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In 1910, Ernest Rutherford made a profound discovery that paved the way for a new understanding of the atom. He realised that the positive electric charge, and hence most of the mass, in an atom is concentrated in a tiny central nucleus, far smaller than the atom itself. Much of the nucleus is empty space!
Hans Geiger and Ernest Marsden were working in Rutherford's laboratory in Manchester University on experiments with thin foils of gold. They fired a beam of alpha particles (helium nuclei) from a radioactive source which struck the foil and were scattered. The number of flashes seen as the a-particles hit a fluorescent screen were then measured at different angles of scattering (q).
J.J. Thomson had discovered in 1897 that atoms contain negatively charged electrons which constitute only about 1/2000 the atomic mass. He proposed that the electrons were embedded in a positively charged lump of matter, like a 'plum pudding'. Rutherford, Geiger and Marsden therefore expected their positively charged alpha particles to be deflected only slightly by the positive charge spread across each of the gold atoms. They found that many of the alpha particles were deflected by a very small amount as expected. But remarkably about 1 in 8000 were scattered through angles greater than 90º. This astounded Rutherford and he commented that it was "as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."
The only way that Rutherford could explain this was to consider the atom as having a very small nucleus containing the positive charge with the electrons orbiting around it. This model of the nucleus meant that if the particle came very close to the nucleus, the incoming positively charged alpha particles would be repelled by a large electrostatic (Coulomb) force due to the positive nucleus, and give large scattering angles.
In this way Rutherford was able to estimate of the size of the gold nucleus which he found to be 6.6 x 10-12cm (see Rutherford's Notebook). This was an overestimate, as we now know it is about 7 x 10-13cm. In a more detailed analysis, Rutherford developed what is now known as the Rutherford Scattering Formula, which shows that the curve of the intensity against scattering angle q is proportional to 1/sin4(q/2).
By 1919 Rutherford had begun to find that his formula did not always work. He had investigated the scattering of alpha-particles in hydrogen gas, where the atomic nuclei have only a single unit of positive charge (unlike gold atoms, which have 79 times as much charge). The electrostatic repulsion between the alpha particle and the other nucleus in this case is much smaller than for gold. Rutherford found that his scattering formula broke down with higher energy, that is higher velocity, alpha particles. He assumed that the critical velocity at which his formula began to break down corresponded to a distance of closest approach to a "point charge", and he calculated calculated this to be 3.5 x 10-13cm. This is much closer to today's values for the size of the nucleus than his first estimates).