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?
Accuracy and Precision
Measured value
Ave
Standard deviation
Std
error
20
25
Ave
Std
error
30
35
Correct value
Wt.% Fe
Low precision, but relatively
accurate
20
25
30
Correct value
Wt.% Fe
High precision, but low
accuracy
35
Accuracy and Precision
Measured value
Ave
Standard deviation
Std
error
20
25
30
35
Correct value
Wt.% Fe
Low precision, but relatively
accurate
20
25
Ave
Ave
Std
error
Std
error
30
Correct value
Wt.% Fe
Precise
High
precision,
and accurate
but low
accuracy
35
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.
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 Xray counts
Counts
The standard deviation is:
99.7% of area
95.4% of area
68.3% of area
3σ
2σ
1σ
1σ
2σ
3σ
Variation in percentage of total counts
= (σC / N)100
6
So to obtain a result to 1% precision,
1-sigma error %
5
Must collect at least 10,000 counts
4
3
2
1
0
0
20000
40000
60000
Counts
80000
100000
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
Multiplication and division
Relative std. deviation
i
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
tb
Total counting time
t
P
Peak counts
B
Background counts
b
t

b
t

b
B  B r  B r 


r


r


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 , j b
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…
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
1
2
3
4
5
6
7
8
9
10
Ave, omitting pt. 7
SD
SD%
Th Ma (cps/nA)
154.6281
155.3082
154.8897
154.8656
156.4651
155.6509
156.8881
155.5401
154.8923
154.8614
155.2334889
0.577232495
0.371847917
X-Ray
Pk-Bg Mean (cps/nA)
Th Ma
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)
6.4992
200.35
4098.57 800
bkg cps
pk-bk
285.0897
3813.483
λe (net intensity for sample)
π
(
β
(
σ
λ
σ
2
p
b
k
g
(
(
s
a
k
e
s
2
e
n
(
i
s
e
i
a
i
t
t
d
t
p
n
t
v
)
)
m
t
s
n
n
l
e
a
e
v
n
r
s
i
a
a
i
t
n
r
y
c
i
a
n
f
o
r
e
c
s
e
t
)
d
1
9
2
0
.
1
.
0
.
.
0
3
3
7
2
6
8
4
5
6
6
5
6
7
2
4
2
2
9
2
9
9
1
4
0
0
0
1
3
6
5
0
6
2
3
3
5
0
0
0
7
)
1
)
0
σe
.
3
3
5
3
5
4
.
7
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:
Wt%
Ni
0
distance (microns)
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
25
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)]1/2
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.%) / 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
Detection limit for Pb
PbMα measured on VLPET
100
90
80
200nA, 800 sec
70
ppm
60
50
40
30
20
10
0
0
100000
200000
300000
current * time (nA - sec)
400000
500000
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….
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 –
Time-integral measurements and sample
effects
0.35
2σ counting statistics
0.30
Cps/nA
0.25
0.20
0.15
0.10
0.05
0.00
0
5
10
15
Time (min)
20
25
30
0.35
0.30
Cps/nA
0.25
0.20
0.15
0.10
0.05
0.00
54000
55000
56000
57000
58000
59000
60000
Wavelength (sinθ)
61000
62000
63000
64000
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)
1.70
VLPET
Series2
Pb Ma
GSC 8153 mzt
Expon. (VLPET)
Linear (Series2)
1.60
intensity (cps/nA)
1.50
1.40
1.30
1.20
1.10
y = 8.520777E+01e -7.237583E-05x
1.00
0.90
0.80
54000
PbMα
56000
58000
60000
wavelength (sin-theta)
62000
64000
actual bkg
lin fit 1
diff
%error
bkg
0.23544
0.24223
0.00679
2.883962
net intensity (Pk-bkg)
0.059155
0.052365
-0.00679
11.47832
14
Becomes 50% error at ~ 0.015 net
intensity
Measured
bkg
net intensity (Pk-bkg)
0.23268
0.0956
0.2426 0.08568
0.00992 -0.00992
4.263366 10.37657
actual bkg
lin fit 1
diff
%error
bkg
0.249807
0.25831
0.008503
3.403828
actual bkg
lin fit 1
diff
%error
bkg
net intensity (Pk-bkg)
0.29714 0.33756
0.30466 0.33004
0.00752 -0.00752
2.530794 2.227752
actual bkg
lin fit 1
diff
%error
bkg
net intensity (Pk-bkg)
0.26367 0.21239
0.27187 0.20419
0.0082
-0.0082
3.109948 3.860822
net intensity (Pk-bkg)
0.131163
0.12266
-0.008503
6.482773
Theoretical based on
run5
12
% error of net intensity
actual bkg
lin fit 1
diff
%error
At ~1000ppm Pb, 10% error can
easily produce an age error of 3540Ma (5 wt.% Th, 4000ppm U)
10
8
6
4
y = 0.679x -1
2
0
0.00
0.10
0.20
0.30
0.40
net intensity (cps/nA)
0.50
0.60
Concentration (ppm)
fitted
bkg.
Ave
SStd
StdErr
Y
19196
3518
1573
linear
bkg.
Ave
SStd
StdErr
19194
3519
1574
TH
48043
3043
1361
Pb
1094
119
53
48007
3039
1359
996
123
55
U Age (Ma)
3984
400.2
1343
8
601
3.6
3958
1340
599
365.4
10.1
4.5