Photons are removed from a narrow beam exponentially: I = I0 x e^(-mu x x), where mu is the linear attenuation coefficient of the shield at that photon energy. The thickness that halves the intensity is the half value layer, HVL = ln2 / mu. Because the law is exponential, each successive HVL halves what is left, so n half value layers transmit 2^-n of the incident beam.

The tenth value layer is the same idea to a factor of ten: TVL = ln10 / mu = HVL x log2(10) = 3.322 x HVL. Shielding designers work in TVLs because the attenuation factors needed on a radiography bay are usually several orders of magnitude, and counting in tens is quicker than counting in halves.

Attenuation depends steeply on both photon energy and the atomic number of the shield. At iridium energies lead has an HVL of about 4.8 mm while steel needs 12.7 mm and concrete 44.5 mm for the same effect. At cobalt energies the whole picture stiffens: lead needs 12.5 mm, and lead’s advantage over steel narrows because Compton scattering, which scales with electron density rather than Z to the fourth or fifth power, takes over from photoelectric absorption.

These values are narrow-beam figures. In a real broad, uncollimated field, photons that scatter within the shield still emerge and contribute to the dose behind it. That effect is the build-up factor, and it means a shield designed on HVL arithmetic alone always performs worse than the calculation says. See the shielding thickness calculator for how much margin to add.

Worked example

Isotope	ir192
Shield material	pb
Shield thickness	24 mm
Unshielded dose rate	1000 uSv/h
Build-up factor B (1 = narrow beam)	1
Half value layer	4.8 mm
Tenth value layer	15.95 mm
Half value layers in this shield	5
Transmission	3.125 %
Dose rate behind shield	31.25 uSv/h

Ir-192 in lead has HVL 4.8 mm, so TVL = 4.8 x 3.3219 = 15.95 mm. A 24 mm shield is 24/4.8 = 5.00 half value layers, transmitting 2^-5 = 0.03125 = 3.125 percent. A 1000 uSv/h incident field therefore leaves 31.25 uSv/h behind the lead, before build-up.

Use at your own risk — verify before you act

These calculators support, and never replace, the judgement of qualified NDT and engineering personnel. Results are provided as is, without warranty of any kind, express or implied, and must be independently verified against the governing code edition named in your contract before being used in any inspection, acceptance, rejection, radiation-safety or fitness-for-service decision. By using them you accept full responsibility for how the results are applied; NDT Inspect, its owners and contributors accept no liability for any loss, damage, injury or death arising from their use or from reliance on them. If a result matters to safety, check it by hand and have it reviewed by a competent person.