1. The Chemistry of Solids
Chapter 11
Miller Indices (l,m,n) are a way of denoting planes in crystal lattices.
The Chemistry of Solids

2. Band Theory
As the half-filled 4s orbitals of an increasing number of Cu atoms overlap, their energies are split into a half-filled valence band.
Electrons can move from the filled half (purple) to the slightly higher energy upper half (red), where they are free to migrate from one empty orbital to another.

3. Metallic Bonds
Band theory is an extension of molecular orbital theory that describes bonding in solids.
Bands of orbitals that are filled or partially filled by valence electrons are called valence bands.
Higher-energy unoccupied bands in which electrons are free to migrate are called conduction bands.

4. Band Gap
The energy gap between the valence and conduction bands is called the band gap.
A semiconductor is a substance whose conductivity can be made to vary over several orders of magnitude by altering its chemical composition.
A n-type semiconductor contains excess electrons contributed by electron-rich dopant atoms.
A p-type semiconductor contains electron-poor dopant atoms.

5. Band Theory
The Fermi Energy (Level) is the energy of the highest occupied state.

6. Band Theory
The Band Gap influences the electrical and optical properties of the material. Semiconductor “doping” can be used to create solar cell (PV), diodes, transistors, etc.

7. Some semiconducting materials and associated bandgaps (eV)

8. n-type semiconductors result from the addition of pentavalent impurities like phosphorus, arsenic and antimony. These donors contribute “extra” electrons.
p-type semiconductors result from the addition of trivalent impurities like boron, aluminum and gallium. These additions create valence “holes” which act as additional levels

10. The “unit cell” is the basic repeating unit of the arrangement of atoms, ions or molecules in a crystalline solid.
The “lattice” refers to the 3-D array of particles in a crystalline solid. One type of atom occupies a “lattice point” in the array.

11. Examples of Unit Cells

12. There are seven crystal systems also knows as Bravais Structures.
In Chapter 11 we will cover only the cubic systems, sc, fcc, and bcc.
Sides:
a = b = c
Angles
a = b = g = 90°

13. . A: Body-centered cubic (bcc) B: Simple cubic (sc)

14. Contributions of Atoms to Cubic Unit Cells
Position of Atoms in Unit Cell
Contribution to Unit Cell
Unit-Cell Type
Center 1
bcc
Face
1/2
Fcc
Corner
1/8
fcc, bcc, simple cubic
How many total atoms are found in a simple cubic unit cell? Face centered cube? Body centered cube?

15. Unit Cells
A body-centered cubic (bcc) unit cell has atoms at the 8 corners of a cube and at the center of the cell
A simple cubic unit cell has atoms only at the 8 corners of a cube.

16. Number Atoms in a Unit Cell
In the simple cubic cell there are only the 8 atoms at the corners.
1/8 x 8 = 1 atom in cell
In bcc, 8 atoms at the corners and 1 in center.
1/8 x x 1 = 2 atoms in the cell

17. Using the figure below, determine the formula for the mineral perovskite, CaxTiyOz.
2 Green & blue = oxygen, Gray = titanium,
Red = calcium

18. Example Problem:
Polonium metal crystallizes in a simple cubic structure. Calculate the density of the polonium metal if the atom radius is 176 pm.
[Based on a literature density of g cm-3, what is the radius of Po? (167 pm)]

20. Problem The radius of the copper atom is 127
Problem The radius of the copper atom is pm, and its’ density is 8.95 g/cm3. Which unit cell is consistent with these data: sc, bcc, or fcc?

23. Calculate the density of lithium (6
Calculate the density of lithium (6.941 g/mol) metal at 20°C if it forms a body centered cubic (bcc) unit cell and the radius of the lithium atom is 1.52 x 10-8 cm. [assume the atoms touch across the long cubic diagonal]

25. Fig. 10.4: In the crystalline solid NaCl, the simplest repeating unit cell is a cube containing eight chloride ions at the corners of the cube, six chloride ions in the faces of the cube, twelve sodium ions on the edges of the cube and one sodium ion in the center of the cube. This arrangement is called a face-centered cube (fcc). Notice the Na+ ion in the center of the unit cell; it has six closet neighbors or a coordination number of 6.
NOTE: Only Cl- atoms define the Unit Cell

26. Metallic Crystals can be thought to form via an efficient packing scheme…

27. Stacking Patterns
A crystalline solid is made of an ordered array of atoms, ions, or molecules.
Hexagonal closest-packed (hcp) describes a crystal structure in which the layers of atoms or ions in hexagonal unit cell have an a-b-a-b-a-b stacking pattern.
Cubic closest packed (ccp) describes a crystal structure in which the layers of atoms, ions, or molecules in face-centered cubic unit cells have an a-b-c-a-b-c-a-b-c stacking pattern.

28. Fig. 10.6: Two efficient ways to pack particles together in a crystal structure are hexagonal closest packed (A, hcp) and cubic closest packed (B, ccp). The difference is in the third layer. In hcp the third layer sits directly above the first layer (ababab…). In ccp the third layer is different from the other two (abcabcabc…). The NaCl crystal structure contains a cubic closest packed arrangement of chloride ions.

29. Fig. 10.13b: Cubic closest packed structure

30. Hexagonal closest packed structure
The hcc (aba) structure has 6 atoms/uc which includes 6 octahedral holes and 12 tetrahedral holes.

31. Summary of Crystal Structures

32. Fig. 10.7: A closest packed structure contains holes that other ions can fit into.
Na+ ions in a NaCl crystal have six closest neighbors or a coordination number of 6. Therefore the Na+ ions sit in octahedral holes in the ccp structure.

33. Fig. 10.12: Cesium chloride (CsCl) crystal structure.

34. Example Problem:
What is the packing efficiency in the simple cubic cell of CsCl? What is the percentage of empty space in the unit cell? The chloride ions are at the corners with the cesium in the middle of the unit cell.
rCl- = 181 pm; rCs+ = 169 pm

38. Fig. 10.8: The location of the octahedral holes occupied by sodium ions are shown by the X’s.

39. Fig. 10.9: The “X” marks the location of the tetrahedral holes in a face centered cubic unit cell

40. Fig : The mineral sphalerite (ZnS) is based on a fcc arrangement of S2- ions with four Zn2+ ions in tetrahedral holes.
Are all the tetrahedral holes occupied?

41. Fig : The mineral fluorite (CaF2) has a fcc arrangement of Ca2+ ions with F ions in all eight of the tetrahedral holes.

42. rCl- = 181 pm; rNa+ = 98 pm; edge dist.NaCl = 562.8 pm
Example Problem
What is the packing efficiency of NaCl? What is the percentage of empty space in the NaCl unit cell? )
rCl- = 181 pm; rNa+ = 98 pm; edge dist.NaCl = pm

45. Atomic Size Ratios and the Location of Atoms in Unit Cells
Packing
Type of Hole
Radius Ratio
hcp or ccp
Tetrahedral
Octahedral
Simple Cubic
Cubic

46. Problem
One of the expected “products” of “cold” fusion is 3He, radius 32 pm.
Could a 3He fit in the octahedral holes form by Pd atoms?

48. Table 10.2 Ion size Ratios and holes in Closest Packed Structures
Packing
Type of holes
Rsmall/Rlarge
hcp/ccp
Tetrahedral
Octahedral
cubic
Cubic
For NaCl the radius ratio = RNa+/RCl- = 98pm/181pm =0.54
From the table data the Na+ ions are “predicted” to occupy Tetrahedral hole.

49. Alloys
An alloy is a blend of a host metal and one or more other elements which are added to change the properties of the host metal.
In a substitutional alloy the atoms of one metal replace atom in the crystal lattice.
Interstitial alloys are formed when hetero atoms occupy interstitial octahedral and tetrahedral holes of the host metal lattice.

50. Bronze is substitutional allow

51. Carbon Steel is an interstitial alloy

52. Effect of Carbon Content on the Properties of Steel
Designation
Properties
Used to Make
Low Carbon
Malleable, ductile
Nails, cables
Medium Carbon
High strength
Construction girders
High Carbon
Hard but brittle
Cutting tools

53. Network Solids
Covalent network solids are made of a rigid, three-dimensional array of covalently bonded atoms.
Crystals of molecular solids are formed by neutral covalently bonded molecules held together by intermolecular attractive forces.

54. Quartz (SiO2), a crystalline solid and the most abundant mineral in the earth’s crust)
Fig. 10.2: Crystalline solids are characterized by highly ordered arrays of atoms, ions or molecules and give distinct X-ray diffraction patterns.
Amorphous solids have no long-range ordering in their structures.
What type of attractive forces are most important in the quartz structure?
Glass (SiO2), has the same chemical composition as quartz but exhibits no extensive, repeating organization.

55. Crystalline Versus Amorphous
Quartz Figure 11.26
Obsidian Figure 11.27

56. Allotropes of Carbon

57. Allotropes of Phosphorus

58. Other Ionic Crystals

59. Pyroxenes
i.e. jadeite
Orthosilicates
i.e. olivine
Cyclic SiO4 units
i.e. beryl
Layered silicate structures in clays
i.e. kaolinite
Because of covalent linkages across unit cells, silicates are an example of a network solid.

60. Other forms of Silica

61. The phase diagram for SiO2 shows that many temperature and pressure dependent solid phases can form. Consequently we find many polymorphs of SiO2 present in nature.

62. Superconductors
A superconductor is a material that has zero resistance to the flow of electric current.
The critical temperature (Tc) is the temperature below which a material becomes a superconductor.
Current superconductors, like Nb3Sn, have to be cooled to 20 K to remain superconducting.

63. High Temperature Superconductors
YBa2Cu3O7 ceramic is superconducting at 77K (just above liquid nitrogen’s boiling point).
The ceramic is structure is called a perovskite unit cell.

64. Yttrium-barium-copper Oxides
These and related materials behave as superconductors because of the formation of electron pairs called Cooper pairs.

65. X-ray Diffraction
X-ray diffraction (XRD) is a technique for determining the arrangement of atoms or ion in a crystal by analyzing the pattern that results when X-rays are scattered after bombarding the crystal.
The Bragg equation relates the angle of diffraction (2) of X-rays to the spacing (d) between the layers of ions or atoms in a crystal: n2dsin.

66. Structure Determination
X-ray Diffraction (XRD).
The Bragg equation
nλ = 2dsin θ
Can be used in conjunction with XRD data to find the spacing between crystal planes (d).
l is the wavelength of the X-rays; n is an integer; d is the distance between crystal planes; θ is the angle of incidence.

67. Peaks result from constructive interference for the reflected X-rays
Fig. 10.3c: This is an example of an XRD spectrum for quartz.

68. Problem (text)
Cobalt(II) oxide is used as a pigment in pottery. It has the same type of crystal structure as NaCl. When exposed to X-rays (l=153 pm) reflections were observed at 42.38°, 65.68°, and 92.60°. Determine the values of n to which these reflections correspond, and calculate the spacing between the crystal layers.
nλ = 2dsin θ