XRF Analysis: identifying and estimating errors - Prolab Systems · 2012-07-15 · Human Errors during calibration • Multi-linear Regression Analysis – The regression software

XRF Analysis: identifying and estimating errors

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Types of Error

• Gross Error e.g. / Outlier – Instrument failure– Sample contamination– Human errors

• Random Error e.g.– Counting statistics– Sample preparation

• Systematic Error e.g.– Errors in calibration (model, constants)– Dead time losses

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To evaluate the systematic error check the calibration against

well known standards

Accurate Precise and AccuratePrecise

Accuracy and precision

Typical sources of errors and their contribution in rel. %

Random Systematic

Sample taking or inhomogeneity 0 -50

Sample preparation 0 - 2 0 - 50

Spectrometer hardware 0.05-0.2 0.05-0.5

Counting statistics (time)

Absorption and enhancement effects 0 -50

Wrong regression parameters 0 - 200

Calibration standards (quality)

Operator mistakes4

Propagation of Random Errors

errorstatisticscountingCSEE

erroralinstrumentinstE

errornpreparatiosamplesmplE

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−

−

−

++= CSEinstsmplRandom EEEE

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Distribution of Random Errors

– Repeated measurement results are distributed around a mean N– A measure of the error is the “RMS”

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Counting statistical error

C:\Documents and Settings\alexander.komelkov\Desktop\Count_rates.xlsx

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( )

tsmeasurementheofnumber

valueaverage

tmeasuremenpartuculartheofvalue

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nxx

xxRMSRMSE n

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222CSEinstsmplRandom EEEE ++=

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Instrumental error

• Instrumental error consist of:– High voltage generator variations– X-ray tube instability – Sample positioning – Other hardware issues

Usually Einst < 0.05 rel.% of the intensity (for WD XRF spectrometers)

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Counting Statistical Error (CSE)

• Rp - gross peak count rate

• Rb - background count rate at the peak wavelength

• Net count rate = Rp - Rb

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• r - count rate (counts per second, cps)

• t - counting time (seconds)

• N - total number of collected counts

trNCSE counts ⋅==)(

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tr

ttr

tN

tCSE

CSE countscountrate =⋅=== )()(

Relative Counting Statistical Error (RCSE)




CSErel decreases with increase of intensity or measurement time!

trNNNCSE countsrel

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Sample preparation errors

• Due to errors of other equipment (scales, pipettes, press,…)

• Due to differences in batches of used chemicals (binders, fluxes, diluents,…)

• Not accurate handling of the samples

• Etc….

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Evaluation of errors contribution

! softwarethebyreportedRMS

relrel RMSE =

222CSEinstsmplRandom EEEE ++=

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Statistic approach of the random error evaluation

1 bead 10 times

10 beads 1 time

Compound TiO2 Na2O TiO2 Na2OMeasure time (s) 8 36 8 36

Mean Raw kcps 7.93 1.03 7.96 1.03RMS rel. Raw kcps (%) 0.53 0.53 0.54 1.28

Mean Bg cor. kcps 7.17 0.84 7.20 0.84RMS rel. Bg cor. kcps (%) 0.60 0.58 0.63 1.53

CSE mean (kcps) 0.031 0.005 0.032 0.005CSE rel. (%) 0.397 0.517 0.396 0.518

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Typical sources of errors and their contribution in rel. %

Random Systematic

Sample taking or inhomogeneity 0 -50

Sample preparation 0 - 2 0 - 50

Spectrometer hardware 0.05-0.2 0.05-0.5

Counting statistics (time)

Absorption and enhancement effects 0 -50

Wrong regression parameters 0 - 200

Calibration standards (quality)

Operator mistakes20

Systematic errors due to the samples

• Absorption effect …-300%

• Enhancement effect …-25%

• Particle size effect …-100%

• Chemical state / mineralogical / metallurgical effect …-5-20 %

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Errors in chemical composition of standards

• Best commercially available standards have a composition determination error up to 0.1% relative; but often worse.

• e.g. SiO2 content 0.5% +/- 0.05% means:between 0.45% and 0.55%

(i.e. ± 10% relative !)

e.g. Ni content 10.2% +/- 0.05% means:

between 10.15% and 10.25%

(i.e. ± 0.5% relative !)

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Regression Analysis

• Multi-linear Regression AnalysisCalculates by minimising errors:

A. slope & intercept of calibration line

B. inter-element correction factors

C. line overlap and background factors

using concentration & count rate data from “standards”

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Uncorrected Ni calibration

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Secondary fluorescence ; path indicated with ‘Alpha’Steel 40-50 %Geology

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Matrix corrections

• Theoretical matrix corrections in SuperQ– Classic alphas

• Based on typical values from standards

– Fundamental parameter• Determined per standards

– Better linearity over large calibration ranges

• Sum of concentrations of standards must be close to 100%!

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Ni Corrected with a’s

Human Errors during calibration

• Multi-linear Regression Analysis– The regression software is very powerful but large errors in

the constants calculated are often caused by:

• Too many correction constants calculated simultaneously

• Poor data fitted with “fictitious” calculated correction constants

• Deleting standards that contain valuable information such as interference from line overlap

• Incomplete data on standards that must be used to correct for spectral interferences and matrix effects

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Typical expected errors in XRF

Concentration range Total relative error, %

2 -100 % 0.1 – 2

0.1 – 2 % 1-10

Traces (100 -1000 ppm) 5-20

< 50 -100 ppm 10-100

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Slide Number 1Types of ErrorAccuracy and precisionTypical sources of errors and their contribution in rel. %Propagation of Random ErrorsDistribution of Random ErrorsCounting statistical errorSlide Number 8Instrumental error Counting Statistical Error (CSE)Counting Statistical Error (CSE)Counting Statistical Error (CSE)Relative Counting Statistical Error (RCSE)Sample preparation errorsEvaluation of errors contributionSlide Number 16Slide Number 17Slide Number 18Statistic approach of the random error evaluationTypical sources of errors and their contribution in rel. %Systematic errors due to the samplesErrors in chemical composition of standardsRegression AnalysisUncorrected Ni calibrationWhat are matrix effects ?Matrix correctionsNi Corrected with a’sHuman Errors during calibrationTypical expected errors in XRF