This is a 2D radial-profile counterpart to the 3D orbit-view Gamow simulator: instead of flying a camera around a nucleus, it plots the actual WKB tunneling mathematics as a function of radius r. The Coulomb barrier V(r) = 2(Z−2)e²/(4πε₀r) is drawn against the fixed decay energy Q, and the classically-forbidden zone R → r_c is shaded. Inside that zone the chart plots the cumulative WKB exponent — how much the wavefunction's log-amplitude has dropped so far at each radius:
κ(r) = √(2m(V(r) − Q))/ħ local decay rate inside the barrier
G(r) = 2·∫[R,r] κ(r′) dr′ cumulative exponent (this sim's own trapezoid integral)
G_total = (2/ħ)√(2mQ)·r_c·[arccos(√x) − √(x(1−x))], x = R/r_c (exact closed form, Gamow 1928)
T = e^(−G_total), f = v/(2R), v = √(2Q/m)
λ = f·T, T₁/₂ = ln(2)/λ
Two independent calculations of the same exponent are cross-checked live: the pointwise trapezoid integral that draws the curve must land within ~0.01% of the exact closed-form G_total used for the headline stats — if they ever disagreed it would mean a bug in the numerical integrator.
The travelling marker moves by real dynamics, not a fixed tween: outside the barrier its radius is integrated frame-by-frame from dr/dt = √(2(Q−V(r))/m) — genuine energy-conserving classical motion — while inside the forbidden zone its crawl speed is throttled by the local κ(r) (it visibly slows exactly where tunneling is least likely). The escape/reflect outcome at r_c is drawn from the real transmission probability T. As in the 3D version, the animation clock is compressed for watchability (the true assault frequency f is of order 10²¹ Hz) — only the relative pacing and the escape odds are physically faithful, not the wall-clock timing.
This two-parameter model (Z, Q — with R fixed by A) reproduces the real half-life range correctly to within a few orders of magnitude across 30+ orders of magnitude of half-lives, from Po-212's 0.3 μs to U-238's 4.5 billion years. Real values also depend on nuclear-structure corrections (spin, parity, preformation factor) this simplified model omits.