1 00:00:00,030 --> 00:00:02,400 The following content is provided under a creative 2 00:00:02,400 --> 00:00:03,840 commons license. 3 00:00:03,840 --> 00:00:06,900 Your support will help MIT OpenCourseWare continue to 4 00:00:06,900 --> 00:00:10,520 offer high-quality educational resources for free. 5 00:00:10,520 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,430 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,430 --> 00:00:18,680 ocw.mit.edu. 8 00:00:21,480 --> 00:00:22,800 PROFESSOR: OK. 9 00:00:22,800 --> 00:00:26,340 We're still on Exam 2 of the fall 2009 class. 10 00:00:26,340 --> 00:00:28,420 We're going to be doing problem number 2 now. 11 00:00:28,420 --> 00:00:30,750 This is the x-ray problem, as I like to call it. 12 00:00:30,750 --> 00:00:32,140 It's kind of exciting. 13 00:00:32,140 --> 00:00:34,690 Let's talk about what we need to know before we actually 14 00:00:34,690 --> 00:00:36,480 reasonably try to attempt this problem. 15 00:00:36,480 --> 00:00:39,480 So I would review these things again. 16 00:00:39,480 --> 00:00:42,300 We want to know emission line nomenclature, how to name 17 00:00:42,300 --> 00:00:43,170 emission lines. 18 00:00:43,170 --> 00:00:45,850 We want to know Moseley's Law. 19 00:00:45,850 --> 00:00:48,440 We want to know Bragg's Law, and I would also emphasize, 20 00:00:48,440 --> 00:00:50,250 I'll get onto it later, you want to know how to derive 21 00:00:50,250 --> 00:00:52,420 Bragg's Law as well. 22 00:00:52,420 --> 00:00:54,690 We want to know, this is pronounced Bremsstrahlung, 23 00:00:54,690 --> 00:00:57,000 it's a German word which means breaking radiation. 24 00:00:57,000 --> 00:00:59,690 So Bremsstrahlung radiation. 25 00:00:59,690 --> 00:01:02,540 And we also want to know some reflection rules, and I'll 26 00:01:02,540 --> 00:01:03,870 sort of elucidate that a bit more. 27 00:01:03,870 --> 00:01:07,020 But let me start you off in that direction. 28 00:01:07,020 --> 00:01:10,390 So getting on to part A, we are told that somebody has 29 00:01:10,390 --> 00:01:14,080 been horsing around with our x-ray machine. 30 00:01:14,080 --> 00:01:15,730 We've all had this problem before. 31 00:01:15,730 --> 00:01:18,380 And they've changed the target. 32 00:01:18,380 --> 00:01:20,490 So let me first tell you what a target is, and how the x-ray 33 00:01:20,490 --> 00:01:21,270 machine looks. 34 00:01:21,270 --> 00:01:22,790 Then we'll be able to understand what exactly it is 35 00:01:22,790 --> 00:01:24,700 we're trying to figure out. 36 00:01:24,700 --> 00:01:28,740 An x-ray machine, basically the x-ray source involves 37 00:01:28,740 --> 00:01:30,970 electrons being accelerated into a 38 00:01:30,970 --> 00:01:32,710 target material, a metal. 39 00:01:32,710 --> 00:01:35,180 And then that gives off a whole bunch of electromagnetic 40 00:01:35,180 --> 00:01:37,700 radiation, so photons. 41 00:01:37,700 --> 00:01:40,610 Some of those photons we'll talk a little about later, in 42 00:01:40,610 --> 00:01:42,530 the Bremsstrahlung spectrum, are very 43 00:01:42,530 --> 00:01:44,800 characteristic of the material. 44 00:01:44,800 --> 00:01:47,800 So I'm going to call, you know, we have long wavelength 45 00:01:47,800 --> 00:01:49,150 photons, short wavelength photons. 46 00:01:49,150 --> 00:01:52,980 Some of the photons have a very characteristic wavelength 47 00:01:52,980 --> 00:01:57,720 or frequency that corresponds to a particular electron 48 00:01:57,720 --> 00:01:58,200 transition. 49 00:01:58,200 --> 00:02:00,320 We're going to talk more about this in a little bit, but I'm 50 00:02:00,320 --> 00:02:03,030 going to call this the K-alpha photon, OK? 51 00:02:03,030 --> 00:02:07,680 So that just happens to be in metals, x-ray. 52 00:02:07,680 --> 00:02:13,310 We know that for particular materials, we have particular 53 00:02:13,310 --> 00:02:15,070 K-alpha wavelengths. 54 00:02:15,070 --> 00:02:17,710 And what's basically happened in this problem is that, you 55 00:02:17,710 --> 00:02:20,940 know, maybe we knew what this material was before, and then 56 00:02:20,940 --> 00:02:22,750 all of a sudden we weren't around, somebody switched it 57 00:02:22,750 --> 00:02:26,110 on us, and now we have to figure out what it is. 58 00:02:26,110 --> 00:02:29,480 Don't you hate when that happens? 59 00:02:29,480 --> 00:02:35,180 So the best way to approach this problem is to understand 60 00:02:35,180 --> 00:02:37,690 Moseley's Law, and then have it on your equation sheet. 61 00:02:37,690 --> 00:02:40,480 We allow all of our students to have an equation sheet, and 62 00:02:40,480 --> 00:02:41,550 also a periodic table. 63 00:02:41,550 --> 00:02:44,780 So don't panic if you can't memorize this equation. 64 00:02:44,780 --> 00:02:47,200 Our students are expected to, not to memorize it, but to 65 00:02:47,200 --> 00:02:48,400 understand it. 66 00:02:48,400 --> 00:02:50,590 So this is Moseley's Law. 67 00:02:50,590 --> 00:02:53,230 And Moseley's Law basically tells us, let me go through 68 00:02:53,230 --> 00:02:55,320 these variables. 69 00:02:55,320 --> 00:02:56,770 We have the wave number. 70 00:02:56,770 --> 00:02:58,950 We have a Rydberg constant. 71 00:02:58,950 --> 00:03:01,220 This is a bunch of physical constants, sort of 72 00:03:01,220 --> 00:03:04,420 agglomerated into one big one, to save us some time. 73 00:03:04,420 --> 00:03:11,310 We have the final energy level, where the electron 74 00:03:11,310 --> 00:03:14,240 ends, and we have the initial energy level where it begins. 75 00:03:14,240 --> 00:03:17,590 And then we have z, which designates our element. 76 00:03:17,590 --> 00:03:19,190 So it's the number of protons we have in the 77 00:03:19,190 --> 00:03:20,540 nucleus, for example. 78 00:03:20,540 --> 00:03:22,890 And then we have this sigma. 79 00:03:22,890 --> 00:03:23,710 OK? 80 00:03:23,710 --> 00:03:26,320 So, you know, in order to solve this problem, we want to 81 00:03:26,320 --> 00:03:27,645 know what element. 82 00:03:35,290 --> 00:03:39,600 So obviously, we're looking for z. 83 00:03:39,600 --> 00:03:41,360 Now the question is, do we know everything else? 84 00:03:45,330 --> 00:03:50,130 So yeah, we definitely do. 85 00:03:50,130 --> 00:03:52,730 Let's just go through each individual thing that we know. 86 00:03:52,730 --> 00:03:57,180 So we know Rydberg constant. 87 00:03:57,180 --> 00:03:59,250 That's just, you know, have that on your equation sheet. 88 00:03:59,250 --> 00:04:01,940 It's a bunch of constants agglomerated together. 89 00:04:01,940 --> 00:04:04,380 We know that, for K-alpha we're told, 90 00:04:04,380 --> 00:04:06,700 this is K-alpha radiation. 91 00:04:06,700 --> 00:04:09,020 Let me draw a cartoon for you here. 92 00:04:12,100 --> 00:04:12,270 OK? 93 00:04:12,270 --> 00:04:15,400 This is our nucleus, OK? 94 00:04:15,400 --> 00:04:18,890 This is n equals 1, this is n equals 2. 95 00:04:18,890 --> 00:04:20,340 I'm going to go kind off the board. 96 00:04:20,340 --> 00:04:21,420 This is 3. 97 00:04:21,420 --> 00:04:23,480 OK? 98 00:04:23,480 --> 00:04:29,120 And K-alpha corresponds to an electron dropping down from n2 99 00:04:29,120 --> 00:04:33,290 to n1 to here, and then giving off a photon. 100 00:04:33,290 --> 00:04:37,260 So I'm going to draw a photon as a squiggly line like this. 101 00:04:37,260 --> 00:04:38,055 OK? 102 00:04:38,055 --> 00:04:39,700 So this is going to be h nu. 103 00:04:39,700 --> 00:04:44,570 This will give you our K-alpha radiation. 104 00:04:44,570 --> 00:04:45,640 So we know all of those things. 105 00:04:45,640 --> 00:04:47,760 We know our wave number here. 106 00:04:47,760 --> 00:04:49,230 So we know pretty much everything. 107 00:04:49,230 --> 00:04:50,520 But what is sigma? 108 00:04:50,520 --> 00:04:54,590 Well, sigma is the correction factor that Moseley found. 109 00:04:54,590 --> 00:04:56,530 Now, if you're looking at this and you're thinking, wait, 110 00:04:56,530 --> 00:04:59,610 this looks a lot like the Rydberg equation, it's because 111 00:04:59,610 --> 00:05:00,640 it's sort of is. 112 00:05:00,640 --> 00:05:02,500 The only difference is that the Rydberg equation is only 113 00:05:02,500 --> 00:05:05,040 for hydrogenic-type atoms. OK? 114 00:05:05,040 --> 00:05:07,170 So we're talking about hydrogen, we're talking about 115 00:05:07,170 --> 00:05:09,670 helium plus 1, lithium plus 2. 116 00:05:09,670 --> 00:05:12,490 We can't assume that our target is hydrogenic, so we 117 00:05:12,490 --> 00:05:14,870 can't use the Rydberg equation. 118 00:05:14,870 --> 00:05:17,260 So we're using Moseley's equation. 119 00:05:17,260 --> 00:05:20,650 And what Moseley found, just to show you where the sigma 120 00:05:20,650 --> 00:05:23,930 sort of comes from, and this is actually reviewed in, I 121 00:05:23,930 --> 00:05:27,630 think, lecture 17 of the online postings. 122 00:05:27,630 --> 00:05:32,990 What Moseley found was that if you plot your wave number 123 00:05:32,990 --> 00:05:38,110 versus your z, you've got points like this. 124 00:05:40,680 --> 00:05:41,930 You've got some sort of line. 125 00:05:46,120 --> 00:05:46,946 OK? 126 00:05:46,946 --> 00:05:49,220 So this is what Moseley found. 127 00:05:49,220 --> 00:05:50,790 And what Moseley basically did, was he fit 128 00:05:50,790 --> 00:05:51,553 an equation to it. 129 00:05:51,553 --> 00:05:54,140 And the equation looks a lot like the Rydberg equation. 130 00:05:54,140 --> 00:05:56,480 And because these don't intersect the origin, we've 131 00:05:56,480 --> 00:05:59,910 got this fitting factor right here. 132 00:05:59,910 --> 00:06:03,920 We know that for a K-alpha transition, we're looking at a 133 00:06:03,920 --> 00:06:05,230 sigma equals 1. 134 00:06:09,430 --> 00:06:12,270 OK? 135 00:06:12,270 --> 00:06:14,250 So we're good. 136 00:06:14,250 --> 00:06:15,060 We know our sigma. 137 00:06:15,060 --> 00:06:15,820 We know our n's. 138 00:06:15,820 --> 00:06:18,150 We know n final is going to be 1. 139 00:06:18,150 --> 00:06:20,840 We know n initial is going to be 2. 140 00:06:20,840 --> 00:06:23,000 We know the Rydberg constant, and we know our wave number, 141 00:06:23,000 --> 00:06:25,030 because we're given our wavelength here. 142 00:06:25,030 --> 00:06:27,390 So we can go ahead and actually just calculate z. 143 00:06:27,390 --> 00:06:28,630 And let's do that right now. 144 00:06:28,630 --> 00:06:31,185 So let me rewrite this equation so that 145 00:06:31,185 --> 00:06:32,435 it makes more sense. 146 00:06:39,120 --> 00:06:41,790 Rydberg constant. 147 00:06:41,790 --> 00:06:42,540 1 over. 148 00:06:42,540 --> 00:06:43,610 our n final. 149 00:06:43,610 --> 00:06:48,090 That's 1 squared minus 1 over, our 150 00:06:48,090 --> 00:06:49,920 initial point is 2 squared. 151 00:06:52,880 --> 00:06:56,920 We're looking for our z, and we're going to subtract off 1, 152 00:06:56,920 --> 00:06:59,010 because we're dealing with a K-alpha transition. 153 00:06:59,010 --> 00:07:01,760 So you're expected to either remember that, or have it on 154 00:07:01,760 --> 00:07:02,760 your question sheet. 155 00:07:02,760 --> 00:07:05,210 There's no way to derive that unless you've got all this 156 00:07:05,210 --> 00:07:07,240 data, which you won't have. 157 00:07:07,240 --> 00:07:10,160 So we've got this squared. 158 00:07:10,160 --> 00:07:14,160 We know everything, so we can basically reduce this. 159 00:07:14,160 --> 00:07:16,090 This is why on the answer sheet, you see 160 00:07:16,090 --> 00:07:17,900 something like this. 161 00:07:17,900 --> 00:07:19,440 You kind of skip all these steps, and we 162 00:07:19,440 --> 00:07:21,570 went right to this. 163 00:07:21,570 --> 00:07:24,990 3/4 z minus 1 squared. 164 00:07:24,990 --> 00:07:26,490 And now you can just solve for z. 165 00:07:26,490 --> 00:07:27,740 No problem. 166 00:07:43,660 --> 00:07:44,070 OK? 167 00:07:44,070 --> 00:07:48,600 So that's 4 over 3 lambda Rydberg constant to 168 00:07:48,600 --> 00:07:50,125 the half plus 1. 169 00:07:50,125 --> 00:07:53,090 And that's going to give you about 23. 170 00:07:53,090 --> 00:07:54,290 You might get 22.9. 171 00:07:54,290 --> 00:07:55,530 You might get 23.1. 172 00:07:55,530 --> 00:07:57,010 We're talking about 23. 173 00:07:57,010 --> 00:07:59,260 That's going to be the vanadium. 174 00:07:59,260 --> 00:08:02,950 That's the answer to the first part of this problem. 175 00:08:02,950 --> 00:08:03,550 OK. 176 00:08:03,550 --> 00:08:05,440 We're going to take a second and clean up, and come right 177 00:08:05,440 --> 00:08:06,690 back and finish the problem. 178 00:08:09,610 --> 00:08:12,130 We're going to start part B now of problem 2 179 00:08:12,130 --> 00:08:14,340 on the second exam. 180 00:08:14,340 --> 00:08:18,780 So we basically have used the Moseley equation to find out 181 00:08:18,780 --> 00:08:19,470 an element. 182 00:08:19,470 --> 00:08:21,200 And now we're going to switch the problem up a little bit 183 00:08:21,200 --> 00:08:25,230 more, and we're going to ask you to find the smallest 184 00:08:25,230 --> 00:08:27,590 defraction angle for a particular situation. 185 00:08:27,590 --> 00:08:29,210 So what's the situation? 186 00:08:29,210 --> 00:08:31,630 Let's go back to our x-ray machine. 187 00:08:31,630 --> 00:08:34,690 The way you actually do x-ray defraction, you know, it's 188 00:08:34,690 --> 00:08:37,210 acronymized XRD. 189 00:08:37,210 --> 00:08:40,130 The way you do it, is you have to generate x-rays. 190 00:08:40,130 --> 00:08:42,790 So we accelerate an electron into some material. 191 00:08:42,790 --> 00:08:46,130 That material gives off a whole bunch of photons. 192 00:08:46,130 --> 00:08:49,320 You then filter all the photons, so you only pick up a 193 00:08:49,320 --> 00:08:50,980 certain wavelength. 194 00:08:50,980 --> 00:08:53,510 And we're going to use the K-alpha wavelength. 195 00:08:53,510 --> 00:08:57,820 And then you get these K-alpha radiation, which then probes 196 00:08:57,820 --> 00:08:58,710 your sample. 197 00:08:58,710 --> 00:08:59,720 So here we go. 198 00:08:59,720 --> 00:09:01,990 This is what we're looking at for this part. 199 00:09:01,990 --> 00:09:06,480 We've got K-alpha radiation coming in, and it's hitting 200 00:09:06,480 --> 00:09:09,460 our structure, which has a crystal, hopefully. 201 00:09:09,460 --> 00:09:11,040 You're going to do XRD. 202 00:09:11,040 --> 00:09:13,660 And then you're getting off some defraction, anything 203 00:09:13,660 --> 00:09:16,480 that's reflecting off of the crystal in 204 00:09:16,480 --> 00:09:17,730 some way, like this. 205 00:09:17,730 --> 00:09:20,900 So the thing you need to know to do part B, is you need to 206 00:09:20,900 --> 00:09:23,570 know Bragg's Law, Bragg's equation. 207 00:09:23,570 --> 00:09:28,280 And that's number 3 on the win list. So I drew this picture 208 00:09:28,280 --> 00:09:31,170 here just to define the variables, but one thing I 209 00:09:31,170 --> 00:09:35,040 would stress for you at home, is to please, go home and, you 210 00:09:35,040 --> 00:09:37,570 know, search online, Google image search even. 211 00:09:37,570 --> 00:09:38,690 Look for a picture like this. 212 00:09:38,690 --> 00:09:41,020 Search Bragg's Law and look for a picture like this. 213 00:09:41,020 --> 00:09:42,840 And it's actually from this. 214 00:09:42,840 --> 00:09:45,110 It's very easy, just using geometry, to 215 00:09:45,110 --> 00:09:46,130 derive Bragg's Law. 216 00:09:46,130 --> 00:09:48,470 Bragg's Law is just the geometric interpretation of 217 00:09:48,470 --> 00:09:49,400 this picture. 218 00:09:49,400 --> 00:09:50,250 OK? 219 00:09:50,250 --> 00:09:54,570 So Bragg's Law, in its traditional format, is n 220 00:09:54,570 --> 00:09:56,650 lambda equals 2d sine theta. 221 00:09:56,650 --> 00:09:58,510 And I've defined, I drew the picture to show 222 00:09:58,510 --> 00:10:00,910 the variables here. 223 00:10:00,910 --> 00:10:02,310 We have n, which is-- 224 00:10:02,310 --> 00:10:03,430 we'll talk about n in a second. 225 00:10:03,430 --> 00:10:06,900 We have lambda, which is our incoming radiation. 226 00:10:06,900 --> 00:10:11,030 We have d, which is the spacing between planes. 227 00:10:11,030 --> 00:10:14,560 Whatever plane orientation you're looking at. 228 00:10:14,560 --> 00:10:18,070 2d is the spacing, is there because you need to come in, 229 00:10:18,070 --> 00:10:18,785 and then come out again. 230 00:10:18,785 --> 00:10:22,510 So you're doing twice the distance between the planes. 231 00:10:22,510 --> 00:10:25,830 Theta is the angle at which you're 232 00:10:25,830 --> 00:10:27,920 incident on the material. 233 00:10:27,920 --> 00:10:30,180 And n is going to be your defraction index. 234 00:10:30,180 --> 00:10:32,030 We're not going to worry about that in this class. 235 00:10:32,030 --> 00:10:35,120 We're going to assume it's one, for this class 236 00:10:35,120 --> 00:10:36,460 and for this exam. 237 00:10:36,460 --> 00:10:39,670 So the question is asking us to find the smallest possible 238 00:10:39,670 --> 00:10:41,010 angle of defraction. 239 00:10:41,010 --> 00:10:43,590 What that means, is that we're getting all this radiation, 240 00:10:43,590 --> 00:10:44,970 it's hitting our material, we're moving 241 00:10:44,970 --> 00:10:45,930 our material around. 242 00:10:45,930 --> 00:10:48,680 The question is, what's the smallest angle at which we're 243 00:10:48,680 --> 00:10:52,710 going to see a peak on our XRD graph? 244 00:10:52,710 --> 00:10:56,120 I rearranged the equation here, so theta equals the 245 00:10:56,120 --> 00:10:58,550 inverse sign of lambda over 2d. 246 00:10:58,550 --> 00:11:02,340 And we're looking for as small a theta as possible. 247 00:11:02,340 --> 00:11:05,070 This question is really just mathematical manipulation. 248 00:11:05,070 --> 00:11:07,340 I need to know a couple important details that 249 00:11:07,340 --> 00:11:11,510 hopefully you've read about in advance. 250 00:11:11,510 --> 00:11:14,160 So we want to find the smallest theta possible. 251 00:11:14,160 --> 00:11:16,660 That's the objective for this problem. 252 00:11:16,660 --> 00:11:20,360 I've drawn the arc sine function here. 253 00:11:20,360 --> 00:11:24,600 To minimize theta, we want to have the argument, which I 254 00:11:24,600 --> 00:11:26,900 call x here, we want the argument to 255 00:11:26,900 --> 00:11:28,450 be as small as possible. 256 00:11:28,450 --> 00:11:32,700 The smaller it is, the closer to 0 you are for y, which is 257 00:11:32,700 --> 00:11:34,910 theta, in our case. 258 00:11:34,910 --> 00:11:38,010 So we want to minimize the term in parentheses. 259 00:11:38,010 --> 00:11:42,710 And the way you do that, small theta means small lambda over 260 00:11:42,710 --> 00:11:46,410 2d, which means you want to have a large d. 261 00:11:46,410 --> 00:11:49,820 And the reason I'm not using lambda, the wavelength, as the 262 00:11:49,820 --> 00:11:52,620 knob, is because lambda is set. 263 00:11:52,620 --> 00:11:56,460 Lambda is predetermined by the target material you've chosen. 264 00:11:56,460 --> 00:12:00,060 So for our problem, we have titanium 265 00:12:00,060 --> 00:12:01,360 as our target material. 266 00:12:01,360 --> 00:12:03,900 Which is, we can't change the lambda K-alpha. 267 00:12:03,900 --> 00:12:05,740 It's set by that material. 268 00:12:05,740 --> 00:12:07,730 So we have lambda K-alpha-- 269 00:12:07,730 --> 00:12:08,960 we have k-Alpha coming off, which is 270 00:12:08,960 --> 00:12:10,980 characteristic of titanium. 271 00:12:10,980 --> 00:12:14,500 So we can't use lambda as a knob. 272 00:12:14,500 --> 00:12:17,430 The only thing we can mess around with here, to figure 273 00:12:17,430 --> 00:12:19,660 out this problem, is we have to look at d. 274 00:12:19,660 --> 00:12:22,800 So we want to have a large d to minimize this, and to 275 00:12:22,800 --> 00:12:25,680 minimize your theta. 276 00:12:25,680 --> 00:12:27,490 To get a large d, we need to know the 277 00:12:27,490 --> 00:12:29,770 equation for planar spacing. 278 00:12:29,770 --> 00:12:31,100 OK? 279 00:12:31,100 --> 00:12:32,620 This is your equation for planar spacing. 280 00:12:32,620 --> 00:12:35,890 And as we sort of talked about in problem 1, in the prior 281 00:12:35,890 --> 00:12:41,205 video, d planar spacing is the distance between one plane of 282 00:12:41,205 --> 00:12:43,120 a particular orientation, and another plane of the same 283 00:12:43,120 --> 00:12:44,670 exact orientation. 284 00:12:44,670 --> 00:12:44,980 OK? 285 00:12:44,980 --> 00:12:48,470 So we're looking at maybe a 011 plane in this unit cell, 286 00:12:48,470 --> 00:12:50,920 and a 011 plane in this unit cell. 287 00:12:50,920 --> 00:12:52,590 What's the distance between the two? 288 00:12:52,590 --> 00:12:54,740 That's what this d is. 289 00:12:54,740 --> 00:12:55,170 OK? 290 00:12:55,170 --> 00:12:58,230 So we want to have a large d. 291 00:12:58,230 --> 00:13:02,050 And let's look at what we have in the equation for d. 292 00:13:02,050 --> 00:13:03,380 We have these hkl. 293 00:13:03,380 --> 00:13:07,350 These are the coefficients the Miller indices of the plane. 294 00:13:07,350 --> 00:13:09,540 And we have a, which is the lattice 295 00:13:09,540 --> 00:13:11,330 constant of the material. 296 00:13:11,330 --> 00:13:12,500 We're looking at-- 297 00:13:12,500 --> 00:13:13,470 what are we looking at here? 298 00:13:13,470 --> 00:13:15,750 We're looking at tantalum. 299 00:13:15,750 --> 00:13:17,900 We can't change the lattice constant for constant 300 00:13:17,900 --> 00:13:19,100 temperature and pressure. 301 00:13:19,100 --> 00:13:20,720 So this is nonnegotiable. 302 00:13:20,720 --> 00:13:22,670 We can't change what a is. 303 00:13:22,670 --> 00:13:26,120 The only things that we can toggle, the only switch we can 304 00:13:26,120 --> 00:13:28,540 toggle now, is h, k, and l. 305 00:13:28,540 --> 00:13:31,580 So the only thing we can do, is look at different plane 306 00:13:31,580 --> 00:13:32,750 orientations in the crystal. 307 00:13:32,750 --> 00:13:35,350 So this is the logic I used to go through the problem. 308 00:13:35,350 --> 00:13:37,850 So you want to have as small an h, k, and l as possible, 309 00:13:37,850 --> 00:13:42,340 because the smaller these are, the larger d is, et cetera. 310 00:13:42,340 --> 00:13:45,020 So the final piece of information where we have to 311 00:13:45,020 --> 00:13:50,610 be clever is knowing how to get the smallest h, k, and l. 312 00:13:50,610 --> 00:13:52,630 We're told that we're looking at tantalum. 313 00:13:52,630 --> 00:13:56,020 And the students in this class have access to a periodic 314 00:13:56,020 --> 00:13:58,130 table, which has a vast quantity of information about 315 00:13:58,130 --> 00:13:59,520 all the elements in it. 316 00:13:59,520 --> 00:14:01,140 One of the things it has is the crystal 317 00:14:01,140 --> 00:14:02,780 structure of pure elements. 318 00:14:02,780 --> 00:14:07,810 So for tantalum, we're looking at a bcc structure. 319 00:14:10,350 --> 00:14:13,020 So we're looking at bcc. 320 00:14:13,020 --> 00:14:17,450 bcc, from your reflection rules, which is the number 5 321 00:14:17,450 --> 00:14:19,170 on the things we need to know to do this problem-- 322 00:14:21,930 --> 00:14:27,430 for bcc, you have to have your h plus your k plus your l-- 323 00:14:27,430 --> 00:14:28,680 let me write it over here-- 324 00:14:31,220 --> 00:14:34,130 h plus k plus l-- 325 00:14:34,130 --> 00:14:35,430 must be even. 326 00:14:38,130 --> 00:14:40,200 That's a rule, OK? 327 00:14:40,200 --> 00:14:41,960 Proving that is a little bit beyond the scope of this 328 00:14:41,960 --> 00:14:44,300 class, but that's a rule that we went over in class, and 329 00:14:44,300 --> 00:14:46,270 it's in the lectures as well. 330 00:14:46,270 --> 00:14:47,860 So we have a couple things now. 331 00:14:47,860 --> 00:14:51,000 We want to minimize h, k, and l, and they have to be even. 332 00:14:51,000 --> 00:14:53,650 So basically, what that means is, we have to go through a 333 00:14:53,650 --> 00:14:54,830 couple permutations. 334 00:14:54,830 --> 00:14:55,540 So let's think about it. 335 00:14:55,540 --> 00:14:56,520 0 0 0. 336 00:14:56,520 --> 00:14:58,940 That's even, but that doesn't really correspond to a plane. 337 00:14:58,940 --> 00:15:02,750 That doesn't make, you know, that's not going to help us. 338 00:15:02,750 --> 00:15:05,190 1 0 0, or 0 1 0. 339 00:15:05,190 --> 00:15:06,590 So let me write some of those down. 340 00:15:06,590 --> 00:15:08,800 Let's look at, you know, 1 0 0. 341 00:15:08,800 --> 00:15:11,410 Well, that's not even, so we can't look at that. 342 00:15:11,410 --> 00:15:14,780 So what's the next smallest thing we could go? 343 00:15:14,780 --> 00:15:16,030 How about 1 1 0? 344 00:15:21,050 --> 00:15:23,540 Those are the smallest coefficients we can have for a 345 00:15:23,540 --> 00:15:25,440 plane which has the rule that the h plus k 346 00:15:25,440 --> 00:15:27,770 plus l must be even. 347 00:15:27,770 --> 00:15:31,200 So this could be 1 1 0, this could be 1 0 1, this 348 00:15:31,200 --> 00:15:32,335 could be 0 1 1. 349 00:15:32,335 --> 00:15:34,790 But what we're basically talking about, if you remember 350 00:15:34,790 --> 00:15:43,300 from the first problem, is the family of 1 1 0 planes. 351 00:15:43,300 --> 00:15:46,130 So I plugged in just 1 1 0 for h, k, and l. 352 00:15:46,130 --> 00:15:52,480 You get a d of 3.31 over the square root of 2. 353 00:15:52,480 --> 00:15:55,670 And then with that d, you know that you've just maximized the 354 00:15:55,670 --> 00:15:59,270 size of your d, which means we've minimize the size of 355 00:15:59,270 --> 00:16:04,640 lambda over 2d, and that means we've minimized our theta. 356 00:16:04,640 --> 00:16:07,200 And now for theta, we can easily calculate that we're 357 00:16:07,200 --> 00:16:11,650 looking at an angle, if we plug in for d, which is 3.31 358 00:16:11,650 --> 00:16:14,180 over the square root of 2, here. 359 00:16:14,180 --> 00:16:21,250 And if we plug in for lambda, which is given to us as 2.75 360 00:16:21,250 --> 00:16:30,160 angstroms, we get an answer of 36 degrees. 361 00:16:30,160 --> 00:16:32,735 Put that right in the middle. 362 00:16:32,735 --> 00:16:35,370 And I put in the middle to emphasize that this is sort of 363 00:16:35,370 --> 00:16:37,990 the logic you have to go through, sort of a, you know, 364 00:16:37,990 --> 00:16:41,470 circular logic, to get to that answer. 365 00:16:41,470 --> 00:16:43,370 So that's the answer to part B. 366 00:16:43,370 --> 00:16:45,060 And remember, this is pretty much math. 367 00:16:45,060 --> 00:16:47,360 Just manipulation until the very end, where we discussed 368 00:16:47,360 --> 00:16:50,380 the reflection rules for bcc. 369 00:16:50,380 --> 00:16:53,470 I want to go on to the last part now, and I'm just going 370 00:16:53,470 --> 00:16:55,690 to erase this. 371 00:16:55,690 --> 00:16:56,940 We only need a little bit of room. 372 00:17:00,180 --> 00:17:02,510 This last question was actually my favorite question 373 00:17:02,510 --> 00:17:04,970 in the entire class. 374 00:17:04,970 --> 00:17:07,340 And actually, we only had 3 students get it 375 00:17:07,340 --> 00:17:09,840 right, out of 480. 376 00:17:09,840 --> 00:17:12,980 So this was a great question, and if you get it right, then 377 00:17:12,980 --> 00:17:14,610 you're doing really well in the class. 378 00:17:14,610 --> 00:17:17,940 So let's look at C. 379 00:17:17,940 --> 00:17:20,400 It's really a chatty question. 380 00:17:20,400 --> 00:17:22,240 We're not looking for a number. 381 00:17:22,240 --> 00:17:27,180 The question basically asks us, we want you to talk about, 382 00:17:27,180 --> 00:17:30,620 to sketch the emission spectrum of an x-ray target 383 00:17:30,620 --> 00:17:33,780 that's bombarded by photons instead of electrons. 384 00:17:33,780 --> 00:17:37,820 So let's go back over to what we said in the beginning. 385 00:17:37,820 --> 00:17:41,920 When we generate x-rays, generally the way we do it, is 386 00:17:41,920 --> 00:17:45,970 we accelerate an electron into a target material. 387 00:17:45,970 --> 00:17:47,750 In part B, the target material was titanium, 388 00:17:47,750 --> 00:17:49,010 but it could be anything. 389 00:17:49,010 --> 00:17:51,350 A very common target material is copper. 390 00:17:51,350 --> 00:17:55,380 copper K-alpha is a very standard target material, or 391 00:17:55,380 --> 00:17:57,490 wavelength to use. 392 00:17:57,490 --> 00:18:00,945 You accelerate your electron, and you generate the spectrum 393 00:18:00,945 --> 00:18:03,080 of electromagnetic radiation. 394 00:18:03,080 --> 00:18:03,490 OK? 395 00:18:03,490 --> 00:18:04,400 So this is our spectrum. 396 00:18:04,400 --> 00:18:08,080 We have large wavelength, we have small wavelength, we have 397 00:18:08,080 --> 00:18:09,580 some wavelengths in the middle. 398 00:18:09,580 --> 00:18:14,010 And generally, what we would see, coming back over here, 399 00:18:14,010 --> 00:18:15,570 I'm going to draw it like this. 400 00:18:15,570 --> 00:18:18,270 This is what you'd normally see, if 401 00:18:18,270 --> 00:18:19,520 you're using electrons. 402 00:18:33,440 --> 00:18:35,630 Professor Sadoway refers to this as the whale-shaped 403 00:18:35,630 --> 00:18:38,530 curve, probably because we're in New England. 404 00:18:38,530 --> 00:18:40,410 But this is what you would normally see. 405 00:18:40,410 --> 00:18:43,110 Now, this y-axis is the intensity. 406 00:18:43,110 --> 00:18:47,040 That's basically the number of photons you count at some 407 00:18:47,040 --> 00:18:49,430 specific wavelength. 408 00:18:49,430 --> 00:18:50,790 And you see these spikes. 409 00:18:50,790 --> 00:18:54,410 And these spikes correspond to K-alpha. 410 00:18:54,410 --> 00:18:54,990 This is K-beta. 411 00:18:54,990 --> 00:18:56,850 We'll talk about what they are in a second. 412 00:18:56,850 --> 00:18:59,110 L-alpha and L-beta. 413 00:18:59,110 --> 00:18:59,590 And et cetera. 414 00:18:59,590 --> 00:19:02,240 You'd have M-alpha, N-beta. 415 00:19:02,240 --> 00:19:03,550 You go all the way down. 416 00:19:03,550 --> 00:19:06,090 But they have very low intensity. 417 00:19:06,090 --> 00:19:08,360 So this is what we would normally see if we use 418 00:19:08,360 --> 00:19:11,270 electrons to hit our sample. 419 00:19:11,270 --> 00:19:13,110 But we're not using electrons. 420 00:19:13,110 --> 00:19:15,960 And this is actually the answer we got for probably 90% 421 00:19:15,960 --> 00:19:18,360 of the solutions from students. 422 00:19:18,360 --> 00:19:21,730 They just drew the Bremsstrahlung radiation, 423 00:19:21,730 --> 00:19:23,890 Bremsstrahlung, breaking radiation. 424 00:19:23,890 --> 00:19:26,770 And they walked away, and thought they had full credit. 425 00:19:26,770 --> 00:19:29,000 But in reality, this is not the correct answer, because 426 00:19:29,000 --> 00:19:30,590 you think about what's actually 427 00:19:30,590 --> 00:19:32,400 causing this whale shape. 428 00:19:32,400 --> 00:19:35,100 Once you understand where the whale shape comes from, then 429 00:19:35,100 --> 00:19:36,350 you understand what happens if we're using 430 00:19:36,350 --> 00:19:39,550 photons instead of electrons. 431 00:19:39,550 --> 00:19:42,090 So let's actually understand where these 432 00:19:42,090 --> 00:19:43,010 characteristics come from. 433 00:19:43,010 --> 00:19:46,260 There's two things to pick up from this plot. 434 00:19:46,260 --> 00:19:47,640 We have a whale shape. 435 00:19:47,640 --> 00:19:50,670 We've also got these spikes. 436 00:19:50,670 --> 00:19:52,750 So first, let's talk about where the 437 00:19:52,750 --> 00:19:54,470 whale shape comes from. 438 00:19:54,470 --> 00:19:55,720 Let me just do it with blue. 439 00:19:58,640 --> 00:20:00,440 We'll talk about the spikes in a second, because they're 440 00:20:00,440 --> 00:20:03,030 actually a different phenomenon. 441 00:20:03,030 --> 00:20:05,960 The whale shape comes from-- 442 00:20:05,960 --> 00:20:07,530 let me draw it for you. 443 00:20:07,530 --> 00:20:09,250 Let's zoom in. 444 00:20:09,250 --> 00:20:11,880 Let's zoom in on our target here. 445 00:20:11,880 --> 00:20:14,460 So here's our target material. 446 00:20:14,460 --> 00:20:15,890 We have an electron hitting it. 447 00:20:15,890 --> 00:20:17,160 It's just a metal. 448 00:20:17,160 --> 00:20:18,530 And we've got photons coming up. 449 00:20:18,530 --> 00:20:20,300 Let's zoom in really close to the surface 450 00:20:20,300 --> 00:20:21,566 of that target material. 451 00:20:21,566 --> 00:20:24,740 Let's go back over here. 452 00:20:24,740 --> 00:20:25,990 So we've got-- 453 00:20:33,430 --> 00:20:36,100 looks something like this-- 454 00:20:36,100 --> 00:20:37,860 we've got an electron, which I'll draw in 455 00:20:37,860 --> 00:20:42,220 yellow, coming in. 456 00:20:45,110 --> 00:20:45,825 So here's our electron. 457 00:20:45,825 --> 00:20:46,970 It's about to hit our material. 458 00:20:46,970 --> 00:20:49,160 Notice I've drawn it with some crystal structure, because 459 00:20:49,160 --> 00:20:51,540 that's what we're going to talk about, really zoomed in, 460 00:20:51,540 --> 00:20:52,670 the atomic level. 461 00:20:52,670 --> 00:20:56,240 Now this electron, we've talked about this in class. 462 00:20:56,240 --> 00:20:58,540 What can happen is that this electron could come in-- 463 00:20:58,540 --> 00:21:00,750 here's its path, normally, if it wasn't going to get 464 00:21:00,750 --> 00:21:01,560 deflected-- 465 00:21:01,560 --> 00:21:03,820 it can come in, and it can be deflected. 466 00:21:03,820 --> 00:21:05,260 OK? 467 00:21:05,260 --> 00:21:07,310 So it can get deflected off-- 468 00:21:07,310 --> 00:21:10,070 let me draw the following path in blue. 469 00:21:10,070 --> 00:21:12,750 So it can go off like this, like this. 470 00:21:12,750 --> 00:21:14,400 It could actually go straight through, without being 471 00:21:14,400 --> 00:21:16,230 deflected at all. 472 00:21:16,230 --> 00:21:22,680 It could actually be reflected back, like this. 473 00:21:22,680 --> 00:21:25,250 And these different reflections correspond to the 474 00:21:25,250 --> 00:21:27,790 electron being accelerated. 475 00:21:27,790 --> 00:21:28,690 It's getting accelerated. 476 00:21:28,690 --> 00:21:33,410 If we define our system like this, so here's our x and our 477 00:21:33,410 --> 00:21:36,210 y, the electron's getting accelerated in the 478 00:21:36,210 --> 00:21:37,410 y-direction. 479 00:21:37,410 --> 00:21:39,800 This is just a simple location, but what happens, is 480 00:21:39,800 --> 00:21:42,880 when you accelerate a charge, you generate radiation. 481 00:21:42,880 --> 00:21:45,150 You generate electromagnetic radiation. 482 00:21:45,150 --> 00:21:46,800 Accelerating charge. 483 00:21:46,800 --> 00:21:49,830 So, you know, here in the first path, where it comes 484 00:21:49,830 --> 00:21:52,630 through, and it gets deflected only by a little bit, you 485 00:21:52,630 --> 00:21:55,970 generate low energy radiation, low energy photons. 486 00:21:55,970 --> 00:21:59,340 So what we're talking about here is a very large 487 00:21:59,340 --> 00:22:02,960 wavelength photons coming out. 488 00:22:02,960 --> 00:22:05,210 So we have an electron. 489 00:22:05,210 --> 00:22:06,170 It gets deflected. 490 00:22:06,170 --> 00:22:07,190 It's accelerated. 491 00:22:07,190 --> 00:22:10,150 And from that point, we're also generating a photon. 492 00:22:10,150 --> 00:22:12,520 Same thing for all these other paths, but for the larger 493 00:22:12,520 --> 00:22:13,490 angle deflection. 494 00:22:13,490 --> 00:22:14,580 So this one here. 495 00:22:14,580 --> 00:22:16,875 And as you actually start deflecting back, you're 496 00:22:16,875 --> 00:22:21,400 generating very high energy electromagnetic radiation. 497 00:22:21,400 --> 00:22:26,300 And this is all because the electron is basically, you 498 00:22:26,300 --> 00:22:27,570 know, is getting accelerated. 499 00:22:27,570 --> 00:22:30,360 You can actually have, you know, the maximum energy 500 00:22:30,360 --> 00:22:33,700 photon you can give off here is a photon that corresponds 501 00:22:33,700 --> 00:22:36,180 to this electron coming in and getting stopped 502 00:22:36,180 --> 00:22:37,870 completely by the atom. 503 00:22:37,870 --> 00:22:40,210 Think electrostatic repulsion. 504 00:22:40,210 --> 00:22:43,580 So it comes in, and it gets stopped dead in it's tracks. 505 00:22:43,580 --> 00:22:46,320 So you have some energy, it was moving some kinetic 506 00:22:46,320 --> 00:22:48,300 energy, and now has 0 kinetic energy. 507 00:22:48,300 --> 00:22:51,420 The energy of your photon that gets given off is basically 508 00:22:51,420 --> 00:22:52,700 the difference in the two. 509 00:22:52,700 --> 00:22:55,290 And that's what we call here, on this plot, the short 510 00:22:55,290 --> 00:22:56,190 wavelength limit. 511 00:22:56,190 --> 00:22:59,480 We're going to write SWL. 512 00:22:59,480 --> 00:23:05,340 Because this is lambda, which means that energy moves in the 513 00:23:05,340 --> 00:23:06,140 other direction. 514 00:23:06,140 --> 00:23:08,420 Lambda goes up, energy goes down. 515 00:23:08,420 --> 00:23:11,830 Lambda goes down, energy goes up. 516 00:23:11,830 --> 00:23:13,680 So this is our short wavelength limit, which means 517 00:23:13,680 --> 00:23:17,050 that is the maximum, the highest energy we can generate 518 00:23:17,050 --> 00:23:20,850 from this setup here. 519 00:23:20,850 --> 00:23:25,370 So all I'm saying is that you can get deflections in any 520 00:23:25,370 --> 00:23:28,520 angle across this way, you can get deflections in angle and 521 00:23:28,520 --> 00:23:32,650 generate any number of wavelengths of photons with 522 00:23:32,650 --> 00:23:35,440 the minimum wavelength being this one, that's generated 523 00:23:35,440 --> 00:23:37,620 from this situation. 524 00:23:37,620 --> 00:23:40,300 So I'll just make it very, very small for you. 525 00:23:43,590 --> 00:23:46,030 So you generate all these wavelengths, you create all 526 00:23:46,030 --> 00:23:48,860 this energy of photons, and that's what basically 527 00:23:48,860 --> 00:23:52,680 corresponds to the whale-shaped curve. 528 00:23:52,680 --> 00:23:55,510 You're more likely to get deflections that correspond to 529 00:23:55,510 --> 00:23:57,390 something around here, because you have higher intensity, and 530 00:23:57,390 --> 00:24:01,670 over here, you're less likely to see those happening. 531 00:24:01,670 --> 00:24:03,230 So that's our whale shape. 532 00:24:03,230 --> 00:24:05,140 And now, let's return to the problem. 533 00:24:05,140 --> 00:24:11,420 Let's ask, when we use photons, what's the situation? 534 00:24:11,420 --> 00:24:15,460 Well, photon means we're no longer looking at an electron. 535 00:24:27,670 --> 00:24:29,330 Our photon is coming in. 536 00:24:29,330 --> 00:24:31,710 And the thing about photons, is that they're not going to 537 00:24:31,710 --> 00:24:33,400 get deflected like that. 538 00:24:33,400 --> 00:24:34,260 They don't have a charge. 539 00:24:34,260 --> 00:24:36,800 There's no electrostatic propulsion, for example. 540 00:24:36,800 --> 00:24:39,583 So a photon will either only be absorbed and re-emitted, or 541 00:24:39,583 --> 00:24:41,910 it will go through the material. 542 00:24:41,910 --> 00:24:45,490 So what that means is that there's no process now that 543 00:24:45,490 --> 00:24:48,110 will accelerate a charge to create 544 00:24:48,110 --> 00:24:49,910 Bremsstrahlung radiation. 545 00:24:49,910 --> 00:24:52,830 And if there's no process to accelerate a charge, then 546 00:24:52,830 --> 00:24:55,430 there's no way to get Bremsstrahlung radiation. 547 00:24:55,430 --> 00:24:58,425 So the first thing you need to do to get most the points on 548 00:24:58,425 --> 00:25:00,450 this problem is to say, look! 549 00:25:00,450 --> 00:25:02,720 There's no Bremsstrahlung radiation anymore. 550 00:25:02,720 --> 00:25:03,970 There's no charge being accelerated. 551 00:25:13,140 --> 00:25:15,120 So that's sort of the first answer. 552 00:25:15,120 --> 00:25:18,230 The second thing to realize is that these spikes still exist. 553 00:25:18,230 --> 00:25:20,090 Because the reason for these spikes is a different 554 00:25:20,090 --> 00:25:22,970 mechanism than it is for the Bremsstrahlung. 555 00:25:22,970 --> 00:25:27,545 These characteristic peaks correspond to the movement of 556 00:25:27,545 --> 00:25:29,670 an electron to a different energy level, and then a 557 00:25:29,670 --> 00:25:30,680 dropping back down. 558 00:25:30,680 --> 00:25:33,670 So it's very characteristic of the material itself. 559 00:25:33,670 --> 00:25:36,190 So a photon can still come in. 560 00:25:36,190 --> 00:25:39,120 It could still liberate or move an electron. 561 00:25:39,120 --> 00:25:40,990 It could just knock an electron out of its shell. 562 00:25:40,990 --> 00:25:43,000 Perhaps n equals 1. 563 00:25:43,000 --> 00:25:45,580 And then the n equals 2 electron will drop down and 564 00:25:45,580 --> 00:25:46,870 fill in the n equals 1 shell 565 00:25:46,870 --> 00:25:49,500 corresponding to K-alpha radiation. 566 00:25:49,500 --> 00:25:54,110 If you had n equals 3 electron dropping down to n equals 1, 567 00:25:54,110 --> 00:25:57,190 you'd have your K-beta radiation. 568 00:25:57,190 --> 00:26:02,180 And likewise from, you know, 3 to 2, 4 to 2, L-alpha, L-beta. 569 00:26:02,180 --> 00:26:05,150 So that's the key take-home message here. 570 00:26:05,150 --> 00:26:08,280 The message is that your Bremsstrahlung, the thing that 571 00:26:08,280 --> 00:26:10,350 you see in the notes in the class all the time, the whale 572 00:26:10,350 --> 00:26:13,670 shape with the spikes, that's two specific mechanisms. The 573 00:26:13,670 --> 00:26:17,380 whale, the Bremsstrahlung, corresponds to the breaking of 574 00:26:17,380 --> 00:26:18,840 an electron. 575 00:26:18,840 --> 00:26:21,180 These peaks correspond to the ejection of an electron, and 576 00:26:21,180 --> 00:26:24,560 an electron's moving around energy levels within the atom. 577 00:26:24,560 --> 00:26:24,910 OK? 578 00:26:24,910 --> 00:26:27,830 So our learning objectives for this problem, the things that 579 00:26:27,830 --> 00:26:30,960 we've taken home from the problem overall, is that we 580 00:26:30,960 --> 00:26:34,650 know, for example, Rydberg equation is only for 581 00:26:34,650 --> 00:26:37,530 hydrogenic-type atoms, and we know how to use the Moseley's 582 00:26:37,530 --> 00:26:38,580 equation now. 583 00:26:38,580 --> 00:26:40,040 That was part 1. 584 00:26:40,040 --> 00:26:43,730 Part 2, we know Bragg's Law, and how to use it, and what 585 00:26:43,730 --> 00:26:45,440 all the variables mean. 586 00:26:45,440 --> 00:26:47,080 That's something you should learn and take home. 587 00:26:47,080 --> 00:26:50,390 And I really highly recommend that you derive Bragg's Law 588 00:26:50,390 --> 00:26:52,540 from the geometric interpretation. 589 00:26:52,540 --> 00:26:59,030 And part C, we really wanted to probe and understand who 590 00:26:59,030 --> 00:27:01,370 conceptually understood what was happening. 591 00:27:01,370 --> 00:27:03,720 And part C tells us, the thing that we learned from it is 592 00:27:03,720 --> 00:27:07,260 that this Bremsstrahlung radiation with the peaks that 593 00:27:07,260 --> 00:27:11,860 we saw in class, that's two separate mechanisms occurring. 594 00:27:11,860 --> 00:27:14,820 So that's problem number 2, and I hope you did well.