How it is calculated
NIST tabulates, for each material on a fixed grid of energies, the mass stopping power split into its parts and the CSDA range (plus, for ions, the projected range). The calculator interpolates linearly in log(energy) and log(value) between neighboring grid points. Lengths come from the material’s density \(\rho\):
Behind the tables, NIST computes electronic stopping of fast ions from Bethe’s theory with shell, Barkas, and Bloch corrections and the density effect, and at low energies fits experimental data; electron collision stopping uses the Bethe theory of ICRU Report 37 and radiative stopping the bremsstrahlung cross sections of Seltzer and Berger. The key material constant is the mean excitation energy \(I\) of the Bethe formula: 75 eV for water, 85.7 eV for dry air, and 173 eV for silicon in these tables.
Energy after a layer, dose, and the Bragg curve
In the continuous slowing-down approximation the residual range after a layer of thickness \(t\) is \(R(E) - \rho t\), so the exit energy is the energy with that range, found by inverting the range table the same way:
The dose uses the total stopping for ions and only the collision part for electrons (bremsstrahlung photons mostly escape a thin layer). The Bragg curve plots \(S\big(R^{-1}(R(E) - \rho x)\big)\) against depth \(x\): the stopping power each ion has when it reaches that depth, with no straggling, so its peak is sharper than a measured one.
Accuracy of the interpolation: removing each grid point and interpolating it from its neighbors (a grid twice as coarse as the real one) reproduces NIST’s values with a median error of 0.03–0.15 % and at most 1.5 % (protons near 1.5 GeV) in water, silicon, and lead. On the real grid the error is about four times smaller, well inside NIST’s own uncertainty.
Typical values
Interpolated from the NIST tables for the listed materials, CSDA ranges:
| Particle and energy | Material | S/ρ (MeV cm²/g) | Range |
|---|---|---|---|
| Proton, 3 MeV | Silicon | 84.3 | 92.8 µm |
| Proton, 10 MeV | Kapton | 41.8 | 0.949 mm |
| Proton, 100 MeV (therapy) | Water | 7.29 | 77.2 mm |
| Alpha, 5.5 MeV (²⁴¹Am) | Dry air | 712 | 41.9 mm |
| Alpha, 5.5 MeV | Silicon | 580 | 27.8 µm |
| Electron, 1 MeV | Water | 1.862 | 4.37 mm |
| Electron, 2.28 MeV (⁹⁰Y beta maximum) | Aluminum | 1.536 | 5.21 mm |
An alpha particle from a smoke detector stops in about 4 cm of air or a sheet of paper; a 100 MeV proton crosses 7.7 cm of tissue; a 1 MeV electron travels a 4.4 mm path in water but, because it scatters, rarely gets deeper than about 4 mm.
Worked example
The default inputs: 100 MeV protons entering liquid water (ρ = 1.000 g/cm³), a 50 mm layer, and a fluence of 10⁹ cm⁻².
- PSTAR gives S/ρ = 7.289 MeV·cm²/g at 100 MeV (a grid point, so no interpolation): 7.289 MeV/cm, or 0.729 keV/µm. The nuclear part is 0.04 % of the total.
- The CSDA range is 7.718 g/cm², so 77.18 mm or 77 180 µm of water; the mean depth (projected range) is 77.07 mm.
- After 50 mm the residual range is 7.718 − 5.000 = 2.718 g/cm², which belongs to an energy of 55.8 MeV: the layer absorbed 44.2 MeV per proton, more than its entrance stopping power times its thickness (36.4 MeV), because the protons slow down and lose energy faster as they go.
- The entrance dose is 1.602 × 10⁻¹⁰ × 10⁹ × 7.289 = 1.17 Gy. The Bragg curve below shows the stopping power climbing steeply over the last few millimeters before 77.2 mm.
Data used
The tables are NIST Standard Reference Database 124 (ESTAR, PSTAR, and ASTAR, version 2.0.1), retrieved on 2026-11-06 with NIST’s default energy grids: 132 energies from 1 keV to 10 GeV for protons, 121 from 1 keV to 1 GeV for alphas, and 81 from 10 keV to 1 GeV for electrons. This page does not reproduce the full database: it carries NIST’s tables only for the materials in its menus (21 elements and 14 compounds and mixtures chosen as the ones most often needed for beams, detectors, windows, and dosimetry, of the 74 materials NIST tabulates for ions and 279 for electrons), and every result is interpolated from NIST’s tables for those materials. The page loads only the table for the particle you choose (60–130 kB, less when compressed), with only the columns used here, and NIST’s four significant figures unchanged. Densities and mean excitation energies are NIST’s, separately for the ion and electron sets. For any other material, or for a table at your own energies, use NIST’s programs directly.
NIST Standard Reference Data. Copyright protection on this compilation of data has been secured by the Secretary of the U.S. Department of Commerce on behalf of the United States in the United States and all countries that are parties to the Universal Copyright Convention, pursuant to Section 290(e) of Title 15 of the United States Code. NIST makes no warranties for the data and is not liable for damage that may result from errors or omissions in them.
Assumptions and limits
- Mean values only. Real particles straggle: the energy loss and range vary from particle to particle by about 1 % of the range for protons in water and far more for electrons, which also scatter so much that their CSDA range is a path length, not a depth.
- CSDA. The exit energy assumes all energy is lost continuously and every particle survives. Nuclear reactions remove about 1 % of protons per centimeter of water at 100 MeV, and an electron can lose a large fraction of its energy in one bremsstrahlung event.
- Uncertainty of the tables. NIST quotes ICRU’s estimates. Protons and alphas: 1–2 % for elements and 1–4 % for compounds at high energies, 2–5 % at 1 MeV, 5–10 % at 100 keV, 10–15 % at 10 keV, and at least 20–30 % at 1 keV. Electron collision stopping: 1–2 % above 100 keV, and 2–3 % (low Z) to 5–10 % (high Z) from 10 to 100 keV; radiative stopping 2–5 %. NIST also notes that the values are accurate to three significant figures, with the fourth given for interpolation.
- Materials. Compounds use NIST’s fixed compositions; for compounds without measurements NIST combines the elements by Bragg additivity. Changing the density keeps the tabulated density-effect correction, which for electrons above about 1 MeV differs between a gas and a condensed material.
- Other ions. For a heavier ion at the same speed, a rough estimate is S ≈ z²eff × Sproton(E/A) with an effective charge below z at low speed; use a dedicated heavy-ion code for accurate values.
- The range is where the particle stops in the material, not where it stops ionizing a detector: detector windows, dead layers, and gas gaps add to the material in the path. For beam trajectories through electrode and magnet fields before the target, use the Ion Optics Workbench.
References
Primary sources and standards
- M. J. Berger, J. S. Coursey, M. A. Zucker, and J. Chang, ESTAR, PSTAR, and ASTAR: Computer Programs for Calculating Stopping-Power and Range Tables for Electrons, Protons, and Helium Ions (version 2.0.1), NIST Standard Reference Database 124, National Institute of Standards and Technology, Gaithersburg, MD, doi:10.18434/T4NC7P, physics.nist.gov/PhysRefData/Star [2026, November 6]. Originally published as M. J. Berger, NISTIR 4999, National Institute of Standards and Technology, Gaithersburg, MD (1993) (every stopping power, range, density, and I value used here).
- ICRU Report 49, Stopping Powers and Ranges for Protons and Alpha Particles, International Commission on Radiation Units and Measurements, Bethesda, MD (1993) (the method behind PSTAR and ASTAR).
- ICRU Report 37, Stopping Powers for Electrons and Positrons, International Commission on Radiation Units and Measurements, Bethesda, MD (1984) (the method behind ESTAR).
- ICRU Report 90, “Key data for ionizing-radiation dosimetry: Measurement standards and applications,” J. ICRU 14(1) (2014), doi:10.1093/jicru/ndw043 (the key dosimetric data NIST added in version 2.0.0, April 2017).
- S. M. Seltzer and M. J. Berger, “Bremsstrahlung energy spectra from electrons with kinetic energy 1 keV–10 GeV incident on screened nuclei and orbital electrons of neutral atoms with Z = 1–100,” At. Data Nucl. Data Tables 35, 345–418 (1986), doi:10.1016/0092-640X(86)90014-8 (the radiative stopping powers).
Free introductions
- D. E. Groom and S. R. Klein, “Passage of particles through matter,” in S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024), pdg.lbl.gov (the Bethe equation, its corrections, the CSDA range, minimum ionization, and radiation length).
- NIST, “Stopping-Power & Range Tables for Electrons, Protons, and Helium Ions,” nist.gov (the programs, their method and uncertainty notes, and the tables for any energy).