Rutherford's Nuclear Model of the Atom and Its Limitations

Interpreting the three observations. Rutherford drew his conclusions about atomic structure directly from the three observations of Section 1.2, reasoning as follows. (1) Since the overwhelming majority of alpha particles passed straight through the gold foil undeflected, MOST of the volume of an atom must be essentially empty space -- there is nothing substantial there to deflect a fast, massive alpha particle. (2) Since a small fraction were deflected through large angles, there must exist, somewhere inside the atom, an extremely concentrated, strongly repulsive source of positive charge, capable of exerting a large enough Coulomb force on a passing alpha particle to turn it through such an angle -- and since this happened only rarely, that source of charge must occupy only a tiny fraction of the atom's total volume (most alpha particles simply never passed close enough to it). (3) Since a very few particles bounced back almost $180^\circ$, this concentrated positive charge must also carry almost the ENTIRE mass of the atom -- an alpha particle, itself fairly massive, cannot be reversed in direction by a target far lighter than itself (exactly as a football is not reversed by colliding with a table-tennis ball, but is reversed by colliding with a wall), so whatever is doing the reversing must be much more massive than the alpha particle.

Rutherford's nuclear model. Putting these three conclusions together, Rutherford proposed, in 1911, that:

1. Almost the entire mass of an atom, and ALL of its positive charge, is concentrated in an extremely small, dense central core, called the nucleus, of radius of order $10^{-15}$ to $10^{-14}\ \text{m}$.
2. The electrons, carrying the atom's negative charge, revolve around this nucleus in circular orbits at comparatively enormous distances, giving the atom as a whole a radius of order $10^{-10}\ \text{m}$ -- roughly $10\,000$ to $100\,000$ times larger than the nucleus itself, which is why the atom is overwhelmingly empty space.
3. The electrostatic (Coulomb) force of attraction between the positively charged nucleus and each negatively charged electron supplies exactly the centripetal force needed to keep that electron moving in its circular orbit: $\dfrac{1}{4\pi\epsilon_0}\cdot\dfrac{Ze^2}{r^2} = \dfrac{mv^2}{r}$, exactly analogous to gravity holding a planet in orbit around the Sun -- which is why this picture is often called the "planetary model" of the atom.

Limitation 1 -- the stability problem. According to classical electromagnetic theory (established well before Rutherford's experiment), ANY accelerating electric charge must continuously radiate electromagnetic energy. An electron moving in a circular orbit is always accelerating -- not because its speed changes, but because its DIRECTION is constantly changing (centripetal acceleration) -- so, classically, it must be continuously losing energy as radiation. As it loses energy, the electron's orbit cannot remain at a fixed radius; instead, standard calculations based on classical theory show the electron would spiral inward along a rapidly shrinking path and crash into the nucleus in an extremely short time, of the order of only $10^{-8}\ \text{s}$ (Exercise 3 works through this reasoning). This flatly contradicts the observed fact that atoms are stable and persist unchanged for immense lengths of time -- ordinary matter, made of atoms, simply does not collapse. …