Volume 117, Issue 5-6

X-ray diffraction measurements on magnesium powder as well as single crystals and on ruthenium powder were performed at 296°K, 90°K and 5°K. The atomic scattering factors do not show systematic deviations from a smooth curve indicating that no electron accumulations exist in the lattices of these metals except at points where atoms are located. Debye characteristic temperatures calculated from diffracted intensities increase at 5°K with respect to their value at higher temperatures. The increase is for Mg in agreement with data obtained from specific heat and elastic constants. It is for both metals in agreement with the refined dynamic theory of crystals.

Bustamite is triclinic, space group F [unk], with cell dimensions a = 15.412 Å, b = 7.157 Å, c = 13.824 Å, α = 89°29′, β = 94°51′, γ = 102°56′. The unit cell ideally contains 12(CaMnSi 2 O 6 ). Three-dimensional intensity data were gathered with the single-crystal Geiger-counter diffractometer. Application of the minimum function to P ( xz ), and comparison of the unit cells of bustamite and wollastonite yielded a trial structure which was refined with least-squares. Planes of approximately close-packed oxygen atoms are oriented parallel to (101). Planes containing Ca and Mn atoms in octahedral coordination alternate with planes of Si atoms in tetrahedral coordination between oxygen planes. Ca and Mn are ordered. Si tetrahedra are linked to form chains parallel to b with a repeat unit of three tetrahedra.

The structure of Mg(NH 4 ) 2 (SO 4 ) 2 · 6H 2 O has been redetermined in order to assign the hydrogen bonds. The crystals are monoclinic, space group P 2 1 / a , with a = 9.324, b = 12.597, c = 6.211 Å, and β = 107.14°. The structure previously reported by H ofmann has nearly correct coordinates for the Mg, S, and N atoms, but incorrect coordinates for the oxygen atoms. Three-dimensional x-ray diffraction data were taken with the General Electric goniostat. The structure, including hydrogen atoms, was determined by Fourier and least-squares methods. The magnesium atom is surrounded by an octahedron of water molecules, each forming two hydrogen bonds to oxygen atoms of the sulfate ions. Three hydrogen atoms of each ammonium ion are bonded to sulfate-oxygen atoms, but the fourth is equidistant from two sulfate-oxygen atoms with the geometry which previous authors have called a “bifurcated hydrogen bond”. Average interatomic distances (uncorrected for thermal motion) are: Mg–O = 2.07 Å, S–O = 1.47 Å, O–O (hydrogen bonded) = 2.77 Å, N–O (hydrogen bonded) = 2.90 Å, and N–O (bifurcated bond) = 3.08 Å. The average S–O bond length after correction for thermal motion is 1.49 Å.

The image of one set of points in another set gives rise to a new set called an image set. Those were discussed in recent communication 1,2 . In the present paper the image set is generalized to an image function, which is the image of one function in another. The Patterson function and the autocorrelation function are variations of the special case of image funtions called self-image functions.

Some of the known properties of image sets are derived in this paper by means of convolutions. In this way, B uerger 's theory of image sets can be extended to structures defined by continuous functions such as the electron density of crystals. Also a few new results are derived.

Bismuth phosphate has long been known to crystallize with the monazite-type structure. It is now possible to report the existence of two other modifications, and to describe their crystal structure. One type is hexagonal, and has the CePO 4 · ½ H 2 O structure. The other is a monoclinic high-temperature form which very slowly converts to the monazite-type structure at ordinary temperatures. The structure of the hexagonal crystal has been determined from Weissenberg single-crystal data, and that of the high-temperature form from powder-diffraction data. The three structures preserve certain features in common, and intercomparison seems to indicate that the transformation from one to the other must be predominantly displacive rather than disruptive in character.

Anthrone is monoclinic with unit-cell dimensions a = 15.80, b = 3.998, c = 7.86 Å, and β = 101°40′, and belongs statistically to the space group C 2 h 5 − P 2 1 / a . Though the molecules are non-centrosymmetrical, there are only two molecules per unit cell; these are arranged statistically in orientations so that, anthrone possesses higher symmetry than would be normally expected from two asymmetric molecules. The crystal structure has been determined by two-dimensional Fourier syntheses, and has been refined by seven cycles of difference syntheses, four cycles on the (010) plane to refine the x and z coordinates, and three on the (001) plane to refine the y coordinates. The molecule is well resolved in the projection on the (010) plane, as is expected from the shortness of the b axis, but the resolution on the planes (100) and (001) is poor.

Use of a modified Bridgman anvil device and a supported beryllium cylinder with both molybdenum and copper radiation has yielded sufficiently accurate diffraction data to warrant a general attempt at indexing of the high-pressure polymorph KNO 3 IV. Indexing is best accomplished on the basis of an orthorhombic cell having a = 11.04 8 Å, b = 8.36 7 Å, c = 7.40 2 Å, and 10 KNO 3 per cell. The larger RbNO 3 -type unit cell established by J amieson (1956) with Z = 18 is open to question because of new diffraction data which do not allow centering on C . Although a complete structure analysis has not yet been carried out, possible positions for the K + ions are suggested having the symmetry of the space group Pmn 2 1 . Rate studies on the transition KNO 3 II ⇋ KNO 3 IV show an over-all approach to first-order kinetics but in detail reveal several growth stages.

The crystal structure of potassium hexatitanate has been determined. The space group is C 2/ m and the cell dimensions are a = 15.582 ± 0.006 Å, b = 3.82 ± 0.01 Å, c = 9.112 ± 0.001 Å. β = 99.764 ± 0.008°. This unit cell contains 2(K 2 Ti 6 O 13 ). Three-dimensional intensity data were collected by means of a single-crystal Geiger-counter diffractometer and the intensities were corrected for Lorentz-polarization factors and absorption. In later stages of the structure determination the three Ti were corrected for anomalous scattering. A three-dimensional Patterson synthesis suggested a structure based on Ti octahedra with their axes parallel to the 2-fold axis. A model structure which gave the correct number of oxygens in this symmetry was confirmed by means of the minimum function. This structure, consisting of chains of Ti octahedra sharing edges and corners, with some tunnels where the K ions can fit, was refined primarily by successive three-dimensional Fourier syntheses and finally by least-squares methods, leading to a discrepancy factor R of 12.4% for all reflections. Individual temperature factors and interatomic distances are in agreement with reported values.

The crystal structure of the anhydrous stannous sulphate, SnSO 4 , has been determined from x-ray powder data obtained with a Norelco diffractometer. The unit cell is orthorhombic with a = 8.799, b = 5.319, c = 7.115 Å, Z = 4, space group D 2 h 16 − Pnma . Stannous sulphate belongs to the barite group. The unknown parameters were determined by trial and error, after taking into account the similarity with the barite group. The S atoms lie at the center of an almost regular tetrahedron of O atoms. The Sn atoms are irregularly surrounded by twelve O atoms belonging to seven different tetrahedra.

The crystal structure of copper-potassium oxalate dihydrate, CuK 2 (C 2 O 4 ) 2 · 2H 2 O has been determined by application of the Patterson method and the linear inequalities of O kaya and N itta to two-dimensional ( hk 0) and ( h 0 l ) x-ray data. The crystal belongs to the triclinic space group P [unk]. There are two formula units in the unit cell of dimensions a = 8.66, b = 10.19, c = 6.86 Å, α = 120°41′, β = 83°52′, γ = 110°18′. The R factors are 0.162 and 0.175 for the ( hk 0) and ( h 0 l ) reflections respectively. The copper atoms Cu 1 and Cu 2 occupy the special positions 0 0 0 and 0½0 and are surrounded coplanarly by four nearest oxalate-oxygen atoms (Cu–O ≈ 2 Å). The distorted octahedral coordination of Cu 1 is completed by two water molecules related by a centre of symmetry (Cu 1 –H 2 O = 2.27 Å), and that of Cu 2 is completed by two more distant oxalate-oxygen atoms (Cu 2 –O = 2.70 Å). The environment of the K + ions consists of eight oxygen neighbours at a mean distance of 2.9 Å. The non-equivalent oxalate groups within the cell are held together by hydrogen bonds and the structure is built up by the linking of these aggregates to their centrosymmetrically adjacent neighbours by alternate Cu–O and K–O interactions.

A 22-layered new polytype of cadmium iodide, designated as 22 H c , with unit cell dimensions a = 4.24 Å, c = 75.18 Å has been discovered. The detailed structure of the polytype has been worked out. It is shown that this polytype is based on a 2 H ideal structure. The polytype 22 H c is so far the largest of the cadmium iodide polytypes whose detailed structure is known. The growth of the polytype has been discussed in terms of screw-dislocation theory and the difficulties which arise in this are pointed out.

The scattering function is calculated for a net of points in the reciprocal space for silver chloride using the measured values of elastic constants. The lines of constant scattering intensity are drawn and compared with the experimental results. In case of zinc, the lines are constructed only in the vicinity of the selective reflections.

The deviation from the ideal octahedral coordination is correlated to the van der Waals radii of the ligand atoms and their distances from the central atom.

The structure of high quartz was determined applying the space groups P 6 2 2 (or P 6 4 2) and, under supposition of twinning, P 3 1 2 (or P 3 2 2) at 650° of The Wyckoff structure could be refined to 25% for relative mean deviation C. intensities ( R ∼ 16.2%). Assuming a submicroscopic twinning of low quartz lead to 19,3 mean deviation ( R ∼ 12.6%). Therefore must be assumed that high quartz is submicroscopically twinned according low-quartz symmetry.

K 2 SnBr 6 is tetragonal, pseudocubic at room temperature and becomes cubic at 126.5°C. The transition is reversible and “displacive” (B uerger ). The expansion coefficients are: λ a = 4.96 · 10 −5 , λ c = 5.65 · 10 −5 for the tetragonal phase between 0°C and 100°C; and λ a = 5.22 · 10 −5 for the cubic phase between 130°C and 250°C. In the transition point no change of volume occurs, indicating a second order transition.