1 00:00:00,840 --> 00:00:03,180 The following content is provided under a Creative 2 00:00:03,180 --> 00:00:04,570 Commons license. 3 00:00:04,570 --> 00:00:06,780 Your support will help MIT OpenCourseWare 4 00:00:06,780 --> 00:00:10,870 continue to offer high-quality educational resources for free. 5 00:00:10,870 --> 00:00:13,440 To make a donation or to view additional materials 6 00:00:13,440 --> 00:00:17,400 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,400 --> 00:00:18,280 at ocw.mit.edu. 8 00:00:22,710 --> 00:00:24,690 MICHAEL SHORT: So I want to do a quick review 9 00:00:24,690 --> 00:00:26,773 of what we did last time, because I know I threw-- 10 00:00:26,773 --> 00:00:29,550 I think we threw the full six boards of math and physics 11 00:00:29,550 --> 00:00:30,600 at you guys. 12 00:00:30,600 --> 00:00:33,240 We started off trying to describe 13 00:00:33,240 --> 00:00:34,830 this general situation. 14 00:00:34,830 --> 00:00:39,870 If you have a small nucleus 1 firing at a large nucleus 2, 15 00:00:39,870 --> 00:00:43,380 something happens, and we didn't specify what that was. 16 00:00:43,380 --> 00:00:45,690 A potentially different nucleus 3 17 00:00:45,690 --> 00:00:49,080 could come shooting off at angle theta, 18 00:00:49,080 --> 00:00:51,690 and a potentially different nucleus 4 goes off 19 00:00:51,690 --> 00:00:53,113 at a different angle phi. 20 00:00:53,113 --> 00:00:55,530 Just to warn you guys, before you start copying everything 21 00:00:55,530 --> 00:00:58,080 from the board, starting last week 22 00:00:58,080 --> 00:01:00,730 I've been taking pictures of the board at the end of class. 23 00:01:00,730 --> 00:01:03,960 So if you prefer to look and listen or just take a few notes 24 00:01:03,960 --> 00:01:05,700 rather than copy everything else down, 25 00:01:05,700 --> 00:01:08,033 I'll be taking pictures of the board at the end of class 26 00:01:08,033 --> 00:01:10,200 from now on and posting them to the Stellar site. 27 00:01:10,200 --> 00:01:12,302 So up to you how you want to do it. 28 00:01:12,302 --> 00:01:14,010 We started off with just three equations. 29 00:01:14,010 --> 00:01:16,710 We conserve mass, energy, and momentum. 30 00:01:16,710 --> 00:01:19,320 Mass and energy-- let's see-- 31 00:01:19,320 --> 00:01:21,270 come from the same equation. 32 00:01:21,270 --> 00:01:28,320 c squared plus T1 plus M2 c squared plus T2 33 00:01:28,320 --> 00:01:37,260 has to equal M3 c squared plus T3 plus M4 c squared plus T4. 34 00:01:37,260 --> 00:01:40,230 We started off making one quick assumption, 35 00:01:40,230 --> 00:01:43,260 that the nucleus 2, whatever we're firing things at, 36 00:01:43,260 --> 00:01:44,670 has no kinetic energy. 37 00:01:44,670 --> 00:01:47,115 So we can just forget that. 38 00:01:47,115 --> 00:01:48,810 What we also said is that we have 39 00:01:48,810 --> 00:01:51,660 to conserve x and y momentum. 40 00:01:51,660 --> 00:01:54,600 So if we say the x momentum of particle 1 41 00:01:54,600 --> 00:02:00,930 would be root 2 M1 T1 plus 0 for particle 2, 42 00:02:00,930 --> 00:02:04,650 because if particle 2 is not moving, it has no momentum. 43 00:02:04,650 --> 00:02:12,090 Has to equal root 2 M3 T3 cosine theta, 44 00:02:12,090 --> 00:02:15,480 because it's the x component of the momentum, 45 00:02:15,480 --> 00:02:21,740 plus root 2 M4 T4 cosine phi. 46 00:02:21,740 --> 00:02:23,880 And the last equation for y momentum-- 47 00:02:23,880 --> 00:02:29,940 we'll call this x momentum, call that mass and energy, 48 00:02:29,940 --> 00:02:33,360 call this y momentum-- 49 00:02:33,360 --> 00:02:37,050 was-- let's say there's no y momentum at the beginning 50 00:02:37,050 --> 00:02:38,350 of this equation. 51 00:02:38,350 --> 00:02:41,550 So I'll just say 0 plus 0. 52 00:02:41,550 --> 00:02:45,450 Equals the y component of particle 3's momentum, root 53 00:02:45,450 --> 00:02:52,230 2 M3 T3 sine theta, minus-- 54 00:02:52,230 --> 00:02:53,730 almost did that wrong-- because it's 55 00:02:53,730 --> 00:03:02,800 going in the opposite direction, root 2 M4 T4 sine phi. 56 00:03:02,800 --> 00:03:09,140 We did something, and we arrived at the Q equation. 57 00:03:09,140 --> 00:03:11,805 I'm trying to make sure we get to something new today. 58 00:03:11,805 --> 00:03:13,430 So the Q equation went something like-- 59 00:03:13,430 --> 00:03:16,110 and I want to make sure that I don't miswrite it at all. 60 00:03:23,650 --> 00:03:25,320 So when we refer to the Q equation, 61 00:03:25,320 --> 00:03:29,700 we're referring to this highly generalized equation relating 62 00:03:29,700 --> 00:03:31,342 all of the quantities that we see here. 63 00:03:31,342 --> 00:03:33,300 So I'm not going to go through all of the steps 64 00:03:33,300 --> 00:03:34,758 from last time, because, again, you 65 00:03:34,758 --> 00:03:36,810 have a picture of the board from last time. 66 00:03:36,810 --> 00:03:47,550 But it went Q equals T1 times M1 over M4 minus 1 plus T3 times 1 67 00:03:47,550 --> 00:04:01,348 plus M3 over M4 minus 2 root M1 M3 T1 T3 cosine theta. 68 00:04:01,348 --> 00:04:03,640 And last time we talked about which of these quantities 69 00:04:03,640 --> 00:04:05,230 are we likely to know ahead of time 70 00:04:05,230 --> 00:04:07,480 and which ones might we want to find out. 71 00:04:07,480 --> 00:04:09,460 Chances are we know all of the masses involved 72 00:04:09,460 --> 00:04:13,210 in these particles, because, well, you guys have 73 00:04:13,210 --> 00:04:15,790 been calculating that for the last 2 and 1/2 weeks or so. 74 00:04:15,790 --> 00:04:17,920 So those would be known quantities. 75 00:04:17,920 --> 00:04:22,360 We'd also know the Q value for the reaction from conservation 76 00:04:22,360 --> 00:04:24,160 of mass and energy up there. 77 00:04:24,160 --> 00:04:27,310 And we'd probably be controlling the energy of particle 1 78 00:04:27,310 --> 00:04:28,060 as it comes in. 79 00:04:28,060 --> 00:04:29,920 Either we know-- if it's a neutron, 80 00:04:29,920 --> 00:04:31,397 we know what energy it's born at. 81 00:04:31,397 --> 00:04:32,980 Or if it's coming from an accelerator, 82 00:04:32,980 --> 00:04:36,040 we crank up the voltage on the accelerator and control that. 83 00:04:36,040 --> 00:04:39,010 And that leaves us with just three quant-- 84 00:04:39,010 --> 00:04:41,080 two quantities that we don't know-- 85 00:04:41,080 --> 00:04:44,800 the kinetic energy of particle 3 and the angle 86 00:04:44,800 --> 00:04:47,030 that it comes off at. 87 00:04:47,030 --> 00:04:50,580 So this was the highly, highly generalized form. 88 00:04:50,580 --> 00:04:55,220 Recognize also that this is a quadratic equation in root 3, 89 00:04:55,220 --> 00:04:56,135 or root T3. 90 00:05:01,940 --> 00:05:09,180 And we did something else, and we 91 00:05:09,180 --> 00:05:16,140 arrived at root T3 equals s plus or minus root s 92 00:05:16,140 --> 00:05:21,700 squared plus t, where s and t-- 93 00:05:21,700 --> 00:05:22,200 let's see. 94 00:05:22,200 --> 00:05:33,980 I believe s is root M1 M3 T1 cosine theta over M3 plus M4. 95 00:05:33,980 --> 00:05:36,510 And we'll make a little bit more room. 96 00:05:36,510 --> 00:05:40,270 t should be-- damn it, got to look. 97 00:05:40,270 --> 00:05:40,770 Let's see. 98 00:05:40,770 --> 00:05:46,232 I believe it's minus M4 Q plus-- 99 00:05:46,232 --> 00:05:50,078 oh, I'll just take a quick look. 100 00:05:50,078 --> 00:05:51,620 All right, I have it open right here. 101 00:05:51,620 --> 00:05:54,141 I don't want to give you a wrong minus sign or something. 102 00:05:56,860 --> 00:05:58,110 I did have a wrong minus sign. 103 00:05:58,110 --> 00:05:58,950 Good thing I looked. 104 00:06:04,175 --> 00:06:12,160 1 times E1 over M3 M4. 105 00:06:12,160 --> 00:06:13,870 And so we started looking at, well, 106 00:06:13,870 --> 00:06:16,960 what are the implications of this solution right here? 107 00:06:16,960 --> 00:06:25,350 For exothermic reactions, where Q is greater than 0, 108 00:06:25,350 --> 00:06:28,710 any energy E1 gets this reaction to occur. 109 00:06:28,710 --> 00:06:32,220 And all that that says, well, it doesn't really say much. 110 00:06:32,220 --> 00:06:34,195 All that it really says is that E3-- 111 00:06:34,195 --> 00:06:40,140 I'm sorry-- T3-- and let me make sure that I don't use any 112 00:06:40,140 --> 00:06:42,420 sneaky E's in there-- 113 00:06:42,420 --> 00:06:46,770 plus T4 has to be greater than the incoming energy T1. 114 00:06:46,770 --> 00:06:48,840 That's the only real implication here, 115 00:06:48,840 --> 00:06:51,780 is that some of the mass from particles 1 and 2 116 00:06:51,780 --> 00:06:55,940 turned into some kinetic energy in particles 3 and 4. 117 00:06:55,940 --> 00:06:59,480 So that one's kind of the simpler case. 118 00:06:59,480 --> 00:07:06,120 For the endothermic case, where Q is less than 0, 119 00:07:06,120 --> 00:07:09,350 there's going to be some threshold energy required 120 00:07:09,350 --> 00:07:14,210 to overcome in order to get this reaction to occur. 121 00:07:14,210 --> 00:07:17,300 So where did we say? 122 00:07:17,300 --> 00:07:19,550 So, first of all, where would we go about deciding 123 00:07:19,550 --> 00:07:22,575 what is the most favorable set of conditions 124 00:07:22,575 --> 00:07:24,200 that would allow one of these reactions 125 00:07:24,200 --> 00:07:27,448 to occur by manipulating parameters in s and t? 126 00:07:27,448 --> 00:07:29,490 What's the first one that you'd start to look at? 127 00:07:35,260 --> 00:07:37,960 Well, let's start by picking the angle. 128 00:07:37,960 --> 00:07:39,610 Let's say if there was a-- if we had 129 00:07:39,610 --> 00:07:41,590 what's called forward scattering, 130 00:07:41,590 --> 00:07:45,730 then this cosine of theta equals 1. 131 00:07:45,730 --> 00:07:48,850 And that probably gives us the highest likelihood 132 00:07:48,850 --> 00:07:52,002 of a reaction happening, or the most energy gone into, 133 00:07:52,002 --> 00:07:53,710 let's say, just moving the center of mass 134 00:07:53,710 --> 00:07:57,025 and not the particles going off in different directions. 135 00:07:57,025 --> 00:07:57,525 Let's see. 136 00:08:02,440 --> 00:08:05,010 Ah, so what it really comes down to is 137 00:08:05,010 --> 00:08:08,670 a balance in making sure that this term right here, 138 00:08:08,670 --> 00:08:11,020 well, it can't go negative. 139 00:08:11,020 --> 00:08:13,440 If it goes negative, then the solution is imaginary 140 00:08:13,440 --> 00:08:15,630 and you don't have anything going on. 141 00:08:15,630 --> 00:08:17,430 So what this implies is that s squared 142 00:08:17,430 --> 00:08:23,070 plus t has to be greater or equal than 0 in order for this 143 00:08:23,070 --> 00:08:25,320 to occur. 144 00:08:25,320 --> 00:08:27,500 Otherwise, you would have, like I said, 145 00:08:27,500 --> 00:08:29,847 a complex solution to an energy. 146 00:08:29,847 --> 00:08:31,430 And energy is not going to be complex. 147 00:08:31,430 --> 00:08:34,570 That means the reaction won't occur. 148 00:08:34,570 --> 00:08:36,760 So this is where we got to last time. 149 00:08:36,760 --> 00:08:37,260 Yes. 150 00:08:37,260 --> 00:08:40,890 AUDIENCE: When you say s squared plus t is greater than 0 151 00:08:40,890 --> 00:08:43,169 or greater than or equal to 0, it's endothermic, 152 00:08:43,169 --> 00:08:45,390 wouldn't it also be greater than or equal to 0 153 00:08:45,390 --> 00:08:46,290 for an exothermic? 154 00:08:46,290 --> 00:08:47,290 MICHAEL SHORT: It would. 155 00:08:47,290 --> 00:08:48,870 But there's a condition here that-- 156 00:08:48,870 --> 00:08:49,860 lets see. 157 00:08:49,860 --> 00:08:52,710 In this case, for exothermic, Q is greater 158 00:08:52,710 --> 00:08:55,230 than 0 and that condition is always satisfied. 159 00:08:55,230 --> 00:08:58,030 For an endothermic reaction, Q is negative. 160 00:08:58,030 --> 00:09:00,310 So that's a good point. 161 00:09:00,310 --> 00:09:06,210 So if endothermic, then Q is less than 0, 162 00:09:06,210 --> 00:09:08,250 and it's all about making sure that that sum, 163 00:09:08,250 --> 00:09:10,720 s squared plus t, is not negative. 164 00:09:10,720 --> 00:09:12,480 What that means is in order to balance out 165 00:09:12,480 --> 00:09:15,360 the fact that you've got a negative Q here, 166 00:09:15,360 --> 00:09:20,610 you have to increase T1 in order to make that sum greater than 167 00:09:20,610 --> 00:09:21,810 or equal to 0. 168 00:09:21,810 --> 00:09:22,310 Yes. 169 00:09:22,310 --> 00:09:23,660 AUDIENCE: So that condition, s squared 170 00:09:23,660 --> 00:09:25,300 plus t is greater than or equal to 0, 171 00:09:25,300 --> 00:09:28,200 is that basically a condition for the endothermic reaction 172 00:09:28,200 --> 00:09:28,770 to occur? 173 00:09:28,770 --> 00:09:30,060 MICHAEL SHORT: That's correct. 174 00:09:30,060 --> 00:09:32,910 If s squared plus t is smaller than 0, which 175 00:09:32,910 --> 00:09:36,660 is to say that this whole sum right here doesn't help you 176 00:09:36,660 --> 00:09:39,220 balance out the negative Q, then the reaction 177 00:09:39,220 --> 00:09:40,650 is not going to happen. 178 00:09:40,650 --> 00:09:42,210 And something else might happen. 179 00:09:42,210 --> 00:09:46,110 So let's say you were looking at a case of inelastic scattering 180 00:09:46,110 --> 00:09:49,050 where a neutron would get absorbed by a nucleus 181 00:09:49,050 --> 00:09:51,420 and be re-emitted at a different energy level. 182 00:09:51,420 --> 00:09:54,300 If the energy is too small for that to occur, 183 00:09:54,300 --> 00:09:56,220 then the neutron is not going to get absorbed. 184 00:09:56,220 --> 00:09:59,930 Instead it might bounce off and undergo elastic scattering. 185 00:09:59,930 --> 00:10:01,410 And as a quick flash forward, I'll 186 00:10:01,410 --> 00:10:07,590 show you a quick plot of elastic and inelastic cross-sections 187 00:10:07,590 --> 00:10:09,210 that kind of hammers this home. 188 00:10:09,210 --> 00:10:11,550 You'll be looking at a lot of these plots, that 189 00:10:11,550 --> 00:10:15,030 are going to be logarithmic in energy space 190 00:10:15,030 --> 00:10:20,450 and probably logarithmic in microscopic cross-section, 191 00:10:20,450 --> 00:10:23,053 to bring back that variable from before. 192 00:10:23,053 --> 00:10:24,470 If you remember, the cross-section 193 00:10:24,470 --> 00:10:26,640 is like the probability that a certain reaction 194 00:10:26,640 --> 00:10:27,860 is going to occur. 195 00:10:27,860 --> 00:10:30,080 The larger the cross-section, the higher 196 00:10:30,080 --> 00:10:33,650 the reaction rate for a given flux of particles. 197 00:10:33,650 --> 00:10:36,300 And let's say we'll split this into two things. 198 00:10:36,300 --> 00:10:40,610 We'll call it sigma elastic and sigma inelastic. 199 00:10:40,610 --> 00:10:42,230 And we'll give them-- 200 00:10:42,230 --> 00:10:44,070 that will be-- oh, we have colors. 201 00:10:44,070 --> 00:10:45,200 Let's just use those. 202 00:10:45,200 --> 00:10:46,970 Even better. 203 00:10:46,970 --> 00:10:48,560 Where's my second color? 204 00:10:48,560 --> 00:10:49,730 Under the paper. 205 00:10:49,730 --> 00:10:51,410 Awesome. 206 00:10:51,410 --> 00:10:53,920 So let's say the elastic cross-section is in red, 207 00:10:53,920 --> 00:10:56,540 and the inelastic cross-section is in green. 208 00:10:56,540 --> 00:11:03,140 And for white, we'll plot sigma total. 209 00:11:03,140 --> 00:11:05,240 Usually, one of these cross-sections for any old 210 00:11:05,240 --> 00:11:05,842 interaction-- 211 00:11:05,842 --> 00:11:07,550 I'm not even being specific on which one. 212 00:11:07,550 --> 00:11:10,310 Let's just say neutrons hitting something big-- 213 00:11:10,310 --> 00:11:12,050 would tend to look like this. 214 00:11:12,050 --> 00:11:15,750 There will be some insanity here, that we'll discuss, 215 00:11:15,750 --> 00:11:18,033 and it might start to increase a little bit 216 00:11:18,033 --> 00:11:19,200 as it goes to high energies. 217 00:11:19,200 --> 00:11:23,070 And this is definitely not to scale. 218 00:11:23,070 --> 00:11:26,340 Just for the purposes of illustration. 219 00:11:26,340 --> 00:11:28,830 The elastic cross-section is going to look something 220 00:11:28,830 --> 00:11:33,183 like exactly this. 221 00:11:33,183 --> 00:11:34,850 See how closely I can draw on top of it. 222 00:11:39,660 --> 00:11:42,300 So at low energies, when a neutron 223 00:11:42,300 --> 00:11:45,270 can't be absorbed and re-emitted at a different energy, 224 00:11:45,270 --> 00:11:48,690 the inelastic scattering process can't happen. 225 00:11:48,690 --> 00:11:51,220 Which is to say that the incoming kinetic energy-- 226 00:11:51,220 --> 00:11:54,720 so this log of E right here, better known 227 00:11:54,720 --> 00:11:57,360 as T1 in the symbols up there-- 228 00:11:57,360 --> 00:12:00,562 it's not high enough to allow inelastic scattering to occur. 229 00:12:00,562 --> 00:12:02,520 So if we want to graph the inelastic scattering 230 00:12:02,520 --> 00:12:06,290 cross-section, it will typically look 231 00:12:06,290 --> 00:12:11,570 like that, where once you reach your threshold 232 00:12:11,570 --> 00:12:16,390 energy, determined by that condition there, 233 00:12:16,390 --> 00:12:18,790 then the inelastic scattering turns on 234 00:12:18,790 --> 00:12:22,460 and it's actually able to proceed. 235 00:12:22,460 --> 00:12:24,960 So this is why we're getting into these threshold reactions, 236 00:12:24,960 --> 00:12:26,840 because it helps you understand why 237 00:12:26,840 --> 00:12:28,970 do some of the cross-sections that we study 238 00:12:28,970 --> 00:12:30,980 have the shapes that they do. 239 00:12:30,980 --> 00:12:33,800 And this holds true for pretty much 240 00:12:33,800 --> 00:12:36,140 every inelastic cross-section I've seen, 241 00:12:36,140 --> 00:12:37,940 is they all-- almost all of them, 242 00:12:37,940 --> 00:12:39,565 if you're starting at the ground state, 243 00:12:39,565 --> 00:12:42,050 require some initial energy input to get going. 244 00:12:42,050 --> 00:12:44,930 Whereas elastic scattering can happen at any energy. 245 00:12:44,930 --> 00:12:47,570 So let's write that last condition right here. 246 00:12:50,870 --> 00:12:56,750 Elastic scattering, which means things just 247 00:12:56,750 --> 00:13:01,490 bounce off like billiard balls, you have Q equals 0. 248 00:13:01,490 --> 00:13:05,210 No energy changes hands, so to speak. 249 00:13:05,210 --> 00:13:07,190 You just get some kinetic energy from 1 250 00:13:07,190 --> 00:13:11,870 being imparted to nucleus 2, but you don't turn any mass 251 00:13:11,870 --> 00:13:12,590 into energy. 252 00:13:15,180 --> 00:13:17,330 So any questions here before we go into-- 253 00:13:17,330 --> 00:13:17,940 yes. 254 00:13:17,940 --> 00:13:19,880 AUDIENCE: If theta isn't 0 for cosine theta, 255 00:13:19,880 --> 00:13:20,922 how would you plug it in? 256 00:13:20,922 --> 00:13:24,433 If you don't know what the angle is, that it's not [INAUDIBLE]?? 257 00:13:24,433 --> 00:13:25,850 MICHAEL SHORT: So the question is, 258 00:13:25,850 --> 00:13:28,400 if you don't know the angle, what do you do about it, right? 259 00:13:28,400 --> 00:13:31,140 If you don't-- so in this case, we've said, 260 00:13:31,140 --> 00:13:33,330 what is the bare minimum threshold for this reaction 261 00:13:33,330 --> 00:13:36,860 to occur, and the best way for that to happen is for theta 262 00:13:36,860 --> 00:13:38,040 to equal 0. 263 00:13:38,040 --> 00:13:39,752 If theta is larger, that will actually 264 00:13:39,752 --> 00:13:42,210 mean that the reaction is not allowed to proceed unless you 265 00:13:42,210 --> 00:13:44,110 get to an even higher energy. 266 00:13:44,110 --> 00:13:46,290 So this condition still holds. 267 00:13:46,290 --> 00:13:53,140 But if cosine-- so let's say if cosine theta is less than 1, 268 00:13:53,140 --> 00:13:58,410 then the value of s goes down, and that makes this condition 269 00:13:58,410 --> 00:14:01,550 harder to satisfy. 270 00:14:01,550 --> 00:14:03,140 So that's a good question. 271 00:14:03,140 --> 00:14:06,350 What that actually means is that for certain nuclear reactions 272 00:14:06,350 --> 00:14:08,060 very close to the threshold energy, 273 00:14:08,060 --> 00:14:09,760 only certain angles are allowed. 274 00:14:09,760 --> 00:14:11,510 I'm not going to get into the nitty-gritty 275 00:14:11,510 --> 00:14:12,677 of which angles are allowed. 276 00:14:12,677 --> 00:14:13,940 I think it's-- 277 00:14:13,940 --> 00:14:16,070 I'll call it minutia for the scope of this class, 278 00:14:16,070 --> 00:14:17,840 but it is in the Yip reading, which 279 00:14:17,840 --> 00:14:19,960 I'll be posting pretty soon. 280 00:14:19,960 --> 00:14:23,590 But suffice to say that the only time you can-- let's say 281 00:14:23,590 --> 00:14:26,050 if s squared plus t equals 0. 282 00:14:26,050 --> 00:14:28,600 The only time that can happen is when theta 283 00:14:28,600 --> 00:14:30,700 equals 0 or cosine equals 1. 284 00:14:30,700 --> 00:14:34,240 And that means that the nucleus can only recoil in a very, very 285 00:14:34,240 --> 00:14:36,130 narrow cone forward. 286 00:14:36,130 --> 00:14:39,550 As that energy increases, the allowable angles 287 00:14:39,550 --> 00:14:42,018 start to increase further. 288 00:14:42,018 --> 00:14:43,310 Does that answer your question? 289 00:14:43,310 --> 00:14:43,720 AUDIENCE: Well, yes. 290 00:14:43,720 --> 00:14:45,095 So you're just saying [INAUDIBLE] 291 00:14:45,095 --> 00:14:48,500 0 is the minimum [INAUDIBLE]. 292 00:14:48,500 --> 00:14:51,310 MICHAEL SHORT: Or you'd say-- let's say if cosine equals 180. 293 00:14:51,310 --> 00:14:52,510 Then-- I'm sorry. 294 00:14:52,510 --> 00:14:54,680 If theta equals 180, cosine would be negative 1. 295 00:14:54,680 --> 00:14:59,330 And that would be, let's say, the least favorable condition. 296 00:14:59,330 --> 00:15:00,404 Yes. 297 00:15:00,404 --> 00:15:04,180 AUDIENCE: Why did you put the sigma total 298 00:15:04,180 --> 00:15:07,490 under sigma elastic? 299 00:15:07,490 --> 00:15:09,920 MICHAEL SHORT: Oh, I'm saying-- so sigma elastic plus-- 300 00:15:09,920 --> 00:15:12,260 sorry-- sigma inelastic would, let's say, 301 00:15:12,260 --> 00:15:14,728 give you the total scattering cross-section. 302 00:15:14,728 --> 00:15:16,520 AUDIENCE: And then what was the green line? 303 00:15:16,520 --> 00:15:19,460 MICHAEL SHORT: The green line-- oh, it is a little hard to see. 304 00:15:19,460 --> 00:15:22,620 The green line is the inelastic cross-section. 305 00:15:22,620 --> 00:15:23,120 Yes. 306 00:15:23,120 --> 00:15:25,280 I can imagine from back there the green 307 00:15:25,280 --> 00:15:27,620 and the white might look a little similar, yes. 308 00:15:27,620 --> 00:15:29,280 OK, cool. 309 00:15:29,280 --> 00:15:29,780 Yes. 310 00:15:29,780 --> 00:15:31,820 Like one of these, they give they give me 311 00:15:31,820 --> 00:15:34,130 an almost black one. 312 00:15:34,130 --> 00:15:36,110 That's about as invisible as it gets. 313 00:15:36,110 --> 00:15:39,020 So make sure to use visible colors. 314 00:15:39,020 --> 00:15:39,950 OK. 315 00:15:39,950 --> 00:15:44,782 So now let's take the case of elastic neutron scattering. 316 00:15:52,500 --> 00:15:56,090 And can anyone tell me how can we simplify the general Q 317 00:15:56,090 --> 00:15:58,610 equation for the case of neutrons 318 00:15:58,610 --> 00:16:00,910 hitting some random nucleus? 319 00:16:00,910 --> 00:16:03,160 What can we start plugging in for some of those values 320 00:16:03,160 --> 00:16:03,952 to make it simpler? 321 00:16:08,073 --> 00:16:08,990 What about the masses? 322 00:16:08,990 --> 00:16:13,505 What's M1 in atomic mass units? 323 00:16:13,505 --> 00:16:14,005 AUDIENCE: 1. 324 00:16:14,005 --> 00:16:17,345 MICHAEL SHORT: M1 is just 1 to a pretty good approximation. 325 00:16:17,345 --> 00:16:22,940 It's actually 1.0087, which we're going to say is 1. 326 00:16:22,940 --> 00:16:25,770 And what about M3? 327 00:16:25,770 --> 00:16:26,640 AUDIENCE: 1. 328 00:16:26,640 --> 00:16:29,010 MICHAEL SHORT: Is also-- yes, also 1. 329 00:16:29,010 --> 00:16:30,660 If this is elastic neutron scattering, 330 00:16:30,660 --> 00:16:33,000 the same neutron goes in and goes out. 331 00:16:33,000 --> 00:16:36,410 So M1 and M3 are the same thing. 332 00:16:36,410 --> 00:16:41,670 What about M2 and M4? 333 00:16:41,670 --> 00:16:45,320 Have we specified what this nucleus is? 334 00:16:45,320 --> 00:16:49,880 So what mass would we give it if it's a general nucleus with N 335 00:16:49,880 --> 00:16:51,160 neutrons and Z protons? 336 00:16:51,160 --> 00:16:51,660 AUDIENCE: A. 337 00:16:51,660 --> 00:16:54,200 MICHAEL SHORT: A, sure. 338 00:16:54,200 --> 00:16:56,630 So A, that's again A for the mass number. 339 00:17:00,620 --> 00:17:01,790 Cool. 340 00:17:01,790 --> 00:17:04,490 So with those in mind, and then the last thing is 341 00:17:04,490 --> 00:17:06,980 we only have T1 and T3. 342 00:17:06,980 --> 00:17:10,369 Just for clarity, let's call T1 Tin, like the neutron energy 343 00:17:10,369 --> 00:17:11,599 going in. 344 00:17:11,599 --> 00:17:14,450 And T3, we'll call Tout. 345 00:17:14,450 --> 00:17:20,960 So let's rewrite the Q equation, the Q-eq, 346 00:17:20,960 --> 00:17:22,819 with these symbols in there. 347 00:17:22,819 --> 00:17:24,829 And the last thing to note, what's 348 00:17:24,829 --> 00:17:28,650 Q for elastic scattering? 349 00:17:28,650 --> 00:17:30,220 0, yes. 350 00:17:30,220 --> 00:17:32,750 Because no mass is turned into energy or vice versa. 351 00:17:32,750 --> 00:17:37,180 So to rewrite the whole Q equation, we'll get 0 352 00:17:37,180 --> 00:17:45,520 equals Tin times M1 over M4, which is just 1 over A, 353 00:17:45,520 --> 00:17:55,600 minus 1 plus Tout times 1 plus M3 over M4. 354 00:17:55,600 --> 00:18:04,670 M3 is also 1, M4 is also A. And minus 2 root M1 M3. 355 00:18:04,670 --> 00:18:07,160 They're both 1. 356 00:18:07,160 --> 00:18:13,690 Tin Tout cosine theta. 357 00:18:13,690 --> 00:18:15,570 So this looks a whole lot simpler. 358 00:18:15,570 --> 00:18:18,480 I'm going to do one quick thing right here and take 359 00:18:18,480 --> 00:18:22,440 the minus sign that's hiding in here outside this equation. 360 00:18:22,440 --> 00:18:26,417 It's going to make the form a lot simpler. 361 00:18:26,417 --> 00:18:28,500 So I'm just multiplying the inside and the outside 362 00:18:28,500 --> 00:18:30,375 of this term by negative 1, but hopefully you 363 00:18:30,375 --> 00:18:31,920 can see that it's the same thing. 364 00:18:31,920 --> 00:18:34,250 It will just make the form a lot nicer in the end. 365 00:18:34,250 --> 00:18:35,850 And so now we want to start asking, 366 00:18:35,850 --> 00:18:39,120 what is the maximum and minimum energy 367 00:18:39,120 --> 00:18:41,895 that the neutron can lose? 368 00:18:41,895 --> 00:18:43,270 So let's start with the easy one. 369 00:18:43,270 --> 00:18:45,118 What is the minimum amount of energy 370 00:18:45,118 --> 00:18:46,285 that the neutron could lose? 371 00:18:50,510 --> 00:18:51,010 Anyone? 372 00:18:51,010 --> 00:18:52,420 I hear some whispers. 373 00:18:52,420 --> 00:18:53,260 AUDIENCE: 0. 374 00:18:53,260 --> 00:18:54,290 MICHAEL SHORT: 0. 375 00:18:54,290 --> 00:18:57,490 And if the neutron comes in-- 376 00:18:57,490 --> 00:19:02,350 if theta equals 0, then you end up 377 00:19:02,350 --> 00:19:08,230 with actually Tin will equal Tout. 378 00:19:08,230 --> 00:19:13,720 And, that way, let's say delta T1 or delta T neutron 379 00:19:13,720 --> 00:19:14,920 could equal 0. 380 00:19:14,920 --> 00:19:19,420 So a neutron can lose at least none of its energy 381 00:19:19,420 --> 00:19:20,810 in an elastic collision. 382 00:19:20,810 --> 00:19:22,310 Hopefully that makes intuitive sense 383 00:19:22,310 --> 00:19:24,530 because we would call that a miss. 384 00:19:24,530 --> 00:19:27,950 Now let's take the other case. 385 00:19:27,950 --> 00:19:30,680 At what angle would you think the neutron 386 00:19:30,680 --> 00:19:36,590 would transfer as much energy as possible to the recoil nucleus? 387 00:19:36,590 --> 00:19:39,360 So if we have a big nucleus of mass A 388 00:19:39,360 --> 00:19:46,230 and we have a little neutron firing at it, at which angle 389 00:19:46,230 --> 00:19:47,850 does it transfer the most energy? 390 00:19:51,730 --> 00:19:52,230 Yes. 391 00:19:52,230 --> 00:19:52,930 AUDIENCE: Pi? 392 00:19:52,930 --> 00:19:53,812 If it's like-- 393 00:19:53,812 --> 00:19:54,770 MICHAEL SHORT: Exactly. 394 00:19:54,770 --> 00:19:56,610 AUDIENCE: [INAUDIBLE]. 395 00:19:56,610 --> 00:19:59,960 MICHAEL SHORT: At theta equals pi, which means this-- 396 00:19:59,960 --> 00:20:01,720 we call this backscattering. 397 00:20:05,480 --> 00:20:08,530 So, yes, good one. 398 00:20:08,530 --> 00:20:10,030 I'll correct your statement, though. 399 00:20:10,030 --> 00:20:11,870 You said if the neutron just stopped 400 00:20:11,870 --> 00:20:13,550 and the nucleus moved forward. 401 00:20:13,550 --> 00:20:16,300 Does not happen in every case. 402 00:20:16,300 --> 00:20:18,640 For example, if you were to-- and I'd say don't try this 403 00:20:18,640 --> 00:20:19,630 at home, kids-- 404 00:20:19,630 --> 00:20:23,800 put on a nice helmet and run charging at a truck, 405 00:20:23,800 --> 00:20:25,750 can you actually just stop cold? 406 00:20:25,750 --> 00:20:28,180 And we're not assuming any bones breaking or anything. 407 00:20:28,180 --> 00:20:30,080 Chances are you'd bounce right back off. 408 00:20:30,080 --> 00:20:30,580 Yes. 409 00:20:30,580 --> 00:20:33,040 That's the analogy I like to give for what happens when 410 00:20:33,040 --> 00:20:34,930 a neutron scatters off uranium. 411 00:20:34,930 --> 00:20:37,160 It's like running at a truck with a helmet on. 412 00:20:37,160 --> 00:20:38,940 It will just bounce back. 413 00:20:38,940 --> 00:20:42,670 So in the case of theta equals pi-- 414 00:20:42,670 --> 00:20:47,760 so we're going to substitute in theta equals pi. 415 00:20:47,760 --> 00:20:51,926 Therefore, cosine theta equals negative 1, 416 00:20:51,926 --> 00:20:54,880 and we have an even easier equation. 417 00:20:54,880 --> 00:21:03,610 0 equals, let's just say, Tout times 1 plus 1 over A. 418 00:21:03,610 --> 00:21:05,770 I'm going to arrange these terms in order 419 00:21:05,770 --> 00:21:10,120 of their exponent for Tout since that's our variable again. 420 00:21:10,120 --> 00:21:15,460 And if this stuff is negative 1, then the 2 minus signs 421 00:21:15,460 --> 00:21:18,890 cancel conveniently. 422 00:21:18,890 --> 00:21:25,550 And we have plus 2 root Tin Tout. 423 00:21:25,550 --> 00:21:26,050 Let's see. 424 00:21:26,050 --> 00:21:27,270 That's it. 425 00:21:27,270 --> 00:21:38,530 And we have minus Tin times 1 over 1 minus A. 426 00:21:38,530 --> 00:21:40,960 Ideally, we'd like to try to simplify this as much 427 00:21:40,960 --> 00:21:42,580 as possible. 428 00:21:42,580 --> 00:21:44,213 So let's combine. 429 00:21:44,213 --> 00:21:45,880 Let's try to get everything in some sort 430 00:21:45,880 --> 00:21:48,070 of a common denominator, because that 431 00:21:48,070 --> 00:21:49,720 would make things a lot easier. 432 00:21:49,720 --> 00:21:53,830 If we multiply each of these 1's by A over A-- 433 00:21:53,830 --> 00:22:03,970 so let's put that as a step, because we can totally do 434 00:22:03,970 --> 00:22:05,770 that-- 435 00:22:05,770 --> 00:22:12,220 we get 0 over Tout times A over A plus 1 436 00:22:12,220 --> 00:22:24,250 over A plus 2 root Tin Tout minus Tin times A 437 00:22:24,250 --> 00:22:28,270 over A minus 1 over A. 438 00:22:28,270 --> 00:22:29,770 At this point, we can-- 439 00:22:29,770 --> 00:22:32,740 well, everything's in common terms, right? 440 00:22:32,740 --> 00:22:37,480 We can just extend that fraction sign and put the sign in here. 441 00:22:37,480 --> 00:22:40,930 Extend the fraction sign, put the sign in here. 442 00:22:43,882 --> 00:22:49,290 And we'll just say that's A. We'll say that's A. 443 00:22:49,290 --> 00:22:52,110 Last step that we'll do is try and isolate Tout 444 00:22:52,110 --> 00:22:54,840 so at least one of our quadratic factors is going to be simple, 445 00:22:54,840 --> 00:22:56,690 like 1. 446 00:22:56,690 --> 00:23:04,570 So next step, divide by A plus 1. 447 00:23:04,570 --> 00:23:10,810 And then we get 0 equals just Tout plus 2 over A plus 448 00:23:10,810 --> 00:23:26,330 one root Tin Tout minus Tin times A minus 1 over A plus 1. 449 00:23:26,330 --> 00:23:29,150 Now we've got a simple-looking quadratic equation, 450 00:23:29,150 --> 00:23:31,490 even though it's quadratic in the square root of Tout. 451 00:23:31,490 --> 00:23:31,990 Yes. 452 00:23:31,990 --> 00:23:35,368 AUDIENCE: What happened to the A from the denominator? 453 00:23:35,368 --> 00:23:36,410 MICHAEL SHORT: Let's see. 454 00:23:36,410 --> 00:23:37,827 AUDIENCE: Could it be Tout over A? 455 00:23:42,062 --> 00:23:43,270 MICHAEL SHORT: What did I do? 456 00:23:43,270 --> 00:23:45,320 Did I miss an A or dividing by A? 457 00:23:45,320 --> 00:23:46,760 AUDIENCE: The last two equations. 458 00:23:46,760 --> 00:23:47,870 MICHAEL SHORT: It's from back here? 459 00:23:47,870 --> 00:23:48,360 AUDIENCE: No, no, no. 460 00:23:48,360 --> 00:23:49,880 It's probably the step you just did. 461 00:23:49,880 --> 00:23:51,260 MICHAEL SHORT: Just these steps. 462 00:23:51,260 --> 00:23:53,480 AUDIENCE: So you divide by A plus 1. 463 00:23:53,480 --> 00:23:54,980 MICHAEL SHORT: Ah, I see. 464 00:23:54,980 --> 00:23:56,720 AUDIENCE: Should it be Tout over A? 465 00:23:56,720 --> 00:23:57,565 [INTERPOSING VOICES] 466 00:23:57,565 --> 00:23:58,940 MICHAEL SHORT: Yes, you're right. 467 00:23:58,940 --> 00:24:00,470 So I want to make sure I didn't skip 468 00:24:00,470 --> 00:24:02,180 a step in dividing an A. Let me just 469 00:24:02,180 --> 00:24:04,172 check something real quick. 470 00:24:04,172 --> 00:24:05,630 AUDIENCE: There should've been an A 471 00:24:05,630 --> 00:24:08,977 in the minus 2 square root. 472 00:24:08,977 --> 00:24:10,310 MICHAEL SHORT: Oh, you're right. 473 00:24:10,310 --> 00:24:14,300 If we go back to our Q equation-- 474 00:24:14,300 --> 00:24:16,190 let's see. 475 00:24:16,190 --> 00:24:19,130 There's an M4 missing, isn't there? 476 00:24:19,130 --> 00:24:21,296 That's it. 477 00:24:21,296 --> 00:24:22,220 Hah. 478 00:24:22,220 --> 00:24:26,150 See, this is what happens when you don't look at your notes. 479 00:24:26,150 --> 00:24:28,550 I'll go back and correct those, because then there 480 00:24:28,550 --> 00:24:35,240 should have been an over A. There should have been an over 481 00:24:35,240 --> 00:24:38,300 A. There should have been an over A. 482 00:24:38,300 --> 00:24:40,710 Thank you for pointing that out. 483 00:24:40,710 --> 00:24:43,100 And there should have been another-- 484 00:24:43,100 --> 00:24:47,190 oh, in this case we can just cancel all of the A's. 485 00:24:47,190 --> 00:24:48,750 I knew it came out nice and clean. 486 00:24:48,750 --> 00:24:51,150 OK, cool. 487 00:24:51,150 --> 00:24:56,360 So at this point, this is a quadratic in root 488 00:24:56,360 --> 00:25:03,350 Tout, where we have-- 489 00:25:03,350 --> 00:25:07,590 what are our a, b, and c terms for this quadratic formula? 490 00:25:07,590 --> 00:25:10,130 So what's a first of all if it's quadratic in root Tout? 491 00:25:14,597 --> 00:25:15,550 AUDIENCE: 1? 492 00:25:15,550 --> 00:25:16,482 MICHAEL SHORT: Just 1. 493 00:25:16,482 --> 00:25:18,440 That was part of the goal of this manipulation, 494 00:25:18,440 --> 00:25:21,440 is to make at least one of these things pretty simple. 495 00:25:21,440 --> 00:25:22,010 What's b? 496 00:25:25,530 --> 00:25:29,048 AUDIENCE: 2 over A plus 1 times radical Tin? 497 00:25:29,048 --> 00:25:29,840 MICHAEL SHORT: Yes. 498 00:25:29,840 --> 00:25:32,610 2 root Tin over A plus 1. 499 00:25:32,610 --> 00:25:34,590 And c is just that whole term right there. 500 00:25:40,440 --> 00:25:42,300 I'll do this up here. 501 00:25:42,300 --> 00:25:46,710 So then we can say root Tout equals negative 502 00:25:46,710 --> 00:25:55,360 b plus or minus the square root of b squared. 503 00:25:55,360 --> 00:25:58,110 So that's 4 Tin over A plus 1 squared. 504 00:26:03,340 --> 00:26:08,140 Minus 4 times a times c, so just minus 4 times c. 505 00:26:08,140 --> 00:26:18,820 So minus 4 times negative Tin A minus 1 over A plus 1. 506 00:26:18,820 --> 00:26:20,420 So let's see what cancels. 507 00:26:20,420 --> 00:26:24,730 So, first of all, those minus signs cancel. 508 00:26:24,730 --> 00:26:27,877 And everything has-- oh, and over 2a. 509 00:26:27,877 --> 00:26:28,960 Don't want to forget that. 510 00:26:28,960 --> 00:26:32,620 Over 2a, which is just 2. 511 00:26:32,620 --> 00:26:36,340 First thing we note is that everything here has a 2 in it, 512 00:26:36,340 --> 00:26:40,850 either directly as a 2 or hiding as a square root of 4. 513 00:26:40,850 --> 00:26:42,930 So we can cancel all of those. 514 00:26:42,930 --> 00:26:46,736 4, 4, 4, 4. 515 00:26:46,736 --> 00:26:50,565 Let me make sure that minus sign is nice and visible. 516 00:26:50,565 --> 00:26:52,190 What else is common to everything here? 517 00:27:00,748 --> 00:27:01,790 Well, I'll tell you what. 518 00:27:01,790 --> 00:27:04,160 I'll write it all out simpler without all 519 00:27:04,160 --> 00:27:06,200 the crossed-out stuff. 520 00:27:06,200 --> 00:27:12,620 Minus root Tin over A plus 1 plus or minus 521 00:27:12,620 --> 00:27:26,380 root Tin over A plus 1 squared plus Tin times A minus 1 522 00:27:26,380 --> 00:27:28,330 over A plus 1. 523 00:27:28,330 --> 00:27:30,220 So with that written a little simpler, 524 00:27:30,220 --> 00:27:33,188 what's also common and can be factored out of everything? 525 00:27:36,540 --> 00:27:37,870 AUDIENCE: Square root of Ti? 526 00:27:37,870 --> 00:27:39,037 MICHAEL SHORT: That's right. 527 00:27:39,037 --> 00:27:40,240 Square root of Tin. 528 00:27:40,240 --> 00:27:43,232 Because there's a root Tin here, and then you can-- 529 00:27:43,232 --> 00:27:45,190 everything's got a Tin inside the square roots. 530 00:27:45,190 --> 00:27:47,590 You can pull that out. 531 00:27:47,590 --> 00:27:50,770 So we have a direct relation between root Tout and root Tin. 532 00:27:54,490 --> 00:27:58,660 Minus 1 over A plus 1 plus or minus 533 00:27:58,660 --> 00:28:10,180 root 1 over A plus 1 squared plus A minus 1 over A plus 1. 534 00:28:10,180 --> 00:28:12,568 What do we do here to simplify all the junk 535 00:28:12,568 --> 00:28:13,360 in the square root? 536 00:28:21,690 --> 00:28:25,273 AUDIENCE: Multiply the right side by A plus 1 over A plus 1. 537 00:28:25,273 --> 00:28:26,440 MICHAEL SHORT: That's right. 538 00:28:26,440 --> 00:28:29,700 You can always multiply by something, better known as 1. 539 00:28:29,700 --> 00:28:31,660 And that gets everything here-- just 540 00:28:31,660 --> 00:28:35,380 like there was a 2 or a root 4 everywhere in the equation, 541 00:28:35,380 --> 00:28:37,870 or there was a root Tin and a root Tin 542 00:28:37,870 --> 00:28:39,490 everywhere else in the equation, we'll 543 00:28:39,490 --> 00:28:42,430 do the same thing to get the A plus 1 out of there. 544 00:28:42,430 --> 00:28:48,078 So we'll multiply this by A plus 1 over A plus 1. 545 00:28:48,078 --> 00:28:49,120 I'll stick it over there. 546 00:28:53,020 --> 00:28:54,190 OK. 547 00:28:54,190 --> 00:28:58,660 And we get root Tout equals-- 548 00:28:58,660 --> 00:29:03,400 now everything has an A plus 1, so let's bring all 549 00:29:03,400 --> 00:29:05,875 of those outside the fraction. 550 00:29:11,780 --> 00:29:18,740 Root Tin over A plus 1 times negative 1 551 00:29:18,740 --> 00:29:27,070 plus or minus the square root of 1 plus A minus 1, A plus 1. 552 00:29:27,070 --> 00:29:29,330 Starting to get a lot simpler. 553 00:29:29,330 --> 00:29:32,250 Let's see how much-- if I run out of space for this one. 554 00:29:32,250 --> 00:29:36,020 So this stuff right here is just A squared 555 00:29:36,020 --> 00:29:40,180 minus A plus A minus 1. 556 00:29:40,180 --> 00:29:44,390 The minus A and the plus A cancel out. 557 00:29:44,390 --> 00:29:47,580 And then the plus 1 and the minus 1 cancel out. 558 00:29:47,580 --> 00:29:50,480 And all that's inside the square root is A squared. 559 00:29:50,480 --> 00:29:53,600 So the only hopefully nonlinear board technique, 560 00:29:53,600 --> 00:29:55,470 I'm going to move to the left. 561 00:29:55,470 --> 00:30:04,010 And we end up with root Tout equals root Tin over A plus 1. 562 00:30:04,010 --> 00:30:08,380 And all that's left there is A minus 1 563 00:30:08,380 --> 00:30:10,390 if we take the positive root. 564 00:30:14,450 --> 00:30:16,840 Almost done. 565 00:30:16,840 --> 00:30:18,318 Just square both sides. 566 00:30:18,318 --> 00:30:20,110 And we should arrive at a result that might 567 00:30:20,110 --> 00:30:22,480 look familiar to some of you. 568 00:30:22,480 --> 00:30:32,790 Tout equals Tin times A minus 1 over A plus 1 squared. 569 00:30:32,790 --> 00:30:34,850 And we've gotten to the point now where 570 00:30:34,850 --> 00:30:38,780 we can determine how much energy the neutron can possibly lose 571 00:30:38,780 --> 00:30:40,610 or the recoil nucleus can possibly 572 00:30:40,610 --> 00:30:43,680 gain in an elastic collision. 573 00:30:43,680 --> 00:30:45,650 It's this factor right here. 574 00:30:45,650 --> 00:30:48,620 I'll use the red since it's more visible. 575 00:30:48,620 --> 00:30:54,860 This is usually referred to in nuclear textbooks as alpha. 576 00:30:54,860 --> 00:30:58,860 It's sort of the maximum amount of energy a neutron can lose 577 00:30:58,860 --> 00:31:01,450 or a recoil nucleus can gain. 578 00:31:01,450 --> 00:31:07,380 So what we've arrived at is a pretty important result, 579 00:31:07,380 --> 00:31:13,730 that, let's say, the energy, the kinetic energy of a neutron, 580 00:31:13,730 --> 00:31:20,600 has to be between its initial kinetic energy and alpha times 581 00:31:20,600 --> 00:31:22,610 its initial kinetic energy. 582 00:31:22,610 --> 00:31:24,890 This right here is one of the ways in which you 583 00:31:24,890 --> 00:31:27,470 choose a moderator or a slowing down 584 00:31:27,470 --> 00:31:30,420 medium for neutrons in reactors. 585 00:31:30,420 --> 00:31:32,880 So it's this alpha factor right here 586 00:31:32,880 --> 00:31:35,230 that really distinguishes what we call a thermal-- 587 00:31:35,230 --> 00:31:36,240 or what is it? 588 00:31:36,240 --> 00:31:38,310 Like a light water reactor or a thermal spectrum 589 00:31:38,310 --> 00:31:41,610 reactor from a fast spectrum reactor. 590 00:31:41,610 --> 00:31:45,020 Let's look at a couple of limiting cases to see why. 591 00:31:45,020 --> 00:31:45,920 Let's see. 592 00:31:45,920 --> 00:31:47,600 Anyone mind if I hide this board here? 593 00:31:47,600 --> 00:31:48,290 Or you have a question? 594 00:31:48,290 --> 00:31:48,873 AUDIENCE: Yes. 595 00:31:48,873 --> 00:31:53,030 Can you explain why you ended up dropping the negative case? 596 00:31:53,030 --> 00:31:54,080 MICHAEL SHORT: Let's see. 597 00:31:54,080 --> 00:31:58,820 If we took the negative case, we'd end up with minus 1 598 00:31:58,820 --> 00:32:05,010 minus A. You just have an A plus 1 on the top. 599 00:32:05,010 --> 00:32:05,510 Yes. 600 00:32:05,510 --> 00:32:07,060 So in that case, you just have-- 601 00:32:07,060 --> 00:32:08,790 let's see. 602 00:32:08,790 --> 00:32:10,040 You just have root Tin, right? 603 00:32:12,930 --> 00:32:13,430 Let me see. 604 00:32:13,430 --> 00:32:15,472 AUDIENCE: Negative root Tin actually. 605 00:32:15,472 --> 00:32:16,430 MICHAEL SHORT: Oh, yes. 606 00:32:16,430 --> 00:32:18,650 So that wouldn't make very much sense, right? 607 00:32:18,650 --> 00:32:19,340 Yes. 608 00:32:19,340 --> 00:32:22,522 So in that case, well, you don't want to have a negative energy. 609 00:32:22,522 --> 00:32:24,230 So that case doesn't make physical sense. 610 00:32:24,230 --> 00:32:26,210 Thanks for making sure we explained that. 611 00:32:26,210 --> 00:32:27,520 And did I see another question? 612 00:32:27,520 --> 00:32:28,020 Yes. 613 00:32:28,020 --> 00:32:28,603 AUDIENCE: Yes. 614 00:32:28,603 --> 00:32:30,320 What happened to the coefficients 615 00:32:30,320 --> 00:32:31,400 you had before Tin? 616 00:32:31,400 --> 00:32:33,125 You had 4. 617 00:32:33,125 --> 00:32:34,900 You needed 4 or 2, but [INAUDIBLE].. 618 00:32:34,900 --> 00:32:35,650 MICHAEL SHORT: Ah. 619 00:32:35,650 --> 00:32:36,150 OK. 620 00:32:36,150 --> 00:32:39,500 So what I did is I took the square root of 4 out of every 621 00:32:39,500 --> 00:32:42,950 term inside the square root and said, OK, they're all 2's. 622 00:32:42,950 --> 00:32:44,930 Just like in the next step, I said, all right, 623 00:32:44,930 --> 00:32:47,600 there's all of these A plus 1's, including 624 00:32:47,600 --> 00:32:50,360 all of these A plus 1's squares inside the square root, 625 00:32:50,360 --> 00:32:51,530 and took that out. 626 00:32:51,530 --> 00:32:52,760 Or I think even over here. 627 00:32:52,760 --> 00:32:55,610 Yes, so the whole thing here has been combine and destroy. 628 00:32:58,197 --> 00:33:00,780 Any other questions on what we did here before I go on to some 629 00:33:00,780 --> 00:33:02,238 of the implications of what we got? 630 00:33:04,600 --> 00:33:05,432 Cool. 631 00:33:05,432 --> 00:33:07,015 Let's look at a couple limiting cases. 632 00:33:09,640 --> 00:33:13,390 I'll rewrite that inequality right there because that's 633 00:33:13,390 --> 00:33:14,640 the important one of the day. 634 00:33:20,760 --> 00:33:23,220 So what is alpha for typical materials? 635 00:33:23,220 --> 00:33:28,940 Let's say for hydrogen. Alpha equals-- 636 00:33:28,940 --> 00:33:34,160 well, it's always A minus 1 over A plus 1 squared. 637 00:33:34,160 --> 00:33:37,990 And for hydrogen, A equals equals 1, equals 1. 638 00:33:37,990 --> 00:33:40,730 And then we have 1 minus 1 in the numerator. 639 00:33:40,730 --> 00:33:42,860 Alpha equals 0. 640 00:33:42,860 --> 00:33:45,620 What this means is that for the case of hydrogen, 641 00:33:45,620 --> 00:33:51,694 you can lose all of the neutron energy in a single collision. 642 00:33:56,140 --> 00:33:59,260 That doesn't mean that you lose all energy in every collision 643 00:33:59,260 --> 00:34:01,215 with a hydrogen atom if you're a neutron, 644 00:34:01,215 --> 00:34:02,590 but it means that you can lose up 645 00:34:02,590 --> 00:34:05,710 to all of your energy in one single collision. 646 00:34:05,710 --> 00:34:08,800 And this is what makes hydrogen such a good moderator 647 00:34:08,800 --> 00:34:11,770 or a slower down of neutrons, is when it undergoes 648 00:34:11,770 --> 00:34:15,820 elastic scattering, especially at energies below an MeV or so, 649 00:34:15,820 --> 00:34:18,790 which is where most of the neutrons in the reactor are, 650 00:34:18,790 --> 00:34:20,050 it just bounces around. 651 00:34:20,050 --> 00:34:22,030 And the more it hits hydrogens, the more 652 00:34:22,030 --> 00:34:25,222 it imparts energy to those hydrogens and slows down. 653 00:34:25,222 --> 00:34:26,889 Why do we want to slow the neutrons down 654 00:34:26,889 --> 00:34:28,460 in the first place? 655 00:34:28,460 --> 00:34:31,219 Well, that has to do with another cross-section, 656 00:34:31,219 --> 00:34:33,219 that I'm going to draw if I can find some chalk. 657 00:34:36,531 --> 00:34:37,989 Like I think I've mentioned before, 658 00:34:37,989 --> 00:34:42,080 every nuclear reaction has its own cross-section. 659 00:34:42,080 --> 00:34:46,219 And this time I'm going to introduce a new one called 660 00:34:46,219 --> 00:34:50,840 sigma fission, the probability, if a nucleus absorbs 661 00:34:50,840 --> 00:34:53,179 a neutron, that it undergoes fission and creates 662 00:34:53,179 --> 00:34:57,540 more neutrons and like 200 MeV of recoil energy. 663 00:34:57,540 --> 00:35:01,760 So in this case, I'll draw it for U235, 664 00:35:01,760 --> 00:35:04,250 since this is the one I pretty much remember from memory. 665 00:35:04,250 --> 00:35:08,240 And it looks something like that. 666 00:35:08,240 --> 00:35:12,650 So what you want is for the neutrons to be at low energies. 667 00:35:12,650 --> 00:35:17,240 So this would be around the thermal energy, better known 668 00:35:17,240 --> 00:35:20,180 as about 0.025 eV. 669 00:35:20,180 --> 00:35:22,490 Your goal is that the more neutrons that you 670 00:35:22,490 --> 00:35:25,020 have in this energy region-- 671 00:35:25,020 --> 00:35:27,890 oh, that chalk erases other chalk. 672 00:35:27,890 --> 00:35:31,010 The more neutrons you have in that energy region, the higher 673 00:35:31,010 --> 00:35:32,720 probability you have a fission. 674 00:35:32,720 --> 00:35:35,270 So this is the basis behind thermal reactors, 675 00:35:35,270 --> 00:35:39,200 is the neutrons all start here. 676 00:35:39,200 --> 00:35:44,740 They're born at around 1 to 10 MeV. 677 00:35:44,740 --> 00:35:48,100 They don't undergo fission very well at 1 to 10 MeV. 678 00:35:48,100 --> 00:35:50,590 So your goal as a thermal reactor designer 679 00:35:50,590 --> 00:35:53,840 is to slow them down as efficiently as possible. 680 00:35:53,840 --> 00:35:57,130 What's the most efficient way to slow down neutrons? 681 00:35:57,130 --> 00:35:59,680 Cram the reactor full of hydrogen. 682 00:35:59,680 --> 00:36:03,810 What's the cheapest and most hydrogenous substance we know? 683 00:36:03,810 --> 00:36:05,250 Water. 684 00:36:05,250 --> 00:36:10,000 This is why water makes such a good reactor moderator. 685 00:36:10,000 --> 00:36:11,020 It's pretty cool. 686 00:36:11,020 --> 00:36:13,770 There's also lots of other reasons that we use water. 687 00:36:13,770 --> 00:36:16,650 It's everywhere, which is another way of saying cheap. 688 00:36:16,650 --> 00:36:18,330 It's pretty chemically inert. 689 00:36:18,330 --> 00:36:20,070 There are corrosion problems in reactors, 690 00:36:20,070 --> 00:36:22,710 but it doesn't just spontaneously combust 691 00:36:22,710 --> 00:36:26,880 when you see air, like sodium does, another reactor coolant. 692 00:36:26,880 --> 00:36:29,490 It takes a lot of energy to heat it up. 693 00:36:29,490 --> 00:36:34,242 So it's specific heat capacity, the CP of water, 694 00:36:34,242 --> 00:36:34,950 if you remember-- 695 00:36:34,950 --> 00:36:42,210 I think it's-- was it 4.184 joules per gram? 696 00:36:42,210 --> 00:36:44,645 That's a point. 697 00:36:44,645 --> 00:36:47,100 One of the highest substances that we know of. 698 00:36:47,100 --> 00:36:49,800 So you can put a lot of that recoil energy 699 00:36:49,800 --> 00:36:52,350 or a lot of heat energy into this water 700 00:36:52,350 --> 00:36:54,000 without raising its temperature as much 701 00:36:54,000 --> 00:36:55,740 as a comparative substance. 702 00:36:55,740 --> 00:36:59,137 Metals can have heat capacity's like three or four times lower. 703 00:36:59,137 --> 00:37:01,470 So you wouldn't necessarily want to use a metal coolant. 704 00:37:01,470 --> 00:37:03,180 Or would you? 705 00:37:03,180 --> 00:37:05,700 In what cases would you want to use a metal coolant 706 00:37:05,700 --> 00:37:06,630 for a reactor? 707 00:37:06,630 --> 00:37:10,140 Has anyone ever heard of liquid metal reactors before? 708 00:37:10,140 --> 00:37:10,900 Just a couple. 709 00:37:10,900 --> 00:37:11,400 Good. 710 00:37:11,400 --> 00:37:13,350 I get to be the first one to tell you. 711 00:37:13,350 --> 00:37:16,980 I did my whole PhD on alloys for the liquid lead reactor. 712 00:37:16,980 --> 00:37:23,230 So let's take a look at lead, which has an A of about-- 713 00:37:23,230 --> 00:37:26,050 let's call it 200. 714 00:37:26,050 --> 00:37:28,390 I think there's some isotopes, like 203 or so. 715 00:37:28,390 --> 00:37:31,390 There's probably an isotope called lead 200. 716 00:37:31,390 --> 00:37:33,790 What would alpha be for lead? 717 00:37:37,380 --> 00:37:39,220 Well, let's just plug in the numbers. 718 00:37:39,220 --> 00:37:41,842 A minus 1 is 199. 719 00:37:41,842 --> 00:37:44,740 A plus 1 is 201. 720 00:37:44,740 --> 00:37:45,520 Square that. 721 00:37:48,250 --> 00:37:51,080 Almost 1. 722 00:37:51,080 --> 00:37:51,580 Almost. 723 00:37:54,160 --> 00:37:57,910 This means that when neutrons hit something like lead, 724 00:37:57,910 --> 00:38:00,040 basically don't slow down. 725 00:38:00,040 --> 00:38:04,240 They can lose at least none and at most almost 726 00:38:04,240 --> 00:38:06,032 none of their energy. 727 00:38:06,032 --> 00:38:07,490 And this is the basis behind what's 728 00:38:07,490 --> 00:38:10,820 called fast reactors if you want to use a coolant that 729 00:38:10,820 --> 00:38:13,190 keeps the neutrons very fast. 730 00:38:13,190 --> 00:38:15,918 Because for uranium 238, there's what's called a-- 731 00:38:15,918 --> 00:38:17,960 well, what you do is you want to capture neutrons 732 00:38:17,960 --> 00:38:22,670 with uranium 238, make plutonium 239, and then breed that. 733 00:38:22,670 --> 00:38:29,493 Or uranium 238 has got its fast fission cross-section. 734 00:38:29,493 --> 00:38:31,910 I don't think I want to get into bringing it on the screen 735 00:38:31,910 --> 00:38:34,590 today since we're almost five of five of. 736 00:38:34,590 --> 00:38:36,590 What I will say is there's lots of other reactor 737 00:38:36,590 --> 00:38:37,950 coolants besides water. 738 00:38:37,950 --> 00:38:39,950 And it sounds to me like almost no one had heard 739 00:38:39,950 --> 00:38:41,450 of a liquid metal reactor. 740 00:38:41,450 --> 00:38:43,370 Why would you want to use a liquid metal 741 00:38:43,370 --> 00:38:48,760 as a coolant besides keeping the neutrons at high energy? 742 00:38:48,760 --> 00:38:49,690 Anyone have any ideas? 743 00:38:53,500 --> 00:38:57,050 What sort of properties do you want out of a coolant? 744 00:38:57,050 --> 00:38:59,790 Not even for a reactor but for anything. 745 00:38:59,790 --> 00:39:00,790 AUDIENCE: Heat transfer. 746 00:39:00,790 --> 00:39:02,470 MICHAEL SHORT: Good heat transfer. 747 00:39:02,470 --> 00:39:05,110 Metals are extremely thermally conductive. 748 00:39:05,110 --> 00:39:07,750 So if you want to get the heat out of the fuel rods 749 00:39:07,750 --> 00:39:09,850 and into the coolant and then out to make steam 750 00:39:09,850 --> 00:39:12,790 for a turbine, liquid metals are a pretty awesome coolant 751 00:39:12,790 --> 00:39:16,540 to use because they conduct heat extremely well. 752 00:39:16,540 --> 00:39:17,080 What else? 753 00:39:19,630 --> 00:39:20,740 Let's try and think now. 754 00:39:20,740 --> 00:39:22,360 If you were a reactor designer, you don't just 755 00:39:22,360 --> 00:39:24,130 have to make the reactor work but you want 756 00:39:24,130 --> 00:39:26,450 to make it avoid accidents. 757 00:39:26,450 --> 00:39:29,710 What sort of thermodynamic properties about metals 758 00:39:29,710 --> 00:39:31,738 could prevent accidents from happening? 759 00:39:31,738 --> 00:39:32,828 AUDIENCE: They solidify. 760 00:39:32,828 --> 00:39:33,620 MICHAEL SHORT: Yes. 761 00:39:33,620 --> 00:39:34,740 So there's one problem. 762 00:39:34,740 --> 00:39:36,170 They could solidify. 763 00:39:36,170 --> 00:39:38,810 So coolants that have been chosen for reactors 764 00:39:38,810 --> 00:39:41,660 have been things like sodium, which melts just 765 00:39:41,660 --> 00:39:44,840 below 100 Celsius, liquid lead-bismuth, which 766 00:39:44,840 --> 00:39:46,550 melts at 123 Celsius. 767 00:39:46,550 --> 00:39:49,180 And I know because I've it in a frying pan before. 768 00:39:49,180 --> 00:39:51,590 So I did four years of research on liquid lead-bismuth, 769 00:39:51,590 --> 00:39:54,110 and if there's anyone that's gotten enough exposure to that, 770 00:39:54,110 --> 00:39:54,800 it's me. 771 00:39:54,800 --> 00:39:56,750 It does not seem to have affected my brain 772 00:39:56,750 --> 00:39:59,780 too much because I only made like two major mistakes 773 00:39:59,780 --> 00:40:01,580 on today's board. 774 00:40:01,580 --> 00:40:03,480 Good enough. 775 00:40:03,480 --> 00:40:03,980 Yes. 776 00:40:03,980 --> 00:40:07,370 So we've got hundreds of pounds of lead-bismuth sitting around. 777 00:40:07,370 --> 00:40:08,990 It's pretty inert stuff. 778 00:40:08,990 --> 00:40:13,500 It's really dense, so it can store a lot of heat. 779 00:40:13,500 --> 00:40:16,030 The other thing is the boiling point. 780 00:40:16,030 --> 00:40:18,280 Anyone know what temperature metals boil at? 781 00:40:21,022 --> 00:40:22,850 [INTERPOSING VOICES] 782 00:40:22,850 --> 00:40:25,130 MICHAEL SHORT: Extremely high, yes. 783 00:40:25,130 --> 00:40:31,250 Sodium boils at approximately exactly 893 degrees Celsius. 784 00:40:31,250 --> 00:40:36,080 Liquid lead-bismuth boils at approximately exactly 1,670 785 00:40:36,080 --> 00:40:37,130 Celsius. 786 00:40:37,130 --> 00:40:39,110 You'll actually melt the steel that the reactor 787 00:40:39,110 --> 00:40:41,780 is made of before you boil off your coolant. 788 00:40:41,780 --> 00:40:43,370 And if you boil your coolant, you 789 00:40:43,370 --> 00:40:44,870 have no way of cooling the reactor, 790 00:40:44,870 --> 00:40:46,760 and that's something you want to avoid. 791 00:40:46,760 --> 00:40:51,510 Water boils at approximately 325-- 792 00:40:51,510 --> 00:40:56,750 let's say 288 to 340 Celsius depending on the pressure 793 00:40:56,750 --> 00:40:58,310 that's used in the reactors. 794 00:40:58,310 --> 00:41:00,470 And that does get reached sometimes, especially 795 00:41:00,470 --> 00:41:01,940 in accident conditions. 796 00:41:01,940 --> 00:41:05,450 So if you want to make something relatively Fukushima-proof, 797 00:41:05,450 --> 00:41:07,490 then you don't want the coolant to boil. 798 00:41:07,490 --> 00:41:11,950 So use a liquid metal, which introduces other problems. 799 00:41:11,950 --> 00:41:14,040 It also introduces some other problems. 800 00:41:14,040 --> 00:41:17,190 I'm going to flash forward a bit to neutrons and reactor design. 801 00:41:17,190 --> 00:41:20,757 Because it does take time for this scattering to happen. 802 00:41:20,757 --> 00:41:22,590 These collisions, they happen pretty quickly 803 00:41:22,590 --> 00:41:25,270 but they do take a finite amount of time. 804 00:41:25,270 --> 00:41:26,820 And in the meantime, you can have 805 00:41:26,820 --> 00:41:29,100 what's called feedback coefficients, 806 00:41:29,100 --> 00:41:32,130 natural bits of physics that help your reactor stay stable 807 00:41:32,130 --> 00:41:35,040 or they don't, depending on whether it's called 808 00:41:35,040 --> 00:41:37,950 negative or positive feedback. 809 00:41:37,950 --> 00:41:42,540 So we can have either negative or positive feedback. 810 00:41:47,680 --> 00:41:49,750 I'll give you one simple example that I'll just 811 00:41:49,750 --> 00:41:52,180 introduce conceptually, and we'll actually 812 00:41:52,180 --> 00:41:53,890 explain it a little mathematically later 813 00:41:53,890 --> 00:41:55,010 in the course. 814 00:41:55,010 --> 00:41:57,400 Let's talk about coolant density. 815 00:41:57,400 --> 00:41:59,650 If you were to heat up your reactor 816 00:41:59,650 --> 00:42:02,923 and the coolant were to get less dense, 817 00:42:02,923 --> 00:42:04,840 what do you think would happen to the reaction 818 00:42:04,840 --> 00:42:09,040 rate of, well, anything-- scattering, fission, 819 00:42:09,040 --> 00:42:10,408 absorption? 820 00:42:10,408 --> 00:42:11,650 AUDIENCE: Go down. 821 00:42:11,650 --> 00:42:12,430 MICHAEL SHORT: It should go down. 822 00:42:12,430 --> 00:42:13,472 Why do you think that is? 823 00:42:13,472 --> 00:42:15,852 AUDIENCE: Because there's not as many particles that's 824 00:42:15,852 --> 00:42:17,732 close together, so [INAUDIBLE]. 825 00:42:17,732 --> 00:42:18,690 MICHAEL SHORT: Exactly. 826 00:42:18,690 --> 00:42:19,570 Yes. 827 00:42:19,570 --> 00:42:22,120 To reintroduce a bit of the cross-section stuff I mentioned 828 00:42:22,120 --> 00:42:25,270 last time, the microscopic cross-section 829 00:42:25,270 --> 00:42:27,760 is the probability that, let's say, one nucleus 830 00:42:27,760 --> 00:42:30,140 hits one other nucleus. 831 00:42:30,140 --> 00:42:32,920 If you then multiply by the number density 832 00:42:32,920 --> 00:42:35,290 or how many of them are there, you 833 00:42:35,290 --> 00:42:38,260 end up with the macroscopic cross-section. 834 00:42:38,260 --> 00:42:43,330 So I'll label this as micro, label this as macro. 835 00:42:43,330 --> 00:42:46,560 And then the macroscopic cross-section 836 00:42:46,560 --> 00:42:52,930 times your neutron flux gives you your reaction rate. 837 00:42:52,930 --> 00:42:56,110 So if you want to get less moderating happening, or less 838 00:42:56,110 --> 00:42:58,570 fission, or less absorption, the simplest way 839 00:42:58,570 --> 00:43:01,270 is-- well, that's a property of the material. 840 00:43:01,270 --> 00:43:03,310 That's whatever your reactor is doing. 841 00:43:03,310 --> 00:43:04,960 You can decrease the number density 842 00:43:04,960 --> 00:43:08,630 by heating things up and decreasing the density. 843 00:43:08,630 --> 00:43:11,600 So this is one of those cases where you can use the reactor 844 00:43:11,600 --> 00:43:14,780 to quickly respond with physics before you could respond 845 00:43:14,780 --> 00:43:16,280 with human intervention. 846 00:43:16,280 --> 00:43:18,590 If you want, let's say, a extra power 847 00:43:18,590 --> 00:43:21,380 transient or a sudden increase in heat 848 00:43:21,380 --> 00:43:24,200 to slow down the nuclear reaction and not speed it up, 849 00:43:24,200 --> 00:43:27,870 you'd pick a moderator that behaves in this way. 850 00:43:27,870 --> 00:43:30,410 So in this case, water would get less dense, 851 00:43:30,410 --> 00:43:34,400 it would moderate less well, and put fewer neutrons 852 00:43:34,400 --> 00:43:37,600 in the high-probability fission region. 853 00:43:37,600 --> 00:43:39,130 Then let's think about what happens 854 00:43:39,130 --> 00:43:42,460 if you're depending on your neutrons to stay fast 855 00:43:42,460 --> 00:43:43,840 or at high energy. 856 00:43:43,840 --> 00:43:46,240 Let's say you were to have a really bad day 857 00:43:46,240 --> 00:43:48,840 and boil your liquid sodium. 858 00:43:48,840 --> 00:43:51,150 All of a sudden, what little moderating power 859 00:43:51,150 --> 00:43:55,350 exists in that sodium disappears or gets even lower, 860 00:43:55,350 --> 00:43:58,380 and that would cause your reactor power to increase. 861 00:43:58,380 --> 00:44:01,530 So one of the dangers of fast spectrum reactors 862 00:44:01,530 --> 00:44:04,740 is positive-- what's called positive void coefficient. 863 00:44:04,740 --> 00:44:08,310 Or if you make a bubble of gaseous sodium, 864 00:44:08,310 --> 00:44:10,830 your power increases rather than decreases. 865 00:44:10,830 --> 00:44:12,120 That would increase the heat. 866 00:44:12,120 --> 00:44:13,620 That would cause the power to go up. 867 00:44:13,620 --> 00:44:14,828 That would increase the heat. 868 00:44:14,828 --> 00:44:16,410 That would cause the power to go up. 869 00:44:16,410 --> 00:44:18,660 Luckily, there are many, many other negative feedback 870 00:44:18,660 --> 00:44:20,610 mechanisms that could be built in 871 00:44:20,610 --> 00:44:23,700 to make sure the overall feedback coefficients are still 872 00:44:23,700 --> 00:44:26,540 negative. 873 00:44:26,540 --> 00:44:29,240 This also lets us understand a little bit about what 874 00:44:29,240 --> 00:44:30,400 went wrong at Chernobyl. 875 00:44:30,400 --> 00:44:32,660 And I'll give you a 1-minute flashforward. 876 00:44:32,660 --> 00:44:36,140 Because the control rods that were made of-- 877 00:44:36,140 --> 00:44:36,860 let's see. 878 00:44:36,860 --> 00:44:40,280 I don't remember what the composition of the control rods 879 00:44:40,280 --> 00:44:42,920 is, but they were neutron absorbers. 880 00:44:42,920 --> 00:44:46,290 The control rods in Chernobyl looked something 881 00:44:46,290 --> 00:44:50,490 like this, where there was the absorber here. 882 00:44:50,490 --> 00:44:54,730 And this was-- they were graphite tipped. 883 00:44:54,730 --> 00:44:58,230 And as you lower that graphite down into the reactor, 884 00:44:58,230 --> 00:45:00,560 you're all of a sudden introducing something. 885 00:45:00,560 --> 00:45:02,930 Well, that's an OK moderator. 886 00:45:02,930 --> 00:45:05,730 For carbon, let's say A equals 12. 887 00:45:05,730 --> 00:45:12,950 So our alpha will be A minus 1 over A plus 1. 888 00:45:12,950 --> 00:45:14,870 I'm not going to write almost equal to 1 889 00:45:14,870 --> 00:45:17,470 because that isn't quite almost equal to 1. 890 00:45:17,470 --> 00:45:18,720 What does this actually equal? 891 00:45:18,720 --> 00:45:19,220 Let's see. 892 00:45:22,330 --> 00:45:24,610 Let's just say definitely less than 1. 893 00:45:24,610 --> 00:45:27,040 There is some moderating power to graphite. 894 00:45:27,040 --> 00:45:29,350 It's also a very bad absorber. 895 00:45:29,350 --> 00:45:32,020 And what this meant, that as you lowered those control 896 00:45:32,020 --> 00:45:34,210 rods into the reactor, you suddenly introduced 897 00:45:34,210 --> 00:45:37,300 a little more moderation when things were already going bad, 898 00:45:37,300 --> 00:45:40,263 and that caused the power level to increase further. 899 00:45:40,263 --> 00:45:41,680 There were other problems, like it 900 00:45:41,680 --> 00:45:44,710 was designed so that if you boiled some of the coolant, 901 00:45:44,710 --> 00:45:46,623 you would have positive feedback. 902 00:45:46,623 --> 00:45:48,790 And that is the sort of 1-minute synopsis as to what 903 00:45:48,790 --> 00:45:50,470 all went haywire at Chernobyl. 904 00:45:50,470 --> 00:45:53,350 But we'll be doing a second by second or, in some cases, 905 00:45:53,350 --> 00:45:55,660 millisecond by millisecond play of what 906 00:45:55,660 --> 00:45:57,662 went wrong with Chernobyl. 907 00:45:57,662 --> 00:45:59,620 And we could probably do the same for Fukushima 908 00:45:59,620 --> 00:46:02,710 now, now that we understand what happened, based on the physics 909 00:46:02,710 --> 00:46:05,740 you'll be learning in this course. 910 00:46:05,740 --> 00:46:08,070 And this is actually a perfect stopping point, 911 00:46:08,070 --> 00:46:09,990 because next up we're going to be looking 912 00:46:09,990 --> 00:46:12,810 at the different processes of radioactive decay, 913 00:46:12,810 --> 00:46:14,910 many of which are a simplification of this Q 914 00:46:14,910 --> 00:46:17,190 equation, and I think some of which 915 00:46:17,190 --> 00:46:18,660 are probably familiar to you guys, 916 00:46:18,660 --> 00:46:22,680 because radiation decay is part of the normal lexicon, 917 00:46:22,680 --> 00:46:24,060 especially nowadays. 918 00:46:24,060 --> 00:46:26,380 So since it's five of five of, do you guys 919 00:46:26,380 --> 00:46:28,380 have any questions on what we've covered so far? 920 00:46:30,640 --> 00:46:31,140 Yes. 921 00:46:31,140 --> 00:46:34,740 AUDIENCE: [INAUDIBLE] if you have water as your coolant 922 00:46:34,740 --> 00:46:36,912 and it gets too hot, [INAUDIBLE].. 923 00:46:39,768 --> 00:46:40,560 MICHAEL SHORT: Yes. 924 00:46:40,560 --> 00:46:40,800 AUDIENCE: Right. 925 00:46:40,800 --> 00:46:42,114 And that will decrease the reaction rate. 926 00:46:42,114 --> 00:46:42,990 MICHAEL SHORT: Um-hm. 927 00:46:42,990 --> 00:46:47,690 AUDIENCE: And that's how like the [INAUDIBLE].. 928 00:46:47,690 --> 00:46:49,005 MICHAEL SHORT: It's actually-- 929 00:46:49,005 --> 00:46:50,380 are you asking about it the water 930 00:46:50,380 --> 00:46:51,760 feedback is part of the backup? 931 00:46:51,760 --> 00:46:54,160 That's your primary line of defense. 932 00:46:54,160 --> 00:46:55,780 Your backup is human intervention, 933 00:46:55,780 --> 00:46:57,280 because, compared to physics, humans 934 00:46:57,280 --> 00:47:00,340 are really, really slow, like many orders of magnitude 935 00:47:00,340 --> 00:47:01,360 slower. 936 00:47:01,360 --> 00:47:04,540 It takes microseconds for things to thermally expand. 937 00:47:04,540 --> 00:47:07,870 It definitely takes more than seconds for a human to respond. 938 00:47:07,870 --> 00:47:10,493 Anyone ever done those tests where you have a light blinking 939 00:47:10,493 --> 00:47:12,910 and you have to hit a button the second you see the light? 940 00:47:12,910 --> 00:47:16,970 What's the fastest any of you guys have ever responded? 941 00:47:16,970 --> 00:47:18,420 Anyone remember? 942 00:47:18,420 --> 00:47:20,646 Anyone beat a second? 943 00:47:20,646 --> 00:47:21,545 AUDIENCE: Maybe. 944 00:47:21,545 --> 00:47:22,420 MICHAEL SHORT: Maybe. 945 00:47:22,420 --> 00:47:24,294 A tenth of a second? 946 00:47:24,294 --> 00:47:28,590 [INTERPOSING VOICES] 947 00:47:28,590 --> 00:47:31,202 MICHAEL SHORT: And there all you have to do is hit a button. 948 00:47:31,202 --> 00:47:32,910 All you have to do is hit the only button 949 00:47:32,910 --> 00:47:34,925 when you see the only light. 950 00:47:34,925 --> 00:47:36,300 What if you're piloting something 951 00:47:36,300 --> 00:47:38,310 that's about as complicated as the space shuttle 952 00:47:38,310 --> 00:47:40,500 but more likely to explode? 953 00:47:40,500 --> 00:47:43,920 What do you think your reaction time will be? 954 00:47:43,920 --> 00:47:44,820 Probably long. 955 00:47:44,820 --> 00:47:48,330 You'll probably have to pull out the manual. 956 00:47:48,330 --> 00:47:51,690 And probably you'll have to RTFM for a little while. 957 00:47:51,690 --> 00:47:54,510 And maybe you'll find out what you have to do and maybe you 958 00:47:54,510 --> 00:47:55,410 won't. 959 00:47:55,410 --> 00:47:57,210 So it's actually operator error that 960 00:47:57,210 --> 00:48:00,660 has caused most of the near misses or actual misses 961 00:48:00,660 --> 00:48:01,560 in nuclear reactors. 962 00:48:01,560 --> 00:48:04,710 The physics, except for the Russian RBMK design 963 00:48:04,710 --> 00:48:07,140 that was for Chernobyl, usually it's 964 00:48:07,140 --> 00:48:10,150 human error that's the downfall of these things. 965 00:48:10,150 --> 00:48:12,130 So by understanding the physics here, 966 00:48:12,130 --> 00:48:15,130 we can rely on it to keep things safe. 967 00:48:15,130 --> 00:48:15,630 Yes. 968 00:48:15,630 --> 00:48:17,330 AUDIENCE: When you have a lead reactor-- 969 00:48:17,330 --> 00:48:23,623 or I don't know [INAUDIBLE] one of these, but what is like-- 970 00:48:23,623 --> 00:48:25,790 how do you cool the lead once it starts getting hot? 971 00:48:25,790 --> 00:48:27,165 MICHAEL SHORT: Ah, good question. 972 00:48:27,165 --> 00:48:28,610 How do you cool the liquid lead? 973 00:48:28,610 --> 00:48:31,010 You can't send liquid led through a turbine, right? 974 00:48:31,010 --> 00:48:33,260 So at some point you've got to make steam and use that 975 00:48:33,260 --> 00:48:34,550 to drive a turbine. 976 00:48:34,550 --> 00:48:36,620 You can use what's called a heat exchanger. 977 00:48:36,620 --> 00:48:42,140 At its simplest, you can think of it like a couple of tubes 978 00:48:42,140 --> 00:48:45,160 where the lead is going through here 979 00:48:45,160 --> 00:48:46,670 and the steam is going through here. 980 00:48:49,520 --> 00:48:52,700 And they have a very thin barrier between them, 981 00:48:52,700 --> 00:48:56,870 so you have all this heat moving from the lead, which is hotter, 982 00:48:56,870 --> 00:48:58,838 to the steam, which is colder. 983 00:48:58,838 --> 00:49:01,130 They actually have built a bunch of these led reactors. 984 00:49:01,130 --> 00:49:01,910 AUDIENCE: Is that real? 985 00:49:01,910 --> 00:49:02,702 MICHAEL SHORT: Yes. 986 00:49:02,702 --> 00:49:05,630 The Russian fast attack subs, the alpha class subs, 987 00:49:05,630 --> 00:49:08,870 were powered by and are powered by liquid led reactors. 988 00:49:08,870 --> 00:49:12,090 They're the only reactor that can outrun a torpedo. 989 00:49:12,090 --> 00:49:14,083 So when you have a liquid lead reactor powering 990 00:49:14,083 --> 00:49:15,750 and you've got a panic button that says, 991 00:49:15,750 --> 00:49:16,950 forget the safety systems. 992 00:49:16,950 --> 00:49:17,880 Outrun a torpedo. 993 00:49:17,880 --> 00:49:21,240 You have a choice between maybe dying in a reactor explosion 994 00:49:21,240 --> 00:49:24,150 and definitely getting shot out of the water with a torpedo. 995 00:49:24,150 --> 00:49:25,620 You do whatever you can. 996 00:49:25,620 --> 00:49:28,950 And these subs only run two or three not slower 997 00:49:28,950 --> 00:49:30,210 than a torpedo. 998 00:49:30,210 --> 00:49:32,460 So just like that old algebra problem, 999 00:49:32,460 --> 00:49:35,730 if this guy leaves Pittsburgh at 8 AM traveling 1000 00:49:35,730 --> 00:49:38,220 40 miles an hour, and I'm trying to get to Boston, 1001 00:49:38,220 --> 00:49:41,640 30 miles an hour, if a torpedo leaves one sub moving 1002 00:49:41,640 --> 00:49:44,370 this velocity and the alpha attack sub senses it 1003 00:49:44,370 --> 00:49:45,870 from this distance and starts moving 1004 00:49:45,870 --> 00:49:48,330 at a similar velocity, chances are the torpedo 1005 00:49:48,330 --> 00:49:51,930 runs out of steam before it reaches the alpha sub. 1006 00:49:51,930 --> 00:49:54,390 And that's only because they can have an extremely 1007 00:49:54,390 --> 00:49:58,237 compact liquid lead nuclear reactor at the power source. 1008 00:49:58,237 --> 00:50:01,060 AUDIENCE: So can you [INAUDIBLE]?? 1009 00:50:01,060 --> 00:50:03,150 How do you move [INAUDIBLE]? 1010 00:50:03,150 --> 00:50:04,810 MICHAEL SHORT: Good question. 1011 00:50:04,810 --> 00:50:05,650 How do you use the-- 1012 00:50:05,650 --> 00:50:07,020 how do you move the liquid lead? 1013 00:50:07,020 --> 00:50:08,760 You can move it by natural convection 1014 00:50:08,760 --> 00:50:10,500 but that's extremely slow. 1015 00:50:10,500 --> 00:50:12,210 So there are multiple ways of moving it. 1016 00:50:12,210 --> 00:50:15,810 One of the cool ones is called an EM or Electromagnetic pump. 1017 00:50:15,810 --> 00:50:18,600 It induces eddy currents in the liquid lead 1018 00:50:18,600 --> 00:50:20,610 because it's also a conductor. 1019 00:50:20,610 --> 00:50:24,690 And those eddy currents couple with the EM field from the EM 1020 00:50:24,690 --> 00:50:27,700 pump and cause the lead to just start moving on its own. 1021 00:50:27,700 --> 00:50:29,670 So it's a no moving parts pump. 1022 00:50:29,670 --> 00:50:33,540 The only problem is it's like 1% or 2% efficient. 1023 00:50:33,540 --> 00:50:34,282 Yes. 1024 00:50:34,282 --> 00:50:35,740 So they only use those on the subs, 1025 00:50:35,740 --> 00:50:40,000 but you can use EM pumps to move conductive coolants. 1026 00:50:40,000 --> 00:50:42,190 So I think it's pretty awesome. 1027 00:50:42,190 --> 00:50:44,580 And there have there have been land submarines. 1028 00:50:44,580 --> 00:50:47,070 In fact, there's a company called AKME Engineering 1029 00:50:47,070 --> 00:50:49,040 in Russia that's trying to commercialize 1030 00:50:49,040 --> 00:50:51,630 a small modular liquid lead reactor. 1031 00:50:51,630 --> 00:50:54,390 The other nice thing about these liquid metal coolants 1032 00:50:54,390 --> 00:50:56,470 is you can make the reactors much, 1033 00:50:56,470 --> 00:50:59,880 much smaller and denser than in a light water reactor. 1034 00:50:59,880 --> 00:51:02,520 In a light water reactor, you're relying on a lot of water 1035 00:51:02,520 --> 00:51:04,290 to cool things and a lot of water 1036 00:51:04,290 --> 00:51:07,382 to be there to moderate your neutrons. 1037 00:51:07,382 --> 00:51:09,840 In a liquid metal reactor, where you don't need moderation, 1038 00:51:09,840 --> 00:51:11,298 well, you don't need-- all you need 1039 00:51:11,298 --> 00:51:13,040 is enough coolant to keep things cool. 1040 00:51:13,040 --> 00:51:15,248 So you can tighten stuff up and make it more compact. 1041 00:51:17,223 --> 00:51:19,640 So that's one of the nuclear startups coming out nowadays. 1042 00:51:19,640 --> 00:51:21,860 This is an awesome time to be in nuclear. 1043 00:51:21,860 --> 00:51:24,800 When I started nuclear, there were approximately exactly 1044 00:51:24,800 --> 00:51:26,480 zero nuclear startups. 1045 00:51:26,480 --> 00:51:28,550 Like TerraPower didn't even exist yet. 1046 00:51:28,550 --> 00:51:32,793 Now there's something like 52 in the US 1047 00:51:32,793 --> 00:51:33,960 and others around the world. 1048 00:51:33,960 --> 00:51:36,240 So like this is the time to be in nuclear if you're 1049 00:51:36,240 --> 00:51:39,360 up for startups and not just working in academia, 1050 00:51:39,360 --> 00:51:40,900 or a lab, or a big corporation. 1051 00:51:40,900 --> 00:51:43,770 There's a lot of little companies now doing 1052 00:51:43,770 --> 00:51:48,385 some crazy things based on some pretty good physics. 1053 00:51:48,385 --> 00:51:50,760 So maybe time for one more question before I let you guys 1054 00:51:50,760 --> 00:51:51,260 go. 1055 00:51:54,650 --> 00:51:56,930 If not, then I'll see you guys on Tuesday 1056 00:51:56,930 --> 00:51:59,053 when we start radioactive decay.