D-T Fusion Reactivity & the Gamow Peak

Fuel & Plasma Conditions

Fuel mixture

Ion temperature T 50 keV

Ion density n 5.0 ×10²⁰ m⁻³

Energy confinement τ_E 1.0 s

Reactivity & Power

Reactivity ⟨σv⟩

–

Power density

–

Gamow peak energy E₀ (most probable fusion energy)

–

Ignition Progress (Lawson n·T·τ_E)

0% of threshold–

Maxwell–Boltzmann × Tunneling = Gamow Peak

Maxwell–Boltzmann f(E) Tunneling P(E) Reaction rate (product)

How it works

Two positively-charged nuclei must overcome their mutual Coulomb repulsion to fuse. In a hot plasma, ion kinetic energies follow the Maxwell–Boltzmann distribution:

f(E) = 2/√π · √E / T^1.5 · e^(−E/T)

Very few ions have enough classical energy to cross the barrier — instead they quantum-tunnel through it, with a probability that rises steeply with energy (the Gamow factor):

P(E) ∝ e^(−B_G / √E),   B_G = π·α·Z₁Z₂·√(2μc²)

Multiplying the two curves gives the Gamow peak: a narrow energy window, far above the thermal average, where almost all real fusion reactions occur. Averaging the fusion cross-section σ(E) over f(E) gives the reactivity ⟨σv⟩ used below, and fusion power density scales as n² ⟨σv⟩:

σ(E) ≈ (S₀/E)·e^(−B_G/√E)
P_fus = ¼ n² ⟨σv⟩ Q

The Lawson criterion compares n·T·τ_E (density × temperature × energy confinement time) against a threshold (~3×10²¹ keV·s·m⁻³ for D-T) needed for the fusion power to sustain the plasma's own heat losses — the bar above tracks progress toward that ignition condition as you move the sliders.

Temperature — shifts and widens f(E), and shifts the Gamow peak to higher energy.
Density & confinement time — set the Lawson triple product and the fusion power density.
Fuel mixture — D-He3 has a higher Coulomb barrier (Z₁Z₂=2) so it needs a much higher temperature to reach the same reactivity as D-T; D-D has the lowest cross-section of the three.

Species A ion Species B ion Fusion flash

Fusion events: 0

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