Nuclear energy: fission and fusion

Where nuclear energy comes from

Narrated video. Transcript:

Look again at the binding energy curve. A long flat middle, near 8 MeV per nucleon, from A equals 30 to 170. Lower on both sides: for light nuclei below 30, and heavy nuclei above 170.

The greater the binding energy, the smaller the total mass of the nucleus. So if nuclei change into more tightly bound nuclei, mass disappears and energy is released.

There are two ways to climb up the curve. Split a heavy nucleus into two middle-sized ones: that is fission. Or join two light nuclei into a heavier one: that is fusion.

Let us estimate the energy from fission. Take a nucleus with A equals 240 and split it into two nuclei of A equals 120.

For A equals 240, E b n is about 7.6 MeV. For A equals 120, it is about 8.5 MeV. Each nucleon gains about 0.9 MeV of binding energy.

There are 240 nucleons, so the total gain is 240 × 0.9, which is 216 MeV. Fission of one heavy nucleus releases roughly 200 MeV.

Compare a chemical reaction. When coal burns, each reaction releases a few electron volts. A nuclear reaction releases millions of electron volts.

So, kilogram for kilogram, nuclear fuel gives about a million times more energy. Burning 1 kilogram of coal gives about 10⁷ joule. Fission of 1 kilogram of uranium gives about 10¹⁴ joule.

In fact, a chemical reaction also loses a little mass, but a million times less than a nuclear reaction. That is why we notice mass–energy conversion only in nuclear physics.

Fission

Narrated video. Transcript:

Fire a slow neutron at a uranium-235 nucleus. It is absorbed, making uranium-236, which wobbles, stretches like a drop of liquid, and splits into two medium-sized fragments.

One possible result: n plus ²³⁵₉₂U gives ²³⁶₉₂U, which breaks into ¹⁴⁴₅₆Ba plus ⁸⁹₃₆Kr plus 3 neutrons.

Check the balance. Nucleons: 1 plus 235 is 236, and 144 plus 89 plus 3 is also 236. Protons: 92 on the left, 56 plus 36, also 92, on the right.

A nuclear equation is not balanced atom by atom like a chemical one: barium and krypton appear from uranium. But the numbers of protons and of neutrons are each conserved.

The same nucleus can split in other ways too: into antimony-133, niobium-99 and 4 neutrons, or into xenon-140, strontium-94 and 2 neutrons.

The fragments have too many neutrons, so they are radioactive. They give out beta particles one after another until they reach stable nuclei.

The energy, about 200 MeV per fission, appears first as kinetic energy of the fragments and neutrons. As they slow down in the surrounding material, it becomes heat.

Each fission also gives out 2 or 3 neutrons, which can split more uranium nuclei. In a nuclear reactor, this fission is kept under control, and the heat makes electricity. In an atom bomb, the fission is uncontrolled.

How much energy is in 1 kilogram of plutonium-239, at 180 MeV per fission? 1000 grams over 239, times 6.023 × 10²³, gives 2.52 × 10²⁴ nuclei.

Times 180 MeV: 4.5 × 10²⁶ MeV. Multiply by 1.6 × 10⁻¹³: about 7.3 × 10¹³ joule, from just one kilogram.

Fusion: the energy of the stars

Narrated video. Transcript:

At the left end of the curve, light nuclei can join into a more tightly bound nucleus. Two protons can make a deuteron, a positron and a neutrino, releasing 0.42 MeV.

Two deuterons can make helium-3 and a neutron, releasing 3.27 MeV. Or they can make a triton and a proton, releasing 4.03 MeV.

But there is a problem. Both nuclei are positive, so they repel. To fuse, they must come close enough for the short-range nuclear force to grab them. They must climb over a Coulomb barrier.

For two protons, the barrier is about 400 keV. If the average kinetic energy, three-halves k T, equals 400 keV, the temperature T is about 3 × 10⁹ kelvin.

Fusion achieved by heating, so that particles move fast enough to beat the repulsion, is called thermonuclear fusion. It powers every star.

The Sun’s core is about 1.5 × 10⁷ kelvin, much cooler than that. So fusion in the Sun happens through the few protons whose energies are far above the average.

In the Sun, hydrogen burns to helium in steps called the proton–proton cycle. Overall, four hydrogen atoms become one helium-4 atom, releasing 26.7 MeV.

When the core’s hydrogen runs low, the core shrinks and heats up. At about 10⁸ kelvin, helium fuses into carbon. Stars build heavier elements this way, but only up to near the peak of the curve.

The Sun is about 5 × 10⁹ years old, and has hydrogen for about 5 billion years more. Then it will swell into a red giant.

On Earth, fusion reactors try to heat fuel to about 10⁸ kelvin. The fuel becomes a plasma of ions and electrons that no container can hold. Several countries, including India, are working on how to confine it.

Try it: Q value of a reaction

Interactive simulation. Things to try:

• Pick a reaction. The arrows show the nuclei moving on the binding energy curve: up the curve means energy out.
• Q = (mass before − mass after) × 931.5 MeV. Positive Q: energy is released. Negative Q: energy must be supplied.
• The ¹⁴⁴Ba + ⁸⁹Kr split gives about 173 MeV at once; the radioactive fragments release more later, making about 200 MeV in all.
• Compare the energy per kg of fuel for fission and for fusion with 10⁷ J for coal.
• Try ⁵⁶Fe → 2 ²⁸Al: iron is at the top of the curve, so splitting it needs energy.

Try it: the Coulomb barrier

Interactive simulation. Things to try:

• The orange curve is the Coulomb energy kZ₁Z₂e²/r. It peaks where the two nuclei touch, at r = 2R.
• The green arrow is the average kinetic energy (3/2)kT. Where it meets the curve, an average nucleus turns back.
• Start at the Sun’s core: the average energy is far below the barrier. Raise T until the average nuclei get over it.
• Try ⁴He + ⁴He: double the charge on each nucleus makes the barrier four times higher.

A kilogram of plutonium

Fission of one ²³⁹Pu nucleus releases about 180 MeV. How much energy is released if all the nuclei in 1 kg of ²³⁹Pu undergo fission? (N_A = 6.023 × 10²³ per mole)

1. 4.5 × 10²³ MeV
2. 4.5 × 10²⁶ MeV
3. 1.1 × 10²⁹ MeV
4. 7.3 × 10¹³ MeV

Can iron split?

Could a ⁵⁶Fe nucleus split into two ²⁸Al nuclei and release energy? Given m(⁵⁶Fe) = 55.93494 u, m(²⁸Al) = 27.98191 u.

1. Yes, Q = +26.9 MeV
2. No, Q = −26.9 MeV
3. No, Q = −0.029 MeV
4. Yes, because every heavy nucleus releases energy in fission

A lamp on deuterium

For how long could a 100 W lamp glow on the energy from fusing 2.0 kg of deuterium, by ²H + ²H → ³He + n + 3.27 MeV? (1 MeV = 1.6 × 10⁻¹³ J, 1 year = 3.154 × 10⁷ s)

1. About 1.6 × 10¹² years
2. About 1 × 10⁵ years
3. About 5 × 10⁴ years
4. About 25 years

Summary

• Q value: Q = (Σm_initial − Σm_final) c²
• Fission of ²³⁵U: n + ²³⁵U → ²³⁶U → ¹⁴⁴Ba + ⁸⁹Kr + 3n, ≈ 200 MeV
• Sun (p-p cycle, net): 4 ¹H → ⁴He + 26.7 MeV
• Coulomb barrier temperature: (3/2)kT ≈ 400 keV ⇒ T ≈ 3 × 10⁹ K

• Energy is released when nuclei become more tightly bound: a heavy nucleus splitting (fission) or light nuclei joining (fusion).
• Nuclear reactions release MeV per reaction against eV in chemistry: about a million times more energy per kg (10¹⁴ J for uranium, 10⁷ J for coal).
• Fission gives two radioactive fragments and 2–3 neutrons. Controlled in a reactor it makes electricity; uncontrolled, it is an atom bomb.
• Fusion needs very high temperatures to beat the Coulomb barrier (thermonuclear fusion). It powers the Sun and stars; fusion reactors must confine a plasma at about 10⁸ K.