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PROFESSOR: OK, guys.

Welcome back.

As you can see, we're not using the screen today.

This is going to be one of those fill-the-board lectures.

But I am going to work you through every single step.

We're going to go through the Q equation

and derive its most general form together,

which, for the rest of this class,

we'll be using simplified or reduced

forms to explain a lot of the ion

or electron-nuclear interactions as well

as things like neutron scattering

and all sorts of other stuff.

We'll do one example.

For any of you that have looked at neutrons slowing down

before, how much energy can a neutron

lose when it hits something?

We'll be answering that question today in a generally

mathematical form.

And then a few lectures later, we'll

be going over some of the more intuitive aspects

to help explain it for everybody.

So I'm going to show you the same situation that we've

been describing sort of intuitively so far,

but we're going to hit it mathematically today.

Let's say there's a small nucleus,

1, that's firing at a large nucleus, 2, and afterwards,

a different small nucleus, 3, and a different large nucleus,

4, come flying out.

And so we're going to keep this as general as possible.

So let's say if we draw angles from their original paths,

particle 3 went off at angle theta

and particle 4 went off at angle phi.

So hopefully those are differentiable enough.

And if we were to write the overall Q equation showing

the balance between mass and energy here,

we would simply have the mass 1 c squared

plus kinetic energy of 1.

So in this case, we're just saying

that the mass and the kinetic energy

of all particles on the left side and the right side

has to be conserved.

So let's add mass 2 c squared plus T2 has to equal mass 3 c

squared plus T3 plus mass 4 c squared plus T4, where,

just for symbols, M refers to a mass,

T refers to a kinetic energy.

And so this conservation of total mass or total energy

has got to be conserved.

And we'll use it again.

Because, again, we can describe the Q, or the energy consumed

or released by the reaction, as either the change in masses

or the change in energies.

So in this case, we can write that Q--

let's just group all of the c squareds together

for easier writing.

If we take the initial masses minus the final masses,

then we get a picture of how much mass was converted

to energy, therefore, how much energy

is available for the reaction, or Q,

to turn it into kinetic energy.

So in this case, we can put the kinetic energy

of the final products minus the kinetic energies--

I'm going to keep with 1--

of the initial products.

And so we'll use this a little later on.

One simplification that we'll make now

is we'll assume that if we're firing particles at anything,

that anything starts off at rest.

So we can start by saying there's no T2.

That's just a simplification that we'll make right now.

And so then the question is, what

quantities of this situation are we likely to know,

which ones are we not likely to know,

and which ones are left to relate together?

So let's just go through one by one.

Would we typically know the mass of the initial particle coming

in?

We probably know what we're shooting at stuff, right?

So we'd know M1.

What about T1, the initial kinetic energy?

Sure.

Let's say we have a reactor whose energy we know,

or an accelerator, or something that we're

controlling the energy, like in problem set one.

We'd probably know that.

We'd probably know what things we're firing at.

And we would probably know what the masses

of the final products are, because you guys have been

doing nuclear reaction analysis and calculating

binding energies and everything for the last couple of weeks.

But we might not know the kinetic energies

of what's coming out.

Let's say we didn't actually even know the masses yet.

We'd have to figure out a way to get both the kinetic energies.

And what about these angles here?

This is the new variable that we're introducing,

is the kinetic energy of particles 3 and 4

is going to depend on what angles they fire off at.

Let me give you a limiting case.

Let's say theta was 0.

What would that mean, physically?

What would be happening to particles 1, 2, 3,

and 4 if theta and phi were 0, if they kept on moving

in the exact same path?

Yeah?

AUDIENCE: Is it a fusion event, or [INAUDIBLE]

PROFESSOR: We don't know.

Well, let's see.

Yeah.

If it was a fusion event-- let's say there was one here and one

standing still--

then the whole center of mass of the system

would have to move that way.

So one example could be a fusion event.

A second example could be absolutely nothing.

It's perfectly valid to say if, let's say, particle 1 scatters

off particle 2 at an angle of 0 degrees,

that's what's known as forward scattering, which

is to say that theta equals 0.

So this is another quantity that we might not know.

We might not know what theta and phi are.

And the problem here is we've got, like, three or four

unknowns and only one equation to relate them.

So what other-- yeah?

Question?

AUDIENCE: For forward scattering,

when you say theta equals 0, do you mean they just sort of move

together forward, kind of like an inelastic collision,

and they just keep moving in the same direction?

PROFESSOR: An inelastic collision would be one.

And since we haven't gone through what inelastic means,

that would mean some sort of collision where--

let's see.

How would I explain this?

I'd say an inelastic collision would be like

if particles 1 and 2 were to fuse, like a capture event,

for example, or a capture and then a re-emission, let's say,

of a neutron.

Yeah.

If it was re-emitted in the forward direction,

then that could be an inelastic scattering event--

AUDIENCE: Oh, OK.

PROFESSOR: --but still in the same direction.

Or an elastic scatter at an angle of theta

equals 0 could be like there wasn't any scattering at all.

Because really in the end, can matter--

let's say if you have a neutron firing at a nucleus,

depends on what angle it bounces off of,

in the billiard ball sense.

If it bounces off at an angle of 0, that means it missed.

We would consider that theta equals 0.

But the point here is that we now

have more quantities unknown than we

have equations to define them.

So how else can we start relating

some of these quantities?

What else can we conserve, since we've

already got mass and energy?

What's that third quantity I always yell out?

AUDIENCE: Momentum.

PROFESSOR: Momentum.

Right.

So let's start writing some of the momentum conservation

equations so we can try and nail these things down.

So I'm going to write each step one at a time.

We'll start by conserving momentum.

That's what we'll do right here.

And we can write the x and the y equations separately.

So what's the momentum of particle 1?

How do we express that?

AUDIENCE: Mass times velocity.

PROFESSOR: Yep.

So it would be like M1 V1.

So we'll have a little box right here for momentum.

We could say mass times velocity--

or, how do we express that in terms of the variables

that we have here, like we did last week?

What about in terms of kinetic energies?

Well, another way of writing mass times velocity

would be root 2MT.

Because in this case, we would have root 2 times

M1 times 1/2 M1 V1 squared.

The 2s here cancel.

Let's see.

You have M1 squared.

You have a V squared.

And the square root of M squared V squared is just MV.

So this is an equivalent way of writing the momentum

in the variables that we're working in already.

And so since that doesn't introduce another variable

like velocity-- which we do know,

but it's kind of confusing to add more symbols-- let's

keep as few as possible.

So what's the x momentum of particle 1?

Just what I've got up there.

2 M1 T1.

What's the x momentum in this frame of particle 2?

0.

We're assuming that it's at rest.

And now, what's the x momentum of particle 3?

AUDIENCE: [INAUDIBLE]

PROFESSOR: I heard a couple of things.

Can you say them louder?

AUDIENCE: Square root of 2 M3 T3?

PROFESSOR: Yep.

Root 2 M3 T3.

But in this case, if we're defining,

let's say, our x-axis here, it also

matters what this angle is.

So you've got to multiply by cosine theta in this case.

And that's the x momentum of particle 3.

And we've also got to account for particle 4.

So we'll say add 2 cosine phi.

Now let's do the same thing for the y momentum.

What's the y momentum of particles 1 and 2?

0.

They're not moving in the y direction to start.

And how about particle 3?

I hear whispers, but nothing vocalized.

AUDIENCE: Sine?

PROFESSOR: Yep.

Same thing, but root 2 M3 T3 sine theta.

And M4-- I almost wrote the wrong sign

there-- has got to be a minus.

If the momentum of the initial particle system in the y

direction is 0, so must the final momentum

in the y direction.

So these two momenta have to be equal and opposite.

And that's times sine phi.

So now we actually have sets of equations that relate

all of our unknown quantities.

We have the mass conservation equation,

we have the Q equation, we have the x momentum,

and we have the y momentum.

And from this point on, it's a matter of algebra

to express some of these quantities

in terms of some of the others.

So let's get started with that.

Because angles are kind of messy,

and theta should uniquely define phi,

let's try and get things in terms of just one angle.

So I'm going to start by separating

the thetas and the phis on either side of the equals sign,

so that hopefully later on we can eliminate one

in a system of equations.

So all I'm going to do is I'm going to subtract or add

the theta terms to the other side of the equation.