Graef M. Introduction to conventional transmission electron microscopy

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278 Getting started

8.2

6.35

10.5

13.5

18.5

660

950

1130

primary

tilt axis

secondary

tilt axis

diffraction mode

image mode

8.2K 23K 60K

93.5

76.5

119.3

-22.1

Fig. 4.17. Illustration of the rotation calibration for a Philips EM420; images

were recorded

on a Gatan DualView camera. The top row shows three images of a calibration line grid

with 2160 lines per mm (463 nm line spacing). The bottom drawing shows the rotation

angles of image and diffraction mode with respect to the primary and secondary tilt axes.

of course!). When you observe a certain image feature, say, a broad dark band

running across the bright ﬁeld image, try to determine how this band behaves when

you change the orientation of the sample. Or, in diffraction mode, observe how the

diffraction pattern changes when you move the selected area aperture across one of

those bands. It is only by experimenting with various imaging conditions that you

will begin to see how the image contrast depends on the microscope parameters and

settings which you as an operator can select. Once you have familiarized yourself

4.6 Basic CTEM observations 279

2 µm

000

002

110

[11.0]

= double diffraction spots

(a)

(b)

Fig. 4.18. Example of a bend center in a Ti thin foil. The image in (a) was obtained on a JEOL

2000EX, operated at 200 kV. The diffraction pattern in (b) was obtained using a selected

area aperture centered on the intersection of bend contours (circle in a). The resulting zone

axis pattern can be indexed as the [01.0] zone axis pattern of Ti. The reﬂections circled with

a black circle have a zero structure factor, but appear because of double diffraction.

with the basic image features common to all four study materials, it will be time

to proceed to the next chapter and explain these observations, starting from the

relativistically corrected Schr¨odinger equation.

4.6.1 Bend contours

In this section, we will assume that you have a polycrystalline metallic thin foil,

either Cu-15 at% Al or Ti. Since most metallic thin foils are locally bent, it should

be easy to ﬁnd a region (in the bright ﬁeld image mode) that looks somewhat similar

to the region shown in Fig. 4.18(a). We observe a large number of dark contours that

intersect each other in a small region. When you tilt the sample, you will ﬁnd that all

of these contours move across the grain. They are known as bend contours, and their

intersection point is a bend center. Place the selected area aperture over the bend

center and switch to diffraction mode. You will observe a symmetric diffraction

pattern, not unlike the one shown in Fig. 4.18(b).

†

This pattern corresponds to a

plane in reciprocal space, tangent to the Ewald sphere, which means that the incident

beam direction is along a crystallographic direction or a zone axis direction. The

pattern is known as a zone axis diffraction pattern. Using the methods discussed in

†

The pattern you obtain will not necessarily be the same as the one shown in the ﬁgure, but it should be a

symmetric pattern with a signiﬁcant number of reﬂections.

280 Getting started

Section 9.2.1 we can assign Miller indices to each of the reﬂections, and using the

cross product of any two independent reﬂections we ﬁnd the indices of the zone

axis. For the case shown in Fig. 4.18(b) we have used a Ti thin foil, and the zone

axis is [11.0] in 4-index notation.

Several reﬂections have been circled in Fig. 4.18(b); the structure factor van-

ishes for all of these reﬂections, but we clearly have signiﬁcant intensity in each

of them. This is our ﬁrst evidence that electron diffraction is not simply the same

as x-ray diffraction with a shorter wavelength. For conventional x-ray diffraction

experiments, a vanishing structure factor would mean zero intensity in the corre-

sponding diffracted beam. The circled reﬂections in Fig. 4.18(b) will not give rise

to diffraction peaks in an x-ray powder pattern, because the presence of the 6

screw

axis in the non-symmorphic space group of Ti forbids these reﬂections from occur-

ring. Yet, the same reﬂections are clearly present in the electron diffraction pattern.

This means that the structure factor is not sufﬁcient to determine the intensity of

diffracted beams in an electron diffraction experiment. Intensity calculations based

on the structure factor are commonly known as kinematical calculations. Calcula-

tions which account for the presence of the “forbidden” reﬂections are known as

dynamical calculations, and we will spend a signiﬁcant portion of this book dis-

cussing the dynamical theory (Chapters 5–7). The forbidden reﬂections are present

through a mechanism known as double diffraction, and we will return to this concept

in Section 9.2.3.

After you have convinced yourself that the intersection of bend contours cor-

responds to a point where the incident beam travels along a zone axis orientation

of the crystal, you should return to image mode (bright ﬁeld) and move the SA

aperture onto one of the bend contours, away from the bend center. Then switch

back to diffraction mode; you will ﬁnd that one single row of reﬂections is promi-

nent, while reﬂections that do not belong to this row are signiﬁcantly weaker than

they were for the zone axis orientation. Since a row of reﬂections corresponds to

a single set of planes ng

hkl

we also ﬁnd that a single bend contour is associated

with a particular set of planes. We could therefore label the bend contour with

the Miller indices hkl of the corresponding reciprocal lattice point. All along the

bend contour the planes (hkl) are strongly diffracting, which means that the Bragg

condition is satisﬁed along the bend contour. We will determine in Section 7.2.1

exactly where along the contour the Bragg condition is satisﬁed. If a single row of

strongly excited reﬂections is present in the diffraction pattern, then we say that

the crystal is in a systematic row orientation. Let us consider this case in somewhat

more detail.

Consider the zone axis pattern shown in Fig. 4.19(a). This pattern can be indexed

as the [01.0] zone axis pattern of Ti. If we tilt the crystal by about 8

◦

around

the row ng

002

then we obtain the systematic row pattern shown in Fig. 4.19(b).

4.6 Basic CTEM observations 281

500 nm

(b)

(d) (sum of e and f )

(f)

(a)

(e)

(c)

[01.0]

000

002

210

002

Fig. 4.19. (a) [01.0] zone axis pattern of Ti; (b) same but tilted about 8

◦

around the g

002

systematic row axis; (c) bright ﬁeld image corresponding to (b); (e) and (f) are centered

dark ﬁeld images for the reﬂections labeled in (b); (d) is the sum of images (e) and (f),

and

is complementary to (c).

The corresponding bright ﬁeld image is shown in Fig. 4.19(c); we observe a bend

contour that is nearly normal to the edge of the foil.

†

Where the contour intersects

†

The local shape of the foil determines the geometry of the bend contour, and it is not possible to tell from the

systematic row diffraction pattern what the orientation of the bend contour will be with respect to other features

in the foil. To ﬁnd a conﬁguration similar to that shown in Fig. 4.19(c), you should search in bright ﬁeld mode

until you ﬁnd a bend contour nearly normal to the sample edge; you may need to tilt the sample back and forth

until you ﬁnd such a bend contour. Then switch to diffraction mode and you will observe a systematic row

pattern.

282 Getting started

the foil edge, intensity oscillations are present, and the ﬁrst fringe is nearly parallel

to the edge. We will have to wait until Chapters 6 and 7 to explain the details of

this image contrast.

Images 4.19(e) and (f) were obtained by means of the centered dark ﬁeld method

explained in Section 4.4.2.5. The corresponding reﬂections are labeled in the sys-

tematic row pattern 4.19(b). It is important to note that the sample orientation was

not changed at all with respect to the bright ﬁeld orientation. Only beam tilts were

used to obtain the two dark ﬁeld images. We observe a large number of fringes that

are nearly parallel to the sample edge close to the edge, but move away from the

edge with increasing sample thickness. At the arrowed locations the fringes are at

a maximum distance from the edge. Note that the locations of the two arrows do

not coincide at the same location in the foil.

If we add the two dark ﬁeld images together we obtain the computed image

shown in Fig. 4.19(d). Note that there is some similarity between this sum and the

bright ﬁeld image; the bright ﬁeld image appears to be the inverse of the sum image.

We say that these two images are complementary images. The details of this fringe

contrast will have to wait until we cover multi-beam dynamical diffraction theory

in Chapter 7.

4.6.2 Tilting towards a zone axis pattern

If you insert a sample in the microscope, turn on the beam, select a random point

on the sample and switch to diffraction mode, you will almost never end up with

a perfectly symmetric zone axis diffraction pattern. Instead, you will most likely

obtain a pattern similar to that shown in Fig. 4.20(a). The reﬂections visible in this

pattern lie along a segment of the Laue circle, corresponding to the intersection of

the Ewald sphere with a plane in reciprocal space. The corresponding zone axis

orientation can be obtained by “closing the circle”. Figs 4.20(b)–(d) show how you

can accomplish this.

It is easy to say “close the circle”, but you will actually have to learn how to do

this. In rare situations, the foil will be oriented such that a simple rotation around

either the primary or secondary tilt axis will do the trick. In most cases you will

have to use both tilt axes simultaneously to close the circle. This requires ﬁrst of

all an understanding of the orientation of the tilt axes with respect to the diffraction

pattern; this you can derive from the calibration measurements described earlier.

Then you will need to keep the sample in the same location during tilting, so that

you always look at the diffraction pattern corresponding to the location you selected

in the image. This can also be tricky, especially when the sample is far from the

eucentric position. If your sample has a small grain size, then it will be even more

difﬁcult to keep the selected grain under the beam during tilting. There are also

4.6 Basic CTEM observations 283

(a)

(b)

[11.0]

Fig. 4.20. Schematic illustration of tilting a foil by closing the “Laue circle”, the intersection

of the Ewald sphere with the reciprocal plane normal to the zone axis direction. The [11.0]

pattern was obtained from Ti, using a JEOL 2000EX microscope.

signiﬁcant differences between the tilting of a double-tilt holder and a rotation-tilt

holder.

This is where bend contours can come in handy. Assuming that you can see

a bend contour near the area you selected in the image, you can tilt the sample

while looking at the image rather than the diffraction pattern. All you need to do

is keep the bend contour at the same location in the sample during tilting. When

you approach the zone axis orientation, a bend center should appear and when

you position this bend center at the selected location, the corresponding zone axis

diffraction pattern will be symmetric, as shown in Fig. 4.18. If the sample is not

signiﬁcantly bent, or the grain size is too small for bend contours to be visible, then

you should tilt until the selected grain (in image mode) becomes very dark with

respect to its neighbors, indicating that a large number of electrons are diffracted

out of the transmitted beam.

284 Getting started

Tilting from one zone axis orientation to another one is similar to the method

described above. In image mode you would keep a single bend contour (corre-

sponding to the plane common to both zone axis orientations) in view while you tilt

the sample. In diffraction mode you need to keep the reﬂection corresponding to

this common plane as bright as possible while tilting. This requires a coordinated

movement of one or both feet (to operate the tilt controls) and both hands (to keep

the selected area under the beam). Since tilting often results in a change of the

specimen height, the image will go out of focus during tilting and you will need to

adjust the image focus periodically. Do not forget to set your sample back to the

eucentric specimen height after you have tilted to the correct orientation! All of

this requires a lot of practice and you should take your time to learn how to tilt a

sample in a controlled way. On modern computer-controlled microscopes some of

the tediousness of maintaining eucentric position can be taken over by the micro-

scope computer. However, the chances are that, as a microscopy student, you will

not have access to such modern machines, so that it will be necessary to practice,

and practice, and practice.

4.6.3 Sample orientation determination

The zone axis diffraction pattern contains signiﬁcant geometrical information about

your crystal. If you obtain two or more zone axis diffraction patterns of a partic-

ular region in your foil, then you can transfer the geometrical information onto a

stereographic projection which will allow you to index all diffraction patterns in a

consistent way. Consider the four bright ﬁeld images shown in Fig. 4.21, obtained

<111> <211>

<211> <011>

1 µm

(a) (b)

Fig. 4.21. Four different zone axis diffraction patterns and corresponding bright ﬁeld images

of a small grain in a Cu thin foil. The region labeled 1 in (a) was used to obtain all four

diffraction patterns. The images were obtained in a Philips EM420, operated at 120 kV,

using a Gatan DualView digital camera. The image orientations are as-acquired and image

rotations have not been compensated for.

4.6 Basic CTEM observations 285

from a ﬁne-grained Cu foil on a Philips EM420 microscope operated at 120 kV. The

central grain in (a) is dark and in zone axis orientation, as shown in the correspond-

ing diffraction pattern. By comparison with precomputed zone axis patterns from

the

xtalinfo.f90 or zap.f90 programs or, alternatively, using the indexing program

indexZAP.f90 described in Chapter 9, it is easy to show that this pattern belongs to

a zone axis from the 111 family. The sample orientation angles were read from

the goniometer and are equal to α = 8.2

◦

and β = 6.1

◦

. Bright ﬁeld images were

acquired at a nominal magniﬁcation of 30 000×, and the diffraction patterns were

recorded at a nominal camera length of 660 mm. The relative orientation of image

and diffraction pattern can be derived from Fig. 4.17.

When the sample was tilted around the dashed line in Fig. 4.21(a) we obtained a

pattern (b) of the 211 family at an orientation α = 28

◦

and β = 12.5

◦

. A second

rotation around the axis labeled in Fig. 4.21(b) resulted in another 211 pattern

at α =−4.7

◦

and β = 20.4

◦

. And a ﬁnal rotation around the labeled axis resulted

in the 011 pattern at α =−27.0

◦

and β =−6.0

◦

, as shown in Fig. 4.21(d). The

question we will answer next is: how can we index all patterns and reﬂections in a

consistent way? In other words, how do we select the actual family members from

the four direction families?

Consider the stereographic projection in Fig. 4.22; the original drawing for this

pattern was made on a transparent foil, mounted on top of a Wulff net as shown

in Fig. 1.11 on page 32. The primary tilt axis is placed horizontally from A to B,

the arrow indicating the direction in which the sample holder is inserted into the

column. From the calibration measurements in Section 4.5.2 we ﬁnd that the angle

between the diffraction pattern obtained at a nominal camera length of 660 mm and

the primary tilt axis is 57

◦

; this is indicated to the left of the center in Fig. 4.22. If we

take the zone axis pattern of Fig. 4.21(d), rotate it so that its bottom line is parallel

to the line on the stereographic projection and the central beam is positioned at the

center of the projection, then we can draw lines through each row of reﬂections

(i.e. from −g to +g for all reﬂections close to the origin). If we extend those

lines across the stereographic projection we obtain (in this case) six intersections

with the projection circle; those points are indicated by small ﬁlled circles. The

zone axis corresponds to the point at the center of the stereographic projection.

The stereographic projections of the reciprocal lattice points in the 011 pattern in

Fig. 4.21(d) are therefore given by the six points on the projection circle in Fig. 4.22,

in the proper relative orientation with respect to the primary tilt axis.

Since the sample was tilted from the α = 0

◦

, β = 0

◦

reference orientation to

obtain the 011 zone axis pattern we must now correct for this tilt. First, we correct

for the α tilt angle by aligning the M



–M



axis of the Wulff net (see Fig. 1.10) with

the primary tilt axis. Then we rotate all six projection points along the arcs centered

on the points M



and M



by an angle opposite to the tilt angle. In this case, the α tilt

286 Getting started

[211]

[111]

[121]

[011]

primary tilt axis

bottom of diffraction pattern

tilt range

(200)

(111)

[011] zone

Fig. 4.22. Construction of the stereographic projection of the 011 zone axis pattern of

Fig. 4.21(d).

angle is −27

◦

,sowetiltby+27

◦

and obtain six new points. We also tilt the central

point of the projection by +27

◦

. Points in the lower hemisphere are indicated by

open circles. The second tilt can be corrected by ﬁrst rotating the Wulff net by

◦

, and then performing the same operation on all points. This time the rotation

is −β = 6.0

◦

The six points obtained in this way lie on a great circle, and we can draw this circle

on the projection, using the Wulff net. The pole of this great circle is the projection

of the zone axis, labeled [011]. If we repeat this procedure for the other three zone

axis patterns of Fig. 4.21 we obtain the four zone axis projections shown near the

center of Fig. 4.22. Next we select the zone axis direction of the 111 family to

be the [111] direction. Comparison of the locations of the other poles with a standard

stereographic projection (for instance, you could use the

stereo.f90 program to draw

4.6 Basic CTEM observations 287

[111] [211]

[121] [011]

022

220

202

111

131

113

111

202

111

200

Fig. 4.23. Consistently indexed diffraction patterns of Fig. 4.21(a)–(d).

a [111] projection) shows that the other poles are consistently indexed by selecting

[111], [211], [121], and [011] for the patterns of Figs. 4.21(a)–(d), respectively.

The last step involves determination of the Miller indices of all reﬂections of

the four patterns. Since we know which reﬂections were used to tilt from one

zone axis orientation to the next one, we can simply use the cross product of

directions to determine the Miller indices of the reﬂection on the rotation axis.

As an example, the plane common to both the [111] and [211] zones is the (0

22)

plane. Figure 4.23 shows the four diffraction patterns again, this time with the

proper consistent Miller and direction indices indicated. Once the 0

22 reﬂection is

labeled, the other reﬂections in this pattern can be identiﬁed readily by comparison

with a precomputed pattern. The same cross product procedure identiﬁes that the

13 reﬂection lies on the rotation axis between the [211] and [121] zones, and the

11 reﬂection is common to [121] and [011]. This allows us to unambiguously and

consistently index every reﬂection of the four patterns.

This completes the indexing procedure. The grain orientation is completely and

unambiguously determined with respect to the primary tilt axis, and hence with

respect to the microscope column. With care you should be able to determine grain

orientations to within a degree or so. The procedure outlined in this section is

important for defect identiﬁcation. As we will see in Chapter 8, for many defect