Three-beam and many-beam theory in electron diffraction and its use for structure-factor phase determination in non-centrosymmetric crystal structures

Closed-form expressions for three-beam dynamical transmission electron diffraction are compared. These are used as a guide to determine the best experimental conditions for the determination of structure-factor phases by convergent-beam electron diffraction in the general non-systematic case. The validity domains of Kambe's [J. Phys. Soc. Jpn (1957), 12, 1-13] 'strong coupling' approximation and Bethe's [Ann. Phys. (Leipzig) (1928), 87, 55-129] second approximation are compared, and these approximations reconciled. A comparison of many-beam calculations with experimental non-systematic CBED patterns is used to determine a three-phase invariant for CdS with an accuracy of ±5° in the electron structure-factor phase. If it is assumed that two of the phases are known exactly, the error in the third (00) X-ray structure-factor phase would be ±0.75°. The accuracy of the method for determining phases, atomic position parameters and bonding charge distributions is discussed.