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face centered cubic, fcc Atoms are arranged in a periodic pattern in a crystal. The atomic arrangement affects the macroscopic properties of a material. Crystals are relatively easy to model. Many important materials (silicon, steel) are crystals Institute of Solid State Physics Crystal Structure Technische Universität Graz body centered cubic, bcc simple cubic

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Crystals unit cell Bravais latticeCrystal = a1a1 a3a3 a2a2

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Primitive Vectors: a1a1 = ½ a Y + ½ a Z a2a2 = ½ a X + ½ a Z a3a3 = ½ a X + ½ a Y Basis Vectors: B1B1 = 0 (Na) B2B2 = ½ a 1 + ½ a 2 + ½ a 3 = ½ aX + ½ aY + ½ aZ (Cl) Example NaCl

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14 Bravais lattices Points of a Bravais lattice do not necessarily represent atoms.

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Unit Cell Choice of unit cell is not unique volume of a unit cell = diamond a1a1 a3a3 a2a2

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Wigner-Seitz Cells bcc fcc Rhombic dodecahedron Truncated octahedron

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Coordination number Number of atoms touching one atom in a crystal Diamond 4 Graphite 3 bcc 8 fcc 12 hcp 12 sc 6

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atomic packing density HCPFCC close packing density = 0.74 random close pack = 0.64 simple cubic = 0.52 diamond = 0.34

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From: Hall, Solid State Physics Fcc conventional unit cell showing close packed plane Primitive unit cellWigner-Seitz cell

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Crystal planes and directions: Miller indices bcc Wigner Seitz cell KOH rapidly etches the Si planes [ ] specific direction family of equivalent directions ( ) specific plane { } family of equivalent planes

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Cementite - Fe 3 C Unit cell cell natom 3 Fe Fe C rgnr 62 Cohenite (Cementite) Fe3 C Asymmetric unit Generated by PowderCell

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Groups Crystals can have symmetries: translation, rotation, reflection, inversion,... Symmetries can be represented by matrices. All such matrices that bring the crystal into itself form the group of the crystal. AB G for A, B G 32 point groups (one point remains fixed during transformation) 230 space groups

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Asymmetric Unit

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simple cubic Po Number: 221 Primitive Vectors: a1a1 = a X a2a2 = a Y a3a3 = a Z Basis Vector: B 1 = 0

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fcc Al, Cu, Ni, Sr, Rh, Pd, Ag, Ce, Tb, Ir, Pt, Au, Pb, Th Primitive Vectors: a1a1 =½ a Y + ½ a Z a2a2 =½ a X + ½ a Z a3a3 =½ a X + ½ a Y Basis Vector: B 1 = 0 Number 225

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hcp Mg, Be, Sc, Ti, Co, Zn, Y, Zr, Tc, Ru, Cd, Gd, Tb, Dy, Ho, Er, Tm, Lu, Hf, Re, Os, Tl

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bcc W Na K V Cr Fe Rb Nb Mo Cs Ba Eu Ta Primitive Vectors: Basis Vector: B 1 = 0 a1a1 = - ½ a X + ½ a Y + ½ a Z a2a2 = + ½ a X - ½ a Y + ½ a Z a3a3 = + ½ a X + ½ a Y - ½ a Z

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NaCl

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CsCl

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perovskite

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ybco

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graphite

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diamond C Si Ge Primitive Vectors: Basis Vectors: Number: 227 a1a1 = ½ a Y + ½ a Z a2a2 = ½ a X + ½ a Z a3a3 = ½ a X + ½ a Y B1B1 = - 1/8 a 1 - 1/8 a 2 - 1/8 a 3 = - 1/8 a X - 1/8 a Y - 1/8 aZ B2B2 = + 1/8 a 1 + 1/8 a 2 + 1/8 a 3 = + 1/8 a X + 1/8 a Y + 1/8 aZ

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zincblende ZnS GaAs InP

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wurtzite ZnO CdS CdSe GaN AlN

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Quartz

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body centered cubic, bcc simple cubic face centered cubic, fcc

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