Graef M. Introduction to conventional transmission electron microscopy

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528 Electron diffraction patterns

1.1), and (

11.1). The space group P6

/mmc hasa6

screw axis along the

c-direction, and also a c-glide plane. In the [11.0] zone axis orientation, the screw

axis is normal to the electron beam, whereas the glide plane contains the beam

direction. This is indicated to the right of Fig. 9.6. In the projected structure, the

axis effectively becomes equivalent to a 2

screw axis, hence the symbol for the

screw.

The presence of the translational symmetry elements enforces certain relations

on the structure factors of different reﬂections. It is not difﬁcult to show

†

that for

all reﬂections in the [11.0] zone we have

h.l

= F

h.l

l = 2n;

h.l

=−F

h.l

l = 2n + 1.

The structure factors for the ±(1

1.0) reﬂections are identical and negative, whereas

the (1

1.1) and (

11.1) reﬂections have structure factors of opposite sign. This is

indicated in Fig. 9.6 by means of encircled minus and plus signs attached to the

reﬂections. In exact zone axis orientation, electrons which are doubly diffracted

along the path a will experience two negative structure factors, whereas the path b

would have one positive and one negative structure factor. The resulting amplitudes

contributing to the (00.1) reﬂections are therefore equal in magnitude but have

opposite signs so that the contributions of the paths a and b will cancel each other.

The same happens for the paths c and d, and, indeed, for every other possible

path. This means that for the exact zone axis orientation the dynamical intensity of

the (00.1) reﬂection will remain zero because of the presence of the translational

symmetry elements. This remains true when the incident beam is tilted so that the

Laue center lies along the g

00.1

row.

If we tilt the incident beam so that the (00.1) reﬂection is in Bragg orientation,

then the Laue center will fall on the dashed vertical line (the perpendicular bisector

of the vector g

00.1

). Contributions from a and c will cancel each other, and again

there will be no intensity in the (00.1) reﬂection. For all other beam orientations,

the cancellations will only be partial so that the (00.1) reﬂection will have some

intensity due to one or more double-diffraction paths. Since most thin foils are not

perfect foils with uniform thickness and no bending, a typical zone axis diffraction

pattern will contain contributions from not just one incident beam orientation, but

a range of orientations. The gray circle in Fig. 9.6 indicates such a range. Only the

two line segments inside the circle would not contribute to the doubly diffracted

beam because of symmetry-induced cancellations. All other beam directions would

have incomplete cancellations, so that the (00.1) reﬂection would be visible, as

it is in Fig. 4.18(b). The complete cancellations for the two line segments are

†

This is related to the symmetry remarks on page 131.

9.2 Spot patterns 529

(a) (b) (c)

[11.0]

Fig. 9.7. (a) [11.0] zone axis pattern for titanium (200 kV). The arrowed reﬂections are

caused by double diffraction; (b) and (c) the same pattern but progressively tilted around

the g

00.1

axis. The intensity in the (00.1) reﬂection is completely absent in (c).

known as dynamical extinctions. Dynamical extinctions can be derived from group

theoretical considerations and we refer the interested reader to [PG81] for more

details.

It is easy to determine experimentally whether or not a diffraction spot in a ZADP

is caused by double diffraction. If we tilt the crystal, such that the excitation errors

for all diffracted beams, except for reﬂections of the type ng

00.1

, become large,

then the probability for the ﬁrst scattering step of the double-diffraction process

will be small. The second step will therefore be even less probable, and the double-

diffraction contribution to the forbidden reﬂection will decrease. If we tilt the crystal

sufﬁciently far from the zone axis orientation, using the plane normal g

00.1

as the

tilt axis, then the 00.1 reﬂection will vanish altogether, a clear sign that it was

caused by double diffraction. This is illustrated in Fig. 9.7 for the (00.1) reﬂection

of titanium; the [11.0] zone axis pattern is shown on the left. The intensity of the

(00.1) reﬂection decreases signiﬁcantly when the sample is tilted around the g

00.1

axis, indicating that its intensity is due to double diffraction.

As a rule of thumb we can say that whenever two allowed reciprocal lattice

vectors sum to a point corresponding to a forbidden reﬂection, double diffraction

may occur if both of those reciprocal lattice points are close to the Ewald sphere.

Reciprocal lattice points from higher-order Laue zones may also contribute to a

forbidden reﬂection. We will return to this situation when we discuss G-M lines in

Section 9.5.

It is important to note that double diffraction can only occur in crystal structures

which show systematic absences in addition to those caused by lattice center-

ing operations (in other words, only in the non-symmorphic space groups). The

face-centered cubic lattice by itself can never have double diffraction for any of

its forbidden reﬂections; when additional extinction conditions arise, usually be-

cause of the presence of glide planes or screw axes, then double diffraction may

530 Electron diffraction patterns

become a possibility.

†

For the titanium example above, the space group contains

screw axis and a c glide plane, which are directly responsible for the ad-

ditional systematic absences. One must therefore ﬁrst consider the space group

of the crystal being studied, before attributing the presence of certain reﬂections

in an experimental pattern to double diffraction. This has obvious consequences

for multi-beam computations, as it is not sufﬁcient to include only those recipro-

cal lattice vectors g for which F

= 0! Computationally, it is easy to determine

whether or not a reciprocal lattice point g should be taken into account: if F

= 0

because of the centering of the Bravais lattice, then g never contributes to the scat-

tering process since there are no two allowed vectors g

and g

that sum up to

g.IfF

= 0 due to the presence of a symmetry operation other than the Bravais

lattice centering, then g must be taken into account in the dynamical computa-

tion, despite its vanishing structure factor. The multi-beam programs discussed

in Chapter 7 exclude reciprocal lattice vectors with zero structure factor due to

lattice centering, but allow all other scattered beams to have a non-zero ampli-

tude. The

xtalinfo.f90 program introduced in Chapter 4 can be used to produce zone

axis diffraction patterns with labeled double diffraction reﬂections, as shown in

Fig. 9.1(b).

9.2.4 Overlapping crystals and Moir´e patterns

Double diffraction as described in the previous paragraphs can occur within a single

grain of a material. The general principle is that a diffracted beam becomes an

incident beam for the remainder of the crystal and can hence give rise to otherwise

forbidden reﬂections. Another type of double diffraction commonly encountered

in materials occurs when two different crystals or grains overlap along the beam

direction; in this case each diffracted beam of the ﬁrst crystal becomes an incident

beam for the second crystal. This second crystal could be a precipitate embedded

in a matrix or a thin ﬁlm on a substrate or a thin, lamellar twin, or even a thin foil

that is heavily bent and curves back on top of itself.

There are two ways to describe this type of double diffraction: (1) we write down

the general dynamical diffraction equations for both crystals, e.g. using the matrix

theory derived in previous chapters, and proceed from there or (2) we can employ

a more intuitive approach. The ﬁrst method involves quite a bit of sophisticated

mathematics (for a good introduction we refer the reader to [Gev62]) and the second

method is quite simple. The disadvantage of the more intuitive method is that we

cannot derive quantitative results for the diffracted intensities, only geometrical

characteristics can be derived.

†

See equation (E2.2) in Chapter 2.

9.2 Spot patterns 531

Let us assume that crystal A is on top of crystal B (i.e. the electron beam “sees”

crystal A ﬁrst). If A is in zone axis orientation for the zone axis [uvw]

, and B

is also in zone axis orientation for some other axis [uvw]

, then each diffracted

beam k

+ g

of crystal A will become an incident beam for B. This essentially

means that the diffraction pattern of B will be repeated around each reﬂection of A.

Since the incident direction k

+ g

is slightly off zone axis for crystal B, we will

observe a Laue circle in the diffraction pattern of crystal B. For each beam g

this

circle is different and this gives rise to a rather complicated intensity distribution

of the diffraction spots for crystal B.

The diffraction pattern of crystal A can be represented by the following

summation:

(q) =



δ(q − g) (9.5)

where g is a reciprocal lattice vector belonging to the zone [uvw]

and q is an

arbitrary reciprocal space vector. Figure 9.8(a) shows a square zone axis pattern for

crystal A. The intensities I

can be derived from the structure factor or from more

extensive dynamical simulations.

(a) (b)

(c)

(d)

Fig. 9.8. (a) and (b) Geometrical representations of two zone axis patterns; the lattice

parameter for B is 10% smaller than for crystal A. (c) Overlap of pattern (b) on two

reﬂections of pattern (a); (d) overlap on the central nine reﬂections.

532 Electron diffraction patterns

Similarly, the diffraction pattern of B can be represented by

(q) =



δ(q − h), (9.6)

where h is a reciprocal lattice vector belonging to the zone [uvw]

. In Fig. 9.8(b)

a square pattern with lattice parameter 10% smaller than in (a) is shown. The

geometry of the total intensity distribution when A and B overlap is given by the

convolution product

†

I (q) =



g,h

δ(q − g) ⊗ δ(q − h). (9.7)

Some simple mathematical manipulations convert this equation into

I (q) =



g,h

δ(q − (g + h)). (9.8)

We will hence observe a diffracted beam at every point q = g + h. The intensi-

ties in this pattern are of course incorrect, since we did not take into account the

proper boundary conditions between the two crystals, and we did not explicitly

solve the dynamical scattering equations. Nevertheless, the pattern described by

equation (9.8) is a reasonable representation of the actual pattern. A more realistic

intensity distribution can be obtained by taking into account the location of the

Laue center for each of the many incident beams from crystal A. It is easy to see

that the Laue center for the incident beams k

+ g

coincides with the center of the

pattern, regardless of the selected beam g

. The intensity of the double-diffraction

reﬂection at g

+ h decreases with increasing distance from this point to the Laue

circle, and this could be incorporated as an additional intensity scaling factor in

equation (9.8). We leave it to the interested reader to implement such an intensity

scaling.

The ION-routine

moire.pro implements equation (9.8) for the overlap between

two crystals; the user enters values for the lengths of two reciprocal lattice vectors

for each structure, and the angle between the two patterns. The routine returns the

computed diffraction pattern and a drawing of the overlap between the correspond-

ing real-space lattices. An example of the output of the

moire.pro routine is shown

in Fig. 9.9.

We note in Fig. 9.9(c) that a number of reﬂections are centered close to the trans-

mitted beam. If the difference in lattice parameters, i.e. the lattice misﬁt, between

the two crystals is small, then the double-diffraction reﬂections will be close to the

central beam. If we insert an aperture around the central beam to obtain a bright

ﬁeld image, the aperture may be too large to exclude the doubly diffracted beams

†

The convolution of the two patterns repeats the pattern from crystal B around each reﬂection of crystal A.

9.2 Spot patterns 533

(b)

(d)

(a)

(g)

(h)

)

Fig. 9.9. (a) and (e) Square crystal

A; (b) crystal B, 10% larger lattice parameter, (f ) 33

.33%

larger; (c) and (g) are computed o

verlap patterns, and (d) and (h)

show the corresponding

two-dimensional real-space lattices.

close to the origin and the image will contain fringes because of the interference

between the transmitted and doubly diffracted beam(s). In the remainder of this

section we will discuss the geometrical characteristics of these fringes; they are

known as Moir

e fringes.

Crystal A is characterized by a set of lattice planes with interplanar distance

and reciprocal lattice vector g

; the corresponding planes in crystal B have

interplanar spacing d

with reciprocal lattice vector g

. Three different situations

can now arise.

(i) The vectors g

and g

are parallel but have different lengths. In this case we can draw

the difference vector (see Fig. 9.10a) g:

= g

+ g.

The length of this difference vector is given by

|g|=|g

− g

− d

(9.9)

and hence we will observe fringes in the image, with fringe spacing D given by

D =

− d

. (9.10)

The fringes (called parallel or compression fringes) are parallel to both sets of planes

and hence they are perpendicular to the difference vector g.

(ii) The vectors g

and g

have the same length but are slightly rotated with respect to each

other by an angle β (Fig. 9.10b). The length of the vector g again determines the

534 Electron diffraction patterns

∆g

(b)

∆g

(c)

(a)

Fig. 9.10. (a) Parallel Moir´e fringes, (b) rotation Moir´e fringes and (c) general Moir´e fringes

and their relation to the difference vector g.

spacing between the fringes, which is no

w given by

D ≈

, (9.11)

with d = d

= d

and β is measured in radians. The fringes are perpendicular to the

difference vector.

(iii) The most general case involves different lengths and a rotation β (Fig. 9.10c). Simple

trigonometry shows that, for small angles β, the fringe spacing is given by

D ≈



− d

)

+ d

. (9.12)

The fringes are again perpendicular to the difference vector g.

The fringe spacing is always larger than either d

or d

and we can deﬁne a mag-

niﬁcation factor M for the average spacing

d =

+ d

D = M

d. (9.13)

For parallel fringes the magniﬁcation is given by M =

d/|d

− d

|, for rotation

fringes M = 1/β. Defects in one of the two lattices will also be magniﬁed.

9.3 Ring patterns 535

It is possible to derive a dynamical two-beam theory for the case of Moir´e fringes.

We refer the interested reader to [Gev62] for extensive details. Before the advent

of dedicated high-resolution instruments, this type of Moir´e imaging was actually

quite popular because it allowed the study of lattice defects at magniﬁcations larger

than those attainable in the microscope. One would intentionally deposit a material

with a selected lattice parameter on the material to be studied to generate Moir´e

fringes. More recently, Moir´e imaging has lost much of its usefulness because with

modern microscopes one can directly image the crystal lattice and most common

defects. It is still useful to understand the origin of Moir´e fringes, because they can

occur readily in materials with complex three-dimensional microstructures, where

the chances of one phase overlapping with another phase are signiﬁcant. Moir´e

fringes are commonly observed in plan-view samples of layered materials. The

reader can experiment with Moir´e fringe images by evaporating a thin gold ﬁlm

onto a thin foil of the study material Cu-15 at% Al; the lattice parameters of gold

and the Cu-alloy are sufﬁciently different so that double-diffraction reﬂections and

Moir´e fringes should be observable in the overlap region.

9.3 Ring patterns

We have used a ring pattern in Chapter 4 to calibrate the camera length of the

microscope. Ring patterns can arise in a number of situations.

(i) If the grains in a polycrystalline material are randomly oriented or weakly textured,

then the normal g to each diffracting plane will be oriented in all possible directions.

Since the length of a particular g vector is a constant, the endpoint of all vectors with

the same length will describe a sphere with radius |g|. The intersection of such a sphere

with the Ewald sphere is a circle, and, therefore, the diffraction pattern will consist of

concentric rings, as shown schematically for GaAs in Figs 9.11(a)–(c). If the grains

are sufﬁciently large, individual reﬂections can be seen in the rings (Fig. 9.11b). For

(a) (b) (c)

5 nm

-1

Fig. 9.11. (a) Six GaAs patterns of Fig. 4.5 superimposed with random rotations; (b) 36

randomly rotated patterns; (c) ring pattern for ﬁne-grained polycrystalline GaAs.

536 Electron diffraction patterns

nanoscale grains the diffraction pattern would look more like that shown in Fig. 9.11(c).

The xtalinfo.f90 and zap.f90 programs can be used to simulate the ring pattern for an

arbitrary crystal structure. The program assumes that the material is non-textured,

meaning that all orientations have the same probability. If texture is present, then one

or more rings may be absent, or the intensity in any particular ring may vary along

the ring. From an analysis of these intensity distributions one can, in principle, derive

information about the thin-foil texture (e.g. [TL96]).

(ii) Purely amorphous materials do not exhibit long-range translational or orientational

order and, hence, there are no Bragg reﬂections. One can, however, deﬁne a statistical

distribution of interatomic spacings (the so-called radial distribution function or RDF;

e.g. [Jan92, Section 1.3]). The Fourier transform of this function gives rise to broad

diffuse rings. It is not always easy to draw the line between a ﬁne polycrystalline

material and a truly amorphous material. Figure 9.12 shows a diffraction pattern of an

amorphous Ge ﬁlm, with small Au particles deposited on top.

A rotationally averaged

and a rotationally integrated proﬁle are shown in the upper right-hand corner; in the

111

002

220

113

Au-rings

222

8.49 nm

-1

4.25 nm

-1

Fig. 9.12. Experimental electron diffraction pattern of an amorphous Ge ﬁlm with Au

particles (200 kV); rotationally integrated (thick line) and rotationally averaged (thin line)

proﬁles along with ﬁve Au lines are shown in the upper right-hand corner.

9.4 Linear features in electron diffraction patterns 537

averaged proﬁle, each value is divided by the circumference of the circle. It is clear

that amorphous Ge has signiﬁcant scattered intensity well beyond 10 nm

−1

, a fact that

is important for the calibration measurements discussed in Chapter 10.

(iii) Structures that are inherently circular, such as carbon nanotubes for instance, can

also give rise to diffraction patterns with partial or complete circular features. A nice

example can be found in [BA96]. Several minerals of the chrysotile family exhibit

curved lattices and the corresponding diffraction patterns show circular or polygonized

diffraction features [DB95].

9.4 Linear features in electron diffraction patterns

9.4.1 Streaks

We have seen in Chapter 2, Section 2.6.3, that the shape of a reciprocal lattice

point is determined by the Fourier transform of the shape function of the crystal.

For a thin foil, which is thin along the beam direction, this gives rise to reciprocal

lattice rods, or relrods. Relrods are “attached” to the crystal, in the same way as the

excess and defect cones giving rise to Kikuchi lines (see Section 9.4.3). The main

consequence of the presence of relrods is that electrons are scattered into diffracted

beams for which the corresponding reciprocal lattice points do not lie precisely

on the Ewald sphere. We have used the excitation error or deviation parame-

ter s

to take this deviation into account in the dynamical scattering equations in

Chapters 5–7.

Because of the orientation of the relrods with respect to the crystal, it is unlikely

that the relrods themselves can be observed directly in a diffraction pattern. This

would require that the sample be tilted nearly 90

◦

to bring the relrods tangent to

the Ewald sphere. There are other situations in which relrod-like features can be

observed in electron diffraction patterns. Consider a cubic structure viewed along

the [001] axis. Let us assume that there exists a planar fault with the (100) plane as

fault plane and a displacement vector of

[010]. The exact nature of the fault is not

important for this discussion. The geometry is shown in Fig. 9.13a, along with the

expected diffraction pattern. Some of the reﬂections in this pattern are elongated

along the direction normal to the fault plane.

If the planar faults are randomly spaced, then the entire crystal can be subdi-

vided into blocks of width w

, measured along g

100

. The reciprocal lattice for each

individual block will have the same geometry as the pattern for the perfect crystal.

Each block now has two small dimensions; one along the foil normal, giving rise

to relrods along the [001] direction, and a second small dimension along the fault

plane normal, giving rise to a second relrod in the [100] direction. This second

relrod is oriented normal to the incident electron beam, and is therefore tangent to

the Ewald sphere. This means that the reciprocal lattice points corresponding to