Nuclear Fusion
Contents
Nuclear Fusion
Nuclear fusion is a nuclear reaction in which two light atomic nuclei combine to form a heavier nucleus, releasing an enormous amount of energy. Fusion powers the Sun and all stars and is the principle behind the most powerful man-made explosions—thermonuclear or hydrogen bombs.
Fusion is also being studied as a clean and nearly limitless energy source.
The Fusion Reaction
The most studied fusion reaction involves isotopes of hydrogen:
![\[{}^2_1\text{H} + {}^3_1\text{H} \rightarrow {}^4_2\text{He} + {}^1_0n + 17.6\ \text{MeV}\]](http://latex.artofproblemsolving.com/9/1/c/91c85cd2f4fabb04d4e8d27f778436e5a90fe8aa.png)
This means:
- Deuterium (
) + Tritium (
) → Helium-4 + neutron + energy.
) + Tritium (
) → Helium-4 + neutron + energy.Each reaction releases about 17.6 MeV of energy, mostly as kinetic energy of the products.
Why Does Fusion Release Energy?
Fusion releases energy because the total mass of the products is less than the mass of the reactants. The mass defect is converted into energy via:
![\[E = \Delta m \cdot c^2\]](http://latex.artofproblemsolving.com/d/a/f/daf0e36f477b3d1d7577011f83fffadc74224c64.png)
Let 
be the mass difference in atomic mass units (u). Since:
![\[1\ \text{u} = 931.5\ \text{MeV}/c^2,\]](http://latex.artofproblemsolving.com/f/2/2/f22e14e44cb2a82e73b5c9e0cddbcfc88fa5c190.png)
A mass loss of 0.0188 u, for example, gives:
![\[E = 0.0188 \cdot 931.5 \approx 17.5\ \text{MeV}.\]](http://latex.artofproblemsolving.com/5/0/d/50d3fc48e27a1d1293b965ac01a36e6d6873585e.png)
Fusion in Stars
In stars, fusion occurs under extreme temperature and pressure. The Sun primarily fuses hydrogen through the proton–proton chain reaction:
1. 
2. 
3. 
Net reaction:
![\[4p \rightarrow {}^4\text{He} + 2e^+ + 2\nu_e + \text{Energy}\]](http://latex.artofproblemsolving.com/7/f/6/7f61143b12af2c2ad7d3ce17b48b2a1f0701c10f.png)
Only a small fraction of the mass is converted, but over billions of years, this powers entire stars.
Conditions for Fusion
Fusion requires:
- High temperature (≥10 million K) → provides kinetic energy to overcome electrostatic repulsion.
- High pressure → increases collision rate.
- Confinement time → the plasma must be stable long enough for enough reactions.
These three requirements form the Lawson criterion, which determines whether a fusion plasma can become self-sustaining.
The Lawson Criterion
The condition for net fusion energy gain is:
![\[n \cdot T \cdot \tau_E > \text{Threshold},\]](http://latex.artofproblemsolving.com/7/6/c/76c8597f3a77fbcc020c04020cf8e3e199cf022d.png)
where:
= particle density,
= temperature,
= energy confinement time.
= particle density,
= temperature,
= energy confinement time.For deuterium–tritium fusion:
![\[nT\tau_E \gtrsim 10^{21}\ \text{keV·s/m}^3\]](http://latex.artofproblemsolving.com/9/3/b/93b40cac17cddb28cf332ccaa90177be9c7855de.png)
This tells us that even if temperature is high, if the plasma escapes quickly, fusion won't be efficient.
Confinement Methods
Magnetic Confinement (Tokamaks)
In devices like the tokamak, plasma is confined using magnetic fields shaped like a torus. Charged particles spiral along field lines, preventing contact with reactor walls.
Equations used:
- Lorentz force:
- Magnetic pressure balance:
Tokamaks like ITER aim to reach "ignition", where the plasma sustains itself.
Inertial Confinement (ICF)
Small pellets of fuel are compressed by lasers or ion beams. The inertia of the imploding fuel confines the reaction for a brief moment—just long enough for fusion to occur.
The energy gain 
is defined as:
![\[Q = \frac{\text{Energy output}}{\text{Energy input}}.\]](http://latex.artofproblemsolving.com/f/c/d/fcd151afb86e4a9c8f3729327459f8dbc19ab0ad.png)
Goal: 
for a break-even fusion device.
Fusion vs. Fission
| Aspect | Fusion | Fission |
|---|---|---|
| Fuel | Light elements (e.g. H, He) | Heavy elements (e.g. U-235, Pu-239) |
| Waste | Minimal | Long-lived radioactive byproducts |
| Energy per kg | Higher than fission | Lower than fusion |
| Conditions needed | Extremely high temperature and pressure | Neutron flux and critical mass |
| Natural occurrence | Stars (like the Sun) | None (rare natural decays only) |
| Example reaction |  |
|
| Applications | Stars, experimental reactors, hydrogen bombs | Nuclear power plants, atomic bombs |
Thermonuclear Weapons
Fusion reactions are used in hydrogen bombs, triggered by a fission explosion that compresses and heats fusion fuel. The sequence: 1. Fission core explodes, 2. X-rays compress fusion fuel, 3. Fusion releases vast energy.
These can reach megaton yields — 1000× more powerful than fission bombs.
Example Calculation
Suppose 1 gram of deuterium–tritium is fused completely.
- Number of reactions:
- Energy per reaction:
- Total energy:
That’s about 133 tons of TNT from 1 gram of fuel.
Challenges
- Sustaining stable plasma,
- Achieving
energy gain, - Withstanding extreme temperatures,
- Developing materials that can handle neutron bombardment.
Recent Advances
- ITER: International fusion project aiming to reach ignition.
- National Ignition Facility (NIF): Achieved net energy gain using ICF.
- Private startups: Working on compact fusion devices and alternative fuels (like p–B fusion).
