GEOL. 40 ELEMENTARY MINERALOGY

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1 CRYSTAL SYMMETRY AND CLASSIFICATION A. INTRODUCTION Solid matter, which possesses ordered internal structure, wherever it may be, is called a crystal. Such order in the internal structure is also manifested in the external morphology and other properties of the crystal. One of the properties, which reflect internal order in a crystal, is symmetry. Crystal symmetry is manifested in the manner that the bounding planes are arranged. It is significant that the symmetry of every crystal is constant (under a given temperature and pressure). This is a form of the Law of Constant Symmetry. The study of thousands of crystals is greatly facilitated by classifying them. Given their symmetry characteristics crystals can be classified into six crystal systems, whereas, each system can be further divided into crystal classes of which there are 32 in all. Conversely, given the crystal system and class, a crystal's symmetry can also be determined. In this exercise, the aim is to develop the facility to analyze crystalline matter by observing its external properties. Specifically, we wish to pursue the following objectives: 1. Understand and become familiar with the common concepts or terms used to analyze or describe crystals. 2. Understand and become familiar with the concepts of symmetry operation and symmetry elements. 3. Analyze or identify and enumerate in proper notation the symmetry elements of a crystal. 4. Classify a crystal into its system and class. 1

2 B. CONCEPTS AND DEFINITIONS In this section we enumerate, define, or illustrate the concepts, which are useful in describing or analyzing crystals. 1. Crystal A crystal may be viewed either from the external morphology or from the internal structure as indicated below: a. Homogenous solid bounded by natural planar surfaces. This is the definition based on the external morphology. b. Homogeneous solid with ordered internal structure. In this view the internal crystal structure is said to be based on the orderly arrangement of the unit cell, a parallelepiped that is the smallest building block of the internal structure. 2. Crystal Plane (Crystallographic Plane) A plane, which bounds a crystal. 3. Crystal Form A group of symmetrically similar crystal planes. There are two types of crystal form: a closed form, which encloses space, and an open form, which does not. 4. Symmetry Regular or systematic geometric pattern of an image or motif, such as a crystal plane (line or point) with respect to some reference plane, line or point in a crystal. (See Figures 1.1 and 1.4). The essence of symmetry is the repetition of similar motifs according to some definite principles or laws. 2

3 A. Figures on a plane GEOL. 40 Circle Square Equilateral Triangle Rectangle B. Figures in space Sphere Right Circular Cone Square Pyramid Square Prism Rectangular Parallelepiped Figure 1.1 Examples of Figures Showing Symmetry 3

4 C. Living organisms GEOL. 40 Bangus Carabao Chick D. Common objects Top Land cruiser Marble E. Crystal Models Cube Octahedron Rhombic Prism Rhombohedron Figure 1.1 (continued) 4

5 Examples of Figures Showing Symmetry 5. Plane of Symmetry A crystal has a plane of symmetry if every point, P, on the crystal on one side of a reference plane has a corresponding point, Q, across the plane. In this case the reference plane is called the symmetry plane, which is sometimes called a mirror plane since the corresponding points or motifs are said to be mirror images. This is illustrated in Figure 1.2. Figure 1.2 Illustrations of Symmetry Planes in Crystals 1. Hexagonal (Hexagonal Division) 4. Monoclinic 2. Hexagonal (Rhombohedral Division) 5. Orthorhombic 3. Isometric 6. Tetragonal 5

6 6. Axis of symmetry GEOL. 40 A motif in a crystal may repeat any number of times around a given line. Such a reference line may be called a symmetry axis. The number of times a motif repeats around a given axis is called the period of the axis. Common symmetry axes in crystals are one-fold, two-fold, four-fold, and six-fold which are illustrated in Figure 1.3. However, geometrically, other kinds of symmetry axes are possible. Figure 1.3 Illustrations of Symmetry Axes in Crystals 1. Triclinic 2. Tetragonal 3. Orthorhombic 4. Hexagonal 5. Isometric 6. Monoclinic 6

7 7. Center of Symmetry GEOL. 40 A crystal has a center of symmetry, O, if for any point, P, on the crystal a corresponding point, Q, can be located in the opposite direction, by connecting point P to point O, and extending the resulting line segment OP until it intersects the crystal at Q, which is equidistant as the first point, P, from the reference point, O, the center of symmetry. 8. Symmetry Operation A process by which a given motif is repeated and symmetrically arranged in space according to some definite rule. Types of Symmetry Operation a. Reflection by a mirror - wherein for any point P in the crystal on one side of a reference plane (mirror), a corresponding point Q on the other side of the reference plane can be determined by drawing a normal to the reference plane through point P, and extending it across the plane. Point P and Q must be equidistant from the reference plane. b. Rotation about an axis - wherein a motif is repeated a certain number of times at different locations in the same plane around a reference direction called the symmetry axis. c. Inversion about a point - wherein a motif in a crystal is repeated by inversion with respect to a reference point, called the center of symmetry. d. Rotoinversion (rotation + inversion) - wherein a point, line, or plane is repeated a certain number of times by combining the processes of rotation and inversion. 9. Elements of Symmetry Symmetry plane, symmetry line, and center of symmetry are called elements of symmetry. 10. Crystal Symmetry The sum total of all the symmetry elements of a crystal is called the symmetry of the crystal. 7

8 11. Symmetry Notations GEOL. 40 Sets of symbols denoting the symmetry elements present in a crystal are termed as symmetry notations. There are two sets of conventional notations we are going to use for our purpose. a. Hurlbut's notation P - denotes a plane of symmetry C - denotes a center of symmetry A n - denotes an axis of symmetry where n (period of the axis) indicates the number of times a motif repeats around the axis. There are only five of these- A 1, A2, A3, A4, and A6. A n + C - denotes a rotoinversion axis. A 1 + inversion is equivalent to a center of symmetry. b. Hermann- Mauguin notation m - denotes a mirror plane C - denotes a center of symmetry 1, 2, 3, 4, 6 - denote axes of symmetry equivalent to A n of Hurlbut and are in themselves indicative of the period. 1, 2, 3, 4, 6, - (read one bar, two bar, and so on...); denote axes of rotoinversion n/m,where n = 1, 2, 1,2.., - denotes an axis of symmetry perpendicular to a mirror plane. nm, where n denotes an axis of symmetry and m denotes a mirror plane parallel to the axis of symmetry, n 12. With the symmetry elements as basis, crystals are classified into classes with common characteristic symmetry features. These classes are then grouped into crystal systems each having unique symmetry feature and crystallographic parameters. Following Hurlbut's classification, the following table shows the 32 crystal classes and their symmetry elements. 8

9 Table 1.1 The 32 Crystal Classes and their Symmetry Symmetry Elements Class System Hurlbut Notation H-M Notation none 1 Pedial Triclinic C 1 Pinacoidal Triclinic 1A 2 2 Sphenoidal Monoclinic 1P m Domatic Monoclinic C, 1A 2, 1P 2/m Prismatic Monoclinic 3A Rhombic - disphenoidal Orthorhombic 1A 2, 2P mm2 Rhombic-pyramidal Orthorhombic C, 3A 2, 3P 2/m 2/m 2/m Rhombic-dipyramidal Orthorhombic 1A 4 4 Tetragonal-pyramidal Tetragonal 1A 4 4 Tetragonal-disphenoidal Tetragonal C, 1A 4, 1P 4/m Tetragonal-dipyramidal Tetragonal 1A 4, 4A Tetragonal trapezohedral Tetragonal 1A 4, 4P 4mm Ditetragonal-pyramidal Tetragonal 1A 4, 2A 2, 2P 4 2m Tetragonal-scalenohedral Tetragonal C, 1A 4, 4A 2, 5P 4/m 2/m 2/m Ditetragonal - dipyramidal Tetragonal 1A 3 3 Trigonal - pyramidal Hexagonal 1A 3 3 Rhombohedral Hexagonal 1A 3, 3A 2 32 Trigonal-trapezohedral Hexagonal 1A 3, 3P 3m Ditrigonal-pyramidal Hexagonal 1A 3, 3A 2, 3P 3 2/m Hexagonal - scalenohedral Hexagonal 1A 6 6 Hexagonal -pyramidal Hexagonal 1A 6 6 Trigonal- dipyramidal Hexagonal C, 1A 6, 1P 6/m Hexagonal - dipyramidal Hexagonal 1A 6, 6A Hexagonal - trapezohedral Hexagonal 1A 6, 6P 6mm Dihexagonal - pyramidal Hexagonal 1A 6, 3A 2, 3P 6 Ditrigonal - dipyramidal Hexagonal C, 1A 6, 6A 2, 7P 6/m 2/m 2/m Dihexagonal - dipyramidal Hexagonal 3A 2, 4A 3 23 Tetratiodal Isometric 3A 2, 3P, 4A 3 2/m 3 Diploidal Isometric 3A 4, 4A 3, 6A Gyroidal Isometric 3A 4, 4A 3, 6P 4 3m Hextetrahedral Isometric 3A 4, 4A 3, 6A 2, 9p 4/m 3 2/m Hexoctahedral Isometric Source: Hurlbut, C. S. and C. Klein (1977) Manual of mineralogy. 19th ed., p.39 9

10 13. Homework Before doing the laboratory exercises, test your knowledge of symmetry in Figure 1.4. In the blanks provided below label, using the Hurlbut notation (section B11), the symmetry elements of the corresponding diagrams in Figure 1.4. State if the symmetry elements are a line, plane, or point. Answers: Panel A Panel B Panel C 10

11 Figure 1.4 Identification of Symmetry Panel A: Hexagon Octagon Pentagon Panel B: Eagle Tree Starfish Panel C: Hexahedron Octahedron Hexagonal prism Rhombic pyramid 11

12 REFERENCES Bishop, A. C. (1967) An outline of crystal morphology. London: Hutchinson and Co. (Publishers) Ltd. pp Buerger, M. J. (1956) Elementary crystallography. New York: John Wiley and Sons, Inc., pp , 3-11, Hurlbut, C. S. and C. Klein (1977) Manual of mineralogy, 19th ed.. New York: John Wiley and Sons, Inc., pp Phillips, F. C. (1949) An introduction to crystallography. 1st ed.. Great Britain: Robert Maclehouse and Co. Ltd., pp Wade, F. A. and R. B. Mattox (1960) Elements of crystallography. New York: Harpers and Brothers Publishers, pp , tms/12/17/97:lab-ex1.doc 12