Hawkes P.W., Spence J.C.H. (Eds.) Science of Microscopy. V.1 and 2

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Chapter 4 Analytical Electron Microscopy 335

[ω/(1 − ω)]

1/4

= A + BZ + CZ

(44)

where the constants for K, L, and M edges have been summarized in

Goldstein et al. (1986) and they give rise to the dependence shown in

Figure 4–49.

Parameterization of the cross sections and the constants found in

Eqs. (37) and (41) have been discussed in detail in Goldstein et al. (1986)

and show that relative errors in k

factors in the range of 1–5% for K

lines and higher for L lines (10–15%) can be expected based on the

various sources of data and estimates of detector parameters as com-

pared with experimental data. Quantiﬁ cation packages in modern

AEM systems provide accurate quantiﬁ cation based on more precise

knowledge of the internal parameters for the detector often not acces-

sible to the end user. Veriﬁ cation of the quantiﬁ cation procedure is,

however, suggested based on standards of known composition to vali-

date the built-in procedures of the EDXS software.

4.1.2 Absorption

The simple expression for the quantiﬁ cation of EDXS spectra with k

factors based on standards, or with the standardless approach dis-

cussed above, assumes that the sample is thin enough that the absorp-

tion and ﬂ uorescence terms can be neglected, i.e., the thin-ﬁ lm criterion

is applicable. Photons generated by the incident electrons, however, are

absorbed by atoms in the sample via electron excitation processes

while traveling from their origin point to the detector. As in the case

of absorption of X-rays occurring in the EDXS detector, the strength of

the absorption process is related to the X-ray absorption coefﬁ cient

(tabulated in Goldstein et al., 1981) for a given X-ray line and a speciﬁ c

absorbing element. The intensity of the incident radiation (and not the

energy) will decrease with an exponential decay (Figure 4–50) while it

travels through a sample. Large absorption coefﬁ cients occur when

low-energy X-ray lines are just above the absorption thresholds of the

absorbing element, i.e., when the photoabsorption cross sections are

high. At the other extreme, small absorption co efﬁ cients occur for high-

Figure 4–49. Fluorescence coefﬁ cient

for families of lines as a function of

with permission from Springer

Science+Business Media.)

336 G. Botton

energy lines far away from the absorption threshold of the absorbing

element. Considering these effects, the thin ﬁ lm criterion stipulates

that the k

factors should not change more than the arbitrary value of

3% (or less stringent 10%) due to absorption.

If the generation of X-rays is uniform throughout the sample thick-

ness, the k

factor is modiﬁ ed by the absorption correction factor

(ACF) term:

kk ACF

ThinFlim

spec

()

(

)

(

)









µρ





−−

(

)

(

)

{}

−−

(

)

(

)

{}

exp cos

µρ αρ

spec













(45)

where k

Thin Film

is the k

factor for the thin ﬁ lm without the absorption

term, µ/ρ is the absorption coefﬁ cient for element A or B in the sample,

and α is the X-ray elevation (or take-off) angle required to calculate the

pathlength of X-rays in the sample (Figure 4–51).

In practical cases, it clear that the 3% limit to the absorption correc-

tion is very stringent and severely limits the practical applications of

the Cliff and Lorimer approach, thus forcing the use of the correction

term and the measurement of the sample thickness. Examples of cal-

culations of the thickness satisfying the thin ﬁ lm criterion [Eq. (45)]

are given in Table 4–4 for the 10% and 3% criteria (Goldstein et al.,

1986). By inspection, it is possible to note that to avoid absorption cor-

rections, extremely thin samples are required when low-energy X-ray

lines are generated in the sample and when there are elements with

absorption thresholds in proximity of the X-ray lines. For higher energy

photons (and larger separation between the photon energy and the

Figure 4–50. Absorption process of

X-rays and variation of the absorp-

tion coefﬁ cient with energy.

Figure 4–51. Absorption length depen-

dence on elevation angle and foil

thickness.

Chapter 4 Analytical Electron Microscopy 337

absorption threshold), absorption can be neglected in typical AEM

samples. To illustrate this effect, Table 4–5 indicates that an NiAl speci-

men of 9 nm thickness would satisfy the 3% criterion, while a Fe–5%

Ni alloy would require an ≈90-nm-thick foil.

The calculation of the absorption correction factor implies that

absorption coefﬁ cients are known, the density of the sample is known,

and that the thickness of the sample can be measured at the point of

analysis so that the pathlength of the X-rays can be determined.

Although tabulated values for the mass absorption coefﬁ cients for pure

elements can be easily found in the literature (e.g., Goldstein et al., 1986),

the knowledge of the actual sample mass absorption coefﬁ cient and

sample density means that the composition of the sample should be

calculated in an iterative process where initial input of composition,

and thus density and mass absorption coefﬁ cient values, is reﬁ ned at

each iteration step. An additional complication resides in the require-

ment to calculate the mass absorption coefﬁ cients based on all elements

present in the sample even if only relative concentrations are of interest.

The mass absorption coefﬁ cient of the sample is determined from

µρ µρ

(

)

(

)

∑

spec

(46)

where the composition C

for each element is used. All elements where

absorption is potentially signiﬁ cant should be considered, particularly

for low-energy peaks in the presence of light elements where absorp-

tion is important. Similar difﬁ culties arise with the determination of

the sample density that must be deduced following a weighed average

based on the concentration of the pure elements (whether the elements

are detected/analyzed or not). Given the complexity of novel synthetic

structures produced by chemists and physicists and materials scien-

tists, extrapolating the density of materials is not a trivial task for some

samples and the best approach to quantiﬁ cation is to avoid thick regions

where absorption correction is required.

An additional parameter required for absorption correction is the

sample thickness that must be determined accurately. Various

approaches based on convergent beam electron diffraction (Kelly et al.,

1975; Williams and Carter, 1996), contamination spot measurements

(Goldstein et al., 1986; Williams and Carter, 1996), and electron energy

loss spectroscopy (Section 8.2 in this chapter) exist to retrieve the

sample thickness. Their accuracy, however, is approximately 5–10% at

Table 4 –5. Thickness criteria for the 10% and 3% absorption limit.

Thickness for

Material 10% limit 3% limit Absorbed X-ray line

Fe–5% Ni 322 89 NiKa

MgO 304 25 MgKa, OKa

NiAl 32 9 AlKa

SiC 13 3 SiKa, CKa

413 6 SiKa, NKa

338 G. Botton

best. An additional difﬁ culty in the evaluation of the absorption path-

length, from the point of origin to the exit point of the sample, is the

knowledge of the sample geometry and the position of the detector rela-

tive to the point of analysis. This is demonstrated in Figure 4–52 where

the sample is tilted for analysis or when the area of analysis is not suit-

ably positioned with respect to the detector (this might be necessary to

achieve a suitable contrast condition in the image or to align an interface

so that the electron beam is parallel to the interface plane). Signiﬁ cant

differences in absorption corrections, and therefore relative concentra-

tion, can occur if careful consideration is not given to these issues

during experiments (Glitz et al., 1981; Goldstein et al., 1986). Errors in

concentration up to a factor of two have been demonstrated if the detec-

tor position is not known during the analysis (Glitz et al., 1981). Simi-

larly, quantitative analysis of interfaces and grain boundaries should be

carried out so that the X-ray path from the analysis point to the detector

and the electron beam are contained in the same plane. This approach

simpliﬁ es all correction procedures since the absorption is considered

within the boundary itself and the X-rays are generated by the electron

beam traveling within the same material (Figure 4–53).

These difﬁ culties in accurately determining the correction coefﬁ -

cients, density, exact geometry of analysis, and the detector efﬁ ciency,

particularly at low energies, suggest that quantitative work on carbides,

nitrides, and oxides is particularly prone to inaccuracies and error

propagation so that complementary quantitative measurements with

EELS should be carried out whenever possible. For higher energy peaks

(such as transition metal K lines) absorption is signiﬁ cantly lower and

Figure 4–52. Absorption

path for two conﬁ gurations

of sample with respect to

the detector. When the

sample is rotated so that

the detector is in position A

(relative to the sample), the

absorption length is sig-

niﬁ cantly longer than

when the detector is in

position B (relative to the

sample). (Adapted from

Goldstein et al., 1986.)

grain

boundary

plane

incident e-

sample

EDXS

detector

x-rays

Figure 4–53. Ideal geome-

try for analysis of inter-

faces and grain boundaries.

The electron beam and the

absorption path of X-rays

from the generated area to

the detector are all con-

tained within the plane of

the boundary.

Chapter 4 Analytical Electron Microscopy 339

can more easily be neglected in TEM measurements on typical thin

foils.

Alternative approaches to retrieve the composition of TEM samples

while considering the absorption effects are given by the extrapolation

techniques developed over the past decade. These approaches involve

sequences of measurements at increasing thicknesses as discussed in

Section 4.1.4.

4.1.3 Fluorescence

Fluorescence occurs when X-rays have sufﬁ cient energy to ionize atoms

and new photons are generated by the deexcitation process. For

example, the FeKα (6.4 keV) can generate ﬂ uorescence of Cr (5.9 keV)

X-rays but cannot excite MnK (6.5 keV) lines. However, the FeKβ (7.0 keV)

line can generate the MnK X-rays (although the contribution would be

small since the intensity of the Kβ is about 14% of the Kα). At low ener-

gies, ﬂ uorescence can be important given the proximity of several lines.

Although it is possible to generate ﬂ uorescence far away from the point

of origin, ﬂ uorescence in thin specimens is not as important as in bulk

samples for the simple reason that there is less material to ﬂ uoresce.

The original developments required to correct for ﬂ uorescence effects

are due to Philibert and Tixier (1975). Their model considered that the

X-rays are generated from the middle of the tin foil and predicted small

changes in intensities due to ﬂ uorescence. Improvements were later

proposed by Nocholds et al. (1979) to consider uniform emission of X-

rays in the sample including tilted sample geometries when element B

causes ﬂ uorescence of element A:

Crr

BB A A

=−

(

)

[]

(

)













ωµρ

00 932.ln sec−

(

)









{}

µρ ρ α

spec

(47)

where

= the ratio of ﬂ uorescence intensity to primary

intensity,

= ﬂ uorescence yield of element B,

= absorption edge jump ratio of element A,

( µ/ρ)

, (µ/ρ)

spec

= mass absorption coefﬁ cients of X-rays from element

B in element A, and the specimen,

, A

= atomic weights of elements A and B, and

, E

= critical excitation energy for the characteristic radia-

tion of A and B.

As in the case of the absorption corrections, prior knowledge of the

sample composition (in this case of element B) is required for the cal-

culations, thus resulting in an iterative process to estimate the correc-

tion term. This expression proved effective in correcting for ﬂ uorescence

in very thick samples by modern AEM standards (Figure 4–54). Addi-

tional corrections for the ﬂ uorescence caused by the continuum radia-

tion are presently considered to be negligible.

4.1.4 Correction Techniques

The k factor expression [Eq. (42)] assumes that the sample is inﬁ nitesi-

mally thin and corrections are not required. The thin-ﬁ lm criterion

340 G. Botton

discussed above states that corrections are not required if the k

factors

do not change by more than about 3%. This implies that estimates of

the absorption correction factors using Eq. (45) are necessary if the

thickness does not satisfy this criterion and that estimates of the sample

thickness for the points of analysis are required. If the ACF is greater

than 1.03 or smaller that 0.97, corrections must be carried out and exact

measurements of the thickness are necessary.

Similarly, ﬂ uorescence corrections are required if the X-rays of one of

the elements generate signiﬁ cant X-ray emission from other elements

present in the sample. The ﬂ uorescence correction term proposed by

Nocholds et al. (1979) with estimates of the thickness is used to calculate

the ratio of ﬂ uoresced intensity to primary intensity I

for element

A (and for the other elements present). When the correction term is

greater than 5%, the thin-ﬁ lm criterion is not met and the correction is

required. If both the thin-ﬁ lm criteria for absorption and ﬂ uorescence

are not satisﬁ ed, quantiﬁ cation requires the related corrections

ACF

()

(48)

where it is assumed that element A is ﬂ uoresced by element B and the

thickness is calculated at each analysis point. As discussed in Sections

4.1.2 and 4.1.3, the calculations require “prior” knowledge of the sample

composition, thus leading to the use of iterative techniques with

initial assumptions of the composition and reﬁ nement of the com-

positions until there is agreement between the input and output

concentrations.

Various approaches have been proposed over the past few years to

simplify the quantitative analysis of samples without prior knowledge

of the sample thickness. These methods rely on the extrapolation of the

composition for a “zero-thickness” sample based on several measure-

ments obtained from areas of different (but unknown) thicknesses.

These measurements would require a constant acquisition time and

electron beam current during the series of measurements and a uniform

sample concentration in the areas of analysis. The original approach

was proposed by Horita et al. (1987) who suggested the use of measure-

Figure 4–54. Effect of ﬂ uo-

rescence correction on the

quantiﬁ cation of spectra as

a function of thickness. The

raw data show strong thick-

ness dependence while cor-

rected data are shown with

Eq. (47) is independent of

thickness.

Chapter 4 Analytical Electron Microscopy 341

ments of k

factors at different sample thickness. By plotting the k

factors as a function of the intensity of one high-energy X-ray line in

the spectrum and extrapolating the curve to zero intensity, an “absorp-

tion and ﬂ uorescence-free” k

factor can be obtained (Figure 4–55).

This approach is based on the assumption that the high-energy X-ray

line is not strongly absorbed and that its intensity is therefore directly

related to the sample thickness [Eq. (37)]. This method can be used to

determine “zero-thickness” k

factors of standard samples and the

composition of unknown samples following the extrapolation of X-ray

line intensity ratios to zero thickness. Although extremely useful for

the quantiﬁ cation of homogeneous samples, the extrapolation tech-

niques are not suitable for the determination of the composition of

structures where the variations in thickness cause overlap of phases

throughout the analyzed area. For example, when the composition of

precipitates in a thin foil is required, only areas where the precipitates

do not overlap with the matrix can be analyzed.

Various modiﬁ cations to the Horita technique were proposed later

by Van Cappellen (1990) for work on alloys, Eibl (1993) and Van

Cappellen and Doukhan (1994) for work on oxides using the charge

neutrality concept, and in light element analysis demonstrated by

Westwood et al. (1992).

4.2 Quantiﬁ cation in EELS

4.2.1 Quantiﬁ cation Procedures

The atomic cross sections (Section 3) describe the probability of excita-

tion of an atom by a fast incident electron. The energy dependence of

these cross sections provides a basic description of the shape of the

edges (without any solid-state and bonding effects) while the partial

cross section integrated for a given collection angle and energy window

allows the intensity recorded in the spectrum to be related to the inci-

dent beam current and the number of atoms excited by the primary

electron beam. The recorded signal for a particular edge, however, is

superimposed on a large background due to lower energy excitations

(including collective excitations and lower energy ionization edges).

Therefore, even for elements present in the sample in large concentra-

tions, the background can constitute the major contribution to the

Figure 4–55. Extrapolation procedure using

the method proposed by Horita et al. (1987). The

factor is plotted as a function of the inte-

grated number of counts of an X-ray line (not

absorbed in the sample) as an indirect indicator

of the thickness of the sample. Extrapolation to

zero counts leads to the determination of the k

factor of a thin sample. The nonabsorbed X-ray

line must be of sufﬁ ciently high energy and far

from any absorption edge.

342 G. Botton

intensity at a given energy loss (Figure 4–10) and must be removed,

prior to the integration of the signal of the characteristic edge, using

an extrapolation technique (Figure 4–56). The extrapolation is based

on a power-law dependence of the background derived from atomic

physics (Section 3.4.2)

(E) = AE

−r

(49)

where A and r can be considered as constants within a small region of

the spectrum and their values can be retrieved based on a ﬁ t of the

spectrum over an energy window prior to the edge of interest. The A

parameter is strongly dependent on the intensity of the spectrum

(hence, the current, recording time, collection efﬁ ciency, etc.) whereas

the r parameter (related to the slope of the spectrum) varies between

the values of 2 and 6 and is mostly dependent on energy loss and

angular collection conditions. The ﬁ t and extrapolation of the back-

ground are carried out over energy windows (Figure 4–56) using least-

squares ﬁ tting after transformation of the intensity and energy to a

logarithmic scale so that the intercept of the ﬁ t relates to the A param-

eter and the slope to the r parameter. Alternatively, nonlinear least

squares can also be used. Once the A and r parameters are found in

the ﬁ tting region, the extrapolation makes it possible to subtract the

background and to extract the characteristic edge (Figure 4–56). This

power-law extrapolation assumes that the contributions from multiple

inelastic losses to the spectrum in the region of interest are small. For

typical samples having thicknesses in the order of 50–100 nm, this

requirement is relatively easy to meet at high energy losses (a few

hundred electronvolts and higher) but low energy edges (<80 eV) such

as AlL

and LiK are particularly difﬁ cult to analyze due to the rapid

variation of r and the contributions of multiple scattering (see Section

8.2). Very thin samples are thus required and more advanced approaches

able to deal with the steep background and rapid variation of the r

parameter for extrapolation are necessary.

Polynomial-based extrapolation and/or constraint power-law

(forcing the background to cross the spectrum at a given energy) can

be used to deal with numerical instabilities arising from the extrapola-

tion. To remove the complications due to multiple scattering when

extracting low-energy edges, deconvolution of the spectrum using the

Fourier-log technique is necessary (see Section 8.2). Fitting approaches

Energy loss

sity

∆

Figure 4–56. Back-

ground extrapolation of

EELS edges. Fitting of

the background is carried

over a window Γ and

integration is carried out

over an energy window

∆.

Chapter 4 Analytical Electron Microscopy 343

using reference spectra and multiple scattering convolution with low-

loss spectra have been developed recently (Verbeeck and Van Aert,

2004) to deal very successfully with weak and/or overlapping edges

(Figure 4–57). Signal extraction using difference techniques (Egerton

and Leapman, 1995) has also been used to analyze very weak signals

for elements present in trace concentrations (Leapman and Newbury,

1993; Shuman and Somlyo, 1987). A variant of this latter method uses

a numerical difference technique (Zaluzec, 1985). This technique is

relatively insensitive to the sample thickness. Following the extraction

of the background, integration of the signal over an energy window ∆

must be carried out (Figure 4–56) to relate the signal to the number of

atoms present under the electron beam.

Assuming that single scattering is dominant (i.e., one single loss event

occurs when the electron travels through the sample), the integrated

edge intensity, obtained for a given collection angle β and integration

window ∆, is related to the number of atoms per unit area N, the unscat-

tered intensity I

(the zero-loss peak is assumed to be nearly equal to the

incident beam in a single scattering ﬁ rst approximation), and the partial

cross section σ

for the same collection and integration conditions

(β,∆) = NI

(β,∆) (50)

Figure 4–57. Model-based quantiﬁ cation of EELS of edges using maximum likelihood estimators of

parameters of spectra and variation of quantiﬁ cation with respect to thickness (bottom panel). Models

include thickness effects and cross sections. The top panel shows that quantiﬁ cation with this tech-

nique appears to be less sensitive to thickness variations based on a series of measurements with

increasing thickness (relative to the inelastic mean free path) on a wedge sample (Reprinted from

Microscopy, 101, J. Verbeeck and S. Van Aert, Model based quantiﬁ cation of EELS spectra, p. 207–224

344 G. Botton

where the index refers to the single scattering assumption. More realisti-

cally, we must consider that plasmon interactions redistribute the inten-

sity in the spectrum causing multiple inelastic processes visible both at

low energy and at the core edges (i.e., core loss and plasmon scattering

occurs as discussed in Section 8.2) (Figure 4–58). Furthermore, elastic

scattering reduces the collected intensity by scattering electrons outside

the collection aperture both for the apparent zero-loss intensity, relative

to the incident beam intensity, and for the edge. In these conditions, we

must integrate the spectrum intensity above the zero-loss peak so as to

include inelastic scattering up to an energy loss ∆ (Figure 4–58).

(β,∆) = NI(β,∆)σ

(β,∆) (51)

Although this equation allows us to measure the absolute number of

atoms per unit area, the more typical approach in EELS quantiﬁ cation

is to use intensity ratios to deduce the relative concentration using

(

)

(

)

(

)

(

)

σβ

∆

(52)

In this case, the indices k and j relate to the fact that different types of

edges can be used for quantiﬁ cation (K, L, or M). When multiple scat-

tering is removed from the spectrum through the use of deconvolution

techniques (Section 8.2) the expression remains the same but the

extracted intensity is obtained from extrapolation of the deconvoluted

spectrum.

4.2.2 Correction for Convergence

Equation (52) assumes that the incident beam is parallel and the angular

distribution of scattering (represented by the Bethe surface in Figure

4–41) is not altered by any convergence of the incident beam. Correc-

tions to account for convergence of the incident beam onto the sample,

however, become necessary when the electron beam is convergent (α ≥

≥0.3β) as in the case of analysis in STEM mode. In STEM conﬁ guration

the convergence angle α typically exceeds this criterion (Figure 4–59).

The relative concentration must be calculated by including convergence

Figure 4–58. Extrapolation of edge signals and the integration windows used

for quantiﬁ cation. The integration of the signal over the window ∆ is carried

out for the signal at the edge and for the low-loss part of the spectrum [based

on Eq. (51)]. G is the value of the gain change (scale change factor) to visualize

the high-energy part of the spectrum.