The 1944 Landau equation modern detectors quietly outgrew
Every particle physicist learns the same story in their first detector course: when a charged particle rips through matter, the energy it dumps follows the Landau distribution — that lopsided curve with the long tail stretching to the right. It is elegant, it is famous, and for a thin silicon sensor it is wrong.
Not “approximately right with caveats.” Measurably, structurally wrong. Strap a modern 50-micrometre silicon detector into a test beam and the spectrum you record does not match Landau’s 1944 formula — the peak sits in the wrong place and the width is off. The Landau distribution is a beautiful approximation that modern thin detectors have outgrown, and the fix — Hans Bichsel’s first-principles straggling function — is the difference between a calibrated tracker and garbage particle identification. I spent years at CERN staring at exactly these spectra, so let me show you where the textbook breaks.
The mean energy loss is a number your detector never shows you
Start with what most people remember: the Bethe-Bloch formula, which gives the mean energy a particle loses per unit length. For silicon at the minimum-ionizing point it predicts about 1.664 MeV·cm²/g (Particle Data Group), which for a 300-micrometre sensor works out to roughly 116 keV of deposited energy.
Here is the catch: your detector almost never records 116 keV. The most probable energy loss — the value at the peak, the one you actually measure event after event — is only about 82 keV. That 34 keV gap is not noise or miscalibration. It is the whole personality of ionization straggling: a handful of violent collisions fling the mean far out into a long high-energy tail, while the vast majority of events pile up near the peak.
In practical terms that 82 keV peak liberates roughly 22,000 electron-hole pairs in silicon, since it costs just 3.6 eV to create each one. When I first wrote up how the Landau distribution describes ionizing particles, this asymmetry was the entire point — the mean is the one statistic a real tracker should almost never trust.
What lives in that long tail has a name: delta rays. Every so often the incoming particle transfers a large chunk of energy to a single atomic electron, knocking it free with enough punch to ionize on its own. Those rare, violent hits are what haul the mean upward, and in a thin sensor a single delta ray can dominate an entire event. That is the physical reason averaging is the wrong instinct — you are letting a few outliers set your calibration.
Source: Particle Data Group, dE/dx|min = 1.664 MeV·cm²/g in silicon; ρ = 2.33 g/cm³.
Landau’s three convenient fictions
Lev Landau derived his distribution in 1944 by making assumptions that were reasonable for the absorbers of his day but quietly fail in thin silicon. First, he let the maximum energy transferable in a single collision run to infinity. Second, he treated atomic electrons as free, ignoring their binding. Third, he assumed enough collisions that the statistics smooth into a universal shape.
In a thick slab of matter those approximations barely matter. In a 300-micrometre wafer they start to bite, and in the thin silicon detectors of just tens of micrometres now common they fail outright — there are too few collisions, and the binding energies of silicon’s atomic shells distort exactly the region near the peak. The measured width comes out around 25 keV, noticeably broader than the pure Landau prediction. I dug into this failure mode in my piece on the straggling function in silicon layers.
The universal Landau curve has a fixed shape: its peak sits at a scaled value of λ ≈ −0.22, and its entire width scales with a single parameter ξ, about 5.34 keV for 300 micrometres of silicon. That rigidity is its charm and its downfall. A one-parameter curve simply cannot know about silicon’s specific electronic structure — and for a 300-micrometre sensor the shape only settles into a clean Landau form for protons above roughly 550 MeV/c. Go thinner or slower, and the real distribution wanders off on its own.
Physicists knew Landau was incomplete long before silicon trackers existed. Pyotr Vavilov generalized the theory in 1957 to handle the finite maximum energy transfer that Landau had blithely sent to infinity, and the Vavilov distribution bridges the thick-absorber Landau limit and the thin-absorber Gaussian limit. It was real progress. But even Vavilov still treats the atomic electrons as free — the binding problem that matters most in thin silicon survived untouched until Bichsel came along.
Source: straggling (Landau) function, Moyal form, scaled to Δp = 82 keV and ξ = 5.34 keV for 300 µm silicon; mean from Bethe-Bloch.
Bichsel rebuilt energy loss from first principles
Hans Bichsel’s answer was to throw out the shortcuts. Instead of assuming free electrons and an unbounded energy transfer, he computed the straggling function from silicon’s real dielectric response — its measured photoabsorption cross sections and generalized oscillator strengths — including the atomic shell corrections Landau ignored (see the shell-correction analysis).
The payoff is accuracy across a wide range of thicknesses and particle velocities. Bichsel’s model reproduces measured silicon spectra where Landau cannot, which is why the Particle Data Group now publishes Bichsel straggling functions rather than the raw Landau curve. It even agrees with Geant4’s photoabsorption-ionization model while running about three times faster — accuracy and speed at once, which almost never happens in detector simulation.
⚡ PHOTON’S TAKE
Physics loves a clean formula, and Landau’s is gorgeous — which is exactly why we keep teaching a 1944 approximation as if it were gospel. It isn’t. Every time I see a thin-silicon spectrum fit blindly to a Landau curve I wince, because the peak lands in the wrong place and the calibration inherits the error. Bichsel did the unglamorous work of computing energy loss from silicon’s actual electronic structure, and he was right. Use his straggling function. Your particle-ID will thank you, and so will your future self.
Why this gap widens every single year
Here is the part that should make any detector nerd sit up: sensors are getting thinner, not thicker. The High-Luminosity LHC upgrades, CMOS monolithic active pixel sensors, and the timing layers going into every major experiment all push toward silicon in the 50-to-150-micrometre range — precisely the regime where the Landau distribution is least trustworthy.
That matters because energy loss per length is a workhorse for particle identification: if your predicted peak is off by a few keV, your particle-ID separation and your energy calibration drift with it. As I argued when I wrote about how silicon sensors are evolving for future colliders, the hardware is being reinvented — and the physics we use to interpret it has to keep pace. The lesson is simple and a little humbling: the most famous curve in detector physics is only a first draft, and the quiet work of people like Bichsel is what makes the next generation of experiments actually count.
