Volume 135, Issue 1-2

Accurate probability distributions of several structure factors and their magnitudes which have been obtained recently are employed here to derive in a systematic way a number of conditional distributions and associated conditional expectation values. The latter have found application in the development of new methods of phase determination.

The thermal expansion of γ Na 2 FeF 4 was measured up to 140°C using a Guinier-Lenné camera. Refinement of the crystal structure at 25° and 74°C indicates a temperature dependence of the U 's not in agreement with the expansion data. However, consideration of the changes in bond lengths at 25° and 74°C and of the values of the root-mean-squared amplitudes allows a qualitative understanding of the phenomenon. The higher value of the root-mean-squared amplitudes along c positively fix the latter as a preferential direction of stress relief upon heating. Expansion along b is controlled by the rigidity of the M(1) polyhedra located at inversion centers and opposing cooperative phenomena of translational motion along this direction. Due to the stability of the tetrahedral configuration controlling the a repeat, expansion along a is the smallest one, in agreement with theoretical expectations. The abrupt increase in α 1 at 120°C is attributed to the formation of a stronger repulsion potential caused by a change from a sp 3 configuration of the tetrahedral bonds to one tending toward lower values of orbital hybridization. This process opposes the close-packed arrangement and at a critical temperature a phase change is expected to take place. This is in agreement with the experimental data since at 187 ± 5°C an inversion to a monoclinic phase was positively identified.

The structure of copiapite, MgFe 4 [(OH) 2 |(SO 4 ) 6 ] · 20H 2 O, is triclinic, space group P [unk], with a = 7.342 Å, b = 18.818 Å, c = 7.389 Å, α = 91.45°, β = 102.15°, and γ = 98.85°. Fe(O,OH,H 2 O) 6 octahedra and sulfate tetrahedra are connected to form one-dimensional infinite complexes along [101]. Mg(H 2 O) 6 octahedra and water molecules not bound to cations lie between the complexes. These structural groups are interconnected only by hydrogen bonds.

4,8,9,10-Tetrachloro-2,6-dithiaadamantane, C 8 H 8 Cl 4 S 2 , crystallizes in the monoclinic space group P 2 1 / c with unit-cell dimensions a = 7.59 7 ± 0.010, b = 13.31 1 ± 0.013, c = 12.77 1 ± 0.015 Å and β = 118.2 6 ° ± 0.09°. There are four molecules in the unit cell. The phase problem was solved by the interpretation of three-dimensional Patterson functions. The final atomic and thermal parameters were obtained by the least-squares refinement. The molecule has an adamantane cage structure and symmetry [unk] along the line connecting two sulfur atoms. The mean lengths of the C–S, the C–C and the C–Cl bonds are 1.82, 1.52 and 1.84 Å respectively. The average bond angle around the sulfur atoms is 96.4° and that around the carbon atoms is almost tetrahedral. Each molecule is surrounded, with van der Waals contacts, by six adjacent molecules.

The type of sphere packing corresponding to the cubic diamond configuration may be tetragonally distorted with symmetry P 4 1 . The Dirichlet domains have been studied for this case. 33 different types of these have been found. Each type is described by the conditions of its existence and by the number and kind of its faces. The various types are illustrated by drawings.

Derivative structures based upon sphalerite or galena often exhibit hexagonal superstructures characteristic of a peculiar thirteenfold multiplicity. A synthetic semiconductor, Cu 7 As 6 Se 13 , is one of the examples. Crystals of Cu 7 As 6 Se 13 are rhombohedral with a = 8.71 Å, α = 105°41′. The space group is R 3. The unit cell referred to hexagonal axes has dimensions a = 14.025 Å, c = 9.61 Å, and there are three formula units per cell. As an example of the application of the partial Patterson method, the superstructure was solved based upon a three-dimensional Patterson function synthesized with superstructure reflections alone. The structure is a distortion structure based upon sphalerite. The maximum distortion from sphalerite is around 0.2 Å. In terms of the theory of derivative crystal structures, it has been shown that the thirteenfold superstructures are smallest probable ones for rhombohedral distortion structures based upon sphalerite or galena.

The difference in electron density between a superstructure and the structure averaged over all subcells of the superstructure is a continuously variable function having positive and negative values. The partial Patterson functions based upon supercell reflections only are the Patterson functions of such a difference electron density. Therefore the geometry of the partial Patterson functions can be investigated by the aid of the vector set of a set of positive and negative unit points. The characteristics of such a vector set have been investigated in terms of the theory of image sets (B uerger , 1961) for a set of n pairs of positive and negative unit points which well represents the distributions of the positive and negative peaks in the difference electron densities of distortion structures. The vector set is characterized by the occurrence of the sets of quadruple images, each consisting of two positive and two negative point images. In the vector space, there are n ( n – 1) such quadruplets and 2 n negative point images around the origin, in addition to 2 n positive images at the origin. The shape defined by a set of quadruple images varies in accordance with the relative orientation of two pairs that define the quadruplet. There are two alternate solutions to the same vector set. For a vector set arising from a centrosymmetric set of pairs, the two solutions are related by an anti-inversion operation. For the noncentrosymmetric case, one solution is negative of the other. The result can readily be applied to superstructures generally.

The crystal structure has been determined with the use of two-dimensional intensity data. The lattice constants are a = 11.12 ± 0.02, b = 14.17 ± 0.03, c = 6.34 ± 0.02 Å, β = 123°45′ ± 10′ and the space group is C 2 h 6 – C 2/ c with Z = 4. The R value for observed reflections is 12,7%. The Sr atom is surrounded by ten O atoms with distances between 2,65 and 2,80 Å. Eight O atoms lie in the corners of a distorted Archimedean antiprism and the other two O atoms are placed above the larger quadratic surface. The polyhedra form an infinit zig-zag chain running parallel to [10[unk]]. The chains form a layer parallel (010).