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1
Transmission electron goniometry and direct
crystallographic analyses
Peter Moeck Department of Physics Portland
State University P.O. Box 751, Portland, OR
97207-0751
Collaborations and contributions with Phillip
B. Fraundorf (U Missouri St Louis), Wentao Qin
(Motorola), Shirley Turner (NIST), Chunfei Li
(PSU), are kindly acknowledged Financial support
from Research Corporation and PSU Foundation is
gratefully acknowledged
2
Outline 1. Crystallography from CBED is limited
  2. Crystallometry, polariscopy, and X-ray
goniometry 3. TEM specimen holders and
goniometers   4. Direct space goniometry direct
crystallographic analyses in TEM   5.
Conclusions
3
1. Crystallography from CBED is limited
Nanocrystals too thin, i.e. kinematical
diffraction conditions which we want for HR-phase
contrast imaging, CBED disk appear featureless,
are void of fine structure, what is too thin?
Less than 1/4 of an extinction distance? for
semiconductors 20 -10 nm? ... to get the most
out of a CBED pattern the specimen should be
thicker than one extinction distance D.B.
Williams and C.B. Carter there is also
nanoprobe diffraction in TEM (Riecke method) down
to a few tens of nm, and rocking beam diffraction
in STEM down to about 5 nm, there is also
tomographic diffractive imaging J.C.H. Spence
et al., J.M. Zuo et al.
4
2. Crystallometry, polariscopy, and X-ray
goniometry
Precision 0.01
1980
Precision 0.1
? 1936
crystal is fixed to a goniometer head that
provides three mutually perpendicular
translations and two tilts (just like a double
tilt holder in TEM), adding an extra rotation
axis allows for full blown crystallometry (even
in TEM at limited tilt range)
Single-circle goniometer with a two-axis
goniometer head, i.e. full but limited range
goniometer
5
To build one from spare parts, bits and pieces
A simple two-axis goniometer head that could
complement the apparatus to the left when the
specimen (S) is fixed to this device and the
loaded goniometer head is attached to the
specimen mount (F)
Charles S. Barrett, Structure of Metals,
crystallographic methods, principles, and data,
McGraw-Hill, New York and London, 1943
6
Goniometer head
four circle X-ray diffractometer
five circle X-ray diffractometer
7
Crystallometry crystal must be fixed to
goniometer head one way or another, that makes
for some restriction of tilting ranges,
nevertheless, some 90 of whole orientation
space is typically experimentally accessible
Wulff net represents the orientation space, it
has a front and a back side, it also represents
the relationship between two coordinate systems
a cartesian and a spherical with r 1, so can be
easily replaced by a set of transformation
matrices
8
Wulff net (left) and stereographic/cyclographic
projection (right) of a crystal with highest
possible symmetry, crystal class m3m, e.g. Cu
type, fcc, due to symmetry, 1/48 is
representative of whole crystal
9
3. TEM specimen holders and goniometers
simple TEM, - eucentric axis - side entry single
tilt holder just a holder not capable of
varying crystal orientations, if ? 80
tomography of biological materials
Commonly 3 types of crystal orientation
devices/goniometers (two axes two degrees of
freedom) either top entry tilt-rotation holder
(non-eucentric), or side entry double-tilt
rotation or double-tilt holders (one eucentric
axis) typically 20 double tilt, but only
2.48 of orientation space accessible NCEMs
ARM 40 each axis, 9.87 of orientation space
accessible
time two as front and back side
Side entry tilt-rotation, 20 tilt, 360
rotation, 22.22 of orientation space accessible
10
Three degrees of freedom to orient crystals now
also for TEM !
side entry double-tilt rotation holder a
goniometer with an extra degree of freedom,
allowing eucentric tilts around chosen
crystallographic axes, similarly to
crystallometry, e.g. 24 around eucentric axis
after up to 360 rotations and up to 24
tilts to orient a diagnostic, low indexed,
crystal zone axis parallel to eucentric axis
for FEI/Philips TEMs, two goniometer axes can be
run by compustage in addition to x,y,z
translation, software compucentricity
compensates for ?x, ?y, ?z shifts
11
CM 30 at NIST Gaithersburg (S. Turner), 45
eucentric, 24 perpendicular, 360 rotation
3 different modes of operation 50 of
orientation space if eucentric axis and rotation
is used for alignment and perpendicular axis
for extra tilt, difficult 26.67 of orientation
space if perpendicular axis and rotation are
employed for alignment and eucentric axis for
extra tilt, less difficult 6.67 of orientation
space if run as a double-tilt holder where
rotation is only used to bring zone axis parallel
to eucentric axis, easily done
Tecnai F-20 at PSU Portland 24 eucentric,
24 perpendicular, 360 rotation same 3
different modes of operation two of them yield
26.67 of orientation space with support from
compucentricity program 3.55 of orientation
space if run as a double-tilt holder where
rotation is only used to bring zone axis parallel
to eucentric axis, most easily done
12
How much of orientation space does one need
? depends on multiplicity of general form/pole,
either 48, 24, 16, 12, 8, 6, 4, 3, 2, (or 1),
depending on point group/crystal class, Laue
group i.e. 2.08 , 4.17 , 6.25, 8.33, 12.5
16.67, 25 , 33.34, 50 , (or 100 ) of
orientation space - must be accessible for
success of crystallographic analyses
20 double-tilt holder (2.47 ) solves the
problem for crystals of 1 class m3m,
multiplicity of general form/pole 48 NCEMs
ARM 40 double-tilt holder (9.87 ) solves
the problem for 12 crystal classes PSUs tecnai
F-20 24 double-tilt 360 rotation in the
inconvenient modes (26.67 ) solves the problem
for 22 crystal classes, in the most convenient
mode (3.55 ) just highest symmetric m3m class
only NISTs 45 24 double-tilt, 360
rotation in the most inconvenient mode (50 )
solves the problem for all crystal classes except
1, in the most convenient mode (6.67 ) still 12
crystal classes out of the possible 32
13
NISTs double-tilt rotation TEM goniometer in
action
zones Angle between 110 and zone Angle between 110 and zone
zones Calculated Measured
140 31.0 31.5
120 18.4 19.0
110 (0) (0)
210 18.4 18.5
410 31.0 30.0
100 45.0 45.0
a
b
(001)
110 zone axis projection
001
a) large rutile, TiO2, nano-particle oriented
in TEM by double-tilt to 110, then rotated so
that (001) is parallel to the eucentric axis of
the sample holder, b) corresponding 110 SAED
pattern of (a) (dashed line represents projection
of holder x axis).
140
31
45
110
100
for rutile, general form/pole multiplicity is 16,
i.e. 6.25 of orientation space needs to be
accessed for full crystallographic analysis
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4. Direct space goniometry direct
crystallographic analyses
Identify the crystallographic phase of an
individual cubic nanocrystal ? Solved by PhD
thesis Direct Space (Nano)Crystallography via
High-Resolution Transmission Electron Microscopy
by Wentao Qin (2002), supervisor Philip B.
Fraundorf W. Qin, P. Fraundorf, Ultramicroscopy
94 (2003) 245 (similar to P. Möck, patents DE
4037346 A1 and DD 301839 A7, priority date 21.
11. 1989, whole field invented by P. Fraundorf,
Ultramicroscopy 6 (1981) 227 7 (1981) 203 22
(1987) 225 only about 50 papers in this field
worldwide So far only demonstrated with double
tilt-holder, 15 around eucentric axis, 10
perpendicular, will be much more viable when
performed with double-tilt rotation holder our
new project
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How direct-space goniometry works
tilt from 9.74º, 15º to -9.74º, -15º, combined
35.3º
WC0.7 only W atoms shown, a ? 0.425 nm
1st step tilting crystals into at least two
different orientations that can be easily
recognized by, e.g. crossing of lattice fringes
in high resolution images or symmetric spots in
their associated Fourier transforms
2nd step at each of adjusted zone axes,
goniometer readings which are by themselves
coordinates of the direct lattice vectors in the
curvilinear coordinate system of the specimen
goniometer are recorded.
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3rd step coordinates of these goniometer
readings are transformed into a cartesian
coordinate system (Eem) that is fixed to electron
microscope. ------------------------------------
-------------- Crystallographic background
(direct space) lattice vectors of any crystal
(denominated by letters A, B, ... which refer to
direct lattice base of the crystals) can always
be expressed in a cartesian coordinates system
(E) as a 3 by 3 matrix that is called crystal
matrix of the direct lattice (ETA) (ASE)-1
which lends itself perfectly to all sorts of
crystallographic analyses that can be performed
directly while working at the microscope (rather
than later on while being back to the office).
cartesian coordinate system E (that makes the
matrix notation of the direct lattice possible)
can be chosen freely, i.e. can be set to be
identical to Eem.
4th step full blown crystallographic analysis,
e.g. phase identification, on basis of direct
space matrices (ETA), (ASE), metric tensor G
(ASE) (ETA) or in reciprocal space on basis of
matrices (ETA) and (ASE)
http//www.physics.pdx.edu/pmoeck/goniometry.htm
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Experimental demonstration
a 0.4248 nm
WC1-x (x 0.3, a 0.425 nm, non-stoichiometric
fcc like structure) nanocrystals in a Philips
EM430 ST TEM at U of Missouri at St. Louis,
Scherzer resolution 0.19 nm, double tilt holder
15 tilt eucentric axis 10 tilt
perpendicular to that axis
(2-20) spacing 0.15 nm too small to be resolved
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error bars in lattice parameters 1.5 for
spacings, 1.5 for angles, sufficient to
identify this phase out of 36 candidate phases
(backed up by powder XRD)
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Theory of lattice fringe visibility in HR-TEM
Reciprocal lattice points are expanded into
reciprocal lattice spheres, the smaller (direct
space) diameter of a nanocrystal, the larger its
corresponding reciprocal sphere
Bragg Diffraction of bulk crystal (XRD)
Nanocrystal in TEM
g k-k0
2 ?
?
2 ?
g k-k0s
Reciprocal lattice point is mathematically
sharp, zero dimensional
Decreasing the acceleration voltage increases ?,
that decreases radius of Ewald sphere, also
decreases its flatness, makes reciprocal lattice
sphere less likely to intersect Ewald sphere
20
two-fold translation symmetry about lattice
planes rotation symmetry of e-beam about lattice
plane, point A (spherical crystal)
? a visibility band
each point in visibility-band intersection of a
radially inward directed e-beam with direct space
nano-crystal sphere, showing lattice fringe
visibility Band width is indirectly proportional
to nanocrystal diameter but directly proportional
to acceleration voltage (size and flatness of
Ewald sphere)
Widths of bands 1/diameter nanocrystal
d 1/?
lower indexed bands are wider
great circle projection of lattice plane
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spherical fringe visibility-band maps visualizing
lattice fringe visibility for spherical 8 nm
diameter nanocrystals left Al (fcc structure,
resScherzer 0.2 nm), middle Si (diamond
structure, resScherzer 0.19 nm), right W (bcc
structure, resScherzer 0.15 nm), for small unit
cell and closest 74 space filling packing, i.e.
aAl 0.405 nm, and current HRTEM only very few
zone axes separated by fairly large tilt
angles
Si
W
2 bands, 4 zones
2 bands, 2 zones
Al
2 bands, 3 zones
001
001
001
(220)
45º
35.3º
45º
(020)
(020)
54.7º
011
011
011
11 2
60º
35.3º
35.3º
(-1-11)
(-1-11)
111
111
60º
19.5º
(1-10)
no 111 pole visible for Al as crossing 220
bands are only 0.143 nm wide
after P. B. Fraundorf, http//www.umsl.edu/fraund
or/
note dominance of crossed 110 fringes at the
three-fold lt111gt zone in the body-centered case,
dominant crossed 111 fringes at the two-fold
lt110gt zone in the face-centered case, and the
wider disparity between largest and
next-to-largest spacings in the diamond
structure, different structures have different
combinations of visible bands and zone axes,
i.e. directly interpretable characteristic for
goniometry
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Generalization for fcc and bcc cubic crystals, in
principle for all crystals

• Utility
• 30? or 35.3? tilts rSch 0.19 nm, applicable to
  all fcc and bcc lattices with a ? 0.38 nm, i.e.
• ? 79 of all fcc and
• 56 of all bcc elements including both low and
  high temperature phases in (total of 65) elements
  listed
• in Structure of Metals, 3rd ed., C. Barrett and
  T.B. Massalski, Pergamon, Oxford, 1993
• future rSch 0.12 nm would be enough for all 65
  listed fcc and bcc elemental structures

http//www.umsl.edu/fraundor/help/imagnxtl.htm
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Direct crystallographic analyses?
On basis of transmission electron goniometry (and
SEM) using two degrees of freedom) P. Fraundorf,
Determining the 3D Lattice Parameters of
Nanometer-sized Single Crystals from Images,
Ultramicroscopy 22, 225-230 (1987). P. Möck,
Verfahren zur Durchführung und Auswertung von
elektronenmikroskopischen Untersuchungen, German
patents DE 4037346 A1 and DD 301839 A7, priority
date 21. 11. 1989. P. Möck, A Direct Method for
Orientation Determination Using TEM (I),
Description of the Method, Cryst. Res. Technol.
26, 653-658 (1991). P. Möck, A Direct Method for
Orientation Determination Using TEM (II),
Experimental Example, Cryst. Res. Technol. 26,
797-801 (1991). P. Möck, A Direct Method for the
Determination of Orientation Relationships Using
TEM, Cryst. Res. Technol. 26, 975-962 (1991).
P. Möck and W. Hoppe, Direkte
kristallographische Analysen mit SEM, Beitr.
Elekronenmikroskop. Direktabb. Oberfl. 23,
275-278 (1990). P. Möck and W. Hoppe, Direkte
Kristallographische Analysen mit
Elektronenmikroskopen, Beitr. Elektronenmikroskop
. Direktabb. Oberfl. 24, 99-104 (1991). P. Möck,
In-situ indexing of Two-Beam Electron
Diffraction Vectors, Cryst. Res. Technol. 26,
K157-K159 (1991). P. Möck and W. Hoppe,
ELCRYSAN A program for direct crystallographic
analyses, Proc. 10th European Conference on
Electron Microscopy Vol. 1 193-194 (1992). P.
Möck, Estimation of Crystal Textures using
Electron Microscopy, Beitr. Elektronenmikroskop.
Direktabb. Oberfl. 28, 31-36 (1995). W. Qin,
Direct space (Nano)crystallography via
high-resolution transmission electron
microscopy, PhD thesis, University of
Missouri-Rolla, 2000. W. Qin and P. Fraundorf,
Correlating Lattice Fringe Visibility with
Nanocrystal Size and Orientation (2002) Los
Alamos Archives http//arXic.org, document
http//xxx.lanl.gov/abs/cond-mat/0212281 W. Qin
and P. Fraundorf, Lattice parameters from
direct-space images at two tilts,
Ultramicroscopy 94, 245-262 (2003).
24
using a double-tilt rotation TEM specimen holder
S. Turner and D.S. Bright, Characterization of
the Morphology of Facetted Particles by
Transmission Electron Microscopy, Mat. Res. Soc.
Symp. Proc. 703, V6.6.1-V6.6.6 (2001). S. Turner,
Systematic Characterization of Reciprocal Space
by SAED Advantages of a Double-Tilt, Rotate
Holder, Microscopy and Microanalysis Proceedings
2002, 668CD.
General approach tested on basis of goniometry of
reciprocal lattice vectors (X-ray diffractometer)
P. Möck, Darstellung und Analyse der
Orientierungsbeziehungen von Epitaxiesystemen
unter Benutzung des Matrizenkalküls am Beispiel
von CdTe auf GaAs, PhD thesis, Humboldt
University Berlin,1992 P. Möck, Complete
characterization of epitaxial systems from the
lattice geometrical point of view, Fundamentals,
J. Cryst. Growth 128, 122-126 (1993). P. Möck,
Complete characterization of epitaxial CdTe on
GaAs from the lattice geometrical point of view,
Mater. Sci. Eng. B16, 165-167 (1993). P. Möck,
Description of the real orientation
relationships of epitaxial samples using
transformation matrices, Inst. Phys. Conf. Ser.
No. 134, 593-596 (1993). H. Berger, P. Möck, and
B. Rosner, Description and Interpretation of
systematic Deviations from Epitaxial Laws of
Overgrowth, Acta Phys. Polon. A84, 279-286
(1993).
25
Current ideas use compustage/double-tilt
rotation specimen holder for developing

• a method that follows a cubic and a hexagonal
  tilt procedure for randomly oriented
  nanocrystalline (colloidal quantum dot) powders
  at a series of preset rotation axis settings in
  order to get interpretable phase contrast images
  of most of the nanocrystals on a TEM grid (when
  looking at powders, non-eucentricity is a smaller
  concern)
• 2) a method that guesses the phase of epitaxial
  quantum dots (or catalytic metal) from possible
  strain minimizing (or epitaxial) orientation
  relationships with the matrix (or substrate),
  using Kikuchi diffraction in order to orient the
  matrix (or substrate) suitably for the quantum
  dot phase (or catalytic metal particle) to be
  identified (when looking at large matrices or
  substrates, non-eucentricity is a smaller
  concern)
• 3) a method that adjusts any zone axis
  automatically as accurately as possible and with
  the lowest electron dose as possible
• 4) test more inconvenient modes of double tilt
  rotation specimen holder for less symmetric
  crystals

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27
modified after H. Rose
20th century
19th century
21st century
Aberration corrected TEM
Bohr radius
0.15 nm
PSU X
0.2 nm
electron phase or Z-contrast imaging for optimal
Cs, large-tilt range goniometer (preferentially
with an extra degree of freedom to tilt) and
on-line power spectra of images will lead to
(discrete) atomic resolution tomography of
nanocrystals image-based nanocrystallography
Transmission Electron Microscope
1 nm
Scherzer (theory)

10 nm
Point-to-point resolution Å-1
100 nm
Light Microscope
far field resolution limit 250 nm
(theory)
1 µm
28
0.053 nm, Bohr radius
29
Numbers of crystal orientations available as a
function of a given point to point resolution for
a given material
0.2 0.15 0.1 0.06 structural
space nm nm nm nm prototype
group Aluminum 2 2 8 ? 32 Cu-type
Fm3m Iron (FCC) 1 2 5 ? 12 Cu-type
Fm3m Silicon 1 4 8 ? 20 diamond
Fd3m Diamond 1 1 8 ? 16 diamond
Fd3m Tungsten 1 3 5 ? 13 W-type
Im3m Iron (BCC) 1 1 4 ? 12 W-type
Im3m
Symmetry increases multiplicity, inverse of
multiplicity is percentage of orientation space
that need to be accessed for direct space
crystallographic analyses all crystals above have
highest symmetric point group m3m, so all these
orientations are accessible with a common 20
double-tilt holder! but one extra degree of
freedom will make it much more feasible
001
111
101
30
(0-22) band
WC1-x a 0.4248 nm
(2-20) band
(200) Band
(020) Band
(200) band
001
011
010
011
111
101
101
111
110
(1-11) band
100
(1,-1,1) Band
Stereographic projections, 0.1 nm Scherzer
resolution, Al sphere of 8 nm diameter, 8 zone
axes visible
110
(2,-2,0) Band just visible!
011
0.15 nm Scherzer resolution
013
Improved resolution, more zone axes accessible
for goniometry with smaller tilt range goniometers
125
233
001
111
112
114
aAl 0.40496 nm
31
001
011
analogous to stereographic projection of
Kikuchi-maps in TEM and sketch Coates- (electron
channeling) maps in SEM
111
Widths of bands 2 ? 1 /d 1 / U kV
higher indexed bands are wider
32
001
001
-112
so increasing resolution by 30 leads to a two
fold increase in visibility of crossed fringes
(zone axes)
1-12
112
011
011
1-11
101
101
111
1-12
121
010
211
010
100
100
110
110
at 0.2 nm Scherzer resolution as crossing 220
bands are only 0.143 nm wide, only lt100gt and
lt110gt poles available i.e. two different
projections are available
with 0.14 nm Scherzer resolution 220 bands and
lt111gt, lt112gt poles become available ! i.e. four
different projections are available
001
(002) band
Increase in resolution (x) 100 (1 x/0.2
nm)
lets look at corresponding stereographic
projections
modified after P. B. Fraundorf,
http//www.umsl.edu/fraundor/help/imagnxtl.htm
33
100 tilt axis
010 tilt axis
full hemispheric stereographic/cyclographic 001
projection of lattice fringe visibility maps for
Al nanocrystals and Scherzer resolution 0.1 nm,
to demonstrate tilting anticlockwise around one
axis, 010 left and 100 right (Al has
structural prototype A1, also know as
face-centered cubic) (left), 8 nm diameter,
(right), 3 nm diameter,
34
-100
-110
112
-101
001
001
111
010
110
101
11-1
11-2
100
full hemispheric stereographic projection of
lattice fringe visibility maps for 3 nm diameter
Al nanocrystals and Scherzer resolution 0.1 nm,
left 001 right 111 the left map is obtained
from the right map by tiling around -110 by
54.7, (Al has structural prototype A1, also know
as face-centered cubic)
35
tilting from 001 to 114 only 19.5, 114 to
112 15.8, 112 to 111 19.5, 111 to
233 10, 233 to 011 25.3, 011 to
013 26.6, 013 to 001 18.4, 114 to
125 8.2, 125 to 013 11, 125 to 011
25.4, 103 to 114 14.3, 223 to 112
16.8, 125 to 112 14.3, 011 to 112
30, so on average only 18.2 from zone axis to
zone axis
tilt around lt100gt
70.6, titling from one lt111gt to another lt111gt
around one of the lt110gt axes, followed by a 60
rotation, then the procedure starts all over again
tilt around lt110gt
full hemispheric stereographic 001 projection
of lattice fringe visibility maps for 3 nm Al
(top right) and 111 projection 3 nm Si (middle
left) nanocrystals and Scherzer resolution 0.1
nm, models from http//cst-www.nrl.navy.mil/latt
ice/
36
Fringe visibility maps looking at 1/48 of
orientation space for a 8 nm diameter spherical
Al crystal for 0.1 nm Scherzer resolution, left,
we have 8 zone axes accessible for goniometry, as
d111 0.234 nm, d200 0.202 nm, d220
0.143 nm, d311 0.122 nm, d400 0.101 nm
bands are all visible
50 increase in resolution ? 22 increase in pole
visibility
70 increase in resolution ? gt 23 increase in
pole visibility
13 more types of fringes visible
011
011
3 more types of fringes visible
(400)
(200)
(0-22)
(31-1)
013
(-1-11)
gt 25 zone axes visible
125
233
(13-1)
(1-31)
(-311)
111
001
111
001
114
(-220)
112
but for 0.06 nm Scherzer resolution, we have for
1/48 of orientation space for the same 8 nm
diameter spherical Al crystal more than 20
accessible zone axes for goniometry !!! after P.
B. Fraundorf, http//www.umsl.edu/fraundor/
Increase in resolution (x) 100 (1 x/0.2
nm)
37
Now, what if the grain size is even smaller?
With reduced nanocrystal diameter, visibility
bands become simply larger (TEM running at same
acceleration voltage), so we get a problem with
locating exactly where zone axis is, but we
could compensate by reducing acceleration voltage
(increasing ?, decreasing radius of Ewald sphere,
reducing fringe visibility i.e. making bands
smaller again Double-tilt rotation holder also
helps as we can adjust zone axis accurately by
eucentric tilting to maximum range
011
111
001
0.06 nm Scherzer resolution, 3 nm crystal diameter
With a double-tilt rotation goniometer, getting
goniometer coordinates of many zone axes becomes
feasible, crystal matrices can then be calculated
by a least squares algorithm How do we do that?
Just follow the tilting road map
38
Given fine tilt control, an elegant protocol
follows
Find large spacing. Maximize it. Tilt
along it to 1st cross fringe set A.
Maximize cross fringe. Tilt along it to 2nd
cross fringe set B. Etc. to C, D, over full
tilt range
011
111
001
39
5. Conclusions
all goniometers work according to same
principles, goniometry of direct or reciprocal
lattice vectors of individual crystals can be
easily treated by matrices and visualized by
stereographic projections
40
hardware exists, crystallographic background
exist, there is a need as CBED is not available
below 10 nm, lets develop computerized
direct-space nano-crystallography
third degree of freedom allows for direct
following of visibility bands, road map as
described procedure with crystal matrices is
general, its applicable to all crystal classes
! a high precision and accuracy double-tilt
rotation goniometer (with compucentricity) would
be able to identify phase of any nanocrystal
regardless of Bravais lattice, point groups /
crystal classes may also be identified