Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 1 Radioactive decay is a random process. Fluctuations. Characterization of Data: A series of measurements for equal periods of time (what if only one?) x 1, x 2, x 3, …, x N Experimental mean Frequency distribution function Counting Statistics and Error Prediction

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 2 Counting Statistics and Error Prediction Same mean, less scatter.

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 3 Counting Statistics and Error Prediction Residual Deviation True mean! Variance In terms of F(x)? Measure of internal scatter, but the true mean can never be known! Appendix B Prepare the same calculations and plots for the narrower distribution set. Large data sets, no difference! HW 18

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 4 Counting Statistics and Error Prediction TrialDefinition of SuccessProbability of Success ≡ p CoinHeads1/2 DiceGetting a “six”1/6 Radioactive decay The nucleus decays during the observation time  t 1 - e - t Statistical Models: Binomial. A general (rarely used model). Poisson. Probability of success ( p ) is small. Gaussian. In addition, Number of successes is large. When are all three models identical? Number of trials  number of nuclei in the sample

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 5 Counting Statistics and Error Prediction Binomial Distribution n = number of trials. (number of nuclei!!!) p = success probability per trial. x = number of successes. P(x) = probability of obtaining x successes. Computational consequences…! What if it was very small?

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 6 Counting Statistics and Error Prediction Dice: success ≡ 3, 4, 5, 6. p = 4/6 = 2/3. n = 10. Work it out to obtain this distribution. Vary p and n and comment on results. HW 19

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 7 Counting Statistics and Error Prediction Poisson Distribution ( p << 1) Success ≡ Birthday today. p = 1/365. n = Low cross section. Weak resonance. Short measurement (compared to t 1/2 ). Appendix C We need to know only the product. HW 20 Asymmetric

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 8 Counting Statistics and Error Prediction Gaussian (Normal) Distribution (p 20) Success ≡ Birthday today. p = 1/365. n = Symmetric and slowly varying Can be expressed as a function of . Can be expressed in a continuous form. HW 21

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 9 Counting Statistics and Error Prediction

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 10 Counting Statistics and Error Prediction Calculate the percentage of the samples that will deviate from the mean by less than: one . two . etc … HW 22

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 11 Counting Statistics and Error Prediction Baseline offset Total area under the curve above the baseline 2 , approximately the width of the peak at half height This model describes a bell-shaped curve like the normal (Gaussian) probability distribution function. The center x 0 represents the "mean", while ½ w is the standard deviation. What is FWHM? Resolution? Peak centroid? HW 23

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 12 Counting Statistics and Error Prediction (continued) Applications Applications 1- Match experiment to model and not because we set Assume a specific distribution (Poisson, Gaussian). Set distribution mean to be equal to experimental mean. Compare variance to determine if distribution is valid for actual data set (Chi- squared test).

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 13 We can’t use Gaussian model for this data set. Why? Qualitative comparison. Is  2 close to s 2 ? Close!? Less fluctuation than predicted! But quantitatively? Chi-squared test. Counting Statistics and Error Prediction Only to guide the eye! Back to our example HW 24

By definition: Thus: or Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 14 Counting Statistics and Error PredictionChi-squared The degree to which  2 differs from (N-1) is a measure of the departure of the data from predictions of the distribution.

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 15 Counting Statistics and Error Prediction (interpolation) or smaller larger either gives  = Conclusion: no abnormal fluctuation. Perfect fit

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 16 Counting Statistics and Error Prediction Single measurement S 2 =  2 ≈ x 68% probability that this interval includes the true average value. What if we want 99%..? Fractional standard deviation Need 1%? Count

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 17 Counting Statistics and Error Prediction A series of “single” measurements.

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 18 Counting Statistics and Error Prediction

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 19 Counting Statistics and Error Prediction Net counts = Gross counts – Background MeasuredDerived Gross counts = 1000 Background counts = 400 Net counts = 600  37 (not 600  24) Count Rate = ?  ? (Error propagation). What about derived quantities? (Error propagation). Compare to addition instead of subtraction. (Count, stop, count).

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 20 Counting Statistics and Error Prediction Mean value of multiple independent counts. Assume we record N repeated counts from a single source for equal counting times: For Poisson or Gaussian distributions: So that

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 21 Counting Statistics and Error Prediction Standard error of the mean To improve statistical precision of a given measurement by a factor of two requires four times the initial counting time.

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 22 Counting Statistics and Error Prediction Optimizing counting time. S = (net) source count rate. B = background count rate. T S+B = time to count source + background. T B = time to count background. To minimize  s : Low-level radioactivity Weak resonance Very strong background High signal- to- background ratio How to divide the limited available beam time? HW 25

Radiation Detection and Measurement, JU, 1st Semester, (Saed Dababneh). 23 Counting Statistics and Error Prediction Minimum detectable amount. False positives and false negatives. Background measurement? Without the “source”. Should include all sources except the “source”. Accelerator applications: background with beam. Rest of Chapter 3.