1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,840 Commons license. 3 00:00:03,840 --> 00:00:06,840 Your support will help MIT Open Courseware continue to 4 00:00:06,840 --> 00:00:10,520 offer high-quality educational resources for free. 5 00:00:10,520 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,430 hundreds of MIT courses, visit MIT Open Courseware at 7 00:00:17,430 --> 00:00:18,680 ocw.mit.edu. 8 00:00:22,230 --> 00:00:23,260 Hi there. 9 00:00:23,260 --> 00:00:23,710 Welcome. 10 00:00:23,710 --> 00:00:24,590 I'm Brian Spatocco. 11 00:00:24,590 --> 00:00:26,570 We're going to be going over exam 2 from 12 00:00:26,570 --> 00:00:29,590 the fall 2009 semester. 13 00:00:29,590 --> 00:00:33,420 The logical place to start on this exam is problem 1. 14 00:00:33,420 --> 00:00:36,690 And what I usually do for all the problems is discuss the 15 00:00:36,690 --> 00:00:38,740 things we need to know before we start it. 16 00:00:38,740 --> 00:00:43,130 So I would recommend reviewing these concepts before 17 00:00:43,130 --> 00:00:44,890 attempting the problem. 18 00:00:44,890 --> 00:00:46,510 And then attempt the problem and then 19 00:00:46,510 --> 00:00:47,960 identify your weak spots. 20 00:00:47,960 --> 00:00:52,030 So here's what I personally found to be important 21 00:00:52,030 --> 00:00:54,300 information to know before attempting the problem. 22 00:00:54,300 --> 00:00:57,810 So I call it what I need, (W I N), if you want to win. 23 00:00:57,810 --> 00:00:58,870 There's four things. 24 00:00:58,870 --> 00:01:02,230 The first are Miller Indices. 25 00:01:02,230 --> 00:01:06,170 This is a crystallographic notation for figuring out and 26 00:01:06,170 --> 00:01:08,840 communicating planes and directions in crystals. 27 00:01:08,840 --> 00:01:10,840 So review Miller Indices 28 00:01:10,840 --> 00:01:12,420 We have plane/direction notation. 29 00:01:12,420 --> 00:01:15,870 I'll explain a little bit more what that is in a second. 30 00:01:15,870 --> 00:01:17,150 Definition of planar density. 31 00:01:17,150 --> 00:01:19,460 And for that matter, linear and 32 00:01:19,460 --> 00:01:21,380 three-dimensional density as well. 33 00:01:21,380 --> 00:01:23,630 And 4, the concept of a crystal basis. 34 00:01:23,630 --> 00:01:25,700 And this is actually the concept that most students had 35 00:01:25,700 --> 00:01:28,250 trouble with on this particular question. 36 00:01:28,250 --> 00:01:31,520 So those are the things I would recommend you look at 37 00:01:31,520 --> 00:01:32,070 before starting. 38 00:01:32,070 --> 00:01:35,280 And if you haven't, check it out now. 39 00:01:35,280 --> 00:01:39,060 So let's start off with part a of problem 1. 40 00:01:39,060 --> 00:01:42,650 And so the question asks us to draw the crystallographic 41 00:01:42,650 --> 00:01:44,410 feature indicated and label it clearly. 42 00:01:44,410 --> 00:01:46,030 It doesn't actually say if we're drawing planes or 43 00:01:46,030 --> 00:01:49,780 directions and that's left to the exam taker to identify. 44 00:01:49,780 --> 00:01:54,090 So the first thing I note on that problem, part a, is that 45 00:01:54,090 --> 00:01:59,580 we're asked to draw 0, 1 bar, 2 bar. 46 00:01:59,580 --> 00:02:01,300 And 1 1 1 bar. 47 00:02:12,060 --> 00:02:14,980 So the first thing I notice about these two particular 48 00:02:14,980 --> 00:02:17,190 crystallographic features is how they're actually 49 00:02:17,190 --> 00:02:17,840 encapsulated. 50 00:02:17,840 --> 00:02:19,720 So the first one is in parentheses. 51 00:02:19,720 --> 00:02:21,030 The next one is in square brackets. 52 00:02:21,030 --> 00:02:24,410 That actually tells us some information about the problem. 53 00:02:24,410 --> 00:02:27,940 The first part, a part one, is 0 1 bar 2 bar. 54 00:02:27,940 --> 00:02:29,940 So you actually need to know what this implies. 55 00:02:29,940 --> 00:02:33,240 Parentheses mean we're looking at a particular plane 56 00:02:33,240 --> 00:02:34,060 orientation. 57 00:02:34,060 --> 00:02:35,400 Now I'm not saying a plane. 58 00:02:35,400 --> 00:02:36,700 I'm saying a plane orientation. 59 00:02:36,700 --> 00:02:38,850 Which means there are infinitely many planes in a 60 00:02:38,850 --> 00:02:41,270 crystal which have this orientation. 61 00:02:41,270 --> 00:02:44,420 OK so that's distinguishing between just 62 00:02:44,420 --> 00:02:46,350 one plane and a crystal. 63 00:02:46,350 --> 00:02:48,710 The second part, b, we have square brackets. 64 00:02:48,710 --> 00:02:50,920 That implies we're either looking at a direction or a 65 00:02:50,920 --> 00:02:52,420 place in a crystal. 66 00:02:52,420 --> 00:02:55,480 We also have things like this, which we're not seeing on the 67 00:02:55,480 --> 00:02:57,200 exam this time but may show up later. 68 00:03:00,280 --> 00:03:02,480 We have these curly brackets and we have 69 00:03:02,480 --> 00:03:04,470 these angled brackets. 70 00:03:04,470 --> 00:03:07,010 The curly brackets denote a family of planes. 71 00:03:07,010 --> 00:03:10,070 And the angled brackets denote a family of directions. 72 00:03:10,070 --> 00:03:14,230 And it's markedly different than a particular or plane 73 00:03:14,230 --> 00:03:16,560 orientation and a particular direction. 74 00:03:16,560 --> 00:03:20,330 So let's do some visuals here to explain what we mean. 75 00:03:20,330 --> 00:03:24,220 So let's start off with a 1. 76 00:03:24,220 --> 00:03:27,250 Now we're asked for 0 1 bar 2 bar. 77 00:03:27,250 --> 00:03:30,170 Now the bars indicate negative signs. 78 00:03:30,170 --> 00:03:33,070 So you would know that from your Miller Indices review. 79 00:03:33,070 --> 00:03:35,430 Now the way to do this problem-- 80 00:03:35,430 --> 00:03:38,540 what I find is the easiest way we reviewed in recitation-- 81 00:03:38,540 --> 00:03:41,430 was to not be afraid to move your origin. 82 00:03:41,430 --> 00:03:43,200 A lot of people are afraid to move their origin around but 83 00:03:43,200 --> 00:03:45,250 we have to remember in a crystal, basically what a 84 00:03:45,250 --> 00:03:48,890 crystal is, it's a unit lattice repeated infinitely in 85 00:03:48,890 --> 00:03:49,460 three dimensions. 86 00:03:49,460 --> 00:03:51,920 So it's OK to move your origin as long as you choose some 87 00:03:51,920 --> 00:03:52,830 reasonable point. 88 00:03:52,830 --> 00:03:55,090 So in this problem we're looking in the back bottom 89 00:03:55,090 --> 00:03:55,600 left corner. 90 00:03:55,600 --> 00:03:58,250 That's our defined origin on the paper. 91 00:03:58,250 --> 00:04:00,020 But to do this problem, I would recommend 92 00:04:00,020 --> 00:04:01,810 choosing a new origin. 93 00:04:01,810 --> 00:04:04,410 I chose this one. 94 00:04:04,410 --> 00:04:06,170 And that's really going to facilitate doing this problem 95 00:04:06,170 --> 00:04:08,130 a lot more easily. 96 00:04:08,130 --> 00:04:10,280 So let's look at what we have to dran now. 97 00:04:10,280 --> 00:04:11,790 0 1 bar 2 bar. 98 00:04:11,790 --> 00:04:15,730 The 0 indicates that we're never actually intersecting 99 00:04:15,730 --> 00:04:17,020 the x-axis. 100 00:04:17,020 --> 00:04:19,680 OK, so at no point will this plane-- 101 00:04:19,680 --> 00:04:21,710 we know it's a plane because the parentheses-- 102 00:04:21,710 --> 00:04:24,500 intersect the x-axis. 103 00:04:24,500 --> 00:04:29,730 The 1 bar indicates that we are intersecting the y-axis at 104 00:04:29,730 --> 00:04:32,130 1 lattice parameter spacing. 105 00:04:35,050 --> 00:04:36,550 Miller Indices are a little tricky because they're 106 00:04:36,550 --> 00:04:37,270 somewhat inverted. 107 00:04:37,270 --> 00:04:38,520 1 implies you are going the full 108 00:04:38,520 --> 00:04:40,530 distance of the unit cell. 109 00:04:40,530 --> 00:04:43,820 2 implies you're going half the distance. 110 00:04:43,820 --> 00:04:46,530 3 implies you are going 1/3 the distance and 111 00:04:46,530 --> 00:04:48,180 so on and so forth. 112 00:04:48,180 --> 00:04:51,353 So for 1, we're going to go the distance of 1. 113 00:04:55,610 --> 00:04:59,060 We're going to move negative 1 in y. 114 00:04:59,060 --> 00:05:00,340 So we're going to move negative 1. 115 00:05:00,340 --> 00:05:01,410 We're going to go to here. 116 00:05:01,410 --> 00:05:03,010 That's in our y. 117 00:05:03,010 --> 00:05:07,120 And when we actually make this a little clearer. 118 00:05:07,120 --> 00:05:08,640 I've chosen a new origin. 119 00:05:08,640 --> 00:05:11,930 So here's my y prime. 120 00:05:11,930 --> 00:05:14,110 Here's my z prime. 121 00:05:14,110 --> 00:05:16,700 Here's my x prime. 122 00:05:16,700 --> 00:05:20,020 So we're moving negative 1 and y. 123 00:05:20,020 --> 00:05:24,320 We're going to move half, negative 1/2 and z. 124 00:05:24,320 --> 00:05:26,090 And we're also told that this plane doesn't 125 00:05:26,090 --> 00:05:27,120 intersect the x-axis. 126 00:05:27,120 --> 00:05:29,270 So it must run parallel to that asix. 127 00:05:29,270 --> 00:05:33,000 Which tells me that it's going to run along the x-axis. 128 00:05:33,000 --> 00:05:37,042 We're going to get a plane like this. 129 00:05:37,042 --> 00:05:38,292 And here's our plane. 130 00:05:40,880 --> 00:05:43,220 That's the answer to part a 1. 131 00:05:43,220 --> 00:05:46,890 Pretty easy and most students got this correct. 132 00:05:46,890 --> 00:05:52,340 For part a 2 we're asked to draw a direction or point to a 133 00:05:52,340 --> 00:05:54,440 particular place in space. 134 00:05:54,440 --> 00:05:57,390 I also moved my origin on this problem. 135 00:05:57,390 --> 00:06:02,100 And I found it easiest to redefine my origin up here. 136 00:06:02,100 --> 00:06:06,370 Which means that my new set of axes are going to 137 00:06:06,370 --> 00:06:09,630 be x prime, y prime. 138 00:06:09,630 --> 00:06:11,900 I'm going to keep z because z didn't move at all. 139 00:06:11,900 --> 00:06:15,170 OK note how, when I'm drawing these things-- 140 00:06:15,170 --> 00:06:18,490 we're going to draw this one now-- 141 00:06:18,490 --> 00:06:23,420 that the way I choose to move my origin generally tracks 142 00:06:23,420 --> 00:06:26,380 with the negative signs that I have. This problem I had a 143 00:06:26,380 --> 00:06:28,860 negative y and negative z, I moved in the negative y, 144 00:06:28,860 --> 00:06:30,600 negative z directions. 145 00:06:30,600 --> 00:06:31,780 Here I had a negative z. 146 00:06:31,780 --> 00:06:32,910 So I just moved up in the negative z. 147 00:06:32,910 --> 00:06:34,230 That kind of told me how to move my origin. 148 00:06:34,230 --> 00:06:36,050 It made life a little easier. 149 00:06:36,050 --> 00:06:38,910 And now it's pretty easy from here. 150 00:06:38,910 --> 00:06:42,742 Now what we're going to do is draw 1 in the x, we'll go 1 in 151 00:06:42,742 --> 00:06:45,450 the y, we'll go negative 1 in z. 152 00:06:45,450 --> 00:06:48,510 We're pointing to this point. 153 00:06:48,510 --> 00:06:49,380 There we go. 154 00:06:49,380 --> 00:06:50,880 That's our direction. 155 00:06:50,880 --> 00:06:51,860 So pretty easy. 156 00:06:51,860 --> 00:06:53,580 Part a 1. 157 00:06:53,580 --> 00:06:54,940 Part a 2. 158 00:06:54,940 --> 00:06:57,460 So we're batting 100 right now. 159 00:06:57,460 --> 00:06:59,630 And just don't be afraid to move your origin. 160 00:06:59,630 --> 00:07:02,990 That's the take-away from this part. 161 00:07:02,990 --> 00:07:05,290 The problem, of course, gets a little harder. 162 00:07:05,290 --> 00:07:08,360 That's something that we pride ourselves on in 3.091. 163 00:07:08,360 --> 00:07:11,375 If it was always this easy, everyone would pass the class 164 00:07:11,375 --> 00:07:12,530 and get 100s. 165 00:07:12,530 --> 00:07:17,200 So the second part, part b, is asking us to calculate the 166 00:07:17,200 --> 00:07:18,710 density of atoms in a plane. 167 00:07:18,710 --> 00:07:20,240 So this gets back to what I was saying before. 168 00:07:20,240 --> 00:07:20,920 We need to understand the 169 00:07:20,920 --> 00:07:22,800 definition of a planar density. 170 00:07:22,800 --> 00:07:24,470 OK, so I've sort of defined it here. 171 00:07:24,470 --> 00:07:25,810 This is rho. 172 00:07:25,810 --> 00:07:28,650 Not p. rho planar. 173 00:07:28,650 --> 00:07:30,770 And rho planar, I define-- 174 00:07:30,770 --> 00:07:32,800 to help me, maybe it'll help you-- 175 00:07:32,800 --> 00:07:36,780 is the number of full atoms in a particular plane-- 176 00:07:36,780 --> 00:07:39,130 in this case we're looking at 0 0 1-- 177 00:07:39,130 --> 00:07:40,650 divided by some area. 178 00:07:40,650 --> 00:07:43,280 Because the plane is denoted by an area not a volume. 179 00:07:43,280 --> 00:07:45,350 Don't be afraid, I mean densities don't always have to 180 00:07:45,350 --> 00:07:46,580 be mass over volume. 181 00:07:46,580 --> 00:07:51,520 It can be some unit over some specific 182 00:07:51,520 --> 00:07:53,330 distance or area or volume. 183 00:07:53,330 --> 00:07:55,750 So that's general density. 184 00:07:55,750 --> 00:07:57,590 So we're looking at the number of full items in a plane 185 00:07:57,590 --> 00:07:58,730 divided by the area. 186 00:07:58,730 --> 00:08:01,420 We're told that we're looking at Dalium, of course named 187 00:08:01,420 --> 00:08:03,770 after Salvador Dali. 188 00:08:03,770 --> 00:08:05,640 And so we're also told that we're looking 189 00:08:05,640 --> 00:08:06,960 at the 0 0 1 plane. 190 00:08:06,960 --> 00:08:10,250 I've drawn a bcc crystal here. 191 00:08:10,250 --> 00:08:14,220 As a scientist or an engineer, when you're given a bcc fcc, 192 00:08:14,220 --> 00:08:17,620 simple cubic, the first thing you put on that paper is the 193 00:08:17,620 --> 00:08:19,500 actual crystal structure. 194 00:08:19,500 --> 00:08:21,860 So I drew the Dalium up here. 195 00:08:21,860 --> 00:08:25,100 Not a real element, but a make-believe one. 196 00:08:25,100 --> 00:08:28,360 And we're going to identify our 0 0 1 plane. 197 00:08:28,360 --> 00:08:30,090 This is good practice. 198 00:08:30,090 --> 00:08:34,670 So there's our x, y and our z is right here. 199 00:08:34,670 --> 00:08:38,030 Our 0 0 1 plane, so we're not intersecting x. 200 00:08:38,030 --> 00:08:39,220 We're not intersecting y. 201 00:08:39,220 --> 00:08:41,090 We're only intersecting z. 202 00:08:41,090 --> 00:08:42,710 So here's our 0 0 1 plane. 203 00:08:46,720 --> 00:08:49,660 Let me sort of bring this down for you. 204 00:08:49,660 --> 00:08:51,500 So we can look down on it from above. 205 00:08:51,500 --> 00:08:55,250 It's a square, with sides a. 206 00:08:55,250 --> 00:09:03,070 And here's our atoms. So now the question is, how many full 207 00:09:03,070 --> 00:09:05,080 atoms do we have and what's the area? 208 00:09:05,080 --> 00:09:07,030 The area's obviously a squared. 209 00:09:07,030 --> 00:09:10,490 And the number of full atoms in this area is we have four 210 00:09:10,490 --> 00:09:11,690 quarters of an atom. 211 00:09:11,690 --> 00:09:14,270 When you're in two-dimensional space an atom looks like a 212 00:09:14,270 --> 00:09:15,280 circle to you. 213 00:09:15,280 --> 00:09:16,470 When you're in three-dimensional space it 214 00:09:16,470 --> 00:09:17,680 looks like a sphere. 215 00:09:17,680 --> 00:09:20,000 So if you lived in the two-dimensional plane, you 216 00:09:20,000 --> 00:09:22,030 would see a quarter of an atom, a quarter of an 217 00:09:22,030 --> 00:09:22,850 atom, and so on. 218 00:09:22,850 --> 00:09:24,810 You'd have one full atom in this plane. 219 00:09:24,810 --> 00:09:26,660 So this is sort of what we're looking for. 220 00:09:26,660 --> 00:09:27,940 And you say, oh, well I'm done. 221 00:09:27,940 --> 00:09:30,180 A lot people left it at that. 222 00:09:30,180 --> 00:09:31,840 But that's not the answer we're looking for. 223 00:09:31,840 --> 00:09:33,340 We want an actual numerical answer. 224 00:09:33,340 --> 00:09:37,550 So the question now becomes, what is a? 225 00:09:37,550 --> 00:09:39,980 So we're going to digress and find a and then we'll know 226 00:09:39,980 --> 00:09:42,780 exactly the answer to the planar density. 227 00:09:42,780 --> 00:09:44,620 So a-- 228 00:09:44,620 --> 00:09:46,500 this is where you have to be a little clever. 229 00:09:46,500 --> 00:09:47,710 How do you find a? 230 00:09:47,710 --> 00:09:50,350 The lattice parameter, the lattice spacing. 231 00:09:50,350 --> 00:09:54,620 So the best way to do this is to think both small scale and 232 00:09:54,620 --> 00:09:55,550 large scale. 233 00:09:55,550 --> 00:09:59,720 So first let's think about a unit cell. 234 00:09:59,720 --> 00:10:02,510 Let me ask you, what's the three-dimensional 235 00:10:02,510 --> 00:10:05,390 density of a unit cell. 236 00:10:05,390 --> 00:10:09,300 So how many atoms do we have per unit cell? 237 00:10:09,300 --> 00:10:11,130 We have our bcc. 238 00:10:11,130 --> 00:10:13,500 We know in bcc there are two atoms. We have eight corner 239 00:10:13,500 --> 00:10:15,050 atoms and they're each 1/8. 240 00:10:15,050 --> 00:10:17,370 We have one body atom. 241 00:10:17,370 --> 00:10:20,640 So there's two atoms. So we have 2 atoms per-- 242 00:10:20,640 --> 00:10:22,010 what's the volume-- 243 00:10:22,010 --> 00:10:23,522 well the volume is a cubed. 244 00:10:27,460 --> 00:10:29,350 Now we got a cubed. 245 00:10:29,350 --> 00:10:31,170 We didn't really make any progress here, we still have 246 00:10:31,170 --> 00:10:31,970 an unknown variable. 247 00:10:31,970 --> 00:10:34,560 But the question is, how do we get solved for a cubed. 248 00:10:34,560 --> 00:10:36,700 So let's zoom out from that unit cell. 249 00:10:36,700 --> 00:10:39,500 Let's zoom out to a mole of material. 250 00:10:39,500 --> 00:10:42,610 OK, so a mole of material, how many atoms are 251 00:10:42,610 --> 00:10:43,700 in a mole of material? 252 00:10:43,700 --> 00:10:46,350 Well by definition, it's Avogadro's Number. 253 00:10:46,350 --> 00:10:49,320 6.02 times 10 to the 23. 254 00:10:49,320 --> 00:10:54,120 And then what's the volume of a mole of material? 255 00:10:54,120 --> 00:10:55,530 Well looking back at the problem, we're 256 00:10:55,530 --> 00:10:56,620 given the molar volume. 257 00:10:56,620 --> 00:10:57,830 Which is something that I would 258 00:10:57,830 --> 00:10:59,230 recommend you write down. 259 00:10:59,230 --> 00:11:01,230 You know all the information you're given, 260 00:11:01,230 --> 00:11:02,780 see what you've got. 261 00:11:02,780 --> 00:11:05,200 The volume of a mole is the molar 262 00:11:05,200 --> 00:11:08,250 volume, which we're given. 263 00:11:08,250 --> 00:11:09,450 So now it's pretty easy. 264 00:11:09,450 --> 00:11:12,510 We know Avogadro's, we know molar volume. 265 00:11:12,510 --> 00:11:13,830 So we can solve for a. 266 00:11:29,000 --> 00:11:30,430 So not too bad. 267 00:11:30,430 --> 00:11:36,570 You'll find that a is about 2.8 times 10 to the negative 8 268 00:11:36,570 --> 00:11:39,250 centimeters. 269 00:11:39,250 --> 00:11:43,020 You could give me 2.85, 2.79. 270 00:11:43,020 --> 00:11:44,420 We're worried about the concepts 271 00:11:44,420 --> 00:11:45,620 here, not the numbers. 272 00:11:45,620 --> 00:11:47,110 So don't go back and waste a lot of time 273 00:11:47,110 --> 00:11:48,700 getting the exact number. 274 00:11:48,700 --> 00:11:49,690 That's our a. 275 00:11:49,690 --> 00:11:52,450 And now that we've got a, we've got our planar density. 276 00:11:52,450 --> 00:11:55,390 Very easy. 277 00:11:55,390 --> 00:11:59,370 I'm going to write rho plan, just to save some time. 278 00:11:59,370 --> 00:12:02,235 And that equals 1 over this number squared. 279 00:12:06,900 --> 00:12:08,350 That's one atom, by the way. 280 00:12:10,960 --> 00:12:12,860 And that's all squared. 281 00:12:12,860 --> 00:12:16,950 And we're going to find in the end that our planar density is 282 00:12:16,950 --> 00:12:22,870 going to be equal to 1.27 times 10 to the 15. 283 00:12:22,870 --> 00:12:25,220 And the units are important. 284 00:12:25,220 --> 00:12:29,335 That's atoms per centimeter, squared. 285 00:12:29,335 --> 00:12:31,670 If you don't give us the units then we have no idea if you're 286 00:12:31,670 --> 00:12:32,710 right or wrong. 287 00:12:32,710 --> 00:12:35,450 So always include your units at the end. 288 00:12:35,450 --> 00:12:36,950 So hopefully that helped. 289 00:12:36,950 --> 00:12:39,070 We're going to take a second and clean off the boards. 290 00:12:39,070 --> 00:12:40,320 And we're going to start with part c. 291 00:12:43,670 --> 00:12:45,390 OK we're back. 292 00:12:45,390 --> 00:12:49,650 We're going to finish problem 1 now from the 2009 exam 2. 293 00:12:49,650 --> 00:12:51,330 We're on part c. 294 00:12:51,330 --> 00:12:55,190 So thus far in part a, we've talked about Miller Indices, 295 00:12:55,190 --> 00:12:58,380 we've talked about the plane and direction notation with 296 00:12:58,380 --> 00:13:00,260 the parentheses and the brackets. 297 00:13:00,260 --> 00:13:01,990 We've also talked about planar density. 298 00:13:01,990 --> 00:13:04,010 And I would invite you to review those things. 299 00:13:04,010 --> 00:13:05,960 Now we're going to go to part c, which is really the concept 300 00:13:05,960 --> 00:13:07,170 of the crystal basis. 301 00:13:07,170 --> 00:13:09,840 And this was the part of the problem that most students 302 00:13:09,840 --> 00:13:10,680 lost points on. 303 00:13:10,680 --> 00:13:12,890 So it's the thing that I would emphasize the 304 00:13:12,890 --> 00:13:15,150 most on this problem. 305 00:13:15,150 --> 00:13:17,530 The problem asks us to-- 306 00:13:17,530 --> 00:13:20,840 and I put a sad face because that's how people felt after 307 00:13:20,840 --> 00:13:22,770 this part-- 308 00:13:22,770 --> 00:13:27,320 the problem asks us to draw a particular 309 00:13:27,320 --> 00:13:29,830 plane in a fcc crystal. 310 00:13:29,830 --> 00:13:32,190 So we're given Magnesia, MgO. 311 00:13:32,190 --> 00:13:35,600 They want us to draw that plane and show the atoms. And 312 00:13:35,600 --> 00:13:38,670 we also want you to show the relative sizes of the atoms. 313 00:13:38,670 --> 00:13:40,650 You know we have anions and cations. 314 00:13:40,650 --> 00:13:43,760 And we actually give you the sizes of those ions, so it 315 00:13:43,760 --> 00:13:45,660 shouldn't be too tricky on that part. 316 00:13:45,660 --> 00:13:49,890 But I think the best way to start is to go over the answer 317 00:13:49,890 --> 00:13:51,400 that was given that was wrong. 318 00:13:51,400 --> 00:13:53,300 And then we're going to talk about why it was wrong and how 319 00:13:53,300 --> 00:13:54,660 to get the right answer. 320 00:13:54,660 --> 00:13:58,690 So to start off, we want you to draw this crystal. 321 00:13:58,690 --> 00:14:02,130 The best place to start is by actually drawing the crystal 322 00:14:02,130 --> 00:14:03,590 in 3-D and then we're going to take the 323 00:14:03,590 --> 00:14:05,650 projection after that. 324 00:14:05,650 --> 00:14:10,320 So people think, MgO, they see fcc, and they say, oh that's 325 00:14:10,320 --> 00:14:11,020 not so bad. 326 00:14:11,020 --> 00:14:12,860 I know what fcc looks like. 327 00:14:12,860 --> 00:14:14,190 Let me draw fcc. 328 00:14:14,190 --> 00:14:19,430 I'm going to use for this problem, I'm going to use this 329 00:14:19,430 --> 00:14:22,235 red chalk to represent magnesium and the blue chalk 330 00:14:22,235 --> 00:14:24,150 to represent oxygen. 331 00:14:24,150 --> 00:14:26,310 OK so magnesium, we're going to have it as 332 00:14:26,310 --> 00:14:28,350 a circle like this. 333 00:14:28,350 --> 00:14:32,362 This is magnesium, 2 plus. 334 00:14:32,362 --> 00:14:34,850 And we're going to have the oxygen-- 335 00:14:34,850 --> 00:14:37,510 it's going to be slightly bigger because it's an anion. 336 00:14:37,510 --> 00:14:41,850 This is O 2 minus. 337 00:14:41,850 --> 00:14:45,190 Maybe I'll color this in too. 338 00:14:45,190 --> 00:14:47,880 People said, OK easy. fcc. 339 00:14:47,880 --> 00:14:50,380 This is a simple problem, easy points. 340 00:14:50,380 --> 00:14:52,895 They drew what they thought to be an fcc crystal. 341 00:14:52,895 --> 00:14:53,960 So they did-- 342 00:14:53,960 --> 00:14:56,575 let's say they put the Mgs on the corners. 343 00:14:59,600 --> 00:15:01,470 And they said, OK now I need my faces. 344 00:15:01,470 --> 00:15:04,040 So maybe the oxygen are the faces. 345 00:15:04,040 --> 00:15:07,840 So I'm going to put in my 6 face atoms. So here are my 1 346 00:15:07,840 --> 00:15:11,490 side here, and I've got the bottom and the top. 347 00:15:11,490 --> 00:15:13,200 There's my 6 faces. 348 00:15:13,200 --> 00:15:14,440 And they said, OK, easy. 349 00:15:14,440 --> 00:15:15,290 I'm done. 350 00:15:15,290 --> 00:15:19,380 And then they took the projection of 0 1 1. 351 00:15:19,380 --> 00:15:22,230 We also know, when you put the axes on here 352 00:15:22,230 --> 00:15:24,720 just to make it clear. 353 00:15:24,720 --> 00:15:30,990 And here's our z, You know 0 1 1, we know it doesn't 354 00:15:30,990 --> 00:15:35,110 intersect the x but it intersects the y and the z at 355 00:15:35,110 --> 00:15:36,580 one lattice spacing. 356 00:15:36,580 --> 00:15:38,370 So x doesn't intersect. 357 00:15:38,370 --> 00:15:41,000 It intersects z here, intersects y here's. 358 00:15:41,000 --> 00:15:44,750 We've got this kind of plane going on here. 359 00:15:44,750 --> 00:15:45,850 We've got something like this. 360 00:15:45,850 --> 00:15:47,200 And they just drew this. 361 00:15:47,200 --> 00:15:49,430 It's a little bit of 3-D visualization But they drew 362 00:15:49,430 --> 00:15:52,010 that and they got 0 points. 363 00:15:52,010 --> 00:15:55,350 The reason that that is not a correct answer is because that 364 00:15:55,350 --> 00:15:58,230 is not what MgO-- it's rock salt structure-- that's not 365 00:15:58,230 --> 00:15:59,340 what it looks like. 366 00:15:59,340 --> 00:16:01,260 And the reason they got it wrong is because they didn't 367 00:16:01,260 --> 00:16:03,650 understand the concept of the basis. 368 00:16:03,650 --> 00:16:06,780 So I'm going to make a statement right now. 369 00:16:06,780 --> 00:16:09,410 This is a really important statements so I urge you to 370 00:16:09,410 --> 00:16:11,760 think about it. 371 00:16:11,760 --> 00:16:21,430 A crystal equals a lattice plus a basis. 372 00:16:24,450 --> 00:16:25,700 Here's the difference. 373 00:16:25,700 --> 00:16:28,110 A lattice is a collection of points. 374 00:16:28,110 --> 00:16:30,670 So you have an fcc lattice, you have a bcc lattice, you 375 00:16:30,670 --> 00:16:32,200 have a simple cubic lattice. 376 00:16:32,200 --> 00:16:33,900 Those are points where atoms-- 377 00:16:33,900 --> 00:16:37,040 I use atoms plurally, perhaps-- 378 00:16:37,040 --> 00:16:38,150 can exist. 379 00:16:38,150 --> 00:16:41,360 OK, a basis I like to think of as a stamp. 380 00:16:41,360 --> 00:16:44,210 A basis represents the most fundamental pattern that you 381 00:16:44,210 --> 00:16:46,180 stamp at every lattice point. 382 00:16:46,180 --> 00:16:51,070 So let's draw an fcc lattice. 383 00:16:51,070 --> 00:16:52,870 So I'm going to drawn an fcc lattice on 384 00:16:52,870 --> 00:16:55,370 this cube right here. 385 00:16:55,370 --> 00:16:56,360 I'm going to put points. 386 00:16:56,360 --> 00:16:57,510 So I'm going to make them white. 387 00:16:57,510 --> 00:17:00,060 They're just points. 388 00:17:00,060 --> 00:17:01,272 I got all the corners. 389 00:17:01,272 --> 00:17:02,130 They're hard to see. 390 00:17:02,130 --> 00:17:03,600 But they're all the corners. 391 00:17:03,600 --> 00:17:05,870 I got the faces. 392 00:17:05,870 --> 00:17:09,200 Side, back, front and bottom. 393 00:17:09,200 --> 00:17:10,620 You're saying to yourself, that's like the same 394 00:17:10,620 --> 00:17:12,050 thing you just drew. 395 00:17:12,050 --> 00:17:13,470 It's really not. 396 00:17:13,470 --> 00:17:16,590 What I drew the first time, I put atoms in. 397 00:17:16,590 --> 00:17:18,350 That's a crystal I drew. 398 00:17:18,350 --> 00:17:20,460 I just a lattice right now. 399 00:17:20,460 --> 00:17:22,440 This isn't a crystal, this is a lattice. 400 00:17:22,440 --> 00:17:24,440 So to actually answer the question, we want to figure 401 00:17:24,440 --> 00:17:28,030 out, we want to draw the 2-D projection of a crystal. 402 00:17:28,030 --> 00:17:29,680 So we've got to draw the crystal eventually. 403 00:17:29,680 --> 00:17:31,280 Now we're dealing with MgO. 404 00:17:31,280 --> 00:17:33,240 We're told MgO is face-centered cubic. 405 00:17:33,240 --> 00:17:37,240 Which means that our basis must have 1 Mg and 1 O atom. 406 00:17:37,240 --> 00:17:39,980 So I'm going to do that right now. 407 00:17:39,980 --> 00:17:42,630 I'm going to draw what I'm going to call to be my stamp. 408 00:17:42,630 --> 00:17:45,020 You can use whatever method you think is best. But I call 409 00:17:45,020 --> 00:17:46,320 this my stamp. 410 00:17:46,320 --> 00:17:47,570 I'm going to draw it like this. 411 00:18:00,810 --> 00:18:02,150 So here's an oxygen atom. 412 00:18:02,150 --> 00:18:03,150 Here's a Magnesium atom. 413 00:18:03,150 --> 00:18:06,000 And this little white dot is the point that's 414 00:18:06,000 --> 00:18:07,140 going to get stamped. 415 00:18:07,140 --> 00:18:07,800 And here's my stamp. 416 00:18:07,800 --> 00:18:10,460 And the idea is I'm going to take the stamp, this basis-- 417 00:18:10,460 --> 00:18:11,920 that's what it is, it's a basis-- 418 00:18:11,920 --> 00:18:14,550 and I'm going to stamp it at every lattice point. 419 00:18:14,550 --> 00:18:17,580 OK so note I have 8 corner lattice points and I have 6 420 00:18:17,580 --> 00:18:21,000 face lattice points so I'm going to be stamping 14 times. 421 00:18:21,000 --> 00:18:25,210 So I'll do the first couple very slowly. 422 00:18:25,210 --> 00:18:28,500 I'll take my stamp and I'll stamp here. 423 00:18:28,500 --> 00:18:30,130 Here is my O. 424 00:18:30,130 --> 00:18:32,630 Here's my Mg. 425 00:18:32,630 --> 00:18:35,200 I'll stamp over at this corner. 426 00:18:35,200 --> 00:18:37,460 Here's my O, here's my Mg. 427 00:18:37,460 --> 00:18:38,890 I'll stamp in the face. 428 00:18:38,890 --> 00:18:40,840 Here's the face, the front face one. 429 00:18:40,840 --> 00:18:44,070 Here's my O, here's my Mg. 430 00:18:44,070 --> 00:18:46,332 So you can see what's happening. 431 00:18:46,332 --> 00:18:48,910 Let me just circle it for you. 432 00:18:48,910 --> 00:18:50,060 I'll circle one of them. 433 00:18:50,060 --> 00:18:51,310 Here's my stamp. 434 00:18:53,720 --> 00:18:55,790 I'm going to go ahead, I'm going to finish 435 00:18:55,790 --> 00:18:56,490 drawing this in. 436 00:18:56,490 --> 00:18:59,030 It's going to get a little messy for a board. 437 00:18:59,030 --> 00:19:01,420 But I hope that you can go home and do it on some graph 438 00:19:01,420 --> 00:19:02,670 paper as well. 439 00:19:07,000 --> 00:19:08,716 All four corners. 440 00:19:08,716 --> 00:19:10,030 The face is here. 441 00:19:13,090 --> 00:19:15,600 I have a tendency to leave off faces sometimes. 442 00:19:15,600 --> 00:19:17,456 Let me make sure I've got them all. 443 00:19:17,456 --> 00:19:18,706 I already did that one. 444 00:19:24,470 --> 00:19:25,380 Yep that's right. 445 00:19:25,380 --> 00:19:25,870 Is that right? 446 00:19:25,870 --> 00:19:28,210 No. 447 00:19:28,210 --> 00:19:33,480 OK, now let me finish with the magnesiums. The magnesiums-- 448 00:19:33,480 --> 00:19:35,440 it's going to be hard to see-- exist at the 449 00:19:35,440 --> 00:19:36,690 center of every edge. 450 00:19:42,120 --> 00:19:44,370 So it's a little tricky but let me explain what we're 451 00:19:44,370 --> 00:19:45,000 looking at. 452 00:19:45,000 --> 00:19:46,210 What you should be looking at. 453 00:19:46,210 --> 00:19:49,050 We have what looks to be oxygen in the face center 454 00:19:49,050 --> 00:19:50,210 cubic arrangement. 455 00:19:50,210 --> 00:19:53,740 And we have magnesium at all the edges. 456 00:19:53,740 --> 00:19:54,910 So you're thinking, this is weird this 457 00:19:54,910 --> 00:19:57,190 isn't face center cubic. 458 00:19:57,190 --> 00:19:58,010 Look at the magnesium, this doesn't 459 00:19:58,010 --> 00:19:59,360 look face center cubic. 460 00:19:59,360 --> 00:20:01,020 Well the thing I would challenge you to do at home is 461 00:20:01,020 --> 00:20:06,060 actually draw this unit cell out 4 more times. 462 00:20:06,060 --> 00:20:10,260 And then you will actually be able to find face center cubic 463 00:20:10,260 --> 00:20:15,790 magnesium lattic cell in that larger unit cell. 464 00:20:15,790 --> 00:20:17,320 So this is actually the answer. 465 00:20:17,320 --> 00:20:19,120 This is what a rock salt structure looks like. 466 00:20:19,120 --> 00:20:21,450 This is MgO. 467 00:20:21,450 --> 00:20:22,690 You have oxygen, looks like your 468 00:20:22,690 --> 00:20:24,240 traditional face center cubic. 469 00:20:24,240 --> 00:20:26,620 And you've got these magnesium stuck on all the edges. 470 00:20:26,620 --> 00:20:28,790 So now we're just going to take our projection. 471 00:20:28,790 --> 00:20:30,870 Let me use white chalk for that. 472 00:20:30,870 --> 00:20:32,420 Or rather we're going to take our plane. 473 00:20:32,420 --> 00:20:36,690 We're looking at the 0 1 1 plane. 474 00:20:36,690 --> 00:20:37,950 Here's our axes again. 475 00:20:37,950 --> 00:20:39,200 Remember it's always right-handed. 476 00:20:42,310 --> 00:20:43,850 We're going to do 0 1 1. 477 00:20:43,850 --> 00:20:45,090 Not intersecting the x and were going to 478 00:20:45,090 --> 00:20:47,530 intersect z and y. 479 00:20:47,530 --> 00:20:51,810 So we're going to come down across the face. 480 00:20:51,810 --> 00:20:52,940 We've got this. 481 00:20:52,940 --> 00:20:56,260 This is pretty terrible looking. 482 00:20:56,260 --> 00:20:58,180 But this is one of the tricks of the problem. 483 00:20:58,180 --> 00:21:00,310 You have to be able to visualize these in 3-D space. 484 00:21:00,310 --> 00:21:04,960 So let's slowly transcribe what this plane here looks 485 00:21:04,960 --> 00:21:07,240 like projected out on the board. 486 00:21:11,120 --> 00:21:16,690 We know that at our corners and also at the faces of this 487 00:21:16,690 --> 00:21:18,600 cube, we've got oxygen. 488 00:21:18,600 --> 00:21:19,580 So here's a corner. 489 00:21:19,580 --> 00:21:21,990 These are our corners. 490 00:21:21,990 --> 00:21:26,820 At the faces, two faces, we've got oxygen. 491 00:21:26,820 --> 00:21:28,030 What else do we have? 492 00:21:28,030 --> 00:21:29,600 We know we must be hitting some magnesium. 493 00:21:29,600 --> 00:21:31,710 Well we know that at all the edges-- 494 00:21:31,710 --> 00:21:33,860 here's an edge, here's an edge-- 495 00:21:33,860 --> 00:21:35,300 we've got some magnesium. 496 00:21:35,300 --> 00:21:36,610 What about the center? 497 00:21:36,610 --> 00:21:40,500 Well the center we do too because we know that when we 498 00:21:40,500 --> 00:21:45,010 made this stamp, the bottom face with this oxygen, we also 499 00:21:45,010 --> 00:21:47,560 stamped halfway up a magnesium. 500 00:21:47,560 --> 00:21:49,670 So there's actually a magnesium right here. 501 00:21:49,670 --> 00:21:50,420 It's there. 502 00:21:50,420 --> 00:21:51,650 It's right there. 503 00:21:51,650 --> 00:21:52,930 Believe me. 504 00:21:52,930 --> 00:21:55,180 There's our magnesium. 505 00:21:55,180 --> 00:21:56,440 And this is what we've got. 506 00:21:56,440 --> 00:21:58,070 This is the answer to the problem. 507 00:21:58,070 --> 00:22:01,690 Notice how I've giving my oxygens larger than my 508 00:22:01,690 --> 00:22:10,880 magnesiums and I've got it looking like this. 509 00:22:10,880 --> 00:22:13,890 In fact let me let me emphasize this even more. 510 00:22:13,890 --> 00:22:15,470 The fact that you could actually have drawn this 511 00:22:15,470 --> 00:22:16,710 differently. 512 00:22:16,710 --> 00:22:18,890 You could have drawn it like this. 513 00:22:18,890 --> 00:22:20,810 And it would have been exactly the same thing. 514 00:22:20,810 --> 00:22:22,220 You would have gotten full credit. 515 00:22:22,220 --> 00:22:23,750 You could have drawn this. 516 00:22:26,460 --> 00:22:32,840 This is the thing that I would recommend you try at home to 517 00:22:32,840 --> 00:22:34,090 prove that this is true. 518 00:22:39,380 --> 00:22:41,780 These two are equivalent It's the same thing. 519 00:22:41,780 --> 00:22:45,290 It all depends on what you choose as your stamp point. 520 00:22:45,290 --> 00:22:46,890 So I could have chosen to put my red at 521 00:22:46,890 --> 00:22:51,210 every point, my magnesium. 522 00:22:51,210 --> 00:22:53,170 So this is the answer to part c. 523 00:22:53,170 --> 00:22:55,360 This part, most people did not get correct. 524 00:22:55,360 --> 00:22:56,920 Less than half the class got this correct. 525 00:22:56,920 --> 00:22:59,080 So think about a little bit. 526 00:22:59,080 --> 00:23:01,830 Identify crystal as lattice plus basis. 527 00:23:01,830 --> 00:23:05,490 And then think about the full scope of the problem. 528 00:23:05,490 --> 00:23:08,260 What do we go over? when went over Miller Indices, went over 529 00:23:08,260 --> 00:23:11,390 designation and crystallographic notation, so 530 00:23:11,390 --> 00:23:13,870 plane and direction notation. 531 00:23:13,870 --> 00:23:16,350 We went over the definition of planar density. 532 00:23:16,350 --> 00:23:18,130 Think about what linear density means, think about 533 00:23:18,130 --> 00:23:19,650 what three-dimensional density means. 534 00:23:19,650 --> 00:23:21,790 Don't be afraid to not use mass. 535 00:23:21,790 --> 00:23:24,410 And the last thing we just covered was the crystal basis. 536 00:23:24,410 --> 00:23:27,700 So review those and hopefully you did well. 537 00:23:27,700 --> 00:23:28,950 Thanks.