Showing papers in "Acta Crystallographica Section A in 1972"

for The Neutron Diffraction Commission). Coherent neutron scattering amplitudes

Abstract: A list is given which summarizes additions and significant changes which have been reported since the publication of a full list of scattering amplitudes in 1972 [Acta Cryst. (1972). A28, 357-358].

Vacancy short‐range order in substoichiometric transition metal carbides and nitrides with the NaCl structure. I. Electron diffraction studies of short‐range ordered compounds

Abstract: The structure of cubic transition metal compounds VC1 − x, NbC1 − x, TiC1 − x, VN1 − x and TiN1 − x has been studied by electron diffraction. These compounds occur with large deviations from the stoichiometric composition and the carbides have been observed with long-range ordered carbon distributions for certain compositions and temperatures. For crystals of other compositions and temperatures electron diffraction patterns frequently contain diffuse bands of intensity which have a periodic distribution with respect to the reciprocal lattice points. The diffuse intensity follows a continuous surface of complex curvature in reciprocal space, is observed for all compounds except VN1 − x, and has the symmetry of the b.c.c. reciprocal lattice. These features indicate a short-range ordered non-stoichiometric vacancy distribution which is analysed in Part II of this series [Sauvage & Parthe (1972), Acta Cryst. A28, 607].

A mathematical procedure for superimposing atomic coordinates of proteins

Abstract: A procedure is given which determines the best rigid-body rotation and translation that matches a given set of measured atomic coordinates to a fixed set of guide coordinates and minimizes the weighted sum of squared deviations.

Revised parameters of the analytic fits for coherent and incoherent scattered X‐ray intensities of the first 36 atoms

Abstract: Analytical parameters have been recalculated for atomic scattering factors and incoherent scattered intensities of the elements of atomic numbers 2 to 36, based on the tables of Tavard, Nicolas & Rouault [J. Chim. Phys. (1967), 64, 540]. The fit has also been checked with the total scattered intensities.

The normalization of the resolution function for inelastic neutron scattering and its application

Abstract: By extending the considerations of Maier-Leibnitz, the normalization of the resolution function is found to be the product of the volumes in reciprocal space of the incident and scattered beams. Each volume is defined by an integration in reciprocal space over the probability of finding a particular k, where k is the wave vector of the neutron. The resolution can be understood as a convolution of these two volumes. For three-axis spectrometers explicit expressions for these volumes are given. The knowledge of the normalization is necessary for numerical unfolding of experimental data. For two cases, which often occur in inelastic neutron scattering, it is possible to directly correct the experimental data without resorting to numerical unfolding. After applying these corrections the data represent the scattering law folded with a resolution function normalized to unity, i.e. the integral over the corrected data is the integral over the scattering law. It is shown that in this case, the unfolding of the corrected data turns out to be a one-dimensional problem.

Vacancy short-range order in substoichiometric transition metal carbides and nitrides with the NaCl structure. II. Numerical calculation of vacancy arrangement

Abstract: Using Babinet's theorem and certain results of the theory of short-range order originally developed for Cu3Au alloys it is shown that the diffuse scattering intensity in the electron diffraction patterns of the substoichiometric transition metal carbides is related to a short-range order of carbon vacancies. Numerical values for the short-range order parameters can be obtained by integrating over the total diffuse intensity within one repetitive unit in reciprocal space. To perform such calculations on VC0.75 an analytical function was identified which describes the shape of the diffuse intensity curves in reciprocal space. Making the simplifying assumption that the value of the diffuse intensity is 1 on the diffuse intensity curve and 0 everywhere else numerical values for the vacancy short-range order in VC0.75 were obtained. The results, represented as average numbers of vacancies in different coordination shells around a vacancy, agree with the general principles of vacancy arrangement found in ordered transition metal carbides and nitrides.

On the terminology of the phenomena of mutual crystal orientation

Abstract: A system for designation of various types of mutual orientation of crystals and the processes during which they are formed is proposed The neutral terms oriented overgrowth, oriented intergrowth and oriented inclusions, and in special cases polycrystal and single-crystal films, are considered to be sufficient for describing the fact of regular joint growth As interpretative and genetic, four basic terms are submitted: epitaxy – for oriented growth of a phase on the crystal surface of another phase; syntaxy – for simultaneous growth of the mutually oriented crystals of two or more phases; topotaxy – for oriented transformation in an open system with a partial alteration in chemical composition of the primary crystal; and endotaxy – for oriented transformation in a closed system, without exchange of components between the system (primary crystal) and its environment For a later oriented overgrowth of crystals of one and the same phase (in parallel or twin position) it is possible to use the term autoepitaxy

Estimation of twinning parameter for twins with exactly superimposed reciprocal lattices

Abstract: In twinned crystals the intensity contribution from the separate individuals can be estimated from the observed intensities without any knowledge of the actual structure, even when the two reciprocal lattices are completely superimposed. Once the separate intensity contributions are known the trial structure can be sought in the usual ways.

n-Beam lattice images. II. Methods of calculation

Abstract: The calculation of amplitudes and phases of beams of electrons diffracted from thin (≤ 100 nm) crystals of W4Nb26O77 is described. These diffraction data are used to compute 11-beam 00l electron microscope lattice images and the effects of instrument aberrations are considered. Several approximations are compared with a more exact 435-beam, two-dimensional computation, and with experimental diffraction data and lattice images. Finally, the projected charge density approximation to image contrast is evaluated.

Molecular packing analysis

Abstract: A general method of molecular packing analysis based on the minimization of the crystal lattice energy is described. The lattice energy was approximated by a pairwise sum over nonbonded atoms in different molecules, using (exp–6–1) potential functions. Increased speed of calculation of the lattice sums was achieved by a convergence acceleration technique. The variables considered were six rigid body rotations and translations for each molecule in the asymmetric unit, and the six lattice constants. Molecular flexibility was allowed in the form of internal rotations about bonds (subrotations). In this event additional subrotation potentials of the cos2 ψ type could be used to allow for conjugation energy, and the subrotation angles were additional variables. An optional thermal correction, based on the mean square thermal amplitudes and the potential anharmonicities, was applied to calculate the anisotropic thermal expansion. Nonbonded potential parameters, including conjugation energies, were found by fitting them to observed crystal structures.

n-Beam lattice images. I. Experimental and computed images from W4Nb26O77

Abstract: A series of through-focus lattice images has been obtained with an electron microscope for single crystals of W4Nb26O77 set at an orientation to give mainly a systematic set of 00l reflexions. Dynamical n-beam computations (n = 435) have been made for a crystal in this orientation to determine the values of the amplitudes and phases of the diffracted beams. Subsequently, the fringe profiles for lattice images have been computed for various thicknesses of crystal and defects of focus. When the experimental factors, including instrument aberrations, are taken into account, good agreement between computed and observed fringes is obtained. An earlier assumption of direct correlation of features in underfocused images with structure in the crystal, is found to be valid, at least for this compound.

Si(111) surface structures by glancing-incidence high-energy electron diffraction

Abstract: The structures which form on the (111) surface of a silicon single crystal have been examined by glancing incidence high-energy electron diffraction under ultra-high vacuum conditions. A quantitative measurement of the intensities of the hhh systematic set of Bragg reflections from the 2nd to the 7th order was observed as a function of azimuthal angle, rocking angle and temperature. In particular, the intensity of the hhh reflections was measured at crystal azimuths which minimize the non-systematic n-beam interference. These `azimuth-independent' rocking curves for the 1 × 1, 7 and √19 structures are compared and discussed, to explore the possibilities of obtaining structural information for both the surface and the bulk by the glancing incidence high energy technique. The data are suitable for comparison with the calculations of an n-beam dynamical theory as it is developed and adapted to the Bragg case. The refractive shift of the Bragg peaks yielded a value of 12.0 ± 0.4 volts for the mean inner potential of silicon.

n-Beam dynamical diffraction of high-energy electrons at glancing incidence. General theory and computational methods

Abstract: The n-beam dynamical theory of high-energy electrons is currently used in transmission (Laue case) for accurate determination of the Fourier components of the crystal potential. The same theory is expected to provide information about the surface potential when used to interpret diffraction patterns in reflection at glancing incidence (Bragg case). Some peculiar aspects are elucidated in detail, insofar that they are different from the transmission case, particularly the boundary conditions. Inelastic scattering effects are introduced by means of a complex potential. Changes of the Fourier components of the potential near the surface are considered, and a model has been developed which incorporates these changes into the theory. A slice treatment developed in the frame of Bethe's theory is presented.

Comparison of experimental and n‐beam calculated intensities for glancing incidence high‐energy electron diffraction

Abstract: Rocking curves for the 2nd to 7th order reflections of the systematic set from the (111) planes of a highly perfect silicon single crystal have been obtained in an ultra-high-vacuum reflection electron diffraction camera operated at 40 keV. These have been compared with the profiles computed with an n-beam dynamic high-energy electron diffraction theory adapted to the Bragg case at glancing incidence. It is shown that, for the 333 reflection, six nodes of the hhh row from {\bar 111} up to 444 are adequate to take into account the systematic multiple beam interference effects. The calculated widths are found to agree with the observed widths within the experimental error for all reflections. The peak reflectivities computed initially for a perfect semi-infinite crystal are systematically higher than the experimental values, by a factor of 7 for the 222 node, decreasing to near agreement for the 777 node. Possible reasons for such discrepancies are discussed. Neither inelastic scattering effects nor reasonable changes in the silicon lattice potential near the surface can bring the observed and calculated intensities into agreement. The discrepancy can be considerably reduced for all orders of reflection, however, when the non-flat nature of the surface is considered.

Polarization phenomena in X-ray diffraction

Abstract: Because of the dielectric properties of most crystals in the X-ray frequency region, polarization phenomena such as birefringence and optical activity are not found with X-rays in simple transmission. It can however be deduced from the dynamical theory that such effects occur in crystal diffraction. An experiment is described which proves that under certain conditions all four branches of the dispersion surface can be excited by linearly polarized incident waves. This means that elliptically polarized X-rays can also be produced by crystal diffraction. Therefore in principle all polarization experiments that can be performed with visible light are also possible with X-rays.

Indétermination sur les dimensions de la maille magnétique dans l'étude par diffraction neutronique sur poudres de corps cubiques ou uniaxes

Abstract: An ambiguity arises in the study by powder neutron diffraction of compounds where the magnetic atoms lie on a uniaxial or cubic Bravais lattice, when the magnetic lines are indexed {{{h}\over{2}}kl} or {{{h}\over{2}}{{k}\over{2}}l}. Many multiaxial configurations, all having the same isotropic coupling energy, are compatible with a given set of lines. For a simple cubic lattice it is shown that every multiaxial configuration is indistinguishable, using powder intensities, from a particular uniaxial one, and that both have the same dipolar energy.

Absolute measurement of structure factors of Si by using X-ray Pendellösung and interferometry fringes

Abstract: The absolute values of crystal structure factors |Fg| of silicon were determined accurately for five low-order lattice planes, with probable errors of less than 0.05%. The principle of the measurement is to take the ratio of |Fg| and |F0|, which are geometrically proportional to the spacings of the Pendellosung and interferometry fringes respectively. Thus, this method is not only based on a principle appropriate for a truly absolute measurement, but the difficulty in determining the proportional factors is essentially eliminated, and the experimental errors can be reduced to about 0.1%. Some geometrical corrections, however, are required to attain an accuracy better than this. These corrections as well as the theoretical ones are discussed and the necessity of taking into account the effect of the nuclear Thomson scattering is pointed out. The consistency of the results was checked with respect to the following points; (i) |Fg| = |F-g|, (ii) |Fg| values being independent of whether the interference fringes in the direct beam or those in the Bragg-reflected beam are used, and (iii) the agreement of |Fg| for different specimens. The atomic scattering factors |Fg| standardized at 20°C are as follows; 111: 10.660, 220: 8.460, 333: 5.839, 440: 5.404, and 444: 4.168. A comparison is made with the values of other authors.

Bestimmung der Ånderung der Gitterkonstanten und des anisotropen Debye–Waller‐Faktors von Graphit mittels Neutronenbeugung im Temperaturbereich von 25 bis 1850°C

Abstract: With a double-axis spectrometer the neutron-scattering cross section dσ/dΩ of polycrystalline graphite has been measured as a function of temperature. The elastic coherent scattering yielded the variations of the lattice constants for the directions parallel and perpendicular to the c axis of the hexagonal graphite lattice. For these lattice directions also the Debye–Waller factors have been determined and from them the mean-square vibration amplitudes of the atoms and the effective Debye temperatures have been calculated.

The theory of X-ray crystal diffraction for finite polyhedral crystals. III. The Bragg–(Bragg)m cases

Abstract: The plane-wave and spherical-wave theories are described for the Bragg-(Bragg)m cases. The treatment is similar to that of Parts I and II [Saka, Katagawa & Kato (1972). Acta Cryst. A28, 102-113, 113-120] for the Laue-(Bragg)m cases. In the plane-wave theory of the Bragg case, a few aspects which up to now have not been well understood, are described to clarify the mathematical structures of the wave field. In particular, emphasis is put on a method for specifying the plane-wave solution by using a Riemann sheet instead of the dispersion surface. In the spherical-wave theory, the reflected vacuum wave and the transmitted crystal wave at the entrance surface can be represented by two Bessel functions. The crystal wave of the Bragg-(Bragg)m case reflected at the exit surface is represented by a combination of two Bessel functions. The transmitted vacuum wave, however, is given by a combination of four Bessel functions. It is shown that the solutions are compatible with those of the Laue-(Bragg)m cases. The solution for finite polyhedral crystals can be constructed by superposing the solutions for individual cases such as of Laue, Laue-(Bragg)m [Kato (1968). J. Appl. Phys. 39, 2225-2230; Parts I and II] and Bragg-(Bragg)m obtained in the present paper. A comparison is made with Uragami's results obtained by another approach [J. Phys. Soc. Japan (1969), 27, 147-154; (1970), 28, 1508-1527].

Direct calculations of crystalline thermal expansion and molecular reorientation from nonbonded interatomic potential anharmonicity and thermal amplitudes

Abstract: Non-bonded interatomic potential functions were altered to allow for the effects of vibration and anharmonicity by relating the mean displacement to the mean-square vibrational amplitude through the anharmonicity constant, K. Vibrational amplitudes were calculated in the harmonic approximation. The repulsive coefficient, B, of the potential was increased sufficiently to cause the minimum of the potential to shift by the amount of the displacement. The apparent lattice energy was then minimized in the usual manner. The method was checked against experiment by calculation of the lattice constants of noble gases at several temperatures, and of anthracene at two temperatures. The method differs from usual calculations of the Gruneisen type in that calculation of molecular-orientation change with temperature is possible. The anthracene molecular orientation was calculated at 95 and 290°K.

Electron microscope study of an homologous series of shear structures based on molybdenum trioxide

Abstract: Electron microscopy and diffraction evidence is presented for the existence of four new molybdenum oxides MonO3n−2, n = 19, 20, 21, and 22.

On least-squares refinement of the phases of crystallographic structure factors

Abstract: A least-squares technique for direct refinement of the phases of structure-factors of crystals which obey the equations ahFh = \sum_{k}FkFh − k is described. It is shown by tests carried out under somewhat idealized conditions that the initial phases need not be of a completeness or accuracy sufficient to resolve the atoms of the structure. The technique may be able to provide a bridge in protein crystallography and structure determination generally between preliminary phasing techniques and final refinement of atom parameters.

A statistical evaluation of absorption

Abstract: This paper reports an empirical method for evaluating the three-dimensional transmission by the statistical treatment of the intensity differences among equivalent reflexions, and the accuracy of the method is discussed.

Normal probability plot analysis of small samples

Abstract: In using normal probability plots for comparing two sets of crystallographic data [Abrahams, S. C. & Keve, E. T. (1971), Acta Cryst. A27, 157] note should be taken of the fact that the expected values of normal order statistics are not given exactly by the percentage points of the normal distribution. This becomes an important consideration only for small samples. Tables of expected ranked exact moduli of normal observations, for sample sizes to 41, are presented: these are useful for half-normal probability plots.

The zellengleichen supergroups of the space groups

Abstract: A complete list of the zellengleichen supergroups of the 230 Fedorov space groups is presented.

Anomalous scattering by oxygen: measurements on (+)‐tartaric acid

Abstract: The structure of (+)-tartaric acid has been used as a test case to study the feasibility of absolute configuration determinations of compounds in which oxygenis the heaviest atom. Based on about 1100 Ni-filtered Cu Kα reflections (−7 ≤ k ≤ 7) measured on a Picker automatic diffractometer the (2R, 3R) and (2S,3S) configurations could be refined to R = 0.0228 and R = 0.0231 respectively. 94 selected enantiomer sensitive reflections gave R(2R, 3R) = 0.0336 and R(2S, 3S) = 0.0374 (Δf′′o = 0.032). Among the 38 most sensitive Bijvoet differences only one had the incorrect sign. A value of 0.041 (4) for Δf”o (Cu Kα) was derived from 36 observed Bijvoet differences. Similar measurements with Cr Kα radiation also permitted the unequivocal assignment of the correct absolute configuration; however, Cr Kα radiation was not found to offer any overall advantage over Cu Kα radiation. The cell dimensions used in this study are a = 7.7291 (5), b = 6.0069 (2), c = 6.2118 (3) A, β = 100.147 (2)° (space group P21; Z = 2).

X‐ray studies on Debye characteristic temperatures of disordered binary substitutional alloys

Abstract: An expression has been derived for the Debye characteristic temperatures of disordered binary substitutional alloys in terms of the Debye characteristic temperatures of the component metals, concentration of the solute atom and an unknown parameter which can be found from the Debye characteristic temperature of the alloy at any particular concentration. Debye temperatures of polycrystalline specimens of copper, nickel and copper-nickel (20, 30, 40, 50, 60, 80 and 90% nickel by weight) alloys have been determined by X-ray diffraction methods. Intensities of different reflexions have been measured with the help of a highly accurate counter-diffractometer technique and corrected for temperature diffuse scattering, diffuse scattering due to random distribution of guest atoms in host sites and difference in sizes of different atoms (in case of alloys only) and incorrect background. Debye temperatures have been plotted against concentration and the variation has been compared with theoretical predictions. Experimental results are found to agree satisfactorily with the predictions obtained during the present investigations.

Properties of crystal lattices: the derivative lattices and their determination

Abstract: Derivative lattices are classified as super, sub and composite, on the basis of the properties of the transformation matrices relating them to the lattice from which they are derived. A method for obtaining the transformation matrices generating these lattices is given. The method has been applied to the derivation of the unique super and sublattices in a few important cases.

On the interpretation of anisotropic temperature factors

Abstract: A lattice-dynamical formulation of the anisotropic vibration tensors of the atoms in a crystal is presented which may contribute to a better understanding of how the interatomic thermal coupling is expressed in the anisotropic temperature factors.