Rutherford Scattering: From Intuition to Precision
Imagine you're in a dark room, and someone tells you there's a heavy object hidden somewhere. You can't see it, but you have a handful of marbles. If you roll marbles toward where you think the object might be, what happens? Most marbles roll straight through. But occasionally, one hits something hard and bounces back at a sharp angle. By studying how the marbles scatter — how many bounce back, and at what angles — you can figure out where the heavy object is, how big it is, and even how hard it is.
That's exactly what Rutherford did — but with alpha particles (tiny, fast, positively charged "marbles") and a thin gold foil (the "dark room").
The Intuition: Why Do Particles Scatter?
Before Rutherford, the atom was thought to be a "plum pudding" — a diffuse, positively charged blob with electrons scattered inside like raisins. If that were true, alpha particles fired at a gold foil should pass through almost undeflected, like bullets through jelly. The positive charge was spread out, so there was no concentrated "hard spot" to push them away.
But in 1909, Geiger and Marsden (working under Rutherford) found something shocking: a tiny fraction of alpha particles bounced back — some even at angles greater than 90°. Rutherford later said, "It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."
The only explanation: the atom's positive charge and most of its mass are concentrated in a tiny, dense nucleus. When an alpha particle gets close to this nucleus, the strong electrostatic repulsion (both are positively charged) violently deflects it.
The Precise Statement
Rutherford scattering describes the elastic scattering of a charged particle (like an alpha particle) by a stationary, point-like nucleus, due to the Coulomb force (electrostatic repulsion). The key result is a formula for the number of particles scattered into a given direction.
N(θ)=16(4πε0)2r2Ek2NintZ12Z22e4⋅sin4(θ/2)1
where:
- N(θ) = number of particles scattered at angle θ
- Ni = number of incident particles
- n = number of target nuclei per unit volume
- t = target thickness
- Z1,Z2 = atomic numbers of projectile and target
- e = elementary charge
- Ek = kinetic energy of incident particle
- r = distance from target to detector
- θ = scattering angle
The most important part is the 1/sin4(θ/2) dependence. This tells you:
- Most particles scatter at very small angles (they pass far from the nucleus and feel only a weak push).
- Very few scatter at large angles (they must come extremely close to the nucleus).
- The number of particles scattered at a given angle drops sharply as the angle increases.
What This Tells Us
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The nucleus is tiny. For gold, the nuclear radius is about 10−14 m, while the atom's radius is about 10−10 m — the nucleus is 10,000 times smaller. That's why only 1 in 8,000 alpha particles bounced back.
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The nucleus is positively charged. The repulsion is electrostatic, so the projectile and nucleus must have the same sign of charge.
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The force is Coulombic. The sin4(θ/2) dependence is a direct consequence of the inverse-square law (F∝1/r2). If the force were different (e.g., short-range nuclear force), the angular distribution would change.
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The scattering is elastic. The alpha particle loses no kinetic energy — it just changes direction. The nucleus recoils slightly, but the total kinetic energy is conserved.
A common mistake: thinking Rutherford scattering applies to any particle hitting a nucleus. It only works when the projectile's energy is low enough that it never gets close enough to feel the strong nuclear force. At higher energies, the particle can penetrate the nucleus, and the scattering becomes inelastic (nuclear reactions occur). The formula above breaks down in that regime.
The Key Assumptions (Why the Formula Works)
Rutherford made several simplifying assumptions that hold for his experiment:
- Single scattering — each alpha particle interacts with only one nucleus (the foil is thin enough).
- Point-like nucleus — the nucleus is much smaller than the distance of closest approach.
- Coulomb force only — no other forces (gravity, nuclear) are significant.
- Target nucleus is stationary — it's much heavier than the alpha particle, so its recoil is negligible.
- Elastic collision — no energy is lost to radiation or excitation. …
