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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