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Solid state, SRH

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Solid State Chemistry
Oxford VR Chemistry: Solid State
Oxford first year inorganic: Structures of Simple Inorganic Solids
Cambridge introduction to structures of some solids

Crystal structures

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A AB AB2
cubic close packed ccp NaCl, β-ZnS CaF2, Na2O, CdCl2
hexagonal close packed hcp NiAs, α-ZnS hcp-CaF2, CdI2, TiO2
body-centred cubic bcc
primitive cubic cubic P CsCl

Lattice energy

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Lattice energy, E, is the energy required to destroy a crystal structure, so for NaCl, E is ΔH for the process NaCl(s) → Na+(g) + Cl(g)

For an idealised ionic solid, the lattice energy can be considered to arise from four effects, each with its own energy term:

  1. Electrostatic interactions (Coulomb energy), EC, based on Coulomb's law
  2. Repulsive interactions, EA
  3. van der Waals (dispersion) interactions, ED
  4. Zero-point energy (vibrational energy), E0
  • The lattice energy is a sum of these terms:
  • Coulomb's law (force between two ions, + and −, separated by a distance d):
  • Energy due to electrostatic interaction between two charged ions (+ and −):
  • The Coulomb energy of the ionic solid is therefore the sum of the energies of all the electrostatic interactions between all possible pairs of ions:
  • Combine the geometrical part (distance of each ion from a reference point) into one term called the Madelung constant, A, which is different for each structure type.


For example, NaCl. Using one Na+ as a reference point, coordinates (0,0,0), there are 6 Cl at a distance , 12 Na+ at a distance , 8 Cl at a distance , and many more ions at greater distances.




For NaCl, the electrostatic energy term in the lattice energy is


An approximate expression for the lattice energy, independent of crystal structure, is

Radius ratio

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  • Cation-anion radius ratio, radius ratio = r2 / r1, where r1 is the radius of the larger ion (usually the anion) and r2 is the radius of the smaller ion (usually the cation)
Radius Ratio Coordination number Type of hole Example
0.225-0.414 4 Tetrahedral ZnS, CuCl
0.414-0.732 6 Octahedral NaCl, MgO
0.732-1.000 8 Cubic CsCl, NH4Br

Pauling's rules

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  • Pauling's rules: Linus Pauling (1929). "The principles determining the structure of complex ionic crystals". J. Am. Chem. Soc. 51 (4): 1010–1026. doi:10.1021/ja01379a006.
1. Coordination polyhedra: "The coordination number of the cation will be maximized subject to the criterion of maintaining cation-anion contact"
2. Electrostatic valence principle: "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"
3. Polyhedral linking: "The stability of structures with different types of polyhedral linking is vertex sharing > edge-sharing > face-sharing"
4. Cation evasion: In a crystal containing different cations, those of high valency and small coordination number tend not to share polyhedron elements with each other
5. Environmental homogeneity or The principle of parsimony The number of different kinds of constituent in a crystal tends to be small


Rule 2 means local electroneutrality must be preserved to minimise repusive Coulombic interactions - i.e. to maximise the Madelung potential
Rule 3 explains why no hcp analogue of the fluorite structure has been observed: the hcp fluorite structure has face-sharing tetrahedra which give rise to strong electrostatic repulsion between ions of like charge at the centres of neighbouring tetrahedra
Rule 5 means crystals tend to have similar environments for chemically similar atoms.

Band theory

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Solid state reactions

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Defects

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One-dimensional conductors

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n[Pt(CN)4]2− → ([Pt(CN)4]1.7−)n
K2[Pt(CN)4].3H2O + 0.15 Br2 → K2[Pt(CN)4]Br0.3·3H2O
Formula xtal colour Pt ox. state Pt d e Pt···Pt separation
K2[Pt(CN)4].3H2O yellow +2.0 d8 3.48 Å
K2[Pt(CN)4]Br0.3·3H2O bronze +2.3 d7.7 2.89 Å
c.f. Pt metal, Pt···Pt 2.77 Å
  • K2[Pt(CN)4].0.3Br.3H2O good conductor along Pt···Pt axes, poor conductor perpendicular to Pt···Pt axes.
  • Conductivity disappears below 150 K
  • Form a band due to overlapping Pt 5d2
    z
    orbitals - band is full in reduced salt K2[Pt(CN)4].3H2O, but partially occupied (loss of 0-0.4 electrons per Pt) in oxidised salt K2[Pt(CN)4]Br0.3·3H2O. Top of band is antibonding, so removal of electrons from it increases Pt-Pt bonding and reduced Pt···Pt separation. Fermi level within the band leads to conductivity along chain.
  • Peierls’ Theorem: "a 1-D chain will distort to split a partially filled band at the Fermi level, thereby reducing the electronic energy and destroying conductivity".
  • Above 150 K, distortion is not apparent due to thermal vibration, but below 150 K, a band gap appears, and the oxidised salt becomes a semiconductor.

Magnetism

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Superconductivity

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