Isotopteknik f07 2011_eng

Radioactive decay and counting statistics

Jørgen Gomme

The decay equation Predictability and chance

Radioactivity measurements and statistics in practice

F07

2011 Isotope techniques, F07 (JG) 2

Outline

The decay equation Stochastic nature of radioactive decay Poisson-distribution Controlling the stability of counting instrumentation Accuracy and precision


An amount of material containing N radioactive atoms has the acitivity A:

Number of decays (nuclear disintegrations) per unit time.

dNAdt

≡

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 Bq1 Bq =2.7027×10-11 Ci

N

Activity


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

M Nrt t

ε∆ ∆= =

∆ ∆

# decays# counts

counting efficiency

counting time


Predictability and chance...


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

Consequences for:

Uncertainty in radioactivity measurements

Check of instrument performance

λ


0tA A e λ−=

The decay equation may be derived from the assump-tion, that each individual nucleus of a given radio-nuclide 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 radio-active substance)…

… or between the activity at two different points in time:

The decay law

dNAdt

≡


Number of decays / counts per unit time– according to the decay law

( )( )

0

0

1

1

t

t

N N e

M N e

λ

λε

− ∆

− ∆

∆ = −

∆ = −

From the decay equation we might expect the same number of decays in succesive periods of constant length, ∆t

Assuming constant counting efficiency ε, also the number of counts per period would be constant

None of these predictions are fulfilled in practice: The radioactive decay is a stochastic phenomenon

Number of decays:

Number of counts:


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

MeanEmpiricalStandarddeviation

0

1

2

3

4

5

6

12-1

3

14-1

5

16-1

7

18-1

9

20-2

1

22-2

3

24-2

5

26-2

7

28-2

9

30-3

1

32-3

3

34-3

5

36-3

7

38-2

9

Mean


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, ε

Repeated measurements on the same sample


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.



Probability theory: Repeated meaurements on the same sample will follow a distribution, that may be described analogously to the binomial distri-bution – 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


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 %)


It may be shown mathematically, that in the limiting case the binomial distribution approaches the Poisson-distribution (when the probability for any nucleus decaying in the observation period is small):

ν is the true value (expectation value for the number of decays in the period ∆t)

P(∆N) is the probability for ∆N decays in the period ∆t

Poisson-distribution– the ”law of rare events”

( )0 1 tN e λν − ∆= −


Poisson error

For the Poisson distribution, the theoretical standard devia-tion (standard error, ”Poisson error”) is always equal to the square root of the true value (the expectation value).


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


λ


Poisson-distribution for different ν

P(∆N)

∆N

ν = 25

ν = 5

ν = 1For ν > 20 the Poisson-distribution approaches a normal distribution


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.


One or several counts on the same sample...?


Repeated determinations – what to expect?

n separate measurements, each of duration ∆t, total counting time n∆t

A single measurement of duration n∆t


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.


Absolute and relative standard error

Mrt

∆=

∆

rMst

∆=

∆

( ) 1r rel

Mts M Mt

∆∆= =∆ ∆∆

Count rate:

Absolut standard error(Poisson error):

Relative standard error:

The relative error depends only on the number of counts accumulated

cps

cps


Relative standard error as a function of the number of counts

Number of counts Relative error

100 10.00

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 0.32

M∆ ( )rel 1/ 100%s M= ∆


Nevertheless: Why sometimes doing repeat counts?

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

Occasionally, a counting period ∆t is divided into n subperiods, each of duration ∆t/n, and for the overall result the mean af nindividual countings is used:


Practical examples, test of stability...


Radioactive sample: 86Rb (T½ = 18.63 d)Counting time (∆t = 5 min), constant for all samples

Observations:

Mean: 31396 counts per 5 min

S.D.: 1591 (5.07 %)

Poisson error:

1: Example from Laboratory 2– counting a series of samples, prepared in the same say

Uncertainty/errors in sample prepa-ration

30040 29105 3084033505 32995 3151529915 31860 3367030515

M∆ =

M∆ =

From sample distribution

From sampling distribution

2

emp

( )

1n

M Ms

n

∆ −∆= =

−

∑

Pois 31396 177 (0.56%)s M= ∆ = =


1: Example from Laboratory 2(cont’d)

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 ε

Errors in sample preparation

Stochastic nature of decay


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.

2: Repeated measurements of the same sample (under identical conditions)


The χ2-test


Error propagation, composite measurements

Rules for estimating the standard error of a composite quantity, based on independent stochastic variables

2 2z x y

z x y

s s s

= ±

= +

Sum or difference:

Product or quotient: /z x y= z xy=eller

22yxz

sssz x y

= +


Example: Correction for background, error estimation

Gross count (radioactive sample + background):

r

MrtMst

∆=

∆∆

=∆

Background:

Net count (radioactive sample – background):

B

BB

Br

MrtM

st

∆=

∆∆

=∆

net B

net B

2 2r r r

r r r

s s s

= −

= +


1536 7.68200

Mrt

∆= = =

∆

BB

B

446 1.12400

Mrt

∆= = =

∆

net B 7.68 1.12 6.56r r r= − = − =

Example: Correction for background, error estimation

r1536 0.196200

Mst

∆= = =

∆

B

Br

B

446 0.053400

Ms

t∆

= = =∆

net B

2 2 2 2 0.196 0.053

0.203r r rs s s= + = +

=

Sample ∆t (s) ∆M (counts)

Unknown 200 1536

Background (B) 400 446

net 6.6 0.2r = ±

cps

cps

cps

cps

cps

cps

cps


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


Example from worksheet (Laboratory 1)

net Br r r= −2 2

, ,r net r r Bs s s= +


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?

Precision and accuracy


Accuracy and precision of measurements

Good accuracy, good precision

Bad accuracy, good precision

Bad accuracy, bad precision

Good accuracy, bad precision


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

About precision and accuracy

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

... 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 Nrt t

ε∆ ∆= =

∆ ∆

N rAt ε

∆= =∆

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/systema-tic) in counting efficiency determination

Instrumental instability

Determining radioactive concentration

•Stochastic nature of radioactive decay (Poisson error)•Error (random/systema-tic) in counting efficiency determination•Error (random/systema-tic) in sample preparation (pipetting error)

Instrumental stability


Education

Isotopteknik f07 2011_eng