Physics
Lattices
In physics, lattices refer to regular arrangements of points or particles in space. These structures are important in the study of crystalline materials and solid-state physics, as they help describe the arrangement of atoms or molecules within a crystal. Lattices play a crucial role in understanding the properties and behavior of materials, such as their electrical, thermal, and mechanical characteristics.
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11 Key excerpts on "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
- 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.
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 - PDF- Jean-claude Toledano(Author)
- 2011(Publication Date)
- World Scientific(Publisher)
Chapter 2 The structure of crystalline solids Main ideas: Algorithm of construction of a crystal (lattice + basis). Geomet-rical implications of the three-dimensional periodicity (lattice planes and rows), restrictions on the values of rotation angles and on the unequivalent types of Bravais Lattices. Different types of unit cells. Simple packings of atoms. 2.1 Introduction Considered at the atomic scale, solids are part, as well as liquids, of the condensed matter systems, in which the atoms are in “contact” with ea-chother. Their well defined external shape, in given conditions of stresses and of temperature, is related to the fact that the equilibrium position of each atom, referred to a frame attached to the solid, is “fixed”. 1 From the standpoint of the spatial configuration of their constituting atoms, several types of solids exist. One distinguishes crystalline solids from non-crystalline ones (amorphous, quasi-crystalline, etc...). For reasons stated in the introductory chapter, it is mainly the crystalline solids which will be described in this chapter. In these systems, observations by means of instruments giving access to the atomic-scale, show the occurence of specific geometrical regularities in the relative positions of the microscopic constituents. 2 Understanding these regular patterns is a necessary step 1 The correctness of the preceding statement requires mentioning that the concerned atomic positions are average positions of the atomic nuclei. Indeed, the atoms in a crys-tal are in constant motion, vibrating about a point which is their average equilibrium position, with amplitudes of the order of 5.10 − 2 ˚ A at room temperature.The measure-ments of these positions (by X-ray or microscopic techniques) are performed on time scales which are very large with respect to the periods of the vibrations. They therefore reveal, consistently with the above definition, the average atomic positions.
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.
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)
In this latter case, such a solid with no long range atomic order, is said to be amorphous. Both amorphous and polycrystalline systems generally exhibit homoge- neous or isotropic physical properties due to the randomness inher- ent in their structure. Crystals, however, frequently show anisotropy in many of their physical properties. It is therefore very important that we consider the physical arrangement of atoms in a solid when we discuss their physical properties. We define crystalline structures in terms of their symmetries and using symmetry operations we can define up to 230 space Lattices, which can then be subdivided into 32 point groups, fourteen Bravais Lattices and seven crystal systems. The complex inter-relationships being defined by symmetry operations, such as translation, rotation, reflection and inversion. As an aid to our understanding and as a useful tool when considering crystals, we define a unit cell, which is made up of a basis and is attached to each lattice point. Each basis has an identical composition and orientation in the crystalline struc- ture, of which the unit cell defines a sub-unit, which is chosen for convenience. A primitive unit cell should have only one basis or lat- tice point. It is worth noting that since there is no limit on how large a basis can be, there can be many more crystal structures than space groups; the known crystal structures run into thousands. When discussing the specifics of crystals, it is useful to be able to refer to certain directions and planes in a given structure. To do this we define planes and directions in terms of the Miller indices, which are three integers specifying these orientations and directions in the crystal and are distinguished by the type of brackets that are used; (hkl) for a plane, [hkl] for a direction, {hkl} for a family of planes and 〈hkl〉 for a family of directions.
eBook - PDFThe Crystal Lattice
Phonons, Solitons, Dislocations, Superlattices
- Arnold M. Kosevich(Author)
- 2006(Publication Date)
- Wiley-VCH(Publisher)
I express many thanks to Alexander Kotlyar for his invaluable help in preparing the figures and electronic ver- sion of the manuscript. The author is grateful to Oksana Charkina for assistance in preparing the manuscript. I would like to thank my wife Dina for her encouragement. Kharkov March 2005 Arnold M. Kosevich Part 1 Introduction 0 Geometry of Crystal Lattice 0.1 Translational Symmetry The crystalline state of substances is different from other states (gaseous, liquid, amor- phous) in that the atoms are in an ordered and symmetrical arrangement called the crystal lattice. The lattice is characterized by space periodicity or translational sym- metry. In an unbounded crystal we can define three noncoplanar vectors a 1 , a 2 , a 3 , such that displacement of the crystal by the length of any of these vectors brings it back on itself. The unit vectors a α , α = 1, 2, 3 are the shortest vectors by which a crystal can be displaced and be brought back into itself. The crystal lattice is thus a simple three-dimensional network of straight lines whose points of intersection are called the crystal lattice 1 . If the origin of the co- ordinate system coincides with a site the position vector of any other site is written as R = R n = R(n) = 3 ∑ α=1 n α a α , n = (n 1 , n 2 , n 3 ), (0.1.1) where n α are integers. The vector R is said to be a translational vector or a transla- tional period of the lattice. According to the definition of translational symmetry, the lattice is brought back onto itself when it is translated along the vector R. We can assign a translation operator to the translation vector R(n). A set of all possible translations with the given vectors a α forms a discrete group of translations. Since sequential translations can be carried out arbitrarily, a group of transformations is commutative (Abelian).
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 - 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 - PDFCrystal Structures
Lattices and Solids in Stereoview
- M Ladd(Author)
- 1999(Publication Date)
- Woodhead Publishing(Publisher)
(Compare with the triply-primitive hexagonal unit cell in Figure 1.35.) Sec. 1.4] Lattices in One, Two and Three Dimensons 15 °FT Oblique Rectangular Square Hexagonal P P>c P P 2 2mm 4mm 6mm Fig. 1.13. Regular array of points, but not a lattice because all points are not in identical environments. It is best considered as a structure based on a rectangular p unit cell with two entities per unit cell, at points such as O and P. Table 1.4 Two-dimensional Lattices and their conventional lattice unit cells System Unitcell/s Symmetry at each Conventional lattice symbol lattice point unit cell parameters a±b;y*90°, 120° a*b 9 y=90° a = b y = 90° fl = 6;Y=120° 1.4.3 Centring We have considered Lattices for which the conventional lattice unit cells are either primitive or centred. Only a centring position is compatible with the definition of a lattice and, therefore, a suitable site for a lattice point. Figure 1.13 shows a regular two-dimensional array of points where the unit cell, origin 0, contains an additional point, at the end of the vector OP. However, the vector OP placed at P does not terminate on another lattice point. The whole array of points does not constitute a lattice: it could, however, be a structure, based on a rectangular p structure unit cell with two entities per unit cell, at the sites O and P. We shall appreciate this situation more fully when studying space groups. 1.4.4 Three-dimensional, or Bravais, Lattices We consider now a series of identical nets a, b arranged regularly at a third spacing c, whereupon we build up a three-dimensional lattice, a Bravais lattice. Figure 1.14 shows such an array of points in stereoview. There are fourteen Bravais Lattices, distributed unequally among the seven crystal systems listed in Table 1.2. Again, we shall find it convenient to represent each lattice by its conventional lattice unit cell, specified by three non-coplanar vectors a, b and c that are chosen along the reference axes x, y and z respectively.
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