If represents the number of free positive electrons in the nucleus, the electronic charge, the known charge on the -particle, namely , and the known kinetic energy of the -particle, then, since the inertias of the negative electrons are quite negligible in comparison with that of the -particle, if the latter suffers an appreciable change in direction in passing through an atom it will be due to the action of the nuclear charge. If represents the closest possible approach of the -particle to the center of the nucleus, namely, that occurring when the collision is “head on,” and the -particle is thrown straight back upon its course, then the original kinetic energy must equal the work done against the electric field in approaching to the distance , i.e.,

Suppose, however, that the collision is not “head on,” but that the original direction of the -particle is such that, if its direction were maintained, its nearest distance of approach to the nucleus would be (Fig. 41). The deflection of the a particle will now be, not 180°, as before, but some other angle . If follows simply from the geometrical properties of the hyperbola and the elementary principles of mechanics that

For let represent the path of the particle and let . Also let = velocity of the particle on entering the atom and its velocity at . Then from the conservation of angular momentum and from conservation of energy Since the eccentricity , and for any conic the focal distance is the eccentricity times one-half the major axis, i.e., , it follows that But from equations (58) and (59) and since the angle of deviation is , it follows that

Now it is evident from the method used in Appendix E that if there are atoms per cubic centimeter of a metal foil of thickness , and if each atom has a radius , then the probability that a particle of size small in comparison with will pass through one of these atoms in shooting through the foil is given by Similarly the probability that it will pass within a distance of the center of an atom is If this probability is small in comparison with unity, it represents the fraction of any given number of particles shooting through the foil which will actually come within a distance of the nucleus of an atom of the foil.

The fraction of the total number which will strike within radii and is given by differentiation as but from equation (57) Therefore the fraction which is deflected between the angles and is given by integration as

It was this fraction of a given number of -particles shot into the foil which Geiger and Marsden found by direct count by the scintillation method to be deflected through the angles included between any assigned limits and . Since and are known, could be at once obtained. It was found to vary with the nature of the atom, being larger for the heavy atoms than for the lighter ones, and having a value for gold of . This is then an upper limit for the size of the nucleus of the gold atom.

As soon as has thus been found for any atom, equation (56) can be solved for , since , , and are all known. It is thus that the number of free positive electrons in the nucleus is found to be roughly half the atomic weight of the atom, and that the size of the nucleus is found to be very minute in comparison with the size of the atom.