Enter a particle, its momentum, an absorber and a thickness. This page runs the same instrument as two of our articles — the mean stopping rate from the Bethe formula, the most probable energy loss from the Landau theory (as given in the Particle Data Group review), the straggling distribution for that case, and, in silicon, the electron-hole pairs it creates per micrometre.
Reference tool
Energy-loss calculator: mean, most probable value and straggling
Your particle, your absorber. Mean rate from the Bethe formula with the density-effect correction, the most probable loss from the Landau theory as given in the PDG review, and the straggling curve for your case.
The calculator needs JavaScript. The reference values below are computed by the same code.
Method, constants and limits
Method. Mean rate: Bethe formula, ⟨−dE/dx⟩ = K z² (Z/A) β⁻² [ ½ ln(2mec²β²γ²Tmax/I²) − β² − δ/2 ], without shell or Barkas terms (they exceed a few percent only below βγ ≈ 0.1). Most probable loss: Δp = ξ [ ln(2mec²β²γ²/I) + ln(ξ/I) + 0.200 − β² − δ ], ξ = (K/2) z² (Z/A) x/β². Regime from κ = ξ/Tmax: Landau below 0.01, Vavilov to 10, Gaussian above. The Landau density is evaluated by direct quadrature of its defining integral, not an approximation. Both formulas and their notation follow the Particle Data Group review Passage of particles through matter.
Electron-hole pairs (silicon only). Pairs per micrometre = energy deposited per micrometre ÷ w, w = 3.62 eV (silicon, room temperature; PDG review / silicon-detector literature, e.g. Knoll, Radiation Detection and Measurement). Two figures are reported, not one, because they answer different questions: the mean rate uses ⟨dE/dx⟩ and does not depend on thickness (≈108/µm for a minimum-ionising particle in silicon); the most probable rate uses Δp at the thickness entered above and does depend on it, because Δp is not linear in thickness — it is what a thin detector actually reads out on a single pass, the same Landau/straggling quantity this tool otherwise reports in keV. Quoting only the mean figure for a thin-detector signal is a common error this tool deliberately avoids.
Constants. I, density, Z/A and δ(βγ) per material from NIST ESTAR (SRD 124); K = 4πNAre²mec² and mec² from CODATA; particle masses from the PDG; w = 3.62 eV for silicon as above — not an ESTAR/XCOM quantity, typed directly in the engine with its source. The Bethe engine reproduces NIST PSTAR proton stopping powers to within 1% from 50 MeV to 10 GeV in every material offered; the deviations are on record in the repository tests.
Limits. Heavy charged particles only (no electrons). βγ from 0.1 to about 2×10⁴. Thin absorbers: when the mean loss exceeds 10% of the kinetic energy the constant-velocity assumption fails and the result says so. The e-h pair figures are silicon-only: no sourced w is offered for any other material.
Reference values: a minimum-ionising muon in silicon
Muon at βγ = 3.5 (369.8 MeV/c), where the mean rate in silicon is at its minimum. Computed by the calculator above. The last two columns are the e-h pair yield per micrometre (w = 3.62 eV): the most-probable column changes with thickness because Δp is not linear in x; the mean column does not.
| Thickness (µm) | Most probable loss Δp (keV) | Mean loss (keV) | FWHM (keV) | Δp / mean | κ | e-h pairs/µm (most probable) | e-h pairs/µm (mean) |
|---|---|---|---|---|---|---|---|
| 50 | 11.31 | 19.33 | 3.876 | 0.585 | 8.0e-5 | 62.48 | 106.8 |
| 100 | 23.96 | 38.65 | 7.751 | 0.62 | 1.6e-4 | 66.17 | 106.8 |
| 150 | 37.11 | 57.98 | 11.63 | 0.64 | 2.4e-4 | 68.34 | 106.8 |
| 200 | 50.58 | 77.3 | 15.5 | 0.654 | 3.2e-4 | 69.87 | 106.8 |
| 300 | 78.22 | 116 | 23.25 | 0.675 | 4.8e-4 | 72.03 | 106.8 |
| 500 | 135.3 | 193.3 | 38.76 | 0.7 | 8.0e-4 | 74.75 | 106.8 |
| 1000 | 284 | 386.5 | 77.51 | 0.735 | 1.6e-3 | 78.44 | 106.8 |
Method: PDG review Passage of particles through matter (Bethe formula, Landau most probable value). Constants: NIST ESTAR silicon (I = 173 eV, ρ = 2.33 g/cm³), CODATA, PDG masses. Pair-creation energy w = 3.62 eV (silicon, room temperature; PDG review / silicon-detector literature) is not an ESTAR/XCOM quantity and is typed directly in the engine.
Minimum ionisation by material
The minimum of the mean rate ⟨−dE/dx⟩ for a singly charged heavy particle, and the βγ at which it occurs.
| Material | ρ (g/cm³) | I (eV) | Z/A | βγ at minimum | ⟨−dE/dx⟩min (MeV cm²/g) | (keV/µm) |
|---|---|---|---|---|---|---|
| Silicon | 2.33 | 173 | 0.4985 | 3.42 | 1.659 | 0.3865 |
| Germanium | 5.323 | 350 | 0.4408 | 3.26 | 1.372 | 0.7301 |
| Carbon (amorphous) | 2 | 81 | 0.4995 | 3.92 | 1.743 | 0.3486 |
| Aluminium | 2.699 | 166 | 0.4818 | 3.42 | 1.611 | 0.4347 |
| Iron | 7.874 | 286 | 0.4656 | 3.44 | 1.45 | 1.142 |
| Copper | 8.96 | 322 | 0.4564 | 3.42 | 1.402 | 1.257 |
| Tungsten | 19.3 | 727 | 0.4025 | 3.16 | 1.144 | 2.207 |
| Lead | 11.35 | 823 | 0.3958 | 3.02 | 1.121 | 1.273 |
| Argon (gas) | 0.001662 | 188 | 0.4506 | 3.16 | 1.519 | 2.524e-4 |
| Water | 1 | 75 | 0.5551 | 4 | 1.998 | 0.1998 |
| Air (dry, sea level) | 0.001205 | 85.7 | 0.4992 | 3.3 | 1.815 | 2.186e-4 |
| Plastic scintillator | 1.032 | 64.7 | 0.5414 | 4.04 | 1.956 | 0.2018 |
| Kapton | 1.42 | 79.6 | 0.5126 | 3.96 | 1.82 | 0.2584 |
| Silicon dioxide | 2.32 | 139.2 | 0.4993 | 3.64 | 1.697 | 0.3937 |
| Concrete (Portland) | 2.3 | 135.2 | 0.5027 | 3.66 | 1.711 | 0.3936 |
| Bone (compact, ICRU) | 1.85 | 91.9 | 0.5301 | 3.92 | 1.849 | 0.3421 |
| Soft tissue (ICRP) | 1 | 72.3 | 0.5512 | 4.02 | 1.985 | 0.1985 |
Bethe formula with the NIST ESTAR density-effect correction; no shell or Barkas terms. Material constants: NIST ESTAR (SRD 124). Checked against NIST PSTAR to within 1% (50 MeV – 10 GeV protons).
Used in
This calculator is embedded, with the same engine and the same numbers, in two articles: The Landau Distribution and Energy Loss For Ionizing Particles, which explains where the most probable loss comes from, and The Straggling Function. Energy Loss Distribution of Charged Particles in Silicon Layers, which works through the silicon-detector case this page’s electron-hole pair figures are built for. Read either one for the physics; use this page for a bare instrument and a link you can share to one exact result.