X-Ray Microanalysis – Precision and Sensitivity Recall… K-ratio Si = [I SiKα (unknown ) / I SiKα (std.) ] x CF CF relates concentration in std to pure element K x 100 = uncorrected wt.%, and … K (ZAF)(100) = corrected wt.%

Precision, Accuracy and Sensitivity (detection limits) Precision:Reproducibility Analytical scatter due to nature of X-ray measurement process Accuracy:Is the result correct? Sensitivity:How low a concentration can you expect to see?

Accuracy and Precision Wt.% Fe Correct value Low precision, but relatively accurate Wt.% Fe Correct value High precision, but low accuracy Measured value Standard deviation Ave Std error Ave Std error

Accuracy and Precision Wt.% Fe Correct value Low precision, but relatively accurate Wt.% Fe Correct value High precision, but low accuracy Measured value Standard deviation Ave Std error Ave Std error Ave Std error Precise and accurate

Characterizing Error What are the basic components of error? 1)Short-term random error (data set) Counting statistics Instrument noise Surface imperfections Deviations from ideal homogeneity 2)Short-term systematic error (session to session) Background estimation Calibration Variation in coating 3) Long-term systematic error (overall systematic errors that are reproducible session-to-session) Standards Physical constants Matrix correction and Interference algorithms Dead time, current measurement, etc.

Frequency of X- ray counts Counts Short-Term Random Error - Basic assessment of counting statistics X-ray production is random in time, and results in a fixed mean value – follows Poisson statistics At high count rates, count distribution follows a normal (Gaussian) distribution

68.3% of area95.4% of area 99.7% of area 3σ 2σ 1σ 1σ 2σ 3σ The standard deviation is:

Variation in percentage of total counts = (σ C / N)100 So to obtain a result to 1% precision, Must collect at least 10,000 counts

Evaluation of count statistics for an analysis must take into account the variation in all acquired intensities Peak (sample and standard) Background (sample and standard) And errors propagated Relative std. deviation Addition and subtraction Multiplication and division

Positive and negative offsets for the background measurement, relative to the peak position r+ et r- Total number of measurements on the peak and on the background j pmax, j bmax index of measurements on the peak and on the background j p, j b Intensity (Peak-Bkgd in cps/nA) of the element in the sample e Element concentration in the sample CeCe Intensity (Peak-Bkgd in cps/nA) of the element in the standard s Element concentration in the standard CsCs Background counts B Peak counts P Total counting time tb tb Counting time on the peak tp tp Current from the Faraday cup i

For the calibration… And standard deviation…

The measured standard deviation can be compared to the theoretical standard deviation … Theo.Dev(%) = 100* S theo / s The larger of the two then represents the useful error on the standard calibration:  ² s = max ((S meas )², or (S theo )²)

For the sample, the variance for the intensity can be estimated as… where

The intensity on the sample is… Or, in the case of a single measurement… Pk – Bkg cps/nA

And the total count statistical error is then (3σ)…

An example Calibration X-RayTh Ma Pk-Bg Mean (cps/nA) Std.Dev (%)0.372 Theo.Dev (%) Sigma (Wt%)0.563 Pk Mean (cps) Bg Mean (cps) Raw cts Mean (cts)61657 Beam (nA)19.87 S meas

Sample Th data Wt%currpk cpspk t(sec)bkg cpspk-bk This is a very precise number

Sensitivity and Detection Limits Ability to distinguish two concentrations that are nearly equal (C and C’) 95% confidence approximated by: N = average counts N B = average background counts n = number of analysis points Actual standard deviation ~ 2σC, so ΔC about 2X above equation If N >> N B, then

Sensitivity in % is then… To achieve 1% sensitivity Must accumulate at least 54,290 counts As concentration decreases, must increase count time to maintain precision

Example gradient: 0 distance (microns)25 Wt% Ni Take 1 micron steps:Gradient = 0.04 wt.% / step Sensitivity at 95% confidence must be ≤ 0.04 wt.% Must accumulate ≥ 85,000 counts / step If take 2.5 micron steps Gradient = 0.1 wt.% / step Need ≥ 13,600 counts / step So can cut count time by 6X

Detection Limits N no longer >> N B at low concentration What value of N-N B can be measured with statistical significance? Liebhafsky limit: Element is present if counts exceed 3X precision of background: N > 3(N B ) 1/2 Ziebold approximation: C DL > 3.29a / [(nτP)(P/B)] 1/2 τ = measurement time n = # of repetitions of each measurement P = pure element count rate B = background count rate (on pure element standard) a = relates composition to intensity

Or 3.29 (wt.%) / I P [(τ i) / I B ] 1/2 I P = peak intensity I B = background intensity τ = acquisition time i = current Ave Z = 79 Ave Z = 14 Ave Z = 14, 4X counts as b

Detection limit for Pb PbMα measured on VLPET 200nA, 800 sec

Can increase current and / or count time to come up with low detection limits and relatively high precision But is it right?

Accuracy All results are approximations Many factors Level 1 quality and characterization of standards precision matrix corrections mass absorption coefficients ionization potentials backscatter coefficients ionization cross sections dead time estimation and implementation Evaluate by cross checking standards of known composition (secondary standards)

Level 2 – the sample Inhomogeneous excitation volume Background estimation Peak positional shift Peak shape change Polarization in anisotropic crystalline solids Changes in Φ(ρZ) shape with time Measurement of time Time-integral effects Measurement of current, including linearity is a nanoamp a nanoamp? Depends on measurement – all measurements include errors!

Time-integral acquisition effects drift in electron optics, measurement circuitry dynamic X-ray production non-steady state absorbed current / charge response in insulating materials beam damage compositional and charge distribution changes surface contamination

Overall accuracy is the combined effect of all sources of variance…. σ T 2 = σ C 2 +σ I 2 +σ O 2 +σ S 2 +σ M 2 σ T = total error σ C = counting error σ I = instrumental error σ O = operational error σ S = specimen error σ M = miscellaneous error Each of which can consist of a number of other summed terms Becomes more critical for more sensitive analyses - trace element analysis

Sources of measurement error – Time-integral measurements and sample effects

Time (min) Cps/nA 2σ counting statistics

Cps/nA Wavelength (sinθ)

Sources of measurement error: Extracting accurate intensities – peak and background measurements Background shape depends on Bremsstrahlung emission Spectrometer efficiency

PbMα