Table Of ContentDiffraction and
Imaging Techniques
in Material Science
Volume I: Electron Microscopy
Second, revised edition
Editors
S. Amelinckx, R. Gevers, J. Van Landuyt
State University of Antwerp, Belgium
NORTH-HOLLAND PUBLISHING COMPANY · AMSTERDAM - NEW YORK - OXFORD
© NORTH-HOLLAND PUBLISHING COMPANY - 1978
All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photo
copying, recording or otherwise, without the prior permission of the Copyright owner.
ISBN Volume I : 0 444 85128 3
Volume II: 0 444 85129 1
Set : 0 444 85130 5
Publishers:
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM - NEW YORK - OXFORD
Sole distributors for the U.S.A. and Canada:
ELSEVIER NORTH-HOLLAND, INC.
52 VANDERBILT AVENUE
NEW YORK, N.Y. 10017
First edition 1970
Second, revised edition 1978
Reprinted 1979
Library of Congress Cataloging in Publication Data
Main entry under title:
Diffraction and imaging techniques in materials science.
Comprises new contributions and revised and updated
papers originally presented at the International Summer
Course on Material Science, Antwerp, 19&9> a dn published
in 1970 under title: Modern diffraction and imaging
techniques in material science.
Includes bibliographical references and index.
CONTENTS: v. 1. Electron microscopy.—v. 2. Imaging
and diffraction techniques.
1. Electron microscopy—Congresses. 2. Electrons—
Diffraction—Congresses. 3. Imaging systems—Congress
es. I. Amelinckx, Severin. II. Gevers, R. III. Lan
duyt, J. van. IV. International Summer Course on
Material Science, Antwerp, I969. Modern diffraction and
imaging techniques in material science.
TA1+17.23.D5U 620.1f12 7 78-22081
ISBN O-IM-8513O-5
PRINTED IN THE NETHERLANDS
PREFACE TO THE FIRST EDITION
This book contains the proceedings of a summer school sponsored by NATO
and held at the University of Antwerp during the period from July 28th to
August 8th, 1969. The objective of the school was to teach at an advanced level
the recent developments in "Diffraction and Imaging Techniques" which are in
creasingly being used in the study of materials.
The school attracted wide interest and a number of applicants had unfortu
nately to be refused in view of the limited accommodation facilities. It was there
fore felt that the proceedings should be published rapidly and in a permanent
form making them accessible not only to the participants but also to people
which had not been able to attend the course so as to give them an opportunity
to benefit from the lectures given by the best experts in their respective fields.
Although the book reflects inevitably the diversity of viewpoints of the dif
ferent authors, the arrangement of the material is such that it will constitute a
consistent treatment requiring a minimum of background knowledge. In most
cases this background knowledge is provided in introductory lectures.
The organizing committee is grateful to the different authors for their colla
boration in editing this book.
We also gratefully acknowledge the financial help of NATO, and the help of
the University of Antwerp in providing the necessary facilities for lecturing and
for housing the students.
The Organizing Committee
ν
PREFACE TO THE SECOND EDITION
The first edition of this book has been very well received by the scientific
community; it has been widely used as a textbook for courses on diffraction
and solid state electron microscopy. However the first edition was completely
sold out a few years ago and many orders have had to be refused.
It was therefore felt that a new, revised and completed edition would be
very much welcomed by the many users.
Nearly all authors have updated and reworked their contributions and new
contributions in recently developed fields have been added so as to maintain
the spirit and scope of the original edition. It has therefore become necessary
to publish the book in two volumes.
The editors wish to express their appreciation to the authors for making a
real effort to make the book an up to date textbook once again.
It is hoped that this second edition will meet with the same success as the
first one.
The Editors
vi
Diffraction and Imaging Techniques in Material Science,
eds. S. Amelinckx, R. Gevers and J. van Landuyt
©North-Holland Publishing Company, 1978
GENERAL REVIEW OF THE EXPERIMENTAL METHODS
FOR THE DETERMINATION OF ATOMIC STRUCTURES
A.GUINIER
Laboratoire de Physique des Solides, Universite Paris-Sud, Orsay, France
The different chapters of this book are devoted to various methods able
to furnish information on the atomic structure of matter at an atomic scale.
Generally speaking, all these methods are based upon some interaction of
matter with radiations of different kinds.
The necessary condition is that the resolving power of any of these de
vices be sufficient to allow the localization of the individual atoms. A gen
eral rule in optics is that the limiting value of the resolving power (i.e. the
shortest distance between two points which can be distinguished) is of the
order of the wave-length of the radiation used. Every atom having a diam
eter of the order of 1 A, that implies that one must use radiations of a wave
length of 1A or shorter.
1, Classification of methods
We can now classify the possible methods of observation by making a list
of radiations within this range of wave-lengths.
(1) Among the electromagnetic radiations, we find X-rays and fortunately
the desired wave-lengths (0.1 —1 A) correspond to radiations which are both
easy to produce, easy to detect and have a suitable interaction with matter.
X-rays of longer wave-length (10 to 100 A) are so easily absorbed in the matter
that they cannot reach the sample to be studied and X-rays of shorter wave
length (0.01 to 0.1 A) require the use of cumbersome and expensive high-
tension generators.
(2) Besides these electromagnetic radiations, we have at our disposal the
radiations associated to beams of various particles, the wave-length of which
is given by the de Broglie formula
3
4 A.GUINIER
λ =— or λ= 7F~ (0
Ε being the kinetic energy of the particle of mass m. Numerically, for elec
trons, with Ε expressed in electron-volts (£=eV) and λ in Angstroms (A)
λ= 12.5/VF.
So for electrons, the 1A wave-length corresponds to electrons accelerated by
a tension of a few hundred volts. Such electrons are stopped by one or two
atomic layers: thus they can only be used for the observation of the struc
tures of surfaces (LEED methods, p. 553). With an increasing acceleration
tension, the electrons are able to pass through layers of 100 to 1000 A
which allows the observation of a three dimensional structure. The wave
length of such electrons is of the order of 0.1 A; it is well known that tech
nically the production and the detection of 10 to 100 kV electrons do not
encounter special difficulties. In fact, electrons are now the agents of very
effective means for the observation of atomic structures (p. 355).
(3) For heavy particles, the same wave-length is obtained, according to
formula (1) when the particles have much lower energy. For instance in a
neutron beam of 1A wave-length, the neutron has a velocity of 4000 m/sec
and an energy of 0.08 eV. This is the energy of particles in thermal equili
brium at a temperature of 600°K. Neutrons have a special interest for the
study of atomic structures because their interaction with the atoms is of a
quite different nature from the interaction of atoms with photons or charged
particles. Thus the neutrons "see" different aspects of the atoms (p. 593). Fur
thermore the absorption of neutrons in matter is very low. Thus an entire speci
men of very large size may be observed (a few centimeters instead of a frac
tion of millimeters for X-rays, or a fraction of microns for electrons). Of
course, the use of neutrons cannot be general in every Solid State Physics
laboratory, since it is restricted to the laboratories attached to the few big
research reactors in operation in the world.
(4) Some charged heavy particles are also utilized but protons have a
serious inconvenience; they produce serious damages in the matter under ex
amination and so the observed structure may not be the primitive one. But
helium ions are used in a very interesting instrument (field-ion microscope,
p. 811).
METHODS FOR THE DETERMINATION OF ATOMIC STRUCTURES 5
2. Image formation and diffraction techniques
The value of the wave-length of the radiation gives an unavoidable limita
tion of the resolving power but the physical properties of the radiation must
make possible the realisation of a practical device able to approach as far as
possible the theoretical limit.
The most obvious method is the formation of an image, as in an optical
instrument, with a magnification high enough to show distinctly the details
which can be separated. But with some of the radiations which enter the
category of the "possible" radiations, no image forming device may be real
ised. Thus X-rays propagate in straight line and can be deviated neither by
reflection nor by refraction at the interface of two material mediums. So,
the rays issued from a point source cannot be focused in a point image and
- at least up to now - no X-ray microscope is available. In fact, reflection of
X-rays indeed occurs on a plane surface, but only if the angle of incidence is
lower than a few minutes and the attempts to construct an X-ray microscope
with an interesting resolving power have not been successful. The situation is
the same for the neutrons.
On the other hand, the trajectory of a charged particle may be altered by
the forces exerted by electrical or magnetic fields. Combination of such fields
may lead to an image-forming system and so microscopes, even with a very
high magnification, are possible. These instruments (electron microscopes and
ion-microscopes) are now essential tools for Solid State physicists (pp. 107
and 811).
The formation of images is not the only way to collect information about
the structure of an object. There is a second possibility, much more general
since it is applicable for all kinds of radiations, the phenomena of diffraction.
The object is bathed in a coherent plane wave; the different atoms receiving
this primary radiation become sources of scattered radiation; the total dif
fracted radiation results from the interferences between the ensemble of the
coherent secondary sources. The repartition of the diffracted radiation in the
different directions of space depends upon the nature of the interaction of
the atoms and the primary beam and on the arrangement of the atoms in the
object. Thus the knowledge of the diffracted radiation gives information on
the atomic structure of the object and, even in some cases, that is sufficient
to build a model of the structure; but this is possible only because the wave
length of the primary beam is short enough, the concept of the resolving power
used in the case of image being also valid for the diffraction phenomena. But
in the general case, the information contained in the diffraction data do not
allow to give an unambiguous structural model. Nevertheless diffraction is in
the present state of the technique the most direct approach to this model
(pp. 593, 374 and 399).
6 A.GUINIER
The scattering due to the individual atoms being provoked by the incident
beam, the energy of the scattered radiation is substracted from the incident en
ergy. The characteristics of the diffraction phenomena depend on the value of
the ratio of these two energies. If the scattered energy is very small relative to
the incident energy, one can admit that the wave-field after diffraction is
simply the addition of the incident unperturbed wave plus the scattered radia
tion. That is called in quantum mechanics the Born approximation and for dif
fraction the "kinematical" case. This approximation is generally valid for X-
rays but there are some exceptions, rare but very interesting: when the dif
fracting body is a large and perfect crystal, i.e. when the atoms, in volumes of
the order of [10 to 100 Mm]3 are located at the nodes of one single lattice,
the intensities of the diffracted beam and of the incident beam become com
parable. Another theory of diffraction — called dynamical — is then valid: the
diffraction phenomena are profoundly altered (pp. 43 and 462). In electron
diffraction, the cases where the dynamical theory has to be applied are much
more frequent.
In electron microscopy, the image formation is not independent from the
diffraction phenomena. The contrast in the image corresponds to variations
from point to point of the transmission factor of the electrons through the
object, and the diffraction of the direct beam is a very important cause of ab
sorption.
In conclusion, table 1 gives the summary of this rapid review.
Table 1
Nature of radiations Methods of observation
Particles Wave-length
X-ray photons 1A diffraction
Electrons ( ^ e ny e l g 0.05A microscopy diffraction
1 low energy 1A diffraction
Neutrons 1-10A diffraction
Ions microscopy
Diffraction and Imaging Techniques in Material Science,
eds. S. Amelinckx, R. Gevers and J. van Landuyt
© North-Holland Publishing Company, 1978
KINEMATICAL THEORY OF ELECTRON DIFFRACTION
R.GEVERS
Laboratoria van het SCK, Mol-Donk, Belgium
1. Introduction
It is well-known that accelerated electrons incident on a sufficiently thin
crystal in a parallel monochromatic beam are not only transmitted without
change in direction, but also emerge at the back surface in a number of dis
crete directions"
The task of the theory is to calculate the relative number of the electrons
in each of the so-called "diffracted beams".
It is also known that this property is shared by X-ray photons and by
neutrons, and that the phenomenon is due to the translation symmetry of the
crystalline matter. Nevertheless it turns out that the details of the observations
are different for the three types of radiation. This is rather fortunate, since it
allows to gather complementary informations about the substance under study
by the use of the three different techniques.
\A.The Bragg-law
The common feature of the diffraction of the three different types of
particles is the direction of the diffracted beam with respect to the incident
direction and the orientation of the crystal plate. This is rather evident since it
is only determined by the translation symmetry of the crystal.
The wave vectors k and k of the incident and the scattered beam satisfy
0
the well-known Bragg law.
k-k = g (1)
0
where g is any reciprocal lattice vector.
9
10 R.GEVERS
Since only elastic scattering is considered, one has
* = |k| = |k|= 1/λ (2)
0 0
where λ is the wave length.
The Bragg law expresses that the scattering amplitudes of the scattering
events by a very large number of scattering centres, arranged at the nodes of
a regular lattice, are perfectly in phase with each other.
1.2. Reflection sphere construction
The Bragg law (1) can be interpreted with the help of the widely used
Ewald or reflection sphere (see fig. 1). The latter is a sphere with radius 1/λ.
If C is its centre and if CO = k , then Ο must be thought to be the origin of
0
the reciprocal lattice. To any reciprocal lattice node G, defined by the reci
procal lattice vector g, lying on the reflection sphere, corresponds then a dif
fracted beam with wave vector k= CG.
The angle between k and k is noted as 20, and 0 is called the Bragg
0 B B
angle.
1.3. Bragg reflection on lattice planes
An alternative way of interpreting (1) is as follows.
The reciprocal lattice vector g can be written as
g=/*A+A:B + /C (3)
where A, B, Care the base vectors of a unit cell of the reciprocal lattice.
c
Fig. 1. Reflection sphere construction.
