Graef M. Introduction to conventional transmission electron microscopy

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378 Two-beam theory in defect-free crystals

Table 6.2. Modulus and phase of the real and imaginary part of the electrostatic

lattice potential and several derived parameters for a few low index planes in

face-centered cubic materials. All parameters were computed using the

Weickenmeier–Kohl scattering factor parameterization for 100 kV electrons. The

Debye–Waller factors are (at 270 K , in nm

): B

= 0.007 194,

= 0.005 073,B

= 0.005 714,B

= 0.005 844,B

= 0.006 288

(sources [VC85, PRDW96, GP99]).



/ξ

1/q

hkl (V) (V) (rad) (nm) (nm) (nm

−1

)

Al 000 20.270 0.895 – 454.0 – i0.002 20

111 7.051 0.238 57.6 1703.729.60.017 35 + i0.000 59

200 5.828 0.216 69.7 1882.627.00.014 34 + i0.000 53

220 3.534 0.167 115.0 2428.421.10.008 69 + i0.000 41

311 2.767 0.147 146.9 2758.518.80.006 81 + i0.000 36

222 2.584 0.142 157.3 2862.318.20.006 36 + i0.000 35

400

.049

.124

198

.3 3265.41

.50.005

+ i0.000

331 1.773 0.114 229.3 3566.715.60.004 36 + i0.000 28

Cu 000 26.965 1.994

– 203.8 – i0.004 91

111 13.49 0.966 30.1 420.614.00.033 19 + i0.002 38

200 11.94 0.908 34.0 447.713.10.029 37 + i0.002 23

220 8.556 0.761 47.5 533.811.20.021 05 + i0.001 87

311 7.098 0.689 57.3 589.910.30.017 46 + i0.001 70

222 6.707 0.668 60.6 608.110.00.016 50 + i0.001 64

400 5.443 0.597 74.7 680.89.10.013 39 + i0.001 47

331 4.722 0.552 86.1 736.38.50.011 62 + i0.001 36

Au 000 35.734 4.300

– 94.5 – i0.010 58

111 22.26 3.516 18.3 115.66.30.054 77 + i0.008 65

200 19.93 3.403 20.4 119.45.90.049 05 + i0.008 37

220 14.38 3.060 28.3 132.84.70.035 38 + i0.007 53

311 11.97 2.859 33.9 142.24.20.029 46 + i0.007 03

222 11.34 2.798 35.8 145.34.00.027 89 + i0.006 88

400 9.317 2.579 43.6 157.63.60.022 92 + i0.006 34

331 8.182 2.433 49.7 167.03.40.020 13 + i0.005 99

GaAs 000 18.116 1.178

– 345.1 – i0.002 90

111 8.348e

−i0.726

0.517e

−i0.729

48.7 785.416.10.020 55 + i0.001 27

200 0.659 0.040 616.4 10136.916.40.001 62 + i0.000 10

220 7.645 0.615 53.2 661.312.40.018 81 + i0.001 51

311 4.559e

i0.750

0.407e

i0.726

89.1 999.611.20.011 24 + i0.001 00

222 0.191 0.034 2122.4 12103.25.70.000 47 + i0.000 08

400 5.201 0.523 78.1 776.39.90.012 80 + i0.001 29

331 3.315e

−i0.771

0.354e

−i0.726

122.6 1149.46 9.40.008 12 + i0.000 87

6.5 Numerical two-beam image simulations 379

Table 6.3. Modulus and phase of the real and imaginary part of the electrostatic

lattice potential and derived parameters for several non-cubic materials. All

parameters were computed using the Weickenmeier–Kohl scattering factor

parameterization for 100 kV electrons, except for the BeO data which was

computed for 80 kV. The Debye–Waller factors are (at 270 K , in nm

= 0.004 (estimated), B

= 0.004 (estimated) in BaTiO

and 0.004 735 in

pure Ti, B

= 0.002 (estimated) in BaTiO

and 0.0028 in BeO, B

= 0.007 844,

= 0.006 787,B

= 0.003 55. (sources [VC85, PRDW96, GP99, SZ92]).



/ξ

1/q

hkl (V) (V) (rad) (nm) (nm) (nm

−1

)

BaTiO

000 29.336 1.494 – 272.0 – i0.003 68

111 6.237e

i0.108

0.430e

i0.044

65.2 944.914.50.015 41 + i0.001 06

200 10.525 0.616 38.6 659.717.10.025 90 + i0.001 52

220 7.240 0.527 56.1 770.813.70.017 82 + i0.001 30

311 3.211e

i0.102

0.333e

i0.035

126.5 1220.19.60.007

+ i0.000

222 5.439e

i0.010

0.472e

−i0.023

74.7 861.211.50.013 42 + i0.001 16

400 4.380 0.434 92.8 936.610.10.010 78 + i0.001 07

331 2.109e

i0.100

0.285e

i0.033

192.7 1426.77.40.005 24 + i0.000 70

Ti 000 28.334 1.086

– 374.3 – i0.002 67

11.0 −5.825 −0.230 69.8 1770.925.40.014 33 − i0.000 56

11.06.320 0.340 64.3 1196.118.60.015 55 + i0.000 84

2.0 −2.600 −0.155 156.3 2625.016.80.006 40 − i0.000 38

11.18.820 −0.372 46.1 1091.323.70.021 70 + i0.000 92

22.14.274 0.262 95.1 1553.316.30.010 52 + i0.000 64

00.2 −10.649 −0.440 38.2 924.724.20.026 20 + i0.001 08

11.23.823 0.187 106.3 2178.720.50.009 41 + i0.000 46

ZnS 000 16.248 1.106

– 367.6 – i0.002 72

11.0 −5.594 −0.295 72.7 1377.718.90.013 76 − i0.000 73

11.06.853 0.467 59.3 870.914.70.016 86 + i0.001 15

2.0 −2.890 −0.216 140.6 1877.713.30.007 11 − i0.000 53

11.13.452e

−i0.562

0.274e

−i1.213

117.7 1485.212.60.008 90 + i0.000 54

22.11.940e

−i0.721

0.219e

−i1.267

209.5 1851.78.80.005 05 + i0.000 46

00.27.634e

−i0.716

0.446e

−i0.372

53.2 911.417.10.018 42 + i0.001 03

11.22.923e

−i0.701

0.198e

−i0.348

139.0 2056.814.80.007 02 + i0.000 46

BeO 000 20.169 1.614

– 223.2 – i0.004 48

[80 kV]

11.0 −4.788 −0.083 75.2 4317.257.40.013 29 − i0.000 23

11.04.809 0.090 74.9 3985.553.20.013 35 + i0.000 25

2.0 −1.924 −0.039 187.2 9205.149.10.005 34 − i0.000 11

11.12.975e

i0.037

0.068e

i0.433

121.1 5274.143.60.008 19 + i0.000 17

22.11.582e

i0.344

0.039e

i0.509

227.7 9204.740.40.004 37 + i0.000 11

11.22.273e

−i1.031

0.043e

−i1.191

158.5 8462.653.40.006 33 + i0.000 12

00.26.505e

−i0.918

0.120e

−i1.155

55.4 3002.054.20.018 13 + i0.000 32

Exp. 00.25.955e

−i0.885

0.110e

−i1.100

52.3 2857.154.60.019 19 + i0.000 34

380 Two-beam theory in defect-free crystals

we ﬁnd that this special solution (T



(s, z), S



(s, z)) can be written as



(

s, z

)

= S

(

−s, z

)

;



(

s, z

)

= T

(

−s, z

)

In other words, the expressions for transmitted and scattered amplitudes are

swapped, and the sign of the excitation error is changed. From equations (6.20)

and (6.21) we know that S(−s, z) = S(s, z), i.e. the dark ﬁeld rocking curve is

symmetric in the excitation error. We will denote T (−s, z)byT

(−)

, T (s, z)byT ,

and S(s, z)byS. The amplitude at the exit plane of a crystal with thickness z

(including absorption) is then given by





z=z

= e

−(π/ξ





TSe

−iθ

iθ

(−)





z=0

. (6.48)

In Section 5.4, we introduced the scattering matrix S. In the centrosymmetric two-

beam case we have

S(s, z

) ≡ e

−(π/ξ





TSe

−iθ

(−)



, (6.49)

and equation (6.48) reads as

(z

) = S(s, z

)(0). (6.50)

The scattering matrix S(s, z) can be written as

S(s, z) = S

(s, z) + iS

(s, z). (6.51)

It is easy to show that the explicit elements of the two matrices are given by

r,11

= e

−(π/ξ



cos(πσ

z) cosh(πσ

−

|σ |

[

sin(πσ

z) cosh(πσ

z) − σ

cos(πσ

z) sinh(πσ

]

;

r,22

= S

r,11

(−s, z);

r,21

−(π/ξ



|σ |

−



(σ

cos β

+ σ

sin β

)

sin(πσ

z) cosh(πσ

6.5 Numerical two-beam image simulations 381

−



(σ

sin β

− σ

cos β

)

cos(πσ

z) sinh(πσ



= S

r,12

;

i,11

=−e

−(π/ξ



sin(πσ

z) sinh(πσ

|σ |

[

sin(πσ

z) cosh(πσ

z) + σ

cos(πσ

z) sinh(πσ

]

;

i,22

= S

i,11

(−s, z);

i,21

−(π/ξ



|σ |

−



(σ

cos β

+ σ

sin β

)

cos(πσ

z) sinh(πσ

−



(σ

sin β

− σ

cos β

)

sin(πσ

z) cosh(πσ



= S

i,12

While these expressions look rather complicated, they actually contain many re-

peating factors which need only be computed once. The expressions above are

implemented in the subroutine

TBCalcSM; the source code has been written to have

the smallest possible number of computations and function evaluations. The routine

returns the real and imaginary parts of the two-beam scattering matrix S. A sec-

ond routine,

TBCalcdz, is provided which applies the transfer matrix to the column

vector 

and returns the updated column vector 

(z + dz) = S

(z). Finally,

the routine

TBCalcInten returns the intensities of the transmitted and diffracted

beams computed directly from equations (6.25) and (6.24). All images shown in

the next section were computed using either the intensity expressions or the matrix

formalism.

The main advantage of the matrix formalism is that matrix multiplication is

much faster computationally than numerically solving the dynamical diffraction

differential equations by Runge–Kutta-type algorithms. The matrix method is even

faster than the direct application of equations (6.25) and (6.24). As an example,

consider the computation of the transmitted and diffracted intensities as a function of

crystal thickness along the centerline of the standard parameter space, i.e. for w = 0.

The thickness varies from z = 0toz = 6ξ

. We can subdivide this interval into, say,

N equidistant steps dz = 6ξ

/N . Equations (6.25) and (6.24) would then require

evaluation of 2N trigonometric, 2N hyperbolic, and N exponential functions, a

potentially expensive computation in terms of processor clock cycles. The matrix

method on the other hand would evaluate the matrix S(0, dz) once (with only seven

calls to trigonometric, hyperbolic, and exponential functions), and then multiply

this matrix N − 1 times by itself to obtain the amplitudes for all thicknesses. A

possible drawback of the matrix method is that only multiples of the thickness

382 Two-beam theory in defect-free crystals

dz can be computed; for intermediate values we must either use a different value

of dz and recompute the matrix product, or interpolate intensities for thicknesses

bracketing the desired thickness.

The bright ﬁeld and dark ﬁeld images for the standard parameter space in Figs 6.3

and 6.4 were computed using the

TBSM.f90 program. This program implements

both the scattering matrix algorithm and the analytical solutions, and compares the

execution time of the algorithms for a BF–DF image pair of 512 × 512 pixels. The

scattering matrix approach is about ﬁve times faster than the direct evaluation of

the analytical solutions.

6.5.3 Numerical (two-beam) Bloch wave calculations

Numerical implementation of a general complex eigenvalue problem is a non-trivial

task, and is best left to mathematicians and experts in linear algebra. For all Bloch

wave computations in this book, we will make use of version 3.0 of the public

domain package

LAPACK, which is available from www.netlib.org for a large

number of computer platforms.

LAPACK is a library of Fortran-77 routines for solv-

ing the most commonly occurring problems in numerical linear algebra [ABB

99].

LAPACK is also available in Fortran-95, Java, and C++. Pre-compiled libraries for

DEC-ALPHA, Linux (RedHat), IBM RS/6000, Solaris, and Windows NT (Visual

Fortran) are available; alternatively, the package can be installed from scratch us-

ing the extensive installation instructions available from the

www.netlib.org

website.

†

LAPACK replaces the older LINPACK [DBMS79] and EISPACK [GBDM77] li-

braries, and makes extensive use of the

BLAS routines or basic linear algebra

subprograms [DDCHH88]. Documentation for

LAPACK is available on-line, or in

book form [ABB

99]. We will assume that the reader has access to the LAPACK

libraries, so that all Fortran source code for Bloch wave computations can be com-

piled and executed.

From a numerical point of view, the Bloch wave formalism requires the solution

of an eigenvalue problem for a complex non-symmetric matrix in the most general

case. Eigenvalues !

( j)

and eigenvectors C

( j)

must be computed, and the eigenvector

matrix C must be inverted to obtain the Bloch wave excitation amplitudes α

( j)

for

a given initial condition at the sample entrance surface.

The basic structure of a Bloch wave program is independent of the number of

beams in the calculation. We will, in fact, use only one subroutine for the two-beam

case, the systematic row case and the zone axis case (the latter two will be discussed

in the next chapter). The relevant

LAPACK routine is CGEEV, which computes the

†

The complete LAPACK package consists of about 805 000 lines of Fortran source code.

6.5 Numerical two-beam image simulations 383

Pseudo Code PC-14 Core of the Bloch wave computation.

Input: dynamical matrix

Output: eigenvalues and eigenvectors

initialize auxiliary arrays

compute eigenvalues and eigenvectors for complex non-symmetric matrix

{

CGEEV (LAPACK)}

rank eigenvalues from most positive to most negative; also rank eigenvectors in

same order {

SPSORT (SLATEC)}

compute LU decomposition of eigenvector matrix {

CGETRF (LAPACK)}

compute inverse of eigenvector matrix {

CGETRI (LAPACK)}

return eigenvalues, eigenvector matrix and its inverse

Pseudo Code PC-15 Outline of complete Bloch wave computation.

Input: crystal data ﬁle

Output: Bloch wave eigenvalues, eigenvectors, excitation amplitudes

ask user for range of contributing reciprocal lattice points

ask user for incident beam range

compute off-diagonal components of dynamical matrix {

CalcVcg}

for each incident beam orientation do

complete diagonal of dynamical matrix (excitation errors) {

Calcsg}

solve the eigenvalue problem {Pseudo Code

PC-14 }

compute amplitudes of all beams {Equation 5.58}

store relevant information in ﬁle

end for

eigenvalues and eigenvectors of an N × N complex non-symmetric matrix. The

eigenvector matrix is then inverted using the routines

CGETRF, which computes

the

LU decomposition of the matrix, and CGETRI, which computes the inverse of

the matrix using the

LU decomposition. The eigenvalues and eigenvectors are then

ranked so that the largest real part corresponds to γ

(1)

. The sorting can be done with

a variety of different algorithms, and we use the

SPSORT routine available in the

SLATEC library. The main ﬂow of the N-beam Bloch wave algorithm BWsolve is

shown in pseudo code

PC-14 .

†

The outline of a complete Bloch wave program is

shown in pseudo code

PC-15 .

†

If the LAPACK package is not available, any other eigenvalue solver may be substituted; the BWsolve subroutine

is then the only routine that must be modiﬁed.

384 Two-beam theory in defect-free crystals

Note that all the off-diagonal elements of the dynamical matrix

A can be pre-

computed, since they do not depend on the sample orientation. The imaginary part

of the diagonal contains only the normal absorption contribution and can also be

precomputed. The real part of the diagonal is computed for each sample (or incident

beam) orientation. After matrix inversion of the eigenvector matrix, the amplitudes

of all beams can be computed using equation (5.58).

For the two-beam case, we have an analytical solution for both the Bloch wave

and the DHW formalisms, so that a numerical Bloch wave solution is not really

needed. Nevertheless, the reader will ﬁnd the program

TBBW.f90 on the website.

The program produces an output ﬁle with eigenvalues, eigenvector matrices, and

excitation amplitudes for all Bloch waves and for each beam orientation. The pro-

gram

BWshow.f90 can then be used to produce bright ﬁeld and dark ﬁeld images,

along with plots of eigenvalues, excitation amplitudes, and absorption parameters

versus beam direction. The resulting drawings are similar to the ones shown in

Figs 6.6 and 6.7(a) and (c).

6.5.4 Example two-beam image simulations

In this section, we will discuss a number of different sample geometries and

illustrate the computation of two-beam bright and dark ﬁeld images. The reader

is encouraged to study and modify the Fortran source code used to compute the

images in Figs 6.11–6.14. The source code can be found in the ﬁle

TBBFDF.f90 and

is described in pseudo code

PC-16 . A second ﬁle, BFDF.routines, is available from

which the relevant lines can be selected and pasted into the

TBBFDF.f90 program

to simulate a number of different sample geometries. All images in the follow-

ing four subsections were obtained using the

TBBFDF.f90 program. The same image

Pseudo Code PC-16 Computation of two-beam bright ﬁeld and dark ﬁeld images

Input: crystal data ﬁle [*.xtal]

Output: PostScript ﬁle(s)

get acceleration voltage and operative reﬂection

compute excitation and absorption length {

Calcqgi}

compute thickness and excitation error arrays

for a given foil geometry {source code from

BFDF.routines}

compute intensities using analytical expressions {

TBCalcInten}

generate PostScript output

6.5 Numerical two-beam image simulations 385

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

w=0w=0 (DF) w=w=

-1

Fig. 6.11. Simulated bright ﬁeld images for the parameters stated in the text. (a) Bright ﬁeld

and (b) dark ﬁeld images for exact Bragg orientation; (c) bright ﬁeld image for w =+1,

and (d) w =−1. The effect of anomalous absorption is clearly seen in (d).

BF DF

Fig. 6.12. Simulated BF/DF pair for a bent wedge in aluminum for the same parameters

stated in Section 6.5.4.1.

simulation examples are also available in the interactive ION routine wedge.pro

on the website.

6.5.4.1 Wedge-shaped foil

Figures 6.11(a) and (b) show a bright ﬁeld/dark ﬁeld image pair for a straight

wedge in aluminum. The microscope acceleration voltage is 200 kV, and the 111

reﬂection is in the Bragg orientation. The thickness varies from 0 to 400 nm,

the extinction distance is ξ

111

= 73.1 nm, the anomalous absorption parameter is



111

= 2742.6 nm, and the normal absorption length is ξ



= 697.2 nm (all based

on the Weickenmeier–Kohl scattering and atomic form factors).

The bright ﬁeld image shows intensity fringes running parallel to the wedge edge.

The fringe spacing can be derived from equation (6.22) (ignoring absorption). In ex-

act Bragg orientation w = 0, the scattered and transmitted intensities are identical

386 Two-beam theory in defect-free crystals

BF DF

Fig. 6.13. Simulated BF/DF pair for a bent aluminum foil with a hole near the edge.

BF DF

Fig. 6.14. Simulated BF/DF pair for a titanium foil with average thickness 100 nm. Random

thickness variations are present, with negative thickness values corresponding to holes in

the foil. The dimensionless orientation parameter w varies from −1to+1 (left to right).

when z

increases from n to n + 1, which indicates that the periodicity of the

fringes is equal to ξ

. In other words, the crystal thickness increases by one ex-

tinction distance ξ

for every thickness fringe. This fact may be used to estimate

the local thickness of the foil. Care must be taken that the foil is indeed in exact

Bragg orientation, and that there are no other strongly excited reﬂections in the

diffraction pattern. Furthermore, the ﬁrst fringe near the edge may not correspond

6.5 Numerical two-beam image simulations 387

to z

= 1. Thicknesses obtained using this method are at best rough estimates of

the true thickness. We refer to Section 6.5.5 for a more accurate local thickness

determination procedure.

Since the fringes follow the contours of constant thickness, they are known as

thickness fringes. When the excitation error increases in magnitude, the fringe

spacing becomes smaller, since the periodicity is now determined by

√

1 + w

When w = 1, the effective extinction distance decreases by a factor of

√

2, and the

fringes are closer together, as seen in the simulated bright ﬁeld image of Fig. 6.11(c).

The effect of anomalous absorption can be seen clearly in the asymmetry between

the w =+1 and −1 (Fig. 6.11d) bright ﬁeld images.

6.5.4.2 Wedge-shaped bent foil

Figure 6.12 illustrates the BF–DF images for a bent foil with imaging parameters

stated in Section 6.5.4.1. This is the standard two-beam rocking curve discussed

earlier in this chapter; the reader may access the

rock.pro routine on the website to

display the rocking curve for a speciﬁc reduced thickness z

6.5.4.3 Wedge-shaped bent foil with hole near edge

During specimen preparation, one occasionally produces many small holes in the

foil, in addition to the main hole. Often those smaller holes are located near the

edge of the foil. Figure 6.13 illustrates how a circular hole with sample thickness

increasing radially away from the edge of the hole perturbs the thickness fringes

around the Bragg orientation. As stated for the ﬁrst example, the fringes represent

contours of constant

√

1 + w

. For a straight wedge, such contours would coin-

cide with constant thickness contours, but in the more general case of a bent foil, the

interpretation of the fringes is somewhat more difﬁcult. The imaging parameters

are identical to those for Fig. 6.12.

6.5.4.4 Planar foil with random thickness variations

The previous examples all dealt with wedge-shaped foils. With the modern speci-

men preparation techniques now available, we can often obtain large thin areas with

a nearly uniform thickness, and it becomes important to understand how random

thickness variations affect the image contrast. Using the

TBBFDF.f program, we

can explore the image contrast caused by such thickness variations by randomly

superimposing a series of cosine waves with random orientations and phases on the

average thickness.

Figure 6.14 shows a simulated BF/DF pair for a titanium foil with an average

thickness of 100 nm. The minimum thickness has been set to a negative value,

which effectively means that regions with a negative thickness are holes in the foil.