Nuclear Fusion

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Two protons are both positive, so they push each other away, right? Yet inside the sun, protons join all the time, and that is why the sun shines. So how do they manage it? Today we learn nuclear fusion: what it is, why it releases energy, and why it needs enormous temperatures. It comes up often in JEE, so let's begin.

0:27Fusion: light nuclei join

So, what exactly is fusion? Two light nuclei join to form one heavier nucleus, as simple as that. Okay? But wait a moment, splitting a heavy nucleus in fission releases energy, and now joining light nuclei also releases energy? How can both be true? The secret is binding. The new nucleus is more tightly bound than the light nuclei we started with, and that extra binding energy comes out. Right? Look at the graph: the binding energy per nucleon, written E with small b n below it, against capital A, the mass number, which is also the small number at the top left of each symbol. From hydrogen two, the curve climbs steeply, with helium four standing out as a dot above it, up to the peak at iron fifty six, about eight point seven five mega electron volts, M e V, per nucleon, the unit on the vertical axis. Fusion moves light nuclei up this steep side, along the fusion arrow. Clear? And what about fission? Splitting a heavy nucleus like uranium two thirty eight, on the right at about seven point six, moves along the fission arrow, again towards the peak. So both release energy because the products lie nearer the peak. The note in the middle, about the flat part of the curve, belongs to another lesson. Got it?

1:45Why fusion needs an enormous temperature

Now, back to our puzzle. Both nuclei are positive, so the Coulomb repulsion, the electric push between them, fights them all the way in. So how can we force them together? We make them rush at each other fast enough to beat the repulsion. They must come within about one femtometre, written f m, which is ten to the power minus fifteen metre. With me? For that, they must be extremely hot: ten to the power seven to ten to the power nine kelvin. That is why we call it thermonuclear fusion. At such temperatures matter is a plasma, a hot gas of charged particles. And that answers our puzzle: the sun's core is hot enough for its protons to meet and join. Okay? Now, fission runs in a reactor without any such heat. So why does it not need these temperatures? Because fission is triggered by a neutron, and a neutron has no charge, so there is no repulsion to fight. Fusion brings two positive nuclei together, so it needs the heat.

2:50Fusion in the sun: the proton–proton cycle

Clear? Inside the sun, the overall fusion is the proton–proton cycle: four hydrogen one nuclei, that is, four protons, become, as the arrow shows, one helium four nucleus. Out come two positrons, e with a small plus sign, the positive antiparticles of electrons, two neutrinos, nu, the Greek letter, and two gamma ray photons, written with the Greek gamma. Okay? The energy released in all is twenty six point seven M e V, and this includes the annihilation of the two positrons. Our first example gets this number straight from the masses. Right?

3:27Fusion for a reactor on Earth

Now, for a reactor, two reactions are of interest. Hydrogen two is deuterium, and two deuterium nuclei give helium three, a neutron, small n, and three point two seven M e V of energy. Okay? Deuterium plus tritium, which is hydrogen three, gives helium four, a neutron, and seventeen point six M e V, much more energy. Right? The reactor must also hold its plasma long enough: the density n, times the confinement time tau, the Greek letter, must exceed a minimum value. This is the Lawson criterion. Careful, this n is the density, not the neutron from the lines above. Clear?

4:03Fusion against fission, per unit mass of fuel

Now here is a common trap. One fission gives about two hundred M e V, the wavy equals sign meaning about, while the whole proton–proton cycle gives only twenty six point seven. So fission is the better fuel, right? Be careful, that compares one reaction with another; for a fuel, we compare per nucleon, that is, per unit mass. Fission gives about zero point nine M e V per nucleon, while hydrogen burning gives twenty six point seven over four, about six point seven. Okay? So per unit mass of fuel, fusion gives several times more energy than fission. Clear?

4:40Example 1: energy released when 4 hydrogen atoms form helium

Now, find the energy released when four hydrogen atoms are converted into a helium atom. Here, m with a small H below it, the mass of a hydrogen atom, is one point zero zero seven eight two five u, where u is the atomic mass unit. And m with small H e below it, the mass of a helium atom, is four point zero zero two six zero three u. Pause the video and try it first. Done? Okay, let's check. Four hydrogen atoms have a mass of four times one point zero zero seven eight two five, which is four point zero three one three u, okay? The helium atom is lighter by delta m, the Greek capital delta, meaning change. Four point zero three one three minus four point zero zero two six zero three gives zero point zero two eight six nine seven u. So where has this mass gone? Yes, exactly, it has turned into energy, and one atomic mass unit is equivalent to nine hundred and thirty one point five M e V, right? So the energy E is zero point zero two eight six nine seven times nine hundred and thirty one point five, which gives about twenty six point seven M e V, exactly the energy of the proton–proton cycle in the sun. Nice, isn't it?

5:54Example 2: temperature for two protons to reach 2 fm

Now a harder one: estimate the temperature, T, at which two protons, each with kinetic energy three over two k T, can approach head-on to a separation of two femtometres. Small k is Boltzmann's constant, eight point six two times ten to the power minus five electron volt per kelvin, which is what K to the power minus one means. Pause the video and try it first. Done? Okay, let's check. The kinetic energy that fights the repulsion, is it one proton's three over two k T, or both? Yes, exactly, both of them, so the total is two times three over two k T, which is three k T, okay? Use only one proton's share, and you get twice the right temperature: that is the trap. Right? Head-on, the protons stop for an instant at closest approach, so all this energy becomes electric potential energy, e squared over four pi epsilon naught r. Small e is the proton's charge, epsilon naught, the Greek epsilon with a small zero, is the permittivity of free space, and r is the separation. With me? Our favourite constant, e squared over four pi epsilon naught, is one point four four M e V femtometre. Divide by two femtometres, and three k T comes out as zero point seven two M e V, right? So the temperature is zero point seven two times ten to the power six electron volts, over three times eight point six two times ten to the power minus five, which gives about two point eight times ten to the power nine kelvin. That is nearly three billion kelvin. It is only a rough estimate, but it shows why fusion needs such enormous temperatures, and why it is called thermonuclear. Clear?

7:36Quick revision: fusion temperature, 26.7 MeV, vs fission

So, quick revision. Fusion: light nuclei join into a more tightly bound nucleus. It needs ten to the power seven to ten to the power nine kelvin, to beat the Coulomb repulsion. In the sun, four hydrogen nuclei become helium four, with twenty six point seven M e V in all. And per unit mass of fuel, fusion gives several times more energy than fission. Now go and try the practice problems on this page. All the best!