Graef M. Introduction to conventional transmission electron microscopy

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348 Two-beam theory in defect-free crystals

6.3 The two-beam case: DHW formalism

6.3.1 The basic two-beam equations

We will consider only the transmitted beam ψ

and one diffracted beam ψ

.We

have two equations of the type (5.18), one for each beam. The summation on the

right-hand side of equation (5.18) also contains only two terms, g



= 0 and g



= g.

This results in the following equations (using equation 5.17):

beam left-hand side g



= 0g



= g

⇓⇓ ⇓ ⇓

dψ

− 2π is

=−



+ iπ

iθ

−g

;

dψ

− 2π is

= iπ

iθ

−













(6.1)

These equations will be the starting point for two different derivations: the kine-

matical two-beam theory and the dynamical two-beam theory.

6.3.2 The two-beam kinematical theory

As a ﬁrst step towards the general solution of the dynamical equations (6.1), let us

make a few simplifying assumptions.

(i) Ignore absorption. This means that all factors 1/ξ



(including the normal absorption

term 1/ξ



) must vanish. Equivalently, we have

g−g



g−g



(ii) The single scattering approximation. The electron either remains in the transmitted

beam and is never scattered, or it is scattered elastically into a diffracted beam and

remains there. Only scattering from the transmitted beam into a diffracted beam is

allowed, which can be expressed mathematically as

g−g



= 0 except for

g−0

This is known as the kinematical approximation.

Since the excitation error s

is equal to zero (the origin of reciprocal space always

lies on the Ewald sphere), we can simplify the two-beam equations (6.1) to their

6.3 The two-beam case: DHW formalism 349

kinematical form:

dψ

= 0; (6.2)

dψ

= 2πis

+ iπ

iθ

. (6.3)

The ﬁrst equation follows because both the extinction distance and the excitation

error of the transmitted beam vanish. It is clear that under the kinematical ap-

proximation the incident beam has a constant amplitude, and therefore a constant

intensity which is usually taken to be I

=|ψ

= 1. It is convenient to also take

= 1.

The above assumptions can also be inserted directly into the DHW equa-

tions (6.1), and we ﬁnd

dψ

− 2π is

= iπ

iθ

. (6.4)

These equations (one for each set of lattice planes g) form the starting point for the

kinematical multi-beam theory.

Equation (6.3), which is the same for every diffracted beam in the multi-beam

case, can be solved with the following substitution:

= Se

2πis

which results in

iπ

iθ

−2πis

Integration from z = 0, with S(z = 0) = 0 (i.e. there is no diffracted amplitude at

the entrance surface) results in

S =

iπ

iθ

−2πis

dz, (6.5)

from which the solution follows:

= e

iθ

iπs

i sin



πs



. (6.6)

The kinematical diffracted intensity I

=|ψ

for a crystal with total thickness z

is given by

sin



πs







. (6.7)

350 Two-beam theory in defect-free crystals

Note that this expression can also be derived by means of the so-called ﬁrst

Born approximation. If we replace the wave function  on the right-hand side of

equation (5.5) by the incident plane wave 

, the resulting equation can be solved

for . The mathematics is similar to that for the dynamical theory, except for the fact

that the resulting differential equations for the diffracted beams are independent of

each other. For a detailed derivation and discussion of the kinematical diffraction

theory we refer to [Gev70]. Although it is intuitively clear that the kinematical

approximation must break down – the contrast variations in the experimental bright

ﬁeld observations in Section 4.6 demonstrate that the transmitted beam cannot have

constant intensity – it is instructive to take a closer look at the solution for the

diffracted intensity.

It is common practice to introduce a dimensionless parameter w, deﬁned by

w ≡ s

. (6.8)

The crystal thickness is often also expressed in units of the extinction distance, and

we introduce the following notation for the dimensionless thickness:

≡

. (6.9)

We will refer to z

as the reduced thickness. Equation (6.7) takes the following

form:

= π

sin

(πwz

)

(πwz

)

= π

sinc

(πwz

Apart from the constant prefactor π

, this expression has the same mathematical

form as the square of the Fourier transformed shape function for a thin foil, derived

on page 119. The deviation parameter s

takes on the role of the variable q

. The

meaning of the shape function now becomes clear: as we tilt the crystal, different

parts of the relrod intersect the Ewald sphere, and the diffracted intensity varies

according to (6.7).

The dimensionless parameters w and z

simplify the task of representing

the scattered intensity as a function of crystal thickness and orientation. For all

illustrations in this and the following sections, we will consider a wedge-shaped,

bent thin foil, similar to the top sketch in Fig. 6.2(a). For simplicity, we take the

curvature of the foil to be such that the excitation error s

varies linearly from

negative on the left to positive on the right. This is consistent with the orientation

of the diffracting plane indicated in the bottom part of Fig. 6.2(a). The lattice

planes are assumed to have a constant orientation with respect to the foil top and

bottom surfaces. The dimensionless parameter w is then taken in the range −4

to +4. The reduced thickness varies linearly from the wedge edge (z

= 0) to

6.3 The two-beam case: DHW formalism 351

0-4

(a) (b)

s< 0 s= 0 s> 0

Fig. 6.2. (a) Schematic of a bent wedge, indicating the regions of positive and negative

excitation error; (b) schematic of the (w, z

) parameter space used for several of the ﬁgures

in this chapter; (c) grayscale representation of the kinematical diffracted intensity for the

intensity range [0.0, 0.1] and (d) black regions indicate those parts of the parameter space

where the kinematical and dynamical theory differ by more than 5%.

the thickest part (z

= 6). The parameter ranges are shown in Fig. 6.2(b). We

are essentially making use of the column approximation, which we introduced in

Section 6.2.

Figure 6.2(c) shows the kinematical diffracted intensity I

for an array of

400 × 400 points (or columns) sampling the (w , z

) parameter space. The grayscale

ranges from intensity 0.0 (black) to 0.1 (white); all values larger than 0.1 are also

represented by white pixels. The value of 0.1 has no particular importance and only

serves to delineate the regions of parameter space where the diffracted intensity is

less than 10% of the (constant) transmitted intensity. Those regions of the (w, z

)

parameter space for which the difference between the kinematical expression for

and the correct, dynamical expression derived in the next section (equation 6.22)

is larger than 5% are represented by black pixels in the image in Fig. 6.2(d). It is

clear that the kinematical approximation is only valid (with less than 5% error) for

352 Two-beam theory in defect-free crystals

small reduced thicknesses z

and for large values of |w|, or, equivalently, far from

the Bragg orientation.

The kinematical theory is a poor approximation near the exact Bragg orientation

for nearly all thicknesses (as is obvious from the dark central band in Fig. 6.2d)

because the diffracted intensity is then proportional to the square of the reduced

thickness,

= 0) = π

, (6.10)

and this intensity can increase without bound, clearly an unphysical situation. It can

be shown that the thickness must be of the order of one-tenth of the extinction dis-

tance near Bragg orientation, otherwise dynamical corrections are needed [Gev70].

Note that the excitation error (or equivalently, the dimensionless parameter w) can

be interpreted as a parameter probing the relrod corresponding to the vector g; when

we tilt the crystal back and forth, i.e. change the value of s

, we are actually “rocking”

the reciprocal lattice point through the Ewald sphere. Any curve representing the

intensity as a function of w is therefore known as a rocking curve.

Summarizing, the kinematical theory is only valid when: (a) the crystal thickness

is very small and/or (b) the crystal is far from Bragg orientation. Note that within

the kinematical approximation there is no distinction between centrosymmetric and

non-centrosymmetric crystals, since the phase θ

does not appear in the ﬁnal inten-

sity expression. The kinematical theory can also be extended to include absorption

effects and it can be shown that this does not substantially alter the validity range in

the (w, z

) parameter space. A correct theory for thicker samples and orientations

close to the Bragg orientation can only be derived by allowing multiple scattering

events for each electron, or, equivalently, by avoiding the ﬁrst Born approximation.

This will be done in the following sections.

6.3.3 The two-beam dynamical theory

6.3.3.1 General derivation

The terms on the right-hand side of equations (6.1) are of two different types: one

term is real and depends on the normal absorption length ξ



and the other term

represents the interaction with the other beam. We can remove the dependence

on the normal absorption length from the equations (as we did on page 311) and

simultaneously render them more symmetric by means of the following substitution:

= T (z)e

αz

; (6.11)

= S(z)e

iθ

αz

, (6.12)

6.3 The two-beam case: DHW formalism 353

with

α = π i





+ s



=−



+ π is

. (6.13)

After some simple mathematics we arrive at the following ﬁnal equations:

+ π is

T =

iπ

−g

S; (6.14)

− π is

S =

iπ

T. (6.15)

These are the dynamical two-beam equations, including absorption, for a perfect

crystal of arbitrary symmetry. For a centrosymmetric crystal, the factor q

−g

equation (6.14) is replaced by q

Since one can only observe intensities in an experiment, we have from equa-

tions (6.11) and (6.12):

bright ﬁeld → I

=|ψ

=|T |

−(2π/ξ



; (6.16)

dark ﬁeld → I

=|ψ

=|S|

−(2π/ξ



. (6.17)

In the following sections we will consider the solutions of equations (6.14) and

(6.15), and write them in a form that is suitable for the numerical computation of

image contrast.

6.3.3.2 Solutions of the dynamical two-beam equations

We will derive the solutions to the dynamical two-beam equations for the most

general situation, including absorption. Then we will make various approximations

to obtain analytical expressions for the transmitted and diffracted intensities.

From equation (6.14) we ﬁnd

S =

−g

iπ

+ s

−g

T ;

and

−g

iπ

+ s

−g

Substitution in equation (6.15) and rearranging terms leads to

+ π

T = 0, (6.18)

with

σ =

−g

. (6.19)

354 Two-beam theory in defect-free crystals

This is the differential equation for a harmonic oscillator and the general solution

is given by

T (z) = A cos(πσz) + B sin(πσz).

From the boundary condition at the entrance surface we have T (z = 0) = 1 and

hence A = 1. The coefﬁcient B can be found by computing dT/dz, substituting

into the equation for S above, and applying the boundary condition S(z = 0) = 0,

which leads to B =−is

/σ , and thus

T (s, z) = cos(πσz) −

sin(πσz); (6.20)

S(s, z) =

sin(πσz). (6.21)

These are the general solutions to the dynamical two-beam equations, including

absorption. Note that the parameter σ is complex. From these expressions we

can now compute the transmitted and diffracted intensities; ﬁrst, we will take the

simplest case without absorption, then we will derive the general expressions.

6.3.3.3 Dynamical intensities, no absorption

In the absence of absorption, the factors q

become real and equal to ξ

. In addition,

= ξ

−g

for all crystal structures, since the extinction distance depends only on the

modulus of the Fourier coefﬁcient of the electrostatic lattice potential, not on its

phase, and hence

= s



1 + w



where we have again introduced the dimensionless parameter w = s

. σ is now

a real number and it is then straightforward to compute the intensities for the

transmitted and diffracted beams for a crystal of thickness z

=|S|

sin



πσz



1 + w

sin



√

1 + w



1 + w

; (6.22)

= 1 − I

+ cos



πσz



1 + w

+ cos



√

1 + w



1 + w

. (6.23)

The last equality in both equations is included to facilitate comparison of these

expressions with the corresponding scattered intensity for the kinematical theory,

equation (6.7). The scattered intensity differs from the kinematical solution by the

fact that the denominator no longer vanishes upon approaching the Bragg condition,

and by the replacement of s

by σ in the argument of the sin-function. The trans-

mitted intensity is no longer constant; in the absence of absorption, I

and I

are

6.3 The two-beam case: DHW formalism 355

Bright field Dark field

(c)

(a)

(b)

= ∞

= 10

Fig. 6.3. Dynamical rocking curves (bright ﬁeld and dark ﬁeld): (a) in the absence of

absorption (inﬁnite absorption lengths); (b) in the presence of normal absorption; and (c)

in the presence of both normal and anomalous absorption. All computations are carried out

for the standard parameter

space.

complementary functions (i.e. they add up to 1) and hence bright ﬁeld and dark

ﬁeld images are complementary. For large deviations from the Bragg orientation

we can ignore factors of order 1 compared to w

(this includes the cos-function in

the transmitted intensity), and we recover equation (6.7) and the fact that I

= 1,

i.e. the kinematical expressions.

Figure 6.3(a) shows the dynamical bright ﬁeld and dark ﬁeld intensities for the

standard parameter space (w, z

) introduced in Figs 6.2(a) and (b). Note that both

356 Two-beam theory in defect-free crystals

bright ﬁeld and dark ﬁeld rocking curves are symmetric in w (because they only

depend on w

When normal absorption is included, the computations are identical to those

above, except that now the intensities are both multiplied by the normal absorp-

tion exponential, as shown in equations (6.16) and (6.17). The effect of normal

absorption on the image intensities is shown in Fig. 6.3(b), for ξ



= 10ξ

, or equiv-

alently, an absorption exponential equal to e

−2π z

/10

. The absorption length ξ



taken to be inﬁnite. It is obvious from the ﬁgure that the intensity oscillations de-

crease with increasing crystal thickness and that the overall intensity level decreases

exponentially.

6.3.3.4 Dynamical intensities including absorption

In this section, we will derive analytical expressions for the intensities of transmitted

and diffracted beams for the most general situation: a non-centrosymmetric crystal

with absorption. Approximate expressions, which are obtained by ignoring factors

of order O[1/ξ

2

], can be found in [GBA66]. We start from equations (6.20) and

(6.21). We can make use of the following relations (e.g. [AS77]):

2 cos z

cos z

= cos

(

− z

)

+ cos

(

+ z

)

;

2 sin z

sin z

= cos

(

− z

)

− cos

(

+ z

)

;

2 sin z

cos z

= sin

(

− z

)

+ sin

(

+ z

)

;

cos z = cosh iz;

sin z =−i sinh iz;

σ − σ

∗

= 2iσ

;

σ + σ

∗

= 2σ

where z

and z

are general complex numbers, and σ = σ

+ iσ

, to derive explicit

expressions for the transmitted and scattered intensities. The computations are

somewhat lengthy but straightforward, and we ﬁnd:

= e

−

2π



cosh

(

2πσ

)



1 +

|σ |



|σ |

sinh

(

2πσ

)

cos

(

2πσ

)



1 −

|σ |



−

|σ |

sin

(

2πσ

)

; (6.24)

−(2π/ξ



2|q

|σ |

[

cosh(2πσ

) − cos

(

2πσ

)

]

. (6.25)

6.3 The two-beam case: DHW formalism 357

The parameter |σ |

= σ

+ σ

is given by

|σ |









−

2



4 cos







The factor |q

can be computed from



+ i

iβ





− i

−iβ





2

−

2 sin β



It is veriﬁed easily that, in the limit of no absorption (σ

→ 0), both expres-

sions (6.24) and (6.25) converge to the simpler expressions (6.23) and (6.22). The

hyperbolic functions give rise to a thickness-dependent background intensity, onto

which a thickness-dependent oscillatory term is superimposed. Both intensities are

subject to normal absorption. The resulting bright ﬁeld and dark ﬁeld rocking curves

are shown in Fig. 6.3(c). Note the asymmetry of the bright ﬁeld image with respect

to the excitation error; the asymmetry increases with increasing foil thickness.

The explicit expressions for the transmitted and diffracted intensities are quite

useful for computer implementations: not every programming language has built-in

support for complex number arithmetic (in particular, the relations between trigono-

metric and hyperbolic functions of complex arguments), whereas equations (6.24)

and (6.25) no longer contain complex quantities and are straightforward to imple-

ment in any programming language.

Next, we will take a closer look at the symmetry of expressions (6.24) and (6.25).

6.3.3.5 Symmetry of the dynamical two-beam amplitudes

In the kinematical approximation, the probability of scattering from the transmit-

ted beam into a diffracted beam g is described by the inverse of the extinction

distance (i.e. by 1/ξ

), or, equivalently, by the Fourier coefﬁcient V

of the lattice

potential, which, in turn, is related to the structure factor F

. The intensity of the scat-

tered beam (equation 6.7) is proportional to the modulus squared of this structure

factor, or

=|F

= F

∗

It is easy to show that F

∗

= F

−g

, and therefore also F

= F

∗

−g

. Combining all

relations we ﬁnd

= F

∗

= F

∗

−g

= I

−g

. (6.26)

This equation states that a kinematical diffraction pattern always shows inversion

symmetry, even when the crystal is non-centrosymmetric. This is commonly known