Isotopteknik f07 2011_eng


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Statistics of nuclear decay

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Isotopteknik f07 2011_eng

  1. 1. Radioactive decay and counting statistics Jørgen Gomme The decay equation Predictability and chance Radioactivity measurements and statistics in practice F07
  2. 2. Outline      The decay equation Stochastic nature of radioactive decay Poisson-distribution Controlling the stability of counting instrumentation Accuracy and precision 2011 Isotope techniques, F07 (JG) 2
  3. 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 2011 Isotope techniques, F07 (JG) 3
  4. 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 2011 # decays ε ∆N ∆t counting efficiency Isotope techniques, F07 (JG) 4
  5. 5. Predictability and chance... 2011 Isotope techniques, F07 (JG) 5
  6. 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 2011 Isotope techniques, F07 (JG) 6
  7. 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: 2011 Isotope techniques, F07 (JG) A≡ dN dt A = A0e − λt 7
  8. 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 2011 Isotope techniques, F07 (JG) 8
  9. 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 2011 Isotope techniques, F07 (JG) 9
  10. 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, 2011 Isotope techniques, F07 (JG) ε 10
  11. 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. 2011 Isotope techniques, F07 (JG) 11
  12. 12. Stochastic nature of radioactive decay   2011 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 Isotope techniques, F07 (JG) 12
  13. 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 %) 2011 Isotope techniques, F07 (JG) 13
  14. 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 2011 Isotope techniques, F07 (JG) ∆t) 14
  15. 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). 2011 Isotope techniques, F07 (JG) 15
  16. 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 2011 Isotope techniques, F07 (JG) 16
  17. 17. Poisson-distribution for different ν P(∆N) For ν > 20 the Poissondistribution approaches a normal distribution ν=1 ν=5 ν = 25 ∆N 2011 Isotope techniques, F07 (JG) 17
  18. 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. 2011 Isotope techniques, F07 (JG) 18
  19. 19. One or several counts on the same sample...? 2011 Isotope techniques, F07 (JG) 19
  20. 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 2011 Isotope techniques, F07 (JG) 20
  21. 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. 2011 Isotope techniques, F07 (JG) 21
  22. 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 2011 Isotope techniques, F07 (JG) 22
  23. 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 2011 10.00 0.32 Isotope techniques, F07 (JG) 23
  24. 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  2011 Isotope techniques, F07 (JG) 24
  25. 25. Practical examples, test of stability... 2011 Isotope techniques, F07 (JG) 25
  26. 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 = 2011 Uncertainty/errors in sample preparation Isotope techniques, F07 (JG) From sample distribution From sampling distribution 26
  27. 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 2011 ε Isotope techniques, F07 (JG) 27
  28. 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. 2011 Isotope techniques, F07 (JG) 28
  29. 29. The χ2-test 2011 Isotope techniques, F07 (JG) 29
  30. 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 2011 2 2 sx + s y eller z = xy  sx   s y    +  x  y Isotope techniques, F07 (JG) 2 2 30
  31. 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): Isotope techniques, F07 (JG) ∆M ∆t ∆M ∆t rnet = r − rB srnet = 2011 r= sr2 + sr2B 31
  32. 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 = 2011 Isotope techniques, F07 (JG) 32
  33. 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 2011 Isotope techniques, F07 (JG) 33
  34. 34. Example from worksheet (Laboratory 1) rnet = r − rB sr= , net 2011 Isotope techniques, F07 (JG) sr2 + sr2, B 34
  35. 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? 2011 Isotope techniques, F07 (JG) 35
  36. 36. Accuracy and precision of measurements Good accuracy, good precision Bad accuracy, bad precision Good accuracy, bad precision Bad accuracy, good precision 2011 Isotope techniques, F07 (JG) 36
  37. 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 2011 Isotope techniques, F07 (JG) 37
  38. 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 2011 Isotope techniques, F07 (JG) 38
  39. 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 2011 Isotope techniques, F07 (JG) 39
  40. 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 2011 Isotope techniques, F07 (JG) 40
  41. 41. 2011 Isotope techniques, F07 (JG) 41