Lichtenberg Figures: Lightning Frozen in Acrylic

From Dust Figures to Captured Lightning

Lichtenberg figures are the branching patterns traced out when a high-voltage discharge propagates through or across an insulator. They are named for Georg Christoph Lichtenberg (1742–1799), who in 1777 sprinkled charged powder onto an insulating resin plate and found that it settled into radiating, star-like figures marking the electrostatic charge he had deposited with an electrophorus.

Lichtenberg noticed that positive and negative charge left different marks — positive figures are long, sharply branched and star-like, while negative figures are rounder, smaller and more blurred. That asymmetry made the dust figure the first practical way to read the sign of a static charge, and it is a direct ancestor of xerography: the photocopier and laser printer develop a latent electrostatic image with charged toner in exactly this spirit.

The modern spectacle — a clear acrylic block holding a three-dimensional fern of glittering cracks — is the same physics scaled up enormously. Instead of dust settling on a surface, the charge is buried inside the solid by a particle accelerator, and its escape is violent enough to carve channels through the plastic. The result is fairly called 'captured lightning,' a genuine if miniature cousin of the atmospheric kind, and the craft was popularized by engineer Bert Hickman using an industrial electron beam.

Charging the Block: MeV Electrons and a Buried Space Charge

To make a bulk figure, a slab of poly(methyl methacrylate) — PMMA, sold as acrylic or Plexiglas — is placed in the beam of an electron accelerator, typically a linac or Dynamitron delivering electrons of about 1–10 MeV (around 5 MeV is common). PMMA is an excellent insulator, so the electrons that plunge into it have nowhere to flow.

An energetic electron does not stop at the surface. It slows by ionizing and exciting molecules along its path (collisional stopping) and by radiating bremsstrahlung, and its practical range in PMMA (density ~1.18 g/cm³) is a couple of centimetres for a few-MeV beam. The electrons therefore pile up in a buried layer roughly 1–2 cm below the irradiated face, forming a sheet of trapped negative space charge.

This buried sheet and the grounded surroundings behave like the plates of a charged capacitor whose dielectric is the acrylic itself. As the beam keeps delivering charge, the trapped surface charge density σ climbs, and with it the internal field, E ≈ σ/(ε₀εᵣ), where εᵣ ≈ 3 for PMMA. The block is being pumped toward a stored potential of hundreds of kilovolts — a loaded spring, waiting for a release.

The Trigger: Field Enhancement, Avalanche, and Streamers

Breakdown occurs when the internal field reaches the dielectric strength of the acrylic, about 20 MV/m (20 kV/mm). It can be set off on demand by touching a grounded metal point to the block. A sharp tip concentrates the field — the field at a conductor scales inversely with its radius of curvature — so the point creates a local region where E exceeds the threshold and a conducting path can nucleate.

From that seed, a free electron accelerated by the field ionizes a molecule, freeing more electrons that ionize still more: a Townsend avalanche. When the avalanche's own space charge distorts the applied field enough to make it self-propagating, it becomes a streamer — a thin, weakly ionized plasma filament advancing at roughly 10⁵–10⁶ m/s that extends the conducting channel into the charged interior. Because the channel is a conductor, its far tip carries essentially the full potential of the trapped charge, so the field at the tip stays high and the streamer keeps racing forward, draining the block in microseconds.

Inside each channel the current density is enormous for that brief instant. Ohmic (Joule) heating drives the filament to plasma temperatures of order 10⁴ K, flash-vaporizing and thermally crazing a hollow micro-tube through the PMMA and producing the sharp crack and bright flash that accompany the event. When it is over, the vaporized and stress-fractured channels remain as a permanent, light-scattering three-dimensional record — the fern you see.

Why It Branches: Laplacian Growth and the Dielectric Breakdown Model

The signature of a Lichtenberg figure is that channels repeatedly split, always at the tips, into ever-finer branches, and the pattern looks statistically the same at every magnification: it is a fractal. The reason lies in how the electric potential is distributed in the still-charged, undischarged plastic ahead of the growing tree.

In that insulating region the potential φ obeys Laplace's equation, ∇²φ = 0, with the conducting tree held at one potential and the trapped charge at another. The tree screens the field behind it and crowds the field lines onto its most protruding tips. A tip that happens to get slightly ahead therefore sees a higher field and grows faster still — a self-reinforcing tip instability. When a single tip's field becomes high enough, it splits, because two diverging fingers relieve the field more efficiently than one; randomness in the medium selects among nearly-equal candidate tips, which is why the tree is irregular yet self-similar.

This is captured quantitatively by the dielectric breakdown model of Niemeyer, Pietronero and Wiesmann (1984), a stochastic Laplacian-growth model in which the probability that the discharge advances to a given adjacent site is proportional to the local field raised to a power, P ∝ |E|^η. With η ≈ 1 the model reproduces real figures, giving a fractal dimension near 1.7 in two dimensions. It is mathematically kin to diffusion-limited aggregation (Witten and Sander, 1981) and to the whole family of Laplacian-growth instabilities — viscous fingering in a Hele-Shaw cell, the dendritic solidification of a crystal, and electrodeposition — all of which grow the same branched, fractal fronts for the same underlying reason.

The Numbers: Fields, Stored Energy, and Fractal Dimension

A few representative numbers make the mechanism concrete. The trapped surface charge density needed to reach breakdown, from σ = ε₀εᵣE with E ≈ 20 MV/m, is of order 10⁻⁴ C/m² (~0.05 µC/cm²). Across a ~1 cm buried layer that corresponds to an internal potential of a few hundred kilovolts, climbing toward a megavolt in thick, heavily dosed blocks.

The stored electrostatic energy density, u = ½εE², is only a few kilojoules per cubic metre, so a fist-sized block holds of order a joule. That is modest energy — but dumped in microseconds it is a kilowatt-to-megawatt burst, which is why the discharge cracks the air and flashes.

The fractal dimension is the pattern's fingerprint. Flat, surface Lichtenberg figures sit around D ≈ 1.7, comfortably below the value of 2 that a space-filling stain would have; fully three-dimensional bulk figures fill more of their volume and run higher, roughly D ≈ 2.3–2.5. These values depend weakly on how fast the block was charged, the field strength, and defects in the material — the same knobs that tune η in the breakdown model — which is why no two figures are ever quite identical, even from the same apparatus.

On Skin, in Cables, and in the Ground

The name also attaches to marks that appear on people struck by lightning: reddish, feathery, fern-like patterns on the skin, sometimes called keraunographic markings or 'lightning flowers.' Crucially, these are not burns and do not trace current flowing through the body's core. They are thought to arise from a superficial flashover of charge across the wet skin, and from the transient rupture or dilation of small blood vessels along that path. They are essentially painless and fade within hours to a couple of days, which makes them a useful but fleeting diagnostic sign of a strike.

The same branching discharge is a slow, destructive nuisance in high-voltage engineering. Electrical treeing in polymer cable insulation — for example cross-linked polyethylene — is a Lichtenberg figure grown over months or years by repeated partial discharges at a defect or a water inclusion; each tiny discharge erodes a little more channel until the tree bridges the insulation and the cable fails. Detecting and interpreting these trees is a standard part of HV asset diagnostics.

Nature makes larger ones too. Lightning striking sandy ground can fuse a branched glassy tube called a fulgurite — related, but a melt cast of the current path rather than a true fractal discharge figure — and scorched figures are sometimes found across turf and soil around a strike point. In every case the recipe is the same: store charge faster than an insulator can leak it away, exceed the dielectric strength, and let the escaping current find the highest-field path, splitting wherever the field piles up.

Lichtenberg-type discharges and their cousins

Phenomenon	Medium	Timescale	Character
Captured lightning (acrylic)	PMMA bulk, buried e⁻ charge	µs discharge	3D fractal fern, D ≈ 2.5, permanent
Natural lightning	Air, cloud-to-ground	~ms	Branched plasma channel, D ≈ 1.7, transient
Skin Lichtenberg figures	Human skin (surface)	Fades in hours–days	Feathery reddening, not a burn
Electrical treeing	HV polymer cable insulation	Months–years	Slow partial-discharge fractal, a failure mode
Fulgurite	Sand / soil	~ms strike	Fused glass tube — a melt cast, not a fractal discharge