1. Crystal Structure: Diamond and HCP structure
MS213
Crystallography and Diffraction
Crystal Structure: Diamond and
HCP structure
Soon Ho Kwon
Computational Materials Science lab.
KAIST
09/25/2013

2. 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

3. 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

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

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

6. 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 × ×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)

7. HCP Structure
<HCP>
<FCC>

8. 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

9. 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 =

10. 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 =

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