Chapter 3
Basic Crystallography and Electron Diffraction from Crystals
Lecture 7
CHEM 793, 2008 Fall
Outline
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The geometry of electron diffraction
Crystallography
Kinetic Theory of Electron diffraction
Diffraction from crystals
For physical issues of diffraction, please refer to: Diffraction physics,
by John M. Cowley, North-Holland Pub. Co, 1984 .
CHEM 793, 2008 Fall
Why use diffraction in the TEM
• the first event occurring between electrons and specimen
• measure the average spacing between layers or rows of atoms;
• determine the orientation of a single crystal or grain;
• find the crystal structure of an unknown material; identify unknown phase
• measure the size, shape and internal stress of small crystalline regions.
• relate the crystallography to the image, giving TEM its great advantage over
SEM and visible-light microscope.
The geometry of electron
diffraction
CHEM 793, 2008 Fall
E-gun
Schematic lens
configuration of a
TEM system
CHEM 793, 2008 Fall
Objective lens and specimen stage
• Heart of TEM
• The objective lens forms an inverted initial
image, which is subsequently magnified.
• In the back focal plane of the objective lens a
diffraction pattern is formed. The objective
aperture can be inserted here.
• The effect of inserting the aperture is shown on
the next page.
• The objective lens would not usually provide a
magnification of more than ×50 and a TEM is
routinely used to view regions of the specimen
which are only a mm or so across. What does
this imply about the diameter of a typical
aperture if it were placed at the back focal plane
as shown in this diagram?
Diffraction (prior knowledge)
Diffraction is an interference effect which leads to the scattering of strong beams of radiation in specific
directions. Diffraction from crystals is described by the Bragg Law
n λ = 2 d sin θ
where n is an integer (the order of scattering), λ is the wavelength of the radiation, d is the spacing between
the scattering entities (e.g. planes of atoms in the crystal) and θ is the angle of scattering.
Electron and X-ray diffraction are both particularly powerful because their wavelengths are smaller than the
typical spacings of atoms in crystals and strong, easily measurable, diffraction occurs.
CHEM 793, 2008 Fall
Example : Identification of material structure
CHEM 793, 2008 Fall
Example
CHEM 793, 2008 Fall
Diffracted Wave
CHEM 793, 2008 Fall
In analogy, for a 3-D crystal, a similar path difference arguments shows that the diffraction of
monochromatic electron beam by the regularly spaced 3-D array gives an interference
patterns of beams.
Kinematical theory of electrons diffraction and assumption
1.
The e-beam is monochromatic, that is the electrons all have the same energy and
wavelength.
2.
The crystal is free from distortion.
3.
Only a negligible fraction of the incident beam is scattered by the crystal, that is every
atom in the crystal receives an incident wave of same the amplitude.
4.
The incident and scattered waves may be treated as plane waves.
5. There is no interaction between the incident beam and scattered wavelets, that is the
refractive index of crystal is unity.
6.
There is no attenuation of e-beam with increasing depth in the crystal, that is no
absorption.
7.
There is no re-scattering of scattered waves.
8. The kinematical approach is satisfactory for general description of diffraction patterns.
9.
Dynamic theory of electron diffraction is more realistic.
CHEM 793, 2008 Fall
Top of thin foil
Crystal plane (hkl)
Bottom of thin foil
The path difference is
∆ = PO ' + O 'Q = d hkl sin θ + d hkl sin θ
∆ = 2 d hkl sin θ
•For in-phase arrival at observation, or constructive interference,
this path length difference must be equal to an integral number
of wavelength, nλ or
• This is called Bragg Law. n labels the various diffraction
orders for a given set of planes.
• high order of n in Bragg Law usually is for spectrometry
CHEM 793, 2008 Fall
• n equal 1 is for diffraction
2dhkl sinθ = nλ
Bragg’s Law is widely applied to the
diffraction of X-rays as well as
electrons. It tells us that we can
expect very few elastically scattered
electrons to emerge from our
specimen unless they are at an angle
ө.
n is the order of diffraction. Usually
only n=1 is considered for
convenience.
ө is Bragg angle. E.g. Also, ө is very
small in electron diffraction
2d hkl sin θ = nλ
2d hkl sin θ = λ
Qθ is samll
∴ λ ≈ 2dθ
In practice, an electron beam will only
be strongly diffracted from planes of
atoms which are almost parallel to the
electron beam. This fact makes the
geometry of electron diffraction
patterns much simpler than that of Xray diffraction patterns for which ө
can be very large.
CHEM 793, 2008 Fall
Diffracted beam
Incident beam
ө
How to get diffraction
operation mode in TEM?
By lowering the focal strength
of the intermediate lens
CHEM 793, 2008 Fall
Objective
aperture
Change focal strength
SAD
aperture
Intermediate lens
The first intermediate lens magnifies the initial image that is formed by the objective lens.
• The lens can be focused on initial image formed by the objective lens, or Diffraction pattern formed in the back focal plane of
the objective lens.
•This determines whether the viewing screen of the microscope
shows
a diffraction
pattern or an image.
CHEM 793,
2008
Fall
Connecting Electron Diffraction Pattern and Image
(a) A single perfect crystal
(b) A small number of
grains- note that even
with three grains the
spots begin to form circle
(c) A large number pf
randomly oriented grainsthe spots have now
merged into rings
CHEM 793, 2008 Fall
Connecting Electron Diffraction Pattern and Image
(a) Electron diffraction pattern
from a thin film of amorphous
carbon.
(b) The variation of intensity with
scattering angle obtained
from a.
(c)
Diffraction pattern from a fine
grained polycrystalline gold
specimen.
(d) Diffraction from a single
crystal of aluminum.
Unlike TEM image,
diffraction pattern is
somewhat difficult to
understand and describe.
So, we need find some
effective ways to
characterize it.
CHEM 793, 2008 Fall
Firstly, we need know
basics of crystallography,
since Bragg’s Law
describes the crystal
diffraction.
d-spacing of crystal planes, only a few of planes satisfy Bragg’s Law, i.e.
diffraction occurs
•If we consider the lattice
of a crystal, then we can
see that many different
planes with different
spacings exist within that
one lattice.
• It is difficult to imagine
how a beam of waves
entering a crystal might
be diffracted with so
many different planes
set at different angles to
the beam, all with
different spacings.
Which planes will satisfy
the Bragg law to diffract
incident beam?
CHEM 793, 2008 Fall
Crystallography Basics (Further reading: Crystallography, W. Borchardt-Ott,
2nd edition, Springer, Berlin, 1995)
Crystal
Greek: χρυσταλλοζ (ice)
Chinese:
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晶
Ideal-Crystal
– Infinite extended 3-dimentional periodic arrangement of atoms
– Anisotropism (properties are dependent upon direction e.g.
pleochroism for optical properties)
Real-Crystal
– Regular faces
– constant angles between faces
– Habitus (e.g prismatic, fibric; depends on growth conditions:
faster growing faces become
smaller)
CHEM 793, 2008 Fall
Crystal properties
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Crystal: homogeneous solid possessing long-range, three-dimensional
internal order.
Crystals are far-ordered (> 20 Å)
1 Ångström= 10-10 m = 0.1 nm = 100 pm
The principles that control the crystal’s atomic structure also affect the
shapes of the crystal faces and the angles between them.
Unit cell: smallest unit of the structure (or lattice) that can be indefinitely
repeated to generate the whole structure (lattice).
amorphous: solid material which lack any ordered internal atomic
arrangement, the term “amorphous” might be replaced by:
– Short-range ordered (2-5 Å e.g. Si-O coordination polyhedral)
– Medium-range ordered (5-20 Å e.g. borosilicate glass structure
CHEM 793, 2008 Fall
Constructing crystal: there are two most compact ways
• close-packed hexagonal (honeycomb) layers of atoms
• square arrangement
Hexagonal layer
Square layer
CHEM 793, 2008 Fall
Mathematical definition
• Crystal structure: a regular
arrangement of atoms decorating a
periodic, 3-D lattice.
• Lattice: the set of points which is
created by all integer linear
combinations of three basis vectors,
a, b, and c.
• Six numbers of lattice parameters
of unit cell ( a,b,c, α,β,γ)
• there are seven primitive lattices
Lattice vector : t = ua + vb + wc
Lattice Property
• all lattice site are equivalent
• there is only one lattice site per unit set
CHEM 793, 2008 Fall
Lattice vector : t = ua + vb + wc
Centering vectors
• we can place additional lattice sites at the endpoints of so-called centering
vectors.
• the possible centers are (0,1/2,1/2), (1/2,0,1/2),(1/2,1/2,0),(1/2,1/2,1/2), body
center ( I-center) and face center (I-center)
CHEM 793, 2008 Fall
• Lattice vector : t = ua + vb +
wc
• Combining primitive lattice
and centered lattice, there are
14 types of lattice, derived by
August Bravais in 1850, called
Bravais lattice
• Detailed crystal system is
illustrated in the International
Tables for crystallography,
Volume A.
• In this class, we only use the
simple structures: simple
cubic, face-centered cubic,
body-centered cubic, and the
close-packed hexagonal
CHEM 793, 2008 Fall
The simple cubic structure (SC)
Concept of Coordination Number
Ex: α-polonium
For Example :
In a Simple Cubic system
a Central Atom is surrounded by 6 atoms that are called
“First nearest neighbors”
Simple Cubic
2r = a
r = Radius of atom
Coordination Numbers (CN =6)
a = Lattice Parameter
a
2r
To illustrate this we need to put together 8 unit cells of Simple
as shown and mark one central atom and show CN
CHEM 793, 2008 Fall
Constructing simple cubic structure
• Simple cubic structure is
obtained by layers directly
stacking on the top of each
other.
• This is an uncommon
structure, ex. α-Polonium
Practice to construct crystal structure: go home and
use tennis or pen pang ball.CHEM 793, 2008 Fall
The body-centered cubic structure
(BCC): ex. Cr, Mo, and Fe
a
In BCC: A Central Atom
a≠2r
is surrounded by 8 atoms that are
the “First nearest neighbors”
CN=8 for
BCC
Hard Ball
Model
Stick and
Ball Model
Pl. Note: CN can be shown with one unit cell
CHEM 793, 2008 Fall
Face Centered Centered Cubic - FCC
Bravais Lattice: FCC
Typical Materials: Au,Ag,Cu,Pt
CHEM 793, 2008 Fall
Closed-packed hexagonal lattice-hcp
• It is not true unit cell
• six fold symmetry
• it is commonly used as unit cell
• ABAB stack model
Unit cell
• has three closed-packed
directions
CHEM 793, 2008 Fall
***CRYSTAL DIRECTIONS - REVIEW
Crystal Direction Rules
[uvw]=[112]
z
[uvw]=[112]
[uvw]=[111]
z
1
1
x
1
y
y
x
Now, you can
operate through
one unit cell.
y
x
1 1 2
2 2 2
CHEM 793, 2008 Fall
1 1 1
2 2
[uvw] = [1 1 1]
= [+1 +1 +1]
Negative Crystal Direction Rules
[uvw] = [1 1 0]
= [+1 +1 0]
z
1
o
1 o
y
1
x
1
[uvw]
= [1 1 1]
= [-1 -1 +1]
o
y
-1
x
1
-1
[uvw] = [ 1 1 0 ]
= [-1 -1 0]
1
O’
o
1
New Origin =O’
CHEM 793, 2008 Fall
O’
New Origin =O’
Crystal Planes (Miller Indices)
Consider a Point in space whose coordinates are [u v w] intercepting the coordinate
system. From Analytical Geometry we have the following:
z
x=u=1/h
[u,v,w]
O
y
y=v=1/k
x
z=w=1/l
x y z
+ + =1
u v w
1
1
1
 u  .x +  v  . y +  w  .z = 1
 
 
 
h .x + y .kl + l .z = 1
1
as   = u
h
CHEM 793, 2008 Fall
1/k =v 1/l =w
Crystal Planes (Miller Indices)
Representation
of (111) Plane
z
1
1 o
x
•Origin = o
1
y
x
1
Representation
of (110) Plane
1
y
x
1
1
y
In the case (110):Please note
that in the z-direction the plane
is parallel to the axis. So the
plane never meets the z axis.
•Origin = o
•Take intercepts along x, y and z. For
•Take intercepts along x, y and z. For
example if the a plane is (111) … 1
example if the a plane is (111) … 1
intercept in x dir., 1 intercept in y
intercept in x dir., 1 intercept in y
direction and 1 intercept in z dir. [these
direction and 1 intercept in z dir. [these
Fall
terminate at the green (colored) dots]CHEM 793, 2008
terminate
at the green (colored) dots]
Crystal Planes (Miller Indices)
So, if you are asked to identify a plane, say (111)
(111) Plane
o 1
1
110 Plane
y
111
111
1
1
y
x
Given the photo of the plane as above, you are asked to
index the planes or you want to know (hkl).
Procedure: Take intercepts along axes
1.x=1, y=1 and z=1. [these terminate at the green
11
1
(colored) dots]
1 1 1inf1 inity
1 111
=
2.A common numerator is
hk l 111
3.The plane = ( h k l) = (1 1 1)
x
Given the photo of the plane as above, you are
asked to index the planes or you want to know
(hkl).
Procedure: Take intercepts along axes
1.x=1, y=1 and z=∞. [these terminate at the
green (colored) dots, except for z-axis]
11
1
2. Plane is:
1 1 inf inity
3.The plane = ( h k l) = (1 1 0)
4. {111} includes a set of planes (111), (1-11),(-111) and
(11-1)
CHEM 793, 2008 Fall
Zone (Axis) Law
•{hkl} planes of a zone are all
parallel to one line: the zone
axis [uvw]
[uvw]
•The normals to {hkl} are
normal to [uvw]
(hkl)
[hkl][vuw]=0
{hkl}
If a plane (hkl) lies in a zone [uvw] (i.e. [uvw]
is parallel to plane (hkl):
hu+kv+lw=0
CHEM 793, 2008 Fall
Hexagonal plane Indices (hkil)
(01-10)
(10-11)
CHEM 793, 2008 Fall
Hexagonal Directions [UVTW]
t=-(u+v)
(10-11)
(01-10)
[100] or [2-1-10]
U=1/3(2u-v); V=1/3(2v-u); T=-1/3(u+v);W=w
CHEM 793, 2008 Fall
Prof. Barrett’s Book