The Solid State Phase diagrams Crystals and symmetry Unit cells and packing Types of solid
Learning objectives Apply phase diagrams to prediction of phase behaviour Describe distinguishing features of the four solid types Describe principles of X-ray diffraction Apply symmetry operations to simple patterns Distinguish packing sequences among the four common types of crystal lattice Use unit cell to determine crystal composition
Phase diagrams Phase diagrams summarize the states of a substance as function of 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 Summed up in Phase Rule: C + 2 = P + F
Phase diagram for water Ice at 1 atm pressure converts to liquid at 0ºC and to gas at 100ºC Below 6 x 10-3 atm ice converts directly to gas (sublimation) At triple point three phases are in equilibrium single T and P Beyond critical point is region of supercritical fluid: Cannot be condensed no matter what the pressure (critical temperature) Cannot be vaporized no matter what the temperature (critical pressure)
Sublimation and the phase diagram
Phase diagrams explain well known phenomena The slope of the solidliquid phase boundary for water is negative, while for CO 2 it is positive A sample of ice under pressure melts A sample of liquid CO 2 under pressure solidifies At 1 atm pressure: Ice melts on heating Solid CO 2 sublimes (dry ice)
Chemistry and Metamucil: Regularity is everything At equilibrium atom tends to occupy position of lowest energy (highest stability) One position more stable than any others If true for one atom then true for all Regular crystal lattice results Not all solids are crystalline Amorphous solids have no regular structure - atoms occupy many sites Tend to be metastable - revert to crystalline form
Four solids of the apocalypse Ionic Ionic bonds Brittle, hard, high m.p. NaCl, K 2 O Molecular Van der Waals Soft, low m.p. CO 2, I 2 Covalent lattice Covalent bonds Hard, v. high m.p. Diamond, SiO 2 Metallic Metallic bonds Variable hardness and m.p. W, Cu, Hg conductors
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
Diffraction and interference Diffraction arises by interference of electromagnetic radiation Constructive interference waves are in phase increase in intensity Destructive interference 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
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
The Bragg condition: it s incurable Waves reflected from adjacent layers must be in phase For constructive interference, the path length difference must then be a whole number of wavelengths n 2d sin
X-ray diffraction and data A crystal or powder is swept through range of angles of θ and positions of reflections are tabulated Analysis of d spacings gives information about type of crystal lattice Deeper analysis of reflection intensities gives complete description of structure collection
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
Common symmetry elements Translation Rotation Reflection After application of a symmetry operation to a set of atoms related by symmetry, the system appears unchanged
Symmetry operations Translation Reflection Rotation
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
Cubic packing In simple cubic the atoms stack directly on top of each other: a a a a Not close packed (very uncommon) Body-centered cubic: denser packing achieved by putting a layer in the depression of the first: a b a b a b Not close packed (quite common)
Close packing Oranges do it Atoms do it
Adding layers: choices Two arrangements achieve a higher density Hexagonal closepacked- abab Face-centered cubic abcabc Both very common for metals
Close packing in metals
Summary of packing arrangements Structure Packing sequence Coordination number Packing density/% Simple cubic a-a-a-a- 6 52 Body centered cubic (BCC) Hexagonal CP (HCP) Face centered cubic (FCC) a b a b 8 68 a-b-a-b 12 74 a-b-c-a-b-c 12 74
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 Unit cell stacking unit cells In each case the basic unit contains all information required to describe structure completely with no gaps, deficiencies or redundancies
Unit cell accountancy Consider a cube Component of cube Number in cell Center Face Edge Corner 1 6 12 8
Sharing in unit cells Place an atom on each of the unit cell locations: center, face, edge, corner How many unit cells share the atom? What fraction of atom is in the unit cell Component Center Shared between 0 Fraction of atom in cell 1 Face Edge Corner 2 4 8 0.5 0.25 0.125
Unit cell contents and composition Count atoms as follows: Ratio of atoms in unit cell must equal composition One atom on: Number of components in unit cell Fraction in cell Number of atoms in cell Center 1 1 Face 6 0.5 Edge 12 0.25 Corner 8 0.125 1 3 3 1
Primitive and body-centered Primitive: 8 atoms on the corners each contribute 1/8 th 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 cells
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
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 = d 8r
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
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
Sodium chloride unit cell Chloride ions (purple) form FCC lattice Sodium ions (green) 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
Covalent networks Ionic lattices are characterized by high coordination numbers and non-directional bondin Covalent lattices have low coordination numbers and highly directional bonding The bonds are formed from hybridized atomic orbitals to use the valence bond model
Diamonds are forever The diamond lattice is a very common covalent lattice It is a three dimensional tetrahedral net Bonds are made from sp 3 hybrid orbitals Very strong covalent bonds make the lattice extremely stable
Compounds also have diamond In GaAs, the Ga and As atoms alternate in the diamond lattice GaAs is an important semiconductor and laser material Many other similar materials lattice
Diamond and graphite Infinite covalent lattices in different dimensions Graphite 2D Diamond 3D Graphite is more stable than diamond, but can be transformed into diamond by application of high pressure Buckyballs Nanotubes
Metallic bonding: adrift on an Why do metals conduct electricity? Too few electrons for covalent bonds Low ionization energies Lattice consists of positive ions held together by delocalized valence electrons electron sea