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The Solid State

The Solid State. Crystals and symmetry Unit cells and packing Types of solid Phase diagrams. Solids tend to be regular. At equilibrium an atom will tend to occupy the position of lowest energy (highest stability) One position will be more stable than any others

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The Solid State

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  1. The Solid State Crystals and symmetry Unit cells and packing Types of solid Phase diagrams

  2. Solids tend to be regular • At equilibrium an atom will tend to occupy the position of lowest energy (highest stability) • One position will be more stable than any others • If one atom occupies this position then all the other atoms will occupy the same position • All atoms occupy the same position and a regular lattice evolves • Amorphous solids have no regular structure and tend to be metastable or unstable

  3. Four solids

  4. Probing crystal structures • Light is scattered by objects that are larger than the wavelength • Crystal lattices are too small for visible light • X-rays have wavelengths on the order of the interatomic distance in crystals • X-rays suffer diffraction by crystals like visible light diffracted by blinds • X-ray diffraction is the most powerful structural tool developed

  5. Diffraction and interference • Diffraction arises by interference of electromagnetic radiation • Constructive interference – the waves are in phase – increase in intensity • Destructive interference – the waves are out of phase – loss of intensity • X-ray beams diffract from a crystal to give a pattern of spots where constructive interference has occurred on a sea of destructive interference

  6. The crystal lattice and Bragg scattering • X-rays are scattered by the electrons in the atoms • The array of atoms in the crystal is like a diffraction grating for X-rays as a set of slits is for visible light • Diffraction only occurs under conditions of constructive interference • The Bragg equation gives the conditions

  7. The Bragg condition • Waves reflected from adjacent layers must be in phase • The path length difference must then be a whole number of wavelengths

  8. X-ray diffraction and data collection • A crystal or powder is swept through the range of angles of θ and the positions where there are reflections are tabulated • Analysis of the d spacings gives information about the type of crystal lattice • Deeper analysis of the reflection intensities gives a complete description of the structure and positions of all of the atoms

  9. Symmetry and crystal structure • Symmetry underlies chemistry • Arrangements of atoms in crystals • Determination of spectra • Mixtures of orbitals in bonds • Symmetry operators relate the positions of the atoms in the unit cell

  10. Common symmetry elements • Translation • Rotation • Reflection • After application of a symmetry operation to a set of atoms related by symmetry, the system appears unchanged

  11. Translation Reflection Rotation Symmetry operations

  12. Packing of spheres and simple structures • Atoms and simple molecules can be treated like spheres • The crystal lattice can be derived from packing together spheres • There are limited possibilities

  13. Cubic packing • In simple cubic the atoms stack directly on top of each other • Not close packed (very uncommon) • Body-centered cubic: denser packing achieved by putting a layer in the depression of the first • Not close packed (quite common)

  14. Close packing • Two arrangements achieve a higher density • Hexagonal close-packed- abab • Face-centered cubic – abcabc • Both are common for metallic elements

  15. Summary of packing arrangements

  16. Building patterns with unit cells • A floor is made from a mosaic of tiles • A wall is made from stacking of bricks • A crystal is made from stacking unit cells • In each case one unit contains all of the information required to describe the structure completely without any gaps or deficiencies

  17. Building lattices with unit cells • Consider a cube

  18. Counting atoms in a unit cell • The ratio of the atoms in a unit cell must equal the composition of the compound • Count atoms as follows:

  19. Primitive and body-centered cells • Alternate views of the primitive and body-centered cell • Primitive: • 8 atoms on the corners each contribute 1/8th to the contents – overall cell contents = 1 • BCC: • 8 atoms on the corners contribute 1 • 1 atom in center contributes 1 • Overall contents = 2; composition AB

  20. Face-centered cube • Two views of the FCC lattice • Also viewed along 3-fold axis which is perpendicular to the close-packed layers • Contents: • 8 at corners = 1 • 6 on faces = 3 • Total contents = 4

  21. Calculations with unit cells • Calculating unit-cell size from atom size • In FCC cell, the atoms touch along the diagonal • Length of diagonal = 4r • Length of edge =

  22. Estimate density • If we know the unit cell size we can calculate the unit cell volume • If we know the unit cell contents we know the total mass of the cell • Density = mass/volume

  23. Simple ionic compounds • In general, anions are larger than cations • Lattices can be described by close packing of anions with cations occupy regular “holes” in the anion lattice

  24. Sodium chloride • Chloride ions form FCC lattice • Sodium ions occupy octahedral holes • Composition check: • Cl: • 8 on corners = 1 + • 6 on faces = 3 • Total = 4 • Na: • 12 on edges = 3 • 1 in center = 1 • Total = 4

  25. Covalent networks • Ionic lattices are characterized by high coordination numbers and non-directional bonding • Covalent lattices have low coordination numbers and highly directional bonding • The bonds are formed from hybridized atomic orbitals to use the valence bond model

  26. Diamonds are forever • The diamond lattice is a very common covalent lattice • It is a three dimensional tetrahedral net • Bonds are made from sp3 hybrid orbitals • Very strong covalent bonds make the lattice extremely stable

  27. Compounds also have diamond lattice • In GaAs, the Ga and As atoms alternate in the diamond lattice • GaAs is an important semiconductor and laser material • Many other similar materials

  28. Diamond and graphite • Infinite covalent lattices in different dimensions • Diamond 3D • Graphite 2D • Graphite is more stable than diamond, but can be transformed into diamond by application of high pressure

  29. Summing it up: phase diagrams • Phase diagrams summarize the states of a substance as a function of the pressure and temperature • They reveal: • The areas where one phase is stable • The lines where two phases are in equilibrium • The points where three phases are in equilibrium

  30. Phase diagram for water • A piece of ice at 1 atm pressure will convert to liquid at 0ºC and into gas at 100ºC • Below at pressure of 6 x 10-3 atm the ice converts directly into a gas (sublimation) • At the triple point the three phases are in equilibrium – a single temperature and pressure • Beyond the critical point is the region of supercritical fluid: • Cannot be condensed no matter what the pressure (critical temperature) • Cannot be vaporized no matter what the temperature (critical pressure)

  31. Phase diagrams explain well known phenomena • The slope of the solid-liquid phase boundary for water is negative, while for CO2 it is positive • A sample of ice under pressure melts • A sample of liquid CO2 under pressure solidifies • At 1 atm pressure: • Ice melts on heating • Solid CO2 sublimes (dry ice)

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