A helium nucleus sits trapped behind a wall of electric force it does not have the energy to climb — yet it gets out. Classically impossible; quantum-mechanically only rare. Gamow (1928) computed the leak: the barrier's thickness sets an exponential penalty, the Gamow factor e−2G, so a small change in the decay energy Q becomes a colossal change in lifetime — the Geiger–Nuttall line. Down the center: Q, Z, A go in, the tunnelling engine computes, the half-life comes out. Blue builds it; red tries to break it.
source G. Gamow, Zur Quantentheorie des Atomkernes, Z. Phys. 51, 204 (1928) — alpha decay by barrier tunnelling — DOI 10.1007/BF01343196 (paywalled; cited author/title/year, AMBER). Rendered, not quoted.
Inside the nucleus the alpha feels attraction; outside, only the Coulomb push V(r)=Zd·2·ke/r. With energy Q below the barrier top there is a classically forbidden gap from the nuclear edge R out to the turning point b=Zd·2·ke/Q.
The WKB action across that gap is the Gamow exponent:
2G = (2/ħc)·√(2μc²/Q)·Zd·2·ke·[arccos√x − √(x(1−x))], x=R/b
Transmission T = e−2G; rate λ = f₀·T with knock-frequency f₀≈10²¹ s⁻¹; half-life t½ = ln2/λ. Thin barrier (thick x, high Q) ⇒ big T. Thick barrier (low Q) ⇒ vanishing T.
Live for the current nucleus:
| 2G | — | barrier width (nat.) | — |
| T = e−2G | — | turning pt b | — |
The alpha does not go over the wall; it goes through it — the same quantum tunnelling through the Coulomb barrier, aimed outward from the nucleus.
Because T is exponential in 2G and 2G scales like 1/√Q, the Geiger–Nuttall line makes log t½ nearly linear in Zd/√Q — half-life is exponentially sensitive to Q. This sphere is one branch of the radioactive decay; each sphere is the next one's premise.
The blue team's live re-check: recompute the Gamow physics and confirm barrier-suppression (lower Q ⇒ smaller T ⇒ longer life) and Geiger–Nuttall linearity. If red flips the sign, this badge catches it.
An alpha emitter is three numbers: proton number Z, mass number A, and decay energy Q (MeV) — the mass lost, released as kinetic energy. The daughter charge Zd=Z−2 sets the barrier height; Q sets how far below the top the alpha starts.
Pick a real emitter or dial Q by hand. Constants: ke=1.43996 MeV·fm, ħc=197.327 MeV·fm, u=931.494 MeV, mα=3727.38 MeV, r₀=1.2 fm.
Nudge Q by a few percent and watch t½ swing across dozens of orders of magnitude — the barrier is that unforgiving.
Left plot: the Coulomb barrier V(r), the level Q, the shaded forbidden gap the alpha tunnels. Right plot: the Geiger–Nuttall line, log t½ vs 1/√Q, with this nucleus marked. Everything computed on the spot — no lookups.
What the machine proves, live: transmission T=e−2G, half-life t½=ln2/(f₀T), the alpha's kinetic energy Eα=Q·(A−4)/A and the daughter recoil Q·4/A (momentum split). Decay: (Z,A) → (Z−2, A−4) + α. Across Q∈[4,8] MeV, t½ spans — orders of magnitude — the Geiger–Nuttall sensitivity.
The blue team's witness (left) confirms the suppression & the line live; the red team (right) tries to invert it.
So the model nails the slope of Geiger–Nuttall (the tunnelling exponent) but not the absolute prefactor — real half-lives scatter around the line by factors of 10–100. It explains why lifetimes span 10²⁴, not the last digit of any one of them.
"The alpha borrows energy to jump the barrier." Cut. No energy is borrowed. The wavefunction is nonzero inside the classically forbidden region; T=e−2G is a standing fact of the Schrödinger equation, not a loan.
"A wider barrier means faster decay." Cut. Backwards. Wider/taller barrier ⇒ larger 2G ⇒ smaller T ⇒ longer life. That inversion is exactly window 6's tamper.
"Geiger–Nuttall is an empirical fit with no theory." Kept, corrected. It was empirical (1911) — Gamow (1928) derived it: the 1/√Q slope is the barrier integral itself.
The red team's move: drop the minus sign in the Gamow exponent — T=e+2G — so a thicker barrier tunnels more and low-Q nuclei decay fastest. The blue team's witness (window 7) is watching.
Flip the sign and the Geiger–Nuttall line tilts the wrong way: lower Q would give a shorter half-life. The witness recomputes, the barrier-suppression check fails, and it turns red. Nothing is faked; the attack is real and it is caught.