Radiation Shielding Calculator
Calculates the transmitted radiation intensity after passing through a shielding material using the Beer-Lambert attenuation law. Used by health physicists and radiation safety officers to design protective barriers.
Last updated: September 2026
Dose Rate Behind Shield (with scatter buildup, conservative)
158.6 mR/h
Formula below · 2 sources (nrc.gov, Wikipedia) · Updated Sep 2026
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About this calculator
Gamma rays are attenuated exponentially, I = I₀ × e^(−μx), but that narrow-beam law counts only photons that never interacted. Behind a real shield, photons scattered inside the material also reach you, so the dose rate is B × I₀ × e^(−μx), where the buildup factor B grows with the number of mean free paths μx. Ignoring B understates the dose — by about 2× for 5 cm of lead and up to 10× or more for thick concrete or water. This calculator therefore includes a conservative buildup estimate: B = 1 + μx for lead, 1 + μx + (μx)²/8 for steel, and 1 + μx + (μx)²/2 for concrete, water and custom materials; these bound the published point-isotropic exposure buildup factors (ANSI/ANS-6.4.3) at 0.5–1.25 MeV. The material μ values are for 1.25 MeV (Co-60), the most penetrating common case; for Cs-137 (0.662 MeV) they overstate transmission, which is safe. For photons above 1.25 MeV, or other materials, enter μ from NIST XCOM as a custom value (it overrides the material). Each HVL (ln 2 / μ) halves only the uncollided beam.
How to use
A Co-60 source gives 500 mR/h. With 5 cm of lead (μ = 0.66 cm⁻¹): μx = 3.3, uncollided fraction e^(−3.3) = 0.0369, buildup B = 1 + 3.3 = 4.3, so the dose rate behind the shield is 500 × 4.3 × 0.0369 = 79.3 mR/h — not the 18 mR/h the bare exponential suggests. With only 2 cm: μx = 1.32, B = 2.32, 500 × 2.32 × 0.267 = 310 mR/h. For a Cs-137 source enter lead's μ at 0.662 MeV (about 1.23 cm⁻¹) as a custom value: 5 cm then gives μx = 6.15 and, with the conservative custom buildup, 500 × (1 + 6.15 + 18.9) × 0.00213 = 27.8 mR/h.
Frequently asked questions
What is the linear attenuation coefficient and how do I find it for my shielding material?
The linear attenuation coefficient μ (cm⁻¹) quantifies how effectively a material attenuates photons per unit thickness. It is the product of the mass attenuation coefficient (cm²/g), available from NIST XCOM tables, and the material density (g/cm³). Values are strongly energy-dependent: lead has μ ≈ 1.23 cm⁻¹ for 0.662 MeV gamma rays but much higher values at lower energies. For your calculation, identify the dominant photon energy of your source and look up the corresponding mass attenuation coefficient from NIST, then multiply by the density of your chosen material.
Why does radiation shielding follow an exponential rather than a linear decay with thickness?
Each thin layer of material has the same probability of removing a photon from the beam, regardless of how many layers came before. This constant fractional removal per unit thickness is the defining property of a Poisson process and leads directly to exponential attenuation. Mathematically, dI/dx = −μI, whose solution is the Beer-Lambert law. This means you can never reduce the intensity to exactly zero with finite thickness; you can only reduce it by successive factors. Each half-value layer (HVL) cuts the intensity in half, so three HVLs give a factor-of-8 reduction.
When should a buildup factor be included in radiation shielding calculations?
Always, when the result is used to judge a dose rate behind a shield. The plain Beer-Lambert formula applies only to narrow-beam (good geometry) conditions where scattered photons are excluded from the detector. In practical shielding, Compton-scattered photons travelling at small angles still pass through and add to the dose, by a buildup factor B that is about 2 for a few mean free paths of lead and can exceed 10 for thick concrete or water. This calculator always includes a conservative B; use the Gamma Ray Attenuation calculator if you need the pure narrow-beam number for a collimated laboratory measurement.
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