The Gamow Peak and Quantum Tunneling
The narrow energy window where the odds of tunneling the Coulomb barrier and finding a fast particle combine.
Two competing factors
Fusion requires two positively charged nuclei to approach closely enough for the strong force to act, against their mutual Coulomb repulsion. Classically almost none succeed at fusion temperatures; quantum tunneling lets them penetrate the barrier. The tunneling probability rises steeply with energy (the Gamow factor), while the number of particles at a given energy falls exponentially (the Maxwell-Boltzmann tail).
rate(E) ~ exp(-E / k_B T) * exp(-b / sqrt(E))The first factor is the falling population; the second is the Gamow tunneling factor with b a constant set by the nuclear charges. Their product peaks at an intermediate energy, the Gamow peak, well above the thermal energy but where enough particles still exist.
Why the peak is narrow
The product of a rising and a falling exponential is sharply peaked. Most fusion reactions occur within a narrow band around the Gamow energy, far out on the Maxwellian tail. This is why fusion rate is so sensitive to temperature: raising T shifts and broadens this window dramatically.
Consequences
- Fusion rate depends on the fast tail, not the bulk, of the distribution
- The steep temperature dependence of reactivity comes directly from the Gamow peak
- Higher-charge fuels have a larger b, so they need higher temperatures
Fuel comparison
D-T has the lowest Coulomb barrier among practical fuels, so its Gamow peak sits at accessible temperatures, making it the easiest to ignite. D-3He, with a higher charge product, has its Gamow peak at much higher energy, requiring far higher temperature. This physics underlies the choice of D-T for the Hyperion breeder and the higher-temperature operating regime of the D-3He Kronos burner, both design studies.