Atomic Spectra and Rutherford's Nucleus

A hot solid glows with a continuous spectrum, but a gas of free atoms excited by a discharge emits light at a set of sharp, discrete wavelengths, and the same element always emits the same set. That fact, easy to state and impossible to explain with classical physics, is the experimental anchor of atomic structure. The wavelengths are so reproducible that they identify an element the way a fingerprint identifies a person; astronomers read stellar composition directly from them. This lesson covers what the spectra were found to obey, why the prevailing model of the atom could not produce them, and the scattering experiment that fixed where the atom's mass and charge actually live.

Three kinds of spectra

Light dispersed by a prism or grating sorts into one of three patterns, depending on the source.¹

• Continuous spectrum: an unbroken band of all wavelengths, emitted by incandescent solids and dense hot gases. This is the blackbody radiation of the previous module.
• Band spectrum: closely packed groups of lines that blur into bands at low resolving power, emitted by molecules.
• Line spectrum: isolated sharp lines at definite wavelengths, emitted by a gas of free atoms of a single element. The line positions are characteristic of the element.

Classical physics could account for the existence of a continuous spectrum, even if not its detailed shape. It could offer no reason at all for sharp lines. An atom that radiates should do so at the frequency of some internal motion, and nothing in classical mechanics restricts that motion to a discrete set of frequencies.

The Balmer and Rydberg-Ritz formulas

In 1885 Johann Balmer found that the four visible hydrogen lines fit a single formula with one integer parameter:²

Each integer $n$ predicts one line, and every predicted line is observed. The lines crowd together toward a short-wavelength series limit at $364.6\text{nm}$, the value approached as $n\to \infty$.

Rydberg and Ritz generalized Balmer's result to a form that works for hydrogen and, with an element-dependent constant, for other single-electron systems. It gives the reciprocal wavelength as a difference of two terms:³

Writing the reciprocal wavelength as a difference of two terms $R/m^2$ and $R/n^2$ is the Ritz combination principle: every spectral line frequency is a difference of two members of a single set of numbers (the terms). That structure is the empirical shadow of discrete energy levels, made explicit in the Bohr model.

The Thomson model and its failure

By 1900 the atom was known to be about $10^{-10}\text{m}$ across, to contain electrons far lighter than the whole atom, and to be electrically neutral. J. J. Thomson's model placed the electrons inside a diffuse sphere of positive charge that carried most of the mass, like raisins in a pudding. Thomson searched the model for stable configurations whose vibration frequencies would match the spectral lines.

The model fails on two counts.⁴

• No matching frequencies. Despite elaborate calculation, Thomson could not extract from the model a set of vibration frequencies matching the observed hydrogen spectrum.
• Radiative collapse. Electrostatic forces alone cannot hold a charge in stable equilibrium (Earnshaw's theorem), so the electrons must move, and a moving charge accelerates and radiates. A continuously radiating atom loses energy and collapses, and its emission would be continuous, not a line spectrum.

Rutherford's alpha-scattering probe

Rutherford had shown that the alpha particles emitted by radioactive sources are doubly ionized helium: their charge-to-mass ratio is half that of a proton, and letting a source decay in an evacuated chamber produces detectable ordinary helium. An energetic, massive alpha particle makes an excellent probe of the atom's interior. Geiger and Marsden directed a collimated alpha beam at a thin gold foil (about 2000 atoms thick) and counted the scintillations produced on a zinc-sulfide screen as a function of scattering angle.⁵

Most alpha particles passed through undeflected or bent by less than $1^{\circ }$, consistent with a diffuse Thomson atom. But a small fraction scattered through $90^{\circ }$ or more, and a few bounced almost straight back. A diffuse charge cloud cannot do this: the maximum force an alpha particle feels inside a Thomson atom is far too weak to reverse it. Rutherford's reaction is the standard quote: as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. Large-angle scattering requires the positive charge to be concentrated in a region much smaller than the atom, so that an alpha passing close feels an enormous Coulomb force from a single, compact target.

The scattering geometry

Treat the nucleus as a fixed point charge $Q=Ze$ at the origin. An alpha particle of charge $q=2e$, mass $m_{\alpha }$, and speed $v$ approaches along a line a perpendicular distance $b$ from a parallel line through the nucleus. The quantity $b$ is the impact parameter. Under the repulsive Coulomb force $F=kqQ/r^2$ the particle follows a hyperbola and departs at the scattering angle $\theta$. Because the potential energy returns to zero far away, conservation of energy makes the outgoing speed equal to the incoming speed. Classical mechanics relates the two:⁶

A small impact parameter means a close approach and a large deflection; a large impact parameter grazes the nucleus and barely bends. All particles with impact parameter less than a given $b$ scatter through an angle greater than the corresponding $\theta$.

From geometry to a countable cross section

The number of alphas scattered by one nucleus through angles greater than $\theta$ equals the number arriving with impact parameter less than $b(\theta )$, which is the number crossing the area $\pi b^2$. That area is the cross section for scattering through angles greater than $\theta$:⁷

For a foil with $n$ nuclei per unit volume and thickness $t$, the beam of area $A$ sees $nAt$ nuclei, and the fraction scattered beyond $\theta$ is

The Rutherford formula and its verification

Rutherford derived the full angular distribution. The number of alphas $\Delta N$ scattered into a detector of area $A_{sc}$ at distance $r$ and angle $\theta$ is⁸

The prediction packs four independent dependences, each a separate experimental test:

• Angle: $\Delta N\propto {\sin }^{-4}(\theta /2)$, falling by four orders of magnitude from small to large angles.
• Nuclear charge: $\Delta N\propto Z^2$.
• Kinetic energy: $\Delta N\propto E_k^{-2}$.
• Foil thickness: $\Delta N\propto t$.

Geiger and Marsden verified every one over four orders of magnitude in $\Delta N$, and the agreement established the nuclear atom as the basis for all later atomic and nuclear physics.

The size of the nucleus

Agreement with the point-charge formula does not by itself prove the nucleus is a point. The same Coulomb law would hold for a charged ball of radius $R_0$ as long as the alpha particle never penetrates it. Penetration would change the force law and break the ${\sin }^{-4}$ prediction, so the alpha energy at which the data first deviate marks the nuclear surface.

For a head-on collision (near $180^{\circ }$) the alpha stops at the distance of closest approach $r_d$, where all its kinetic energy has become potential energy:⁹

$r_d$ is an upper limit on the nuclear radius: the alpha turns around before reaching the surface, so the surface lies inside $r_d$.

The convenient unit for nuclear sizes is the fermi or femtometer, $1\text{fm}=10^{-15}\text{m}$. Nuclear radii run from about $1$ to $10\text{fm}$ across the periodic table, a scale developed in the nuclear physics module.

Rutherford's experiment settles where the charge and mass are but says nothing about the electrons: the model is silent on how they are arranged or why the atom does not radiate itself to death. Two empirical facts remain unexplained — the sharp line spectra and the stability of the atom. The Bohr model takes the nuclear atom as given and adds the quantum postulates that produce both.