Keith D. McCroan US EPA National Air and Radiation Environmental Laboratory

Issue  Most rad-chemists learn early to estimate “counting uncertainty” by square root of the count C.  They are likely to learn that this works because C has a “Poisson” distribution.  They may not learn why that statement is true, but they become comfortable with it.

“The standard deviation of C equals its square root. Got it.”

The Poisson distribution  What’s special about a Poisson distribution?  What is really unique is the fact that its mean equals its variance: μ = σ 2  This is why we can estimate the standard deviation σ by the square root of the observed value – very convenient.  What other well-known distributions have this property? None that I can name.

The Poisson distribution in Nature  How does Nature produce a Poisson distribution?  The Poisson distribution is just an approximation – like a normal distribution.  It can be a very good approximation of another distribution called a binomial distribution.

Binomial distribution  You get a binomial distribution when you perform a series of N independent trials of an experiment, each having two possible outcomes (success and failure).  The probability of success p is the same for each trial (e.g., flipping a coin, p = 0.5).  If X is number of successes, it has the “binomial distribution with parameters N and p.” X ~ Bin(N, p)

Poisson approximation  The mean of X is Np and the variance is Np(1 − p).  When p is tiny, the mean and variance are almost equal, because (1 − p) ≈ 1.  Example: N is number of atoms of a radionuclide in a source, p is probability of decay and counting of a particular atom during the counting period (assuming half- life isn’t short), and C is number of counts.

Poisson counting  In this case the mean of C is Np and the variance is also approximately Np.  We can treat C as Poisson: C ~ Poi(Np)

Poisson – Summary  In a nutshell, the Poisson distribution describes occurrences of relatively rare (very rare) events (e.g., decay and counting of an unstable atom)  Where significant numbers are observed only because the event has so many chances to occur (e.g., very large number of these atoms in the source)

Violating the assumptions  Imagine measuring 222 Rn and progeny by scintillation counting – Lucas cell or LSC.  Assumptions for the binomial/Poisson distribution are violated. How?  First, the count time may not be short enough compared to the half-life of 222 Rn.  The binomial probability p may not be small.  If you were counting just the radon, you might need the binomial distribution and not the Poisson approximation.

More importantly...  We actually count radon + progeny.  We may start with N atoms of 222 Rn in the source, but we don’t get a simple “success” or “failure” to record for each one.  Each atom might produce one or more counts as it decays.  C isn’t just the number of “successes.”

Lucas 1964  In 1964 Henry Lucas published an analysis of the counting statistics for 222 Rn and progeny in a Lucas cell.  Apparently many rad-chemists either never heard of it or didn’t fully appreciate its significance.  You still see counting uncertainty for these measurements being calculated as.

Radon decay  Slightly simplified decay chain:  A radon atom emits three α-particles and two β-particles on its way to becoming 210 Pb (not stable but relatively long-lived).  In a Lucas cell we count just the alphas – 3 of them in this chain.

Thought experiment  Let’s pretend that for every 222 Rn atom that decays during the counting period, we get exactly 3 counts (for the 3 α-particles that will be emitted).  What happens to the counting statistics?

Non-Poisson counting  C is always a multiple of 3 (e.g., 0, 3, 6, 9, 12,...).  That’s not Poisson – A Poisson variable can assume any nonnegative value.  More important question to us: What is the relationship between the mean and the variance of C?

Index of dispersion, J  The ratio of the variance V(C) to the mean E(C) is called the index of dispersion.  Often denoted by D, but Lucas used J.  That’s why this factor is sometimes called a “J factor”  For a Poisson distribution, J = 1.  What happens to J when you get 3 counts per decaying atom?

Mean and variance  Say D is the number of radon atoms that decay during the counting period and C is the number of counts produced.  Assume D is Poisson, so V(D) = E(D). C = 3 × D So, E(C) = 3 × E(D) V(C) = 9 × V(D) J = V(C) / E(C) = 3 × V(D) / E(D) = 3

Index of dispersion  So, the index of dispersion for C is 3, not 1 which we’re accustomed to seeing.  This thought experiment isn’t realistic.  You don’t really get exactly 3 counts for each atom of analyte that decays.  It’s much trickier to calculate J.

Technique  Fortunately you really only have to consider a typical atom of the analyte (e.g., 222 Rn) at the start of the analysis.  What is the index of dispersion J for the number of counts C that will be produced by this hypothetical atom as it decays?  Easiest approach involves a statistical technique called conditioning.

Conditioning  Consider the possible ways the atom can decay, and group them into mutually exclusive alternative cases (“events”) that together cover all the possibilities.  It is convenient to define the events in terms of the states the atom is in at the beginning and end of the counting period.  Calculate the probability of each event.

Conditioning - Continued  For each event, calculate the conditional expected values of C and C 2 given the event (i.e., assuming the event occurs).  Next calculate the overall expected values E(C) and E(C 2 ) as probability-weighted averages of the conditional values.  Calculate V(C) = E(C 2 ) − E(C) 2.  Finally, J = V(C) / E(C).  Details left to the reader.

Radium-226  Sometimes you measure radon to quantify the parent 226 Ra.  Let J be the index of dispersion for the number of counts produced by a typical atom of the analyte 226 Ra – not radon.  Technique for finding J (conditioning) is the same, but the details are different.  Value of J is always > 1 in this case.  Bounds: 1 < J ≤ 1 + CF × 2 / 3

Thorium-234  If you beta-count a sample containing 234 Th, you’re counting both 234 Th and the short-lived decay product 234m Pa.  With ~50 % beta detection efficiency, you have non-Poisson statistics here too.  The counts often come in pairs.  The value of J doesn’t tend to be as large as when counting radon in a Lucas cell or LSC (less than 1.5)

Gross alpha/beta?  If you don’t know what you’re counting, how can you estimate J?  You really can’t.  Probably most methods implicitly assume J = 1.  But who really knows?

Testing for J > 1  You can test J > 1 with a χ 2 test, but you may need a lot of measurements.

Important points  Suspect non-Poisson counting if:  One atom can produce more than one count as it decays through a series of short-lived states  Detection efficiency is high  Together these effects tend to give you on average more than one count per decaying atom.

Questions