Graef M. Introduction to conventional transmission electron microscopy
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468 Defects in crystals
e
x
m
e
y
m
e
z
m
=
e
z
f
=
e
x
f
=
e
y
f
(a)
e
x
m
e
y
m
e
z
m
(b)
+α
P
α
P
α
P
+α
S
α
S
α
S
e
z
f
e
y
f
e
x
f
e
x
m
e
y
m
e
z
m
(c)
+α
P
α
P
α
P
+α
S
e
z
f
α
S
α
S
e
x
f
e
y
f
Fig. 8.2. (a) Definition of the microscope and foil reference frames for the sample reference
orientation (no tilts); (b) sign conventions for positive rotations around the primary and
secondary tilt axes (for a double-tilt holder); and (c) similarly for a rotation holder.
The equations above define the transformation matrix α
fc
from the Cartesian crystal
reference frame (old) to the foil reference frame (new):
e
f
i
= α
fc
ij
e
c
j
. (8.9)
This matrix satisfies all the transformation rules described in Table 1.6 on page 51.
The next reference frame describes how the foil is oriented with respect to the
microscope column. There are a number of ways in which this reference frame can
be defined. We will take the z-direction of the microscope reference frame, labeled
with a superscript m, as opposite to the beam direction. The x-axis is taken to be
along the primary tilt axis, pointing towards the specimen airlock.
†
If the goniometer
tilt angles are equal to zero, i.e. the sample holder is in its reference orientation,
then we can define two rotation matrices, which express the microscope basis
vectors in terms of the foil basis vectors for a double-tilt and rotation-tilt holder,
respectively. Consider the reference frames shown in Fig. 8.2. In the reference
sample holder orientation, the microscope and foil reference frames coincide, as
shown in Fig. 8.2(a). For a double-tilt holder, the two rotation axes have positive
rotations (counterclockwise when viewing towards the origin), α
P
(primary axis)
and α
S
(secondary tilt angle). The sign conventions are shown in Fig. 8.2(b). Since
the primary tilt axis is attached to the microscope and the secondary tilt axis is
attached to the sample holder (i.e. changes orientation with the primary tilt), the
order of the two rotations is unimportant. The rotation matrix α
fm
connecting the
microscope and foil reference frames is given by
e
f
x
e
f
y
e
f
z
=
cos α
S
sin α
S
sin α
P
−sin α
S
cos α
P
0 cos α
P
sin α
P
sin α
S
−cos α
S
sin α
P
cos α
S
cos α
P
e
m
x
e
m
y
e
m
z
. (8.10)
†
This assumes that the microscope is equipped with a side-entry goniometer. For a top-entry goniometer a different
x-direction must be selected.
8.3 Numerical simulation of defect contrast images 469
A similar rotation matrix can be defined for a rotation-tilt holder. The secondary
rotation angle is defined as α
R
, measured from a suitable reference direction.
e
f
x
e
f
y
e
f
z
=
cos α
R
sin α
R
cos α
P
−sin α
S
cos α
P
−sin α
R
cos α
R
cos α
P
cos α
R
sin α
P
0 −sin α
P
cos α
P
e
m
x
e
m
y
e
m
z
. (8.11)
The resulting transformation matrix α
fm
is defined by
e
f
i
= α
fm
ij
e
m
j
. (8.12)
We will need an additional reference frame to describe the defect itself. The
nature of this reference frame, labeled d, depends on the defect, and may simply
be the crystal reference frame in the case of an isotropic defect. For instance, for a
straight dislocation it appears natural to select the line direction u to be one of the
reference directions, say, e
d
z
. For a planar fault we could select the normal to the
fault plane to be the e
d
z
-axis. Whatever the choice might be, we will need to define
another rotation matrix α
dc
, which transforms the Cartesian crystal reference basis
vectors to the defect basis vectors:
e
d
i
= α
dc
ij
e
c
j
. (8.13)
The final reference frame is that of the simulated image itself, which locates each
pixel in terms of the integer pair (i, j ). If we define the image basis vectors e
i
x,y
to
lie in the lower left-hand corner along the horizontal and vertical directions of the
image (i.e. of the electron micrograph), then the relation between the microscope
reference frame e
m
x,y,z
and the image can be expressed in terms of the angle β
P
between the primary tilt axis and, say, the bottom edge of the negative:
e
m
x
e
m
y
e
m
z
=
cos β
P
−sin β
P
0
sin β
P
cos β
P
0
001
e
i
x
e
i
y
e
i
z
.
primary tilt axis
β
p
e
y
i
e
x
i
j
i
micrograph
(8.14)
It may seem that the introduction of several reference frames makes things more
complicated than necessary. However, it is useful to incorporate the microscope
reference frame in the simulations, because it then becomes a trivial matter to
simulate what happens to the image contrast when the crystal foil is tilted by a
small angle around the primary or secondary tilt axis, which is what one would do
470 Defects in crystals
experimentally. Furthermore, the explicit introduction of the microscope reference
frame allows one to compute the image contrast for situations where multiple defects
in different orientations with respect to the crystal lattice are present in the field
of view. Each defect j would then be described in its own reference frame, the
origin of which is located at a position vector r
i
j
with respect to the main origin,
which is conveniently chosen to be the lower left-hand corner of the image. The
displacement fields due to all defects can then be added together (assuming a linear
elastic description holds), so that images of arrays of defects or overlapping defects
can be computed.
8.3.2 Example of the use of the various reference frames
A concrete example will illustrate the various coordinate transformations introduced
in the previous section. Consider a face-centered cubic material, which contains
a perfect edge dislocation with Burgers vector b =
1
2
[110] and line direction u =
[1
¯
12]. The dislocation lies in the (
¯
111) glide plane. The foil normal is taken to be
along F = [
¯
1
¯
13], and the foil thickness is uniform and equal to z
0
. The center of
the dislocation line (at the center of the foil) corresponds to the origin of the defect
reference frame e
d
i
, and the e
d
z
direction is taken to be parallel to the Burgers vector.
The geometry is shown schematically in the stereographic projection of Fig. 8.3.
For a cubic crystal, the Cartesian crystal reference frame coincides with the
Bravais reference frame, but is normalized, so that we have
e
c
i
=
a
i
a
,
with a the lattice parameter. The direct structure matrix is equal to a
ij
= aδ
ij
, and
the reciprocal structure matrix is given by b
ij
=
1
a
δ
ij
. The transformation matrix,
α
dc
, relating the crystal and defect reference frames, is derived from a definition
of the defect reference frame: the dislocation line direction is taken along the
e
d
z
-direction. The e
d
y
-direction lies in the plane formed by the beam direction B and
the line direction u, and e
d
x
completes the right handed reference frame. We leave it
as an exercise for the reader to show that, for the configuration of Fig. 8.3, we have
e
d
x
e
d
y
e
d
z
=
0.082 51 0.907 27 0.412 38
−0.909 13 −0.100 98 0.404 08
0.408 25 −0.408 25 0.816 50
e
c
x
e
c
y
e
c
z
. (8.15)
Each row is normalized, so that this matrix is a unitary matrix.
Next, we consider the orientation of the thin foil in the microscope column. We
will assume that the g
031
plane normal lies along the primary tilt axis, which itself
is at an angle of, say, β
P
=+96
◦
with respect to the e
i
x
-axis; i.e. the bottom edge
8.3 Numerical simulation of defect contrast images 471
[]
[111]
[11 1]
[111]
[011]
[101]
[101]
[011]
[
1
1 3
]
[122]
[212]
[122]
[21 2]
[ 122]
[
212]
[12 2]
[211]
[211]
[121]
[112]
[112]
[11 2]
[112]
[ 021]
[ 102]
[102]
[ 201]
[012]
[012]
[001]
B
[32
5
]
F
u
g
111
P
[ 031]
primary tilt axis
secondary tilt axis
bottom edge micrograph
β
p
= 96
α
P
= 11
α
S
= 3
Fig. 8.3. Partial stereographic projection of a cubic structure, projected along the foil normal
F. Other vectors indicated are: dislocation line direction u, diffraction vector g, and the beam
direction B. The tilt angles α
P
and α
S
, along with β
P
, are also indicated.
of the computed micrograph. This means that
α
mi
=
−0.104 528 −0.994 522 0
0.994 522 −0.104 528 0
001
. (8.16)
The foil reference frame transformation matrix α
fc
is given by
α
fc
=
0
3
√
10
1
√
10
−10
√
110
1
√
110
−3
√
110
−1
√
11
−1
√
11
3
√
11
, (8.17)
with the e
f
y
-direction along [
¯
10 1
¯
3]. Again the middle row follows from the cross
product between the third and first rows, and all rows are normalized. Both defect
and foil reference frames are now completely defined.
472 Defects in crystals
Assume, next, that the foil is mounted in a double-tilt holder, and that an image
is obtained for the orientation α
P
= 11.6
◦
, α
S
= 3
◦
.
†
The following questions must
then be answered as part of the image simulation.
r
What is the length of each image column (referring to the column approximation)? This
length determines the integration interval for the dynamical equations.
r
What is the defect displacement field along the centerline of the column (i, j)?
r
What is the derivative dR/dz along an arbitrary direction in the defect reference frame
(for the case of a defect with a continuous displacement field)?
The first question is easy to answer: the foil normal e
f
z
is rotated by the angles
α
P
and α
S
, and, therefore, the crystal thickness measured along the beam direction,
which is described by the vector e
m
z
,isgivenby
z
b
=
z
0
e
f
z
· e
m
z
=
z
0
cos α
P
cos α
S
= 1.022z
0
.
A point along the column corresponding to pixel (i, j ) has coordinates (Li, Lj, z)
with respect to the main origin (lower left-hand corner of the simulated image). The
defect origin expressed in the image reference frame is given by (Li
d
, Lj
d
, z
d
); the
components i
d
and j
d
are not necessarily integers. The transformation matrix from
image space to defect space, α
di
, is given by the product
α
di
= α
dc
α
cf
α
fm
α
mi
, (8.18)
and, since all matrices are known, the product can be worked out. The matrix α
cf
is equal to the transpose of α
fc
, since rotation matrices are unitary matrices. The
result is
α
di
=
0.007 94 0.999 97 0.000 00
−0.596 01 0.004 73 0.802 96
0.802 93 −0.006 38 0.596 03
.
Now that we have the transformation matrix relating the image and defect basis
vectors
e
d
k
= α
di
kl
e
i
l
,
we can apply the transformation rules listed in Table 1.6 to convert image space
coordinates to defect space coordinates. A vector with components r
i
k
in im-
age space is converted to defect space
‡
by post-multiplication with the inverse
†
These tilt angles will bring the beam direction B = [
¯
3
¯
25] to the center of the stereographic projection in Fig. 8.3.
‡
Remember that the superscripts indicate the reference frame with respect to which the components are
expressed.
8.3 Numerical simulation of defect contrast images 473
matrix (α
di
)
−1
= α
id
,or
r
d
l
= r
i
k
α
id
kl
. (8.19)
The column (Li, Lj, z) in image space, corresponding to pixel (i, j) of the simulated
image, corresponds to the straight line of length z
b
between the top (t) and bottom
(b) points with defect space coordinates
x
d
t
, y
d
t
, z
d
t
=
)
L
(
i − i
d
)
, L
(
j − j
d
)
,
z
b
2
*
α
id
;
x
d
b
, y
d
b
, z
d
b
=
)
L
(
i − i
d
)
, L
(
j − j
d
)
, −
z
b
2
*
α
id
.
Finally, it is rather easy to derive an explicit expression for the derivative dR/dz
at an arbitrary location r along a unit direction vector
ˆ
k
d
expressed in the defect
reference frame. This direction vector is parallel to the incident beam direction,
which is given by [00
¯
1] in the image reference frame, or, more conventionally, by
the components k
∗
m
in the crystal reciprocal reference frame. In the defect reference
frame the incident beam direction can be expressed by first transforming the com-
ponents k
∗
m
to the crystal Cartesian reference frame, using the reciprocal structure
matrix b
im
, and subsequently transforming to the defect reference frame using the
matrix α
dc
:
k
d
j
= α
dc
ji
b
im
k
∗
m
. (8.20)
The incident beam direction is then described by the unit vector
ˆ
k
d
.
The directional derivative of a vector field R(r) along the direction
ˆ
k
d
is given
by [BT79, page 165]:
dR(r)
dz
=
ˆ
k
d
·∇
d
R(r), (8.21)
where the differential operator ∇
d
operates in the defect reference frame. This
derivative can be rewritten in a form useful for numerical implementation:
dR
d
dz
= k
d
i
∂ R
d
j
∂x
i
#
#
#
#
#
r
e
d
j
. (8.22)
The dot products g · R and g · dR/dz require that the directional derivative be
expressed with respect to the direct space basis vectors a
m
, which can be computed
using the inverse of the direct structure matrix (see Chapter 1):
dR(r)
dz
= k
d
i
∂ R
d
j
∂x
i
#
#
#
#
#
r
α
dc
jl
b
lm
a
m
. (8.23)
474 Defects in crystals
where we have used the fact that b
T
= a
−1
. The dot products of g and R or its
derivative along the beam direction are then given by
g · R(r) = R
d
j
(r)α
dc
jl
b
lm
g
m
;
g ·
dR(r)
dz
= k
d
i
∂ R
d
j
∂x
i
#
#
#
#
#
r
α
dc
jl
b
lm
g
m
.
This completes the computation of the defect geometry. We can now proceed
with the solution of the dynamical equations for every image column (i, j); it is
clear from the above discussion that we will need to have an efficient algorithm to
compute the displacement vector R or its derivative at any point along the column
center line. Depending on the defect being considered, this may be an analytical
expression, or perhaps a discrete set of displacement vectors obtained from a finite
element computation.
If more than one defect is present in the field of view, then the image contrast
due to the combined defects may be calculated. In this case, we must change the
definition of the defect origin to coincide with one of the defects, and introduce a
translation vector relating each defect to the one at the origin. This is not difficult and
is left as an exercise for the reader. If the foil thickness is position dependent, then
the top and bottom points of each column may be different, and the computation
becomes slightly more involved; this is again left as an exercise.
8.3.3 Dynamical multi-beam computations for a column
containing a displacement field
In this section, we use the modified DHW equations as a starting point for the
numerical simulation of defect contrast. Equations (8.5) are valid for the general
multi-beam case, and for any defect. For numerical purposes, it is useful to distin-
guish two cases.
r
Displacement field R(r) is not differentiable
The following substitution can be used to rewrite the modified DHW equations in a form
useful for numerical computations:
ψ
g
= S
g
e
iθ
g
e
αz
with α =−
π
ξ
0
. (8.24)
A simple computation then results in
dS
g
dz
= 2πis
g
S
g
+ iπ
g
e
−iα
g−g
(r)
q
g−g
S
g
, (8.25)
where the prime on the summation indicates that the term g
= g is excluded.
8.3 Numerical simulation of defect contrast images 475
r
Displacement field R(r) is differentiable
We have already derived the relevant equation in Section 8.2; if we also include
substitution (8.24), then the resulting equations are given by
dS
g
dz
= 2πi
s
g
+ g ·
dR(r)
dz
S
g
+ iπ
g
1
q
g−g
S
g
. (8.26)
Now we are set to compute the image contrast for any arrangement of defects,
with mixed differentiable and non-differentiable displacement fields.
†
A simple
example of such a combination would be a pair of partial dislocations (continuous
and differentiable R) separated by a stacking fault (discontinuous R). All continuous
displacement fields contribute to the dR/dz part of the effective excitation error,
while the discontinuous displacement fields contribute to the right-hand side of the
equation in the phase shift term. The modified DHW equations for a configuration of
N
c
defects with continuous displacement fields and N
d
defects with discontinuous
displacement fields are then given by
dS
g
dz
=
2πis
g
+ i
dα
c
g
(r)
dz
S
g
+ iπ
g
e
−iα
d
g−g
(r)
q
g−g
S
g
, (8.27)
with
α
c
g
(r) ≡ 2πg ·
N
c
j=1
R
j
(r);
α
d
g
(r) ≡ 2πg ·
N
d
k=1
R
k
(r).
In most defect image contrast simulations, only one single defect must be consid-
ered, and the equations above are simplified somewhat. It is useful, however, to
have the general equations, because they open the way to the simulation of more
complex defect images, such as faults which change orientation, or intersecting
faults. In the remainder of this chapter, we will make use of the equations derived
above to compute the image contrast of a number of defects. We will introduce
analytical expressions for defect displacement fields whenever they are available.
In some cases, analytical solutions to the modified DHW equations are possible
for the two-beam case, in particular for planar defects such as stacking faults and
anti-phase boundaries. We will see that the crystal transfer matrix formalism intro-
duced in Chapter 5 is useful to compute rapidly the contrast of defects and defect
configurations.
†
It is assumed throughout that linear elasticity theory is valid, which means that the total displacement due to an
arrangement of defects is equal to the sum of the individual displacements.
476 Defects in crystals
The solution to equation (8.27) can always be obtained numerically by means
of an appropriate differential equation solver. We have seen in Chapter 7 that such
a solution method is typically slower than other algorithms, such as the scattering
matrix approach or the Bloch wave formalism. It is possible, however, to use the
scattering matrix formalism for the systematic row case in the presence of defects,
and the algorithm is described in pseudo code
PC-19 . Since most defect contrast
observations are carried out for systematic row conditions, there is no real need for
a defect simulation program for the zone axis case.
The simulation method relies on the fact that all defects, with continuous or
discontinuous displacement fields, give rise to phase shifts of the Fourier coefficients
of the electrostatic lattice potential and that such phase shifts are periodic functions.
This effectively means that it is not the magnitude of g · R that matters, but only
g · R mod 1. Consider the phase factor
θ
gg
(r) ≡ e
−2πi(g−g
)·R(r)
.
For a systematic row, all vectors g are multiples of a single reciprocal lattice vector
G, and we can write
θ
nn
(r) = e
−2πi(n−n
)G·R(r)
.
Since n and n
are integers, it is easy to see that only values G · R mod 1 are of
importance and this suggests that we could precompute the possible phase factors
θ
nn
and employ the scattering matrix formalism to implement systematic row defect
simulations. This would include two-beam and weak-beam (g–3g) simulations.
The scattering matrix formalism relies on the crystal transfer matrix A, as defined
in Chapter 5. In the presence of a defect the crystal transfer matrix depends on
location in the crystal. In particular, only the off-diagonal members of A depend
on position:
A
nn
=
π
q
nn
θ
nn
(r).
From A we can compute S() for a thin slice of thickness , using the Taylor
expansion algorithm described in Section 7.2.2.3. We can repeat this computation
for an array of values for the dot product g · R. The memory requirements for
storing, say, N = 10 000 scattering matrices are not unreasonable: for a nine-beam
systematic row the total number of complex numbers that need to be stored is 9
2
×
10 000, which amounts to about 13 Mbytes of storage space for double-precision
arithmetic (16 bytes per complex number).
The selection of the appropriate scattering matrix is straightforward: for a
given location in the foil the sum of all local defect displacement vectors R(r) =
-
M
i=1
R
i
(r) is computed. The number of the corresponding scattering matrix is
8.3 Numerical simulation of defect contrast images 477
then given by the integer nearest to N (G · R mod 1). The program SRdefect.f90,
available from the
website, implements the entire algorithm. The number of slices
needed for each column will depend on how fast the displacement field varies along
the column; the simplest approach to determining the number of necessary slices
would be to repeat the computation for half the slice thickness and compare the
resulting images. If there is no significant difference between the two images, then
the larger slice thickness is sufficiently small. All systematic row defect image
simulations in this chapter were performed with the
SRdefect.f90 program.
Pseudo Code PC-19 Systematic row defect contrast computation.
Input: PC-2 , PC-3 , elastic data
Output: bright field and dark field defect contrast images
ask for microscope voltage and systematic row indices
define slice thickness, beam orientation
define foil and defect configuration
initialize the perfect crystal dynamical matrix
for each value of G · Rdo
compute and store scattering matrix S()
end for
for each image pixel do
determine contributing defects
initialize incident wave
determine the number of slices
for each slice in column (top to bottom) do
compute total G · R mod 1
select appropriate slice matrix and multiply wave function
end for
convert to intensities
end for
display bright field and dark field images
It is straightforward to include a bent foil shape in this systematic row simulation
by including an additional displacement field. A bent foil can be described by a
position-dependent excitation error s
G
(x, y). If we introduce a displacement field
of the form
R(r) = zs
G
(x, y)G
∗
, (8.28)