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  • 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.