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John Spence Overview of electron diffraction' Erice June 9 2004'

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... TED data, be aware of two important differences between XRD and ... C6Br6 exptl data. Solve structure of C6Br6 by flipping, using experimental X-ray data. ... – PowerPoint PPT presentation

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Title: John Spence Overview of electron diffraction' Erice June 9 2004'


1
John Spence Overview of electron
diffraction. Erice June 9 2004.
1. Pulsed gas diffraction. LEED. RHEED.
Photoelectron diffraction, holography. 2. Spot
(SAD) patterns. Lattice, cell consts. Organics,
inorganics Semi-quantitative
TED HREM. Examples from recent work of Dorset,
Sinkler, Gjonnes, Zou et al. 3. CBED.
Space-group determination, Cell constants,
Strain mapping, Charge-density measurement, HV,
Debye-Waller. Elastic filtering.
Quantitative 4. LACBED. CBIM. Dislocation
Burger's vectors. 5. Coherent Nanodiffraction.
Planar fault vectors. Ronchigrams for aligning
corrected STEMs 6. Diffuse scattering. Phase
transitions, phonons, defect analysis. Isolated
nanostructures ? (Zuo's nanotube) 7. UHV TED
from reconstructed surfaces of thin films.
Quantitative 8. Cryomicroscopy in biology.
(Most successful area of EDImaging).
Solving proteins - organic monolayer
xtals at 0.3nm resolution.
Quantitative 9. Phase identification Combine
CBED (space group), SAD (lattice), EDX ELS
(composition). 10. Advanced topics. HiO
inversion, Precession camera, Dynamical
inversion, Ptychography, in-line STEM
holography, imaging. ELNES QCBED. Serial
Xtallog - protein beam diffn.
2
  • Textbooks and software
  • Introductory
  • "Electron diffraction in the TEM" by P.
    Champness. Royal Micros Soc
  • ISBN 1859961479 (Google that !). Official Erice
    Text ?
  • (inexpensive paperback, unified presentation)
  • Advanced.
  • "High Energy Electron Diffraction and
    Microscopy" L.-M Peng ,
  • S.L. Dudarev and M.J. Whelan. OUP. 2004.
  • 3. Software
  • WebEMAPS-Google that !

Specialised or advanced texts by Morniroli,
Reimer, Hirsch et al, Spence and Zuo, Lorreto,
Williams and Carter, Cowley, Spence, Agar,
Kirkland, Fultz and Howe, Dorset, Egerton, De
Graef and much more (eg free software) at
http//www.public.asu.edu/jspence/ElectrnDiffn.ht
ml
3
Why is electron micro-diffraction a good idea ?
  • It provides the strongest continuous signal from
    the smallest volume of matter known to science.
    Hence good for nanoscience.
  • Field emission sources are the brightest
    continuous particle sources in physics.
  • The interaction is the strongest of all the
    long-lived particles.Hence more signal.
  • Ratio of elastic to inelastic scattering more
    favorable than X-rays (less damage for bio)
  • TED is uniquely sensitive to ionicity.
  • The synthesis of many new nanostructured
    materials demands new methods for structure
    analysis using nanoprobes.

4
Diffraction modes in the TEM/STEM.
1. Source conjugate to detector. SAD.
2. Source conjugate to sample. CBED.
S
Angular resolution limited by source
size. Spatial resolution limited by SA aperture
and Cs.
Fine detail in disks comes from outside probe
? Spatial resolution limited by probe size.
5
1. C2 may be coherently or incoherently
filled. 2. CBED provides maximum info from
smallest volume. 3. For refinement, strain
mapping first - calibrate pattern. 4. We say
that P and P are conjugate points.
6
V
i
r
t
u
a
l

A
p
e
r
t
u
r
e
S
p
e
c
i
m
e
n
P
P

B
a
c
k

F
o
c
a
l

P
l
a
n
S
A

A
p
e
r
t
u
r
e
a) Selected Area Electron Diffraction
Fig. 1 Three modes of electron diffraction. Both
a) selected area electron diffraction (SAED) and
b) nanoarea electron diffraction (NED, Koehler)
use parallel illumination. SAED limits the sample
volume contributing to electron diffraction by
using an aperture in the image plane of the image
forming lens (objective). NED achieves a very
small probe by imaging the condenser aperture on
the sample using a third condenser lens (eg M
1/20). Convergent beam electron diffraction
(CBED) uses a focused probe.
7
Elementary electron diffraction and xtallography.
Scattering geometry.
1/dhkl sqrt(h2 k2 l2)/a q/l g for
cubic system, cell const a. Bragg's Law for HEED
(small angle approx) q 2qB l /dhkl
where q is the scattering angle. Electron
de-Broglie wavelength is l 1.22643 / (V
0.97845E-6 V2)1/2 with V the accelerating
voltage, and l in nm.
dhkl
q
qB
Ewald sphere construction ensures conservation of
energy, xtal momentum. (But ! Apply uncertainty
principle in beam dirn. for "elastic" scattering
. Dt.DKz 1, where Dt thickness t. DKz
DSg 1/t Hence expect strong Bragg beams
for a small range of incident beam directions
within excitation error DSg.)
1/l
Sg
g
8
Phase identification of unknown structures by
CBED and SAD.
  • Find space-group by CBED.
  • 2 Use EDX and ELS to determine atoms present and
    approximate stoichiometery. (Can Nb2O5 be
  • distinguished from NbO2 by EDX ? Probably.
  • 3. Find Bravais lattice from spot patterns with
    HOLZ ring. Use dlL/R, L camera length, R spot
    radius.

uH
  • Find relative position of FOLZ projected onto
  • HOLZ. This gives centering. (eg I, body centered
    in recip space, fcc in real)
  • From FOLZ ring we get c axis dimension. Sg 1/c
    lU2H/2
  • From spot measurements get approx cell consts.
  • More accurate Bravais lattice and cell consts
  • can be obtained by method of Zuo Ultramic 52,
    459 "A new method of
  • Bravais Lattice determination" using HOLZ lines
    in central CBED disk.

FOLZ
4. Use the indexed patterns given in Spence
and Zuo book ?. (HOLZ points indicated on ZOLZ).
ORSimulate any pattern (WebEMAPS-Google
that !) with random atom posns in known space
group. 5 Use Electron Diffraction data base
http//www.nist.gov/srd/nist15.htm and/or Powder
Diffraction File.
9
NIST Standard Reference Database
15 NIST/Sandia/ICDD Electron Diffraction
Database Designed for phase characterization
obtained by electron diffraction methods, this
database and associated software permit highly
selective identification procedures for
microscopic and macroscopic crystalline
materials. The database contains chemical,
physical, and crystallographic information on a
wide variety of materials (over 81,534) including
minerals, metals, intermetallics, and general
inorganic compounds. The Electron Diffraction
Database has been designed to include all the
data required to identify materials using
computerized d-spacing/formula matching
techniques. The data for each entry include
the conventional cell reduced cell
lattice type space group calculated or
observed d-spacings chemical name chemical
and empirical formula material class
indicators references. . . This database and
search software are available in magnetic tape
format and in CD-ROM format.
Can one Google a set of d spacings ? (Try it now
for Si on wireless web)
10
Ewald sphere in more detail We can associate
an excitation error (in Ang-1) with every
Bragg beam in a TED pattern.
The Bragg condition corresponds to Sg 0. In
the zone-axis (symmetric) orientation, the
excitation error is Sg l g2 / 2
t
FT
11
WebEMAPS-Google that ! Kirkland's book has source
code for ms
Approximations for diffracted amplitudes.
Kinematic. F01 (single
scattering)
Two beam F01 - Fg
Thick phase grating
Multislice
Bloch-waves.
Spence and Zuo, "Electron Microdiffraction" Plenum
1992 gives Fortran listings and
full documentation of multislice and Bloch wave.
12
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13
Comparison of N-beam, PGA, kinematic approx. for
NbO
PGAFT exp(if(R)t)
KinVg t sin(pSgt)/(pSgt)
Phase
Exact (N-beam)
Note kin intensity goes as t2 Dyn intens is
"sinusiodal". kin phase is linear. X-ray
amplitudes dont involve t at all !
PGA
Lynch, Moodie, O'Keeffe Acta A31, 300
Kinematic
Amplitude, not intensity.
Conclude Need t lt 8 nm Inorg t lt 20 nm
Org Layer structures help (projection)
kin
Thickness
14
Failure of kinematic approx.
111
000
At x1 (b), central beam is strong Image consists
mainly of interference between central beam and
each beam, producing a faithful image.
Increasing thickness
Kinematic t2
000
Thickness (A)
At x5 (c), central beam is weak. Image consists
mainly of interference between beams, producing
an image of half-spacings, and rotated.
15
How do we know these multiple-scattering
calcns are accurate ? Because they precisely
fit experimental data, as here. (Zuo J. Elec Mic
47, 121)
16
Conclusion Accurate quantification of ED data is
only possible if multiple scattering is included
in calculations But. for light-element
inorganic nanostuctures, kinematic theory is
accurate enough to solve structures especially if
good data can be obtained in 3D !
MgO (111) systematics CBED fitted at 120 kV
using elastic imaging energy filter. (Zuo 97)
17
  • Conditions for quantifiable TED.
  • Data must be collected under known conditions.
    Experimental parameters (thickness,
  • wavelength, etc) must be accurately known if not
    kinematic.(All beams t2. Normalise)
  • Xtalline samples must have no defects, uniform
    thickness, no bending.
  • Scattering must be known to be weak if kinematic
    theory used.
  • 4. Phase problem may need to be solved.
    Three-dimensional data may be needed.
  • How these conditions are fulfilled for the 3
    quantitative methods.
  • For cryo-em of proteins Sample in C,N,O,H. All
    light atoms in ice. Thickness lt 20nm. 3D data.
    Not atomic resolution. Xtal quality problem.
    Thickness known because one monolayer !
  • For QCBED. Probe is so small that xtal is
    perfect. Multiple scattering theory used.
  • UHV TED of Surfaces. Scattering from monolayer
    is weak. Ignore bulk interaction. "Thickness"
    known.
  • For HREM TED.
  • Use observation of forbidden reflections (screw,
    glide, away from Bragg condition) as test for
    single scattering. (Find space-group by CBED to
    indicate which are forbidden ?). At Bragg,
    forbiddens remain forbidden (black cross) even if
    dynamical.

18
  • Tests for kinematic scattering.
  • Use Friedel's law. If a crystal is known to be
    acentric, but I(g) I(-g), there is no
    multiple
  • scattering. (for acentrics, Vg V-g, but
    V(r) is real). This works for unknown structures.
  • Blank discs. If the intensity variation across
    CBED discs is constant, this suggests weak
    scattering. (Wu, Spence Acta A58, 580.
    "Kinematic and dynamical CBED for solving thin
    organic films at helium temperature. Tests with
    anthracene").
  • Study organic monolayers. Thickness equals known
    molecular size. Problem - make self-supporting
    sample ! (eg use ice). Poor sample quality. LB
    films- mols lie on side.
  • Use wedge-shaped sample. Match image against
    thickness. Similar for CBED.
  • Inclined, planar-fault on known plane can be
    used to give thickness on wedge.
  • Accurate fitting to CBED pattern with multiple
    scattering program will calibrate thickness
    within lt 4 A
  • Find space-group by CBED to locate true
    forbiddens (use primitive cell),index pattern.
    Then forbidden reflections in SAD from thin, bent
    areas suggest no multiple scattering.
  • Credibility requires the same structure to be
    determined from the same sample by XRD and TED !
    Are the TED intensities consistent and
    reproducible ? Can you record the same set of TED
    diffracted intensities twice, from different
    areas of the same sample ?
  • Remember- modern XRD software always gives some
    answer ! (R factor ?Potl vs Rho).

(screw, glide)
19
Elementary Xtallography.
centering was used to give orthogonal coords.
  • Xtal Bravais lattice molecule.
  • There are only 14 possible Bravais lattices, s.t.
    the surroundings of every point are identical.
  • Seven primitive cells exist (seven xtal systems
    cubic, hexagonal, triclinic ,monoclinic,
    orthorom, tetragonal, trigonal )
  • Get 7 more from centering, P, I, F, C.
    (primitive, body-centered, centered on every
    face, centered on two)
  • The xtal system is defined by the symmetry of the
    xtal. (eg any xtal with four 3-folds is cubic).
  • From the cell constants a,b,c, a,b,g the
    reciprocal lattice is generated by FT. eg a
    b X c /W.
  • The point group is based on mirror, inversion,
    and rotation symetries alone. There are 32. (eg
    for mols).
  • Translational symmetry adds the possibility of
    screw and glide symmetries. There are 230 space
    groups.
  • There are 2 types of truly forbidden reflections
    - Space group (screw,glide axes). Non-spherical
    atoms.
  • Third type "forbidden" due to centering is
    historical artifact, due to method of indexing.
    Use primitive cell !
  • The f'bdns generated by a centered cell were
    never there ! Physically, electrons diffract from
    the primitive cell.

A list of absent refln indices due to centering,
screw, glide, and many indexed patterns
(including HOLZ) is given in Spence and Zuo,
Electron Microdiffraction, 1992 . Plenum. Use Int
Tables for Crystallography.
20
Solutions to the phase problem.
If diffraction patterns can be collected under
kinematic conditions, we must solve phase
problem. (Phases can be obtained from FT
of HREM image if wpo approx holds. /- if
centric).
Modern PX used MAD. (Heavy atom, abs. Most
proteins dont diffract to atomic
resolution) Modern inorganic xtallog uses Direct
Methods, requires atomic resolution data, lt1000
(?) atoms. Then the powerful atomicity
constraint can be used. Recently, the simple
"charge flipping" algorithm has been developed. (
Oszlanyi and Suto, Acta A60, p.134 (2004) )
Recently the phase problem for continuous
scattering from non-periodic objects has been
solved experimentally with electrons and X-rays
by new iterative methods. ("lensless
imaging") see - Weierstall et al Ultramic 90,
171 and Zuo et al Science 300, p. 1419 for
electrons and Marchesini et al Phys
Rev B68 (140101) (2003) for soft X-rays. These
iterative algorithms are developed from work by
Fienup, Gerchberg, Saxton and others.
If you use XRD software for TED data, be aware of
two important differences between XRD and TED
data 1. For XRD, all reflections are collected
at the Bragg condition. Not so for zone-axis
TED. 2. For XRD, reflections are
angle-integrated. Not so for TED or, usually,
CBED.
21
A new algorithm for xtal phase problem "Charge
flipping" Oszlányi Acta A60 134, '04.
  • Start with measured Fhkl, random phases.
  • Transform.
  • 3 Set Im(r)0. Find x20 of largest positive
    r, keep them. (Support estimate)
  • 4 Reverse sign (flipping) of all other r(r)
    below threshold x.
  • 5. Go to 2.

r
1
R factor
C6Br6 exptl data
Unlike direct methods does not need known scatt
factors ! flipping finds support ! Converges
to same structure indep of initial phases.
x was reduced here
Iteration number
22
Solve structure of C6Br6 by flipping, using
experimental X-ray data. (Wu, O'Keeffe,
Spence, Groy. Acta A In press 04. Use fractional
a, since F0 unknown)
010 400th iteration, R 0.029
24 peaks with the highest density
  • How do we know if a peak is an atom ? If
    stoichiometery, space group known, choose n
    largest.
  • If not, repeat with different starting random
    phases. Add results. Noise peaks decrease, atoms
    dont.
  • Dioxane (SnBr4) also solved. Atomic numbers
    given by height of peaks (if normalised).
  • Gives correct atomic coords.
  • Needs atomic resolution data. If not, see heavy
    atoms. Works for non-periodics to find support,
    like shrinkwrap.
  • Wont work in 2D for xtals (not enough
    zero-density). Does work with oversampled 2D
    non-periodic (more zero density)

23
Imaging theory - needed because collection of 3D
diffraction data is difficult Flipping doesnt
work in 2D (and requires atomic resolution data).
Oops ! Double diffraction
Electron beam
(Lens phase shift)
1. For a perfect lens, image is magnified copy of
exit face wave-function. 2. This is a faithful
representation of the projected potential only if
kinematic, and flat Ewald sphere.(wpo) 3. FT of
such an exit face wave wave gives "structure
factors" Vg. propn to diff. pattern spot
intensities. 4. If multiple scattering inside
sample, FT of exit face wave still gives d.p. But
spots are not stucture factors.
24
Electron/specimen interaction refraction with
locally varying index
Incident wave. Plane of constant phase is flat.

electrons go faster along atoms, with higher
kinetic energy, shorter wavelength, more wiggles.
sample
Exit wave Phase front has wobble
-i




















kin
Representation on argand diagram Wave has a
range of phase values In kinematic theory,
phase is 90o wrt (000)
(000)
1
25
The dynamical exit-face wave.
Kinematic scattering
FT
plane wave
Thin !
Image (copy of exit face wavefunction)
Diffraction pattern (Structure factors)
Exit face wavefunction (Faithful projection)
FT
Multiple (dynamic) scattering
plane wave
Thick !
Image
Diffraction pattern (Not structure factors !)
Exit face wavefunction (Not faithful projection.
Correct period)
Note FT relationship between exit face and
diffraction pattern is preserved with multiple
scattering
26
HREM imaging theory for weak phase objects.
Exit face wavefield
Weak scattering
Diffracted amplitude
Add lens, aperture
4
Image amplitude
Image intensity
Notes dark atoms predicted on bright background
at Scherzer focus, where - 90 degree lens
phase-shift adds to - 90 degree scattering phase
(i exp i p/2). Then replace P(u,v) exp() in 4
by -i to give image I(x,y) 1 - 2 s fp(-x,-y)
for ideal lens without aperture.
27
Intensity in image
1
A
B
Position
WPO
Not WPO
Contrast A/B must be small for WPO.
High contrast - not a weak phase object !
"Multiple scattering is the usual condition in
electron diffraction"
Kisielowski et al NCEM LBL Ultram 2001.
28
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29
Diffractograms for wpo. Finding structure factor
phases from HREM images.
The complex Fourier transform of the WPO image
intensity can be obtained in a computer as
where
2
where D, C are constants. Hence if c is known,
the phase of the fourier coefficients of the
projected potential can be found. These phases
are 0 or 180 degrees (wrt some origin) for
centric projections
30
From "HREM" by JCHS, 3rd Edn 2003. OUP
31
  • Solving structures from point electron
    diffraction patterns alone.
  • Examples. 1. Inorganic.
  • Weirich et al. Acta A56, p.29 (2000). Ti2Se
    Octahedra. Short b axis along beam.
  • Direct Methods.SIR97 SAD patterns from areas
    thin enough to show no forbidden
  • reflections. Symmetry related merging. Assume
    FhklIhkl for "extinction" model.
  • (Cowley 1992). Unlike XRD, peak heights in
    potential map are not proportional
  • to atomic number. Note xtal thickness neither
    known nor enters analysis. No
  • corrections for Ewald sphere curvature. EDX used
    for stoichiometry. R24.
  • Film, CCD. Five other fine-grained structures
    solved. Atom position error 0.2 Ang.

32
Solving structures from point electron
diffraction patterns alone. Examples. 2.
Organics (Proteins are another story
!) Dorset. Acta A54, p. 750 (1998).
Linear chain polymers require view
normal to chain - difficult ! Caprolactane was
xtallized in two orientations. Direct
methods. Solved.
Example 3. The precession camera. (Vincent and
Midgely 1994, Marks 03 JMCFest., Gjonnes
et al 1998). AlmFe intermetallic. Patterson and
Direct Methods. 3D. No short axis.
Precession data is "more kinematic" since
angle-integrated.(Direct beam precesses around
ZOLZ). Blackman, 1939 gives
angle-integrated 2-beam soln. Ig Ug2
t for small t, but remarkably Ig Ug
for large t. Structure solved with 3 Fe
and 12 Al atoms.
Definition of R factor
33
Solving xtal structures by combining HREM images
and TED patterns. Example Inorganic. Zou et
al. Acta A59, p.526 (2003).
AlCrFe. 3D "quasixtal". a4nm !. 13 different
TED Zones. 124 atoms found. Film WPO assumed, and
HREM images corrected for defocus, Cs. Phases
obtained from these. Magnitudes from TED
patterns. Some projections are centro. Xtal is
not. Symmetry merging. CRISP used. No correction
for Ewald sphere curvature. (Same method as
proteins). No thickness
34
  • Space-group determination by CBED.
  • Find point group hence crystal system (cubic,
    hexagonal, trigonal, tetragonal, orthorhombic,
    monoclinic, triclinic).
  • Find Bravais lattice and centering. Index.
  • Find screw and glide elements. Find space group
    by adding these translational symmetries to the
    point group laid down at every point of Bravais
    lattice.

BeO (wurtzite) CBED down c. 100kV . Elastic
imaging filter. Sixfold axis. Projection
diffraction group 6mm1R
35
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36
c
Bragg condition is along line B
Dynamically forbidden reflections in BeO
(hexagonal, wurtzite) shows 21 (contained in 63 )
screw axis along c and/or c glide axis.
from Zuo and Spence, Microdiffraction, Plenum, 92
37
Electrons, X-rays and Neutrons
Electrons
X-ray
Neutrons
Mn
Mn3
Mn3
  • Compared to X-rays and Neutrons, Electrons
  • Interact strongly with matter
  • Have bright source
  • Are very sensitive to charge states at small
    scattering angles
  • Are scattered predominantly by nuclei at high
    scattering angles
  • Have the smallest probe 1 Å

Jim Zuo/MATSE
38
(cf Tanaka, Tsuda, Saunders, Bird, Nufer, Mayer,
Fox) .
Seeing Bonds
39
TED of diffuse scattering from phonon modes in
silicon - beyond the Einstein model.
I(q,w) e.g q-2 w-2 for these transverse
acoustic modes phonones, where e (along
-1,1,0,1,1,0) is phonon polarization, q
(along 110,-1,1,0) is phonon
wavevector and w phonon freq. Recorded
in Koehler mode at 105K on Fuji Image Plates
Hoier, Kim, Zuo, Spence Shindo, MSA 95
40
The atomic mechanism of metal-insulator phase
transition in Magnetite
a
b
(400)
(000)
(220)
Diffuse scattering in Magnetite at
metal-insulator Verwey transition (about
120K). a) 130K b)326 K. Inset (800) left (880)
right. Elastically filtered scattering recorded
using Image Plates. (Leo Omega 912). Zuo,
Pacaud, Hoier, Spence. Micron, 31, p. 527 (2000).
41
v
p
154 K
129 K
386 K
184 K
Temperature dependence of diffuse around (800)
Bragg spots compared with theory for polaron
(Yamada).
42
The monoclinic cell of our low temperature charge
ordering model (model 2, B-sites only). The
yellow tetrahedrons (molecular polarons) form a
linked chain with alternating opposite dipoles
(red arrows). The chains are separated by
molecular polarons of different orientations
(blue). The left shows the schematic arrangement
of the spiral chains (shown as bars) above and
below Verwey transition.
43
First atomic-resolution diffractive image
reconstruction.
Double-walled Nanotube
SAD TEM
Image reconstructed from electron-diffraction
pattern by HiO
J.M.Zuo et al Science 300, 1420 (2003).
44
First atomic- resolution image of a DWNT by any
method ?. Obtained by HiO. Aberration-free. Give
s ID, OD, Chiral vectors. Zuo et al 2003. What
limits resolution ?
45
The challenge of electron crystallography.
In the age of nanoscience, there is an urgent
need for a method of rapidly solving new ,
inorganic nanostructured materials. Many are
fine-grained, light element, crystallites which
cannot be solved by XRD. The unsolved challenge
of 50 years is..
To collect three-dimensional diffraction data
under kinematic conditions. Then not restricted
to short c axis. Note centers of each CBED disc
in a small-probe pattern provide a point pattern
! Blank disks suggest single-scattering
conditions. To solve the same unknown
structure first by TED and then by XRD, get same
result.
  • The difficulties are
  • Goniometers. Loss of area during tilt.
  • Contamination
  • Automation of tilt and data collection.
  • Radiation damage.

GO TO http//www.public.asu.edu/jspence/ElectrnDi
ffn.html For free software ! Useful books,
papers, group web pages , conferences etc !!!!!
OR Google IUCr
46
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47
(0 2 0)
(4 0 0)
(0 0 0)
Convergent Beam Electron Diffraction displays a
rocking curve simultaneously in every diffracted
order. The rocking curve is the change in
intensity of a particular diffracted beam with
variation of the incident beam direction. Each
point in the (000) central disc represents a
different incident beam direction. One such point
is coupled to conjugate points in all the other
discs differing by reciprocal lattice vectors.
48
Dynamically forbidden reflections.
Screw axis along c, k. Alternate reflections
along c extinguished kinematically.
Scattering paths arranged in pairs which cancel
on black lines, cross, for all thickness and
voltage ! Black cross appears on disk at Bragg
condition. Black lines on other kinematically
forbiddens
Gjonnes and Moodie, Acta A19, p. 65 (1965).
49
Reciprocity.
S
B
A
Formally, in systematics orientation, F(h,g)
F(h, -g2h) where g is at Bragg. Here let h1,
g2 in A, then expect F(1,2) F(1, 0),
and in B.
Multislice testgold,(111) systematics, 200 kV.
-2
-1
0
1
2
g
0.00086
0.0423
Amplitude of Bragg beams.
0.423
0.913
0.000869
Orientation A
0.0423
0.0215
Orientation B
0.0000131
0.0102
0.924
In these two different orientations, one from
each family of diffracted beams has identical
complex amplitude This holds for all thicknesses
! Every beam has different intensity in the two
orientations, except 1 !
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