Inorganic Crystal Structures

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
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