ALBERT

Notes: A direct method for the determination of polytype structures of SiC, ZnS and similar substances from X-ray data is described. It is based on the values of a Patterson-like function (the `Pattersonian') which only depends on the stacking of the translationally equivalent layers of the structure. The way of obtaining the Pattersonian function from the experimental intensities is described and an algorithm given by which the sequence of layers may be deduced. This sequence is conveniently characterized by the sequence of the digits of a binary number. The influence of experimental errors in the intensities on the possibilities of determining the real sequence is discussed.

Notes: Seventeen sets of measurements of structure factors of D(+)-tartaric acid, within the range (sin θ)/λ 〈 0.5 Å−1, were provided by the participants in the International Union of Crystallography Single Crystal Intensity Measurement Project. Each participant used a different crystal, all being derived from a single crystallization batch. The results in the Project are representative of those from a wide variety of currently used diffractometers and techniques. The instruments included four-circle, normal-beam and equi-inclination diffractometers. Cu and Mo radiations were used – unfiltered, with single and balanced filters, and with crystal monochromators. The aims of the project were twofold: (a) to provide an estimate of the spread of F values associated with the range of variables involved in the project and (b) to locate, if possible, the sources of error. A number of agreement indices were used to measure the spread of F values both for equivalent reflections within any one experiment and for comparisons between experiments. In an attempt to allocate errors to certain plausible sources, an analysis-of-variance was applied to the weighted deviations of individual values of F from the set of mean values. The variables specified were intensity I, a θ angle factor d* and the Miller indices h,k,1. From the values of the agreement indices and the interaction curves from the analysis-of-variance, it was possible to recognize outlier sets that differ considerably from the mean and to isolate these, where necessary, before arriving at an estimate of the error spread of the main group. In this project, there is no one simple figure of merit which provides a ready assessment of the accuracy of measurement of structure factors. Rather, there are several ways of indicating the probable accuracy. One way is to present the spread of values of Rij(Σ(|Fi| − |Fj|)/Σ ½(|Fi| + |Fj|). This shows that two scaled experimental sets of structure factors, measured under circumstances similar to those of the project, will most probably differ by 6%, agree no better than 3%, and usually no worse than 10% except in cases of extreme systematic error where it may rise to 50% or more. From the analysis-of-variance, inferences are drawn concerning the concordance of results derived from the different types of diffractometer, on features of technique associated with the diffractometers and on other aspects, including 2 dependence, monochromaticity, count rates and extinction in the crystals. It is concluded that other sources of error may be present and that future projects should be designed to reveal these.

Notes: Any crystal structure may be described in terms of a sublattice of points, each of which represents a certain fraction of the electron density. Multiplying this sublattice by a density function f(x) and applying a shift function s(x), which brings the atoms into the right positions, the correct crystal structure can be given in many different ways. It is shown that the shift function s(x) yields phase relations between the structure factors F(h), which may be evaluated directly, if the coefficients of the Fourier representation of s(x) converge rapidly. This behaviour is demonstrated for the case of a one-dimensional acentric model structure consisting of 50 atoms. Complete information on the structure may be obtained by routine methods with the aid of 5 given phases of the structure factor. This procedure may also be applied to three-dimensional structures, if the corresponding computer programs are available.

Notes: The analysis in part I is generalized to any crystal containing rigid molecules which undergo anisotropic translational and librational motion about a site fixed by symmetry. The treatment is correct to terms in (ui2)2 and (ωi2)2, where (ui2) is the mean-square translational displacement of the molecule along the ith axis and (ωi2) is the mean-square angular libration about the same axis. The first-order treatment to terms in (ui2) and (ωi2) is shown to be equivalent to the rigid-body theory in current use.

Notes: It is shown that the concept of the molecular centre of libration in the description of the average rigid-body thermal behaviour of molecules is a useful approximation. A restriction is made on the full theory, making the centre of libration model a constrained version of the complete model. The relationship between the models is discussed. Some examples are chosen to demonstrate the closeness of the approximation. These results show that for data reaching a reliability factor of no better than 7% an analysis using the full theory might yield a meaningless result. Some data of much higher accuracy does, however, show the need for the full theory.

Notes: The number of independent non-vanishing constants required to describe the six known physical properties, involving a polar vector, has been determined by the character method of Bhagavantam & Suryanarayana for each one of the 58 double-coloured point groups.

Notes: The lattice parameters of ammonium chloride have been measured at 31.5 and 54.0°C. The coefficient of linear expansion is found to be 58.5 × 10−6°C−1.

Notes: The recent X-ray diffraction data front Mg2Si, obtained by Panke & Wölfel (Z. Kristallogr. (1969), 129, 9) show systematic differences in the structure factors from those predicted by a model which assumes harmonic thermal vibration of the atoms. These differences indicate an anharmonic component in the thermal vibration of the magnesium atoms, consistent with their tetrahedral site symmetry. An analysis of these data has therefore been carried out in terms of a model which includes anharmonic thermal vibration and a least-squares refinement gave a value of the anharmonicity parameter (βMg = − 2.39 × 10−12 erg.Å−3) which is similar in magnitude to those found for other systems with the fluorite structure.

Notes: A diffraction theory is developed for the model of ordered zones embedded in an otherwise disordered binary alloy. Evaluation of the resulting lattice sums gives an equation for the diffuse scattering everywhere in reciprocal space except under the fundamental lines. From the experimental diffuse intensity one can calculate the relative volume of the ordered zones in the whole crystal, their mean sizes along each of the crystallographic axes, and the distribution of these sizes. A set of relations between the Warren short-range-order parameters αlmn and the size of the ordered zones is also obtained. Comparison with experimental αlmn given in the literature show that a 50 at.% CuAu alloy quenched from 500°C contains ordered zones in a disordered matrix. Similar results were obtained for a 50 at.% CuPt alloy held at 890°C or quenched from 930°C. In a Cu3Au alloy held at 450°C the calculations show the existence of ordered zones with a size of (3 × 3 × 3) unit cells, in accordance with the results of computer simulation work published in the literature.

Notes: The counter method with a time basis has the disadvantage that it includes the systematic error resulting from the variation of the source. The counter method with a monitor counter gets rid of this systematic error but introduces a statistical error originating in the uncertainty of the flux received by the monitor counter. The author proposes a formula which is a compromise between the two methods and which partially retains the advantages of the two methods.

Notes: An attempt is made to differentiate between the three previously proposed models of the structure of tetragonal BaTiO3. In the new model, the serious parameter-interaction problem was avoided by the use of both neutron and X-ray diffraction data collected from c-domain single crystals. The result of the refinement gave the displacement of the Ti atom from its cubic position as 0.0135 ± 0.0004 Å; the R value was 0.027 and 0.024 for the neutron and X-ray data respectively. The analysis also showed that the temperature parameters, B33, along the c axis are approximately independent of mass; B33 = 0.32 ± 0.03 Å2 from X-ray and 0.33 ± 0.04 Å2 from neutron diffraction. The average value obtained for the B11 temperature parameters, B11 = 0.50 Å2, is considerably higher than the B33 value, which thus clearly indicates the existence of low frequency optic modes at room temperature.

Notes: Changes of particle size and strain during annealing of cold-worked aluminum powders have been determined by measuring X-ray line breadths. Aluminum powders ground under special conditions show a decrease in particle size as well as a reduction of strains within certain ranges of temperature. This is due to polygonization processes and allows conclusions to be drawn regarding the limits of coherent regions of X-ray diffraction. The X-ray measurements were supplemented by electron microscope studies.

Notes: The crystal packing of diketopiperazine may be predicted from the approximate molecular geometry and space group symmetry. The van der Waals and hydrogen bonding energies were computed as a function of three rotational degrees of freedom (space group P21/a and Z = 2). The Eulerian angles, relative to the deepest minimum, correspond to a good approximation to the real structure. The calculations were carried out using potential functions tested in some crystals and in the conformational analysis of synthetic and biological macromolecules. A potential function proposed by Stockmayer was chosen to describe the hydrogen bond formation. The rigid-body translations of the molecules in the crystal, known from an analysis of the anisotropic atomic thermal parameters, were qualitatively checked. For this purpose the van der Waals potential energy was computed by moving one chain of molecules and leaving the surrounding chains fixed. The good results obtained again confirm the validity of this method for the solution of the phase problem. It is hoped that this approach may be useful in crystal structure determinations of organic molecules of biological interest.

Notes: The Christian–Gevers theory of X-ray diffraction from hexagonal close-packed crystals with deformation stacking faults is extended to include the effect of change in layer spacing at the faults. The results show that integral breadths as well as integrated intensities remain unaffected to a first approximation. The principal effect is to introduce peak shifts, the magnitude and direction of which depend on the reflexion.

Notes: The elastic constants of certain single crystals may be obtained by the measurement of thermal diffuse X-ray scattering variation in regions close to reciprocal-lattice points. A method has been given which allows for second-order contributions to the total diffuse intensity by use of a least-squares procedure. The refinement was simplified by equating two functions, each depending on the elastic properties of the crystal, the nearby lattice point to the positions of observation and the direction along which the intensity variation is measured. For cubic crystals, there is an exact equality between appropriate powers of these functions for several important crystallographic directions, while for other directions some approximation results. The present paper includes a simple method of allowing for the deviation from equality in the latter cases.

Notes: The lattice parameters of zinc telluride and mercury selenide were measured with a Unicam 19 cm high-temperature powder camera; the following equations represent the results: ZnTe, at = 6.1016 + 54.63 × 10−6t + 6.82 × 10−9t2 + 5.28 × 10−12t3; HgSe, at = 6.0854 + 28.61 × 10−6t + 4.93 × 10−9t2 + 3.74 × 10−12t3. The expressions for thermal expansion coefficients are also given.

Notes: X-ray diffuse scattering with an intensity distribution of one-dimensionally elongated shape in reciprocal space is observed for a single-crystal of AuCu3, which is in a transient state around the order-disorder transition point. The analysis is carried out on the basis of Wilson's model for the disordered structure of AuCu3. The probability that a unit cell at a given position is displaced by half the diagonal of the unit cell cube is derived as a function of the position of unit cell, and is found to be expressed by an exponential function.

Notes: The coherent neutron scattering amplitude of rubidium has been remeasured using powders of RbCl, RbBr, and RbI. A consistent value, (0.68 5 ± 0.01) × 10−12 cm, was obtained, differing significantly from earlier measurements. The new value is believed to be more reliable. The bound coherent scattering cross section is 5.9 ± 0.2 barns.

Notes: A corrected electron density expression is given for space groups P4122 (No. 91) and P4322 (No. 95).

Notes: From X-ray measurements at room and low temperatures it was found that the characteristic temperature Θ of the system Ag-Cd decreases by about 17°K in the concentration range from 0 to 30 at.% Cd. For the system Ag-Zn no systematic change of Θ with increasing concentration of Zn was observed.

Notes: The advantages are discussed of using a ball model to determine the arrangement of lattice points in a given lattice plane and for determining the stacking properties of such planes. It is shown that the ball model can be considered as a simple analogue computer for solving the Diophantine equations involved. To date, such ball models have been used only for cubic and hexagonal crystals, but they can be constructed for many other structures.

Notes: One hundred and eighteen crystals of cadmium iodide have been investigated by X-ray diffraction methods to verify the disorder theory of polytypism in these crystals. The degree of one-dimensional disorder has been evaluated for eighty-two polytypes by intensity measurements on oscillation photographs. The frequency distribution curve of relative abundance of polytypes versus their degree of disorder shows two maxima, one for α1 ∼ 0.06 and the other for α2 ∼ 0.26. The results are in agreement with those obtained earlier by Jagodzinski in silicon carbide crystals and provide more information regarding the origin of polytypism in crystals. Fifty-three new polytypes of cadmium iodide, forty-four hexagonal and nine rhombohedral, have also been discovered during the investigation.

Notes: The accuracy in X-ray diffraction intensities obtained by the use of an automatic IBM 1800-controlled SAAB film scanner is discussed. The random errors have been found to be around 4% of the intensities for most reflexions. By comparison with data obtained from a Joyce–Loebl microdensitometer it has been found that there are no serious systematic errors in the film-scanner data. Measurements from Weissenberg and precession films are discussed.

Notes: A new set of relations between structure factors corresponding to an isomorphous pair is established: inequalities, convolution equations, probabilities. These relations may enhance the power of direct methods for the determination of phases.

Notes: The atomic scattering factor of germanium has been obtained from measurements of the Pendellösung period in wedge-shaped single crystals with a three-crystal X-ray spectrometer. For the 220 reflexion the value is 23.78±0.35, in good agreement with the theoretical value of 23.76. Fine structure in the profile of the rocking curves has been observed in thickness regions close to Pendellösung minima.

Notes: The description of n-dimensional space by a basis of (n + 1) vectors, ai (i = 1, . . . , n + 1), is discussed, with particular reference to the Miller–Bravais system of indexing hexagonal crystals. It is shown that if a hyperplane with normal h makes intercepts hi−1 on the ai and a vector u has components ui relative to the ai, then h. u = Σhiui without any further restrictions on the hi or ui. Furthermore, it is possible to find a basis a†i(i = 1, . . . , n + 1) for reciprocal space such that h = Σhia†i is always true and indeed there are n degrees of freedom available for choosing such a basis. Criteria which may lead to a unique choice of a†i are discussed.

Notes: The high temperature X-ray diffraction study of the monoclinic tetragonal phase transition in ZrO2 showed that it is spread over a temperature range 930–1220°C. Anomalous intensity changes are observed in the pretransformation region 930–1100°C. Coexistence of phases through hybrid crystal formation in the region 1100–1220 °C and the mechanism of transition are discussed. The orientation relationship between the monoclinic (m) and the tetragonal (t) crystal structures consists in the parallelism of the (100)m plane to (110)t and of the bm axis to the ct axis. A drastic change in a small temperature range during the tetragonal–monoclinic transition is interpreted as a cooperative change in both short and long range interactions. The large thermal hysteresis is attributed to the difference in the mechanism of transition during heating and cooling.

Notes: The small angle X-ray scatter from crystalline polymers is evaluated by using a one-dimensional model where the density between crystal and amorphous phases changes linearly over a finite transition range t, and where the sizes of the crystal and amorphous regions fluctuate according to independent Gaussian distributions. The calculation is based on the general model formulated in part I. Approximate expressions are derived for the width and area of the diffraction peaks. The dependence on t occurs in the factor sin2 πst/(πst)2 which affects only the peak intensities. An analysis is made to find how the factor will modify theoretical interpretations based on experimental intensity measurements.

Notes: Takagi's equations of X-ray diffraction by an imperfect crystal are solved in the most general case by the method used in a previous paper in the particular case of a perfect crystal. It is shown that, for small strain gradients, the amplitude inside the crystal is the sum of two generalized wavefields obtained by convolution of the Hankel functions H10 and H20 with a `source distribution' depending on local deformations. This is in agreement with the results given by other authors (Kato, Penning, Malgrange). This corresponds to the approximation of geometrical optics for X-ray propagation in crystals. A quantitative criterion of validity for this approximation is given. When this criterion is not fulfilled in some part of the crystal, the generalized wavefields are diffracted. It is shown that this implies creation of new generalized wavefields. This provides a physical interpretation of the occurrence of this phenomenon in the vicinity of a dislocation line and properties of the contrast of images in X-ray topography.

Notes: Problems in the collection and analysis of accurate diffraction data have been explored in a careful study of urea. Initial neutron measurements on non-spherical crystals show that little confidence can be placed in the values of extinction parameters derived for a non-spherical crystal of a hydrogenous material used in a neutron-diffraction experiment. It is also shown that multiple Bragg scattering is an important source of error. Uncertainties in the absorption and extinction corrections can be overcome by using a spherical crystal with μR ∼ 0.5. Corrections for the effects of multiple Bragg scattering were made by correlating complete sets of data measured at three different wavelengths. A number of models based on rigid-molecule vibrations and anharmonic interactions between atoms were used to treat the effects of thermal vibration. The best results were obtained from a refinement of the spherical-crystal data using general thermal parameters plus corrections for curvilinear motion, but a rigid-molecule treatment, with fewer vibrational parameters, was not much worse. Anharmonic effects did not seem to be appreciable. To look for possible `bonding features' in urea, difference syntheses computed from X-ray data and calculated X-ray structure factors based on the neutron results were plotted. It was not possible to remove the deep negative differences at the atomic positions by adjusting a scale factor and a thermal diffuse scattering correction, and it is probable that a real discrepancy exists between the observed and calculated structure factors.

Notes: The range of validity of an approximation for the F distribution is investigated. The R factor ratio as used by crystallographers is calculated and compared with tabulated values. This is done for the range of degrees of freedom generally encountered in crystallographic problems.

Notes: The rotation and translational parameters of a known molecular rigid group in an unknown crystal structure have been determined using reciprocal-space methods based on the Patterson function. Various refinements of the rotation function have been introduced in order to increase the speed and sensitivity of this method.

Notes: A practical and fast method for the determination of the layer sequence of high-order polytypes is described. Experimentally, the method involves only the determination of the relative order of reflexion intensities and does not employ actual intensity measurements. Auxiliary information, such as the percentage of hexagonality and the cyclicity of the polytype involved, considerably shortens the computer time needed. The method was used to identify a large number of ZnS polytypes and it is applicable to other polytypic material as well.

Notes: The thermal diffuse scattering (TDS) contribution to X-ray Bragg peaks is discussed in terms of its two experimental components, the included and the background parts. The discussion is primarily directed toward situations where detailed elastic constant information and a large computer are not available. Graphs are displayed which, in the spherical average approximation, allow an accurate assessment of the influence of all the experimental variables on the TDS contribution. It is shown that, even in the general case where spherical averaging is a poor approximation, the included TDS varies linearly with length of scan for small scans. A technique for determining the constant of proportionality is suggested. Thus the included TDS may be evaluated even though the elastic constants are not known.

Notes: Primary and secondary extinction are studied using the dynamical theory of X-rays diffracted by imperfect crystals. The transition from dynamical to kinematical scattering is explained in terms of fundamental processes in diffraction. Contrary to existing extinction theories, where the intensities diffracted dynamically by single coherent domains of a mosaic are combined using an ad hoc assumption of mosaic distributions, the present theory permits the dynamical amplitudes to change in response to disturbances of the dynamical interactions by imperfections. Neither the mosaic block model nor the statistical treatment of imperfections is used. The extinction of diffracted intensities is thereby treated as caused solely by inhomogeneous strains in a single coherent domain.

Notes: The correspondence of the thermal movement in crystals and the Bragg-reflexion intensity distributions is treated in terms of a theory developed for describing the effect of static mechanical distortions. The basic functions appearing in that theory are replaced here by the corresponding time averages. The dependence of the temperature factor on the crystal size is discussed. The effect that additional static mechanical distortions have on the temperature factor is also discussed. Moreover it appears from the given expressions that there is a broadening in the Bragg reflexions as a consequence of the crystal distortions due to thermal movement. In the kinematical approximation, the integral intensity of a Bragg reflexion is temperature independent. If the integral intensity decreases as the temperature increases this is either a dynamical effect or a consequence of a measurement that takes only into account the intensity in a limited region around the maximum of the Bragg reflexion. In line profile analysis, the thermal broadening effect can be separated from the mechanical distortion broadening effect by means of Stokes's method if a reference powder, of the same substance (in an undistorted constitution) as the powder subjected to analysis is used.

Notes: Simple physical models are taken to calculate thermal and electronic contributions to absorption coefficients in Al. N-beam systematic calculations are interpreted by means of 2-beam theory, making possible a comparison with the experimental values of Watanabe. The mean absorption coefficient agrees to within 25 per cent. Spherical aberration and thermal diffuse scattering within the objective aperture are shown to explain the poorer agreement for the anomalous absorption coefficient. It is concluded that weak beams are significant in the measurement of absorption coefficients. Weak systematic beams are shown to have a significant effect on the positions of the thickness fringe maxima. A brief comparison of 2- and N-beam rocking curves is given.

Notes: The potential energy in molecular crystals of α-nitrogen, cyanogen and octachlorocyclobutane is calculated as a function of the unit-cell parameters. Some available potential functions for nitrogen-nitrogen and chlorine-chlorine interactions between non-bonded atoms are tested. The results show a good agreement between the unit-cell parameters obtained by the X-ray diffraction method and those corresponding to the deepest minimum of the potential energy, when suitable potential functions are used.

Notes: Many crystals produce a diffuse scattering of X-rays which is localized in a series of relplanes. It is shown that the corresponding linear disorder in the crystal may have various origins. In silicon, the scattering is due to thermal vibrations and is well explained by the elastic properties of the crystal. In neutron irradiated quartz the radiation damage is responsible for the major part of the scattering. The case of BaTiO3 and KNbO3 is discussed in detail. A linear disorder is proposed which accounts better for the different distributions of the scattering in the 4 allotropic phases than the alternative explanation of the soft mode. In spite of some neutron inelastic scattering results it is not yet possible to distinguish between static and dynamic disorder.

Notes: Accurate neutron-diffraction measurements from crystals suffering from severe extinction have been used to test the recent general theory of extinction of Zachariasen (Acta Cryst. (1967). 23, 558). Analysis of these measurements indicates that certain of the approximations made in the theory are not generally valid and result in systematic deviations between theory and experiment, namely a marked angle-dependent effect and an inadequacy of the theory for strong extinction. The original theory is therefore extended to take these factors into account and to give agreement with the observed data.

Notes: A new method of determining X-ray scattering factors by dynamical interference is described. The theoretical background to the interference effect is discussed in detail and an expression for relating the fringe period to the structure factor is developed. The method relies on anomalous transmission and is therefore most suitable for measurements on nearly perfect crystals of high atomic weight. It also has the attractive property of being insensitive to slowly varying lattice strains. Applying the method to the 220 reflexion of silicon a value of 8.487 ± 0.017 for the atomic scattering factor has been obtained using Mo Kα1 radiation. This value is in excellent agreement with the author's previous results using the Pendellösung method.

Notes: The line profiles of X-ray reflexions from different MnxFe3 − xO4 samples were investigated. Sample I (x = 1.66) is cubic with no anomalies; small paracrystalline distortions, g 〈 0.1%, cannot be excluded and are calculated to be of this magnitude from considerations of a simple volume effect of the larger Mn3+ ions. Samples II–IV (x = 1.88) show interesting effects which are closely connected with the development of the tetragonal structure, observed when x 〉 1.80. Sample II (quenched) shows paracrystalline distortions which can be quantitatively explained on the basis of the Jahn–Teller effect: the tragonally deformed single Mn3+O2−6 octahedra are statistically oriented and statistically distributed over B sites and have a mean tetragonality {\bar \epsilon} = 0.014. In sample III (cooling rate 20°C.min−1) about 26% of the volume consists of microdomains which result from a correlation between the orientations and positions of Jahn–Teller octahedra; quantitive agreement with the theory can be obtained by introducing a correlation factor γ = 2. In sample IV (cooling rate 7°C.min−1) this correlation proceded in such a way that the crystal consisted of 74% tetragonal matrix which had already attained a value of c/a = 1.056, and about 22% of tetragonal microdomains oriented in two other directions with a value c/a 〈 1.056. In addition about 3 to 5% of a cubic phase is present. If the mean tetragonality of the microdomains is taken to be {\bar \epsilon} = 0.020 the experiments correspond quantitatively with the theory. The detailed study of sample III proves H¼ ¼ ¼ are the paracrystalline coordination statistics with a fluctuation α = 27 XU in the direction [111] and β = 38 XU perpendicular to it. This is explained by a model of the spinel structure having two kinds of blocks: a tetrahedral A-structure and Jahn–Teller affected octahedral subcubes at B sites; the separation distance is ¼ √3a = 3.7 Å.

Notes: A statistical theory for the X-ray diffuse scattering from disordered binary alloys is developed. It conveys the probability of finding ordered cells as a function of the distance from a given cell. These probabilities are related to the Warren short-range-order parameters. It is also shown that if part of the volume of the crystal is completely disordered, then its size can be calculated from the usual diffuse scattering measurements. Comparison with experiment shows clustering of ordered cells in disordered CuAu. For alloys which obey the Ornstein–Zernike pair correlation function just above Tc, such as β-brass, there is a clustering of ordered cells, but not an anti-phase domain structure.

Notes: An appropriate choice of the function minimized permits linearization of the least-squares determination of the matrix which transforms the diffraction indices into the components of the reciprocal vector in the diffractometer φ-axis system of coordinates. The coefficients of the least-squares equations are based on diffraction indices and measured diffractometer angles of three or more non-coplanar setting reflexions.

Notes: X-ray and neutron diffraction measurements have shown that Mn2Au is a compound with a tetragonal unit-cell of dimensions a = 3.328 and c = 8.539 Å and that it can be assigned to the same space group as MnAu2, namely I4/mmm (D4h17). The neutron diffraction pattern of 74.2 at.% MnAu can be explained only if the ordered structure of this alloy is identical with that of Mn2Au. Thus the compound Mn3Au apparently does not exist.

Notes: A new method for evaluating slit-smeared small-angle scattering data is discussed. It is applicable to any primary beam-intensity distribution. The propagation of measuring errors is examined; expressions for the computation of the precision of all derived pinhole scattering curve properties are given. There is no need to smooth the measured curve before applying the evaluation procedures. The method turns out to be very suitable for practical use.

Notes: Das Gupta & Welch have introduced the concept of `coherent crystal radiation' to explain their experiments. As a consequence of their interpretation the fundamental X-ray line widths taken with the double-crystal spectrometer are in error. As shown in the present paper, the results of Das Gupta & Welch can be explained by means of the existing theories of the double-crystal spectrometer and triple-crystal diffractometer.

Notes: An electron microscope equipped with conventional electron optics can be used as a `diffractometer' for structure research on individual aperiodic objects. The resolution limit of the electron microscope is not the resolution limit of the `diffractometer' and electron structure research with atomic resolution is possible. The advantage is that a diffractometer of this type can measure amplitudes and phases. The theory is developed in the language of the crystallographer, mainly using the concept of modifying functions. It is shown that, as in crystallography, redundancies in structures can be used in the analysis.

Notes: To calculate the magnitude of the correction of measured neutron structure factors for thermal diffuse scattering, it is necessary to know the ratio of the sound velocity in the crystal, cs, to the neutron velocity, vn. For β(= cs/vn) 〈 1, the correction is independent of vn and is evaluated using exactly the same procedure as for X-ray structure factors. For β 〉 1, the correction depends on vn and requires a more complicated analysis than for X-rays, although under certain conditions (discussed in the text) there is no correction at all to the measured intensities. Accurate neutron structure factor measurements are conveniently made with neutrons which are faster than the maximum sound velocity in the crystal; for a `soft' material such as lead this condition implies that the neutron wavelength does not exceed 1.70 Å, whereas for tungsten, a `hard' material, the upper limit is 0.77 Å.

Notes: A formulation of the scattered intensity from molecular crystals explains the appearance of a molecular scattering effect at large scattering-parameter values.

Notes: Solution-grown single crystals of cadmium bromide have been found to exhibit arcing phenomena on their oscillation photographs and to give rise to closed rings of various shapes and sizes on Laue photographs. These have been explained in terms of the formation, during crystal growth, of tilt boundaries consisting of edge dislocations created by simultaneous slip along more than one close-packed direction on different basal and non-basal planes. The measurement of arc lengths on these oscillation photographs enables an estimate of the density of dislocations within the boundaries to be made.

Notes: A method is described for determining, with few measurements and calculations, the bounding planes of a crystal.

Notes: A correction to a previously published result for neutron scattering is given.

Notes: The deficiencies of the conventional structure-factor formalism based on spherical atoms are clearly seen if atomic site symmetry is included in the expressions. Fluorite, cuprite, iron nitride and the h.c.p. structure are given as important examples of the failure of the conventional expressions.

Notes: A neutron diffraction study of deuterated potassium ferrocyanide trihydrate (KFCT) has been made above the ferroelectric Curie temperature. The monoclinic form, space group A2/a (No. 15), has a = 9.42, b = 17.01, c = 9.38 Å, and β= 90.03° at 20°C. Since all of the large crystals of KFCT were twinned by a 90° rotation about the b axis, data from both overlapped and nonoverlapped reflections were used in the determination of the structure. Refinement by a modified least-squares method that accounted for the overlapped data led to a weighted R value on F2 (3678 reflections) of 4.8%. The corresponding R on F is 4.9%. The disordered water molecules are of two types: one is disordered into two positions related by a twofold axis parallel to b: the other is disordered into two positions that are related by an approximate 90° rotation in the plane of the molecule. Both types of water molecules show O–D ...O or O–D...N bonding. The Fe(CN)4−6 octahedra are regular and the potassium ions are loosely surrounded by O and N atoms. This analysis has elucidated the role of the water molecules in the ferroelectric switching mechanism. The [{\bar 1}01] ferroelectric axis in this crystal can be explained on the basis of the dipole moment and orientation of the water molecule dipoles.

Notes: Detailed measurements have been made of the reflected and transmitted intensities for scattering of X-rays from perfect germanium crystals for μt values from 0.6 (`thin' crystal) to 20 (`thick' crystal). Both radiation from emission lines and monochromated `white' radiation have been used. Very good agreement with the dynamical theory of X-ray scattering was found.

Notes: The atomic scattering factor for iron has been measured on five low-order reflexions from pure iron at eighteen wavelengths in the range 0.63 to 2.53 Å; measurements have also been made on nickel for Cu Kα radiation to give additional checks with results of other workers. The variation of f for iron follows the Hönl theory for K-electrons quite closely, except for the 110 reflexion at wavelengths just short of the absorption edge. The dispersion corrections were independent of angle. The limiting values of f at high frequencies indicated by the results agree well with other theoretical and experimental values, except for the 110 reflexion.

Notes: An X-ray study of the arcing phenomena in polytypic crystals of CdI2, PbI2 and CdBr2 has revealed that various polytypes are formed by the creation of stacking faults at different reqular intervals during crystal growth.

Notes: The role of the dispersion correction in the analysis of X-ray diffraction data of lithium fluoride has been considered by applying the recent theoretical and experimental values of the dispersion corrections.

Notes: Equations suitable for obtaining values of the coherent and total intensities of X-ray radiation scattered by N-electron atomic systems are developed from the 1- and 2-particle density matrices. Although electron scattering within the first Born approximation can be treated in the same fashion, a separate consideration of this problem is not required since the scattered electron and X-ray intensities are related by equations well established in the literature. The general formalism presented here is such that it easily allows the computation of X-ray scattering factors and intensities to extend beyond the Hartree–Fock approximation, thereby incorporating electron correlation effects. Evaluation of the relevant equations requires the natural orbitals or the natural spin orbitals and the natural geminals or the natural spin geminals. These can always be obtained from the 1- and 2-particle density matrix analyses of the N-electron wavefunction Ψg characterizing the atomic state. In the event that Ψg is approximated by a single Slater determinant, the equation for the total scattered X-ray intensity reduces to the familiar Waller–Hartree result.

Notes: A numerical calculation is made on the intensity distribution of the beam obtained from the X-ray collimator system of monolithic silicon crystal, in which double 422 asymmetric Bragg-case diffractions of Cu Kα1 take place successively. It is concluded that a weak subsidiary peak found in the rocking curve obtained with a double-crystal diffractometer of parallel setting using a collimator of the above type is caused by the relative angular shift of the range of total reflexion for each crystal component due to the refraction effect.

Notes: The X-ray thermal diffuse scattering of crystalline anthracene was studied to show how it is possible to calculate the diffusion caused by lattice vibrations. From dispersion curves of low frequency phonons, calculated for the rigid molecule model with Kitaigorodsky `6-exp' potentials, the diffusion by phonons is computed at first and second order. The results compare well with experimental values in view of the simplicity of the model. It may be possible to separate diffusion by phonon from diffusion by other defects for studying these defects.

Notes: The intensity of the light scattered by a Ca2+ doped NaCl single-crystal was measured. It may be attributed to three types of scattering units (impurity precipitates): (1) Spherical precipitates. (2) Isotropic cylindrical precipitates which lie along the (110) directions. They are arranged in pairs building four arm crosses in the {100} planes. (3) Anisotropic cylindrical precipitates which lie approximately along the (100) directions. They build six arm stars. The dimensions of those scattering units strongly depend on thermal treatments.

Notes: A practical method of obtaining the best helical parameters p/P and p, where p is the rise per unit length along the screw axis and P is the pitch of the helix, is given, which consists of a graphical method followed by a least-squares analysis, and is particularly useful when the helix is non-integral and when the diffraction pattern is complicated by the existence of other crystalline or amorphous phases. The extension of the method to a coiled coil is also described.

Notes: Absorption corrections for cylindrical and spherical crystals have been evaluated by numerical integration for values of μR in the range 0 to 1.0, where μ is the mean linear absorption coefficient and R is the radius of the crystal. For this range of μR, which is that normally required for neutron diffraction data, interpolation from existing tables is unsatisfactory. The transmission factor Ahkl is tabulated as a function of μRand sin2θ, accurate to four decimal places. The intervals chosen for the tabulation are such that linear interpolation may be used for intermediate values. Analytical expressions, which may be used for calculating the transmission factor when a lower accuracy is acceptable, are also given.

Notes: A periodic polytype 66R was found to constitute part of a ZnS crystal. Oscillation photographs were taken with Cu Kα radiation. A test was applied to these data showing that the crystal was better approximated by the assumption of a perfect crystal (I ̃|F|), than by that of a mosaic crystal (I ̃|F|2). The method described by Dornberger–Schiff and Farkas–Jahnke in part I is applied to the experimental data and the result discussed. The homometric pair of polytypes with Zhdanov symbols (7753)3 and (7735)3 gives a much better fit to the experimental data (traditional R value equal to 0.147) than any other polytype.

Notes: A formal expression for the X-ray dynamical scattering amplitude of an imperfect crystal is obtained. The resultant expression includes the dynamical line broadening effect on the diffracted X-rays caused by imperfections. The effects due to absorption in the crystal are also taken into account.

Notes: The relation between an anomalous intensity effect in patterns from high energy electron diffraction by crystals, and the so-called resonance effect in LEED, is discussed mainly on the basis of experimental observations with electrons in both energy ranges, using crystals of zincblende, magnesium oxide and gallium arsenide. It is concluded that these two effects are essentially of the same nature. It is pointed out, that the intensity enhancement of the specular spot and other Laue-zone spots, which occurs under conditions of resonance, is particularly conspicuous when the specular spot is also a Bragg reflection spot. The nature of the wavefield generated in the crystal at resonance is discussed in connexion with other phenomena which appear under the resonance condition, such as the overall intensity enhancement of electron scattering, the increase of secondary electron emission and the variation of X-ray emission yield.

Notes: Published X-ray powder measurements on diamond and three different sets of published X-ray data on silicon have been re-analysed for bonding features using full-matrix least-squares refinement combined with a statistical analysis of the results obtained. In all cases a highly significant improvement in the fit between observed and calculated structure factors was obtained by introducing a tetrahedral distortion of the spherical `prepared' charge distribution, but subsequent introduction of a fourth-order cubic distortion proved to be highly significant only for the diamond data and one set of the silicon data. Hartree–Fock calculations gave a better fit to the diamond measurements than calculations based on Hartree–Fock–Slater wavefunctions. The necessity for placing restrictions on the form of the radial functions associated with the non-spherical distortions, the large estimated standard deviations of the distortion parameters and the dependence of the parameter values on the set of basis wavefunctions chosen to describe the spherical `prepared' charge distribution indicate the need for exercising caution in analysing the experimental measurements for bonding features.

Notes: Several calculations of atomic Compton profiles have shown that this physical property is very sensitive to the choice of the wavefunctions used to characterize the scattering system when calculating its kinetic energy as well as the mean values of its momentum and radius of charge distribution. The same method related to Dumond theory was applied to the case of molecules isoelectronic with neon and argon atoms. The results obtained give useful information about the charge distribution in the scattering system.

Notes: An analysis is given of the relation between the reduced cells defined by Niggli and the cells obtained by applying Buerger's algorithm. It is shown that in many instances a cell based on the shortest three non-coplanar translations must be transformed to obtain the reduced cell. The required transformations for all cases have been derived and are presented in this paper.

Notes: It is found that absorption of X-rays by the target material can result in beam inhomogeneities of more than 30% at low take-off angles.