Rutherford's model

In 1911, acting on Rutherford's advice, his students Hans Geiger and Ernest Marsden carried out what became one of the most famous experiments in physics: firing alpha particles at a thin gold foil and observing how they scattered.

Setup. A radioactive source of alpha particles (such as polonium) sits inside a thick lead box with a fine hole, so a narrow beam of alpha particles emerges and passes through a second fine hole in a lead screen before striking a thin gold foil. A movable zinc-sulphide-coated fluorescent screen, able to swing through scattering angles from 0 to 180 degrees, registers a flash of light wherever an alpha particle lands after being scattered, observed through a microscope.

Results. (a) The great majority of alpha particles passed straight through the foil essentially undeflected. (b) Some were deflected through small angles. (c) A small fraction - roughly one in a thousand - were deflected through angles greater than 90 degrees. (d) A very few were scattered straight back, deflected by a full 180 degrees. This last result was the shocking one: it is only possible if the atom's positive charge and almost all of its mass are concentrated in an extremely small, dense region rather than spread through the whole atom as Thomson's model assumed.

Rutherford's conclusion. An atom consists mostly of empty space, with a tiny, positively charged, massive core - the nucleus, of size of order $10^{-14}$ m - at its centre, surrounded by negatively charged electrons. Since a static arrangement of charges cannot be stable, Rutherford proposed that the electrons must instead be revolving around the nucleus in circular orbits, in the same way planets orbit the Sun. …

When an alpha particle is aimed exactly at the centre of a nucleus (a head-on collision), it slows down continuously as the repulsive Coulomb force from the positively charged nucleus grows stronger, until at some minimum distance $r_0$ it momentarily comes to rest and then reverses, travelling directly back out along the same line. This minimum separation $r_0$ is called the distance of closest approach (or contact distance).

At the instant the alpha particle is momentarily at rest, all of its original kinetic energy $E_k$ has been converted into electrostatic potential energy between the alpha particle (charge $+2e$) and the nucleus (charge $+Ze$):

Solving for $r_0$,

$r_0=\frac{1}{4\pi\varepsilon_0}\frac{2Ze^2}{\tfrac{1}{2}m\upsilon_0^2}=\frac{1}{4\pi\varepsilon_0}\frac{2Ze^2}{E_k}$ …

Not every alpha particle in Geiger and Marsden's beam was aimed exactly at a gold nucleus's centre - most missed by some offset distance, which is what produces the whole range of observed scattering angles rather than only straight-through or straight-back trajectories. The impact parameter $b$ is defined as the perpendicular distance between the nucleus's centre and the straight-line direction the alpha particle's velocity vector would have followed if it were very far from the nucleus (i.e. its aim, ignoring any deflection).

Geometrically, an alpha particle aimed with impact parameter $b=0$ is the special head-on case that produces the distance-of-closest-approach scenario of the previous section (180-degree back-scattering); an alpha particle with a very large $b$ passes far from the nucleus and is barely deflected at all. The relationship between the impact parameter and the resulting scattering angle $\theta$ is

$b=K\cot\!\left(\frac{\theta}{2}\right),\qquad K=\frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{m\upsilon_0^2}\qquad (8.13)$ …