1. Radius Ratio Rule

2. In an ionic structure each cation tends to surround itself with anions; the number that can be grouped around it will depend on the relative size of the cations and anions.
The Coordination Number (CN) is defined as the number of anions that can fit around a cation.
This number increases as the radius ratio increases. The number of anions that can ‘fit’ around a cation is related to the relative size difference between the ions, and this size difference can be described using the radius ratio, which is given by:
r cation /r anion

3. When this number is small, then only a few anions can fit around a cation. When this number is large, then more anions can fit around a cation. When CN is 4, it is known as tetrahedral coordination; when it is 6, it is octahedral; and when it is 8, it is known as cubic coordination. See the following table.

5. 3 4 6 8 12 Corners of equilateral triangle 0.15 - 0.22
Coordination Number (CN)
Arrange of Anions around the cation
Radius Ratio 3
Corners of equilateral triangle 4
Corners of a tetrahedron 6
Corners of an octahedron
0.41 – 2.42 8
corners of a cube
0.73 – 1.37 12
Midpoints of cube edges 1

6. Holes in which positive ions pack
Radius ratio
Coordinate number
Holes in which positive ions pack
0.225 – 0.414 4
Tetrahedral holes
ZnS,CuCl
0.414 – 2.42 6
Octahedral holes
NaCl,MgO 1 12
Dodecahedral
0.732 – 1.37 8
Cubic holes
CsCl,NH4Br
N.B. the C.N. = 12 not found in simple ionic crystals but in the complicated metal oxides

7. IONIC CRYSTAL TYPES Ionic crystal type Co-ordination number A X
Structure type AX
AX2
AX3
NaCl
CsCl
Rutile(TiO2)
Fluorite (CaF2)
ReO3

8. Example ZnS in which rZn2+ / rS2- = 74 / 184 = 0.4
By using the calculations of radius ratio we can predict the coordination number .
Example
ZnS in which rZn2+ / rS2- = 74 / 184 = 0.4
From this ratio we can expect that Zn2+ ions prefer to fill the tetrahedral holes in the crystal lattice of S2- ions

9. By the same method we can predict that Na+ ions will prefer the octahedral holes in the close pack crystal of Cl- ions (rNa+ / rCl-) = 95 / 181 = The crystal structure of NaCl is of C.N.=6

10. Holes in which positive ions pack
Radius ratio
Coordinate number
Holes in which positive ions pack
0.225 – 0.414 4
Tetrahedral holes
ZnS,CuCl
0.414 – 2.42 6
Octahedral holes
NaCl,MgO 1 12
Dodecahedral
0.732 – 1.37 8
Cubic holes
CsCl,NH4Br

11. IONIC CRYSTAL TYPES Ionic crystal type Co-ordination number A X
Structure type AX
AX2
AX3
NaCl
CsCl
Rutile(TiO2)
Fluorite (CaF2)
ReO3

19. rcation / ranion ranion / rcation
In this case there is two types of calculations :
rcation / ranion
ranion / rcation 1
e.g. CaF2
C.N.=8
ranion / rcation = / 0.1= 1.33 2
C.N.=8
But the no. of F- ions must be twice of Ca2+ so the C.N. of Ca2+ ions twice that of F- ions so C.N. of Ca2+ = 8 and C.N. of F- = 4

21. The structure of SnO2 is the same as the structure of Rutile (TiO2)
Example :
SnO2
rSn2+ / rO2- = 69 / 140 = 0.49
C.N.=6
ro2- / rSn2+ = 140 / 69 = 2.03
C.N.=6
The structure of SnO2 is the same as the structure of Rutile (TiO2)
But the no. of O2- ions must be twice of Sn2+ so the C.N. of Sn2+ ions twice that of O2- ions so C.N. of Sn2+ = 6 and C.N. of O2- = 3

22. Example : Rb2O C.N.=8 C.N.=8 rRb+ / rO2- = 148 / 140 = 1.06
ro2- / rRb+ = 140 / 148 = 0.95
C.N.=8
The structure of this oxide is inverse that of fluorite which means that C.N. of O2- = 8 and C.N. of Rb+ = 4

24. Exceptions

25. This radius ratio rule allows to predict the correct coordination no
This radius ratio rule allows to predict the correct coordination no. of ions in the crystal lattice.
BUT
1.This rule must be used under precautions , specially in covalent bonding , e.g. CdS , we can find the radius ratio = 0.53 which leads to C.N.=6
Although the structure of this compound is like the structure of ZnS in which C.N.=4 .
This means that the covalent bonding between Cd and S prefer C.N.=4

26. 2. LiI, LiBr and LiCl have radius ratio less than 0
2. LiI, LiBr and LiCl have radius ratio less than although all of them have the NaCl structure .
Covalent Bonding
Not pure Ionic Bonding less than 100% ionic bonding
3. The alkali metals of covalent bonding .

27. As the size of positive ion decreases as the negativity of lattice energy increases which mean more stability in crystal lattice .
Ex.
ZnS which has the zinc Blende structure in which ZnS has lattice energy more than that if take any crystal structure else

28. Predicting of Structure of complex of ionic compounds

29. This part is dealing with the structure of complex ionic compounds
Here we are dealing with two types of complex ionic compounds :
1. Compounds of general formula A2BO4 , since A is the metal and B is another metal of higher valence than A or could be nonmetal .
Spinel Structure

30. 2. Perovskite structure AB2O3