1 00:00:00,935 --> 00:00:03,260 The following content is provided under a Creative 2 00:00:03,260 --> 00:00:04,650 Commons license. 3 00:00:04,650 --> 00:00:06,860 Your support will help MIT OpenCourseWare 4 00:00:06,860 --> 00:00:10,950 continue to offer high quality educational resources for free. 5 00:00:10,950 --> 00:00:13,490 To make a donation or to view additional materials 6 00:00:13,490 --> 00:00:17,450 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,450 --> 00:00:18,350 at ocw.mit.edu. 8 00:00:21,400 --> 00:00:23,150 MICHAEL SHORT: Want to give a quick review 9 00:00:23,150 --> 00:00:26,030 so that we can launch into some more technical stuff today. 10 00:00:26,030 --> 00:00:29,840 We started talking about this reaction for boron neutron 11 00:00:29,840 --> 00:00:33,430 capture therapy was the focus of today's lecture. 12 00:00:33,430 --> 00:00:35,180 And I want to get this one up on the board 13 00:00:35,180 --> 00:00:39,170 since I'm going to move to different slides later. 14 00:00:39,170 --> 00:00:42,650 We had boron 10 captures a neutron 15 00:00:42,650 --> 00:00:48,980 and becomes lithium 7, helium 4. 16 00:00:48,980 --> 00:00:50,360 There's a gamma ray. 17 00:00:50,360 --> 00:00:54,560 And there's going to be some q value, or energy 18 00:00:54,560 --> 00:00:57,470 either released or consumed, in this case, released 19 00:00:57,470 --> 00:01:01,010 in the form of the kinetic energy of the recoil products 20 00:01:01,010 --> 00:01:03,410 as well as the gamma ray energy. 21 00:01:03,410 --> 00:01:07,710 So let's also give a quick review of the Q equation 22 00:01:07,710 --> 00:01:10,010 since I think we covered that last week. 23 00:01:10,010 --> 00:01:12,770 If you remember, if you have a general system of, let's say, 24 00:01:12,770 --> 00:01:19,100 a small initial nucleus i firing into a larger nucleus capital 25 00:01:19,100 --> 00:01:24,650 I, after the reaction, off comes some small final nucleus 26 00:01:24,650 --> 00:01:27,480 and some large final nucleus. 27 00:01:27,480 --> 00:01:29,210 If I were to draw these arrows to scale, 28 00:01:29,210 --> 00:01:33,220 the little one would probably be moving faster. 29 00:01:33,220 --> 00:01:37,370 Then we can figure out how much mass and kinetic energy 30 00:01:37,370 --> 00:01:40,310 each of these nuclei are by just conserving everything. 31 00:01:40,310 --> 00:01:43,400 So we can write, let's say, the mass of nucleus i, 32 00:01:43,400 --> 00:01:46,820 c squared plus the kinetic energy of i 33 00:01:46,820 --> 00:01:50,000 plus the mass of big I c squared. 34 00:01:50,000 --> 00:01:55,850 Plus the kinetic energy of big I has to equal the mass of little 35 00:01:55,850 --> 00:02:00,920 f c squared plus the kinetic energy of little f 36 00:02:00,920 --> 00:02:07,160 plus the mass of big F c squared plus its kinetic energy. 37 00:02:07,160 --> 00:02:10,250 And what this tells us is if the total amount of mass, 38 00:02:10,250 --> 00:02:12,500 or the total kinetic energy change, 39 00:02:12,500 --> 00:02:15,900 they've got to exchange energy kind of equally. 40 00:02:15,900 --> 00:02:20,780 So we can write the difference in mass or energy by just, 41 00:02:20,780 --> 00:02:22,760 let's say, taking all of the mass terms 42 00:02:22,760 --> 00:02:24,620 and putting them on one side. 43 00:02:24,620 --> 00:02:26,670 So we can say that-- 44 00:02:26,670 --> 00:02:29,360 let's just say everything here is multiplied by c squared. 45 00:02:29,360 --> 00:02:37,730 We have M i plus M I minus M little f minus M big F 46 00:02:37,730 --> 00:02:44,210 has got to equal the sum of the final kinetic energies 47 00:02:44,210 --> 00:02:53,390 minus the initial ones, which we also call this Q value. 48 00:02:53,390 --> 00:02:56,300 And so by getting the difference in the masses 49 00:02:56,300 --> 00:02:58,400 or the kinetic energies at the end, 50 00:02:58,400 --> 00:03:00,080 you can figure out whether this reaction 51 00:03:00,080 --> 00:03:02,360 is exothermic or endothermic. 52 00:03:02,360 --> 00:03:06,010 If you remember, we said if Q is greater than 0, 53 00:03:06,010 --> 00:03:07,370 it's exothermic. 54 00:03:07,370 --> 00:03:11,660 If Q is less than zero, it's endothermic. 55 00:03:11,660 --> 00:03:14,390 In a nuclear reaction like a chemical reaction, 56 00:03:14,390 --> 00:03:17,000 you've got to input extra energy into the system 57 00:03:17,000 --> 00:03:19,700 beyond just the rest masses of the particles 58 00:03:19,700 --> 00:03:22,880 to make an endothermic reaction happen. 59 00:03:22,880 --> 00:03:26,340 For example, since we found out that this BNCT reaction is 60 00:03:26,340 --> 00:03:29,510 endothermic, you can make it go the other way, 61 00:03:29,510 --> 00:03:32,570 but you have to impart kinetic energy to one 62 00:03:32,570 --> 00:03:38,000 or both of those nuclei to overcome the Q value that-- 63 00:03:38,000 --> 00:03:42,170 let's see-- yeah, to overcome the Q value that you'd get. 64 00:03:42,170 --> 00:03:44,720 And I want to put up a couple of other terms just 65 00:03:44,720 --> 00:03:46,470 to quickly review. 66 00:03:46,470 --> 00:03:50,120 We started looking at the table of nuclides. 67 00:03:50,120 --> 00:03:52,430 We learned how to read it specifically to find things 68 00:03:52,430 --> 00:03:54,950 like excess mass and binding energy, 69 00:03:54,950 --> 00:03:58,080 the definitions for which I want to leave up here on the board. 70 00:03:58,080 --> 00:04:03,230 So we had the excess mass, which, again, doesn't 71 00:04:03,230 --> 00:04:04,910 have a physical significance. 72 00:04:04,910 --> 00:04:08,210 It's the difference between the actual mass of a nucleotide 73 00:04:08,210 --> 00:04:10,790 and the integer approximation of its mass 74 00:04:10,790 --> 00:04:13,640 just from the number of protons and neutrons, 75 00:04:13,640 --> 00:04:15,440 and the binding energy. 76 00:04:15,440 --> 00:04:18,230 Since some of you asked for a nucleus uniquely 77 00:04:18,230 --> 00:04:23,060 defined by its total mass number a and number of protons z. 78 00:04:23,060 --> 00:04:25,340 So let's say this is a functional quantity 79 00:04:25,340 --> 00:04:27,590 because you can have any nucleus with a certain number 80 00:04:27,590 --> 00:04:31,220 of protons and a certain number of protons plus neutrons 81 00:04:31,220 --> 00:04:33,740 would be the sum-- 82 00:04:33,740 --> 00:04:43,970 let's see-- the sum of its individual nucleons 83 00:04:43,970 --> 00:04:48,590 minus its actual mass, which also 84 00:04:48,590 --> 00:04:51,650 is a function of z and a, or its proton 85 00:04:51,650 --> 00:04:54,210 number and its total atomic mass number. 86 00:04:54,210 --> 00:04:58,710 So let's leave these things up on the board 87 00:04:58,710 --> 00:05:00,400 while we move into some new stuff. 88 00:05:05,750 --> 00:05:08,750 So back to where we were on Tuesday. 89 00:05:08,750 --> 00:05:10,640 We wanted to figure out, well, how do we 90 00:05:10,640 --> 00:05:12,140 calculate this Q value? 91 00:05:12,140 --> 00:05:14,430 You can do it any of three ways. 92 00:05:14,430 --> 00:05:16,880 You can use the masses, like we have over here. 93 00:05:16,880 --> 00:05:21,080 So the masses in amu, or atomic mass units, times c squared 94 00:05:21,080 --> 00:05:23,360 gives you the Q value in MeV. 95 00:05:23,360 --> 00:05:25,490 Or the difference in kinetic energies 96 00:05:25,490 --> 00:05:27,890 gives you the Q value in MeV, although you don't usually 97 00:05:27,890 --> 00:05:30,350 just know these off the bat, especially 98 00:05:30,350 --> 00:05:31,970 for the final products. 99 00:05:31,970 --> 00:05:34,040 Or you can use the binding energies. 100 00:05:34,040 --> 00:05:36,710 Because the binding energy is directly related 101 00:05:36,710 --> 00:05:40,580 to the mass additively, you can substitute in binding energies 102 00:05:40,580 --> 00:05:43,040 here and you'll get the same thing. 103 00:05:43,040 --> 00:05:45,620 And so if we have the table of nuclides, 104 00:05:45,620 --> 00:05:48,860 you can either use the atomic mass 105 00:05:48,860 --> 00:05:50,570 of any nucleus or its binding energy 106 00:05:50,570 --> 00:05:52,130 and just look it up directly. 107 00:05:52,130 --> 00:05:53,780 The thing with the fewest steps is just 108 00:05:53,780 --> 00:05:55,988 to use the binding energies because those are already 109 00:05:55,988 --> 00:05:57,590 given in MeV or keV. 110 00:05:57,590 --> 00:06:00,010 And then it's just an addition subtraction problem. 111 00:06:00,010 --> 00:06:02,000 If you use the masses, don't forget 112 00:06:02,000 --> 00:06:03,540 to multiply by c squared. 113 00:06:03,540 --> 00:06:07,160 And remember our conversion formula, 114 00:06:07,160 --> 00:06:17,940 which should not be rounded, which is this right here. 115 00:06:17,940 --> 00:06:22,330 Again, 931.49, not 931, not even 931.5, or else 116 00:06:22,330 --> 00:06:24,545 you're off by 10 kiloelectron volts. 117 00:06:24,545 --> 00:06:26,170 And so then the question is, all right, 118 00:06:26,170 --> 00:06:29,320 let's try to calculate the Q value from this reaction. 119 00:06:29,320 --> 00:06:32,860 I have the individual energies and kinetic energies up here. 120 00:06:32,860 --> 00:06:34,990 But just so we have a worked out example, 121 00:06:34,990 --> 00:06:36,950 let's actually do this. 122 00:06:36,950 --> 00:06:40,210 So the Q value of this reaction should 123 00:06:40,210 --> 00:06:48,550 be the binding energy of lithium 7 plus the binding energy of-- 124 00:06:48,550 --> 00:06:55,060 what do we have-- helium 4 minus the binding energy of boron 10 125 00:06:55,060 --> 00:06:59,620 minus the binding energy of a neutron. 126 00:06:59,620 --> 00:07:00,920 Now first of all, the easy one. 127 00:07:00,920 --> 00:07:03,670 What's the binding energy of a low neutron? 128 00:07:03,670 --> 00:07:04,860 Let's call it out. 129 00:07:04,860 --> 00:07:07,260 0, yeah. 130 00:07:07,260 --> 00:07:10,630 A lone nucleon is not bound to anything, so that's easy. 131 00:07:10,630 --> 00:07:11,472 That's 0. 132 00:07:11,472 --> 00:07:13,180 And we can just use the table of nuclides 133 00:07:13,180 --> 00:07:14,590 to look up the other three. 134 00:07:14,590 --> 00:07:18,650 Luckily I've got it live over here. 135 00:07:18,650 --> 00:07:20,980 So let's just punch these in. 136 00:07:20,980 --> 00:07:24,450 We'll have boron 10. 137 00:07:24,450 --> 00:07:36,880 And our binding energy right here is 64.75 MeV. 138 00:07:36,880 --> 00:07:38,385 Let's look up helium 4. 139 00:07:42,230 --> 00:07:42,730 Good. 140 00:07:42,730 --> 00:07:45,230 That's showing up on the screen. 141 00:07:45,230 --> 00:07:47,540 I think I'll make it bigger so it's easier for everyone 142 00:07:47,540 --> 00:07:49,010 to see. 143 00:07:49,010 --> 00:07:57,680 The binding energy is 28.296 MeV. 144 00:07:57,680 --> 00:07:59,310 And notice everything's already in MeV, 145 00:07:59,310 --> 00:08:02,250 so this is nice and easy to deal with. 146 00:08:02,250 --> 00:08:04,740 And lithium 7. 147 00:08:07,440 --> 00:08:22,340 39.245 plus minus minus 0 MeV. 148 00:08:22,340 --> 00:08:23,060 Let's see. 149 00:08:23,060 --> 00:08:29,790 Actually, I think my sines for exo and endo are backwards. 150 00:08:29,790 --> 00:08:33,960 Let me fix that right now because the idea is 151 00:08:33,960 --> 00:08:37,140 if you release energy-- 152 00:08:37,140 --> 00:08:38,442 there we go. 153 00:08:38,442 --> 00:08:39,150 Sorry about that. 154 00:08:44,110 --> 00:08:45,210 No, wait a minute. 155 00:08:45,210 --> 00:08:48,240 Let's calculate this out, figure what we get. 156 00:08:48,240 --> 00:08:50,280 And again, I can't do six digits in my head, 157 00:08:50,280 --> 00:08:53,430 but I will do it as fast as I can here. 158 00:08:53,430 --> 00:08:59,432 245 plus 28.296 minus 0.75. 159 00:08:59,432 --> 00:09:01,550 Ah, indeed. 160 00:09:01,550 --> 00:09:04,340 2.79 MeV. 161 00:09:04,340 --> 00:09:05,860 I had it right the first time. 162 00:09:05,860 --> 00:09:08,300 So good. 163 00:09:08,300 --> 00:09:09,550 Shouldn't second guess myself. 164 00:09:15,990 --> 00:09:16,490 Cool. 165 00:09:20,810 --> 00:09:25,120 So now we know the total Q value of this reaction. 166 00:09:25,120 --> 00:09:27,325 And I'll bring the reaction back up here. 167 00:09:27,325 --> 00:09:29,450 We also know, which you can find from measurements, 168 00:09:29,450 --> 00:09:38,630 that the gamma ray comes off with an energy of 0.48 MeV, 169 00:09:38,630 --> 00:09:40,310 leaving-- 170 00:09:40,310 --> 00:09:42,290 let's see how much-- 171 00:09:42,290 --> 00:09:48,770 leaving 2.13 MeV for the sum of the kinetic energies 172 00:09:48,770 --> 00:09:51,410 of the lithium and the helium nucleus. 173 00:09:51,410 --> 00:09:58,460 So let's say Tli 7 plus t for helium. 174 00:09:58,460 --> 00:09:59,960 Now the question is how did I get 175 00:09:59,960 --> 00:10:03,890 to those numbers for the split between the kinetic energies 176 00:10:03,890 --> 00:10:06,045 of those two? 177 00:10:06,045 --> 00:10:06,920 Anyone have any idea? 178 00:10:10,420 --> 00:10:11,582 Yeah. 179 00:10:11,582 --> 00:10:12,947 AUDIENCE: Related to their mass? 180 00:10:12,947 --> 00:10:13,780 MICHAEL SHORT: Yeah. 181 00:10:13,780 --> 00:10:16,690 So it's definitely related to their relative masses. 182 00:10:16,690 --> 00:10:18,910 More specifically, we have this conservation 183 00:10:18,910 --> 00:10:21,340 of energy equation, but we've still 184 00:10:21,340 --> 00:10:23,320 got two variables and one equation. 185 00:10:23,320 --> 00:10:24,670 We need a second equation. 186 00:10:24,670 --> 00:10:26,170 That certainly does relate the mass. 187 00:10:26,170 --> 00:10:26,670 Yep. 188 00:10:26,670 --> 00:10:28,003 AUDIENCE: Conservation momentum. 189 00:10:28,003 --> 00:10:30,080 MICHAEL SHORT: You can relate their momentum. 190 00:10:30,080 --> 00:10:34,870 So we can say that if this initial kinetic energy of boron 191 00:10:34,870 --> 00:10:37,870 and of the neutron was approximately 0, then 192 00:10:37,870 --> 00:10:40,150 it's like these two nuclei, lithium and helium, 193 00:10:40,150 --> 00:10:42,520 were kind of standing still and all of a sudden 194 00:10:42,520 --> 00:10:44,710 moved off in opposite directions, which 195 00:10:44,710 --> 00:10:48,260 means they've got to have equal and opposite momenta. 196 00:10:48,260 --> 00:10:52,150 So let's say the absolute value of the momentum of lithium 197 00:10:52,150 --> 00:10:53,890 has got to equal the absolute value 198 00:10:53,890 --> 00:10:57,580 of the momentum of helium. 199 00:10:57,580 --> 00:10:59,350 This is our second equation, where 200 00:10:59,350 --> 00:11:01,390 we'll use the first one to figure out what are 201 00:11:01,390 --> 00:11:03,550 the relative kinetic energies. 202 00:11:03,550 --> 00:11:06,010 And so we're going to use a quick trick to say if momentum 203 00:11:06,010 --> 00:11:10,060 equals mass times velocity, we can also 204 00:11:10,060 --> 00:11:14,470 say it equals the kinetic energy, which 205 00:11:14,470 --> 00:11:18,490 is 1/2 mv squared, and then multiply by things in order 206 00:11:18,490 --> 00:11:19,990 to make it mv. 207 00:11:19,990 --> 00:11:22,690 So we can multiply by 2. 208 00:11:22,690 --> 00:11:26,590 Let's make a little bit of space here. 209 00:11:26,590 --> 00:11:32,020 If we take the kinetic energy, multiply by 2, multiply by m, 210 00:11:32,020 --> 00:11:35,800 and take the square root, we have t. 211 00:11:35,800 --> 00:11:37,210 Let's see, that would give us 2 m 212 00:11:37,210 --> 00:11:41,170 squared v squared inside the square root gives us mv. 213 00:11:41,170 --> 00:11:42,730 So now we can take these expressions 214 00:11:42,730 --> 00:11:49,390 and we can say that the root 2 mass lithium, t lithium, 215 00:11:49,390 --> 00:11:56,000 equals root 2 mass helium t helium. 216 00:11:56,000 --> 00:11:59,600 And they both have a square root of 2. 217 00:11:59,600 --> 00:12:03,050 We can square both sides of both equations. 218 00:12:03,050 --> 00:12:08,240 And we end up with mass of lithium t lithium 219 00:12:08,240 --> 00:12:14,060 equals mass of helium t helium. 220 00:12:14,060 --> 00:12:17,360 Now we take this equation, rearrange it a little bit. 221 00:12:17,360 --> 00:12:21,830 Let's just call that Q to keep things in variable space. 222 00:12:21,830 --> 00:12:31,270 And we can say that t helium equals Q minus t lithium. 223 00:12:31,270 --> 00:12:36,960 We can take this t helium, stick it in here. 224 00:12:36,960 --> 00:12:40,920 We end up with m lithium t lithium 225 00:12:40,920 --> 00:12:49,000 equals the mass of helium times Q minus t lithium. 226 00:12:49,000 --> 00:12:50,700 There's a missing h right there. 227 00:12:50,700 --> 00:12:51,960 There we go. 228 00:12:51,960 --> 00:12:53,040 And then from here-- 229 00:12:53,040 --> 00:12:53,540 oh good. 230 00:12:53,540 --> 00:12:55,950 We've got some blank space right here. 231 00:12:55,950 --> 00:12:56,450 Let's see. 232 00:12:56,450 --> 00:12:58,200 So we'd have-- I'll do out all the steps-- 233 00:13:00,930 --> 00:13:08,600 equals mass of helium times Q minus mass of helium times 234 00:13:08,600 --> 00:13:11,160 t lithium. 235 00:13:11,160 --> 00:13:16,860 And we're solving for t lithium, so we can put this term over 236 00:13:16,860 --> 00:13:18,640 on the other side. 237 00:13:18,640 --> 00:13:23,610 So let's say t lithium times the mass of lithium 238 00:13:23,610 --> 00:13:31,610 plus the mass of helium equals Q times the mass of helium. 239 00:13:31,610 --> 00:13:33,710 Then we can just divide both sides 240 00:13:33,710 --> 00:13:45,510 by the sum of those masses 241 00:13:45,510 --> 00:13:47,640 Those two terms cancel, and we're 242 00:13:47,640 --> 00:13:53,120 left with the expression for the kinetic energy of lithium, 243 00:13:53,120 --> 00:13:56,060 which is the mass of helium over the sum of the masses 244 00:13:56,060 --> 00:13:58,750 times the total Q value. 245 00:13:58,750 --> 00:14:00,500 Looks like the two of those might actually 246 00:14:00,500 --> 00:14:02,390 be backwards, huh? 247 00:14:02,390 --> 00:14:06,340 Because the lithium one should be smaller. 248 00:14:06,340 --> 00:14:10,070 I'll correct that for the notes when I put them up online 249 00:14:10,070 --> 00:14:14,180 because this ratio should be smaller than 0.5 Q. 250 00:14:14,180 --> 00:14:16,880 But at any rate, this is how you actually get them. 251 00:14:16,880 --> 00:14:18,590 And I want to point out something 252 00:14:18,590 --> 00:14:21,860 a little flash forward to some of the decay 253 00:14:21,860 --> 00:14:24,800 that you're going to be looking at in terms of nuclear decays. 254 00:14:24,800 --> 00:14:27,440 Let's talk for a second about alpha decay. 255 00:14:27,440 --> 00:14:30,890 In alpha decay, you have a low nucleus sitting around. 256 00:14:30,890 --> 00:14:33,890 And then all of a sudden it emits a helium nucleus as well 257 00:14:33,890 --> 00:14:35,540 as a recoil nucleus. 258 00:14:35,540 --> 00:14:38,240 And one of the questions that came up a lot last year 259 00:14:38,240 --> 00:14:41,420 is why isn't the Q value of an alpha decay 260 00:14:41,420 --> 00:14:46,070 reaction-- let's just write one up on the board, possibly one 261 00:14:46,070 --> 00:14:50,780 that you'll be dealing with very soon, hands on. 262 00:14:50,780 --> 00:14:58,390 Uranium 235 can spontaneously go to a helium nucleus. 263 00:14:58,390 --> 00:15:06,490 That's 92 plus looks like 90 to 31. 264 00:15:06,490 --> 00:15:08,200 What comes before uranium? 265 00:15:08,200 --> 00:15:11,050 I think that is thorium, although I wouldn't-- yeah, 266 00:15:11,050 --> 00:15:12,423 I think that's thorium. 267 00:15:12,423 --> 00:15:13,840 And the idea here is that there is 268 00:15:13,840 --> 00:15:16,780 some Q value associated with the kinetic energy of both 269 00:15:16,780 --> 00:15:18,260 of these. 270 00:15:18,260 --> 00:15:21,380 But it's not the alpha kinetic energy 271 00:15:21,380 --> 00:15:25,190 because the thorium nucleus would have to take away 272 00:15:25,190 --> 00:15:26,630 some of that kinetic energy. 273 00:15:26,630 --> 00:15:29,520 And I want to show you this on the diagrams. 274 00:15:29,520 --> 00:15:34,040 So let's look at U 235. 275 00:15:34,040 --> 00:15:38,880 And we can see that it has an alpha decay to thorium 231. 276 00:15:38,880 --> 00:15:39,380 Awesome. 277 00:15:39,380 --> 00:15:41,040 Got the symbol right. 278 00:15:41,040 --> 00:15:46,170 And it has a decay energy of 4.679 MeV. 279 00:15:46,170 --> 00:15:47,610 Let's take a look at the diagram, 280 00:15:47,610 --> 00:15:50,910 which actually lists all of the possible alpha decay energies, 281 00:15:50,910 --> 00:15:54,220 of which there are many. 282 00:15:54,220 --> 00:15:56,584 Had to zoom out a little bit there. 283 00:15:56,584 --> 00:15:57,970 Yeah? 284 00:15:57,970 --> 00:15:59,590 Anyone have a question? 285 00:15:59,590 --> 00:16:01,010 OK. 286 00:16:01,010 --> 00:16:06,680 So notice that the difference in energy levels is 4.676 MeV. 287 00:16:06,680 --> 00:16:08,730 And if we look at the highest energy alpha ray, 288 00:16:08,730 --> 00:16:11,422 it's less than that. 289 00:16:11,422 --> 00:16:13,380 And that's because just like we showed up here, 290 00:16:13,380 --> 00:16:15,180 the thorium nucleus has to take away 291 00:16:15,180 --> 00:16:17,970 some of that kinetic energy to conserve both energy 292 00:16:17,970 --> 00:16:19,630 and momentum. 293 00:16:19,630 --> 00:16:22,430 So this is a question that came up quite a few times last year, 294 00:16:22,430 --> 00:16:25,180 and I want to make sure you guys don't get tripped up like this. 295 00:16:25,180 --> 00:16:26,610 So again, I think I've said it every single day, 296 00:16:26,610 --> 00:16:28,027 and I'll say it again because it's 297 00:16:28,027 --> 00:16:30,360 the next day, is make sure to conserve mass energy 298 00:16:30,360 --> 00:16:30,930 and momentum. 299 00:16:30,930 --> 00:16:32,910 That's the whole theme of this class. 300 00:16:36,960 --> 00:16:37,972 Yes. 301 00:16:37,972 --> 00:16:39,430 AUDIENCE: How do you get that value 302 00:16:39,430 --> 00:16:42,710 for [INAUDIBLE] mega electron volts? 303 00:16:42,710 --> 00:16:44,750 MICHAEL SHORT: Should have been 2.79 minus-- oh, 304 00:16:44,750 --> 00:16:48,180 did I do a little mental math mistake? 305 00:16:48,180 --> 00:16:49,070 That should be-- oh. 306 00:16:49,070 --> 00:16:49,590 Yeah, no. 307 00:16:49,590 --> 00:16:51,010 A little dyslexia thing. 308 00:16:51,010 --> 00:16:53,520 3, 1. 309 00:16:53,520 --> 00:16:54,350 Yeah. 310 00:16:54,350 --> 00:16:55,080 There we go. 311 00:16:55,080 --> 00:16:55,580 Thank you. 312 00:16:58,460 --> 00:17:01,240 Cool. 313 00:17:01,240 --> 00:17:02,930 OK. 314 00:17:02,930 --> 00:17:05,290 Is everyone clear on how to calculate Q values 315 00:17:05,290 --> 00:17:08,859 from nuclear reactions using either kinetic energies, which 316 00:17:08,859 --> 00:17:12,040 you won't typically know, or masses or binding energies, 317 00:17:12,040 --> 00:17:15,109 which you can look up directly from the table of nuclides? 318 00:17:15,109 --> 00:17:15,730 Yeah. 319 00:17:15,730 --> 00:17:18,849 AUDIENCE: So what did you do down there in the bottom right 320 00:17:18,849 --> 00:17:20,145 corner of the chalkboard? 321 00:17:20,145 --> 00:17:21,270 MICHAEL SHORT: Of this one? 322 00:17:21,270 --> 00:17:22,047 AUDIENCE: Yeah. 323 00:17:22,047 --> 00:17:22,839 MICHAEL SHORT: Yep. 324 00:17:22,839 --> 00:17:26,950 So I took this expression right here, which 325 00:17:26,950 --> 00:17:28,720 is to say the Q value has got to be 326 00:17:28,720 --> 00:17:30,940 the sum of the kinetic energies of the lithium 327 00:17:30,940 --> 00:17:35,760 and helium nuclei, rearranged it thusly to isolate-- 328 00:17:35,760 --> 00:17:37,490 ah, there we go-- 329 00:17:37,490 --> 00:17:41,240 yep, to isolate the helium kinetic energy, 330 00:17:41,240 --> 00:17:45,230 and then substituted that expression in here 331 00:17:45,230 --> 00:17:46,760 to get this one right there. 332 00:17:46,760 --> 00:17:47,697 AUDIENCE: OK. 333 00:17:47,697 --> 00:17:48,530 MICHAEL SHORT: Yeah. 334 00:17:48,530 --> 00:17:49,883 So this way, we have-- 335 00:17:49,883 --> 00:17:51,800 in this case, we had, let's say, two variables 336 00:17:51,800 --> 00:17:52,592 and three unknowns. 337 00:17:52,592 --> 00:17:54,948 But because we have this equation relating them, 338 00:17:54,948 --> 00:17:56,990 we're left down with two variables, two unknowns, 339 00:17:56,990 --> 00:17:59,250 and we can actually solve this thing. 340 00:17:59,250 --> 00:18:00,771 Yep. 341 00:18:00,771 --> 00:18:04,260 Yeah, good question. 342 00:18:04,260 --> 00:18:05,420 Yes. 343 00:18:05,420 --> 00:18:08,110 AUDIENCE: The energy of the gamma, is that just a known? 344 00:18:08,110 --> 00:18:09,097 Like the .48 MeV? 345 00:18:09,097 --> 00:18:10,680 MICHAEL SHORT: That's something either 346 00:18:10,680 --> 00:18:13,500 I tell you or you would measure, let's say. 347 00:18:13,500 --> 00:18:15,640 That's just for completeness to say all right, 348 00:18:15,640 --> 00:18:17,680 this reaction actually gives off a gamma, 349 00:18:17,680 --> 00:18:21,570 and I want to give the right value for the kinetic energies. 350 00:18:21,570 --> 00:18:24,230 And we'll get into what gamma transitions are allowed 351 00:18:24,230 --> 00:18:26,960 and then how you measure them in the next couple of weeks, 352 00:18:26,960 --> 00:18:28,430 actually. 353 00:18:28,430 --> 00:18:29,810 Yes. 354 00:18:29,810 --> 00:18:33,110 AUDIENCE: So do we refer to Q as the Q you calculated up there, 355 00:18:33,110 --> 00:18:36,110 or that 2.31? 356 00:18:36,110 --> 00:18:38,732 MICHAEL SHORT: They're the same one, actually. 357 00:18:38,732 --> 00:18:39,440 AUDIENCE: The 2.7 358 00:18:39,440 --> 00:18:42,190 MICHAEL SHORT: Oh, I see. 359 00:18:42,190 --> 00:18:44,270 That's a good point. 360 00:18:44,270 --> 00:18:46,210 So this wouldn't really be Q, would it? 361 00:18:46,210 --> 00:18:49,300 But it is the sum of the kinetic energies. 362 00:18:49,300 --> 00:18:54,380 This is like Q minus the gamma ray energy. 363 00:18:54,380 --> 00:18:55,520 Let me stick that in there. 364 00:19:00,000 --> 00:19:01,980 L, i. 365 00:19:01,980 --> 00:19:02,580 Yep. 366 00:19:02,580 --> 00:19:03,080 Yeah. 367 00:19:03,080 --> 00:19:03,580 Good point. 368 00:19:08,420 --> 00:19:09,500 OK. 369 00:19:09,500 --> 00:19:12,250 Any other questions before I move on? 370 00:19:12,250 --> 00:19:15,040 We're going to get into a universal formula 371 00:19:15,040 --> 00:19:18,940 to predict in a so-so way what the binding energy of any given 372 00:19:18,940 --> 00:19:21,490 nucleus will be and start looking at stability trends 373 00:19:21,490 --> 00:19:24,885 so you can predict, just from the number of protons 374 00:19:24,885 --> 00:19:27,490 and the number of neutrons, how stable a nucleus will 375 00:19:27,490 --> 00:19:31,840 be with a few exceptions, which we will go over. 376 00:19:31,840 --> 00:19:34,270 And this is what's referred to as the semi-empirical mass 377 00:19:34,270 --> 00:19:35,510 formula. 378 00:19:35,510 --> 00:19:38,590 So I'm going to erase some stuff. 379 00:19:38,590 --> 00:19:43,510 Has everyone got the notes on this bottom board right here? 380 00:19:43,510 --> 00:19:44,170 OK. 381 00:19:44,170 --> 00:19:47,350 Let me know when you're ready, and let's see. 382 00:19:49,790 --> 00:19:50,290 Yeah. 383 00:19:50,290 --> 00:19:52,870 I want to make sure I move at your guys' pace. 384 00:20:06,610 --> 00:20:13,400 Let's say going to have a graph of binding energy per nucleon 385 00:20:13,400 --> 00:20:17,027 on versus nucleons. 386 00:20:17,027 --> 00:20:18,610 Well, anyway, I'll leave that up there 387 00:20:18,610 --> 00:20:20,650 and I'll do the work on this board right here. 388 00:20:20,650 --> 00:20:23,890 So let's say we wanted to figure out a weighted graph 389 00:20:23,890 --> 00:20:27,340 or to predict the binding energy per nucleon. 390 00:20:27,340 --> 00:20:29,170 So I have this binding energy term 391 00:20:29,170 --> 00:20:32,380 over a, where a is the total number of nucleons, 392 00:20:32,380 --> 00:20:35,050 as a function of the number of nucleons-- 393 00:20:35,050 --> 00:20:36,100 in a generalized way. 394 00:20:36,100 --> 00:20:40,420 Not accounting for magic numbers or anything else that we'll 395 00:20:40,420 --> 00:20:41,530 get into pretty soon. 396 00:20:41,530 --> 00:20:43,720 And I don't like the term magic numbers, 397 00:20:43,720 --> 00:20:46,390 but that is the parlance that's used in this field 398 00:20:46,390 --> 00:20:48,170 so I'm going to stick with it. 399 00:20:48,170 --> 00:20:51,520 Let's try and think about if you imagine the nucleus 400 00:20:51,520 --> 00:20:53,290 as a kind of drop of liquid-- and one 401 00:20:53,290 --> 00:20:55,660 of the other words for the semi-empirical mass formula 402 00:20:55,660 --> 00:20:59,650 is called the liquid drop formula or the liquid drop 403 00:20:59,650 --> 00:21:02,420 model. 404 00:21:02,420 --> 00:21:05,030 It assumes that the nucleus takes the shape 405 00:21:05,030 --> 00:21:06,980 roughly of a liquid drop, and you can kind of 406 00:21:06,980 --> 00:21:11,147 treat some of the energy terms accordingly. 407 00:21:11,147 --> 00:21:13,480 This is why I have all the different colors of chalk out 408 00:21:13,480 --> 00:21:14,660 for this. 409 00:21:14,660 --> 00:21:18,210 Makes it a little visually easier to see. 410 00:21:18,210 --> 00:21:20,660 So let's start writing a general expression 411 00:21:20,660 --> 00:21:24,268 for the binding energy as a function of a and z, 412 00:21:24,268 --> 00:21:26,060 and start thinking about what sort of terms 413 00:21:26,060 --> 00:21:28,940 would add to or decrease the stability 414 00:21:28,940 --> 00:21:32,540 of a given liquid drop nucleus, where all the nucleons are just 415 00:21:32,540 --> 00:21:35,660 kind of there in some sort of floating, crazy, coulombic, 416 00:21:35,660 --> 00:21:38,000 strong nuclear force soup. 417 00:21:38,000 --> 00:21:40,430 First of all, as you add nucleons 418 00:21:40,430 --> 00:21:42,620 to a given nucleus, what tends to happen 419 00:21:42,620 --> 00:21:44,600 to the binding energy, in general? 420 00:21:47,280 --> 00:21:50,030 Without knowing anything else. 421 00:21:50,030 --> 00:21:53,390 Assemble more nucleons, you convert more mass to energy. 422 00:21:53,390 --> 00:21:56,870 And you end up increasing the binding energy. 423 00:21:56,870 --> 00:21:59,240 So let's call this the volume term. 424 00:21:59,240 --> 00:22:02,310 As you increase the volume of this liquid drop, 425 00:22:02,310 --> 00:22:05,700 its total binding energies start to increase. 426 00:22:05,700 --> 00:22:07,550 So let's say there's some term that's 427 00:22:07,550 --> 00:22:13,190 going to be proportional to A, the number of nucleons that's 428 00:22:13,190 --> 00:22:15,027 in this liquid drop. 429 00:22:15,027 --> 00:22:15,860 And we'll draw this. 430 00:22:15,860 --> 00:22:17,970 Let's see if I can do the trick right. 431 00:22:17,970 --> 00:22:18,920 Yes. 432 00:22:18,920 --> 00:22:21,700 I love doing that. 433 00:22:21,700 --> 00:22:23,990 If we were to graph binding energy per nucleon 434 00:22:23,990 --> 00:22:27,620 as a function of number of nucleotides for this term, 435 00:22:27,620 --> 00:22:30,230 it would just be a flat line because it's related to A. 436 00:22:30,230 --> 00:22:35,820 And there's going to be some constant, which we're 437 00:22:35,820 --> 00:22:39,335 going to call the volume constant, that says, well, 438 00:22:39,335 --> 00:22:40,710 there's going to be some relation 439 00:22:40,710 --> 00:22:42,690 between the actual amount of stability gained 440 00:22:42,690 --> 00:22:44,100 and the number of nucleons. 441 00:22:44,100 --> 00:22:45,840 We don't know what it is yet, but what 442 00:22:45,840 --> 00:22:48,090 we're really concerned with is the functional 443 00:22:48,090 --> 00:22:50,040 form of this thing. 444 00:22:50,040 --> 00:22:52,720 It's proportional to A. 445 00:22:52,720 --> 00:22:56,410 Next up, what also happens to a liquid drop 446 00:22:56,410 --> 00:22:57,917 as you increase its volume? 447 00:22:57,917 --> 00:22:59,500 What other parameters do you increase? 448 00:23:02,755 --> 00:23:03,880 AUDIENCE: The surface area? 449 00:23:03,880 --> 00:23:04,838 MICHAEL SHORT: Exactly. 450 00:23:04,838 --> 00:23:05,740 The surface area. 451 00:23:05,740 --> 00:23:08,440 The idea here is that if this liquid drop 452 00:23:08,440 --> 00:23:10,810 is made of all sorts of different nucleons-- 453 00:23:13,660 --> 00:23:15,920 and let's pretend that they're like atoms in a crystal 454 00:23:15,920 --> 00:23:19,460 and they're all binding to each other-- 455 00:23:19,460 --> 00:23:24,860 the ones on the outside aren't bound to as many nucleons 456 00:23:24,860 --> 00:23:26,990 as the ones on the inside. 457 00:23:26,990 --> 00:23:29,870 And so the more nuclei there are near the surface 458 00:23:29,870 --> 00:23:33,380 as opposed to inside the liquid droplets, say, 459 00:23:33,380 --> 00:23:36,020 inside some little radius where all it sees around 460 00:23:36,020 --> 00:23:39,470 it are other nucleons, then they're not quite as bound. 461 00:23:39,470 --> 00:23:42,020 And how does the surface area of a liquid drop 462 00:23:42,020 --> 00:23:44,833 scale with its volume? 463 00:23:44,833 --> 00:23:46,375 To what function or to what exponent? 464 00:23:51,520 --> 00:23:52,780 2/3. 465 00:23:52,780 --> 00:23:56,800 I mean, let's take a quick look at the volume of a sphere 466 00:23:56,800 --> 00:24:00,050 is 4/3 pi r cubed. 467 00:24:00,050 --> 00:24:04,070 And the surface area of a sphere is 4 pi r squared. 468 00:24:04,070 --> 00:24:06,160 So if you want to get some expression for how 469 00:24:06,160 --> 00:24:11,080 does area scale as volume, I said cube, 470 00:24:11,080 --> 00:24:12,720 then I wrote squared. 471 00:24:12,720 --> 00:24:15,460 It's going to end up looking like something times 472 00:24:15,460 --> 00:24:18,220 r to the 2-- let's see. 473 00:24:18,220 --> 00:24:18,790 Oh, yeah. 474 00:24:18,790 --> 00:24:20,998 I'm sorry, that's not the expression I want to write. 475 00:24:20,998 --> 00:24:24,680 But the idea here is it's going to scale with r to the 2/3. 476 00:24:24,680 --> 00:24:28,300 So let's pick a different color and say 477 00:24:28,300 --> 00:24:33,940 we're going to have some surface term times number of nucleons 478 00:24:33,940 --> 00:24:35,830 to the 2/3. 479 00:24:35,830 --> 00:24:39,670 And if we then adjust this formula to also take 480 00:24:39,670 --> 00:24:42,220 into account this surface area term-- 481 00:24:42,220 --> 00:24:44,930 which is to say for very small nuclei, 482 00:24:44,930 --> 00:24:48,070 there's a lot of nucleotides near the surface, 483 00:24:48,070 --> 00:24:50,040 and as the nucleus gets bigger and bigger, 484 00:24:50,040 --> 00:24:52,540 more and more of them are in the juicy center and don't know 485 00:24:52,540 --> 00:24:54,130 they're near the surface-- 486 00:24:54,130 --> 00:24:59,710 we'd have some modification that looks like that. 487 00:24:59,710 --> 00:25:01,960 Now I'm going to erase the stuff over here because I'm 488 00:25:01,960 --> 00:25:02,793 running out of room. 489 00:25:05,900 --> 00:25:10,460 Now these nucleons aren't just untagged, anonymous nucleons. 490 00:25:10,460 --> 00:25:12,440 They're either protons or neutrons. 491 00:25:15,618 --> 00:25:17,910 And what happens when you try and cram a lot of protons 492 00:25:17,910 --> 00:25:19,413 into one space? 493 00:25:19,413 --> 00:25:21,080 AUDIENCE: They want to repel each other. 494 00:25:21,080 --> 00:25:21,872 MICHAEL SHORT: Yep. 495 00:25:21,872 --> 00:25:24,390 They want to repel each other by coulombic forces. 496 00:25:24,390 --> 00:25:26,220 And so every proton-- 497 00:25:26,220 --> 00:25:29,800 let's pick a different color for the coulombic forces-- 498 00:25:29,800 --> 00:25:31,550 and that should be a minus if I want 499 00:25:31,550 --> 00:25:34,950 to stick with all the notation. 500 00:25:34,950 --> 00:25:37,650 There's going to be some other term to account for the fact 501 00:25:37,650 --> 00:25:40,560 that the nuclei, specifically the protons, 502 00:25:40,560 --> 00:25:42,668 are trying to repel each other. 503 00:25:42,668 --> 00:25:45,210 So in this case, it's going to be proportional to, let's say, 504 00:25:45,210 --> 00:25:48,450 the number of protons that we have. 505 00:25:48,450 --> 00:25:51,300 And every proton should feel a repulsive force 506 00:25:51,300 --> 00:25:52,325 from every other proton. 507 00:25:55,800 --> 00:25:59,260 So let's say it's times z times z minus 1, 508 00:25:59,260 --> 00:26:02,220 so that every proton feels the force of every other proton 509 00:26:02,220 --> 00:26:04,290 except for itself. 510 00:26:04,290 --> 00:26:08,310 And that's going to be mediated by the total number 511 00:26:08,310 --> 00:26:09,040 of nucleons. 512 00:26:09,040 --> 00:26:11,160 So if there are more neutrons in the way, 513 00:26:11,160 --> 00:26:13,280 it won't be quite as bad. 514 00:26:13,280 --> 00:26:14,850 And there's going to be some other-- 515 00:26:14,850 --> 00:26:18,430 we'll call it a C for the coulombic term-- 516 00:26:18,430 --> 00:26:21,040 and that will say that as you make a bigger and bigger 517 00:26:21,040 --> 00:26:23,200 nucleus, you start to get more and more coulombic 518 00:26:23,200 --> 00:26:26,060 repulsion trying to rip it apart. 519 00:26:26,060 --> 00:26:29,350 So if we were to then modify the purple curve-- 520 00:26:29,350 --> 00:26:30,100 oops. 521 00:26:30,100 --> 00:26:35,700 Trying to get it to go the same as the nucleus gets bigger. 522 00:26:35,700 --> 00:26:37,590 I want to make sure it's really to scale-ish. 523 00:26:42,310 --> 00:26:44,080 As the nuclei get bigger and bigger, 524 00:26:44,080 --> 00:26:46,485 it's going to be a little less stable. 525 00:26:46,485 --> 00:26:47,860 And already we're starting to get 526 00:26:47,860 --> 00:26:51,822 a curve that is getting close to looking like the binding energy 527 00:26:51,822 --> 00:26:52,780 curve from the reading. 528 00:26:52,780 --> 00:26:55,570 But there are a couple more terms to reckon with. 529 00:26:55,570 --> 00:26:58,623 So let's pick a fourth color. 530 00:26:58,623 --> 00:27:00,040 What other sort of trends that you 531 00:27:00,040 --> 00:27:02,623 notice in the reading about the stability of different nuclei? 532 00:27:06,210 --> 00:27:09,060 Let's say you were to take a common nucleus like carbon-12. 533 00:27:09,060 --> 00:27:11,850 It's got six protons and six neutrons, 534 00:27:11,850 --> 00:27:13,830 and it's exceptionally stable. 535 00:27:13,830 --> 00:27:15,600 What about carbon 6? 536 00:27:15,600 --> 00:27:18,930 A nucleus of just 6 protons? 537 00:27:18,930 --> 00:27:19,620 Doesn't exist. 538 00:27:19,620 --> 00:27:21,600 Exceptionally unstable. 539 00:27:21,600 --> 00:27:23,560 What about carbon 24? 540 00:27:23,560 --> 00:27:25,500 18 neutrons, six protons. 541 00:27:29,020 --> 00:27:31,120 Sound stable or not? 542 00:27:31,120 --> 00:27:32,050 Not at all. 543 00:27:32,050 --> 00:27:35,140 So there's some sort of asymmetry term going on. 544 00:27:35,140 --> 00:27:37,030 When the number of protons and neutrons 545 00:27:37,030 --> 00:27:40,480 is roughly in balance, especially for light nuclei, 546 00:27:40,480 --> 00:27:42,580 the nucleus tends to be more stable. 547 00:27:42,580 --> 00:27:44,410 So we can write some sort of term-- 548 00:27:44,410 --> 00:27:48,340 let's call it an asymmetry term-- 549 00:27:48,340 --> 00:27:51,970 that relates to the number of neutrons 550 00:27:51,970 --> 00:27:54,840 minus the number of protons. 551 00:27:54,840 --> 00:27:58,780 And in this case, for reasons I'm not going to get into, 552 00:27:58,780 --> 00:28:01,050 but are derived in a reference in your reading, 553 00:28:01,050 --> 00:28:02,050 there's a squared on it. 554 00:28:02,050 --> 00:28:04,930 But suffice to say if the number of neutrons and number 555 00:28:04,930 --> 00:28:07,630 of protons are equal, then the nucleus is 556 00:28:07,630 --> 00:28:09,490 predicted to be pretty stable. 557 00:28:09,490 --> 00:28:11,560 And this works out quite well for light nuclei. 558 00:28:11,560 --> 00:28:16,090 It starts to break down a little bit for heavier nuclei. 559 00:28:16,090 --> 00:28:21,070 And then divide by the number of nuclei that there are. 560 00:28:21,070 --> 00:28:25,230 I also see a missing 1/3 because let's say 561 00:28:25,230 --> 00:28:28,230 this nucleus has a volume that scales with roughly 562 00:28:28,230 --> 00:28:29,880 the number of nucleons. 563 00:28:29,880 --> 00:28:31,920 Then the distance of that coulombic force 564 00:28:31,920 --> 00:28:34,890 is going to be like A to the 1/3 or the radius 565 00:28:34,890 --> 00:28:36,810 of this nuclear drop. 566 00:28:36,810 --> 00:28:40,100 So let's take the asymmetry term. 567 00:28:40,100 --> 00:28:44,630 That's going to give us a further modification slightly 568 00:28:44,630 --> 00:28:46,690 downward. 569 00:28:46,690 --> 00:28:49,030 And finally, there's what's called the pairing term. 570 00:28:49,030 --> 00:28:50,800 What's the last color I haven't used? 571 00:28:55,860 --> 00:28:57,210 This pairing term delta. 572 00:28:57,210 --> 00:28:58,710 And this is not a smooth function. 573 00:28:58,710 --> 00:29:01,140 It's a piecewise function that depends on 574 00:29:01,140 --> 00:29:04,260 whether you have an odd or an even number 575 00:29:04,260 --> 00:29:07,513 of each type of nucleon, protons or neutrons. 576 00:29:07,513 --> 00:29:08,930 And so what this means, it's going 577 00:29:08,930 --> 00:29:11,370 to add a little bit of jaggedness 578 00:29:11,370 --> 00:29:16,630 to the beginning of the curve and equal out in the end 579 00:29:16,630 --> 00:29:18,730 because this delta term can be something 580 00:29:18,730 --> 00:29:23,320 like plus, let's call it an A pairing. 581 00:29:23,320 --> 00:29:30,220 And it scales with the square root of A, or minus, or 0, 582 00:29:30,220 --> 00:29:32,860 depending on if the nuclei are odd odd, 583 00:29:32,860 --> 00:29:35,680 like odd number of protons, odd number of neutrons, 584 00:29:35,680 --> 00:29:39,460 even even, or odd even. 585 00:29:39,460 --> 00:29:42,160 Now I know that the derivation is a little hand wavy. 586 00:29:42,160 --> 00:29:44,380 That's why we call it semi-empirical. 587 00:29:44,380 --> 00:29:46,450 We're taking each of these additive terms 588 00:29:46,450 --> 00:29:50,920 and saying it kind of comes from a fairly OK, a little poor 589 00:29:50,920 --> 00:29:52,630 approximation of the nucleus. 590 00:29:52,630 --> 00:29:55,570 But what we end up with is a formula 591 00:29:55,570 --> 00:30:00,580 whose constants are fit, whose terms, the actual variables, 592 00:30:00,580 --> 00:30:04,360 are derived somewhat from physical intuition. 593 00:30:04,360 --> 00:30:07,772 These constants were then fit later by some other folks, 594 00:30:07,772 --> 00:30:09,730 and the references for this are in the reading. 595 00:30:09,730 --> 00:30:13,630 They're all in MeV, and this gives you a binding energy 596 00:30:13,630 --> 00:30:17,210 in MeV for a given nucleus. 597 00:30:17,210 --> 00:30:20,820 Now it works some places and it doesn't work in other places. 598 00:30:20,820 --> 00:30:21,320 Yeah. 599 00:30:21,320 --> 00:30:24,054 AUDIENCE: So the lower case a [INAUDIBLE] 600 00:30:24,054 --> 00:30:26,030 no matter what the nucleus looks like? 601 00:30:26,030 --> 00:30:27,260 MICHAEL SHORT: That's right. 602 00:30:27,260 --> 00:30:31,340 So this is the universal, semi-empirical, usually works 603 00:30:31,340 --> 00:30:35,480 formula for the binding energy of a nucleus. 604 00:30:35,480 --> 00:30:38,390 So the constants don't change because the variables here 605 00:30:38,390 --> 00:30:41,840 are z, [INAUDIBLE] a, and n. 606 00:30:41,840 --> 00:30:43,520 And don't forget-- because you'll 607 00:30:43,520 --> 00:30:45,260 need to remember this on the homework-- 608 00:30:45,260 --> 00:30:48,830 that A equals z plus n. 609 00:30:48,830 --> 00:30:52,790 So for example, if you want to express 610 00:30:52,790 --> 00:30:55,700 what is the most stable nucleus, you 611 00:30:55,700 --> 00:30:58,040 could take the derivative of this formula with respect 612 00:30:58,040 --> 00:30:59,450 to A or z. 613 00:30:59,450 --> 00:31:03,520 And don't forget that you can substitute this expression 614 00:31:03,520 --> 00:31:06,020 into there. 615 00:31:06,020 --> 00:31:09,190 That's giving you guys a hint for the homework. 616 00:31:09,190 --> 00:31:10,940 And let's look at what this actually looks 617 00:31:10,940 --> 00:31:14,480 like as far as theory compared to experiment. 618 00:31:14,480 --> 00:31:17,240 So the red points are theoretical predictions. 619 00:31:17,240 --> 00:31:19,940 The black points are experimental predictions. 620 00:31:19,940 --> 00:31:22,770 And all of the different nuclei are shown here. 621 00:31:22,770 --> 00:31:25,460 First of all, the curve looks quite a bit like the one 622 00:31:25,460 --> 00:31:28,100 that we just hand wavy made on the board right here. 623 00:31:28,100 --> 00:31:31,010 And second of all, there aren't too many exceptions. 624 00:31:31,010 --> 00:31:32,810 It's hard to see what the exceptions are, 625 00:31:32,810 --> 00:31:35,300 so it's a little easier to draw them 626 00:31:35,300 --> 00:31:36,590 in terms of relative error. 627 00:31:36,590 --> 00:31:38,780 So you can see where does this formula work 628 00:31:38,780 --> 00:31:40,910 and where does this formula not? 629 00:31:40,910 --> 00:31:43,430 So if you notice, for the small nuclei, 630 00:31:43,430 --> 00:31:45,320 approximating it as a liquid drop 631 00:31:45,320 --> 00:31:47,690 is not a very good approximation because you 632 00:31:47,690 --> 00:31:50,570 can't treat this as a homogenized, 633 00:31:50,570 --> 00:31:51,800 smeared liquid drop. 634 00:31:51,800 --> 00:31:53,630 It's much more-- well, there's either 635 00:31:53,630 --> 00:31:57,320 two or three nucleons, and very few protons and neutrons 636 00:31:57,320 --> 00:31:58,340 in each. 637 00:31:58,340 --> 00:32:00,650 But then as you get to larger and larger nuclei, 638 00:32:00,650 --> 00:32:05,435 it starts to hit very close with a few exceptions 639 00:32:05,435 --> 00:32:08,980 that I want to point out right now. 640 00:32:08,980 --> 00:32:10,870 If you zoom in on that part, you can actually 641 00:32:10,870 --> 00:32:14,350 see that at certain neutron numbers, or certain proton 642 00:32:14,350 --> 00:32:17,560 numbers, there is an exceptionally high stability 643 00:32:17,560 --> 00:32:18,700 of a lot of those nuclei. 644 00:32:18,700 --> 00:32:21,010 And that's as you start to approach 645 00:32:21,010 --> 00:32:22,510 these what's called magic numbers, 646 00:32:22,510 --> 00:32:26,110 or numbers of nucleons which, say, fill all energy 647 00:32:26,110 --> 00:32:27,880 levels at a certain level. 648 00:32:27,880 --> 00:32:30,850 And again, it's not for every nucleus as a function 649 00:32:30,850 --> 00:32:31,700 of neutron number. 650 00:32:31,700 --> 00:32:34,370 But even drawing an envelope around this curve, 651 00:32:34,370 --> 00:32:37,060 you can see that the nuclei around 82, 652 00:32:37,060 --> 00:32:41,110 around 50, around 28, are a whole lot more stable 653 00:32:41,110 --> 00:32:42,190 than the ones in between. 654 00:32:42,190 --> 00:32:45,700 And this pattern kind of repeats with larger and larger 655 00:32:45,700 --> 00:32:47,200 periodicity. 656 00:32:47,200 --> 00:32:48,790 And it kind of looks like right here, 657 00:32:48,790 --> 00:32:51,400 at the edge of our knowledge of nuclei, 658 00:32:51,400 --> 00:32:53,935 we haven't quite gotten to the next peak yet. 659 00:32:53,935 --> 00:32:56,560 This is something we're going to talk about Friday on the quest 660 00:32:56,560 --> 00:32:59,110 for super heavy elements, or SHE's, 661 00:32:59,110 --> 00:33:00,640 as you'll see in the reading. 662 00:33:00,640 --> 00:33:01,190 Yeah. 663 00:33:01,190 --> 00:33:05,680 AUDIENCE: So the most stable nuclei peaks were closest to 0. 664 00:33:05,680 --> 00:33:08,260 MICHAEL SHORT: Closest to 0 is the closest agreement 665 00:33:08,260 --> 00:33:11,290 between experiment and theory. 666 00:33:11,290 --> 00:33:13,470 So the ones that are exceptionally stable, 667 00:33:13,470 --> 00:33:17,040 which are not predicted by this very simple formula, 668 00:33:17,040 --> 00:33:19,680 are up here at the peaks of these magic numbers. 669 00:33:19,680 --> 00:33:22,410 And actually, I want you to take a look right here at some 670 00:33:22,410 --> 00:33:24,190 of these very small nuclei. 671 00:33:24,190 --> 00:33:27,390 Like helium 4 is probably way up here somewhere, all 672 00:33:27,390 --> 00:33:28,680 the way over on the right. 673 00:33:28,680 --> 00:33:30,990 It's an exceptionally stable nucleus 674 00:33:30,990 --> 00:33:33,390 that is not very well approximated by liquid drop 675 00:33:33,390 --> 00:33:35,460 model because it's got four nucleons. 676 00:33:35,460 --> 00:33:36,750 They're all on the surface. 677 00:33:36,750 --> 00:33:41,100 There's nothing on the inside of a helium nucleus, let's say. 678 00:33:41,100 --> 00:33:43,210 And then if you look at stability trends in terms 679 00:33:43,210 --> 00:33:46,990 of are the nuclei more stable if they have odd numbers 680 00:33:46,990 --> 00:33:50,050 or even numbers, you can graph the two separately 681 00:33:50,050 --> 00:33:52,690 and look at the number of stable nuclei 682 00:33:52,690 --> 00:33:56,020 that have an odd total mass number or an even total mass 683 00:33:56,020 --> 00:33:56,890 number. 684 00:33:56,890 --> 00:33:58,880 And there's a few things to note here. 685 00:33:58,880 --> 00:34:01,480 One of them is the even numbers tend to have 686 00:34:01,480 --> 00:34:03,310 a lot more stable nuclei. 687 00:34:03,310 --> 00:34:06,050 This is something I mentioned on the second day of class. 688 00:34:06,050 --> 00:34:10,030 If you look at the [INAUDIBLE] table of nuclides-- 689 00:34:10,030 --> 00:34:12,370 let's go to their home page-- 690 00:34:12,370 --> 00:34:17,210 and just look at it sort of in a color way. 691 00:34:17,210 --> 00:34:19,550 The blue colors are stable nuclei, 692 00:34:19,550 --> 00:34:21,770 and you notice that every other row of pixels 693 00:34:21,770 --> 00:34:25,219 here has a whole lot more stable ones. 694 00:34:25,219 --> 00:34:28,190 And that's the same thing that we're seeing right here, 695 00:34:28,190 --> 00:34:32,480 is that there's a lot more even nuclei that are more stable. 696 00:34:32,480 --> 00:34:36,540 If I jump back to our semi-empirical maths formula, 697 00:34:36,540 --> 00:34:40,162 notice that this binding energy goes up for even, even nuclei. 698 00:34:40,162 --> 00:34:41,870 So when there's an even number of protons 699 00:34:41,870 --> 00:34:45,380 and an even number of neutrons, the semi-empirical mass formula 700 00:34:45,380 --> 00:34:48,710 does predict an increase in stability, 701 00:34:48,710 --> 00:34:53,909 which you can actually see on the table of nuclides, 702 00:34:53,909 --> 00:34:56,630 and on this sort of stability trend. 703 00:34:56,630 --> 00:34:58,020 And so let's look a little closer 704 00:34:58,020 --> 00:35:03,150 and see how many nuclei for each proton number or each neutron 705 00:35:03,150 --> 00:35:04,860 number are actually stable. 706 00:35:04,860 --> 00:35:07,622 And we graphed the odd and the even ones separately. 707 00:35:07,622 --> 00:35:09,330 And what's important to note here is one, 708 00:35:09,330 --> 00:35:11,550 the odd is way lower than the even. 709 00:35:11,550 --> 00:35:16,590 There's usually either 2 1 or 0 stable nuclei at that number. 710 00:35:16,590 --> 00:35:19,800 And what other sort of features do you guys notice about this? 711 00:35:24,390 --> 00:35:26,603 It's not smooth, first of all. 712 00:35:26,603 --> 00:35:27,520 Where are those peaks? 713 00:35:27,520 --> 00:35:29,603 Where do you tend to find that most stable nuclei? 714 00:35:33,210 --> 00:35:33,710 4? 715 00:35:33,710 --> 00:35:37,367 Where do you tend to find the least? 716 00:35:37,367 --> 00:35:38,700 What about these two right here? 717 00:35:38,700 --> 00:35:41,530 No stable nuclei at these proton numbers. 718 00:35:41,530 --> 00:35:44,340 And remember, proton number uniquely defines an element. 719 00:35:44,340 --> 00:35:47,643 Anyone know what these two might be? 720 00:35:47,643 --> 00:35:48,560 What sort of elements? 721 00:35:48,560 --> 00:35:50,268 Look to the back of the room if you want. 722 00:35:50,268 --> 00:35:52,940 There's a periodic table in the back wall. 723 00:35:52,940 --> 00:35:55,280 And you can see, except for the super heavy things 724 00:35:55,280 --> 00:35:56,840 down at the bottom, there's a couple 725 00:35:56,840 --> 00:36:00,150 of elements that have no stable isotopes. 726 00:36:00,150 --> 00:36:03,920 These are technetium and promethium, which 727 00:36:03,920 --> 00:36:05,750 are relatively light isotopes-- 728 00:36:05,750 --> 00:36:08,570 I say relatively light compared to things like uranium-- 729 00:36:08,570 --> 00:36:10,490 with no stable isotopes. 730 00:36:10,490 --> 00:36:13,460 They're also fairly far away from these so-called magic 731 00:36:13,460 --> 00:36:17,480 numbers or other regions, where you tend to have a spike 732 00:36:17,480 --> 00:36:20,360 in the number of stable nuclei due to-- 733 00:36:20,360 --> 00:36:22,700 well, things that you'll learn in 22.02, 734 00:36:22,700 --> 00:36:26,540 in terms of nuclear shell occupancy and stability. 735 00:36:26,540 --> 00:36:30,700 But you see the same thing when you graph the neutron number. 736 00:36:30,700 --> 00:36:32,290 You can see a couple of sudden spikes 737 00:36:32,290 --> 00:36:38,693 right here at 20, 28, 50, 82, and 126. 738 00:36:38,693 --> 00:36:41,110 When everything gets really stable, all of a sudden you've 739 00:36:41,110 --> 00:36:45,010 got one last gasp of a stable isotope 740 00:36:45,010 --> 00:36:48,600 before you go off into nowhere land. 741 00:36:48,600 --> 00:36:51,540 So let's start looking at relative stabilities of nuclei, 742 00:36:51,540 --> 00:36:54,510 let's say, for a given mass number or a given proton 743 00:36:54,510 --> 00:36:56,340 number. 744 00:36:56,340 --> 00:36:57,840 Anyone mind if I cover this board 745 00:36:57,840 --> 00:36:59,210 because you can't roll it up. 746 00:36:59,210 --> 00:37:01,581 You all got the notes from here? 747 00:37:01,581 --> 00:37:04,510 Cool. 748 00:37:04,510 --> 00:37:07,840 That one has less to erase. 749 00:37:07,840 --> 00:37:10,590 And I want to keep these formulas up for our reference. 750 00:37:22,210 --> 00:37:24,075 Let's say I pose this problem. 751 00:37:24,075 --> 00:37:26,700 I want to find out, make sure I solve the right one-- actually, 752 00:37:26,700 --> 00:37:28,780 I'm going to check my notes real quick-- 753 00:37:28,780 --> 00:37:33,250 that what is the-- for a given A, or for a given mass number-- 754 00:37:38,295 --> 00:37:40,045 what is the most stable number of protons? 755 00:37:47,980 --> 00:37:50,890 for A, given A. 756 00:37:50,890 --> 00:37:53,350 This is the question that I like to answer here. 757 00:37:53,350 --> 00:37:54,940 How would you approach this question 758 00:37:54,940 --> 00:37:57,700 using the semi-empirical mass formula 759 00:37:57,700 --> 00:37:59,830 that-- well, you can't see here, so I will bring it 760 00:37:59,830 --> 00:38:02,384 up back on the screen. 761 00:38:02,384 --> 00:38:02,884 Here. 762 00:38:05,540 --> 00:38:06,707 How would you find this out? 763 00:38:16,950 --> 00:38:20,170 Well, let's say for a given mass number 764 00:38:20,170 --> 00:38:24,000 A, for a given poor approximation of the total mass 765 00:38:24,000 --> 00:38:28,110 of a nucleus, the more binding energy it has, the more stable 766 00:38:28,110 --> 00:38:29,220 it is. 767 00:38:29,220 --> 00:38:37,500 Therefore, if you want to find the minimum of a mass 768 00:38:37,500 --> 00:38:43,540 for A and Z, given a fixed A, that 769 00:38:43,540 --> 00:38:45,310 will give you the most stable nucleus 770 00:38:45,310 --> 00:38:48,760 because it will tell you which value of Z 771 00:38:48,760 --> 00:38:50,530 gives you the smallest m, or rather, 772 00:38:50,530 --> 00:38:52,810 the most tightly bound nucleus that 773 00:38:52,810 --> 00:38:55,387 has the most binding energy. 774 00:38:55,387 --> 00:38:56,720 So let's start writing this out. 775 00:38:56,720 --> 00:38:59,690 First of all, we can use one of the two equations 776 00:38:59,690 --> 00:39:02,660 we already have up here, a relation between the mass 777 00:39:02,660 --> 00:39:04,200 and the binding energy. 778 00:39:04,200 --> 00:39:07,040 The second one, well, we have it right here. 779 00:39:07,040 --> 00:39:11,510 So let's substitute in our binding energy equation 780 00:39:11,510 --> 00:39:15,590 and express it in terms of mass. 781 00:39:15,590 --> 00:39:25,970 So let's say our mass of a nucleus A and Z is equal to Z 782 00:39:25,970 --> 00:39:34,360 times the mass of hydrogen plus A minus Z times the mass 783 00:39:34,360 --> 00:39:39,222 of a neutron minus the binding energy 784 00:39:39,222 --> 00:39:41,180 because in this case, what I've done right here 785 00:39:41,180 --> 00:39:44,510 is I've added mass to each side of the equation, subtracted 786 00:39:44,510 --> 00:39:46,690 binding energy from each side of the equation, 787 00:39:46,690 --> 00:39:48,770 and we can just take negative that expression 788 00:39:48,770 --> 00:39:50,610 and write it all out together. 789 00:39:50,610 --> 00:40:04,510 So we have minus AVA plus A surface A to the 2/3 plus AC z 790 00:40:04,510 --> 00:40:09,670 times z minus 1 over A to the 1/3, 791 00:40:09,670 --> 00:40:22,690 and plus AA symmetry and minus z squared over A and minus delta. 792 00:40:22,690 --> 00:40:26,220 So we've got one expression for the total mass. 793 00:40:26,220 --> 00:40:28,200 We've fixed the value of A because we're 794 00:40:28,200 --> 00:40:29,580 going to take some fixed-- 795 00:40:29,580 --> 00:40:31,620 we're going to choose some fixed value of A. 796 00:40:31,620 --> 00:40:33,945 Let's say A equals 93. 797 00:40:33,945 --> 00:40:36,570 And that's the example that I've kind of worked out in my head. 798 00:40:36,570 --> 00:40:40,680 It so happens that niobium has a stable isotope that a mass of A 799 00:40:40,680 --> 00:40:41,745 equals 93. 800 00:40:41,745 --> 00:40:43,620 And we just found out in some of our research 801 00:40:43,620 --> 00:40:45,700 that niobium doesn't stick to chromium very well. 802 00:40:45,700 --> 00:40:47,200 That's why I've got it on the brain. 803 00:40:47,200 --> 00:40:49,710 So this is what I was thinking about this morning. 804 00:40:49,710 --> 00:40:52,770 So for a fixed A equals 93, we want 805 00:40:52,770 --> 00:40:56,280 to find what is the most stable A. How do we do that? 806 00:41:00,690 --> 00:41:02,160 Anyone have an idea? 807 00:41:07,550 --> 00:41:08,050 Yeah. 808 00:41:08,050 --> 00:41:09,050 AUDIENCE: Differentiate? 809 00:41:09,050 --> 00:41:10,280 MICHAEL SHORT: Differentiate. 810 00:41:10,280 --> 00:41:11,580 Sure. 811 00:41:11,580 --> 00:41:16,418 Let's take, we'll just say the derivative of-- oh, no, 812 00:41:16,418 --> 00:41:17,960 it is a partial derivative because we 813 00:41:17,960 --> 00:41:19,080 have two variables here. 814 00:41:22,010 --> 00:41:25,670 Take the derivative as a function of z, 815 00:41:25,670 --> 00:41:27,140 set it equal to 0. 816 00:41:27,140 --> 00:41:29,960 This will give us the z number that gives us 817 00:41:29,960 --> 00:41:32,690 the minimum mass for a fixed A. So let's actually 818 00:41:32,690 --> 00:41:34,850 do this right now. 819 00:41:34,850 --> 00:41:36,140 So let's see. 820 00:41:36,140 --> 00:41:38,870 This gives us mh. 821 00:41:38,870 --> 00:41:46,880 And this term expanded out is AMN minus ZMN. 822 00:41:46,880 --> 00:41:49,240 So we have a minus MN. 823 00:41:49,240 --> 00:41:51,410 I'm going to make one quick correction right here. 824 00:41:51,410 --> 00:41:54,140 I want to make sure everything's in the same units. 825 00:41:54,140 --> 00:41:58,570 All of these semi-empirical mass terms are in MeV. 826 00:41:58,570 --> 00:42:01,960 These right here are in AMU, atomic mass units. 827 00:42:01,960 --> 00:42:07,344 What do we have to add in order to get these all in MeV? 828 00:42:07,344 --> 00:42:11,405 AUDIENCE: [INAUDIBLE] the conversion factor. 829 00:42:11,405 --> 00:42:13,030 MICHAEL SHORT: Conversion factor, yeah. 830 00:42:13,030 --> 00:42:16,640 Or in this case, we'll just stick a C squared. 831 00:42:16,640 --> 00:42:18,390 I'll make a little bit of room so I can 832 00:42:18,390 --> 00:42:21,510 stick the C squared in there. 833 00:42:21,510 --> 00:42:22,660 C squared. 834 00:42:22,660 --> 00:42:24,120 Now everything's in MeV. 835 00:42:24,120 --> 00:42:26,140 We're all in the same units. 836 00:42:26,140 --> 00:42:31,140 So let's say we have mh minus mn. 837 00:42:31,140 --> 00:42:33,240 These are in C squared. 838 00:42:33,240 --> 00:42:48,750 And minus VC plus 2/3 AS times A to the negative 1/3 plus-- 839 00:42:48,750 --> 00:42:54,240 I'm going to expand this out to call it z squared minus z. 840 00:42:54,240 --> 00:43:00,270 I'm also going to stick in n equals A minus Z 841 00:43:00,270 --> 00:43:06,420 so that this expands out to A minus 2Z squared. 842 00:43:06,420 --> 00:43:07,980 Is everyone with me here? 843 00:43:07,980 --> 00:43:08,630 Yeah. 844 00:43:08,630 --> 00:43:11,673 AUDIENCE: Shouldn't the A terms, like the AVA and the AS, 845 00:43:11,673 --> 00:43:12,400 stay at 2/3? 846 00:43:12,400 --> 00:43:13,798 MICHAEL SHORT: Oh. 847 00:43:13,798 --> 00:43:14,340 You're right. 848 00:43:14,340 --> 00:43:18,000 I'm deriving with respect to the wrong variable. 849 00:43:18,000 --> 00:43:19,270 Thank you. 850 00:43:19,270 --> 00:43:19,770 Yep. 851 00:43:19,770 --> 00:43:21,840 So we want to do this as a function of z. 852 00:43:21,840 --> 00:43:25,260 So that term disappears. 853 00:43:25,260 --> 00:43:27,630 That term disappears. 854 00:43:27,630 --> 00:43:28,752 Thank you. 855 00:43:28,752 --> 00:43:30,210 Let's work on these ones right now. 856 00:43:30,210 --> 00:43:34,590 So we have AC time z squared over A to the 1/3. 857 00:43:34,590 --> 00:43:43,260 So that will give us A plus AC over A to the 1/3 times 2z. 858 00:43:46,830 --> 00:43:50,090 And then we have-- 859 00:43:50,090 --> 00:43:54,960 let's see- minus AC over A to the 1/3. 860 00:43:57,900 --> 00:44:00,270 That's it, actually. 861 00:44:00,270 --> 00:44:03,600 Oh, times plus 1, OK? 862 00:44:03,600 --> 00:44:05,943 And then we have the A minus 2z. 863 00:44:05,943 --> 00:44:08,610 So let's expand this out just so we can see it all on the board. 864 00:44:08,610 --> 00:44:15,750 So we have AA over A times A minus 4z squared 865 00:44:15,750 --> 00:44:24,810 plus 4 minus 4 AZ plus 4z squared. 866 00:44:24,810 --> 00:44:28,170 So let's take the derivative with respect to z of that. 867 00:44:28,170 --> 00:44:30,900 That term goes away. 868 00:44:30,900 --> 00:44:33,840 That becomes 4A. 869 00:44:33,840 --> 00:44:39,720 So we have plus AA over A times 4A. 870 00:44:39,720 --> 00:44:44,830 And there's a minus sign there. 871 00:44:44,830 --> 00:44:46,580 And that becomes 8z. 872 00:44:49,470 --> 00:44:56,070 So we have plus AA symmetry over A times 8z. 873 00:44:56,070 --> 00:44:58,470 And the delta term goes away because there's 874 00:44:58,470 --> 00:45:00,090 no z dependence. 875 00:45:00,090 --> 00:45:03,690 And what we end up here is the solution for what is the most 876 00:45:03,690 --> 00:45:06,120 stable z as a function of A? 877 00:45:06,120 --> 00:45:08,040 This is a linear equation. 878 00:45:08,040 --> 00:45:09,840 There's only one solution for it. 879 00:45:09,840 --> 00:45:14,640 If we actually want to graph this m as a function of A 880 00:45:14,640 --> 00:45:23,700 and Z, we end up with what's called a mass parabola, which 881 00:45:23,700 --> 00:45:26,700 is to say you can graph the binding energy per nucleon, 882 00:45:26,700 --> 00:45:31,500 or the mass, or pretty much similar things, of a nucleus, 883 00:45:31,500 --> 00:45:35,640 of all nuclei with A given as a function of z. 884 00:45:35,640 --> 00:45:38,670 Think I can do this on the remaining space right here. 885 00:45:38,670 --> 00:45:44,940 So let's say 4A equals 93 if this is Z 886 00:45:44,940 --> 00:45:49,170 and this is m as a function of A and Z. Let's actually 887 00:45:49,170 --> 00:45:51,100 look at a concrete example. 888 00:45:51,100 --> 00:45:54,450 So let's go live to the chart of nuclides 889 00:45:54,450 --> 00:45:57,735 and start looking at things with a mass number of 93. 890 00:46:00,380 --> 00:46:02,070 Looks like I clicked a little too high. 891 00:46:05,000 --> 00:46:06,780 There it is. 892 00:46:06,780 --> 00:46:07,610 Let's see. 893 00:46:07,610 --> 00:46:11,620 Moly 93 I was looking at, and that 894 00:46:11,620 --> 00:46:16,790 becomes niobium 93, which is the stable isotope I was thinking. 895 00:46:16,790 --> 00:46:20,050 So let's put niobium right here. 896 00:46:20,050 --> 00:46:22,000 I haven't given an actual scale to this 897 00:46:22,000 --> 00:46:24,800 because I just want to show you in sort of relative terms. 898 00:46:24,800 --> 00:46:27,710 So let's say niobium is the stable one. 899 00:46:27,710 --> 00:46:30,370 So it's going to have the lowest actual mass, even though it's 900 00:46:30,370 --> 00:46:32,650 got an A number of 93. 901 00:46:32,650 --> 00:46:35,650 If you look here on the chart of nuclides, 902 00:46:35,650 --> 00:46:38,080 you can see it could have come from a couple 903 00:46:38,080 --> 00:46:41,110 of different places, either from zirconium 93 904 00:46:41,110 --> 00:46:43,120 or from molybdenum 93. 905 00:46:43,120 --> 00:46:44,950 And now is a good time to start introducing 906 00:46:44,950 --> 00:46:49,190 these different modes of decay so you can figure out, well, 907 00:46:49,190 --> 00:46:52,810 how would a nucleus decay to get to the most stable place? 908 00:46:52,810 --> 00:46:58,240 Let's say it came from zirconium 93 and-- 909 00:46:58,240 --> 00:47:02,580 let's see, miobium has a proton number of 41. 910 00:47:02,580 --> 00:47:05,670 So if we go to zirconium, it beta 911 00:47:05,670 --> 00:47:11,310 decays to niobium 93 with an energy of .091 MeV. 912 00:47:11,310 --> 00:47:13,560 Very, very close. 913 00:47:13,560 --> 00:47:16,710 So we'll draw it slightly higher. 914 00:47:16,710 --> 00:47:19,560 That's about 91 keV. 915 00:47:19,560 --> 00:47:24,300 Zirconium 93 could have come from the beta decay of yttrium 916 00:47:24,300 --> 00:47:27,250 93 you can see right here. 917 00:47:27,250 --> 00:47:30,840 So let's go up the mass parabola and keep exploring. 918 00:47:30,840 --> 00:47:37,590 And now we see that yttrium can decay by beta decay with 3 MeV. 919 00:47:37,590 --> 00:47:43,860 So if we put yttrium on this graph, it would be way higher. 920 00:47:43,860 --> 00:47:46,960 Yttrium itself could have come from-- 921 00:47:46,960 --> 00:47:50,410 well, let's see, strontium 93 with the decay 922 00:47:50,410 --> 00:47:52,720 energy of 4.1 MeV. 923 00:47:52,720 --> 00:47:55,930 So let's put strontium here. 924 00:47:55,930 --> 00:48:00,190 And I think 4 MEV would be, like, off the chart. 925 00:48:00,190 --> 00:48:00,880 But whatever. 926 00:48:00,880 --> 00:48:02,170 That's the way we drew it. 927 00:48:02,170 --> 00:48:05,140 Already we've got the makings of a parabola. 928 00:48:05,140 --> 00:48:10,010 And each one of these can decay by beta decay, 929 00:48:10,010 --> 00:48:11,950 or does decay by beta decay, in order 930 00:48:11,950 --> 00:48:15,640 to get to the most stable nucleus. 931 00:48:15,640 --> 00:48:17,350 So let's write the nuclear reaction 932 00:48:17,350 --> 00:48:20,470 for beta decay of one of these, let's say, from zirconium 933 00:48:20,470 --> 00:48:21,970 to niobium. 934 00:48:21,970 --> 00:48:27,610 So we'd have 9340 zirconium spontaneously 935 00:48:27,610 --> 00:48:36,730 goes to 9341 niobium plus a beta and 936 00:48:36,730 --> 00:48:39,332 plus an electron anti-neutrino. 937 00:48:39,332 --> 00:48:41,290 That's the part I don't expect you to know yet. 938 00:48:41,290 --> 00:48:44,590 But that's the whole energy conservation thing. 939 00:48:44,590 --> 00:48:46,330 A little bit of a flash forward. 940 00:48:46,330 --> 00:48:49,012 The beta decay energy is not necessarily 941 00:48:49,012 --> 00:48:50,470 the energy of the electron that you 942 00:48:50,470 --> 00:48:52,137 will measure because some of that energy 943 00:48:52,137 --> 00:48:54,700 is taken away by the anti-neutrino. 944 00:48:54,700 --> 00:48:58,640 But we'll get into how those relate probably next week. 945 00:48:58,640 --> 00:49:01,090 So let's now look on the other side of the parabola 946 00:49:01,090 --> 00:49:03,790 and confirm that the semi-empirical mass 947 00:49:03,790 --> 00:49:05,680 formula, which predicts something 948 00:49:05,680 --> 00:49:08,920 parabolic here and here with respect to z, 949 00:49:08,920 --> 00:49:11,750 actually checks out. 950 00:49:11,750 --> 00:49:16,640 So let's back up to niobium 93 and notice 951 00:49:16,640 --> 00:49:20,370 that it could also have come from electron capture 952 00:49:20,370 --> 00:49:22,650 from molybdenum 93. 953 00:49:22,650 --> 00:49:26,160 So let's put molybdenum right here. 954 00:49:26,160 --> 00:49:33,420 And it decays with an energy of 0.4 MeV 955 00:49:33,420 --> 00:49:38,650 into niobium, which let's say it's around here. 956 00:49:38,650 --> 00:49:40,270 Let's keep going through the chain. 957 00:49:40,270 --> 00:49:43,950 Anyone have any questions so far while we keep going? 958 00:49:43,950 --> 00:49:44,930 Cool. 959 00:49:44,930 --> 00:49:47,160 Let's trace it back up the chain. 960 00:49:47,160 --> 00:49:50,060 Technetium 93 can beget molybdenum 93 961 00:49:50,060 --> 00:49:55,917 by electron capture with a much higher energy, 3.201. 962 00:49:55,917 --> 00:49:58,250 I'm going to extend our graph because we need the space. 963 00:50:01,220 --> 00:50:05,280 Technetium, another 3 MeV. 964 00:50:05,280 --> 00:50:08,630 And let's go one more back. 965 00:50:08,630 --> 00:50:11,950 Technetium can be made by electron capture from rubidium 966 00:50:11,950 --> 00:50:16,293 93 with an even higher energy, which 967 00:50:16,293 --> 00:50:18,460 means a higher difference in mass between these two. 968 00:50:18,460 --> 00:50:24,820 So so far-- let's see, what's 6 MeV for rubidium here? 969 00:50:24,820 --> 00:50:27,670 It would be like there, I guess. 970 00:50:30,280 --> 00:50:33,492 There's our mass parabola, right from the data. 971 00:50:33,492 --> 00:50:35,950 So I like doing this better than just showing you a diagram 972 00:50:35,950 --> 00:50:38,200 because you can actually try it for yourself. 973 00:50:38,200 --> 00:50:41,470 Pick a fixed A, change A, and construct 974 00:50:41,470 --> 00:50:43,540 the mass parabolas yourself. 975 00:50:43,540 --> 00:50:47,800 Now the question is how could these decay into niobium 93, 976 00:50:47,800 --> 00:50:49,730 which is the stable isotope? 977 00:50:49,730 --> 00:50:51,340 I have negative one minutes left, 978 00:50:51,340 --> 00:50:55,150 so I'm very quickly going to tell you for large energy 979 00:50:55,150 --> 00:50:58,760 changes, it can either be positron decay or electron 980 00:50:58,760 --> 00:50:59,260 capture. 981 00:50:59,260 --> 00:51:03,770 And we'll go over what these modes of decay are next week. 982 00:51:03,770 --> 00:51:08,440 This can be, again, positron or electron capture. 983 00:51:08,440 --> 00:51:11,800 And for small amounts of decay energy, 984 00:51:11,800 --> 00:51:17,740 it can only be electron capture because in order for positron 985 00:51:17,740 --> 00:51:22,060 decay to happen, you have to be able to create the positron. 986 00:51:22,060 --> 00:51:24,460 And the positron plus the extra electron 987 00:51:24,460 --> 00:51:32,040 ejected to balance charge has got to be 1.022 MeV, 988 00:51:32,040 --> 00:51:35,490 or same thing as what's known as two times the rest 989 00:51:35,490 --> 00:51:38,700 mass of the electron. 990 00:51:38,700 --> 00:51:40,500 I'm going to stop there, and we'll 991 00:51:40,500 --> 00:51:43,920 pick up with lots of examples and questions tomorrow. 992 00:51:43,920 --> 00:51:46,500 The last thing-- well, I'll go over the next problem 993 00:51:46,500 --> 00:51:47,167 set tomorrow. 994 00:51:47,167 --> 00:51:48,750 I want to make sure everyone's seen it 995 00:51:48,750 --> 00:51:50,460 seven days before it's due. 996 00:51:50,460 --> 00:51:53,540 And the best way to do that is to show it on the board.