Direct-method treatment of pseudo symmetry

        Pseudo symmetry is due to the symmetry of a partial structure being higher than that of the whole structure. There are two categories of pseudo symmetry [1]:

• Pseudo-translation symmetry and translational phase ambiguity
          Pseudo-translation symmetry is due to part of the crystal structure having a higher translational symmetry (a shorter translation vector) than that of the whole structure. Such part of the structure will then have no contribution to structure factors of certain index group, leading to the translational phase ambiguity of the corresponding reflections. This obstructs completion of the structure by Fourier recycling based on the partial structure.
          Modified Sayre equations [2-4, see also 5] relate structure factors of different index groups. This enables resolving the translational phase ambiguity by phasing reflections of certain index groups on the base of known phases of reflections from other index groups.
          Pseudo-translation symmetry and translational phase ambiguity can be found also in incommensurate modulated structures. "Multi-dimensional direct methods" developed on the base of modified Sayre equations can solve the problem (see details).
• Pseudo centrosymmetry and enantiomorphous phase ambiguity
          Pseudo centrosymmetry is due to the centric arrangement of a partial structure in noncentrosymmetric crystal structures. Fourier maps phased by such a partial structure will not reveal the true structure but rather a superimposed image of the true structure and its enantiomorph, leading to the enantiomorphous ambiguity. From the superimposed image, by setting the unit-cell origin at the pseudo-inverse center, one can calculate for each structure factor the real component and derive the absolute value of the imaginary component, while the sign of which remains unknown. That is the enantiomorphous phase ambiguity.
          Component relations [6] relate real parts with imaginary parts of structure factors. They can be used to resolve the enantiomorphous phase ambiguity by deriving signs of imaginary parts on the base of known signs of the real parts of structure factors [7, see also 5].
          Enantiomorphous phase ambiguity can be found also in protein-structure determination. Direct methods developed on the base of component relations can be used to solve the problem (see details).

References

        [1] Fan Hai-fu & Zheng Qi-tai (1982) Acta Phys Sin. 31 191-198. (In Chinese with English abstract)
        [2] Fan Hai-fu (1965) Acta Phys. Sin. 21, 1105-1114. (In Chinese with English abstract)
        [3] Fan Hai-fu (1975) Acta Phys. Sin. 24, 57-60. (In Chinese)
        [4] Fan Hai-fu, He Luo, Qian Jin-zi & Liu Shi-xiang (1978) Acta Phys. Sin. 27, 554-558. (In Chinese with English abstract)
        [5] Fan Hai-fu (1984) The Rigaku Journal vol. 1, no. 2, 15-21.
        [6] Fan Hai-fu (1965) Acta Phys. Sin. 21, 114-1118. (In Chinese with English abstract)
        [7] Fan Hai-fu & Zheng Qi-tai, (1978) Acta Phys. Sin. 27, 169-174. (In Chinese with English abstract)