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PASI Santiago, Chile July 20061Eades / Dynamical theory
Dynamical Theory
(without equations)
PASI Santiago, Chile July 20062Eades / Dynamical theory
Dynamical Theory
• This lecture aims to communicate some intuitive bases for thinking dynamically
• Some familiarity with basic ideas of diffraction (for example, Bragg’s law and the reciprocal lattice) will be assumed.
PASI Santiago, Chile July 20063Eades / Dynamical theory
Kinematical vs Dynamical theory.
• Any sample more than a few atoms thick requires dynamical theory.
• A simple argument suggests that if the sample thickness is greater than about ξg/8, then dynamical theory is necessary
PASI Santiago, Chile July 20064Eades / Dynamical theory
Bragg’s Law
Bragg’s Law
2d sin θ = λ
still tells us where there are diffracted beams.
PASI Santiago, Chile July 20065Eades / Dynamical theory
Rocking Curves
How far from the Bragg angle do we get diffraction?
For thick samplesΔθ = 4/gξg ξg is the extinction distance
For thin samplesΔθ = 1/gt t is the sample thickness
PASI Santiago, Chile July 20066Eades / Dynamical theory
ReimerTransmission Electron Microscopy
PASI Santiago, Chile July 20067Eades / Dynamical theory
Channeling
Electrons do not travel through crystals with a uniform distribution.
When electrons are incident on a crystal close to the Bragg angle, the electrons are channeled down the atomic planes.
Indeed, Bragg reflection may be thought of as the result of electrons coupling into channeling states.
PASI Santiago, Chile July 20068Eades / Dynamical theory
Channeling II
The channeling states are also called Bloch states or Bloch waves.
Bloch waves consist of electrons traveling parallel to the planes but the electrons are not uniformly distributed. The Bloch waves may be channeled onto the atoms or between the atoms.
Defect contrast arises because the defects scatter electrons from one Bloch state to another.
PASI Santiago, Chile July 20069Eades / Dynamical theory
PASI Santiago, Chile July 200610Eades / Dynamical theory
PASI Santiago, Chile July 200611Eades / Dynamical theory
PASI Santiago, Chile July 200612Eades / Dynamical theory
To understand dynamical diffraction it is convenient to use a new graphical construction. We replace the Ewald sphere construction with a dispersion surface construction.
For kinematical diffraction they give the same result but it is easier to extend the dispersion surface construction to describe dynamical diffraction
PASI Santiago, Chile July 200613Eades / Dynamical theory
Constructions
PASI Santiago, Chile July 200614Eades / Dynamical theory
PASI Santiago, Chile July 200615Eades / Dynamical theory
PASI Santiago, Chile July 200616Eades / Dynamical theory
PASI Santiago, Chile July 200617Eades / Dynamical theory
PASI Santiago, Chile July 200618Eades / Dynamical theory
PASI Santiago, Chile July 200619Eades / Dynamical theory
PASI Santiago, Chile July 200620Eades / Dynamical theory
PASI Santiago, Chile July 200621Eades / Dynamical theory
PASI Santiago, Chile July 200622Eades / Dynamical theory
PASI Santiago, Chile July 200623Eades / Dynamical theory
PASI Santiago, Chile July 200624Eades / Dynamical theory
PASI Santiago, Chile July 200625Eades / Dynamical theory
PASI Santiago, Chile July 200626Eades / Dynamical theory
PASI Santiago, Chile July 200627Eades / Dynamical theory
PASI Santiago, Chile July 200628Eades / Dynamical theory
PASI Santiago, Chile July 200629Eades / Dynamical theory
End
• Dynamical diffraction explains everything.
PASI Santiago, Chile July 200630Eades / Dynamical theory
PASI Santiago, Chile July 200631Eades / Dynamical theory
PASI Santiago, Chile July 200632Eades / Dynamical theory
PASI Santiago, Chile July 200633Eades / Dynamical theory
PASI Santiago, Chile July 200634Eades / Dynamical theory
PASI Santiago, Chile July 200635Eades / Dynamical theory
PASI Santiago, Chile July 200636Eades / Dynamical theory
Two-beam dynamical