8.4 Nuclear energy and E = mc² — Learn

Prerequisites: work, energy and the conservation of energy from Section 3.3; the nucleus, nucleon number and isotopes from Section 8.3; the electronvolt and the atomic mass unit from Section 1.1; photon energy from Section 8.1; powers and scientific notation.

By the end of this section you can: state the mass-energy relation $E = mc^2$ ; define mass defect and binding energy; use the atomic mass unit and its energy equivalent; calculate the energy released in a nuclear reaction or an annihilation; and explain why fission and fusion release energy.

1. Mass and energy

In 1905 Einstein showed that mass and energy are two forms of the same thing:

E = mc^2

The relation says that a mass $m$ has an energy equivalent $mc^2$ , and that an energy $E$ has a mass equivalent $\dfrac{E}{c^2}$ . The total mass-energy of an isolated system is conserved.

The energies in nuclear physics are large. One atomic mass unit, $1\,\text{u} = 1.66 \times 10^{-27}\,\text{kg}$ , is equivalent to

1\,\text{u} = 931.5\,\text{MeV}

This number is on the data sheet. It lets you work in atomic mass units and megaelectronvolts, which keeps the numbers manageable.

2. Mass defect and binding energy

If you could pull a nucleus apart into its separate protons and neutrons, you would find a surprise. The mass of the separate nucleons is larger than the mass of the nucleus they formed. The difference is the mass defect, $\Delta m$ .

The energy released when the nucleus formed is the binding energy. It is the same as the energy needed to pull the nucleus apart again:

E = \Delta m\,c^2

A larger binding energy means a more tightly bound nucleus. To compare different nuclei, divide by the number of nucleons to get the binding energy per nucleon.

Worked example 1. A helium-4 nucleus, $^{4}_{2}\text{He}$ , has a mass of 4.0015 u. A proton has a mass of 1.00728 u and a neutron a mass of 1.00867 u. Find the mass defect and the binding energy of the helium-4 nucleus.

Find the total mass of the separate nucleons. The nucleus has 2 protons and 2 neutrons.

\begin{aligned} m_{\text{separate}} &= 2(1.00728) + 2(1.00867) \\ &= 2.01456 + 2.01734 = 4.03190\,\text{u} \end{aligned}

\Delta m = m_{\text{separate}} - m_{\text{nucleus}} = 4.03190 - 4.00150 = 0.03040\,\text{u}

Now use the energy equivalent of the atomic mass unit.

E = 0.03040 \times 931.5 = 28.3\,\text{MeV}

The helium-4 nucleus has a mass defect of 0.03040 u and a binding energy of 28.3 MeV. Its binding energy per nucleon is $\dfrac{28.3}{4} = 7.08\,\text{MeV}$ per nucleon.

Worked example 2. A nuclear reaction releases 200 MeV. Find the mass defect in kilograms.

Rearrange $E = \Delta m\,c^2$ and change the energy to joules.

\begin{aligned} \Delta m &= \frac{E}{c^2} = \frac{200 \times 10^{6} \times 1.60 \times 10^{-19}}{(3.00 \times 10^{8})^2} \\ &= \frac{3.20 \times 10^{-11}}{9.00 \times 10^{16}} = 3.56 \times 10^{-28}\,\text{kg} \end{aligned}

The mass defect is $3.56 \times 10^{-28}\,\text{kg}$ . In atomic mass units it is $\dfrac{200}{931.5} = 0.215\,\text{u}$ .

3. The binding energy curve

The binding energy per nucleon is different for every nuclide. A graph of it against nucleon number has a clear shape.

It rises steeply for the lightest nuclei, such as hydrogen and helium.
It reaches a broad maximum near iron-56, at about 8.8 MeV per nucleon.
It falls slowly for the heaviest nuclei, such as uranium.

The maximum is the most stable region. A nucleus near the peak needs the most energy per nucleon to pull it apart, so it is the hardest to break.

This curve explains where nuclear energy comes from. Both fission and fusion move a nucleus towards the peak, and a more tightly bound nucleus has less mass. The mass that disappears is released as energy.

Fission splits a very heavy nucleus into two lighter ones. The products have a higher binding energy per nucleon, so energy is released.
Fusion joins two light nuclei into a heavier one. The product has a higher binding energy per nucleon, so energy is released.

Fusion powers the Sun. Fission powers a nuclear reactor. Both are applications of $E = mc^2$ .

4. Annihilation

When a particle meets its antiparticle, the two can annihilate. Their mass disappears and becomes photon energy. The electron's antiparticle is the positron, which has the same mass as an electron and a positive charge.

For a positron and an electron, the total rest energy is $2 \times 0.511 = 1.022\,\text{MeV}$ . Two photons are produced, each of energy 0.511 MeV. Two photons are needed because momentum must be conserved: the collision has almost no momentum, so the two photons must travel in opposite directions and share the energy equally.

Worked example 3. In the hospital tracer, a positron annihilates with an electron. Each particle has a rest energy of 0.511 MeV. Find the total energy released, in joules, and the energy of each photon.

\begin{aligned} E_{\text{total}} &= 2 \times 0.511 = 1.022\,\text{MeV} \\ &= 1.022 \times 10^{6} \times 1.60 \times 10^{-19} \\ &= 1.64 \times 10^{-13}\,\text{J} \end{aligned}

Two photons are produced, so each carries half of the energy: $0.511\,\text{MeV} = 8.18 \times 10^{-14}\,\text{J}$ . The gamma-ray photons from the tracer have the energy found in Section 8.1, because they come from this annihilation.

Watch out. The mass defect is not a mass that has been lost. Mass and energy are two forms of the same quantity, and the total is conserved. The mass that seems to disappear has reappeared as kinetic energy of the products or as photon energy. A second trap is to mix the two routes: use the energy equivalent of 1 u (931.5 MeV) when the mass is in atomic mass units, and use $E = mc^2$ when the mass is in kilograms. Mixing them gives an answer that is wrong by a large factor.

5. Try it: the binding energy curve

The figure draws the binding energy per nucleon against the nucleon number. A control moves a marker along the curve and shows the value for that mass number.

Predict first. Do not move the control yet. Write down your answers.

Is the binding energy per nucleon larger for helium-4 or for iron-56?
A uranium nucleus is near the right-hand end of the curve. Is its binding energy per nucleon higher or lower than iron-56?
Why does moving towards the peak of the curve release energy?

Test. Move the marker to a light nucleus, then to iron-56, then to a heavy nucleus. Read the value at each position. Watch how the curve rises to the peak and then falls.

Explain. Why do both fission and fusion release energy? Write two or three sentences. Use the idea that a higher binding energy per nucleon means a more tightly bound nucleus with less mass.

Progress check

Return to the hospital tracer. The positron from a fluorine-18 decay annihilates with an electron.

Find the total energy released in the annihilation, in joules.
Find the energy of each of the two gamma photons, in joules.
Explain why two photons are produced rather than one.
The team checks the energy of the gamma photons. Each has an energy of 511 keV. Show that this is consistent with the rest energy of an electron.

If question 1 goes wrong, use $E = mc^2$ with the combined rest mass, or add the two rest energies. If question 2 goes wrong, the energy is shared equally. If question 3 goes wrong, think about momentum. If question 4 goes wrong, remember that $1\,\text{MeV} = 10^{3}\,\text{keV}$ .

You should now be able to

state the mass-energy relation $E = mc^2$ ;
use the energy equivalent of the atomic mass unit, $1\,\text{u} = 931.5\,\text{MeV}$ ;
define mass defect and binding energy;
calculate a mass defect and a binding energy from nuclear masses;
find the binding energy per nucleon;
describe the shape of the binding energy curve;
explain why fission and fusion release energy;
describe annihilation, and calculate the energy released when a positron meets an electron;
explain why two photons are produced in the annihilation.