Symmetry in Crystals

9 Key excerpts on "Symmetry in Crystals"

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  Modern X-Ray Analysis on Single Crystals

  A Practical Guide

  • Peter Luger(Author)
  • 2014(Publication Date)
  • De Gruyter

    (Publisher)

  4 Crystal symmetry

  4.1 Symmetry operations in a crystal lattice

  4.1.1 Introduction

  It was already mentioned in the Introduction, Section 1.1 , that crystals have outstanding macroscopic properties. From optical inspection more or less pronounced symmetry of the crystal faces can be recognized, causing a beauty which makes crystals so attractive that some of them are admired as precious stones. In this respect crystals belong to the most expensive solid materials. One carat (0.2 g) of a good quality single crystal of diamond costs about 15 to 20 thousand Euros, which is much more than, for example, one carat of gold or platinum, which is three orders of magnitude cheaper. The symmetry which is macroscopically visible must exist also microscopically in the crystal lattice. First indications were already seen on the various film exposures we described in Chapter 3 , where the diffraction pattern showed a symmetry which must have its correspondence in the crystal symmetry. So we have to consider which symmetry can occur in a crystal. Then we have to examine the consequences of crystal symmetry for the intensity of the diffraction pattern. It is important to care for that question, because investigators are interested in the minimum number of necessary measurements. We have learned already that the number of reflections which can be recorded is a fixed number given by equation (2.11) . It is therefore quite useful to know how the symmetry properties of the crystal can reduce the amount of experimental data required and also the number of atomic parameters to be determined.

  Figure 4.1 illustrates two different arrangements in a crystal. In Figure 4.1(a) we have a pattern where the only symmetry is periodicity. The unit cell is the smallest nonperiodic volume element consisting of one motif. In Figure 4.1 (b), however, the situation is quite different. Here the motif is present in four different orientations. The identical motif reappears first after having passed another one in both directions. The lattice constants have twice the magnitude of those of Figure 4.1

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  Solid State Physics

  • Gerald Burns(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  5 Crystal Symmetry and Physical Properties (S) 5-1 Introduction 5-2 Neumann's Principle 5-3 Tensors 5-4 Crystal Symmetry and Physical Properties a Pyroelectricity b Polarizability c Piezoelectricity d Elastic coefficients 5-5 Nonlinear Optics Notes Problems This page intentionally left blank CRYSTAL SYMMETRY AND PHYSICAL PROPERTIES All things intertwine one with another, in a holy bond: scarce one thing is disconnected from another. In due coor-dination they combine for one and the same order. M. Aurelius, 7o Himself u One of the ways in which crystalline solids differ from gases, liquids, and glasses is that in crystals the physical properties can vary a great deal depending on the direction of measurement with respect to the crystalline axes. By rather elementary use of symmetry, relations between the values in the various directions can be determined. Fur-ther, in certain directions, for certain physical properties, zero must be obtained; this too can be determined by symmetry. In this chapter we discuss these aspects of crystals. This chapter also serves as a general introduction to certain con-cerns of solid state science. However, it is of specialized interest and need not be covered on a first reading of this book. 5-1 Introduction Most physical properties of crystals, the constants of the crys-tal, can be defined in terms of a relationship between two measurable quantities. For example, the density of a crystal is defined as the mass divided by the volume of the crystal. Since both mass and volume are scalars (zero-rank tensors) their magnitudes do not depend on direc-tions in the crystal, so density ( = mass/volume) is independent of direction, or isotropic. On the other hand, a quantity such as the polarization, P, of a crystal can depend on the direction in which the electric field, E, is applied. In fact, most properties of single crystals will depend on the direction in which they are measured; such a prop-erty is said to be anisotropic.

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  Solid State Physics

  • Mircea S. Rogalski, Stuart B. Palmer(Authors)
  • 2000(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 2 L attice S ym m etry 2.1. SYM METRY OPERATIONS 2.2. CRYSTAL LATTICES 2.1. S ym m etry O perations Just as the properties of atoms are determined by their electronic configuration, we should expect that the properties of solids, which consist of spatial arrangements of atoms, reflect both the electronic structure of the separate atoms and the existing order in the atomic arrangements. An understanding of the solid is considerably simplified if it possesses a crystal structure characterized by a regular arrangement of identical structural units that each contains one or more atoms. Each of these units is called a basis and if the basis position is represented by a point in space, we obtain a periodic array of points called a lattice . Although the concept of a lattice only provides information about the geometry of the structure and has no physical reality, it provides a convenient frame for discovering the intrinsic regularities of the atomic arrangement in crystals. These regularities can lead to simplification of the physical models employed. The basis at each lattice point defines the atomic environment and allows the crystal structure to be built up by translation. The position vector of a lattice point is defined in terms of one, two or three primitive translation vectors, for one-, two- or three-dimensional lattices respectively, as: Rpqs= p a + q b + sc (2.1) where the numbers p , q and s are integers, so that the lattice is invariant under any translation that consists of multiples of the individual primitive vectors a,b,c. This implies that we ignore the boundaries of the crystal, assuming that the average size of elementary translation vectors is very small with respect to the crystal dimension. In 52 Lattice Symmetry 53 other words we neglect the surface effects on the bulk properties of the crystal. The primitive vectors are often omitted and the lattice vectors, Eq.(2.1) written by convention [pqs].

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  Liquid Crystals, Laptops and Life

  • Michael R Fisch(Author)
  • 2004(Publication Date)
  • WSPC

    (Publisher)

  This may be stated as follows: “The symmetry elements of any physical property of a crystal must include the symmetry elements of the point group of the crystal.” At this stage this has many undefined terms, as this chapter unfolds this statement will become clearer. It is also essential to understanding their various molec- ular organizations or phases. Many of the molecules of life have a certain handedness. This chapter will explore some aspects of symmetry that we will need later in this book. The Greek root of the word symmetry is metron, which means to mea- sure, and the prefix sym means together. Combined, they convey the sense of an object having the same measure or “look.” The equality of certain dimensions or measured quantities is the essence of symmetry. 3.3 Representation of a pattern To illustrate symmetry we must first pick a motif or pattern that is unsym- metrical. For example, consider the shoe shown in Fig. 3.1 Fig. 3.1 A shoe, an example of a n asymmetrical object. Of course, shoes come in two forms, one for the right foot and one for the left foot, as illustrated in Fig. 3.2. Fig. 3.2 Left and right shoes, examples of enantiomorphic objects. To distinguish between these two forms, we call one left-handed and the SYMMETRY 19 other right-handed. Any asymmetric object can exist in both left-handed and right-handed forms. Pairs of objects that are related by having right and left-handed forms are said to be enantiomorphous. Enantiomorphic objects are mirror images of each other. If you hold your right hand up to a mirror, it looks like your reflection is holding up its left hand. Thus your left hand is said to be the mirror image of your right hand and vice-versa. Hence the terms left-handed and right-handed. It is important to note that enantiomorphic objects are not identical to each other. One test of the identicalness of two objects is the ability to superimpose one on top of the other.

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  Problems And Solutions In Group Theory For Physicists

  • Zhong-qi Ma, Xiao-yan Gu(Authors)
  • 2004(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 5 SYMMETRY OF CRYSTALS 5.1 Symmetric Operations and Space Groups * The fundamental character of a crystal is the spatial periodic array of the atoms composing the crystal, called the crystal lattice. By the periodic boundary condition, the crystal is invariant in the following translation: where [is called the vector of the crystal lattice. Three fundamental periods aj of the crystal lattice, which are not coplanar, are taken to be the basis vectors of the crystal lattice, or briefly called the lattice bases. The lattice bases are said to be primitive if any vector of the crystal lattice is an integral combination of the lattice bases. For simplicity, we only use the primitive lattice bases if without special indication. * The multiplication of two translations is defined to be a translation where two translation vectors are added. The set of all translations T ( 4 which leave the crystal invariant forms the Abelian translation group T of the crystal. Usually, in addition to the translation symmetry, a crystal has some other symmetric operations which leave the crystal invariant. A general symmetric operation may be a combinative operation composed of the spatial inversion, the rotation and the translation. Denoted by g(R, 6) a general symmetric operation, where R is a proper or improper rotation, R E0(3), and d is a translation vector, not necessary an integral combina- tion of the lattice bases aj: g(R,G)r = Rr + 6. ( 5 4 Moving out the vector of the crystal lattice lfrom G, we express the general 173 174 Problems and Solutions in Group Theory symmetric operation as g(R,a') = T(i+)g(R, t), t = C ajtj, a' = e'+ t 3 o 5 tj < 1.

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  Structure of Materials

  An Introduction to Crystallography, Diffraction and Symmetry

  • Marc De Graef, Michael E. McHenry(Authors)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  8 Symmetry in crystallography Mathematics possesses not only cold truth but supreme beauty, a beauty cold and austere, like that of sculpture, sublimely pure, and capable of a stern perfection, such as only the greatest art can show. Bertrand Russell In this chapter, we will discuss the concept of symmetry in great detail. We will begin with the description of symmetry operations as coordinate transformations, followed by a discussion of the difference between passive and active operators. Then we introduce rotations, and we determine which rotations are compatible with the 14 Bravais lattices. After a discussion of operators of the first (rotation, translation) and second (mirror, inver-sion) kinds, we will generate combinations of symmetry operators, which will lead to glide planes and screw axes. Along the way, we will also introduce the time-reversal operator and study how it can be combined with the regular symmetry operators. We conclude the chapter with the definition of point symmetry. 8.1 Symmetry of an arbitrary object • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Many objects encountered in nature display some form of symmetry, in many cases only an approximate symmetry; e.g. the human body shows an approximate mirror symmetry between the left and right halves, many flowers have five-or seven-fold rotational sym-metry, . . . In the following paragraphs, we will discuss the classical theory of symmetry, which is the theory of symmetry transformations of space into itself. If an object can be (1) rotated, (2) reflected, or (3) displaced, without changing the distances between its material points and so that it comes into self-coincidence, then that object is symmetric.

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  Minerals

  Their Constitution and Origin

  • Hans-Rudolf Wenk, Andrey Bulakh(Authors)
  • 2016(Publication Date)
  • Cambridge University Press

    (Publisher)

  Hahn, T. (ed.) (2006). International Tables for Crys-tallography. Volume A: Space-group symmetry . International Union of Crystallography. Phillips, F. C. (1963). An Introduction to Crystallog-raphy . Longmans, London. Weyl, H. (1989). Symmetry. Princeton University Press, Princeton, NJ. The concept of a lattice and description of crystal structures 80 8 Crystal symmetries: point-groups and space-groups The lattice geometry discussed in the previous chapter restricts the possible combinations of symmetry elements that can exist in crystals. There are 32 possible combinations of rotations, mirror planes, and inversions that are expressed in the crystal morphology (point-groups) and 230 combinations of rotations, mirror planes, inversions, and translations that are expressed in the crystal structure (space-groups). The chapter also introduces ef fi cient graphical representations of crystal forms. Spherical projections have wide applications in structural geology and materials science, well beyond mineralogy. ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... 8.1 Introduction In the previous chapter we have become familiar with the extraordinary regularity of the internal structure of crystals. The local balancing of bonding forces between atoms leads to a periodic repetition of elem-entary units. We have seen that these unit cells and the corresponding lattice arrays are diagnostic for speci fi c minerals and have classi fi ed them according to their symmetry. Symmetry emerged as a central feature of minerals and crystals.

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  Course of Theoretical Physics

  • L. D. Landau, E. M. Lifshitz(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Such substances may be regarded as consist-ing of freely moving plane layers at equal distances apart. In each layer the molecules have an ordered orientation, but the configuration of their centres of mass is random. It has been shown in § 137 that structures with a one-dimensional perio-dicity of the density function are smoothed out by thermal fluctuations. The divergence of these fluctuations is, however, only logarithmic. Although this excludes the possibility of a one-dimensional periodicity extended to arbitrarily large distances, it does not exlude its existence over comparatively small but nevertheless macroscopic regions of space, as already noted at the end of § 137. Finally, it may be mentioned that in ordinary isotropic liquids also there are two different types of symmetry. If the liquid consists of a substance which does not have stereoisomers, it is completely symmetrical not only t In the other groups with axial symmetry (C M and C^) the two directions along the axis are not equivalent. Such liquid crystals would in general be pyroelectric. SP -ο» 440 The Symmetry of Crystals under a rotation through any angle about any axis but also under a reflection in any plane, i.e. its symmetry group is the complete group of rotations about a point, together with a centre of symmetry (group K h ). If the substance has two stereoisomeric forms, however, and the liquid contains different num-bers of molecules of the two isomers, it will not possess a centre of symmetry and therefore will not allow reflections in planes. Its symmetry group is just the complete group of rotations about a point (group K). § 140. Nematic and cholesteric liquid crystals The orientational symmetry of nematic liquid crystals is uniaxial: at every point in the liquid there is only one preferred direction of orientation of the molecules, namely that of axial symmetry.

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  Manual of Mineral Science

  • Cornelis Klein, Barbara Dutrow(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  They display the same chemical and physical prop- erties because all are underlain by like atoms in the same geometrical arrangement (see Fig. 6.22). On crystal drawings, each face that belongs to the same form is labeled with the same letter (Fig. 6.33). The relationship between form and the symmetry content of a crystal is important to understand. By se- lecting one individual face in a form and performing all of the symmetry operations of the crystal class, one can generate the remaining faces in that form. For exam- ple, the full form development for the unit face (111), on the basis of the symmetry content of two crystal classes, (in the triclinic system) and 4/m 2/m (in the isometric system), is shown in Fig. 6.34. In the case of 3 1 CRYSTALLOGRAPHIC NOTATION FOR PLANES 135 FIG. 6.31 Derivation of the four-digit Bravais–Miller index from the intercepts of three different crystal faces in the hexagonal system. FIG. 6.32 Crystal zones and zone axes indicated by Miller indices in square brackets. Intersections of faces m, a, m, and b belong to the [001] zone and those of faces r, c, r, and b belong to the [100] zone. the symmetry (which is equivalent to a center of sym- metry), one additional face is generated by inverting the original face (111) through the origin of the three crystallographic axes. This additional face will have in- dices This form consists of two parallel faces only, and is known as a pinacoid. For the (111) face in the isometric system, symmetry elements for 4/m 2/m will generate seven additional faces equivalent to 3 (1 1 1 ). 1 (111) with indices ( 11) (1 1) (11 ) ( 1) ( 1 ) (1 ) and Each face cuts all three crystallo- graphic axes at equal lengths and, therefore, has the same relationship to the symmetry elements. This form is known as an octahedron. Therefore, the num- ber of faces that belong to a form is a function of the symme- try content of the crystal class (or point group).

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