1 00:00:10,010 --> 00:00:11,260 PROFESSOR: Why don't we get started. 2 00:00:15,866 --> 00:00:17,070 Good afternoon-- 3 00:00:17,070 --> 00:00:22,060 to stretch the literal meaning of the word considerably. 4 00:00:22,060 --> 00:00:23,690 What a miserable day out there. 5 00:00:23,690 --> 00:00:29,470 I get to and from the boonies via the commuter rail. 6 00:00:29,470 --> 00:00:33,480 And this morning, we sat there for an hour part way in 7 00:00:33,480 --> 00:00:38,140 because a tree had fallen across the railroad track. 8 00:00:38,140 --> 00:00:40,550 I don't know who went out to do it, but somebody had to get 9 00:00:40,550 --> 00:00:43,860 a saw and saw the tree up and take it off the tracks before 10 00:00:43,860 --> 00:00:47,580 we could proceed into Boston. 11 00:00:47,580 --> 00:00:49,440 Even with that, I was sort of reluctant 12 00:00:49,440 --> 00:00:51,520 to get off the train. 13 00:00:51,520 --> 00:00:53,600 I saw how it was pouring outside. 14 00:00:53,600 --> 00:00:57,360 So I am highly flattered that you decided to trudge through 15 00:00:57,360 --> 00:01:00,810 the puddles and show up this afternoon. 16 00:01:00,810 --> 00:01:04,060 All right, this is a momentous occasion because we are 17 00:01:04,060 --> 00:01:07,770 exactly at this point, halfway through the term. 18 00:01:07,770 --> 00:01:11,600 And this is when we are about to change gears from our 19 00:01:11,600 --> 00:01:15,520 discussion of symmetry theory and switch over to physical 20 00:01:15,520 --> 00:01:19,530 properties of crystals and the way in which symmetry impacts 21 00:01:19,530 --> 00:01:22,250 on those properties. 22 00:01:22,250 --> 00:01:25,690 What I wanted to do primarily today, though, is to say a 23 00:01:25,690 --> 00:01:30,580 little bit in a very fast and very simple way about the 24 00:01:30,580 --> 00:01:33,910 nature of crystal structures because that is one of the 25 00:01:33,910 --> 00:01:36,360 primary uses of symmetry theory. 26 00:01:36,360 --> 00:01:39,520 It's to describe the periodic arrangements and symmetrical 27 00:01:39,520 --> 00:01:41,700 arrangements of atoms in crystals. 28 00:01:41,700 --> 00:01:44,410 So it's the language that's necessary to describe such 29 00:01:44,410 --> 00:01:48,210 arrangements, particularly when they're beyond the red 30 00:01:48,210 --> 00:01:50,710 balls at the corner of the cube and black balls in the 31 00:01:50,710 --> 00:01:52,370 middle of the faces level of structure. 32 00:01:56,540 --> 00:02:01,850 Let me first, for your amusement and edification, 33 00:02:01,850 --> 00:02:03,740 pass out another problem set. 34 00:02:03,740 --> 00:02:05,730 This has only two problems on it. 35 00:02:05,730 --> 00:02:09,840 You are not really equipped to do it yet because we haven't 36 00:02:09,840 --> 00:02:11,520 talked about space groups. 37 00:02:11,520 --> 00:02:13,970 So don't worry about doing that now. 38 00:02:13,970 --> 00:02:18,510 But I wanted to get it in your hand so that you 39 00:02:18,510 --> 00:02:20,770 could look it over. 40 00:02:20,770 --> 00:02:25,310 It's a good problem set on which to end this part of the 41 00:02:25,310 --> 00:02:25,840 discussion-- 42 00:02:25,840 --> 00:02:27,650 but there's another one that I'll be 43 00:02:27,650 --> 00:02:30,100 passing out next week-- 44 00:02:30,100 --> 00:02:33,930 because this is an example of how you can use the notions of 45 00:02:33,930 --> 00:02:37,610 symmetry that we have established to say some fairly 46 00:02:37,610 --> 00:02:41,320 profound and non-obvious things about the nature of 47 00:02:41,320 --> 00:02:42,850 atomic arrangements. 48 00:02:42,850 --> 00:02:44,960 So that's the purpose of this problem set. 49 00:02:44,960 --> 00:02:48,050 It is to give you an opportunity to see the 50 00:02:48,050 --> 00:02:52,230 surprising things that can fall out of our material that 51 00:02:52,230 --> 00:02:55,170 we now have at our disposal. 52 00:02:55,170 --> 00:02:59,140 Let me begin by telling you where we left off last time. 53 00:02:59,140 --> 00:03:07,670 We had completed deriving the so-called Bravais lattices, 54 00:03:07,670 --> 00:03:10,940 the three-dimensional space lattices. 55 00:03:10,940 --> 00:03:14,310 The process by which we did this was to take our 56 00:03:14,310 --> 00:03:17,460 two-dimensional space groups, the plane groups, and say 57 00:03:17,460 --> 00:03:20,610 that, if we add a third translation, we will have 58 00:03:20,610 --> 00:03:22,510 generated a space lattice. 59 00:03:22,510 --> 00:03:26,210 But these different nets in the base of the cell are 60 00:03:26,210 --> 00:03:29,600 special and determined by the fact that there are symmetry 61 00:03:29,600 --> 00:03:30,710 in these nets. 62 00:03:30,710 --> 00:03:35,300 So if we take, for example, a net that is exactly square-- 63 00:03:35,300 --> 00:03:38,120 it's square because there's a fourfold axis there. 64 00:03:38,120 --> 00:03:41,520 And that fourfold axis, in a space lattice, pokes not only 65 00:03:41,520 --> 00:03:44,760 into the base of the cell but extends up through space. 66 00:03:44,760 --> 00:03:48,050 So you must pick a third translation such that the 67 00:03:48,050 --> 00:03:49,840 symmetry elements line up. 68 00:03:49,840 --> 00:03:53,960 We would take each of the crystallographic plane groups 69 00:03:53,960 --> 00:03:57,000 and pick a third translation. 70 00:03:57,000 --> 00:04:03,260 You end up with almost all of the lattices that you're going 71 00:04:03,260 --> 00:04:07,570 to get except for the ones that involve cubic symmetries 72 00:04:07,570 --> 00:04:10,400 that are inherently three-dimensional. 73 00:04:10,400 --> 00:04:14,810 And the reason for that is that the more complex 74 00:04:14,810 --> 00:04:18,660 three-dimensional point groups are, for the most part, 75 00:04:18,660 --> 00:04:25,340 obtained by adding inversion to the simple symmetries that 76 00:04:25,340 --> 00:04:26,690 exist in two dimensions. 77 00:04:26,690 --> 00:04:30,290 But inversion doesn't require anything of a lattice. 78 00:04:30,290 --> 00:04:34,850 So you can immediately say that 2M and 2/M are compatible 79 00:04:34,850 --> 00:04:39,640 with the same sort of space lattice because the symmetry 80 00:04:39,640 --> 00:04:43,730 2/M is nothing more than what you get when you add inversion 81 00:04:43,730 --> 00:04:47,220 to either a twofold axis or a mirror plane. 82 00:04:47,220 --> 00:04:52,290 And, finally, another way of classifying crystals, and a 83 00:04:52,290 --> 00:04:56,650 broader way of classifying them, is according to the 84 00:04:56,650 --> 00:04:59,480 coordinate systems that are necessary to 85 00:04:59,480 --> 00:05:00,730 describe the lattices. 86 00:05:05,050 --> 00:05:07,305 And these are called the crystal systems. 87 00:05:13,400 --> 00:05:19,270 And, as a mnemonic device for keeping straight what the 88 00:05:19,270 --> 00:05:23,370 crystal systems are as opposed to the crystal classes, which 89 00:05:23,370 --> 00:05:25,920 is another word that we haven't used but it's another 90 00:05:25,920 --> 00:05:29,640 word for the point groups, the crystal systems goes with 91 00:05:29,640 --> 00:05:32,510 coordinate systems. 92 00:05:32,510 --> 00:05:37,200 And it's simply a word to describe the shape of the 93 00:05:37,200 --> 00:05:40,480 lattices, family of lattices, that we've described. 94 00:05:40,480 --> 00:05:44,450 And we distributed a handout last time that gave the names 95 00:05:44,450 --> 00:05:48,470 of these systems and the relations that exist between 96 00:05:48,470 --> 00:05:52,620 the edges of the unit cell, which we use as the basis for 97 00:05:52,620 --> 00:05:54,670 our coordinate system and also the labels 98 00:05:54,670 --> 00:05:57,090 that we put on them. 99 00:05:57,090 --> 00:05:59,030 I have another sheet, which I'll pass around, which is 100 00:05:59,030 --> 00:05:59,980 nothing new. 101 00:05:59,980 --> 00:06:01,720 It just assembles-- 102 00:06:01,720 --> 00:06:03,780 I'll give you two sheets, which I'll pass around. 103 00:06:03,780 --> 00:06:08,410 The first is an assembly of the crystallographic point 104 00:06:08,410 --> 00:06:12,130 groups in a somewhat neater and tidier way than we had on 105 00:06:12,130 --> 00:06:16,020 the sheet that we passed around last time. 106 00:06:16,020 --> 00:06:19,480 So this is a picture of all 32 of the point groups. 107 00:06:19,480 --> 00:06:23,630 Again, beware of the fact that mirror planes and guides to 108 00:06:23,630 --> 00:06:30,100 the eye that split things up into 90 degree segments or 60 109 00:06:30,100 --> 00:06:32,560 degree segments come out deceptively 110 00:06:32,560 --> 00:06:34,530 similar on these sheets. 111 00:06:34,530 --> 00:06:37,540 But there with the representative pattern are the 112 00:06:37,540 --> 00:06:40,050 arrangements of symmetry elements, a representative 113 00:06:40,050 --> 00:06:44,750 pattern of motifs, and the international and [INAUDIBLE] 114 00:06:44,750 --> 00:06:46,990 symbol for all 32 of the 115 00:06:46,990 --> 00:06:48,520 three-dimensional point groups. 116 00:06:51,300 --> 00:06:53,490 And then the next sheet I'll pass around takes the point 117 00:06:53,490 --> 00:06:57,160 groups and groups them together with the lattices 118 00:06:57,160 --> 00:07:00,660 that can accommodate them and the crystal system that 119 00:07:00,660 --> 00:07:05,740 describes the arrangement of axes in the lattice. 120 00:07:05,740 --> 00:07:08,730 So this is, in a nutshell, everything that we've covered 121 00:07:08,730 --> 00:07:11,090 up to this point. 122 00:07:11,090 --> 00:07:16,880 And, at this point, I have to say that there is some 123 00:07:16,880 --> 00:07:22,160 disagreement on how many crystal systems exist. 124 00:07:22,160 --> 00:07:26,180 And even though people who work with symmetry and 125 00:07:26,180 --> 00:07:33,350 mathematics are usually rational, almost to a fault 126 00:07:33,350 --> 00:07:38,190 logical individuals, there are several junctures as you build 127 00:07:38,190 --> 00:07:41,300 up a body of knowledge where you could go either way or 128 00:07:41,300 --> 00:07:42,680 adopt either convention. 129 00:07:42,680 --> 00:07:45,910 You sort of got to pay your money and take your choice. 130 00:07:45,910 --> 00:07:50,700 And this occurs in our arrangement of point groups 131 00:07:50,700 --> 00:07:53,460 and lattices among the crystal systems. 132 00:07:53,460 --> 00:07:56,080 This occurs in the hexagonal crystal system. 133 00:07:58,580 --> 00:08:02,980 Symmetry 3 and the other point groups that can be derived 134 00:08:02,980 --> 00:08:09,010 from it can be placed either in a hexagonal lattice, that 135 00:08:09,010 --> 00:08:13,890 is to say, a third translation C that is exactly at right 136 00:08:13,890 --> 00:08:17,760 angles to a pair of translations, A1 and A2, which 137 00:08:17,760 --> 00:08:20,460 are 120 degrees apart. 138 00:08:20,460 --> 00:08:24,840 Or a three-fold symmetry can go into this funny double-body 139 00:08:24,840 --> 00:08:28,980 centered hexagonal lattice, which could be defined, if we 140 00:08:28,980 --> 00:08:32,159 were masochistic and enjoyed working in an oblique 141 00:08:32,159 --> 00:08:35,340 coordinate system, could be defined instead in terms of a 142 00:08:35,340 --> 00:08:38,000 rhombohedral lattice, which had the special 143 00:08:38,000 --> 00:08:41,860 characteristics that three translations are identical by 144 00:08:41,860 --> 00:08:46,360 symmetry, called therefore A1, A2, and A3; and the angles 145 00:08:46,360 --> 00:08:50,080 between the more equal alpha 1, alpha 2, alpha 3; and they 146 00:08:50,080 --> 00:08:52,730 could be anything we wished. 147 00:08:52,730 --> 00:08:57,230 So that is a unique sort of crystal system and unique sort 148 00:08:57,230 --> 00:08:58,730 of coordinate system. 149 00:08:58,730 --> 00:09:03,830 There are those who would say that if this is the case, we 150 00:09:03,830 --> 00:09:06,600 really have seven crystal systems-- 151 00:09:06,600 --> 00:09:09,760 triclinic; monoclinic; orthorhombic; tetragonal; 152 00:09:09,760 --> 00:09:13,790 hexagonal; and then this one that is compatible only with a 153 00:09:13,790 --> 00:09:17,870 three-fold axis and that is called the trigonal system or, 154 00:09:17,870 --> 00:09:20,410 by some folks, the rhombohedral system; and then, 155 00:09:20,410 --> 00:09:22,590 finally, the cubic. 156 00:09:22,590 --> 00:09:24,870 Well isn't that a rational and logical thing to do? 157 00:09:24,870 --> 00:09:25,840 Yeah, question at this point? 158 00:09:25,840 --> 00:09:27,605 AUDIENCE: Do you have any extra copies of the sheet 159 00:09:27,605 --> 00:09:29,350 [INAUDIBLE] 160 00:09:29,350 --> 00:09:31,680 PROFESSOR: Oh, yes. 161 00:09:31,680 --> 00:09:35,270 There should be more than enough floating around. 162 00:09:35,270 --> 00:09:36,866 This one here? 163 00:09:36,866 --> 00:09:37,364 [? AUDIENCE: The ?] 164 00:09:37,364 --> 00:09:37,862 [? chart ?] 165 00:09:37,862 --> 00:09:38,858 AUDIENCE: Yeah, [? not the ?] 166 00:09:38,858 --> 00:09:39,360 [? chart ?]. 167 00:09:39,360 --> 00:09:40,860 PROFESSOR: OK. 168 00:09:40,860 --> 00:09:42,020 I'll give you-- 169 00:09:42,020 --> 00:09:43,220 I think there are more back there. 170 00:09:43,220 --> 00:09:44,610 But why don't you take one for now? 171 00:09:47,270 --> 00:09:50,230 Yeah, so it's a perfectly reasonable thing to do is to 172 00:09:50,230 --> 00:09:55,500 set up the trigonal system as a separate coordinate system 173 00:09:55,500 --> 00:09:57,650 to describe the rhombohedral lattice. 174 00:10:00,380 --> 00:10:05,070 The problem that creates, however, is that if you do 175 00:10:05,070 --> 00:10:09,800 this, there is no longer any one to one correspondence 176 00:10:09,800 --> 00:10:13,430 between the point groups and the crystal systems. 177 00:10:13,430 --> 00:10:17,590 A crystal with symmetry 3, for example, can fit into it 178 00:10:17,590 --> 00:10:22,620 either in a hexagonal lattice or a trigonal lattice. 179 00:10:22,620 --> 00:10:26,380 US So sometimes a crystal with point group 3 would have to be 180 00:10:26,380 --> 00:10:29,770 called a hexagonal crystal, sometimes a trigonal system. 181 00:10:29,770 --> 00:10:30,760 But which is it? 182 00:10:30,760 --> 00:10:32,170 Well it can be either. 183 00:10:32,170 --> 00:10:36,770 So admitting the trigonal system as a separate crystal 184 00:10:36,770 --> 00:10:40,550 system breaks down the one to one correspondence between 185 00:10:40,550 --> 00:10:45,720 point groups and crystal systems. 186 00:10:45,720 --> 00:10:50,330 The very pragmatic reason for not admitting the trigonal 187 00:10:50,330 --> 00:10:53,530 system as a separate crystal system 188 00:10:53,530 --> 00:10:58,150 is that, although you can in this funny triple hexagonal 189 00:10:58,150 --> 00:11:01,590 cell define a primitive trigonal cell that has a 190 00:11:01,590 --> 00:11:04,900 special geometry, nobody in their right mind, as I 191 00:11:04,900 --> 00:11:07,310 indicated earlier, would want to work in such 192 00:11:07,310 --> 00:11:08,710 a coordinate system. 193 00:11:08,710 --> 00:11:11,540 Better to take the triple cell, pay the price of 194 00:11:11,540 --> 00:11:15,650 three-fold redundancy, and yet have two interaxial angles 195 00:11:15,650 --> 00:11:19,100 that are 90 degrees and one that's specialized at 120. 196 00:11:19,100 --> 00:11:24,620 That's a much more convenient reference system for our 197 00:11:24,620 --> 00:11:26,270 description of crystal properties. 198 00:11:29,620 --> 00:11:32,520 One final thing I'll comment on but won't say anything 199 00:11:32,520 --> 00:11:34,350 about very much-- 200 00:11:34,350 --> 00:11:35,690 I always do that. 201 00:11:35,690 --> 00:11:38,400 I say I won't say anything about it but, and then launch 202 00:11:38,400 --> 00:11:40,470 into a 10 minute digression. 203 00:11:40,470 --> 00:11:44,120 Off in the right hand corner is something called the Laue 204 00:11:44,120 --> 00:11:48,340 group of which there are 11. 205 00:11:48,340 --> 00:11:52,330 The Laue groups are those of the point groups that contain 206 00:11:52,330 --> 00:11:55,940 the operation of inversion. 207 00:11:55,940 --> 00:11:58,690 And the reason they're called the Laue groups is that one of 208 00:11:58,690 --> 00:12:01,950 the primary ways, other than looking at crystal morphology 209 00:12:01,950 --> 00:12:08,140 to determine the symmetries of a crystal, is to look at the 210 00:12:08,140 --> 00:12:11,090 way it diffracts x-rays. 211 00:12:11,090 --> 00:12:15,010 So if we had, for example a tetragonal crystal and there 212 00:12:15,010 --> 00:12:19,010 were lattice planes in here that were related by 90 degree 213 00:12:19,010 --> 00:12:23,960 rotations, then if we brought in an x-ray beam at such an 214 00:12:23,960 --> 00:12:28,700 angle to satisfy Bragg's law and saw some sort of intensity 215 00:12:28,700 --> 00:12:35,620 that came off, if we rotated the x-ray beam 90 degrees, we 216 00:12:35,620 --> 00:12:38,420 ought to see diffraction at the same angle and with 217 00:12:38,420 --> 00:12:41,300 exactly the same intensity. 218 00:12:41,300 --> 00:12:44,650 So either by moving the x-ray beam, or simpler thing would 219 00:12:44,650 --> 00:12:48,550 be to leave the x-ray beam fixed, since it's attached to 220 00:12:48,550 --> 00:12:53,500 an x-ray generator that weighs a ton or more, instead move 221 00:12:53,500 --> 00:12:57,240 the crystal relative to the x-ray beam. 222 00:12:57,240 --> 00:13:03,490 Now what would you do in this way of thinking to determine 223 00:13:03,490 --> 00:13:09,530 whether a crystal had the operation of inversion in it? 224 00:13:09,530 --> 00:13:14,870 If this is the set of planes with indices HKL, you, again, 225 00:13:14,870 --> 00:13:20,680 would bring in an x-ray beam at an angle theta, in out. 226 00:13:20,680 --> 00:13:24,790 Look at the intensity from the set of points HKL. 227 00:13:24,790 --> 00:13:28,370 And then look at diffraction, both in terms of the Bragg 228 00:13:28,370 --> 00:13:31,290 angle theta and in terms of the intensity. 229 00:13:31,290 --> 00:13:33,830 Look at that diffraction from the planes that were related 230 00:13:33,830 --> 00:13:36,640 to HKL by symmetry. 231 00:13:36,640 --> 00:13:39,460 Well what plane is related to HKL by symmetry? 232 00:13:39,460 --> 00:13:43,490 It's these planes down here which have indices minus H, 233 00:13:43,490 --> 00:13:46,130 minus K, minus L. 234 00:13:46,130 --> 00:13:49,430 So if we did the corresponding experiment to what I did to 235 00:13:49,430 --> 00:13:52,450 the left, we'd bring in an x-ray beam like this. 236 00:13:52,450 --> 00:13:55,310 It would be diffracted at some angle theta. 237 00:13:55,310 --> 00:13:58,560 And we'd look at the intensity for the set of planes minus H, 238 00:13:58,560 --> 00:14:02,870 minus K, minus L. 239 00:14:02,870 --> 00:14:05,334 Is that going to be any different? 240 00:14:05,334 --> 00:14:06,170 No. 241 00:14:06,170 --> 00:14:08,690 Right-minded person would say, why should this 242 00:14:08,690 --> 00:14:09,870 intensity be different? 243 00:14:09,870 --> 00:14:14,680 All we're doing is bouncing the x-ray beam off the 244 00:14:14,680 --> 00:14:18,800 opposite side of one in the same set of lattice planes. 245 00:14:18,800 --> 00:14:22,040 So why should the intensity be any different? 246 00:14:22,040 --> 00:14:26,400 In this case, clearly if the set of lattice points is 247 00:14:26,400 --> 00:14:31,500 rotated by some finite angle, we are not going to see 248 00:14:31,500 --> 00:14:33,410 equivalent diffraction from planes that 249 00:14:33,410 --> 00:14:36,850 are 90 degrees apart. 250 00:14:36,850 --> 00:14:42,460 So the upshot is that the diffraction symmetry of a 251 00:14:42,460 --> 00:14:46,900 crystal, the way in which planes with indices that are 252 00:14:46,900 --> 00:14:50,250 related by symmetry diffract in terms of position of the 253 00:14:50,250 --> 00:14:54,110 beam and intensity, the diffraction symmetry of a 254 00:14:54,110 --> 00:14:58,340 crystal always looks as though that crystal possessed an 255 00:14:58,340 --> 00:15:00,600 inversion center. 256 00:15:00,600 --> 00:15:01,880 So, therefore, to-- 257 00:15:01,880 --> 00:15:05,650 and I qualify this by saying, to a good approximation-- 258 00:15:05,650 --> 00:15:08,200 the only point groups that you could determine and 259 00:15:08,200 --> 00:15:13,000 distinguish from one another by x-ray diffraction are the 260 00:15:13,000 --> 00:15:17,570 point groups that contain an inversion center. 261 00:15:17,570 --> 00:15:19,950 Almost, but not quite. 262 00:15:19,950 --> 00:15:23,010 If we look a little more closely at what is doing the 263 00:15:23,010 --> 00:15:25,490 scattering, it is not the Bragg planes. 264 00:15:25,490 --> 00:15:28,560 These are imaginary, shimmering, absolutely planar 265 00:15:28,560 --> 00:15:32,970 things that are constructs that we use to interpret the 266 00:15:32,970 --> 00:15:34,315 process called diffraction. 267 00:15:36,880 --> 00:15:40,090 Inside the crystal what is actually doing the diffraction 268 00:15:40,090 --> 00:15:45,380 is a collection of atoms, each of which are scattering the 269 00:15:45,380 --> 00:15:45,980 x-radiation. 270 00:15:45,980 --> 00:15:50,410 And that physically is what causes the diffraction. 271 00:15:50,410 --> 00:15:54,810 And the resultant intensity is the addition of all the little 272 00:15:54,810 --> 00:15:59,200 wavelets scattered by these separate atoms. 273 00:15:59,200 --> 00:16:02,800 So in terms of the atomistics of what is involved, to look 274 00:16:02,800 --> 00:16:08,850 at diffraction from the set of planes HKL and contrast that 275 00:16:08,850 --> 00:16:11,570 with the diffraction for the set of planes minus H, minus 276 00:16:11,570 --> 00:16:15,870 K, minus L, we're coming in from opposite directions. 277 00:16:15,870 --> 00:16:18,300 And here the beam would hit-- 278 00:16:18,300 --> 00:16:22,320 let's call the atoms, if I did not do so already, let's call 279 00:16:22,320 --> 00:16:27,155 them A and B. A for a and b for big. 280 00:16:30,740 --> 00:16:33,620 Diffraction from the set of points HKL would pass through 281 00:16:33,620 --> 00:16:36,730 the sheets of A atoms first and then impinge on the B 282 00:16:36,730 --> 00:16:40,910 atoms, whereas in the opposite direction, the x-ray beams 283 00:16:40,910 --> 00:16:44,120 would impinge upon the B atoms first and 284 00:16:44,120 --> 00:16:46,420 then hit the A atoms. 285 00:16:46,420 --> 00:16:48,860 Does that make a difference? 286 00:16:48,860 --> 00:16:49,380 Aha. 287 00:16:49,380 --> 00:16:51,440 Believe it or not, it does. 288 00:16:51,440 --> 00:16:54,030 And I can't explain why without going into a very 289 00:16:54,030 --> 00:16:56,210 detailed discussion of diffraction. 290 00:16:56,210 --> 00:16:59,570 But if you compare the intensities that are scattered 291 00:16:59,570 --> 00:17:07,220 by these two sets of planes, which are related by inversion 292 00:17:07,220 --> 00:17:10,940 and therefore just opposite sides of one another, it does 293 00:17:10,940 --> 00:17:15,380 make a difference which atom the x-ray 294 00:17:15,380 --> 00:17:17,060 beam encounters first. 295 00:17:17,060 --> 00:17:23,700 And the difference in intensity is on the order of 296 00:17:23,700 --> 00:17:29,720 3% to 5%, although by picking a radiation appropriately for 297 00:17:29,720 --> 00:17:32,360 the particular chemical species in the crystal, you 298 00:17:32,360 --> 00:17:36,140 could increase this perhaps to 10% or 12%. 299 00:17:36,140 --> 00:17:40,000 But that is something that you could not do until synchrotron 300 00:17:40,000 --> 00:17:43,990 radiation came along, which could actually tune the 301 00:17:43,990 --> 00:17:47,660 wavelength to a particular value that you wish to use and 302 00:17:47,660 --> 00:17:51,130 not use something that was spewed out by an x-ray tube 303 00:17:51,130 --> 00:17:56,550 that had a target such as molybdenum or copper or 304 00:17:56,550 --> 00:17:59,330 chromium, perhaps. 305 00:17:59,330 --> 00:18:02,060 OK, so you can with this careful measurement of 306 00:18:02,060 --> 00:18:07,140 intensity distinguished between the opposite sides of 307 00:18:07,140 --> 00:18:10,410 the given set of Bragg planes, unless, of course, the crystal 308 00:18:10,410 --> 00:18:11,970 did have inversion in it. 309 00:18:11,970 --> 00:18:14,120 And then there would be no difference at all. 310 00:18:14,120 --> 00:18:17,800 It requires a very careful experiment, though, because 311 00:18:17,800 --> 00:18:19,180 there are many things that could make 312 00:18:19,180 --> 00:18:20,470 the intensities different. 313 00:18:20,470 --> 00:18:23,605 For example, absorption by the crystal. 314 00:18:23,605 --> 00:18:26,250 If the crystal has an irregular shape, the 315 00:18:26,250 --> 00:18:30,290 absorption, which can easily be factors 10, 20, or even 316 00:18:30,290 --> 00:18:34,260 more, the absorption would be different for x-ray beams 317 00:18:34,260 --> 00:18:36,330 coming in those two orientations. 318 00:18:36,330 --> 00:18:38,860 So in any case, as I say, I won't say anything about it. 319 00:18:38,860 --> 00:18:40,860 But 10 minutes later, that's the 320 00:18:40,860 --> 00:18:42,020 meaning of the Laue groups. 321 00:18:42,020 --> 00:18:44,440 These are the point groups that have inversion in them. 322 00:18:53,570 --> 00:18:55,970 That effect that I just described, by the way, is 323 00:18:55,970 --> 00:18:59,570 called anomalous scattering or anomalous dispersion. 324 00:18:59,570 --> 00:19:01,930 There's really nothing anomalous about at all. 325 00:19:01,930 --> 00:19:05,460 It has to do with the way electromagnetic radiation 326 00:19:05,460 --> 00:19:10,180 interacts with the clouds of electrons that are distributed 327 00:19:10,180 --> 00:19:11,770 about the nucleus of each atom. 328 00:19:17,060 --> 00:19:24,520 Where we will go from here to finish up symmetry theory will 329 00:19:24,520 --> 00:19:30,150 be to now take the 32 crystallographic point groups 330 00:19:30,150 --> 00:19:37,530 and to drop those point groups into each of the distinct 14 331 00:19:37,530 --> 00:19:39,170 Bravais lattices. 332 00:19:39,170 --> 00:19:42,450 And then we'll need combination theorems to deduce 333 00:19:42,450 --> 00:19:46,360 where the additional symmetry elements arrive. 334 00:19:46,360 --> 00:19:49,450 And when we go through that process, if we were to do this 335 00:19:49,450 --> 00:19:54,310 exhaustively, we would find that there are 230 distinct 336 00:19:54,310 --> 00:19:57,880 three-dimensional symmetries that involve point group 337 00:19:57,880 --> 00:20:01,410 symmetry operations and lattice type. 338 00:20:05,150 --> 00:20:07,200 I saw several faces blanch. 339 00:20:07,200 --> 00:20:08,110 Don't worry. 340 00:20:08,110 --> 00:20:10,040 We're not going to do that. 341 00:20:10,040 --> 00:20:14,380 What we will do next time is do a small handful and then 342 00:20:14,380 --> 00:20:18,590 conclude by showing you how this information is tabulated 343 00:20:18,590 --> 00:20:20,490 in the international tables. 344 00:20:20,490 --> 00:20:23,010 And the quick, immediate answer to that question is 345 00:20:23,010 --> 00:20:26,630 that the problem set that I passed around has two pages 346 00:20:26,630 --> 00:20:29,800 for two different point groups that show the way that this 347 00:20:29,800 --> 00:20:33,360 information is tabulated in the international tables. 348 00:20:33,360 --> 00:20:34,840 So that's where we'll end up. 349 00:20:34,840 --> 00:20:37,960 And we'll come pretty close to completing that when we meet 350 00:20:37,960 --> 00:20:39,210 on Thursday. 351 00:20:41,180 --> 00:20:42,710 I heard no sighs of relief. 352 00:20:42,710 --> 00:20:43,670 I heard no cheers. 353 00:20:43,670 --> 00:20:46,610 So you are presumably still engaged in 354 00:20:46,610 --> 00:20:49,320 this material somewhat. 355 00:20:49,320 --> 00:20:51,750 I have two things I would like to do. 356 00:20:51,750 --> 00:20:55,110 We talked about crystal systems and point groups. 357 00:20:55,110 --> 00:21:00,670 And let me now ask, do you think there is a crystal known 358 00:21:00,670 --> 00:21:05,590 for every one of the point groups? 359 00:21:05,590 --> 00:21:06,840 Think so? 360 00:21:09,240 --> 00:21:12,510 I saw some skeptical shakes of the head. 361 00:21:12,510 --> 00:21:16,920 The answer is yes, but just barely for some 362 00:21:16,920 --> 00:21:18,355 of the point groups. 363 00:21:21,150 --> 00:21:25,760 One of the very rare point groups, in terms of structures 364 00:21:25,760 --> 00:21:31,150 that populate it, is our high symmetries but which lack 365 00:21:31,150 --> 00:21:33,360 inversion or mirror planes. 366 00:21:33,360 --> 00:21:38,290 One very, very rare point group is 432-- 367 00:21:38,290 --> 00:21:41,785 just axes arranged in a cubic arrangement with no mirror 368 00:21:41,785 --> 00:21:43,550 planes or inversion centers. 369 00:21:43,550 --> 00:21:47,580 And my rationalization of that is that if an arrangement of 370 00:21:47,580 --> 00:21:51,490 atoms has to conform to all of these rotational symmetries, 371 00:21:51,490 --> 00:21:55,960 it's pretty hard for it to do it and have that assemblage of 372 00:21:55,960 --> 00:21:59,210 atoms assume the lowest possible energy state if it 373 00:21:59,210 --> 00:22:02,950 doesn't pick up in the process inversion or mirror planes as 374 00:22:02,950 --> 00:22:06,090 additional symmetry element. 375 00:22:06,090 --> 00:22:08,440 Sort of satisfying for me. 376 00:22:08,440 --> 00:22:09,880 But that's not a rigorous argument. 377 00:22:12,490 --> 00:22:16,320 OK, let me now take a poll. 378 00:22:16,320 --> 00:22:20,160 Do you think, using the broader cut of the crystal 379 00:22:20,160 --> 00:22:24,360 systems, do you think there's a crystal known for every 380 00:22:24,360 --> 00:22:25,420 crystal system? 381 00:22:25,420 --> 00:22:27,850 Are there triclinic, monoclinic, orthorhombic, 382 00:22:27,850 --> 00:22:30,070 tetragonal, trigonal, hexagonal, 383 00:22:30,070 --> 00:22:32,770 cubic crystals known? 384 00:22:32,770 --> 00:22:33,750 Yeah. 385 00:22:33,750 --> 00:22:40,010 That's a pretty realistic expectation. 386 00:22:40,010 --> 00:22:41,835 And that is indeed the case. 387 00:22:41,835 --> 00:22:45,550 But now let me cut this a little finer. 388 00:22:45,550 --> 00:22:49,430 Which do you think is the most popular crystal systems for 389 00:22:49,430 --> 00:22:52,920 real materials and which do you think is the rarest? 390 00:22:58,560 --> 00:22:58,980 OK. 391 00:22:58,980 --> 00:23:04,880 Who thinks triclinic crystals are the most common? 392 00:23:04,880 --> 00:23:07,500 OK, there are pessimists. 393 00:23:07,500 --> 00:23:10,370 What is the definition of a pessimist? 394 00:23:10,370 --> 00:23:13,600 The pessimist is somebody you should borrow money from 395 00:23:13,600 --> 00:23:15,740 because they'll never expect to be paid back. 396 00:23:18,640 --> 00:23:22,182 Who would say cubic crystals? 397 00:23:22,182 --> 00:23:25,640 There are naive [? optimists. ?] 398 00:23:25,640 --> 00:23:28,560 Now I will reveal all. 399 00:23:28,560 --> 00:23:33,150 The answer is, it depends. 400 00:23:33,150 --> 00:23:36,640 Depends upon whether you're talking about, interestingly, 401 00:23:36,640 --> 00:23:39,630 inorganic materials or whether you're talking 402 00:23:39,630 --> 00:23:40,985 about organic materials. 403 00:23:43,980 --> 00:23:52,860 If you look at inorganic materials, the most popular 404 00:23:52,860 --> 00:23:56,570 crystal systems are-- 405 00:23:56,570 --> 00:24:01,740 and the results are, first of all, cubic. 406 00:24:04,240 --> 00:24:07,870 Is this not a benign world in which we live. 407 00:24:07,870 --> 00:24:09,560 Most inorganic materials-- 408 00:24:09,560 --> 00:24:12,020 if you're a material scientist who works with metals and with 409 00:24:12,020 --> 00:24:15,260 ceramics, you're going to work with cubic materials, thank 410 00:24:15,260 --> 00:24:18,710 God, most of the time. 411 00:24:18,710 --> 00:24:20,780 The second most popular-- 412 00:24:20,780 --> 00:24:23,295 by a considerable margin but there are still a lot of 413 00:24:23,295 --> 00:24:24,545 them-- is orthorhombic. 414 00:24:31,250 --> 00:24:39,140 And the third most popular, shortly behind orthorhombic 415 00:24:39,140 --> 00:24:42,065 and almost in a tie, monoclinic and tetragonal. 416 00:24:52,120 --> 00:24:55,470 We'll launch into a quick overview of crystal symmetry. 417 00:24:55,470 --> 00:25:02,500 And we'll see why inorganic materials have high symmetry. 418 00:25:02,500 --> 00:25:08,250 It's true, in particular, when the material is held together 419 00:25:08,250 --> 00:25:11,680 by ionic bonding. 420 00:25:11,680 --> 00:25:16,020 If the bonding is ionic, it's non-directional and atoms of 421 00:25:16,020 --> 00:25:19,040 opposite charge will therefore try to get as close as 422 00:25:19,040 --> 00:25:23,440 possible to the atoms of opposite charge. 423 00:25:23,440 --> 00:25:26,750 And because atoms of like charge repel one another, the 424 00:25:26,750 --> 00:25:28,940 surrounding atoms will try to be as widely 425 00:25:28,940 --> 00:25:30,660 separated as possible. 426 00:25:30,660 --> 00:25:33,970 And this grouping of atoms, positive around negative, 427 00:25:33,970 --> 00:25:35,770 tends to have a lot of symmetry. 428 00:25:35,770 --> 00:25:38,340 And, therefore, the way in which these groups are linked 429 00:25:38,340 --> 00:25:41,840 together has a lot of symmetry. 430 00:25:41,840 --> 00:25:45,820 OK, I said organic materials. 431 00:25:45,820 --> 00:25:49,940 And I'll pass around some actual numbers that were 432 00:25:49,940 --> 00:25:52,960 assembled a long time ago but are still, in terms of 433 00:25:52,960 --> 00:25:54,780 proportion, still valid. 434 00:25:54,780 --> 00:25:58,410 The organic compounds are broken down into whether these 435 00:25:58,410 --> 00:26:01,240 are compounds based on rings or 436 00:26:01,240 --> 00:26:04,330 extended, lopsided molecules. 437 00:26:04,330 --> 00:26:06,740 And the numbers are very different depending on that. 438 00:26:06,740 --> 00:26:12,170 But, for organic molecules [? material, ?] 439 00:26:12,170 --> 00:26:22,030 monoclinic is the most common, then orthorhombic. 440 00:26:30,550 --> 00:26:37,200 And shortly after orthorhombic comes way, way behind is 441 00:26:37,200 --> 00:26:41,440 tetragonal and triclinic, which are almost tied. 442 00:26:52,860 --> 00:26:56,130 And I think I can rationalize this observation, as well. 443 00:26:59,890 --> 00:27:04,760 Organic materials contain discrete molecules. 444 00:27:04,760 --> 00:27:07,630 And if we've got something-- and I'm going to, since I know 445 00:27:07,630 --> 00:27:09,960 I'm among friends, display my abysmal 446 00:27:09,960 --> 00:27:12,070 ignorance of organic chemistry. 447 00:27:12,070 --> 00:27:18,080 I am unapologetically and unabashedly inorganic and even 448 00:27:18,080 --> 00:27:21,000 non-metallic in my research interests. 449 00:27:21,000 --> 00:27:23,270 So let's suppose we have something that's composed of 450 00:27:23,270 --> 00:27:27,720 various strange rings with little side groups sticking 451 00:27:27,720 --> 00:27:31,690 off like this, another one off like this, and then maybe 452 00:27:31,690 --> 00:27:36,460 another ring up like this. 453 00:27:36,460 --> 00:27:41,900 That, to me, looks like a reasonable molecule in terms 454 00:27:41,900 --> 00:27:44,520 of my ignorance of organic chemistry. 455 00:27:44,520 --> 00:27:47,520 So if there are these finite groups and you're going to put 456 00:27:47,520 --> 00:27:49,130 them together in a crystal, what are you 457 00:27:49,130 --> 00:27:50,210 going to try to do? 458 00:27:50,210 --> 00:27:53,380 They're going to be held together by very weak van der 459 00:27:53,380 --> 00:27:56,510 Waals forces, which are non-directional. 460 00:27:56,510 --> 00:27:59,640 So think of the outline of this molecule as being some 461 00:27:59,640 --> 00:28:02,300 ugly thing like a ham hock. 462 00:28:02,300 --> 00:28:05,290 And then your problem, if the forces between these molecules 463 00:28:05,290 --> 00:28:09,980 are non-directional, how do you close pack ham hocks? 464 00:28:09,980 --> 00:28:15,350 The way you would do it would be to put down a row of them 465 00:28:15,350 --> 00:28:18,800 in a close-packed sequence. 466 00:28:18,800 --> 00:28:21,580 So we'll put down things that look like this. 467 00:28:21,580 --> 00:28:25,810 And then, maybe to make them fit together, dovetail another 468 00:28:25,810 --> 00:28:27,660 one in here. 469 00:28:27,660 --> 00:28:32,360 And maybe something like that. 470 00:28:32,360 --> 00:28:35,930 And then for the next row down, and you would maybe put 471 00:28:35,930 --> 00:28:39,000 the nose of one in here. 472 00:28:39,000 --> 00:28:43,540 And then put the other one in here, the nose of one here, 473 00:28:43,540 --> 00:28:46,800 and something like that-- 474 00:28:46,800 --> 00:28:49,050 being very schematic. 475 00:28:49,050 --> 00:28:53,610 The lattice that describes this packing is not going to 476 00:28:53,610 --> 00:28:54,170 be orthogonal. 477 00:28:54,170 --> 00:28:57,040 It's not going to be square with high symmetry or even 478 00:28:57,040 --> 00:28:59,140 have two orthogonal translations. 479 00:28:59,140 --> 00:29:02,570 It's going to be something that has this as translations. 480 00:29:02,570 --> 00:29:05,270 There's going to be an A and a B that is not at 481 00:29:05,270 --> 00:29:07,040 right angles to it. 482 00:29:07,040 --> 00:29:09,780 And the B translation will have different links, simply 483 00:29:09,780 --> 00:29:12,100 because the thing that you're putting together in close 484 00:29:12,100 --> 00:29:15,750 packing has a very irregular shape. 485 00:29:15,750 --> 00:29:19,390 So I think that is the reason why molecules, discrete 486 00:29:19,390 --> 00:29:22,800 molecules that try to achieve something that is as close 487 00:29:22,800 --> 00:29:26,140 packed as possible, will have translations in them that are 488 00:29:26,140 --> 00:29:29,210 of unequal length and making arbitrary angles with respect 489 00:29:29,210 --> 00:29:31,440 to one another. 490 00:29:31,440 --> 00:29:34,660 So having kept you guessing for so long, let me pass out 491 00:29:34,660 --> 00:29:35,690 another sheet. 492 00:29:35,690 --> 00:29:39,810 This was a compilation that was done a long time ago. 493 00:29:39,810 --> 00:29:44,480 And there were only 9,000 crystals for which complete 494 00:29:44,480 --> 00:29:46,510 crystallographic data had been obtained. 495 00:29:46,510 --> 00:29:50,170 This was way back in 1967. 496 00:29:50,170 --> 00:29:55,350 As I say, there's much more data now, 497 00:29:55,350 --> 00:29:58,920 easily in order of magnitude. 498 00:29:58,920 --> 00:30:00,280 But I don't think the proportions 499 00:30:00,280 --> 00:30:01,480 will change very much. 500 00:30:01,480 --> 00:30:03,031 They'll stay about the same. 501 00:30:03,031 --> 00:30:04,815 Let me pass out these sheets. 502 00:30:08,690 --> 00:30:11,390 So make sure you memorize them because the first question on 503 00:30:11,390 --> 00:30:17,000 the next quiz is, how many out of 8,759 crystals 504 00:30:17,000 --> 00:30:18,250 are actually triclinic? 505 00:30:20,800 --> 00:30:22,050 I would never do that. 506 00:30:27,077 --> 00:30:30,320 All right, any comments or questions? 507 00:30:30,320 --> 00:30:30,798 Yes, sir. 508 00:30:30,798 --> 00:30:33,188 AUDIENCE: Why would organic molecules of such odd shapes 509 00:30:33,188 --> 00:30:34,622 even be crystalline to begin with? 510 00:30:34,622 --> 00:30:37,490 Why wouldn't they just be amorphous? 511 00:30:37,490 --> 00:30:37,790 PROFESSOR: OK. 512 00:30:37,790 --> 00:30:39,980 Clearly the interatomic forces are very 513 00:30:39,980 --> 00:30:43,140 directional and very strong. 514 00:30:43,140 --> 00:30:46,130 And that's going to hold the molecule into a fixed 515 00:30:46,130 --> 00:30:47,486 configuration. 516 00:30:47,486 --> 00:30:49,230 AUDIENCE: Didn't you say there were van der Waals forces? 517 00:30:49,230 --> 00:30:50,710 PROFESSOR: But they're these-- 518 00:30:50,710 --> 00:30:56,180 even for something that is not ionic or metallic, there still 519 00:30:56,180 --> 00:30:59,690 are weak attractive forces between these molecules. 520 00:30:59,690 --> 00:31:00,220 Why? 521 00:31:00,220 --> 00:31:03,590 Well, the charges are not uniformly distributed over 522 00:31:03,590 --> 00:31:05,620 these molecules. 523 00:31:05,620 --> 00:31:09,250 So there are going to be some van der Waals forces that tend 524 00:31:09,250 --> 00:31:10,220 to hold them together. 525 00:31:10,220 --> 00:31:13,810 And a van der Waals force is essentially non-directional. 526 00:31:13,810 --> 00:31:15,770 So if you have this weak force, you're going to be able 527 00:31:15,770 --> 00:31:18,070 to squish the crystal into something 528 00:31:18,070 --> 00:31:20,020 amorphous very easily. 529 00:31:20,020 --> 00:31:23,710 And organic materials are usually very subject to 530 00:31:23,710 --> 00:31:25,140 plastic deformation. 531 00:31:25,140 --> 00:31:29,110 But if you assemble the molecules carefully from 532 00:31:29,110 --> 00:31:32,970 solution, or by vaporization, condensation, they're going to 533 00:31:32,970 --> 00:31:38,080 want the order if they are solidified with enough 534 00:31:38,080 --> 00:31:41,340 mobility to rearrange themselves in the lowest 535 00:31:41,340 --> 00:31:42,590 energy configuration. 536 00:31:45,350 --> 00:31:48,190 I think what you're saying is these are not going to be very 537 00:31:48,190 --> 00:31:51,130 strongly bonded crystals. 538 00:31:51,130 --> 00:31:52,980 They're going to melt at low temperatures. 539 00:31:52,980 --> 00:31:55,580 But, nevertheless, prepare them properly, and the lowest 540 00:31:55,580 --> 00:31:56,730 energy state will be ordered. 541 00:31:56,730 --> 00:31:57,724 AUDIENCE: I'm sure [? they're ordered. ?] 542 00:31:57,724 --> 00:31:59,712 But would it be such that you have [? translational ?] 543 00:31:59,712 --> 00:32:01,700 symmetry? 544 00:32:01,700 --> 00:32:04,682 Who's to say it's going to have a definite pattern that 545 00:32:04,682 --> 00:32:06,680 will repeat itself? 546 00:32:06,680 --> 00:32:10,880 PROFESSOR: Yeah, if there are attractive forces-- 547 00:32:10,880 --> 00:32:12,600 let me again use a hand-waving argument. 548 00:32:12,600 --> 00:32:15,030 If you've got a bunch of particles and they aggregate 549 00:32:15,030 --> 00:32:19,340 here in the lowest energy configuration, that is going 550 00:32:19,340 --> 00:32:21,320 to be the lowest energy configuration. 551 00:32:21,320 --> 00:32:23,190 And a similar group of atoms will want to do 552 00:32:23,190 --> 00:32:24,850 the same thing here. 553 00:32:24,850 --> 00:32:27,780 And if there are interactions between these groups, they're 554 00:32:27,780 --> 00:32:30,395 going to arrange themselves in an order configuration. 555 00:32:32,980 --> 00:32:34,340 That is simply saying-- 556 00:32:34,340 --> 00:32:36,610 this is a very weak qualitative argument-- says if 557 00:32:36,610 --> 00:32:39,080 you have a configuration which is the lowest energy 558 00:32:39,080 --> 00:32:43,290 arrangement here, any other place in the system where 559 00:32:43,290 --> 00:32:47,160 those atoms get together and congeal are going to assume 560 00:32:47,160 --> 00:32:49,801 that same configuration. 561 00:32:49,801 --> 00:32:52,540 OK? 562 00:32:52,540 --> 00:32:53,610 Other comments or questions? 563 00:32:53,610 --> 00:32:58,760 This is all very qualitative but actually concerns matters 564 00:32:58,760 --> 00:33:02,620 that we are describing a language to describe. 565 00:33:05,950 --> 00:33:07,200 AUDIENCE: [INAUDIBLE] 566 00:33:09,420 --> 00:33:10,000 Yes, there are. 567 00:33:10,000 --> 00:33:13,020 If you look at this table that came around, if you look at 568 00:33:13,020 --> 00:33:15,780 cubic organic materials out of-- 569 00:33:22,370 --> 00:33:23,990 there-- 570 00:33:23,990 --> 00:33:26,160 what is the number? 571 00:33:26,160 --> 00:33:31,440 117 out of what looks to be several thousand. 572 00:33:31,440 --> 00:33:34,280 Yes, there are cubic organic crystals, 573 00:33:34,280 --> 00:33:35,720 but very few in number. 574 00:33:35,720 --> 00:33:37,130 Hexagonal-- 575 00:33:37,130 --> 00:33:41,760 56 out of a rather large number. 576 00:33:49,920 --> 00:33:54,970 All right, when we took a poll a week and a half or so ago 577 00:33:54,970 --> 00:33:59,160 and said, how many of you had seen any material and crystal 578 00:33:59,160 --> 00:34:03,800 chemistry in solid state chemistry in a previous class 579 00:34:03,800 --> 00:34:09,600 on structure and bonding or crystal chemistry, and only a 580 00:34:09,600 --> 00:34:13,880 few people tentatively twitched a hand slightly. 581 00:34:13,880 --> 00:34:17,449 So what I'd like to do for the rest of this session, since 582 00:34:17,449 --> 00:34:23,739 this is what symmetry theory is developed to describe, I'd 583 00:34:23,739 --> 00:34:28,040 like to say something about structures and why they form 584 00:34:28,040 --> 00:34:30,159 the way they do to give some qualitative insight. 585 00:34:32,750 --> 00:34:35,409 This is a fat pack of notes. 586 00:34:35,409 --> 00:34:38,639 And I'll go over it quickly with you and lead you by the 587 00:34:38,639 --> 00:34:40,650 hand through it. 588 00:34:40,650 --> 00:34:44,739 But this is not material for which I'll hold you 589 00:34:44,739 --> 00:34:46,929 responsible. 590 00:34:46,929 --> 00:34:50,420 In fact, I'll turn my back and those who want to get home 591 00:34:50,420 --> 00:34:52,380 early because of the rain can tiptoe out. 592 00:34:52,380 --> 00:34:56,290 And I won't even notice who you are unless everybody goes. 593 00:34:56,290 --> 00:34:58,460 And then I'll feel very hurt because I wrote these out. 594 00:34:58,460 --> 00:34:59,710 And it took a lot of trouble. 595 00:35:07,040 --> 00:35:08,420 All right. 596 00:35:08,420 --> 00:35:15,140 I have to apologize and say that most of what is described 597 00:35:15,140 --> 00:35:20,400 in these notes applies to ionic bonding and, to a lesser 598 00:35:20,400 --> 00:35:23,980 extent but still in large measure, to metallic 599 00:35:23,980 --> 00:35:25,920 structures. 600 00:35:25,920 --> 00:35:27,720 And what is the reason for that? 601 00:35:27,720 --> 00:35:31,850 It's because the metallic bond, to a fair approximation, 602 00:35:31,850 --> 00:35:36,140 an ionic bond rigorously, if you believe Coulomb's law, is 603 00:35:36,140 --> 00:35:38,500 a non-directional sort of bonding. 604 00:35:38,500 --> 00:35:41,380 So the structures that are assumed, the structures that 605 00:35:41,380 --> 00:35:48,390 are the lowest energy configuration turn out to be 606 00:35:48,390 --> 00:35:53,790 determined by interatomic distances and ionic or 607 00:35:53,790 --> 00:35:57,290 metallic sizes. 608 00:35:57,290 --> 00:36:02,930 The Coulombic interaction that holds ionic compounds together 609 00:36:02,930 --> 00:36:04,800 is a central force. 610 00:36:04,800 --> 00:36:09,650 And one of the features of electromagnetism is that, if 611 00:36:09,650 --> 00:36:13,450 you have some distribution of charge about center and 612 00:36:13,450 --> 00:36:16,290 another distribution of charge of a different sort about a 613 00:36:16,290 --> 00:36:20,870 another center, the cohesive force between, let's say, a 614 00:36:20,870 --> 00:36:27,300 positive ion and a negative ion is, along a line, joining 615 00:36:27,300 --> 00:36:29,520 their centers. 616 00:36:29,520 --> 00:36:32,970 And, secondly, you can take, once you're outside the 617 00:36:32,970 --> 00:36:37,760 distribution of charge, you can clump all the charge 618 00:36:37,760 --> 00:36:40,920 together at the center of the distribution. 619 00:36:40,920 --> 00:36:43,700 So that's what makes the ionic bond so simple. 620 00:36:43,700 --> 00:36:46,155 And that's why everybody who talks about bonding and 621 00:36:46,155 --> 00:36:49,350 structure discusses ionic bonding because organic 622 00:36:49,350 --> 00:36:51,535 chemistry is just too complex in comparison. 623 00:36:54,750 --> 00:36:56,715 So let's look at a given cation. 624 00:36:56,715 --> 00:37:01,700 And cations generally tend to be smaller than anions because 625 00:37:01,700 --> 00:37:04,640 you've removed electrons from the outer configuration. 626 00:37:04,640 --> 00:37:07,980 We'll look at some effective sizes of ions in just a bit. 627 00:37:07,980 --> 00:37:10,650 But you've stripped electrons off to make a positively 628 00:37:10,650 --> 00:37:11,550 charged ion. 629 00:37:11,550 --> 00:37:14,340 You've added electrons on to make a negatively charged ion. 630 00:37:14,340 --> 00:37:17,710 So when you add electrons, the ion functions has [? always ?] 631 00:37:17,710 --> 00:37:20,020 had a larger radius. 632 00:37:20,020 --> 00:37:28,820 So let's bring up a B minus ion as a neighbor. 633 00:37:28,820 --> 00:37:36,660 And for those two species, there will be, if we put 634 00:37:36,660 --> 00:37:40,360 energy as a function of separation D, there's going to 635 00:37:40,360 --> 00:37:46,360 be a Coulombic attractive force that goes as 1/D. 636 00:37:46,360 --> 00:37:50,350 But when the electron configurations get in to close 637 00:37:50,350 --> 00:37:53,970 proximity, at small separations, the electrons 638 00:37:53,970 --> 00:37:56,940 will start to push one another, beginning with the 639 00:37:56,940 --> 00:38:00,240 outer electrons first, into higher energy states. 640 00:38:00,240 --> 00:38:05,160 And the energy of interaction, due to the interaction of 641 00:38:05,160 --> 00:38:08,480 orbitals, will increase the energy of the system. 642 00:38:08,480 --> 00:38:15,240 And the result then is that the net potential, as a 643 00:38:15,240 --> 00:38:18,310 function of distance for an isolated ion pair-- 644 00:38:18,310 --> 00:38:20,660 I was looking for some colored chalk and it's gone-- is 645 00:38:20,660 --> 00:38:24,750 something that rises steeply at low separations and then 646 00:38:24,750 --> 00:38:28,230 levels out more gradually at larger separations. 647 00:38:28,230 --> 00:38:33,390 So this then would be an equilibrium separation, D0, 648 00:38:33,390 --> 00:38:35,360 for that ionic pair. 649 00:38:39,750 --> 00:38:42,942 If we got energy out, and for that isolated ionic pair it'll 650 00:38:42,942 --> 00:38:46,130 be just exactly this much energy, E0. 651 00:38:46,130 --> 00:38:49,880 If it weren't for a pair of ions-- 652 00:38:49,880 --> 00:38:54,890 let's take another A and bring it into the system. 653 00:38:54,890 --> 00:38:59,960 And there's a lot of room to put still another A around. 654 00:38:59,960 --> 00:39:02,200 And we can add that to the system. 655 00:39:02,200 --> 00:39:04,920 And every time we bring up an additional neighbor, we get 656 00:39:04,920 --> 00:39:09,080 approximately this much energy out of the system. 657 00:39:09,080 --> 00:39:13,860 But not quite because there will also be repulsive 658 00:39:13,860 --> 00:39:20,890 interactions between the ions of like charge. 659 00:39:20,890 --> 00:39:21,850 OK? 660 00:39:21,850 --> 00:39:25,290 But to a good approximation to within about 10% to be 661 00:39:25,290 --> 00:39:28,530 quantitatively, every time we bring up another neighbor, we 662 00:39:28,530 --> 00:39:31,290 get out the same amount of energy again by 663 00:39:31,290 --> 00:39:34,060 forming this group. 664 00:39:34,060 --> 00:39:37,895 And all that works splendidly until-- 665 00:39:40,460 --> 00:39:46,090 let me now look at the arrangement of anions about 666 00:39:46,090 --> 00:39:48,560 the little cation because they will have to 667 00:39:48,560 --> 00:39:50,640 have a shell as well. 668 00:39:50,640 --> 00:39:53,980 So if we have a small cation and we bring up a big fat 669 00:39:53,980 --> 00:39:59,150 anion, there will be, first of all, two consequences. 670 00:39:59,150 --> 00:40:03,490 These ions of like charge will interact repulsively. 671 00:40:03,490 --> 00:40:08,320 So the lowest energy situation is going to be one in which 672 00:40:08,320 --> 00:40:16,990 these neighboring ions arrange themselves so that the 673 00:40:16,990 --> 00:40:20,260 repulsive energy is minimized. 674 00:40:20,260 --> 00:40:23,730 And we're going to get the maximum separation between all 675 00:40:23,730 --> 00:40:28,240 of these anions and get the minimum energy when they 676 00:40:28,240 --> 00:40:31,750 arrange themselves on the corners of a regular 677 00:40:31,750 --> 00:40:33,670 polyhedron. 678 00:40:33,670 --> 00:40:37,540 And that's where symmetry starts to come in to the 679 00:40:37,540 --> 00:40:39,650 importance of ionic structures. 680 00:40:39,650 --> 00:40:45,080 That, for this particular configuration of anions, the B 681 00:40:45,080 --> 00:40:47,920 ions might be on the corners. 682 00:40:47,920 --> 00:40:50,710 And I am not drawing them in contact now just so it'll be 683 00:40:50,710 --> 00:40:52,090 easier to draw. 684 00:40:52,090 --> 00:40:57,590 The B ions will be equidistant and tend to arrange themselves 685 00:40:57,590 --> 00:40:59,285 on the corner of the regular polyhedron. 686 00:41:02,080 --> 00:41:04,770 If we got energy up by bringing up extra neighbors, 687 00:41:04,770 --> 00:41:11,430 we will reach a point where the B ions will essentially be 688 00:41:11,430 --> 00:41:13,280 in contact. 689 00:41:13,280 --> 00:41:20,770 And the A ion is rattling around in a hole. 690 00:41:20,770 --> 00:41:23,020 It can touch perhaps two neighbors. 691 00:41:23,020 --> 00:41:25,170 And we will get the full amount of energy on. 692 00:41:25,170 --> 00:41:27,310 But for the other two neighbors, we're going to be 693 00:41:27,310 --> 00:41:28,400 out here on this curve. 694 00:41:28,400 --> 00:41:30,500 And we won't get nearly as much energy out. 695 00:41:30,500 --> 00:41:34,545 But at the same time, we're paying the price energetically 696 00:41:34,545 --> 00:41:39,600 of all of these repulsive interactions between the 697 00:41:39,600 --> 00:41:42,610 surrounding neighbors of like charge. 698 00:41:42,610 --> 00:41:47,960 So the second consideration is that, as we said, first of 699 00:41:47,960 --> 00:41:51,380 all, this group of atoms, which is referred to as a 700 00:41:51,380 --> 00:42:03,980 coordination polyhedron, it first 701 00:42:03,980 --> 00:42:05,975 of all will be symmetric. 702 00:42:10,670 --> 00:42:18,020 And then the number of neighbors, the number of ions 703 00:42:18,020 --> 00:42:23,780 of one charge that are coordinating the others-- 704 00:42:23,780 --> 00:42:26,690 and that's something called the coordination number, 705 00:42:26,690 --> 00:42:28,470 introduce some jargon-- 706 00:42:35,570 --> 00:42:45,200 has an upper limit that is going to be determined by the 707 00:42:45,200 --> 00:42:46,450 size of the ions. 708 00:42:59,250 --> 00:43:03,930 So these are the two basic tenets of 709 00:43:03,930 --> 00:43:07,110 ionic crystal chemistry. 710 00:43:07,110 --> 00:43:09,890 So having established those points, we can do some 711 00:43:09,890 --> 00:43:19,320 geometry and assume that the ions are hard spheres or at 712 00:43:19,320 --> 00:43:23,330 least elastic spheres with some stiffness because it's 713 00:43:23,330 --> 00:43:26,190 going to take energy to perturb the orbitals of the 714 00:43:26,190 --> 00:43:32,440 electrons and, just in terms of simple geometry, calculate 715 00:43:32,440 --> 00:43:38,030 the range of radius ratios RA/RB where A is the central 716 00:43:38,030 --> 00:43:41,680 ion in the polyhedron and B is the surrounding ion, and do 717 00:43:41,680 --> 00:43:44,860 this for different coordination numbers, picking 718 00:43:44,860 --> 00:43:48,570 polyhedra in which the surrounding ions of like 719 00:43:48,570 --> 00:43:52,070 charge are as widely separated as possible. 720 00:43:52,070 --> 00:43:54,850 So starting at the beginning, coordination number one would 721 00:43:54,850 --> 00:43:57,440 be a pair of ions. 722 00:43:57,440 --> 00:44:00,190 Obviously no constraints whatsoever on 723 00:44:00,190 --> 00:44:01,390 the relative sizes. 724 00:44:01,390 --> 00:44:07,010 RA/RB can be anywhere it likes between zero and infinity. 725 00:44:07,010 --> 00:44:10,420 For coordination number two, again the surrounding ions 726 00:44:10,420 --> 00:44:13,510 want to be as widely separated as possible. 727 00:44:13,510 --> 00:44:15,780 So it'll be a linear chain. 728 00:44:15,780 --> 00:44:18,880 But, again, the central ion can become as small or as 729 00:44:18,880 --> 00:44:19,730 large as it like. 730 00:44:19,730 --> 00:44:23,450 So the range of radii is between zero and infinity. 731 00:44:23,450 --> 00:44:26,000 For coordination number three, the polyhedron will be an 732 00:44:26,000 --> 00:44:27,680 equilateral triangle. 733 00:44:27,680 --> 00:44:32,110 And if you calculate the geometry, when the B ions are 734 00:44:32,110 --> 00:44:38,220 exactly in contact, you have a little 30, 60, 90 triangle in 735 00:44:38,220 --> 00:44:40,860 which one edge of the triangle is RB and the 736 00:44:40,860 --> 00:44:42,850 other is RA plus RB. 737 00:44:42,850 --> 00:44:45,350 And you can manipulate that into the form where you can 738 00:44:45,350 --> 00:44:50,030 show that there will be a lower limit RA/RB, which is 2 739 00:44:50,030 --> 00:44:51,800 over the square root of 3 minus 1. 740 00:44:51,800 --> 00:44:56,940 And that turns out to be a magic number, 0.155. 741 00:44:56,940 --> 00:44:59,830 Going to coordination number four, you'd say, aha. 742 00:44:59,830 --> 00:45:01,460 That's going to be square group. 743 00:45:01,460 --> 00:45:05,360 No, a square group would not keep the surrounding ions of 744 00:45:05,360 --> 00:45:08,540 like charge as widely separated as possible. 745 00:45:08,540 --> 00:45:12,010 And the geometry which would keep them as widely separated 746 00:45:12,010 --> 00:45:15,300 as possible but yet in contact with the central sphere would 747 00:45:15,300 --> 00:45:17,670 be a tetrahedron. 748 00:45:17,670 --> 00:45:20,440 The Bs would be arranged on alternating 749 00:45:20,440 --> 00:45:22,260 vertices of a cube. 750 00:45:22,260 --> 00:45:28,480 The magic number there turns out to be 0.225. 751 00:45:28,480 --> 00:45:31,120 Coordination five-- 752 00:45:31,120 --> 00:45:34,570 very, very rare. 753 00:45:34,570 --> 00:45:38,530 Five is a crystallographic abomination because five-fold 754 00:45:38,530 --> 00:45:41,200 symmetry does not conform to a lattice. 755 00:45:41,200 --> 00:45:42,930 But there would be, in principle, 756 00:45:42,930 --> 00:45:44,285 one five-fold group. 757 00:45:47,710 --> 00:45:55,630 And that would be a trigonal prism where you have three Bs 758 00:45:55,630 --> 00:45:59,520 arranged on an equilateral triangle about the central A. 759 00:45:59,520 --> 00:46:03,730 And then you have one A up here, another B up here, and 760 00:46:03,730 --> 00:46:05,790 another B down here. 761 00:46:05,790 --> 00:46:11,770 So this, then, is a bipyramid, a triangular bipyramid. 762 00:46:11,770 --> 00:46:17,550 And you can derive a range of radius ratios for this. 763 00:46:17,550 --> 00:46:21,050 And this is a dandy problem to give on a problem set. 764 00:46:21,050 --> 00:46:27,330 But you find that the range of radius ratios RA/RB is the 765 00:46:27,330 --> 00:46:36,350 same as for six-fold coordination, which is the 766 00:46:36,350 --> 00:46:39,630 next one considered on the handout. 767 00:46:39,630 --> 00:46:44,850 So we six coordination with the corners on a regular 768 00:46:44,850 --> 00:46:47,300 arrangement would be an octahedron. 769 00:46:47,300 --> 00:46:52,480 And the permitted range there is 0.732 to infinity. 770 00:46:52,480 --> 00:46:55,700 Five coordination, the trigonal prism, leads to 771 00:46:55,700 --> 00:46:59,770 exactly the same range of radii. 772 00:46:59,770 --> 00:47:04,410 And the reason is, if you look at the geometries, this 773 00:47:04,410 --> 00:47:08,490 triangle here is the same for both an octahedral 774 00:47:08,490 --> 00:47:14,410 coordination and a trigonal pyramidal coordination. 775 00:47:14,410 --> 00:47:18,350 Notice as we chug on that the lower limit to the radius 776 00:47:18,350 --> 00:47:21,980 ratio is gradually increasing. 777 00:47:21,980 --> 00:47:25,440 And if we go to coordination number eight, cubic 778 00:47:25,440 --> 00:47:29,110 coordination, it goes up to 1 to infinity. 779 00:47:29,110 --> 00:47:34,650 And for coordination number 12, RA/RB has to 780 00:47:34,650 --> 00:47:36,850 be exactly 1 period. 781 00:47:36,850 --> 00:47:38,580 And that would be what you achieved in the 782 00:47:38,580 --> 00:47:39,830 close-packed number. 783 00:47:47,950 --> 00:47:50,390 OK, turning the page. 784 00:47:50,390 --> 00:47:54,540 There is also going to be a range of radius ratios that's 785 00:47:54,540 --> 00:48:01,170 determined by the number of As that can be packed around a 786 00:48:01,170 --> 00:48:11,940 central B. And doing exactly the same calculation, if, for 787 00:48:11,940 --> 00:48:19,270 example, four-fold coordination permitted in 788 00:48:19,270 --> 00:48:29,240 terms of RA/RB, the range is 0.225 to infinity. 789 00:48:29,240 --> 00:48:35,220 If we want to know the limits for packing of A around B with 790 00:48:35,220 --> 00:48:41,300 tetrahedral coordination, this would give us numbers RA/RB 791 00:48:41,300 --> 00:48:46,840 that were exactly the same range of numbers because we've 792 00:48:46,840 --> 00:48:48,700 just changed the labels. 793 00:48:48,700 --> 00:48:53,280 If we want to put everything on the same basis, in terms of 794 00:48:53,280 --> 00:48:57,450 one particular radius ratio, if we do anion to cation 795 00:48:57,450 --> 00:49:03,930 radius ratio, the range would be 0.225 to infinity. 796 00:49:03,930 --> 00:49:09,170 If we take the reciprocals of these numbers for four-fold 797 00:49:09,170 --> 00:49:12,520 coordination of B, this would be 1 over 798 00:49:12,520 --> 00:49:17,460 0.225 to 1 over infinity. 799 00:49:17,460 --> 00:49:20,880 So this will give us-- 800 00:49:20,880 --> 00:49:21,880 this is zero. 801 00:49:21,880 --> 00:49:22,960 This is infinity. 802 00:49:22,960 --> 00:49:31,840 So there'll be a range of radius ratios, 0.225 less than 803 00:49:31,840 --> 00:49:40,970 RA/RB, less than 1 over 0.225, which turns out to be 4.44. 804 00:49:40,970 --> 00:49:46,160 So this would be the range of permitted radius ratios for 805 00:49:46,160 --> 00:49:53,560 CNA equals 4 and CNB equals the same thing, 4. 806 00:49:53,560 --> 00:49:56,640 So there's going to be an upper limit that's imposed by 807 00:49:56,640 --> 00:50:00,810 the packing of As around B and a lower limit that's imposed 808 00:50:00,810 --> 00:50:08,450 by the packing of Bs around A. 809 00:50:08,450 --> 00:50:10,420 But what are going to be the coordination 810 00:50:10,420 --> 00:50:12,270 numbers of A and B? 811 00:50:12,270 --> 00:50:13,290 Let me finish this up. 812 00:50:13,290 --> 00:50:15,700 And then we can take our break. 813 00:50:19,480 --> 00:50:24,180 I'm going to now restrict things rather drastically. 814 00:50:24,180 --> 00:50:39,370 I'm going to consider binary compounds in which all cations 815 00:50:39,370 --> 00:50:53,560 have the same coordination and all anions with the same 816 00:50:53,560 --> 00:50:54,810 coordination. 817 00:51:03,410 --> 00:51:06,210 And the implication there is there's only one type of 818 00:51:06,210 --> 00:51:08,582 coordination number for both. 819 00:51:08,582 --> 00:51:10,720 And that's not always true in compounds. 820 00:51:10,720 --> 00:51:14,630 Sometimes you find a given species that has two different 821 00:51:14,630 --> 00:51:17,995 coordination numbers in a more complex compound. 822 00:51:21,370 --> 00:51:28,970 If that's the case, if this is the A ion and this is the B 823 00:51:28,970 --> 00:51:33,790 ion, let me consider the chemical 824 00:51:33,790 --> 00:51:35,185 composition of a bond. 825 00:51:43,350 --> 00:51:45,410 And that's going to translate into the same 826 00:51:45,410 --> 00:51:46,600 thing as the compound. 827 00:51:46,600 --> 00:51:50,270 So let's suppose there's a number of Bs around the As, 828 00:51:50,270 --> 00:51:53,335 which is, by definition, the coordination number of A, and 829 00:51:53,335 --> 00:51:56,670 a number of As around a B and that is, by definition, the 830 00:51:56,670 --> 00:52:00,990 coordination number of B. So let me look at a wedge that's 831 00:52:00,990 --> 00:52:03,260 associated with this bond. 832 00:52:03,260 --> 00:52:07,130 And this is an amount of an A atom, which is 1 over the 833 00:52:07,130 --> 00:52:09,220 coordination number of A. 834 00:52:09,220 --> 00:52:14,390 And if I do the same thing for the B ion the part of the B 835 00:52:14,390 --> 00:52:17,900 ion associated with one bond is going to be 1 over the 836 00:52:17,900 --> 00:52:21,070 coordination number of B. So we can say then that the 837 00:52:21,070 --> 00:52:32,500 composition of a bond is going to be A subscript 1 over the 838 00:52:32,500 --> 00:52:36,970 coordination number of A, B, 1 over the coordination number 839 00:52:36,970 --> 00:52:38,910 of B. 840 00:52:38,910 --> 00:52:42,290 The chemists abhor fractional subscripts on a chemical 841 00:52:42,290 --> 00:52:43,510 composition. 842 00:52:43,510 --> 00:52:46,820 So let's multiply this by the product of the two 843 00:52:46,820 --> 00:52:48,070 coordination numbers. 844 00:52:52,200 --> 00:52:55,620 And what we'll find is that the formula of the compound 845 00:52:55,620 --> 00:53:01,620 with integer subscripts is going to be A, C, and B; B 846 00:53:01,620 --> 00:53:10,350 coordination number of A. And that is a subject to a very 847 00:53:10,350 --> 00:53:15,800 drastic set of assumptions that each species A and B has 848 00:53:15,800 --> 00:53:17,230 just one coordination number. 849 00:53:23,660 --> 00:53:26,740 OK, so doing this in terms of composition, just turning this 850 00:53:26,740 --> 00:53:27,610 result around. 851 00:53:27,610 --> 00:53:29,170 And then we'll quit and take a stretch. 852 00:53:32,230 --> 00:53:39,850 This says that if we look at a particular composition, AnBm, 853 00:53:39,850 --> 00:53:47,230 something like TiO2 or Al2O3, then the coordination number 854 00:53:47,230 --> 00:53:51,100 of A in proportion to the coordination number of B is 855 00:53:51,100 --> 00:54:01,490 going to be equal to the ratio of M to N. 856 00:54:01,490 --> 00:54:06,500 And the more ions of one charge that we can pack around 857 00:54:06,500 --> 00:54:09,200 the other, the lower the energy will be. 858 00:54:09,200 --> 00:54:16,480 And, therefore, the coordination numbers will be 859 00:54:16,480 --> 00:54:33,560 the maximum permitted values of CNA and CNB, subject to the 860 00:54:33,560 --> 00:54:37,510 constraints of radius ratio. 861 00:54:37,510 --> 00:54:40,540 In other words, to conclude very quickly with one specific 862 00:54:40,540 --> 00:54:46,590 example, if we look at a compound like Al2O3, an 863 00:54:46,590 --> 00:54:50,040 important ceramic and a gemstone, this says that the 864 00:54:50,040 --> 00:54:55,080 coordination number of Al to the coordination number of the 865 00:54:55,080 --> 00:54:59,110 oxygen has to be equal to three to two. 866 00:54:59,110 --> 00:55:07,500 So possible structures might be aluminum with coordination 867 00:55:07,500 --> 00:55:10,850 number three, oxygen with coordination 868 00:55:10,850 --> 00:55:14,820 number equal to two. 869 00:55:14,820 --> 00:55:19,410 Or aluminum could have six coordination. 870 00:55:19,410 --> 00:55:22,010 The oxygen could have four coordination. 871 00:55:22,010 --> 00:55:25,560 Or the aluminum could have 12 coordination. 872 00:55:25,560 --> 00:55:29,390 And the oxygen could have eight coordination. 873 00:55:32,370 --> 00:55:34,940 Those three possibilities. 874 00:55:34,940 --> 00:55:38,530 Next we have to use the ionic sizes to see what constraints 875 00:55:38,530 --> 00:55:41,530 there are that are going to limit the maximum coordination 876 00:55:41,530 --> 00:55:44,720 numbers because the greater the number of neighbors, the 877 00:55:44,720 --> 00:55:46,030 lower the energy. 878 00:55:46,030 --> 00:55:49,820 And, in the case of aluminum oxide, it turns out that it's 879 00:55:49,820 --> 00:55:51,460 six to four. 880 00:55:51,460 --> 00:55:54,500 And this is determined by the radius ratio. 881 00:56:01,450 --> 00:56:01,810 And you know what? 882 00:56:01,810 --> 00:56:06,220 If we used the ionic radii for Al3 plus and O2 minus, we were 883 00:56:06,220 --> 00:56:07,970 right one the nose. 884 00:56:07,970 --> 00:56:12,740 That is the nature of the structure of aluminum oxide-- 885 00:56:12,740 --> 00:56:17,780 aluminum with octahedral coordination by oxygen and 886 00:56:17,780 --> 00:56:24,200 oxygen with a tetrahedral coordination of aluminum. 887 00:56:24,200 --> 00:56:27,850 And that is the actual structure. 888 00:56:27,850 --> 00:56:32,720 The three-dimensional linkage would have to be such that 889 00:56:32,720 --> 00:56:35,615 ions of like charge have the maximum possible separation. 890 00:56:40,310 --> 00:56:43,050 And to keep you beyond your patience for just a little 891 00:56:43,050 --> 00:56:44,220 bit, what is that going to be? 892 00:56:44,220 --> 00:56:46,960 We can use very simple considerations there. 893 00:56:46,960 --> 00:56:50,120 If we have coordination polyhedra and we want to join 894 00:56:50,120 --> 00:56:53,920 them together, the worst possible situation is to have 895 00:56:53,920 --> 00:57:00,140 them share edges because the central ions then had minimum 896 00:57:00,140 --> 00:57:01,390 separation. 897 00:57:03,270 --> 00:57:09,940 The better situation would be to have to the polyhedra share 898 00:57:09,940 --> 00:57:16,180 corners and have the ions in the center of these triangles 899 00:57:16,180 --> 00:57:20,740 form a straight line with the shared ions. 900 00:57:20,740 --> 00:57:23,460 For a three-dimensional polyhedron, the sequence of 901 00:57:23,460 --> 00:57:26,950 increasing repulsive energy would be shared faces for 902 00:57:26,950 --> 00:57:30,240 something like cubes, shared edges-- 903 00:57:30,240 --> 00:57:33,440 that would be lower energy but not the best you could do. 904 00:57:33,440 --> 00:57:36,990 Shared corners would be the best possible arrangement. 905 00:57:36,990 --> 00:57:39,690 As you make each of those transitions, you take the 906 00:57:39,690 --> 00:57:42,960 central ion and keep the separation between the 907 00:57:42,960 --> 00:57:46,480 neighboring ion of like charge as low as possible. 908 00:57:46,480 --> 00:57:48,865 The interaction energy would be as low as possible. 909 00:57:53,130 --> 00:57:54,620 All right, so let me stop there. 910 00:57:54,620 --> 00:57:57,150 I think we're probably about halfway through. 911 00:57:59,890 --> 00:58:03,930 We can then determine ranges of radius ratio for various 912 00:58:03,930 --> 00:58:05,310 stoichiometries. 913 00:58:05,310 --> 00:58:09,940 And all this is just a fantasy unless we can say, do ions 914 00:58:09,940 --> 00:58:15,380 have sizes that are the same from compound to compound? 915 00:58:15,380 --> 00:58:18,750 And, if so, how do we determine them? 916 00:58:18,750 --> 00:58:24,290 I hope that intriguing pair of questions will keep you here, 917 00:58:24,290 --> 00:58:29,790 dry and warm, rather than heading off early. 918 00:58:29,790 --> 00:58:35,630 OK, so let's take our usual 10-minute break and resume to 919 00:58:35,630 --> 00:58:36,880 wrap this up.