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Basic Crystallography
Crystals and Bragg’s Law
Amy Sarjeant
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Crystallography – what is it?
• Some semantics (yes, it’s important)
– Spectroscopy/Spectrometry vs. Diffraction
• Spectrum: an array of entities, such as light waves or particles, ordered in accordance with the magnitudes of a common physical property, such as wavelength or mass.
• Diffraction: the breaking up of an incoming wave by some sort of geometrical structure – for example, a series of slits – followed by reconstruction of the wave by interference.
– Diffraction is NOT energy dispersive
• (that would be XPS, EDS, XRF)
• The meaning of peaks…
– NMR/MS/IR etc. peaks have information about specific chemical moieties
– Each XRD peak has information about EVERYTHING
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Crystallography – what can it do?
• Solid state structure determination
– Connectivity
– Molecular morphology
– Interactions
– Packing
– Surfaces
– Porosity
• Phase transitions
– Temperature / Pressure dependent
• Charge Density
• Modulation
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Crystallography – what can’t it do?
• Solid state structure determination
– No gas/liquid phase information
– Limited dynamics
– Usually energetically minimized
– No ab initio elemental analysis
• Synthetic limitations
– Must have a crystalline material
– Must be large enough
– Must be stable
• Can overcome temperature/humidity/oxidation/light within limits
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Some History
• X-Rays Discovered in 1895 by Wilhelm Röntgen
• First diffraction experiment theorized by Max von Laue
• Carried out by Friedrich and Knipping in 1912.
• Nobel Prizes:
– 1901 – Röntgen
– 1914 – von Laue
– 1915 – Bragg’s
– … 29 in all (http://www.iucr.org/people/nobel-prize)
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Why Crystals?
• What is a crystal?
– Regularly shaped
– High degree of long-range order and repetition
– Yields a sharp diffraction pattern
• What is a good crystal?
– Single
– No re-entrant faces
– Appropriately sized
• Typically 0.05-0.6mm
– Not just pretty on the outside
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Crystals – how do you get them?
• Crystal growth
– Directly from reaction
– Slow and steady (literally)
– Solvent/solution based
• Slow evaporation
• Slow cooling
• Vapor diffusion
• Liquid diffusion
– Sublimation
http://xray.chem.uwo.ca/Guides.html
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Unit Cells
• Smallest unique part of a crystal that can be translated through space to reproduce the entire crystal
• Parallelepiped defined by 3 non-coplanar vectors
– Magnitudes = a, b, c
– Angles = α,β, γ
– Atomic coordinates = x, y, z
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Unit cells
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Miller Indices
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Diffraction
• The bending of a wave front around an object, as with light passing through a suitably small slit.
• Constructive interference
• Destructive interference
• The Envelope and Sampling Regions
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Diffraction gratings
• Two Rays (1 & 2) are diffracted by a grating.
– AB = CD (on the way in)
– FG = EH (on the way out)
• Difference in pathlengths
– DE – BF = nλ
• Using geometry
– DE = a cosαo
– BF = a cosα
– Therefore: nλ = a(cosαo – cosα)
• Laue Equations
– a(cosαo – cosα) = hλ
– b(cosβo – cosβ) = kλ
– c(cosγo – cosγ) = lλ
C
A
B
D
1 2
E
F
G
H
aαo
α
B
D
Eαo
a
α
a
F
EB
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Bragg’s Law
– William Henry (father) and William Lawrence Bragg (son)
• Lawrence did most of the work
• Both won the Nobel Prize in 1915
– Conceptualized diffraction as a reflection
d
θ d
θA
B
C
AB + BC = nλ
2AB = nλ
2(d sinθ) = nλ
nλ = 2d sin θ
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Reciprocal Space
• Rearrange Bragg’s Law:
Diffraction pattern is reciprocal of crystal lattice
Reflection from planes (hkl) is the r.l. point hkl at a distance 1/dhkl
from the origin and perpendicular to the planes
What is the relationship between the crystal (real, direct) lattice
and the diffraction pattern (reciprocal lattice)?
nλ = 2d sin θdhkl