1. Paulings model

2. Radius ratio RC RA + Rc RA
In ionic solids, cations try to maximize the number of neighboring anions. The maximum number of
anions which have simultaneous contact to the central cation depends on the ratio between the two ion radii.
Example:
Lower limiting radius ratio for triangular coordination C.N. 3 :
minimum radius ratio for triangular coordination:
RA RA
RA + Rc Rc
= = cos 30° RC
= (cos30 )-1 -1 RC RA
RA + Rc RA RC RA
= 0.155
For RC RA
< => “rattling” cation

3. Coordination polyhedra
The number of nearest neighbors of an ion is called coordination number C.N. Usually the coordination numbers for cations are given. For example, the titanium cation in rutile (TiO2) is bonded to six oxygens e.g. its coordination number is 6. The lines connecting nearest oxygen atoms depicts an polyhedron, which has the shape of an octahedron. The titanium is, therefore, coordinated octahedrally. In representation of crystal structures, instead of showing each individual atom, coordination polyhedra are shown:
The rutile structure depicted atom by atom. The cluster consisting of the 6 nearest oxygens to every titanium atom describes an octahedron. Replacing all the clusters by such octahedra (= polyhedral repres-entation) facilitates the "reading" of the structure (top)

4. The rutile structure
Juxtaposition of the rutile structure projected down the c-axis: atom by atom with the the ion radii at right scale (left) and the polyhedral representation (right).

5. Coordination polyhedra II

6. Pauling rules I 1. Pauling Rule: Coordination Polyhedra
Pauling’s Rules make predicition about the arrangement of anions and cations in a ionic structure:
1. Pauling Rule: Coordination Polyhedra
A coordination polyhedron of anions is formed around every cation (and vice-versa) - a cation will try to be in simultaneous contact with the maximum number of anions.The maximum number = probable coordination number (coordination polyhedron) is given by the ratio rule:
Rc /Ra:
< : pair
: triangle
: tetrahedron : octahedron
: cube
> : cuboctahedron

7. Pauling rules II
2. Pauling Rule: Electrostatic Valence Principle : “ Bond Strength”
In a stable ionic structure the charge on an cation is balanced by the sum of the electrostatic
bond strengths to the anions in the coordination polyhedron, i.e. a stable ionic structure must
be arranged to preserve local electroneutrality.
Electrostatic bond strength (e.b.s) of a M (cat.) - X (an.) bond
Mm+ coordinated by n Xx- => e.b.s of M:
The second Pauling‘s rule is followed when
Xx- coordinated by p Mm+ => p x = x m n m n
Prewitt's addendum: Given that the chemical formula for a crystal is charge balanced, then the sum of the coordination numbers of the cations must equal the sum of the coordination numbers of the anions.

8. Pauling rules III
Example for the second Pauling rule: Rutile TiO2
4/6
4/6
4/6
4/6
Each titanium ion Ti4+ (charge = +4) is bonded
to 6 oxygens ions (O2- ), cation coordination
number n = 6
Each oxygen ion O2- (charge = -2) is bonded to
3 titanium ion Ti4+ , anion coordination
number p = 3
bond strength of one Ti4+ - O2- bond: 4/6
bond strength of one Ti4+ - O2- bond: 4/6
Pauling second rule:
p * = x m n
3 * = 2 fullfilled! 4 6
Pauling‘s second rules allows predictions of how individual polyhedra are linked together.

9. Bond valence
Although Pauling‘s second rule works well for rutile, it is not a general rule. Bond strength does not only depend on CN and the charge of the ions but also on bond length. Brown and Shannon (1973) have derived a semi-empirical expression for the bond strength that does take into account the bond length:
S0, R0 and n are characteristic of each cation – anion pair. Universal bond valence curves have been given by Brown (1981). The variables R0 and n are characteristic for isoelectronic cations. For cation – oxygen pairs the parameters are:
Cations
No of el.
R1 (A) n H+
0.86
2.170
Li+,Be2+,B3+ 2
1.378
4.065
Na+,Mg2+,Al3+,Si4+,P5+,S6+ 10
1.622
4.290
K+,Ca2+,Sc3+,Ti4+,V5+,Cr6+ 18
1.799
4.483
Mn2+,Fe3+, 23
1.760
5.117
Zn2+,Ga3+,Ge4+,As5+ 28
1.746
6.050

10. Pauling rule IV 3. Pauling Rule: Polyhedral Linking
The stability of different polyhedral linkings is
corner-sharing >
edge-sharing >
face-sharing
- effect is largest for cations with high charge and low coordination number.
- especially large when r+ /r- approaches the lower limit of the polyhedral stability.
corner sharing
edge sharing
face sharing
- Sharing edges/faces brings ions at the centre of each polyhedron closer together,
hence increasing electrostatic repulsions.
Why?
- Sharing edges/faces lowers the screening of the negative charges
lower cation
not visible
lower cation
visible
lower cation
visible

11. Pauling rules V
4. Pauling Rule: Cation Evasion In a crystal containing different cations those of high valency and small coordination number tend to share the mininum number of polyhedral elements with each other.
Example: Perovskite structure CaTiO3
Ti4+ cation in octahedral (6) coordination
Ca2+ cation in cuboctahedral (12) coordination
=> corner shared
=> face shared
5. Pauling Rule: Environmental Homogeneity
The number of essentially different kinds of constituent elements in a crystal tend to be small.

12. The rocksalt structure
Stoichiometry: NaCl
Ion charge: Na: +1 (m) Cl: -1 (x)
Each sodium cation
is coordinated by 6 (n) chlorine anions. The octahedra share edges.
Each chlorine anion
is coordinated by 6 (p) sodium atoms.
sodium cation
chlorine anion
Pauling rules:
1. rule fulfilled rule fulfilled
Ion radius: Na+(IV): 0.99Å e.b.s of Na - Cl bond: 1/6
Na+(VI): 1.02Å C.N. of anion: * 1/6 = anion charge
Na+(VIII): 1.18Å
Cl-: rule not fulfilled: octahedra share edges
Radius ratio: IV: 0.54 too large for four-fold coord.
VI: 0.56 in the range for six fold coord.
VIII:0.65 too small for eight-fold coord.
Minerals with rocksalt structure: uraninite KCl, MgO

13. The fluorite structure CaF2
Cation distribution in a cube layer of the fluorite structure. The filled cubes share edges with each other.
Perspective view of
two unit-cells
Calcium cations are
coordinated by 8 fluorine anions.
The fluorine anions are
coordinated by 4 calcium
cations. Ca
The fluorine cube adjacent
to the first one is empty. F
Cations are coordinate by 8 anions forming a cube.
Anions are tetrahedrally coordinated.
Pauling’s rules:
1. rule: does predict cube coordination for calcium.
2. rule: e.b.s of Ca - F bond: 2/8 = 1/4, C.N. of anion: 4 4* 1/4 = 1 fulfilled!
3. rule: not fulfilled!
Minerals with fluorite structure: uraninite UO2

14. The perovskite structure I
B-cations are coordinated by
6 oxygen anions
A-cation 12-fold coordinated
B-cation, octahedrally coord.
Oxygen
A- cations:
1+: Na, K
2+: Ca, Sr, Ba, Pb
3+: La, Y
B- cations:
5+: Nb
4+: Ti, Zr, Sn, Ce, Th, Pr
3+: Al, Fe, Cr
A-B cation radii relationship:
t: tolerance factor, for ideal perovskite structure t = 1.0
for t perovskite structure can be expected,
but slightly distorted.
A-cations are coordinated by
12 oxygen anions

15. The perovskite structure II
Pauling’s rules:
1.rule Ion radius: Ca2+(X): 1.23Å Ti4+(IV): 0.42Å
Ca2+(XII): 1.34Å Ti4+(VI): 0.605Å
Radius ratio: Ca/O: X: Ti/O: VI: in the range for six-fold coord.
XII:0.95 too small VIII: too small for eight-fold coor.
but close
2. rule: e.b.s of Ca - O bond: 2/12 = 1/6
e.b.s of Ti - O bond: 4/6
Coordination of oxygen: 2 Ti and 4 Ca cations => 2*4/6 + 4*1/6 = 2
3. and 4. rule: partially fulfilled
Highly charged, small Ti is in Low charged, large Ca
corner sharing octahedra is in face-sharing cuboctahedra
Minerals with perovskite structure: perovskite CaTiO3, p MgSiO3 under mantel conditions

16. Structural mapping
Sorting of structures based on ionic radii and other parameters such as ionicity, electron negativity etc.
Structural map as function of radius ratios for AB compounds (pm: picometer).
Structural map as function of radius ratios for A2BO4 compounds.

17. Packing of spheres I 1. Dense sphere packings in 2-D
tetragonal dense packed hexagonal dense packed
2. Dense sphere packings in 3-D
Stacking of tetragonal dense packed layers
A-layer
A-layer
B-layer
Projection
Projection
A-layer
A-layer
2. layer on top of 1. layer
2. layer displaced by a0 /2
3. layer = 1.
cubic primitive stacking
interstitial void: cube
cubic body centered lattice
interstitial voids: octahedra

18. Packing of spheres II Stacking of hexagonal dense packed layers
Two sets
of interstitials:
blue and green
The spheres of the next layer can either beplaced in the green or in the blue interstitials.
Hexagonal closest packed (hcp)
Cubic closest packed (ccp) array of spheres
1. layer A
2.layer B in
blue interstitials
2.layer B in blue
inter stitials
3.layer C in green interstitials
4. layer on top of 1. layer
3. layer A on top
of first layer

19. Packing of spheres III Shape of interstitial voids C-layer B-layer
A-layer
B-layer
C-layer
tetrahedron
[111] direction is perpendicular to closed packed chlorine layers
The rocksalt structure: Chlorine forms a ccp array and sodium (black circles) fills all octahedral voids
octahedron