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Distance is the cheapest shield: the inverse-square law in practice

Use inverse-square estimates to compare distance and shielding, understand model limits and distinguish calculated contours from surveyed boundaries.

Practical guide

By Steven Michael Barber · Published October 17, 2026 · Updated November 3, 2026 · 6 min read

Of the three ways to cut the dose from a gamma source, time, distance and shielding, distance is the one that costs nothing and takes effect the moment you step back. Our guide to the dose-rate formula explains where the inverse-square law comes from. This one is about using it: setting a boundary around a source, deciding whether to move back or to shield, and knowing when the law stops being true.

The rule, and the two numbers to remember

For the point-source model, Ẋ = Γ × A ÷ d². With the source's activity A and gamma constant Γ unchanged, only distance d changes. Two consequences make the rule useful:

• Double the distance, quarter the rate. Halve it and the rate goes up four times. Ten times the distance cuts the rate by a hundred.
• The rate at 1 meter is Γ × A. Everything else is that number divided by d². For a 1 curie cobalt-60 source that is 1.32 R/h at 1 meter, 330 mR/h at 2 meters, 82.5 mR/h at 4 meters.

Because the law is a square, the first steps back matter most. Moving from 30 centimeters to 1 meter cuts the rate eleven times; moving from 10 meters to 11 cuts it by 17 percent.

Worked example: a portable moisture-density gauge

A portable nuclear gauge of the kind used on road and compaction jobs holds a cesium-137 source of about 8 millicuries (0.008 Ci, or 296 MBq) for its density measurement, plus an americium-241:beryllium neutron source that this gamma estimate does not cover. With Γ = 0.33 R·m²/(Ci·h), the rate at 1 meter is 0.33 × 0.008 = 0.00264 R/h, which is 2.64 mR/h. The inverse-square law gives the rest:

These are unshielded, open-air figures for the source in its measuring position. In the gauge's stored position the source sits behind tungsten or lead inside the housing, and the licensee's survey records, not this table, give the real numbers. The pattern is still the lesson: a minute at 30 centimeters gives about the same exposure as three quarters of an hour at 2 meters.

Setting a boundary: solve for distance

Turned around, the law tells you how far back a line has to be for the rate to fall to a target: d = √(Γ × A ÷ Ẋ). Federal rules limit the dose in an unrestricted area to 2 millirem in any one hour, alongside a 100 millirem annual limit for a member of the public (10 CFR 20.1301). With the photon convention 1 R ≈ 1 rem, the examples below solve for the 2 mR/h contour. That contour is arithmetic, not a boundary: a licensee places a restricted area from surveys, the expected occupancy and the cumulative dose, and controls access to it (10 CFR 20.1302).

For the gauge: d = √(0.00264 ÷ 0.002) = √1.32 ≈ 1.15 meters. A 0.5 mR/h line sits at 2.3 meters and a 0.1 mR/h line at 5.1 meters.

Now change the source. An industrial radiography camera may carry 50 curies of iridium-192 (Γ = 0.48). At 1 meter from the bare source the rate is 0.48 × 50 = 24 R/h, not milliroentgen but roentgen, about 9,100 times the gauge's rate at the same distance. The same arithmetic puts the 2 mR/h contour at √(24 ÷ 0.002) = √12,000 ≈ 110 meters, and a 100 mR/h line at 15.5 meters. That is why a radiography site is roped off at a distance that looks absurd for the size of the camera, and why the crew's own protection comes from the camera's shielding and from never being near the source while it is out.

The square root is doing something useful here. A rate about 9,100 times higher at 1 meter puts the same contour only about 95 times farther away, because √9,091 ≈ 95. Distance scales gently; that is what makes it practical.

Step back or shield?

Shielding and distance can be traded against each other, and the exchange rate is simple. Doubling the distance cuts the rate by four; so do two half-value layers of shield. For cesium-137 the half-value layer in lead is 0.65 cm, so one doubling of distance is worth about 1.3 cm of lead (the calculator's tenth-value model gives a transmission of 0.24 for 1.3 cm, the half-value model exactly 0.25).

Take the gauge again, with someone who has to work at 30 centimeters from the extended source, where the rate is 29.3 mR/h. To bring that to 2 mR/h in place, the shield has to pass only 0.068 of the radiation, which the broad-beam tables turn into:

Three steps back do the work of an inch of lead. The same comparison for the radiography source shows where shielding wins instead: at 10 meters the rate is 240 mR/h, and holding a 2 mR/h line there would take 4.2 cm of lead or 31 cm of concrete around the whole source, which is roughly what a shielded exposure room provides. In the field, where there is no room, the model puts the line about 110 meters out, and surveys set where it actually goes. Distance is cheap when you have space; shielding is cheap when you have a wall.

Where the law stops being true

The inverse-square law describes a point. Real sources have size, and real sites have walls.

• Close to the source. The rule of thumb is that a source behaves as a point once you are at least three times its largest dimension away. A gauge source pellet a few millimeters long is a point at 2 centimeters; a 1 meter length of contaminated pipe is not a point until you are 3 meters off. Inside that range the rate rises more slowly than d² predicts as you approach, and the formula overstates it.
• Lines and planes. Near a long line source the rate falls as 1/d, not 1/d²: doubling the distance halves it. Near a wide contaminated surface the rate barely changes with distance at all until you are well past the edge. For those geometries the point-source formula gives the wrong slope, not just the wrong number.
• Scatter. Floors, walls, equipment and the ground return some of the photons that left the source in another direction. Inside a room the measured rate at a few meters is commonly 10 to 30 percent above the open-air figure, and behind a shield the scattered component can dominate. The broad-beam layers in the shielding tables include scatter within the shield for the conditions of those tables; they do not include the room.
• Air. Over a few meters, air attenuation can usually be neglected; over a hundred meters it cannot. At normal air density, NIST's attenuation coefficients give a loss of about 1.8 percent per meter at 100 keV, counting photons that interact at all, and over 110 meters only about a quarter of 300 keV photons, a third of 600 keV photons and half of 1.25 MeV photons pass without an interaction. Scattered photons put some of that back, so how much the total rate changes takes a transport and geometry model to say. The 110 meter figure above is a model contour, not a validated boundary.

A calibrated survey meter settles all of this. The formula tells you what to expect and where to put the first stake; the meter tells you where the boundary actually is. If the two disagree by more than a factor of two, something about the source, its housing or the geometry is not what you assumed, and that is worth finding out before anyone works there.