Ch 7 The Crystalline Solid State

1
Ch 7 The Crystalline Solid State

• Formulas and Structures
• Simple Structures
• Unit Cell simplest repeating unit of the
  regular crystalline array
• Bravais Lattices 14 possible basic crystal
  structure unit cell types

2

• Atoms on corners and edges are shared between
  unit cells
• Rectangular corners shared by 8 unit cells (8 x
  1/8 1 total atom in cell)
• Other corners shared unequally, but still
  contribute 1 atom to unit cell
• Edges shared by 4 cells, contribute (4 x ¼ 1
  atom to unit cell)
• Faces shared by 2 cells, contribute ½ atom to
  each
• Angles and Dimensions can vary triclinic has all
  different lengths and angles
• Lattice Points positions of atoms needed to
  generate the whole crystal
• Body Centered Cubic (0,0,0) origin and (½ ,
  ½, ½) center
• All other atoms can be generated from these 2 by
  moving them exactly one cell length
• Cubic Structures
• Primitive Cubic is the simplest type
• To fully describe length of side, 90o angles,
  (0,0,0) lattice point
• 8 x 1/8 1 atom in the unit cell
• each atom surrounded by 6 others (Coordination
  Number CN 6)
• Not efficiently packed only 52.4 of volume is
  occupied (74.1 max)
• Vacant space at center with CN 8 0.73r
  sphere would fit here

3

• Body-Centered Cubic (bcc)
• One more atom is added to the center of the cube
• Size of unit cell must increase over simple cubic
• Diagonal across center 4r (r radius of one
  atom)
• Corner atoms not in contact with each other due
  to cell size expansion
• Side 2.31r (Calculate this in Ex. 7-1)
• Unit cell contains 1(1) 8(1/8) 2 atoms
• Lattice Points (0,0,0) and ( ½ , ½ , ½ )
• Close-Packed Structures
• Spheres will arrange to take up the least space,
  not cubic or bcc
• Two structures have almost identical packing
  efficiency (74.1)
• Hexagonal Close Packing (hcp)
• Cubic Close Packing (ccp) Face-Centered Cubic
  (fcc)
• Both have CN 12 for each atom 6 in its layer,
  3 above, and 3 below
• HCP has 3rd layer in identical position as first
  ABA second layer above holes CCP has 3rd layer
  displaced so above holes in first layer ABC
• Both have 2 tetrahedral (Td) holes per atom (CN
  4) formed by 3 atoms in one layer and one atom
  above/below
• Both have 1 octahedral (Oh) hole per atom (CN
  6) formed by 3 atoms in one layer and 3 atoms
  above/below

APF 68.0 CN 8
4r
l
l
4
(No Transcript)
5

• HCP unit cell is smaller than the hexagonal prism
• Take four touching atoms in 1st layer and extend
  lines up to 3rd layer
• 8(1/8) 1 2 atoms in unit cell
• Dimensions 2r, 2r, and 2.83r
• Angles 120o, 90o, and 90o
• Lattice Points (0,0,0,) and ( 1/3, 2/3, 1/2)

6

• CCP unit cell
• One corner from layer 1, opposite corner from
  layer 4
• 6 atoms each from layers 2 and 3 in the unit cell
• Unit cell is fcc
• 8(1/8) 6(1/2) 4 atoms in the unit cell
• Lattice Points (0,0,0), ( ½ , 0 , ½), ( ½ , ½,
  0), (0 , ½ , ½)

7

• Holes
• 2 Td and 1 Oh hole per atom in both (hcp) and
  (ccp)
• If ionic compound, small cations can fill these
  holes
• Td hole 0.225r Oh hole 0.414r
• NaCl has Cl- in (ccp) with Na (ccp) but also in
  the Oh holes
• Na 0.695r, so it forces the Cl- to be farther
  apart, but still CN 6
• Metallic Crystals
• Most metals crystallize in (bcc), (ccp), or (hcp)
  structures
• Changing pressure or temperature can interchange
  these forms for a metal
• Must consider bonding, not just geometric packing
  only
• Soft and malleable metals usually have (ccp)
  structure (copper)
• Harder and more brittle metals usually have the
  (hcp) structure (zinc)
• Most metals can be bent due to non-directional
  bonding
• Weak bonding to all neighbors, not strong bonding
  to any single one
• Atoms can slide past each other and then realign
  into crystal form
• Dislocations imperfections in lattice make it
  easier to bend
• Impurities other elements allow slippage of
  layers
• Work Hardening hammer until impurities are
  together

A summary
8
FCC elements
HCP elements
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• Diamond
• Each atom is tetrahedrally bonded to 4 other
  carbon atoms
• Directional single bonds, unlike metals, cause
  hardness
• Binary Compound Structures
• Simplest structures just have 2nd element in
  holes of the first elements lattice
• Small cations can fit in Td/Oh holes of large
  anions
• Large cations may only be able to fit in Oh holes
• Even larger cations may cause changes in
  structure if they dont fit in holes
• Relative Number of Cations/Anions
• M2X wont allow close-packing of anion lattice
  with cations in Oh holes because there are more
  cations than holes
• Alternatives cations in Td holes, vacancies, not
  close-packed

10

• Sodium Chloride Structure NaCl
• Na in (fcc) and Cl- in (fcc)
• Offset by ½ unit cell length
• Na are centered in edges of Cl- lattice (or vice
  versa)
• Most alkali halides have this same structure
• Large size difference of ions facilitate this
  structure
• Each Cl- has CN 6 Na Each Na has CN 6
  Cl-
• Cesium Chloride Structure CsCl
• Cs in simple cubic structure with Cl- in center
  (or vice versa)
• Cl- 0.83Cs size (0.73r in center is ideal)
• Rare structure, need big cation (Cs, Tl only
  cations known with this structure)

11

• Zinc Blende Structure ZnS
• Same as diamond structure with alternating Zn and
  S atoms
• Alternate Zn and S each in (fcc) lattices
  combined so each ion is in a Td hole of the other
  lattice
• Stoichiometry only ½ of the Td holes are
  occupied and ½ are vacant
• Wurtzite Structure ZnS
• Rarer than Zinc Blende structure for ZnS formed
  at higher Temperatures
• Zn and S each in (hcp) lattices combined so each
  ion is in a Td hole of the other lattice
• Again, ½ of the Td holes are vacant

12

• Fluorite Structure CaF2
• Ca2 in (ccp) lattice with 8 F- surrounding each
  and occupying all Td holes
• Alternate F- in simple cubic lattice with Ca2
  in alternate body centers
• Nearly perfect radius fits for this structure
• Antifluorite Structure
• Reverse stoichiometry compounds like Na2O
• Every Td hole in the anion lattice is occupied by
  a cation

13

• Nickel Arsenide Structure NiAs
• As atoms in close packed layers exactly above
  each other
• Ni atoms in all the Oh holes
• Both Ni and As have CN 6
• Alternate Ni atoms occupy all Oh holes of (hcp)
  As lattice
• Usual for MX compounds where X Sn, As, Bi, S,
  Se, Te
• Rutile Structure TiO2
• Distorted TiO6 octahedra forming columns by
  sharing edges
• Ti CN 6 O CN 3
• Adjacent columns connected by sharing corners of
  octahedra
• Unit cell has Ti at corners and in the body
  center, 4 O in the faces, and 2 O in the plane of
  the body center Ti
• MgF2, ZnF2 are other examples

14

• The Radius Ratio
• We can crudely predict CN by using the ratio
  r/r-
• This assumes atoms are just packing as hard
  spheres (not really all that occurs)
• Examples
• NaCl r/r- 113/167 (CN 4) 0.667
• r/r- 116/167 (CN 6) 0.695 Fits best
  with CN 6
• ZnS r/r- 74/170 (CN 4) 0.435 Fits best
  with CN 4
• r/r- 88/170 (CN 6) 0.518 CN 4 in
  actual structure
• 4) Exercise 7-2 CaF2 has fluoride ions in a
  simple cubic array and calcium ions in alternate
  body centers, with r/r- 0.97. What are the
  coordination numbers of the two ions predicted by
  r/r- ? What are the coordination numbers
  observed? Predict coordination numbers of Ca2
  in CaCl2 and CaBr2.

Appendix B1