Physics
Fundamental Lattices
Fundamental lattices are the smallest repeating patterns in a crystal lattice, representing the arrangement of atoms or molecules in a crystal structure. They are defined by a set of basis vectors that can be used to generate the entire lattice through translation. Fundamental lattices play a crucial role in understanding the physical and mechanical properties of crystalline materials.
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10 Key excerpts on "Fundamental Lattices"
eBook - PDF- H.P. Myers(Author)
- 1997(Publication Date)
- CRC Press(Publisher)
2 Crystallography 2.1 Lattices We are most familiar with condensed matter in the form of solid crystalline substances and, although interest in liquids and amorphous solids has grown considerably in the past few years, the physics of condensed matter is to a very great extent the physics of crystals. To begin, we therefore need to learn how to describe the regular geometrical arrangement of atoms in space that is the essence of crystallinity. Crystals are finite regular arrangements of atoms in space. In practice the atomic arrangement is never perfect, but in crystallography we neglect this aspect and describe crystals by reference to perfect infinite arrays of geometrical points called lattices. A lattice is an infinite array of points in space so arranged that every point has identical surroundings. All lattice points are geometrically equivalent. A lattice therefore exhibits perfect translational symmetry and, relative to an arbitrarily chosen origin, at a lattice point, any other lattice point has the position vector (2.1) The numbers n are necessarily integral and the vectors a, b and c are fundamental units of the translational symmetry; the latter are arbitrary, but a sensible choice is usually that which gives the shortest vectors or the highest symmetry to the unit cell. On the other hand, by definition, the volume associated with a single lattice point is unique, but since there is a choice regarding the vectors a, b and c, it may take a variety of shapes as illustrated for a two-dimensional example in Figs 2.1 and 2.2. The volume associated with a single lattice point is called the primitive cell, and this usually takes one of two forms.
- Eduardo C. Marino(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
Part I Condensed Matter Physics 1 Independent Electrons and Static Crystals The expression “condensed matter” refers to materials that are either in a solid or in a liquid state. Soon after the atomic theory was established, the structure of matter in these condensed forms became the object of study under that new perspective. These early investigations already revealed that a large amount of the solids, interestingly, exhibit a peculiar structure, which is called a crystal. These rich forms of matter surprisingly assemble their constituent atoms or molecules in such a way that the most stable configuration has a periodic character, namely, there exists a basic unit that repeats itself along the whole sample. The specific geometric form of the periodic crystalline structure is determined by the spatial orientation of the atomic or molecular valence orbitals of the basic components of each crystal material. The existence of this periodic geometric array exerts a profound influence upon the physical properties of the material. These include the energy spectrum, charge and heat transport, specific heat, magnetic and optical properties. The study of crystal lattices, consequently, is of fundamental importance in the physics of condensed matter. 1.1 Crystal Lattices The mathematical concept that most closely describes an actual crystal lattice is that of a Bravais lattice, a set of mathematical points corresponding to the discrete positions in space given by {R| R = n 1 a 1 + n 2 a 2 + n 3 a 3 ; n i ∈ Z}, (1.1) where a i , i = 1, 2, 3 are the so-called primitive vectors in three-dimensional space. The corresponding structure in one(two)-dimensional space would be analogous to (1.1), but having only one(two) primitive vector(s). We can see that the points in the Bravais lattice form a pattern that repeats itself periodically. A characteristic feature of this type of mathematical structure is that it looks exactly the same from the perspective of any of its points R. 3
eBook - PDF- Nevill Gonzalez Szwacki, Teresa Szwacka(Authors)
- 2016(Publication Date)
- Jenny Stanford Publishing(Publisher)
Chapter 1 One-and Two-Dimensional Crystal Lattices 1.1 Introduction Many of the materials surrounding us (metals, semiconductors, or insulators) have a crystalline structure. That is to say, they represent a set of atoms distributed in space in a particular way. Strictly speaking, this is the case when the atoms occupy their equilibrium positions. Obviously, in the real case they are vibrating. Below we will see examples of crystal structures, beginning with one-dimensional cases. 1.2 One-Dimensional Crystal Structures A one-dimensional crystal structure is formed by a set of atoms or groups of them distributed periodically in one direction. In Fig. 1.1 there are three examples of one-dimensional crystal structures. In all three cases, the whole crystal structure may be obtained by placing atoms (or groups of them), at a distance a = a from one another, along a straight line. When we translate an infinite structure by vector a we obtain the same structure. The same will occur if we Basic Elements of Crystallography (2nd Edition) Nevill Gonzalez Szwacki and Teresa Szwacka Copyright c 2016 Pan Stanford Publishing Pte. Ltd. ISBN 978-981-4613-57-6 (Hardcover), 978-981-4613-58-3 (eBook) www.panstanford.com 2 One-and Two-Dimensional Crystal Lattices Figure 1.1 Three different one-dimensional crystal structures: (a) periodic repetition of identical atoms, (b) periodic repetition of a building block composed of two different atoms, and (c) periodic repetition of a building block composed of two identical atoms. translate the structure by a vector equal to the multiple of vector a, that is, n a, where n ∈ Z . The vector a is called a primitive translation vector. A clear difference can be seen between the crystal structure from Fig. 1.1a and the other two structures in this figure. In the structure from Fig. 1.1a, all the atoms have equivalent positions in space, while in the case of structures from Figs.
Available until 25 Jan |Learn more- Maureen M. Julian(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
1 C H A P T E R 1 Lattices The first step in the analysis of a crystal is the identification of the lattice. A lattice is a regu-lar array of points in space. Above is a two-dimensional set of lattice points. Ideally these points go on forever, filling the entire plane. CONTENTS Chapter Objectives 2 1.1 Introduction 2 1.2 Two-Dimensional Lattices 3 1.3 Two-Dimensional Basis Vectors and Unit Cells 6 1.3.1 Handedness of Basis Vectors 6 1.3.2 Describing the Lattice Mathematically 7 1.3.3 The Unit Cell 8 1.4 Two-Dimensional Transformations between Sets of Basis Vectors 9 1.5 Three-Dimensional Basis Vectors, Unit Cells, and Lattice Transformations 11 1.5.1 Basis Vectors 11 1.5.2 Unit Cell 12 1.5.3 Three-Dimensional Transformations between Sets of Basis Vectors 12 1.6 Conversion into Cartesian Coordinates 14 1.6.1 Two-Dimensional Conversion into Cartesian Coordinates 14 1.6.2 Three-Dimensional Conversion into Cartesian Coordinates 16 1.7 A Crystal: Hexamethylbenzene 17 1.7.1 Definition of Angstrom 18 1.7.2 Two-Dimensional Unit Cell for HMB 18 2 ◾ Lattices CHAPTER OBJECTIVES • Recognize a crystal as a repeating pattern. • Associate a unique lattice with every repeating pattern. • Describe a lattice with its basis vectors. • Distinguish the handedness of basis vectors. • Mathematically describe a lattice. • Realize that the basis vectors need not be orthogonal. • Describe a lattice with more than one set of basis vectors. • Associate a unit cell with basis vectors. • Transform between sets of basis vectors. • Expand concepts of lattices from two to three dimensions. • Construct a transformation matrix between crystallographic and Cartesian coordi-nates for use in computer programs. • Construct, using a computer program, a unit cell of any crystal given the lattice parameters from the literature. • Observe variations in lattice parameters with temperature and pressure. 1.1 INTRODUCTION The ideal crystal has perfect periodicity.
eBook - ePub- A. Srikantha Phani, Mahmoud I. Hussein, A. Srikantha Phani, Mahmoud I. Hussein(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
Chapter 1 Introduction to Lattice Materials A. Srikantha Phani 1 and Mahmoud I. Hussein 2 1 Department of Mechanical Engineering, University of British Columbia, Vancouver, Canada 2 Department of Aerospace Engineering Sciences, University of Colorado Boulder, USA 1.1 Introduction The word “lattice” implies a certain ordered pattern characterized by spatial periodicity, and hence symmetry. In crystalline solids, for example, atoms are arranged in a spatially periodic pattern or a lattice. Such a crystal lattice is specified by a unit cell and the associated basis vectors defining the directions of tessellation [1, 2]. Spatially repetitive patterns are not unique to atomic length scales. They appear over a wide range of length scales, spanning several disciplines and areas of application; see Figure 1.1 for a representative list. Carbon nanotubes [3] and single-layer graphene sheets [4] are periodic materials with nanoscale features. Microelectromechanical systems (MEMS) for radio frequency applications use microscale periodic architectures to form mechanical filters [5]. Biomedical implants such as cardiovascular stents are periodic cylindrical mesh structures [6, 7]. At macro and mega scales, periodic structural construction is widely used in composites in materials engineering [8, 9], turbomachinery in aerospace engineering [10, 11], and bridge and tower structures in civil engineering [12]. Aircraft surfaces typically use a skin-stinger configuration in the form of a uniform shell, reinforced at regular spatial intervals by identical stiffener/stingers. Similarly, rib-skin aircraft structural components, used in tails and fins, comprise two skins (plates) interconnected by ribs [13]. Interested readers are referred to the book by Gibson and Ashby [14] for further studies on lattice materials and the reviews by Mead and by Hussein et al
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 - ePub- Ruslan P. Ozerov, Anatoli A. Vorobyev(Authors)
- 2007(Publication Date)
- Elsevier Science(Publisher)
preserving the cubic symmetry. In a similar manner, the so-called face-centered cubic (FCC) with four atoms in the unit cell is obtained. Copper, for instance, has a FCC structure. Such cells are called cells with basis (or Bravais lattices).Crystallographic directions are characterized by identity distances (periods), i.e., the shortest distance between identical atoms in a given direction. Parallel crystallographic planes are also identical to each other. The distance between two adjacent planes with indexes h, k, l, or what amounts to the same things, the distance from the origin to the nearest crystallographic plane with the same indexes are called interplanar spacingdh,k,l. Exactly this value falls into the Bragg equation (refer to Sections 6.3.5 ).Periods of lattices a , b , c together with angles α , β , γ , plane indexes (h, k, l ) and interplanar spacing d are bound by a so-called quadratic form. For the simplest case of a cubic crystal, the quadratic form is presented by the equation(9.1.2)1=d 2.h 2+k 2+l 2a 2(9.1.2)Structure and crystal lattice define the positions of atomic CM in the unit cell. In addition, atoms in crystals perform thermal oscillations, significantly influencing the crystal’s physical properties. These oscillations are characterized by root mean square displacements.The main method of crystal structure determination is X-ray diffraction analysis (refer to Section 6.3.5
eBook - PDFCrystallography and Surface Structure
An Introduction for Surface Scientists and Nanoscientists
- Klaus Hermann(Author)
- 2016(Publication Date)
- Wiley-VCH(Publisher)
This means that, for a given definition of a crystal, one can always find an infinite number of alternatives that describe the same crystal. While this ambiguity may be considered a drawback at first glance, it allows choosing crystal representa- tions according to additional constraints, for example, those given by symmetry, physical, or chemical properties. Here, one can distinguish between alternative descriptions that affect the crystal basis but not its lattice representation and those where both the lattice representation and the basis are affected. 12 2 Bulk Crystals: Three-Dimensional Lattices 2.2.1 Alternative Descriptions Conserving the Lattice Representation Examples of alternative crystal descriptions that do not affect the crystal lattice are given by elemental or compound decompositions of a crystal. Here, the basic idea is to decompose the basis inside the unit cell of a complex crystal into components and consider (fictitious) crystals of these components with the same periodicity as that of the initial crystal, given by its lattice. This decomposition is of didactic value but may also help to understand details of chemical binding inside the crystal, for example, discriminating between intra- and inter-molecular binding in molecular crystals. In the simplest case, a crystal with p atoms in its primitive unit cell can be considered alternatively as a combination of p crystals with the same lattice but with only one atom in their primitive unit cells. The origins of the corresponding p crystals can be set at positions given by the lattice basis vectors r i of the complete non-primitive crystal. As a very simple example, the cubic cesium chloride, CsCl, crystal, shown in Figure 2.2, is defined by a simple cubic (sc) lattice with lattice vectors R 1 , R 2 , R 3 given by Eq.
eBook - ePubCharacterization of Condensed Matter
An Introduction to Composition, Microstructure, and Surface Methods
- Yujun Song, Qingwei Liao(Authors)
- 2023(Publication Date)
- Wiley-VCH(Publisher)
3 ).Source: From [1] /John Wiley & Sons.3.2.2 Lattice and Crystal Structure
The periodicity of atoms or groups of atoms in crystals can be represented by a number of points that are regularly distributed in space. The distance between adjacent points in either direction is the period of the crystal along that direction. A collection of such points constitutes a spatial lattice, and each point is called a node of the lattice. Since the lattice is only a geometric abstraction representing the arrangement of atoms or groups of atoms, a point in the lattice does not necessarily represent the exact position of an atom. In other words, there may be exactly one atom or a group of atoms at each lattice point. In any case, the environment around each lattice point (including the atom types and distributions) is always the same, i.e. the lattice points are all identical.In mineralogy and crystallography, crystal structure describes 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 a characteristic 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. A crystalline structure is similar to a kind of three‐dimensional wallpaper, an infinite repetition of some entity (such as a series of atoms or molecules). The procedure for creating a pattern consists of performing the point group operations that define it, such as rotation, reflection, and inversion. Making the wallpaper requires a movement (with or without rotation and reflection) to create a lattice that is a complete structure. In practice, crystalline structures can consist of single lattices or multiple grids to synthesize complex crystalline molecules. As long as it is repeatable, X‐ray diffraction can be used to determine its structure.
eBook - PDF- Steven M. Girvin, Kun Yang(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
3 Lattices and Symmetries Many solids exist in the form of crystals . The shapes, colors, and sheer beauty of naturally occurring crystals have fascinated humans from the earliest times to the present day. It was recognized, both by the ancient Greeks and by scientists struggling in the nineteenth century to come to grips with the concept of atoms, that the angles of facets on the surface of crystals could be explained by the existence of atoms and their simple geometric packing into periodic and mathematically beautiful structures that have well-defined cleavage planes. Crystals are the thermodynamic equilibrium state of many elements and compounds at room tem-perature and pressure. Yet, the existence of natural mineral crystals seems quite miraculous because this requires special non-equilibrium geological processes in the earth to concentrate particular ele-ments in one place and arrange the atoms so perfectly. Without these non-equilibrium processes that somehow managed to put lots of copper atoms in northern Michigan, iron atoms in North Dakota, tin atoms in Britain, gold atoms in California, carbon atoms (diamonds) in South Africa, and lithium atoms in Chile, human civilization would be vastly different than it is today. Laboratory synthesis of nearly perfect crystals is an art, some would even say a black art. Billions of dollars have been spent perfecting the synthesis of silicon crystals that are nearly atomically perfect across their 300 mm diameter so that wafers can be sliced from them and enter a 10 billion dollar processing plant to make computer chips. It is now possible to make diamond crystals which are easily distinguishable from naturally occuring ones because they are much more nearly perfect. In this chapter we will develop the mathematical framework needed for studying crystal lattices and the periodic arrangements of atoms in them.
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