The possibility of solving by direct methods the difference structure 6 of a superstructure using only the intensities of the superstructure reflections h is shown. The refinement of the phases • of the strong superstructure E values, which are normalized separately from the E values of the main or substructure reflections, is carried out maximizing the function Z~p, = y]h(Eh-(Eh))Ch(~), where Ch(~b ) is the amplitude of the structure factor of 63 expressed in terms of ~. The maximization is performed with a new tangent formula that only differs from that given previously [Rius (1993). Acta Cryst. A49, 406-409] by an extra summation, i.e. the phase information is now derived from quartets and negative quintets instead of triplets and negative quartets. A preliminary test calculation demonstrates the capability of this tangent formula to solve the difference structure of the mineral wermlandite using only the measured superstructure intensities. Although more tests covering a variety of situations are still required to allow for a generalization, this result seems to confirm the viability of determining the internal structure of reconstructed surfaces by interpreting the corresponding three-dimensional difference Patterson function by direct methods. Access to this function is now possible, since, with the advent of intense synchrotron sources, not only in-plane intensity data but also the corresponding diffraction rods can be measured.is a subgroup of index n of the space group of the average structure. If the subgroup is maximal and if the reduction of symmetry operations affects the number of lattice points, the resulting superstructure is called klassengleich (Hermann, 1929). The reduction of lattice points causes the appearance of superstructure reflections (h), which are systematically weaker than the main reflections (H). The different mean intensity of the two reflection sets is due to the different amounts of electron density that contribute to each set. This can be best illustrated with the one-dimensional superstructure in Fig. l(a) consisting of two heavy atoms with form factors fp placed at the origin of the 'subcells' (at x = 0, 1/2) and of two lighter atoms with form factors fL at x = +x L. The structure-factor expression for the main (m) and superstructure (s) reflections are, respectively,
The structure of wermlandite, [Mg 7 (Al 0 .5 7 Fe 0 .43) 2 (OH) 18 ] 2 + [(Ca<). 6 , Mgo. 4 ) (S0 4 ) 2 (H 2 0) 12 ] 2 -has been determined and refined to an R value of 7.2% based on 779 symmetry-independent reflections, of which 224 are unobserved (weighted R=6.1%). The cell dimensions are a = ¿ = 9.303(3), c = 22.57(l)A, the space group is Phc\ with Z=2 and D x = 1.96 cm -3 .The double layer structure of wermlandite is similar to those of pyroaurite and sjögrenite. In these compounds the excess positive charge in the brucitelike main layer is compensated by additional anions in the interlayers. Main and interlayer are connected by O-H.. .0 hydrogen bonds >2.91 Â. The network of hydrogen bonds could be evaluated.The reflections of type h,k = 3n, /=2n are especially strong and indicate the existence of an eighteenfold trigonal superstructure. The superstructure was solved using partial Fourier and Patterson syntheses.
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