Crystallography: Crystal Symmetry PDF

Lec 2: Crystallography Geol 103: Elementary Mineralogy Outline Introduction Minerals as crystalline solid Symmetry Operations and Elements Crystal Morphology Crystallographic Axes Crystal Systems  Crystallography Study of crystalline solids and the principles that govern t...

Lec 2: Crystallography Geol 103: Elementary Mineralogy Outline Introduction Minerals as crystalline solid Symmetry Operations and Elements Crystal Morphology Crystallographic Axes Crystal Systems  Crystallography Study of crystalline solids and the principles that govern their growth, external shape, and internal structure  Crystal Crystalline Minerals or other solids with All solid materials with an definite chemical composition and internal ordered arrangement of external shape bounded by atoms or ions in a 3D pattern smooth plane surfaces Amethyst: violet quartz variety Amorphous Minerals as a Crystalline Solid Minerals as a Crystalline Solid Euhedral Minerals as a Crystalline Solid Subhedral Minerals as a Crystalline Solid Anhedral Minerals as a Crystalline Solid Euhedral Minerals as a Crystalline Solid Subhedral Minerals as a Crystalline Solid Anhedral  Crystallography: Crystal Symmetry What’s next? What’s next? Which cannot be formed based on unfolded cube? Which cannot be formed based on unfolded cube? How many blocks are there? Symmetry Operations and Elements Internal order AKA crystal structure Group of atoms repeated on a periodic array of points in space May be reflected in the crystal’s external shape Symmetry Operations and Elements Symmetry Operations and Elements Symmetry Operation an operation that can be performed either physically or imaginatively that results in no change in the appearance of an object. Symmetry Operations and Elements Motif: when repeated creates the whole Crystal pattern (what) symmetry Operation: Some act that reproduces the motif to create the pattern (how) Element: Operation located at a particular point in space (where) Symmetry Operations and Elements Symmetry Operations and Elements Simple: produce repetition of motif Symmetry using a single type of operation operation Compound: produce repetition of motif using a combination of 2 types of operation ROTOINVERSION Symmetry Operations and Elements ROTATIONAL SYMMETRY Symmetry MIRROR SYMMETRY or REFLECTION Elements CENTER OF SYMMETRY or INVERSION ROTATION-INVERSION or ROTOINVERSION COMBINATION Symmetry Operations and Elements Rotation (n): motif is rotated and repeats Simple itself in appearance once or several times during a complete rotation symmetry Reflection (m): repeats the motif in an operations equidistant projection perpendicular to the mirror plane Inversion (i): produces a pattern when a motif is inverted through an inversion center Symmetry Operations and Elements Rotation (n) – motif is rotated and repeats itself in appearance once or several times during a complete rotation 1-fold – it appears identical after a rotation of 360° 2-fold – it appears identical after a rotation of 180° 3-fold – it appears identical after a rotation of 120° 4-fold – it appears identical after a rotation of 90° 6-fold – it appears identical after a rotation of 60° Assignment Is 5- fold rotational axis possible in crystals? Symmetry Operations and Elements Reflection and mirror plane (m) repeats the motif in an equidistant projection perpendicular to the mirror plane Symmetry Operations and Elements Reflection (m) Enantiomorphic: motifs are related by a mirror but cannot be superimposed on each other Symmetry Operations and Elements Inversion and symmetry center (i) produces a pattern when a motif is inverted through an inversion center Symmetry Operations and Elements Rotoinversion (𝑛) ത  1- fold Rotoinversion axis (𝟏) ഥ Similar to the presence of a center of symmetry AKA center of symmetry or i for inversion Symmetry Operations and Elements 2-fold Rotoinversion axis – involves first rotating the object by 180° then inverting it through a symmetry center  2- fold Rotoinversion axis (𝟐 ഥ) Equivalent to the reflection operation Lec 2: Crystal Symmetry Symmetry Operations and Elements 3-fold Rotoinversion axis –involves rotating the object by 120° (360/3 = 120), and inverting through a center  3- fold Rotoinversion axis (𝟑) ഥ Equivalent to a 3-fold symmetry and a center of symmetry  4- fold rotoinversion axis (𝟒 ഥ) Unique operation involves rotation of the object by 90° then inverting through a center 4- fold rotoinversion axis (𝟒 ഥ) Unique operation  6- fold rotoinversion axis (𝟔 ഥ) Equivalent to a 3- fold rotation with a mirror plane perpendicular to the rotation axis  6- fold rotoinversion axis (𝟔 ഥ) Equivalent to a 3- fold rotation with a mirror plane perpendicular to the rotation axis Combinations of Symmetry Operations  Rectangular-shaped sides with a square- shaped top and bottom. The square-shaped top indicates that there must be a 4-fold rotation axis perpendicular to the square shaped face. The rectangular shaped face on the left side of the crystal must have a 2-fold rotation axis that intersects it The square-shaped top of the crystal also suggests that there must be a 2-fold axis that cuts diagonally through the crystal Furthermore, the square-shaped top and rectangular-shaped front of the crystal suggest that a plane of symmetry is present The square top further suggests that there must be a mirror plane cutting the diagonal through the crystal. Finally, there is another mirror plane that cuts through the center of the crystal parallel to the top and bottom faces. Thus, this crystal has the following symmetry elements: 1 - 4-fold rotation axis (A4) 4 - 2-fold rotation axes (A2), 2 cutting the faces & 2 cutting the edges. 5 mirror planes (m), 2 cutting across the faces, 2 cutting through the edges, and one cutting horizontally through the center. Note also that there is a center of symmetry (i). The symmetry content of this crystal is thus: i, 1A4, 4A2, 5m The symmetry content of this crystal is thus: i, 1A4, 4A2, 5m This belongs to crystal class 4/m2/m2/m. 32 Point groups or 32 Crystal Symmetry Classes -represent the 32 possible combinations of the 10 symmetry operations – Divided into 6 crystal system 10 symmetry operations: 1, 2, 3, 4, 6, i (or c), m, 3ത , 4ത , 6ത 6 Crystal Systems Tricilinic - has only 1-fold rotation or 1-fold rotoinversion axes. Monoclinic - has only mirror plane(s) or a single 2-fold axis. Orthorhombic - has only two fold axes or a 2- fold axis and 2 mirror planes. Tetragonal - has either a single 4-fold rotation axis or a 4-fold rotoinversion axis. Hexagonal - has no 4-fold axes, but has at least 1 6-fold or 3-fold axis. Isometric - has either four 3-fold axes or four 3-fold rotoinversion axes. 6 Crystal Systems Crystal Morphology  - Defines the basic building blocks of the crystal Unit Cell – A group of atoms or molecules in a lattice – Must contain all the necessary points on the lattice that can be translated to repeat itself in an infinite array  -the ordered internal arrangement of Lattice atoms in a crystal structure 4 Points to Remember about lattice 1. Crystal faces develop along planes defined by the points in the lattice BRAVAIS LAW states that a face is more commonly developed in a crystal if it intersects a larger number of lattice points. 4 Points to Remember about lattice 2. The angle between crystal faces is controlled by the spacing between lattice points. 4 Points to Remember about lattice 3. Since all crystals of the same substance will have the same spacing between lattice points (they have the same crystal structure), the angles between corresponding faces of the same mineral will be the same What is this Law? 4 Points to Remember about lattice 4. The symmetry of the lattice will determine the angular relationships between crystal faces. Crystallographic Axes – are imaginary lines that can be drawn within the lattice – defines a coordinate system within the crystal Crystal axes  c Crystal axes b a 𝛼 = 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑏 𝑎𝑛𝑑 𝑐 𝛽 = 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑎 𝑎𝑛𝑑 𝑐 𝛾 = 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑎 𝑎𝑛𝑑 𝑏 6 Crystal Systems In crystallographic axes, two important points to remember are: 1. The lengths of the crystallographic axes are controlled by the dimensions of the unit cell upon which the crystal is based. 2. The angles between the crystallographic axes are controlled by the shape of the unit cell “It never gets easier. You just get better.” – Jordan Hoechlin Questions?