Showing papers in "Acta Crystallographica Section A in 1969"
TL;DR: In this article, the amplitude and phase information can also be obtained from a generalized diffraction experiment (in spite of the inability of the detector to observe phases directly) on condition that the primary ray can be controlled in a defined way.
Abstract: In addition to the amplitude information, phase information can also be obtained from a generalized diffraction experiment (in spite of the inability of the detector to observe phases directly) on condition that the primary ray can be controlled in a defined way. A single diffraction pattern suffices for the determination of the phases except in the case of a centre of symmetry in real space, and the combination of at least two patterns removes this ambiguity.
288 citations
TL;DR: In this paper, the lattice parameter of pure pure Cu is 3.61491 A (corrected for refraction), and thermal expansion coefficient α = 14.87 × 10−6°C−1 between 15 and 55°C.
Abstract: Spectroscopically pure Cu has a lattice parameter a25 = 3.61491 A (corrected for refraction), and a thermal expansion coefficient α = 14.87 × 10−6°C−1 between 15 and 55°C. The measured density d25 is 8.9314 ± 0.0002 g.cm−3 in agreement with the calculated value dx = 8.9316. In the α solid solution region additions of In increase the lattice parameter of Cu according to ax = 3.6149 + 0.0091x up to x = 10.4 (x = atomic % In, balance Cu). The thermal expansion coefficients between 15 and 65°C of the homogeneous alloys increase from 14.87 (pure Cu) to 17.2 × 10−6°C−1 at the solid solubility limit (10.4 atomic % In, quenched from 650°C). With the increase of In content the experimental densities become increasingly lower than the calculated ones because of void formation. Upon cold rolling the voids close and the differences disappear. The α phase represents a substitutional solid solution without structural defects. Alloys quenched from the liquid state do not show any microporosity; the voids appear after homogenization at 800°C. Micropore formation is explained by differential shrinkage of the various crystalline fractions formed during solidification, giving rise to internal stresses in the solid alloy. Relief of stresses results in vacancies or micropores, which coalesce into voids upon heat treatment.
258 citations
TL;DR: It is shown that a suitable curve fitting procedure can reduce the standard deviation of intensity measurements made with a scanning diffractometer except when the intensity is large in relation to the background level.
Abstract: It is shown that a suitable curve fitting procedure can reduce the standard deviation of intensity measurements made with a scanning diffractometer except when the intensity is large in relation to the background level The procedure has been implemented with an Arndt–Phillips linear diffractometer, on-line to a Ferranti Argus 312 computer The shape of the profile which is fitted to the observed profile is learnt and continually revised In typical cases the standard deviation of the intensity, as determined from counting statistics alone, is about 04–06 times that given by alternative methods In addition to profile fitting, a straightforward peak-minus-background treatment is always done, and the two results are compared This enables a large measure of internal consistency checking to be done, so that certain types of error condition can be detected Criteria are also developed whereby the better of the two results may be chosen in each case, having regard to such parameters as the peak to background ratio and the closeness of fit accomplished in profile fitting
198 citations
TL;DR: In this article, the effect of thermal diffuse scattering (TDS) on X-ray and neutron structure factors is considered, with the assumption that each atom vibrates in a potential field whose symmetry conforms with the site symmetry.
Abstract: Two basic assumptions are usually made by crystallographers in the evaluation of X-ray or neutron structure factors. The first is the validity of the harmonic approximation in deriving the Debye–Waller factors which account for the effect of lattice vibrations in reducing the intensities of the Bragg reflexions. The second assumption is that the contribution to the measured intensity of thermal diffuse scattering (TDS), which rises to a peak at the reciprocal lattice point, can be ignored. Anharmonicity and TDS can each give rise to appreciable intensity effects, and so must be allowed for in accurate work. Anharmonicity can be taken into account by treating the crystal as a system of independent anharmonic oscillators, with each atom vibrating in a potential field whose symmetry conforms with the site symmetry of the atom. The effect of TDS can be calculated approximately using first-order (one-phonon) harmonic scattering theory, together with a knowledge of the elastic constants of the crystal. Calculations of both types of correction are considered in detail for a cubic crystal, and are applied to the analysis of X-ray diffraction measurements on KCl and of neutron measurements on BaF2.
136 citations
TL;DR: The general features, terminology, and method of the dynamical theory of X-ray diffraction are discussed in this paper, stressing the analogy with the general theory of small oscillations of a mechanical system.
Abstract: The general features, terminology, and method of the dynamical theory of X-ray diffraction are discussed, stressing the analogy with the general theory of small oscillations of a mechanical system.
110 citations
TL;DR: In this article, the principles established for periodic objects in parts I and II are extended to non-periodic objects, and the basis for the extension lies in the possibility of introducing a virtual lattice, which can be treated as in parts II and III when working with a primary wave field that has zero amplitude outside a given spatial region.
Abstract: The principles established for periodic objects in parts I and II are extended to non-periodic objects. The basis for the extension lies in the possibility of introducing a virtual lattice, which can be treated as in parts I and II when working with a primary wave field that has zero amplitude outside a given spatial region. Thus it becomes possible to form an image without the use of a lens in a manner similar to holography but based on different principles. A reference wave (as used in holography) is not required. The new principle can be applied in a number of variations, some of which are discussed.
108 citations
TL;DR: In this article, the use of an inhomogeneous distribution of the primary beam is used to determine the phase of a laser beam in a centric and an acentric two-dimensional lattice.
Abstract: Optical diffraction experiments with the use of an inhomogeneous distribution of the primary beam as described in part I indicate that the determination of phases is still possible even under unfavourable conditions. This method has been applied to both a centric and an acentric two-dimensional lattice.
93 citations
TL;DR: The Cumulantexpansion model for thermal motion as mentioned in this paper is a statistical model without kinematic constraints and provides an unbiased estimate for the skewness of the density function of thermal motion.
Abstract: The usual crystallographic structure-factor equation, with three positional and six anisotropic-temperature-factor coefficients, assumes that the thermal-motion probability density function is centrosymmetric. However, phenomena such as libration and anharmonic vibration can cause skewness. In this study, ten more coefficients per atom representing the third cumulant of the probability density function for thermal-motion are added to the structure-factor equation to permit a determination of the nature of the skewness. The Edgeworth series expansion based on the normal probability density function is used to analyze the results. The equations are generalized to include also the fourth cumulant, which describes kurtosis. The `cumulant-expansion model' for thermal motion is a statistical model without kinematic constraints and provides an unbiased estimate for the skewness of the density function for thermal motion. Application of the model to neutron diffraction data from crystals containing methyl groups (which are undergoing torsional oscillation) confirms the assumption that the density functions for the hydrogen atoms of a methyl group are skewed as an arc about the axis of torsional oscillation. The model has not been applied with X-ray diffraction data; if it were, the resulting parameters would describe the skewness of the combined electron and thermal-motion probability density functions.
86 citations
TL;DR: Some recent developments leading to improved knowledge of the distribution of oscillator strength in the photo-ionization continua of atoms are briefly reviewed in this article, and a comparison between the experiments and calculations are indicated.
Abstract: Some recent developments leading to improved knowledge of the distribution of oscillator strength in the photo-ionization continua of atoms are briefly reviewed. Selective comparisons between the experiments and calculations are indicated. Estimates for attenuation by compounds and solids, insofar as they depart from the mixture rule, require an understanding of X-ray fine structure or, at least, of its limiting form, which is not yet at hand. Some of the mechanisms leading to fine structure in solids and molecules are briefly reviewed.
85 citations
TL;DR: In this paper, the repulsive lattice energy is approximated by a sum of quadratic nonbonded interatomic potential functions and the lattice sum is minimized by full-matrix least squares.
Abstract: A novel method of calculating the molecular position in a lattice of known dimensions is presented. The repulsive lattice energy is approximated by a sum of quadratic nonbonded interatomic potential functions and the lattice energy sum is minimized by full-matrix least squares. The convergence range from arbitrary trial models is greater than the previously used steepest descent method using (exp-6) nonbonded potentials. Greatly increased speed of convergence is also obtained because of the inclusion of off-diagonal terms and the small number of repulsive interactions which are considered. The calculated packing models are sufficiently accurate to serve as a starting point for structure factor least-squares refinement based on diffraction data.
57 citations
TL;DR: In this paper, the experimental determination of the transmission surface, which represents an approximation to the transmission of X-rays in crystals, is described, and the experimental transmission surfaces are compared with values calculated from the crystal shape.
Abstract: The experimental determination of the transmission surface, which represents an approximation to the transmission of X-rays in crystals, is described Experimental transmission surfaces are compared with values calculated from the crystal shape Also structural parameters of some crystal structures obtained from data with and without experimental absorption correction are compared
Journal Article•
TL;DR: In this article, the frequency-wave-vector dispersion relation for the normal modes of vibration of terbium at room temperature has been measured by means of slow-neutron inelastic scattering techniques.
Abstract: The frequency-wave-vector dispersion relation for the normal modes of vibration of terbium at room temperature has been measured by means of slow-neutron inelastic scattering techniques. The triple-axis spectrometer at the Oak Ridge high flux isotope reactor was used, mostly in the constant-$Q$ mode of operation. Phonon frequencies for wave vectors along the principal symmetry directions have been determined and, in addition, measurements of phonon frequencies along the boundaries of the Brillouin zone and along a more general direction are reported. The data have been fitted with a Born-von K\'arm\'an force model which includes interactions out to the eighth nearest neighbor. The interactions have been assumed to be general (tensor) out to the fourth neighbor and axially symmetric beyond. The model has been used to calculate a frequency distribution function $g(\ensuremath{
u})$ and related quantities such as the lattice specific heat and Debye temperature.
TL;DR: In this article, the optical rotatory power for general directions in the (001) plane has been observed for the first time in a crystal of the non-enantiomorphous class \bar 4 CdGa2S4.
Abstract: Optical activity has been observed for the first time in a crystal of the non-enantiomorphous class \bar 4 CdGa2S4 is optically isotropic at 4872 A (20°C) and this allows observation of optical rotation for propagation in directions away from the c axis. The magnitude of the optical rotatory power is 17.3 deg.mm−1 along (100) and 11.6 deg.mm−1 along (110). The form of the optical rotatory power for general directions in the (001) plane has the predicted symmetry, showing positive and negative lobes. The refractive indices and birefringence are temperature-dependent and the point of isotropy shifts from 4825 A at 0°C to 5780 A at 400°C.
TL;DR: In this paper, the structure of Sm(BrO 3 ) 3 ·9H 2 O has been solved by the neutron anomalous dispersion method, with the use of data collected at two wavelengths.
Abstract: The structure of Sm(BrO 3 ) 3 ·9H 2 O has been solved by the neutron anomalous-dispersion method, with the use of data collected at two wavelengths. Five out of eight atoms in the asymmetric unit were located from the 'anomalous difference Patterson' synthesis, and the remaining three from a Fourier synthesis. The structure is centrosymmetric, and the signs of 78% of the reflections were determined correctly from the differences in the Fobs values at the two wavelengths.
TL;DR: In this paper, the sensitivity of δ2Fmeas to errors in Fmeas is examined under both real and hypothetical conditions, and it is shown that δ 2FMEas is sensitive to systematic intensity error at a level in which the maximum intensity error in 97% of the data reaches 5%.
Abstract: Objective estimation of the error (σFmeas) in each structure factor (Fmeas), by a procedure such as that outlined in Acta Cryst. (1964). 17, 1327, allows the indicator \cal uα = [Σ (Δ2Fmeas/σ2Fmeas) ÷ χ2m − n,α]1/2 to be evaluated, where ΔFmeas = |Fmeas| − |Fcalc|, χ2m − n,α is given by the χ2 distribution at the α-significance level, and the model for which ΣΔ2Fmeas is minimized contains m independent Fmeas and n variables. The sensitivity of \cal uα to errors in Fmeas is examined under both real and hypothetical conditions. For the real case, the range in \cal u0.01 for eight recently measured inorganic crystals indicates the average minimum residual error in σFmeas to be about 1%, the maximum error to be about 21%. The hypothetical case is considered by propagating several types of error to a set of 772 independent Fmeas corresponding to a model consisting of 10 independent atoms undergoing isotropic thermal vibration. The indicator \cal u0.01 detects a systematic intensity error at a level in which the maximum intensity error in 97% of the data reaches 5%. Long term drift in the experiment is indicated with considerable sensitivity by \cal u0.01. Systematic error as a function of scattering angle is detectable only if the σFmeas contain a component due to the suspected error.
TL;DR: The theoretical basis and experimental procedures of the Pendellosung method are described in this paper, and the experimental results so far obtained on Si, Ge and α-quartz are briefly reviewed.
Abstract: The theoretical basis and the experimental procedures of the Pendellosung method are described. Structure factors |Fg| can be determined on the absolute scale by measuring the fringe spacing Λg on the diffraction topographs and the angles involved in a geometrical factor Φg. The experimental results so far obtained on Si, Ge and α-quartz are briefly reviewed in comparison with the results obtained by other methods. Effects of absorption and of lattice distortions are estimated to be of about 0.1%. The errors in measuring Λg are less than 0.1% in favourable cases. At present, the accuracy in |Fg| is limited by the difficulty in determining Φg accurately. It is about 1%. The ratio of structure factors can be determined with an accuracy of about 0.1% by taking the ratio of the fringe spacings. The following examples are described: (i) |Fhk, l|/|Fhk,_{{\bar l}}| of a α-quartz, (ii) the ratio of |Fg| of Si at low and room temperatures and (iii) |Fg|/|F0| of Si. Through the experiment (ii), the increase of the Debye temperature at low temperatures (≃ 40°K) was confirmed. In experiment (iii), the Pendellosung fringes and X-ray interferometry fringes are used, whose spacings are proportional to |Fg| and |F0|, respectively. Since |F0| is essentially the total charge Z in the unit cell, the |Fg| value determined by this method is the truly absolute value.
TL;DR: In this paper, a precise X-ray determination of the lattice constants of ruthenium dioxide, RuO2, has been made in the temperature range 30° to 702°C using a Unicam 19 cm high-temperature powder camera.
Abstract: A precise X-ray determination of the lattice constants of ruthenium dioxide, RuO2, has been made in the temperature range 30° to 702°C using a Unicam 19 cm high-temperature powder camera. It has been observed that the c parameter decreases with increasing temperature as in the case of FeF2 and CrO2 which also have a ruffle-type structure.
TL;DR: In this article, a method of determining the atom form factors from the values of accelerating voltage for which the second order Kikuchi line disappears is applied to aluminum, iron, nickel and copper.
Abstract: A new method of determining the atom form factors from the values of accelerating voltage for which the second order Kikuchi line disappears is applied to aluminum, iron, nickel and copper. The accuracy of the method is briefly discussed and the numerical results of the atom form factors are given.
TL;DR: In this article, it has been shown that the Debye-Waller coefficients of the ions in alkali halide crystals have one prominent feature: the square of the relative vibration amplitude of an individual ion, which is proportional to the ratio B/a2 (B is the debeau-waller coefficient and a the length of the edge of the unit cell), increases when the number of electrons of its company increases.
Abstract: Results of X-ray powder intensity measurements and spherical harmonic expansions have been used for investigating deformations of the ions in NaCl and AgCl crystals. It is found that within the limits of experimental accuracy the ions in crystalline NaCl are spherically symmetric. Compared with free ions, the charge around Ag in AgCl has spread farther and a part of it is located inside the Cl sphere. The electron clouds inside the Ag and Cl spheres are, however, almost spherically symmetric. It has been shown that the results of recent X-ray measurements concerning the Debye–Waller coefficients of the ions in alkali halide crystals have one prominent feature: the square of the relative vibration amplitude of an individual ion, which is proportional to the ratio B/a2 (B is the Debye–Waller coefficient and a the length of the edge of the unit cell), increases when the number of electrons of its company increases. It is proposed that this dependence is mainly due to the repulsive forces between the next-nearest-neighbours.
TL;DR: The Mossbauer spectrum of ilvaite has been measured at room temperature, and indicates the presence of both high-spin ferrous and ferric iron as discussed by the authors, with the proportion of Fe2+ ions differing by about 20%.
Abstract: The Mossbauer spectrum of ilvaite has been measured at room temperature, and indicates the presence of both high-spin ferrous and ferric iron. Two distinguishable ferrous sites are detected, with the proportion of Fe2+ ions differing by about 20%. The ferric site is highly distorted.
TL;DR: In this article, the authors used tilt boundaries formed by vertical alignment of edge dislocations, created during crystal growth, for the estimation of dislocation densities inside the crystals.
Abstract: X-ray oscillation and Laue photographs of solution-grown single crystals of cadmium iodide frequently show extension of the diffraction spots into small continuous or discontinuous arcs. The phenomenon is satisfactorily explained in terms of tilt boundaries formed by vertical alignment of edge dislocations, created during crystal growth. Measurement of arc lengths affords a convenient method for the estimation of dislocation densities inside the crystals. Calculations made on several crystals give the value of dislocation density, ~ 105 – 106 dislocations cm−1.
TL;DR: In this paper, a new method of correction of X-ray powder diffraction line profiles by deconvolution is developed, in which the convolution equation, representing the phenomenon of deformation of a true diffusion line profile, is approached with the help of a quadrature formula by a system of linear equations.
Abstract: A new method of correction of X-ray powder diffraction line profiles by deconvolution is being developed. The convolution equation, representing the phenomenon of deformation of a true diffraction line profile, is approached with the help of a quadrature formula by a system of linear equations. The bad conditioning of this system has led to the use of a stabilization method enabling one to obtain a result converging to the true solution which satisfies the physical phenomenon. The quadrature formulae which are used are described. This method has been programmed in Fortran II for the IBM 1620.
TL;DR: In this article, the determination of an absolute scale for X-ray intensities involves the experimental evaluation of a number of constraints in the intensity equation whose values are not normally required.
Abstract: The determination of an absolute scale for X-ray intensities involves the experimental evaluation of a number of constraints in the intensity equation whose values are not normally required. The absolute intensities so obtained will not be useful, in the present context, unless the overall accuracy can be maintained at about 1% or better. Some of the difficulties which can occur in making the measurements to the necessary accuracy are discussed.
TL;DR: In this paper, the authors discuss the nature of systematic error in crystallographic experiments and in crystal structure refinement models and present statistical methods for testing the significance of the difference between the parameters derived in two crystallographic measurements.
Abstract: A number of crystal structures have been precisely analyzed by both X-ray and neutron diffraction. Comparison between the results can lead to some understanding of the type and magnitude of error present in one or both methods. We discuss here the nature of systematic error in crystallographic experiments and in crystal structure refinement models. Statistical methods are presented for testing the significance of the difference between the parameters derived in two crystallographic experiments. These methods are applied to existing neutron and X-ray diffraction data on oxalic acid dihydrate, hydroxyapatite, s-triazine, potassium hydrogen diaspirinate and methylglyoxal bisguanylhydrazone. These tests show that there are strong systematic differences in thermal parameters for heavy atoms and in all parameters for hydrogen atoms. Differences in positional parameters for heavy atoms are marginal. The pattern of results strongly suggests that the differences between neutron and X-ray experiments have some physical bases rather than being due entirely to systematic error in one or both experiments. The results may also be interpreted as indicating that both position parameters and root-mean-square amplitudes of vibration may with care be determined to a precision of 0.001 A in structures with a moderate number of atoms in the asymmetric unit.
TL;DR: In this article, a new method for obtaining the resolution function for a triple-axis neutron spectrometer is described, involving a combination of direct measurement and analytical calculation, and Gaussian or experimentally determined probability distributions may be used.
Abstract: A new method for obtaining the resolution function for a triple-axis neutron spectrometer is described, involving a combination of direct measurement and analytical calculation. All factors which contribute to the finite resolution of the instrument may be taken into account, and Gaussian or experimentally determined probability distributions may be used. The application to the study of the dispersion relation for excitations in a crystal is outlined.
TL;DR: In this paper, the intensity of an X-ray (Bragg) reflection from a mosaic crystal plate under the conditions of multiple diffractions is discussed theoretically. And the set of simultaneous differential equations is solved exactly for the two-beam and the three-beam cases.
Abstract: The intensity of an X-ray (Bragg) reflection from a mosaic crystal plate under the conditions of multiple diffractions is discussed theoretically. The set of simultaneous differential equations is solved exactly for the two-beam and the three-beam cases. A second order approximation is given for the multiple beam case, and a third order approximation for the triple beam case. The fact that the ratio R of the peak intensity of a single diffracted beam to the intensity of the same beam under conditions of multiple diffraction depends on the mosaic spread η of the crystal provides a method to obtain this magnitude from the experiment. Preliminary measurements were performed on Si and Ge single-crystal plates under conditions of multiple diffraction. From them experimental values of R were obtained for different planes, which then were used to calculate the mosaic spread of the crystals. In this method only relative intensity measurements have to be used and most of the corrections found to be necessary in other procedures used to determine mosaic spread are not needed. The polarization factor for the case of a double reflection preceded by a monochromator is analyzed and the results are given in an Appendix.
TL;DR: In this paper, a replication electron microscopy study of etched fracture surfaces of opal revealed four distinctive patterns of packing of the constituent SiO2 spheres These patterns could be accounted for by consideration of appropriate stacking faults or modifications in a cubic closest packed structure.
Abstract: A replication electron microscopy study of etched fracture surfaces of opal revealed four distinctive patterns of packing of the constituent SiO2 spheres These patterns could be accounted for by consideration of appropriate stacking faults or modifications in a cubic closest packed structure These were (1) intrinsic stacking faults (2) extrinsic stacking faults (3) twinning and (4) rhombohedral polytypism The success in predicting some of the observed packing patterns by assuming stacking modifications of a cubic closest packed structure indicates the existence of a basic, though frequently faulted, fcc opal structure Screw dislocation-spiral growth is proposed as a possible mechanism for explaining the occurrence of the ordered opal structures
TL;DR: The theory of Cowley & Pogany is used for the numerical calculation of thermal diffuse electron scattering from thin monatomic crystals, along the line of a set of systematic reflexions as discussed by the authors.
Abstract: The theory of Cowley & Pogany (Acta Cryst. (1968) A24,1 09) is used for the numerical calculation of thermal diffuse electron scattering from thin monatomic crystals, along the line of a set of systematic reflexions. Dynamical interactions of 15 Bragg and 25 diffuse beams are considered for up to 183 A of gold (200) systematics, treating the range of coherence of the interaction as a parameter of calculation. A one-phonon Debye model is used. It is found that thermal diffuse scattering will not produce strong thickness fringe contrast, but will yield Kikuchi bands and lines, with little dependence on the range of coherent interaction. The unindexed Kikuchi line at the centre of the bands is predicted. Thermal scattering tends to increase in the region of strong Bragg beams for tilted crystals. Compared to that predicted kinematically, dynamical thermal scattering is greater for thin crystals near principal orientations, but will in general be less for highly tilted and particularly for thick crystals; it is more spread out in the diffraction pattern, and far stronger in the first Brillouin zone. There is an indication that thermal streak patterns from thin crystals should be stronger than expected kinematically.
TL;DR: In this paper, a simple fast method was described for producing three dimensional ''ball-and-stick'' representations of molecules, using a digital plotter; any viewpoint at a reasonable distance away is suitable.
Abstract: A simple fast method is described for producing three dimensional `ball-and-stick' representations of molecules, using a digital plotter; any viewpoint at a reasonable distance away is suitable.