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 concentration
Incident 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
Excited
volume
Pure A
Mixture of A and B
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 (E0 –
/ρ
ρ = density
E0 = accelerating potential
EC = excitation potential
1.5
EC1.5)
X-ray distribution from a point source…
Example:
Si in fayalite at 15keV
ρ = 4.39
X
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
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 a 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
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)
)
m
t
s
n
n
l
e
a
e
v
n
r
s
i
a
a
i
t
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r
y
c
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o
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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