1 00:00:00,850 --> 00:00:03,220 The following content is provided under a Creative 2 00:00:03,220 --> 00:00:04,610 Commons license. 3 00:00:04,610 --> 00:00:06,820 Your support will help MIT OpenCourseWare 4 00:00:06,820 --> 00:00:10,910 continue to offer high quality educational resources for free. 5 00:00:10,910 --> 00:00:13,480 To make a donation or to view additional materials 6 00:00:13,480 --> 00:00:17,440 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,440 --> 00:00:18,313 at ocw.mit.edu. 8 00:00:22,597 --> 00:00:25,180 MICHAEL SHORT: So today is going to be the last day of neutron 9 00:00:25,180 --> 00:00:25,630 physics. 10 00:00:25,630 --> 00:00:27,422 As promised, we're going to talk about what 11 00:00:27,422 --> 00:00:30,520 happens as a function of time when you perturb the reactor, 12 00:00:30,520 --> 00:00:32,619 like you all did about a month ago. 13 00:00:32,619 --> 00:00:35,410 Did any of you guys notice the old-fashioned analog panel 14 00:00:35,410 --> 00:00:38,710 meter that said, reactor period, when you were doing your power 15 00:00:38,710 --> 00:00:39,460 manipulations? 16 00:00:39,460 --> 00:00:40,750 We're going to do that today. 17 00:00:40,750 --> 00:00:42,792 And you're going to explore that on the homework. 18 00:00:42,792 --> 00:00:45,790 So I'm arranging for all of your actual power manipulation 19 00:00:45,790 --> 00:00:47,240 traces to be sent to you. 20 00:00:47,240 --> 00:00:50,005 So each one, you'll have your own reactor data. 21 00:00:50,005 --> 00:00:51,880 You'll be able to describe the reactor period 22 00:00:51,880 --> 00:00:56,200 and see how well it fits our infinite medium single group 23 00:00:56,200 --> 00:00:58,330 equations, which it turns out is not very well. 24 00:00:58,330 --> 00:01:01,670 But that's OK, because you'll get to explain the differences. 25 00:01:01,670 --> 00:01:03,760 First, before we get into transients 26 00:01:03,760 --> 00:01:09,130 I wanted to talk a bit about criticality and perturbing it. 27 00:01:09,130 --> 00:01:13,135 So let's say we had our old single group kit criticality 28 00:01:13,135 --> 00:01:13,635 relation. 29 00:01:22,420 --> 00:01:25,510 And I'd like to analyze, just intuitively or mentally 30 00:01:25,510 --> 00:01:28,100 with you guys, a few different situations. 31 00:01:28,100 --> 00:01:30,910 Let's say we're talking about a light water 32 00:01:30,910 --> 00:01:37,180 reactor or a thermal reactor, like the MIT reactor, or pretty 33 00:01:37,180 --> 00:01:40,470 much all the reactors we have in this country. 34 00:01:40,470 --> 00:01:43,830 What sort of things could you do to perturb it? 35 00:01:43,830 --> 00:01:47,430 And how would that affect criticality? 36 00:01:47,430 --> 00:01:49,800 For example, let's say you shoved in a control rod. 37 00:01:49,800 --> 00:01:52,530 Let's take the simplest scenario. 38 00:01:52,530 --> 00:01:53,670 Control rods in. 39 00:01:59,800 --> 00:02:01,600 What would happen to each of the terms 40 00:02:01,600 --> 00:02:02,950 in the criticality condition? 41 00:02:02,950 --> 00:02:05,860 And then, what would happen to k effective? 42 00:02:05,860 --> 00:02:07,450 So let's just go one by one. 43 00:02:07,450 --> 00:02:10,479 Does nu ever change, ever? 44 00:02:10,479 --> 00:02:12,280 Actually, yeah, it does. 45 00:02:12,280 --> 00:02:14,638 Over time, you'll start-- 46 00:02:14,638 --> 00:02:16,180 that nu right there, remember, that's 47 00:02:16,180 --> 00:02:19,120 a nu bar, number of neutrons produced per fission. 48 00:02:19,120 --> 00:02:27,040 As you start to consume U238 add neutrons. 49 00:02:27,040 --> 00:02:30,370 And as you guys saw through a complicated chain of events 50 00:02:30,370 --> 00:02:34,100 on the exam, eventually make plutonium 239, 51 00:02:34,100 --> 00:02:36,370 which is a fissile fuel. 52 00:02:36,370 --> 00:02:42,380 The nu for 238 is actually different than the nu for 239. 53 00:02:42,380 --> 00:02:45,137 So I don't want to say that nu never changes. 54 00:02:45,137 --> 00:02:47,470 It's just that shoving the control rods into the reactor 55 00:02:47,470 --> 00:02:49,300 is not going to change nu. 56 00:02:49,300 --> 00:02:53,050 But it does change slowly over time as you build up plutonium. 57 00:02:53,050 --> 00:02:55,660 What about sigma fission? 58 00:02:55,660 --> 00:02:59,330 If this were a blended homogeneous reactor 59 00:02:59,330 --> 00:03:03,340 or a reactor in a blender, what would happen to sigma fission 60 00:03:03,340 --> 00:03:06,010 as you then shove in an absorbing material? 61 00:03:10,430 --> 00:03:11,210 Does it change? 62 00:03:14,661 --> 00:03:16,140 AUDIENCE: No. 63 00:03:16,140 --> 00:03:17,370 MICHAEL SHORT: You say, no. 64 00:03:17,370 --> 00:03:20,580 And I'm going to add here homogeneous. 65 00:03:24,490 --> 00:03:27,730 So in this case, remember if we define the average sigma 66 00:03:27,730 --> 00:03:29,920 fission as a sum-- 67 00:03:29,920 --> 00:03:34,210 I'll add bits to it-- 68 00:03:34,210 --> 00:03:38,230 of each material's volume fraction, or let's say 69 00:03:38,230 --> 00:03:45,490 atomic fraction, times each material's sigma fission, 70 00:03:45,490 --> 00:03:48,010 if we throw nu materials into the reactor, 71 00:03:48,010 --> 00:03:49,780 then this homogeneous sigma fission 72 00:03:49,780 --> 00:03:52,420 does change when we put materials in 73 00:03:52,420 --> 00:03:55,360 or take materials out. 74 00:03:55,360 --> 00:03:59,260 So you guys want to revise your idea? 75 00:03:59,260 --> 00:03:59,950 AUDIENCE: Yes. 76 00:03:59,950 --> 00:04:00,750 MICHAEL SHORT: Yes, thank you. 77 00:04:00,750 --> 00:04:02,000 There's only one other choice. 78 00:04:02,000 --> 00:04:05,770 Now the question is, by how much? 79 00:04:05,770 --> 00:04:09,370 If you put in a control rod where let's say the control 80 00:04:09,370 --> 00:04:14,200 rod's sigma fission would be equal to zero, 81 00:04:14,200 --> 00:04:18,430 but volume would be equal to small. 82 00:04:18,430 --> 00:04:20,373 Can't be any more specific than that. 83 00:04:20,373 --> 00:04:21,790 How much of an effect do you think 84 00:04:21,790 --> 00:04:23,960 you'll have on sigma fission? 85 00:04:23,960 --> 00:04:25,137 AUDIENCE: Small. 86 00:04:25,137 --> 00:04:26,220 MICHAEL SHORT: Very small. 87 00:04:26,220 --> 00:04:30,560 So let's say a little down arrow like that. 88 00:04:30,560 --> 00:04:34,320 What about sigma absorption? 89 00:04:34,320 --> 00:04:37,950 The volume is still small, but a control rod by definition sigma 90 00:04:37,950 --> 00:04:39,480 absorption equals huge. 91 00:04:43,320 --> 00:04:44,195 So what do you think? 92 00:04:44,195 --> 00:04:45,570 AUDIENCE: It's going to increase. 93 00:04:45,570 --> 00:04:46,950 MICHAEL SHORT: It's going to increase a little or a lot? 94 00:04:46,950 --> 00:04:48,420 AUDIENCE: A lot. 95 00:04:48,420 --> 00:04:50,790 MICHAEL SHORT: Quite a bit. 96 00:04:50,790 --> 00:04:53,370 Now let's look at the diffusion constant. 97 00:04:53,370 --> 00:04:58,200 And remember that the diffusion constant is 1 over 3 sigma 98 00:04:58,200 --> 00:05:03,990 total, minus the average cosine scattering angle 99 00:05:03,990 --> 00:05:07,230 sigma scattering. 100 00:05:07,230 --> 00:05:10,080 What do you think is going to happen to the neutron diffusion 101 00:05:10,080 --> 00:05:13,020 coefficient as you throw in an absorbing material? 102 00:05:18,280 --> 00:05:22,580 Something that's got an enormous absorption cross-section 103 00:05:22,580 --> 00:05:29,240 is also going to have an enormous total cross-section, 104 00:05:29,240 --> 00:05:34,160 because sigma total is sigma absorption plus sigma 105 00:05:34,160 --> 00:05:35,710 scattering. 106 00:05:35,710 --> 00:05:38,750 And sigma scattering doesn't change that much. 107 00:05:38,750 --> 00:05:43,100 But if sigma absorption goes up, sigma total goes up. 108 00:05:43,100 --> 00:05:46,160 If sigma total goes up, then what happens 109 00:05:46,160 --> 00:05:47,793 o the diffusion coefficient? 110 00:05:50,751 --> 00:05:51,740 AUDIENCE: Decrease. 111 00:05:51,740 --> 00:05:53,407 MICHAEL SHORT: Yep, it's got a decrease. 112 00:05:57,675 --> 00:06:00,050 And how does inserting a control rod change the geometry? 113 00:06:02,665 --> 00:06:04,520 AUDIENCE: It doesn't. 114 00:06:04,520 --> 00:06:06,728 MICHAEL SHORT: Very, very close. 115 00:06:06,728 --> 00:06:07,520 Yeah, you're right. 116 00:06:07,520 --> 00:06:10,700 The control rod better not change the geometry, 117 00:06:10,700 --> 00:06:12,890 but what I do want to remind you of is 118 00:06:12,890 --> 00:06:16,940 that this buckling term includes-- let's say, 119 00:06:16,940 --> 00:06:20,420 this was a one dimensional infinite slab 120 00:06:20,420 --> 00:06:21,365 Cartesian reactor. 121 00:06:27,500 --> 00:06:31,425 That little hot over there means we have some extrapolation 122 00:06:31,425 --> 00:06:31,925 distance. 123 00:06:40,600 --> 00:06:44,040 Remember, if we were to draw our infinite reactor 124 00:06:44,040 --> 00:06:46,320 with the thickness A and we wanted 125 00:06:46,320 --> 00:06:50,180 to draw a flux profile on top of that, 126 00:06:50,180 --> 00:06:54,090 it would have to be symmetric about the middle. 127 00:06:54,090 --> 00:07:01,200 And let's say we had our axis of this is x and this is flux. 128 00:07:01,200 --> 00:07:05,470 Flux can't go to zero right at the edge of the reactor, 129 00:07:05,470 --> 00:07:07,800 because that would mean that no neutrons were literally 130 00:07:07,800 --> 00:07:08,890 leaking out. 131 00:07:08,890 --> 00:07:11,370 So there's going to be some small extrapolation 132 00:07:11,370 --> 00:07:14,580 distance equal to about two times 133 00:07:14,580 --> 00:07:17,890 the diffusion coefficient. 134 00:07:17,890 --> 00:07:19,590 So the geometric buckling is actually 135 00:07:19,590 --> 00:07:24,820 pi over the reactor geometry, plus 2 times 136 00:07:24,820 --> 00:07:26,980 the diffusion coefficient. 137 00:07:26,980 --> 00:07:31,360 And if the diffusion coefficient goes down, 138 00:07:31,360 --> 00:07:33,940 but it's also very, very small compared 139 00:07:33,940 --> 00:07:38,290 to the geometric buckling, how much does the buckling change 140 00:07:38,290 --> 00:07:39,740 and by how much? 141 00:07:39,740 --> 00:07:40,685 And in what direction? 142 00:07:40,685 --> 00:07:42,227 AUDIENCE: It increases very slightly. 143 00:07:42,227 --> 00:07:45,070 MICHAEL SHORT: Increases very slightly. 144 00:07:45,070 --> 00:07:48,370 So the buckling might increase very slightly. 145 00:07:48,370 --> 00:07:52,043 What's the overall net effect on k effective? 146 00:07:52,043 --> 00:07:53,893 AUDIENCE: It goes down. 147 00:07:53,893 --> 00:07:55,810 MICHAEL SHORT: Should go down, you would hope. 148 00:07:55,810 --> 00:07:57,640 If you put a control rod in, it should 149 00:07:57,640 --> 00:07:59,350 make k effective go down, because there's 150 00:07:59,350 --> 00:08:02,377 a little decrease here. 151 00:08:02,377 --> 00:08:03,710 Things kind of cancel out there. 152 00:08:03,710 --> 00:08:08,050 But the big one is putting an absorption material, 153 00:08:08,050 --> 00:08:11,283 like a control rod in, should make k effective go down. 154 00:08:11,283 --> 00:08:12,700 And that's the most intuitive one, 155 00:08:12,700 --> 00:08:15,520 but you can work out one term at a time what's 156 00:08:15,520 --> 00:08:17,560 generally going to happen. 157 00:08:17,560 --> 00:08:23,670 So let's now look at some other scenarios 158 00:08:23,670 --> 00:08:25,811 for the same criticality condition. 159 00:08:28,700 --> 00:08:31,650 I'll just rewrite it so that we can mess it all up again. 160 00:08:38,179 --> 00:08:46,900 Now we want to go for the case of boil or void your coolant. 161 00:08:49,900 --> 00:08:52,450 And now we're getting into the concept of different feedback 162 00:08:52,450 --> 00:08:53,200 mechanisms. 163 00:08:53,200 --> 00:08:55,720 We've already talked once about how 164 00:08:55,720 --> 00:08:57,280 raising the temperature of something 165 00:08:57,280 --> 00:09:02,620 tends to increase cross-sections in certain ways. 166 00:09:02,620 --> 00:09:04,120 But now let's say, what would happen 167 00:09:04,120 --> 00:09:05,680 if you boil your coolant? 168 00:09:05,680 --> 00:09:08,950 If things got really hot and the water started to boil. 169 00:09:08,950 --> 00:09:12,515 What do you want to happen to k effective? 170 00:09:12,515 --> 00:09:13,515 You want it to increase? 171 00:09:13,515 --> 00:09:14,190 AUDIENCE: Decrease. 172 00:09:14,190 --> 00:09:15,648 MICHAEL SHORT: Decrease, thank you. 173 00:09:15,648 --> 00:09:17,970 You want it to decrease, or else you'd get a Chernobyl. 174 00:09:17,970 --> 00:09:21,750 And we'll talk about how that happened in a week or two. 175 00:09:21,750 --> 00:09:25,240 So now let's reason through each one of these. 176 00:09:25,240 --> 00:09:28,350 Let's assume that nu doesn't change 177 00:09:28,350 --> 00:09:30,120 when you boil the coolant. 178 00:09:30,120 --> 00:09:32,955 What about sigma fission of the whole reactor? 179 00:09:41,690 --> 00:09:45,190 You're taking a little bit of material out of the reactor 180 00:09:45,190 --> 00:09:47,440 by taking liquid water, which is fairly dense, 181 00:09:47,440 --> 00:09:50,950 and making it gaseous water, which is less dense. 182 00:09:50,950 --> 00:09:55,270 So overall, there are more fissile atoms in the reactor 183 00:09:55,270 --> 00:09:58,640 proportionately when the coolant is boiled away 184 00:09:58,640 --> 00:10:01,000 than when it's not. 185 00:10:01,000 --> 00:10:04,870 So what happens to sigma fission? 186 00:10:04,870 --> 00:10:08,290 The average sigma fission for the reactor 187 00:10:08,290 --> 00:10:11,020 will go up ever so slightly. 188 00:10:11,020 --> 00:10:12,820 Probably not enough to matter. 189 00:10:12,820 --> 00:10:16,010 What about sigma absorption? 190 00:10:16,010 --> 00:10:17,210 If the coolant disappears. 191 00:10:21,010 --> 00:10:22,910 AUDIENCE: It goes down. 192 00:10:22,910 --> 00:10:25,310 MICHAEL SHORT: Yeah, water is an absorber. 193 00:10:25,310 --> 00:10:28,130 Hydrogen and oxygen-- really just hydrogen-- 194 00:10:28,130 --> 00:10:31,940 have some pretty non-negligible absorption coefficients. 195 00:10:31,940 --> 00:10:35,510 And if those go away, then you're 196 00:10:35,510 --> 00:10:38,917 losing a bit of absorber, aren't you? 197 00:10:38,917 --> 00:10:40,000 Actually it's interesting. 198 00:10:40,000 --> 00:10:44,620 Oxygen is the lowest thermal cross-section of any element. 199 00:10:44,620 --> 00:10:48,010 So we can treat it as pretty much transparent. 200 00:10:48,010 --> 00:10:50,770 Now how about the diffusion coefficient? 201 00:10:50,770 --> 00:10:53,380 We've got the formula for it up there. 202 00:10:53,380 --> 00:10:56,170 If all of a sudden your neutrons don't 203 00:10:56,170 --> 00:10:58,590 have much to moderate from-- 204 00:10:58,590 --> 00:11:00,430 there's not much to moderate your neutrons. 205 00:11:00,430 --> 00:11:00,660 Yeah. 206 00:11:00,660 --> 00:11:02,410 AUDIENCE: Your scattering just disappears. 207 00:11:02,410 --> 00:11:06,560 MICHAEL SHORT: Your scattering just disappears, right? 208 00:11:06,560 --> 00:11:09,140 But so does some of your total cross-section. 209 00:11:09,140 --> 00:11:10,790 So chances are, those neutrons are 210 00:11:10,790 --> 00:11:14,000 going to go farther before they undergo any given 211 00:11:14,000 --> 00:11:16,720 collision because there's no water in the way. 212 00:11:16,720 --> 00:11:22,470 So you'd expect neutron diffusion to go up. 213 00:11:22,470 --> 00:11:23,940 And what about geometric buckling? 214 00:11:27,050 --> 00:11:31,350 Diffusion goes up, then the geometric buckling-- 215 00:11:31,350 --> 00:11:33,830 I'm just going to make it really small. 216 00:11:33,830 --> 00:11:38,960 But the net effect here, once again, k effective goes down. 217 00:11:38,960 --> 00:11:42,240 We didn't talk about anything to do with the actual temperature 218 00:11:42,240 --> 00:11:43,490 effects on the cross-sections. 219 00:11:43,490 --> 00:11:46,950 This is just a density thing on the coolant itself. 220 00:11:46,950 --> 00:11:49,100 So let's now look at that. 221 00:11:49,100 --> 00:11:54,560 What about if you have some power spike, 222 00:11:54,560 --> 00:11:56,840 raised fuel temperature? 223 00:11:56,840 --> 00:11:59,330 I'll write it again, so we can mess it up again. 224 00:12:10,500 --> 00:12:13,930 So let's say you raise the fuel temperature. 225 00:12:13,930 --> 00:12:17,330 And that's going to cause every cross-section effectively 226 00:12:17,330 --> 00:12:20,840 to increase if you're doing this average scenario. 227 00:12:20,840 --> 00:12:22,280 Let's talk a little bit about why. 228 00:12:22,280 --> 00:12:23,960 It's not as simple as just saying, 229 00:12:23,960 --> 00:12:27,030 the cross-sections go up. 230 00:12:27,030 --> 00:12:29,580 So let's say we had two different temperatures, 231 00:12:29,580 --> 00:12:31,980 cold and hot. 232 00:12:31,980 --> 00:12:35,430 So this would be your sigma fission cold. 233 00:12:35,430 --> 00:12:39,150 And this would be your sigma fission hot. 234 00:12:39,150 --> 00:12:44,250 For cold, sigma fission looks something like that. 235 00:12:44,250 --> 00:12:46,380 And as the temperature goes up, these 236 00:12:46,380 --> 00:12:49,110 resonances, which I'll just label right here-- 237 00:12:54,440 --> 00:12:56,690 resonances being specific energy is 238 00:12:56,690 --> 00:12:59,120 where the absorption suddenly goes up, suddenly goes 239 00:12:59,120 --> 00:13:03,170 down will actually decrease in height. 240 00:13:03,170 --> 00:13:06,950 But they'll start to spread out more. 241 00:13:06,950 --> 00:13:10,310 That's about as well as I can draw it very crudely. 242 00:13:10,310 --> 00:13:14,870 And same thing goes not just for sigma fission, 243 00:13:14,870 --> 00:13:18,920 but for sigma anything, including absorption, including 244 00:13:18,920 --> 00:13:20,790 total whatever you want. 245 00:13:20,790 --> 00:13:23,510 And so if your goal is to get your neutrons 246 00:13:23,510 --> 00:13:25,910 from the fast region where they're 247 00:13:25,910 --> 00:13:30,890 born into the thermal region where you get fission, 248 00:13:30,890 --> 00:13:34,160 broadening these cross-sections makes it more likely 249 00:13:34,160 --> 00:13:36,890 that if the neutron loses any amount of energy, 250 00:13:36,890 --> 00:13:40,160 it's going to hit one of these big resonance regions 251 00:13:40,160 --> 00:13:44,330 and get absorbed or taken away before it gets a chance 252 00:13:44,330 --> 00:13:46,920 to go to the fission region. 253 00:13:46,920 --> 00:13:48,420 So what this is really going to do-- 254 00:13:48,420 --> 00:13:51,800 it's kind of funny to say it in terms of a one group 255 00:13:51,800 --> 00:13:55,610 criticality relation, but your fission cross-section 256 00:13:55,610 --> 00:13:58,250 is actually going to go down. 257 00:13:58,250 --> 00:14:01,580 One reason is that the fuel physically spreads out. 258 00:14:01,580 --> 00:14:04,520 And so just from the density modification, 259 00:14:04,520 --> 00:14:07,550 you're not going to get as much. 260 00:14:07,550 --> 00:14:09,230 But then you've also got that effect 261 00:14:09,230 --> 00:14:14,690 of increasing fission from these resonance regions spreading 262 00:14:14,690 --> 00:14:15,440 out. 263 00:14:15,440 --> 00:14:17,990 The question is, which one is a bigger effect? 264 00:14:17,990 --> 00:14:20,930 Can't answer that with a simple statement. 265 00:14:20,930 --> 00:14:22,920 You'll go over a lot more of that in 22.05 266 00:14:22,920 --> 00:14:25,610 when you talk about what actually defines a resonance 267 00:14:25,610 --> 00:14:27,290 region, how do you calculate them, 268 00:14:27,290 --> 00:14:30,680 and how do they Doppler broaden or broaden with temperature. 269 00:14:30,680 --> 00:14:32,390 How about sigma absorption? 270 00:14:35,270 --> 00:14:36,278 AUDIENCE: It goes down. 271 00:14:36,278 --> 00:14:38,070 MICHAEL SHORT: Yeah, sigma absorption, it's 272 00:14:38,070 --> 00:14:40,770 going to go down because things spread out. 273 00:14:40,770 --> 00:14:45,930 But it might also go up because the cross-sections spread out, 274 00:14:45,930 --> 00:14:47,405 or the resonances spread out. 275 00:14:47,405 --> 00:14:49,530 What's really going to happen though is the reactor 276 00:14:49,530 --> 00:14:52,200 atoms are effectively spreading themselves apart. 277 00:14:52,200 --> 00:14:53,915 The coolant's less dense. 278 00:14:53,915 --> 00:14:56,040 The structural materials in the fuel and everything 279 00:14:56,040 --> 00:14:57,042 are still there. 280 00:14:57,042 --> 00:14:58,500 They're less dense, but there's not 281 00:14:58,500 --> 00:15:00,120 fewer of them in the reactor. 282 00:15:00,120 --> 00:15:02,560 But there is going to be less coolant in the reactor, 283 00:15:02,560 --> 00:15:06,750 because it has the ability to sparsify or get 284 00:15:06,750 --> 00:15:08,730 less dense, and kind of squeeze out the inlet 285 00:15:08,730 --> 00:15:10,128 and outlet of the reactor. 286 00:15:10,128 --> 00:15:11,670 So what's really going to happen here 287 00:15:11,670 --> 00:15:14,680 is, we know diffusion is going to go up, 288 00:15:14,680 --> 00:15:17,280 which might cause a corresponding change 289 00:15:17,280 --> 00:15:18,540 in buckling. 290 00:15:18,540 --> 00:15:22,610 And the net effect, as we would hope, 291 00:15:22,610 --> 00:15:25,135 k effective would go down. 292 00:15:25,135 --> 00:15:26,760 And so what we've talked about now here 293 00:15:26,760 --> 00:15:29,340 is directly controlling reactivity 294 00:15:29,340 --> 00:15:32,910 with control rods, what's called a void coefficient, 295 00:15:32,910 --> 00:15:35,430 where you actually want to have a negative void coefficient. 296 00:15:35,430 --> 00:15:37,910 So if you boil your coolant too much, 297 00:15:37,910 --> 00:15:39,310 k effective should go down. 298 00:15:39,310 --> 00:15:42,210 And that's one of the mechanisms that a light water 299 00:15:42,210 --> 00:15:45,290 or a thermal reactor can help stabilize itself. 300 00:15:45,290 --> 00:15:48,540 And you can see that now from just a really simplified one 301 00:15:48,540 --> 00:15:51,557 group criticality relation. 302 00:15:51,557 --> 00:15:53,140 And if you raise the fuel temperature, 303 00:15:53,140 --> 00:15:55,290 let's say the fuel gets really hot because there's 304 00:15:55,290 --> 00:15:57,360 been some power spike, you also want 305 00:15:57,360 --> 00:16:00,330 the reactor to shut itself down, which you can see that it does. 306 00:16:03,880 --> 00:16:05,380 Let's make things a little trickier. 307 00:16:08,230 --> 00:16:10,060 Let's now talk about a sodium reactor. 308 00:16:13,510 --> 00:16:14,470 Fast reactor. 309 00:16:18,220 --> 00:16:29,710 This one relies a lot more on fast fission of U238. 310 00:16:29,710 --> 00:16:36,120 So if we were to draw the two cross-sections of 235 sigma 311 00:16:36,120 --> 00:16:40,980 fission and 238 sigma fission-- 312 00:16:40,980 --> 00:16:43,200 remember, uranium 235 looked like the one 313 00:16:43,200 --> 00:16:52,030 that we drew before, whereas U238 goes something like that, 314 00:16:52,030 --> 00:16:54,190 with no actual scale given. 315 00:16:54,190 --> 00:16:56,350 I'm not going to even go there. 316 00:16:56,350 --> 00:17:00,430 But uranium 238 does not need moderation for the neutrons 317 00:17:00,430 --> 00:17:02,860 to induce more fission. 318 00:17:02,860 --> 00:17:07,329 So let's now write the same criticality reaction, which, 319 00:17:07,329 --> 00:17:10,470 again, is a super simplified view of things, but that's OK. 320 00:17:17,230 --> 00:17:20,319 What would happen to each of these terms in a sodium 321 00:17:20,319 --> 00:17:23,319 fast reactor if you void the coolant? 322 00:17:29,790 --> 00:17:32,588 So nu won't change. 323 00:17:32,588 --> 00:17:33,630 What about sigma fission? 324 00:17:40,450 --> 00:17:44,300 Well, if the coolant goes away, then on average 325 00:17:44,300 --> 00:17:46,210 there is fissile materials contributing 326 00:17:46,210 --> 00:17:48,870 more to that cross-section, but not that much. 327 00:17:48,870 --> 00:17:52,120 So if you want to get technical, might 328 00:17:52,120 --> 00:17:56,900 be the slightest of increases, but doesn't matter that much. 329 00:17:56,900 --> 00:18:01,240 What really matters, though, is the stuff on the bottom. 330 00:18:01,240 --> 00:18:04,660 Sodium does have a low, but non-negligible absorption 331 00:18:04,660 --> 00:18:06,230 cross-section. 332 00:18:06,230 --> 00:18:09,410 So if the sodium were to boil away, 333 00:18:09,410 --> 00:18:12,290 then the absorption would go down 334 00:18:12,290 --> 00:18:15,510 by a non-negligible amount. 335 00:18:15,510 --> 00:18:17,480 And then what about diffusion? 336 00:18:26,427 --> 00:18:28,260 Well, we've got the formula for it up there. 337 00:18:32,970 --> 00:18:34,720 If there's not as much coolant in the way, 338 00:18:34,720 --> 00:18:38,230 then the neutrons are going to be 339 00:18:38,230 --> 00:18:39,880 able to get further on average. 340 00:18:39,880 --> 00:18:42,047 Let's say, they're not going to be scattering around 341 00:18:42,047 --> 00:18:43,250 with as much of the sodium. 342 00:18:43,250 --> 00:18:46,360 So there might be a small increase in diffusion 343 00:18:46,360 --> 00:18:49,090 and corresponding small increase in buckling. 344 00:18:49,090 --> 00:18:54,190 But this is where the one group kind of fails. 345 00:18:54,190 --> 00:18:55,990 What the sodium is actually doing 346 00:18:55,990 --> 00:18:59,080 is providing a little bit of moderation, 347 00:18:59,080 --> 00:19:02,740 so that some of those neutrons when they bounce off of sodium 348 00:19:02,740 --> 00:19:05,030 leave the fast fission region and get absorbed. 349 00:19:05,030 --> 00:19:07,360 And that's part of the balance of the reactor. 350 00:19:07,360 --> 00:19:10,060 If all of the neutrons are then born fast 351 00:19:10,060 --> 00:19:13,100 and don't really slow down and just get absorbed, 352 00:19:13,100 --> 00:19:20,140 then you might have an overall positive void coefficient. 353 00:19:20,140 --> 00:19:22,720 So this would tell you that in a fast reactor where you're 354 00:19:22,720 --> 00:19:24,670 depending on your coolant not just to cool 355 00:19:24,670 --> 00:19:27,580 the reactor, but to absorb somewhat 356 00:19:27,580 --> 00:19:30,310 and to moderate somewhat, you don't 357 00:19:30,310 --> 00:19:33,450 want to boil the coolant in a fast reactor. 358 00:19:33,450 --> 00:19:35,860 And is a lot of the reason why most fast reactor 359 00:19:35,860 --> 00:19:38,950 coolants tend to have extremely high boiling points. 360 00:19:38,950 --> 00:19:42,850 Sodium is approximately 893 Celsius. 361 00:19:42,850 --> 00:19:47,350 Lead bismuth is approximately 1,670 Celsius. 362 00:19:47,350 --> 00:19:50,240 Molten salt, about 1,400 Celsius. 363 00:19:50,240 --> 00:19:53,337 So all those coolant, except for the sodium one, 364 00:19:53,337 --> 00:19:55,420 you'll melt the steel that the reactor is made out 365 00:19:55,420 --> 00:19:57,370 of before your boil the coolant. 366 00:19:57,370 --> 00:20:00,700 So boiling the coolant is a bad day in a fast reactor, 367 00:20:00,700 --> 00:20:03,340 because then things will go from bad to worse, 368 00:20:03,340 --> 00:20:07,270 because in this case, the feedback coefficient can 369 00:20:07,270 --> 00:20:10,750 be positive for voiding the coolant. 370 00:20:10,750 --> 00:20:11,990 That's no good. 371 00:20:11,990 --> 00:20:13,793 So you want to keep the reactor submerged. 372 00:20:13,793 --> 00:20:16,210 And that's another reason why a lot of these fast reactors 373 00:20:16,210 --> 00:20:18,460 are what's called, pool-type reactors. 374 00:20:18,460 --> 00:20:20,230 The reactor is not a vessel with a bunch 375 00:20:20,230 --> 00:20:23,350 of piping under it that can break and fail, 376 00:20:23,350 --> 00:20:29,500 but instead it's designed as a huge pool of liquid sodium. 377 00:20:29,500 --> 00:20:34,030 And then the core is somewhere in here with a bunch of pumps 378 00:20:34,030 --> 00:20:38,230 sending the coolant in and back out, or through some heat 379 00:20:38,230 --> 00:20:39,830 exchange or something. 380 00:20:39,830 --> 00:20:42,100 So there's not really any penetrations 381 00:20:42,100 --> 00:20:43,810 on the bottom up this pool. 382 00:20:43,810 --> 00:20:46,300 And you make sure that you maintain, 383 00:20:46,300 --> 00:20:50,230 either when you have sodium or lead bismuth 384 00:20:50,230 --> 00:20:54,640 eutectic, or liquid lead, or some other fast reactor 385 00:20:54,640 --> 00:20:56,792 coolant. 386 00:20:56,792 --> 00:20:58,750 So these are some kind of interesting scenarios 387 00:20:58,750 --> 00:20:59,390 to think about. 388 00:20:59,390 --> 00:21:01,930 I think one of them that I put in the homework was 389 00:21:01,930 --> 00:21:04,030 imagine you have the MIT reactor and replace 390 00:21:04,030 --> 00:21:06,250 the coolant with molten sodium. 391 00:21:06,250 --> 00:21:07,810 What's going to happen? 392 00:21:07,810 --> 00:21:09,880 Well, let's say you got all the water out first 393 00:21:09,880 --> 00:21:12,010 and it wouldn't just blow up. 394 00:21:12,010 --> 00:21:14,538 What would actually happen to the criticality relation? 395 00:21:14,538 --> 00:21:16,330 That's something I want you to think about, 396 00:21:16,330 --> 00:21:18,288 because one of the big problems on the homework 397 00:21:18,288 --> 00:21:20,410 is doing exactly this for scenarios 398 00:21:20,410 --> 00:21:23,167 that have happened to the MIT reactor, 399 00:21:23,167 --> 00:21:24,250 except for the sodium one. 400 00:21:24,250 --> 00:21:26,500 That's never happened and hopefully never will. 401 00:21:29,430 --> 00:21:31,010 I can't even imagine. 402 00:21:31,010 --> 00:21:32,760 But now let's talk a little bit about when 403 00:21:32,760 --> 00:21:35,910 you perturb a reactor by doing something to it, 404 00:21:35,910 --> 00:21:38,130 putting the control rods in, or pulling them out, 405 00:21:38,130 --> 00:21:39,750 or doing whatever you want. 406 00:21:39,750 --> 00:21:42,750 You're by definition going to take 407 00:21:42,750 --> 00:21:49,830 one of our first assumptions about how the neutron diffusion 408 00:21:49,830 --> 00:21:52,860 equation works and throw it out the window. 409 00:21:52,860 --> 00:21:55,575 So we're now moving into the transient regime. 410 00:21:59,100 --> 00:22:02,100 So to study what happens in a reactor transient 411 00:22:02,100 --> 00:22:04,770 or when something changes as a function of time, 412 00:22:04,770 --> 00:22:11,700 let's first go from k effective to what we call k infinity. 413 00:22:11,700 --> 00:22:17,183 The multiplication factor for an infinite medium. 414 00:22:17,183 --> 00:22:18,600 We're only doing this because it's 415 00:22:18,600 --> 00:22:20,370 analytically easier to understand 416 00:22:20,370 --> 00:22:22,480 and still gets the point across. 417 00:22:22,480 --> 00:22:25,980 So we'll say that our k infinity is still 418 00:22:25,980 --> 00:22:29,868 a balance between production and destruction. 419 00:22:32,860 --> 00:22:36,020 The difference is if we have an infinite medium, 420 00:22:36,020 --> 00:22:37,540 there's no leakage. 421 00:22:37,540 --> 00:22:40,030 You can't leak out of an infinitely sized reactor, 422 00:22:40,030 --> 00:22:41,980 should one ever exist. 423 00:22:41,980 --> 00:22:46,090 And so it just comes out as nu sigma fission 424 00:22:46,090 --> 00:22:47,800 over sigma absorption. 425 00:22:47,800 --> 00:22:50,270 A much simpler form. 426 00:22:50,270 --> 00:22:52,400 And so now we can write what would 427 00:22:52,400 --> 00:22:59,540 happen to the flux in the reactor as a function of time. 428 00:22:59,540 --> 00:23:01,680 In this case, it's going to be one over velocity. 429 00:23:01,680 --> 00:23:06,920 I'm going to make this a very obvious wide v. That change 430 00:23:06,920 --> 00:23:09,830 in the reactor flux is going to just be proportional 431 00:23:09,830 --> 00:23:13,790 to the imbalance now in the number of neutrons produced 432 00:23:13,790 --> 00:23:15,110 and destroyed. 433 00:23:15,110 --> 00:23:17,520 So the number of neutrons produced 434 00:23:17,520 --> 00:23:21,350 will be proportional to very sharp nu sigma 435 00:23:21,350 --> 00:23:24,410 fission minus the number of neutrons destroyed, 436 00:23:24,410 --> 00:23:29,310 sigma absorption, times phi is a function of t. 437 00:23:32,050 --> 00:23:34,270 Y'all with me so far? 438 00:23:34,270 --> 00:23:38,500 So this right here is a change, which 439 00:23:38,500 --> 00:23:46,470 is proportional to an imbalance between production 440 00:23:46,470 --> 00:23:49,830 and destruction, times the actual flux that you 441 00:23:49,830 --> 00:23:52,750 have in some given time. 442 00:23:52,750 --> 00:23:55,710 So to make this simpler, let's multiply everything 443 00:23:55,710 --> 00:24:01,520 by v. Where's my green substitute color? 444 00:24:01,520 --> 00:24:05,390 Multiply everything by v. And the only unfortunate 445 00:24:05,390 --> 00:24:07,980 situation is we have a v and a nu next to each other. 446 00:24:07,980 --> 00:24:11,390 I'm going to try to keep them looking really different. 447 00:24:11,390 --> 00:24:12,290 Those go away. 448 00:24:15,090 --> 00:24:23,500 And then we end up with, if we divide by phi, 449 00:24:23,500 --> 00:24:26,170 then those phi's go away. 450 00:24:26,170 --> 00:24:36,460 And we have phi prime over phi, equals v nu sigma fission, 451 00:24:36,460 --> 00:24:41,410 minus v sigma absorption. 452 00:24:41,410 --> 00:24:43,030 And now we can start to define things 453 00:24:43,030 --> 00:24:45,730 in terms of our k infinity factor 454 00:24:45,730 --> 00:24:49,060 and a new quantity I'd like to introduce 455 00:24:49,060 --> 00:24:53,080 called the prompt lifetime. 456 00:24:53,080 --> 00:24:56,910 It's a measure of how long a given neutron tends to live 457 00:24:56,910 --> 00:24:58,290 before something happens to it. 458 00:24:58,290 --> 00:25:01,620 Before it's either absorbed or leaks out, well, not 459 00:25:01,620 --> 00:25:03,580 from our infinite reactor. 460 00:25:03,580 --> 00:25:07,900 And so we can define this as 1 over their neutron velocity, 461 00:25:07,900 --> 00:25:09,700 times sigma absorption. 462 00:25:09,700 --> 00:25:11,220 And just to check the units here-- 463 00:25:11,220 --> 00:25:15,110 velocities in meters per second. 464 00:25:15,110 --> 00:25:19,810 Macroscopic cross-sections are in 1 over meters. 465 00:25:19,810 --> 00:25:23,590 So those cancel out, and we're left 466 00:25:23,590 --> 00:25:26,650 with a total units of seconds. 467 00:25:26,650 --> 00:25:27,190 That's nice. 468 00:25:27,190 --> 00:25:30,490 We would want a mean neutron lifetime, 469 00:25:30,490 --> 00:25:36,310 or a prompt lifetime to have of seconds or time, at least. 470 00:25:36,310 --> 00:25:37,017 Yep. 471 00:25:37,017 --> 00:25:39,309 AUDIENCE: Can you say again why the [INAUDIBLE] squared 472 00:25:39,309 --> 00:25:40,117 went away? 473 00:25:40,117 --> 00:25:42,200 MICHAEL SHORT: Why the d phi dt squared went away? 474 00:25:42,200 --> 00:25:43,910 AUDIENCE: No, the [INAUDIBLE] square. 475 00:25:43,910 --> 00:25:44,840 MICHAEL SHORT: Oh, OK. 476 00:25:44,840 --> 00:25:47,030 So that's because we assume we're 477 00:25:47,030 --> 00:25:49,460 going to be analyzing an infinite medium. 478 00:25:49,460 --> 00:25:53,990 So right here, this to relabel these terms, 479 00:25:53,990 --> 00:25:55,790 this would be the total production term. 480 00:25:58,660 --> 00:26:00,430 That right there represents absorption. 481 00:26:03,440 --> 00:26:07,100 And that right there represents leakage. 482 00:26:07,100 --> 00:26:09,440 But if we're analyzing an infinite medium, 483 00:26:09,440 --> 00:26:12,980 you can't leak out, because it takes up the entire universe 484 00:26:12,980 --> 00:26:17,810 and beyond, depending on what you believe metaphysically. 485 00:26:17,810 --> 00:26:19,850 That's different costs. 486 00:26:19,850 --> 00:26:30,420 So this right here, we can rewrite as 1 over lifetime. 487 00:26:30,420 --> 00:26:32,400 That makes it easier. 488 00:26:32,400 --> 00:26:39,580 And this right here, if we note that nu-- 489 00:26:39,580 --> 00:26:40,270 that's a nu. 490 00:26:40,270 --> 00:26:42,895 I'm going to be really explicit about that. 491 00:26:46,700 --> 00:26:49,740 Nu sigma fission over sigma absorption. 492 00:26:49,740 --> 00:26:55,750 This kind of looks like this is looking to be like k-- 493 00:26:55,750 --> 00:26:58,350 wrong color. 494 00:26:58,350 --> 00:27:03,030 --like our k infinity over lp. 495 00:27:03,030 --> 00:27:06,710 So all of a sudden we of a much simpler relation. 496 00:27:06,710 --> 00:27:15,600 We have 5 prime over 5 equals k infinity minus 1 497 00:27:15,600 --> 00:27:18,180 over the prompt neutron lifetime. 498 00:27:18,180 --> 00:27:20,980 So if we solve this, this is just an exponential. 499 00:27:20,980 --> 00:27:27,170 So we end up with our phi as a function of t is-- 500 00:27:27,170 --> 00:27:30,540 whatever flux we started at, like for your power 501 00:27:30,540 --> 00:27:33,380 in manipulations, it would be whatever the neutron flux was 502 00:27:33,380 --> 00:27:38,760 before you touch the control rod, times e to the t, 503 00:27:38,760 --> 00:27:47,370 or e to the that stuff, k infinity minus 1 over lp, 504 00:27:47,370 --> 00:27:56,718 times t, which we can rewrite as t over capital T. 505 00:27:56,718 --> 00:27:58,510 We're going to define this symbol as what's 506 00:27:58,510 --> 00:27:59,650 called the reactor period. 507 00:28:08,450 --> 00:28:10,530 What the reactor period actually says 508 00:28:10,530 --> 00:28:24,410 is how long before the flux increases by a factor of e. 509 00:28:29,580 --> 00:28:32,610 And so this is actually what that meter 510 00:28:32,610 --> 00:28:33,860 was measuring on the reactor. 511 00:28:33,860 --> 00:28:36,120 It's the reactor period or the time 512 00:28:36,120 --> 00:28:38,190 it would then take for the reactor's power 513 00:28:38,190 --> 00:28:43,280 to increase by a factor of e because it's an exponential. 514 00:28:43,280 --> 00:28:45,540 To tell you what these typical reactor periods tend 515 00:28:45,540 --> 00:28:52,310 to be for a thermal reactor, t is 516 00:28:52,310 --> 00:28:58,850 about 0.1 seconds corresponding to an average prompt neutron 517 00:28:58,850 --> 00:29:04,800 lifetime of 10 to the minus 4 seconds. 518 00:29:04,800 --> 00:29:06,780 Seems fast, doesn't it? 519 00:29:06,780 --> 00:29:08,400 Like, really fast. 520 00:29:08,400 --> 00:29:11,040 So the question I asked you guys is, why don't reactors 521 00:29:11,040 --> 00:29:13,089 just blow up? 522 00:29:13,089 --> 00:29:14,390 AUDIENCE: [INAUDIBLE]. 523 00:29:14,390 --> 00:29:16,130 MICHAEL SHORT: Yes, there is something 524 00:29:16,130 --> 00:29:17,480 we've neglected from here. 525 00:29:17,480 --> 00:29:19,380 It's like what Sarah said. 526 00:29:19,380 --> 00:29:20,630 And it deserves its own board. 527 00:29:23,450 --> 00:29:25,535 There is a fraction of delayed neutrons. 528 00:29:35,460 --> 00:29:38,250 We'll give that fraction the symbol, beta. 529 00:29:38,250 --> 00:29:43,530 And for a uranium 235, it equals about 0.0064. 530 00:29:43,530 --> 00:29:46,890 So there's less than a percent of all the neutrons coming out 531 00:29:46,890 --> 00:29:50,190 of a reactor have some delay to them, because they're not 532 00:29:50,190 --> 00:29:54,450 made directly from fission in the 10 to the minus 14 seconds 533 00:29:54,450 --> 00:29:58,430 that we talked about in the timeline. 534 00:29:58,430 --> 00:30:01,220 But they come out of radioactive decay processes 535 00:30:01,220 --> 00:30:12,070 with delayed lifetimes ranging from about 0.2 seconds 536 00:30:12,070 --> 00:30:13,585 to about 54 seconds. 537 00:30:17,100 --> 00:30:20,370 This is the whole reason why reactors don't just blow up. 538 00:30:20,370 --> 00:30:23,700 So you can actually make a reactor go super critical. 539 00:30:23,700 --> 00:30:33,270 But if the k effective is less than 1 plus beta, 540 00:30:33,270 --> 00:30:38,147 then the reactor is not what we call prompt super critical. 541 00:30:45,110 --> 00:30:48,080 And so the reason for that is, let's say you raise the reactor 542 00:30:48,080 --> 00:30:56,340 power by some amount and the k effective goes up to 1.005, 543 00:30:56,340 --> 00:30:59,940 there's still this fraction 0.0064 of the neutrons 544 00:30:59,940 --> 00:31:01,970 are not going to be released immediately. 545 00:31:01,970 --> 00:31:05,370 They're going to be released not in 10 to the minus 14 seconds, 546 00:31:05,370 --> 00:31:07,350 but in 10 to the 2 seconds. 547 00:31:07,350 --> 00:31:10,170 So a measly 15 orders of magnitude 548 00:31:10,170 --> 00:31:12,660 slower, meaning that there's actually 549 00:31:12,660 --> 00:31:16,890 some ability for this reactor to raise its power level. 550 00:31:16,890 --> 00:31:18,810 And these delayed neutrons, even though that's 551 00:31:18,810 --> 00:31:21,990 such a small fraction, takes the reactor period 552 00:31:21,990 --> 00:31:27,350 from its t infinity value of about 0.1 seconds 553 00:31:27,350 --> 00:31:31,550 to about 100 seconds. 554 00:31:31,550 --> 00:31:33,260 So the same reactor when you account 555 00:31:33,260 --> 00:31:36,440 for the delayed neutrons increases in power 556 00:31:36,440 --> 00:31:37,343 by a factor of e. 557 00:31:37,343 --> 00:31:39,260 And it takes it about 100 seconds, which means 558 00:31:39,260 --> 00:31:42,002 this is totally controllable. 559 00:31:42,002 --> 00:31:43,460 Now I have a question for you guys. 560 00:31:43,460 --> 00:31:46,670 Would you guys like me to derive this formula, 561 00:31:46,670 --> 00:31:48,740 or do you want to go into more of the intuitive 562 00:31:48,740 --> 00:31:49,730 implications of it? 563 00:31:49,730 --> 00:31:52,960 Because we can go either way. 564 00:31:52,960 --> 00:31:55,120 There is a formula that will tell you 565 00:31:55,120 --> 00:31:58,240 what the reactor period and time dependence will be. 566 00:31:58,240 --> 00:32:01,590 And you will hit it in 22.05 probably. 567 00:32:01,590 --> 00:32:03,850 I can't guarantee it because I'm not teaching it. 568 00:32:03,850 --> 00:32:06,370 Or we can talk a little bit more about some of the intuition 569 00:32:06,370 --> 00:32:08,610 behind delayed neutrons. 570 00:32:08,610 --> 00:32:11,680 So a bit of choose your own adventure. 571 00:32:11,680 --> 00:32:12,730 Math or intuition? 572 00:32:12,730 --> 00:32:13,720 AUDIENCE: Intuition. 573 00:32:13,720 --> 00:32:14,762 MICHAEL SHORT: Intuition. 574 00:32:14,762 --> 00:32:16,450 OK, that's fine. 575 00:32:16,450 --> 00:32:17,740 Good. 576 00:32:17,740 --> 00:32:22,300 So that was the derivation. 577 00:32:22,300 --> 00:32:24,790 I'll post that anyway, if you guys want to see. 578 00:32:24,790 --> 00:32:28,270 I think in the Yip reading it says, let's account 579 00:32:28,270 --> 00:32:29,950 for the delayed neutrons. 580 00:32:29,950 --> 00:32:33,340 Intuitively we find that the answer ends up being-- 581 00:32:33,340 --> 00:32:35,350 so I'll skip the derivation. 582 00:32:35,350 --> 00:32:44,270 And it comes out to phi naught e to the beta minus 1, 583 00:32:44,270 --> 00:32:54,740 times k minus 1 over lt, plus beta phi naught over beta minus 584 00:32:54,740 --> 00:33:01,940 1k, minus 1, times 1 minus e to the beta minus 1k, minus 1 585 00:33:01,940 --> 00:33:03,110 over l. 586 00:33:03,110 --> 00:33:07,145 OK, so left as an exercise to the reader-- 587 00:33:07,145 --> 00:33:08,270 AUDIENCE: That's intuitive. 588 00:33:08,270 --> 00:33:10,490 MICHAEL SHORT: Yeah, that's intuitive. 589 00:33:10,490 --> 00:33:13,098 But let's actually talk about how intuitive it is. 590 00:33:13,098 --> 00:33:15,140 I do want to give you the starting and the ending 591 00:33:15,140 --> 00:33:15,530 equation. 592 00:33:15,530 --> 00:33:16,750 And we will not go through the rest. 593 00:33:16,750 --> 00:33:17,240 Yeah, Charlie? 594 00:33:17,240 --> 00:33:18,590 AUDIENCE: Should we copy that down? 595 00:33:18,590 --> 00:33:19,530 MICHAEL SHORT: No, you shouldn't. 596 00:33:19,530 --> 00:33:20,947 I'm going to scan it for you guys. 597 00:33:20,947 --> 00:33:22,610 So don't bother copying it down. 598 00:33:22,610 --> 00:33:24,530 Let's talk about where it comes from. 599 00:33:24,530 --> 00:33:26,990 And the answer may astound you because we're 600 00:33:26,990 --> 00:33:29,540 going to bring right back the idea of series radioactive 601 00:33:29,540 --> 00:33:30,690 decay. 602 00:33:30,690 --> 00:33:35,450 So let's say you want to relate the change 603 00:33:35,450 --> 00:33:41,020 in the number in the neutron flux to a 1 minus-- 604 00:33:41,020 --> 00:33:43,700 I'm going to take a quick look at the original equation 605 00:33:43,700 --> 00:33:46,717 because I don't want to screw that up. 606 00:33:46,717 --> 00:33:48,800 That's the first page, and that's the one we want. 607 00:33:51,132 --> 00:33:53,340 Let's say we had some equations that looked something 608 00:33:53,340 --> 00:33:53,970 like this. 609 00:34:00,802 --> 00:34:05,710 Phi plus phi naught times beta. 610 00:34:05,710 --> 00:34:07,750 This is the original differential equation 611 00:34:07,750 --> 00:34:09,403 from whence it came. 612 00:34:09,403 --> 00:34:11,320 And the intuitive part that I want you to note 613 00:34:11,320 --> 00:34:16,060 is that the jump from changing k effective 614 00:34:16,060 --> 00:34:20,719 is moderated by this term right here, 1 minus beta. 615 00:34:20,719 --> 00:34:29,090 So that's the fraction of prompt neutrons, 616 00:34:29,090 --> 00:34:31,100 that as soon as you pull the control rod out, 617 00:34:31,100 --> 00:34:33,281 that's your instantaneous feedback. 618 00:34:33,281 --> 00:34:35,239 By instantaneous, I mean on the order of, like, 619 00:34:35,239 --> 00:34:36,980 10 to the minus 4 seconds, or something 620 00:34:36,980 --> 00:34:38,929 that you can't really control. 621 00:34:38,929 --> 00:34:41,104 This right here represents the delayed fraction. 622 00:34:44,873 --> 00:34:47,040 This is as mathy as it's going to get because you've 623 00:34:47,040 --> 00:34:48,150 chosen intuition. 624 00:34:48,150 --> 00:34:49,929 I think you have chosen wisely. 625 00:34:49,929 --> 00:34:51,570 It's going to be a more fun. 626 00:34:51,570 --> 00:34:54,060 So what this represents right here 627 00:34:54,060 --> 00:35:02,090 is your kind of instant change, because whatever you change 628 00:35:02,090 --> 00:35:04,040 k effective to, it's going to be moderated 629 00:35:04,040 --> 00:35:08,000 by the prompt fraction, how long the neutrons tend to take 630 00:35:08,000 --> 00:35:09,090 to undergo that feedback. 631 00:35:09,090 --> 00:35:09,590 Yes, Sara? 632 00:35:09,590 --> 00:35:11,087 AUDIENCE: Was that the average? 633 00:35:11,087 --> 00:35:12,420 MICHAEL SHORT: The average what? 634 00:35:12,420 --> 00:35:14,625 AUDIENCE: Average neutron lifetime. 635 00:35:14,625 --> 00:35:17,000 MICHAEL SHORT: Yes, this is the average neutron lifetime. 636 00:35:17,000 --> 00:35:20,300 So let's define the average neutron lifetime 637 00:35:20,300 --> 00:35:27,580 as simply 1 minus beta times the prompt neutron lifetime, 638 00:35:27,580 --> 00:35:36,680 plus the beta times some delayed neutron lifetime. 639 00:35:36,680 --> 00:35:40,160 So what no book I've ever seen actually says, 640 00:35:40,160 --> 00:35:43,910 this is what's referred to as a Maxwell mixing model. 641 00:35:50,070 --> 00:35:52,030 It's just the simplest thing to say, oh, 642 00:35:52,030 --> 00:35:54,030 if you want to get the average of some variable, 643 00:35:54,030 --> 00:35:58,080 take the fraction of one species times its variable, 644 00:35:58,080 --> 00:36:00,750 plus the fraction of the other species, times its variable. 645 00:36:00,750 --> 00:36:03,420 Folks do the same thing with electrical resistivity, 646 00:36:03,420 --> 00:36:06,180 thermal conductivity, or any sort of other material 647 00:36:06,180 --> 00:36:07,140 property. 648 00:36:07,140 --> 00:36:09,570 And it is or isn't good in some situations. 649 00:36:09,570 --> 00:36:14,840 Like, if you had a piece of material made out 650 00:36:14,840 --> 00:36:17,030 of two different things-- 651 00:36:17,030 --> 00:36:19,550 let's say this had thermal conductivity k1, 652 00:36:19,550 --> 00:36:21,750 and it had thermal conductivity K2. 653 00:36:21,750 --> 00:36:24,290 Would a Maxwell mixing model be appropriate to describe 654 00:36:24,290 --> 00:36:27,080 the flow of heat across this thing? 655 00:36:27,080 --> 00:36:28,550 Probably not. 656 00:36:28,550 --> 00:36:30,000 But in the case of neutrons where 657 00:36:30,000 --> 00:36:33,260 they're flying about like crazy and their mean free path is 658 00:36:33,260 --> 00:36:36,470 much larger than the distance between atoms, 659 00:36:36,470 --> 00:36:38,090 this works great. 660 00:36:38,090 --> 00:36:50,870 So we can define this mean neutron lifetime 661 00:36:50,870 --> 00:36:53,400 and use that in this equation right here. 662 00:36:53,400 --> 00:36:55,580 So this term right here describes 663 00:36:55,580 --> 00:36:56,840 the instantaneous change. 664 00:36:56,840 --> 00:36:59,510 You pull the control rods out, and fraction 1 665 00:36:59,510 --> 00:37:03,740 minus beta neutrons respond immediately. 666 00:37:03,740 --> 00:37:06,970 What about that fraction of neutrons? 667 00:37:06,970 --> 00:37:10,240 Those are being produced with a fraction beta 668 00:37:10,240 --> 00:37:12,610 depending on what the flux was before, 669 00:37:12,610 --> 00:37:15,640 because they're still waiting to decay from the old power level. 670 00:37:21,780 --> 00:37:25,980 Does anyone notice anything suspiciously familiar 671 00:37:25,980 --> 00:37:30,810 about the final form of this equation for flux? 672 00:37:30,810 --> 00:37:33,420 You've seen it before with a couple 673 00:37:33,420 --> 00:37:35,070 of constants changed around. 674 00:37:39,570 --> 00:37:42,324 What about the form of this differential equation? 675 00:37:42,324 --> 00:37:43,710 [INTERPOSING VOICES] 676 00:37:43,710 --> 00:37:46,860 It is exactly the same as series radioactive decay. 677 00:37:46,860 --> 00:37:49,920 So the horrible derivation I was going to do for you guys 678 00:37:49,920 --> 00:37:53,910 and we're not anymore is, use an integrating factor. 679 00:37:53,910 --> 00:37:56,430 You solve it in exactly the same way. 680 00:37:56,430 --> 00:37:59,650 You bring everything to one side of the equation. 681 00:37:59,650 --> 00:38:03,000 Find some factor mu, that makes this a product rule. 682 00:38:03,000 --> 00:38:04,590 Do a lot of algebra. 683 00:38:04,590 --> 00:38:08,610 And you end up with a very suspiciously similar looking 684 00:38:08,610 --> 00:38:09,390 equation. 685 00:38:09,390 --> 00:38:13,350 So it's exactly the same posing and solution 686 00:38:13,350 --> 00:38:16,350 as series radioactive decay, with the difference being, 687 00:38:16,350 --> 00:38:18,240 that's the constant in front of everything, 688 00:38:18,240 --> 00:38:21,090 instead of a bunch of lambdas and fluxes. 689 00:38:21,090 --> 00:38:24,600 So what this says here is that the flux as a function of time, 690 00:38:24,600 --> 00:38:27,270 this is the prompt feedback right here, 691 00:38:27,270 --> 00:38:29,040 which says that if-- let's graph it, 692 00:38:29,040 --> 00:38:31,350 since we're going intuitive. 693 00:38:31,350 --> 00:38:33,430 There's no room. 694 00:38:33,430 --> 00:38:36,980 Even those all boards are full. 695 00:38:36,980 --> 00:38:39,340 OK, here we go. 696 00:38:39,340 --> 00:38:44,200 If we graft time and flux right here, 697 00:38:44,200 --> 00:38:45,993 what that part right there says is 698 00:38:45,993 --> 00:38:47,410 that you're going to get some sort 699 00:38:47,410 --> 00:38:51,690 of instantaneous exponential feedback. 700 00:38:51,690 --> 00:38:54,870 But it's going to be moderated by this one minus exponential 701 00:38:54,870 --> 00:38:57,920 on top. 702 00:38:57,920 --> 00:39:04,250 So you're going to end up with a little bit of prompt feedback, 703 00:39:04,250 --> 00:39:05,230 this stuff right here. 704 00:39:11,690 --> 00:39:15,150 And then-- have to draw longer because it's going to take 705 00:39:15,150 --> 00:39:17,970 forever-- 706 00:39:17,970 --> 00:39:22,010 you'll have some delayed feedback, 707 00:39:22,010 --> 00:39:24,550 because you've got to wait 100 or so seconds, 708 00:39:24,550 --> 00:39:26,690 or whatever that new reactor period is, 709 00:39:26,690 --> 00:39:28,685 for the delayed neutrons to take effect. 710 00:39:28,685 --> 00:39:31,310 And that's the whole reason you could pull the control rods out 711 00:39:31,310 --> 00:39:35,540 at almost any speed you wanted and the reactor doesn't just 712 00:39:35,540 --> 00:39:36,800 explode. 713 00:39:36,800 --> 00:39:39,500 If you pull the control rods out fast enough, such 714 00:39:39,500 --> 00:39:43,370 that the change in k effective is greater than beta, 715 00:39:43,370 --> 00:39:46,370 then the reactor goes prompt super critical, 716 00:39:46,370 --> 00:39:49,190 which means you don't have any delayed neutrons slowing down 717 00:39:49,190 --> 00:39:50,360 the feedback. 718 00:39:50,360 --> 00:39:53,010 And you've kind of turned your reactor into a weapon. 719 00:39:53,010 --> 00:39:56,630 A very poor, terrible weapon, but 720 00:39:56,630 --> 00:40:00,550 a prompt super critical nuclear device, nonetheless. 721 00:40:00,550 --> 00:40:03,300 Did anybody pull out the control rods too fast 722 00:40:03,300 --> 00:40:05,747 and the controls took over for you? 723 00:40:05,747 --> 00:40:07,080 What about you guys in training? 724 00:40:07,080 --> 00:40:09,810 Did you ever do things when you watched the automatic control 725 00:40:09,810 --> 00:40:11,640 take over? 726 00:40:11,640 --> 00:40:12,388 No? 727 00:40:12,388 --> 00:40:13,762 AUDIENCE: It'll just take over. 728 00:40:13,762 --> 00:40:16,060 AUDIENCE: Yeah, it'll kick you off if you don't pay attention. 729 00:40:16,060 --> 00:40:17,477 MICHAEL SHORT: That's what I mean. 730 00:40:17,477 --> 00:40:20,020 The machine takes over and it will kick you off 731 00:40:20,020 --> 00:40:21,280 and stop responding to you. 732 00:40:21,280 --> 00:40:22,780 AUDIENCE: [INAUDIBLE] horrible noise 733 00:40:22,780 --> 00:40:24,520 and so we don't want that. 734 00:40:24,520 --> 00:40:26,663 It's more to avoid an annoying alarm. 735 00:40:26,663 --> 00:40:27,550 MICHAEL SHORT: I see. 736 00:40:27,550 --> 00:40:29,170 But the annoying alarm is to stop you 737 00:40:29,170 --> 00:40:31,690 from doing something like that, like, making the reactor go 738 00:40:31,690 --> 00:40:32,607 prompt super critical. 739 00:40:32,607 --> 00:40:33,912 AUDIENCE: [INAUDIBLE] 740 00:40:33,912 --> 00:40:35,620 MICHAEL SHORT: OK, so that's what I would 741 00:40:35,620 --> 00:40:37,090 call the machine taking over. 742 00:40:37,090 --> 00:40:38,942 AUDIENCE: Oh, I see. 743 00:40:38,942 --> 00:40:41,400 AUDIENCE: It'll kick you on to manual, and then [INAUDIBLE] 744 00:40:41,400 --> 00:40:43,547 still don't do anything [INAUDIBLE].. 745 00:40:43,547 --> 00:40:44,380 MICHAEL SHORT: Yeah. 746 00:40:44,380 --> 00:40:46,920 So if your blood alcohol level is above beta 747 00:40:46,920 --> 00:40:49,930 and you try and, let's say, increase the reactor 748 00:40:49,930 --> 00:40:52,750 reactivity too much, it will then take over, insert 749 00:40:52,750 --> 00:40:55,420 a control rod, make a horrible noise, and say, go home, 750 00:40:55,420 --> 00:40:57,330 you're drunk. 751 00:40:57,330 --> 00:40:58,250 Something like that. 752 00:40:58,250 --> 00:41:01,660 OK, that makes sense to me. 753 00:41:01,660 --> 00:41:04,500 So what did your guys' reactor power traces look like? 754 00:41:04,500 --> 00:41:07,295 Did they look something like this, 755 00:41:07,295 --> 00:41:09,670 where there was an initial rise as you pulled the control 756 00:41:09,670 --> 00:41:10,180 right out? 757 00:41:10,180 --> 00:41:13,810 And then after you pulled the control rod out the power kept 758 00:41:13,810 --> 00:41:16,098 rising just a smidge, right? 759 00:41:16,098 --> 00:41:18,390 And what happened when you put the control rod back in? 760 00:41:21,840 --> 00:41:23,960 Let's say you put the control rod back in. 761 00:41:23,960 --> 00:41:26,990 You're going to get another prompt drop, not equal 762 00:41:26,990 --> 00:41:29,000 to the same prompt gain that you got, 763 00:41:29,000 --> 00:41:31,900 because now the reactor's at a different flux, 764 00:41:31,900 --> 00:41:35,280 and then some asymptotic feedback like that. 765 00:41:35,280 --> 00:41:37,900 And so this is why to those who don't understand 766 00:41:37,900 --> 00:41:41,200 neutron physics, reactor feedback is very non-intuitive. 767 00:41:41,200 --> 00:41:42,607 It's not a linear system. 768 00:41:42,607 --> 00:41:45,190 You can't just pull the control right out and change the power 769 00:41:45,190 --> 00:41:46,340 accordingly. 770 00:41:46,340 --> 00:41:49,600 This is why there's automated controls in systems to stop you 771 00:41:49,600 --> 00:41:51,610 in case, like I said, if your blood alcohol 772 00:41:51,610 --> 00:41:55,227 content's above beta, which is very low, by the way. 773 00:41:55,227 --> 00:41:57,060 Though you shouldn't be drinking on the job, 774 00:41:57,060 --> 00:41:59,010 especially at a nuclear reactor. 775 00:41:59,010 --> 00:42:01,870 Plus, you're all under 21, so what am I even saying? 776 00:42:01,870 --> 00:42:03,800 AUDIENCE: What is alcohol? 777 00:42:03,800 --> 00:42:05,450 MICHAEL SHORT: That's right. 778 00:42:05,450 --> 00:42:06,470 Good answer. 779 00:42:06,470 --> 00:42:08,333 What is alcohol? 780 00:42:08,333 --> 00:42:10,500 AUDIENCE: Is that going to be covered in [INAUDIBLE] 781 00:42:10,500 --> 00:42:11,760 MICHAEL SHORT: That'll be on the exam, yeah. 782 00:42:11,760 --> 00:42:13,380 AUDIENCE: What is alcohol? 783 00:42:13,380 --> 00:42:15,380 MICHAEL SHORT: Yeah, cool. 784 00:42:15,380 --> 00:42:18,260 So that's all I want to go into for the intuitive stuff. 785 00:42:18,260 --> 00:42:20,360 And it's about 5 of 5 of. 786 00:42:20,360 --> 00:42:23,240 So I'd like to stop here and see if you guys have any questions 787 00:42:23,240 --> 00:42:25,790 on neutron physics at a whole. 788 00:42:25,790 --> 00:42:27,790 Noting that we're going to take Thursday's class 789 00:42:27,790 --> 00:42:30,040 and turn into a recitation. 790 00:42:30,040 --> 00:42:32,800 So I would like all of you guys to look at the problem set, 791 00:42:32,800 --> 00:42:34,510 because it is posted. 792 00:42:34,510 --> 00:42:36,860 It is hard. 793 00:42:36,860 --> 00:42:38,050 Trust me. 794 00:42:38,050 --> 00:42:40,582 This one's a doozy. 795 00:42:40,582 --> 00:42:42,290 So I want to warn you guys because you've 796 00:42:42,290 --> 00:42:43,640 got seven days to work on it. 797 00:42:43,640 --> 00:42:46,820 But I want you to look at it so that we can start formulating 798 00:42:46,820 --> 00:42:50,150 strategies for the problems together on Thursday, 799 00:42:50,150 --> 00:42:53,210 because there are some tricks to it. 800 00:42:53,210 --> 00:42:55,160 You guys know me by now, right? 801 00:42:55,160 --> 00:42:56,660 There's always some sort of a trick. 802 00:42:56,660 --> 00:42:58,610 Like, do you have to integrate every energy 803 00:42:58,610 --> 00:42:59,720 to get the stopping power? 804 00:42:59,720 --> 00:43:02,103 No, you actually don't have to do any integrals at all. 805 00:43:02,103 --> 00:43:03,770 But you can if you want, and your answer 806 00:43:03,770 --> 00:43:05,210 will be more accurate and correct. 807 00:43:05,210 --> 00:43:07,220 It'll just take longer to get to. 808 00:43:07,220 --> 00:43:10,340 So there's a lot of diminishing returns on these problems sets. 809 00:43:10,340 --> 00:43:12,140 If you're willing to take an hour 810 00:43:12,140 --> 00:43:13,670 and think about how can I do this 811 00:43:13,670 --> 00:43:16,640 simpler and with fewer decimal points, 812 00:43:16,640 --> 00:43:18,670 you're probably onto something. 813 00:43:18,670 --> 00:43:21,190 And we'll work on those strategies together. 814 00:43:21,190 --> 00:43:23,420 AUDIENCE: Is this due next Monday as well? 815 00:43:23,420 --> 00:43:24,890 MICHAEL SHORT: Yes. 816 00:43:24,890 --> 00:43:27,453 So I posted it yesterday at around noon or whatever 817 00:43:27,453 --> 00:43:28,370 the Stellar site says. 818 00:43:32,150 --> 00:43:36,710 I'll also teach you guys explicitly how to use Janus. 819 00:43:36,710 --> 00:43:40,070 So we got a comment in from the anonymous feedback saying, 820 00:43:40,070 --> 00:43:41,813 we have to use a lot of software. 821 00:43:41,813 --> 00:43:43,730 Can we have some sort of tutorial for dummies? 822 00:43:43,730 --> 00:43:45,563 Well, you guys aren't dummies, but you still 823 00:43:45,563 --> 00:43:46,500 deserve a tutorial. 824 00:43:46,500 --> 00:43:49,520 So I will show you how to export the data you'll 825 00:43:49,520 --> 00:43:51,530 need from this problem set for Janus. 826 00:43:51,530 --> 00:43:53,960 So you can focus on the intuition and the physics 827 00:43:53,960 --> 00:43:56,450 and not get frustrated with getting data out of a computer. 828 00:44:00,160 --> 00:44:03,370 So any questions on anything from the neutron diffusion 829 00:44:03,370 --> 00:44:04,420 equation? 830 00:44:04,420 --> 00:44:05,050 Yeah, Luke. 831 00:44:05,050 --> 00:44:07,816 AUDIENCE: I'm not real clear on [INAUDIBLE] neutrons are 832 00:44:07,816 --> 00:44:10,300 and how those are different from the prompt neutrons? 833 00:44:10,300 --> 00:44:11,717 MICHAEL SHORT: The prompt neutrons 834 00:44:11,717 --> 00:44:12,950 come right out of fission. 835 00:44:12,950 --> 00:44:15,010 If we looked at that timeline of, 836 00:44:15,010 --> 00:44:18,350 let's say the fission event happens here. 837 00:44:18,350 --> 00:44:23,870 Two fission products are released in about 10 838 00:44:23,870 --> 00:44:26,870 to the minus 14 seconds. 839 00:44:26,870 --> 00:44:30,090 They move a little further apart. 840 00:44:30,090 --> 00:44:32,420 And then some of them just boil off neutrons, 841 00:44:32,420 --> 00:44:36,170 because they're so neutron heavy, after around 10 842 00:44:36,170 --> 00:44:39,500 to the minus 13 seconds or so. 843 00:44:39,500 --> 00:44:42,740 These right here are prop neutrons, 844 00:44:42,740 --> 00:44:46,310 coming directly from the immediate decay of neutron 845 00:44:46,310 --> 00:44:47,900 rich fission products. 846 00:44:47,900 --> 00:44:51,560 Some of the delayed neutrons come from radioactive decay, 847 00:44:51,560 --> 00:44:54,050 but of the much later fission products 848 00:44:54,050 --> 00:44:57,830 with much less likely occurrences, 849 00:44:57,830 --> 00:44:59,840 which is why the fraction is very low. 850 00:44:59,840 --> 00:45:02,630 But also, because it's much longer half life, 851 00:45:02,630 --> 00:45:05,150 those delayed neutrons take seconds, instead 852 00:45:05,150 --> 00:45:07,115 of pico seconds, to show up. 853 00:45:07,115 --> 00:45:10,310 And that's the whole basis behind easier control 854 00:45:10,310 --> 00:45:12,170 and feedback a reactor. 855 00:45:12,170 --> 00:45:14,840 Good question. 856 00:45:14,840 --> 00:45:17,360 So anything starting from neutron transport 857 00:45:17,360 --> 00:45:19,670 to simplifying to neutron diffusion, 858 00:45:19,670 --> 00:45:22,340 to getting to this criticality condition, 859 00:45:22,340 --> 00:45:24,770 making the two group criticality condition if you 860 00:45:24,770 --> 00:45:28,160 want to have fast and thermal, or any of the time dependent 861 00:45:28,160 --> 00:45:31,630 stuff that we intuited today. 862 00:45:31,630 --> 00:45:32,130 Yeah? 863 00:45:32,130 --> 00:45:34,088 AUDIENCE: So for that cross-section [INAUDIBLE] 864 00:45:34,088 --> 00:45:36,885 you have there, so you have one for 235 and one for 238. 865 00:45:36,885 --> 00:45:41,348 235, it has to be thermal neutrons [INAUDIBLE] fast? 866 00:45:41,348 --> 00:45:42,140 MICHAEL SHORT: Yep. 867 00:45:42,140 --> 00:45:44,182 AUDIENCE: And you said that [INAUDIBLE] different 868 00:45:44,182 --> 00:45:45,760 [INAUDIBLE] as well it had-- 869 00:45:45,760 --> 00:45:49,653 if you have [INAUDIBLE] into the-- 870 00:45:49,653 --> 00:45:51,320 MICHAEL SHORT: Was it on the other board 871 00:45:51,320 --> 00:45:52,139 or from a different day? 872 00:45:52,139 --> 00:45:53,597 AUDIENCE: It was a different board. 873 00:45:53,597 --> 00:45:55,840 MICHAEL SHORT: OK. 874 00:45:55,840 --> 00:45:59,018 AUDIENCE: Yeah, so could you explain that graph? 875 00:45:59,018 --> 00:45:59,810 MICHAEL SHORT: Yes. 876 00:45:59,810 --> 00:46:02,442 So in this case-- 877 00:46:02,442 --> 00:46:06,650 let me get a finer chalk. 878 00:46:06,650 --> 00:46:08,930 This blue one would be for low temperature, 879 00:46:08,930 --> 00:46:12,990 and this red one would be for high temperature. 880 00:46:12,990 --> 00:46:14,910 So this blue graph, there are resonances, 881 00:46:14,910 --> 00:46:18,630 which have very high values, but they're very narrow. 882 00:46:18,630 --> 00:46:21,900 And because the width of a resonance doesn't matter, 883 00:46:21,900 --> 00:46:23,460 it doesn't affect the probability 884 00:46:23,460 --> 00:46:27,030 that a neutron scatters up here and moves some distance 885 00:46:27,030 --> 00:46:28,680 down the energy spectrum. 886 00:46:28,680 --> 00:46:32,020 Thinner resonances tend to get passed over, 887 00:46:32,020 --> 00:46:34,510 especially if your reactor's full of hydrogen. Some 888 00:46:34,510 --> 00:46:37,300 of those neutrons will be born and immediately 889 00:46:37,300 --> 00:46:39,730 jump into the thermal region, where 890 00:46:39,730 --> 00:46:42,580 it's easy to tell how much fission they'll undergo. 891 00:46:42,580 --> 00:46:44,800 As you go up in temperature, you undergo 892 00:46:44,800 --> 00:46:53,990 what's called Doppler broadening, which 893 00:46:53,990 --> 00:46:57,680 causes these resonances to spread out and also go down 894 00:46:57,680 --> 00:46:58,830 in value. 895 00:46:58,830 --> 00:47:01,250 So the actual value of the cross-section 896 00:47:01,250 --> 00:47:05,700 at these residences is lower, but the widths are larger. 897 00:47:05,700 --> 00:47:08,630 So there's a higher probability that a neutron scattering 898 00:47:08,630 --> 00:47:10,850 around and losing energy will hit 899 00:47:10,850 --> 00:47:13,610 one of these higher cross-section regions, called 900 00:47:13,610 --> 00:47:18,340 a resonance, at a higher temperature. 901 00:47:18,340 --> 00:47:20,740 That's the difference there, is these two plots show 902 00:47:20,740 --> 00:47:25,450 the same cross-section at low and high temperature. 903 00:47:25,450 --> 00:47:29,410 These plots show the difference between uranium 235 904 00:47:29,410 --> 00:47:33,190 and uranium 238. 905 00:47:33,190 --> 00:47:34,390 Good question. 906 00:47:34,390 --> 00:47:34,960 Anyone else? 907 00:47:41,680 --> 00:47:42,550 Cool. 908 00:47:42,550 --> 00:47:44,670 OK, for the first time in history, 909 00:47:44,670 --> 00:47:47,220 I'll let you out a minute early. 910 00:47:47,220 --> 00:47:49,410 Bring all your questions on Thursday. 911 00:47:49,410 --> 00:47:51,870 So we'll start off with a Janus tutorial. 912 00:47:51,870 --> 00:47:54,756 And then we'll start attacking this problem set together.