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Crystalline Solids
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
cells are identical.
Ionic radii and coordination number
Cation
Anion
Large cation & Large anion
Small cation & Large anion
r(Cation)/r(Anion) ~0.72
r(Cation)/r(Anion) ~0.3
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
Radius Ratio Rules
r(cation)/r(anion)
Coordination
Number
Coordination
Geometry
 0.73
8
Cube
 0.41
6
Octahedron
 0.22
4
Tetrahedron
 0.15
3
Trigonal 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.
Ionic Crystal Structures
(a) CsCl
(b) NaCl
(c) Zinc blende
Decreasing coordination number
Decreasing r(cation)/r(anion)
Cell Edge vs. Radii
Body centered
cubic metal (i.e. Fe)
Face centered
cubic metal (i.e. Ag)
diag
ReO3 structure
a
diag
a
a
4rFe = √3 a
Atoms touch along
the body diagonal of
the cubic unit cell
4rAg = √2 a
Atoms touch along a
face diagonal of the
cubic unit cell
a
2rRe + 2rO = a
Atoms touch along
an edge of the cubic
unit cell
Optical Diffraction Experiment
Diffraction
Spots
Diffraction
Slide
X
Laser
L
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”
• For crystals X-rays have a
wavelength comparable to
spacings of atoms
Single Crystal X-ray Diffraction
Bragg’s Law
A
B
d
The extra path length traveled by wave B (shown in red) is equal to
2d sin q, where d is the spacing between planes. Constructive
interference will only occur if this distance is an integer multiple of
the wavelength (nl).
X-ray Powder Diffraction
Divergence
Slit
Horizontal
Diffraction
Circle
Sample (Vertical
Flat Plate)
q
Divergent
X-ray
Source
2q
Antiscatter
Slit
Receiving Slit
Horizontal
Soller Slits
Detector
X-ray Powder Diffraction Pattern