How Crystals Form: Nucleation, Bravais Lattices & DLA

Q: Why does nucleation happen almost instantaneously above a threshold?

Classical Nucleation Theory shows the nucleation rate depends exponentially on the free energy barrier ΔG*, which itself falls as the inverse square of ln(supersaturation). A two-fold increase in supersaturation can raise the nucleation rate by ten orders of magnitude, which is why clear solutions can cloud almost instantly once disturbed past a critical threshold.

Q: How many possible crystal lattice structures are there?

Exactly 14 Bravais lattices across 7 crystal systems exhaust every possibility for 3D periodic symmetry. Combined with the 32 point groups of rotation and reflection symmetry, they generate exactly 230 space groups — the complete catalogue of 3D crystal symmetry, identified experimentally through X-ray diffraction via Bragg's law.

Q: Why is every snowflake different if they're all hexagonal?

Ice's hexagonal lattice gives every snow crystal six-fold symmetry, and all six arms of a single flake experience the same temperature and humidity at any moment since they are only micrometres apart — so the arms grow identically. But each snowflake follows a unique micro-trajectory of temperature and supersaturation through the cloud, so no two flakes share the same growth history.

Supersaturation and the nucleation barrier

Crystallization occurs when the chemical potential of molecules in solution exceeds that in the crystal phase, driven by supersaturation S = c/c*. Before a crystal can grow, a stable nucleus must form — energetically costly, since new crystal-liquid interface must be created. Classical Nucleation Theory balances the bulk free energy gain against the interfacial cost:

ΔG(r) = −(4/3)πr³·Δg_v + 4πr²·γ

r* = 2γΩ / (kT·ln S)              (critical radius)
ΔG* = (16π/3)·γ³Ω² / (kT·ln S)²    (nucleation barrier)
J = A·exp(−ΔG*/kT)                (nucleation rate)

Because ΔG* depends on ln(S) squared in the denominator, nucleation rate is an extremely sensitive function of supersaturation — a two-fold increase in supersaturation can raise the nucleation rate by ten orders of magnitude, which is why clear solutions can cloud almost instantly once disturbed past threshold.

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Bravais lattices: 14 possibilities, no more

Every crystal is a periodic arrangement of atoms describable by a unit cell — the smallest repeating block, characterised by three edge lengths and three angles. There are exactly 7 crystal systems and 14 Bravais lattices that exhaust all possibilities for 3D periodic symmetry — cubic (NaCl, diamond), hexagonal (quartz, ice, graphite), and five others. Combined with the 32 point groups of rotation and reflection symmetry, they generate exactly 230 space groups, the complete catalogue of 3D crystal symmetry, identified experimentally by X-ray diffraction via Bragg's law: nλ = 2d·sinθ.

Snowflakes and diffusion-limited aggregation

Ice crystallizes in the hexagonal system, so every snow crystal inherits six-fold symmetry — and because all six arms of a falling flake are only micrometres apart, they experience identical temperature and humidity at every instant and grow synchronously. Yet no two snowflakes are alike, because each follows a unique micro-trajectory of temperature and supersaturation through the cloud (the Nakaya diagram maps 41 identified morphologies). At high supersaturation, growth becomes diffusion-limited rather than reaction-limited, and corners and tips grow faster than flat faces — producing the branching, fractal-like arms modelled by diffusion-limited aggregation (DLA): release a random walker, let it wander until it touches the cluster and sticks, repeat. The resulting fractal has dimension D ≈ 1.71 in 2D, and the same mathematics — Laplacian growth — governs mineral dendrites, lightning channels and viscous fingering in a Hele-Shaw cell.

Frequently asked questions

Why does nucleation happen almost instantaneously above a threshold?

Nucleation rate depends exponentially on the free energy barrier, which falls as the inverse square of ln(supersaturation) — a two-fold increase in supersaturation can raise the rate by ten orders of magnitude.

How many possible crystal lattice structures are there?

Exactly 14 Bravais lattices across 7 crystal systems. Combined with the 32 point groups, they generate exactly 230 space groups — the complete catalogue of 3D crystal symmetry.

Why is every snowflake different if they're all hexagonal?

The six arms of one flake grow identically since they share the same instantaneous conditions, but each flake follows a unique temperature/humidity path through the cloud, so no two share a growth history.

Try it live

Everything above runs in your browser — open Crystal Growth and watch random-walk particles stick to a growing fractal crystal across cubic, hexagonal and FCC lattices.

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