Quantitative X-ray Spectrometry in TEM/STEM

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 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) 307-316

21. z-factor zfactor 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 = 100-200°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) 25-34.

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