Joint ICTP-IAEA School on Novel Experimental Methodologies for Synchrotron Radiation Applications in Nano-science and Environmental Monitoring Quantitative XRF Analysis algorithms and their practical use Piet Van Espen piet.vanespen@uantwerpen.be 20 Nov 2014 1
Content • Quantitative analysis • Relation between intensity and concentration • Consequences of this relation • The fundamental parameter method • Calibration curves • Dealing with detection limits • Some final remarks 2
Quantitative analysis in XRF The NET intensity of the characteristic x-ray lines is proportional to the concentration NET = background corrected and interference free Use mainly Ka or La lines of elements 3
Quantitative results Only with major elements (conc. range 100% - 5%) homogeneous samples uncertainty of < 1% relative i.e. 23.40 +- 0.12 % Cu (0.5 % relative error) minor elements (conc. range 5% - 0.1%) uncertainty of 5% relative trace elements (<0.1%) uncertainty of >5 % Semi-quantitative results The real situation in uncertainties between 5 and 30% relative many XRF applications Qualitative results The reality for the presence/absence of elements analysis of culturale heritage samples 4
The fundamental parameter relation Derivation of the relation between concentration and X-ray measured intensity: the Sherman equation 5
3.1 The fundamental parameter relation Derivation of the relation between Concentration and X-ray measured intensity: the Sherman equation Mono-energetic excitation Sample 95% Al, 5% Fe The measured intensity (cps) of the Fe K α x-rays depend on… (1) How many primary x-ray reach the Sample: 95 % Al, 5% Fe sample at a certain depth d (2) How many Fe K-vacancies are x + dx produced and how many of them cause the (2) x (1) emission of K α photons Θ 2 Θ 1 (3) How many of those Fe K α photons Ω 1 Ω 2 I o , E o can leave the sample and get detected (3) I i , E Fe K α Detector X-ray source 6
1) Number of primary x-ray that reach at depth x: d Path traveled: Sample: 95 % Al, 5% Fe x l = sin Θ 1 x + dx x x l X-ray intensity impinging on depth x : Θ 1 I x = I 0 ( E 0 ) exp[ − µ M ( E 0 ) ρ M x/ sin Θ 1 ] ρ M density of the sample (matrix) µ M ( E 0 ) mass att. coeff. of the matrix for the primary radiation 7
2) Number of Fe K α photons emitted from dx : Number of Fe vacancies created in the layer dx at depth x dx τ Fe ( E 0 ) ρ I x sin Θ 1 ρ Fe - "density" of Fe, gram Fe per cm 3 [g/cm 3 ] � τ Fe ( E 0 ) – fraction of photons that are absorbed and create vacancies in any shell photo-electric mass absorption coefficient of Fe [cm 2 /g] ✓ ◆ 1 − 1 x Fraction of K shell vacancies: (J K - K-edge jump ratio of Fe) τ K , Fe ( E 0 ) = τ Fe ( E 0 ) × J K x Fraction emitted as K photons: ×ω K ( ω K - K-shell fluorescence yield of Fe ) � ⇥ (f K α - K α to total K (K α +K β ) ratio) x Fraction emitted as K α photons: × f K α � 1 − 1 ⇥ dx dI Fe = f K α ω K τ Fe ( E 0 ) ρ Fe I x J K sin Θ 1 8
3) Number of Fe K α that reach the detector � ⇥ Path traveled: d x Sample: 95 % Al, 5% Fe l = sin Θ 2 x + dx x l x × Attenuation of Fe K α x-ray from layer at depth x : Θ 2 I i , E Fe K α exp[ − µ M ( E Fe K α ) ρ M x/ sin Θ 2 ] µ M ( E Fe K α ) mass att. coeff. of the matrix for Fe K α Ω 2 × Fraction viewed by the detector: 4 ⇤ × Attenuation in air path, detector windows… � ( E Fe K α ) (detector efficiency) dI Fe K α = Ω 2 4 ⇤� ( E Fe K α ) exp[ − µ M ( E Fe K α ) ⌅ M x/ sin Θ 2 ] dI Fe � ⇥ 9
Combination of the 3 terms � ⇥ dI Fe K α = Ω 2 4 ⇤� ( E Fe K α ) exp[ − µ M ( E Fe K α ) ⌅ M x/ sin Θ 2 ] dI Fe � 1 − 1 ⇥ dx � ⇥ dI Fe = f K α ⌃ K ⇧ Fe ( E 0 ) ⌅ Fe I x J K sin Θ 1 x I x = I 0 ( E 0 ) exp[ − µ M ( E 0 ) ⌅ M x/ sin Θ 1 ] � ⇥ 1 − 1 K Fe = f K α ⌃ K Define “fundamental” constants J K Ω 2 G = geometrie factor 4 π sin Θ 1 χ M ( E Fe K α , E 0 ) = µ M ( E Fe K α ) + µ M ( E 0 ) absorption term sin Θ 2 sin Θ 1 detected intensity of Fe K α from a layer dx at depth x dI Fe K α dx = G � ( E Fe K α ) K Fe ⌅ Fe ⇧ Fe ( E 0 ) exp[ − ⌃ M ( E Fe K α , E 0 ) ⌅ M x ] dxI 0 10
Intensity from the entire sample: integration over thickness d ⌃ x = d I Fe K α = G � ( E Fe K α ) K Fe ⌅ Fe ⇧ Fe ( E 0 ) I 0 exp[ − ⌃ M ⌅ M x ] dx x =0 0 � exp[ − ⌃ M ⌅ M x ] � I Fe K α = G � ( E Fe K α ) K Fe ⌅ Fe ⇧ Fe ( E 0 ) I 0 � ⌃ M ⌅ M � d ⌅ 1 − e ( − χ M ρ M d ) ⇧ I Fe K α = G � ( E Fe K α ) K Fe ⌅ Fe ⇧ Fe ( E 0 ) I 0 ⌃ M ⌅ M but ρ Fe / ρ M = w Fe weight fraction of Fe in the sample Relation between intensity of the K α line and weight fraction of element i for mono-energetic excitation of a sample of thickness d I i = G � ( E i ) K i w i ⇧ i ( E 0 )1 − e − χ M ( E i ,E 0 ) ρ M d I 0 ⌃ M ( E i , E 0 ) Or if we consider the sample as “infinitely” thick 1 I i = G � ( E i ) K i w i ⇧ i ( E 0 ) ⌃ M ( E i , E 0 ) I 0 11
Consequences of this relation Define the “sensitivity” for element i S i = ✏ ( E i ) K i ⌧ i ( E 0 ) sensitivity depends on the photo-electric cross section, thus of the excitation energy ( E 0 ) � = µ M ( E i ) + µ M ( E 0 ) The absorption term is sin ✓ 2 sin ✓ 2 Intensity of element i having a weight fraction w i 1 − exp( − �⇢ d ) for a “intermediate thick” sample I i = I 0 GS i w i � 1 for a “infinity thick” sample I i = I 0 GS i w i � 12
Depth of analysis is small and depends on the element analyzed Variation with thickness 1.200 1.000 0.800 I(d)/I(inf) Al 0.600 Fe 0.400 0.200 0.000 0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 rd g/cm2 100 µm at density of 1 g/cm 3 13
The relation between is not necessary linear and depends on the element and the matrix if the absorption is nearly constant linear calibration lines can be used 1.000 0.900 0.800 Fe 0.700 0.600 Intensity 0.500 0.400 0.300 Al 0.200 0.100 0.000 0.00 0.20 0.40 0.60 0.80 1.00 Conc Wi 14
The fundamental parameter method Standardless FP method All elements in the sample give characteristic lines in the spectrum Set of n equation with n +1 unknowns I 0 G , w i 1 = I Al I 0 GS Al w Al � Al 1 = I Fe I 0 GS Fe w Fe � Fe With w Al µ Al ( E Al ) + w Fe µ Fe ( E Al ) + w Al µ Al ( E 0 ) + w Fe µ Fe ( E 0 ) = � Al sin ✓ 2 sin ✓ 1 w Al µ Al ( E Fe ) + w Fe µ Fe ( E Fe ) + w Al µ Al ( E 0 ) + w Fe µ Fe ( E 0 ) = � Fe sin ✓ 2 sin ✓ 1 Need one more equation w Al + w Fe = 1 Can be solved iteratively 15
Table 3: Results obtained on NIST 1108 Naval Brass CRM Line Compound Estim. Conc. Stdev Certified value Mn-Ka Mn 470ppm 90ppm 0.025% Ni-Ka Ni <219.8 ppm 0.033% Fe-Ka Fe 670ppm 70ppm 0.05% Cu-Ka Cu 66.9% 0.1% 64.95% Zn-Ka Zn 32.92% 0.07% 34.43% Table 4: Results obtained on NIST 1156 Steel CRM Line Compound Estim. Conc. Stdev Certified value Mo-Ka Mo 2.86% 0.01% 3.1% Cu-Ka Cu 0.11% 0.02% 0.025% Fe-Ka Fe 70.7% 0.2% 69.7% Ni-Ka Ni 17.8% 0.1% 19.0% Cr-Ka Cr 0.22% 0.02% 0.2% Mn-Ka Mn 0.27% 0.03% 0.21% Co-Ka Co 7.82% 0.07% 7.3% Ti-Ka Ti 0.25% 0.05% 0.21% 16
Standard FP method use at least one standard to determine I 0 G No normalisation ∑ w i =1, check for correctness possible Problem with FP method concentration of ALL elements must be estimated to do the absorption correction χ • Metals often ok for metals • Geological material (stone, sediments, pottery...) contains oxygen from stochiometry Al 2 O 3 , CaO, Fe 2 O 3 (FeO?) • Organic material Missing C, O, N 17
Calibration curves If the matrix remains more or less constant then the absorption term χ remains also constant 1 I i = I 0 GS i w i � or I i = b 1 × w i I i = b 0 + b 1 × w i or better Straight line equation y = b 0 + b 1 × x 18 18
Works for e.g. organic material Concentration range is always limited Standards and unknown must be measured under the same conditions and intensities corrected for measuring time and tube current Calib Curve Br in indolinone 12000 10000 y = 77.613x + 23.137 8000 I Br Ka / 500s 6000 4000 2000 0 0 20 40 60 80 100 120 140 160 -2000 Br ppm 19 19
Calibration using the incoherent scattered radiation As the matrix changes also the amount of Compton scattering changes. Normalising with the intensity of the Compton peak helps I i = b 0 + b 1 w i I Inc Useful for quantitative analysis of geological material 20 20
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