95                     Materials and              3. Materials and Their Characteristics: Overview                        ...
96           Part A   The Materials Measurement System                      well-defined SI-Units (see Chapt. 1 of this han...
Materials and Their Characteristics: Overview   3.1 Basic Features of Materials   97structure: grains composed of the same...
98           Part A   The Materials Measurement System                           Nanoscale                               M...
Materials and Their Characteristics: Overview   3.1 Basic Features of Materials   99                                     P...
100          Part A   The Materials Measurement System                               Raw materials                        ...
Materials and Their Characteristics: Overview   3.2 Classification of Materials Characterization Methods   101Table 3.1 App...
102        Part A   The Materials Measurement System                            Composition, Structure                    ...
17                          Materials Scie   2. Materials Science for the Experimental MechanistCraig S. Hartley          ...
18        Part A   Solid Mechanics Topics                      a material depend on its structure at one or more lev-     ...
Materials Science for the Experimental Mechanist                                                                          ...
20         Part A   Solid Mechanics Topics                      rates atoms of other elements into its structure without  ...
Materials Science for the Experimental Mechanist    2.1 Structure of Materials   21The resulting attraction forms a weak b...
22       Part A   Solid Mechanics Topics                      ductility, and high formability by a variety of tech-       ...
Materials Science for the Experimental Mechanist     2.1 Structure of Materials   23    Crystal Structures of Elements and...
24        Part A   Solid Mechanics Topics                           In the solid state, patterns of atoms and molecules   ...
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
1 introduction to material science
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1 introduction to material science

  1. 1. 95 Materials and 3. Materials and Their Characteristics: Overview 3.1 Basic Features of Materials .................... 96 In its most general context, the term mater- 3.1.1 Nature of Materials....................... 96 ials measurements denotes principles, techniques 3.1.2 Types of Materials ........................ 97 and operations to distinguish qualitatively and 3.1.3 Scale of Materials ......................... 99 to determine quantitatively the characteristics 3.1.4 Processing of Materials ................. 99 of materials. As materials comprise all natu- 3.1.5 Properties of Materials .................. 99 ral and synthetic substances and constitute the 3.1.6 Application of Materials ................ 100 physical matter of products and systems, such Part A 3 3.2 Classification of Materials as Characterization Methods...................... 101 • machines, devices, commodities References .................................................. 102 • power plants and energy supplies • means of habitation, transport, and commu- the science of measurement, are outlined, in nication, this chapter an overview on the basic features of it is clear that materials characterization methods materials is given as a basis for the classification of have a wide scope and impact for science, tech- the various methods used to characterize materials nology, the economy and society. Whereas in the by analysis, measurement, testing, modelling and preceding chapters the principles of metrology, simulation.Materials measurements are aimed at characterizing the Generally speaking, measurement begins with a def-features of materials quantitatively; this is often closely inition of the measurand, the quantity that is to berelated to the analysis, modelling and simulation, and measured [3.2], and it always involves a comparison ofthe qualitative characterization of materials through test- the measurand with a known quantity of the same kind.ing [3.1], see Fig. 3.1. Whereas the general metrological system is based on the Measurement Materials Testing Set of operations Natural or synthetic Technical procedure for the purpose of substances; consisting of the determining the value physical matter of determination of attributes, of a quantity products in accordance with a specified procedure Result of analysis, measurement, testing, modelling, simulation: characterization of materials by quantities and attributesFig. 3.1 Scheme for the characterization of materials
  2. 2. 96 Part A The Materials Measurement System well-defined SI-Units (see Chapt. 1 of this handbook), or thermal conductivity; some are Boolean, such as the for materials there is a broad spectrum of “material mea- ability to be recycled; some, like resistance to corrosion, surands”. This is due to the variety of materials, their may be expressed as a ranking (poor, adequate, good, intrinsic chemical and physical nature, and the many at- for instance) and some can only be captured with text tributes, which are related to materials with respect to and images [3.3]. As a background for the materials composition, structure, scale, synthesis, properties and measurement system and the classification of mater- applications. Some of these attributes can be expressed ials characterization methods, in this chapter the basic – in a metrological sense – as numbers, like density features of materials are briefly reviewed. 3.1 Basic Features of Materials Materials can be of natural origin or synthetically • Metallic bonds occur between elements with lowPart A 3.1 processed and manufactured. According to their chem- electron-negativities, so that the electrons are only ical nature they are broadly grouped traditionally loosely attracted to the ionic nuclei. A metal is into inorganic and organic materials. Their physical thought of as a set of positively charged ions em- structure can be crystalline, or amorphous. Compos- bedded in an electron sea. ites are combinations of materials assembled together • Van der Waals bonds are due to the different inter- to obtain properties superior to those of their sin- nal electronic polarities between adjacent atoms or gle constituents. Composites are classified according molecules leading to weak (secondary) electrostatic to the nature of their matrix: metal, ceramic or dipole bonding forces. polymer composites, often designated MMCs, CMCs and PMCs, respectively. Figure 3.2 illustrates with Materials Spatial Atomic Structure. The amorphous or characteristic examples the spectrum of materials be- crystalline arrangement of atoms (or molecules) in crys- tween the categories natural, synthetic, inorganic, and talline structures is characterized by unit cells which are organic. the fundamental building blocks or modules repeated many times in space within a crystal. 3.1.1 Nature of Materials Grains. Crystallites made up of identical unit cells re- From the view of materials science [3.4], the fundamen- peated in space, separated by grain boundaries. tal features of a solid material are described as follows. Phases. Homogeneous aggregations of matter with re- Materials Atomic Nature. The atomic elements of the spect to chemical composition and uniform crystal periodic table which constitute the chemical composi- tion of a material. Materials Atomic Bonding. The type of cohesive elec- Natural tronic interactions between the atoms (or molecules) in a material, empirically categorised into the following Minerals Wood basic classes: Paper • Inorganic Organic Ionic bonds form between chemical elements Composites with very different electron-negativity (tendency MMC, CMC, to gain electrons), resulting in electron transfer PMC and the formation of anions and cations. Bond- ing occurs through electrostatic forces between the Metals ions. Ceramics Polymers • Covalent bonds form between elements that have similar electron-negativities, the electrons are lo- Synthetic calised and shared equally between the atoms, leading to spatially directed angular bonds. Fig. 3.2 Classification of materials
  3. 3. Materials and Their Characteristics: Overview 3.1 Basic Features of Materials 97structure: grains composed of the same unit cells are for the good ductility and formability of metals. Met-the same phase. als and metallic alloys are the most important group of the so-called structural materials (see below) whoseLattice Defects. Deviations of an ideal crystal structure: special features for engineering applications are their mechanical properties, e.g. strength and toughness.• Point defects or missing atoms: vacancies• Line defects or rows of missing atoms: dislocations Semiconductors. Semiconductors have an intermedi-• Area defects: grain boundaries ate position between metals and inorganic non-metallic• Volume defects: cavities materials. Their most important representatives are the elements silicon and germanium, possessing covalentMicrostructure. The microscopic collection of grains, bonding and diamond structure and the similarly struc-phases, lattice defects and grain boundaries. tured III–V-compounds, like gallium arsenide (GaAs). Together with bulk material characteristics, surface Being electric non-conductors at absolute zero, semi- Part A 3.1and interface phenomena have to considered. conductors can be made conductive through thermal energy input or atomic doping which leads to the3.1.2 Types of Materials creation of free electrons contributing to electrical conductivity. Semiconductors are important functionalIt has been estimated that there are between 40 000 and materials (see below) for electronic components and80 000 materials which are used or can be used in today’s applications.technology [3.3]. Figure 3.3 lists the main conventionalfamilies of materials together with examples of classes, Inorganic Non-Metallic Materials, Ceramics. Atomsmembers, and attributes. For the examples of attributes, in these materials are held together by covalent andsufficient characterization methods are named. ionic bonding. As covalent and ionic bonding ener- From a technological point of view, the materials gies are much higher than metallic bonds, inorganiccategorized in Fig. 3.3 as families have different charac- non-metallic materials, like ceramics have high hard-teristics relevant for engineering applications [3.5]: ness and high melting temperatures. These materials are basically brittle and not ductile: In contrast toMetallic Materials; Alloys. In metals, the grains as the the metallic bond model, a displacement of atomisticbuildings blocks are held together by the electron gas. dimensions theoretically already breaks localised cova-The free valence electrons of the electron gas account lent bonds or transforms anion–cation attractions intofor the high electrical and thermal conductivity and anion–anion or cation–cation repulsions. Because ofthe optical gloss of metals. The metallic bonding – missing free valence electrons, inorganic non-metallicseen as an interaction between the sum total of atomic materials are poor conductors for electricity and heat,nuclei and the electron gas – is not significantly influ- this qualifies them as good insulators in engineeringenced by a displacement of atoms. This is the reason applications. Subject Family Class Member Attributes Natural Steels CuBeCo Composition: Chemical analysis Ceramics Cast iron CuCd Density: Polymers Al-alloys CuCr Measurement Materials Grain size: Metals Cu-alloys CuPb Computational modelling Semiconductors Ni-alloys Bronze Wear resistance: 3-body-systems-testing Composites Ti-alloys CuTe Reliability: Biomaterials Zn-alloys CuZr Probabilistic simulationFig. 3.3 Materials types with examples of materials attributes and characterization methods (after [3.3])
  4. 4. 98 Part A The Materials Measurement System Nanoscale Microscale Macroscale % Atomic, molecular % Microelectromechanical % Continuum engineering systems systems (MEMS) systems % Electronic, quantum % Microstructures % Bulk, components structures of materials joined structures 10 – 9 10 – 6 10 –3 10 0 10 3(m) Fig. 3.4 Scale of materials: systems and structuresPart A 3.1 Organic Materials; Polymers, Blends. Organic mater- sen so that the properties of one constituent enhance the ials whose technologically most important represen- deficient properties of the other. Usually, a given prop- tatives are the polymers, consist of macromolecules erty of a composite lies between the values for each containing carbon covalently bonded with itself and with constituent, but not always. Sometimes, the property of elements of low atomic number (e.g. H, N, O, S). Inti- a composite is clearly superior to those of either of the mate mechanical mixtures of several polymers are called constituents. The potential for such a synergy is one rea- blends. In thermoplastic materials, the molecular chains son for the interest in composites for high-performance have long linear structures and are held together through applications. However, because manufacturing of com- (weak) intermolecular (van der Waals) bonds, leading to posites involves many steps and is labour intensive, low melting temperatures. In thermosetting materials the composites may be too expensive to compete with met- chains are connected in a network structure and do not als and polymers, even if their properties are superior. In melt. Amorphous polymer structures (e.g. polystyrene) high-tech applications of advanced composites it should are transparent, whereas the crystalline polymers are also be borne in mind that they are usually difficult to translucent to opaque. The low density of polymers recycle. gives them a good strength-to-weight ratio and makes them competitive with metals in structural engineering Natural Materials. Natural materials used in engineer- applications. ing applications are classified into natural materials of mineral origin, e.g. marble, granite, sandstone, mica, Composites. Generally speaking, composites are hybrid sapphire, ruby, diamond, and those of organic origin, creations made of two or more materials that maintain e.g. timber, India rubber, natural fibres, like cotton and their identities when combined. The materials are cho- wool. The properties of natural materials of mineral Subject Family Class Member Attributes ... Casting Material type Deformation ... Compression Joining Moulding Size range Rotation Process Shaping Composite Shape Injection Tolerance Surfacing Powder Blow Roughness Rapid prototyping ... Batch quantity Fig. 3.5 Hierarchy of processing of materials
  5. 5. Materials and Their Characteristics: Overview 3.1 Basic Features of Materials 99 Processing Matter Materials Manufacturing % Structural Materials % Solids, Liquids Machining → mechanical, thermal tasks Forming % Functional Materials → electrical, magnetic, optical tasks % Atoms, Molecules Nanoscale manipulation % Smart Materials → sensor + actuator task AssemblyFig. 3.6 Materials and their characteristics result from the processing of matter Part A 3.1origin, such as for example high hardness and good materials measurement methods have to characterizechemical durability, are determined by strong cova- materials with respect to thelent and ionic bonds between their atomic or molecular 1. Nanoscale, sizes of about 1 to 100 nanometers [3.6],constituents and stable crystal structures. Natural mater- 2. Microscale, relevant for micro-devices and micro-ials of organic origin often possess complex structures systems having sizes of typically 1 to 1000 micro-with direction-dependent properties. Advantageous ap- meters [3.7],plication aspects of natural materials are recycling and 3. Macroscale materials have the dimensions of all cus-sustainability. tomary products, devices and plants, ranging from the millimeter to the kilometer scale [3.8].Biomaterials. Biomaterials can be broadly definedas the class of materials suitable for biomedical ap- Figure 3.4 gives an overview on materials scales withplications. They may be synthetically derived from some key words.non-biological or even inorganic materials or theymay originate in living tissues. The products that 3.1.4 Processing of Materialsincorporate biomaterials are extremely varied and in-clude artificial organs; biochemical sensors; disposable For their use, materials have to be engineered by pro-materials and commodities; drug-delivery systems; den- cessing and manufacture in order to fulfil their purposetal, plastic surgery, ear and ophthalmological devices; as the physical basis of products designed for the needsorthopedic replacements; wound management aids; of the economy and society. There are the followingand packaging materials for biomedical and hygienic main technologies to transform matter into engineereduses. materials [3.9]: For the application of biomaterials the understand-ing of the interactions between synthetic substrates and • Machining, i. e. shaping, cutting, drilling, etc. ofbiological tissues are of crucial importance to meet the solids,needs of clinical requirements. However, medical and • Net forming of suitable matter, e.g. liquids, moulds,clinical aspects of biomaterials are not treated in this • Nanotechnology assembly of atoms or molecules.Handbook. In addition to these methods, there are also fur-3.1.3 Scale of Materials ther technologies, like surfacing and joining, which are applied to process, shape and assemble mater-The geometric length scale of materials has more than ials and products. The design of materials may alsotwelve orders of magnitude. The scale ranges from be supported by computational methods [3.10]. Itthe nanometer dimensions of quantum-well structures – has been estimated that there are at least 1000 dif-with novel application potentials for advanced commu- ferent ways to produce materials [3.3]. Figure 3.5nications technologies – to the kilometer-long. structures lists some of the families of processing materialsof bridges for public transport, pipelines and oil-drilling together with examples of classes, members, andplatforms for the energy supply of society. Accordingly, attributes.
  6. 6. 100 Part A The Materials Measurement System Raw materials Engineering materials % ores % metals % natural substances % ceramics % coal % polymers % chemicals % structural materials % oil % functional materials Recycling RecyclingPart A 3.1 Technical Products The earth % scrap Deposition % waste Performance % refuse Fig. 3.7 The materials cycle 3.1.5 Properties of Materials 3.1.6 Application of Materials According to their properties, materials can be broadly For the application of materials, their quality, safety and classified into the following groups [3.11]: reliability as constituents of products and engineered components and systems are of special importance. This • Structural materials: engineered materials with spe- adds performance attributes to the characteristics to be cific mechanical or thermal properties determined by materials measurement and testing. In • Functional materials: engineered materials with this context the materials cycle must be considered. specific electrical, magnetic or optical properties Figure 3.7 illustrates that all materials (accompa- • Smart materials: engineered materials with intrin- nied by the necessary flow of energy and information) sic or embedded “sensors” and “actuators” which move in cycles through the techno-economic system: are able to react in response to external loading, from raw materials to engineering materials and techni- aiming at optimising the materials’ behaviour ac- cal products, and finally, after the termination of their cording to given requirements for the materials task and performance, to deposition or recycling. From performance [3.12]. the materials cycle, which applies to all branches of tech- nology, it is obvious that materials and their properties – It must be emphasized that the characteristics of to be determined through measurement and testing – are engineered structural, functional, and smart materials of crucial importance for the performance of technical depend essentially on their processing and manufac- products. This is illustrated in Table 3.1 for some exam- ture, as illustrated in a highly simplified manner in ples of products and technical systems from the energy Fig. 3.6. sector [3.13].
  7. 7. Materials and Their Characteristics: Overview 3.2 Classification of Materials Characterization Methods 101Table 3.1 Application examples of materials in energy systems and relevant materials properties [3.13] Application Materials properties Mechanical Thermal Electrical Magnetic Optical Heat engine High-temperature strength Electricity generator High-temperature strength Nuclear pressure vessel Resistance to crack growth Solar energy Heat absorption Photoelectricity Reflectance Part A 3.2 Superconductor Ductility; strength High current Magnetic capacity quenching Conservation Light weight; Thermal insulation; Semiconductivity Magnetic Low strength high-temperature efficiency transmission resistance loss3.2 Classification of Materials Characterization MethodsFrom a realization concerning the application of all are described in detail in the following parts of thismaterial, a classification of materials characterization book:methods can be outlined in a simplified manner: Whenever a material is being created, developed, • Methods to analyze the composition and structure of materials with respect to chemical composition,or produced the properties or phenomena the mater- nanoscopic architecture and microstructure, surfacesial exhibits are of central concern. Experience shows and interfaces are compiled in Part B .that the properties and performance associated witha material are intimately related to its composition • Methods to measure the mechanical, thermal, elec- trical, magnetic and optical material properties areand structure at all levels, including which atoms are described in Part C .present and how the atoms are arranged in the mater-ial, and that this structure is the result of synthesis, • Methods of testing material performance through the determination of mechanisms which are detrimentalprocessing and manufacture. The final material must to materials integrity, like corrosion, wear, biode-perform a given task and must do so in an eco- terioration, materials-environment interactions, arenomical and socially acceptable manner. These main outlined in Part D , which also contains the de-elements: scription of methods for performance control and• composition and structure, condition monitoring.• properties, • Methods of modelling and simulation by mathemat-• performance ical and computational approaches – ranging from Molecular Dynamics Modelling to Monte Carlo sim-and the interrelationship among them define the main ulation – are described in Part E .categories of materials characterization methods to beapplied to these elements, see Fig. 3.8. Supporting the presentation of the materials characteri- Figure 3.8 illustrates that the materials charac- zation methods, in the Appendix relevant Internationalterization methods comprise analysis, measurement, Standards of Materials Measurement Methods are com-testing, modelling, and simulation. These methods piled.
  8. 8. 102 Part A The Materials Measurement System Composition, Structure Properties % Chemistry % Mechanical % Microstructure Analysis % Thermal % Surfaces and Measurement % Electrical interfaces Testing % Magnetic Modelling % Optical SimulationPart A 3 Performance Materials failure mechanisms: % corrosion % friction and wear % biogenic impact % materials-environment interactions And performance control by condition monitoring methods: % non-destructive evaluation % lifetime predictions % characterization of safety and reliability Fig. 3.8 Categories of materials characterization methods References 3.1 BIPM: International Vocabulary of Basic and Gen- 3.6 Springer Handbook of Nanotechnology, ed. by eral Terms in Metrology (Bureau International Poids B. Bhushan (Springer, Berlin, Heidelberg 2004) Mesures, Paris 1993) 3.7 S. D. Senturia: Microsystem Design (Kluwer, Boston 3.2 H. Czichos, W. Daum: Measurement methods and 2001) sensors. In: Dubbel Taschenbuch für den Maschi- 3.8 Dubbel Taschenbuch für den Maschinenbau, ed. by nenbau, ed. by W. Beitz, K.-H. Grote (Springer, W. Beitz, K.-H. Grote (Springer, Berlin, Heidelberg Berlin, Heidelberg 2004) (in German) 2004) 3.3 M. F. Ashby, Y. J. M. Brechet, D. Cebon, L. Salvo: Se- 3.9 M. P. Groover: Fundamentals of Modern Manufac- lection strategies for materials and processes, Mater. turing (Wiley, New York 2002) Design 25, 51–67 (2004) 3.10 Computational Materials Design, ed. by T. Saito 3.4 Encyclopedia of Materials: Science and Technology, (Springer, Berlin, Heidelberg 1999) ed. by K. H. J. Buschow, R. W. Cahn, M. C. Flem- 3.11 N. A. Waterman, M. F. Ashby: The Materials Selector, ings, B. Ilschner, E. J. Kramer, S. Mahajan (Elsevier, 2nd edn. (Chapman, London 1996) Amsterdam 2001) 3.12 M. Schwartz: Encyclopedia of Smart Materials (Wiley, 3.5 H. Czichos (Ed.): Materials. In: HÜTTE Das Inge- New York 2002) nieurwissen (Springer, Berlin, Heidelberg 2004) (in 3.13 Britannica Editors: Materials. In: Encyclopedia Bri- German) tannica, 2001 edn. (Britannica, Chicago 2001)
  9. 9. 17 Materials Scie 2. Materials Science for the Experimental MechanistCraig S. Hartley Part A 2 2.1 Structure of Materials ........................... 17 This chapter presents selected principles of ma- 2.1.1 Atomic Bonding ........................... 18 terials science and engineering relevant to the 2.1.2 Classification of Materials .............. 21 interpretation of structure–property relationships. 2.1.3 Atomic Order ............................... 22 Following a brief introduction, the first section 2.1.4 Equilibrium and Kinetics ............... 28 describes the atomic basis for the description of 2.1.5 Observation and Characterization structure at various size levels. Types of atomic of Structure ................................. 31 bonds form a basis for a classification scheme of materials as well as for the distinction between 2.2 Properties of Materials .......................... 33 amorphous and crystalline materials. Crystal struc- 2.2.1 The Continuum Approximation ...... 34 tures of elements and compounds are described. 2.2.2 Equilibrium Properties .................. 35 The second section presents the thermodynamic 2.2.3 Dissipative Properties ................... 38 and kinetic basis for the formation of microstruc- 2.2.4 Transport Properties of Materials .... 43 tures and describes the use of phase diagrams 2.2.5 Measurement Principles for determining the nature and quantity of equi- for Material Properties .................. 46 librium phases present in materials. Principal methods for the observation and determination of References .................................................. 47 structure are described. The structural foundations for phenomenological descriptions of equilibrium, dissipative, and transport properties are described. these properties. In conclusion the chapter The chapter includes examples of the relation- presents several useful principles for experimen- ships among physical phenomena responsible for tal mechanists to consider when measuring and various mechanical properties and the values of applying values of material properties.2.1 Structure of MaterialsEngineering components consist of materials having components affect both the choice of experimental tech-properties that enable the items to perform the func- niques and the interpretation of results. In measuringtions for which they are designed. Measurements of static behavior, it is important to know whether relevantthe behavior of engineering components under various properties of the constituent materials are independentconditions of service are major objectives of experimen- of time. Similarly, measurements of dynamic behav-tal mechanics. Validation and verification of analytical ior require information on the dynamic and dissipativemodels used in design require such measurements. properties of the materials. At best, the fundamental na-All models employ mathematical relationships that re- ture of materials, which is the ultimate determinant ofquire knowledge of the behavior of materials under their behavior, forms the basis of these models. Thea variety of conditions. Assumptions such as isotropy, extent to which such assumptions represent the actualhomogeneity, and uniformity of materials affect both physical situation limits the accuracy and significanceanalytical calculations and the interpretation of experi- of results.mental results. Regardless of the scale or purpose of the The primary axiom of materials science and engi-measurements, properties of materials that comprise the neering states that the properties and performance of
  10. 10. 18 Part A Solid Mechanics Topics a material depend on its structure at one or more lev- came widespread among scientists in the 19th and 20th els, which in turn is determined by the composition century. Atomic theory of matter led to the discovery and the processing, or thermomechanical history of the of primitive units of matter known as electrons, pro- material. The meaning of structure as employed in ma- tons, and neutrons and laws that govern their behavior. terials science and engineering depends on the scale of Although discoveries through research in high-energy reference. Atomic structure refers to the number and ar- physics constantly reveal more detail about the struc- rangement of the electrons, protons, and neutrons that ture of the atom, the planetary model proposed in 1915Part A 2.1 compose each type of atom in a material. Nanostructure by Niels Bohr, with some modifications due to later dis- refers to the arrangement of atoms over distances of the coveries of quantum mechanics, suffices to explain most order of 10−9 m. Analysis of the scattering of electrons, of the important aspects of engineering materials. In this neutrons, or x-rays is the principal tool for measure- model, atoms consist of a nucleus, containing protons, ments of structure at this scale. Microstructure refers to which have a positive electrical charge, and an approx- the spatial arrangement of groups of similarly oriented imately equal number of electrically neutral neutrons, atoms as viewed by an optical or electron microscope each of which has nearly the same mass as a proton. at resolutions in the range 10−6 –10−3 m. Macrostruc- Surrounding this nucleus is an assembly of electrons, ture refers to arrangements of groups of microstructural which are highly mobile regions of concentrated nega- features in the range of 10−3 m or greater, which can tive charge each having substantially smaller mass than be viewed by the unaided eye or under low-power a proton or neutron. The number of electrons is equal to optical magnification. Structure-insensitive properties, the number of protons in the nucleus, so each atom is such as density and melting point, depend principally electrically neutral. on composition, or the relative number and types of Elements differ from one another primarily through atoms present in a material. Structure-sensitive proper- the atomic number, or number of protons in the nu- ties, such as yield strength, depend on both composition cleus. However, many elements form isotopes, which and structure, principally at the microscale. are atoms having identical atomic numbers but different This survey will acquaint the experimental mecha- numbers of neutrons. If the number of neutrons differs nist with some important concepts of materials science excessively from the number of protons, the isotope is and engineering in order to provide a basis for in- unstable and either decays by the emission of neutrons formed selections and interpretations of experiments. and electromagnetic radiation to form a more stable The chapter consists of a description of the princi- isotope or fissions, emitting electromagnetic radiation, pal factors that determine the structure of materials, neutrons, and assemblies of protons and neutrons that including techniques for quantitative measurements of form nuclei of other elements. structure, followed by a phenomenological description The Periodic Table, shown in Fig. 2.1, classifies of representative material properties with selected ex- elements based on increasing atomic number and a pe- amples of physically based models of the properties. riodic grouping of elements having similar chemical A brief statement of some principles of measurement characteristics. The manner in which elements inter- that acknowledge the influence of material structure on act chemically varies periodically depending on the properties concludes the chapter. Additional informa- energy distribution of electrons in the atom. The ba- tion on many of the topics covered in the first two sis for this grouping is the manner in which additional sections can be found in several standard introductory electrons join the atom as the atomic numbers of the el- texts on materials science and engineering for engi- ements increase. Quantum-mechanical laws that govern neers [2.1–4]. Since this introduction can only briefly the behavior of electrons require that they reside in the survey the complex field of structure–property relation- vicinity of the nucleus in discrete spatial regions called ships, each section includes additional representative orbitals. Each orbital corresponds to a specific energy references on specific topics. state for electrons and is capable of accommodating two electrons. Electron orbitals can have a variety of spatial 2.1.1 Atomic Bonding orientations, which gives a characteristic symmetry to the atom. Four quantum numbers, arising from solutions The Periodic Table to the Schrödinger wave equation, governs the behavior The realization that all matter is composed of a fi- of the electrons: the principal quantum number n, which nite number of elements, each consisting of atoms with can have any integer value from 1 to infinity; the az- a characteristic arrangement of elementary particles, be- imuthal quantum number , which can have any integer
  11. 11. Materials Science for the Experimental Mechanist 2.1 Structure of Materials 19value from 0 to (n − 1); the magnetic quantum number the periodic table, have nearly full orbitals and tend tom , which can have any integer value between − and interact with other atoms by accepting electrons to form+ ; and the spin quantum number m s which has val- a negatively charged entity called an anion. The neg-ues ±1/2. The Pauli exclusion principle states that no ative charge arises since electrons join the originallytwo electrons in a system can have the same four quan- neutral atom. Electropositive elements occupy columnstum numbers. As the number of electrons increases with towards the left on the periodic table and ionize byincreasing atomic number, orbitals are filled beginning yielding electrons from their outer orbitals to form pos- Part A 2.1with those having the lowest electron energy states and itively charged cations.proceeding to the higher energy states. Broadly speaking, elements are metals, metalloids, Elements with electrons in full, stable orbitals are and nonmetals. The classification proceeds from thechemically inert gases, which occupy the extreme right most electropositive elements on the left of the peri-column of the periodic table (group 8). Electronegative odic table to the most electronegative elements on theelements, which occupy columns towards the right of right. A metal is a pure element. A metal that incorpo- 1A 8A 1 2 H H 1s1 1s2 hydrogen helium 1.008 2A 3A 4A 5A 6A 7A 4.003 3 4 5 6 7 8 9 10 Li Be B C N O F Ne [He]2s1 [He]2s2 [He]2s22p1 [He]2s22p2 [He]2s22p3 [He]2s22p4 [He]2s22p5 [He]2s22p6 lithium beryllium boron carbon nitrogen oxygen fluorine neon 6.941 9.012 10.81 12.01 14.01 16.00 19.00 20.18 11 12 13 14 16 16 17 18 Na Mg Al Si P S Cl Ar [Ne]3s1 [Ne]3s2 [Ne]3s23p1 [Ne]3s23p2 [Ne]3s23p3 [Ne]3s23p4 [Ne]3s23p5 [Ne]3s23p6 sodium magnesium aluminum silicon phosphorus sulfur chlorine argon 22.99 24.31 3B 4B 5B 6B 7B 8B 11B 12B 26.98 28.09 30.97 32.07 35.45 39.95 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr [Ar]4s1 [Ar]4s2 [Ar]4s23d1 [Ar]4s23d2 [Ar]4s23d3 [Ar]4s13d5 [Ar]4s23d5 [Ar]4s23d6 [Ar]4s23d7 [Ar]4s23d8 [Ar]4s13d10 [Ar]4s23d10 [Ar]4s23d104p1 [Ar]4s23d104p2 [Ar]4s23d104p3 [Ar]4s23d104p4 [Ar]4s23d104p5 [Ar]4s23d104p6 pollassium calcium scandium titanium vanadium chromium manganese iron cobalt nickel copper zinc gallium germanium arsenic selenium bromine krypton 39.10 40.08 44.96 47.88 50.94 52.00 55.94 55.85 58.93 58.69 63.55 65.39 69.72 72.58 74.92 78.96 79.90 83.80 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 52 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe [Kr]5s1 [Kr]5s2 [Kr]5s24d1 [Kr]5s24d2 [Kr]5s14d4 [Kr]5s14d5 [Kr]5s24d5 [Kr]5s14d7 [Kr]5s14d8 [Kr]4d10 [Kr]5s14d10 [Kr]5s24d10 [Kr]5s24d105p1 [Kr]5s24d105p2 [Kr]5s24d105p3 [Kr]5s24d105p4 [Kr]5s24d105p5 [Kr]5s24d105p6 nubidium strontium yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon 85.47 87.62 88.91 91.22 92.91 95.94 (98) 101.1 102.9 106.4 107.9 112.4 114.8 118.7 121.8 127.6 126.9 131.3 55 57 57 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 Cs Ba La* Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn [Xe]6s 1 [Xe]6s2 [Xe]6s25d1 [Xe]6s24f145d2 [Xe]6s24f145d3 [Xe]6s24f145d4 [Xe]6s24f145d5 [Xe]6s24f145d6 [Xe]6s24f145d7 [Xe]6s14f145d9 [Xe]6s14f145d10 [Xe]6s24f145d10 [Xe]6s24f145d106p1 [Xe]6s24f145d106p2 [Xe]6s24f145d106p3 [Xe]6s24f145d106p4 [Xe]6s24f145d106p5 [Xe]6s24f145d106p6 casium barium lanthanum hafnium tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon 132.9 137.3 138.9 178.5 180.9 183.9 186.2 190.2 190.2 195.1 197.0 200.5 204.4 207.2 208.9 (209) (210) (222) 87 88 89 104 105 106 107 108 109 110 111 112 114 116 118 Fr Ra Ac ~ Rf Db Sg Bh Hs Mt Ds Uuu Uub Uuq Uuh Uuo [Rn] 7s1 [Rn]7s2 [Rn]7s26d1 [Rn]7s25f146d2 [Rn]7s25f146d3 [Rn]7s25f146d4 [Rn]7s25f146d5 [Rn]7s25f146d6 [Rn]7s25f146d7 [Rn]7s15f146d9 francium radium actinium rutherfordium dubnium seaborgium bohrium hassium meitnerium darmstadtium (223) (226) (227) (257) (260) (263) (262) (265) (266) (271) (272) (277) (296) (298) (?) 58 59 60 61 62 63 64 65 66 67 68 69 70 71 Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Lanthanide series* [Xe]6s24f15d1 [Xe]6s24f3 [Xe]6s24f4 [Xe]6s24f5 [Xe]6s24f6 [Xe]6s24f7 [Xe]6s24f75d1 [Xe]6s24f9 [Xe]6s24f10 [Xe]6s24f11 [Xe]6s24f12 [Xe]6s24f13 [Xe]6s24f14 [Xe]6s24f145d1 cerium praseodymium neodymium promethium samarium europium gadolinium terbium dysprosium holmium erbium thulium ytterbium lutetium 140.1 140.9 144.2 (147) (150.4) 152.0 157.3 158.9 162.5 164.9 167.3 168.9 173.0 175.0 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Actinide series ~ Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr [Rn]7s26d2 [Rn]7s25f26d1 [Rn]7s25f36d1 [Rn]7s25f46d1 [Rn]7s25f6 [Rn]7s25f7 [Rn]7s25f76d1 [Rn]7s25f9 [Rn]7s25f10 [Xe]6s24f11 [Rn]7s25f12 [Rn]7s25f13 [Rn]7s25f14 [Rn]7s25f146d1 thorium protactinium uranium neptunium plutonium americium curium berkelium californium einsteinium fermium mendelevium nobelium lawrencium 232.0 (231) (238) (237) (242) (43) (247) (247) (249) (254) (253) (256) (254) (257) Liquids at room temperature Gases at room temperature Solids at room temperatureFig. 2.1 The Periodic Table of the elements. Elements named in blue are liquids at room temperature. Elements named in red aregases at room temperature. Elements named in black are solids at room temperature
  12. 12. 20 Part A Solid Mechanics Topics rates atoms of other elements into its structure without are strongly directional since the orbitals involved are changing its essential metallic character forms an al- typically nonspherical. In both ionic and covalent bonds loy, which is not a metal since it is not a pure element. nearest-neighbor ions are most strongly involved and The major differences in materials have their origins in the valence electrons are highly localized. the nature of the bonds formed between atoms, which Metallic bonds occur in strongly electropositive are determined by the manner in which electrons in the elements, which surrender their valence electrons to highest-energy orbitals interact with one another and by form a negatively charged electron gas or distributionPart A 2.1 whether the centers of positive and negative charge of of highly nonlocalized electrons that moves relatively the atoms coincide. The work required to remove an freely throughout the substance. The positively charged ion from the substance in which it resides is a measure ions repel one another but remain relatively stationary of the strength of these bonds. At suitable tempera- because the electron gas acting as glue holds them to- tures and pressures, all elements can exist in all states gether. Metallic bonds are relatively nondirectional and of matter, although in some cases this is very difficult the ions are approximately spherical. A major difference to achieve experimentally. At ambient temperature and between the metallic bond and the ionic and covalent pressure, most elements are solids, some are gases, and bonds is that it does not involve an exchange or sharing a few are liquid. of electrons with nearest neighbors. The bonds in many substances closely approximate Primary Bonds the pure bond types described above. However, mix- Primary bonds are the strongest bonds that form among tures of these archetypes occur frequently in nature, atoms. The manner in which electrons in the highest and a substance can show bonding characteristics that energy levels interact produces differences in the kinds resemble more than one type. This hybrid bond situa- of primary bonds. Valence electrons occupy the highest tion occurs most often in substances that exhibit some energy levels of atoms, called the valence levels. Va- characteristics of directional covalent bonds along with lence electrons exhibit three basic types of behavior: nondirectional metallic or ionic bonds. atoms of electropositive elements yield their valence electrons relatively easily; atoms of electronegative el- Secondary Bonds ements readily accept electrons to fill their valence Some substances are composed of electrically neutral levels; and elements between these extremes can share clusters of ions called molecules. Secondary bonds ex- electrons with neighboring atoms. The valence of an ion ist between molecules and are weaker than primary is the number of electrons yielded, accepted or shared bonds. One type of secondary bond, the van der Waals by each atom in forming the ion. Valence is positive bond, is due to the weak electrostatic interaction be- or negative according to whether the ion has a positive tween molecules in which the instantaneous centers of (cation) or negative (anion) charge. positive and negative charge do not coincide. A mol- The behavior of valence electrons gives rise to three ecule consisting of a single ion of an electropositive types of primary bonds: ionic, covalent and metallic. element and a single ion of an electronegative element, Ionic bonds occur between ions of strongly electropos- such as a molecule of HCl gas, is a simple example itive elements and strongly electronegative elements. of a diatomic molecule. The center of negative charge Each atom of the electropositive element surrenders one of the system coincides with the nucleus of the chlo- or more electrons to one or more atoms of the elec- rine ion, but the center of positive charge is displaced tronegative element to form oppositely charged ions, from the center of the ion because of the presence of the which attract one another by the Coulomb force be- smaller, positively charged hydrogen ion (a single pro- tween opposite electrical charges. This exchange of ton), which resides near the outer orbital of the chlorine electrons occurs in such a manner that the overall ion. This results in the formation of an electrical dipole, structure remains electrically neutral. To a good approx- which has a short-range attraction to similar dipoles, imation, ions involved in ionic bonds behave as charged, such as other HCl molecules, at distances of the order of essentially incompressible, spheres, which have no the molecular dimensions. However, no long-range at- characteristic directionality. In contrast, covalent bonds traction exists since the overall charge of the molecule is involve sharing of valence electrons between neigh- zero. The example given is a permanent dipole formed boring atoms. This type of bonding occurs when the by a spatial separation of centers of charge. A tempo- valence energy levels of the atoms are partially full, cor- rary dipole can occur when the instantaneous centers responding to valences in the vicinity of 4. These bonds of charge separate because of the motion of electrons.
  13. 13. Materials Science for the Experimental Mechanist 2.1 Structure of Materials 21The resulting attraction forms a weak bond at small free electron gas that permeates the lattice of ionsdistances and is typical of van der Waals bonds. causes these materials to exhibit high electrical and The other secondary bond, the hydrogen bond, in- thermal conductivity. In addition they possess rela-volves the single valence electron of hydrogen. In tively high yield strengths, high moduli of elasticity,materials science and engineering, the most important and melting points ranging from nearly room temper-type of hydrogen bond is that which occurs in polymers, ature to > 3200 K. Although generally malleable andwhich consist of long chains or networks of chemically ductile, they can exhibit extreme brittleness, depending Part A 2.1identical units called mers. When the composition of on structure and temperature. One of the most use-a mer includes hydrogen, it is possible for the hydrogen ful features of metallic materials is their ability to beatom to share its valence electron with identical mers in formed into complex shapes using a variety of thermo-neighboring chains, so that the hydrogen atom is partly mechanical processes, including melting and casting,in one chain and partly in another. This sharing of the hot working in the solid state, and a combination ofhydrogen atom creates a hydrogen bond between the cold working and annealing. All of these processes pro-chains. The bond is relatively weak but is an important duce characteristic microstructures that lead to differentfactor in the behavior of polymeric materials. combinations of physical and mechanical properties. Applications that require complex shapes having both2.1.2 Classification of Materials strength and fracture resistance with moderate resis- tance to environmental degradation employ metallicIt is useful to categorize engineering materials in terms materials.either of their functionality or the dominant type ofatomic bonding present in the material. Since most ma- Metalloidsterials perform several functions in a component, the Metalloids are elements in groups III–V of the peri-classification scheme described in the following sec- odic table and compounds formed from these elements.tions takes the latter approach. The nature and strength Covalent bonding dominates both the elements andof atomic bonding influences not only the arrangement compounds in this category. The name arises fromof atoms in space but also many physical properties the fact that they exhibit behavior intermediate be-such as electrical conductivity, thermal conductivity, tween metals and ceramics. Many are semiconductors,and damping capacity. that is, they exhibit an electrical conductivity lower Ceramics than metals, but useable, which increases rather thanCeramic materials possess bonding that is primarily decreases with temperature like metals. These mater-ionic with varying amounts of metallic or covalent char- ials exhibit high elastic moduli, relatively high meltingacter. The dominant features on the atomic scale are points, low ductility, and poor formability. Commer-the localization of electrons in the vicinity of the ions cially useful forms of these materials require processingand the relative incompressibility of atoms, leading by solidification directly from the molten state followedto structures that are characterized by the packing of by solid-state treatments that do not involve signifi-rigid spheres of various sizes. These materials typically cant deformation. Metalloids are useful in a variety ofhave high melting points (> 1500 K), low thermal and applications where sensitivity and response to electro-electrical conductivities, high resistance to atmospheric magnetic radiation are important.corrosion, and low damping capacity. Mechanical prop-erties of ceramics include high moduli of elasticity, high Polymersyield strength, high notch sensitivity, low ductility, low Polymeric materials, also generically called plastics, areimpact resistance, intermediate to low thermal shock assemblies of complex molecules consisting of molecu-resistance, and low fracture toughness. Applications lar structural units called mers that have a characteristicthat require resistance to extreme thermal, electrical or chemical composition and, often, a variety of spatialchemical environments, with the ability to absorb me- configurations. The assemblies of molecules generallychanical energy without catastrophic failure a secondary take the form of long chains of mers held together byissue, typically employ ceramics. hydrogen bonds or networks of interconnected mers. Most structural polymers are made of mers with an Metals organic basis, i. e., they contain carbon. They are char-Metallic materials include pure metals (elements) and acterized by relatively low strength, low thermal andalloys that exhibit primarily metallic bonding. The electrical conductivity, low melting points, often high
  14. 14. 22 Part A Solid Mechanics Topics ductility, and high formability by a variety of tech- terials can exhibit more than one crystalline form, called niques. These materials are popular as electrical and allotropes, depending on the temperature and pres- thermal insulators and for structural applications that sure. It is this property of iron with small amounts do not require high strength or exposure to high tem- of carbon dissolved that is the basis for the heat peratures. Their principal advantages are relatively low treatment of steel, which provides a wide range of cost, high formability, and resistance to most forms of properties. atmospheric degradation. At the other extreme of atomic arrangement arePart A 2.1 amorphous materials. These materials can exhibit lo- Composites cal order of structural units, but the arrangement a large Composite materials consist of those formed by in- number of such units is haphazard or random. There timate combinations of the other classes. Composites are two principal categories of amorphous structures: combine the advantages of two or more material classes network structures and chain structures. The molecules by forming a hybrid material that exhibits certain of network structures lie at the nodes of an irregular desirable features of the constituents. Generally, one network, like a badly constructed jungle gym. Never- type of material predominates, forming a matrix con- theless, the network has a high degree of connectivity taining a distribution of one or more other types on and if the molecules are not particularly mobile, the a microscale. A familiar example is glass-reinforced network can be very stable. This type of structure plastic (GRP), known by the commercial name of is characteristic of most glasses. Materials possessing Fibreglass R . In this material, the high elastic modulus this structure possess a relatively rigid mechanical re- of the glass fibers (a ceramic) reinforces the tough- sponse at low temperatures, but become more fluid ness and formability of the polymeric matrix. Other and deformable at elevated temperatures. Frequently the classes of composites have metal matrices with ceramic transition between the relatively rigid, low-temperature dispersions (metal matrix composites, MMC), ceramic form and the more fluid high-temperature form oc- matrices with various types of additions (ceramic matrix curs over a narrow temperature range. By convention composites, CMC), and polymeric matrices with metal- the midpoint of this transition range defines the glass- lic or ceramic additions. The latter, generically known transition temperature. as organic matrix composites (OMC) or polymer matrix Linear chain structures are characteristic of poly- composites (PMC), are important structural materials meric materials made of long chains of mers. Relatively for aerospace applications. weak hydrogen bonds and/or van der Waals bonds hold these chains together. The chains can move past one 2.1.3 Atomic Order another with varying degrees of difficulty depending on the geometry of the molecular arrangement along Crystalline and Amorphous Materials the chain and the temperature. An individual chain can The structure of materials at the atomic level can possess short-range order, but the collections of chains be highly ordered or nearly random, depending on that comprise the substance sprawl haphazardly, like the nature of the bonding and the thermomechan- a bowl of spaghetti. Under certain conditions of forma- ical history. Pure elements that exist in the solid tion, however, the chains can arrange themselves into state at ambient temperature and pressure always ex- a pattern with long-range order, giving rise to crystalline hibit at least one form that is highly ordered in the forms of polymeric materials. In addition, some ele- sense that the surroundings of each atom are identi- ments, specific to the particular polymer, can bond with cal. Crystalline materials exhibit this locally ordered adjacent chains, creating a three-dimensional network arrangement over large distances, creating long-range structure. The addition of sulfur to natural rubber in the order. The formal definition of a crystal is a sub- process called Vulcanizing R is an example. stance in which the structure surrounding each basis Materials that can exist in both the crystalline unit, an atom or molecule, is identical. That is, if and amorphous states can also have intermediate, one were able to observe the atomic or molecular metastable structures in which these states coexist. arrangement from the vantage point of a single struc- Glass that has devitrified has microscopic crystalline tural unit, the view would not depend on the location regions dispersed in an amorphous network matrix. or orientation of the structural unit within the ma- Combinations of heat treatment and mechanical defor- terial. All metals and ceramic compounds and some mation can alter the relative amounts of these structures, polymeric materials have crystalline forms. Some ma- and the overall properties of the material.
  15. 15. Materials Science for the Experimental Mechanist 2.1 Structure of Materials 23 Crystal Structures of Elements and Compounds metric compound contains ions exactly in the ratiosBecause ionic bonds require ions of at least two ele- that produce electrical neutrality of the substance. Inments, either metallic or covalent bonds join ions of binary (two-component) compounds, the ratio of thepure elements in the solid state, although the con- number of ions of each kind present is the inversedensed forms of highly electronegative elements and of the ratio of the absolute values of their valences.the inert gases exhibit weak short-range bonding typ- For example, Na2 O has two sodium atoms for eachical of van der Waals bonds. Chemical compounds, oxygen atom. Since the valence of sodium is +1 Part A 2.1which can exhibit ionic bonding as well as the other and that of oxygen is −2, the 2 : 1 ratio of sodiumtypes of strong bonds, form when atoms of two or to oxygen ions produces electrical neutrality of themore elements combine in specific ratios. A stoichio- structure. Cubic Simple Body-centered Face-centered Tetragonal Monoclinic Simple Body-centered Simple End-centered Orthohombic Simple Body-centered Face-centered End-centered Rhombohedral Hexagonal TriclinicFig. 2.2 Bravais lattices and crystal systems
  16. 16. 24 Part A Solid Mechanics Topics In the solid state, patterns of atoms and molecules three integers having no common factor that are in the form lattices, which are three-dimensional arrays of same ratio as the direction cosines, relative to the co- points having the property that the surroundings of each ordinate axes, of such a vector characterizes the lattice lattice point are identical to those of any other lattice direction. Square brackets, e.g., [100], denote specific point. There are only 14 unique lattices, the Bravais crystallographic directions, while the same three inte- lattices, shown in Fig. 2.2. Each lattice possesses three gers enclosed by carats, e.g., 100 , describe families of non-coplanar, non-collinear axes and a characteristic, directions. Directions are crystallographically equiva-Part A 2.1 unique array of lattice points occupied by structural lent if they possess an identical arrangement of lattice units, which can be individual atoms or identical clus- points. Families of directions in the cubic crystal system ters of atoms, depending on the nature of the substance. are crystallographically equivalent, but those in noncu- The relative lengths of the repeat distance of lattice bic crystals may not be because of differences in the points along each axis and the angles that the axes make lattice parameters. with one another define the seven crystal systems. Fig- The Miller indices, another set of three integers de- ure 2.2 also shows the crystal system for each of the termined in a different manner, specify crystallographic Bravais lattices. planes. The notation arose from the observation by 19th Each of the illustrations in Fig. 2.2 represents the century crystallographers on naturally occurring crys- unit cell for the lattice, which is the smallest arrange- tals that the reciprocals of the intercepts of crystal faces ment of lattice points that possesses the geometric with the principal crystallographic axes occurred in the characteristics of the extended structure. Repeating one ratios of small, whole numbers. To determine the Miller of the figures in Fig. 2.2 indefinitely throughout space indices of a plane, first obtain the intercepts of the plane with an appropriately chosen structural unit at each lat- with each of the principal crystallographic directions. tice point defines a crystal structure. Lattice parameters Then take the reciprocals of these intercepts and find include the angles between coordinate axes, if variable, the three smallest integers with no common factor that and the dimensions of the unit cell, which contains have the same ratios to one another as the reciprocals one or more lattice points. To determine the number of of the intercepts. Enclosed in parentheses, these are the points associated with a unit cell, count 1/8 for each Miller indices of the plane. For example, the (120) plane corner point, 1/2 for each point on a face, and 1 for has intercepts of 1, 1/2, and ∞, in units of the lattice each point entirely within the cell. A primitive unit cell parameters, along the three principal crystallographic contains only one lattice point (one at each corner). directions. Families of planes are those having the same The coordination number Z is the number of nearest three integers in different permutations, including neg- neighbors to a lattice point. atives, as their Miller indices. Braces enclose the Miller One of the most important characteristics of crys- indices of families, e.g., {120}. Crystallographically tal lattices is symmetry, the property by which certain equivalent planes have the same density and distribution rigid-body motions bring the lattice into an equivalent of lattice points. In cubic crystals, families of planes are configuration indistinguishable from the initial config- crystallographically equivalent. uration. Symmetry operations occur by rotations about Although there are examples of all of the crystal an axis, reflections across a plane or a combination of structures in naturally occurring materials, a relatively rotations, and translations along an axis. For example, few suffice to describe common engineering materials. a plane across which the structure is a mirror image All metals are either body-centered cubic (bcc), face- of that on the opposite side is a mirror plane. An axis centered cubic (fcc) or hexagonal close-packed (hcp). about which a rotation of 2π/n brings the lattice into The latter two structures consist of different stacking coincidence forms an n-fold axis of symmetry. This sequences of closely packed planes containing identical characteristic of crystals has profound implications on spheres or ellipsoids, representing the positive ions in certain physical properties. the metallic lattice. Figure 2.3 shows a plane of spheres The geometry of the lattice provides a natural co- packed as closely as possible in a plane. ordinate system for describing directions and planes An identical plane fitted as compactly as possible using the axes of the unit cell as coordinate axes and on top of or below this plane occupies one of two possi- the lattice parameters as units of measure. Principal ble locations, corresponding to the depressions between crystallographic axes and directions are those paral- the spheres. These locations correspond to the upright lel to the edges of the unit cell. A vector connecting and inverted triangular spaces between spheres in the two lattice points defines a lattice direction. A set of figure. The same option exists when placing a third