Perfect Crystal

7 Key excerpts on "Perfect Crystal"

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  Crystallography and Crystal Defects

  • Kevin M. Knowles, Anthony Kelly(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Part I Perfect Crystals 1 Lattice Geometry 1.1 The Unit Cell Crystals are solid materials in which the atoms are regularly arranged with respect to one another. This regularity of arrangement can be described in terms of symmetry opera- tions; these operations determine the symmetry of the physical properties of a crystal. For example, the symmetry operations show in which directions the electrical resistance of a crystal will be the same. Many naturally occurring crystals, such as halite (sodium chlo- ride), quartz (silica), and calcite (calcium carbonate), have very well-developed external faces. These faces show regular arrangements at a macroscopic level, which indicate the regular arrangements of the atoms at an atomic level. Historically, such crystals are of great importance because the laws of crystal symmetry were deduced from measurements of the interfacial angles in them; measurements were first carried out in the seventeenth century. Even today, the study of such crystals still possesses some heuristic advantages in learning about symmetry. Nowadays the atomic pattern within a crystal can be studied directly by techniques such as high-resolution transmission electron microscopy. This atomic pattern is the fundamental pattern described by the symmetry operations and we shall begin with it. In a crystal of graphite the carbon atoms are joined together in sheets. These sheets are only loosely bound to one another by van der Waals forces. A single sheet of such atoms provides an example of a two-dimensional crystal; indeed, recent research has shown that such sheets can actually be isolated and their properties examined. These single sheets are now termed ‘graphene’. The arrangement of the atoms within a sheet of graphene is shown in Figure 1.1a. In this representation of the atomic pattern, the centre of each atom is represented by a small dot, and lines joining adjacent dots represent bonds between atoms.

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  Solid State Chemistry and its Applications

  • Anthony R. West(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  2 Crystal Defects, Non‐stoichiometry and Solid Solutions

  2.1 Perfect and ImPerfect Crystals

  In a Perfect Crystal, all the atoms are at rest on their correct lattice positions. Such a Perfect Crystal can be obtained, hypothetically, only at absolute zero; at all real temperatures, crystals are imperfect. Atoms vibrate, which may be regarded as a form of defect, but also a number of atoms are inevitably misplaced. In some crystals, the number of defects may be very small, ≪1%, as in, e.g., high‐purity diamond or quartz. In others, high defect concentrations may be present. In highly defective crystals, the question often arises as to whether the defects themselves should be regarded as forming a fundamental part of the structure rather than as some imperfection in an otherwise ideal structure.

  Crystals are invariably imperfect because the presence of defects up to a certain concentration leads to a reduction of free energy, Fig. 2.1 . Let us consider the effect on the free energy of a Perfect Crystal of creating a single defect, say a vacant cation site. This requires a certain amount of energy, ΔH, but causes a considerable increase in entropy, ΔS, because of the large number of positions which this defect can occupy. Thus, if the crystal contains 1 mol of cations, there are ~1023 possible positions for the vacancy. The entropy gained is called configurational entropy and is given by the Boltzmann equation:

  (2.1)

  where the probability, W, is proportional to 1023

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  Solid State Physics

  From the Material Properties of Solids to Nanotechnologies

  • David Schmool(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  CHAPTER 4

  IMPERFECTIONS IN CRYSTALLINE ORDER

  “Even imperfection itself may have its ideal or perfect state.”

  —Thomas de Quincey

  “If someone is too perfect they won’t look good. Imperfection is important.”

  —Eric Cantona

  4.1 INTRODUCTION

  Thus far, we have only considered solids as being a perfectly arranged periodic array of atoms. In reality the order in solids is far from perfect. The principal properties of solids does indeed come, for the most part, from this ordered portion of the sample. However, it would be a gross oversight to ignore the effects of imperfections in crystalline order. Of particular importance are mechanical and electronic properties, which are greatly affected by structural disorder. The consideration of perfect order is an important aid to the theoretical modelling of physical properties and conceptualisation of solids. Many would consider this as a first approximation to the understanding of the physical properties of solids. However, as we have stated, we do need to consider the departures from this perfect image of the crystal/solid to gain a fuller insight in to their physical properties.

  Structural defects in solids can take a number of forms and generally we classify defects as to their dimensionality. That is to say, whether they are 0, 1, 2 or 3 dimensional defects. In the following sections, we will follow more or less this order in our overview of the defects that can occur in real solids. In the first case, zero-dimensional disorder consists of point defects, which pertain to single atomic positions. Following this we have one dimensional disorders, which occur along a crystal direction, and are called disclocations. Two-dimensional disorder is a planar defect, such as a slip plane and even surfaces in a crystal. Finally, a three dimensional defect will be some form of volume imperfection such as granular

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  Physics for Chemists

  • Ruslan P. Ozerov, Anatoli A. Vorobyev(Authors)
  • 2007(Publication Date)
  • Elsevier Science

    (Publisher)

  9

  Solid State Physics

  Publisher Summary

  This chapter outlines the crystalline state of a solid. A crystal can be characterized both by the unit cell (the carrier of the chemical composition and atomic structure) and by three translation vectors a , b, and c. The three translation vectors can be used to build the whole crystal from original cells. Each of these three vectors corresponds to a symmetry operation because these operations superpose a crystal with themselves. Each atom (supposed to be a point) moves over to a similar atom in the nearby unit cell. Any point can be transferred to another crystal point, identical to the first, by operation of a translation. Translation is an attribute of the crystalline state because it translates a given atom to a similar one in the neighboring unit cell but does not combine an atom with itself. It is important to note that the crystal is considered to be of infinite size; this is a good approximation because crystal size is usually many orders of larger value than the unit cell.

  Solids are mostly subdivided into crystal and amorphous substances. In this chapter the crystalline state is predominantly considered although some features of the amorphous and liquid states will also be briefly touched on.

  9.1 CRYSTAL STRUCTURE, CRYSTAL LATTICE

  A crystal is characterized by a three-dimensional, regular, periodic array of particles—atoms and/or molecules. Modern experimental techniques allow us to see the structure of large molecules in crystals using an electron microscope. Figure 9.1 is an electron photograph of a crystal of tobacco mosaic virus, showing the regular packing of the molecules in a crystal.

  Figure 9.1 Image of the tobacco mosaic crystal as seen in the electron microscope (size of molecules is approximately 25 nm).

  Remember that by “molecule” we usually mean the smallest part of a substance that can exist alone and retain the characteristics of that substance. Imagine that we can divide a crystal until it becomes the “brick,” which retains the main (but, certainly, not all) characteristics of the whole crystal. The contents and form of this “brick” defines the crystal structure: the relative amount and mutual disposition of atoms (molecules), the chemical composition of crystalline material, the interatomic distances and valence angles (i.e., chemical bonding), etc. The smallest part of the crystal that retains the specified characteristics (the “brick”) is referred to as a unit cell. Using the property of periodicity one can build the whole crystal by regular repetition of unit cells along coordinate axes, as shown in Figure 9.2

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  An Introduction to Materials Science

  • Wenceslao González-Viñas, Héctor L. Mancini(Authors)
  • 2015(Publication Date)
  • Princeton University Press

    (Publisher)

  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. Solids whose structures have spatial regularity or periodicity are known as crystalline; solids that have no order are called amorphous (the experimental determination of these structures is briefly treated in section 2.6). There are also intermediate types of materials, as shown in section 1.1. A complete theory of these materials must correlate macroscopic properties like elasticity and hardness, electrical and thermal conductivity, and optical reflectivity with their spatial structures. Some general properties are due to the kind of bond between the solid constituents. This is the initial criterion for the classification that we use. 1. Ionic materials (ionic crystals). A regular distribution of positive and negative ions results when some valence electrons are transferred from one component to another (fig- ure 2.1), which is the case for NaCl, KCl, and CsCl, among other materials. The atoms are distributed in a stable way because of the very strong electric interactions. For example, the distance between Na + and Cl − in NaCl is 2.81 × 10 −10 m. Ionic materials are poor conductors of heat and electricity, hard, brittle, and have a high melting point. The atoms can absorb energy in the far infrared (< 1 eV), creating, for example, a vibrating mode in the crystal lattice. Figure 2.1 Scheme of an ionic solid. 2. Covalent materials. As we see later (figure 2.8), there is a continuous gradation between ionic and covalent bonds. Nevertheless, a pure covalent bond joins atoms direc- tionally, as with covalent molecules (figure 2.2).

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  Imperfections and Active Centres in Semiconductors

  International Series of Monographs on Semiconductors, Vol. 6

  • R. G. Rhodes, Heinz K. Henisch(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  approaching that uf the insulators. In real crystals, in contrast to idealized perfect structures, it is the physical defects and the imperfections in the arrangement of their atoms, as well as the inevitable presence of foreign atoms (impurities) in them, which have a significant effect on their physical properties. In addition to the electrical conduction, the diffusion and precipitation of impurity atoms, the plastic deformation, optical and thermal properties, crystal growth phenomena, and numerous other physical and chemical effects all depend, for a satisfactory explanation, on the assumption of basic defect structures in the crystal lattice. In fact, it may be argued that, since all crystals are imperfect from this point of view, their properties can only be completely understood by taking into account the influences of their defects. A theory based entirely on a perfect lattice structure must inevitably be regarded as inadequate. The aim of the present work is to describe the more significant properties and behaviour of semiconductors and to consider these, both experimentally and theoretically, in relation to the defect content of the crystal. The main emphasis will, therefore, be on these imperfec-tions and the electrically active impurity centres, their origin, distri-bution, behaviour and influence on the properties of germanium and silicon crystals. The principal concern of the semiconductor technologist FUNDAMENTAL CONCEPTS OF THE SEMICONDUCTOR CRYSTAL 3 is the control of the crystal properties, and hence, by implication, that of the physical defects and the foreign atoms in the crystal which in-fluence them. A knowledge of the basic concepts of the defects is not only essential for the design and fabrication of solid state devices, but it is also necessary both to the theoretical physicist in his interpretation of the semiconductor behaviour and to the experimentalist in the prepara-tion of good single crystals.

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  Macromolecular Physics V1

  • Bernhard Wunderlich(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R I V The Defect Crystal 4.1 Macroscopic Recognition of Defects The elucidation of the microscopic structure of a crystal (Chapter II) and its connection to the morphology (Chapter III) forms the basis of under-standing of the crystalline solid state. For crystals other than those of linear macromolecules this understanding was developed to a high degree by the 1930s. Most of the crystals studied at that time were highly perfect. Quantities like density, refractive index, heat of fusion, and modulus could be correlated with what one might call the ideal crystal structure. However, it became increasingly apparent by that time that not all properties even of seemingly Perfect Crystals could be understood on this basis. The computed ultimate strength of crystals is, for example, usually 100-1000 times larger than the actually measured strength. The deformation, electrolytical conductivity, and chemical reactivity behavior which are based on coordinated motion or diffusion of motifs in the crystal are much larger than expected from an ideal crystal structure. Such properties were said to be structure-sensitive, in contrast to the structure-insensitive properties listed above (Smekal, 1929). The study of the defect structure of crystals of relatively high perfection was the next stage of development of knowledge about the solid state.j t For a summary of the history of development of knowledge on crystal defects see, for example, Nabarro (1967) or Hirth and Lothe (1968). 380 4.1 Macroscopic Recognition of Defects 381 The electrolytic conductivity of salts was recognized to depend upon the presence of interstitial ions or vacant lattice sites generated as a result of thermal fluctuations (Frenkel, 1926; Schottky and Wagner, 1930; Schottky, 1935).

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