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Maximum nuclear size estimate

Rutherford begins his analysis by considering a head-on collision between the alpha particle and atom. This will establish the minimum distance between them, a value which will be used throughout his calculations.

Assuming there are no external forces and that initially the alpha particles are far from the nucleus, the inverse-square law between the charges on the alpha particle and nucleus gives the potential energy gained by the particle as it approaches the nucleus. For head-on collisions between alpha particles and the nucleus, all the kinetic energy of the alpha particle is turned into potential energy and the particle stops and turns back.

Where the particle stops, a distance r min {\displaystyle r_{\text{min}}} the potential energy matches the original kinetic energy:

1 2 m v 2 = k q 1 q 2 r min {\displaystyle {\frac {1}{2}}mv^{2}=k{\frac {q_{1}q_{2}}{r_{\text{min}}}}}

where

k = 1 4 π ϵ 0 {\displaystyle k={\frac {1}{4\pi \epsilon _{0}}}}

Rearranging: r min = k 2 q 1 q 2 m v 2 {\displaystyle r_{\text{min}}=k{\frac {2q_{1}q_{2}}{mv^{2}}}}

For an alpha particle:

The distance from the alpha particle to the center of the nucleus (r) at this point is an upper limit for the nuclear radius.Substituting these in gives the value of about 2.7 × 10 m, or 27 fm. (The true radius is about 7.3 fm.) The true radius of the nucleus is not recovered in these experiments because the alphas do not have enough energy to penetrate to more than 27 fm of the nuclear center, as noted, when the actual radius of gold is 7.3 fm.

Rutherford's 1911 paper started with a slightly different formula suitable for head-on collision with a sphere of positive charge:

1 2 m v 2 = N e E ⋅ (1 b − 3 2 R + b 2 2 R 3) {\displaystyle {\frac {1}{2}}mv^{2}=NeE\cdot \left({\frac {1}{b}}-{\frac {3}{2R}}+{\frac {b^{2}}{2R^{3}}}\right)}

In Rutherford's notation, e is the elementary charge, N is the charge number of the nucleus (we now know this to be equal to the atomic number), and E is the charge of an alpha particle. The convention in Rutherford's time was to measure charge in electrostatic units, distance in centimeters, force in dynes, and energy in ergs. The modern convention is to measure charge in coulombs, distance in meters, force in newtons, and energy in joules. Using coulombs requires using the Coulomb constant (k) in the equation. Rutherford used b as the turning point distance (called r above) and R is the radius of the atom. The first term is the Coulomb repulsion used above. This form assumes the alpha particle could penetrate the positive charge. At the time of Rutherford's paper, Thomson's plum pudding model proposed a positive charge with the radius of an atom, thousands of times larger than the r found above. Fig. 1 shows how concentrated this potential is compared to the size of the atom.Many of Rutherford's results are expressed in terms of this turning point distance r, simplifying the results and limiting the need for units to this calculation of turning point.