ALBERT

Notes: Observations are reported on two effects of specimen motion upon neutron diffraction processes. These include the necessary realignment of a single-crystal in maintaining Bragg reflection conditions and the shift in the long wavelength cut-off characteristic of the transmission through a polycrystalline sample. The observations are compared with those to be expected from an analysis of Doppler and velocity compounding effects.

Notes: Relativistic versions of dynamical theories are surveyed and it is shown that electron diffraction phenomena become more dynamical at very high voltages. The results of diffraction experiments obtained with a 500 kV electron microscope are outlined. Specimens were magnesium oxide wedges and molybdenite films. The dynamical many-beam interactions were enhanced at 500 kV. The variation of Bragg width with the accelerating voltage was in agreement with the relativistic theory. The intensity reversal of Kikuchi bands, which is not remarkable below 100 kV, was clearly observed at 500 kV. The 3n + 1,0,0 reflections of molybdenite were found to be vanishingly weak, although it is not yet clear whether or not this is due to a dynamical effect. The appearance of many diffraction spots at high voltages seems to be due mainly to the decrease of the background intensity.

Notes: A wave mechanical treatment of the spatial dependence of the intensity for positrons and electrons emitted from a radioactive source embedded in a crystal is developed. The predicted intensity pattern is found to be characterized by Bragg angles and Bragg widths and as such cannot be derived from a classical treatment.

Notes: By the use of the X-ray Pendellösung fringes in α quartz including a single Dauphiné twin, the ratios of the structure factors having the forms of hk, l and hk, \bar l were determined with an accuracy of about a few tenths of one per cent. In the most favourable case, namely, the case of |F10,1|/|F_{10,{\bar 1}}|, the agreement between independent measurements for the same specimen was better than 0.1%. The values of the ratios for the extinction-corrected structure factors |Fcorr| and the structure factors |Fcal| calculated from the proposed structure in recent investigations (Young & Post, Acta Cryst. (1962), 15, 337; Zachariasen & Plettinger, Acta Cryst. (1965), 18, 710) are in accordance with the present ones within the mean deviation of about a few per cent, whereas the ratios for the observed values |Fobs| differ from ours more than ten percent. Some critical remarks on the Pendellösung method of determining structure factors, particularly by means of traverse-type topographs, are put forward.

Notes: In nearly perfect crystals of KDP standing waves of thermal neutrons are produced, causing an anomalous weak or strong incoherent scattering by hydrogen nuclei in the 004 reflexion. At the double crystal diffractometer in Rossendorf for the observation of anomalous incoherent scattering rocking curves in the Laue-case were measured in the reflexion- and transmission directions. The observed effect is interpreted and computed by the dynamic interference theory. The results of the experiments show that the incoherent scattering gives a contribution to the imaginary part of the scattering amplitude; its value is a function of the chemical bonding of the H atoms.

Notes: Vector methods are used to derive relationships between angles that occur in the direct and the reciprocal lattices, to give pedagogically useful demonstrations of the power of vector algebra and the reciprocal lattice concept. Several relationships involve the unit-cell volumes of the direct or the reciprocal lattices.

Notes: Ewald's and von Laue's dynamical theories of X-ray diffraction are discussed. Ewald derived the dispersion equations with a microscopic theory (classical electron theory), whereas von Laue used a phenomenological macroscopic approach (Maxwell's theory). As is to be expected, the dispersion equations of the two theories agree if one chooses the same physical model for the electromagnetic resonators, i.e. point dipoles or atoms.

Notes: The X-ray intensity of Zn Kα radiation emitted from clean cleavage surfaces (110) of zincblende crystals excited by fast electrons, which impinge on the surfaces in the same way as in ordinary electron diffraction experiments, was measured under various diffraction conditions of exciting electrons. The electron energy was about 30 keV. It was found that the X-ray intensity decreases at Bragg conditions by about 10–20%, in an asymmetric way with respect to the glancing angle of the electrons. Some of features of such an anomaly in X-ray intensity have been discussed on the basis of a two-wave dynamical theory of electron diffraction taking account of the phenomenological absorption.

Notes: The angular and voltage dependence of the diffracted intensity for electrons in the energy range of 150–900 volts from a clean tungsten (110) surface has been measured with a single-crystal diffractometer. Selecting the angle of incidence and the electron wavelength to satisfy the Bragg condition the crystal is rotated about its normal, and a Renninger plot of diffracted intensity versus azimuthal angle is made. Strong intensity variations are observed when the plane of diffraction is parallel to those low index crystallographic planes which contain relatively dense atomic rows. Additional structure occurs for each intersection of an extended reciprocal lattice point with the Ewald sphere. For some orientations the intensity is reduced to half the value within one degree, which is the beam divergence. When simultaneous diffraction does not occur, the intensities of the Bragg maxima are close to the background. The integral order Bragg maxima observed in electron diffraction are thus shown to have their origin in multiple diffraction. The frequently observed fractional order Bragg maxima are predicted to have the same origin. Renninger plots for such maxima show this to be the case. In addition the appearance of the fractional order peaks should depend only on the geometry for multiple diffraction. Intensity versus voltage curves for the beam incident along the [hkl] direction are predicted to have maxima of order (h2 + k2 + l2)−1, which is verified. The implication of these observations in terms of previous two-beam models is discussed.

Notes: In the analysis of spin structures a `natural' point of view looks for the set of symmetry operations which leave the magnetic structure invariant and has led to the development of magnetic or Shubnikov groups. A second point of view presented here simply asks for the transformation properties of a magnetic structure under the classical symmetry operations of the 230 conventional space groups and allows one to assign irreducible representations of the actual space group to all known magnetic structures. The superiority of representation theory over symmetry invariance under Shubnikov groups is already demonstrated by the fact proven here that the only invariant magnetic structures describable by magnetic groups belong to real one-dimensional representations of the 230 space groups. Representation theory on the other hand is richer because the number of representations is infinite, i.e. it can deal not only with magnetic structures belonging to one-dimensional real representations, but also with those belonging to one-dimensional complex and even to two-dimensional and three-dimensional representations associated with any k vector in or on the first Brillouin zone. We generate from the transformation matrices of the spins a representation Γ of the space group which is reducible. We find the basis vectors of the irreducible representations contained in Γ. The basis vectors are linear combinations of the spins and describe the structure. The method is first applied to the k = 0 case where magnetic and chemical cells are identical and then extended to the case where magnetic and chemical cells are different (k ≠ 0) with special emphasis on k vectors lying on the surface of the first Brillouin zone in non-symmorphic space groups. As a specific example we consider several methods of finding the two-dimensional irreducible representations and its basis vectors associated with k = ½ b2 = [0½0] in space group Pbnm (D162h). We illustrate the physical context of representation theory by constructing an effective spin Hamiltonian H invariant under the crystallographic space group and under spin reversal. H is even in the spins and automatically invariant under the (isomorphous) magnetic group. We show by the example of CoO that the invariants in H, formed with the help of basis vectors, give information on the nature of spin coupling as for instance isotropic (Heisenberg–Néel) coupling, vectorial (Dzialoshinski–Moriya) and anisotropic symmetric couplings. Magnetic structures, cited in the text to show the implications of the representation theory of space groups are ErFeO3, ErCrO3, TbFeO3, TbCrO3, DyCrO3, YFeO3, V2CaO4, β-CoSO4, Er2O3, CoO and RMn2O5 (R = Bi, Y or rare earth). Representation theory of magnetic groups must be considered when the Hamiltonian contains terms which are odd in the spins. The case may occur when the magnetic energy is coupled with other forms of energy as for instance in the field of magneto-electricity. Here again representation theory correctly predicts the couplings between magnetic and electric polarizations as shown on LiCoPO4 and (previously) on FeGaO3.