Graef M. Introduction to conventional transmission electron microscopy

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358 Two-beam theory in defect-free crystals

as Friedel’s law (e.g. [MM86, page 199]). As a direct consequence of Friedel’s law,

a kinematical diffraction pattern must belong to one of the 11 centrosymmetric

point groups. All point groups which become a given centrosymmetric point group

when an inversion element is added belong to a so-called Laue class; there are

11 Laue classes, and the point group dependences are described in Appendix A4.

We will now see that under dynamical electron diffraction conditions, Friedel’s law

may be violated.

The symmetry of the dynamical two-beam scattering process can be analyzed

in a number of different ways. First of all, let us consider equations (6.20) and

(6.21). The transmitted amplitude depends on both the excitation error s

and the

lattice potential through the quantity σ . We have already seen that the parameters

)

−1

and (q

−g

)

−1

describe the scattering from 0 to g, and from g back to 0,

respectively. The parameter σ depends on the product of both qs. This product

expresses the fact that electrons in the transmitted beam are either not scattered at

all, or scattered twice, or four times, and so on. Thus, electrons in the transmitted

beam have been scattered an even number of times. Therefore, we expect that the

transmitted amplitude ψ

for a given excitation error s

will be equal to the amplitude

for the opposite orientation s

−g

= s

. Friedel’s law is therefore valid for the

transmitted beam, for both centrosymmetric and non-centrosymmetric crystals.

Electrons in the scattered beam have been scattered an odd number of times,

which is reﬂected in the presence of an extra factor (q

)

−1

in equation (6.21).

Since for a non-centrosymmetric crystal the numbers (q

)

−1

and (q

−g

)

−1

are not

necessarily equal, we ﬁnd that the scattered amplitudes for the beams g and −g

for the same excitation error may be different for a non-centrosymmetric crystal.

This means that Friedel’s law is not valid for the scattered beam under dynamical

diffraction conditions. For a centrosymmetric crystal, for which q

= q

−g

, Friedel’s

law is not violated.

This behavior can also be derived from the reciprocity theorem (equation 5.62), as

applied to the transmitted beam 0. In the two-beam case, the non-centrosymmetric

crystal must have type I symmetry, since there are no other reﬂections and

the projection approximation is valid. This means that we must have (from

equation 5.63)

(⊗, r, C) = ψ

(⊗, −r, C). (6.27)

We ﬁnd that the transmitted amplitude (and therefore intensity) is symmetric in the

excitation error s

±g

, and hence Friedel’s law is valid. This can also be seen from

the explicit equation (6.24), when s

is replaced by s

−g

The situation is quite different when we consider the symmetry with respect to

the sign of the excitation error ±s

. The scattered amplitude (6.25) only depends on

6.3 The two-beam case: DHW formalism 359

Table 6.1. Intensity symmetry relations for centrosymmetric and

non-centrosymmetric crystal structures under dynamical two-beam

conditions with absorption.

Centrosymmetric Non-centrosymmetric Comment

(+s

) = I

(+s

−

) I

(+s

) = I

(+s

−

) Friedel’s law

(+s

) = I

−g

(+s

−

) I

(+s

) = I

−g

(+s

−

) Friedel’s law

(+s

) = I

(−s

) I

(+s

) = I

(−s

) Asymmetric bright ﬁeld

(+s

) = I

(−s

) I

(+s

) = I

(−s

) Symmetric dark ﬁeld

, and hence the scattered rocking curve is symmetric in the excitation error. The

transmitted intensity (6.24) depends linearly on the excitation error, and therefore

is not symmetric in ±s

. Both of these statements are valid for centrosymmetric

and non-centrosymmetric crystal structures.

Table 6.1 summarizes the various symmetry relations; the shorthand notation

±s

is used to denote the excitation error (positive or negative according to the

ﬁrst ± sign) for either the g or −g reﬂection (indicated by the second ± sign). The

magnitude of the excitation error is the same for all entries in the table. The trans-

mitted intensity is hence asymmetric in the excitation error, but satisﬁes Friedel’s

law for all crystal structures; the scattered intensity is symmetric in the excitation

error, but violates Friedel’s law in non-centrosymmetric crystals. In the absence

of absorption (σ

→ 0), the bright ﬁeld rocking curve becomes symmetric in the

excitation error.

The BF–DF rocking curves for a centrosymmetric crystal (β

= 0) are shown in

Fig. 6.4 for a range of crystal thicknesses and orientations. The BF asymmetry is

clearly present. It appears that there is a second type of absorption process which is

only visible for negative excitation errors; this process depends on the value of the

absorption length ξ



and is known as anomalous absorption.

†

When we talk about

Bloch waves in Section 6.4, we will discuss the physics underlying anomalous

absorption.

For a non-centrosymmetric crystal, there is no a priori reason why the phases

of the real and imaginary potentials should be identical, and β

can take on any

value. This has profound consequences. Figure 6.5 shows how the bright ﬁeld and

dark ﬁeld contrast for a foil thickness of z

= 2ξ

(see Fig. 6.4) varies with the

parameter β

. The bright ﬁeld contrast reverses completely when β

= π, and is

symmetric in the excitation error for β

= π/2 and 3π/2. The bright ﬁeld contrast

†

One can also think of this phenomenon in terms of enhanced transmission for the opposite sign of the excitation

error. Both terminologies are in use.

360 Two-beam theory in defect-free crystals

Bright field

Dark field

-4 -2024

z =

0.5

z = 1.5

z = 4.0

z = 1.0

z = 2.0

z = 6.0

Fig.

6.4. Bright

ﬁeld

and dark

ﬁeld

rocking curves for the range

−4 ≤ w ≤ 4,

for crystal

thicknesses z

= 0.5, 1.0, 1.5, 2.0, 4.0, and 6.0. Note the asymmetric bright ﬁeld curve.

Other parameters are: β

= 0,ξ



= 10ξ

= ξ

6.4 The two-beam case: Bloch wave formalism 361

Bright field Dark field

π/2

3π/2

2π

--20 2 4

Fig. 6.5. Bright ﬁeld and dark ﬁeld intensities as a function of dimensionless excitation

error w and β

, for the case z

= 2ξ

of Fig. 6.4.

is also symmetric in ±β

. The dark ﬁeld contrast remains symmetric in w for all

values of β

, but is asymmetric in ±β

The reader may ﬁnd it useful to enter various parameters into the ION-routine

betag.pro on the website. This routine displays the rocking curves for a non-

centrosymmetric crystal, based on equations (6.20) and (6.21). The absorption

ratio is deﬁned as ξ



/ξ, so that low absorption implies a large absorption ra-

tio. A second ION-routine called

rock.pro is also available on the website; this

routine displays the bright ﬁeld and dark ﬁeld rocking curves for the standard

parameter space for a centrosymmetric crystal; the user can then click on any

point in the bright ﬁeld image to generate a curve similar to those shown in

Fig. 6.4.

6.4 The two-beam case: Bloch wave formalism

The purpose of this section is to compare the Bloch wave approach to the dif-

ferential equation approach of the previous section, for a case where the entire

computation can be performed analytically. We will rederive the solution of the

two-beam dynamical scattering problem for the case of a non-centrosymmetric

crystal with absorption. We will also introduce a graphical method to solve the

problem; this is known as the dispersion surface construction, and is equivalent

to the Ewald sphere construction of the kinematical scattering problem. Where

appropriate, we will compare the results with those from the differential equation

approach.

362 Two-beam theory in defect-free crystals

6.4.1 Mathematical solution

For the two-beam case, the eigenvalue equation (5.51) reads as



0 U

−g





( j)



= 2k

( j)



( j)



. (6.28)

This results in the following quadratic determinantal equation:

( j)

− (2k

)

( j)

− U

−g

= 0, (6.29)

with solutions

( j)

= k



+ U

−g

For a centrosymmetric crystal, we have U

−g

= U

, and also k

/|U

|=ξ

, and we

can rewrite the solutions as

( j)

=|U

)

w ±



1 + w

, (6.30)

where w = s

is the dimensionless excitation error. The difference between the

two eigenvalues γ

(1)

and γ

(2)

is given by

γ = γ

(1)

− γ

(2)



1 + w

= σ,

where we have used the deﬁnition of the variable σ on page 354. We ﬁnd that in the

exact Bragg orientation, when w = 0, the difference between the two eigenvalues

is precisely equal to the inverse of the extinction distance ξ

. Figure 6.6 shows the

graphical representation of the two eigenvalues ξ

( j)

in equation (6.30), for w

from −3to+3.

-3 -2 -10123

-1

-2

ξ γ

(1)

(2)

Fig. 6.6. Graphical representation of the two-beam eigenvalues γ

(1)

and γ

(2)

(multiplied

by the extinction distance ξ

) as a function of the dimensionless parameter w = s

. The

straight lines represent the asymptotes of the empty crystal approximation.

6.4 The two-beam case: Bloch wave formalism 363

The gap between the two dispersion surface branches is the inverse of the extinc-

tion distance and is, therefore, proportional to the potential Fourier coefﬁcient. This

is very similar to solid-state band theory, where the band-gap is also proportional

to the Fourier coefﬁcient. Furthermore, whereas band theory assumes parabolic

branches close to the Brillouin zone boundary, electron diffraction is described by

hyperbolic branches.

At this point, we have sufﬁcient information to determine the eigenvectors. From

equation (6.28), we ﬁnd that

(1)

)

w +



1 + w

−iθ

;

(2)

)

w −



1 + w

−iθ

If we introduce the notation cot β = w , then we can make use of the following

relations [AS77]:

cos β + 1

sin β

cos (β/2)

sin (β/2)

and

cos β − 1

sin β

=−

sin (β/2)

cos (β/2)

. (6.31)

Combining these expressions with the normalization relations (5.54) and (5.55) we

ﬁnd for the components of the eigenvectors:

(1)

= sin

; C

(2)

= cos

iθ

;

(1)

= cos

−iθ

; C

(2)

=−sin



(6.32)

For the two-beam case we then have

) = C

(1)

2πiγ

(1)

+ C

(2)

2πiγ

(2)

;

) = C

(1)

2πiγ

(1)

+ C

(2)

2πiγ

(2)

For the general case with absorption, we must explicitly invert the dynamical matrix

to obtain the excitation amplitudes α

( j)

. In the absence of absorption, the two-beam

Bloch wave excitation amplitudes are given by (using α

( j)

= C

( j)∗

(1)

= sin

;

(2)

= cos

−iθ

The excitation amplitudes are shown in Fig. 6.7(a) as a function of the dimensionless

parameter w. When the reciprocal lattice point g is inside the Ewald sphere (positive

excitation error), then the second Bloch wave contributes most to the diffracted

364 Two-beam theory in defect-free crystals

0.0

0.2

0.4

0.6

0.8

1.0

Excitation Amplitudes

(1)

(2)

-4 -2024

0.0

1.0

0.2

0.4

0.6

0.8

Absorption Factors q

(j)

(1)

(2)

-4 -2024

w=0

w=2

w=4

| |

(1)

Ψ | |

(2)

(a)

(b)

(c)

Fig. 6.7. Two-beam Bloch wave parameters: (a) excitation amplitudes α

( j )

; (b) type I

and II Bloch wave intensities for w = 0, 2, 4, relative to the atom locations (spheres);

intensity. When the reciprocal lattice point is outside the Ewald sphere, then the

ﬁrst Bloch wave is more strongly excited.

It is now straightforward to compute the intensities of the transmitted and

diffracted beams for the two-beam case without absorption. We will only com-

pute the dark ﬁeld intensity as an example. Inserting all eigenvectors and Bloch

wave excitation amplitudes into the expression for ψ

(z) we ﬁnd:

(

)

sin β

−iθ

2πiγ

(1)

− e

2πiγ

(2)

;

= ie

−iθ

2πiγ

(1)

−πiσ z

sin β sin

(

πσz

)

;

|ψ

= sin

β sin

(πσz

);

sin

(πσz

)

1 + w

6.4 The two-beam case: Bloch wave formalism 365

which is identical to the solution (6.22) on page 354. We leave it to the reader to

verify the expression for the transmitted intensity.

One particular advantage of the Bloch wave formulation is that it provides the

answer to the question: where in the crystal do the electrons travel? Since there are

two Bloch wave vectors inside the crystal, each of those waves must be associated

with a different potential energy, and must therefore travel at a different location in

the crystal. The two Bloch waves for the two-beam case are given by



( j)

(r) = C

( j)

2πik

( j )

·r

+ C

( j)

2πi(k

( j )

+g)·r

It is a simple exercise to show that the intensities associated with the Bloch waves

are given by

|

(1)

(r)|

= 1 + sin β cos



2πg · r − θ



; (6.33)

|

(2)

(r)|

= 1 − sin β cos



2πg · r − θ



. (6.34)

If we take θ

= 0, and the x-direction is normal to the planes g, then 2π g · r =

2πgx, and we ﬁnd:

|

(1)

(r)|

= 1 +

cos 2πgx

√

1 + w

;

|

(1)

(r)|

= 1 −

cos 2πgx

√

1 + w

The resulting intensity proﬁles are shown in Fig. 6.7(b): the wave 

(1)

on the left

has its maximum intensity on the planes x = nd, where d = 1/g is the interplanar

spacing. The second Bloch wave has its maxima in between the atomic planes, at

x = (n +

)d.

We shall call a Bloch wave with maxima at the atom positions a type I wave;

a Bloch wave with maxima in between the atom positions is then a type II wave.

Electrons in the type I wave spend much of their time near the atom cores, and are

hence likely to undergo inelastic scattering events, which essentially means that

they are absorbed according to the phenomenological description of absorption

on page 122. Electrons in a type II wave spend most of their time in between

the atom columns and are less likely to undergo inelastic scattering events. We

have already seen that for negative excitation errors, the ﬁrst Bloch wave has the

strongest excitation amplitude α

(1)

(Fig. 6.7a). Conversely, for a positive excitation

error the second Bloch wave is more strongly excited. Since the ﬁrst Bloch wave

is more strongly absorbed, this generates an asymmetry between the transmitted

intensities for positive and negative excitation errors. We have called this asymmetry

anomalous absorption (see page 359).

Two sound waves with a slightly different wavelength will give rise to an in-

terference sound (a “beating” sound) at the difference frequency. A similar thing

366 Two-beam theory in defect-free crystals

happens for electrons in a solid: for a two-beam situation, the two allowed waves

interfere and give rise to interference fringes, which we can observe as thickness

fringes or Pendell

osung fringes. Note that the variable time (for sound waves) is

replaced by depth in the crystal (for electrons).

It is rather straightforward to include absorption explicitly into the two-beam

equations. The resulting eigenvalue equation reads as





−g

+ iU



−g

+ iU



+ iU







( j)



= 2k

( j)



( j)



. (6.35)

For a centrosymmetric crystal we have U

= U

−g

and U



= U



−g

, and the determi-

nantal equation becomes

( j)

− 2k

( j)



+ 2iU





− (U

+ iU



)

= 0.

Ignoring quadratic terms in U



/U we ﬁnd for the solutions:

( j)

≈|U

w + i



1 + w

+ i

(wU



+ U



)

The imaginary components of these equations are small, and we can use the fol-

lowing Taylor expansion to separate the real and imaginary components of the

eigenvalues:



x + iy ≈

√

x +

∂

√

x + iy

∂y

y=0

y =

√

x +

√

Introducing the expressions for the absorption distances we ﬁnd

( j)

2ξ

)

w ±



1 + w

; (6.36)

( j)





√

1 + w



. (6.37)

The real part of the eigenvalues is identical to the case without absorption. The

imaginary components for the type I and type II Bloch waves in exact Bragg

orientation w = 0 are given by

(1)







; (6.38)

6.4 The two-beam case: Bloch wave formalism 367

(2)





−





. (6.39)

The scaled absorption parameters ξ

( j)

are shown as a function of w in Fig. 6.7(c);

the ratio ξ/ξ



was taken to be equal to 1. Since the type I wave is more strongly

excited (Fig. 6.7a) for negative excitation errors, the preferential absorption will

be more noticeable than in the case of a positive excitation error and results in

anomalous absorption.

Equation (6.39) provides us with an important restriction on the allowed values

of the normal and anomalous absorption lengths. Since q

(2)

must be positive,

†

must have



≥



which means that ξ



must be greater than ξ



[HHW62]. Experiments show that

typically ξ



≈ 2ξ



and also that ξ

≈ 0.1ξ



(e.g. [Has64]). We have seen in Fig. 6.5

that, for a non-centrosymmetric crystal, anomalous absorption may occur for pos-

itive excitation errors, as a consequence of the parameter β

, which expresses the

phase difference between the Fourier coefﬁcients of the real and imaginary parts

of the electrostatic lattice potential. We leave it to the reader to derive the explicit

equations (6.24) and (6.25) from the Bloch wave formalism.

6.4.2 Graphical solution

In this section, we will derive a graphical method to determine the Bloch wave

vectors inside the crystal. The method is the dynamical equivalent of the kinematical

Ewald sphere construction discussed in Chapter 2, and is known as the dispersion

surface construction. This section is based on [Met75] and [SZ92]. Unless stated

otherwise, a centrosymmetric crystal is assumed.

Consider the drawing in Fig. 6.8(a); it shows the Ewald sphere construction

for a reciprocal lattice point g, which falls somewhat inside the sphere (i.e. with

a positive excitation error s

). The diffracted wave vector is given by k



= k

g + s

, where e

is a unit vector along the incident beam direction. This is a

purely kinematical description, which only deals with the geometry of Bragg’s

law. We can now use equation (6.29) to derive the eigenvalues γ

( j)

for the so-

called empty crystal approximation. In this approximation, all Fourier coefﬁcients

vanish, except for the mean inner potential which is taken into account in the

refraction-corrected wave vector k

. The solutions of the determinantal equation are

†

A negative value for q

(2)

would give rise to an increase with crystal thickness z

in the number of electrons in

Bloch wave (2).