SolidsSolids

Ch.13Ch.13

SolidsSolids

• Fixed, immobile (so to speak)Fixed, immobile (so to speak)

• SymmetrySymmetry

• CrystalsCrystals

• So what’s the inner order?So what’s the inner order?

Unit CellsUnit Cells

• Unit cellUnit cell = smallest repeating unit = smallest repeating unit containing all symmetry characteristicscontaining all symmetry characteristics

• Unit cell reflects stoichiometry of solidUnit cell reflects stoichiometry of solid

• Several unit cell types possible, but Several unit cell types possible, but atoms or ions placed at atoms or ions placed at lattice pointslattice points or corners of geometric objector corners of geometric object

Crystal LatticesCrystal Lattices

• 3D unit cells built like legos 3D unit cells built like legos • Crystal LatticeCrystal Lattice = arrangement of units = arrangement of units

cellscells• seven 3D units cells foundseven 3D units cells found• Simplest = Simplest = Cubic Unit CellCubic Unit Cell (equal length (equal length

edges meeting at 90edges meeting at 90° angles)° angles)

• Each face part of 2 cubesEach face part of 2 cubes• Each edge part of 4 cubesEach edge part of 4 cubes• Each corner part of 8 cubesEach corner part of 8 cubes

Cubic Unit CellCubic Unit Cell

• 3 types:3 types:

• 1)1) PrimitivePrimitive or or Simple Cubic (SC)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.)Cubic Unit Cell (cont.)

• Similarity:Similarity:

• Same ions/atoms/molecules at each cornerSame ions/atoms/molecules at each corner

• Difference:Difference:

• BCC/FCC have more items at other BCC/FCC have more items at other locationslocations

• BCC has same item in center of cubeBCC has same item in center of cube

• FCC has same item centered on each side FCC has same item centered on each side of cubeof cube

What do they look like?What do they look like?

• BCC: BCC:

• FCC:FCC:

• SC:SC:

Which metals have which Which metals have which crystal lattices?crystal lattices?

• Simple cubic: PoSimple cubic: Po

• BCC: GI, 3B, 4B, Ba, Ra, FeBCC: GI, 3B, 4B, Ba, Ra, Fe

• FCC: VIIIB, IB, Al, In, PbFCC: VIIIB, IB, Al, In, Pb

How many atoms per unit How many atoms per unit cell?cell?

• SC: each atom shared by 8 cubesSC: each atom shared by 8 cubes

• 8 corners of cube 8 corners of cube 1/8 of each 1/8 of each corner atom w/in unit cell = corner atom w/in unit cell = 1 net 1 net atom/unit cellatom/unit cell

More on SCMore on SC

• Each atom touches one another along Each atom touches one another along edgeedge

• Thus, each Thus, each edge = 2redge = 2r

• Coordination numberCoordination number (# of atoms (# of atoms with which each atom is in direct with which each atom is in direct contact) = contact) = 66

• Packing efficiency Packing efficiency = fraction of = fraction of volume occupied = volume occupied = 52%52%

How many atoms per unit cell? How many atoms per unit cell? (cont.)(cont.)

• BCC: BCC: 2 net atoms2 net atoms w/in unit cell (SC w/in unit cell (SC + 1 in center)+ 1 in center)

• FCC: 6 faces of cube FCC: 6 faces of cube ½ atom w/in ½ atom w/in unit cell = 3 atoms + 1 atom (SC) = unit cell = 3 atoms + 1 atom (SC) = 4 net4 net

More on BCCMore on BCC

• Each atom Each atom does not does not touch another along edgetouch another along edge• However, atoms touch along However, atoms touch along internalinternal diagonal diagonal• Thus, each Thus, each edge length = 4r/edge length = 4r/33• Let’s derive this…Let’s derive this…

• Coordination numberCoordination number (# of atoms with (# of atoms with which each atom is in direct contact) = which each atom is in direct contact) = 88– Central atom touches 8 atomsCentral atom touches 8 atoms

• Packing efficiency Packing efficiency = fraction of volume = fraction of volume occupied = occupied = 68%68%

More on FCCMore on FCC

• Each atom Each atom does not does not touch another along edgetouch another along edge• However, atoms touch along However, atoms touch along faceface diagonal diagonal• Thus, each Thus, each edge length = (2edge length = (22)r2)r• Let’s derive this…Let’s derive this…

• Coordination numberCoordination number (# of atoms with (# of atoms with which each atom is in direct contact) = which each atom is in direct contact) = 1212

• Packing efficiency Packing efficiency = fraction of volume = fraction of volume occupied = occupied = 74%74%

Problems Problems

• Eu is used in TV screens. Eu has a BCC Eu is used in TV screens. Eu has a BCC structure. Calculate the radius of a structure. Calculate the radius of a europium atom given a MW = 151.964 europium atom given a MW = 151.964 g/mol, a density of 5.264 g/cmg/mol, a density of 5.264 g/cm33..

• Iron has a BCC unit cell with a cell Iron has a BCC unit cell with a cell dimension of 286.65 pm. The density of dimension of 286.65 pm. The density of iron is 7.874 g/cmiron is 7.874 g/cm33 and its MW = 55.847 and its MW = 55.847 g/mol. Calculate Avogadro’s number. g/mol. Calculate Avogadro’s number.

CCP and HCP: Efficiency in CCP and HCP: Efficiency in StackingStacking

• CCP = Cubic Close-Packing (it’s FCC) CCP = Cubic Close-Packing (it’s FCC)

• HCP = Hexagonal Close-PackingHCP = Hexagonal Close-Packing

• 74% packing efficiency74% packing efficiency

Structures of ionic solidsStructures of ionic solids

• Take a SC or FCC lattice of larger Take a SC or FCC lattice of larger ionsions

• Place smaller ions in holes w/in Place smaller ions in holes w/in latticelattice

• Smallest repeating unit = Smallest repeating unit = unit cellunit cell

CsClCsCl

• SC unit cellSC unit cell• CsCs++ in center of cube in center of cube Cubic hole Cubic hole• Surrounded by 1 ClSurrounded by 1 Cl-- (in 8 parts) (in 8 parts)

– 1 Cs1 Cs++ : 1 Cl : 1 Cl--

• Coordination # = Coordination # = 88• Why SC and not BCC?Why SC and not BCC?• Because ion in center different from lattice Because ion in center different from lattice

pt ionspt ions

LiClLiCl

• Notice: LiNotice: Li++ has octahedral geometry has octahedral geometry

• Thus, cation in Thus, cation in octahedral hole octahedral hole (between 6 ions)(between 6 ions)– Coordination # = Coordination # = 66

• FCCFCC

NaClNaCl

• FCCFCC• Lattice has net Lattice has net 4 Cl4 Cl--/unit cell/unit cell

– (8x1/8)+(6x1/2) = 4(8x1/8)+(6x1/2) = 4

• 1 Na1 Na++ in center of unit cell in center of unit cell• 3 Na3 Na++ along edges of unit cell along edges of unit cell

– (12x1/4) = 3(12x1/4) = 3– Thus, net total of Thus, net total of 4 Na4 Na++ ions ions

• Total 4 ClTotal 4 Cl- - : 4 Na: 4 Na+ + 1:1 1:1

Tetrahedral holesTetrahedral holes

• Each ion surrounded by 4 other oppositely-Each ion surrounded by 4 other oppositely-charged ionscharged ions

• Unit cell: 4 of each ion Unit cell: 4 of each ion total 8 ions total 8 ions• Coordination # = Coordination # = 44• 8 tetrahedral holes in FCC unit cell8 tetrahedral holes in FCC unit cell

– 4 by Zn4 by Zn2+2+ and 4 by S and 4 by S2-2-

• ZnZn2+2+ occupies ½ of tetrahedral holes and occupies ½ of tetrahedral holes and surrounded by 4 Ssurrounded by 4 S2-2-

• SS2-2- forms FCC unit cell forms FCC unit cell

ZnSZnS

ZnSZnS

Other Types of Solids: Network Other Types of Solids: Network SolidsSolids

• Array of covalently bonded atomsArray of covalently bonded atoms

• Graphite, diamond, and siliconGraphite, diamond, and silicon

• The latter two The latter two sturdy, hard, & high sturdy, hard, & high m.p.’sm.p.’s

Graphite and diamondGraphite and diamond

Other Types of Solids: Other Types of Solids: Amorphous SolidsAmorphous Solids

• Glass & plasticsGlass & plastics

• No regular No regular structurestructure– Break in all sorts of Break in all sorts of

shapesshapes

• Long range of Long range of m.p.’sm.p.’s