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Unformatted text preview: Crystals, Crystal Structures &
Crystallography 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
brick”
cells are identical. 1 Translational Symmetry in 2D
b
a Unit Cell Arbitrary
Lattice Point
2a + 3b • a and b are the lattice vectors, every lattice point can be defined by
and
adding these two vectors
• The unit cell is defined by the lattice vectors, all space can be filled
be
by tiling unit cells 2D Bravais Lattices
b γ a Centered
Unit Cell b
a Rectangular (a ≠ b, γ = 90°) γ Primitive
Unit Cell Centered Rectangular
(a ≠ b, γ = 90°) b
a b γ Square (a = b, γ = 90°) a γ Hexagonal (a = b, γ = 120°) Including Oblique (a ≠ b, γ ≠ 90°), shown on the previous slide,
there are 4 crystal systems (oblique, rectangular, square and
hexagonal) and 5 Bravais lattices in 2D. 2 Cubic 3D Crystal
Systems (7)
Cubic
Tetragonal
Hexagonal
Rhombohedral
Orthorhombic
Monoclinic
Triclinic
14 Bravais
Lattices
The crystal systems each
have distinctive symmetry
and unit cell dimensions Xray Diffraction 3 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 Bragg’s Law A
B d The extra path length traveled by wave B (shown in red) is equal to
2d sin θ, where d is the spacing between planes. Constructive
interference will only occur if this distance is an integer multiple of
the wavelength (nλ). 4 XRay Powder Pattern 20 30 20 40 50 60 2Theta (Degrees) There are many different planes of atoms in a crystal. In an Xray powder
diffraction pattern we see many peaks, each one corresponding to scattering
from different planes of atoms. The numbers in the above diagram are
called Miller Indices, they identify different planes of atoms in the crystal. Miller Indices in 3D
Miller
The distance between planes is
given by the following formula (for
an orthorhombic lattice): Plane that
goes
through the
origin 1/d2 = h2/a2 + k2/b2 + l2/c2
For a cubic lattice this reduces to: 1/d2 = (h2 + k2 + l2)/a2
The next plane is the one
used to calculate hkl In a 3D system there are three Miller Indices, h, k and l. The
values of h, k and l are integers whose values are determined as
follows:
h = 1/(xintercept)
1/(xk = 1/(yintercept)
1/(yl = 1/(zintercept)
1/(z h = a/(1a) = 1
k = b/(1b) = 1
l = c/(∞) = 0 110 plane 5 XRay Powder Diffraction Pattern
(Ex. Lead Sulfide, PbS)
(Ex. Int.
350 020 planes 300 250 200 220 planes 150 100 50 0
20 30 40 2Theta 50 60 70 Each peak corresponds to scattering from a different
set of lattice planes. Two planes are shown above for
PbS, which has the same structure as NaCl. 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 6 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 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) 7 ...
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This note was uploaded on 06/03/2011 for the course CHM 123 taught by Professor Woodward during the Spring '08 term at Ohio State.
 Spring '08
 WOODWARD
 Crystallography, pH

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