Graef M. Introduction to conventional transmission electron microscopy

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558 Electron diffraction patterns

structure can often also be regarded as an interface modulated structure. We will describe

such structures in Section 9.6.2.3.

In many of the cases just introduced it is possible to deﬁne one or more modulation

wave vectors q

(n)

. In general, those vectors are described in reciprocal space, which

means that they can be written as a linear combination of the reciprocal basis

vectors a

∗

(n)

= q

(n)

∗

. (9.19)

If all of the numbers q

(n)

are rational numbers, then it can be shown that there

exists an integer p such that the vector pq

(n)

is again a reciprocal lattice vector.

This means that the modulated structure contains an integer number of unit cells

of the underlying lattice. In such a case, we say that the modulation is commen-

surate with the underlying lattice. A modulation with wave vector q =

[310] is

a commensurate modulation since 5q is a reciprocal lattice vector. If one or more

of the components q

(n)

are irrational, then such an integer p does not exist, and

the resulting structure is incommensurate, i.e. the modulation wavelength is not

an integer multiple of any lattice spacing.

†

In the following subsections, we will

describe several possible commensurate and incommensurate modulations in more

detail.

9.6.2 Commensurate modulations

Before dealing with general cases, let us ﬁrst consider a simple example of a

displacively modulated structure. For simplicity, we will work with a 1D modulation

with wave vector q =

∗

. The atom positions will then be shifted by a sinusoidal

function:





= r

+ A sin(2π q · r

where A indicates the direction and amplitude of the modulation. We will take

A = Aa, i.e., a longitudinal modulation shown in Fig. 9.28(a). If the position vectors

of the atoms are given by r

= ja, then we have for the argument of the structure

factor exponential:

g · R





= hj + hAsin



2π



†

The distinction between commensurate and incommensurate modulations is an academic one, since it is not

possible to measure a modulation wavelength and prove that it is an irrational number. However, in most

materials with incommensurate modulations, the modulation wavelength depends on an external parameter,

often the temperature. If the modulation wavelength varies continuously with temperature, then the modulation

is considered to be incommensurate.

9.6 Diffraction effects in modulated crystals 559

(a)

(b)

Fig. 9.28. Two examples of displacive modulations of a 2D square lattice: (a) q =

∗

A = 1.8a (longitudinal); (b) q =

∗

, A = 1.2b (transverse).

If A is small, then we can expand the exponential in a Taylor series and only keep

the ﬁrst two terms:





2πihj

hAi sin

(

2π j/

)

≈



2πihj



1 + 2π ihAsin



2π



= F



πhA

2πi(h+j/ )

− e

−2πi(h−j/ )

;

i.e. in addition to the normal diffraction amplitude described by the structure factor

of the undistorted lattice, there will be satellite spots ﬂanking each fundamen-

tal spot at a distance of ±j/ . The transmitted or zero order beam does not have

560 Electron diffraction patterns

satellite reﬂections because the second term of the modulated structure factor is pre-

multiplied by h. The diffraction pattern, therefore, consists of the regular diffraction

spots at positions g and satellite reﬂections at the positions g ± q.

9.6.2.1 Displacive modulations

Consider a sinusoidal displacive modulation with amplitude and direction A and

wave vector q, so that [AVD93]

R(r

) = r

+ A sin(2π q · r

This modulation could be longitudinal (A  q) or transverse (A ⊥ q). The structure

factor evaluated at a location h in reciprocal space is given by



2πih·[r

+A sin(2πq·r

)]

;



2πih·r

2πih·A sin(2πq·r

)

Using the Jacobi–Anger relation

†

and changing the order of the summations we

can rewrite this expression as

+∞



n=−∞

(2πh · A)



2πi(h+nq)·r

The second summation can only be different from zero if h + nq equals a

reciprocal lattice vector g,orh = g − nq. From this expression we can derive

the following observations:

(i) reciprocal lattice points for which h · A = 0 will have satellite reﬂections along the

direction of the wave vector q;

(ii) for a given value of the argument x = 2πh · A, the Bessel function J

(x) rapidly

decreases with increasing n, indicating that the amplitude of the satellites decreases

rapidly with increasing n – typically, only the ﬁrst three or four satellites will be

observable;

(iii) for a longitudinal modulation reciprocal lattice points normal to q will have no satel-

lites, since the argument of the Bessel function vanishes. For a transverse modulation

reﬂections parallel to q will have no satellites. Because of double-diffraction effects

satellites may be present around all reciprocal lattice points.

†

If a and b are real numbers, then the following equality holds:

ia sin b

+∞



n=−∞

inb

(a),

where J

are the Bessel functions of the ﬁrst kind [AS77].

9.6 Diffraction effects in modulated crystals 561

Figure 9.28 shows two examples of structures with 1D displacive modulations;

both examples use a 2D square undistorted lattice as starting point. Fig. 9.28(a) is a

longitudinal modulation along the x-direction, along with its computed diffraction

pattern. Satellites can be found around all reciprocal lattice points, except for the

row normal to q. Figure 9.28(b) shows a transverse modulation with direction along

b and the same wave vector as in (a). In this case, the central horizontal row of

reﬂections does not have any satellites.

The two-dimensional modulation can be regarded as two 1D modulations ap-

plied in turn. The ﬁrst modulation produces sets of satellites, and then the second

modulation produces satellites on all reﬂections present after the ﬁrst modulation.

This is illustrated in Figs 9.29(a) and (b), for the indicated modulation amplitude

(a)

(b)

Fig. 9.29. Two-dimensional displacive modulations of a 2D square lattice: (a) q

∗

, A

= 1.8b, q

∗

, A

= 1.2a; (b) q

∗

+ 2

∗

), A

= 0.9(2a − b), q

∗

− b

∗

), A = 0.6(a + b).

562 Electron diffraction patterns

and wave vectors. The amplitudes are somewhat exaggerated to emphasize the 1D

and 2D arrays of satellite reﬂections.

If an experimental diffraction pattern shows 1D or 2D arrays of satellite reﬂec-

tions, then a careful measurement of the positions of all satellites with respect to

the fundamental reciprocal lattice sites along with the directions of the modulations

provides sufﬁcient information to deduce the character of the displacive modula-

tions. If the pattern of satellites is different for different fundamental reﬂections, i.e.

if the satellite positions are different with respect to each of the reciprocal lattice

points, then this may point to the presence of interface modulations, which we will

discuss in Section 9.6.2.3. Displacive modulations occur in several of the high-T

superconductor materials. The reader may ﬁnd examples of electron diffraction

studies of commensurate and incommensurate displacive modulations in the fol-

lowing references: [VTZA88,HII

91, ZWT99].

The ION routine

modulation.pro on the website can be used to produce images

similar to those of Figs 9.28 and 9.29 for 1D and 2D displacive modulations of a

square lattice. The reader can experiment with different parameters, to see how the

resulting diffraction patterns reﬂect the presence of the modulated structure.

9.6.2.2 Compositional modulations

For simplicity, we will continue to use the 2D square lattice to illustrate the effect of

compositional modulations on the diffraction pattern. Consider a binary alloy AB,

with a local concentration c

of atoms of the type A (0 ≤ c

≤ 1). The average

concentration of A atoms is represented by

. The electron scattering factor at

position r

can then be written as

= c

A, j

+ c

B, j

= c

A, j

+ (1 − c

A, j

) f

In a compositionally modulated structure the concentration of A atoms varies from

site to site. For a sinusoidal modulation we have

A, j

+ c

sin(2πq · r

), (9.20)

with c

the maximum deviation from the average value, and q the modulation

wave vector. If we convert the sin-function to complex exponential notation, it is

easy to show that the structure factor is given by

[

(

1 −

)

]



2πih·r

(

− f

)

c



2πi(h+q)·r

− e

2πi(h−q)·r

9.6 Diffraction effects in modulated crystals 563

(a)

(b)

Fig. 9.30. (a) Sinusoidal and (b) square wave compositional modulations on a 2D square

lattice. Simulation parameters are

= 0.6, c

= 0.4 and q =

∗

This expression is only different from zero if one of the vectors h, h + q,orh − q

is equal to a reciprocal lattice vector. In other words, every reﬂection g, including

the transmitted beam, is surrounded by two satellite reﬂections at g + q and g − q.

The intensities of those reﬂections depend on the difference between the scattering

factors for the particular elements A and B, and on the magnitude c

of the

compositional modulation. Figure 9.30(a) shows an example of such a modulation

for

= 0.6, c

= 0.4, and q =

∗

If the compositional proﬁle deviates from a purely sinusoidal proﬁle, then addi-

tional satellite reﬂections will be present, depending on the actual proﬁle. It is easy

to see that the concentration proﬁle in equation (9.20) is a Fourier series with only

one term. For any other proﬁle the Fourier series would contain additional terms,

which would then correspond to additional satellite reﬂections at multiples of q.

For a square wave modulation, the Fourier series would contain only odd multiples

564 Electron diffraction patterns

γd

hkl

(hkl)

Fig. 9.31. Schematic drawing of a crystallographic shear plane defect.

of q, and the corresponding diffraction pattern is shown in Fig. 9.30(b) for the

same parameters as in (a). The presence or absence of certain satellites provides

information on the Fourier coefﬁcients of the concentration proﬁle, provided this

proﬁle has a single basic periodicity described by q. If more than one wavevector

is needed, the satellite patterns become more complicated and analysis is more

difﬁcult. Compositional modulations can be found in many material systems. The

most notable type of compositional modulation is perhaps spinodal decomposition.

We refer the interested reader to [HGWA84] for details and examples.

9.6.2.3 Interface modulations: crystallographic shear planes

Consider the crystal shown in Fig. 9.31. We slice the crystal in plane-parallel blocks

of thickness pd

hkl

, where p is an integer and d

hkl

the interplanar spacing of the planes

parallel to the cut. At this cut, we remove a thin layer of material with thickness

t = γ d

hkl

, with 0 <γ <1. This is the component of the displacement vector R

perpendicular to the cut plane. For a conservative fault, γ = 0. We then displace

the second block by the vector R and close the gap. We have now created a new

unit cell with lattice parameter A

= pa

+ R. This is a so-called crystallographic

shear fault.

In many oxides, the crystal structure can be described by geometric coordination

shapes, such as octahedra and tetrahedra. Often, the structure consists of a three-

dimensional arrangement of corner, edge, or face-sharing polyhedra, with oxygen

at the corners and metal ions at the centers of the polyhedra. In response to local

chemical variations, say, an oxygen deﬁciency, many oxides adapt by the introduc-

tion of crystallographic shear planes. These defects are to oxide structures what

anti-phase boundaries are for ordered intermetallics.

Since this defect can occur in a periodic fashion, we ask the question: what

will we observe in the diffraction pattern? Based on the discussion of streaks in

Section 9.4.1, we anticipate that the diffraction information will be located along the

9.6 Diffraction effects in modulated crystals 565

normal to the shear planes; we will denote the unit vector along this normal by e

We expect a pattern of satellite reﬂections, with a spacing related to the inverse of the

thickness of the individual blocks of the shear structure. In the following paragraphs,

we will derive an expression for the location and intensity of the satellite reﬂections.

The method used is very general and can be applied to any type of periodic planar

defect.

We take the origin to be in the center of one of the blocks and the x-axis in

the direction of the normal to the shear planes. The thickness of the blocks is thus

described by d = ( p − γ )d

hkl

. We deﬁne the function U (x) to be the shape function

of the block, i.e.

U (x) = 1 for −

< x <

and zero elsewhere. Let the potential of the basic structure be V (r). We will assume

that the lattice potential of the sheared structure, V

(r), is at least approximately

described by the perfect crystal potential; this is a good approximation inside the

blocks but may become questionable close to the shear planes [VDBMA87]. The

shear potential at position r is thus given by

(r) =



(r − n(a + R)),

where we have deﬁned a = pa

. The block potential V

is deﬁned by

(r) = V (r) U (x).

The diffraction pattern associated with this potential can be computed by means of

standard Fourier transformation techniques. The Fourier coefﬁcients of the shear

potential are given by

(h) =

...

crystal

(r)e

2πih·r

dr,

with h (for now) an arbitrary vector in reciprocal space. Redeﬁning the vector r as

r = r



+ n(a + R), we ﬁnd

(h) =



...

crystal

V (r



)U(x)e

2πih·[r



+n(a+R)]





2πinh·(a+R)

...

block

V (r



2πih·r



= I(g)





...

block

2πi(h−h



)·r



566 Electron diffraction patterns

where we have introduced the Fourier expansion of the perfect crystal potential.

Since we expect the satellite reﬂections to occur on rows parallel to e

,wedeﬁne

a new vector in reciprocal space:

h = h



+ ue

= h



+ s.

The magnitude of s remains to be determined. We can then rewrite the equation as

(h) = I(h)





...

block

2πis·r



= I(h)





sin πud

πud

The prefactor I(h) can be computed as follows:

I(h) =



2πinh·(a+R)



2πin(h



+s)·(a+R)



2πin(R·h



+ud)

where we have used the fact that a

· h



is always an integer number and (a + R) ·

= d. In the limit n →∞this can be rewritten as

I(h) =





u −

(k − R · h



)



In other words, the satellite reﬂections will occur at the positions

= h



(k − R · h



, (9.21)

with k taking on all integer values. The Fourier coefﬁcients of the shear potential

at these locations are given by

= V



sin πud

πud

with u =

(k − R · h



), (9.22)

as is easily veriﬁed.

Let us consider a 2D example. The upper drawing in Fig. 9.32 shows a square crys-

tal with lattice parameter a. A periodic crystallographic shear plane is introduced

with fault plane (10) (in two dimensions) and displacement vector R =

b. The

block thickness is equal to d = 8a. The modulus squared of the Fourier transform

9.6 Diffraction effects in modulated crystals 567

Fig. 9.32. One-dimensional (a) and two-dimensional (b) crystallographic shear plane struc-

tures for the parameters described in the text, along with their kinematical diffraction

patterns.

of this pattern (i.e. the kinematic diffraction pattern) is shown on the right. The

unit shear plane normal is e

= a/a = aa

∗

= a(10). From the equations above, we

expect satellite reﬂections at the positions

= h





k − h





Consider as an example the reﬂection h



= (0 1); for this reﬂection we calculate

that

= (0 1) +

4k − 1

(1 0).

In Fig. 9.32(a), we indeed ﬁnd the satellite reﬂections at distances −

, and

from the initial position of the (01) reﬂection for the unfaulted structure. For other

reﬂections we ﬁnd different patterns of satellites; in particular, for the central row

of fundamental reﬂections h



= (n 0) there are no satellites since the dot product