X-Ray Microanalysis – Precision and Sensitivity

Recall…

wt.fraction I = ISiKα (unknown) / ISiKα (pure std.)

K-ratio I = [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.%

Weight Percent?

X-ray intensities are related to mass concentration, not atom concentrationIncident electrons penetrate a constant mass of material which will differ as the composition differs

Electrons interact with orbital electrons of target atoms

lose kinetic energy

number of electrons proportional to atomic mass

Example:

Elements A and B

B is heavier than A

Pure A Mixture of A and B

Excited volume

If atomic concentration of A = nA

the mass concentration is:

CA = nAAA / [nAAA+(1-nA)AB]

Where: AA = atomic weight of A

AB = atomic weight of B

# of excited atoms in pure A = Nm / AA

Where: N is Avogadro’s number

m is mass penetrated by incident electrons

In the compound:

# of A atoms excited is

= nANm / [nAAA+(1-nA)AB]

The X-ray intensity ratio (proportional to the number of excited atoms) is then

= {nANm / [nAAA+(1-nA)AB]} / (Nm / AA)

Which is equal to the expression for CA, the mass concentration of A

Spatial Resolution

D = 0.077 (E01.5 – EC

1.5) / ρ

ρ = density

E0 = accelerating potential

EC = excitation potential

Example:

Si in fayalite at 15keV

ρ = 4.39

E0 = 15 keV

EC = 1.840 keV for SiKα

d = 0.98 μm

3σ = 2.9 μm diameter

volume containing 99% of X-ray productions

X-ray distribution from a point source…

X

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

20 25 30 35

Correct value

Low precision, but relatively accurate

Wt.% Fe

20 25 30 35

Correct value

High precision, but low accuracy

Measured value

Standard deviationAve

Std error

Ave

Std error

Accuracy and Precision

Wt.% Fe

20 25 30 35

Correct value

Low precision, but relatively accurate

Wt.% Fe

20 25 30 35

Correct value

High precision, but low accuracy

Measured value

Standard deviationAve

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 a 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 area99.7% of area

3σ 2σ 1σ 1σ 2σ 3σ

The standard deviation is:

0

1

2

3

4

5

6

0 20000 40000 60000 80000 100000

Counts

1-s

igm

a e

rro

r %

Variation in percentage of total counts

= (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

Relative std. deviation

Addition and subtraction

Multiplication and division

rrrBrBB

tttbbb

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

jpmax, jbmax

index of measurements on the peak and on the backgroundjp, jb

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

Element concentration in the sampleCe

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

Element concentration in the standardCs

Background countsB

Peak counts P

Total counting timetb 

Counting time on the peaktp 

Current from the Faraday cupi

For the calibration…

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…

Or, in the case of a single measurement…

Pk – Bkg cps/nA

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

An example

Calibration

Point Th Ma (cps/nA)1 154.62812 155.30823 154.88974 154.86565 156.46516 155.65097 156.88818 155.54019 154.8923

10 154.8614

Ave, omitting pt. 7 155.2334889SD 0.577232495SD% 0.371847917

X-Ray Th Ma

Pk-Bg Mean (cps/nA) 155.2335

Std.Dev (%) 0.372

Theo.Dev (%) 0.136

3 Sigma (Wt%) 0.563

Pk Mean (cps) 3119.686

Bg Mean (cps) 34.455

Raw cts Mean (cts) 61657

Beam (nA) 19.87

S meas 0.57746862

Sample Th data

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

6.4992 200.35 4098.57 800 285.0897 3813.483

λe (net intensity for sample) 19.0337268

π (peak int) 20.45665672β (bkg int) 1.422929914

σ2e (sample variance) 0.000136506

λs (net intensity for std) 155.2335σ2s (std variance) 0.333470007

σe 0.073511882

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

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…

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