Diffraction and Crystal Structure

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Diffraction and Crystal Structure

• NANO Workshop 2006

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Outline

• Light and X-rays
• Diffraction basics
• Determining crystal structures
• Activities!

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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

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Anatomy of a Wave

• Wavelength ?
• Frequency f
• of cycles per unit time
• Speed v ?f
• Amplitude A

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Electromagnetic Waves

• Speed 3 ? 105 km/sec (about 186,000 mi/sec)

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The Electromagnetic Spectrum
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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

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Visible Light

• 400440 nm Violet
• 440480 nm Blue
• 480530 nm Green
• 530590 nm Yellow
• 590630 nm Orange
• 630700 nm Red

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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)

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X-Ray Diffraction

• X-rays have wavelengths comparable to atomic
  sizes and spacings, about 1010 m
• Crystals and molecules reflect X-rays in specific
  patterns depending on their structures

X-ray diffraction pattern of myoglobin
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Interaction of X-Rays with Atoms

• Involves the electrons, primarily

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Fraunhofer Diffraction

• Fraunhofer incident, outgoing rays parallel
• Start with single plane of atoms (a grating)
• Wave in phase incident crests and troughs
  aligned

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Fraunhofer Diffraction

• Extra distance traveled by 2 is d sin?
• If this is an integer number of wavelengths, we
  get constructive interference

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d
?
2
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Condition for Maxima

• where n 0, 1, 2,
• If ? is known, measuring ? tells us d
• Or if d is known we can get ?, etc.

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Diffraction Maxima
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Many Diffraction Centers

• All combine constructively at the specified
  angles
• Nice sharp maxima!

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Bragg Diffraction
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Braggs Law

• W. H. Bragg and W. L. Bragg, 1913 (Nobel 1915)
• Condition for constructive interference
• Diffraction from different sets of planes in the
  crystal gives a picture of the overall structure

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More Information

• Intensities of diffraction maxima can vary more
  information about detailed structure
• Symmetry of the crystal structure is reflected in
  the diffraction pattern

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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
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Optical Transform Exercises

• Uses visible light, ? 400-700 nm
• Diffracting patterns created with features on
  this scale
• Illuminate with laser pointer coherent light
  source

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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

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Group Exercise

• Given a known wavelength of light (its shown on
  the laser pointer), determine the line spacing of
  the Discovery Slide pattern a.

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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?