What is the half-value layer (HVL)?

The HVL is the thickness of a material that reduces the intensity of a narrow photon beam to half its initial value. It depends on photon energy and on the material's composition and density: lead has an HVL of about 0.011 cm at 100 keV but about 1.04 cm at the 1.25 MeV average energy of Co-60.

What is the half-value layer formula?

HVL = ln(2) / μ ≈ 0.693 / μ, where μ is the linear attenuation coefficient in cm⁻¹. It comes from I/I₀ = e^(−μx): setting I/I₀ = 0.5 gives x = ln 2 / μ. If you have the mass attenuation coefficient μ/ρ (cm²/g), multiply by density first: μ = (μ/ρ) × ρ.

What is the tenth-value layer (TVL)?

The TVL is the thickness that reduces intensity to 10%. TVL = ln(10) / μ ≈ 2.303 / μ, so in narrow-beam conditions TVL = 3.32 × HVL. Shielding work uses TVLs because the count is easy: a 1000× reduction needs 3 TVLs (about 10 HVLs).

How many half-value layers reduce radiation by 1000?

log₂(1000) ≈ 9.97 HVLs, or exactly 3 TVLs. In general the number of HVLs for a reduction factor F is log₂(F) = ln(F) / ln(2), and the number of TVLs is log₁₀(F).

What is the HVL of lead for Cs-137, Co-60 and Ir-192?

From NIST narrow-beam data, lead has an HVL of about 0.55 cm at 662 keV (Cs-137) and 1.04 cm at 1.25 MeV (Co-60). Published practical values that include scatter are larger: OSHA lists 0.7 cm for Cs-137 and 1.2 cm for Co-60, and QSA Global gives about 5.1 mm for Ir-192. Use the practical values for real shields and confirm them with a qualified expert.

Why are published HVLs larger than the calculated ones?

Calculated values assume good (narrow-beam) geometry, where any scattered photon is removed from the beam. Around a real source, photons scattered inside the shield still reach the detector, which is described by a buildup factor greater than 1. The effective broad-beam HVL is therefore thicker, especially for concrete and steel at high energy.

Where do the attenuation coefficients come from?

From the NIST X-Ray Mass Attenuation Coefficients tables (Hubbell and Seltzer, NIST Standard Reference Database 126), which list total μ/ρ with coherent scattering. Densities used: lead 11.35, tungsten 19.3, steel 7.85, ordinary concrete 2.30, aluminum 2.699 and water 1.00 g/cm³; you can edit the density.

Does this account for scatter, buildup or beam hardening?

No. It uses the narrow-beam exponential model for a single photon energy. Polyenergetic X-ray beams harden as they pass through a filter, so the second HVL is thicker than the first, and broad beams need buildup factors. Use NCRP Report 147 or 151 methods, or a qualified medical or health physicist, for barrier design.

Half-Value Layer Calculator

HVL = ln 2 ÷ μ and TVL = ln 10 ÷ μ, where μ = (μ/ρ) × ρ. Pick a material and photon energy to use NIST mass attenuation data, or enter μ, a known HVL or two readings, then get the thickness for any transmission or reduction factor.

The half-value layer is HVL = ln 2 ÷ μ ≈ 0.693 ÷ μ, the thickness that halves a narrow photon beam; the tenth-value layer is TVL = ln 10 ÷ μ ≈ 3.32 × HVL.

Material & Radiation Input

Input mode

Material & energy (NIST) Attenuation coefficient (μ) Known HVL Beam measurement

Material

Photon energy

μ/ρ (cm²/g)

Density ρ (g/cm³)

NIST XCOM total μ/ρ. Energies between table points use log-log interpolation.

Shield thickness & transmission

Enter either thickness OR a target transmission / reduction factor — the other is computed.

Shield thickness (x)

Target transmission or reduction

Reduction × 1000 means the dose rate falls to 1/1000 (0.1%).

Important: Ideal narrow-beam, single-energy, homogeneous-material model only. It excludes scatter, buildup, leakage, beam hardening, spectrum and geometry effects. Do not use it alone for shielding design, compliance, or exposure decisions.

Enter parameters and click Calculate

Shielding Parameters

μ (cm⁻¹)

—

HVL

—

TVL

—

TVL/HVL

3.322

constant

Attenuation vs. Thickness

HVLs	Thickness (cm)	Transmission (%)	Attenuation (%)

Formulas used

μ = (μ/ρ) × ρ

I/I₀ = e^(−μx)

HVL = ln(2) / μ ≈ 0.6931 / μ

TVL = ln(10) / μ ≈ 2.3026 / μ

x = ln(1/T) / μ (for target T)

HVLs = log₂(F), TVLs = log₁₀(F)

Narrow-beam (good geometry) model — no buildup factor.

Narrow-beam HVL / TVL from NIST data (cm)

HVL / TVL computed from NIST μ/ρ and the listed density. Click a cell to load it.

Material, ρ (g/cm³)	100 keV	662 keV (Cs-137)	1.25 MeV (Co-60)
Lead (Pb), ρ 11.35	0.0110 / 0.0366	0.550 / 1.83	1.039 / 3.45
Tungsten (W), ρ 19.3	0.0081 / 0.0269	0.365 / 1.21	0.644 / 2.14
Steel (iron μ/ρ), ρ 7.85	0.238 / 0.79	1.202 / 3.99	1.650 / 5.48
Concrete, ordinary, ρ 2.3	1.734 / 5.76	3.826 / 12.71	5.190 / 17.24
Aluminum (Al), ρ 2.699	1.507 / 5.01	3.443 / 11.44	4.673 / 15.52
Water, ρ 1	4.061 / 13.49	8.091 / 26.88	10.962 / 36.42

Source: NIST X-Ray Mass Attenuation Coefficients (SRD 126). 662 keV is log-log interpolated; 1.25 MeV is the mean of the 1.173 and 1.332 MeV Co-60 lines.

Published practical HVLs for gamma sources (cm)

These include scattered radiation (broad beam), so they are thicker than the narrow-beam values.

Source (MeV)	Lead	Steel	Concrete	Source
Cs-137 (0.662)	0.7	1.6	4.8	OSHA; TVL 2.1 / 5.3 / 15.7
Co-60 (1.17, 1.33)	1.2	2.1	6.6	OSHA; TVL 4.0 / 6.9 / 20.8
Ir-192 (0.206–0.612)	0.51	1.30	4.32	QSA Global

Sources: OSHA shielding layer examples; QSA Global Ir-192 (lead 11.34, steel 7.80, concrete 2.35 g/cm³). Ir-192 emits many lines, so a single-energy narrow-beam calculation does not represent it.

Quick answer

The half-value layer is HVL = ln 2 ÷ μ ≈ 0.693 ÷ μ, the thickness that halves a narrow photon beam; the tenth-value layer is TVL = ln 10 ÷ μ ≈ 3.32 × HVL. The linear attenuation coefficient μ is the mass attenuation coefficient μ/ρ from NIST times the material density ρ. To cut a beam to a fraction T you need x = ln(1/T) ÷ μ, which is log₂(1/T) HVLs or log₁₀(1/T) TVLs.

Examples

Material, energy	μ/ρ (cm²/g)	ρ (g/cm³)	μ (cm⁻¹)	HVL	TVL
Lead, 662 keV (Cs-137)	0.1110	11.35	1.260	0.55 cm	1.83 cm
Lead, 1.25 MeV (Co-60)	0.05876	11.35	0.667	1.04 cm	3.45 cm
Concrete, 1.25 MeV (Co-60)	0.05807	2.30	0.134	5.19 cm	17.2 cm
Water, 100 keV	0.1707	1.00	0.171	4.06 cm	13.5 cm

Choose a material and energy above, then enter a thickness or a reduction factor such as 1000.

Summary

The half-value layer (HVL) is the thickness of a material that reduces a narrow photon beam to 50% of its intensity; the tenth-value layer (TVL) reduces it to 10%. Both follow from the linear attenuation coefficient μ, which is the tabulated mass attenuation coefficient μ/ρ multiplied by the density. This calculator includes NIST XCOM μ/ρ values for lead, tungsten, iron/steel, ordinary concrete, aluminum and water from 50 keV to 10 MeV, so you can get HVL, TVL and the thickness needed for a target transmission or reduction factor without looking anything up. It applies the ideal narrow-beam exponential model; published "practical" HVLs for Cs-137, Co-60 and Ir-192 are larger because they include scattered photons (buildup), so this is not a barrier-design or compliance tool.

How it works

Choose an input: material and photon energy (NIST data), a linear attenuation coefficient μ, a known HVL, or two beam readings through a test filter.
For a material, μ = (μ/ρ) × ρ. μ/ρ comes from the NIST XCOM table; energies between table points (Cs-137 at 662 keV) use log-log interpolation. You can change the density.
The tool computes HVL = ln 2 ÷ μ and TVL = ln 10 ÷ μ, and shows them in cm, mm and inches.
Enter a shield thickness to get the transmitted fraction I/I₀ = e^(−μx), or enter a target transmission (%) or reduction factor (×) to get the thickness x = ln(1/T) ÷ μ.
Compare with the server-rendered NIST table and the published broad-beam HVLs for Cs-137, Co-60 and Ir-192 below.

Use cases

Look up the narrow-beam HVL and TVL of lead, steel or concrete at a given photon energy.
Convert a NIST mass attenuation coefficient (cm²/g) into μ, HVL and TVL for any density.
Find how many HVLs or TVLs give a 100× or 1000× dose-rate reduction, and the thickness that takes.
Estimate an effective attenuation coefficient from two readings through a known filter.
Teach exponential attenuation and why broad-beam (buildup) HVLs exceed narrow-beam values.
Build an attenuation schedule by number of HVLs for a lab report or exam problem.

Frequently Asked Questions

Last updated: 2026-10-10 · Reviewed by Nham Vu