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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.

• 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

• 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

• 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 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.