Landau distribution calculator
Most probable energy loss, width, mean and probability density for your detector.
| Most probable loss | 83.41 keV |
|---|---|
| Landau FWHM | 21.49 keV |
| Mean | 150.3 keV |
| MPV / mean | 0.5551 |
| Landau scale ξ | 5.348 keV |
| κ | 1.119e-6 |
| Electron-hole pairs (MPV) | 23040 |
Landau regime: the thin-absorber Landau shape applies.
This model is for heavy charged particles. Electrons are excluded. Thickness is the traversed path; shell and Barkas corrections, charge collection and electronic noise are not modelled.
PDG RPP 2024, 34.2.9; Landau (1944); Kölbig and Schorr (1984); Bichsel (1988)
The Landau distribution is the probability distribution of the energy a fast charged particle loses in one pass through a thin layer. It is asymmetric, with a long high-energy tail from rare hard collisions that produce delta rays, so the most probable loss lies well below the mean. Its width is set by the scale ξ.
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Use this in your AI assistant
This calculator is available to Claude, ChatGPT and other assistants through ScienceShot’s free connector.
https://scienceshot.com/mcpConnector tool: landau_distribution
Example: What is the Landau distribution for a 10 GeV/c muon in 300 micrometres of silicon?
The formulas
| The formulas | |
|---|---|
| Δp | ξ [ln(2 m c2 β2 γ2 / I) + ln(ξ / I) + j − β2 − δ(βγ)] |
| ξ | (K/2) (Z/A) z2 x / β2 |
| j; K | 0.200; 0.307075 MeV mol−1 cm2 |
| FWHM w | 4.018 ξ (4.0186 ξ) |
| φ(λ) | (1/π) ∫0∞ exp(−t ln t − λt) sin(πt) dt |
| λpeak | −0.2228 |
Here m is the electron mass, I the mean excitation energy, z the charge number, and x the areal density in g/cm2 (density times path length). Δ, ξ and I must use the same energy unit. Δp is the peak, not the location parameter. The curve ignores escape of delta rays from the sensitive volume.
Reference values
Minimum-ionising muon, βγ = 3.5, momentum 369.803 MeV/c. Losses and Landau FWHM are in keV. The mean is the finite-transfer Bethe estimate, not a moment of the ideal Landau density, whose mean diverges. Widths in the Vavilov regime are Landau estimates.
| Material | Most probable loss (keV) | Landau FWHM (keV) | Mean (keV) | MPV / mean | Regime |
|---|---|---|---|---|---|
| Silicon, 300 µm | 78.19 | 23.24 | 115.9 | 0.6746 | Landau |
| Silicon, 50 µm | 11.30 | 3.874 | 19.32 | 0.5852 | Landau |
| Germanium, 1 mm | 550.7 | 156.6 | 730.4 | 0.7539 | Landau |
| Argon (gas), 1 cm | 1.240 | 0.4998 | 2.528 | 0.4904 | Landau |
| Plastic scintillator, 1 cm | 1675 | 372.9 | 2023 | 0.8282 | Landau |
| Water, 1 cm | 1655 | 370.5 | 2001 | 0.8272 | Landau |
| Lead, 1 mm | 981.5 | 299.8 | 1277 | 0.7685 | Landau |
The standard Landau function
| λ | φ(λ) |
|---|---|
| -3 | 0.00067373 |
| -2 | 0.043985 |
| -1 | 0.15139 |
| -0.22280 | 0.18066 |
| 0 | 0.17885 |
| 1 | 0.14521 |
| 2 | 0.10491 |
| 3 | 0.074248 |
| 5 | 0.039163 |
| 10 | 0.011976 |
| 20 | 0.0030050 |
In ROOT TMath::Landau(x, mpv, sigma), mpv is the location parameter, not the peak: the peak lies 0.22278 sigma below it. Here λ = (Δ – Δp)/ξ – 0.22278 and f(Δ) = φ(λ)/ξ.
When the Landau curve fails
- κ above 0.01: finite maximum energy transfer changes the shape. Use the Vavilov distribution.
- Thick layers, κ above 10: use a Gaussian about the mean, with the finite-transfer variance. If velocity changes appreciably, use stepwise particle transport.
- Very thin layers and gases: atomic binding and discrete collisions matter. Use Bichsel straggling functions or a microscopic collision model; the Landau width is too narrow.
Related reading: The Landau distribution and energy loss; The Bethe-Bloch formula; Energy-loss calculator; Detector signal calculator; Materials library.
Cite this
ScienceShot. Landau distribution calculator. https://scienceshot.com/landau-distribution-calculator.
Sources: PDG RPP 2024, Passage of particles through matter, 34.2.9; L. Landau, J. Phys. USSR 8 (1944) 201; K. S. Kölbig and B. Schorr, Comput. Phys. Commun. 31 (1984) 97; H. Bichsel, Rev. Mod. Phys. 60 (1988) 663.
Reviewed by a physicist with CERN and data-centre industry experience
ScienceShot calculation.
Cite this page
ScienceShot. (2026). Landau distribution calculator [Online calculator]. https://scienceshot.com/landau-distribution-calculator
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author = {{ScienceShot}},
title = {{Landau distribution calculator}},
year = {2026},
note = {Online calculator},
url = {https://scienceshot.com/landau-distribution-calculator}
}