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 Note: atoms are in contact along face diagonals (close packed) 1. 2. 3. 4.

3. Octahedral Sites Coordinate ½, ½, ½ Distance = a/2 Coordinate 0, ½, 0 [=1, ½, 0] Distance = a/2 In a face centred cubic anion array, cation octahedral sites at: ½ ½ ½, ½ 0 0, 0 ½ 0, 0 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. Bond lengths important dimensions in a cube Face diagonal, fd (fd) =  (a 2 + a 2 ) = a  2 Body diagonal, bd (bd) =  (2a 2 + a 2 ) = a  3

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

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:000½½00½½½0½ 4 octahedral sites at:½½½00½½000½0 4 tetrahedral T + sites at:¼¼¼¾¾¼¾¼¾¼¾¾ 4 tetrahedral T - sites at:¾¼¼¼¼¾¼¾¼¾¾¾ 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 discussed energy considerations in relation to close packing Rough estimate - C, N, O occupy 20Å 3 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. Unit cell side 2a 2 = (4r) 2 a = 2r  2 Volume = (16  2) r 3 Face centred cubic - so number of atoms per unit cell =corners + face centres = (8  1/8) + (6  1/2) = 4 So the face of the unit cell looks like:

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 a = 2r a 3 = 8r 3 No. of atoms = (8 x 1/8) = 1 = 0.52

16. 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