January 18th, 2021 - Newton's Inverse Square Law as Applied to Radioactivity
Newton developed the Inverse Square Law as it applies to gravity. However, this exponential relationship applies to many diverse phenomena including an electric field, light, sound and radiation. For radiation the Inverse Square Law relates the intensity of the radioactive source to the detector or person by the inverse of the distance squared. Therefore, as the detector or person moves away from a radioactive source the intensity will fall off exponentially, that is, inversely by the distance (d) squared or, 1÷d2. The intensity can be in the units of Roentgens, rads, sieverts [1] or rems.
Practical Applications
There are a number of aspects to consider when applying the Inverse Square Law. Notice that the diagram below only illustrates the principle of the inverse square law. In practice the radiation from a point source is spherical (4π) and the detector is measuring a solid angle impinging on its surface. Therefore, when this relationship is applied to a real application the detector must be in the shape of a right circular cylinder and the source should be positioned directly in front of the detector (on a centerline with the detector to avoid errors in geometry). In addition, for accurate equivalent dose rate (biological dose rate) avoid positioning the detector no closer to the source than five times the diameter of the detector. Photon energy and solid angle can become so large that the Inverse Square Law measurement may be somewhat inaccurate.
When adhering to the above precautions dose-rate measurements can become quite accurate. Many first-line responders are familiar with the Inverse Square Law but fail to realize the last few inches obeys the same law and is often very important. If you are 10 inches away from a source and moving to 1 inch away, this can increase the intensity by a factor of 100. Many radioisotope identifiers (RIIDs) are very sensitive so the tendency is to stop searching as soon as a source is indicated. However, a few more inches may be needed for accurate isotope identification. For spectroscopy, it is always paramount to get good statistics for precise identification even when a close distance may give some error in the dose-rate measurement.
Be aware that some detectors have shapes that don’t conform to the inverse square law when it comes to accurate dose rate measurements. For example, a detector measuring 4x4x16 inches cannot accurately absorb all photons in a solid angle from a source at any distance. For these non-traditional detectors the factory calibration is usually set for compliance with NORM background.
[1] The sievert is the SI (International Unit) of exposure and is equal to 100 rem.
Examples with Solutions
- Safety requirements for workers near radioactive material use the acronym ALARA (as low as reasonably achievable). This is illustrated by a worker receiving an equivalent dose rate of 50 mrem/hr while standing 1 foot (d1) from a radioactive source. What will be the equivalent dose rate (H) if the worker moves back 8 feet (d2)?
The solution will be in the form of: H x d12/d22 Answer: Squaring both distances gives: 50 x 1/64 = 0.78 mrem/hr
Notice that the worker could be safe with a smaller change in distance but also be aware that ALARA is only satisfied when the exposure is as low as practical.
- A RIID is held 15 inches from a source with an exposure rate of 10 uR/hr. How close to the source would give an exposure rate of 90 uR/hr? 10(152/x2) = 90, then solve for x (Answer is 5 inches).
