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Physical Methods for Materials Characterisation Second Edition
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Series in Materials Science and Engineering
Physical Methods for Materials Characterisation Second Edition
P E J Flewitt Nuclear Electric plc and
R K Wild (formerly Nuclear Electric plc) Interface Analysis Centre, University of Bristol
Institute of Physics Publishing Bristol and Philadelphia
Copyright © 2003 IOP Publishing Ltd.
# IOP Publishing Ltd 2003 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, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0808 7 Library of Congress Cataloging-in-Publication Data are available
Series Editors: B Cantor and M J Goringe Commissioning Editor: Tom Spicer Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing: Nicola Newey and Verity Cooke Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset by Academic+Technical, Bristol Printed in the UK by MPG Books Ltd, Bodmin, Cornwall
Copyright © 2003 IOP Publishing Ltd.
To our wives Ann & Gillian
Copyright © 2003 IOP Publishing Ltd.
‘‘Whence it is that nature does nothing in vain; and whence arises all that order and beauty which we see in the world’’ Isaac Newton 1642–1727
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Contents
Preface to second edition
x
Preface
xi
1 Introduction 1.1 Introduction 1.2 Atom bonding 1.3 Ceramics 1.4 Semiconductors 1.5 Glasses 1.6 Metals and alloys 1.7 Polymers 1.8 Composite materials 1.9 Microstructure 1.10 References
1 1 1 6 8 9 11 11 13 14 18
2 Interaction of radiation with materials 2.1 Radiation sources 2.2 Penetration depths 2.3 Material damage 2.4 Resolution 2.5 Loss processes 2.6 Atom and ion processes 2.7 Effect of high electric fields 2.8 Acoustic phenomena 2.9 References
21 21 22 33 38 38 47 48 49 50
3 Vacuum systems 3.1 Introduction 3.2 Kinetic theory of gases 3.3 Production of vacuum 3.4 Vacuum pumps
53 53 53 55 60 vii
Copyright © 2003 IOP Publishing Ltd.
viii
Contents 3.5 3.6 3.7 3.8
Pressure measurement Leak detection Specimen handling References
68 73 74 74
4
Diffraction 4.1 Electromagnetic radiation 4.2 Photons 4.3 X-ray diffraction 4.4 Electron diffraction 4.5 References
76 76 78 89 129 171
5
Photo/electromagnetic sources 5.1 Introduction 5.2 Resolution 5.3 Lens defects 5.4 Light microscopy 5.5 Laser microscopy 5.6 Acoustic microscopy 5.7 Infrared microscopy 5.8 X-ray microscopy 5.9 X-ray topography 5.10 X-ray photoelectron spectroscopy 5.11 Autoradiography 5.12 Mo¨ssbauer spectroscopy 5.13 Nuclear magnetic resonance 5.14 Total reflection X-ray fluorescence spectroscopy 5.15 References
176 176 177 179 182 199 213 223 226 233 236 254 256 263 266 268
6
Electron sources 6.1 Introduction 6.2 Scanning electron microscopy 6.3 Electron probe microanalysis 6.4 Transmission electron microscopy 6.5 Electron energy-loss spectrometry 6.6 Auger electron spectroscopy 6.7 References
274 274 274 297 324 393 411 440
7
Atom/ion sources 7.1 Introduction 7.2 Ion scattering spectroscopy 7.3 Rutherford backscattering 7.4 Proton backscattering 7.5 Secondary ion mass spectroscopy
451 451 452 457 463 463
Copyright © 2003 IOP Publishing Ltd.
Contents 7.6 7.7 7.8 7.9 7.10 7.11
Sputtered neutral mass spectroscopy Field ion microscopy Scanning probe microscopy Particle induced X-ray emission Glow discharge spectroscopy References
ix 489 493 501 513 517 519
8 Application of computers 8.1 Introduction 8.2 Instrument control 8.3 Computer aided instruction 8.4 Data acquisition 8.5 Data processing and analysis 8.6 Image quantification 8.7 Data bases 8.8 Data transfer 8.9 Expert systems 8.10 Computer simulation 8.11 Future developments 8.12 References
523 523 524 525 527 530 542 556 558 558 562 568 570
Appendix 1
574
List of symbols used in book
Appendix 2.1
Commonly used conversion factors
576
Appendix 2.2
Wavelength of selected radiation sources
577
Appendix 3
Physical constants
578
Appendix 4
Acronyms for techniques
574
Appendix 5
Electron structure of elements
583
Index
Copyright © 2003 IOP Publishing Ltd.
587
Preface to second edition
Over the period since this book was first published in 1994 there have been, as anticipated, advances in the field of the physical methods used to characterise materials. These developments have been made to reflect the requirements and needs to understand the interrelationship between the microstructure and the physical, mechanical and chemical properties of materials. In particular the emphasis from the electronic industry on nanoscale materials has resulted in significant developments in the scanning probe techniques originally proposed by Binnig and Roher, and by Gerber and Weibel in 1982. As a consequence, for this second edition of the book the authors have attempted to reflect incremental developments as well as the more significant techniques that have emerged over the past years. However, as the readers will appreciate, the principles and physics upon which many of the techniques describe are based remain unaltered and as a consequence the basic format of the book remains unaltered. A chapter that again has been subject to consideration in this context is chapter 8, where the emergence of computer technologies have a significant impact across all sections of the techniques addressed in this book.
Acknowledgments to second edition In the preparation of this second edition the authors would like to acknowledge the interaction with colleagues at Bristol University both in the Interface Analysis Centre and the Department of Physics. P E J Flewitt would like thank BNFL Magnox Generation for the secondment to Bristol University. Also to considerable positive interactions with a range of colleagues including Professors L M Brown FRS, J F Knott FRS, R Faulkner, D Bacon and G W Greenwood FRS.
Copyright © 2003 IOP Publishing Ltd.
Preface
This book was initiated as a consequence of discussions that led to the conclusion that over the past decade significant advances have been made on the range of techniques now available for interrogating the microstructure of materials. In some cases these developments have been a consequence of the flexibility offered by the ability to interface small but powerful computers to instruments to effect both instrument control and acquisition of data. This has been coupled with the ability to process the acquired data rapidly in such a way that even small signals contained within a large background noise can be used and interpreted. The developments have been promoted by the need to evaluate the microstructure of material to fulfil various technological needs such that it is necessary to ensure reproduction of the properties and to interpret features that lead to departures as a consequence of defects, such as those in the crystal structure or the fine scale chemistry. The interrelationship between the physical and mechanical properties of materials and their microstructure is being progressively developed. It is certainly evident that the properties for which materials are selected for a particular application depend upon the microstructure which, in itself, can be considered to extend to the atomic level. Microstructure is a generic term which has been used to describe the constitution of a material that can be visualised from a range of techniques extending from simple optical microscopy to those capable of atom resolution and indeed even indirect techniques such as X-ray diffraction. The interaction of electromagnetic radiation with crystalline solids is now understood in considerable detail, so it can be exploited to provide the necessary information. The concepts related to the use of light and electron imaging together with electron and X-ray diffraction are common to a range of microstructural evaluation techniques. The factors controlling both image formation and wave diffraction are described. The penetration depth of high-energy electrons, the dispersion of electrons through foils and the mean free path of slow electrons has been established for a variety of systems. This has paved the way for accurate quantitative determination of microstructural features within materials.
Copyright © 2003 IOP Publishing Ltd.
The book is directed primarily to senior undergraduate students and postgraduate workers to facilitate an appreciation of the underlying theory, the selection and application of the range of techniques available to examine a microstructural feature. It is clear that on many occasions more than one technique can be selected to provide the appropriate microstructural information; indeed it is often desirable to select a combination of complementary techniques to provide this detail. Many of the techniques described have now reached a stage of development where they are appropriate, not only to the dedicated research worker, but are of equal importance to applied research and those undertaking development used to promote and support a range of industrial and commercial activities. The content of the book has been structured to allow the reader to acquire both a background to the microstructure of materials (chapter 1) and an appreciation of the principles (chapter 2) which underline many of the techniques described in the subsequent chapters. In view of the importance that it has in many of the techniques presented, in chapter 3 we have described the control of instrument environment. Chapters 4 to 7 set out a range of techniques divided on the basis of those using diffraction (chapter 4), photon and electromagnetic sources (chapter 5) electron sources (chapter 6) and atom/ion sources (chapter 7). Finally, in view of the emphasis placed upon computers, chapter 8 is devoted, albeit briefly, to the application of computers. Throughout the book we have attempted to provide clear and simple diagrams to assist the understanding of the techniques and support this with selected examples to illustrate their use and application. In this way we hope we provide the reader with an appreciation of those techniques and procedures currently available which enable the microstructure of materials to be characterised. We would like to acknowledge the help of our colleagues with Technology Division of Nuclear Electric and Dr D A Dominey for encouraging the production of this book. P E J Flewitt would like to acknowledge the interaction afforded by the Department of Physics, University of Surrey and to express personal gratitude to Professor A G Crocker. In addition, P E J Flewitt is grateful for invaluable collaboration with Dr P Doig, Mr D Lonsdale and Mr R A Stevens over a number of years. R K Wild would like to express his thanks to all at the Interface Analysis Centre at the University of Bristol for their help and encouragement and in particular Professor J Steeds, Professor G C Allen, Dr J Day, Mr I T Brown, and Dr K Hallam. R K Wild would also like to acknowledge the help and collaboration over many years of Dr P A Tempest. Finally we would like to thank Mrs Rita Pollock for editorial assistance. P E J Flewitt and R K Wild
Copyright © 2003 IOP Publishing Ltd.
Chapter 1 Introduction 1.1
Introduction
This book provides a guide to those techniques and procedures which enable the microstructure of materials to be completely classified and characterised. As a consequence, it is appropriate to those studying and working in the interrelated fields of metallurgy, materials science, ceramics, polymer science and solid state physics. Material is the generic term used to describe physical matter in the solid state which occurs naturally or is manufactured to achieve particular physical properties and characteristics. Materials have been classified in various ways, but perhaps the simplest and most complete classification divides into two categories (table 1.1) (Bever (1986)), one based upon the nature of the material and the other upon the application. Such a classification is flexible, accommodating existing materials and perceived future materials. It is not appropriate to address each of the materials set out under the heading of nature in table 1.1 in detail, but rather to consider briefly how their atomic and molecular structure influences the mechanical and physical properties associated with some of the more important of these. It is to this nanoscale level that microstructure has to be resolved ultimately, although there are essentially many lower-resolution techniques covering the meso and microscale that assist this understanding.
1.2
Atom bonding
There is an attractive force between atoms and a repulsive force which prevents them from approaching beyond a minimum distance. The stable position for the atoms is best addressed by considering how the potential energy of a pair of atoms varies with their separation. The repulsive force gives rise to a positive potential energy which results in work being done on the system to bring the atoms closer together; this energy varies as an inverse power of the atomic separation r as A=rn . The attractive force gives a negative potential energy of the form B=rm which tends to zero when the
Copyright © 2003 IOP Publishing Ltd.
2
Introduction
Table 1.1. Classification of materials based on nature and applications (Bever (1986)). Nature
Applications
Ceramics Glasses Metals and alloys Other inorganic materials (including semiconductors) Polymers Elastomers Fibres Composite materials Wood Paper and paperboard Other biological materials
Industrial materials Electrical materials Electronic materials Superconducting materials Magnetic materials Nuclear materials Materials for other energy applications Optical materials Biomedical materials Dental materials Building materials
atoms are widely separated and increases negatively as they are brought together. The combined curve in potential energy, A=rn B=rm , as a function of the interatomic spacing is shown in figure 1.1 and this passes through a minimum. The atomic separation r0 at which this minimum potential energy occurs is the stable spacing for the pair of atoms; the negative and positive forces balance. This minimum in the potential energy arises
Figure 1.1. The potential energy of two atoms as a function of their separation r. The minimum in the potential energy at a separation r0 corresponds to the equilibrium separation.
Copyright © 2003 IOP Publishing Ltd.
Atom bonding
3
Table 1.2. A summary of physical and mechanical properties associated with interatomic bonds. Property
Ionic
Covalent
Nondirectional; Directional; Structures of high structures of low coordination coordination and low density
Metallic
Van der Waals
Nondirectional structures of high coordination and high density
Analogous to metallic bond
Mechanical Strong, hard crystals
Strong, hard crystals
Variable crystals
Weak, soft crystals
Thermal
High melting point, low expansion coefficient
High melting point, low expansion coefficient
Range of melting points, extended liquidus range
Low melting point, large expansion coefficient
Electrical
Weak insulator, Insulator in solid conduction by ion and liquid state transport when liquid
Conduction by Insulator electron transport
Optical
Absorption and other properties mainly of the individual ions
Opaque, with similar properties in liquid state
High refractive index, absorption different in solid or gas
Properties of individual molecules
because the power n in the repulsive term is greater than m in the attractive term. If m is less than n an unstable situation would develop and all the atoms collapse together. Since the mechanical and physical properties of materials are a direct consequence of their interatomic force (Cottrell (1967), Mott and Jones (1958) and Mott (1976)) (table 1.2), it is appropriate to consider the different bonding configurations that are associated with various materials and the potential crystalline and molecular structures that can form. However, it has to be remembered that the bond descriptions given here are simple and idealised. Moreover, many materials used for practical applications are in the form of either polycrystalline arrays or aggregates, rather than simple single crystals. It is the specific type of interatomic bond that leads, in solid crystals, to the development of atoms or molecules into particular periodic arrangements in three dimensions and thus specific materials. Crystals differ from liquids and gases because the atomic arrangements in the latter do not possess this periodicity. However, not all solids are crystalline; some are amorphous, such as glasses, a state that does not have any periodic arrangement of atoms. The regularity of the array can be described in terms of symmetry elements (Kelly and Groves (1970) and Barrett and
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4
Introduction
Figure 1.2. A simple point lattice defining a unit cell (Cullity (1979) (reproduced with permission of John Wiley and Sons).
Massalski (1986)) and these elements determine the directionality of the physical properties of crystals. For example, the symmetry elements reveal directions where electrical resistance in a crystal will be similar. Figure 1.2 shows a simple crystal lattice where all the cells are identical. The size and shape of the outlined unit cell can be described by three vectors, a, b and c, which define the crystallographic axes (figure 1.3). The lengths a, b and c and the angles between them , and , are the constants which describe the crystal uniquely. Figure 1.4 shows the 14 possible lattice arrangements, Bravais lattice, for crystalline materials. Figure 1.5 shows the five types of interatomic bond that can exist for all materials, either individually or in combinations. These are: (a) ionic, (b) covalent, (c) metallic, (d) molecular and (e) hydrogen. In the case of the ionic bond, the atoms either gain or lose an electron so that their outer electron shell is complete. As a consequence, the atoms are electrically charged,
Figure 1.3. The unit cell can be described by three vectors a, b and c; the lattice constants are a, b and c and , and (Cullity (1979) (reproduced with permission of John Wiley and Sons).
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Atom bonding
5
Figure 1.4. The fourteen crystal systems for all crystalline solids. Generally pure metals adopt either the body centred cubic (bcc), face centred cubic (fcc), or hexagonal close packed (hcp) packing arrangements.
either positively or negatively, and thereby attract atoms of opposite charge. For the covalent bond, pairs of atoms share outer electrons to fill the outer electron shells; this differs from the metallic bond where all atoms share the valence electrons. The molecular bond (van der Waals) arises from the displacement of charge within electrically neutral atoms or molecules
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6
Introduction
Figure 1.5. Schematic diagram showing the five types of atomic bond in materials (a) ionic, (b) covalent, (c) metallic, (d) molecular and (e) hydrogen, together with examples of each.
producing a weak attractive force between them. The hydrogen bond is weak and mediated by the hydrogen atom. It arises because hydrogen is a small atom and the charge is easily displaced.
1.3
Ceramics
The term ceramic describes those products that are made from inorganic crystalline materials and have non-metallic properties. Natural stone is a ceramic that was one of the first solid materials used by man. Indeed, stone has the characteristic properties associated with a ceramic of high hardness and strength, brittleness, low thermal and electrical conductivity together with a resistance to chemical attack. The constitution of a ceramic is usually a combination of one or more metals with a non-metallic element, usually oxygen. As a result, the atoms in a ceramic crystal are linked by a combination of ionic and covalent bonds. The combination of oxygen atoms with the metal atoms provides a strong ionic bond because oxygen, with two electron vacancies in the outer electron shell, effectively borrows two electrons from the neighbouring metal atoms. The associated ionisation of both atom species, one negatively and one positively, provides
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Ceramics
7
Figure 1.6. A silica unit, which forms the basic building block of many ceramics, consists of a silicon atom surrounded by four oxygen atoms.
strong electrostatic attraction. It is this combination of atomic bonds that establishes the stability and strength associated with a ceramic. Simple examples are ionically bonded magnesia, MgO and covalently bonded silicon carbide, SiC. These have the sodium chloride and diamond crystal structures respectively. The physical and mechanical properties of ceramics are controlled by the crystal structure and the chemical composition (Davidge (1980) and Kingery et al (1976)). This is demonstrated by considering the important but varied structures generated by silica (SiO2 ). The silicon atom, like carbon, has four valence electrons and forms a tetrahedral grouping with the oxygen atoms positioned so that four oxygen atoms surround each silicon atom (figure 1.6), and it is these groups of atoms that can link together in various ways. If attached end to end by one of the oxygen atoms a chain is formed giving fibrous asbestos (figure 1.7), whereas if built-up into sheets they produce layer minerals such as talc or mica. However, this tetrahedral grouping can link to produce a three-dimensional network, an arrangement that results in the quartz crystal. The versatility of these silica tetrahedrons in forming bonds with one another and, indeed, with other groups of atoms, explains how silica serves as the bonding material for clay particles in bricks and earthenware and bonds a glaze to porcelains.
Figure 1.7. Each atom of silicon has four valence electrons passed to the surrounding oxygen atoms, leaving the outer shell one electron short. The linking into a chain is one basic grouping leading to, for example, the asbestos fibre structure.
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8
Introduction
If ceramic crystals were of perfectly organised structures and uniform microstructure, these materials would have mechanical properties that exceed those achieved. Indeed, failure of a ceramic is generally a consequence of a microstructural defect, or combination of defects, such as inclusions, pores, voids and distributions of irregular size grains. Mechanical failure occurs from pre-existing flaws: high mechanical stresses which exceed the local tensile strength effect crack propagation from flaws followed by rupture. Apart from their known high-temperature applications, some polycrystalline electronic ceramics are used extensively by communications, electronic and appliance industries. Among the best known of the ceramics for these applications are the ZnO varistors, boundary layer capacitors, ferrites and positive temperature coefficient devices. These owe their unusual electrical properties to the presence and character of their grain boundaries since the single crystals of these materials do not exhibit the same phenomena as the polycrystals. A single defect in a ceramic capacitor can cause electrical breakdown and short circuiting and similarly for piezoceramics, where during the ensuing polarisation the electrical breakdown can be followed by mechanical failure. As in other materials, crystal lattice imperfections, in particular lattice vacancies and dislocations, influence thermal conductivity, electrical and magnetic properties.
1.4
Semiconductors
Many of the traditional semiconductor materials have crystal structures that are related to the simple diamond cubic lattice where each atom is tetrahedrally coordinated, but the local atomic environment is not identical for all atoms. Table 1.3 gives the values of basic physical parameters of some commonly encountered semiconductor materials. Most semiconductor compounds and alloys are designed to keep the average electron to atom ratio to a value of four. The simplest illustration is given by the range of AB-type semiconductor materials formed between Group III and Group V elements in the Periodic Table of Elements. These so-called III to V semiconductors include GaAs and have a sphalerite superlattice structure, whereas the Types II and VI and IV and VI compounds usually have crystal structures of the sphalerite and rock salt respectively. Here the lattice constants of these materials lie in the range 0.50 to 0.65 nm and are generally larger than for metallic elements. This makes it generally easier to obtain details of the crystalline defects. Moreover the binary semiconductor compounds have band gaps in the range 0 to 3 eV. Although covering the band gap range for applications to microelectronic devices it is not possible to prepare such devices from this limited range of materials. This is due to (i) difficulties in preparing suitable defect-free pure materials and (ii) the need to have precisely controlled band gaps to optimise the performance of
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Glasses
9
Table 1.3. Crystal structure and basic electrical properties of some important semiconductor elements and compounds (after Grovenor (1989)). Crystal structure
Lattice spacing (nm)
Band-gap width (at 300 K) (eV)
D D
0.5431 0.5646
1.12 0.66
III–V compounds GaAs GaP GaSb InAS InP InSb AlAs AlSb
S S S S S S S S
0.5653 0.5451 0.6096 0.6058 0.5869 0.6479 0.5661 0.6136
1.42 2.26 0.72 0.36 1.35 0.17 2.16 1.58
II–VI compounds CdS CdSe CdTe ZnS ZnSe ZnTe HgTe
S/W S S S/W S S S
0.5832/a ¼ 0:416, c ¼ 0:6756 0.605 0.6482 0.542/a ¼ 0:382, c ¼ 0:626 0.5669 0.6089 0.644
2.42 1.7 1.56 3.68 2.7 2.2 0
Chalcopyrite CuInSe2
S
a ¼ 0:5782, c ¼ 1:1564
1.04
IV–VI compounds PbS PbSe PbTe SnTe
R R R R
0.594 0.612 0.646 0.632
0.41 0.27 0.31 0.18
Element Si Ge
D is diamond cubic, S is sphalerite, W is wurtzite and R is the rocksalt lattice.
particular devices. Hence the need to create materials by a combination of binary semiconductor compounds to give ternary and even quaternary semiconductors (Pollock et al (1982)).
1.5
Glasses
Glass is a class of material that does not crystallise when cooled from the molten state and, therefore, does not have long-range periodicity within the atomic structure (Hlavac (1983)). A pure oxide glass consists of a random three-dimensional network of atoms where each oxygen atom
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10
Introduction
Figure 1.8. Pure oxide glass consisting of a random three-dimensional network in which each oxygen atom is bonded to two metal atoms.
(figure 1.8) is bonded to two atoms of a metal, such as boron, and each metal atom is bonded with three oxygen atoms. However, there are many types of glass, and in the case of silica glass each metal atom is bonded with four oxygen atoms producing a more complex atomic configuration. The addition of fluxing atoms such as sodium reduces the number of bond cross links (figure 1.9). The major constituents of glasses are contained in two widely
Figure 1.9. Flux containing glass consists of a random three-dimensional network where the flux atoms such as sodium have reduced the number of cross-links.
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Polymers
11
separated regions of the Periodic Table, Group VI and Groups I and II. The chief glass-forming elements in Group VI are oxygen, silicon, selenium and tellurium. However, it is the neighbouring elements that enter into the chain-forming structures that result in the different types of glass. Elements from within Groups I and II are used primarily as fluxes and indeed, they control the viscosity and viscoelastic properties of glasses.
1.6
Metals and alloys
Metals and alloys are opaque, lustrous and relatively heavy, easily fabricated and shaped, have good mechanical strength and high thermal and electrical conductivity. All these properties are a consequence of the metallic bond in which all the atoms share their electrons in the outer electron shell forming an electron cloud and the bonding is by Coulomb attraction. Changes in the strength of this metallic bond cause differences in optical, electrical, mechanical and thermal properties of various metals and alloys. The simple, regular crystalline structures of metals and alloys result from the metallic bond which retains atoms in close packed arrangements, so that pure metals, in general, have one of the face centred cubic (fcc), body centred cubic (bcc) or hexagonal close packed (hcp) structures of the 14 crystal systems (figure 1.4). With these crystal structures, metals and alloys have relatively high ductility since they are resistant to tensile stresses and less resistant to shearing forces. However, the overall mechanical properties of metals and alloys are controlled by the crystal lattice defects, such as dislocations and vacancies (Nabarro (1967), Bollman (1970) and Honeycombe (1968)). Mechanical and chemical properties can be modified by the addition of alloying elements in varying proportion which are used to advantage in a range of commercial alloys. In many, alloy systems compositions and heat treatments are selected that produce complex distributions of phases to give the required properties (Barrett and Massalski (1986)).
1.7
Polymers
Polymers are by definition materials composed of long-chain molecules, typically 10 to 20 nm, that have developed as a consequence of the linking of many smaller molecules, monomers (Odian (1970)). Polymers which can be either natural or synthetic and have a wide range of characteristic physical properties such as strength, flexibility and the ability to soften when heated. Indeed, it is the particular combination of tensile strength and flexibility that make these materials attractive. If the molecular chains are packed side by side (figure 1.10) the molecules form an array with a crystalline structure. However, naturally occurring polymeric materials typically have a complex
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12
Introduction
Figure 1.10. Schematic diagrams of the molecular structures of polymeric materials. (a) A polymer with few branched chains; chains are regularly packed with extensive crystalline regions. (b) A polymer with many branched chains; chains are regularly packed and largely amorphous with few crystalline regions. (c) A polymer with extensive cross-linking.
microstructure comprising a mixture of crystalline and amorphous material. Generally, if the crystalline structure predominates, the material is relatively rigid with a higher tensile strength and is more resistant to heat than a material which contains a greater proportion of amorphous material. In the synthetic polymers, the proportion of crystalline to amorphous material is controlled and depends upon the chemical composition, molecular arrangement and the processing conditions used to produce the material (Ward (1971) and Kinlock and Young (1983)). In the case of polymeric materials, the interatomic bonds between molecular chains are the weak van der Waals forces, but in the crystalline structures, the chains are closer together over comparatively large distances so that the contribution of intermolecular forces has the effect of producing a more rigid material. The production of a crystalline structure is one of two methods used to develop stronger, more rigid, polymers such as polyethylene and nylon; the other is the formation of a strong covalent bond between the molecular chains by cross linking. A typical example of the latter is the established process of vulcanising raw rubber by heating with the controlled addition of sulphur atoms. Under these conditions, a proportion of these sulphur atoms cross-link between adjacent rubber molecules to increase both stiffness and strength. As the heating time is increased, more crosslinks are developed, and the rubber further stiffens, leading ultimately to the hard material ebonite. These materials are the thermosetting plastics which retain comparatively high tensile strength until excessive heating leads to breakdown of the cross-links and chemical deposition. By comparison in thermoplastics, only weak van der Waals forces bond the molecular chains together and these materials are softened by heating and, if necessary, can be remoulded. The heat treatment can be repeated provided the temperature is kept below that affecting chemical decomposition since at that stage the covalent bonds bind the atoms together in a long chain break-down.
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Composite materials
1.8
13
Composite materials
A composite material was originally considered to be a combination of two materials but now this class of material is regarded as any combination which has particular physical and mechanical properties (Kelly (1973) and Hale (1976)). The concept of composite materials has led to the design and manufacture of a new range of structural materials that are generally lighter, stiffer and stronger than anything previously manufactured. Figure 1.11 shows a simple schematic representation of the various ways of combining constituent materials to make a composite. Like synthetic polymeric materials, composite materials are often designed to replicate naturally occurring materials. Wood, for example, is a composite consisting of cellulose and lignin. The cellulose fibres have a high tensile strength and flexibility whereas the lignin provides the matrix for binding these fibres and adds the property of stiffness. Bone is another composite material, comprising the strong, but soft, protein collagen and the hard, brittle mineral apatite. The developed synthetic composite materials attempt to achieve similar total properties to naturally occurring materials by combining individual properties such as strong fibres of a material, for example carbon, in a soft matrix, such as an epoxy resin. Thus, the microstructure of significance in these materials spans the simple, relatively macrofeatures associated with the
Figure 1.11. Schematic representation of the various types of composite materials.
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14
Introduction
overall distribution of the mixture down to the microstructure of the individual components and their actions in concert. As we have discussed in this chapter, the basic chemical reason metals and alloys and polymeric materials are so much more resistant to cracking than ceramics is that interatomic forces in metals and alloys and intermolecular forces in polymers do not depend critically upon a particular directional alignment to achieve tensile strength. Moreover, the atomic bonds of metals and alloys and polymers are essentially unsaturated and are capable, therefore, of forming new bonds. Ceramics have highly oriented interatomic forces and saturated atomic bonds. The large amounts of plastic strain, more usually accommodated in metals and alloys and polymers as a result of plastic flow, provide a better resistance to crack extension than in a ceramic or glass. To overcome such a limitation of a ceramic or glass, but to make use of the potential strength of this class of material, modern composite materials divide the ceramic or glass into small pieces and bond them in a matrix. Thus, any inherent or developing cracks do not find a continuous, easy path through the total material. The ceramic or glass is often introduced into the composite in the form of fibres. However, the properties of the associated matrix are also extremely important. Under such circumstances, the matrix would be required to have specific properties so that it must (i) not cause mechanical or chemical change to the fibres that would introduce cracks, (ii) have sufficient plasticity to allow the transmission of stress to the fibres and should be adhesive with the fibres and (iii) have appropriate elastic and fracture properties. Fortunately there are other stiff materials with fibres that are covalently bonded and indeed, examples of these, boron and carbon, also have high melting points. These two physical properties are associated with the covalent bond which requires a high energy to break it. Therefore, materials that replace glass fibre because of their greater stiffness also, in many cases, overcome temperature limitations. Stiff fibres of graphite, boron and silicon carbide are used for a wide range of commercial composite materials.
1.9
Microstructure
It is obvious from the brief descriptions of the various materials given in the preceding sections of this chapter that to understand the physical properties and the response of materials to static, dynamic and cyclic stresses, various environments and temperatures, it is essential to be able to describe the ‘total microstructure’. For this, it may be necessary to combine a knowledge of the distribution, proportion and types of phases present, the chemical composition and state, and crystal and defect structure. On many occasions the term microstructure is still confined to describing objects that are visible by optical light microscopy (Saltykov (1974) and
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Microstructure
15
Table 1.4. A classification of microstructural elements in metal alloys after Hornbogen (1984) based on Euclidean dimensions and specific energies. Geometrical
Type of elements
Density, densities
Specific energy (U)
0 1 2 3
Vacancy Dislocation Grain Boundary Dispersed particle, pore or void
m3 m2 m1 m0
J J m1 J m2 J m3
Underwood (1970)). In reality the dimensions of microstructural elements that are significant in the investigation of materials commence at the level of the atomic spacing and, therefore, size alone is an insufficient definition of microstructure. Microstructure can be considered simply as the identical arrangement in three-dimensional space of atoms and all types of nonequilibrium defects and, therefore, both single phase and multiphase materials have a microstructure. Hornbogen (1984) considered the basic elements of microstructure for metal alloys and put these into a systematic order based upon geometric dimensions (Table 1.4). This quantitative characterisation of microstructure starts with information on structure characterised by the Euclidean dimension (Stanley and Ostrowsky (1986)) assigned zero to three for discontinuities in the phase structure followed by information on the density, . The dimension of these densities varies with geometric dimension of the defects so that for: P N=V ¼ ½m3 0-dimensional: 0 ¼ v ¼ P 1-dimensional: 1 ¼ d ¼ L=V ¼ ½m2 ð1:1Þ P 2-dimensional: 2 ¼ b ¼ A=V ¼ ½m1 P 3-dimensional: 3 ¼ p ¼ Vp =V ¼ ½m0 where N, L, A and V are number, length, area and volume and the subscripts refer to the dimension of the defect type: v ¼ vacancy, d ¼ dislocation, b ¼ boundary, p ¼ particle or void. The relationship between density i and average spacing Si of defects is given by Sv 3 v ¼ ½m Sd 2 d ¼ ½m Sb 1 b ¼ ½m 1=3
S f 1=3 f
ð1:2Þ
¼ ½m
where refers to the volume fraction of the phase which for a discrete dispersion of second phase particles 0 < f < 1. The units for the density of these elements depend upon the dimension 0 d 3. The product of
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16
Introduction
density and specific energy gives a bulk microstructural energy with the unit of an energy density (J m3 ) which describes reactions such as recovery, recrystallisation and particle coarsening (Hornbogen (1989)). Thus elements may transform during certain solid state reactions, an example being when vacancies in a crystal lattice with d ¼ 0, condense to dislocation loops with d ¼ 1. Thus transformations of this type imply a change in dimension. In addition to spacing, both geometrical and statistical functions are required to describe microstructure; these include the distribution of crystal orientation, the local distribution, the shape of one-, two- and three-dimensional elements and the orientation of these elements in space (Hornbogen (1984). This concept of microstructure may be sufficient to describe metal alloys and ceramic materials, but in the case of polymeric materials as discussed here, the appropriate structural level is the molecule. Certainly the configurations of the molecules and chains are necessary supplements to the microstructure (Ward (1971)). These concepts have to be further added to for other materials such as concrete where pores, fissures and structural gradients play an important role (Huang (1982)). Furthermore, there is the need to address the simple and complex periodic sequences of layers produced, for example, by sputtering techniques (Gibson and Davidson (1985)). The thickness of each layer can be controlled at minimum scatter and a very high degree of microstructural order, and the resulting diffraction effects can be obtained from such artificially produced microstructures. As a consequence there is no strict and unified definition of microstructure. Fractal analysis offers a way forward to quantify microstructures that are not in thermodynamic equilibria and cannot be easily classified or described by parameters such as particle size and spacing (Mandlebrot (1983)). This is briefly discussed in chapter 8 but the reader is directed to the review by Hornbogen (1989) who discusses this approach and draws upon experience derived from work undertaken in the area of the chemistry and physics of surfaces. The basic approaches to the investigation of the microstructure of materials were laid over a hundred years ago by Henry Clifford Sorby with his development of a preparation method and etching treatment to allow metal, mainly steel, specimens to be viewed under a reflected light microscope (Quarrell (1963)). Indeed this technique, progressively refined, remains a powerful tool for establishing essential microstructural features such as grain size and shape and distributions of phases, to the limit of resolution of the optical microscope which is approximately 300nm. The development in the early 1950s of a theoretical understanding of the principles controlling the strength of materials resulted in a need to apply techniques with a resolution approaching that of the interatomic spacing. This led to the development and use of electron based techniques where the shorter wavelengths of the electrons enable the resolution of atom dimensions to be achieved
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Microstructure
17
with current generation electron microscopes. The electron microscope, developed in 1931, was initially used to study biological systems (Ruska (1962)), but techniques for specimen replication and the preparation of thin metal foils in the mid-1950s enabled microstructural investigations to be undertaken on metals and alloys (Ruska (1962), Hirsch (1954) and Thomas (1962)) and later ceramic and polymeric materials (Thomas (1984)). Since then the resolution has been improved and the accelerating voltages increased (Isaacson et al (1979) and Jouffrey (1976)). An area of advance has been to extend the atomic, structural and crystallographic information obtained by X-ray diffraction techniques to the small-scale features contained within thin foil specimens using methods based on electron diffraction. In the early 1960s the introduction of the electron probe microanalyser enabled relatively high spatial resolution chemical analyses, using characteristic X-ray emissions, to be obtained from features down to approximately 1 mm diameter. However, since that time it has become evident that chemical changes over distances approaching atomic dimensions have a profound effect on mechanical and chemical properties of materials (Hondros and Seah (1977), McMahon (1980) and Flewitt and Wild (2001)). For example, the segregation of trace impurities to grain boundaries in polycrystalline metals to give single atom layer coverage can drastically modify the properties of metal alloys and fine scale distributions of impurity elements control the properties of many semiconductor devices (Doig and Flewitt (1987) and Grovenor (1989)). To study such fine scale distribution of elements, a number of techniques have been developed with good depth resolution for surface analysis such as X-ray photoelectron spectroscopy, Auger electron spectroscopy and secondary ion mass spectrometry. These together with the high spatial resolution techniques of scanning and transmission electron microscopy used in conjunction with energy dispersive X-ray and electron energy loss spectroscopy have further improved knowledge by examining both the chemical composition and state (Newbery and Williams (2000)). Over the past decade scanning tunnelling microscopy has stimulated an entire family of instruments referred to generically as scanning probe microscopes (Binnig et al (1982), Saiid (1991) and Wiesendanger (1994)). Since these instruments are capable of measuring a range of microstructural parameters on the nanoscale they have extended the understanding of materials. The innovation in the approach is that these new microscopes are not limited by wavelength since their resolution is controlled by the size of the interacting probe. As a consequence they come under the general class of super-resolution or near field scanning probe microscopes (Wickramasinghe (2000)). In this book, we review the techniques which permit the complete characterisation of the microstructures of materials. However, before proceeding to that stage we consider, in chapter 2, the interaction of various particles and radiations, including photons, electrons, atoms and ions with materials. It is
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18
Introduction
these interactions that provide many of the signals that are used subsequently to evaluate the microstructure of materials. Certainly, many techniques are reaching, or indeed have reached, a stage where it is possible to establish the information necessary to correlate existing theoretical models which describe high and low temperature deformation and fracture, corrosion, oxidation, environmentally assisted fracture, electrical and other physical properties with the microstructure of the material. After considering the basic theory behind the techniques that may be used to investigate the microstructure of materials, we address the specific techniques, the underlying theory, their benefits and their application in the succeeding chapters. Many of the techniques described have reached a stage where they are widely available as tools for use by research scientists although not necessarily suitable for routine applications without either a clear understanding and ability to interpret the data or modifications to the hardware. In this book, certain basic techniques are assumed and as a result less attention given, for example, to optical microscopy although this remains a simple, but powerful technique for investigating the microstructure of materials. Indeed, the advent of computer-based image analysis and pattern recognition techniques (Bruggins (1983), Saxton (1978) and Horne and Markham (1973)), have advanced the quantitative evaluation of certain microstructural parameters using optical methods. It is here that the material has to be investigated and understood over a multi-dimensional range of scale that spans the atomic dimension, nanoscale, through the meso range to the microscale. As a consequence it is important to appreciate that to establish this understanding it is unlikely that one particular technique will provide all the necessary information. Indeed, it is generally only by the use of several techniques in the correct combination that this will be achieved. By presenting in this book the range of techniques now available, together with a description of their use and how the information is processed, the reader is provided with the basis to select the correct combination of techniques. To assist with this selection, we have provided applications of techniques to demonstrate how essential information can be extracted and interpreted.
1.10
References
Barrett C S and Massalski T B 1986 Structure of Metals (New York: McGraw-Hill) (third edition) Bever M B (ed) 1986 Encyclopedia of Materials, Science and Engineering vol 1 ed R W Cahn (Oxford: Pergamon) Binnig G, Roher H, Gerber Ch and Weibel E 1982 Phys. Rev. Lett. 49 57 Bollman W 1970 Crystal Defects and Crystalline Interfaces (Berlin: Springer) Bruggins D 1983 Opt. Electron Microsc. 3 9 Cottrell A H 1967 An Introduction to Metallurgy (London: Edward Arnold) Cullity B D 1979 Elements of X-ray Diffraction (New York: Addison-Wesley)
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References
19
Davidge R W 1980 Mechanical Behaviour of Ceramics (Cambridge: Cambridge University Press) Doig P and Flewitt P E J 1987 Met. Trans. 18A 399 Flewitt P E J and Wild R K 2001 Grain Boundaries: Their Microstructure and Chemistry (Chichester: Wiley) Gibson J M and Davidson R L 1985 Layered Structures, Epitaxy and Interfaces, Mat. Res. Soc. Symposium 37 Grovenor C R M 1989 Microelectronic Materials (Bristol: Adam Hilger) Hale D K 1976 J. Mat. Sci. 11 2105 Hirsch P B 1954 Proceeding of the Third International Conference on Electron Microscopy (London) p 231 Hlavac J 1983 Technology of Glass and Ceramics (Amsterdam: Elsevier) Hondros E D and Seah M P 1977 Met. Rev. 22 262 Honeycombe R W K 1968 The Plastic Deformation of Metals (London: Edward Arnold) Hornbogen E 1984 Acta Metall. 32 615 Hornbogen E 1989 Int. Met. Rev. 34 277 Horne R W and Markham R 1973 Application of optical microscopy in Practical Methods in Electron Microscopy ed A A M Glauert (Amsterdam: North-Holland) p 327 Huang G 1982 Concrete and Reinforced Concrete (China) 5/6 Isaacson M, Ohtsuki M and Utlaut M 1979 Electron microscopy of individual atoms in Introduction to Analytical Electron Microscopy ed J J Hren, J I Goldstein and D C Joy (New York: Plenum Press) Jouffrey B 1976 Electron Microscopy in Materials Science part III ed E Ruedl and U Valdre´ (Brussels: Commission of European Communities) p 981 Kelly A 1973 Strong Solids (Oxford: Clarendon Press) Kelly A and Groves G W 1970 Crystallography and Crystal Defects (London: Longmans) Kingery W D, Bowen H K and Uhlmann D R 1976 Introduction to Ceramics (New York: Wiley) Kinlock A J and Young R J 1983 Fracture Behaviour of Polymers (London: Elsevier) McMahon C J 1980 Mat. Sci. Eng. 42 215 Mandlebrot B 1983 The Fractal Geometry of Nature (San Francisco: W H Freeman) Mott N F 1976 The Solid State, Scientific American p 80 Mott N F and Jones H 1958 The Theory of the Properties of Metals and Alloys (New York: Dover) Nabarro F R N 1967 Theory of Crystal Dislocations (Oxford: Clarendon Press) Newbery D G and Williams D B 2000 Acta Mater. 48 323 Odian G 1970 Principles of Polymerisation (New York: McGraw-Hill) Pollock G A, Deline V A and Furman B K 1982 Grain Boundaries in Semiconductors ed H J Leamy, G E Pike and C H Seager (New York: North-Holland) Quarrell A G 1963 15th Hatfield Memorial Lecture, ‘Metallography’, ISI Special Report No 80 (London: Eyre and Spottiswoode) p 1 Ruska E 1962 Fifth Int. Congress for Electron Microscopy; Philadelphia ed S S Bleese (London: Academic Press) Saiid D 1991 Scanning Force Microscopy with Applications to Electric, Magnetic and Atomic Forces (New York: Oxford University Press) Saltykov S A 1974 Stereometrische Metalloghie (Leipzig: VEB Dt Verlag fu¨r Grundstoff Industrie)
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20
Introduction
Saxton W O 1978 Computer techniques for image processing in Electron Microscopy (New York: Academic Press) Stanley H E and Ostrowsky N 1986 On Growth and Form (Boston: Martinus Nijhoff) Thomas E L 1984 Structure of crystalline polymers in Transmission Electron Microscopy of Polymers ed I H Hall (London: Chapman and Hall) ch 3 Thomas G 1962 Transmission Electron Microscopy of Metals (New York: Wiley) Underwood E E 1970 Quantitative Stereology (Reading: Addison-Wesley) Ward I M 1971 Mechanical Properties of Solid Polymers (London: Wiley-Interscience) Weisendanger R 1994 Scanning Probe Microscopy and Spectroscopy: Methods and Applications (Cambridge: Cambridge University Press) Wickramasinghe H K 2000 Acta Mater. 48 347
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Chapter 2 Interaction of radiation with materials 2.1
Radiation sources
To characterise a microstructure it is necessary to perturb the material by interacting in some way with it. Indeed in order to see a surface it is necessary to bombard that surface with photons of wavelengths within the visible range and this in itself may alter the material. A typical example of damage caused by photons is the response of a photographic film. To achieve higher resolution and thereby magnification it is possible to use, for example, a scanning electron microscope where the photon source is replaced with electrons with an energy in the region of 10 to 30 keV. These are more damaging than photons since they penetrate a considerable distance on the atomic scale into the material. Many modern analytical instruments require high spatial resolution, while at the same time needing high sensitivity for the detection of elements within the material. Often this involves bombarding the surface with ionised atoms of high energy which, although extremely damaging, provides microstructural information that outweighs this disadvantage. With any characterisation of a material the objective must be to obtain the maximum information whilst incurring the least amount of damage to the specimen. Thus, in general, initial examinations of a surface should be carried out using a low intensity beam of low energy photons. To obtain more information the source energy may have to be increased, for example with the use of X-rays initially and progressively through electrons to finally ions. There are, of course, situations where this simplistic approach may not hold; in the technique of ion scattering spectroscopy (ISS), the ions reflect from the surface and do not perturb it as much as a high energy photon or electron, so care must be taken when deciding how to examine a material. It is the purpose of this chapter to summarise the processes that occur when photons, electrons, ions and particles interact with materials. The various sources available will be considered, their properties described, and their potential uses outlined.
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2.2
Interaction of radiation with materials
Penetration depths
The penetration depth or mean free path of the incident beam determines the depth and volume of material that will be sampled. In many cases one is probing with one type of radiation but detecting a second type. This occurs in X-ray photoelectron spectroscopy (XPS) where the incident probe is a beam of X-ray photons but emitted electrons are detected, whereas this is reversed for the technique of energy dispersive X-ray (EDX) analysis. Generally the particle or radiation which has the shortest mean free path in the material will determine the volume analysed. Whatever beam is selected we must be aware of its interaction with the material, what photons, electrons or other particles are ejected and how they, in turn, interact with the material. Only in this way can we use the emitted signals to gain an understanding of the material being examined. 2.2.1
Photons
Photons are discrete quanta of electromagnetic radiation. The photon is identified by the wavelength, , energy, E, and frequency, , all of which are related by the equation h ¼ E ¼ hc=
ð2:1Þ
where h is the Planck constant and c the velocity of light. The electromagnetic spectrum spans a vast range with wavelengths varying from 106 m down to 1014 m. The frequency, energy and wavelengths of the different types of electromagnetic radiation are illustrated in figure 2.1. If we are to use electromagnetic radiation for microstructural characterisation of materials a photon wavelength is needed that is of comparable size to the features being studied. This means that photon wavelengths greater than 104 m would result in an inadequate spatial resolution and we do not require radiation less than about 1010 m. The penetration of photons shows considerable and dramatic variations between different types of material and photon energy or wavelength. It is not possible or instructive to go into any detail regarding penetration depths over the whole of the electromagnetic spectrum, but only some specific wavelengths that are important for interrogating the microstructure of materials. The long wavelength infrared radiation is used to characterise materials by determining how specific wavelengths are absorbed, visible light is used in a variety of instruments mainly to obtain a visual image of the surface while at the shorter wavelength ultraviolet radiation is often used to obtain information concerning the electron distribution in the surface atoms. Some materials are opaque while others are transparent to this range of wavelengths. However, even the most opaque or highly reflecting of these materials will allow the radiation to penetrate at least a fraction
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Penetration depths
23
Figure 2.1. The electromagnetic spectrum illustrating the relationship between energy, frequency, wavelength and wave number.
of a wavelength below the surface. In the case of visible light, where the wavelength is approximately 500 nm this penetrates an average of between 50 to 300 nm into the bulk so that any analysis performed or image obtained will average over several hundred atom layers. After visible light, X-rays are probably the most utilised photon source for investigating the microstructure of materials. The whole subject of X-rays and their interaction with matter has been thoroughly treated by Cullity (1979). X-rays are produced by bombarding a metal target with high energy electrons to produce a band of ‘white’ radiation. The intensity of the X-rays within this band varies with the wavelength determined by the energy of electrons incident on the target material (figure 2.2(a)). Superimposed on the ‘white’ X-radiation are a series of discrete maxima whose wavelength and intensity is determined by the electron binding energies of the atom making up the metal target being bombarded. These characteristic X-ray photons, shown for a copper target in figure 2.2(b), result from electrons falling into holes created in core electron levels by the incident electron beam with the emission of a photon whose energy is
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Interaction of radiation with materials
Figure 2.2. The X-ray spectrum produced by bombarding metals with electrons: (a) the relationship between the X-ray intensity and wavelength for electron beam energies from 20 keV to 50 keV and (b) the spectrum from copper showing the characteristic K and K peaks.
given by replacing E in equation (2.1) by E1 E2 , the energy difference between the electron shells. The penetration of X-rays into a material shows less variation from one material to another than visible light and is easier to predict. The penetration distance varies both with wavelength and material and is typically several micrometres. The absorption coefficient, , which increases with atomic number determines the depth of penetration. The intensity of transmitted radiation, I, through a layer of material of thickness, t, is given by Peiser et al (1960): I ¼ I0 expðtÞ
ð2:2Þ
where I0 is the intensity of the incident X-ray beam. Gamma rays have very high energies in the region of 50 keV to 50 MeV and wavelengths that are considerably less than X-rays, and would typically be in the region of 102 nm (Seigbahn (1965)). When a beam of gamma rays passes through a material photons are removed from the beam in individual events, thus the number of gamma ray photons removed is proportional to the thickness traversed. Therefore the intensity of the gamma ray decays as I ¼ I0 expðxÞ
ð2:3Þ
where is the absorption coefficient and x is the distance traversed by the beam. Gamma rays can penetrate considerable distances through materials but the penetration distance tends to vary inversely with the atomic number. However, gamma rays will pass through the bulk of almost all practical specimens.
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Penetration depths 2.2.2
25
Electrons
The penetration depth of electrons varies dramatically with both the energy of the electron and the atomic number of the material that is being examined. Figure 2.3(a) reproduces the mean free path of electrons in stainless steel as a function of incident beam energy (Castaing (1960)). The mean free path length increases from a fraction of a micrometre, at energies in the region of 10 keV up to 2 mm at 30 keV. In figure 2.3(b) the mean free path of electrons is plotted as a function of atomic number for three incident electron energies—10, 20 and 30 keV. Here, even more dramatic changes can be seen: the mean free path of electrons in elements of low atomic number is very large and can be as great as 10 mm for elements with atomic number below 20, while elements with high atomic numbers greater than 40 have short electron mean free paths generally less than 2 mm. This clearly has important consequences for any microstructural characterisation since materials will invariably be composed of elements with different atomic numbers; there may be precipitates such as carbides with low atomic mass in a matrix with a high mean atomic number and this will modify the images for each constituent. A situation often encountered when examining a metal alloy in the scanning electron microscope is that the surface images differently as the interrogating beam
Figure 2.3. The mean free path length of electrons (a) in stainless steel as a function of electron energy and (b) as a function of atomic number of the material being probed for 10, 20 and 30 keV electrons (after Castaing (1960)).
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26
Interaction of radiation with materials
Figure 2.4. Secondary electron images from a specimen of colloidal silver and colloidal carbon on an aluminium stub recorded using an incident electron beam of (a) 20 keV and (b) 900 eV (reproduced by permission of ETP Ltd (1990)).
energy is changed and, indeed, different analyses can be obtained using the emitted characteristic X-rays. This is because the metal alloy specimen almost certainly has a thin oxide surface layer which can on occasions be further covered with a layer of carbon as a result of contamination. An example of this effect is shown in figure 2.4 where two images are obtained with incident beam energies of 900 eV and 20 keV (ETP (1990)) from colloidal silver and carbon on an aluminium substrate. The high energy electron beam produces an image in which the silver appears very bright and the carbon is poorly imaged. The low energy beam, on the other hand, produces a clear image from the carbon but a mottled image from the silver. The differences in the two images can be explained in terms of both the penetration of the electrons into the bulk and the backscattering of electrons by atoms of different atomic number. The low energy image is different from that obtained using the higher energy incident beam because the thin layer of carbon is penetrated only by the higher energy electrons. We have so far only, in general, considered the penetration of relatively high energy electrons, above 10 keV. However, many techniques detect electrons with energies much lower than this in the region 0 to 2 keV, where the effect of the material on the mean free path of the electrons is much reduced (figure 2.5) (Seah and Dench (1979)). Clearly the mean free path is very short over the whole of this energy region, varying from approximately 0.4 to 300 nm, which is a hundred to a thousand times less than for high energy electrons. Moreover the mean free path of electrons for elements with low atomic number is essentially the same as for elements with high atomic number and the mean free path increases, to a first approximation, as the square root of the electron energy over the range 0.1 to 2 keV. These changes in electron mean free path can be used in many ways to obtain additional microstructural information concerning a surface, but it also indicates the great care that must be exercised when using electrons to probe a material.
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Penetration depths
27
Figure 2.5. Electron mean free path lengths in materials for electrons in the energy range 0 to 2000 eV (Seah and Dench (1979)) (reproduced with permission of John Wiley and Sons).
Since an incident high energy electron beam is scattered as it penetrates a material (figure 2.6), the resolution will be influenced by the spread of electrons around the incident beam. Figure 2.7 is a plot of the intensity of the secondary electrons as a function of distance from the centre of 5 and
Figure 2.6. Schematic diagram illustrating the volume of material that is probed by an incident electron beam together with the volumes from which X-rays and backscattered, Auger and secondary electrons emanate.
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Interaction of radiation with materials
Figure 2.7. The scattered electron distribution following bombardment of gold and aluminium by electron beams of 5 and 50 nm diameter (Seah (1986)) (reproduced with permission of John Wiley and Sons).
50 nm electron beams incident on aluminium and gold. The majority of the electrons come from the area of the incident beam, but a fraction emanate from an area around the incident beam (Seah (1986)). This is the result of electron processes taking place within the material; the scattered secondary electrons are detected up to 2 mm distant from the centre of the electron beam in the aluminium specimen but only 0.2 mm from the centre for the gold. However, the intensity of these scattered electrons is lower in aluminium than gold. In general the scattered electrons will not significantly degrade the image except where the intensity of the scattered electrons makes a significant contribution to the total, as for gold with a 50 nm incident electron beam. However, the scattering of secondary electrons places an ultimate spatial resolution on images that can be obtained in the scanning electron microscope and it is for this reason that the images obtained have generally lower spatial resolution than images obtained using transmission electron microscopy. At the same time a large number of electrons are produced with relatively low energy as the result of atoms being ionised by the removal of electrons from the valence band. In addition, a smaller fraction of atoms are ionised by the ejection of core level electrons and these atoms can rearrange and eject either photons (X-rays) or Auger electrons (see chapter 6). Finally a number of incident electrons may be scattered back towards the specimen surface without losing a significant amount of energy. The Auger electrons are generally confined to energies in the range 0 to 2 keV and escape from the surface only if they emanate from within the top few
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Penetration depths
29
atom layers and laterally not beyond the incident beam diameter. Secondary electrons also have relatively low energies, and although produced by many of the electrons travelling within the subsurface volume those that escape are restricted to the surface volume and to a relatively small distance laterally outside the diameter of the incident beam. It is these electrons that are used to form an image in the scanning electron microscope and as a result the image has a resolution essentially defined by the incident electron beam diameter. The backscattered electrons have high energies, large mean free paths and can originate from a greater depth in the material. They are scattered with components normal to the direction of the incident beam and, therefore, define a volume of diameter that is much larger than that of the incident beam. Finally, the X-rays produced can penetrate much greater distances than electrons and potentially all the X-rays produced which travel towards the surface can escape. Thus X-rays originate from any point that the scattered electrons reach and this defines a volume of excitation. 2.2.3
Neutrons
Although a neutron is approximately one thousand times the mass of an electron and as a consequence is more particle-like, it still possesses sufficient wave character to be diffracted by materials. However, since it does not have an electric charge it is not affected by the electron cloud surrounding the nucleus and on passing through a material effectively interacts only with the atom nucleus. As a consequence neutron penetration distances are much greater than for electrons and even X-rays. The precise penetration depth depends on the atomic species being examined but for most materials neutrons will penetrate distances of several millimetres (Hutchings and Windsor (1986)). Neutrons can be used to study the microstructure within the bulk of a material. 2.2.4
Protons
The interaction of a proton beam with a material has many similarities to the electron but there are some important differences. The proton being charged is influenced by the electrostatic forces within the material but because the mass is 1836 times that of the electron a proton of a few MeV energy has a much greater momentum than electrons of say 50 keV. The proton loses a small fraction of momentum in each atom collision and will not be deviated significantly from the incident beam direction. Therefore protons will travel much farther into a material than electrons of the equivalent energy with little scattering. The stopping power, S, is the term which defines the depth to which protons penetrate a material. The stopping power decreases with increasing proton energy and with increasing atomic number. A 2.5 MeV proton has a range of 55 mm (S ¼ 123 keV mg1 cm2 ) in carbon and a
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Interaction of radiation with materials
Figure 2.8. The stopping power for protons entering aluminium, nickel and silver as a function of incident energy (Ziegler et al (1985)) (reproduced with permission of Pergamon Press).
range of 28 mm (S ¼ 56 keV mg1 cm2 ) in silver. Figure 2.8 shows the stopping power as a function of proton incident energy for aluminium, nickel and silver (Ziegler et al (1985)). Protons are frequently used to excite X-rays in a technique known as particle induced X-ray emission (PIXE) (Johansson and Campbell (1988)) (see chapter 7). 2.2.5
Ions/atoms
It is natural to move on from protons to consider ions. Invariably if ions penetrate a material so much damage occurs that it is more accurate to address the stopping distance rather than a penetration distance. It is perhaps instructive to describe what happens when either an atom or an ion impinges on a surface. At very low energies of a few eV an atom is simply reflected from the surface. When a primary ion of mass M1 and energy E1 impinges on a surface of atoms mass M2 it will be reflected with a kinetic energy E2 determined by the relative masses of the incident and surface atoms and the angle between the incident and reflected atom. Kinetic energy is transferred to the surface atom M2 but the impinging ion does not penetrate into the surface (see chapter 7). At higher energies the atom burrows into the material, causing atoms, atom clusters, ions and ion clusters to be ejected from the surface while, at the same time, atoms are knocked farther into the material (figure 2.9). Here the incident ion of relatively high energy knocks surface atoms farther into the material. These in turn collide with other atoms, establishing a cascade process where atoms collide with one another, and atoms move in both forward and backward directions. Some atoms, and atom clusters, will be ejected both in the
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Penetration depths
31
Figure 2.9. Schematic diagram illustrating the sputtering of ions from a surface during ion bombardment.
ionised and neutral state together with some electrons. The original ion will either come to rest within the body of the material or may be ejected as part of the scattering process. The distance penetrated is determined by the kinetic energy of the incident ion, the atomic number of the ion and the atomic number of the material. Figure 2.10 shows the penetration distance for krypton ions of energies from 5 to 50 keV impinging on germanium (Littmark and Hofer (1980)). Considerable effort has been devoted to the
Figure 2.10. The penetration distance of krypton ions with energies from 5 to 50 keV in germanium (Littmark and Hofer (1980)) (reproduced with permission of North-Holland Publishing Company).
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Interaction of radiation with materials
Figure 2.11. Schematic diagram illustrating the penetration of ions into a solid. R ¼ range and x ¼ penetration depth.
study of implantation of specific atom species into semiconductor materials because of the importance in silicon chip technology. In this example the krypton ions with energies of 5 keV do not penetrate deeper than 13 nm whereas those with 50 keV energy can be implanted to depths greater than 50 nm. Penetration distance and damage cannot be separated for ions. When an ion enters a polycrystalline material it will follow a path which is not necessarily normal to the surface and travel a distance before coming to rest at a point (figure 2.11). The distance travelled by the ion is greater than the range R or the penetration depth x, but cannot readily be measured. It is therefore customary to define the penetration depth, x. The range along the direction of the incident beam is defined as the projected range Rp . Naturally this will vary with the ion and the material but for ions in the energy range 0:002 E 0:1 keV (Schiøt (1972)) is given by 2=3 2=3 2=3 Z1 þ Z2 E ð2:4Þ Rp ¼ C1 ðÞM2 Z1 Z2 where M2 is the atomic mass of the material, E the energy in keV, Z the atomic number and C1 ðÞ is obtained from experimental values shown in figure 2.12. However for ions with energies 0:5 E 10 keV 2=3 2=3 2=3 ðZ1 þ Z2 Þ1=2 E : ð2:5Þ Rp ¼ C1 ðÞM2 Z1 Z2 If the ion enters a single crystal in a direction close to a low index crystallographic axis then the ion will be channelled into that direction so that there will be less deviation from the incident direction and the penetration distance will be considerably enhanced compared with the polycrystalline case.
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Material damage
33
Figure 2.12. Experimental values for C1 ðÞ as a function of for (a) M1 > M2 and (b) M1 < M2 where M1 and M2 are the respective masses of the ions and the parent material (Schiøtt (1972)) (reproduced with permission of Gordon and Breach Science Publishers Ltd).
2.3 2.3.1
Material damage Photons
Generally a photon source is regarded as the least damaging of the analytical probes, but the degree of damage is never zero and can in some instances be quite severe. The photon wave packet will have a momentum determined by either the energy or wavelength. This momentum is clearly small for light quanta and other radiation sources of longer wavelengths such as infrared radiation, microwaves etc. However, one only has to look at the results of leaving the Christmas pudding in a microwave oven too long to realise that the damage is not negligible. In general the damage caused by photons is the result of heating and the degree and extent is determined by the penetration of the photon source into the material, the energy of the radiation and the photon flux (Smith (1971)). X-ray beams can cause the surfaces of certain oxides to be reduced and laser beams can burn holes through metal by heating to temperatures that result in the instantaneous melting and evaporation in the immediate vicinity of the beam. Indeed this is the basis of one technique (laser induced mass analyser (LIMA)) where a small volume of the material surface is vaporised by a pulsed laser beam and the evaporated material is then mass analysed. Figure 2.13 shows the effect of a laser beam used in a laser induced mass analyser on a metal surface. However, most photon sources selected cause very little damage and the surface being
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Interaction of radiation with materials
Figure 2.13. Laser damage on a metal specimen in a laser induced mass analyser (courtesy of M D Crapper).
studied does not alter over very long periods of exposure. As a general rule, if the results for a microstructural investigation can be obtained using a photon source, then this should be used. 2.3.2
Electrons
While electrons are readily described as having a dual wave particle character, their mass allows a considerable momentum, particularly when accelerated to several hundred keV in the transmission electron microscope, to be transferred. Again the resultant damage is related to the amount of energy or heat transferred to the material and to the thermal conductivity of the material. At low incident beam energies atom bonds do not break in the target material, so in general metals and alloys can be examined without any significant degree of damage taking place. However, in the case of oxides and polymeric materials the damage can be considerable. Indeed it is not usually possible to obtain a secondary electron image in the electron microscope from most polymeric materials before they degrade. One approach to obtaining images of these materials can be obtained is to coat the surface with a conducting material such as gold, but this renders any chemical determination almost impossible. Oxides are also damaged by the electron beam although if the oxide layer is thin and in contact with a metal substrate the damage is rarely so great that an image cannot be obtained. Figure 2.14 is an Auger electron spectrum obtained from the surface of a stainless steel which initially contained a thin silicon oxide, SiO2 , layer (Wild (1985)). The top spectrum (a) was recorded using an electron beam of 10 keV and approximately 100 nA current focused into a spot size of 100 nm but rastered over an area of 200 mm 200 mm. The spectrum is typical of that expected from SiO2 with two peaks at 62 and 77 eV. The rastering of the beam was then turned off
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Figure 2.14. The effect of a 10 keV electron beam focused on an SiO2 layer on stainless steel showing reduction of the oxide to silicon (a) rastered beam (b) static beam (Wild (1985)) (reproduced with permission of Pergamon Press).
so that it was stationary when the second spectrum (b) was obtained. Here the spectrum is essentially that from silicon with a peak at 92 eV indicating that most of the oxide has been reduced by the influence of the electron beam. This result indicates oxide reduction caused by electron beam heating and reducing the oxide; the beam does not cause atom bond breaking. A further dramatic demonstration of the heating effect of electrons is shown in figure 2.15 where a metal surface is examined under similar conditions to the previous example. The metal surface has oxide particles loosely adhering and the thermal conductivity between the particles and the metal is poor. In figure 2.15(a) the particle in the centre is rectangular in shape but after this image was recorded the electron beam was focused on the particle
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Interaction of radiation with materials
Figure 2.15. The heating effect of a 10 keV, 5 109 A˚ electron beam on an oxide particle when there is poor conductivity between the particle and the substrate.
to obtain a chemical analysis. Following analysis, the secondary electron image in (b) was recorded and this indicates that the oxide particle, which was a chromium iron oxide, had melted and had been heated to a temperature in excess of 2800 K. We have been considering the damaging effect of electrons in conventional instruments where the incident beam energy does not normally exceed 100 to 200 keV. However, there exist electron beam instruments which use considerably more energetic electrons and these can cause atoms to be displaced from normal lattice positions by the transfer of momentum (Madden et al (1979)). Such damage occurs in the million electron volt transmission electron microscope. Indeed such is the effect at this energy that the microscope is often used to simulate the damage that is caused by fast neutrons in nuclear reactors. Figure 2.16 shows damage that has occurred in a thin foil specimen of stainless steel where voids and dislocation loops have formed by electron interactions. 2.3.3
Ions and atoms
When ions or atoms penetrate a material they either interact in essentially a totally non-damaging manner as in ion scattering spectroscopy (ISS) (Niehus and Baner (1975)) where they interact elastically with the surface or they cause severe damage. Ion damage is effected by displacing atoms from their normal lattice positions and a minimum energy is required by the ion to exceed the binding energy of the atom. In addition a certain amount of energy is required to displace the atom and this varies with the direction of the incident ion relative to the crystallographic directions of the material. However, the threshold displacement energy is some ten times the energy required to break the atom bonds. If the ion has sufficient energy to displace an atom then the total damage caused will be related to the ion energy and
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Figure 2.16. Damage produced in stainless steel foils by 1 MeV electron beams (courtesy J Buswell).
the flux. At a low flux the damage regions are isolated one from the other since the ion produces a region of amorphous material surrounded by regions containing large numbers of defects. As the flux increases so these regions overlap and an amorphous layer is found. The processes involved in ion cascade events are described by Benninghoven et al (1987) and Sigmund (1981). An example of the damage caused by ions penetrating a material is shown in figure 2.17. Here a transmission electron micrograph is reproduced
Figure 2.17. Damage caused in Type 316 stainless steel following bombardment with 4 MeV iron ions at 813 K. Each incident ion displaces five stainless steel ions on average (Ward and Fisher (1992)) (courtesy A Ward).
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Interaction of radiation with materials
showing the damage in Type 316 stainless steel following bombardment with 4 MeV iron ions at 813 K. Here each iron ion causes five displacements in the stainless steel. Ion penetration depth varies with ion energy and with ion species. Ions will be stopped over a range of distances and the penetration distance is normally defined as the maximum in the implantation profile, i.e. the stopping distance for the largest number of ions.
2.4
Resolution
In the earlier sections of this chapter we discussed the penetration of various types of radiation into materials. This determines the depth resolution and will have a distinct bearing on the spatial resolution that is obtainable for investigating microstructural features. The resolution normal to the direction of the incident beam, frequently referred to as spatial resolution, is influenced by the diameter of the incident beam, the wavelength of the incident radiation and the mean free path of the incident beam in the material. An image of an object can be obtained in two basic ways. One method is to illuminate the object over its entire surface by using a suitable source of radiation (photons, electrons or ions) and then use a lens arrangement to form an image by focusing the radiation that is either reflected or emitted from the object. This is achieved such that a point on the object is focused to an equivalent point on an image plane. In such systems the spatial resolution is determined by the lens system and the wavelength of the emitted or reflected radiation. In general optical microscopes, certain X-ray microscopes and some ion microscopes operate in this way. The second method is to direct a very narrow beam of radiation on to the object and to detect either the absorbed or reflected radiation. The incident beam is rastered over the object surface and changes that occur in absorption and reflection allow an image of the surface to be built up. In these cases the spatial resolution is determined by the diameter of the incident beam, the wavelength of the incident radiation, and the scattering of the incident radiation within the object surface. Most electron and ion optical instruments form their image in this way although the advent of lasers has meant that some light microscopes also use this method. Details of the resolution achieved for the specific techniques is discussed in the following chapters where each technique is discussed.
2.5
Loss processes
The previous part of this chapter has been concerned primarily with processes which describe how photons, electrons, neutrons, atoms and ions interact with, and can be used to give images of, the surface and bulk
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materials. We now move on to describe the interactions of these particles with the material and how processes transfer energy to the material under investigation. This energy transfer is used to determine information concerning the type of atom, its environment or chemical state. Any process that involves the incident beam surrendering some energy is described as a loss process. We will attempt to identify the loss processes that are currently utilised to characterise the microstructure of a material. Most, although not all, of the techniques mentioned briefly here will be described in more detail in later chapters. 2.5.1
Photons
Very long wavelengths (>1 mm) (including radio and micro-waves) A molecule with either a magnetic nucleus or an unpaired electron will have nuclear and electron energy levels that can be influenced by a magnetic field. The magnetic field, B, causes the electron to take up new quantized values of ð12Þðh=2pÞ, where h is the Plank constant, and with each of these is associated an energy level, one above and the other below the original energy level. The separation of these energy levels is B, where is the magnetic dipole moment, and this is linearly dependent on the magnitude of the applied magnetic field, B. By combining a magnetic field with an appropriate electromagnetic radiation, transitions between the two energy levels can be induced. With magnetic fields that can be applied routinely to materials it is necessary to use radio-frequency waves to excite the nuclear magnetic resonance (NMR) (Akitt (1983) and Cudby and Williamson (1990)) and micro-waves to excite the electron spin resonance (ESR) (Symons (1978)) and electron paramagnetic resonance (EPR) (Thomson (1990)). The magnetic moments of certain nuclei also interact with the unpaired electron to produce additional fine structure on the major resonance. This technique has been used extensively to study kinetic processes in organic material reactions and to follow catalytic reactions. NMR has been used to determine the structure of organic materials, and degradation in microstructure of resins, rubbers and other hydrocarbons under certain conditions. Long wavelengths As considered in chapter 1, materials are composed of atoms bonded together where the distance between the ions is determined by a balance between the attractive long range interactions of the ions with charge þq and the repulsive short range interactions between the ion cores. If given sufficient energy the atoms are able to vibrate and this process can be visualised by imagining the atoms as hard spheres connected by springs. The simplest case is to consider two atoms connected together and, by applying Hookes’s law for elastic expansion under a force, the frequency
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Interaction of radiation with materials
of vibration of the two atoms, , is given by 1 Fc 1=2 ¼ 2c M0
ð2:6Þ
where c is the velocity of light, Fc is the force constant of the atom bond and M0 is the reduced mass of the system given by M0 ¼
M1 M 2 M1 þ M 2
ð2:7Þ
where M1 , M2 are the masses of the two atoms. When a material is illuminated with light then the wavelengths which correspond to the vibrational frequencies are absorbed. This simplistic approach produces surprisingly good agreement between theory and experiment with the vibration frequencies for, say, a hydrogen atom bound to carbon being fairly well predicted. As the complexity of the molecule increases so the number of vibration frequencies increases. A nonlinear molecule containing n atoms has 3n degrees of freedom and 3n-6 vibrational modes each with a characteristic band frequency. As the molecule becomes more complicated with atoms bound to more than one atom and when there are nonlinear chains of atoms, vibrations can take place in directions with components normal to the bond direction. The vibrations in the direction of the bond are referred to as stretching vibrations while those normal to the bond direction are known as bending or deformation vibrations. Figure 2.18 illustrates some of the different ways a molecule can vibrate (Cross (1960)). These vibrations may be determined by observing the absorption of infrared radiation either
Figure 2.18. Some of the possible molecular vibrations for two identical atoms bonded to a third dissimilar atom.
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in the transmission or reflection mode. This, in turn, permits substances to be identified and molecular structure to be determined and allows reaction kinetics to be studied. Infrared spectroscopic techniques (Herzberg (1945) and Wilson et al (1955)) use radiation with wavelengths from approximately 1 mm (wave number 104 cm1 ) to 1 mm (wave number 10 cm1 ) to study these vibrational absorption bands. The wave number has been quoted here because it is conventional, in infrared spectroscopy, to refer to the wave number rather than the wavelength; the wavelength, , and wave number, , are related by ¼ 1=. There are now many cases where lasers are being used to provide the source of infrared radiation with the advantage that high intensities are confined to relatively small areas with the associated improved spatial resolution. Intermediate wavelengths (including visible and ultraviolet light) As wavelength is decreased so the energy available to excite an atom increases until a stage is reached where it becomes possible to raise electrons from their ground state to higher electron orbitals (Rao (1961)). The binding energies of electrons in atoms are specific to a particular element and by determining the difference in energy between two electron levels, by measuring absorption lines, it is possible to identify the type of atom. Consider the hydrogen atom illustrated in figure 2.19 which contains a series of energy levels where electrons may be present or absent. The energy levels are filled from the lowest level in pairs, one with spin up and one with spin down, until all the lowest levels are filled. Energy may then be given to the electrons in the outermost orbits which may be excited to higher levels. The effect of
Figure 2.19. Energy levels and possible transitions in the hydrogen atom.
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Interaction of radiation with materials
Figure 2.20. The hydrogen spectrum showing characteristic absorption bands.
this on a beam of white light is to absorb light of frequency, , since: h ¼ E1 E2
ð2:8Þ
where E1 and E2 are the initial and final electron levels. Thus a beam of white light when passed through a gas or material will have a series of missing wavelengths corresponding to the difference in electron binding energies of the atoms. By measuring the wavelengths of the absorption band the atom type can be identified. Absorption bands have been catalogued according to the lowest energy level which takes place in the absorption. Those in which the quantum number of the lowest level is 1 are referred to as the Lyman series, for n ¼ 2 the Balmer, n ¼ 3 the Paschen and n ¼ 4 the Brackett. The atom that has been excited by an electron being transferred to a higher orbital will subsequently decay to the original ground state by the emission of light. The decay may be by a direct transition to the original electron energy level or it may be by a series of transitions. Thus a spectrum is observed which contains a number of discrete lines. Figure 2.20 shows an absorption spectrum for hydrogen and shows the Balmer series of absorption bands which gradually close together as the quantum number n increases. In practice visible and ultraviolet radiation is used to study the electron energy levels of the outer shell electrons because the energy supplied by the incident radiation is insufficient to excite core level electrons from most elements to the next highest state. When light is incident on a material certain resonance frequencies are absorbed in raising the atom to an excited state. When the atom decays that same frequency may be re-emitted in a random direction and not necessarily in the direction of the incident beam (Baranska et al (1987), Clark and Hester (1983–5) and Andrews and McCoustra (1990)). This is known as Rayleigh scattering. However, the material illuminated will contain electron energy levels at both higher and lower energies than the energy level of the electron that was initially excited. These energy levels may be unfilled because they too may have been excited to higher levels. The atom may, therefore, decay by the excited electron falling into one of these other energy shells effecting emission of radiation at both higher and lower frequencies than the Rayleigh line. These lines are known as Stokes (at lower energies than Rayleigh) and anti-Stokes (at higher energies than
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Figure 2.21. Processes involved in Rayleigh and Raman radiation and the production of Stokes and anti-Stokes lines.
the Rayleigh line) while the effect is known as the Raman effect (figure 2.21). The effect is most easily observed if a material is illuminated with an intense beam of radiation. Lasers are therefore used to illuminate materials in the technique of laser Raman spectroscopy, which provides information concerning the electron energy levels in atoms which constitute the material with relatively good spatial resolution (Baranska et al (1987)). Short wavelengths (1012 to 109 m) (including ultraviolet excitation) In the previous section we considered the absorption of infrared and light radiation which has an intermediate wavelength and hence low energy and does not have sufficient energy to excite core-level electrons to higher orbits. However, as the wavelength of the incident radiation is decreased so the number of energy levels available for excitation increases and electrons positioned closer to the nucleus of the atom may be excited. The energy of light and ultraviolet radiation is sufficient to excite energy levels in materials of low atomic number but for higher atomic number elements it is capable of exciting only outer shell and valence electrons. However, if the material is bombarded with X-rays then many of the core shell electrons can be excited and will, in most cases, be given sufficient energy to be ejected from the atom (see the photoelectric effect below). The atom then rearranges, with the electrons falling into the hole created by the initial excitation, and energy is released as a photon or by the emission of an Auger electron (see below). The energy of this photon is determined by the difference in the
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Interaction of radiation with materials
electron energy levels E1 , E2 by the expression h ¼ E1 E2 . This is the basis of X-ray fluorescence (XRF) spectroscopy where the material is bombarded with a beam of X-rays and the emitted X-ray energy is measured using a either a wavelength dispersive or energy dispersive analyser. In the previous section we have described how photons may excite electrons to higher energy levels. However, as the energy of the photon increases it may supply sufficient energy to the electron such that it overcomes the work function of the material and eject the electron into the vacuum. This is known as the photoelectric effect. The energy of the ejected photon is given by E ¼ h EB
ð2:9Þ
where EB is the electron binding energy of the ejected photoelectron and is the work function of the material. If the energy of the incident photon is known, the energy of the photoelectron is measured then the work function allows the binding energy of the electron in the atom to be determined. Therefore an atom can be identified and this is the basis of ultraviolet photoelectron spectroscopy (UPS) (Williams (1977)) and X-ray photoelectron spectroscopy (XPS) (Briggs and Seah (1990) and Rivie`re (1990)) described in chapter 5. Following ionisation of the atom by the incident photon, the atom will decay to the ground state either by the emission of a photon or by ejecting an electron. In the first case an electron in a higher orbital will fall into the hole created by the initial ionisation event with the emission of a photon of a wavelength determined by the difference in energy of the two electron energy levels, while in the second case an Auger electron is ejected. Emissions of longer wavelengths resulting from a sequence of decays is also possible as the atom rearranges. The process in which an Xray ionises the atom and an X-ray photon is emitted is known as X-ray fluorescence (XRF). In 1925 Pierre Auger (1925) was studying cosmic ray tracks in a Wilson cloud chamber and realised that certain tracks could be explained only if the ionised atom was decaying by emitting another electron. This process has since become known as the Auger effect and the emitted electron as the Auger electron. Here the atom is ionised and rearranges with an electron from an outer electron shell falling into the hole created by the initial ionisation, but, instead of the energy being emitted as a photon, it is transferred to a third outer shell electron that is then ejected. The energy of the ejected Auger electron is determined by the binding energies of the electrons which take part in the process and is given approximately by the equation EAuger ¼ E1 E2 E3
ð2:10Þ
where E1 , E2 , E3 are the energies of the electron shells taking part and is the work function.
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Very short wavelengths (<1012 m gamma rays) In 1958 R L Mo¨ssbauer (1958, 1964) discovered nuclear gamma ray resonance in a solid and this technique has become a popular method for studying both composition and crystal structure. The basic principle of the method involves processes that occur when an atom emits a gamma ray (Long (1984), Frauenfelder (1962) and Wertheim (1964)). When atoms decay by the emission of either a photon or an electron, momentum is transferred to the emitted particle and to conserve momentum in the system the atom must recoil in the opposite direction. This is manifested by induced lattice vibrations or heating within the material. However, the quantum theory states that the energy states within an atom are not continuous but increase in discrete steps. If an atom is, say, in a ground state it can increase to the next level only if it receives an amount of energy equal to or greater than the difference between the ground state and the next energy level. The momentum of the photon emitted from the atom may be less than the quantum of energy required to excite the lowest energy level to the next highest. As a consequence there is a finite probability that the gamma ray will not lose energy and the event will be recoil-free. Similarly a gamma ray has a finite probability that it will be absorbed in a recoil free event by an atom. If energy is not lost by emission or absorption then the associated lines may overlap to allow nuclear gamma ray resonance. In the event that the emission and absorption lines do not overlap it is possible to bring them into coincidence by moving the emitter or absorber relative to one another and utilising the Doppler shift to vary the frequency of the gamma ray. This technique, described in more detail in chapter 5, is used to identify the environment of atoms in materials in which the Mo¨ssbauer effect occurs. While iron is the most commonly studied metal there are many others which have been utilised.
2.5.2
Electrons
In many respects the loss processes involving electrons have many similarities with higher energy photon loss processes. When an electron impinges on an atom in a material it may interact with that atom to either excite an electron in the atom to a higher orbital or give an electron in the atom sufficient energy to escape. In the former case the incident electron will lose a discrete amount of energy equal to the difference in binding energy of the two electron energy shells involved. By measuring the amount of energy the incident electron has lost, the difference in electron binding energies may be determined. This is utilised to identify the atoms causing the energy loss in a technique known as electron energy loss spectroscopy (EELS) (Ibach and Mills (1982), Joy (1979)). If the electron is ejected, however, the incident
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Interaction of radiation with materials
electron will lose energy that is equal to or greater than the binding energy of the electron in the atom. Thus the energy loss spectrum of the incident electron appears as a series of edges with tails to higher electron energies. It is frequently used in conjunction with the transmission electron microscope to determine the presence and concentration of elements of low atomic number. Ionisation with photon emission We have described the effect of excitation or ionisation on the incident electron, but following the initial excitation, processes take place within an atom which are identical to those described in the section dealing with photons. Clearly in the excited or the ionised condition the atom contains an inner electron shell with a hole and the atom can relax by an electron from an outer orbital falling into this hole. When this occurs an amount of energy is released, equal to E1 E2 , as a photon. By determining the energy of the photon the type of atom may be identified. The photon energy may be measured using a Li drifted silicon crystal in a technique known as energy dispersive X-ray (EDX) spectroscopy (chapter 6) or by using single crystals to determine the wavelength and hence wavelength dispersive X-ray (WDX) spectroscopy (Gilfrich (1974). Ionisation with Auger electron emission There is another way in which the ionised atom can decay, that is with the emission of an Auger electron in exactly the same manner as described for the photon ionised atom. In this process the atom relaxes by an electron from a higher orbital falling into the hole created by the initial ionisation
Figure 2.22. The probability that an atom will decay with the emission of a photon or an Auger electron for K and L shell electrons.
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process in the same manner as electron fluorescence. However, instead of the energy being released as a photon, it is transferred to an electron in an outer orbital which may have sufficient energy to escape from the atom. This electron is known as the Auger electron and has an energy given by equation (2.10). The probability, W, that the atom will decay with the emission of a photon or an electron is given by (Burhop (1952)) W ¼ ð1 aZ 4 Þ1
ð2:11Þ
where a is a constant and Z is the atomic number. This function is plotted in figure 2.22 for both K and L shell electrons. For elements of low atomic number the predominant form of decay is by Auger emission but as the atomic number of the element increases so the likelihood of Auger emission decreases and photon emission increases.
2.6
Atom and ion processes
Finally we turn to the interaction of atoms and ions with materials. Compared with the photon and electron beams, an atom or ion is massive and the wave nature may effectively be ignored and the atom or ion can be treated by Newtonian mechanics. 2.6.1
Scattering
Ions with very low energy can be elastically scattered from a material surface as shown schematically in figure 2.23. If the incident ion has a mass M1 and an energy E1 and is scattered through an angle such that, after deflection by atoms of mass M2 , it will lose energy. The energy of the deflected ion, E2 , is, for the case when M2 > M1 , given by E2 ¼
½M1 cos þ ðM22 M12 sin2 Þ1=2 2 E1 : M2 þ M 1
ð2:12Þ
When ions of higher energy impinge on a surface, considerable damage occurs by inelastic scattering. The ion will embed into the surface, knocking atoms of the material in random directions. Some of the initial momentum will be transferred to ions, or ion clusters, in a backwards direction. A proportion of these are ejected from the surface together with electrons. The electron current can be detected in a similar manner to the scanning electron microscope and an image built up or the ions can be mass analysed and the composition and chemical form of the material determined. Atoms will be ionised and will decay with the emission of photons and Auger electrons which may be detected and analysed in exactly the same manner as for photon and electron ionisation described above.
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Interaction of radiation with materials
Figure 2.23. Elastic scattering of ions of mass M1 by a surface composed of atoms of mass M2 .
2.7
Effect of high electric fields
There are a few techniques that utilise the effects of high electric fields to obtain microstructural information concerning materials. In order for electrons to leave an atom they have to surmount a potential barrier, V. If an electrostatic potential is applied to the specimen both the shape and height of the barrier are modified as shown in figure 2.24. The field at certain points can be made very large by appropriate specimen design such that electrons can be induced to leave the surface in a technique known as field emission microscopy (FEM) (Muller (1956)) and in related techniques atoms near the surface may be ionised and accelerated to form an image in field ion microscopy (FIM) (Kane (1979), Muller and Tsong (1969) and Panitz (1982)) while individual atoms may be induced to desorb from the surface and are subsequently identified using a mass spectrometer in atom probe microscope (Muller et al (1968)). When two conducting materials are separated by an insulator the insulator acts as a barrier to the flow of electrons. This is because the electron shells in the insulator are completely full or empty and the requirement for conduction that one or more electron shells be only partially full is not satisfied. If an electron from a full shell can surmount the potential barrier then the material acts as a conductor. There is a finite probability that an electron that does not have sufficient energy to surmount the potential barrier may ‘tunnel’ through it. A similar phenomenum occurs with electron tunnelling through a thin insulating layer. If the distance between two conductors is
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Figure 2.24. Potential energy curves for an atom (a) and an atom in the presence of a field (b).
sufficiently small, typically 1 nm, then there is a significant probability that the electron will pass through the barrier. This phenomenon, known as electron tunnelling, is utilised in the scanning tunnelling microscope (STM) (Bessenbacher et al (1989)) and can be used to identify surface features to a sub-atomic spatial resolution.
2.8
Acoustic phenomena
Before leaving this chapter acoustic waves should be mentioned. These waves travel through a solid medium by exciting vibrations in the material. Imperfections in the lattice, such as strain fields, particles, voids and cracks, will interact with the acoustic wave and, by detecting the scattered wave, information relating to the imperfection can be obtained. Acoustic waves range from the relatively long sound waves to the much shorter ultrasonic waves. The long wavelength waves will penetrate many centimetres into a solid and can detect, non-destructively, defects buried in the bulk. However, the long wavelength results in poor spatial resolution. On the other hand, the short wavelength ultrasonic energy can have much better spatial resolution but is absorbed more rapidly. Acoustic phenomena are often combined with other techniques to obtain additional information concerning a solid (Smith (1986) and Briggs (1985)). An example is the use of a laser to scan a surface while the reflected light is detected and simultaneously the acoustic wave transmitted through the solid defects below the surface may be detected
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Interaction of radiation with materials
Figure 2.25. A secondary electron image (SEI) and its corresponding electron acoustic image (SEAM) (reproduced by permission of Cambridge Technology Ltd).
in a technique known as scanning laser acoustic microscopy (SLAM). Similarly when an electron beam is rastered over a material surface, as in the scanning electron microscope, acoustic waves are generated and travel through the material. By detecting the acoustic waves, flaws in the solid may be identified. This technique is known as scanning acoustic electron microscopy (SEAM) and is used to identify regions in the material for study by other techniques. Figure 2.25 shows a secondary electron image together with the electron acoustic image from the same area where subsurface defects have been resolved.
2.9
References
Akitt J W 1983 NMR and Chemistry 2nd edition (London: Chapman and Hall) Andrews D L and McCoustra M R S 1990 in Perspectives in Modern Chemical Spectroscopy (Berlin: Springer) ch 8 Auger P 1925 J. Phys. Radium 6 205 Baranska H, Labudzinska and Terpinsk J 1987 Laser Raman Spectroscopy; Analytical Applications (Chichester: Harwood) Benninghoven A, Rudenauer F G and Werner H H 1987 Secondary Ion Mass Spectrometry (New York: Wiley) Bessenbacher F, Laegsgaard E and Stensgaard I 1989 Microscopy and Analysis July 17 Briggs D and Seah M P 1990 Practical Surface Analysis 2nd edition vol 1 (Chichester: Wiley) Briggs G A D 1985 An Introduction to Scanning Acoustic Microscopy (Oxford: Oxford University Press) Burhop E S 1952 The Auger Effect (Cambridge: Cambridge University Press) Castaing R 1960 Adv. Electron. Electron Phys. 13 317 Cazaux J 1982 J. Appl. Surf. Sci. 10 124
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References
51
Clark R J H and Hester R E ed 1983–5 Advances in Infra-Red and Raman Spectroscopy (Chichester: Wiley) Cross A D 1960 Practical Infra-Red Spectroscopy (London: Butterworths) Cudby M E A and Williamson D J 1990 Multinuclear High Resolution NMR in Solids, in Perspectives in Modern Chemical Spectroscopy ed D L Andrews (Berlin: Springer) ch 7 Cullity B D 1979 Elements of X-ray Diffraction (Reading, MA: Addison-Wesley) ETP Semra Pty. Ltd. 1990 Technical Bulletin March 3 Frauenfelder H 1962 The Mo¨ssbauer Effect (New York: W A Benjamin) Gilfrich K V 1974 in Characterisation of Solid Surfaces ed P F Kane and G B Larrabee (New York: Plenum Press) ch 12 Hertzberg G 1945 Infra-Red and Raman Spectra of Polyatomic Molecules (New York: Van Nostrand) Hutchings M T and Windsor C G 1986 in Neutron Scattering ed K Skold and D L Price (London: Academic Press) ch 25 Ibach H and Mills D L 1982 Electron Energy Loss Spectroscopy and Surface Vibrations (New York: Academic) Johansson S A E and Campbell J L 1988 PIXE: A Novel Technique for Elemental Analysis (Chichester: Wiley) Joy D C 1979 in Introduction to Analytical Electron Microscopy ed J J Hren, J I Goldstein and D C Joy (New York: Plenum Press) Kane P F 1979 in Introduction to Analytical Electron Microscopy ed J J Hren, J I Goldstein and D C Joy (New York: Plenum Press) ch 6 Littmark U and Hofer W O 1980 Nucl. Instrum. Meth. 170 177 Long G J ed 1984 Mo¨ssbauer Spectroscopy Applied to Inorganic Chemistry vol 1 (New York: Plenum Press) Madden P K, Buswell J T and Fisher S B 1979 CEGB Report RD/B/N4568 Mo¨ssbauer R L 1958 Z. Phys. 151 124 Mo¨ssbauer R L 1964 Recoiless nuclear resonance absorption of gamma radiation in Nobel Lectures 1961 Laureates, Biographies and Presentation Speeches (Amsterdam: Elsevier) p 583 Muller E W 1956 Field emission microscopy in Physical Methods in Chemical Analysis vol III ed W G Bel (New York: Academic Press) p 135 Muller E W, Panitz J A and McLane S B 1968 Rev. Sci. Instrum. 39 83 Muller E W and Tsong T T 1969 Field Ion Microscopy (New York: Elsevier) Niehus H and Baner E 1975 Surf. Sci. 47 222 Panitz J A 1982 J. Phys. E: Sci. Instrum. 15 1281 Peiser H S, Rooksby H P and Wilson A J C 1960 X-ray Diffraction by Polycrystalline Materials (London: Chapman and Hall) Rao C N R 1961 Ultra-Violet and Visible Spectroscopy (London: Butterworths) Rivie`re J C 1990 Surface Analytical Techniques (Oxford: Clarendon) Schiøt H E 1972 Radiat. Eff. 14 39 Seah M P and Dench W A 1979 Surf. Interface Anal. 1 2 Seah M P 1986 Surf. Interface Anal. 9 85 Siegbahn K ed 1965 Alpha-, Beta- and Gamma-Ray Spectroscopy (Amsterdam: NorthHolland) Sigmund P 1981 in Topics in Applied Physics 47 Sputtering by Particle Bombardment I ed R Behrisch (Berlin: Springer) Smith D P 1971 Surf. Sci. 25 335
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Interaction of radiation with materials
Smith G C 1986 Mater. Sci. Technol. 2 881 Symons M 1978 Electron Spin Resonance Spectroscopy (New York: Van Nostrand Reinhold) Thomson A J 1990 Electron paramagnetic resonance and electron nuclear double resonance spectroscopy in Perspectives in Modern Chemical Spectroscopy ed D L Andrews (Berlin: Springer) ch 12 Wertheim G K 1964 The Mo¨ssbauer Effect: Principles and Applications (New York: Academic Press) Ward A and Fisher S B 1992 Proceedings of the 15th International Symposium on Effects of Radiation on Materials (Nashville) eds R E Stoller, A S Kumar and D S Gellos (Philadelphia: ASTM) Wild R K 1985 Spectrochimica Acta 40B 827 Williams P M 1977 in Handbook of X-Ray and Ultraviolet Photoelectron Spectroscopy ed D Briggs (London: Heyden) ch 9 Wilson E B, Decius J C and Cross P C 1955 Molecular Vibrations—The Theory of InfraRed and Raman Spectra (London: McGraw-Hill) Ziegler J F, Biersack J P and Littmark U 1985 The Stopping Range of Ions in Solids vol 1 (New York: Pergamon Press)
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Chapter 3 Vacuum systems 3.1
Introduction
Within the wide variety of techniques for characterising the microstructure of materials, some operate using a relatively simple environment such as the normal atmosphere whereas others require sophisticated containment systems usually to produce extremely good vacuums. For example, simple electron microscopes operate with a vacuum that is less than 102 Pa and, indeed, with the addition of sophisticated analytical attachments such as windowless spectrometers there is now a trend to achieve pressures of 105 Pa or better. Any technique that is required to analyse a surface either chemically or crystallographically is compelled to operate with vacuums in the region 107 to 109 Pa. Figure 3.1 indicates some of the techniques that are currently available for microstructural evaluation and their vacuum requirements. Since so many methods of characterising material microstructure require the production of high and ultra-high vacuums (UHV) we will describe the various methods for producing, containing and measuring vacuums (O’Hanlon (1989), Leybold (1987)).
3.2
Kinetic theory of gases
We do not propose to describe the kinetic theory of gases in detail, but since a number of conclusions of the theory are essential for determining the requirements for various instruments we will set out the basic elements of the theory. For a more thorough description the reader is referred to Dushman (1962), Weber (1968) and Diels and Jaeckel (1966). In essence the kinetic theory of gases is the result of theoretical considerations concerning the movement of discrete particles between which no forces are acting. The number of particles, atoms or molecules, per unit volume or density, , their speed, c, and their molecular mass, M, determine the pressure, p, within a system where the pressure is related to the number of molar particles, n, in unit volume, V, and the temperature, T, by the
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Vacuum systems
Figure 3.1. Pressure regions based upon existing terminology and their relationship to some common microstructural evaluation techniques.
ideal gas equation PV ¼
nkT NA
ð3:1Þ
where k is the Boltzman constant and NA is the Avogadro number (k ¼ R=NA ; where R is the molar gas constant). Two measures of the speed of the particles are used: the mean particle velocity, v, and the mean square velocity, v2 , which are given by rffiffiffiffiffiffiffiffiffi 8kT ð3:2Þ v ¼ M and v2 ¼
3kT : M
ð3:3Þ
The pressure in the system is given by v2 ¼ 13 nM v2 : P ¼ 13
ð3:4Þ
The particles travel in straight lines until they collide with the walls of the vessel or other particles. The collision rate, n_ , is defined as the number of collisions per second and the mean free path, l, is the distance a particle travels, on average, between two collisions. These are given by equations n_ ¼ v=l ð3:5Þ l ¼
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2
1=2
1 nð2rÞ2
ð3:6Þ
Production of vacuum
55
where r is the particle radius. Thus the mean free path length is inversely proportional to the number density and to the pressure. The impingement rate is of considerable interest when evacuating a system since it determines the contamination rate of a surface of the containing vacuum chamber or that of the specimen being examined. The impingement rate, N_ , of particles on to a surface is given by n v ð3:7Þ N_ ¼ : 4 The length of time that is required to contaminate a specimen is often referred to as the time to form a monolayer of containing environment particles on a clean surface. If all the particles that strike a surface stick to that surface, i.e. the sticking probability is unity, and if the number of free spaces per unit of surface area, A, then the time to form a monolayer ðÞ is ¼
A 4A ¼ n v N_
ð3:8Þ
Substituting in the above gives the monolayer time in terms of pressure, molecular mass and temperature. If the pressure, , is measured in Pascals the time, in seconds, to completely cover the surface with a single layer of oxygen at room temperature is ¼
5 104
ð3:9Þ
Thus if a specimen is examined in a vacuum of 5 104 Pa, a surface will be completely covered by the containing environment particles in one second. In practice to obtain a chemical analysis of a surface it is necessary to keep the specimen under investigation relatively free from contamination for a period of say one hour. To achieve this it is necessary to produce a vacuum that is in the region of 108 to 109 Pa. Typical growth rates for various gases as a function of gas pressure assuming a sticking coefficient of unity are reproduced in figure 3.2. In practice sticking coefficients are often less than unity, particularly for inert gases and for active gases on oxide surfaces. In addition many analytical techniques sample from many thousands of atom layers, and in such cases the build up of one or two atomic layers of contaminant would not degrade the results significantly. Thus UHV conditions, whilst required for certain surface analytical techniques, are not used in all cases because a balance is required between the practicality of achieving the vacuum and the ability to produce the required accuracy of the analysis.
3.3
Production of vacuum
There are two stages in the production of a vacuum. First, a vessel must be constructed from materials that do not outgas significantly compared with
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Vacuum systems
Figure 3.2. Rate of contamination of a surface, assuming a sticking coefficient of unity, as a function of pressure for some common gasses.
the speed of the vacuum pumps. Second, design is such that it does not permit gas to enter from the outside atmosphere, either by diffusion through the walls or by leaking passed seals. The chamber must then be evacuated to the desired pressure by pumps that do not introduce any deleterious gases into the chamber whilst removing the original atmosphere. We will first describe the construction of vacuum chambers and then typical pumping systems available. 3.3.1
Construction of vessels
Materials used in the construction of vacuum systems must be readily available, capable of being machined and have low rates of outgassing under the conditions of use. The outgassing rates for a number of common metals as a function of temperature are given in figure 3.3. Materials may have to operate at elevated temperatures and these must clearly have a very low saturation vapour pressure at the temperature of operation. Cadmium and magnesium have relatively high vapour pressures and should not be introduced into systems that operate above room temperature. Indium, copper and gold are all ductile materials and are often used as seals between components, although indium should only be used at room temperature or below. Gold is clearly preferable to indium from the outgassing properties but indium is cheaper and normally acceptable. Iron and steels all have low vapour pressures at temperatures between room temperature and 600 K and are suitable for construction of vessels. Many applications, particularly electron optical and others, where the beam used to interrogate the material is influenced by a magnetic field, require that the containment vessel be manufactured from non-magnetic material. In such cases an
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Production of vacuum
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Figure 3.3. Outgassing rates of some common elements as a function of temperature.
austenitic stainless steel (usually AISI Type 304) is adequate although for specialised applications other materials, for example Invar, which has a low coefficient of expansion and has magnetic properties, is used where the effect of external magnetic fields must be reduced. Filaments for ionisation gauges are normally made from tungsten, which may be thoriated to increase the electron emission yield while reducing the operating temperature. Windows are manufactured from quartz or silica and when it is necessary to mount these in stainless steel a graded seal is added between the main part of the window and the metal. A graded seal is constructed of materials that have thermal expansion coefficients ranging from that of the window to that of the steel to prevent the build-up of thermal stresses when the system is heated. Electrical connections are frequently manufactured from tungsten and set in a ceramic graded seal that is bonded to the steel flange. Outgassing of components is reduced by giving the steel a fine ideally polished surface finish. This effectively reduces the surface area and hence the area available for adsorption of gases. A low pressure oxidation to produce a thin chromium-rich surface oxide on stainless steel can also help to reduce the outgassing rates. Transfer of heat is best carried out using components constructed from copper and insulated using a combination of ceramics and stainless steel. The production of ultra-high vacuum requires the entire vacuum system to be baked to a temperature between 400 and 500 K to drive off adhered gases from the component surfaces and chamber walls and hence reduce the outgassing rate when the system is operated at room temperature. Vacuums better than 106 Pa can rarely be achieved without baking and, therefore, considerable thought has to be given to the choice of materials used in their construction.
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58 3.3.2
Vacuum systems Medium to high vacuum (102 to 105 Pa)
These are vessels which normally contain equipment designed primarily for bulk specimen examination and analysis. This would involve the analysis of layers greater than a few hundred nanometres thick where a few atom layers of contamination is not a serious problem. Transmission electron microscopes, scanning electron microscopes, ion source and some plasma research instruments are in this category. The bulk of these instruments are constructed from a range of steels including stainless and often contain large electromagnetic lenses for focusing the incident electron beams which travel down a narrow column. These components are, in general, not capable of being heated and as a result outgassing is relatively high. Sections of the instrument are connected by ‘O’ rings made from a synthetic rubber compound such as Viton. The ‘O’ rings may require sealing with a vacuum grease which, in turn, may outgas hydrocarbons into the system.
3.3.3
Ultra-high vacuum (106 to 109 Pa)
Ultra-high vacuum is obtainable using current technology and is required for all the surface analytical techniques such as low energy electron diffraction (LEED), X-ray photoelectron spectroscopy (XPS), Auger electron spectroscopy (AES), secondary ion mass spectroscopy (SIMS), secondary neutral mass spectroscopy (SNMS), field ion microscopy (FIM) and the atom probe. The objective is to create an environment with as low a pressure as possible and with the residual partial pressures being composed primarily from inert gases. Such vessels are manufactured from a Type 304 stainless steel and the whole vessel would be capable of being baked at temperatures about 500 K to drive off any adsorbed water vapour and reduce outgassing. Thus ‘O’ rings that require grease cannot be used and Viton ‘O’ rings are avoided wherever possible. Sections of the system that require connecting together are joined in one of two ways shown in figure 3.4. The cheapest and easiest is to use the knife-edge copper gasket system (figure 3.4(a)). In this method a knife-edge shaped in the form of a tilted ‘V’ is machined on to each of the faces to be joined in such a way that the top of the knifeedge is approximately 1 mm below the level of the flange edges. An annealed copper gasket is then positioned between the two knife-edges which are tightened using a series of bolts. The knife-edge cuts into the copper gasket making a good vacuum seal. The alternative method is to machine flat surfaces on to the flange and to position an annealed gold, or in some cases an aluminium or indium, ‘O’ ring between the flanges. The two flanges are then tightened down, compressing the ‘O’ ring and making a good vacuum seal. The gold ‘O’ ring approach is generally used where the flange diameter is large, normally 400 mm or greater, or in situations where the seal is intended to remain undisturbed. The copper gasket type
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Figure 3.4. Methods for connecting rigid components to construct UHV vessels: (a) copper gasket, (b) gold ‘O’ ring.
of seal has the advantage that it is cheap and easy to replace, particularly where the flange is vertical or at an angle to the horizontal. Components, such as specimen stages and x–y–z position manipulators, require to be moved within the vacuum chamber and the motion must be transmitted through the vacuum walls. This is effected by a series of edge-welded stainless steel bellows (figure 3.5). This method of construction of ultra-high vacuum systems leads to a characteristic design of instruments.
Figure 3.5. The use of edge-welded stainless steel bellows to translate motion to a UHV system.
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Vacuum systems
Figure 3.6. A typical UHV system constructed in stainless steel and fitted with all-metal seals.
One such system (figure 3.6) is a multipurpose instrument designed to use the techniques of Auger electron spectroscopy, X-ray photoelectron spectrscopy and secondary ion mass spectroscopy to study material surfaces. The characteristic feature of the construction is the few interconnecting chambers with many removable ports. Instruments tend to be made with the maximum number of optional ports included, since the cost of incorporation at the manufacturing stage is low but at a later date would prove difficult and expensive and this offers maximum flexibility for future applications. In the example shown the right-hand chamber houses all the analytical techniques while the chamber to the left contains the preparation equipment. This particular instrument is fitted with two fracture stages, one for impact fracture at liquid nitrogen temperatures and one for tensile fracture. Specimens are moved from the atmosphere to the preparation chamber via an introduction chamber using a transfer probe. After fracture, followed possibly by ion cleaning, the specimen can be moved on to an x–y–z manipulator in the analytical chamber situated to the right-hand side of the system.
3.4
Vacuum pumps
3.4.1
Pumping media
While many vacuum pumps used in modern systems can be regarded as clean, employing no liquids or greases, a large number rely on oils to
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Figure 3.7. Saturation vapour pressure of some frequently used pump oils and mercury.
produce the vacuum. The saturation vapour pressure of the liquid will determine the ultimate pressure that can be attained. Figure 3.7 gives the saturation vapour pressure of a number of oils as a function of temperature. Mercury, once a common diffusion pump liquid, has a vapour pressure greater than all the oils listed and if this is to be used as a pump liquid to attain low pressures a liquid nitrogen cold trap must be used. However, for reasons of safety and health this liquid is rarely used now. There is a large range of vapour pressures between the different oils with almost seven orders of magnitude between the highest and the lowest at room temperature. The choice of pump oil will be determined by the vacuum required and the conditions under which it has to operate. Diffusion pumps tend to use mineral oils, silicone oils and oils based on polyphenyl ethers. Silicone oils are more resistant to air than mineral oils and can withstand higher temperatures. Certain polyphenyl ether based oils and silicone oils can have extremely low vapour pressures and are recommended where robust oils are required to achieve low ultimate pressures. 3.4.2
Low to medium vacuum pumps
These are pumps used to reduce the system from atmospheric pressure to a rough vacuum or 101 Pa, a condition that would permit high vacuum pumps to take over. They include rotary pumps, sorption pumps and turbomolecular pumps, although the latter can operate over a much wider vacuum range.
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Vacuum systems
Rotary pumps There exists a large number of types and many different designs of rotary pump (Bode (1960), Flecken (1966), and Ba¨chler and Knobloch (1971)). Some are more efficient than others but they all work on the same basic principle of compressing gas in one region of the pump, and then moving it to a second region from which it is then expelled. Gas from the chamber being evacuated then moves into the first region of the pump and the process of compression and evacuation begins again. Figure 3.8 shows the pumping stages of a trochoid rotary pump which consists of an elliptical piston driven in eccentric rotation by toothed wheels fixed to the drive shaft and piston. Consider the pump in position (1) with a small volume of gas on the input side of the pump and a large volume on the output side. As the piston rotates, the gas on the output side is compressed and at the same time evacuated, while gas enters the input side from the chamber (stages 2 to 4). The piston is shaped such that a part of it is always in contact with the pump wall at position P, ensuring that no gas can travel from the output side to the input. There are many variations on this pump. One common version is the rotary vane pump which has a circular cylinder with vanes on either side that are arranged to be always in contact with the pump walls. Other pumps have two stages with two rotating vanes and some have two pistons rotating in one chamber. The rotary pump will pump out chambers at atmospheric pressure and reduce the pressure to 1–102 Pa with an efficiency which varies with the gas
Figure 3.8. The stages (1 to 4) in one cycle of a rotary pump (reproduced with permission of Leybold AG).
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Vacuum pumps
63
being pumped. They are, in general, very efficient at handling large quantities of gas, tolerate corrosive gases and are the workhorses of vacuum technology. They pump air and inert gases such as argon and nitrogen very well but are less well equipped to pump the compressible gases such as CO2 . Adsorption pumps These pumps operate by the physical adsorption of gases on to the surface of molecular sieves. Zeolite, an alkali alumino-silicate, has a very large surface area for a given mass, approximately 103 m2 for each gramme of material. It has a pore diameter of 1.3 nm which is comparable with the diameters of most atmospheric gases such that a gramme of this material is capable, at liquid nitrogen temperature, of adsorbing 133 mbar per litre of nitrogen gas. However, the adsorption of gas on the zeolite surface varies with temperature and is, in general, three to four orders of magnitude less at room temperature than at liquid nitrogen temperature (77 K). A schematic diagram of this type of pump is shown in figure 3.9. The pump is first isolated from the chamber to be evacuated and cooled by immersing in liquid nitrogen, and when equilibrium is reached a valve to the chamber is opened and the gases allowed to adsorb on the zeolite surface. The ultimate pressure that can be reached is dependent on the amount of gas to be pumped and the volume of zeolite in the adsorption pump; as the surface sites become filled the pumping speed will decrease. However, if sufficient zeolite is used the ultimate pressure attainable should be in the region of 102 Pa. The advantage of this type of pump is that it is very clean since there are no oils present to backstream and contaminate. It is, therefore, primarily
Figure 3.9. A schematic of a typical sorption pump. 1, inlet port; 2, degassing port (safety outlet); 3, support; 4, pump body; 5, thermal conducting vanes; 6, adsorption material (e.g. zeolite) (reproduced with permission of Leybold AG).
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Vacuum systems
used as the first stage of pumping UHV systems that must be kept free from contamination. Moreover, it has the secondary advantage that no moving parts are involved and maintenance is minimal. The drawbacks are that pumping is not continuous so that after the chamber has been evacuated the pump must be isolated from the chamber and allowed to warm up to room temperature to release the adsorbed gas through a release valve before it can be cooled and used again. If large quantities of water vapour are to be pumped it is advisable to heat the pump to 500 K before reusing. In addition while it pumps most large compressible gases such as oxygen, nitrogen, water vapour and CO2 well it is poor at pumping inert gases such as helium or neon. Turbomolecular pumps Turbomolecular pumps (Frank (1972), Fle´cher (1977) and Henning and Knorr (1980)) (figure 3.10) work on the principle that a gas molecule striking a moving surface will be given a component of momentum in the direction of the moving surface, the principle used by fans to circulate air. However, it is only comparatively recently that this principle has been used to attain
Figure 3.10. A diagram showing a cross-section through a turbomolecular pump. 1, stator blades; 2, rotor body; 3, intake flange; 4, blades of the suction stage; 5, blades of the compression stage; 6, drive shaft; 7 and 8, ball bearings; 9, high frequency rotor (reproduced with permission of Leybold AG).
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Vacuum pumps
65
relatively high vacuums since the rotors must move at high speeds and tolerances between the rotors and the walls must be small to prevent gas backstreaming. Gas enters from the chamber being pumped through the large port (3) to a series of large rotor/stator blades designed to capture the gas (4) which is then compressed in stages by the smaller rotor/stator blades (5) before being exhausted. Turbomolecular pumps rotate the blades typically at 500 Hz and are capable of pumping from relatively high pressures down to ultra-high vacuum. Whilst they are capable of pumping from atmospheric pressure this puts considerable strain on the pump and it is advisable to use some other form of pumping to achieve pressures of approximately 102 Pa before bringing in the turbomolecular pump. These pumps are very reliable but may cause some vibration in the system which can be reduced to a minimum by using magneto-bearings. There is also a danger that oil, used to lubricate the bearings, may creep past and enter the vacuum system. To reduce this risk it is advisable to install a valve close to the top of the turbomolecular pump and to close it at all times when the pump is not in use. Diffusion pumps Diffusion pumps are used to remove large volumes of gases at high speeds over the pressure range 101 to 109 Pa (Noller (1966)). They are simple and robust and resilient to pumping corrosive gases. Figure 3.11(a) shows schematically the design and operation of a typical diffusion pump. The pump fluid, usually oil, is heated at the base of the pump and the vapour rises up the chimneys and emerges at supersonic speeds from the nozzles. These jets travel towards the sides, which are water cooled, where they condense and flow back to the reservoir. Gases in the region of the top of the diffusion pump get entrapped by the oil stream and are taken down to the outlet. In this region the oil is reheated to about 400 K to drive off the entrapped gases, which are removed by a rotary backing pump maintained at 101 Pa. Oil does not only travel in the direction of the pump walls, but some of the oil will make its way into the chamber being evacuated where it can become a source of contamination. This is a very serious problem for UHV systems and it is necessary to incorporate a series of devices to reduce this risk. Figure 3.11(b) shows the standard arrangement to prevent oil backstreaming into the vacuum chamber. It consists of a baffle arrangement, an anti-creep barrier and a liquid nitrogen cold trap placed above the oil diffusion pump. In normal operation this system works well and little oil contamination reaches the main system. However, it relies on the liquid nitrogen cold trap operating at all times that the pump is running and the danger of contamination is always present. If it is important that a chamber is to be kept free from contamination such oil diffusion and rotary pumps should not be used.
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Vacuum systems
Figure 3.11. The operation of oil diffusion pumps. (a) Diagram showing oil streaming to effect pumping: 1, heater; 2, boiler; 3, pump body; 4, cooling coil; 5, high vacuum flange connection; 6, gas particles; 7, vapour jet; 8, backing vacuum connection port; A–D, nozzles. (b) Addition of a cold trap arrangement to reduce backstreaming: 1, diffusion pump; 2, shell or chevron baffle; 3, anti-creep barrier; 4, sealing gasket; 5, bearing ring; 6, cold trap (chilled with liquid nitrogen); 7, vessel (reproduced with permission of Leybold AG).
3.4.3
High to ultra-high vacuum pumps
Sputter-ion pumps These pumps operate by ionising the gas to be evacuated and then utilising electrostatic and magnetic fields to remove the ions (Wutz (1969)). One such pump, known as a diode ion pump (figure 3.12), consists of a cathode constructed from titanium with the anode and chamber walls made of stainless steel. The pump is surrounded by large magnets to produce a magnetic field, B, in the direction indicated (figure 3.12(a)). These pumps operate by producing a cold cathode discharge which ionises gas atoms in the body of the pump. These are accelerated to the cathode where they sputter titanium ions from the cathode walls (figure 3.12(b)). The ionised gas atoms become buried in the cathode, titanium atoms trap other gas atoms as they impinge on the cathode and anode walls and also pump active gas atoms by reacting with them. The pumping speed of these pumps varies with the type of gas being pumped. There is more than one mechanism by which active gases can be pumped but inert gases can be pumped only by ionisation followed by burial in the
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Vacuum pumps
67
Figure 3.12. Schematic diagrams showing the principle and operation of a diode ion pump. B is the direction of the magnetic field (reproduced with permission of Leybold AG).
cathode wall. In general most gases pump at rates similar to that for air or nitrogen, but argon and helium are pumped at rates approximately three times slower. In addition, the inert gases that are trapped in the cathode will eventually be released by further ion implants and the effectiveness for pumping these gases will decrease. Triode ion pumps have a different geometrical arrangement which makes them more efficient for pumping inert gases. Sputter ion pumps are very clean and there is no risk of contamination of the vacuum chamber. They perform best at low pressures, <104 Pa, but can be started at 102 Pa and if operated at pressures better than 108 Pa for most of the time will last up to ten years before it is necessary to replace the cathodes. They should not be used where there is a requirement to pump large volumes of gas, particularly if the gas is inert. Cryopumps Cryopumps operate by condensing the gas to be pumped on to a cold surface (Klipping (1974) and Schafer (1978)). Indeed, any ‘cold finger’ cooled to liquid nitrogen temperature (77 K) is a form of cryopump. Cryopumping takes place in three ways. A gas which has a boiling point higher than the cold surface will condense on that surface. A gas with a boiling point lower than that of the cold surface will be trapped by a high boiling point gas as it condenses (cryotrapping). Gases with a very low boiling point may bond to a gas with a high boiling point and thus be pumped (cryosorption) and hydrogen can be pumped in this way using ammonia. Whilst liquid
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Vacuum systems
nitrogen can be used as a coolant for cryopumps, high and ultra-high vacuums can be achieved only if liquid helium is used. The loss of liquid helium is reduced by surrounding the helium reservoirs with the less expensive liquid nitrogen. In a typical cryopump liquid helium is pumped through a heat exchanger to provide the cold surface. The rate at which the helium is pumped will determine the temperature of the heat exchanger surface. The helium which evaporates during this process is used to cool a surface around the heat exchanger and thus reduce heat flow into the heat exchanger. These pumps clearly require large volumes of liquid gases and are expensive to operate but provide the cleanest vacuum available with no unwanted gas atoms involved. Titanium sublimation pumps This pump is simply a titanium filament enclosed in a stainless steel chamber which is heated by passing a large current through it to evaporate titanium on to the chamber walls. The evaporated titanium pumps active gases by combining chemically with them. In most cases the walls of the pump are maintained at room temperature but in some cases are cooled by liquid nitrogen to improve the efficiency. Generally these pumps are used in conjunction with ion pumps to improve the speed or to attain lower base pressures. The filaments need to be changed on a regular basis and to increase the time between changes multiple titanium filaments are mounted in one chamber.
3.5
Pressure measurement
It is important to be able to measure the pressure in a vacuum chamber and on occasion to be able to identify and measure the partial pressures of the constituent gases. We will describe vacuum gauges that are commonly installed in analytical instruments and not the more fundamentally based instruments against which many of these gauges are calibrated. 3.5.1
Pirani or thermal conductivity gauges
As the gas pressure in a system falls so the number of atoms per unit volume decreases and the mean free path increases. This causes the thermal conductivity of the gas to vary with the pressure and it is this property that is used to determine the pressure in the range 1–101 Pa using gauges known as Pirani gauges. The Pirani gauge has a metal filament through which a small current is passed, causing it to heat up. The filament can lose heat by conduction, convection and radiation but at pressures between 102 and 101 Pa heat is lost primarily by conduction. By making the filament one arm of a Wheatstone bridge circuit the change in resistance can be determined and this, in turn, is calibrated to measure the pressure. At pressures above 100 Pa the major heat loss is
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Pressure measurement
69
Figure 3.13. A schematic diagram showing the cross-section of a Penning pressure measurement gauge (reproduced with permission of Leybold AG).
through convection and below 101 Pa it is lost by radiation and since both are independent of pressure this limits the range of operation for the gauge. 3.5.2
Ionisation gauges
The number of atoms in the vacuum chamber varies with the pressure. By ionising a fraction of these atoms, a current can be made to flow to a detecting electrode. The magnitude of the current will then be proportional to the pressure (Meinke and Reich (1967) and Beek and Reich (1974)). In the Penning or cold-cathode ionisation gauge, ionisation of atoms is produced by a cold discharge similar to the process adopted for sputter-ion pumps. In a Penning gauge (figure 3.13), the cold discharge is produced by applying a potential of approximately 2 kV between two electrodes. To make the mean free path of the electrons as long as possible, a magnet is added to the gauge in a position so that the magnetic lines of force are normal to the lines of electrical potential. This causes the electrons to travel in a spiral path and increases the probability of ionising an atom. The current produced is then calibrated to measure the pressure to an accuracy limited to about 50%. These gauges do not operate at pressures above 1 Pa because the gas forms a glow discharge but they can operate down to ultra-high vacuum and are cheap, simple to operate, and reliable. In the hot-cathode ionisation gauge shown schematically in figure 3.14, electrons are produced by a heated filament. These electrons are accelerated by a grid at a positive potential of a few hundred volts (figure 3.14(a)) and thereby ionize gas atoms in the region between the grid and the ion collector. The collector is negatively biased with respect to the grid and so reflects electrons and collects ions; the ion current is proportional to the pressure. These gauges operate below approximately 1 Pa down to ultra-high
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Vacuum systems
Figure 3.14. Principle of a hot cathode ionisation gauge together with some practical arrangements for pressure measurement.
vacuum, where they are limited by the photon (X-ray) emission caused by electrons impinging on the grid. Some of the photons that strike the collector in turn produce photoelectrons which travel to the grid, which is identical to a small positive charge hitting the collector, and this limits the pressure measurement range for these gauges. Several versions of the hot cathode ionisation gauge exist. In addition to the type shown in figure 3.14(b) there is the Bayard–Alpert ionisation gauge which attempts to reduce the X-ray effect by separating the cathode from the anode grid and making the collector as small as possible. Several alternative arrangements for the hot cathode ionisation gauge are shown in the figure ((c)–(f )). The probability that an atom will be ionized varies with the type of gas such that two different gases, present at the same partial pressure, will be measured as different pressures by the gauge. Most gauges are calibrated with respect to nitrogen and the difference between this and air is small, but helium will give a reading six times greater while xenon will be a factor of three lower. Therefore care must be exercised in placing too much reliance on an ionisation gauge reading without also determining the partial pressures of the different gases in the chamber. 3.5.3
Partial pressure measurement
This ideal method of pressure measurement has the objective of determining the pressure of each of the constituent gases making up the total gas pressure in the vacuum chamber. To do this requires the use of mass analysers and two commonly encountered types will be briefly described. Magnetic sector mass spectrometers A magnetic sector mass spectrometer (Polaschegg (1976)) is one of the less frequently used for partial pressure determination in analytical instruments
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Figure 3.15. Schematic diagram of a magnetic sector mass spectrometer for partial pressure measurement. 1, flange of the ion source unit; 2, cathodes; 3, anode (heated); 4, shield tube; 5, diaphragm (also used as ion collector for measuring total pressure); 6, intermediate diaphragm; 7, magnetic field; 8, cylindrical condensor; 9, diaphragm; 10, ion collector; 11, flange of the deflector unit (reproduced with permission of Leybold AG).
because it is bulky and can produce stray magnetic fields which may influence the analysis system (figure 3.15). It operates by a hot filament producing electrons which are accelerated across the entrance slit of 180 sector magnetic detector. A fraction of the atoms are ionised and accelerated to the entrance slit. Only those atoms with the correct velocity, mass and charge will travel around the spectrometer and pass through the exit slit and be collected by the detector. By varying the accelerating potential on the entrance slit a spectrum can be produced giving peaks at different mass numbers that allow the gas to be identified. Knowledge of the relative ionisation efficiencies of the different gases together with the characteristics of the mass spectrometer enable the partial pressures of the gases within the chamber to be determined. The quadrupole mass spectrometer The quadrupole mass spectrometer (Knoopmann (1980)), shown in figure 3.16, is now the most common source of partial pressure measurement in analytical systems. It is small, often less than 10 cm long and 5 cm in diameter, can be mounted on small flanges and provides a total pressure measurement in addition to partial pressures. Again the ions are produced in the source by electrons from the hot filament and are the accelerated to the entrance of the quadrupole. This consists of four electrodes positioned as shown in figure 3.16(b), to which a constant high frequency voltage, 2.5 MHz, and a d.c. voltage are applied. The ions travel along the axis of the rods oscillating in the high frequency field. Only those ions with the correct charge to mass ratio for the specific conditions will reach the detector
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Figure 3.16. (a) A typical example of a quadrupole mass spectrometer together with (b) a schematic diagram showing the operating principle (reproduced with permission of Leybold AG).
and be collected. The others oscillate increasingly towards the electrodes and eventually hit the sides. Both the high-frequency and the d.c. voltages are increased ensuring that the ratio between them is kept constant and in this way a mass spectrum (figure 3.17) is obtained. Quadrupole mass spectrometers are cheap, small and easy to fit to existing systems and enhance control over the vacuum, particularly when attempting to find a leak.
Figure 3.17. Two mass spectra obtained from the same environment using a quadrupole mass spectrometer (a) with constant peak width and (b) with constant sensitivity (reproduced with permission of Leybold AG).
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Leak detection
3.6
73
Leak detection
A major problem with all vacuum systems is the location of leaks, which if very large are easy to find but if small may be difficult to confirm and locate. When a leak in the system is suspected it must first be confirmed. The best way to do this is to switch off the pumping system and watch the pressure rise on the ionisation gauge or better still the mass spectrometer (figure 3.18(a)). In any vacuum system there will be constant outgassing from the chamber walls and other components. If there is no leak then the pressure rise will be initially linear but will soon achieve a limiting value (figure 3.18(b)). If there is a leak there will be a constantly rising component which will continue albeit at a slower rate (figure 3.18(c)). If a mass spectrometer is fitted to the system with no leak present then the outgassing components water vapour (18 amu) CO (28 amu), H2 (2 amu) will all increase but if there is a leak then oxygen (16 amu) and nitrogen (28 amu) will both increase with an O2 :N2 ratio of 1 :4. Here it is important not to confuse CO with nitrogen as both have the same atomic mass unit. Having confirmed that a leak exists the next stage is to locate its position. Clearly seals between components must be suspect, particularly if they have recently been changed. If the system has an ionisation gauge fitted then the change in sensitivity of the gauge to different gases can be utilised. Helium is six times more sensitive than air and by applying a small jet of this gas to all possible areas where there may be a leak should produce a rise in pressure on the gauge when the leak is found. It is important to start at the top of the system because the spent helium will rise from the area just tested. If the system has a mass spectrometer fitted, adopt the same procedure but adjust the mass spectrometer to sit on the helium peak position (4 amu) and watch for a rise in the partial pressure. As an alternative
Figure 3.18. Pressure changes observed on turning the pumps off in the presence of (a) a leak, (b) outgassing and (c) a leak with outgassing.
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to a source of helium, apply a small quantity of acetone to the suspect areas—but this should only be used in a well ventilated area.
3.7
Specimen handling
All materials outgas in a vacuum by the amount determined by the history and form of the specimen. Tomkins (1998) has reviewed ultra-high vacuum techniques and the vacuum compatibility of materials, while Lindfors (1998) deals with aspects of the handling, cleaning and processing of specimens. Thus it is necessary to avoid touching the specimen, since hands contain many oils with high vapour pressures and will degrade the vacuum for some considerable time. Always wear cotton gloves and use tweezers to handle the specimen. If the specimen has been in a solution, if possible, wash in distilled water then displace the water with a wash in iso-propanol. If it can be warmed in an oven without risk of damage, this will drive off adsorbed vapour and reduce outgassing. If specimens are to be stored this should be in a dry environment, ideally dry nitrogen, but failing this in a desiccator with silica gel. In some cases it is desirable to retain them in a vacuum chamber prior to transferring to the instrument for examination. Adopting these practices will ensure fast pump downtimes to achieve the desired vacuum and it will also reduce the amount of servicing necessary on the pumps. The subject of specimen handling, preparation and treatment is comprehensively covered by Czanderna et al (1998).
3.8
References
Bachler W and Knobloch D 1971 J. Vac. Sci. Technol. 9 402 Beek U and Reich G 1974 Vacuum 24 27 Beek U and Reich G 1975 Vacuum 25 223 Bode H 1960 6th Nat. Symp. on Vacuum Technology Transitions (London: Pergamon Press) p 268 Czanderna A W, Powell C J and Madey T E ed 1998 Specimen Handling, Preparation and Treatments in Surface Characterisation (New York: Kluwer Academic/Plenum Publishers) Diels K and Jaeckel R 1966 Leybold Vacuum Handbook (London: Pergamon Press) Dushman S 1962 Scientific Foundation of Vacuum Technique 2nd edition ed I M Laferty (New York: Wiley) Flecher P 1977 Proc. 7th Intern. Vac. Congress (Vienna) 1 25 Flecken F A 1966 Uber die Anwendung von Verdvangerpumpen in der Kabelindustrie Draht-Fachzeitschrift 17 3 Frank R 1972 Suppl. de la Revue le Vide 157 257 Henning H H and Knorr C 1980 Proc. 8th Intern. Vac. Congress (Cannes) Suppl. ‘Le Vide’ 2 283
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Klipping G 1974 Japan J. Appl. Phys. Suppl. 2 81 Knoopmann G 1980 Proc. 8th Int. Vac. Congress (Cannes) Suppl. ‘Le Vide’ 2 215 Meinke C and Reich G 1967 J. Vac. Sci. Tech. 4 356 Leybold AG 1987 Vacuum Technology; its Foundations, Formulae and Tables (Ko¨ln: Leybold) Lindfors P A 1998 Specimen Handling, Preparation and Treatments in Surface Characterisation ed A W Czanderna, C J Powell and T E Madey (New York: Kluwer Academic/Plenum Publishers) p 45 Noller H G 1966 Theory of vacuum diffusion pumps in Handbook of Vacuum Physics vol 1 ed A H Beck (London: Pergamon Press) O’Hanlon J F 1989 A User’s Guide to Vacuum Technology 2nd Edition (Chichester: Wiley) Polaschegg H D 1976 Appl. Phys. 9 223 Schafer G 1978 Vacuum 28 399 Tomkins H G 1998 Specimen Handling, Preparation and Treatments in Surface Characterisation ed A W Czanderna, C J Powell and T E Madey (New York: Kluwer Academic/Plenum Publishers) p 1 Weber F 1968 Dictionary of High Vacuum Science and Technology (Amsterdam: Elsevier) Wutz M 1969 Vacuum 19 1
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Chapter 4 Diffraction 4.1
Electromagnetic radiation
The wave nature of electromagnetic radiation provides a basis to describe a medium in which transverse waves propagate and from which the phenomena of interference, diffraction and polarisation can be derived from Maxwell’s electromagnetic theory of light in 1880 (Ditchburn (1952) and Jenkins and White (1951)). The basic equations of physical optics are derived from the differential form of the Maxwell equations and will be used here (Carlson (1977), Kline and Kay (1965), Lipson and Lipson (1969), Stratton (1941) and Tatarski (1961)). 4.1.1
The wave equation
A wave may be represented as a perturbation " which is described as " ¼ a sinð! þ Cx þ Þ þ a sin
ð4:1Þ
where ! is the frequency, is the phase which is a function of distance x and time , is the phase angle, and a and C are constants. More generally ¼ ! Cðx þ y þ zÞ þ
ð4:2Þ
where , , are constants such that 2 þ 2 þ 2 ¼ 1. Differentiation of equation (4.1) gives 2 "= 2 ¼ !2 "
ð4:3Þ
2 "=x2 ¼ C 2 2 ":
ð4:4Þ
Combining these plus differentials with respect to y and z gives 2 " 2 " 2 " C 2 2 " 1 2 " þ þ ¼ ¼ x2 y2 z2 !2 2 b2 2 which is the general equation of propagation in three dimensions.
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Figure 4.1. Vector representation of the addition of a series of waves.
Consider the situation where we have a series of waves "1 , "2 , "3 , . . . which if combined produce a resultant wave " ¼ "1 þ "2 þ "3 þ . This is known as the ‘principle of superposition’ which states that the disturbance due to a number of waves is equal to the algebraic sum of the disturbances produced by individual waves. If the waves "1 , "2 , "3 , . . . are represented by vectors then the resultant wave is simply the sum of these vectors (figure 4.1). 4.1.2
Huygens’ principle
Huygens postulated that each point on an advancing wavefront is a source of wave motion and that the resultant wavefront at some later time is a consequence of the superposition of the point sources. This principle allows topics such as reflection and refraction to be treated by simply considering the point sources on a surface and deriving the resultant wave in the new medium or direction. This approach is illustrated for reflection and refraction in figure 4.2, where the velocity of light in medium 1 is v1 and in medium 2 is v2 . A plane wave incident on the surface XOZ reaches O at time ¼ 0 when it becomes the source of the secondary wave. The side B of the plane wave would reach R at a later time given by ¼
distance OB R OR sin 1 ¼ ¼ v1 velocity v1
ð4:6Þ
The spherical wave leaving O reaches OP at a time ¼
OP OR sin 01 ¼ v1 v1
ð4:7Þ
or sin 01 ¼ OP=OR ¼ sin 1
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ð4:8Þ
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Diffraction
Figure 4.2. Reflection and refraction of light through a surface.
where 1 and 2 are defined in figure 4.2. In other words the angle of incidence equals the angle of reflection.
4.2
Photons
It is worth briefly describing some of the properties of light before going on to describe diffraction and interference. 4.2.1
Reflection and refraction
When light is incident upon a plane surface it may be absorbed, reflected or transmitted. Most surfaces of materials and media such as glass, water etc. reflect and transmit a proportion with the remainder being absorbed. The amount of light reflected is determined by the type of surface, the angle of incidence and the wavelength of the light. Consider again figure 4.2 in which light is incident on a plane surface at an angle 1 to the surface normal. As we have shown above light, will be reflected at an angle 01 where 1 ¼ 01 . Generally a fraction of the incident light will also be refracted through the surface at an angle 2 . Light travels at different velocities in different media. If we consider the case of light from points O and R in the figure the light incident on the surface will reach R at a time OR sin 1 =C1 later than O. The refracted light in the second medium will then travel a distance c2 in this time, where c1 and c2 are the velocities of light in the mediums 1 and 2 respectively. Since c2 ¼ OR sin 2 we can rearrange to give the ratio of the refractive indices in the two media sin 1 c1 ¼ : sin 2 c2
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ð4:9Þ
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Photons
Figure 4.3. The change in refractive index of fused quartz with wavelength at 291 K.
Since the velocity of light in a particular medium is also a function of the wavelength the refractive index will vary with wavelength. Figure 4.3 shows the change in refractive index as a function of wavelength in the visible region of the spectrum for fused quartz at a temperature of 291 K. When light passes from a medium of low refractive index to one of high refractive index it is bent towards the surface normal and conversely away from the surface normal on passing from a medium of high refractive index, n1 , to that of a low refractive index, n2 . In the latter case as the angle of incidence increases the refracted beam is bent towards the plane of the surface. Eventually a point is reached when n1 sin 1 ¼ n2 sinð=2Þ
ð4:10Þ
or sin 1 ¼
n2 : n1
ð4:11Þ
No light is then refracted and all the light is reflected (figure 4.4), which is known as total internal reflection and is the basis for many applications in
Figure 4.4. Total internal reflection given by 3–30 .
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Figure 4.5. Constructive and destructive interference of two waves. (a) and (b) are in phase and (c) and (d) are out of phase.
modern technology. Light pipes, fibre optics all rely on total internal reflection to transmit light and hence information over large distances. 4.2.2
Interference
Light can be considered either as continuous waves or discrete quanta of electromagnetic energy, photons. The quanta are in fact discrete bundles of waves and the wavelength determines the energy that our eye recognises as colour. Light of a single colour contains quanta with identical wavelengths, while white light is made up from quanta with a range of wavelengths from 200 to 800 nm. When quanta reach a point they interfere either constructively to form brighter regions or destructively to form darker regions. The result of this interference is determined by the difference in the phase of the different waves. Extreme cases are illustrated in figure 4.5, where two waves of the same wavelength (a) and (b) are exactly in phase, i.e. their phase difference is zero or a multiple of 2, and the resultant wave has the intensity of (a) plus (b) while waves (c) and (d) are out of phase by or multiples of and the resultant wave intensity is the intensity of (c) minus (d) which for two waves of equal amplitude results in complete destructive interference. This interference effect is illustrated by considering (figure 4.6) a beam of monochromatic light of wavelength incident on the slit positioned at S0 which causes waves to emanate from S0 , radially but in phase. If these waves then impinge on the second surface which has slits at S1 and S2 , spaced equally from the axis, then two new radial in phase waves will emanate from these slits. The image formed on a final screen is the result of interference of these waves with each other at points along the screen. Consider the point C, a distance y from the axis, and with the screen
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Figure 4.6. Interference from two slits, S1 and S2 .
placed a distance D from the slits S1 and S2 which are themselves separated by a distance d. The condition for a maximum at C is that the path difference between the rays r1 and r2 be an integral number of wavelengths, that is S2 C S1 C ¼ n . For this to be achieved, to a first approximation, n ¼ d sin
ð4:12Þ
where d D and the integer n ¼ 0; 1; 2; 3; . . . and where is the angle between the slits and the axis normal and the point C. The effect is illustrated by using the well known example of Newton’s rings observed when light passes through a lens resting on a flat reflective plate as shown in figure 4.7. The rings are in fact maxima in the interference between the reflected waves. Consider a ray of light incident on the centre of the plate and one incident a distance r from the centre. If R is the radius of the curved surface then the ray at a distance r from the centre will travel a distance 2d greater than the central ray in reaching its starting point. For
(a)
(b)
Figure 4.7. (a) Schematic representation and (b) an example of Newton’s rings.
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Figure 4.8. Use of interference to identify and measure the height of slip bands in copper– aluminium alloy (a) using white light and (b) monochromatic light.
the intensity to be a maximum 2d ¼ ðn þ 12Þ
ð4:13Þ
where n and are defined above. The 12 arises from the phase change of the light suffers when reflected. Thus the path difference is given by d ¼ R ðR2 r2 Þ1=2 ¼ R Rð1 ðr=RÞ2 Þ1=2 :
ð4:14Þ
If r=R 1 then d ¼ R Rð1 12 ðr=RÞ2 þ Þ ’ r2 =2R
ð4:15Þ
r ¼ ððn þ 12Þ RÞ1=2 :
ð4:16Þ
or An example of the use that can be made of this effect is in the interference fringes that are used in the light microscope, when used as an interferometer, to detect and measure small changes in height. This is particularly useful when determining the height of deformation bands or slip traces. An example is shown in figure 4.8, taken from deformation bands in single crystal copper–aluminium alloy (Mitchell et al (1968)). 4.2.3
Polarised light
The light that has been considered so far can be described by its wavelength, phase and amplitude and the wave by scalar quantities with planes equivalent. There is, however, another property of light which requires that the wave be specified by a vector and both direction as well as magnitude must be identified because the light behaves differently in different planes. Such light is known as polarised light and there are a number of forms of polarised light (figure 4.9). The vector describing the polarised light varies in amplitude
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Figure 4.9. Polarised light: (a) Plane polarised, two beams in phase and polarised normal to one another. (b) Circularly polarised, two beams of equal amplitude polarised normal to one another separated in phase by p=2.
and direction as the phase changes. When the direction remains fixed and only the magnitude changes the light is referred to as plane polarised (figure 4.9(a)). When the amplitude remains constant but the vector direction changes in a uniform way so that it describes a circle, then the light is said to be circularly polarised (figure 4.9(b)). When the magnitude and the vector direction change such that the vector describes an ellipse the light is said to be elliptically polarised. If two plane polarised waves have their vectors inclined at an angle to one another this is equivalent to a third plane polarised ray with a vector that is the vector sum of the two vectors describing the original waves. Circularly polarised light can be produced by combining two plane polarised waves of equal magnitude but out of phase by /4 or
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Figure 4.10. The various methods of producing polarised light using calcite crystals (a) Rochon prism, (b) Wollaston prism (dots indicate optic axis normal to lines), (c) Nicol prism.
3 /4. Here the plane of polarisation for the wave rotates with position along the ray. Elliptically polarised light can be produced by combining rays of equal magnitude but out of phase by /8 or 3 /8. A beam of unpolarised light may be regarded as the resultant of two beams, polarised in two different planes with no phase relationship. The simplest method of polarising ordinary light is to reflect it from the surface of a transparent medium such as glass or water. Light that has been reflected by a sheet of glass will be polarised in a direction normal to the surface of the glass. Here polarisation can be confirmed by reflecting this beam by a second sheet of glass: it will be strongly reflected in one direction but weakly reflected at an angle /2 relative to the strong direction. It is the normal convention that the plane in which the beam is most strongly reflected is known as the plane of polarisation. Light is polarised also when it is transmitted through a transparent medium unless it is incident normally on that surface. In this case the plane of polarisation of the transmitted light is normal to the plane of polarisation of the reflected light. Production of polarised light by reflection or refraction is, however, very inefficient. Certain transparent crystals such as calcium carbonate (calcite or Iceland Spar) are optically anisotropic and when a beam of light passes through such a crystal it is split into two parts which are refracted in different directions. The two refracted beams are each plane polarised with the plane of polarisation normal to each other, thus removing one of the beams produces a single plane polarised beam. In practice this is done by combining two prisms such that the second prism blocks one ray while allowing the other to pass the beam through (figure 4.10(a) and (b)). An alternative method is to use two prisms, cemented together, but arranged such that one of the polarised beams will be totally reflected from the second prism surface while the other is allowed to pass as in the Nicol prism (figure 4.10(c)).
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Figure 4.11. The use of polarised light to detect strain fields in a flat diamond plate.
Polarised light has many uses in the study of materials, particularly for cases when crystallographic materials are examined using light microscopes. Polarised light can be used to identify grains with different orientations or to detect deformation twins, because the reflection of the polarised light is dependent on grain orientation. An example of the use of polarised light to study the plastic deformation in an anisotropic material is shown in figure 5.10. Here polarised light has been used to study the strains that are put into cadmium as a result of deformation. Polarised light may also be used to show up strain patterns in transparent media. A beam of plane polarised light will have the direction of the plane of polarisation rotated by a strain field. By viewing the material through a second polariser placed normal to the plane of polarisation of the incident beam, areas where the plane of polarisation have been rotated will appear bright. Thus regions of strain appear as light areas on a dark background. Strain fields in a single crystal of diamond have been revealed by this process and a typical example is given in figure 4.11 (Wild et al (1967)). 4.2.4
Diffraction
Light is diffracted on passing through a narrow slit so that when monochromatic light passes through a single slit of width AB ¼ d, two diffraction cases can be envisaged (figure 4.12). In the first, light from a point source near to the screen produces a point at a screen and in the second a parallel beam from a source infinitely distant from the screen produces a parallel beam on the far side of the screen. The first case is known as Fresnel diffraction (figure 4.12(a)) and the second as Fraunhofer diffraction (figure 4.12(b)). Fraunhofer diffraction can be produced by using a lens combination to produce a parallel beam of light from a point source and to focus the beam on to a screen after passing through the slits. Diffraction can be described if we consider the slit to be split into two halves with a continuum of waves passing through from the top to the centre. Consider the case of Fresnel
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Diffraction
Figure 4.12. Conditions leading to (a) Fresnel and (b) Fraunhofer diffraction.
diffraction with a wave (1) starting from the edge of the slit at D and a wave (2) from the centre of the slit. If d, the width of the slit, is very much less than D, the distance from the slit to the screen, then the path difference between rays (1) and (2) is d=2 sin . The condition for a minimum is that the path difference be /2. If is chosen so that d=2 sin ¼ /2 then every ray from the top half of the slit will be cancelled by a ray from the bottom half. Thus the resultant intensity will be zero and we have the position of the first minimum. Extending the argument to the general case, the condition for the nth minimum is d sin ¼ n where n is an integer. If we now turn to the case where there are a series of equally spaced slits separated by a distance d and the parallel beam is focused on to a screen at a distance D from the focusing lens as illustrated in figure 4.13. The series of parallel slits constitutes a diffraction grating and the principal maxima is given by d sin ¼ n . The principal maxima do not change position with the number of slits, N, in the grating but rather the sharpness of the maxima increases with number. It can be shown that the position of zero intensity occurs, on either side of the central principal maximum, when ¼
d: N
ð4:17Þ
Thus as N increases so decreases and the peaks become narrow. For visible light is about 300 nm and thus to obtain a diffraction pattern where the first minimum is at an angle of 18 to the direction of the incident
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Figure 4.13. Diffraction using multiple slits separated by a distance d and total number N.
beam would require d to be of the order of 0.1 mm. It is certainly possible to produce mechanically diffraction gratings that can have a thousand slits and hence of a few micrometres. 4.2.5
Optical diffraction in transmission electron microscopy
Fourier realised that any waveform can be reconstructed from a series of sine waves. He and others have utilised the method to describe complicated waveforms in terms of the simple sine wave and the technique has become known as Fourier transform. A diffraction pattern, which is also a sinusoidal variation of intensity with distance, can be used to determine the form of the diffracting object. This is because a diffraction pattern is a Fourier transform of the diffracting object and this can be most easily visualised by considering the Fraunhofer diffraction pattern from a single slit. The ‘reciprocal space’ of the Fraunhofer diffraction pattern can be used to reproduce the spatial data. A simple example is given in figure 4.14, which is a Fraunhofer diffraction pattern from a rectangular aperture (Mulvey (1991)). The symmetry indicates
Figure 4.14. Fraunhofer diffraction pattern from a rectangular aperture. The difference between the vertical and horizontal components allows the size of the rectangle to be determined (Mulvey (1991)) (reproduced by permission of Rolston Gordon Communications).
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Diffraction
Figure 4.15. High resolution (0.4 nm) TEM focal series of a carbon support foil and associated optical diffraction pattern at various focal settings. Top left: over focus. Top right: Gauss focus. Bottom left: Scherzer focus. Bottom right: Under focus. (Courtesy of the Director of the Institute for Solid State Physics and Electron Microscopy, Halle (Saale), Germany.)
that the diffracting object has two-fold symmetry but the difference between the horizontal and vertical components reveals that the diffracting object is rectangular with the dimensions that can be determined from the intensity variation with distance. This method is used to quantitatively determine the size, spacing and symmetry of objects imaged in the transmission electron microscope (TEM) and, in particular, to give a quantitative measure of the ultimate spatial resolution. A transmission electron micrograph is first obtained and a laser optical system used to form a Fraunhofer diffraction pattern from the micrograph. If the image in the micrograph is formed from particles uniformly spaced, then a ring pattern will be obtained similar to that observed in Fresnel diffraction patterns. The fringe separation is then a direct measure of the best resolution that can be achieved. Figure 4.15 shows four micrographs, together with their associated diffraction pattern, from a carbon support foil at various focal settings. These indicate high spatial resolution of 0.4 nm (Mulvey (1991)). If the TEM has sufficient
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X-ray diffraction
89
resolution to resolve individual atom positions the optical diffraction method may be utilised to detect changes in the structure of the material. For example, in a study of electron beam damage in rutile (TiO2 ) (Buckett et al (1989)), optical diffraction patterns obtained at different points indicate that -TiO2 has been formed near to the surface as the result of electron bombardment but that -Ti2 O3 is present below a depth of 100 nm. Optical diffraction methods of determining size and spatial resolution in the TEM are described by Thon (1966, 1971) and Chapman (1986) and are to be preferred because they remove the subjective bias that is difficult to avoid with many other methods.
4.3 4.3.1
X-ray diffraction Simple theory
X-ray diffraction has been treated extensively in many texts, including those by Cullity (1979), Peiser et al (1960), Klug and Alexander (1954), Barrett and Massalski (1967), Kalbe (1968), Warren (1969) and Hammond (1997). Xrays are photons with a wavelength of the order of a fraction of a nanometre compared with the hundreds of nanometres of light waves. It is impractical to consider making a grating to diffract X-rays because the size and spacing would be too small. A separation of the order of 0.1 to 0.5 nm would be required. This condition, however, is inbuilt into crystalline materials where rows of atoms have a spacing of 0.3 nm. Thus an X-ray beam incident on a material penetrates many micrometres into the bulk and the direction of the diffracted beam intensity is determined by the periodicity of the atom planes in the crystalline solid; we will later meet cases where in electron diffraction it is possible for the periodicity of atoms in the topmost surface layer to determine the diffracted beam intensity. In chapter 2 we described the source of white X-rays and characteristic Xrays. Consider the case where a beam of characteristic X-rays of wavelength is incident on a single crystal surface at an angle (figure 4.16) which
Figure 4.16. X-ray diffraction from atoms in a crystalline material.
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produces a diffracted beam at an angle . The ray from the second row of atoms travels a distance (DA þ AC) greater than the ray from the top row. The two rows of atoms are separated by a distance d. For this to produce a diffraction maxima the path difference must be an integral number of wavelengths. But DA and AC both equal d sin and thus the condition for diffraction maxima is 2d sin ¼ n :
ð4:18Þ
This is the now famous Bragg equation for diffraction. By measuring , and knowing , d may be determined and the crystal spacing identified. Specific atom species will have an influence on the phase of the diffracted beam and dissimilar atoms in specific positions can cause certain diffraction peaks to be absent. A large library of expected diffraction patterns has been accumulated which allows most compounds to be identified from the observed diffraction positions and intensities. The application of this approach is now developed more fully. 4.3.2
The reciprocal lattice
Before proceeding further with X-ray diffraction it is appropriate to consider the reciprocal lattice (Peiser et al (1960)) that is constructed to aid the interpretation of diffraction from crystal lattices. In real space crystal planes are defined by their intercepts on coordinate axes, usually with axis units being defined as integral multiples of the unit cell dimensions (figure 1.2). For example the plane that intercepts the x, y and z axes at 3a0 , 2b0 and 4c0 , where a0 , b0 and c0 are unit cell dimensions in the x, y and z directions, would be referred to as the 3, 2, 4 plane. Planes with intercepts h, k, l have families of planes nh, nk, nl that are parallel to h, k, l and contribute to a diffracted beam. These planes are separated by a distance dh;k;l =n. The planes in real space can be represented by a point in reciprocal space. The reciprocal lattice is constructed for a defined crystal lattice by drawing a line from the origin, normal to the lattice plane h, k, l. This will be of length dh;k;l and is equal to the reciprocal of the interplanar spacing dh;k;l . The construction of part of a reciprocal lattice for a face centred cubic lattice is shown in figure 4.17. The reciprocal lattice points correspond both to planes with Miller indices h, k, l and those with indices nh, nk, nl where n ¼ 1; 2; 3; 4 . . . which also contribute to diffraction. Thus, the reciprocal lattice defines a range of potential lattice sites that may lead to diffraction. A particular lattice type may be characterised by ‘absent’ diffraction positions and the corresponding points in the reciprocal lattice will be missing, for example a face centred cubic Bravais lattice is equivalent to a body centred cubic reciprocal lattice and vice versa. The diffraction of an X-ray beam can be predicted from the reciprocal lattice using the Ewald construction illustrated in figure 4.18. The incident
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Figure 4.17. The reciprocal lattice construction.
X-ray beam is considered to pass through the origin in both real space and reciprocal space. A sphere is then drawn with a radius of 1/ , where is the wavelength of the X-ray photon, with its centre on the incident beam direction and position such that the surface of the sphere passes through the origin. This sphere is known as the Ewald sphere. Diffraction of the X-ray beam will occur if the Ewald sphere passes through a reciprocal lattice point. The direction of the diffracted beam is then given by k1 the vector from the centre of the Ewald sphere, to rh;k;l , the point where the sphere passes through a reciprocal lattice point. Thus, the diffracting planes have a reciprocal point at rh;k;l which satisfies the equation g ¼ k1 k0
ð4:19Þ
Figure 4.18. Use of the reciprocal lattice to predict diffraction: Ewald construction.
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where g is the reciprocal lattice vector corresponding to the diffracting planes, k0 is the incident wave vector and k1 is the diffracted wave vector. Diffraction can be guaranteed by either rotating the crystal and hence the reciprocal lattice points or replacing the single crystal with a powder giving an infinite variety of orientation of planes. Clearly no reciprocal lattice points outside a sphere of radius 2/ can pass through the Ewald sphere and therefore cannot diffract the X-ray beam so that this is the limiting sphere. 4.3.3
Intensity of diffracted X-ray beams
The problem of calculating the intensity of a particular diffraction peak is related simply to adding sine waves of different amplitude and phase but of the same wavelength. X-ray diffraction has proved to be invaluable in the evaluation of the microstructure of materials because it is possible to predict the diffracted beam positions with considerable accuracy together with their relative intensities. Therefore it is possible to compare with the measured values to establish the lattice parameters and, therefore, the types of crystal planes present. Thus the libraries of diffraction patterns utilise both position and intensity to determine the crystal structure under investigation (Cullity (1979), Peiser et al (1960), Kalbe (1968) and Warren (1969)). It should be pointed out that in this section we will concern ourselves with the intensity of diffracted beams resulting from atoms at various positions within the crystal lattice. The direction of a diffracted beam is not affected by the type of atom at a particular site, and two unit cells of the same size but with differing arrangements of atoms will diffract X-rays in precisely the same directions. However, the intensities of those diffracted beams may vary and, indeed, the intensities of certain beams may be zero. The approach adopted to determine the intensity of a diffracted beam involves the following three steps: (a) scattering of X-rays from a single electron, (b) scattering of X-rays from a single atom using the scattering from a single electron, and (c) diffraction of X-rays from a unit cell using the scattering from a single atom. Scattering by an electron Since an X-ray photon is an electromagnetic wave it will be strongly scattered by an electron, either coherently or incoherently. For coherent scattering the wavelength of the incident X-ray beam is unaltered by the interaction and there is a phase difference between the incident and scattered beams, /2. This difference is the same for all electron interactions and led J J Thomson to determine that an electron would scatter an X-ray beam of intensity I0 so that the intensity emitted at a distance r from an electron of charge e and mass M is given by I ¼ I0
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e4 sin2 2 r M 2 c4 2
ð4:20Þ
X-ray diffraction
93
Figure 4.19. Scattering of X-rays by an atom, electrons at positions A and B are considered.
where c is the velocity of light and 2 is the scattering direction. The incoherent, Compton, scattering arises because on impact with an electron the X-ray beam loses energy, h , so that the wavelength of the scattered beam is greater than that of the incident beam (see equation (2.1)). Since there is no fixed phase relationship between the incident and scattered X-rays the contribution is to the ‘background’. Scattering by an atom An atom consists of a relatively massive positively charged nucleus surrounded by a number of negatively charged electrons. The incident Xray beam will be scattered by both the nucleus and the electrons. However, an examination of equation (4.20) shows that the large mass of the nucleus, which is several thousand times that of the electron, results in negligible scattering of the incident beam by the nucleus. The total scattering from the atom is therefore essentially the result of the scattering from all the individual electrons (figure 4.19). As shown in figure 4.19 since the atom has a finite size waves scattered by electrons located at positions A and B have different path lengths. For those X-rays, however, scattered in the forward direction on reaching plane XX 0 they will be in phase because each wave has travelled the same total distance before and after scattering. As the angle of diffraction increases so the path difference, CB AD, for the X-rays scattered from electrons A and B increases with a consequent fall in the amplitude of the resultant wave at YY 0 . The total scattering by an atom is known as the electron density distribution. A shorter incident X-ray wavelength will produce a greater decrease in amplitude. However, it is the quantity f , known as the atomic scattering factor, that describes the efficiency of scattering in a
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Figure 4.20. The variation of the atomic scattering factors, f , for copper as a function of sin = .
particular direction where f ¼
amplitude of the wave scattered by an atom : amplitude of the wave scattered by one electron
Since f is equal to the number of electrons in the atom for ¼ 0, then f equals the atomic number, Z, but the value decreases as increases and decreases. A plot of f as a function of sin = for copper in figure 4.20 is an example of how f decreases from atomic number 29 as the value of sin = increases. Diffraction from a unit cell We now move on to consider the effect of the position of the atoms in the unit cell on the amplitude of the scattered wave. Since the unit cell is the smallest repeat constituting the crystal this is the last step in determining the intensity of the diffracted beam. The situation is similar to scattering by electrons at different positions in the atom except that here we have phase differences as a result of scattering by atoms at different positions in the unit cell. If we consider first two atoms A and B with atom A positioned at the origin and atom B at a point u; v; w (figure 4.21), then the phase difference between the wave scattered by atom B and that scattered by atom A for an h; k; l reflection is ¼ 2ðhu þ kv þ lwÞ:
ð4:21Þ
If the atoms are of the same kind, the amplitudes of the wave scattered by each atom will be the same; but in the case of unit cells containing more
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Figure 4.21. Diffraction of X-rays by atoms at positions A and B.
than one type of atom, waves of different amplitude as well as phase must be combined to arrive at the resultant intensity. The amplitude of the scattered wave may be described as a complex exponential function A ei . The intensity of this wave is simply the square of the amplitude, i.e. jA ei j2 , and the amplitude is the value of f for the scattering of the atom under consideration. Thus, using the expression for the phase we have A ei ¼ f e2iðhu þ kv þ lwÞ :
ð4:22Þ
The resultant wave from the unit cell is simply the sum of the waves from all the atoms and is referred to as the Structure Factor, F. It is the sum of the waves of amplitude A1 , A2 , A3 , . . . and phase angle 1 , 2 , 3 , . . . in the unit cell. If the amplitude of each wave is represented by the length of a vector and the phase by the direction of a vector, then the diffracted beam is the vector sum of the waves (figure 4.22). Thus the wave from the jth atom is resolved into horizontal and vertical components of lengths fj cos j and fj sin j respectively. These produce a triangle of sides with a
Figure 4.22. Vector addition of X-rays diffracted from atoms.
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hypotenuse of F where jFj2 equals the intensity. Thus X 2 X 2 jFj ¼ fj cos 2ðhuj þ kvj þ lwj Þ þ fj sin 2ðhuj þ kvj þ lwj Þ j
j
ð4:23Þ or jFhkl j ¼
n X
fn e2iðhun þ kvn þ lwn Þ
ð4:24Þ
1
since the power series is given by eix ¼ cos X þ i sin X. Equation (4.23) can be simplified by considering when a crystal has a centre of symmetry which is possible if atoms in the unit cell are divided into sets that are related by the symmetry of the space group. For centrosymmetric crystals the second term in equation (4.23) is zero since sin 2 ¼ 0 and for planes where the intensity is zero no diffraction occurs. For commonly occurring crystal lattice planes in simple symmetry crystals, the conditions for the presence or absence of a diffracted X-ray beam may be established (Cullity (1979)) such that for a simple cubic crystal all planes diffract. For a body centred cubic crystal diffraction occurs if h þ k þ l is an even integer but not if odd and for a face centred cubic crystal h, k and l are unmixed for diffraction and mixed for no diffraction. Other extinction rules apply to crystal lattices that do not fall into this category. An example is the application to the caesium chloride crystal which has the B2 ordered structure based upon the body centred cubic structure, with a caesium ion at 000 and a chlorine ion at 12, 12, 12. Here the f factors for the two ions are not the same, so that the lattice can no longer be regarded as simply body centred cubic and the structure factor is given by F 2 ¼ ½ fCs þ fCl cos ðh þ k þ lÞ2 þ ½ fCl sin ðh þ k þ lÞ2 2
ð4:25Þ 2
which reduces to ( fCs þ fCl ) when (h þ k þ l) is even and ( fCs fCl ) when (h þ k þ l) is odd. Since fCs is not equal to fCl the latter diffraction intensity is not zero as it is for the simple body centred cubic structure. The structure factor can be determined for a unit cell provided the positions and atomic scattering factors for each atom are known. For example, a face centred cubic material has atoms at 000, 12, 12, 0; 12, 0, 12; and 0, 12, 12 positions in the unit cell. If all the atoms are of the same type then the structure factor will be given by: F ¼ f e2ið0Þ þ f e2iðh=2 þ k=2Þ þ f e2iðh=2 þ l=2Þ þ f e2iðk=2 þ l=2Þ iðh þ kÞ
F ¼ f ½1 þ e
þe
iðh þ lÞ
þe
iðk þ lÞ
ð4:26Þ ð4:27Þ
If all the indices h; k; l are odd or even then the exponentials will each equal one and F ¼ 4f , here enI ¼ ð1Þn . However, if h; k; l are mixed odd and even then the sum of the exponentials will be equal to 1 and F ¼ 0. Thus
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X-ray diffraction Table 4.1. Diffraction peaks present and absent in some common Bravais lattices. Bravais lattice
Diffraction present
Diffraction absent
Simple Base-centred Body-centred Face-centred
All h and k not mixed ðh þ k þ lÞ even h, k and l not mixed
None h and k mixed ðh þ k þ lÞ odd h, k and l mixed
diffraction peaks of the type (111), (200) and (220) will be present but of the type (100), (110) and (112) will be absent. Table 4.1 summarises the diffraction peaks that will be observed for some of the common Bravais lattices. The structure factor establishes the conditions simply for the presence or absence of a diffracted peak, but it does not provide information on the relative intensity of these peaks. There is a need to determine the proportionality constant in the condition I / jFj2 :
ð4:28Þ
This will depend upon the particular application or method used and these various options are described in subsequent sections 4.3.5 and 4.3.6 in particular. For single crystals, white X-rays are used with a range of wavelengths and this calculation is complex and rarely invoked. However, for those methods that adopt monochromatic X-ray sources such as the powder method, section 4.3.6, five factors, in addition to the structure factor, have to be taken into account to derive the relative intensities: (i) Lorenz, (ii) polarisation, (iii) multiplicity, (iv) absorption and (v) temperature. It is usual to consider the contribution for the Lorenz and polarisation factors together. An X-ray source produces an unpolarised beam but the vibrating atoms in the crystal do not scatter these waves in all directions with equal efficiency. This results in a partially polarised diffracted beam with a decrease in intensity given by I ¼ 12 ð1 þ cos2 2Þ:
ð4:29Þ
Thus the greatest reduction in diffracted intensity is a 2 ¼ /2 and the least reduction at 2 ¼ 0 or . In addition there are further geometrical contributions, one related to the relative time a given set of crystal planes will be sampled and the second to the proportion of the diffracted cone of the X-rays sampled. These can be combined to give a geometrical factor equal to (4 sin2 cos )1 . Hence the Lorenz polarisation factor is given by Lp ¼
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1 þ cos2 2 : sin2 cos
ð4:30Þ
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Figure 4.23. The effect of increasing temperature on f .
The overall effect of this factor is to reduce the intensity of the diffracted X-ray beam at the intermediate 2 values compared with the low and high values. By comparison the multiplicity factor, p, accommodates the number of crystal planes which contribute to a given diffraction peak. These can be generalised for groups of planes in a given crystal system (see Table 4.1). As discussed in section 2.2.1 the intensity of an X-ray beam after passing through material of a given thickness depends upon the linear absorption coefficient (see equation (2.2)). As a consequence there is a need to accommodate the absorption of the particular specimen, A(), for the particular the method of diffraction selected; for a flat specimen of the type used in a diffractometer system the relative intensity can be taken as unity since it is independent of 2. Values of absorption factors for different specimen geometries are given in International Tables. The final contribution temperature, T, is addressed, in part, via the atomic scattering factor f whereby thermal vibrations lead to an increase in the size of the atoms. Hence in figure 4.20, f will decrease more rapidly for the idealised stationary electron model of the atom. It can be shown theoretically and experimentally that f is given by f ¼ f0 eM
ð4:31Þ
where f0 is the value for an atom at rest and M is given by M ¼ Bðsin = Þ2 where B ¼ 82 2 . Here is the mean square amplitude of atomic vibration. Figure 4.23 shows the effect of increasing temperature on f . As a consequence the relative intensity of a diffracted peak for polycrystalline material subject to monochromatic X-rays is given by IRel ¼ jFj2 Lp pAðÞT:
ð4:32Þ
To give absolute intensities of X-ray diffracted peaks it is necessary to include the specific instrumental conditions adopted so that I ¼ I0 ½X½Y
ð4:33Þ
where X is the instrumental term and Y is described by equation (4.32).
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X-ray diffraction 4.3.4
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X-ray production
X-rays can most easily be produced by bombarding a material surface with relatively high energy electrons. When a high energy electron impinges on the material X-rays are produced in two ways. All charged particles emit electromagnetic radiation when they are accelerated. As the incident electron is decelerated it generates a continuous spectrum or bremsstrahlung. The intensity of the X-rays at a specific energy is a function of the electron energies and is of the form IðEÞ ¼ CZðE0 EÞ=E
ð4:34Þ
where E0 is the incident electron energy, E is the energy of the electron following deceleration and Z is the atomic number of the target and C is a constant. The incident electron also causes the surface atoms to be ionised by the removal of an inner shell electron. As the atom rearranges with electrons from outer orbitals falling into the hole created (e.g. an L shell electron falling into a hole in a K shell) energy is released in the form of a photon, EX-ray ¼ EK EL :
ð4:35Þ
The probability that these characteristic X-rays will be emitted is called the Xray fluorescence yield and increases with atomic number and is larger for K line emissions than for L line emissions. In practice it is normal to use metals to produce X-rays for use in X-ray diffraction instruments. This is because an intense beam of X-rays is desired and the good thermal conductivity of the metal allows the heat produced during bombardment with an intense high energy electron beam to be readily removed thus avoiding damage to the source. When a metal is bombarded with electrons the X-ray spectrum produced is similar to that shown in figure 4.24. There is a continuous band of X-rays emitted from an edge which corresponds to the energy of the incident electron to much longer wavelengths. Superimposed on this background are characteristic peaks resulting from specific transitions within the metal atoms. Normally the characteristic peaks, which may be enhanced by monochromators are used to produce diffraction patterns but in some instances the ‘white’ background radiation may be employed. Since electromagnetic radiation is emitted when a charged particle is accelerated, X-rays can be produced by deliberately accelerating electrons. Very high energy electrons are made to follow circular orbits by confining the electron beam using magnets. Such a device is known as a synchrotron. Here the energy or wavelength of the emitted X-ray is determined by the electron energy and the radius of curvature. The synchrotron produces a narrow cone of X-rays tangential to the electron path. While such an Xray source is highly specialised and very expensive it does have many advantages over conventional sources.
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Figure 4.24. The characteristic X-ray spectrum from a metal illustrating K and L peaks superimposed on the continuous bremsstrahlung background.
4.3.5
Single crystal diffraction
In figure 4.25 we show a schematic diagram of the experimental arrangement for Laue back diffraction together with an example of a typical back diffraction pattern. A beam of white X-radiation is incident on a single crystal specimen. The incident X-ray beam passes through a hole in a photographic plate and the diffracted beams are detected as spots on the film. This method of recording diffraction patterns is less common today and present-day techniques will be described later. The white X-radiation provides the range of wavelengths necessary to ensure that the Bragg Law is satisfied for all planes. With the experimental arrangement shown in figure 4.25 the film can be positioned to record the back diffraction patterns as hyperbola or a transmission pattern can be recorded as ellipses as shown in figure 4.26. Therefore a Laue pattern provides details of the crystal
Figure 4.25. Location of Laue spots on hyperbolas in the back diffraction of X-rays together with an example of a Laue back diffraction pattern.
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Figure 4.26. Location of Laue spots on ellipses in the transmission method of X-ray diffraction together with a typical pattern for a {110} orientation niobium single crystal.
symmetry and the orientation of the single crystals. Moreover, if the grain size of a polycrystal exceeds the incident collimated beam diameter <0.5 mm, then for a conventional experimental system individual grains may be oriented (Cullity (1979)). However, the diffracted spots contain additional information concerning grain size and crystal perfection. This technique can be particularly useful for the examination of microstructure since from the orientation of a given grain it is possible to establish using (a) single and (b) two surface trace analysis procedures the habit plane for a range of precipitates and second phase transformation products e.g. martensite plates, together with their orientation relationship with the parent matrix (Crocker and Flewitt (1984)). A serial sectioning, single surface technique determines the angle that a feature, such as an interphase boundary, makes with a surface by measuring the shift in position as the surface is polished to remove known depths of material (figure 4.27(a)). Depth measurements may be achieved by determining the change in dimension of hardness or similar reference indentations. For this it is necessary that the hardness impressions have planar sides and the indentation images can be superimposed to achieve register. The precision of depth measurement is approximately 5% which gives a possible error in establishing the feature angular relationship of <58 (Bevis and Swindells (1967)). In the case of two surface trace analysis (figure 4.27(b)), if the features have a relatively high aspect ratio the sample may be ground and polished to produce an apex (included) angle of approximately 1608 rather than the 908 which is more usually used (Cullity (1979)). The angular relationship between the two measurement surfaces can be determined by using a tilting stage on an optical microscope. If one surface is oriented by X-ray diffraction, both surfaces can be used to establish trace angles and, indeed, these procedures may be applied to a range of samples and microstructural features. The results of such two surface trace analyses to establish the habit plane for 1 plates formed in binary Cu–40%Zn and ternary Cu–9%Au–40%Zn alloys are shown in figure 4.27(c) (Doig and Flewitt (1983)). Although limited spatial resolution restricts the application of X-ray diffraction,
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(a)
(b)
(c)
Figure 4.27. Trace-trace analysis. (a) Single surface; microstructure and the stereographic projection on matrix plane (hkl). (b) Two surface; traces where the included angle between the surfaces is ( ) together with the corresponding stereographic projection. (c) Measured and predicted 1 plate habit planes for binary Cu–40%Zn and ternary CuAu9 Zn40 aged metastable 0 phase (Doig and Flewitt (1983)) (reproduced with permission of the Institute of Materials).
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(a)
103
(b)
Figure 4.28. Slip in copper (fcc) single crystals. (a) Slip traces in the surface of a deformed copper single crystal. (b) Lattice rotation during deformation of copper single crystal with an initial orientation given by the pole marked 1.
these procedures for microstructure evaluation can be applied equally to other higher resolution methods such as Kossel X-ray diffraction and electron diffraction. However, using the X-ray Laue back diffraction technique it is possible to analyse, for example, slip traces developed on prepolished surfaces of deformed single crystals (figure 4.28(a)), and establish the slip processes which effect the deformation. Moreover, changes in orientation which occur can be followed stereographically as shown in figure 4.28(b) (Cullity (1979) and Honeycombe (1968)). Here the initial tensile axis for the fcc crystal is given by the pole 1, but with the progressive deformation the related extensions cause the position of the axis to move through 2 to 4 along the [ 101] slip direction. Since the slip plane is (111) the lattice orientation causes both the slip plane and direction, in that plane, to rotate towards the stress axis. The size of grains and subgrains in polycrystalline metals and alloys may be established from Laue back diffraction patterns. In conventional Laue back diffraction patterns the rings become discontinuous when the size of either the grains or subgrains exceed a certain dimension, the value of which depends upon the diameter of the incident X-ray beam (Hirsch (1955) and Lonsdale and Flewitt (1984)). This technique, therefore, provides a means of establishing the dimensions of microstructural features within a single phase region of a material simply from a measure of the number of diffraction spots contained within a specific diffraction ring. The method has the inherent advantage that statistical accuracy is high since many grains or subgrains are sampled within the analysed volume. Stephens and Barnes (1937) established an empirical relationship between the number of diffraction spots in a given ring observed on a photographic film and the known grain size for a series of aluminium samples. The method depends
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on Bragg diffraction of X-rays from a given set of {hkl} planes within individual subgrains contributing to the final ‘spotty’ diffraction pattern. As a consequence the greater the number of subgrains sampled by the incident X-ray beam the larger the number of diffraction spots produced. There is an upper and lower limit to the size of subgrains measured since if the number is too large (small grains) the ring becomes continuous and if too small (large grains) the number of diffraction spots is insufficient for statistical confidence to be placed on the measured value (Lonsdale and Flewitt (1984)). Assuming cuboidal shaped subgrains, the subgrain size, d , is given by (Andrews and Johnson (1959)): dhkl ¼ ½V pB cos =2DN1=3
ð4:36Þ
where B is the effective diameter of the incident X-ray beam, is the diffraction angle, p is the multiplicity factor, D is the film to specimen distance, N is the number of diffraction spots given for {hkl} diffraction planes and V is the volume irradiated such that V ¼ =4B2 di , where di is the depth of penetration of the X-ray beam (a function of the absorption coefficient, , for the material). Unfortunately this simple procedure fails to account for contributions from either sectioned subgrains at the surface or the larger grains they are contained within. Several procedures have been proposed to overcome this restriction. One is the double exposure method (Andrews and Johnson (1959)) which relies on the exponential relationship between diffracted X-ray intensity and depth within the sampled volume. A film exposed for a short period, 1 , contains information derived from the near surface volume defined by the depth d1 in figure 4.29, and a second exposure for a substantially longer period, 2 , produces diffraction spots representative of the volume given by the depth d2 . Clearly the difference between the two defines the volume irradiated (d2 d1 ) and eliminates surface and grain size contributions and equation (4.36) becomes: ¼ ½B3 p cos lnð2 =1 Þ=8DN ð1 sec 21=3 : dhkl
ð4:37Þ
This predicts the variation of number of diffraction spots contained within a ring fh k lg for a subgrain size specimen. A measure of the subgrain size is important, for example in creep deformation studies, since it is inversely proportional to the applied stress (Lonsdale and Flewitt (1984)). The technique has been rigorously examined by Hirsch and Keller (1951, 1952) who devised procedures for extending the lower size limit evaluated to below the 0.5 mm obtainable with conventional back diffraction cameras by using microbeam techniques where the incident X-ray beam is collimated by a capillary tube. Laue diffraction spots from a perfect crystal are sharp and their size is determined by the diameter and divergence of the X-ray beam and the film to specimen distance. From bent crystal planes or misorientations associated with a dislocation substructure the spots become elongated in a radial
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Figure 4.29. Subgrain size determination showing interaction of an incident X-ray beam of diameter, b, with specimen; d1 and d2 represent depth of sampling for two exposure times 1 and 2 , where 2 1 (Lonsdale and Flewitt (1984)) (reproduced by permission of Pergamon Press).
direction (asterism) (Cullity (1979) and Masing et al (1959)). The direction and amount of elongation is a function of the orientation of the bending axis and/or the range of orientation of the lattice planes sampled by the incident beam of X-rays. Typically Laue diffraction can be used to follow the kinetics of recrystallisation by observing the progressive formation of sharp diffraction spots derived from newly formed, strain free grains. 4.3.6
Powder methods
The method devised by Debye and Scherrer is the most widely applied powder X-ray technique (Cullity (1979)). Here an incident beam of monochromatic X-radiation interacts with a specimen which is in the form of either a small non-diffracting filament containing bonded powder or a polycrystalline fibre of very small grain size. These specimens must contain sufficient particles with the correct orientation to allow diffraction from all possible diffracting planes when rotated in the X-ray beam (figure 4.30(a)). The angle between the incident and diffracted X-ray beam is 2 and consequently each set of crystal planes produce X-rays of semi-angle 2. A film placed around the specimen intersects the different conics and produces curves concentric with the entrance and exit apertures. Figure 4.30(a) shows the arrangement of the powder sample, film and X-ray beam while figure 4.30(b) shows the equipment with the light-tight cover removed. The position of intensity peaks in the diffracted beam is characteristic of the material being examined and unknown phases may be identified by comparison with
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Figure 4.30. Powder diffraction. (a) Schematic diagram of the Debye–Scherrer method. (b) A Debye–Scherrer camera.
standards (JCPDS Powder Diffraction File). From the Bragg Law the peak positions provide both the crystal structure and the lattice parameter for each phase contained in the powder sample. The diffracted beam intensity provides a measure of the distribution and position of atoms within the crystal. Precise lattice parameter determination is important in many areas of microstructural evaluation such as the study of equilibrium phase diagram boundaries, thermal expansion coefficients, density determinations, variation of properties with composition, precipitation processes and solid state phase transformations (Cullity (1979), Warren (1969), Kalbe (1968), Andrews and Johnson (1959), Barrett and Massalki (1966), Donachie and Krieg (1972), Goldschmidt (1967) and Pearson (1967, 1972)). To overcome limitations associated with photographic recording, the diffracted X-ray beam may be detected using a counter tube, usually a proportional or scintillation counter, linked with associated electronic circuitry (Cullity (1979)). This is incorporated into an X-ray diffractometer
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Figure 4.31. Powder diffraction. (a) Geometry of a conventional diffractometer. (b) A modern diffractometer (courtesy Siemens Ltd).
(figure 4.31(b)), where a divergent beam of either filtered or monochromatic X-radiation is incident on a powder specimen which is rotated at half the angular speed of the receiving slit to maintain constant angular relationship between the incident and diffracted beams. A receiving slit mounted in front of the counter tube is attached to a carrier tube arm behind which is located a scatter slit to ensure that only radiation is received from the area of the specimen exposed to the incident X-ray beam. A typical modern X-ray diffractometer is shown in figure 4.31(b) where the specimen is placed in the centre. This is contained in an enclosure which is safety interlocked to prevent X-rays being generated while the specimen is being handled. The X-ray source is visible to the left of the specimen while the detector is to the right hand side. The intensity of the diffracted beam is automatically recorded either on a chart or as digitised data which can be processed by a small computer to enable direct output of 2 and d values. Indeed with the current generation instruments, further direct interaction with the computer is achieved so that standards of known substances stored on the JPDS powder diffraction file can be directly accessed and the specimen identified without the need for operator interaction. Figure 4.32 shows diffraction traces obtained for specimens containing mixtures of carbides and Laves phases extracted from Type 316 austenitic stainless steel. This is a typical application of this method for the quantitative determination of the weight fraction of phases (Lai and Galbraith (1980)). To enhance the speed of data acquisition using a diffractometer, the counter can be replaced either with a linear position-sensitive proportional counter (Go¨bel (1978)) or an energy dispersive spectrometer (Bivas et al (1978)). The former counter accepts X-ray counts over a fixed angular range, typically 108 but this can be as high as 1608 for a large exposure counter, at a high rate so the detector can then be precessed rapidly over the required 2 range.
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Figure 4.32. X-ray diffractometer traces obtained from precipitates extracted from three Type 316 stainless steels variously aged, showing the proportion of -phase (D8b complex tetragonal), -phase (hcp) and M23 C6 type carbide (fcc) established using an internal standard ZnO calibration (Lai and Galbraith (1980)) (reproduced by permission of the Institute of Materials).
There continue to be developments associated mainly with the optics of X-ray diffractometer systems to improve the collimation and focusing of the X-rays. The slit system in the goniometer shown in figure 4.31 has a limitation since it rejects most of the output of the X-rays from the source and hence reduces the flux of X-radiation to the specimen. Several approaches have been adopted to overcome this. One uses hollow glass capillaries of 3 to 5 mm inner diameter so that the X-rays that undergo total internal reflection at the capillary surface can be directed more closely. Another makes use of X-ray mirrors that consist of a stack of nanometre-thick layers of nickel and carbon or tungsten and silicon, Go¨bel mirrors, to form a parabolic mirror for collimating and an elliptical mirror for focusing these beams (Mai (1999) and Dietsch et al (1998)). The current generation of X-ray diffractometers are designed to be used in different operating modes. For example the preferred orientation (texture) which arises from deformation processes, recrystallisation and the orientation of surface films such as oxide layers or vaporised electrolytic deposition
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Table 4.2. Proportion of -phase, -phase and M23 C6 type carbide in X-ray diffractometer traces in figure 4.32.
Specimen
Total amount of precipitate (wt%)
(wt%)
M23 C6 (wt%)
(wt%)
1 2 3
4.55 3.3 1.1
0.91 0.25 0.01
0.86 0.53 1.1
2.78 2.52 0.01
products can be investigated. Transmission and back diffraction methods may be used by means of a method originally developed by Schultz (Schultz (1949)). The principle of this is shown in figure 4.33 where the angles of rotation and are defined. The vertical height of the beam is restricted because the -rotation destroys the focusing condition except along the AA0 direction. One advantage of the technique is that no absorption correction is required for the various - and - measurements. The development of a deformation texture is shown in figure 4.34 for a low interstitial content, ferritic stainless steel which has been cold rolled to given thickness reductions, 35, 75 and 95% (Lewis and Pickering (1982)). After 35% reduction the texture is mainly {111}h112i with {100}h011i with no evidence of rotational freedom about the strip normal and rolling direction. Increasing the cold rolling reduction to 75% causes the {111}h11 2i texture to be replaced by {1 11}h1 10i and the {112}h110i components together with an increase in the {111}h01 1i component. At the highest reduction the major component becomes the {100}h011i with a minor component from {112}h1 10i.
Figure 4.33. Schematic arrangement of the diffractometer technique for determining the rolling texture of a sheet specimen together with the method of plotting the intensity results on a pole figure.
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Figure 4.34. Development of deformation textures in low interstitial 17% Cr stainless steel, (a) 35%, (b) 75% and (c) 95% reduction (Lewis and Pickering (1982)) (reproduced by permission of the Institute of Materials).
4.3.7
Layer thickness measurements using X-ray diffraction
X-ray diffracion is used to determine the relative thickness of layers present on a bulk substrate. This can be particularly useful when the thickness is changing with time and it is necessary to monitor the relative thickness non-destructively. If we have two layers of thickness tS and tR on a stainless steel substrate A (figure 4.35), the relative intensities of peaks from the two layers is given by (Tempest and Wild (1982)): 2 S tS ½1 expð2 S tS = sin S Þ ð4:38Þ IS =IR ¼ AS =AR exp sin R ½1 expð2 R tR = sin R Þ where AS ¼
jFS j2 sin S 1 þ cos2 2S p S 2 S sin2 S cos S VS2
ð4:39Þ
with a similar expression for the rhombohedral equivalent term, AR , and where FS and FR are the structure factors, VS and VR are the volumes of the unit cell, pS and pR the multiplicity factors, S and R the linear absorption coefficients and S and R are the Bragg angles for the spinel and rhombohedral phases respectively.
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Figure 4.35. Schematic of method for using X-ray diffraction to determine thicknesses wS and wR of layers S and R. Peaks are chosen such that the Bragg angle for the diffraction from two layers are similar (Tempest and Wild (1982)) (reproduced by permission of Plenum Press).
In practice it is often possible to select two peaks such that S R in which case equation (4.38) simplifies to expð2 S tS = sin Þ 1 : ð4:40Þ IS =IR ¼ AS =AR 1 expð2 R tR = sin Þ This method has been used to study the growth of oxide layers on austenitic stainless steel. At high temperature and pressure, stainless steel oxidises in CO2 gas by first forming a spinel oxide layer but gradually a more protective rhombohedral oxide, rich in chromium, forms between the spinel layer and the parent steel. Figure 4.36 shows three diffraction peaks from an oxidised surface, two from the rhombohedral oxide phase and one from the spinel oxide phase. The intensity of these peaks varies with time exposed to the gas at 1123 K such that after 100 h exposure the spinel peak is the largest but by 200 h the peaks are of approximately equal intensity and after 500 h the rhombohedral peaks are dominant. Thus the relative thicknesses of the two layers can be determined (figure 4.37). The combined oxide thickness is made up from the spinel layer which grows at a parabolic rate and the underlying rhombohedral oxide which grows linearly with time over the period of the experiment. This method can be applied to any layer system where the two layers give separate diffraction peaks provided the total thickness is within the penetration depth of the X-rays.
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Figure 4.36. X-ray diffraction peaks from spinel and rhombohedral oxide layers on austenitic stainless steel as a function of time exposed to CO2 –2%CO at 1123 K (Tempest and Wild (1982)) (reproduced by permission of Plenum Press).
4.3.8
Macrostress and microstress measurement by diffraction
The changes in interplanar spacing in a crystal when subject to an elastic strain can be used as an internal strain gauge through a knowledge of the Bragg equation, equation (4.18). Here the change in the interplanar spacing, d, is given by differentiating the Bragg equation: " ¼ d=d ¼ cot :
ð4:41Þ
This effect on the position of the recorded diffraction peak is shown schematically in figure 4.38. It is necessary to have a measure of the stressfree interplanar spacing and then the strain can be converted to stress using elasticity theory (Hauk (1997) and Winholz and Kranitz (1996)). The fundamental relationship for evaluating a general strain state using diffraction is " ¼ "11 cos2 sin2 þ "12 sin2 sin2
þ "22 sin2 sin2 þ "13 cos sin2
þ "33 cos2 þ "23 sin sin 2
ð4:42Þ
where " is the strain in the direction defined in figure 4.39. The strain components on the right of equation (4.42) are related to the coordinate
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Figure 4.37. Thicknesses of the oxide layers formed on austenitic stainless steel as a function of time exposed to CO2 –2%CO at 1123 K determined by X-ray diffraction (Tempest and Wild (1982)) (reproduced by permission of Plenum Press).
system shown in figure 4.39. Hence the stress is given by ij ¼ Ehkl =ð1 þ hkl Þð"ij þ ½ hkl =ð1 2 hkl Þ"kk ij Þ
ð4:43Þ
where E is the elastic modulus for the particular crystal planes being considered, is the Poisson ratio and ij is the Kro¨necker delta function (Noyan and Cohen (1987)). In many cases the principal stress directions can be deduced by symmetry considerations so that three strain values are
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Figure 4.38. The effect changes in interplanar spacing, d, on the position of the recorded diffraction peak.
sufficient to establish three principal stresses: ij ¼ fEhkl ½ð1 hkl Þð1 2 hkl Þgfð1 hkl Þ"11 þ hkl ð"22 "33 Þg
etc: ð4:44Þ
In addition it is possible to determine the near surface biaxial stress by considering the simplification that 11 ¼ 22 and 33 ¼ 0. X-ray diffraction The wavelength of X-rays is between 0.1 and 0.2 nm and as such the penetration into materials is limited to about 10 mm so that the material sampled is confined to the surface layers where the stress state can be considered to be biaxial. Hence the interrelationship between the measured change in the diffraction peak position and the stress is given by 2 ¼ ½2 ð1 þ hkl Þ tan =Ehkl ½sin2
ð4:45Þ
where in radians and is the angle of the specimen with respect to the incident X-ray beam. Therefore a plot of 2 versus sin2 results in a straight
Figure 4.39. The strain ellipsoid where the principal strains are "1 , "2 and "3 .
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Figure 4.40. The strain components on the right of equation 4.42 related to the coordinate system .
line with a slope equal to the first term in equation (4.45) (figure 4.40). Here the term (1 þ hkl )/Ehkl is specific to the particular material and diffraction planes examined and is determined by prior calibration. The basic relationships given by equation (4.45) reveals requirements for the X-ray diffraction conditions (Doig et al (1985)). The sensitivity for stress measurement increases both as and tend to /2. A practical limit to is /3 and this is controlled by the spread of the X-ray beam and focusing for most diffraction systems. Moreover since the value of tan increases rapidly as tends to /2 it is preferable to select an appropriate diffraction peak, free from overlap, at a large diffraction angle. In general to achieve an accuracy of 10 MPa in evaluated stress the position of the diffraction peak has to be measured to better than 0.018. For stress measurement both laboratory and dedicated transportable computer controlled X-ray systems can be used. Figure 4.41 shows an example of a transportable ganimeter fitted with a proportional counter. Also this ganimeter can be fitted with a linear position sensitive detector in the side inclination orientation. The results of the measured residual stresses across a repair weld made in a low alloy ferritic CrMoV steel plate with C– Mn steel weld metal is shown in figure 4.42 (McDonald et al (2002)) using Cr K X-radiation that gives a diffraction peak at a position of 2 ¼ 1568. Here the results, figure 4.42(a), are compared with a stress profile calculated by finite element analysis, figure 4.42(b). Stresses in thin films can be measured using a glancing incident arrangement where the depth of X-ray penetration is reduced by using incident beam angles of 18. This technique has been applied to the measurement of stresses in diamond like films (Mohrbrachen et al (1996)). An extension of this technique to incident angles of <0.58 leads to the X-ray beam forming an evanescent wave which penetrates 3 nm: grazing incidence diffraction (Marra et al (1979)). Here the detector is positioned at 2 to the incident
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Figure 4.41. An example of a transportable X-ray stress measurement system fitted with a proportional counter measuring stresses on a turbine alternator end ring.
beam, just above the specimen surface such that the diffraction plane normal is almost within the surface. In this way a direct measure of the in-plane strain is obtained. In general the shape of the diffracted X-ray peak is a convolution of line profiles arising from several sources of imperfection, but in particular crystallite size and microstrain (Suryanarayana and Grant-Norton (1998)). Certainly X-ray diffraction peak profile analysis has been widely used to evaluate the contribution of these different microstructural parameters (Sommers and Mittemeijer (1995) and Bilger et al (1996)). To study these contributions it is necessary to have a measure of the integral breadth and shape parameter of the peaks. In addition, there is a contribution from instrumental broadening that has to be separated from the peak broadening. Crystallite size: X-ray diffraction peak broadening arising from crystallite size is inversely proportional to the domain thickness, t, perpendicular to the particular diffraction plane. Hence the integral breadth of the peak due to the thickness is given by S / 1=t :
ð4:46Þ
Microstrain: This refers to the condition where the internal strains in a polycrystal vary from grain to grain. Here the elastic strain is given by " ¼ cot d
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Figure 4.42. The results of the measured residual stresses across a repair weld made in a low alloy ferritic CrMoV steel plate with C–Mn steel weld material (McDonald et al (2002)) using Cr K X-radiation that gives a diffraction peak at a position of 2 ¼ 1568 (a) compared with strain gauge measurements and (b) a finite element calculation of the stresses.
hence the peak broadening is given by D ¼ 2"= cot ¼ 2" tan
ð4:48Þ
and in terms of sin = ¼ "d where d is the reciprocal lattice distance. The total broadening for a given diffraction peak less the instrumental contribution is given by ¼ S þ D
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ð4:49Þ
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Figure 4.43. A plot of integral breadth versus d ð1=dÞ has a gradient proportional to the microstrain and an intercept proportional to the crystallite size.
and / 1=t þ "d :
ð4:50Þ
As a consequence a plot of integral breadth versus d ð1=dÞ has a gradient proportional to the microstrain and an intercept proportional to the crystallite size (figure 4.43). Synchrotron X-ray diffraction The use of diffraction techniques based upon hard X-ray emissions from synchrotron sources has evolved more recently with the advent of the third generation of sources such as the European Synchrotron Research Facility in Grenoble, France (Lebrun et al (1995), Daymond and Withers (1996), Webster et al (1996a,b) and Withers and Webster (2001)). Such sources produce X-ray beams of high energy (50–150 keV) and intensity. The latter is increased by up to about three orders of magnitude compared with the more conventional laboratory X-ray sources. This leads to a combination of improved resolution with this high intensity. As a consequence the technique has been applied to the measurement of strain within bodies based upon the approaches described earlier in this section. Certainly the potential to obtain data at fast acquisition rates and in materials with a significant strain gradient that can be readily resolved has resulted in increased use of this approach to measure strain. To date three main methods summarised in figure 4.44 have been used (Withers and Bhadeshia (2001)). These are (i) traditional 2 scanning
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Figure 4.44. Three main methods using hard X-rays. (a) 2 scanning; (b) low angle transmission; (c) energy dispersive.
(Withers and Webster (2001)), (ii) high energy low angle two-dimensional transmission diffraction (Poulsen et al (1997)) and (iii) whole beam high energy dispersive methods (Reimens et al (1998)). In each case small volumes of material 20 mm 1 mm can be sampled. Neutron scattering and diffraction Neutrons, with no charge, do not interact with the electrons surrounding the nucleus and are perturbed only by the atomic nucleus in a ball-type collision (Slattery and Windsor (1983) and Allen et al (1985)). As a result the neutrons penetrate many millimetres into most solids, which is an advantage in that little or no specimen preparation is required, but is a disadvantage when small specimens need to be examined. A major problem with the technique is obtaining a source of neutrons. An experimental nuclear reactor is required and these are few and far between. In the UK to overcome this limitation neutrons are produced at the ISIS facility by the spallation process (Withers and Webster (2001)). In this facility a heavy metal target is bombarded with pulses of high energy protons from a synchrotron accelerator. This ejects neutrons from the nuclei of the target atoms and produces an intense neutron pulse with a relatively small level of heat production in the target material. The neutrons produced have high energies and are slowed to speeds that then result in wavelengths suitable for diffraction experiments and other investigations. This is achieved by placing hydrogenous moderators around the target. As a consequence the source of neutrons is appropriate for investigating a range of material parameters.
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Figure 4.45. A neutron diffraction pattern from a bent mild steel bar (Allen et al (1985)) (reproduced by permission of Taylor and Francis).
Neutron beams can be monochromated by diffraction from a germanium single crystal and collimated using cadmium slits before impinging on the specimen. The diffracted beam is detected using counters of a type similar to the proportional gas counters used in X-ray diffraction but with BF3 as the gas. Several counters are required and these are placed around the diffracting zone. Neutron diffraction takes the same basic form as Xray, electron and photon diffraction. A monochromatic beam of neutrons incident on a material will be diffracted by the atoms according to the Bragg equation. A typical diffraction pattern obtained from a bent mild steel bar is shown in figure 4.45 and illustrates the good energy resolution that can be achieved. The neutron scattering amplitude, which is defined as the intensity of the diffracted beam at a distance of 1 cm from the nucleus as a fraction of the incident beam intensity, appears to vary in a random manner throughout the Periodic Table. This is unlike electrons and X-rays where the amplitude increases systematically with atomic number. This response by neutrons has certain advantages for the technique. For example, in certain ordered alloy systems such as Cr–Ni alloys the superlattice reflection intensities are a function of the difference in scattering amplitudes of the Cr and Ni atoms. This leads to significant superlattice neutron diffraction peaks, whereas using X-ray diffraction such peaks
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Figure 4.46. Neutron diffraction patterns of Yba2 Cu3 O7 recorded at room temperature for (a) fresh specimen, (b) after a gap of 100 h and (c) 175 h. Dashed line indicates room background (Dasannacharya and Sequeira (1988)).
would be negligible. Furthermore it allows the location of light elements in the presence of heavier atoms or distinquishing between adjacent elements in the Periodic Table. Neutron diffraction allows lattice parameter measurement, the identification of phases, the detection of preferred orientation and the determination of residual stress. Particularly neutron diffraction is important for investigating cation ordering in alloys, mixed oxides and minerals, microstructural changes involving low atomic mass elements during phase transformations in a variety of hydrogeneous materials and ceramics. Hydrogen has a large scattering cross-section to neutrons and a feature used to investigate moisture absorption in high Tc superconducting ceramics. Figure 4.46 shows neutron diffraction patterns obtained from YBa2 Cu3 O7 which has a 1 :1 :3 stacked perovskite crystal structure (Dasannacharya and Sequeira (1988)). It has been noted that the superconducting properties change with environmental changes and this is related to changes in the ordered structure as shown in figure 4.46. A benefit of neutron diffraction is that it can be used to probe, nondestructively, the state of strain within a polycrystalline material which for typical engineering components can be to a depth of many centimetres. By limiting the irradiation volume and the field of view of the detector, it is possible to sample a small volume, 1 mm3 , within the body of a specimen. In general, one of two techniques are used for these measurements: conventional
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/2 scanning as described for X-ray diffraction or a time-of-flight approach (Withers and Bhadeshia (2001), Winholz and Kravitz (1996), Yagchi et al (2000) and Todd et al (1997)). The former is well suited to neutron beams produced by reactor sources whereas the latter is appropriate to the pulsed beam generated from a spallation source. In the time-of-flight method the Bragg angle is held constant, usually at 2 ¼ /2, and the incident wavelength is varied. In this case each pulse of neutrons leaving the moderated spallation target has a range of neutron energies. The most energetic neutrons arrive at the target in advance of the least energetic so that the energy and, hence, the wavelength of each detected neutron can be obtained from the time that has elapsed since the neutron pulse was produced. In this case the strain " is given by " ¼ t=t
ð4:51Þ
where t is the time of flight. As strain resolution depends upon the accuracy of the measurement of t, high resolution instruments invoke large flight paths, 100 m. An approach adopted to analyse these diffraction spectra is to use the Rieveld refinement (Webster (2000) and Young (1993)) to derive a single value of the lattice spacing by simultaneously fitting a curve to the intensity profile obtained from all the diffraction peaks obtained within the time of flight measurements (Daymond et al (1997)). This value is weighted towards the most intense peaks. The approach has been experimentally and theoretically demonstrated to provide a good representation of the bulk elastic response as compared to the relative insensitivity to tensile and compressive shifts of the various diffraction peaks. A complication when interpreting neutron diffraction data is the contribution of apparent strain arising as a result of diffraction from internal and external surfaces because the diffracting volume is only partially filled. These contributions are accommodated by including the diffraction geometry and attenuation. An alternative is to use the so-called z scan geometry where the surface is approached by bringing the specimen into the gauge volume vertically to eliminate lateral displacement of the centre of gravity of the scattering volume, thereby eliminating geometrical peak shift (Spooner and Wang (1997)). Additional information can be obtained from small angle neutron scattering (SANS) of long wavelength neutrons (Hutchings and Windsor (1986) and Windsor et al (1984)). Long wavelength neutrons are not diffracted by periodic arrays of atoms but are diffracted by fluctuations in the scattering density of the matrix as a result of the presence of particles, precipitates, voids etc. Defects in the range a few nanometres to 50 nm are detected in specimens up to 20 mm thick. Neutron diffraction is expressed in terms of a macroscopic scattering cross-section: the effective area A which scatters neutrons into a solid angle T. If, for example, particles have a mean diameter D then for a low volume fraction V the macroscopic
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Figure 4.47. Plot of dA=d versus Q2 for a collection of spherical defects (reproduced by permission of the Institute of Materials).
cross-section is given by dA VD3 ðp m Þ2 expðQ2 D2 =20Þ ¼ 6 d
ð4:52Þ
where is the neutron scattering solid angle, Q is a scattering vector and p , m , are scattering length densities of the particle and the matrix. If ln (dA=d) is plotted against Q2 , a straight line is be obtained with a slope of D2 /20 from which the particle size can be obtained (figure 4.47). The technique has been applied to the study of gamma prime precipitation in nickel based superalloys. It is possible to determine the size of these particles by TEM but it is more difficult to establish their volume fraction by this technique. By comparison, SANS measures both the mean particle size and the volume fraction in material and has been compared with TEM values (Windsor et al (1984)) (figure 4.48). The SANS and TEM results agree for all gamma prime particles detected and there is agreement between the two techniques for the volume fraction for ageing times up to 1000 h at 700 8C, but above this time SANS gives a decreasing volume fraction whereas TEM indicates the volume fraction to be essentially constant. Here the poor resolution of SANS is responsible for the divergence. However, it should be remembered that SANS is a relatively straightforward technique and results can be obtained quickly, whereas there is considerable laborious and time consuming work required in producing thin specimens for TEM. An extension of neutron diffraction is neutron interferometry. Figure 4.49 shows a typical three crystal interferometer constructed from a perfect silicon single crystal to give three identical thickness parallel slabs (Dasannacharya and Sequeira (1988)). This provides an emerging neutron beam with a sinusoidal interference as a function of angular position of the phase shifts. Figure 4.50 shows typical measured patterns such that the periodic of modulation is directly related to the average scattering length of the nuclei
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Figure 4.48. Mean particle size and volume fraction of gamma-prime precipitates in alloy PE16 determined by SANS and TEM (reproduced by permission of the Institute of Materials).
distributed in the phase shifter. Here estimates of small quantities of hydrogen in vanadium have been undertaken (Ranch and Seidli (1987)). Proton scattering Proton scattering is dealt with in more detail in chapter 7 on atom/ion sources and will only be briefly mentioned here. The proton has the same mass as the neutron but has a charge equal in magnitude but of opposite sign to the electron. It will therefore interact with the electron cloud but, although the forces exerted by the electrons will be of similar magnitude to interactions between electrons, the effect on the proton motion will be less marked because of the vastly greater momentum of the proton. Protons therefore penetrate much farther into materials than electrons but the diffraction of protons has
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Figure 4.49. A schematic picture of a typical three crystal neutron interferometer (reproduced by permission of North-Holland Publishing Company).
similarities to both electron and neutron diffraction. Considerable channelling of protons takes place down suitable crystallographic directions and in this case the number of scattered particles is reduced. Figure 4.51 illustrates the effect of orientation on scattering with a silicon model oriented in (a) a random direction, (b) with planes aligned and (c) with an axis aligned. Proton and ion scattering can be recorded either by using a setup with a specimen placed in front of a photographic plate or by detecting the scattered ions in specific directions. Figure 4.52 shows schematically the arrangement for production of channel patterns while figure 4.53 shows schematically the spectra (Rutherford backscattering (RBS)) that would be obtained for a layer of amorphous silicon on a single crystal of silicon containing a buried impurity.
Figure 4.50. Neutron interference patterns resulting from small quantities of hydrogen (0.004 to 0.035) in vanadium (reproduced by permission of North-Holland Publishing Company).
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Figure 4.51. Model of silicon aligned (a) randomly, (b) with a plane parallel to and (c) with an axis parallel to the ion beam (Courtesy AEA Technology). # UKAEA 1992.
It is worth noting here that ion scattering can take a similar form to proton scattering, but as the mass of the ion increases so the damaging potential becomes greater and diffraction or channelling is less likely. As yet this technique is little used.
Figure 4.52. Schematic diagram for obtaining a proton channelling pattern.
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Figure 4.53. Random alignment and channelling spectra for a structure containing an amorphous surface layer and a substitutional impurity (courtesy AEA Technology). # UKAEA 1992.
Kossel diffraction In scanning electron optical instruments the electron beam has sufficient energy, >15 keV, to generate characteristic X-rays from a specimen, but as discussed in chapter 2 the electron interaction volume is small giving a point source typically 2 mm diameter of generated X-rays when the beam is stationary. When Bragg diffraction is satisfied, a divergent beam produces an X-ray diffraction pattern called a Kossel pattern (Kossel (1936), Morris (1968), Swindells (1979) and Rowlands et al (1968)) (figure 4.54) which
Figure 4.54. Geometry of diffraction and absorption Kossel line conics derived from a single crystal (after Swindells (1979)).
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Figure 4.55. Kossel diffraction pattern from a martensite plate in a 32% Ni steel (b) together with the stereographic projection showing the martensite habit planes, (a), and a corresponding optical micrograph (c) (Rowlands et al (1968)) (reproduced by permission of the Institute of Materials).
provides a measure of interplanar spacing and lattice orientation. The pattern can be recorded on a photographic emulsion as a back diffraction pattern, with the lines showing dark against a lighter background, or by transmission through thin samples as absorption or deficiency patterns. The advantage of generating these patterns with a scanning electron beam instrument is that the beam can be located accurately within specific microstructural regions. As a result individual phases in a multiphase material can be examined by X-ray diffraction on a scale comparable with the resolution in an optical microscope. The pattern is related directly to the specimen since it is not processed by the instrument and is independent of the incident electron beam direction. Kossel patterns provide orientation data which can be combined with twosurface trace analysis (section 4.3.5) to define crystallographic features observed by the optical and scanning electron microscopy of a few micrometres in size. Many applications exist which demonstrate the ability to provide a complete crystallographic description of the microstructure. Figure 4.55 shows a diffraction pattern obtained from a martensite plate in a 32% Ni steel that enables the habit plane to be established in the associated stereographic plot (Rowlands et al (1968)). Kossel X-ray diffraction from bulk crystalline materials is now becoming a less specialised technique since patterns which are easily interpreted can be obtained from specimens with grain sizes <5 mm. Swindells (1979) has
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pointed out that the availability of this technique should lead to a reappraisal of the use of more conventional X-ray diffraction. For example, there is little point in reducing a polycrystalline substance to powder to establish lattice constants by the Debye–Scherrer method if information can be obtained to the required precision using Kossel microdiffraction. Moreover, with polycrystalline multiphase specimens each phase can be studied separately instead of as a mixture.
4.4 4.4.1
Electron diffraction Wave nature
For many years following its discovery the electron was considered to be a particle carrying a negative charge. It was Davisson and Germer (1927) who conclusively demonstrated that electrons also have a wave character by producing a back electron diffraction pattern from mica. The dual particle/wave nature of the electron is firmly established and for many purposes such as diffraction, the behaviour parallels that of the photon. Indeed two-slit diffraction has been demonstrated by Tonomura et al (1989) by firing individual electrons at each slit. In the sequence shown in figure 4.56 each spot on the television screen represents the arrival of one electron. After a few hundred electrons passed through the slits (figure 4.56(a–c)), the pattern remains apparently random but when a few thousand electrons had arrived the characteristic fringe diffraction pattern is developed (figure 4.56(d,e)), demonstrating the wave characteristic. The wavelength of the electron is determined by its energy. If the electron beam is characterised by a propagation vector K0 then the wavelength is given by the expression ¼
2 : K0
ð4:53Þ
The interrelationship between the wavelength and the accelerating voltage of the electron beam is also discussed in chapter 6. However, the relativistic corrected relationship is given by ¼
1:226 ½Eð1 þ 0:96 106 EÞ1=2
ð4:54Þ
where is in nm and E is the electron energy in volts. The variation of the wavelength with relativistic accelerating voltage is shown in table 5.2. We now move on to consider the diffraction of electrons by crystalline materials. While the basic principles involved are similar to those described earlier in this chapter there are a variety of situations where specific differences are encountered. The wave nature of the electrons means that they can be used
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Figure 4.56. Demonstration of double-slit diffraction for electrons (Tonomura et al (1989)). Each spot represents the arrival of a single electron. When a few hundred electrons have reached the screen, (a), (b) and (c), the pattern appears to be random but interference effects can be observed after the arrival of a few thousand electrons, (d) and (e) (reproduced by permission of the American Association of Physics Teachers).
to study materials by observing electrons that are scattered back from a surface or by allowing them to be transmitted through thin foils of material. In back diffraction the observed effects are strongly dependent on the energy of the incident electron beam. In this section we will separately consider diffraction in transmission, high energy back diffraction, low energy back diffraction and the corresponding high energy processes. 4.4.2
Transmission of electrons
The number of diffraction methods that have become available for studying materials to a high spatial resolution has increased significantly over the
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past 10 to 20 years with the development of electron microscopies that have improved vacuum systems to reduce specimen contamination and the ability to produce convergent electron beams and indeed various beam operating modes including scanning and rocking. This capability provides the opportunity to evaluate fine-scale phases and lattice defects more uniquely than by other techniques that sample larger volumes of material. In the transmission electron microscope the incident electron beam passes through a thin foil of material and depending on the lens conditions selected an image of the foil or a diffraction pattern is observed (Hirsch et al (1965), Amelinckx et al (1978), Loretto (1984), Jones (1988), Edington (1976), Warren (1979) and Beeston et al (1983)). Normally a transmission electron microscope produces a diffraction pattern by a method known as selected area diffraction (figure 6.36). For this condition the beam of incident electrons is parallel at the specimen surface and part of this beam passes without diffraction, but a small fraction of the electrons is diffracted out of the beam in directions determined by the Bragg diffraction condition. These beams are focused by the objective lens to form a diffraction pattern on the back focal plane and an image at the fluorescent screen. The image results from a beam defined by the diameter of the selecting aperture. By altering the conditions of the intermediate lenses it is possible to focus the diffraction pattern on to either the fluorescent screen or photographic plate. As the electron beam travels through the thin foil so a fraction of the electrons will be diffracted out of the primary incident beam. Thus the primary beam will lose intensity on passing through successive atom layers of the specimen while the diffracted beams will become increasingly intense. Similarly the diffracted beams will further diffract and lose part of their intensity and indeed a fraction will be diffracted back into the main beam. Electron diffraction in transmission is described essentially by the kinematical and dynamical theories. In the kinematical theory it is assumed that the incident beam does not lose intensity in passing through the foil and that there is no subsequent interaction between diffracted beams. However, the dynamical theory accounts for these effects and is therefore a more complete description, albeit more complicated. Kinematical theory of electron diffraction This theory assumes that the intensity of the main beam remains constant as it travels through the foil and ignores the scattering of diffracted beams back into the main beam. Therefore it is essentially the same theory as used to describe X-ray diffraction (Cullity (1979)). Initially an incident plane electron wave of the form eikz is assumed to be scattered by point sources in a regular lattice which identifies the positions of intensity maxima and minima. This is then modified by considering the positions of the atoms with respect to the array and by including a term to account for the scattering of electrons by
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Figure 4.57. Bragg conditions for electron diffraction where g is the reciprocal lattice spacing and LO ¼ 1= .
individual atoms. As for the treatment of X-ray diffraction an atomic scattering factor f is derived (Mott and Massey (1976)) which is given by Me2 2 ðZ f Þ ð4:55Þ f ¼ 2 8h "0 sin2 where Z is the atomic number, f the X-ray atomic scattering factor and "0 is the permittivity of vacuum. The amplitude of the scattered wave is described by ¼
eik:r X X fi expf2i½ðmi þ ii Þgi þ g r m i ¼ eik:r =rFL
where F ¼ structure factor ¼ L ¼ Lattice factor ¼
X
X
ð4:56Þ ð4:57Þ
fi expf2iði1 g1 þ i2 g2 þ i3 g3 Þg
ð4:58Þ
expf2iðm1 g1 þ m2 g2 þ m3 g3 Þg:
ð4:59Þ
i
m
F depends on the distribution of atoms in the unit cell while L depends on the type, shape and size of the crystal. The intensity is therefore ¼ 1=r2 jFj2 jL0 j2 :
ð4:60Þ
The maximum in L occurs when gi is a scalar with integer value when g ¼ ðS S0 Þ= ; S, S0 are unit vectors in the direction of the incident and diffracted electron beams (figure 4.57(a)). Since g is the reciprocal of the lattice spacing d and LO is constructed to be 1/ (figure 4.57(b)). sin ¼ OG=2OL ¼ g=2ð1= Þ ¼ ð1=dÞ=ð1= Þ
ð4:61Þ
d sin ¼ :
ð4:62Þ
or
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The theory here assumes that there is an average of one electron per unit volume in the beam. If v is the electron velocity then from equation (4.61) the number of electrons in the diffracted beam which cross an area dS, perpendicular to the beam, and at a distance r from the crystal surface is v dS ¼ v=r2 F 2 L2 dS
ð4:63Þ
Since dS=r2 is the solid angle subtended at the crystal by dS, the number of electrons scattered into unit solid angle per unit time is vF 2 L2 . Thus the scattering cross-section of the crystal, , is given by the number of electrons diffracted per unit solid angle/unit time divided by the number of electrons in incident beam crossing unit area/unit time F 2 L2 . It has been assumed to this point that the incident beam is parallel, has a single incident energy and is not distorted. Also crystals will not be perfectly formed, there will be some elastic deformation, dislocations and crystal misorientations. These will result in the cross-section having a value which depends on the mean value of L2 . It can be shown that (Rymer (1970)) the observed scattering cross section will have a value, given by ¼ ½V=v1 NF 2
ð4:64Þ
where V is the volume of the unit cell, v1 is the volume of the diffracting shell, N is the total number of unit cells in the crystal and F is the structure factor. The term V=v1 depends on the experimental conditions and in an experiment would be constant. Thus the intensity of the diffracted beam is determined by F 2. Consider a face centred cubic crystal with atoms at positions i1 , i2 , i3 ¼ 0, 0, 0; 0, 12, 12; 12, 0, 12; 12, 12, 0 (see chapter 1). From equation (4.59) substituting F 2 ¼ f ½1 þ expfiðg1 þ g2 Þg þ expfiðg2 þ g3 Þg þ expfiðg3 þ g1 Þg ð4:65Þ since the g are integers and the exponential terms are 1, there are two possible situations: (i) the values for g are either all even or all odd and the terms are all þ1 so that F 2 ¼ 4f ; (ii) the values for g are mixed, two 1 and one þ1, when F 2 ¼ 0. Therefore the diffracted beams are associated with these specific locations on the reciprocal lattice and are identified by g1 , g2 , g3 known as Laue indices, and are identical with the Miller indices specifying the direction of the atom planes leading to the diffraction. To determine the intensity of the diffracted beam it is appropriate to consider the general case where there are many crystallites with all possible orientations and the diffraction pattern will be a continuous ring. If the incident and diffracted beams have a small divergence then g will vary over a small range and the volume occupied by the diffracted beam, V1 , will be V1 ¼ 4g2 g
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ð4:66Þ
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Figure 4.58. Projection of the electron diffraction pattern onto a photographic plate.
The pattern observed, if recorded on a photographic plate will be magnified in proportion O0 G0 =OG ¼ L (figure 4.58), and the radius, R, and width, , of the diffracted ring are related by V1 ¼ 4R2 R=ð LÞ3
ð4:67Þ
which if I0 is the intensity of the incident beam then the number of electrons, Ne , diffracted into unit solid angle per unit time is Ne ¼
I0 VNF 2 ð LÞ3 : 4R2 R
ð4:68Þ
The area of the ring is 2R R=L2 therefore the number of electrons diffracted into the ring is I0 VNF 2 3 L=2R. If I0 is the incident electron beam current, I the diffracted electron beam current and A the area of the specimen sampled then I=I0 ¼ F 2 2 A=2R2 :
ð4:69Þ
In principle all the quantities except F can be measured; F is determined by calculation or experiment. The dynamical theory of electron diffraction In the more complete dynamical theory of electron diffraction, account is taken of incident electron beam intensity losses by diffraction as it passes through successive crystal layers, but at the same time gains intensity by
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diffraction of the initially diffracted beams back into the direction of the incident beam. After the incident electron beam has passed through a certain distance of the crystal, an equilibrium or dynamical situation occurs in which the ratio of the intensity of the incident beam to the intensity of the diffracted beam is a constant for all thicknesses greater than this minimum value. In electron diffraction this distance is approximately 10 nm compared with about 0.1 mm in X-ray diffraction, and is the reason why the kinematical theory can accurately predict X-ray intensities while the dynamical theory is required to describe electron diffraction. The theory considers the diffraction of electrons through either a wedgeshaped crystal, which is the case for most specimens thinned for examination in the transmission electron microscope, or a parallel sided crystal by obtaining a solution to the Schro¨dinger equation representing the primary and diffracted wave. A significant difference between the predictions of the kinematical and dynamical diffraction theories concerns the diffracted beam intensity when the Ewald sphere passes close to a reciprocal lattice point. In the kinematical theory, a diffracted intensity is predicted only if the Ewald sphere actually passes through the reciprocal lattice point. However, the dynamical theory predicts a significant intensity for the diffracted beam if the Ewald sphere is close to the reciprocal lattice point. Indeed, it is sufficient that the reciprocal lattice point be within 103 of the radius of the Ewald sphere to give a diffracted intensity. It is possible to compare the predictions of diffracted intensity by the dynamical and kinematical theories. Essentially the two theories give similar results if the crystal thickness does not exceed one extinction distance, that is the distance between two diffraction maxima. However as crystal thickness increases so the accuracy of the kinematical theory decreases and it becomes increasingly necessary to employ the dynamical theory (Cowley (1986)). 4.4.3
Electron diffraction with static beams
As described more fully in chapter 6, the evacuated column of an electron microscope contains an electron source or gun together with an assembly of lenses (Agar (1974), Cosslet (1970) and Grundy and Jones (1976)). On leaving the gun the electrons are formed by the lens into a crossover and this demagnified source image is projected on to the specimen by condenser lenses. The first condenser forms a demagnified image of the source about 1 mm diameter (figure 6.34), that is subsequently projected on to the specimen by a second condenser lens with a magnification of about two. The final illumination electron beam which is incident on the specimen may be as small as 2 mm in diameter which is sufficient to fill the screen at highest magnifications. The current density at the specimen depends upon the electron source characteristics and the divergence angle (Hirsch et al (1965) and Gronsky (1980)); however, typically the condenser aperture and
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the operating divergence angle would be 400 mm in diameter and 103 rad respectively for a 100 keV instrument. This second lens gives both fine control over the area illuminated, thereby reducing contamination from neighbouring areas and a beam of low divergence for an equivalent amount of defocusing. A smaller value of reduces the effective electron source size, and increases the coherence length of the beam, thereby providing improved contrast and resolution in images and diffraction patterns. On passing through a thin foil specimen the electrons enter an objective lens whose design and aberrations critically affect the performance of the microscope. Figure 6.34(b) shows coherent Bragg diffracted electron beams leaving the specimen to form an intermediate low magnification image I1 . The intermediate lens produces a second intermediate image I2 which is magnified at the viewing screen by the final projector lens. For a parallel incident electron beam the diffracted beams leaving the specimen are focused on the back focal plane of the objective lens (figure 6.36). This diffraction pattern is observed if the back focal plane is projected on to the viewing screen by reducing the excitation of the first projector lens (figure 6.36). Interchangeable objective apertures varying in size from 50 mm to 200 mm are located close to the back focal plane of the objective lens enhance contrast. Figure 4.59(a) shows a typical single crystal transmission electron diffraction pattern obtained from an M23 C6 type carbide precipitate. For 100 keV electrons the radius of the Ewald sphere (figure 4.57), is 2.5 nm and the Bragg angles are small, 102 radians, so that the diffraction sphere approximates to a plane. When the electron beam is parallel to a prominent zone axis in the diffracting crystal, several reciprocal lattice points intersect the diffracting sphere giving a projection of the prominent zone in the reciprocal lattice. Increasing the accelerating voltage of the electrons from 100 keV to 1000 keV allows diffraction patterns to be obtained from thicker regions of a foil specimen, and since the radius of the diffracting sphere is increased Laue zones are more extensive. Typical zones for a bcc crystal are shown in figure 4.59(b). To interpret an unknown pattern the diffraction spots or rings (if polycrystalline) it is necessary to assign Miller indices corresponding to the diffracting lattice planes. Figure 4.60 shows the diffraction pattern from a thin foil of a nickel based PE16 alloy, a face centred cubic (fcc) matrix, which shows the pattern of superlattice spots from the 0 phase. The terminology used to describe these patterns includes incident beam divergence, angular resolution and the spatial resolution. The divergence is the half apex angle of the cone of the incident electron beam, . Divergence can be either measured from an electron diffraction pattern or calculated from a knowledge of the particular electron microscope geometry. The angular resolution of the diffraction pattern is simply the angular width of the smallest intensity fluctuation in the pattern that can be attributed to the specimen.
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Figure 4.59. Electron diffraction (a) extracted carbide ðxÞ from 2.25%Cr–1%Mo steel is a composite of two carbides; corresponding electron diffraction pattern showing [2 33] fcc zone pattern where splitting of diffraction spots corresponds to lattice spacing for M23 C6 and M6 C carbides (courtesy D Lonsdale). (b) Commonly occurring diffraction patterns for bcc crystals.
Figure 4.60. Diffraction pattern obtained from thin foil of fcc matrix (PE16 alloy) showing superlattice spots from the 0 phase (courtesy C Baker).
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Selected area The advantage of selected area electron diffraction in the transmission electron microscope is that a small area of foil specimen about 1 mm diameter can be identified and from this structural information recorded rapidly. By comparison the minimum area selected using X-ray diffraction is approximately two orders of magnitude greater and requires commensurately longer exposure times. The basic information obtained from either a single or a sequence of electron diffraction patterns is the orientation of the planes of atoms with respect to the electron beam direction and the image and, therefore, in the case of a multiphase system the individual crystal structures and orientation relationship of one phase to another. To achieve this, a selecting aperture of appropriate size is used to identify the feature of interest and then the aperture and image are set to ensure they have a coincident focal plane. A single crystal in the electron beam will produce essentially a regular and symmetrical arrangement of diffraction spots, each with a value of h; k; l. Spots equidistant from the centre and diametrically opposite are associated with the same crystal planes of indices h; k; l and h; k; l respectively. Their simultaneous appearance is a consequence of the small wavelength of the electrons, the small divergence angle and other instrumental parameters. To establish the solution of these patterns requires that the distances and angles can be related. This is achieved by determining the camera constant for the particular instrument where ðDdÞ ¼ CL
ð4:70Þ
where L is the effective camera length, is the wavelength, D is the ring diameter of a standard electron diffraction pattern and d is the interplanar spacing; CL is the camera constant. Various methods are used to establish this constant and these have been described by Andrews et al (1971). Figure 4.61 shows the diffraction pattern obtained from an area of foil containing 0 plates (hcp) in a niobium 40% Zr alloy (the TEM image of this is illustrated in figure 6.46). This pattern contains diffraction spots showing the foil normal corresponds to the {111} plane for the bcc phase matrix. However, the additional diffraction spots match with two variants of the [00.1] zone for the hcp 0 plates and, indeed, there is streaking in the h100i directions. From this and similar electron diffraction patterns it is possible to show that the orientation relationships between the 0 plates and the matrix satisfies the Burgers relationship where (110) ||(00.1) and the [111] ||[11.0] (Burgers (1934)). High resolution To form a high resolution diffraction pattern the specimen is mounted on a special stage positioned below the final projector lens of the transmission electron microscope (figure 4.62(a)), at a distance of 300 mm from the
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Figure 4.61. Selected area electron diffraction pattern taken from an area of phase niobium 40% zirconium alloy containing 0 plates. Foil normally corresponds to the [111] and shows two variants of the Burgers orientation relationship for the 0hcp and bcc product and parent phases.
photographic recording plate. Under these conditions all the lenses are used to focus the electron beam to a small spot, thereby reducing both the aperture angle and interference from chromatic aberration and lens instabilities. Such a procedure is suited for determining both lattice parameter and crystal structure and hence identification of phases. A typical application is the determination of the various types of carbide precipitates in steels. Figure 4.62(b) shows the ring pattern produced from fine grained gold and a G-silicide ( fcc, a ¼ 1 nm) in an iron alloy containing silicon and tantalum (Brown and Whiteman (1969)). Micro area There are obvious advantages to obtaining diffraction patterns from areas less than 500 nm in diameter in multiphase phase alloys (Warren (1979)). The region of interest may be a small second phase precipitate, a particular grain in a small grain size material, a sub-grain developed in a mechanically deformed material or a region of the strain field of a second phase precipitate. Prior to the introduction of the STEM the lens configuration used for diffraction was limited to areas greater than 500 nm by the sperical aberration of the objective lens. Nano-area electron diffraction in a scanning transmission electron microscope is extensively applied to a range of
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(a)
Figure 4.62. High resolution electron diffraction. (a) The standard mode of operation. (b) Typical pattern obtained from fine grain size gold and G-silicide in an iron base alloy containing silicon and tantalum (Brown and Whiteman (1969)).
specimens, in particular semiconductors (Cowley (1986)). A nanometre-size electron beam is formed by the two condenser lenses and the pre-field of the objective lens of the electron microscope. The resultant electron diffraction patterns from areas down to about 2 nm diameter are then recorded on
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Figure 4.63. Bright field image of a wedge of the Ga0:63 Al0:37 As/GaAs superlattice, (a) showing the equal-thickness fringes and (b) nano-diffraction patterns from the Ga0:63 Al0:37 As and GaAs regions (Tanaka and Mikama (1988)).
high sensitivity film or continuously in a real time cathode ray tube system (Tanaka (1989) and Tanaka and Mikama (1988)). Figure 4.63 shows the application of this technique to a superlattice of Ga0:63 Al0:37 As/GaAs. The 2 nm diameter electron beam is traversed across the bright field image contrast of the interface of interest as shown in figure 4.63(a). The pattern obtained from the Ge0:63 Al0:37 As compared with that of the GaAs shows that the 200 reflection becomes weak and almost disappears in the GaAs due to the small structure factor for this diffraction condition in this material. Certainly it is now possible to undertake high resolution electron diffraction to a scale comparable with the resolution of the images that can be obtained. Kikuchi diffraction A special type of diffraction pattern is obtained from thin foils specimens of greater than a few hundred nanometres thick (figure 4.64). As the thickness of the crystal increases so the intensity of the diffraction spots decreases until eventually they are masked by the background. On this background are bright and dark lines oriented with directions related to the crystal symmetry.
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Figure 4.64. Kikuchi lines observed in transmission foils.
These lines were first discovered by Kikuchi in 1928 and his name has since been used to describe them. They arise from Bragg diffraction of elastically scattered electrons as the incident beam penetrates the foil (figure 4.65). The angular spread of these electrons is large and leads to elastic scattering by the crystal lattice oriented at the Bragg angle. Therefore, electrons inelastically scattered in the direction OA will be diffracted into direction OB and vice versa. If the numbers of electrons diffracted in these two directions are equal there is no gain or loss of electrons travelling parallel to the OA and OB directions. However, as the scattering angle increases so fewer electrons are scattered. Following electron scattering more electrons follow OA than OB but on Bragg diffraction there is an overall loss of electrons travelling parallel to OA and a gain in electrons following OB. Scattering is such that conic sections of bright and dark Kikuchi lines are developed within the overall diffraction pattern (figure 4.65). These lines provide an accurate determination of specimen orientation since, while diffraction spots only change intensity when a crystal is tilted, the Kikuchi line pairs (bright plus dark) traverse the diffraction pattern as though rigidly fixed to the specimen (figure 4.65). The angle is small in Kikuch line formation and we may approximate the Bragg condition to 2d ¼ , where d is the interplanar spacing. If the distance between the Kikuchi lines is D and the specimen to photographic plate distance is L then D=L ¼ 2:
ð4:71Þ
Dd ¼ L:
ð4:72Þ
Thus The Kikuchi lines move as the crystal is tilted and only those Kikuchi lines corresponding to planes which make a small angle to the beam direction are viewed. As the specimen orientation is changed the visible Kikuchi lines alter depending upon the crystallographic axis of rotation. Twodimensional Kikuchi maps can be produced either by calculation or experimentally (figure 4.66). The latter figure shows a schematic Kikuchi
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Figure 4.65. (a) The origin of Kikuchi lines. (b) Schematic diagram showing Kikuchi line positions near (200) orientation for a fcc crystal: beam exactly parallel to [200], crystal tilted by about [002] direction into exact position for 020 diffraction, crystal tilted so that 020, 002 and 022 spots are simultaneously in exact diffracting condition. (c) The variation of the dislocation cell misorientation (angle) with strain (Lonsdale and Flewitt (1978)).
map for a bcc crystal centred on a [001] electron beam direction. An important parameter to determine is the precise deviation from the Bragg angle of crystal planes which give rise to spot diffraction patterns. This deviation parameter, Sg , is defined in figure 4.67 as the distance from the Ewald sphere in a direction parallel to the beam direction of the reciprocal lattice point giving rise to a diffraction spot. The simplest way to establish, Sg , is to measure the displacement of a Kikuchi line from the corresponding diffraction spot. When the precise Bragg condition is satisfied, the bright Kikuchi line will pass exactly through the directly transmitted beam and the corresponding dark line through the associated diffraction spot. The
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Figure 4.66. Schematic Kikuchi map for a bcc crystal centred on [001].
value of Sg at a particular deviation from the Bragg condition is given by geometry to be Sg ¼ g2 =2
ð4:73Þ
where g is the diffraction vector (see figure 4.67). Thus Kikuchi diffraction provides an invaluable method of establishing the electron beam direction, a measure of the deviation parameter and an ability to orient specific microstructural features accurately. An accuracy of 0.18 for orientation determination may be achieved from Kikuchi line measurements (Ball (1981) and Bendersky et al (1982)). The method is well suited for determining the small misorientations between sub-grains developed, for example, during
Figure 4.67. (a) Schematic diagram illustrating the method for determining S at the symmetry position. (b) Schematic diffraction pattern which would be obtained at the symmetry position.
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creep deformation. Indeed it has been possible to follow the progressive change of misorientation with creep strain in a-iron (Lonsdale and Flewitt (1978)) (figure 4.65(c)). Convergent beam patterns This class of diffraction patterns was first demonstrated in 1939 by Kossel and Mollenstedt but their use and application has only recently become widespread following the ability to produce electron beams with a small diameter in the TEM and STEM. The effect has many similarities to Kikuchi diffraction. In convergent beam electron diffraction the operating conditions in the microscope are arranged such that the electron beam is focused on to the crystal specimen. Thus the specimen is traversed by a cone of electron beams. Convergent beam diffraction differs from Kikuchi diffraction only in that for the convergent beam the cone of electrons is produced externally while in Kikuchi diffraction the cone of electrons arises from the diffuse scattering within the crystal. The principle is illustrated in figure 4.68 where the cone of electrons is incident on the specimen surface at the plane BB0 , the objective lens then focuses the diffracted beams on to the back focal plane at CC 0 and DD0 . The ray diagrams for the production of convergent beam diffraction (CBD) is shown in figure 4.69 for both the TEM and STEM modes. Here, the
Figure 4.68. Ray diagrams illustrating the formation of convergent beam patterns (CBPs). The direct, zero order, and one diffracted spot are shown.
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Figure 4.69. Ray diagrams for convergent beam diffraction (CBD) using a condenser objective lens: (a) TEM mode, (b) STEM mode.
second condenser aperture determines the convergence angle and a large number with varying diameters is required. The objective lens focuses the electron beam on to the back focal plane after passing through the specimen. For a more detailed description of the technique the reader is referred to Steeds (1979) and Tanaka and Terauchi (1985). These patterns display the variation in intensity of transmitted or diffracted beams as a function of the angle between the incident electron beams and the crystal. These patterns have been given the generic name ‘Tanaka patterns’ and a typical bright field pattern is shown in figure 4.70 for a stainless steel specimen. This is essentially a map of the intensity of the direct beam variation with angle, over a range of angles in the general region of the [001] zone axis (Eades (1988). Specimens greater than 100 nm thick can be examined by this technique but the beam is broadened as it passes through the foil and this limits the spatial resolution obtained (Kyser and Geiss (1979) and Hutchins et al (1979)) (see figure 2.7). CBD patterns are very sensitive to strain fields from dislocations, precipitates or surface films and it is essential that such regions are avoided. Contamination of the specimen can also cause problems so that an area studied over long periods of time with the potential for contamination, which results in consequent degradation. To achieve the best from the technique, steps must be taken to reduce contamination, usually by obtaining the best possible vacuum in the microscope and here a microscope operating at UHV pressures is invaluable (see chapter 3). A
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Figure 4.70. Zone axis diffraction (Steeds (1979)). (a) Zone axis pattern for stainless steel [110] at 100 keV. (b) [001] convergent beam pattern of a phase precipitate in Type 316 austenitic stainless steel taken at 100 keV. (c) Image of M23 C6 precipitates where the convergent beam pattern is obtained from the central carbide, the pattern is a [111] axis at 100 keV. (d) Nickel [111] and 316 stainless steel zero order convergent beam patterns showing higher order Laue zone (HOLZ) lines at 100 keV. Using a standard, it is possible to deduce a ¼ 0:3516 nm for nickel and a ¼ 0:3589 nm for Type 316 austenitic stainless steel (courtesy J W Steeds).
cold finger, cooled to liquid nitrogen temperatures, and placed close to the specimen, will reduce contamination. Higher order Laue zones The Ewald sphere for electrons of energy of the order of 100 kV has a diameter that is much greater than the reciprocal lattice spacing (figure 4.71). As a result diffraction occurs at the origin and for some distance away from the origin until the Ewald sphere is sufficiently distant from the zero
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Figure 4.71. Ewald sphere construction in plan and section showing the formation of zero order and higher order Laue zones. d is the distance from the origin to the Ewald sphere.
reciprocal lattice plane for the effect to be negligible. At a greater distance from the origin the next layer of reciprocal lattice points will be encountered, giving rise to a ring of diffraction spots for the first order Laue zone (FOLZ). As the distance increases, further higher order Laue zones (HOLZ) are encountered but the intensity decreases with distance from the pattern centre. Figure 4.72 is a CBD pattern obtained from silicon carbide and is a good example of diffraction from the zero order, the central spot with its associated diffraction effects, the first order Laue zone, the next six spots and three higher order Laue zones. The zero order zones are shown in greater detail in figure 4.73. Applications Since convergent beam patterns provide information to establish small differences in both crystal structure and lattice parameter, they can be used to identify precipitates and, indeed, to measure small changes in lattice parameter between coherent precipitates and the parent matrix. Figure 4.70(c) shows [111] zone axis convergent beam diffraction patterns from M23 C6 and M6 C carbides. Steeds (1979) has pointed out that a powerful way to use these patterns is a ‘fingerprint’ of the relevant zone axes of commonly encountered materials and phases, and identification of over sixty different materials has been achieved. For this approach a reference accelerating voltage has to be adopted and, to accommodate changes resulting from specimen thickness variations, a series of patterns have been generated to cover a range of foil thicknesses. Additional information may be obtained from the HOLZ lines contained within the zero order disc of the convergent beam pattern (figure
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Figure 4.72. Convergent beam pattern from silicon carbide showing the zero order and higher order Laue zones (HOLZ) (courtesy N Tanaka) (reproduced by permission of JEOL).
Figure 4.73. Detail contained within the zero order direct beam CBD from silicon carbide (courtesy N Tanaka) (reproduced by permission of JEOL).
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4.70(d)). These lines in electron diffraction are the equivalent of Kossel lines in X-ray diffraction. HOLZ lines do not occur in all convergent beam diffraction patterns but are present usually in those produced by large unit cell metals and alloy phases, particularly nitrides, carbides and borides. The position of the HOLZ lines can be used to determine local crystal lattice constants with an accuracy of about 0.2%. The detailed symmetry information further aids the interpretation of the ‘fingerprint’ of different phases. A major application of CBD is the determination of lattice parameters and small changes in lattice parameters that can arise from local lattice strain. A small change in the lattice parameter will alter the position of the reciprocal lattice point, which will in turn move the HOLZ line. For a cubic material because the change in lattice parameter, a, produces a change in Bragg angle, , then = ¼ a=a:
ð4:74Þ
This method may be extended to determine strains in the lattice, particularly at planar interfaces and around large precipitates. If the strain is caused by a change in the chemical composition then measurement of the strain can be used to determine the concentration of the elements. The technique has been applied to determine the local concentration of aluminium in copper-aluminium alloys (Merton-Lyn (1977)) where it is possible to determine the concentration of aluminium to an accuracy of 1 at%. In addition, this technique provides an extremely accurate method for evaluating the thickness of foils. This is discussed in chapter 6. 4.4.4
Electron diffraction from rocking beams
Prior to the introduction of the STEM, instrument lens configuration diffraction was limited by spherical aberration of the objective lens to areas greater than 500 nm. It is possible to improve the spatial resolution of the areas of electron images selected for diffraction by operating a STEM instrument in a rocking diffraction mode (Spence and Carpenter (1986)). A number of different rocking incident electron beam techniques have been developed for various applications (Geiss (1976)). Ray diagrams corresponding to some of these techniques are given in figure 4.74(a)–(c). Figure 4.74(a) shows rocking selected area diffraction which is the reciprocal of conventional selected area diffraction. Here the beam is focused at the front of the focal plane of the objective lens, producing an approximately parallel electron beam of the specimen at the lens image plane. The selected area aperture is positioned at the lens object plane to be conjugate to the specimen. The scan coils produce a translation of the crossover at the front focal plane, resulting in a rocking beam at a point on the specimen. This produces a diffraction pattern which can be recorded on the electronic detector. A method of reducing the size of the area selected is shown in figure
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(c)
Figure 4.74. Ray diagrams for rocking beam microdiffraction. (a) Rocking selected area diffraction. This technique is reciprocal to conventional SAD and is often used in DSTEM instruments. (b) The rocking beam method. (c) The double-rock (or rock– unrock) method. The double-rock method can be visualised in either direction, provided reciprocity for detector and collection apertures is considered. Rocking angle is . (d) Transmission electron micrograph and corresponding diffraction patterns of evaporated gold particles on a carbon substrate (courtesy JEOL).
4.74(b). Here the instrument is operated in the image mode so that is the image is conjugate with the viewing detector. Certainly it is possible to produce diffraction patterns from areas as small as 3 nm using large diameter, relatively parallel incident electron beams which minimise radiation effects within the specimen and therefore in the resulting diffraction patterns (Geiss (1976), Chevalier and Craven (1977) and Fujimoto and Lehampfuhl (1974)).
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Diffraction
Figure 4.75. Central portion of an energy filtered Eades double-rocked, wide angle CBED pattern from thin silicon, with beam direction [111]. The field of view increases with rocking angle. This pattern was recorded serially, through an electron spectrometer that collected only elastically scattered electrons with E 4 eV (Spence and Carpenter (1986)) (reproduced by permission of Plenum Press).
A technique, the double rock, is shown in figure 4.74(c) which is useful for investigating the symmetry of perfect crystals (Eades (1980), Tanaka et al (1980) and (Spence and Carpenter (1986)). Figure 4.75 shows a typical energy filtered double rocked, wide angle
