1 00:00:00,700 --> 00:00:03,040 The following content is provided under a Creative 2 00:00:03,040 --> 00:00:04,460 Commons license. 3 00:00:04,460 --> 00:00:06,670 Your support will help MIT OpenCourseWare 4 00:00:06,670 --> 00:00:10,760 continue to offer high quality educational resources for free. 5 00:00:10,760 --> 00:00:13,300 To make a donation or to view additional materials 6 00:00:13,300 --> 00:00:17,260 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,260 --> 00:00:18,622 at ocw.mit.edu. 8 00:00:22,183 --> 00:00:24,600 PROFESSOR: I wanted to do a quick review of all the photon 9 00:00:24,600 --> 00:00:26,767 interactions, because I've released problems at 5:00 10 00:00:26,767 --> 00:00:28,380 for you guys. 11 00:00:28,380 --> 00:00:29,701 It involves your banana data. 12 00:00:29,701 --> 00:00:32,159 So we're going to be taking a second look at all the banana 13 00:00:32,159 --> 00:00:35,850 data and the problem statement for the lab part is simple. 14 00:00:35,850 --> 00:00:37,680 Identify all the peaks. 15 00:00:37,680 --> 00:00:38,888 Tell me where they came from. 16 00:00:38,888 --> 00:00:40,805 And tell me all the peaks that should be there 17 00:00:40,805 --> 00:00:42,000 that you don't see and why. 18 00:00:42,000 --> 00:00:46,090 And that's like a quarter of the problem set or something. 19 00:00:46,090 --> 00:00:47,670 So just to review the three effects 20 00:00:47,670 --> 00:00:51,070 that we talked about sort of in order of what energies 21 00:00:51,070 --> 00:00:55,350 they're important in, my spelling I'm sure 22 00:00:55,350 --> 00:00:57,090 will be slower than usual today. 23 00:01:00,660 --> 00:01:03,300 I spent like 3 and 1/2 days listening 24 00:01:03,300 --> 00:01:09,030 to Russian presentations with an English translator microphone. 25 00:01:09,030 --> 00:01:13,008 Russian scientific presentation is really similar to English. 26 00:01:13,008 --> 00:01:14,550 All the technical words are the same, 27 00:01:14,550 --> 00:01:16,967 but as soon as you start trying to talk to a two-year-old, 28 00:01:16,967 --> 00:01:17,910 you're just lost. 29 00:01:17,910 --> 00:01:19,230 But it's pretty cool. 30 00:01:19,230 --> 00:01:21,700 So we went over the photoelectric effect, 31 00:01:21,700 --> 00:01:28,818 Compton scattering, and pair production. 32 00:01:32,730 --> 00:01:34,720 And so in addition to knowing what these three 33 00:01:34,720 --> 00:01:38,020 mechanisms actually are and how to tell what they would look 34 00:01:38,020 --> 00:01:40,270 like on a given detector spectrum, 35 00:01:40,270 --> 00:01:42,953 the other two important things we wanted to remember 36 00:01:42,953 --> 00:01:44,620 are what are the cross-sections, so what 37 00:01:44,620 --> 00:01:47,290 are the relative probabilities of each happening 38 00:01:47,290 --> 00:01:50,860 as a function of their energy the photon and the material 39 00:01:50,860 --> 00:01:54,310 that they're going in and then filling in this map of if you 40 00:01:54,310 --> 00:01:59,337 have energy and z, where are these effects most prevalent? 41 00:01:59,337 --> 00:02:00,670 Does anyone want to kick me off? 42 00:02:00,670 --> 00:02:03,250 Does anyone remember the general form of the cross-sections 43 00:02:03,250 --> 00:02:06,048 for any of these effects? 44 00:02:06,048 --> 00:02:08,090 Or does anyone remember what this map looks like? 45 00:02:12,760 --> 00:02:13,260 Yeah? 46 00:02:13,260 --> 00:02:14,142 Chris. 47 00:02:14,142 --> 00:02:15,560 STUDENT: [INAUDIBLE] 48 00:02:15,560 --> 00:02:16,437 PROFESSOR: Indeed. 49 00:02:16,437 --> 00:02:18,690 STUDENT: [INAUDIBLE] 50 00:02:18,690 --> 00:02:19,690 PROFESSOR: That's right. 51 00:02:19,690 --> 00:02:21,370 So pair production-- 52 00:02:21,370 --> 00:02:23,830 I think we gave it the symbol kappa 53 00:02:23,830 --> 00:02:26,230 to go along with a reading that you guys have-- 54 00:02:26,230 --> 00:02:27,510 was around here. 55 00:02:27,510 --> 00:02:31,745 That's because it's not going to happen below 1.022 MeV, 56 00:02:31,745 --> 00:02:34,120 because you need the energy to make the positron electron 57 00:02:34,120 --> 00:02:34,900 pair. 58 00:02:34,900 --> 00:02:37,570 And indeed, the more electrons there are in each atom, 59 00:02:37,570 --> 00:02:39,690 the more likely pair production is, 60 00:02:39,690 --> 00:02:43,390 this happens when the photon gets near the nucleus, which 61 00:02:43,390 --> 00:02:46,690 is going to have a higher charge for higher z and so on. 62 00:02:46,690 --> 00:02:50,560 And so pair production, it was proportional to-- 63 00:02:50,560 --> 00:02:52,590 you'll never need to know the exact things 64 00:02:52,590 --> 00:02:53,590 that the cross-sections. 65 00:02:53,590 --> 00:02:54,880 That's what books are for. 66 00:02:54,880 --> 00:02:57,190 But it was proportional to-- 67 00:02:57,190 --> 00:03:02,770 let's see, I think it was like z to the third or fourth, 68 00:03:02,770 --> 00:03:05,050 pretty strong like that. 69 00:03:05,050 --> 00:03:06,370 What about Compton scattering? 70 00:03:06,370 --> 00:03:08,764 Where does that lie on this map? 71 00:03:08,764 --> 00:03:10,942 STUDENT: [INAUDIBLE] 72 00:03:10,942 --> 00:03:12,400 PROFESSOR: Yup, high z, low energy. 73 00:03:12,400 --> 00:03:14,840 So in this general region. 74 00:03:14,840 --> 00:03:17,550 We'll just give it a C for Compton scattering. 75 00:03:17,550 --> 00:03:19,510 And in that one, the cross-section 76 00:03:19,510 --> 00:03:21,610 was proportional to something like 1 77 00:03:21,610 --> 00:03:25,510 over the energy or h bar omega in your reading. 78 00:03:25,510 --> 00:03:28,520 That's the same thing as saying photon energy. 79 00:03:28,520 --> 00:03:31,960 And then what about photoelectric effect? 80 00:03:31,960 --> 00:03:34,732 Well, that's the only place there is left, right? 81 00:03:34,732 --> 00:03:35,940 We'll give it the symbol tau. 82 00:03:35,940 --> 00:03:37,210 So I'll put these up here. 83 00:03:40,680 --> 00:03:42,790 I don't quite know why they chose those symbols. 84 00:03:42,790 --> 00:03:45,100 But I'll just stick to the notation in the reading. 85 00:03:45,100 --> 00:03:47,710 And then the idea here is this was proportional to something 86 00:03:47,710 --> 00:03:53,190 like z to the fifth over-- 87 00:03:53,190 --> 00:03:56,280 what is it, like energy to the like 7/2. 88 00:03:56,280 --> 00:03:59,640 So significantly low energy, significantly high z. 89 00:03:59,640 --> 00:04:01,320 And does anyone remember at what energy 90 00:04:01,320 --> 00:04:03,390 does the photoelectric effect start to kick in? 91 00:04:09,010 --> 00:04:10,720 Very close to zero. 92 00:04:10,720 --> 00:04:14,700 So here, the energy has got to be greater than or equal to, 93 00:04:14,700 --> 00:04:18,349 but what is the photoelectric effect physically? 94 00:04:18,349 --> 00:04:20,695 STUDENT: [INAUDIBLE] 95 00:04:20,695 --> 00:04:21,320 PROFESSOR: Yep. 96 00:04:21,320 --> 00:04:22,910 A gamma gets absorbed, or any photon 97 00:04:22,910 --> 00:04:24,800 gets absorbed that knocks out an electron. 98 00:04:24,800 --> 00:04:27,748 So how energetic does it have to be to knock out the electron? 99 00:04:27,748 --> 00:04:30,040 STUDENT: Binding energy of the electrons. 100 00:04:30,040 --> 00:04:34,320 PROFESSOR: The binding energy of the lowest bound electron, 101 00:04:34,320 --> 00:04:36,770 which we give that symbol phi or the work function. 102 00:04:41,218 --> 00:04:43,510 The idea here is that as soon as you have enough energy 103 00:04:43,510 --> 00:04:45,130 to eject the outermost electron, which 104 00:04:45,130 --> 00:04:49,330 is super low for the alkali metals, like sodium, potassium, 105 00:04:49,330 --> 00:04:53,090 cesium, then you can exceed the work function 106 00:04:53,090 --> 00:04:55,370 and get the photoelectric effect going. 107 00:04:55,370 --> 00:04:57,200 And for this one, we said the energy here 108 00:04:57,200 --> 00:05:00,740 has to be greater than or equal to 2 times the rest 109 00:05:00,740 --> 00:05:06,126 mass of the electron c squared, better known as 1.022 MeV. 110 00:05:06,126 --> 00:05:11,130 Is there a minimum energy for Compton scattering? 111 00:05:11,130 --> 00:05:12,180 Photons can scatter. 112 00:05:12,180 --> 00:05:14,580 They don't have to have any energy to scatter. 113 00:05:14,580 --> 00:05:16,700 Certainly. 114 00:05:16,700 --> 00:05:19,410 And let's see the two interesting bits of technology 115 00:05:19,410 --> 00:05:20,790 we talked about related to these, 116 00:05:20,790 --> 00:05:23,370 one was called a Compton camera, where you could actually 117 00:05:23,370 --> 00:05:24,680 use two detectors. 118 00:05:24,680 --> 00:05:29,130 Let's say you're looking for a tiny source in a big box 119 00:05:29,130 --> 00:05:30,910 somewhere. 120 00:05:30,910 --> 00:05:33,410 You can have one detector. 121 00:05:33,410 --> 00:05:35,290 And you can have a second detector, 122 00:05:35,290 --> 00:05:38,430 so that this source is sending out gammas in all directions. 123 00:05:38,430 --> 00:05:41,420 And let's say one of them interacts with detector one, 124 00:05:41,420 --> 00:05:45,800 bounces off, and interacts in detector two. 125 00:05:45,800 --> 00:05:47,660 At that point, you've constrained 126 00:05:47,660 --> 00:05:50,750 sort of the angle between these detectors, 127 00:05:50,750 --> 00:05:53,570 so that you know what energy the gamma came from. 128 00:05:53,570 --> 00:05:57,260 And you know generally where it came from physically, which 129 00:05:57,260 --> 00:06:00,290 is a cool piece of equipment. 130 00:06:00,290 --> 00:06:01,898 I'm going to try to find pictures 131 00:06:01,898 --> 00:06:03,940 of one of these actual things, because I actually 132 00:06:03,940 --> 00:06:05,270 haven't seen one myself. 133 00:06:05,270 --> 00:06:07,437 I've just heard it described physically and it seems 134 00:06:07,437 --> 00:06:08,130 to make sense. 135 00:06:08,130 --> 00:06:10,280 And the second one that we touched upon 136 00:06:10,280 --> 00:06:13,760 at the very end of last class had to do with this thing right 137 00:06:13,760 --> 00:06:14,260 here. 138 00:06:14,260 --> 00:06:18,220 Does anyone remember thermionic devices? 139 00:06:18,220 --> 00:06:20,110 Well, the work function for some materials, 140 00:06:20,110 --> 00:06:23,410 like for cesium, the work function 141 00:06:23,410 --> 00:06:26,170 is a little less than an eV. 142 00:06:26,170 --> 00:06:28,930 It's like 0.7 electron volts, which 143 00:06:28,930 --> 00:06:31,750 means when you get things up to about 2,000 Celsius 144 00:06:31,750 --> 00:06:34,348 or so, the temperature of the atoms themselves 145 00:06:34,348 --> 00:06:36,640 exceeds the work function, and the outer electrons just 146 00:06:36,640 --> 00:06:37,900 boil off. 147 00:06:37,900 --> 00:06:45,220 So if you have two pieces of material, probably in a vacuum, 148 00:06:45,220 --> 00:06:48,700 and one of them is like 2,000 C, and one of them is, let's say, 149 00:06:48,700 --> 00:06:54,130 room temperature, you end up with this net flux of electrons 150 00:06:54,130 --> 00:06:56,035 boiling off the hot one to the cold one. 151 00:06:56,035 --> 00:06:57,910 And this has been one of the methods proposed 152 00:06:57,910 --> 00:06:59,800 to directly convert heat to electricity 153 00:06:59,800 --> 00:07:01,900 for ultra high temperature applications, 154 00:07:01,900 --> 00:07:03,940 like space reactors or other things that 155 00:07:03,940 --> 00:07:06,190 can get super crazy hot. 156 00:07:06,190 --> 00:07:08,560 So it's one of those energy conversion mechanisms-- did 157 00:07:08,560 --> 00:07:11,180 anyone ever hear about this one in high school? 158 00:07:11,180 --> 00:07:13,810 Highly, highly doubt that it would ever be mentioned. 159 00:07:13,810 --> 00:07:16,930 One of the professors in our department, Elias Gyftopoulos 160 00:07:16,930 --> 00:07:20,290 was one of the folks that came up with this whole idea. 161 00:07:20,290 --> 00:07:24,520 And in my senior design course, we actually 162 00:07:24,520 --> 00:07:27,280 designed a space reactor that uses thermionics, 163 00:07:27,280 --> 00:07:29,550 and he showed up in the audience by surprise. 164 00:07:29,550 --> 00:07:31,258 And that's probably the dumbest I've ever 165 00:07:31,258 --> 00:07:33,250 looked at a presentation, explaining something 166 00:07:33,250 --> 00:07:34,600 that someone invented. 167 00:07:34,600 --> 00:07:37,960 They knew every single mistake and everything that was wrong. 168 00:07:37,960 --> 00:07:41,500 So since then I've kind of boned up on thermionics knowledge. 169 00:07:41,500 --> 00:07:44,800 But that's enough for the photon stuff. 170 00:07:44,800 --> 00:07:48,130 Now we want to start getting into ion nuclear interactions 171 00:07:48,130 --> 00:07:50,230 and in today's reading, it started off-- 172 00:07:50,230 --> 00:07:52,060 I think the first paragraph went something 173 00:07:52,060 --> 00:07:54,550 like, the formula for stopping power 174 00:07:54,550 --> 00:07:55,780 can be expressed as follows. 175 00:08:01,240 --> 00:08:04,790 Squared times log of-- 176 00:08:04,790 --> 00:08:07,660 squared over mean ionization energy. 177 00:08:07,660 --> 00:08:11,480 And I find this explanation to be unsatisfactory. 178 00:08:11,480 --> 00:08:13,730 I'm not a fan of the kind of books that just say, 179 00:08:13,730 --> 00:08:14,540 here's a formula. 180 00:08:14,540 --> 00:08:16,530 For practice, plug things in and use them. 181 00:08:16,530 --> 00:08:18,440 So instead, I'm going to skip ahead 182 00:08:18,440 --> 00:08:20,750 to a little bit of next week's reading 183 00:08:20,750 --> 00:08:23,180 or the [INAUDIBLE] reading and actually derive it. 184 00:08:23,180 --> 00:08:25,790 So when I just throw up a formula like this, 185 00:08:25,790 --> 00:08:28,310 it's like how the hell do you remember that, right? 186 00:08:28,310 --> 00:08:31,370 Well, it's going to make a lot more sense once we actually 187 00:08:31,370 --> 00:08:32,600 derive it. 188 00:08:32,600 --> 00:08:34,940 So let's set up this problem. 189 00:08:34,940 --> 00:08:40,490 You have a charged particle with charge little z times e. 190 00:08:40,490 --> 00:08:42,230 Little z, we'll say, is the number 191 00:08:42,230 --> 00:08:46,190 of protons in this nucleus or the charge 192 00:08:46,190 --> 00:08:49,070 on an electron if you want it, times the unit 193 00:08:49,070 --> 00:08:51,090 charge of an electron. 194 00:08:51,090 --> 00:08:55,430 And it's firing at some other electron 195 00:08:55,430 --> 00:08:57,630 somewhere else in the material. 196 00:08:57,630 --> 00:09:01,580 So the basis of any sort of ion electron interaction 197 00:09:01,580 --> 00:09:04,993 has to start with the ion being either struck or repelled-- 198 00:09:04,993 --> 00:09:06,410 I'm sorry, with the electron being 199 00:09:06,410 --> 00:09:08,720 struck a repelled by the ion. 200 00:09:08,720 --> 00:09:12,220 And so let's say that this ion exists. 201 00:09:12,220 --> 00:09:14,920 We'll draw kind of a unit cylinder 202 00:09:14,920 --> 00:09:16,940 around this physical situation. 203 00:09:19,650 --> 00:09:22,450 And if we draw this distance right here, 204 00:09:22,450 --> 00:09:24,000 it's one of those rare cases where 205 00:09:24,000 --> 00:09:27,450 the nomenclature is the same in pretty much every reading. 206 00:09:27,450 --> 00:09:30,551 There's this distance b, which we call the impact parameter. 207 00:09:34,045 --> 00:09:35,420 It's kind of a funny name for it, 208 00:09:35,420 --> 00:09:37,810 but it just means by how close does your particle 209 00:09:37,810 --> 00:09:40,150 get to that electron when it undergoes 210 00:09:40,150 --> 00:09:42,280 this single interaction. 211 00:09:42,280 --> 00:09:45,550 And so this particle is moving quite fast with some speed 212 00:09:45,550 --> 00:09:49,468 v towards and then away from this electron. 213 00:09:49,468 --> 00:09:51,010 And what's going to happen is there's 214 00:09:51,010 --> 00:09:52,540 going to be some sort of a Coulomb 215 00:09:52,540 --> 00:09:58,660 force between this charged particle and this electron. 216 00:09:58,660 --> 00:10:01,090 So let's say you're firing an electron at an electron. 217 00:10:01,090 --> 00:10:03,340 There's going to be some negative repulsion. 218 00:10:03,340 --> 00:10:05,873 Or if you're firing an ion at an electron, 219 00:10:05,873 --> 00:10:07,540 there might be some positive attraction. 220 00:10:07,540 --> 00:10:10,570 But at any rate, there's going to be some deflection. 221 00:10:10,570 --> 00:10:13,000 So let's just say it's a negatively charged particle. 222 00:10:13,000 --> 00:10:15,960 If we draw its actual trajectory, 223 00:10:15,960 --> 00:10:18,250 it's actually going to go off kind of barely 224 00:10:18,250 --> 00:10:19,840 in that direction, right? 225 00:10:19,840 --> 00:10:21,250 Two charges passing in the night. 226 00:10:21,250 --> 00:10:22,930 They know each other's there. 227 00:10:22,930 --> 00:10:24,920 And they kind of repel each other. 228 00:10:24,920 --> 00:10:26,710 So what we want to do is figure out 229 00:10:26,710 --> 00:10:29,080 what is the total amount of force in the x 230 00:10:29,080 --> 00:10:31,180 and the y direction. 231 00:10:31,180 --> 00:10:34,680 Let's just define our axes to make sure we're all 232 00:10:34,680 --> 00:10:36,180 on the same page. 233 00:10:36,180 --> 00:10:39,120 And can we resolve that into a total amount of energy 234 00:10:39,120 --> 00:10:42,060 lost per unit distance? 235 00:10:42,060 --> 00:10:46,460 This quantity right here was for referred to as stopping power. 236 00:10:50,180 --> 00:10:52,040 Before we launch into it, does anyone 237 00:10:52,040 --> 00:10:57,420 know why I put a negative sign on this quantity? 238 00:10:57,420 --> 00:10:59,578 STUDENT: This is all [INAUDIBLE].. 239 00:10:59,578 --> 00:11:00,370 PROFESSOR: Exactly. 240 00:11:00,370 --> 00:11:00,913 Yep. 241 00:11:00,913 --> 00:11:02,830 If you're changing the energy in the particle, 242 00:11:02,830 --> 00:11:04,205 unless it's, let's say, following 243 00:11:04,205 --> 00:11:06,870 it to some gravitational field, which we're not covering today 244 00:11:06,870 --> 00:11:10,390 or ever then, it's any sort of interaction 245 00:11:10,390 --> 00:11:13,730 is going to cause the particle to lose some energy. 246 00:11:13,730 --> 00:11:16,450 So this quantity right here is going to be negative, 247 00:11:16,450 --> 00:11:19,610 and so this quantity right here is going to be positive. 248 00:11:19,610 --> 00:11:21,870 We stick a minus sign in front of it. 249 00:11:21,870 --> 00:11:23,970 But let's get back to the basics then. 250 00:11:23,970 --> 00:11:27,900 What is the force between this charged particle 251 00:11:27,900 --> 00:11:30,148 and the electron from 802? 252 00:11:33,812 --> 00:11:34,945 This Coulomb force. 253 00:11:40,558 --> 00:11:41,600 STUDENT: It's a constant. 254 00:11:41,600 --> 00:11:43,660 PROFESSOR: There is a constant. 255 00:11:43,660 --> 00:11:46,365 Let's just call it k0, because the reading calls it k0. 256 00:11:46,365 --> 00:11:49,290 STUDENT: And then it'd be z e [INAUDIBLE].. 257 00:11:49,290 --> 00:11:51,410 PROFESSOR: Yes. 258 00:11:51,410 --> 00:11:53,130 Yeah. 259 00:11:53,130 --> 00:11:55,740 So this is like your q1 and your q1, right? 260 00:11:55,740 --> 00:11:58,153 Your charge 1 and your charge 2 over? 261 00:11:58,153 --> 00:11:59,070 STUDENT: The distance. 262 00:11:59,070 --> 00:12:00,910 PROFESSOR: The distance squared. 263 00:12:00,910 --> 00:12:03,450 Let's call that the distance between the two particles. 264 00:12:03,450 --> 00:12:06,570 And so now we can say if this is the distance away 265 00:12:06,570 --> 00:12:11,340 in the x direction, then we know that r is 266 00:12:11,340 --> 00:12:14,980 root x squared plus b squared. 267 00:12:14,980 --> 00:12:17,400 So we can stick that in there, and we 268 00:12:17,400 --> 00:12:25,490 know that our Coulomb force would then be this k0 little z 269 00:12:25,490 --> 00:12:28,550 e squared over root-- 270 00:12:28,550 --> 00:12:34,363 I'm sorry not square root, just x squared plus b squared. 271 00:12:34,363 --> 00:12:35,780 So like we've done with everything 272 00:12:35,780 --> 00:12:37,000 so far in the class-- 273 00:12:37,000 --> 00:12:38,675 it's kind of dark in the back. 274 00:12:38,675 --> 00:12:40,550 Like we've done with everything in the class, 275 00:12:40,550 --> 00:12:43,080 let's split this up into x and y-forces. 276 00:12:43,080 --> 00:12:44,990 So if we assume that the electron basically 277 00:12:44,990 --> 00:12:48,350 doesn't move, what's the net amount 278 00:12:48,350 --> 00:12:51,080 of force in the x direction that this particle is 279 00:12:51,080 --> 00:12:53,690 going to feel when it goes from minus infinity, 280 00:12:53,690 --> 00:12:57,100 so over here, to plus infinity like over here. 281 00:12:57,100 --> 00:12:57,943 STUDENT: Zero. 282 00:12:57,943 --> 00:12:58,610 PROFESSOR: Zero. 283 00:12:58,610 --> 00:12:59,443 Why do you say that? 284 00:12:59,443 --> 00:13:02,900 STUDENT: Because [INAUDIBLE]. 285 00:13:02,900 --> 00:13:04,100 PROFESSOR: Exactly. 286 00:13:04,100 --> 00:13:06,830 Whatever force it feels repelling it from here, 287 00:13:06,830 --> 00:13:09,530 as soon as it hits this midpoint, 288 00:13:09,530 --> 00:13:11,972 it gets that same amount of propulsion 289 00:13:11,972 --> 00:13:12,930 in the other direction. 290 00:13:12,930 --> 00:13:16,610 So your net force, if you integrate from minus infinity 291 00:13:16,610 --> 00:13:22,340 to infinity, of your x force as a function of t, 292 00:13:22,340 --> 00:13:25,310 that comes out to zero, which makes our life a lot easier. 293 00:13:25,310 --> 00:13:28,100 All we have to worry about is the total integral 294 00:13:28,100 --> 00:13:31,760 of the y force to figure out how much net deflection do 295 00:13:31,760 --> 00:13:33,770 we get in that direction. 296 00:13:33,770 --> 00:13:38,640 This integral is also better known as a momentum. 297 00:13:38,640 --> 00:13:41,510 Anyone recognize where with this comes from? 298 00:13:41,510 --> 00:13:44,070 If you take the integral of a force, 299 00:13:44,070 --> 00:13:48,770 it's like the integral of a mass times an acceleration, which 300 00:13:48,770 --> 00:13:51,230 is like mass times the integral of acceleration, which 301 00:13:51,230 --> 00:13:56,610 is like mv, which is a momentum this is where some 302 00:13:56,610 --> 00:13:59,700 of our particle wave stuff is going to get funky, 303 00:13:59,700 --> 00:14:02,250 because we're going to start throwing in expressions 304 00:14:02,250 --> 00:14:04,835 for particle momentums in wave equations 305 00:14:04,835 --> 00:14:06,960 when we start to determine, well, if this is really 306 00:14:06,960 --> 00:14:08,070 an electron. 307 00:14:08,070 --> 00:14:10,335 There's some limitations on how we can treat it, 308 00:14:10,335 --> 00:14:12,460 where it kind of loses its character as a particle. 309 00:14:12,460 --> 00:14:15,370 So I just want to warn you that that's coming up. 310 00:14:15,370 --> 00:14:19,950 So now, let's make an expression for the y force. 311 00:14:19,950 --> 00:14:23,130 If we were to say what is the y momentum imparted, which 312 00:14:23,130 --> 00:14:27,645 is an integral of the y component of the force dt, 313 00:14:27,645 --> 00:14:29,520 we already have the expression for the force, 314 00:14:29,520 --> 00:14:31,630 like you guys derived. 315 00:14:31,630 --> 00:14:36,930 K0 time as little ze times e over r 316 00:14:36,930 --> 00:14:42,225 squared, which is x squared plus b squared. 317 00:14:42,225 --> 00:14:44,100 And then how do we get the y component of it? 318 00:14:49,790 --> 00:14:52,360 Well, we've got to define an angle. 319 00:14:52,360 --> 00:14:54,023 That's our angle theta. 320 00:14:54,023 --> 00:14:55,565 What's the y component of that force? 321 00:15:00,510 --> 00:15:01,920 STUDENT: [INAUDIBLE] 322 00:15:01,920 --> 00:15:08,620 PROFESSOR: Yeah, it's just times cosine theta dt. 323 00:15:08,620 --> 00:15:10,690 What's the expression for cosine theta 324 00:15:10,690 --> 00:15:12,590 in this physical situation? 325 00:15:16,020 --> 00:15:17,500 STUDENT: [INAUDIBLE] 326 00:15:17,500 --> 00:15:20,230 PROFESSOR: Close. 327 00:15:20,230 --> 00:15:22,120 b over r. 328 00:15:22,120 --> 00:15:27,220 And in this case, r is root x squared plus b squared. 329 00:15:27,220 --> 00:15:29,770 And the last thing we want is because we have things in terms 330 00:15:29,770 --> 00:15:33,850 of x and b, b's a constant, x is a variable, 331 00:15:33,850 --> 00:15:35,930 t's kind of the wrong variable. 332 00:15:35,930 --> 00:15:38,500 So we can do a variable change and say 333 00:15:38,500 --> 00:15:41,380 this is equivalent to the velocity of the particle over-- 334 00:15:41,380 --> 00:15:47,500 I'm sorry-- to dx over v. We're just using this whole like 335 00:15:47,500 --> 00:15:52,810 velocity equals, what is it, distance times time, 336 00:15:52,810 --> 00:15:55,234 so our whole, what is it-- 337 00:15:55,234 --> 00:15:57,430 yeah. 338 00:15:57,430 --> 00:15:59,890 STUDENT: [INAUDIBLE] 339 00:15:59,890 --> 00:16:01,660 PROFESSOR: Thank you. 340 00:16:01,660 --> 00:16:03,590 Distance equals velocity times. 341 00:16:03,590 --> 00:16:04,840 Thinks, [INAUDIBLE]. 342 00:16:04,840 --> 00:16:07,300 OK, anyway. 343 00:16:07,300 --> 00:16:08,977 So luckily, I had the expression right 344 00:16:08,977 --> 00:16:10,060 and the explanation wrong. 345 00:16:10,060 --> 00:16:10,602 So thank you. 346 00:16:10,602 --> 00:16:12,470 Was that Luke or Jared's voice? 347 00:16:12,470 --> 00:16:12,970 Awesome. 348 00:16:12,970 --> 00:16:13,720 OK. 349 00:16:13,720 --> 00:16:15,850 So let's put this whole expression in, 350 00:16:15,850 --> 00:16:19,438 keep that little embarrassment behind us. 351 00:16:19,438 --> 00:16:20,980 We have the integral from negative to 352 00:16:20,980 --> 00:16:28,670 plus infinity of k0 little z e squared times b 353 00:16:28,670 --> 00:16:32,067 over x squared plus b squared times 354 00:16:32,067 --> 00:16:33,150 the square root of itself. 355 00:16:33,150 --> 00:16:38,440 So let's just say x squared plus b squared to the 3/2, 356 00:16:38,440 --> 00:16:41,930 and there's a v on the bottom dx. 357 00:16:41,930 --> 00:16:44,030 This is finally valuable. 358 00:16:44,030 --> 00:16:45,568 So we're getting closer. 359 00:16:45,568 --> 00:16:47,360 Let's take all the constants and shove them 360 00:16:47,360 --> 00:16:49,070 outside the integral. 361 00:16:49,070 --> 00:17:00,090 So we have a k0 z squared eV over b squared b over velocity 362 00:17:00,090 --> 00:17:07,910 times the integral of just 1 over x squared 363 00:17:07,910 --> 00:17:13,310 plus b squared to the 3/2 dx. 364 00:17:13,310 --> 00:17:16,119 Not remembering the formula off the top of my head, I-- yeah? 365 00:17:16,119 --> 00:17:18,442 STUDENT: So we can treat the velocity as a constant 366 00:17:18,442 --> 00:17:20,329 even though it's losing energy. 367 00:17:20,329 --> 00:17:22,940 PROFESSOR: Yes, that's-- well, we'll call it a crude 368 00:17:22,940 --> 00:17:23,537 derivation. 369 00:17:23,537 --> 00:17:25,579 But if we're assuming that the electron basically 370 00:17:25,579 --> 00:17:28,099 doesn't change position, that it changes so little, 371 00:17:28,099 --> 00:17:30,557 then we're going to assume that also the velocity basically 372 00:17:30,557 --> 00:17:32,390 doesn't change, that one collision 373 00:17:32,390 --> 00:17:36,350 for a high enough velocity doesn't lose that much energy. 374 00:17:36,350 --> 00:17:38,708 So that's what we're going with for now. 375 00:17:38,708 --> 00:17:40,250 And we'll actually be able to compare 376 00:17:40,250 --> 00:17:42,500 this kind of crude derivation to one 377 00:17:42,500 --> 00:17:45,378 done from quantum mechanics, and they look pretty similar. 378 00:17:45,378 --> 00:17:47,420 There's like an extra factor of two or something. 379 00:17:47,420 --> 00:17:50,000 But as I showed you guys in preparing for the test, when 380 00:17:50,000 --> 00:17:53,330 I said 9 equals about 10, it therefore 381 00:17:53,330 --> 00:17:56,240 follows that 1 equals about 2, and as long 382 00:17:56,240 --> 00:17:58,640 as we get the constants and orders of magnitude right, 383 00:17:58,640 --> 00:18:00,765 we're going to gain the physical intuition for what 384 00:18:00,765 --> 00:18:01,640 we're looking at. 385 00:18:01,640 --> 00:18:04,190 I'll leave it up there. 386 00:18:04,190 --> 00:18:07,280 Anyway, evaluated this integral, and it came out to something 387 00:18:07,280 --> 00:18:09,710 like 2 over b squared. 388 00:18:09,710 --> 00:18:20,120 So this just comes out to k0 ze squared b with a 2 over 2 vb 389 00:18:20,120 --> 00:18:21,530 squared. 390 00:18:21,530 --> 00:18:25,200 Cancel the b's. 391 00:18:25,200 --> 00:18:27,010 I don't know where that 2 came from. 392 00:18:27,010 --> 00:18:27,510 Whatever. 393 00:18:31,000 --> 00:18:33,280 Yeah, that's what we have for the stopping 394 00:18:33,280 --> 00:18:37,270 power for this sort of one particle hitting one electron. 395 00:18:37,270 --> 00:18:41,110 Now, we have-- well, sorry, that's the momentum equation. 396 00:18:41,110 --> 00:18:43,602 But we're interested in the change in energy. 397 00:18:43,602 --> 00:18:45,310 So what's that equation we've used before 398 00:18:45,310 --> 00:18:48,210 to go from momentum to energy? 399 00:18:48,210 --> 00:18:49,820 Our kinetic energy t. 400 00:18:52,840 --> 00:18:54,560 STUDENT: Square root of q [INAUDIBLE].. 401 00:18:57,333 --> 00:18:58,500 PROFESSOR: Other way around. 402 00:18:58,500 --> 00:19:04,950 So let's do it that way, right, so p equals root 2 mT. 403 00:19:04,950 --> 00:19:06,680 OK, so square both sides. 404 00:19:09,272 --> 00:19:09,980 Yeah, you got it. 405 00:19:12,760 --> 00:19:20,410 And we have our energy T is p squared over 2m. 406 00:19:20,410 --> 00:19:26,080 So let's take this small little mess, stick it in here, 407 00:19:26,080 --> 00:19:32,710 and then we end up with 4k0 squared little z squared e 408 00:19:32,710 --> 00:19:41,830 to the fourth over 2mv squared b. 409 00:19:41,830 --> 00:19:43,060 Cool. 410 00:19:43,060 --> 00:19:46,390 And so this gives us the little differential energy change 411 00:19:46,390 --> 00:19:48,560 from one electron collision. 412 00:19:48,560 --> 00:19:49,110 Yeah? 413 00:19:49,110 --> 00:19:50,128 STUDENT: [INAUDIBLE] 414 00:19:50,128 --> 00:19:52,170 PROFESSOR: I think we cancel one of those, right? 415 00:19:52,170 --> 00:19:55,480 STUDENT: Yeah, but then when you square [INAUDIBLE].. 416 00:19:55,480 --> 00:19:57,250 PROFESSOR: Oh yeah, you're right. 417 00:19:57,250 --> 00:19:58,600 Thank you. 418 00:19:58,600 --> 00:19:59,680 Comes back. 419 00:19:59,680 --> 00:20:00,790 b squared b squared. 420 00:20:00,790 --> 00:20:01,540 Yep, you're right. 421 00:20:01,540 --> 00:20:02,860 Thank you. 422 00:20:02,860 --> 00:20:05,680 So now we've only accounted for the ion hitting 423 00:20:05,680 --> 00:20:09,580 a single electron as it moves through this hollow cylinder 424 00:20:09,580 --> 00:20:11,180 of whatever medium it's going through. 425 00:20:11,180 --> 00:20:13,380 So this is when we can kind of take things back from abstract 426 00:20:13,380 --> 00:20:15,370 to reality and say, all right, it's moving 427 00:20:15,370 --> 00:20:17,720 through some actual material. 428 00:20:17,720 --> 00:20:19,720 And we have to describe its electron density 429 00:20:19,720 --> 00:20:21,880 in this cylindrical shell. 430 00:20:21,880 --> 00:20:25,300 So the electron density in the cylindrical shell depends on-- 431 00:20:25,300 --> 00:20:27,820 well, the number density of the material 432 00:20:27,820 --> 00:20:33,070 itself, just how many atoms are there times big Z, 433 00:20:33,070 --> 00:20:34,930 the number of protons in that nucleus 434 00:20:34,930 --> 00:20:38,530 and therefore the number of electrons in each nucleus, 435 00:20:38,530 --> 00:20:40,885 and the volume of the cylindrical shell. 436 00:20:40,885 --> 00:20:42,510 So what's the expression for the volume 437 00:20:42,510 --> 00:20:43,552 of the cylindrical shell? 438 00:20:45,900 --> 00:20:47,120 In differential form? 439 00:20:47,120 --> 00:20:49,790 Yeah, I started hearing? 440 00:20:49,790 --> 00:20:50,540 I heard a 2. 441 00:20:50,540 --> 00:20:52,130 That's correct. 442 00:20:52,130 --> 00:20:52,630 Keep going. 443 00:20:56,200 --> 00:21:01,510 Well, 2 pi b gives us the circumference 444 00:21:01,510 --> 00:21:05,600 of the circle on the outside of the cylinder. 445 00:21:05,600 --> 00:21:09,850 And if we add a little db there, some differential thickness 446 00:21:09,850 --> 00:21:14,680 element, and we add on a little dx for some differential 447 00:21:14,680 --> 00:21:17,920 distance down the cylinder, we end up 448 00:21:17,920 --> 00:21:27,290 with 2 pi b dv dx, multiplied by this stuff. 449 00:21:27,290 --> 00:21:32,490 And we get some differential change in energy scales like-- 450 00:21:32,490 --> 00:21:37,700 let's say there is a 4 and a 2 there. 451 00:21:37,700 --> 00:21:39,620 So we end up with 4. 452 00:21:39,620 --> 00:21:41,580 That's not a 4. 453 00:21:41,580 --> 00:21:48,130 pi k0 squared, little z squared, big Z 454 00:21:48,130 --> 00:22:01,690 e to the fourth b dv dx over mv squared b squared. 455 00:22:01,690 --> 00:22:04,020 Now those other b's cancel. 456 00:22:04,020 --> 00:22:05,400 We can divide everything by dx. 457 00:22:08,220 --> 00:22:11,250 And we've already almost got our stopping power expression. 458 00:22:11,250 --> 00:22:12,837 We're getting pretty close. 459 00:22:12,837 --> 00:22:14,670 Anyone see some similarities between the one 460 00:22:14,670 --> 00:22:19,220 I just threw out of my head and what we've got so far? 461 00:22:19,220 --> 00:22:22,220 We've almost got the makings of it. 462 00:22:22,220 --> 00:22:24,020 So now if we want to account for the fact 463 00:22:24,020 --> 00:22:26,360 that our charged particle is probably not 464 00:22:26,360 --> 00:22:29,460 shooting through the center of a perfect hollow cylinder, 465 00:22:29,460 --> 00:22:32,270 but we're just firing it into like actual matter, 466 00:22:32,270 --> 00:22:34,340 we have to account for every possible impact 467 00:22:34,340 --> 00:22:38,200 parameter in every possible cylindrical shell 468 00:22:38,200 --> 00:22:40,360 that it would be moving through. 469 00:22:40,360 --> 00:22:44,638 So in this case, we can integrate this. 470 00:22:44,638 --> 00:22:46,180 We've already got our db right there. 471 00:22:46,180 --> 00:22:48,700 That's our integrating variable. 472 00:22:48,700 --> 00:22:52,030 And now here's where things get a little tricksy. 473 00:22:52,030 --> 00:22:57,520 Can we actually integrate this from an impact parameter of 0? 474 00:23:01,310 --> 00:23:03,972 And this is not an easy question actually. 475 00:23:03,972 --> 00:23:04,930 What do you guys think? 476 00:23:07,708 --> 00:23:09,100 STUDENT: [INAUDIBLE] 477 00:23:09,100 --> 00:23:11,660 PROFESSOR: So Luke says no, why? 478 00:23:11,660 --> 00:23:13,591 STUDENT: [INAUDIBLE] 479 00:23:20,810 --> 00:23:22,970 PROFESSOR: We actually have an over b. 480 00:23:22,970 --> 00:23:26,950 We have a v squared, but that's not our integrating variable. 481 00:23:26,950 --> 00:23:29,010 Yeah, so we have like a 1 over b looking-- 482 00:23:29,010 --> 00:23:31,116 STUDENT: [INAUDIBLE] 483 00:23:31,116 --> 00:23:32,324 PROFESSOR: Yeah, that's fine. 484 00:23:32,324 --> 00:23:34,930 STUDENT: [INAUDIBLE] 485 00:23:34,930 --> 00:23:36,957 PROFESSOR: That's true. 486 00:23:36,957 --> 00:23:38,790 There's another more physical reason though. 487 00:23:38,790 --> 00:23:40,960 But you're right mathematically. 488 00:23:40,960 --> 00:23:45,950 Can you know precisely the location of an electron ever? 489 00:23:45,950 --> 00:23:47,450 Now I see a lot of people saying no. 490 00:23:47,450 --> 00:23:49,100 Why do you say that? 491 00:23:49,100 --> 00:23:51,340 STUDENT: [INAUDIBLE] 492 00:23:51,340 --> 00:23:52,340 PROFESSOR: That's right. 493 00:23:52,340 --> 00:23:55,980 There's this thing-- the De Broglie uncertainty principle. 494 00:24:03,040 --> 00:24:05,815 It's kind of the punchline of a lot of quantum mechanics jokes. 495 00:24:05,815 --> 00:24:07,690 You never know where something is going to be 496 00:24:07,690 --> 00:24:08,720 or where it's going. 497 00:24:08,720 --> 00:24:11,200 We used to say this about some of the older professors 498 00:24:11,200 --> 00:24:12,580 in this department. 499 00:24:12,580 --> 00:24:14,320 If you call them and say, I'm on my way, 500 00:24:14,320 --> 00:24:15,778 I'm getting there as fast as I can, 501 00:24:15,778 --> 00:24:17,325 they could be anywhere in the world. 502 00:24:17,325 --> 00:24:19,450 And if they say, don't worry, I'm three miles away. 503 00:24:19,450 --> 00:24:22,040 You don't know how long it's going to take them to get here. 504 00:24:22,040 --> 00:24:23,540 Same thing with me and getting here. 505 00:24:23,540 --> 00:24:25,420 Although I was on MIT standard time, 506 00:24:25,420 --> 00:24:26,980 which means five minutes late. 507 00:24:26,980 --> 00:24:28,240 Not bad. 508 00:24:28,240 --> 00:24:31,660 So in this case, we have to ascribe the electron 509 00:24:31,660 --> 00:24:33,398 some sort of a wavelength. 510 00:24:33,398 --> 00:24:35,440 So in this case, we can't just treat the electron 511 00:24:35,440 --> 00:24:38,320 like a particle whose position we know. 512 00:24:38,320 --> 00:24:40,240 We're going to go with our original equation 513 00:24:40,240 --> 00:24:45,760 for a photon energy, which looks like hc over lambda. 514 00:24:45,760 --> 00:24:48,520 Rearrange that so that we'll have some lambda wavelength 515 00:24:48,520 --> 00:24:54,810 equals hc over E. 516 00:24:54,810 --> 00:24:55,310 I'm sorry. 517 00:24:55,310 --> 00:24:57,320 This is a momentum thing, not an energy thing. 518 00:25:02,280 --> 00:25:04,580 And what's the momentum of the electron? 519 00:25:09,130 --> 00:25:10,740 From the classical definition? 520 00:25:14,030 --> 00:25:17,200 It's just mass times velocity, right? 521 00:25:17,200 --> 00:25:21,260 So we'll just stick in the mass of the electron 522 00:25:21,260 --> 00:25:23,510 times the velocity right there. 523 00:25:23,510 --> 00:25:26,120 And this wavelength right here, the De Broglie wavelength 524 00:25:26,120 --> 00:25:29,210 of the electron is as close as we can specify that impact 525 00:25:29,210 --> 00:25:30,290 parameter. 526 00:25:30,290 --> 00:25:34,430 And it turns out to be pretty significant, like on the order 527 00:25:34,430 --> 00:25:36,170 of 0.1 to 0.2 angstroms. 528 00:25:36,170 --> 00:25:40,010 You can't tell where an electron is going to be finer than that. 529 00:25:40,010 --> 00:25:42,710 So we're going to have this b minimum. 530 00:25:42,710 --> 00:25:48,750 I'll just write that in there and some b maximum, 531 00:25:48,750 --> 00:25:54,157 where are b minimum is the same as our De Broglie 532 00:25:54,157 --> 00:25:55,740 wavelength of the electron, because we 533 00:25:55,740 --> 00:25:58,830 can't define its position any better than that, 534 00:25:58,830 --> 00:26:03,960 which is just Planck's constant over its mass times velocity. 535 00:26:07,560 --> 00:26:10,620 For b max, it comes out to something 536 00:26:10,620 --> 00:26:15,780 like hv over this quantity, I bar 537 00:26:15,780 --> 00:26:18,426 what's called the mean ionization potential. 538 00:26:27,670 --> 00:26:30,790 What this quantity physically represents 539 00:26:30,790 --> 00:26:34,180 is that if your impact parameter is too large, 540 00:26:34,180 --> 00:26:36,640 then the electron will-- or the charged particle 541 00:26:36,640 --> 00:26:40,150 will feel so little force, that it won't eject an electron 542 00:26:40,150 --> 00:26:42,100 and won't really be deflected. 543 00:26:42,100 --> 00:26:45,460 And the farthest away it can be corresponds 544 00:26:45,460 --> 00:26:48,910 to the minimum amount of energy to create an average ionization 545 00:26:48,910 --> 00:26:50,670 in the material. 546 00:26:50,670 --> 00:26:55,710 And this mean ionization potential scales 547 00:26:55,710 --> 00:26:58,770 with something like this constant k times 548 00:26:58,770 --> 00:27:05,670 z, where k is on the order of like 30 to 35-- 549 00:27:05,670 --> 00:27:06,530 think it's like eV. 550 00:27:09,950 --> 00:27:13,700 But remarkably tight constant, so picking a mean value 551 00:27:13,700 --> 00:27:16,240 like that is no problem. 552 00:27:16,240 --> 00:27:18,060 And there we have our b min and b max. 553 00:27:18,060 --> 00:27:22,550 Those are our limits of integration. 554 00:27:22,550 --> 00:27:26,610 I think I planned this just to fill up the boards today. 555 00:27:26,610 --> 00:27:29,490 So let's write out the final integral that we have 556 00:27:29,490 --> 00:27:31,220 and see what we get. 557 00:27:31,220 --> 00:27:34,400 So we have our stopping power, should 558 00:27:34,400 --> 00:27:48,200 be integral from b min h over mv to b max jv over I bar of 4 pi 559 00:27:48,200 --> 00:27:54,860 k0 squared little z squared big Ze to the fourth over mv 560 00:27:54,860 --> 00:27:58,240 squared b db. 561 00:27:58,240 --> 00:27:59,910 And like I think it was Sarah that you 562 00:27:59,910 --> 00:28:02,953 mentioned that we'd have a log. 563 00:28:02,953 --> 00:28:04,120 I forget who's mentioned it. 564 00:28:04,120 --> 00:28:04,620 Sorry. 565 00:28:04,620 --> 00:28:05,548 That was Luke, OK? 566 00:28:05,548 --> 00:28:06,090 You're right. 567 00:28:06,090 --> 00:28:07,715 So it ends up just being a natural log. 568 00:28:07,715 --> 00:28:10,980 It's like all this junk on the outside times 569 00:28:10,980 --> 00:28:12,810 the integral of 1 over b. 570 00:28:12,810 --> 00:28:15,960 So this just comes out to 4 pi k0 571 00:28:15,960 --> 00:28:22,570 squared little z squared big Ze to the fourth over mv 572 00:28:22,570 --> 00:28:32,570 square times the natural log of dv max over b min. 573 00:28:37,760 --> 00:28:40,200 The h's cancel. 574 00:28:40,200 --> 00:28:42,550 We get a v squared. 575 00:28:42,550 --> 00:28:45,470 And so all this stuff inside just 576 00:28:45,470 --> 00:28:50,630 becomes the natural log of mv squared over mean ionization 577 00:28:50,630 --> 00:28:52,060 potential. 578 00:28:52,060 --> 00:28:55,190 And we've arrived basically at the same equation 579 00:28:55,190 --> 00:28:57,530 that we have over there, that I took care 580 00:28:57,530 --> 00:28:59,940 to memorize on the plane. 581 00:28:59,940 --> 00:29:02,490 So great that we've gotten here through the math. 582 00:29:02,490 --> 00:29:04,195 Let's actually see what this means, 583 00:29:04,195 --> 00:29:06,570 and we're going to go into some of the limits of validity 584 00:29:06,570 --> 00:29:10,050 like Luke was saying, where you can't have a natural log of 0. 585 00:29:10,050 --> 00:29:14,520 So the stopping power formula isn't quite going to work at 0. 586 00:29:14,520 --> 00:29:17,390 Nor will it work at super low energies. 587 00:29:17,390 --> 00:29:23,000 So if you want to write what this should be proportional to. 588 00:29:23,000 --> 00:29:25,500 I kind of see some constants here that we don't really care. 589 00:29:25,500 --> 00:29:27,060 They don't vary at all. 590 00:29:27,060 --> 00:29:30,480 But this looks kind of like a kinetic energy term, 591 00:29:30,480 --> 00:29:32,920 doesn't it? 592 00:29:32,920 --> 00:29:34,190 Like kinetic energy terms. 593 00:29:34,190 --> 00:29:38,290 So it's kind of proportional to this function 1 over t 594 00:29:38,290 --> 00:29:40,958 times the natural log of t. 595 00:29:40,958 --> 00:29:42,500 When you get rid of all the constants 596 00:29:42,500 --> 00:29:44,600 and just express it in terms of the variables, 597 00:29:44,600 --> 00:29:47,420 it looks a whole lot simpler. 598 00:29:47,420 --> 00:29:50,900 And so let's see what this would look like if we 599 00:29:50,900 --> 00:29:53,780 started to graph it out. 600 00:29:53,780 --> 00:29:57,080 And this is pretty universal for any charged particle 601 00:29:57,080 --> 00:29:58,400 stopping power. 602 00:29:58,400 --> 00:30:01,340 So if this was the kinetic energy t, 603 00:30:01,340 --> 00:30:07,690 and this was our stopping power, we've got this 1 over T term 604 00:30:07,690 --> 00:30:10,980 that's going to look something like this. 605 00:30:10,980 --> 00:30:15,430 And we have this natural log of T term, which 606 00:30:15,430 --> 00:30:17,930 is going to look something like this that actually goes down 607 00:30:17,930 --> 00:30:19,448 to infinity. 608 00:30:19,448 --> 00:30:20,990 So like Luke was saying, if these two 609 00:30:20,990 --> 00:30:22,615 are multiplied by each other, we're not 610 00:30:22,615 --> 00:30:25,370 going to have negative infinity as a stopping power, which 611 00:30:25,370 --> 00:30:28,550 would physically mean that once the particle hit zero energy, 612 00:30:28,550 --> 00:30:30,990 it speeds up to infinite speed, and that 613 00:30:30,990 --> 00:30:32,850 doesn't make any sense. 614 00:30:32,850 --> 00:30:35,510 But we can start to draw what the curve would 615 00:30:35,510 --> 00:30:38,550 look like with this general envelope. 616 00:30:38,550 --> 00:30:41,970 And so at low energies, the stopping power kind of scales 617 00:30:41,970 --> 00:30:44,840 like 1 over e. 618 00:30:44,840 --> 00:30:46,490 Let's now start drawing another graph 619 00:30:46,490 --> 00:30:48,620 with a little more physical intuition, 620 00:30:48,620 --> 00:30:50,560 the range of the particle. 621 00:30:50,560 --> 00:30:53,060 So while stopping power might be kind of a new quantity that 622 00:30:53,060 --> 00:30:55,730 represents the differential amount of energy lost 623 00:30:55,730 --> 00:30:58,130 as a function of distance-- that's kind of a mouthful-- 624 00:30:58,130 --> 00:31:03,590 the range is pretty simple, just how far it goes. 625 00:31:03,590 --> 00:31:06,825 So to get the range from the stopping power, 626 00:31:06,825 --> 00:31:08,200 you can integrate-- let's say you 627 00:31:08,200 --> 00:31:12,460 fire a particle into a bunch of matter at some energy T. 628 00:31:12,460 --> 00:31:15,370 So you start off an energy T, and you want to see 629 00:31:15,370 --> 00:31:19,060 how far it gets at distance 0. 630 00:31:19,060 --> 00:31:24,890 Well, you can just integrate the stopping power 631 00:31:24,890 --> 00:31:27,840 as a function of T, or you can switch 632 00:31:27,840 --> 00:31:30,490 your limits of integration. 633 00:31:30,490 --> 00:31:31,850 Let's see. 634 00:31:31,850 --> 00:31:32,390 I'm sorry. 635 00:31:32,390 --> 00:31:34,515 I'm not going to switch those limits of integration 636 00:31:34,515 --> 00:31:44,200 yet, which is like saying from 0 to T of dt dx dt, which 637 00:31:44,200 --> 00:31:55,182 is like saying from 0 to T of dt dx to the minus 1 dx. 638 00:31:55,182 --> 00:31:57,107 Much simpler expression. 639 00:31:57,107 --> 00:31:58,940 And when you forget all the crazy constants, 640 00:31:58,940 --> 00:32:02,870 and you just take this kind of form 641 00:32:02,870 --> 00:32:07,033 as the variable part of the expression for stopping power, 642 00:32:07,033 --> 00:32:08,450 unless your energy is really high, 643 00:32:08,450 --> 00:32:12,470 and this natural log counts at all, your range kind of scales 644 00:32:12,470 --> 00:32:16,880 like the integral of just 1 over T. 645 00:32:16,880 --> 00:32:20,100 I'm sorry, that to the minus 1, which 646 00:32:20,100 --> 00:32:24,870 is like the integral of T, which is like T squared, which 647 00:32:24,870 --> 00:32:28,590 means that this are pretty interesting intuitive result, 648 00:32:28,590 --> 00:32:31,230 that the range of the particle increases 649 00:32:31,230 --> 00:32:32,898 with the square of its energy. 650 00:32:32,898 --> 00:32:34,440 So this gives you a good hint to say, 651 00:32:34,440 --> 00:32:36,960 if I increase the particle by a certain amount, 652 00:32:36,960 --> 00:32:42,010 I'll increase the range by the square root of that increase. 653 00:32:42,010 --> 00:32:44,290 So anyway let's start drawing this range 654 00:32:44,290 --> 00:32:46,780 curve as a function of x. 655 00:32:46,780 --> 00:32:49,390 What this says right here is that if we start our particle 656 00:32:49,390 --> 00:32:54,400 at some high energy, and we're firing into the material, 657 00:32:54,400 --> 00:32:56,650 and it's losing energy as it goes, 658 00:32:56,650 --> 00:33:00,010 and we track this value of the stopping power 659 00:33:00,010 --> 00:33:05,660 to figure out how far it's going to go, change that in a second, 660 00:33:05,660 --> 00:33:08,150 for the first little while as this particle loses 661 00:33:08,150 --> 00:33:10,010 more and more energy, its stopping power 662 00:33:10,010 --> 00:33:13,910 stays mostly constant, and it loses a pretty constant amount 663 00:33:13,910 --> 00:33:17,000 of energy as a function of time. 664 00:33:17,000 --> 00:33:19,970 As its energy gets lower and lower, 665 00:33:19,970 --> 00:33:23,420 it loses more and more as a function of position. 666 00:33:23,420 --> 00:33:26,300 What this actually means is that as the velocity goes down 667 00:33:26,300 --> 00:33:28,700 or as the particle's energy goes down, 668 00:33:28,700 --> 00:33:32,170 it spends more time in the vicinity of the electron 669 00:33:32,170 --> 00:33:33,802 and gets deflected more. 670 00:33:33,802 --> 00:33:35,510 It's just that kind of a simple argument. 671 00:33:35,510 --> 00:33:38,270 Like the more time it spends near this electron, 672 00:33:38,270 --> 00:33:40,660 the more it feels the push. 673 00:33:40,660 --> 00:33:43,500 And so it will lose more and more energy 674 00:33:43,500 --> 00:33:47,490 as its energy gets lower and lower until you hit the point 675 00:33:47,490 --> 00:33:50,297 where this curve breaks down. 676 00:33:50,297 --> 00:33:51,630 Where do you guys think that is? 677 00:33:55,960 --> 00:33:57,220 Even mathematically speaking. 678 00:34:01,978 --> 00:34:04,270 Well, what happens if your natural log term is negative 679 00:34:04,270 --> 00:34:05,495 here? 680 00:34:05,495 --> 00:34:07,120 Then you get a negative stopping power, 681 00:34:07,120 --> 00:34:09,850 which would be like the particle picks up energy. 682 00:34:09,850 --> 00:34:13,489 That's not quite physical at all. 683 00:34:13,489 --> 00:34:16,510 So in reality, we know that at some point 684 00:34:16,510 --> 00:34:20,739 it's going to taper off, and the stopping power at 0 685 00:34:20,739 --> 00:34:23,310 should be 0. 686 00:34:23,310 --> 00:34:28,880 This maximum right here occurs around 500 times 687 00:34:28,880 --> 00:34:31,997 the mean ionization potential, which is a pretty low energy, 688 00:34:31,997 --> 00:34:34,580 but what this actually says is that when the particle's moving 689 00:34:34,580 --> 00:34:38,000 really slow, it's moving so slow that once in a while, 690 00:34:38,000 --> 00:34:40,219 it will capture one of those electrons, 691 00:34:40,219 --> 00:34:44,690 like if you fire in a proton or a positively charged heavy ion, 692 00:34:44,690 --> 00:34:47,090 if it's moving so slow that it can feel the pull, 693 00:34:47,090 --> 00:34:49,580 it will just partially neutralize. 694 00:34:49,580 --> 00:34:53,929 And that becomes the next mechanism of energy loss. 695 00:34:53,929 --> 00:34:56,150 And so if we keep following this curve, once we 696 00:34:56,150 --> 00:34:58,850 hit some sort of a maximum, then it's 697 00:34:58,850 --> 00:35:01,430 going to lose less and less and less energy, 698 00:35:01,430 --> 00:35:05,120 do less and less damage, and you end up with the same curve, 699 00:35:05,120 --> 00:35:09,140 this kind of brag peak curve that we did together 700 00:35:09,140 --> 00:35:12,330 when we used the SRIM code. 701 00:35:12,330 --> 00:35:15,050 Did I go through the stopping range of ions in matter 702 00:35:15,050 --> 00:35:15,670 with you guys? 703 00:35:15,670 --> 00:35:17,550 Did I show you this on the screen? 704 00:35:17,550 --> 00:35:20,120 Remember the curve of the-- 705 00:35:20,120 --> 00:35:22,310 let's say damage events per distance or number 706 00:35:22,310 --> 00:35:25,390 of implanted ions as a function of distance. 707 00:35:25,390 --> 00:35:29,270 You end up with the exact same thing. 708 00:35:29,270 --> 00:35:32,770 That's what the SR stands for in SRIM is stopping, 709 00:35:32,770 --> 00:35:35,270 what is it, stopping power and range or something like that. 710 00:35:38,300 --> 00:35:40,230 Something range of ions and matter. 711 00:35:40,230 --> 00:35:42,510 Is it the stopping power and range of ions and matter? 712 00:35:42,510 --> 00:35:45,750 I don't know, but all SRIM is a gigantic stopping power 713 00:35:45,750 --> 00:35:48,510 database and a big Monte Carlo engine. 714 00:35:48,510 --> 00:35:51,270 So it takes an equation just like this one or-- 715 00:35:54,120 --> 00:35:56,400 yeah, just like this one, that one, 716 00:35:56,400 --> 00:36:00,025 whichever one you want and decides, well, 717 00:36:00,025 --> 00:36:01,650 how often is the particle going to lose 718 00:36:01,650 --> 00:36:06,070 how much energy depending on where it happens to be? 719 00:36:06,070 --> 00:36:08,910 And that's all there is to it. 720 00:36:08,910 --> 00:36:11,430 This point here, we would call the range 721 00:36:11,430 --> 00:36:14,040 or like the average range at which-- well, that's 722 00:36:14,040 --> 00:36:14,820 not quite right. 723 00:36:18,132 --> 00:36:19,840 There'd be some average range around here 724 00:36:19,840 --> 00:36:21,670 where the particles actually stop, 725 00:36:21,670 --> 00:36:23,620 when their energy goes to zero. 726 00:36:23,620 --> 00:36:26,380 And in reality, because every one of these processes 727 00:36:26,380 --> 00:36:29,020 is random in nature, the impact parameter 728 00:36:29,020 --> 00:36:30,580 is going to be kind of random. 729 00:36:30,580 --> 00:36:33,250 Not every particle will stop at the same place, 730 00:36:33,250 --> 00:36:35,710 because all the electrons are moving around in the atoms. 731 00:36:35,710 --> 00:36:40,980 So there's going to be some sort of a range of ranges, which 732 00:36:40,980 --> 00:36:46,430 we call straggling, which is to say 733 00:36:46,430 --> 00:36:49,310 that not all of the particles end up at the exact same range, 734 00:36:49,310 --> 00:36:53,250 but they end up pretty close. 735 00:36:53,250 --> 00:36:55,000 I think I want to pause here for a seconds 736 00:36:55,000 --> 00:36:58,110 and see if there's any questions from this four board 737 00:36:58,110 --> 00:37:00,330 derivation or any intuition questions 738 00:37:00,330 --> 00:37:02,110 that you guys might have. 739 00:37:02,110 --> 00:37:03,040 Yeah. 740 00:37:03,040 --> 00:37:05,620 STUDENT: You didn't initially have the negative d2 over dx. 741 00:37:05,620 --> 00:37:06,650 PROFESSOR: Oh yeah, where'd that go? 742 00:37:06,650 --> 00:37:07,790 Let's trace this through. 743 00:37:07,790 --> 00:37:09,415 STUDENT: What happened to the negative? 744 00:37:09,415 --> 00:37:11,350 [INTERPOSING VOICES] 745 00:37:12,510 --> 00:37:15,335 STUDENT: But we didn't actually derive it. 746 00:37:15,335 --> 00:37:16,460 PROFESSOR: Yeah, let's see. 747 00:37:20,300 --> 00:37:24,230 So the change in energy should have been 748 00:37:24,230 --> 00:37:26,600 a negative change in energy. 749 00:37:26,600 --> 00:37:29,120 So if we-- it went missing there. 750 00:37:29,120 --> 00:37:32,150 So there we go. 751 00:37:32,150 --> 00:37:35,270 That's the only other place that seems to be missing. 752 00:37:35,270 --> 00:37:37,106 OK, great. 753 00:37:37,106 --> 00:37:38,445 Cool. 754 00:37:38,445 --> 00:37:40,820 Then if you want to start looking at the number of damage 755 00:37:40,820 --> 00:37:45,140 events that this particle will incur, 756 00:37:45,140 --> 00:37:47,090 we'll call this the number of ion pairs, which 757 00:37:47,090 --> 00:37:49,400 might look suspiciously familiar if you guys remember 758 00:37:49,400 --> 00:37:51,200 the Chadwick paper, he was talking 759 00:37:51,200 --> 00:37:54,410 about this proton of this energy should make this many ion 760 00:37:54,410 --> 00:37:55,970 pairs at this distance. 761 00:37:55,970 --> 00:37:57,470 Now you guys actually have the tools 762 00:37:57,470 --> 00:38:00,110 to find out what that number should be, 763 00:38:00,110 --> 00:38:09,310 because it's going to be 1 over some ion pair energy, 764 00:38:09,310 --> 00:38:16,150 usually around 30 to 35 eV, depending on the material times 765 00:38:16,150 --> 00:38:21,830 dt vx, which is to say when the stopping power is higher, 766 00:38:21,830 --> 00:38:23,930 you're going to have more ion pairs produced 767 00:38:23,930 --> 00:38:25,850 as a function of distance. 768 00:38:25,850 --> 00:38:28,250 So the real label for this y-axis 769 00:38:28,250 --> 00:38:36,630 here should be like ion pairs or damage or defects or anything 770 00:38:36,630 --> 00:38:39,090 like that that refers to the same kind of thing as damages 771 00:38:39,090 --> 00:38:42,360 to the material, either by ionization or even 772 00:38:42,360 --> 00:38:46,383 by similar nuclear processes. 773 00:38:46,383 --> 00:38:48,050 And so that's what results in those SRIM 774 00:38:48,050 --> 00:38:50,490 curves that we showed from before, where you have, 775 00:38:50,490 --> 00:38:53,550 let's say, a bunch of protons entering a material here 776 00:38:53,550 --> 00:38:55,290 at some high energy. 777 00:38:55,290 --> 00:38:58,100 They don't lose very much energy when they go in, 778 00:38:58,100 --> 00:39:00,440 but as soon as they get to a low enough energy 779 00:39:00,440 --> 00:39:02,840 where they're stopping power reaches the maximum, 780 00:39:02,840 --> 00:39:05,590 they dump most of their energy in there. 781 00:39:05,590 --> 00:39:09,620 And this is the basis behind proton cancer therapy, which 782 00:39:09,620 --> 00:39:13,350 I mentioned to you guys in the first or second day of class. 783 00:39:13,350 --> 00:39:16,560 Now that we know both exponential attenuation 784 00:39:16,560 --> 00:39:20,250 and stopping power, we can explain theoretically 785 00:39:20,250 --> 00:39:23,500 why proton therapy is a more effective treatment. 786 00:39:23,500 --> 00:39:28,930 So let's say this is the person that contains a tumor. 787 00:39:28,930 --> 00:39:30,470 Say it's right there. 788 00:39:30,470 --> 00:39:35,670 And you have a choice between firing in an X-ray 789 00:39:35,670 --> 00:39:39,430 or firing in a proton. 790 00:39:39,430 --> 00:39:41,920 What is the dose to this person, not just to the tumor, 791 00:39:41,920 --> 00:39:46,760 but through the whole person going to look like for X-rays? 792 00:39:46,760 --> 00:39:47,650 Get another board. 793 00:39:50,260 --> 00:39:52,630 So if we look at the number of ion pairs, and let's 794 00:39:52,630 --> 00:39:58,300 say this is the thickness of the person, 795 00:39:58,300 --> 00:40:03,230 and the tumor is in this range. 796 00:40:03,230 --> 00:40:08,830 And if you send in your X-rays, or you 797 00:40:08,830 --> 00:40:12,130 send in your protons, what is the number of ion 798 00:40:12,130 --> 00:40:14,980 pairs produced from X-ray or from proton going 799 00:40:14,980 --> 00:40:16,488 to look like in either case? 800 00:40:16,488 --> 00:40:18,280 So first of all, who wants do the X-ray one 801 00:40:18,280 --> 00:40:19,405 or tell me what it will be? 802 00:40:24,623 --> 00:40:25,540 You guys-- yeah, Luke? 803 00:40:25,540 --> 00:40:26,970 STUDENT: Would it be pretty flat? 804 00:40:26,970 --> 00:40:28,720 PROFESSOR: It'll be fairly flat, but there 805 00:40:28,720 --> 00:40:30,020 would be some decay to it. 806 00:40:30,020 --> 00:40:33,880 So let's say we defined some initial intensity, X-rays just 807 00:40:33,880 --> 00:40:36,890 get attenuated exponentially. 808 00:40:36,890 --> 00:40:39,790 So you're going to do a whole lot of damage to the person 809 00:40:39,790 --> 00:40:42,700 before the X-rays reach the tumor, which 810 00:40:42,700 --> 00:40:45,280 is why when you do X-ray therapy, 811 00:40:45,280 --> 00:40:48,880 you have to send in x-rays from a bunch of different locations 812 00:40:48,880 --> 00:40:51,080 so that the tumor gives out the most, 813 00:40:51,080 --> 00:40:53,020 and the rest of the person in any location 814 00:40:53,020 --> 00:40:55,780 doesn't get that much dose. 815 00:40:55,780 --> 00:40:57,730 For proton therapy, it's quite different. 816 00:40:57,730 --> 00:40:59,673 It looks just like this. 817 00:40:59,673 --> 00:41:01,840 So you might do a little bit of damage as you go in, 818 00:41:01,840 --> 00:41:06,787 and you tune the energy of those protons, 819 00:41:06,787 --> 00:41:08,620 so that they do all the damage in the tumor, 820 00:41:08,620 --> 00:41:11,548 and then they stop in the tumor or just beyond it, 821 00:41:11,548 --> 00:41:13,090 so that they don't do any more damage 822 00:41:13,090 --> 00:41:16,310 to the rest of the person, and they do very little going in. 823 00:41:16,310 --> 00:41:18,190 And so that's why proton therapy centers 824 00:41:18,190 --> 00:41:19,930 are popping up all over the world 825 00:41:19,930 --> 00:41:21,950 because it's a more effective treatment. 826 00:41:21,950 --> 00:41:27,512 It's also more expensive because you need a proton accelerator 827 00:41:27,512 --> 00:41:28,970 so then, here's a question for you. 828 00:41:28,970 --> 00:41:33,390 This is something they actually do in the lab. 829 00:41:33,390 --> 00:41:35,820 Say, here's your human. 830 00:41:35,820 --> 00:41:37,900 There's your tumor. 831 00:41:37,900 --> 00:41:44,110 There is your proton gun at a fixed 250 MeV, 832 00:41:44,110 --> 00:41:46,610 firing protons out. 833 00:41:46,610 --> 00:41:48,960 How do you change the range of those protons 834 00:41:48,960 --> 00:41:50,210 without changing their energy? 835 00:41:52,965 --> 00:41:54,590 STUDENT: The distance it has to travel? 836 00:41:54,590 --> 00:41:55,110 PROFESSOR: Is what? 837 00:41:55,110 --> 00:41:56,900 STUDENT: The distance it has to travel the other way. 838 00:41:56,900 --> 00:41:59,420 PROFESSOR: The distance it has to travel specifically? 839 00:41:59,420 --> 00:42:02,790 I mean, if they travel in a vacuum, do they lose energy? 840 00:42:02,790 --> 00:42:03,290 No. 841 00:42:03,290 --> 00:42:04,420 So what can you do? 842 00:42:04,420 --> 00:42:07,545 STUDENT: [INAUDIBLE] 843 00:42:07,835 --> 00:42:09,210 PROFESSOR: You could deflect them 844 00:42:09,210 --> 00:42:10,620 and change their direction. 845 00:42:10,620 --> 00:42:14,580 But as we'll get into on Tuesday, if you deflect them, 846 00:42:14,580 --> 00:42:16,180 they're going to emit lots of X-rays 847 00:42:16,180 --> 00:42:17,430 in the form of Bremmstrahlung. 848 00:42:17,430 --> 00:42:20,040 So that's probably not what we want to do. 849 00:42:20,040 --> 00:42:21,630 You can put stuff-- 850 00:42:21,630 --> 00:42:24,690 and I can't be any more specific than that-- 851 00:42:24,690 --> 00:42:27,720 in between the proton beam and the patient, 852 00:42:27,720 --> 00:42:31,950 because if the stopping power for 250 MeV protons and 50 MeV 853 00:42:31,950 --> 00:42:34,320 protons basically doesn't change, 854 00:42:34,320 --> 00:42:36,150 then you just put things in the way. 855 00:42:36,150 --> 00:42:38,595 So let's say this is the thickness of the person. 856 00:42:42,150 --> 00:42:46,650 You just put some tissue equivalent stuff, 857 00:42:46,650 --> 00:42:50,100 or what they'll call a phantom, so some tissue equivalent gel 858 00:42:50,100 --> 00:42:53,220 or water or some other stuff to lower the proton 859 00:42:53,220 --> 00:42:56,220 energy without deflecting the beam that much. 860 00:42:56,220 --> 00:42:58,860 So as you guys saw in the SRIM simulation, 861 00:42:58,860 --> 00:43:02,430 if you track the 3D positions of these protons 862 00:43:02,430 --> 00:43:05,880 as they enter into the material, they all go pretty straight, 863 00:43:05,880 --> 00:43:08,550 and then they start getting funny. 864 00:43:12,000 --> 00:43:14,550 Computer can fly the ions faster than I can. 865 00:43:14,550 --> 00:43:16,650 But no matter what it goes through here, 866 00:43:16,650 --> 00:43:18,220 while the protons have high energy, 867 00:43:18,220 --> 00:43:19,890 they don't get deflected much, they 868 00:43:19,890 --> 00:43:21,600 don't lose that much energy. 869 00:43:21,600 --> 00:43:25,620 And you can very finely tune the amount of stuff in the way. 870 00:43:25,620 --> 00:43:29,010 This would be the stuff section before entering the person. 871 00:43:32,080 --> 00:43:33,330 So this is why it's so useful. 872 00:43:35,962 --> 00:43:37,420 So let me check the time, because I 873 00:43:37,420 --> 00:43:40,000 haven't checked at all. 874 00:43:40,000 --> 00:43:41,060 The clock's broken. 875 00:43:41,060 --> 00:43:43,420 Oh, we have like 10 minutes. 876 00:43:43,420 --> 00:43:46,060 So now is a good time to stop for questions 877 00:43:46,060 --> 00:43:48,940 and see if you guys have any questions from the derivation 878 00:43:48,940 --> 00:43:52,030 or the sort of physical meaning of stopping power and matter. 879 00:43:55,690 --> 00:43:56,300 Yeah. 880 00:43:56,300 --> 00:44:00,540 STUDENT: What was the nz sort of [INAUDIBLE]?? 881 00:44:00,540 --> 00:44:11,182 PROFESSOR: Yes, the nz, n is the number density of atoms. 882 00:44:11,182 --> 00:44:13,640 So if you're traveling through some actual block of matter, 883 00:44:13,640 --> 00:44:16,500 it depends how many atoms are in the way. 884 00:44:16,500 --> 00:44:21,830 So if the total stopping power decreases with decreasing 885 00:44:21,830 --> 00:44:24,410 density, like if you're going through tungsten, 886 00:44:24,410 --> 00:44:27,410 but it happens to be a tungsten gas, that tungsten gas will not 887 00:44:27,410 --> 00:44:30,470 have nearly as much stopping power as tungsten metal, 888 00:44:30,470 --> 00:44:32,680 because there's just more tungsten than the weight. 889 00:44:32,680 --> 00:44:34,460 Or less, I'm sorry. 890 00:44:34,460 --> 00:44:39,890 And the z right here is the charge per atom, 891 00:44:39,890 --> 00:44:42,500 to say if you're firing electrons into something, 892 00:44:42,500 --> 00:44:43,947 the strength of the Coulomb force 893 00:44:43,947 --> 00:44:46,280 that they'll feel, or let's say, the number of electrons 894 00:44:46,280 --> 00:44:50,030 that they can smack into is the same with the number of protons 895 00:44:50,030 --> 00:44:54,110 in that nucleus if we're not using ionized materials. 896 00:44:54,110 --> 00:44:56,210 And we're typically not firing anything 897 00:44:56,210 --> 00:44:57,290 into ionized materials. 898 00:44:57,290 --> 00:45:00,710 It's just normal neutral matter. 899 00:45:00,710 --> 00:45:02,352 Does that make more sense? 900 00:45:02,352 --> 00:45:03,570 Cool. 901 00:45:03,570 --> 00:45:04,810 Yeah, Luke. 902 00:45:04,810 --> 00:45:07,618 STUDENT: Where did that n go? 903 00:45:07,618 --> 00:45:09,160 PROFESSOR: It should have been there. 904 00:45:09,160 --> 00:45:11,320 Thank you. 905 00:45:11,320 --> 00:45:13,755 pi n. 906 00:45:13,755 --> 00:45:14,880 Should absolutely be there. 907 00:45:19,540 --> 00:45:21,290 Anything else? 908 00:45:21,290 --> 00:45:21,910 Yeah, Dan. 909 00:45:21,910 --> 00:45:22,830 STUDENT: [INAUDIBLE] 910 00:45:22,830 --> 00:45:23,756 PROFESSOR: OK. 911 00:45:23,756 --> 00:45:25,390 STUDENT: [INAUDIBLE] 912 00:45:26,850 --> 00:45:28,680 PROFESSOR: The charge per atom is 913 00:45:28,680 --> 00:45:32,130 big Z. The charge on the particle 914 00:45:32,130 --> 00:45:35,320 is little z, because both of them actually matter. 915 00:45:35,320 --> 00:45:37,950 So little z tells you the strength 916 00:45:37,950 --> 00:45:41,220 of the interaction between the particle and each electron. 917 00:45:41,220 --> 00:45:44,730 Big Z tells you how many electrons are there per atom. 918 00:45:44,730 --> 00:45:46,720 Big N tells you how many atoms are in the way. 919 00:45:46,720 --> 00:45:48,720 And in that way, you have a complete description 920 00:45:48,720 --> 00:45:50,530 of the material. 921 00:45:50,530 --> 00:45:53,170 Curious that the mass of the charged particle 922 00:45:53,170 --> 00:45:56,290 is absent from this formula. 923 00:45:56,290 --> 00:45:58,390 Isn't it? 924 00:45:58,390 --> 00:45:59,220 Yeah. 925 00:45:59,220 --> 00:46:01,280 The mass doesn't matter. 926 00:46:01,280 --> 00:46:04,190 You will certainly change the momentum of, 927 00:46:04,190 --> 00:46:06,320 let's say, a charged particle less. 928 00:46:06,320 --> 00:46:10,250 But in the end, it's just non-contact Coulomb forces 929 00:46:10,250 --> 00:46:12,230 that determine the energy transfer 930 00:46:12,230 --> 00:46:14,810 between the electrons in there and the charged particles 931 00:46:14,810 --> 00:46:17,730 slowing down in the medium. 932 00:46:17,730 --> 00:46:21,680 So that is a curious thing to look at, but it is intentional. 933 00:46:21,680 --> 00:46:25,110 For the case of this ionization or electronic stopping power, 934 00:46:25,110 --> 00:46:30,250 the mass does not enter into it at least in this formula. 935 00:46:30,250 --> 00:46:33,220 There is another version derived that's in your reading 936 00:46:33,220 --> 00:46:35,380 that they just kind of plop it in front of you 937 00:46:35,380 --> 00:46:37,720 that's got the mass somewhere in the natural long term 938 00:46:37,720 --> 00:46:39,095 somewhere where it really doesn't 939 00:46:39,095 --> 00:46:41,845 change much at all, except for really high energies. 940 00:46:45,500 --> 00:46:47,160 So if you want to think about, well, 941 00:46:47,160 --> 00:46:49,130 what do I want you to know, I would 942 00:46:49,130 --> 00:46:52,340 want you to be able to go through this derivation again, 943 00:46:52,340 --> 00:46:55,670 so that I can know you can go from a intuitive example 944 00:46:55,670 --> 00:46:58,520 to an actual equation you can use, graph 945 00:46:58,520 --> 00:47:01,010 what that equation should look like and talk 946 00:47:01,010 --> 00:47:05,210 about where it really matters and where it breaks down. 947 00:47:05,210 --> 00:47:08,750 Like mathematically speaking, if this natural long term 948 00:47:08,750 --> 00:47:12,110 is negative, you're not going to have a negative stopping power. 949 00:47:12,110 --> 00:47:15,560 Something else has got to occur, and what's happening here 950 00:47:15,560 --> 00:47:16,640 is neutralization. 951 00:47:23,690 --> 00:47:25,720 And that's why the stopping power curve diverges 952 00:47:25,720 --> 00:47:27,470 for really low energies, because sometimes 953 00:47:27,470 --> 00:47:28,428 electrons get captured. 954 00:47:31,817 --> 00:47:33,400 Any other questions on stopping power? 955 00:47:38,490 --> 00:47:39,320 Cool. 956 00:47:39,320 --> 00:47:42,220 This is a good place to stop for now.