1 Crystallographic Concepts GLY 4200 Fall, 2012

2 Atomic Arrangement Minerals must have a highly ordered atomic arrangement The crystal structure of quartz is an example

3 Quartz Crystals The external appearance of the crystal may reflect its internal symmetry

4 Quartz Blob Or the external appearance may show little or nothing of the internal structure

5 Building Blocks A cube may be used to build a number of forms

6 Fluorite Fluorite may appear as octahedron (upper photo) Fluorite may appear as a cube (lower photo), in this case modified by dodecahedral crystal faces

7 Crystal Growth Ways in which a crystal can grow:  Dehydration of a solution  Growth from the molten state (magma or lava)  Direct growth from the vapor state

8 Unit Cell Simplest (smallest) parallel piped outlined by a lattice Lattice: a two or three (space lattice) dimensional array of points

9 Lattice Requirements Environment about all lattice points must be identical Unit cell must fill all space, with no “holes”

10 Auguste Bravais Found fourteen unique lattices which satisfy the requirements Published Études Crystallographiques in 1849

11 Isometric Lattices P = primitive I = body-centered (I for German innenzentriate) F = face centered a = b = c, α = β = γ = 90 ̊

12 Tetragonal Lattices a = b ≠c α = β = γ = 90 ̊

13 Tetragonal Axes The tetragonal unit cell vectors differ from the isometric by either stretching the vertical axis, so that c > a (upper image) or compressing the vertical axis, so that c < a (lower image)

14 Orthorhombic Lattice a ≠ b ≠c α = β = γ = 90 ̊ C - Centered: additional point in the center of each end of two parallel faces

15 Orthorhombic Axes The axes system is orthogonal Common practice is to assign the axes so the the magnitude of the vectors is c > a > b

16 Monoclinic Lattice a ≠ b ≠c α = γ = 90 ̊ (β ≠ 90 ̊ )

17 Monoclinic Axes The monoclinic axes system is not orthogonal

18 Triclinic Lattice a ≠ b ≠c α ≠ β ≠ γ ≠ 90 ̊

19 Triclinic Axes None of the axes are at right angles to the others Relationship of angles and axes is as shown

20 Hexagonal Some crystallographers call the hexagonal group a single crystal system, with two divisions  Rhombohedral division  Hexagonal division Others divide it into two systems, but this practice is discouraged

21 Hexagonal Lattice a = b ≠ c α = γ = 90 ̊ β = 120 ̊

22 Rhombohedral Lattice a = b = c α = β = γ ≠ 90 ̊

23 Hexagonal Axes The hexagonal system uses an ordered quadruplicate of numbers to designate the axes a 1, a 2, a 3, c

24 Arrangement of Ions Ions can be arranged around the lattice point only in certain ways These are known as point groups

25 Crystal Systems The six different groups of Bravais lattices are used to define the Crystal Systems The thirty-two possible point groups define the crystal classes

26 Point Group Point indicates that, at a minimum, one particular point in a pattern remains unmoved Group refers to a collection of mathematical operations which, taken together, define all possible, nonidentical, symmetry combinations