Objectives

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. 1. 2. 3. 4. 0, 0, 0 ½, ½, 0 ½, 0, ½ 0, ½, ½ Fractional coordinates Used to locate atoms within unit cell 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 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) = (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. Summaryf.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 =0.74 - 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