what is diffraction?

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what is diffraction?

• Diffraction the spreading out of waves as they
  encounter a barrier.

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• What is a Diffraction pattern?
• an interference pattern that results from the
  superposition of waves.
• Mathematically, this process can be described by
  Fourier transform, if the diffraction is
  kinematic (electron or X-ray has been scatted
  only once inside the object).

Laser diffraction pattern of a thin grating
films, where the size of holes is closed to the
wavelength of the laser (Ruby red light 594 um) .
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Fourier transform of regular lattices
Reciprocal space
Real space
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What is a crystal ?
DNA single crystal
single crystal of quartz

• Same structural unit (an atom, many atoms,
  molecule) unit cell.
• The units are packed periodically in a infinite
  space lattice.
• Unit cell contains all the necessary points on
  the lattice that can be translated to repeat
  itself in an infinite array.  In other words, the
  unit cell defines the basic building blocks of
  the crystal, and the entire crystal is made up of
  repeatedly translated unit cells.

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A crystal structure is composed of unit cell,
periodically repeated in three dimensions on a
lattice. Lattice parameter the spacing between
unit cells in various directions. They are
parameters to describe the unit cell of a
crystal. Crystal can be classified by its
symmetry. According to the axial system used to
describe their lattice, there are 7 crystal
systems cubic, tetragonal, rhombohedral,
hexagonal, orthorhombic, monoclinic and
triclinic. With the Bravais lattice, lattice
plane and direction can be defined.
d
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The wavelength of high energy electron is about
0.0037 nm at 100 keV The bond of atoms
(distance of two adjacent atoms) is about 0.08
0.2 nm. The crystal is the best barrier to
observe the diffraction of electron and X-ray!
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William Henry Bragg 1862 1942 Nobel
Prize in Physics 1915
X-ray diffraction in a crystal. Like an electron
beam an X-ray has its own wavelength which is
proportional to its energy (10 0.01 nm).
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If a known wavelength l is used and the Braggs
angle can be measured or inferred then the
d-spacing of a crystal of unknown composition can
be calculated.
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This is the principle behind X-ray diffraction
(XRD) in which an X-ray of known wavelength is
focussed onto a crystal that can be aligned until
a diffraction pattern is created. A blanker on
the optical access blocks the transmitted
wavelengths.
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The atomic structure can be deduced by performing
a Fast Fourier Transform (FFT) on the resultant
diffraction pattern once the phase is known.
Phase problem can be solved by direct method!
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Structure determination method X-ray
crystallography
Purified protein
Crystal
X-ray Diffraction
Electron density
3D structure
Biological interpretation
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To operate the TEM in diffraction mode the
objective aperture is removed from the beam path
and the scope is adjusted to focus an image of
the back focal plane of the objective lens, not
the image plane.
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This is most easily accomplished by adjusting the
strength of the objective lens so that an image
of the back focal plane is projected onto the
viewing screen.
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The result is an electron diffraction (ED)
pattern. The pattern one obtains is completely
dependent on the d-spacing and composition of the
crystal that is being analyzed.
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If an ED is made of an amorphous structure (i.e.
no crystalline formation) then one simply gets a
central bright spot comprised of transmitted
electrons and a single ring of randomly forward
scattered electrons.
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If an ED is made of field of many crystals, some
of which are oriented at the Braggs angle while
others are not, a pattern with well defined
concentric rings, but not spots, will result.
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Selected Area Electron Diffraction (SAED)
SAED use parallel illumination and limits the
sample volume by an aperture in the image plane
of the low object lens.
An SAED pattern of a crystal.
Lattice plane have spacing of d
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ELECTRON DIFFRACTION PATTERNS
MOSAIC SINGLE CRYSTAL
PLATELIKE TEXTURE
POLYCRYSTAL
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POLYCRYSTAL-TYPE ELECTRON DIFFRACTION PATTERN
Electron diffraction patterns from samples
containing very large number of small randomly
distributed crystals consist of continuous
rings. The radii of the rings are inversely
proportional to the interplanar spacings dhkl of
a lattice planes of crystals. The formula
rhkl dhkl L ?, (r- radius of the
ring) is used.
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The relationships between the axis and anglesin
unit cells
Triclinic a ? b ? c ????? ?
? Monoclinic a ? b ? c ?? ?
??? Orthorhombic a ? b ? c ????
?900 Hexagonal a b ? c ????
900 ? ? Tetragonal a b ? c ????
? 900 Cubic a b c
???? ? 900
Having only one plane of reciprocal lattice for
unknown crystal we cant determined the 3D
lattice parameter. Because we do not have the
information perpendicular to the plane.
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Tilting the crystal to have three patterns of
different zones
Schematic representation of the tilting method
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So that individual crystals can be oriented to
the appropriate Braggs angle one uses a double
tilt specimen holder which allows for positioning
in X, Y, and Z directions.
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Unit cell determination for unknown crystal
Many sections of the reciprocal lattice of a
crystal can be obtained by tilting of a crystal
in electron microscope. The lattice type and
parameters can be determined if the relationship
of these 2D sections is known.
A quarter of stereographic projection of a cubic
crystal.
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Unit cell determination 1. Tilt the unknown
crystal to obtain the first pattern with
low-order index. 2. Tilt the crystals along a
certain direction and collect at least 3
patterns. Write down the tilt angels. 3.
Reciprocal lattice reconstruction. 4. Index the
pattern and check the tilt angles (experimental
and calculated).
A new thin-film phase of pentacene
Four electron diffraction patterns obtained by
tilting a pentacene crystal in the film deposited
on the (100) NaCl with a high deposited rate at
room temperature followed by annealing at 200ºC
for 2 hours. The reconstructed lattice is a
triclinic one with a6.08 Å, b7.63 Å, c15.3 Å,
?80.7º, ?84.5º and ?89.5º, with the (001)
d-spacing is 15.1 Å. The experimental angles
(without parentheses) and calculated ones (in
parentheses) were labeled.
Pentacene is the most important organic
semiconductor which has been used in the
fabrication of high-performance organic thin film
transistors.
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Orientation Relationship
A composite electron diffraction can be used to
determine orientation relationship between
crystals.
Orientation relationship can be identified as
(0001)MgB2//(0001)SiC and 110MgB2//110SiC
(a) A cross-section image of MgB2/SiC interface
taken along 100 direction. (b). Composite
electron diffraction pattern taken at the
interface consisting the diffraction spots from
the substrate and the film.
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P Photographic plane L distance of specimen
from P T Forward scattered beam O point
where T strikes P S Bragg diffracted beam G
point where S strikes P R vector distance from
O to G
R / L tan 2q and from Braggs law we know that
2dsinq l. Thus R / L 2q 2l / 2d which
simplifies to R l L / d If we can measure R
and both l and L are constants then d can be
calculated.
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Reciprocal lattice vs Crystalline lattice in real
space
Crystal structure in real space
Diffraction in reciprocal space
Thus, the distance, R on the pattern between the
spot G (hkl ) and the spot O (000) is related to
the interplanar spacing between the hkl planes of
the crystal, dhkl, by the equation R l L / d.
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Summary
Electron diffraction is a technique which allows
users to determine the atomic arrangement of
crystals. When combined with other analytical
techniques such as EDS it can aid in the
identification of unknown crystals and/or
determine the d-spacing of newly described
crystals.
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Space Group Determination
Systematic extinction in the diffraction pattern
can be used to determine space group for an
unknown crystal.
1. (001) pattern, all hk0 spots satisfying
hkodd disappear n-glide plane parallel to the
(001) plane. 2. (010) pattern, all h00 spots
with hodd are extinct 21 screw axis along the
100 all 00l spots with lodd disappear 42
screw axis. 3. (110) pattern, h-hl
reflections with lodd are disappear c glide
plane parallel to (110). space group is P 42/n
21 c , (No. 137).
Diffraction patterns of a tetragonal Ga-Mn phase
with a1.25 nm and c2.50 nm.