1 00:00:01,050 --> 00:00:03,420 The following content is provided under a Creative 2 00:00:03,420 --> 00:00:04,810 Commons license. 3 00:00:04,810 --> 00:00:07,020 Your support will help MIT OpenCourseWare 4 00:00:07,020 --> 00:00:11,110 continue to offer high quality educational resources for free. 5 00:00:11,110 --> 00:00:13,680 To make a donation or to view additional materials 6 00:00:13,680 --> 00:00:17,640 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,640 --> 00:00:18,526 at ocw.mit.edu. 8 00:00:22,065 --> 00:00:23,690 MICHAEL SHORT: I think I might actually 9 00:00:23,690 --> 00:00:27,350 use all 16 colors today. 10 00:00:27,350 --> 00:00:29,330 Oh no, this is the most satisfying day. 11 00:00:29,330 --> 00:00:33,260 Whereas Tuesday was probably the most mathematically intense, 12 00:00:33,260 --> 00:00:35,632 because we developed this equation right here, 13 00:00:35,632 --> 00:00:37,340 today is going to be the most satisfying, 14 00:00:37,340 --> 00:00:40,280 because we are going to cancel out just about every term, 15 00:00:40,280 --> 00:00:43,940 leaving a homogeneous, infinite reactor criticality condition. 16 00:00:43,940 --> 00:00:45,860 So we will go over today, how do you 17 00:00:45,860 --> 00:00:49,970 go from this, to what is criticality in a reactor? 18 00:00:49,970 --> 00:00:54,680 So I want to get a couple of variables up over here 19 00:00:54,680 --> 00:00:57,440 to remind you guys. 20 00:00:57,440 --> 00:01:00,330 We had this variable flux of r, e, 21 00:01:00,330 --> 00:01:05,300 omega, t in the number of neutrons 22 00:01:05,300 --> 00:01:09,020 per centimeter squared per second traveling 23 00:01:09,020 --> 00:01:11,120 through something. 24 00:01:11,120 --> 00:01:14,060 And we also had its corresponding non-angular 25 00:01:14,060 --> 00:01:18,980 dependent term, on just r, e, t, if we don't care 26 00:01:18,980 --> 00:01:21,110 what angle things go through. 27 00:01:21,110 --> 00:01:23,990 We've got a corresponding variable called current. 28 00:01:23,990 --> 00:01:27,290 So I'll put this as flux. 29 00:01:27,290 --> 00:01:38,720 We have current j, r, e, omega, t, and its corresponding, 30 00:01:38,720 --> 00:01:40,390 we don't care about angle form. 31 00:01:43,210 --> 00:01:45,060 And today, what we're going to do 32 00:01:45,060 --> 00:01:47,280 is first go over this equation again so that we 33 00:01:47,280 --> 00:01:48,690 understand all of its parts. 34 00:01:48,690 --> 00:01:51,150 And there are more parts here than are in the reading. 35 00:01:51,150 --> 00:01:52,710 If you remember, that's because I 36 00:01:52,710 --> 00:01:56,100 wanted to show you how all of these terms are created. 37 00:01:56,100 --> 00:01:57,600 Just about every one of these terms, 38 00:01:57,600 --> 00:02:01,620 except the external source and the flow through some surface, 39 00:02:01,620 --> 00:02:08,669 has the form of some multiplier, times the integral 40 00:02:08,669 --> 00:02:12,930 over all possible variables that we care about, 41 00:02:12,930 --> 00:02:18,750 times a reaction rate d stuff, where this reaction 42 00:02:18,750 --> 00:02:24,660 rate is always going to be some cross-section, times some flux. 43 00:02:24,660 --> 00:02:27,480 So when you look at this equation using that template, 44 00:02:27,480 --> 00:02:29,310 it's actually not so bad. 45 00:02:29,310 --> 00:02:32,130 So let's go through each of these pieces right here. 46 00:02:32,130 --> 00:02:34,320 And then we're going to start simplifying things. 47 00:02:34,320 --> 00:02:36,180 And this board's going to look like some sort of rainbow 48 00:02:36,180 --> 00:02:36,720 explosion. 49 00:02:36,720 --> 00:02:40,080 But all that's going to be left is a much simpler form 50 00:02:40,080 --> 00:02:42,940 of the neutron diffusion equation. 51 00:02:42,940 --> 00:02:45,810 So we've got our time-dependent term right here, 52 00:02:45,810 --> 00:02:48,300 where I've stuck in this variable flux, 53 00:02:48,300 --> 00:02:50,700 instead of the number of neutrons n, 54 00:02:50,700 --> 00:02:54,360 because we know that flux is the number of neutrons, 55 00:02:54,360 --> 00:02:57,600 times the speed at which they're moving. 56 00:02:57,600 --> 00:02:59,850 And just to check our units, flux 57 00:02:59,850 --> 00:03:04,450 should be in neutrons per centimeters squared per second. 58 00:03:04,450 --> 00:03:11,100 And n is in neutrons per cubic centimeter. 59 00:03:11,100 --> 00:03:17,120 And velocity is in centimeters per second. 60 00:03:17,120 --> 00:03:18,160 So the units check out. 61 00:03:18,160 --> 00:03:20,380 That's why I made that substitution right there. 62 00:03:20,380 --> 00:03:24,830 And this way, everything is in terms of little fi, the flux. 63 00:03:24,830 --> 00:03:26,240 We have our first term here. 64 00:03:26,240 --> 00:03:29,120 I think I'll have a labeling color. 65 00:03:29,120 --> 00:03:32,720 That'll make things a little easier to understand. 66 00:03:32,720 --> 00:03:35,450 Which is due to regular old fission. 67 00:03:35,450 --> 00:03:39,650 In this case, we have new, the number of neutrons created 68 00:03:39,650 --> 00:03:44,570 per fission, times chi, the sort of fission birth spectrum, 69 00:03:44,570 --> 00:03:47,840 or at what energy the neutrons are born. 70 00:03:47,840 --> 00:03:50,630 Over 4pi would account for all different angles in which they 71 00:03:50,630 --> 00:03:54,680 could go out, times the integral over our whole control volume, 72 00:03:54,680 --> 00:03:57,265 and all other energies in angles. 73 00:03:57,265 --> 00:03:58,640 If you remember now, we're trying 74 00:03:58,640 --> 00:04:03,500 to track the number of neutrons in some small energy group, e, 75 00:04:03,500 --> 00:04:06,940 traveling in some small direction, omega. 76 00:04:06,940 --> 00:04:09,110 And those have little vector things on it 77 00:04:09,110 --> 00:04:12,417 at some specific position as a function of time. 78 00:04:12,417 --> 00:04:14,750 So in order to figure out how many neutrons are entering 79 00:04:14,750 --> 00:04:16,820 our group from fission, we need to know, 80 00:04:16,820 --> 00:04:18,589 what are all the fissions happening 81 00:04:18,589 --> 00:04:20,310 in all the other groups? 82 00:04:20,310 --> 00:04:22,730 I've also escalated this problem a little bit 83 00:04:22,730 --> 00:04:26,060 to not assume that the reactor's homogeneous. 84 00:04:26,060 --> 00:04:29,330 So I've added an r, or a spatial dependence 85 00:04:29,330 --> 00:04:31,528 for every cross-section here, which 86 00:04:31,528 --> 00:04:33,320 means that as you move through the reactor, 87 00:04:33,320 --> 00:04:35,030 you might encounter different materials. 88 00:04:35,030 --> 00:04:37,250 You almost certainly will, unless your reactor 89 00:04:37,250 --> 00:04:38,470 has been in a blender. 90 00:04:38,470 --> 00:04:40,790 So except for that case, you would actually 91 00:04:40,790 --> 00:04:44,390 have different cross sections in different parts of the reactor. 92 00:04:44,390 --> 00:04:45,890 So all of a sudden, this is starting 93 00:04:45,890 --> 00:04:48,830 to get awfully interesting, or messy depending on what 94 00:04:48,830 --> 00:04:50,150 you want to think about it. 95 00:04:50,150 --> 00:04:52,760 There is the external source, which is actually 96 00:04:52,760 --> 00:04:55,910 a real phenomenon, because reactors 97 00:04:55,910 --> 00:04:59,430 do stick in those californium kickstarter sources. 98 00:04:59,430 --> 00:05:01,040 So for some amount of time, there 99 00:05:01,040 --> 00:05:03,140 is an external source of neutrons, 100 00:05:03,140 --> 00:05:06,050 giving them out with some positional energy angle 101 00:05:06,050 --> 00:05:08,260 and time dependence. 102 00:05:08,260 --> 00:05:10,270 So let's call this the kickstarter source. 103 00:05:14,650 --> 00:05:19,720 There's this term right here, the nin reactions. 104 00:05:19,720 --> 00:05:25,550 So these are other reactions where it's absorb a neutron, 105 00:05:25,550 --> 00:05:29,420 and give off anywhere between 2 and 4 neutrons. 106 00:05:29,420 --> 00:05:31,460 Beyond that, it's just not energetically 107 00:05:31,460 --> 00:05:33,230 possible in a fission reactor. 108 00:05:33,230 --> 00:05:34,880 But don't undergo fission. 109 00:05:34,880 --> 00:05:37,730 They have their own cross sections, their own birth 110 00:05:37,730 --> 00:05:38,720 spectrum. 111 00:05:38,720 --> 00:05:40,610 And I've stuck in something right here, 112 00:05:40,610 --> 00:05:43,460 if we have summing over all possible i, where you have 113 00:05:43,460 --> 00:05:48,920 this reaction be n in reaction, where 1 neutron goes in, 114 00:05:48,920 --> 00:05:50,790 and i neutrons come out. 115 00:05:50,790 --> 00:05:53,450 You've got to multiply by the number of neutrons 116 00:05:53,450 --> 00:05:54,840 per reaction. 117 00:05:54,840 --> 00:05:57,060 For fission, that was new. 118 00:05:57,060 --> 00:05:59,875 For an nin reaction, that's just i. 119 00:05:59,875 --> 00:06:01,500 But otherwise, the term looks the same. 120 00:06:01,500 --> 00:06:05,490 You have your multiplier, your birth spectrum, your 4pi, 121 00:06:05,490 --> 00:06:08,730 your integral over stuff, your unique cross-section, 122 00:06:08,730 --> 00:06:10,530 and the flux. 123 00:06:10,530 --> 00:06:12,920 And these two together give you a reaction rate. 124 00:06:12,920 --> 00:06:15,740 I've just written all of the differentials as d stuff, 125 00:06:15,740 --> 00:06:17,840 because it takes a lot of time to write those 126 00:06:17,840 --> 00:06:19,450 over and over again. 127 00:06:19,450 --> 00:06:26,050 And then we have our photo fission term, 128 00:06:26,050 --> 00:06:28,520 where gamma rays of sufficiently high energy 129 00:06:28,520 --> 00:06:32,000 can also induce fission external to the neutrons. 130 00:06:32,000 --> 00:06:34,140 The term looks exactly the same. 131 00:06:34,140 --> 00:06:36,350 There's going to be some new for photofission, 132 00:06:36,350 --> 00:06:38,510 some birth spectrum for photofission, 133 00:06:38,510 --> 00:06:41,090 some cross section for photofission, and the same flux 134 00:06:41,090 --> 00:06:43,430 that we're using everywhere else. 135 00:06:43,430 --> 00:06:53,120 Then we had what's called the in scattering term, where neutrons 136 00:06:53,120 --> 00:06:55,760 can undergo scattering, lose some energy, 137 00:06:55,760 --> 00:06:58,880 and enter our group from somewhere else. 138 00:06:58,880 --> 00:07:01,400 That's why we have those en omega primes, 139 00:07:01,400 --> 00:07:03,110 because it's some other energy group. 140 00:07:03,110 --> 00:07:06,320 And we have to account for all of those energy groups. 141 00:07:06,320 --> 00:07:08,570 That's why we have this integral there. 142 00:07:08,570 --> 00:07:10,470 And it looks very much the same. 143 00:07:10,470 --> 00:07:12,530 There's a scattering cross-section. 144 00:07:12,530 --> 00:07:16,080 And that should actually be an e prime right there. 145 00:07:16,080 --> 00:07:17,570 Make sure I'm not missing any more 146 00:07:17,570 --> 00:07:21,110 of those inside the integral. 147 00:07:25,460 --> 00:07:26,480 That's all good. 148 00:07:26,480 --> 00:07:29,000 That's a prime, good. 149 00:07:29,000 --> 00:07:30,020 There's also a flux. 150 00:07:30,020 --> 00:07:32,180 And then there was this probability function 151 00:07:32,180 --> 00:07:35,780 that a given neutron starting off an energy 152 00:07:35,780 --> 00:07:38,926 e prime omega prime, ends up scattering into r 153 00:07:38,926 --> 00:07:40,460 energy, e and omega. 154 00:07:40,460 --> 00:07:44,130 So this would be the other one. 155 00:07:44,130 --> 00:07:47,100 And this would be r group. 156 00:07:47,100 --> 00:07:50,530 But otherwise, the term looks very much the same. 157 00:07:50,530 --> 00:07:53,310 And that takes care of all the possible gains of neutrons 158 00:07:53,310 --> 00:07:54,630 into r group. 159 00:07:54,630 --> 00:07:56,680 The losses are a fair bit simpler. 160 00:07:56,680 --> 00:08:00,490 There is reaction of absolutely any kind. 161 00:08:00,490 --> 00:08:08,770 Let's say this would be the total cross-section, which 162 00:08:08,770 --> 00:08:11,380 says that if a neutron undergoes any reaction at all, 163 00:08:11,380 --> 00:08:16,120 it's going to lose energy and go out of our energy group d, e. 164 00:08:16,120 --> 00:08:17,770 Notice here that these are all-- 165 00:08:17,770 --> 00:08:20,320 these energies and omega is our all r group, 166 00:08:20,320 --> 00:08:22,750 because we only care about how many neutrons in r group 167 00:08:22,750 --> 00:08:24,940 undergo a reaction and leave. 168 00:08:24,940 --> 00:08:26,710 And the form is very simple-- 169 00:08:26,710 --> 00:08:31,130 integrate over volume, energy, and direction, 170 00:08:31,130 --> 00:08:33,700 times a cross-section, times a flux, 171 00:08:33,700 --> 00:08:35,559 just like all the other ones. 172 00:08:35,559 --> 00:08:37,420 Then the only difference in one right here 173 00:08:37,420 --> 00:08:38,545 is what we'll call leakage. 174 00:08:42,750 --> 00:08:46,120 These are neutrons moving out of whatever control 175 00:08:46,120 --> 00:08:47,370 surface that we're looking at. 176 00:08:47,370 --> 00:08:53,790 And this can be some arbitrarily complex control surface in 3D. 177 00:08:53,790 --> 00:08:55,770 I don't really know how to draw a blob in 3D. 178 00:08:55,770 --> 00:08:58,380 But at every point on that blob, there's 179 00:08:58,380 --> 00:09:00,510 going to be a normal vector. 180 00:09:00,510 --> 00:09:05,340 And you can then take the current of neutrons traveling 181 00:09:05,340 --> 00:09:08,280 out that normal vector, and figure out how much of that 182 00:09:08,280 --> 00:09:11,340 is actually leaving our surface ds. 183 00:09:11,340 --> 00:09:13,620 The one problem we had is that everything here 184 00:09:13,620 --> 00:09:18,000 is in terms of volume, volume, volume, surface. 185 00:09:18,000 --> 00:09:19,770 So we don't have all the same terms, 186 00:09:19,770 --> 00:09:22,410 because once we have everything in the same variables 187 00:09:22,410 --> 00:09:26,190 we can start to make some pretty crazy simplifications. 188 00:09:26,190 --> 00:09:33,830 The last thing we did is we invoked the divergence theorem, 189 00:09:33,830 --> 00:09:39,100 that says that the surface integral of some variable FdS 190 00:09:39,100 --> 00:09:43,750 is the same as the volume integral of the divergence 191 00:09:43,750 --> 00:09:47,167 of that variable dV. 192 00:09:47,167 --> 00:09:49,250 So I remember there was some snickering last time, 193 00:09:49,250 --> 00:09:51,125 because you probably haven't seen this since, 194 00:09:51,125 --> 00:09:53,140 was it 1801 or 1802? 195 00:09:53,140 --> 00:09:56,480 1802, OK, that makes sense, because divergence usually 196 00:09:56,480 --> 00:09:59,060 has more than one variable associated with it. 197 00:09:59,060 --> 00:10:00,560 I'll include the dot, because that's 198 00:10:00,560 --> 00:10:02,450 what makes that divergence. 199 00:10:02,450 --> 00:10:05,790 So we can then rewrite this term. 200 00:10:05,790 --> 00:10:08,120 Let's start our simplification colors. 201 00:10:08,120 --> 00:10:11,210 That's our divergence theorem. 202 00:10:11,210 --> 00:10:14,890 So let's get rid of it in this form, 203 00:10:14,890 --> 00:10:19,150 and call it minus integral over all that stuff. 204 00:10:21,970 --> 00:10:30,140 Then we'll have dot, little j, to be 205 00:10:30,140 --> 00:10:40,790 careful, r, e, omega, t, d soft. 206 00:10:40,790 --> 00:10:43,290 So for every step, I"m going to use a different color so you 207 00:10:43,290 --> 00:10:48,407 can see which simplification led to how much crossing stuff out. 208 00:10:48,407 --> 00:10:49,990 And so like I said, this board's going 209 00:10:49,990 --> 00:10:51,330 to look like a rainbow explosion. 210 00:10:51,330 --> 00:10:52,872 But then we'll rewrite it at the end. 211 00:10:52,872 --> 00:10:54,820 And it's going to look a whole lot simpler. 212 00:10:54,820 --> 00:10:59,437 So now, let's start making some simplifications. 213 00:10:59,437 --> 00:11:01,270 Let's say you're an actual reactor designer, 214 00:11:01,270 --> 00:11:04,000 and all you care about is how many neutrons are here. 215 00:11:04,000 --> 00:11:05,650 Of the variables here, which one do you 216 00:11:05,650 --> 00:11:07,060 think we care the least about? 217 00:11:10,270 --> 00:11:12,732 Angle, I mean do we really care which direction 218 00:11:12,732 --> 00:11:13,690 the neutrons are going? 219 00:11:13,690 --> 00:11:15,820 No, we pretty much care, where are they? 220 00:11:15,820 --> 00:11:18,400 And are they causing fission or getting absorbed? 221 00:11:18,400 --> 00:11:20,970 So let's start our simplification board. 222 00:11:20,970 --> 00:11:26,200 And in blue, we'll neglect angle. 223 00:11:26,200 --> 00:11:28,120 This is where it starts to get fun. 224 00:11:28,120 --> 00:11:32,440 So in this case, we'll just perform the omega integral 225 00:11:32,440 --> 00:11:33,880 over all angles. 226 00:11:33,880 --> 00:11:36,610 We just neglect angle here. 227 00:11:36,610 --> 00:11:41,170 We forget the omega integral, forget omega there. 228 00:11:41,170 --> 00:11:44,830 Away goes the 4pi, because we've integrated overall 4pi star 229 00:11:44,830 --> 00:11:47,200 radians, or all solid angle. 230 00:11:47,200 --> 00:11:48,400 Let's just keep going. 231 00:11:48,400 --> 00:11:51,100 Forget the 4pi, forget the omega, 232 00:11:51,100 --> 00:11:57,490 forget the omega, forget the 4pi and the omega, and the omega. 233 00:11:57,490 --> 00:12:01,690 Same thing here-- forget the omega in the scattering kernel, 234 00:12:01,690 --> 00:12:04,420 forget it in the flux, forget it there, 235 00:12:04,420 --> 00:12:09,300 forget it there, and there as well. 236 00:12:09,300 --> 00:12:12,775 OK, we've now completely eliminated one variable. 237 00:12:12,775 --> 00:12:14,650 And all we had to do is ditch the 4pi and one 238 00:12:14,650 --> 00:12:17,670 of the integrals. 239 00:12:17,670 --> 00:12:18,300 What next? 240 00:12:20,970 --> 00:12:23,370 We're tracking right now every possible position, 241 00:12:23,370 --> 00:12:26,280 every possible energy, at every possible time. 242 00:12:26,280 --> 00:12:29,280 If you want to know, what is your flux 243 00:12:29,280 --> 00:12:33,400 going to be in the reactor at steady state, what 244 00:12:33,400 --> 00:12:35,440 variable do you attack next? 245 00:12:35,440 --> 00:12:36,520 Time. 246 00:12:36,520 --> 00:12:38,950 So let's just say this reactor is at steady state. 247 00:12:43,580 --> 00:12:45,140 That's going to invoke a few things. 248 00:12:45,140 --> 00:12:50,250 For one, it's going to ditch the entire steady state term. 249 00:12:50,250 --> 00:12:54,240 We're going to get rid of all the ts in all the fluxes. 250 00:12:54,240 --> 00:12:58,220 This shouldn't take too long to do. 251 00:12:58,220 --> 00:13:00,960 I think that's all of them. 252 00:13:00,960 --> 00:13:03,930 And the third thing is if this reactor is at steady state, 253 00:13:03,930 --> 00:13:07,420 chances are we've taken our kickstarter source out, 254 00:13:07,420 --> 00:13:09,230 because we just needed it to get it going. 255 00:13:09,230 --> 00:13:11,710 But the reactor should be self-sustaining 256 00:13:11,710 --> 00:13:13,310 once it's at steady state. 257 00:13:13,310 --> 00:13:17,530 So let's just get rid of our source term. 258 00:13:17,530 --> 00:13:19,700 I just want to make sure I didn't miss any here. 259 00:13:19,700 --> 00:13:26,080 OK, next up, let's go with green. 260 00:13:26,080 --> 00:13:28,590 What else do you think we can simplify about this problem? 261 00:13:34,710 --> 00:13:37,370 Well, if you look far enough away from the reactor, 262 00:13:37,370 --> 00:13:40,700 we can make an assumption that the reactor 263 00:13:40,700 --> 00:13:42,500 is roughly homogeneous. 264 00:13:45,560 --> 00:13:47,960 In some cases, it's not so good of an assumption, 265 00:13:47,960 --> 00:13:51,950 like very close to anything that has a huge absorption 266 00:13:51,950 --> 00:13:53,200 cross-section. 267 00:13:53,200 --> 00:13:55,490 Now, I want to explain the physics behind this. 268 00:13:55,490 --> 00:13:57,890 If the neutrons travel a very long distance 269 00:13:57,890 --> 00:14:01,640 through any group of materials, then those materials 270 00:14:01,640 --> 00:14:05,900 will appear to be roughly homogeneous to the neutrons. 271 00:14:05,900 --> 00:14:08,900 If, however, the neutrons travel through something that's 272 00:14:08,900 --> 00:14:11,510 very different from the materials around it, 273 00:14:11,510 --> 00:14:14,500 then that homogeneous assumption breaks down. 274 00:14:14,500 --> 00:14:17,810 So in what locations in a nuclear reactor do you think 275 00:14:17,810 --> 00:14:22,770 you cannot treat the system as homogeneous? 276 00:14:22,770 --> 00:14:24,270 Where do the properties of materials 277 00:14:24,270 --> 00:14:26,260 suddenly change by a huge amount? 278 00:14:26,260 --> 00:14:26,760 Yeah, Luke? 279 00:14:26,760 --> 00:14:27,420 AUDIENCE: Control rods. 280 00:14:27,420 --> 00:14:29,130 MICHAEL SHORT: Control rods, right, 281 00:14:29,130 --> 00:14:34,515 so let's say it's bad for control rods. 282 00:14:39,140 --> 00:14:39,740 Where else? 283 00:14:47,780 --> 00:14:49,538 How about the fuel? 284 00:14:49,538 --> 00:14:51,330 All of a sudden, you're moving from a bunch 285 00:14:51,330 --> 00:14:54,420 of structural materials where sigma fission equals 0, 286 00:14:54,420 --> 00:14:57,180 to the fuel where sigma fission, like you saw on the test, 287 00:14:57,180 --> 00:14:59,940 can be like 500 barns, which even though it's 288 00:14:59,940 --> 00:15:02,080 got a very small exponent in front of it, 289 00:15:02,080 --> 00:15:04,500 10 to the minus 22 centimeters squared, 290 00:15:04,500 --> 00:15:06,910 it's still pretty significant. 291 00:15:06,910 --> 00:15:10,020 So this assumption breaks down around the control rods 292 00:15:10,020 --> 00:15:11,325 and around the fuel. 293 00:15:11,325 --> 00:15:12,450 But we can get around this. 294 00:15:12,450 --> 00:15:15,960 Let's analyze the simplest, craziest possible reactor, 295 00:15:15,960 --> 00:15:18,270 which would be a molten salt fueled reactor. 296 00:15:18,270 --> 00:15:21,360 It's just a blob of 700 Celsius goo 297 00:15:21,360 --> 00:15:25,140 that's got its fuel, coolant, and control rods all built in. 298 00:15:25,140 --> 00:15:28,740 So if we assume that the reactor is homogeneous, which 299 00:15:28,740 --> 00:15:31,410 is a pretty good assumption for molten salt 300 00:15:31,410 --> 00:15:33,870 fueled reactors, because the fuel's 301 00:15:33,870 --> 00:15:35,460 dissolved in the coolant. 302 00:15:35,460 --> 00:15:37,590 And it builds up its own fission product poison. 303 00:15:37,590 --> 00:15:41,250 So it's got some of its own control rods kind of built in. 304 00:15:41,250 --> 00:15:44,010 Usually, we'll have other extra ones too, but whatever. 305 00:15:44,010 --> 00:15:47,480 Then we can start to really simplify things. 306 00:15:47,480 --> 00:15:52,200 If we get rid of any homogeneity assumptions, 307 00:15:52,200 --> 00:15:55,860 we cannot necessarily get rid of the r in the flux, 308 00:15:55,860 --> 00:15:58,890 because even if the reactor's homogeneous it still might have 309 00:15:58,890 --> 00:16:00,330 boundaries. 310 00:16:00,330 --> 00:16:02,100 So you might be able to approximate it 311 00:16:02,100 --> 00:16:09,060 as just a cylinder or a slab of uniform materials. 312 00:16:12,490 --> 00:16:17,610 But if we were to get rid of the r's in the flux term, that 313 00:16:17,610 --> 00:16:22,260 would mean that as we graph flux as a function of distance, 314 00:16:22,260 --> 00:16:28,052 it would look like that, including infinitely far away 315 00:16:28,052 --> 00:16:28,760 from the reactor. 316 00:16:28,760 --> 00:16:30,970 Now, is was that true? 317 00:16:30,970 --> 00:16:36,200 Absolutely not, so I don't want to leave that up for anyone. 318 00:16:36,200 --> 00:16:40,640 We'll fill in what these graphs look like a little later, 319 00:16:40,640 --> 00:16:43,820 just leave them there for now. 320 00:16:43,820 --> 00:16:47,460 We can get rid of some of the other r's though, 321 00:16:47,460 --> 00:16:49,530 like these cross sections. 322 00:16:49,530 --> 00:16:51,780 If the reactor is actually homogeneous, 323 00:16:51,780 --> 00:16:54,630 then the cross section is the same everywhere 324 00:16:54,630 --> 00:16:57,000 because the materials are the same everywhere. 325 00:16:57,000 --> 00:17:01,290 So we can get rid of the r's here, the r's here, 326 00:17:01,290 --> 00:17:04,109 and there, and there. 327 00:17:04,109 --> 00:17:07,500 And that's it, I think. 328 00:17:07,500 --> 00:17:09,510 I don't think I missed any, good. 329 00:17:12,369 --> 00:17:18,220 Next up-- if this reactor is homogeneous, 330 00:17:18,220 --> 00:17:21,650 then does it really matter at which location 331 00:17:21,650 --> 00:17:25,060 we're taking this balance? 332 00:17:25,060 --> 00:17:27,609 Does it really matter which little volume 333 00:17:27,609 --> 00:17:30,070 element we're looking at? 334 00:17:30,070 --> 00:17:32,320 We say these equations are-- 335 00:17:32,320 --> 00:17:39,490 we'll call them volume identical, which 336 00:17:39,490 --> 00:17:41,830 means if this same equation is satisfied 337 00:17:41,830 --> 00:17:44,200 at any point in the reactor, we don't 338 00:17:44,200 --> 00:17:47,127 need to do the volume integral over the whole reactor. 339 00:17:47,127 --> 00:17:49,210 It's not like it's going to change anywhere we go. 340 00:17:49,210 --> 00:17:54,140 So forget the volume integrals. 341 00:17:54,140 --> 00:17:56,800 Hopefully, you guys see where I'm going with this. 342 00:17:56,800 --> 00:18:00,730 And I've never tried teaching it like this rainbow explosion 343 00:18:00,730 --> 00:18:01,230 before. 344 00:18:01,230 --> 00:18:04,470 But I'm kind of excited to see how it turns out. 345 00:18:04,470 --> 00:18:09,700 So already like 2/3 of the stuff that we had written are gone. 346 00:18:09,700 --> 00:18:13,930 What's the only variable left that we can go after? 347 00:18:13,930 --> 00:18:17,410 What's the only color left that I haven't really used? 348 00:18:17,410 --> 00:18:22,450 Energy, so we can make a couple of different assumptions. 349 00:18:22,450 --> 00:18:27,070 This equation as it is not yet really analytically solvable, 350 00:18:27,070 --> 00:18:30,040 because a lot of these energy dependent terms 351 00:18:30,040 --> 00:18:32,290 don't have analytical solutions, or even 352 00:18:32,290 --> 00:18:34,900 forms like the cross sections. 353 00:18:34,900 --> 00:18:37,015 But we can start attacking energy. 354 00:18:41,180 --> 00:18:43,430 Hopefully, this is different enough from white. 355 00:18:45,980 --> 00:18:48,880 Yeah, is that big enough difference for you guys to see? 356 00:18:48,880 --> 00:18:52,687 Good, OK, we can start doing this in a few different ways. 357 00:18:52,687 --> 00:18:54,020 I want to mention what they are. 358 00:18:54,020 --> 00:18:56,410 And then we're going to do the easiest one. 359 00:18:56,410 --> 00:18:59,620 So the way it's done for real, like in the computational 360 00:18:59,620 --> 00:19:01,540 reactor physics group, is you can 361 00:19:01,540 --> 00:19:15,750 discretize the energy into a bunch of little energy groups. 362 00:19:15,750 --> 00:19:19,890 So you can write this equation for every little energy group, 363 00:19:19,890 --> 00:19:21,990 and assume that along this energy scale, 364 00:19:21,990 --> 00:19:27,600 ranging from your maximum energy to probably thermal energy-- 365 00:19:32,410 --> 00:19:36,305 025, let's do this clearly with thick chalk. 366 00:19:40,010 --> 00:19:42,100 There we go. 367 00:19:42,100 --> 00:19:47,680 You can then discretize into some little energy group. 368 00:19:47,680 --> 00:19:55,950 Let's say that's egi, that's egi plus 1, and so on, and so on. 369 00:19:55,950 --> 00:19:58,450 And depending on the type of reactor that you're looking at, 370 00:19:58,450 --> 00:20:00,760 and the energy resolution that you need, 371 00:20:00,760 --> 00:20:03,610 you choose the number of energy groups accordingly. 372 00:20:03,610 --> 00:20:05,840 Does anyone happen to know for a light water reactor, 373 00:20:05,840 --> 00:20:07,810 how many energy groups do you think we need 374 00:20:07,810 --> 00:20:09,600 to model a light water reactor? 375 00:20:13,720 --> 00:20:17,060 The answer might surprise you. 376 00:20:17,060 --> 00:20:19,640 It's just two, actually. 377 00:20:19,640 --> 00:20:21,350 All we care about-- 378 00:20:21,350 --> 00:20:25,730 so let's say this would be for the general case. 379 00:20:25,730 --> 00:20:28,720 All we care about for a light water reactor 380 00:20:28,720 --> 00:20:32,635 is, are your neutrons thermal? 381 00:20:35,440 --> 00:20:37,985 Or are they not? 382 00:20:37,985 --> 00:20:39,360 Because the neutrons that are not 383 00:20:39,360 --> 00:20:42,705 thermal are not contributing to fission that much. 384 00:20:42,705 --> 00:20:43,830 They are just a little bit. 385 00:20:43,830 --> 00:20:45,360 And you can account for those. 386 00:20:45,360 --> 00:20:46,950 But pretty much, they're not. 387 00:20:46,950 --> 00:20:50,640 Once the neutron slowdown down to get thermal, in the range 388 00:20:50,640 --> 00:20:55,620 from, let's say, about an EV to that temperature-- 389 00:20:55,620 --> 00:20:58,650 took a surprising amount of time to write with sidewalk chalk-- 390 00:20:58,650 --> 00:21:01,350 then you've got things that are about 500 or 1,000 times 391 00:21:01,350 --> 00:21:03,360 more likely to undergo fission. 392 00:21:03,360 --> 00:21:06,810 And so all you care about is the neutrons are all born. 393 00:21:06,810 --> 00:21:08,850 They're all born right about here. 394 00:21:08,850 --> 00:21:10,800 And they scatter, and bounce around. 395 00:21:10,800 --> 00:21:12,510 And you don't care, because they're just 396 00:21:12,510 --> 00:21:14,215 in this not thermal region. 397 00:21:14,215 --> 00:21:15,840 And when they enter the thermal region, 398 00:21:15,840 --> 00:21:18,173 you start tracking them, because those are the ones that 399 00:21:18,173 --> 00:21:20,020 really count for fission. 400 00:21:20,020 --> 00:21:23,320 And if you actually look up the specifications for the AP 1000, 401 00:21:23,320 --> 00:21:25,450 this is a modern reactor under construction 402 00:21:25,450 --> 00:21:27,580 in many different places in the world. 403 00:21:27,580 --> 00:21:30,370 When you see, how do they do the neutron analysis? 404 00:21:30,370 --> 00:21:32,150 Two group approximation. 405 00:21:32,150 --> 00:21:34,150 So this isn't just an academic exercise 406 00:21:34,150 --> 00:21:36,610 to make it easier for sophomores to understand. 407 00:21:36,610 --> 00:21:39,310 This is actually something that's done for real reactors. 408 00:21:39,310 --> 00:21:41,830 So if you ever felt like I'm making it too simple, 409 00:21:41,830 --> 00:21:44,940 no, no, no, I'm simplifying it down to what's really done. 410 00:21:44,940 --> 00:21:46,810 And I will get you that specification so 411 00:21:46,810 --> 00:21:49,270 you can see what Westinghouse says, like, 412 00:21:49,270 --> 00:21:50,980 this is how we design the reactor. 413 00:21:50,980 --> 00:21:54,980 We made a two group simplification in many cases. 414 00:21:54,980 --> 00:21:57,730 So you can discretize. 415 00:21:57,730 --> 00:22:02,400 You can forget it, which we're going to call the one group 416 00:22:02,400 --> 00:22:05,660 approximation. 417 00:22:05,660 --> 00:22:08,630 Or you can-- let's say two group is the other one that we're 418 00:22:08,630 --> 00:22:12,280 actually going to tackle. 419 00:22:12,280 --> 00:22:15,530 We're going to do this one, forget energy. 420 00:22:15,530 --> 00:22:18,440 But we're not really going to forget energy, 421 00:22:18,440 --> 00:22:22,340 because you can't just pick an energy, 422 00:22:22,340 --> 00:22:25,760 and pick a cross-section, and say, OK, 423 00:22:25,760 --> 00:22:28,190 that's the cross section we're going to use. 424 00:22:28,190 --> 00:22:34,430 If most cross sections have the following form-- 425 00:22:34,430 --> 00:22:38,510 if this is log of energy, and this is log of sigma, 426 00:22:38,510 --> 00:22:46,420 and it goes something like that, what energy do you pick? 427 00:22:46,420 --> 00:22:46,920 Go ahead. 428 00:22:46,920 --> 00:22:47,420 Tell me. 429 00:22:47,420 --> 00:22:48,810 Which energy do you pick? 430 00:22:48,810 --> 00:22:52,190 Anyone want to wager a guess? 431 00:22:52,190 --> 00:22:55,210 AUDIENCE: The ones before or after the big squigglies. 432 00:22:55,210 --> 00:22:57,917 MICHAEL SHORT: The ones before or after the big squiggles. 433 00:22:57,917 --> 00:23:00,250 I don't think that's correct, because if you do it this, 434 00:23:00,250 --> 00:23:02,320 then you're going away underestimate fission. 435 00:23:02,320 --> 00:23:04,120 If you do it here, you're going to way overestimate fission, 436 00:23:04,120 --> 00:23:05,950 or whatever other reaction you have. 437 00:23:05,950 --> 00:23:08,110 We didn't say which reaction this is. 438 00:23:08,110 --> 00:23:10,360 The rest of you who are silent and afraid to speak up, 439 00:23:10,360 --> 00:23:12,680 you're actually correct. 440 00:23:12,680 --> 00:23:15,440 I wouldn't actually pick any single value here. 441 00:23:15,440 --> 00:23:20,320 What you need to do is find some sort of average cross-section 442 00:23:20,320 --> 00:23:22,900 for whatever reaction that accurately 443 00:23:22,900 --> 00:23:24,700 represents the number of reactions 444 00:23:24,700 --> 00:23:26,470 happening in the system. 445 00:23:26,470 --> 00:23:29,290 And in order to do that, you have to come up with some 446 00:23:29,290 --> 00:23:32,500 average cross-section for whatever reaction you have 447 00:23:32,500 --> 00:23:39,370 by integrating over your whole energy range of the energy 448 00:23:39,370 --> 00:23:45,930 dependent cross-section as a function of energy, 449 00:23:45,930 --> 00:23:50,800 times your flux de over-- 450 00:23:50,800 --> 00:23:52,660 does this look familiar from 1801 or 2 451 00:23:52,660 --> 00:23:57,240 as well, what's the average value of some function? 452 00:23:57,240 --> 00:23:58,280 Little bit? 453 00:23:58,280 --> 00:24:00,350 Well, we'll bring it back here now. 454 00:24:03,470 --> 00:24:05,930 So retrieve it from cold storage in your memories, 455 00:24:05,930 --> 00:24:09,590 because this is how actual cross sections are averaged. 456 00:24:09,590 --> 00:24:11,915 For whatever energy range you're picking-- 457 00:24:11,915 --> 00:24:13,790 I'm going to make this a little more general. 458 00:24:13,790 --> 00:24:15,470 I won't say zero. 459 00:24:15,470 --> 00:24:18,110 I'll just say your minimum energy for your group. 460 00:24:21,240 --> 00:24:26,460 So now, this equation is general for the multi group and the one 461 00:24:26,460 --> 00:24:28,560 group and two group method. 462 00:24:28,560 --> 00:24:31,160 For whatever cross-section you want to pick 463 00:24:31,160 --> 00:24:33,120 and whatever energy range you're looking at, 464 00:24:33,120 --> 00:24:37,820 you take the actual data and perform an average 465 00:24:37,820 --> 00:24:40,640 for the fast and thermal delineation, 466 00:24:40,640 --> 00:24:45,800 where, let's say this is fast and this is thermal, 467 00:24:45,800 --> 00:24:47,450 you would have two different averages. 468 00:24:47,450 --> 00:24:50,287 Maybe this average would be right there. 469 00:24:50,287 --> 00:24:50,870 You know what? 470 00:24:50,870 --> 00:24:55,880 Let's use white so it actually has some contrast. 471 00:24:55,880 --> 00:24:59,210 So this would be one value of the cross-section. 472 00:24:59,210 --> 00:25:02,000 And maybe the next average would be right there. 473 00:25:02,000 --> 00:25:06,140 So you simplify this absolutely non analytical form 474 00:25:06,140 --> 00:25:08,100 of your complicated cross-section 475 00:25:08,100 --> 00:25:10,070 to just a couple of values. 476 00:25:10,070 --> 00:25:15,290 Maybe we'll call that average sigma fast. 477 00:25:15,290 --> 00:25:19,925 And we'll call that average sigma thermal. 478 00:25:24,200 --> 00:25:28,580 So using this analogy and this color, 479 00:25:28,580 --> 00:25:31,310 we can then say, we're going to take 480 00:25:31,310 --> 00:25:35,900 an average new, an average chi, get rid of the energies, 481 00:25:35,900 --> 00:25:38,030 because we can perform the same energy average 482 00:25:38,030 --> 00:25:42,000 integration on every quantity with energy dependence. 483 00:25:42,000 --> 00:25:43,910 So all we do is we put a bar there, 484 00:25:43,910 --> 00:25:47,270 ditch the energies, ditch the energies. 485 00:25:47,270 --> 00:25:50,910 And let's just say that flux is going to be what it is. 486 00:25:50,910 --> 00:25:55,330 Same thing here-- yeah, same thing there, 487 00:25:55,330 --> 00:26:00,710 and there, and there, there, there, there, here, and here, 488 00:26:00,710 --> 00:26:01,832 and here. 489 00:26:01,832 --> 00:26:03,040 And there is a cross-section. 490 00:26:03,040 --> 00:26:04,480 There is an energy. 491 00:26:04,480 --> 00:26:06,450 There is an energy. 492 00:26:06,450 --> 00:26:07,550 There is a cross-section. 493 00:26:07,550 --> 00:26:09,470 We don't care about those anymore. 494 00:26:09,470 --> 00:26:13,250 And there's a couple of other implications of this energy 495 00:26:13,250 --> 00:26:14,601 simplification. 496 00:26:26,150 --> 00:26:28,960 What is the birth spectrum now? 497 00:26:28,960 --> 00:26:30,700 What's the probability that a neutron 498 00:26:30,700 --> 00:26:35,430 is born in our energy group which contains all energies? 499 00:26:35,430 --> 00:26:43,700 1, OK, so forget the chi, and that one, and that one. 500 00:26:43,700 --> 00:26:45,380 And what about this scattering kernel? 501 00:26:45,380 --> 00:26:48,740 What's the probability that a neutron scatters 502 00:26:48,740 --> 00:26:51,260 from any other energy which is already in our group 503 00:26:51,260 --> 00:26:54,988 into our group, which contains all energies? 504 00:26:54,988 --> 00:26:56,856 AUDIENCE: 1. 505 00:26:56,856 --> 00:26:58,900 MICHAEL SHORT: Yeah, scattering no longer 506 00:26:58,900 --> 00:27:01,570 matters when you do the one group approximation, 507 00:27:01,570 --> 00:27:03,970 because if the neutron loses some of its energy, 508 00:27:03,970 --> 00:27:06,520 it's still in our energy group, because our energy 509 00:27:06,520 --> 00:27:08,680 group contains all energies. 510 00:27:08,680 --> 00:27:14,490 So forget the scattering kernel. 511 00:27:14,490 --> 00:27:18,910 And forget the energy integrals. 512 00:27:18,910 --> 00:27:21,920 What are we actually left with? 513 00:27:21,920 --> 00:27:23,750 Not much. 514 00:27:23,750 --> 00:27:25,810 There's no green in here yet. 515 00:27:25,810 --> 00:27:27,560 Good, because I need to do one more thing. 516 00:27:30,380 --> 00:27:31,400 There is no more green. 517 00:27:31,400 --> 00:27:32,210 Oh, we did green. 518 00:27:32,210 --> 00:27:32,870 We did time. 519 00:27:32,870 --> 00:27:37,650 OK, green, red, orange-- is this the orange I used? 520 00:27:37,650 --> 00:27:38,900 Dammit. 521 00:27:38,900 --> 00:27:40,650 OK, we use those. 522 00:27:40,650 --> 00:27:42,120 Purple, no, we've used it. 523 00:27:42,120 --> 00:27:43,650 Oh my God. 524 00:27:43,650 --> 00:27:46,230 We've used both blues. 525 00:27:46,230 --> 00:27:47,600 Bright yellow. 526 00:27:47,600 --> 00:27:48,340 Yeah? 527 00:27:48,340 --> 00:27:49,215 AUDIENCE: [INAUDIBLE] 528 00:27:49,215 --> 00:27:50,370 MICHAEL SHORT: Yes. 529 00:27:50,370 --> 00:27:52,590 Chi is the fission birth spectrum, 530 00:27:52,590 --> 00:27:56,250 the probability that a neutron is born at any given energy. 531 00:27:56,250 --> 00:27:59,250 But because all neutrons are born in our energy group, which 532 00:27:59,250 --> 00:28:02,632 contains all energies, then that just becomes one and goes away. 533 00:28:02,632 --> 00:28:04,590 There's no birth spectrum, because they're just 534 00:28:04,590 --> 00:28:05,730 born in our group. 535 00:28:05,730 --> 00:28:07,380 Does that makes sense? 536 00:28:07,380 --> 00:28:11,310 OK, I think I found the actual only color, besides black 537 00:28:11,310 --> 00:28:15,180 on a black chalkboard, and white which we already have, 538 00:28:15,180 --> 00:28:16,440 that I have left. 539 00:28:16,440 --> 00:28:18,343 We also have a slightly darker shade of gray. 540 00:28:18,343 --> 00:28:19,260 But I'm literally out. 541 00:28:19,260 --> 00:28:23,607 This worked out awesome, because there's one more thing 542 00:28:23,607 --> 00:28:24,690 that we want to deal with. 543 00:28:28,730 --> 00:28:30,470 What do we even have left? 544 00:28:30,470 --> 00:28:33,410 All right, what is the one term that 545 00:28:33,410 --> 00:28:35,900 is not in all of the same variables as the others? 546 00:28:38,750 --> 00:28:41,660 That current, that j. 547 00:28:41,660 --> 00:28:43,040 What do we do about that? 548 00:28:46,450 --> 00:28:47,880 So we're going-- sorry? 549 00:28:47,880 --> 00:28:50,362 AUDIENCE: The F e to e. 550 00:28:50,362 --> 00:28:51,570 MICHAEL SHORT: The F e to e-- 551 00:28:51,570 --> 00:28:54,720 so, actually, I'll recreate some of our variables 552 00:28:54,720 --> 00:28:57,690 here, because there's a lot of them. 553 00:28:57,690 --> 00:29:02,610 So our F of e prime to e is what's 554 00:29:02,610 --> 00:29:03,960 called the scattering kernel. 555 00:29:10,160 --> 00:29:12,050 And that's the probability that a neutron 556 00:29:12,050 --> 00:29:14,690 scatters from some other energy group, e prime, 557 00:29:14,690 --> 00:29:18,110 into ours in e about de. 558 00:29:18,110 --> 00:29:22,790 And chi of e is the fission birth spectrum. 559 00:29:30,180 --> 00:29:34,070 And just for completeness, knew of e 560 00:29:34,070 --> 00:29:41,474 is our neutron multiplier, or neutrons per fission. 561 00:29:41,474 --> 00:29:44,040 And I think that gives a pretty complete explanation 562 00:29:44,040 --> 00:29:45,450 of what's up here. 563 00:29:45,450 --> 00:29:48,120 So now, let's figure out how to deal with the current term. 564 00:29:48,120 --> 00:29:51,450 This is when we make one of the biggest approximations 565 00:29:51,450 --> 00:29:54,120 here, and go from what's called the neutron transport 566 00:29:54,120 --> 00:29:56,550 equation, which is a fully accurate physical model 567 00:29:56,550 --> 00:30:00,100 of what's really going on, to the neutron diffusion equation. 568 00:30:06,990 --> 00:30:09,180 And this is where it gets really fun. 569 00:30:09,180 --> 00:30:12,750 You don't assume that neutrons are subatomic particles that 570 00:30:12,750 --> 00:30:15,630 are whizzing about and knocking off of everything else. 571 00:30:15,630 --> 00:30:18,930 You then treat the neutrons kind of like a gas, 572 00:30:18,930 --> 00:30:20,520 or like a chemical. 573 00:30:20,520 --> 00:30:23,870 And you just say that it follows the laws of diffusion. 574 00:30:23,870 --> 00:30:27,240 Again, this works out very well, except for places 575 00:30:27,240 --> 00:30:30,390 where cross sections suddenly change, like near control 576 00:30:30,390 --> 00:30:31,890 rods or near fuel. 577 00:30:31,890 --> 00:30:33,750 But for most of the reactor, especially 578 00:30:33,750 --> 00:30:36,240 if we have a molten salt fuel reactor, 579 00:30:36,240 --> 00:30:39,190 we can invoke what's called Fick's law. 580 00:30:39,190 --> 00:30:42,330 Does this sound familiar to anyone? 581 00:30:42,330 --> 00:30:49,020 Fick's law diffusion, 3091 or 5111. 582 00:30:49,020 --> 00:30:51,480 It's the change of a chemical down a density 583 00:30:51,480 --> 00:30:52,720 or a concentration gradient. 584 00:30:52,720 --> 00:30:55,950 So, yeah, you've got the idea. 585 00:30:55,950 --> 00:30:59,190 What Fick's law says is that the current-- 586 00:30:59,190 --> 00:31:01,410 or let's say the diffusion current or the neutron 587 00:31:01,410 --> 00:31:02,470 current-- 588 00:31:02,470 --> 00:31:05,160 is going to be equal to some diffusion 589 00:31:05,160 --> 00:31:09,500 coefficient, times the gradient of whatever 590 00:31:09,500 --> 00:31:11,090 chemical concentration you've got. 591 00:31:11,090 --> 00:31:14,830 Let me put the c in there. 592 00:31:14,830 --> 00:31:16,585 So right here, this would be the current. 593 00:31:16,585 --> 00:31:18,730 I'll label it in a different color. 594 00:31:21,780 --> 00:31:26,200 This would be your variable of interest. 595 00:31:26,200 --> 00:31:33,180 Maybe c is for concentration, or phi is for flux. 596 00:31:33,180 --> 00:31:37,070 Oh, that reminds me, where are those bars on our flux? 597 00:31:37,070 --> 00:31:38,080 Which term did we do? 598 00:31:38,080 --> 00:31:42,740 Energy-- where's my slightly lighter blue over here? 599 00:31:42,740 --> 00:31:48,880 All of these phi's become capital, 600 00:31:48,880 --> 00:31:51,280 because we've gotten rid of all the angular, and energy, 601 00:31:51,280 --> 00:31:52,840 and everything dependence. 602 00:31:52,840 --> 00:31:55,950 Oh, angular dependence-- neglect omega, 603 00:31:55,950 --> 00:31:57,390 that should be dark blue. 604 00:32:02,549 --> 00:32:05,810 Omega goes away. 605 00:32:05,810 --> 00:32:07,340 And the fluxes become capital. 606 00:32:10,000 --> 00:32:12,940 So many terms to keep track of. 607 00:32:12,940 --> 00:32:15,430 Luckily, you will never have to. 608 00:32:15,430 --> 00:32:18,120 And then the j becomes a capital J. 609 00:32:18,120 --> 00:32:21,630 Did I miss any phi's here? 610 00:32:21,630 --> 00:32:25,300 No, because that one was already gone, cool. 611 00:32:25,300 --> 00:32:27,730 All right, so we can use Fick's law, 612 00:32:27,730 --> 00:32:31,893 and transform the current into something related to flux. 613 00:32:31,893 --> 00:32:34,060 And what we're saying here is that we're getting rid 614 00:32:34,060 --> 00:32:36,620 of the true physics, which is that there's some fixed neutron 615 00:32:36,620 --> 00:32:37,120 current. 616 00:32:37,120 --> 00:32:40,720 And we're saying that neutrons behave kind of like a gas, 617 00:32:40,720 --> 00:32:42,730 or a chemical in solution. 618 00:32:42,730 --> 00:32:49,120 And so in yellow, we can ditch our current related term, 619 00:32:49,120 --> 00:32:50,920 and rewrite it. 620 00:32:50,920 --> 00:32:56,780 We don't have any integrals left, as negative del squared 621 00:32:56,780 --> 00:32:58,790 phi. 622 00:32:58,790 --> 00:33:07,220 I think the only variable left is r, not too bad. 623 00:33:07,220 --> 00:33:10,780 Now, we have a second order linear differential 624 00:33:10,780 --> 00:33:14,250 equation describing the flow of neutrons in the system. 625 00:33:14,250 --> 00:33:15,700 And so we actually have something 626 00:33:15,700 --> 00:33:18,760 that we can solve for flux. 627 00:33:18,760 --> 00:33:21,620 I think it's time to rewrite it. 628 00:33:21,620 --> 00:33:22,880 Wouldn't you say? 629 00:33:22,880 --> 00:33:24,260 This has been fun. 630 00:33:24,260 --> 00:33:28,640 So let's rewrite what's left. 631 00:33:31,830 --> 00:33:34,740 Make sure you guys can actually see everything there. 632 00:33:34,740 --> 00:33:36,910 We'll write it in boring old white. 633 00:33:36,910 --> 00:33:41,040 So we have no transient dependence. 634 00:33:41,040 --> 00:33:50,630 We have left sigma fission, times flux, as a function of r. 635 00:33:50,630 --> 00:34:01,890 No source, and we have our neutron nin reactions of-- 636 00:34:01,890 --> 00:34:05,790 oh, we forgot our new sigma fission. 637 00:34:05,790 --> 00:34:12,954 Then we have our i sigma fission from nin, times flux. 638 00:34:15,929 --> 00:34:20,340 Next term, we have photofission. 639 00:34:20,340 --> 00:34:24,420 So we have a new, from gamma rays, 640 00:34:24,420 --> 00:34:28,580 times sigma fission from gamma rays, times flux. 641 00:34:31,179 --> 00:34:32,350 Next up, we have-- 642 00:34:32,350 --> 00:34:35,770 well, last simplification to make. 643 00:34:35,770 --> 00:34:37,230 We have scattering. 644 00:34:37,230 --> 00:34:40,300 And we have total cross-section. 645 00:34:40,300 --> 00:34:43,030 When we said, forget about energy, 646 00:34:43,030 --> 00:34:45,110 and our scattering kernel becomes one-- 647 00:34:45,110 --> 00:34:47,429 and that's light blue-- 648 00:34:47,429 --> 00:34:51,270 got to make one more modification to this board. 649 00:34:55,409 --> 00:35:00,750 Do we care about scattering at all anymore whatsoever? 650 00:35:00,750 --> 00:35:03,160 Because scattering doesn't change the number of neutrons 651 00:35:03,160 --> 00:35:03,660 left. 652 00:35:03,660 --> 00:35:09,340 So we can then take these two terms 653 00:35:09,340 --> 00:35:18,230 and just call it sigma absorption, times flux, 654 00:35:18,230 --> 00:35:22,130 because if we take scattering, minus the total cross-section, 655 00:35:22,130 --> 00:35:24,380 it's like saying, all that's left if you don't scatter 656 00:35:24,380 --> 00:35:25,910 is you absorb. 657 00:35:25,910 --> 00:35:30,210 And if you remember, I'll add to the energy pile, 658 00:35:30,210 --> 00:35:33,420 we said that our total cross section 659 00:35:33,420 --> 00:35:36,150 is scattering, plus absorption. 660 00:35:36,150 --> 00:35:44,460 And absorption could be fission and capture. 661 00:35:44,460 --> 00:35:46,920 And capture could be-- 662 00:35:46,920 --> 00:35:49,650 let's say, capture with nothing happens, 663 00:35:49,650 --> 00:35:55,080 plus these nin reactions, plus any other capture 664 00:35:55,080 --> 00:35:56,970 reaction that does something. 665 00:35:56,970 --> 00:35:59,640 So we're going to use this cross-section identity right 666 00:35:59,640 --> 00:36:03,330 here with a couple of minus signs on it. 667 00:36:03,330 --> 00:36:06,120 And say, well, scattering minus total, 668 00:36:06,120 --> 00:36:12,175 leaves you with negative absorption, to simplify terms. 669 00:36:15,250 --> 00:36:19,340 I'll leave that up there for everyone to see. 670 00:36:19,340 --> 00:36:22,930 So then we have scattering and total just becomes 671 00:36:22,930 --> 00:36:31,200 minus sigma absorption, times phi of r. 672 00:36:31,200 --> 00:36:32,730 And we're left with-- 673 00:36:32,730 --> 00:36:33,420 what was that? 674 00:36:33,420 --> 00:36:35,820 Current, that becomes plus. 675 00:36:38,810 --> 00:36:40,950 There is a d missing in there, isn't there? 676 00:36:45,560 --> 00:36:51,250 A yellow d, minus d. 677 00:36:51,250 --> 00:36:55,210 OK, and that's it. 678 00:36:55,210 --> 00:36:55,710 Yes? 679 00:36:55,710 --> 00:36:57,040 AUDIENCE: What is d? 680 00:36:57,040 --> 00:36:59,160 MICHAEL SHORT: d is the diffusion coefficient 681 00:36:59,160 --> 00:37:00,450 right here. 682 00:37:00,450 --> 00:37:04,110 So we're assuming that neutrons diffuse 683 00:37:04,110 --> 00:37:11,690 like a gas or a chemical with some diffusion coefficient. 684 00:37:14,560 --> 00:37:16,150 And so we'll define what that is, oh, 685 00:37:16,150 --> 00:37:18,580 probably next class, because we have seven minutes. 686 00:37:18,580 --> 00:37:19,625 Yeah, Luke? 687 00:37:19,625 --> 00:37:20,803 AUDIENCE: [INAUDIBLE] 688 00:37:20,803 --> 00:37:21,720 MICHAEL SHORT: Uh-huh. 689 00:37:21,720 --> 00:37:25,325 AUDIENCE: [INAUDIBLE] 690 00:37:25,325 --> 00:37:26,950 MICHAEL SHORT: The c right here, that's 691 00:37:26,950 --> 00:37:29,440 whatever variable we're tracking. 692 00:37:29,440 --> 00:37:32,140 So let's call that flux. 693 00:37:32,140 --> 00:37:35,710 Or let's call it n, the number of neutrons, 694 00:37:35,710 --> 00:37:41,390 because flux is just number of neutrons times velocity. 695 00:37:41,390 --> 00:37:43,270 So let's say that the concentration was 696 00:37:43,270 --> 00:37:45,300 the concentration of neutrons. 697 00:37:45,300 --> 00:37:47,500 And we just multiply by their velocity to get flux. 698 00:37:47,500 --> 00:37:50,050 So it's almost like we can say that the concentration 699 00:37:50,050 --> 00:37:53,292 of neutrons is directly related to the flux. 700 00:37:53,292 --> 00:37:55,000 And that way, we have everything in flux. 701 00:37:55,000 --> 00:37:59,590 And that's the entire neutron diffusion equation. 702 00:38:08,590 --> 00:38:13,050 Yeah, this is for one group with all the assumptions we 703 00:38:13,050 --> 00:38:18,710 made right here, homogeneous. 704 00:38:22,132 --> 00:38:23,590 What other assumptions did we make? 705 00:38:23,590 --> 00:38:32,050 Steady state, and we already neglected that. 706 00:38:32,050 --> 00:38:35,120 And I think that's enough qualifiers for this. 707 00:38:35,120 --> 00:38:37,070 But it's directly from this equation 708 00:38:37,070 --> 00:38:39,440 right here that we can develop what's called 709 00:38:39,440 --> 00:38:41,450 our criticality condition. 710 00:38:41,450 --> 00:38:46,750 Under what conditions is the reactor critical? 711 00:38:46,750 --> 00:38:48,880 So in this case, by critical, we're 712 00:38:48,880 --> 00:38:53,380 going to have some variable called k effective, 713 00:38:53,380 --> 00:38:56,290 which defines the number of neutrons 714 00:38:56,290 --> 00:39:00,100 produced over the number of neutrons consumed. 715 00:39:00,100 --> 00:39:04,480 And if k effective equals 1, then we 716 00:39:04,480 --> 00:39:05,890 say that the reactor is critical. 717 00:39:09,470 --> 00:39:12,050 That means that exactly the number of neutrons produced 718 00:39:12,050 --> 00:39:16,850 by regular fission, nin reactions, and photofission 719 00:39:16,850 --> 00:39:19,070 equals exactly the number of neutrons 720 00:39:19,070 --> 00:39:23,180 absorbed in the anything, and that leak out. 721 00:39:23,180 --> 00:39:27,980 So let's relabel our terms in the same font that we did here. 722 00:39:27,980 --> 00:39:31,850 So this would be the fission term. 723 00:39:31,850 --> 00:39:35,350 This would be nin reactions. 724 00:39:35,350 --> 00:39:36,580 This would be photofission. 725 00:39:40,750 --> 00:39:42,010 This would be absorption. 726 00:39:44,948 --> 00:39:45,865 This would be leakage. 727 00:39:48,730 --> 00:39:51,820 How many neutrons get out of our finite boundary? 728 00:39:51,820 --> 00:39:54,560 And if you remember when we started out, 729 00:39:54,560 --> 00:39:58,450 we said we were going to make the neutron balance equation 730 00:39:58,450 --> 00:40:00,355 equal to gains minus losses. 731 00:40:04,140 --> 00:40:06,630 And through our rainbow explosion simplification, 732 00:40:06,630 --> 00:40:07,800 we've done exactly that. 733 00:40:10,490 --> 00:40:11,525 These are your gains. 734 00:40:15,138 --> 00:40:16,055 These are your losses. 735 00:40:18,770 --> 00:40:22,130 When gains minus losses equals zero, 736 00:40:22,130 --> 00:40:23,670 the reactor's in perfect balance. 737 00:40:23,670 --> 00:40:24,170 Yep? 738 00:40:24,170 --> 00:40:28,370 AUDIENCE: How does leakage come out to be negative? 739 00:40:28,370 --> 00:40:30,620 MICHAEL SHORT: Leakage comes out to be negative, 740 00:40:30,620 --> 00:40:32,400 despite the plus sign here. 741 00:40:32,400 --> 00:40:34,340 And that's actually intentional. 742 00:40:34,340 --> 00:40:36,740 That's because neutrons traveled down 743 00:40:36,740 --> 00:40:38,700 the concentration gradient. 744 00:40:38,700 --> 00:40:41,360 So let's say we're going to draw an imaginary flux 745 00:40:41,360 --> 00:40:43,460 spectrum that's going to be quite correct. 746 00:40:43,460 --> 00:40:45,710 And I'm doing all of those features for a reason. 747 00:40:45,710 --> 00:40:47,990 But let's look at the concentration gradient 748 00:40:47,990 --> 00:40:50,000 right here. 749 00:40:50,000 --> 00:40:59,280 Leakage is positive when your flux gradient is negative. 750 00:41:02,300 --> 00:41:04,850 That's why the sign is flipped right there. 751 00:41:04,850 --> 00:41:07,160 So a positive diffusion term means 752 00:41:07,160 --> 00:41:11,300 you have neutrons leaking out down a negative concentration 753 00:41:11,300 --> 00:41:15,710 gradient, because if you look at the slope here, the change in x 754 00:41:15,710 --> 00:41:17,450 is positive. 755 00:41:17,450 --> 00:41:19,970 And the change in flux is negative. 756 00:41:19,970 --> 00:41:21,590 So the slope is negative. 757 00:41:21,590 --> 00:41:23,450 Concentration gradient is negative. 758 00:41:23,450 --> 00:41:25,070 That's why the sign is the opposite 759 00:41:25,070 --> 00:41:26,698 of what you may expect. 760 00:41:26,698 --> 00:41:28,740 And the same thing goes for chemical, or gaseous, 761 00:41:28,740 --> 00:41:30,122 or any other kind of diffusion. 762 00:41:30,122 --> 00:41:31,830 I'm glad you asked, because that's always 763 00:41:31,830 --> 00:41:35,370 a point of confusion, is, why is there that plus sign? 764 00:41:35,370 --> 00:41:37,330 That's intentional And that's correct. 765 00:41:37,330 --> 00:41:37,830 Cool. 766 00:41:40,650 --> 00:41:41,370 Yeah, Shawn? 767 00:41:41,370 --> 00:41:42,910 AUDIENCE: So in that case, if you 768 00:41:42,910 --> 00:41:44,550 were to explicitly right out losses, 769 00:41:44,550 --> 00:41:48,512 would it be minus absorption, plus leakage? 770 00:41:48,512 --> 00:41:50,220 MICHAEL SHORT: Let's put some parentheses 771 00:41:50,220 --> 00:41:58,360 on here, equals zero, and a minus. 772 00:41:58,360 --> 00:42:01,910 And when we say plus leakage, we have that plus sign in there. 773 00:42:01,910 --> 00:42:03,910 So I'm not going to put any parentheses up here, 774 00:42:03,910 --> 00:42:05,590 because that wouldn't be correct. 775 00:42:05,590 --> 00:42:10,300 But what I can say is that gains minus losses 776 00:42:10,300 --> 00:42:15,610 have to be in perfect balance to have a k effective equal to 1. 777 00:42:15,610 --> 00:42:17,110 Does anyone else have any questions, 778 00:42:17,110 --> 00:42:19,496 before I continue the explanation? 779 00:42:19,496 --> 00:42:21,740 Cool. 780 00:42:21,740 --> 00:42:24,720 Let's say you're producing more neutrons 781 00:42:24,720 --> 00:42:26,630 than you're destroying. 782 00:42:26,630 --> 00:42:28,310 That's what we call supercritical. 783 00:42:32,120 --> 00:42:33,930 So I just did an interview for this K 784 00:42:33,930 --> 00:42:36,450 through 12 outreach program. 785 00:42:36,450 --> 00:42:38,730 And they said, should people be afraid 786 00:42:38,730 --> 00:42:42,090 when something, quote unquote, goes critical? 787 00:42:42,090 --> 00:42:44,140 Sounds scary emotionally, right? 788 00:42:44,140 --> 00:42:45,550 And the answer is absolutely not. 789 00:42:45,550 --> 00:42:47,370 If you're reactor goes critical, it's turned on. 790 00:42:47,370 --> 00:42:48,537 And it's in perfect balance. 791 00:42:48,537 --> 00:42:50,010 That's exactly what you want. 792 00:42:50,010 --> 00:42:52,230 So going critical is not a scary thing. 793 00:42:52,230 --> 00:42:53,850 It means we have control. 794 00:42:53,850 --> 00:42:56,790 If something goes supercritical, it doesn't necessarily 795 00:42:56,790 --> 00:42:58,560 mean it's out of control. 796 00:42:58,560 --> 00:43:01,440 Reactors can be very slightly supercritical 797 00:43:01,440 --> 00:43:03,120 and still in control, because of what's 798 00:43:03,120 --> 00:43:05,880 called delayed neutrons, which I will not introduce today, 799 00:43:05,880 --> 00:43:07,650 because we have two minutes. 800 00:43:07,650 --> 00:43:10,980 If a reactor has a k effective of less than one, 801 00:43:10,980 --> 00:43:12,300 we call that subcritical. 802 00:43:15,730 --> 00:43:18,730 So it's important to note that the nuclear terminology that's 803 00:43:18,730 --> 00:43:22,060 kind of leaked out into our vernacular 804 00:43:22,060 --> 00:43:24,970 is not physically correct, in the way that it's used. 805 00:43:24,970 --> 00:43:29,500 Words like critical are used to incite emotions, and bring 806 00:43:29,500 --> 00:43:30,160 about fear. 807 00:43:30,160 --> 00:43:32,670 When to a nuclear engineer critical means, 808 00:43:32,670 --> 00:43:35,590 in perfect control, in balance, like 809 00:43:35,590 --> 00:43:37,100 you would expect, or in equilibrium. 810 00:43:37,100 --> 00:43:38,860 That all sounds kind of nice, makes 811 00:43:38,860 --> 00:43:40,660 you calm down a little bit. 812 00:43:40,660 --> 00:43:44,490 Yeah, so we can put one last term 813 00:43:44,490 --> 00:43:48,670 in front of our criticality condition. 814 00:43:48,670 --> 00:43:53,330 We can take either the gains or the losses, move the equal sign 815 00:43:53,330 --> 00:44:00,750 and zero over a little bit, and put a 1 over k effective here. 816 00:44:00,750 --> 00:44:05,070 This, then, perfectly describes the difference 817 00:44:05,070 --> 00:44:08,610 between the gains and the losses in a reactor. 818 00:44:08,610 --> 00:44:13,050 So if the gains equal the losses, then k effective 819 00:44:13,050 --> 00:44:14,170 must equal 1. 820 00:44:14,170 --> 00:44:16,440 And the reactor has got to be in balance. 821 00:44:16,440 --> 00:44:19,390 If there are more gains than losses, 822 00:44:19,390 --> 00:44:21,930 which means if you are producing more neutrons then 823 00:44:21,930 --> 00:44:24,570 you're consuming, than k effective must 824 00:44:24,570 --> 00:44:30,600 be greater than 1 for this equation to still equal zero, 825 00:44:30,600 --> 00:44:33,530 because this equation must be satisfied. 826 00:44:33,530 --> 00:44:35,630 So if you're making more neutrons, 827 00:44:35,630 --> 00:44:38,280 your k effective has got to be greater than 1. 828 00:44:38,280 --> 00:44:41,360 So you have a less than 1 multiplier in front. 829 00:44:41,360 --> 00:44:45,380 And on the opposite side, if you're losing more neutrons 830 00:44:45,380 --> 00:44:48,890 then you're gaining, your k effective has to be less than 1 831 00:44:48,890 --> 00:44:51,680 to make this equation balanced. 832 00:44:51,680 --> 00:44:56,380 Going along with all these definitions right here. 833 00:44:56,380 --> 00:44:58,870 So it's exactly 5 of 5 of. 834 00:44:58,870 --> 00:45:02,470 I've given you delivered promised blackboard of Lucky 835 00:45:02,470 --> 00:45:03,760 Charms. 836 00:45:03,760 --> 00:45:06,400 And we've hit a perfect spot, which 837 00:45:06,400 --> 00:45:10,060 is the one group homogeneous steady state neutron diffusion 838 00:45:10,060 --> 00:45:12,250 equation, from which we can develop 839 00:45:12,250 --> 00:45:15,220 our criticality conditions and solve this much 840 00:45:15,220 --> 00:45:19,270 simpler equation to get the flux profiles that I've 841 00:45:19,270 --> 00:45:20,812 started to draw here. 842 00:45:20,812 --> 00:45:23,020 So I want to stop here, and take any questions on any 843 00:45:23,020 --> 00:45:24,450 of the terms you see here. 844 00:45:24,450 --> 00:45:24,950 Yeah? 845 00:45:24,950 --> 00:45:27,700 AUDIENCE: Didn't we talk a lot about the different energies, 846 00:45:27,700 --> 00:45:31,320 like the one-group, two-group, or the discrete distributed 847 00:45:31,320 --> 00:45:32,403 discretized energy groups? 848 00:45:32,403 --> 00:45:33,820 So when we're doing the one group, 849 00:45:33,820 --> 00:45:35,890 you're actually just treating them fast together? 850 00:45:35,890 --> 00:45:36,310 MICHAEL SHORT: We are. 851 00:45:36,310 --> 00:45:36,852 That's right. 852 00:45:36,852 --> 00:45:38,920 AUDIENCE: To know that, like you said, 853 00:45:38,920 --> 00:45:41,925 the reactors do two group in the actual analysis. 854 00:45:41,925 --> 00:45:44,050 MICHAEL SHORT: So a lot a lot of reactors, at least 855 00:45:44,050 --> 00:45:46,660 thermal reactors where you only care if neutrons are thermal 856 00:45:46,660 --> 00:45:48,820 or not, two group is enough. 857 00:45:48,820 --> 00:45:51,590 When you have a one group or a two group equation, 858 00:45:51,590 --> 00:45:54,718 these are fairly analytically solvable things. 859 00:45:54,718 --> 00:45:56,260 You get to any more groups than that, 860 00:45:56,260 --> 00:45:57,940 and, yes, they're analytically solvable. 861 00:45:57,940 --> 00:45:58,870 But it gets horrible. 862 00:45:58,870 --> 00:46:00,940 And that's why we have computers to do 863 00:46:00,940 --> 00:46:03,190 the sorts of repetitive calculations over and over 864 00:46:03,190 --> 00:46:04,210 again. 865 00:46:04,210 --> 00:46:06,800 Once we've solved the one group equation, 866 00:46:06,800 --> 00:46:10,180 I'll then show you intuitive ways to write, but not solve, 867 00:46:10,180 --> 00:46:15,280 the equations for multi group equations. 868 00:46:15,280 --> 00:46:16,280 Sure. 869 00:46:16,280 --> 00:46:17,531 Any other questions? 870 00:46:21,932 --> 00:46:23,890 So like I promised, we didn't stay complex for, 871 00:46:23,890 --> 00:46:25,740 long because there's basically nothing left. 872 00:46:25,740 --> 00:46:26,440 Yeah? 873 00:46:26,440 --> 00:46:28,930 AUDIENCE: What is the 2n over 2t? 874 00:46:28,930 --> 00:46:30,770 Are we saying that? 875 00:46:30,770 --> 00:46:33,430 MICHAEL SHORT: Oh, that's a partial derivative. 876 00:46:33,430 --> 00:46:37,520 Yeah, there we go. 877 00:46:37,520 --> 00:46:42,630 So this is saying a change in neutron population, 878 00:46:42,630 --> 00:46:45,480 or the partial derivative of n with respect to t, 879 00:46:45,480 --> 00:46:48,210 because n varies with space, energy, angle, time, 880 00:46:48,210 --> 00:46:52,070 and anything else you could possibly think about, 881 00:46:52,070 --> 00:46:53,750 equals the gains minus the losses. 882 00:46:58,590 --> 00:47:01,920 I think this is worthy of a t-shirt. 883 00:47:01,920 --> 00:47:04,170 If any of you guys would like to update the department 884 00:47:04,170 --> 00:47:07,110 shirts to properly take into account photofission 885 00:47:07,110 --> 00:47:09,840 external sources and nin reactions, 886 00:47:09,840 --> 00:47:13,860 I think it would make for a much more impressive thing, 887 00:47:13,860 --> 00:47:18,840 because we kind of printed an oversimplification before. 888 00:47:18,840 --> 00:47:19,680 It's too bad. 889 00:47:19,680 --> 00:47:21,180 We definitely had room on the shirt. 890 00:47:21,180 --> 00:47:23,180 There was room on the sides, and on the sleeves. 891 00:47:25,470 --> 00:47:26,687 Yeah, keep going. 892 00:47:26,687 --> 00:47:28,020 It might have to be long sleeve. 893 00:47:28,020 --> 00:47:29,478 I think that would be pretty sweet. 894 00:47:32,950 --> 00:47:38,080 Yeah, OK, if no one else has any immediate questions, 895 00:47:38,080 --> 00:47:39,780 you'll have plenty of time tomorrow, 896 00:47:39,780 --> 00:47:41,447 because the whole goal tomorrow is going 897 00:47:41,447 --> 00:47:42,760 to be to solve this equation. 898 00:47:42,760 --> 00:47:45,350 That's only going to take like 20 minutes. 899 00:47:45,350 --> 00:47:48,310 So we can do a quick review of the simplification 900 00:47:48,310 --> 00:47:50,140 of the neutron transport equation, 901 00:47:50,140 --> 00:47:52,355 solve the neutron diffusion equation. 902 00:47:52,355 --> 00:47:53,980 If you have questions, we'll spend time 903 00:47:53,980 --> 00:47:55,210 to answer them there. 904 00:47:55,210 --> 00:47:57,910 And if you don't, we'll move on to writing multi group 905 00:47:57,910 --> 00:47:59,200 equations. 906 00:47:59,200 --> 00:48:02,500 And also Friday for recitation, it's electron microscope time. 907 00:48:02,500 --> 00:48:05,440 So now that you guys have learned different electron 908 00:48:05,440 --> 00:48:07,962 interactions with matter, you're going to see them. 909 00:48:07,962 --> 00:48:10,420 So we're going to be analyzing a couple of different pieces 910 00:48:10,420 --> 00:48:13,717 of materials that a couple of you are going to get to select. 911 00:48:13,717 --> 00:48:15,550 And we're going to image them with electrons 912 00:48:15,550 --> 00:48:19,600 to show how you can beat the wavelength of light imaging 913 00:48:19,600 --> 00:48:21,067 limit, like I told you before. 914 00:48:21,067 --> 00:48:22,900 We're going to produce our own X-ray spectra 915 00:48:22,900 --> 00:48:24,650 to analyze them elementally, where 916 00:48:24,650 --> 00:48:25,900 you'll see the bremsstrahlung. 917 00:48:25,900 --> 00:48:27,730 You'll see the characteristic peaks. 918 00:48:27,730 --> 00:48:30,990 And you'll see a couple of other features that I'll explain too. 919 00:48:30,990 --> 00:48:34,416 So get ready for some SEM tomorrow.