Graef M. Introduction to conventional transmission electron microscopy

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398 Systematic row and zone axis orientations

(a) (b)

000 11.111.1

Fig. 7.2. (

11.1) systematic row CBED pattern for Ti acquired at 200 kV with a beam

convergence angle of 5.5 mrad. Pattern (a) is taken in the symmetric orientation and in

(b) the (

11.1) reﬂection is in the Bragg orientation.

sphere, indicating that there is a substantial probability for electrons in the incident

beam to be scattered into the beam 2g. How many beams are involved depends on

the acceleration voltage used for the observations and on the local orientation of

the foil. A higher voltage means a ﬂatter Ewald sphere, which increases the prob-

ability that many reciprocal lattice points are simultaneously close to the Ewald

sphere. Consequently, even a contrast feature as simple as a bend contour requires

a multi-beam computation rather than a two-beam computation.

This is also shown in the systematic row convergent beam pattern of Fig. 7.2.

This pattern was obtained using the (

11.1) systematic row of Ti at 200 kV. The

beam convergence angle is 5.5 mrad. It is clear that the symmetric orientation

shown in (a) would require at least three beams, whereas the Bragg orientation

pattern in (b) could be reasonably well approximated by a two-beam simulation. In

Section 7.2.2.7 we will see how systematic row CBED patterns can be simulated.

7.2.2 Theory and simulations for the systematic row orientation

7.2.2.1 The N -beam dynamical equations

When it comes to the computation of diffracted amplitudes, there is no intrinsic

difference between the systematic row case and the zone axis case (which we will

treat in Section 7.3). The crystal transfer matrix A, introduced in Chapter 5, has

one row entry for each contributing reciprocal lattice point g, so that we can simply

number all the contributing beams, starting from 1 for the transmitted beam.

In Section 5.4, we have seen that the Darwin–Howie–Whelan equations can be

written as

= iπ









δ(n − n



) +

1 − δ(n − n



)

n−n







, (7.1)

where n labels the individual beams. In matrix notation we have

= iAS. (7.2)

7.2 The systematic row case 399

A general multi-beam simulation program will then proceed along the following

lines.

(i) Setup. Read the crystal data ﬁle, compute all symmetry operations, compute all atom

positions, and determine the electron wavelength, corrected for refraction.

(ii) Diffraction geometry. Ask the user for the incident beam direction k and the foil normal

F; for a systematic row, also ask for g.

(iii) Determine contributing beams. For the systematic row case, these are simply multiples

of g. For the zone axis orientation, we will need an algorithm based on the symmetry

of the zone axis. Either way, we will end up with a list of N contributing beams.

(iv) Compute the off-diagonal part of A. We will use the subroutine

CalcUcg (introduced

in Chapter 2) to compute the real and imaginary parts of the off-diagonal elements of

A. The routine

CalcDynMat is provided to compute the dynamical matrix for either

the Darwin–Howie–Whelan approach or the Bloch wave approach.

(v) Beam orientations. Determine the range of incident beam directions, the number of

pixels in the ﬁnal image (this will determine the number of individual multi-beam

computations

to be performed), and the thickness of the foil (either constant thickness

or thickness gradient). For each incident beam direction, we will need to compute the

diagonal of A. We will also make use of symmetry relations to reduce the total number

of beam directions for which the computation will be carried out.

(vi) Solve the equations. Use your favorite solution method to obtain the amplitudes S

of all N beams, for each incident beam direction and/or foil thickness. Conversion to

bright ﬁeld and dark ﬁeld images is then straightforward.

In the following subsections, we will provide a more detailed description of the

most important steps and algorithms for the systematic row case. The reader may

download the program

SR.f90 from the website; this program implements various

solution methods for the general systematic row case, as described below.

7.2.2.2 The differential equation approach

We will use the numerical package

rksuite90, available from www.netlib.org,

to solve the system of ﬁrst-order differential equations. We have already used this

package when we discussed the

lens.f90 program in Chapter 3. The reader may

consult the source code of the

SRIntegrate routine in the SR.f90 program on the

website. Pseudo code for the systematic row case is shown in PC-17 . The calling

sequence of the

rksuite90 routines is as follows:

call setup(comm,t_start,YS,t_end,tol,thres,method='M', &

task='R',message=.FALSE.)

call range_integrate(comm,f,t_want,t_got,y_got,yderiv_got,

& flag=flag)

call statistics(comm,num_succ_steps=steps)

call collect_garbage(comm)

400 Systematic row and zone axis orientations

Pseudo Code PC-17 Solution of the dynamical scattering equations for the

N -beam systematic row case

SR.f90.

Input: PC-2

Output: multi-beam wave function

ask user for beam direction, foil normal, and g-vector

ask for extent of integration (z

and k

t,max

)

precompute off-diagonal part of dynamical matrix {

CalcUcg}

for each incident beam orientation do

complete diagonal of dynamical matrix {

Calcsg}

∗

integrate the system of equations

compute transmitted and scattered intensities for required thickness(es)

end for

prepare PostScript/TIFF output

The ﬁrst routine (setup) deﬁnes the integration parameters: starting thickness

(

t start), ending thickness (t end), initial values (YS), integration tolerance

(

tol), and error threshold (thres); the method (M) indicates that either fourth-

or ﬁfth-order Runge–Kutta integration will be used. The task parameter identi-

ﬁes that the next call will be to the

range integrate subroutine, and messages

are turned off, to prevent all but the most serious of errors and warnings being

shown. The

range integrate subroutine does the actual work: the variable f

is a function that implements the actual differential equation (7.2); t want is the

requested thickness,

t got the actual ending thickness, and the resulting vector is

returned in

y got, along with its derivative in yderiv got. The flag parameter

provides information about the success or failure of the integration. The remaining

two procedure calls are optional:

statistics can be used to retrieve statistical

information about the integration, and

collect garbage cleans up all dynam-

ically allocated memory. For more details on each of these routines we refer the

interested reader to the

rksuite90 manual [BG94].

7.2.2.3 The scattering matrix approach

The scattering matrix approach, introduced on page 312, uses a Taylor expansion

for the matrix S of the following form:

S(z

) = 1 +

∞



n=1

(iA)

7.2 The systematic row case 401

Table 7.1. Comparison of the computation times for the SR.f90 program for a

(2N + 1)-beam systematic row using the g

200

reﬂection of aluminum. The

microscope acceleration voltage is 200 kV, the beam orientation varies from

=−3g to +3g, and the foil thickness increases from zero to 200 nm. The

computational array consists of 512 × 512 pixels. All computations were

performed on a 666 MHz Compaq Digital UNIX workstation, computational

times are scaled relative to the ﬁve-beam (N = 2) Bloch wave computation.

Computational method N = 24 6 8

Runge–Kutta integration 61.1 211.7 483.6 1031.5

Runge–Kutta computation for S()3.013.629.960.0

Taylor

expansion

S 1.99.922.140.1

Bloch wave approach 1.01.42.64.1

For a small thickness z

= , the series expansion will converge after only a few

terms. The general term in the expansion can be written as

= A

n−1

, (7.3)

and this relation can be used in an efﬁcient algorithm (

CalcSMslice and SMloop)

implemented in the

SR.f90 program. The algorithm computes the term on the right-

hand side of equation (7.3) and multiplies it with itself until the largest change

in any element of A

is smaller than a preset threshold. The slice thickness  is

typically of the order of the unit cell size, i.e. a few tenths of a nanometer. The

larger the slice thickness, the more terms are needed in the Taylor expansion. The

wave function S at any thickness n can be computed by multiplication of S()

with the incident wave S(0).

A combination of the differential equation and scattering matrix methods is

also implemented in the

SR.f90 program; the scattering matrix for the ﬁrst slice

of thickness  is computed using the

rksuite90 routines instead of the Taylor ex-

pansion above, and then the resulting scattering matrix is multiplied with itself as

before. Table 7.1 compares the computation times of the pure differential equation

approach, the scattering matrix approach, and the mixed approach, along with the

Bloch wave approach to be discussed next. It is clear that the differential equation

approach is signiﬁcantly slower than any of the other methods.

7.2.2.4 The Bloch wave approach

The Bloch wave approach, using the

BWsolve routine introduced in the previous

chapter, is implemented in the program

SR.f90. In addition to the bright ﬁeld and

402 Systematic row and zone axis orientations

dark ﬁeld images, which obviously are identical to those computed with the other

three methods, the Bloch wave approach provides much more detailed information

about the scattering process. In Section 7.2.2.6 we will discuss the representation

of Bloch wave eigenvalues, excitation amplitudes, and Bloch wave intensities with

respect to the atom positions in the unit cell. First, we will analyze the symmetry

of the systematic row case.

7.2.2.5 Symmetry of the systematic row case

In Section 5.8.3, where we introduced the reciprocity theorem, we have seen that a

thin foil can have two different symmetries, type I or type II symmetry, correspond-

ing to a mirror plane through the center of the foil, parallel to the foil surfaces,

and an inversion center located in the middle of the foil, respectively. For the sys-

tematic row case the entire crystal is considered to be one-dimensional (only one

row of reciprocal lattice vectors is important), so that all crystals must have at least

type I symmetry in the systematic row orientation. This means that the diffracted

amplitude ψ

satisﬁes the following relation:

)

⊗, −

+ r, C

= ψ

)

⊗, −

− r, C

using the notation of Chapter 5. This in turn means that the diffracted beam g belongs

to the diffraction group 1

. The dark ﬁeld bend contour image must therefore have

this internal symmetry. If the crystal is non-centrosymmetric, then the −g dark

ﬁeld bend contour need not have the same intensity distribution as the +g contour,

provided both do not belong to the same family.

†

For a centrosymmetric space group the systematic row has, in addition to the

type I symmetry, also type II symmetry, which means that

)

⊗, −

+ r, C

= ψ

−g

)

⊗,

+ r, C

This, in turn, means that the diffraction group is 21

, so that the g and −g dark

ﬁeld bend contour images are identical.

A careful analysis of bend contour bright ﬁeld and dark ﬁeld images may in

principle be used to determine whether a crystal structure has inversion symmetry

or not. However, the reader should bear in mind that the corresponding asymmetries

in the dark ﬁeld intensities may be very small, so that they are not detected easily

†

In a non-centrosymmetric space group, the reciprocal lattice vectors g and −g, in general, do not belong to

the same family. It is possible, however, that certain families of planes contain both g and −g. As an example,

consider the non-centrosymmetric space group Pm (# 6): all reciprocal lattice vectors normal to the mirror

plane belong to families of the type {g, −g}, whereas all other vectors belong to families with only one single

member {g}.

7.2 The systematic row case 403

(a)

-g

-2g

(b)

Fig. 7.3. (a) Within the column approximation a bent foil with parallel incident illumination

is equivalent to a planar foil with conical incident illumination; (b) schematic illustrating

the use of the Laue center to determine the orientation of the incident beam.

in experimental images. Furthermore, experimental bend contours are rarely as

straight and perfect as simulated ones, so that it may become even more difﬁcult to

analyze small intensity differences between g and −g dark ﬁeld images.

7.2.2.6 Simulation examples

It is common practice to compute the excitation errors of the reﬂections in the

systematic row by tilting the incident wave vector, rather than changing the foil ori-

entation. Within the column approximation, these two descriptions are equivalent,

as shown in Fig. 7.3(a). It is much easier computationally to change the orientation

of the incident wave vector; this corresponds to specifying the location of the Laue

center, the projection of the center of the Ewald sphere onto the systematic row.

The Laue center position is measured in units of |g|, as shown in Fig. 7.3(b). If we

denote the component of k

along the systematic row by k

, then the reﬂection N g

is in exact Bragg orientation when k

=−

N g. The Laue center positions for the

two incident beam directions in Fig. 7.3(b) would then be −1.2 for the beam on

the left and 0.5 for the beam on the right. All the Fortran routines in this chapter

employ this method to determine the relative orientation of crystal foil and incident

beam.

Figure 7.4 shows the bright ﬁeld and ﬁve dark ﬁeld images for the (00.2) system-

atic row in titanium, for a microscope acceleration voltage of 200 kV, a thickness

varying from 0 to 200 nm, an incident beam tilt between k

=−3g and +3g and

a total of 11 beams (−5g ···+5g). The images were computed on a grid of 512 ×

512 pixels using the transfer matrix series expansion (

SR.f90 program). Since the Ti

structure is centrosymmetric, the ±g dark ﬁeld images are identical. The images are

404 Systematic row and zone axis orientations

00.0 00.2 00.4

00.6 00.8 00.10

Fig.

7.4. Titanium (00

.2)

systematic row for the parameters stated in the text. The arrows at

the top indicate the locations for which one of the reﬂections N g

00.2

is in Bragg orientation.

The images

are shown as optical densities, i.e. as

−log(I ), with I = 0.0001

corresponding

to white and I = 1.0 to black.

displayed as optical densities to bring out small intensity variations; this display

mode also makes it easier to compare them directly to electron negatives. It is

obvious that with increasing order of reﬂection, the dark ﬁeld bend contour con-

trast is conﬁned to a narrower region. The straight lines in the bright ﬁeld image

(arrowed) correspond to the locations where the ng

00.2

reﬂections are in exact Bragg

orientation. Those lines are known as Bragg lines [Ead92].

Figure 7.5 shows a comparison between the calculated bright ﬁeld and (00.2) dark

ﬁeld images, displayed as optical densities, and experimental images obtained on a

JEOL 2000EX microscope and displayed with inverted contrast (i.e. as negatives).

The simulated images used a perfect linear wedge-shaped foil, whereas the foil

shape for the experimental images is not a perfectly linear wedge. Instead, the

thickness increases faster than the theoretical wedge close to the edge of the foil,

resulting in more closely spaced fringes. There is good qualitative agreement be-

tween the two sets of images; in the dark ﬁeld image, there are several locations at

which two fringes merge, and these merge points can also be observed in the exper-

imental dark ﬁeld image (black arrows). The vertical white lines in the simulated

bright ﬁeld image, labeled 1 and 2, can also be found in the experimental image.

7.2 The systematic row case 405

(a) (b)

500 nm

Bright Field Dark Field

Fig. 7.5. Comparison of experimental and computed bright ﬁeld and (00.2) dark ﬁeld images

for a Ti foil (inverted contrast, same parameters as in Fig. 7.4, except for the maximum

thickness, which was taken to be 300 nm).

Because of the local shape of the foil, the lines in the experimental image are not

perfectly straight.

It is rather straightforward to compute systematic row bright and dark ﬁeld images

for a foil with an irregular shape. The only input parameters that are needed are

the local thickness and the local value of k

/g for each column (i.e. pixel) of the

image. Figure 7.6 shows an example of the results of such a simulation: the local

value of k

/g is shown as a contour plot in Fig. 7.6(a), along with a contour plot of

the local thickness in (b). The irregular shape on the lower right is a hole in the foil.

The other simulation parameters are: Ti foil, 200 kV, nine beams, g

00.2

systematic

row, Bloch wave formalism. The resulting bright and dark ﬁeld images are shown

in (c) and (e)–(l), respectively. The corresponding diffraction pattern is shown in

Fig. 7.6(d). The simulation was carried out for a grid of 512 × 512 columns, and

took about 2 min on a 666 MHz workstation. We leave it to the interested reader to

adapt the

SR.f90 program for the simulation of systematic row images similar to

those shown in Fig. 7.6.

406 Systematic row and zone axis orientations

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

000

002

002 004 006 008

002

004006008

000

Fig. 7.6. Example systematic row simulation for an irregularly shaped foil with a k

contour map (a) and thickness contour map (b). The simulation used a Ti foil, 200 kV,

nine-beam g

00.2

systematic row Bloch wave calculation. The bright ﬁeld image is shown in

(c), along with the diffraction pattern (d) and all eight dark ﬁeld images.

The Bloch wave approach has a signiﬁcant advantage over the other simulation

methods in that more detailed information about the scattering process is available.

Bloch wave eigenvalues !

( j)

and excitation amplitudes α

( j)

can be displayed as

a function of incident beam orientation, and the position of the individual Bloch

waves with respect to the crystal reference frame can also be determined. Let us

consider two examples: the g

202

systematic row of copper and the g

101

systematic

row of BaTiO

. In both cases we will use 200 kV electrons, nine beams, and the

incident beam orientation varies in the range −3 < k

/g < 3.

Figures 7.7(a) and 7.8(a) show the eigenvalues γ

( j)

of the dynamical matrix for

both Cu and BaTiO

systematic rows. An alternative representation of the eigen-

values in terms of k

( j)

− k

is shown in Figs 7.7(b) and 7.8(b). The eigenvalues

inside the ﬁrst Brillouin zone, which runs from k

/g =−0.5to+0.5, are repeated

periodically in the so-called extended zone representation. For the largest beam

tilts the periodicity is no longer obvious from the ﬁgure, and this indicates that the

computation did not take into account a sufﬁcient number of beams; the beam tilt is

so large that higher-order reﬂections are close enough to the Ewald sphere to give

7.2 The systematic row case 407

-1

-2

-3

- 4

-3 -2 -10123

-1

-2

-3

-4

-5

-3 -2 -10123

1.0

0.8

0.6

0.4

0.2

0.0

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

-3 -2 -1 0123 -3 -2 -10123

(j)

(nm

)α

(j)

- k

(nm

ⴛ10

(nm

)

(c)

(a) (b)

(d)

((

Fig. 7.7. (a) and (b) eigenvalues γ

( j )

of the dynamical matrix for pure Cu, 200 kV, nine-beam

202

systematic row simulation. (c) Bloch wave excitation coefﬁcients and (d) absorption

coefﬁcients (( j) = 1–5).

rise to signiﬁcant scattering. When we repeat the computation with a larger number

of beams the periodicity is fully restored.

The modulus of the Bloch wave excitation coefﬁcients α

( j)

is shown in Figs 7.7(c)

and 7.8(c). The excitation coefﬁcients are not periodic along the systematic row,

since they are determined by the boundary conditions at the entrance plane of the

crystal, i.e. by the incident beam direction. These curves are quite similar to the

ones for the two-beam case in Fig. 6.7(a): at the Bragg orientation of a particular

reﬂection the amplitude of two of the excitation coefﬁcients changes rapidly. In