Fusion Reaction Energy

In the D-T reaction, two deuterium nuclei (2.0141 u each) and one tritium (3.0160 u) total reactants = 5.0302 u. Products: helium-4 (4.0026 u) + neutron (1.0087 u) = 5.0113 u. The difference 0.0189 u is about 0.37% of the reactant mass. In kilograms, 0.0189 u = 3.14×10⁻²⁹ kg. That's roughly the mass of 19 electrons. When converted using E = mc², this tiny mass yields 17.6 MeV, which is enormous per particle—enough to power a 100W light bulb for about 2.8×10⁻¹⁶ seconds per reaction, but when multiplied by Avogadro's number (6×10²³ reactions per gram), it gives ~80,000 kWh per gram of fuel.

Burning one hydrogen molecule (H₂ + ½O₂ → H₂O) releases about 1.23 eV per molecule. The D-T fusion reaction releases 17.6 MeV, which is 14 million times larger. The reason is that chemical reactions involve only the rearrangement of electron orbitals (∼eV per bond), while fusion involves the nuclear strong force and changes in nuclear binding energy (∼MeV per nucleon). A single fusion reaction releases enough energy to break roughly 15 million chemical bonds. This is why fusion fuel has an energy density millions of times higher than oil or coal.

Both the alpha and the neutron are products of the reaction; their kinetic energies sum to 17.6 MeV due to momentum conservation. The full energy is eventually deposited in the reactor: the alpha particle stays in the plasma via magnetic confinement and provides self-heating (important for ignition), while the neutron escapes the plasma and heats the blanket, driving the steam cycle. The total thermal energy available to the reactor is indeed 17.6 MeV per reaction (minus any losses from escaping neutrinos, but those are zero for D-T as it's a two-body final state). So, when calculating fusion power, we use the total Q-value of the reaction.

The mass defect is simply the difference between the total mass of the reactants and the total mass of the products, using precise atomic masses. For the triple-alpha process (3⁴He → ¹²C), the mass of three helium-4 nuclei is 3 × 4.002603 u = 12.007809 u, while a carbon-12 atom has mass 12.000000 u (by definition). The mass defect is 0.007809 u, which corresponds to about 7.27 MeV per reaction. The formula E = Δm c² applies to any nuclear reaction; one just sums masses appropriately, ensuring that the binding energy differences yield the Q-value.

D-D has two equally probable branches: D + D → T (1.01 MeV) + p (3.02 MeV) with Q = 4.03 MeV, and D + D → He-3 (0.82 MeV) + n (2.45 MeV) with Q = 3.27 MeV. The branching ratio is about 50-50. The total kinetic energy of each branch is fixed, but the energy appears in different partitions. D-T has only one reaction channel, yielding a monoenergetic neutron (14.1 MeV) and alpha (3.5 MeV). So D-T gives a predictable, fixed Q-value, which is why it's preferred for reactor design—no branching uncertainties.

A 1 GW(e) plant with 33% thermal efficiency needs ~3 GW(th) of fusion power. At 17.6 MeV per reaction, one reaction yields 2.82×10⁻¹² J. So the required reaction rate is 3×10⁹ W / 2.82×10⁻¹² J = 1.06×10²¹ reactions per second. Over a day (~86,400 s), that's 9.2×10²⁵ reactions. Using Avogadro's number, that's about 153 moles of fuel mixture (D+T), or ~0.38 kg of combined D+T per day (since the atomic mass of D is 2 and T is 3, average mixture mass ~2.5 g/mol, but actually per reaction one D and one T are consumed, so 153 moles corresponds to 153×2g + 153×3g = 765g total per day, roughly 0.77 kg). This illustrates the minuscule fuel mass compared to coal (about 10,000 tons/day for a coal plant).

The p-¹¹B reaction: p + ¹¹B → 3⁴He + 8.7 MeV. Unlike D-T, which has a single neutron and alpha, p-¹¹B yields three alphas, each carrying roughly 2.9 MeV (plus a small recoil). Because there are multiple charged particles, their kinetic energies are distributed and not monoenergetic, which complicates direct energy conversion. Furthermore, 8.7 MeV per reaction is about half of D-T's 17.6 MeV, meaning you need twice as many reactions for the same power. However, p-¹¹B is aneutronic (no significant neutron yield), which is attractive for reducing activation, but the lower energy per reaction and much higher required temperature (T > 50 keV) make it challenging.

The mass defect (Δm) is the mass equivalent of the binding energy released. When reactants fuse into more tightly bound products, the total rest mass decreases. This mass difference is converted into kinetic energy of the products via conservation of energy. The kinetic energy appears as relative motion of the helium and neutron (or alphas) after the reaction. So E = Δm c² gives the total available kinetic energy in the center-of-mass frame. The energy doesn't appear as 'heat' instantly; it's initially in the form of fast particles that subsequently thermalize via collisions, eventually becoming heat. The formula is the fundamental source: the difference in nuclear binding energy is what powers the reaction.

The Q-value is the total energy released in a specific nuclear reaction (e.g., 17.6 MeV for D-T). The 'mass excess' (or mass defect relative to ¹²C) of a nuclide is the difference between its actual mass and its mass number in atomic mass units, multiplied by 931.5 MeV. The Q-value for a reaction is calculated from the sum of mass excesses of reactants minus products (or vice versa). For example, the mass excesses: D = 13.136 MeV, T = 14.950 MeV, He-4 = 2.425 MeV, n = 8.071 MeV. Q = (13.136 + 14.950) - (2.425 + 8.071) = 17.59 MeV. So Q is a derived quantity from mass excesses, but it's directly the energy released per reaction.

For light elements, fusion is exothermic (releases energy). As you fuse up to iron, the binding energy per nucleon increases, releasing energy. The Q-value per reaction for carbon burning (e.g., ¹²C + ¹²C → ²⁰Ne + α) is about 4.6 MeV, for oxygen burning ~7.9 MeV, and for silicon burning to iron it's around 8-10 MeV per reaction. These are all lower than D-T's 17.6 MeV because the fusion of heavier nuclei produces less binding energy gain per nucleon. Stellar cores use these reactions at high densities and temperatures (billions K) because they have exhausted lighter fuels; they release enough energy to sustain the star, but the energy per reaction is lower, requiring higher reaction rates.

At 15 keV, the average thermal kinetic energy per particle is about 1.5 kT = 22.5 keV (since kT = 15 keV). The fusion reaction releases 17.6 MeV = 17,600 keV. The ratio is ~780. This means each fusion event deposits an energy equivalent to the kinetic energy of about 780 particles at that temperature. This huge energy release per reaction is why a small number of fusion events can sustain a plasma. However, only 3.5 MeV (20%) goes to the alpha particle, which heats the plasma; the neutron escapes. The alpha energy (3.5 MeV) is still ~155 times the average particle energy. This large ratio is what makes ignition possible: if the alphas deposit their energy efficiently, a single alpha can heat many particles, potentially leading to a self-sustaining burn.

Modern atomic mass measurements have fractional uncertainties of about 10⁻¹⁰ to 10⁻¹². For D-T, the mass defect is known to better than 0.1 eV out of 17.6 MeV (relative uncertainty 5×10⁻⁹). This translates to an energy uncertainty of less than 1 eV, which is negligible for engineering. Reactor design calculations require precision at the percent level; thus the Q-value is known with far greater accuracy than needed. The main uncertainty in energy output is not the Q-value but the reaction rates, neutron flux measurements, and thermal conversion efficiencies.