---
title: Atomic Spectra and Rutherford's Nucleus
module: Early Atomic Models and the Old Quantum Theory
moduleNumber: 1
lessonNumber: 1
order: 101
summary: >
  Atoms emit light only at sharp, reproducible wavelengths, and by 1890 those
  wavelengths were captured by the Rydberg-Ritz formula. Neither empirical
  regularity had a mechanical explanation. Rutherford's alpha-scattering
  experiment supplied the missing structure: the atom's positive charge and
  nearly all its mass sit in a tiny central nucleus, with the electrons far
  outside.
topics: [Early Atomic Models and the Old Quantum Theory]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 4 — The Nuclear Atom; §4-1 Atomic Spectra"
  - book: Tipler & Llewellyn
    ref: "§4-2 Rutherford's Nuclear Model; Geiger and Marsden's Results"
  - book: Tipler & Mosca
    ref: "Ch. 36 — Atoms; §36-2 The Bohr Model of the Hydrogen Atom"
draft: false
---

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.[^tl-spectra]

- **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](/quantum-mechanics/old-quantum-theory/blackbody-radiation-and-the-planck-quantum).
- **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.

$$
% caption: A slit collimates light from the source; the prism bends shorter
% wavelengths through a larger angle, spreading the beam into separated lines
% on the screen.
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$$

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:[^tl-balmer]

$$
\lambda = 364.6\,\text{nm}\;\frac{n^2}{n^2 - 4}, \qquad n = 3, 4, 5, \dots
$$

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:[^tl-rydberg]

> **Definition (Rydberg-Ritz formula).** For a hydrogen-like atom the observed
> reciprocal wavelengths are
> $$
> \frac{1}{\lambda} = R\left(\frac{1}{m^2} - \frac{1}{n^2}\right), \qquad n > m,
> $$
> where $m$ fixes a **series** and $n = m+1, m+2, \dots$ runs over its lines.
> The **Rydberg constant** for hydrogen is $R_\text{H} = 1.0967757 \times 10^7\,\text{m}^{-1}$.

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](/atomic-physics/early-models-and-old-quantum-theory/bohr-model-hydrogen).

| Series | $m$ | Lines $n$ | Region | Series limit $\lambda = m^2/R$ |
| --- | --- | --- | --- | --- |
| Lyman | 1 | 2, 3, 4, … | ultraviolet | 91.2 nm |
| Balmer | 2 | 3, 4, 5, … | visible / near UV | 364.6 nm |
| Paschen | 3 | 4, 5, 6, … | infrared | 820.6 nm |
| Brackett | 4 | 5, 6, 7, … | infrared | 1459 nm |
| Pfund | 5 | 6, 7, 8, … | infrared | 2280 nm |

$$
% caption: The Balmer series of hydrogen. Lines crowd toward the short-wavelength
% series limit near 365 nm; the longest-wavelength line at 656 nm is the red
% one seen in a discharge tube.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
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$$

> **Example (Balmer, Lyman, and Paschen first lines).** The first Balmer line
> ($m=2, n=3$) has
> $$
> \frac{1}{\lambda} = R_\text{H}\left(\frac{1}{4} - \frac{1}{9}\right)
> = \frac{5}{36}R_\text{H} = 1.523 \times 10^6\,\text{m}^{-1},
> \quad \lambda = 656.5\,\text{nm}.
> $$
> The first Lyman line ($m=1, n=2$) gives $\frac{1}{\lambda} = \frac34 R_\text{H}$,
> so $\lambda = 121.6\,\text{nm}$ (ultraviolet). The first Paschen line
> ($m=3, n=4$) gives $\frac{1}{\lambda} = \frac{7}{144}R_\text{H}$, so
> $\lambda = 1876\,\text{nm}$ (infrared). Only the Balmer series falls mostly in
> the visible.

## 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.[^tl-thomson]

- **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.

$$
% caption: Thomson spreads the positive charge through the whole atom with
% electrons embedded in it; Rutherford concentrates the charge and mass in a
% tiny central nucleus, leaving the atom mostly empty.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
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\node[below, align=center] at (0,-1.85) {Thomson model\\(charge spread out)};
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\node[below, align=center] at (0,-1.85) {Rutherford model\\(charge concentrated)};
\end{scope}
\end{tikzpicture}
$$

## 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.[^tl-rutherford]

$$
% caption: The Geiger-Marsden apparatus. A collimated alpha beam strikes a thin
% gold foil; the scintillation screen and microscope rotate about the foil to
% count particles scattered through each angle.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
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\end{tikzpicture}
$$

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.

| | Thomson model | Rutherford model |
| --- | --- | --- |
| Positive charge | spread through the whole atom | concentrated in a central nucleus |
| Mass | spread with the charge | almost entirely in the nucleus |
| Max deflection per encounter | far below $1^\circ$ | up to $180^\circ$ |
| Large-angle scattering | negligible | single close encounter |

## 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:[^tl-geometry]

$$
b = \frac{kqQ}{m_\alpha v^2}\cot\frac{\theta}{2}.
$$

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 $\theta$.

$$
% caption: An alpha particle approaches with impact parameter b, is repelled by
% the point nucleus along a hyperbola, and leaves deflected through the
% scattering angle; smaller b gives a larger angle.
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\end{tikzpicture}
$$

## 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 $\theta$:[^tl-cross]

> **Definition (Cross section).** The cross section $\sigma$ for a scattering
> process is the number of events per target nucleus per unit time divided by
> the incident intensity $I_0$ (particles per unit area per unit time). For
> Rutherford scattering through angles greater than $\theta$,
> $\sigma = \pi b^2$, an effective target area.

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

$$
f = \pi b^2\, n t.
$$

$$
% caption: Two alpha particles of equal energy: the one with the smaller impact
% parameter passes closer and scatters through the larger angle. The disk of
% radius b is the cross section for scattering beyond the corresponding angle.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\fill[acc] (0,0) circle (2.8pt);
\node[acc, below, font=\scriptsize] at (0,-0.15) {nucleus};
% inner (small b, large angle)
\draw[->, black, thick] (-4,0.5) -- (-1.1,0.5);
\draw[black, thick] (-1.1,0.5) .. controls (-0.1,0.35) and (0.2,0.9) .. (-0.4,1.9);
\draw[->, black, thick] (-0.4,1.9) -- (-0.65,2.5);
\node[black!70, font=\scriptsize, anchor=west] at (-1.05,0.62) {small $b$};
% outer (large b, small angle)
\draw[->, black, thick] (-4,1.7) -- (-1.4,1.7);
\draw[black, thick] (-1.4,1.7) .. controls (0.2,1.5) and (1.4,1.75) .. (3.2,2.35);
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\node[black, font=\scriptsize, anchor=west] at (2.4,2.0) {large $b$};
\end{tikzpicture}
$$

> **Example (Expected fraction beyond $90^\circ$).** For $5\,\text{MeV}$ alphas
> on gold ($Z=79$) foil $10^{-6}\,\text{m}$ thick, the number density is
> $n = \rho N_A / M = 5.90 \times 10^{28}\,\text{m}^{-3}$, and the impact
> parameter for $\theta = 90^\circ$ is
> $$
> b = \frac{(2)(79)ke^2}{2E_k}\cot 45^\circ = 2.28 \times 10^{-14}\,\text{m}.
> $$
> Then $f = \pi b^2 n t \approx 9.6 \times 10^{-5}$, about $1$ in $10^4$. Geiger
> and Marsden measured roughly $1$ in $8000$, confirming the nuclear model.

## 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 $\theta$
is[^tl-formula]

$$
\Delta N = \left(\frac{I_0\, A_{sc}\, n t}{r^2}\right)
\left(\frac{kZe^2}{2E_k}\right)^2 \frac{1}{\sin^4(\theta/2)}.
$$

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.

$$
% caption: The Rutherford angular distribution. Counts fall steeply with
% increasing angle, dropping four orders of magnitude across the measured range;
% the vertical axis is logarithmic.
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\draw[->, black] (0,0) -- (0,4.7) node[above, font=\scriptsize] {counts (log)};
\draw[acc, very thick] plot[smooth] coordinates
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$$

> **Example (Moving the detector from $45^\circ$ to $90^\circ$).** Since
> $\Delta N \propto \sin^{-4}(\theta/2)$ and all other factors are unchanged,
> $$
> \frac{\Delta N(90^\circ)}{\Delta N(45^\circ)}
> = \frac{\sin^4(22.5^\circ)}{\sin^4(45^\circ)}.
> $$
> Starting from $450$ counts per minute at $45^\circ$, the detector at
> $90^\circ$ registers $450\,\sin^4(22.5^\circ)/\sin^4(45^\circ) \approx 39$
> counts per minute.

## 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:[^tl-size]

$$
\frac{kqQ}{r_d} = \tfrac{1}{2}m_\alpha v^2
\quad\Longrightarrow\quad
r_d = \frac{kqQ}{\tfrac{1}{2}m_\alpha v^2}.
$$

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

> **Example (Upper limit on the gold nucleus).** For $7.7\,\text{MeV}$ alphas on
> gold ($Z=79$),
> $$
> r_d = \frac{(2)(79)(1.44\,\text{eV}\cdot\text{nm})}{7.7 \times 10^6\,\text{eV}}
> = 3 \times 10^{-5}\,\text{nm} = 3 \times 10^{-14}\,\text{m}.
> $$
> The gold nucleus is therefore no larger than about $3 \times 10^{-14}\,\text{m}$,
> roughly $10^{-4}$ of the atomic radius. Rutherford's naturally radioactive
> sources could not supply alphas energetic enough to reach the gold surface, so
> he used lighter targets like aluminum, where the data deviate from the
> $\sin^{-4}$ law and pin the radius to about $10^{-14}\,\text{m}$.

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](/nuclear-physics/nuclear-properties/nuclear-constituents-nuclide-chart).

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](/atomic-physics/early-models-and-old-quantum-theory/bohr-model-hydrogen) takes the nuclear
atom as given and adds the quantum postulates that produce both.

[^tl-spectra]: **Tipler & Llewellyn**, _Modern Physics_, §4-1 — the three classes of spectra (continuous, band, line) and the historical role of spectroscopy.
[^tl-balmer]: **Tipler & Llewellyn**, §4-1, Eq. 4-1 — Balmer's empirical formula for the visible hydrogen lines and the series limit.
[^tl-rydberg]: **Tipler & Llewellyn**, §4-1, Eq. 4-2 — the Rydberg-Ritz formula, the Rydberg constant, and the Ritz combination principle; Example 4-1 for the Lyman and Paschen first lines.
[^tl-thomson]: **Tipler & Llewellyn**, §4-2 — Thomson's model and its two failures: no matching vibration frequencies and radiative instability.
[^tl-rutherford]: **Tipler & Llewellyn**, §4-2 — the identification of alpha particles as doubly ionized helium and the Geiger-Marsden scattering apparatus.
[^tl-geometry]: **Tipler & Llewellyn**, §4-2, Eq. 4-3 — the hyperbolic trajectory and the relation between impact parameter and scattering angle.
[^tl-cross]: **Tipler & Llewellyn**, §4-2, Eqs. 4-4, 4-5 — cross section, number density of foil nuclei, and the scattered fraction; Example 4-2.
[^tl-formula]: **Tipler & Llewellyn**, §4-2, Eq. 4-6 — the full Rutherford angular distribution and its verification by Geiger and Marsden; Example 4-3.
[^tl-size]: **Tipler & Llewellyn**, §4-2, Eq. 4-11 — distance of closest approach as an upper limit on nuclear size; Examples 4-3 through 4-5, and the fermi unit.