Band 115, Heft 1-2

Generalized point groups, the so-called simple crypto-symmetries, have been described earlier 1 . They were obtained by attributing a variable quality to the points of a point group, in such a way that the alterations of the quality form a group homomorphic to the point group. In this paper the attribution of several qualities is studied, each of which fulfills the conditions for a simple crypto-symmetry (multiple crypto-symmetries). The connection between multiple crypto-symmetries and reducible representations of the point group is analogous to the one between simple crypto-symmetries and irreducible representations. A simple way of characterizing the crypto-symmetries is their description by means of permutations.

Zinc hydroxychloride II [Zn 5 (OH) 8 Cl 2 · 1 H 2 O] was synthesized from ZnO and ZnCl 2 . The symmetry is rhombohedral with a H = 6.34 ± 0.01, c H = 23.64 ± 0.02 Å; space group D 3 d 5 – R [unk] m ; Z = 3 formula units per threefold-primitive hexagonal cell. Patterson and Fourier projections yielded the atomic positions: Zn I = 9 e , Zn II = 6 c ( z = 0.0721 ± 0.0004), Cl = 6 c ( z = 0.1708 ± 0.0013), O I H = 18 h ([unk] = 0.179 ± 0.005, z = 0.0532 ± 0.0013), O II H = 6 c ( z = 0.3820 ± 0.0013), H 2 O III = 3 b with R 1 (0 k[unk]l l ) = 0.154 and R 1 hk[unk] 0) = 0.123. – The structure is of the layer type and may be derived from a hypothetical C 6-Zn(OH) 2 structure. Within one composite layer, 60% of the Zn atoms are octahedrally (Zn I ) and 40% tetrahedrally (Zn II , above and below the Zn I plane) coordinated; coordination and interatomic distances indicate that the zinc octahedral bonds (Zn–O = 2.16, 2.17 Å) are predominantly ionic in nature and that the tetrahedral bonds (Zn–O = 2.02 Å, Zn–Cl = 2.33 Å) contain an appreciable amount of covalent character. The structure consists of these composite layers following the rhombohedral repeat scheme; they are held together by O–H … O = 2.79 Å and O–H … Cl = 3.09 Å bonds (all ± 0.05 Å).

The well-known but little used concept of space filling proves to be an excellent method for presenting geometrical relationships of different simple crystal structures. By introducing the radius ratio into the equation for space filling it is possible to express space filling in a way which is independent of the particular size of the atoms involved, but is instead characteristic for each structure type. The shape of the space-filling curves allows one to distinguish homeotect structure types, structure types with commutative partial structures and structure types with identical partial structures. The features of space-filling curves can be related to the values in L aves ' construction formula. The space-filling concept can be used to find geometrical correlations which have heretofore been obscured by the differing methods used to describe various structures. By comparing space-filling values one can predict the possible pressure structures of compounds. Finally the space-filling diagram offers a convenient way to present geometrical structure arguments. This is demonstrated for the known relations applicable to valence compounds and Hägg compounds. This discussion is complemented by a further demonstration of the use of space-filling diagrams in respect to certain new relations found to be valid for monosilicides and borides.

Formulae are derived which concern the half-width of single crystal reflections of zero and higher levels, and which determine the “aperture” of the counter tube needed for the measurement of integral intensities. It is investigated how far reflections of higher levels can be measured.

A platinum-wound resistance furnace for the Buerger precession camera has been designed. Temperatures up to 1300°C can be attained. The crystal is mounted directly on a thermocouple attached to the goniometer head. The position of the crystal in the furnace affects inversely the cone angle of the emergent x-rays and the temperature gradient across the crystal; a compromise between these two must be made.

The detailed x-ray analysis of the structure of copper perchlorate hexahydrate has been carried out by Fourier and least-squares methods aided by chemical considerations of ionic packing. The monoclinic crystals, space group P 2 1 / c , have six formula units in a cell of dimensions a = 5.14 Å, b = 23.173 Å, c = 14.147 Å with β = 90°. The structure consists of octahedral groups of water molecules around the cupric ion, and tetrahedral perchlorate groups. The Cu-(H 2 O) 6 octahedron with a mean Cu–O length of 2.18 Å and the ClO 4 tetrahedron with a mean Cl–O length of 1.48 Å are only slightly deformed from the ideal structures. The hydrogen bonds which link the water and perchlorate oxygens are puckered six-membered rings having the ‘boat’ and ‘chair’ shapes unlike those in the structure of magnesium perchlorate hexahydrate proposed by W est where these rings have always a chair-shaped structure. As the two structures are otherwise similar in all respects, W est 's analysis may require revision.

The crystal structure of vanadium pentoxide, V 2 O 5 , has been redetermined with three-dimensional data obtained by photographic and by counter methods, and has been refined by differential syntheses. The results are in closer agreement with those of B yström , W ilhelmi and B rotzen than with those of K etelaar . The probable space group is Pmnm and the unit-cell dimensions are a = 11.510 Å, b = 4.369 Å, c = 3.563 Å. The structure is built up from distorted trigonal bipyramidal coordination polyhedra of O around V, which share edges to form zigzag double chains along [001] and are cross-linked along [100] through shared corners, thus forming sheets in the xz plane.

Single-crystal photographs of hauyne from the Eifel district showed beside the cubic reflections of the space group P [unk]3 n numerous additional ones which can be interpreted as reflections of a superstructure. They show a rod-like distribution along [110] directions in the reciprocal lattice. Beside these reflections diffuse blackening of the films is visible along the same directions. This indicates an additional one-dimensional disorder. The reflections of the superstructure do not disappear on heating, but in the range between 1000–1100°C a rearrangement of these reflections occurs. Some hauyne from Vesuvius shows no superstructure. Fourier syntheses of the hauyne with and without superstructures provide evidence that the superstructure is called forth by SO 4 groups and by a part of the (Ca, Na) ions.

This paper deals with investigations on the crystal structure analysis of tetrachlor-1-keto-naphthalene-dihydride. The method of folded molecules was applied in conjunction with the method of diffuse scattering and by methods of “the haevy atom technique”. A structure model has been derived the refinement of which being under consideration.

The Q , α criterion is put forward in order to give the significant features of tables of values more-or-less with one term. Its significance for x-ray structure determination is discussed and its advantage is emphasized compared with the customary R factor, which is only applicable as a necessary (but not sufficient) criterion.