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LECTURE 8.1 Lecture Outline Weekly deadlines Symmetry Name Structure 3: Bonding , Crystal Structures and Propertie s Lectures Lecture 8.1 Symm etry of Patterns Lecture 8.2 Recogn izing Symm etry Requi red Book Reading 1 (For the end-of-UNIT qui z) Part D: Chapters 22, 23, and 24, plus the Epil ogue to Part D Animations Ian H arrison: "PolymerSyn thF" “StateMatterF” Biographies Mineral of the Week Assignments due this week Assignment 5, A Materia ls' Essay; Part III, is due Wednesday mi dnigh t Prac tice Quiz Practice qui z que stions are ava il able on ANGEL, in the Lesson 08 folders. End of Unit Quiz Quiz 8 will cons ist of ~ twenty (20) que stions for a total of fifty (50) points. Quizze s are “individua li zed” , but with the que stion s taken from a large da tabase. Material cove red: Book Reading and Anim ations OUTLINE Symmetry and the Crystalline State Mirror Symmetry Center of Symmetry Rotational Symmetry Translational Symmetry M. C. Escher Symmetry of Graphite MIRROR SYMMETRY Mirror symmetry reflects left to right. There is a vertical mirror line, midway between the two eyes MIRROR SYMMETRY Mirror symmetry reflects left to right. There is a vertical mirror line (m), midway between the two hands CENTER OF SYMMETRY A center of symmetry “inverts through a point”. A hand, with thumb “up”, and “palm-side” facing out. Transforms to a hand with thumb “down” and back-side facing out. ROTATIONAL SYMMETRY The “motif” or pattern is repeated every 360/n˚, where n = 1, 2, 3, 4,… To the left, n = 3, and the motif (hand) is reproduced every 120˚ The symbol for the “triad” axis is an equilateral triangle. FIVE-FOLD ROTATIONAL SYMMETRY IN THE PLANT KINGDOM CACTUS PLANT The “motif” (a petal) repeats every 360/5 = 72˚. There is a “five-fold (pentad)” rotation axis perpendicular to the flower-head. FIVE-FOLD ROTATIONAL SYMMETRY IN THE PLANT KINGDOM MORNING GLORY The “motif” (a petal) repeats every 360/5 = 72˚. There is a “five-fold (pentad)” rotation axis perpendicular to the flower-head. TRANSLATIONAL SYMMETRY All crystalline materials must display “TRANSLATIONAL SYMMETRY”, or: All crystals must be describable in terms of a lattice which consists of an infinite number of lattice points, each of which has identical surroundings. TRANSLATIONAL SYMMETRY The lattice is described by a “unit cell”, which defines the pattern shape. In two-dimensions, the unit cell is described by two lengths, a and b, and an angle, g In two-dimensions, there are five pattern shapes. In three-dimensions, three lengths (lattice parameters) and three angles define the unit cell. In three-dimensions, there are fourteen pattern shapes THE FIVE PLANAR PATTERNS a) Parallelogram (can always be found. Only used if no other pattern can be used) b) Rectangle c) Centered Rectangle (always preferred to a parallelogram) d) Square (the two sides must be equal in length, and the two interior angles must be 90˚ Always preferred to a centered rectangle) e) Hexagon (all six sides must be equal in length, and all interior angles must be 120˚ Always preferred to a centered rectangle) SYMMETRY OF THE CLOSEST PACKED PLANE IN BCC METALS MIRROR SYMMETRY The atoms are in contact along the two red lines. The angle ABC = 70˚32’ AB and BC are not mirror lines The horizontal and vertical lines are mirror lines TWO-DIMENSIONAL LATTICES a) is possible: indeed it is always possible to describe any planar pattern by a parallelogram. However, it should only be used if no other pattern shape is applicable. b) is the conventional pattern shape. The centered rectangle better reflects the “symmetry” of the pattern, than does the parallelogram. c) is incorrect. The “hexagon” is not regular! AN ESCHER PRINT ESCHER PRINT II ESCHER PRINT III ESCHER PRINT IV: SQUARE ESCHER PRINT IV: SQUARE ESCHER PRINT V: HEXAGON ESCHER PRINT V: HEXAGON ESCHER PRINT VI: RECTANGLE ESCHER PRINT VI: RECTANGLE THE STRUCTURE, LATTICE AND SYMMETRY OF GRAPHITE I In a), a two-dimensional sheet of carbon atoms, in the graphite structure is shown. The structure consists of a hexagonal distribution of carbon atoms. A possible lattice site is indicated by the asterisk at A. If A is taken to be the original lattice site, then all such positions are also lattice sites as shown in b). THE STRUCTURE, LATTICE AND SYMMETRY OF GRAPHITE II In c), the lattice is outlined by the broken lines. The pattern shape is hexagonal. In d), the rotational symmetries associated with points A, B, and C on a) are shown to be six-fold, three-fold, and two-fold respectively. The mirror lines that are associated with the three positions are also indicated.