ALARA – as low as reasonably achievable – is not a slogan, it is an arithmetic exercise done before the source leaves the pot. Three multiplicative factors set the dose anyone receives, and each can be traded against the others: activity (which decays with time), distance (inverse square), and shielding (exponential). Multiply them together and the field at any position is D = Gamma x A0 x 2^(-dt/T_half) / d^2 x 2^(-x/HVL).

Seeing the three terms side by side is what makes the planning decision obvious. Distance is free and works in every direction, so it is always the first lever – moving from 5 m to 10 m removes 75 percent of the dose at no cost but time. Shielding is expensive and heavy but works where distance is unavailable, such as an adjacent occupied room. Decay works for you slowly, but it also means a source nearing replacement gives a smaller barrier and a longer exposure at the same time.

Dose is then simply rate times occupancy. A planner that shows the dose for the actual time a technician stands at the barrier, rather than a worst-case hour, is what turns a barrier calculation into a defensible dose assessment. The complementary figure is stay time: how long the position can be occupied before a chosen budget is spent.

Everything here is a free-air point-source calculation with narrow-beam attenuation. Scatter adds to the result and build-up in a thick shield adds more. Use it to plan the layout and predict the survey, never to replace it.

Worked example

Isotope	ir192
Activity at assay	100 Ci
Days since assay	73.83 d
Distance to work position	10 m
Shield material	pb
Shield thickness in the beam path	24 mm
Shield build-up factor B	1
Planned occupancy at this position	30 min
Barrier dose-rate limit	7.5 uSv/h
Dose budget for the task	100 uSv
Activity today	50 Ci
Unshielded rate at the work position	2102.4 uSv/h
Shield transmission	3.125 %
Dose rate at the work position	65.7 uSv/h
Dose in the planned occupancy	32.85 uSv
Stay time to spend the budget	91.32 min
Unshielded barrier radius	167.43 m

One half-life has passed, so A = 50.00 Ci. Unshielded at 10 m: 0.48 x 50 / 100 = 0.24 R/h = 0.24 x 8760 = 2102.40 uSv/h. 24 mm of lead is 5 HVL, transmitting 2^-5 = 3.125 percent, giving 65.70 uSv/h. Over 30 min that is 65.70 x 0.5 = 32.85 uSv. The 100 uSv budget lasts 100/65.70 h = 91.32 min. The unshielded 7.5 uSv/h barrier sits at sqrt(0.48 x 50 x 8760 / 7.5) = 167.43 m.

Use at your own risk — verify before you act

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