1 00:00:16,181 --> 00:00:19,319 But I want to pick up where we left off on Monday. 2 00:00:19,319 --> 00:00:21,755 We were talking about the three Bravais lattices. 3 00:00:21,755 --> 00:00:24,824 There a couple more things that I want to say about them. 4 00:00:24,824 --> 00:00:28,194 What we captured, the key essence, 5 00:00:28,194 --> 00:00:30,363 apart from they're being these cubic crystals 6 00:00:30,363 --> 00:00:33,299 and the different elements that form them, 7 00:00:33,299 --> 00:00:39,639 is this idea that you can go from the unit cell, 8 00:00:39,639 --> 00:00:45,845 once you know the unit cell, you can get the atomic radius. 9 00:00:45,845 --> 00:00:47,280 So that's kind of cool. 10 00:00:47,280 --> 00:00:50,083 Because you know information about things 11 00:00:50,083 --> 00:00:52,619 like, for example, packing. 12 00:00:52,619 --> 00:00:54,954 So then we had the packing fraction, the atomic packing 13 00:00:54,954 --> 00:00:55,588 fraction. 14 00:00:55,588 --> 00:00:58,358 Calculated that. 15 00:00:58,358 --> 00:01:03,730 The maximum, the max packing fraction. 16 00:01:03,730 --> 00:01:04,497 Packing fraction. 17 00:01:07,400 --> 00:01:12,572 We calculated things like the closed pack direction. 18 00:01:12,572 --> 00:01:14,674 Well, you can also go to things like the density. 19 00:01:14,674 --> 00:01:17,343 Once you know how to go back and forth. 20 00:01:17,343 --> 00:01:22,749 And so for example, if I want the density of a material, 21 00:01:22,749 --> 00:01:25,785 that's grams per centimeter cubed, right? 22 00:01:25,785 --> 00:01:28,021 That's the density. 23 00:01:28,021 --> 00:01:30,890 But you could write the density in terms of the things 24 00:01:30,890 --> 00:01:32,125 we talked about on Monday. 25 00:01:32,125 --> 00:01:33,759 So again, you know, I want you to be 26 00:01:33,759 --> 00:01:34,860 able to go back and forth. 27 00:01:34,860 --> 00:01:39,833 Once you know the unit cell, and you can know something 28 00:01:39,833 --> 00:01:42,836 about the radius of the atom, then you can get the density. 29 00:01:42,836 --> 00:01:46,439 For example, here, the density could 30 00:01:46,439 --> 00:01:50,676 be defined as the number of atoms 31 00:01:50,676 --> 00:01:59,219 in the cell, times the grams per mole of that element, 32 00:01:59,219 --> 00:02:06,292 divided by the volume of the cell, of the unit cell, 33 00:02:06,292 --> 00:02:08,328 times Avogadro. 34 00:02:12,365 --> 00:02:14,134 Now, you can go through the units 35 00:02:14,134 --> 00:02:19,005 and you can work out that this is grams per centimeter cubed. 36 00:02:19,005 --> 00:02:23,877 So just as an example, if I had, let's see, 37 00:02:23,877 --> 00:02:25,578 the example I have is copper. 38 00:02:25,578 --> 00:02:30,116 So example, copper. 39 00:02:30,116 --> 00:02:33,720 So you know, everything comes from the periodic table. 40 00:02:33,720 --> 00:02:34,420 Everything. 41 00:02:34,420 --> 00:02:37,957 Life, class, everything comes from the periodic table. 42 00:02:37,957 --> 00:02:38,558 So I go-- 43 00:02:38,558 --> 00:02:40,326 I say, I want to know-- 44 00:02:40,326 --> 00:02:42,629 I go to the periodic table. 45 00:02:42,629 --> 00:02:44,297 And instead of looking up the density, 46 00:02:44,297 --> 00:02:45,598 I look up other things. 47 00:02:45,598 --> 00:02:48,434 So I look up, for example, that it's FCC. 48 00:02:51,304 --> 00:03:00,880 And then I look up that it's 63.55 grams per mole. 49 00:03:00,880 --> 00:03:03,116 And I also look up the atomic radius. 50 00:03:03,116 --> 00:03:08,488 So I look up these things, the atomic radius. 51 00:03:08,488 --> 00:03:12,525 And that's equal to 1.3 angstroms. 52 00:03:12,525 --> 00:03:14,427 So I look all this stuff up. 53 00:03:14,427 --> 00:03:16,763 And the point is, I can now go back and forth. 54 00:03:16,763 --> 00:03:19,532 I can use the information, because I 55 00:03:19,532 --> 00:03:22,835 know if it's a crystal and I know the crystal type, 56 00:03:22,835 --> 00:03:25,572 FCC, then I can use this information 57 00:03:25,572 --> 00:03:27,874 to get other aspects, right? 58 00:03:27,874 --> 00:03:28,942 So other properties. 59 00:03:28,942 --> 00:03:34,781 So for example, I know that the radius of the atom for an FCC, 60 00:03:34,781 --> 00:03:39,652 if it's close packed, if it's-- you know, FCC one element, 61 00:03:39,652 --> 00:03:43,022 remember it grows out until they touch. 62 00:03:43,022 --> 00:03:45,825 And that got us how we get the radius of the atom. 63 00:03:45,825 --> 00:03:50,897 And it's a root 2 over 4. 64 00:03:50,897 --> 00:03:55,868 But I also know that there's four atoms per unit cell 65 00:03:55,868 --> 00:03:58,605 because it's FCC. 66 00:03:58,605 --> 00:04:01,407 I also know that the volume of the cell 67 00:04:01,407 --> 00:04:04,844 is simply the lattice constant cubed 68 00:04:04,844 --> 00:04:06,646 because it's a cubic cell. 69 00:04:06,646 --> 00:04:08,348 All the ones here are cubic. 70 00:04:08,348 --> 00:04:10,316 So the volume of these cells are cubic. 71 00:04:10,316 --> 00:04:14,454 From this information, I can compute the density. 72 00:04:14,454 --> 00:04:17,790 So again, it's going back and forth. 73 00:04:17,790 --> 00:04:21,728 The radius of the atoms packed in, the lattice constants, 74 00:04:21,728 --> 00:04:23,329 the number of atoms in the cell, these 75 00:04:23,329 --> 00:04:25,632 are the things we talked about on Monday. 76 00:04:25,632 --> 00:04:29,636 And as you go back and forth, you can do things like this. 77 00:04:29,636 --> 00:04:32,238 All right, good. 78 00:04:32,238 --> 00:04:36,342 Now we talked about the-- 79 00:04:36,342 --> 00:04:41,281 we talked about the Bravais lattice there, right? 80 00:04:41,281 --> 00:04:43,716 And those are three Bravais lattices that I 81 00:04:43,716 --> 00:04:45,918 want you to know. 82 00:04:45,918 --> 00:04:49,122 Bravais lattices. 83 00:04:49,122 --> 00:04:53,493 But the thing is that there's something else 84 00:04:53,493 --> 00:04:58,931 you need to know in order to know what crystal you have. 85 00:04:58,931 --> 00:05:02,368 And so this is the FCC Bravais lattice. 86 00:05:02,368 --> 00:05:03,136 There it is. 87 00:05:03,136 --> 00:05:05,004 It's over there on the right. 88 00:05:05,004 --> 00:05:06,673 It's the face-centered cubic. 89 00:05:06,673 --> 00:05:08,574 OK, gesundheit. 90 00:05:08,574 --> 00:05:11,311 And I'm going to pass this around. 91 00:05:11,311 --> 00:05:13,479 All right, oh yeah, I'm starting over here 92 00:05:13,479 --> 00:05:16,482 because I keep on starting over there and it's not fair. 93 00:05:16,482 --> 00:05:17,417 That's not right. 94 00:05:17,417 --> 00:05:19,752 Yeah, all right. 95 00:05:19,752 --> 00:05:23,156 Got a couple-- 96 00:05:23,156 --> 00:05:24,757 OK, so that's FCC, right? 97 00:05:24,757 --> 00:05:26,125 So we're going to write down FCC. 98 00:05:26,125 --> 00:05:29,529 But the thing is is that see, this is our stamp. 99 00:05:29,529 --> 00:05:36,803 Remember, it's a stamp of points that are equivalent. 100 00:05:36,803 --> 00:05:39,038 That's what the lattice vectors are. 101 00:05:39,038 --> 00:05:42,808 They are not the things that you put at that point. 102 00:05:42,808 --> 00:05:44,043 That is something else. 103 00:05:44,043 --> 00:05:46,646 Now, on Monday, we put the things there were 104 00:05:46,646 --> 00:05:49,182 atoms, spherical particles. 105 00:05:49,182 --> 00:05:50,582 And they're all the same. 106 00:05:50,582 --> 00:05:53,386 That was their introduction to crystallography. 107 00:05:53,386 --> 00:06:03,463 But the basis can be more than just a single atom. 108 00:06:03,463 --> 00:06:05,164 It can be more. 109 00:06:05,164 --> 00:06:07,500 So how do we think about the basis? 110 00:06:07,500 --> 00:06:08,568 So here's a lattice. 111 00:06:08,568 --> 00:06:10,103 This is a lattice of points. 112 00:06:10,103 --> 00:06:11,804 The way you want to think about a lattice 113 00:06:11,804 --> 00:06:15,241 is a stamp without putting things there yet. 114 00:06:15,241 --> 00:06:17,543 The basis is what you put there. 115 00:06:17,543 --> 00:06:20,446 Let's write that down because it's so important. 116 00:06:20,446 --> 00:06:26,886 So the lattice is all we're going-- 117 00:06:26,886 --> 00:06:32,024 oh, I'm going all caps, how to repeat. 118 00:06:32,024 --> 00:06:42,068 And the basis is what to repeat. 119 00:06:44,971 --> 00:06:48,074 So if this is my lattice and I decide 120 00:06:48,074 --> 00:06:50,610 that I want to get a million downloads on YouTube, 121 00:06:50,610 --> 00:06:52,311 I'm going to put a cap there. 122 00:06:52,311 --> 00:06:54,580 And this is my crystal. 123 00:06:54,580 --> 00:06:56,516 It's a crystal of cats. 124 00:06:56,516 --> 00:06:57,517 That's OK. 125 00:06:57,517 --> 00:07:01,020 I have done nothing wrong. 126 00:07:01,020 --> 00:07:03,222 Because I have simply, according to the rules 127 00:07:03,222 --> 00:07:05,691 of crystallography, I've had a lattice and a basis. 128 00:07:05,691 --> 00:07:07,660 And I've just defined my basis as a cat. 129 00:07:07,660 --> 00:07:10,430 And that's the crystal you get. 130 00:07:10,430 --> 00:07:13,299 But see, the basis could also have been, for example, 131 00:07:13,299 --> 00:07:15,935 two atoms colored differently to show 132 00:07:15,935 --> 00:07:17,170 that they are different atoms. 133 00:07:17,170 --> 00:07:20,072 And then this would be my crystal. 134 00:07:20,072 --> 00:07:23,876 So the actual crystal that you get 135 00:07:23,876 --> 00:07:27,113 may be different than the lattice structure. 136 00:07:27,113 --> 00:07:28,448 All right, so let's take a look. 137 00:07:28,448 --> 00:07:32,552 So for example, over here, you know, 138 00:07:32,552 --> 00:07:39,759 if my basis is a single atom, nickel, copper, gold, 139 00:07:39,759 --> 00:07:45,431 platinum, well, I showed you these from the periodic table 140 00:07:45,431 --> 00:07:46,199 on Monday, right? 141 00:07:46,199 --> 00:07:48,267 We highlighted them when we talked about FCC. 142 00:07:48,267 --> 00:07:54,207 Then the crystal that you get is FCC. 143 00:07:54,207 --> 00:07:56,808 These are the ones we did on Monday. 144 00:07:56,808 --> 00:07:58,711 Those are FCC crystals. 145 00:07:58,711 --> 00:08:01,814 But at the same time, if my basis-- 146 00:08:01,814 --> 00:08:03,816 check this out. 147 00:08:03,816 --> 00:08:11,090 Oh, this is salt. When you guys pour salt on your food, 148 00:08:11,090 --> 00:08:14,527 do not see it any more except as this. 149 00:08:14,527 --> 00:08:19,932 This is salt. This is salt. It is-- 150 00:08:19,932 --> 00:08:21,400 let's see-- simple cubic. 151 00:08:25,838 --> 00:08:27,106 Is it, though? 152 00:08:27,106 --> 00:08:31,110 Because a lattice, a simple cubic-- 153 00:08:31,110 --> 00:08:32,578 remember the definition. 154 00:08:32,578 --> 00:08:38,484 A simple cubic lattice would mean a stamp of a simple cube 155 00:08:38,484 --> 00:08:42,554 where every corner is exactly the same, exactly the same. 156 00:08:42,554 --> 00:08:43,856 That's what a lattice is. 157 00:08:43,856 --> 00:08:45,992 So this cannot be a simple cube. 158 00:08:45,992 --> 00:08:49,428 This is not simple cubic because they're different. 159 00:08:49,428 --> 00:08:51,264 They're different. 160 00:08:51,264 --> 00:08:53,432 At the corners, I've got two blues and then 161 00:08:53,432 --> 00:08:55,101 two red and then two other blues. 162 00:08:55,101 --> 00:08:56,702 So they're not the same. 163 00:08:56,702 --> 00:08:59,472 It cannot be a simple cubic structure. 164 00:08:59,472 --> 00:09:01,741 This is an FCC lattice. 165 00:09:01,741 --> 00:09:03,109 This is an FCC. 166 00:09:03,109 --> 00:09:04,644 Yeah, I'm starting over here again. 167 00:09:04,644 --> 00:09:06,445 I'm trying to even it out. 168 00:09:06,445 --> 00:09:15,221 That's an FCC lattice, but with a basis of sodium and chlorine. 169 00:09:15,221 --> 00:09:16,656 So it's an FCC. 170 00:09:16,656 --> 00:09:18,691 You can call it an FCC lattice. 171 00:09:18,691 --> 00:09:21,961 But the basis is a sodium chloride pair. 172 00:09:21,961 --> 00:09:24,030 And this has a name. 173 00:09:24,030 --> 00:09:27,967 It's called the rock salt structure. 174 00:09:27,967 --> 00:09:29,569 It's called the rock salt structure. 175 00:09:29,569 --> 00:09:33,573 It's an FCC lattice with this two atom basis where, look, 176 00:09:33,573 --> 00:09:35,308 you can see the FCC lattice here. 177 00:09:35,308 --> 00:09:36,876 Look at those blue points, right? 178 00:09:36,876 --> 00:09:39,278 Blue outlines an FCC lattice. 179 00:09:39,278 --> 00:09:42,782 But now the basis is one atom here and one there. 180 00:09:42,782 --> 00:09:47,119 So now I take that, this pair, and I put that 181 00:09:47,119 --> 00:09:49,121 at all the lattice points. 182 00:09:49,121 --> 00:09:49,789 You see that? 183 00:09:49,789 --> 00:09:52,358 You put that in all the lattice points of FCC 184 00:09:52,358 --> 00:09:55,394 and you've got salt. And that is how you must 185 00:09:55,394 --> 00:09:58,431 see table salt from now on. 186 00:09:58,431 --> 00:09:59,665 Well, we talked about diamond. 187 00:10:02,201 --> 00:10:03,869 This is diamond. 188 00:10:03,869 --> 00:10:06,305 Diamond is the same Bravais lattice. 189 00:10:06,305 --> 00:10:09,041 It doesn't look like it at first. 190 00:10:09,041 --> 00:10:10,443 It doesn't look like it at first. 191 00:10:10,443 --> 00:10:17,583 But see here, I have a different pair. 192 00:10:17,583 --> 00:10:19,852 Here I have a carbon-carbon pair. 193 00:10:19,852 --> 00:10:22,355 Or it could be, or silicon-silicon. 194 00:10:22,355 --> 00:10:28,527 There are actually a number of pairs of that 195 00:10:28,527 --> 00:10:32,164 would form this diamond. 196 00:10:32,164 --> 00:10:34,400 In fact, we specify because there are different types. 197 00:10:34,400 --> 00:10:37,470 This is called cubic diamond crystal. 198 00:10:40,606 --> 00:10:42,540 This is the cubic diamond crystal. 199 00:10:42,540 --> 00:10:43,843 What is the lattice? 200 00:10:43,843 --> 00:10:45,911 It's FCC. 201 00:10:45,911 --> 00:10:46,479 It's FCC. 202 00:10:46,479 --> 00:10:48,314 Now, let's take a look at that. 203 00:10:48,314 --> 00:10:50,983 Look, there is my FCC. 204 00:10:50,983 --> 00:10:52,818 Here, here, here, here. 205 00:10:52,818 --> 00:10:56,389 But inside of the FCC, I've got this pair of carbon atoms. 206 00:10:56,389 --> 00:10:58,357 Now they're the same type of atom. 207 00:10:58,357 --> 00:11:00,826 And instead of one being kind of in the middle 208 00:11:00,826 --> 00:11:04,263 of this edge, as it was with sodium and chlorine, now 209 00:11:04,263 --> 00:11:08,300 the other atom is kind of off in a diagonal. 210 00:11:08,300 --> 00:11:10,736 But it's still just a pair of atoms. 211 00:11:10,736 --> 00:11:12,471 It's a pair of carbon atoms. 212 00:11:12,471 --> 00:11:14,840 But if you look carefully at diamond, 213 00:11:14,840 --> 00:11:17,276 and this doesn't go on in this picture, 214 00:11:17,276 --> 00:11:21,547 but it does in real life, this is an FCC lattice 215 00:11:21,547 --> 00:11:24,684 of these pairs, repeating everywhere. 216 00:11:24,684 --> 00:11:26,886 That's what diamond is. 217 00:11:26,886 --> 00:11:28,454 It's an FCC lattice. 218 00:11:28,454 --> 00:11:32,091 So the symmetry is FCC. 219 00:11:32,091 --> 00:11:37,029 But the basis is a carbon-carbon dimer, right? 220 00:11:37,029 --> 00:11:39,999 And that gives you cubic diamond. 221 00:11:39,999 --> 00:11:43,736 OK, so there's the Bravais lattice 222 00:11:43,736 --> 00:11:45,905 and there's the resulting crystal structure. 223 00:11:45,905 --> 00:11:48,340 Now, speaking of diamond. 224 00:11:48,340 --> 00:11:52,411 And this is a graph that I showed you a while ago 225 00:11:52,411 --> 00:11:55,314 when we were talking about the differences between diamond 226 00:11:55,314 --> 00:11:57,349 and graphite. 227 00:11:57,349 --> 00:12:00,086 One of the things that you can see that's so important, 228 00:12:00,086 --> 00:12:02,755 this is the same element, two different crystal structures. 229 00:12:02,755 --> 00:12:05,024 One of the things that you can see that is so important 230 00:12:05,024 --> 00:12:09,962 is how direction is going to be an important property. 231 00:12:09,962 --> 00:12:14,333 You need to know how to identify direction 232 00:12:14,333 --> 00:12:17,436 because you know, in this case, it looks like maybe if I point 233 00:12:17,436 --> 00:12:19,105 in different directions, the thing would 234 00:12:19,105 --> 00:12:23,743 look kind of the same, or maybe have similar properties along-- 235 00:12:23,743 --> 00:12:24,910 but look at this. 236 00:12:24,910 --> 00:12:27,980 This looks seriously directionally dependent. 237 00:12:27,980 --> 00:12:30,449 It looks like if I did something like 238 00:12:30,449 --> 00:12:34,186 tried to carry charge or thermal energy this way, 239 00:12:34,186 --> 00:12:35,855 then it would be very, very different 240 00:12:35,855 --> 00:12:37,890 than if I carried it this way. 241 00:12:37,890 --> 00:12:40,159 Well, we have a word for that. 242 00:12:40,159 --> 00:12:42,061 And it's called anisotropy. 243 00:12:42,061 --> 00:12:48,367 And so this is another important word, anisotropy. 244 00:12:53,472 --> 00:12:56,208 If it's anisotropic, then the property, 245 00:12:56,208 --> 00:13:00,279 well, it could be like electrical or thermal 246 00:13:00,279 --> 00:13:02,615 conductivity, for example. 247 00:13:02,615 --> 00:13:06,786 Thermal conductivity could be like how it breaks, 248 00:13:06,786 --> 00:13:08,654 the fracture. 249 00:13:08,654 --> 00:13:10,623 And so on. 250 00:13:10,623 --> 00:13:14,627 But whatever-- but the property is-- 251 00:13:14,627 --> 00:13:15,761 depends. 252 00:13:15,761 --> 00:13:18,164 I'll just say it's directionally dependent. 253 00:13:18,164 --> 00:13:23,035 Directionally dependent. 254 00:13:23,035 --> 00:13:28,240 So the reason I'm telling you this is that I now need a way-- 255 00:13:28,240 --> 00:13:33,179 we've come up with sort of ways to talk about crystals. 256 00:13:33,179 --> 00:13:38,050 But I now need a way to specify where I am in a crystal. 257 00:13:38,050 --> 00:13:40,519 And maybe how the crystal cuts. 258 00:13:40,519 --> 00:13:43,322 I need a way to talk about directionality 259 00:13:43,322 --> 00:13:44,924 in these crystals. 260 00:13:44,924 --> 00:13:46,625 And that's what I want you to-- 261 00:13:46,625 --> 00:13:49,862 that's the topic for today. 262 00:13:49,862 --> 00:13:51,664 So the first question-- 263 00:13:51,664 --> 00:13:53,766 well, to jump to the punch line is 264 00:13:53,766 --> 00:13:55,634 there are these things called Miller indices. 265 00:13:55,634 --> 00:13:57,937 And that's what we're going to learn today. 266 00:13:57,937 --> 00:14:01,807 Miller indices and they describe direction and cuts, planes. 267 00:14:01,807 --> 00:14:05,845 OK, and the first question you can ask is, where are you? 268 00:14:05,845 --> 00:14:06,712 Where are you? 269 00:14:06,712 --> 00:14:13,319 Now it turns out that there are these people called 270 00:14:13,319 --> 00:14:14,987 crystallographers. 271 00:14:14,987 --> 00:14:18,324 And they came up with crystallography over the years. 272 00:14:18,324 --> 00:14:22,628 And they have very strong ideas about notation. 273 00:14:22,628 --> 00:14:25,130 So we'll be talking about crystallographer notation. 274 00:14:25,130 --> 00:14:28,200 And we'll be very careful not to do anything 275 00:14:28,200 --> 00:14:31,470 that would upset a crystallographer because that 276 00:14:31,470 --> 00:14:34,340 is not something you want to see. 277 00:14:34,340 --> 00:14:38,878 But the first question that you could ask is, where are you? 278 00:14:38,878 --> 00:14:42,114 So I'm going to draw the planes and we're going to-- the axes-- 279 00:14:42,114 --> 00:14:44,783 and we're always going to use the same notation, 280 00:14:44,783 --> 00:14:47,052 the same axis directions, just for simplicity. 281 00:14:47,052 --> 00:14:48,420 In this class, we're going to say 282 00:14:48,420 --> 00:14:50,623 that as we're talking about crystal structures, 283 00:14:50,623 --> 00:14:53,459 that's y, that's z, and that's x. 284 00:14:53,459 --> 00:14:55,628 x is coming out of the board. 285 00:14:55,628 --> 00:14:58,664 y is it going that way and z is going that way. 286 00:14:58,664 --> 00:15:04,603 Now, if I want to know where I am, then what I can do 287 00:15:04,603 --> 00:15:05,905 is draw the cube. 288 00:15:05,905 --> 00:15:08,240 Oh boy, here we go. 289 00:15:08,240 --> 00:15:11,277 OK, there it is. 290 00:15:11,277 --> 00:15:13,112 And I can just start looking at points. 291 00:15:13,112 --> 00:15:16,749 But instead of having all sorts of different numbers, 292 00:15:16,749 --> 00:15:20,419 crystallographers like to work with simple numbers. 293 00:15:20,419 --> 00:15:23,656 So what we do is we define the position 294 00:15:23,656 --> 00:15:29,728 as the fraction of the cell edge. 295 00:15:32,765 --> 00:15:36,101 OK, so what that means is if, you know, OK first of all, 296 00:15:36,101 --> 00:15:38,070 the origin is 0, 0, 0. 297 00:15:38,070 --> 00:15:39,238 So that's 0, 0, 0. 298 00:15:42,374 --> 00:15:45,577 But it also means that if I had this point here, 299 00:15:45,577 --> 00:15:48,681 that's going to be 1, 0, 0. 300 00:15:48,681 --> 00:15:50,950 Doesn't matter what a is. 301 00:15:50,950 --> 00:15:53,485 Remember, a is the lattice constant. 302 00:15:53,485 --> 00:15:57,389 It's the edge of the cube, which has a real value 303 00:15:57,389 --> 00:15:58,691 in different crystals. 304 00:15:58,691 --> 00:15:59,758 There's copper. 305 00:15:59,758 --> 00:16:03,963 There's a in copper, which we could compute. 306 00:16:03,963 --> 00:16:07,299 But no, when we specify positions in crystal, 307 00:16:07,299 --> 00:16:09,902 we do it as a fraction of that length. 308 00:16:09,902 --> 00:16:12,404 OK, so that-- so this point, for example, 309 00:16:12,404 --> 00:16:16,742 would be 1, 1, 1 out there. 310 00:16:16,742 --> 00:16:17,943 All right. 311 00:16:17,943 --> 00:16:19,244 And let's do one more. 312 00:16:19,244 --> 00:16:20,312 This point here. 313 00:16:20,312 --> 00:16:22,648 All right, well that's going to be-- gesundheit-- that's 314 00:16:22,648 --> 00:16:25,851 going to be over a half now. 315 00:16:25,851 --> 00:16:31,090 So it's a half and then 1, 0. 316 00:16:31,090 --> 00:16:34,159 It went over a half, over 1, up 0. 317 00:16:34,159 --> 00:16:35,227 So it's a 1/2. 318 00:16:35,227 --> 00:16:35,894 OK, good. 319 00:16:35,894 --> 00:16:36,996 This is just where we are. 320 00:16:36,996 --> 00:16:38,197 We're just getting warmed up. 321 00:16:38,197 --> 00:16:42,134 But really what we need to know is where we're going. 322 00:16:42,134 --> 00:16:44,703 And that goes from a point to a vector. 323 00:16:44,703 --> 00:16:49,540 OK, now again, there are some fairly straightforward rules 324 00:16:49,540 --> 00:16:53,445 that we follow not that we don't know what vectors are. 325 00:16:53,445 --> 00:16:54,580 We know what vectors are. 326 00:16:54,580 --> 00:16:57,950 The point is, do we know how to specify a vector in a way 327 00:16:57,950 --> 00:17:00,652 that a crystallographer would be happy with? 328 00:17:00,652 --> 00:17:02,254 That's what we have to learn. 329 00:17:02,254 --> 00:17:06,525 Because again, we don't want to make them upset. 330 00:17:06,525 --> 00:17:11,597 So there is a set of very simple rules that you can follow. 331 00:17:11,597 --> 00:17:13,098 And I've got this for the direction 332 00:17:13,098 --> 00:17:14,665 and I've got this for the planes. 333 00:17:14,665 --> 00:17:16,801 So here are the rules for the direction. 334 00:17:16,801 --> 00:17:18,503 Here's a vector in a crystal. 335 00:17:18,503 --> 00:17:21,540 Origin o and here are different vectors. 336 00:17:21,540 --> 00:17:23,409 And what we do is we, OK, we position 337 00:17:23,409 --> 00:17:24,510 it to start at the origin. 338 00:17:24,510 --> 00:17:26,744 And then we read off the projections 339 00:17:26,744 --> 00:17:29,348 in terms of the unit cell dimensions. 340 00:17:29,348 --> 00:17:32,384 a is x. 341 00:17:32,384 --> 00:17:34,586 b is y and c is z. 342 00:17:34,586 --> 00:17:37,523 Now, we know that those of the same length in a cubic crystal. 343 00:17:40,459 --> 00:17:42,628 Adjust to the smallest integer values, 344 00:17:42,628 --> 00:17:43,695 that's really important. 345 00:17:43,695 --> 00:17:45,397 And enclose in square brackets. 346 00:17:45,397 --> 00:17:47,433 No commas. 347 00:17:47,433 --> 00:17:49,568 So let's do a few. 348 00:17:49,568 --> 00:17:59,645 So oa, here are the xyz read out. 349 00:17:59,645 --> 00:18:02,915 Well you know, this is 1. 350 00:18:02,915 --> 00:18:04,550 This is 0. 351 00:18:04,550 --> 00:18:05,884 This is 0. 352 00:18:05,884 --> 00:18:11,123 So this would give us this as the notation for the oa vector. 353 00:18:11,123 --> 00:18:13,325 You see that there, right? 354 00:18:13,325 --> 00:18:16,061 Notice I've already done step two. 355 00:18:16,061 --> 00:18:17,629 So the second bullet there, read off 356 00:18:17,629 --> 00:18:19,631 projections in terms of unit cell dimensions, 357 00:18:19,631 --> 00:18:22,634 a, b, and c, because it went all the way to the edge. 358 00:18:22,634 --> 00:18:24,136 So it's a 1. 359 00:18:24,136 --> 00:18:25,170 It didn't go halfway. 360 00:18:25,170 --> 00:18:26,505 It's not 1/2. 361 00:18:26,505 --> 00:18:27,506 OK, good. 362 00:18:27,506 --> 00:18:36,615 So ob, OK, well that's going to be 1 along x, 1 along y, and 0. 363 00:18:36,615 --> 00:18:40,686 So that direction would be written as the 110. 364 00:18:40,686 --> 00:18:45,457 And oc would be 111. 365 00:18:45,457 --> 00:18:48,861 This is feeling kind of boring. 366 00:18:48,861 --> 00:18:50,395 Oh, gesundheit. 367 00:18:50,395 --> 00:18:52,498 But it gets exciting in just a sec. 368 00:18:52,498 --> 00:19:01,039 Because all we're doing here is we're going OK, 100, 110, 111. 369 00:19:01,039 --> 00:19:03,642 OK, good. 370 00:19:03,642 --> 00:19:06,145 But now we go to od. 371 00:19:06,145 --> 00:19:10,349 Now, od is one of these cases where we have to be careful. 372 00:19:10,349 --> 00:19:12,551 So od, you see that? 373 00:19:12,551 --> 00:19:14,887 It's going from here to there. 374 00:19:14,887 --> 00:19:18,090 So along the x direction, it's still 375 00:19:18,090 --> 00:19:20,392 going a full one of the edge. 376 00:19:20,392 --> 00:19:22,728 Along the y direction, it's 0. 377 00:19:22,728 --> 00:19:25,764 But along the z direction, it's 1/2. 378 00:19:25,764 --> 00:19:29,434 And so because crystallographers don't like fractions 379 00:19:29,434 --> 00:19:33,071 when they talk about directions, we have to scale that up. 380 00:19:33,071 --> 00:19:34,373 And so we just get rid of it. 381 00:19:34,373 --> 00:19:36,074 We multiply everything by 2. 382 00:19:36,074 --> 00:19:37,242 We put it in brackets. 383 00:19:37,242 --> 00:19:39,478 And everyone is happy. 384 00:19:39,478 --> 00:19:41,914 201. 385 00:19:41,914 --> 00:19:44,550 That is the 201 direction. 386 00:19:44,550 --> 00:19:46,351 In this crystal. 387 00:19:46,351 --> 00:19:50,389 That is the 201 direction. 388 00:19:50,389 --> 00:19:52,558 They don't like fractions. 389 00:19:52,558 --> 00:19:55,227 Now, they don't like negative signs either. 390 00:19:55,227 --> 00:19:56,962 They don't like negative signs. 391 00:19:56,962 --> 00:19:59,164 But so if I were doing-- 392 00:19:59,164 --> 00:20:01,833 if I were doing oe, OK, that's just 393 00:20:01,833 --> 00:20:03,869 going in the other direction. 394 00:20:03,869 --> 00:20:09,174 So oe would be-- 395 00:20:09,174 --> 00:20:11,109 let's see-- minus-- 396 00:20:11,109 --> 00:20:14,179 OK, well if I just jumped to what 397 00:20:14,179 --> 00:20:16,148 we think it might be given what we just saw, 398 00:20:16,148 --> 00:20:17,149 it would look like that. 399 00:20:17,149 --> 00:20:20,385 But crystallographers don't like having negative, you 400 00:20:20,385 --> 00:20:22,788 know, these minuses inside of their brackets. 401 00:20:22,788 --> 00:20:25,057 So we write a bar. 402 00:20:25,057 --> 00:20:26,191 So we write it like this. 403 00:20:26,191 --> 00:20:28,860 0, 1 bar, 0. 404 00:20:28,860 --> 00:20:30,862 And that is the notation for going 405 00:20:30,862 --> 00:20:32,598 in the negative direction. 406 00:20:32,598 --> 00:20:36,368 OK, 01bar0. 407 00:20:36,368 --> 00:20:37,269 That would be oe. 408 00:20:37,269 --> 00:20:41,273 Did I get that right? oe, Yeah. 409 00:20:41,273 --> 00:20:45,244 And you can see that of, if you have the same kind of fun, 410 00:20:45,244 --> 00:20:51,416 you would see that of is the 112 direction, that that's 411 00:20:51,416 --> 00:20:53,885 how you would write the of vector right 412 00:20:53,885 --> 00:21:00,425 because you got 1/2, 1/2 and 1, but they don't like the 1/2. 413 00:21:00,425 --> 00:21:03,328 So you multiply through by 2, you get the 112. 414 00:21:03,328 --> 00:21:05,297 These are vectors in crystals. 415 00:21:05,297 --> 00:21:10,369 Now, there are equivalences here because this 416 00:21:10,369 --> 00:21:13,538 is a cubic lattice. 417 00:21:13,538 --> 00:21:17,242 So if I look at this and say, well, OK, if I went this way 418 00:21:17,242 --> 00:21:21,546 and I went that way, where I wind up are equivalent points. 419 00:21:21,546 --> 00:21:25,117 Right, where I wind up is, you know, 420 00:21:25,117 --> 00:21:28,053 you're winding up at the same place in a way in the crystal 421 00:21:28,053 --> 00:21:30,656 because the definition of that stamp in a cubic lattice 422 00:21:30,656 --> 00:21:32,424 is that those are all equivalent. 423 00:21:32,424 --> 00:21:35,894 Doesn't matter what's on your basis. 424 00:21:35,894 --> 00:21:38,930 Because it's defined by the Bravais lattice. 425 00:21:38,930 --> 00:21:40,766 That's to stamp. 426 00:21:40,766 --> 00:21:44,202 So we have a way of writing that, too. 427 00:21:44,202 --> 00:21:45,437 So the 101. 428 00:21:45,437 --> 00:21:50,042 That's the 101 direction and the 110 direction, cool. 429 00:21:50,042 --> 00:21:52,377 If I make it a little simpler, you know, 430 00:21:52,377 --> 00:22:00,952 I can say that the 100 and the 010 and the 001, oh, 431 00:22:00,952 --> 00:22:02,387 let's do them all. 432 00:22:02,387 --> 00:22:11,863 And the 1bar00 and the 01bar0, and the 001bar all get me 433 00:22:11,863 --> 00:22:13,432 to equivalent places. 434 00:22:13,432 --> 00:22:17,102 And we can call this a family, just because we don't 435 00:22:17,102 --> 00:22:18,670 want to keep writing them. 436 00:22:18,670 --> 00:22:20,806 If they're all equivalent, we can say 437 00:22:20,806 --> 00:22:24,409 this is a family of directions. 438 00:22:24,409 --> 00:22:26,411 And you've got to be careful here 439 00:22:26,411 --> 00:22:29,514 because you can't use a bracket anymore because it would just 440 00:22:29,514 --> 00:22:30,782 be one of the directions. 441 00:22:30,782 --> 00:22:35,420 Instead, we use-- sorry, you can't use the square brackets. 442 00:22:35,420 --> 00:22:37,022 Instead, we use this kind of bracket 443 00:22:37,022 --> 00:22:40,892 and we say it's the 100 family. 444 00:22:40,892 --> 00:22:44,896 So if you see a direction written with brackets 445 00:22:44,896 --> 00:22:51,303 like that, then it means all of these directions, 446 00:22:51,303 --> 00:22:54,139 this family of directions. 447 00:22:54,139 --> 00:22:56,875 But if you see it written with a bracket like this, 448 00:22:56,875 --> 00:22:58,243 it means that one vector. 449 00:23:01,847 --> 00:23:04,783 Crystallographers, you've got to keep them happy. 450 00:23:04,783 --> 00:23:06,418 You've got to keep them happy. 451 00:23:06,418 --> 00:23:10,689 All right, now, that is directions. 452 00:23:10,689 --> 00:23:12,858 That is direction. 453 00:23:12,858 --> 00:23:14,826 But what about cuts? 454 00:23:14,826 --> 00:23:16,528 What about planes? 455 00:23:16,528 --> 00:23:18,930 So we've got our vectors, now we need our cuts. 456 00:23:18,930 --> 00:23:20,999 We've got to be able to specify these things. 457 00:23:20,999 --> 00:23:25,837 And you know, as we'll see, as we'll see, 458 00:23:25,837 --> 00:23:31,810 the properties of materials, of crystals, especially 459 00:23:31,810 --> 00:23:37,883 the more anisotropic they are, depend very heavily 460 00:23:37,883 --> 00:23:41,086 on what's in what direction. 461 00:23:41,086 --> 00:23:44,623 So for example, if I have a plane in a crystal that 462 00:23:44,623 --> 00:23:47,025 doesn't have a high density of atoms, 463 00:23:47,025 --> 00:23:48,493 and then I got another plane that's 464 00:23:48,493 --> 00:23:50,862 got a lot of density of atoms in that plane, 465 00:23:50,862 --> 00:23:54,499 well you might expect those to have different responses 466 00:23:54,499 --> 00:23:57,969 to, for example, mechanical stress, strain. 467 00:23:57,969 --> 00:23:58,570 You push it. 468 00:23:58,570 --> 00:23:59,271 You pull it. 469 00:23:59,271 --> 00:23:59,805 You break it. 470 00:23:59,805 --> 00:24:01,540 You hammer it. 471 00:24:01,540 --> 00:24:03,542 Maybe one of those planes, if it doesn't 472 00:24:03,542 --> 00:24:08,413 have as many bonds in it, might break first. 473 00:24:08,413 --> 00:24:09,481 This is just one example. 474 00:24:09,481 --> 00:24:11,917 So we've got to know how to talk about these things. 475 00:24:11,917 --> 00:24:14,719 We've got to know how to talk about these things. 476 00:24:14,719 --> 00:24:17,722 So now with planes, I have another recipe 477 00:24:17,722 --> 00:24:22,294 to make crystallographers happy. 478 00:24:22,294 --> 00:24:25,197 Now first read off-- 479 00:24:25,197 --> 00:24:27,365 oh, well let's put an example up. 480 00:24:27,365 --> 00:24:28,800 So I've got an example. 481 00:24:28,800 --> 00:24:29,968 I'll do a few. 482 00:24:29,968 --> 00:24:33,972 And then as always, you guys need to do more to practice. 483 00:24:33,972 --> 00:24:34,840 So I'll do a few. 484 00:24:34,840 --> 00:24:36,675 So I'm going to draw a plane and I'm 485 00:24:36,675 --> 00:24:38,710 going to make it a nice simple one. 486 00:24:38,710 --> 00:24:44,149 These are my axes, x, y, z. 487 00:24:44,149 --> 00:24:47,619 And I'm going to draw my cubic crystal-- 488 00:24:47,619 --> 00:24:49,721 my cubic unit cell, sorry. 489 00:24:49,721 --> 00:24:52,090 OK, here we go. 490 00:24:52,090 --> 00:24:54,493 Almost a cube. 491 00:24:54,493 --> 00:24:57,996 OK, now I want to take a plane here. 492 00:24:57,996 --> 00:24:59,531 I'm going to take this top one. 493 00:24:59,531 --> 00:25:02,701 So I'm going to take this and I'm going to go through those-- 494 00:25:02,701 --> 00:25:04,569 I want to know, how do I describe 495 00:25:04,569 --> 00:25:06,805 that plane that cuts that way? 496 00:25:06,805 --> 00:25:13,912 OK, one, these correspond to those, crystal plane algorithm. 497 00:25:17,015 --> 00:25:21,286 I hear a few of you like algorithms and computing 498 00:25:21,286 --> 00:25:21,920 in course six. 499 00:25:25,323 --> 00:25:27,692 We got one whistle. 500 00:25:27,692 --> 00:25:34,032 So here's the thing, if I have a plane like this and I ask you, 501 00:25:34,032 --> 00:25:36,268 where does it intercept? 502 00:25:36,268 --> 00:25:38,703 Where does it intercept the axis? 503 00:25:38,703 --> 00:25:44,876 Right, well, it's never going to intercept the x and y axes, 504 00:25:44,876 --> 00:25:45,377 never. 505 00:25:48,013 --> 00:25:52,717 It's never going to intercept them, no matter what I do. 506 00:25:52,717 --> 00:25:55,387 But you say, but what if I put it in the xy plane, then 507 00:25:55,387 --> 00:25:57,255 it's always intercept-- no, that's 508 00:25:57,255 --> 00:26:00,392 not what we mean by intercept. 509 00:26:00,392 --> 00:26:03,461 What I mean by intercept is, it has to reach them eventually 510 00:26:03,461 --> 00:26:07,232 no matter where it is, no matter where I position it. 511 00:26:07,232 --> 00:26:10,068 But this never does. 512 00:26:10,068 --> 00:26:11,670 This never does. 513 00:26:11,670 --> 00:26:18,176 And so the intercepts, you know, for the ABC intercepts, 514 00:26:18,176 --> 00:26:20,111 so we can write those here. 515 00:26:20,111 --> 00:26:25,450 OK, I'll write the same way, x, y, z intercepts, it's infinite. 516 00:26:25,450 --> 00:26:27,919 And it's infinite. 517 00:26:27,919 --> 00:26:31,590 All right, and then for the z, OK here, 518 00:26:31,590 --> 00:26:34,726 I'm intercepting at this part of z. 519 00:26:34,726 --> 00:26:36,928 So I'm intercepting at 1. 520 00:26:36,928 --> 00:26:39,497 But the thing is, oh man, crystallographers 521 00:26:39,497 --> 00:26:41,533 don't like infinity either. 522 00:26:41,533 --> 00:26:42,968 They don't. 523 00:26:42,968 --> 00:26:45,637 And so what they do is they just take 1 over. 524 00:26:45,637 --> 00:26:48,139 That's a nice way to fix infinity. 525 00:26:48,139 --> 00:26:55,046 And so step two, this would become 00 and 1 over 1 is 1. 526 00:26:55,046 --> 00:26:59,250 Right, 1 over infinity is 0. 527 00:26:59,250 --> 00:27:03,188 Now we reduce to integer values and there's 528 00:27:03,188 --> 00:27:05,323 no work to do there. 529 00:27:05,323 --> 00:27:09,494 And then we are very careful about this. 530 00:27:09,494 --> 00:27:11,429 We use parentheses. 531 00:27:11,429 --> 00:27:14,399 No more brackets, parentheses. 532 00:27:14,399 --> 00:27:18,670 Do not get this wrong because you might be talking 533 00:27:18,670 --> 00:27:20,839 about a vector by accident. 534 00:27:20,839 --> 00:27:22,907 You don't want to talk about a vector by accident. 535 00:27:22,907 --> 00:27:27,545 So this is the 001 plane, no commas. 536 00:27:27,545 --> 00:27:31,116 Remember the dislike of commas. 537 00:27:31,116 --> 00:27:34,753 It's the 001 plane in this crystal. 538 00:27:34,753 --> 00:27:36,354 That is a cut. 539 00:27:36,354 --> 00:27:37,822 OK, good. 540 00:27:37,822 --> 00:27:39,224 And that's called a Miller plane. 541 00:27:39,224 --> 00:27:44,729 Well, if I do another example, do one more, 542 00:27:44,729 --> 00:27:47,265 so that is this one. 543 00:27:47,265 --> 00:27:48,299 I have it. 544 00:27:48,299 --> 00:27:50,502 I've got a bunch of examples here. 545 00:27:50,502 --> 00:27:53,204 And they can draw it, you know, better with the shading 546 00:27:53,204 --> 00:27:53,872 and stuff there. 547 00:27:53,872 --> 00:27:55,407 But you see, that's what we just did. 548 00:27:55,407 --> 00:27:58,243 We did the 001 plane, that one there. 549 00:27:58,243 --> 00:28:01,079 All right, there it is, nice and big. 550 00:28:01,079 --> 00:28:02,947 The 001 plane. 551 00:28:02,947 --> 00:28:04,049 What about that one? 552 00:28:04,049 --> 00:28:07,652 That's the 110. 553 00:28:07,652 --> 00:28:12,957 The 110, if you do the 110, 110, I'm 554 00:28:12,957 --> 00:28:14,392 not going to try to draw it again, 555 00:28:14,392 --> 00:28:19,964 what you get is step one, you get 11infinity. 556 00:28:19,964 --> 00:28:23,101 Step two, you get 110. 557 00:28:23,101 --> 00:28:25,603 Step three, you get 110. 558 00:28:25,603 --> 00:28:28,373 and step four, you get 110. 559 00:28:28,373 --> 00:28:32,110 Those are the steps to identifying and labeling 560 00:28:32,110 --> 00:28:34,279 a crystal plane. 561 00:28:34,279 --> 00:28:36,147 Let's make it a little more complicated. 562 00:28:36,147 --> 00:28:39,350 We still haven't really done something 563 00:28:39,350 --> 00:28:43,655 where we take advantage of the rule of step three. 564 00:28:43,655 --> 00:28:45,156 So let's make it harder. 565 00:28:45,156 --> 00:28:46,324 Let's bump it up. 566 00:28:46,324 --> 00:28:47,926 So there's a slice. 567 00:28:47,926 --> 00:28:52,464 Remember, I can slice a crystal in any way I want. 568 00:28:52,464 --> 00:28:56,134 And I got to be able to write it down. 569 00:28:56,134 --> 00:28:58,536 I got to be able to represent it on paper. 570 00:28:58,536 --> 00:28:59,604 That's what this is about. 571 00:28:59,604 --> 00:29:02,040 That's what the Miller indices and the Miller planes 572 00:29:02,040 --> 00:29:04,409 let us do. 573 00:29:04,409 --> 00:29:05,944 So this one I'm going to try to draw. 574 00:29:05,944 --> 00:29:08,747 So here, I've got my axes. 575 00:29:08,747 --> 00:29:10,548 And that's y. 576 00:29:10,548 --> 00:29:11,382 And that's x. 577 00:29:11,382 --> 00:29:12,817 And that's z. 578 00:29:12,817 --> 00:29:13,485 Here we go. 579 00:29:19,424 --> 00:29:21,092 It's almost a cube. 580 00:29:21,092 --> 00:29:25,964 Now, I've gone up here to 3/4 high. 581 00:29:25,964 --> 00:29:27,732 So I've gone up about 3/4 there. 582 00:29:27,732 --> 00:29:30,201 I've gone here to 1/2. 583 00:29:30,201 --> 00:29:33,238 That's 1/2 of the cell length. 584 00:29:33,238 --> 00:29:36,174 This is 3/4 of the cell length. 585 00:29:36,174 --> 00:29:40,044 And then over, I've gone all the way to 1. 586 00:29:40,044 --> 00:29:44,949 So now if I go through my steps on this one, I've got a 1/2, 587 00:29:44,949 --> 00:29:48,953 and I've got 1, and I've got 3/4. 588 00:29:48,953 --> 00:29:52,056 OK, I'm going to take the reciprocals. 589 00:29:52,056 --> 00:29:56,594 OK, that becomes 21 4/3. 590 00:29:56,594 --> 00:29:59,864 Oh, not happy yet. 591 00:29:59,864 --> 00:30:01,166 Not happy yet. 592 00:30:01,166 --> 00:30:02,200 We don't like fractions. 593 00:30:02,200 --> 00:30:03,501 We got to get rid of fractions. 594 00:30:03,501 --> 00:30:07,539 So in step three, we simply multiply through by 3. 595 00:30:07,539 --> 00:30:11,709 And we get 634. 596 00:30:11,709 --> 00:30:13,678 And then we can write down that point. 597 00:30:13,678 --> 00:30:15,346 That is 634 plane. 598 00:30:19,717 --> 00:30:21,719 And so on and so on. 599 00:30:21,719 --> 00:30:24,689 Now, let's go back to-- 600 00:30:24,689 --> 00:30:25,423 let me erase this. 601 00:30:25,423 --> 00:30:28,860 There's a couple of things about the planes 602 00:30:28,860 --> 00:30:31,963 that once we know what the plane is, 603 00:30:31,963 --> 00:30:35,066 we know some other things about it. 604 00:30:35,066 --> 00:30:39,704 So let's talk about that now. 605 00:30:39,704 --> 00:30:41,806 All right, if I-- 606 00:30:44,475 --> 00:30:50,315 first of all, a Miller plane is not just one plane. 607 00:30:50,315 --> 00:30:53,051 A Miller plane is not just one plane. 608 00:30:53,051 --> 00:30:56,588 A Miller plane is an infinite set of planes. 609 00:30:56,588 --> 00:31:08,266 OK so when I say a Miller plane, well, very often what you do 610 00:31:08,266 --> 00:31:11,569 is what we've been doing, which is 611 00:31:11,569 --> 00:31:15,173 you define the plane that's kind of closest, 612 00:31:15,173 --> 00:31:16,975 somehow, to the origin. 613 00:31:16,975 --> 00:31:18,810 In the unit cell, the one that cuts it. 614 00:31:18,810 --> 00:31:21,012 You define it that way. 615 00:31:21,012 --> 00:31:23,281 But actually, what Miller planes are 616 00:31:23,281 --> 00:31:26,050 are infinite sets of planes. 617 00:31:26,050 --> 00:31:28,119 And so I'm going to put-- 618 00:31:28,119 --> 00:31:31,856 Now, I'm not a crystallographer so I can write infinity. 619 00:31:31,856 --> 00:31:32,991 I'm OK with that. 620 00:31:32,991 --> 00:31:37,028 It's an infinite set of planes. 621 00:31:37,028 --> 00:31:39,831 OK. 622 00:31:39,831 --> 00:31:49,240 They are equally spaced and here's the key, 623 00:31:49,240 --> 00:31:51,142 one should always be going through the origin. 624 00:31:51,142 --> 00:31:53,845 If you want to look at it as a set of planes, 625 00:31:53,845 --> 00:31:55,413 you've got to put one at the origin. 626 00:31:55,413 --> 00:31:59,517 That will help you think through what I mean. 627 00:31:59,517 --> 00:32:07,659 OK, so if one goes through the origin, 628 00:32:07,659 --> 00:32:15,700 then what you see right away is how they repeat. 629 00:32:15,700 --> 00:32:20,905 Because you know, it's not just that I go-- in that case, 630 00:32:20,905 --> 00:32:24,242 let's take that one, the 001. 631 00:32:24,242 --> 00:32:28,579 Now you say, where's the next 1 in this infinite set of planes? 632 00:32:28,579 --> 00:32:32,283 OK, well, put that one at the origin and make another one. 633 00:32:32,283 --> 00:32:34,719 Then put that one at the origin again and make another one. 634 00:32:34,719 --> 00:32:38,089 That's how to think about this infinite set of planes. 635 00:32:38,089 --> 00:32:39,557 OK? 636 00:32:39,557 --> 00:32:43,995 So if I had one that went half way, 637 00:32:43,995 --> 00:32:50,001 a 200 plane or an 002 plane, as it would be if it's properly 638 00:32:50,001 --> 00:32:54,305 noted with Miller indices, well then you say, well, OK, 639 00:32:54,305 --> 00:32:56,474 I've got one half way. 640 00:32:56,474 --> 00:32:58,009 You know, isn't that it? 641 00:32:58,009 --> 00:32:59,077 No. 642 00:32:59,077 --> 00:33:02,680 The set of infinite planes also would have one 643 00:33:02,680 --> 00:33:05,183 at the origin, one half way, one at the top. 644 00:33:05,183 --> 00:33:07,986 Because each time you put it back and you do your next one. 645 00:33:07,986 --> 00:33:13,758 That's how to think about this infinite set of Miller planes. 646 00:33:13,758 --> 00:33:19,998 So you get this infinite set of planes that have-- 647 00:33:19,998 --> 00:33:22,467 I'm going to go to this picture. 648 00:33:22,467 --> 00:33:26,371 There it is, a 100, that just go on forever. 649 00:33:26,371 --> 00:33:28,172 Those are the Miller planes. 650 00:33:28,172 --> 00:33:30,975 And what they have is a spacing between them. 651 00:33:30,975 --> 00:33:33,111 And this is really important. 652 00:33:33,111 --> 00:33:35,313 So this is the next thing I want to tell you. 653 00:33:35,313 --> 00:33:38,016 All right, so I've got-- so a Miller plane isn't just one, 654 00:33:38,016 --> 00:33:39,150 it's infinite. 655 00:33:39,150 --> 00:33:41,319 It's an infinite number of them. 656 00:33:41,319 --> 00:33:46,657 And what that means is that when I give you-- 657 00:33:46,657 --> 00:33:55,033 when I give you this, right, or that, or any Miller plane, 658 00:33:55,033 --> 00:33:58,102 I know what the spacing is between them. 659 00:33:58,102 --> 00:34:00,505 That is enough information. 660 00:34:00,505 --> 00:34:16,687 Because the distance between planes is-- 661 00:34:16,687 --> 00:34:18,856 actually, here it's very easy to see. 662 00:34:18,856 --> 00:34:22,927 The distance between these planes is a. 663 00:34:22,927 --> 00:34:29,033 But the difference between the 200 planes is a over 2. 664 00:34:29,033 --> 00:34:31,601 And in fact, for a cubic cell, there's 665 00:34:31,601 --> 00:34:34,672 a formula that you can use for any two planes. 666 00:34:34,672 --> 00:34:38,643 The distance between, well, let's see. 667 00:34:38,643 --> 00:34:40,110 I'll write the notation first. 668 00:34:40,110 --> 00:34:46,717 The distance and the plane there is equal to a. 669 00:34:46,717 --> 00:34:49,920 And the distance of a 200 plane between planes 670 00:34:49,920 --> 00:34:51,722 is equal to a over 2. 671 00:34:51,722 --> 00:34:57,027 But there's a general formula of any HKL. 672 00:34:57,027 --> 00:34:59,797 HKL, the Miller indices. 673 00:34:59,797 --> 00:35:01,199 Those are just variables. 674 00:35:01,199 --> 00:35:04,702 But if those are the variables of the Miller plane, 675 00:35:04,702 --> 00:35:07,038 then the distance between two of them 676 00:35:07,038 --> 00:35:11,642 is the length of the edge of the unit cell, a, 677 00:35:11,642 --> 00:35:15,780 divided by the square root of the squares added together, 678 00:35:15,780 --> 00:35:22,553 a squared plus k squared plus l squared. 679 00:35:22,553 --> 00:35:26,324 So you see, OK, if it's 100, that a. 680 00:35:26,324 --> 00:35:27,091 That works. 681 00:35:27,091 --> 00:35:29,927 If it's 200, that works. 682 00:35:29,927 --> 00:35:31,729 And it's general. 683 00:35:31,729 --> 00:35:34,699 So now, just by knowing, but because I figured out 684 00:35:34,699 --> 00:35:37,869 how to write a plane in HKL notation, 685 00:35:37,869 --> 00:35:42,073 I also know the distance between these infinite planes. 686 00:35:42,073 --> 00:35:44,175 That becomes very, very important 687 00:35:44,175 --> 00:35:47,078 once we shine x-rays on crystals next week. 688 00:35:47,078 --> 00:35:48,846 That becomes very important. 689 00:35:48,846 --> 00:35:50,081 OK? 690 00:35:50,081 --> 00:35:55,119 OK, so what else can we know from the Miller planes? 691 00:35:55,119 --> 00:35:58,856 OK, so that's the distance between them. 692 00:35:58,856 --> 00:36:02,460 Now the other thing that actually becomes fairly 693 00:36:02,460 --> 00:36:05,096 self-evident when you play around with this a little bit, 694 00:36:05,096 --> 00:36:07,832 and that was on the slide before, is that-- 695 00:36:13,237 --> 00:36:14,472 let's see. 696 00:36:14,472 --> 00:36:38,829 The plane and direction with the same Miller indices 697 00:36:38,829 --> 00:36:40,665 are orthogonal. 698 00:36:47,338 --> 00:36:50,942 And you can see this from this very simple picture. 699 00:36:50,942 --> 00:36:53,744 There's the 001 plane. 700 00:36:53,744 --> 00:36:58,983 But that's also the 001 vector, just as we defined it today. 701 00:36:58,983 --> 00:37:02,887 So the 001 plane and the 001 vector are orthogonal. 702 00:37:02,887 --> 00:37:07,992 So this is an example, the 001 plane is orthogonal to, oh, 703 00:37:07,992 --> 00:37:11,629 now gotta get this right, the 001 vector. 704 00:37:11,629 --> 00:37:14,298 Oh, no commas. 705 00:37:14,298 --> 00:37:19,270 Brackets, parentheses, no negatives, no fractions, 706 00:37:19,270 --> 00:37:21,105 no infinities. 707 00:37:21,105 --> 00:37:23,374 I followed the rules. 708 00:37:23,374 --> 00:37:25,343 And because of that, these indices 709 00:37:25,343 --> 00:37:30,848 have meaning as I'm showing you, because of that. 710 00:37:30,848 --> 00:37:32,250 Now, you can play around with this 711 00:37:32,250 --> 00:37:34,418 but this can also be fairly straightforwardly 712 00:37:34,418 --> 00:37:36,320 derived, things like this. 713 00:37:36,320 --> 00:37:38,389 But you don't need to know that derivation 714 00:37:38,389 --> 00:37:40,791 so I'm just telling it to you. 715 00:37:40,791 --> 00:37:44,262 OK, so these are the kinds of things that you 716 00:37:44,262 --> 00:37:47,698 can do with Miller indices. 717 00:37:47,698 --> 00:37:48,799 As I said in the beginning. 718 00:37:48,799 --> 00:37:51,002 One of the things that you would want 719 00:37:51,002 --> 00:37:54,071 to know what planes you're talking about 720 00:37:54,071 --> 00:37:57,275 is because you want to know what the packing in that plane is. 721 00:37:57,275 --> 00:38:01,612 Just like we talked about packing, right-- 722 00:38:01,612 --> 00:38:05,016 well, we talked about it before on Monday, the atomic packing 723 00:38:05,016 --> 00:38:09,720 fraction, there it is, the max packing fraction in a volume. 724 00:38:09,720 --> 00:38:13,691 We also want to know, especially if a crystal is anisotropy, 725 00:38:13,691 --> 00:38:17,628 we want to know about packings in planes. 726 00:38:17,628 --> 00:38:22,166 And so, you know, if you take a very simple example 727 00:38:22,166 --> 00:38:26,570 like a simple cubic, there's a simple cubic lattice. 728 00:38:26,570 --> 00:38:28,506 And now we're all good with our notation. 729 00:38:28,506 --> 00:38:29,774 There's a simple cubic crystal. 730 00:38:32,376 --> 00:38:33,611 The basis is one atom. 731 00:38:33,611 --> 00:38:35,079 I don't have anything else in here. 732 00:38:35,079 --> 00:38:36,247 Those are the atoms. 733 00:38:36,247 --> 00:38:39,417 It must be polonium. 734 00:38:39,417 --> 00:38:42,520 And there's the 100 plane, the 110 plane, 735 00:38:42,520 --> 00:38:48,426 and the 111 plane, 111 plane, 110 plane, 100 plane. 736 00:38:48,426 --> 00:38:53,030 Now as you can see, the density of atoms in these planes 737 00:38:53,030 --> 00:38:54,198 is different. 738 00:38:54,198 --> 00:38:56,267 The density of atoms in these planes is different. 739 00:38:56,267 --> 00:39:02,239 And that has a real serious impact on the properties. 740 00:39:02,239 --> 00:39:04,742 So if we look at this-- 741 00:39:04,742 --> 00:39:06,677 so I'll do a couple of quick examples. 742 00:39:06,677 --> 00:39:10,081 If we look at this example of simple cubic, 743 00:39:10,081 --> 00:39:13,117 that's a very easy case. 744 00:39:13,117 --> 00:39:22,026 And so for a simple cubic with a lattice-- 745 00:39:22,026 --> 00:39:25,229 this a is called the lattice constant. 746 00:39:25,229 --> 00:39:26,864 I've mentioned it a few times. 747 00:39:26,864 --> 00:39:28,699 Lattice constant a. 748 00:39:28,699 --> 00:39:33,571 Then you know, if we look at the 100 plane, 749 00:39:33,571 --> 00:39:38,309 then the area is a squared. 750 00:39:38,309 --> 00:39:40,511 The area of that plane is a squared. 751 00:39:40,511 --> 00:39:44,482 But the question is, how many atoms are in that a squared. 752 00:39:44,482 --> 00:39:47,284 Well, you do the same thing that we did before. 753 00:39:47,284 --> 00:39:49,820 So I'm looking at a plane here. 754 00:39:49,820 --> 00:39:52,990 And I'm trying to figure out how many atoms are in this-- well, 755 00:39:52,990 --> 00:39:56,660 you know this is being shared in this whole big plane that 756 00:39:56,660 --> 00:39:57,395 goes on and on. 757 00:39:57,395 --> 00:40:00,998 It's being shared by four other squares. 758 00:40:00,998 --> 00:40:03,167 And so there's a fourth of each atom in there. 759 00:40:03,167 --> 00:40:06,604 So there's one atom, effectively, in that plane. 760 00:40:06,604 --> 00:40:10,107 Again, this is in one of the simplest cases. 761 00:40:10,107 --> 00:40:19,417 But the number atoms equal to 1. 762 00:40:22,153 --> 00:40:28,325 And so the density of atoms per area 763 00:40:28,325 --> 00:40:31,462 is equal to 1 over a squared. 764 00:40:31,462 --> 00:40:33,597 That's a pretty easy case. 765 00:40:33,597 --> 00:40:38,169 If I did the 110 direction, it's a little bit different 766 00:40:38,169 --> 00:40:39,970 because the area is now different. 767 00:40:39,970 --> 00:40:48,446 So the area in this case is root 2 a squared. 768 00:40:48,446 --> 00:40:49,547 You see that? 769 00:40:49,547 --> 00:40:53,818 The area in that plane is different. 770 00:40:53,818 --> 00:40:58,055 So the density of atoms in the plane is different. 771 00:41:01,292 --> 00:41:04,228 So the density in the plane affects properties. 772 00:41:04,228 --> 00:41:09,133 And I want to give you just a quick example of that 773 00:41:09,133 --> 00:41:10,034 and why this matters. 774 00:41:10,034 --> 00:41:12,736 Because in fact, this is one of the things, 775 00:41:12,736 --> 00:41:16,073 all the way back to Hook and his cannonballs, 776 00:41:16,073 --> 00:41:19,910 and even way before that, people noticed this stuff. 777 00:41:19,910 --> 00:41:22,480 I broke something, but it always seems 778 00:41:22,480 --> 00:41:25,049 to slice in a certain way. 779 00:41:25,049 --> 00:41:25,649 Why is that? 780 00:41:25,649 --> 00:41:29,687 Why would something always break in the same way every time? 781 00:41:29,687 --> 00:41:31,522 With the same kind of-- 782 00:41:31,522 --> 00:41:33,190 what does way mean here? 783 00:41:33,190 --> 00:41:35,259 It means like the angles that form. 784 00:41:35,259 --> 00:41:39,129 So why are those angles always kind of the same? 785 00:41:39,129 --> 00:41:41,332 Well, it has to do with the strength, 786 00:41:41,332 --> 00:41:44,168 the relative strength, of the planes 787 00:41:44,168 --> 00:41:46,704 inside of these crystals. 788 00:41:46,704 --> 00:41:49,073 And in fact, here's a beautiful piece 789 00:41:49,073 --> 00:41:51,242 of work on stretching a wire. 790 00:41:51,242 --> 00:41:55,446 All right, so here you're taking a wire that is crystalline 791 00:41:55,446 --> 00:41:56,947 and you're pulling it. 792 00:41:56,947 --> 00:41:58,816 And you can see how it breaks. 793 00:41:58,816 --> 00:42:00,484 You can actually see how it breaks. 794 00:42:00,484 --> 00:42:03,521 It's breaking along the weakest plane. 795 00:42:03,521 --> 00:42:05,122 It's breaking along the weakest plane. 796 00:42:05,122 --> 00:42:07,424 So if you want to think, then, about well, OK, 797 00:42:07,424 --> 00:42:11,161 how do I know how to make this stronger, for example? 798 00:42:11,161 --> 00:42:13,230 So it stops breaking-- you know, the weakest link 799 00:42:13,230 --> 00:42:15,533 is the weakest link. 800 00:42:15,533 --> 00:42:16,133 That was deep. 801 00:42:19,837 --> 00:42:21,805 So then that's the plane of the crystal 802 00:42:21,805 --> 00:42:22,940 that I need to think about. 803 00:42:22,940 --> 00:42:27,945 How are bonding together or packing in in that plane? 804 00:42:27,945 --> 00:42:29,013 This is real stuff. 805 00:42:29,013 --> 00:42:32,716 This comes about in many of the properties, not just 806 00:42:32,716 --> 00:42:38,022 the fracture, but in many of the properties of these materials. 807 00:42:38,022 --> 00:42:42,359 The property, as I said before, depends heavily on which plane 808 00:42:42,359 --> 00:42:43,193 you're in. 809 00:42:43,193 --> 00:42:45,663 And we'll see that more and more. 810 00:42:45,663 --> 00:42:52,970 As one last example of this, you could go a little-- 811 00:42:52,970 --> 00:42:56,073 So again, you go back to your periodic table. 812 00:42:56,073 --> 00:42:58,776 And you take-- there was copper. 813 00:42:58,776 --> 00:43:00,611 Let's do nickel. 814 00:43:00,611 --> 00:43:04,615 All right, you take nickel. 815 00:43:04,615 --> 00:43:11,689 And let's see, I know that the atomic radius of nickel 816 00:43:11,689 --> 00:43:15,125 is 1.52 angstroms. 817 00:43:15,125 --> 00:43:17,227 I've looked that up. 818 00:43:17,227 --> 00:43:19,330 But I also know that it's BCC. 819 00:43:19,330 --> 00:43:20,497 And this gives it away. 820 00:43:20,497 --> 00:43:21,732 This gives me what I need. 821 00:43:21,732 --> 00:43:25,636 The crystal symmetry, the lattice, gives me what I need. 822 00:43:25,636 --> 00:43:27,438 Because once I know that it's BCC, 823 00:43:27,438 --> 00:43:30,608 then I also know that the lattice edge 824 00:43:30,608 --> 00:43:34,011 goes as 4r over root 3. 825 00:43:34,011 --> 00:43:38,482 Remember, for an element like nickel, 826 00:43:38,482 --> 00:43:41,552 the packing direction is along the body diagonal. 827 00:43:41,552 --> 00:43:45,823 That's what gave me this relationship, back and forth. 828 00:43:45,823 --> 00:43:50,361 Unit cell edge, radius of the atom. 829 00:43:50,361 --> 00:43:54,231 So that means that in nickel, it's 3.5 angstroms. 830 00:43:54,231 --> 00:44:00,504 And once I know what a is, well then I can know what like the-- 831 00:44:00,504 --> 00:44:03,440 you know, in one of the planes I say, which plane is that? 832 00:44:03,440 --> 00:44:06,010 I don't know, let's look at the densities in the plane. 833 00:44:06,010 --> 00:44:09,146 So for the 100 plane, it's-- 834 00:44:09,146 --> 00:44:10,781 I'm just going to give you the answer-- 835 00:44:10,781 --> 00:44:16,520 0.59 is the packing fraction. 836 00:44:16,520 --> 00:44:19,390 Packing density. 837 00:44:19,390 --> 00:44:21,892 This is the-- sorry, that's the fraction. 838 00:44:26,930 --> 00:44:28,899 All I did is I used the periodic table 839 00:44:28,899 --> 00:44:30,968 to do what I just did for the simple cubic case 840 00:44:30,968 --> 00:44:32,102 but now for BCC. 841 00:44:32,102 --> 00:44:33,537 And it's because of all the things 842 00:44:33,537 --> 00:44:36,140 that we've learned in the last two days, in the last two 843 00:44:36,140 --> 00:44:38,609 lectures that we're able to do this kind of calculation. 844 00:44:38,609 --> 00:44:40,911 And I can compare this plane with another plane 845 00:44:40,911 --> 00:44:43,380 with another plane. 846 00:44:43,380 --> 00:44:45,716 And think about how the atoms are packed differently 847 00:44:45,716 --> 00:44:48,385 within the anisotropy of the crystal. 848 00:44:48,385 --> 00:44:51,555 OK, Friday, bring parents. 849 00:44:51,555 --> 00:44:53,023 Bring parents.