Mineralogy Problem Set Crystal Systems, Crystal Classes

Transcription

1 Mineralogy Problem Set Crystal Systems, Crystal Classes (1) For each of the three accompanying plane patterns: (a) Use a ruler to draw solid lines to show where there are mirror planes on the pattern. (b) Where mirror planes intersect, show the required rotation axis symbols. (c) If there are no mirror planes look for rotation axes. (d) If there are no mirror planes or rotation axes, look for glide planes only. (e) Using the symmetry elements you have found, identify the plane pattern group (see attached description of the 17 plane patterns). (f) Show a unit cell for the pattern that has symmetry elements located in the same places as the unit cell shown for the plane pattern in the plane pattern reference diagram. Add any symmetry elements that you missed in steps (a)-(d) that must be present for the plane pattern group. Be sure to add the glide planes as dashed lines (glide planes were not asked for above and are not needed to identify the plane pattern except for pattern pg). (2) For the lattice accompanying this assignment: (a) Show three different unit cells (not just offset!) including one primitive and one centered unit cell. Make sure that there are lattice points only at corners and possibly inside your unit cell; there should be no lattice points on the edges of your unit cell. Label the unit cells clearly. (b) Draw a vector in the direction [250]. Be sure to show which unit cell you are using to define the coordinate system that you are using to locate this vector. (c) Show the lines (vertical planes) with Miller indices (340) and (12 0). Use the same coordinate system (unit cell) that you used to answer (b). (3) For each of the five numbered wooden blocks (record the version examined, e.g. 5a or 5b): (a) Identify the crystal system; (b) Identify the crystal class; (c) List the forms present. (4) Determine the axial ratios (a:b:c) for the mineral celestite using the crystals and goniometers (contact or reflection) available in the lab. Please show all of your calculations and reference them to cross-section diagrams of the crystal(s) measured. Draw carefully. Make your diagrams accurate. Be sure to label all faces with the appropriate Miller indices on your diagrams. Celestite has an orthorhombic unit cell with α = β = γ = 90. It is isostructural with barite, with similar crystal forms. The most common faces are c (001), m, o (011), and d. You may also see b (010) or l (102). See attached diagram. (5) Real crystals are rarely as perfect as models. The symmetry of their shapes does not always match their crystal class because of the circumstances of crystal growth. Nevertheless, the angles between faces must be consistent with the mineral s crystal system. Some good natural crystals are also in the tray with the wooden blocks. Determine the crystal system for each of the eight examples of real crystals (149, 602, 1101, 2502, 2506, 2807, 2837).

2 The 17 Plane Patterns p1 p2 pm pg cm p2mm p2mg p2gg c2mm p3 p3m1 p31m p6 p6mm p4 p4mm p4gm 2-fold axis 3-fold axis 4-fold axis 6-fold axis mirror plane glide plane

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6 Stereographic Projections of the Symmetry Elements in the 32 Crystal Classes Triclinic Monoclinic Orthorhombic Trigonal Hexagonal Tetragonal Isometric One 1-fold or 1 axis One 2-fold or 2 axis Three 2-fold or 2 axes One 3-fold or 3 axis One 6-fold or 6 axis One 4-fold or 4 axis Four 3-fold or 3 axes = i m = = 3/m /m 6/m 4/m (2nd Setting) 2mm 3m 6mm 4mm 43m Symmetry Element Symbols inversion center 2-fold rotation axis 3-fold rotation axis 4-fold rotation axis 6-fold rotation axis mirror plane (= 2 axis) 3-fold rotoinversion axis 4-fold rotoinversion axis 6-fold rotoinversion axis 3 2 m 6 m 2 = 3 m m 2 42m 2 m 3 Lattice Constraints: α = γ = m m m α = β = γ = m m m a = b a = b α = β = 90, γ = 120 α = β = 90, γ = m m m a = b α = β = γ = 90 4 m 3 2m a = b = c α = β = γ = 90

7 SEVERAL WAYS OF DESCRIBING THE 32 CRYSTAL CLASSES Rogers (1937) Schoenflies Full Hermann-Mauguin SHAPE (Example) TRICLINIC Pedial C calcium thiosulfate Pinacoidal C i 1 B1 albite MONOCLINIC Sphenoidal C clinohedrite Domatic C s m m tartaric acid Prismatic C 2h 2/m 2/m gypsum ORTHORHOMBIC Rhombic-pyramidal C 2v m m 2 mm2 hemimorphite Rhombic-disphenoidal D epsomite Rhombic-dipyramidal D 2h 2/m 2/m 2/m mmm barite TRIGONAL Trigonal-pyramidal C sodium periodite Rhombohedral C 3i 3 B3 phenacite Trigonal-trapezohedral D low quartz Ditrigonal-pyramidal C 3v 3m 3m tourmaline Hexagonal-scalenohedral D 3d 3 2/m B3m calcite TETRAGONAL Tetragonal-pyramidal C wulfenite Tetragonal-disphenoidal S 4 4 B Tetragonal-dipyramidal C 4h 4/m 4/m scheelite Tetragonal-trapezohedral D nickel sulfate Ditetragonal-pyramidal C 4v 4 m m 4mm iodosuccinimide Tetragonal-scalenohedral D 2d 42m B42m chalcopyrite Ditetragonal-dipyramidal D 4h 4/m 2/m 2/m 4/mmm zircon HEXAGONAL Hexagonal-pyramidal C nepheline Trigonal-dipyramidal C 3h 6 B6 disilverorthophosphate Hexagonal-dipyramidal C 6h 6/m 6/m apatite Hexagonal-trapezohedral D high quartz Dihexagonal-pyramidal C 6v 6 m m 6mm zincite Ditrigonal-dipyramidal D 3h 6 m 2 B6m2 benitoite Dihexagonal-dipyramidal D 6h 6/m 2/m 2/m 6/mmm beryl ISOMETRIC Tetartoidal T ullmanite Diploidal T h 2/m 3 mb3 pyrite Hextetrahedral T d 4 3 m B43m tetrahedrite Gyroidal O cuprite Hexoctahedral O h 4/m 3 2/m m3m galena Notes: 1. In the SHAPE program, a B preceeding a number indicates an rotary inversion axis. Thus B3 is equivalent to 3. SHAPE usage is in column 4, and is a variant of the short Hermann-Mauguin symbols. 2. This list is modified from Dana's System of Mineralogy, Edition 7, volume 1 page 8. A listing of more alternative descriptions can be found in Dana's Textbook of Mineralogy, Fourth Edition (1932) pages "Type minerals" or chemicals for most classes are given in column 5, as listed in Dana's Textbook. Ironically, some of them are no longer considered to belong to the class for which they are the supposed type minerals.

8 Celestite (SrSO 4 ) (001) (001) Common faces: c (001) m o (011) d b (010) l (102) (001) (001)