1. 3/2003 Rev 1 II.3.15b – slide 1 of 19 IAEA Post Graduate Educational Course Radiation Protection and Safe Use of Radiation Sources Part IIQuantities and Measurements Module 3Principles of Radiation Detection and Measurement Detection and Measurement Session 15bSpectral Analysis Session II.3.15b

2. 3/2003 Rev 1 II.3.15b – slide 2 of 19 Objectives Upon completion of this session, the student will be able to:  Describe how curve fitting is used to identify peaks in spectral analysis  Understand that a Gaussian analysis is used  Describe how background is modeled to aid in peak identification

3. 3/2003 Rev 1 II.3.15b – slide 3 of 19 Objectives  Understand how chi-squared is used to determine the least squares fit methodology  Describe acceptable values of  Explain how the peak channel and area are used to identify the radionuclide and the corresponding activity 2222N

4. 3/2003 Rev 1 II.3.15b – slide 4 of 19 Curve Fitting Curve fitting is a standard technique in which the parameters of a fitting function are varied to best describe the data. The photopeaks for the NaI and Ge detectors are well described by a Gaussian or Normal, distribution. This can be seen in the next slide.

5. 3/2003 Rev 1 II.3.15b – slide 5 of 19 Curve Fitting The solid curve is a Gaussian plus background fit to data taken from the NaI MCA/detector system that is used in laboratories. The error bars are equal to the square-root of the counts in the various channels.

6. 3/2003 Rev 1 II.3.15b – slide 6 of 19 Curve Fitting The approach taken in curve fitting is to assume that the data follow a certain function which contain a number of unknown parameters. Then the parameters are varied to “best fit” the data. We will be fitting the photopeaks in our gamma spectra, and assume that the photopeak has the shape of a Gaussian function due to the gamma ray plus a background. We will limit our fit to the data around the photopeak where these assumptions apply. A Gaussian function plus background can be described by 5 parameters:

7. 3/2003 Rev 1 II.3.15b – slide 7 of 19 The function Y(C) represents the number of counts in channel C for the theoretical fitting function. where: C is the channel number p is the peak center h is the height of the Gaussian function  is related to the width of the Gaussian shape Curve Fitting Y(C) = he + background (b 1, b 2 ) C-p  -2

8. 3/2003 Rev 1 II.3.15b – slide 8 of 19 Background Description The background function that we use is a flat plateau before the peak of height b 1, a flat plateau after the peak of height b 2, and a line connecting the two plateaus. The plateau before the peak stops at channel p-2 , and the plateau after the peak starts at channel p+2 . So the line starts at channel number p-2  with height b 1, and ends at channel number p+2  with height b 2.

9. 3/2003 Rev 1 II.3.15b – slide 9 of 19 Best Fit Technique The “best fit” to the data is determined by varying the 5 parameters in the fitting function Y(C) so that Y(C) comes as close to the data as possible. Mathematically this is accomplished by defining a chi-square function,  2, as follows: [Y(C) – Exp(C)] 2 Exp(C)  CfCfCfCf CiCiCiCi  2 

10. 3/2003 Rev 1 II.3.15b – slide 10 of 19 Curve Fitting Model Exp(C) is the experimental value for the number of counts in channel C. The statistical uncertainty of Exp(C) is from our analysis of statistical uncertainty. Thus the uncertainty squared is just Exp(C), which is the denominator in the fraction above. For a particular channel C, (Y(C) – Exp(C)) is just the difference between the fitting function and the data. One squares this difference, to make it positive, then divides by the uncertainty squared.

11. 3/2003 Rev 1 II.3.15b – slide 11 of 19 Chi-Squared Analysis The chi-square function is just the sum of the sum of the squares of the difference between the fitting function, Y(C), and the data divided by the error from an initial channel C i to a final channel C f. The smaller the value of the  2 function, the better the curve Y(C) fits the data.

12. 3/2003 Rev 1 II.3.15b – slide 12 of 19 Curve Fitting Parameters The function Y(C) and  2 contain 5 parameters: h, p, , b 1, and b 2. The “best fit” is determined by finding values for these 5 parameters which make  2 as small as possible. When the function  2 is minimized, the curve Y(C) will be as close to the data as possible. This technique is called chi-square minimization, and is used in many areas of data analysis.

13. 3/2003 Rev 1 II.3.15b – slide 13 of 19 Curve Fitting In the laboratory, a computer program will do all the calculations for us. We will only need to supply the initial channel C i and final channel C f for the Gaussian fit. The computer program will vary the 5 parameters to find values that make the  2 function as small as possible.

14. 3/2003 Rev 1 II.3.15b – slide 14 of 19 Acceptable Values for Chi-Squared How small should  2 be? It is best to divide  2 by the number of data points. This number is referred to as the chi-square per data point, and tells us how many standard deviations (on the average) the fit is away from each data 2222N point. The chi-square per data point,, should be less than 2.0 for an acceptable fit.

15. 3/2003 Rev 1 II.3.15b – slide 15 of 19 Ideally should be between 1.0 and 1.5. The computer program will print the to let the user know the quality of the fit. The users main task is to supply the channel window for the fit, C i and C f. Acceptable Values for Chi-Squared 2222N 2222N

16. 3/2003 Rev 1 II.3.15b – slide 16 of 19 Curve Fitting You want to be sure that you include enough of the photopeak and flat background, but not too much extraneous background in choosing the window for the fit. Your final results should not be too sensitive (hopefully) to the choice of channel window.

17. 3/2003 Rev 1 II.3.15b – slide 17 of 19 Curve Fitting Two parameters from the fit will be of interest to us: the peak center p and the area under the peak A. The accuracy which we can extract these from the data depend on our knowledge about the shape of the peak and background. Fortunately the peaks are very close to a Gaussian function, and the background is well parameterized by the model.

18. 3/2003 Rev 1 II.3.15b – slide 18 of 19 Peak and Activity Determination  The peak center enables identification of the channel number, which is correlated to the energy of the photon. Knowing the energy, and library is evaluated by the software which identifies the radionuclide  The peak area is used to determine the activity present in the sample analyzed at the time the count was conducted

19. 3/2003 Rev 1 II.3.15b – slide 19 of 19  http://www.canberra.com/literature/basic_ principles/spectrum.htm  http://www.canberra.com/literature/techni cal_ref/gamma/ref_gamma.htm  Knoll, G.T., Radiation Detection and Measurement, 3 rd Edition, Wiley, New York (2000) Where to Get More Information