Download - Quantitative X-ray Spectrometry in TEM/STEM
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Quantitative X-ray Spectrometry in TEM/STEM
Charles LymanLehigh UniversityBethlehem, PA
Based on presentations developed for Lehigh University semester courses and for the Lehigh Microscopy School
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Quantitative X-ray Analysis of Thin
Specimens Aim of quantitative analysis: to transform the intensities in the X-ray
spectrum into compositional values, with known precision and accuracy
Cliff-Lorimer method:
Precision: collect at least 10,000 counts in the smallest peak to obtain a counting error of less than 3%
Accuracy: measure kAB on a known standard and find a way to handle x-ray absorption effects
How much of each element is present?
€
CA
CB
= kAB
IA
IB
and CA + CB =1
CA = concentration of element AIA = x-ray intensity from element AkAB = Cliff-Lorimer sensitivity factor
What could be simpler?
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Assumptions in Cliff-Lorimer Method
Basic assumptions» X-ray intensities for each element are measured simultaneously» Ratio of intensities accounts for thickness variations» Specimen is thin enough that absorption and fluorescence can be ignored
– the “thin-film criterion”– We would like to handle absorption in a better way!
Cliff-Lorimer equation:
» CA and CB are weight fractions or atomic fractions (choose one, be consistent)» kAB depends on the particular TEM/EDS system and kV (use highest kV)
– k-factor is most closely related to the atomic number correction» Can expand to measure ternaries, etc. by measuring more k-factors
€
CA
CB
= kAB
IA
IB
and CA + CB =1
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Steps in Quantitative Analysis
Remove background intensity under peaks Integrate counts in peaks Determine k-factors (or -factors) Correct for absorption (if necessary)
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Calculate Background, the Subtract
Gross-Net Method» Draw line at ends of window
covering full width of peak» Impossible with peak overlap» Should work better above 2
keV where background changes slowly
Three-Window Method» Set window with FWHM (or even better
1.2 FWHM)
» Average backgrounds B1 and B2
» Subtract Bave from peak
» Requires well-separated peaks
Background Modeling» Mathematical model of background
as function of Z and E» Useful when peaks are close
together
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Digital Filtering
Convolute spectrum with “top-hat” filter
» Multiply channels of top-hat filter times each spectrum channel
» Place result in central channel
» Step filter over each spectrum channel
Background becomes zero
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Digital Filtering
Spectrum before filteringNote MgK, AlK, and SiK
Spectrum after filteringPositive lobes are
proportional to peak intensities
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Obtaining k-factors
Requirements for standard specimen for k-factor measurement» Single phase (stoichiometric composition helpful)» Homogeneous at the nanometer scale» Thinned to electron transparency without composition change (microtome)» Insensitive to beam damage
Measure k-factors on a known standard:
» Usually kASi or kAFe
» Measure k-factors at various thicknesses and extrapolate to zero thickness
Other ways» Calculate k-factors (when standards are not available)» Use literature values at same kV for x-rays 5-15keV (not recommended)» Use kAB = kAC/kBC (use only when necessary - errors add)
€
kAB =CA
CB
IB
IA
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Why Collect 10,000 Counts?
There is a 99% chance that a single measurement is within 3N1/2 of the true value
The relative counting error =
Thus, for 10,000 counts the relative counting error =
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Experimental k-factors
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Calculated k-factors
When suitable standard is not available» When a modestly accurate analysis is acceptable
Most EDS system software can calculate k-factors» But errors can be up to 20%
Simple expression:
€
kAB =Qωa( )B
AA
Qωa( )AAB
εA
εB
but Q not known well which leads to error
Q = ionization cross-section = fluorescence yielda = relative transition probability =A = atomic weight = detector efficiency
€
IntensityKα
Intensity(Kα + Kβ )
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Calculated k-factors
Calculated kAFe-factors using different ionization cross-sections
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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kAFe for K-series
Errors of calculated versus standards ~ 4%
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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kAFe for L-series
Errors of calcuated versus standards up to 20%
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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The Absorption Problem
k-factors measured at different specimen thicknesses will be different» X-rays from some elements will be
absorbed more than others» “Thin-film criterion” breaks down if high
accuracy required» We need a better way to handle
absorption effects
What to do:1. Measure unknown and standard at the
same thickness (impractical)2. Extrapolate all k-factors to zero-
thickness, then apply absorption correction to each measurement (but we need to know the specimen thickness)
3. Use -factors
aa
Incident beamα
XEDStt cosec α
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Extrapolate to the Zero-Thickness k-factor
Zero-thickness k-factor
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
Horita et al. (1987) and van Cappellan (1990) methods
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Obtaining the Zero-thickness k-factor
Thin standard of known composition
Pt-13wt% Rh thermocouple wire
Thickness measured by EELS log-ratio method
€
kPtRh=CPt
CRh
IRhI Pt
=0.870.13
I RhI Pt
€
kPtRh=1.079
R. E. Lakis, C. E. Lyman, and H. G. Stenger, J. Catal. 154 (1995) 261-275.
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Absorption Correction
Effective sensitivity factor kAB* = kAB(ACF)
€
ACF =
μ
ρ
⎡
⎣ ⎢
⎤
⎦ ⎥Spec
A
μ
ρ
⎡
⎣ ⎢
⎤
⎦ ⎥Spec
B
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
1− exp −μ
ρ
⎡
⎣ ⎢
⎤
⎦ ⎥Spec
B
ρt cosec α( ) ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1− exp −μ
ρ
⎡
⎣ ⎢
⎤
⎦ ⎥Spec
A
ρt cosec α( ) ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
Equation 35.29:
Zero-thickness k-factor
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Original -factor method
Absorption correction contains t» foil thickness t must be determined at analysis point» specimen density for composition at analysis point)
-factor method» assume x-ray intensity t» then
» subsititute into absorption equation:
€
t = ζ A
IA
CA
€
CA
CB
= kAB
IA
IB
⎛
⎝ ⎜
⎞
⎠ ⎟
μ
ρ
⎡
⎣ ⎢
⎤
⎦ ⎥Spec
A
μ
ρ
⎡
⎣ ⎢
⎤
⎦ ⎥Spec
B
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
1− exp −μ
ρ
⎡
⎣ ⎢
⎤
⎦ ⎥Spec
B
ζ A
IA
IB
⎛
⎝ ⎜
⎞
⎠ ⎟cosec α( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1− exp −μ
ρ
⎡
⎣ ⎢
⎤
⎦ ⎥Spec
A
ζ A
IA
IB
⎛
⎝ ⎜
⎞
⎠ ⎟cosec α( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
We can determine both absorption-corrected compositions and t if kAB and known from measurements on standard
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Modified -factor method
Measure the -factor for both elements:
Assume CA + CB = 1 for binary system and rearrange:
Determine CA, CB, and t simultaneously from three equations in three unknowns
t can be determined if density is known
€
t = ζ A
IA
CA
€
t = ζ B
IB
CB
€
CA =ζ AIA
ζ AIA + ζ BIB
, CB =ζ BIB
ζ AIA + ζ BIB
, ρt = ζ AIA + ζ BIB
M. Watanabe and D.B. Williams, Z. Metalkd. 94 (2003) 307-316
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-factor
factor is dependent on • x-ray energy• accelerating voltage• beam current
factor is independent of• specimen thickness• specimen composition• specimen density
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Quantitative analysis by factor method
Lucadamo et al. (1999)
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Effect of kV on Beam Spreading
Elastic scattering broadens the beam as it traverses the specimen
Beam broadening is less for
» Higher kV» Lighter materials» Smaller thicknesses
Goldstein-Reed Eqn.
b=7.21x105 ZE0
1/2
ρA( )
32t
b
bfrom Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Spatial Resolution vs. Analytical Sensitivity
Conditions that favor high spatial resolution (thinnest specimen) result in poorer analytical sensitivity and vice versa. For example to obtain equivalent analytical sensitivity in an AEM to an EPMA, the X-ray generation and detection efficiency would have to be improved by a factor of 108
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Composition Profiles Across an Interphase Interface
The change in Mo and Cr composition across the interface can be used to determine the compositions of the phases either side of the interface which, in turn, give the tie lines on the Ni-Cr-Mo phase diagram.
Courtesy R. Ayer
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Measurement of Low T Diffusion Data
Measurement of composition profiles with high spatial resolution permits extraction of low- temperature diffusion data because the small diffusion distances at low T are detectable by AEM X-ray microanalysis. Here Zn profiles across a 200 nm wide precipitate-free zone in Al-Zn are used to determine values of the Zn diffusivity at T = 100-200°C. Courtesy A.W. Nicholls
Low-temp data
High-temp data
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Predicted Phase Separation Observed in Nanoparticles
Dotted misibility gap was predicted from other similar systems --> only observed in nanoparticles
Two phases observed
Pt-rich phase
Rh-rich phase
C. E. Lyman, R. E. Lakis, and H. G. Stenger, Ultramicroscopy 58 (1995) 25-34.
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Summary
Know the question you are trying to answer
Know the precision and accuracy required to answer the question
Accumulate enough counts in the spectrum to achieve the required precision (> 10,000 counts in the smallest peak)
Know the precision and accuracy of your k-factor
Measure zero-thickness k-factors and apply an absorption correction (need t at analysis point) or use -factors where t is not needed
Spatial resolution vs. detectability: » You cannot achieve the highest spatial resolution and the best analytical sensitivity
under the same experimental conditions