ALBERT

Notes: Experiments measuring the intensity of three reflexions from a `very thick' Ge crystal (Laue case) were made, while the temperature of the crystal was lowered from 293 to 5°K. The variation of the intensity agreed with the dynamic theory, when the Debye temperature Θ, used for the evaluation of the Debye–Waller factor, was assumed to be 290°K throughout the temperature range. This result may suffer some slight correction in the future, but there is no variation of Θ at 20°K. Θ = constant constrasts with the specific heat data of Θ but essentially agrees with Batterman & Chipmans's prediction.

Notes: The 220 Bragg reflexion of silicon has been studied in considerable detail. By the Pendellösung fringe method we have measured the atomic scattering factor with an internal consistency of better than 0.1%. Particular care was taken to exclude systematic errors which might arise from elastic strain, X-ray absorption and X-ray polarization effects. The crystal was cut parallel to the Bragg planes at the points of observation so that its thickness could be directly measured with a travelling microscope. At the value of sinθ/λ corresponding to the 220 Bragg reflexion, the experimental atomic scattering factors (20°C) were: f = 8.478 ± 0.008 for Mo Kα1 radiation; f = 8.448 ± 0.012 for Ag Kα1 radiation, and f (Mo Kα1)/f (Ag Kα1) = 1.0035 ± 0.0007.

Notes: An improved method of including accidentally absent reflexions in the least-squares refinement of crystal structures, so as to accelerate the initial convergence of the procedure, is discussed. It is shown that when the calculated structure factors of these reflexions are less than the observed threshold value they determine nothing more about the structure, and should not be included in expressions for either the standard deviations or the shifts during refinement of atomic parameters.

Notes: Three Kikuchi lines not belonging to the same zone, which nearly intersect at the same point, can be utilized for determination of the ratio between lattice parameter and electron wavelength. The method given is analogous to a method based on Kossel line intersections in the X-ray case. One example is given in which the electron wavelength is determined with an uncertainty of ca. 0.1%.

Notes: X-ray structure factors and Compton line shape measurements provide complementary information on charge and momentum density, the latter measurement evidencing very much greater sensitivity to the valence electrons. Theoretical calculations of structure factors and Compton line shapes for solids have been sparse and have probable errors in excess of 3%. When these errors are compounded with experimental errors of several per cent the cases for which comparison between theory and experiment is meaningful are presently limited to diamond, silicon, germanium, MgO, LiH, Li, Be, Mg, Al and perhaps a few others. In diamond the 111 and 222 structure factors are in agreement with theory as are the 111 structure factors of LiH and MgO. The experimental momentum densities in Li, Be, Mg, and Al do not agree with conventional band calculations since these are unable at present to account for a significant electron-electron correlation effect.

Notes: The accuracy of the determination of X-ray intensities, and hence structure factor F-values, is of crucial importance to studies of the solid state, particularly at the present time when computational facilities allow exhaustive analysis of experimental data against theoretical models. Experimental errors are far less easy to estimate than appears from consideration of published individual experiments at a first, or even more careful, consideration; methods of both estimating and correcting for them need very careful elucidation. By far the best way to do this – and probably the only way that will yield reliable information as to where we are in this ill-defined field – is to organize group projects specifically to allow estimation of the magnitude of overall error, the identification of individual sources of error where possible and hence, the detection of specific experimental features, which should be either carefully assessed in each experiment to allow the magnitudes of errors to be kept as low as possible, or actually physically corrected. It is helpful in this regard to allow error-sources to be thought of as stabilized or variable according to the type of project planned. It is obvious that careful planning of a series of projects with different characteristics will be an evan more powerful tool for the investigation of error-sources in depth. In this paper, two projects are considered in some detail – one originated by the IUCr the other by the ACA – both having considerable similarities but also instructive differences. A method of comparing and also of editing the projects is considered using the correlation R factor |Rij| where |Rij| = Σ |Fi − Fj| /½ Σ |Fi + Fj|. This is seen to allow simple but important deductions.

Notes: 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.

Notes: Spectroscopically pure Cu has a lattice parameter a25 = 3.61491 Å (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.

Notes: The authors' work on the lattice complexes of plane groups is related to the earlier publications of N. L. Smirnova and her co-workers.

Notes: A calculation of the incoherent scattering by bromine has been made by use of the Waller–Hartree formulation including electron exchange and Hartree–Fock–Slater wave functions.

Notes: The structure factor is shown to be a basis for a one-dimensional representation of the point group, and the following property of non-primitive translations of most of the space groups is derived: the sum of non-primitive translations is a primitive one.

Notes: An attempt is made to show that the Q-functions [Tollin, Acta Cryst. (1966). 21, 613] and the translation function [Crowther & Blow, Acta Cryst. (1967). 23, 544] are virtually identical, and the modifications to the translation function proposed by Crowther & Blow are discussed.

Notes: A matrix formalism is developed for calculating the elastically scattered waves diffracted by an infinite plane parallel crystal. Introduction of projection operators makes it possible to cover both Laue and Bragg reflected waves under the same formalism.

Notes: An up to date Table of coherent neutron scattering amplitudes is presented.

Notes: The absorption coefficient of X-rays in germanium was measured for 32 wavelengths in the interval 2.3 Å 〉 λ 〉 0.3 Å (λK = 1.12 Å). A detailed analysis is given, including a comparison with theoretical expressions and a calculation of the oscillator strength for the K-electrons. Agreement between theory and experiment is remarkable. Hydrogen-type wavefunctions give gK = 1.28 for the dipole oscillator strength. A calculation based on the Thomas–Reiche–Kuhn sum rule and Hartree–Fock type wavefunctions gives gK = 1.29, while the authors' experimental value is gK = 1.27± 0.02.

Notes: The theory of Cowley & Pogany (Acta Cryst. (1968). A24, 109) 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 Å 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.

Notes: The molecular susceptibilities and anisotropies of some aromatic molecules have been deduced from their crystal diamagnetic susceptibilities and their molecular orientations. The molecules investigated are 10,10′- dianthronyl, bi(anthracene-9,10-dimethylene) photo-isomer and potassium acid phthalate. The molecular anisotropy can usually be calculated with a good degree of accuracy even in cases where the mean absolute susceptibility is not known with certainty.

Notes: The determination of real one-dimensional representations of a group is equivalent to the determination of its invariant subgroups of index two. This idea is used to construct and classify the magnetic space groups.

Notes: A survey of the present status of the error problem concerning the experimental structure factor values (F-values) is given, with special reference to the fluctuation, or the reproducibility, of original data obtained by various methods under different experimental conditions. It is pointed out that in most methods the reproducibility of F-values approaches the range of 1% ̃0.5% in favourable cases; the absolute accuracy is lower than this in all but a few examples.

Notes: Doubling of Kikuchi lines inside strong bands has been reported previously in electron diffraction patterns from MgO. The `doubling' is now explained as being due to two different lines, which at the particular wavelength employed in the experiment are very close. Owing to strong enhancement from multiple beam interactions inside the band, one of the lines is visible only in this region. Further examples of such doubling are shown in patterns from MgO and natural spinel. The observed patterns are compared with theoretical calculations.

Notes: The very strong absorption of low energy electrons by matter has some consequences on the symmetry of diffraction patterns. The symmetry of a pattern is that of one or several structural planes according to the depth reached by the electrons. It may differ from the symmetry of the crystal unit cell. When the symmetry of the crystal is lower than that of the structural planes, the symmetry of the pattern corresponds to that of the diffracting medium whereas if the symmetry of the crystal is higher than that of the structural planes, the symmetry of the pattern is lower than that of the diffracting medium.

Notes: Applying Jahn's method of reduction of a representation and using the authors' method of construction of the magnetic symmetry groups, the significant features of the second order piezomagnetic coefficients for the 90 magnetic crystal classes have been described.

Notes: In the paper published under the above title (Donnay & Kennard, 1964), please restore omitted words as follows: Page 1340, column 1, line 3 up: after the word aspect, insert P31, 2** into. Note also that the space-group symbols used in this paper are those of the 1935 edition of the International Tables. Any screw axis that results from intersecting symmetry planes need not be explicitly symbolized, although this was done in the 1952 edition. Examples: P4mc = P42mc, I4md = I41md, Pbc2 = Pbc21. The subscripts can easily be added by those who prefer the more explicit symbolism. We feel an explanation is in order, in view of the correspondence we received on this particular point.

Notes: In order to resolve differences in the reported data on thermal expansion, the lattice parameters of pure indium have been redetermined as functions of temperature over the range 27° to 106°C. The work was done independently on two cameras with good agreement in the results. The principal coefficient of thermal expansion along the a and b directions of the tetragonal structure is found to be independent of temperature and equal to 60.5 × 10−6 °C−1. A negative coefficient, which increases in magnitude with temperature, is obtained for the expansion along the tetragonal axis. The results are compared with those from earlier reports and discussed in relation to the available data on the elastic behaviour of the metal.

Notes: A chart is presented on which the appropriate zero-level Weissenberg photograph of a pseudo-merohedral twin may be superimposed. This enables the separation of twin-pairs of reflexions at any point on the film to be related to the twin obliquity angle. A separate chart which may be used with the zero-level Weissenberg photograph of any crystal, twinned or untwinned, gives the film-to-reciprocal-lattice magnification ratio corresponding to any short interval measured along a festoon.

Notes: This paper is devoted to the study of the directions in a crystal along which the acoustical Poynting vector and the wavevector are parallel. Theorems relevant to all crystalline systems are demonstrated and all such directions are determined for cubic, hexagonal and tetragonal systems.

Notes: 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.

Notes: The effects of simultaneous reflections on the intensities of reflections used in crystal structure analysis have been discussed by Zachariasen (1965) and by Moon & Shull (1964) for cases where only secondary extinction need be considered. In the present investigation the intensities of a number of reflections from a variety of crystals have been monitored in the course of azimuthal scans about the S vectors (θ constant). These include several space group 'forbidden' reflections (of the type 010 in P21, etc.) and others of weak to medium intensity. Measurements were made with a PAILRED automatic diffractometer using crystal monochromatized Mo K and Cu K reflections. The `mosaic' crystals investigated included ammonium oxalate, anthrone, Ca2VO4Cl, 1,2-trans-cyclohexanedicarboxylic acid and L-alanyl, D-alanyl-2,5-diketopiperazine. The effects observed vary widely from crystal to crystal and are often difficult to explain in terms of secondary extinction alone. Although in a few cases the incidence of simultaneous diffraction effects was surprisingly low, it is evident that the intensities of very weak reflections are often systematically overestimated because of simultaneous diffraction effects. The effects of unit-cell size, incident beam divergence, X-ray wavelength and other variables on the magnitudes of simultaneous diffraction effects will be discussed. Results observed with highly perfect crystals will be compared with those encountered when ordinary imperfect crystals are used.

Notes: Correction of X-ray intensities for thermal diffuse scattering (TDS) is necessary, though difficult in practice. Existing calculations for the TDS correction are reviewed and the hitherto neglected resolution function, R(q), of the diffractometer is discussed. It is concluded that overcorrection for inelastic scattering results when R(q) is ignored. Computation of accurate corrections requires a knowledge of the elastic constants of the material, experimental measurement of R(q) for several reflexions and considerable machine programming and computing time.

Notes: Currently used techniques for studying the perfection of large single crystals often fail to show any evidence of mosaicity when the dislocation density is very small. This mosaicity may be revealed by the use of relatively high-energy γ-sources. In single and also double crystal spectrometers corrections must be made for the fluorescent radiation: this cannot be eliminated by pulse-height discrimination techniques.

Notes: 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.

Notes: The methods normally used for photographic intensity measurement are briefly reviewed and the evidence given for the accuracy claimed ( ̃1% error) for the method of choice – photometry of integrating Weissenberg photographs taken with Cu Kα radiation. Examples are given of the use of the method in critical cases and the advantages and requirements of the photographic method are stated.

Notes: The interest of photographic recording of X-ray single crystal data is again gradually increasing. Film methods in combination with automatic film readers offer at the moment the fastest and most economical data collection system for large molecules. The paper describes some features of such a system.

Notes: The assessment of the overall accuracy in powder intensity measurement involves the estimation of the error of virtually every term in the intensity equation. Problems related to the accuracy of those quantities which are needed for the determination of relative structure factors are discussed. An example of the consideration of errors in an absolute measurement is given.

Notes: The problem of the interatomic bond in molecules and crystals is first of all that of a fairly precise determination of the electron distribution. The electron density distribution in crystals can be considered as the distribution of the square of the wave function. Thus an experimental determination of structure factors is one of the most direct methods of experimental determination of wave functions. Calculations of physical and physical-chemical properties of crystals can be made by using quantum mechanical methods, making use of wave functions obtained experimentally. The question arises of the necessary limits of accuracy of experimental F2 values for solving various quantum mechanical problems as well as for comparing them with theoretical values. It should be noted that the theoretical structure factors are approximate since the exact solutions of the Schrödinger equation is possible for the hydrogen atom only. Experimental determination of F2 values on powders poses very complicated problems including the measurement of absolute X-ray scattering intensities. Results of experimental determinations of F2 values on powder of metals, alloys and compounds with covalent character, including semiconductors, are analysed. Possible reasons for deviations, including the role of extinction, particle size of the powders under investigation, possibility of preferred orientation, change of state and composition of the surface, the role of polarization of the monochromated beam and lattice dynamics, are discussed. The difference between the data of different authors, which seems to be due to the degree of the accuracy of scaling to absolute intensity values, is also discussed. Different parts of the f-curves give different information on the electron density distribution of the crystal. The amount of information which F2values contain on the outer electron distribution depends to a great extent on the type and the parameters of the crystal lattice, and on the degree of anisotropy. There are some differences mainly in absolute values between the data of different authors. Some problems of reduction of the reflexion intensities are also discussed. the results of some measurements made on powders of the elements and on the semiconducting-type compounds with covalent bonds are analysed as well. A large number of measurements were made on AIIIBV compounds. The results show the differences in F2 values due to the change of the effective charge of ions. In conclusion some experimental determinations of f-curves are quoted and the necessary accuracy and possibilities of correction of the measured intensity of Debye reflexions are evaluated.

Notes: 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.

Notes: 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.

Notes: A computerized procedure based on the Evjen method for the direct summation of Madelung energies of centrosymmetric structures is described. The values obtained are strictly according to the definition of the Madelung function and the accuracy depends only on the number of terms taken into consideration. Madelung energies for a variety of structures calculated by this method are compared with values obtained by various other methods.

Notes: Neutron and X-ray intensities of NH4F were measured at −196°C and −155°C respectively. The wurtzite type structure and space group P63mc were confirmed. The displacement of the two h.c.p. sublattices, formed by each of the F−- and NH+4- ions, is such that all bond-distances are equivalent. The N–H bond distances as found from the X-ray data are about 0.1 Å shorter than those obtained from the neutron data. The final weighted R values were 1.9% and 2.6% for the X-ray and neutron data respectively.

Notes: The conditions which give high-resolution monochromatization of neutrons by multiple Bragg reflection in hexagonal close-packed crystals have been studied theoretically. The role of reflections forbidden by special atomic positions has been clarified, and, contrary to a previous conclusion, it is found that these `forbidden' reflections may be useful for high-resolution purposes. Detailed calculations of the necessary crystal orientations have been carried out for beryllium, magnesium and zinc. On the basis of somewhat arbitrary criteria the calculations show that these crystals can reflect a total of approximately 200 wavelengths in the range 0.7 to 4.9 Å (neutron energy 0.003 to 0.17 eV). Several orientations have been found at which mosaic crystals (with a sacrifice of angular resolution) should produce beams that are both intense and highly monochromatic.

Notes: A least-squares method of analysing joint neutron and X-ray diffraction data from the same crystal is described. The procedure is applied to hexamethylenetetramine, where it is shown that a more satisfactory refinement is obtained than by analysing each set of data independently.

Notes: The general analytic expression is given for the integral reflexion coefficient of X-rays from thick ideal absorbing crystals.

Notes: The method of factorizing the derivatives matrix when applying constraints is discussed in terms of three similar examples. First `symmetry constraint', in which molecules in the crystal must conform to a given molecular symmetry. Second `identical molecule constraint', in which molecules of the same chemical type but unrelated by symmetry are required to have the same shape. Finally the well established `shape constraint', where parts of a structure are constrained to a definite well known shape. A flow diagram for one of the constrained refinements is given, showing how the addition of the constraint can be done by additional programming rather than large scale alteration of the basic program. This suggests a program system with a different philosophy from that in present use, giving greater flexibility and efficiency than heretofore.

Notes: 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, f.c.c. opal structure. Screw dislocation-spiral growth is proposed as a possible mechanism for explaining the occurrence of the ordered opal structures.

Notes: The crystal energies of TiO2, GeO2 and SnO2 have been calculated by both the Born–Haber cycle and Born–Mayer equations. The agreement between the results suggests strongly ionic character in the M–O bonds.

Notes: An estimate of the variance in the electron density for P1 has been derived by consideration of the errors in the observed and calculated structure factors for accurate diffractometer experiments. It is concluded that the variance in the electron density is substantially dependent on the errors in the form-factor curves for the better quality instruments available.

Notes: It is shown that the measurement of X-ray diffraction integrated intensities allows the determination of a periodic function, Pa(r), which depends on the way in which the intensities are measured. When the crystal is perfectly ordered, Pa(r) is the usual Patterson function. When the crystal is only partially ordered, the derivation of Pa(r) is more involved but it can be calculated starting from the electronic density. From simple models the relationship between the actual structure and the Pa(r) deconvolution result is demonstrated. This function can be used in the interpretation of the shape of X-ray diffraction lines.

Notes: The recently introduced Edgeworth map is exactly the same as the smearing function for an atom. The most generally understood name for this function is the probability density function. The identity above is demonstrated for the specific case of BaF2, which is cubic. Demonstration for the general case is straightforward but is not given.

Notes: It is shown, on the basis of Jahn's method, that the conjecture made by one of us that there should be n nth order elastic coefficients of an isotropic solid does not hold good.

Notes: It is believed that there is an error in the paper by W. H. Zachariasen [Acta Cryst. (1967) 23, 558] and attention is drawn to relevant work published elsewhere.

Notes: The effect of including first-order and second-order thermal diffuse scattering contributions on the theoretical treatment for integrated Bragg intensities of single crystals is given for crystals belonging to any crystal system and with any number of atoms per unit cell. With certain restricting approximations and assumptions, the modified integrated intensity is found to be a product of the Bragg scattering and a factor, exp (C sin2 θ0/λ2), where C is a simple function of the X-ray wavelength, the temperature, the angular peak-width, and a quantity related to the elastic properties of the crystal. For cubic crystals, C is found to be approximately constant and thus confirms the result obtained by Nilsson (1957, Ark. Fys. 12, 247) for simple cubic crystals. The theory is applied to Al and KCl crystals, and calculated values for the ratio of uncorrected to corrected integrated Bragg intensities show reasonable agreement with some experimental values obtained using the Mössbauer effect.

Notes: 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 Å (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 Å at 0°C to 5780 Å at 400°C.

Notes: The two-phonon X-ray scattering from hexamethylenetetramine is calculated, and this is compared with the one-phonon result, here independently recalculated. The details of the calculation procedure are fully described, and possible savings in computation discussed. In a few regions of reciprocal space within the Cu Kα reflecting sphere the two-phonon scattering is larger than the one-phonon scattering. It is therefore essential when making quantitative comparisons with experiment that the two-phonon scattering be considered.

Notes: X-ray diffraction measurements have been made of the intensities of Bragg reflexions from a single-crystal of MgO which had been neutron irradiated in order to overcome the effects of extinction. Least-squares refinement of the data in terms of a number of spherical `prepared' models shows that the best agreement is obtained for the Mg2+, O2− (+ 2 well) atomic scattering factors combined with the two isotropic temperature factors BMg = 0.346 (0.009) Å2 and Bo = 0.315 (0.010) Å2. Three different sets of powder data for MgO have been re-examined and for two of these sets of measurements the best agreement is obtained with a combination of Mg2+ and O2− (+ 2 well) atomic scattering factors, while the Mg2+, O2− (Tokonami) scattering factors gives the best fit to the third set of data. The above sets of X-ray data give various descriptions of the thermal motion in MgO and these results are compared with vibration amplitudes calculated with the use of a simple shell model which yielded the values BMg = 0.287 Å2 and Bo = 0.350 Å2 for the isotropic temperature factors.

Notes: The point symmetry of the hexagonal lattice, 6/m 2/m 2/m, possesses 3 kinds of symmetry directions (Blickrichtungen); that of the rhombohedral lattice, {\bar 3} 2/m, has only two. The proposed extension of the Hermann–Mauguin symbolism rests on this fact. Examples: P311 = P3, P31 = R3. The dummy 1s stand for lattice-symmetry directions, as in the well known symbols P3m1, P31m, etc.

Notes: Considerable progress is foreseen toward the precise determination of integrated X-ray reflection intensities from small single crystals. High on the list of presently-limiting problems are multiple reflection, extinction, thermal diffuse scattering, unnecessary other background, counting statistics, effective integration and specimen change. Specific recommendations for equipment design and use are made which will partially alleviate each of them. Opportunities for better utilization of existing equipment through improved strategies for data collection and employment of small on-line computers are pointed out.

Notes: 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.

Notes: For the purpose of accurate intensity measurements the use of transmission specimens is recommended. Apart from special advantages with respect to certain features e.g. surface roughness, they allow the measurement of the orientation distribution over a large range, using off-symmetry azimuth values. Correction for preferred orientation should then be possible, provided a simple texture model – a cylindrico-symmetrical needle-type texture – can be assumed. Some computing schemes for carrying out the elimination of the texture parameters are discussed.

Notes: 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.

Notes: A commercial pole figure goniometer (Philips) is operated by a process control computer (IBM 1800) with a digital voltmeter to digitize the ratemeter output. The pole density data are stored on disk, and, after processing, are recorded in the pole figure by a plotter attached to the computer. Automatic 2&thgr;–&thgr; powder scans can also be performed without hardware modification.

Notes: The use of the Rachinger correction in line-broadening studies is discussed. A simplified Stokes correction is shown to be generally valid.

Notes: A description is given of a fully automatic program, written in ALGOL 60, that finds the constants of the reciprocal lattice from powder data. The progress of the program is illustrated with the (nearly) complete computer output for one selected case. Planes through the origin of the reciprocal lattice (zones) are found first. After evaluating these, the program selects pairs of zones with a common row in order to find reciprocal lattices, which are then reduced in a simple way. Each solution is compared with the experimental data and a figure of merit is calculated. The program is most suited for compounds of orthorhombic or lower symmetry.

Notes: A portable apparatus for cooling specimens used in X-ray and neutron diffraction studies is described. Temperatures in the + 50 to −1 185 °C range can be automatically and continuously maintained within 1 ° of the desired temperature for several weeks. Cold gas is obtained from liquid nitrogen which is boiled off at a rate sufficient to maintain a constant pressure in the sealed Dewar. The temperature-determining gas-flow rate is controlled by a pneumatically operated cryogenic valve which responds to a thermocouple signal.

Notes: The X-ray powder diffraction pattern and a precise determination of the lattice parameter of U2C3 (ao = 8.0899±0.0002 Å) are given. Further analysis indicates that U2C3 appears to be a stoichiometric compound.

Notes: A horizontal diffractometer has been set up for transmission studies of liquid samples. An asymmetric, singly bent LiF monochromator in the primary X-ray beam focuses the radiation on the receiving slit of the diffractometer. The sample rotates at half the angular speed of that of the X-ray detector to retain focusing at all angles of diffraction. A high temperature furnace assembly (camera) has been constructed which fits on the diffractometer. Its heating element consists of a thin-walled, pyrolitic graphite tube 38 mm in diameter. A special sample holder for the liquids is mounted on a boron nitride stool bolted to the bottom plate of the camera. The entire set-up allows the measurement of the diffracted intensity between 2&thgr; = 2° and 2&thgr; = 90° corresponding to K = 4πsin&thgr;/λ = 0.3 Å−1 at 12.4 Å−1 when Mo Kα radiation is used. Correction procedures to obtain the scattered intensity per atom from the raw data are discussed, and a comparison is made between the transmission and reflection methods with liquid tin.

Notes: The strain induced phase transformations produced in lead monoxide powders by ball milling at temperatures ranging from −196 to 200°C have been studied by X-ray diffraction methods. It has been found that either of the two polymorphs, litharge and massicot, was partially converted to the other at all the temperatures investigated, and that, whichever polymorph was the starting material, the end product, after a sufficiently long time, was a mixture of the two polymorphs, both highly strained. The percentage of each phase present was found to be dependent on the temperature of deformation. The results have been interpreted by consideration of the production of the microstrain energy required to overcome the energy barrier to the transformation, and of the shear generated in the mill on the pressure-temperature phase diagram. It is considered that shear stresses can affect the nature of the pressure-temperature phase diagram, in a manner theoretically postulated by Kumazawa. The commercial importance of the results has been briefly outlined.

Notes: The theory of the direct comparison X-ray method of phase analysis is extended to correct for preferred orientation effects. Texture parameters are defined to assess the type and intensity of preferred orientation using data from diffractometer patterns. The analysis is illustrated with results obtained on three austenitic stainless steels.

Notes: The theory of single and multiple small-angle scattering of fibres with a two-phase system of densities is revised. Luzzati's fundamental treatment of multiple scattering has been applied to systems with cylindrical symmetry. It is shown that the results of measurements with an `infinite' slit parallel to the fibre axis cannot give the specific internal surface directly, but that this information can be obtained by a combination of `infinite' slit measurement and measurements with a pin-hole system in the case of needle-like pores with preferred orientation. The experimental results obtained indicate that the scattering at very small angles is a multiple scattering determined by the filament cross-section. It is shown that parameters related to the size and the shape of the filament cross-section as wall as the specific surface of the filaments can be computed from this scattering. The scattering at somewhat wider angles is essentially a single scattering and can contain two components. The predominant component is a scattering of needle-like pores with diameters in the 10–20 Å range and lengths of at least 200–300 Å. The porosity is around 30%. The needle axes show a preferred orientation parallel to the fibre axis. The angular distribution of the needle axes is very similar to that of the graphitic layers. A second component can be observed, which is produced by density fluctuations in the parallel stacking of graphitic layers. Studies on powdered fibres indicate changes in the microstructure due to the grinding.

Notes: For the measurement of crystallite orientation in regenerated cellulosic fibers, Hermans and co-workers defined a parameter K in terms of the intensity distribution in the (10\overline 1, 021) sickle. Their theory is extended in the present study to mercerized fibers. It is found from the experimental observations that K is itself dependent on the X-ray orientation factor, and possible reasons for this are advanced. The dependence of K on the ratio of the number and diffractive powers of (021) and (10\overline 1) planes is, however, confirmed. A regression equation based on the correlation of K with the readily observed relative peak height of 021 is found to be of practical interest in estimating the crystallite orientation.

Notes: A new study of the structure of vitreous silica has been made under greatly improved conditions. Using Rh Kα radiation with the method of fluorescence excitation, reliable intensity values were measured to 4π sin&thgr;/λ = 20. The interpretation was in terms of pair functions, thereby eliminating the approximations in earlier work. Each silicon is tetrahedrally surrounded by 4 oxygen atoms, with a Si–O distance which is closely 1.62 Å. Each oxygen atom is bonded to 2 silicon atoms. The Si–O–Si bond angle α shows a distribution V(α) extending all the way from 120° to 180°, with a maximum at α = 144°. This wide variation in α is an important distinction between the vitreous and the crystalline forms of silica, and it furnishes an important criterion for any proposed model of vitreous silica. Good agreement with the measured pair function distribution curve was obtained by assuming a random orientation about the Si–O bond directions. The interpretation leads to the familiar random network model, but with the new results the model is more precise.

Notes: The use of a BECM-4 computer for interpreting X-ray powder patterns of polycrystals belonging to the orthorhombic system is described. By modifying the program for the intermediate crystalline systems the proper program is obtained for the direct indexing of X-ray powder patterns in the orthorhombic system after Neskuchaev's method, and the unit-cell dimensions are computed. The program has been tested and has been used for the calculation of the powder pattern of lead oxide (yellow massicot), PbO. A block diagram showing the complete procedure for the computations is given.

Notes: Electron microscope experiments show that the mechanism of electron-induced decomposition of tin(IV) sulfide differs from that of thermal decomposition. Hexagonal tin(IV) sulfide is converted to tetragonal tin in three equivalent orientations with [100]Sn parallel to [100]SnS2 and [001]Sn parallel to [001]SnS2. Experiments on epitaxic growth of tin on tin(IV) sulfide (001) surfaces and theoretical considerations of relative interface energies show that the reaction is controlled by epitaxy and not by a three-dimensional relation between the two lattices.

Notes: An X-ray reflexion topography camera has been designed to study structural defects near the surface of monocrystalline semiconductors. Large surfaces can be covered by means of successive fixed exposures. A surface of 30 mm x 10 mm for example can be studied in less than ten minutes on nuclear emulsions. The conditions for the observations of dislocations in gallium arsenide have been examined: the choice of reflecting planes is limited by the depth of penetration of X-rays, so that the effective crystal thickness must be of the order of the width of a dislocation image (2-5 microns). Studies to which these observations can be applied are numerous: structural defects induced by diffusion, lattice strains, orientation of dislocations near the surface, structural evolution of silicon during integrated circuits elaboration, structure of cleavage surfaces, lapping and polishing defects, structural defects of epitaxic layers.

Notes: The practical importance of tungsten monocarbide prompted an investigation into its wear mechanism. Initial studies on ball-milled tungsten carbide powders revealed considerable plastic deformation, coupled with a reduction in crystallite size. Deformation stacking faults, involving slip on the basal plane, were observed. The work was extended to sintered blocks, tool tips and single crystals. Studies on the worn surfaces of these revealed plastic deformation that extended to a depth of a few microns below the surface. The plastic deformation is an important factor in the wear of tungsten carbide.

Notes: Slightly twinned and untwinned single crystals of Ba2NaNb5O15 have been studied by X-ray techniques. Oscillation and full rotation X-ray photographs show the c axis to be twice the previously reported value; therefore, there are 8 formula units per unit cell. It was also found that Cmm2 (C2v11) is not the correct space group assignment for the crystals studied, and that Ccc2 (C2v13) appears to be likely.