Alpha Decay Quantum Tunneling

Isotope

Nucleus Parameters

Mass number A 226

Atomic number Z 88

Decay energy Q (MeV) 4.87

Result

Computed half-life T₁/₂

—

Barrier height V(R)

— MeV

Turning point r_c

— fm

Gamow factor G

—

Transmission T

—

Assault freq. f

— /s

Escaped / attempts

0 / 0

How it works

An alpha particle sits inside the nucleus behind a Coulomb barrier built by the repulsion from the daughter's Z−2 protons. Classically it cannot escape — its energy Q is below the barrier peak — but quantum mechanically it tunnels through with a probability set by the WKB approximation (Gamow, 1928):

V(r) = 2(Z−2)e²/(4πε₀ r)      Coulomb barrier outside R = r₀A^(1/3)
r_c: V(r_c) = Q                 classically-forbidden zone is R → r_c
G = (2/ħ)·√(2m)·∫[R,r_c] √(V(r) − Q) dr     Gamow factor (WKB integral)
T = e^(−G)                      transmission probability per assault
f = v/(2R),  v = √(2Q/m)        assault frequency on the barrier wall
λ = f·T,   T₁/₂ = ln(2)/λ        decay constant → half-life

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 — the entire spread comes from Q sitting inside an exponential. Real values also depend on nuclear-structure corrections (spin, parity, preformation factor) this simplified model omits.

Escape animation: attempts are shown on a fixed demo clock (not the real assault frequency f, which is ~10²¹ Hz — far too fast to render). The escape/reflect outcome of each shown attempt is drawn from a probability rescaled from the real Gamow factor G, so a thinner barrier (lower G) still escapes more often here, just as it would in reality — only the timescale is compressed to be watchable.

Proton Neutron Alpha particle (α) Coulomb barrier V(r)

attempts: 0
escaped: 0
reflected: 0

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Alpha Decay Quantum Tunneling (Gamow Theory)