1 00:00:00,995 --> 00:00:03,320 The following content is provided under a Creative 2 00:00:03,320 --> 00:00:04,710 Commons license. 3 00:00:04,710 --> 00:00:06,920 Your support will help MIT OpenCourseWare 4 00:00:06,920 --> 00:00:11,010 continue to offer high quality educational resources for free. 5 00:00:11,010 --> 00:00:13,580 To make a donation or to view additional materials 6 00:00:13,580 --> 00:00:17,540 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,540 --> 00:00:18,420 at ocw.mit.edu. 8 00:00:22,505 --> 00:00:25,600 MICHAEL SHORT: Hey guys, hope you enjoyed the brief break 9 00:00:25,600 --> 00:00:27,210 from the heavy technical stuff. 10 00:00:27,210 --> 00:00:29,085 Because we're going to get right back into it 11 00:00:29,085 --> 00:00:31,222 and develop the neutron transport equation today, 12 00:00:31,222 --> 00:00:32,680 the one that you see on everybody's 13 00:00:32,680 --> 00:00:34,470 t-shirts here in the department. 14 00:00:34,470 --> 00:00:36,550 So I think multiple years folks have 15 00:00:36,550 --> 00:00:39,940 used this equation on the back of t-shirts just be like, 16 00:00:39,940 --> 00:00:40,540 we're awesome. 17 00:00:40,540 --> 00:00:42,940 And we do difficult math. 18 00:00:42,940 --> 00:00:45,563 Well, this is what you're going to start to do it. 19 00:00:45,563 --> 00:00:47,230 In fact, it's big enough and hard enough 20 00:00:47,230 --> 00:00:50,770 that we're going to spend all day today developing it, 21 00:00:50,770 --> 00:00:53,080 like actually writing out the terms of the equation 22 00:00:53,080 --> 00:00:55,390 and understanding what it actually means. 23 00:00:55,390 --> 00:00:57,490 Before, on Thursday and Friday, we're 24 00:00:57,490 --> 00:01:00,580 going to reduce it down to a much simpler equation, 25 00:01:00,580 --> 00:01:02,680 something that you can actually solve and do 26 00:01:02,680 --> 00:01:05,080 some simple reactor calculations with. 27 00:01:05,080 --> 00:01:09,100 We started off the whole idea of the neutron transport equation 28 00:01:09,100 --> 00:01:13,710 as a way to track some population of neutrons. 29 00:01:13,710 --> 00:01:17,770 Let's see, I'm going to have our variable list up here. 30 00:01:17,770 --> 00:01:19,763 What I'll probably do is on Thursday and Friday 31 00:01:19,763 --> 00:01:21,430 I'll just have it back up on the screens 32 00:01:21,430 --> 00:01:23,097 so that we don't have to write it twice. 33 00:01:23,097 --> 00:01:25,985 But there's going to be a lot of variables in this equation. 34 00:01:25,985 --> 00:01:27,610 I'm going to do my best, again, to make 35 00:01:27,610 --> 00:01:31,240 the difference between V and nu very obvious, and anything else 36 00:01:31,240 --> 00:01:32,230 like that. 37 00:01:32,230 --> 00:01:36,940 But the goal is to track some population of neutrons, 38 00:01:36,940 --> 00:01:40,330 at some position, at some energy, 39 00:01:40,330 --> 00:01:45,110 traveling in some direction omega, as a function of time. 40 00:01:45,110 --> 00:01:49,060 And the 3D representation of what we're looking at here 41 00:01:49,060 --> 00:01:54,250 is let's say we had some small volume element, which 42 00:01:54,250 --> 00:01:57,370 we'll call that our dV. 43 00:01:57,370 --> 00:02:00,100 That's got normal vectors sticking out of it. 44 00:02:00,100 --> 00:02:03,550 We'll call those n-hats in all directions. 45 00:02:03,550 --> 00:02:08,530 And inside there, let's say if this is our energy scale, 46 00:02:08,530 --> 00:02:12,850 we're tracking the population of neutrons 47 00:02:12,850 --> 00:02:18,540 that occupies some small energy group dE, 48 00:02:18,540 --> 00:02:23,700 and is also traveling in some small direction 49 00:02:23,700 --> 00:02:25,092 that we designate as d-omega. 50 00:02:27,930 --> 00:02:29,660 So that's the goal of this whole equation 51 00:02:29,660 --> 00:02:32,970 is to track the number of neutrons at any given position. 52 00:02:32,970 --> 00:02:37,730 So let's call this distance, or the vector r. 53 00:02:37,730 --> 00:02:41,390 In this little volume, traveling in some direction omega, 54 00:02:41,390 --> 00:02:46,550 with some infinitesimally small energy group d. 55 00:02:46,550 --> 00:02:49,100 That's going to be the goal of the whole thing. 56 00:02:49,100 --> 00:02:51,900 And what we'll do is write this to say 57 00:02:51,900 --> 00:02:56,220 the change in the population of neutrons at a given distance, 58 00:02:56,220 --> 00:03:03,910 energy, angle, and time, over time 59 00:03:03,910 --> 00:03:06,440 is just going to be a sum of gain and loss terms. 60 00:03:11,110 --> 00:03:13,190 And what I think we'll take all day today to do 61 00:03:13,190 --> 00:03:15,440 is to figure out what are the actual physical things 62 00:03:15,440 --> 00:03:18,800 that neutrons can do in and out of this volume, 63 00:03:18,800 --> 00:03:20,990 and how do we turn those into math, something 64 00:03:20,990 --> 00:03:23,980 that we can abstract and solve? 65 00:03:23,980 --> 00:03:26,020 There's a couple other terms that we're 66 00:03:26,020 --> 00:03:27,710 going to put up here. 67 00:03:27,710 --> 00:03:31,210 We'll say that the flux of neutrons, which is usually 68 00:03:31,210 --> 00:03:34,960 the variable that we actually track, 69 00:03:34,960 --> 00:03:40,120 is just the velocity times the neutron population. 70 00:03:40,120 --> 00:03:46,540 And also let's define some angularly independent terms. 71 00:03:50,608 --> 00:03:52,150 Because in the end we've been talking 72 00:03:52,150 --> 00:03:55,480 about what's the probability of some neutron 73 00:03:55,480 --> 00:03:58,000 or particle interacting with some electron going out 74 00:03:58,000 --> 00:03:59,230 at some angle. 75 00:03:59,230 --> 00:04:01,840 But as we're interested in how many neutrons are there 76 00:04:01,840 --> 00:04:04,690 in the reactor, we usually don't care in which direction 77 00:04:04,690 --> 00:04:05,440 they're traveling. 78 00:04:05,440 --> 00:04:08,350 So the first simplification that we will do 79 00:04:08,350 --> 00:04:10,812 is get rid of any sort of angular dependence, 80 00:04:10,812 --> 00:04:12,520 getting rid of two of the seven variables 81 00:04:12,520 --> 00:04:13,995 that we're dealing with here. 82 00:04:13,995 --> 00:04:15,370 So all these variables right here 83 00:04:15,370 --> 00:04:17,170 will be dependent on angle. 84 00:04:17,170 --> 00:04:18,670 And all these variables right here 85 00:04:18,670 --> 00:04:24,930 will be angularly independent. 86 00:04:29,460 --> 00:04:31,590 So there'll be some corresponding capital 87 00:04:31,590 --> 00:04:36,260 N, or number of neutrons as a function of r, E, and t. 88 00:04:38,970 --> 00:04:41,320 We'll call this Flux. 89 00:04:41,320 --> 00:04:42,340 We'll call this Number. 90 00:04:45,813 --> 00:04:47,730 There's going to be a number of cross sections 91 00:04:47,730 --> 00:04:48,897 that we need to worry about. 92 00:04:48,897 --> 00:04:53,820 So we'll refer to little sigma as a function of energy, 93 00:04:53,820 --> 00:05:03,390 as our micro-cross-section, and big sigma of E 94 00:05:03,390 --> 00:05:05,220 as a macroscopic cross-section. 95 00:05:11,690 --> 00:05:16,355 Then you want to remember the relation between these two. 96 00:05:16,355 --> 00:05:17,840 AUDIENCE: Solid angle? 97 00:05:17,840 --> 00:05:20,660 MICHAEL SHORT: The solid angle, not quite. 98 00:05:20,660 --> 00:05:22,598 That's, let's see. 99 00:05:22,598 --> 00:05:24,140 There's a difference between-- and so 100 00:05:24,140 --> 00:05:26,030 what these physically mean is little sigma 101 00:05:26,030 --> 00:05:28,977 means the probability of interaction with one particle. 102 00:05:28,977 --> 00:05:30,560 And this is just the total probability 103 00:05:30,560 --> 00:05:32,893 of interaction with all the particles that may be there. 104 00:05:32,893 --> 00:05:33,748 So yeah, Chris? 105 00:05:33,748 --> 00:05:34,790 AUDIENCE: Number density? 106 00:05:34,790 --> 00:05:37,550 MICHAEL SHORT: There's number density. 107 00:05:37,550 --> 00:05:39,770 Already we have another variable conflict. 108 00:05:39,770 --> 00:05:42,930 How do we want to resolve this? 109 00:05:42,930 --> 00:05:43,440 Let's see. 110 00:05:46,460 --> 00:05:48,962 We'll have to change the symbol somehow. 111 00:05:48,962 --> 00:05:52,195 Let's make it cursive. 112 00:05:52,195 --> 00:05:53,320 Don't know what else to do. 113 00:05:53,320 --> 00:05:55,195 I don't want to give it a number other than n 114 00:05:55,195 --> 00:05:57,030 since we're talking about neutrons, 115 00:05:57,030 --> 00:05:59,155 or that right here it's going to be number density. 116 00:06:03,750 --> 00:06:06,350 And in the end, we're worried about some sort of reaction 117 00:06:06,350 --> 00:06:11,150 rate, which is always going to equal some flux, 118 00:06:11,150 --> 00:06:15,470 or let's just stick with some angularly dependent flux, 119 00:06:15,470 --> 00:06:23,330 that r, E omega t, times some cross-section 120 00:06:23,330 --> 00:06:24,800 as a function of energy. 121 00:06:24,800 --> 00:06:26,460 And it's these reaction rates that 122 00:06:26,460 --> 00:06:30,690 are the rates of gains and losses of neutrons 123 00:06:30,690 --> 00:06:33,300 out of this volume, out of this little angle, 124 00:06:33,300 --> 00:06:35,970 out of this energy group, and out of that space, 125 00:06:35,970 --> 00:06:39,600 or into that volume energy group and space. 126 00:06:39,600 --> 00:06:42,780 So let's see, other terms that we'll want to define 127 00:06:42,780 --> 00:06:48,000 include nu, like last time. 128 00:06:48,000 --> 00:06:50,280 We'll call this neutron multiplication. 129 00:06:56,523 --> 00:06:57,940 In other words, this is the number 130 00:06:57,940 --> 00:07:01,690 of neutrons made on average during each fission event. 131 00:07:01,690 --> 00:07:03,790 And we give it energy dependence because as we 132 00:07:03,790 --> 00:07:08,170 saw on the Janis libraries on Friday, I think it was. 133 00:07:08,170 --> 00:07:09,320 What's today, Tuesday? 134 00:07:09,320 --> 00:07:10,690 I don't even know anymore. 135 00:07:10,690 --> 00:07:13,960 I think as we saw on Friday, that depends on energy 136 00:07:13,960 --> 00:07:15,615 for the higher energy levels. 137 00:07:15,615 --> 00:07:19,550 And there's also going to be some Kai 138 00:07:19,550 --> 00:07:30,010 spectrum, or some neutron birth spectrum, 139 00:07:30,010 --> 00:07:33,130 which tells you the average energy at which neutrons 140 00:07:33,130 --> 00:07:34,180 are born from fission. 141 00:07:34,180 --> 00:07:38,980 So regardless of what energy goes in to cause the fission, 142 00:07:38,980 --> 00:07:42,470 there's some probability distribution 143 00:07:42,470 --> 00:07:46,280 of a neutron being born at a certain energy. 144 00:07:46,280 --> 00:07:53,910 And it looks something like this, where that's about 1 MeV. 145 00:07:53,910 --> 00:07:56,160 That's about 10 MeV. 146 00:07:56,160 --> 00:08:00,158 And that average right there is around 2 MeV. 147 00:08:00,158 --> 00:08:01,950 And so it's important to note that neutrons 148 00:08:01,950 --> 00:08:03,360 are born at different energies. 149 00:08:03,360 --> 00:08:05,850 Because we want to track every single possible dE 150 00:08:05,850 --> 00:08:08,340 throughout this control volume, which 151 00:08:08,340 --> 00:08:10,400 we'll also call a reactor. 152 00:08:10,400 --> 00:08:12,930 Let's see, what other terms will we need to know? 153 00:08:12,930 --> 00:08:14,520 The different types of cross sections, 154 00:08:14,520 --> 00:08:16,320 or the different interactions that neutrons 155 00:08:16,320 --> 00:08:17,520 can have with matter. 156 00:08:17,520 --> 00:08:19,645 What are some of the ones that we had talked about? 157 00:08:23,440 --> 00:08:25,590 What can neutrons do when they run into stuff? 158 00:08:25,590 --> 00:08:26,350 AUDIENCE: Scatter. 159 00:08:26,350 --> 00:08:27,683 MICHAEL SHORT: They can scatter. 160 00:08:27,683 --> 00:08:31,111 So there's going to be some scattering cross-section. 161 00:08:33,701 --> 00:08:35,409 And when they scatter, the important part 162 00:08:35,409 --> 00:08:37,159 here is they're going to change in energy. 163 00:08:42,820 --> 00:08:44,610 What else can they do? 164 00:08:44,610 --> 00:08:45,110 Yeah? 165 00:08:45,110 --> 00:08:45,820 AUDIENCE: Absorbed. 166 00:08:45,820 --> 00:08:46,820 MICHAEL SHORT: They can be absorbed. 167 00:08:46,820 --> 00:08:48,640 So we'll have some sigma absorption. 168 00:08:55,080 --> 00:08:57,210 What are some of the various things that can happen 169 00:08:57,210 --> 00:08:58,395 when a neutron is absorbed? 170 00:09:02,196 --> 00:09:03,200 AUDIENCE: Fission. 171 00:09:03,200 --> 00:09:05,810 MICHAEL SHORT: Yeah, so one of them is fission. 172 00:09:11,680 --> 00:09:14,656 What are some of the other ones? 173 00:09:14,656 --> 00:09:15,406 AUDIENCE: Capture. 174 00:09:15,406 --> 00:09:16,590 MICHAEL SHORT: Yep, capture. 175 00:09:23,232 --> 00:09:25,190 What were some of the ones that we talked about 176 00:09:25,190 --> 00:09:27,740 during the Chadwick paper? 177 00:09:27,740 --> 00:09:29,180 AUDIENCE: Neutron [INAUDIBLE]. 178 00:09:29,180 --> 00:09:32,260 MICHAEL SHORT: Yep, so there can be some-- 179 00:09:32,260 --> 00:09:38,710 we'll call it n,in, which means one neutron goes in, i neutrons 180 00:09:38,710 --> 00:09:47,420 come out, so 1 to i neutrons, sure. 181 00:09:47,420 --> 00:09:48,100 Anything else? 182 00:09:51,320 --> 00:09:52,935 Encompassed in absorption? 183 00:09:58,980 --> 00:10:02,385 Well when we refer to scatter here, what type of scattering 184 00:10:02,385 --> 00:10:03,260 are we talking about? 185 00:10:07,688 --> 00:10:09,660 AUDIENCE: Compton's scatter? 186 00:10:09,660 --> 00:10:11,385 MICHAEL SHORT: Compton's for photons. 187 00:10:11,385 --> 00:10:13,260 It's OK. 188 00:10:13,260 --> 00:10:15,420 Was it elastic or inelastic scattering? 189 00:10:15,420 --> 00:10:16,170 AUDIENCE: Elastic. 190 00:10:16,170 --> 00:10:17,640 MICHAEL SHORT: Elastic scattering. 191 00:10:17,640 --> 00:10:20,430 So another thing you could call an absorption event, depending 192 00:10:20,430 --> 00:10:30,150 on what bin you put things in, is inelastic scattering, 193 00:10:30,150 --> 00:10:31,590 which is that kind of-- 194 00:10:31,590 --> 00:10:34,110 we call it scattering because one neutron goes in, 195 00:10:34,110 --> 00:10:35,610 one neutron comes out. 196 00:10:35,610 --> 00:10:38,510 But in reality, you have a compound nucleus forming 197 00:10:38,510 --> 00:10:40,920 and a neutron emitted from a different energy level. 198 00:10:40,920 --> 00:10:44,760 So it doesn't follow the simple ballistic laws of, 199 00:10:44,760 --> 00:10:47,850 and kinematic laws of inelastic scattering. 200 00:10:47,850 --> 00:10:49,050 What else can neutrons do? 201 00:10:52,092 --> 00:10:54,050 Now we're getting into the real esoteric stuff. 202 00:10:54,050 --> 00:10:55,883 But I want to see if you guys have any idea. 203 00:10:58,770 --> 00:11:02,640 Did you know that neutrons can decay? 204 00:11:05,410 --> 00:11:08,080 A low neutron is actually not a stable particle. 205 00:11:08,080 --> 00:11:11,130 If you look up on the Kyrie table of nuclides, 206 00:11:11,130 --> 00:11:13,460 it's got a half life of 12 minutes. 207 00:11:13,460 --> 00:11:15,170 So if you happen to be able to have 208 00:11:15,170 --> 00:11:17,570 neutrons in a bottle or something, which we actually 209 00:11:17,570 --> 00:11:18,640 can do. 210 00:11:18,640 --> 00:11:21,320 There's centers for ultra-cold neutrons and atoms. 211 00:11:21,320 --> 00:11:24,933 There's one at North Carolina State where they actually cool 212 00:11:24,933 --> 00:11:27,350 down neutrons to cryogenic temperatures to the point where 213 00:11:27,350 --> 00:11:29,150 they can actually confine them. 214 00:11:29,150 --> 00:11:32,540 They only live on average 12 minutes. 215 00:11:37,212 --> 00:11:38,920 And then there would also be what we call 216 00:11:38,920 --> 00:11:40,810 a neutron-neutron interactions. 217 00:11:44,180 --> 00:11:47,330 There is a finite, non-zero but very small 218 00:11:47,330 --> 00:11:50,870 probability that neutrons can hit other neutrons. 219 00:11:50,870 --> 00:11:55,970 But the mean-free path for these is on the order of 10 220 00:11:55,970 --> 00:11:59,495 to the 8th centimeters. 221 00:11:59,495 --> 00:12:01,370 So this is not something we have to consider. 222 00:12:01,370 --> 00:12:03,350 But it's interesting to know that yes, neutrons 223 00:12:03,350 --> 00:12:05,090 can run into other neutrons. 224 00:12:05,090 --> 00:12:07,280 And these sorts of things have been measured. 225 00:12:07,280 --> 00:12:09,680 We won't have to worry about this. 226 00:12:09,680 --> 00:12:11,810 We won't have to worry about neutron decay. 227 00:12:11,810 --> 00:12:14,060 But it's interesting to note that a low neutron is not 228 00:12:14,060 --> 00:12:15,460 a stable particle. 229 00:12:15,460 --> 00:12:20,000 It will spontaneously undergo beta decay, 230 00:12:20,000 --> 00:12:22,520 into a proton and an electron. 231 00:12:22,520 --> 00:12:24,510 Pretty neat, huh? 232 00:12:24,510 --> 00:12:30,220 Anyway, if we sum up all these possible interactions, 233 00:12:30,220 --> 00:12:33,340 we have one other cross-section, which 234 00:12:33,340 --> 00:12:37,330 we're going to call the total cross-section, the probability 235 00:12:37,330 --> 00:12:40,600 of absolutely any interaction occurring at all. 236 00:12:40,600 --> 00:12:44,260 Because any sort of interaction of that neutron 237 00:12:44,260 --> 00:12:49,520 is going to cause removal from this group of energy position, 238 00:12:49,520 --> 00:12:52,450 angle, location, whatever. 239 00:12:52,450 --> 00:12:55,570 Whether it's absorption, or fission, or elastic scattering, 240 00:12:55,570 --> 00:12:58,300 or inelastic scattering, any sort of event-- 241 00:12:58,300 --> 00:13:01,870 except for forward scattering, which means nothing happens-- 242 00:13:01,870 --> 00:13:05,530 is going to result in this neutron either leaving 243 00:13:05,530 --> 00:13:06,160 the volume. 244 00:13:06,160 --> 00:13:09,100 So it might scatter out of our little volume. 245 00:13:09,100 --> 00:13:13,180 Or it might change direction, scatter out of our d-omega. 246 00:13:13,180 --> 00:13:17,080 Or it will lose some energy, or gain 247 00:13:17,080 --> 00:13:20,920 some energy, in some cases, leaving our little dE, which 248 00:13:20,920 --> 00:13:23,057 is what we're trying to track. 249 00:13:23,057 --> 00:13:24,640 Because we're actually tracking what's 250 00:13:24,640 --> 00:13:27,580 the population of neutrons in this little dE, 251 00:13:27,580 --> 00:13:31,210 in this direction, in this position, at this time. 252 00:13:31,210 --> 00:13:33,760 And supposedly if we know this term fully, 253 00:13:33,760 --> 00:13:35,200 we can solve for all the neutrons 254 00:13:35,200 --> 00:13:38,408 everywhere, anywhere in the reactor with full information. 255 00:13:38,408 --> 00:13:40,200 So what we'll spend the rest of today doing 256 00:13:40,200 --> 00:13:44,300 is figuring out what are all the possible gain and loss terms. 257 00:13:44,300 --> 00:13:48,940 So let's start just putting them out physically, or in words. 258 00:13:48,940 --> 00:13:50,740 And then we'll put them to math. 259 00:13:50,740 --> 00:13:54,160 So what are some of the ways in which neutrons 260 00:13:54,160 --> 00:13:58,630 can enter our little group of volume, angle, and energy? 261 00:14:01,414 --> 00:14:04,940 How are neutrons created? 262 00:14:04,940 --> 00:14:05,580 Yeah, Luke? 263 00:14:05,580 --> 00:14:08,830 AUDIENCE: From fission, or a neutron emission. 264 00:14:08,830 --> 00:14:12,010 MICHAEL SHORT: From fission, yeah, so that's one big source. 265 00:14:12,010 --> 00:14:14,650 So we'll call this a gains. 266 00:14:14,650 --> 00:14:17,550 This is a losses. 267 00:14:17,550 --> 00:14:18,900 And you said a neutron source. 268 00:14:18,900 --> 00:14:19,942 Can you be more specific? 269 00:14:19,942 --> 00:14:22,260 AUDIENCE: A neutron emission, like [INAUDIBLE].. 270 00:14:22,260 --> 00:14:29,190 MICHAEL SHORT: OK, so we'll say n,in reactions, right? 271 00:14:29,190 --> 00:14:31,170 OK, cool. 272 00:14:31,170 --> 00:14:32,505 How else can we gain neutrons? 273 00:14:37,164 --> 00:14:38,600 AUDIENCE: Fusion? 274 00:14:38,600 --> 00:14:39,800 MICHAEL SHORT: Fusion? 275 00:14:39,800 --> 00:14:41,390 OK. 276 00:14:41,390 --> 00:14:43,010 That is true. 277 00:14:43,010 --> 00:14:44,690 Although fusion reactors don't really 278 00:14:44,690 --> 00:14:48,380 operate on the principle of neutron criticality, or neutron 279 00:14:48,380 --> 00:14:49,312 balance. 280 00:14:49,312 --> 00:14:50,770 So this discussion for now is going 281 00:14:50,770 --> 00:14:52,310 to be limited to fission reactors. 282 00:14:52,310 --> 00:14:53,690 But yeah, good point. 283 00:14:53,690 --> 00:14:56,540 Fusion does make neutrons. 284 00:14:56,540 --> 00:14:57,080 What else? 285 00:15:02,210 --> 00:15:02,710 Yeah? 286 00:15:02,710 --> 00:15:07,225 AUDIENCE: They could enter from one of the adjacent volume? 287 00:15:07,225 --> 00:15:09,600 MICHAEL SHORT: Yeah, they could come from somewhere else, 288 00:15:09,600 --> 00:15:10,620 right? 289 00:15:10,620 --> 00:15:12,415 Let's just call that an external source. 290 00:15:15,760 --> 00:15:17,300 In the books and in your reading, 291 00:15:17,300 --> 00:15:19,750 you'll just see them treat this external source 292 00:15:19,750 --> 00:15:28,900 as some variable s of r, E, omega, t. 293 00:15:33,600 --> 00:15:35,700 So you'll just see this treated as s, a source, 294 00:15:35,700 --> 00:15:36,990 with no further explanation. 295 00:15:36,990 --> 00:15:38,490 It's like, oh, math says that there 296 00:15:38,490 --> 00:15:39,845 could be external sources. 297 00:15:39,845 --> 00:15:41,970 But I want to tell you where they really come from. 298 00:15:41,970 --> 00:15:44,190 Most reactors nowadays don't just 299 00:15:44,190 --> 00:15:46,860 start up when you throw a bunch of uranium into a pool 300 00:15:46,860 --> 00:15:48,510 and pull out the control rods. 301 00:15:48,510 --> 00:15:50,580 You actually have to stick in-- 302 00:15:50,580 --> 00:15:53,530 if this is your little reactor right here-- 303 00:15:53,530 --> 00:15:57,630 you actually have to stick in a little piece of californium-- 304 00:15:57,630 --> 00:16:00,630 I think the isotope is 252-- 305 00:16:00,630 --> 00:16:03,080 as what we call a kickstarter source. 306 00:16:03,080 --> 00:16:09,120 So californium is made mostly in the HFIR, or the High Flux 307 00:16:09,120 --> 00:16:13,980 Isotope Reactor, at the Oakridge National Lab in Tennessee, 308 00:16:13,980 --> 00:16:16,980 where they have a really, really high power reactor. 309 00:16:16,980 --> 00:16:18,780 It's 85 megawatts. 310 00:16:18,780 --> 00:16:22,050 It's about that big around and this tall. 311 00:16:22,050 --> 00:16:23,490 It's really, really small. 312 00:16:23,490 --> 00:16:26,580 For reference, that's about the size of the MIT reactor, 313 00:16:26,580 --> 00:16:28,530 except our reactor's 6 megawatts. 314 00:16:28,530 --> 00:16:30,510 Theirs is 85 megawatts. 315 00:16:30,510 --> 00:16:33,690 And it's designed to be an incredibly high flux, 316 00:16:33,690 --> 00:16:36,630 to go by neutron capture, and neutron capture reactions, 317 00:16:36,630 --> 00:16:41,430 to build up californium 252, which is spontaneously 318 00:16:41,430 --> 00:16:44,110 giving off neutrons like crazy. 319 00:16:44,110 --> 00:16:48,800 And this right here, that's your external source. 320 00:16:48,800 --> 00:16:50,790 And this helps get reactors going. 321 00:16:50,790 --> 00:16:52,560 Because you can either very slowly 322 00:16:52,560 --> 00:16:54,690 wait for the fission reaction to build up 323 00:16:54,690 --> 00:16:55,950 in a controlled manner. 324 00:16:55,950 --> 00:16:58,620 Or you can give it a kick in the pants and get it going. 325 00:16:58,620 --> 00:17:00,960 This HFIR reactor is pretty cool. 326 00:17:00,960 --> 00:17:03,750 Like I said, it's 85 megawatts. 327 00:17:03,750 --> 00:17:05,940 And it's about as dense as it can get. 328 00:17:05,940 --> 00:17:08,880 The fuel is actually made by explosively bonding sheets 329 00:17:08,880 --> 00:17:14,190 of uranium in a certain sort of semi-cylindrical configuration. 330 00:17:14,190 --> 00:17:17,369 And it produces so much decay heat in so little space 331 00:17:17,369 --> 00:17:19,230 that if it were to lose cooling, the reactor 332 00:17:19,230 --> 00:17:21,206 would melt in 8 seconds. 333 00:17:21,206 --> 00:17:23,530 You usually have days or so before 334 00:17:23,530 --> 00:17:26,200 that happens in a conventional reactor 335 00:17:26,200 --> 00:17:28,840 because the power density just isn't that high. 336 00:17:28,840 --> 00:17:30,970 So you can actually see down to the tank 337 00:17:30,970 --> 00:17:34,090 that contains HFIR if you go for a tour at Oakridge National 338 00:17:34,090 --> 00:17:34,870 Lab. 339 00:17:34,870 --> 00:17:37,630 And it's way down below this gigantic like, not quite 340 00:17:37,630 --> 00:17:40,883 Olympic, but getting there sized pool of water, 341 00:17:40,883 --> 00:17:42,550 just to make sure that there is adequate 342 00:17:42,550 --> 00:17:43,910 cooling for this thing. 343 00:17:43,910 --> 00:17:45,250 It's intense. 344 00:17:45,250 --> 00:17:48,380 But that's just a notice that these external sources, 345 00:17:48,380 --> 00:17:51,520 these are real things that we use in power reactors 346 00:17:51,520 --> 00:17:53,960 to get them going. 347 00:17:53,960 --> 00:17:56,750 What are some other ways that one could make neutrons, 348 00:17:56,750 --> 00:18:00,170 or that neutrons could enter into our energy group? 349 00:18:04,515 --> 00:18:06,640 And the silence is expected because this is usually 350 00:18:06,640 --> 00:18:09,490 the hardest part of developing this equation. 351 00:18:09,490 --> 00:18:10,720 And I want to introduce it. 352 00:18:10,720 --> 00:18:11,466 Yeah, Luke? 353 00:18:11,466 --> 00:18:13,090 AUDIENCE: [INAUDIBLE] scattering, too. 354 00:18:13,090 --> 00:18:14,465 MICHAEL SHORT: That's exactly it. 355 00:18:14,465 --> 00:18:15,580 They can scatter in. 356 00:18:15,580 --> 00:18:18,910 So when we develop this neutron transport equation, 357 00:18:18,910 --> 00:18:21,370 we're not just tracking the neutrons 358 00:18:21,370 --> 00:18:25,450 in this little energy group dE, direction, d-omega, and volume 359 00:18:25,450 --> 00:18:26,290 dV. 360 00:18:26,290 --> 00:18:29,320 You actually have to know what's the population of neutrons 361 00:18:29,320 --> 00:18:31,030 in every single group. 362 00:18:31,030 --> 00:18:34,210 Because you might have a neutron at a higher energy level that 363 00:18:34,210 --> 00:18:40,245 undergoes scattering from some different energy, 364 00:18:40,245 --> 00:18:43,650 E-prime into our energy group. 365 00:18:43,650 --> 00:18:49,030 So continuing with our gigantic list of variables, 366 00:18:49,030 --> 00:18:51,730 we're going to call E is, we'll say r energy. 367 00:18:55,290 --> 00:18:57,408 And this vector omega is our direction, 368 00:18:57,408 --> 00:18:58,575 the one that we're tracking. 369 00:19:03,100 --> 00:19:05,500 And E-prime is going to be some other energy. 370 00:19:09,780 --> 00:19:13,260 And omega-vector-prime is going to be some coming 371 00:19:13,260 --> 00:19:14,670 from some other direction. 372 00:19:18,350 --> 00:19:20,010 And like Luke said, this is what we 373 00:19:20,010 --> 00:19:30,740 would refer to as in-scattering, which means some neutron comes, 374 00:19:30,740 --> 00:19:32,960 that was going in a different direction, that 375 00:19:32,960 --> 00:19:35,000 did have a different energy, and has now 376 00:19:35,000 --> 00:19:37,832 entered into the single group that we're tracking. 377 00:19:37,832 --> 00:19:40,040 Eventually we're going to integrate over all energies 378 00:19:40,040 --> 00:19:42,270 to track all energy groups. 379 00:19:42,270 --> 00:19:44,960 So that's where we're going. 380 00:19:44,960 --> 00:19:48,530 And there's one more term that I want to introduce right now. 381 00:19:48,530 --> 00:19:50,330 It's what's called the scattering kernel. 382 00:19:55,880 --> 00:19:57,440 Don't ask me why it's called kernel. 383 00:19:57,440 --> 00:20:01,400 But this is just the terminology I want you guys to get used to. 384 00:20:01,400 --> 00:20:04,370 And there's going to be some sort of probability function 385 00:20:04,370 --> 00:20:08,210 where a neutron starts off at a different energy, E-prime, 386 00:20:08,210 --> 00:20:10,170 and in a different direction, omega-prime. 387 00:20:10,170 --> 00:20:17,920 And it enters into our group energy E and direction omega. 388 00:20:17,920 --> 00:20:20,430 Right now we'll leave it as a highly general function. 389 00:20:20,430 --> 00:20:22,990 What we're going to find later is there's just 390 00:20:22,990 --> 00:20:26,440 some sort of simple line to it. 391 00:20:26,440 --> 00:20:31,140 If you guys remember, if some neutron starts off, 392 00:20:31,140 --> 00:20:37,170 let's see, probability of entering into some energy 393 00:20:37,170 --> 00:20:38,190 group. 394 00:20:38,190 --> 00:20:40,110 If you notice, if you remember from last time, 395 00:20:40,110 --> 00:20:43,650 the neutron, when it undergoes any sort of scattering 396 00:20:43,650 --> 00:20:47,460 reaction, can end up with any energy 397 00:20:47,460 --> 00:20:53,280 between its original energy for the case of theta equals 0, 398 00:20:53,280 --> 00:20:55,710 and this parameter, alpha energy, for the case theta 399 00:20:55,710 --> 00:21:01,220 equals pi, where alpha is A minus 1, 400 00:21:01,220 --> 00:21:05,870 over A plus 1 squared, where A is the atomic mass. 401 00:21:09,927 --> 00:21:12,010 You guys remember this from back in the Q equation 402 00:21:12,010 --> 00:21:14,320 days, when we were finding out what's 403 00:21:14,320 --> 00:21:17,320 the probability that a neutron coming in with energy E 404 00:21:17,320 --> 00:21:18,580 ends up at any energy E-prime? 405 00:21:21,180 --> 00:21:24,480 Actually I'll just write this as the scattering kernel. 406 00:21:24,480 --> 00:21:27,120 What it ends up looking like, in most cases, 407 00:21:27,120 --> 00:21:28,260 is just a flat line. 408 00:21:28,260 --> 00:21:32,550 There's an equal probability of the neutron ending up anywhere 409 00:21:32,550 --> 00:21:37,070 between energy E and anywhere between energy 410 00:21:37,070 --> 00:21:39,960 alpha-E. It's actually a pretty simple function. 411 00:21:39,960 --> 00:21:44,517 It's just a constant value here and 0 everywhere else. 412 00:21:44,517 --> 00:21:46,100 What that means is that if, let's say, 413 00:21:46,100 --> 00:21:48,890 a neutron hits a uranium atom, there is no way in hell 414 00:21:48,890 --> 00:21:51,050 that it can transfer all of its energy 415 00:21:51,050 --> 00:21:56,720 to a uranium atom because of conservation of energy 416 00:21:56,720 --> 00:22:00,250 and momentum, like we've been harping on for kind 417 00:22:00,250 --> 00:22:02,150 of this whole class. 418 00:22:02,150 --> 00:22:04,940 What's the only time that this alpha-E could actually 419 00:22:04,940 --> 00:22:09,110 extend all the way to 0? 420 00:22:09,110 --> 00:22:11,790 What case would that be? 421 00:22:11,790 --> 00:22:13,130 AUDIENCE: [INAUDIBLE]. 422 00:22:13,130 --> 00:22:14,838 AUDIENCE: You're hitting another neutron. 423 00:22:14,838 --> 00:22:17,047 MICHAEL SHORT: You're hitting another neutron, which, 424 00:22:17,047 --> 00:22:18,660 as we said, is a very rare event. 425 00:22:18,660 --> 00:22:19,770 That is true. 426 00:22:19,770 --> 00:22:21,312 Or what else could you be hitting? 427 00:22:21,312 --> 00:22:22,790 AUDIENCE: A proton? 428 00:22:22,790 --> 00:22:24,930 MICHAEL SHORT: A proton, hydrogen. That's right. 429 00:22:24,930 --> 00:22:28,110 So it can only be, let's say you can only 430 00:22:28,110 --> 00:22:30,690 have the probability of the neutron ending up 431 00:22:30,690 --> 00:22:39,130 with any energy for the case of hydrogen. Incidentally, 432 00:22:39,130 --> 00:22:41,260 this is why we fill light water reactors 433 00:22:41,260 --> 00:22:43,420 with hydrogen. The goal is to get 434 00:22:43,420 --> 00:22:47,080 the neutrons as slow as possible as quick as possible. 435 00:22:47,080 --> 00:22:49,060 Interesting sentence to say there, right? 436 00:22:49,060 --> 00:22:52,420 We want the neutrons to be as low energy as possible 437 00:22:52,420 --> 00:22:54,340 as rapidly as possible. 438 00:22:54,340 --> 00:22:56,770 And the best way to do that is to fill the reactor 439 00:22:56,770 --> 00:23:00,070 with hydrogen because then any collision could, in theory, get 440 00:23:00,070 --> 00:23:02,260 the neutron down to zero energy. 441 00:23:02,260 --> 00:23:05,680 Without water, or something with the same mass as a neutron, 442 00:23:05,680 --> 00:23:09,600 like another neutron, there is no way 443 00:23:09,600 --> 00:23:11,688 that that neutron can slow down by very much. 444 00:23:11,688 --> 00:23:13,230 So even though we're going to keep it 445 00:23:13,230 --> 00:23:16,260 as this generalized function, note that in reality it's 446 00:23:16,260 --> 00:23:18,270 this pretty simple function. 447 00:23:18,270 --> 00:23:20,190 It changes a little bit, as there 448 00:23:20,190 --> 00:23:24,030 can be a forward scattering bias for some neutron reactions. 449 00:23:24,030 --> 00:23:27,000 But we are not going to deal with that this year. 450 00:23:27,000 --> 00:23:30,890 You will deal with that next year in 22.05. 451 00:23:30,890 --> 00:23:32,840 So I've been saying a lot, oh, well, we're 452 00:23:32,840 --> 00:23:35,423 not going to go into this topic because you're going to see it 453 00:23:35,423 --> 00:23:37,220 in 22.02, which is quantum. 454 00:23:37,220 --> 00:23:39,050 Now I switched gears to say, we're 455 00:23:39,050 --> 00:23:41,630 not going to go into the way this function changes 456 00:23:41,630 --> 00:23:43,940 because you'll see it next year in 22.05, 457 00:23:43,940 --> 00:23:45,500 which is neutron physics. 458 00:23:45,500 --> 00:23:49,490 But for now I want you to be prepared for 22.05. 459 00:23:49,490 --> 00:23:53,315 So we'll put on in-scattering as one of our gains. 460 00:23:56,702 --> 00:23:58,660 There's a last one I want to make you aware of. 461 00:23:58,660 --> 00:24:00,143 We very briefly touched upon it. 462 00:24:00,143 --> 00:24:02,560 But I wouldn't be surprised if no one remembers because it 463 00:24:02,560 --> 00:24:04,935 was for like 10 seconds. 464 00:24:04,935 --> 00:24:06,310 It's what's called photo fission. 465 00:24:10,617 --> 00:24:13,200 What this means is you have some reaction that would, in comes 466 00:24:13,200 --> 00:24:18,190 a gamma, and out goes fission. 467 00:24:18,190 --> 00:24:22,310 This actually does start to happen around 3 or 4 MeV, 468 00:24:22,310 --> 00:24:25,030 for isotopes like uranium 235. 469 00:24:25,030 --> 00:24:29,800 And in our reactor, whatever shape we decide it is, 470 00:24:29,800 --> 00:24:32,950 there are tons of gamma rays flying about in all directions 471 00:24:32,950 --> 00:24:34,020 at very high energy. 472 00:24:34,020 --> 00:24:35,770 Does anyone remember where they come from? 473 00:24:42,820 --> 00:24:45,930 Anyone remember the fission timeline 474 00:24:45,930 --> 00:24:48,760 that we drew on Friday? 475 00:24:48,760 --> 00:24:51,240 So what we said there was right away, 476 00:24:51,240 --> 00:24:53,440 let's say fission happens. 477 00:24:53,440 --> 00:24:58,520 And almost instantly, you get your fission product one 478 00:24:58,520 --> 00:25:00,750 and fission product two. 479 00:25:00,750 --> 00:25:03,100 And they move around for a little while. 480 00:25:03,100 --> 00:25:05,925 And then some of them will emit some neutrons. 481 00:25:09,270 --> 00:25:13,110 And then some of them will start to emit gamma rays, betas, 482 00:25:13,110 --> 00:25:19,160 and whatever else they're going to do until they finally 483 00:25:19,160 --> 00:25:22,370 lose all their kinetic energy and stop in the surrounding 484 00:25:22,370 --> 00:25:26,330 fuel, creating the heat that actually powers the turbine 485 00:25:26,330 --> 00:25:29,210 and make steam to make electricity. 486 00:25:29,210 --> 00:25:31,400 And so it's from these gammas, as well as 487 00:25:31,400 --> 00:25:35,150 any of the gammas from the decay products of the fission 488 00:25:35,150 --> 00:25:38,990 products that lead to a huge flux of gamma rays firing out 489 00:25:38,990 --> 00:25:40,640 from all sides in the reactor. 490 00:25:40,640 --> 00:25:42,807 That's one of the main things that you actually have 491 00:25:42,807 --> 00:25:44,750 to shield in a nuclear reactor. 492 00:25:44,750 --> 00:25:47,570 Since we talked about all sorts of different shielding, 493 00:25:47,570 --> 00:25:50,450 and all sorts of ways that you have to shield things, 494 00:25:50,450 --> 00:25:53,630 you know from seeing the MIT reactor-- which you all did-- 495 00:25:53,630 --> 00:25:56,270 that there's like six feet of lead and concrete 496 00:25:56,270 --> 00:25:58,100 shielding around the reactor. 497 00:25:58,100 --> 00:26:00,470 It's not there to shield the alphas and the betas, 498 00:26:00,470 --> 00:26:03,090 because those don't really make it out of the water. 499 00:26:03,090 --> 00:26:05,750 It's not there to shield the soft X-rays that betas 500 00:26:05,750 --> 00:26:08,120 make from bremsstrahlung. 501 00:26:08,120 --> 00:26:10,630 It's also not there to shield the neutrons 502 00:26:10,630 --> 00:26:12,380 because the neutrons don't really get out. 503 00:26:12,380 --> 00:26:15,300 They bounce around, or get absorbed in the water, 504 00:26:15,300 --> 00:26:16,840 or the fuel, the reflector. 505 00:26:16,840 --> 00:26:19,820 It's there to shield the high-energy gamma rays. 506 00:26:19,820 --> 00:26:21,200 Because the only thing that stops 507 00:26:21,200 --> 00:26:23,750 high energy gamma rays is lots of mass 508 00:26:23,750 --> 00:26:26,480 in between the source and you. 509 00:26:26,480 --> 00:26:29,350 So we know there's tons of gammas all about. 510 00:26:29,350 --> 00:26:36,020 So let's say there's also going to be some gamma ray flux. 511 00:26:36,020 --> 00:26:38,510 There'll be some gamma ray energy. 512 00:26:38,510 --> 00:26:42,740 And there'll be some cross-section for photo fission 513 00:26:42,740 --> 00:26:48,240 as a function of the incoming gamma ray energy spectrum. 514 00:26:48,240 --> 00:26:51,660 Now I'm adding terms to the ones that you'll see in the reading 515 00:26:51,660 --> 00:26:55,350 because drawing them out in math is actually fairly instructive. 516 00:26:55,350 --> 00:26:56,810 They all follow the same pattern. 517 00:26:56,810 --> 00:26:58,560 So instead of just showing you one of each 518 00:26:58,560 --> 00:27:01,715 and saying memorize, we'll develop a whole lot of these. 519 00:27:01,715 --> 00:27:03,090 And you'll find out that they all 520 00:27:03,090 --> 00:27:04,660 actually look almost the same. 521 00:27:07,220 --> 00:27:11,257 Can anyone else think of any possible gains of neutrons? 522 00:27:11,257 --> 00:27:12,590 Where else could they come from? 523 00:27:15,700 --> 00:27:16,340 Yeah? 524 00:27:16,340 --> 00:27:18,407 AUDIENCE: Neutron birth spectrum, is that? 525 00:27:18,407 --> 00:27:20,240 MICHAEL SHORT: So the neutron birth spectrum 526 00:27:20,240 --> 00:27:22,280 is included in fission. 527 00:27:22,280 --> 00:27:24,950 So our nu is in there. 528 00:27:24,950 --> 00:27:26,480 Our chi of E is in there. 529 00:27:26,480 --> 00:27:29,300 And that's a nu of E. That's all accounted 530 00:27:29,300 --> 00:27:30,440 for in the fission term. 531 00:27:30,440 --> 00:27:32,813 And we'll see how we put that together to math. 532 00:27:32,813 --> 00:27:34,730 And if no one else has any ideas, that's good. 533 00:27:34,730 --> 00:27:36,530 Because neither do I. 534 00:27:36,530 --> 00:27:38,630 Now what about the lost terms? 535 00:27:38,630 --> 00:27:40,700 There aren't too many of these, as long 536 00:27:40,700 --> 00:27:43,260 as you lump them correctly. 537 00:27:43,260 --> 00:27:45,170 So what sort of ways could neutrons 538 00:27:45,170 --> 00:27:48,608 be lost from our energy group? 539 00:27:48,608 --> 00:27:49,480 Yep? 540 00:27:49,480 --> 00:27:50,910 AUDIENCE: Scatter out. 541 00:27:50,910 --> 00:27:53,112 MICHAEL SHORT: Scatter out, yep. 542 00:27:56,570 --> 00:27:58,700 They can undergo any kind of scattering reaction. 543 00:27:58,700 --> 00:28:01,220 And they will probably change direction and energy. 544 00:28:01,220 --> 00:28:03,920 What else? 545 00:28:03,920 --> 00:28:08,080 Well, we've got to list up on the board right there, right? 546 00:28:08,080 --> 00:28:13,930 Capture, fission, because in order 547 00:28:13,930 --> 00:28:19,340 to undergo a vision you actually have to lose a neutron, 548 00:28:19,340 --> 00:28:20,810 and so on, and so on, and so on. 549 00:28:20,810 --> 00:28:25,360 What I want to do to simplify things is this. 550 00:28:28,750 --> 00:28:33,130 It's a lot simpler just to track the total cross-section, 551 00:28:33,130 --> 00:28:34,930 the probability of any interaction 552 00:28:34,930 --> 00:28:37,570 at all whatsoever, because any interaction 553 00:28:37,570 --> 00:28:41,260 will cause the neutron to either change energy and angle, 554 00:28:41,260 --> 00:28:44,860 or disappear, even if it makes some other ones. 555 00:28:44,860 --> 00:28:49,830 So we can simplify this to just the total cross-section term. 556 00:28:49,830 --> 00:28:51,850 And there's only one other way that neutrons 557 00:28:51,850 --> 00:28:56,428 can leave our energy angle and volume group. 558 00:28:56,428 --> 00:28:57,220 What would that be? 559 00:29:01,270 --> 00:29:04,690 So any reaction takes care of energy and angle. 560 00:29:04,690 --> 00:29:06,902 What about volume? 561 00:29:06,902 --> 00:29:08,610 How do neutrons leave the control volume? 562 00:29:12,470 --> 00:29:14,740 It's simpler than it may sound. 563 00:29:14,740 --> 00:29:15,820 They just go. 564 00:29:15,820 --> 00:29:17,050 They just move. 565 00:29:17,050 --> 00:29:18,990 The neutrons are always moving, right? 566 00:29:18,990 --> 00:29:19,990 We'll call that leakage. 567 00:29:23,650 --> 00:29:30,380 Because every neutron's got a speed, like we showed up here, 568 00:29:30,380 --> 00:29:32,720 where the flux of neutrons, the number 569 00:29:32,720 --> 00:29:36,200 of neutrons moving through some surface per second, 570 00:29:36,200 --> 00:29:39,380 is just their velocity times the number that are there. 571 00:29:39,380 --> 00:29:40,880 For there to be a neutron flux there 572 00:29:40,880 --> 00:29:43,790 has to be a velocity, which means the neutrons are moving. 573 00:29:43,790 --> 00:29:46,670 So the neutrons, even without undergoing any reaction, 574 00:29:46,670 --> 00:29:49,353 could just move out of our control volume. 575 00:29:49,353 --> 00:29:50,270 And then they're gone. 576 00:29:53,490 --> 00:29:56,050 And that's all there is for gain and loss terms. 577 00:29:56,050 --> 00:30:00,210 So let's see if we can do this all on one board. 578 00:30:00,210 --> 00:30:04,710 I want to start putting this table right here into math 579 00:30:04,710 --> 00:30:08,090 that we'll be able to abstract, simplify, and then solve, 580 00:30:08,090 --> 00:30:10,590 but not today, not solve today. 581 00:30:10,590 --> 00:30:12,300 So if we want to track the change 582 00:30:12,300 --> 00:30:21,270 in the number of neutrons as a function of time, 583 00:30:21,270 --> 00:30:24,240 let's start writing down the gain terms. 584 00:30:24,240 --> 00:30:26,490 So how do we describe the number of neutrons 585 00:30:26,490 --> 00:30:30,012 produced from fission? 586 00:30:30,012 --> 00:30:31,720 What sort of terms do we have to include? 587 00:30:35,027 --> 00:30:37,360 And Jared started kicking us off, so what would you say? 588 00:30:39,940 --> 00:30:40,940 AUDIENCE: Neutron birth? 589 00:30:40,940 --> 00:30:43,030 MICHAEL SHORT: Yep, so the neutron birth spectrum, 590 00:30:43,030 --> 00:30:46,510 there's going to be some probability that a neutron is 591 00:30:46,510 --> 00:30:49,840 born in our energy group E. Because we're 592 00:30:49,840 --> 00:30:54,580 tracking how many neutrons are in our little dE energy group. 593 00:30:54,580 --> 00:30:59,320 What else matters in terms of fission? 594 00:31:02,645 --> 00:31:04,550 AUDIENCE: Number of fissions? 595 00:31:04,550 --> 00:31:06,270 MICHAEL SHORT: Yep, number of fissions. 596 00:31:06,270 --> 00:31:08,020 So if we want to write number of fissions, 597 00:31:08,020 --> 00:31:10,930 we have to write that as a reaction rate. 598 00:31:10,930 --> 00:31:13,910 So let's take those two terms right there. 599 00:31:13,910 --> 00:31:18,230 So we'll have sigma fission. 600 00:31:18,230 --> 00:31:22,610 In this case, we're going to write E-prime times flux 601 00:31:22,610 --> 00:31:29,590 of r E-prime, omega-prime, t. 602 00:31:29,590 --> 00:31:31,810 Why did I write E and omega prime here? 603 00:31:35,640 --> 00:31:36,930 Just from a physical reason. 604 00:31:36,930 --> 00:31:37,110 Yeah? 605 00:31:37,110 --> 00:31:38,950 AUDIENCE: So you're going to be coming from another energy 606 00:31:38,950 --> 00:31:39,870 group. 607 00:31:39,870 --> 00:31:41,340 MICHAEL SHORT: Precisely. 608 00:31:41,340 --> 00:31:41,910 That's right. 609 00:31:41,910 --> 00:31:45,750 So the neutrons are going to be produced from some other energy 610 00:31:45,750 --> 00:31:46,500 group. 611 00:31:46,500 --> 00:31:49,380 For example, the fission birth spectrum right here 612 00:31:49,380 --> 00:31:50,603 starts out-- 613 00:31:50,603 --> 00:31:51,270 where did it go? 614 00:31:51,270 --> 00:31:54,150 I knew I drew it somewhere-- at one MeV. 615 00:31:54,150 --> 00:31:57,420 But most of the neutrons that cause fission to happen 616 00:31:57,420 --> 00:32:00,210 are way down below 1 eV. 617 00:32:00,210 --> 00:32:02,400 So it's different energy neutrons 618 00:32:02,400 --> 00:32:05,520 that cause neutrons to be born in our energy group. 619 00:32:05,520 --> 00:32:07,680 That's why we're using E-prime and not E. 620 00:32:07,680 --> 00:32:11,160 It's some other energy group. 621 00:32:11,160 --> 00:32:16,810 And so we also have to account for all possible other energy 622 00:32:16,810 --> 00:32:17,502 groups. 623 00:32:17,502 --> 00:32:19,210 So if we want to write this, right, we'll 624 00:32:19,210 --> 00:32:28,010 say this could be as low as 0 eV, to our maximum energy. 625 00:32:28,010 --> 00:32:35,980 And there's going to be some d-omega-prime, dE-prime, dV. 626 00:32:35,980 --> 00:32:41,920 We'll also have to account for all possible angles 627 00:32:41,920 --> 00:32:47,770 and integrate over our entire volume. 628 00:32:47,770 --> 00:32:49,700 It's going to look ugly quick, but it's all 629 00:32:49,700 --> 00:32:51,860 going to be understandable. 630 00:32:51,860 --> 00:32:53,360 So what this says is that we have 631 00:32:53,360 --> 00:32:56,360 to account for the reaction rate of fission 632 00:32:56,360 --> 00:33:00,650 from all other energy neutrons inside our volume 633 00:33:00,650 --> 00:33:02,660 from other energies and other angles, 634 00:33:02,660 --> 00:33:05,480 and account for every other possible energy. 635 00:33:05,480 --> 00:33:08,630 Because they can all make fission happen. 636 00:33:08,630 --> 00:33:11,000 What else is missing in terms of describing the number 637 00:33:11,000 --> 00:33:14,051 of neutrons made from fission? 638 00:33:14,051 --> 00:33:15,426 AUDIENCE: Neutron multiplication. 639 00:33:15,426 --> 00:33:17,009 MICHAEL SHORT: Yep, there's the number 640 00:33:17,009 --> 00:33:18,940 of neutrons made per fission. 641 00:33:18,940 --> 00:33:23,690 So we have to put in our neutron multiplication factor. 642 00:33:23,690 --> 00:33:26,040 And in this case, normalize-- 643 00:33:26,040 --> 00:33:28,910 I think someone had mentioned solid angle-- we normalize over 644 00:33:28,910 --> 00:33:32,570 all possible angles with an over 4 pi in there. 645 00:33:32,570 --> 00:33:35,460 And this right here is the fission term. 646 00:33:35,460 --> 00:33:37,490 So this tells us the number of neutrons 647 00:33:37,490 --> 00:33:40,460 gained in terms of a reaction rate, 648 00:33:40,460 --> 00:33:44,750 times the number of neutrons for each of those reactions, 649 00:33:44,750 --> 00:33:46,430 times the probability that there just 650 00:33:46,430 --> 00:33:48,140 happened to be born in the energy group 651 00:33:48,140 --> 00:33:50,900 that we're tracking. 652 00:33:50,900 --> 00:33:54,380 So is there any term here that's unclear to folks? 653 00:33:54,380 --> 00:33:55,010 Yeah? 654 00:33:55,010 --> 00:33:57,110 AUDIENCE: So what's the lower bound on the first integral? 655 00:33:57,110 --> 00:33:58,652 MICHAEL SHORT: On the first integral? 656 00:33:58,652 --> 00:33:59,570 That 0 electron volts. 657 00:33:59,570 --> 00:34:00,308 AUDIENCE: Oh, OK. 658 00:34:00,308 --> 00:34:01,850 MICHAEL SHORT: Because supposedly you 659 00:34:01,850 --> 00:34:04,490 could have a neutron at 0 eV, which 660 00:34:04,490 --> 00:34:06,380 has a very high cross-section. 661 00:34:06,380 --> 00:34:08,750 So it should probably induce fission. 662 00:34:08,750 --> 00:34:11,929 In reality, there might be some actual minimum temperature. 663 00:34:11,929 --> 00:34:14,630 But there is a non-zero probability 664 00:34:14,630 --> 00:34:16,708 that you could have a neutron at rest. 665 00:34:16,708 --> 00:34:17,750 It's just not very large. 666 00:34:17,750 --> 00:34:18,770 AUDIENCE: And the top bound? 667 00:34:18,770 --> 00:34:20,728 MICHAEL SHORT: The top bound as E max, whatever 668 00:34:20,728 --> 00:34:22,760 your maximum neutron energy is. 669 00:34:22,760 --> 00:34:28,670 This is usually around 10 MeV, for most fission reactors. 670 00:34:28,670 --> 00:34:33,600 That E max is going to be this point right here, the highest 671 00:34:33,600 --> 00:34:38,750 energy at which neutrons can be born by any process. 672 00:34:38,750 --> 00:34:41,150 And so this term right here is going 673 00:34:41,150 --> 00:34:45,762 to serve as a template for all the other gain and loss terms. 674 00:34:45,762 --> 00:34:47,679 So I think this is the hardest one that we had 675 00:34:47,679 --> 00:34:49,630 to develop from the beginning. 676 00:34:49,630 --> 00:34:54,230 Now let's develop the term for, let's just 677 00:34:54,230 --> 00:34:57,030 go with external sources, pretty easy. 678 00:34:57,030 --> 00:35:00,065 There's going to be some source making neutrons. 679 00:35:05,500 --> 00:35:07,900 It's something that you would just impose. 680 00:35:07,900 --> 00:35:10,660 Like say, all right, I have a californium source giving off 681 00:35:10,660 --> 00:35:11,902 this many neutrons. 682 00:35:11,902 --> 00:35:14,110 Well then you know how many neutrons it's giving off. 683 00:35:14,110 --> 00:35:15,070 And that one's done. 684 00:35:15,070 --> 00:35:16,640 That's easy. 685 00:35:16,640 --> 00:35:19,490 So we've done fission. 686 00:35:19,490 --> 00:35:20,460 We've done external. 687 00:35:23,690 --> 00:35:28,190 Now that we've done fission let's tackle photo fission. 688 00:35:28,190 --> 00:35:30,790 So what would be photo fission cross-section look like? 689 00:35:39,840 --> 00:35:42,293 It's going to look awfully similar. 690 00:35:42,293 --> 00:35:43,710 So what sort of things do you need 691 00:35:43,710 --> 00:35:45,168 to know if it's a fission reaction? 692 00:35:50,460 --> 00:35:51,710 Well, what do we have up here? 693 00:35:51,710 --> 00:35:52,960 Just start reading things off. 694 00:35:56,392 --> 00:35:57,100 I heard a murmur. 695 00:35:57,100 --> 00:35:57,810 What was that? 696 00:35:57,810 --> 00:35:59,260 AUDIENCE: [INAUDIBLE] flux. 697 00:35:59,260 --> 00:36:00,760 MICHAEL SHORT: Yeah, so you're going 698 00:36:00,760 --> 00:36:02,900 to have to have some flux. 699 00:36:02,900 --> 00:36:07,310 In this case, we want to know what's the flux of gamma rays 700 00:36:07,310 --> 00:36:09,700 because photo fission starts off with a gamma, 701 00:36:09,700 --> 00:36:11,390 then ends up with a fission. 702 00:36:11,390 --> 00:36:14,600 And it's also going to be in our volume. 703 00:36:14,600 --> 00:36:18,110 It's going to matter what the energy of those gammas is. 704 00:36:18,110 --> 00:36:22,250 They'll all be traveling in some direction at some time. 705 00:36:22,250 --> 00:36:25,330 What else do we need? 706 00:36:25,330 --> 00:36:27,452 AUDIENCE: [INAUDIBLE] the 4 pi [INAUDIBLE].. 707 00:36:27,452 --> 00:36:28,910 MICHAEL SHORT: Yeah, if we're going 708 00:36:28,910 --> 00:36:34,010 to be going over all angles, you need the 4 pi. 709 00:36:34,010 --> 00:36:34,550 What else? 710 00:36:38,230 --> 00:36:41,174 Do we have a reaction rate yet? 711 00:36:41,174 --> 00:36:42,020 AUDIENCE: No. 712 00:36:42,020 --> 00:36:44,452 MICHAEL SHORT: No, well what's missing? 713 00:36:44,452 --> 00:36:46,323 AUDIENCE: The cross-section. 714 00:36:46,323 --> 00:36:47,490 MICHAEL SHORT: That's right. 715 00:36:47,490 --> 00:36:49,280 We need a cross-section. 716 00:36:49,280 --> 00:36:51,120 And in this case, instead of just fission, 717 00:36:51,120 --> 00:36:54,750 or neutron fission, we'll put in the gamma fission 718 00:36:54,750 --> 00:36:57,560 cross-section. 719 00:36:57,560 --> 00:37:01,100 And so now we have a reaction rate for a single reaction. 720 00:37:01,100 --> 00:37:04,670 We've got to integrate over all of our gamma ray 721 00:37:04,670 --> 00:37:10,710 energies, over all angles, over our volume. 722 00:37:10,710 --> 00:37:26,780 What else is missing besides our d-omega, dE gamma, d-Volume. 723 00:37:26,780 --> 00:37:29,420 It should look awfully similar because the terms are basically 724 00:37:29,420 --> 00:37:31,743 exactly the same, with just different cross sections 725 00:37:31,743 --> 00:37:32,660 and energies in there. 726 00:37:32,660 --> 00:37:35,600 So what's missing between the photo fission and the neutron 727 00:37:35,600 --> 00:37:37,006 fission one? 728 00:37:37,006 --> 00:37:38,697 AUDIENCE: [INAUDIBLE] 729 00:37:38,697 --> 00:37:41,030 MICHAEL SHORT: Sure, there might be some different birth 730 00:37:41,030 --> 00:37:43,736 spectrum for gammas. 731 00:37:43,736 --> 00:37:45,800 And there might be some different multiplication 732 00:37:45,800 --> 00:37:49,570 factor for gammas between neutron fission and photo 733 00:37:49,570 --> 00:37:50,250 fission. 734 00:37:50,250 --> 00:37:52,320 But these terms should look exactly the same 735 00:37:52,320 --> 00:37:55,350 because in every case you're looking at some reaction 736 00:37:55,350 --> 00:37:58,770 rate between either the neutrons and fission or the gamma 737 00:37:58,770 --> 00:38:00,150 and fission. 738 00:38:00,150 --> 00:38:02,670 And you need to know at what energy they're born, 739 00:38:02,670 --> 00:38:05,640 how many are made, all the angles, 740 00:38:05,640 --> 00:38:08,600 and integrate overall the variables that we care about. 741 00:38:08,600 --> 00:38:10,920 And this is part of why I'm adding these extra terms 742 00:38:10,920 --> 00:38:13,800 because they end up looking all exactly the same. 743 00:38:13,800 --> 00:38:18,280 It's the integral of a reaction rate times some stuff. 744 00:38:18,280 --> 00:38:21,230 That's all that every single one of these terms is going to be. 745 00:38:21,230 --> 00:38:23,050 So we've got photo fission. 746 00:38:23,050 --> 00:38:26,910 Now let's tackle in-scattering. 747 00:38:26,910 --> 00:38:28,830 So how do we represent scattering? 748 00:38:31,580 --> 00:38:35,160 In the same way that we represented fission, what do we 749 00:38:35,160 --> 00:38:38,130 start with inside the integral? 750 00:38:38,130 --> 00:38:39,130 AUDIENCE: Reaction rate? 751 00:38:39,130 --> 00:38:40,900 MICHAEL SHORT: Reaction rate, yes. 752 00:38:40,900 --> 00:38:44,845 So we're going to have some scattering cross-section. 753 00:38:44,845 --> 00:38:46,220 And if it's in the scattering, it 754 00:38:46,220 --> 00:38:48,710 means it's coming from a different energy, hence 755 00:38:48,710 --> 00:38:50,177 the prime. 756 00:38:50,177 --> 00:38:51,010 We'll have our flux. 757 00:38:57,840 --> 00:39:01,830 And here's where we're going to bring in our scattering kernel. 758 00:39:01,830 --> 00:39:04,710 Because there's some probability that the neutron 759 00:39:04,710 --> 00:39:07,200 scatters in from a different group. 760 00:39:15,180 --> 00:39:21,505 And then we'll have our d-omega, dE-prime, dV. 761 00:39:21,505 --> 00:39:22,380 Is this complete yet? 762 00:39:28,630 --> 00:39:32,700 We've now accounted for one other energy, E-prime. 763 00:39:32,700 --> 00:39:35,300 Now how do we account for all possible other energies 764 00:39:35,300 --> 00:39:37,213 scattering into our energy group? 765 00:39:37,213 --> 00:39:38,140 AUDIENCE: Integrals. 766 00:39:38,140 --> 00:39:40,680 MICHAEL SHORT: Yep, again, same integrals. 767 00:39:40,680 --> 00:39:45,470 We integrate over all possible energies, 768 00:39:45,470 --> 00:39:51,190 over all possible angles, and over our volume. 769 00:39:51,190 --> 00:39:53,350 Hopefully these terms are looking very similar. 770 00:39:53,350 --> 00:39:57,820 In every case it's a volume, angle, energy integral 771 00:39:57,820 --> 00:39:59,365 of a reaction rate. 772 00:39:59,365 --> 00:40:01,990 And all that's saying is there's some rate that these reactions 773 00:40:01,990 --> 00:40:05,770 are occurring, which is either a gain rate or a loss rate. 774 00:40:05,770 --> 00:40:08,080 We integrate over whatever volume, energy, and angle 775 00:40:08,080 --> 00:40:08,800 we're tracking. 776 00:40:08,800 --> 00:40:10,840 And that's all there is to it. 777 00:40:10,840 --> 00:40:12,463 So we've got in-scattering. 778 00:40:16,540 --> 00:40:20,722 Now let's tackle the n,in reactions. 779 00:40:20,722 --> 00:40:22,680 There's only going to be one little difference. 780 00:40:22,680 --> 00:40:24,763 But I want you guys to tell me what sort of things 781 00:40:24,763 --> 00:40:27,030 are going to be the same as all the other terms 782 00:40:27,030 --> 00:40:27,780 that we have here. 783 00:40:31,660 --> 00:40:34,760 So what do we start off with inside the integral? 784 00:40:34,760 --> 00:40:35,760 AUDIENCE: Reaction rate. 785 00:40:35,760 --> 00:40:37,052 MICHAEL SHORT: A reaction rate. 786 00:40:37,052 --> 00:40:38,610 And how do we write that? 787 00:40:38,610 --> 00:40:39,610 AUDIENCE: [INAUDIBLE]. 788 00:40:43,110 --> 00:40:45,505 MICHAEL SHORT: Yep, a cross-section, 789 00:40:45,505 --> 00:40:47,130 there's going to be some cross-section, 790 00:40:47,130 --> 00:40:53,000 for, let's call it n,in reaction as a function of energy, 791 00:40:53,000 --> 00:40:53,710 times a flux. 792 00:40:58,290 --> 00:41:02,970 There'll be our normal integrated over all angles, 793 00:41:02,970 --> 00:41:06,940 all energies, and our volume. 794 00:41:06,940 --> 00:41:08,200 So we have a reaction rate. 795 00:41:08,200 --> 00:41:11,290 What do we have to then integrate that reaction right 796 00:41:11,290 --> 00:41:12,910 over to get all the neutrons in? 797 00:41:16,375 --> 00:41:19,850 AUDIENCE: [INAUDIBLE] same things. 798 00:41:19,850 --> 00:41:22,400 MICHAEL SHORT: Same things as everything else, exactly. 799 00:41:22,400 --> 00:41:26,050 Integrate over all possible energies, 800 00:41:26,050 --> 00:41:27,880 integrate over all possible angles, 801 00:41:27,880 --> 00:41:31,630 integrate over our volume, looks quite similar. 802 00:41:31,630 --> 00:41:36,260 The only thing we haven't dealt with is this i term right here. 803 00:41:36,260 --> 00:41:45,790 Because there can be, there are actually n,2n reactions, n,3n, 804 00:41:45,790 --> 00:41:49,560 n,4n, and so on. 805 00:41:49,560 --> 00:41:52,070 So there are probabilities that, let's say 806 00:41:52,070 --> 00:41:55,130 you, in goes 1 neutron, out comes 3 neutrons. 807 00:41:55,130 --> 00:41:56,810 But no fission actually happened. 808 00:41:56,810 --> 00:41:58,760 You just blast a few of them out. 809 00:41:58,760 --> 00:42:04,775 So I think all we'd really have to do is sum over i equals 1. 810 00:42:04,775 --> 00:42:07,715 Oh, I'm sorry, i equals 2, because a neutron 811 00:42:07,715 --> 00:42:09,590 going in and then the same neutron going out, 812 00:42:09,590 --> 00:42:11,000 that's just scattering. 813 00:42:11,000 --> 00:42:15,050 And what would be the maximum? 814 00:42:15,050 --> 00:42:17,440 Probably 4. 815 00:42:17,440 --> 00:42:19,870 Because the probability of an increasing i 816 00:42:19,870 --> 00:42:24,400 or more neutrons coming out gets lower and lower as you go. 817 00:42:24,400 --> 00:42:27,640 In fact, these reactions don't even turn on until-- 818 00:42:27,640 --> 00:42:32,260 the n,2n reaction turns on and around like 1 MeV. 819 00:42:32,260 --> 00:42:36,070 This one turns on and around like 5 MeV. 820 00:42:36,070 --> 00:42:37,930 This one turns on at like 12 MeV. 821 00:42:37,930 --> 00:42:42,410 I was just looking up these cross-sections before class. 822 00:42:42,410 --> 00:42:45,010 So if you have a reaction that doesn't happen 823 00:42:45,010 --> 00:42:47,410 beyond your highest neutron energy, 824 00:42:47,410 --> 00:42:49,655 you probably don't need to worry about it. 825 00:42:49,655 --> 00:42:52,030 But the reason I had us write all these extra equations-- 826 00:42:52,030 --> 00:42:55,510 and I think that the t-shirt for this department needs some 827 00:42:55,510 --> 00:42:57,830 updating to include these extra terms-- 828 00:42:57,830 --> 00:43:00,100 is because they're all the same term. 829 00:43:00,100 --> 00:43:02,980 It is in every case, it's an integral 830 00:43:02,980 --> 00:43:13,860 over all of our stuff of a reaction rate, 831 00:43:13,860 --> 00:43:18,090 d-stuff, times a multiplier. 832 00:43:21,590 --> 00:43:23,900 Every single term in this equation 833 00:43:23,900 --> 00:43:27,360 follows the exact same pattern. 834 00:43:27,360 --> 00:43:30,870 So what I hope, and I would expect out of you guys, 835 00:43:30,870 --> 00:43:32,790 is that if I were to give you this table 836 00:43:32,790 --> 00:43:35,250 of possible reactions, you would be 837 00:43:35,250 --> 00:43:39,510 able to recreate this neutron transport equation using 838 00:43:39,510 --> 00:43:42,030 this template to know that every single reaction is just 839 00:43:42,030 --> 00:43:45,450 multiplier, times integral of stuff of a reaction rate, 840 00:43:45,450 --> 00:43:49,890 d-stuff, where the reaction rate is just a cross-section times 841 00:43:49,890 --> 00:43:51,450 a flux. 842 00:43:51,450 --> 00:43:54,500 That's all there is to it. 843 00:43:54,500 --> 00:43:56,450 Not bad when you see that everything follows 844 00:43:56,450 --> 00:43:58,740 the same pattern, right? 845 00:43:58,740 --> 00:44:01,080 That's the basis behind most of the hideous equations 846 00:44:01,080 --> 00:44:02,940 that you see in all of physics everywhere, 847 00:44:02,940 --> 00:44:05,700 is if there are additive or subtractive terms, 848 00:44:05,700 --> 00:44:07,363 they'd better be in the same units. 849 00:44:07,363 --> 00:44:09,030 And so they're going to follow some sort 850 00:44:09,030 --> 00:44:11,430 of a similar template. 851 00:44:11,430 --> 00:44:14,380 Not too scary when you think of it that way. 852 00:44:14,380 --> 00:44:16,335 So let's now come up with the loss terms. 853 00:44:20,270 --> 00:44:22,650 I should have planned these boards better. 854 00:44:22,650 --> 00:44:25,350 Keep these ones here so we keep a template. 855 00:44:25,350 --> 00:44:28,890 And we'll have a minus, well, how 856 00:44:28,890 --> 00:44:35,770 do we write the anything reaction using this template? 857 00:44:41,160 --> 00:44:43,250 How many neutrons undergo a reaction 858 00:44:43,250 --> 00:44:45,000 when one neutron undergoes a reaction? 859 00:44:48,250 --> 00:44:48,982 Yeah, 1. 860 00:44:48,982 --> 00:44:49,940 So our multiplier is 1. 861 00:44:49,940 --> 00:44:51,630 We don't have to worry about it. 862 00:44:51,630 --> 00:44:53,440 We have our integral of stuff. 863 00:44:53,440 --> 00:44:58,190 So we'll have to integrate over all possible volumes, angles, 864 00:44:58,190 --> 00:44:59,540 and energy. 865 00:44:59,540 --> 00:45:02,485 And what's on the inside? 866 00:45:02,485 --> 00:45:05,300 AUDIENCE: [INAUDIBLE] total cross-section. 867 00:45:05,300 --> 00:45:09,220 MICHAEL SHORT: Yep, total cross-section 868 00:45:09,220 --> 00:45:22,460 as a function of energy times the flux, d-stuff to save time. 869 00:45:22,460 --> 00:45:25,100 So don't worry, even though the boards are laid out funny, 870 00:45:25,100 --> 00:45:26,933 on the pictures of the blackboard that we'll 871 00:45:26,933 --> 00:45:29,308 put on the Stellar site, I'll Photoshop these and arrange 872 00:45:29,308 --> 00:45:30,850 them so that they're all in sequence. 873 00:45:30,850 --> 00:45:32,180 And you can see everything. 874 00:45:32,180 --> 00:45:35,390 And then there's the last term to account for. 875 00:45:35,390 --> 00:45:37,540 That's the leakage term. 876 00:45:37,540 --> 00:45:39,200 This one is a little different. 877 00:45:39,200 --> 00:45:42,970 It's the only one that's a little different. 878 00:45:42,970 --> 00:45:46,750 And in this case, we're going to say that our little volume 879 00:45:46,750 --> 00:45:53,140 element also has a surface to it. 880 00:45:53,140 --> 00:45:57,480 And if the neutrons leave the surface, 881 00:45:57,480 --> 00:45:59,470 then they leave the volume. 882 00:45:59,470 --> 00:46:05,770 So in this case, we'll have a surface integral of our neutron 883 00:46:05,770 --> 00:46:10,860 flux, say our neutron flux dS. 884 00:46:10,860 --> 00:46:14,160 Because there's no reaction happening when neutrons just 885 00:46:14,160 --> 00:46:15,840 move, right. 886 00:46:15,840 --> 00:46:17,520 They just go. 887 00:46:17,520 --> 00:46:23,652 And so, well, we'd also have to multiply by our normal vector. 888 00:46:23,652 --> 00:46:25,110 Because every flux is going to have 889 00:46:25,110 --> 00:46:28,210 a certain number of neutrons moving in a certain direction. 890 00:46:28,210 --> 00:46:31,860 Let's say we were tracking the flow of neutrons 891 00:46:31,860 --> 00:46:34,310 through this surface right here. 892 00:46:34,310 --> 00:46:42,830 And if we had a flux going in exactly this direction, 893 00:46:42,830 --> 00:46:47,600 through this surface, and this is the normal vector, 894 00:46:47,600 --> 00:46:52,800 in this case, flux, which is a vector 895 00:46:52,800 --> 00:46:55,950 dotted with the normal vector, is just the flux. 896 00:46:55,950 --> 00:46:58,200 Which is to say that if the flux and the normal vector 897 00:46:58,200 --> 00:47:01,080 are aligned in the same way, then every neutron going 898 00:47:01,080 --> 00:47:04,740 through the surface is tracked as going through the surface. 899 00:47:04,740 --> 00:47:08,760 To take the opposite example, what about the situation where 900 00:47:08,760 --> 00:47:15,120 you have a surface here and you have a mono-directional flux 901 00:47:15,120 --> 00:47:18,340 of neutrons in this direction. 902 00:47:18,340 --> 00:47:20,320 And that is your surface normal. 903 00:47:25,670 --> 00:47:28,510 What does the flux dotted with the surface normal vector 904 00:47:28,510 --> 00:47:30,280 equal? 905 00:47:30,280 --> 00:47:35,840 0, it's just a dot product between the direction 906 00:47:35,840 --> 00:47:39,410 that your neutrons are moving and the normal vector saying, 907 00:47:39,410 --> 00:47:41,390 does it go out of the surface at all? 908 00:47:41,390 --> 00:47:45,990 So for these two limiting cases, in this case, the fluxes just 909 00:47:45,990 --> 00:47:47,540 let's say, what is it, the number 910 00:47:47,540 --> 00:47:49,820 of neutrons leaving the surface is the flux. 911 00:47:49,820 --> 00:47:51,830 In this case, no neutrons leave the surface 912 00:47:51,830 --> 00:47:54,680 because they're not actually going through the surface. 913 00:47:54,680 --> 00:47:58,820 It's a good time to mention, again, that these units of flux 914 00:47:58,820 --> 00:48:03,590 are in neutrons per centimeter squared per second, which 915 00:48:03,590 --> 00:48:05,690 is to say the number of particles traveling 916 00:48:05,690 --> 00:48:09,655 through this area in centimeter squared every second. 917 00:48:09,655 --> 00:48:11,030 I know we've gone over it before, 918 00:48:11,030 --> 00:48:13,622 but I want you to keep these units in mind. 919 00:48:13,622 --> 00:48:15,830 Because now they actually have a little more physical 920 00:48:15,830 --> 00:48:18,010 significance. 921 00:48:18,010 --> 00:48:22,590 And that's why we have this flux times normal vector dS. 922 00:48:22,590 --> 00:48:24,210 That describes the number of neutrons 923 00:48:24,210 --> 00:48:26,050 that get through the surface. 924 00:48:26,050 --> 00:48:30,730 The last thing we'll do, because everything else is a volume 925 00:48:30,730 --> 00:48:33,850 integral, we want this to be a volume integral because we're 926 00:48:33,850 --> 00:48:35,710 going to simplify this in terms of getting 927 00:48:35,710 --> 00:48:37,157 rid of all the volume stuff. 928 00:48:37,157 --> 00:48:39,490 We're going to use what's called the divergence theorem. 929 00:48:42,170 --> 00:48:44,810 I hear some snickering. 930 00:48:44,810 --> 00:48:46,340 Because I remember this is probably 931 00:48:46,340 --> 00:48:50,080 something where you were told in 1801 or 1802, this exists. 932 00:48:50,080 --> 00:48:51,230 Use it in a few problems. 933 00:48:51,230 --> 00:48:52,190 Moving on. 934 00:48:52,190 --> 00:48:53,965 That sound about right? 935 00:48:53,965 --> 00:48:55,340 This is when you actually use it. 936 00:48:58,820 --> 00:49:02,110 So the divergence theorem says that the integral 937 00:49:02,110 --> 00:49:07,310 of some variable F dS, through some volume element of surface, 938 00:49:07,310 --> 00:49:11,970 is the same as the volume integral of-- 939 00:49:11,970 --> 00:49:12,800 how does this go-- 940 00:49:16,810 --> 00:49:19,160 del dot F dV. 941 00:49:19,160 --> 00:49:22,060 And this is going to be quite important because one, 942 00:49:22,060 --> 00:49:25,370 it gives us a volume integral. 943 00:49:25,370 --> 00:49:33,630 So this is, it will be a volume integral of our del dot flux 944 00:49:33,630 --> 00:49:36,940 dV, so now everything's in the same units. 945 00:49:36,940 --> 00:49:40,330 And if we were to say forget about our little volume 946 00:49:40,330 --> 00:49:40,830 element. 947 00:49:40,830 --> 00:49:43,230 Let's just assume an infinite reactor. 948 00:49:43,230 --> 00:49:47,070 Every single volume integral in term just instantly disappears. 949 00:49:47,070 --> 00:49:49,650 Because we wrote these equations to be 950 00:49:49,650 --> 00:49:55,080 identical for any dV anywhere inside this reactor. 951 00:49:55,080 --> 00:49:59,340 If the reactor is then infinite, then all of those volume terms 952 00:49:59,340 --> 00:49:59,952 disappear. 953 00:49:59,952 --> 00:50:01,410 And that's the first simplification 954 00:50:01,410 --> 00:50:05,030 that we'll make in the next class. 955 00:50:05,030 --> 00:50:09,380 But right here on these five boards, 956 00:50:09,380 --> 00:50:11,610 we've developed the neutron transport equation, 957 00:50:11,610 --> 00:50:15,140 which is the absolute, most general, highest escalated 958 00:50:15,140 --> 00:50:17,390 form of how do you track neutrons 959 00:50:17,390 --> 00:50:21,560 through any volume, any direction, any energy, 960 00:50:21,560 --> 00:50:23,060 at any time. 961 00:50:23,060 --> 00:50:25,850 And we'll spend Thursday and Friday simplifying this 962 00:50:25,850 --> 00:50:28,190 to something that we can solve. 963 00:50:28,190 --> 00:50:32,040 The other reason that we use this divergence theorem 964 00:50:32,040 --> 00:50:34,040 is because we're going to make an approximation. 965 00:50:34,040 --> 00:50:36,440 This crazy looking thing right here, 966 00:50:36,440 --> 00:50:38,570 we will make an approximation called the diffusion 967 00:50:38,570 --> 00:50:40,940 approximation where we assume that neutrons 968 00:50:40,940 --> 00:50:43,580 are like a gas that just diffuse away from each other. 969 00:50:43,580 --> 00:50:46,910 And that's going to make solving this really, really easy. 970 00:50:46,910 --> 00:50:48,650 It's going to go from some second order 971 00:50:48,650 --> 00:50:51,740 differential or differential integral equation 972 00:50:51,740 --> 00:50:55,090 to just the equation that you can solve with algebra. 973 00:50:55,090 --> 00:50:56,190 Yep? 974 00:50:56,190 --> 00:50:59,470 AUDIENCE: Do you need the dE d-omega for the last term? 975 00:50:59,470 --> 00:51:05,470 MICHAEL SHORT: Probably, yeah, over all E, over all omega. 976 00:51:05,470 --> 00:51:12,280 And that flux is going to be of r, E omega, t. 977 00:51:15,190 --> 00:51:16,755 Absolutely. 978 00:51:16,755 --> 00:51:18,880 Just to make sure, everything is in the same units, 979 00:51:18,880 --> 00:51:21,610 every term has a fairly similar template. 980 00:51:21,610 --> 00:51:26,260 The only difference is leakage, there's no reaction here. 981 00:51:29,790 --> 00:51:32,940 Every single other term constitutes a reaction. 982 00:51:32,940 --> 00:51:34,650 And they all follow this template. 983 00:51:37,280 --> 00:51:39,750 So I will stop here because it is five of. 984 00:51:39,750 --> 00:51:41,350 See if anyone has any quick questions 985 00:51:41,350 --> 00:51:42,980 on what you've got here. 986 00:51:42,980 --> 00:51:46,180 I'll make sure to get all of this on the board images 987 00:51:46,180 --> 00:51:47,950 so you guys can take a look at it. 988 00:51:47,950 --> 00:51:49,450 And I'll projected up on the screen 989 00:51:49,450 --> 00:51:51,700 so that we can make some simplifications based on what 990 00:51:51,700 --> 00:51:54,440 we see here on Thursday. 991 00:51:54,440 --> 00:51:55,070 Yeah? 992 00:51:55,070 --> 00:51:57,582 AUDIENCE: What is the neutron birth spectrum? 993 00:51:57,582 --> 00:51:59,290 MICHAEL SHORT: The neutron birth spectrum 994 00:51:59,290 --> 00:52:01,900 says that if you have any old fission event, what's 995 00:52:01,900 --> 00:52:04,000 the probability of those neutrons being 996 00:52:04,000 --> 00:52:06,080 born at different energies? 997 00:52:06,080 --> 00:52:09,250 What this says is that they're born between 1 and 10 MeV, 998 00:52:09,250 --> 00:52:11,342 with a peak at around 2 MeV. 999 00:52:11,342 --> 00:52:13,300 But if you want to track the number of neutrons 1000 00:52:13,300 --> 00:52:15,675 in every energy group, you need to know where they begin. 1001 00:52:19,020 --> 00:52:19,620 Good question. 1002 00:52:19,620 --> 00:52:22,823 So if any of the terms here are unclear what they physically 1003 00:52:22,823 --> 00:52:24,990 mean, because that's what I'm most interested in you 1004 00:52:24,990 --> 00:52:28,020 guys knowing, please do ask either on Piazza, 1005 00:52:28,020 --> 00:52:29,620 on email, on Thursday. 1006 00:52:29,620 --> 00:52:30,120 Yeah? 1007 00:52:30,120 --> 00:52:30,810 AUDIENCE: What's the difference between the big N 1008 00:52:30,810 --> 00:52:32,610 and the little n again? 1009 00:52:32,610 --> 00:52:35,490 MICHAEL SHORT: The big N and the little n, which one? 1010 00:52:35,490 --> 00:52:38,054 The cursive one, or this one? 1011 00:52:38,054 --> 00:52:40,350 AUDIENCE: The little n up top there and then 1012 00:52:40,350 --> 00:52:41,340 the non-cursive one. 1013 00:52:41,340 --> 00:52:44,220 MICHAEL SHORT: OK, the little n and the non-cursive one. 1014 00:52:44,220 --> 00:52:47,700 The little n is the number of neutrons in a volume, 1015 00:52:47,700 --> 00:52:51,290 at a certain energy, going in a direction, at a certain time. 1016 00:52:51,290 --> 00:52:54,270 Big N right here is just number density, number 1017 00:52:54,270 --> 00:53:01,030 of atoms per centimeter cubed. 1018 00:53:01,030 --> 00:53:04,210 Cursive n is the number of neutrons at an energy, 1019 00:53:04,210 --> 00:53:05,020 in a volume. 1020 00:53:05,020 --> 00:53:07,270 We don't care where they're going. 1021 00:53:07,270 --> 00:53:10,870 And the reason I write these terms up here is we 1022 00:53:10,870 --> 00:53:13,750 are going to switch from lowercase to capital, 1023 00:53:13,750 --> 00:53:16,480 or angularly-dependent to angularly-independent 1024 00:53:16,480 --> 00:53:18,290 by making a simple approximation to say, 1025 00:53:18,290 --> 00:53:20,082 we don't care what direction they're going. 1026 00:53:20,082 --> 00:53:22,560 We just care if they're there. 1027 00:53:22,560 --> 00:53:25,027 But in real complex neutron physics problems, 1028 00:53:25,027 --> 00:53:27,360 like the one solved at the computational reactor physics 1029 00:53:27,360 --> 00:53:29,670 group, you need to know all the angles. 1030 00:53:29,670 --> 00:53:31,170 And you need to know the probability 1031 00:53:31,170 --> 00:53:33,960 or the cross-section that a neutron coming in at this angle 1032 00:53:33,960 --> 00:53:36,230 leaves at that angle and imparts a certain energy. 1033 00:53:36,230 --> 00:53:37,480 Because they're all different. 1034 00:53:37,480 --> 00:53:39,450 For the purposes of this class, I just 1035 00:53:39,450 --> 00:53:41,252 want you to know that they exist. 1036 00:53:41,252 --> 00:53:43,460 And the first thing we will do is simplify them away. 1037 00:53:45,717 --> 00:53:47,300 But this way, you'll be fully prepared 1038 00:53:47,300 --> 00:53:49,730 for 22.05 and a lifetime of reactor physics, 1039 00:53:49,730 --> 00:53:50,540 if you so choose. 1040 00:53:53,230 --> 00:53:55,580 Who here is done a year op in the computational reactor 1041 00:53:55,580 --> 00:53:56,690 physics group? 1042 00:53:56,690 --> 00:53:58,186 Just one, OK. 1043 00:53:58,186 --> 00:53:59,033 I recommend more. 1044 00:53:59,033 --> 00:54:01,200 They tend to be the biggest group in the department. 1045 00:54:01,200 --> 00:54:02,930 They've got like 20 grad students 1046 00:54:02,930 --> 00:54:05,880 and probably more year ops than that. 1047 00:54:05,880 --> 00:54:07,310 So try it out. 1048 00:54:07,310 --> 00:54:10,430 It's what makes us us, us nukes, right, 1049 00:54:10,430 --> 00:54:16,400 is neutrons and tracking them to ridiculous proportions. 1050 00:54:16,400 --> 00:54:19,350 OK, definitely want to let you guys go it's one of. 1051 00:54:19,350 --> 00:54:21,461 So I'll see you all on Thursday.