On this page
Almost every atomic structure chapter starts with the same picture: alpha particles fired at gold foil, most going straight through, a few bouncing back. It is easy to memorise and forget. The real story is more useful, because it shows how a small, unexpected count changed physics, and it gives you a calculation that appears in JEE and NEET papers. This post covers what was measured, what Rutherford concluded and how to work out how close an alpha particle gets to a nucleus.
What did Geiger and Marsden find in 1909?
Ernest Rutherford, born in Nelson, New Zealand, in 1871, was already a Nobel laureate when the experiment began. His 1908 Nobel Prize in Chemistry was "for his investigations into the disintegration of the elements, and the chemistry of radioactive substances". He was then at Manchester.
The leading model of the atom at the time was J. J. Thomson's: positive charge spread through a sphere, with electrons inside. In that model, an alpha particle passing through an atom should be nudged only slightly.
Hans Geiger and Ernest Marsden tested this. In On a diffuse reflection of the α-particles (1909), they sent alpha particles at metal plates and counted flashes on a screen on the same side as the source. Heavier metals reflected more: lead, gold and platinum gave far more counts than aluminium. Using a platinum plate, "three different determinations showed that of the incident α-particles about 1 in 8000 was reflected".
One in 8000 sounds tiny. In Thomson's model it should have been far smaller still.
How did Rutherford explain the bounce-back?
In The scattering of α and β particles by matter and the structure of the atom (1911), Rutherford quoted Geiger and Marsden's observation that about 1 in 20,000 alpha particles were turned through an average angle of 90 degrees by a gold foil only about 0.00004 cm thick. Many small nudges could not add up to that. A single encounter had to do it.
So he considered an atom with a charge concentrated at its centre, which for distances less than 10⁻¹² cm could be treated as a point. He noted that his main results did not depend on whether that central charge was positive or negative, and assumed positive for convenience. The model predicted how the number of scattered particles should change with angle, foil thickness and the alpha particle's speed. In 1913 Geiger and Marsden published a longer set of measurements testing these laws, titled The laws of deflexion of α particles through large angles.
Worked example: how close does an alpha particle get to a gold nucleus?
An alpha particle with kinetic energy 5.0 MeV heads straight at a gold nucleus (Z = 79). Find the distance of closest approach.
At closest approach in a head-on collision, the alpha particle stops for an instant, so all its kinetic energy has become electrical potential energy:
K = k(2e)(Ze)/d, so d = k(2e)(Ze)/K.
Convert energy: K = 5.0 × 10⁶ × 1.6 × 10⁻¹⁹ J = 8.0 × 10⁻¹³ J.
Numerator: 9 × 10⁹ × 2 × 79 × (1.6 × 10⁻¹⁹)² = 9 × 10⁹ × 158 × 2.56 × 10⁻³⁸ ≈ 3.64 × 10⁻²⁶ J m.
Divide: d ≈ 3.64 × 10⁻²⁶ / 8.0 × 10⁻¹³ ≈ 4.6 × 10⁻¹⁴ m, about 45 femtometres.
A quicker route uses ke² ≈ 1.44 MeV fm: d = 1.44 × 158 / 5.0 ≈ 45.5 fm.
An atom is about 10⁻¹⁰ m across, so the alpha particle turns back at a distance roughly two thousand times smaller than the atom. The nucleus must be smaller still, because the particle never touches it.
Common mistakes to avoid:
Using e instead of 2e for the alpha particle's charge. That halves your answer.
Leaving energy in MeV while using k in SI units. Convert to joules, or use the 1.44 MeV fm shortcut consistently.
Calling d the radius of the nucleus. It is an upper limit. Faster alpha particles get closer: doubling K halves d.
If you want to check each step of problems like this, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. Try the problem yourself first.
What came next?
Rutherford's model had a problem: an electron circling a positive centre should radiate energy and spiral in. Niels Bohr's answer is the subject of our post on Bohr's model of the atom. For the quantum picture that replaced orbits, see our guide to quantum numbers and atomic structure.
What can a student learn from how Rutherford worked?
Take the rare result seriously. One in 8000 could have been dismissed as error. It was instead the clue.
Turn a picture into a number. Rutherford did not stop at "the charge is concentrated". He predicted how counts should vary with angle, and that made the idea testable.
Check your answer against scale. A distance of 45 fm only means something when you compare it with the size of an atom.
Practice this week: redo the closest-approach calculation for a 7.7 MeV alpha particle and for a silver target (Z = 47), and explain in one sentence why each answer moves the way it does.
