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Alpha decay

Gamow's picture: an alpha particle rattles inside the nucleus and leaks out through the Coulomb barrier. The leak is exponentially small, so the half-life follows Q_α over thirty decades.

Parent nucleus

What if Q_α were

Barrier and half-life

Rendering

What is drawn: the landscape is the potential of the radial equation the drawn wave solves, −V₀′ inside R and the Coulomb potential outside, each plus the centrifugal term ħ²l(l+1)/2μr² (zero for l = 0), as a solid, height in MeV and radius in fm each scaled to fit (scales differ per nucleus), warm where it exceeds Q (classically forbidden), cool where it is below Q, with 5 MeV contours and 10 fm rings. It is cut open towards you, the cut turning with the camera, so the well and the inside of the barrier stay visible; the cut faces are the same profile, hatched above Q, and the amber line across them at height Q, from R to b, is the tunnelling path (for l > 0 it also crosses the centrifugal core near the centre, classically forbidden as well, where the regular wave falls off as r^l+1; that stretch is not part of the escape path). The glass sheet is the level Q. The wave rides on Q with height ∝ log₁₀|u| over 40 decades (linear inside R); its ribbon runs just in front of the two cut faces. The well floor is drawn at the depth V₀′ the drawn wave solves (see Model), not at the 35 MeV of the half-life, and for l > 0 it carries the centrifugal term inside R as well, so the drawn wave solves the drawn profile for every l; that term diverges at r = 0, so the profile is cut flat at 1.2 times its height at R, above which nothing is drawn. The alpha particle (2 p + 2 n) is not to size, and its bounce is not to time. Materials, lights, glow and the stand are decoration.

Model, exactly

Q_α from AME2020 (Q(α) of the rct1 table, with its uncertainty, which carries the correlation between parent and daughter masses; Z ≥ 50). Below Z = 50, Q = B(Z−2, N−2) + B(⁴He) − B(Z, N) from the NUBASE2020 binding energies, whose 4-decimal B/A leave each B uncertain by up to A × 5×10⁻⁵ MeV (that bound is the error bar there). The Q error and its effect on log T are shown. Electron screening: Q is raised by (65.3 Z^7/5 − 80.0 Z^2/5) eV of the parent. The alpha and the daughter are a two-body system in the centre-of-mass frame with their reduced mass, so recoil is exact and needs no extra correction. Sharp surface at R = r₀(A_d^1/3 + 4^1/3), r₀ = 1.20 fm. Inside, a flat well, alpha kinetic energy Q + V₀, V₀ = 35 MeV (25 or 50 MeV moves the even-even rms by < 0.001). Outside, V = 2Z_de²/r + ħ²l(l+1)/2μr². The drawn wave: inside R the regular solution u_l(K′r) = K′r j_l(K′r) of the flat well with its centrifugal term, −V₀′ + ħ²l(l+1)/2μr² (the profile drawn inside R), joined in value and slope to the exact outgoing Coulomb wave at R. A 35 MeV well generally has no level exactly at Q, while the formula below assumes one (for l = 0, sin²KR = K²/(K² + κ_R²)); the well that has it, depth V₀′ (39.6 MeV for ²¹²Po, 41.5 MeV for ²³⁸U), is the one drawn and solved. The half-life keeps V₀ = 35 MeV, where it enters only through K. λ = S ν P, ν = ħK/(2μR) = v_in/2R, P = [4Kκ_R/(K² + κ_R²)] exp(−2∫_R^bκ dr), Langer (l + ½)². This is the WKB limit of the decay width of the quasi-stationary state with the outgoing Coulomb wave G + iF; the exact form, with 1/(F_l² + G_l²) and G′/G at kR, is computed too and differs by at most 0.008 in log T over all 624 decays. S = 0.041 (log S = −1.391) is the one fitted constant, the mean of log(T_calc/T_exp) over the even-even set (187 decays, ²⁶⁴Hs left out, see below); rms 0.371. It is not a measured preformation probability: r₀ and S are nearly degenerate (any r₀ from 1.0 to 1.3 fm fits with rms 0.34 to 0.43 once S is refitted), and S absorbs the nuclear attraction that a sharp surface leaves out of the barrier. T_exp = T_½/BR with BR the total alpha branch of NUBASE2020, which includes decays to excited daughter states, while the calculation is ground state to ground state. The ground-state branch alone is smaller (²³⁸U: 79%), so the true partial g.s. half-life is longer than T_exp: by about 0.1 decade for an even-even parent like ²³⁸U, and by more for odd ones, whose g.s. transition is often hindered. l: 0 for even-even (0⁺ → 0⁺); otherwise the lowest value allowed by the ground-state J^π of parent and daughter (angular momentum and parity), 0 when either is unknown (flagged tentative). Where no l satisfies both (²¹⁰Bi 1⁻ → 0⁻, ²³⁸Am 1⁺ → (0⁺), ²⁵⁸Db 0⁻ → 4⁺), there is no g.s. → g.s. alpha decay: these parents are left out of the comparison.

Masses, half-lives, J^π, alpha branching: NUBASE2020, Kondev et al., Chin. Phys. C 45, 030001 (2021). Q_α: AME2020, Wang et al., Chin. Phys. C 45, 030003 (2021). Coulomb functions under the barrier agree with an independent arbitrary-precision evaluation to 4×10⁻⁹.
Flagged, kept as tabulated, left out of the statistics: ²⁶⁴Hs (hollow point). NUBASE2020 lists T_1/2 = 0.7(3) s with a 70% alpha branch, and the model gives 3.3 decades shorter; Sato et al., J. Phys. Soc. Jpn. 80, 094201 (2011) report the half-life in milliseconds (of order 0.6 ms), which the model would match to 0.3 decades. A unit slip in the table is likely but not established here. Even-even rms: 0.371 without it, 0.443 with the tabulated 0.7 s, 0.371 if it were 0.7 ms.

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