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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
Where
Detection limit (Ancey)
α : Risk of considering that C>0 when it is in fact C=0β : Risk of considering that C=0 when it is in fact C>0
By default, we will use α = β = 5%.
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 = σC
2+σI2+σO
2+σS2+σM
2
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
0.00
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)
130
131
132
133
134
135
136
137
138
0 500 1000 1500 2000 2500 3000 3500
time (sec)
ab
s. c
urr
en
t (n
A)
130
131
132
133
134
135
136
137
138
0 500 1000 1500 2000 2500 3000 3500
time (sec)
ab
s. c
urr
en
t (n
A)
130
131
132
133
134
135
136
137
138
0 500 1000 1500 2000 2500 3000 3500
time (sec)
ab
s. c
urr
en
t (n
A)
ideal
Beam damage
Monazite
LGG246-5 Lower Granite Gorge - Grand Canyon
15kV, 200nA, 30 min
Sources of measurement error:
Current and dead-time
SP2 1-12-04After picommeter adjustment
0.97
0.98
0.99
1
1.01
1.02
0 50 100 150 200 250 300
current (nA)
I /
I 15
Initial
Initial
After adjustment
After adjustment
Sources of measurement error:
Extracting accurate intensities – peak and background measurements
Background shape depends on
Bremsstrahlung emission
Spectrometer efficiency
PHA effects relating to trace analysis
LaPO4 and monazite
0.0
2.0
4.0
6.0
8.0
10.0
12.0
1 2 3 4 5 6
Pressure (bars)
La
La
1 (
2) /
Au
LaPO4
Low Th monazite
Detector gas type and pressure
GdPO4 Pb region (PET)sp3
y = 8.520777E+01e -7.237583E-05x
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
54000 56000 58000 60000 62000 64000
wavelength (sin-theta)
inte
ns
ity
(c
ps
/nA
)
VLPET
Expon. (VLPET)
Linear (Series2)
PbMα
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
Pb Ma bkg (PET LGG 246-5 m1)
0.19
0.20
0.21
0.22
0.23
59000 59500 60000 60500 61000 61500
Wavelength (sin)
I (c
ps
/nA
)
PbMa
High Th Bkg.
Low Th Bkg.
7.3 wt% Th
4.0 wt% Th
exp. lin.
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
bkg net intensity (Pk-bkg)actual bkg 0.23268 0.0956lin fit 1 0.2426 0.08568diff 0.00992 -0.00992%error 4.263366 10.37657
bkg net intensity (Pk-bkg)actual bkg 0.249807 0.131163lin fit 1 0.25831 0.12266diff 0.008503 -0.008503%error 3.403828 6.482773
bkg net intensity (Pk-bkg)actual bkg 0.29714 0.33756lin fit 1 0.30466 0.33004diff 0.00752 -0.00752%error 2.530794 2.227752
bkg net intensity (Pk-bkg)actual bkg 0.26367 0.21239lin fit 1 0.27187 0.20419diff 0.0082 -0.0082%error 3.109948 3.860822
Pb
0
2000
4000
6000
8000
10000
12000
14000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
net intensity (Pk-bkg cps/nA)
pp
m
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
Sources of measurement error - Peak shifts
Wavelength may shift between standard and sample (or between samples) if X-ray transition involves valance electrons
Example: Y Lγ2,3
transitions from N2 (4p1/2) and N3 (4p3/2) levels
Critical in Pb measurement in monazite: PbMα - YLγ overlap
YPO4 (tetragonal) Monazite (monoclinic)
O coordination 8-fold 9-fold
Ave REE-O bond length (Å) 2.346 2.559
YPO4
Calculate peak wavelength shift of -4.5x10-3Å in monazite relative to YPO4
Results in overestimation of overlap:
36ppm Pb / wt.%Y