3/2003 rev 1 ii.3.15b – slide 1 of 19 iaea post graduate educational course radiation protection...
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3/2003 Rev 13/2003 Rev 1 II.3.15b – slide II.3.15b – slide 11 of 19 of 19IAEA Post Graduate Educational CourseIAEA Post Graduate Educational Course
Radiation Protection and Safe Use of Radiation SourcesRadiation Protection and Safe Use of Radiation Sources
Part IIPart II Quantities and MeasurementsQuantities and Measurements
Module 3Module 3 Principles of RadiationPrinciples of Radiation Detection and MeasurementDetection and Measurement
Session 15bSession 15b Spectral AnalysisSpectral Analysis
Session II.3.15bSession II.3.15b
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ObjectivesObjectives
Upon completion of this session, the student Upon completion of this session, the student will be able to:will be able to:
Describe how curve fitting is used to identify Describe how curve fitting is used to identify peaks in spectral analysispeaks in spectral analysis
Understand that a Gaussian analysis is usedUnderstand that a Gaussian analysis is used
Describe how background is modeled to aid Describe how background is modeled to aid in peak identificationin peak identification
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ObjectivesObjectives
Understand how chi-squared is used to Understand how chi-squared is used to determine the least squares fit methodologydetermine the least squares fit methodology
Describe acceptable values ofDescribe acceptable values of
Explain how the peak channel and area are Explain how the peak channel and area are used to identify the radionuclide and the used to identify the radionuclide and the corresponding activitycorresponding activity
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Curve FittingCurve Fitting
Curve fitting is a standard technique in which Curve fitting is a standard technique in which the parameters of a fitting function are varied to the parameters of a fitting function are varied to best describe the data. The photopeaks for the best describe the data. The photopeaks for the NaI and Ge detectors are well described by a NaI and Ge detectors are well described by a Gaussian or Normal, distribution. This can be Gaussian or Normal, distribution. This can be seen in the next slide.seen in the next slide.
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Curve FittingCurve Fitting
The solid The solid curve is a curve is a Gaussian plus Gaussian plus background background fit to data fit to data taken from the taken from the NaI NaI MCA/detector MCA/detector system that is system that is used in used in laboratories. laboratories.
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Channel Number
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The error bars are equal to the square-root of the The error bars are equal to the square-root of the counts in the various channels.counts in the various channels.
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Curve FittingCurve Fitting
The approach taken in curve fitting is to assume The approach taken in curve fitting is to assume that the data follow a certain function which contain that the data follow a certain function which contain a number of unknown parameters. Then the a number of unknown parameters. Then the parameters are varied to “best fit” the data. We will parameters are varied to “best fit” the data. We will be fitting the photopeaks in our gamma spectra, and be fitting the photopeaks in our gamma spectra, and assume that the photopeak has the shape of a assume that the photopeak has the shape of a Gaussian function due to the gamma ray plus a Gaussian function due to the gamma ray plus a background. We will limit our fit to the data around background. We will limit our fit to the data around the photopeak where these assumptions apply. A the photopeak where these assumptions apply. A Gaussian function plus background can be Gaussian function plus background can be described by 5 parameters:described by 5 parameters:
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The function Y(C) represents the number of The function Y(C) represents the number of counts in channel C for the theoretical fitting counts in channel C for the theoretical fitting function.function.
where:where:C is the channel numberC is the channel numberp is the peak centerp is the peak centerh is the height of the Gaussian functionh is the height of the Gaussian function is related to the width of the Gaussian shapeis related to the width of the Gaussian shape
Curve FittingCurve Fitting
Y(C) = he + background (bY(C) = he + background (b11, b, b22))C-pC-p--
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Background DescriptionBackground Description
The background function that we use is a flat The background function that we use is a flat plateau before the peak of height bplateau before the peak of height b11, a flat , a flat
plateau after the peak of height bplateau after the peak of height b22, and a line , and a line
connecting the two plateaus. The plateau connecting the two plateaus. The plateau before the peak stops at channel p-2before the peak stops at channel p-2, and the , and the plateau after the peak starts at channel p+2plateau after the peak starts at channel p+2. . So the line starts at channel number p-2So the line starts at channel number p-2 with with height bheight b11, and ends at channel number p+2, and ends at channel number p+2
with height bwith height b22..
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Best Fit TechniqueBest Fit Technique
The “best fit” to the data is determined by The “best fit” to the data is determined by varying the 5 parameters in the fitting function varying the 5 parameters in the fitting function Y(C) so that Y(C) comes as close to the data as Y(C) so that Y(C) comes as close to the data as possible. Mathematically this is accomplished possible. Mathematically this is accomplished by defining a chi-square function, by defining a chi-square function, 22, as , as follows:follows:
[Y(C) – Exp(C)][Y(C) – Exp(C)]22
Exp(C)Exp(C)CCff
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Curve Fitting ModelCurve Fitting Model
Exp(C) is the experimental value for the number of Exp(C) is the experimental value for the number of counts in channel C. The statistical uncertainty of counts in channel C. The statistical uncertainty of Exp(C) is from our analysis of statistical uncertainty. Exp(C) is from our analysis of statistical uncertainty. Thus the uncertainty squared is just Exp(C), which is Thus the uncertainty squared is just Exp(C), which is the denominator in the fraction above. For a particular the denominator in the fraction above. For a particular channel C, (Y(C) – Exp(C)) is just the difference channel C, (Y(C) – Exp(C)) is just the difference between the fitting function and the data. One squares between the fitting function and the data. One squares this difference, to make it positive, then divides by the this difference, to make it positive, then divides by the uncertainty squared. uncertainty squared.
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Chi-Squared AnalysisChi-Squared Analysis
The chi-square function is just the sum of the The chi-square function is just the sum of the sum of the squares of the difference between sum of the squares of the difference between the fitting function, Y(C), and the data divided the fitting function, Y(C), and the data divided by the error from an initial channel Cby the error from an initial channel C ii to a final to a final
channel Cchannel Cff. The smaller the value of the . The smaller the value of the 22
function, the better the curve Y(C) fits the data. function, the better the curve Y(C) fits the data.
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Curve Fitting ParametersCurve Fitting Parameters
The function Y(C) and The function Y(C) and 22 contain 5 parameters: contain 5 parameters: h, p, h, p, , b, b11, and b, and b22. The “best fit” is determined . The “best fit” is determined
by finding values for these 5 parameters which by finding values for these 5 parameters which make make 22 as small as possible. When the as small as possible. When the function function 22 is minimized, the curve Y(C) will be is minimized, the curve Y(C) will be as close to the data as possible. This as close to the data as possible. This technique is called chi-square minimization, technique is called chi-square minimization, and is used in many areas of data analysis. and is used in many areas of data analysis.
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Curve FittingCurve Fitting
In the laboratory, a computer program will do In the laboratory, a computer program will do all the calculations for us. We will only need to all the calculations for us. We will only need to supply the initial channel Csupply the initial channel Cii and final channel and final channel
CCff for the Gaussian fit. The computer program for the Gaussian fit. The computer program
will vary the 5 parameters to find values that will vary the 5 parameters to find values that make the make the 22 function as small as possible. function as small as possible.
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Acceptable Values for Acceptable Values for Chi-SquaredChi-Squared
How small should How small should 22 be? It is best to divide be? It is best to divide 22 by the number of data points. This number is by the number of data points. This number is referred to as the chi-square per data point, referred to as the chi-square per data point, and tells us how many standard deviations (on and tells us how many standard deviations (on the average) the fit is away from each datathe average) the fit is away from each data
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NNpoint. The chi-square per data point, , point. The chi-square per data point, ,
should be less than 2.0 for an acceptable fit. should be less than 2.0 for an acceptable fit.
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Ideally should be between 1.0 and 1.5. The Ideally should be between 1.0 and 1.5. The
computer program will print the to let the computer program will print the to let the
user know the quality of the fit. The users main user know the quality of the fit. The users main
task is to supply the channel window for the fit, task is to supply the channel window for the fit,
CCii and C and Cff. .
Acceptable Values for Acceptable Values for Chi-SquaredChi-Squared
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Curve FittingCurve Fitting
You want to be sure that you include enough You want to be sure that you include enough of the photopeak and flat background, but not of the photopeak and flat background, but not too much extraneous background in choosing too much extraneous background in choosing the window for the fit. Your final results the window for the fit. Your final results should not be too sensitive (hopefully) to the should not be too sensitive (hopefully) to the choice of channel window.choice of channel window.
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Curve FittingCurve Fitting
Two parameters from the fit will be of interest Two parameters from the fit will be of interest to us: the peak center p and the area under the to us: the peak center p and the area under the peak A. The accuracy which we can extract peak A. The accuracy which we can extract these from the data depend on our knowledge these from the data depend on our knowledge about the shape of the peak and background. about the shape of the peak and background. Fortunately the peaks are very close to a Fortunately the peaks are very close to a Gaussian function, and the background is well Gaussian function, and the background is well parameterized by the model.parameterized by the model.
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Peak and Activity DeterminationPeak and Activity Determination
The peak center enables identification of the The peak center enables identification of the channel number, which is correlated to the channel number, which is correlated to the energy of the photon. Knowing the energy, energy of the photon. Knowing the energy, and library is evaluated by the software and library is evaluated by the software which identifies the radionuclidewhich identifies the radionuclide
The peak area is used to determine the The peak area is used to determine the activity present in the sample analyzed at activity present in the sample analyzed at the time the count was conductedthe time the count was conducted
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http://www.canberra.com/literature/http://www.canberra.com/literature/basic_principles/spectrum.htmbasic_principles/spectrum.htm
http://www.canberra.com/literature/http://www.canberra.com/literature/technical_ref/gamma/ref_gamma.htmtechnical_ref/gamma/ref_gamma.htm
Knoll, G.T., Knoll, G.T., Radiation Detection and Radiation Detection and MeasurementMeasurement, 3, 3rdrd Edition, Wiley, New Edition, Wiley, New York (2000)York (2000)
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