Joint ICTP-IAEA School on Novel Experimental Methodologies for Synchrotron Radiation Applications in Nano-science and Environmental Monitoring Evaluation of XRF Spectra from basics to advanced systems Piet Van Espen [email protected] 20 Nov 2014 1 Content 1. Introduction: some concepts 2. Simple peak integration 3. Method of least squares 4. Fitting of x-ray spectra 5. Improvements to the model 6. Final remarks 2 Some basic concepts Nature and science: a personal view Nature = Signal + Noise Science = Model + Statistics 3 Why spectrum evaluation? element concentrations ⇔ net intensity of fluorescence lines interference free continuum corrected But: frequent peak overlap presence of a continuum Especially in energy-dispersive spectra 4 4 Information content of a spectrum Spectrum contains • Information: energy and intensity of x-rays the signal • Amplitude noise: due to Poisson statistics ► fluctuations in the spectrum the noise • Energy noise: finite resolution of the detector ► nearly Gaussian peaks with a width of ~160 eV 5 Amplitude noise Counting events involves Poisson statistics Poisson probability density function: The probability to observe N counts if the true number is µ Property: Poisson distribution Normal distribution ≅ µ µ and σ2 = µ approximation is very good for µ (or N) ≥ 9 Poisson : P(N | µ=3) Normal : P(x | µ=3 σ2 = 3) 6 2 Spectrum analysis Energy Noise Resolution of ED-XRF spectrometers 2.35⇥ F E (9) Full Width at Half Maximum× (FWHM)× of a peak 2 2 2 FWHMPeak = FWHMElec + FWHMDet (10) 2 Spectrum analysis Electronic noise Intrinsic contribution ~100 eV 2.35⇥ F E (9) × × FWHM2 = FWHM2 +Det2 (10) Peakϵ energy to createElec e-h pairPealk 3.85 eV F Fano factor ~0.114 E x-ray energy in eV Mn Kα @ 5.895 keV FWHMDet = 120 eV FWHMElec = 100 eV => FWHMPeak = 156 eV 7 2 2 Energy Noise Resolution of ED-XRF spectrometers Cr – Mn – Fe overlap at ~20 eV Cr – Mn – Fe overlap at ~160 eV 8 Without amplitude noise (counting statistics) there would be NO PROBLEM But it is part of the nature We can only measure longer or with a more efficient system Without energy noise there would be LITTLE PROBLEM The natural line width of x-rays is only a few eV!!! The observed peak width is the result of the detection process with a fundamental limitation imposed by the Fano factor 9 Information content of a spectrum If no energy noise or no amplitude noise ► could determine the “information” unambiguously Need methods to extract information in a optimum way These methods rely on “addition” information (knowledge) to extract the useful information Not the method itself is important (if implemented correctly) but the correctness of the additional information. 10 10 2. Simple peak integration 11 Simple peak integration Estimate Uncertainty As good as is can get We have to make assumptions if the assumptions (model) are correct! integration limits linear background no interference 12 3. Method of Least Squares 13 Method of least squares Need to “estimate” the net peak area with highest possible • correctness (no systematic error) • precision (smallest random error) Least-squares estimation (fitting): • unbiased • minimum variance Limiting factors: • counting statistical fluctuations (precision) • accuracy of the fitting model 14 Method of least squares, straight line 2 SpectrumData: analysis {xi,yi}, i=1, 2, …, N 2 SpectrumModel: y(i analysis) = b + b x noise 2.35p0 1F i E (9) ⇥ ⇥ 2.35p F E (9) Fitting the model: ⇥ ⇥ estimating2 b0 and2 b1 2 FWHMPeak = FWHMElec + FWHMDet (10) 2 2 2 FWHMPeak = FWHMElec + FWHMDet (10) Criterion: SS = [y y(i)]2 = [y b b x ]2 = min (11) i − i − 0 − 1 i i SS = [y y(ii)]2 = [y b b x ]2 = min (11) X i − X i − 0 − 1 i Xi Xi ⇤SS Set of 2 equations in 2 unknowns b and b =0⇤SS yi = b0n + b1 xi (12) 0 1 ⇤b0 ! =0i yi = b0in + b1 xi Normal equations(12) ⇤b X ! X 0 i i ⇤SS X X 2 =0 xiyi = b0 xi + b1 x (13) ⇤b ⇤SS! i Direct2 analytical solution 1 =0i xiyi = b0 xii + b1 xi (13) ⇤b1 X ! X X Xi X Xi 15 2 2 4. Fitting X-ray Spectra 16 Least-squares estimate of x-ray spectrum parameters position area 1 (x x )2 Peak described by a Gaussian y(i)=b + A exp i − p σp2⇡ 2σ2 linear parameter continuum width non-linear n2 2 1 1 2 Criterion, agreement between model and data χ = [y(i) yi] (1) ⌫ wi − iX=n1 y(i)=yCont(i)+ Aj RijP (i, Eij) (2) " # Minimum:Xj Xk No direct analytical solution 2 Gain 2 1 E(i) Ejk P = G(i, Ejk)=Search χ exp for minimum − (3) σ p2⇡ −2 σjk jk " ✓ ◆ # E(i)=Zero + Gain i (4) ⇥ 17 1/2 Noise 2 σ = + ✏ FanoE (5) jk p jk "✓2 2 ln 2◆ # Gain E(i) Ejk E(i) Ejk 1 T (i, Ejk)= exp − erfc − + 2βσ exp 1 βσ σp2 p2β − 2β2 h i Gain E(i) Ejk S(i, Ejk)= erfc − 2Ejk σp2 P (i, Ejk)=G(i, Ejk)+fSS(i, Ejk)+fT T (i, Ejk) (6) fS(Ejk)=µDet(Ejk)(a0 + a1Ejk) fTK↵(Ejk)=b0 + b1µDet(Ejk) fTKβ(Ejk)=c0 + c1µDet(Ejk) β(Ejk)=d0 + d1Ejk Gain E1(i) E ↵ K − jk , L σjkp2⇡ σjkp2 2σjkp2! K(x, y)=Re exp( z2)erfc( iz) z = x + iy (7) − − ⇥ ⇤ 1 We can still apply the concept of least squares minimising the square of the differences between the model and the data 2 2 1 1 2 χ = χ (b, h, x0,w)= [yi y(xi,b,h,x0,w)] ⌫ i yi − P The sum of squares is a function of the values of the parameters and for a given set of values should be minimum In this case SS describes a 4 dimensional hyper-surface in a 5- dimension space We can only “see” in 3-dimensions but mathematically we can search in a higher dimensional space to locate the minimum h Starting from some initial values we can modify the parameter values until the minimum is reached. w 18 General form of such a search algorithm 1. Select starting values for all parameters bj χ = χ(b) and calculate the ch-square 2. Obtain (calculate, guess...) a change (increment or decrement) Δbj such that one χ(b +∆b) <χ(b) moves towards the minimum: 3. Replace the old parameter values with the new ones b ← b + Δb 4. repeat step 2 until the “true” minimum is found Iterative process AXIL = Analysis of X-ray spectra by Iterative Least-squares 19 n2 2 1 1 2 χ = [y(i) yi] (1) ⌫ wi − iX=n1 Analytically important parameters: net peak areas y(i)=yCont(i)+ Aj RijP (i, Eij) (2) " # Xj Xk Statistical optimal estimate: 2 using correct weight (PoissonGain statistics w1 i =E yi()i) Ejk P = G(i, Ejk)= exp − (3) σ p2⇡ −2 σjk jk " ✓ ◆ # In general y(i) is non-linear → Marquardt – Leverberg algorithm Gradient search ↔ linearisationE(i)=Zero + Gain i (4) Reliable error estimated ⇥ But unstable 1/2 Noise 2 σ = + ✏ FanoE (5) jk p jk "✓2 2 ln 2◆ # Gain E(i) Ejk E(i) Ejk 1 T (i, Ejk)= exp − erfc − + 2βσ exp 1 βσ20 σp2 p2β − 2β2 h i Gain E(i) Ejk S(i, Ejk)= erfc − 2Ejk σp2 P (i, Ejk)=G(i, Ejk)+fSS(i, Ejk)+fT T (i, Ejk) (6) fS(Ejk)=µDet(Ejk)(a0 + a1Ejk) fTK↵(Ejk)=b0 + b1µDet(Ejk) fTKβ(Ejk)=c0 + c1µDet(Ejk) β(Ejk)=d0 + d1Ejk Gain E(i) E ↵ K − jk , L σjkp2⇡ σjkp2 2σjkp2! K(x, y)=Re exp( z2)erfc( iz) z = x + iy (7) − − ⇥ ⇤ 1 10 peaks ⇒ > 30 parameters !!!! WANT WORK in practice!!!! Need parsimony!!! But we can do better 2 We known the energies of the x-ray lines (in most cases)1 (xi xp) y(i)=b +1 A (expx x )−2 Where they are depends on the energy calibration σp2⇡ i p 2σ2 y(i)=b + A exp − 2 2 (the same applies to the width of the peaks: resolutionσp2⇡ 1 calibration)2(σxi xp) y(i)=b + A exp − 2 σp2⇡ 2σ 2 Gain (Ei E(i)) ⇒ Add additional information to the modelG(i, E)= exp − 2 Gainσ(E)p2⇡(Ei E(2iσ))2(E) G(i, E)= Gainexp −(E E)2 Gaussian peak shape G(i,σ E(E)=)p2⇡ exp2σ2(Ei −) σ(E)p2⇡ 2σ2(E) E(i)=Zero + Gain i ⇥ Energy relation: E(i)=Zero + Gain i ⇥ 1/2 Noise 2 Resolution relation: σ(E)= 2 + ✏FanoE1/2 Noise" 2p2 ln 2 # σ(E)= ✓ + ✏◆FanoE p Only 4 non-linear parameters"✓ 2 2 ln 2◆ # For 10 peaks only 14 parameters 21 1 1 1 1 (x x )2 y(i)=b + A exp i − p σp2⇡ 2σ2 2 1 (xi xp) 2 y(i)=bGain+ A exp(E E−(i))2 G(i, E)= σpexp2⇡ i − 2σ σ(E)p2⇡ 2σ2(E) Gain (E E(i))2 G(i, E)= exp i − E(i)=σ(ZeroE)p2+⇡Gain i 2σ2(E) ⇥ Already better, but we know more 1/2 E(iNoise)=Zero2 + Gain i σ(E)= + ✏FanoE⇥ p We know (to some extend)"✓2 2 lnthe 2 ◆ratio between# lines of an element 1/2 Noise 2 σ(E)=IK↵2 IKβ + ✏FanoE ,p ... I"K✓↵12 2I lnK↵ 2◆ # I I We can group lines togetherK↵2 (“peakgroup”), Kβ ... with one “area” and fixed intensity ratiosIK↵1 IK↵ y(i)=yCont(i)+ Aj RjkP (i, Ejk) ( ) Xj Xk Continuum Area Line Peak function ratio shape for j elements (or peak groups) 10 elements ⇒ 10 Area’s + 4 calibration parameters 22 1 1 Further refinements: escape peaks Known position (energy) intensity (escape probability) 23 Sum peaks Known position relative intensity 24 And more Different background models polynomial exponential polynomial Bremsstrahlung background filter background Parameter constraining and more..