1 00:00:17,417 --> 00:00:22,956 Today, clearly, I am a noble gas. 2 00:00:25,025 --> 00:00:29,029 Thanks to-- it was actually the TAs' idea. 3 00:00:29,029 --> 00:00:31,731 And I was like, of course, that sounds awesome. 4 00:00:31,731 --> 00:00:36,870 So I hope you guys have a fantastic Halloween. 5 00:00:36,870 --> 00:00:40,306 Obviously, what you want to do is any kind of trick 6 00:00:40,306 --> 00:00:43,243 or treating or partying, you know the drill, 7 00:00:43,243 --> 00:00:47,113 you bring your periodic table, because you never know. 8 00:00:47,113 --> 00:00:48,882 And that's what happened-- 9 00:00:48,882 --> 00:00:53,887 so on Sunday right before the exam, I got this. 10 00:00:53,887 --> 00:00:56,623 And so this is Charles and Raymond. 11 00:00:56,623 --> 00:00:59,359 And they're saying we wanted to celebrate the Sox 12 00:00:59,359 --> 00:01:03,496 victory in Boston, but because the midterm was the next day, 13 00:01:03,496 --> 00:01:05,899 we brought along our periodic table to study. 14 00:01:05,899 --> 00:01:07,634 There were so many people moving around it 15 00:01:07,634 --> 00:01:10,970 felt like we were in a sea of electrons. 16 00:01:10,970 --> 00:01:12,472 And of course, the only thing that 17 00:01:12,472 --> 00:01:14,140 could have improved on that was actually 18 00:01:14,140 --> 00:01:15,341 like a little dance video. 19 00:01:15,341 --> 00:01:16,509 But that will probably come. 20 00:01:16,509 --> 00:01:18,678 I'm sure there on that next. 21 00:01:18,678 --> 00:01:21,414 But you see, they brought it not just to study for the exam, 22 00:01:21,414 --> 00:01:24,084 because they didn't know if it was going to be important 23 00:01:24,084 --> 00:01:25,618 or not at the rally. 24 00:01:25,618 --> 00:01:27,854 You never know. 25 00:01:27,854 --> 00:01:31,558 So make sure you have your periodic table with you. 26 00:01:31,558 --> 00:01:37,697 Now, OK, oh, yeah, they mentioned the exam. 27 00:01:37,697 --> 00:01:41,101 So speaking of the exam. 28 00:01:41,101 --> 00:01:42,502 Here are the results. 29 00:01:42,502 --> 00:01:47,774 And you can see that there is a pretty wide distribution. 30 00:01:47,774 --> 00:01:50,009 You can also see the averages is 77. 31 00:01:50,009 --> 00:01:52,812 And just to remind you-- that's about here-- 32 00:01:52,812 --> 00:01:57,684 just to remind you, these marks, 85, that's the A range. 33 00:01:57,684 --> 00:01:59,085 This is the B range. 34 00:01:59,085 --> 00:02:02,288 So the average was in the solid B range territory. 35 00:02:03,556 --> 00:02:05,925 And so it goes. 36 00:02:05,925 --> 00:02:10,497 The standard deviation was 12. 37 00:02:10,497 --> 00:02:16,369 Now I could tell-- you know, the exam 1, 38 00:02:16,369 --> 00:02:19,772 the topics of exam 1 many of you had already seen. 39 00:02:19,772 --> 00:02:22,942 And I think in this exam, some of you had not 40 00:02:22,942 --> 00:02:24,477 seen some of these topics, especially 41 00:02:24,477 --> 00:02:27,013 like the crystallography, but also 42 00:02:27,013 --> 00:02:30,617 even the molecular orbitals, the band structure. 43 00:02:30,617 --> 00:02:37,090 And I could tell that, you know, some people had to work harder, 44 00:02:37,090 --> 00:02:41,394 maybe a little bit stressed, and I could sense, some of that. 45 00:02:41,394 --> 00:02:44,564 But when I get stressed what I do 46 00:02:44,564 --> 00:02:47,033 is I need to kind of de-stress somehow. 47 00:02:47,033 --> 00:02:49,102 And sometimes some people listen to music, 48 00:02:49,102 --> 00:02:50,203 some of the other things. 49 00:02:50,203 --> 00:02:54,908 I always like comedy as a way of relaxing. 50 00:02:54,908 --> 00:02:56,543 And so what I do is I'll Google like 51 00:02:56,543 --> 00:02:59,012 for articles about Harvard. 52 00:02:59,012 --> 00:03:04,117 And so I found this one. 53 00:03:04,117 --> 00:03:07,187 This is actually an article in The Crimson 54 00:03:07,187 --> 00:03:08,521 from a few years ago. 55 00:03:08,521 --> 00:03:11,691 And the most common grade at Harvard 56 00:03:11,691 --> 00:03:16,529 is actually an A, a solid A, the most common grade. 57 00:03:19,632 --> 00:03:22,435 Suspicion that the college employs a softer grading 58 00:03:22,435 --> 00:03:24,637 standard than many of its peer institutions. 59 00:03:24,637 --> 00:03:26,206 You think? 60 00:03:26,206 --> 00:03:31,277 I mean, if everyone's getting an A, you think? 61 00:03:31,277 --> 00:03:35,048 So what I wanted to point out is there's a difference. 62 00:03:35,048 --> 00:03:38,484 And it's not just about your privileged, therefore, 63 00:03:38,484 --> 00:03:42,522 you deserve an A, whereas we know that that actually 64 00:03:42,522 --> 00:03:43,790 takes work. 65 00:03:43,790 --> 00:03:45,992 That's not the point I want to make. 66 00:03:45,992 --> 00:03:47,894 The point I want to make is that we know also 67 00:03:47,894 --> 00:03:51,030 what to do when we miss things. 68 00:03:51,030 --> 00:03:54,067 We know what to do when we don't get things right. 69 00:03:54,067 --> 00:03:56,536 That's when you do the work. 70 00:03:56,536 --> 00:03:58,605 So wherever you are on this curve, 71 00:03:58,605 --> 00:04:01,140 go back and figure out what didn't work. 72 00:04:01,140 --> 00:04:02,141 What did you miss? 73 00:04:02,141 --> 00:04:05,812 Figure that out, because that's where you learn. 74 00:04:05,812 --> 00:04:08,881 Thomas Edison who said, I never failed, I just 75 00:04:08,881 --> 00:04:13,453 did 10,000 experiments that didn't work. 76 00:04:13,453 --> 00:04:14,754 You've got to get through that. 77 00:04:14,754 --> 00:04:16,555 You've got to know that it takes hard work. 78 00:04:16,555 --> 00:04:18,224 And that's the thing that we know here. 79 00:04:18,224 --> 00:04:19,158 That's the difference. 80 00:04:19,158 --> 00:04:22,328 So please make sure you do that with exam 2. 81 00:04:22,328 --> 00:04:25,598 We are all here to help you continue learning. 82 00:04:25,598 --> 00:04:26,332 I'm a noble gas. 83 00:04:30,003 --> 00:04:34,707 And speaking of continuing to learn, where were we? 84 00:04:34,707 --> 00:04:35,207 X-rays. 85 00:04:39,846 --> 00:04:43,549 On Friday, we had sort of some other things going on. 86 00:04:43,549 --> 00:04:45,852 But I also was trying to teach you guys about X-rays. 87 00:04:49,422 --> 00:04:52,825 And what we did is we learned how they're generated. 88 00:04:52,825 --> 00:04:53,393 Remember that? 89 00:04:53,393 --> 00:04:56,963 The Roentgen experiments. 90 00:04:56,963 --> 00:04:59,932 And so I want to pick up here with the kinds of X-rays 91 00:04:59,932 --> 00:05:00,533 that we have. 92 00:05:00,533 --> 00:05:01,801 There are two kinds of X-rays. 93 00:05:04,637 --> 00:05:08,408 And we learned about those on Friday. 94 00:05:08,408 --> 00:05:10,910 And I want to just remind you about those. 95 00:05:10,910 --> 00:05:17,116 And I also want to show you a video to kind of recapture 96 00:05:17,116 --> 00:05:19,185 what they are. 97 00:05:19,185 --> 00:05:24,123 The first kind of X-ray, remember, 98 00:05:24,123 --> 00:05:26,459 we plotted this as intensity. 99 00:05:26,459 --> 00:05:31,664 And I'm not going to draw the cathode ray 100 00:05:31,664 --> 00:05:34,834 tube again and the experiment that Roentgen did and all that. 101 00:05:34,834 --> 00:05:37,270 But I'm going to just jump to the two kinds of x-rays 102 00:05:37,270 --> 00:05:39,005 that you get. 103 00:05:39,005 --> 00:05:42,608 And so if you plot the wavelength 104 00:05:42,608 --> 00:05:46,512 of the X-ray versus the intensity of the X-ray, 105 00:05:46,512 --> 00:05:49,582 then one kind is the Bremsstrahlung, 106 00:05:49,582 --> 00:05:54,354 which comes from that that electron getting slowed down. 107 00:05:54,354 --> 00:05:57,357 And if it slows down, it emits radiation 108 00:05:57,357 --> 00:05:59,058 in a continuous spectrum. 109 00:06:02,895 --> 00:06:06,065 And remember, we sort of drew these. 110 00:06:09,068 --> 00:06:15,441 And this would be like the incident electron energy. 111 00:06:15,441 --> 00:06:21,881 Let's say incident energy of the electron. 112 00:06:21,881 --> 00:06:24,617 Maybe like that's, OK, I don't know, like 10 keV. 113 00:06:27,653 --> 00:06:30,790 And then up here the incident energy of the electron 114 00:06:30,790 --> 00:06:35,161 was you know something like maybe 30 keV. 115 00:06:35,161 --> 00:06:40,133 And so you see that as you hit that anode, remember what 116 00:06:40,133 --> 00:06:44,003 Roentgen did, he took a cathode ray tube 117 00:06:44,003 --> 00:06:45,938 and he upped the voltage. 118 00:06:45,938 --> 00:06:47,440 So he really cranked up the voltage. 119 00:06:47,440 --> 00:06:49,709 So those electrons coming off the cathode 120 00:06:49,709 --> 00:06:52,478 are going really fast. 121 00:06:52,478 --> 00:06:56,249 And then what he did is he put a piece of metal in their way. 122 00:06:56,249 --> 00:07:00,019 And what happened is those really high energy electrons, 123 00:07:00,019 --> 00:07:03,489 they see those metal atoms and sometimes they 124 00:07:03,489 --> 00:07:06,025 get inside the electron cloud and they turn. 125 00:07:06,025 --> 00:07:09,996 And that's when they give off this continuous radiation. 126 00:07:09,996 --> 00:07:14,066 But we also know that there is a limit here. 127 00:07:14,066 --> 00:07:14,867 Remember that? 128 00:07:14,867 --> 00:07:16,002 We talked about that? 129 00:07:16,002 --> 00:07:28,247 And that this limit is set by this maximum. 130 00:07:28,247 --> 00:07:33,619 It's just the maximum amount of energy 131 00:07:33,619 --> 00:07:35,288 that you could get out of a photon being 132 00:07:35,288 --> 00:07:38,957 emitted this way would be equal to the incoming electron. 133 00:07:38,957 --> 00:07:42,361 The incoming electron transferred all of its energy 134 00:07:42,361 --> 00:07:43,362 to the photon. 135 00:07:43,362 --> 00:07:45,031 So that's why there is a maximum. 136 00:07:45,031 --> 00:07:49,635 And it's also why it increases-- remember, shorter wavelength, 137 00:07:49,635 --> 00:07:51,170 higher energy-- 138 00:07:51,170 --> 00:07:53,973 so it increases as you increase the energy 139 00:07:53,973 --> 00:07:55,107 of the instant an electron. 140 00:07:57,844 --> 00:07:59,278 That is Bremsstrahlung. 141 00:07:59,278 --> 00:08:01,147 Let's watch a video, because this is actually 142 00:08:01,147 --> 00:08:05,284 a very nicely done video that captures it with animation. 143 00:08:09,021 --> 00:08:12,024 So OK, here's your CRT. 144 00:08:12,024 --> 00:08:13,359 Now here they come. 145 00:08:13,359 --> 00:08:15,061 Those are those electrons. 146 00:08:15,061 --> 00:08:16,762 And that voltage is high. 147 00:08:16,762 --> 00:08:17,663 So they're coming out. 148 00:08:17,663 --> 00:08:20,266 Look at them coming out really fast. 149 00:08:20,266 --> 00:08:23,503 Lots of kinetic energy. 150 00:08:23,503 --> 00:08:25,037 And then they go. 151 00:08:25,037 --> 00:08:26,973 And then this is what Roentgen did. 152 00:08:26,973 --> 00:08:29,108 He put a little piece of metal in there. 153 00:08:29,108 --> 00:08:31,677 So now those electrons are hitting the metal. 154 00:08:31,677 --> 00:08:33,846 So those are the two things that he did differently. 155 00:08:33,846 --> 00:08:37,884 Remember, the room lit up even when all the lights were off. 156 00:08:37,884 --> 00:08:38,985 There they are. 157 00:08:38,985 --> 00:08:39,818 Nice. 158 00:08:39,818 --> 00:08:40,520 Good electrons. 159 00:08:40,520 --> 00:08:41,587 Oh, and here they go. 160 00:08:41,587 --> 00:08:44,457 And there's a metal atom. 161 00:08:44,457 --> 00:08:46,058 And oh look at that. 162 00:08:46,058 --> 00:08:47,627 Now what are these? 163 00:08:47,627 --> 00:08:49,328 Those are x-rays coming off. 164 00:08:49,328 --> 00:08:51,097 Those are the electrons hitting the metal. 165 00:08:51,097 --> 00:08:53,232 And here's the metal atom. 166 00:08:53,232 --> 00:08:55,167 And as the electron comes in, remember 167 00:08:55,167 --> 00:08:58,204 it sees the charge of the nucleus. 168 00:08:58,204 --> 00:09:00,873 And it gets deflected. 169 00:09:00,873 --> 00:09:03,509 And that deflection loses energy. 170 00:09:03,509 --> 00:09:05,578 And that loss of energy goes into a photon. 171 00:09:05,578 --> 00:09:08,381 Now, because these electrons have such high energy 172 00:09:08,381 --> 00:09:17,056 to start with, the energies of the photons are very high. 173 00:09:17,056 --> 00:09:17,823 They're x-rays. 174 00:09:21,994 --> 00:09:23,496 And so here's the wavelength. 175 00:09:23,496 --> 00:09:24,463 And you can see that-- 176 00:09:24,463 --> 00:09:26,799 well, it's sort of a little hard to see this blue range. 177 00:09:26,799 --> 00:09:29,068 But see higher energy, shorter wavelength. 178 00:09:29,068 --> 00:09:30,870 Lower energy, longer wavelength. 179 00:09:30,870 --> 00:09:31,804 Very nice. 180 00:09:31,804 --> 00:09:35,575 So that's the animation of Bremsstrahlung. 181 00:09:35,575 --> 00:09:36,275 OK. 182 00:09:36,275 --> 00:09:37,209 Good. 183 00:09:37,209 --> 00:09:39,679 Now the thing is what we also learned 184 00:09:39,679 --> 00:09:42,648 is that there is another type of X-ray. 185 00:09:42,648 --> 00:09:44,250 There's another type of X-ray. 186 00:09:44,250 --> 00:09:49,255 And in fact, if you crank this up high enough, 187 00:09:49,255 --> 00:09:50,423 you can get that other type. 188 00:09:50,423 --> 00:09:51,790 So now, we go higher. 189 00:09:51,790 --> 00:09:53,626 Oh, we didn't see it here. 190 00:09:53,626 --> 00:09:55,695 It just looks like that and that and that. 191 00:09:55,695 --> 00:10:01,167 And then all of a sudden we go up to 40 keV and we see this. 192 00:10:06,439 --> 00:10:07,239 Why? 193 00:10:07,239 --> 00:10:08,874 What happened here? 194 00:10:08,874 --> 00:10:11,277 And what happened here is a totally different mechanism 195 00:10:11,277 --> 00:10:13,212 for generating X-rays. 196 00:10:13,212 --> 00:10:15,915 And that was the second type that we talked about on Friday. 197 00:10:15,915 --> 00:10:19,919 Those are called characteristic. 198 00:10:19,919 --> 00:10:25,124 And the reason is that-- remember, we have these levels, 199 00:10:25,124 --> 00:10:28,327 which now that we are talking about X-rays, 200 00:10:28,327 --> 00:10:31,597 we give them letters, K, L, M, N. 201 00:10:31,597 --> 00:10:33,532 But it's just the quantum numbers. 202 00:10:33,532 --> 00:10:35,167 n equals 1. 203 00:10:35,167 --> 00:10:38,204 n equals 2. 204 00:10:38,204 --> 00:10:39,939 n equals 3. 205 00:10:39,939 --> 00:10:42,408 n equals 4. 206 00:10:42,408 --> 00:10:44,810 And what we said is that the weight characteristic 207 00:10:44,810 --> 00:10:48,914 X-rays are labeled is that if an electron is 208 00:10:48,914 --> 00:10:55,588 excited from this lower level, if an electron is knocked out 209 00:10:55,588 --> 00:10:58,391 of here, then there's a place for an electron from here 210 00:10:58,391 --> 00:11:00,426 to go down. 211 00:11:00,426 --> 00:11:02,061 Maybe the electron is excited. 212 00:11:02,061 --> 00:11:03,829 Maybe the electron is kicked out. 213 00:11:03,829 --> 00:11:06,532 And then something here can come down. 214 00:11:06,532 --> 00:11:13,172 And when that happens, just like in the Bohr model, 215 00:11:13,172 --> 00:11:15,741 you get radiation. 216 00:11:15,741 --> 00:11:19,412 But now, unlike hydrogen 13.6 electrons, 217 00:11:19,412 --> 00:11:21,380 these are keV of energy. 218 00:11:21,380 --> 00:11:23,382 These are very high energies. 219 00:11:23,382 --> 00:11:23,883 Why? 220 00:11:23,883 --> 00:11:26,619 Because it's the one S electron. 221 00:11:26,619 --> 00:11:30,322 And we know that once you get down to those metals, 222 00:11:30,322 --> 00:11:32,290 those one s electrons, you go further 223 00:11:32,290 --> 00:11:35,294 down the periodic table, 1s electrons are seriously 224 00:11:35,294 --> 00:11:37,997 tightly bound. 225 00:11:37,997 --> 00:11:41,834 And those levels have a difference down there 226 00:11:41,834 --> 00:11:43,069 that is pretty high in energy. 227 00:11:43,069 --> 00:11:45,204 It's X-ray high in energy. 228 00:11:45,204 --> 00:11:46,472 That's the point. 229 00:11:46,472 --> 00:11:50,242 So now, when that cascade happens, 230 00:11:50,242 --> 00:11:53,179 we call that a k alpha. 231 00:11:53,179 --> 00:11:56,048 And if it were to have come from here, 232 00:11:56,048 --> 00:11:57,483 it would be called a k beta. 233 00:12:01,954 --> 00:12:07,526 And those are transitions, that unlike this continuous energy, 234 00:12:07,526 --> 00:12:11,797 those transitions only happen at very specific energies, 235 00:12:11,797 --> 00:12:13,265 delta energies. 236 00:12:13,265 --> 00:12:16,469 The change in energy from L to K-- 237 00:12:16,469 --> 00:12:20,740 or for K beta it would be going from M to K. 238 00:12:20,740 --> 00:12:23,142 It's k alpha, k beta. 239 00:12:23,142 --> 00:12:26,612 We use the k because that's the final place the electron goes 240 00:12:26,612 --> 00:12:29,782 when it decays down. 241 00:12:29,782 --> 00:12:33,552 So if I just showed you these, well, that 242 00:12:33,552 --> 00:12:38,224 would be like k alpha, and that would be like k beta 243 00:12:38,224 --> 00:12:42,128 because you know the k beta is going to be a higher energy 244 00:12:42,128 --> 00:12:45,998 photon because it came up from a higher level. 245 00:12:45,998 --> 00:12:49,935 Well, you would also have on here some other peaks. 246 00:12:49,935 --> 00:12:53,439 You would have the L peaks. 247 00:12:53,439 --> 00:13:02,748 So you'd have like L alpha, L beta. 248 00:13:02,748 --> 00:13:06,118 So as you crank the energy up, then you 249 00:13:06,118 --> 00:13:08,220 can knock out those core electrons 250 00:13:08,220 --> 00:13:11,624 and these cascades happen and you get these discrete peaks. 251 00:13:11,624 --> 00:13:16,629 Notice, they will only come when you have enough energy 252 00:13:16,629 --> 00:13:20,199 in that incident electron to knock this electron out 253 00:13:20,199 --> 00:13:21,534 from the core. 254 00:13:21,534 --> 00:13:23,769 So that's why they don't appear until you 255 00:13:23,769 --> 00:13:26,505 get to a certain incident electron energy. 256 00:13:26,505 --> 00:13:29,775 They don't appear until you get to that certain energy. 257 00:13:29,775 --> 00:13:35,915 And so we have a video on that, which also I will narrate. 258 00:13:39,351 --> 00:13:40,419 Is this it? 259 00:13:40,419 --> 00:13:42,221 There we go. 260 00:13:42,221 --> 00:13:43,222 So there it is. 261 00:13:43,222 --> 00:13:44,123 It's a metal atom. 262 00:13:44,123 --> 00:13:47,459 I don't know which one. 263 00:13:47,459 --> 00:13:48,994 OK, there's the incident electron. 264 00:13:48,994 --> 00:13:49,695 You fired it. 265 00:13:49,695 --> 00:13:50,429 And look at that. 266 00:13:50,429 --> 00:13:52,665 It knocked out a core 1s electron, 267 00:13:52,665 --> 00:13:55,301 because it had enough energy, very high energy. 268 00:13:55,301 --> 00:13:56,869 That's what Roentgen did, cranked up 269 00:13:56,869 --> 00:13:58,571 the voltage, higher keV. 270 00:13:58,571 --> 00:13:59,338 And there it is. 271 00:13:59,338 --> 00:14:01,974 A cascade down and an X-ray comes out. 272 00:14:01,974 --> 00:14:02,508 You see that? 273 00:14:06,679 --> 00:14:07,513 Is that all? 274 00:14:07,513 --> 00:14:09,181 Oh, yeah, and then it's going to draw-- 275 00:14:09,181 --> 00:14:11,717 because of those, you get these characteristic peaks. 276 00:14:11,717 --> 00:14:13,752 Now, we call them characteristic, 277 00:14:13,752 --> 00:14:15,321 because now you see why. 278 00:14:15,321 --> 00:14:19,291 So unlike the continuous radiation, 279 00:14:19,291 --> 00:14:23,162 these peaks depend on the atom, because they 280 00:14:23,162 --> 00:14:26,198 depend on the energy levels of the atom. 281 00:14:26,198 --> 00:14:28,434 And so that's why like if you looked this up, 282 00:14:28,434 --> 00:14:31,337 you say, well, OK, let's look at the k alpha radiation. 283 00:14:31,337 --> 00:14:33,639 Let's look at the k alpha peaks that 284 00:14:33,639 --> 00:14:36,141 come out of different atoms. 285 00:14:36,141 --> 00:14:37,409 They're going to be different. 286 00:14:39,945 --> 00:14:44,783 So you have very sharp lines of x-rays 287 00:14:44,783 --> 00:14:47,019 at very specific energies. 288 00:14:47,019 --> 00:14:49,488 For copper, it's 8 keV. 289 00:14:49,488 --> 00:14:51,857 For molybdenum it's 17.5. 290 00:14:51,857 --> 00:14:53,626 Silver tungsten, it changes. 291 00:14:53,626 --> 00:14:59,465 And you can see that it goes up, as that 1s electron is 292 00:14:59,465 --> 00:15:02,034 lower and lower energy because I'm adding all these protons. 293 00:15:02,034 --> 00:15:05,671 So it all makes sense from the concepts we've learned. 294 00:15:05,671 --> 00:15:09,008 Oh but see, now that's really useful. 295 00:15:09,008 --> 00:15:11,677 That is really useful, because now I've 296 00:15:11,677 --> 00:15:15,447 got a way to have a source of X-rays 297 00:15:15,447 --> 00:15:17,616 that is super well defined. 298 00:15:17,616 --> 00:15:18,317 It's super clear. 299 00:15:18,317 --> 00:15:21,186 It's always this-- that's so cool. 300 00:15:21,186 --> 00:15:25,190 As long as I have the same metal, it's always the same. 301 00:15:25,190 --> 00:15:26,992 I can increase or decrease-- 302 00:15:26,992 --> 00:15:28,527 well, I can't go below the threshold. 303 00:15:28,527 --> 00:15:29,995 But I can't go above it. 304 00:15:29,995 --> 00:15:34,066 And that peak is characteristic of the metal. 305 00:15:34,066 --> 00:15:35,367 So it doesn't change. 306 00:15:35,367 --> 00:15:39,605 That's really useful because I've got now a flashlight. 307 00:15:39,605 --> 00:15:42,241 I've got an X-ray flashlight where 308 00:15:42,241 --> 00:15:44,543 the energy that I'm sending out is always 309 00:15:44,543 --> 00:15:46,478 exactly what I know it to be. 310 00:15:46,478 --> 00:15:47,613 I can predict what it is. 311 00:15:47,613 --> 00:15:50,582 And it always will be that depending on which medal 312 00:15:50,582 --> 00:15:52,084 I put in there. 313 00:15:52,084 --> 00:15:55,621 So that's a useful thing. 314 00:15:55,621 --> 00:15:57,456 Why is that useful? 315 00:15:57,456 --> 00:16:01,527 Well, that gets to the topic that is the topic of today 316 00:16:01,527 --> 00:16:05,497 and of Friday, which is what are we doing with these X-rays? 317 00:16:05,497 --> 00:16:06,999 Well, first, we're generating them. 318 00:16:06,999 --> 00:16:09,635 So that's what we've talked about so far. 319 00:16:09,635 --> 00:16:12,571 But now, we're going to actually use 320 00:16:12,571 --> 00:16:16,375 them to determine the crystal that we have. 321 00:16:16,375 --> 00:16:17,743 We're going to actually use them. 322 00:16:17,743 --> 00:16:20,913 We're going to use that flashlight. 323 00:16:20,913 --> 00:16:23,649 And so you can see why this would be useful, 324 00:16:23,649 --> 00:16:26,385 because this is the range-- 325 00:16:26,385 --> 00:16:28,120 we've showed this before-- 326 00:16:28,120 --> 00:16:29,088 of X-rays. 327 00:16:29,088 --> 00:16:31,290 So they have these energies of keV. 328 00:16:31,290 --> 00:16:34,693 And they have wavelengths right around a few Angstrom. 329 00:16:34,693 --> 00:16:37,563 That is a little less than an Angstrom, maybe 2 Angstroms. 330 00:16:37,563 --> 00:16:39,131 See where those wavelengths are? 331 00:16:39,131 --> 00:16:42,167 Well, those are atomic spacings. 332 00:16:42,167 --> 00:16:46,972 Those are like distances between layers. 333 00:16:46,972 --> 00:16:49,675 And so if we could shine these on a crystal 334 00:16:49,675 --> 00:16:53,645 and somehow figure out what it is with that light-- 335 00:16:53,645 --> 00:16:55,547 oh, there's a way we can do that. 336 00:16:55,547 --> 00:16:57,983 It's called diffraction. 337 00:16:57,983 --> 00:16:59,952 Because what happens is-- 338 00:16:59,952 --> 00:17:02,287 and we know this from many fields. 339 00:17:02,287 --> 00:17:06,791 You can think about this just as a water wave, any way. 340 00:17:06,791 --> 00:17:13,665 If the wavelength is similar in size to the features, 341 00:17:13,665 --> 00:17:17,603 then you get constructive and destructive interference 342 00:17:17,603 --> 00:17:20,472 as a result of the interaction between the wave 343 00:17:20,472 --> 00:17:22,207 and the features. 344 00:17:22,207 --> 00:17:24,910 So that's called diffraction. 345 00:17:24,910 --> 00:17:27,846 And you can see it here with this very simple picture 346 00:17:27,846 --> 00:17:29,648 of say a water-- this could be like a water 347 00:17:29,648 --> 00:17:30,849 wave, a sound wave. 348 00:17:30,849 --> 00:17:31,750 And there it is. 349 00:17:31,750 --> 00:17:34,887 And it's interfering both constructively 350 00:17:34,887 --> 00:17:39,491 and destructively along these lines. 351 00:17:39,491 --> 00:17:41,360 You can do this test yourself. 352 00:17:41,360 --> 00:17:43,328 You can take-- 353 00:17:43,328 --> 00:17:45,564 I highly recommend this-- take a laser pointer. 354 00:17:49,501 --> 00:17:51,437 Now if I just had a piece of metal-- 355 00:17:51,437 --> 00:17:54,640 I don't have a piece of metal, but if I did, 356 00:17:54,640 --> 00:17:56,341 then I would shine it on it. 357 00:17:56,341 --> 00:17:58,777 And what you'd see is that the dot would just 358 00:17:58,777 --> 00:18:01,213 reflect off the piece of metal. 359 00:18:01,213 --> 00:18:03,582 So I just would get the dot back. 360 00:18:03,582 --> 00:18:06,618 But now I've got this thing here. 361 00:18:06,618 --> 00:18:09,488 Many of you may not know what this. 362 00:18:09,488 --> 00:18:10,756 This is called a CD. 363 00:18:13,859 --> 00:18:17,596 But it turns out that a CD has features in it. 364 00:18:17,596 --> 00:18:21,934 It's got trenches that are like a little less than a micron 365 00:18:21,934 --> 00:18:24,069 apart, 100 nanometers. 366 00:18:24,069 --> 00:18:26,438 And this is 500 nanometer light. 367 00:18:26,438 --> 00:18:28,974 So you would expect there to be diffraction. 368 00:18:28,974 --> 00:18:32,010 You would expect there to be constructive and destructive 369 00:18:32,010 --> 00:18:32,744 interference. 370 00:18:32,744 --> 00:18:35,180 And when I bounce it off of this, look at that. 371 00:18:35,180 --> 00:18:36,582 There it is. 372 00:18:36,582 --> 00:18:39,351 This never gets old. 373 00:18:39,351 --> 00:18:43,889 I'm not getting just one reflection here. 374 00:18:43,889 --> 00:18:47,426 I'm getting a whole scatter of them that have constructively 375 00:18:47,426 --> 00:18:50,229 interfered, because of the feature sizes being 376 00:18:50,229 --> 00:18:54,233 the same as the wavelength. 377 00:18:54,233 --> 00:18:56,301 But now, I want to do that with X-rays. 378 00:18:56,301 --> 00:19:00,305 And I want to do it onto crystals. 379 00:19:00,305 --> 00:19:02,274 So how do we do that? 380 00:19:02,274 --> 00:19:03,742 So let's think about that. 381 00:19:03,742 --> 00:19:05,811 And we're going to think about it in terms of what 382 00:19:05,811 --> 00:19:09,148 the Bragg father-son pair, who won the Nobel Prize and are 383 00:19:09,148 --> 00:19:10,149 on a stamp. 384 00:19:10,149 --> 00:19:14,419 That's what you get when you win a Nobel Prize I guess. 385 00:19:14,419 --> 00:19:18,857 And what they did is they figured out how to do this. 386 00:19:18,857 --> 00:19:22,127 So let's go through that just so we understand it conceptually. 387 00:19:26,064 --> 00:19:32,371 So I'm going to say that I have a set of atoms. 388 00:19:32,371 --> 00:19:37,276 Now, I'm not going to worry about what they are. 389 00:19:37,276 --> 00:19:40,345 But I'm just going to say that there's some plane of atoms 390 00:19:40,345 --> 00:19:40,979 here. 391 00:19:40,979 --> 00:19:44,917 And there's another plane of atoms beneath that. 392 00:19:44,917 --> 00:19:48,720 So there is another one and so on and so on. 393 00:19:48,720 --> 00:19:53,358 And now, these would be Miller planes. 394 00:19:53,358 --> 00:19:55,327 These would be Miller planes in the crystal. 395 00:19:55,327 --> 00:19:57,729 And let's just assume they're very, very simple, 396 00:19:57,729 --> 00:19:58,964 this plane and that plane. 397 00:19:58,964 --> 00:20:01,733 And now what I'm going to do is I'm going to have some x-rays. 398 00:20:01,733 --> 00:20:05,470 I'm going to have some light shining on this. 399 00:20:05,470 --> 00:20:07,873 And it's going to be incident. 400 00:20:07,873 --> 00:20:09,908 And it's going to be reflected. 401 00:20:09,908 --> 00:20:14,179 But see, I'm going to have another wave here. 402 00:20:14,179 --> 00:20:18,116 OK, let's see if I can get through this drawing, 403 00:20:18,116 --> 00:20:22,054 sort of, almost, kind of. 404 00:20:22,054 --> 00:20:26,258 Now, here's the deal, these are waves. 405 00:20:29,328 --> 00:20:32,297 So if I want to draw this as a wave, 406 00:20:32,297 --> 00:20:33,599 I might draw it like that. 407 00:20:33,599 --> 00:20:39,905 And if these waves are constructively interfering, 408 00:20:39,905 --> 00:20:46,545 let's just complete that, then this wave would look like that. 409 00:20:46,545 --> 00:20:49,381 If they're in phase, then that's what they would look like. 410 00:20:49,381 --> 00:20:52,117 Now, this one's getting reflected off the surface. 411 00:20:52,117 --> 00:20:53,852 So I'm going to do that reflection here. 412 00:20:56,622 --> 00:21:00,192 And if I wanted to come back off of the surface, 413 00:21:00,192 --> 00:21:03,295 and this one got through the first layer, 414 00:21:03,295 --> 00:21:04,396 this didn't get reflected. 415 00:21:04,396 --> 00:21:05,230 It's going down. 416 00:21:05,230 --> 00:21:06,999 And if I want it to come back up, 417 00:21:06,999 --> 00:21:11,203 then this one must also look like that to be in phase. 418 00:21:11,203 --> 00:21:13,739 They must be in phases they come out, 419 00:21:13,739 --> 00:21:16,475 or else they're not going to interfere constructively. 420 00:21:16,475 --> 00:21:17,175 You see that? 421 00:21:17,175 --> 00:21:18,577 So those are my X-rays. 422 00:21:18,577 --> 00:21:19,578 They're waves. 423 00:21:19,578 --> 00:21:21,713 They're just waves. 424 00:21:21,713 --> 00:21:25,017 Oh, but this is the whole secret, 425 00:21:25,017 --> 00:21:33,725 because if this angle here is theta, then what that means 426 00:21:33,725 --> 00:21:35,027 is that this angle is theta. 427 00:21:38,030 --> 00:21:41,433 And if that's true, then this distance 428 00:21:41,433 --> 00:21:47,339 is d sine theta, where this is d. 429 00:21:47,339 --> 00:21:51,843 That's just some simple geometry. 430 00:21:51,843 --> 00:21:55,347 So what you know then is, OK, now we're 431 00:21:55,347 --> 00:21:59,818 getting somewhere, because you know if I had a wave, 432 00:21:59,818 --> 00:22:03,288 this is one wavelength. 433 00:22:03,288 --> 00:22:07,459 So if I had a wave come in like this and one 434 00:22:07,459 --> 00:22:09,594 of them is going to get reflected off of this lower 435 00:22:09,594 --> 00:22:12,230 surface and the other one got reflected on this, 436 00:22:12,230 --> 00:22:14,666 but I don't want them to interfere in any way 437 00:22:14,666 --> 00:22:20,105 but constructively, that's what Bragg said, Bragg and Bragg. 438 00:22:20,105 --> 00:22:21,139 That's what they said. 439 00:22:21,139 --> 00:22:23,675 Then the only way for that to happen 440 00:22:23,675 --> 00:22:26,878 is if this distance plus that distance-- 441 00:22:26,878 --> 00:22:29,748 so d sine theta plus d sine theta 442 00:22:29,748 --> 00:22:32,651 is equal to some multiple of the wavelength. 443 00:22:32,651 --> 00:22:34,653 It has to be. 444 00:22:34,653 --> 00:22:38,657 And so what you get is-- well, that's what they have there, 445 00:22:38,657 --> 00:22:40,692 n lambda, some multiple of the wavelength, 446 00:22:40,692 --> 00:22:46,398 equals 2d sine theta, where theta is the incident 447 00:22:46,398 --> 00:22:49,968 angle of that X-ray. 448 00:22:49,968 --> 00:22:51,937 So this is incident-- 449 00:22:51,937 --> 00:22:55,774 just to be clear-- incident X-ray. 450 00:22:55,774 --> 00:23:01,380 And these are reflected X-rays. 451 00:23:04,883 --> 00:23:07,419 Now to keep it simple in this class, 452 00:23:07,419 --> 00:23:16,194 we're just going to say n equals 1 for 3.091, just 453 00:23:16,194 --> 00:23:19,931 to keep things simple, because I want to grasp 454 00:23:19,931 --> 00:23:22,567 the basic concepts here. 455 00:23:22,567 --> 00:23:25,470 We're not using X-rays to get the structure of DNA. 456 00:23:25,470 --> 00:23:28,340 But we're going to use X-rays to figure out cubic crystal 457 00:23:28,340 --> 00:23:29,408 structures. 458 00:23:29,408 --> 00:23:30,942 I'm going to show how that works. 459 00:23:33,779 --> 00:23:38,049 So this must be true for-- 460 00:23:38,049 --> 00:23:39,985 so this is in parentheses-- 461 00:23:39,985 --> 00:23:42,554 for constructive interference. 462 00:23:48,293 --> 00:23:51,363 Those dots that you saw interface-- 463 00:23:51,363 --> 00:23:55,867 interface, no interference, interference. 464 00:23:59,671 --> 00:24:02,140 This is true for constructive interference. 465 00:24:02,140 --> 00:24:04,176 Of course, you could write any equation you want. 466 00:24:04,176 --> 00:24:07,012 But if you want them to be constructively interfering 467 00:24:07,012 --> 00:24:08,847 when they come out, that has to be true. 468 00:24:08,847 --> 00:24:10,182 And that's what the Braggs said. 469 00:24:10,182 --> 00:24:13,051 But we're not there yet right, because now we've 470 00:24:13,051 --> 00:24:15,821 got to do experiments. 471 00:24:15,821 --> 00:24:17,122 So we've got to do experiments. 472 00:24:17,122 --> 00:24:19,558 So what does this mean? 473 00:24:19,558 --> 00:24:28,733 Well, OK, I'm going to take X-rays of some lambda. 474 00:24:28,733 --> 00:24:33,572 And I'm going to shine them on a sample. 475 00:24:33,572 --> 00:24:35,474 And I'm going to measure. 476 00:24:35,474 --> 00:24:37,108 So what I'm going to do-- 477 00:24:37,108 --> 00:24:38,643 well, I think I have a picture-- 478 00:24:38,643 --> 00:24:39,811 I'm going to measure-- 479 00:24:39,811 --> 00:24:40,745 there it is. 480 00:24:40,745 --> 00:24:45,417 This is what an X-ray diffraction experiment 481 00:24:45,417 --> 00:24:47,052 looks like. 482 00:24:47,052 --> 00:24:49,387 So I've got some sample. 483 00:24:49,387 --> 00:24:50,856 And I shine x-rays. 484 00:24:50,856 --> 00:24:52,023 I've got a source of x-rays. 485 00:24:52,023 --> 00:24:56,161 We now know how to make that source. 486 00:24:56,161 --> 00:25:00,465 And we'll just filter out one of these lines. 487 00:25:00,465 --> 00:25:04,236 So I've got that source, and I can change the angle. 488 00:25:04,236 --> 00:25:06,171 And then I've got a detector. 489 00:25:06,171 --> 00:25:09,241 And I can measure did I get interference or not? 490 00:25:09,241 --> 00:25:11,843 It's just like the dots. 491 00:25:11,843 --> 00:25:17,582 So now, we know that if I do that and I say I scan-- 492 00:25:22,320 --> 00:25:24,589 so I've got intensity. 493 00:25:24,589 --> 00:25:27,792 And now I'm plotting it with angle. 494 00:25:27,792 --> 00:25:32,697 So now I'm moving the angle around and I'm changing it. 495 00:25:32,697 --> 00:25:34,699 And, bam, I get interference. 496 00:25:34,699 --> 00:25:36,768 And I see a spot in my detector. 497 00:25:36,768 --> 00:25:39,471 So literally it would look like this. 498 00:25:39,471 --> 00:25:42,841 You would get some angle where there's interference. 499 00:25:42,841 --> 00:25:45,176 And the detector would say ding, ding, ding, ding, ding, 500 00:25:45,176 --> 00:25:48,146 I see a lot of x-rays coming off. 501 00:25:48,146 --> 00:25:49,981 And now you change the angle a few degrees 502 00:25:49,981 --> 00:25:52,918 and I don't see anything, because it's all 503 00:25:52,918 --> 00:25:55,754 destructive from these crystal planes. 504 00:25:58,924 --> 00:26:04,763 So it seems like then if I just vary theta, am I there yet? 505 00:26:04,763 --> 00:26:10,201 The problem is that I might not know d. 506 00:26:10,201 --> 00:26:13,438 A, ha, but we do know d. 507 00:26:13,438 --> 00:26:16,641 We do know d, because we learned about d. 508 00:26:16,641 --> 00:26:19,711 For cubic crystals, we know d. 509 00:26:19,711 --> 00:26:23,148 Because for cubic systems, we know 510 00:26:23,148 --> 00:26:30,221 that d of hkl of any plane, the distance between those planes 511 00:26:30,221 --> 00:26:35,126 is equal to a over the square root of h squared 512 00:26:35,126 --> 00:26:37,629 plus k squared plus l squared. 513 00:26:37,629 --> 00:26:39,230 And that's something that we learned. 514 00:26:39,230 --> 00:26:51,276 This is distance between Miller planes in cubic crystal. 515 00:26:54,346 --> 00:26:55,146 I'm saving time. 516 00:26:55,146 --> 00:26:57,148 I wrote xtal. 517 00:26:57,148 --> 00:26:59,484 I saved a lot of time, which I just 518 00:26:59,484 --> 00:27:01,453 wasted by being so proud of it. 519 00:27:05,557 --> 00:27:06,157 I know d. 520 00:27:06,157 --> 00:27:07,792 Well, OK, so what does that mean? 521 00:27:07,792 --> 00:27:10,562 Well, let's take a look, because now I'm 522 00:27:10,562 --> 00:27:16,234 going to go back my equation. 523 00:27:16,234 --> 00:27:19,537 And I'm going to say, OK, lambda equals 2d-- 524 00:27:19,537 --> 00:27:21,906 and I'm saying n equals 1. 525 00:27:21,906 --> 00:27:23,475 And I'm not going to be very specific. 526 00:27:23,475 --> 00:27:28,013 This is now a d that comes from the spacing between planes 527 00:27:28,013 --> 00:27:30,849 that are specified by the Miller indices 528 00:27:30,849 --> 00:27:35,220 times the sine of theta. 529 00:27:35,220 --> 00:27:40,158 And I'm putting hkl on the theta as well, and you'll see why. 530 00:27:40,158 --> 00:27:45,163 Because here's the thing, now I've got constants. 531 00:27:45,163 --> 00:27:47,365 Now I've got constants, because look, 532 00:27:47,365 --> 00:27:53,972 this is fixed by the source. 533 00:27:53,972 --> 00:27:56,508 So this is a constant. 534 00:27:56,508 --> 00:28:00,945 If I have copper, then it's 1.54 Angstroms. 535 00:28:00,945 --> 00:28:02,781 This is a constant. 536 00:28:02,781 --> 00:28:05,483 This is fixed by the crystal. 537 00:28:05,483 --> 00:28:07,619 That's also a constant. 538 00:28:07,619 --> 00:28:13,358 Fixed by the crystal, because that's also 539 00:28:13,358 --> 00:28:15,326 a constants, because it's the lattice constant. 540 00:28:15,326 --> 00:28:16,961 We're not changing that. 541 00:28:16,961 --> 00:28:20,732 So for a given set of planes that these waves 542 00:28:20,732 --> 00:28:26,271 are bouncing off of and maybe constructively interfering 543 00:28:26,271 --> 00:28:31,109 with, that depending on the theta, 544 00:28:31,109 --> 00:28:33,411 then these are constants. 545 00:28:36,081 --> 00:28:39,684 So if I regroup then-- so I'm going to regroup them. 546 00:28:39,684 --> 00:28:42,353 And so I'm going to say that-- 547 00:28:42,353 --> 00:28:47,225 let's see, d-- so I'm going to substitute in that expression 548 00:28:47,225 --> 00:28:50,495 up there down here. 549 00:28:50,495 --> 00:28:52,931 And I'm going to use a copper source. 550 00:28:52,931 --> 00:28:55,366 So I'm going to say-- well, let me go through this one step 551 00:28:55,366 --> 00:28:55,867 at a time. 552 00:28:55,867 --> 00:29:01,372 So d equals 1.54 Angstroms over-- 553 00:29:04,209 --> 00:29:10,148 let's see-- 2 sine theta hkl. 554 00:29:10,148 --> 00:29:11,816 Now what have I done here? 555 00:29:11,816 --> 00:29:15,453 This is for a copper source. 556 00:29:15,453 --> 00:29:20,992 So I'm fixing in the constant. 557 00:29:20,992 --> 00:29:22,260 So that's the Bragg condition. 558 00:29:22,260 --> 00:29:26,898 But I also know that d is equal to-- 559 00:29:29,467 --> 00:29:30,869 I don't want to write it again. 560 00:29:30,869 --> 00:29:33,471 It's also equal to that. 561 00:29:33,471 --> 00:29:35,173 OK, good. 562 00:29:35,173 --> 00:29:37,008 OK, so let's put that together. 563 00:29:37,008 --> 00:29:38,576 We'll do a little division. 564 00:29:38,576 --> 00:29:48,119 And what we get is 1.54 Angstroms over 2a squared, 565 00:29:48,119 --> 00:29:49,120 the whole thing squared. 566 00:29:49,120 --> 00:29:51,456 I'm squaring it-- I don't want the square root-- 567 00:29:51,456 --> 00:29:58,029 equals sine squared theta hkl over h 568 00:29:58,029 --> 00:30:02,500 squared plus k squared plus l squared for-- 569 00:30:02,500 --> 00:30:14,078 let's be very specific-- for constructive interference 570 00:30:14,078 --> 00:30:16,681 and a copper source. 571 00:30:16,681 --> 00:30:18,483 So now, I'm getting specific because this 572 00:30:18,483 --> 00:30:20,852 is how experiments with X-ray diffraction are done. 573 00:30:24,389 --> 00:30:27,659 But now the last time I checked, if you 574 00:30:27,659 --> 00:30:31,996 got something equal to a constant, 575 00:30:31,996 --> 00:30:35,033 then that something also is a constant. 576 00:30:35,033 --> 00:30:37,902 And this is a constant. 577 00:30:37,902 --> 00:30:42,240 And so what I need to do now is figure out I'm 578 00:30:42,240 --> 00:30:47,045 going to measure these data is where I get a signal. 579 00:30:47,045 --> 00:30:49,547 Remember, I'm going to now change theta so 580 00:30:49,547 --> 00:30:52,617 that I see where I get signals. 581 00:30:52,617 --> 00:31:00,024 Now, those thetas divided by the hkl 582 00:31:00,024 --> 00:31:03,428 that they are bouncing off of must be a constant. 583 00:31:03,428 --> 00:31:04,929 They must not change. 584 00:31:04,929 --> 00:31:08,132 That is at the heart of X-ray diffraction. 585 00:31:08,132 --> 00:31:11,035 That is at the heart of it. 586 00:31:11,035 --> 00:31:16,074 And so we're going to do that with a specific example 587 00:31:16,074 --> 00:31:17,242 today and Friday. 588 00:31:17,242 --> 00:31:19,611 But before we do that, there's another thing 589 00:31:19,611 --> 00:31:20,445 that we can observe. 590 00:31:20,445 --> 00:31:22,213 There's another thing that we can observe. 591 00:31:22,213 --> 00:31:23,848 Oh, this is what it would look like. 592 00:31:23,848 --> 00:31:26,117 So here it is. 593 00:31:26,117 --> 00:31:29,020 So now I I've changed-- now, why do we do 2 theta? 594 00:31:29,020 --> 00:31:30,455 It's kind of historical. 595 00:31:30,455 --> 00:31:34,626 You plot X-ray diffraction spectra. 596 00:31:34,626 --> 00:31:37,862 So this is an XRD, X-ray diffraction spectrum. 597 00:31:37,862 --> 00:31:40,531 This is the intensity of the peaks. 598 00:31:40,531 --> 00:31:42,333 And these are the peaks. 599 00:31:42,333 --> 00:31:43,434 This is a beautiful thing. 600 00:31:43,434 --> 00:31:45,603 I'm seeing a crystal here. 601 00:31:45,603 --> 00:31:46,571 I'm seeing a crystal. 602 00:31:46,571 --> 00:31:49,707 And by Friday, you will be seeing a crystal. 603 00:31:49,707 --> 00:31:50,708 Those aren't just peaks. 604 00:31:50,708 --> 00:31:53,444 Those are planes in a crystal. 605 00:31:53,444 --> 00:31:55,980 Those are planes in a crystal, which tells me 606 00:31:55,980 --> 00:31:58,082 not only that I have these planes, 607 00:31:58,082 --> 00:31:59,984 but it tells me what crystal I have. 608 00:31:59,984 --> 00:32:02,120 But that's not how we start. 609 00:32:02,120 --> 00:32:04,289 The way we start is we do these measurements 610 00:32:04,289 --> 00:32:05,857 and we just read off angles. 611 00:32:05,857 --> 00:32:09,193 So we got to get from there to there, to crystal structure. 612 00:32:09,193 --> 00:32:12,697 So what I want to determine is the crystal structure 613 00:32:12,697 --> 00:32:13,731 and the lattice constant. 614 00:32:13,731 --> 00:32:15,033 That's my goal. 615 00:32:15,033 --> 00:32:17,001 What I have is a spectrum that looks 616 00:32:17,001 --> 00:32:18,503 like this where all I've done here 617 00:32:18,503 --> 00:32:22,006 is put these specific angles here. 618 00:32:22,006 --> 00:32:25,009 And you have to be given-- 619 00:32:25,009 --> 00:32:27,979 so this is the aluminum XRD spectrum. 620 00:32:27,979 --> 00:32:31,282 So if you shine X-rays on aluminum, 621 00:32:31,282 --> 00:32:34,419 this is what you get if you know also 622 00:32:34,419 --> 00:32:37,188 that those X-rays are from copper, which 623 00:32:37,188 --> 00:32:39,958 means that lambda is fixed. 624 00:32:39,958 --> 00:32:44,462 So this should be like the information you get to start. 625 00:32:44,462 --> 00:32:47,098 You'd be given this spectrum, given these peaks. 626 00:32:47,098 --> 00:32:48,900 And you've be giving this information here. 627 00:32:48,900 --> 00:32:50,401 It's a copper target. 628 00:32:50,401 --> 00:32:52,637 And from that, we can determine the crystal structure 629 00:32:52,637 --> 00:32:53,638 in the lattice constant. 630 00:32:53,638 --> 00:32:59,911 Now, there's something-- oh, why do we do 2 theta? 631 00:32:59,911 --> 00:33:01,312 Well, it's historical. 632 00:33:01,312 --> 00:33:04,882 It could have been theta to make all those dividing by 2s 633 00:33:04,882 --> 00:33:05,350 go away. 634 00:33:05,350 --> 00:33:09,020 But instead, you can see that as I rotate this, 635 00:33:09,020 --> 00:33:10,788 this changes by theta. 636 00:33:10,788 --> 00:33:12,957 The detector changes by 2 theta. 637 00:33:12,957 --> 00:33:16,561 So that's why extra spectra are given in 2 theta. 638 00:33:16,561 --> 00:33:19,964 There's no other real good reason for it, 639 00:33:19,964 --> 00:33:22,600 even though in the Bragg condition, it's not 2 theta, 640 00:33:22,600 --> 00:33:23,601 it's theta. 641 00:33:23,601 --> 00:33:27,205 This comes from geometry of the planes. 642 00:33:27,205 --> 00:33:31,275 And this just comes from historical setups 643 00:33:31,275 --> 00:33:34,045 and how you move the detector. 644 00:33:34,045 --> 00:33:38,282 So what's measured and plotted is the 2 theta. 645 00:33:38,282 --> 00:33:41,319 But before we go, before we do this transformation, 646 00:33:41,319 --> 00:33:43,654 where we take an X-ray spectrum like this 647 00:33:43,654 --> 00:33:47,158 and we get the information we want, there's one more thing. 648 00:33:47,158 --> 00:33:51,829 And that is not all reflections are allowed. 649 00:33:51,829 --> 00:33:53,698 Not all reflections are allowed. 650 00:33:53,698 --> 00:33:57,301 And so let's talk about that, and then we'll 651 00:33:57,301 --> 00:34:00,071 come back to the spectrum. 652 00:34:00,071 --> 00:34:03,074 Now, you can kind of understand this 653 00:34:03,074 --> 00:34:07,578 by looking at just a simple kind of comparison here. 654 00:34:10,380 --> 00:34:11,482 So these are the hkl's. 655 00:34:11,482 --> 00:34:14,786 Remember, that's the hkl for a Miller plane. 656 00:34:14,786 --> 00:34:17,021 This is h squared plus k squared plus l squared. 657 00:34:17,021 --> 00:34:18,356 Why do we put that? 658 00:34:18,356 --> 00:34:20,091 Because we know we're going to need it. 659 00:34:20,091 --> 00:34:21,492 There it is right there. 660 00:34:21,492 --> 00:34:24,562 So we know we're going to need it. 661 00:34:24,562 --> 00:34:29,199 But if you look at the simple cubic, that's simple cubic. 662 00:34:29,199 --> 00:34:32,670 Any combination of hkl is OK. 663 00:34:32,670 --> 00:34:38,042 There is no combination that would give you interference 664 00:34:38,042 --> 00:34:41,245 along those plains stacking. 665 00:34:41,245 --> 00:34:44,581 You may say, well, OK, yeah, what are you talking about? 666 00:34:44,581 --> 00:34:45,983 Why are you even bringing this up? 667 00:34:45,983 --> 00:34:48,485 Well, when you see the other two crystal structures 668 00:34:48,485 --> 00:34:50,321 you'll see what I mean. 669 00:34:50,321 --> 00:34:53,558 So now we have the case of BCC-- 670 00:34:53,558 --> 00:34:56,527 when you see this, you see BCC and FCC. 671 00:34:56,527 --> 00:34:59,764 And what I'm showing you here isn't the 100 plane. 672 00:34:59,764 --> 00:35:01,599 It's the 200 plane. 673 00:35:01,599 --> 00:35:02,633 So this is the family-- 674 00:35:02,633 --> 00:35:04,435 remember, the family-- of 200 planes. 675 00:35:04,435 --> 00:35:05,069 There they are. 676 00:35:08,206 --> 00:35:12,877 But now you see that what happens-- 677 00:35:12,877 --> 00:35:14,479 and I have a picture here to show you, 678 00:35:14,479 --> 00:35:15,513 but I'll tell you first-- 679 00:35:15,513 --> 00:35:18,182 what happens is the light comes in. 680 00:35:18,182 --> 00:35:20,618 So there's those squiggly X-rays. 681 00:35:20,618 --> 00:35:21,519 It comes in. 682 00:35:21,519 --> 00:35:24,288 And there is, OK, d, which depends 683 00:35:24,288 --> 00:35:26,023 on the lattice constant, is related 684 00:35:26,023 --> 00:35:27,024 to the lattice constant. 685 00:35:27,024 --> 00:35:27,625 But look. 686 00:35:27,625 --> 00:35:30,261 Now there's another plane in between. 687 00:35:30,261 --> 00:35:31,629 There's another plane in between. 688 00:35:31,629 --> 00:35:33,698 And in fact, with the 200 planes, 689 00:35:33,698 --> 00:35:36,367 that plane in between exactly cancels out 690 00:35:36,367 --> 00:35:37,602 the constructive interference. 691 00:35:37,602 --> 00:35:39,070 Here it is. 692 00:35:39,070 --> 00:35:42,907 So there's what I would have had. 693 00:35:42,907 --> 00:35:48,246 If you want to think about this as the 100, there's the 200s. 694 00:35:48,246 --> 00:35:53,784 But notice when I go from 100 to 200s, I add this plane in here. 695 00:35:53,784 --> 00:35:57,221 And because that plane has atoms in it, 696 00:35:57,221 --> 00:36:00,725 because that plane has atoms in it, it acts like a mirror, 697 00:36:00,725 --> 00:36:04,295 and it can also reflect. 698 00:36:04,295 --> 00:36:07,665 And so what happens is I would have had this nice-- there 699 00:36:07,665 --> 00:36:08,166 it is. 700 00:36:08,166 --> 00:36:09,200 There's a picture I drew. 701 00:36:09,200 --> 00:36:11,502 There's that first X-ray bouncing off. 702 00:36:11,502 --> 00:36:13,104 There's the second one bouncing off. 703 00:36:13,104 --> 00:36:14,672 And those are nicely in phase. 704 00:36:14,672 --> 00:36:16,307 So I would see that. 705 00:36:16,307 --> 00:36:18,042 If that's the angle that gave that to me, 706 00:36:18,042 --> 00:36:19,443 I'd see that in the detector. 707 00:36:19,443 --> 00:36:24,348 But now, for BCC or FCC, I've got something in the middle. 708 00:36:24,348 --> 00:36:27,952 And that something in the middle is exactly canceling out. 709 00:36:27,952 --> 00:36:28,653 You see that? 710 00:36:28,653 --> 00:36:30,288 So now, it cancels that out. 711 00:36:30,288 --> 00:36:32,623 In fact, that's called forbidden. 712 00:36:32,623 --> 00:36:33,624 You won't see a signal. 713 00:36:36,694 --> 00:36:40,731 And so these are called selection rules. 714 00:36:40,731 --> 00:36:45,903 And for a simple cubic, you can see there's nothing inside. 715 00:36:45,903 --> 00:36:49,774 So there's nothing in this unit cell that could do this. 716 00:36:49,774 --> 00:36:50,875 So everything's allowed. 717 00:36:50,875 --> 00:36:53,344 Whether there's a plane in there or not, it doesn't matter. 718 00:36:53,344 --> 00:36:57,548 The selection rule is whether it's ever allowed. 719 00:36:57,548 --> 00:36:59,483 And for a simple cubic, everything's 720 00:36:59,483 --> 00:37:01,886 fine, because nothing would cancel out. 721 00:37:01,886 --> 00:37:04,722 But in here, you see in this 200 case, 722 00:37:04,722 --> 00:37:06,924 you can see very clearly from that picture 723 00:37:06,924 --> 00:37:07,725 how it cancels out. 724 00:37:07,725 --> 00:37:12,730 But there's many other kinds of angles or planes 725 00:37:12,730 --> 00:37:14,999 that might also do that. 726 00:37:14,999 --> 00:37:16,434 And so I'm going to just give you 727 00:37:16,434 --> 00:37:17,635 what the selection rules are. 728 00:37:17,635 --> 00:37:21,105 We won't go through and derive them all. 729 00:37:21,105 --> 00:37:26,577 But let's see, they are actually quite simple. 730 00:37:26,577 --> 00:37:28,879 And so I'm going to write them down. 731 00:37:28,879 --> 00:37:39,390 So if we look at allowed reflections, 732 00:37:39,390 --> 00:37:47,164 and then we look at forbidden reflections-- 733 00:37:47,164 --> 00:37:50,868 so this is what the selection rules tell us-- 734 00:37:50,868 --> 00:38:02,313 so if it's simple cubic, then it's any h, k, and l. 735 00:38:02,313 --> 00:38:06,217 And there's no forbidden reflections. 736 00:38:06,217 --> 00:38:13,624 But if we go through BCC and FCC, then what we find 737 00:38:13,624 --> 00:38:16,027 is that for BCC the selection rule 738 00:38:16,027 --> 00:38:20,865 is that h plus k plus l equals even. 739 00:38:20,865 --> 00:38:24,268 If that is even-- and we won't derive these, 740 00:38:24,268 --> 00:38:26,404 but it comes from the same very simple picture 741 00:38:26,404 --> 00:38:27,571 I just showed you. 742 00:38:27,571 --> 00:38:30,174 If something is in there that can cancel out 743 00:38:30,174 --> 00:38:34,211 the constructive interference, it's going to be forbidden. 744 00:38:34,211 --> 00:38:37,281 Otherwise, it can constructively interfere. 745 00:38:37,281 --> 00:38:38,616 And that's what this tells us. 746 00:38:38,616 --> 00:38:41,385 For BCC, it turns out to be h plus k plus l. 747 00:38:41,385 --> 00:38:44,021 And so here, what's forbidden for BCC 748 00:38:44,021 --> 00:38:47,158 is h plus k plus l is odd. 749 00:38:47,158 --> 00:38:57,435 And for FCC it's h, k, l, all odd or all even. 750 00:39:00,104 --> 00:39:10,881 And the forbidden FCC is h, k, l, mixed odd/even. 751 00:39:10,881 --> 00:39:12,116 These are the selection rules. 752 00:39:16,921 --> 00:39:23,260 So if I were to give you the planes that you see, then 753 00:39:23,260 --> 00:39:26,797 right away from an X-ray spectrum, 754 00:39:26,797 --> 00:39:28,933 you could just use these selection rules 755 00:39:28,933 --> 00:39:32,837 right away to know something, to know something about it. 756 00:39:32,837 --> 00:39:34,572 And if you work this out and you look at, 757 00:39:34,572 --> 00:39:37,908 OK, so we have simple cubic. 758 00:39:37,908 --> 00:39:43,114 So h squared plus k squared plus 1, 100, BCC, FCC, you're 759 00:39:43,114 --> 00:39:44,982 not going to see it. 760 00:39:44,982 --> 00:39:48,552 That doesn't mean that there is no 100 plane in those crystals. 761 00:39:48,552 --> 00:39:50,955 It just means that if you shine X-rays on it, 762 00:39:50,955 --> 00:39:51,822 you will not see it. 763 00:39:54,892 --> 00:39:58,329 OK, so the 110, though, now here, OK, 764 00:39:58,329 --> 00:40:03,167 so we see our-- by the way, mixed, even, odd, and it 765 00:40:03,167 --> 00:40:04,301 adds to odd. 766 00:40:04,301 --> 00:40:06,237 That's why neither one of these works. 767 00:40:06,237 --> 00:40:09,073 Here we go, adds to even, BCC OK, 768 00:40:09,073 --> 00:40:13,077 but it's still mixed even odd, won't be FCC. 769 00:40:13,077 --> 00:40:20,518 3, it's not mixed, so it can be a allowable reflection 770 00:40:20,518 --> 00:40:24,054 for FCC, but not for BBC. 771 00:40:24,054 --> 00:40:27,158 Because if you add them up, it's on odd, 772 00:40:27,158 --> 00:40:28,526 and so forth and so forth. 773 00:40:28,526 --> 00:40:31,395 And look it, 7 doesn't exist, because you can't do it. 774 00:40:31,395 --> 00:40:34,732 No matter how hard you try, you can't get 7. 775 00:40:34,732 --> 00:40:35,866 That's OK. 776 00:40:35,866 --> 00:40:37,701 And there's 8 and 9 and so forth. 777 00:40:37,701 --> 00:40:41,505 So 9 also-- so here it's allowed, 778 00:40:41,505 --> 00:40:43,841 but it's not allowed in either of these, 779 00:40:43,841 --> 00:40:53,150 because you can't get either of these to be satisfied. 780 00:40:53,150 --> 00:40:55,319 This is what selection rules give us. 781 00:40:55,319 --> 00:41:00,858 And it comes, again, from simple-- 782 00:41:00,858 --> 00:41:02,726 oh, OK, well, that that's another thing. 783 00:41:02,726 --> 00:41:05,429 I said a simple, I have to tell you something, 784 00:41:05,429 --> 00:41:11,635 because the Bragg condition, it relies on an assumption 785 00:41:11,635 --> 00:41:13,604 that's mostly true. 786 00:41:13,604 --> 00:41:26,350 But the Bragg condition requires that the reflection 787 00:41:26,350 --> 00:41:28,419 is independent-- 788 00:41:28,419 --> 00:41:31,922 some of you may be thinking I did I draw it onto the atom 789 00:41:31,922 --> 00:41:33,724 or did I draw it in between atoms? 790 00:41:33,724 --> 00:41:34,825 Where did that thing-- 791 00:41:34,825 --> 00:41:36,427 does it have to reflect off an atom? 792 00:41:36,427 --> 00:41:37,928 We're not going there. 793 00:41:37,928 --> 00:41:44,869 With the Bragg condition, it's independent of the atom 794 00:41:44,869 --> 00:41:46,704 positions in a plane. 795 00:41:52,843 --> 00:41:54,778 And the second thing which is what 796 00:41:54,778 --> 00:41:59,517 I've been sort of alluding to is that the atomic planes 797 00:41:59,517 --> 00:42:00,351 are mirror like. 798 00:42:10,794 --> 00:42:12,897 I mean, this is sort of an obvious assumption 799 00:42:12,897 --> 00:42:15,799 since I've been assuming they've been mirrors. 800 00:42:15,799 --> 00:42:22,239 But if you start thinking about atom position, 801 00:42:22,239 --> 00:42:26,076 you might go back to that selection rule picture 802 00:42:26,076 --> 00:42:28,712 and say, well, wait a second, does it 803 00:42:28,712 --> 00:42:30,214 always have to hit the atom? 804 00:42:30,214 --> 00:42:32,550 What if this one was over or something like that? 805 00:42:32,550 --> 00:42:36,020 No, no, we assume it's just one continuous plane 806 00:42:36,020 --> 00:42:40,124 if there are atoms in it. 807 00:42:40,124 --> 00:42:44,828 If there's no atom in it, then it's not a reflective plane. 808 00:42:44,828 --> 00:42:47,398 But it's continuous in the assumption of the Bragg 809 00:42:47,398 --> 00:42:48,332 condition. 810 00:42:52,903 --> 00:42:54,638 Now we go back to our picture. 811 00:42:54,638 --> 00:43:00,077 So what we want to do is, again, our goal, 812 00:43:00,077 --> 00:43:08,385 should we choose, our goal is to go from this spectrum knowing 813 00:43:08,385 --> 00:43:09,920 this information-- 814 00:43:09,920 --> 00:43:13,824 it's a copper target and being able to read off the peak-- 815 00:43:13,824 --> 00:43:16,493 our goal is to determine the crystal structure 816 00:43:16,493 --> 00:43:18,062 and the lattice constant. 817 00:43:18,062 --> 00:43:20,864 That's our goal. 818 00:43:20,864 --> 00:43:23,300 And let me let me just write this again, 819 00:43:23,300 --> 00:43:25,903 because it's extremely important. 820 00:43:25,903 --> 00:43:29,540 So maybe I'll keep that one. 821 00:43:29,540 --> 00:43:32,576 I'm going to erase this and put it right in the middle, 822 00:43:32,576 --> 00:43:38,582 because this is what drives XRD. 823 00:43:38,582 --> 00:43:42,019 This is what drives XRD, which is 824 00:43:42,019 --> 00:44:07,978 that our goal is figure out what makes this constant always. 825 00:44:07,978 --> 00:44:10,214 Now, you say what is this constant always? 826 00:44:12,783 --> 00:44:17,488 This is the expression, which I'm going to write again. 827 00:44:17,488 --> 00:44:21,091 So what I have if it's a copper source 828 00:44:21,091 --> 00:44:26,330 is 1.54 Angstroms divided by 2 a squared. 829 00:44:26,330 --> 00:44:29,566 That's a constant. 830 00:44:29,566 --> 00:44:36,774 That equals sine squared of the theta for some plane divided 831 00:44:36,774 --> 00:44:42,312 by h squared plus k squared plus l squared. 832 00:44:42,312 --> 00:44:44,148 So I'm just kind of repeating what I've said 833 00:44:44,148 --> 00:44:46,417 and what I've written elsewhere, but that is really it. 834 00:44:46,417 --> 00:44:49,386 That is what we do in X-ray diffraction. 835 00:44:49,386 --> 00:44:51,088 What makes this? 836 00:44:51,088 --> 00:44:57,127 Well, by this what I mean is this term on the right. 837 00:44:57,127 --> 00:45:01,799 How do I make sure that this never changes its value? 838 00:45:01,799 --> 00:45:05,803 Because the thing on the left never changes its value. 839 00:45:05,803 --> 00:45:08,572 And just making sure that you've got that concept, 840 00:45:08,572 --> 00:45:10,307 that is this, oh, yeah. 841 00:45:14,144 --> 00:45:20,684 And it turns out I got a recipe for you to follow to do this. 842 00:45:20,684 --> 00:45:22,486 And it's on the previous slide. 843 00:45:22,486 --> 00:45:24,988 And so we'll start thinking about it now. 844 00:45:24,988 --> 00:45:26,490 And we've got five more minutes. 845 00:45:26,490 --> 00:45:29,626 And then on Friday, we will finish this and then 846 00:45:29,626 --> 00:45:34,264 talk about what to do with those continuous X-rays. 847 00:45:34,264 --> 00:45:36,700 So what do I do? 848 00:45:36,700 --> 00:45:38,535 Well, the way you do this is systematically. 849 00:45:43,173 --> 00:45:45,142 And the first thing you do is you read off 850 00:45:45,142 --> 00:45:50,481 the 2 theta values that generate a set of sine squared value. 851 00:45:50,481 --> 00:45:53,183 So that the X-ray spectrum measures 2 theta. 852 00:45:53,183 --> 00:45:56,386 But you know from here that I need sine squared. 853 00:45:56,386 --> 00:46:00,057 So I'm going to write down the sine squared values. 854 00:46:00,057 --> 00:46:01,358 So that's a first step. 855 00:46:08,465 --> 00:46:14,037 So for the first one, I've got the first peak. 856 00:46:14,037 --> 00:46:16,440 So I would start to log my data. 857 00:46:16,440 --> 00:46:23,480 The first peak and the 2 theta is 38.43. 858 00:46:23,480 --> 00:46:27,417 And the sine squared theta is-- 859 00:46:27,417 --> 00:46:28,919 and I'm just going to do that math-- 860 00:46:28,919 --> 00:46:32,623 0.1083. 861 00:46:32,623 --> 00:46:33,524 OK, good. 862 00:46:33,524 --> 00:46:40,130 And the next one, the next peak is 44.67. 863 00:46:40,130 --> 00:46:46,003 And the sine square of that is 0.1444 so on. 864 00:46:46,003 --> 00:46:49,406 So you read off all the peaks and you make those columns. 865 00:46:49,406 --> 00:46:50,407 So I've gone through. 866 00:46:50,407 --> 00:46:52,576 Now the second one, normalize the sine 867 00:46:52,576 --> 00:46:56,013 squared theta values by the smallest value. 868 00:46:56,013 --> 00:46:57,514 You say, so why am I doing that? 869 00:46:57,514 --> 00:47:00,450 Trust me, this will achieve our goal. 870 00:47:00,450 --> 00:47:02,052 This will get us there. 871 00:47:02,052 --> 00:47:04,955 This is a nice simple recipe to achieve our goal. 872 00:47:04,955 --> 00:47:07,591 So the next column would simply be 873 00:47:07,591 --> 00:47:20,304 sine squared theta divided by sine squared theta min. 874 00:47:20,304 --> 00:47:22,072 Well, I ran out of room there, but that's 875 00:47:22,072 --> 00:47:24,541 going to be obviously 1. 876 00:47:24,541 --> 00:47:30,948 And this would be 1.333 and so on. 877 00:47:30,948 --> 00:47:31,915 So that's the next one. 878 00:47:31,915 --> 00:47:37,287 So I'm just setting that the top row to 1. 879 00:47:37,287 --> 00:47:41,091 I'm setting the top row to 1. 880 00:47:41,091 --> 00:47:41,592 Good. 881 00:47:41,592 --> 00:47:43,327 Now, I'm going to clear fractions. 882 00:47:43,327 --> 00:47:46,163 So I've normalized, and now I'm going to clear fractions. 883 00:47:46,163 --> 00:47:50,500 And if I do that, I clear fractions. 884 00:47:50,500 --> 00:47:54,571 Well, you say, what does clearing fractions mean? 885 00:47:54,571 --> 00:47:56,006 It means just what it implies. 886 00:47:56,006 --> 00:47:58,475 I don't want fractions anymore. 887 00:47:58,475 --> 00:48:01,945 It means I just need to multiply the whole set of numbers 888 00:48:01,945 --> 00:48:04,982 by something that gets rid of the fractions. 889 00:48:04,982 --> 00:48:08,685 And it turns out that in this case it is 3. 890 00:48:08,685 --> 00:48:18,829 So 3 times sine squared theta over sine squared theta min, 891 00:48:18,829 --> 00:48:21,932 and that is going to be 3. 892 00:48:21,932 --> 00:48:27,337 And that is going to be 4 and so on. 893 00:48:31,975 --> 00:48:34,077 We're almost there. 894 00:48:34,077 --> 00:48:34,912 We're almost there. 895 00:48:34,912 --> 00:48:36,413 And we're not going to finish today, 896 00:48:36,413 --> 00:48:38,415 but we're getting real close. 897 00:48:38,415 --> 00:48:39,082 That's good. 898 00:48:39,082 --> 00:48:44,121 It leaves us with a sense of anticipation and excitement. 899 00:48:44,121 --> 00:48:46,823 So, OK, I got clear fractions. 900 00:48:46,823 --> 00:48:51,361 What values of h, k, l, if I have this h squared 901 00:48:51,361 --> 00:48:53,530 plus k squared plus l squared, would give me 902 00:48:53,530 --> 00:48:55,465 the sequence of clear fractions? 903 00:48:55,465 --> 00:48:57,167 What values of h squared? 904 00:48:57,167 --> 00:49:00,304 Now, remember, you say, why am I doing this? 905 00:49:00,304 --> 00:49:02,372 Again, go back to this. 906 00:49:02,372 --> 00:49:04,441 This is why. 907 00:49:04,441 --> 00:49:08,912 I have a simple recipe for you to accomplish this goal. 908 00:49:08,912 --> 00:49:11,949 That's where we're going with this. 909 00:49:11,949 --> 00:49:15,018 So I'm now going to see with those clear fractions, 910 00:49:15,018 --> 00:49:17,888 now it's very easy to see what h, k, 911 00:49:17,888 --> 00:49:22,025 l's would give me-- h squared plus k squared plus l squared 912 00:49:22,025 --> 00:49:25,228 equals that clear fraction value. 913 00:49:25,228 --> 00:49:30,167 So for example, here, well, if this is h 914 00:49:30,167 --> 00:49:33,270 squared plus k squared plus l squared, maybe this is 111. 915 00:49:33,270 --> 00:49:43,580 And This might be 200, for example. 916 00:49:43,580 --> 00:49:45,315 So we're almost there. 917 00:49:45,315 --> 00:49:46,817 We're so close. 918 00:49:46,817 --> 00:49:48,719 What we're going to do is I'm going 919 00:49:48,719 --> 00:49:51,221 to start on Friday I'll put this up on the board 920 00:49:51,221 --> 00:49:55,392 and we'll finish filling it out and going from this matrix 921 00:49:55,392 --> 00:49:58,028 to the crystal structure and the lattice constant. 922 00:49:58,028 --> 00:50:00,497 OK, have a great Halloween.