Crystalline SolidsCrystalline Solids
• Crystal LatticeCrystal Lattice – A 3D array of points where each – A 3D array of points where each point has an identical environment.point has an identical environment.
• Unit CellUnit Cell – The repeating unit (a unit cell is to a – The repeating unit (a unit cell is to a crystal, like a “brick” is in a house). In a given crystal, like a “brick” is in a house). In a given crystal all unit cells are identical.crystal all unit cells are identical.
CaFCaF22 Unit Cell Unit Cell
Large cation & Large anion
r(Cation)/r(Anion) ~0.72
Small cation & Large anion
r(Cation)/r(Anion) ~0.3
Cation Anion
Anions and cations want to stay in contact with each other, but when the cation
becomes much smaller than the anion, anion-anion contacts prevent this.
The solution is to lower the coordination number
Ionic radii and coordination Ionic radii and coordination numbernumber
Radius Ratio RulesRadius Ratio Rules
rr(cation)/(cation)/rr(anion)(anion)
Coordination Coordination NumberNumber
Coordination Coordination GeometryGeometry
0.730.73 88 CubeCube
0.410.41 66 OctahedronOctahedron
0.220.22 44 TetrahedronTetrahedron
0.150.15 33 Trigonal PlanarTrigonal Planar
These are only meant to be used as rough guidelines, but the coordination number does
goes down as the radius ratio [r(cation)/r(anion)] decreases.
(a) CsCl (b) NaCl (c) Zinc blende
Ionic Crystal StructuresIonic Crystal Structures
Decreasing coordination numberDecreasing r(cation)/r(anion)
diag
a
diag
a
a
a
4rAg = √2 a 2rRe + 2rO = a4rFe = √3 a
Cell Edge vs. RadiiCell Edge vs. Radii
Body centered cubic metal (i.e. Fe)
Face centered cubic metal (i.e. Ag)
ReO3 structure
Atoms touch along the body diagonal of
the cubic unit cell
Atoms touch along a face diagonal of the
cubic unit cell
Atoms touch along an edge of the cubic
unit cell
X
L
Laser
Diffraction Slide
Diffraction Spots
Optical Diffraction ExperimentOptical Diffraction Experiment
Diffraction Diffraction DemoDemo
Take home messageTake home message
• The diffraction pattern is The diffraction pattern is related but not equal to the related but not equal to the grid patterngrid pattern
• Diffraction is most effective Diffraction is most effective for monochromatic light for monochromatic light whose wavelength is similar whose wavelength is similar to the spacing of “slits”to the spacing of “slits”
• For crystals X-rays have a For crystals X-rays have a wavelength comparable to wavelength comparable to spacings of atomsspacings of atoms
Single Crystal X-ray DiffractionSingle Crystal X-ray Diffraction
Bragg’s LawBragg’s Law
The extra path length traveled by wave B (shown in red) is equal to 2d sin , where d is the spacing between planes.
Constructive interference will only occur if this distance is an integer multiple of the wavelength (n).
AA
BB
dd
Horizontal Diffraction
Circle
Divergent X-ray Source
Detector
Divergence Slit
Antiscatter Slit
Receiving Slit
HorizontalSoller Slits
Sample (Vertical Flat
Plate)
X-ray Powder DiffractionX-ray Powder Diffraction
X-ray Powder Diffraction PatternX-ray Powder Diffraction Pattern
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