Practical Electron Microscopy and Database

An Online Book, Second Edition by Dr. Yougui Liao (2006)

Practical Electron Microscopy and Database - An Online Book

Chapter/Index: Introduction | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z | Appendix

Multiple Linear Least Squares (MLLS) Fitting in EELS Analysis

Multiple Linear Least Squares (MLLS) Fitting in EELS analysis matchs the measured spectrum to several components. However, in this case, the amount of additional data needed is much more than a single number. The MLS procedure has become available in commercial software and deserves to be used more frequently than in the past [5]. Multiple linear least squares (MLLS) fit technique has many applications:
        i) To evaluate the atomic percentages of each element by computing the relative fit weights and integrating the references over an energy range.
        ii) To interpret the elemental maps and/or EEL spectra due to overlapping edges (e.g. Table 1389a). In EELS quantification, if the edges from different elements are too close and thus the peaks overlap, then the conventional edge intensity extraction is not possible. In this case, the MLLS deconvolution can be used to fit suitable standard (reference) spectra to all overlapping edges, and thus the overlapping edges can be effectively separated. However, if the edge onsets are extremely close (e.g., for the case of Ga/Si in Table 1389), the measurement accuracy will be affected by the change of bonding environment.

Table 1389a. Deconvolution examples of EELS overlap peaks by using MLLS fit technique. The energy range is used to perform MLLS fit.
Edge separation between edges (eV)
Overlap peaks Energy range (eV) Reference
Element A Element B
Symbol
Edge (eV)
Window used for signal (eV)
Symbol
Edge (eV)
Window used for signal (eV)
42 Pb N7,6: 138 145-240 Zr M5,4: 180 180-240 145-240 [2]
43 Cr L3,2: 575   O K: 532     [3]
4 Si L3,2: 99   Ga M3,2: 103      

        iii) To analyze the fine structure of EELS SI (spectrum image) in order to map out the relative intensities associated with a number of chemical states of core loss edges, e.g. Cu (Cu0+ and Cu2+) and Mn (Mn3+ and Mn4+). The advantage of the MLLS fit technique is that it is able to differentiate the small separation between the primary peaks from different valence states.
        iv) To improve the EELS detection limit and weak signal extraction. For instance, by using the MLLS fit technique instead of conventional background subtraction, Cr concentrations as low as 0.03% in Al2O3 was possibly detected at a particular microscope setting. [1]

For the single scattering spectrum, the overall intensity of the spectrum can be given by a power-law background and core-loss profile for each edge:
          I(E) = AE-r + C1IC1(E) + C2IC2(E) + ... + CNICN(E) ---------------------------- [1389a]
where,
         I(E): The recorded intensity as a function of energy E. This represents the spectrum obtained from the EELS measurement.
         AE-r: Power-law background fit.
         C1 - CN: Weightings of the contribution of each element. They are the coefficients that represent the contribution of each reference spectrum to the overall measured spectrum.
         I1(E) - I N(E): Intensity of the contribution of each element at an energy E. They are the reference spectra corresponding to different components (e.g., different elements, chemical states, or phases).

Equation 1389b is the linear combination of all the reference spectra. The core-loss profiles can be calculated using elemental standards of known thickness and composition. Or, a more accurate way is that the profiles are measured experimentally. In multiple least-squares (MLS) fitting, the weightings of the contribution of each element are adjusted by and their values converted to relative concentrations.

Figure 1389a illustrates the intensity distribution as a function of energy, incorporating multiple Gaussian core loss spectra for different elements. In this plot, the background term has been removed. In this modeling, we assume the core loss peaks are perfectly Gaussian distributions. That is, the individual Gaussian peaks correspond to the core loss energies for different elements. For example, Element 1 shows peaks at 500 eV, 510 eV, and 520 eV, indicating the presence of multiple energy states or transitions. Element 2 has a single peak at 520 eV, while Element 3 has peaks at 530 eV and 540 eV, and Element 4 has a peak at 550 eV. The total intensity is the sum of these contributions, shown by the black line in the plot. This visual representation allows for the analysis of how different elements and their respective energy states contribute to the overall spectrum, providing insight into the material's composition and the energy transitions occurring within it.

probability of multiple scattering

Figure 1389a. Intensity distribution as a function of energy, incorporating multiple Gaussian core loss spectra for different elements.

In reality, the Electron Energy Loss Spectroscopy (EELS) core loss peaks are not perfectly Gaussian but are instead more complex in shape, typically characterized by an asymmetric profile. The core loss peaks in EELS arise from inelastic scattering events where incident electrons lose energy by exciting core-level electrons in the atoms of the material. These excitations typically correspond to transitions of electrons from core energy levels (like K, L, M shells) to unoccupied states above the Fermi level. Unlike the idealized Gaussian shape, EELS core loss peaks tend to be asymmetric. The peak often has a sharp rise on the low-energy side (onset of the core excitation) followed by a more gradual fall-off on the high-energy side. This asymmetry is due to the distribution of available final states for the excited electrons, which increases with energy. The width of the core loss peak is influenced by several factors, including the lifetime broadening of the core-excited state, instrumental broadening, and the energy resolution of the spectrometer. This means that the peaks are often broader than a simple Gaussian. A single core loss edge can contain multiple features, including the main peak, which corresponds to the direct transition, and additional fine structure known as the Extended Energy Loss Fine Structure (EXELFS) or Near-Edge Structure (ELNES). These features provide detailed information about the chemical environment, bonding, and local electronic structure of the material. The position of the core loss peak can be slightly shifted depending on the chemical state of the element (chemical shift). For instance, the core loss peak of an element in a higher oxidation state may appear at a higher energy than in a lower oxidation state due to the higher binding energy of the core electrons. In addition to the core loss peaks, there is often a significant background in the EELS spectrum, which includes contributions from plural scattering and other inelastic processes. This background needs to be carefully subtracted to accurately quantify the core loss features.

In practice, such standard spectra used in the MLLS fit technique for Equation 1389a are acquired from reference specimens. The selection of proper references is important to the success of the MLLS fit method. Furthermore, for accurately quantitative analysis, spectra obtained from different valence states should be used as references. Therefore, MLLS depends
on fitting a sum of reference spectra, usually a Power law background (or any background model) and one or more isolated ionization edges, to a measured spectrum.

As an example, the procedure of using MLLS fit to deconvolute O K edge and Cr L2,3 edges is:
        i) Standard (reference) spectra with background, oxygen, chromium signals, obtained from alumina and chromium films, are measured under identical conditions to the source spectra and are normal ized to the integrated counts within 100 eV above the edge onset. In this case, the fitting coefficients obtained with the MLLS fit are equal to the estimated signals in the source spectra.
        ii) The background in front of the O K edge was modeled using power-law fit, starting immediately in front of the edge.
        iii) The MLLS fit is performed including the background function and reference edges.

Figure 1389b shows the oxygen (O) elemental maps obtained with both power-law fit and multiple linear least-squares fit. The measured materials is Ca/TiO. It shows that the extracted net intensities in the area on right-hand side differ by as much as 30%, and the O map obtained with MLLS fit can differentiate the oxygen levels in different areas clearer than that with power-fit. .

EELS of Ca/TiO

Figure 1389b. Oxygen elemental maps obtained with power-law fit (a) and multiple linear least-squares fit (b). The extracted net intensities differ by as much as 30% (c). The measured materials is Ca/TiO. [4]

However, it is necessary to note that inaccuracy from MLLS fit can be introduced because uncertainties, e.g. shape difference between the reference spectrum and the analyzing spectrum, exist over the fitting region which is not necessarily the same region used to extract the signal.

 

 

 

 

 

 

 

 

 

 

 

 

[1] Riegler, K.; Kothleitner, G.: EELS detection limits revisited: Ruby — a case study, Ultramicroscopy 110 (2010) , S. 1004 – 1013.
[2] Harkins, P. and MacKenzie, M. and Craven, A.J. and McComb, D.W. (2008) Quantitative electron energy-loss spectroscopy (EELS) analyses of lead zirconate titanate. Micron, 39 (6). pp. 709-716. ISSN 0968-4328.
[3] Katharina Riegler, and Gerald Kothleitner, EELS detection limits revisited: Ruby — a case study, Ultramicroscopy 110, 1004–1013, (2010).
[4] Gerald Kothleitner and Ferdinand Hofer, Elemental occurrence maps: a starting point for quantitative EELS spectrum image processing, Ultramicroscopy 96, 491–508 (2003).
[5] R.F. Egerton, New techniques in electron energy-loss spectroscopy and energy-filtered imaging, Micron 34 (2003) 127–139.