Graef M. Introduction to conventional transmission electron microscopy

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478 Defects in crystals

with G

∗

a vector such that G · G

∗

= 1,

†

then it is easy to see that the contribution

of this “defect” to the effective excitation error is given by

G ·

= s

(x, y). (8.29)

This is similar to the technique used to obtain Fig. 7.6 in Chapter 7. This method

approximates the foil excitation error, and its accuracy increases with increasing

acceleration voltage. At lower acceleration voltage, this approach overestimates the

excitation error of the higher-order reﬂections.

Weak beam images can also be simulated using the systematic row program.

All that is required is to provide the correct orientation of the systematic row with

respect to the incident beam direction. This is most easily accomplished by entering

the value of the excitation error s

. The following cases are of importance and can

all be derived from the solution to problem 2.9, Chapter 2:

≈−n

|G|



2 cos ω + nλ|G|

1 + nλ|G|cos ω



, (8.30)

with ω the angle between the incident beam and G (see Fig. 2.13).

Symmetric systematic row orientation: ω = π/2, so that

=−

|G|

. (8.31)

Weak beam orientation:ifs

= 0, we have cos ω =−

λ|G| and

2λ|G|

2 − 3λ

|G|

. (8.32)

−G in Bragg orientation:ifs

−G

= 0, we have cos ω =

|G| and

=−

2λ|G|

2 + λ

|G|

. (8.33)

8.4 Image contrast for selected defects

In the following sections, we will describe two-beam and systematic row image

contrast for selected defects: coherent precipitates, dislocations, stacking faults,

anti-phase boundaries, inversion boundaries, permutation boundaries, and grain

boundaries. Most of the simulations will involve the two-beam theory. For some

defects we will also describe weak beam contrast, which involves the systematic row

†

In other words, G

∗

= G/G

8.4 Image contrast for selected defects 479

solution. For dislocations and planar defects the website has several ION routines

that the reader can use to obtain bright ﬁeld and dark ﬁeld images for a variety of

input parameters.

8.4.1 Coherent precipitates and voids

8.4.1.1 Crystallography

Consider a cubic second phase particle which has a lattice parameter a

that is

nearly equal to that of the matrix, a

, in which the particle is embedded. The lattice

misﬁt δ is then deﬁned by

δ =

− a

. (8.34)

When a

< a

, the particle puts the surrounding matrix in tension, and δ<0.

The displacement ﬁeld for a spherical particle with radius r

and the associated

diffraction contrast were ﬁrst derived by Ashby and Brown [AB63a, AB63b]:

R(r) = C(δ)

(

|r| < r

)

|r|

r, (8.35)

where the symbol (a < b) means that the smaller one of a and b must be used. C(δ)

is a constant that depends on the elastic properties of the isotropic matrix and the

lattice misﬁt, and is given by

C =

3K δ(1 + ν)

3K (1 + ν) + 2E

, (8.36)

where K is the bulk modulus of the particle, and E and ν are Young’s modulus

and Poisson’s ratio of the matrix, respectively. If ν =

and K ≈ E, then we have

C =

δ. Hence, inside the particle the displacement is proportional to r, whereas

the displacements outside the particle decay as 1/r

The displacement ﬁeld (8.35) is valid for small isotropic coherent inclusions in

an isotropic matrix. Small means that the particle diameter should be signiﬁcantly

smaller than one extinction distance. For larger, misﬁtting particles, Mader [Mad87]

has described the diffraction contrast based on a Bloch wave approach.

For a void or a gas bubble, it is a good approximation to assume that the cavity

is empty, which means that the scattering potential U + iU



vanishes. The only

components of the scattering matrix that are non-zero are, hence, the diagonal

components, and it is easy to see that the scattering matrix of a thin empty slice is

equal to the vacuum propagator, described in Section 5.6.2.

480 Defects in crystals

8.4.1.2 Image contrast

It is straightforward to implement equation (8.35). The

SRdef

ect.f90

program in-

cludes this displacement ﬁeld in its standard set of displacement ﬁelds. An example

simulation for an array of nine inclusions of radius 10 nm in a 200 nm thick Cu foil

is shown in Fig. 8.4. The top illustration shows the distance of the center of each in-

clusion to the top surface, as a percentage of the total foil thickness. The simulations

were carried out for the g

200

systematic row with seven beams (200 kV) and for three

different systematic row conditions: (a) symmetric orientation, (b) s

= 0 Bragg

orientation, and (c) g–3g weak beam orientation. The images on each individual

row have a common intensity scale.

102030

40 50 60

70 80 90

200

000 200 400 600200

(a)

(b)

(c)

Fig. 8.4. Seven-beam systematic row simulation (200 kV) for nine spherical inclusions

in a 200 nm Cu foil. The operating reﬂection is g

200

. (a) Systematic orientation (s

−0.0383 nm

−1

), (b) Bragg orientation (s

= 0.0nm

−1

), (c) weak beam orientation (s

0.076 75 nm

−1

8.4 Image contrast for selected defects 481

The displacement ﬁeld of equation (8.35) is strictly only valid for the case where

the inclusion sits in the middle of the foil. Inclusions closer to the foil surfaces

must have contributions from so-called mirror-inclusions, to guarantee that the

surfaces are stress-free. Such mirror-inclusions were not taken into account for the

simulations in Fig. 8.4.

The bright ﬁeld contrast on the central row displays characteristic diffraction

contrast: the inclusion at the center of the foil has a symmetrical dark-lobed image,

with a straight no-contrast line normal to the g

200

vector. The inclusion near the top

(lower left) displays an asymmetric lobe contrast, which is identical to the contrast

in the dark ﬁeld image. For the inclusion near the bottom foil surface, this bright

ﬁeld/dark ﬁeld symmetry is reversed. This contrast reversal remains even when the

mirror-inclusions are taken into account. From a measurement of the extent of the

contrast lobes, and a knowledge of the extinction distance and particle diameter,

the sign and magnitude of the misﬁt parameter δ can be derived. We refer the reader

to [AB63a] for further details.

There are several other lattice defects that give rise to nearly identical diffraction

contrast. Small coherent precipitates, small dislocation loops, voids, and vacancy

clusters can be difﬁcult to distinguish from each other, and the reader should consult

[Edi76] and [JK01] for a detailed discussion of experimental procedures.

8.4.2 Line defects

The study of dislocations forms an important part of a more general study of the

mechanical properties of a material. Identiﬁcation of line directions, Burgers vec-

tors, slip planes, dislocation interaction mechanisms, pinning sites, etc., is needed in

order to explain plasticity, work hardening, creep, and many other mechanical phe-

nomena. In this section, we will ﬁrst describe the geometry and crystallography of

dislocations and then introduce image simulation methods to compute dislocation

image contrast.

8.4.2.1 Crystallography

A dislocation is a linear lattice defect and is characterized by a Burgers vector b

and a line direction u. The line direction may be constant along the line, in which

case we refer to the dislocation as a straight dislocation, or it may vary randomly

for a curved dislocation or change along a closed loop for a dislocation loop.A

dislocation line segment has edge character when b · u = 0, screw character when

b  u, and mixed character in all other cases. For an extensive introduction to

dislocations we refer the reader to the books by Hull and Bacon [HB84], and Hirth

and Lothe [HL68].

482 Defects in crystals

In addition to the Burgers vector and line direction, the nature of the displacement

ﬁeld of a dislocation is also determined by the elastic properties of the material.

The second-order elastic moduli tensor c

ijkl

relates strain 

to stress σ

(assuming

that Hooke’s law is valid):

= c

ijkl



and the strain is related to the displacement through





∂ R

∂x

∂ R

∂x



It can be shown, and we refer the reader to [HL68], that for an elastically isotropic

medium, the displacement ﬁeld of a straight dislocation is given by

R =

2π

bθ + b

sin 2θ

4(1 − ν)

+ b × u



1 − 2ν

2(1 − ν)

ln r +

cos θ

4(1 − ν)

>

, (8.37)

where ν is Poisson’s ratio and (r,θ) are polar coordinates in a plane perpendicular

to the line direction u. The angle θ is conventionally measured from the slip plane.

is the edge component of the Burgers vector.

In the case of an elastically anisotropic medium, the situation is more complicated

and a general analytical solution does not exist. We simply state the result and refer

the reader to [HL68] and [Str58] for more details. The dislocation line direction is

taken to be the z = x

-direction; the displacement ﬁeld then only depends on the

coordinates (x

, x

) normal to the dislocation line.

2π



α=1

kα

αj

ln(x

+ p

)

, (8.38)

where "[] denotes the real part of the enclosed expression and

iα

= (c

i2k1

+ p

i2k2

kα

;

= L

−1

;



iα

αj

− A

∗

iα

∗

αi

);

= B

−1

The vectors A

kα

are the solution vectors to the following system of equations

(summation over k):

i1k1

+ (c

i1k2

+ c

i2k1

)p + c

i2k2

= 0(i = 1,... ,3).

8.4 Image contrast for selected defects 483

This gives rise to a sextic determinantal equation for the coefﬁcients p:

i1k1

+ (c

i1k2

+ c

i2k1

)p + c

i2k2

|=0,

and one can show that the six solutions occur in three complex conjugate pairs

,α = 1,...,3. For each solution p

there is also a corresponding solution vec-

tor A

kα

. The computation of the displacement components does not depend on

the absolute values of the elastic moduli; only relative values are of importance.

The expressions for the displacement ﬁeld have been implemented in the routine

makedislocation, which is part of the SRdefect.f90 program.

8.4.2.2 Image contrast

For a screw dislocation in an isotropic medium, the edge component b

vanishes

and the Burgers vector is parallel to the line direction u. Hence, the displacement

ﬁeld (8.37) reduces to

R =

bθ

2π

. (8.39)

The visibility criterion for pure screw dislocations is g · R = g · b = 0, or, alter-

natively, planes containing the line direction remain undistorted. This leads to a

simple recipe for the determination of the direction of the Burgers vector of a screw

dislocation: ﬁnd two vectors g

for which the dislocation contrast vanishes. Then

take the cross product between the two g-vectors to ﬁnd the direction of b.

†

For an edge dislocation in an isotropic medium the situation is slightly more

complex: the edge component b

= b and the vector b × u never vanishes. A pure

edge dislocation is hence only invisible if simultaneously g · b = 0 and g · (b ×

u) = 0. This only happens for one single reﬂection, i.e. only for g parallel to the

line direction u. This is again consistent with the intuitive observation that, for an

edge dislocation, planes perpendicular to the line direction remain undistorted.

For a general dislocation in an anisotropic matrix, neither of the above cases

is exactly satisﬁed and no generally valid visibility criterion can be used. One

has to solve the DHW differential equations, using the expression (8.38) for the

displacement ﬁeld. In addition, the observed contrast will depend on other factors,

such as the beam direction B, the foil normal F, the extinction distance, the overall

excitation error, the crystal thickness, and the anomalous absorption factor. We

conclude that dislocation contrast is a complex function of many variables and

detailed image simulations may be needed to unambiguously identify a dislocation.

†

The length of b is more difﬁcult to determine from images; often, crystallographic arguments can be used to list

possible values for the length.

484 Defects in crystals

For a dislocation in an elastically anisotropic matrix, all lattice planes are distorted

and hence the visibility criterion should be used with caution, even when pure screw

or edge dislocations are considered.

8.4.2.3 Experimental example

Figure 8.5(a)–(f) show seven sets of bright ﬁeld/dark ﬁeld images, along with

diffraction patterns, for the two-beam orientations indicated. The images were

BF DF

(a)

(b)

(c)

(d)

200

trace (111) & (111)

1µm

220

020

220

Fig. 8.5. BF-DF image pairs along with two-beam diffraction patterns of a series of dis-

locations and a stacking fault in Cu-15 at% Al. The images were recorded at 200 kV in a

Philips CM20.

8.4 Image contrast for selected defects 485

(e)

(f)

(g)

311

131

Fig. 8.5 (cont.).

obtained at 200 kV, using a thick foil of Cu-15 at% Al. Two-beam conditions

were obtained by tilting from the [001], [

114], and [013] zone axis orientations.

All dark ﬁeld images are centered dark ﬁeld (CDF) images. The images show a

sequence of dissociated perfect dislocations (lower center), and a stacking fault

(top, arrowed in a). The trace of the stacking fault is indicated by a solid line, and

corresponds to either the (111) or the (11

1) plane.

From the contrast rules for stacking faults, described in Section 8.4.3.3, we ﬁnd

that the left-hand side of the stacking fault corresponds to the bottom of the foil,

and hence the fault plane is the (111) plane. The row of dislocations then lies in

the (11

1) glide plane. There are three two-beam conditions for which the stacking

fault is invisible: (

220), (3

11), and (

31). This gives rise to three equations for the

components of the displacement vector:

−2R

+ 2R

= 0, ±1,... ;

− R

+ R

= 0, ±1,... ;

−R

− 3R

+ R

= 0, ±1,... ,

486 Defects in crystals

where the right-hand side should be an integer (but not necessarily the same integer

for every equation). It is straightforward to solve these equations, and we ﬁnd that

= R

= 1 and R

=−2. The resulting displacement vector is then R =

[11

2],

which corresponds to the Burgers vector of the partial dislocation that created

this fault (the leading partial). The trailing partial is then either a b

or a b

[

1] Schockley partial dislocation. Analysis of the image contrast in

Fig. 8.5(g) shows that the trailing partial is visible for the g

reﬂection; this leaves

only the Burgers vector b

, since g

· b

= 0. The original dislocation which

dissociated into the two partials and created the stacking fault is therefore a

[01

dislocation.

The analysis in the preceding paragraphs is straightforward for the particular

images shown in Fig. 8.5, since sufﬁcient information was available concerning the

orientation of the foil, and the various active reﬂections. In general, the identiﬁcation

of dislocations can be a complex problem, in particular if there are many possi-

bilities to choose from. The more crystallographic information can be determined,

the easier identiﬁcation will become. Several worked out examples of dislocation

characterization can be found in Chapters 5 and 8 in [HHC

73]. The reader is en-

couraged to study these methods, and to apply them to dislocation characterization

in the four study materials. In many cases, image simulations will be needed to

distinguish between several possible dislocation models. In the next section, we

will describe the basics of dislocation contrast simulations for the two-beam and

systematic row cases.

8.4.2.4 Image simulations

Dislocation image simulations have a long history, starting with the well-known

ONEDIS and TWODIS programs, developed at the University of Melbourne, and

described in detail in the book Computed Electron Micrographs and Defect Iden-

tiﬁcation [HHC

73]. On the website, the reader will ﬁnd the Fortran-77 source

code for the program

hh.f, which is based on the original dislocation programs. The

program was modiﬁed extensively by P. Skalicky and coworkers of the Technical

University of Vienna, Austria, and is made available as public domain source code

with their permission. The modiﬁed program has been augmented to deal with up

to four parallel dislocations and three stacking faults, and also takes into account

the effects of piezoelectric charges along the dislocation line.

The program is written in standard Fortran-77, and makes use of the so-called

generalized cross-section. Up to four parallel dislocations connected by stacking

faults can be simulated. The generalized cross-section reduces the total number of

integrations that need to be carried out to compute the complete dislocation image.

Consider the straight dislocation in Fig. 8.6: the dislocation line PQ runs from the

bottom left to the top right of the thin foil, which itself is tilted with respect to

8.4 Image contrast for selected defects 487

(a) (b)

, b

Fig. 8.6. (a) Schematic illustration of a straight dislocation in a tilted foil; neighboring

columns in the plane ABCD have identical displacements on lines parallel to the dislocation

line; (b) generalized cross-section. (Figure based on Figs 4.2, 4.4, and 4.5 in [HHC

73].)

the incident electron beam. The integration columns are parallel to the beam. If we

consider two columns in the plane ABCD then the displacements are identical along

a line parallel to the dislocation line. This means that we only need to consider one

single plane, normal to the plane ABCD: the plane EFGH. If we draw the dislocation

line normal to the plane of the drawing (central circle in Fig. 8.6b), then each vertical

line corresponds to an entire plane of the type ABCD. The column with bottom c

and top c

corresponds to a segment of the line XY in Fig. 8.6(b). The column

–c

corresponds to a different segment of the line XY, so that we only need

to evaluate the displacements along the entire line XY to cover the entire plane

ABCD. This leads to a signiﬁcant reduction of the number of integration steps to

be carried out. For a detailed description of the generalized cross-section we refer

the interested reader to Chapter 4 in [HHC

73].

Because of the geometry of the generalized cross-section, the

hh.f program can-

not deal with dislocations for which u ⊥ B or u  B. The expression (8.38) for

the displacement components was efﬁciently encoded in Fortran-77 in the routine

ANCALC of the original “Head–Humble” code. On the website, the hh.f program is

available in two different forms. The standard form takes input in

namelist format

†

and produces an ASCII output ﬁle which contains the bright ﬁeld and dark ﬁeld

intensities for the entire image; this ﬁle must then be converted to a more standard

image ﬁle format (GIF, TIFF, JPEG, PostScript, etc.) for display. The second format

of the program links the

hh executable directly into the ION framework, and all

variables are directly passed from IDL to the executable; the resulting images are

†

Namelist format is a standard Fortran input format in which all variables in a common block can be read in from

a textﬁle. Examples of the correct namelist format for the

hh.f program are available on the website.