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The great scattering experiments

Hans Geiger was one of Rutherford’s students at Manchester University. He had been trying to make a workable detector to count alpha particles. During his investigations, he found that the alpha particles were deflected when they passed through a mica film. He told Rutherford of this effect. 
 
Rutherford encouraged Geiger and Ernest Marsden, an undergraduate student, to investigate the deflections. They counted scattered alpha particles by the faint scintillations they make on a screen.

Deflections

Only a few bounced back. Rutherford is widely quoted presenting this as an amazig result, saying 'it was as if you fired a 15" shell at a piece of tissue paper and it came back to hit you'. From Geiger and Marsden's results, Rutherford devised his new model of the atom: a very small massive nucleus with electrons so far out, and so light, that the alpha particles which were seriously deflected met the full force of a bare nucleus. He assumed that the nucleus carried a charge of +Ze, where Z is the serial number in the periodic table. Using this theory, the force between the alpha particle, itself a helium nucleus, and a gold nucleus is the inverse-square law Coulomb force of electrostatic repulsion. 
 
Geiger and Marsden went on to make a great series of measurements of the deflections of a narrow beam of alpha particles that hits gold leaf in a vacuum. Their measurements served to test his new model. 

The new model

Alpha particles making a very close approach to another nucleus are deflected through a large angle. Those missing the target widely are deflected through a small angle. Measurements over a big range of angles serve to investigate the field of force inside the scattering atoms over a large range of distances from the centre. 
 
Rutherford assumed an inverse-square law of repulsion between the big electric charge on the massive nucleus of the gold atom and the smaller charge on the alpha particle flying past it. That is equivalent to Newton’s assumption of an inverse-square law attraction between the massive Sun and a planet. Instead of the simple circular orbits which serve approximately for planets, the change to a repulsive force predicts a different shape: hyperbolas. The alpha particle sails in, bends around, and sails out again on another almost straight track in a new direction. The simple calculation with circular orbits that predicts Kepler III becomes more complicated. 
 
Instead of measuring the orbits of a few planets, Rutherford had to use hordes of little alpha particles to give him a statistical test. He made his theory predict the number of particles that an observer would count on a receiving screen in various directions, in some standard time. In calculating that prediction he simply used an inverse-square law of repulsive force and Newton’s laws of motion. 
 
The table of actual measurements of scattered alpha particles for various angles (taken from Geiger and Marsden’s original paper) shows how the numbers counted fit the predictions for an inverse-square law of force. 

Scattering of Alpha Particles by Gold

Geiger and Marsden's table of results compared with predictions using Rutherford's model. 

* Of path of alpha-particles. 
† Number of scintillations seen, for deflection A°, in a standard time. 

Note: In the actual experiments Geiger and Marsden made one set of measurements for the larger angles of deflection, and another set, with a much smaller radioactive source, for the smaller angles. To make one complete set in the table above, the numbers for smaller angles have been multiplied up to fit the set for larger angles. The multiplying factor was provided by experiment because counting was done for 30° in both data sets. 

 
Finding the charge on the nucleus 
Rutherford’s theory also predicted the way the count on a fixed screen would depend on the speed of the alpha particles: 
 
N ∝ 1/v4 
and on the electric charge of the scattering nucleus: 
 
N ∝ (Ze)2 
 
Chadwick used this to measure the charge on the nuclei of a number of elements. He used thin sheets of copper, silver and platinum instead of gold and measured the scattering of alpha particles from each. From his counts, with Rutherford’s theory, he calculated the charge on the nucleus of each of those nuclei. 
 
His results were: copper 29.3 electron charges, silver 46.3 electron charges, platinum 77.4 electron charges, with expected errors of about 1%. The serial numbers of those elements, arranged in order of atomic weights and placed in the period table are: 29, 47, 78. Chadwick’s measurements showed that the nuclear charge is the atomic number. 
 
Nowadays the charge of a nucleus is understood in terms of the proton number, and its value is measured in electron charges. Originally, from Geiger and Marsden's scattering experiments, it was deduced that the nucleus had a charge of about half the atomic weight multiplied by the electron charge. 
 
Back-scattering to measure the size of the nucleus 
From the known mass and speed of the alpha particles, Rutherford could calculate the distance of closest approach to a nucleus. This is the distance from a gold atom’s centre at which an alpha particle making a rare head-on collision would come to rest momentarily and bounce straight back. 
 
Rutherford tried different energy alpha particles, and found some for which the measured number deviated from the predicted number. He suggested that, at this energy, the alpha particles were reaching the nucleus and being assimilated into it. This, he said, gave an indication of the radius of the nucleus. That radius turned out to be 10,000 times smaller than the radius of the atom. 

Thanks to David Baum for pointing out an error on this page, now corrected. Editor

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