Graef M. Introduction to conventional transmission electron microscopy

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338 Dynamical electron scattering in perfect crystals

2mm

Fig. 5.9. Graphical representation of the 31 diffraction groups (based on [BESR76]).

symmetry of the corresponding disk is higher than that of the general disk. It is

clear from the ﬁgures that there are only four possible symmetries for an individ-

ual diffraction disk: 1, 2, m, and 2mm, as shown schematically in Fig. 5.10. The

symmetry of an individual diffraction disk is generally known as the dark ﬁeld

symmetry, and we distinguish between the general dark ﬁeld symmetry (1 or 2),

and the special dark ﬁeld symmetry (m or 2mm), which can only occur in those

diffraction groups that contain mirror planes.

If the projection approximation is valid, then the highest-order diffraction group

in each set must be used. This is the so-called projection diffraction group; the

5.8 Important diffraction geometries and diffraction symmetry 339

441

4mm

4 mm

4mm1

661

6mm

6m 6 mm

6mm1

Fig. 5.9 (cont.).

2mmm21

Fig. 5.10. The four possible symmetries

of an individual diffraction disk. Symmetries

and 2mm are only possible if the diffraction disk lies on a mirror plane of the parent crystal

structure.

schematic for this group is surrounded by a thicker frame in Figs. 5.9 and is marked

by the letter [P] in Table 5.1. All projection diffraction group symbols end in 1

reﬂecting the mirror symmetry of the projection approximation.

It is now a simple, but somewhat lengthy task to determine the diffraction group

for every possible zone axis orientation for each crystallographic point group. This

was ﬁrst done by Buxton et al. [BESR76], and their Table 3 is reproduced in

Fig. 5.11. The table lists for each crystallographic point group which diffraction

groups are possible. We will make use of this table when we discuss convergent

340 Dynamical electron scattering in perfect crystals

Table 5.1. Diffraction pattern symmetries (from Table 2 in [BESR76]).

Dark ﬁeld

Diffraction group Bright ﬁeld Whole pattern General Special

1111None

[P] 21 2None

2221None

11 1None

[P] 22 2None

m1 1m

mmm1m

[P] 2mm m 2 2mm

2mm 2 1 m

2mm 2mm 2mm 1 m

mm 1m

2mm1

[P] 2mm 2mm 2 2mm

4441None

42 1None

[P] 44 2None

4mm 4 1 m

4mm 4mm 4mm 1 m

4mm 2mm 1 m

4mm1

[P] 4mm 4mm 2 2mm

3331None

[P] 63 2None

3m 3 1 m

3m 3m 3m 1 m

3m1

[P] 6mm 3m 2 2mm

6661None

33 1None

[P] 66 2None

6mm 6 1 m

6mm 6mm 6mm 1 m

3m 3m 1 m

6mm1

[P] 6mm 6mm 2 2mm

beam electron diffraction in Chapter 9. For completeness we also reproduce in

Table 5.1 a portion of a related table (Table 2, [BESR76]), which lists the various

symmetries for each diffraction group. This table will be discussed in more detail

in Chapter 9.

5.9 Concluding remarks and recommended reading

The literature on the dynamical theory of electron diffraction is extensive. The

theory can be traced back to the initial Bloch wave treatment of electron diffrac-

tion by Bethe [Bet28] in 1928, followed in 1940 by a more detailed theory by

5.9 Concluding remarks and recommended reading 341

2/m

222

mm2

mmm

4/m

422

4mm

42m

4/mmm

6/m

622

6mm

6m2

6/mmm

432

43m

m3m

6mm1

3m1

6mm

4mm1

4mm

2mm1

2mm

Fig. 5.11. Table representing the relation between the 31 diffraction groups and the 32

crystallographic point groups. (Table 3 in [BESR76].)

342 Dynamical electron scattering in perfect crystals

MacGillavry [Mac40]. In 1949, Heidenreich [Hei49] published an important paper

on the theory and application of electron diffraction. This paper can be consid-

ered to be the ﬁrst didactical exposition, and although the notation is somewhat

different from that used in this book, the reader is strongly encouraged to study

this publication. In the 1950s, electron microscopes became more widely available;

several research groups worked on the development of the dynamical theory and

the formalism became ﬁrmly rooted in quantum mechanics. Among the most im-

portant advances or reviews are the papers by: Cowley and Moodie (introduction of

slice methods, 1957, [CM57]); Hirsch, Howie, and Whelan (kinematical theory of

defect images, 1960, [HHW60]); Howie and Whelan (ﬁrst derivation of DHW equa-

tions (5.18), 1961, [HW61]); Tournarie (matrix solution similar to equation (5.24),

1961, [Tou61]); Sturkey (scattering matrix formalism, 1962, [Stu62]); Gevers,

Blank, and Amelinckx (extension of DHW equations to non-centrosymmetric crys-

tals, 1966, [GBA66]); Metherell (review of the Bloch wave method, 1975, [Met75]);

Humphreys (review of Bloch wave method, 1979, [Hum79]); Bird (review of the-

ory and application to convergent beam electron diffraction, 1989, [Bir89]). This

list is by no means exhaustive, and the reader should consult the references in these

papers for additional publications.

The relativistic nature of high-energy electron diffraction was analyzed in detail

by Fujiwara in 1961 [Fuj61]. The equivalence of the various theories developed

in the 1950s and 1960s was, at ﬁrst, not obvious; in 1974 Goodman and Moodie

[GM74] reviewed the different formalisms and described their equivalence in detail.

In 1975 Van Dyck [VD75] showed that all of the dynamical theory formulations

could be derived from a path integral formulation of the scattering process.

The theory presented in this chapter is based entirely on plane waves. The electron

wave functions are written as linear combinations of plane waves, with wave vectors

given by the Bragg equation (for the DHW formulation), or determined by the

solution of an eigenvalue problem (for the Bloch wave method). Since the high-

energy elastic scattering problem is equivalent to a standard band-structure problem

in solid-state physics, we might consider changing from a plane wave description to

a description in terms of other basis functions. We have already seen in Chapter 2

that a large number of plane waves (Fourier coefﬁcients) is needed to describe

accurately the variations of the electrostatic lattice potential near atoms. Basis

functions which are better suited to describing such variations are well known

in the solid-state physics community, and several alternative basis sets have been

proposed in the literature, among which are the following.

Combined basis algorithm (CB). Tochilin and Whelan [TW93] introduced a so-called

combined basis approach to reduce the size of the dynamical matrix. The basis functions

Exercises 343

consist of a combination of plane waves and localized functions describing electrons chan-

neling along atomic strings. Such a description results in a fast algorithm and sometimes

allows for an analytical solution of a multi-beam conﬁguration.

Orthogonalized plane waves (OPW). Barreteau and Ducastelle [BD95] have suggested

the use of an OPW expansion to reduce the size of the scattering matrix. For zone axis ori-

entations this method, combined with the use of symmetry considerations (see Chapter 7),

results in a signiﬁcantly faster algorithm.

Atomic column approximation (ACA). Van Dyck and Chen [VDC99] have proposed an

analytical solution for the exit wave function, based on an atomic column approximation.

They show that the contribution of each column of atoms parallel to a zone axis can be

parameterized with a single parameter, which depends on the “projected weight” of the

column.

While all of these alternative methods do succeed in reducing the size of the compu-

tational problem, they are not commonly available as public-domain or commercial

algorithms, nor have they been implemented for the most general crystal structure

case. It will be up to the interested and adventurous reader to experiment with these

and other alternative approaches.

To conclude this chapter on the multi-beam theory of elastic electron scattering,

we list a few other sources which the reader may wish to consult for additional

information:

Modern Diffraction

and Imaging Techniques in Material Science

, edited by S. Amelinckx,

R. Gevers, G. Renault, and J. Van Landuyt [AGRVL70].

Electron

Microscopy of Thin Crystals

, P. Hirsch,

A. Howie, R.B. Nicholson, D.W. Pashley,

and M.J. Whelan [HHN

77].

Electron Microdiffraction, J.C.H. Spence and J.M. Zuo [SZ92].

Transmission Electron Microscopy, Physics of Image Formation and Microanalysis,L.

Reimer [Rei93].

Principles of Electron Optics. Volume 3: Wave Optics, P.W. Hawkes and E. Kasper

[HK94].

Transmission Electron Microscopy: a Textbook for Materials Science, D.B. Williams and

C.B. Carter [WC96].

Advanced Computing in Electron Microscopy, E.J. Kirkland [Kir98].

Exercises

5.1 Derive equations (5.54) and (5.55).

5.2 Write down the explicit equations (5.24) for a three-beam case with the following

beams:

(a) −g, 0, g;

(b) 0, g, 2g.

344 Dynamical electron scattering in perfect crystals

Next, write down the corresponding equations (5.61) for the Bloch wave formu-

lation.

5.3 Show that equation (5.31) is indeed incorrect.

5.4 Show that a wedge-shaped foil can have at the most the point group symmetry m.

5.5 For a crystal with point group symmetry 4/mmm determine which zone axis orien-

tations [uvw] correspond to each of the allowed diffraction groups.

Two-beam theory in defect-free crystals

6.1 Introduction

Most x-ray diffraction experiments are designed to increase the probability that a

reciprocal lattice point will fall on the Ewald sphere. This is usually not a problem

for electron diffraction, since the radius of the Ewald sphere is much larger, as we

have seen in Chapter 2. It is, therefore, ironic that the most basic TEM technique

involves minimizing the number of lattice points on the Ewald sphere! Indeed, the

most popular conventional TEM technique relies on the use of only two electron

beams, the transmitted beam and a single scattered beam. In many cases, we have to

work hard at getting only one diffracted beam. The shorter the wavelength, the more

reﬂections will fall on or close to the Ewald sphere for any given crystal orientation.

Furthermore, if the direct space lattice has a large unit cell, then reciprocal space

will be ﬁlled densely with lattice points and hence two-beam microscopy becomes

progressively more difﬁcult with increasing unit cell size.

There is a good reason for the use of two-beam techniques: the theory becomes

mathematically more tractable and closed-form solutions to the dynamical scatter-

ing equations are known. Intuitively, it makes a lot of sense to study a crystal by

looking at one set of lattice planes at a time. For instance, when studying dislo-

cations, the displacement ﬁeld around the dislocation core causes lattice planes to

distort. For a given dislocation type, certain lattice planes may remain undistorted

and the two-beam technique allows identiﬁcation of these undistorted lattice planes,

thus providing valuable information concerning the displacement ﬁeld of the dis-

location. Before we can begin to study defects, however, we need to understand

elastic electron scattering in a defect-free crystal.

We will see in this chapter that perfect, defect-free crystals give rise to image

contrast features (in two-beam mode), which are essentially independent of the

crystal structure. In other words, the contrast features observed for a perfect crystal

are the same whether we study metals, semiconductors, ceramics, or any other

345

346 Two-beam theory in defect-free crystals

crystalline material. In addition, the general image features are qualitatively the

same at all acceleration voltages, for all microscopes. We will assume that we have

a defect-free crystal for which we can compute the electrostatic lattice potential,

as described in Section 2.6. We will start from the general multi-beam elastic

scattering equations derived in the previous chapter, and apply them to the non-

centrosymmetric two-beam case. This approach will permit us to derive explicit

mathematical expressions for the intensities of transmitted and diffracted beams.

Once the general expressions are derived, it will be straightforward to apply them

to speciﬁc cases.

All theoretical computations will be compared with experimental images ob-

tained from the four study materials. The reader should by now have prepared thin

foils of one or more of the study materials, following the instructions in Chapter 4.

Several of the experimental images of Chapter 4 will be analyzed in the present

chapter and compared with theoretical predictions. First, we will simplify the DHW

equations (5.18) to the two-beam situation. After deriving the solutions for a perfect

defect-free crystal, we will discuss different ways to compute the image contrast nu-

merically and compare the simulations with the observations on the study materials.

We will introduce the column approximation and apply it to the two-beam case.

In the second half of this chapter we will repeat the two-beam analysis, this time

using the Bloch wave formalism (equation 5.51). We will introduce the dispersion

surface construction, which is the dynamical equivalent to the kinematical Ewald

sphere construction. The Bloch wave formalism will provide physical insight into

the phenomenon of anomalous absorption. We will conclude the chapter with

examples of two-beam image simulations, along with Fortran-90 source code and

ION-routines available from the

website.

6.2 The column approximation

Since scattering angles for electron diffraction are small compared with the angles

in x-ray diffraction, electron diffraction is essentially a forward scattering process.

This means that, even after being scattered several times, most electrons will travel

nearly parallel to the incident electron beam. Let us assume that the maximum cu-

mulative scattering angle is, say, 2

◦

; then an electron which travels through a 100 nm

foil will end up at most 3.5 nm away from the projection of the entrance point on

the exit plane. The higher the acceleration voltage, the smaller the diffraction angle,

which means that the lateral spread of the electron wave decreases. It is then a good

approximation to assume that an electron which enters the foil at one point will

never leave a cylindrical column centered on this point; this column is parallel to the

incident beam direction, and has a diameter of only a few nanometers (depending

on the foil thickness and the acceleration voltage). For numerical purposes it is more

6.2 The column approximation 347

Sample

Image

Fig. 6.1. Schematic drawing of a column in a crystal, and how that column relates to the

image pixels.

convenient to consider rectangular columns of width L

= 5–10 nm, as shown in

Fig. 6.1. This is called the column approximation, ﬁrst introduced around 1957

by M. Whelan and coworkers [WHHB57]. It is assumed that neighboring columns

do not interact, i.e. no electrons are exchanged between columns, so that the dynam-

ical equations can be solved for each column in turn. The column approximation

is essentially equivalent to the high-energy approximation (see page 306).

Consider a rectangular sample area (perpendicular to the electron beam) of di-

mensions L

by L

(distances are measured in nanometers). A schematic drawing

is shown in Fig. 6.1. We can subdivide this area into square elements with edge

length L

(typically 5–10 nm). The total number of squares N

is then given by

= N

× N

where N

) is the number of squares along the x (y) direction. Each square

can be considered to be the top of a column, and is labeled by the pair of integers

(i, j). The length of the column (the local sample thickness z

(i, j)) and the crystal

orientation along the column (the local excitation error s

(i, j)) are known, and

then it is a simple matter of solving the dynamical diffraction equations for those

parameters to obtain the transmitted and scattered intensities at the exit plane of

the column. These intensities can then be arranged into arrays of N

× N

pixels,

where each pixel is labeled by two integers i and j , as shown in Fig. 6.1. In this way,

we can calculate numerically the bright ﬁeld and multiple dark ﬁeld images for a

given sample geometry. We will make extensive use of this basic image simulation

method throughout this and the following chapter.