A  mathematical equation provides useful input to prediction of radiological terror. This is because radioactive decay proceeds as a natural, inexorable process, affecting the ability of terrorists to amass a stockpile.

Markov chains model radioactive decay, providing precise predictions of how long a radioactive substance will remain potent, resulting in the simple equation presented here. Decay proceeds with the precision of an atomic clock, and cannot be slowed by “proper storage” or any other treatment by man. (Markov chains will also be used in a future discussion of power projection.)

Use of dirty bombs has long been anticipated, yet the threat has not so far materialized. To understand why, and to offer an insight as to when it might, we must consider the terrorist problem, which is to accumulate a stockpile prior to use. The more radioactive material can be assembled in one place, the more harmful the effect will be.

A stockpile of a radioactive isotope undergoes exponential decay, decreasing in potency by half every half-life. The half-life of a particular isotope is precisely known and unchangeable. The physical bulk of a source remains, but containing an ever-increasing proportion of inert, stable isotopes. Isotopes used in hospitals have half-lives between days and years.

• Most radioactive substances  in hospitals are used as tracers. Injected into the body, a tracer concentrates in particular tissue, providing diagnostic information. Tracer isotopes have short half-lives, decaying rapidly, avoiding persistence of radioactivity in the patient. Tracers have little terror potential.
• In brachytherapy, a tumor is irradiated from a source placed inside the body. These sources are more powerful and longer lasting than tracers. Cesium-137 is  used as an implant to a tumor. With a half life of 30.17 years, it must be removed from the body after treatment. The discarded implants  have terror potential. Amassing a stockpile requires collection of many discarded doses.
• The most dangerous isotopes used by hospitals are found in external radiation machines, which direct an intense beam of radiation into the body. The standard isotope is cobalt-60,  contained within a thick lead capsule with a bore hole at one end. Cobalt-60 has a half-life of 5.27 years.

Cesium-137 and cobalt-60 do not fit well into  the conventional definitions of low or high level waste. While a single hospital source is far less radioactive than a single spent reactor fuel rod, collected discarded sources over time can accrete to massive levels. But there is a limit, determined by the half life of the isotope.

Suppose some  individuals in Peshawar, perhaps a nuclear medicine technician or physician and an analytical chemist,  put themselves in the illicit business of collecting, assaying, and reselling radioactive sources to terrorists.  Suppose that, on the average, they manage to collect a fixed amount of a particular isotope per year, perhaps a few ounces. Every source in their stockpiles decays relative to when they acquired it. For this example, let’s consider cobalt-60.

After some years of collecting, the total radioactivity in their possession will almost plateau, as an asymptote. They more isotope they have, the more is decaying. At some year relative to start of their enterprise, the amount lost to decay is merely balanced by the amount of their annual collection. The stockpile continues to increase in bulk, becoming ever harder to concentrate for dirty bombs, yet no more radioactive.

This is the cash-out point. Some time relative to it, both the terrorist and the technicians  make a deal. We cannot predict the actual event of use  but we can estimate the cash-out point, when the stockpile reaches a point of diminishing returns. Nothing can be done to change the rate of decay.

Where

h = half life of the isotope

t = the elapsed time from start of collection

R = rate of collecting discarded sources, in whatever unit of radioactivity you prefer

ln is the natural logarithm

Q is the  quantity of isotope in the stockpile

Q_T is the maximum amount of isotope that can be accumulated given the rate of collection R

we have

Q = [h*R/ln(2) ] * [1  – 2  (to the power of (- t/h))]

while the maximum size of the stockpile (the asymptotic value) is

Q_T = h*R/ln(2)

For cobalt-60, the point of futility, the cash-out point, when the amount collected is about 84% of Q_T, with the bulk ever increasing, is about 15.5 years. Cesium-137 can accumulate for a much longer time.

This calculation is independent of how much isotope is collected annually, assuming only the same amount each year. If the collecting started in the early 2000’s,  the time is about now.