Graef M. Introduction to conventional transmission electron microscopy

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548 Electron diffraction patterns

When the excitation errors for g and −g are equal (and therefore negative), both

lines are located halfway between the origin and the reciprocal lattice point. The

intensities of both lines are brighter than the surrounding background; in fact, the

entire region between the two lines has a higher-than-background intensity, and

one can clearly identify a Kikuchi band. Kikuchi bands can be used to navigate

in reciprocal space. On the stereographic projection sphere, each Kikuchi band

corresponds to the region enclosed by the defect and excess cones. If we tilt the

sample, so that one particular band remains in the same location in the diffraction

pattern, then we are effectively rotating the sample around the plane normal g.

The intersection of multiple Kikuchi bands corresponds to a zone axis orientation;

this is similar to the intersection of bend contours in a bright ﬁeld image, which also

indicates a zone axis orientation. One can use the Kikuchi bands as a roadmap of

reciprocal space. They are particularly useful to tilt from one zone axis orientation

to another one, while observing the diffraction pattern. On older microscopes with

manual specimen attitude controls, this may require considerable manual skill,

as both specimen translation controls must be used to keep the sample location

constant, and both specimen tilt axes to keep the Kikuchi band at a constant location.

On modern microscopes with computer-controlled eucentric goniometer stages, this

type of reciprocal space navigation may be easier, as the computer takes care of

maintaining the eucentric sample position.

9.4.3.2 Kinematical simulation of Kikuchi lines

The location of a Kikuchi line or line pair can easily be computed using the same

mathematical formalism as that used for the HOLZ line computations. The only

difference is that two lines must be considered instead of just one. The mathematical

formalism derived in Sections 9.4.2 and 3.9.5 provides the excitation error of a

reﬂection for a given beam direction, foil normal, and beam tilt (Laue center).

From the excitation error we can determine the location of the defect line via

equations (9.14) and (9.15). The excess line is then located at a distance |g| away

from the defect line. The program

kikuchi.f90 can be used to compute diffraction

patterns with superimposed Kikuchi lines. The program structure is similar to that

of the

holz.f90 program described in Chapter 3. The mathematical equations derived

for Fig. 9.16(b) can be applied in this case as well.

Figure 9.22(a) shows a [110] zone axis pattern for aluminum. In the zone axis

orientation the defect cone for g coincides with the excess cone for −g, and both

Kikuchi lines have the same intensity, typically bright. This is indicated by bright

lines against a gray background. If the crystal is tilted so that the Laue center is

located at (

0.80.8 1), as shown in Fig. 9.22(b), then the entire pattern of Kikuchi

lines shifts rigidly while the intensities of the underlying Bragg reﬂections change.

The excess lines (in this case the lines corresponding to the smallest excitation

9.4 Linear features in electron diffraction patterns 549

2 2

5 nm

-1

2 2 4

-1 1 3

2 2 2

-2 2 2

1 1 1

-3 3 1

-2 2 0

1 1

-3 3 -1

0 0 -2

0 0

-2

2 2

-1 1 -3

0 0 -4

0 0

-4

(a)

(b)

Fig. 9.22. [110] zone axis pattern for Al, with Kikuchi lines for (a) the exact zone axis

orientation and (b) for an orientation with the Laue center (+sign) at the location (

0.80.8 1).

error) are indicated in white, the defect lines in black. The line width is indicative

of the magnitude of the Fourier coefﬁcient V

†

When we introduced higher-order Laue zones (HOLZ) in Section 3.9.5, we con-

cluded that dynamical electron diffraction is a three-dimensional process, with

reﬂections from all HOLZ planes participating in the scattering event. When study-

ing thick crystals, we may observe the effects of the HOLZ reﬂections in the form

of extra Kikuchi lines which cannot be explained in terms of the regular reﬂections

of the zone axis pattern. We know that, in zone axis orientation, the bright and

dark Kikuchi line pair is replaced by a Kikuchi band with an above-background

intensity, centered on the central beam. The edge of every Kikuchi band either lies

halfway between the central beam and one of the diffracted beams, or it crosses

a diffracted beam (which simply means that the band lies halfway between 0 and

2g). Often one can observe additional dark lines, without their bright partners, at

locations which do not correspond to any of the Bragg reﬂections in the ZOLZ.

These lines can be explained by taking the HOLZ reﬂections into account.

HOLZ lines, as deﬁned in Chapter 3, are caused by elastic scattering of electrons

from ZOLZ reﬂections into HOLZ reﬂections. Because of the slope of the Ewald

sphere where it intersects the HOLZ layers (see Fig. 9.15), the result of this elastic

scattering process is the appearance of dark lines in the ZOLZ diffraction disks. The

corresponding bright lines can be observed along the HOLZ rings. Quite often, the

dark and bright HOLZ lines continue outside of the diffraction disks. Here, they

†

The kinematic theory can only predict the location of the Kikuchi lines, not their intensities. The full dynamical

theory including inelastic scattering must be used to predict the intensity proﬁle of a Kikuchi band. This involves

solving the Yoshioka equations (Section 2.6.5), which is beyond the scope of this book. The interested reader

may consult [Wan95].

550 Electron diffraction patterns

can no longer be caused by elastic scattering processes, and they are hence Kikuchi

lines. The continuation of bright and dark HOLZ lines outside the diffraction disks

is a consequence of inelastic scattering processes. Since the dynamical simulation

of HOLZ lines is already more complicated than a normal multi-beam simulation

using the projection approximation, the inclusion of inelastic scattering effects to

compute the intensity distribution of Kikuchi lines due to HOLZ reﬂections poses

a very complex problem. We refer the interested reader to [Wan95] for a more de-

tailed introduction to inelastic scattering theory. It is possible to compute dynam-

ical Kikuchi patterns, starting from the Yoshioka equations of inelastic scattering

(page 124). Omoto, Tsuda, and Tanaka [OTT02] have derived a full dynamical

theory for inelastic scattering from a perfect crystal using the Bloch wave theory.

Using the kinematic theory, we can predict the locations of the HOLZ Kikuchi

lines; in fact, these lines are simply the extensions of the HOLZ lines that we have

already implemented in the program

holz.f90. The kikuchi.f90 program uses the

HOLZ geometry introduced in Chapter 3 for most of its computations. This means

that only reﬂections belonging to a particular zone and its related HOLZ layers are

considered. When Kikuchi bands are used for navigation, a representation of all

major Kikuchi bands drawn on the asymmetric part of a stereographic projection

can be useful. Such a map can be constructed from a set of experimental Kikuchi

maps, or the entire pattern can be calculated and displayed on a stereogram. Such a

computation only generates the geometry of the Kikuchi bands, not their intensities.

The program

kikmap.f90 can be used to generate sections of the asymmetric unit

of the stereographic projection for an arbitrary crystal structure. The program uses

purely geometrical information to determine the location of all Kikuchi bands

within a conical region of opening angle α around the incident beam. This angle

can be changed experimentally, by varying the diffraction camera length. For each

set of planes, g, we compute the angle β between the plane normal and the incident

beam direction (see Fig. 9.23). If one or both of the angles β ± θ

is smaller than

α, then a portion or the entire Kikuchi band will be visible.

The angles are converted readily into distances from the origin of reciprocal

space, and the equations derived for Fig. 9.16(b) can be used to compute the loca-

tions of both edges of the Kikuchi band. Typical output of the

kikmap.f90 program

is shown in Fig. 9.24 for the [001] zone axis orientation of GaAs. For each band

the edges of the band and the center line are indicated. Zone axis orientations occur

where multiple bands intersect.

9.5 Convergent beam electron diffraction

We have seen in Chapter 5 that the diffraction process is highly sensitive to the

crystal symmetry. Convergent beam electron diffraction patterns provide a direct

9.5 Convergent beam electron diffraction 551

(a) (b)

Fig. 9.23. Schematic representation of a Kikuchi band caused by the defect and excess

cones intersecting the viewing screen. The band in (a) falls entirely within the viewing area

corresponding to the cone with opening angle α. The band in (b) is only partially visible.

visualization of the various symmetries present in the crystal. By carefully exam-

ining the symmetry of multiple CBED patterns of a given crystal

structure, it is

possible to determine both the point group and the space group of that structure.

In the following subsections, we will describe brieﬂy how one can determine the

symmetry groups for the study materials Cu-15 at% Al and GaAs. It is left as an

exercise for the reader to determine the symmetry groups of study materials Ti and

BaTiO

(see exercises).

9.5.1 Point group determination

The combined use of Tables 5.1 (page 340) and 7.2 (page 438), along with Fig. 5.11

(page 341), allows one to determine nearly all the point groups. Let us ﬁrst con-

sider the Cu-15 at% Al study material. We know that the space group is Fm

3m.

Figures 9.25 and 9.26 show [001] and [111] CBED patterns acquired in a Philips

CM20, operated at 80 kV. The patterns are digital combinations of individual ex-

posures; the range of intensities in CBED patterns is often too large to capture in a

single exposure. The bright ﬁeld (BF) symmetry of the [001] pattern in Fig. 9.25(a)

is given by 4mm, which is also equal to the whole pattern (WP) symmetry. Accord-

ing to Table 5.1, this means that the diffraction group is either 4mm or 4mm1

The 200 and 220 reﬂections are both special reﬂections, because they lie on the

mirror planes of the group 4mm. Accordingly, the special dark ﬁeld symmetry

should be equal to 2mm. This is veriﬁed easily by tilting the incident beam so that

the Bragg condition is satisﬁed for any one of these reﬂections. This is shown for

the (200) reﬂection in Fig. 9.25(b), and for (220) in Fig. 9.25(c). In both cases, the

552 Electron diffraction patterns

2 2 0

6 4 0

4 2 0

6 2 0

2 0 0

6 -2 0

4 -2 0

4 0

2 -2 0

5 5 1

-5 -5 1

5 3 1

-5 -3 1

5 1 1

-5 -1 1

5 -1 1

-5 1 1

5 -3 1

-5 3 1

5 -5 1

-5 5 1

4 6 0

6 0

2 4 0

3 5 1

-3 -5 1

3 -5 1

-3 5 1

2 -4 0

2 6 0

2 -6 0

1 5 1

-1 -5 1

1 -5 1

1 5 1

0 2 0

Fig. 9.24. Kikuchi band pattern for the [001] orientation of GaAs, computed with the

kikmap.f90

program. Each band has a thin central line; intersections of multiple thin lines

indicate the locations of zone axes.

orthogonal mirror planes are clearly visible. General reﬂections, such as the 240

reﬂection, have symmetry 2. The dark ﬁeld symmetries allow us to choose between

the two possible diffraction groups, and it is clear from Table 5.1 that the diffraction

group of the [001] zone axis is 4mm1

According to Fig. 5.11, there are only two possible point groups that can give rise

to the diffraction group 4mm1

: 4/mmm and m

3m. Therefore, we need another

zone axis CBED pattern to distinguish between these two point groups. Consider

the [111] CBED pattern, shown in Fig. 9.26. The BF symmetry is 3m, which is

also equal to the WP symmetry (note the ﬁne HOLZ lines in the central disk). This

is consistent with only two diffraction groups: 3m and 6

. Only 6

consistent with one of the two possibilities for the point group, so that the point

group of Cu-15 at% Al is equal to m

3m.

9.5 Convergent beam electron diffraction 553

(a)

(b)

(c)

200

020

220

Fig. 9.25. [100] zone axis CBED patterns for Cu-15 at% Al: (a) whole

pattern, (b) 200 and

mirror planes indicated as short line segments (patterns

recorded at 80 kV).

(a)

(b)

Fig. 9.26. [111] zone axis CBED pattern for Cu-15 at% Al, recorded at 80 kV. Both the

whole pattern (a)

and bright

ﬁeld (b) symmetries

are equal to

3m.

Next, let us consider GaAs as a second example. We have already determined the

BF and WP symmetry of the [110] zone axis from the patterns shown in Fig. 4.25.

The bright ﬁeld symmetry is 2mm, and the whole pattern symmetry is m . From

Table 5.1, we see that there is only one possibility for the diffraction group: m1

There are six point groups consistent with m1

: mm2, 4mm,

42m, 6mm,

6m2,

and

43m. A second zone axis CBED pattern is needed; the [111] CBED pattern,

acquired at 80 kV, is shown in Fig. 9.27. Both the BF and WP symmetries are

equal to 3m, as is derived easily from the HOLZ lines in the central reﬂection. The

554 Electron diffraction patterns

(a) (b)

Fig. 9.27. [111] CBED patterns of GaAs, recorded at 80 kV in a Philips CM20 microscope.

Both the whole pattern (a) and bright ﬁeld (b) symmetries are equal to 3m (sample courtesy

of J. Neethling).

corresponding diffraction groups are then 3m (with point groups

3m and m

3m) and

(with point groups 3m and

43m). The only point group that is common

between the possible groups for both zone axes is the non-centrosymmetric cubic

point group

43m.

The determination of the point group of an unknown crystal structure requires, in

general, several CBED patterns, for all the highest symmetry zone axis orientations.

For each zone axis you will need to record:

the central bright ﬁeld disk;

the entire pattern, including HOLZ reﬂections (this typically requires a small camera

length);

individual diffraction disks, with the crystal (or beam) tilted so that each reﬂection in turn

is in Bragg orientation;

pairs of ±g diffraction disks.

For a detailed description of the procedure, we refer to the paper by Steeds and

Vincent [SV83]. Examples can be found in [WC96, Chapter 21], [FH01, Section

6.5.4] and [Man84]. The three volumes on CBED by Tanaka and coworkers [TT85,

TTK88, TTT94] are mandatory reading for anyone who plans to make extensive

use of CBED techniques.

9.5.2 Space group determination

We have seen in Chapter 1 that space groups are constructed by combining a Bravais

lattice with a crystal point group. For both study materials considered in the previous

section, the point groups belong to the cubic crystal system, which means that

9.5 Convergent beam electron diffraction 555

there are three possible Bravais lattices, cP, cI, and cF. We can determine which

centering we have in two different ways.

Index several different zone axis patterns for a given grain or location in your material. If

you do this correctly, then you will ﬁnd that all reﬂections of a particular kind are missing.

For the examples above, only reﬂections for which all Miller indices have the same parity

are allowed, which means that the lattices are face-centered.

Obtain zone axis patterns of high symmetry with a small camera length (and possibly

a lower acceleration voltage) to determine the location of the HOLZ layer reﬂections

with respect to the ZOLZ reﬂections (see e.g. [WC96, Section 21.2]). This also provides

information about the lattice centering.

For Cu-15 at% Al, we have the cF Bravais lattice combined with the point group

3m, which leads to the following possible space groups (see Tables A4.2 and

A4.3): Fm

3m, Fm

3c, Fd

3m, and Fd

3c. For GaAs, we end up with the possible

space groups F

43m and F

43c. In each case, we have to choose between a symmor-

phic and one or more non-symmorphic space groups.

We know from structure factor considerations (e.g. [MM86] and the examples in

Section 9.2.3) that the presence of glide planes and/or screw axes causes additional

forbidden reﬂections. Such reﬂections may appear in zone axis diffraction patterns

because of double-diffraction effects. In CBED patterns, the phases of the various

structure factors contributing to a zone axis pattern may be such that the diffraction

disks corresponding to kinematically forbidden reﬂections show Gj¨onnes–Moodie

(G-M) lines, or dynamical extinctions [GM65], as discussed in Section 9.2.3. The

presence of G-M lines (or crosses) is usually quite obvious, since their presence is

independent of foil thickness and acceleration voltage, and can be used to determine

whether or not glide planes or screw axes are present. G-M lines, originating from

dynamical extinctions due to ZOLZ reﬂections, are typically broad lines, which are

known as A

and B

lines. The subscript 2 denotes 2D diffraction (i.e. the projection

approximation); A lines are parallel to the g vector of the reﬂection, whereas B lines

are normal to the g vector. Double diffraction interactions with HOLZ reﬂections

result in narrower G-M lines, labeled A

and B

lines (3, because of the 3D nature

of the diffraction process).

Tanaka and coworkers have determined the occurrence of G-M lines for all high

symmetry zone axis orientations of all 230 space groups. The corresponding tables

can be found in [TSN83, TT85]. The tables list, for each space group, and for each

zone axis orientation, which reﬂections are expected to have G-M lines, and which

type of lines (A

2,3

and/or B

2,3

). We refer the reader to the many examples in the

volumes [TT85, TTK88, TTT94] for more details on the determination of glide

planes and screw axes. The G-M lines allow for the determination of 185 out of the

230 different space groups.

556 Electron diffraction patterns

Examination of CBED patterns of the high-symmetry zone axis orientations of

Cu-15 at% Al and GaAs reveals that there are no G-M lines, and that, therefore,

both space groups are symmorphic: Fm

3m for Cu-15 at% Al, and F

43m for GaAs.

The reader is encouraged to repeat the procedures described in this and the previ-

ous section to study materials Ti and BaTiO

. A worked-out example for Ti can be

found in [FH01, Sections 6.5.4 and 6.5.5]. Further examples of the structure deter-

mination of σ , χ , and R intermetallic phases using CBED methods can be found

in [RM94].

9.6 Diffraction effects in modulated crystals

9.6.1 Modulation types

In previous chapters, we have seen that there are two different approaches to elec-

tron diffraction: the kinematical model, in which |F

hkl

describes the intensity of a

diffracted beam, and the dynamical theory, which permits electrons to scatter mul-

tiple times. While the kinematical theory has only limited validity when it comes

to the prediction of images, the theory is quite useful for the description of the

geometry and approximate intensity distributions in diffraction experiments. We

have deﬁned the structure factor F

hkl

as a summation over all atoms in the unit cell

of the electron scattering factors, appropriately phase shifted (see Chapter 2). The

unit cell is repeated throughout the volume of the crystal, which determines the

shape of the reciprocal lattice points. In this section, we will no longer consider

the unit cell as the essential building block of the crystal. Instead, we will redeﬁne

the structure factor as a summation over all atoms in the entire crystal. In other

words,

(t) =



j=1

f (r

, t)e

2πig·R(r

,t)

. (9.17)

The electron scattering factor f (r

, t) depends on the position of the atom; the

vector function R(r

, t) returns the position of atom j at time t, and N is the total

number of atoms in the crystal. This is a very general expression which will allow

us to introduce a large variety of modulated and disordered structures. Time t is a

possible variable; for instance, by averaging the atom displacements due to thermal

vibrations we can derive an expression for the intensity distribution due to thermal

diffuse scattering (TDS), as done explicitly in Section 6.5.1. Diffusion processes

that occur when a thin foil is heated inside the microscope column may also give

rise to a time dependent intensity distribution. For the remainder of this section,

however, we will ignore time as a variable.

9.6 Diffraction effects in modulated crystals 557

The functions f and R completely deﬁne the state of the crystal, provided they

are known for each lattice site. We can use the behavior of these functions to deﬁne

different types of modulated crystals.

Compositional modulations. All atoms are located on their equilibrium lattice sites

(R(r

) = r

), and the electron scattering factors f (r

) vary periodically in one or more

directions. In such a case, we may expand the scattering factor into a Fourier series with

respect to the reciprocal lattice vectors q of the modulated structure:

f (r

) =



2πiq·r

. (9.18)

It is, of course, possible for the modulated structure to have the same basis vectors as

the underlying

lattice (or an integer multiple of those basis vectors), in

which case we

end up with an ordered structure. The vectors q needed to describe this ordered structure

are commonly referred to as the ordering wave vectors, and the corresponding complex

exponentials are known as concentration waves [Kha83].

Alternatively, the function f may vary slowly but periodically, so that the structure

displays compositional modulations on a length scale larger than the underlying unit cell.

Spinodal

decomposition results in structures that display compositional

modulations with

well-deﬁned periodicities.

Disordered and partially ordered structures. All atoms are located at their normal

lattice sites as before, but now the scattering factor f is not a periodic function of posi-

tion. This does

not necessarily mean that there is no order at all. There could be small

regions with a particular kind of ordering, but there is no long-range correlation with

other such regions. Such structures are known as short-range ordered structures, and

we will deﬁne the mathematical behavior of f and the resulting diffraction patterns in

Section 9.7.

Displacive modulations. All atom sites have a well-deﬁned composition (i.e. atom type),

but the atoms are not necessarily located at their normal lattice sites. If the function R(r

)

varies periodically in one or more directions, we end up with a displacively modulated

structure. Such a structure may arise, for instance, during cooling if a particular phonon

mode becomes “frozen in”. In essence, this means that the displacement pattern of the

phonon becomes locked into the crystal structure. Such a process is thought to be of im-

portance for martensitic phase transformations. We will describe displacively modulated

structures and their effects on diffraction patterns in Section 9.6.2.1.

Interface modulations. In Chapter 8, we have introduced planar defects with displace-

ment vector R. In certain crystal systems, it may be energetically favorable for such

defects to organize themselves in a periodic fashion. The crystal is, therefore, divided

into perfect regions, called blocks, which are shifted with respect to each other at regular

intervals. The function R(r

) is then a periodic function that is constant inside the blocks,

and changes nearly discontinuously at each block boundary. Crystallographic shear planes

and anti-phase boundaries can give rise to such interface modulated structures. If a crystal

structure is built from a regular stacking of two or more building blocks, the resulting