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X-Ray Microanalysis – Precision and SensitivityPowerPoint Presentation

X-Ray Microanalysis – Precision and Sensitivity

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X-Ray Microanalysis – Precision and Sensitivity

Recall…

K-ratio Si = [ISiKα (unknown) / ISiKα (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?

Measured value

Ave

Std error

Standard deviation

Ave

Std error

20 25 30 35

20 25 30 35

Correct value

Correct value

Wt.% Fe

Wt.% Fe

Low precision, but relatively accurate

High precision, but low accuracy

Std error

Ave

Std error

Accuracy and Precision

Measured value

Ave

Std error

Standard deviation

20 25 30 35

20 25 30 35

Correct value

Correct value

Wt.% Fe

Wt.% Fe

Low precision, but relatively accurate

High precision, but low accuracy

Precise and accurate

- Characterizing Error
- What are the basic components of error?
- Short-term random error (data set)
- Counting statistics
- Instrument noise
- Surface imperfections
- Deviations from ideal homogeneity
- 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.

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

Frequency of X-ray counts

Counts

Variation in percentage of total counts statistics

= (σC / 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

Addition and subtraction

Relative std. deviation

Multiplication and division

i into account the variation in all acquired intensities

Current from the Faraday cup

tp

Counting time on the peak

r+ et r-

Positive and negative offsets for the background measurement, relative to the peak position

Total counting time

tb

P

Peak counts

Background counts

B

Cs

Element concentration in the standard

s

Intensity (Peak-Bkgd in cps/nA) of the element in the standard

Ce

Element concentration in the sample

e

Intensity (Peak-Bkgd in cps/nA) of the element in the sample

jp, jb

index of measurements on the peak and on the background

jpmax, jbmax

Total number of measurements on the peak and on the background

For the calibration… into account the variation in all acquired intensities

And standard deviation…

The measured standard deviation can be compared to the theoretical standard deviation …

Theo.Dev(%) = 100* Stheo/s

The larger of the two then represents the useful error on the standard calibration:

²s = max ((Smeas)², or (Stheo)²)

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

where

The intensity on the sample is… estimated as…

Or, in the case of a single measurement…

Pk – Bkg cps/nA

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

An example estimated as…

Calibration

Sample Th data estimated as…

Wt% curr pk cps pk t(sec) bkg cps pk-bk

6.4992 200.35 4098.57 800 285.0897 3813.483

This is a very precise number

Sensitivity and Detection Limits estimated as…

Ability to distinguish two concentrations that are nearly equal (C and C’)

95% confidence approximated by:

N = average counts

NB = average background counts

n = number of analysis points

Actual standard deviation ~ 2σC, so ΔC about 2X above equation

If N >> NB, then

Sensitivity in % is then… estimated as…

To achieve 1% sensitivity

Must accumulate at least 54,290 counts

As concentration decreases,

must increase count time to maintain precision

Example gradient: estimated as…

Wt%

Ni

0 distance (microns) 25

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 estimated as…

N no longer >> NB at low concentration

What value of N-NB can be measured with statistical significance?

Liebhafsky limit:

Element is present if counts exceed 3X precision of background:

N > 3(NB)1/2

Ziebold approximation:

CDL > 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 estimated as…3.29 (wt.%) / IP[(τ i) / IB]1/2

IP = peak intensity

IB = background intensity

τ = acquisition time

i = current

Ave Z = 79

Ave Z = 14

Ave Z = 14, 4X counts as b

Can increase current and / or count time to come up with low detection limits and relatively high precision

But is it right?

Accuracy detection limits and relatively high precision

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 detection limits and relatively high precision

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 detection limits and relatively high precision

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….

σT2 = σC2+σI2+σO2+σS2+σM2

σT= total error

σC= counting error

σI= instrumental error

σO= operational error

σS= specimen error

σM= 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 – variance….

Time-integral measurements and sample effects

Cps/nA variance….

Wavelength (sinθ)

Sources of measurement error: variance….

Extracting accurate intensities – peak and background measurements

Background shape depends on

Bremsstrahlung emission

Spectrometer efficiency

PbM variance….α

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