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MICHAEL SHORT: I think I might actually

use all 16 colors today.

Oh no, this is the most satisfying day.

Whereas Tuesday was probably the most mathematically intense,

because we developed this equation right here,

today is going to be the most satisfying,

because we are going to cancel out just about every term,

leaving a homogeneous, infinite reactor criticality condition.

So we will go over today, how do you

go from this, to what is criticality in a reactor?

So I want to get a couple of variables up over here

to remind you guys.

We had this variable flux of r, e,

omega, t in the number of neutrons

per centimeter squared per second traveling

through something.

And we also had its corresponding non-angular

dependent term, on just r, e, t, if we don't care

what angle things go through.

We've got a corresponding variable called current.

So I'll put this as flux.

We have current j, r, e, omega, t, and its corresponding,

we don't care about angle form.

And today, what we're going to do

is first go over this equation again so that we

understand all of its parts.

And there are more parts here than are in the reading.

If you remember, that's because I

wanted to show you how all of these terms are created.

Just about every one of these terms,

except the external source and the flow through some surface,

has the form of some multiplier, times the integral

over all possible variables that we care about,

times a reaction rate d stuff, where this reaction

rate is always going to be some cross-section, times some flux.

So when you look at this equation using that template,

it's actually not so bad.

So let's go through each of these pieces right here.

And then we're going to start simplifying things.

And this board's going to look like some sort of rainbow

explosion.

But all that's going to be left is a much simpler form

of the neutron diffusion equation.

So we've got our time-dependent term right here,

where I've stuck in this variable flux,

instead of the number of neutrons n,

because we know that flux is the number of neutrons,

times the speed at which they're moving.

And just to check our units, flux

should be in neutrons per centimeters squared per second.

And n is in neutrons per cubic centimeter.

And velocity is in centimeters per second.

So the units check out.

That's why I made that substitution right there.

And this way, everything is in terms of little fi, the flux.

We have our first term here.

I think I'll have a labeling color.

That'll make things a little easier to understand.

Which is due to regular old fission.

In this case, we have new, the number of neutrons created

per fission, times chi, the sort of fission birth spectrum,

or at what energy the neutrons are born.

Over 4pi would account for all different angles in which they

could go out, times the integral over our whole control volume,

and all other energies in angles.

If you remember now, we're trying

to track the number of neutrons in some small energy group, e,

traveling in some small direction, omega.

And those have little vector things on it

at some specific position as a function of time.

So in order to figure out how many neutrons are entering

our group from fission, we need to know,

what are all the fissions happening

in all the other groups?

I've also escalated this problem a little bit

to not assume that the reactor's homogeneous.

So I've added an r, or a spatial dependence

for every cross-section here, which

means that as you move through the reactor,

you might encounter different materials.

You almost certainly will, unless your reactor

has been in a blender.

So except for that case, you would actually

have different cross sections in different parts of the reactor.

So all of a sudden, this is starting

to get awfully interesting, or messy depending on what

you want to think about it.

There is the external source, which is actually

a real phenomenon, because reactors

do stick in those californium kickstarter sources.

So for some amount of time, there

is an external source of neutrons,

giving them out with some positional energy angle

and time dependence.

So let's call this the kickstarter source.

There's this term right here, the nin reactions.

So these are other reactions where it's absorb a neutron,

and give off anywhere between 2 and 4 neutrons.

Beyond that, it's just not energetically

possible in a fission reactor.

But don't undergo fission.

They have their own cross sections, their own birth

spectrum.

And I've stuck in something right here,

if we have summing over all possible i, where you have

this reaction be n in reaction, where 1 neutron goes in,

and i neutrons come out.

You've got to multiply by the number of neutrons

per reaction.

For fission, that was new.

For an nin reaction, that's just i.

But otherwise, the term looks the same.

You have your multiplier, your birth spectrum, your 4pi,

your integral over stuff, your unique cross-section,

and the flux.

And these two together give you a reaction rate.

I've just written all of the differentials as d stuff,

because it takes a lot of time to write those

over and over again.

And then we have our photo fission term,

where gamma rays of sufficiently high energy

can also induce fission external to the neutrons.

The term looks exactly the same.

There's going to be some new for photofission,

some birth spectrum for photofission,

some cross section for photofission, and the same flux

that we're using everywhere else.

Then we had what's called the in scattering term, where neutrons

can undergo scattering, lose some energy,

and enter our group from somewhere else.

That's why we have those en omega primes,

because it's some other energy group.

And we have to account for all of those energy groups.

That's why we have this integral there.

And it looks very much the same.

There's a scattering cross-section.

And that should actually be an e prime right there.

Make sure I'm not missing any more

of those inside the integral.

That's all good.

That's a prime, good.

There's also a flux.

And then there was this probability function

that a given neutron starting off an energy

e prime omega prime, ends up scattering into r

energy, e and omega.

So this would be the other one.

And this would be r group.

But otherwise, the term looks very much the same.

And that takes care of all the possible gains of neutrons

into r group.

The losses are a fair bit simpler.

There is reaction of absolutely any kind.

Let's say this would be the total cross-section, which

says that if a neutron undergoes any reaction at all,

it's going to lose energy and go out of our energy group d, e.

Notice here that these are all--

these energies and omega is our all r group,

because we only care about how many neutrons in r group

undergo a reaction and leave.

And the form is very simple--

integrate over volume, energy, and direction,

times a cross-section, times a flux,

just like all the other ones.

Then the only difference in one right here

is what we'll call leakage.

These are neutrons moving out of whatever control

surface that we're looking at.

And this can be some arbitrarily complex control surface in 3D.

I don't really know how to draw a blob in 3D.

But at every point on that blob, there's

going to be a normal vector.

And you can then take the current of neutrons traveling

out that normal vector, and figure out how much of that

is actually leaving our surface ds.

The one problem we had is that everything here

is in terms of volume, volume, volume, surface.

So we don't have all the same terms,

because once we have everything in the same variables

we can start to make some pretty crazy simplifications.

The last thing we did is we invoked the divergence theorem,

that says that the surface integral of some variable FdS

is the same as the volume integral of the divergence

of that variable dV.

So I remember there was some snickering last time,

because you probably haven't seen this since,

was it 1801 or 1802?

1802, OK, that makes sense, because divergence usually

has more than one variable associated with it.

I'll include the dot, because that's

what makes that divergence.

So we can then rewrite this term.

Let's start our simplification colors.

That's our divergence theorem.

So let's get rid of it in this form,

and call it minus integral over all that stuff.

Then we'll have dot, little j, to be

careful, r, e, omega, t, d soft.

So for every step, I"m going to use a different color so you

can see which simplification led to how much crossing stuff out.

And so like I said, this board's going

to look like a rainbow explosion.

But then we'll rewrite it at the end.

And it's going to look a whole lot simpler.

So now, let's start making some simplifications.

Let's say you're an actual reactor designer,

and all you care about is how many neutrons are here.

Of the variables here, which one do you

think we care the least about?

Angle, I mean do we really care which direction

the neutrons are going?

No, we pretty much care, where are they?

And are they causing fission or getting absorbed?

So let's start our simplification board.

And in blue, we'll neglect angle.

This is where it starts to get fun.

So in this case, we'll just perform the omega integral

over all angles.

We just neglect angle here.

We forget the omega integral, forget omega there.

Away goes the 4pi, because we've integrated overall 4pi star

radians, or all solid angle.

Let's just keep going.

Forget the 4pi, forget the omega,

forget the omega, forget the 4pi and the omega, and the omega.

Same thing here-- forget the omega in the scattering kernel,

forget it in the flux, forget it there,

forget it there, and there as well.

OK, we've now completely eliminated one variable.

And all we had to do is ditch the 4pi and one

of the integrals.

What next?

We're tracking right now every possible position,

every possible energy, at every possible time.

If you want to know, what is your flux

going to be in the reactor at steady state, what

variable do you attack next?

Time.

So let's just say this reactor is at steady state.

That's going to invoke a few things.