Transcript Non-Poisson Counting Uncertainty, or “What’s this J Factor

Keith D. McCroan
US EPA National Air and Radiation Environmental Laboratory
Radiobioassay and Radiochemical Measurements Conference
October 29, 2009
NON-POISSON COUNTING
UNCERTAINTY, OR
“WHAT’S THIS J FACTOR ALL
ABOUT?”
Counting uncertainty
 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.
2
“The standard deviation of C
equals its square root. Got it.”
3
The Poisson distribution
 What’s special about a Poisson
distribution?
 What is really unique is the fact that its
mean equals its variance:
μ = σ2
 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.
4
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.
5
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)
6
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 halflife isn’t short), and C is number of counts.
7
Poisson counting
 In this case the mean of C is Np and the
variance is also approximately Np.
 We can consider C to be Poisson:
C ~ Poi(μ)
where μ = Np
8
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)
9
Violating the assumptions
 Imagine measuring 222Rn 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 222Rn.
 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.
10
More importantly...
 We actually count radon + progeny.
 We may start with N atoms of 222Rn 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.”
11
Lucas 1964
 In 1964 Henry Lucas published an
analysis of the counting statistics for 222Rn
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
.
12
Radon decay
 Slightly simplified decay chain:
 A radon atom emits three α-particles and
two β-particles on its way to becoming
210Pb (not stable but relatively long-lived).
 In a Lucas cell we count just the alphas –
3 of them in this chain.
13
Thought experiment
 Let’s pretend that for every 222Rn 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?
14
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?
15
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?
16
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).
 By assumption, 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
17
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 correctly.
18
Technique
 Fortunately you really only have to
consider a typical atom of the analyte
(e.g., 222Rn) 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.
19
Conditioning
 Consider all the possible histories for the
atom – i.e., all the different ways the atom
can decay.
 It is convenient to define the histories in
terms of the states the atom is in at the
beginning and end of the counting period.
 Calculate the probability of each history
 typically using Bateman equations
20
Conditioning - Continued
 For each history, calculate the conditional




expected values of C and C2 given that
history (i.e., assuming it occurs).
Next calculate the overall expected values
E(C) and E(C2) as probability-weighted
averages of the conditional values.
Calculate V(C) = E(C2) − E(C)2 .
Finally, J = V(C) / E(C).
Details left to the reader.
21
Radium-226
 Sometimes you measure radon to quantify
the parent 226Ra.
 Let J be the index of dispersion for the
number of counts produced by a typical
atom of the analyte 226Ra – not radon.
 Technique for finding J (conditioning) is
the same, but the details are different.
 Value of J is always > 1 in this case.
22
Thorium-234
 If you beta-count a sample containing
234Th,
you’re counting both 234Th and the
short-lived decay product 234mPa.
 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).
23
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?
24
Simplification
 Assume every radiation of the decaying
atom has detection efficiency ε or 0. Then
where
m1 is expected number of detectable radiations from
an atom of analyte during the counting interval
m2 is expected square of this number
25
Bounds for J
 m1 ≤ m2 ≤ Nm1, where N is the maximum
number of counts per atom. So,
1 − ε × m1 ≤ J ≤ 1 + ε × (N − m1 − 1)
 In many situations m1 is very small. Then
1 ≤ J ≤ 1 + ε × (N − 1)
 E.g., for 226Ra measured by 222Rn in a
Lucas cell, N = 3. So,
1 ≤ J ≤ 1 + 2ε
26
Remember
 Suspect non-Poisson counting if:
 One atom can produce more than one count
(N > 1) as it decays through a series of shortlived states
 Detection efficiency (ε) is high
 Together these effects tend to give you on
average more than one count per decaying
atom.
 In many cases, 1 ≤ J ≤ 1 + ε × (N − 1).
27
Questions?
28
Reference
 Lucas, H.F., Jr., and D.A. Woodward.
1964. Journal of Applied Physics 35:452.
29
Testing for J > 1
 You can test J > 1 with a χ2 test, but you
may need a lot of measurements.
Minimum Detectable J > 1
5.50
5.00
4.50
4.00
J
3.50
3.00
2.50
2.00
1.50
1.00
10
100
1000
Number of measurements (log scale)
30