Mass Attenuation Coefficient Calculator

NIST photon attenuation data as a reference: μ/ρ and μen/ρ for eight materials from 1 keV to 20 MeV, with the linear coefficients, the mean free path, every absorption edge and a derived split of the three interaction processes.

This is narrow-beam attenuation through uniform material: the physics, not a shielding design. Real geometry scatters, and the scattered photons a narrow-beam calculation ignores can add tens of per cent to the dose behind a thick shield — that is what a build-up factor is for, and where a page omits one it says so on the page. A shield arrived at from a figure here has not been designed: shielding is specified by a qualified expert against the regulation that applies, and then verified by measurement.

One coefficient per gram, three processes inside it, and a step where a shell switches on

μ/ρ at six source energies, with μen/ρ and the ratio at caesium-137

Every absorption edge in the eight materials, with the size of the step

Where Compton scattering takes over, and where pair production takes it back

Four coefficients that get mistaken for each other

Two coefficients that get swapped, one step where a shell switches on, and three processes in three regions

There are two coefficients on this page and they answer different questions. μ/ρ counts every interaction that removes a photon from the beam, including a Compton scatter that deflects it by two degrees and carries 99 per cent of its energy onward. μ_en/ρ counts the energy that actually stays behind. In lead at caesium-137 the ratio is 0.539 and in water it is 0.380 — so a water shield passes on 62 per cent of what it takes out of the beam. Transmission, half-value layers and shield thicknesses all want μ/ρ. Absorbed dose, air kerma rate constants and detector response all want μ_en/ρ. Swapping them is a factor of two to three, and the direction of the error is not the one people expect: μ_en/ρ is the SMALLER, so putting it in the exponential under-states how much each centimetre attenuates and asks for too much shielding — in water at caesium-137 it would ask for 2.6 times the thickness actually needed. Expensive rather than dangerous, which is exactly why the mistake survives. The ratio is not a constant you can remember either, and it is not even monotone: in water it is 0.997 at 1.5 keV, bottoms out at 0.141 at 80 keV, and climbs back to 0.76 at 20 MeV.

The K edge is why lead is wonderful at 100 keV and poor at 85. Photoelectric absorption needs the photon to eject a bound electron, and each shell has a threshold. Below 88.0045 keV a photon cannot touch lead’s K shell at all, so the atom is absorbing with its L and M shells only; a few electronvolts above, the most tightly bound shell in the atom switches on and the coefficient QUADRUPLES — 1.910 cm²/g below, 7.683 above, a factor of 4.02 across a step that is essentially vertical. The half-value layer falls from 0.293 mm at 85 keV to 0.084 mm at 90 keV. This is the one place on the whole attenuation curve where a HIGHER photon energy is easier to shield, and it has real consequences: a filtered 150 kV X-ray beam behaves in lead like a monoenergetic 84 keV beam, sitting just under the edge where lead is at its weakest in that band, and tungsten — whose K edge is at 69.5 keV — beats it between 69.5 and 88 keV for exactly that reason. No smooth fit to this data can represent an edge, which is why the lookup behind these pages interpolates between NIST’s tabulated points instead.

Three processes, three regions, and the shape of the curve is the argument. On the left, photoelectric absorption: falling roughly as the cube of the energy and rising as the fourth or fifth power of atomic number, which is where lead is worth fifteen to thirty times its weight in concrete and where the edges live. In the middle, Compton scattering off individual electrons, which depends on the number of electrons per gram and essentially not at all on which element they belong to — so in this region all eight materials converge to within a factor of 1.3 of one another per gram, and lead’s advantage evaporates completely. On the right, pair production, which cannot happen below 1.022 MeV and then grows as the square of the atomic number, so every curve has a minimum and the heavy elements reach theirs first: 4 MeV for lead and tungsten, 8 for iron, and past the end of the table for water. The energy at which Compton takes over is the single most useful number on the page, and it rises from 29 keV in water to 587 keV in lead.

Lead has fewer electrons per gram than water, and that is not a trick. NIST’s ⟨Z/A⟩ column gives lead 0.39575 electrons per nucleon and water 0.55508, so a gram of water holds 40 per cent MORE electrons than a gram of lead. Wherever attenuation is Compton scattering and nothing else, that is the whole story: per unit mass, lead beats water below 1.012 MeV, LOSES to it from 1.012 to 2.471 MeV, and wins again above that as pair production takes over. The often-quoted claim that lead is “20 to 30 per cent better than an equal mass of low-Z material” is true at diagnostic energies and false at cobalt-60. What is never in doubt is the DENSITY: lead is 11.35 g/cm³ against water’s 1.00, so even where it loses per gram it wins per centimetre by a factor of ten. Lead buys space.

The interaction split on this page is derived, not tabulated, and that is stated rather than implied. NIST’s table carries the total μ/ρ and the total μ_en/ρ, not the five partial cross-sections its methodology section sums. So the Compton column here is computed from the Klein–Nishina cross-section per FREE electron — which reproduces the two standard check values, 0.6652 barn in the Thomson limit and 0.2112 barn at 1 MeV — multiplied by NIST’s own ⟨Z/A⟩, and the remainder is whatever is left over. Free electrons are the wrong model at low energy, where binding suppresses incoherent scattering, so below about 100 keV the Compton share printed here is an UPPER bound and the remainder a lower one. Through the Compton region, which is where the split is actually read, it is good to a few per cent. If you need the partial coefficients properly, NIST’s XCOM program publishes them.

What this page is for, and what it is not. It is the reference the other shielding pages on this site read from: a half-value layer, a shield thickness and a layered stack are all this coefficient with a density and an exponential attached. It is also the page to come to when a number elsewhere looks wrong, because it prints the raw quantity rather than a result derived from it. What it is not is a dose calculation, a shield design or a statement about anything being safe: a coefficient is a property of a material, and every number on this page would be the same whether the beam were a millisievert an hour or a thousand. The thicknesses it prints are narrow-beam, like everything in this category, and a real shield needs 11 to 33 per cent more than ln2/μ for the reason the half-value layer page sets out at length.

Frequently asked questions

Related calculators

References

1. J. H. Hubbell and S. M. Seltzer, Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients, NIST Standard Reference Database 126, physics.nist.gov/PhysRefData/XrayMassCoef/ (read 7 October 2026). Every coefficient on these pages comes from here: eight materials, 1 keV to 20 MeV, 369 tabulated rows. A work of the United States Government and therefore free of domestic copyright, which is the reason this vertical exists in the form it does — the alternative sources for the same numbers are copyrighted standards. Three things were checked against it before anything was written: every absorption edge sits at its published energy (lead K at 88.0045 keV, tungsten K at 69.525, iron K at 7.112); μ/ρ at 1 MeV reproduces the published spot value for all eight materials to the last printed digit; and μ_en/ρ is at or below μ/ρ in every one of the 369 rows, as it must be.
2. Table 1: Values of <Z/A>, I and Densities for Elemental Media and Table 2: … for Compounds and Mixtures, NIST X-Ray Mass Attenuation Coefficients (read 7 October 2026). Two things are taken from these. The <Z/A> column, which is what turns a Klein–Nishina cross-section per electron into a coefficient per gram: lead 0.39575, tungsten 0.40250, iron 0.46556, aluminium 0.48181, ordinary concrete 0.50932, dry air 0.49919, ICRU-44 soft tissue 0.54996, water 0.55508. And the density column, which matters because a mass attenuation coefficient is per gram and cannot become a thickness without one: 11.35, 19.30, 7.874, 2.699, 2.300, 1.205×10⁻³, 1.060 and 1.000 g/cm³. Note what the <Z/A> column shows on its own: lead carries 0.396 electrons per nucleon against water’s 0.555, so a gram of lead holds 29 per cent FEWER electrons than a gram of water. Where attenuation is Compton scattering and nothing else, which for these two materials is 1.01 MeV to 2.47 MeV, that makes lead the worse material per gram, and only its density rescues it.
3. X-Ray Mass Attenuation Coefficients — Section 2: Methodology, NIST (read 7 October 2026). Cited for the composition of the total cross-section, which is the structure of the interaction-share figures on the coefficient page: NIST’s equation sums five processes — the photoelectric effect, coherent (Rayleigh) scattering, incoherent (Compton) scattering, electron–positron pair production and triplet production — and notes that photonuclear absorption, which is not in the sum, “can contribute as much as 5 % to 10 % to the total photon interaction cross section in a fairly narrow energy region usually occurring somewhere between 5 MeV and 40 MeV”. It also says that the photoelectric cross-sections were calculated over 1 keV to 1.5 MeV, which is why these pages treat the residual above a couple of MeV as pair production. The partial coefficients themselves are NOT in the table these pages use, and the split printed on the coefficient page is therefore derived rather than tabulated — which the page states.
4. CODATA recommended values of the fundamental physical constants, as adopted in the 2019 SI. Used for three numbers in the Klein–Nishina cross-section on the coefficient page: the classical electron radius r_e = 2.8179403262×10⁻¹³ cm, the electron rest energy m_ec² = 0.510 998 950 MeV, and the Avogadro constant N_A = 6.022 140 76×10²³ mol⁻¹, which is now exact by definition. The cross-section computed from them reproduces the two standard check values: the Thomson limit 8πr_e²/3 = 0.6652 barn as the energy goes to zero, and 0.2112 barn at 1 MeV.
5. Ionizing Radiation — Shielding Layer Examples, Occupational Safety and Health Administration, osha.gov/ionizing-radiation/introduction/shielding-layer-examples (read 7 October 2026). A US Government work and reproduced here. Two rows, six materials-worth of numbers, all in centimetres: caesium-137 at 0.66 MeV, HVL 4.8 concrete / 1.6 steel / 0.7 lead and TVL 15.7 / 5.3 / 2.1; cobalt-60 at 1.17 and 1.33 MeV, HVL 6.6 / 2.1 / 1.2 and TVL 20.8 / 6.9 / 4.0. These are BROAD-BEAM figures and every one of them is larger than ln2/μ from the NIST coefficient. At caesium-137 the excess is 25 per cent for concrete, 33 for steel and 27 for lead; at cobalt-60 it is 27, 28 and 15. The ratio of TVL to HVL in the table is also worth reading: narrow-beam it would be exactly ln10/ln2 = 3.322 for every entry, and here it runs 3.27, 3.31, 3.00, 3.15, 3.29 and 3.33, which is the rounding of one-significant-figure entries rather than physics.
6. Lead shielding thickness — what are the lead codes?, IonActive Consulting radiation protection resource hub (read 7 October 2026). The source for the six BS EN 12588 rolled lead sheet codes and for the practical advice around them: Code 3 is 1.32 mm, Code 4 1.80, Code 5 2.24, Code 6 2.65, Code 7 3.15 and Code 8 3.55, with “a ± 5% tolerance”. Three statements are taken from it directly. Sheet lead codes “tend to be used for relatively low energy applications (i.e. up to about 150 kVp)”. “Where significantly thick lead is required (e.g. 10’s – 100’s mm), lead bricks are often the better option”. And “since lead codes relate to specific thickness dimensions there is often a compromise required, so you may need to use a lead code above what you actually require” — which is the reason the shield-thickness page prints the code at or above the answer rather than the nearest one. It also gives a useful anchor at the diagnostic end: “at 150 kV it is often reported that the TVT is 0.95 mm lead (so practically 1 mm lead)”.
7. Gamma ray, Wikipedia (read 7 October 2026). Read for the shielding section and quoted for one comparison that is worth having because it is in the units people actually think in: “gamma rays that require 1 cm (0.4 inch) of lead to reduce their intensity by 50% will also have their intensity reduced in half by 4.1 cm of granite rock, 6 cm (2.5 inches) of concrete, or 9 cm (3.5 inches) of packed soil”, and for the statement that “a lead (high Z) shield is 20–30% better as a gamma shield than an equal mass of a low-Z shielding material”. Both are consistent with the NIST coefficients at around 1 MeV, with one correction. The energy at which lead’s narrow-beam half-value layer is exactly 1.000 cm is 1.194 MeV, and concrete’s there is 5.07 cm, not 6 — so the quoted pairing is about 18 per cent generous to concrete, which is roughly the size of a build-up correction and may be where it came from. The second statement needs more care still: per unit MASS lead beats water below 1.012 MeV, LOSES to it from 1.012 to 2.471 MeV, and wins again above that as pair production takes over. “20–30% better” is true at diagnostic energies and false at cobalt-60.