1 00:00:16,015 --> 00:00:19,252 Big picture-- I showed you this big picture before. 2 00:00:19,252 --> 00:00:20,520 We're now going to add to it. 3 00:00:20,520 --> 00:00:22,155 Look at what we've done. 4 00:00:22,155 --> 00:00:22,956 Ha! 5 00:00:22,956 --> 00:00:26,693 Electronic structure all the way up through [INAUDIBLE],, 6 00:00:26,693 --> 00:00:29,462 [? Louis, ?] molecular orbital theory, 7 00:00:29,462 --> 00:00:31,965 how to handle multi-electrons, all these different kinds 8 00:00:31,965 --> 00:00:32,899 of bonding-- 9 00:00:32,899 --> 00:00:34,834 check. 10 00:00:34,834 --> 00:00:36,503 We've covered every single one of these, 11 00:00:36,503 --> 00:00:39,239 and this is solid-state chemistry. 12 00:00:39,239 --> 00:00:41,107 OK, but now here we are, right? 13 00:00:41,107 --> 00:00:43,910 Now OK, remember we also talked about this 14 00:00:43,910 --> 00:00:45,912 on the very first lecture. 15 00:00:45,912 --> 00:00:47,814 When you classify solids, there are two things 16 00:00:47,814 --> 00:00:50,316 that are really important. 17 00:00:50,316 --> 00:00:52,752 One is the bonding type. 18 00:00:52,752 --> 00:00:56,556 How do the atoms in the solid want to talk to each other? 19 00:00:56,556 --> 00:00:58,258 How do they interact? 20 00:00:58,258 --> 00:01:01,428 And that is something that we have covered in a lot of ways. 21 00:01:01,428 --> 00:01:04,497 But the next is how they come together, 22 00:01:04,497 --> 00:01:07,067 the actual arrangement of those atoms. 23 00:01:07,067 --> 00:01:08,835 And that is what we're going to talk 24 00:01:08,835 --> 00:01:11,838 about today and on Wednesday. 25 00:01:11,838 --> 00:01:16,408 And then after exam two as we learn about X-rays, that's 26 00:01:16,408 --> 00:01:18,411 a way to characterize the arrangement 27 00:01:18,411 --> 00:01:20,780 so that you can see how they're ordered, OK? 28 00:01:20,780 --> 00:01:22,549 We're going to use X-rays and do something 29 00:01:22,549 --> 00:01:24,451 called X-ray diffraction. 30 00:01:24,451 --> 00:01:28,988 But for this week I want to just classify it. 31 00:01:28,988 --> 00:01:32,892 I want to talk about the tools that we 32 00:01:32,892 --> 00:01:36,696 use to understand how a solid forms, 33 00:01:36,696 --> 00:01:40,600 how to describe the formation of a solid, how 34 00:01:40,600 --> 00:01:42,168 the atoms can be described. 35 00:01:42,168 --> 00:01:44,170 And OK, so, of course, a solid is 36 00:01:44,170 --> 00:01:47,340 that which is dimensionally stable, simple definition, 37 00:01:47,340 --> 00:01:47,874 right? 38 00:01:47,874 --> 00:01:52,679 But we can break down how these atoms arrange 39 00:01:52,679 --> 00:01:56,348 into two basic categories-- 40 00:01:56,348 --> 00:02:00,453 atomic arrangement, the order, and what we call it, right? 41 00:02:00,453 --> 00:02:03,957 In the first category, it's regular. 42 00:02:03,957 --> 00:02:07,360 So what that means is that everywhere I look, I 43 00:02:07,360 --> 00:02:10,263 have the same thing repeating. 44 00:02:10,263 --> 00:02:11,164 And you know what? 45 00:02:11,164 --> 00:02:14,734 It repeats over a long, long way-- 46 00:02:14,734 --> 00:02:18,338 like really long, OK? 47 00:02:18,338 --> 00:02:24,344 So that kind of regular, long range order is a crystal. 48 00:02:24,344 --> 00:02:26,546 That's a crystalline solid. 49 00:02:26,546 --> 00:02:30,617 And that is going to be the topic of this week, 50 00:02:30,617 --> 00:02:31,851 crystalline solids. 51 00:02:31,851 --> 00:02:35,555 Now, there are also solids that you can 52 00:02:35,555 --> 00:02:37,991 make where there isn't order. 53 00:02:37,991 --> 00:02:40,560 Or maybe there's some but not much. 54 00:02:40,560 --> 00:02:41,961 So, things are kind of jumbled up. 55 00:02:41,961 --> 00:02:45,298 They're not regularly repeating. 56 00:02:45,298 --> 00:02:48,101 They might be random, the atomic arrangement. 57 00:02:48,101 --> 00:02:51,771 Or maybe there's repetition, but it's very short-range. 58 00:02:51,771 --> 00:02:54,707 So things look ordered, but only for a few bond lengths, not 10 59 00:02:54,707 --> 00:02:58,344 to the 23rd bond lengths, right? 60 00:02:58,344 --> 00:03:04,350 Those are called amorphous solids, or also glass. 61 00:03:04,350 --> 00:03:07,754 Now, as I mentioned, this is the topic of this week, 62 00:03:07,754 --> 00:03:10,557 and this will be covered on exam two. 63 00:03:10,557 --> 00:03:15,461 This is the topic after exam two next week, OK? 64 00:03:15,461 --> 00:03:20,667 So we'll talk about both of these types of solids as we go. 65 00:03:20,667 --> 00:03:26,105 All right, now, OK, it turns out that nature 66 00:03:26,105 --> 00:03:30,610 has been thinking about order for a long, long time. 67 00:03:30,610 --> 00:03:32,545 And there is a reason for this. 68 00:03:32,545 --> 00:03:36,216 Because nature wants to be efficient. 69 00:03:36,216 --> 00:03:39,719 This plant is like, well, I'm only going to grow this big, 70 00:03:39,719 --> 00:03:42,488 and I got to put as many seeds as I can 71 00:03:42,488 --> 00:03:44,156 on the surface of this thing. 72 00:03:44,156 --> 00:03:45,858 How do I fit them-- 73 00:03:45,858 --> 00:03:47,860 how do I pack them in? 74 00:03:47,860 --> 00:03:50,463 Well, I could fit them randomly, but it 75 00:03:50,463 --> 00:03:52,932 doesn't seem like I could get as many seeds on there. 76 00:03:52,932 --> 00:03:56,736 What if I packed them in in some regular pattern? 77 00:03:56,736 --> 00:03:59,973 What if I packed them in in some regular way? 78 00:03:59,973 --> 00:04:02,308 Well, you can pack a lot more in. 79 00:04:02,308 --> 00:04:04,244 And so nature has already been thinking 80 00:04:04,244 --> 00:04:08,281 about symmetry and order and even long-range order 81 00:04:08,281 --> 00:04:10,350 for a long time. 82 00:04:10,350 --> 00:04:13,720 People have been thinking about this, too. 83 00:04:13,720 --> 00:04:18,391 Robert Hooke, the same person from Hooke's law, in the 1600s 84 00:04:18,391 --> 00:04:19,826 he was all about cannonballs. 85 00:04:19,826 --> 00:04:22,662 And he was like, how can I stack cannonballs? 86 00:04:22,662 --> 00:04:24,264 And he tried this, right? 87 00:04:24,264 --> 00:04:29,402 He's like, well, maybe if I just put them like this, 88 00:04:29,402 --> 00:04:31,671 is that a good way to stack them? 89 00:04:31,671 --> 00:04:33,673 I don't know. 90 00:04:33,673 --> 00:04:37,977 And then he sort of put them all together. 91 00:04:37,977 --> 00:04:41,614 But now, he tried this, and then he tried another arrangement, 92 00:04:41,614 --> 00:04:43,449 and he'd be like, well, but how-- 93 00:04:43,449 --> 00:04:45,118 first of all, are they stable? 94 00:04:45,118 --> 00:04:46,319 Do they fall over easily? 95 00:04:46,319 --> 00:04:47,320 Because you don't want cannonballs 96 00:04:47,320 --> 00:04:48,554 falling all over the place. 97 00:04:48,554 --> 00:04:50,323 But then also, did you pack as much 98 00:04:50,323 --> 00:04:52,358 in this way as another way? 99 00:04:52,358 --> 00:04:53,960 So he was thinking about it. 100 00:04:53,960 --> 00:04:57,764 And he figured out that you could pack more cannonballs 101 00:04:57,764 --> 00:05:02,902 in the same area or volume if you sort of switched it up, 102 00:05:02,902 --> 00:05:04,203 right? 103 00:05:04,203 --> 00:05:07,507 And then he thought, well, maybe other things have packing, too. 104 00:05:07,507 --> 00:05:09,208 And he thought about all sorts of things. 105 00:05:09,208 --> 00:05:11,311 He looked at salt crystals, and he 106 00:05:11,311 --> 00:05:15,014 said, they must also pack in some way, 107 00:05:15,014 --> 00:05:18,051 because cannonballs do. 108 00:05:18,051 --> 00:05:23,389 So people have been thinking about this for a long time, OK? 109 00:05:23,389 --> 00:05:28,127 Now, people are also inspired by regular order, right? 110 00:05:28,127 --> 00:05:31,564 So if you look at, for example, an Escher painting, 111 00:05:31,564 --> 00:05:35,268 you can see that there is a repetitive pattern here. 112 00:05:35,268 --> 00:05:39,305 This is a crystal of ducks, right? 113 00:05:42,008 --> 00:05:45,345 OK, so now, when you look at patterns like this, 114 00:05:45,345 --> 00:05:46,279 you have to-- 115 00:05:46,279 --> 00:05:49,282 well, you don't have to, but we have to in this class. 116 00:05:49,282 --> 00:05:53,219 We've got to figure out what is the repeating unit. 117 00:05:53,219 --> 00:05:56,856 What is it that is repeating everywhere? 118 00:05:56,856 --> 00:05:58,691 So that's the first question we need to ask. 119 00:05:58,691 --> 00:06:01,060 And you can see if you look at this picture, 120 00:06:01,060 --> 00:06:02,662 you can draw a box like that. 121 00:06:02,662 --> 00:06:05,765 There's a box around a light colored, dark colored duck. 122 00:06:05,765 --> 00:06:10,470 And now you see if you take this box and you move it over, 123 00:06:10,470 --> 00:06:13,806 you translate it over there, you get exactly the same thing. 124 00:06:13,806 --> 00:06:16,342 If you take that box and you put it up there, 125 00:06:16,342 --> 00:06:17,643 you get exactly the same thing. 126 00:06:17,643 --> 00:06:19,312 So that is a repeating unit. 127 00:06:21,881 --> 00:06:24,350 Oh, we have a definition. 128 00:06:24,350 --> 00:06:26,252 It's a unit cell. 129 00:06:26,252 --> 00:06:29,355 It's called a unit cell. 130 00:06:29,355 --> 00:06:37,630 And it's the repeating unit in a crystal. 131 00:06:43,569 --> 00:06:44,604 It's a repeating unit. 132 00:06:44,604 --> 00:06:48,541 Now, it's not always obvious. 133 00:06:48,541 --> 00:06:53,112 So, there's one way that I can repeat and take 134 00:06:53,112 --> 00:06:55,715 this and make the picture by repeating it everywhere. 135 00:06:55,715 --> 00:06:56,549 But check this out-- 136 00:06:56,549 --> 00:06:58,351 I could have done that. 137 00:06:58,351 --> 00:07:01,754 If you stare at that, you can see that that is also 138 00:07:01,754 --> 00:07:03,956 a repeating unit, right? 139 00:07:03,956 --> 00:07:06,259 That is also repeating unit. 140 00:07:06,259 --> 00:07:09,862 If I take this and I move it down here, 141 00:07:09,862 --> 00:07:11,197 I get exactly the same thing. 142 00:07:11,197 --> 00:07:15,501 And I move it over there, and I can tile all of space. 143 00:07:15,501 --> 00:07:19,439 I can tile all of space. 144 00:07:19,439 --> 00:07:21,073 And something is really important here. 145 00:07:21,073 --> 00:07:25,077 I can take these repeating units and stamp them. 146 00:07:25,077 --> 00:07:26,813 Oh, I like that word. 147 00:07:26,813 --> 00:07:31,417 And I can tile all of space and recreate this picture 148 00:07:31,417 --> 00:07:32,985 without any voids, right? 149 00:07:32,985 --> 00:07:34,086 I didn't miss anything. 150 00:07:34,086 --> 00:07:37,457 I filled all of space. 151 00:07:37,457 --> 00:07:38,658 I filled all of space. 152 00:07:38,658 --> 00:07:41,394 And that is what a crystal system is, right? 153 00:07:41,394 --> 00:07:43,930 A crystal system-- let's write it here-- 154 00:07:43,930 --> 00:07:45,898 crystal system. 155 00:07:45,898 --> 00:07:56,108 Crystal system-- that is a way of enumerating-- 156 00:07:56,108 --> 00:08:09,121 enumerates ways that space can be filled, OK? 157 00:08:09,121 --> 00:08:12,325 Enumerates ways that space can be filled with no voids. 158 00:08:17,997 --> 00:08:19,499 That's a crystal system. 159 00:08:19,499 --> 00:08:22,835 OK, well, let's go back to the canons for a minute, 160 00:08:22,835 --> 00:08:26,339 because we can think about this first in 3D before we-- 161 00:08:26,339 --> 00:08:29,876 we're just warming up before we get to 3D, OK? 162 00:08:32,411 --> 00:08:33,645 Oh, there it is. 163 00:08:33,645 --> 00:08:34,947 That's a unit cell. 164 00:08:34,947 --> 00:08:38,618 OK, now the thing is, so I've drawn a unit cell. 165 00:08:38,618 --> 00:08:40,086 And notice what I've done. 166 00:08:40,086 --> 00:08:45,625 I've taken this, and I've drawn it this way, OK? 167 00:08:45,625 --> 00:08:49,762 And those vectors define the unit cell, right? 168 00:08:49,762 --> 00:08:51,731 Those vectors define the unit cell. 169 00:08:51,731 --> 00:08:54,500 So those are my stamps, the vectors there. 170 00:08:54,500 --> 00:08:56,435 So these things here-- 171 00:08:56,435 --> 00:08:58,404 well, I went a little far there, but they 172 00:08:58,404 --> 00:09:01,407 should go to the center of the next cannonball. 173 00:09:01,407 --> 00:09:02,808 So now these are vectors. 174 00:09:02,808 --> 00:09:06,178 If I just take these, and I put them-- 175 00:09:06,178 --> 00:09:08,981 now, if I translate them over to there, 176 00:09:08,981 --> 00:09:11,284 then I get the next unit cell and the next one. 177 00:09:11,284 --> 00:09:12,952 And I go up, and I go over, and I get 178 00:09:12,952 --> 00:09:15,454 all the tiling of all space. 179 00:09:15,454 --> 00:09:17,590 And so those vectors are really important, 180 00:09:17,590 --> 00:09:18,491 and they have a name. 181 00:09:18,491 --> 00:09:20,026 Those are called the lattice vectors. 182 00:09:24,530 --> 00:09:26,566 Those are the lattice vectors. 183 00:09:26,566 --> 00:09:33,172 I really like to think of them as my stamps, or my stamp. 184 00:09:33,172 --> 00:09:34,807 It is a stamp. 185 00:09:34,807 --> 00:09:35,308 Mm! 186 00:09:35,308 --> 00:09:36,175 Mm! 187 00:09:36,175 --> 00:09:36,943 Mm! 188 00:09:36,943 --> 00:09:40,846 And no voids-- these reactors are my stamp. 189 00:09:40,846 --> 00:09:42,848 Now, I'm stamping all of space. 190 00:09:42,848 --> 00:09:45,851 I didn't really say what I'm putting there. 191 00:09:45,851 --> 00:09:47,420 I mean, here there's cannonballs. 192 00:09:47,420 --> 00:09:49,555 In Escher there's ducks. 193 00:09:49,555 --> 00:09:51,958 Doesn't matter right now, right? 194 00:09:51,958 --> 00:09:53,893 It's a stamp that fills all of space. 195 00:09:53,893 --> 00:10:00,032 That is how we're building up our knowledge of crystals. 196 00:10:00,032 --> 00:10:04,370 OK, well you can also ask the question, 197 00:10:04,370 --> 00:10:08,574 which is, how much could you fill in there? 198 00:10:08,574 --> 00:10:11,510 And that's a very important question that we want-- 199 00:10:11,510 --> 00:10:14,947 ah-- --that we want to know. 200 00:10:14,947 --> 00:10:18,551 OK, mm! 201 00:10:18,551 --> 00:10:21,554 OK, now, so I've got my lattice vectors. 202 00:10:21,554 --> 00:10:23,856 But I want to know now what am I-- 203 00:10:23,856 --> 00:10:26,492 because I could put there-- 204 00:10:26,492 --> 00:10:27,093 look at this. 205 00:10:27,093 --> 00:10:29,495 I could take a lattice vector like this. 206 00:10:29,495 --> 00:10:32,131 This is a square lattice. 207 00:10:32,131 --> 00:10:34,300 OK, if it's a square lattice, then those 208 00:10:34,300 --> 00:10:36,402 are the same length, right? 209 00:10:36,402 --> 00:10:37,937 A is like the length of the-- 210 00:10:37,937 --> 00:10:42,508 this is like the length of the lattice vector. 211 00:10:46,779 --> 00:10:51,984 Now, if I took this lattice, this stamp, and I say, 212 00:10:51,984 --> 00:10:54,120 OK, now I'm going to decide what to put there-- 213 00:10:54,120 --> 00:10:56,055 I'm going to put circles. 214 00:10:56,055 --> 00:10:58,658 OK, so I'm going to put a circle there, circle there. 215 00:10:58,658 --> 00:11:03,863 And what you do then is you put a circle everywhere 216 00:11:03,863 --> 00:11:07,233 that you stamp, right? 217 00:11:07,233 --> 00:11:10,970 OK, so I have now taken my stamp, 218 00:11:10,970 --> 00:11:12,338 and I've stamped everywhere. 219 00:11:12,338 --> 00:11:15,307 And I've put something at each place-- 220 00:11:15,307 --> 00:11:15,975 a circle. 221 00:11:15,975 --> 00:11:17,543 But I might ask a question. 222 00:11:17,543 --> 00:11:19,145 And this is a very important question, 223 00:11:19,145 --> 00:11:24,750 which is, if I have circles, what is the maximum 224 00:11:24,750 --> 00:11:28,054 that they can pack in for this lattice? 225 00:11:28,054 --> 00:11:29,889 How much can I pack? 226 00:11:29,889 --> 00:11:31,824 Well, you can just visualize that now. 227 00:11:31,824 --> 00:11:34,193 Let's grow them. 228 00:11:34,193 --> 00:11:38,831 A stamp means it's the same thing everywhere, right? 229 00:11:38,831 --> 00:11:41,300 So now if I make one of them bigger, 230 00:11:41,300 --> 00:11:44,303 well, I've made them all bigger, right? 231 00:11:44,303 --> 00:11:46,706 Because they're all the same everywhere. 232 00:11:46,706 --> 00:11:48,540 And if I make them all big enough, 233 00:11:48,540 --> 00:11:52,511 eventually they're going to touch like that. 234 00:11:52,511 --> 00:11:54,847 That's a special thing, because now I've 235 00:11:54,847 --> 00:11:58,017 grown the shape that I'm putting there to the point where it's 236 00:11:58,017 --> 00:12:00,252 the maximum packing, right? 237 00:12:00,252 --> 00:12:03,255 And so you can imagine this. 238 00:12:03,255 --> 00:12:10,930 The area of maximum packing is a very important parameter. 239 00:12:10,930 --> 00:12:12,932 And it's something that we'll talk about today 240 00:12:12,932 --> 00:12:17,670 with 3D crystals, which is the solids of elements 241 00:12:17,670 --> 00:12:20,873 in the periodic table. 242 00:12:20,873 --> 00:12:24,376 But if you say with this square analogy-- 243 00:12:24,376 --> 00:12:25,778 so my lattice is a square. 244 00:12:25,778 --> 00:12:27,313 My stamp is a square. 245 00:12:27,313 --> 00:12:29,548 My sides are equivalent. 246 00:12:29,548 --> 00:12:31,016 And what I put there is a circle. 247 00:12:31,016 --> 00:12:32,885 Then the area of maximum packing is something 248 00:12:32,885 --> 00:12:35,921 you can calculate, right? 249 00:12:35,921 --> 00:12:38,224 Because the circle itself-- 250 00:12:38,224 --> 00:12:41,026 in this case, if I-- 251 00:12:41,026 --> 00:12:41,794 [STUDENT SNEEZES] 252 00:12:41,794 --> 00:12:44,130 Gezuntheit. 253 00:12:44,130 --> 00:12:45,798 If I look at it, then-- 254 00:12:45,798 --> 00:12:49,168 oh, that's not a good circle. 255 00:12:49,168 --> 00:12:52,938 But anyway, you'll get the point here, which is that-- 256 00:12:52,938 --> 00:12:57,409 oh, boy-- which is that if this is the lattice vector 257 00:12:57,409 --> 00:13:02,915 is a, like I drew there and I'm looking at the maximum packing, 258 00:13:02,915 --> 00:13:07,386 then you know already that the radius of the circle 259 00:13:07,386 --> 00:13:08,921 is a half a. 260 00:13:08,921 --> 00:13:13,159 So radius equals 1/2 a. 261 00:13:17,763 --> 00:13:22,635 If the radius of the circle is 1/2 a-- 262 00:13:22,635 --> 00:13:24,837 if the radius of a circle is 1/2 a, 263 00:13:24,837 --> 00:13:26,839 then you also know what the area is. 264 00:13:26,839 --> 00:13:28,274 I'm in 2D right now. 265 00:13:28,274 --> 00:13:29,842 I'm in 2D. 266 00:13:29,842 --> 00:13:32,311 So if I've maximized the packing of these circles 267 00:13:32,311 --> 00:13:42,188 on the square lattice, then the area of the circle 268 00:13:42,188 --> 00:13:49,461 is pi times r squared. 269 00:13:49,461 --> 00:13:57,436 So 1/2 a squared equals pi a-squared over 4. 270 00:13:57,436 --> 00:14:00,506 But see, I also know how much I packed it in, right? 271 00:14:00,506 --> 00:14:03,008 And I'm doing this slowly on purpose, 272 00:14:03,008 --> 00:14:06,912 because we'll go faster when we do some of the other ones. 273 00:14:06,912 --> 00:14:08,981 But I want to do the first one slow, 274 00:14:08,981 --> 00:14:12,985 because then we'll all be able to see what we're talking 275 00:14:12,985 --> 00:14:14,553 about with the simplest case. 276 00:14:14,553 --> 00:14:22,261 The area of the square is a-squared, right? 277 00:14:22,261 --> 00:14:34,139 And so the area of max packing is equal to pi a squared over 278 00:14:34,139 --> 00:14:36,675 a-squared-- 279 00:14:36,675 --> 00:14:39,612 pi over 4. 280 00:14:39,612 --> 00:14:41,513 because the a-squared is cancelled. 281 00:14:41,513 --> 00:14:42,314 So it's 78%. 282 00:14:47,786 --> 00:14:51,190 This is what Hooke was-- but did I max it out, 283 00:14:51,190 --> 00:14:52,591 or could I do something different? 284 00:14:52,591 --> 00:14:54,793 Could I pick a different lattice? 285 00:14:54,793 --> 00:14:59,265 What if we made cannonballs into rectangles? 286 00:14:59,265 --> 00:15:01,867 Bad idea. 287 00:15:01,867 --> 00:15:04,803 No, he stuck with spheres, but how do you pack them in? 288 00:15:04,803 --> 00:15:09,275 How can you arrange them in a way to change the max packing? 289 00:15:09,275 --> 00:15:15,114 That's what nature does with atoms, as we'll see. 290 00:15:15,114 --> 00:15:19,985 So for example, if I had a 2D hexagonal lattice, instead of 291 00:15:19,985 --> 00:15:22,888 my stamp being a square-- 292 00:15:22,888 --> 00:15:30,296 so if it were a 2D hexagonal lattice-- 293 00:15:30,296 --> 00:15:32,798 I won't do this one out. 294 00:15:32,798 --> 00:15:41,941 But then the max packing goes all the way up to 91%. 295 00:15:41,941 --> 00:15:44,276 That's a lot more per area-- 296 00:15:44,276 --> 00:15:45,110 max packing. 297 00:15:49,815 --> 00:15:52,117 So this is something that we care a lot about. 298 00:15:52,117 --> 00:15:56,355 It's something that we care about because, 299 00:15:56,355 --> 00:15:59,091 when atoms see each other in a solid, what's 300 00:15:59,091 --> 00:16:01,060 the first thing they want to determine? 301 00:16:01,060 --> 00:16:03,696 The first thing they want to do is they want to go back in here 302 00:16:03,696 --> 00:16:08,267 and they want to be like, which one of those can we do? 303 00:16:08,267 --> 00:16:12,204 Can we find some way to bond? 304 00:16:12,204 --> 00:16:13,706 That's what atoms do. 305 00:16:13,706 --> 00:16:14,974 That's what everyone does. 306 00:16:14,974 --> 00:16:17,776 How can we bond together? 307 00:16:17,776 --> 00:16:21,613 What kind of relationship are we going to have? 308 00:16:21,613 --> 00:16:24,550 But they want to max that out, right? 309 00:16:24,550 --> 00:16:27,786 And so they want to try to pack in, given the bond that they're 310 00:16:27,786 --> 00:16:29,488 going to have. 311 00:16:29,488 --> 00:16:31,156 And that is what's going to dictate 312 00:16:31,156 --> 00:16:35,060 the kind of arrangement they can have in the crystal, OK? 313 00:16:35,060 --> 00:16:40,199 So OK, so here we are. 314 00:16:40,199 --> 00:16:42,201 And we're going from 2D to 3D. 315 00:16:42,201 --> 00:16:49,108 Now, the thing is, this is not going 316 00:16:49,108 --> 00:16:52,177 to be an in-depth dive into symmetry and space 317 00:16:52,177 --> 00:16:55,514 groups and group theory. 318 00:16:55,514 --> 00:16:58,417 Those are all great topics. 319 00:16:58,417 --> 00:17:00,452 But I'm just going to tell you-- 320 00:17:00,452 --> 00:17:01,920 and it's actually kind of cool. 321 00:17:01,920 --> 00:17:06,458 In 3D, there are only seven unique crystal systems. 322 00:17:06,458 --> 00:17:09,060 There are only seven ways that you can do this. 323 00:17:09,060 --> 00:17:12,464 You can only pack in and have no voids-- 324 00:17:12,464 --> 00:17:15,800 and have no voids-- you can only pack in these seven 325 00:17:15,800 --> 00:17:19,104 different systems where, as you can see, what's changing here? 326 00:17:19,104 --> 00:17:20,472 Well, it's Cartesian coordinates, 327 00:17:20,472 --> 00:17:23,942 so you've got the length of each lattice vector 328 00:17:23,942 --> 00:17:26,478 and their relative angles, right? 329 00:17:26,478 --> 00:17:28,414 That's all you're changing. 330 00:17:28,414 --> 00:17:33,218 Well, sometimes you change all the angles at once, OK? 331 00:17:33,218 --> 00:17:35,354 Sometimes you change only-- 332 00:17:35,354 --> 00:17:37,423 well, sometimes you change only a few of them. 333 00:17:37,423 --> 00:17:38,957 Sometimes you make them all the same. 334 00:17:38,957 --> 00:17:40,359 Sometimes you change a few. 335 00:17:40,359 --> 00:17:42,895 Sometimes you have all of them be even. 336 00:17:42,895 --> 00:17:45,030 And sometimes some of them are equal, 337 00:17:45,030 --> 00:17:46,732 some of the lattice lengths are equal 338 00:17:46,732 --> 00:17:50,235 but another one is different, all right? 339 00:17:50,235 --> 00:17:53,272 These are the seven ways that we can fill all of space. 340 00:17:53,272 --> 00:17:55,507 And there are only seven. 341 00:17:55,507 --> 00:17:58,410 In this class, we're only going to focus on one, 342 00:17:58,410 --> 00:18:01,580 and that's the cubic system. 343 00:18:01,580 --> 00:18:05,217 And the reason is that this gives us plenty 344 00:18:05,217 --> 00:18:08,654 to work with to understand crystallography 345 00:18:08,654 --> 00:18:11,523 and how chemistry relates to crystallography. 346 00:18:11,523 --> 00:18:16,195 And there is the benefit of the fact that a lot of elements 347 00:18:16,195 --> 00:18:17,629 take one of these three-- 348 00:18:17,629 --> 00:18:19,932 actually, only two. 349 00:18:19,932 --> 00:18:22,668 But they take cubic symmetry. 350 00:18:22,668 --> 00:18:23,602 What does that mean? 351 00:18:23,602 --> 00:18:30,776 Well, it means that all the lattice vectors are the same. 352 00:18:30,776 --> 00:18:33,445 Each one has a length a. 353 00:18:33,445 --> 00:18:36,815 They're orthogonal to each other, which means 354 00:18:36,815 --> 00:18:39,184 that the angle is 90 degrees. 355 00:18:39,184 --> 00:18:40,953 So we've got a cube. 356 00:18:40,953 --> 00:18:45,023 Now, it turns out that now we say, well, how do you pack? 357 00:18:45,023 --> 00:18:51,563 How can I pack things together in the cubic system? 358 00:18:51,563 --> 00:18:55,567 And there's three different ways to do it. 359 00:18:55,567 --> 00:19:01,073 So what this is is it's more symmetry and more group theory 360 00:19:01,073 --> 00:19:05,611 and math that goes into this that Bravais figured out. 361 00:19:05,611 --> 00:19:09,047 And so we call these Bravais lattices. 362 00:19:09,047 --> 00:19:10,849 Now, what Bravais figured out-- 363 00:19:10,849 --> 00:19:13,318 and again, you don't need to know all the math behind this, 364 00:19:13,318 --> 00:19:17,956 but I want you to know these three cubic lattices. 365 00:19:17,956 --> 00:19:21,493 What you figured out is that from the seven crystal systems, 366 00:19:21,493 --> 00:19:24,696 there are only 14 ways to pack. 367 00:19:24,696 --> 00:19:26,431 There are 14 ways to pack. 368 00:19:26,431 --> 00:19:28,700 That's it, no more. 369 00:19:28,700 --> 00:19:29,635 That's pretty cool. 370 00:19:29,635 --> 00:19:31,803 There's no 15. 371 00:19:31,803 --> 00:19:33,005 There's no 16. 372 00:19:33,005 --> 00:19:34,706 There's 14 exactly-- 373 00:19:34,706 --> 00:19:39,511 14 ways that you can, by having a lattice, which means a stamp, 374 00:19:39,511 --> 00:19:41,346 there are only 14 ways you can do it 375 00:19:41,346 --> 00:19:42,581 for the seven crystal systems. 376 00:19:42,581 --> 00:19:45,651 And for the cubic system, there's only three. 377 00:19:45,651 --> 00:19:48,887 So I have three types of stamp. 378 00:19:48,887 --> 00:19:51,790 That's what it means, three types of stamps 379 00:19:51,790 --> 00:19:54,793 where I leave no voids, right? 380 00:19:54,793 --> 00:19:55,494 I leave no voids. 381 00:19:55,494 --> 00:19:58,630 So I can tile all of 3D space, right? 382 00:19:58,630 --> 00:20:05,504 OK, and it forms a cubic unit cell, cubic unit cell. 383 00:20:05,504 --> 00:20:06,805 These are the three ways-- 384 00:20:06,805 --> 00:20:10,342 a simple cubic, a body-centered cubic, 385 00:20:10,342 --> 00:20:11,610 and a face-centered cubic. 386 00:20:11,610 --> 00:20:14,513 These are the three types of lattices 387 00:20:14,513 --> 00:20:15,914 that have cubic symmetry. 388 00:20:15,914 --> 00:20:17,482 They are the only three. 389 00:20:17,482 --> 00:20:21,653 So in a Bravais lattice, the difference 390 00:20:21,653 --> 00:20:27,859 is in a Bravais lattice what Bravais did was he enumerated-- 391 00:20:27,859 --> 00:20:30,128 so here we enumerate the ways space can be filled. 392 00:20:32,965 --> 00:20:42,774 He enumerated the packing, all the ways 393 00:20:42,774 --> 00:20:46,745 that the space can be packed with, 394 00:20:46,745 --> 00:20:50,949 for example, packing within the unit cell. 395 00:20:54,553 --> 00:20:59,424 So there's three distinct ways to pack within a cubic unit 396 00:20:59,424 --> 00:21:01,093 cell, OK? 397 00:21:01,093 --> 00:21:03,595 That's what we're going to bring to life today 398 00:21:03,595 --> 00:21:06,164 and we'll talk more about on Wednesday as well. 399 00:21:08,900 --> 00:21:13,305 And what I want to do is I want to go through each one. 400 00:21:13,305 --> 00:21:15,040 So we're going to go through simple cubic, 401 00:21:15,040 --> 00:21:17,409 body-centered cubic, face-centered cubic. 402 00:21:17,409 --> 00:21:21,380 And then with each one, I'm going to talk about it, 403 00:21:21,380 --> 00:21:23,382 and then I'm going to show you a little video so 404 00:21:23,382 --> 00:21:26,652 that we can really see how the unit cell gets kind of sliced 405 00:21:26,652 --> 00:21:27,152 up. 406 00:21:27,152 --> 00:21:29,655 And there's like a razor that comes in and slices it. 407 00:21:29,655 --> 00:21:30,589 It's really cool. 408 00:21:30,589 --> 00:21:34,526 And so you can really see and feel what we're talking about. 409 00:21:34,526 --> 00:21:37,362 And speaking of seeing and feeling, so I have these. 410 00:21:37,362 --> 00:21:42,100 And I had wanted for these to be what go in your goody bag. 411 00:21:42,100 --> 00:21:46,672 But apparently like $100,000 is too much or whatever. 412 00:21:46,672 --> 00:21:48,040 [LAUGHTER] 413 00:21:48,040 --> 00:21:51,176 But so you have smaller versions of these. 414 00:21:51,176 --> 00:21:52,411 And I encourage you-- 415 00:21:52,411 --> 00:21:55,614 there's no better way to think about crystallography 416 00:21:55,614 --> 00:21:58,250 than to build stuff and see it. 417 00:21:58,250 --> 00:21:59,151 You've got to do that. 418 00:21:59,151 --> 00:22:00,252 It's really cool. 419 00:22:00,252 --> 00:22:03,922 And so this is the simple cubic. 420 00:22:03,922 --> 00:22:05,457 OK, so I'll pass that around. 421 00:22:05,457 --> 00:22:08,026 And this is the body-centered cubic. 422 00:22:08,026 --> 00:22:11,063 And here you have spheres, right? 423 00:22:11,063 --> 00:22:11,897 That's fine. 424 00:22:11,897 --> 00:22:16,201 But all we've done so far is set up a lattice. 425 00:22:16,201 --> 00:22:17,436 This could be anything. 426 00:22:17,436 --> 00:22:20,372 It's a repeat lattice. 427 00:22:20,372 --> 00:22:21,673 Well, that's what a lattice is. 428 00:22:21,673 --> 00:22:25,977 The lattice just defines that this is the same as this. 429 00:22:25,977 --> 00:22:29,648 Whatever is here is the same, is the same as this, 430 00:22:29,648 --> 00:22:30,716 is the same as this. 431 00:22:30,716 --> 00:22:34,186 That's what we're defining here, OK? 432 00:22:34,186 --> 00:22:42,661 OK, all right, now let's talk about these three. 433 00:22:42,661 --> 00:22:45,163 This is a picture from [? April. ?] 434 00:22:45,163 --> 00:22:47,466 These are the three systems I just talked about, 435 00:22:47,466 --> 00:22:49,768 the three Bravais lattices, three different ways 436 00:22:49,768 --> 00:22:55,707 to pack in a cubic symmetric cell. 437 00:22:55,707 --> 00:22:57,976 And notice there's three ways of looking at it, right? 438 00:22:57,976 --> 00:23:01,113 So here is sort of a ball-and-stick model, 439 00:23:01,113 --> 00:23:04,683 which makes it easy to see the bonding and the atoms. 440 00:23:04,683 --> 00:23:07,586 But the space-filling model is really important here, 441 00:23:07,586 --> 00:23:09,821 because the space-filling model lets us think 442 00:23:09,821 --> 00:23:12,591 about the maximum packing. 443 00:23:12,591 --> 00:23:13,592 That's really important. 444 00:23:13,592 --> 00:23:15,827 Remember the maximum packing, because remember 445 00:23:15,827 --> 00:23:17,963 these are bonding. 446 00:23:17,963 --> 00:23:19,398 Really? 447 00:23:19,398 --> 00:23:22,768 These are bonding together, right? 448 00:23:22,768 --> 00:23:26,304 So drawing them all the way up to where they touch, 449 00:23:26,304 --> 00:23:29,441 that's going to help us understand what 450 00:23:29,441 --> 00:23:30,809 their maximum packing can be. 451 00:23:30,809 --> 00:23:33,111 So let's look at that for the simple cubic, OK? 452 00:23:33,111 --> 00:23:36,014 So for the simple cubic, we'll do that here-- 453 00:23:36,014 --> 00:23:36,848 simple cubic. 454 00:23:39,918 --> 00:23:46,825 OK, simple cubic-- oh, let's just call it SC for short. 455 00:23:46,825 --> 00:23:47,793 Why not? 456 00:23:47,793 --> 00:23:49,728 Now, one of the things you want to think about 457 00:23:49,728 --> 00:23:52,130 is, how many nearest neighbors do I have? 458 00:23:52,130 --> 00:23:53,064 And I lost the-- 459 00:23:53,064 --> 00:23:54,032 so you can see. 460 00:23:54,032 --> 00:23:55,600 Look at look at the edge and think 461 00:23:55,600 --> 00:23:57,402 about how many nearest neighbors-- well, you 462 00:23:57,402 --> 00:23:58,770 know you got six, right? 463 00:23:58,770 --> 00:24:02,240 If you're in the corner there of the cube in your simple cubic, 464 00:24:02,240 --> 00:24:03,642 you've got six nearest neighbors. 465 00:24:03,642 --> 00:24:05,243 OK, that's important, right? 466 00:24:05,243 --> 00:24:13,385 Six nearest neighbors-- why is it important? 467 00:24:13,385 --> 00:24:17,255 Well, it's important because that's 468 00:24:17,255 --> 00:24:25,130 how many other atoms you could bond with in this solid, OK? 469 00:24:25,130 --> 00:24:28,700 But the other thing we want to know is the packing. 470 00:24:28,700 --> 00:24:34,272 Now, again, what you want to do to figure out 471 00:24:34,272 --> 00:24:37,709 the maximum packing is you want to think about it 472 00:24:37,709 --> 00:24:39,044 in the same way we did here. 473 00:24:39,044 --> 00:24:43,415 Think about putting something at each of the lattice points 474 00:24:43,415 --> 00:24:46,251 and then growing it out until they 475 00:24:46,251 --> 00:24:47,686 can't grow any more, right? 476 00:24:47,686 --> 00:24:49,754 And that's a good way to think about it. 477 00:24:49,754 --> 00:24:52,224 And in this case, we're going to take spheres. 478 00:24:52,224 --> 00:24:54,626 We're going to be taking spheres because that's how 479 00:24:54,626 --> 00:24:56,161 we're going to represent atoms. 480 00:24:56,161 --> 00:24:57,896 So we're going to be taking spheres. 481 00:24:57,896 --> 00:25:01,366 And we're going to grow them out and see what the packing is. 482 00:25:01,366 --> 00:25:05,370 Yeah, but I've got to do the same trick I did here. 483 00:25:05,370 --> 00:25:07,105 I've got to do the same trick I did here. 484 00:25:07,105 --> 00:25:11,076 I've got to write the radius of the atom in terms of the unit 485 00:25:11,076 --> 00:25:12,210 cell. 486 00:25:12,210 --> 00:25:15,313 So if the unit cell-- 487 00:25:15,313 --> 00:25:16,448 oh, boy. 488 00:25:16,448 --> 00:25:19,451 I'm going to be drawing a 3D structure. 489 00:25:19,451 --> 00:25:20,218 Danger! 490 00:25:20,218 --> 00:25:25,490 a, a, a-- that's a cube. 491 00:25:25,490 --> 00:25:27,826 All the sides are a. 492 00:25:27,826 --> 00:25:31,062 I made myself very happy. 493 00:25:31,062 --> 00:25:31,897 It's a cube. 494 00:25:31,897 --> 00:25:34,499 Now-- oh, but we have a cube. 495 00:25:34,499 --> 00:25:36,601 Now, OK, here we go. 496 00:25:36,601 --> 00:25:40,171 This one-- oh, but see, I want maximum packing. 497 00:25:40,171 --> 00:25:42,541 So I'm going to grow them out. 498 00:25:42,541 --> 00:25:48,346 And you can see what's going to happen as I do that. 499 00:25:51,750 --> 00:25:55,153 And so you can see that in the case of a simple cubic lattice, 500 00:25:55,153 --> 00:25:57,055 it's just like in the square lattice. 501 00:25:57,055 --> 00:25:59,491 It's just like in the square lattice, right? 502 00:25:59,491 --> 00:26:04,362 When I've grown out the sphere, the radius of that sphere, 503 00:26:04,362 --> 00:26:09,501 the radius is r equals-- 504 00:26:09,501 --> 00:26:10,802 now, oh, wait a second. 505 00:26:10,802 --> 00:26:13,738 OK, so I have 4/3 pi r-- 506 00:26:13,738 --> 00:26:15,507 OK, but that doesn't matter yet. 507 00:26:15,507 --> 00:26:16,408 Don't do volume. 508 00:26:16,408 --> 00:26:17,909 Think about the radius 509 00:26:17,909 --> 00:26:21,279 OK, the radius is just going to be 1/2 the lattice, right? 510 00:26:21,279 --> 00:26:23,248 So radius is 1/2 a. 511 00:26:23,248 --> 00:26:30,155 Volume is 4/3 pi 1/2 a cubed. 512 00:26:33,058 --> 00:26:34,392 Now, this is important. 513 00:26:34,392 --> 00:26:42,767 That is at the max packing. 514 00:26:42,767 --> 00:26:46,037 OK, so I've grown these volumes out, 515 00:26:46,037 --> 00:26:48,106 and I've seen what the packing can be. 516 00:26:48,106 --> 00:26:50,075 Now, we actually call this-- 517 00:26:50,075 --> 00:26:53,378 so now I've got the volume of an atom. 518 00:26:53,378 --> 00:26:55,513 I'm pretending that these are atoms now. 519 00:26:55,513 --> 00:26:57,315 No, I'm not pretending anymore. 520 00:26:57,315 --> 00:26:58,984 It's real. 521 00:26:58,984 --> 00:27:01,953 These spheres are now atoms. 522 00:27:01,953 --> 00:27:06,891 And OK, so that's the volume of the atom, right? 523 00:27:06,891 --> 00:27:19,871 And so we can write atomic packing fraction 524 00:27:19,871 --> 00:27:26,044 as the volume of the atom over the volume of the cell. 525 00:27:26,044 --> 00:27:29,214 That's exactly what I did up here, right? 526 00:27:29,214 --> 00:27:33,118 That's how we got the area of the square, 527 00:27:33,118 --> 00:27:36,655 the area of the circle, right? 528 00:27:36,655 --> 00:27:39,324 But there's one more thing that's really important, 529 00:27:39,324 --> 00:27:47,399 which is it has to be times the number of atoms in the cell. 530 00:27:47,399 --> 00:27:50,301 When I say cell, I mean unit cell. 531 00:27:50,301 --> 00:27:52,704 It has to be times the number of atoms in the unit cell. 532 00:27:55,807 --> 00:27:58,043 So it's however many atoms I have 533 00:27:58,043 --> 00:28:00,045 times the volume of the atom. 534 00:28:00,045 --> 00:28:01,046 OK, that's good. 535 00:28:01,046 --> 00:28:03,148 That's how much is inside of the cell 536 00:28:03,148 --> 00:28:04,749 divided by the volume of the cell, 537 00:28:04,749 --> 00:28:08,386 and that gives me the atomic packing fraction. 538 00:28:08,386 --> 00:28:12,424 So in the case of a simple cubic, 539 00:28:12,424 --> 00:28:16,928 the APF, the Atomic Packing Fraction, 540 00:28:16,928 --> 00:28:25,470 is equal to pi over 6, which is 52%. 541 00:28:25,470 --> 00:28:29,974 That's not very good, all right? 542 00:28:29,974 --> 00:28:32,944 That's not-- well, I mean, it's relative. 543 00:28:32,944 --> 00:28:33,678 We don't know yet. 544 00:28:33,678 --> 00:28:35,413 But it feels kind of low-- 545 00:28:35,413 --> 00:28:36,214 52%. 546 00:28:36,214 --> 00:28:39,117 So with a simple cubic lattice, all 547 00:28:39,117 --> 00:28:45,423 I can do in my wildest dreams is cover 52% of that volume. 548 00:28:45,423 --> 00:28:49,094 That's not very-- that doesn't feel like a lot. 549 00:28:49,094 --> 00:28:53,398 And in fact, when you look at the periodic table, what 550 00:28:53,398 --> 00:28:56,201 you find is that there's only one element-- 551 00:28:56,201 --> 00:28:58,903 one-- that takes the simple cubic lattice. 552 00:28:58,903 --> 00:29:01,239 Anybody know what it is? 553 00:29:01,239 --> 00:29:03,541 Of course-- it's polonium. 554 00:29:03,541 --> 00:29:04,876 [LAUGHTER] 555 00:29:04,876 --> 00:29:06,544 Of course. 556 00:29:06,544 --> 00:29:09,180 That's a really good conversation starter. 557 00:29:09,180 --> 00:29:11,282 I'm just saying. 558 00:29:11,282 --> 00:29:14,018 That is a great way to meet people. 559 00:29:14,018 --> 00:29:14,619 Which one? 560 00:29:14,619 --> 00:29:16,354 It's polonium? 561 00:29:16,354 --> 00:29:17,055 Why? 562 00:29:17,055 --> 00:29:19,691 Relativity. 563 00:29:19,691 --> 00:29:24,462 In polonium, those electrons are so high energy-- 564 00:29:24,462 --> 00:29:25,029 literally. 565 00:29:25,029 --> 00:29:26,631 They're so high energy they're like close 566 00:29:26,631 --> 00:29:27,565 to the speed of light. 567 00:29:27,565 --> 00:29:28,733 They're relativistic. 568 00:29:28,733 --> 00:29:29,801 Their mass is heavier. 569 00:29:29,801 --> 00:29:32,570 All sorts of interesting things happen. 570 00:29:32,570 --> 00:29:36,407 Polonium is the only one that goes simple cubic. 571 00:29:36,407 --> 00:29:37,509 I promised a video. 572 00:29:37,509 --> 00:29:39,043 Oh, there's the-- oh. 573 00:29:39,043 --> 00:29:40,912 Ha. 574 00:29:40,912 --> 00:29:43,982 a-- a-- a. 575 00:29:43,982 --> 00:29:46,417 There is a diagonal. 576 00:29:46,417 --> 00:29:47,786 There is another diagonal. 577 00:29:47,786 --> 00:29:48,353 Oh! 578 00:29:48,353 --> 00:29:51,956 We're going to come back to this. 579 00:29:51,956 --> 00:29:53,191 Let's draw that. 580 00:29:53,191 --> 00:29:55,126 Because that might be important, right? 581 00:29:55,126 --> 00:29:59,097 So if I have a cube, then this diagonal, 582 00:29:59,097 --> 00:30:04,269 if that's a and that's a, then that is a root 2. 583 00:30:04,269 --> 00:30:06,538 And the body diagonal-- 584 00:30:06,538 --> 00:30:07,906 that's the body diagonal-- 585 00:30:11,242 --> 00:30:14,979 is a root 3-- 586 00:30:14,979 --> 00:30:18,116 equals a root 3, right? 587 00:30:18,116 --> 00:30:21,419 OK, we'll come back-- 588 00:30:21,419 --> 00:30:24,756 oh, I didn't need that, because this is just a over 2. 589 00:30:24,756 --> 00:30:27,225 But whoa, hold on. 590 00:30:27,225 --> 00:30:28,026 We're coming to it. 591 00:30:28,026 --> 00:30:29,561 But I promised a video first. 592 00:30:29,561 --> 00:30:32,530 So here's a video with the laser coming in. 593 00:30:32,530 --> 00:30:34,065 And you're going to really see-- this 594 00:30:34,065 --> 00:30:35,266 will come to life a little bit. 595 00:30:35,266 --> 00:30:35,934 [VIDEO PLAYBACK] 596 00:30:35,934 --> 00:30:38,736 - [INAUDIBLE] of a cube. 597 00:30:38,736 --> 00:30:41,606 The simple, or primitive cubic unit cell 598 00:30:41,606 --> 00:30:44,876 has particles at the corners only. 599 00:30:44,876 --> 00:30:47,812 In reality, the particles lie as close 600 00:30:47,812 --> 00:30:50,148 to each other as possible. 601 00:30:50,148 --> 00:30:53,651 Note that the particles touch along the cube edges 602 00:30:53,651 --> 00:30:58,056 but not along a diagonal in the face or along a diagonal 603 00:30:58,056 --> 00:30:59,924 through the body. 604 00:30:59,924 --> 00:31:03,294 By slicing away parts that belong to neighboring unit 605 00:31:03,294 --> 00:31:06,831 cells, we see that the actual unit cell consists 606 00:31:06,831 --> 00:31:10,368 of portions of the particles. 607 00:31:10,368 --> 00:31:14,038 When the cells pack next to each other in all three dimensions, 608 00:31:14,038 --> 00:31:16,574 we obtain the crystal. 609 00:31:16,574 --> 00:31:20,311 If we fade the others out, you can see the original group 610 00:31:20,311 --> 00:31:23,381 of eight particles within the array and the unit cell 611 00:31:23,381 --> 00:31:24,949 within that group. 612 00:31:24,949 --> 00:31:29,454 We find the number of particles in one unit cell by combining 613 00:31:29,454 --> 00:31:32,190 all the particles' portions. 614 00:31:32,190 --> 00:31:36,194 In the simple cubic unit cell, eight corners, each of which 615 00:31:36,194 --> 00:31:41,599 is 1/8 of a particle, combine to give one particle. 616 00:31:41,599 --> 00:31:43,735 A key feature of a crystal structure 617 00:31:43,735 --> 00:31:47,572 is its coordination number, the number of the nearest neighbors 618 00:31:47,572 --> 00:31:49,874 surrounding each particle. 619 00:31:49,874 --> 00:31:52,944 In a simple cubic array, any given particle 620 00:31:52,944 --> 00:31:56,581 has a neighboring particle above, below, to the right, 621 00:31:56,581 --> 00:31:59,250 to the left, in front, and in back of it, 622 00:31:59,250 --> 00:32:03,154 for a total of six nearest neighbors. 623 00:32:03,154 --> 00:32:05,723 The body-centered cubic unit cell has a-- 624 00:32:05,723 --> 00:32:07,759 [END PLAYBACK] 625 00:32:07,759 --> 00:32:10,628 OK, so you see-- that was so cool, right? 626 00:32:10,628 --> 00:32:13,064 Like unh, unh, and you're figuring out how many 627 00:32:13,064 --> 00:32:17,735 atoms are in the cell, right? 628 00:32:17,735 --> 00:32:20,004 Well, why didn't you just put the atom in the middle? 629 00:32:20,004 --> 00:32:24,309 You could do that, but then it's not at the point, right? 630 00:32:24,309 --> 00:32:27,312 So you've got to understand, the concept of the lattice 631 00:32:27,312 --> 00:32:30,982 is that whatever I put here, wherever my vectors go, 632 00:32:30,982 --> 00:32:35,186 that's what I put there and there and there. 633 00:32:35,186 --> 00:32:37,689 And so we often as a standard, we say, well, OK, 634 00:32:37,689 --> 00:32:38,556 what did I put there? 635 00:32:38,556 --> 00:32:41,926 I put an atom and then I put another atom. 636 00:32:41,926 --> 00:32:44,696 But then your unit cell is that cube. 637 00:32:44,696 --> 00:32:46,831 And so we cut the unit cell, and you're like, well, 638 00:32:46,831 --> 00:32:48,499 how many atoms do I have in it? 639 00:32:48,499 --> 00:32:49,167 Well, let's see. 640 00:32:49,167 --> 00:32:53,004 I got one here, but it's shared with all these other cubes, 641 00:32:53,004 --> 00:32:55,740 so there's an 1/8 of an atom there and 1/8 of an atom there. 642 00:32:55,740 --> 00:32:58,109 We add them all up, and you get one atom right back, 643 00:32:58,109 --> 00:32:59,911 which you knew-- 644 00:32:59,911 --> 00:33:00,912 which you knew. 645 00:33:00,912 --> 00:33:04,549 And by the way, this was not 1 over a. 646 00:33:04,549 --> 00:33:05,350 It was. 647 00:33:05,350 --> 00:33:07,885 I wrote it mistakenly 1 over a. 648 00:33:07,885 --> 00:33:13,257 The radius is 1/2 a over there, which is correct here. 649 00:33:13,257 --> 00:33:14,192 OK, good. 650 00:33:14,192 --> 00:33:17,895 Now, next crystal structure-- oh, look. 651 00:33:17,895 --> 00:33:22,467 How do you know what crystal structure-- 652 00:33:22,467 --> 00:33:24,969 there it is right there! 653 00:33:24,969 --> 00:33:27,972 It's the periodic table! 654 00:33:27,972 --> 00:33:29,907 To the rescue again-- he had it right there. 655 00:33:29,907 --> 00:33:30,408 [APPLAUSE] 656 00:33:30,408 --> 00:33:30,942 Thank you. 657 00:33:30,942 --> 00:33:33,311 [CHEERING AND APPLAUSE] 658 00:33:33,311 --> 00:33:42,053 That means T-shirts and T-shirts and T-shirts and T-shirts. 659 00:33:42,053 --> 00:33:43,755 [CHATTER] 660 00:33:43,755 --> 00:33:49,260 All right, I'm going all the way up there and up there. 661 00:33:49,260 --> 00:33:50,628 We'll bring more. 662 00:33:50,628 --> 00:33:53,264 You bring a periodic table like that-- 663 00:33:53,264 --> 00:33:54,499 what is a crystal symmetry? 664 00:33:57,902 --> 00:33:59,971 I know. 665 00:33:59,971 --> 00:34:02,607 Which one is simple cubic? 666 00:34:02,607 --> 00:34:03,908 Polonium. 667 00:34:03,908 --> 00:34:05,610 Where's-- it's only polonium. 668 00:34:05,610 --> 00:34:06,778 Oh, there it is-- 669 00:34:06,778 --> 00:34:07,879 polonium. 670 00:34:07,879 --> 00:34:09,414 Thank you for being representative 671 00:34:09,414 --> 00:34:11,315 of simple cubic crystals. 672 00:34:11,315 --> 00:34:13,351 But now we're talking about the next kind, 673 00:34:13,351 --> 00:34:15,987 which is body-centered cubic. 674 00:34:15,987 --> 00:34:19,857 And look, a lot more have a body-centered cubic. 675 00:34:19,857 --> 00:34:28,299 And here, OK, so if it's BCC, Body-Centered Cubic, 676 00:34:28,299 --> 00:34:35,239 well, now you know if I take these atoms at these corners, 677 00:34:35,239 --> 00:34:37,442 and I add now-- there's a lattice 678 00:34:37,442 --> 00:34:39,911 that takes me to the middle. 679 00:34:39,911 --> 00:34:44,181 Remember, my lattice is giving me this symmetry. 680 00:34:44,181 --> 00:34:45,449 My lattice is giving me this. 681 00:34:45,449 --> 00:34:49,187 So it means whatever I put here, I put in all the corners 682 00:34:49,187 --> 00:34:51,755 and I put in the middle, right? 683 00:34:51,755 --> 00:34:52,790 That's BCC. 684 00:34:52,790 --> 00:34:55,426 That's what BCC means. 685 00:34:55,426 --> 00:34:57,428 Well, and yeah, if you grow that-- 686 00:34:57,428 --> 00:35:00,331 OK, so first of all, eight nearest neighbors-- 687 00:35:03,267 --> 00:35:04,802 eight nearest neighbors-- well, you 688 00:35:04,802 --> 00:35:07,405 can see that, because I've got the six, 689 00:35:07,405 --> 00:35:09,707 but then I've got these ones here, right? 690 00:35:09,707 --> 00:35:13,744 I've got actually it's not the six. 691 00:35:13,744 --> 00:35:15,813 I've got the one along the diagonal, 692 00:35:15,813 --> 00:35:18,716 which is kind of the point I wanted to make next. 693 00:35:18,716 --> 00:35:20,384 Oh, this is BCC. 694 00:35:20,384 --> 00:35:23,554 This is BCC, right? 695 00:35:23,554 --> 00:35:27,258 Now, how do you see that it's BCC? 696 00:35:27,258 --> 00:35:32,130 Because you stare at it and you stare at it and you build it. 697 00:35:32,130 --> 00:35:34,565 You get together with friends. 698 00:35:34,565 --> 00:35:37,502 You get together with friends, and you build even more. 699 00:35:37,502 --> 00:35:40,104 And you see how much you can build, and then you look, 700 00:35:40,104 --> 00:35:41,706 and you see, look here it is. 701 00:35:41,706 --> 00:35:43,908 There is my square, right? 702 00:35:43,908 --> 00:35:45,209 OK, and there's another square. 703 00:35:45,209 --> 00:35:46,811 And that's my cube. 704 00:35:46,811 --> 00:35:48,479 And you go like this. 705 00:35:48,479 --> 00:35:50,281 And then you're like, oh, look, and there's 706 00:35:50,281 --> 00:35:51,849 an atom in the middle. 707 00:35:51,849 --> 00:35:52,683 That's BCC. 708 00:35:52,683 --> 00:35:53,851 And they're all the same. 709 00:35:53,851 --> 00:35:55,553 That's BCC, right? 710 00:35:55,553 --> 00:35:58,956 But see, because this is defined by the lattice, 711 00:35:58,956 --> 00:36:01,392 I could have made my square anywhere. 712 00:36:01,392 --> 00:36:02,994 This didn't have to be the middle atom. 713 00:36:02,994 --> 00:36:04,896 I could move it over, and then this 714 00:36:04,896 --> 00:36:08,366 would be the middle atom or this, right? 715 00:36:08,366 --> 00:36:09,467 They're all equivalent. 716 00:36:09,467 --> 00:36:11,235 It's a lattice. 717 00:36:11,235 --> 00:36:18,609 Now, if you look at this and you take that square or that cube, 718 00:36:18,609 --> 00:36:21,345 then what you'll find is that if you go along 719 00:36:21,345 --> 00:36:24,115 the body diagonal-- see, now I'm going to take each of these, 720 00:36:24,115 --> 00:36:26,250 and I'm going to grow them. 721 00:36:26,250 --> 00:36:28,886 Where do the atoms touch? 722 00:36:28,886 --> 00:36:31,355 If I grow the atoms in a BCC crystal, 723 00:36:31,355 --> 00:36:35,860 they touch along the body diagonal first. 724 00:36:35,860 --> 00:36:37,094 Now, that's really important. 725 00:36:37,094 --> 00:36:38,763 Let's pass this one around, too. 726 00:36:38,763 --> 00:36:42,700 That's really important, OK? 727 00:36:42,700 --> 00:36:51,242 Because now we have this concept that is also very important, 728 00:36:51,242 --> 00:36:53,144 which is the close-packed direction-- 729 00:36:56,447 --> 00:37:02,453 close-packed direction. 730 00:37:06,290 --> 00:37:09,627 And for the BCC structure, it's the body diagonal. 731 00:37:15,266 --> 00:37:17,768 Over here it's the cube edge. 732 00:37:17,768 --> 00:37:21,105 For a simple cubic, it's the cube edge. 733 00:37:21,105 --> 00:37:25,776 Cube edge is the close-packed-- 734 00:37:25,776 --> 00:37:27,278 OK, how is that going to work? 735 00:37:27,278 --> 00:37:28,145 I'll do this. 736 00:37:30,781 --> 00:37:33,317 No, that's going to be confusing. 737 00:37:33,317 --> 00:37:39,590 Cube edge close-packed-- oh, that's really small. 738 00:37:39,590 --> 00:37:44,528 But you get my point, close-packed direction. 739 00:37:44,528 --> 00:37:46,764 Close-packed direction, body diagonal-- 740 00:37:46,764 --> 00:37:48,332 what that means is that-- 741 00:37:48,332 --> 00:37:52,069 and you can see it right here. 742 00:37:52,069 --> 00:37:56,007 If you go all the way back to this-- there it is. 743 00:37:56,007 --> 00:37:57,975 As I grow these-- these are the same. 744 00:37:57,975 --> 00:38:01,245 They're colored blue just for clarity, just you can see. 745 00:38:01,245 --> 00:38:04,282 But these are the same because it's a BCC lattice. 746 00:38:04,282 --> 00:38:06,851 And as I grow them, these all have the same radius. 747 00:38:06,851 --> 00:38:09,687 These are the ones that touch first along the diagonal, not 748 00:38:09,687 --> 00:38:11,555 those, all right? 749 00:38:11,555 --> 00:38:13,357 Not those. 750 00:38:13,357 --> 00:38:16,894 And that feeds into what the radius is of the atom. 751 00:38:16,894 --> 00:38:27,138 I do the same trick, and the APF is going to be 4/3 pi. 752 00:38:27,138 --> 00:38:32,310 And now I've got my a root 3 over 4. 753 00:38:32,310 --> 00:38:32,910 Why? 754 00:38:32,910 --> 00:38:36,847 Because a is the length of the cube. 755 00:38:36,847 --> 00:38:40,651 And you can see here I've got four radii, right? 756 00:38:40,651 --> 00:38:45,456 Two diameters of the sphere go down the body diagonal. 757 00:38:45,456 --> 00:38:49,293 So r cubed, 4/3 pi r cubed, is going 758 00:38:49,293 --> 00:38:54,598 to be the volume of a maximally-packed sphere 759 00:38:54,598 --> 00:38:57,501 in a BCC lattice. 760 00:38:57,501 --> 00:39:03,040 So this has to be divided by the whole volume, which is a-cubed. 761 00:39:03,040 --> 00:39:08,045 But I'm missing something, because there's not just one. 762 00:39:08,045 --> 00:39:08,946 There's not just one. 763 00:39:08,946 --> 00:39:10,448 There's another one. 764 00:39:10,448 --> 00:39:13,684 There's actually two atoms in a unit cell. 765 00:39:13,684 --> 00:39:14,552 There's two of these. 766 00:39:14,552 --> 00:39:15,653 You can see that one in the middle 767 00:39:15,653 --> 00:39:17,621 and then that eighth, eighth, eighth, eighth. 768 00:39:17,621 --> 00:39:18,522 You've got two. 769 00:39:18,522 --> 00:39:25,162 So I've got to multiply it by 2, and it gives you 68%, 770 00:39:25,162 --> 00:39:26,764 is the APF. 771 00:39:26,764 --> 00:39:31,302 And that's why so many more structures, especially metals, 772 00:39:31,302 --> 00:39:33,170 are happy to be BCC. 773 00:39:33,170 --> 00:39:36,107 Let's watch the video again, or a different video, 774 00:39:36,107 --> 00:39:36,874 this one on BBC. 775 00:39:36,874 --> 00:39:37,541 [VIDEO PLAYBACK] 776 00:39:37,541 --> 00:39:40,644 - --particle at each corner and one in the center, 777 00:39:40,644 --> 00:39:44,148 which is colored pink to make it easier to see. 778 00:39:44,148 --> 00:39:46,550 With full-size spheres, you can see 779 00:39:46,550 --> 00:39:49,320 that the particles don't touch along the edges of the cube. 780 00:39:49,320 --> 00:39:49,887 Close-packing 781 00:39:49,887 --> 00:39:54,992 - But each corner particle does touch the one in the center. 782 00:39:54,992 --> 00:39:58,195 The actual unit cell consists of portions 783 00:39:58,195 --> 00:40:02,199 of the corner particles and the whole one in the center. 784 00:40:02,199 --> 00:40:04,935 Eight eighths give one particle. 785 00:40:04,935 --> 00:40:07,037 And the one in the center gives another, 786 00:40:07,037 --> 00:40:10,274 for a total of two particles. 787 00:40:10,274 --> 00:40:13,944 In this tiny portion of a body-centered cubic array, 788 00:40:13,944 --> 00:40:16,213 you can see that any given particle has 789 00:40:16,213 --> 00:40:19,350 four nearest neighbors above and four 790 00:40:19,350 --> 00:40:23,621 below, for a total of eight nearest neighbors. 791 00:40:23,621 --> 00:40:24,221 [END PLAYBACK] 792 00:40:24,221 --> 00:40:26,323 OK, now it's really important, right? 793 00:40:26,323 --> 00:40:28,893 Just so you see conceptually, visually how we got this, 794 00:40:28,893 --> 00:40:33,497 the close-packed direction for BCC is four times the radius. 795 00:40:33,497 --> 00:40:38,135 I want to write that, four times the radius. 796 00:40:38,135 --> 00:40:40,438 And it's that thing that we're doing now 797 00:40:40,438 --> 00:40:43,941 a couple of times, where you've got the radius of this sphere 798 00:40:43,941 --> 00:40:46,977 that you're packing in with a given symmetry. 799 00:40:46,977 --> 00:40:49,113 But I've got to write that in terms 800 00:40:49,113 --> 00:40:52,950 of the lattice vectors, which are in this case all a, right? 801 00:40:52,950 --> 00:40:53,918 So it's really easy. 802 00:40:53,918 --> 00:40:56,053 But they're all a, so I've got to write it in terms 803 00:40:56,053 --> 00:40:57,254 of the length of the lattice. 804 00:40:57,254 --> 00:41:01,459 That's how you get the atomic packing, OK? 805 00:41:01,459 --> 00:41:04,295 Good, now the last one is FCC. 806 00:41:04,295 --> 00:41:06,897 Look at how many elements take the FCC form. 807 00:41:06,897 --> 00:41:07,665 How do you know? 808 00:41:07,665 --> 00:41:11,068 You bring out the periodic table, and you look it up. 809 00:41:11,068 --> 00:41:15,473 Now, many of these elements can take different crystal 810 00:41:15,473 --> 00:41:16,674 symmetries. 811 00:41:16,674 --> 00:41:19,610 The ones that are listed are the ones that are often-- 812 00:41:19,610 --> 00:41:22,146 well, always the most stable one, right. 813 00:41:22,146 --> 00:41:23,314 They're the most stable one. 814 00:41:23,314 --> 00:41:26,484 But sometimes they can take many different symmetries 815 00:41:26,484 --> 00:41:30,321 if you do something like add temperature or pressure 816 00:41:30,321 --> 00:41:32,456 or other things to the system. 817 00:41:32,456 --> 00:41:34,859 What's in the periodic table is the most stable one. 818 00:41:34,859 --> 00:41:42,099 Now, the FCC is the last one that we're going to care about. 819 00:41:42,099 --> 00:41:47,338 And it is the face-centered cubic. 820 00:41:47,338 --> 00:41:54,812 Now, OK, so on this one, this is the face, right? 821 00:41:54,812 --> 00:41:59,316 So we've got these atoms that are sharing across the face. 822 00:41:59,316 --> 00:42:03,320 And then we've got these atoms here at the corners. 823 00:42:03,320 --> 00:42:07,725 There's nothing in the middle now, but they're on the faces-- 824 00:42:07,725 --> 00:42:11,328 sorry, the corners and the faces, all right? 825 00:42:11,328 --> 00:42:15,799 And so if you look at this, what you 826 00:42:15,799 --> 00:42:19,703 can see from looking at this-- and you've got to look at it. 827 00:42:19,703 --> 00:42:21,305 Where's my cube? 828 00:42:21,305 --> 00:42:22,573 Where's my unit cell? 829 00:42:22,573 --> 00:42:25,609 You now know it's called the unit cell, right? 830 00:42:25,609 --> 00:42:26,477 Oh, and I see. 831 00:42:26,477 --> 00:42:28,312 And then there's the face atom, right? 832 00:42:28,312 --> 00:42:30,414 There's a square there. 833 00:42:30,414 --> 00:42:32,683 Then there's the atom in the face. 834 00:42:32,683 --> 00:42:34,418 And then there's nothing in the middle. 835 00:42:34,418 --> 00:42:35,986 It's not BCC, right? 836 00:42:35,986 --> 00:42:40,057 But if I look at another edge of that, 837 00:42:40,057 --> 00:42:43,994 there's the atom in that face and so on and so on, right? 838 00:42:43,994 --> 00:42:47,197 Now, what's really cool about FCC 839 00:42:47,197 --> 00:42:49,900 is that it's got a lot of neighbors. 840 00:42:49,900 --> 00:42:52,369 And it's got more packing. 841 00:42:52,369 --> 00:42:58,742 So for FCC, this is the close-packed direction. 842 00:42:58,742 --> 00:43:07,418 Close-packed direction is the face, 843 00:43:07,418 --> 00:43:09,587 is the-- did I write down? 844 00:43:09,587 --> 00:43:11,422 I didn't write down. 845 00:43:11,422 --> 00:43:13,057 OK, that's fine. 846 00:43:13,057 --> 00:43:23,000 It's the face diagonal, which is a root 2 in length. 847 00:43:23,000 --> 00:43:27,171 And so the nearest neighbors is 12. 848 00:43:27,171 --> 00:43:35,479 And the atomic packing fraction for FCC is 4/3 pi times-- 849 00:43:35,479 --> 00:43:37,881 and now here we go, right? 850 00:43:37,881 --> 00:43:41,085 When I expand them, it's along the face 851 00:43:41,085 --> 00:43:43,354 that they're going to touch, right? 852 00:43:43,354 --> 00:43:45,356 So that's the close-packed direction. 853 00:43:45,356 --> 00:43:48,525 Face diagonal, that's this one here, right? 854 00:43:48,525 --> 00:43:55,232 So it's a root 2 over 4, right? 855 00:43:55,232 --> 00:43:58,769 Because again, I've got a radius and another 856 00:43:58,769 --> 00:44:00,604 and another and another. 857 00:44:00,604 --> 00:44:04,475 There's two diameters worth that go across that face. 858 00:44:04,475 --> 00:44:09,079 And so that's going to be cubed, OK, divided by a-cubed. 859 00:44:09,079 --> 00:44:10,748 Now, is there something missing? 860 00:44:10,748 --> 00:44:11,782 Yes. 861 00:44:11,782 --> 00:44:14,752 The number of atoms. 862 00:44:14,752 --> 00:44:19,189 If you look at a FCC crystal, and you look at the corners, 863 00:44:19,189 --> 00:44:22,593 each one of those shares with eight other unit cells, 864 00:44:22,593 --> 00:44:23,227 so there's 1/8. 865 00:44:23,227 --> 00:44:26,664 That gives you the one, like in our simple cubic. 866 00:44:26,664 --> 00:44:29,767 But now you've got all these ones sharing across faces. 867 00:44:29,767 --> 00:44:33,504 You've got six of those, each one sharing 1/2 atom. 868 00:44:33,504 --> 00:44:34,772 So that's three more. 869 00:44:34,772 --> 00:44:35,539 So there's 4. 870 00:44:38,342 --> 00:44:45,516 And this gives you a very nice packing of 74%-- 871 00:44:45,516 --> 00:44:46,617 74%. 872 00:44:46,617 --> 00:44:49,687 Let's see what our video guy says about FCC. 873 00:44:53,190 --> 00:44:53,857 [VIDEO PLAYBACK] 874 00:44:53,857 --> 00:44:57,361 - The face-centered cubic unit cell has a particle at each 875 00:44:57,361 --> 00:45:01,699 corner and in each face, which are colored yellow here, 876 00:45:01,699 --> 00:45:03,434 but none in the center. 877 00:45:03,434 --> 00:45:06,336 The corner particles don't touch each other. 878 00:45:06,336 --> 00:45:09,606 But each corner does touch a particle in the face. 879 00:45:09,606 --> 00:45:13,310 And those in the faces touch each other as well. 880 00:45:13,310 --> 00:45:15,979 The actual unit cell consists of portions 881 00:45:15,979 --> 00:45:19,450 of particles at the corners and in the faces. 882 00:45:19,450 --> 00:45:22,853 Eight-eighths at the corners gives one particle. 883 00:45:22,853 --> 00:45:25,489 And half a particle in each of six faces 884 00:45:25,489 --> 00:45:29,993 gives three more, for a total of four particles. 885 00:45:29,993 --> 00:45:33,797 In this tiny portion of a face-centered cubic array, 886 00:45:33,797 --> 00:45:37,634 notice that a given particle has four nearest neighbors 887 00:45:37,634 --> 00:45:41,138 around it, four more above, and four more below, 888 00:45:41,138 --> 00:45:45,008 for a total of 12 nearest neighbors. 889 00:45:45,008 --> 00:45:49,012 Stacking spheres shows how the three cubic unit cells arise. 890 00:45:49,012 --> 00:45:50,114 Now we talk about packing. 891 00:45:50,114 --> 00:45:54,284 - Arranged a layer of spheres in horizontal and vertical rows. 892 00:45:54,284 --> 00:45:58,622 Note the large, diamond-shaped space among the particles. 893 00:45:58,622 --> 00:46:02,192 Placing the next layer directly over the first 894 00:46:02,192 --> 00:46:06,563 gives a structure based on the simple cubic unit cell. 895 00:46:06,563 --> 00:46:11,135 Those larger spaces mean an inefficient use of space. 896 00:46:11,135 --> 00:46:14,972 In fact, only 52% of the available volume 897 00:46:14,972 --> 00:46:18,675 is actually occupied by spheres. 898 00:46:18,675 --> 00:46:21,645 Because of this inefficiency, the simple cubic unit cell 899 00:46:21,645 --> 00:46:23,547 is seen rarely in nature. 900 00:46:23,547 --> 00:46:24,648 Polonium! 901 00:46:24,648 --> 00:46:26,450 - A more efficient stacking occurs 902 00:46:26,450 --> 00:46:29,586 if we place the second layer over the spaces formed 903 00:46:29,586 --> 00:46:32,122 by the first layer and the third layer 904 00:46:32,122 --> 00:46:34,892 over the spaces formed by the second. 905 00:46:34,892 --> 00:46:39,630 That simple change leads to 68% of the available volume 906 00:46:39,630 --> 00:46:42,032 occupied by the spheres and a structure 907 00:46:42,032 --> 00:46:45,702 based on the body-centered cubic unit cell. 908 00:46:45,702 --> 00:46:49,206 Many metals, including all the alkali metals, 909 00:46:49,206 --> 00:46:51,675 adopt this arrangement. 910 00:46:51,675 --> 00:46:53,243 For the most efficient stacking-- 911 00:46:53,243 --> 00:46:53,877 Now watch this. 912 00:46:53,877 --> 00:46:57,447 - --shift every other row in the first layer so the large, 913 00:46:57,447 --> 00:47:01,618 diamond-shaped spaces become smaller triangular spaces, 914 00:47:01,618 --> 00:47:04,922 and place the second layer over them. 915 00:47:04,922 --> 00:47:07,691 Then the third layer goes over the holes 916 00:47:07,691 --> 00:47:10,994 visible through the first and second layers. 917 00:47:10,994 --> 00:47:15,065 In this arrangement, called cubic closest packing, 918 00:47:15,065 --> 00:47:18,902 spheres occupy 74% of the volume. 919 00:47:18,902 --> 00:47:20,204 Note that it is based-- 920 00:47:20,204 --> 00:47:21,238 [END PLAYBACK] 921 00:47:21,238 --> 00:47:24,741 --on the FCC crystal, Bravais lattice. 922 00:47:24,741 --> 00:47:26,176 That's what he meant to say there. 923 00:47:26,176 --> 00:47:28,478 And you really see it come to life, right? 924 00:47:28,478 --> 00:47:30,781 I wish I could do this somehow with the 3D models. 925 00:47:30,781 --> 00:47:33,250 But you stare at the 3D models and you see that-- you know, 926 00:47:33,250 --> 00:47:36,053 what's interesting is the way you put it-- 927 00:47:36,053 --> 00:47:38,522 you slide-- this is what Hooke did with the cannonballs. 928 00:47:38,522 --> 00:47:40,190 You slide that base layer, and then you 929 00:47:40,190 --> 00:47:41,592 can stack them differently. 930 00:47:41,592 --> 00:47:45,462 But it's still a cubic system, FCC, all right? 931 00:47:45,462 --> 00:47:48,198 We'll pick up on this on Wednesday.