Primitive and conventional cells Structure=Lattice+basis basis Primitive cell- parallelepiped defined by primitive with one lattice point per cell Conventional cell may have more than one lattice point per cell. Body centered cubic (bcc): conventional ¹primitive cell Fractional coordinates: 000, 100, 010, 001, 110,101, 011, 111, ½ ½ ½ Simple cubic (sc): primitive cell= conventional cell Fractional coordinates of lattice points: 000, 100, 010, 001, 110,101, 011, 111

Primitive and conventional cells Face centered cubic (fcc): conventional ¹ primitive cell Fractional coordinates: 000, 100, 010, 001, 110,101, 011, 111, ½ ½ 0, ½ 0 ½, 0 ½ ½ , ½1 ½ , 1 ½ ½ , ½ ½ 1 Body centered cubic (bcc): primitive (rombohedron) ¹conventional cell Fractional coordinates: 000, 100, 101, 110, 110,101, 011, 211, 200

Primitive and conventional cells points of primitive cell 120o Hexagonal close packed cell (hcp): conventional =primitive cell Fractional coordinates: 100, 010, 110, 101,011, 111,000, 001

Packing B C Close pack A B A Sequence AAAA…- simple cubic Sequence ABABAB..- hexagonal close pack Sequence ABCABCAB..-face centered cubic

Simple Cubic lattice Atomic position projected on a cube face Crystals: Po 1 height above plane (in terms of a ) a -lattice parameter 8 corner atoms shared by 8 unit cells -> one atom per unit cell

Body Centred Cubic lattice Na V Cr Fe K Nb Mo Rb Ta W Cs Atomic position projected on a cube face 1 1 1/2 1 1 8 corner atoms shared by 8 cells+ one centre atom -> two atoms per cell (0,0,0) and (½, ½, ½)

Face Centred Cubic lattice Ni Cu Al Pb Ne Pd Ag Ar Pt Au Kr Xe Atomic position projected on a cube face ½ 1 1 ½ 0 1 ½ 1 ½ 1 8 corner atoms shared by 8 cells+ 6 face- centred atoms each shared by two cells -> four atoms per cell (0,0,0), (0, ½, ½), (½, 0, ½) and (½, ½, 0)

Diamond structure=2FCC lattices Atomic position projected on a cube face FCC lattice +basis of 2 atoms (2 FCC displaced by ( ¼, ¼, ¼)) Crystals: diamond, Si, Ge, grey Sn, ZnS, III-V semiconductors (GaAs, InSb,…) 01 ½ ¼ ¾ 8 atoms per cubic cell (0,0,0) -> ( ¼, ¼, ¼) (0, ½, ½) ( ¼, ¾, ¾) (½, 0, ½) ( ¾, ¼ , ¾) (½, ½, 0) ( ¾, ¾, ¼) L tetrad screw axis (left and right) R

NaCl structure Atomic position projected on a cube face Crystals: LiH, MgO, Mn0, AgBr, PbS, KCl Atomic position projected on a cube face 2 FCC lattices displaced by ( ½ , ½, ½) 0½1 8 ions per cubic cell Na+ ions Cl- ions (0,0,0) (½,½,½) (0,½,½) (½,1,1) (½,0,½) (1,½,1) (½,½,0) (1,1,½) Cl- Na+

CsCl structure Atomic position projected on a cube face Crystals: BeCu, AlNi, CuPd, CuZn, AgMg Atomic position projected on a cube face 2 SC lattices displaced by ( ½ , ½, ½) 01 01 2 ions per cubic cell Cs+ ions Cl- ions (0,0,0) (½,½,½) Cl- Cs+ ½ 01 01

Fluorite structure Atomic position projected on a cube face Crystals: CaF2 Atomic position projected on a cube face 3 FCC lattices at (0,0,0) and displaced by ( 1/4 ,1/4, 1/4) and (3/4, 3/4, 3/4) (all 8 tetrahedral sites are occupied) 01 ½ 01 ¼ ¾ 12 ions per cubic cell ¼ ¾ Ca+ ions F- ions (0,0,0) (¼,¼,¼ ) (¾, ¾, ¾) (0, ½, ½) (¼, ¾,¾) (¾, ¼, ¼,) (½,0, ½) (¾,¼, ¾) (¼, ¾, ¼,) (½,½,0) (¾, ¾,¼) (¼, ¼, ¾) 01 ½ ½ ¼ ¾ ¼ ¾ 01 ½ 01