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

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Chapter 4 Analytical Electron Microscopy 325

show the increase of expected signal with increases of collection angle

and the variation with atomic number. Note the asymptotic behavior

of the cross sections for large scattering angles approaching 100 mrad.

This effect suggests that by using large collection angles above a few

tens of millirad, the collected intensity does not increase signiﬁ cantly.

This asymptotic behavior affects the optimization of signals and signal-

to-background ratios as discussed in relation to the detection limits

(Section 7). This integration over scattering angles and energy window

is demonstrated in the hatched area in Figure 4–41b.

For the calculation of the probability of X-ray signal generation fol-

lowing ionization, the collection angle of the primary electrons is irrel-

evant and it is necessary to consider all possible scattering angles of

the incident electron. The integration must be carried out up to β = π,

which includes possible backscattering. This integration leads to the

total inelastic cross section for core shell ionization that is used in X-ray

quantiﬁ cation (Section 4.1.1)

≅ 4πa

(R/T)(R/E

) ln (c

T/E

) (33)

where N

is a number related to the occupancy in a particular shell (2,

8, 18 for K, L, M shells, respectively) and b

and c

are factors deduced

from theory or experiments [(Inokuti, 1971, 1978; Powell, 1976), also

tabulated in Goldstein et al. (1986)] and T is the kinetic energy of the

electron.

It is useful to compare the magnitude of the cross sections for elastic

and various inelastic events such as K and L shell excitation, outer shell

(including excitations of plasmons), and generation of secondary elec-

trons (Figure 4–43) as a function of energy. Based on the inelastic cross

sections it is possible to deﬁ ne an inelastic mean free path (mean dis-

tance between inelastic scattering of events) in a fashion similar to the

elastic mean free path (Section 3.1). For an inelastic scattering event i,

= (n

)

−1

(34)

where n

is the number of atoms per unit volume and σ

is the cross

section for an inelastic event i (either total inelastic scattering or K, L

shell excitation).

Figure 4–43. Comparison of cross

section for various scattering

processes for Al as a function of

incident electron energy. The com-

parison shows the strongest inelas-

tic scattering (dominated by plasmon

losses P), followed by elastic scatter-

ing (E), L shell ionization (L), K shell

ionization (K), fast secondary elec-

trons (FSE), and secondary electrons

(SE). The generations of FSE and SE

are not discussed in this chapter.

mission from Springer Science+

Business Media.)

326 G. Botton

3.5 Shape of EELS Edges

There are two factors that contribute to the shape of inner-shell absorp-

tion edges observed in energy loss spectra. The atomic calculations of

the cross sections (hydrogenic of Hartree–Slater) give, to a ﬁ rst good

approximation, the expected shape of the edge related to transitions

from a core state to free continuum states. Although the general shape

is reproduced particularly well at high energies from the threshold,

details arising from transitions to discrete unoccupied bound states

can be important. These effects are visible both in terms of the inte-

grated edge intensity (particularly for transition metal L edges) and in

terms of the ﬁ ne details arising from electronic structure and the

bonding environment in the solid state. These latter modulations are

called ELNES. The dipole selection rules for the transitions from an

initial core level to a ﬁ nal state (see Section 8.1) imply that there must

be a change in angular momentum quantum number ∆艎 = ±1. This

implies that the angular character of the ﬁ nal states (i.e., the symmetry)

is determined by the core state quantum numbers. Making use of the

nomenclature presented in Figure 4–11, the following rules can be

summarized. For K edges, the principal quantum number is n = 1, the

angular momentum quantum number of the core state is s (i.e., 艎 = 0),

and the ﬁ nal state character is p (艎 = 1). For L

edges, the core level is

2p and the ﬁ nal states probed are s (艎 = 0) and d (艎 = 2). For M

edges,

the core level is 3d and the ﬁ nal states will be f or p. Spectroscopically

the K, L, M, N, and O edges therefore reﬂ ect the main quantum number

of the core level n = 1, 2, 3, 4, 5, respectively, and each edge presents

sublevels due to the angular momentum quantum number 艎 and spin-

orbit coupling due to the j quantum number (Figures 4–11 and 4–4). L

edges thus present transitions to three sublevels for 2s, 2p

3/2

, and 2p

1/2

giving rise to the L

, L

, and L

edges, respectively, while M edges have

ﬁ ve sublevels and so forth as summarized in ﬁ gure 4–11.

A summary of the types of shapes is reﬂ ected in Figures 4–44 and

4–45 along with a table (Table 4.4) showing the expected edges visible

for energy losses up to 2000 eV. These edges can be detected in typical

TEM experiments. K edges are observed for light elements up to Si

Figure 4–44. Schematic diagram of edge shapes. (Adapted from Colliex, 1984,

and Hofer, 1995.)

Chapter 4 Analytical Electron Microscopy 327

Figure 4–45. (a) Trends in K edges for light elements and (b) sp electron

metals.

(1800 eV) and generally present a sawtooth shape. The overall edge

shapes are relatively well predicted by the hydrogenic and Hartree–

Slater cross section models although, within the ﬁ rst 30 eV from the

edge threshold, ﬁ ne variations due to the bound ﬁ nal states of p-

character in the ELNES are present. L edges are observed for a larger

range of elements and the shape of these varies with the atomic number.

The L

and L

edges (also labeled as L

edges) are the strongest in the

series due to the large cross sections and the selection rules imposing

transitions to unoccupied levels with large density of states above the

Fermi energy (if there are 3d empty states). The L

edges are hardly

visible in typical acquisition conditions favoring the dipole transitions

(i.e., small collection angle). The edges of elements Na to Ar present

328 G. Botton

the so-called “delayed edges” (Figure 4–44b) with a broad maximum

of the edge intensity 10–20 eV above the edge threshold. This delayed

shape is predicted by cross-sectional calculations and is due to a

maximum in the effective atomic potential resulting in a “centrifugal

barrier” that prevents the overlap between the core and the broad ﬁ nal

Figure 4–45. Continued (c) Trends in L

edge intensities (called white lines)

for the transition metal series. It is clearly visible that the strong peaks at the

edge threshold decrease in intensity with respect to the continuum after the

edge threshold. This effect reﬂ ects the ﬁ lling of the 3d transition metal band

with electrons. (d) Shape of delayed M edges (Zaluzec, 1982, 1984). (Courtesy

of N. Zaluzec.)

Chapter 4 Analytical Electron Microscopy 329

state wavefunctions at low energy from the threshold. This poor overlap

results in low values of the matrix element near the edge threshold (see

Section 8.1). When the ﬁ nal state wavefunctions are very localized, as

in the case of the 3d electrons, they are contained within the barrier

and there is therefore a strong overlap with the core wavefunctions.

This results in strong features at the edge threshold for the L

edges

in the series K to Ni where the 3d band is being ﬁ lled and transitions

to empty 3d electron states are possible. In these cases, sharp features,

known as “white-lines,” are clearly visible leading to noticeable trends

in the transition metal series (Figure 4–45). For the Cu L

edge, where

the 3d band is full with 10d electrons, the delayed maximum is again

observed without strong peaks at the edge threshold. When solid-state

effects result in changes in the electronic conﬁ guration, however, for

example, when Cu loses 3d electrons when bonding to O (such as CuO

or CuO

), white lines do appear in the spectra (Figure 4–46). In Zn and

its oxide ZnO, however, the electrons affected by bonding are of s

character and the changes with bonding are not visible in the L edge

of Zn. White line features in L

edges are observed for the 4d transition

metals (the series Rb to Pd) at higher energy losses.

The M

and M

edges are the strongest in the M series with large

cross sections. These edges exhibit the same trends with delayed

Figure 4–46. Cu L

edge in metallic Cu,

CuO, Cu

O. The pres-

ence of the white line

in CuO and Cu

O (as

compared to metallic

Cu) reﬂ ects the transfer

of electrons from the 3d

band to O atoms in the

oxidation.

Table 4 – 4. Summary of Expected Edges.

Edge Level Elements Shapes

K 1 s Li-Si a

2 p

1/2

, 2 p

3/2

Mg-Ar b

K-Ni c

Cu-B b

3 p

1/2

, 3 p

3/2

K-Cu d

3 d

3/2

, 3 d

5/2

Se-Kr d

Rb-I b

Cs-Yb c

Lu-Au b

4 d

3/2

, 4 d

5/2

Cs-Yb d

5 d

3/2

, 5 d

5/2

U,Th d

330 G. Botton

maxima and strong white lines depending on the atomic number

(Figure 4–47). For the series Rb to In, a delayed maximum is observed

due to transitions to delocalized empty f states well above the Fermi

level while strong white lines are visible for Cs to Yb since the f states

are bound and are being ﬁ lled in the series. The very strong white lines

make these edges very suitable for detection of trace quantities of ele-

ments in samples. For Lu to Au, as the f band is completely ﬁ lled, very

broad delayed edges are present and their detection, due to the back-

ground in the spectra, is often difﬁ cult even for elements present in

samples in large concentrations. M

edges are detected for the K to Zn

series in the energy range between 30 and 100 eV with plasmon-like

features represented in Figure 4–44d. These edges overlap with the

strongly varying background in the low portion of the energy loss

spectrum and are thus difﬁ cult to extract from the spectra. Due to the

large cross sections, however, elemental maps can be easily obtained

with very high spatial resolution (Freitag & Mader, 1999).

3.6 The Background in EELS and EDXS

The background in core level energy loss spectra (Figure 4–10) is domi-

nated by lower energy loss processes such as lower core losses and

plasmon/outer shell losses and the characteristic signal of edges must

be extracted with procedures described in Section 4.2.1. In the case of

Figure 4–47. Nb M

and Ba M

edges. While the Ba edge shows strong peaks

and a lower continuum past the edge onset, the Nb M

edge shows a strong

rising intensity at the postedge threshold.

Chapter 4 Analytical Electron Microscopy 331

EDXS, the continuum background depicted in Figure 4–6c arises from

the “bremsstrahlung” radiation (or braking radiation) as the electrons

are subjected to centripetal acceleration forces in the Coulomb potential

of the atoms in the solid as their trajectories are modiﬁ ed. The X-rays are

generated over a continuous energy range from 0 eV up to the energy of

the incident electrons. The angular distribution of the emitted X-ray due

to the bremsstrahlung radiation is not isotropic as in the case of the

characteristic radiation emitted following ionization and deexcitation.

The distribution is almost forward-peaked with maxima in intensity a

few degrees from the forward direction and very little intensity emitted

in the backscattered direction. Hence the peak over background ratio is

dependent on the location of the EDXS detector (lowest peak-over-

background with higher elevation angles) although additional difﬁ cul-

ties in the acquisition of spectra arise due to the solid angle limitations

and the instrumental contribution of the spectra (see Section 7.2). Expres-

sions describing the continuum spectrum intensity (I

) demonstrate

the dependence on the incident current and the average atomic number

–

of the sample irradiated by the electron beam (Kramers, 1923)

IE iZ

cm b

(

)

∝

−





)

(35)

where i

is the beam current, E

is the incident energy of the electrons,

and E

is the energy of the emitted radiation. At lower energy, the

recorded spectrum does not show the increase in the intensity expected

from Eq. (35) due to the absorption in the sample and in the detector.

Although the generated continuum spectrum does not show sharp

features, in crystalline solids well-deﬁ ned peaks can occur due to

coherent bremsstrahlung reported by Reese et al. (1984) when electrons

travel close to a zone axis. The coherent bremsstrahlung energy peak

positions E

are related to the spacing of the atomic planes in the

direction of the electron beam (L), the ratio of the velocity of the elec-

trons/velocity of light (v/c), and the elevation angle α

(Figure 4–23) as

described by Spence et al. (1983) and Reese et al. (1984)

EkeV

Lvc

()

(

)

−

(

)

(

)

[]

12 4

190

cos α

(36)

These peaks can be mistaken for minor constituent phases when

they overlap in energy with characteristic peaks and can be minimized

by carrying out analyses when the samples are oriented far from major

zone axes.

4 Quantiﬁ cation

4.1 EDXS Microanalysis

4.1.1 Quantiﬁ cation in EDXS

As indicated in Figure 4–7, t her e a r e several processes that need to be

considered in the measurements of concentration based on the detected

signals. First, X-rays generated by the excited atoms must travel through

the specimen and are therefore subjected to absorption. These X-rays

332 G. Botton

can also cause possible ﬂ uorescence of secondary X-ray radiation

within the sample, which generates additional signals further away

from the site irradiated by the primary electrons. The subsequent

detection process includes transmission through a detector window,

various detector layers, and generation of electron-hole pairs in the

solid-state detector device. All these effects need to be considered in

the quantiﬁ cation procedure. We shall consider the different processes

starting with the generation of X-rays in the sample assuming that the

characteristic signals in the spectrum can be extracted from the con-

tinuum background by simple interpolation of the background inten-

sity under the characteristic peak from the background on the low and

high energy sides of the characteristic peak.

As electrons travel through a sample of thickness t and are scattered

at various angles according to the elastic cross section (Sections 3.1 and

3.2) losing energy as dictated by the inelastic cross sections and the

stopping power (Sections 3.1, 3.3, and 3.4) they generate photons in the

X-ray portion of the electromagnetic spectrum. The generated X-ray

intensity for element A is

⬀ (C

t)/A

(37)

where

= weight fraction of element A,

= ﬂ uorescence yield for the K, L, or M line,

= fraction of the total K, L, or M line intensity that is measured,

= atomic weight of element A,

= total ionization cross section for a shell K, L, or M for atom A in

the specimen, and

t = the sample thickness.

The total ionization cross section derived in Section 3.4 can be

expressed in terms of the overvoltage U = (E

) relative to the ioniza-

tion energy E

and incident energy E

commonly used in the AEM lit-

erature where the constants have been included into one single term

and N, b, c are the same constants found in Equation (33) with sub-

scripts related to the shell s, i.e.,

σ= =

()

−

Nb cU

ss s

651 10

(38)

When the thickness of the sample satisﬁ es the thin-ﬁ lm criterion

(Section 4.1.2), the absorption and ﬂ uorescence in the sample can be

neglected. In this condition, the X-ray intensity still remains to be cor-

rected for the detection process that includes absorption in the detec-

tor’s various layers and for the efﬁ ciency of the detector active layer in

generating electron-hole pairs from the impinging X-rays (Sections

2.3.1 and 2.3.2). For each particular X-ray peak, the intensity detected

must therefore include a detector efﬁ ciency term that describes how

the intensity is absorbed

= I

(39)

where

Chapter 4 Analytical Electron Microscopy 333

= {exp[−(µ/ρ)

− (µ/ρ)

]}

{1 − exp[−(µ/ρ)

]} (40)

and

( µ/ρ) = the mass absorption coefﬁ cients for elements A in the window

material W, the Au contact layer, and the Si,

ρ = the densities of the window material W, Au surface contact

layer, and Si dead layer,

X = the thickness of the window W, the Au layer, and the Si dead

layer, and

= the EDXS detector active layer thickness,

The second term accounts for the detector thickness that generates

the signal based on the electron-hole pairs created by the incident

photons (the active layer thickness).

Although these equations can, in principle, be easily calculated, exact

values of the various constants, sample thickness t, and detector layer

thicknesses are not known to sufﬁ cient accuracy and alternative

approaches have been suggested to eliminate the inﬂ uence of these

parameters on quantiﬁ cation. The calculation of the relative concentra-

tion of elements (C

) makes it possible to include all the constant

terms both in the cross sections and detector response into one single

term and thus eliminates the need to know the sample thickness

(

)

(

)













σω ε

(41)

The term within brackets is a constant for a given set of X-ray peaks

for elements A and B, detector, and accelerating voltage. This term

allows the user to relate the intensity ratio for two peaks to the relative

concentration of the elements in the sample. The k

factor approach

suggested by Cliff and Lorimer (1975) makes it possible to simplify the

calculation of relative concentration of elements provided that stan-

dards can be obtained so that

(42)

For binary systems, the concentration can be retrieved assuming C

+ C

= 1 and for ternary systems a similar assumption can be made. In

principle, the k

term in Eq. (41) can be calculated based on knowledge

of the terms within the bracket but the signiﬁ cant advantage of the

Cliff–Lorimer approach resides in the fact that the k

factors can be

obtained from a standard of known composition C

and C

related to

one reference element. As a convention, k factors can be expressed in

terms of a standard with respect to Si and Fe (i.e., k

ASi

BSi

or k

AFe

, k

BFe

etc.) so that any set of k

factors can be retrieved by

ASi

BSi

(43)

Reference k factors can be developed with respect to any elements

but references to Si and Fe were historically developed for users in the

334 G. Botton

geological science and metallurgy communities. Examples of k

AFe

factors are shown in Figure 4–48 and can be used as a good approxima-

tion to quantify measurements, although the exact values are depen-

dent on a particular detector (and in particular detector efﬁ ciency) and

accelerating voltage. Some tabulated values for a range of elements can

be found in Williams and Carter (1996). Users of EDXS in AEM typi-

cally develop their own standards based on samples of known compo-

sition C

, C

to quantify unknown samples containing the same

elements. These k

factors would therefore account for the speciﬁ cs of

the microscope-detector system. Standardless k

factors can be calcu-

lated based on the knowledge of all the terms expressed in Eq. (41)

making use of constants tabulated in the literature.

The ﬂ uorescence term can be derived based on the parameterization

Figure 4–48. Experimental k

AFe

factors for a range of elements (top) for Kα

lines and (bottom) for Lα lines. The limits reﬂ ect the range of theoretical values

obtained with different cross-sectional parameterizations. (From Williams