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 p2⇡ 2 2 linear parameter continuum width non-linear
n2 2 1 1 2 Criterion, agreement between model and data = [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) p2⇡ 2 jk jk " ✓ ◆ #
E(i)=Zero + Gain i (4) ⇥
17 1/2 Noise 2 = + ✏ FanoE (5) jk p jk "✓2 2 ln 2◆ #
Gain E(i) Ejk E(i) Ejk 1 T (i, Ejk)= exp erfc + 2 exp 1 p2 p2 2 2 h i
Gain E(i) Ejk S(i, Ejk)= erfc 2Ejk p2
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)
fTK (Ejk)=c0 + c1µDet(Ejk)
(Ejk)=d0 + d1Ejk
Gain E1(i) E ↵ K jk , L jkp2⇡ jkp2 2 jkp2!
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 = (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 = (b) and calculate the ch-square
2. Obtain (calculate, guess...) a change (increment or decrement) Δbj such that one (b + b) < (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 = [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) p2⇡ 2 jk 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 = + ✏ FanoE (5) jk p jk "✓2 2 ln 2◆ #
Gain E(i) Ejk E(i) Ejk 1 T (i, Ejk)= exp erfc + 2 exp 1 20 p2 p2 2 2 h i
Gain E(i) Ejk S(i, Ejk)= erfc 2Ejk p2
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)
fTK (Ejk)=c0 + c1µDet(Ejk)
(Ejk)=d0 + d1Ejk
Gain E(i) E ↵ K jk , L jkp2⇡ jkp2 2 jkp2!
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 p2⇡ i p 2 2 y(i)=b + A exp 2 2 (the same applies to the width of the peaks: resolution p2⇡ 1 calibration)2( xi xp) y(i)=b + A exp 2 p2⇡ 2 2 Gain (Ei E( i)) ⇒ Add additional information to the modelG(i, E)= exp 2 Gain (E)p2⇡(Ei E(2i ))2(E) G(i, E)= Gainexp (E E)2 Gaussian peak shape G(i, E(E)=)p2⇡ exp2 2(Ei ) (E)p2⇡ 2 2(E) E(i)=Zero + Gain i ⇥ Energy relation: E(i)=Zero + Gain i ⇥ 1/2 Noise 2 Resolution relation: (E)= 2 + ✏FanoE1/2 Noise" 2p2 ln 2 # (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 p2⇡ 2 2 2 1 (xi xp) 2 y(i)=bGain+ A exp(E E (i))2 G(i, E)= pexp2⇡ i 2 (E)p2⇡ 2 2(E) Gain (E E(i))2 G(i, E)= exp i E(i)= (ZeroE)p2+⇡Gain i 2 2(E) ⇥
Already better, but we know more 1/2 E(iNoise)=Zero2 + Gain i (E)= + ✏FanoE⇥ p We know (to some extend)"✓2 2 lnthe 2 ◆ratio between# lines of an element 1/2 Noise 2 (E)=IK↵2 IK + ✏FanoE ,p ... I"K✓↵12 2I lnK↵ 2◆ #
I I We can group lines togetherK↵2 (“peakgroup”), K ... 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... 25 26 Highly flexible method • Fit individual lines, multiplets, elements… • Different parametric and non-parametric continuum models • Include escape and sum peaks Quality criteria • Chi-square of fit • uncertainty estimate of parameters Statistically correct • unbiased, minimum variance estimate of the parameters
“Resolving power” is only limited by the noise (counting statistic) BUT THE MODEL MUST BE ACCURATE
27 5. Improvements to the model
28 Incorrectness of the model
Not all peaks follow the energy calibration relation - incoherent (Compton) scatter peaks - spurious peaks (diffraction, γ-rays) - even the relation might not be linear
Not all peaks follow the resolution calibration relation - incoherent scatter peaks (are wider) - spurious peaks
Peaks are certainly not perfect Gaussians - shelf (step) due to detector effects (incomplete charge collection) - tailing due to radiative effects and detector effects - deviation due to natural line width (Lorentzian)
29 Incorrect fitting model ⇒ biased results
especially for trace elements
Solution Adapt the model Very inconvenient (fitting region, which lines to include...) when analyzing for each particular case many spectra
30 y(i)=yCont(i)+ Aj RijP (i, Eij) (1) " # Xj Xk
2 Gain 1 E(i) Ejk P = G(i, Ejk)= exp (2) p2⇡ 2 jk jk " ✓ ◆ #
E(i)=Zero + Gain i (3) ⇥
2 1/2 Noisey(i)=yCont(i)+ Aj RijP (i, Eij) (1) jk = + ✏ FanoEjk (4) p j " k # "✓2 2 ln 2◆ X # X y(i)=yCont(i)+ Aj RijP (i, Eij) (1) " # 2 Xj Xk Gain 1 E(i) Ejk Gain P = G(i, EEjk(i))=Ejk expE(i) Ejk 1 (2) T (i, Ejk)= exp jkperfc2⇡ " 2 + jk # 1 2 p2 ✓ p2 ◆ 2 exp Gain2 2 1 E(i) Ejk P = G(i, Ejk)= exp (2) h jkpi2⇡ " 2 jk # E✓(i)=Zero◆+ Gain i (3) Gain E(i) E ⇥ ImprovementsS(i, E )= erfc jk E(i)=jkZero2+EGain i p (3) jk ⇥ Improve 2 the peak profile 1/2 Noise 2 jk = + ✏ FanoEjk (4) Peak P (i, Ejk)=G(i, Ejk2 )+fSS(i, Ejk)+1/2fT T (i, Ejk) (5) Noise " 2p2 ln 2 # = + ✏ FanoE✓ ◆ (4) jk p jk "✓2 2 ln 2◆ #
Gain E(i) Ejk E(i) Ejk 1 Tail T (i, Ejk)= exp erfc + Gain E(i) Ejk1 E(i) Ejk 1 p2 p2 T (i, Ejk)= exp2 exp 2 2erfc + 2 exp 1 p2 p2 2 2 h i h i Gain E(i) Ejk Gain S(i,E E(ijk) )=Ejk erfc Step S(i, Ejk)= erfc 2Ejk p2 2Ejk p2
P (i, Ejk)=G(i, Ejk)+fSS(i, Ejk)+fT T (i, Ejk) (5) Adding 1 non-linear and 2 linear parameters for each peak!!!
Where is my parsimony gone!!!
1 31
1
1 Step parameterisation
Step fraction fS is related to the MAC of the detector crystal
Step is a fundamental aspect of the detector (charge loss by photo-electrons near the surface of the detector)
Tail fraction parameterisation
Tail fraction fT is related to the MAC of the detector and the type of radiation (Kα and Kβ)
The tail has a component due to the detector and a radiative component
32 y(i)=yCont(i)+ Aj RijP (i, Eij) (1) " # Xj Xk
2 Gain 1 E(i) Ejk P = G(i, Ejk)= exp (2) p2⇡ 2 jk jk " ✓ ◆ #
E(i)=Zero + Gain i (3) ⇥
1/2 Noise 2 = + ✏ FanoE (4) jk p jk Tail width parameteris"ation✓2 2 ln 2◆ #
similar magnitude overGain the entire E(i) Ejk E(i) Ejk 1 T (i, E )= exp erfc + energyjk range 2 exp 1 p2 p2 2 2 h i
Gain E(i) Ejk S(i, Ejk)= erfc 2Ejk p2
P (i, Ejk)=G(i, Ejk)+fSS(i, Ejk)+fT T (i, Ejk) (5)
⇒ Fitting parameters a0, a1, b0, b1, c0, c1, d0, d1
fS(Ejk)=µDet(Ejk)(a0 + a1Ejk)
fTK↵(Ejk)=b0 + b1µDet(Ejk)
fTK (Ejk)=c0 + c1µDet(Ejk)
(Ejk)=d0 + d1Ejk
(compare to Zero, Gain, Noise and Fano)
33
1 improvement
Fit of a NIST SRM 1106 Brass spectrum (SpecTrace 5000, Rh tube)
34 2 1 (xi xp) y(i)=b + A exp p 2 2 2⇡
2 Gain (Ei E(i)) G(i, E )= exp p 2 2(E) (E) 2⇡ 1 (x x )2 y(i)=b + A exp i p p2⇡ 2 2 E(i)=Zero+ Gain i ⇥ Gain (E E(i))2 G(i, E)= exp i 2 (E)p21⇡/2 2 2(E) Noise (E)= + ✏FanoE p "✓2 2 ln 2 ◆ # E(i)=Zero + Gain i ⇥ I I K↵2 , K ... IK↵1 IK↵ 1/2 To account for peak shift and peak broadeningNoise 2 (E)= + ✏FanoE p "✓2 2 ln 2◆ # y(i)=yCont(i)+ Aj RjkP (i, E jk) Need to make the peak profile still a bit more complicated ) j ( k I I X X K↵2 , K ... IK↵1 IK↵ 2 Gain (Ei E(i)+ E) G(i, E )= exp p 2 2(E) (E) 2⇡ y(i)=yCont(i)+ Aj RjkP (i, Ejk) j ( k ) δE peak shift Eparameterini (normallyE E 0.00) Eini + X E X Gain (E E(i)+ E)2 γ peak broadening parameterG(i, E(normally)= 1.00) exp i p 2 ini(E+) 2⇡ 2 (E) ini
Eini E E Eini + E These parameters are constrained to vary within a certain range + ini ini
35
1
1 y(i)=yCont(i)+ Aj RijP (i, Eij) (1) y(i)=yCont(i)+ Aj RijP (i, Eij) (1) j "" k ## XXj XXk
2 GainGain 11 EE((ii)) EEjkjk P P==G(Gi,( Ei,jk Ejk)=)= expexp (2)(2) jk pp2⇡2⇡ 22 jkjk jk "" ✓✓ ◆ #
EE(i()=i)=ZeroZero++GainGain ii (3)(3) ⇥⇥
2 11//22 NoiseNoise 2 jk jk== ++✏✏FanoEFanoEjkjk (4)(4) " p2p2 ln 2 # "✓✓2 2 ln 2◆◆ #
GainGain EE(i()i) EEjk EE((ii)) Ejk 11 The last step T (i, Ejk)= exp jk erfc jk + T (i, Ejk)= 1 exp erfc p 2 + p2 2 exp 12 2 p2 p2 2 exp 2 2 For high Z elements the natural h linei width becomes substantial relative to the detector resolutionh i Gain E(i) Ejk S(i, Ejk)=Gain erfc E(i) Ejk S(i, Ejk)= 2Ejkerfc p2 2Ejk p2
P (i, Ejk)=G(i, Ejk)+fSS(i, Ejk)+fT T (i, Ejk) (5) P (i, Ejk)=G(i, Ejk)+fSS(i, Ejk)+fT T (i, Ejk) (5)
fS(Ejk)=µDet(Ejk)(a0 + a1Ejk) fS(Ejk)=µDet(Ejk)(a0 + a1Ejk) fTK↵(Ejk)=b0 + b1µDet(Ejk) fTK↵(Ejk)=b0 + b1µDet(Ejk) fTK (Ejk)=c0 + c1µDet(Ejk) f (E )=c + c µ (E ) TK (Ejkjk)=0d0 + 1d1EDetjk jk Replace Gaussian with the convolution (Ejk of)= a Gaussiand0 + d1 Ewithjk a Lorentzian = Voigt profile Gain E(i) E ↵ K jk , L Gain p2⇡ E(i ) pE2 2 ↵ p2 jk K jk jk , jkL ! p2⇡ p2 2 p2 with K(...) the complex error functionjk jk jk !
K(x, y)=Re exp( z2)erfc( iz) z = x + iy (6) ⇥36 ⇤
1
1 Natural line width at high Z elements becomes important e.g. W K ~ 50 eV
Gaussians Voigts
37 Original fit of a geological standard (JG1)
38 Improved fit of the geological standard (JG1)
Mo secondary target No background
39 NIST SRM 1155
40 NIST SRM 1247
41 More Details
Handbook of X-ray Spectrometry R. Van Grieken, A. Markowicz Marcel Decker, N.Y. 2002 ISBN: 0-8247-0600-5
Chapter 4: Spectrum Evaluation
42 Some final remarks: The future
Non-linear least-squares works if you have a good parsimonious model if you have TIME
X-ray fluorescence imaging: 256 x 256 image = 65536 x-ray spectra
@ 1 s / spectrum = 65536 seconds = 1092 minutes = 18 hours !!!!
Need to explore new methods Linear models? Multivariate models?
43 6. Final remarks
44 Thanks for your attention
45