Solids Ch.13

Solids Fixed, immobile (so to speak) Fixed, immobile (so to speak) Symmetry Symmetry Crystals Crystals So what’s the inner order? So what’s the inner order?

Unit Cells Unit cell = smallest repeating unit containing all symmetry characteristics Unit cell = smallest repeating unit containing all symmetry characteristics Unit cell reflects stoichiometry of solid Unit cell reflects stoichiometry of solid Several unit cell types possible, but atoms or ions placed at lattice points or corners of geometric object Several unit cell types possible, but atoms or ions placed at lattice points or corners of geometric object

Crystal Lattices 3D unit cells built like legos  3D unit cells built like legos  Crystal Lattice = arrangement of units cells Crystal Lattice = arrangement of units cells seven 3D units cells found seven 3D units cells found Simplest = Cubic Unit Cell (equal length edges meeting at 90° angles) Simplest = Cubic Unit Cell (equal length edges meeting at 90° angles) Each face part of 2 cubes  Each face part of 2 cubes  Each edge part of 4 cubes  Each edge part of 4 cubes  Each corner part of 8 cubes  Each corner part of 8 cubes 

Cubic Unit Cell 3 types: 3 types: 1) Primitive or Simple Cubic (SC) 1) Primitive or Simple Cubic (SC) 2) Body-Centered Cubic (BCC) 2) Body-Centered Cubic (BCC) 3) Face-Centered Cubic (FCC) 3) Face-Centered Cubic (FCC)

Cubic Unit Cell (cont.) Similarity: Similarity: Same ions/atoms/molecules at each corner Same ions/atoms/molecules at each corner Difference: Difference: BCC/FCC have more items at other locations BCC/FCC have more items at other locations BCC has same item in center of cube BCC has same item in center of cube FCC has same item centered on each side of cube FCC has same item centered on each side of cube

What do they look like? BCC: BCC: FCC: FCC: SC: SC:

Which metals have which crystal lattices? Simple cubic: Po Simple cubic: Po BCC: GI, 3B, 4B, Ba, Ra, Fe BCC: GI, 3B, 4B, Ba, Ra, Fe FCC: VIIIB, IB, Al, In, Pb FCC: VIIIB, IB, Al, In, Pb

How many atoms per unit cell? SC: each atom shared by 8 cubes SC: each atom shared by 8 cubes 8 corners of cube  1/8 of each corner atom w/in unit cell = 1 net atom/unit cell 8 corners of cube  1/8 of each corner atom w/in unit cell = 1 net atom/unit cell

More on SC Each atom touches one another along edge Each atom touches one another along edge Thus, each edge = 2r Thus, each edge = 2r Coordination number (# of atoms with which each atom is in direct contact) = 6 Coordination number (# of atoms with which each atom is in direct contact) = 6 Packing efficiency = fraction of volume occupied = 52% Packing efficiency = fraction of volume occupied = 52%

How many atoms per unit cell? (cont.) BCC: 2 net atoms w/in unit cell (SC + 1 in center) BCC: 2 net atoms w/in unit cell (SC + 1 in center) FCC: 6 faces of cube  ½ atom w/in unit cell = 3 atoms + 1 atom (SC) = 4 net FCC: 6 faces of cube  ½ atom w/in unit cell = 3 atoms + 1 atom (SC) = 4 net

More on BCC Each atom does not touch another along edge Each atom does not touch another along edge However, atoms touch along internal diagonal However, atoms touch along internal diagonal Thus, each edge length = 4r/  3 Thus, each edge length = 4r/  3 Let’s derive this… Let’s derive this… Coordination number (# of atoms with which each atom is in direct contact) = 8 Coordination number (# of atoms with which each atom is in direct contact) = 8 –Central atom touches 8 atoms Packing efficiency = fraction of volume occupied = 68% Packing efficiency = fraction of volume occupied = 68%

More on FCC Each atom does not touch another along edge Each atom does not touch another along edge However, atoms touch along face diagonal However, atoms touch along face diagonal Thus, each edge length = (2  2)r Thus, each edge length = (2  2)r Let’s derive this… Let’s derive this… Coordination number (# of atoms with which each atom is in direct contact) = 12 Coordination number (# of atoms with which each atom is in direct contact) = 12 Packing efficiency = fraction of volume occupied = 74% Packing efficiency = fraction of volume occupied = 74%

Problems Eu is used in TV screens. Eu has a BCC structure. Calculate the radius of a europium atom given a MW = g/mol, a density of g/cm 3. Eu is used in TV screens. Eu has a BCC structure. Calculate the radius of a europium atom given a MW = g/mol, a density of g/cm 3. Iron has a BCC unit cell with a cell dimension of pm. The density of iron is g/cm 3 and its MW = g/mol. Calculate Avogadro’s number. Iron has a BCC unit cell with a cell dimension of pm. The density of iron is g/cm 3 and its MW = g/mol. Calculate Avogadro’s number.

CCP and HCP: Efficiency in Stacking CCP = Cubic Close-Packing (it’s FCC) CCP = Cubic Close-Packing (it’s FCC) HCP = Hexagonal Close-Packing HCP = Hexagonal Close-Packing 74% packing efficiency 74% packing efficiency

Structures of ionic solids Take a SC or FCC lattice of larger ions Take a SC or FCC lattice of larger ions Place smaller ions in holes w/in lattice Place smaller ions in holes w/in lattice Smallest repeating unit = unit cell Smallest repeating unit = unit cell

CsCl SC unit cell SC unit cell Cs + in center of cube  Cubic hole Cs + in center of cube  Cubic hole Surrounded by 1 Cl - (in 8 parts) Surrounded by 1 Cl - (in 8 parts) –1 Cs + : 1 Cl - Coordination # = 8 Coordination # = 8 Why SC and not BCC? Why SC and not BCC? Because ion in center different from lattice pt ions Because ion in center different from lattice pt ions

LiCl Notice: Li + has octahedral geometry Notice: Li + has octahedral geometry Thus, cation in octahedral hole (between 6 ions) Thus, cation in octahedral hole (between 6 ions) –Coordination # = 6 FCC FCC

NaCl Lattice has net 4 Cl - /unit cell Lattice has net 4 Cl - /unit cell –(8x1/8)+(6x1/2) = 4 1 Na + in center of unit cell 1 Na + in center of unit cell 3 Na + along edges of unit cell 3 Na + along edges of unit cell –(12x1/4) = 3 –Thus, net total of 4 Na + ions Total 4 Cl - : 4 Na +  1:1 Total 4 Cl - : 4 Na +  1:1

Tetrahedral holes Each ion surrounded by 4 other oppositely- charged ions Each ion surrounded by 4 other oppositely- charged ions Unit cell: 4 of each ion  total 8 ions Unit cell: 4 of each ion  total 8 ions Coordination # = 4 Coordination # = 4 8 tetrahedral holes in FCC unit cell 8 tetrahedral holes in FCC unit cell –4 by Zn 2+ and 4 by S 2- Zn 2+ occupies ½ of tetrahedral holes and surrounded by 4 S 2- Zn 2+ occupies ½ of tetrahedral holes and surrounded by 4 S 2- S 2- forms FCC unit cell S 2- forms FCC unit cell

ZnS

ZnS

Other Types of Solids: Network Solids Array of covalently bonded atoms Array of covalently bonded atoms Graphite, diamond, and silicon Graphite, diamond, and silicon The latter two  sturdy, hard, & high m.p.’s The latter two  sturdy, hard, & high m.p.’s

Graphite and diamond

Other Types of Solids: Amorphous Solids Glass & plastics Glass & plastics No regular structure No regular structure –Break in all sorts of shapes Long range of m.p.’s Long range of m.p.’s