Reciprocal Lattice Space Research Articles

Assessment of Image Formation by Three-Dimensional Power Spectrum in Transmission Electron Microscopy

The three-dimensional property of the image contrast obtained with transmission electron microscope was investigated through the three-dimensional power spectrum by taking the defocus value as the third axis. It has been confirmed for the first time that the three-dimensional power spectrum of the image intensity (intensity of interference) of a weakly scatterig object is concentrated only on the limited area of surfaces of the Ewald spheres in contact with each other at the origin of the reciprocal lattice space

Magnetic pairing in a lattice of Kondo ions: Application to UPt3.

Recently, several groups have calculated possible superconducting solutions for ${\mathrm{UPt}}_{3}$ from magnetic fluctuation pairing, by utilizing the band-structure Fermi surface and experimental dynamic susceptibility as input. The resulting solutions do not have proper periodicity in reciprocal-lattice space, though. In this paper, a more fundamental derivation of the pairing interaction is given by treating a lattice of interacting Kondo ions. Proper inclusion of umklapp processes leads to odd-parity solutions for ${\mathrm{UPt}}_{3}$, with order parameters of ${A}_{1u}$ or ${E}_{1u}$ symmetry preferred.

Correlation and scattering functions in a lattice model of oil–water–amphiphile solutions

The scattering function is calculated in mean-field approximation for two different thermodynamic states of a lattice model of oil–water–amphiphile solutions, one a disordered state and the other a one-dimensionally ordered (lamellar) state. The total scattering function for the former, and the fluctuation contribution to the scattering function for the latter, have maxima near q=1, where q is the magnitude of the scattering vector q in units of the reciprocal lattice spacing. For q up to and a little beyond the maximum the scattering functions and their spherical averages (averaged over the surface of a sphere of radius q in q space) are of the Teubner–Strey form I(q)=I(0)/(1−bq2+cq4) with b and c both positive, as in microemulsions. It implies a correlation function that is an exponentially damped sinusoid, of wavelength d and attenuation length, determined by b and c.

The thickness dependence of the X-ray diffuse scattering intensity for crystals with microdefects at laue-case diffraction

The theory of the dynamical X-ray diffuse scattering (DS) in single crystals with homogeneously distributed microdefects (point defect clusters, inclusions, new phase nuclei, dislocation loops, etc.) is developed. The thickness dependence of the DS intensity distribution in reciprocal lattice space is investigated by constructing both, iso-intensity contours and plane projections of three-dimensional images. Possibilities to obtain a detailed information concerning various-kind microdefects characteristics by using distribution patterns of the DS intensity are discussed. [Russian Text Ignored].

Synchroton x-ray diffraction study of an interfacial super-structure: a-Si on Ge 0.2Si 0.8 (111)-5 × 5

The interfacial structure between amorphous Si (a-Si) and Ge 0.2Si 0.8 is investigated by grazing incidence X-ray diffraction with the use of a synchroton radiation X-ray source. The sample, a-Si/Ge 0.2Si 0.8/Si(111) was prepared by molecular beam epitaxy. An a-Si film of 100 Å in thickness was deposited on the surface of Ge 0.2Si 0.8/Si(111), whose structure was 5 × 5 as confirmed by a RHEED pattern. The Bragg peaks of the X-ray diffraction wereobserved at ( 4 5 , 4 5 ) and (− 4 5 , 8 5 ) in the two-dimensional reciprocal lattice space. This result indicates that the period of the 5 × 5 reconstructed surface structure is preserved even when the surface is covered with a-Si.

Synchrotron X-ray diffraction study of an interfacial super-structure: a-Si on Ge 0.2Si 0.8 (111)-5×5

The interfacial structure between amorphous Si (a-Si) and Ge 0.2 Si 0.8 is investigated by grazing incidence X-ray diffraction with the use of a synchrotron radiation X-ray source. The sample, a-Si/Ge 0.2 Si 0.8 /Si(111) was prepared by molecular beam epitaxy. An a-Si film of 100 Å in thickness was deposited on the surface of Ge 0.2 Si 0.8 /Si(111), whose structure was 5×5 as confirmed by a RHEED pattern. The Bragg peaks of the X-ray diffraction were observed at (4/5, 4/5) and (−4/5, 8/5) in the two-dimensional reciprocal lattice space. This result indicates that the period of the 5×5 reconstructed surface structure is preserved even when the surface is covered with a-Si.

Finite-size effects on the critical structure factor of the two-dimensional Ising model

Explicit calculations of the critical structure factor of the planar Ising model on square lattices with a variety of boundaries are compared with the results of conformal invariance. Monte Carlo methods are used for square, circular and 2*1 rectangular geometries with free boundary conditions. Transfer matrices are used for an infinite strip with periodic boundary conditions. Good agreement is found for wavevectors less than a quarter of a reciprocal lattice spacing.

Polytypic behaviour of zirconolite

A polytypic family is described whose simplest member is zirconolite, (CaZrTi 2 O 7 ). The parent structure consists of calcium and zirconium atoms inserted between planes of titanium atoms, the latter being arranged in an hexagonal tungsten bronze (h. t. b.) type motif. Polytypes are generated by modifying the relative orientation of successive h. t. b. layers. Hettotypes are uniquely described by stacking formulae that are similar to those encountered in mica polytypes. Each stacking formula is com­posed of an interlayer stacking vector and a rotation. Zirconolite as a com­ponent of synroc (a proposed high level waste disposal matrix) was examined by selected area electron diffraction (s. a. d.) and high resolution electron microscopy (h. r. e. m.). Coherent polytypic intergrowths were observed which gave rise to laminar zirconolite domains. In crystallites that contain a large number of narrow domains, diffraction patterns contained sharp superlattice reflections of extremely small (100–200 Å) reciprocal lattice spacing; however, no superstructures with perfect translational periodicity were observed. (1 Å = 10 -10 m.) It is proposed that the intergrowth of zirconolite polytypes on a unit cell scale provides a means of incorporating cationic impurities in the parent structure.

Multi-analyser crystal spectrometer (max) for pulsed neutron source and its application to various problems

The multi-analyser crystal spectrometer (MAX) is an inverted type TOF spectrometer recently installed at KENS (spallation neutron source at KEK) for inelastic neutron scattering. Fifteen counters in the spectrometer are able to make time-of-flight scans along a desired direction in reciprocal lattice space, covering a wide range of the energy transfers corresponding to the fifteen analyser crystals. The magnetic excitations in antiferromagnetic γFeMn alloy were measured by the MAX at various temperatures up to 1.5 T N and the results are demonstrated as the contour map of the intensity in the ηω- Q space. The high energy excitations from the metallic hydrides such as ZrH 2 and VH 2 have also been measured.

Kronig–Penney model in reciprocal lattice space

Kronig–Penney model in reciprocal lattice space

Lattice Vibrations in CdGa2S4

Raman scattering and polarized infrared reflection spectra are presented for CdGa2S4. CdGa2S4 has the defect chalcopyrite structure and is treated as a perturbation of the zincblende structure found in binaries such as GaP and ZnS. The points Γ, X and W in the reciprocal lattice space of the zincblende structure coincide with the point Γ of the chalcopyrite structure. Then, the characteristics observed in the spectra are interpreted as resulting from the transformation of the phonons at the points Γ X and W of the zincblende structure. Twelve of thirteen modes expected from irreducible representation are assigned and the static and optical dielectric constants are also calculated.

New instabilities in Wigner-Dyson lattices

Binary NaCl and CsCl Wigner-Dyson lattices have been found to exhibit a new instability (mode-softening) at critical values of the charge ratio of the two species. These occur for transverse-acoustic vibrational modes with wave vector in the [110] direction in CsCl structure (and polarization in the [001] direction) at the first Brillouin-zone boundary and in the [100] direction in NaCl structure at the zone boundary (point $M$ for CsCl and point $X$ for NaCl in reciprocal-lattice space). These instabilities leave a gap in the charge-ratio parameter range in which the only currently known locally stable structure consists of segregated bcc lattices for the two species.

Neutron scattering from Rb2CocMg1-cF4: observation of two-dimensional magnetic long-range order

The elastic magnetic scattering from a series of compounds exhibiting two-dimensional Ising-like antiferromagnetic behaviour, Rb2CocMg1-cF4 (c=1.00, 0.89, 0.85 and 0.70), has been examined below their Neel temperatures. The magnetic scattering from Rb2Co0.7Mg0.3F4 takes the form of almost smooth elastic Bragg rods along the c direction rather than points in reciprocal lattice space, and provides the best approximation yet found to an ideal two-dimensional long-range order. This two-dimensional magnetic long-range order is found to depend strongly on the heat treatment of the crystals and on the concentration of Mg ions, becoming less pronounced as the concentration decreases.

Critical scattering of X-rays in some ferroelectrics

Abstract The experimental results on critical scattering are usually interpreted in terms of a cross-section which is spherically symmetric in the reciprocal lattice space, e.g. the Ornstein-Zernike cross-section.

Systematic Solution of the Mean Field Equation for Liquid Crystals

Abstract Starting with a pairwise, spatially and orientationally dependent intermolecular potential of the Kobayashi-McMillan form, we carry out a systematic solution of the mean field equation for liquid crystals. The mean field equation, presented as first of a hierarchy of BBGKY equations, is first reduced to a set of coupled integrodifferential equations by means of expanding the distribution function f (r, δ) and In f (r, δ) in Legendre polynomials and the reciprocal lattice space. In the first level of approximation, the expansion retains only the lowest-order coefficients, permitting a complete decoupling of the equations. In the second level of approximation, the leading coefficient which couples spatial order to orientational order is included. In the third level of approximation two more higher order coefficients are included. At each level, the free energy functional is evaluated to determine the equilibrium phase at given temperatures and chainlengths of a homologous series. It is shown that t...

X-ray critical scattering in BaTiO 3

Critical scattering of X-rays in a BaTiO 3 monocrystal was measured around the [005] reciprocal lattice point. The constant intensity curves in reciprocal lattice space are exactly analogous with those determined by Yamada et al. around the [200] point.

Anisotropy in the reciprocal lattice space of critical X-ray scattering in TGS

Measurement of the anisotropy of critical scattering of X-rays in TGS, revealed pronounced scattering for the scattering vector close to the c ∗ as well as b ∗ directions.

An energy-independent method of band-structure calculation for transition metals

The hybrid nearly-free-electron tight-binding first-principle schemes of Hubbard and Jacobs are investigated further, using the Korringa, Kohn and Rostoker equations as starting point. An improved energy-independent model Hamiltonian is presented, which is both more rapidly convergent in reciprocal lattice space and much less sensitive to the choice of splitting parameter than that proposed recently by Hubbard and Dalton. The validity of the approximations made is then tested by calculating the band structures of face-centred cubic copper and iron from first principles. The energy levels of Burdick and Wood are found to be reproduced with a root-mean-square accuracy of better than 0?01 ryd in both cases, with the use of only a 9 ? 9 model Hamiltonian matrix. When a hybrid nearly-free-electron tight-binding model Hamiltonian is used as an empirical interpolation scheme, the expressions derived here show explicitly how the parameters describing both the hybridization and the tight-binding overlap integrals may be expressed in terms of only two constants, namely the energy 0 and width ? of the resonance associated with the d bands.

High-Energy Bremsstrahlung and Electron Pair Production in Thin Crystals

A survey of theoretical and experimental methods now available for calculating, producing, and measuring high-energy coherent bremsstrahlung (HEB) and electron pair production (EPP) is presented.After an introduction in which the historical development of the subject matter is sketched, a preliminary theoretical approach is outlined. A rough classical argument is shown, and a formal expression of the Laue-Bragg law suitable for the next calculation is deduced, together with some fundamental kinematics of the HEB and EPP processes. The structure factors for the cubic, face-centered cubic, and diamond lattice are deduced and some qualitative features of the interferential cross sections are shown.A complete calculation of the HEB and EPP cross sections is carried out. HEB polarized and unpolarized cross sections as functions of recoil momentum are obtained and then integrated over all the reciprocal lattice space. Corresponding results for EPP cross sections are expressed and numerical calculation results are shown.The last part of the paper deals with experimental methods and techniques used in different laboratories in order to produce and measure high-energy coherent bremsstrahlung suitable for photoproduction experiments by polarized photons. Experimental apparatus and results are described in some detail. Finally, concluding remarks are made concerning the topics omitted.

Direct Determination of the Reciprocal Lattice Spacing and the Radial Interference Distribution by the Fourier Method

AbstractA procedure for extracting the interference distribution from the X-ray diffraction line using measured instrumental and wavelength distributions has been applied to the study of residual stress in copper following extension.In the direct unfolding method the principal instrumental distributions due to the source and to specimen absorption are unfolded by the Fourier method. The remaining instrumental aberrations are extracted using the centroid displacements. The resultant scattering distribution is transformed by the relations of Mitchell and de Wolff and the spectral distribution unfolded. Conditions limiting the Fourier coefficients are applied to preserve stability. Measurements have been carried out with a diffractometer equipped with a diffracted beam monochromator.The 420 and 331 interference distributions in copper following 10% extension have been compared with those of stress - free material. The distributions are symmetric both in the annealed specimen and after extension. In extension the distributions are displaced by an amount equivalent to a surface compressive stress of 6 kg/mm2, The symmetry of the interference distributions indicates that the displacement is due to a macrostress distribution.