Rutherford and Bohr atomic models

In Rutherford's experiment (1909–1911) $\alpha$-particles passing through a thin gold foil were occasionally deflected through large angles; this means that the positive charge and nearly all the mass of the atom are concentrated in a nucleus about $10^{-14}$ m across, while the atom itself is $\sim 10^{-10}$ m across. Rutherford's planetary model could not explain the stability of the atom: an electron moving with acceleration must radiate continuously, fall into the nucleus within $\sim 10^{-10}$ s and produce a continuous spectrum. In 1913 Bohr postulated the existence of stationary orbits on which the electron does not radiate, together with quantization of angular momentum $m v r = n\hbar$, where $n = 1, 2, 3, \dots$. Emission and absorption occur only in transitions between stationary states, and the photon energy equals the difference of the level energies: $h\nu = E_n - E_m$ — this uses the relations $E = h\nu$ and $p = h/\lambda$ you already know. For the hydrogen atom Bohr's postulates give $r_n = n^2 a_0$ (where $a_0 = 0.0529$ nm) and $E_n = -13.6/n^2$ eV, which reproduces exactly the observed lines of the Lyman, Balmer and Paschen series.

Find the energy and wavelength of the photon emitted by a hydrogen atom in the transition from level $n = 3$ to level $n = 2$ (the H$\alpha$ line of the Balmer series).

For a hydrogen atom in the ground state ($n = 1$) calculate the orbital radius and the electron speed, then check the quantization postulate by comparing the electron de Broglie wavelength with the length of the orbit.

A hydrogen atom in the ground state absorbs a photon and makes a transition to level $n = 2$. Find the wavelength of the absorbed photon and then estimate the recoil speed of the atom using the photon momentum $p = h/\lambda$. The mass of a hydrogen atom is $m_H = 1.67\cdot10^{-27}$ kg.

Self-study plan on a computer (safe: no radiation sources and no vacuum apparatus).

1. Open the PhET simulation "Models of the Hydrogen Atom" ("Experiment" mode) or any similar interactive model.
2. Select Rutherford's model and fire photons of different wavelengths at the atom: verify that absorption is possible at any wavelength and that the spectrum comes out continuous.
3. Switch to Bohr's model and repeat the experiment: record that the atom absorbs a photon only at certain wavelengths.
4. Switch to "Spectrum" mode and record 3–4 observed lines of the Balmer series.
5. Check them against the Rydberg formula $\frac{1}{\lambda} = R_\infty\left(\frac{1}{2^2} - \frac{1}{n^2}\right)$, $R_\infty = 1.097\cdot10^{7}$ m$^{-1}$, and estimate the relative discrepancy with the simulation result.

Do not work with real discharge tubes or UV lamps: observing ultraviolet lines and handling high voltage are hazardous outside a laboratory, so all observation is done in the simulation.

Show that discrete energy levels explain the line spectrum and the stability of the atom, whereas a classical orbit would give a continuous spectrum.

A two-part diagram.

Left: energy-level diagram of hydrogen — a vertical energy axis (eV) with horizontal lines at $E_n = -13.6/n^2$ for $n = 1, 2, 3, 4, \infty$; downward transition arrows are labelled with $\lambda$ (Balmer series lines: $3\to2$, $4\to2$), and above $n = \infty$ the continuum (ionization) is shown.

Right: a sketch of Rutherford's experiment — an $\alpha$-particle beam, the foil and trajectories: mostly straight lines, with a few trajectories deflected through large angles; an inset shows the nucleus ($10^{-14}$ m) and the electron cloud ($10^{-10}$ m) drawn to scale.

A home smoke detector contains a tiny americium-241 source that emits alpha particles with kinetic energy about $5.5\,\text{MeV}$. In air at normal pressure each alpha travels only about $4\,\text{cm}$ and creates on the order of $1.6\times10^5$ ion pairs before stopping. The atom is roughly $10^{-10}\,\text{m}$ across, while the nucleus is only about $10^{-15}\,\text{m}$ across.

This is the Rutherford scattering picture: an alpha particle is a helium nucleus with charge $+2e$, so it interacts mainly with the tiny positive nucleus. Because the atom is mostly empty space, most alphas pass almost straight through; a rare close approach to a nucleus can deflect one through a large angle. The observed backscattering is what forced the nuclear model.

A neon sign in a sports arena has electrodes with a voltage of about $120\,\text{V}$ across the tube. Electrons accelerated by this voltage collide with neon atoms and can raise a bound electron to a higher Bohr level with an energy difference of roughly $18.7\,\text{eV}$. When the electron falls back, the sign emits red light with a wavelength near $640\,\text{nm}$, so each emitted photon has energy $E=hc/\lambda \approx 1.94\,\text{eV}$.

This uses the Bohr part of the model: only discrete orbits with fixed energies are allowed, and a photon is emitted when an electron jumps down by an energy difference $\Delta E = hf$. The Rutherford nucleus supplies the central positive charge that binds those orbits; the $18.7\,\text{eV}$ excitation and the $1.94\,\text{eV}$ photon show that the level spacing, not the total atomic energy, sets the light colour.

During a geomagnetic storm, electrons from the solar wind with energies of a few $\text{keV}$ enter the atmosphere near $100\,\text{km}$ altitude and collide with oxygen atoms. Excited oxygen atoms emit a green line at $557.7\,\text{nm}$ and a red line at $630.0\,\text{nm}$. The corresponding photon energies are $E=hc/\lambda \approx 2.22\,\text{eV}$ and $1.97\,\text{eV}$.

The aurora exercises Bohr's quantized energy levels: collisional excitation moves an oxygen electron into a higher allowed level, and the downward jump emits a photon whose energy equals the level difference. The colours are discrete because the level differences are discrete, exactly as in the Bohr model; Rutherford's nuclear atom gives the compact nucleus around which those levels are arranged.