Power-law Fit: a Background Fitting Model
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In quantitative EELS elemental analysis, the core-loss intensities are extracted by straightforward subtraction of the background intensity from the acquired signal. The background removal procedure is normally performed by an extrapolation method suggested by Egerton [1]. The inverse power law can be used to fit the pre-edge region (i.e. immediately prior to the onset of the edges), which is given by,
IB(E) = AE-r ----------------------- [3419]
where,
IB -- Intensity in the channel of energy loss E.
A, r -- Fitting constants for a particular curve fit (A: broad, r : 2 - 5).

The goodness of fit is defined by linear-squares fit to the experimental spectrum ( χ2). The fit is subtracted from the total spectrum intensity and is extrapolated beyond the edge to give the core-loss signal. Note that the power law form in Equation 3419 is widely used to remove the background because it contains only two parameters to be adjusted and it yields a good fit in many cases. Different ways of adjusting the two parameters (A and r) are normally used, for instance, the 2-area method [1], ravine-search [2] and simplex [3 - 4] procedures.

In the cases that very thick TEM samples are used in EELS or EFTEM measurements, the increase of plural scattering effects induces two main problems: i) The background models, e.g. the form of power law, are not convenient to represent the actual background; ii) The low energy edge onsets are embedded in the background, meaning some edge becomes invisible.

Furthermore, in many cases, the background window cannot be too big since the energy peaks of other possible elements in the specimen are very close to the core-loss of the interesting element. For instance, for a core loss of the interesting element is about 348 eV, a small background window (e.g. 20 eV) is used to fit the spectrum in order to avoid the contribution from the carbon edge (284 eV), so that we only pick up the signal generated by the interesting element for the elemental evaluation.

In general, if the TEM specimen is too thick (t/λ > 0.4), a deconvolution process must be employed to remove the effect of plural scattering, since the increase of plural scattering intensity in the higher energy region of an ionization edge can cause some artifacts:
ii) Make the background signal on subsequent edges deviate significantly from the power law model.

In this case, in order to deconvolute the core-loss spectrum, the Fourier-ratio method is applied. The deconvolution procedures then are:
i) Collect both the low- and core-loss spectra from the same region of the specimen under the same conditions (including eV/change, convergence and collection semiangles).
ii) Isolate the edge of interest and remove the background intensity.
iii) Fourier-transfer the low-loss spectrum and background-subtracted edge.
iv) Divide the core-loss spectrum Fourier transform by the low-loss Fourier transform.
v) Inverse the Fourier transform to yield the desired deconvolved spectrum.

However, in most cases, recording low-loss and core-loss spectra under the same conditions is extremely challenging, since the acquisiton time required for a good SNR (signal to noise ratio) in the core-loss spectrum is usually not short enough to avoid saturation of the signal from the ZLP (zero-loss peak). Therefore, in practice, it is necessary to sacrifice the SNR in the core-loss signal, or utilize a spectrometer system that has an ultrafast electrostatic shutter installed.

[1] Egerton R.F., Electron Energy Loss Spectroscopy in the Electron Microscope (NewYork, Plenum Press, 1986).
[2] Bevington P.R., Data reduction and error analysis in the physical science (McGraw-Hill, New York, 1969), p. 105.
[3] Colliex C., Jeanguillaume C. and Trebbia P., Microprobe Analysis of Biological Systems, Hutchinson and Somlyo Eds. (Acad. Press, New york, 1981), p. 251.
[4] De Bruijn W., Ketelaars D., Gelsema E. and Sorber L., Microsc. Microanal. Microstruct. 2 (1991) 281.

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