1. Objectives By the end of this section you should:
know how atom positions are denoted by fractional coordinates
be able to calculate bond lengths for octahedral and tetrahedral sites in a cube
be able to calculate the size of interstitial sites in a cube
know what the packing fraction represents
be able to define and derive packing fractions for 2 different packing regimes

2. Fractional coordinates
Used to locate atoms within unit cell
0, 0, 0
½, ½, 0
½, 0, ½
0, ½, ½ 1. 2. 3. 4.
Note 1: atoms are in contact along face diagonals (close packed)
Note 2: all other positions described by positions above (next unit cell along)

3. Octahedral Sites Coordinate ½, ½, ½ Distance = a/2
In a face centred cubic anion array, cation octahedral sites at:
½ ½ ½, ½ 0 0, ½ 0, ½

4. Tetrahedral sites Relation of a tetrahedron to a cube:
i.e. a cube with alternate corners missing and the tetrahedral site at the body centre

5. Can divide the f.c.c. unit cell into 8 ‘minicubes’ by bisecting each edge; in the centre of each minicube is a tetrahedral site

6. So 8 tetrahedral sites in a fcc

7. important dimensions in a cube
Bond lengths
important dimensions in a cube
Face diagonal, fd
(fd) = (a2 + a2) = a 2
Body diagonal, bd
(bd) = (2a2 + a2) = a 3

8. Bond lengths:
Octahedral:
half cell edge, a/2
Tetrahedral:
quarter of body diagonal, 1/4 of a3
Anion-anion:
half face diagonal,
1/2 of a2

9. Sizes of interstitials
fcc / ccp
Spheres are in contact along face diagonals
octahedral site, bond distance = a/2
radius of octahedral site = (a/2) - r
tetrahedral site, bond distance = a3/4
radius of tetrahedral site = (a3/4) - r

10. Summary f.c.c./c.c.p anions
4 anions per unit cell at: 000 ½½0 0½½ ½0½
4 octahedral sites at: ½½½ 00½ ½00 0½0
4 tetrahedral T+ sites at: ¼¼¼ ¾¾¼ ¾¼¾ ¼¾¾
4 tetrahedral T- sites at: ¾¼¼ ¼¼¾ ¼¾¼ ¾¾¾
A variety of different structures form by occupying T+ T- and O sites to differing amounts: they can be empty, part full or full.
We will look at some of these later.
Can also vary the anion stacking sequence - ccp or hcp

11. Packing Fraction
We (briefly) mentioned energy considerations in relation to close packing (low energy configuration)
Rough estimate - C, N, O occupy 20Å3
Can use this value to estimate unit cell contents
Useful to examine the efficiency of packing - take c.c.p. (f.c.c.) as example

12. So the face of the unit cell looks like:
Calculate unit cell side in terms of r:
2a2 = (4r)2
a = 2r 2
Volume = (162) r3
Face centred cubic - so number of atoms per unit cell =corners + face centres = (8  1/8) + (6  1/2) = 4

13. Packing fraction
The fraction of space which is occupied by atoms is called the “packing fraction”, , for the structure
For cubic close packing:
The spheres have been packed together as closely as possible, resulting in a packing fraction of 0.74

14. Group exercise:
Calculate the packing fraction for a primitive unit cell

15. Primitive

16. Close packing Cubic close packing = f.c.c. has =0.74
Calculation (not done here) shows h.c.p. also has = equally efficient close packing
Primitive is much lower: Lots of space left over!
A calculation (try for next time) shows that body centred cubic is in between the two values.
THINK ABOUT THIS! Look at the pictures - the above values should make some physical sense!

17. Summary
By understanding the basic geometry of a cube and use of Pythagoras’ theorem, we can calculate the bond lengths in a fcc structure
As a consequence, we can calculate the radius of the interstitial sites
we can calculate the packing efficiency for different packed structures
h.c.p and c.c.p are equally efficient packing schemes