Technology & Engineering
Crystalline Lattice Structure
A crystalline lattice structure refers to the arrangement of atoms or molecules in a repeating pattern within a crystal. This structure is characterized by well-defined geometric shapes and angles, resulting in a highly organized and symmetrical arrangement. The lattice structure determines many of the physical properties of the material, such as its strength, conductivity, and optical properties.
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10 Key excerpts on "Crystalline Lattice Structure"
eBook - PDFMaterials Science and Engineering
An Introduction
- William D. Callister, Jr., David G. Rethwisch(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
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. For those that do not crystallize, this long-range atomic order is absent; these noncrystalline or amorphous materials are discussed briefly at the end of this chapter. Some of the properties of crystalline solids depend on the crystal structure of the material, the manner in which atoms, ions, or molecules are spatially arranged. There is an extremely large number of different crystal structures all having long-range atomic order; these vary from relatively simple structures for metals to exceedingly complex ones, as displayed by some of the ceramic and polymeric materials. The present dis- cussion deals with several common metallic crystal structures. Chapters 12 and 14 are devoted to crystal structures for ceramics and polymers, respectively. When crystalline structures are described, atoms (or ions) are thought of as being solid spheres having well-defined diameters. This is termed the atomic hard-sphere model in which spheres representing nearest-neighbor atoms touch one another. An example of the hard-sphere model for the atomic arrangement found in some of the common elemental metals is displayed in Figure 3.1c. In this particular case all the atoms are identical. Sometimes the term lattice is used in the context of crystal structures; in this sense lattice means a three-dimensional array of points coinciding with atom posi- tions (or sphere centers). crystal structure lattice (a) (b) (c) Figure 3.1 For the face-centered cubic crystal structure, (a) a hard-sphere unit cell representation, (b) a reduced- sphere unit cell, and (c) an aggregate of many atoms.
eBook - PDFMaterials Science and Engineering, P-eBK
An Introduction
- William D. Callister, Jr., David G. Rethwisch, Aaron Blicblau, Kiara Bruggeman, Michael Cortie, John Long, Judy Hart, Ross Marceau, Ryan Mitchell, Reza Parvizi, David Rubin De Celis Leal, Steven Babaniaris, Subrat Das, Thomas Dorin, Ajay Mahato, Julius Orwa(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Senese, Chemistry: Matter and Its Changes, 4th edition. Copyright © 2004 by John Wiley & Sons, Hoboken, NJ. Reprinted by permission of John Wiley & Sons, Inc. (d) © William D. Callister, Jr. (e) iStockphoto 3.1 Introduction We have previously been concerned primarily with the various types of atomic bonding, which are determined by the electron structures of the individual atoms. The present discussion is devoted to the next level of the structure of materials, specifically, to some of the arrangements that may be assumed by atoms in the solid state. Within this framework, concepts of crystallinity and noncrystallinity are introduced. For crystalline solids, the notion of crystal structure is presented, specified in terms of a unit cell. The three common crystal structures found in metals are then detailed, along with the scheme by which crystallographic points, directions, and planes are expressed. Single crystals, polycrystalline materials, and noncrystalline materials are considered. Another section of this chapter briefly describes how crystal structures are determined experimentally using x‐ray diffraction techniques. CRYSTAL STRUCTURES 3.2 Fundamental concepts 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 in a repetitive three‐dimensional pattern, in which each atom is bonded to its nearest‐neighbour atoms. All metals, many ceramic materials, and certain polymers form crystalline structures under normal solidification conditions. For those that do not crystallise, this long‐range atomic order is absent; these noncrystalline or amorphous materials are discussed briefly at the end of this chapter.
- William D. Callister, Jr., David G. Rethwisch(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Sometimes the term lattice is used in the context of crystal structures; in this sense lattice means a three-dimensional array of points coinciding with atom posi- tions (or sphere centers). crystal structure lattice (a) (b) (c) Figure 3.1 For the face-centered cubic crystal structure, (a) a hard-sphere unit cell representation, (b) a reduced- sphere unit cell, and (c) an aggregate of many atoms. [Figure (c) adapted from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. I, Structure, John Wiley & Sons, 1964. Reproduced with permission of Janet M. Moffatt.] The atomic order in crystalline solids indicates that small groups of atoms form a repeti- tive pattern. Thus, in describing crystal structures, it is often convenient to subdivide the structure into small repeat entities called unit cells. Unit cells for most crystal structures are parallelepipeds or prisms having three sets of parallel faces; one is drawn within the aggregate of spheres (Figure 3.1c), which in this case happens to be a cube. A unit cell is unit cell 3.3 UNIT CELLS 52 • Chapter 3 / The Structure of Crystalline Solids chosen to represent the symmetry of the crystal structure, wherein all the atom positions in the crystal may be generated by translations of the unit cell integral distances along each of its edges. Thus, the unit cell is the basic structural unit or building block of the crystal structure and defines the crystal structure by virtue of its geometry and the atom positions within. Convenience usually dictates that parallelepiped corners coincide with centers of the hard-sphere atoms. Furthermore, more than a single unit cell may be chosen for a particular crystal structure; however, we generally use the unit cell having the high- est level of geometrical symmetry. The atomic bonding in this group of materials is metallic and thus nondirectional in nature.
- James A. Newell(Author)
- 2012(Publication Date)
- Wiley(Publisher)
| Microstructure | The third level of structure in materials, describing the sequencing of crystals at a level invisible to the human eye. | Macrostructure | The fourth and final level of structure in materials, describing how the microstructures fit together to form the material as a whole. | Amorphous Materials | Materials whose order extends only to nearest neighbor atoms. | Crystal Structure | The size, shape, and arrangement of atoms in a three-dimensional lattice. | Bravais Lattices | The 14 distinct crystal structures into which atoms arrange themselves in materials. | Simple Cubic | A Bravais lattice that has one atom in each corner of the unit cell. 2.2 | Levels of Order 33 Figure 2-1 Structure- Properties-Processing Relationship 34 Chapter 2 | Structure in Materials Body-centered cubic (BCC) unit cells also have one atom in each of the eight corners, but an additional atom is present in the center of the cube. Face- centered cubic (FCC) unit cells have one atom in each of the eight cor- ners, plus one atom in each of the six cube faces. Of the noncubic lattices, hexagonal close-packed (HCP) is the most common. The top and bottom of the lattice consist of six atoms forming a hexagon that surrounds a single atom Table 2-1 Levels of Order Applied to Salt Crystals Atomic Structure Sodium (Na) and chlorine (Cl) atoms ionically bond. Atomic Arrange- ment Multiple NaCl molecules bond together to form a face-centered cubic lattice. Micro- structure The edge of the lattice is called a grain boundary. These boundaries are visible under a microscope. Macro- structure The plain eye sees NaCl crystals as clear solids, though they can be colored with impurities. Cl Na | Body-Centered Cubic (BCC) | One of the Bravais lattices that contains one atom in each corner of the unit cell as well as one atom in the center of the unit cell.
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 - 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 - 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 - PDFCrystallography and Surface Structure
An Introduction for Surface Scientists and Nanoscientists
- Klaus Hermann(Author)
- 2016(Publication Date)
- Wiley-VCH(Publisher)
7 2 Bulk Crystals: Three-Dimensional Lattices This section deals with the geometric properties of three-dimensional bulk crystals, which are described, in their perfect structure, by atom arrangements that are periodic in three dimensions. As an example, Figure 2.1 shows a section of a tetragonal YBa 2 Cu 3 O 7 crystal, where vectors R 1 , R 2 , R 3 (lattice vectors) indicate the mutually perpendicular directions of periodicity. Further, the basis of the crystal structure consists of 13 atoms (1 × yttrium, 2 × barium, 3 × copper, 7 × oxygen) inside a rectangular block (unit cell) that is repeated periodically inside the crystal. The building unit is shown to the left of the figure. In this section, all basic definitions used for a quantitative description of structural properties of perfect three-dimensionally periodic crystals will be provided. Here, the crystals are considered not only in terms of their translational symmetry, that is, periodicity, but also by their different point symmetry elements, such as inversion points, mirror planes, or rotation axes, which characterize the positions of all atoms inside a crystal. While the definitions and general properties are rather abstract and mathematical, they can be quite relevant for theoretical studies of real three-dimensional crystals. As an example, lattice representations of crystals are required as input to any electronic structure calculation for solid crystalline material. Further, the theoretical treatment of three-dimensional crystals serves as a foundation to study the surfaces of single crystals, as will be discussed in Chapters 4, 5, and 6. 2.1 Basic Definition The basic definition of a perfect three-dimensional bulk crystal becomes quite clear by considering a simple example. Figure 2.2a shows a section of the cubic CsCl crystal, which is periodic in three perpendicular directions.
eBook - PDF- Wei Cai, William D. Nix(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
The description of perfect crystal structures includes the necessary mathematical methods for describing the positions of atoms in the crystals as well as nomenclature (e.g. Miller indices) for describing the directions and planes in crystals. The geometrical description of crystal structures also allows for the calculation of atomic volumes that play an important role in understanding imperfections. The control of imperfections in crystalline materials lies at the heart of materials science. Indeed, the very definition of materials science can be stated as: the synthesis of useful engineering materials and the control of their properties through the control of composition and microstructure . For crystalline materials the control of microstructure involves the control of crystal imperfec-tions. The hypothetical properties of perfect crystals, without defects, are shown to be mostly uninteresting. Important material properties as diverse as alloying and diffusion, strength and plasticity, magnetic permeability of ferromagnetic crystals, electronic properties of compound semiconductors, and color of ceramic crystals, all depend critically on the presence and behavior of crystal imperfections. Imperfections can be classified according to their dimensionality. This scheme describes atomic point defects, such as solute atoms and lattice vacancies, as zero-dimensional defects with lattice distortions confined to within a few atomic dimensions in any direction. Atomic point defects are critically important in alloying and diffusion in crystalline solids. Dislocations, 16 INTRODUCTION which are responsible for both plastic flow and strengthening of metals, are one-dimensional or line defects that can extend indefinitely along a line in the crystal. They may be surrounded by perfect crystal in the case of perfect dislocations or be connected to two-dimensional defects called stacking faults in the case of partial dislocations.
eBook - PDF- Wenceslao González-Viñas, Héctor L. Mancini(Authors)
- 2015(Publication Date)
- Princeton University Press(Publisher)
Yet now we can design materials that exploit the properties of different types of chemical components, including arrangements or matrices of materials that define the spatial structure. Such chemical elements can be introduced either as key ingredients or as impurities. They enable one to come up with materials whose properties were previously chosen. Probably the most typical example in the twentieth century is the semiconductor, whose dominant role is unquestioned. Its discovery has enhanced the nature of communications, the automatic production of goods by means of robots, the matériel of the military, and the intellectual support that occurs with computers and calculators. Technological improvements have opened the doors to smaller and more powerful semiconductor devices, which in turn have transformed our homes into places replete with microwave ovens, sound systems, televisions, and video recorders. If the chemical structure of atoms and molecules is essential to the properties of solids, it is even more important for liquids and gases, where the chemical structure is the sole determinant of all other properties. Hence the science of materials focuses more on the relation between the spatial structures of materials and their properties in the solid state than in liquids and gases. In a solid, atoms or molecules are close packed and are maintained in positions fixed by means of electromagnetic forces that have the same order of magnitude as those in molecular bonds. In solids, atoms or molecules are not like isolated entities. On the contrary, their 1 Also, there is the Bose-Einstein condensate state. CRYSTALLINE SOLIDS 7 properties are modified by their proximity to other atoms or molecules, which modify the energy levels of their outer electrons.
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