1 00:00:00,600 --> 00:00:02,730 PROFESSOR: OK. 2 00:00:02,730 --> 00:00:05,670 Let me also remind you that we slightly changed the schedule 3 00:00:05,670 --> 00:00:06,330 for next week. 4 00:00:06,330 --> 00:00:11,925 On one of the older schedules, Tuesday appears as a lab date, 5 00:00:11,925 --> 00:00:13,050 but it's actually Thursday. 6 00:00:13,050 --> 00:00:15,540 So Tuesday will be a regular lecture here 7 00:00:15,540 --> 00:00:17,250 and Thursday will be the lab. 8 00:00:17,250 --> 00:00:19,740 And for the lab, we meet in 1115. 9 00:00:19,740 --> 00:00:22,680 That's on the handout for the first day. 10 00:00:22,680 --> 00:00:26,400 Also, if you're not registered for class, 11 00:00:26,400 --> 00:00:30,330 you don't have automatic access to Stellar, the website. 12 00:00:30,330 --> 00:00:35,020 So we can manually add you, but you'll have to let us know. 13 00:00:35,020 --> 00:00:38,640 Just send an email to Professor [? Marzari ?] and myself. 14 00:00:38,640 --> 00:00:42,280 And even better is if you send an email to both of us. 15 00:00:42,280 --> 00:00:44,280 And then we'll manually add that. 16 00:00:44,280 --> 00:00:49,170 And we'll put links there to things like articles and things 17 00:00:49,170 --> 00:00:52,880 that may have restricted access, copies of the lecture notes, 18 00:00:52,880 --> 00:00:55,520 things like that will appear there. 19 00:00:55,520 --> 00:00:57,180 OK. 20 00:00:57,180 --> 00:01:00,530 So what I want to go do is go back to their potential models. 21 00:01:00,530 --> 00:01:02,190 And what we're going to do today, 22 00:01:02,190 --> 00:01:06,270 is talk a little more in detail about the sort 23 00:01:06,270 --> 00:01:09,960 of formal or conceptual failures of pair potential models, 24 00:01:09,960 --> 00:01:15,130 and then how to get addressed with other empirical models. 25 00:01:15,130 --> 00:01:19,210 But before I do that, I wanted to-- 26 00:01:19,210 --> 00:01:26,493 let's see if I can get this into place, guess not-- 27 00:01:30,650 --> 00:01:32,390 go through a few practical issues, 28 00:01:32,390 --> 00:01:35,000 since the first lab is actually on using 29 00:01:35,000 --> 00:01:38,580 pair potentials to model things so you get a bit of an idea. 30 00:01:38,580 --> 00:01:43,820 So in a potential model, your energy is written pairwise. 31 00:01:43,820 --> 00:01:45,500 And so essentially what you need to do 32 00:01:45,500 --> 00:01:47,510 between every pair of atoms-- 33 00:01:47,510 --> 00:01:50,547 every couple iJ-- essentially to evaluate 34 00:01:50,547 --> 00:01:52,130 the distance between them and then put 35 00:01:52,130 --> 00:01:54,270 that in some kind of energy function. 36 00:01:54,270 --> 00:01:59,240 So this is clearly an N squared operation 37 00:01:59,240 --> 00:02:01,040 where N is the number of atoms, since you 38 00:02:01,040 --> 00:02:03,170 have to look for the distance between every atom 39 00:02:03,170 --> 00:02:04,610 and every other atom. 40 00:02:04,610 --> 00:02:07,472 So if systems get extremely big-- 41 00:02:07,472 --> 00:02:08,930 and that's often the reason why you 42 00:02:08,930 --> 00:02:11,180 want to use the pair potentials rather than, say, 43 00:02:11,180 --> 00:02:12,800 a quantum mechanical method-- 44 00:02:12,800 --> 00:02:14,870 N squared will get very large. 45 00:02:14,870 --> 00:02:17,360 And people do kind of tricks. 46 00:02:17,360 --> 00:02:19,730 If you have millions and millions of atoms 47 00:02:19,730 --> 00:02:23,690 in your simulation, if you know ahead of time that some of them 48 00:02:23,690 --> 00:02:27,560 are very far from the atom on which J you want to calculate-- 49 00:02:27,560 --> 00:02:28,893 energies or forces-- 50 00:02:28,893 --> 00:02:31,060 then you really never want to look further distance. 51 00:02:31,060 --> 00:02:34,280 So people do things like keeping neighbor lists. 52 00:02:34,280 --> 00:02:37,490 So literally, for every atom you would know, 53 00:02:37,490 --> 00:02:42,160 well, these other items or possibly 54 00:02:42,160 --> 00:02:45,310 in its neighbor environment. 55 00:02:45,310 --> 00:02:46,810 So then you really only have to look 56 00:02:46,810 --> 00:02:49,550 for distances with those atoms. 57 00:02:49,550 --> 00:02:51,190 So if you have a solid, that's great. 58 00:02:51,190 --> 00:02:54,400 Because if you have a solid, then the topological relation 59 00:02:54,400 --> 00:02:57,070 between atoms will not change much during the simulation. 60 00:02:57,070 --> 00:02:59,800 Of course, if you're simulating a gas or a liquid, they will. 61 00:02:59,800 --> 00:03:02,650 And then what you have to do is essentially update the neighbor 62 00:03:02,650 --> 00:03:04,810 lists regularly. 63 00:03:04,810 --> 00:03:07,510 And so there's overhead associated with that. 64 00:03:07,510 --> 00:03:09,760 But the benefit you get is that you essentially 65 00:03:09,760 --> 00:03:16,300 have an order N system rather than an N squared system. 66 00:03:16,300 --> 00:03:21,310 I'll show you in a second some large scale simulations. 67 00:03:21,310 --> 00:03:25,190 Very often, you will want to relax a system-- 68 00:03:25,190 --> 00:03:26,920 so get it to its lowest energy. 69 00:03:26,920 --> 00:03:30,130 And very often, that's done by calculating the forces on atoms 70 00:03:30,130 --> 00:03:32,545 and literally stepping down the force. 71 00:03:32,545 --> 00:03:34,670 If you want to do dynamics, Professor [? Marzari ?] 72 00:03:34,670 --> 00:03:36,400 will later teach you molecular dynamics. 73 00:03:36,400 --> 00:03:38,290 You also need the force because the force 74 00:03:38,290 --> 00:03:41,950 will tell you how much atoms accelerate or decelerate. 75 00:03:41,950 --> 00:03:44,080 You will need to calculate the force, which 76 00:03:44,080 --> 00:03:45,340 is a potential derivative. 77 00:03:45,340 --> 00:03:46,870 So you're essentially calculating 78 00:03:46,870 --> 00:03:51,130 the derivative of the total energy with respect 79 00:03:51,130 --> 00:03:56,710 to a given position. 80 00:03:56,710 --> 00:03:57,400 Same deal. 81 00:03:57,400 --> 00:04:00,130 That's an N squared operation. 82 00:04:00,130 --> 00:04:02,950 You have to sum over pairs of atoms. 83 00:04:02,950 --> 00:04:06,580 We have to sum over 1 all atoms for the force on a given atom. 84 00:04:06,580 --> 00:04:10,900 But since they're N atoms, it's an N squared operation. 85 00:04:10,900 --> 00:04:12,760 In a lot of codes, the minimization 86 00:04:12,760 --> 00:04:14,710 is done by very standard schemes-- things 87 00:04:14,710 --> 00:04:18,519 like conjugate gradient, Newton-Raphson, sometimes often 88 00:04:18,519 --> 00:04:21,820 trivial line minimizations where you literally just calculate 89 00:04:21,820 --> 00:04:25,990 the force and assume some kind of elasticity 90 00:04:25,990 --> 00:04:27,920 along the force path. 91 00:04:27,920 --> 00:04:30,640 And so if you know the force, you 92 00:04:30,640 --> 00:04:32,260 assume you know what the curvature is 93 00:04:32,260 --> 00:04:33,593 of the energy in that direction. 94 00:04:33,593 --> 00:04:35,830 You kind of can make an approximation to its minimum 95 00:04:35,830 --> 00:04:39,520 when you do that iteratively. 96 00:04:39,520 --> 00:04:47,560 So you can do that with very large simulations. 97 00:04:47,560 --> 00:04:51,100 And I wanted to show you sort of one of the leading edge ones. 98 00:04:51,100 --> 00:04:53,020 I got to get out of PowerPoint for that 99 00:04:53,020 --> 00:04:56,820 if I can make that work. 100 00:04:56,820 --> 00:05:00,100 OK, here we go. 101 00:05:00,100 --> 00:05:04,510 So what I'm going to show you is a order million atom 102 00:05:04,510 --> 00:05:05,100 simulation. 103 00:05:05,100 --> 00:05:12,060 This is essentially a cube of 1,000 by 1,000 by 1,000 atoms. 104 00:05:12,060 --> 00:05:15,840 That would make more than a million. 105 00:05:15,840 --> 00:05:19,750 I forgot to write down what it was. 106 00:05:19,750 --> 00:05:22,380 Yeah, I wrote down 1,000 by 1,000 by 1,000 atoms. 107 00:05:22,380 --> 00:05:24,420 That would be a billion atoms. 108 00:05:24,420 --> 00:05:26,130 And essentially what's being done-- 109 00:05:26,130 --> 00:05:28,500 and what you see is a notch on top and on bottom 110 00:05:28,500 --> 00:05:31,590 because there's so many atoms you don't see the discrete 111 00:05:31,590 --> 00:05:34,048 resolution anymore [? of ?] [? the ?] [? atoms, ?] [? so ?] 112 00:05:34,048 --> 00:05:37,380 this thing starts to look like a continuum. 113 00:05:37,380 --> 00:05:39,220 And it's being pulled from the side. 114 00:05:39,220 --> 00:05:42,870 So you'll see the cracked grow and dislocation spread. 115 00:05:42,870 --> 00:05:46,082 This is a simulation from Farid Abraham when he was at IBM. 116 00:05:46,082 --> 00:05:47,790 And of course, if you have big computers, 117 00:05:47,790 --> 00:05:50,200 you can do big simulations. 118 00:05:50,200 --> 00:05:51,870 So let me show it to you. 119 00:05:55,603 --> 00:05:57,020 So what they actually did was they 120 00:05:57,020 --> 00:05:59,210 showed just the dislocations by counting 121 00:05:59,210 --> 00:06:02,240 under-coordinated or over-coordinated atoms. 122 00:06:02,240 --> 00:06:05,450 So what you're actually seeing is the dislocation lines. 123 00:06:05,450 --> 00:06:07,400 I mean, if we plotted all the atoms, 124 00:06:07,400 --> 00:06:08,750 you wouldn't see anything. 125 00:06:08,750 --> 00:06:10,610 You'd just kind of see green everywhere. 126 00:06:10,610 --> 00:06:13,970 And that's sort of zoom up. 127 00:06:13,970 --> 00:06:15,840 So this is a molecular dynamic simulation. 128 00:06:15,840 --> 00:06:18,257 And you learn later more about that, where you essentially 129 00:06:18,257 --> 00:06:22,490 just do Newtonian mechanics on atoms. 130 00:06:26,252 --> 00:06:28,210 So this is one of these very large scale things 131 00:06:28,210 --> 00:06:33,980 that you can actually do with potentials. 132 00:06:40,970 --> 00:06:43,747 Now, you could ask yourself, what did I learn from this? 133 00:06:47,090 --> 00:06:49,580 This is the kind of work that my personal opinion 134 00:06:49,580 --> 00:06:52,070 is scientifically not necessarily all that relevant. 135 00:06:52,070 --> 00:06:54,200 But it's pioneering work, you know. 136 00:06:54,200 --> 00:06:56,900 This is like Lewis and Clark who went to the West. 137 00:06:56,900 --> 00:06:59,720 You could always say, what did we learn from them? 138 00:06:59,720 --> 00:07:02,780 Maybe not a lot, except that you could get the Pacific. 139 00:07:02,780 --> 00:07:04,250 But it's kind of pioneering work. 140 00:07:04,250 --> 00:07:06,397 And when people do these kind of simulations, 141 00:07:06,397 --> 00:07:08,480 it's not as much for the science as it is probably 142 00:07:08,480 --> 00:07:10,160 for pushing the envelope and seeing 143 00:07:10,160 --> 00:07:13,760 what can you do with enormous computational resources. 144 00:07:13,760 --> 00:07:16,820 You really have to fine tune your algorithms, 145 00:07:16,820 --> 00:07:18,400 your parallelization scheme. 146 00:07:18,400 --> 00:07:21,922 So it's kind of like flying to the moon which 147 00:07:21,922 --> 00:07:23,130 may have not brought us much. 148 00:07:23,130 --> 00:07:25,920 But there was a lot of technology fine tuned for that. 149 00:07:25,920 --> 00:07:28,820 So that's one of the reasons I wanted to show it. 150 00:07:33,310 --> 00:07:34,460 OK, let's go back. 151 00:07:37,490 --> 00:07:39,980 Another very practical issue that I'm sure a lot of you 152 00:07:39,980 --> 00:07:42,680 are familiar with. 153 00:07:42,680 --> 00:07:44,660 Even when you have a billion atoms, 154 00:07:44,660 --> 00:07:46,320 it's actually still very small. 155 00:07:46,320 --> 00:07:49,710 I actually think if you do the calculation, a million atoms 156 00:07:49,710 --> 00:07:53,030 is, for a lot of materials, a cube of just about 200 157 00:07:53,030 --> 00:07:56,120 Angstrom on the side so you're still, 158 00:07:56,120 --> 00:07:59,840 if these were finite systems, would be very small systems. 159 00:07:59,840 --> 00:08:01,310 So to get rid of boundary effects, 160 00:08:01,310 --> 00:08:03,518 you almost always use what's called periodic boundary 161 00:08:03,518 --> 00:08:07,550 conditions in simulations in as many directions as you can. 162 00:08:07,550 --> 00:08:09,950 And of course, what a periodic boundary condition 163 00:08:09,950 --> 00:08:16,290 means is that if you have some unit, if you have some unit, 164 00:08:16,290 --> 00:08:18,080 you essentially repeat it next to it. 165 00:08:18,080 --> 00:08:21,250 So this end here is the same as that end, 166 00:08:21,250 --> 00:08:24,120 and this end here is the same as that end. 167 00:08:24,120 --> 00:08:26,220 So people always ask you how many 168 00:08:26,220 --> 00:08:28,140 atoms do you have in your simulation, 169 00:08:28,140 --> 00:08:30,810 and the answer is infinite. 170 00:08:30,810 --> 00:08:33,960 It's just that you only have a finite number of degrees 171 00:08:33,960 --> 00:08:38,669 of freedom because it's these atoms here do 172 00:08:38,669 --> 00:08:41,460 exactly the same as these here. 173 00:08:41,460 --> 00:08:43,299 But the answer is infinite. 174 00:08:43,299 --> 00:08:45,690 So we don't really work with small systems. 175 00:08:45,690 --> 00:08:46,530 Or, rarely. 176 00:08:46,530 --> 00:08:50,260 I mean, sometimes we work with finite systems. 177 00:08:50,260 --> 00:08:52,470 So what's the repeat unit you use? 178 00:08:52,470 --> 00:08:54,990 Well, if you're studying a periodic material, 179 00:08:54,990 --> 00:08:56,177 then it's easy. 180 00:08:56,177 --> 00:08:58,260 If you study, say, a perfect crystalline material, 181 00:08:58,260 --> 00:09:01,710 here's a perovskite unit, so that's what you repeat. 182 00:09:01,710 --> 00:09:05,165 Because the material already has periodicity implied in it. 183 00:09:05,165 --> 00:09:06,540 So at that point, you're actually 184 00:09:06,540 --> 00:09:09,240 not making any approximation. 185 00:09:09,240 --> 00:09:11,820 You're not forcing anything on the material-- 186 00:09:11,820 --> 00:09:14,850 not when you do a static calculation that it already 187 00:09:14,850 --> 00:09:16,165 doesn't have. 188 00:09:16,165 --> 00:09:17,790 Of course, when you do dynamics, you're 189 00:09:17,790 --> 00:09:19,207 enforcing something because you're 190 00:09:19,207 --> 00:09:24,330 saying that in dynamics, the translational symmetry is 191 00:09:24,330 --> 00:09:27,810 temporarily broken because this atom can vibrate independent 192 00:09:27,810 --> 00:09:29,190 of that one. 193 00:09:29,190 --> 00:09:32,370 And so, then, when you improve impulse periodicity, 194 00:09:32,370 --> 00:09:35,440 you are imposing that these vibrate together. 195 00:09:35,440 --> 00:09:38,430 So in some sense, you're imposing a cutoff 196 00:09:38,430 --> 00:09:40,560 on the frequencies of vibrations that can occur. 197 00:09:43,490 --> 00:09:46,490 If systems don't have periodicity, 198 00:09:46,490 --> 00:09:48,410 you typically tend to enforce it. 199 00:09:48,410 --> 00:09:52,250 If I took this perovskite and I made one defect-- 200 00:09:52,250 --> 00:09:55,070 say I wanted to study oxygen vacancy so I take out 201 00:09:55,070 --> 00:09:59,570 the oxygen vacancy, I lose all translational periodicity now. 202 00:09:59,570 --> 00:10:02,420 But what you do is you tend to impose it back 203 00:10:02,420 --> 00:10:03,560 by repeating the defect. 204 00:10:08,410 --> 00:10:10,610 It's going to be slow. 205 00:10:10,610 --> 00:10:16,150 So let's say I've taken out an oxygen here. 206 00:10:16,150 --> 00:10:19,600 I repeat the defect so that, in essence, 207 00:10:19,600 --> 00:10:21,130 I just have a bigger unit cell. 208 00:10:21,130 --> 00:10:22,810 But it's one with a defect in it. 209 00:10:22,810 --> 00:10:25,000 This is now my unit cell. 210 00:10:25,000 --> 00:10:28,910 And that's called a supercell approximation. 211 00:10:28,910 --> 00:10:32,410 So you make a cell that's big enough so that you 212 00:10:32,410 --> 00:10:34,700 hope that these defects-- 213 00:10:34,700 --> 00:10:35,950 I don't know how you call it-- 214 00:10:35,950 --> 00:10:42,190 don't interact because then, if you have N cells, 215 00:10:42,190 --> 00:10:45,910 your calculation just has the energy of N defects. 216 00:10:45,910 --> 00:10:46,480 OK? 217 00:10:46,480 --> 00:10:48,490 And the critical assumption is, of course, 218 00:10:48,490 --> 00:10:49,918 that they don't interact. 219 00:10:49,918 --> 00:10:51,460 Because if they interact, then you're 220 00:10:51,460 --> 00:10:53,020 not modeling an isolated defect. 221 00:10:53,020 --> 00:10:56,900 You're modeling a defect that's interacting with other ones. 222 00:10:56,900 --> 00:10:59,920 So how do they interact typically? 223 00:10:59,920 --> 00:11:01,315 It's kind of important to know. 224 00:11:01,315 --> 00:11:03,940 This is one of these things that you're going to do in the lab. 225 00:11:03,940 --> 00:11:06,600 You should always check conversions if you can. 226 00:11:06,600 --> 00:11:08,350 The problem is that often you already work 227 00:11:08,350 --> 00:11:10,900 at your computational limits for size of supercells, 228 00:11:10,900 --> 00:11:13,072 so it's easier said than done. 229 00:11:13,072 --> 00:11:14,530 In an ideal world, you say, well, I 230 00:11:14,530 --> 00:11:18,400 calculate bigger and bigger, and if the answer doesn't change, 231 00:11:18,400 --> 00:11:19,270 I take it. 232 00:11:19,270 --> 00:11:22,095 But usually, you work so much at your computational edge 233 00:11:22,095 --> 00:11:24,220 already that you can't make them bigger and bigger. 234 00:11:24,220 --> 00:11:26,220 A lot of people forget that [? to ?] [? check ?] 235 00:11:26,220 --> 00:11:28,720 [? conversions, ?] you also can make it smaller. 236 00:11:28,720 --> 00:11:30,160 And if it doesn't change much, you 237 00:11:30,160 --> 00:11:32,740 were probably already converged. 238 00:11:32,740 --> 00:11:36,190 But it's useful to know how these things usually 239 00:11:36,190 --> 00:11:37,340 interact with each other. 240 00:11:37,340 --> 00:11:40,780 So that gives you some sense of how far you have to go. 241 00:11:40,780 --> 00:11:43,300 There's often the direct sort of interaction. 242 00:11:43,300 --> 00:11:44,890 If you work with a pair potential, 243 00:11:44,890 --> 00:11:48,320 then you can actually see the range of the pair potential. 244 00:11:48,320 --> 00:11:50,170 That's usually not the problem. 245 00:11:50,170 --> 00:11:53,200 Rarely do you have potentials that are relevant, large, 246 00:11:53,200 --> 00:11:55,720 and over large distances that you need to make 247 00:11:55,720 --> 00:11:57,700 really large supercells. 248 00:11:57,700 --> 00:11:59,755 But there can be other energetic effects. 249 00:11:59,755 --> 00:12:02,432 If you have electrostatics in your system, 250 00:12:02,432 --> 00:12:04,390 that will give you very long range interaction. 251 00:12:04,390 --> 00:12:05,860 If that's a charge defect-- 252 00:12:05,860 --> 00:12:08,350 which it is in a perovskite-- 253 00:12:08,350 --> 00:12:10,150 you will have a very strong one. 254 00:12:10,150 --> 00:12:14,200 Actually, if we truly did a charge defect 255 00:12:14,200 --> 00:12:15,910 in that supercell, what would happen? 256 00:12:20,780 --> 00:12:23,240 If that supercell became charged-- 257 00:12:23,240 --> 00:12:25,070 so let's say I took out an oxygen 258 00:12:25,070 --> 00:12:27,350 2 minus, which means the supercell now 259 00:12:27,350 --> 00:12:29,180 has a net positive charge-- 260 00:12:31,910 --> 00:12:34,140 what would happen? 261 00:12:34,140 --> 00:12:37,790 You'd have an infinite energy because you would essentially 262 00:12:37,790 --> 00:12:43,460 have a positive charge interacting with only other net 263 00:12:43,460 --> 00:12:45,260 positive charges. 264 00:12:45,260 --> 00:12:47,840 And at any distance, if you do an infinite system, 265 00:12:47,840 --> 00:12:50,763 that gives you an infinite energy. 266 00:12:50,763 --> 00:12:52,180 So that's something that you would 267 00:12:52,180 --> 00:12:54,450 have to explicitly take out. 268 00:12:54,450 --> 00:12:58,150 Now you say, well, I'll compensate 269 00:12:58,150 --> 00:13:03,290 with a negative charge because that's what happens in reality. 270 00:13:03,290 --> 00:13:05,683 If I take out an oxygen, those two electrons 271 00:13:05,683 --> 00:13:07,600 of that or O2 minus are going to go somewhere, 272 00:13:07,600 --> 00:13:10,040 so it's going to create a net positive charge. 273 00:13:10,040 --> 00:13:12,990 So then you have a dipole. 274 00:13:12,990 --> 00:13:15,750 So let's say I have sort of plus, minus, 275 00:13:15,750 --> 00:13:18,060 and I have a dipole now. 276 00:13:18,060 --> 00:13:20,160 So now I have dipoles interacting. 277 00:13:24,690 --> 00:13:26,880 Dipoles will not give you infinite energy, 278 00:13:26,880 --> 00:13:31,120 but they will give you a fairly long range energy tail. 279 00:13:31,120 --> 00:13:33,160 And if you're smart about it-- 280 00:13:33,160 --> 00:13:36,360 so the brute force way to convert supercell calculation 281 00:13:36,360 --> 00:13:39,840 is always to push him as hard as you can and hope 282 00:13:39,840 --> 00:13:42,545 and sort of see the energy converge of the defect. 283 00:13:42,545 --> 00:13:43,920 But if you're smart about it, you 284 00:13:43,920 --> 00:13:46,050 can often do much smaller cell calculations 285 00:13:46,050 --> 00:13:48,900 and explicitly take out the term that's 286 00:13:48,900 --> 00:13:51,990 the defect-defect interaction. 287 00:13:51,990 --> 00:13:54,720 Which is, if you know that this is a dipole, 288 00:13:54,720 --> 00:13:59,340 you know what the form is of it, and you can take it out. 289 00:13:59,340 --> 00:14:02,940 Other things that often cause havoc is relaxation. 290 00:14:02,940 --> 00:14:05,460 If I were to put a really big ion 291 00:14:05,460 --> 00:14:08,580 in the center of that perovskite unit cell, 292 00:14:08,580 --> 00:14:12,030 I would build up a strain field and the different defects 293 00:14:12,030 --> 00:14:15,060 would interact through their strain field. 294 00:14:15,060 --> 00:14:16,980 So, rarely do you actually interact 295 00:14:16,980 --> 00:14:19,180 through the direct energetics. 296 00:14:19,180 --> 00:14:23,460 You often interact through secondary effects. 297 00:14:23,460 --> 00:14:27,960 Another way they often interact which is very indirect. 298 00:14:27,960 --> 00:14:29,970 If you put in your defect and you 299 00:14:29,970 --> 00:14:36,140 relax the volume of the supercell, 300 00:14:36,140 --> 00:14:38,720 then that will interfere with the relaxation 301 00:14:38,720 --> 00:14:41,340 of the supercell next to it. 302 00:14:41,340 --> 00:14:44,570 So you get a strain field just through the homogeneous. 303 00:14:44,570 --> 00:14:46,430 You get an extra energy term just 304 00:14:46,430 --> 00:14:48,393 through the homogeneous strain interaction. 305 00:14:48,393 --> 00:14:50,060 So these are all things to consider when 306 00:14:50,060 --> 00:14:52,720 you do supercell calculations. 307 00:14:55,910 --> 00:14:59,030 We're trying something new where we save the annotations 308 00:14:59,030 --> 00:15:02,270 we make on the slides, but it seems to be really slow. 309 00:15:02,270 --> 00:15:06,770 So I may not be able to keep that up. 310 00:15:06,770 --> 00:15:11,720 OK, so this I went over-- the kind of convergence. 311 00:15:11,720 --> 00:15:14,540 You know, here's a typical example. 312 00:15:14,540 --> 00:15:16,400 I don't know if this is typical. 313 00:15:16,400 --> 00:15:18,592 This is the vacancy formation energy in aluminum. 314 00:15:18,592 --> 00:15:20,300 Now, this is done with quantum mechanics, 315 00:15:20,300 --> 00:15:22,730 versus the number of atoms in the supercell. 316 00:15:22,730 --> 00:15:26,450 And you know, this is one of these 317 00:15:26,450 --> 00:15:30,402 that you really don't know if you're converged. 318 00:15:30,402 --> 00:15:32,360 I mean, it depends on how accurate you want it. 319 00:15:32,360 --> 00:15:36,710 If you really only care that it's between 0.7 and 0.75, 320 00:15:36,710 --> 00:15:38,100 you're probably OK. 321 00:15:38,100 --> 00:15:40,410 But if you did this point and that point, 322 00:15:40,410 --> 00:15:42,387 you'd probably think you're converged. 323 00:15:42,387 --> 00:15:43,970 But then, if you look at the next one, 324 00:15:43,970 --> 00:15:46,012 you're actually going back up a little bit again. 325 00:15:46,012 --> 00:15:48,980 So it's sometimes very hard to figure out whether you're 326 00:15:48,980 --> 00:15:50,314 actually converged. 327 00:15:54,030 --> 00:15:55,955 OK. 328 00:15:55,955 --> 00:15:58,080 So that was a few practical issues to sort of start 329 00:15:58,080 --> 00:16:00,163 gearing you up for the lab. 330 00:16:00,163 --> 00:16:01,830 What I want to do now is talk explicitly 331 00:16:01,830 --> 00:16:04,620 about some of the formal failures of pair potentials 332 00:16:04,620 --> 00:16:06,250 and go on to correct them. 333 00:16:06,250 --> 00:16:08,490 And the first one is the vacancy formation energy 334 00:16:08,490 --> 00:16:13,500 which is, if you look at historically when people 335 00:16:13,500 --> 00:16:19,470 started doing simulations late 1950s, 1960s, into the '70s, 336 00:16:19,470 --> 00:16:22,170 people thought that whenever there was something 337 00:16:22,170 --> 00:16:24,720 wrong with the outcome that they just needed better 338 00:16:24,720 --> 00:16:27,810 parameters in the potential. 339 00:16:27,810 --> 00:16:31,992 And it's actually rather recent-- sort of 1980s-- 340 00:16:31,992 --> 00:16:34,200 that people figured out that there were really formal 341 00:16:34,200 --> 00:16:37,110 problems and that there were certain things you could not 342 00:16:37,110 --> 00:16:38,760 get right-- never-- 343 00:16:38,760 --> 00:16:41,130 with a potential because the problem of the form 344 00:16:41,130 --> 00:16:42,780 rather than of the parameters. 345 00:16:42,780 --> 00:16:45,630 And a critical number in that work 346 00:16:45,630 --> 00:16:47,940 was actually the vacancy formation energy which 347 00:16:47,940 --> 00:16:49,530 nobody could ever get right. 348 00:16:49,530 --> 00:16:50,940 And I'll show in a second why. 349 00:16:53,640 --> 00:16:55,110 Here's an FCC crystal. 350 00:16:55,110 --> 00:16:59,280 So let's calculate the pair approximation to the vacancy 351 00:16:59,280 --> 00:17:01,080 formation energy. 352 00:17:01,080 --> 00:17:03,720 What do I do when I make a vacancy? 353 00:17:03,720 --> 00:17:08,579 What I do is that I take this central atom 354 00:17:08,579 --> 00:17:12,339 and I move it somewhere else in the bulk. 355 00:17:12,339 --> 00:17:16,020 So in some sense, I make my system one side bigger. 356 00:17:16,020 --> 00:17:19,300 So let's calculate the energy balance of that. 357 00:17:19,300 --> 00:17:21,390 So the vacancy formation energy. 358 00:17:24,520 --> 00:17:25,020 OK. 359 00:17:25,020 --> 00:17:26,339 What do I destroy? 360 00:17:26,339 --> 00:17:31,440 I destroy 12 bonds because this atom is coordinated 12-fold 361 00:17:31,440 --> 00:17:32,790 in FCC. 362 00:17:32,790 --> 00:17:34,140 So I destroy 12 bonds. 363 00:17:36,790 --> 00:17:41,310 I lose 12 times the bond energy. 364 00:17:41,310 --> 00:17:44,270 But then, of course, I put it back somewhere in the bulk. 365 00:17:44,270 --> 00:17:49,660 And so I gained the cohesive energy of one atom. 366 00:17:49,660 --> 00:17:50,590 OK? 367 00:17:50,590 --> 00:17:54,130 And what's the cohesive energy per atom in FCC? 368 00:17:54,130 --> 00:17:57,310 It's 6 times the bond energy because there 369 00:17:57,310 --> 00:17:59,740 are 12 bonds around an atom, but every bond 370 00:17:59,740 --> 00:18:01,430 is shared between two. 371 00:18:01,430 --> 00:18:01,930 OK? 372 00:18:01,930 --> 00:18:04,240 So there's six per atom. 373 00:18:04,240 --> 00:18:07,870 So the vacancy formation is really minus 12 times 374 00:18:07,870 --> 00:18:10,960 the bond energy plus 6 times the bond energy. 375 00:18:13,990 --> 00:18:19,890 So what it is is minus 6 times the bond the energy. 376 00:18:19,890 --> 00:18:25,260 So essentially, it's minus the cohesive energy. 377 00:18:25,260 --> 00:18:29,120 The cohesive energy per atom. 378 00:18:29,120 --> 00:18:31,100 And that's essentially what you will always 379 00:18:31,100 --> 00:18:33,270 find repair potentials. 380 00:18:33,270 --> 00:18:35,690 Now, I did a static approximation. 381 00:18:35,690 --> 00:18:37,100 I assumed that all of these bonds 382 00:18:37,100 --> 00:18:39,770 were kind of unchanged as I took the atom out. 383 00:18:39,770 --> 00:18:42,500 In reality, if I take that central atom out, 384 00:18:42,500 --> 00:18:44,570 the atoms around it will relax somewhat 385 00:18:44,570 --> 00:18:46,760 and change their bond energy. 386 00:18:46,760 --> 00:18:49,190 But you'll see this when you do this in the lab. 387 00:18:49,190 --> 00:18:52,550 That will make it from one time the cohesive energy 388 00:18:52,550 --> 00:18:56,330 to something like 0.9 or 0.95 times the cohesive energy. 389 00:18:56,330 --> 00:18:58,340 Essentially, you find in pair potentials 390 00:18:58,340 --> 00:19:00,350 that the vacancy formation energy 391 00:19:00,350 --> 00:19:03,090 is about the cohesive energy. 392 00:19:03,090 --> 00:19:06,450 If you look, on the other hand, at experimental results 393 00:19:06,450 --> 00:19:07,550 for the cohesive energy-- 394 00:19:11,240 --> 00:19:17,130 [INAUDIBLE]---- here's some experimental data. 395 00:19:17,130 --> 00:19:20,240 So this is for FCC metals, and this is for Leonard-Jones, 396 00:19:20,240 --> 00:19:21,230 and for noble gases. 397 00:19:21,230 --> 00:19:24,590 The ratio of the vacancy formation energy 398 00:19:24,590 --> 00:19:27,170 to the cohesive energy. 399 00:19:27,170 --> 00:19:31,590 And what you see is that, in the noble gases-- 400 00:19:31,590 --> 00:19:33,710 which you think of as well-described by pair 401 00:19:33,710 --> 00:19:35,000 potentials-- 402 00:19:35,000 --> 00:19:37,550 you're reasonably close. 403 00:19:37,550 --> 00:19:39,410 Because remember, noble gases are just 404 00:19:39,410 --> 00:19:42,230 inert shells interacting through Van der Waals interaction, 405 00:19:42,230 --> 00:19:45,470 and that's how we came up with the Leonard-Jones potential. 406 00:19:45,470 --> 00:19:49,760 But if you look at the metals, the vacancy formation energy 407 00:19:49,760 --> 00:19:52,220 is actually only a small fraction 408 00:19:52,220 --> 00:19:55,370 of the cohesive energy. 409 00:19:55,370 --> 00:19:58,220 And we can never get that low with pair potentials. 410 00:19:58,220 --> 00:19:59,330 Never. 411 00:19:59,330 --> 00:19:59,900 OK? 412 00:19:59,900 --> 00:20:04,100 Unless you maybe really have the most odd form 413 00:20:04,100 --> 00:20:06,530 of a pair potential. 414 00:20:06,530 --> 00:20:09,470 But the reason is because there are certain physics missing 415 00:20:09,470 --> 00:20:11,052 from pair potentials. 416 00:20:20,840 --> 00:20:28,220 So actually, let me go back for a second to the pair 417 00:20:28,220 --> 00:20:30,380 potentials. 418 00:20:30,380 --> 00:20:34,850 What's the physics here of why the vacancy formation energy is 419 00:20:34,850 --> 00:20:39,440 so low in real materials? 420 00:20:39,440 --> 00:20:43,070 Remember what I said before, that bonding 421 00:20:43,070 --> 00:20:47,330 in metals, the bonding energy goes not linear 422 00:20:47,330 --> 00:20:49,100 in the coordination number. 423 00:20:49,100 --> 00:20:51,560 It's more something like a rough approximation 424 00:20:51,560 --> 00:20:54,980 would be the square root of the coordination number. 425 00:20:54,980 --> 00:20:59,020 But essentially, when you go from 1 to 2 bonds, 426 00:20:59,020 --> 00:21:00,770 your energy goes down a lot more than when 427 00:21:00,770 --> 00:21:04,250 you go from 11 to 12 bonds. 428 00:21:04,250 --> 00:21:07,850 So why is the vacancy formation energy so low? 429 00:21:07,850 --> 00:21:10,100 Well, if you think about what you do when you take out 430 00:21:10,100 --> 00:21:14,990 an atom, you make a lot of the atoms around it 431 00:21:14,990 --> 00:21:19,260 go from 12 coordination to 11. 432 00:21:19,260 --> 00:21:27,498 So they lose a lot less than 1/12 of the cohesive energy 433 00:21:27,498 --> 00:21:29,290 because the cohesive energy you could think 434 00:21:29,290 --> 00:21:32,830 of as the average of all bonds. 435 00:21:32,830 --> 00:21:36,430 But around the vacancy, you are just reducing the coordination 436 00:21:36,430 --> 00:21:37,840 from 12 to 11. 437 00:21:37,840 --> 00:21:41,230 You can actually make a simple model 438 00:21:41,230 --> 00:21:47,680 if you said that the cohesive energy was 439 00:21:47,680 --> 00:21:54,800 some constant times the square root of the coordination. 440 00:21:54,800 --> 00:21:57,020 So rather than linear in the coordination, which 441 00:21:57,020 --> 00:22:01,370 is what we would get in pair of potentials, if you calculate 442 00:22:01,370 --> 00:22:03,800 the vacancy formation energy in this model-- 443 00:22:08,970 --> 00:22:10,410 let's do it for FCC. 444 00:22:10,410 --> 00:22:11,820 In FCC, Z is 12. 445 00:22:15,120 --> 00:22:20,840 The vacancy formation energy, what would it be? 446 00:22:20,840 --> 00:22:25,480 Well, if we put the atom somewhere back in, 447 00:22:25,480 --> 00:22:31,070 we would gain Z times the square root of 12. 448 00:22:31,070 --> 00:22:34,820 And then, as we take it out, what do we lose? 449 00:22:34,820 --> 00:22:41,540 For 12 atoms, we go from coordination 12 to 11. 450 00:22:44,720 --> 00:22:46,250 So this is the atom in the bulk. 451 00:22:49,920 --> 00:22:53,010 This is actually the change around where we made the hole. 452 00:22:58,430 --> 00:23:01,900 And if you actually calculate that-- 453 00:23:01,900 --> 00:23:04,690 the vacancy formation energy over the cohesive energy-- 454 00:23:09,380 --> 00:23:15,530 you find that that's 1 plus 12 times square root of 11 455 00:23:15,530 --> 00:23:18,590 over square root of 12 minus 1. 456 00:23:18,590 --> 00:23:21,680 And it's about 0.49. 457 00:23:21,680 --> 00:23:23,330 So in a square root model, you're 458 00:23:23,330 --> 00:23:25,970 already much closer to the experiment 459 00:23:25,970 --> 00:23:29,540 where the vacancy formation energy 460 00:23:29,540 --> 00:23:32,660 is a much smaller fraction of the cohesive energy. 461 00:23:42,640 --> 00:23:44,890 Another critical piece of information 462 00:23:44,890 --> 00:23:47,890 was surface relaxation, which people 463 00:23:47,890 --> 00:23:52,270 had been seeing with [? LEEDs. ?] If you look 464 00:23:52,270 --> 00:23:55,960 at a metal, typically if you create a surface, 465 00:23:55,960 --> 00:23:59,830 the difference between the first and the second layer 466 00:23:59,830 --> 00:24:01,790 becomes smaller. 467 00:24:01,790 --> 00:24:05,360 So in a metal, the first layer tends to relax inward. 468 00:24:08,320 --> 00:24:11,973 And can somebody explain to me why that happens? 469 00:24:11,973 --> 00:24:13,390 And typically, you don't see this. 470 00:24:13,390 --> 00:24:16,330 In pair potentials, you very often 471 00:24:16,330 --> 00:24:18,200 have about the same distance. 472 00:24:18,200 --> 00:24:20,530 So the first to second neighbor distance 473 00:24:20,530 --> 00:24:25,090 is often bulk-like and in some cases actually relaxes outwards 474 00:24:25,090 --> 00:24:27,010 because of competing forces. 475 00:24:27,010 --> 00:24:32,820 But in almost all real systems, it tends to relax inward. 476 00:24:32,820 --> 00:24:35,360 Why do you think that is? 477 00:24:35,360 --> 00:24:36,500 Yes, [INAUDIBLE]? 478 00:24:36,500 --> 00:24:39,106 AUDIENCE: [INAUDIBLE] 479 00:24:44,893 --> 00:24:45,560 PROFESSOR: True. 480 00:24:45,560 --> 00:24:49,180 But I'm not sure that that necessarily 481 00:24:49,180 --> 00:24:52,570 makes it relax inward. 482 00:24:52,570 --> 00:24:54,820 The reason it relaxes inward is because the bonds 483 00:24:54,820 --> 00:24:57,228 between the first and the second layer strengthen. 484 00:24:57,228 --> 00:24:58,770 That's really what is a sign of them. 485 00:24:58,770 --> 00:25:00,960 And why do they strengthen? 486 00:25:00,960 --> 00:25:03,630 Because the surface is under-coordinated. 487 00:25:03,630 --> 00:25:06,090 Remember, if you go to lower coordination, 488 00:25:06,090 --> 00:25:07,560 you're effectively-- 489 00:25:07,560 --> 00:25:09,270 although you're losing energy and that's 490 00:25:09,270 --> 00:25:10,740 why there's surface energy-- 491 00:25:10,740 --> 00:25:15,300 but the energy per bond is actually becoming stronger. 492 00:25:15,300 --> 00:25:17,850 Remember that I showed you the potentials for copper? 493 00:25:17,850 --> 00:25:25,090 Remember that for the copper II molecule-- 494 00:25:25,090 --> 00:25:29,410 so where you just had one bond-- had the strongest potential. 495 00:25:29,410 --> 00:25:31,170 The equation of state gave you where 496 00:25:31,170 --> 00:25:34,270 your were 12-fold coordinate gave you much weaker potential. 497 00:25:34,270 --> 00:25:37,080 So you see that in surfaces as well. 498 00:25:37,080 --> 00:25:41,700 If you cut away the bonds here, the remaining ones strengthen 499 00:25:41,700 --> 00:25:43,110 and they pull the layer in. 500 00:25:49,580 --> 00:25:51,890 The other problem that's common in pair potential 501 00:25:51,890 --> 00:25:53,473 which I won't say much about is what's 502 00:25:53,473 --> 00:25:55,040 called the Cauchy problem. 503 00:25:55,040 --> 00:26:01,430 If you write down the relation between the stress 504 00:26:01,430 --> 00:26:03,950 and the strain tensor and so this 505 00:26:03,950 --> 00:26:05,810 is the matrix of elastic constant that 506 00:26:05,810 --> 00:26:11,130 for pair potential C12 and C44 are always the same. 507 00:26:11,130 --> 00:26:13,400 And this has something to do with there actually being 508 00:26:13,400 --> 00:26:15,170 a zero-energy strain mold. 509 00:26:15,170 --> 00:26:16,880 Remember that I showed you this? 510 00:26:16,880 --> 00:26:19,460 It's related to the same issue that in a cubic 511 00:26:19,460 --> 00:26:23,330 or in a square model, you can actually kind of slide it 512 00:26:23,330 --> 00:26:25,370 without any energy change. 513 00:26:25,370 --> 00:26:27,410 And that actually puts a constraint 514 00:26:27,410 --> 00:26:29,450 on the elastic constant, which essentially 515 00:26:29,450 --> 00:26:33,110 translates to C12 being C44. 516 00:26:36,230 --> 00:26:38,930 And the last one-- 517 00:26:38,930 --> 00:26:40,880 if you're a practicing material scientist-- 518 00:26:40,880 --> 00:26:43,730 may be one of the important ones. 519 00:26:43,730 --> 00:26:47,270 But you can essentially not predict crystal structure 520 00:26:47,270 --> 00:26:50,900 in anything that's covalent. 521 00:26:50,900 --> 00:26:52,880 And metals are covalent. 522 00:26:52,880 --> 00:26:55,160 I know they teach you there's metallic, covalent, then 523 00:26:55,160 --> 00:26:56,030 ionic bonding. 524 00:26:56,030 --> 00:27:00,450 But metallic is just delocalized covalent. 525 00:27:00,450 --> 00:27:02,188 And the reason is-- 526 00:27:02,188 --> 00:27:03,980 I think you can already see-- first of all, 527 00:27:03,980 --> 00:27:07,160 potentials tend to go to close packing always 528 00:27:07,160 --> 00:27:10,880 because they just they want to have as many atoms around them 529 00:27:10,880 --> 00:27:12,680 as possible because that's how you 530 00:27:12,680 --> 00:27:14,240 bring the cohesive energy down. 531 00:27:14,240 --> 00:27:17,540 And they don't count that cost, that if you bring more 532 00:27:17,540 --> 00:27:21,290 around you, your bound strength weakens. 533 00:27:21,290 --> 00:27:23,550 And they have no directional dependence, 534 00:27:23,550 --> 00:27:27,860 so they miss those two important components 535 00:27:27,860 --> 00:27:29,378 to get crystal structure right. 536 00:27:29,378 --> 00:27:31,170 You can actually do some theory about this, 537 00:27:31,170 --> 00:27:32,720 which was done already in the '70s 538 00:27:32,720 --> 00:27:36,020 that you actually need what's called 539 00:27:36,020 --> 00:27:39,620 at least the fourth moment of the density of states 540 00:27:39,620 --> 00:27:41,030 to get crystal structure right. 541 00:27:41,030 --> 00:27:44,100 And that turns out to be at least a four-body effect. 542 00:27:44,100 --> 00:27:47,970 So if you actually thought of potentials as an expansion-- 543 00:27:47,970 --> 00:27:52,020 so I'm going to write the energy as a constant plus an expansion 544 00:27:52,020 --> 00:27:54,570 in two-body terms, three-body terms, four-body terms, et 545 00:27:54,570 --> 00:27:56,900 cetera, which we don't even know would converge-- 546 00:27:56,900 --> 00:27:59,840 you would have to go at least up to the fourth order 547 00:27:59,840 --> 00:28:02,360 to get crystal structure differences right. 548 00:28:06,180 --> 00:28:09,270 OK, so how do you fix the problem? 549 00:28:09,270 --> 00:28:11,610 There are essentially, if you look 550 00:28:11,610 --> 00:28:18,630 at the literature of empirical models, two avenues to go. 551 00:28:18,630 --> 00:28:23,850 You can either make what's called cluster potentials. 552 00:28:23,850 --> 00:28:26,760 And so, if you don't like a two-body potential, 553 00:28:26,760 --> 00:28:28,482 you can make a three-body potential. 554 00:28:28,482 --> 00:28:29,940 And the advantage of that, once you 555 00:28:29,940 --> 00:28:35,850 have a three-body potential, you can see angular dependence. 556 00:28:35,850 --> 00:28:41,490 Because if I have three atoms and they don't interact 557 00:28:41,490 --> 00:28:44,400 just pairwise, but there's a [? two ?] three-body, 558 00:28:44,400 --> 00:28:48,330 I can include this bonding angle. 559 00:28:48,330 --> 00:28:52,560 A bonding angle is truly a three-body concept. 560 00:28:52,560 --> 00:28:56,040 So you can start building an angular dependence. 561 00:28:56,040 --> 00:29:01,590 The other direction to go is go to pair functionals. 562 00:29:01,590 --> 00:29:05,160 So you still want to write the [? energies ?] pairwise-- 563 00:29:05,160 --> 00:29:06,970 some pairwise summation. 564 00:29:06,970 --> 00:29:09,100 You just don't want the cohesive energy per atom 565 00:29:09,100 --> 00:29:12,510 to go linear with what you sum pairwise 566 00:29:12,510 --> 00:29:16,980 or what you sum as its environment pairwise. 567 00:29:16,980 --> 00:29:19,020 So we're going to deal with both in class. 568 00:29:19,020 --> 00:29:22,236 Today, I'm going to mainly do pair functionals. 569 00:29:22,236 --> 00:29:27,240 Pair functionals tends to be most applicable to metals. 570 00:29:27,240 --> 00:29:29,790 Where I would say the square root depends, 571 00:29:29,790 --> 00:29:33,570 that effect is strongest, whereas cluster potentials 572 00:29:33,570 --> 00:29:35,490 or many body potentials in general 573 00:29:35,490 --> 00:29:39,270 tend to be more favored in the organic literature-- 574 00:29:39,270 --> 00:29:42,510 both organic chemistry and polymer chemistry. 575 00:29:42,510 --> 00:29:44,970 And so, going to that. 576 00:29:54,060 --> 00:29:56,730 So there's a whole class of methods 577 00:29:56,730 --> 00:29:59,108 which are called effective medium theories. 578 00:29:59,108 --> 00:30:00,900 The version of it that I'm going to discuss 579 00:30:00,900 --> 00:30:03,400 is what's called the embedded atom method, which is probably 580 00:30:03,400 --> 00:30:06,510 the most popular one. 581 00:30:06,510 --> 00:30:09,150 This was developed by [? Mike ?] [? Baskis ?] [? and ?] 582 00:30:09,150 --> 00:30:14,460 [? Murray ?] [? Dahl ?] I think going back to the '80s. 583 00:30:14,460 --> 00:30:16,630 Mid '80s, or something like that. 584 00:30:16,630 --> 00:30:19,500 And it's probably now the most popular 585 00:30:19,500 --> 00:30:21,780 of the sort of empirical energy schemes. 586 00:30:21,780 --> 00:30:23,340 If you look back at those papers, 587 00:30:23,340 --> 00:30:27,090 those papers have now over 1,000 citations. 588 00:30:27,090 --> 00:30:29,265 And the idea is extremely simple. 589 00:30:29,265 --> 00:30:31,590 It's, if you understand the source of the problem-- 590 00:30:31,590 --> 00:30:33,120 that our energy should not be linear 591 00:30:33,120 --> 00:30:35,483 with coordination-- that's what you're going to fix. 592 00:30:35,483 --> 00:30:36,900 So in these models, what you do is 593 00:30:36,900 --> 00:30:41,460 you write the energy per atom as a function of the number 594 00:30:41,460 --> 00:30:43,200 of bonds around it. 595 00:30:43,200 --> 00:30:46,650 It's just that F should not be linear. 596 00:30:46,650 --> 00:30:50,850 And so that's why they call it energy functionals rather than 597 00:30:50,850 --> 00:30:52,650 a pair potential. 598 00:30:52,650 --> 00:30:56,832 The question is, how do you measure number of bonds? 599 00:30:56,832 --> 00:30:59,040 And if you sort think about it, if you think yourself 600 00:30:59,040 --> 00:31:01,635 back in the '80s, how do you want 601 00:31:01,635 --> 00:31:03,570 to measure number of bonds? 602 00:31:03,570 --> 00:31:06,900 You don't really want to do it discreetly. 603 00:31:06,900 --> 00:31:08,970 Because you could say, well, maybe I'm 604 00:31:08,970 --> 00:31:12,780 just going to count atoms within a two-Angstrom radius, 605 00:31:12,780 --> 00:31:14,100 or some radius. 606 00:31:14,100 --> 00:31:16,590 The problem is, as soon as you measure it discretely, 607 00:31:16,590 --> 00:31:19,620 you're going to have to deal with discontinuities. 608 00:31:19,620 --> 00:31:21,630 As atoms move in and out of that radius, 609 00:31:21,630 --> 00:31:26,070 suddenly you've got 14 things around you instead of 15. 610 00:31:26,070 --> 00:31:29,310 So you would always have super large forces 611 00:31:29,310 --> 00:31:32,100 because of those discontinuities around your [? cutoff ?] 612 00:31:32,100 --> 00:31:32,800 interface. 613 00:31:32,800 --> 00:31:37,860 So what you need is a smooth count of coordination. 614 00:31:37,860 --> 00:31:42,150 And the one they came up with in the embedded atom method 615 00:31:42,150 --> 00:31:44,670 is literally, you measure the electron 616 00:31:44,670 --> 00:31:49,110 density being projected on you from your neighbors. 617 00:31:49,110 --> 00:31:50,070 It's ironic. 618 00:31:50,070 --> 00:31:52,590 I'll show you later the formal justification 619 00:31:52,590 --> 00:31:55,050 for the embedded atom method which has a lot more 620 00:31:55,050 --> 00:31:57,210 to do with quantum mechanics. 621 00:31:57,210 --> 00:32:01,000 But that was actually a justification after the fact. 622 00:32:01,000 --> 00:32:03,080 The original idea came up with sort of, 623 00:32:03,080 --> 00:32:06,450 you really want to have a continuous measure of bumbling 624 00:32:06,450 --> 00:32:07,420 around you. 625 00:32:07,420 --> 00:32:11,280 And so the idea is that you're going to do something like, 626 00:32:11,280 --> 00:32:13,530 I have a central atom. 627 00:32:13,530 --> 00:32:17,250 I have other atoms around me. 628 00:32:17,250 --> 00:32:20,520 How do I measure how much these contribute to my neighborhood? 629 00:32:20,520 --> 00:32:24,840 Well, these have some electron density functions around them. 630 00:32:24,840 --> 00:32:30,770 Their atomic electron density I'm sort of drawing. 631 00:32:30,770 --> 00:32:33,320 And they all project onto that central atom. 632 00:32:33,320 --> 00:32:36,650 You count up how much electron density there 633 00:32:36,650 --> 00:32:40,162 and that's somehow a measurement of your neighborhood. 634 00:32:40,162 --> 00:32:41,870 And you see, that's going to roughly have 635 00:32:41,870 --> 00:32:42,662 the right behavior. 636 00:32:42,662 --> 00:32:43,760 It's continuous. 637 00:32:43,760 --> 00:32:47,270 As atoms move farther away, they still contribute, 638 00:32:47,270 --> 00:32:48,650 but they contribute less. 639 00:32:48,650 --> 00:32:50,090 So it is a continuous measure. 640 00:32:50,090 --> 00:32:53,780 And the more density you project, 641 00:32:53,780 --> 00:32:57,840 either the closer by the atoms are or the more of them 642 00:32:57,840 --> 00:32:58,340 there are. 643 00:33:04,610 --> 00:33:05,960 OK. 644 00:33:05,960 --> 00:33:07,820 So this is the idea of the electron density. 645 00:33:07,820 --> 00:33:10,430 You're going to actually calculate the electron density 646 00:33:10,430 --> 00:33:14,060 on a site as a measurement in some sense of the coordination 647 00:33:14,060 --> 00:33:18,080 as a sum of electron densities coming from its neighboring 648 00:33:18,080 --> 00:33:19,890 atoms, J. OK? 649 00:33:23,230 --> 00:33:28,600 What electron densities you use is kind of less relevant. 650 00:33:28,600 --> 00:33:31,330 In the original embedded atom and in most embedded atom codes 651 00:33:31,330 --> 00:33:34,930 still, they use essentially the atomic densities. 652 00:33:34,930 --> 00:33:38,950 So people have calculated the atomic electron densities 653 00:33:38,950 --> 00:33:40,180 with [INAUDIBLE] methods. 654 00:33:40,180 --> 00:33:42,700 And they're actually tabulated in these Clementi and Roetti 655 00:33:42,700 --> 00:33:43,200 tables. 656 00:33:46,520 --> 00:33:48,270 And you know, I'm just sort of showing you 657 00:33:48,270 --> 00:33:50,480 that they're basically parameterized 658 00:33:50,480 --> 00:33:53,960 by a bunch of Gaussians, but you can get those. 659 00:33:53,960 --> 00:33:56,970 So you can get those functions. 660 00:33:56,970 --> 00:33:58,040 We skipped through that. 661 00:33:58,040 --> 00:33:59,070 OK. 662 00:33:59,070 --> 00:34:01,590 So remember that if you have-- 663 00:34:01,590 --> 00:34:02,615 yes, sir. 664 00:34:02,615 --> 00:34:05,530 AUDIENCE: [INAUDIBLE]? 665 00:34:05,530 --> 00:34:06,970 PROFESSOR: Correct. 666 00:34:06,970 --> 00:34:09,100 Correct. 667 00:34:09,100 --> 00:34:11,110 Well, it's a distance dependent function. 668 00:34:11,110 --> 00:34:13,670 AUDIENCE: [INAUDIBLE] 669 00:34:13,670 --> 00:34:14,510 PROFESSOR: Exactly. 670 00:34:14,510 --> 00:34:14,929 Yeah. 671 00:34:14,929 --> 00:34:16,929 So it's sort of what people in quantum mechanics 672 00:34:16,929 --> 00:34:19,159 think of as sort of non self-consistent methods. 673 00:34:22,772 --> 00:34:25,400 You literally treat the electron density 674 00:34:25,400 --> 00:34:29,150 of the crystal as the overlap of the atomic densities 675 00:34:29,150 --> 00:34:31,400 of the atoms. 676 00:34:31,400 --> 00:34:33,650 The reason that saves you here is that you're actually 677 00:34:33,650 --> 00:34:36,043 not going to do quantum mechanics on that density, 678 00:34:36,043 --> 00:34:37,460 and you're just using that density 679 00:34:37,460 --> 00:34:40,260 as some measure of coordination. 680 00:34:40,260 --> 00:34:44,924 So [INAUDIBLE] I skipped some slides. 681 00:34:51,322 --> 00:34:52,739 Somehow, I lost the slide in here. 682 00:34:52,739 --> 00:34:53,449 Sorry about that. 683 00:34:59,800 --> 00:35:00,300 OK. 684 00:35:00,300 --> 00:35:00,883 You know what? 685 00:35:00,883 --> 00:35:02,163 I put it in the wrong place. 686 00:35:02,163 --> 00:35:03,070 Here we go. 687 00:35:03,070 --> 00:35:08,940 So the way you write the energy in the embedded atom method 688 00:35:08,940 --> 00:35:12,360 is you write it as the embedding energy, which 689 00:35:12,360 --> 00:35:15,780 is you sum over every atom. 690 00:35:15,780 --> 00:35:19,890 And you're looking for it's cohesive energy, which 691 00:35:19,890 --> 00:35:25,680 is some function F of the neighborhood 692 00:35:25,680 --> 00:35:27,750 measurements, which is the electron 693 00:35:27,750 --> 00:35:29,280 density at the site of i. 694 00:35:29,280 --> 00:35:32,190 And that's your summing from these functions F 695 00:35:32,190 --> 00:35:33,600 which are the projected densities 696 00:35:33,600 --> 00:35:38,580 from all the other atoms J. 697 00:35:38,580 --> 00:35:43,440 So this is the nonlinear part of the energy. 698 00:35:43,440 --> 00:35:49,350 Typically, that is supplemented with a pair potential. 699 00:35:49,350 --> 00:35:51,840 So the standard embedded atom method 700 00:35:51,840 --> 00:35:53,760 is a pair potential which is often 701 00:35:53,760 --> 00:35:57,930 just used for having a repulsive part and a nonlinear 702 00:35:57,930 --> 00:35:59,920 embedding energy. 703 00:35:59,920 --> 00:36:02,880 So that means we have to solve three problems, 704 00:36:02,880 --> 00:36:04,230 really-- the one we've solved. 705 00:36:04,230 --> 00:36:06,420 You need to know what the electron densities are. 706 00:36:06,420 --> 00:36:08,170 Take the Clementi and Roetti tables. 707 00:36:08,170 --> 00:36:10,170 You need to know what the embedding function is, 708 00:36:10,170 --> 00:36:12,690 and you need to know what the pair potential is. 709 00:36:12,690 --> 00:36:14,240 The pair potential, to be honest, 710 00:36:14,240 --> 00:36:15,850 you can use almost anything. 711 00:36:15,850 --> 00:36:19,440 It's mainly used for the hardcore repulsion part. 712 00:36:19,440 --> 00:36:21,900 You don't want the atoms to fly into each other. 713 00:36:21,900 --> 00:36:26,700 Typically, they use some screened electrostatic form 714 00:36:26,700 --> 00:36:30,990 which is essentially the product of charge distribution 715 00:36:30,990 --> 00:36:32,740 functions on atoms. 716 00:36:32,740 --> 00:36:35,820 But you know, nothing would stop you if you really wanted to 717 00:36:35,820 --> 00:36:37,350 from using a Leonard-Jones. 718 00:36:37,350 --> 00:36:39,000 I'll show you in a second that there 719 00:36:39,000 --> 00:36:44,010 is ambiguity about the division of energy between the embedding 720 00:36:44,010 --> 00:36:47,305 part and the pair potential is not unique. 721 00:36:47,305 --> 00:36:49,680 So that means, every time you change your pair potential, 722 00:36:49,680 --> 00:36:51,597 you'll have to change your embedding function. 723 00:36:51,597 --> 00:36:53,933 I'll show that in a second. 724 00:36:53,933 --> 00:36:55,725 What do you use for the embedding function? 725 00:37:01,250 --> 00:37:04,620 There are two ways that it can be done. 726 00:37:04,620 --> 00:37:08,750 People have written down analytical forms 727 00:37:08,750 --> 00:37:10,760 with some theoretical justification 728 00:37:10,760 --> 00:37:12,890 that I'm not going to go into. 729 00:37:12,890 --> 00:37:14,960 There's actually an excellent review article 730 00:37:14,960 --> 00:37:16,490 on the embedded atom methods which 731 00:37:16,490 --> 00:37:19,880 we post just for internal use on the website. 732 00:37:19,880 --> 00:37:22,820 It's a 1993 Materials Science and Engineering report. 733 00:37:22,820 --> 00:37:26,970 It's a really long paper, but it's really an excellent paper. 734 00:37:26,970 --> 00:37:29,688 So there are analytical forms. 735 00:37:29,688 --> 00:37:30,980 And here's a few of them given. 736 00:37:30,980 --> 00:37:32,647 And people have come up with other ones. 737 00:37:32,647 --> 00:37:34,430 More and more now, what people simply do 738 00:37:34,430 --> 00:37:37,160 is tabulate the embedding function. 739 00:37:37,160 --> 00:37:39,560 And really what they do is they tabulate it 740 00:37:39,560 --> 00:37:42,840 so that you exactly reproduce the equation of state. 741 00:37:42,840 --> 00:37:44,090 What is the equation of state? 742 00:37:44,090 --> 00:37:46,940 The equation of state is energy versus volume or lattice 743 00:37:46,940 --> 00:37:48,440 parameter. 744 00:37:48,440 --> 00:37:51,560 So say at every level parameter, you could try to get that 745 00:37:51,560 --> 00:37:55,250 energy with a better method-- say, quantum mechanics-- 746 00:37:55,250 --> 00:37:57,060 and then fit your embedding function 747 00:37:57,060 --> 00:37:59,910 so that you exactly reproduce that. 748 00:37:59,910 --> 00:38:04,280 And so, often, the embedding function is tabulated. 749 00:38:04,280 --> 00:38:10,030 But commonly it has this form. 750 00:38:10,030 --> 00:38:14,630 It's decaying, it's sort of decreasing convex, 751 00:38:14,630 --> 00:38:17,210 decreasing in energy and convex, and it 752 00:38:17,210 --> 00:38:18,950 needs both those properties. 753 00:38:18,950 --> 00:38:21,920 Well, why does it need to be decreasing in energy? 754 00:38:21,920 --> 00:38:26,060 So this is the embedding density and this is the value 755 00:38:26,060 --> 00:38:27,690 of the embedding function. 756 00:38:27,690 --> 00:38:30,650 Of course, as you put more density on a site, 757 00:38:30,650 --> 00:38:31,970 that comes from more neighbors. 758 00:38:31,970 --> 00:38:34,863 So your cohesive energy should be going down. 759 00:38:34,863 --> 00:38:37,280 That's why the energy of the embedding function goes down. 760 00:38:37,280 --> 00:38:41,450 But it needs to be convex because convexity gives you 761 00:38:41,450 --> 00:38:43,400 sort of the correct behavior. 762 00:38:43,400 --> 00:38:46,700 Convexity really tells you that adding a bond going from here 763 00:38:46,700 --> 00:38:47,450 to here-- 764 00:38:47,450 --> 00:38:50,540 let's say a bond is adding 0.01. 765 00:38:50,540 --> 00:38:56,150 So if we added a bond here, we'd get this change 766 00:38:56,150 --> 00:38:58,010 in embedding energy. 767 00:38:58,010 --> 00:39:03,220 If we added it here, we'd only get that change. 768 00:39:03,220 --> 00:39:05,520 So the function has to be convex in some sense. 769 00:39:05,520 --> 00:39:09,290 Decrease has to be per unit, and electron density 770 00:39:09,290 --> 00:39:12,260 has to be less for higher densities than for lower. 771 00:39:12,260 --> 00:39:15,080 And that's the definition of convex. 772 00:39:15,080 --> 00:39:19,640 The slope of it has to be increasing of this function. 773 00:39:19,640 --> 00:39:23,630 And you see, that's going to by construction now get 774 00:39:23,630 --> 00:39:26,300 some of our problems right because this shape 775 00:39:26,300 --> 00:39:28,310 of the function, the convexity is 776 00:39:28,310 --> 00:39:31,880 what tells you that the low coordination 777 00:39:31,880 --> 00:39:34,700 bonds are stronger than the high coordination bonds. 778 00:39:37,553 --> 00:39:38,970 So you can already guess from this 779 00:39:38,970 --> 00:39:40,887 that we're going to get a lot of things right. 780 00:39:48,380 --> 00:39:50,220 I'll show you some examples in a second. 781 00:39:53,860 --> 00:39:57,438 What's the physical concept that sort of justifies 782 00:39:57,438 --> 00:39:58,480 the embedded atom method? 783 00:40:03,190 --> 00:40:06,700 There are things you can do formally 784 00:40:06,700 --> 00:40:11,800 in ab initio theory showing that in fairly 785 00:40:11,800 --> 00:40:13,430 simple tight binding models. 786 00:40:13,430 --> 00:40:16,630 So these are things that have the structure 787 00:40:16,630 --> 00:40:20,560 of a proper quantum mechanical Hamiltonian, 788 00:40:20,560 --> 00:40:23,590 but they are parameterized forms of a Hamiltonian. 789 00:40:23,590 --> 00:40:25,270 But still, what they all tend to show 790 00:40:25,270 --> 00:40:29,440 is that the cohesive energy in highly delocalized systems 791 00:40:29,440 --> 00:40:31,390 to first order goals like the square root 792 00:40:31,390 --> 00:40:33,470 of the coordination. 793 00:40:33,470 --> 00:40:35,020 So you can get some justification 794 00:40:35,020 --> 00:40:37,450 for this nonlinear behavior. 795 00:40:37,450 --> 00:40:41,050 A somewhat intuitive concept of where the bonding energy comes 796 00:40:41,050 --> 00:40:46,510 from is that actually, if you put an atom 797 00:40:46,510 --> 00:40:52,570 in a metallic solid, where does it get a cohesive energy from? 798 00:40:52,570 --> 00:40:55,570 For the most part, it gets its cohesive energy 799 00:40:55,570 --> 00:40:59,410 from the delocalization of its electrons. 800 00:40:59,410 --> 00:41:03,292 And that's largely a kinetic energy effect. 801 00:41:03,292 --> 00:41:05,500 Remember that the kinetic energy in quantum mechanics 802 00:41:05,500 --> 00:41:07,930 is the curvature of the wave function. 803 00:41:07,930 --> 00:41:15,070 It's d squared phi, dx squared if you it in one dimension. 804 00:41:15,070 --> 00:41:19,690 So highly curved wave functions have high kinetic energy. 805 00:41:19,690 --> 00:41:22,420 So if you can actually delocalize, 806 00:41:22,420 --> 00:41:24,940 your wave function becomes, in some sense, 807 00:41:24,940 --> 00:41:27,190 much smoother with much less curvature, 808 00:41:27,190 --> 00:41:30,500 and you have lower kinetic energy. 809 00:41:30,500 --> 00:41:34,060 So if you believe that a lot of bonding cohesive energy 810 00:41:34,060 --> 00:41:36,280 and delocalized source comes from that, 811 00:41:36,280 --> 00:41:38,110 then the measure of cohesive energy 812 00:41:38,110 --> 00:41:42,520 is essentially how well you can delocalize it. 813 00:41:42,520 --> 00:41:47,380 So what do you measure with the embedding density? 814 00:41:47,380 --> 00:41:50,380 If you measure the electron density coming 815 00:41:50,380 --> 00:41:53,170 from atoms, the reason this works is that that's probably 816 00:41:53,170 --> 00:41:55,420 a very good measure over the number of states 817 00:41:55,420 --> 00:41:57,550 you can delocalize over. 818 00:41:57,550 --> 00:41:59,063 Things with higher electron density 819 00:41:59,063 --> 00:42:00,730 tend to have a higher number of states-- 820 00:42:00,730 --> 00:42:02,380 higher densities of states. 821 00:42:02,380 --> 00:42:04,780 And so you can delocalize over more states. 822 00:42:04,780 --> 00:42:10,210 And that's the somewhat intuitive explanation 823 00:42:10,210 --> 00:42:12,010 for why the embedded atom works. 824 00:42:12,010 --> 00:42:14,450 I mean, people have given somewhat other explanations. 825 00:42:14,450 --> 00:42:18,160 They could say, well, the electron density 826 00:42:18,160 --> 00:42:20,710 that I see from a neighboring atom measures 827 00:42:20,710 --> 00:42:23,600 the number of electrons I can bond with. 828 00:42:23,600 --> 00:42:27,130 But that's not totally true because if those electrons come 829 00:42:27,130 --> 00:42:31,810 from a filled shell, there's nothing I can do with them. 830 00:42:31,810 --> 00:42:34,120 So somehow, I tend to believe more 831 00:42:34,120 --> 00:42:36,580 it's really a measure of the delocalization you see. 832 00:42:49,240 --> 00:42:51,070 The embedded atom method is-- yes, sir? 833 00:42:51,070 --> 00:42:53,500 AUDIENCE: Did you say [INAUDIBLE]?? 834 00:43:00,790 --> 00:43:01,762 PROFESSOR: Correct. 835 00:43:01,762 --> 00:43:02,637 AUDIENCE: [INAUDIBLE] 836 00:43:02,637 --> 00:43:03,390 PROFESSOR: Yeah. 837 00:43:03,390 --> 00:43:03,890 Yeah. 838 00:43:03,890 --> 00:43:07,010 So in an atom, you actually have a much higher kinetic energy 839 00:43:07,010 --> 00:43:09,650 because your state is much more localized. 840 00:43:09,650 --> 00:43:11,150 The embedded atom method is probably 841 00:43:11,150 --> 00:43:14,360 the most popular one of all these effective medium 842 00:43:14,360 --> 00:43:15,050 theories. 843 00:43:15,050 --> 00:43:18,020 But you'll see other ones with other names 844 00:43:18,020 --> 00:43:21,350 that are in spirit very much the same-- things like the glue 845 00:43:21,350 --> 00:43:22,640 model. 846 00:43:22,640 --> 00:43:24,560 Finnis-Sinclair potentials, which 847 00:43:24,560 --> 00:43:27,560 slightly predate the embedded atom method, which 848 00:43:27,560 --> 00:43:30,390 are also non-linear potentials. 849 00:43:30,390 --> 00:43:33,920 So they're really particle potential pair functionals. 850 00:43:33,920 --> 00:43:36,230 And the equivalent crystal models Smith and Banerjee. 851 00:43:36,230 --> 00:43:40,940 These are all exactly in the same spirit. 852 00:43:40,940 --> 00:43:43,820 The brilliance of the embedded atom method 853 00:43:43,820 --> 00:43:47,300 was essentially that it was something for nothing. 854 00:43:47,300 --> 00:43:49,370 The computational cost is essentially 855 00:43:49,370 --> 00:43:52,430 the same as for doing pair potentials because you 856 00:43:52,430 --> 00:43:54,950 sum things only pairwise. 857 00:43:54,950 --> 00:43:57,410 If you think of what you need to do an embedded atom 858 00:43:57,410 --> 00:44:02,330 calculation, you have your pair potential. 859 00:44:02,330 --> 00:44:03,950 That's a pairwise sum. 860 00:44:03,950 --> 00:44:07,670 But your embedding energy is also a pairwise sum. 861 00:44:07,670 --> 00:44:09,800 Because what you have to do per atom, 862 00:44:09,800 --> 00:44:11,390 you just have to sum the electron 863 00:44:11,390 --> 00:44:16,430 density coming from a bunch of other atoms around you. 864 00:44:16,430 --> 00:44:17,773 So that's a pairwise sum. 865 00:44:17,773 --> 00:44:19,190 And then you just have to stick it 866 00:44:19,190 --> 00:44:21,860 in F, which is an extra evaluation. 867 00:44:21,860 --> 00:44:27,690 But embedded atom fundamentally scales only like N squared. 868 00:44:27,690 --> 00:44:29,660 So that was the brilliant part of it. 869 00:44:29,660 --> 00:44:32,270 Or in the end, that's what made it successful. 870 00:44:32,270 --> 00:44:36,890 You could essentially do much better things 871 00:44:36,890 --> 00:44:39,590 with almost the same effort as pair potentials. 872 00:44:39,590 --> 00:44:41,600 And that's why it sort of took off as a method. 873 00:44:41,600 --> 00:44:43,130 And pretty much anybody in metals 874 00:44:43,130 --> 00:44:46,310 now would use embedded atom if they 875 00:44:46,310 --> 00:44:47,840 want to get any physics right. 876 00:44:47,840 --> 00:44:50,570 If you want to do demonstration projects or things 877 00:44:50,570 --> 00:44:52,340 like want to do billions of atoms, 878 00:44:52,340 --> 00:44:55,010 there can be reasons to go through much simpler forces 879 00:44:55,010 --> 00:44:56,960 still to evaluate. 880 00:44:56,960 --> 00:44:59,360 But in metals, there's really no reason 881 00:44:59,360 --> 00:45:03,368 not to use EM over simple pair potentials. 882 00:45:06,950 --> 00:45:08,620 What do people fit to? 883 00:45:08,620 --> 00:45:11,650 People fit to anything, like I told you before. 884 00:45:11,650 --> 00:45:14,440 The original EM potentials were fit to a standard set 885 00:45:14,440 --> 00:45:17,110 of data, which was lattice parameter, sublimation 886 00:45:17,110 --> 00:45:22,180 energy, elastic constants, and vacancy formation energies. 887 00:45:22,180 --> 00:45:25,870 Now you'll regularly see papers in the literature claiming, 888 00:45:25,870 --> 00:45:28,390 I have a better potential than you for aluminum, 889 00:45:28,390 --> 00:45:30,670 and I have a better potential than you for nickel. 890 00:45:30,670 --> 00:45:32,210 And some of that is true. 891 00:45:32,210 --> 00:45:34,390 So people are not in the game of kind of super 892 00:45:34,390 --> 00:45:37,900 optimizing these potentials for what it's worth. 893 00:45:37,900 --> 00:45:40,990 But the original database was a set of standard potentials-- 894 00:45:40,990 --> 00:45:43,510 largely unknowable metal alloys. 895 00:45:43,510 --> 00:45:47,350 People have spent a lot of time finding better potentials 896 00:45:47,350 --> 00:45:50,380 on technologically important alloys like nickel aluminum. 897 00:45:50,380 --> 00:45:53,320 You'll find at least five potentials 898 00:45:53,320 --> 00:45:56,500 who all claim to be better than somebody else's potential 899 00:45:56,500 --> 00:45:58,090 on the nickel aluminum system. 900 00:46:02,770 --> 00:46:05,550 So what I want to do now is just show you some results. 901 00:46:05,550 --> 00:46:07,900 And we're not going to analyze these in great deal. 902 00:46:07,900 --> 00:46:10,440 But I want to show you so you get some idea of what 903 00:46:10,440 --> 00:46:12,750 people do with this stuff. 904 00:46:12,750 --> 00:46:17,700 Here's linear thermal expansion for a bunch of late transition 905 00:46:17,700 --> 00:46:22,020 metals calculated in EM and calculated in experiment. 906 00:46:22,020 --> 00:46:25,690 And these agree pretty well. 907 00:46:25,690 --> 00:46:30,090 That's maybe not a big surprise in close packed solids. 908 00:46:30,090 --> 00:46:32,250 The thermal expansion is largely a measure 909 00:46:32,250 --> 00:46:35,650 of the asymmetry of your equation of state. 910 00:46:35,650 --> 00:46:37,750 I mean, in complex materials, that's not true. 911 00:46:37,750 --> 00:46:40,270 But in simple, close packed metals. 912 00:46:40,270 --> 00:46:43,760 Remember, your equation of state is slightly asymmetric. 913 00:46:43,760 --> 00:46:48,630 It curves less towards the high lattice parameters 914 00:46:48,630 --> 00:46:49,880 and the low lattice parameter. 915 00:46:49,880 --> 00:46:55,950 So if you draw energy versus volume, 916 00:46:55,950 --> 00:46:58,500 it's asymmetric around the minimum. 917 00:46:58,500 --> 00:47:00,365 And that's why you have thermal expansion. 918 00:47:00,365 --> 00:47:00,990 Think about it. 919 00:47:00,990 --> 00:47:03,780 It's slightly easier to fluctuate to the higher volume 920 00:47:03,780 --> 00:47:07,350 than to the lower volume. 921 00:47:07,350 --> 00:47:08,285 Yes, sir? 922 00:47:08,285 --> 00:47:12,380 AUDIENCE: [INAUDIBLE] 923 00:47:12,380 --> 00:47:14,390 PROFESSOR: Oh, good point. 924 00:47:14,390 --> 00:47:16,370 There is, of course, no temperature dependence 925 00:47:16,370 --> 00:47:18,090 in the potential itself. 926 00:47:18,090 --> 00:47:22,370 So to do something like thermal expansion, 927 00:47:22,370 --> 00:47:26,240 you either have to do a dynamic simulation like molecular 928 00:47:26,240 --> 00:47:29,210 dynamics-- and we'll come to that-- 929 00:47:29,210 --> 00:47:32,090 or you have to explicitly calculate 930 00:47:32,090 --> 00:47:34,970 the entropic factors that give you thermal expansion, which 931 00:47:34,970 --> 00:47:37,430 is essentially the dependence of the phonon 932 00:47:37,430 --> 00:47:39,290 frequencies on volume. 933 00:47:39,290 --> 00:47:41,540 So you can indirectly calculate the thermal expansion. 934 00:47:41,540 --> 00:47:44,525 But we sort of haven't got into-- we 935 00:47:44,525 --> 00:47:45,770 will go into micro dynamics. 936 00:47:45,770 --> 00:47:47,437 But of course, in the potential, there's 937 00:47:47,437 --> 00:47:49,347 no temperature dependence. 938 00:47:58,410 --> 00:48:02,260 Here's activation barriers for self-diffusion in metals. 939 00:48:02,260 --> 00:48:05,020 These are almost too good to believe it. 940 00:48:05,020 --> 00:48:08,570 You actually don't get them this good with quantum mechanics. 941 00:48:08,570 --> 00:48:12,580 So it's almost surprising to me that they're this good, 942 00:48:12,580 --> 00:48:14,080 if you compare EM to external. 943 00:48:14,080 --> 00:48:18,980 So this is the activation barrier for-- 944 00:48:18,980 --> 00:48:19,720 I forget. 945 00:48:19,720 --> 00:48:22,020 But I thought it was interstitials self-diffusion, 946 00:48:22,020 --> 00:48:25,180 but I'm not for sure whether it was interstitial or vacancy 947 00:48:25,180 --> 00:48:26,395 self-diffusion. 948 00:48:30,360 --> 00:48:34,350 You know, here's a bunch of surface energies. 949 00:48:34,350 --> 00:48:38,580 These are not quite as good as the experiment. 950 00:48:38,580 --> 00:48:44,020 If you, for example, look at copper, 951 00:48:44,020 --> 00:48:50,480 here's the experimental number in ergs per centimeter squared. 952 00:48:50,480 --> 00:48:54,220 And here is the calculation on different facets. 953 00:48:54,220 --> 00:48:56,680 Because of course, often experimentally 954 00:48:56,680 --> 00:48:59,560 you measure some average of different facets. 955 00:48:59,560 --> 00:49:01,810 Unless you do careful single crystal work, 956 00:49:01,810 --> 00:49:04,653 you have some average. 957 00:49:04,653 --> 00:49:06,820 On the calculations you have to calculate the energy 958 00:49:06,820 --> 00:49:07,900 on a specific surface. 959 00:49:07,900 --> 00:49:09,520 And that's actually one of the things 960 00:49:09,520 --> 00:49:11,630 you're going to do in the lab. 961 00:49:11,630 --> 00:49:14,590 So these are a little lower. 962 00:49:14,590 --> 00:49:17,350 And that seems to be a rather consistent problem that is, 963 00:49:17,350 --> 00:49:20,670 for example, also in platinum. 964 00:49:20,670 --> 00:49:25,290 These are quite a bit lower than the experiment. 965 00:49:25,290 --> 00:49:28,950 So that may tell you that either there 966 00:49:28,950 --> 00:49:32,310 are more complicated quantum mechanical effects going on 967 00:49:32,310 --> 00:49:36,360 that you don't capture with these simple energy models. 968 00:49:36,360 --> 00:49:39,360 But sometimes, it also means the experiment is wrong. 969 00:49:39,360 --> 00:49:42,840 I think in this case, platinum is a fairly easy metal 970 00:49:42,840 --> 00:49:45,090 to deal with clean. 971 00:49:45,090 --> 00:49:46,980 But surface energies of materials 972 00:49:46,980 --> 00:49:50,750 is sort of notoriously difficult to get experimental. 973 00:49:54,590 --> 00:49:56,910 Here's a phonon dispersion curve for copper. 974 00:49:56,910 --> 00:49:58,620 Again, don't worry about the details. 975 00:49:58,620 --> 00:50:01,280 I just wanted to show you the kind of things people do. 976 00:50:01,280 --> 00:50:05,510 Calculations are the lines and the points 977 00:50:05,510 --> 00:50:07,280 are the measurements. 978 00:50:07,280 --> 00:50:10,310 So, reasonably good at some of these. 979 00:50:10,310 --> 00:50:12,890 High frequencies there seems to be some problems with. 980 00:50:17,200 --> 00:50:18,610 Melting point. 981 00:50:18,610 --> 00:50:21,070 Again, this would have been done with molecular dynamics 982 00:50:21,070 --> 00:50:23,050 and then free energy integration. 983 00:50:23,050 --> 00:50:27,840 And again, for most materials, these come out remarkably well. 984 00:50:27,840 --> 00:50:29,620 So, look at this. 985 00:50:29,620 --> 00:50:32,945 Except again for the very late transition metals 986 00:50:32,945 --> 00:50:34,570 like palladium and platinum, there just 987 00:50:34,570 --> 00:50:37,240 seems to be some amount of problems. 988 00:50:37,240 --> 00:50:40,840 I can tell you in a second about the limitations 989 00:50:40,840 --> 00:50:41,800 of these methods. 990 00:50:41,800 --> 00:50:48,080 And they may be related to the unusual electronic structure 991 00:50:48,080 --> 00:50:49,450 of palladium and platinum. 992 00:50:52,210 --> 00:50:54,280 You know, here's a pair correlation function 993 00:50:54,280 --> 00:50:56,290 in a liquid. 994 00:50:56,290 --> 00:50:58,150 [? People, ?] [? we're ?] [? good? ?] 995 00:50:58,150 --> 00:51:01,090 A grain boundary. 996 00:51:01,090 --> 00:51:02,890 This is something you simply could not 997 00:51:02,890 --> 00:51:05,590 do with a pair potential in metals 998 00:51:05,590 --> 00:51:08,020 because when you calculate a grain boundary, 999 00:51:08,020 --> 00:51:12,790 you really are dealing with different coordinated 1000 00:51:12,790 --> 00:51:14,020 environments. 1001 00:51:14,020 --> 00:51:16,120 And the painful thing about a grain boundary 1002 00:51:16,120 --> 00:51:20,477 is that you don't know ahead of time what kind of coordinations 1003 00:51:20,477 --> 00:51:21,310 you're going to see. 1004 00:51:21,310 --> 00:51:24,258 And you're going to see multiple coordinations. 1005 00:51:24,258 --> 00:51:26,050 See, but people with potentials always get, 1006 00:51:26,050 --> 00:51:29,070 if I know I'm going to deal with a surface, 1007 00:51:29,070 --> 00:51:31,290 I can make a pair potential that's 1008 00:51:31,290 --> 00:51:34,650 pretty good for those coordinations at the surface. 1009 00:51:34,650 --> 00:51:36,960 If I deal with the bulk, I make a pair potential 1010 00:51:36,960 --> 00:51:39,373 that's pretty good with bulk coordinations. 1011 00:51:39,373 --> 00:51:41,040 It's when you work with grain boundaries 1012 00:51:41,040 --> 00:51:43,980 that you kind of end up with all kinds of coordinations 1013 00:51:43,980 --> 00:51:45,930 because the atoms in the grain boundary, 1014 00:51:45,930 --> 00:51:48,930 some are high coordinated, some are low coordinated, 1015 00:51:48,930 --> 00:51:51,630 depending on how the grains contact each other. 1016 00:51:55,660 --> 00:52:00,070 So it's a great method, but it still has its limitations. 1017 00:52:00,070 --> 00:52:04,440 One is that the bonding is spherical. 1018 00:52:04,440 --> 00:52:07,950 You literally sum all the electron density 1019 00:52:07,950 --> 00:52:12,670 coming from around you, and you really just sum it up. 1020 00:52:12,670 --> 00:52:14,820 You don't keep track of its orientation dependence. 1021 00:52:14,820 --> 00:52:19,140 So having one atom there next to me and one atom on that side 1022 00:52:19,140 --> 00:52:22,693 is the same as having both straight in front of me 1023 00:52:22,693 --> 00:52:24,735 because I just sum up the electron density coming 1024 00:52:24,735 --> 00:52:27,810 from them So it's purely spherical. 1025 00:52:27,810 --> 00:52:30,330 That can be fixed. 1026 00:52:30,330 --> 00:52:34,380 About in the '90s, [? Mike ?] [? Baskis ?] developed 1027 00:52:34,380 --> 00:52:38,460 what's called a MEAM, and that just stands for modified 1028 00:52:38,460 --> 00:52:42,660 embedded atom method, which basically has embedding 1029 00:52:42,660 --> 00:52:47,160 functions that keep track of the angular dependence 1030 00:52:47,160 --> 00:52:49,110 of the electron density around you, 1031 00:52:49,110 --> 00:52:51,600 essentially by mapping the problem on to spherical 1032 00:52:51,600 --> 00:52:53,100 harmonics. 1033 00:52:53,100 --> 00:52:57,360 MEAM is not nearly as popular as the standard EM 1034 00:52:57,360 --> 00:53:02,310 because it adds a level of complexity to the method. 1035 00:53:02,310 --> 00:53:05,790 And it's not totally clear that you're 1036 00:53:05,790 --> 00:53:09,360 making things work because the physics is better 1037 00:53:09,360 --> 00:53:11,922 or you just have more fitting parameters. 1038 00:53:11,922 --> 00:53:14,130 [? Mike ?] [? Baskis ?] originally developed the MEAM 1039 00:53:14,130 --> 00:53:17,850 to work on silicon where angular dependence clearly seems 1040 00:53:17,850 --> 00:53:19,450 to play an important role. 1041 00:53:19,450 --> 00:53:21,950 Yes, sir? 1042 00:53:21,950 --> 00:53:25,304 AUDIENCE: [INAUDIBLE] 1043 00:53:37,652 --> 00:53:38,860 PROFESSOR: Well, your force-- 1044 00:53:38,860 --> 00:53:39,220 AUDIENCE: [INAUDIBLE] 1045 00:53:39,220 --> 00:53:39,887 PROFESSOR: Yeah. 1046 00:53:39,887 --> 00:53:41,860 Your force is directional dependent 1047 00:53:41,860 --> 00:53:45,490 because you're taking the derivative of energy 1048 00:53:45,490 --> 00:53:49,330 with respect to a vector, which is some distance. 1049 00:53:49,330 --> 00:53:50,277 But-- 1050 00:53:50,277 --> 00:53:53,546 AUDIENCE: [INAUDIBLE] 1051 00:54:00,130 --> 00:54:03,130 PROFESSOR: But again, the force will be directional dependent. 1052 00:54:03,130 --> 00:54:04,520 Let me show you an example. 1053 00:54:04,520 --> 00:54:07,720 Let's say I have a central atom here 1054 00:54:07,720 --> 00:54:11,950 and I got four atoms around it. 1055 00:54:11,950 --> 00:54:16,080 The force will be directional dependent because-- 1056 00:54:16,080 --> 00:54:17,380 what is the force? 1057 00:54:17,380 --> 00:54:20,050 The force is how the energy of an atom 1058 00:54:20,050 --> 00:54:23,535 changes as I move it into a particular direction. 1059 00:54:23,535 --> 00:54:24,910 So the force along this direction 1060 00:54:24,910 --> 00:54:27,730 is the energy change as I move the atom along this direction 1061 00:54:27,730 --> 00:54:30,880 because it is the derivative of the total energy with respect 1062 00:54:30,880 --> 00:54:35,000 to some unit vector in that direction. 1063 00:54:35,000 --> 00:54:36,652 See, if I change in this direction, 1064 00:54:36,652 --> 00:54:38,110 I'm going to significantly increase 1065 00:54:38,110 --> 00:54:40,480 the overlap with that atom. 1066 00:54:40,480 --> 00:54:44,980 But if I change it, say, in this direction, 1067 00:54:44,980 --> 00:54:49,030 that's not necessarily the same derivative 1068 00:54:49,030 --> 00:54:51,360 than in this direction. 1069 00:54:51,360 --> 00:54:55,080 So the force will be directional dependent just because 1070 00:54:55,080 --> 00:54:58,530 of the crystallography, but the energy is not. 1071 00:54:58,530 --> 00:55:01,530 So what I mean with that is that-- 1072 00:55:01,530 --> 00:55:08,570 let's call this some unit length L, and they're all at L. 1073 00:55:08,570 --> 00:55:15,750 If I do this, and here's my central atom, 1074 00:55:15,750 --> 00:55:22,400 and these are all L, that gives me exactly the same embedding 1075 00:55:22,400 --> 00:55:24,585 energy as in the first problem. 1076 00:55:24,585 --> 00:55:26,460 Of course, the total energy will be different 1077 00:55:26,460 --> 00:55:29,480 because these guys interact very closely. 1078 00:55:29,480 --> 00:55:31,370 But the bonding energy of the central atom 1079 00:55:31,370 --> 00:55:32,570 is exactly the same. 1080 00:55:32,570 --> 00:55:34,850 And that's what I mean with no angular dependence. 1081 00:55:37,590 --> 00:55:41,130 It doesn't in the end see what orbitals that electron 1082 00:55:41,130 --> 00:55:44,450 density comes from. 1083 00:55:44,450 --> 00:55:46,700 And when you think about angular dependence, 1084 00:55:46,700 --> 00:55:50,630 in the end, what it is, it is polarization of orbitals. 1085 00:55:50,630 --> 00:55:52,980 Think of sp3 bonding. 1086 00:55:52,980 --> 00:55:54,940 What is that really telling you? 1087 00:55:54,940 --> 00:55:58,010 If I do an sp3 hybrid along this direction, 1088 00:55:58,010 --> 00:56:00,290 I'm really starting to polarize the orbitals along-- 1089 00:56:00,290 --> 00:56:00,790 what is it-- 1090 00:56:00,790 --> 00:56:03,300 109 degrees in other directions. 1091 00:56:03,300 --> 00:56:08,190 So I really need to try to get my bonds in those directions. 1092 00:56:08,190 --> 00:56:11,090 So that's essentially what angular dependence comes from. 1093 00:56:13,640 --> 00:56:14,425 Does that help? 1094 00:56:14,425 --> 00:56:16,050 AUDIENCE: [? I ?] [? think ?] [? so. ?] 1095 00:56:16,050 --> 00:56:16,768 PROFESSOR: OK. 1096 00:56:16,768 --> 00:56:17,310 You think so. 1097 00:56:17,310 --> 00:56:20,540 Well, let me know if it doesn't. 1098 00:56:20,540 --> 00:56:22,970 OK. 1099 00:56:22,970 --> 00:56:24,710 The other problem is-- which is more 1100 00:56:24,710 --> 00:56:27,542 a technical problem-- is that the potential is not unique. 1101 00:56:27,542 --> 00:56:29,000 And this is something you just have 1102 00:56:29,000 --> 00:56:30,770 to be careful with if you start messing 1103 00:56:30,770 --> 00:56:33,410 around with potential files yourself. 1104 00:56:33,410 --> 00:56:38,420 If you're going to use it as a black box, you're probably OK. 1105 00:56:38,420 --> 00:56:40,100 Let me get you some notes on that. 1106 00:56:48,982 --> 00:56:51,190 The way you can see that the potential is not unique. 1107 00:56:54,230 --> 00:56:58,400 If I have an embedding function F of rho, 1108 00:56:58,400 --> 00:57:01,100 if I define another embedding function-- we 1109 00:57:01,100 --> 00:57:04,160 call it G of rho-- they're always the electron density. 1110 00:57:04,160 --> 00:57:09,650 And I say, that's F of rho plus some linear term in rho. 1111 00:57:12,380 --> 00:57:17,230 You could make the total energy function exactly the same 1112 00:57:17,230 --> 00:57:18,980 with either of the two embedding functions 1113 00:57:18,980 --> 00:57:20,660 just by changing the pair potential. 1114 00:57:20,660 --> 00:57:22,100 And let me show you that. 1115 00:57:22,100 --> 00:57:30,260 If I write the energy as the sum over atoms, 1116 00:57:30,260 --> 00:57:36,600 the embedding function on that atom, plus some air 1117 00:57:36,600 --> 00:57:37,500 potential part. 1118 00:57:48,130 --> 00:57:51,920 Let me substitute this in. 1119 00:57:51,920 --> 00:57:56,050 So I get that this is sum i. 1120 00:57:56,050 --> 00:58:13,850 Fi rho plus 1/2 sum iJ psi iJ, RiJ, plus-- 1121 00:58:13,850 --> 00:58:26,310 let me-- sum i k sum J [? not equal ?] i of f 1122 00:58:26,310 --> 00:58:36,090 rj where rho i was sum j f. 1123 00:58:36,090 --> 00:58:37,390 OK. 1124 00:58:37,390 --> 00:58:41,310 Essentially, if you want to look through the math, 1125 00:58:41,310 --> 00:58:43,380 don't worry too much about it. 1126 00:58:43,380 --> 00:58:46,320 If you do a linear transformation on the embedding 1127 00:58:46,320 --> 00:58:52,380 function, you see this term is linear in things coming 1128 00:58:52,380 --> 00:58:54,960 from the atoms around it. 1129 00:58:54,960 --> 00:58:58,990 It's linear in these electron density functions. 1130 00:58:58,990 --> 00:59:02,220 So anything that's linear in functions 1131 00:59:02,220 --> 00:59:05,593 that I sum from the other atoms is just like a pair potential 1132 00:59:05,593 --> 00:59:07,260 because that's what a pair potential is. 1133 00:59:07,260 --> 00:59:09,900 I sum contributions, OK? 1134 00:59:09,900 --> 00:59:12,720 I add up contributions from the things around me. 1135 00:59:12,720 --> 00:59:15,580 So I'm linear in that contribution. 1136 00:59:15,580 --> 00:59:19,200 So any linear part of the embedding function 1137 00:59:19,200 --> 00:59:20,600 just looks like a potential. 1138 00:59:20,600 --> 00:59:21,870 This is a pair potential. 1139 00:59:21,870 --> 00:59:24,300 This is the sum over [? 2n. ?] This is i an J, 1140 00:59:24,300 --> 00:59:25,950 and I add something up. 1141 00:59:25,950 --> 00:59:30,600 So this I could just add the psi. 1142 00:59:30,600 --> 00:59:34,320 So if I add psi to this and use the embedding function F, 1143 00:59:34,320 --> 00:59:37,440 I have exactly the same as using the embedding function G 1144 00:59:37,440 --> 00:59:39,600 and using just the potential psi. 1145 00:59:39,600 --> 00:59:43,320 So the linear part you can move arbitrarily 1146 00:59:43,320 --> 00:59:48,150 between the embedding function and the pair potential. 1147 00:59:48,150 --> 00:59:53,170 So that tells you something else that's kind of important. 1148 00:59:53,170 --> 00:59:57,160 When is a pair potential alone good in metals? 1149 01:00:02,520 --> 01:00:05,610 It's when the embedding part doesn't change. 1150 01:00:05,610 --> 01:00:09,970 If my atoms stay roughly at the same density, 1151 01:00:09,970 --> 01:00:13,900 then all the changes come from the pair potential part. 1152 01:00:13,900 --> 01:00:15,010 OK? 1153 01:00:15,010 --> 01:00:19,090 So what would you do-- 1154 01:00:19,090 --> 01:00:42,560 [INAUDIBLE]---- if you're having an embedding function 1155 01:00:42,560 --> 01:00:44,970 as a function of rho. 1156 01:00:44,970 --> 01:00:49,940 If you're working, say, at this density? 1157 01:00:49,940 --> 01:00:51,690 Let's sort of say the density of the bulk, 1158 01:00:51,690 --> 01:00:53,790 and you're studying bulk problems. 1159 01:00:53,790 --> 01:00:58,600 You could take the slope of this, take the linear term, 1160 01:00:58,600 --> 01:01:01,170 make a linear approximation to that 1161 01:01:01,170 --> 01:01:02,910 and add that to the pair potential, 1162 01:01:02,910 --> 01:01:05,160 and now you wouldn't have to worry about the embedding 1163 01:01:05,160 --> 01:01:06,900 function. 1164 01:01:06,900 --> 01:01:09,720 So at constant density, pair potentials 1165 01:01:09,720 --> 01:01:11,860 are actually pretty good. 1166 01:01:11,860 --> 01:01:13,380 It's when the density around you-- 1167 01:01:13,380 --> 01:01:15,013 the electron density around you changes 1168 01:01:15,013 --> 01:01:17,430 which is, in essence, a measure of the coordination change 1169 01:01:17,430 --> 01:01:19,770 that you run into trouble. 1170 01:01:19,770 --> 01:01:22,470 And so, people have gone into that game of making 1171 01:01:22,470 --> 01:01:26,138 density dependent potentials. 1172 01:01:26,138 --> 01:01:27,930 And so you would have a different potential 1173 01:01:27,930 --> 01:01:30,480 when you're near a surface than when you're near the bulk. 1174 01:01:40,460 --> 01:01:42,945 So my summary on effective medium theories 1175 01:01:42,945 --> 01:01:44,570 which I showed by embedded atom methods 1176 01:01:44,570 --> 01:01:47,198 is, in essence, that there's no reason not to use them. 1177 01:01:47,198 --> 01:01:48,740 If you've decided you're going to use 1178 01:01:48,740 --> 01:01:53,990 an empirical model for metals, there's really no reason at all 1179 01:01:53,990 --> 01:01:56,420 to use pair potentials. 1180 01:01:56,420 --> 01:02:01,393 Embedded atom basically does better at very little cost. 1181 01:02:01,393 --> 01:02:02,810 Saying that it does better doesn't 1182 01:02:02,810 --> 01:02:06,180 say that the does everything. 1183 01:02:06,180 --> 01:02:08,810 And I remember talking to people like [? Mike ?] [? Baskis ?] 1184 01:02:08,810 --> 01:02:09,860 who invented this. 1185 01:02:09,860 --> 01:02:13,530 They're embarrassed for kind of things people now do 1186 01:02:13,530 --> 01:02:14,780 with the embedded atom method. 1187 01:02:14,780 --> 01:02:17,900 Like with any successful method, people push it way too hard 1188 01:02:17,900 --> 01:02:21,080 and try to calculate all kinds of subtle energy effects 1189 01:02:21,080 --> 01:02:23,840 that it just never was made for. 1190 01:02:23,840 --> 01:02:25,460 And you see people write papers on, 1191 01:02:25,460 --> 01:02:29,810 if I change my constant by 1,000 here, 1192 01:02:29,810 --> 01:02:31,760 I can get this phase correct. 1193 01:02:31,760 --> 01:02:35,150 But in the end, it gives you these gross coordination 1194 01:02:35,150 --> 01:02:38,900 effects right, but it does not give you very subtle 1195 01:02:38,900 --> 01:02:41,570 hybridization effects right. 1196 01:02:41,570 --> 01:02:46,550 Why, for example, were platinum and palladium not so good? 1197 01:02:46,550 --> 01:02:50,800 Because platinum and palladium have almost filled or filled 1198 01:02:50,800 --> 01:02:53,180 D bands. 1199 01:02:53,180 --> 01:02:56,150 And as a function of the environment around platinum 1200 01:02:56,150 --> 01:02:59,240 and palladium, there is very subtle electron transfer 1201 01:02:59,240 --> 01:03:01,710 between the D band and the S band. 1202 01:03:01,710 --> 01:03:02,380 OK? 1203 01:03:02,380 --> 01:03:04,380 And when you're just measuring electron density, 1204 01:03:04,380 --> 01:03:06,200 that all looks the same, but those states 1205 01:03:06,200 --> 01:03:08,750 behave very differently. 1206 01:03:08,750 --> 01:03:11,420 So there will be always subtle electronic change 1207 01:03:11,420 --> 01:03:12,650 that you simply won't get. 1208 01:03:12,650 --> 01:03:17,770 And if your system is really influenced strongly by those, 1209 01:03:17,770 --> 01:03:19,300 you won't get them right. 1210 01:03:19,300 --> 01:03:23,480 For example, people try to get complex crystal structure 1211 01:03:23,480 --> 01:03:27,320 differences in metals with EM. 1212 01:03:27,320 --> 01:03:29,230 You will just never get there, period. 1213 01:03:31,770 --> 01:03:33,500 Let's say you make an alloy-- 1214 01:03:33,500 --> 01:03:38,990 palladium ruthenium, whatever-- the crystal structure energy 1215 01:03:38,990 --> 01:03:41,000 difference in that [? material, ?] like in most 1216 01:03:41,000 --> 01:03:45,260 metals over the order of 5 to 10 millielectron [? volt. ?] You 1217 01:03:45,260 --> 01:03:48,500 cannot get things right on that scale with these kind of simple 1218 01:03:48,500 --> 01:03:50,340 models, so you shouldn't try too hard. 1219 01:03:50,340 --> 01:03:52,340 But on the other hand, if you're going to study, 1220 01:03:52,340 --> 01:03:58,010 say, I don't know, fracture in a material in a simple element 1221 01:03:58,010 --> 01:04:01,160 like one of these big movies I showed you, 1222 01:04:01,160 --> 01:04:03,830 that's to a large extent a topological event. 1223 01:04:06,790 --> 01:04:10,515 You will get a lot of essential pieces of that right. 1224 01:04:10,515 --> 01:04:11,390 Because what is that? 1225 01:04:11,390 --> 01:04:13,760 That's a coordination change event, largely. 1226 01:04:13,760 --> 01:04:17,278 You will get a lot of that right by doing EM over potentials. 1227 01:04:23,270 --> 01:04:25,250 This is a great website I only added 1228 01:04:25,250 --> 01:04:27,380 that this year to the lecture I found. 1229 01:04:27,380 --> 01:04:29,330 There's a group in Japan that called 1230 01:04:29,330 --> 01:04:34,820 themselves the [? EAMers. ?] They even have a blog now. 1231 01:04:34,820 --> 01:04:37,570 And so there's a lot of cool stuff you can download. 1232 01:04:37,570 --> 01:04:39,890 They have potential files. 1233 01:04:39,890 --> 01:04:44,120 You can get EM code from the Sandia site. 1234 01:04:44,120 --> 01:04:46,050 It's a little hard to get to, because Sandia 1235 01:04:46,050 --> 01:04:50,240 being a national lab, they're totally 1236 01:04:50,240 --> 01:04:53,630 paranoid about downloading anything. 1237 01:04:53,630 --> 01:04:56,480 But there are sources where you can get the EM code. 1238 01:04:56,480 --> 01:04:59,360 It is technically freely available. 1239 01:04:59,360 --> 01:05:02,120 It's just not necessarily trivial to get to it. 1240 01:05:06,920 --> 01:05:12,120 So we talked about pair functionals, 1241 01:05:12,120 --> 01:05:13,870 which was this direction. 1242 01:05:13,870 --> 01:05:15,390 So the other direction is obviously 1243 01:05:15,390 --> 01:05:18,240 to go to cluster potentials. 1244 01:05:18,240 --> 01:05:20,880 Well, you know, I don't like my expansion 1245 01:05:20,880 --> 01:05:22,740 in just two-body terms. 1246 01:05:22,740 --> 01:05:25,430 I'll start adding three-body terms. 1247 01:05:25,430 --> 01:05:26,920 It's the obvious extension. 1248 01:05:26,920 --> 01:05:29,730 And if you don't like that, you can go to four-body terms. 1249 01:05:35,450 --> 01:05:37,280 That's definitely done. 1250 01:05:37,280 --> 01:05:42,110 And for reasons I'll go into in the next lecture on Tuesday, 1251 01:05:42,110 --> 01:05:43,730 it's extremely popular in the world 1252 01:05:43,730 --> 01:05:46,610 of organics, where it's actually really, really powerful. 1253 01:05:46,610 --> 01:05:49,940 And we'll go into why that is the case. 1254 01:05:49,940 --> 01:05:51,740 But a typical material in condensed matter 1255 01:05:51,740 --> 01:05:54,500 where obviously this was done was silicon. 1256 01:05:54,500 --> 01:05:57,320 Silicon is a low coordinated solid-- only fourfold 1257 01:05:57,320 --> 01:05:59,390 coordinated in the diamond cubic structure. 1258 01:05:59,390 --> 01:06:01,830 It's extremely hard to get that with a pair potential 1259 01:06:01,830 --> 01:06:06,860 because again, your systems always try to form FCC or HGP, 1260 01:06:06,860 --> 01:06:09,000 essentially. 1261 01:06:09,000 --> 01:06:13,550 So what you can do is build a three-body potential 1262 01:06:13,550 --> 01:06:18,240 that favors these 190-degree tetrahedral angles. 1263 01:06:18,240 --> 01:06:20,480 And as soon as you do that, if you actually 1264 01:06:20,480 --> 01:06:22,520 make that force strong enough, you 1265 01:06:22,520 --> 01:06:25,813 can do almost nothing else than form diamond cubic. 1266 01:06:25,813 --> 01:06:27,230 Because diamond cubic is basically 1267 01:06:27,230 --> 01:06:31,745 the way to pack atoms and put them under all these 109-- 1268 01:06:31,745 --> 01:06:36,210 what is it-- .4 or something degree angles. 1269 01:06:36,210 --> 01:06:38,780 So if you make a strong enough three-body force, 1270 01:06:38,780 --> 01:06:45,740 you will by definition end up in the diamond cubic structure. 1271 01:06:45,740 --> 01:06:49,190 If you work with three coordinates, 1272 01:06:49,190 --> 01:06:51,290 you can also transform that and instead 1273 01:06:51,290 --> 01:06:54,500 work with two distances and an angle. 1274 01:06:54,500 --> 01:06:57,600 So if I have three atoms, the coordinates of them, 1275 01:06:57,600 --> 01:07:00,800 I can also always say that to have the relation between them, 1276 01:07:00,800 --> 01:07:03,890 I just need to have these two distances and this angle that 1277 01:07:03,890 --> 01:07:06,560 fully defines the three-body problem. 1278 01:07:06,560 --> 01:07:08,600 And that's usually how it's done. 1279 01:07:08,600 --> 01:07:12,020 People have pair potentials that take care of the distance 1280 01:07:12,020 --> 01:07:13,500 dependence of that. 1281 01:07:13,500 --> 01:07:15,740 And then they have an angular potential 1282 01:07:15,740 --> 01:07:17,960 which takes care of the angle between these. 1283 01:07:17,960 --> 01:07:20,300 And often, the prefactor of the angular potential 1284 01:07:20,300 --> 01:07:22,610 will have some distance depends in it. 1285 01:07:22,610 --> 01:07:24,380 Because of course, maybe you want 1286 01:07:24,380 --> 01:07:26,810 to enforce this 190-degree angle when 1287 01:07:26,810 --> 01:07:28,700 your two other atoms are close. 1288 01:07:28,700 --> 01:07:32,300 But when they're very far, you don't really care anymore. 1289 01:07:32,300 --> 01:07:35,720 What's the possible choices for angular potentials? 1290 01:07:35,720 --> 01:07:40,700 The simplest one is simply a quadratic-- a harmonic function 1291 01:07:40,700 --> 01:07:42,650 in the angle. 1292 01:07:42,650 --> 01:07:45,710 If you want to impose an angle theta naught, 1293 01:07:45,710 --> 01:07:50,970 you make a potential that's literally 1294 01:07:50,970 --> 01:07:54,140 a quadratic around theta naught. 1295 01:07:54,140 --> 01:07:57,450 So, extremely simple to evaluate. 1296 01:07:57,450 --> 01:08:00,150 So that function's actually often used. 1297 01:08:00,150 --> 01:08:01,940 You can also use-- 1298 01:08:01,940 --> 01:08:06,120 the disadvantage of this one is that it's not periodic. 1299 01:08:06,120 --> 01:08:08,145 Angle measurements should be periodic. 1300 01:08:08,145 --> 01:08:10,020 So you can kind of get into trouble with this 1301 01:08:10,020 --> 01:08:11,980 if you go too far from equilibrium. 1302 01:08:11,980 --> 01:08:14,940 So another thing you can use is a cosine function which 1303 01:08:14,940 --> 01:08:16,950 has the proper periodicity. 1304 01:08:16,950 --> 01:08:18,540 If you take-- 1305 01:08:18,540 --> 01:08:19,560 I don't know-- 1306 01:08:19,560 --> 01:08:22,890 109 degrees or whatever angle you want, or you 1307 01:08:22,890 --> 01:08:24,470 take its complement-- 1308 01:08:24,470 --> 01:08:28,359 360 minus that-- you should get the same answer. 1309 01:08:28,359 --> 01:08:31,380 And so, cosine functions will do that right. 1310 01:08:31,380 --> 01:08:34,890 And this one, cosine theta plus 1/3, this part 1311 01:08:34,890 --> 01:08:41,628 is minimal when theta is 109 with tetrahedral angle. 1312 01:08:41,628 --> 01:08:43,439 [CLEARS THROAT] Excuse me. 1313 01:08:43,439 --> 01:08:44,760 Losing my voice. 1314 01:08:44,760 --> 01:08:47,370 So it works working with three-body potentials. 1315 01:08:47,370 --> 01:08:49,300 It's, of course, more work. 1316 01:08:49,300 --> 01:08:51,930 It's not [? order ?] N cubed. 1317 01:08:51,930 --> 01:08:54,810 Because, for every atom, you now have 1318 01:08:54,810 --> 01:08:58,500 to look at distance and angles with pairs of atoms. 1319 01:08:58,500 --> 01:09:02,370 OK so it becomes an N cubed operation, so it's more work. 1320 01:09:02,370 --> 01:09:05,760 And calculating the forces as the derivatives of the energy 1321 01:09:05,760 --> 01:09:07,450 is also quite a bit more work. 1322 01:09:17,710 --> 01:09:20,800 The most famous three-body potential is probably 1323 01:09:20,800 --> 01:09:24,609 the Stillinger Webber potential which, as you can see, 1324 01:09:24,609 --> 01:09:27,189 has this cosine theta form for the-- 1325 01:09:27,189 --> 01:09:30,130 this is the angular part. 1326 01:09:30,130 --> 01:09:34,029 This is the distance dependence of the angular part. 1327 01:09:34,029 --> 01:09:38,529 So how much should you penalize angular deviations [INAUDIBLE].. 1328 01:09:38,529 --> 01:09:40,500 And then this is the two-body part. 1329 01:09:40,500 --> 01:09:44,500 Just sort of a more [? slight ?] potential. 1330 01:09:44,500 --> 01:09:48,042 Originally, Stillinger Webber, when they actually 1331 01:09:48,042 --> 01:09:49,750 wrote down the function, they didn't even 1332 01:09:49,750 --> 01:09:52,069 fit it very carefully. 1333 01:09:52,069 --> 01:09:55,150 I mean, it was a great form, but they just kind of did 1334 01:09:55,150 --> 01:09:56,260 a rough fit. 1335 01:09:56,260 --> 01:09:58,510 And so, the original Stillinger Webber parameters 1336 01:09:58,510 --> 01:10:00,738 and the ones people now use sometimes more recently 1337 01:10:00,738 --> 01:10:01,780 are not the same anymore. 1338 01:10:01,780 --> 01:10:03,190 The potential has the same form. 1339 01:10:03,190 --> 01:10:05,650 But some people have done recently better fits. 1340 01:10:05,650 --> 01:10:09,220 So when somebody says that Stillinger Webber potential, 1341 01:10:09,220 --> 01:10:11,740 there are variations of it for the constants. 1342 01:10:19,700 --> 01:10:23,110 So what do you get right with a [? sale ?] 1343 01:10:23,110 --> 01:10:27,190 something like a Stillinger Webber potential? 1344 01:10:27,190 --> 01:10:30,070 Well, you can kind of guess why people did this. 1345 01:10:30,070 --> 01:10:31,620 Because silicon is so important, they 1346 01:10:31,620 --> 01:10:33,120 wanted to be able to model silicon right. 1347 01:10:33,120 --> 01:10:35,770 So it's important to understand which pieces you get right. 1348 01:10:38,770 --> 01:10:42,910 One of the critical issues was surface reconstruction. 1349 01:10:42,910 --> 01:10:44,836 If you take silicon 111-- 1350 01:10:44,836 --> 01:10:47,650 sorry, 100. 1351 01:10:47,650 --> 01:10:51,400 If you truncate it, it actually undergoes 1352 01:10:51,400 --> 01:10:53,530 a 2 times 1 reconstruction. 1353 01:10:53,530 --> 01:11:00,170 What essentially happens if you take this top layer atoms, 1354 01:11:00,170 --> 01:11:03,550 they're only bonded to two atoms below them, 1355 01:11:03,550 --> 01:11:06,070 and so they dimerize on the surface. 1356 01:11:06,070 --> 01:11:08,860 So basically, these move together 1357 01:11:08,860 --> 01:11:11,570 in sort of double rows. 1358 01:11:11,570 --> 01:11:13,590 I need a better color. 1359 01:11:13,590 --> 01:11:16,560 And then these will move together. 1360 01:11:16,560 --> 01:11:19,950 And so they double the periodicity in one direction. 1361 01:11:19,950 --> 01:11:22,040 And so you have these dimers that form. 1362 01:11:22,040 --> 01:11:24,080 And because of that, the surface atom 1363 01:11:24,080 --> 01:11:26,510 is bonded to three other atoms-- 1364 01:11:26,510 --> 01:11:29,622 two below it, one all the way in the surface. 1365 01:11:29,622 --> 01:11:30,830 And so they're better bonded. 1366 01:11:30,830 --> 01:11:33,290 And that's the 2 by 1 reconstruction. 1367 01:11:33,290 --> 01:11:36,290 The 2 by 1 reconstruction you get right with the Stillinger 1368 01:11:36,290 --> 01:11:37,370 Webber potential. 1369 01:11:37,370 --> 01:11:39,860 You'd actually get it right with almost anything. 1370 01:11:39,860 --> 01:11:43,370 It's essentially a topological reconstruction. 1371 01:11:43,370 --> 01:11:47,030 You know, atoms like to have more atoms around them to bond, 1372 01:11:47,030 --> 01:11:48,110 period. 1373 01:11:48,110 --> 01:11:51,560 And so that's what drives a 2 by 1 reconstruction. 1374 01:11:51,560 --> 01:11:55,970 At the top layer, atoms now are at least threefold coordinated 1375 01:11:55,970 --> 01:11:59,960 and they're actually almost fourfold coordinated instead 1376 01:11:59,960 --> 01:12:04,350 of twofold coordinated at just an unreconstructed terminated 1377 01:12:04,350 --> 01:12:04,850 surface. 1378 01:12:14,310 --> 01:12:18,270 But here's the reconstruction on the 111 surface. 1379 01:12:18,270 --> 01:12:22,500 The reconstruction on the 111 surface is 7 by 7. 1380 01:12:22,500 --> 01:12:27,480 So it forms a much larger unit cell on the surface. 1381 01:12:27,480 --> 01:12:30,090 And that is not reproduced by the Stillinger Webber 1382 01:12:30,090 --> 01:12:30,780 potential. 1383 01:12:30,780 --> 01:12:32,870 So clearly, that's a more subtle effect. 1384 01:12:42,300 --> 01:12:45,480 Silicon, being such an important material, 1385 01:12:45,480 --> 01:12:48,870 has a multitude of potentials. 1386 01:12:48,870 --> 01:12:52,140 Everybody sort of has made one and claims 1387 01:12:52,140 --> 01:12:55,080 theirs is better than yours. 1388 01:12:55,080 --> 01:12:57,940 And I just show this graph not for the details, 1389 01:12:57,940 --> 01:13:02,520 but to show you that this isn't even a tenth 1390 01:13:02,520 --> 01:13:03,990 of the potentials made for silicon, 1391 01:13:03,990 --> 01:13:05,230 but some of the famous ones. 1392 01:13:05,230 --> 01:13:07,530 SW-- Stillinger Webber-- 1393 01:13:07,530 --> 01:13:10,450 [? bismuth, ?] [? Harman, ?] and the Tersoff potentials, 1394 01:13:10,450 --> 01:13:12,970 which are quite famous for silicon, are not even on here. 1395 01:13:12,970 --> 01:13:17,070 But if you see, this is the angular part 1396 01:13:17,070 --> 01:13:18,060 of these potentials. 1397 01:13:18,060 --> 01:13:20,760 It's very different. 1398 01:13:20,760 --> 01:13:25,470 They all give the same answer, unlike static properties 1399 01:13:25,470 --> 01:13:28,660 like the elastic properties of silicon, the cohesive energy, 1400 01:13:28,660 --> 01:13:30,420 the lattice parameter. 1401 01:13:30,420 --> 01:13:33,720 But it's sort of divided differently between pair parts 1402 01:13:33,720 --> 01:13:34,590 and angular parts. 1403 01:13:37,230 --> 01:13:39,030 This one clearly wouldn't give you 1404 01:13:39,030 --> 01:13:42,570 the same stiffness of bond bending 1405 01:13:42,570 --> 01:13:45,750 around the typical dihedral angle. 1406 01:13:45,750 --> 01:13:46,920 And that shows up. 1407 01:13:46,920 --> 01:13:50,680 There's what I thought was a really nice paper. 1408 01:13:50,680 --> 01:13:51,990 Oh, and I'm sorry, I-- oh, no. 1409 01:13:51,990 --> 01:13:52,820 The reference [? isn't ?] here. 1410 01:13:52,820 --> 01:13:54,945 This is the [? Norman ?] [? and ?] [? paper ?] from 1411 01:13:54,945 --> 01:13:59,280 Physical Review B. They did what I thought was brilliant. 1412 01:13:59,280 --> 01:14:01,320 They did exactly the same simulation 1413 01:14:01,320 --> 01:14:04,140 with three different potentials. 1414 01:14:04,140 --> 01:14:06,210 Left is the Stillinger Webber potential, 1415 01:14:06,210 --> 01:14:09,510 and then middle column is Tersoff [? 2. ?] This is 1416 01:14:09,510 --> 01:14:12,120 the Tersoff potential, and then Tersoff [? 3. ?] So this is 1417 01:14:12,120 --> 01:14:15,240 exactly the same Monte Carlo simulation with the three 1418 01:14:15,240 --> 01:14:16,210 potentials. 1419 01:14:16,210 --> 01:14:20,310 And this is the [? 001 ?] or [? 100 ?] surface of silicon. 1420 01:14:20,310 --> 01:14:24,450 So remember, that's the one I showed you that dimerizes. 1421 01:14:24,450 --> 01:14:28,443 And so the top line is at low temperature. 1422 01:14:32,070 --> 01:14:35,560 Tersoff [? 3 ?] doesn't dimerize. 1423 01:14:35,560 --> 01:14:38,760 Tersoff 2 does sort of dimerize. 1424 01:14:38,760 --> 01:14:40,410 And so does Stillinger Webber, but they 1425 01:14:40,410 --> 01:14:41,410 look slightly different. 1426 01:14:41,410 --> 01:14:42,993 But these are in reasonable agreement. 1427 01:14:42,993 --> 01:14:48,180 If you go higher in temperature, then Tersoff [? 3 ?] dimerizes. 1428 01:14:48,180 --> 01:14:49,710 So does the Stillinger Webber. 1429 01:14:49,710 --> 01:14:53,580 But Tersoff [? 2 ?] starts kind of looking quite different 1430 01:14:53,580 --> 01:14:54,420 already. 1431 01:14:54,420 --> 01:14:55,920 So these are potentials that, if you 1432 01:14:55,920 --> 01:14:59,790 looked at how they reproduce static properties, 1433 01:14:59,790 --> 01:15:03,000 they all do extremely well. 1434 01:15:03,000 --> 01:15:06,390 But it's extremely hard to fit to dynamical properties 1435 01:15:06,390 --> 01:15:09,480 because you don't have that as quantitative information 1436 01:15:09,480 --> 01:15:10,770 very often. 1437 01:15:10,770 --> 01:15:14,130 Like, first of all, we don't know 1438 01:15:14,130 --> 01:15:16,830 what the right answer is here. 1439 01:15:16,830 --> 01:15:21,720 But even if we did, let's say, somebody does brilliant STM 1440 01:15:21,720 --> 01:15:24,540 and we know the right answer, it's very hard to feed 1441 01:15:24,540 --> 01:15:27,280 that back into a potential. 1442 01:15:27,280 --> 01:15:29,010 And so the reason I'm telling you 1443 01:15:29,010 --> 01:15:31,980 this is not to dissuade you from using [INAUDIBLE] 1444 01:15:31,980 --> 01:15:32,850 when you need them. 1445 01:15:32,850 --> 01:15:35,865 It's just to make sure you're cautious. 1446 01:15:39,300 --> 01:15:42,450 There's a big reason that people do ab initio calculations 1447 01:15:42,450 --> 01:15:43,650 these days. 1448 01:15:43,650 --> 01:15:46,270 It's not because they necessarily like it, 1449 01:15:46,270 --> 01:15:47,760 but it's because the results tend 1450 01:15:47,760 --> 01:15:51,090 to be more reliable, or at least predictable. 1451 01:15:51,090 --> 01:15:53,370 The errors are more predictable, let's put it 1452 01:15:53,370 --> 01:15:56,430 that way, which is a useful thing to have. 1453 01:15:56,430 --> 01:15:57,720 Having error is one thing. 1454 01:15:57,720 --> 01:16:01,320 Having unpredictable error is a lot worse. 1455 01:16:01,320 --> 01:16:05,130 But still, there will always be problems where they're just 1456 01:16:05,130 --> 01:16:07,770 too complex, or you want to do them fast where you 1457 01:16:07,770 --> 01:16:09,630 want to use pair potentials. 1458 01:16:09,630 --> 01:16:11,970 Just don't always believe everything you see. 1459 01:16:14,856 --> 01:16:17,280 Try to fit as close to where you're 1460 01:16:17,280 --> 01:16:19,770 going to explore the physics that's essentially 1461 01:16:19,770 --> 01:16:25,680 [? meaning. ?] 1462 01:16:25,680 --> 01:16:27,820 OK, here's a bunch of references. 1463 01:16:27,820 --> 01:16:30,960 I'll actually put these on the Stellar website. 1464 01:16:30,960 --> 01:16:36,390 We'll finish the empirical potential lecture. 1465 01:16:36,390 --> 01:16:38,460 We'll go into organics and a little bit 1466 01:16:38,460 --> 01:16:41,058 of polymers on Tuesday. 1467 01:16:41,058 --> 01:16:42,600 And then Professor [? Marzari ?] will 1468 01:16:42,600 --> 01:16:46,080 start giving you introduction to basic quantum mechanics. 1469 01:16:46,080 --> 01:16:49,512 And then we'll start sort of ab initio energy methods, 1470 01:16:49,512 --> 01:16:51,220 and Professor [? Marzari ?] will do that. 1471 01:16:51,220 --> 01:16:53,140 So I'll be still lecturing on Tuesday. 1472 01:16:53,140 --> 01:16:57,170 So have a good weekend, and I'll see you on Tuesday.