Statistics of nuclear decay

1. Radioactive decay and counting statistics
Jørgen Gomme
The decay equation
Predictability and chance
Radioactivity measurements and statistics in practice
F07

2. Outline





The decay equation
Stochastic nature of radioactive decay
Poisson-distribution
Controlling the stability of counting instrumentation
Accuracy and precision
2011
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3. Activity
An amount of material containing
acitivity A:
N radioactive atoms has the
Number of decays (nuclear disintegrations) per unit time.
N
dN
A≡
dt
The fundamental SI-unit for activity is the becquerel (Bq):
1 Bq ≡ 1 disintegration per second
The old unit (still occasionally used) is the curie (Ci):
1 Ci =3.7000×1010 Bq
1 Bq =2.7027×10-11 Ci
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4. Counting efficiency and counting rate
 ∆N decays occur in the radioactive sample in a period of
length ∆t

The detector (and the associated instrumentation) only
records a fraction of these decays (ε = counting efficiency)
 ∆M impulses are recorded during the period ∆t
 The counting rate (pulse rate) is r = ∆M/∆t
# counts
∆M
= =
r
∆t
counting time
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# decays
ε
∆N
∆t
counting
efficiency
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5. Predictability and chance...
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6. Radioactive decay is a stochastic phenomenon

Impossible to predict when a radioactive atom
decays, but the probability of decay in a given period
of time is known

Radioactive decay follows a Poisson-distribution
λ
Consequences for:

Uncertainty in radioactivity measurements

Check of instrument performance
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7. The decay law
The decay equation may be
derived from the assumption, that each individual
nucleus of a given radionuclide has a well-defined
probability (λdt) for decay in
the time interval dt
The decay equation provides
an exact relation between the
activity and the number of
atoms (amount of the radioactive substance)…
… or between the activity at
two different points in time:
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A≡
dN
dt
A = A0e − λt
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8. Number of decays / counts per unit time
– according to the decay law

From the decay equation we might expect the same number
of decays in succesive periods of constant length,

Assuming constant counting efficiency ε, also the number of
counts per period would be constant
Number of decays:
Number of counts:

∆t
∆N N 0 (1 − e − λ∆t )
=
∆=
M
ε N 0 (1 − e−λ∆t )
None of these predictions are fulfilled in practice: The radioactive
decay is a stochastic phenomenon
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9. Varying number of decays in succesive periods ∆t

The decay equation predicts a constant number of decays
per unit time

However, the actual number of decays in succesive periods
of the same length vary, i.e. radioactive decay is a stochastic
phenomenon
Mean
6
5
4
3
2
1
38-29
36-37
34-35
32-33
30-31
28-29
26-27
24-25
22-23
20-21
18-19
16-17
14-15
12-13
0
Empirical
Standard
deviation
Mean
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10. Repeated measurements on the same sample




Repeated measurements of sample counting rate: Variation
around a mean value
In practice, two different sources of variation:
– Stochastic nature of radioactive decay
– Instrumental variability – instability in detector and
electronics
The stochastic nature of radioactive decay may be related to
the decay constant λ (probability of decay in unit time)
Instrumental variability may be seen as a phenomenon, that
is reflected in fluctuating counting efficiency,
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ε
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11. Sources of variation
 How to perform radioactivity measurements, so that the
results are representative of the ”true activity” of the
sample?
 Differentiating between the variability due to:
 Stochastic nature of radioactive decay
 Instrumental instability
In practical work: Uncertainty and errors in sample preparation represents
an additional source of variation – to be discussed later.
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12. Stochastic nature of radioactive decay


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Probability theory: Repeated meaurements on the
same sample will follow a distribution, that may
be described analogously to the binomial distribution – a situation corresponding to ”throwing
dice”
The binomial distribution gives an expression for
the probability P(∆N) of observing ∆N decays
during the period ∆t when the total number of
radioactive atoms is N0
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13. Radioactive decay, compared to throwing dice
The probability of getting exactly 10 sixes when doing 50 throws with
a single die is: 0.116 (11.6 %)
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14. Poisson-distribution
– the ”law of rare events”
It may be shown mathematically, that in the limiting
case the binomial distribution approaches the Poissondistribution (when the probability for any nucleus
decaying in the observation period is small):
P(∆N) is the probability for
∆N decays in the period ∆t
= N 0 (1 − e − λ∆t )
ν
ν is the true value
(expectation value for the number of decays in the period
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∆t)
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15. Poisson error
For the Poisson distribution, the theoretical standard deviation (standard error, ”Poisson error”) is always equal to the
square root of the true value (the expectation value).
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16. Predictability and chance
Two sides of the same coin:
 The decay law – with its exact relation between the
number of radioactive atoms, and the number of
disintegrations per unit time
 Stochastic nature of radioactive decay
λ
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17. Poisson-distribution for different ν
P(∆N)
For ν > 20 the Poissondistribution approaches
a normal distribution
ν=1
ν=5
ν = 25
∆N
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18. Counting a radioactive sample
∆M counts are recorced in the time ∆t, giving
the count rate r = ∆M/∆t
With the counting efficiency ε, this will
correspond to the disintegration rate
∆N/∆t – to be taken as the best estimate
of the true value ν
Estimate of standard error on r: sr
The standard error sr (”Poisson error”)
is a measure of the variation in the
results, that may be explained by the
stochastic nature of radioactive decay.
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19. One or several counts on the same sample...?
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20. Repeated determinations – what to expect?
n separate measurements,
each of duration ∆t, total
counting time n∆t
A single measurement of
duration n∆t
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21. One or several counting periods?

From a statistical point of view, we obtain the same information
when counting a sample in the following two situations:
– Counting in a single period of length ∆t
– Counting in n periods, each of length ∆t/n

It is the the total counting time (or rather: the total number of
counts collected, ∆M), that determines the ”information”
obtained, or the ”uncertainty” associated with the measurement.
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22. Absolute and relative standard error
∆M
∆t
Count rate:
r=
Absolut standard error
(Poisson error):
∆M
sr =
∆t
=
( sr )rel
Relative standard error:
cps
cps
∆M
∆t
=
∆M
∆t
1
∆M
The relative error depends only on
the number of counts accumulated
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23. Relative standard error as a function of the
number of counts
Number of counts
∆M
Relative error
=
srel
(1/
)
∆M 100%
100
200
7.07
500
4.47
1000
3.16
2000
2.24
5000
1.41
10000
1.00
20000
0.71
50000
0.45
100000
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10.00
0.32
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24. Nevertheless: Why sometimes doing
repeat counts?
Occasionally, a counting period ∆t is divided into n subperiods,
each of duration ∆t/n, and for the overall result the mean af n
individual countings is used:
As seen above, this does not provide additional information
about the radioactive sample, …
However:
 Repeated measurements on the same sample gives an
opportunity to evaluate the stability of the counting equipment,
or:
 It is possible to estimate the following sources of variation:
– The stochastic nature of radioactive decay (cf. λ)
– Instability of the instrumentation; may be seen as a
variation of ε

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25. Practical examples, test of stability...
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26. 1: Example from Laboratory 2
– counting a series of samples, prepared in the same say
Radioactive sample:
86Rb
(T½ = 18.63 d)
Counting time (∆t = 5 min), constant for all samples
Observations: ∆M =
30040
29105
30840
33505
32995
31515
29915
31860
33670
30515
Mean:
∆M =
S.D.:
semp
=
31396 counts per 5 min
∑ (∆M − ∆M )
2
=
n −1
n
1591 (5.07 %)
177 (0.56 %)
Poisson error: sPois = ∆M = 31396 =
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Uncertainty/errors
in sample preparation
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From sample
distribution
From sampling
distribution
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27. 1: Example from Laboratory 2
(cont’d)
Stochastic
nature of decay
Errors in sample
preparation

The measurement result (count rate per 25 µl sample) is
subject to the following sources of variation:
– The stochastic nature of radioactive decay
– Instrument instability ?
– Errors (stochastic / systematic) in sample preparation
(pipetting)

If we want the result expressed as activity (Bq) per 25 µl
sample, we must add:
– Error (uncertainty) in the determination of the counting
efficiency
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ε
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28. 2: Repeated measurements of the same sample
(under identical conditions)
What is best:
1. Counting the sample once in 10 min?
2. 10 repeated measurements, each of duration 1 min, i.e.
total counting time = 10 min?
If the stochastic nature of radioactive decay is the only source of
variation, procedures 1 and 2 will give the same information.
Selecting procedure 2 makes it possible to test the stability of the
counting instrumentation.
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29. The χ2-test
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30. Error propagation, composite measurements
Rules for estimating the standard error of a composite quantity,
based on independent stochastic variables

Sum or difference:
z= x ± y
=
sz

Product or quotient:
z = x/ y
sz
=
z
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2
2
sx + s y
eller
z = xy
 sx   s y 
  + 
x  y
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2
2
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31. Example:
Correction for background, error estimation
Gross count (radioactive sample + background):
Background:
sr =
∆M B
rB =
∆t
∆M B
srB =
∆t
Net count
(radioactive sample – background):
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∆M
∆t
∆M
∆t
rnet = r − rB
srnet
=
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r=
sr2 + sr2B
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32. Example:
Correction for background, error estimation
∆t (s)
∆M (counts)
Unknown
200
1536
Background (B)
400
446
Sample
=
r
∆M 1536
= = 7.68 cps
∆t
200
=
rB
∆M B 446
= = 1.12 cps
∆tB
400
∆M
=
∆t
=
sr
=
srB
rnet =r − rB =7.68 − 1.12 =6.56 cps
1536
= 0.196 cps
200
∆M B
=
∆tB
srnet =
sr2 + sr2B =
446
= 0.053 cps
400
0.1962 + 0.0532
= 0.203 cps
rnet 6.6 ± 0.2 cps
=
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33. Reporting the counting results
Since radioactive decay is a stochastic phenomenon, it is
meaningsless to report the results from pulse countings
without appending an estimate of the associated error
(uncertainty).
E.g., it is senseless to report the result of pulse counting as
6.6 cps.
An estimate of the error of measurement must always be
appended :
6.6 ± 0.2 ips
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34. Example from worksheet (Laboratory 1)
rnet = r − rB
sr=
, net
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sr2 + sr2, B
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35. Precision and accuracy
To every physical measurement is associated a true value.
When describing the validity of measurement data, we must
differentiate between:

Precision (reproducibility): How close are the measurements
to each other?
(repeated measurements of the same quantity)

Accuracy: How close are the measurements to the true value?
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36. Accuracy and precision of measurements
Good accuracy,
good precision
Bad accuracy, bad
precision
Good accuracy,
bad precision
Bad accuracy,
good precision
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37. About precision and accuracy

The precision of a method (measurement technique) may be
explored by statistical tools

The accuracy of a method can only be ascertained from a
detailed knowledge of its systematic sources of error —
statistical tools cannot be used for this purpose.
Good accuracy = absence of systematic errors
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38. What is a measurement?
(cf. questions to the participants during Laboratory 1)
Measurement: Process of experimentally obtaining one or more
quantity values that can reasonable be attributed to a
quantity
Quantity: Property of a phenomenon, body or substance, where
the property has a magnitude that can be expressed as a
number and a reference
Quantity value: Number and reference together expressing the
magnitude of a quantity
Measurement principle: Phenomenon serving as a basis of a
measurement
Definitions from: International vocabulary of metrology – Basic and general concepts
and associated terms (VIM), ISO/IEC Guide 99:2007
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39. ... and what that means in practice

Measurement principle: Detecting the number of counts
(pulses) in a given period of time in a detector designed to
absorb and detect particles or photons of a given kind of
ionizing radiation.

Determination of count rate (r = ∆M/∆t)

If the counting efficiency (E) is also known, the count rate
may be used to compute the activity of a radioactive sample:
∆M
= =
r
∆t
ε
∆N
∆t
∆N r
= =
A
∆t ε
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40. Sources of error in radioactivity measurements
– cf. Laboratory 2
Type of
measurement
Error source always
present
Possible error
source
Determining count rate of
single sample
•Stochastic nature of
radioactive decay (Poisson
error)
Instrumental instability
(stability test, χ2-test,
repeat count of sample)
Determining activity of
single sample
•Stochastic nature of
radioactive decay (Poisson
error)
•Error (random/systematic) in counting efficiency
determination
Instrumental instability
Determining radioactive
concentration
•Stochastic nature of
radioactive decay (Poisson
error)
•Error (random/systematic) in counting efficiency
determination
•Error (random/systematic) in sample preparation
(pipetting error)
Instrumental stability
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