Muon radiography · real Geant4
A shape made of lead sits inside a 20 cm block. You can't see it and neither can this page.
But muons — heavy cousins of the electron, made when cosmic rays hit the atmosphere — are falling through you right now, about one per square centimetre per minute. They pass through almost anything, and they get knocked slightly off course by dense material. So measure only how far each one was knocked, never look inside, and the hidden shape draws itself.
Fire muons until you can tell what's in there. Your score is how few you needed — which is exactly how real muon scanners are judged, because muons per picture means hours of waiting in the field.
Hold your nerve: a few thousand muons is pure static. Most shapes become guessable somewhere around 5,000–10,000.
Give it around 10,000 muons before judging the picture — under a few thousand everything looks like static, and that is the honest answer rather than a loading bar. Editing the shape starts the count over, because muons already fired measured the old one.
One number per muon: the angle between the way it was going before, and the way it is going after. That is the whole measurement. Nothing looks inside the block.
Then the picture is built in three steps, repeated for every muon:
After enough muons, every patch holds a pile of angles. Take the typical one — patches where the muons bent hard are drawn bright, patches where they sailed through are dark. That is the image. It is a map of how much stuff is in the way, and nothing else.
A muon crossing matter is knocked sideways by thousands of tiny scatters off atomic nuclei. The spread of that wobble is given by the Highland formula:
Denser, higher-Z material has a shorter radiation length X₀, so it bends muons
harder. Lead's is 0.56 cm against concrete's 11.6 cm — so 8 cm of the hidden lead scatters
about three times as hard as the entire 20 cm block of concrete around it. That contrast
is what you are looking at. Fast muons bend less (the 1/p), which is why a muon's momentum
matters and why this uses a real measured momentum spectrum.
Simulating every shape somebody might draw is impossible. But a muon travels almost straight, so its deflection depends only on what is in its column — lead, or no lead. That means two simulations cover every shape there is: one of plain concrete, one of concrete with lead all the way across. To image a drawing, take a muon, check whether the cell it crossed is filled, and draw its deflection from the matching pool.
Every angle used is still a real Geant4 muon that really crossed that thickness of material. The approximation is at the edges, where an angled muon can clip a filled cell and an empty one and gets credited to only one. That was measured rather than assumed: composed images land within 1–3% of the noise floor that two identical full simulations set for each other, and recover the shape just as well (0.94 against 0.94). The whole thing is 364 KB.
The drawing mode also uses the sea-level cosmic-ray muon spectrum (Guan et al. 2015) rather than the ATLAS one — mean 4.7 GeV instead of 15.8. Real cosmic muons are softer, and since deflection goes as 1/p they scatter about three times harder, so the picture builds faster.
Squared, and typical rather than average. θ² because that is the quantity that adds up along a path. Typical (the median) rather than the average, because the distribution has a long tail — one muon that happens to bounce off a nucleus head-on would otherwise burn a false bright spot into whatever patch it landed in.
Follow the track, do not use the entry point. Muons arrive at all angles. One coming in at 30° crosses the object almost 6 cm from where it entered the block — two patches over. Bin it by where it entered and the picture smears into nothing, which is exactly what happened the first time this was tried.
Geant4 (QBBC physics list) transported every muon through the actual geometry. Their momenta were sampled from muons genuinely measured by the ATLAS experiment at the LHC in run 310341 — which matters, because the deflection goes as 1/p, so the real momentum mix sets the contrast. Nothing here is drawn by hand or fitted to a textbook curve.
The muons were fired in advance rather than while you wait: Geant4 needs 2.2 GB of physics data and about a minute per image. Only three numbers per muon are shipped — where it entered and how far it bent — so the page reshuffles them into a different image every play.
There is no detector. The two trackers in the diagram are where a real apparatus spends its money and its error budget; here every muon's direction is known exactly. A real tracker resolves about 1 mrad, which is comparable to the concrete signal itself — so real images are worse than these, and real exposures longer. A real system also only uses muons that cross both planes, and that acceptance is not modelled either.
This is a 2D shadow: muon radiography, not tomography. There is no depth information. Recovering depth needs muons arriving from many angles plus a maximum-likelihood reconstruction — the straightforward approach fails at it, and the source repo documents that failure rather than hiding it. Real systems work in 3D. This one does not.
The void found inside the Great Pyramid in 2017, locating the melted cores at Fukushima, screening cargo containers for shielded nuclear material, imaging the insides of volcanoes — all with muons that arrive free, from space, everywhere on Earth. At sea level about one crosses every square centimetre every minute, which is why the muon counts here are also quoted as minutes of just standing there.