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

Crystals unit cell Bravais latticeCrystal = a1a1 a3a3 a2a2

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

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

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

Wigner-Seitz Cells bcc fcc Rhombic dodecahedron Truncated octahedron

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

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

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

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

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

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

Asymmetric Unit

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

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

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

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

NaCl

CsCl

perovskite

ybco

graphite

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

zincblende ZnS GaAs InP

wurtzite ZnO CdS CdSe GaN AlN

Quartz

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