NUC-16

Gamow tunnelling factor

P ~ exp(−2π η), η = Z₁ Z₂ e² / (ħ v) = Z₁ Z₂ α c / v. Barrier penetration.

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NuclearGamow

Governing equation

P\sim e^{-2\pi\eta},\quad\eta=Z_1 Z_2\alpha c/v

where

Z_1: Z₁ (—)
Z_2: Z₂ (—)
E: CM energy (keV)
\mu: Reduced mass (u)
\eta: Sommerfeld (—)
P: Penetration (—)

Lecture brief

Historical brief

Rutherford, Chadwick, the semi-empirical mass formula, fission (Hahn–Strassmann 1938) and fusion Q-values, then Compton and Bethe–Bloch stopping, are the nuclear toolkit. Decay, binding and dose start here. This sheet (NUC-16 — Gamow tunnelling factor) is the form associated with Gamow. Working symbols: $Z_1$ , $Z_2$ , $E$ , $\mu$ $\rightarrow$ $\eta$ , $P$ . Why stars burn slowly: the Gamow tail of the Maxwellian tunnels through Coulomb.

Purpose

Purpose: compute

\eta

P

from

Z_1

Z_2

E

\mu

in Nuclear physics via

P\sim e^{-2\pi\eta},\quad\eta=Z_1 Z_2\alpha c/v

P ~ exp(−2π η), η = Z₁ Z₂ e² / (ħ v) = Z₁ Z₂ α c / v. Barrier penetration. Use it when a real nuclear physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

Z_1 = 1.000\,\mathrm{—}

Z_2 = 1.000\,\mathrm{—}

E = 10.000\,\mathrm{keV}

\mu = 0.500\,\mathrm{u}

, the governing relation

P\sim e^{-2\pi\eta},\quad\eta=Z_1 Z_2\alpha c/v

yields

\eta = 1.114\,\mathrm{—}

P = 9.149e-4\,\mathrm{—}

. A Coulomb wall, a leaking exponential. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Z₁ Z_11 —Z₂ Z_21 —CM energy E10 keVReduced mass \mu0.5 u

Outputs

Sommerfeld \eta1.114 —
Penetration P0.000915 —

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Narration of this film

A Coulomb wall, a leaking exponential.

Why stars burn slowly: the Gamow tail of the Maxwellian tunnels through Coulomb.

Reading speed

Watch on YouTube