Physics
Ordered Structure
An ordered structure refers to a system that has a specific arrangement or pattern. In physics, this can refer to the arrangement of atoms or molecules in a crystal lattice, which determines the properties of the material. The ordered structure can also affect the behavior of waves and particles within the material.
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8 Key excerpts on "Ordered Structure"
eBook - PDFFundamentals of Condensed Matter and Crystalline Physics
An Introduction for Students of Physics and Materials Science
- David L. Sidebottom(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
In the third chapter, we pause to examine the inter-particle forces that provide the mortar necessary for condensed matter to form. There we survey the fundamental types of bonds and discuss how each can in fl uence the resulting structure. In our fi nal chapter on the topic of structures, we look at magnetic materials. Although the atoms that compose these materials may be arranged in an ordered manner, their magnetic moments can either be oriented randomly or, like the aligned straws of a thatched roof, assume an ordered con fi guration. 1 Crystal structure Introduction We often think of crystals as the gemstones we give to a loved one, but most metals (e.g. copper, aluminum, iron) that we encounter daily are common crystals too. In this chapter, we will examine the structure of crystalline matter in which particles are arranged in a repeating pattern that extends over very long distances. This long-range order is formally described by identifying small local groupings of particles, known as a basis set, that are identically af fi xed to the sites of a regularly repeating space lattice . As it happens, most crystals found in nature assume one of a limited set of special space lattices known as Bravais lattices. These lattices are special by virtue of their unique symmetry properties wherein only discrete translations and rotations allow the lattice to appear unchanged. Chief among these Bravais lattices are the cubic and hexagonal lattice structures that appear most frequently in nature. We focus extra attention on both to provide a useful introduction to coordination properties and packing fractions. 1.1 Crystal lattice Crystals have a decided advantage because of the inherent repeating pattern present in their structure. In an ideal (perfect) crystal, this repeating pattern extends inde fi nitely.
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.
eBook - PDF- R. H. Tredgold(Author)
- 1994(Publication Date)
- Cambridge University Press(Publisher)
In conventional solid state physics it is usual to view a perfect single crystal as the ideal ordered system and a state related to this perfect crystal but containing some dislocations and localised lattice defects as the most highly ordered state attainable in practice. In the materials that we are to discuss, such a degree of perfection is rarely achieved. The 14 2.2 Methods of measuring order 15 nearest approach to true three-dimensional long range order that we are likely to arrive at would be obtained by epitaxial growth on the surface of a macroscopic three-dimensional crystal. At the time of writing, very limited progress in this direction has been made using organic molecules. This is hardly surprising as it is difficult to obtain as a substrate three- dimensional crystalline material which can be cleaved to produce a step- free surface, which is not destroyed by the process of deposition and which has a suitable lattice constant to accommodate organic materials. Most of the thin films of organic materials achieved to date consist either of polycrystalline structures in which true three-dimensional crystalline structure exists only over very small microscopic areas or are really best described as frozen liquid crystals. Indeed, some Langmuir-Blodgett films are true smectic liquid crystals even at room temperature. As a broad generalisation, the systems which we are here concerned with can be thought of as ordered in three different ways. (a) Multilayers repeat their structure in a more or less regular way in a direction normal to the plane of the layers and this regularity can be investigated by X-ray diffraction and also by neutron diffraction. (b) Within a particular layer it is sometimes possible to obtain a two- dimensional crystalline structure, but a hexatic structure is more usually found and this form of structure is discussed in Section 3.4. (c) The degree to which rod-like molecules all share the same axial direction is also of major interest.
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- Richard C. Ropp(Author)
- 2003(Publication Date)
- Elsevier Science(Publisher)
31 Chapter 2 Determining the Structure of Solids This chapter will present more advanced topics than those of the first chapter in terms of determining the structure of solids. Consequently, you will gain some knowledge of how one goes about determining the structure of a solid, even if you never have to do it. 2.1-SCIENTIFIC DETERMINATION OF THE STRUCTURE OF SOLIDS In this section, we will present the basis developed to explain the structure of solids. That is, the concepts that were perfected in order to accurately describe how atoms or ions fit together to form a solid phase. This work was accomplished by many prior workers who established the rationale used to define the structure of a symmetrical solid. As you will recall, we said that the basic difference between a gas, liquid and that of a solid lay in the orderliness of the solid, compared to the other phases of the same material. We have already indicated that solids can have several forms or symmetries. To elucidate the structure of solids in more detail, at least three postulates apply: First, the formation of a solid results from a symmetrical stacking of atoms to near infinity from atoms or molecules with spacings is much smaller than those found in the liquid or gaseous state. Secondly, if we wish to gain an insight into how these atoms are arranged in the solid, we need to determine what kind of pattern they form while in close proximity. Thirdly, we can then relate our pattern to other define the symmetry of solids in general. structures and thus One way to approach a solution of the last two postulates is to define the 32 structure of any given solid in terms of its lattice points, What this means is that by substituting a point for each atom(ion) composing the structure, we find that these points constitute a latticework, i.e.- three-dimensional solid, having certain symmetries (Examples of the symmetries to which we refer are given in 1.3.2. of Chapter 1).
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.
eBook - PDFAperiodic Structures in Condensed Matter
Fundamentals and Applications
- Enrique Macia Barber(Author)
- 2008(Publication Date)
- CRC Press(Publisher)
The same basic principle essentially applies to any piece of con-densed matter: it is composed of a huge number (of the order of magnitude of the Avogadro’s number) of building blocks at the atomic/molecular scale. The way these building blocks are arranged through the space constitutes one of the most fundamental notions in solid state physics and it allows for the introduction of the useful notion of a crystal lattice: a mathematical set of material points whose positions remain …xed through the space. In addition to its undeniable mathematical convenience for a rigorous description of crystals symmetry within the framework of group theory concepts, it turns out that atoms usually behave as point-like ideal particles when they interact among themselves or with propagating energy …elds. In fact, the regular structure of crystals at the atomic level can be properly characterized by means of x-ray di/raction experiments in which the appear-ance of sharp spots indicates the existence of long-range translational order between di/erent atomic distributions. The idea that x-rays might be electro-magnetic waves with wavelengths of the order of magnitude of the distance between layers of atoms in crystals was originally put forward by Max von Laue (1879-1960) at the beginning of the XX century. In a series of exper-iments a beam of x-rays passed through a crystal of copper sulfate, and a number of spots (corresponding to di/raction peaks maxima) were observed on the photographic plates set up around the crystal. Laue then developed a set of equations, relating the direction in which di/raction maxima occur to the structure of the crystal.
eBook - PDF- Michael P. Marder(Author)
- 2010(Publication Date)
- Wiley(Publisher)
Ding, M. R. Norman, J. C. Campuzano, et al. (1996), Angle-resolved photoe- mission spectroscopy study of the superconducting gap anisotropy in BÌ2Sr2CaCu20s +x , Physical Review B, 54(14), R9678-R9681 , Figure 2, ©1996, American Physical Society. This page intentionally left blank Parti ATOMIC STRUCTURE This page intentionally left blank 1. The Idea of Crystals 1.1 Introduction From the point of view of the physicist, a theory of matter is a policy rather than a creed; its object is to connect or co-ordinate apparently diverse phenomena, and above all to suggest, stimulate and direct exper- iment. —Thomson (1907), p. 1 The goal of condensed matter physics is to understand how underlying laws unfold themselves in objects of the natural world. Because the complexity of con- densed matter systems is so enormous, the number of atoms they involve so great, and the possibility of solving all underlying equations in full detail so remote, the laws of greatest importance are principles of symmetry. A first step is to describe how atoms are arranged. As a mental image of ar- rangement, the idea of the crystal has emerged out of an obscure class of minerals to dominate thought about all solids. Here is symmetry with a vengeance. A small group of atoms repeats a simple pattern endlessly through the stretches of a macro- scopic body. The most precise experiments and the most detailed theories of solids are all carried out in perfect crystals. Yet the world is neither a collection of crys- tals, nor a collection of solids wishing to be crystals but falling short of perfection. Principles of symmetry more general than crystalline order still function in struc- tures bearing no resemblance to the perfect lattice, while a rigid insistence upon considering only solids in crystalline form would force one to abandon most natu- rally occurring substances and technologically important materials.
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