### Abstract

A new approach to the inversion problem of dynamical transmission electron diffraction is described, based on the method of generalized projections in set theory. An algorithm is described that projects between two sets of constrained scattering matrices. This iterative process can be shown to converge, giving the required structure factors (for some choice of origin) if the sets are convex. For the dynamical inversion problem, the set topology is that of an N^{2} torus, the sets are not convex, and traps are therefore sometimes encountered. These can be distinguished from solutions, allowing the algorithm to be restarted until a solution is found. Examples of successful inversion from simulated multiple-scattering data are given, which therefore solve the phase problem of electron diffraction for centrosymmetric or noncentrosymmetric crystal structures. The method may also be useful for the three-beam X-ray diffraction problem.

Original language | English (US) |
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Pages (from-to) | 112-118 |

Number of pages | 7 |

Journal | Acta Crystallographica Section A: Foundations of Crystallography |

Volume | 55 |

Issue number | 2 PART I |

State | Published - Mar 1 1999 |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Structural Biology

### Cite this

*Acta Crystallographica Section A: Foundations of Crystallography*,

*55*(2 PART I), 112-118.

**Dynamic inversion by the method of generalized projections.** / Spence, John; Calef, B.; Zuo, J. M.

Research output: Contribution to journal › Article

*Acta Crystallographica Section A: Foundations of Crystallography*, vol. 55, no. 2 PART I, pp. 112-118.

}

TY - JOUR

T1 - Dynamic inversion by the method of generalized projections

AU - Spence, John

AU - Calef, B.

AU - Zuo, J. M.

PY - 1999/3/1

Y1 - 1999/3/1

N2 - A new approach to the inversion problem of dynamical transmission electron diffraction is described, based on the method of generalized projections in set theory. An algorithm is described that projects between two sets of constrained scattering matrices. This iterative process can be shown to converge, giving the required structure factors (for some choice of origin) if the sets are convex. For the dynamical inversion problem, the set topology is that of an N2 torus, the sets are not convex, and traps are therefore sometimes encountered. These can be distinguished from solutions, allowing the algorithm to be restarted until a solution is found. Examples of successful inversion from simulated multiple-scattering data are given, which therefore solve the phase problem of electron diffraction for centrosymmetric or noncentrosymmetric crystal structures. The method may also be useful for the three-beam X-ray diffraction problem.

AB - A new approach to the inversion problem of dynamical transmission electron diffraction is described, based on the method of generalized projections in set theory. An algorithm is described that projects between two sets of constrained scattering matrices. This iterative process can be shown to converge, giving the required structure factors (for some choice of origin) if the sets are convex. For the dynamical inversion problem, the set topology is that of an N2 torus, the sets are not convex, and traps are therefore sometimes encountered. These can be distinguished from solutions, allowing the algorithm to be restarted until a solution is found. Examples of successful inversion from simulated multiple-scattering data are given, which therefore solve the phase problem of electron diffraction for centrosymmetric or noncentrosymmetric crystal structures. The method may also be useful for the three-beam X-ray diffraction problem.

UR - http://www.scopus.com/inward/record.url?scp=4244066414&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4244066414&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:4244066414

VL - 55

SP - 112

EP - 118

JO - Acta Crystallographica Section A: Foundations and Advances

JF - Acta Crystallographica Section A: Foundations and Advances

SN - 0108-7673

IS - 2 PART I

ER -