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Unformatted text preview: Structures of Solids
& Xray Diffraction
Chemistry 123
Dr. Patrick Woodward
Supplemental Lecture 4 Crystalline Solids
Crystalline CaF2 Unit Cell
• Crystal Lattice – A 3D array of points where each point
has an identical environment.
• Unit Cell – The repeating unit (a unit cell is to a crystal,
like a “brick” is in a house). In a given crystal all unit
brick”
cells are identical. 1 Crystal
Systems Cubic
Tetragonal
Hexagonal
Rhombohedral
Orthorhombic
Monoclinic
Triclinic
The crystal systems each
have distinctive symmetry
and unit cell dimensions Close Packed Array of Spheres
The gray spheres represent a 2D
Close Packed Array.
In 3D the next layer of spheres
could sit on the depressions
marked in red (B) or those
marked in blue (C).
(C). AB Stacking AC Stacking 2 Cubic and Hexagonal Close Packing
Hexagonal Close Packing
(ABAB…)
(ABAB… ABAB Stacking Cubic Close Packing
(ABCABC…)
(ABCABC…
ABCABC Stacking Close Packed Spheres Hexagonal Close Packing
(ABAB…)
(ABAB… Cubic Close Packing
(ABCABC…)
(ABCABC… 3 Hexagonal Close
Packing Body Centered
Cubic Packing Cubic Close
Packing CsCl Structure Cubic or Hexagonal Close Packing
Coordination Number = 12
Packing Efficiency = 74%
Body Centered Cubic Packing
Coordination Number = 8
Packing Efficiency = 68% 4 Eutactic Structures
Many ionic structure types can be described as a
close packing of anions with cations filling voids
or holes in the structure. Generally we will
consider two types of holes (for the cations)
• Octahedral holes  Voids are surrounded by 6 anions and
lead to octahedral coordination of the cation
• Tetrahedral holes  Voids are surrounded by 4 anions
and lead to tetrahedral coordination of the cation Octahedral Holes
Start with a close packed layer of anions (A) Insert cations in the triangular depressions (c) The resulting cation coordination is an octahedron
Add another anion layer (B) 5 Tetrahedral Holes
Start with a close packed layer of anions (A) Insert cations in the triangular depressions (b) The resulting cation coordination is a tetrahedron
Add another anion layer (B) Eutactic Structures
Structures obtained by
filling Octahedral Holes
Structure Fraction
Packing
Holes
Filled Structures obtained by
filling Tetrahedral Holes
Structure Fraction
Packing
Holes
Filled NaCl ccp Fluorite‡ 1 ccp NiAs 1 hcp Sphalerite 1/2 ccp CdCl2 1/2 ccp Wurtzite 1/2 hcp CdI2 1/2 hcp TiO2† 1/2 hcp Al2O3
† The 1 2/3 hcp ‡In fluorite (i.e. CaF2) the cations are
close packed and the anions fill the
tetrahedral holes. The opposite is true
of the antifluorite structure (Na2O) hcp anion layers are buckled in rutile.
rutile. 6 Cubic close packed (CCP) anion array
Rock salt structure (NaCl)
(NaCl)
(Octahedral Holes) Space Group = Fm3m
Atom
Site
x
Anion
4a
0
Oct Hole 4b
½ y
0
½ z
0
½ Antifluorite structure (Na2O)
(Tetrahedral Holes) Space Group = Fm3m
Atom
Site
x
Anion
4a
0
Tetr Hole 8c
¼ y
0
¼ z
0
¼ CCP Anion Array & Tetrahedral Holes
Zinc Blende Structure (ZnS)
(ZnS)
(50% Tetrahedral Holes) Space Group = F43m
Atom
Site
x
Anion
4a
0
Tetr Hole 4b
¼ y
0
¼ z
0
¼ Antifluorite structure (Na2O)
(100% Tetrahedral Holes) Space Group = Fm3m
Atom
Site
x
Anion
4a
0
Tetr Hole 8c
¼ y
0
¼ z
0
¼ 7 Sphalerite (ZnS) (ccp, 50% Tetr. Holes Filled)
ccp,
Tetr. Space Group = F43m
FAtom
Site
x
Zn
4a
0
S
4c
¼ y
0
¼ z
0
¼ Cation Coord. → Tetrahedron
Coord.
Anion Coord. → Tetrahedron
Coord.
Connectivity → Corner sharing Oct. Wurtzite (ZnO) (hcp, 50% Tetr. Holes Filled)
hcp,
Tetr. Space Group = P63mc
Atom
Site
x
1/3
Zn
2b
1/3
O
2b
1/3 y 2/3
2/3 z 0
0.38 Cation Coord. → Tetrahedron
Coord.
Anion Coord. → Tetrahedron
Coord.
Connectivity → Corner sharing Oct. Hexagonal Close Packed Anion Array
Nickel Arsenide Structure (Octahedral Holes) Space Group = P63/mmc
Atom
Site
x
y
1/3
2/3
Anion
2c
Anion
Oct Hole 2a
0
0 z 1/4 0 8 HCP Anion Array  Tetrahedral Holes
No such structure exists Space Group = P63/mmc
Atom
Site
x
y
1/3
2/3
Anion
2c
Anion
Tetr Hole 4f
1/3
2/3
1/3 z 1/4
z z ~ 0.632 Xray Diffraction 9 Diffraction Demo
Take home message
• The diffraction pattern is
related but not equal to the
grid pattern
• Diffraction is most effective
for monochromatic light whose
wavelength is similar to the
spacing of “slits”
slits”
• For crystals Xrays have a
Xwavelength comparable to
spacings of atoms Powder Diffractometer
Divergence
Slit Horizontal
Diffraction
Circle
Sample (Vertical
Flat Plate) Antiscatter
Slit
Receiving Slit θ Divergent
Xray
Source 2θ
Horizontal
Soller Slits
Detector 10 Single Crystal
Diffraction Powder
Diffraction Diffracted
Beam Diffracted
Beam
Incident
Incident
Beam Incident
Beam
In powder diffraction only a
small fraction of the crystals
(shown in blue) are correctly
oriented to diffract. Bragg’s Law 11 Xray Powder Pattern Bond Distance from XRD Pattern
(Ex. PbS, Rock Salt Structure)
PbS, 1. Determine the 2theta value, 2θ, and hkl values for a diffraction
peak
Int. 2θ = 38.7° 20 h=2, k=2, l=0 00 80 60 40 20 0
20 25 30 Bragg’s Law: 35 40 2Theta 45 50 55 60 65 λ = 2dhkl sin θhkl 12 Bond Distance from XRD (Cont.)
(Ex. PbS, Rock Salt Structure)
PbS, 2. Use Bragg’s Law and the wavelength of radiation (typically λ =
1.541 Å) to calculate dhkl λ = 2dhkl sin θhkl
dhkl = λ /(2 sin θhkl)
dhkl = 1.541 Å /{2 sin (38.7°/2)} = 2.10 Å
3. The interplanar spacing, dhkl, is related to the unit cell size. For a
cubic crystal: a = (h2 + k2 + l2)1/2 dhkl
a = (22 + 22 + 02)1/2(2.10 Å) = 5.94 Å Bond Distance from XRD (Cont.)
(Ex. PbS, Rock Salt Structure)
PbS, 4. Now that we know the unit cell size, the PbS distance can be
determined from the unit cell using simple geometry. dist (PbS) = a/2
dist (PbS) = 5.94 Å/2
dist (PbS) = 2.97 Å 13 Peak Positions
Bragg’s Law: λ = 2dhkl sin θhkl The distance between different planes of atoms in a crystal, dhkl,
where h, k and l are integers that correspond to different planes Cubic:
1/d2 = (h2 + k2 + l2)/a2
Tetragonal:
1/d2 = {(h2 + k2)/a2} + (l2/c2)
Orthorhombic:
1/d2 = (h2/a2) + (k2/b2) + (l2/c2)
Hexagonal:
1/d2 = (4/3){(h2 + hk + k2)/a2} + (l2/c2) 14 ...
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 Spring '08
 WOODWARD
 Chemistry

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