Subtitles section Play video Print subtitles The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. 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