Mass–Energy at Work: Binding Energy, Nuclear Power and Antimatter

Where the mass goes

Remember from Grade 11: in Work, Energy and Power you found that the products of deuterium–tritium fusion have about 3.1 × 10⁻²⁹ kg less mass than the reactants, released as about 2.8 × 10⁻¹² J, and in Thermal and Nuclear Energy you learned how fission, fusion and CANDU reactors work. In this lesson you will use the tools of nuclear and particle physicists (atomic mass units and MeV) to explain why some reactions release energy and others need it, and see the most complete conversion of mass to energy there is: matter meeting antimatter.

Mass defect and binding energy

A helium-4 nucleus has 2 protons and 2 neutrons. Surprisingly, it has less mass than 2 free protons plus 2 free neutrons. Putting the nucleons together released energy (the strong nuclear force pulled them in), and by mass–energy equivalence, the nucleus lost mass Δm = E/c². To pull the nucleus apart again, you would have to supply that energy back: this is the binding energy.

Nuclear masses are listed in atomic mass units: 1 u = 1.6605 × 10⁻²⁷ kg, and

1 u × c² = 931.5 MeV

So a mass defect in u, multiplied by 931.5, gives the energy in MeV. Tables usually list atomic masses (including electrons), so we use the mass of a hydrogen-1 atom (1.007825 u) for each proton; the electron masses then cancel. The neutron mass is 1.008665 u.

The curve that explains nuclear energy

Plot binding energy per nucleon against mass number and you get one of the most important graphs in physics. It rises steeply for light nuclei, peaks near iron-56 (about 8.8 MeV per nucleon) and falls slowly for heavy nuclei.

• Fusion: joining very light nuclei (hydrogen isotopes into helium) moves up the steep left side. The products are more tightly bound, have less mass, and energy is released. This powers the Sun.
• Fission: splitting a very heavy nucleus (uranium-235) into two medium nuclei moves up the gentle right side, from about 7.6 to about 8.5 MeV per nucleon. With 235 nucleons, that gain of roughly 0.9 MeV each releases about 200 MeV per fission.

Either way, energy is released when nuclei move toward iron. Going the other way (splitting iron, or fusing beyond iron) needs energy.

Nuclear power: E = mc² on the grid

A CANDU reactor, like those at Ontario's Bruce and Darlington stations, uses natural uranium fuel and heavy water to slow neutrons so they keep the chain reaction going. Each fission's 200 MeV becomes thermal energy, which boils water to drive turbines. Ontario gets roughly half of its electricity from nuclear stations.

How much mass actually 'disappears'? A reactor producing 900 MW of electrical power at about 33% efficiency produces thermal energy at about 2.7 × 10⁹ W. In a year (3.16 × 10⁷ s) that is 8.5 × 10¹⁶ J, which corresponds to Δm = E/c² ≈ 1 kg of mass converted into energy per year, from many tonnes of fuel. The social and environmental side of nuclear power is discussed in the next lesson, Relativity in Technology and Society.

Antimatter: total conversion

Every type of particle has an antiparticle with the same mass and opposite charge. The electron's antiparticle is the positron (e⁺). When an electron and a positron meet, they annihilate: both disappear and their entire rest energy becomes photons:

e⁺ + e⁻ → γ + γ, each photon 0.511 MeV (if the particles were nearly at rest).

Two photons, not one: the pair had (almost) zero total momentum, so the photons must fly off in opposite directions with equal momenta.

The reverse also happens: a photon with at least 2 × 0.511 = 1.022 MeV passing near a nucleus can create an electron–positron pair (pair production). Energy becomes matter. In particle accelerators such as CERN's Large Hadron Collider, collision energy creates new, heavier particles, such as the Higgs boson (about 125 GeV/c²) discovered in 2012. Mass–energy is conserved in all of these processes; mass alone is not.

PET scanners

Positron emission tomography (PET) is E = mc² in a hospital. A patient receives a small dose of a tracer, such as a sugar-like molecule tagged with fluorine-18, which emits positrons. Active tissue (for example, a growing tumour) takes up more of the sugar. Each positron travels only a millimetre or so before annihilating. A ring of detectors catches pairs of 0.511 MeV photons arriving at the same time, and a computer reconstructs where the annihilations happened. The tracer isotopes are made in cyclotrons, including hospital cyclotrons and TRIUMF in Vancouver.