Crystal Structure: Diamond and HCP structure MS213 Crystallography and Diffraction Crystal Structure: Diamond and HCP structure Soon Ho Kwon Computational Materials Science lab. KAIST honest@kaist.ac.kr 09/25/2013

Diamond Structure Tetrahedral bonding ! FCC with alternate tetrahedral interstices filled e.g. (¼, ¼, ¾), (¾, ¼, ¼), (¼, ¾, ¼), (¾, ¾, ¾) Each atom has four nearest neighbors (CN=4) 8 atoms per unit cell Examples: C(diamond), Si, Ge, α-Sn

Diamond Structure Face Centered Cubic and Diamond Crystal Structure It can be described as two interpenetrating FCC structures, one at co-ordinates (0,0,0) and the other at (1/4,1/4,1/4). Cf.) Zinc blend structure It has a cubic close packed (face centered) array of A and the B sit in tetrahedral (1/2 occupied) sites in the lattice

Diamond Structure Atomic packing factor (APF) a a a a

Diamond Structure a a XY2 = XW2 + WY2 + YZ2 = # of atoms in unit cell: 8 (= 1 8 ×8+ 1 2 ×6+4) APF = v = = APF = = 0.34 ( 34% ) Thus it is a loosely packed structure. But XZ = 2r (2r)2 = 4r2 = r2= Atomic radius r =

HCP Structure Hexagonal system: a=b≠c and α=β=90°, γ=120° Two lattice parameters a and c. Ideal ratio c/a =1.633 CN=12 APF = 74% # of atoms in unit cell: 6 (= 1 6 ×12+ 1 2 ×2+3) Example: Mg, Ti, Zn, Be, Co, Zr, Cd * Stacking sequence Layers stacked ABABAB… → hexagonal close-packed (HCP) Layers stacked ABCABC… → cubic close-packed (CCP)

HCP Structure <HCP> <FCC>

Atomic packing factor (APF) AB = AC = BO = ‘a’. CX = where c  height of the hcp unit cell. Area of the base = 6  area of the triangle – ABO = 6  1/2  AB  OO = 6  1/2  a  OO In triangle OBO APF = v = 6  4/3 r3 Substitute r = , v = 6  4/3   v = a3

a a a V = Area of the base × height V =  c APF = cos30º =  OO = a cos 30º = a Now, substituting the value of OO, Area of the base = 6   a  a =

Calculation of c/a ratio In the triangle AXC, AC2 = AX2 + CX2 Substituting the values of AC, AX and CX, a2 = Calculation of c/a ratio In the triangle ABA, cos 30º = AA = AB cos 30º = a But AX = AA = i.e. AX =

Now substituting the value (a/c ratio) of to calculate APF of an hcp unit cell, Packing Fraction = 74%