The Crystalline Solid State

1. The Crystalline Solid State Chapter 7

2. Crystalline Solid State • Many more “molecules” in the solid state. • We will focus on crystalline solids composed of atoms or ions. • Unit cell – structural component that, when repeated in all directions, results in a macroscopic (observable) crystal. • 14 possible crystal structures (Bravais lattices) • Discuss positions of atoms in the unit cell.

3. The Cubic Unit Cell (or Primitive) • 1 atom per unit cell (how?). • What is the coordination number? Volume occupied? • Let’s calculate the length of the edge. What size of sphere would fit into the hole?

4. The Body-Centered Cubic • How many atoms per unit cell? • What is the length of the edge? This is a more complicated systems than the simple cubic.

5. Close-Packed Structures • How many atoms is each atom surrounded by in the same plane? • What is the coordination number? • Hexagonal close packing (hcp) – discuss the third layer (ABA). • Cubic close packing (ccp) or face-centered cubic (fcc) – discuss the third layer (ABC). • Two tetrahedral holes and one octahedral hole per atom. Can you see them?

6. Close-Packed Structures • The hcp has hexagonal prisms sharing vertical faces (Figure). • How many atoms per unit cell in the hcp structure? • What is the length of the cell edge? • The unit cell for the ccp or fcc is harder to see. • Need four close-packed layers to complete the cube. • What is the length of the cell edge? • In both close-packed structures, 74.1% of the total volume is occupied.

7. Ionic Crystals • The tetrahedral and octahedral holes can have varying occupancies. • Holes are generally filled by smaller ions. • Tetrahedral holes • Octahedral holes • NaCl structure

8. Metallic Crystals • Most crystalize in bcc, ccp, and hcp structures. • Hard sphere model does not work well. • Depends on electronic structure. • Properties • Conductivity • Dislocations

9. Diamond • Each carbon atom is bonded tetrahedrally to four nearest neighbors (Figure). • Essentially the same strength in all directions.

10. Structures of Binary Compounds • Close-packed structures are generally defined by the larger ions (usually anions). The oppositely-charged ions occupy the holes. • Two important factors in considering the structure • Radius ratio (r+/r-) • Relative number of combining cations and anions.

11. NaCl Crystal Structure • Face-centered cubes of both ions offset by a half a unit cell in one direction. • Many alkali metals have this same geometry. • What is the coordination number (nearest neighbor)?

12. CsCl Crystal Structure • Chloride ions form simple cubes with cesium ions in the center (Figure 7-7). • The cesium ion is able to fit in to center hole. How? • Other crystal structures.

13. TiO2 (the rutile structure) • Distorted TiO6 octahedra. • Ti has a C.N. of 6, octahedral coordination • O has a C.N. of 3

14. Rationalization of Structure of Crystalline Solids • Predicting coordination number from radius ratio (r+/r-). • A hard sphere treatment of the ions. • Treats bonding as purely ionic. • Simply, as as the M+ ratio increases, more anions can pack around it. • Table 7-1. Let’s look at a few (NaCl, CaF2, and CaCl2).

15. Thermodynamics of Ionic Crystal Formation • A compound tends to adopt the crystal structure corresponding to lowest Gibbs energy. M+(g) + X-(g)  MX(s) G = H - TS (standard state), 2nd term can be ignored • Lattice enthalpy MX(s)  M+(g) + X-(g) HL (standard molar enthalpy change) Currently, we are interested in lattice formation.

16. The Born-Haber Cycle • A special thermodynamic cycle that includes lattice formation as one step. • The cycle has to sum up to zero if written appropriately. • Write down values for KCl.

17. The Born-Haber Cycle • Calculate the lattice enthalpy for MgBr2. • A discrepancy between this value and the real value may indicate the degree of covalent character. • We have assumed Coulombic interactions between ions. • The actual values for KCl and MgBr2 are 701 and 2406 kJ/mol (versus 720 and 2451).

18. Lattice Enthalpy Calculations • Considering only Coulombic contributions • The electrostatic potential energy between each pair. zA, zB = ionic charges in electron units r0 = distance between ion centers e = electronic charge 4o = permittivity of a vacuum e2/ 4o = 2.307  10-28 J m Calculation would be performed on each cation/anion pair (nearest neighbor).

19. Lattice Enthalpy Calculations • A more accurate equation depicts the Coulombic interactions over the entire crystal. NA = Avogadro’s constant A = Madelung’s constant, value specific to a crystal type (in table). This is a sum of all the geometric factors carried out until the interaction become infinitesimal.

20. Lattice Enthalpy Calculations • Repulsions between ions in close proximity term. C’ = constant (will cancel out when finding the minimum) • = compressibility constant, ~ 30 pm • Combining terms

21. Lattice Enthalpy Calculations • Finding the minimum energy • dU/dr0 = O • A negative of this value may be defined as the lattice enthalpy.

22. Lattice Enthalpy Calculations • As the polarizability of the resultant ions increase the agreement with this ionic model worsens. • Polarizibility generally indicates more covalent character. Calculations NaCl and CaBr2

23. Molecular Orbitals in Solids • A very large number of atoms are used to generate molecular orbitals. • One-dimensional model. • Creation of bands that are closely spaced. • Factors affecting the width of the band. This would be called an ‘s band’. A similar model can be constructed for the p-orbitals and d-orbitals.

24. The Bonding Picture in Solids

25. Molecular Orbitals in Solids • Band gap – separation between bands in which no MOs exist (Figure 7-13). • Valence band – highest energy band containing electrons. • Conduction band – the band immediately above the valence band in energy.

26. Metals and Insulators • Metals • Partially filled valence band (e.g. s band) • Electrons move to slightly higher energy levels by applying a small voltage. Electrons and ‘holes’ are both free to move in the metal. • Overlapping bands (e.g. s and p bands) • If the bands are close enough in energy (or overlapping) an applied voltage can cause the electrons to jump into the next band (conduction band).

27. Density of States • Concentration of energy levels within a band. • Helps to describe bonding/reactivity in solids.

28. Conductivity of Solids Versus Temperature • Metals – decrease with temperature. • Semiconductors – increase with temperature. • Insulators – increase with temperature (if measurable).

29. Semiconductor Types • Intrinsic semiconductors – pure material having semiconductive properties. • Doped semiconductors – semiconductors that are fabricated by adding a small amount of another element with energy levels close to the pure state material. • n-type semiconductors • p-type semiconductors (look at figure)

30. Semiconductors • Fermi-level (semiconductor) – the energy at which an electron is equally likely to be in each of two levels (Figure). • Effects of dopants on the Fermi level. • n-type and p-type.

31. Diodes (creating p-n junctions) • Migration of electrons from the n-type material to the p-type material. • Equilibrium is established due to charge transfer. • Application of a negative potential to the n-type material and a positive potential to the p-type material. • Discuss (Figure 7-16).

32. Superconductivity • No resistance to flow of electrons. • Currents started in a loop will continue to flow indefinitely. • Type I superconductors – expel all magnetic fields below a critical temperature, Tc (Meisner effect). • Type II superconductors – below a critical temperature exclude all magnetic fields completely. Between this temperature and a second critical temperature, they allow partial penetration by the magnetic field. • Levitation experiment works well.

33. Theory of Superconducting • Cooper pair theory • Bardeen, Cooper, and Schrieffer • Electrons travel through the material in pairs. • The formation and propagation of these pairs is assisted by small vibrations in the lattice. • discuss

34. YBa2Cu3O7 High-Temperature Superconductors • Discovered in 1987 and has a Tc of 93 K. • N2(l) can be used • Type II superconductor. • Difficult to work with. • Possesses copper oxide planes and chains.

35. Bonding in Solid State Structures • The hard-sphere model is too simplistic. • Deviations are observed in ion sizes. • Sharing of electrons (or transfer back to the cation) can vary depending upon the polarizability. • LiI versus NaCl (which structure would exhibit more covalent character?)

36. Bonding in TiO2 • The crystal has a rutile structure. • Each titanium has ___ nearest neighbors and each oxygen atom has ___ nearest neighbors. • There is no effective O···O or Ti···Ti interactions (only Ti···O interactions). Why? • The structure consists of TiO6 fragments (discuss).

37. Bonding in TiO2 For a TiO6 monomer (no significant -bonding). An approximation of the ‘bands in the solid structure.

38. Bonding in TiO2 • The calculated DOS curve in 3-d space is slightly more complicated. • The O 2s, O2p, Ti t2g, and eg bands are well separate. The separation predicts that this material has ‘insulator-like’ properties.

39. Bonding in TiO • Several of the 3d monoxides illustrate high conductivity that decreases with temperature. • TiO and VO (positioning in the table). • TiO adopts the rocksalt structure (NaCl). • Discuss geometry and consequences on bonding.

40. Bonding in TiO • The titanium atoms are close enough to form a ‘conduction’ band. • Overlap of t2g orbitals of the metal ions in neighboring octahedral sites. • Illustrated for dxy orbitals.

41. Bonding in TiO • The calculated DOS curve for TiO reveals that the bonds aren’t well separated. • Diffuse bands indicate more conductive behavior. • Why is TiO2 different than TiO?

42. Bonding in TiO • MnO, FeO, CoO, and NiO do not conduct, but they have the same basic structure. Why?

43. Imperfections in Solids • All crystalline solids possess imperfections. • Crystal growth occurring at many sites causes boundaries to form. • Vacancies and self-interstitials • Substitutions • Dislocations

44. Silicates • The earth’s crustal rocks (clays, soils, and sands) are composed almost entirely (~95%) of silicate minerals and silica (O, Si, and Al). • There exist many structural types with widely varying stoichiometries (replacement of Si by Al is common). Consequences? • Common to all: • SiO4 tetrahedra units • Si is coordinated tetrahedrally to 4 oxygens http://www.soils.wisc.edu/virtual_museum/displays.html http://mineral.galleries.com/minerals/silicate/class.htm

45. The Tetrahedral SiO4 Unit Cheetham and Day

46. Structures with the SiO4 Unit • Discrete structural units which commonly contain cations for charge balance. • Corner sharing of O atoms into larger units. • O lattice is usually close-packed (near) • Charge balance is obtained by presence of cations. Individual units, chains, multiple chains (ribbons), rings, sheets and 3-d networks.

47. Structure Containing Discrete Units • Nesosilicates – no O atoms are shared. • Contain individual SiO44- units. • ZrSiO4 (zircon) – illustrate with softwares • Stoichiometry dictates 8-fold coordination of the cation. • (Mg3 or Fe3)Al2Si3O12 (garnet) – illustrate with softwares • 8-fold coordination for Mg or Fe and 6-fold coordination for the Al.

48. Structure Containing Discrete Units • The sorosilicates (disilicates) – 1 O atom is shared. • Contain Si2O76- units • Show Epidote (Ca2FeAl2(SiO4)(Si2O7)O(OH)) with softwares. • Epidote contains SiO44- and Si2O76- units • Near linear Si-O-Si bond angle between tetrahedra.

49. Cyclosilicates (discrete cyclic units) • Each SiO4 units shares two O atoms with neighboring SiO4 tetrahedra. • Formula – SiO32- or [(SiO3)n]2n- (n=3-6 are the most common. • Beryl – six-linked SiO4 tetrahedra (show with softwares). • Be3Al2(SiO3)6 – contains Si6O1812- cyclic units • The impurities produce its colors. • Wadeite – three-linked SiO4 tetrahedra (don’t have an actual picture) • K2ZrSi3O9

50. Silicates with Chain or Ribbon Structures • Corner sharing of SiO4 tetrahedra (SiO32-) • Very common (usually to build up more complicated silicate structures). • Differing conformations can be adopted by linked tetrahedra. • Changes the repeat distance. • The 2T structure is the most common (long).