WEBVTT 00:00:10.010 --> 00:00:11.260 PROFESSOR: Why don't we get started. 00:00:15.866 --> 00:00:17.070 Good afternoon-- 00:00:17.070 --> 00:00:22.060 to stretch the literal meaning of the word considerably. 00:00:22.060 --> 00:00:23.690 What a miserable day out there. 00:00:23.690 --> 00:00:29.470 I get to and from the boonies via the commuter rail. 00:00:29.470 --> 00:00:33.480 And this morning, we sat there for an hour part way in 00:00:33.480 --> 00:00:38.140 because a tree had fallen across the railroad track. 00:00:38.140 --> 00:00:40.550 I don't know who went out to do it, but somebody had to get 00:00:40.550 --> 00:00:43.860 a saw and saw the tree up and take it off the tracks before 00:00:43.860 --> 00:00:47.580 we could proceed into Boston. 00:00:47.580 --> 00:00:49.440 Even with that, I was sort of reluctant 00:00:49.440 --> 00:00:51.520 to get off the train. 00:00:51.520 --> 00:00:53.600 I saw how it was pouring outside. 00:00:53.600 --> 00:00:57.360 So I am highly flattered that you decided to trudge through 00:00:57.360 --> 00:01:00.810 the puddles and show up this afternoon. 00:01:00.810 --> 00:01:04.060 All right, this is a momentous occasion because we are 00:01:04.060 --> 00:01:07.770 exactly at this point, halfway through the term. 00:01:07.770 --> 00:01:11.600 And this is when we are about to change gears from our 00:01:11.600 --> 00:01:15.520 discussion of symmetry theory and switch over to physical 00:01:15.520 --> 00:01:19.530 properties of crystals and the way in which symmetry impacts 00:01:19.530 --> 00:01:22.250 on those properties. 00:01:22.250 --> 00:01:25.690 What I wanted to do primarily today, though, is to say a 00:01:25.690 --> 00:01:30.580 little bit in a very fast and very simple way about the 00:01:30.580 --> 00:01:33.910 nature of crystal structures because that is one of the 00:01:33.910 --> 00:01:36.360 primary uses of symmetry theory. 00:01:36.360 --> 00:01:39.520 It's to describe the periodic arrangements and symmetrical 00:01:39.520 --> 00:01:41.700 arrangements of atoms in crystals. 00:01:41.700 --> 00:01:44.410 So it's the language that's necessary to describe such 00:01:44.410 --> 00:01:48.210 arrangements, particularly when they're beyond the red 00:01:48.210 --> 00:01:50.710 balls at the corner of the cube and black balls in the 00:01:50.710 --> 00:01:52.370 middle of the faces level of structure. 00:01:56.540 --> 00:02:01.850 Let me first, for your amusement and edification, 00:02:01.850 --> 00:02:03.740 pass out another problem set. 00:02:03.740 --> 00:02:05.730 This has only two problems on it. 00:02:05.730 --> 00:02:09.840 You are not really equipped to do it yet because we haven't 00:02:09.840 --> 00:02:11.520 talked about space groups. 00:02:11.520 --> 00:02:13.970 So don't worry about doing that now. 00:02:13.970 --> 00:02:18.510 But I wanted to get it in your hand so that you 00:02:18.510 --> 00:02:20.770 could look it over. 00:02:20.770 --> 00:02:25.310 It's a good problem set on which to end this part of the 00:02:25.310 --> 00:02:25.840 discussion-- 00:02:25.840 --> 00:02:27.650 but there's another one that I'll be 00:02:27.650 --> 00:02:30.100 passing out next week-- 00:02:30.100 --> 00:02:33.930 because this is an example of how you can use the notions of 00:02:33.930 --> 00:02:37.610 symmetry that we have established to say some fairly 00:02:37.610 --> 00:02:41.320 profound and non-obvious things about the nature of 00:02:41.320 --> 00:02:42.850 atomic arrangements. 00:02:42.850 --> 00:02:44.960 So that's the purpose of this problem set. 00:02:44.960 --> 00:02:48.050 It is to give you an opportunity to see the 00:02:48.050 --> 00:02:52.230 surprising things that can fall out of our material that 00:02:52.230 --> 00:02:55.170 we now have at our disposal. 00:02:55.170 --> 00:02:59.140 Let me begin by telling you where we left off last time. 00:02:59.140 --> 00:03:07.670 We had completed deriving the so-called Bravais lattices, 00:03:07.670 --> 00:03:10.940 the three-dimensional space lattices. 00:03:10.940 --> 00:03:14.310 The process by which we did this was to take our 00:03:14.310 --> 00:03:17.460 two-dimensional space groups, the plane groups, and say 00:03:17.460 --> 00:03:20.610 that, if we add a third translation, we will have 00:03:20.610 --> 00:03:22.510 generated a space lattice. 00:03:22.510 --> 00:03:26.210 But these different nets in the base of the cell are 00:03:26.210 --> 00:03:29.600 special and determined by the fact that there are symmetry 00:03:29.600 --> 00:03:30.710 in these nets. 00:03:30.710 --> 00:03:35.300 So if we take, for example, a net that is exactly square-- 00:03:35.300 --> 00:03:38.120 it's square because there's a fourfold axis there. 00:03:38.120 --> 00:03:41.520 And that fourfold axis, in a space lattice, pokes not only 00:03:41.520 --> 00:03:44.760 into the base of the cell but extends up through space. 00:03:44.760 --> 00:03:48.050 So you must pick a third translation such that the 00:03:48.050 --> 00:03:49.840 symmetry elements line up. 00:03:49.840 --> 00:03:53.960 We would take each of the crystallographic plane groups 00:03:53.960 --> 00:03:57.000 and pick a third translation. 00:03:57.000 --> 00:04:03.260 You end up with almost all of the lattices that you're going 00:04:03.260 --> 00:04:07.570 to get except for the ones that involve cubic symmetries 00:04:07.570 --> 00:04:10.400 that are inherently three-dimensional. 00:04:10.400 --> 00:04:14.810 And the reason for that is that the more complex 00:04:14.810 --> 00:04:18.660 three-dimensional point groups are, for the most part, 00:04:18.660 --> 00:04:25.340 obtained by adding inversion to the simple symmetries that 00:04:25.340 --> 00:04:26.690 exist in two dimensions. 00:04:26.690 --> 00:04:30.290 But inversion doesn't require anything of a lattice. 00:04:30.290 --> 00:04:34.850 So you can immediately say that 2M and 2/M are compatible 00:04:34.850 --> 00:04:39.640 with the same sort of space lattice because the symmetry 00:04:39.640 --> 00:04:43.730 2/M is nothing more than what you get when you add inversion 00:04:43.730 --> 00:04:47.220 to either a twofold axis or a mirror plane. 00:04:47.220 --> 00:04:52.290 And, finally, another way of classifying crystals, and a 00:04:52.290 --> 00:04:56.650 broader way of classifying them, is according to the 00:04:56.650 --> 00:04:59.480 coordinate systems that are necessary to 00:04:59.480 --> 00:05:00.730 describe the lattices. 00:05:05.050 --> 00:05:07.305 And these are called the crystal systems. 00:05:13.400 --> 00:05:19.270 And, as a mnemonic device for keeping straight what the 00:05:19.270 --> 00:05:23.370 crystal systems are as opposed to the crystal classes, which 00:05:23.370 --> 00:05:25.920 is another word that we haven't used but it's another 00:05:25.920 --> 00:05:29.640 word for the point groups, the crystal systems goes with 00:05:29.640 --> 00:05:32.510 coordinate systems. 00:05:32.510 --> 00:05:37.200 And it's simply a word to describe the shape of the 00:05:37.200 --> 00:05:40.480 lattices, family of lattices, that we've described. 00:05:40.480 --> 00:05:44.450 And we distributed a handout last time that gave the names 00:05:44.450 --> 00:05:48.470 of these systems and the relations that exist between 00:05:48.470 --> 00:05:52.620 the edges of the unit cell, which we use as the basis for 00:05:52.620 --> 00:05:54.670 our coordinate system and also the labels 00:05:54.670 --> 00:05:57.090 that we put on them. 00:05:57.090 --> 00:05:59.030 I have another sheet, which I'll pass around, which is 00:05:59.030 --> 00:05:59.980 nothing new. 00:05:59.980 --> 00:06:01.720 It just assembles-- 00:06:01.720 --> 00:06:03.780 I'll give you two sheets, which I'll pass around. 00:06:03.780 --> 00:06:08.410 The first is an assembly of the crystallographic point 00:06:08.410 --> 00:06:12.130 groups in a somewhat neater and tidier way than we had on 00:06:12.130 --> 00:06:16.020 the sheet that we passed around last time. 00:06:16.020 --> 00:06:19.480 So this is a picture of all 32 of the point groups. 00:06:19.480 --> 00:06:23.630 Again, beware of the fact that mirror planes and guides to 00:06:23.630 --> 00:06:30.100 the eye that split things up into 90 degree segments or 60 00:06:30.100 --> 00:06:32.560 degree segments come out deceptively 00:06:32.560 --> 00:06:34.530 similar on these sheets. 00:06:34.530 --> 00:06:37.540 But there with the representative pattern are the 00:06:37.540 --> 00:06:40.050 arrangements of symmetry elements, a representative 00:06:40.050 --> 00:06:44.750 pattern of motifs, and the international and [INAUDIBLE] 00:06:44.750 --> 00:06:46.990 symbol for all 32 of the 00:06:46.990 --> 00:06:48.520 three-dimensional point groups. 00:06:51.300 --> 00:06:53.490 And then the next sheet I'll pass around takes the point 00:06:53.490 --> 00:06:57.160 groups and groups them together with the lattices 00:06:57.160 --> 00:07:00.660 that can accommodate them and the crystal system that 00:07:00.660 --> 00:07:05.740 describes the arrangement of axes in the lattice. 00:07:05.740 --> 00:07:08.730 So this is, in a nutshell, everything that we've covered 00:07:08.730 --> 00:07:11.090 up to this point. 00:07:11.090 --> 00:07:16.880 And, at this point, I have to say that there is some 00:07:16.880 --> 00:07:22.160 disagreement on how many crystal systems exist. 00:07:22.160 --> 00:07:26.180 And even though people who work with symmetry and 00:07:26.180 --> 00:07:33.350 mathematics are usually rational, almost to a fault 00:07:33.350 --> 00:07:38.190 logical individuals, there are several junctures as you build 00:07:38.190 --> 00:07:41.300 up a body of knowledge where you could go either way or 00:07:41.300 --> 00:07:42.680 adopt either convention. 00:07:42.680 --> 00:07:45.910 You sort of got to pay your money and take your choice. 00:07:45.910 --> 00:07:50.700 And this occurs in our arrangement of point groups 00:07:50.700 --> 00:07:53.460 and lattices among the crystal systems. 00:07:53.460 --> 00:07:56.080 This occurs in the hexagonal crystal system. 00:07:58.580 --> 00:08:02.980 Symmetry 3 and the other point groups that can be derived 00:08:02.980 --> 00:08:09.010 from it can be placed either in a hexagonal lattice, that 00:08:09.010 --> 00:08:13.890 is to say, a third translation C that is exactly at right 00:08:13.890 --> 00:08:17.760 angles to a pair of translations, A1 and A2, which 00:08:17.760 --> 00:08:20.460 are 120 degrees apart. 00:08:20.460 --> 00:08:24.840 Or a three-fold symmetry can go into this funny double-body 00:08:24.840 --> 00:08:28.980 centered hexagonal lattice, which could be defined, if we 00:08:28.980 --> 00:08:32.159 were masochistic and enjoyed working in an oblique 00:08:32.159 --> 00:08:35.340 coordinate system, could be defined instead in terms of a 00:08:35.340 --> 00:08:38.000 rhombohedral lattice, which had the special 00:08:38.000 --> 00:08:41.860 characteristics that three translations are identical by 00:08:41.860 --> 00:08:46.360 symmetry, called therefore A1, A2, and A3; and the angles 00:08:46.360 --> 00:08:50.080 between the more equal alpha 1, alpha 2, alpha 3; and they 00:08:50.080 --> 00:08:52.730 could be anything we wished. 00:08:52.730 --> 00:08:57.230 So that is a unique sort of crystal system and unique sort 00:08:57.230 --> 00:08:58.730 of coordinate system. 00:08:58.730 --> 00:09:03.830 There are those who would say that if this is the case, we 00:09:03.830 --> 00:09:06.600 really have seven crystal systems-- 00:09:06.600 --> 00:09:09.760 triclinic; monoclinic; orthorhombic; tetragonal; 00:09:09.760 --> 00:09:13.790 hexagonal; and then this one that is compatible only with a 00:09:13.790 --> 00:09:17.870 three-fold axis and that is called the trigonal system or, 00:09:17.870 --> 00:09:20.410 by some folks, the rhombohedral system; and then, 00:09:20.410 --> 00:09:22.590 finally, the cubic. 00:09:22.590 --> 00:09:24.870 Well isn't that a rational and logical thing to do? 00:09:24.870 --> 00:09:25.840 Yeah, question at this point? 00:09:25.840 --> 00:09:27.605 AUDIENCE: Do you have any extra copies of the sheet 00:09:27.605 --> 00:09:29.350 [INAUDIBLE] 00:09:29.350 --> 00:09:31.680 PROFESSOR: Oh, yes. 00:09:31.680 --> 00:09:35.270 There should be more than enough floating around. 00:09:35.270 --> 00:09:36.866 This one here? 00:09:36.866 --> 00:09:37.364 [? AUDIENCE: The ?] 00:09:37.364 --> 00:09:37.862 [? chart ?] 00:09:37.862 --> 00:09:38.858 AUDIENCE: Yeah, [? not the ?] 00:09:38.858 --> 00:09:39.360 [? chart ?]. 00:09:39.360 --> 00:09:40.860 PROFESSOR: OK. 00:09:40.860 --> 00:09:42.020 I'll give you-- 00:09:42.020 --> 00:09:43.220 I think there are more back there. 00:09:43.220 --> 00:09:44.610 But why don't you take one for now? 00:09:47.270 --> 00:09:50.230 Yeah, so it's a perfectly reasonable thing to do is to 00:09:50.230 --> 00:09:55.500 set up the trigonal system as a separate coordinate system 00:09:55.500 --> 00:09:57.650 to describe the rhombohedral lattice. 00:10:00.380 --> 00:10:05.070 The problem that creates, however, is that if you do 00:10:05.070 --> 00:10:09.800 this, there is no longer any one to one correspondence 00:10:09.800 --> 00:10:13.430 between the point groups and the crystal systems. 00:10:13.430 --> 00:10:17.590 A crystal with symmetry 3, for example, can fit into it 00:10:17.590 --> 00:10:22.620 either in a hexagonal lattice or a trigonal lattice. 00:10:22.620 --> 00:10:26.380 US So sometimes a crystal with point group 3 would have to be 00:10:26.380 --> 00:10:29.770 called a hexagonal crystal, sometimes a trigonal system. 00:10:29.770 --> 00:10:30.760 But which is it? 00:10:30.760 --> 00:10:32.170 Well it can be either. 00:10:32.170 --> 00:10:36.770 So admitting the trigonal system as a separate crystal 00:10:36.770 --> 00:10:40.550 system breaks down the one to one correspondence between 00:10:40.550 --> 00:10:45.720 point groups and crystal systems. 00:10:45.720 --> 00:10:50.330 The very pragmatic reason for not admitting the trigonal 00:10:50.330 --> 00:10:53.530 system as a separate crystal system 00:10:53.530 --> 00:10:58.150 is that, although you can in this funny triple hexagonal 00:10:58.150 --> 00:11:01.590 cell define a primitive trigonal cell that has a 00:11:01.590 --> 00:11:04.900 special geometry, nobody in their right mind, as I 00:11:04.900 --> 00:11:07.310 indicated earlier, would want to work in such 00:11:07.310 --> 00:11:08.710 a coordinate system. 00:11:08.710 --> 00:11:11.540 Better to take the triple cell, pay the price of 00:11:11.540 --> 00:11:15.650 three-fold redundancy, and yet have two interaxial angles 00:11:15.650 --> 00:11:19.100 that are 90 degrees and one that's specialized at 120. 00:11:19.100 --> 00:11:24.620 That's a much more convenient reference system for our 00:11:24.620 --> 00:11:26.270 description of crystal properties. 00:11:29.620 --> 00:11:32.520 One final thing I'll comment on but won't say anything 00:11:32.520 --> 00:11:34.350 about very much-- 00:11:34.350 --> 00:11:35.690 I always do that. 00:11:35.690 --> 00:11:38.400 I say I won't say anything about it but, and then launch 00:11:38.400 --> 00:11:40.470 into a 10 minute digression. 00:11:40.470 --> 00:11:44.120 Off in the right hand corner is something called the Laue 00:11:44.120 --> 00:11:48.340 group of which there are 11. 00:11:48.340 --> 00:11:52.330 The Laue groups are those of the point groups that contain 00:11:52.330 --> 00:11:55.940 the operation of inversion. 00:11:55.940 --> 00:11:58.690 And the reason they're called the Laue groups is that one of 00:11:58.690 --> 00:12:01.950 the primary ways, other than looking at crystal morphology 00:12:01.950 --> 00:12:08.140 to determine the symmetries of a crystal, is to look at the 00:12:08.140 --> 00:12:11.090 way it diffracts x-rays. 00:12:11.090 --> 00:12:15.010 So if we had, for example a tetragonal crystal and there 00:12:15.010 --> 00:12:19.010 were lattice planes in here that were related by 90 degree 00:12:19.010 --> 00:12:23.960 rotations, then if we brought in an x-ray beam at such an 00:12:23.960 --> 00:12:28.700 angle to satisfy Bragg's law and saw some sort of intensity 00:12:28.700 --> 00:12:35.620 that came off, if we rotated the x-ray beam 90 degrees, we 00:12:35.620 --> 00:12:38.420 ought to see diffraction at the same angle and with 00:12:38.420 --> 00:12:41.300 exactly the same intensity. 00:12:41.300 --> 00:12:44.650 So either by moving the x-ray beam, or simpler thing would 00:12:44.650 --> 00:12:48.550 be to leave the x-ray beam fixed, since it's attached to 00:12:48.550 --> 00:12:53.500 an x-ray generator that weighs a ton or more, instead move 00:12:53.500 --> 00:12:57.240 the crystal relative to the x-ray beam. 00:12:57.240 --> 00:13:03.490 Now what would you do in this way of thinking to determine 00:13:03.490 --> 00:13:09.530 whether a crystal had the operation of inversion in it? 00:13:09.530 --> 00:13:14.870 If this is the set of planes with indices HKL, you, again, 00:13:14.870 --> 00:13:20.680 would bring in an x-ray beam at an angle theta, in out. 00:13:20.680 --> 00:13:24.790 Look at the intensity from the set of points HKL. 00:13:24.790 --> 00:13:28.370 And then look at diffraction, both in terms of the Bragg 00:13:28.370 --> 00:13:31.290 angle theta and in terms of the intensity. 00:13:31.290 --> 00:13:33.830 Look at that diffraction from the planes that were related 00:13:33.830 --> 00:13:36.640 to HKL by symmetry. 00:13:36.640 --> 00:13:39.460 Well what plane is related to HKL by symmetry? 00:13:39.460 --> 00:13:43.490 It's these planes down here which have indices minus H, 00:13:43.490 --> 00:13:46.130 minus K, minus L. 00:13:46.130 --> 00:13:49.430 So if we did the corresponding experiment to what I did to 00:13:49.430 --> 00:13:52.450 the left, we'd bring in an x-ray beam like this. 00:13:52.450 --> 00:13:55.310 It would be diffracted at some angle theta. 00:13:55.310 --> 00:13:58.560 And we'd look at the intensity for the set of planes minus H, 00:13:58.560 --> 00:14:02.870 minus K, minus L. 00:14:02.870 --> 00:14:05.334 Is that going to be any different? 00:14:05.334 --> 00:14:06.170 No. 00:14:06.170 --> 00:14:08.690 Right-minded person would say, why should this 00:14:08.690 --> 00:14:09.870 intensity be different? 00:14:09.870 --> 00:14:14.680 All we're doing is bouncing the x-ray beam off the 00:14:14.680 --> 00:14:18.800 opposite side of one in the same set of lattice planes. 00:14:18.800 --> 00:14:22.040 So why should the intensity be any different? 00:14:22.040 --> 00:14:26.400 In this case, clearly if the set of lattice points is 00:14:26.400 --> 00:14:31.500 rotated by some finite angle, we are not going to see 00:14:31.500 --> 00:14:33.410 equivalent diffraction from planes that 00:14:33.410 --> 00:14:36.850 are 90 degrees apart. 00:14:36.850 --> 00:14:42.460 So the upshot is that the diffraction symmetry of a 00:14:42.460 --> 00:14:46.900 crystal, the way in which planes with indices that are 00:14:46.900 --> 00:14:50.250 related by symmetry diffract in terms of position of the 00:14:50.250 --> 00:14:54.110 beam and intensity, the diffraction symmetry of a 00:14:54.110 --> 00:14:58.340 crystal always looks as though that crystal possessed an 00:14:58.340 --> 00:15:00.600 inversion center. 00:15:00.600 --> 00:15:01.880 So, therefore, to-- 00:15:01.880 --> 00:15:05.650 and I qualify this by saying, to a good approximation-- 00:15:05.650 --> 00:15:08.200 the only point groups that you could determine and 00:15:08.200 --> 00:15:13.000 distinguish from one another by x-ray diffraction are the 00:15:13.000 --> 00:15:17.570 point groups that contain an inversion center. 00:15:17.570 --> 00:15:19.950 Almost, but not quite. 00:15:19.950 --> 00:15:23.010 If we look a little more closely at what is doing the 00:15:23.010 --> 00:15:25.490 scattering, it is not the Bragg planes. 00:15:25.490 --> 00:15:28.560 These are imaginary, shimmering, absolutely planar 00:15:28.560 --> 00:15:32.970 things that are constructs that we use to interpret the 00:15:32.970 --> 00:15:34.315 process called diffraction. 00:15:36.880 --> 00:15:40.090 Inside the crystal what is actually doing the diffraction 00:15:40.090 --> 00:15:45.380 is a collection of atoms, each of which are scattering the 00:15:45.380 --> 00:15:45.980 x-radiation. 00:15:45.980 --> 00:15:50.410 And that physically is what causes the diffraction. 00:15:50.410 --> 00:15:54.810 And the resultant intensity is the addition of all the little 00:15:54.810 --> 00:15:59.200 wavelets scattered by these separate atoms. 00:15:59.200 --> 00:16:02.800 So in terms of the atomistics of what is involved, to look 00:16:02.800 --> 00:16:08.850 at diffraction from the set of planes HKL and contrast that 00:16:08.850 --> 00:16:11.570 with the diffraction for the set of planes minus H, minus 00:16:11.570 --> 00:16:15.870 K, minus L, we're coming in from opposite directions. 00:16:15.870 --> 00:16:18.300 And here the beam would hit-- 00:16:18.300 --> 00:16:22.320 let's call the atoms, if I did not do so already, let's call 00:16:22.320 --> 00:16:27.155 them A and B. A for a and b for big. 00:16:30.740 --> 00:16:33.620 Diffraction from the set of points HKL would pass through 00:16:33.620 --> 00:16:36.730 the sheets of A atoms first and then impinge on the B 00:16:36.730 --> 00:16:40.910 atoms, whereas in the opposite direction, the x-ray beams 00:16:40.910 --> 00:16:44.120 would impinge upon the B atoms first and 00:16:44.120 --> 00:16:46.420 then hit the A atoms. 00:16:46.420 --> 00:16:48.860 Does that make a difference? 00:16:48.860 --> 00:16:49.380 Aha. 00:16:49.380 --> 00:16:51.440 Believe it or not, it does. 00:16:51.440 --> 00:16:54.030 And I can't explain why without going into a very 00:16:54.030 --> 00:16:56.210 detailed discussion of diffraction. 00:16:56.210 --> 00:16:59.570 But if you compare the intensities that are scattered 00:16:59.570 --> 00:17:07.220 by these two sets of planes, which are related by inversion 00:17:07.220 --> 00:17:10.940 and therefore just opposite sides of one another, it does 00:17:10.940 --> 00:17:15.380 make a difference which atom the x-ray 00:17:15.380 --> 00:17:17.060 beam encounters first. 00:17:17.060 --> 00:17:23.700 And the difference in intensity is on the order of 00:17:23.700 --> 00:17:29.720 3% to 5%, although by picking a radiation appropriately for 00:17:29.720 --> 00:17:32.360 the particular chemical species in the crystal, you 00:17:32.360 --> 00:17:36.140 could increase this perhaps to 10% or 12%. 00:17:36.140 --> 00:17:40.000 But that is something that you could not do until synchrotron 00:17:40.000 --> 00:17:43.990 radiation came along, which could actually tune the 00:17:43.990 --> 00:17:47.660 wavelength to a particular value that you wish to use and 00:17:47.660 --> 00:17:51.130 not use something that was spewed out by an x-ray tube 00:17:51.130 --> 00:17:56.550 that had a target such as molybdenum or copper or 00:17:56.550 --> 00:17:59.330 chromium, perhaps. 00:17:59.330 --> 00:18:02.060 OK, so you can with this careful measurement of 00:18:02.060 --> 00:18:07.140 intensity distinguished between the opposite sides of 00:18:07.140 --> 00:18:10.410 the given set of Bragg planes, unless, of course, the crystal 00:18:10.410 --> 00:18:11.970 did have inversion in it. 00:18:11.970 --> 00:18:14.120 And then there would be no difference at all. 00:18:14.120 --> 00:18:17.800 It requires a very careful experiment, though, because 00:18:17.800 --> 00:18:19.180 there are many things that could make 00:18:19.180 --> 00:18:20.470 the intensities different. 00:18:20.470 --> 00:18:23.605 For example, absorption by the crystal. 00:18:23.605 --> 00:18:26.250 If the crystal has an irregular shape, the 00:18:26.250 --> 00:18:30.290 absorption, which can easily be factors 10, 20, or even 00:18:30.290 --> 00:18:34.260 more, the absorption would be different for x-ray beams 00:18:34.260 --> 00:18:36.330 coming in those two orientations. 00:18:36.330 --> 00:18:38.860 So in any case, as I say, I won't say anything about it. 00:18:38.860 --> 00:18:40.860 But 10 minutes later, that's the 00:18:40.860 --> 00:18:42.020 meaning of the Laue groups. 00:18:42.020 --> 00:18:44.440 These are the point groups that have inversion in them. 00:18:53.570 --> 00:18:55.970 That effect that I just described, by the way, is 00:18:55.970 --> 00:18:59.570 called anomalous scattering or anomalous dispersion. 00:18:59.570 --> 00:19:01.930 There's really nothing anomalous about at all. 00:19:01.930 --> 00:19:05.460 It has to do with the way electromagnetic radiation 00:19:05.460 --> 00:19:10.180 interacts with the clouds of electrons that are distributed 00:19:10.180 --> 00:19:11.770 about the nucleus of each atom. 00:19:17.060 --> 00:19:24.520 Where we will go from here to finish up symmetry theory will 00:19:24.520 --> 00:19:30.150 be to now take the 32 crystallographic point groups 00:19:30.150 --> 00:19:37.530 and to drop those point groups into each of the distinct 14 00:19:37.530 --> 00:19:39.170 Bravais lattices. 00:19:39.170 --> 00:19:42.450 And then we'll need combination theorems to deduce 00:19:42.450 --> 00:19:46.360 where the additional symmetry elements arrive. 00:19:46.360 --> 00:19:49.450 And when we go through that process, if we were to do this 00:19:49.450 --> 00:19:54.310 exhaustively, we would find that there are 230 distinct 00:19:54.310 --> 00:19:57.880 three-dimensional symmetries that involve point group 00:19:57.880 --> 00:20:01.410 symmetry operations and lattice type. 00:20:05.150 --> 00:20:07.200 I saw several faces blanch. 00:20:07.200 --> 00:20:08.110 Don't worry. 00:20:08.110 --> 00:20:10.040 We're not going to do that. 00:20:10.040 --> 00:20:14.380 What we will do next time is do a small handful and then 00:20:14.380 --> 00:20:18.590 conclude by showing you how this information is tabulated 00:20:18.590 --> 00:20:20.490 in the international tables. 00:20:20.490 --> 00:20:23.010 And the quick, immediate answer to that question is 00:20:23.010 --> 00:20:26.630 that the problem set that I passed around has two pages 00:20:26.630 --> 00:20:29.800 for two different point groups that show the way that this 00:20:29.800 --> 00:20:33.360 information is tabulated in the international tables. 00:20:33.360 --> 00:20:34.840 So that's where we'll end up. 00:20:34.840 --> 00:20:37.960 And we'll come pretty close to completing that when we meet 00:20:37.960 --> 00:20:39.210 on Thursday. 00:20:41.180 --> 00:20:42.710 I heard no sighs of relief. 00:20:42.710 --> 00:20:43.670 I heard no cheers. 00:20:43.670 --> 00:20:46.610 So you are presumably still engaged in 00:20:46.610 --> 00:20:49.320 this material somewhat. 00:20:49.320 --> 00:20:51.750 I have two things I would like to do. 00:20:51.750 --> 00:20:55.110 We talked about crystal systems and point groups. 00:20:55.110 --> 00:21:00.670 And let me now ask, do you think there is a crystal known 00:21:00.670 --> 00:21:05.590 for every one of the point groups? 00:21:05.590 --> 00:21:06.840 Think so? 00:21:09.240 --> 00:21:12.510 I saw some skeptical shakes of the head. 00:21:12.510 --> 00:21:16.920 The answer is yes, but just barely for some 00:21:16.920 --> 00:21:18.355 of the point groups. 00:21:21.150 --> 00:21:25.760 One of the very rare point groups, in terms of structures 00:21:25.760 --> 00:21:31.150 that populate it, is our high symmetries but which lack 00:21:31.150 --> 00:21:33.360 inversion or mirror planes. 00:21:33.360 --> 00:21:38.290 One very, very rare point group is 432-- 00:21:38.290 --> 00:21:41.785 just axes arranged in a cubic arrangement with no mirror 00:21:41.785 --> 00:21:43.550 planes or inversion centers. 00:21:43.550 --> 00:21:47.580 And my rationalization of that is that if an arrangement of 00:21:47.580 --> 00:21:51.490 atoms has to conform to all of these rotational symmetries, 00:21:51.490 --> 00:21:55.960 it's pretty hard for it to do it and have that assemblage of 00:21:55.960 --> 00:21:59.210 atoms assume the lowest possible energy state if it 00:21:59.210 --> 00:22:02.950 doesn't pick up in the process inversion or mirror planes as 00:22:02.950 --> 00:22:06.090 additional symmetry element. 00:22:06.090 --> 00:22:08.440 Sort of satisfying for me. 00:22:08.440 --> 00:22:09.880 But that's not a rigorous argument. 00:22:12.490 --> 00:22:16.320 OK, let me now take a poll. 00:22:16.320 --> 00:22:20.160 Do you think, using the broader cut of the crystal 00:22:20.160 --> 00:22:24.360 systems, do you think there's a crystal known for every 00:22:24.360 --> 00:22:25.420 crystal system? 00:22:25.420 --> 00:22:27.850 Are there triclinic, monoclinic, orthorhombic, 00:22:27.850 --> 00:22:30.070 tetragonal, trigonal, hexagonal, 00:22:30.070 --> 00:22:32.770 cubic crystals known? 00:22:32.770 --> 00:22:33.750 Yeah. 00:22:33.750 --> 00:22:40.010 That's a pretty realistic expectation. 00:22:40.010 --> 00:22:41.835 And that is indeed the case. 00:22:41.835 --> 00:22:45.550 But now let me cut this a little finer. 00:22:45.550 --> 00:22:49.430 Which do you think is the most popular crystal systems for 00:22:49.430 --> 00:22:52.920 real materials and which do you think is the rarest? 00:22:58.560 --> 00:22:58.980 OK. 00:22:58.980 --> 00:23:04.880 Who thinks triclinic crystals are the most common? 00:23:04.880 --> 00:23:07.500 OK, there are pessimists. 00:23:07.500 --> 00:23:10.370 What is the definition of a pessimist? 00:23:10.370 --> 00:23:13.600 The pessimist is somebody you should borrow money from 00:23:13.600 --> 00:23:15.740 because they'll never expect to be paid back. 00:23:18.640 --> 00:23:22.182 Who would say cubic crystals? 00:23:22.182 --> 00:23:25.640 There are naive [? optimists. ?] 00:23:25.640 --> 00:23:28.560 Now I will reveal all. 00:23:28.560 --> 00:23:33.150 The answer is, it depends. 00:23:33.150 --> 00:23:36.640 Depends upon whether you're talking about, interestingly, 00:23:36.640 --> 00:23:39.630 inorganic materials or whether you're talking 00:23:39.630 --> 00:23:40.985 about organic materials. 00:23:43.980 --> 00:23:52.860 If you look at inorganic materials, the most popular 00:23:52.860 --> 00:23:56.570 crystal systems are-- 00:23:56.570 --> 00:24:01.740 and the results are, first of all, cubic. 00:24:04.240 --> 00:24:07.870 Is this not a benign world in which we live. 00:24:07.870 --> 00:24:09.560 Most inorganic materials-- 00:24:09.560 --> 00:24:12.020 if you're a material scientist who works with metals and with 00:24:12.020 --> 00:24:15.260 ceramics, you're going to work with cubic materials, thank 00:24:15.260 --> 00:24:18.710 God, most of the time. 00:24:18.710 --> 00:24:20.780 The second most popular-- 00:24:20.780 --> 00:24:23.295 by a considerable margin but there are still a lot of 00:24:23.295 --> 00:24:24.545 them-- is orthorhombic. 00:24:31.250 --> 00:24:39.140 And the third most popular, shortly behind orthorhombic 00:24:39.140 --> 00:24:42.065 and almost in a tie, monoclinic and tetragonal. 00:24:52.120 --> 00:24:55.470 We'll launch into a quick overview of crystal symmetry. 00:24:55.470 --> 00:25:02.500 And we'll see why inorganic materials have high symmetry. 00:25:02.500 --> 00:25:08.250 It's true, in particular, when the material is held together 00:25:08.250 --> 00:25:11.680 by ionic bonding. 00:25:11.680 --> 00:25:16.020 If the bonding is ionic, it's non-directional and atoms of 00:25:16.020 --> 00:25:19.040 opposite charge will therefore try to get as close as 00:25:19.040 --> 00:25:23.440 possible to the atoms of opposite charge. 00:25:23.440 --> 00:25:26.750 And because atoms of like charge repel one another, the 00:25:26.750 --> 00:25:28.940 surrounding atoms will try to be as widely 00:25:28.940 --> 00:25:30.660 separated as possible. 00:25:30.660 --> 00:25:33.970 And this grouping of atoms, positive around negative, 00:25:33.970 --> 00:25:35.770 tends to have a lot of symmetry. 00:25:35.770 --> 00:25:38.340 And, therefore, the way in which these groups are linked 00:25:38.340 --> 00:25:41.840 together has a lot of symmetry. 00:25:41.840 --> 00:25:45.820 OK, I said organic materials. 00:25:45.820 --> 00:25:49.940 And I'll pass around some actual numbers that were 00:25:49.940 --> 00:25:52.960 assembled a long time ago but are still, in terms of 00:25:52.960 --> 00:25:54.780 proportion, still valid. 00:25:54.780 --> 00:25:58.410 The organic compounds are broken down into whether these 00:25:58.410 --> 00:26:01.240 are compounds based on rings or 00:26:01.240 --> 00:26:04.330 extended, lopsided molecules. 00:26:04.330 --> 00:26:06.740 And the numbers are very different depending on that. 00:26:06.740 --> 00:26:12.170 But, for organic molecules [? material, ?] 00:26:12.170 --> 00:26:22.030 monoclinic is the most common, then orthorhombic. 00:26:30.550 --> 00:26:37.200 And shortly after orthorhombic comes way, way behind is 00:26:37.200 --> 00:26:41.440 tetragonal and triclinic, which are almost tied. 00:26:52.860 --> 00:26:56.130 And I think I can rationalize this observation, as well. 00:26:59.890 --> 00:27:04.760 Organic materials contain discrete molecules. 00:27:04.760 --> 00:27:07.630 And if we've got something-- and I'm going to, since I know 00:27:07.630 --> 00:27:09.960 I'm among friends, display my abysmal 00:27:09.960 --> 00:27:12.070 ignorance of organic chemistry. 00:27:12.070 --> 00:27:18.080 I am unapologetically and unabashedly inorganic and even 00:27:18.080 --> 00:27:21.000 non-metallic in my research interests. 00:27:21.000 --> 00:27:23.270 So let's suppose we have something that's composed of 00:27:23.270 --> 00:27:27.720 various strange rings with little side groups sticking 00:27:27.720 --> 00:27:31.690 off like this, another one off like this, and then maybe 00:27:31.690 --> 00:27:36.460 another ring up like this. 00:27:36.460 --> 00:27:41.900 That, to me, looks like a reasonable molecule in terms 00:27:41.900 --> 00:27:44.520 of my ignorance of organic chemistry. 00:27:44.520 --> 00:27:47.520 So if there are these finite groups and you're going to put 00:27:47.520 --> 00:27:49.130 them together in a crystal, what are you 00:27:49.130 --> 00:27:50.210 going to try to do? 00:27:50.210 --> 00:27:53.380 They're going to be held together by very weak van der 00:27:53.380 --> 00:27:56.510 Waals forces, which are non-directional. 00:27:56.510 --> 00:27:59.640 So think of the outline of this molecule as being some 00:27:59.640 --> 00:28:02.300 ugly thing like a ham hock. 00:28:02.300 --> 00:28:05.290 And then your problem, if the forces between these molecules 00:28:05.290 --> 00:28:09.980 are non-directional, how do you close pack ham hocks? 00:28:09.980 --> 00:28:15.350 The way you would do it would be to put down a row of them 00:28:15.350 --> 00:28:18.800 in a close-packed sequence. 00:28:18.800 --> 00:28:21.580 So we'll put down things that look like this. 00:28:21.580 --> 00:28:25.810 And then, maybe to make them fit together, dovetail another 00:28:25.810 --> 00:28:27.660 one in here. 00:28:27.660 --> 00:28:32.360 And maybe something like that. 00:28:32.360 --> 00:28:35.930 And then for the next row down, and you would maybe put 00:28:35.930 --> 00:28:39.000 the nose of one in here. 00:28:39.000 --> 00:28:43.540 And then put the other one in here, the nose of one here, 00:28:43.540 --> 00:28:46.800 and something like that-- 00:28:46.800 --> 00:28:49.050 being very schematic. 00:28:49.050 --> 00:28:53.610 The lattice that describes this packing is not going to 00:28:53.610 --> 00:28:54.170 be orthogonal. 00:28:54.170 --> 00:28:57.040 It's not going to be square with high symmetry or even 00:28:57.040 --> 00:28:59.140 have two orthogonal translations. 00:28:59.140 --> 00:29:02.570 It's going to be something that has this as translations. 00:29:02.570 --> 00:29:05.270 There's going to be an A and a B that is not at 00:29:05.270 --> 00:29:07.040 right angles to it. 00:29:07.040 --> 00:29:09.780 And the B translation will have different links, simply 00:29:09.780 --> 00:29:12.100 because the thing that you're putting together in close 00:29:12.100 --> 00:29:15.750 packing has a very irregular shape. 00:29:15.750 --> 00:29:19.390 So I think that is the reason why molecules, discrete 00:29:19.390 --> 00:29:22.800 molecules that try to achieve something that is as close 00:29:22.800 --> 00:29:26.140 packed as possible, will have translations in them that are 00:29:26.140 --> 00:29:29.210 of unequal length and making arbitrary angles with respect 00:29:29.210 --> 00:29:31.440 to one another. 00:29:31.440 --> 00:29:34.660 So having kept you guessing for so long, let me pass out 00:29:34.660 --> 00:29:35.690 another sheet. 00:29:35.690 --> 00:29:39.810 This was a compilation that was done a long time ago. 00:29:39.810 --> 00:29:44.480 And there were only 9,000 crystals for which complete 00:29:44.480 --> 00:29:46.510 crystallographic data had been obtained. 00:29:46.510 --> 00:29:50.170 This was way back in 1967. 00:29:50.170 --> 00:29:55.350 As I say, there's much more data now, 00:29:55.350 --> 00:29:58.920 easily in order of magnitude. 00:29:58.920 --> 00:30:00.280 But I don't think the proportions 00:30:00.280 --> 00:30:01.480 will change very much. 00:30:01.480 --> 00:30:03.031 They'll stay about the same. 00:30:03.031 --> 00:30:04.815 Let me pass out these sheets. 00:30:08.690 --> 00:30:11.390 So make sure you memorize them because the first question on 00:30:11.390 --> 00:30:17.000 the next quiz is, how many out of 8,759 crystals 00:30:17.000 --> 00:30:18.250 are actually triclinic? 00:30:20.800 --> 00:30:22.050 I would never do that. 00:30:27.077 --> 00:30:30.320 All right, any comments or questions? 00:30:30.320 --> 00:30:30.798 Yes, sir. 00:30:30.798 --> 00:30:33.188 AUDIENCE: Why would organic molecules of such odd shapes 00:30:33.188 --> 00:30:34.622 even be crystalline to begin with? 00:30:34.622 --> 00:30:37.490 Why wouldn't they just be amorphous? 00:30:37.490 --> 00:30:37.790 PROFESSOR: OK. 00:30:37.790 --> 00:30:39.980 Clearly the interatomic forces are very 00:30:39.980 --> 00:30:43.140 directional and very strong. 00:30:43.140 --> 00:30:46.130 And that's going to hold the molecule into a fixed 00:30:46.130 --> 00:30:47.486 configuration. 00:30:47.486 --> 00:30:49.230 AUDIENCE: Didn't you say there were van der Waals forces? 00:30:49.230 --> 00:30:50.710 PROFESSOR: But they're these-- 00:30:50.710 --> 00:30:56.180 even for something that is not ionic or metallic, there still 00:30:56.180 --> 00:30:59.690 are weak attractive forces between these molecules. 00:30:59.690 --> 00:31:00.220 Why? 00:31:00.220 --> 00:31:03.590 Well, the charges are not uniformly distributed over 00:31:03.590 --> 00:31:05.620 these molecules. 00:31:05.620 --> 00:31:09.250 So there are going to be some van der Waals forces that tend 00:31:09.250 --> 00:31:10.220 to hold them together. 00:31:10.220 --> 00:31:13.810 And a van der Waals force is essentially non-directional. 00:31:13.810 --> 00:31:15.770 So if you have this weak force, you're going to be able 00:31:15.770 --> 00:31:18.070 to squish the crystal into something 00:31:18.070 --> 00:31:20.020 amorphous very easily. 00:31:20.020 --> 00:31:23.710 And organic materials are usually very subject to 00:31:23.710 --> 00:31:25.140 plastic deformation. 00:31:25.140 --> 00:31:29.110 But if you assemble the molecules carefully from 00:31:29.110 --> 00:31:32.970 solution, or by vaporization, condensation, they're going to 00:31:32.970 --> 00:31:38.080 want the order if they are solidified with enough 00:31:38.080 --> 00:31:41.340 mobility to rearrange themselves in the lowest 00:31:41.340 --> 00:31:42.590 energy configuration. 00:31:45.350 --> 00:31:48.190 I think what you're saying is these are not going to be very 00:31:48.190 --> 00:31:51.130 strongly bonded crystals. 00:31:51.130 --> 00:31:52.980 They're going to melt at low temperatures. 00:31:52.980 --> 00:31:55.580 But, nevertheless, prepare them properly, and the lowest 00:31:55.580 --> 00:31:56.730 energy state will be ordered. 00:31:56.730 --> 00:31:57.724 AUDIENCE: I'm sure [? they're ordered. ?] 00:31:57.724 --> 00:31:59.712 But would it be such that you have [? translational ?] 00:31:59.712 --> 00:32:01.700 symmetry? 00:32:01.700 --> 00:32:04.682 Who's to say it's going to have a definite pattern that 00:32:04.682 --> 00:32:06.680 will repeat itself? 00:32:06.680 --> 00:32:10.880 PROFESSOR: Yeah, if there are attractive forces-- 00:32:10.880 --> 00:32:12.600 let me again use a hand-waving argument. 00:32:12.600 --> 00:32:15.030 If you've got a bunch of particles and they aggregate 00:32:15.030 --> 00:32:19.340 here in the lowest energy configuration, that is going 00:32:19.340 --> 00:32:21.320 to be the lowest energy configuration. 00:32:21.320 --> 00:32:23.190 And a similar group of atoms will want to do 00:32:23.190 --> 00:32:24.850 the same thing here. 00:32:24.850 --> 00:32:27.780 And if there are interactions between these groups, they're 00:32:27.780 --> 00:32:30.395 going to arrange themselves in an order configuration. 00:32:32.980 --> 00:32:34.340 That is simply saying-- 00:32:34.340 --> 00:32:36.610 this is a very weak qualitative argument-- says if 00:32:36.610 --> 00:32:39.080 you have a configuration which is the lowest energy 00:32:39.080 --> 00:32:43.290 arrangement here, any other place in the system where 00:32:43.290 --> 00:32:47.160 those atoms get together and congeal are going to assume 00:32:47.160 --> 00:32:49.801 that same configuration. 00:32:49.801 --> 00:32:52.540 OK? 00:32:52.540 --> 00:32:53.610 Other comments or questions? 00:32:53.610 --> 00:32:58.760 This is all very qualitative but actually concerns matters 00:32:58.760 --> 00:33:02.620 that we are describing a language to describe. 00:33:05.950 --> 00:33:07.200 AUDIENCE: [INAUDIBLE] 00:33:09.420 --> 00:33:10.000 Yes, there are. 00:33:10.000 --> 00:33:13.020 If you look at this table that came around, if you look at 00:33:13.020 --> 00:33:15.780 cubic organic materials out of-- 00:33:22.370 --> 00:33:23.990 there-- 00:33:23.990 --> 00:33:26.160 what is the number? 00:33:26.160 --> 00:33:31.440 117 out of what looks to be several thousand. 00:33:31.440 --> 00:33:34.280 Yes, there are cubic organic crystals, 00:33:34.280 --> 00:33:35.720 but very few in number. 00:33:35.720 --> 00:33:37.130 Hexagonal-- 00:33:37.130 --> 00:33:41.760 56 out of a rather large number. 00:33:49.920 --> 00:33:54.970 All right, when we took a poll a week and a half or so ago 00:33:54.970 --> 00:33:59.160 and said, how many of you had seen any material and crystal 00:33:59.160 --> 00:34:03.800 chemistry in solid state chemistry in a previous class 00:34:03.800 --> 00:34:09.600 on structure and bonding or crystal chemistry, and only a 00:34:09.600 --> 00:34:13.880 few people tentatively twitched a hand slightly. 00:34:13.880 --> 00:34:17.449 So what I'd like to do for the rest of this session, since 00:34:17.449 --> 00:34:23.739 this is what symmetry theory is developed to describe, I'd 00:34:23.739 --> 00:34:28.040 like to say something about structures and why they form 00:34:28.040 --> 00:34:30.159 the way they do to give some qualitative insight. 00:34:32.750 --> 00:34:35.409 This is a fat pack of notes. 00:34:35.409 --> 00:34:38.639 And I'll go over it quickly with you and lead you by the 00:34:38.639 --> 00:34:40.650 hand through it. 00:34:40.650 --> 00:34:44.739 But this is not material for which I'll hold you 00:34:44.739 --> 00:34:46.929 responsible. 00:34:46.929 --> 00:34:50.420 In fact, I'll turn my back and those who want to get home 00:34:50.420 --> 00:34:52.380 early because of the rain can tiptoe out. 00:34:52.380 --> 00:34:56.290 And I won't even notice who you are unless everybody goes. 00:34:56.290 --> 00:34:58.460 And then I'll feel very hurt because I wrote these out. 00:34:58.460 --> 00:34:59.710 And it took a lot of trouble. 00:35:07.040 --> 00:35:08.420 All right. 00:35:08.420 --> 00:35:15.140 I have to apologize and say that most of what is described 00:35:15.140 --> 00:35:20.400 in these notes applies to ionic bonding and, to a lesser 00:35:20.400 --> 00:35:23.980 extent but still in large measure, to metallic 00:35:23.980 --> 00:35:25.920 structures. 00:35:25.920 --> 00:35:27.720 And what is the reason for that? 00:35:27.720 --> 00:35:31.850 It's because the metallic bond, to a fair approximation, 00:35:31.850 --> 00:35:36.140 an ionic bond rigorously, if you believe Coulomb's law, is 00:35:36.140 --> 00:35:38.500 a non-directional sort of bonding. 00:35:38.500 --> 00:35:41.380 So the structures that are assumed, the structures that 00:35:41.380 --> 00:35:48.390 are the lowest energy configuration turn out to be 00:35:48.390 --> 00:35:53.790 determined by interatomic distances and ionic or 00:35:53.790 --> 00:35:57.290 metallic sizes. 00:35:57.290 --> 00:36:02.930 The Coulombic interaction that holds ionic compounds together 00:36:02.930 --> 00:36:04.800 is a central force. 00:36:04.800 --> 00:36:09.650 And one of the features of electromagnetism is that, if 00:36:09.650 --> 00:36:13.450 you have some distribution of charge about center and 00:36:13.450 --> 00:36:16.290 another distribution of charge of a different sort about a 00:36:16.290 --> 00:36:20.870 another center, the cohesive force between, let's say, a 00:36:20.870 --> 00:36:27.300 positive ion and a negative ion is, along a line, joining 00:36:27.300 --> 00:36:29.520 their centers. 00:36:29.520 --> 00:36:32.970 And, secondly, you can take, once you're outside the 00:36:32.970 --> 00:36:37.760 distribution of charge, you can clump all the charge 00:36:37.760 --> 00:36:40.920 together at the center of the distribution. 00:36:40.920 --> 00:36:43.700 So that's what makes the ionic bond so simple. 00:36:43.700 --> 00:36:46.155 And that's why everybody who talks about bonding and 00:36:46.155 --> 00:36:49.350 structure discusses ionic bonding because organic 00:36:49.350 --> 00:36:51.535 chemistry is just too complex in comparison. 00:36:54.750 --> 00:36:56.715 So let's look at a given cation. 00:36:56.715 --> 00:37:01.700 And cations generally tend to be smaller than anions because 00:37:01.700 --> 00:37:04.640 you've removed electrons from the outer configuration. 00:37:04.640 --> 00:37:07.980 We'll look at some effective sizes of ions in just a bit. 00:37:07.980 --> 00:37:10.650 But you've stripped electrons off to make a positively 00:37:10.650 --> 00:37:11.550 charged ion. 00:37:11.550 --> 00:37:14.340 You've added electrons on to make a negatively charged ion. 00:37:14.340 --> 00:37:17.710 So when you add electrons, the ion functions has [? always ?] 00:37:17.710 --> 00:37:20.020 had a larger radius. 00:37:20.020 --> 00:37:28.820 So let's bring up a B minus ion as a neighbor. 00:37:28.820 --> 00:37:36.660 And for those two species, there will be, if we put 00:37:36.660 --> 00:37:40.360 energy as a function of separation D, there's going to 00:37:40.360 --> 00:37:46.360 be a Coulombic attractive force that goes as 1/D. 00:37:46.360 --> 00:37:50.350 But when the electron configurations get in to close 00:37:50.350 --> 00:37:53.970 proximity, at small separations, the electrons 00:37:53.970 --> 00:37:56.940 will start to push one another, beginning with the 00:37:56.940 --> 00:38:00.240 outer electrons first, into higher energy states. 00:38:00.240 --> 00:38:05.160 And the energy of interaction, due to the interaction of 00:38:05.160 --> 00:38:08.480 orbitals, will increase the energy of the system. 00:38:08.480 --> 00:38:15.240 And the result then is that the net potential, as a 00:38:15.240 --> 00:38:18.310 function of distance for an isolated ion pair-- 00:38:18.310 --> 00:38:20.660 I was looking for some colored chalk and it's gone-- is 00:38:20.660 --> 00:38:24.750 something that rises steeply at low separations and then 00:38:24.750 --> 00:38:28.230 levels out more gradually at larger separations. 00:38:28.230 --> 00:38:33.390 So this then would be an equilibrium separation, D0, 00:38:33.390 --> 00:38:35.360 for that ionic pair. 00:38:39.750 --> 00:38:42.942 If we got energy out, and for that isolated ionic pair it'll 00:38:42.942 --> 00:38:46.130 be just exactly this much energy, E0. 00:38:46.130 --> 00:38:49.880 If it weren't for a pair of ions-- 00:38:49.880 --> 00:38:54.890 let's take another A and bring it into the system. 00:38:54.890 --> 00:38:59.960 And there's a lot of room to put still another A around. 00:38:59.960 --> 00:39:02.200 And we can add that to the system. 00:39:02.200 --> 00:39:04.920 And every time we bring up an additional neighbor, we get 00:39:04.920 --> 00:39:09.080 approximately this much energy out of the system. 00:39:09.080 --> 00:39:13.860 But not quite because there will also be repulsive 00:39:13.860 --> 00:39:20.890 interactions between the ions of like charge. 00:39:20.890 --> 00:39:21.850 OK? 00:39:21.850 --> 00:39:25.290 But to a good approximation to within about 10% to be 00:39:25.290 --> 00:39:28.530 quantitatively, every time we bring up another neighbor, we 00:39:28.530 --> 00:39:31.290 get out the same amount of energy again by 00:39:31.290 --> 00:39:34.060 forming this group. 00:39:34.060 --> 00:39:37.895 And all that works splendidly until-- 00:39:40.460 --> 00:39:46.090 let me now look at the arrangement of anions about 00:39:46.090 --> 00:39:48.560 the little cation because they will have to 00:39:48.560 --> 00:39:50.640 have a shell as well. 00:39:50.640 --> 00:39:53.980 So if we have a small cation and we bring up a big fat 00:39:53.980 --> 00:39:59.150 anion, there will be, first of all, two consequences. 00:39:59.150 --> 00:40:03.490 These ions of like charge will interact repulsively. 00:40:03.490 --> 00:40:08.320 So the lowest energy situation is going to be one in which 00:40:08.320 --> 00:40:16.990 these neighboring ions arrange themselves so that the 00:40:16.990 --> 00:40:20.260 repulsive energy is minimized. 00:40:20.260 --> 00:40:23.730 And we're going to get the maximum separation between all 00:40:23.730 --> 00:40:28.240 of these anions and get the minimum energy when they 00:40:28.240 --> 00:40:31.750 arrange themselves on the corners of a regular 00:40:31.750 --> 00:40:33.670 polyhedron. 00:40:33.670 --> 00:40:37.540 And that's where symmetry starts to come in to the 00:40:37.540 --> 00:40:39.650 importance of ionic structures. 00:40:39.650 --> 00:40:45.080 That, for this particular configuration of anions, the B 00:40:45.080 --> 00:40:47.920 ions might be on the corners. 00:40:47.920 --> 00:40:50.710 And I am not drawing them in contact now just so it'll be 00:40:50.710 --> 00:40:52.090 easier to draw. 00:40:52.090 --> 00:40:57.590 The B ions will be equidistant and tend to arrange themselves 00:40:57.590 --> 00:40:59.285 on the corner of the regular polyhedron. 00:41:02.080 --> 00:41:04.770 If we got energy up by bringing up extra neighbors, 00:41:04.770 --> 00:41:11.430 we will reach a point where the B ions will essentially be 00:41:11.430 --> 00:41:13.280 in contact. 00:41:13.280 --> 00:41:20.770 And the A ion is rattling around in a hole. 00:41:20.770 --> 00:41:23.020 It can touch perhaps two neighbors. 00:41:23.020 --> 00:41:25.170 And we will get the full amount of energy on. 00:41:25.170 --> 00:41:27.310 But for the other two neighbors, we're going to be 00:41:27.310 --> 00:41:28.400 out here on this curve. 00:41:28.400 --> 00:41:30.500 And we won't get nearly as much energy out. 00:41:30.500 --> 00:41:34.545 But at the same time, we're paying the price energetically 00:41:34.545 --> 00:41:39.600 of all of these repulsive interactions between the 00:41:39.600 --> 00:41:42.610 surrounding neighbors of like charge. 00:41:42.610 --> 00:41:47.960 So the second consideration is that, as we said, first of 00:41:47.960 --> 00:41:51.380 all, this group of atoms, which is referred to as a 00:41:51.380 --> 00:42:03.980 coordination polyhedron, it first 00:42:03.980 --> 00:42:05.975 of all will be symmetric. 00:42:10.670 --> 00:42:18.020 And then the number of neighbors, the number of ions 00:42:18.020 --> 00:42:23.780 of one charge that are coordinating the others-- 00:42:23.780 --> 00:42:26.690 and that's something called the coordination number, 00:42:26.690 --> 00:42:28.470 introduce some jargon-- 00:42:35.570 --> 00:42:45.200 has an upper limit that is going to be determined by the 00:42:45.200 --> 00:42:46.450 size of the ions. 00:42:59.250 --> 00:43:03.930 So these are the two basic tenets of 00:43:03.930 --> 00:43:07.110 ionic crystal chemistry. 00:43:07.110 --> 00:43:09.890 So having established those points, we can do some 00:43:09.890 --> 00:43:19.320 geometry and assume that the ions are hard spheres or at 00:43:19.320 --> 00:43:23.330 least elastic spheres with some stiffness because it's 00:43:23.330 --> 00:43:26.190 going to take energy to perturb the orbitals of the 00:43:26.190 --> 00:43:32.440 electrons and, just in terms of simple geometry, calculate 00:43:32.440 --> 00:43:38.030 the range of radius ratios RA/RB where A is the central 00:43:38.030 --> 00:43:41.680 ion in the polyhedron and B is the surrounding ion, and do 00:43:41.680 --> 00:43:44.860 this for different coordination numbers, picking 00:43:44.860 --> 00:43:48.570 polyhedra in which the surrounding ions of like 00:43:48.570 --> 00:43:52.070 charge are as widely separated as possible. 00:43:52.070 --> 00:43:54.850 So starting at the beginning, coordination number one would 00:43:54.850 --> 00:43:57.440 be a pair of ions. 00:43:57.440 --> 00:44:00.190 Obviously no constraints whatsoever on 00:44:00.190 --> 00:44:01.390 the relative sizes. 00:44:01.390 --> 00:44:07.010 RA/RB can be anywhere it likes between zero and infinity. 00:44:07.010 --> 00:44:10.420 For coordination number two, again the surrounding ions 00:44:10.420 --> 00:44:13.510 want to be as widely separated as possible. 00:44:13.510 --> 00:44:15.780 So it'll be a linear chain. 00:44:15.780 --> 00:44:18.880 But, again, the central ion can become as small or as 00:44:18.880 --> 00:44:19.730 large as it like. 00:44:19.730 --> 00:44:23.450 So the range of radii is between zero and infinity. 00:44:23.450 --> 00:44:26.000 For coordination number three, the polyhedron will be an 00:44:26.000 --> 00:44:27.680 equilateral triangle. 00:44:27.680 --> 00:44:32.110 And if you calculate the geometry, when the B ions are 00:44:32.110 --> 00:44:38.220 exactly in contact, you have a little 30, 60, 90 triangle in 00:44:38.220 --> 00:44:40.860 which one edge of the triangle is RB and the 00:44:40.860 --> 00:44:42.850 other is RA plus RB. 00:44:42.850 --> 00:44:45.350 And you can manipulate that into the form where you can 00:44:45.350 --> 00:44:50.030 show that there will be a lower limit RA/RB, which is 2 00:44:50.030 --> 00:44:51.800 over the square root of 3 minus 1. 00:44:51.800 --> 00:44:56.940 And that turns out to be a magic number, 0.155. 00:44:56.940 --> 00:44:59.830 Going to coordination number four, you'd say, aha. 00:44:59.830 --> 00:45:01.460 That's going to be square group. 00:45:01.460 --> 00:45:05.360 No, a square group would not keep the surrounding ions of 00:45:05.360 --> 00:45:08.540 like charge as widely separated as possible. 00:45:08.540 --> 00:45:12.010 And the geometry which would keep them as widely separated 00:45:12.010 --> 00:45:15.300 as possible but yet in contact with the central sphere would 00:45:15.300 --> 00:45:17.670 be a tetrahedron. 00:45:17.670 --> 00:45:20.440 The Bs would be arranged on alternating 00:45:20.440 --> 00:45:22.260 vertices of a cube. 00:45:22.260 --> 00:45:28.480 The magic number there turns out to be 0.225. 00:45:28.480 --> 00:45:31.120 Coordination five-- 00:45:31.120 --> 00:45:34.570 very, very rare. 00:45:34.570 --> 00:45:38.530 Five is a crystallographic abomination because five-fold 00:45:38.530 --> 00:45:41.200 symmetry does not conform to a lattice. 00:45:41.200 --> 00:45:42.930 But there would be, in principle, 00:45:42.930 --> 00:45:44.285 one five-fold group. 00:45:47.710 --> 00:45:55.630 And that would be a trigonal prism where you have three Bs 00:45:55.630 --> 00:45:59.520 arranged on an equilateral triangle about the central A. 00:45:59.520 --> 00:46:03.730 And then you have one A up here, another B up here, and 00:46:03.730 --> 00:46:05.790 another B down here. 00:46:05.790 --> 00:46:11.770 So this, then, is a bipyramid, a triangular bipyramid. 00:46:11.770 --> 00:46:17.550 And you can derive a range of radius ratios for this. 00:46:17.550 --> 00:46:21.050 And this is a dandy problem to give on a problem set. 00:46:21.050 --> 00:46:27.330 But you find that the range of radius ratios RA/RB is the 00:46:27.330 --> 00:46:36.350 same as for six-fold coordination, which is the 00:46:36.350 --> 00:46:39.630 next one considered on the handout. 00:46:39.630 --> 00:46:44.850 So we six coordination with the corners on a regular 00:46:44.850 --> 00:46:47.300 arrangement would be an octahedron. 00:46:47.300 --> 00:46:52.480 And the permitted range there is 0.732 to infinity. 00:46:52.480 --> 00:46:55.700 Five coordination, the trigonal prism, leads to 00:46:55.700 --> 00:46:59.770 exactly the same range of radii. 00:46:59.770 --> 00:47:04.410 And the reason is, if you look at the geometries, this 00:47:04.410 --> 00:47:08.490 triangle here is the same for both an octahedral 00:47:08.490 --> 00:47:14.410 coordination and a trigonal pyramidal coordination. 00:47:14.410 --> 00:47:18.350 Notice as we chug on that the lower limit to the radius 00:47:18.350 --> 00:47:21.980 ratio is gradually increasing. 00:47:21.980 --> 00:47:25.440 And if we go to coordination number eight, cubic 00:47:25.440 --> 00:47:29.110 coordination, it goes up to 1 to infinity. 00:47:29.110 --> 00:47:34.650 And for coordination number 12, RA/RB has to 00:47:34.650 --> 00:47:36.850 be exactly 1 period. 00:47:36.850 --> 00:47:38.580 And that would be what you achieved in the 00:47:38.580 --> 00:47:39.830 close-packed number. 00:47:47.950 --> 00:47:50.390 OK, turning the page. 00:47:50.390 --> 00:47:54.540 There is also going to be a range of radius ratios that's 00:47:54.540 --> 00:48:01.170 determined by the number of As that can be packed around a 00:48:01.170 --> 00:48:11.940 central B. And doing exactly the same calculation, if, for 00:48:11.940 --> 00:48:19.270 example, four-fold coordination permitted in 00:48:19.270 --> 00:48:29.240 terms of RA/RB, the range is 0.225 to infinity. 00:48:29.240 --> 00:48:35.220 If we want to know the limits for packing of A around B with 00:48:35.220 --> 00:48:41.300 tetrahedral coordination, this would give us numbers RA/RB 00:48:41.300 --> 00:48:46.840 that were exactly the same range of numbers because we've 00:48:46.840 --> 00:48:48.700 just changed the labels. 00:48:48.700 --> 00:48:53.280 If we want to put everything on the same basis, in terms of 00:48:53.280 --> 00:48:57.450 one particular radius ratio, if we do anion to cation 00:48:57.450 --> 00:49:03.930 radius ratio, the range would be 0.225 to infinity. 00:49:03.930 --> 00:49:09.170 If we take the reciprocals of these numbers for four-fold 00:49:09.170 --> 00:49:12.520 coordination of B, this would be 1 over 00:49:12.520 --> 00:49:17.460 0.225 to 1 over infinity. 00:49:17.460 --> 00:49:20.880 So this will give us-- 00:49:20.880 --> 00:49:21.880 this is zero. 00:49:21.880 --> 00:49:22.960 This is infinity. 00:49:22.960 --> 00:49:31.840 So there'll be a range of radius ratios, 0.225 less than 00:49:31.840 --> 00:49:40.970 RA/RB, less than 1 over 0.225, which turns out to be 4.44. 00:49:40.970 --> 00:49:46.160 So this would be the range of permitted radius ratios for 00:49:46.160 --> 00:49:53.560 CNA equals 4 and CNB equals the same thing, 4. 00:49:53.560 --> 00:49:56.640 So there's going to be an upper limit that's imposed by 00:49:56.640 --> 00:50:00.810 the packing of As around B and a lower limit that's imposed 00:50:00.810 --> 00:50:08.450 by the packing of Bs around A. 00:50:08.450 --> 00:50:10.420 But what are going to be the coordination 00:50:10.420 --> 00:50:12.270 numbers of A and B? 00:50:12.270 --> 00:50:13.290 Let me finish this up. 00:50:13.290 --> 00:50:15.700 And then we can take our break. 00:50:19.480 --> 00:50:24.180 I'm going to now restrict things rather drastically. 00:50:24.180 --> 00:50:39.370 I'm going to consider binary compounds in which all cations 00:50:39.370 --> 00:50:53.560 have the same coordination and all anions with the same 00:50:53.560 --> 00:50:54.810 coordination. 00:51:03.410 --> 00:51:06.210 And the implication there is there's only one type of 00:51:06.210 --> 00:51:08.582 coordination number for both. 00:51:08.582 --> 00:51:10.720 And that's not always true in compounds. 00:51:10.720 --> 00:51:14.630 Sometimes you find a given species that has two different 00:51:14.630 --> 00:51:17.995 coordination numbers in a more complex compound. 00:51:21.370 --> 00:51:28.970 If that's the case, if this is the A ion and this is the B 00:51:28.970 --> 00:51:33.790 ion, let me consider the chemical 00:51:33.790 --> 00:51:35.185 composition of a bond. 00:51:43.350 --> 00:51:45.410 And that's going to translate into the same 00:51:45.410 --> 00:51:46.600 thing as the compound. 00:51:46.600 --> 00:51:50.270 So let's suppose there's a number of Bs around the As, 00:51:50.270 --> 00:51:53.335 which is, by definition, the coordination number of A, and 00:51:53.335 --> 00:51:56.670 a number of As around a B and that is, by definition, the 00:51:56.670 --> 00:52:00.990 coordination number of B. So let me look at a wedge that's 00:52:00.990 --> 00:52:03.260 associated with this bond. 00:52:03.260 --> 00:52:07.130 And this is an amount of an A atom, which is 1 over the 00:52:07.130 --> 00:52:09.220 coordination number of A. 00:52:09.220 --> 00:52:14.390 And if I do the same thing for the B ion the part of the B 00:52:14.390 --> 00:52:17.900 ion associated with one bond is going to be 1 over the 00:52:17.900 --> 00:52:21.070 coordination number of B. So we can say then that the 00:52:21.070 --> 00:52:32.500 composition of a bond is going to be A subscript 1 over the 00:52:32.500 --> 00:52:36.970 coordination number of A, B, 1 over the coordination number 00:52:36.970 --> 00:52:38.910 of B. 00:52:38.910 --> 00:52:42.290 The chemists abhor fractional subscripts on a chemical 00:52:42.290 --> 00:52:43.510 composition. 00:52:43.510 --> 00:52:46.820 So let's multiply this by the product of the two 00:52:46.820 --> 00:52:48.070 coordination numbers. 00:52:52.200 --> 00:52:55.620 And what we'll find is that the formula of the compound 00:52:55.620 --> 00:53:01.620 with integer subscripts is going to be A, C, and B; B 00:53:01.620 --> 00:53:10.350 coordination number of A. And that is a subject to a very 00:53:10.350 --> 00:53:15.800 drastic set of assumptions that each species A and B has 00:53:15.800 --> 00:53:17.230 just one coordination number. 00:53:23.660 --> 00:53:26.740 OK, so doing this in terms of composition, just turning this 00:53:26.740 --> 00:53:27.610 result around. 00:53:27.610 --> 00:53:29.170 And then we'll quit and take a stretch. 00:53:32.230 --> 00:53:39.850 This says that if we look at a particular composition, AnBm, 00:53:39.850 --> 00:53:47.230 something like TiO2 or Al2O3, then the coordination number 00:53:47.230 --> 00:53:51.100 of A in proportion to the coordination number of B is 00:53:51.100 --> 00:54:01.490 going to be equal to the ratio of M to N. 00:54:01.490 --> 00:54:06.500 And the more ions of one charge that we can pack around 00:54:06.500 --> 00:54:09.200 the other, the lower the energy will be. 00:54:09.200 --> 00:54:16.480 And, therefore, the coordination numbers will be 00:54:16.480 --> 00:54:33.560 the maximum permitted values of CNA and CNB, subject to the 00:54:33.560 --> 00:54:37.510 constraints of radius ratio. 00:54:37.510 --> 00:54:40.540 In other words, to conclude very quickly with one specific 00:54:40.540 --> 00:54:46.590 example, if we look at a compound like Al2O3, an 00:54:46.590 --> 00:54:50.040 important ceramic and a gemstone, this says that the 00:54:50.040 --> 00:54:55.080 coordination number of Al to the coordination number of the 00:54:55.080 --> 00:54:59.110 oxygen has to be equal to three to two. 00:54:59.110 --> 00:55:07.500 So possible structures might be aluminum with coordination 00:55:07.500 --> 00:55:10.850 number three, oxygen with coordination 00:55:10.850 --> 00:55:14.820 number equal to two. 00:55:14.820 --> 00:55:19.410 Or aluminum could have six coordination. 00:55:19.410 --> 00:55:22.010 The oxygen could have four coordination. 00:55:22.010 --> 00:55:25.560 Or the aluminum could have 12 coordination. 00:55:25.560 --> 00:55:29.390 And the oxygen could have eight coordination. 00:55:32.370 --> 00:55:34.940 Those three possibilities. 00:55:34.940 --> 00:55:38.530 Next we have to use the ionic sizes to see what constraints 00:55:38.530 --> 00:55:41.530 there are that are going to limit the maximum coordination 00:55:41.530 --> 00:55:44.720 numbers because the greater the number of neighbors, the 00:55:44.720 --> 00:55:46.030 lower the energy. 00:55:46.030 --> 00:55:49.820 And, in the case of aluminum oxide, it turns out that it's 00:55:49.820 --> 00:55:51.460 six to four. 00:55:51.460 --> 00:55:54.500 And this is determined by the radius ratio. 00:56:01.450 --> 00:56:01.810 And you know what? 00:56:01.810 --> 00:56:06.220 If we used the ionic radii for Al3 plus and O2 minus, we were 00:56:06.220 --> 00:56:07.970 right one the nose. 00:56:07.970 --> 00:56:12.740 That is the nature of the structure of aluminum oxide-- 00:56:12.740 --> 00:56:17.780 aluminum with octahedral coordination by oxygen and 00:56:17.780 --> 00:56:24.200 oxygen with a tetrahedral coordination of aluminum. 00:56:24.200 --> 00:56:27.850 And that is the actual structure. 00:56:27.850 --> 00:56:32.720 The three-dimensional linkage would have to be such that 00:56:32.720 --> 00:56:35.615 ions of like charge have the maximum possible separation. 00:56:40.310 --> 00:56:43.050 And to keep you beyond your patience for just a little 00:56:43.050 --> 00:56:44.220 bit, what is that going to be? 00:56:44.220 --> 00:56:46.960 We can use very simple considerations there. 00:56:46.960 --> 00:56:50.120 If we have coordination polyhedra and we want to join 00:56:50.120 --> 00:56:53.920 them together, the worst possible situation is to have 00:56:53.920 --> 00:57:00.140 them share edges because the central ions then had minimum 00:57:00.140 --> 00:57:01.390 separation. 00:57:03.270 --> 00:57:09.940 The better situation would be to have to the polyhedra share 00:57:09.940 --> 00:57:16.180 corners and have the ions in the center of these triangles 00:57:16.180 --> 00:57:20.740 form a straight line with the shared ions. 00:57:20.740 --> 00:57:23.460 For a three-dimensional polyhedron, the sequence of 00:57:23.460 --> 00:57:26.950 increasing repulsive energy would be shared faces for 00:57:26.950 --> 00:57:30.240 something like cubes, shared edges-- 00:57:30.240 --> 00:57:33.440 that would be lower energy but not the best you could do. 00:57:33.440 --> 00:57:36.990 Shared corners would be the best possible arrangement. 00:57:36.990 --> 00:57:39.690 As you make each of those transitions, you take the 00:57:39.690 --> 00:57:42.960 central ion and keep the separation between the 00:57:42.960 --> 00:57:46.480 neighboring ion of like charge as low as possible. 00:57:46.480 --> 00:57:48.865 The interaction energy would be as low as possible. 00:57:53.130 --> 00:57:54.620 All right, so let me stop there. 00:57:54.620 --> 00:57:57.150 I think we're probably about halfway through. 00:57:59.890 --> 00:58:03.930 We can then determine ranges of radius ratio for various 00:58:03.930 --> 00:58:05.310 stoichiometries. 00:58:05.310 --> 00:58:09.940 And all this is just a fantasy unless we can say, do ions 00:58:09.940 --> 00:58:15.380 have sizes that are the same from compound to compound? 00:58:15.380 --> 00:58:18.750 And, if so, how do we determine them? 00:58:18.750 --> 00:58:24.290 I hope that intriguing pair of questions will keep you here, 00:58:24.290 --> 00:58:29.790 dry and warm, rather than heading off early. 00:58:29.790 --> 00:58:35.630 OK, so let's take our usual 10-minute break and resume to 00:58:35.630 --> 00:58:36.880 wrap this up.