Graef M. Introduction to conventional transmission electron microscopy

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Electron diffraction patterns

9.1 Introduction

In this chapter, we will take a closer look at the geometry of electron diffraction pat-

terns. First we will discuss spot patterns and how to index them. Superpositions of

spot patterns and double diffraction are explained in detail, as well as Moir´e patterns

and the corresponding diffraction effects. After a brief discussion of ring patterns

we introduce linear features, such as streaks, Kikuchi lines, and HOLZ lines. Con-

vergent beam electron diffraction and symmetry determination (both point group

and space group) form the topic of Section 9.5. Diffraction effects in modulated

structures are introduced; displacive and compositional modulations as well as in-

terface modulations are analyzed, and the difference between commensurate and

incommensurate structures is highlighted. Various types of diffuse intensity distri-

butions are introduced next, and we conclude this chapter with a discussion of the

shape function of polyhedral particles.

9.2 Spot patterns

9.2.1 Indexing of simple spot patterns

Indexing a spot pattern is perhaps one of the most important tasks of an electron

microscopist and it is hence useful to outline a possible procedure. The procedure

below can be used if the crystal structure of the material is known a priori. Needless

to say, it is important to obtain diffraction patterns in the correct experimental

conditions (eucentric height, focused SAD aperture and image, focused spots), so

that the calibrated value for the camera constant can be used.

(i) Obtain a zone axis pattern with a high density of spots (i.e., short distances between

the spots). Document the sample orientation (tilt angles) and the area from which you

took the pattern (bright ﬁeld image).

(ii) Measure the distance to the origin (in mm) for three reﬂections closest to the origin.

518

9.2 Spot patterns 519

(iii) Measure the angles between these three reﬂections.

(iv) Convert the distances into d-spacings, using the calibrated camera length L and the

wavelength λ (equation 3.71 on page 206).

(v) Compute a table of d-spacings for the known crystal structure and compare those

values with the experimentally measured ones. Assign Miller indices to the reﬂections

(in most cases you will only be able to assign a family {hkl} to each reﬂection).

(vi) Compute the angles between all the members of the different families and compare

them with the measured angles. Assign the correct Miller indices to the measured

reﬂections, making sure that all three angles (1–2, 1–3, and 2–3) are correct.

(vii) Take the cross product between two reﬂections (counterclockwise) to obtain the zone

axis of the diffraction pattern.

As an example, consider the Ti zone axis pattern shown in Fig. 4.18(b) on

page 279. The pattern was obtained on a JEOL 2000EX microscope operated at

200 kV, with a calibrated camera length of L = 1490 mm. To identify this zone

axis, we measure the distance between the origin and three different reﬂections. It

is typically more accurate to measure the distance between −ng and +ng and then

divide this distance by 2n. The measured distances are shown in Fig. 9.1(a) (dis-

tances shown were measured on the original negative). We also measure the angles

between the three reﬂections. It is usually best to select the shortest independent

reciprocal lattice vectors.

Using the measured lattice parameters (Table 4.2 on page 241) we can calculate

a table of angles between reciprocal lattice vectors. The calculated angles between

61.5°

90.0°

= 8.0 mm

= 14.6 mm

= 16.5 mm

Zone axis [ 1 1 . 0]

Multiplicity 6

5 nm

−1

000

110

1 10

002 002

1 111 11

111111112

1 12

112

1 12

113113

1 13 1 13

220

2 202 21

221

2 21

221

00004

2 22 2 22

222222

(a) (b)

Fig.

9.1. Example of a manual indexing procedure, using the pattern for titanium shown in

Fig. 4.18(b). Distances were measured on the original micrograph and are slightly magniﬁed

in this schematic.

520 Electron diffraction patterns

Table 9.1. Calculated angles between low-index plane normals for

the structure of titanium. The lengths of the reciprocal space vectors

were computed for a 200 kV microscope with a camera length of

1490 mm and are ranked in increasing order. All angles are in

degrees; the values 0

◦

and 180

◦

are not shown.

{10.0}{10.1}{10.2}{11.0}

{00.1}

90.0

61.4 , 118.6

42.5, 137.5 90.0

7.98 mm

{10.0} 60.0, 120.0 28.6, 64.0, 47.5, 70.3, 30.0, 90.0,

14.63 mm 116.0, 151.4 109.7, 132.5 150.0

52.1, 57.2, 18.9, 49.5,

{10.1} 81.0, 99.0, 76.1, 86.7, 40.5, 90.0,

16.66 mm 103.9, 130.5 93.2, 103.9, 139.5

130.5, 161.1

{10.2} 39.5, 71.6, 54.2, 90.0,

21.65 mm 85.0,

95.0, 125.8

108.4, 140.5

{11.0} 60.0, 120.0

25.33 mm

plane normals are shown in Table 9.1 for the lowest-order reﬂections; the data for

this table were generated by the Fortran program

tabangle.f90. The algorithm is

nearly identical to that used for the program

indexZAP.f90, which will be discussed

in the following paragraphs. For one member of each independent family of re-

ﬂections we calculate the angles between the plane normal for that member and all

other plane normals. The number of possible angles can become quite large, es-

pecially when higher-order reﬂections and/or low symmetry crystal structures are

considered.

First, we ﬁnd that the top three rows in the table correspond to the three measured

distances r

in Fig. 9.1(a). The angles between the families of plane normals are

also consistent with the measured values: the angles are highlighted in the ﬁrst row

of the table. We select reﬂection 1 ({00.1}) to be the (00.1) reﬂection; reﬂection 2

({10.0}) is then equal to, say, the (1

1.0) reﬂection. The last reﬂection 3 must be equal

to the vector sum of the other two, i.e. (1

1.1). The zone index is then computed from

the cross product between (00.1) and (1

1.0) and is equal to [110] = [11

20] (see

Table 1.1 on page 25). The calculated pattern is shown in Fig. 9.1(b); note that the

reﬂections of the type (00.l) with l odd are forbidden due to the non-symmorphic

nature of the space group P6

/mmc. In the experimental pattern those reﬂections

are present because of double diffraction (see Section 9.2.3).

9.2 Spot patterns 521

While the manual procedure works in principle, in practice it is found to be

useful only for high-symmetry crystal structures, such as cubic and hexagonal.

In lower symmetry crystals, many zone axis patterns look very similar, and an

accurate measurement of the positions of the reﬂections may not always be suf-

ﬁcient for an unambiguous manual indexing of the pattern. In addition, tables

such as the one shown in Table 9.1 become very large and unwieldy for low-

symmetry crystal structures. In multi-phase materials, indexing is complicated by

the fact that it may not always be easy to tell from which phase the pattern was

obtained. Precomputed patterns, such as those produced by the

xtalinfo.f90 pro-

gram, may be useful if they are printed at the correct scale, so that negatives may

be placed directly on the prints for visual comparison. It is also possible to use

the measured values of d-spacings and angles to determine a “best-ﬁt” pattern;

it is not too difﬁcult to write a program that will compare computed d-spacings

and angles with experimental ones and identify a best ﬁt, based on a least-squares

criterion or another appropriate statistical measure. Such a program would use

a look-up table of all possible angles and d-spacings, along with an evaluation

criterion to select the most likely solution. It is obvious that symmetry argu-

ments should be used to reduce the number of possibilities to the smallest number

possible.

The program

indexZAP.f90 implements a simple algorithm for indexing zone axis

diffraction patterns from an arbitrary (but known) crystal structure. The program

outline is described in pseudo code

PC-20 . The user must provide a crystal struc-

ture, the microscope accelerating voltage and camera length, a truncation radius in

reciprocal space, and ﬁve measured quantities for each zone axis pattern. Those ﬁve

quantities are the lengths of three reciprocal lattice vectors g

, g

, and g

+ g

, the

angle α

between g

and g

, and the angle α

between g

and g

+ g

, as shown

schematically in Fig. 9.2. Lengths should be measured in millimeters, angles in

degrees. All angles should be less than or equal to 90

◦

, and distances should be

< 90∞

< α

| < |g

Fig. 9.2. Schematic illustration of the input parameters needed for the indexZAP.f90 program.

The parameter restrictions are shown on the left of the ﬁgure.

522 Electron diffraction patterns

Pseudo Code PC-20 Indexing of a zone axis diffraction pattern.

Input: PC-2 , PC-3

Output: best-ﬁt zone axis pattern identiﬁcation

ask for a truncation radius in reciprocal space

compute all independent reﬂections g

hkl

within this radius

rank them in order of increasing length

for every independent reﬂection do

compute all g-vectors that enclose less than 90

◦

with selected reﬂection

eliminate doubles from this list

compute lengths of vectors and sum vector, along with angles α

and α

store data in linked list

end for

repeat

ask user for experimental data

scale measured distances using the microscope camera constant λL

compute 

for every combination in the linked list {9.1}

determine smallest 

list solution

until no further patterns to be indexed

ranked from smallest to largest. It is always possible to ﬁnd three reﬂections that

satisfy these conditions.

The algorithm ﬁrst determines one reﬂection from each of the families that have

their reciprocal d-spacing within the truncation radius. The algorithm then con-

structs a list of all possible combinations of two independent reﬂections satisfying

the conditions described in the previous paragraph. Each of these combinations

represents a possible zone axis orientation. A linked list is used to store the com-

puted lengths of the three vectors g

, g

, and g

+ g

and the two angles α

and

. If these ﬁve numbers are represented by the symbol d

, with j = 1,...,5,

and i = 1,...,N, where N is the total number of possible zone axis orientations,

then we can deﬁne a difference parameter 











j=1



− d



. (9.1)

The parameters e

, j = 1,...,5, represent the experimentally measured quantities.

The zone axis orientation for which 

is smallest is the most likely pattern matching

the experimental data.

9.2 Spot patterns 523

One complication arises when the experimental diffraction pattern contains so-

called “double-diffraction” reﬂections. We will discuss the origin of these reﬂec-

tions in Section 9.2.3. The algorithm in the program

indexZAP.f90 recognizes poten-

tial double-diffraction reﬂections and is capable of indexing experimental patterns

that contain them.

The

indexZAP.f90 program should function well if the microscope camera length

has been accurately determined, and if the experimental parameters are accurately

measured. Accuracies of a few percent are required. It is possible to extend the

algorithm to take into account weighting factors to compensate for inaccuracies in

distance and angle measurements. It is left to the reader to improve the algorithm

for the case where the microscope camera length is not exactly known; one could

even consider the camera length as a ﬁtting parameter, to be determined by the

algorithm. It is also left as an exercise for the reader to expand the algorithm to

include multiple crystal structures. The algorithm should then be able to select

the most likely crystal structure and zone axis out of a set of possible crystal

structures.

The indexing problem in electron diffraction is similar to the indexing problem

encountered in electron back-scattered diffraction or EBSD. EBSD is used to obtain

orientational information from a crystalline material; the technique is known as

orientation imaging microscopy, and is usually carried out in a scanning electron

microscope (SEM) equipped with a dedicated camera system. From the EBSD

pattern, the spatial orientation of several (typically around six or seven) plane

normals is determined. From the orientational information, the Miller indices of all

planes must then be determined. There is extensive literature on this problem and

the interested reader may ﬁnd more information in [SKA00].

9.2.2 Zone axis patterns and orientation relations

Obtaining low-index zone axis diffraction patterns and correctly indexing them is

an important skill for the microscope user. Often, however, the most interesting

diffraction patterns consist of multiple individual patterns. Let us consider a simple

example: a twin in a face-centered cubic lattice. The twin geometry is shown in

the [110] projection of Fig. 9.3. This projection clearly shows the ABC stacking

sequence. The twin plane is a mirror plane (labeled m) parallel to the close-packed

planes. Taking an A-plane as the twin plane, the stacking sequence on the bottom

crystal is the mirror image of the sequence on the top. The twin plane is common

to both twin variants, and therefore we expect the g

111

reciprocal lattice vector to

be common to both lattices.

We have already described in Section 1.9.5 how we can express the relative

orientation of the two lattices by means of an orientation relation. For the fcc twin

524 Electron diffraction patterns

ABCABCA

Fig. 9.3. Schematic illustration of two mirror twins in a face-centered cubic lattice in a [110]

projection. The twin planes mirror the ABC stacking sequence into the sequence CBA.

this OR is given by

(111)

 (111)

;

10]

 [

110]

While this relation was used in Chapter 1 to compute the combined stereographic

projection of the two crystals, one can also use the same mathematical expressions to

compute the electron diffraction pattern for the twin. After all, a zone axis diffraction

pattern contains those reciprocal lattice points that lie along the equatorial circle of

the stereographic projection.

Figure 9.4(a) shows experimental zone axis patterns for the Cu grain of Fig. 4.21

on page 284. The zone axis pattern for location 1 (top left in Fig. 9.4a) is a [110]

pattern. When the selected area aperture is placed across variants 1 and 2, the

pattern on the top right of Fig. 9.4(a) is obtained. The central nearly vertical row of

reﬂections g

is common to both variants; the twin plane is normal to this direction

and indicated by a dotted line in Fig. 9.4(b). All reﬂections of variant 1 are mirrored

across this plane and are shown as open circles. Variants 2 and 3 result in the pattern

shown on the lower left-hand side of Fig. 9.4(a). The common reﬂections now lie

along the g

vector, and the mirror plane is again at 90

◦

from the common row of

reﬂections. When the selected area aperture is placed across the three variants, the

resulting pattern is a superposition of the three diffraction patterns. The individual

twin mirror planes are now no longer prominent, and instead the pattern shows two

new mirror planes, indicated by dashed lines in Fig. 9.4(b). In order to index all

9.2 Spot patterns 525

1 + 2

2 + 3

1 + 2 + 3

(a)

1 + 2

2 + 3

1 + 2 + 3

(b)

Fig. 9.4. Zone axis diffraction patterns of twins in the Cu grains shown in Fig. 4.21. The

experimental patterns across several twin boundaries (a) are shown as simulated patterns

in (b).

reﬂections in this overlap pattern and assign them to the correct twin variant, we

must record the individual variant patterns as well as the overlap pattern. This may

require the use of a small selected area aperture, or, in the case of a ﬁne-grained

material, microdiffraction techniques. The geometry of the three twin variants is

shown in Fig. 9.3. The angle between the two twin planes is equal to the angle

between two 111 directions, i.e. 70.5

◦

or 109.5

◦

The Fortran program

zapOR.f90 can be used to simulate zone axis diffraction

patterns for a given orientation relation. The second phase can either be identical

to or different from the ﬁrst phase. The output of the program is similar to that of

the

zap.f90 program, shown in Fig. 4.5(c).

9.2.3 Double diffraction

Consider the diamond crystal structure, shown in Fig. 9.5(a). The structure can be

regarded as two interpenetrating face-centered cubic lattices, displaced by a vector

(

). Consequently, the structure factor squared for this cell may be written as

hkl

= 4 f

1 + (−1)

h+k

+ (−1)

h+l

+ (−1)

k+l

cos

(h + k + l)

. (9.2)

This structure factor vanishes for mixed indices h, k and l, as in the fcc case,

and for h + k + l = 4n + 2, with n an integer (i.e. h + k + l = 2, 6, 10, 14,...

are forbidden). Figure 9.5(b) shows the [110] zone axis pattern for the diamond

526 Electron diffraction patterns

000

111

111111

111

220

113

113 113

113

004004

224224

224 224

002

222

002

222

(a) (b)

Fig. 9.5. (a) Diamond crystal structure and (b) [110] zone axis pattern.

structure. The reﬂections of the type {002}, {222}, etc. (indicated by squares in the

ﬁgure) are forbidden because their structure factor vanishes. Yet, in many electron

diffraction patterns taken along this zone axis, weak reﬂections are observed at

these positions. How can there be intensity at a reciprocal lattice point with a

vanishing structure factor, i.e. a reciprocal lattice point for which the kinematical

theory predicts zero intensity?

The full dynamical scattering equation for the (002) reﬂection in diamond ori-

ented along the [110] zone axis is given by

dψ

002

− 2π is

002

= iπ



···+

iθ

002

000

+···+

iθ

111

+···

···+

iθ

111

+···



. (9.3)

The ﬁrst term on the right-hand side is zero, since q

002

=∞. The next two terms

correspond to the (1

11) and (

111) reﬂections, both allowed for the diamond struc-

ture, so their contributions will be non-zero! It is easy to understand how this

happens: when we introduced the coefﬁcients q

hkl

we stated that they are inversely

proportional to the Fourier coefﬁcient V

hkl

of the electrostatic lattice potential. The

second term on the right of the equation above contains the factor q

along with

the diffracted amplitude ψ

111

. Electrons in this beam have a non-zero probability,

described by q

(or V

), to be scattered dynamically by the (1

11) planes. The

resulting electrons will have been scattered by two planes (

111) and (1

11), and they

will therefore travel in the direction given k + (

111) + (1

11) = k + (002). We thus

conclude that, although no electrons are directly scattered into the 002 diffracted

beam, electrons which are scattered twice, once by the (

111) planes and subse-

quently by the (1

11) planes, do end up traveling in the direction of the 200 beam.

This phenomenon is therefore known as double diffraction.

9.2 Spot patterns 527

The Fourier coefﬁcients V

and V

111

are identical so that the two doubly

diffracted paths a and b in Fig. 9.5(b) will reinforce each other. This is not al-

ways the case. Consider the titanium crystal structure, shown in Fig. 4.3. It is easy

to show (exercise) that the structure factor squared for this cell may be written as

hkl

= 2 f

1 + cos π



(k − h) + l)

>

. (9.4)

This structure factor vanishes whenever the cosine term is equal to −1, or, equiva-

lently, when

2(k − h) = 6n + 3(1 −l), n integer.

Figures 9.1 and 4.18(b) show the [11.0] zone axis pattern for the titanium struc-

ture. The reﬂections of the type {00l} and (3

3l) with l = 2n + 1, etc. (indicated by

squares in Fig. 9.1 and encircled in Fig. 4.18b) are forbidden because their structure

factor vanishes. There are now several doubly diffracted paths that will contribute

to the (00.1) reﬂection; four possible paths are indicated by the letters a, b, c, and d

in Fig. 9.6. The four reﬂections participating in those paths are (

11.0), (1

1.0),

110

1 10

002 002

1 111 11

111111112

1 12

112

1 12

001001

and

Fig. 9.6. Schematic illustration of four double diffraction paths contributing to the (00.1)

reﬂection ([11.0] zone axis orientation). The paths a and b cancel each other for the exact

zone axis orientation and for all beam directions that have their Laue center on the g

00.1

row. For all incident beam directions with Laue center along the vertical dashed line the

contributions of a and c cancel each other and there will be no double-diffracted intensity

in the kinematically forbidden (00.1) reﬂection.