Transcript Convergent Beam Electron Diffraction & It’s Applications

Convergent Beam Electron
Diffraction & It’s Applications
John F. Mansfield
University of Michigan
Electron Microbeam Analysis
Laboratory
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Outline 1
Outline
Introduction to CBED
Applications Examples
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Outline 2
Outline
Introduction to CBED
Applications Examples
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Why CBED? Why not SAD?
Limits of Conventional SAD
Conventional SAD uses an aperture to define the area from which
the pattern is to be recorded. The aperture is placed in the image
plane of the objective lens to create a virtual aperture in the
specimen plane (Le Poole 1947). The spatial resolution in SAD is
limited by both spherical aberration and the ability of the operator
to focus the aperture of the and the image in the same plane. The
error in area selection U is given by:
U=Cs(2qB)3+D2qB
where:
Cs= spherical aberration coefficient
qB= Bragg angle
D= minimum focus step.
The result is that the theoretical lower limit of area
selection is ~0.5µm (in practice governed by aperture
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size).
Why CBED
J.B. Le Poole, Philips Tech. Rundsch 9
(1947) 33.
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Selected Area Diffraction
SAD Example
Parallel Electron Beam
Sample
Objective Lens
Diffraction Plane
Selected Area Diffraction
Parallel Illumination.
Lens Aberration limits
resolution to ~1 µm.
Silicon <100> zone axis
pattern. Recorded with
Gatan 622 wide-angle
CCD TV camera
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CBED
Example
Convergent Beam
Electron Diffraction
Convergent
Electron
Beam
Sample
Objective Lens
M23 X 6 <111> zone
axis pattern. Recorded
at 200kV with Gatan 622
wide-angle TV camera.
Diffraction Plane
Array of discs in diffraction plane,
c.f. spots in SAD.
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Angle of incidence maps on to position
with the diffracted discs
Points in discs
CBED Discs.
More information
than simple spot
patterns
Direction equivalent to
position of X in disc
Edge of cone of
illumination is
equivalent to the
aperture edge in
the pattern
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Convergence
Angle
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How to obtain a
CBED pattern
How to obtain
CBED patterns
1. Focus image at eucentric height.
2. Excite C1 to yield a small spot size.
3. Focus probe with C2 on to the area of
interest.
4. Press Diffraction button.
5. Optimize pattern.
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Poor PatternEven
Example
poor looking
patterns can be
useful!
[1123] Laves zone axis pattern.
Even poor quality ZAPs are often
useful in phase identification
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Learning the CBED Technique.
Begin with a straightforward
problem, a relatively
Learning CBED
large particle (~1-5µm) or a larger grain of material .
Learn to tilt around reciprocal space and recognize
the Kikuchi lines. Try and navigate around by
viewing the shadow image to save constant
switching from diffraction to image and back again.
Try and use a tilt-rotate holder. It may be initially
more difficult to use than a double-tilt holder, but it
allows access to a greater area of reciprocal space.
Take lots of pictures. Compared with the
time it will take you to get the zone axes
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that you need film is cheap (even grad
student time!).
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Electron Beam
Zone axis
of crystal
Sample, when
initially placed
Sample
into
microscope,
is usually at
Orientation
some random
orientation.
Sample
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Viewing the shadow image in CBED mode
1.
1. C2 heavily
under-focussed.
2. C2 approaching
focus.
2. 3. C2 at focus
(CBED Pattern).
4. C2 overoverfocussed.
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3.
4.
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Final pattern
Final Pattern Centering
centering is
Cone of
Illumination
Ideal alignment of
illumination and
zone axis
Actual orientation of
zone axis w.r.t
illumination
usually performed
by moving the C2
aperture a small
distance. This
effectively tilts the
beam through the
desired small
angle.
Sample
Centering of C2 with respect to the
zone axis of the crystal.
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Cr <111> XZAP
<111> Cr ZAP
Convergent Beam Electron
Diffraction (CBED) Pattern
recorded at 120kV.
The ring pattern and bright
spot at the centre of the
direct disc are characteristic
of a near critical voltage.
Pattern symmetry is 6mm
(projection symmetry).
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X-ray case, no special orientation
Sphere
X-ray Ewald
Ewald
X-ray Ewald Sphere
k'
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k
k''
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High Energy Electron Case.
High Energy Electron Ewald
Sphere
Very
Large Radius
Ewald Sphere.
k
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Ewald Sphere is a tangent to the
dotted line of reciprocal lattice points.
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HOLZ Example
NbSe3 CBED Pattern with
multiple HOLZ rings
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Formation of
HOLZ rings.
If H2  0
HOLZ spacings
G1  2kH
G2  2 kH
k=1/
G2
SOLZ
FOLZ
ZOLZ
H
Ewald
Sphere
G1
H
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ZOLZ or Zero Layer.
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FOLZ
Excess/DeficitHOLZ lines
Zero
Layer
Deficit in ZOLZ
Excess in HOLZ
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Deficit
Direct disc
Excess
HOLZ reflection
Excess
HOLZ reflection
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Direct Disc Si <111> ZAP
Si <111> ZAP
Direct disc of a silicon <111>
zone axis pattern (ZAP).
Pattern recorded at 200kV.
Sample cooled to in a liquid
nitrogen cold stage to reduce
the thermal diffuse scattering
and sharpen the higher order
Laue zone (HOLZ) line detail.
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Problems?
Contamination.
•Cool specimen (liquid N2 stage).
•Clean specimen (bake out, plasma etch).
•Handle all items that go into the microscope
vacuum with gloves, including film & plate holders.
•Keep specimen in vacuum desiccator
Problems getting patterns
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Problems?
More Problems?
No HOLZ lines visible.
Record long and short camera micrographs,
examine for any traces of HOLZ information.
Try an alternate zone axis, where layers may be
closer together.
Try a different (lower) accelerating voltage.
Examine specimen carefully for obvious defects.
Cool the specimen (reduce Debye-Waller factor).
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Microscope Related Criteria.
To do successful CBED one needs:
Microscope Related Criteria
A microscope capable of forming a small probe (~100 to
~2nm) with a large convergence angle.
Microscope design should be such that the diffraction plane
can be viewed out to at least 5° without interference from the
lens pole-pieces.
Sample stages that allow either tilting in 2 orthogonal
directions or tilting in one direction and rotation in the plane
of the specimen cup (the latter is to be preferred).
A microscope that has a very clean vacuum system,
preferably ion-pumped so that the probe may be focussed
on to the specimen to have the ability to vary the microscope
voltage continuously.
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Specimen Related Criteria.
To do
successful
CBED one
needs:
Specimen
Related
Criteria
Specimens that are, as far as possible, free from
hydrocarbon contamination. Remember contamination rate
proportional to 1/d2, where d is the probe diameter.
Wash specimens thoroughly to remove any kind of adhesive
used in preparation.
Always handle with tweezers.
Clean with Plasma Etcher/Bake-out.
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Outline 3
Outline
Introduction to CBED
Applications Examples
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CBED Applications
Examples
CBED Applications Examples
Phase Identification
Symmetry Determination - point &
space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter
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Measurement.
Structure Factor Determination.
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CBED Applications
Examples
Phase Identification
Phase Identification
Symmetry Determination - point &
space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter
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Measurement.
Structure Factor Determination.
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Partial ZAP Map of a Laves Phase particle
from an Al-SiC Composite
Laves Partial Zap Map
__
[2113]
_
[1012]
_
[1123]
_
[0111]
_
[2023]
_
[1011]
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Poor Quality Laves ZAP
[1123] Laves zone axis pattern.
Even poor quality ZAPs are often
useful in phase identification
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Stererogram for Laves
Standard Hexagonal
Stereogram
Patterns in question
were recorded from
the shaded area
[0001]
_
[1101]
_
[1012]
_
[0111]
_
[1011]
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Stereogram generated
by Diffract™
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Kikuchi Map of Laves Phase
CalculatedPartial
Kikuchi
Map for Laves
for comparison with experimental
CBED ZAP Map.
__
[2113]
_
[1012]
_
[0111]
_
[1123]
[¡ 5¢3]
_
[2023]
_
[1011]
Kikuchi Map
generated by Diffract™
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4000
Zn
Spectrum from a Laves Phase
embedded in an Al-SiC Composite.
Live Time=554 Secs
Laves XEDS spectrum
COUNTS
3000
Mg
2000
Zn
Cu
Al
1000
Zn
Si
P
Cu
Ar
0
0.000
1.000
2.000
3.000
4.000
5.000
6.000
ENERGY
7.000
keV
8.000
9.000
10.000
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Laves Particle
0.5µm
Laves phase precipitate in
the aluminium matrix of an
Al-SiC composite.
Particle was thick and
faulted, however, it was
easily analyzable by CBED.
Laves
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CBED Applications
Examples
Symmetry Determination
Phase Identification
Symmetry Determination - point &
space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter
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Measurement.
Structure Factor Determination.
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[001] Er2O3
Point Group
PointDetermination
group 1
m
m
m
m
m
m
2mm
2mm
ZOL
Z
2mm
WP
BF
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Pattern Symmetries
Buxton et al tables
Originally from
Buxton et al (1976)
Phil. Trans. Roy. Soc.
281, 181.
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Relation between the diffraction groups and
the crystal point groups
Buxton et al tables 2
Originally from Buxton
et al (1976) Phil. Trans.
Roy. Soc. 281, 181.
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[011] Er2O3
Point Group
PointDetermination
group 2
m
m
m
m
2mm
ZOL
Z
m
WP
Note the dark bar in this disc, it is important and will
be discussed later.
BF
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[111] Er2O3
Point Group
PointDetermination
group 3
3
6
ZOL
Z
3
WP
BF
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Er2O3
Point Group
Point GroupDetermination
Determination
Zone Axis
ZOLZ
WP
BF
Possible
Diffraction
groups
[001]
2mm
2mm
2mm
[011]
[111]
2mm
6
2mm or
2mm1R
2RmmR
6R
m
3
_
Point Group m3
m
3
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Possible reasons for Symmetry to deviate
Symmetry
Deviations
from
that which
is expected
Crystal Defects - Point defects, dislocations,
stacking faults
Element not in mid-plane
Glide or Screw out of surface
Probe smaller than unit cell
Heavily tilted sample
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Dynamic Absences
1
Dynamic Absences.
In thin samples
these occur as
missing or dim
reflections.
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Missing or "dim" reflections
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Dynamic AbsencesDynamic
2
Absences.
In thicker samples
double diffraction
puts intensity into
forbidden discs.
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Double Diffraction puts intensity into the forbidden
reflections, except along the lines where the the
intensities are equal and opposite.
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Dynamic
Dynamic Absences
3
Absences.
Dark bars form
along the centers
of the forbidden
discs .
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Dark Bars, Dynamic Absences,
Gjønnes-Moodie lines.
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Dark Bars
Dynamic Absences 4
Gjønnes-Moodie
Lines
Dynamic
Absences
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Dynamic Absences 5
Glide plane
parallel to
zone axis
Bright field m
Orientation of
absences with respect
to symmetry elements
indicates whether
space group operator
is screw axis or glide
plane.
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Screw axis
perpendicular
to mirror
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Dark Bars, Dynamic Absences
or Gjønnes-Moodie
Lines
What Dynamic
Absences
Mean
A bright field mirror line parallel to a
line of dynamic absences indicates
that there is a glide plane parallel to the
incident beam.
A bright field mirror line orthogonal to
a line of dynamic absences indicates
that there is a 21 screw axis or its
equivalent perpendicular the mirror
line. (The 41, 43, 61, 63 and 65 screw
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axes all include the 21 operation).
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[011] Er2O3
Bright field
mirror
(from
previous
pattern).
Space Group
Space
Group
Determination
Phase determined to be body
_ centered by analysis of the
033 [001] ZOLZ to FOLZ spacing.
_ Dark bars indicate a glide
011 parallel to the beam direction
in the [011] pattern.
Examination of tables in
Tanaka & Terauchi (1985)
reveals
_
the space group to be
Ia3.
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m
“Convergent Beam Electron
Diffraction”, Tanaka & Terauchi (1985),
JEOL Ltd.
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Dark Bars, Dynamic Absences or
Gjønnes-Moodie Lines
More
on
Dark
Bars
Dynamic absences only occur along the principal axes of a zone
axis pattern when the crystal is accurately aligned on a zone axis
direction.
Alternate reflections along a systematic row in a given layer MUST
all show the characteristic line of absence.
The lines of absence should become narrower as the thickness
increases.
The absences should occur for all thicknesses and microscope
accelerating voltages.
Upon satisfying the Bragg condition for any particular order that
contains an absence, a second line of absence will be observed
orthogonal to the first line (the “black cross”).
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This condition is only strictly obeyed in the ZOLZ
where three dimensional diffraction is weak.
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CBED Applications
Examples
Phase Fingerprinting
Phase Identification
Symmetry Determination - point &
space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter
EMAL
Measurement.
Structure Factor Determination.
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Phase Fingerprinting Example:
M23 X6 <110> ZAP
Certain ZAPs in alloy systems
are so characteristic that they
may be identified merely by
comparing them to a standard
"fingerprint". This phase,
based on a chromium carbide,
exists in a great variety of
compositions, however, the
pattern is always characteristic
enough to identify the phase.
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Phase
Fingerprinting Atlas
 Phase
Fingerprinting Atlas
Collection of ZAP,
ZAP Maps and
XEDS Spectra
from phases seen
in steels and
superalloys
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CBED Applications
Examples
Thickness Measurement
Phase Identification
Symmetry Determination - point &
space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter
EMAL
Measurement.
Structure Factor Determination.
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Theoretical Considerations for
Thickness Measurement
Thickness Measurement Theory
Convergent beam diffraction discs are maps
1 of incident
of diffracted intensity as a function
wave angle and therefore have a direct
correspondence to a rocking curve. In the
two-beam approximation the rocking curve
2
for the diffracted intensity  is given by
(Hirsch et al. 1965):
2
(1)
  sin2 kz
Where:
2
1
(s
)
1
g
1
) k 
  tan (
s g
g
s is the deviation parameter,g is the
extinction distance and z the foil thickness.
"Electron Microscopy of Thin Crystals",
Hirsch et al (1965).
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Differentiation with respect to s reveals that
the minima of  in equation (1) obey the
2
Thickness Measurement Theory
2
relationship:
kz  integer (2a)
and the maxima obey the relationship:
tan kz  kz (2b)
Also, s=0 is always either a maximum or
minimum. Kelly et al. (1975) expressed
equation (2a) as:
s
1 1
1
( i )2  ( ) 2 ( )2  ( )2
(3a)
nk
z
 nk
s
It is evident that a plot of ( i )2 against
nk
1
( )2 in a two-beam condition yields a
nk
1
straight line with intercept ( )2 and slope
z
1
of ( )2 . This is the basis of the CBED

thickness measurement technique that is
now well known.
P.M. Kelly et al.,
Phys. Stat. Sol.
(1975)A31, 771.
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In 1981, Allen noted that the equation 3a can
be rewritten thus:
si
(
xk'
Thickness Measurement Theory
3
1
1
1
)  ( ) ( )  ( )
(3b)
2
2

2
xk'
2
z
Equations 3a and 3b are equations of the
same straight line and the accuracy of the
thickness measurement can be nearly
doubled by the using both sets of fringes.
The subscripts of these two equations need
to be carefully noted since the values ofsi
are labeled separately for maxima and
minima.
The values of nk are a sequence of integers
and the values of xk are a sequence of non
integers.
S.M. Allen, Phil. Mag. (1981)A43 325.
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2
2
2
s 2
( n i ) =-( 1 ) ( n1 ) +( 1
)
z
k
k

Thickness Measurement Theory
Graph
Points for minima open squares
Intercept
1
z2
Schematic
of
Slope = - 1
2
thickness plot

2
2
2
s 2
( x i ' ) =-( 1 ) ( x1 ' ) +( 1
)
z
k
k

Points for the maxima filled circles
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2
2
2
s 2
( x i ' ) =-( 1 ) ( x1 ' ) +( 1
)
z
k
k

Thickness Measurement Theory
Diagram
Equation for sequence of
maxima where:
 i
s i = 12
d 2 d
 d
 4
 3
 2
 1
1
2
3
4
5
2
2
2
s 2
( n i ) =-( 1 ) ( n1 ) +( 1
)
z
k
k

Equation for sequence of
minima where:
s i = 12 i
d 2 d
Schematic
of
measurements needed
for
thickness plot
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Thickness Measurement in NIH
Image 1
•Two-beam convergent beam electron diffraction
pattern acquired into a modified version of NIHImage.
•Extra code added to the application is accessed
from the custom “Thickness” menu.
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entry
Thickness Measurement in•Data
NIH
dialog allows
the user to
Image 2
enter the
information
necessary to
determine the
thickness.
•Default
supplied
values are for
Silicon 220 at
200kV.
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Thickness Measurement in NIH
Image 3
•It is necessary to identify the spacing of the discs by
clicking at equivalent points in the 000 and g reflections.
•The floating dialog “Bragg Angle Measurements”
prompts the user for the next required action.
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Thickness Measurement in NIH
Image 4
• The fringes are identified by the user clicking on each
one and then clicking on the calculate button.
• Both dark and bright fringes may be measured, or
each set individually.
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Thickness Measurement in NIH
Image 5 • When all the fringes
have been entered,
the calculate button
is clicked.
• The application
plots the curve
assuming that the
first fringe is n=1.
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Thickness Measurement
in user
NIH
• The
can simple
iterate through
possible values of n
Image 6
or select one.
• The best fit plot may
be saved as an
image.
• The thickness and
extinction distance
determined from the
plot are included on
the graph.
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CBED Applications
Strain &Examples
lattice Parameter
Measurement
Phase Identification
Symmetry Determination - point &
space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter
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Measurement.
Structure Factor Determination.
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<001> CBED ZAPs from Si & SiGe Alloys
Si
Si2%Ge
Si3%Ge
CBED ZAPs from Si, Si2%Ge & Si3%Ge cross section samples.
Center of direct disc. Patterns recorded at 150kV, with a beam
convergence of ~8mrad. Probe diameter ~30nm. Samples were held at
liquid nitrogen temperature to reduce the thermal diffuse scattering. The
orthorhombic distortion of the lattice is most obvious in the
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Si2%Ge sample.
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Calculated Patterns
Si
Si2%Ge
Si3%Ge
CBED Simulations based on the kinematical approximation for Si, Si2%Ge
and Si3%Ge. <001> zone axis pattern, convergence angle approximately
10mrad. Voltage set to 149.5kV by comparison with the Si pattern.
Lattice parameters:
Silicon a=0.5429nm, Si2%Ge a=0.5434nm b= 0.5429nm c= 0.5430nm
Si3%Ge a= 0.5445nm b= 0.5440nm c= 0.5440nm
HOLZ Plots
generated by Diffract™
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CBED Applications
Examples
Structure Factor Determination
Phase Identification
Symmetry Determination - point &
space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter
EMAL
Measurement.
Structure Factor Determination.
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Structure Factor
Determination 1
CBED
Crystal
The ultimate goal here is to develop a technique
that allows us to determine a completely unknown
crystal structure from convergent beam electron
diffraction patterns
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Experimental Zone Axis CBED patterns are compared to
calculated patterns. A number of parameters are varied
to minimize the differences between the real and
calculated patterns.
Structure
Factor
Structure
Factor
Determination 2
Determination 2
Real CBED
Calculated CBED
Crystal Structure
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Energy Filter
Patterns
Unfiltered
Filtered
Comparison of
an unfiltered and
zero-loss only
silicon <110>
zone axis
pattern. Note the
large amount of
thermal diffuse
scattering
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Sample: Copper <110> ZAP.
Voltage: 118kV.
Liquid Nitrogen Cooled.
C2 Aperture: 200µm.
Pattern: 512x400 pixels.
Contrast enhanced.
Si <110> filtered pattern
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Fitting Routine.
Compares the intensities calculated from the Bloch-wave
coefficients of the many beam equations.
Uses a Quasi-Newton method to minimize the sum-ofsquares function:
 
2
1
N

d
Nd
( I expt  cI theory  B)
I
2
expt
where: Iexpt is the experimental intensity, Itheory is the calculated
intensity, c is a normalization coefficient, Ni is the number of
data points and B is the background level.
Quasi-Newton method chosen because:
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A. approximately linear scaling in time with number
of parameters
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B. rapid convergence, 4 or 5 iterations.
Fitting Parameters.
A number of parameters are needed for the fit:
Crystal Structure. Known in this case. For an unknown this would need
to be determined by CBED.
Lattice Parameters. Again these are known but would need to be
determined for a real unknown structure.
Debye-Waller Factor. Estimated from the mean square vibrational
amplitude in Radi (Acta Cryst. A26(1970) p41).
Specimen Thickness. Determined by an initial thickness scan of the
data.
Normalization constant c is essentially a scaling factor for the
theoretical intensities and it is determined from thickness scan.
Thickness scan calculates  for each thickness using neutral atom
structure factors.
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Fitting Variables.
Vary the low order structure factors (Ug) and fix all the others
at their neutral atom values.
Starting values for the structure factors are the neutral atom
values, Doyle & Turner (Acta Cryst. A24 (1968) p390) for
the elastic parts and Bird and King (Acta Cryst. A46 (1968)
p202) for the absorptive parts.
Sample thickness and normalization coefficient obtained
from a thickness scan.
Background levels (Bn). Vary as a function of scattering
angle.
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1
2
Fit Results.
1. “Raw” data extract from CBED
discs.
2. Fitted pattern.
3. Residual map
Specimen: Copper.
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Thickness: 1052nm.
3
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X-ray and Electron Structure Factors for Cu
X-ray and Electron Structure
Factors for Cu
Parameter
2
{111}
{002}
{220}
{113}
t (in nm)
Data Set 1
Neutral Atom
Data Set 2
Ug values Ug values fx values
fx values
3.883(3)
3.426(4)
2.422(12)
2.035(11)
105.2
21.53(2)
20.18(3)
16.41(10)
14.15(13)
3.873(6)
3.403(6)
2.371(19)
2.005(17)
65.2
-rays
1981
X-Rays
1982
Schneider Takama
et al.
& Sato
{111}
{200}
{220}
{311}
Data Set 1
21.51(5)
20.22(4)
16.45(5)
14.54(4)
21.51(1)
20.13(2)
16.16(7)
13.97(10)
Data Set
21.76
20.33
16.18
14.05
Electron Diffraction
1980
1988
1993
Smart &
Fox &
Humphries Fisher
21.80(6) 21.786
20.28(11) 20.454
16.75(8) 16.696
14.74(4)
21.72(4)
20.45(4)
16.68(8)
14.76(7)
Mansfield et al
Thick Sample
[21.51(1)]
[20.33(2)]
[16.16(7)]
[13.97(10)]
Estimated 2 values in parentheses.
Values in brackets are low temperature and the others are room
temperature.
Top table: The fx values are X-ray structure factors at near liquid nitrogen
temperature derived for the fitted Ugs using the Mott formula. Neutral
atom structure factors derived from Doyle and Turner.
EMAL
U of M
Abbreviations.
AEM
(C)TEM
CB(E)D(P)
SAD(P)
ZAP
HOLZ
FOLZ
C2
WP
CL
Analytical Electron Microscopy
(Conventional) Transmission Electron
Microscopy
Convergent Beam (Electron) Diffraction
(Pattern)
Selected Area Diffraction (Pattern)
Zone Axis Pattern
Higher Order Laue Zone
First Order Laue Zone
(S=Second, T=Third)
Second Condenser Aperture
EMAL
Whole Pattern
Abbreviations
Camera Length
U of M