Graef M. Introduction to conventional transmission electron microscopy

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538 Electron diffraction patterns

(a)

(b)

Fig. 9.13. (a) Perfect square structure with a single (100) fault plane indicated, along with

the [001] ZADP (inverted contrast); (b) multiple randomly spaced blocks of width w

, and

the resulting diffraction pattern.

block i have an associated relrod of length proportional to 1/w

along the [100]

direction, as shown in Fig. 9.13(a).

†

If the crystal consists of many blocks with a range of block widths w

, then the

corresponding diffraction pattern will have the same geometry as that of the perfect

crystal, with a set of relrods of varying lengths through each reciprocal lattice point.

The resulting pattern is shown in Fig. 9.13(b). Nearly continuous lines of diffuse

intensity are observed in between the Bragg reﬂections; the direction of the lines is

parallel to the fault plane normal g

100

. Such lines are commonly known as streaks.

An experimental example is shown in Fig. 9.14. Co

NiGa is a ferromagnetic

shape memory alloy with a cubic room-temperature structure [WJLC01]. When

it is cooled down the structure transforms martensitically to a ﬁnely twinned

structure, shown in Fig. 9.14(a). The corresponding diffraction pattern, with the

†

The absence of a streak along the central row of reﬂections in Fig. 9.13 is due to the particular choice of fault.

In general, the central row will also show a streak.

9.4 Linear features in electron diffraction patterns 539

400 nm

(a) (b)

Fig. 9.14. (a) Finely twinned martensitic Co

NiGa foil and (b) the corresponding zone

axis diffraction pattern. The reciprocal unit cells of the two twin variants are outlined.

The streaks are normal to the twin planes. Each reciprocal lattice point has two pairs of

modulation satellites which also give rise to streaks (thin foil courtesy of Y. Kichin).

twin boundaries oriented parallel to the incident beam, is shown in Fig. 9.14(b).

The 2D reciprocal unit cells corresponding to the two twin variants are outlined

in white and black. Continuous streaks are present normal to the twin plane. Each

reciprocal lattice point is surrounded by two pairs of satellite reﬂections caused by

a modulation of the martensitic phase;

†

the streaks are also present for the satellites,

giving rise to groups of three parallel streaks (see inset).

In general, any planar feature which divides crystal space into separate regions

of varying widths will give rise to some sort of streak in the diffraction pattern.

The direction of the streak is, of course, perpendicular to the planar feature. Planar

Guinier–Preston zones in, for example, Al–Cu alloys give rise to streaks along the

principal cubic directions. Streaks can give rise to misleading features in diffraction

patterns; if the crystal of Fig. 9.13 were oriented away from the [001] zone axis,

then each streak in the reciprocal lattice would still intersect the Ewald sphere at

some point. This can easily be recognized experimentally: if the diffraction spots

change location while tilting the sample, then it is likely that there are streaks and

therefore disordered faults present in the crystal.

We have seen that a high density of randomly placed planar faults gives rise

to streaks in reciprocal space. If those planar faults were separated by the same

distance, i.e. w

takes on only one value, then we can deﬁne a new unit cell, with

a c-axis of length w

, and the diffraction pattern will not show streaks. Instead,

additional reﬂections will appear, as we will discuss in Section 9.6.2.3.

†

See Section 9.6 for a discussion of modulated structures and their effects on diffraction patterns.

540 Electron diffraction patterns

HOLZ

FOLZ

Bright line

Dark line

Fig. 9.15. Schematic illustration of the origin of dark HOLZ lines in the central disk: the

intersection of the Ewald sphere, shown here with exaggerated curvature, with a diffraction

disk in the FOLZ layer is a curved line segment that will show up as a bright line in the

diffraction pattern. The corresponding part of the central disk will become dark.

9.4.2 HOLZ lines

In Section 3.9.5, we introduced the concept of higher-order Laue zones. HOLZ

reﬂections can typically be observed in high-index zone axis diffraction patterns

obtained with a small camera length (less than about 500 mm). If we increase the

beam convergence angle θ

to obtain a convergent beam pattern, then each recip-

rocal lattice point becomes a disk, with each point in the disk corresponding to a

different incident beam direction. The HOLZ reﬂections also become disks that

are parallel to the HOLZ layers (see Fig. 9.15). The intersection of these disks

with the Ewald sphere, which is inclined with respect to the HOLZ layer, is a

(curved) line segment across the disk. For the beam orientations corresponding

to this line segment, electrons will be dynamically scattered out of the transmit-

ted beam and into the HOLZ beam. This leaves a local deﬁcit of electrons in

the transmitted beam at the corresponding beam orientations. For realistic accel-

eration voltages, the curvature of the intersection between the Ewald sphere and

the HOLZ diffraction disk is very small, and the resulting line segments will be

nearly straight. The dark lines across the central disk are known as deﬁcient HOLZ

lines. They are caused by dynamical elastic scattering from HOLZ reﬂections un-

der convergent beam conditions. Their bright counter parts are known as excess

HOLZ lines.

Because the HOLZ lines correspond to HOLZ reﬂections that are described

by long g vectors, the position of the lines will be very sensitive to the lattice

9.4 Linear features in electron diffraction patterns 541

90−θ

Zone

axis

(a) (b)

Fig. 9.16. (a) Schematic representation of a HOLZ reﬂection in Bragg orientation;

(b) reference frame used to ﬁnd the intersections T

of the HOLZ line with the diffraction

disk perimeter.

parameters. Small strains in the lattice, due to deformations around defects, will

shift the lines from where they would be located normally. For the zone axis case,

the location of a HOLZ line with respect to the center of the transmitted disk can

be computed as follows: Fig. 9.16(a) shows a HOLZ reﬂection denoted by g. This

reﬂection is located close to the Ewald sphere. In order to bring this reﬂection onto

the Ewald sphere, the incident beam (which initially is parallel to the vertical zone

axis direction) must be tilted over an angle φ. The magnitude of this angle can be

computed from

φ + (90

◦

− θ

) + α = 90

◦

where θ

is the Bragg angle associated with the reﬂection g. Using the Bragg

equation we ﬁnd

†

φ = θ

− α

= sin

−1



λg



− sin

−1





where g

is the component of g parallel to the zone axis (i.e. the separation NH

between the planes of the reciprocal lattice normal to the beam direction). If the

beam is oriented along the zone axis direction, then the deﬁcient HOLZ line corre-

sponding to the HOLZ reﬂection g must be located at an angular distance φ from

the origin; converting this angular distance into a real distance in reciprocal space

we ﬁnd the location of the deﬁcient HOLZ line. If the radius of the central disk

is given by R (in millimeters, see Fig. 9.16b), and the convergence angle is θ

(in

†

Other expressions can be derived for the angle φ. We refer the reader to [EMPH93] for a comparison of three

different approximations.

542 Electron diffraction patterns

radians), then the distance between the HOLZ line and the center of the diffraction

disk is given by

|OQ|=R

(in mm).

If the Cartesian coordinates of the HOLZ reciprocal lattice point projection (based

on equation 3.77 on page 215) are given by (P

, P

), then it is left to the reader

to show that the coordinates of the intersections of the HOLZ line T

with the

diffraction disk perimeter are given by

x±



+ Q

tan α



1 ±

√



1 + tan

; (9.14)

y±

= Q

−

x±

− Q

tan α

, (9.15)

with

D = 1 −



1 + tan



1 −



R tan α

+ Q

tan α



, Q

) =|OQ|(cos α, sin α), and tan α = P

. The special cases of P

= 0

and P

= 0 are also left as an exercise. These equations have been implemented in

the

holz.f90 program, introduced in Chapter 3.

Figure 9.17 shows an experimental example of a zone axis pattern for the [10.4]

zone axis orientation of Ti, recorded at 200 kV with a small camera length. The ﬁrst

and second order HOLZ rings are clearly visible. The bright line segments across

the HOLZ disks are the excess HOLZ lines. The lines in between the diffraction

disks are Kikuchi lines and will be discussed in the next section. The CBED pattern

in Fig. 9.17(b) represents the central region of the pattern in (a), and shows faint

deﬁcit HOLZ lines inside the transmitted disk. The beam convergence angle is 5.1

mrad, measured from the radius of the disk and the distance between disks. The

disks are not completely circular, a sign that the shape of the second condensor

aperture was not perfectly circular.

The computations shown in Figs 9.17(c) and (d) were carried out with the

holz.f90

program. On the left, the diffraction pattern including the ﬁrst two HOLZ layers is

shown. It is clear that for every HOLZ reﬂection in the computed pattern, there is an

excess HOLZ line in the experimental pattern. The excess lines are all located along

the intersection of the Ewald sphere with the HOLZ plane, so that they line up along

the HOLZ ring. In the computed pattern, the positions of the HOLZ reﬂections do

not necessarily coincide with the HOLZ ring.

The computed HOLZ line pattern is shown in Fig. 9.17(d). Only the lines cor-

responding to the experimental lines are shown, along with the indices of the

9.4 Linear features in electron diffraction patterns 543

(a) (b)

Convergence angle (mrad) 5.1

0 5

3 3

4 1 -2

5 -1 -2

Zone axis [ 1 0 . 4]

HOLZ radii (nm

-1

)

1 20.26

28.64

5 nm

-1

( 1-2 . 0)

( 2 0 . 1)

Fig. 9.17. (a) Experimental zone axis pattern for the [10.4] zone axis of Ti (200 kV).

(b) Central region of (a), showing HOLZ lines inside the transmitted disk. The beam con-

vergence angle is 5.1 mrad. (c) Simulated HOLZ diffraction pattern showing the most

intense reﬂections. The reference frame to the left shows the two basis vectors and the

location of the FOLZ with respect to the HOLZ layer.

(d) Simulated 000 disk; only the lines

corresponding to the experimental pattern in (b) are shown. The normals to the HOLZ lines

are shown as dotted lines in (a), indicating the corresponding HOLZ reﬂection.

corresponding HOLZ reﬂection. The normals to these six lines are shown as dotted

lines in Fig. 9.17(a); the HOLZ reﬂections giving rise to each HOLZ line can then

be identiﬁed easily. The

holz.f90 program predicts many more HOLZ lines than are

visible in the experimental pattern. This is probably due to the fact that the foil is not

necessarily strain-free. The weaker lines are often only visible when the sample is

cooled down to low temperature, and when zero-loss energy ﬁltering is employed.

544 Electron diffraction patterns

a : 0.5653

b : 0.5653

c : 0.5653

α : 90.0

β : 90.0

γ : 90.0

a : 0.5653

b : 0.5653

c : 0.5681

α : 90.0

β : 90.0

γ : 90.0

a : 0.5653

b : 0.5653

c : 0.5653

α : 89.5

β : 89.5

γ : 89.5

Fig. 9.18. [112] HOLZ line patterns for the ﬁrst two HOLZ layers in GaAs at 100 kV:

(a) nominal lattice parameters, (b) tetragonal elongation, (c) rhombohedral distortion.

HOLZ line positions can be used to determine small changes of the lattice pa-

rameters around defects. Figure 9.18 shows three HOLZ line patterns for the [112]

zone axis orientation of GaAs, for the nominal lattice parameters, and for two small

distortions: a 0.5% tetragonal elongation along [001], and a 0.5

◦

rhombohedral dis-

tortion along [111]. Additionally, if the lattice parameters are accurately known

from other diffraction experiments, then the position of the HOLZ lines can be

used to determine the electron wavelength and hence the accelerating voltage.

The symmetry of the bright ﬁeld disk, including the pattern of HOLZ lines, is

determined by the diffraction group, as explained in Chapter 5. The symmetry of

the [10.4] zone axis pattern is described by the diffraction group symbol 2

as can be seen from Table 7.2 on page 438. Table 5.1 on page 340 then shows that

both the bright ﬁeld and the whole pattern symmetry are equal to m. The single

mirror plane is clearly present in both Figs 9.17(a) and (b). We will say more about

symmetry and convergent beam electron diffraction patterns in Section 9.5.

9.4.3 Kikuchi lines

Figure 9.19 shows a sequence of electron diffraction patterns for the [10.2] zone

axis orientation of titanium. The thickness of the foil increases from left to right.

The leftmost pattern was taken close to the edge of the thin foil, and the rightmost

pattern corresponds to a rather thick region of the foil. The foil was tilted away

from the exact zone axis orientation so that the Laue center lies at the position

(

30.3) (indicated by a small cross). As the foil thickness increases, so does the

probability of inelastic scattering. This results in an increased intensity in between

9.4 Linear features in electron diffraction patterns 545

(a) (b) (c)

[10.2]

30.3

12.0

Fig. 9.19. Sequence of titanium [10.2] zone axis patterns for increasing foil thickness from

(a) to (c) (200 kV).

the Bragg reﬂections. In addition, straight lines appear, both brighter and darker

than the background intensity. These lines are known as Kikuchi lines and in this

section we will explore their origin and geometry.

9.4.3.1 Origin and geometry of Kikuchi lines

From Fig. 2.2(b) in Chapter 2 we know that, for a given set of planes, the Bragg

condition is satisﬁed for all electrons traveling along a conical surface with open-

ing angle π/2 − θ. In particular, all inelastically scattered electrons traveling in

directions on this conical surface may be scattered elastically

†

by an angle 2θ . Let

us now consider a particular set of planes g, and all inelastically scattered electrons

which travel towards a point P on this plane, as shown schematically in Fig. 9.20(a).

Those inelastically scattered electrons which travel on the conical surface along the

line OPQ are in the proper orientation to be scattered elastically by the planes g.

They will, therefore, be scattered onto the same conical surface, moving away along

the direction PR. We will therefore have an excess of electrons in the direction PR,

and a deﬁcit of electrons along the direction PQ. Since the direction OP could lie

anywhere on the conical surface, the top cone will have an excess of electrons and

is therefore known as the excess cone; the bottom cone is then known as the deﬁcit

cone. The two cones are also known as Kossel cones.

The intersection of the Ewald sphere with the excess and defect cones gives rise

to a pair of (curved) lines, one brighter than the background and one darker than the

background intensity. The curvature of the lines is rather small and may not always

be observable with commonly used camera lengths. The curvature decreases with

increasing microscope accelerating voltage. The lines are known as Kikuchi lines,

after S. Kikuchi who discovered them in 1928 [Kik28] (before the invention of

†

Inelastically scattered electrons have an energy which is different from the primary beam energy. This corre-

sponds to a different wavelength and therefore a different Bragg angle. It is a good ﬁrst-order approximation to

ignore the difference between these “elastic” and “inelastic” Bragg angles.

546 Electron diffraction patterns

(a)

(b)

Fig. 9.20. (a) Geometry of excess and defect cones; (b) single frame from a QuickTime

movie available from the

website, illustrating the geometry of the defect and excess cones.

the TEM!). The distance between the two lines corresponds to the Bragg angle for

elastic scattering from the planes g, as shown in the rendered drawing of Fig. 9.20(b).

This ﬁgure is a single frame from a 200-frame QuickTime movie available from the

website. The movie illustrates the geometry of the Kossel cones in relation to the

reciprocal lattice points and the Ewald sphere.

It is important to realize that the excess and defect cones are “attached” to the

diffracting planes g; when we change the orientation of the crystal, both cones move

along with the sample, and hence the positions of the bright and dark lines change

with respect to the reciprocal lattice point g. Kikuchi lines, therefore, provide us

with a very sensitive way to determine the orientation of the crystal with respect

to the incident beam. For each pair of reciprocal lattice points g and −g, there are

two Kikuchi lines, corresponding to both sides of the plane. Which of those lines is

bright depends on the relative excitation errors of the two reﬂections. Figure 9.21

shows the relation between crystal orientation and position of the Kikuchi lines

for the reﬂection g. The defect (D) and excess (E) cones are shown as thick, short

lines attached to the lattice plane g. In exact Bragg orientation, we have s

= 0,

and the bright Kikuchi line intersects the reciprocal lattice point (Fig. 9.21a). When

the crystal is tilted by a small angle +α, such that the excitation error s

becomes

negative, then the bright Kikuchi line moves towards the origin of reciprocal space

(Fig. 9.21c). A specimen tilt in the opposite direction −α results in a positive

excitation error s

and a bright Kikuchi line on the opposite side of the reciprocal

lattice point (Fig. 9.21e).

9.4 Linear features in electron diffraction patterns 547

< 0

= 0 s

> 0

(b)

(c)

D E

k+g

D E

+α

k+s+g

D E

−α

k+s+g

(a)

(d)

(e)

x (<0)

)

Fig. 9.21. Location of Kikuchi lines with respect to reciprocal lattice points for (a) s

= 0,

< 0, and (e) s

> 0. (b), (d) and (f ) are experimental images for the (10.3) reﬂection

of Ti, recorded at 200 kV.

Figure 9.21(b), (d) and (f) show experimental Kikuchi patterns for the (10.3)

reﬂection of Ti, recorded at 200 kV. From the position of the Kikuchi line with

respect to the reciprocal lattice point and the origin, we can derive a numerical

value for the excitation error. It is easy to show (exercise) that the relation between

the excitation error and the position of the bright Kikuchi line is given by

λ|g|

, (9.16)

where x and R are indicated in Fig. 9.21(d). Since |g

10.3

|=7.5097 nm

−1

and x/R =−1.568/21.244 =−0.073 81 we have for the excitation error s =

−0.01 nm

−1

. If we change the position of the bright line by, say, one-tenth of

the length of the g-vector, then this corresponds to a specimen tilt of 2θ/10, which

is a small angle.