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UNIVERSITY OF SOUTH ALABAMA GY 302: Crystallography & Mineralogy Lecture 3: Miller Indices & Point Groups Last Time Rotoinversion Translational Symmetry 3. Bravais Lattices 1. 2. Rotoinversion A combination of rotation with a center of inversion. e.g., 4-fold Rotoinversion - This involves rotation of the object by 90o then inverting through a center. Note that an object possessing a 4- fold rotoinversion axis will have two faces on top and two identical faces upside down on the bottom, if the axis is held in the vertical position. http://www.cartage.org.lb Symmetry in Crystals Translation: Repetition of points by lateral displacement. Consider 2 dimensional translations: b a Unit Mesh or Plane Lattice Symmetry in Crystals Symmetry in Crystals Symmetry in Crystals The 14 Bravais Lattices Unit Cells NaCl Source: www.chm.bris.ac.uk (Halite) + -Na -Cl Face-centered isometric crystal Today’s Agenda Miller Indices 2. Point Groups (32 of them) Hermann-Mauguin Class Symbols 1. 3. Miller Indices • Crystal facies can be identified using a set of coordinates. Miller Indices • • Crystal facies can be identified using a set of coordinates. The most widely used scheme is that by Miller (Miller Indices) Miller Indices • • Crystal facies can be identified using a set of coordinates. The most widely used scheme is that by Miller (Miller Indices) Miller Indices •Consider the plane in pink (a, ∞, ∞) Miller Indices •Consider the plane in pink. •It’s actually one of an infinite number of parallel planes each a consistent distance from the origin (a, ∞, ∞) Miller Indices •Consider the plane in pink. •It’s actually one of an infinite number of parallel planes each a consistent distance from the origin e.g., 1a, 2a, 3a… 1a (a, ∞, ∞) 2a 3a Miller Indices •In the x direction, the first plane terminates at point 1a. It continues indefinitely in the y and z directions (1a, ∞, ∞) Miller Indices •This plane can be designated (1a, ∞, ∞) or better yet (1, ∞, ∞) (1a, ∞, ∞) Miller Indices •Likewise, this plane in yellow can be designated ( ∞, 1, ∞) And the plane in green can be designated ( ∞,∞, 1) ( ∞, ∞, 1) ( ∞, 1, ∞) ( 1, ∞, ∞) Miller Indices By convention, Miller Indices are reciprocals of the parameters of each crystal face ( ∞, ∞, 1) ( ∞, 1, ∞) ( 1, ∞, ∞) Miller Indices By convention, Miller Indices are reciprocals of the parameters of each crystal face Pink Face = 1/1, 1/∞, 1/∞ 1, 0, 0 Yellow Face = 1/∞, 1/1, 1/∞ 0, 1, 0 ( ∞, ∞, 1) ( ∞, 1, ∞) ( 1, ∞, ∞) Green Face = 1/∞, 1/∞, 1/1 0, 0, 1 Miller Indices Miller Indices are placed in parentheses with no commas and no fractions* Pink Face = (1 0 0) Yellow Face = (0 1 0) Green Face = (0 0 1) * e.g., if you got (1 ¾ ½), you would convert this to (4 3 2) Miller Indices The opposite sides of each face are designated with negative signs (-1 0 0) Pink Face = (-1 0 0) Yellow Face = (0 -1 0) Green Face = (0 0 -1) (0 -1 0) (0 0 -1) Miller Indices •This time, the plane of interest cuts two of the crystallographic axes. •The Miller Index? Miller Indices •This time, the plane of interest cuts two of the crystallographic axes. •The Miller Index? (1 1 0) Miller Indices •This plane cuts all three crystallographic axes. •The Miller Index? Miller Indices •This plane cuts all three crystallographic axes. •The Miller Index? (1 1 1) Miller Indices •Tricky; this plane cuts two of the crystallographic axes, but not equidimensionally • Miller Indices •Tricky; this plane cuts two of the crystallographic axes, but not equidimensionally •The coordinates of the plane are: (1/2, 1, 0) Miller Indices •Tricky; this plane cuts two of the crystallographic axes, but not equidimensionally •The coordinates of the plane are: (1/2, 1, 0) Multiple by 2 to get Miller Indices = (1 2 0) Miller Indices Miller Indices Ness, W.D., 2000. Introduction to Mineralogy. Oxford University Press, New York, 442p Miller Indices Isometric crystal forms related to Miller Indices Ness, W.D., 2000. Introduction to Mineralogy. Oxford University Press, New York, 442p Miller Indices Hexagonal crystal forms related to Miller Indices Ness, W.D., 2000. Introduction to Mineralogy. Oxford University Press, New York, 442p The Point Groups The Point Groups •There are 32 possible combinations of symmetry operations (the point groups or crystal classes) The Point Groups •There are 32 possible combinations of symmetry operations (the point groups or crystal classes) •Each point group will have crystal faces that define the symmetry of the class (the crystal forms) The Point Groups •There are 32 possible combinations of symmetry operations (the point groups or crystal classes) •Each point group will have crystal faces that define the symmetry of the class (the crystal forms) •The point groups are best appreciated through the use of stereo net projections (Thursdays Lecture… Oh Boy!) The Point Groups •There are 5 possible isometric Point Groups; all either have 4 3-fold rotational axes or 4 3-fold-rotoinversion axes Hermann-Mauguin class symbol; more on this shortly The Point Groups •There are 5 possible isometric Point Groups; all either have 4 3-fold rotational axes or 4 3-fold-rotoinversion axes Symmetry Parameters: A2 = 2 fold rotational axes; A3 = 3 fold rotational axes; A4 = 4 fold rotational axes; m = mirror planes The Point Groups •There are 5 possible isometric Point Groups; all either have 4 3-fold rotational axes or 4 3-fold-rotoinversion axes Name of the crystal form The Point Groups •There are 5 possible isometric Point Groups; all either have 4 3-fold rotational axes or 4 3-fold-rotoinversion axes The Point Groups •There are 12 possible hexagonal and trigonal Point Groups; the former has at least one 6-fold rotational axis, the later at least one 3-fold rotational axis The Point Groups •There are 7 possible tetragonal Point Groups; all either have a single 4-fold rotational axis or a 4 foldrotoinversion axis The Point Groups •There are 3 possible orthorhombic Point Groups; all only have either 2-fold rotational axes or 2 foldrotational axes and mirror planes The Point Groups •There are also 3 possible monoclinic Point Groups; all only have a single 2-fold rotational axis or a single mirror plane The Point Groups •Lastly we have the 2 triclinic Point Groups. They only contain 1-fold rotational axes or 1 fold-rotoinversion axes Hermann-Mauguin Class Symbols Each symmetry operation has a symbol: Hermann-Mauguin Class Symbols Each symmetry operation has a symbol… m - mirror planes 1, 2, 3, 4, 6 - rotational axes (1-fold, 2-fold, 3-fold….etc.) 1, 2, 3, 4, 6 - rotoinversion axes (1-fold, 2-fold, ...etc.) i - inversion … which are used to classify and name the Point Groups. Hermann-Mauguin Class Symbols Example 1: Orthorhombic crystal Hermann-Mauguin Class Symbols Example 1: Orthorhombic crystal •3 2-fold rotational axes (A2) •3 mirror planes (m) •center of inversion Hermann-Mauguin Class Symbols Example 1: Orthorhombic crystal Step 1: Write down a number representing each unique rotational axis 222 Hermann-Mauguin Class Symbols Example 1: Orthorhombic crystal Step 2: Write an “m” for every unique mirror plane* 2m 2m 2m * those not produced by other symmetry operations Hermann-Mauguin Class Symbols Example 1: Orthorhombic crystal Step 3: Mirror planes perpendicular to rotational axes are put in a denominator position relative to the rotational axes 2/m 2/m 2/m Hermann-Mauguin Class Symbols Example 2: Orthorhombic crystal Hermann-Mauguin Class Symbols Example 2: Orthorhombic crystal •1 2-fold rotational axes (A2) •2 mirror planes (m) Hermann-Mauguin Class Symbols Example 2: Orthorhombic crystal Step 1: Write down a number representing each unique rotational axis 2 Hermann-Mauguin Class Symbols Example 2: Orthorhombic crystal Step 2: Write an “m” for every unique mirror plane 2mm Hermann-Mauguin Class Symbols Example 2: Orthorhombic crystal Step 3: Mirror planes perpendicular to rotational axes? No 2mm Hermann-Mauguin Class Symbols Example 3: Tetragonal crystal Hermann-Mauguin Class Symbols Example 3: Tetragonal crystal •1 4-fold rotational axes (A4) •4 2-fold rotational axes (A2) •5 mirror planes (m) •center of inversion Hermann-Mauguin Class Symbols Example 3: Tetragonal crystal Step 1: Write down a number representing each unique rotational axis* 422 * here 2 of the 2-fold rotational axis are generated by 4 fold rotation; they are not unique Hermann-Mauguin Class Symbols Example 3: Tetragonal crystal Step 2: Write an “m” for every unique mirror plane* 4 m 2 m 2m * here 2 of the 5 mirror planes are not unique. They are generated by 4 fold rotation Hermann-Mauguin Class Symbols Example 3: Tetragonal crystal Step 3: Mirror planes perpendicular to rotational axes? Yes 4/m 2/m 2/m Hermann-Mauguin Class Symbols Example 4: Isometric crystal Hermann-Mauguin Class Symbols Example 4: Isometric crystal •3 4-fold rotational axes (A4) •4 3-fold rotoinversion axes (A3) •6 2-fold rotational axes (A2) •9 mirror planes (m) •center of inversion Hermann-Mauguin Class Symbols Example 4: Isometric crystal Step 1: Write down a number representing each unique rotational axis* 432 * in high symmetry crystals, most axes are not unique. Here only 1 of each axes is unique. Hermann-Mauguin Class Symbols Example 4: Isometric crystal Steps 2/3: Write an “m” for every unique mirror plane. Determine if they are perpendicular to the axes* 4/m 3 2/m * none of the mirror planes is perpendicular to the 3-fold rotoinversion axes Thursday’s Lecture 1. Stereoprojections (another assignment) 2. Point Group Projections Tuesday’s Lab 1. Isometric/Hexagonal models