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14 – 1
Lecture 14: Statistics • Accuracy & precision • Counting statistics • Detection limit (sensitivity)
GEOL-4700/5700 – Summer 2015
14 – 2
X-Ray Microanalysis – Precision and Sensitivity Recall…
K-ratio Element = [IElement)in)unknown)/)IElement)in)std.])x)CF CF relates concentration in std to pure element
K x 100 = uncorrected wt.%, and…
K x (ZAF) x 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?
14 – 3
Accuracy and Precision
14 – 4
Accuracy and Precision
Wt.% Fe
20 25 30 35
Correct value Wt.% Fe
20 25 30 35
Correct value
Measured value
Standard deviation
Low precision, but relatively accurate
Avg
Std error
High precision, but low accuracy
Avg
Std error
14 – 5
Accuracy and Precision
Wt.% Fe
20 25 30 35
Correct value
Low precision, but relatively accurate
Wt.% Fe
20 25 30 35
Correct value
Avg
Std error
Avg
Std error
Precise and accurate
Measured value
Standard deviation
14 – 6
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.
14 – 7
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
14 – 8
68.3% of area
95.4% of area
99.7% of area
3σ 2σ 1σ N 1σ 2σ 3σ
The standard deviation is:
σ theoretical = N
14 – 9
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
1-si
gma
erro
r %
Counts
Variation in percentage of total counts
=100 N / N =100(σ theoretical / N )
So to obtain a result to 1% precision, must collect at least 10,000 counts.
…and for 0.5% precision, must collect around 40,000 counts.
…and for 0.1% precision = 1,000,000!
14 – 10
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
σC = σ 2A +σ
2B
εC = ε 2A +ε2B εC =
σx
14 – 11
i Current from the Faraday cup
tp Counting time of the peak (p)
r+, r– Positive and negative offsets for the background measurement, relative to the peak position
tb Total counting time on background
P(n) Peak counts (for n-th measurement)
B(n) Background counts: - For n-th measurement; - Assume LINEAR background.
Cs Element concentration in the standard
Is Intensity (Peak-background) in cps/nA of the element in the standard
Ce Element concentration in the sample
Ie Intensity (Peak-background) in cps/nA of the element in the sample
np, nb Index of measurement on the peak and on the background
np_max, nb_max Total number of measurements on the peak and on the background
B = (B+r− −B−r+ )r− − r+
tb = tb− + tb
+
Abbreviations used in the following cps = counts per second cps/nA = count per second per nanoamp
14 – 12
For the calibration…
And standard deviation…
Mean =(Is )n
np
∑
np_max
σ measure =
(Pn −Bn −P +B)2
np_max −1np
∑#
$%%
&
'((
1/2
itp
Mean and standard deviation for a series of n measurements on STANDARD
NOTE: Pn, Bn = individual n-th measurement P, B = average of n-th measurements
Sum of the intensities divided by the number of measurement
Values are normalized to time (tp) and current (i)!
14 – 13
Measured standard deviation
compared to… Theoretical standard deviation…
Theo.Dev(%) = 100 * σtheoretical/Is
σ theoretical =
1np_max
Pn(tp )nnp
∑"
#
$$
%
&
''
2
+1
nb_maxBn(tb )nnb
∑"
#$$
%
&''
2
i
σ measure =
(Pn −Bn −P +B)2
np_max −1np
∑#
$%%
&
'((
1/2
itp
The larger of the two represents the useful error on the standard calibration:
σ2s = max((σmeasure)2,
(σtheoretical)2)
14 – 14
For the sample, the variance for the intensity can be estimated as…
σ 2Ie=
π e
(tp )n in1
nmax
∑+
βe
(tb )n in1
nmax
∑
The intensity on the sample is…
Or, in the case of a single measurement…
Pk – Bkg [cps/nA] Ie =
Pn(tp )n
−Bn(tb )nn
"
#$$
%
&''1
nmax
∑
innmax
Mean and standard deviation for a series of n measurements on SAMPLE
βe =
Bn(tb )n in1
nmax
∑nb_max
π e =
Pn(tp )n in1
nmax
∑
np_maxWith…
N.b.: standard deviation = square root of variance …or… (standard deviation)2 = variance
14 – 15
The TOTAL count statistical error is then (3σ)…
σCe= 3 Ce
Ie
!
"#
$
%& σ Ie
2 +IeIs
!
"#
$
%&
2
σ Is2
Standard deviation from SAMPLE measurement
(counts)
Standard deviation from STANDARD
measurement (counts)
Standard deviation from SAMPLE concentration
14 – 16
An example of calibration
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 SD measure 0.5775
Point Th Ma
(cps/nA) 1 154.6281 2 155.3082 3 154.8897 4 154.8656 5 156.4651 6 155.6509 7 156.8881 8 155.5401 9 154.8923
10 154.8614 Average,
omitting #7 155.2335 SD 0.5772
SD% 0.3718 154$
155$
156$
157$
0$ 2$ 4$ 6$ 8$ 10$
Complete statistics given by software (e.g. Cameca “PeakSight”)
14 – 17
This is a very precise number
An example of sample analysis Sample Th data @ ~200 nA
Wt% Current Peak cps Peak t (sec) Bkg cps Pk-bkg cps 6.4992 200.35 nA 4098.57 800 285.0897 3813.483
Ie (net intensity for sample) 19.0337 πe (peak int) 20.4567 βe (bkg int) 1.4229 σ2
e (sample variance) 0.000137 Is (net intensity for std) 155.2335 σ2
s (std variance) 0.3335
σe
0.0735 (~ 0.39%)
σCe= 3 6.50
19.03!
"#
$
%& (0.0735)22 + 19.03
155.23!
"#
$
%&2
(0.578)2 = 0.105Error on the
concentration (including std)
14 – 18
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…
ΔC =C −C ' ≥ 2.33n
σC
N − NB
$
%&
'
()
ΔC =C −C ' ≥ 2.33nN
Sensitivity in % is then… ΔC /C(%) = 2.33
nNx100
Ziebold (1967)
14 – 19
Example gradient:
• If take 1 micron steps o gradient = 0.04 wt.% / step o Sensitivity at 95% confidence must be ≤ 0.04 wt.% o Must accumulate ≥ 85,000 counts / step
• If take 2.5 micron steps o Gradient = 0.1 wt.% / step o Must accumulate ≥ 13,600 counts / step ! So can cut count time by 6x
To achieve 1% sensitivity, must accumulate at least 54,290 counts.
As concentration decreases, must increase count time to maintain precision.
0 distance (microns) 25
Wt%
Ni
1%
14 – 20
Detection Limits • At low concentration N is no longer much larger than NB (rather N ≈ NB) • 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
ntpPPB
tp = measurement time n = # of repetitions of each measurement P = pure element count rate B = background count rate on pure element a = relates composition to intensity
14 – 21
Or…
Ave Z = 79
Ave Z = 14
Ave Z = 14, 4X counts as b
CDL =3.29(wt%)IP (tpi) / IB
IP = peak intensity IB = background intensity tp = acquisition time i = current
14 – 22
Detection limit & counts You can decrease the detection limit by increasing the total number of counts!
Several options to increase the number of counts:
• Change the acceleration voltage to reach an optimum, both in term of spatial resolution (volume increase with increasing voltage!) and overvoltage (optimum usually around 2-3x);
• Increase the current;
• Increase the counting time;
HOWEVER, keep in mind that your sample might suffer from beam exposure, in this case, you need to lower the energy density:
• Increase the analytical volume:
o Increase the acceleration voltage (loss of depth resolution);
o Increase the beam diameter (loss of lateral resolution);
• Decrease the current and/or the counting time.
14 – 23
Microprobe dating of monazite • Monazite is a U-Th-bearing REE phosphate: can be dated by
measuring the quantity (NOT the isotopes!) of U, Th and Pb. This assume no common lead (Pb present when the crystal form), which is a reasonable assumption (typically ~2-3 ppm common Pb):
• Pb in monazite is often less than a couple thousands ppm of Pb • Age accuracy will depend on how accurate the Pb-content (and
actinide) can be measured ! TRACE element analysis…
Pb = Th232
(eλ232*t −1)
"
#$%
&'* 208
+U
2380.9928(eλ
238*t −1)"
#$%
&'* 206
+U
2350.0072(eλ
235*t −1)"
#$%
&'* 207
+ 204Pb
208Pb from 232Th
206Pb from 238U
207Pb from 235U
Common lead (assumed = 0) ~ a few ppm in Mnz
14 – 24
0
10
20
30
40
50
60
70
80
90
100
0 100000 200000 300000 400000 500000
ppm
current * time (nA * sec)
Detection limit for Pb Pb Mα 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?
14 – 25
Accuracy All results are approximations. Many factors… Level 1 • Quality and characterization of standards • Precision • Matrix corrections
o Mass absorption coefficients o Ionization potentials o Backscatter coefficients o Ionization cross sections
• Dead time estimation and implementation Level 2 – inaccuracy due to sample • Inhomogeneous excitation volume • Background estimation • Change in peak position (peak shift) or change in peak shape • Polarization in anisotropic crystalline solids • Changes in Φ(ρZ) shape with time
! Evaluate by cross checking standards of known composition (secondary standards)
14 – 26
ALL measurements include errors! • Measurement of time • Time-integral effects (*) • Measurement of current, including linearity, is a nanoamp a
nanoamp? Depends on measurement…
(*) Time-integral acquisition effects
• Drift in electron optics, measurement circuitry • Dynamic X-ray production • Non-steady state absorbed current / charge response in
insulating materials o Beam damage o Compositional and charge distribution changes o Surface contamination
14 – 27
Overall accuracy 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
14 – 28
Sources of measurement error: Time-integral measurements and sample effects
Sample damage (especially with focus beam) can alter the count rate! Change is apparently correlated
with abnormal behavior of the absorbed current
14 – 29
Sources of measurement error: Extracting accurate intensities: peak and background measurements Background shape depends on… • Bremsstrahlung emission (background continuum) • Spectrometer efficiency
14 – 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
Wavelength (sinθ)
Sources of measurement error: Background curvature, example in the Pb Mα region (PET crystal)
Cps
/ nA
How (non-)linear is a background? 2σ counting statistics
14 – 31
Pb Mα
Exponential regression y = 74.4951 e-8.465 E-05 x
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.54 0.56 0.58 0.60 0.62 0.64
Inte
nsity
(cps
/nA
)
Wavelength (sinθ)
Pb-M region on GdPO4
Gd Lβ2 3rd order 59847 7.1 KeV
Background: linear or not?
14 – 32
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
% e
rror
of n
et in
tens
ity
Net intensity (cps/nA)
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 0.2354 0.0592 Linear fit 1 0.2422 0.0524
Difference 0.0068 -0.0068
% error 2.884 11.478 Actual bkg 0.2327 0.0956 Linear fit 2 0.2426 0.0857
Difference 0.0100 -0.0100
% error 4.263 10.377 Actual bkg 0.2498 0.1312 Linear fit 3 0.2583 0.1227
Difference 0.0085 -0.0085
% error 3.404 6.483 Actual bkg 0.2971 0.3376 Linear fit 4 0.3047 0.3300
Difference 0.0075 -0.0075
% error 2.531 2.228 Actual bkg 0.2637 0.2124 Linear fit 5 0.2719 0.2042
Difference 0.0082 -0.0082
% error 3.110 3.861
Background & Net Intensity: linear vs. exponential bkg…
14 – 33
Challenge of trace element analysis: Background & peak interferences
We will come back later on the peak interference issue…
14 – 34
Concentration (ppm) Y Th Pb U Age (Ma)
fitted bkg.
Ave 19196 48043 1094 3984 400.2 S Std 3518 3043 119 1343 8.0
Std Err 1573 1361 53 601 3.6
linear bkg.
Ave 19194 48007 996 3958 365.4 S Std 3519 3039 123 1340 10.1
Std Err 1574 1359 55 599 4.5
Choosing the wrong background…
…can results in an error of 10’s to 100’s ppm! For U-Th-Pb dating of monazite, this can represents 10’s millions of years!
✔ ✗
Isotopic age: 394 Ma