[QMSE] W02 - Diffraction

• waves tend to spread out as they propagate

diffraction

• variety of wave interference effects during wave propagation

point source

• a point wave source generates circular waves

aperture source

• waves splitting through an aperture can be assumed to have several point sources along the plane of the aperture

wavefronts

• several wavefronts are formed at a distance ahead in the propagation path due to interference

• the waves get weaker by spreading out as they move away from the aperture
• the angle of the spread of the wave is the diffraction angle
  • diffraction angle $\theta \propto \frac{\lambda }{d}$
  • where:
    • $\lambda$: wavelength
    • $d$: size of aperture opening
• larger the opening, smaller the diffraction angle and vice-verse
  • lesser the pinching of waves into a point-source like opening

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diffraction from periodic structures

• a diffraction grating is a device with a piece of glass (or other light transmitter) with a set of closely spaced lines scratched onto it
• crystals naturally have periodically spaced atoms in them
  • crystals are just periodic array of atoms
• x-rays (and other short wavelength lights) shined through crystals tell us a lot about the crystal structure

periodic scatterers

• one scatter will give a set of phase fronts called concentric circles

• multiple equally spaced scatterers give multiple sets of concentric circles

• these scatterers all add in phase for particular directions
  • ‘straight ahead’ called a zeroth order diffraction
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  • also, two more are obtained - ‘upward’ and ‘downward’
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• at large distances from the scatterer, a multiple beam diffraction pattern in obtained
  • which looks like a set of points on a screen
• larger scatterer separation gives beams closer in angle
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• the angle $\theta$ of these diffracted waves is given by simple geometry
  • 
  • $\sin \theta =\frac{\lambda }{s}$
    • $\lambda$: wavelength
    • $s$: separation between scatterers
  • larger spacing $s$ multiple diffraction ‘orders’ are possible
    • $\sin {\theta }_1=\frac{\lambda }{s}$
      • is a first order diffraction
    • $\sin {\theta }_2=\frac{2\lambda }{s}$
      • is a second order diffraction
• generalizing, for a 1D array of scatterers
  • the diffraction angle is ${\theta }_n={\sin }^{-1}\left(\frac{n\lambda }{s}\right)$

multiple slits

• for an electron beam incident on an array of multiple slits
  • you will see a pattern with the same period as that for an array of two slits
• however, as $n$ increases, the sharpness of the intensity peaks will increase
  • so the pattern will look more and more like a series of discrete spots