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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∼e−2πη,η=Z1Z2αc/vP\sim e^{-2\pi\eta},\quad\eta=Z_1 Z_2\alpha c/v

E∈[0.5,400]E \in \left[0.5, 400\right]

where

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

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NUC-16 · quantum
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①

Governing law — red

P∼e−2πη,η=Z1Z2αc/vP\sim e^{-2\pi\eta},\quad\eta=Z_1 Z_2\alpha c/v

E∈[0.5,400]E \in \left[0.5, 400\right]

E=0.5E = 0.5η=2.58×10−14\eta = 2.58\times 10^{-14}

②

Domain of the variable — blue

Emin⁡=0.5,Emax⁡=400E_{\min} = 0.5,\quad E_{\max} = 400

E∈[0.5,400]E \in \left[0.5, 400\right]

E=0.5E = 0.5E=0.5E = 0.5

③

Response curve — green

P∼e−2πη,η=Z1Z2αc/vP\sim e^{-2\pi\eta},\quad\eta=Z_1 Z_2\alpha c/v

E∈[0.5,400]E \in \left[0.5, 400\right]

E=0.5E = 0.5η=2.58×10−14\eta = 2.58\times 10^{-14}

④

Rate of change — amber

dηdE\frac{\mathrm{d}\eta}{\mathrm{d}E}

E∈[0.5,400]E \in \left[0.5, 400\right]

E=3.013E = 3.013η′=1.63×10−5\eta^{\prime} = 1.63\times 10^{-5}

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: Z1Z_1, Z2Z_2, EE, μ\mu →\rightarrow η\eta, PP. Why stars burn slowly: the Gamow tail of the Maxwellian tunnels through Coulomb.

Purpose

Purpose: compute η\eta, PP from Z1Z_1, Z2Z_2, EE, μ\mu in Nuclear physics via P∼e−2πη,η=Z1Z2αc/vP\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 Z1=1.000 —Z_1 = 1.000\,\mathrm{—}, Z2=1.000 —Z_2 = 1.000\,\mathrm{—}, E=10.000 keVE = 10.000\,\mathrm{keV}, μ=0.500 u\mu = 0.500\,\mathrm{u}, the governing relation P∼e−2πη,η=Z1Z2αc/vP\sim e^{-2\pi\eta},\quad\eta=Z_1 Z_2\alpha c/v yields η=1.114 —\eta = 1.114\,\mathrm{—}, P=9.149e−4 —P = 9.149e-4\,\mathrm{—}. A Coulomb wall, a leaking exponential. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Sommerfeld \eta1.114 —
  • Penetration P0.000915 —
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NUC-16 · quantum
00:0 / 00:08

Narration of this film

A Coulomb wall, a leaking exponential.

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

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