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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