• Pixel-art sticker of Scattro, personifying Closest Approach, with the caption “Head-On, Straight Back.”

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Scattro

Closest Approach Physics

\[d = \frac{k q_1 q_2}{K}\]
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Meet Scattro, an alpha particle who charged straight at a gold nucleus and came bouncing back. In a head-on hit, all his kinetic energy becomes electric potential energy. At 5 MeV he gets within about 45 fm of gold, showing Rutherford in 1911 that the nucleus is tiny and dense. Pixel-art kiss-cut sticker in 2, 3, 4 and 6 inches; equation, symbols and sources in the physics panel below.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

Meet Scattro, an alpha particle who charged straight at a gold nucleus and came bouncing back.

In 1909 Hans Geiger and Ernest Marsden fired alpha particles at very thin gold foil. Most went straight through, but a very few bounced almost straight back. Ernest Rutherford realised this meant the positive charge of an atom is packed into a tiny, dense nucleus.

For a head-on hit, the alpha particle slows as it climbs the electric repulsion until all its kinetic energy has become potential energy. That gives the distance of closest approach, d = kq₁q₂/K. An alpha particle has charge 2e and gold has 79e. With K = 5 MeV, and ke² ≈ 1.44 MeV·fm, d ≈ 1.44 × 2 × 79 / 5 ≈ 45 fm.

So the gold nucleus must be smaller than about 45 femtometres, tens of thousands of times smaller than the whole atom.

Rutherford published his nuclear model in 1911, and it became the starting point for every later picture of the atom.

Equations

\[d = \frac{k q_1 q_2}{K}\]
SymbolMeaningUnit
\(d\) distance of closest approach \(\mathrm{m}\)
\(k\) Coulomb constant \(\mathrm{N·m²/C²}\)
\(q_1\) charge of the alpha particle \(\mathrm{C}\)
\(q_2\) charge of the nucleus \(\mathrm{C}\)
\(K\) initial kinetic energy of the alpha particle \(\mathrm{J}\)

Sources

  1. OpenStax University Physics Vol. 3 — Bohr's Model of the Hydrogen Atom
  2. Wikipedia — Rutherford scattering experiments
  3. NobelPrize.org — Ernest Rutherford facts

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