• Ernest Rutherford Gold Foil pixel-art physics sticker, 3" × 3" / White

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Ernest Rutherford Sticker

Some Came Back | Physics

\[\frac{d\sigma}{d\Omega} = \left(\frac{zZe^2}{16\pi\varepsilon_0 E}\right)^2\frac{1}{\sin^4(\theta/2)}\]
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How did we learn that atoms have a nucleus? Ernest Rutherford’s lab fired alpha particles at thin gold foil. Almost all passed through, but a few bounced straight back. In 1911 Rutherford explained this: an atom’s positive charge and nearly all its mass sit in a tiny nucleus. This is a 3" pixel-art physicist sticker from Pixelated Physics.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

Rutherford fired alpha particles at gold foil and some bounced almost straight back. His formula, with its steep dependence on 1/sin⁴(θ/2), only works if the positive charge sits in a tiny, dense nucleus.

In 1909 Hans Geiger and Ernest Marsden, working in Ernest Rutherford’s lab, fired alpha particles at a thin gold foil. Almost all went straight through. A very few came back.

In 1911 Rutherford explained why. Nearly all the atom’s mass and positive charge sit in a tiny nucleus. An alpha particle aimed far from it barely bends; one aimed almost dead-on is turned right round. The angle follows tan(θ/2) = d/2b.

Every path in the art is integrated from the Coulomb force and matches that formula to within 0.05°. The nucleus is drawn hugely magnified.

Rutherford played rugby for his school, Nelson College in New Zealand, so here he takes the catch himself.

Some came back.

Equations

\[\frac{d\sigma}{d\Omega} = \left(\frac{zZe^2}{16\pi\varepsilon_0 E}\right)^2\frac{1}{\sin^4(\theta/2)}\]
\[\tan\frac{\theta}{2} = \frac{d}{2b},\qquad d = \frac{zZe^2}{4\pi\varepsilon_0 E}\]
SymbolMeaningUnit
\(d\sigma/d\Omega\) differential cross section per solid angle \(\mathrm{m^2/sr}\)
\(z\) charge number of the alpha particle (2) \(\mathrm{1}\)
\(Z\) charge number of the target nucleus (gold: 79) \(\mathrm{1}\)
\(e\) elementary charge \(\mathrm{C}\)
\(\varepsilon_0\) vacuum permittivity \(\mathrm{F/m}\)
\(E\) kinetic energy of the alpha particle \(\mathrm{J}\)
\(\theta\) scattering angle \(\mathrm{rad}\)
\(b\) impact parameter: how far off-centre the alpha is aimed \(\mathrm{m}\)
\(d\) distance of closest approach in a head-on collision \(\mathrm{m}\)

Sources

  1. Rutherford (1911) The scattering of α and β particles by matter and the structure of the atom, Phil. Mag. 21, 669
  2. Geiger & Marsden (1909) On a diffuse reflection of the α-particles, Proc. R. Soc. A 82, 495
  3. Te Ara, Dictionary of New Zealand Biography: Rutherford, Ernest

Source Rutherford, E. (1911). The scattering of α and β particles by matter and the structure of the atom. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 21, 669–688.doi.org/10.1080/14786440508637080

Field notes

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