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Diffraction and
Crystal Structure
NANO Workshop 2006
Outline
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Light and X-rays
Diffraction basics
Determining crystal structures
Activities!
What is Light?
• Particle or wave?
Both!
• Photons: particles or
“quanta” of light
E = hf
• Many photons →
wavelike
• A disturbance that
propagates
• Usually requires a
medium
Anatomy of a Wave
crest
2 x amplitude
• Wavelength 
• Frequency f
wavelength
trough
direction of wave motion
– # of cycles per unit
time
• Speed v = f
• Amplitude A
Electromagnetic Waves
• Speed = 3  105 km/sec (about 186,000 mi/sec)
The Electromagnetic Spectrum
Visible Light
• The color of visible light
is determined by its
wavelength
• White light is a mixture
of all colors
• We can separate out
individual colors with a
prism
Visible Light
400–440 nm
440–480 nm
480–530 nm
530–590 nm
590–630 nm
630–700 nm
Violet
Blue
Green
Yellow
Orange
Red
Superposition and Interference
• When multiple waves
pass through the same
place, the total wave is
obtained by adding
together the individual
wave displacements
(Principle of
Superposition)
X-Ray Diffraction
• X-rays have wavelengths
comparable to atomic
sizes and spacings,
about 10–10 m
• Crystals and molecules
reflect X-rays in specific
patterns depending on
their structures
X-ray diffraction pattern of myoglobin
Interaction of X-Rays with Atoms
• Involves the electrons, primarily
Fraunhofer Diffraction
• “Fraunhofer” = incident,
outgoing rays parallel
• Start with single plane of
atoms (a grating)
• Wave “in phase” –
incident crests and
troughs aligned
1
2
d
Fraunhofer Diffraction
1
d
2

• Extra distance traveled by 2 is d sin
• If this is an integer number of wavelengths,
we get constructive interference
Condition for Maxima
d sin   n
where n = 0, 1, 2, …
• If  is known, measuring  tells us d
• Or if d is known we can get , etc.
Diffraction Maxima
Many Diffraction Centers
• All combine
constructively
at the specified
angles
• Nice sharp
maxima!
Bragg Diffraction
Bragg’s Law
• W. H. Bragg and W. L. Bragg, 1913 (Nobel
1915)
• Condition for constructive interference:
2d sin   n
• Diffraction from different sets of planes in the
crystal gives a picture of the overall structure
More Information
• Intensities of diffraction
maxima can vary – more
information about detailed
structure
• Symmetry of the crystal
structure is reflected in the
diffraction pattern
Electron Diffraction
• Can be done with
particles too, due to
their wave nature!
• Direct test of the De
Broglie relation
 = h/p
(Davisson and Germer,
Thomson)
Electron diffraction image
Optical Transform Exercises
• Uses visible light,  ~ 400-700 nm
• Diffracting patterns created with
features on this scale
• Illuminate with laser pointer – coherent
light source
Discovery Slide
• Horizontal line patterns (a and c)
• Diffraction pattern is a set of vertical
dots – why?
• Check spacing of dots for different line
spacings
– “Reciprocal lattice effect”
• Check spacing of dots for different
wavelengths
Group Exercise
• Given a known wavelength of light (it’s
shown on the laser pointer), determine
the line spacing of the Discovery Slide
pattern a.
Other Exercises
• Explore the symmetry relations between
diffraction patterns and the “atomic”
arrangements
• Look for variations in intensity for
diffraction maxima – when does this
occur?