1. Quantitative X-ray Spectrometry in TEM/STEM
Charles Lyman
Lehigh University
Bethlehem, PA
Based on presentations developed for Lehigh University semester courses and for the Lehigh Microscopy School

2. Quantitative X-ray Analysis of Thin Specimens
How much of each element is present?
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
CA = concentration of element A
IA = x-ray intensity from element A
kAB = Cliff-Lorimer sensitivity factor
What could be simpler?

3. 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

4. Steps in Quantitative Analysis
Remove background intensity under peaks
Integrate counts in peaks
Determine k-factors (or z-factors)
Correct for absorption (if necessary)

5. 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

6. 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

7. Digital Filtering Spectrum before filtering Spectrum after filtering
Note MgK, AlK, and SiK
Spectrum after filtering
Positive lobes are proportional to peak intensities
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996

8. 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)

9. 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 =

10. Experimental k-factors
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996

11. 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:
but Q not known well which leads to error
Q = ionization cross-section
= fluorescence yield
a = relative transition probability =
A = atomic weight
e = detector efficiency

12. Calculated k-factors
Calculated kAFe-factors using different ionization cross-sections
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996

13. kAFe for K-series Errors of calculated versus standards ~ 4%
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996

14. kAFe for L-series Errors of calcuated versus standards up to 20%
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996

15. 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:
Measure unknown and standard at the same thickness (impractical)
Extrapolate all k-factors to zero-thickness, then apply absorption correction to each measurement (but we need to know the specimen thickness)
Use z-factors
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996

16. Extrapolate to the Zero-Thickness k-factor
Horita et al. (1987) and van Cappellan (1990) methods
Zero-thickness k-factor
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996

17. Obtaining the Zero-thickness k-factor
Thin standard of known composition
Pt-13wt% Rh thermocouple wire
Thickness measured by EELS log-ratio method

18. Absorption Correction
Effective sensitivity factor kAB* = kAB(ACF)
Zero-thickness k-factor
Equation 35.29:
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996

19. 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:
We can determine both absorption-corrected compositions and t if kAB and zA known from measurements on standard

20. Modified -factor method
Measure the z-factor for both elements:
Assume CA + CB = 1 for binary system and rearrange:
Determine CA, CB, and rt simultaneously from three equations in three unknowns
t can be determined if density is known
M. Watanabe and D.B. Williams, Z. Metalkd. 94 (2003)

21. z-factor z factor is dependent on x-ray energy accelerating voltage
beam current
z factor is independent of
specimen thickness
specimen composition
specimen density

22. Quantitative analysis by z factor method
Lucadamo et al. (1999)

23. 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 b
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996

24. 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

25. 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

26. Measurement of Low T Diffusion Data
Low-temp data
High-temp 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 = °C.
Courtesy A.W. Nicholls
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996

27. Predicted Phase Separation Observed in Nanoparticles
Two phases observed
Pt-rich phase
Rh-rich phase
Dotted misibility gap was predicted from other similar systems --> only observed in nanoparticles
C. E. Lyman, R. E. Lakis, and H. G. Stenger, Ultramicroscopy 58 (1995)

28. 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 z-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