Where Colour Comes From

Ruby and emerald contain the same coloured ion. The difference between them is one number — and that number is predicted by an algebra of four-object recoupling whose central symbol is a tetrahedron.

The observation
Why this covers most of the colours you see
The algebra behind it
1. Why it is a tetrahedron
Does it actually predict the number?
The Tanabe–Sugano diagram
What this does and does not show
Further reading

The observation

Ruby is red. Emerald is green. Both get their colour from the same impurity: a Cr³⁺ ion, sitting in a lattice of otherwise colourless host material — Al₂O₃ for ruby, beryl for emerald. Remove the chromium and both are clear.

The same ion produces two different colours because the surroundings differ. This is not a chemical difference — the chromium is in the same oxidation state, with the same three d-electrons, in both. It is a difference in how strongly the neighbouring oxygens split the d-orbital energies.

That splitting has a name, 10Dq, and it is essentially the only parameter that changes between the two gemstones.

	10Dq (cm⁻¹)	Broad absorption	Colour transmitted
Ruby (Cr³⁺ in Al₂O₃)	≈ 18,000	≈ 556 nm — green	red
Emerald (Cr³⁺ in beryl)	≈ 16,300	≈ 613 nm — orange	green

A 9% change in one number moves the absorption band by 57 nm, and that is the whole difference between the two most famous gemstones in the world.

Why this covers most of the colours you see

Transition-metal d-electron transitions, and the closely related charge-transfer transitions, account for the colour of a remarkable fraction of the inorganic world:

Gemstones — ruby, emerald, sapphire (Fe/Ti charge transfer), alexandrite, turquoise, peridot, garnet
Pigments — cobalt blue, chrome green, cadmium yellow, Prussian blue, ultramarine, the ochres and umbers
Biology — the red of haemoglobin and the green of chlorophyll are porphyrin π→π* transitions modulated by a central metal (Fe, Mg); the blue of haemocyanin is Cu
Glass and glaze — essentially all traditional colouring is a transition-metal oxide
Rust, verdigris, patina — the visible chemistry of weathering

The exceptions are real but bounded: organic dyes work by extended π conjugation, and structural colour (butterfly wings, opal, peacock feathers) is interference rather than absorption.

The algebra behind it

Here is the part that connects to everything else on this site.

A d³ ion like Cr³⁺ has three electrons distributed over five d-orbitals. The question “what are the allowed energy levels?” is a question about coupling angular momenta — each electron carries orbital and spin angular momentum, and they must be combined into total states.

Combining two angular momenta is the Clebsch–Gordan problem, solved in the 1930s. Combining three is still unambiguous. But as soon as you ask how the answer changes when you couple them in a different order, you need a new object: the amplitude relating one coupling scheme to another.

That amplitude is Wigner’s 6j symbol, and Racah built the systematic theory of atomic spectra on it between 1942 and 1949.

Why it is a tetrahedron

A 6j symbol has six arguments. They are not six independent things — they are the six edges of a tetrahedron, whose four faces are the four triangle conditions the arguments must satisfy:

        j₁ ────── j₂
         │ ╲    ╱ │
         │  j₁₂  │        six edges  = six arguments
         │ ╱    ╲ │        four faces = four triangle conditions
        j₃ ────── j

The 6j symbol’s 24 symmetries are exactly the symmetries of the tetrahedron. This is why the same object appears in Ponzano–Regge quantum gravity as the amplitude for a spacetime tetrahedron, and in Racah’s theory as the amplitude for recoupling four angular momenta. It is the same symbol.

Four is the threshold. Two objects combine trivially. Three combine associatively, with no choice to make. Four is where recoupling becomes a real question with a non-trivial answer — and where a tetrahedron appears to carry it.

Does it actually predict the number?

Yes, and this is checkable in a few lines.

For a d³ ion, the ²E state — the one responsible for ruby’s sharp red fluorescence line, the transition that made the first laser work — has a term energy given in the strong-field limit by

\[E(^2E) = 9B + 3C - \frac{24B^2}{10Dq}\]

where B and C are the Racah parameters: two numbers that summarise all the electron–electron repulsion within the d-shell. They are combinations of Slater–Condon integrals, and the reason there are exactly two of them is a group-theoretic fact about SO(3) recoupling.

Using published values for Cr³⁺ in Al₂O₃ (B = 640, C = 3250, 10Dq = 18,000 cm⁻¹):

\[E(^2E) = 14{,}964 \text{ cm}^{-1} \quad\Rightarrow\quad 668 \text{ nm}\]

The observed ruby R₁ line is at 14,403 cm⁻¹, or 694 nm — an error of 3.9%, from an algebraic formula with two fitted parameters and no wavefunction calculation at all.

The Tanabe–Sugano diagram

Tanabe and Sugano’s 1954 diagrams plot every term energy of a d^n ion against 10Dq/B. They remain in every inorganic chemistry textbook, and they are pure recoupling algebra — no Hamiltonian is solved, no integral is evaluated numerically.

One feature deserves attention. Most terms slope steeply with 10Dq: change the ligand field and the energy moves. But the d³ ²E term is almost flat. Its energy barely depends on the crystal field at all, which is why ruby’s R-line is sharp where the other absorptions are broad — and why it works as a laser transition.

That flatness is visible in the formula above: 10Dq enters only through a small correction term, while B and C dominate. The geometry of the diagram encodes which spectroscopic features will be sharp and which broad, before any computation.

What this does and does not show

What it shows. A substantial part of visible chemistry is governed by recoupling algebra rather than by solving the Schrödinger equation. The allowed levels, the selection rules, the sharp-versus-broad character of each transition, and the term energies to a few percent all follow from group theory plus two empirical parameters.

What it does not show. The Racah parameters B and C are fitted to experiment, not derived. Predicting them from first principles requires real integrals over real wavefunctions — the Hamiltonian half of chemistry. So the algebra tells you the structure of the spectrum and gets the numbers close; it does not give you the numbers for free.

This is the general pattern set out in Diagrammatic Chemistry: symmetry fixes which states exist and how they are labelled, and a computation is still needed to say where they sit.