In each parentheses, the first term is the attraction due to Cl- and the second term is the repulsion due to the next neighbour Na+. Note that each term in parentheses is positive and the sum of the series is found to be |
|
| U ionic = - 1.4 e2 / r0 |
(16.2) |
This arrangement is more stable than a single pair of Na+ and Cl-. This is the reason why lattices of ions or other objects are stabler than pairs, triplets or quartets. In three dimensions, the summation such as eq (16.1) leads to a value of Coulombic energy given by |
| |
| UCoulombic = - Me2 / r0,
|
(16.3) |
| |
| where r0 is the shortest distance between the + and the - ions and M is called the Madelung constant. The values of M for various lattices is given in Table 16.4. |
| |
Molecule |
Lattice |
M |
M/nf |
Coordination Number |
| |
|
|
|
|
NaCl |
Rock salt |
1.7476 |
0.8738 |
6 |
| |
|
|
|
|
CsCl |
Calsium chloride |
1.7627 |
0.8813 |
8 |
| |
|
|
|
|
ZnS |
Wurtzite |
1.641 |
0.820 |
4 |
| |
|
|
|
|
TiO2 |
Rutile |
2.408 |
0.803 |
4 |
|
| |
| In the Table nf is the number of ions per molecular formula unit (e.g., nf for CaCl2 is 3) and the coordination number is the average number of neighbours around an ion. Madelung constants are in the range 1.5 to 2.5 and the total Coulomb energy for N ion pairs is given by |
| |
| Ucoul = - NM z + z - e 2 / r0, |
(16.4) |
| where z+ and z- are the magnitudes of charges on the ions. |
| In eq (16.4) we have taken into account only the attractive interaction between ions. The repulsive interactions may be expressed as Be -r /r0 (exponential form) or Cr-n (power law repulsion). The values of B or C and n are to be determined for each lattice. |
| Taking the second form for the repulsive term, the total ionic lattice energy is |
| |
| UL = -Me2/ r + C / rn |
(16.5) |
| |
| |
