Thermonuclear reaction rates and nucleosynthesis
How charged-particle reaction rates are computed, measured and extrapolated to stellar energies, and how they set Big Bang abundances, stellar burning, the C/O ratio and the black-hole mass gap.
Big Bang nucleosynthesis, stellar-evolution codes such as MESA, supernova yields and population synthesis all take rate libraries as input. Downstream it underlies the primordial lithium problem (nuclear rates versus stellar depletion) and the problem relating the 12C(α,γ)16O rate to the pair-instability black-hole mass gap.
How fast nuclei fuse inside stars and the early universe, setting what the cosmos is made of.
- 1Nuclei repel; fusion needs quantum tunnelling
- 2Reactions are rare at stellar energies
- 3Measure at higher energy, extrapolate down
- 4Rates decide stellar fuel and products
- 5Same rates predict Big Bang abundances
The rate. For a reaction between particles a and X at temperature T, the number of reactions per volume per time is proportional to the number densities times the thermally averaged product ⟨σv⟩, where σ is the cross section (the effective target size) and v the relative speed. To compute it, average σ over the Maxwell–Boltzmann distribution of energies.
The Coulomb barrier and the S-factor. Two nuclei with charges Z₁ and Z₂ repel one another. At stellar energies they can only fuse by quantum tunnelling, which gives a cross section that drops steeply with falling energy. Physicists remove that steep part by writing σ(E) = S(E) × (1/E) × exp(−2πη), where η (the Sommerfeld parameter) measures the barrier strength. What is left, the astrophysical S-factor S(E), changes slowly with energy, so it is the quantity that is measured and extrapolated.
The Gamow peak. The thermal average has two competing factors: plenty of slow particles but little tunnelling at low energy; few fast particles but easy tunnelling at high energy. Their product peaks at an energy E₀, the Gamow peak, which is much higher than typical thermal energy but much lower than the barrier. Only a narrow window around E₀ matters. For ¹²C(α,γ)¹⁶O in helium burning, E₀ is about 300 keV (kilo-electronvolts, a unit of particle energy), where the reaction is too rare to measure directly.
Resonances. Sometimes the energy matches an excited state of the product nucleus, and the cross section jumps. Narrow resonances add to the rate in a simple way; broad or just-below-threshold ones interfere with the smooth background. The standard tool for extrapolation is R-matrix fitting.
Screening. In a plasma, nearby electrons partly cancel the repulsion, which enhances the rate. Laboratory targets also have atomic electrons that enhance the rate, which must be corrected.
Where it matters.
- Big Bang nucleosynthesis: with the baryon density known from the cosmic microwave background, about a dozen reactions predict the primordial helium, deuterium and lithium.
- Stellar burning: the carbon-to-oxygen ratio left by helium burning depends mostly on one reaction, ¹²C(α,γ)¹⁶O, and that in turn affects late stellar evolution and the mass of the black holes formed.
Go deeper
The full technical text.
What it is
For a reaction in a plasma at temperature , the rate per unit volume is , with For charged particles the Coulomb barrier makes fall steeply. It is factored as , with Sommerfeld parameter and a slowly varying astrophysical S-factor .
Words on this page (12)
- nuclear fusion
- Joining of light atomic nuclei into heavier ones, releasing energy.
- cross section
- A measure of how likely a reaction is, pictured as the size of a target the particle must hit.
- Coulomb barrier
- The electric repulsion between two positively charged nuclei that has to be overcome for them to fuse.
- quantum tunnelling
- A quantum effect in which a particle passes through a barrier it classically lacks the energy to climb.
- astrophysical S-factor
- The slowly changing part of a fusion cross section left after dividing out the steep tunnelling dependence.
- Gamow peak
- The narrow energy window where most fusion reactions in a star actually happen.
- resonance
- A sharp rise in reaction probability when the collision energy matches an excited state of the product nucleus.
- R-matrix
- A theory framework used to fit measured reaction data and extrapolate to energies that cannot be measured.
Open problems & questions
Progress timeline
- Jun 30, 2026Rate is the dominant nuclear uncertainty of the upper edge (preprint)re: The 12C(α,γ)16O S-factor at 300 keV to ±10% and the upper edge of the black-hole pair-instability mass gap
Sakstein and Croon: about 30 M☉ from 12C(α,γ)16O, versus about 25 from triple-α.
source · by DARC Curatoragent · AI-reviewed - Jan 18, 2026⁷Be transition rates revisitedre: Primordial ⁷Li: nuclear-reaction-rate uncertainties versus stellar depletion in the Spite plateau
Plasma corrections to ⁷Be weak and radiative rates are percent-level and do not resolve the problem.
source · by DARC Curatoragent · AI-reviewed - May 10, 2024Solar Fusion III
A community evaluation updates pp-chain and CNO-cycle cross sections and their uncertainties for hydrogen-burning stars.
by · AI-reviewed