,080 So it's got a and a as the edge. 700 00:35:55,080 --> 00:35:59,920 And now, the rotational axis, it's about a 701 00:35:59,920 --> 00:36:03,790 normal rotational axis. 702 00:36:03,790 --> 00:36:07,470 A normal rotational axis. 703 00:36:07,470 --> 00:36:10,840 So what I'm going to ask is how many degrees do I have to 704 00:36:10,840 --> 00:36:14,270 go through before I get this same image back. 705 00:36:14,270 --> 00:36:17,270 You can see that you go 90 degrees, it'll come back. 706 00:36:17,270 --> 00:36:20,110 If I stop at 45 degrees, it's going to look like this. 707 00:36:20,110 --> 00:36:22,120 You're going to say I know he moved the specimen. 708 00:36:22,120 --> 00:36:25,290 If I go 90 degrees, you can't tell it apart. 709 00:36:25,290 --> 00:36:29,330 So we define fold. 710 00:36:29,330 --> 00:36:35,220 The fold is equal to 360 divided by the 711 00:36:35,220 --> 00:36:38,630 basic angle of rotation. 712 00:36:38,630 --> 00:36:42,240 So we would say that this plane here exhibits 4-fold 713 00:36:42,240 --> 00:36:47,070 symmetry, OK? 714 00:36:47,070 --> 00:36:49,000 Let's do a second one. 715 00:36:49,000 --> 00:36:51,630 The second one is 0 1 1. 716 00:36:51,630 --> 00:36:55,000 So if I look at 0 1 1, 0 1 1 at this plane. 717 00:36:58,480 --> 00:37:02,100 0 1 1 goes across the diagonal. 718 00:37:02,100 --> 00:37:05,490 So I'm going to take the diagonal here and 719 00:37:05,490 --> 00:37:06,880 plot it like so. 720 00:37:06,880 --> 00:37:09,980 So it's going to have an edge of a. 721 00:37:09,980 --> 00:37:11,850 And it's going to have the diagonal, which is 722 00:37:11,850 --> 00:37:13,940 root 2 times a. 723 00:37:13,940 --> 00:37:17,830 And if I put a rotational axis in the center of that, clearly 724 00:37:17,830 --> 00:37:21,020 if I go only 90 degrees, I'm going to tell that the thing 725 00:37:21,020 --> 00:37:21,980 is rotated. 726 00:37:21,980 --> 00:37:25,190 I have to go 180 degrees before I can't tell that 727 00:37:25,190 --> 00:37:26,940 there's been any disturbance. 728 00:37:26,940 --> 00:37:29,410 So this one here has 2-fold symmetry. 729 00:37:36,210 --> 00:37:36,990 We'll do one more. 730 00:37:36,990 --> 00:37:38,850 Because there's only three major ones that we 731 00:37:38,850 --> 00:37:40,180 have to deal with. 732 00:37:40,180 --> 00:37:42,540 And that is going to be the 1 1 1 plane. 733 00:37:42,540 --> 00:37:44,650 And the 1 1 1 plane. 734 00:37:44,650 --> 00:37:47,450 And the 1 1 1 plane, I think I've got an image of it here. 735 00:37:47,450 --> 00:37:51,630 You can see the 1 1 1 plane drawn in this cube. 736 00:37:51,630 --> 00:37:54,540 So what's the face of 1 1 1 look like? 737 00:37:54,540 --> 00:37:57,990 The face of 1 1 1 looks like this. 738 00:37:57,990 --> 00:37:59,920 That's face of 1 1 1. 739 00:37:59,920 --> 00:38:03,940 And if you go through this analysis, this is root 3a 740 00:38:03,940 --> 00:38:07,750 because it's a diagonal of a difference persuasion. 741 00:38:07,750 --> 00:38:10,180 So this is 3-fold symmetry. 742 00:38:10,180 --> 00:38:14,610 This has 3-fold rotational symmetry. 743 00:38:14,610 --> 00:38:17,210 And that means if I take a crystal such 744 00:38:17,210 --> 00:38:19,010 as this, for example. 745 00:38:19,010 --> 00:38:22,780 I've told you before, this is my silicon wafer, and this 746 00:38:22,780 --> 00:38:26,310 silicon wafer has been cut from a single crystal. 747 00:38:26,310 --> 00:38:31,180 So I'm looking at the edge of an atomic plane. 748 00:38:31,180 --> 00:38:33,790 The question is, which one is it? 749 00:38:33,790 --> 00:38:36,150 The crystal didn't come with a label on the side. 750 00:38:36,150 --> 00:38:38,530 I don't know what the crystal growth axis was. 751 00:38:38,530 --> 00:38:41,320 I started with a single crystal, and I dipped it into 752 00:38:41,320 --> 00:38:42,630 molten silicon. 753 00:38:42,630 --> 00:38:45,730 And just like rock mountain candy, I drop the heating 754 00:38:45,730 --> 00:38:49,350 coils and cause the liquid to solidify 755 00:38:49,350 --> 00:38:50,490 around the seed crystal. 756 00:38:50,490 --> 00:38:53,790 And I grow this salami that's about 8 inches in diameter, 757 00:38:53,790 --> 00:38:55,630 about 2 meters long. 758 00:38:55,630 --> 00:38:58,020 So now I go and I cut these things with the diamond wheel. 759 00:38:58,020 --> 00:39:01,860 So I'm cutting them normal to the growth axis, but I still 760 00:39:01,860 --> 00:39:05,020 don't know what plane I'm looking at. 761 00:39:05,020 --> 00:39:08,910 So if I use the Laue technique and put the crystal like so, 762 00:39:08,910 --> 00:39:11,840 bombard it with white x-rays, and I 763 00:39:11,840 --> 00:39:13,000 look at the spot pattern. 764 00:39:13,000 --> 00:39:15,160 The spot pattern is going to give me one of those 765 00:39:15,160 --> 00:39:17,000 symmetries. 766 00:39:17,000 --> 00:39:19,060 And if I get a 3-fold symmetry, I know I'm 767 00:39:19,060 --> 00:39:20,120 looking at 1 1 1. 768 00:39:20,120 --> 00:39:22,490 I mean, it's not going to be some wacko plane. 769 00:39:22,490 --> 00:39:26,380 It's going to be either face, edge, or diagonal. 770 00:39:26,380 --> 00:39:28,940 And on the basis of the symmetry in the Laue pattern, 771 00:39:28,940 --> 00:39:33,830 I can tell which plane I'm looking at. 772 00:39:33,830 --> 00:39:35,420 They use this stuff. 773 00:39:35,420 --> 00:39:38,160 They use it to make the devices that are in your cell 774 00:39:38,160 --> 00:39:39,160 phone and your computer. 775 00:39:39,160 --> 00:39:41,260 I'm not just telling you a story. 776 00:39:41,260 --> 00:39:41,640 All right. 777 00:39:41,640 --> 00:39:42,610 So let's look at some others. 778 00:39:42,610 --> 00:39:45,050 We'll go back to the Escher prints. 779 00:39:45,050 --> 00:39:47,060 Everything has symmetry. 780 00:39:47,060 --> 00:39:50,220 OK, so what's this one? 781 00:39:50,220 --> 00:39:51,620 See this? 782 00:39:51,620 --> 00:39:52,870 What's the symmetry? 783 00:39:56,220 --> 00:39:58,776 4-fold. 784 00:39:58,776 --> 00:40:02,040 How about this one? 785 00:40:02,040 --> 00:40:03,840 What do you think? 786 00:40:03,840 --> 00:40:05,090 3-fold. 787 00:40:05,090 --> 00:40:09,670 I even went into Photoshop just to show you 788 00:40:09,670 --> 00:40:10,670 how dedicated I am. 789 00:40:10,670 --> 00:40:11,830 So I took this image. 790 00:40:11,830 --> 00:40:15,430 And I told Photoshop to give me 120 degrees rotation. 791 00:40:15,430 --> 00:40:16,520 And I got this. 792 00:40:16,520 --> 00:40:19,020 You can even see the fold in the book there. 793 00:40:19,020 --> 00:40:20,760 So I'm really doing it. 794 00:40:20,760 --> 00:40:21,630 It's coming back. 795 00:40:21,630 --> 00:40:24,790 It's 120 degrees rotation. 796 00:40:24,790 --> 00:40:27,310 How about this one? 797 00:40:27,310 --> 00:40:30,340 That's 3. 798 00:40:30,340 --> 00:40:32,040 Now these are real Laue patterns. 799 00:40:32,040 --> 00:40:33,630 This is 4-fold. 800 00:40:33,630 --> 00:40:34,930 This is obviously 3-fold. 801 00:40:34,930 --> 00:40:36,080 This one is hard to see. 802 00:40:36,080 --> 00:40:37,770 But it turns out it's 2-fold. 803 00:40:37,770 --> 00:40:39,400 The one axis is just a little bit longer 804 00:40:39,400 --> 00:40:40,650 than the other axis. 805 00:40:43,630 --> 00:40:46,740 What about this one? 806 00:40:46,740 --> 00:40:49,180 That's our simple cubic puppy. 807 00:40:49,180 --> 00:40:51,790 Yes, 1-fold. 808 00:40:51,790 --> 00:40:55,190 Translational symmetry without rotational symmetry. 809 00:40:55,190 --> 00:40:57,700 You can make this by just moving them sideways. 810 00:40:57,700 --> 00:41:01,510 But you have to go all the way around. 811 00:41:01,510 --> 00:41:04,070 So that's 1-fold symmetry translation. 812 00:41:04,070 --> 00:41:05,320 Look at this one? 813 00:41:08,250 --> 00:41:09,960 What's going on here? 814 00:41:09,960 --> 00:41:11,360 Is that one of the Bravais lattices? 815 00:41:16,600 --> 00:41:20,330 This is called Penrose tile, again rotational symmetry 816 00:41:20,330 --> 00:41:22,010 without translational symmetry. 817 00:41:22,010 --> 00:41:26,230 You can take a patch of this and move it rotationally. 818 00:41:26,230 --> 00:41:28,690 But you can't take and cover a wall with it. 819 00:41:28,690 --> 00:41:31,980 So that's rotational without translational. 820 00:41:31,980 --> 00:41:32,930 See? 821 00:41:32,930 --> 00:41:34,960 You can go around this way. 822 00:41:34,960 --> 00:41:38,780 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and reproduce this. 823 00:41:38,780 --> 00:41:43,510 But you can't use this as a unit cell and keep it hopping 824 00:41:43,510 --> 00:41:46,280 laterally and tile a wall with it. 825 00:41:49,740 --> 00:41:53,190 Here's another one, not a Bravais lattice. 826 00:41:57,020 --> 00:41:57,390 All right. 827 00:41:57,390 --> 00:42:02,080 So I told you at the beginning of this unit that there are 828 00:42:02,080 --> 00:42:04,090 ordered solids and disordered solids. 829 00:42:04,090 --> 00:42:06,560 We've been focusing on ordered solids. 830 00:42:06,560 --> 00:42:07,520 They have a unit cell. 831 00:42:07,520 --> 00:42:08,350 They're periodic. 832 00:42:08,350 --> 00:42:10,180 And we call them crystals. 833 00:42:10,180 --> 00:42:13,950 And next week we're going to start looking 834 00:42:13,950 --> 00:42:15,920 at disordered solid. 835 00:42:15,920 --> 00:42:18,390 So they have no building block, no long range order. 836 00:42:18,390 --> 00:42:20,920 And we call them glasses. 837 00:42:20,920 --> 00:42:24,020 For a long time, people thought that solis have to 838 00:42:24,020 --> 00:42:26,970 fall into one box or the other box. 839 00:42:26,970 --> 00:42:30,390 And then the best thing that can happen in 840 00:42:30,390 --> 00:42:32,860 science is not, oh, yeah. 841 00:42:32,860 --> 00:42:34,370 That's exactly what I expected. 842 00:42:34,370 --> 00:42:37,370 It's, I wonder what that means. 843 00:42:37,370 --> 00:42:40,570 And there was one of these moments in 1982. 844 00:42:40,570 --> 00:42:43,980 In 1982, a man by the name of Danny Shechtman from Technion 845 00:42:43,980 --> 00:42:47,250 in Israel, was over here in the United States working at 846 00:42:47,250 --> 00:42:50,370 the National Institute of Standards and Technology, 847 00:42:50,370 --> 00:42:52,800 which used to be National Bureau of Standards down in 848 00:42:52,800 --> 00:42:55,240 Gaithersburg, Maryland. 849 00:42:55,240 --> 00:42:59,400 And he was looking at a set of aluminum-manganese alloys. 850 00:42:59,400 --> 00:43:00,560 So aluminum is a metal. 851 00:43:00,560 --> 00:43:01,780 Manganese is a metal. 852 00:43:01,780 --> 00:43:05,010 You can make a solid solution which we call an alloy. 853 00:43:05,010 --> 00:43:07,250 And these are highly ordered. 854 00:43:07,250 --> 00:43:13,850 And what he found in his Laue measurements, were rotational 855 00:43:13,850 --> 00:43:16,440 symmetries that are impossible in a crystal. 856 00:43:16,440 --> 00:43:19,000 He was getting 5-fold symmetry. 857 00:43:19,000 --> 00:43:20,810 You can't get 5-fold symmetry. 858 00:43:20,810 --> 00:43:23,290 There is no set of planes here that will give 859 00:43:23,290 --> 00:43:25,710 you a 5-fold symmetry. 860 00:43:25,710 --> 00:43:27,440 They lack translational symmetry. 861 00:43:27,440 --> 00:43:29,010 They were called aperiodic. 862 00:43:29,010 --> 00:43:33,390 And the popular name for them was quasicrystals. 863 00:43:33,390 --> 00:43:37,850 So here's the Laue pattern of one of Danny Shechtman's 864 00:43:37,850 --> 00:43:39,300 aluminum-manganese specimens. 865 00:43:39,300 --> 00:43:41,250 I think this is 25% manganese and 866 00:43:41,250 --> 00:43:43,400 aluminum if I'm not mistaken. 867 00:43:43,400 --> 00:43:49,160 So let's count, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 868 00:43:49,160 --> 00:43:53,170 You can't have 10 or 5. 869 00:43:53,170 --> 00:43:54,460 Atoms don't work that way. 870 00:43:54,460 --> 00:43:56,140 But he got it. 871 00:43:56,140 --> 00:44:01,470 He got it, 5- fold, 72 degrees. 872 00:44:01,470 --> 00:44:03,870 Now this is really, really exciting. 873 00:44:03,870 --> 00:44:06,420 This is a shot of his lab book. 874 00:44:06,420 --> 00:44:08,430 You're looking over his shoulder. 875 00:44:08,430 --> 00:44:11,000 These are the specimen numbers. 876 00:44:11,000 --> 00:44:18,000 And this is aluminum 25% manganese, April 8, 1982. 877 00:44:18,000 --> 00:44:21,640 So it's just another day at work. 878 00:44:21,640 --> 00:44:22,890 You could be doing this. 879 00:44:26,310 --> 00:44:27,850 And he can't believe what he's getting. 880 00:44:31,200 --> 00:44:33,310 He's wondering what's going on. 881 00:44:33,310 --> 00:44:36,890 You can't contribute this to a calibration error on the 882 00:44:36,890 --> 00:44:38,030 instrument. 883 00:44:38,030 --> 00:44:43,320 This thing is on unassailably 5-fold symmetry. 884 00:44:43,320 --> 00:44:47,720 The joke is that this is not far from the Pentagon. 885 00:44:47,720 --> 00:44:50,480 Only within shouting distance of the Pentagon would you 886 00:44:50,480 --> 00:44:53,760 discover 5-fold symmetry. 887 00:44:53,760 --> 00:44:55,030 So there it is. 888 00:44:55,030 --> 00:44:57,370 So where have we come with all of this? 889 00:44:57,370 --> 00:45:01,710 We've come with the ability to start with some very simple 890 00:45:01,710 --> 00:45:04,040 ideas about x-ray generation. 891 00:45:04,040 --> 00:45:06,460 And we've come to the point where we can characterize 892 00:45:06,460 --> 00:45:10,010 crystals, get quantitative measurements of their lattice 893 00:45:10,010 --> 00:45:12,300 constants, and by using Laue 894 00:45:12,300 --> 00:45:15,210 techniques, investigate symmetry. 895 00:45:15,210 --> 00:45:15,810 OK. 896 00:45:15,810 --> 00:45:17,040 Let's adjourn. 897 00:45:17,040 --> 00:45:18,960 We'll see you on Friday.