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Chapter 4 Analytical Electron Microscopy 395

properties of the dielectric function and, in particular, the Kramers–

Kronig relations (Wooten, 1972) that link the real and imaginary parts

of the dielectric function with the help of the Kramers–Kronig analysis

(KKA): if the loss function Im{−1/ε} is known from the measurement

of a spectrum,

then Re{1/ε} can be determined by KKA. Based on this

relationship it is possible to retrieve both ε

and ε

and gain full knowl-

edge of the dielectric function of a material (Figure 4–97). Once the

Figure 4–96. Variation of the plasmon energy for a series of pure elements.

“x” symbols are experimental values and full circles are theoretical values

calculated with the free electron model of Eq. (90). (Reprinted from Colliex

Figure 4–97. Real and imaginary part of the dielectric function based on

Kramers–Kronig analysis of energy loss spectra for (a) TiN and (b) VC.

Springer Science+Business Media.)

To retrieve the loss function from a spectrum, the single scattering

distribution must be retrieved as demonstrated below.

396 G. Botton

dielectric function is known, the results can be compared to optical

measurements at low energies and can be used to associate peaks to

features in electronic transitions. One important advantage of the

dielectric function measurement with EELS (as compared to optical

techniques) is in the energy range of the measurements. With EELS, it

is possible to analyze energies up to few hundred electronvolts cover-

ing, with the use of monochromator energies as low as the infrared

region of the spectrum (down to about 0.5 eV), the visible and the UV

up the soft X-ray region (few 100 eV). Examples of KKA with EELS to

study the dielectric function of material is given in Moreau and Cheynet

(2003) and Launay et al. (2004). Systematic work on the modeling of

low-loss spectra using ﬁ rst principles and a discussion of the limitation

of this approach has been presented by Keast (2005).

Another very useful application of low-energy losses is related to the

strong dependence of the spectra on the thickness of the samples

(Figure 4–98). As the sample thickness increases, the probability of

multiple inelastic losses increases. For example, the incident electron

can excite one or more plasmon losses when traveling in the sample.

If the mean distance between inelastic losses is λ, the probability of

multiple loss events n is given by

(

)

−

(

)

exp

λλ

(92)

If we consider the intensity of the zero-loss peak I

(corresponding

to zero plasmon loss events, i.e., n = 0) with respect to the total intensity

in the spectrum I

, we can deduce the probability P

for n = 0 and

Figure 4–98. Example of the variation of the low-loss spectrum in Al for two

different relative thicknesses. The increase in thickness (from t/λ = 0.2 to t/λ

= 2.2) causes multiple plasmon losses visible at energy values of multiple single

losses (i.e., n times E

). For broad plasmon peaks, multiple peaks are not well

resolved. (Courtesy of N. Braidy, McMaster University.)

Chapter 4 Analytical Electron Microscopy 397

== −

(

)

exp

(93)

This expression is typically used as

Iλ





)

(94)

with the mean free path λ values either tabulated in the literature (for

a standard collection condition) or based on measurements with stan-

dards of known thickness or local measurements using complemen-

tary techniques (such as convergent beam or contamination spot

measurements) (Williams and Carter, 1996; Kelly et al., 1975). Measure-

ments of the intensities for I

and I

are based on the integration of the

spectra over windows as demonstrated in Figure 4–99. The measure-

ments of thickness have been demonstrated to be reliable for thickness

up to t/λ = 4 (Botton et al., 1995b). Values of the inelastic mean free

path are dependent on the energy of the primary electrons, the mate-

rial, and the angular collection conditions. Estimates of λ can be

obtained based on the work of Malis et al. (1988).

The use of a ﬁ nite collection angle β results in scattering outside the

collection aperture and affects the thickness measurements. For mul-

tiple inelastic losses, the total angular distribution emerging from the

sample is broader as compared to the single scattering distribution due

to multiple scattering events and the convolution of the inelastic

scattering distributions. A ﬁ nite collection angle therefore results in

loss in collection efﬁ ciency of electrons having lost higher energies by

multiple events and leads to changes in the apparent mean free path.

Approaches to convert mean free paths from one aperture collection

condition to another have been presented by Botton et al. (1995b), while

iterative methods and computer programs have been developed by

Figure 4–99. Features used for the determination of the relative thickness t/λ

according to Eq. (94).

398 G. Botton

Egerton (1996). Mean free paths for 100 keV electrons and β = 10 mrad

are in the range of 115 nm for C, 100 nm for Al, 110 for Si, and 75 nm

for Fe and have been tabulated in Egerton (1996) and Malis et al. (1988).

The mean free path can be estimated with some basic assumptions on

the sample composition if β << (E/E

) where E is the energy loss and

is the incident energy (so as to satisfy the dipole condition). If these

conditions are respected, values of mean free paths can be calculated

using (Malis et al., 1988)

λβ

()

≈

(

)

(

)

106

FE E

(95)

with E

values given in kiloelectronvolts, λ in nanometers, and β in

millirad. F is a relativistic factor that can be calculated as

(

)

1 1022

1 511

(96)

and E

is the mean energy loss in electronvolts. E

can be approxi-

mated as E

≈ 7. 5 Z

0.36

without taking into account any contributions

from changes in density or the crystallographic structure. For com-

pounds containing elements of known atomic fraction f

, a mean effec-

tive atomic number can be calculated by summing over each element

i as

eff

∑

(97)

so that the effective atomic number Z

eff

can be used to determine

the mean energy loss E

. Using the Kramers–Kronig analysis of

the scattering distribution, the absolute sample thickness can also be

calculated (Yang and Egerton, 1995) without prior knowledge of the

sample.

From the recorded spectra containing multiple inelastic losses (e.g.,

Figure 4–98), the single scattering distribution can be retrieved using

deconvolution methods. These techniques are founded on the principle

that the recorded spectrum is based on the sum of the single scattering

distribution, the double scattering distribution, . . . , the i scattering

distribution, up to the nth scattering distribution. Each i distribution

is obtained by the convolution of the single scattering distribution with

itself i times. This sum of convolutions can be analyzed with Fourier

techniques to retrieve, from the experimental spectrum containing

multiple inelastic losses, the original single scattering distribution.

These methods are available in commercial EELS analysis packages as

standard functions. In this context, the Fourier-log technique is used

to process the full spectrum (Egerton, 1996) from zero electronvolts to

few hundred electronvolts. The analysis of the spectrum to retrieve the

distribution representative of single scattering is necessary for all

quantitative work related to the Kramers–Kronig analysis and the mea-

surement of absolute thickness discussed in Egerton (1996) and Egerton

and Cheng (1987). The additional application of the deconvolution

method is to remove multiple scattering effects from core loss spectra.

Chapter 4 Analytical Electron Microscopy 399

The probability of multiple scattering including one core loss and

plasmon losses increases in thick samples and the effect is visible as

an increased intensity further away from the edge threshold: the

spectra appear as a convolution of the low-loss spectra with the expected

edge proﬁ le (Figure 4–100). These multiple scattering effects must be

removed by deconvolution methods (in this context using the Fourier

ratio method) to correctly interpret modulations in the spectra at high

energy from the threshold. Structures within a few electronvolts from

the edge threshold, however, are less sensitive to these multiple

scattering effects.

Acknowledgments. I am grateful for the patience and understanding of

Profs. Peter Hawkes and John Spence who allowed me to complete the

work with on-going academic commitments, conference organi zation,

and setting up of a new facility. I am indebted to members of my group

for providing some ﬁ gures and for feedback, in particular, N. Braidy,

G. Radtke, M. Couillard, C. Maunders, and Y. Zhu. I want to thank

several collaborators and friends who have provided, over the years,

interesting samples, motivating discussions, and moral support.

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