Graef M. Introduction to conventional transmission electron microscopy

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328 Dynamical electron scattering in perfect crystals

Starting from equation (5.5), and proceeding along the same path as before, we ﬁnd

for the eigenvalue equation (including absorption):









−g

+ iU



−g

... U

−h

+ iU



−h

+ iU



+ iU



... U

g−h

+ iU



g−h

+ iU



h−g

+ iU



h−g

... 2k

+ iU















( j)







= 2k

( j)







( j)







(5.61)

We have made use of the high-energy approximation (ignoring factors of order

( j)

)

), and also U



= 0. The resulting eigenvalue problem is a general complex

eigenvalue problem, which is a non-trivial numerical problem. We will return to this

equation in Chapter 6 when we discuss the two-beam case, and in Chapter 7 where

we will introduce a standard numerical procedure for determining the eigenvalues

and eigenvectors of this equation.

5.8 Important diffraction geometries and diffraction symmetry

5.8.1 Diffraction geometries

In Chapter 4, we have seen that interesting image features are observed for a number

of different sample orientations. The most important orientations are as follows.

The two-beam orientation. In this orientation a single diffracted beam is strongly excited,

while all other beams are weak. While this orientation is somewhat artiﬁcial in that

it is nearly impossible to obtain for crystal structures with a large unit cell, it is of

importance from a theoretical point of view; the dynamical diffraction equations for the

two-beam case have an analytical solution. The two-beam orientation is important for

defect characterization and we will discuss the experimental and theoretical details in

Chapters 6 and 8.

The systematic row orientation. The systematic row orientation is similar to the two-

beam orientation, except that now additional reﬂections along the row g are included in

the image formation process. In an actual experiment, we can, of course, not avoid con-

tributions from higher-order reﬂections, such as 2g and −g. For crystals with a relatively

small unit cell, it is rather easy to orient the sample in a two-beam orientation; this is no

longer the case for materials with larger unit cells, or for observations at higher acceler-

ation voltages, for which the Ewald sphere is ﬂatter. For a two-beam sample orientation

far away from any zone axis, an entire row of reﬂections may be visible in the diffraction

pattern. If the most intense reﬂections in a diffraction pattern can be expressed as multi-

ples of a single reciprocal lattice vector g, then we say that the sample is in a systematic

row orientation. The reciprocal lattice points can be written as ng, with n

min

≤ n ≤ n

max

5.8 Important diffraction geometries and diffraction symmetry 329

The number of active reﬂections depends on the unit cell size, the sample orientation and

the microscope acceleration voltage.

The zone axis orientation. If the incident beam is parallel to a crystal zone axis, then the

diffraction pattern in the objective lens back focal plane will be a two-dimensional pattern.

This is the so-called zone axis orientation, and the diffraction pattern is known as a zone

axis pattern or ZAP. If the zone axis is given by [uvw], then we can employ the duality

between real space and reciprocal space and the diffraction pattern will correspond to the

plane (uvw)

∗

in reciprocal space.

Each reﬂection in a ZAP can be indexed as an integer linear combination of two short

reciprocal lattice vectors, g

and g

. The number of contributing reﬂections need not be

the same along these two directions, and hence the general diffracted beam is described by

g = mg

+ ng

with m

min

≤ m ≤ m

max

and n

min

≤ n ≤ n

max

. In exact zone axis orientation we usually

have m

max

=−m

min

and similarly for n, so that the total number of diffraction spots is

equal to N = (2m

max

+ 1)(2n

max

+ 1). For materials with a large unit cell, there may

well be several hundred diffracted beams.

We can use the magnitude of the excitation error as a selection criterion for which

beams contribute to the multiple scattering process (i.e. all reﬂections for which

|≤s

max

contribute). A general multi-beam simulation program must then allow for

the number of beams to depend on the actual beam orientation. This, in turn, obviates

the need for an adaptive algorithm that automatically determines which beams need to

be included in the simulation.

For both systematic row orientation and zone axis orientation, the relative orientation

of the beam and the crystal is commonly described by the Laue center, i.e. the projection

of the center of the Ewald sphere onto the reciprocal plane (uvw)

∗

In the following chapters, we will discuss the elastic scattering theory for all

three sample orientations, and, when possible, provide analytical expressions for

the beam intensities. Before we attempt to solve the general dynamical equations,

however, we must ﬁrst discuss the symmetry of the diffraction process. As was

the case with crystallography, discussed in Chapter 1, the use of symmetry may

provide useful insights into the diffraction process, and may also allow us to create

fast algorithms to solve the dynamical equations.

The following sections are based on two landmark papers:

(i) Pogany and Turner [PT68]: Reciprocity in Electron Diffraction and Microscopy;

(ii) Buxton, Eades, Steeds, and Rackham [BESR76]: The Symmetry of Electron Diffraction

Zone Axis Patterns.

The reader is encouraged to read these original papers, as they provide the foun-

dation for all diffraction symmetry work. We will closely follow both papers, with

minor changes in notation where appropriate.

330 Dynamical electron scattering in perfect crystals

5.8.2 Thin-foil symmetry

We have seen in Chapter 2 that the ﬁnite nature of a thin foil has consequences

for the shape of the reciprocal lattice points. The reciprocal lattice point is no

longer a mathematical point with zero volume, but, instead, is related to the Fourier

transform of the thin-foil shape function. A second important consequence is that

the symmetry of a thin-foil sample is no longer equal to the space group symmetry

of the crystal structure! This is surprising at ﬁrst, but it is rather easy to understand.

Space groups are inﬁnite groups, and the full space group symmetry, in particular the

translational symmetry of the underlying Bravais lattice, cannot apply to a planar

foil with a ﬁnite thickness. Let us consider the various symmetry elements (we will

refer to the plane through the center of the foil and parallel to both surfaces as the

center plane).

Bravais lattice translations. Only translation vectors which lie in the plane of the sample

can survive the introduction of the surfaces of the thin foil. All translation vectors with a

component along the foil normal are lost.

Rotation axes. All rotation axes which are inclined with respect to the foil normal disap-

pear. Only rotations around the foil normal, and two-fold rotations around an axis in the

center plane survive.

Mirror planes. All mirror planes which are inclined with respect to the foil normal dis-

appear. Only mirror planes which contain the foil normal, or the mirror plane coinciding

with the center plane survive.

Inversion operation. Only inversion centers in the center plane can survive the introduction

of the sample surfaces.

Glide planes and screw axes. Since the only surviving translations are those with a

vanishing component along the foil normal, nearly all glide planes and screw axes will

vanish. The only surviving glide planes are those for which the mirror plane coincides

with the center plane, or glide planes which contain the beam direction and have glide

components in the center plane. The only surviving screw axis is the two-fold screw

in the center plane (and higher-order screw axe

s which reduce to the same projected

symmetry).

Out of the surviving symmetry elements, there are only two which interchange the

top and bottom surfaces of the foil: the mirror plane coinciding with the center

plane, and the inversion center in the center plane. These two symmetry elements

will become important in the following section.

If the foil is inclined with respect to the beam direction, then there are even

fewer surviving symmetry elements; we leave it to the reader to verify that the

highest possible point group symmetry of such an inclined parallel foil is 2/m,

with the two-fold axis normal to the beam direction. For a wedge-shaped foil, the

5.8 Important diffraction geometries and diffraction symmetry 331

only surviving symmetry element is a mirror plane containing the beam and normal

to the wedge edge (point group symmetry m).

While the symmetry reductions imposed by the presence of the sample surfaces

are signiﬁcant, things are not as bad as they appear to be. There is a strong probability

for electron scattering, so that even a small number of unit cells along the beam

direction will impose some of the crystal symmetry on the scattering process. In

many cases, the presence of the surfaces can be considered as a perturbation of the

perfect space group symmetry. Similarly, internal defects in the crystal lattice will

change the space group symmetry, but the perfect material surrounding the defect

will impose the space group symmetry on the scattering process. The space group

symmetry is therefore a good starting point, even in the case of a thin foil with

non-parallel surfaces.

A simple argument, based on equation (5.35), can illustrate

†

how the crystal

symmetry inﬂuences the symmetry of the electron wave function at the crystal exit

plane. Consider a symmetry operator O that belongs to the space group of the

electrostatic lattice potential V (r), i.e. O[V (r)] = V (r). In the derivation of the

phase grating equation (Section 5.6.1) we have seen that the phase of the electron

wave is modiﬁed by the projected potential V

(equation 5.37). The projected

potential is a two-dimensional function which exhibits the projected space group

symmetry. This means that only a subset of the space group symmetry operators

O survives the projection operation. We will denote such operators by O

.Ifthe

wave function ψ satisﬁes equation (5.35), and we apply the projected operator O

we ﬁnd



= [O

(

ψ) + O

(

V ψ)],

= (

O

ψ + O

V O

ψ),

= (

 + O

V )O

ψ,

= (

 +

V )O

ψ.

Both ψ and O

ψ satisfy the same equation, which means that the wave function

has the symmetry of the projected crystal space group.

5.8.3 The reciprocity theorem

It has been known for a long time that the principle of reciprocity holds for

Maxwell’s theory of the electromagnetic ﬁeld (e.g. [Wan79, page 532]). Figure 5.4

†

This argument is intuitive and not very rigorous. For a rigorous approach to crystal symmetry and its effect on

the diffraction process we refer the interested reader to Section 2b in [BESR76].

332 Dynamical electron scattering in perfect crystals

Source

Fig. 5.4. Schematic illustration of the reciprocity theorem.

illustrates the principle:

The amplitude at B of a wave originating from a source

A, and

scattered by P, is equal to the scattered amplitude at A due to the

same source placed at B.

Following von Laue’s pioneering work on x-ray ﬂuorescence in single crystals

in 1935 [vL35], Kainuma [Kai55] and Fukuhara [Fuk66] applied the reciprocity

principle to the theory of Kikuchi lines and to multi-beam electron diffraction,

respectively. Pogany and Turner [PT68] then formally stated the reciprocity theorem

as it applies to electron diffraction.

The meaning of the reciprocity theorem can be made clear by the following

example [PT68]. Consider a sample with parallel top and bottom surfaces. A beam

is incident at an angle α with respect to the foil normal, such that the reciprocal lattice

point g

hkl

is in exact Bragg orientation. Multiples of g

hkl

, labeled by the integer

n, will also give rise to diffracted beams, as shown schematically in Fig. 5.5(a).

The reciprocity theorem then states that the amplitude of the hkl scattered beam

for a given incident amplitude along k

is the same as the amplitude that would

be measured in the −k

-direction for an incident beam along −(k

+ g

hkl

), shown

schematically in Fig. 5.5(b). The amplitudes of the other beams will in general be

very different for these two incident beam directions.

The reciprocity theorem thus states that for a given diffracted beam ψ

, there are

two different incident beam directions which will give rise to the same amplitude.

The two incident beam directions are on opposite sides of the crystal, which makes

it a bit inconvenient to check the validity of the theorem in the microscope. It can be

shown (e.g. [PT68,Wan95]) that the reciprocity theorem is also valid for intensities,

even in the presence of inelastic scattering processes.

5.8 Important diffraction geometries and diffraction symmetry 333

−1

(

)(

)

Fig. 5.5. (a) Schematic beam conﬁguration for an incident beam and g

hkl

reciprocal lattice

point in exact Bragg orientation; (b) conﬁguration equivalent to (a) by reciprocity.

Laue circle

-g/2

k¢

Fig. 5.6. Graphical deﬁnition of the various vectors used to derive the effect of the reci-

procity theorem on the symmetry of the amplitudes of diffracted beams.

It is useful to formalize the reciprocity theorem, and to study its relationship

to other symmetry properties of the crystalline sample. Following Pogany and

Turner [PT68], but with slightly different notation, we will write the incident

wave vector k

as the sum of two vectors; k

along the z-direction (which we will

assume to be normal to the foil surfaces), and k

which is tangential to the foil surface

(and normal to the z-direction). Figure 5.6 shows a graphical representation of the

decomposition.

334 Dynamical electron scattering in perfect crystals

We also introduce the vector k

, which is the tangential component of the scat-

tered wave vector k



. The following relations are now valid:

= k

+ k

;



= k

+ g = k

+ k

;

= k

+ g.

Since the length of the wave vectors is constant for elastic scattering processes,

we only need to keep track of the magnitude of the tangential component; the

normal component is then calculated easily from the wave number and the tangential

component. The only information we need to keep about the normal component is

whether the beam is incident from above or from below the sample. We will denote

a wave vector incident from above the sample by the components (⊗, k

), and a

wave vector incident from below by the components (, k

). The amplitude of the

diffracted beam g is then written as

= ψ

(⊗, k

, C),

where the last argument C indicates the crystal itself; the meaning of this last

argument will be made clear shortly.

Application of the reciprocity theorem is equivalent to reversing the beam direc-

tion to be along the vector −k



, which can be decomposed into −k

− (k

+ g) =

(, −(k

+ g)). The corresponding amplitude of the diffracted beam from the same

set of planes g is then given by

(, −(k

+ g), C).

The second argument of this expression is the tangential component, which does

not necessarily have the same magnitude as k

. We recall from Chapter 2 that in

exact Bragg orientation the tangential component of the incident wave vector must

be equal to −g/2, and therefore we introduce a new vector r, which describes the

deviation from the Bragg condition. Decomposing the tangential component k

into

−g/2 + r (as deﬁned in Fig. 5.6), and applying the reciprocity theorem, we have

for the diffracted amplitudes:

)

⊗, −

+ r, C

= ψ

)

, −

− r, C

(5.62)

This equation states explicitly the reciprocity relation for a given set of diffracting

planes, and is valid regardless of the underlying crystal symmetry. It is also valid

regardless of the number of diffracted beams involved.

The schematic drawing in Fig. 5.7 illustrates the meaning of the reciprocity

theorem: the conﬁgurations in a and b are equivalent by reciprocity. The reciprocity

5.8 Important diffraction geometries and diffraction symmetry 335

-r

-k

-r

-k

mirror

-g

inversion

(a) (b)

Type I symmetry Type II symmetry

Reciprocity

Fig. 5.7. Parts (a) and (b) are equivalent by reciprocity; (c) is derived from (b) by application

of a mirror operation in the plane of the drawing. Part (d) follows from (b) by application

of an inversion operation.

theorem is particularly useful if the

crystal foil has symmetry elements which relate

the top and bottom surfaces to each other. On page 330 we have found that the only

crystal symmetry elements that survive the introduction of the foil surfaces and

which interchange the top and bottom of the crystal foil are:

a mirror plane through the center of the foil, parallel to the foil surfaces, and

an inversion center located in the middle of the foil.

We will denote the crystal foil obtained from C by application of the mirror plane

by C

(m)

. The mirror operation changes the sign of the normal component of every

vector. Similarly, the crystal foil obtained from C by application of the inversion

operation is denoted by C

(−)

. The inversion operation replaces every vector by its

opposite.

A crystal with a mirror plane through the center, normal to the z-axis, is said to

have type I symmetry; the reciprocity relation (5.62) then reads as

)

⊗, −

+ r, C

= ψ

)

⊗, −

− r, C

(m)

. (5.63)

This relation is depicted graphically in Fig. 5.7(c). A crystal with an inversion

center in the center plane is said to exhibit type II symmetry, and application of the

reciprocity relation results in

)

⊗, −

+ r, C

= ψ

−g

)

⊗,

+ r, C

(−)

. (5.64)

This relation is depicted graphically in Fig. 5.7(d).

336 Dynamical electron scattering in perfect crystals

Let us consider these two symmetry types in more detail. For a crystal with type

I symmetry (i.e. C = C

(m)

), the relation (5.63) reads as

)

⊗, −

+ r, C

= ψ

)

⊗, −

− r, C

This relation states that, in the presence of a center mirror plane, the diffracted

amplitudes (and hence intensities) corresponding to the deviations from Bragg

orientation r and −r must be equal. This implies that the diffraction disk for

the beam corresponding to g has at least this internal symmetry. Since an in-

version operation in two dimensions is actually a rotation of 180

◦

around the

normal to the plane of the drawing, we ﬁnd that in the presence of a horizon-

tal mirror, each diffraction disk must have a two-fold rotation axis through the

location of the exact Bragg orientation. The operation which rotates a diffrac-

tion disk around its own center by 180

◦

is commonly denoted by the symbol

R [BESR76].

As an example, consider a crystal with point group symmetry 1, i.e. no symmetry

elements. If the incident beam lies along a zone axis, then a plane of reciprocal lattice

points will be tangent to the Ewald sphere. If only the reciprocal lattice points in

this plane contribute to the scattering process, then the plane itself becomes a mirror

plane (all points not in the plane are ignored, and therefore the mirror symmetry

is valid). This is known as the projection approximation. For a converged incident

beam, each diffraction spot will be a disk with radius determined by the beam

convergence angle. The presence of the mirror plane (both in reciprocal and direct

space) implies that each diffraction disk has the 180

◦

symmetry around the location

of exact Bragg orientation. The resulting symmetry is written as 1

, and this is

one of the 31 so-called diffraction groups. A diffraction group combines the point

group symmetry of the crystal with the type I and type II symmetry operations of

the crystal foil.

The graphical depiction of the diffraction group 1

can be constructed by com-

bining Figs 5.7(a) and (c) into a single drawing, as shown in Fig. 5.8(a). We can

then simplify this ﬁgure by only drawing those components which are essential:

the origin of reciprocal space is indicated by a plus-sign, the diffraction disk by a

circle. The direction of the vector g is along the line between the origin and the

center of the disk, and is represented by a solid line across the disk. The disk center,

corresponding to the exact Bragg orientation, is indicated by a short line segment.

Points in the disk which are related to each other by a symmetry operation are

indicated by small open circles. The resulting schematic drawing for diffraction

group 1

is then shown in Fig. 5.8(b).

5.8 Important diffraction geometries and diffraction symmetry 337

(a)

(b)

Fig. 5.8. Parts (a) and (b) derivation of the 1

diffraction group drawing; (c) and (d) the

same for the diffraction group 2

Next, we consider a crystal foil with type II symmetry (C = C

(−)

). The rela-

tion (5.64) reads as

)

⊗, −

+ r, C

= ψ

−g

)

⊗,

+ r, C

This implies a relation between amplitudes (and hence intensities) in opposite

diffraction disks g and −g. We can again determine the resulting symmetry by

combining Figs. 5.7(a) and (d) into a single drawing, shown in Fig. 5.8(c). The

resulting diffraction group is 2

, i.e. a rotation of 180

◦

around the origin followed

by a 180

◦

rotation around the disk center. An important implication of this particular

diffraction symmetry is that, if the g and −g disks do not show this relation, then

the crystal structure can not have inversion symmetry.

We can repeat this same procedure for all possible orientations of all 32 crystal-

lographic point groups, and it can be shown (e.g. [BESR76]) that this leads to 31

diffraction groups. Alternatively, one can use the 10 two-dimensional point groups

†

and combine each of them with the crystal foil symmetries of type I and type II.

The graphical representations of the 31 diffraction groups are given in Figs. 5.9

(adapted from Table 1 in [BESR76]). The diffraction groups derived from a given

two-dimensional point group are displayed in 10 sets of two, three or four groups.

The short straight line segments in some of the drawings indicate mirror planes;

if the reciprocal lattice point lies on one of those mirror planes, then the internal

†

The two-dimensional crystallographic point groups (planar groups) are 1, 2, m, 2mm, 4, 4mm, 3, 3m, 6, and

6mm.