Subtitles section Play video Print subtitles The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. MICHAEL SHORT: Hey guys, hope you enjoyed the brief break from the heavy technical stuff. Because we're going to get right back into it and develop the neutron transport equation today, the one that you see on everybody's t-shirts here in the department. So I think multiple years folks have used this equation on the back of t-shirts just be like, we're awesome. And we do difficult math. Well, this is what you're going to start to do it. In fact, it's big enough and hard enough that we're going to spend all day today developing it, like actually writing out the terms of the equation and understanding what it actually means. Before, on Thursday and Friday, we're going to reduce it down to a much simpler equation, something that you can actually solve and do some simple reactor calculations with. We started off the whole idea of the neutron transport equation as a way to track some population of neutrons. Let's see, I'm going to have our variable list up here. What I'll probably do is on Thursday and Friday I'll just have it back up on the screens so that we don't have to write it twice. But there's going to be a lot of variables in this equation. I'm going to do my best, again, to make the difference between V and nu very obvious, and anything else like that. But the goal is to track some population of neutrons, at some position, at some energy, traveling in some direction omega, as a function of time. And the 3D representation of what we're looking at here is let's say we had some small volume element, which we'll call that our dV. That's got normal vectors sticking out of it. We'll call those n-hats in all directions. And inside there, let's say if this is our energy scale, we're tracking the population of neutrons that occupies some small energy group dE, and is also traveling in some small direction that we designate as d-omega. So that's the goal of this whole equation is to track the number of neutrons at any given position. So let's call this distance, or the vector r. In this little volume, traveling in some direction omega, with some infinitesimally small energy group d. That's going to be the goal of the whole thing. And what we'll do is write this to say the change in the population of neutrons at a given distance, energy, angle, and time, over time is just going to be a sum of gain and loss terms. And what I think we'll take all day today to do is to figure out what are the actual physical things that neutrons can do in and out of this volume, and how do we turn those into math, something that we can abstract and solve? There's a couple other terms that we're going to put up here. We'll say that the flux of neutrons, which is usually the variable that we actually track, is just the velocity times the neutron population. And also let's define some angularly independent terms. Because in the end we've been talking about what's the probability of some neutron or particle interacting with some electron going out at some angle. But as we're interested in how many neutrons are there in the reactor, we usually don't care in which direction they're traveling. So the first simplification that we will do is get rid of any sort of angular dependence, getting rid of two of the seven variables that we're dealing with here. So all these variables right here will be dependent on angle. And all these variables right here will be angularly independent. So there'll be some corresponding capital N, or number of neutrons as a function of r, E, and t. We'll call this Flux. We'll call this Number. There's going to be a number of cross sections that we need to worry about. So we'll refer to little sigma as a function of energy, as our micro-cross-section, and big sigma of E as a macroscopic cross-section. Then you want to remember the relation between these two. AUDIENCE: Solid angle? MICHAEL SHORT: The solid angle, not quite. That's, let's see. There's a difference between-- and so what these physically mean is little sigma means the probability of interaction with one particle. And this is just the total probability of interaction with all the particles that may be there. So yeah, Chris? AUDIENCE: Number density? MICHAEL SHORT: There's number density. Already we have another variable conflict. How do we want to resolve this? Let's see. We'll have to change the symbol somehow. Let's make it cursive. Don't know what else to do. I don't want to give it a number other than n since we're talking about neutrons, or that right here it's going to be number density. And in the end, we're worried about some sort of reaction rate, which is always going to equal some flux, or let's just stick with some angularly dependent flux, that r, E omega t, times some cross-section as a function of energy. And it's these reaction rates that are the rates of gains and losses of neutrons out of this volume, out of this little angle, out of this energy group, and out of that space, or into that volume energy group and space. So let's see, other terms that we'll want to define include nu, like last time. We'll call this neutron multiplication. In other words, this is the number of neutrons made on average during each fission event. And we give it energy dependence because as we saw on the Janis libraries on Friday, I think it was. What's today, Tuesday? I don't even know anymore. I think as we saw on Friday, that depends on energy for the higher energy levels. And there's also going to be some Kai spectrum, or some neutron birth spectrum, which tells you the average energy at which neutrons are born from fission. So regardless of what energy goes in to cause the fission, there's some probability distribution of a neutron being born at a certain energy. And it looks something like this, where that's about 1 MeV. That's about 10 MeV. And that average right there is around 2 MeV. And so it's important to note that neutrons are born at different energies. Because we want to track every single possible dE throughout this control volume, which we'll also call a reactor. Let's see, what other terms will we need to know? The different types of cross sections, or the different interactions that neutrons can have with matter. What are some of the ones that we had talked about? What can neutrons do when they run into stuff? AUDIENCE: Scatter. MICHAEL SHORT: They can scatter. So there's going to be some scattering cross-section. And when they scatter, the important part here is they're going to change in energy. What else can they do? Yeah? AUDIENCE: Absorbed. MICHAEL SHORT: They can be absorbed. So we'll have some sigma absorption. What are some of the various things that can happen when a neutron is absorbed? AUDIENCE: Fission. MICHAEL SHORT: Yeah, so one of them is fission. What are some of the other ones? AUDIENCE: Capture. MICHAEL SHORT: Yep, capture. What were some of the ones that we talked about during the Chadwick paper? AUDIENCE: Neutron [INAUDIBLE]. MICHAEL SHORT: Yep, so there can be some-- we'll call it n,in, which means one neutron goes in, i neutrons come out, so 1 to i neutrons, sure. Anything else? Encompassed in absorption? Well when we refer to scatter here, what type of scattering are we talking about? AUDIENCE: Compton's scatter? MICHAEL SHORT: Compton's for photons. It's OK. Was it elastic or inelastic scattering? AUDIENCE: Elastic. MICHAEL SHORT: Elastic scattering. So another thing you could call an absorption event, depending on what bin you put things in, is inelastic scattering, which is that kind of-- we call it scattering because one neutron goes in, one neutron comes out. But in reality, you have a compound nucleus forming and a neutron emitted from a different energy level. So it doesn't follow the simple ballistic laws of, and kinematic laws of inelastic scattering. What else can neutrons do? Now we're getting into the real esoteric stuff. But I want to see if you guys have any idea. Did you know that neutrons can decay? A low neutron is actually not a stable particle. If you look up on the Kyrie table of nuclides, it's got a half life of 12 minutes. So if you happen to be able to have neutrons in a bottle or something, which we actually can do. There's centers for ultra-cold neutrons and atoms. There's one at North Carolina State where they actually cool down neutrons to cryogenic temperatures to the point where they can actually confine them. They only live on average 12 minutes. And then there would also be what we call a neutron-neutron interactions. There is a finite, non-zero but very small probability that neutrons can hit other neutrons. But the mean-free path for these is on the order of 10 to the 8th centimeters. So this is not something we have to consider. But it's interesting to know that yes, neutrons can run into other neutrons. And these sorts of things have been measured. We won't have to worry about this. We won't have to worry about neutron decay. But it's interesting to note that a low neutron is not a stable particle. It will spontaneously undergo beta decay, into a proton and an electron. Pretty neat, huh? Anyway, if we sum up all these possible interactions, we have one other cross-section, which we're going to call the total cross-section, the probability of absolutely any interaction occurring at all. Because any sort of interaction of that neutron is going to cause removal from this group of energy position, angle, location, whatever. Whether it's absorption, or fission, or elastic scattering, or inelastic scattering, any sort of event-- except for forward scattering, which means nothing happens-- is going to result in this neutron either leaving the volume. So it might scatter out of our little volume. Or it might change direction, scatter out of our d-omega. Or it will lose some energy, or gain some energy, in some cases, leaving our little dE, which is what we're trying to track. Because we're actually tracking what's the population of neutrons in this little dE, in this direction, in this position, at this time. And supposedly if we know this term fully, we can solve for all the neutrons everywhere, anywhere in the reactor with full information. So what we'll spend the rest of today doing is figuring out what are all the possible gain and loss terms.