Objectives • Written and graphic symbols of symmetry elements

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Presentation transcript:

Objectives • Written and graphic symbols of symmetry elements • Basic symmetry elements and symmetry operations • Written and graphic symbols of symmetry elements • Crystal systems and Miller indices • Lattices and unit cell

Periodic array in a crystal: Example 1 STM (Scanning Tunneling Microscope) image of a platinum surface IBM Research Almaden Research Center

Periodic array in a crystal: Example 2 TopoMetrix Corporation Interconnected 6-membered rings of graphite and the triangular geometry about each carbon atom.

Symmetrical crystal forms

Non-isometric forms Fig. 5.38, Klein pg. 205

Isometric forms Fig. 5.38, Klein pg. 206

Rotation A Symmetrical Pattern 6 6

6 6 Two-fold rotation 360˚/2 Motif Element Operation The axis The plane (perpendicular to the axis) The terms: motif, symmetry element, symmetry operation, the pattern 2-fold, 360/2=180 The symbal 6 Operation

Three-fold rotation 360o/3 6 step 1 6 step 3 6 step 2

d 9 n-fold Rotation a Z t 6 6 6 6 6 1-fold 2-fold 3-fold 4-fold 6-fold identity

6 6 Inversion In 2D, inversion = 2-fold rotation Role play: difference between In 2D, inversion = 2-fold rotation In 3D, inversion ≠ 2-fold rotation

Rotation + Inversion 3

Rotation + Inversion 3 1

Rotation + Inversion 3

Rotation + Inversion 3 1 2

Rotation + Inversion 3

Rotation + Inversion 3

Rotation + Inversion 3 1 2 3

Rotation + Inversion 3 1 2 3 4

Rotation + Inversion 3 1 2 5

Rotation + Inversion 3 3 5 1 4 2 6

Crystal systems: length/angle relations Klein Fig. 5.27, pg. 196

Crystal System - Symmetry Characteristics

Crystal system - Symmetry characteristics Klein Fig. 5.25, pg. 193

Lattice, lattice point, unit cell

Escher Print: Equivalent points

Escher Print: Which Unit Cell?

Plane lattices (nets): 5 unique types Fig. 5.50, Klein, pg 218

Bravais lattices (14 unique types) Triclinic ¹ b ¹ g ¹ c c c Fig. 5.63 Klein, pg 232 Table 5.9 Klein, pg 233 b b P I = C a a Monoclinic g o a = = 90 ¹ b a ¹ b ¹ c c b a P C F I Orthorhombic o ¹ a b c b g a = = = 90

Bravais lattices (14 unique types) 2 1 P I Tetragonal 2 a 1 = a ¹ c a = b = g = 90 o a 3 a 2 Fig. 5.63 Klein, pg 232 Table 5.9 Klein, pg 233 a 1 P F I Isometric 1 2 a = a 3 a = b = g = 90 o