User:Benjah-bmm27/degree/2/SRH

Solid state, SRH

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

• Close-packing of spheres, HCP, CCP
• Unit cell, Miller indices

• Common structure types: Cubic diamond, NaCl, CsCl, NiAs, β-ZnS, α-ZnS, CdCl₂, CdI₂, TiO₂, CaF₂, CaTiO₃, ReO₃

	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

• Molybdenum disulfide: layer sulfide - weak interactions between layers, so they slip past each other easily - makes MoS₂ a useful lubricant
• Lattice energy: Born-Landé equation, Madelung constant, Kapustinskii equation

Lattice energy

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), E_C, based on Coulomb's law
2. Repulsive interactions, E_A
3. van der Waals (dispersion) interactions, E_D
4. Zero-point energy (vibrational energy), E₀

• The lattice energy is a sum of these terms: $\ E=E_{C}+E_{A}+E_{D}+E_{0}$

• Coulomb's law (force between two ions, + and −, separated by a distance d): $F_{+-}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {z^{+}z^{-}}{d^{2}}}$

• Energy due to electrostatic interaction between two charged ions (+ and −): $E_{+-}={1 \over 4\pi \varepsilon _{0}}{z^{+}z^{-} \over d}\$

• 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: $E_{C}=-{\frac {N_{\text{A}}e^{2}}{4\pi \varepsilon _{0}}}\sum {\frac {z^{+}z^{-}}{d}}$

• 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 $\ d$, 12 Na⁺ at a distance ${\frac {d}{\sqrt {2}}}$, 8 Cl⁻ at a distance ${\frac {d}{\sqrt {3}}}$, and many more ions at greater distances.

$E_{C}=-{\frac {N_{\text{A}}e^{2}}{4\pi \varepsilon _{0}}}{\frac {z^{+}z^{-}}{d}}\left[{-6\left(1\right)+12\left({\frac {1}{\sqrt {2}}}\right)-8\left({\frac {1}{\sqrt {3}}}\right)+\ldots }\right]$

$A=\left[{-6\left(1\right)+12\left({\frac {1}{\sqrt {2}}}\right)-8\left({\frac {1}{\sqrt {3}}}\right)+\ldots }\right]\to 1.748$

For NaCl, the electrostatic energy term in the lattice energy is $E_{C}=-{\frac {\nu N_{\text{A}}z^{+}z^{-}e^{2}}{4\pi \varepsilon _{0}\left({r^{+}+r^{-}}\right)}}\left({\frac {1.748}{2}}\right)$

An approximate expression for the lattice energy, independent of crystal structure, is $L_{0}={\frac {-\left({1.214\times 10^{5}}\right)\nu z^{+}z^{-}}{\left({r^{+}+r^{-}}\right)}}$

Radius ratio

• 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

• 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

${\text{electrostatic bond strength of a cation-anion bond}}={\frac {\text{charge on cation}}{\text{number of neighbouring anions}}}$

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

• Free electron model, Density of states
• Band theory
• Fermi energy, Fermi level

Solid state reactions

• Solid-state synthesis: ceramic method (heat and beat, shake and bake)
• Coprecipitation, Solid-state reaction, fr:Chimie douce, Redox, Comproportionation, Electrolysis
• Intercalation: potassium graphite
• Ionic conductivity: fast ion conductors

Defects

• Lattice defects:
  • Schottky defects
  • Frenkel defects
  • Dislocations:
    • Edge dislocations
    • Screw dislocations
    • Burgers vector describes orientation and magnitude (perpendicular to dislocation line for edge dislocations, but parallel to line for screw dislocations)
    • Dislocations lead to work hardening (so many dislocations present that they impede each others' movement)
    • Annealing: heating a crystal to remove dislocations

One-dimensional conductors

n[Pt(CN)₄]²⁻ → ([Pt(CN)₄]^1.7−)_n

• Atomic wires - cyanoplatinates - Jmol models:
  • Crystal structure of K₂[Pt(CN)₄].3H₂O: Inorg. Chem. (1976) 15, 74–78
  • Crystal structure of K₂[Pt(CN)₄]Br_0.3·3H₂O: Mater. Res. Bull. (1975) 10, 411-415 (plus this and this)
  • Crystal structure of K₂[Pt(CN)₄]: Z. Anorg. Allg. Chem. (2004) 630, 1462-1468
  • Crystal structure of K₂[Pt(CN)₄Br₂]: Z. Naturforsch. B (2004) 59b, 567-572

K₂[Pt(CN)₄].3H₂O + 0.15 Br₂ → K₂[Pt(CN)₄]Br_0.3·3H₂O

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]Br_0.3·3H₂O. 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

• Electron magnetic dipole moment

• Magnetic structure: magnet, diamagnetism, paramagnetism, magnetic susceptibility, Curie–Weiss law, ferromagnetism, antiferromagnetism, ferrimagnetism, antiferrimagnetism, Néel temperature

• Other magnetism stuff: spin glass, photomagnetism, metamagnetism, superparamagnetism

Superconductivity

• Superexchange: MnO, FeO, CoO, NiO;
• Superconductivity: zero resistance + Meissner effect, Heike Kamerlingh Onnes, high-temperature superconductivity, yttrium barium copper oxide

• Semiconductors