Graef M. Introduction to conventional transmission electron microscopy

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578 Electron diffraction patterns

where # = 0,... ,L

parameterizes the edge. The contribution of the edge e to the

shape amplitude is now given by the integral

(q) =

2πq

q · n

−2πiq·r

d#.

Replacing r by the parameterization of the edge we ﬁnd

(q) =

2πq

q · n

−2πiq·ξ

−2πiq·t

d#,

and this integral can be solved readily. If we express the edge with respect to the

center point ξ

= ξ

+ L

/2, then we ﬁnd for the contribution of the edge e to

the shape amplitude of the polygon:

(q) =

2πq

q · n

sin(πq · t

)

πq · t

−2πiq·ξ

. (9.46)

The complete shape amplitude is then given by

D(q) =



e=1

(q).

For the case of the three-dimensional polyhedral particle the computations are

carried out in very much the same way as for the 2D polygon. For a complete

derivation of the equations we refer to the original paper by Komrska [Kom87].

The shape amplitude for a 3D polyhedron can be expressed in terms of a summation

over edges and faces, edges and vertices, or faces and vertices. For numerical com-

putations, the expressions for faces and edges are most easily implemented. While

the equations appear to be quite complicated, they are really rather straightforward,

and the reader is encouraged to work through the derivation in [Kom87]. The shape

amplitude of a polyhedral particle with E edges and F faces is given by

D(q) =−

(2πq)



f =1

q · n

− (q · n

)



e=1

q · n

sin(πq · t

)

πq · t

−2πiq·ξ

(9.47)

This equation is only valid if the ﬁrst denominator is non-zero. If q =±qn

(in

other words, if q is parallel to any one of the face normals), then the contribution

of that particular face (or faces) must be replaced by

(q) =

2π

q · n

−2πiq·n

, (9.48)

9.8 Diffraction effects from polyhedral particles 579

where P

is the surface area of the face f , and d

the distance between the origin

and the face f . The symbols in equation (9.47) are deﬁned as:

, unit outward normal to face f ;

, length of the eth edge of the fth face;

, unit vector along the eth edge of the fth face, deﬁned by

× N

where N

is the unit outward normal on the face which has the edge e in common with

the face f ;

†

, unit outward normal in the face f on the edge e deﬁned by n

= t

× n

The input parameters needed to complete this computation for an arbitrary poly-

hedron are the V vertex coordinates ξ

and a list of which vertices make up each

face (counterclockwise when looking towards the polyhedron center). All other

quantities can be computed from these parameters. In particular, the perpendicular

distance d

of a face to the origin is equal to the dot product of the position vector

ξ of any one of the vertices of that face with the plane normal n

, i.e. d

= ξ · n

The unit face normals can be computed from the cross product of any two edge

vectors for a given face.

The surface area P

of a face can be computed as follows: consider the poly-

gon shown in Fig. 9.36. There are E

vertices, numbered counterclockwise, with

(Cartesian) coordinates ξ

, e = 1,... ,E

. The ﬁrst step involves transforming the

vertex coordinates to a two-dimensional reference frame, with origin located at the

ﬁrst vertex. The e

vector of this reference frame is equal to the unit edge vector

f 1

along the ﬁrst edge of the polygon, as deﬁned above. The e

vector is equal to

the opposite of the unit outward normal n

f 1

. It is then straightforward to compute

the 2D coordinates (x

, y

), e = 1,... ,E

of all vertices:

, y

) = ((ξ

− ξ

) · t

f 1

, −(ξ

− ξ

) · n

f 1

). (9.49)

The surface area of the polygon can then be computed by adding the areas of the

individual triangles (shaded in Fig. 9.36) which make up the polygon. It is easy to

verify that the area is then given by



e=2

e+1

− y

e+1

). (9.50)

†

An alternative deﬁnition of the unit edge vectors could be given in terms of the vertex vectors ξ:

e+1

− ξ

|ξ

e+1

− ξ

580 Electron diffraction patterns

e = t

e = n

Fig. 9.36. Illustration of the computation of the surface area of an arbitrary 2D polygon.

The origin of a Cartesian reference frame is placed at the ﬁrst vertex, and the entire polygon

is subdivided into triangular regions.

The e = 1 term does not contribute because the origin of the 2D reference frame is

located at the ﬁrst vertex. The ﬁrst terms in this summation correspond to the areas

of the individual triangles, and the second terms rotate the coordinate system to the

base of each subsequent triangle. This relation is easily implemented in a computer

program.

The Fortran routine

shape.f90 implements the above formalism for the shape

amplitude of an arbitrary polyhedral particle. The pseudo code

PC-21 describes

the computational steps involved. The equations can be simpliﬁed for the case of

a centrosymmetric polyhedron, in which case the shape amplitude is real. Explicit

equations for the centrosymmetric case are given in [Kom87].

As an example we will compute the shape amplitudes for a tetrahedron, an octa-

hedron, and a rhombic dodecahedron, using the input data presented in Table 9.3.

The computed images for |D(q)|

are shown in Fig. 9.37 for the viewing directions

[100], [110], and [111]. The intensities in the images are displayed on a logarithmic

scale, since the secondary maxima are about an order of magnitude weaker than

the center maximum (which is arbitrarily scaled to intensity 1). The computations

were carried out for a particle with unit dimensions; the size of the reciprocal space

volume will change with changing particle size, but the shape of the volume will

remain the same. A wire frame sketch of the three polyhedra is shown to the left of

the shape amplitudes; numbers refer to the vertex coordinates in Table 9.3.

An interactive web-based implementation of the shape amplitude program is

available on the

website in the form of an ION routine called shape.pro. The user

9.8 Diffraction effects from polyhedral particles 581

Pseudo Code PC-21 Compute the shape amplitude of a polyhedral particle.

Input: input data describing polyhedron

Output: shape amplitude in planar section

read polyhedron data from ﬁle

compute vertex centers and edge lengths

compute edge and outward unit normals for each edge and face

compute face distances and surface areas

ask for viewing direction

ask for offset along viewing direction

for each vector q normal to viewing direction do

for each face f do

compute denominator q

− (q · n

)

{9.47}

if denominator equals zero then

compute D

(q) {9.48}

else

for each vertex of face f do

compute contribution to shape amplitude {9.47}

end for

end if

end for

add all contributions together

end for

write real and imaginary part to ﬁle

can select one of a number of polyhedral shapes,

†

the viewing direction, and at which

point along the viewing direction a cross-section through the shape amplitude is

computed (the default cross-section contains the origin). The program then returns

the calculated intensity, using a logarithmic intensity scale. The user will notice

that with increasing number of polyhedral facets, the shape amplitude increasingly

resembles that for a spherical particle (a set of concentric intensity shells).

In addition to the interactive shape amplitude computation, an animated GIF

movie is made available for each of the polyhedra (click on the polyhedron’s name),

which shows a rendered image of the iso-intensity surface for intensity values

between 0.01 and 0.25. Each movie is created for the same viewing direction, and

†

The following polyhedra are available: tetrahedron, truncated tetrahedron, octahedron, truncated octahedron,

cube, snub cube, truncated cube, cuboctahedron, rhombicuboctahedron, rhombitruncated cuboctahedron, do-

decahedron, snub dodecahedron, truncated dodecahedron, rhombic dodecahedron, icosahedron, truncated icosa-

hedron, icosidodecahedron, rhombicosidodecahedron, rhombitruncated icosidodecahedron, and a ﬂat plate.

582 Electron diffraction patterns

Table 9.3. Vertex coordinates and face edges for the tetrahedron, octahedron, and

rhombic dodecahedron. The vertex list is in a counterclockwise direction when

viewed along the face normal to the center of the polyhedron. Coordinates

refer to a standard Cartesian reference frame (see Fig. 9.37).

N Vertices Face edges N Vertices Face edges

Tetrahedron Rhombic dodecahedron

1(1,

1, 1) 1,3,2 1 (1, 0, 0) 1,4,3,7

1, 1, 1) 4,2,3 2 (0, 1, 0) 2,5,3,4

3(1, 1,

1) 3,1,4 3 (0, 0, 1) 6,3,5,8

1) 4,1,2 4 (1/2, 1/2, 1/2) 9,7,3,6

5(−1/2, 1/2, 1/2) 2,4,1,11

Octahedron

6(−1/2, −1/2, 1/2) 8,5,2,14

7(1/2, −1/2, 1/2) 9,6,8,13

1(1, 0, 0) 1,2,5 8 (

1, 0, 0) 1,7,9,12

2(0, 1, 0) 2,3,5 9 (0,

1, 0) 1,12,10,11

1, 0, 0) 3,4,5 10 (0, 0,

1) 14,2,11,10

4(0,

1, 0) 4,1,5 11 (1/2, 1/2, −1/2) 13,8,14,10

5(0, 0, 1) 2,1,6 12 (1/2, −1/2, −1/2) 12,9,13,10

6(0, 0,

1) 3,2,6 13 (−1/2, −1/2, −1/2) –

7 – 4,3,6 14 (−1/2, 1/2, −1/2) –

8 – 1,4,6

the computational volume contains 128 × 128 × 128 pixels. The computation of

D(q) over a 3D volume is carried out using the program

shape3D.f90, also provided

on the

website. This program takes quite a bit more time to run than the 2D section

program

shape.f90; the only difference between the two programs is the range of

the vector q. An example calculation for the tetrahedron, octahedron, and rhombic

dodecahedron is shown in Fig. 9.38; these images are snapshots from the movies

available on the

website.

The shape amplitude arguments are valid for all diffraction experiments, regard-

less of the nature of the radiation used. For instance, Gragg and Cohen [GC71]

have analyzed the structure of Guinier–Preston zones in Al-5 at% Ag by means of

x-rays, and deduced from the shape of the reciprocal lattice points that the zones

have an octahedral shape.

The effect of the shape amplitude on a diffraction pattern is not necessarily

restricted to the shape of the reciprocal lattice points. The shape amplitude can also

become important in molecular crystals (i.e. crystals which are formed by the regular

stacking of large molecules, in which the molecules do not lose their identity), and

a particularly interesting example of such an effect is present in solid C

. The

Buckminster-fullerene molecule crystallizes in a face-centered cubic structure with

9.8 Diffraction effects from polyhedral particles 583

Fig. 9.37. Modulus squared of the shape amplitudes for the polyhedra shown in the left-

hand column, for viewing directions [100], [110], and [111] (Cartesian reference frame).

The intensities are shown on a logarithmic scale, to bring out the secondary maxima.

Fig. 9.38. Iso-intensity surfaces of the shape amplitudes for the polyhedra shown in the

left-hand column of Fig. 9.37. The intensities are logarithmically scaled. These images are

single frames taken from animations, which can be found on the website.

a lattice parameter of 1.4 nm. The radius of the “buckyball” molecule is 0.354 nm,

or very nearly 1/4 of the fcc lattice parameter. If one approximates the molecule

(which at room temperature spins rapidly) by a spherical shell with a ﬁnite thickness

and a uniform or Gaussian charge density, then one can show [AVHVDVT92] that

the shape amplitude (Fourier transform of the charge density) has zero crossings

at distances corresponding to the h00 (h even) reciprocal lattice spacings of the

584 Electron diffraction patterns

fcc lattice. Therefore, the h00 (h even) reﬂections have a signiﬁcantly reduced

intensity from that calculated from the standard structure factor. This has been

observed by means of x-ray diffraction, and also by electron diffraction (although

care must be taken to properly interpret double-diffraction effects [AVHVDVT92],

see Section 9.2.3).

One could incorporate the shape amplitude algorithm into a diffraction simulation

program, or combine the point symmetry routines described in the ﬁrst chapter of

this book with the algorithm and use them to automatically generate the vertex

coordinates and faces of arbitrary regular polyhedra. Neither of these algorithms are

difﬁcult to implement and the computer-oriented reader is encouraged to experiment

with different algorithms.

Exercises

9.1 Consider the algorithm for the indexZAP.f90 program. Adapt the algorithm shown in

pseudo code

PC-20 to include the microscope camera length as a ﬁtting parameter.

9.2 Modify the

indexZAP.f90 program to include

multiple crystal structures. The new

algorithm should be able to distinguish between several crystal structures and decide

which one is most likely, along with the identiﬁcation of the correct zone axis.

9.3 Determine the possible double diffraction reﬂections for the diamond structure in the

[011] zone axis orientation. Write down three pairs of reﬂections that will contribute

to the forbidden 200 reﬂection.

9.4 Derive equations (9.14) and (9.15).

9.5 Show that the excitation error is related to the position of the bright Kikuchi line

described in equation (9.16).

9.6 Obtain a zone axis orientation for a thick region in one of the study materials. From

your precomputed diffraction patterns using the

zap.f90 program you should be able

to identify the pattern. Select one of the Kikuchi bands and tilt the sample so that this

band remains on the viewing screen. Continue tilting until you reach a new zone axis

orientation. Select a second Kikuchi band nearly perpendicular to the ﬁrst and tilt to a

third zone axis. For each orientation record the diffraction patterns and the tilt angles.

Then, consistently index all three patterns and the Kikuchi bands.

9.7 Show experimentally (i.e. by recording CBED patterns) that the symmetry of the

{220} diffraction disk in Cu-15 at% Al is equal to 1. Show also that the (220) and

(

20) diffraction disks are related to each other by the 2

symmetry operation.

9.8 Determine the point group of study material Ti. You should work at low acceleration

voltage (80–100 kV), to have a more strongly curved Ewald sphere. Since you already

know the point group, ﬁrst determine which zone axis CBED pattern(s) you must

acquire to make an unambiguous determination. Then record CBED patterns and

analyze their symmetries.

9.9 Repeat the previous exercise for the study material BaTiO

Phase contrast microscopy

10.1 Introduction

In the preceding chapters, we have studied how fast electrons interact with a crys-

talline thin foil. We have introduced the multi-beam dynamical diffraction equa-

tions in Chapter 5, solved the two-beam equations in Chapter 6, provided numerical

methods to compute the multi-beam amplitudes for systematic row and near zone

axis orientations in Chapter 7, and explained various defect contrast features and

diffraction effects in Chapters 8 and 9. What we have not done yet is to describe

how the microscope affects (i.e. changes) the electron wave function as it travels

down the column. Actually, we did describe part of the inﬂuence of the microscope

when we talked about two-beam microscopy; in a typical bright ﬁeld–dark ﬁeld ob-

servation, we introduce an aperture in the OL back focal plane, and this determines

which component of the wave function reaches the observation screen. Moving the

aperture around (or, more practically, tilting the incident electron beam) changes

the image contrast, and this is indeed an example of how the microscope settings

affect the image contrast. Another example is the excitation current of the objective

lens: when we change the current, the image goes from under-focus to over-focus,

and at the in-focus condition we observe an interpretable image. It is important to

realize that the same information is present in all images, regardless of the focus

condition; the human brain is just better at interpreting the contrast of the in-focus

image, and we do not usually care too much about the out-of-focus condition. We

will see in this chapter that there are observation conditions for which the in-focus

image is not necessarily the easiest image to interpret, and, indeed, does no longer

hold a special position in a through-focus series of images.

We will describe how the microscope affects the electron wave function as it

travels through the objective lens. We will see how various lens aberrations inﬂuence

both the amplitude and phase of the exit wave function. Then we will describe a

method to determine the magnitude of these aberrations. In the remainder of this

585

586 Phase contrast microscopy

chapter we will describe the simulation of images acquired with two important

observation modes: high-resolution mode and Lorentz mode. We conclude this

ﬁnal chapter with a few general remarks about currently active areas of research in

the TEM ﬁeld.

10.1.1 A simple experimental example

Figure 10.1(d) shows an example of a high-resolution electron micrograph for the

[100] zone axis orientation of BaTiO

. The image was recorded on a JEOL 4000EX,

010

001

(a) (b) (c)

(d)

4 nm

010

001

Fig. 10.1. (a) Schematic [100] zone axis pattern for BaTiO

; (b) experimental pattern with

an outline of the diffraction aperture; (c) computed diffractogram of the high-resolution

image in (d). The aperture is represented as a white circle, and several diffraction spots can

be seen outside the circle (e.g. 022 reﬂection), indicating non-linear image contributions.

Images acquired at 400 kV in a JEOL 4000EX.

10.2 The microscope as an information channel 587

operated at 400 kV. The experimental diffraction pattern is shown in Fig. 10.1(b),

along with a circle indicating the diffraction aperture (radius about 6 nm

−1

). All

diffracted beams inside the circle contribute to the image. The interference between

individual beams gives rise to an interference pattern, Fig. 10.1(d). The microscope

operator has some control over the details of this pattern through, primarily, the

microscope defocus. The image is very sensitive to several external parameters,

including: the precise zone axis orientation, the alignment of the incident beam

along the coma-free axis, the foil thickness, the presence of astigmatism, sample

drift, vibrations, and so on. The main challenge of experimental phase contrast

observations is, then, to create an environment in which all residual aberrations

(to be deﬁned in Section 10.2.5) can be kept within reasonable bounds, so that

the image can be interpreted, using the theory and modeling described in this

chapter.

The computed diffractogram in Fig. 10.1(c) is essentially the modulus-squared

of the Fourier transform of the image. This pattern shows which spatial frequencies

are present in the image. It is clear that there are higher-order frequencies, larger

than the aperture radius, in this image. This is due to non-linear image contributions,

which we will discuss in Section 10.2.2. The diffractogram is an important tool in

HREM observations, as it can be used to analyze the conditions under which the

micrograph was recorded. Several aberrations and related artifacts can be visualized

using diffractograms, as discussed in Sections 10.2.4 and 10.2.5.2.

One of the basic questions which arises in the interpretation of Fig. 10.1(d) is:

where are the atoms with respect to the bright and dark contrast features in this

image? In other words, do the brightest dots correspond to the locations of the

Ba atoms? Or do they coincide with the Ti positions? Can we resolve the oxygen

positions? What does “resolve” mean in the context of wave interference? What is

the smallest distance that we can observe with a given microscope? Can we obtain

an image in which all projected atom positions are visible? We will attempt to

answer all of these questions, and several more, in Sections 10.2.1.4 and 10.2.2.

There are many excellent texts covering the basics of high-resolution elec-

tron microscopy: [Spe88], [BCE88], [Hor94], [WC96], and [FH01]. We refer the

reader to these citations for more details on the experimental aspects of HREM

observations. We will be concerned mostly with the underlying image formation

theory, and the simulation of phase contrast images.

10.2 The microscope as an information channel

The experimental high-resolution (or multi-beam) image shown in the introduction

cannot be interpreted simply by direct visual comparison with a crystal structure

model. In many cases, the crystal or defect structure may not be known a priori,