Half Value Layer and Transmission Calculator

The half-value layer, the tenth-value layer and the transmission through any thickness, from NIST photon attenuation data for eight materials — with the published broad-beam figure printed beside the calculated one, because build-up makes them different numbers.

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 exponential, two named thicknesses, and the correction that is not in it

The number this page prints against the number a shielding table prints

Narrow-beam half-value layers: eight materials, six sources

A second published source, and where it disagrees with the first

What n half-value layers actually buys

Narrow beam against broad beam, half-value layers instead of millimetres, and why the tables are read in log space

The one thing to take from this page: the half-value layer it prints is smaller than the one in a shielding table, and the difference is build-up. ln2/μ is the half-value layer of a pencil beam passing through a uniform slab with every scattered photon counted as lost. A real shield is a wall with a room on the other side of it, and photons that Compton-scatter inside the wall arrive at the far face travelling in directions the beam never pointed in. Published half-value layers include those photons and the ones on this page do not, which is why every published figure for lead, concrete, iron and water at caesium-137 and cobalt-60 is 11 to 33 per cent larger. The table above prints both columns and the ratio between them. Lead at caesium-137 is 5.50 mm here and 6.5 mm in a shielding table; if you take the first number to a supplier you will have ordered a shield a sixth too thin, and the error is in the direction nobody notices until a survey meter says so.

Count half-value layers, not millimetres. The transmission is an exponential and nobody has a feel for one. The number of half-value layers is n = x/HVL and the answer is 2⁻ⁿ, which anybody can do: one layer leaves 50 per cent, two leave 25, four leave 6.25, ten leave 0.098 and twenty leave a millionth. Every figure on this page is that, with a material and a density attached. It is also the sanity check — if the calculator says one thing and 2⁻ⁿ says another, one of the inputs is in the wrong unit — and it is why the thickness in half-value layers is the first row of the results rather than buried in them. The tenth-value layer is the same idea in decades: ln10/μ, exactly 3.3219 half-value layers, and the unit shielding people actually talk in because a design requirement is usually a factor of a thousand rather than a factor of eight.

Why the tables are read in log-log space, which is not a detail. NIST tabulates μ/ρ at a set of energies and anything in between has to be interpolated. The curve is steeply convex on linear axes, so a straight line drawn between two tabulated points sits ABOVE the real curve, and the interpolated coefficient comes out too high — which makes the half-value layer come out too LOW. For lead at 662 keV, between the 600 and 800 keV rows, the linear reading gives μ/ρ = 0.113609 and a half-value layer of 5.375 mm; the log-log reading gives 0.111050 and 5.499 mm. Two and a third per cent, in the unsafe direction, and both answers are entirely plausible for lead at caesium. This page interpolates the logarithm of the coefficient against the logarithm of the energy, which is how the data is meant to be read, and prints what the linear reading would have added so the size of the trap is visible.

Lead is the default for its atomic number, not its density, and that stops being true above about 1 MeV. Below a few hundred keV attenuation in high-Z material is dominated by the photoelectric effect, whose cross-section climbs roughly as the fourth or fifth power of Z, so a gram of lead is worth many grams of concrete. Around 1 MeV almost all of the attenuation is Compton scattering, which depends on the number of ELECTRONS per gram and on nothing else — and 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. The consequence is sharp: per unit mass, lead beats water below 1.012 MeV, loses to it between 1.012 and 2.471 MeV, and wins again above that as pair production takes over. What never stops being true is the density. At caesium-137, concrete needs seven times the THICKNESS of lead for the same attenuation but only 41 per cent more MASS. Lead buys space, not weight, and that is usually the thing in short supply.

What this page cannot do, said plainly. It models one uniform slab in a parallel beam, so it has no build-up factor, no scatter, no leakage round the shield, no door, no duct and no floor. It takes one photon energy, so it is a convention rather than a calculation for iridium-192’s nine lines and it cannot see beam hardening — the spectrum that survives a thick shield is harder than the one that entered, so the effective half-value layer grows with depth and a single-energy answer is increasingly optimistic. It says nothing about the bremsstrahlung a beta emitter makes in its own shield, which is the reason a yttrium-90 source wants perspex before lead rather than lead alone. And it renders no verdict: there is no thickness on this page that is adequate, compliant or permitted, because adequacy is a dose limit, an occupancy factor and a regulator, and none of those is a coefficient.

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. 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.
4. J. Eakins, An MCNP-4C2 Determination of Gamma Source Shielding, HPA-RPD-030, Health Protection Agency, Centre for Radiation, Chemical and Environmental Hazards, Radiation Protection Division (September 2007; read 7 October 2026). Crown copyright, quoted briefly. It is a Monte Carlo calculation done explicitly for BROAD-beam geometry — its stated aim is transmission “corresponding to a source that is considered plane parallel and effectively infinite in extent” — which makes it the cleanest available statement of how much a narrow-beam answer under-states. For 511 keV photons it gives lead a half-value thickness and tenth-value thickness of “approximately 5 mm and 17 mm”, against ln2/μ = 3.90 mm and ln10/μ = 12.96 mm from the NIST coefficient: ratios of 1.28 and 1.31 — larger than any of the shielding-table ratios, which is the point. A build-up factor belongs to a GEOMETRY and not to a material, and an infinite plane source is a harsher geometry than the collimated room a shielding table has in mind. The report’s iron figures are read off a plotted curve rather than tabulated and are used here only for that qualitative statement: “approximately 21 mm and 55 mm” against ln2/μ = 10.57 and ln10/μ = 35.12 give ratios of 1.99 and 1.57, and a pair that inconsistent cannot both be right.
5. V. Steiner, A. Malki, T. Ben-Yehuda and M. Moinester, Concrete and Lead Shielding Requirements for PET Facilities, arXiv:2407.12991 (read 7 October 2026). The one source found that MEASURES the narrow-beam and wide-beam cases side by side on the same wall, which is what makes it worth citing above any table. Through 20 cm of Israeli B30 concrete (density 2.30 g/cm³) at 511 keV, the narrow-beam transmission was “T = (3.0±1.0)%, corresponding to a B = 1.4±0.5 buildup factor” and the wide-beam transmission “T = (8.8±1.8)%, corresponding to a buildup factor B = 4.0±0.8, consistent with Monte Carlo (MC) simulations B = 3.9±0.6”. It uses μ_m = 0.0833 cm²/g, μ = 0.196 cm⁻¹ and a mean free path of 5.10 cm, where NIST’s ordinary concrete interpolates to 0.08831 cm²/g — 6.0 per cent apart, which is two different concretes rather than an error. It also states that “a 3.3 cm lead wall results in <0.5% transmission”, which the NIST coefficient reproduces at 0.29 per cent.
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. A. Pearce, NPL Report IR 6: Recommended Nuclear Decay Data, National Physical Laboratory. Cited, not reproduced — Crown copyright. Used here through `_nuclide_data.py` for the gamma line energies and emission probabilities behind the energy presets, and for the two numbers that show why a single-energy preset is a convention: iridium-192’s nine lines run from 205.8 to 612.5 keV and sum to 2.135 photons per decay, giving a yield-weighted mean of 371.6 keV and an energy-weighted mean of 397.8; cobalt-60’s two lines at 1173.2 and 1332.5 keV, at essentially one photon each, give a yield-weighted mean of 1252.9 keV against the conventional 1.25 MeV.
8. 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.