Structures of Simple Inorganic Solids

Dr S.J. Heyes

Third of Four Lectures in the 1st Year Inorganic Chemistry Course

Hilary Term 2000

If you have any comments please contact stephen.heyes@chem.ox.ac.uk 

Lecture 3. Rationalization of 'Ionic' Structures.

1. Principles of Laves

2. Ionic model

• Pauling's rules
  • Rule 1, Coordination Polyhedra
  • Rule 2, Bond Strength
  • Rule 3, Polyhedral Linking
  • Rule 4, Cation Evasion
  • Rule 5, Environmental Homogeneity
• Radius ratio rules
• Structure Maps - ionic radius-based
• Lattice Energy - Madelung constants
• Electronic Bond Strength Calculations

3. Specific Interactions stabilising some structures (e.g. NiAs, PbO, PdO, NH₄F)

4. Directed Bonding/Covalency/Polarization - trends in dimensionality

• Fajan's rules
• Ketelaar's Triangle
• Mooser-Pearson Plots - Electronegativity
• Phillips-van Vechten Plots - Critical Ionicity

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Aims of this Lecture

After studying this lecture you should be able to:-

1. List the Space-filling Principles of Laves

2. Derive the Radius Ratio Rules for Structure Prediction and use them in an intelligent manner

3. Utilize the Concept of Electrostatic Bond Strength

4. Identify specific bonding effects that influence structure

5. Identify polarization/covalency effects and place compounds on a Ketelaar Triangle

6. Predict the likely structure of a compound or rationalize the known structure of a compound using the above-listed principles

Principles of Laves

1. Space Principle: Space is used most efficiently
2. Symmetry Principle: Highest possible symmetry is adopted
3. Connection Principle: There will be the most possible "connections" between components

   (i.e. coordination numbers are maximised)

• Followed by metals and inert gases - close-packed structures
• Deviations include:- BCC metals
• Preferred interactions:-
  • lower the symmetry
  • reduce the coordination number
  • decrease the space-filling efficiency
• 'Ionic' compounds strive to follow the principles, but to an extent modified by any specific directional bonding interactions, size differentials and stoichiometry concerns.

Considering Solids as Ionic - useful?

In D.M. Adams' opinion (D.M. Adams, Inorganic Solids, Wiley, 1974):

"Ionic Theory is a good starting place for getting some general guidance.... Ionic theory has had a good run (> 50 years) and is still heavily over-emphasized: SO FAR AS DETAILED CONSIDERATIONS OF CRYSTAL STRUCTURES ARE CONCERNED IT IS TIME IT WAS INTERRED"

We need to examine why despite Adams's reservations with regard to detailed explanations, an ionic model approach has generally been used for simple prediction and rationalization of many simple inorganic structures.

We will also see how a qualitative consideration of non-ionic interactions can help improve the utility of the ionic model approach.

The 'Ionic Model' (Goldschmidt')

"Ions are essentially Charged, Incompressible, Non-Polarizable Spheres"

More sophisticated models assume ions are composed of two parts:

• a central hard, unperturbable core, where most electron density is concentrated
• a soft, polarizable outer sphere, which contains very little electron density

Pauling's Rules

• Goldschmidt's structural principles for IONIC crystals were summarized by Pauling in a series of Rules.
• Historically Pauling's Rules have been widely used, & are still useful in many situations.

Biographical Information about Linus Pauling, one of the most influential of all Chemists.

Pauling Rule 1: Coordination Polyhedra

"A coordination polyhedron of anions is formed around every cation (and vice-versa) - it will only be stable if the cation is in contact with each of its neighbours.

• Ionic crystals may thus be considered as sets of linked polyhedra.
• The cation-anion distance is regarded as the sum of the ionic radii."

Common coordination polyhedra are:-

Molecular Materials

• Absolute coordination numbers are controlled by valency (VSEPR)

Non-Molecular Materials

• Valency has only an indirect bearing on coordination number

  e.g. Na^ICl, Mg^IIO, Sc^IIIN, Ti^IVC all have the Rock Salt (6:6) Structure despite change in valency and from predominantly ionic to covalent bonding

• Ionic Size does influence coordination number

The Coordination Number of the Cation will be Maximized subject to the criterion of Maintaining Cation-Anion Contact

Determined by comparison of the ratio of the ionic radii, r₊/r_- with values derived from the geometric contact criterion

The Radius Ratio Rules

Limiting Radius Ratios - anions in the coordination polyhedron of cation are in contact with the cation and with each other {if cation were to shrink further (i.e. r₊/r_- decrease) cation-anion contact would be lost in contravention of Pauling's 1st Rule}.

Radius Ratio	Coordination no.	Binary (AB) Structure-type
r+/r- = 1	12	none known
1 > r+/r- > 0.732	8	CsCl
0.732 > r+/r- > 0.414	6	NaCl
0.414 > r+/r- > 0.225	4	ZnS

What's the Numerical Value of a specific Ionic Radius?

• Ionic Radii in most scales do not generally meet at experimental electron density minima, because of polarization of the anion by the cation
• The various scales are designed to be self-consistent in reproducing r_o = r₊ + r_-
• Ionic radii change with coordination number
  • r₈ > r₆ > r₄ {use the appropriate one!}
• Use the same scale for cation and anion

Do the Radius Ratio Rules Work?

Test with Structures of Alkali Halides

• Graph compares structures (CsCl / NaCl) with predictions by radius ratio rules from r₊/r_- {r_-/r₊ if cation is larger}
• For Li⁺ and Na⁺ salts, ratios calculated from both r₆ and r₄ are indicated
• Radius ratios suggest adoption of CsCl structure more than is observed in reality
• NaCl structure is observed more than is predicted
• Radius ratios are only correct ca. 50% of the time, not very good for a family of archetypal ionic solids - random choice might be just as successful as radius ratio rules and saying that all adopt the NaCl structure more so!
• Is the Goldschmidt cation-anion Contact Criterion all there is to it? - No!

Lattice Energy

Qu. Why is highest possible Coordination adopted?

Ans. Greater Madelung Potentials!

Structure Type	Madelung Constant
CsCl	1.763
NaCl	1.748
ZnS (Wurtzite)	1.641
ZnS (Zinc Blende)	1.638

Plots of Coulombic Madelung Energy vs Radius Ratio

• Madelung energy rises as r₊/r_- falls until the structure can no longer support cation-anion contact.
• At the geometric limiting radius ratio, the Madelung energy remains constant as r₊/r_- falls, because r_o ­ r₊ + r_- but instead is limited by "anion-anion contact".
• Plot indicates structural transitions do not occur at limiting radius ratios
  • NaCl CsCl transition at r₊/r_- Å 0.71
  • ZnS NaCl transition at r₊/r_- Å 0.32
• Taking into account that r₈ > r₆ indicates that the CsCl structure is never favourable (dotted line)
• CsCl structure is only adopted where 8:8 coordination maximizes Dispersion Forces
• NaCl structure is highly favoured by Covalency best utilization of Cl^- p³ orbitals

Pressure-dependence of structures?

All Expect CsCl structure to be favoured at high pressures

e.g. RbCl undergoes NaCl CsCl structural transition at 5~20 kbar

When the Radius Ratio Rules Work do they do so for the Right Reasons?

• Radius ratios might appear to work well for predicting structures of oxides, MO₂

  e.g. r(Ti⁴⁺)/r(O^2-)

  • = 75 pm / 126 pm (Shannon & Prewitt) = 0.60

    = 61 pm / 140 pm (Pauling) = 0.44

  6:3 Coordination, as is observed in practice for TiO₂

  But...

• Ti⁴⁺ is very polarizing Covalency probably predominates in bonding?!
• r(O^2-) was determined in systems like MgO, CaO etc...

  Perhaps we really should use Covalent Radii?

  • r_cov(O)/r_cov(Ti) = 73 pm / Å 95 pm = 0.77

  8:4 Coordination, i.e. Fluorite structure - NOT what's observed!

Alternative reasoning?

Ti in TiO₂ is 6-coordinate because this maximizes covalent bonding, not because the radius ratio rules are necessarily correct!

Beware!

Use radius ratios as a simple Predictional Tool, not as a means of rationalizing structures

Is There Any General Influence of Ionic Radii on Structure?

Structure Maps

• Separation of structure types is achieved on structure diagrams - but, the boundaries are complex
• Conclusion:- Size does matter, but not necessarily in any simple way!

Pauling Rule 2: Electrostatic Valence Principle ("Bond Strength")

"In a stable ionic structure the charge on an ion is balanced by the sum of electrostatic bond strengths to the ions in its coordination polyhedron"

i.e. A stable ionic structure must be arranged to preserve Local Electroneutrality

(ions in a crystal are surrounded by ions of opposite charge so as not to produce large volumes of similar charge in the crystal - this maximizes Madelung potential!)

Electrostatic Bond Strength (e.b.s.)

• For a cation M^m+ surrounded by n anions X^x- the electrostatic bond strength of the cation is defined as:-

• For each anion (cation) the sum of the electrostatic bond strengths of the surrounding cations (anions) must balance the negative (positive) charge on the anion (cation)

For a binary compound A_xB_y the coordination numbers of A and B are in the ratio y:x

e.g. Fluorite, CaF₂ Ca²⁺ (8-coordinate), F^- (4-coordinate)

• Can allow Structure Prediction/Rationalization

  e.g. In Perovskite, CaTiO₃

  • Ca²⁺ is 12-coordinated by O^2- Ca-O bond has e.b.s. = ²/₁₂ = ¹/₆
  • Ti⁴⁺ is 6-coordinated by O^2- Ti-O bond has e.b.s. = ⁴/₆ = ²/₃

{Green = Ca; Blue = Ti; Red = O}

• O^2- has a total valency of 2 satisfied by {4 x Ca²⁺(¹/₆)} +{2 x Ti⁴⁺(²/₃)}

  Each Oxygen must be common to 4 CaO₁₂ cuboctahedra & 2 TiO₆ octahedra

• This information suffices to define the idealized structural arrangement Uniquely

Bond Valence Calculations in 'Real' Solids

Coordination polyhedra are not always regular - precise structures often reveal a range of bond distances.

For each bond:

Valency =

CaCrF₅: has very distorted CrF₆ octahedra

View a Quicktime CaCrF₅ Movieor Quicktime CaCrF₅ VR scene

 

Cr 2{F(1) 194 pm} + 2{F(2) 191.8 pm} + 2{F(3) 184.8 pm}

e.b.s. 2{0.42} + 2{0.46} + 2{0.65} = 3.06

Ca {F(1) 249 pm} + 2{F(2) 229.1 pm} + 2{F(2) 239.1 pm} + 2{F(3) 224.4 pm}

e.b.s. {0.20} + 2{0.29} + 2{0.24} + 2{0.33} = 1.92

F(1) 2{Cr 194 pm} + {Ca 249 pm}

e.b.s. 2{0.42} + {0.20} = 1.02

F(2) {Cr 191.8 pm} + {Ca 229.1 pm} + {Ca 239.1 pm}

e.b.s. {0.46} + {0.29} + {0.24} = 0.99

F(3) {Cr 184.8 pm} + {Ca 224.4 pm}

e.b.s. {0.65} + {0.33} = 0.98

• Electrostatic Bond Strength Summations are used to check aspects of structure determinations

  e.g. That sites in complex structures have been assigned to the correct ions

Pauling Rule 3: Polyhedral Linking

"The stability of structures with different types of polyhedral linking is vertex-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

Why? Sharing edges/faces brings ions at the centre of each polyhedron closer together, hence increasing electrostatic repulsions

i.e. disposition of ions of similar charge will be such as to minimize the Electrostatic Energy between them

Exceptions?

• Obeyed by compounds of high polarity, e.g. fluorides/oxides
• Not obeyed by compounds of lower polarity

  e.g. SiO₂ = vertex-linked Tetrahedra but SiS₂ = edge-linked Tetrahedra

• Flouted by some compounds
  • e.g. for NiAs, face-sharing promotes Ni···Ni bonding

Pauling Rule 4: Cation Evasion in >Binaries

"In a crystal containing different cations those of high valency and small coordination number tend not to share polyhedron elements with each other"

e.g. In Perovskite, CaTiO₃

Ca^II 12-coordinate CaO₁₂ cuboctahedra share FACES

Ti^IV 6-coordinate TiO₆ octahedra share only VERTICES

Pauling Rule 5: Environmental Homogeneity

"The number of essentially different kinds of constituent in a crystal tend to be small"

i.e. as far as possible, similar environments for chemically similar atoms

e.g. Treating the mineral Garnet Ca₃Al₂Si₃O₁₂ as an ionic crystal

	Ca2+	Al3+	Si4+
coordination	8	6	4
e.b.s.	1/4	1/2	1

• O^2- bond strength of 2 is satisfied by a number of alternative combination of bonds
• Pauling Rule 5 Each O^2- would prefer the same environment
• Only one possible arrangement:

  

• This defines the Garnet structure uniquely

  Rule 5 is, however, often not obeyed

When Pauling's Rules are NOT Obeyed

Special structural influences in the bonding

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Indirect evidence that the structure is NOT ionic

Increasing Polarization in bonding

low-dimensionality - layers/chains

Crystalmaker Files for CaF₂, HgI₂, SiS₂

Fajan's Rules (Polarization)

Polarization will be increased by:-

1. High charge and small size of the cation

Ionic potential Å Z₊/r₊ (= polarizing power)

2. High charge and large size of the anion

The polarizability of an anion is related to the deformability of its electron cloud (i.e. its "softness")

3. An incomplete valence shell electron configuration

Noble gas configuration of the cation better shielding less polarizing power

i.e. charge factor in (1) should be effective nuclear charge

e.g. Hg²⁺ (r₊ = 102 pm) is more polarising than Ca²⁺ (r₊ = 100 pm)

BOND DIRECTIONALITY / Extent of Covalency

Can this be used to rationalize / predict structures?

• Electronegativity: defined by Pauling as "The power of an atom in a molecule to attract electrons to itself".
• Bond type depends on the electronegativity difference between the elements involved.

Ketelaar's Triangle

Ketelaar's Triangle finds different structure types in different regions

i.e. according to the electronegativities of A and B

Mooser-Pearson Plots

Combining Bond Directionality, Size and Electronegativity in a Structure Map

• x-axis is Electronegativity Difference, (partially related to bond ionicity)
• y-axis is the average principal quantum number

  • c_i = no. of atoms of i per formula unit

    is related to bond directionality

• larger larger orbitals

  • less directionality

• sharp borderlines observed a critical ionicity which determines AB structures
• boundaries are curved - not so helpful!
• Indicates the increased ionicity of Wurtzite over Zinc Blende
  • (N.B. higher Madelung Constant of Wurtzite)

Phillips-van Vechten Plots

• Like Mooser-Pearson Plots based on Critical Ionicity
• Phillips & van Vechten used a better, theoretically-determined concept of ionicity
• HOMO LUMO transition of A^NB^8-N compounds is:-

 

• Clean Straight-line separation between 6:6 and 4:4 structures with a critical ionicity of

Connoisseurs of Structure-Type prediction will want to familiarise themselves with the work of Oxford's Prof. Paul Madden on the role of Induction & Dispersion Forces in determining structures of solids.

see e.g. P.A. Madden, Chem. Soc. Reviews, 1997, 25, p. 339, "Covalent Effects in 'Ionic' Crystals"

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