Bringing The Power Source of the Stars Down to Earth

How Fusion Reactions Work


Fusion of light (low-mass) elements releases energy, as does fission of heavy (high-mass) elements.

Binding Energy per Nucleon as a Function of Nuclear Mass
Full binding energy


The relation E = mc2 states the equivalence of mass and energy. In a fusion reaction, some reactant mass energy is converted to kinetic energy of the products. Binding energy is the energy equivalent of the mass difference between a whole nucleus and its individual constituent protons and neutrons. For energy release in fusion or fission, the products need to have a higher binding energy per nucleon (proton or neutron) than the reactants. As the graph above shows, fusion only releases energy for light elements and fission only releases energy for heavy elements.

The actual fusion reaction occurs when two nuclei approach within about 1.0E-15 m, so that the attraction, via the residual strong interaction between the nuclei, overcomes the electrical repulsion between the protons. Such close encounters only occur when nuclei collide with sufficient kinetic energy. Only at high temperatures do enough energetic particles exist for there to be many fusion reactions.

Binding Energies (Low-Mass Elements Only)
Light element binding 
energy figure not loaded.

Reaction Energy Ef = k*(mi-mf)*c2

This equation follows from Einstein's E = m*c2. The change in energy Ef of the system is proportional to the mass difference (mi-mf) between the reactants and the products. In the equation above,
  • Ef = Energy per reaction
  • mi = total initial (reactant) mass
  • mf = total final (product) mass
  • The conversion factor k equals 1 in SI units, or 931.466 MeV/c2 in "natural units" where E is in MeV and m is in atomic mass units, u.

Useful Nuclear Masses

(The electron's mass is 0.000549 u.)

Species Symbols Mass (u)*
n Neutron 1.008665
p (H-1) Proton 1.007276
D (H-2) Deuteron 2.013553
T (H-3) Triton 3.015500
He-3 Helium-3 3.014932
He-4 (alpha) Helium-4 4.001506

* Note: 1 u = 1 atomic mass unit = 1.66054 x 10-27 kg = 931.466 MeV/c2

Fusion Rate Coefficients
Rate Coefficients figure not 

Plasma Fusion Reaction Rate = R * n1 * n2

n1,n2 = Densities of reacting species (particles/m3); R = Rate Coefficient (m3/s).
Multiply by Ef to get fusion power density.


To calculate the rate of reactions per unit volume, multiply the rate coefficient, R, by the particle densities of the two reacting species (divide by two if there is only one species, in order to avoid double-counting the reaction possibilities). The p + p => D reaction rate coefficient in the sun is much lower than that achievable with a deuterium-tritium fuel mix, because the p + p reaction proceeds by the weak nuclear interaction. Despite the sun's high density, the low rate coefficient means a proton in the sun will exist for an average of billions of years before it fuses. By comparison, a deuteron in a magnetic fusion power plant would only exist for about 100 seconds, and a deuteron in an imploding, fully-burned inertial confinement pellet only for 1.0E-9 seconds.

More Information:

* The Particle Adventure: All about the constituents of the nucleus and the fundamental particles they're made of - but don't forget to come back!

* Continue the Tour

Page originally created by students Jason Edson and Hannah Cohen.

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