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