Three-beam convergent-beam electron diffraction for measuring crystallographic phases

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Under almost all circumstances, electron diffraction patterns contain information about the phases of structure factors, a consequence of the short wavelength of an electron and its strong Coulombic interaction with matter. However, extracting this information remains a challenge and no generic method exists. In this work, a set of simple analytical expressions is derived for the intensity distribution in convergent-beam electron diffraction (CBED) patterns recorded under three-beam conditions. It is shown that these expressions can be used to identify features in three-beam CBED patterns from which three-phase invariants can be extracted directly, without any iterative refinement processes. The octant, in which the three-phase invariant lies, can be determined simply by inspection of the indexed CBED patterns (i.e. the uncertainty of the phase measurement is ±22.5°). This approach is demonstrated with the experimental measurement of three-phase invariants in two simple test cases: Centrosymmetric Si and non-centrosymmetric GaAs. This method may complement existing structure determination methods by providing direct measurements of three-phase invariants to replace 'guessed' invariants in ab initio phasing methods and hence provide more stringent constraints to the structure solution.

Original languageEnglish
Pages (from-to)753-764
Number of pages12
Publication statusPublished - 1 Jan 2018


  • convergent-beam electron diffraction
  • crystallographic phase problem
  • dynamical studies
  • enantiomorph ambiguity
  • multiple scattering
  • nanocrystals
  • structure determination
  • three-phase invariants

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