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

X-Ray Microanalysis – Precision and Sensitivity

Recall…

K-ratio Si = [ISiKα (unknown) / ISiKα (std.)] x CFCF 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

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 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 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 countsNB = average background countsn = 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

Detection Limits

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)]1/2τ = measurement timen = # of repetitions of each measurementP = pure element count rateB = background count rate (on pure element standard)a = relates composition to intensity

Or 3.29 (wt.%) / IP[(τ i) / IB]1/2IP = peak intensityIB = background intensityτ = acquisition timei = current

Ave Z = 79

Ave Z = 14

Ave Z = 14, 4X counts as b

0

10

20

30

40

50

60

70

80

90

100

0 100000 200000 300000 400000 500000

current * time (nA - sec)

pp

m

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 linearityis 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 materialsbeam damagecompositional and charge distribution changessurface contamination

Overall accuracy is the combined effect of all sources of variance….σT2 = σC2+σI2+σO2+σS2+σM2σT = total errorσC = counting errorσI = instrumental errorσO = operational errorσS = specimen errorσM = 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

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Time (min)

Cp

s/n

A

2σ counting statistics

0 5 10 15 20 25 30

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

54000 55000 56000 57000 58000 59000 60000 61000 62000 63000 64000

Cp

s/n

A

Wavelength (sinθ)

Sources of measurement error:

Extracting accurate intensities – peak and background measurements

Background shape depends on

Bremsstrahlung emission

Spectrometer efficiency

UMass sp3 GdPO4 and GSC 8153 (VLPET)

y = 8.520777E+01e-7.237583E-05x

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

54000 56000 58000 60000 62000 64000

wavelength (sin-theta)

inte

ns

ity

(c

ps

/nA

)

VLPET

Series2

Pb Ma

GSC 8153 mzt

Expon. (VLPET)

Linear (Series2)

PbMα

y = 0.679x-1

0

2

4

6

8

10

12

14

0.00 0.10 0.20 0.30 0.40 0.50 0.60

net intensity (cps/nA)

% e

rro

r o

f n

et in

ten

sity

Measured

Theoretical based on run5

At ~1000ppm Pb, 10% error can easily produce an age error of 35-40Ma (5 wt.% Th, 4000ppm U)

Becomes 50% error at ~ 0.015 net intensity

bkg net intensity (Pk-bkg)actual bkg 0.23544 0.059155lin fit 1 0.24223 0.052365diff 0.00679 -0.00679%error 2.883962 11.47832





Concentration (ppm)Y TH Pb U Age (Ma)

fitted Ave 19196 48043 1094 3984 400.2bkg. SStd 3518 3043 119 1343 8

StdErr 1573 1361 53 601 3.6

linear Ave 19194 48007 996 3958 365.4bkg. SStd 3519 3039 123 1340 10.1

StdErr 1574 1359 55 599 4.5

Documents

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