Graef M. Introduction to conventional transmission electron microscopy

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568 Electron diffraction patterns



· b/4 = 0. In reality, there will be satellites on this row for two reasons: ﬁrst,

the shear lattice potential is inadequately described near the shear planes. A more

accurate description does introduce satellites along the central row [VDBMA87].

Secondly, there will be double-diffraction effects, as discussed in Section 9.2.3.

Crystallographic shear planes can also occur in more than one direction, as shown

in the example of Fig. 9.32(b). The sheared structure has a ﬁrst set of shear planes

with the same parameters as in Fig. 9.32(a). A second set with parameters R = a/3,

d = 4a, and e

= bb

∗

is superimposed on the faulted structure. As a consequence

of the second series of faults, both fault planes are rotated away from the original

orientation, and the rows of satellite reﬂections remain normal to the rotated fault

planes, as is obvious from the computed diffraction pattern. For the reﬂection (11),

the satellite reﬂections occur at locations of the type

k,l

= (11) +

3k − 1

(01) +

4l − 1

(10),

which generates a two-dimensional pattern of satellites around the original reﬂec-

tion (11).

The kinematical diffraction patterns of Figs 9.28, 9.30, and 9.32 can be computed

using the interactive ION routine

modulation.pro, available on the website. The

routine creates a structure on a 1024 × 1024 grid, multiplies it with a Hanning

window to eliminate edge effects in the subsequent Fourier transform, and then

displays the central 512 × 512 pixels of the kinematical diffraction pattern using

a logarithmic intensity scale. A Fortran-90 version of the same program is also

available as

modulation.f90.

9.6.2.4 Interface modulations: long period anti-phase boundary structures

A material with a completely ordered lattice, say A

B, can undergo an additional

ordering transformation by the introduction of a periodic array of defects in the or-

dering. The simplest ordering defect is the anti-phase boundary (APB), introduced

in the previous chapter. Let us consider an example: the L1

-structure (Fig. 8.10b).

Since the B atom can be located on any of the four sublattices of the fcc structure,

there will be four translation variants in the crystal (assuming heterogeneous nu-

cleation of the ordered state). These domains are separated by APBs. Across an

APB, the underlying fcc lattice is continuous, but the ordered superlattice is not.

Often, the APB energy is strongly anisotropic, meaning that the APB will prefer

a certain plane above all others. The simplest APB in this type of structure is then

the conservative APB on the (001) plane, with displacement vector R =

[110].

This type of defect has been observed in many alloys, e.g. Cu

Pd [BVTVL

88],

Pt [MMC

96], CuAu [ST61], Cu

Al [BDGVT

87], etc. The spacing between

the APBs determines the size of the tetragonal unit cell; the smallest spacing would

9.6 Diffraction effects in modulated crystals 569

M == 1 (= D0

)

M ==

(a)

(b)

(c)

200

020

220

Fig. 9.33. (a) M = 1orDO

structure, (b) M =

structure, (c) experimental [010] zone

axis pattern for an M =

modulated structure in a Cu-23 at% Al alloy.

correspond to the distance a. Fig. 9.33(a) illustrates the resulting structure, which

is known as DO

. This type of structure is generally known as a long-period anti-

phase boundary lattice or LPAPB and can be characterized by the ratio of two

integers: M = p/q, where p is the number of unit cells in a repeat distance and

q the number of APBs within that same repeat distance. The L1

structure has

M =∞and for the DO

structure M =

= 1. Fractional values for M are also

possible.

These periodic defect structures give rise to superlattice reﬂections along the

[001]-direction; it can be shown [VDBMA87] that the satellite reﬂections will arise

at positions of the type (2h + 1, 0, l ± m/2M), where m is an integer (this involves

computation of the structure factor for an arbitrary value of M). An experimental

diffraction pattern ([010] zone axis pattern) for M =

is shown in Fig. 9.33(c),

along with the corresponding structure in Fig. 9.33(b). The value of 1/M can be

determined from the spacing between the superlattice reﬂections, as indicated.

9.6.3 Incommensurate modulations and quasicrystals

All modulation types introduced so far have in common that all reﬂections (fun-

damental and modulation satellites) can be described by the following expres-

sion [Jan86]:

g =



i=1

∗



j=1

3+j

∗

3+j

, (9.23)

570 Electron diffraction patterns

where g

and g

3+j

are integers, and the vectors a

∗

3+j

describe the modulation periods

and directions; i.e. they are the modulation wave vectors. For a 1D modulation, we

have d = 1, while for a 2D modulation d = 2, and so on. Since the crystal is a

3D object, the additional basis vectors must be linear combinations of the original

three:

∗

3+j



i=1

∗

. (9.24)

For the modulation types discussed so far, the coefﬁcients κ

are rational numbers,

so that the modulation period is commensurate with the underlying lattice.

†

If one or more of the coefﬁcients κ

is irrational, then the modulation period

becomes incommensurate with the underlying lattice. A general reﬂection position

for a 1D modulated structure is then given by

g =



i=1

+ pκ

∗

, (9.25)

with p an integer labeling the satellite reﬂections. Strictly speaking, such an in-

commensurate structure no longer satisﬁes the Bravais lattice translation vectors,

and therefore cannot be described by any of the 230 3D space groups. It is possible

to consider such a structure in a (3 + d)-dimensional space, and there is a signiﬁ-

cant amount of literature on the concept of superspace groups [De 74,JJ77, JJ79].

Experimental examples of incommensurate phases studied by electron diffrac-

tion include NbTe

and TaTe

[vBM86,Bvd

87], K

0.3−x

MoO

[Col89], and TiSe

[Fun85].

For many incommensurately modulated structures, it is straightforward to select

the ﬁrst three basis vectors, corresponding to the unmodulated structure. This struc-

ture can always be described by one of the 14 Bravais lattices, and, therefore, one

of the 3D space groups. If the vectors a

∗

, with i = 1,... ,3, do not describe one

of the Bravais lattices, then the resulting structures are known as quasi-crystals.

The ﬁrst experimental evidence for the existence of quasi-crystals was found in

1984 in an Al–Mn alloy by Shechtmann, Blech, Gratias, and Cahn [SBGC84]. This

discovery was made by means of electron diffraction observations, and a careful

analysis of the angles between two-, three-, and ﬁve-fold zone axes. Soon after

this discovery, many more materials were found to exhibit either the icosahedral

orientational symmetry, or octagonal, decagonal, and more exotic symmetries. The

ﬁeld of “quasi-crystallography” was born.

†

Note that in the hexagonal system we can choose κ

= (−1, −1, 0) to describe the fourth reciprocal basis vector

(which we called A

∗

in equation 1.34). The hexagonal lattice can, hence, be regarded as a 1D commensurately

modulated lattice.

9.7 Diffuse intensity due to short range ordering 571

For an excellent introduction to quasi-crystals, we refer the interested reader to

the book by Janot [Jan92]. A detailed description of the indexing of diffraction pat-

terns from quasi-crystals can be found in [CSG86]. The icosahedral quasi-crystals

can be described in terms of a 6D space, in which a primitive, body-centered, or

face-centered cubic lattice is decorated with atoms. When this 6D structure is pro-

jected onto 3D space along an irrational projection axis, a 3D structure is created

which has no long-range translational order, but only long-range orientational order.

The reader should consult the journals Philosphical Magazine and Philosophical

Magazine Letters for papers on many different quasi-crystalline systems.

9.7 Diffuse intensity due to short range ordering

We know from structure factor computations that an ordered structure gives rise

to superlattice reﬂections, i.e. reﬂections on positions that are not allowed in the

disordered structure. Often, the structure factor associated with these reﬂections is

proportional to the difference of two (for a binary alloy) atomic scattering factors.

In this section, we will consider what happens when the ordering state of the crystal

is not perfect, i.e. if the material is only partially ordered.

Let us consider a binary alloy with a primitive disordered lattice. The atoms

of types A and B have atomic fractions c

and c

, respectively. We deﬁne a site

occupation operator, σ

, such that

1 if position j occupied by an A atom;

0 if position j occupied by a B atom;

(9.26)

can be deﬁned in the same way, such that

+ σ

= c

+ c

= 1 (9.27)

for each lattice site j. Using these operators, the atomic scattering factor of a site

j may be rewritten as

= f

+ f

. (9.28)

As we have seen in Chapter 2, in the kinematical diffraction theory the amplitude

of a diffracted beam is proportional to the structure factor for that beam, i.e.

A(q) =



−2πiq·r





+ f



−2πiq·r

, (9.29)

with q a vector in reciprocal space. It is customary to introduce a new operator, the

scalar Flynn operator:

¯σ

= c

− σ

= σ

− c

> c

). (9.30)

572 Electron diffraction patterns

The Flynn operator is simply a different way of writing the site occupation parame-

ters. Using this operator, the scattered amplitude may be split into two contributions:

A(q) = A

(q) + A

(q), (9.31)

with

(q) =

(

+ c

)



−2πiq·r

; (9.32)

(q) =

(

− f

)



¯σ

−2πiq·r

. (9.33)

The ﬁrst amplitude is only different from zero at the Bragg positions q = g;it

represents the average disordered structure. The second term is zero at the Bragg

positions (because the average value of the Flynn operator over the whole crystal

is zero) but different from zero in between the Bragg reﬂections. We thus ﬁnd that

(q)A

(q) = 0 everywhere in reciprocal space. (9.34)

We conclude from the above equations that any deviation from perfect disorder will

give rise to diffuse intensity in between the Bragg reﬂections. This diffuse intensity

is written as

(q) =|f

− f



k,l

¯σ

−2πiq·(r

−r

)

. (9.35)

For a primitive unit cell, the difference between the vectors r

and r

is again a

lattice vector, say, r

, and the intensity for a crystal with N unit cells is then written

(q) = Nc

| f

− f



0, j

−2πiq·r

. (9.36)

The coefﬁcients α

0, j

are the Warren–Cowley short-range order parameters

[Cow67], deﬁned by

0, j

¯σ

¯σ

l+j



. (9.37)

These parameters represent the probability that, if position l is occupied by an A

atom, position l + j is also occupied by an A atom. The ···sign indicates a

spatial average over the whole crystal.

9.7 Diffuse intensity due to short range ordering 573

Table 9.2. Atom coordinates (×2) and corresponding SRO

parameters α

0, j

derived from diffuse neutron scattering

measurements (see the second footnote on this page).

Position α

0, j

Position α

0, j

Position α

0, j

(0, 1, 1) −0.172 (0, 2, 4) 0.003 (0, 0, 6) 0.003

(0, 0, 2) 0.101 (2, 3, 3) 0.007 (1, 1, 6) 0.002

(1, 1, 2) 0.036 (2, 2, 4) −0.005 (2, 3, 5) 0.003

(0, 2, 2) 0.010 (1, 3, 4) 0.005 (0, 2, 6) 0.000

(0, 1, 3) −0.027 (0, 1, 5) −0.007 (1, 4, 5) 0.002

(2, 2, 2) −0.029 (1, 2, 5) −0.004 (2, 2, 6) −0.001

(1, 2, 3) 0.003 (0, 4, 4) 0.002 (1, 3, 6) −0.003

(0, 0, 4) 0.021 (0, 3, 5) −0.007 (4, 4, 4) −0.007

(0, 3, 3) −0.007 (3, 3, 4) −0.002 (0, 1, 7) 0.004

(1, 1, 4) 0.012 (2, 4, 4) −0.004 (3, 4, 5) 0.001

For perfect disorder, all parameters are zero,

†

except α

0,0

= 1. The diffuse scat-

tered intensity away from the reciprocal lattice points g is then given by

(q) = Nc

| f

− f

for q = g. (9.38)

This is a monotonically varying background intensity, known as monotonic Laue

scattering. The variation is caused by the |q| dependence of the atomic scattering

factors, as illustrated in Fig. 2.9 on page 113.

As an example, Table 9.2 lists the Warren–Cowley SRO parameters for Cu-15

at% Al, measured by neutron diffraction [DG89].

‡

The parameters were determined

on cylindrical single crystals after a heat treatment of 4 days at 505 K. Note that

the value of α

0,1

nearly reaches the limit value of −0.18, indicating that there are

no ﬁrst nearest-neighbor Al–Al pairs.

Experimental electron diffraction patterns, showing the SRO diffuse intensity

in Cu-15 at% Al, can be obtained by annealing the material for a few days at

500 K, slowly cooling to room temperature (furnace cool), and preparing the thin

foil as described in Chapter 4. The patterns for the [001], [110], and [112] zone

axis orientations, recorded at 200 kV, are shown in Fig. 9.34(a). The patterns were

obtained with long exposure times (60 s); none of the diffuse intensity was visible

†

Actually, a detailed calculation shows that for perfect disorder α

0,l

=−1/(N − 1) for l = 0. This number

approaches zero for a sufﬁciently large number of unit cells.

‡

Diffuse neutron scattering experiments on [001]- and [110]-oriented single crystals of Cu-15 at% Al were carried

out at the “Centre d’Etudes Nucl´eaires, Laboratoire L´eon Brillouin” at Saclay, near Paris, in the Spring of 1988.

The neutrons were generated by the cold-source Orph´ee reactor, and a wavelength of 0.259 nm was used. More

details can be found in [DG89] and [CF83].

574 Electron diffraction patterns

000

200

020

002

022

222

202

111

(a) (b)

(c)

[001]

[110]

[112]

Fig. 9.34. (a) Experimental (200 kV) and (b) simulated diffuse intensity zone axis patterns

for Cu-15 at% Al, annealed for 4 days at 505K. Part (c) is a 3D rendering of the iso-intensity

surface of the diffuse component I

. The simulation uses the 30 shells listed in Table 9.2.

to the naked eye on the viewing screen.

†

Simulated diffuse intensity patterns, using

the SRO parameters of Table 9.2, are shown in Fig. 9.34(b) for the same zone

axis orientations. The computations were carried out with the ION routine

sro.pro,

available from the

website. The qualitative agreement between experimental and

simulated patterns is rather good. It is possible to use experimental diffuse intensity

patterns to derive the SRO parameters via a least-squares ﬁtting routine.

‡

The diffuse

intensity in the Cu-15 at% Al system is an indicator of the various ordered alloy

phases that are found for higher Al contents [BDGVT

87, BVTVL

89]. The trend

of no ﬁrst nearest-neighbor Al–Al pairs is also found at higher Al concentrations,

and gives rise to a rich set of long-period anti-phase boundary (LPAPB) structures.

In many alloy systems, the diffuse intensity is located near or on a geometric

surface in reciprocal space. This surface, with equation χ(q) = 0, is periodic with

†

It is possible to obtain a stronger diffuse intensity by annealing the alloy at a higher temperature, around 800 K,

and quenching to room temperature. This, however, does not produce an equilibrium SRO state.

‡

The reader should be warned that any combination of SRO parameters can be entered into the sro.pro program, but

that not every combination is physically realizable. Conversely, there are many different atomic conﬁgurations

which give rise to the same set of SRO parameters.

9.8 Diffraction effects from polyhedral particles 575

the same periodicity as the underlying reciprocal lattice. The diffuse amplitude

(q) (equation 9.33) is only different from zero on the surface, so that we have

(q)χ(q) = 0. (9.39)

If we write χ as a Fourier sum,

χ(q) =



2πiq·r

, (9.40)

then we can rewrite equation (9.39) (using equation 9.33) as



¯σ

−2πiq·(r

−r

)







¯σ

j+k



−2πiq·r

= 0. (9.41)

This is only possible if, for every j,wehave



¯σ

j+k

= 0, (9.42)

which is a linear relation between the Flynn operators. If we can determine the

coefﬁcients ω

from the experimental diffuse intensity patterns (and typically there

are only a few coefﬁcients needed to represent the surface), then the linear relation

leads to the identiﬁcation of a cluster of lattice sites with identical composition.

For examples of the application of this cluster model we refer the interested reader

to [DRVTVDA76, DRVDVTA77, VDDRVTA77].

There is a very large amount of literature dealing with short-range order, order–

disorder transformations, and long-range ordered structures. Theoretical models

range from the pair-correlation Gorsky–Bragg–Williams model [CDGD

89], to

complex multi-cluster models in the cluster variation theory [kik51, Fin87], and the

statistical “axial next-nearest-neighbor interaction” model, known as the ANNNI

model [FS81, Sel88]. It is possible, in principle, to use diffuse scattering infor-

mation to determine the long-range ordering potential. This is often based on the

Krivoglaz–Clapp–Moss theory, which relates the diffuse scattering surface to the

Fourier transform of the ordering potential [CM66, CDGD

89]. The reader is re-

ferred to the proceedings of the various conferences on solid–solid phase transfor-

mations for many examples of short-range order and order–disorder transforma-

tions [ALSW84, Tsa84, Lor87, JHLS94, KOM99].

9.8 Diffraction effects from polyhedral particles

In the ﬁnal section of this chapter, we will consider diffraction features caused by

polyhedral inclusions in a matrix. We have seen in Chapter 2 (p. 119) that, for a

ﬁnite crystal, each reciprocal lattice point assumes a shape described by the Fourier

576 Electron diffraction patterns

transform D(q) of the shape function D(r). We shall call this reciprocal shape

function the shape amplitude:

D(q) =

...

D(r)e

−2πig·r

dr =

...

−2πig·r

dr,

with V the (ﬁnite) volume of the crystal. Consider a particle with a regular polyhedral

shape function, consisting of F faces, V vertices, and E edges. One can show that

in 3D space the following relation is valid:

V + F − 2 = E; (9.43)

this is known as Euler’s theorem. We will now explicitly work out the Fourier

integral above for an arbitrary polyhedral shape.

Following von Laue [vLR36] and Komrska [Kom87] (but with a slightly different

notation) we will use the so-called Abbe transform to convert the volume integral

into a sum of surface integrals over the faces of the polyhedron. We start from the

Helmholtz equation (2.36) in Chapter 2:

 + k

 = 0,

 =−

.

Consider then the integral of  over a ﬁnite region V in an N -dimensional space:

(N )

 d

r =−

(N )

 d

The second integral can be rewritten using the theorem of Gauss (e.g. [Wre72,

pp. 297–298]), which reduces the dimension of the integration space by 1 to N − 1:

(N )

∇·A d

r =

(N −1)

A · n dS,

where S is the surface bounding the region V, and n the unit outward normal on

the surface S. If we select A =∇, then we arrive at the Abbe transform:

(N )

 d

r =−

(N −1)

∇ · n dS. (9.44)

This relation is valid for every “well-behaved” function  that satisﬁes the

Helmholtz equation, and in particular it is valid for the phase factor in the Fourier

transform integral. It is easy to show that the function

 = e

−2πiq·r

9.8 Diffraction effects from polyhedral particles 577

satisﬁes the Helmholtz equation with k

= (2πq)

, with q the length of the vector

q. The Abbe transform of this function is then readily obtained as

(N )

−2πiq·r

r =

2πq

(N −1)

−2πiq·r

q · n dS. (9.45)

If the bounding surface S consists of planar sections, as is the case for a polyhedral

particle, then the surface integral may be replaced by a sum of surface integrals, one

over each face of the polyhedron. In such a case, the normal vector n is constant

across a face and the dot product q · n may be taken out of the integral. The Abbe

transform may then be carried out once again, this time to reduce the dimension from

N = 2 (a plane) to a N = 1 (a line). It is the repeated use of this transform which

permits explicit algebraic computation of the shape amplitude for a polyhedral

inclusion.

An example computation for a 2D polygon will illustrate the general procedure.

Consider the three-sided polygon shown in Fig. 9.35. There are E = 3 edges, and

V = 3 vertices, and Euler’s relation in this case reduces to E = V . We number the

vertices from e = 1,... ,E, and the vertex position vectors are denoted by ξ

with

respect to an arbitrary origin O. It is convenient, but not necessary, to select O at

the center of the polygon. The vectors n

denote the unit outward normals to the

edges, and the edge length is denoted by L

. We also deﬁne the unit vector t

along

the edge e, pointing in counterclockwise direction. An arbitrary position along the

edge e can then be written as

r = ξ

+ #t

Fig. 9.35. Illustration of the notation for edges and unit outward normals for a two-

dimensional three-sided polygon. The location of the origin is arbitrary.