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Aims
On completion of this TLP you should:


Understand why the spots on an electron diffraction pattern appear where they do.
Know how to index a diffraction pattern from a sample with a known lattice.
Introduction:
Electrons can act as waves as well as particles; this is a consequence of
quantum mechanics. A series of electrons hitting an object is exactly
equivalent to a beam of electron waves hitting the object and it produces a
diffraction pattern in the same way as a beam of X-rays does.
The two important differences between electron and X-ray diffraction are that
(1) electrons have a much smaller wavelength than X-rays, and (2) the sample
is very thin in the direction of the electron beam (of the order of 100 nm or less)
- it has to be thin so that enough electrons can get through to form a diffraction
pattern without being absorbed. These factors conspire to have a fortunate
effect on the Ewald sphere construction (see The Ewald sphere in the X-ray
Diffraction TLP) and diffraction pattern:
1. The thin sample makes the reciprocal lattice points longer in the
reciprocal direction corresponding to the real-space dimension in which
the sample is thin:
It should be noted that there is not necessarily always a particular plane
oriented like this. However, it is usual for identification of crystalline
phases in a sample to orient the sample so that the electron beam is
parallel to a low index lattice direction, as this makes the electron
diffraction pattern easier to interpret.
2. The small electron wavelength makes the radius of the Ewald sphere
very large (recall its radius is 1/). The small electron wavelength also
makes the diffraction angles  small (1-2°); this can be seen by
substituting a wavelength of 2.51 x 10-12 m into the Bragg equation (see
The Bragg law in the X-ray Diffraction TLP).
These make the Ewald sphere diagram look like this so that whole layers of
the reciprocal lattice end up projected onto the film or screen:
Note that the large (strong) spot in the middle is the straight-through beam (the
beam which has passed through the sample without diffracting). This always
has the index 000.
Caution 1: systematic (kinetic) absences appear in electron diffraction patterns
just as in X-ray diffraction patterns, for the same reason: the various features
of the lattice or motif diffract electrons in the same direction but the phase
factors from the various features cancel, leaving an absence.
Caution 2: sometimes where there should be a systematic absence, the spot
appears to be still there. This is because of the strong interaction between
electrons and atoms: there is a small but significant probability that an electron
will be diffracted twice, from two planes one after another - i.e. in two different
reciprocal lattice directions one after another. These two directions can add up
so that the twice-diffracted electron may arrive at a position in reciprocal space
where there is a systematic absence. As an example, the diagram below is a
schematic of the [011] electron diffraction pattern of silicon: the 200 type
reflections are systematically absent. The intensity at the 200 reflections is
caused by double diffraction (arising from the addition of the two reciprocal
lattice vectors shown). Thus, in words, intensity can occur in the 200 reflection
from, firstly, diffraction from the 11 planes, followed by, secondly, difraction
by the 1 1 planes as the electron wave passes throught the specimen.
Mathematics relating the real space to the electron
diffraction pattern
The distance, rhkl, on the pattern between the spot hkl and the spot 000 is
related to the interplanar spacing between the hkl planes of atoms, dhkl, by
the following equation:
(Derivation)
where L is the distance between the sample and the film/screen.
We can therefore say that the diffraction pattern is a projection of the
reciprocal lattice with projection factor L, because reciprocal lattice vectors
have length 1/dhkl.
Relation 2
Since the diffraction pattern is a projection of the reciprocal lattice, the angle
between the lines joining spots h1k1l1 and h2k2l2 to spot 000 is the same as
the angle between the reciprocal lattice vectors [h1k1l1]* and [h2k2l2]*. This is
also equal to the angle between the (h1k1l1) and (h2k2l2) planes, or
equivalently the angle between the normals to the (h1k1l1) and (h2k2l2)
planes. This angle is  in the diagram below.
Using these two relations between the diffraction pattern and the reciprocal
lattice, we are now able to index the electron diffraction pattern from a
specimen of a known crystal structure.
The two pages linked to here refer only to indexing the central region of the
diffraction pattern - the rest will be dealt with later.
Indexing with the orientation of the electron beam
back
known
From the Ewald sphere diagram, we know that the zero order Laue zone
(ZOLZ) contains reflections hkl where hu + kv + lw  0 (the Weiss zone law).
This ZOLZ can be identified by finding two reciprocal lattice vectors in the
ZOLZ. Suppose these two reciprocal lattice vectors are h1a* + k1b* + l1c*
and h2a* + k2b* + l2c*. Then we know
h1u + k1v + l1w  0
and
h2u + k2v + l2w  0
and that the angle between these reciprocal lattice vectors is the angle
between the h1k1l1 and h2k2l2 planes.
Other reflections in the electron diffraction pattern can then be deduced from
simple vector addition, with the proviso that the indices of the reciprocal
lattice vectors are integers and that they are not forbidden by the lattice. The
pattern can then be built up manually or by computer.
Example of indexing with a known electron beam
back
orientation
Suppose the material under examination is copper, and suppose the electron
beam direction is [211]. Copper has a cubic close packed structure with a
lattice parameter, a, of 0.361 nm. Allowed reflections must have h,k,l either
all even or all odd. Thus the planes with the highest interplanar spacings
(and hence those that give rise to reflections with the smallest rhkl values) are
{111}, {200}, {220}, {311}, {222}, etc.
Looking at the {111} planes, it is apparent that the Weiss zone law is obeyed
for ( 11) when [uvw]  [211]. Hence 11 is a possible reciprocal lattice
vector.
No {200} plane will obey the Weiss zone law for [uvw]  [211], but of the
{220} planes it is apparent that (02 ) will. Hence 02 is a second possible
reciprocal lattice vector.
The angle between the 11 and 02 reciprocal lattice vectors is 90° - the
dot product of these two reciprocal lattice vectors is zero. The ratio of the
lengths of these two reciprocal lattice vectors is
. These are the two
shortest reciprocal lattice vectors in the [211] electron diffraction pattern.
Thus the pattern looks like:
Indexing with the orientation of the electron beam
back
unknown
If we do not know the beam orientation, it is rather more difficult to find which
reciprocal plane is the one projected down onto the film.
One approach is to consult tables of angles and distance ratios for the low
index reflections for the structure of the crystal we are imaging. Again, we will
use copper as an example.
Table of angles
The angles in this table are the angles between the reciprocal lattice vectors
given at the sides of the table in the appropriate row and column. Such a
table can be extended to include planes with negative indices.
111 200
220
113
222
133
111 -
54.7° 35.3° 29.5° collinear 22.0°
200 -
-
45.0° 72.5° 54.7°
76.7°
220 -
-
-
64.7° 35.3°
50.0°
113 -
-
-
-
29.5°
26.0°
222 -
-
-
-
-
22.0°
133 -
-
-
-
-
-
Table of distance ratios
The dimensionless numbers in the central portion are the ratios of the 1/dhkl
values for the planes given at the sides of the table (column/row).
1/dhkl (nm-1) 111 200 220 113
1/dhkl (nm-1)
222 133
4.8 5.54 7.83 9.19 9.60 12.07
111
4.8
1
1.15 1.63 1.91 2.00 2.52
200
5.54
-
1
1.41 1.66 1.73 2.18
220
7.83
-
-
1
1.17 1.22 1.54
113
9.19
-
-
-
1
1.04 1.31
222
9.60
-
-
-
-
1
1.26
133
12.07
-
-
-
-
-
1
When making such tables spots forbidden by the lattice type should be
excluded. Thus for copper, which has an F lattice, the reflections are those
with h,k,l all even or all odd.
Now we pick two spots on the diffraction pattern and measure the angle
between them and the ratio of their distances from the 000 spot - and see if
they correspond to any of the values in the tables.
View an example
Once we know for sure what two of the non-collinear dots are, we can index
the rest of the pattern by vector addition.
Example of indexing with an unknown electron beam
orientation
If  were 72.5° and we were to measure the ratio x/y and find it to be
back
numerically equal to 1.66 then we could be reasonably convinced that the
dot at Y could be labelled as the 200 spot, and the dot at X could be
labelled as 113. In this case our predicted electron beam direction is in the
direction common to the 200 and 113 planes, i.e. [03 ]. This particular
electron diffraction pattern has a central rectangular repeat. If this is
correct, further spots on this diffraction pattern can be indexed in a
self-consistent manner by vector addition.
Laue zone
So far we have been looking at the central region of the diffraction pattern. This
is only a part of the total diffraction pattern. If we look again at the Ewald
sphere construction, we have:
We have been indexing the portion in the middle with the 000 spot in it.
However, there are also areas of diffraction spots at the edges of the film,
caused by the Ewald sphere intersecting points in an adjacent parallel plane
containing reciprocal lattice points. (If the film was small or the camera length
large it is possible that it did not catch these spots at the side, so that we
sometimes only have the middle part.)
These outlying parts of the diffraction pattern are called Higher Order Laue
Zones (HOLZs). Each of the HOLZs can be described by an equation of the
general form
hu + kv + hw  N
where:
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

N is always an integer, and is called the order of the Laue zone.
[uvw] is the direction of the incident electron beam.
hkl are the co-ordinates of an allowed reflection in the Nth order Laue zone.
The middle part of the diffraction pattern, with 000 in it, is the zero order Laue
zone (ZOLZ), because it comes from the plane for which N0: an allowed
reflection hkl in the ZOLZ is joined to the origin 000 by a reciprocal lattice
vector that lies in the ZOLZ. For the ZOLZ the electron beam [uvw] and the
allowed reflection hkl satisfy the Weiss zone law hu + kv + lw  0. The next
layer up has a value N1, then N 2, and so on, as shown.
From the geometry of the way in which the Ewald sphere intersects the HOLZs,
the radius of the Nth HOLZ ring, Rn, in reciprocal space, is given to a very good
approximation by the formula
assuming that the wavelength of the electrons is much less than the modulus
|uvw| of the direction [uvw] in the crystal parallel to the electron beam direction.
Thus, HOLZs are seen more easily at lower voltages (e.g. 100 kV rather than
300 kV) and when the electron beam is parallel to a relatively high index
direction in a crystal.
It is possible to index the reflections in the HOLZs on a diffraction pattern.
Examples of such indexing are given in the book Transmission Electron
Microscopy of Materials by D B Williams and C B Carter.
Kikuchi lines
Kikuchi lines often appear on electron diffraction patterns:
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An example of a "two-beam" electron diffraction pattern with a number of Kikuchi lines.
A pair of Kikuchi lines is arrowed.
[The term "two-beam" denotes the fact that the straight-through beam, 000, and one
diffraction spot are both diffracting very strongly. The intensity of all spots in this
electron diffraction pattern are significantly weaker by comparison with these two
beams.]
(Click on image to view larger version)
We will not learn to index the Kikuchi lines in this TLP. Instead, we will explain
their origin and behaviour with the help of the following animation.
Kikuchi lines are interesting because of what they do when the crystal is
moved in the beam. Diffraction spots fade or become brighter when the crystal
is rotated or tilted, but stay in the same places; the Kikuchi lines move across
the screen.
The difference in behaviour can be explained by the position of the effective
source of the electrons that are Bragg-scattered to produce the two
phenomena. The diffraction spots are produced directly from the electron
beam, which either hits or misses the Bragg angle for each plane; so the spot
is either present or absent depending on the orientation of the crystal. The
source of the electrons that are Bragg-scattered to give Kikuchi lines is the set
of inelastic scattering sites within the crystal. When the crystal is tilted the
effective source of these inelastically scattered electrons is moved, but there
are always still some electrons hitting a plane at the Bragg angle - they merely
emerge at an angle different to the one that they did before the crystal was
tilted.
Using polycrystalline materials in the TEM
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Just as with X-rays, a completely isotropic fine-grained polycrystalline sample
will give a diffraction pattern of concentric rings in the zero order Laue zone
(ZOLZ), as the many small crystals at random orientations produce a
continuous angular distribution of hkl spots at distance 1/dhkl from the 000 spot
- a ring of radius 1/dhkl around the 000 spot for each allowed reflection. The
rings are then indexed according to the order of allowed reflections within the
ZOLZ.
As the grain size increases, the rings within the diffraction pattern break up into
discontinuous rings containing discrete reflections. If there is any texture
(preferred orientation) within the specimen, arcs may be seen instead of
complete rings.
Convergent beam electron diffraction (CBED)
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When a convergent beam is used instead of a parallel beam of electrons, the
rays converge to a point within the specimen and come out the other side
inverted like a camera. However, we do not look at the inverted image; we look
at the diffraction pattern, with the spots magnified:
Depending on the camera length chosen, either the zero order Laue zone can
be examined or the zero order Laue zone and higher order Laue zones. Two
examples of CBED images are shown below. The symmetry seen is such
patterns can be related to the space group symmetry of the specimen.
Examples of CBED images:
Diffraction pattern showing a zero order Laue
zone
with a mirror in the pattern as shown
(Click on image to view larger version)
Diffraction pattern showing a first order Laue
zone
(Click on image to view larger version)
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Using other methods in conjunction with electron
diffraction
Electron diffraction is a powerful technique - but other techniques must be
used with it to put the results in context. This is a brief synopsis of how other
methods can be used to help.
Optical imaging
This is a very important way of analysing a specimen. Using the naked eye
and optical microscopes we can determine down to a point-to-point resolution
limited by the wavelength of light how many phases there are and how they
relate to one another. We can also infer what type of material they are likely to
be and how they may have been processed.
Chemical analysis
A wide range of chemical techniques can be used to find out what components
are present in the different phases and in what proportions. This will narrow the
field of possible elements that we need to consider when analysing our
diffraction results. These techniques range from simple chemical tests, through
infrared spectroscopy of organic samples, to a wide variety of chemical
characterisation techniques that can often be performed within the
transmission electron microscope.
TEM imaging
Using the TEM to image the same area of sample that is being used to
produce the diffraction pattern is an invaluable technique:
Nitrided surface layer of austenitic stainless
steel
(Click on image to view larger version)
Diffraction pattern from nitrided surface layer
of austenitic stainless steel
(Click on image to view larger version)
Using the image to verify that the double dots in the diffraction pattern are
being caused by the two crystal structures either side of the twin boundary, we
can index the pattern and determine the twin plane and the crystal structures
either side of it.
Summary
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In this teaching and learning package we have considered how electron
diffraction patterns are formed in the transmission electron microscope. The
principles of how to index spot electron diffraction patterns have been
discussed in some detail. Although we have considered how to index
electron diffraction patterns from relatively simple crystal structures to
illustrate the basic principles, these principles are generic and can therefore
be applied to any crystal structure. We have also considered other features
of electron diffraction patterns such as the formation of Kikuchi lines, the
formation of convergent beam electron diffraction patterns and the
formation of higher index Laue zones.