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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