Physics
Crystal Structure
Crystal structure refers to the arrangement of atoms or molecules in a crystalline material. It is characterized by a repeating, three-dimensional pattern that gives crystals their unique shape and properties. The arrangement of these building blocks determines the crystal's symmetry, density, and other physical properties.
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12 Key excerpts on "Crystal Structure"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
___________________________ WORLD TECHNOLOGIES ________________________ Chapter- 4 Crystal Structure In mineralogy and crystallography, Crystal Structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A Crystal Structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry. Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. The symmetry properties of the crystal are embodied in its space group. A crystal's structure and symmetry play a role in determining many of its physical properties, such as cleavage, electronic band structure, and optical transparency. Insulin crystals Unit cell The Crystal Structure of a material or the arrangement of atoms within a given type of Crystal Structure can be described in terms of its unit cell. The unit cell is a small box containing one or more atoms, a spatial arrangement of atoms. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. The Crystal Structure has a three dimensional shape. The unit cell is given by its lattice ___________________________ WORLD TECHNOLOGIES ________________________ parameters which are the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions ( x i , y i , z i ) measured from a lattice point. Simple cubic (P) Body-centered cubic (I) Face-centered cubic (F) Miller indices Planes with different Miller indices in cubic crystals
eBook - PDFStructure of Materials
An Introduction to Crystallography, Diffraction and Symmetry
- Marc De Graef, Michael E. McHenry(Authors)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
3 What is a Crystal Structure? In mathematics, if a pattern occurs, we can go on to ask, Why does it occur? What does it signify? And we can find answers to these questions. In fact, for every pattern that appears, a mathematician feels he ought to know why it appears. W. W. Sawyer, mathematician At the atomic length scale, most solids can be described as regular arrangements of atoms. In this chapter we take a closer look at the framework that underlies such periodic arrange-ments: the “space lattice.” We will introduce the standard nomenclature to describe lattices in both 2-D and 3-D, as well as some mathematical tools (mostly based on vectors) that are used to provide unambiguous definitions. Then we will answer the question: how many uniquely different lattices are there? This will lead to the concepts of crystal systems and Bravais lattices. We will explore a few other ways to describe the lattice periodicity, and we conclude this chapter with a description of magnetic time-reversal symmetry, and how the presence of magnetic moments complicates the enumeration of all the space lattices. 3.1 Periodic arrangements of atoms • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • In this section, we will analyze the various components that make up a Crystal Structure . We will proceed in a rather pragmatic way, and begin with a loose “definition” of a Crystal Structure that most of us could agree on: a Crystal Structure is a regular arrangement of atoms or molecules. We have some idea of what atoms and molecules are – at least, we think we do . . . And we also have some understanding of the words “regular arrangement.” The word “reg-ular” could imply the existence of something that repeats itself, whereas “arrangement” would imply the presence of a pattern .
eBook - PDFMaterials Science and Engineering
An Introduction
- William D. Callister, Jr., David G. Rethwisch(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
For crystalline solids, the notion of Crystal Structure is presented, specified in terms of a unit cell. The three com- mon Crystal Structures found in metals are then detailed, along with the scheme by which crystallographic points, directions, and planes are expressed. Single crystals, polycrys- talline materials, and noncrystalline materials are considered. Another section of this chapter briefly describes how Crystal Structures are determined experimentally using x-ray diffraction techniques. 3.1 INTRODUCTION Solid materials may be classified according to the regularity with which atoms or ions are arranged with respect to one another. A crystalline material is one in which the atoms are situated in a repeating or periodic array over large atomic distances—that is, long-range order exists, such that upon solidification, the atoms will position themselves crystalline 3.2 FUNDAMENTAL CONCEPTS Crystal Structures The properties of some materials are directly related to their Crystal Structures. For example, pure and undeformed magnesium and beryllium, having one Crystal Structure, are much more brittle (i.e., fracture at lower degrees of deformation) than are pure and undeformed metals such as gold and silver that have yet another Crystal Structure (see Section 7.4). Furthermore, significant property differences exist between crystalline and noncrystalline materials having the same composition. For example, noncrystalline ceramics and polymers normally are optically transparent; the same materials in crystalline (or semicrystalline) form tend to be opaque or, at best, translucent. 50 • Chapter 3 / The Structure of Crystalline Solids in a repetitive three-dimensional pattern, in which each atom is bonded to its nearest- neighbor atoms. All metals, many ceramic materials, and certain polymers form crystal- line structures under normal solidification conditions.
- William D. Callister, Jr., David G. Rethwisch(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
For crystalline solids, the notion of Crystal Structure is presented, specified in terms of a unit cell. The three com- mon Crystal Structures found in metals are then detailed, along with the scheme by which crystallographic points, directions, and planes are expressed. Single crystals, polycrys- talline materials, and noncrystalline materials are considered. Another section of this chapter briefly describes how Crystal Structures are determined experimentally using x-ray diffraction techniques. 3.1 INTRODUCTION Solid materials may be classified according to the regularity with which atoms or ions are arranged with respect to one another. A crystalline material is one in which the atoms are situated in a repeating or periodic array over large atomic distances—that is, long-range order exists, such that upon solidification, the atoms will position themselves crystalline 3.2 FUNDAMENTAL CONCEPTS Crystal Structures The properties of some materials are directly related to their Crystal Structures. For example, pure and undeformed magnesium and beryllium, having one Crystal Structure, are much more brittle (i.e., fracture at lower degrees of deformation) than are pure and undeformed metals such as gold and silver that have yet another Crystal Structure (see Section 7.4). Furthermore, significant property differences exist between crystalline and noncrystalline materials having the same composition. For example, noncrystalline ceramics and polymers normally are optically transparent; the same materials in crystalline (or semicrystalline) form tend to be opaque or, at best, translucent. 3.3 Unit Cells • 51 in a repetitive three-dimensional pattern, in which each atom is bonded to its nearest- neighbor atoms. All metals, many ceramic materials, and certain polymers form crystal- line structures under normal solidification conditions.
eBook - PDF- Jean-claude Toledano(Author)
- 2011(Publication Date)
- World Scientific(Publisher)
They therefore reveal, consistently with the above definition, the average atomic positions. 2 The main scientists who have analyzed these regularities are the French Ha¨ uy and Bravais, the Germans Hermann and Schoenflies, and the Russian Fedorov. 13 14 The structure of crystalline solids in the study of the mechanical properties of solids. The present chapter is devoted to the description of these regularities which are also termed symmetries . 2.2 Crystal geometry 2.2.1 Ideal crystal The structure of a solid is defined as the configuration in space of its con-stituting atoms. Figure 2.1 Starting with a “basis” of atoms (here a group of three atoms), a crystal is constructed by placing an identical basis at the end of each vector of the form ( n 1 − → a 1 + n 2 − → a 2 + n 3 − → a 3 ). This configuration can be specified by the set of coordinates of the nu-clei of all the atoms. It is useful, in a first approach of description of “real solids”, to define the concept of ideal crystal . Such a crystal is an infinitely extended solid, the structure of which possesses a three-dimensional spa-tial periodicity . By this statement, one implies that the structure can be built by replicating a given set of atoms, through displacements defined by translations − → T of the form: − → T = n 1 − → a 1 + n 2 − → a 2 + n 3 − → a 3 (2.1) where the coefficients n 1 ,n 2 ,n 3 are integers (positive, negative, or equal to zero), and where the three vectors − → a 1 , − → a 2 , − → a 3 are three non-coplanar vectors. These vectors define the periodicity of the crystal. They are its primitive (or elementary) translations . The set of atoms, the replication of which generates the crystal, is the basis of the structure. For many crystals of common use, such as metals, the basis contains a single atom or a few atoms.
- James A. Newell(Author)
- 2012(Publication Date)
- Wiley(Publisher)
Understanding material structures provides the gateway to understanding both the material properties that these structures spawn and the processing procedures that can be used to alter the structures and, as a result, the properties of the material. Figure 2-1 provides a graphical representation of the unbreakable interrelationship among structure, properties, and processing. The development of structure of materials provides a perfect point of entry into the broader realm of materials science and engineering. The properties of any material are determined by its structure at four distinct levels: 1. Atomic structure. What atoms are present and what properties do they possess? 2. Atomic arrangement. How are the atoms positioned relative to each other, and what type of bonding, if any, exists between them? 3. Microstructure. What sequencing of crystals exists at a level too small to be seen with the eye? 4. Macrostructure. How do the microstructures fit together to make the larger material? Table 2-1 shows how these levels of order apply to salt crystals. The prop- erties of a material are determined by the combined effects of all four levels and can be altered using a variety of processing techniques. This chapter focuses on the development of structure in crystalline materials and includes a relatively detailed examination of X-ray diffraction. This examination serves as a tool to clarify the real meaning and relevance of the crystallographic terms in the chapter. 2.2 LEVELS OF ORDER M ost of the materials in this text possess significant order, but that is not true for all materials. The lowest level of order involves mon- atomic gas molecules randomly filling space, which have limited relevance in the study of materials science. Instead, most materials have at least some short-range order. Water molecules, shown in Figure 2-2, pro- vide a classic example of such short-range order.
eBook - PDFThe Crystal Lattice
Phonons, Solitons, Dislocations, Superlattices
- Arnold M. Kosevich(Author)
- 2006(Publication Date)
- Wiley-VCH(Publisher)
To do this efficiently, the fundamental properties of the simplest forms of solids, i. e., single crystals must be understood. Not so long ago, materials science implied the development, experimental investi- gation, and theoretical description, of primarily construction materials with given elas- tic, plastic and resistive properties. In the last few decades, however, new materials, primarily crystalline, have begun to be viewed differently: as finished, self-contained devices. This is particularly true in electronics and optics. To understand the properties of a crystal device it is not only necessary to know its structure but also the dynamics of physical processes occurring within it. For example, to describe the simplest displacement of the crystal atoms already requires a knowl- edge of the interatomic forces, which of course, entails a knowledge of the atomic positions. The dynamics of a crystal lattice is a part of the solid-state mechanics that studies intrinsic crystal motions taking into account structure. It involves classical and quan- tum mechanics of collective atomic motions in an ideal crystal, the dynamics of crystal lattice defects, a theory of the interaction of a real crystal with penetrating radiation, the description of physical mechanisms of elasticity and strength of crystal bodies. In this book new trends in dislocation theory and an introduction to the nonlinear dynamics of 1D systems, that is, soliton theory, are presented. In particular, the dis- location theory of melting of 2D crystals is briefly discussed. We also provide a new treatment of the application of crystal lattice theory to physical objects and phenomena whose investigation began only recently, that is, quantum crystals, electron crystals on a liquid-helium surface, lattices of cylindrical magnetic bubbles in thin-film ferromag- netics, and second sound in crystals.
eBook - PDFMetals and Materials
Science, Processes, Applications
- R. E. Smallman, R J Bishop(Authors)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
Furthermore, in comparing the two degrees of order-ing of Figures 2.1a and 2.1b, one can appreciate why the structures of comparatively highly-ordered crystalline substances, such as chemical compounds, minerals and metals, have tended to be more amenable to scientific investigation than glasses. 2.2 Crystal lattices and structures We can rationalize the geometry of the simple repre-sentation of a Crystal Structure shown in Figure 2.1a by adding a two-dimensional frame of reference, or space lattice, with line intersections at atom centres. Extending this process to three dimensions, we can construct a similar imaginary space lattice in which triple intersections of three families of parallel equidistant lines mark the positions of atoms (Figure 2.2a). In this simple case, three reference axes (*, y, z) are oriented at 90° to each other and atoms are 'shrunk', for convenience. The orthogonal lattice of The terms glassy, non-crystalline, vitreous and amorphous are synonymous. Figure 2.2a defines eight unit cells, each having a shared atom at every corner. It follows from our recognition of the inherent order of the lattice that we can express the geometrical characteristics of the whole crystal, containing millions of atoms, in terms of the size, shape and atomic arrangement of the unit cell, the ultimate repeat unit of structure. 2 We can assign the lengths of the three cell param-eters (Λ, b, c) to the reference axes, using an inter-nationally-accepted notation (Figure 2.2b). Thus, for the simple cubic case portrayed in Figure 2.2a, x - y - z -90°; a-b - c. Economizing in symbols, we only need to quote a single cell parameter (a) for the cubic unit cell. By systematically changing the angles (α, β, y) between the reference axes, and the cell parameters (a, b, c), and by four skewing operations, we derive the seven crystal systems (Figure 2.3). Any crystal, whether natural or synthetic, belongs to one or other of these systems.
eBook - PDFFundamentals Of Imaging, The: From Particles To Galaxies
From Particles to Galaxies
- Michael Mark Woolfson(Author)
- 2011(Publication Date)
- ICP(Publisher)
Chapter 15 Images of Atoms 15.1. The Nature of Crystals Crystalline materials are commonplace in everyday life, ranging from household chemicals such as common salt, sugar and washing soda, industrial materials such as corundum and germanium to precious stones such as emeralds and diamonds. A characteristic of crystals is that they are bounded by facets — plane surfaces of mirror per-fection. Plane surfaces are not very common in nature; they occur as the surfaces of liquids, but this is due to the influence of gravity on the liquid, a force that would not affect the form of a small rigid solid. It is easy to verify that the significance of the planar surfaces is not just confined to the surface of crystals but is also related to their internal structure. Many crystals can be cleaved along preferred directions to form new facets and even when a crystal is crushed into tiny fragments so that it seems to be a powder, examination through a microscope shows that each grain is a tiny crystal bounded by plane surfaces. Another feature of crystals is that all the crystals of a particular substance tend to look alike — all plates or all needles, for example. The overall conclusion from these characteristics of crystals is that the arrangement of atoms within a crystal governs its overall appear-ance. The flatness of crystal facets can be interpreted as due to the presence of planar layer of atoms forming a surface of low energy and hence one that is intrinsically stable. 303 304 The Fundamentals of Imaging: From Particles to Galaxies (a) (b) Figure 15.1 (a) An alum crystal. (b) A quartz crystal. 15.1.1. The shapes of crystals Many crystals are very regular in shape and show a great deal of symmetry. Figure 15.1 shows two very regular shapes, an alum crystal with the form of an octahedron (a solid with surface a symmetrical arrangement of eight equilateral triangles) and a quartz crystal with a cross-section that is a perfect hexagon.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 4 Solid-state Physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from their atomic-scale properties. Thus, solid-state physics forms the theoretical basis of materials science. It also has direct applications, for example in the technology of transistors and semiconductors. Introduction Solid materials are formed from densely-packed atoms, with intense interaction forces between them. These interactions are responsible for the mechanical (e.g. hardness and elasticity), thermal, electrical, magnetic and optical properties of solids. Depending on the material involved and the conditions in which it was formed, the atoms may be arranged in a regular, geometric pattern (crystalline solids, which include metals and ordinary water ice) or irregularly (an amorphous solid such as common window glass). The bulk of solid-state physics theory and research is focused on crystals, largely because the periodicity of atoms in a crystal — its defining characteristic — facilitates mathe-matical modeling, and also because crystalline materials often have electrical, magnetic, optical, or mechanical properties that can be exploited for engineering purposes. The forces between the atoms in a crystal can take a variety of forms. For example, in a crystal of sodium chloride (common salt), the crystal is made up of ionic sodium and chlorine, and held together with ionic bonds. In others, the atoms share electrons and form covalent bonds. In metals, electrons are shared amongst the whole crystal in metallic bonding. Finally, the noble gases do not undergo any of these types of bonding.
eBook - PDF- Bernhard Wunderlich(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
C H A P T E R I I The Microscopic Structure of Crystals 2.1 Discovery and Proof of the Lattice Theory Crystals have been treasured since prehistoric times. Their brilliance, color, and symmetry has not only fascinated people,! but also led to the belief that crystals may possess magical powers. Even to date evidence of such super-stitions can be found. The known roots of rational explanations of Crystal Structure go back to the speculations of the ancient Greeks. From the beginning, one can find elements of a space lattice hypothesis, i.e., the assumption of a regular arrange-ment of fundamental particles in crystals. The atomists asserted matter to consist of qualitatively identical, indivisible atoms which are different in size and shape, and move ceaselessly through the void. Various arrangements were to account for the macroscopic appearances and properties. The hardest material was thought to be the most compacted. Lucretius thought diamond to be composed of branchlike atoms which intertwine with one another. Plato describes stones as earth, one of the four elements, thrust together by air so as not to be soluble by water. Since earth is given cubic form by Plato, t For outstanding collections of color and black and white reproductions of crystals see for example: Deribere, M. (1956). Belles roches; beaux cristaux. Larousse, Paris (92 photographs, many in color); deMichele, V. (1969). Kristalle; ein farbenfrohes Bild geheimnisvoller und gesetzmassiger Kunstformen der Natur. Sudwestverlag, Munchen (125 color reproductions); Schrocke, H. (1967). Mineralien, 162 Farbtafeln nach Originalen von Claus Caspari. Kronen Verlag E. Cramer, Hamburg. 21 22 2.1 Discovery and Proof of the Lattice Theory stones should be aggregates of cubic particles. Aristotle saw bodies solidify either on the action of heat when composed of earth, or on the action of cold when composed of water.
No longer available |Learn moreSolid State Physics
From the Material Properties of Solids to Nanotechnologies
- David Schmool(Author)
- 2016(Publication Date)
- Mercury Learning and Information(Publisher)
This is due to the scattering of waves by the atoms in the crystal, the strength of this scattering depends in most cases on the element involved and has a specific form factor. The structure factor, which can be seen to be a Fourier transform, permits us to assess the form of the reciprocal lattice of a crystal. It has the added advantage of giving the relative intensity of the points in reciprocal space, which is related to the intensity variations observed in diffrac- tion patterns. We described some of the more important experimental diffrac- tion techniques used. These concerned the use of x-rays, electrons and neutrons as incident radiation. The differences between these techniques are related to the nature of the radiation and how they interact with (and scatter from) the atoms in a crystalline solid. Spe- cific information can be gleaned from each type of experiment. REFERENCES AND FURTHER READING Basic Texts ● ● J. S. Blakemore, Solid State Physics, Cambridge University Press, Cambridge (1985) ● ● H. P. Myers, Introductory Solid State Physics, Taylor and Francis, London (1998) Advanced Texts ● ● M. A. Wahab, Solid State Physics: Structure and Properties of Materials, Alpha Science International Ltd., Harrow (2007) ● ● N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saun- ders College, Philadelphia (1976) ● ● G. E. Bacon, Neutron Diffraction, Oxford University Press, (1975) 92 • Solid State Physics EXERCISES Q1. Use the reciprocal lattice vectors, expressed in Equations (3.1) or (3.2), to determine the reciprocal lattice for the body-centered tetragonal structure. Q2. Explain why Equations (3.1) and (3.2) are equivalent. Q3. Use Equation (3.13) to derive the general interplanar separation for cubic crystals, c.f. Equation (2.11). Q4. Prove that a 1 ⋅ (a 2 × a 3 ) = a 2 ⋅ (a 3 × a 1 ) = a 3 ⋅ (a 1 × a 2 ). What do these represent? Q5. Calculate the energies for x-rays, electrons and neutrons if they have a wavelength of 0.75 Å.
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