June 26, 2016


The famous Rutherford gold foil experiment is actually a series of Geiger-Marsden experiments.

They were performed between 1908 to 1913 by Hans Geiger and Ernest Marsden under the guidance of Ernest Rutherford at the University of Manchester.

It was conducted to verify the existing model of atom prevalent at that time which was proposed by the great J. J. Thomson (who turned to be inaccurate).

J J Thomson had proposed:
"... the atoms of the elements consist of a number of negatively electrified corpuscles in a sphere of uniform positive electrification..."

This is popularly known as the plum pudding model.

The Japanese physicist Hantaro Nagaoka had completely rejected this atomic model on the grounds that the opposite charges are impenetrable.

Instead, Nagaoka made 2 predictions how an atom could be:

1. A very massive atomic center

2. Electrons revolving around the electrons bound by electrostatic forces.

Rutherford decided to test it out.

According to his theoretical calculations, at the atomic scale, the concept of "solid matter" is meaningless.

Since alpha particles are tiny submicroscopic, positively charged particles, the alpha particles would not bounce of the atom like a marble.

(Remember, the thin gold foil is roughly 400 atoms thick).

The alpha particles would primarily be affected by the electric fields of atoms in the gold foil.

And as per the Thomson's model, the electrical field of gold atoms would be too weak to deflect a passing very fast moving alpha particles.

In fact, according to their calculations, the deflection should be just 0.000326 radians or 0.0186° (a small fraction of a degree).

This was the kind of precision recording that physicists had begun to expect even in early 1900s.

We shall continue to follow these experiments.

I wish to show you some simple calculations that the Rutherford team performed before starting their experiments.

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As a worked example, consider an alpha particle passing tangentially to a Thomson gold atom, where it will experience the electric field at its strongest and thus experience the maximum deflection θ. Since the electrons are very light compared to the alpha particle, their influence can be neglected^[6] and the atom can be seen as a heavy sphere of positive charge.

Q_n = positive charge of gold atom = 79 e = 1.266×10⁻¹⁷ C

Q_α = charge of alpha particle = 2 e = 3.204×10⁻¹⁹ C

r = radius of a gold atom = 1.44×10⁻¹⁰ m

v_α = velocity of alpha particle = 1.53×10⁷ m/s

m_α = mass of alpha particle = 6.645×10⁻²⁷ kg

k = Coulomb's constant = 8.998×10⁹ N·m²/C²

Using classical physics, the alpha particle's lateral change in momentum Δp can be approximated using the impulse of force relationship and the Coulomb force expression:

${\displaystyle \Delta p=F\Delta t=k\cdot {\frac {Q_{\alpha }Q_{n}}{r^{2}}}\cdot {\frac {2r}{v_{\alpha }}}}$

${\displaystyle \theta \approx {\frac {\Delta p}{p}}<k\cdot {\frac {2Q_{\alpha }Q_{n}}{m_{\alpha }rv_{\alpha }^{2}}}=8.998\cdot 10^{9}\times {\frac {2\times 3.204\cdot 10^{-19}\times 1.266\cdot 10^{-17}}{6.645\cdot 10^{-27}\times 1.44\cdot 10^{-10}\times (1.53\cdot 10^{7})^{2}}}}$

${\displaystyle \theta <0.000326~\mathrm {rad} ~(\mathrm {or} ~0.0186^{\circ })}$