To reduce a beam from I0 to a target I, the required attenuation factor is I0/I. Since each half value layer halves the beam, the number of half value layers needed is log2(I0/I) and the thickness is x = HVL x log2(I0/I). In tenth value layers the same result reads x = TVL x log10(I0/I), which is how shielding reports are usually written.

This arithmetic assumes a narrow, well-collimated beam in which any photon that scatters is gone for good. A real shield sits in a broad field, and photons that Compton-scatter inside the shield can still emerge on the far side travelling forwards. The correction is the build-up factor B, defined so that I = B x I0 x e^(-mu x). For lead at iridium energies B is around 2 to 4 at five to ten mean free paths, and for concrete it can exceed 10. Build-up always works against you, so a thickness calculated on half value layers alone is optimistic and the real dose behind the shield will be higher than predicted.

The practical allowance is to add one extra half value layer, which buys a factor of two of margin and covers a build-up factor up to about 2. For deep shields, high-energy sources or low-Z materials such as concrete, that is not enough – a full broad-beam calculation with tabulated build-up factors, or a survey of the completed shield, is required. Whichever route is taken, the finished shield must be proved by survey meter before it is relied on.

Weight matters as much as thickness. Lead at 11.34 g/cm3 gets the job done in the fewest millimetres but loads the structure heavily, which is why bay walls are concrete and only the local collimators and pots are lead or tungsten.

Worked example

Isotope	ir192
Shield material	pb
Unshielded dose rate	1000 uSv/h
Target dose rate behind shield	7.8125 uSv/h
Build-up factor B	2
Half value layer	4.8 mm
Attenuation factor required	128
Half value layers required	7
Shield thickness (narrow beam)	33.6 mm
Recommended thickness with build-up allowance	38.4 mm

The required attenuation is 1000 / 7.8125 = 128, and log2(128) = 7 exactly. With an Ir-192 lead HVL of 4.8 mm the narrow-beam thickness is 7 x 4.8 = 33.60 mm. Adding log2(2) = 1 extra HVL for the default build-up factor B = 2 gives a recommended 38.40 mm of lead.

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