Electron diffraction is a unique tool for analysing the crystal structures of very small crystals. In particular, precession electron diffraction has been shown to be a useful method for ab initio structure solution. In this work it is demonstrated that precession electron diffraction data can also be successfully used for structure refinement, if the dynamical theory of diffraction is used for the calculation of diffracted intensities. The method is demonstrated on data from three materials - silicon, orthopyroxene (Mg,Fe)(2)Si(2)O(6) and gallium-indium tin oxide (Ga,In)(4)Sn(2)O(10). In particular, it is shown that atomic occupancies of mixed crystallographic sites can be refined to an accuracy approaching X-ray or neutron diffraction methods. In comparison with conventional electron diffraction data, the refinement against precession diffraction data yields significantly lower figures of merit, higher accuracy of refined parameters, much broader radii of convergence, especially for the thickness and orientation of the sample, and significantly reduced correlations between the structure parameters. The full dynamical refinement is compared with refinement using kinematical and two-beam approximations, and is shown to be superior to the latter two.
- Klíčová slova
- dynamical diffraction, orthopyroxene, precession electron diffraction, site occupancy,
- Publikační typ
- časopisecké články MeSH
Following a brief overview of current knowledge on lattices of subgroups of the space groups, it is shown that in the case of reducible space groups those lattices contain sublattices which are lattice isomorphic to the lattices of subgroups of layer and rod groups. Both sublattices involve the sublattice consisting of equitranslational subgroups.
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- časopisecké články MeSH
The charge-flipping algorithm in its band-flipping variant is capable of ab initio reconstructions of scattering densities with positive and negative values. It is shown that the method can be applied to reconstructions of difference electron densities of superstructures, i.e. densities obtained as a difference between the true scattering density and the average density over two or more subcells of the true structure. The amplitudes of reflections lying on the reciprocal lattice of the subcell are not required for the procedure. A series of examples shows applications of the method to the solution of superstructures in periodic crystals or quasicrystals as well as the application to ab initio solution of modulation of an incommensurately modulated structure from satellite reflections only and solution of a structure from a crystal twinned by reticular pseudomerohedry. The method is especially suited for solving pseudosymmetry problems occurring frequently in superstructures.
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- časopisecké články MeSH
Although encoded in theoretical works, relationships between the diffraction symmetry of magnetic structures and magnetic space/superspace groups are often ignored in practical applications. It is shown that magnetic symmetry operations have a direct impact through the rotation parts in the diffraction symmetry of the crystal and can be used to simplify calculations of magnetic structure factors. Besides, the translation parts can introduce specific systematic extinctions of magnetic reflections. Another point is that the efficiency and stability of refinement of magnetic crystal structures, in analogy with the refinement of nuclear structures, depend on direct application of the magnetic symmetry in the structure-factor formula. Magnetic symmetry also allows diffractionally independent magnetic domains and their mutual spatial orientations to be recognized. The selection of one irreducible representation can provide extra relationships between magnetic moments which do not directly affect diffraction symmetry. Thus the combination of both methods seems to be the most effective way to analyze and refine magnetic structures.
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- časopisecké články MeSH
The mineral kettnerite, CaBi(OFCO(3)), is a rare example of an order-disorder (OD) structure with a quadratic net. The lattice parameters of the simplest possible 1O polytype are a = 5.3641 (1), b = 5.3641 (1), c = 13.5771 (2) A, and the space group is Pbaa. There are three kinds of OD layers, identical to structure-building layers. Two of them are non-polar: the Bi-O and Ca-F at z = 0 and z = 1/2, respectively, with the layer-group symmetry C2/m2/m(4/a,b)2(1)/m2(1)/m. The third kind of OD layer of CO(3) groups (located between the Bi-O and Ca-F layers) is polar, with alternating sense of polarity. The layer group is Pba(4)mm. Triangular CO(3) groups are parallel to (110) or (110) planes with one O atom oriented towards the Bi-O layer and the remaining two O atoms oriented towards the Ca-F layer. The orientations of CO(3) groups alternate along the [110] and [110] directions. As a result, each group parallel to (110) is surrounded by four nearest neighbors parallel to (110) and vice versa. These positions can be interchanged by an (a + b)/2 shift or by pi/2 rotation; thus stacking of the layer onto adjacent ones is ambiguous. Instead of OD layers, the polytypes are generated by stacking of OD packets, comprising the whole CO(3) layers and adjacent halves of the Bi-O and Ca-F layers. They are polar, with alternating sense of polarity; the layer group is Pba(4)mm. Stacking sequences are expressed by ball-and-stick models, with the aid of symbolic figures, and by sequences of orientational characters. There are two maximum-degree-of-order (MDO) polytypes, 1O (really found and described, see lattice parameters and space group above) and 2O, with doubled c parameter and space group Ibca (not yet found). The derivation of the MDO generating operations of both polytypes is presented in this paper. The stacking rule also allows another, non-MDO, polytype with doubled c, i.e. the 2Q polytype, space group P4(2)bc (tetragonal, not yet found). Various kinds of domains can exist: (i) out-of-step domains shifted by (a + b)/2, (ii) twin domains rotated by pi/2 around local tetrads of odd or even packets, and (iii) upside-down domains in the polar 2Q polytype. Stacking sequences of 16 possible domains of the polytypes mentioned above are listed. Also 60 domains of four distinct six-packet polytypes are theoretically possible.
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- časopisecké články MeSH
Similarly to atomic positions in a crystal being fixed, or at least constrained by the space group of that crystal, the displacements of atoms in a domain wall are determined or constrained by the symmetry of the wall given by the sectional layer group of the corresponding domain pair. The sectional layer group can be interpreted as comprised of operations that leave invariant a plane transecting two overlapping structures, the domain states of the two domains adhering to the domain wall. The procedure of determining the sectional layer groups for all orientations and positions of a transecting plane is called scanning of the space group. Scanning of non-magnetic space groups has been described and tabulated. It is shown here that the scanning of magnetic groups can be determined from that of non-magnetic groups. The information provided by scanning of magnetic space groups can be utilized in the symmetry analysis of domain walls in non-magnetic crystals since, for any dichromatic space group, which expresses the symmetry of overlapped structures of two non-magnetic domains, there exists an isomorphic magnetic space group. Consequently, a sectional layer group of a magnetic space group expresses the symmetry of a non-magnetic domain wall. Examples of this are given in the symmetry analysis of ferroelectric domain walls in non-magnetic perovskites.
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- časopisecké články MeSH
The system of Hermann-Mauguin symbols for space and subperiodic Euclidean groups in two and three dimensions is extended to groups with continuous and semicontinuous translation subgroups (lattices). An interpretation of these symbols is proposed in which each symbol defines a quite specific Euclidean group with reference to a crystallographic basis, including the location of the group in space. Symbols of subperiodic (layer and rod) groups are strongly correlated with symbols of decomposable space groups on the basis of the factorization theorem. Introduction of groups with continuous and semicontinuous lattices is connected with a proposal for several new terms that describe the properties of these groups and with a proposal to amend the meaning of space groups and of crystallographic groups. Charts of plane, layer and space groups describe variants of these groups with the same reducible point group but various types of lattices. Examples of such charts are given for plane, layer and space groups to illustrate the unification principle for groups with decomposable lattices.
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The theory of domain states is reviewed as a prerequisite for consideration of tensorial distinction of domain states. It is then shown that the parameters of the first domain in a ferroic phase transition from a set of isomorphic groups of the same oriented Laue class can be systematically and suitably represented in terms of typical variables. On replacing these variables by actual tensor components according to the previous paper, we can reveal the tensorial parameters associated with each particular symmetry descent. Parameters are distinguished by the ireps to which they belong and this can be used to determine which of them are the principal parameters that distinguish all domain states, in contrast to secondary parameters which are common to several domain states. In general, the parameters are expressed as the covariant components of the tensors. A general procedure is described which is designed to transform the results to Cartesian components. It consists of two parts: the first, called the labelling of covariants, and its inverse, called the conversion equations. Transformation of parameters from the first domain state to other states is now reduced to irreducible subspaces whose maximal dimension is three in contrast with higher dimensions of tensor spaces. With this method, we can explicitly calculate tensor parameters for all domain states. To find the distinction of pairs of domain states, it is suitable to use the concept of the twinning group which is briefly described.
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- časopisecké články MeSH
This article is a roadmap to a systematic calculation and tabulation of tensorial covariants for the point groups of material physics. The following are the essential steps in the described approach to tensor calculus. (i) An exact specification of the considered point groups by their embellished Hermann-Mauguin and Schoenflies symbols. (ii) Introduction of oriented Laue classes of magnetic point groups. (iii) An exact specification of matrix ireps (irreducible representations). (iv) Introduction of so-called typical (standard) bases and variables -- typical invariants, relative invariants or components of the typical covariants. (v) Introduction of Clebsch-Gordan products of the typical variables. (vi) Calculation of tensorial covariants of ascending ranks with consecutive use of tables of Clebsch-Gordan products. (vii) Opechowski's magic relations between tensorial decompositions. These steps are illustrated for groups of the tetragonal oriented Laue class D(4z) -- 4(z)2(x)2(xy) of magnetic point groups and for tensors up to fourth rank.
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Diffraction on a crystalline slab formed by point-like scattering centres is treated as a multiple scattering problem based on the Ewald equations. Using general results expressed in a lucid matrix form, the two-beam solution for both coplanar and non-coplanar cases valid near and far from Bragg peaks is found and a detailed comparison of the final formulae obtained with those following from Laue's theory is performed.
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