Friedel's Law Research Articles

Anharmonicity in GaAs0.065P0.935

Inclusion of dispersion corrections in the generalized structure factor expression for the zinc blende structure give rise to the breakdown of Friedel's law for all even index reflections, an effect due to the presence of anharmonicity. Accurate X-ray diffraction intensity measurements have been obtained using a spherical single crystal of GaAs0.065P0.935 at 298 K. Least-squares refinement yield the Debye-Waller factors, BGa = 0.52 ± 0.01 Å2 and BAs/P = 0.39 ± 0.02 Å2, also the individual anharmonic thermal parameter βGa = −1.8 ± 1.3 × 10−17 J Å−3 and βAs/P = −0.8 ± 0.6 × 10−17 J Å−3. The residual index for the discrepancy between measured and calculated structure factors in the present work is R = 0.0172. The Bijvoet inequality between inverse reflections, reveals good agreement with the calculated values for a large number of reflections.

A systematic analysis of HREM imaging of sphalerite semiconductors

The interpretation of high-resolution electron micrographs of semiconductor compounds having the sphalerite structure has been investigated both theoretically using extensive computer simulations and experimentally. The images display considerably more complexity and diversity than those of the diamond-structured elemental semiconductors, mainly due to the absence of a crystallographic center of symmetry. A systematic study of image contrast for these compounds reveals a dependence on the degree of their non-centrosymmetry. Subtle contrast effects result from the differences in both amplitude and phase of Bijvoet pairs of reflections that appear with increasing thickness, due to the breakdown of Friedel's law under dynamical diffraction conditions. A strategy is suggested for identifying the absolute direction of the polar axis in low-resolution images. It is shown that first- and second-order scattering effects can be used to determine the crystal polarity from the fine detail visible in experimental micrographs of sufficiently high resolution.

A systematic analysis of HREM imaging of Wurtzite semiconductors

The interpretation of high-resolution electron micrographs of materials with wurtzite structure has been investigated in a comparative study of the 〈11 2 0〉-oriented semiconductors cadmium selenide (CdSe) and cadmium sulphide (CdS). Trends in scattering and imaging behaviour are shown to be similar to those exhibited by the related (cubic) sphalerite structure, in particular, the implications for image formation of the breakdown of Friedel's law under dynamical scattering conditions and the importance of both low- and high-resolution imaging for the determination of crystal thickness, defocus and polarity. Experimental images of 〈11 2 0〉 CdSe have been recorded at 100 kV and 400 kV. A major limitation to image interpretation is shown to be the extreme sensitivity of materials with the wurtzite structure to the effects of beam or crystal misalignment due to the presence of dynamically forbidden reflections (Gjønnes-Moodie lines) in the projection investigated.

Backscatter Kikuchi diffraction in the SEM for identification of crystallographic point groups

AbstractA preliminary account is given of the application of backscatter Kikuchi diffraction patterns (BKP) to point group determination in the SEM. Studies carried out on bulk specimens selected from the 32‐point groups indicate that 27 of the 32‐point groups can be determined unambiguously using BKPs, though distinction between 6,6,4,4 and 3 and 3 is difficult and requires special procedures. Evidence for breakdown of Friedel's law was observed in BKPs from GaSb, thus confirming the noncentrosymmetric point group 43m. Differences in intensities observed between the reflections hkl and hk̄l̄ reflections in GaSb which arise from the sensitivity of dynamical interactions to phases of the structure factors, could not be found in GaAs, illustrating that point group determination using BKPs should be considered with caution.

Does magnetite lack a centre of symmetry?

Doubts concerning the conventionally assumed centrosymmetry of the cubic crystal structure of spinel ferrites, including magnetite, have existed for many years, chiefly because dielectric constants as high as 103-104 have been observed at low frequencies. The authors show that evidence neglected in a published X-ray diffraction study of magnetite favors a non-centrosymmetrical structure. They also report new measurements of X-ray diffraction intensity which reveal that, contrary to the behaviour expected of a centrosymmetrical structure. Friedel's law is violated by magnetite when anomalous scattering is present.

On the breakdown of Friedel's law in electron backscattering channelling patterns

The theory for electron channelling patterns has been investigated. Calculations based on existing intensity expressions have been found inadequate for explaining recently observed deviations from Friedel's law. This is found to be the case even if the calculations are based on the full non-Hermitian eigenvalue matrix. The theory has hence been reinvestigated and a new intensity expression has been derived which includes the inter Bloch-wave coupling terms and is valid for non-centrosymmetric polyatomic crystals. The expression explains the observed asymmetries for GaSb which make it possible to determine unambiguously the correct non-centrosymmetric point group for this crystal. It is further found that the same effects in GaAs should be very weak in accordance with the non-observed deviation from Friedel's law in this case.

Determination of crystal symmetry from electron channelling patterns

The application of selected area channelling patterns in crystal symmetry determination has been investigated. In addition to rotation axes and mirror planes diffraction effects have been observed which may be ascribed to violation of Friedel's law. It has thus been possible for the first time with this technique to distinguish uniquely between centrosymmetric and non-centrosymmetric point groups. This is demonstrated with examples taken from the cubic and the hexagonal crystal systems, of which some show very detailed zone-axis absorption HOLZ patterns. Owing to the large tilt angle represented in each diagram the directions of double scattering may easily be localized. It is shown that this may be utilized in determination of the extinctions and the space group. The sensitivity to deviation from centrosymmetry is discussed and found to be somewhat less than for the convergent-beam technique. Crystal sizes from micrometres to bulk may be used. The strength of the channelling method is its simplicity and in particular the possible use of bulk specimens which may also be directly studied by other techniques, for example by X-rays.

Properties of electron microdiffraction patterns from amorphous materials

Properties of electron microdiffraction patterns obtained in a scanning transmission electron microscope (STEM) from thin specimens of amorphous germanium using a high-angular-resolution detector are described as a function of defocus, specimen thickness and astigmatism. It is shown that Friedel's law (centro-symmetry) always fails in the microdiffraction plane, even for an ideal phase-grating specimen. Calculations which account for the coherence function (source size) agree with the experimental results.

THE APPLICATIONS OF ANOMALOUS DIFFRACTION IN POWDER DIFFRACTOMETRY

An atom scatters X-ray with an amplitude and a phase that can be represented by the complex number ; ftof (K,E) = fo (K) + f' (K,E)+ if (K, E) . fo(K) represents the atomic scattering factor for the radiations with frequencies smaller than the absorption edge frequencies. The anomalous scattering terms f' and f, which are functions of the wavelength, describe the in-phase an out phase components of the change due to finite binding energies of the electrons .in the atom. Near an absorption edge of an atom, where the x-ray energy is close to the binding energy of an inner shell electron both f' and vary rapidly with wavelength , jumps rapidly up to a maximum while f' dips and then rises again as the x-ray energy goes from below the edge to above it. The term results in breaking of Friedel's law, this effect has long been used to find the solution of the crystallographic phase problem. Very few applications were performed to show the useful aspect of anomalous dispersion on the powder dif f ractometry . Even though the principle is simple and straight forword the implementation was not a trivia tasK. The main difficulty was to obtain reliable data. As the energy dependence of diffraction intensity is usually small then a very high accurancy of measurments were required. The recent availability of synchrotron radiation as a tunable source of x-ray, with very high intensity made it possible to use this effect to obtain structural information from a powder diffractometry. For this purpose a powder automated diffractometer has been designed to be adapted to such new sources (L.U.R.E. DCI) . This diffractometer consistes of three main parts :

Friedel'S law breakdown and interpretation of stacking fault images in tetrahedral semiconductors

Images of <11> oriented crystals with diamond structure (i.e. C,Si,Ge) are dominated by white spot contrast which, depending on thickness and defocus, can correspond to either atom-pair columns or tunnel sites. Olsen and Spence have demonstrated a method for identifying the correspondence which involves the assumed structure of a stacking fault and the preservation of point-group symmetries by correctly aligned and stigmated images. For an intrinsic stacking fault, a two-fold axis lies on a row of atoms (not tunnels) and the contrast (black/white) of the atoms is that of the {111} fringe containing the two-fold axis. The breakdown of Friedel's law renders this technique unsuitable for the related, but non-centrosymmetric binary compound sphalerite materials (e.g. GaAs, InP, CdTe). Under dynamical scattering conditions, Bijvoet related reflections (e.g. (111)/(111)) rapidly acquire relative phase differences deviating markedly from thin-crystal (kinematic) values, which alter the apparent location of the symmetry elements needed to identify the defect.

An X-ray analysis of anomalous dispersion effects, anharmonicity and thermal vibrations in gallium phosphide

Thermal vibrations of gallium and phosphorus atoms in gallium phosphide have been determined from the integrated intensities of Bragg reflections measured at ambient temperature. The thermal parameter values, obtained from the least-squares refinement, are B Ga = 0.66 ± 0.06 Å 2 and B p = 0.94 ± 0.10 Å 2 . Anharmonic thermal vibration effect is found to be small. The inclusion of dispersion corrections and anharmonicity in the structure factor expressions for GaP is found to give rise to the breakdown of Friedel's law for all even index reflections also. The observed Bijvoet ratios for several reflections are in general agreement with the calculated values.

Mirror-symmetric and non-mirror-symmetric glide planes in CuAsSe 1−xS x: I. A convergent beam electron diffraction study

The question of a breakdown of Friedel's law occuring in electron diffraction from parallel-plate crystals purely due to the symmetry-breaking effect of translation glide-plane vectors has recently been the source of some debate in the literature. Convergent beam diffraction patterns from cleavage crystals of the compounds CuAsSe 0.8S 0.2 and CuAsSe, having the space group Pbna, are used to illustrate the point that the symmetry-breaking effect described is amplified in certain crystals settings close to the principal zone and can be used to distinguish b- and n-glides in orthorhombic structures in a practical way. Calculations show a similar effect at the exact zone setting which, however, would be less accessible experimentally. These results lend support to the long-held belief that CBED pattern symmetries are the transformed symmetries of a parallel-sided crystal slab, in accordance with standard N-beam transmission theory, and that this result is of positive value in space group and structure determination.

Determination of the odd part of the texture function by anomalous scattering

The texture function f(g) of a polycrystalline material may be split into two parts \tilde{f}(g) and \tilde{f}(g), where the first one contains the terms of even order l in the series expansion, whereas the second one consists of the odd-order terms. In the case of centrosymmetric crystals, or by virtue of Friedel's law in noncentrosymmetric crystals, the contribution of {\approx}\atop{f} to experimental pole figures is identically zero. Hence it is impossible to determine {\approx}\atop{f} from pole figures obtained by `normal' polycrystal diffraction experiments. It is suggested therefore that anomalous scattering of X-rays or neutrons should be used to determine {\approx}\atop{f} from pole figure measurements directly and the corresponding formulae are given.

The role of the inversion centre in texture analysis

Polycrystalline samples of non-centrosymmetrical enantiomorphic materials may be considered as two-phase materials consisting of a right-handed and a left-handed phase, each of which has its own texture. Centrosymmetrical crystals may be considered as right- or left-handed at the same time. If the right- and left-handed nature of the crystals is fully to be taken into account then the statistical sample symmetry is correctly to be described by black–white point groups rather than by the ordinary ones. If the sample symmetry does not belong to one of the `grey' groups then the series expansion of the texture functions contains terms of odd order which are blotted out in polycrystal diffraction experiments by virtue of the inversion centre in the crystal symmetry or by Friedel's law. In certain cases it seems possible, however, to obtain an approximation to the odd part by making use of the non-negativity condition in the zero ranges of the texture function. Many important physical properties are centrosymmetric (even if the crystals themselves are not). In these cases the odd part of the texture function does not enter into the expression relating the polycrystal property to the corresponding one of the single crystals.

Anharmonic temperature factors of zinc selenide determined by X-ray diffraction from an extended-face crystal

Generalized structure factor expressions are derived for the zinc blende structure. Inclusion of dispersion corrections is shown to give rise to the breakdown of Friedel's law for all-even-index reflections, an effect due to the presence of bonding or anharmonicity. X-ray intensity measurements from an extended-face crystal are interpreted within the generalized structure factor formalism to yield the B factors BZn = 1.020 ± 0.005 A2 and BSe = 0.739 ± 0.008 A2 and an effective cubic anharmonic thermal parameter {\cal B}ZnSe = βZn/α3Zn - βSe/α3Se = --(6.8 ± 1.1) × 1036 J-2 A3 at 296 K. These are compared with values for other zinc blende compounds and with theoretically calculated B factors.

A new type of partial diffraction enhancement of symmetry found in CdI2 polytype*

Abstract A new trigonal polytype, 20 H7, of CdI2 was found to exhibit a peculiar type of partial diffraction enhancement of symmetry; the diffraction symmetry of hkl reflections with l ≢ 0 (mod 10) is 6/mmm, while that with l ≡ 0 (mod 10) is [unk]m. To explain this type of composite diffraction symmetry, the structure was assumed to be built up by superposition of two substructures: an idealized one-layer CdI2 polytype called the base, and a structure called the modifier and having in it, besides ordinary atoms, fictitious ‘negative’ Cd and I atoms so that it may convert appropriate layers in the base into those in the 20 H7 structure. Then, in view of the facts that twofold rotation symmetry appears in the vector set of the modifier which is governed by dichroic twofold rotation symmetry and that the base makes no contribution to hkl reflections with l ≢ 0 (mod 10), it was concluded that the diffraction symmetry of hkl reflections with l ≢ 0 (mod 10) is 6/mmm when Friedel's law holds. On the other hand, hk · 10 l′ reflections correspond to the hkl′ reflections from a new structure derived by the convolution of the original 20 H7 structure with a lattice with basis vectors a, b, and c/10, where a, b, and c represent the basis vectors for 20 H7. Consideration of this ‘convoluted’ structure led us to the conclusion that the symmetry of hkl reflections with l ≡ 0 (mod 10) is 6/mmm or [unk]m since the number of layers in the modifier is half the number of layers in the unit cell of the entire structure.

Breakdown of Friedel's Law in the Kikuchi patterns of tellurium

Evidence of the breakdown of Frieders law in the Kikuchi patterns of tellurium crystals has been observed with a standard transmission electron microscope (TEM). These observations give a direct method of identifying enantiomorphously related phases in the TEM. An interpretation of the observed deviations from Frieders law is proposed on the basis of the model of Thomas & Humphreys [Phys. Status Solidi A (1970). 3, 599-615] for Kikuchi patterns. The possible differences between the phases of the Fourier coefficients of the crystal potential and of the absorption potential have to be taken into account. Some light is thrown on the physical significance of the phenomenological parameters.

Some consequences of the presence of antisymmetric atomic features in noncentrosymmetric structures

Noncentrosymmetric structures are considered in terms of a generalized substructure formulation. Inclusion of dispersion, bonding and anharmonicity leads to generalized expressions for Bijvoet ratios. Methods of numerically estimating the effects due to bonding and anharmonicity upon the Bijvoet ratios are suggested. Allowance for bonding and anharmonicity is shown to have wide implications for the breakdown of Friedel's law. Subsets of reflections that obey Friedel's law in the conventional approximation are now shown to violate it. These violations are of considerable importance for measuring antisymmetric contributions due to bonding and anharmonic thermal vibrations. The possibility that bonding effects will result in the appearance of Bijvoet differences in noncentrosymmetric structures of elements is explored.

A prediction of Bijvoet intensity differences in the noncentrosymmetric structures of selenium and tellurium

The presence of an antisymmetric component of the non-spherical atomic charge distribution results in a more general breakdown of Friedel's law in polyatomic noncentrosymmetric structures. That this is also true in the case of noncentrosymmetric structures of elements is demonstrated directly for the case of the trigonal structures of Se and Te.

Calcul numérique de la fonction harmonique sphérique généralisée dotée des propriétés de symétrie du cristal cubique ou hexagonal

The three-dimensional quantitative analysis of the distribution function f(g) of crystalline orientations requires knowledge of the numerical values of the generalized spherical harmonic function endowed with crystal and sample symmetries. A Fortran program has been written in order to compute this function for cubic or hexagonal crystal symmetry, without a limiting assumption about sample symmetry – except Friedel's law which imposes only a centre of symmetry on the sample. The entire utilization of the relations arising from symmetries allows one to limit the amount of calculation and to obtain the results quickly and easily. This program can be used on a 32-bit-word computer of which the compiler does not work with complex numbers.