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Objectives

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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

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)

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

Relation of a tetrahedron to a cube:

i.e. a cube with alternate corners missing and the tetrahedral site at the body centre

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

Bond lengths

important dimensions in a cube

Face diagonal, fd

(fd) = (a2 + a2) = a 2

Body diagonal, bd

(bd) = (2a2 + a2) = a 3

Bond lengths:

Octahedral:

half cell edge, a/2

Tetrahedral:

quarter of body diagonal, 1/4 of a3

Anion-anion:

half face diagonal,

1/2 of a2

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 = a3/4

radius of tetrahedral site = (a3/4) - r

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

- 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

So the face of the unit cell looks like:

Calculate unit cell side in terms of r:

2a2 = (4r)2

a = 2r 2

Volume = (162) r3

Face centred cubic - so number of atoms per unit cell =corners + face centres = (8 1/8) + (6 1/2) = 4

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

Group exercise:

Calculate the packing fraction for a primitive unit cell

Primitive

- 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!

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