1 00:00:01,000 --> 00:00:03,340 The following content is provided under a Creative 2 00:00:03,340 --> 00:00:04,760 Commons license. 3 00:00:04,760 --> 00:00:06,970 Your support will help MIT OpenCourseWare 4 00:00:06,970 --> 00:00:11,060 continue to offer high-quality educational resources for free. 5 00:00:11,060 --> 00:00:13,600 To make a donation or to view additional materials 6 00:00:13,600 --> 00:00:17,560 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,560 --> 00:00:18,920 at ocw.mit.edu. 8 00:00:21,967 --> 00:00:22,800 PROFESSOR: OK, guys. 9 00:00:22,800 --> 00:00:23,428 Welcome back. 10 00:00:23,428 --> 00:00:25,470 As you can see, we're not using the screen today. 11 00:00:25,470 --> 00:00:27,845 This is going to be one of those fill-the-board lectures. 12 00:00:27,845 --> 00:00:30,373 But I am going to work you through every single step. 13 00:00:30,373 --> 00:00:32,040 We're going to go through the Q equation 14 00:00:32,040 --> 00:00:34,673 and derive its most general form together, 15 00:00:34,673 --> 00:00:36,090 which, for the rest of this class, 16 00:00:36,090 --> 00:00:37,950 we'll be using simplified or reduced 17 00:00:37,950 --> 00:00:40,440 forms to explain a lot of the ion 18 00:00:40,440 --> 00:00:43,050 or electron-nuclear interactions as well 19 00:00:43,050 --> 00:00:44,902 as things like neutron scattering 20 00:00:44,902 --> 00:00:46,110 and all sorts of other stuff. 21 00:00:46,110 --> 00:00:47,280 We'll do one example. 22 00:00:47,280 --> 00:00:50,610 For any of you that have looked at neutrons slowing down 23 00:00:50,610 --> 00:00:52,410 before, how much energy can a neutron 24 00:00:52,410 --> 00:00:54,072 lose when it hits something? 25 00:00:54,072 --> 00:00:56,280 We'll be answering that question today in a generally 26 00:00:56,280 --> 00:00:57,270 mathematical form. 27 00:00:57,270 --> 00:00:59,320 And then a few lectures later, we'll 28 00:00:59,320 --> 00:01:01,320 be going over some of the more intuitive aspects 29 00:01:01,320 --> 00:01:03,130 to help explain it for everybody. 30 00:01:03,130 --> 00:01:06,060 So I'm going to show you the same situation that we've 31 00:01:06,060 --> 00:01:09,480 been describing sort of intuitively so far, 32 00:01:09,480 --> 00:01:11,910 but we're going to hit it mathematically today. 33 00:01:11,910 --> 00:01:13,410 Let's say there's a small nucleus, 34 00:01:13,410 --> 00:01:20,250 1, that's firing at a large nucleus, 2, and afterwards, 35 00:01:20,250 --> 00:01:24,870 a different small nucleus, 3, and a different large nucleus, 36 00:01:24,870 --> 00:01:26,940 4, come flying out. 37 00:01:26,940 --> 00:01:30,340 And so we're going to keep this as general as possible. 38 00:01:30,340 --> 00:01:34,410 So let's say if we draw angles from their original paths, 39 00:01:34,410 --> 00:01:36,840 particle 3 went off at angle theta 40 00:01:36,840 --> 00:01:39,570 and particle 4 went off at angle phi. 41 00:01:39,570 --> 00:01:43,780 So hopefully those are differentiable enough. 42 00:01:43,780 --> 00:01:47,640 And if we were to write the overall Q equation showing 43 00:01:47,640 --> 00:01:51,560 the balance between mass and energy here, 44 00:01:51,560 --> 00:01:56,280 we would simply have the mass 1 c squared 45 00:01:56,280 --> 00:01:58,043 plus kinetic energy of 1. 46 00:01:58,043 --> 00:01:59,460 So in this case, we're just saying 47 00:01:59,460 --> 00:02:01,230 that the mass and the kinetic energy 48 00:02:01,230 --> 00:02:03,810 of all particles on the left side and the right side 49 00:02:03,810 --> 00:02:05,790 has to be conserved. 50 00:02:05,790 --> 00:02:12,750 So let's add mass 2 c squared plus T2 has to equal mass 3 c 51 00:02:12,750 --> 00:02:21,570 squared plus T3 plus mass 4 c squared plus T4, where, 52 00:02:21,570 --> 00:02:25,800 just for symbols, M refers to a mass, 53 00:02:25,800 --> 00:02:27,630 T refers to a kinetic energy. 54 00:02:33,700 --> 00:02:36,790 And so this conservation of total mass or total energy 55 00:02:36,790 --> 00:02:37,860 has got to be conserved. 56 00:02:37,860 --> 00:02:38,820 And we'll use it again. 57 00:02:38,820 --> 00:02:43,220 Because, again, we can describe the Q, or the energy consumed 58 00:02:43,220 --> 00:02:47,180 or released by the reaction, as either the change in masses 59 00:02:47,180 --> 00:02:48,972 or the change in energies. 60 00:02:48,972 --> 00:02:50,555 So in this case, we can write that Q-- 61 00:02:53,680 --> 00:02:55,870 let's just group all of the c squareds together 62 00:02:55,870 --> 00:02:57,910 for easier writing. 63 00:02:57,910 --> 00:03:07,560 If we take the initial masses minus the final masses, 64 00:03:07,560 --> 00:03:09,870 then we get a picture of how much mass was converted 65 00:03:09,870 --> 00:03:11,910 to energy, therefore, how much energy 66 00:03:11,910 --> 00:03:14,190 is available for the reaction, or Q, 67 00:03:14,190 --> 00:03:16,000 to turn it into kinetic energy. 68 00:03:16,000 --> 00:03:18,150 So in this case, we can put the kinetic energy 69 00:03:18,150 --> 00:03:23,480 of the final products minus the kinetic energies-- 70 00:03:23,480 --> 00:03:25,770 I'm going to keep with 1-- 71 00:03:25,770 --> 00:03:27,770 of the initial products. 72 00:03:27,770 --> 00:03:30,480 And so we'll use this a little later on. 73 00:03:30,480 --> 00:03:32,340 One simplification that we'll make now 74 00:03:32,340 --> 00:03:35,670 is we'll assume that if we're firing particles at anything, 75 00:03:35,670 --> 00:03:38,070 that anything starts off at rest. 76 00:03:38,070 --> 00:03:41,880 So we can start by saying there's no T2. 77 00:03:41,880 --> 00:03:45,940 That's just a simplification that we'll make right now. 78 00:03:45,940 --> 00:03:47,880 And so then the question is, what 79 00:03:47,880 --> 00:03:51,480 quantities of this situation are we likely to know, 80 00:03:51,480 --> 00:03:53,070 which ones are we not likely to know, 81 00:03:53,070 --> 00:03:55,470 and which ones are left to relate together? 82 00:03:55,470 --> 00:03:57,630 So let's just go through one by one. 83 00:03:57,630 --> 00:04:00,930 Would we typically know the mass of the initial particle coming 84 00:04:00,930 --> 00:04:03,012 in? 85 00:04:03,012 --> 00:04:05,220 We probably know what we're shooting at stuff, right? 86 00:04:05,220 --> 00:04:07,140 So we'd know M1. 87 00:04:07,140 --> 00:04:10,960 What about T1, the initial kinetic energy? 88 00:04:10,960 --> 00:04:11,460 Sure. 89 00:04:11,460 --> 00:04:14,100 Let's say we have a reactor whose energy we know, 90 00:04:14,100 --> 00:04:15,990 or an accelerator, or something that we're 91 00:04:15,990 --> 00:04:18,540 controlling the energy, like in problem set one. 92 00:04:18,540 --> 00:04:20,339 We'd probably know that. 93 00:04:20,339 --> 00:04:24,130 We'd probably know what things we're firing at. 94 00:04:24,130 --> 00:04:27,510 And we would probably know what the masses 95 00:04:27,510 --> 00:04:29,880 of the final products are, because you guys have been 96 00:04:29,880 --> 00:04:33,780 doing nuclear reaction analysis and calculating 97 00:04:33,780 --> 00:04:36,990 binding energies and everything for the last couple of weeks. 98 00:04:36,990 --> 00:04:42,030 But we might not know the kinetic energies 99 00:04:42,030 --> 00:04:42,940 of what's coming out. 100 00:04:42,940 --> 00:04:45,270 Let's say we didn't actually even know the masses yet. 101 00:04:45,270 --> 00:04:48,040 We'd have to figure out a way to get both the kinetic energies. 102 00:04:48,040 --> 00:04:51,210 And what about these angles here? 103 00:04:51,210 --> 00:04:53,730 This is the new variable that we're introducing, 104 00:04:53,730 --> 00:04:56,310 is the kinetic energy of particles 3 and 4 105 00:04:56,310 --> 00:04:59,160 is going to depend on what angles they fire off at. 106 00:04:59,160 --> 00:05:01,010 Let me give you a limiting case. 107 00:05:01,010 --> 00:05:05,110 Let's say theta was 0. 108 00:05:05,110 --> 00:05:07,550 What would that mean, physically? 109 00:05:10,370 --> 00:05:12,770 What would be happening to particles 1, 2, 3, 110 00:05:12,770 --> 00:05:16,220 and 4 if theta and phi were 0, if they kept on moving 111 00:05:16,220 --> 00:05:19,293 in the exact same path? 112 00:05:19,293 --> 00:05:19,793 Yeah? 113 00:05:19,793 --> 00:05:22,150 AUDIENCE: Is it a fusion event, or [INAUDIBLE] 114 00:05:22,150 --> 00:05:23,192 PROFESSOR: We don't know. 115 00:05:23,192 --> 00:05:23,950 Well, let's see. 116 00:05:23,950 --> 00:05:24,450 Yeah. 117 00:05:24,450 --> 00:05:27,490 If it was a fusion event-- let's say there was one here and one 118 00:05:27,490 --> 00:05:28,668 standing still-- 119 00:05:28,668 --> 00:05:30,460 then the whole center of mass of the system 120 00:05:30,460 --> 00:05:31,627 would have to move that way. 121 00:05:31,627 --> 00:05:33,820 So one example could be a fusion event. 122 00:05:33,820 --> 00:05:36,800 A second example could be absolutely nothing. 123 00:05:36,800 --> 00:05:40,220 It's perfectly valid to say if, let's say, particle 1 scatters 124 00:05:40,220 --> 00:05:43,130 off particle 2 at an angle of 0 degrees, 125 00:05:43,130 --> 00:05:54,200 that's what's known as forward scattering, which 126 00:05:54,200 --> 00:05:56,120 is to say that theta equals 0. 127 00:05:58,790 --> 00:06:01,190 So this is another quantity that we might not know. 128 00:06:01,190 --> 00:06:04,910 We might not know what theta and phi are. 129 00:06:04,910 --> 00:06:07,160 And the problem here is we've got, like, three or four 130 00:06:07,160 --> 00:06:09,980 unknowns and only one equation to relate them. 131 00:06:09,980 --> 00:06:10,990 So what other-- yeah? 132 00:06:10,990 --> 00:06:11,515 Question? 133 00:06:11,515 --> 00:06:12,890 AUDIENCE: For forward scattering, 134 00:06:12,890 --> 00:06:16,280 when you say theta equals 0, do you mean they just sort of move 135 00:06:16,280 --> 00:06:19,260 together forward, kind of like an inelastic collision, 136 00:06:19,260 --> 00:06:21,260 and they just keep moving in the same direction? 137 00:06:21,260 --> 00:06:23,218 PROFESSOR: An inelastic collision would be one. 138 00:06:23,218 --> 00:06:26,240 And since we haven't gone through what inelastic means, 139 00:06:26,240 --> 00:06:29,300 that would mean some sort of collision where-- 140 00:06:29,300 --> 00:06:30,020 let's see. 141 00:06:30,020 --> 00:06:31,776 How would I explain this? 142 00:06:31,776 --> 00:06:34,040 I'd say an inelastic collision would be like 143 00:06:34,040 --> 00:06:38,300 if particles 1 and 2 were to fuse, like a capture event, 144 00:06:38,300 --> 00:06:41,570 for example, or a capture and then a re-emission, let's say, 145 00:06:41,570 --> 00:06:42,390 of a neutron. 146 00:06:42,390 --> 00:06:42,890 Yeah. 147 00:06:42,890 --> 00:06:44,660 If it was re-emitted in the forward direction, 148 00:06:44,660 --> 00:06:46,743 then that could be an inelastic scattering event-- 149 00:06:46,743 --> 00:06:47,490 AUDIENCE: Oh, OK. 150 00:06:47,490 --> 00:06:49,365 PROFESSOR: --but still in the same direction. 151 00:06:49,365 --> 00:06:51,740 Or an elastic scatter at an angle of theta 152 00:06:51,740 --> 00:06:55,010 equals 0 could be like there wasn't any scattering at all. 153 00:06:55,010 --> 00:06:57,650 Because really in the end, can matter-- 154 00:06:57,650 --> 00:07:01,100 let's say if you have a neutron firing at a nucleus, 155 00:07:01,100 --> 00:07:02,930 depends on what angle it bounces off of, 156 00:07:02,930 --> 00:07:05,240 in the billiard ball sense. 157 00:07:05,240 --> 00:07:08,210 If it bounces off at an angle of 0, that means it missed. 158 00:07:08,210 --> 00:07:10,490 We would consider that theta equals 0. 159 00:07:10,490 --> 00:07:12,110 But the point here is that we now 160 00:07:12,110 --> 00:07:14,120 have more quantities unknown than we 161 00:07:14,120 --> 00:07:16,460 have equations to define them. 162 00:07:16,460 --> 00:07:18,200 So how else can we start relating 163 00:07:18,200 --> 00:07:19,280 some of these quantities? 164 00:07:19,280 --> 00:07:21,500 What else can we conserve, since we've 165 00:07:21,500 --> 00:07:24,420 already got mass and energy? 166 00:07:24,420 --> 00:07:26,410 What's that third quantity I always yell out? 167 00:07:26,410 --> 00:07:27,210 AUDIENCE: Momentum. 168 00:07:27,210 --> 00:07:28,140 PROFESSOR: Momentum. 169 00:07:28,140 --> 00:07:28,750 Right. 170 00:07:28,750 --> 00:07:31,980 So let's start writing some of the momentum conservation 171 00:07:31,980 --> 00:07:35,730 equations so we can try and nail these things down. 172 00:07:35,730 --> 00:07:37,860 So I'm going to write each step one at a time. 173 00:07:37,860 --> 00:07:39,510 We'll start by conserving momentum. 174 00:07:42,700 --> 00:07:44,680 That's what we'll do right here. 175 00:07:44,680 --> 00:07:47,650 And we can write the x and the y equations separately. 176 00:07:47,650 --> 00:07:50,590 So what's the momentum of particle 1? 177 00:07:50,590 --> 00:07:53,590 How do we express that? 178 00:07:53,590 --> 00:07:54,840 AUDIENCE: Mass times velocity. 179 00:07:54,840 --> 00:07:55,480 PROFESSOR: Yep. 180 00:07:55,480 --> 00:07:58,290 So it would be like M1 V1. 181 00:07:58,290 --> 00:08:02,490 So we'll have a little box right here for momentum. 182 00:08:02,490 --> 00:08:04,920 We could say mass times velocity-- 183 00:08:04,920 --> 00:08:07,290 or, how do we express that in terms of the variables 184 00:08:07,290 --> 00:08:09,540 that we have here, like we did last week? 185 00:08:15,083 --> 00:08:16,750 What about in terms of kinetic energies? 186 00:08:20,620 --> 00:08:22,620 Well, another way of writing mass times velocity 187 00:08:22,620 --> 00:08:26,610 would be root 2MT. 188 00:08:26,610 --> 00:08:32,669 Because in this case, we would have root 2 times 189 00:08:32,669 --> 00:08:39,000 M1 times 1/2 M1 V1 squared. 190 00:08:39,000 --> 00:08:41,100 The 2s here cancel. 191 00:08:41,100 --> 00:08:42,175 Let's see. 192 00:08:42,175 --> 00:08:43,500 You have M1 squared. 193 00:08:43,500 --> 00:08:45,120 You have a V squared. 194 00:08:45,120 --> 00:08:51,370 And the square root of M squared V squared is just MV. 195 00:08:51,370 --> 00:08:54,330 So this is an equivalent way of writing the momentum 196 00:08:54,330 --> 00:08:56,543 in the variables that we're working in already. 197 00:08:56,543 --> 00:08:58,710 And so since that doesn't introduce another variable 198 00:08:58,710 --> 00:09:00,540 like velocity-- which we do know, 199 00:09:00,540 --> 00:09:02,790 but it's kind of confusing to add more symbols-- let's 200 00:09:02,790 --> 00:09:04,560 keep as few as possible. 201 00:09:04,560 --> 00:09:06,900 So what's the x momentum of particle 1? 202 00:09:10,640 --> 00:09:13,510 Just what I've got up there. 203 00:09:13,510 --> 00:09:16,360 2 M1 T1. 204 00:09:16,360 --> 00:09:20,680 What's the x momentum in this frame of particle 2? 205 00:09:20,680 --> 00:09:21,440 0. 206 00:09:21,440 --> 00:09:24,340 We're assuming that it's at rest. 207 00:09:24,340 --> 00:09:27,140 And now, what's the x momentum of particle 3? 208 00:09:31,044 --> 00:09:33,447 AUDIENCE: [INAUDIBLE] 209 00:09:33,447 --> 00:09:35,030 PROFESSOR: I heard a couple of things. 210 00:09:35,030 --> 00:09:36,080 Can you say them louder? 211 00:09:36,080 --> 00:09:38,635 AUDIENCE: Square root of 2 M3 T3? 212 00:09:38,635 --> 00:09:39,260 PROFESSOR: Yep. 213 00:09:39,260 --> 00:09:43,340 Root 2 M3 T3. 214 00:09:43,340 --> 00:09:45,560 But in this case, if we're defining, 215 00:09:45,560 --> 00:09:49,100 let's say, our x-axis here, it also 216 00:09:49,100 --> 00:09:50,820 matters what this angle is. 217 00:09:50,820 --> 00:09:56,780 So you've got to multiply by cosine theta in this case. 218 00:09:56,780 --> 00:09:59,200 And that's the x momentum of particle 3. 219 00:09:59,200 --> 00:10:02,050 And we've also got to account for particle 4. 220 00:10:02,050 --> 00:10:10,410 So we'll say add 2 cosine phi. 221 00:10:10,410 --> 00:10:13,170 Now let's do the same thing for the y momentum. 222 00:10:13,170 --> 00:10:17,990 What's the y momentum of particles 1 and 2? 223 00:10:17,990 --> 00:10:20,000 0. 224 00:10:20,000 --> 00:10:24,220 They're not moving in the y direction to start. 225 00:10:24,220 --> 00:10:25,480 And how about particle 3? 226 00:10:30,590 --> 00:10:32,760 I hear whispers, but nothing vocalized. 227 00:10:32,760 --> 00:10:33,495 AUDIENCE: Sine? 228 00:10:33,495 --> 00:10:34,120 PROFESSOR: Yep. 229 00:10:34,120 --> 00:10:42,190 Same thing, but root 2 M3 T3 sine theta. 230 00:10:42,190 --> 00:10:45,310 And M4-- I almost wrote the wrong sign 231 00:10:45,310 --> 00:10:47,470 there-- has got to be a minus. 232 00:10:47,470 --> 00:10:50,380 If the momentum of the initial particle system in the y 233 00:10:50,380 --> 00:10:54,550 direction is 0, so must the final momentum 234 00:10:54,550 --> 00:10:56,440 in the y direction. 235 00:10:56,440 --> 00:10:59,300 So these two momenta have to be equal and opposite. 236 00:11:03,440 --> 00:11:06,600 And that's times sine phi. 237 00:11:06,600 --> 00:11:09,690 So now we actually have sets of equations that relate 238 00:11:09,690 --> 00:11:11,970 all of our unknown quantities. 239 00:11:11,970 --> 00:11:14,400 We have the mass conservation equation, 240 00:11:14,400 --> 00:11:17,150 we have the Q equation, we have the x momentum, 241 00:11:17,150 --> 00:11:18,780 and we have the y momentum. 242 00:11:18,780 --> 00:11:20,970 And from this point on, it's a matter of algebra 243 00:11:20,970 --> 00:11:23,490 to express some of these quantities 244 00:11:23,490 --> 00:11:27,240 in terms of some of the others. 245 00:11:27,240 --> 00:11:29,780 So let's get started with that. 246 00:11:29,780 --> 00:11:32,150 Because angles are kind of messy, 247 00:11:32,150 --> 00:11:36,320 and theta should uniquely define phi, 248 00:11:36,320 --> 00:11:39,360 let's try and get things in terms of just one angle. 249 00:11:39,360 --> 00:11:41,900 So I'm going to start by separating 250 00:11:41,900 --> 00:11:45,740 the thetas and the phis on either side of the equals sign, 251 00:11:45,740 --> 00:11:48,470 so that hopefully later on we can eliminate one 252 00:11:48,470 --> 00:11:50,240 in a system of equations. 253 00:11:50,240 --> 00:11:53,090 So all I'm going to do is I'm going to subtract or add 254 00:11:53,090 --> 00:11:56,060 the theta terms to the other side of the equation. 255 00:11:56,060 --> 00:11:57,585 So let's say we'll separate angles. 256 00:12:03,660 --> 00:12:14,850 So we'll have root 2 M1 T1 and minus root 2 M3 T3 cosine 257 00:12:14,850 --> 00:12:15,930 theta. 258 00:12:15,930 --> 00:12:18,335 I'll be depending on you guys to check for sign errors 259 00:12:18,335 --> 00:12:19,710 here because those will be messy. 260 00:12:19,710 --> 00:12:21,450 I do have notes in case, but I'm hoping 261 00:12:21,450 --> 00:12:23,190 I won't have to look at them. 262 00:12:23,190 --> 00:12:24,780 And all we have left on this side 263 00:12:24,780 --> 00:12:33,390 is root 2 M4 T4 cosine phi. 264 00:12:33,390 --> 00:12:35,280 So that's the x momentum equation. 265 00:12:35,280 --> 00:12:38,970 Let's do the same thing with the y momentum equation. 266 00:12:38,970 --> 00:12:42,360 So all we'll do is take the theta term 267 00:12:42,360 --> 00:12:44,820 and stick it to the left of the equals sign. 268 00:12:44,820 --> 00:12:54,960 So that would give us minus root 2 M3 T3 sine theta equals 269 00:12:54,960 --> 00:13:02,540 minus root 2 M4 T4 sine phi. 270 00:13:02,540 --> 00:13:06,020 Right away, we can see that the minus signs can cancel out, 271 00:13:06,020 --> 00:13:08,120 just for simplicity. 272 00:13:08,120 --> 00:13:12,810 And what else is common to these that we can get rid of? 273 00:13:12,810 --> 00:13:13,310 Yep? 274 00:13:13,310 --> 00:13:14,435 AUDIENCE: Square root of 2. 275 00:13:14,435 --> 00:13:17,410 PROFESSOR: Everything here has a square root of 2. 276 00:13:17,410 --> 00:13:21,140 So we'll just get rid of all of the square root of 2s 277 00:13:21,140 --> 00:13:23,920 to simplify as much as possible. 278 00:13:23,920 --> 00:13:27,060 And now we look a little stuck. 279 00:13:27,060 --> 00:13:29,930 But now is the time to remember those trigonometric identities 280 00:13:29,930 --> 00:13:31,680 back from high school that I don't think-- 281 00:13:31,680 --> 00:13:34,310 has anyone used these since? 282 00:13:34,310 --> 00:13:37,820 In 1801 or 1802, anyone used a trig identity? 283 00:13:37,820 --> 00:13:38,620 A little bit? 284 00:13:38,620 --> 00:13:39,590 OK. 285 00:13:39,590 --> 00:13:41,390 I would hope so. 286 00:13:41,390 --> 00:13:43,798 But I don't know what other people are teaching nowadays. 287 00:13:43,798 --> 00:13:46,340 At least this way I'll make sure you remember the high school 288 00:13:46,340 --> 00:13:47,390 stuff. 289 00:13:47,390 --> 00:13:50,020 We're going to rely on the fact that we already 290 00:13:50,020 --> 00:13:52,070 have got a cosine and a sine. 291 00:13:52,070 --> 00:13:54,260 We have a set of simultaneous equations. 292 00:13:54,260 --> 00:13:57,440 If we can add them together and destroy the angles somehow, 293 00:13:57,440 --> 00:13:59,640 that will make things a lot easier. 294 00:13:59,640 --> 00:14:02,420 So for the thetas, we have a cosine, a sine, 295 00:14:02,420 --> 00:14:05,930 and an unangled term that looks kind of messy. 296 00:14:05,930 --> 00:14:09,500 Here we have a cosine and a sine. 297 00:14:09,500 --> 00:14:11,030 Anyone have any idea where we could 298 00:14:11,030 --> 00:14:12,875 go next to destroy one of these angles? 299 00:14:16,050 --> 00:14:19,110 Anyone remember any handy cosine or sine trig identities? 300 00:14:19,110 --> 00:14:20,610 AUDIENCE: If you squared both terms, 301 00:14:20,610 --> 00:14:23,010 you could get square root of cosine squared, square root 302 00:14:23,010 --> 00:14:23,510 of-- sorry. 303 00:14:23,510 --> 00:14:24,800 You get cosine squared and sine squared 304 00:14:24,800 --> 00:14:27,050 and then you factor out the square root of M4T4 305 00:14:27,050 --> 00:14:29,618 and then cosine squared plus sine squared equals 1. 306 00:14:29,618 --> 00:14:30,410 PROFESSOR: Exactly. 307 00:14:30,410 --> 00:14:32,270 So we can rely on the fact that if we 308 00:14:32,270 --> 00:14:36,860 can square both sides of both equations and add them up, 309 00:14:36,860 --> 00:14:40,310 we would have a cosine squared of phi 310 00:14:40,310 --> 00:14:46,930 plus a sine squared of phi, which also equals 1. 311 00:14:46,930 --> 00:14:51,180 So we can destroy this phi angle and make things a lot simpler. 312 00:14:51,180 --> 00:14:54,150 So we'll start by squaring both sides. 313 00:15:00,590 --> 00:15:03,020 Let's start with the x momentum equation. 314 00:15:03,020 --> 00:15:05,170 So if we have-- 315 00:15:05,170 --> 00:15:08,520 let's see-- root M1T1. 316 00:15:08,520 --> 00:15:12,000 So we're going to take that stuff squared. 317 00:15:12,000 --> 00:15:14,490 And that squared is not too hard. 318 00:15:17,090 --> 00:15:19,350 Neither are those. 319 00:15:19,350 --> 00:15:22,360 So we'll have root M1 T1 squared, which just gives us 320 00:15:22,360 --> 00:15:31,690 M1 T1, minus root M1 T1 times root M3 T3 cosine theta. 321 00:15:31,690 --> 00:15:38,470 Let's just lump those terms together as root M1 M3 T1 T3 322 00:15:38,470 --> 00:15:41,170 cosine theta. 323 00:15:41,170 --> 00:15:42,670 Also, anyone, raise your hand or let 324 00:15:42,670 --> 00:15:44,180 me know if I'm going too fast. 325 00:15:44,180 --> 00:15:46,060 I'm trying to hit every single step. 326 00:15:46,060 --> 00:15:48,550 But in case I skip one, please slow me down. 327 00:15:48,550 --> 00:15:50,940 That's what class is for. 328 00:15:50,940 --> 00:15:52,080 OK. 329 00:15:52,080 --> 00:15:53,740 And then we've got another one. 330 00:15:53,740 --> 00:15:57,480 Let's just stick a 2 in front of there 331 00:15:57,480 --> 00:16:01,470 and plus that term squared. 332 00:16:01,470 --> 00:16:06,120 So we'll have M3 T3. 333 00:16:08,990 --> 00:16:09,670 Let's see. 334 00:16:12,780 --> 00:16:13,590 Yeah. 335 00:16:13,590 --> 00:16:17,960 Looks like cosine squared of theta. 336 00:16:17,960 --> 00:16:20,010 Yep. 337 00:16:20,010 --> 00:16:22,990 Equals-- this one's easier-- 338 00:16:22,990 --> 00:16:29,740 M4 T4 cosine squared of phi. 339 00:16:29,740 --> 00:16:31,110 OK. 340 00:16:31,110 --> 00:16:34,200 Now we'll do the same thing for the y momentum equation. 341 00:16:34,200 --> 00:16:36,405 Much easier because there's no addition anywhere. 342 00:16:36,405 --> 00:16:43,796 And we have M3 sine squared theta-- 343 00:16:43,796 --> 00:16:51,880 over here-- equals M4 T4 sine squared phi. 344 00:16:51,880 --> 00:16:53,060 So this is quite nice. 345 00:16:53,060 --> 00:16:55,040 Now if we add these equations together, 346 00:16:55,040 --> 00:16:58,820 we get rid of all of the cosine and sine squared terms. 347 00:16:58,820 --> 00:17:00,310 So let's add them up. 348 00:17:03,572 --> 00:17:05,510 Let's see. 349 00:17:05,510 --> 00:17:06,950 We'll add the two equations. 350 00:17:13,310 --> 00:17:15,762 Add equations. 351 00:17:15,762 --> 00:17:17,720 And let's try and group all the terms together. 352 00:17:17,720 --> 00:17:29,215 So we have M1 T1 minus 2 root M1 M3-- 353 00:17:29,215 --> 00:17:32,270 it's getting hard to write over the lip of the chalkboard 354 00:17:32,270 --> 00:17:32,770 here-- 355 00:17:37,890 --> 00:17:40,200 cosine theta. 356 00:17:40,200 --> 00:17:47,910 And we have M3 T3 cosine squared theta plus M3 T3 357 00:17:47,910 --> 00:17:48,860 sine squared theta. 358 00:17:58,530 --> 00:18:06,340 Equals M4 T4 cosine squared theta plus sine squared theta-- 359 00:18:06,340 --> 00:18:06,960 or phi. 360 00:18:06,960 --> 00:18:07,460 I'm sorry. 361 00:18:07,460 --> 00:18:12,810 Cosine squared phi plus sine squared phi. 362 00:18:12,810 --> 00:18:13,310 OK. 363 00:18:13,310 --> 00:18:16,250 Hopefully that's as low as I'll have to write. 364 00:18:16,250 --> 00:18:18,350 And like we saw before, cosine squared 365 00:18:18,350 --> 00:18:20,150 plus sine squared equals 1. 366 00:18:20,150 --> 00:18:24,150 So that goes away. 367 00:18:24,150 --> 00:18:25,820 That goes away. 368 00:18:25,820 --> 00:18:28,440 And let's keep going over on this side of the board. 369 00:18:28,440 --> 00:18:31,290 I told you this would be a fill-the-board day. 370 00:18:31,290 --> 00:18:32,790 Let's see if we actually get all six 371 00:18:32,790 --> 00:18:34,170 instead of just the four visible. 372 00:18:34,170 --> 00:18:38,190 But I think we'll finish this derivation in four boards. 373 00:18:38,190 --> 00:18:40,130 So let's write what we've got left. 374 00:18:40,130 --> 00:18:41,030 Let's see. 375 00:18:41,030 --> 00:18:41,720 Remaining. 376 00:18:46,030 --> 00:18:53,730 So we have M1 T1 minus 2 root M1 M3-- 377 00:18:53,730 --> 00:18:55,680 so much easier to write standing up-- 378 00:18:58,310 --> 00:19:05,070 cosine theta equals M4 T4. 379 00:19:05,070 --> 00:19:06,210 Quite a bit simpler. 380 00:19:06,210 --> 00:19:07,192 AUDIENCE: [INAUDIBLE] 381 00:19:07,192 --> 00:19:08,400 PROFESSOR: Did I miss a term? 382 00:19:08,400 --> 00:19:09,092 AUDIENCE: The M3 T3. 383 00:19:09,092 --> 00:19:09,720 PROFESSOR: Ah. 384 00:19:09,720 --> 00:19:10,150 Thank you. 385 00:19:10,150 --> 00:19:10,500 You're right. 386 00:19:10,500 --> 00:19:11,042 You're right. 387 00:19:13,930 --> 00:19:17,650 And we had a plus M3 T3. 388 00:19:17,650 --> 00:19:18,900 Yeah, that would be important. 389 00:19:18,900 --> 00:19:20,090 Thank you. 390 00:19:20,090 --> 00:19:23,570 Equals M4 T4. 391 00:19:23,570 --> 00:19:26,900 So we now have a relation between the masses, 392 00:19:26,900 --> 00:19:30,410 the energies, and one angle, which is getting a lot better. 393 00:19:30,410 --> 00:19:35,210 We still have one more variable than we can deal with. 394 00:19:35,210 --> 00:19:36,830 So let's say if we're-- 395 00:19:36,830 --> 00:19:38,568 let's see. 396 00:19:38,568 --> 00:19:40,110 Which of these variables do you think 397 00:19:40,110 --> 00:19:43,027 we can eliminate using any of the equations you see, 398 00:19:43,027 --> 00:19:44,860 let's go with, on that top board over there? 399 00:19:52,280 --> 00:19:55,610 Well, what other quantities are we 400 00:19:55,610 --> 00:19:59,195 likely to know about this nuclear reaction? 401 00:19:59,195 --> 00:20:00,320 Let's bring this back down. 402 00:20:02,900 --> 00:20:05,618 Are we likely to know the Q value? 403 00:20:05,618 --> 00:20:06,470 AUDIENCE: Yeah. 404 00:20:06,470 --> 00:20:08,138 PROFESSOR: Probably. 405 00:20:08,138 --> 00:20:09,680 Because like you guys have been doing 406 00:20:09,680 --> 00:20:12,380 on problem sets one and two, if you know, let's say, 407 00:20:12,380 --> 00:20:15,500 the binding energies, or the masses, or the excess masses, 408 00:20:15,500 --> 00:20:18,320 or the kinetic energies of all your products, 409 00:20:18,320 --> 00:20:20,540 any combination of those can get you 410 00:20:20,540 --> 00:20:23,150 the Q value of that reaction. 411 00:20:23,150 --> 00:20:25,840 And if you just look up those reactions like, let's say, 412 00:20:25,840 --> 00:20:28,660 radioactive decay reactions, on the table of nuclides, 413 00:20:28,660 --> 00:20:30,370 it just gives you the Q value. 414 00:20:30,370 --> 00:20:34,240 So chances are we can express some of these kinetic energies 415 00:20:34,240 --> 00:20:36,230 in terms of Q. 416 00:20:36,230 --> 00:20:40,680 And all we've got left is T1, T3, and T4. 417 00:20:40,680 --> 00:20:42,530 So which of these are we most likely to be 418 00:20:42,530 --> 00:20:44,720 able to know or measure? 419 00:20:44,720 --> 00:20:49,530 T1, we probably fixed it by cranking up our particle 420 00:20:49,530 --> 00:20:52,280 accelerator to a certain energy. 421 00:20:52,280 --> 00:20:55,590 T3 or T4, what do you guys think? 422 00:20:55,590 --> 00:20:57,450 Let's say we had a very small nucleus 423 00:20:57,450 --> 00:20:59,520 firing at a very big one. 424 00:20:59,520 --> 00:21:01,020 Which one do you think would be more 425 00:21:01,020 --> 00:21:02,850 likely to escape this system and get 426 00:21:02,850 --> 00:21:06,934 detected by us standing a couple feet away with a detector? 427 00:21:06,934 --> 00:21:07,900 Yep? 428 00:21:07,900 --> 00:21:08,890 AUDIENCE: T3. 429 00:21:08,890 --> 00:21:11,320 PROFESSOR: Probably T3, the smaller particle. 430 00:21:11,320 --> 00:21:13,570 We've just arbitrarily chosen that. 431 00:21:13,570 --> 00:21:15,250 But for intuitive sake, let's say, yeah. 432 00:21:15,250 --> 00:21:21,100 Why don't we try and get T4 in terms of Q T1 and T3? 433 00:21:21,100 --> 00:21:25,300 That's not too hard, since it's addition. 434 00:21:25,300 --> 00:21:27,040 So our next step will be substitute. 435 00:21:32,240 --> 00:21:35,450 And we'll say that Q equals-- 436 00:21:35,450 --> 00:21:38,070 I'm just going to copy it up from there-- 437 00:21:38,070 --> 00:21:43,052 T3 plus T4 minus T1. 438 00:21:43,052 --> 00:21:54,280 So we can isolate T4 and say T4 equals Q plus T1 minus T3. 439 00:21:57,270 --> 00:21:59,190 And continue substituting. 440 00:22:06,600 --> 00:22:09,800 I usually don't like to have my back to the class this much. 441 00:22:09,800 --> 00:22:12,860 But when you're writing this much, it can be a little hard. 442 00:22:12,860 --> 00:22:16,310 So let's stick this T4 in right here and rewrite the equation 443 00:22:16,310 --> 00:22:18,270 as we've got it. 444 00:22:18,270 --> 00:22:31,350 M1 T1 minus 2 root M1 M3 T1 T3 cosine theta plus M3 T3 equals 445 00:22:31,350 --> 00:22:37,050 M4 times Q plus T1 minus T3. 446 00:22:37,050 --> 00:22:41,080 I anticipate us needing to see this side of the board soon. 447 00:22:41,080 --> 00:22:43,020 I also apologize for the amount of time 448 00:22:43,020 --> 00:22:44,410 it takes to write these things. 449 00:22:44,410 --> 00:22:46,110 There's another strategy one can use 450 00:22:46,110 --> 00:22:48,960 at the board which is defining intermediate symbols. 451 00:22:48,960 --> 00:22:50,970 And here's why I'm not doing that. 452 00:22:50,970 --> 00:22:53,245 When I was a freshman, back in-- 453 00:22:53,245 --> 00:22:55,650 whoa-- 2001. 454 00:22:55,650 --> 00:22:58,150 Who here was born after 2001? 455 00:22:58,150 --> 00:22:58,650 Nobody. 456 00:22:58,650 --> 00:22:59,150 OK. 457 00:22:59,150 --> 00:23:00,450 Thank god. 458 00:23:00,450 --> 00:23:02,750 I don't feel so old. 459 00:23:02,750 --> 00:23:06,000 I was in 18023, which was math with applications, 460 00:23:06,000 --> 00:23:08,940 which was better known as math with extra theory. 461 00:23:08,940 --> 00:23:11,920 And in one class, not only did we fill nine boards, 462 00:23:11,920 --> 00:23:14,250 but we ran out of English letters-- 463 00:23:14,250 --> 00:23:17,040 symbols-- and we ran out of Greek letter symbols, 464 00:23:17,040 --> 00:23:19,020 and we moved on to Hebrew. 465 00:23:19,020 --> 00:23:21,420 Because they were distinct enough from English and Greek. 466 00:23:21,420 --> 00:23:24,200 And being, I think, the only Hebrew speaker in the class, 467 00:23:24,200 --> 00:23:26,242 I was the only one that could follow the symbols, 468 00:23:26,242 --> 00:23:27,990 but I couldn't follow the math anymore. 469 00:23:27,990 --> 00:23:30,780 So I am not going to define intermediate symbols for this 470 00:23:30,780 --> 00:23:32,970 and just keep it understandable, even 471 00:23:32,970 --> 00:23:36,050 if it takes longer to write. 472 00:23:36,050 --> 00:23:37,140 OK. 473 00:23:37,140 --> 00:23:40,110 So let's start off by dividing by M4. 474 00:23:44,770 --> 00:23:47,200 Our goal now is to try to isolate Q. 475 00:23:47,200 --> 00:23:49,900 Because this is something that we would know or measure. 476 00:23:49,900 --> 00:23:52,690 And it will relate all of the other quantities, only one 477 00:23:52,690 --> 00:23:55,150 of which we won't really know yet. 478 00:23:55,150 --> 00:23:57,490 So let's divide everything by M4. 479 00:23:57,490 --> 00:24:07,900 So we have T1 times M1 over M4 minus 2 over M4 times root 480 00:24:07,900 --> 00:24:20,250 of all that stuff plus T3 times M3 over M4 481 00:24:20,250 --> 00:24:24,970 equals Q plus T1 minus T3. 482 00:24:24,970 --> 00:24:27,870 And we've almost isolated Q. I'll call this step just 483 00:24:27,870 --> 00:24:30,990 add and subtract. 484 00:24:30,990 --> 00:24:32,740 And I'm going to group the terms together. 485 00:24:32,740 --> 00:24:35,650 So let's, for example, group all the T1s together 486 00:24:35,650 --> 00:24:37,610 and group all the T3s together. 487 00:24:37,610 --> 00:24:46,240 So if I subtract T1, I get T1 times M1 over M4 minus 1, 488 00:24:46,240 --> 00:24:57,550 minus 2 over M4 root M1 M3 T1 T3 cosine theta, plus-- 489 00:24:57,550 --> 00:25:09,140 and if I add T3, then I would get M3 over M4 plus 1 equals Q. 490 00:25:09,140 --> 00:25:12,710 So this is a good place to stop, turn around, and see you guys, 491 00:25:12,710 --> 00:25:15,230 and now ask you, which of the remaining quantities 492 00:25:15,230 --> 00:25:16,880 do we probably not know? 493 00:25:16,880 --> 00:25:21,628 So let's just go through them one by one, 494 00:25:21,628 --> 00:25:22,670 just to remind ourselves. 495 00:25:22,670 --> 00:25:25,960 Are we likely to know what T1 is? 496 00:25:25,960 --> 00:25:27,380 Probably. 497 00:25:27,380 --> 00:25:31,075 How about the masses M1 and M4? 498 00:25:31,075 --> 00:25:32,700 If we know what particles are reacting, 499 00:25:32,700 --> 00:25:35,660 we can just look those up, or measure them, or whatever. 500 00:25:35,660 --> 00:25:36,600 We know M4. 501 00:25:36,600 --> 00:25:37,650 We know our masses. 502 00:25:37,650 --> 00:25:39,240 We know T1. 503 00:25:39,240 --> 00:25:39,850 What about T3? 504 00:25:43,260 --> 00:25:45,510 We don't necessarily know yet. 505 00:25:45,510 --> 00:25:47,550 So T3 is a question mark. 506 00:25:47,550 --> 00:25:49,710 How about cosine theta or theta? 507 00:25:52,470 --> 00:25:55,170 We haven't said yet. 508 00:25:55,170 --> 00:25:56,760 And T3 we don't know. 509 00:25:56,760 --> 00:25:57,930 And the masses we know. 510 00:25:57,930 --> 00:25:59,940 And the Q we know. 511 00:25:59,940 --> 00:26:02,410 So finally, to solve for-- 512 00:26:02,410 --> 00:26:06,280 well, we only have two variables left, T3 and theta. 513 00:26:06,280 --> 00:26:10,830 So this here-- this is actually called the Q equation in its 514 00:26:10,830 --> 00:26:12,060 most complete form-- 515 00:26:16,260 --> 00:26:18,780 describes the relationship between the kinetic energy 516 00:26:18,780 --> 00:26:23,650 of the outgoing particle and the angle at which it comes off. 517 00:26:23,650 --> 00:26:24,710 How do we solve this? 518 00:26:24,710 --> 00:26:26,377 How do we get one in terms of the other? 519 00:26:29,330 --> 00:26:31,820 Anyone recognize what kind of equation we have here? 520 00:26:36,500 --> 00:26:38,320 It's a little obscure. 521 00:26:38,320 --> 00:26:39,290 Well, it's not obscure. 522 00:26:39,290 --> 00:26:40,538 But it's a little bit hiding. 523 00:26:40,538 --> 00:26:42,080 But it should be a very familiar one. 524 00:26:42,080 --> 00:26:43,890 Think back to high school again. 525 00:26:43,890 --> 00:26:44,390 Yes. 526 00:26:44,390 --> 00:26:50,990 AUDIENCE: Is it the cosine angle for the triangle [INAUDIBLE] 527 00:26:50,990 --> 00:26:53,580 PROFESSOR: Let's see. 528 00:26:53,580 --> 00:26:56,160 Certainly, there's probably some trig involved in here, 529 00:26:56,160 --> 00:26:58,050 in terms of, yeah, if you know the cosine, 530 00:26:58,050 --> 00:27:00,240 then you know, let's say, the x or the y 531 00:27:00,240 --> 00:27:02,710 component of the momentum. 532 00:27:02,710 --> 00:27:04,630 But there's something simpler, something that 533 00:27:04,630 --> 00:27:05,838 doesn't require trigonometry. 534 00:27:05,838 --> 00:27:06,338 Yep. 535 00:27:06,338 --> 00:27:07,450 AUDIENCE: Is it quadratic? 536 00:27:07,450 --> 00:27:08,158 PROFESSOR: It is. 537 00:27:08,158 --> 00:27:10,090 It's a quadratic-- so who saw that? 538 00:27:10,090 --> 00:27:15,110 It's actually a quadratic equation, where the variable 539 00:27:15,110 --> 00:27:18,960 is the square root of T3. 540 00:27:18,960 --> 00:27:23,040 That's the trick here, is you have something without T3, 541 00:27:23,040 --> 00:27:25,267 you have something with square root T3, 542 00:27:25,267 --> 00:27:26,850 and you have something with T3, better 543 00:27:26,850 --> 00:27:31,000 known as root T3 squared. 544 00:27:31,000 --> 00:27:32,830 And there. 545 00:27:32,830 --> 00:27:34,760 So this is actually a quadratic equation. 546 00:27:41,098 --> 00:27:42,640 Despite the fact that it may not have 547 00:27:42,640 --> 00:27:45,210 looked that way in the first place, there we go. 548 00:27:47,760 --> 00:27:50,210 So now, someone who remembers from high school, 549 00:27:50,210 --> 00:27:54,890 tell me, what are the roots of a quadratic equation? 550 00:27:54,890 --> 00:27:59,170 Let's say if we have the form y equals ax squared plus bx 551 00:27:59,170 --> 00:28:03,842 plus c, what does x equal? 552 00:28:03,842 --> 00:28:04,550 Just call it out. 553 00:28:04,550 --> 00:28:05,960 AUDIENCE: Negative b-- 554 00:28:05,960 --> 00:28:06,962 PROFESSOR: Yeah. 555 00:28:06,962 --> 00:28:09,330 AUDIENCE: [INAUDIBLE] square root-- 556 00:28:09,330 --> 00:28:10,040 PROFESSOR: Yep. 557 00:28:10,040 --> 00:28:13,292 AUDIENCE: --b squared minus 4ac-- 558 00:28:13,292 --> 00:28:14,000 PROFESSOR: Over-- 559 00:28:14,000 --> 00:28:14,840 AUDIENCE: 2a. 560 00:28:14,840 --> 00:28:16,040 PROFESSOR: 2a. 561 00:28:16,040 --> 00:28:21,170 And in this case, a is that stuff. 562 00:28:21,170 --> 00:28:24,500 b is that stuff without the T3. 563 00:28:24,500 --> 00:28:26,600 And c is that stuff. 564 00:28:26,600 --> 00:28:28,833 Because we have, like, 15 minutes 565 00:28:28,833 --> 00:28:30,500 before I want to open it up to questions 566 00:28:30,500 --> 00:28:33,380 and I don't think we have to repeat the quadratic formula 567 00:28:33,380 --> 00:28:36,020 stuff, I will skip ahead. 568 00:28:36,020 --> 00:28:37,160 Skip ahead. 569 00:28:40,218 --> 00:28:42,760 This is when I'd normally say it's an exercise to the reader. 570 00:28:42,760 --> 00:28:43,443 But no. 571 00:28:43,443 --> 00:28:44,860 It's not the phrase I like to use. 572 00:28:44,860 --> 00:28:45,670 It's boring. 573 00:28:45,670 --> 00:28:49,420 And I can just tell you guys what it ends up as. 574 00:28:49,420 --> 00:28:53,890 It ends up with root T3 equals-- and this is the one time I am 575 00:28:53,890 --> 00:28:56,470 going to define new symbols because it's just easier 576 00:28:56,470 --> 00:28:58,000 to parse-- 577 00:28:58,000 --> 00:29:02,130 ends up being, we'll call it s, plus or minus root 578 00:29:02,130 --> 00:29:08,500 s squared plus t, where s-- 579 00:29:08,500 --> 00:29:11,720 let's see if I can remember this without looking it up. 580 00:29:11,720 --> 00:29:12,220 No. 581 00:29:12,220 --> 00:29:14,270 I have to look at my notes. 582 00:29:14,270 --> 00:29:17,380 I don't want to get it wrong and have you all write it down 583 00:29:17,380 --> 00:29:19,300 incorrectly because of me. 584 00:29:19,300 --> 00:29:21,730 There we go. 585 00:29:21,730 --> 00:29:26,660 The remaining stuff in the square root, E1 586 00:29:26,660 --> 00:29:31,520 cosine theta over M3 plus M4. 587 00:29:31,520 --> 00:29:36,850 And t equals M4 Q-- 588 00:29:36,850 --> 00:29:38,030 is it a minus? 589 00:29:38,030 --> 00:29:38,645 It is a plus. 590 00:29:44,540 --> 00:29:49,290 Over M3 plus M4. 591 00:29:49,290 --> 00:29:51,660 So these are the roots of this equation. 592 00:29:51,660 --> 00:29:53,970 This is how you can actually relate 593 00:29:53,970 --> 00:29:55,800 the kinetic energy of the outgoing particle 594 00:29:55,800 --> 00:29:59,520 directly to the angle. 595 00:29:59,520 --> 00:30:01,620 So I want to let that sink in just for a minute, 596 00:30:01,620 --> 00:30:03,330 stop here, and check to see if there's 597 00:30:03,330 --> 00:30:05,580 any questions on the derivation before we 598 00:30:05,580 --> 00:30:08,340 start to use it to do something a little more concrete. 599 00:30:08,340 --> 00:30:09,041 Yep. 600 00:30:09,041 --> 00:30:10,885 AUDIENCE: Where did the E come from? 601 00:30:10,885 --> 00:30:12,240 PROFESSOR: The E. Oh, I'm sorry. 602 00:30:12,240 --> 00:30:14,760 That's a T. Thank you. 603 00:30:14,760 --> 00:30:15,810 Kinetic energy. 604 00:30:15,810 --> 00:30:17,790 Again, we should be consistent with symbols. 605 00:30:17,790 --> 00:30:19,740 And I think-- 606 00:30:19,740 --> 00:30:21,670 I don't see any other hanging Es. 607 00:30:21,670 --> 00:30:22,170 Good. 608 00:30:22,170 --> 00:30:24,043 Thank you. 609 00:30:24,043 --> 00:30:26,460 So any other questions on the derivation as we've done it? 610 00:30:26,460 --> 00:30:28,430 We managed to do it in less than four boards. 611 00:30:28,430 --> 00:30:31,190 There we go. 612 00:30:31,190 --> 00:30:31,690 OK. 613 00:30:31,690 --> 00:30:33,820 Since I don't see any questions, let's 614 00:30:33,820 --> 00:30:36,820 get into a couple of the implications of this. 615 00:30:36,820 --> 00:30:44,410 So let's now look at what defines an exothermic reaction 616 00:30:44,410 --> 00:30:47,620 where we say if Q is greater than 0-- 617 00:30:47,620 --> 00:30:52,510 which is to say that some of the mass becomes kinetic energy-- 618 00:30:52,510 --> 00:30:56,770 if an exothermic reaction is energetically possible, then 619 00:30:56,770 --> 00:30:59,600 what is the minimum T1? 620 00:30:59,600 --> 00:31:00,400 Ah. 621 00:31:00,400 --> 00:31:01,540 That's why I brought it. 622 00:31:01,540 --> 00:31:04,510 What's the minimum T1 up here to make that exothermic reaction 623 00:31:04,510 --> 00:31:05,010 happen? 624 00:31:10,930 --> 00:31:12,430 We'll put a condition on T1. 625 00:31:15,670 --> 00:31:17,260 So if the reaction's exothermic, which 626 00:31:17,260 --> 00:31:18,790 means it will happen spontaneously, 627 00:31:18,790 --> 00:31:22,063 how much extra kinetic energy do you have to give to the system 628 00:31:22,063 --> 00:31:23,230 to make the reaction happen? 629 00:31:29,840 --> 00:31:32,030 Let's think of it in the chemical sense. 630 00:31:32,030 --> 00:31:35,150 If you have an exothermic chemical reaction, 631 00:31:35,150 --> 00:31:38,220 is it spontaneous or is it not? 632 00:31:38,220 --> 00:31:39,480 It is spontaneous. 633 00:31:39,480 --> 00:31:41,340 Same thing in the nuclear world. 634 00:31:41,340 --> 00:31:44,310 If you have an exothermic nuclear reaction, 635 00:31:44,310 --> 00:31:46,530 do you need any kinetic energy to start with 636 00:31:46,530 --> 00:31:48,340 to make it happen? 637 00:31:48,340 --> 00:31:48,840 No. 638 00:31:48,840 --> 00:31:49,080 OK. 639 00:31:49,080 --> 00:31:49,630 There we go. 640 00:31:49,630 --> 00:31:51,240 So that's kind of the analogy. 641 00:31:51,240 --> 00:31:54,870 So T1 has to be greater than or equal to 0. 642 00:31:54,870 --> 00:31:57,660 It's pretty much not a condition, right? 643 00:31:57,660 --> 00:31:59,650 It happens all the time. 644 00:31:59,650 --> 00:32:03,420 So if we were to say T1 were to equal 0-- 645 00:32:03,420 --> 00:32:07,410 let me get my crossing out color again. 646 00:32:07,410 --> 00:32:10,950 If T1 were to equal 0, then s could equals 0. 647 00:32:13,970 --> 00:32:17,000 And T1 is 0 here. 648 00:32:17,000 --> 00:32:18,635 And then you just get-- 649 00:32:18,635 --> 00:32:27,530 that's an s-- t equals M4 Q over M3 plus M4. 650 00:32:29,910 --> 00:32:31,660 And this just kind of gives you a relation 651 00:32:31,660 --> 00:32:35,800 between the relative kinetic energies of the two particles. 652 00:32:35,800 --> 00:32:37,900 Another way of writing this relation 653 00:32:37,900 --> 00:32:43,120 would just be that E3 plus E4 has to be 654 00:32:43,120 --> 00:32:46,530 greater than or equal to E1. 655 00:32:46,530 --> 00:32:47,030 AUDIENCE: T? 656 00:32:47,030 --> 00:32:48,113 PROFESSOR: All this-- hmm? 657 00:32:48,113 --> 00:32:48,854 AUDIENCE: T? 658 00:32:48,854 --> 00:32:49,710 PROFESSOR: Ah. 659 00:32:49,710 --> 00:32:50,210 Thank you. 660 00:32:52,958 --> 00:32:55,830 Because Es will be used in a different point of this class. 661 00:32:55,830 --> 00:32:58,060 So we'll stick with T for kinetic energy. 662 00:32:58,060 --> 00:32:59,560 Thank you. 663 00:32:59,560 --> 00:33:02,140 So all that this condition says is 664 00:33:02,140 --> 00:33:05,362 that if mass has been converted to energy, 665 00:33:05,362 --> 00:33:06,820 then that kinetic energy at the end 666 00:33:06,820 --> 00:33:09,940 has to be greater than at the beginning. 667 00:33:09,940 --> 00:33:10,917 And that's all it is. 668 00:33:10,917 --> 00:33:12,750 So it makes this equation quite a lot easier 669 00:33:12,750 --> 00:33:15,810 to solve for an exothermic reaction. 670 00:33:15,810 --> 00:33:18,240 You can also start to look to say, well, what happens 671 00:33:18,240 --> 00:33:21,510 as we vary this angle theta? 672 00:33:21,510 --> 00:33:22,980 What does the kinetic energy do? 673 00:33:25,760 --> 00:33:29,362 Let's take the case of an endothermic reaction. 674 00:33:29,362 --> 00:33:30,695 Now we are running out of space. 675 00:33:35,220 --> 00:33:42,120 For an endothermic reaction where Q is less than 0, 676 00:33:42,120 --> 00:33:45,160 you would have to have T1 to be greater than 0. 677 00:33:48,980 --> 00:33:51,470 Otherwise the reaction can't occur. 678 00:33:51,470 --> 00:33:54,500 So you have to impart additional energy into the system 679 00:33:54,500 --> 00:33:55,910 to get it going. 680 00:33:55,910 --> 00:34:01,218 And it also means that not every angle of emission is possible. 681 00:34:01,218 --> 00:34:03,260 You might wonder, why do we care about the angle, 682 00:34:03,260 --> 00:34:05,010 because the reaction still happens anyway? 683 00:34:05,010 --> 00:34:07,130 Well, it doesn't happen at every angle. 684 00:34:07,130 --> 00:34:09,320 And reactions have different probabilities 685 00:34:09,320 --> 00:34:12,920 of occurring depending on the angle at which the things come 686 00:34:12,920 --> 00:34:14,320 out. 687 00:34:14,320 --> 00:34:17,980 So you could see here that as you vary T1 688 00:34:17,980 --> 00:34:19,780 and as you vary cosine theta, you still 689 00:34:19,780 --> 00:34:22,840 have to make sure that this quantity on the inside here-- 690 00:34:26,679 --> 00:34:28,690 so, s squared plus t-- 691 00:34:28,690 --> 00:34:33,100 always has to be greater than or equal to zero or else 692 00:34:33,100 --> 00:34:36,370 the roots of this are imaginary and you don't have a solution. 693 00:34:40,017 --> 00:34:42,100 So it's kind of nice that this came out quadratic. 694 00:34:42,100 --> 00:34:44,725 Because it lets you take some of the knowledge you already know 695 00:34:44,725 --> 00:34:48,100 and now apply it to say, when or when are nuclear reactions not 696 00:34:48,100 --> 00:34:48,949 or are they allowed? 697 00:34:48,949 --> 00:34:49,449 Wait. 698 00:34:49,449 --> 00:34:50,679 Let me rephrase that. 699 00:34:50,679 --> 00:34:53,560 When are nuclear reactions allowed or not allowed? 700 00:34:53,560 --> 00:34:56,760 You can now tell, depending on the angle of emission 701 00:34:56,760 --> 00:35:00,640 and the incoming energy and the masses, which are all things 702 00:35:00,640 --> 00:35:03,546 that you would tend to know. 703 00:35:03,546 --> 00:35:05,810 So is everyone clear on the implications here? 704 00:35:08,357 --> 00:35:09,190 If not, let me know. 705 00:35:09,190 --> 00:35:10,714 Because that's what this class is for. 706 00:35:10,714 --> 00:35:11,158 AUDIENCE: Yeah. 707 00:35:11,158 --> 00:35:12,741 Can you just go over it one more time? 708 00:35:12,741 --> 00:35:13,600 PROFESSOR: Yes. 709 00:35:13,600 --> 00:35:17,560 So, for exothermic reactions where Q is greater than 0, 710 00:35:17,560 --> 00:35:20,590 all that says from our initial part of the Q equation, 711 00:35:20,590 --> 00:35:24,010 if Q is greater than 0, then we have this thing right here, 712 00:35:24,010 --> 00:35:27,190 where the final kinetic energies have to be larger 713 00:35:27,190 --> 00:35:28,690 than the initial one. 714 00:35:28,690 --> 00:35:30,940 Which is to say that some mass has turned 715 00:35:30,940 --> 00:35:33,350 into extra kinetic energy. 716 00:35:33,350 --> 00:35:36,260 And the solution to these is pretty easy 717 00:35:36,260 --> 00:35:39,170 because you don't need any kinetic energy to make 718 00:35:39,170 --> 00:35:41,330 an exothermic reaction happen. 719 00:35:41,330 --> 00:35:45,530 So you can just set T1 equal to 0, which makes s equal to 0, 720 00:35:45,530 --> 00:35:48,300 because they're all multiplied here. 721 00:35:48,300 --> 00:35:51,750 And then it simplifies lowercase t 722 00:35:51,750 --> 00:35:55,110 as just a ratio of those masses times the Q equation, 723 00:35:55,110 --> 00:35:58,590 which will tell you pretty much how much kinetic energy is 724 00:35:58,590 --> 00:36:04,480 going to be sent off to particle 3 right here. 725 00:36:04,480 --> 00:36:05,200 Up there. 726 00:36:05,200 --> 00:36:07,310 Particle 3. 727 00:36:07,310 --> 00:36:12,560 Because then we have this condition, if root T3 equals s 728 00:36:12,560 --> 00:36:16,610 plus square root of s squared plus t, 729 00:36:16,610 --> 00:36:21,550 and we've decided that s equals 0, 730 00:36:21,550 --> 00:36:30,420 that just means that T3 equals lowercase t, which equals that. 731 00:36:30,420 --> 00:36:33,170 So then you've uniquely defined the kinetic energy 732 00:36:33,170 --> 00:36:36,550 for an exothermic reaction, as long 733 00:36:36,550 --> 00:36:39,010 as you have no incoming kinetic energy. 734 00:36:39,010 --> 00:36:42,260 For the case of an endothermic reaction, first of all, 735 00:36:42,260 --> 00:36:44,170 we know that the incoming kinetic energy 736 00:36:44,170 --> 00:36:46,750 has to be greater than 0. 737 00:36:46,750 --> 00:36:48,370 It's like the excess energy that you 738 00:36:48,370 --> 00:36:50,430 need to get a chemical reaction going. 739 00:36:50,430 --> 00:36:52,330 Has anyone here ever played with-- 740 00:36:52,330 --> 00:36:55,715 what's the one, a striking one here? 741 00:36:55,715 --> 00:36:57,340 Well, has anyone ever lit anything on-- 742 00:36:57,340 --> 00:36:59,080 no, that's-- yeah. 743 00:36:59,080 --> 00:36:59,890 Of course you have. 744 00:36:59,890 --> 00:37:02,164 And that's not a good explanation. 745 00:37:02,164 --> 00:37:03,120 Hmm. 746 00:37:03,120 --> 00:37:07,552 What's a good, striking endothermic chemical reaction? 747 00:37:07,552 --> 00:37:09,090 Can anyone think of one? 748 00:37:09,090 --> 00:37:09,590 Yeah? 749 00:37:09,590 --> 00:37:12,410 AUDIENCE: When you put tin foil in Liquid-Plumr and it 750 00:37:12,410 --> 00:37:12,910 releases-- 751 00:37:12,910 --> 00:37:15,207 PROFESSOR: And it's a hydrogen generator? 752 00:37:15,207 --> 00:37:16,040 AUDIENCE: Let's see. 753 00:37:16,040 --> 00:37:17,060 I guess that's an explosion. 754 00:37:17,060 --> 00:37:18,080 PROFESSOR: I think that happen-- yeah. 755 00:37:18,080 --> 00:37:19,330 That's more like an explosion. 756 00:37:19,330 --> 00:37:22,540 That's, like, the intuitive definition of exothermic. 757 00:37:22,540 --> 00:37:23,040 Yeah. 758 00:37:23,040 --> 00:37:24,730 Actually, there's a fun one you can do, too. 759 00:37:24,730 --> 00:37:26,120 This is great that it's on video. 760 00:37:26,120 --> 00:37:29,990 You do that plus put manganese dioxide in hydrogen peroxide 761 00:37:29,990 --> 00:37:32,090 and you have an oxygen generator. 762 00:37:32,090 --> 00:37:35,178 And then you have the purest, beyond glacially pure, 763 00:37:35,178 --> 00:37:35,720 spring water. 764 00:37:35,720 --> 00:37:37,850 You just mix H and O directly. 765 00:37:37,850 --> 00:37:39,020 Just don't get near it. 766 00:37:39,020 --> 00:37:41,390 Because it tends to be pretty loud. 767 00:37:41,390 --> 00:37:44,150 We do this for our RTC or reactor technology 768 00:37:44,150 --> 00:37:48,063 course, where I've got to teach a bunch of CEOs enough 769 00:37:48,063 --> 00:37:50,480 basic high school chemistry so they can understand reactor 770 00:37:50,480 --> 00:37:51,630 water chemistry. 771 00:37:51,630 --> 00:37:54,140 And the way I make sure that they're paying attention 772 00:37:54,140 --> 00:37:56,530 is with a tremendous explosion. 773 00:37:56,530 --> 00:38:00,550 So folks come here, pay about $25 grand apiece for me 774 00:38:00,550 --> 00:38:03,140 to fire water-powered bottle rockets at them. 775 00:38:03,140 --> 00:38:04,750 It's a pretty sweet job. 776 00:38:04,750 --> 00:38:06,787 So if you guys are interested in academia, 777 00:38:06,787 --> 00:38:08,370 you know, these things happen in life. 778 00:38:08,370 --> 00:38:10,100 It's pretty cool. 779 00:38:10,100 --> 00:38:10,600 Yeah. 780 00:38:10,600 --> 00:38:10,780 All right. 781 00:38:10,780 --> 00:38:13,330 Since I can't think of any endothermic chemical reactions 782 00:38:13,330 --> 00:38:14,705 off the top of my head, I'll have 783 00:38:14,705 --> 00:38:16,450 to keep it general and abstract and say, 784 00:38:16,450 --> 00:38:18,070 if you have an endothermic reaction, 785 00:38:18,070 --> 00:38:20,770 you have to add energy in the form of heat 786 00:38:20,770 --> 00:38:22,690 to get the reaction going. 787 00:38:22,690 --> 00:38:24,910 In an endothermic nuclear reaction, 788 00:38:24,910 --> 00:38:26,950 heating up the material does not impart 789 00:38:26,950 --> 00:38:28,600 very much kinetic energy. 790 00:38:28,600 --> 00:38:31,030 You might raise it from a fraction of an electron 791 00:38:31,030 --> 00:38:33,495 volt to maybe a couple of electron volts 792 00:38:33,495 --> 00:38:34,870 if things are so hot that they're 793 00:38:34,870 --> 00:38:36,670 glowing in the ultraviolet. 794 00:38:36,670 --> 00:38:38,620 That doesn't cut it for nuclear. 795 00:38:38,620 --> 00:38:42,970 So you have to impart kinetic energy to the incoming particle 796 00:38:42,970 --> 00:38:46,540 such that the kinetic energy plus the rest masses 797 00:38:46,540 --> 00:38:50,920 is enough to create the rest masses of the final particles. 798 00:38:50,920 --> 00:38:53,020 And that's the general explanation I'd give. 799 00:38:55,870 --> 00:38:57,580 I forget who had asked the question. 800 00:38:57,580 --> 00:38:59,830 But does that help explain it a bit? 801 00:38:59,830 --> 00:39:00,810 AUDIENCE: [INAUDIBLE] 802 00:39:00,810 --> 00:39:01,477 PROFESSOR: Cool. 803 00:39:01,477 --> 00:39:02,380 OK. 804 00:39:02,380 --> 00:39:04,190 I'll take five minutes. 805 00:39:04,190 --> 00:39:06,970 And let's do a severely reduced case 806 00:39:06,970 --> 00:39:09,940 of this, the case of elastic neutron scattering. 807 00:39:09,940 --> 00:39:12,790 It's kind of a flash forward to what we'll be 808 00:39:12,790 --> 00:39:14,470 doing in the next month or so. 809 00:39:14,470 --> 00:39:16,953 Does everyone have what's behind this board here? 810 00:39:16,953 --> 00:39:18,620 I know that was, like, three boards ago. 811 00:39:18,620 --> 00:39:20,010 So I hope so. 812 00:39:20,010 --> 00:39:23,073 So let's take the case of elastic neutron scattering. 813 00:39:30,468 --> 00:39:33,760 Remember I told you that after we developed this highly 814 00:39:33,760 --> 00:39:35,412 general solution to the Q equation, 815 00:39:35,412 --> 00:39:37,120 everything else that we're going to study 816 00:39:37,120 --> 00:39:39,040 is just a reduction of that. 817 00:39:39,040 --> 00:39:41,560 And this is about as reduced as it gets. 818 00:39:41,560 --> 00:39:44,080 So in elastic neutron scattering, 819 00:39:44,080 --> 00:39:47,010 we can say that M1-- 820 00:39:47,010 --> 00:39:50,670 well, what's the mass of a neutron in AMU? 821 00:39:50,670 --> 00:39:55,050 And let's forgive our six decimal points' precision 822 00:39:55,050 --> 00:39:55,763 for now. 823 00:39:55,763 --> 00:39:56,430 What's it about? 824 00:39:56,430 --> 00:39:57,190 AUDIENCE: 1. 825 00:39:57,190 --> 00:39:58,160 PROFESSOR: 1. 826 00:39:58,160 --> 00:40:00,621 So we can say that M1 equals 1. 827 00:40:00,621 --> 00:40:03,270 And in the case of elastic scattering, 828 00:40:03,270 --> 00:40:05,670 the particles bounce into each other 829 00:40:05,670 --> 00:40:08,790 and leave with their original identities. 830 00:40:08,790 --> 00:40:10,290 So that also equals M3. 831 00:40:13,710 --> 00:40:18,180 If we're shooting neutrons at an arbitrary nucleus, what's M2? 832 00:40:22,720 --> 00:40:23,220 Yep? 833 00:40:23,220 --> 00:40:23,850 AUDIENCE: A? 834 00:40:23,850 --> 00:40:26,860 PROFESSOR: Just A, the mass number. 835 00:40:26,860 --> 00:40:29,100 Same as M4. 836 00:40:29,100 --> 00:40:32,470 Now, we don't have M2 in this equation. 837 00:40:32,470 --> 00:40:34,910 Whatever. 838 00:40:34,910 --> 00:40:36,590 But the point is, yeah. 839 00:40:36,590 --> 00:40:38,740 We're going to use these two. 840 00:40:38,740 --> 00:40:40,650 We're going to use these two. 841 00:40:40,650 --> 00:40:42,000 So let's substitute that in. 842 00:40:42,000 --> 00:40:44,790 Oh, and one last other thing I mentioned. 843 00:40:44,790 --> 00:40:49,506 What is the Q value for elastic scattering? 844 00:40:49,506 --> 00:40:50,622 AUDIENCE: [INAUDIBLE] 845 00:40:50,622 --> 00:40:51,330 PROFESSOR: Right. 846 00:40:51,330 --> 00:40:52,390 0. 847 00:40:52,390 --> 00:40:55,060 Because the Q value is the difference in the rest 848 00:40:55,060 --> 00:40:58,000 masses of the ingoing and outgoing particles. 849 00:40:58,000 --> 00:41:00,540 If the ingoing and outgoing particles are the same, 850 00:41:00,540 --> 00:41:06,010 M1 equals M3, M2 equals M4, that sum equals 0. 851 00:41:06,010 --> 00:41:07,300 Therefore, Q equals 0. 852 00:41:09,810 --> 00:41:11,610 So let's use these three things right here 853 00:41:11,610 --> 00:41:15,930 and rewrite the general Q equation in those terms. 854 00:41:15,930 --> 00:41:18,200 Which board is it on? 855 00:41:18,200 --> 00:41:19,690 Right there. 856 00:41:19,690 --> 00:41:22,010 So let's copy that down. 857 00:41:22,010 --> 00:41:31,900 So let's say we have T1 times M1 is 1 over M4 is A minus 1 858 00:41:31,900 --> 00:41:37,560 minus 2 over M4 is A. This is where it gets nice and easy. 859 00:41:37,560 --> 00:41:41,640 M1 and M3 are just 1. 860 00:41:41,640 --> 00:41:46,228 So 1 times 1 times T1. 861 00:41:46,228 --> 00:41:47,520 We don't know what that is yet. 862 00:41:47,520 --> 00:41:52,650 So let's call it the Tn, T of the neutron coming in. 863 00:41:55,530 --> 00:41:56,160 How about this? 864 00:41:56,160 --> 00:42:04,550 We'll call it T in and T out for ease of understanding. 865 00:42:04,550 --> 00:42:05,750 Cosine theta. 866 00:42:05,750 --> 00:42:08,040 What do we have left? 867 00:42:08,040 --> 00:42:12,160 Plus T out. 868 00:42:12,160 --> 00:42:16,740 And let's make T1 into an in right there. 869 00:42:16,740 --> 00:42:19,680 Times M3 over M4. 870 00:42:19,680 --> 00:42:20,955 M3 was 1. 871 00:42:20,955 --> 00:42:27,150 M4 is A. Plus 1 equals Q, equals 0. 872 00:42:27,150 --> 00:42:29,920 This is quite a simpler equation to solve. 873 00:42:29,920 --> 00:42:33,392 So let's group this all together. 874 00:42:33,392 --> 00:42:34,850 There's a couple of tricks that I'm 875 00:42:34,850 --> 00:42:36,392 going to apply right now to make sure 876 00:42:36,392 --> 00:42:38,590 that everything has A in the denominator 877 00:42:38,590 --> 00:42:40,180 to make stuff easier. 878 00:42:40,180 --> 00:42:44,780 We can call 1 A over A here. 879 00:42:44,780 --> 00:42:49,740 We can call 1 A over A there. 880 00:42:49,740 --> 00:42:52,830 That lets us combine our denominators 881 00:42:52,830 --> 00:42:55,370 and stick the sine right there. 882 00:42:55,370 --> 00:42:58,430 That becomes an A. Same thing here. 883 00:42:58,430 --> 00:43:02,930 I'll just connect the dashes and stick the minus sign there, 884 00:43:02,930 --> 00:43:05,380 leaving an A right there. 885 00:43:05,380 --> 00:43:07,150 Now we can just multiply everything 886 00:43:07,150 --> 00:43:11,050 by A, both sides of the equation. 887 00:43:11,050 --> 00:43:15,010 So the As go away there. 888 00:43:15,010 --> 00:43:16,835 We have a much simpler equation. 889 00:43:16,835 --> 00:43:20,440 0 equals T in of-- 890 00:43:20,440 --> 00:43:27,240 let's see-- 1 minus A over 1. 891 00:43:27,240 --> 00:43:27,740 OK. 892 00:43:27,740 --> 00:43:36,850 We'll just call it 1 minus A. Minus 2 root T in T 893 00:43:36,850 --> 00:43:47,240 out cosine theta plus T out A plus 1. 894 00:43:47,240 --> 00:43:50,300 And, OK, it's 10 minutes of, or it's five minutes 895 00:43:50,300 --> 00:43:51,800 of five minutes of. 896 00:43:51,800 --> 00:43:53,420 So I'm going to stop this right here 897 00:43:53,420 --> 00:43:55,092 at a fairly simple equation. 898 00:43:55,092 --> 00:43:56,300 We'll pick it up on Thursday. 899 00:43:56,300 --> 00:43:58,430 And I want to open the last five minutes to any questions 900 00:43:58,430 --> 00:43:59,180 you guys may have. 901 00:44:02,370 --> 00:44:04,370 Since that request came in on the anonymous rant 902 00:44:04,370 --> 00:44:06,700 forum, which hopefully you all know now exists. 903 00:44:06,700 --> 00:44:07,435 Yep. 904 00:44:07,435 --> 00:44:10,405 AUDIENCE: So what exactly is forward scattering? 905 00:44:10,405 --> 00:44:12,890 I didn't really get that before. 906 00:44:12,890 --> 00:44:14,900 PROFESSOR: So let's look at elastic scattering 907 00:44:14,900 --> 00:44:15,720 as an example. 908 00:44:15,720 --> 00:44:18,290 So in elastic scattering, two particles 909 00:44:18,290 --> 00:44:20,450 bounce off each other like billiard balls. 910 00:44:20,450 --> 00:44:23,390 In forward elastic scattering, the neutron, 911 00:44:23,390 --> 00:44:27,560 after interacting somehow with particle 2, 912 00:44:27,560 --> 00:44:30,840 keeps moving forward unscathed. 913 00:44:30,840 --> 00:44:32,460 So in the elastic scattering sense, 914 00:44:32,460 --> 00:44:36,422 forward scattering is also known as missing. 915 00:44:36,422 --> 00:44:37,400 AUDIENCE: [INAUDIBLE] 916 00:44:37,400 --> 00:44:38,940 PROFESSOR: You can have other reactions, 917 00:44:38,940 --> 00:44:40,815 let's say, where you have a particle at rest, 918 00:44:40,815 --> 00:44:45,320 another particle slams into it, and the whole center of mass 919 00:44:45,320 --> 00:44:46,185 moves together. 920 00:44:46,185 --> 00:44:47,810 I don't know if you'd call that forward 921 00:44:47,810 --> 00:44:49,310 scattering as much as, let's say, 922 00:44:49,310 --> 00:44:52,410 capture or fusion or something. 923 00:44:52,410 --> 00:44:55,700 But in this case, scattering means that two particles go in, 924 00:44:55,700 --> 00:44:57,710 two particles leave. 925 00:44:57,710 --> 00:44:59,330 Whether it's elastically, which means 926 00:44:59,330 --> 00:45:01,730 with no transfer of energy into rest mass, 927 00:45:01,730 --> 00:45:03,770 or inelastically, where, let's say, 928 00:45:03,770 --> 00:45:07,280 a neutron is absorbed and then re-emitted 929 00:45:07,280 --> 00:45:08,690 from a different energy level. 930 00:45:08,690 --> 00:45:11,260 And that's something we'll get into in, like, a month. 931 00:45:11,260 --> 00:45:13,780 So you can have forward elastic or inelastic scattering. 932 00:45:13,780 --> 00:45:16,810 In this case, I'm talking about elastic scattering, which 933 00:45:16,810 --> 00:45:20,470 is the simple case of, like, the billiard balls miss each other. 934 00:45:20,470 --> 00:45:23,080 Which is technically a case that can be treated by this. 935 00:45:23,080 --> 00:45:25,930 Because all you have to do is plug in theta equals 0 936 00:45:25,930 --> 00:45:28,600 and you have the case for how much energy 937 00:45:28,600 --> 00:45:32,483 do you think the neutron would lose if it misses particle 2. 938 00:45:32,483 --> 00:45:33,400 AUDIENCE: [INAUDIBLE]. 939 00:45:33,400 --> 00:45:33,850 PROFESSOR: Yeah. 940 00:45:33,850 --> 00:45:34,770 It wouldn't lose any energy. 941 00:45:34,770 --> 00:45:34,970 Right? 942 00:45:34,970 --> 00:45:36,510 It would have the same energy. 943 00:45:36,510 --> 00:45:38,380 So that's the case for forward scattering. 944 00:45:38,380 --> 00:45:42,080 A neutron, when it interacts somehow with another particle, 945 00:45:42,080 --> 00:45:45,320 can lose as little as none of its energy. 946 00:45:45,320 --> 00:45:47,990 If it misses, no one said it had to lose any energy. 947 00:45:47,990 --> 00:45:49,830 And by solving this equation here, 948 00:45:49,830 --> 00:45:51,650 which we'll do on Thursday, we'll 949 00:45:51,650 --> 00:45:54,500 see what the maximum amount of energy that neutron can lose 950 00:45:54,500 --> 00:45:58,830 is, which is the basis for neutrons slowing down 951 00:45:58,830 --> 00:46:02,350 or moderation in reactors. 952 00:46:02,350 --> 00:46:03,330 Yeah. 953 00:46:03,330 --> 00:46:07,130 AUDIENCE: Are T in and T out equal there, in which case 954 00:46:07,130 --> 00:46:10,012 that equation is used to solve for theta? 955 00:46:10,012 --> 00:46:13,660 PROFESSOR: T in and T out are not always equal. 956 00:46:13,660 --> 00:46:17,590 But in the case of forward elastic scattering, 957 00:46:17,590 --> 00:46:18,990 they would be. 958 00:46:18,990 --> 00:46:21,540 Because the neutron comes in with energy T in 959 00:46:21,540 --> 00:46:24,390 and it leaves with energy T in. 960 00:46:24,390 --> 00:46:27,780 For any other case in which the neutron comes off 961 00:46:27,780 --> 00:46:30,060 of particle 2 at a different angle, 962 00:46:30,060 --> 00:46:34,290 it will have bounced off of particle 2, moving particle 2 963 00:46:34,290 --> 00:46:36,180 at some other angle phi, and giving it 964 00:46:36,180 --> 00:46:38,540 some of its energy elastically. 965 00:46:38,540 --> 00:46:41,180 The total amount of that kinetic energy will be conserved. 966 00:46:41,180 --> 00:46:46,000 So let's say-- what did we call it? 967 00:46:46,000 --> 00:46:46,500 What is it? 968 00:46:46,500 --> 00:46:47,000 Yeah. 969 00:46:47,000 --> 00:46:51,320 So T1 would have to be the same as T3 and T4 970 00:46:51,320 --> 00:46:53,520 together for this Q equation where 971 00:46:53,520 --> 00:46:56,090 Q equals 0 to be satisfied. 972 00:46:56,090 --> 00:46:58,500 So what you said can happen. 973 00:46:58,500 --> 00:47:02,097 But it's only the case for forward scattering. 974 00:47:02,097 --> 00:47:02,930 Any other questions? 975 00:47:02,930 --> 00:47:04,780 Yep. 976 00:47:04,780 --> 00:47:07,650 AUDIENCE: In the case of an exothermic reaction, 977 00:47:07,650 --> 00:47:10,258 we assume that T1 equals 0. 978 00:47:10,258 --> 00:47:13,103 Can you re-explain why we made that assumption? 979 00:47:13,103 --> 00:47:15,520 PROFESSOR: So the question was, in an exothermic reaction, 980 00:47:15,520 --> 00:47:17,710 why did we say T1 equals 0? 981 00:47:17,710 --> 00:47:19,270 It's not always the case. 982 00:47:19,270 --> 00:47:22,300 But it provides the simplest case for us to analyze. 983 00:47:22,300 --> 00:47:26,560 So an exothermic reaction can happen when T1 equals 0. 984 00:47:26,560 --> 00:47:29,790 It can also happen when T1 is greater than 0. 985 00:47:29,790 --> 00:47:31,970 So we're not putting any restrictions on that. 986 00:47:31,970 --> 00:47:36,650 But in the case that T1 equals 0, s is destroyed 987 00:47:36,650 --> 00:47:39,470 and the harder part of T is destroyed, 988 00:47:39,470 --> 00:47:42,200 making the solution to this equation very 989 00:47:42,200 --> 00:47:43,970 simple and intuitive. 990 00:47:43,970 --> 00:47:47,120 Which is to say that if you just have two particles that 991 00:47:47,120 --> 00:47:51,320 are kind of at rest and they just merge and fire off 992 00:47:51,320 --> 00:47:53,930 two different pieces in opposite directions, 993 00:47:53,930 --> 00:47:56,180 their energies are proportional to the ratio 994 00:47:56,180 --> 00:47:59,583 of their single mass to the total mass. 995 00:47:59,583 --> 00:48:01,250 So that's like a center of mass problem. 996 00:48:01,250 --> 00:48:04,810 You'll notice also I'm not using center of mass coordinates. 997 00:48:04,810 --> 00:48:08,626 Center of mass coor-- who here has used those in 801 or 802? 998 00:48:08,626 --> 00:48:10,920 And who here enjoyed the experience? 999 00:48:10,920 --> 00:48:11,460 Oh. 1000 00:48:11,460 --> 00:48:11,960 Wow. 1001 00:48:11,960 --> 00:48:13,860 No hands whatsoever. 1002 00:48:13,860 --> 00:48:17,540 So center of mass coordinates and laboratory coordinates 1003 00:48:17,540 --> 00:48:20,330 are different ways of expressing the same thing. 1004 00:48:20,330 --> 00:48:22,460 Usually you can write simpler equations 1005 00:48:22,460 --> 00:48:24,243 in center of mass coordinates. 1006 00:48:24,243 --> 00:48:26,660 But for most people-- and I'm going to go with all of you, 1007 00:48:26,660 --> 00:48:29,465 since none of you raised your hand-- it's not that intuitive. 1008 00:48:29,465 --> 00:48:30,590 That's the same way for me. 1009 00:48:30,590 --> 00:48:32,360 So that's why I've made a decision 1010 00:48:32,360 --> 00:48:34,910 to show things in laboratory coordinates, 1011 00:48:34,910 --> 00:48:37,250 so you have a fixed frame of reference 1012 00:48:37,250 --> 00:48:39,650 and not a moving frame of reference of the center of mass 1013 00:48:39,650 --> 00:48:41,240 of the two particles. 1014 00:48:41,240 --> 00:48:44,480 But the center of mass idea does kind of make sense here. 1015 00:48:44,480 --> 00:48:47,210 If you have two particles that are almost touching 1016 00:48:47,210 --> 00:48:50,900 and then they touch and they break into pieces and fly off, 1017 00:48:50,900 --> 00:48:54,650 the total amount of momentum of that center of mass was 0. 1018 00:48:54,650 --> 00:48:56,540 And it has to remain 0. 1019 00:48:56,540 --> 00:48:59,840 And so each of these particles will take a differing ratio 1020 00:48:59,840 --> 00:49:01,700 of their masses away. 1021 00:49:01,700 --> 00:49:04,720 We already looked at this for the case of alpha decay, 1022 00:49:04,720 --> 00:49:07,450 where if you have one nucleus just sitting here-- let's 1023 00:49:07,450 --> 00:49:09,190 say there was no T1. 1024 00:49:09,190 --> 00:49:12,290 There was just some unstable T2 that was about to explode 1025 00:49:12,290 --> 00:49:13,375 and then it did. 1026 00:49:13,375 --> 00:49:15,130 Remember how we talked about how the Q 1027 00:49:15,130 --> 00:49:18,400 value of an alpha reaction is not the same energy that you 1028 00:49:18,400 --> 00:49:20,660 see the alpha decay at? 1029 00:49:20,660 --> 00:49:22,010 Same thing right here. 1030 00:49:22,010 --> 00:49:25,700 So this Q equation describes that same situation. 1031 00:49:25,700 --> 00:49:28,290 Notice there's no hint of M1. 1032 00:49:28,290 --> 00:49:30,720 There was really no M1 in the end. 1033 00:49:30,720 --> 00:49:34,590 We don't care what the initial mass of the particle that 1034 00:49:34,590 --> 00:49:35,640 made alpha decay is. 1035 00:49:35,640 --> 00:49:39,990 All we care about is what are the mass ratios and energy 1036 00:49:39,990 --> 00:49:44,470 ratios of the alpha particle and its recoil nucleus. 1037 00:49:44,470 --> 00:49:46,590 So it all does tie together. 1038 00:49:46,590 --> 00:49:48,340 That's the neat thing, is this universal Q 1039 00:49:48,340 --> 00:49:51,910 equation can be used to describe almost everything we're 1040 00:49:51,910 --> 00:49:54,550 going to talk about. 1041 00:49:54,550 --> 00:49:57,040 So this is as complex as it gets. 1042 00:49:57,040 --> 00:49:59,530 And from now on, we'll be looking at simpler reductions 1043 00:49:59,530 --> 00:50:01,850 and specific cases of each one. 1044 00:50:01,850 --> 00:50:02,760 So it's five of. 1045 00:50:02,760 --> 00:50:05,177 I want to actually make sure to get you to your next class 1046 00:50:05,177 --> 00:50:05,695 on time. 1047 00:50:05,695 --> 00:50:08,790 And I'll see you guys on Thursday.