1 00:00:00,970 --> 00:00:02,900 Let's get going. Welcome back. 2 00:00:02,980 --> 00:00:09,640 This is the 11th lecture of the condensed matter, of course, where we left off last time we were studying reciprocal space. 3 00:00:10,360 --> 00:00:22,570 Reciprocal space. And we came to this understanding that the reciprocal lattice is somehow the Fourier transform of the direct lattice, 4 00:00:22,870 --> 00:00:27,699 and that the reason we want to study reciprocal space is because it's going to make our life easier when we come to studying waves. 5 00:00:27,700 --> 00:00:33,909 So it's a rather important thing to to understand well, and sometimes it's hard to understand for a transfer as well, 6 00:00:33,910 --> 00:00:39,520 because they're not something that we think about in everyday life a little bit complicated. 7 00:00:39,910 --> 00:00:45,940 So it turns out that there's a much nicer geometric way to understand a lot of reciprocal space, which is what we're going to discuss today. 8 00:00:46,480 --> 00:00:50,050 So I need some definitions. As usual, let's start with a simple definition. 9 00:00:50,620 --> 00:01:13,659 A lattice plane equals a plane containing containing three three, non linear, non linear, linear, 10 00:01:13,660 --> 00:01:30,700 and therefore linear and therefore an infinite number and therefore are an infinity of lattice points. 11 00:01:34,420 --> 00:01:39,610 Okay, so a lattice plane is a plane containing three non linear and therefore an infinite number of lattice points. 12 00:01:39,850 --> 00:01:44,290 So here are some pictures of lattice planes here. On the left we have a BK lattice. 13 00:01:44,530 --> 00:01:50,110 There's a plane cutting through three points. Therefore, if you continued out to infinity, it will cut to an infinity number of points. 14 00:01:50,470 --> 00:01:54,250 That's a lattice plane. There's a on the right is a simple cubic lattice. 15 00:01:54,370 --> 00:02:00,670 There's a plane that cuts the three last points. If you continue out to infinity, it cuts through an infinite number of lattice points. 16 00:02:00,910 --> 00:02:04,030 Another definition, a family of lattice planes. 17 00:02:05,860 --> 00:02:12,070 Of lattice planes, mama lattice plane. 18 00:02:12,160 --> 00:02:17,230 Baby lattice plane. No, I'm joking. That's not what no one ever laughs in this class. 19 00:02:17,440 --> 00:02:22,389 And again, it is families. 20 00:02:22,390 --> 00:02:39,580 Large planes equals an infinite, infinite, infinite and finite set of equally spaced equals spaced lattice planes. 21 00:02:44,580 --> 00:02:55,920 Such that. Such that all lattice points, all lad points are included. 22 00:02:56,220 --> 00:03:08,790 Are included in one of the planes in one of the planes. 23 00:03:11,040 --> 00:03:19,260 Let me know if my riding starts getting to be too messy to read. So here's a simple cubic lattice and a couple of families of lattice planes. 24 00:03:19,560 --> 00:03:22,650 Each one of these is a lattice planes that cut through an infinite number of lattice points, 25 00:03:23,010 --> 00:03:28,380 and they're equally spaced, and they include every single point in the lattice when taken all together. 26 00:03:28,800 --> 00:03:34,710 This is a BK lattice and there's a family of lattice plans for the BK lattice cut through all the lattice points in the BC lattice. 27 00:03:35,070 --> 00:03:40,709 This, however, is not a family of lattice planes for the BK lattice, even though they're lattice planes. 28 00:03:40,710 --> 00:03:45,660 What's drawn? They are equally spaced, but they do not include every lattice point they miss. 29 00:03:45,900 --> 00:03:51,240 These guys here in the middle of the conventional media cell. Therefore, it is not a family of lattice planes. 30 00:03:52,050 --> 00:03:55,770 Okay. So the connections, the reciprocal lattice claim. 31 00:04:00,600 --> 00:04:05,610 A family of lattice planes. A family of lattice planes. 32 00:04:09,940 --> 00:04:26,610 It is perpendicular to our reciprocal lattice vector back, which I called G and further and the distance, 33 00:04:26,610 --> 00:04:50,220 the spacing between between neighbour planes, between neighbour planes planes is d equals to pi over absolute d men. 34 00:04:51,140 --> 00:04:56,040 We're g in the g men equals shortest. 35 00:05:00,330 --> 00:05:04,000 Recep. Lattice vector. 36 00:05:04,000 --> 00:05:07,660 Lat vec. Indirection. 37 00:05:08,620 --> 00:05:16,030 Indirection. Of G OC. 38 00:05:17,090 --> 00:05:18,260 This is important statement. 39 00:05:18,530 --> 00:05:24,170 Every time we have a family of lattice planes orthogonal to this family of lattice planes, there is a reciprocal lattice vector. 40 00:05:24,350 --> 00:05:28,760 And if you want to know the spacing between the lattice plane in the family of lattice planes, 41 00:05:28,970 --> 00:05:33,379 you find the shortest reciprocal lattice vector pointing in that direction to pi over the 42 00:05:33,380 --> 00:05:38,450 absolute value of that shortest reciprocal lattice factor gives you the spacing between planes. 43 00:05:38,780 --> 00:05:41,980 Okay. This is a big statement. We haven't shown it. 44 00:05:41,990 --> 00:05:46,309 It's useful to actually prove this statement because actually in the proof, 45 00:05:46,310 --> 00:05:51,590 there's a lot of important geometry that we're going to need in coming up in a moment. 46 00:05:51,600 --> 00:05:58,640 So let's go through the proof of that. This is true. So, first of all, so this is just a proof by geometry. 47 00:06:00,140 --> 00:06:02,200 So a little bit of geometry that you probably remember. 48 00:06:03,230 --> 00:06:18,200 First of all, it's define a plane pick of our vector g, g points X equals a constant C defines a plane defines a plane of points of points, 49 00:06:18,470 --> 00:06:27,500 points X, perpendicular to G, perpendicular to g, where. 50 00:06:30,140 --> 00:06:34,190 Okay, let me actually draw this over here. So here's the point zero. 51 00:06:34,820 --> 00:06:50,180 Here's the vector g, here's the plane perpendicular to G, which I'm calling X is the plane of points X and the distance from zero to the plane x. 52 00:06:50,180 --> 00:07:01,820 Let's call that distance X-Men here where X-Men the minimum distance to the plane equals C over absolute G. 53 00:07:02,930 --> 00:07:06,500 Does that geometry look familiar from before it look reasonable? 54 00:07:07,280 --> 00:07:15,560 Yeah. So the X-Men here is C over absolute G for defining the plane by G that x equals equals C. 55 00:07:16,640 --> 00:07:19,670 Okay, so that's how are you going to define a plane now? 56 00:07:19,670 --> 00:07:32,450 We're going to define a set of planes. Set of planes, of planes we define by each the I got x equals one. 57 00:07:32,870 --> 00:07:46,520 So why does that define a set of planes? Well, of the i g that x equals one is the same thing as saying g dot x equals 2 p.m. where m is an integer. 58 00:07:48,110 --> 00:07:51,800 There's an element of integers, right? So. 59 00:07:53,120 --> 00:08:00,199 So that means we have a whole bunch of planes whose minimum distance to the origin is given by 2 p.m. 60 00:08:00,200 --> 00:08:05,990 So X-Men equals two pi am over absolute g. 61 00:08:07,250 --> 00:08:12,770 So space in between these planes. Spacing between planes. 62 00:08:15,780 --> 00:08:23,230 Plains D is two pi over absolute g. 63 00:08:24,000 --> 00:08:27,990 Okay, let's actually I have a better picture of that than I can draw. 64 00:08:28,500 --> 00:08:32,430 So the geometry we're interested in is if the object x equals one. 65 00:08:32,640 --> 00:08:39,840 It defines a set of parallel planes. One of the planes will be g that x equals 2 p.m. another plane will be God. 66 00:08:39,840 --> 00:08:45,660 X equals two pi and plus one. The distance to the first plane is two pi am over absolute g. 67 00:08:45,840 --> 00:08:49,860 The second distance the second plane is two pi and plus one over absolute g. 68 00:08:50,100 --> 00:08:54,420 So the distance between the two is two pi over G. Does that geometry look pretty good? 69 00:08:55,050 --> 00:09:06,870 Yeah. Okay, good. All right. So now recall that if G is a reciprocal lattice vector, so g is reciprocal added factor, reciprocal lat. 70 00:09:07,950 --> 00:09:17,099 Back then e to the i g r equals one for all four. 71 00:09:17,100 --> 00:09:20,490 All are indirect. Lattice indirect. 72 00:09:24,520 --> 00:09:26,410 That's a definition of the reciprocal lattice. 73 00:09:27,790 --> 00:09:36,220 So that what that tells us is that every lattice point is going to be included inside this set of planes 74 00:09:36,910 --> 00:09:42,160 because NPR will have to satisfy the IG that our equals one if these are simply reciprocal last factor. 75 00:09:42,170 --> 00:09:47,500 So all of our lattice points in the direct lattice are going to be included in the set of planes that we just defined. 76 00:09:47,920 --> 00:09:52,210 However, it does not mean that every plane cuts through lattice points. 77 00:09:52,540 --> 00:09:56,560 Okay. Why not? Okay, here's a here's a picture to explain that. 78 00:09:57,080 --> 00:10:03,040 So here we have a set of planes defined by some reciprocal last vector, which I'll happen to call G-men. 79 00:10:03,460 --> 00:10:08,560 And here we have a set of planes divided by defined by twice that reciprocal lattice vector. 80 00:10:08,560 --> 00:10:13,990 The spacing between the lattice planes is half as big because the spacing is always two pi over g. 81 00:10:14,410 --> 00:10:21,820 You'll notice that in the second picture we have a plane which is defined by this reciprocal aspect of G, which doesn't intersect any lattice points. 82 00:10:22,120 --> 00:10:27,550 What we're guaranteed by this law is that all direct lattice points must be included in one of the planes. 83 00:10:27,790 --> 00:10:31,840 But we're not guaranteed that every plane has to cut through a lattice point. 84 00:10:32,920 --> 00:10:40,690 So if we want to get a set of planes in which everyone every plane cuts through the lattice points, we have to choose. 85 00:10:41,680 --> 00:10:56,350 So to get family of lattice planes, of lattice planes, and to be a lattice plane, it has to has to cut through a lattice point. 86 00:10:56,560 --> 00:11:01,720 So if we want a set of planes where every plane cuts through a lattice point, we must choose. 87 00:11:02,200 --> 00:11:11,510 Must choose. Game in the shortest reciprocal lattice vector in that given direction. 88 00:11:11,530 --> 00:11:17,890 If it was not the minimum reciprocal last vector in that direction, you could make it smaller, 89 00:11:18,340 --> 00:11:22,239 cut it in half, for example, and get something which has fewer planes. 90 00:11:22,240 --> 00:11:29,770 And yet all of the lattice points should still be included. So you should take the minimum reciprocal lattice vectors so that you get enough planes. 91 00:11:29,830 --> 00:11:36,230 There's guaranteed that all all your lattice points in the direct lattice will be included in those planes, but you don't get any extra OC. 92 00:11:36,820 --> 00:11:43,450 Good. All right. So that's the the proof of that of that rather important theorem. 93 00:11:43,960 --> 00:11:51,160 Now, it's it's useful to actually come up with some some way to describe these reciprocal lattice planes. 94 00:11:51,160 --> 00:11:55,600 Are these are these family of lattice planes or these reciprocal lattice vectors. 95 00:11:55,600 --> 00:12:01,330 And this is a type of notation that we will use very frequently known as Miller Indices. 96 00:12:04,620 --> 00:12:08,160 For describing. For describing. 97 00:12:09,250 --> 00:12:26,220 Describe families of planes, planes, planes or vectors in respect of space vectors and risk space. 98 00:12:30,060 --> 00:12:34,980 And the idea of a mirror index is that you write in parentheses. 99 00:12:35,010 --> 00:12:39,870 HK Well, to mean actually, maybe, maybe before I do that. 100 00:12:40,140 --> 00:12:48,920 All right, we describe families of planes or vectors of reciprocal space in terms of terms of reference vectors, 101 00:12:49,620 --> 00:12:57,509 reference vex b, sub i these bits of i's very important. 102 00:12:57,510 --> 00:13:01,470 They may be primitive, primitive, reciprocal lattice vectors or they may not be. 103 00:13:02,580 --> 00:13:07,440 Generally what we do is we write as we're starting to write a second go HK in 104 00:13:07,440 --> 00:13:15,780 parentheses to mean h times b one plus k times b two plus l times b three. 105 00:13:16,590 --> 00:13:26,730 And this notation should look a little bit familiar. We can compare it to the notation we use in real space, indirect space, direct space. 106 00:13:27,630 --> 00:13:41,910 We use the notation u v w to mean u times A1 plus v times A2 plus w times A3. 107 00:13:44,340 --> 00:13:50,489 So the important thing to take away here is that if we use curved parentheses here, we mean reciprocal space. 108 00:13:50,490 --> 00:13:53,700 If we use square brackets, we mean direct space. 109 00:13:53,700 --> 00:14:01,020 So don't confuse those with each other. Now, in both cases, we may or may not mean these A's of these B's to be primitive. 110 00:14:01,380 --> 00:14:06,690 Okay, that's very important, because a lot of the questions that come up on exams. 111 00:14:06,930 --> 00:14:10,740 Exactly reference this subtlety that sometimes you work with primitive lattice 112 00:14:11,190 --> 00:14:14,670 reciprocal lattice vectors are primitive lattice vectors and sometimes you don't. 113 00:14:15,090 --> 00:14:19,999 Why don't you? Sometimes because sometimes the primitive lattice vectors are primitive, reciprocal. 114 00:14:20,000 --> 00:14:25,590 Large vectors are ugly angles. And if you choose to use a conventional unit cell rather than primitive unit cell, 115 00:14:25,770 --> 00:14:30,030 you can use orthogonal vectors rather than using things that weird angles. 116 00:14:30,240 --> 00:14:34,650 So if you can use orthogonal, you usually do, even if it means you're not using primitive lattice vectors. 117 00:14:34,830 --> 00:14:42,720 Okay, so let's be be careful here being careful, being careful about our definitions. 118 00:14:44,880 --> 00:14:55,950 The rules kind of go like this. First, choose direct real space vectors, direct space vectors, direct space vectors, 119 00:14:56,370 --> 00:15:02,489 a sub I that's going to be these things which may or may not be primitive. 120 00:15:02,490 --> 00:15:09,060 I'll even write that down. May or may not, may not be p elves. 121 00:15:09,990 --> 00:15:12,420 P elves. Now for us, 122 00:15:12,420 --> 00:15:19,140 you'll remember we had this rule that we're not going to be responsible for lattices in three dimensions where the axes are not orthogonal. 123 00:15:19,530 --> 00:15:25,980 Okay. For us, for us in three D, we always choose. 124 00:15:27,840 --> 00:15:34,500 Choose orthogonal or orthogonal A's. 125 00:15:35,770 --> 00:15:41,309 Okay? Sometimes in two D, they'll ask a question where they're not orthogonal, 126 00:15:41,310 --> 00:15:45,570 but in three D we always choose to choose orthogonal A's That means for simple cubic, 127 00:15:46,230 --> 00:15:52,110 simple cubic cubic, these can be these are primitive lattice vectors players. 128 00:15:52,110 --> 00:16:01,409 But for FCC, FCC and BC, these are not not pelvises, but they have the advantage of being orthogonal. 129 00:16:01,410 --> 00:16:06,560 So they make your life a lot easier because it's a lot easier to work with things that are orthogonal and things that are not orthogonal. 130 00:16:06,570 --> 00:16:12,630 Okay. So given that we've defined these A's, we then construct B's. 131 00:16:12,930 --> 00:16:31,350 So can so call that to construct construct B's by v i talked a j equals to pi delta i j and we've seen this equation before. 132 00:16:31,920 --> 00:16:43,410 If, if, and only if is our pelvis, then B's will be reciprocal. 133 00:16:43,440 --> 00:16:50,610 Are reciprocal. PVS are reciprocal. That reciprocal lat plays primitive lattice vectors. 134 00:16:51,960 --> 00:16:56,510 Okay. Again for us. 135 00:16:56,930 --> 00:17:04,520 For us, since we are using in 3D, in 3D, since we are always using orthogonal axes in 3D, 136 00:17:04,850 --> 00:17:13,130 it will always turn out to be the case that the absolute value of BI equals to pi over the absolute value of A.I. because axes are orthogonal. 137 00:17:13,640 --> 00:17:20,220 Axes orthogonal. Okay. 138 00:17:21,160 --> 00:17:22,930 So that makes life a lot easier. 139 00:17:25,570 --> 00:17:43,660 So generally when we write down R, g equals L, we mean, whereas this thing over here, h, b, one plus k and we already again we mean hb1 plus k, 140 00:17:44,080 --> 00:17:51,100 b two plus lb3 where the B's have been constructed in this way, 141 00:17:51,610 --> 00:18:07,510 but always g defines a set of plain defines a set of plains, a set of plains plains via the idea that x equals one. 142 00:18:07,520 --> 00:18:11,350 Either the i g dot x equals one. 143 00:18:11,470 --> 00:18:18,130 So we can always define a set of plains for any vector g by either the x equals one, 144 00:18:18,310 --> 00:18:29,050 and the spacing of these plains spacing is d h l equals to pi over absolute g. 145 00:18:29,920 --> 00:18:43,120 Or I guess in assuming our orthogonal axes, assume orthogonal B's, orthogonal A's orthogonal to A's and B's are orthogonal or orthogonal. 146 00:18:43,810 --> 00:18:54,460 Orthogonal. Then we have we can write that as two pi over our absolute value of h squared, 147 00:18:55,450 --> 00:19:03,760 absolute b one squared plus k squared, absolute b two squared plus l squared, absolute b three squared. 148 00:19:05,020 --> 00:19:15,849 Okay. And for cubic lattice for cubic for cubic where A1, A2 and a3 are the same length and orthogonal to each other, 149 00:19:15,850 --> 00:19:20,230 therefore b1 b2 b three are the same length and are orthogonal to each other. 150 00:19:20,440 --> 00:19:27,700 The spacing d h kcl is a the length of the primitive, 151 00:19:28,270 --> 00:19:36,760 the length of the conventional used cell in indirect space over a squared plus k squared plus elsewhere. 152 00:19:38,110 --> 00:19:41,560 Okay. And maybe I'll note this one more time. 153 00:19:42,220 --> 00:19:49,990 If the A's our pelvis is our pelvis, then the bees are pelvis. 154 00:19:50,650 --> 00:20:03,610 Then these pelvis, the right that above B's pelvis. And that means that hk l is a reciprocal lattice factor receptor laterally. 155 00:20:05,830 --> 00:20:10,000 But it's not always the case if the A's are not pelvis to start with. 156 00:20:10,390 --> 00:20:20,020 Okay. All right. So supposing someone hands you a set of planes and asks you for the major indices. 157 00:20:20,230 --> 00:20:22,630 That sounds like it might be hard to figure out, but in fact, 158 00:20:22,960 --> 00:20:28,600 there's an easy way to figure out geometrically what the Miller Indices are for a set of planes. 159 00:20:28,780 --> 00:20:32,920 And this is known as the method of intercepts. Method of intercepts. 160 00:20:39,640 --> 00:20:43,760 It goes kind of like kind of like this. 161 00:20:43,770 --> 00:20:51,320 So suppose we have, uh, so here's three axes X axis, the y axis, and the z axis. 162 00:20:51,560 --> 00:20:58,730 Let's assume orthodontics are the primitive lattice vectors in the three directions aren't necessarily the same. 163 00:20:58,730 --> 00:21:11,330 So call this a and two A2 and this one will be, I guess, a one to a131 for a one here and then a three to a three and three. 164 00:21:11,330 --> 00:21:15,140 A three. So suppose that we have a plane going through the origin. 165 00:21:15,440 --> 00:21:22,100 So imagine there's a plane cutting through the origin here and then also cuts through the axes like this. 166 00:21:23,470 --> 00:21:26,830 Okay. So you take the first two parallel planes. One of them cuts to the origin. 167 00:21:27,020 --> 00:21:30,140 It's parallel to this plane, which cuts through the axes like this. 168 00:21:30,620 --> 00:21:35,689 The rule is that hk l right this way. 169 00:21:35,690 --> 00:21:53,300 H is to k is to l as a proportionality is the same as a one over the x intercept is to a two over the y intercept is to a three over the Z intercept. 170 00:21:54,620 --> 00:22:03,979 So in this picture here, this would be a one over the x intercept is one over 4h2 is over, 171 00:22:03,980 --> 00:22:10,510 the y intercept is one over two and a three over the z intercept is one over three. 172 00:22:11,240 --> 00:22:17,150 So it should have the same ratios as this. If we multiply by the least common multiple, least common multiple, 173 00:22:17,540 --> 00:22:30,380 we get three comma six come a four as Miller Indices that would be appropriate for this this plane and generally write it as 364 without any commas, 174 00:22:30,620 --> 00:22:34,459 at least as long as there's no confusion as to where you should put the commas. 175 00:22:34,460 --> 00:22:37,580 If you go over ten, then you have to indicate where the commas are. 176 00:22:37,580 --> 00:22:42,110 Otherwise it's not clear where it would be. All right. 177 00:22:42,620 --> 00:22:56,509 Now, if and this and what we mean by three, six, four here is that 364 is three B, one plus six, B two plus four, B three. 178 00:22:56,510 --> 00:22:58,040 And that's what we mean by the Miller Indices. 179 00:22:58,040 --> 00:23:05,360 And that that set of Miller Indices is gives a reciprocal last vector, which is normal to this to this plane. 180 00:23:06,500 --> 00:23:09,740 There's a some cases which are a little more complicated. 181 00:23:10,010 --> 00:23:23,210 If a plane if a plane does not intersect, an axis plane does not not intersect and axis not intersect, intersect an axis. 182 00:23:25,430 --> 00:23:35,600 Then you get a miller index of zero, you get miller index index equals zero. 183 00:23:36,740 --> 00:23:41,210 Let's see if we can explain what that is. So here again, three axes. 184 00:23:43,190 --> 00:23:47,780 This is the A1 axis out here. Maybe this would be a two, this would be two, a two. 185 00:23:48,230 --> 00:23:53,000 And maybe here we have three, a three. 186 00:23:53,240 --> 00:23:57,650 And suppose we have an axis. Axis, always on a plane, always cutting through zero. 187 00:23:57,830 --> 00:24:01,490 And then the next plane cuts here. 188 00:24:01,910 --> 00:24:06,950 But it's parallel to the A1 axis here to the x axis, so it never intercepts it. 189 00:24:07,280 --> 00:24:13,849 So what do we do with the Miller Indices? Well, we say H is two, K is two, L as well. 190 00:24:13,850 --> 00:24:17,240 Okay. So the first question is, where do we intercept the X axis? 191 00:24:17,510 --> 00:24:24,230 Well, if it doesn't intercept the x axis at all, you can say that is really like intersecting the x axis at infinity. 192 00:24:24,440 --> 00:24:30,829 So we write one over infinity here and then k while intersects the y axis at two. 193 00:24:30,830 --> 00:24:34,670 So one over two and we intersect the z axis of three. 194 00:24:34,670 --> 00:24:47,180 So we write one over three and multiply by the least common multiple and we get one over infinity 0332 So the Miller indices here would be 032. 195 00:24:48,110 --> 00:24:51,560 Okay. So let me show you a couple examples of this. 196 00:24:52,760 --> 00:24:59,690 Here's a set of Miller Indices that are 100 because it intersects X at one and doesn't intersect Z or Y. 197 00:25:00,320 --> 00:25:06,170 Here's one. That's one one, two, because it intersects exec one Y at one and Z at one half. 198 00:25:06,170 --> 00:25:09,170 The reciprocal of one half is two. 199 00:25:09,170 --> 00:25:12,020 So it's one, one, two. So you can have a situation. 200 00:25:13,010 --> 00:25:34,669 R can have you can have Miller index less than zero index negative like one minus one zero, which you typically write as one one with a bar over it. 201 00:25:34,670 --> 00:25:38,800 Zero. Okay. So the. R bar means negative. 202 00:25:40,150 --> 00:25:49,460 Means negative. That's just the notation that's conventionally used now. 203 00:25:49,480 --> 00:25:55,610 Okay, so here is a great big apology for the UN coordination of our community. 204 00:25:56,060 --> 00:26:02,570 So depending on which side of the ocean you come from, the way you say this out loud is different. 205 00:26:02,930 --> 00:26:07,250 If you're from the United States, you say one one, bar zero. 206 00:26:07,520 --> 00:26:10,760 If you're from England, you say one bar one zero. 207 00:26:11,210 --> 00:26:16,010 I don't know why it makes things incredibly confusing. When it's written, it's totally unambiguous. 208 00:26:16,010 --> 00:26:22,730 The bar on top means negative, but when you say it out loud, it's totally confusing and no one knows what you mean. 209 00:26:22,910 --> 00:26:26,000 So I recommend just writing it down, and then everyone knows what you're talking about. 210 00:26:26,600 --> 00:26:29,790 Okay, let me show you an example where we have a negative index. 211 00:26:30,110 --> 00:26:36,290 So this plane here actually is maybe it's not so clear, but but it's supposed to you supposed to sort of be able to extrapolate down with 212 00:26:36,290 --> 00:26:40,849 your with your eyes and see that this plane is sort of starts up here and goes down. 213 00:26:40,850 --> 00:26:47,389 It would intersect the z-axis at minus one half. And so its intercepts are one one minus one half. 214 00:26:47,390 --> 00:26:52,170 So it's more indices or 112 with the bar over it. Negative two. 215 00:26:52,930 --> 00:26:59,100 Okay, so here's a whole bunch of different Miller Indices for a bunch of different planes. 216 00:26:59,100 --> 00:27:04,050 He's 001100010101, so forth and so on. 217 00:27:04,290 --> 00:27:07,410 And you see that they actually they actually kind of come in families. 218 00:27:07,680 --> 00:27:13,360 So if you look at the family across the top row, across the top row, you'll see that 001 and 100. 219 00:27:13,380 --> 00:27:17,970 They kind of look similar and 010 looks kind of similar. And if you imagined actually having a sample, 220 00:27:18,450 --> 00:27:23,220 it's a simple cubic lattice and you picked up the simple cubic lattice and you looked at it from one side, 221 00:27:23,520 --> 00:27:29,190 100 would look exactly like 010, depending on if you picked it up and turned it sideways. 222 00:27:29,490 --> 00:27:33,090 It's exactly the same. It just depends on which side you picked up and looked at first. 223 00:27:33,360 --> 00:27:39,300 Similarly, with 101 or 110, if you picked it up and you looked at it sideways, it would look exactly the same. 224 00:27:39,900 --> 00:27:43,200 The spacing between planes in those families would be exactly the same. 225 00:27:43,380 --> 00:27:46,770 The way it cuts diagonally through the through the unit cell would look the same. 226 00:27:46,860 --> 00:27:56,040 It just matters which way you picked it up in the first place. So we have a notation for equivalent sets of indices, equivalent sets. 227 00:27:59,260 --> 00:28:08,230 Of Miller Indices, indices which we notate with curly brackets. 228 00:28:08,680 --> 00:28:25,030 Okay, that is a terrible crummy bracket like that h cal, which means equivalent sets of 12 sets of h cal and regular parentheses brackets. 229 00:28:25,300 --> 00:28:43,510 So for example, in this picture, curly brackets, 100 would mean 100 or 010 or 001 or one bar zero zero or zero one bar zero or 001 bar. 230 00:28:45,430 --> 00:28:53,169 And coincidentally, these planes look like the surface of a cube, and you can pick up the surface of a cube and look at it six different ways. 231 00:28:53,170 --> 00:28:57,280 And it would look equivalent from all of those six different angles. 232 00:28:57,520 --> 00:29:12,819 We also have what's known as the multiplicity of each cal, which is the number equals the number of parentheses. 233 00:29:12,820 --> 00:29:16,330 H cal in bracket. 234 00:29:16,420 --> 00:29:31,780 H cal. So in this case we would have multiplicity of 100 is six because it includes all those six different sets of indices that I wrote down. 235 00:29:32,110 --> 00:29:40,990 Okay. So for the first candy bar of the day, what is the multiplicity of one, two, three? 236 00:29:42,300 --> 00:29:46,980 Now. Well, thanks. Now, who was it? 237 00:29:48,040 --> 00:29:51,790 Someone said something. 48 Yeah, that's who said that. 238 00:29:52,510 --> 00:29:55,540 48 This one's yours. Very good. I can't throw it that far. 239 00:29:55,780 --> 00:30:03,370 48 The answer is 48 So let's see why the multiplicity of this is 48 First of all, one, two and three can be in any order. 240 00:30:03,850 --> 00:30:07,990 So that six back that's three factorial gives you six. 241 00:30:08,230 --> 00:30:13,629 One, two, three, one, three, two, two, three. One, two, one, three, three, one, two, three, two, one, six. 242 00:30:13,630 --> 00:30:18,970 Possible ways of ordering them. But then in addition, any one of those numbers can have a bar over it or not. 243 00:30:19,240 --> 00:30:23,950 So this one has two possibilities. This has two possibilities. This two has two possibilities that makes eight possibilities. 244 00:30:24,220 --> 00:30:32,020 So it's eight times six gives you 48. So multiplicity of one, two, three equals 48, which is kind of interesting. 245 00:30:32,030 --> 00:30:37,090 It means if you have the family of lattice planes associated with a one, two, three Miller Indices, 246 00:30:37,390 --> 00:30:42,190 you can pick up a simple cubic lattice and look at it from 48 different directions 247 00:30:42,400 --> 00:30:45,490 and have lattice planes with exactly the same spacings looking at you. 248 00:30:47,200 --> 00:30:55,359 Sort of surprising. But. But that's true. Okay, so now why is it we care so much about these these lattice planes besides the fact that 249 00:30:55,360 --> 00:31:00,339 they're going to tell us a lot about reciprocal space and we're interested in waves and waves, 250 00:31:00,340 --> 00:31:07,749 you know, reciprocal spaces, language waves. Well, the reason one reason we might be interested in these families of lattice planes in reciprocal 251 00:31:07,750 --> 00:31:13,000 space is because they actually tell you a lot about the macroscopic structure of the crystal. 252 00:31:13,210 --> 00:31:18,100 It's kind of interesting that all of this information about lattice planes and reciprocal lattices, 253 00:31:18,100 --> 00:31:23,829 this was all worked out in the middle of the 1800s before people even knew that there was such thing as atoms. 254 00:31:23,830 --> 00:31:27,580 And the way it was worked out is because people knew that minerals, 255 00:31:27,580 --> 00:31:35,410 nice crystal and minerals broke and so they fractured only along certain axes, and that enabled them to deduce all of this information. 256 00:31:35,620 --> 00:31:36,370 So, for example, 257 00:31:36,370 --> 00:31:46,420 with a simple cubic lattice you'll discover or any type of cubic lattice that the lattice will will break the material will break along simple, 258 00:31:47,440 --> 00:31:55,570 simple sets of Miller indices. So like the 100 face here or the 010 phase here is where the material will will break most easily. 259 00:31:55,780 --> 00:32:00,750 This is a law it's not a hard and fast law, but it's a very, very good rule of thumb by Broadway, 260 00:32:00,820 --> 00:32:04,480 known as Bravo's Law, the same guy who worked out all the lattices. 261 00:32:06,220 --> 00:32:12,280 And it actually tells you that when you when you take a cubic crystal and you break it, you get only the 100 phases. 262 00:32:12,280 --> 00:32:17,350 Occasionally you'll get the 110 faces and even less occasionally you'll get one one, too. 263 00:32:17,530 --> 00:32:21,100 So here's a picture of sodium chloride, which is a cubic crystal. 264 00:32:21,260 --> 00:32:29,260 It's actually FCC crystal, and it breaks almost exclusively along the 100 are lattice planes. 265 00:32:29,260 --> 00:32:33,910 You can see that these these breaks are entirely at 90 degrees quartz. 266 00:32:34,360 --> 00:32:37,540 Okay. We don't study quartz this year. It's just a very nice picture. 267 00:32:37,540 --> 00:32:42,969 So I put it in there and we don't study it this year because the axes are not orthogonal to each other. 268 00:32:42,970 --> 00:32:49,510 It's a tricky little crystal, but the axes very clearly, the faces are very clearly at very fixed angles to each other, 269 00:32:49,660 --> 00:32:53,110 which are dictated by how the lattice planes are arranged. 270 00:32:54,040 --> 00:33:01,540 Okay, so I'm going to emphasise this one fact one more time because it shows up on exam so frequently. 271 00:33:02,410 --> 00:33:13,090 So this is important fact. If as our please then the of if and only if A's are PVS, then bees are PVS. 272 00:33:13,090 --> 00:33:19,209 I think I've written it down three times today that bees are pelvic responses. 273 00:33:19,210 --> 00:33:25,720 Reciprocal pelvis, reciprocal pelvis, for example. 274 00:33:25,720 --> 00:33:39,040 This is true for simple cubic, which means any h kcl is hb1 plus kb2. 275 00:33:39,640 --> 00:33:41,830 This is the definition of HK I'll be three. 276 00:33:42,130 --> 00:33:53,200 So any of these is a receptor lattice vector receptor lat vec because integer sums are primitive lattice vectors give you an element of the lattice. 277 00:33:54,580 --> 00:34:06,910 If we want a family of lattice planes, a family of lattice planes, of flat planes, we should have, should have, that should be shortest. 278 00:34:07,600 --> 00:34:13,100 The shortest. In a given direction. 279 00:34:13,340 --> 00:34:20,150 Shortest recap flat vec in a given direction. 280 00:34:25,820 --> 00:34:34,130 So for example, if we're considering the direction to four six, the shortest reciprocal lattice for the simple cubic lattice, 281 00:34:34,460 --> 00:34:39,920 the shortest reciprocal lattice vector in that direction is one, two, three. 282 00:34:40,400 --> 00:34:47,390 So you divide by divide by greatest common divisor. 283 00:34:49,670 --> 00:34:56,690 So if you want to describe a set of lattice planes, you should always choose the reciprocal lattice factor, which is shortest in that given direction. 284 00:34:57,260 --> 00:35:14,149 However, however, if the A's are not pelvises, then and this is then these are not PVS, not reciprocal. 285 00:35:14,150 --> 00:35:37,250 These. Which means that not all hk l r recep lattice factors are recep lattice factors, and this is the source of most most good exam problems. 286 00:35:37,640 --> 00:35:43,130 Okay, so let me show you exactly what we mean by this. 287 00:35:43,460 --> 00:35:49,280 So recall this picture and the lattice planes that the family of last planes have drawn in this picture. 288 00:35:49,290 --> 00:35:57,480 Here is the 100 family of lattice planes and and the minimum reciprocal last vector in that direction is, in fact, 100. 289 00:35:57,500 --> 00:36:04,160 This is simple cubic lattice. So this is definitely a are a family of lattice planes and a reciprocal lattice vector. 290 00:36:04,370 --> 00:36:11,510 Now, twice that 200 is a reciprocal lattice factor, but it's not a family of lattice planes because you have extra planes, 291 00:36:11,660 --> 00:36:15,470 you have a plane that doesn't cut to any lattice points, half the spacing of planes. 292 00:36:15,710 --> 00:36:18,530 So it's not a family of last planes, but it is a reciprocal lattice vector. 293 00:36:18,770 --> 00:36:25,309 Now, let's think about what happens when we go to either a BK or an FCC lattice with a boxy and FCC lattice, 294 00:36:25,310 --> 00:36:29,210 we're still going to use the same three orthogonal A's. 295 00:36:29,480 --> 00:36:35,180 We're going to choose to use the conventional unit cell to define our A's, just like we did with a simple cubic lattice. 296 00:36:35,420 --> 00:36:39,080 And we're doing this. They're not primitive lattice vectors, and that's a problem, 297 00:36:39,260 --> 00:36:43,820 but we're doing it because we want to keep everything orthogonal and life is just a lot easier if you keep everything orthogonal. 298 00:36:44,180 --> 00:36:50,959 Okay, so here's a picture with a bunch of Miller Indices, a bunch of lattice planes drawn for the simple cubic lattice. 299 00:36:50,960 --> 00:36:59,630 Here's the 100 set of last planes just like I drew before. Now, in the back case, if you added one more point in the centre of that unit cell, 300 00:36:59,840 --> 00:37:05,420 in the centre of that convention, in the cell, it would not be intersected by the 100 set of planes. 301 00:37:05,750 --> 00:37:13,130 It is intersected by the 200 set of planes. So 200 is a correct family of lattice planes for BK. 302 00:37:13,520 --> 00:37:23,480 It is not for four. Simple cubic is 100 isn't even a reciprocal lattice vector for BK because it does not intersect every lattice point. 303 00:37:24,230 --> 00:37:29,750 Okay, that's that's the subtlety. And we're going to come back to this subtlety over and over for a BK. 304 00:37:29,960 --> 00:37:35,870 You have to choose Miller Indices to make sure that every one of your lattice points is included in one of those planes. 305 00:37:35,870 --> 00:37:43,580 Same for FCC. And the reason this is a problem is because we're using lattice vectors a which are not the primitive lattice vectors. 306 00:37:43,790 --> 00:37:47,330 We're using orthogonal vectors because they're orthogonal makes your life easier, believe it or not. 307 00:37:47,660 --> 00:37:56,690 Same with FCC. If we consider the FCC lattice and you consider the 111 planes, these planes here, we would not intersect some of these lattice points. 308 00:37:56,690 --> 00:38:01,640 These would be missing. So 100 is not a reciprocal lattice vector for FCC. 309 00:38:02,060 --> 00:38:15,379 200 is okay. Similarly, while we can look at 111 here, so 111 is a perfectly good set of family of lattice planes for a simple cubic. 310 00:38:15,380 --> 00:38:20,540 It would not be a family of lattice planes for BK because it would miss the point in the centre of the unit cell. 311 00:38:20,750 --> 00:38:28,190 But actually it is a family of lattice planes for FCC because it just happens to catch all those points in the centre of the faces. 312 00:38:28,490 --> 00:38:34,520 Okay, now later on, we're actually going to derive a condition which will tell us when we have Miller Indices, 313 00:38:34,700 --> 00:38:43,700 which are family of lattice planes for a particular lattice. But it's important to realise not every set of not every set of Miller Indices will 314 00:38:43,700 --> 00:38:48,919 give you a reciprocal lattice vector or a family of lattice planes for an FCC lattice. 315 00:38:48,920 --> 00:38:54,829 And the origin of this problem is we're not using primitive lattice vectors to describe our convention itself. 316 00:38:54,830 --> 00:38:58,850 We're using the edges of the convention yourself. Is that clear? 317 00:38:59,090 --> 00:39:02,360 Have I warned you enough? All right, good. 318 00:39:02,960 --> 00:39:09,620 All right, so that's everything we have to say about reciprocal space. 319 00:39:09,620 --> 00:39:13,159 And now we can get on to some applications of reciprocal space. 320 00:39:13,160 --> 00:39:18,680 And perhaps the most important application we're going to run into is scattering experiments. 321 00:39:20,180 --> 00:39:27,320 Scattering experiments, which will be the subject of the rest of this lecture and probably much of the next two lectures. 322 00:39:27,740 --> 00:39:32,180 So the general scattering experiment starts with a sample sample. 323 00:39:34,640 --> 00:39:37,250 You send in some wave with wave vector K, 324 00:39:37,940 --> 00:39:45,320 some of it goes through and scattered comes out at wave vector K and some of it comes out at wave vector K prime. 325 00:39:45,740 --> 00:39:52,040 By observing what comes out at wave vector K prime, you're supposed to be able to deduce what is in the sample. 326 00:39:52,310 --> 00:39:57,500 Now, it's hard to overstate the importance of this type of experiment. 327 00:39:57,500 --> 00:39:59,990 You may have studied it in other courses, especially subatomic, 328 00:40:00,350 --> 00:40:06,050 but it is not an over is not an overstatement to say that almost everything we know about the microscopic world, 329 00:40:06,230 --> 00:40:10,520 everything the fact that we know that there are nuclei, the fact that we know the structure of atoms, 330 00:40:10,670 --> 00:40:13,610 the fact that we know the structure of materials, the fact that we know the structure of the proton. 331 00:40:13,940 --> 00:40:17,880 All of these things come from this kind of experiment, this type. A scattering experiment. 332 00:40:18,120 --> 00:40:23,370 You know, we grew up in a world where even kids know what atoms look like and they know what DNA looks like, 333 00:40:23,370 --> 00:40:27,240 this double stranded helix, they know what water molecule looks like. They may even know what salt looks like. 334 00:40:27,480 --> 00:40:33,060 Sodium chloride, sodium chloride. All of this stuff that we know and we take for granted comes from doing this kind of experiment. 335 00:40:33,840 --> 00:40:36,899 The only thing that's different when we do this experiment first is when the high 336 00:40:36,900 --> 00:40:41,610 energy physicist doing this experiment is what wave we put in to start with. 337 00:40:41,810 --> 00:40:46,860 Okay, now we're interested in seeing atoms. And atoms are on the scale of about an angstrom. 338 00:40:47,040 --> 00:40:52,920 It turns out to be a really good idea to choose a wave whose wavelength is about the same scale of the thing you want to see. 339 00:40:53,280 --> 00:41:01,470 So for us, they're going to be a couple of options. One is x rays, electromagnetic waves whose length scale is about an angstrom. 340 00:41:01,650 --> 00:41:05,610 So that will enable us to see things roughly of the size of an angstrom. 341 00:41:07,530 --> 00:41:12,510 But, you know, we live in this quantum mechanical world, and we know that waves and particles are the same thing. 342 00:41:12,690 --> 00:41:17,280 So we're entitled to use not only things we usually think of as ways, but things we usually think of as particles. 343 00:41:17,640 --> 00:41:22,980 So a particle which is very convenient to do scattering experiments with as neutrons, 344 00:41:24,780 --> 00:41:27,330 neutrons, you know, I think of it in particles, but they're also waves. 345 00:41:27,450 --> 00:41:33,420 It turns out to be fairly easy to get neutrons to have wavelengths which are on about the same scale of about an angstrom, 346 00:41:33,570 --> 00:41:37,410 which makes it easy to see things which are on the scale of roughly an angstrom. 347 00:41:37,710 --> 00:41:41,670 These are pretty much the two types of waves that we're going to be responsible for this year. 348 00:41:41,910 --> 00:41:44,940 I should mention that people frequently use electron waves as well, 349 00:41:47,460 --> 00:41:51,840 but we don't study them this year because they have some complications, which we don't want to delve into. 350 00:41:52,260 --> 00:41:58,350 So whatever type of wave we send into our material, 351 00:41:58,380 --> 00:42:04,110 the thing that we're supposed to imagine is that the atoms in sample atoms in the sample 352 00:42:07,140 --> 00:42:23,520 provide a potential a potential V of our that scatters that scatters wave scatters the wave. 353 00:42:25,320 --> 00:42:31,800 And our objective is actually a bit of an inverse problem where instead of trying to calculate 354 00:42:32,190 --> 00:42:38,639 what is the potential due to the wave much more frequently we're told what the wave has done. 355 00:42:38,640 --> 00:42:42,990 We've measured what the wave has done, and we're trying to deduce the potential. 356 00:42:43,200 --> 00:42:46,500 The potential is going to tell us where the atoms are in the material. 357 00:42:46,620 --> 00:42:50,879 And we're trying to deduce that from observing what physically came out of our sample. 358 00:42:50,880 --> 00:42:52,620 Now, just a little bit of a caveat. 359 00:42:54,360 --> 00:42:59,250 It's easy to think about a potential for neutrons and neutron moves around it, feel some potential as it moves around. 360 00:42:59,580 --> 00:43:04,380 It's a little bit less natural to think about of potential for for x rays, which are electromagnetic waves. 361 00:43:04,710 --> 00:43:08,010 You can if you want, you can kind of think of it as a potential for photons, 362 00:43:08,670 --> 00:43:18,540 but much more appropriate is you really should be thinking of it as spatially dependent optical index or spatially dependent dielectric constant. 363 00:43:18,720 --> 00:43:24,510 But you won't go too far wrong to think about it as just a potential for particles as a particle move through your material. 364 00:43:25,050 --> 00:43:30,209 So the way we're going to actually scat, we're going to actually calculate how much get scattered out in some way. 365 00:43:30,210 --> 00:43:33,420 Vector. K prime is to use. Fermi's Golden rule. 366 00:43:34,470 --> 00:43:40,530 Fermi's Golden Rule rule condensed matter joke. 367 00:43:40,710 --> 00:43:49,350 This is a symbol for gold anywhere near one. So I made that up on the spot. 368 00:43:49,350 --> 00:43:53,790 It was because I was trying to save my arm from writing. All right, never mind. 369 00:43:53,790 --> 00:43:58,320 I'm not going to be a comedian. That's why I'm a physicist. Okay. So the scattering rate, 370 00:43:59,820 --> 00:44:11,760 the scattering rate brings rates from K to K prime is given by you probably probably seen from his golden rule to PI over h bar. 371 00:44:12,030 --> 00:44:26,220 There's a matrix element K prime the K that gets squared and an energy preserving delta function vectors are energy preserving delta function. 372 00:44:26,490 --> 00:44:40,559 Now to simplify your life, we are going to assume we assume elastic scattering, elastic scatter, which means that e equals e prime. 373 00:44:40,560 --> 00:44:47,640 The energy coming in and the energy coming out are the same. That will also mean the absolute value of K equals the absolute value of K prime. 374 00:44:48,070 --> 00:44:51,990 There's several reasons why we assume elastic scattering. First of all, it's easier. 375 00:44:52,230 --> 00:44:57,510 Secondly, when you actually do these experiments, most of the scattering that you measure actually is elastic. 376 00:44:57,750 --> 00:45:00,479 It's actually quite interesting to look at inelastic scattering as well. 377 00:45:00,480 --> 00:45:04,350 I might mention that a couple of lectures further on with inelastic scattering, 378 00:45:04,500 --> 00:45:09,329 you actually get you leave some energy behind in the sample and that actually allows you to figure 379 00:45:09,330 --> 00:45:14,640 out what the energy of the possible excitations are that you can excite in the sample as well. 380 00:45:14,880 --> 00:45:18,230 But that's a little bit more complicated. So we're not going to worry about that for now. 381 00:45:19,670 --> 00:45:26,120 So we're just going to assume elastic scattering. And the only thing that's difficult in that expression for me is golden rule is the matrix element. 382 00:45:26,130 --> 00:45:32,150 So let's write out the matrix element. So we have K prime v k equals what? 383 00:45:32,180 --> 00:45:35,570 Well, we'll use the, the real space representation of these cats. 384 00:45:35,990 --> 00:45:50,950 It's integral D and d are of either the signs in the i k prime dot r v of our times into the i kkr have the signs, right? 385 00:45:50,960 --> 00:45:55,760 Yeah. Okay. And if you were careful, we would normalise this by one over square root of volumes. 386 00:45:55,760 --> 00:45:59,120 This is the the way vector. This is the wave function k prime. This is the wave. 387 00:45:59,120 --> 00:46:01,190 Function K, there's v r in the middle. 388 00:46:01,400 --> 00:46:16,400 And we realise that what that is is all right this way d three are either minus i k prime minus k dotted with r v of r that is just well we'll 389 00:46:16,400 --> 00:46:29,390 write it as v twiddle of k minus k prime k minus k prime where v twiddle is means for a transform v twiddle equals 48 transform of v of r. 390 00:46:30,020 --> 00:46:37,730 This is a very, very general rule that the scattering amplitude from a K prime is just the 40 transform of the scattering potential. 391 00:46:38,030 --> 00:46:41,150 Now in the last lecture. 392 00:46:41,330 --> 00:46:45,500 And so that actually always, almost always holds true for anything that you're going to do. 393 00:46:45,500 --> 00:46:49,909 Scattering experiments on that, the scattering of using Fermi's golden rule, 394 00:46:49,910 --> 00:46:55,880 the scattering amplitude from some K2 k prime is just the four transform whatever the scattering potential is now. 395 00:46:56,420 --> 00:47:02,120 In the last lecture, we considered 48 transforms of periodic functions. 396 00:47:02,120 --> 00:47:09,590 So we're going to assume we have a crystal. Let's assume a periodic periodic scattering potential the of R. 397 00:47:09,650 --> 00:47:13,790 So if we have anything that's crystalline, the potential will be periodic. 398 00:47:14,210 --> 00:47:20,540 And so we can use the the law that we derived in the last lecture that the 40th transform of periodic 399 00:47:20,540 --> 00:47:31,730 potential came on as k prime equals equals a sum over our reciprocal lattice vectors g of a delta function, 400 00:47:31,790 --> 00:47:35,240 three dimensional delta function of k minus k prime. 401 00:47:36,110 --> 00:47:41,690 Right that again k minus k primaries, g times structure factor of g. 402 00:47:42,080 --> 00:47:47,959 I guess there's some normalisation factor to pi cubed over the volume of unit cell volume. 403 00:47:47,960 --> 00:47:57,030 And so. And the structure factor here as of g o coming out of space, 404 00:47:57,570 --> 00:48:16,290 r s of g equals the integral is equal to the 40 transform within a unit cell integral d three are in the i i g r v r r right the rs the same way, 405 00:48:17,070 --> 00:48:20,729 and it's integral over the unit cell. Okay, a couple of important. 406 00:48:20,730 --> 00:48:27,960 So this is a law that we derived last time and a couple of important facts about this. 407 00:48:28,200 --> 00:48:34,650 This law we mentioned last time. First of all, the free transfer of this periodic function is zero. 408 00:48:34,980 --> 00:48:40,770 Unless we're looking at values of K, which are elements of the reciprocal lattice factor. 409 00:48:41,040 --> 00:48:57,350 So point one scattering is zero is zero and less and less k minus K prime is a reciprocal lattice vector g. 410 00:48:58,020 --> 00:49:02,640 This is important, but the box around it, this is known as the Laue condition. 411 00:49:05,520 --> 00:49:14,400 After Max von Laue, who won a Nobel Prize for discovering this in X-ray scattering Nobel Prize was actually exactly 100 years ago now. 412 00:49:16,470 --> 00:49:22,740 And then there's this complication about whether it should be called the Laue condition or the Von Laue condition because he was born Max Laue. 413 00:49:22,950 --> 00:49:27,900 But then his father was elevated to Noble Hood much later, and he took the name Von Laue. 414 00:49:28,080 --> 00:49:30,180 But when he discovered this, his name was still Laue. 415 00:49:30,390 --> 00:49:34,500 So it's usually called the Laue condition, not the found life condition, but other people will disagree. 416 00:49:34,770 --> 00:49:39,690 So anyway, it's either Laue condition of the Von Laue condition, and but we've seen this thing before. 417 00:49:39,690 --> 00:49:45,900 What this is, is conservation of conservation of crystal momentum. 418 00:49:49,940 --> 00:49:55,880 You only scatter if you can conserve crystal momentum, right? 419 00:49:56,420 --> 00:50:00,229 F k minus k prime is a reciprocal lattice vector that conserves crystal momentum. 420 00:50:00,230 --> 00:50:03,290 It doesn't change the crystal momentum of the wave. 421 00:50:03,560 --> 00:50:11,000 The second important fact that we're going to use later is that the intensity of scattering, intensity of scattering, 422 00:50:12,700 --> 00:50:24,410 scatter is the structure factor s of well, delta delta k squared where delta k is the reciprocal lattice factor G which is scattered by. 423 00:50:24,680 --> 00:50:31,670 Okay. So the point is that if you measure the intensity of the various scattering, you can figure out the structure factor. 424 00:50:32,210 --> 00:50:35,990 What you're measuring is the structure factor at given reciprocal lattice factors. 425 00:50:36,260 --> 00:50:40,520 The structure factor of giving your reciprocal lattice factors is the for transform of the scattering potential. 426 00:50:40,700 --> 00:50:44,330 So in principle, you're measuring the scattering potential. 427 00:50:44,630 --> 00:50:51,890 That's the point. By measuring the what is scattered, you're actually measuring the scattering potential now. 428 00:50:52,790 --> 00:50:58,279 So I'm going to go over about one minute here. It may seem a little bit surprising this condition, 429 00:50:58,280 --> 00:51:08,420 one that you only get scattering under certain conditions only if crystal momentum is satisfied, only if you if satisfy this lousy condition. 430 00:51:08,600 --> 00:51:14,270 Now, that might seem a little bit weird, but I'm going to convince you that this is something that you know very well. 431 00:51:14,480 --> 00:51:17,540 So imagine you have a crystal like this. 432 00:51:18,290 --> 00:51:23,720 Then you think of there's really this is a bunch of lattice planes, it's a family of lattice planes. 433 00:51:24,320 --> 00:51:28,610 And you send some wave into a family of lattice planes and it bounces off. 434 00:51:28,610 --> 00:51:34,430 But it can bounce off this layer, too. And you know very well that if you bounce off planes like this, you can get diffraction. 435 00:51:34,670 --> 00:51:37,850 And diffraction only occurs at certain angles. 436 00:51:37,850 --> 00:51:40,790 You only get constructive interference at certain angles. 437 00:51:40,790 --> 00:51:47,179 And what we're going to see is that this Laue condition is actually exactly the same condition as having constructive interference. 438 00:51:47,180 --> 00:51:50,000 And we'll do that next time I see you Wednesday.