Acta Crystallographica Section A
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Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.enCopyright (c) 2019 International Union of Crystallography2019-04-30International Union of CrystallographyInternational Union of Crystallographyhttp://journals.iucr.orgurn:issn:2053-2733Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.text/htmlActa Crystallographica Section A: Foundations and Advances, Volume 75, Part 3, 2019textweekly62002-01-01T00:00+00:003752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section A: Foundations and Advances411urn:issn:2053-2733med@iucr.orgApril 20192019-04-30Acta Crystallographica Section Ahttp://journals.iucr.org/logos/rss10a.gif
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Still imageThe transformation matrices (distortion, orientation, correspondence), their continuous forms and their variants
http://scripts.iucr.org/cgi-bin/paper?ae5057
The crystallography of displacive/martensitic phase transformations can be described with three types of matrix: the lattice distortion matrix, the orientation relationship matrix and the correspondence matrix. Given here are some formulae to express them in crystallographic, orthonormal and reciprocal bases, and an explanation is offered of how to deduce the matrices of inverse transformation. In the case of the hard-sphere assumption, a continuous form of distortion matrix can be determined, and its derivative is identified to the velocity gradient used in continuum mechanics. The distortion, orientation and correspondence variants are determined by coset decomposition with intersection groups that depend on the point groups of the phases and on the type of transformation matrix. The stretch variants required in the phenomenological theory of martensitic transformation should be distinguished from the correspondence variants. The orientation and correspondence variants are also different; they are defined from the geometric symmetries and algebraic symmetries, respectively. The concept of orientation (ir)reversibility during thermal cycling is briefly and partially treated by generalizing the orientation variants with n-cosets and graphs. Some simple examples are given to show that there is no general relation between the numbers of distortion, orientation and correspondence variants, and to illustrate the concept of orientation variants formed by thermal cycling.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Cayron, C.2019-04-10doi:10.1107/S205327331900038XInternational Union of CrystallographyThree transformation matrices (distortion, orientation and correspondence) define the crystallography of displacive phase transformations. This article explains how to calculate them and their variants, and why they should be distinguished.ENphase transformationsmartensitic transformationtransformation matricesvariantsdistortionorientationcorrespondenceThe crystallography of displacive/martensitic phase transformations can be described with three types of matrix: the lattice distortion matrix, the orientation relationship matrix and the correspondence matrix. Given here are some formulae to express them in crystallographic, orthonormal and reciprocal bases, and an explanation is offered of how to deduce the matrices of inverse transformation. In the case of the hard-sphere assumption, a continuous form of distortion matrix can be determined, and its derivative is identified to the velocity gradient used in continuum mechanics. The distortion, orientation and correspondence variants are determined by coset decomposition with intersection groups that depend on the point groups of the phases and on the type of transformation matrix. The stretch variants required in the phenomenological theory of martensitic transformation should be distinguished from the correspondence variants. The orientation and correspondence variants are also different; they are defined from the geometric symmetries and algebraic symmetries, respectively. The concept of orientation (ir)reversibility during thermal cycling is briefly and partially treated by generalizing the orientation variants with n-cosets and graphs. Some simple examples are given to show that there is no general relation between the numbers of distortion, orientation and correspondence variants, and to illustrate the concept of orientation variants formed by thermal cycling.text/htmlThe transformation matrices (distortion, orientation, correspondence), their continuous forms and their variantstext3752019-04-10Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers411437Automatic calculation of symmetry-adapted tensors in magnetic and non-magnetic materials: a new tool of the Bilbao Crystallographic Server
http://scripts.iucr.org/cgi-bin/paper?lk5043
Two new programs, MTENSOR and TENSOR, hosted on the open-access website known as the Bilbao Crystallographic Server, are presented. The programs provide automatically the symmetry-adapted form of tensor properties for any magnetic or non-magnetic point group or space group. The tensor is chosen from a list of 144 known tensor properties gathered from the scientific literature or, alternatively, the user can also build a tensor that possesses an arbitrary intrinsic symmetry. Four different tensor types are considered: equilibrium, transport, optical and nonlinear optical susceptibility tensors. For magnetically ordered structures, special attention is devoted to a detailed discussion of the transformation rules of the tensors under the time-reversal operation 1′. It is emphasized that for non-equilibrium properties it is the Onsager theorem, and not the constitutive relationships, that indicates how these tensors transform under 1′. In this way it is not necessary to restrict the validity of Neumann's principle. New Jahn symbols describing the intrinsic symmetry of the tensors are introduced for several transport and optical properties. In the case of some nonlinear optical susceptibilities of practical interest, an intuitive method is proposed based on simple diagrams, which allows easy deduction of the action of 1′ on the susceptibilities. This topic has not received sufficient attention in the literature and, in fact, it is usual to find published results where the symmetry restrictions for such tensors are incomplete.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Gallego, S.V.Etxebarria, J.Elcoro, L.Tasci, E.S.Perez-Mato, J.M.2019-04-30doi:10.1107/S2053273319001748International Union of CrystallographyTwo new tools hosted on the Bilbao Crystallographic Server are presented. The programs permit the automatic calculation of symmetry-adapted forms of tensor properties for magnetic and non-magnetic groups. The cases of equilibrium, transport, optical and nonlinear optical susceptibility tensors are studied separately.ENequilibrium tensorstransport tensorsoptical tensorsnonlinear optical susceptibility tensorsmagnetic groupstime-reversal symmetryOnsager relationshipsTwo new programs, MTENSOR and TENSOR, hosted on the open-access website known as the Bilbao Crystallographic Server, are presented. The programs provide automatically the symmetry-adapted form of tensor properties for any magnetic or non-magnetic point group or space group. The tensor is chosen from a list of 144 known tensor properties gathered from the scientific literature or, alternatively, the user can also build a tensor that possesses an arbitrary intrinsic symmetry. Four different tensor types are considered: equilibrium, transport, optical and nonlinear optical susceptibility tensors. For magnetically ordered structures, special attention is devoted to a detailed discussion of the transformation rules of the tensors under the time-reversal operation 1′. It is emphasized that for non-equilibrium properties it is the Onsager theorem, and not the constitutive relationships, that indicates how these tensors transform under 1′. In this way it is not necessary to restrict the validity of Neumann's principle. New Jahn symbols describing the intrinsic symmetry of the tensors are introduced for several transport and optical properties. In the case of some nonlinear optical susceptibilities of practical interest, an intuitive method is proposed based on simple diagrams, which allows easy deduction of the action of 1′ on the susceptibilities. This topic has not received sufficient attention in the literature and, in fact, it is usual to find published results where the symmetry restrictions for such tensors are incomplete.text/htmlAutomatic calculation of symmetry-adapted tensors in magnetic and non-magnetic materials: a new tool of the Bilbao Crystallographic Servertext3752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers438447Fast analytical evaluation of intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density. II. The Fourier transform method
http://scripts.iucr.org/cgi-bin/paper?ae5056
The Fourier transform method for analytical determination of the two-center Coulomb integrals needed for evaluation of the electrostatic interaction energies between pseudoatom-based charge distributions is presented, and its Fortran-based implementation using the 128-bit floating-point arithmetic in the XDPROP module of the XD software is described. In combination with mathematical libraries included in the Lahey/Fujitsu LF64 Linux compiler, the new implementation outperforms the previously reported Löwdin α-function technique [Nguyen et al. (2018). Acta Cryst. A74, 524–536] in terms of precision of the determined individual Coulomb integrals regardless of whether the latter uses the 64-, 80- or 128-bit precision floating-point format, all the while being only marginally slower. When the Löwdin α-function or Fourier transform method is combined with a multipole moment approximation for large interatomic separations (such a hybrid scheme is called the analytical exact potential and multipole moment method, aEP/MM) the resulting electrostatic interaction energies are evaluated with a precision of ≤5 × 10−5 kJ mol−1 for the current set of benchmark systems composed of H, C, N and O atoms and ranging in size from water–water to dodecapeptide–dodecapeptide dimers. Using a 2012 4.0 GHz AMD FX-8350 computer processor, the two recommended aEP/MM implementations, the 80-bit precision Löwdin α-function and 128-bit precision Fourier transform methods, evaluate the total electrostatic interaction energy between two 225-atom monomers of the benchmark dodecapeptide molecule in 6.0 and 7.9 s, respectively, versus 3.1 s for the previously reported 64-bit Löwdin α-function approach.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Nguyen, D.Volkov, A.2019-04-30doi:10.1107/S2053273319002535International Union of CrystallographyNumerical implementations of the presented Fourier transform method and the previously reported Löwdin α-function approach for analytical determination of the two-center Coulomb integrals that appear in calculations of the electrostatic interaction energies between pseudoatom-based charge distributions are carefully examined in terms of precision and speed. The refined Fortran-based computer code allows a fast evaluation of electrostatic interaction energies with a precision of 5 × 10−5 kJ mol−1 or better using either of the two techniques.ENelectrostatic interaction energycharge densitypseudoatom modelFourier transformLöwdin α-functionThe Fourier transform method for analytical determination of the two-center Coulomb integrals needed for evaluation of the electrostatic interaction energies between pseudoatom-based charge distributions is presented, and its Fortran-based implementation using the 128-bit floating-point arithmetic in the XDPROP module of the XD software is described. In combination with mathematical libraries included in the Lahey/Fujitsu LF64 Linux compiler, the new implementation outperforms the previously reported Löwdin α-function technique [Nguyen et al. (2018). Acta Cryst. A74, 524–536] in terms of precision of the determined individual Coulomb integrals regardless of whether the latter uses the 64-, 80- or 128-bit precision floating-point format, all the while being only marginally slower. When the Löwdin α-function or Fourier transform method is combined with a multipole moment approximation for large interatomic separations (such a hybrid scheme is called the analytical exact potential and multipole moment method, aEP/MM) the resulting electrostatic interaction energies are evaluated with a precision of ≤5 × 10−5 kJ mol−1 for the current set of benchmark systems composed of H, C, N and O atoms and ranging in size from water–water to dodecapeptide–dodecapeptide dimers. Using a 2012 4.0 GHz AMD FX-8350 computer processor, the two recommended aEP/MM implementations, the 80-bit precision Löwdin α-function and 128-bit precision Fourier transform methods, evaluate the total electrostatic interaction energy between two 225-atom monomers of the benchmark dodecapeptide molecule in 6.0 and 7.9 s, respectively, versus 3.1 s for the previously reported 64-bit Löwdin α-function approach.text/htmlFast analytical evaluation of intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density. II. The Fourier transform methodtext3752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers448464Solving the disordered structure of β-Cu2−xSe using the three-dimensional difference pair distribution function
http://scripts.iucr.org/cgi-bin/paper?vk5036
High-performing thermoelectric materials such as Zn4Sb3 and clathrates have atomic disorder as the root to their favorable properties. This makes it extremely difficult to understand and model their properties at a quantitative level, and thus effective structure–property relations are challenging to obtain. Cu2−xSe is an intensely studied, cheap and non-toxic high performance thermoelectric, which exhibits highly peculiar transport properties, especially near the β-to-α phase transition around 400 K, which must be related to the detailed nature of the crystal structure. Attempts to solve the crystal structure of the low-temperature phase, β-Cu2−xSe, have been unsuccessful since 1936. So far, all studies have assumed that β-Cu2−xSe has a three-dimensional periodic structure, but here we show that the structure is ordered only in two dimensions while it is disordered in the third dimension. Using the three-dimensional difference pair distribution function (3D-ΔPDF) analysis method for diffuse single-crystal X-ray scattering, the structure of the ordered layer is solved and it is shown that there are two modes of stacking disorder present which give rise to an average structure with higher symmetry. The present approach allows for a direct solution of structures with disorder in some dimensions and order in others, and can be thought of as a generalization of the crystallographic Patterson method. The local and extended structure of a solid determines its properties and Cu2−xSe represents an example of a high-performing thermoelectric material where the local atomic structure differs significantly from the average periodic structure observed from Bragg crystallography.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Roth, N.Iversen, B.B.2019-04-30doi:10.1107/S2053273319004820International Union of CrystallographyUsing three-dimensional difference pair distribution function analysis of single-crystal diffuse X-ray scattering, the disordered structure copper selenide (β-Cu2−xSe) at room temperature is solved. The structure is ordered in two dimensions but disordered in the third.ENthree-dimensional difference pair distribution function3D-ΔPDFsingle-crystal diffuse X-ray scatteringdisordercopper selenidethermoelectricsHigh-performing thermoelectric materials such as Zn4Sb3 and clathrates have atomic disorder as the root to their favorable properties. This makes it extremely difficult to understand and model their properties at a quantitative level, and thus effective structure–property relations are challenging to obtain. Cu2−xSe is an intensely studied, cheap and non-toxic high performance thermoelectric, which exhibits highly peculiar transport properties, especially near the β-to-α phase transition around 400 K, which must be related to the detailed nature of the crystal structure. Attempts to solve the crystal structure of the low-temperature phase, β-Cu2−xSe, have been unsuccessful since 1936. So far, all studies have assumed that β-Cu2−xSe has a three-dimensional periodic structure, but here we show that the structure is ordered only in two dimensions while it is disordered in the third dimension. Using the three-dimensional difference pair distribution function (3D-ΔPDF) analysis method for diffuse single-crystal X-ray scattering, the structure of the ordered layer is solved and it is shown that there are two modes of stacking disorder present which give rise to an average structure with higher symmetry. The present approach allows for a direct solution of structures with disorder in some dimensions and order in others, and can be thought of as a generalization of the crystallographic Patterson method. The local and extended structure of a solid determines its properties and Cu2−xSe represents an example of a high-performing thermoelectric material where the local atomic structure differs significantly from the average periodic structure observed from Bragg crystallography.text/htmlSolving the disordered structure of β-Cu2−xSe using the three-dimensional difference pair distribution functiontext3752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers465473Experimentally obtained and computer-simulated X-ray asymmetric eight-beam pinhole topographs for a silicon crystal
http://scripts.iucr.org/cgi-bin/paper?wo5031
In this study, experimentally obtained eight-beam pinhole topographs for a silicon crystal using synchrotron X-rays were compared with computer-simulated images, and were found to be in good agreement. The experiment was performed with an asymmetric all-Laue geometry. However, the X-rays exited from both the bottom and side surfaces of the crystal. The simulations were performed using two different approaches: one was the integration of the n-beam Takagi–Taupin equation, and the second was the fast Fourier transformation of the X-ray amplitudes obtained by solving the eigenvalue problem of the n-beam Ewald–Laue theory as reported by Kohn & Khikhlukha [Acta Cryst. (2016), A72, 349–356] and Kohn [Acta Cryst. (2017), A73, 30–38].Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Okitsu, K.Imai, Y.Yoda, Y.Ueji, Y.2019-04-30doi:10.1107/S2053273319001499International Union of CrystallographyExperimentally obtained eight-beam pinhole topographs for a silicon crystal were compared with computer simulations based on the n-beam Takagi–Taupin equation and Ewald–Laue theory.ENX-ray diffractiondynamical theorymultiple reflectioncomputer simulationn-beam reflectionphase problemsiliconprotein crystallographyIn this study, experimentally obtained eight-beam pinhole topographs for a silicon crystal using synchrotron X-rays were compared with computer-simulated images, and were found to be in good agreement. The experiment was performed with an asymmetric all-Laue geometry. However, the X-rays exited from both the bottom and side surfaces of the crystal. The simulations were performed using two different approaches: one was the integration of the n-beam Takagi–Taupin equation, and the second was the fast Fourier transformation of the X-ray amplitudes obtained by solving the eigenvalue problem of the n-beam Ewald–Laue theory as reported by Kohn & Khikhlukha [Acta Cryst. (2016), A72, 349–356] and Kohn [Acta Cryst. (2017), A73, 30–38].text/htmlExperimentally obtained and computer-simulated X-ray asymmetric eight-beam pinhole topographs for a silicon crystaltext3752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers474482Experimentally obtained and computer-simulated X-ray non-coplanar 18-beam pinhole topographs for a silicon crystal
http://scripts.iucr.org/cgi-bin/paper?wo5032
Non-coplanar 18-beam X-ray pinhole topographs for a silicon crystal were computer simulated by fast Fourier transforming the X-ray rocking amplitudes that were obtained by solving the n-beam (n = 18) Ewald–Laue dynamical theory (E-L&FFT method). They were in good agreement with the experimentally obtained images captured using synchrotron X-rays. From this result and further consideration based on it, it has been clarified that the X-ray diffraction intensities when n X-ray waves are simultaneously strong in the crystal can be computed for any n by using the E-L&FFT method.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Okitsu, K.Imai, Y.Yoda, Y.2019-04-30doi:10.1107/S2053273319002936International Union of CrystallographyExperimentally obtained non-coplanar 18-beam pinhole topographs were compared with computer simulations based on the Ewald–Laue theory.ENX-ray diffractiondynamical theorymultiple reflectionn-beam reflectionphase problemprotein crystallographyNon-coplanar 18-beam X-ray pinhole topographs for a silicon crystal were computer simulated by fast Fourier transforming the X-ray rocking amplitudes that were obtained by solving the n-beam (n = 18) Ewald–Laue dynamical theory (E-L&FFT method). They were in good agreement with the experimentally obtained images captured using synchrotron X-rays. From this result and further consideration based on it, it has been clarified that the X-ray diffraction intensities when n X-ray waves are simultaneously strong in the crystal can be computed for any n by using the E-L&FFT method.text/htmlExperimentally obtained and computer-simulated X-ray non-coplanar 18-beam pinhole topographs for a silicon crystaltext3752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers483488Identification of the impurity phase in high-purity CeB6 by convergent-beam electron diffraction
http://scripts.iucr.org/cgi-bin/paper?lk5041
The rare earth hexaborides are known for their tendency towards very high crystal perfection. They can be grown into large single crystals of very high purity by inert gas arc floating zone refinement. The authors have found that single-crystal cerium hexaboride grown in this manner contains a significant number of inclusions of an impurity phase that interrupts the otherwise single crystallinity of this prominent cathode material. An iterative approach is used to unequivocally determine the space group and the lattice parameters of the impurity phase based on geometries of convergent-beam electron diffraction (CBED) patterns and the symmetry elements that they possess in their intensity distributions. It is found that the impurity phase has a tetragonal unit cell with space group P4/mbm and lattice parameters a = b = 7.23 ± 0.03 and c = 4.09 ± 0.02 Å. These agree very well with those of a known material, CeB4. Confirmation that this is indeed the identity of the impurity phase is provided by quantitative CBED (QCBED) where the very close match between experimental and calculated CBED patterns has confirmed the atomic structure. Further confirmation is provided by a density functional theory calculation and also by high-angle annular dark-field scanning transmission electron microscopy.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Peng, D.Nakashima, P.N.H.2019-03-11doi:10.1107/S2053273319000354International Union of CrystallographyThe impurity phase in high-purity CeB6 is unequivocally identified by first determining its space group, using an iterative convergent-beam electron diffraction approach, and following this with an atomic structure confirmation by quantitative convergent-beam electron diffraction and high-angle annular dark-field scanning transmission electron microscopy.ENspace groupsconvergent-beam electron diffractionimpurity phasesquantitative convergent-beam electron diffractionGjønnes–Moodie linescerium hexaboride (CeB6)cerium tetraboride (CeB4)The rare earth hexaborides are known for their tendency towards very high crystal perfection. They can be grown into large single crystals of very high purity by inert gas arc floating zone refinement. The authors have found that single-crystal cerium hexaboride grown in this manner contains a significant number of inclusions of an impurity phase that interrupts the otherwise single crystallinity of this prominent cathode material. An iterative approach is used to unequivocally determine the space group and the lattice parameters of the impurity phase based on geometries of convergent-beam electron diffraction (CBED) patterns and the symmetry elements that they possess in their intensity distributions. It is found that the impurity phase has a tetragonal unit cell with space group P4/mbm and lattice parameters a = b = 7.23 ± 0.03 and c = 4.09 ± 0.02 Å. These agree very well with those of a known material, CeB4. Confirmation that this is indeed the identity of the impurity phase is provided by quantitative CBED (QCBED) where the very close match between experimental and calculated CBED patterns has confirmed the atomic structure. Further confirmation is provided by a density functional theory calculation and also by high-angle annular dark-field scanning transmission electron microscopy.text/htmlIdentification of the impurity phase in high-purity CeB6 by convergent-beam electron diffractiontext3752019-03-11Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers489500A hidden Markov model for describing turbostratic disorder applied to carbon blacks and graphene
http://scripts.iucr.org/cgi-bin/paper?ib5063
A mathematical framework is presented to represent turbostratic disorder in materials like carbon blacks, smectites and twisted n-layer graphene. In particular, the set of all possible disordered layers, including rotated, shifted and curved layers, forms a stochastic sequence governed by a hidden Markov model. The probability distribution over the set of layer types is treated as an element of a Hilbert space and, using the tools of Fourier analysis and functional analysis, expressions are developed for the scattering cross sections of a broad class of disordered materials.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Hart, A.G.Hansen, T.C.Kuhs, W.F.2019-04-10doi:10.1107/S2053273319000615International Union of CrystallographyA mathematical framework for analysing aperiodic crystals encompassing a wide range of turbostratic disorders, including disorder of the first and second type, is presented. The framework uses the theory of hidden Markov models and is applied to carbon blacks and graphene.ENturbostratic disordercarbon blackshidden Markov modelscattering cross sectionsA mathematical framework is presented to represent turbostratic disorder in materials like carbon blacks, smectites and twisted n-layer graphene. In particular, the set of all possible disordered layers, including rotated, shifted and curved layers, forms a stochastic sequence governed by a hidden Markov model. The probability distribution over the set of layer types is treated as an element of a Hilbert space and, using the tools of Fourier analysis and functional analysis, expressions are developed for the scattering cross sections of a broad class of disordered materials.text/htmlA hidden Markov model for describing turbostratic disorder applied to carbon blacks and graphenetext3752019-04-10Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers501516Dependence of X-ray asymmetrical Bragg case plane-wave rocking curves on the deviation from exact Bragg orientation in and perpendicular to the diffraction plane
http://scripts.iucr.org/cgi-bin/paper?td5058
For the asymmetrical Bragg case the X-ray plane-wave reflection coefficients and rocking-curve dependences on the deviation angles from the exact Bragg orientation in the diffraction plane and in the direction perpendicular to the diffraction plane are analysed. The region of total reflection and its size dependence on two deviation angles are analysed as well. New peculiarities of the rocking curves are obtained.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Balyan, M.K.2019-04-10doi:10.1107/S205327331900161XInternational Union of CrystallographyThe rocking-curve dependence on the deviation of an incident X-ray plane wave from the exact Bragg orientation in and perpendicular to the diffraction plane for the asymmetrical Bragg case is investigated.ENdynamical diffractionBragg case diffractiondeviation angleX-ray rocking curvesFor the asymmetrical Bragg case the X-ray plane-wave reflection coefficients and rocking-curve dependences on the deviation angles from the exact Bragg orientation in the diffraction plane and in the direction perpendicular to the diffraction plane are analysed. The region of total reflection and its size dependence on two deviation angles are analysed as well. New peculiarities of the rocking curves are obtained.text/htmlDependence of X-ray asymmetrical Bragg case plane-wave rocking curves on the deviation from exact Bragg orientation in and perpendicular to the diffraction planetext3752019-04-10Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers517526The characteristic radiation of copper Kα1,2,3,4
http://scripts.iucr.org/cgi-bin/paper?lk5042
A characterization of the Cu Kα1,2 spectrum is presented, including the 2p satellite line, Kα3,4, the details of which are robust enough to be transferable to other experiments. This is a step in the renewed attempts to resolve inconsistencies in characteristic X-ray spectra between theory, experiment and alternative experimental geometries. The spectrum was measured using a rotating anode, monolithic Si channel-cut double-crystal monochromator and backgammon detector. Three alternative approaches fitted five Voigt profiles to the data: a residual analysis approach; a peak-by-peak fit; and a simultaneous constrained method. The robustness of the fit is displayed across three spectra obtained with different instrumental broadening. Spectra were not well fitted by transfer of any of three prior characterizations from the literature. Integrated intensities, line widths and centroids are compared with previous empirical fits. The novel experimental setup provides insight into the portability of spectral characterizations of X-ray spectra. From the parameterization, an estimated 3d shake probability of 18% and a 2p shake probability of 0.5% are reported.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Melia, H.A.Chantler, C.T.Smale, L.F.Illig, A.J.2019-04-10doi:10.1107/S205327331900130XInternational Union of CrystallographyThe characterization of Cu Kα1,2,3,4 radiation is presented, including the 2p satellite. The details are robust enough to be transferable to other experiments for calibration and reference.ENX-ray characteristic radiationCu Kαprofile analysisX-ray spectroscopyshake probabilityA characterization of the Cu Kα1,2 spectrum is presented, including the 2p satellite line, Kα3,4, the details of which are robust enough to be transferable to other experiments. This is a step in the renewed attempts to resolve inconsistencies in characteristic X-ray spectra between theory, experiment and alternative experimental geometries. The spectrum was measured using a rotating anode, monolithic Si channel-cut double-crystal monochromator and backgammon detector. Three alternative approaches fitted five Voigt profiles to the data: a residual analysis approach; a peak-by-peak fit; and a simultaneous constrained method. The robustness of the fit is displayed across three spectra obtained with different instrumental broadening. Spectra were not well fitted by transfer of any of three prior characterizations from the literature. Integrated intensities, line widths and centroids are compared with previous empirical fits. The novel experimental setup provides insight into the portability of spectral characterizations of X-ray spectra. From the parameterization, an estimated 3d shake probability of 18% and a 2p shake probability of 0.5% are reported.text/htmlThe characteristic radiation of copper Kα1,2,3,4text3752019-04-10Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers527540The polytopes of the H3 group with 60 vertices and their orbit decompositions
http://scripts.iucr.org/cgi-bin/paper?eo5092
The goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H3, i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter–Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H3 is given and presented as a `pancake structure'.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Bourret, E.Grabowiecka, Z.2019-04-30doi:10.1107/S2053273319000640International Union of CrystallographyA description of polytopes with 60 vertices generated by the finite reflection group H3 is given and a decomposition of their vertices into orbits of lower-symmetry groups is provided.ENCoxeter groupspolytopesorbit decompositionThe goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H3, i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter–Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H3 is given and presented as a `pancake structure'.text/htmlThe polytopes of the H3 group with 60 vertices and their orbit decompositionstext3752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers541550The chromatic symmetry of twins and allotwins
http://scripts.iucr.org/cgi-bin/paper?eo5091
The symmetry of twins is described by chromatic point groups obtained from the intersection group {\cal H}^* of the oriented point groups of the individuals {\cal H}_i extended by the operations mapping different individuals. This article presents a revised list of twin point groups through the analysis of their groupoid structure, followed by the generalization to the case of allotwins. Allotwins of polytypes with the same type of point group can be described by a chromatic point group like twins. If the individuals are all differently oriented, the chromatic point group is obtained in the same way as in the case of twins; if they are mapped by symmetry operation of the individuals, the chromatic point group is neutral. If the same holds true for some but not all individuals, then the allotwin can be seen as composed of twinned regions described by a twin point group, that are then allotwinned and described by a colour identification group; the allotwin is then described by a chromatic group obtained as an extension of the former by the latter, and requires the use of extended symbols reminiscent of the extended Hermann–Mauguin symbols of space groups. In the case of allotwins of polytypes with different types of point groups, as well as incomplete (allo)twins, a chromatic point group does not reveal the full symmetry: the groupoid has to be specified instead.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Nespolo, M.2019-04-30doi:10.1107/S2053273319000664International Union of CrystallographyThe chromatic symmetry of twins is extended to the case of allotwins through a groupoid analysis.ENallotwinninggroupoidschromatic symmetrytwin point groupstwinningThe symmetry of twins is described by chromatic point groups obtained from the intersection group {\cal H}^* of the oriented point groups of the individuals {\cal H}_i extended by the operations mapping different individuals. This article presents a revised list of twin point groups through the analysis of their groupoid structure, followed by the generalization to the case of allotwins. Allotwins of polytypes with the same type of point group can be described by a chromatic point group like twins. If the individuals are all differently oriented, the chromatic point group is obtained in the same way as in the case of twins; if they are mapped by symmetry operation of the individuals, the chromatic point group is neutral. If the same holds true for some but not all individuals, then the allotwin can be seen as composed of twinned regions described by a twin point group, that are then allotwinned and described by a colour identification group; the allotwin is then described by a chromatic group obtained as an extension of the former by the latter, and requires the use of extended symbols reminiscent of the extended Hermann–Mauguin symbols of space groups. In the case of allotwins of polytypes with different types of point groups, as well as incomplete (allo)twins, a chromatic point group does not reveal the full symmetry: the groupoid has to be specified instead.text/htmlThe chromatic symmetry of twins and allotwinstext3752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers551573On two special classes of parallelohedra in E6
http://scripts.iucr.org/cgi-bin/paper?ae5048
The cone of positive-definite quadratic forms is subdivided into aggregates of parallelohedra having certain properties. In E6, the Σ-subcones are investigated and, in particular, the Σ0-subcone will be described which first occurs in E6 and which is governed by the group {\cal G}_{E_6^*}. Further the \Sigma_{d+1 \choose 2}\-subcone which is governed by the group {\cal G}_{A_d^*} will be discussed for d = 6.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Engel, P.2019-04-30doi:10.1107/S2053273319001359International Union of CrystallographyThe structures of quasicrystals can be considered as sections of lattices of translations in higher dimensions, which has greatly stimulated the investigation of lattices in arbitrary dimensions. The structure of the cone of positive definite quadratic forms in Ed×d is investigated for d = 6. A partition of the cone in Φ and Σ domains is performed and its local symmetries are determined.ENtranslation latticesparallelohedracone of positive quadratic formsΣ-subconesThe cone of positive-definite quadratic forms is subdivided into aggregates of parallelohedra having certain properties. In E6, the Σ-subcones are investigated and, in particular, the Σ0-subcone will be described which first occurs in E6 and which is governed by the group {\cal G}_{E_6^*}. Further the \Sigma_{d+1 \choose 2}\-subcone which is governed by the group {\cal G}_{A_d^*} will be discussed for d = 6.text/htmlOn two special classes of parallelohedra in E6text3752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers574583Gröbner–Shirshov bases for non-crystallographic Coxeter groups
http://scripts.iucr.org/cgi-bin/paper?eo5097
For the group algebra of the finite non-crystallographic Coxeter group of type H4, its Gröbner–Shirshov basis is constructed as well as the corresponding standard monomials, which describe explicitly all symmetries acting on the 120-cell and produce a natural operation table between the 14400 elements for the group.Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Lee, J.-Y.Lee, D.-I.2019-04-30doi:10.1107/S2053273319002092International Union of CrystallographyA Gröbner–Shirshov basis and the corresponding standard monomials for the non-crystallographic Coxeter group H4 are constructed.ENCoxeter groupsGröbner–Shirshov basisstandard monomialsFor the group algebra of the finite non-crystallographic Coxeter group of type H4, its Gröbner–Shirshov basis is constructed as well as the corresponding standard monomials, which describe explicitly all symmetries acting on the 120-cell and produce a natural operation table between the 14400 elements for the group.text/htmlGröbner–Shirshov bases for non-crystallographic Coxeter groupstext3752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers584592A space for lattice representation and clustering
http://scripts.iucr.org/cgi-bin/paper?ae5061
Algorithms for quantifying the differences between two lattices are used for Bravais lattice determination, database lookup for unit cells to select candidates for molecular replacement, and recently for clustering to group together images from serial crystallography. It is particularly desirable for the differences between lattices to be computed as a perturbation-stable metric, i.e. as distances that satisfy the triangle inequality, so that standard tree-based nearest-neighbor algorithms can be used, and for which small changes in the lattices involved produce small changes in the distances computed. A perturbation-stable metric space related to the reduction algorithm of Selling and to the Bravais lattice determination methods of Delone is described. Two ways of representing the space, as six-dimensional real vectors or equivalently as three-dimensional complex vectors, are presented and applications of these metrics are discussed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)Copyright (c) 2019 International Union of Crystallographyurn:issn:2053-2733Andrews, L.C.Bernstein, H.J.Sauter, N.K.2019-04-30doi:10.1107/S2053273319002729International Union of CrystallographyAlgorithms for defining the difference between two lattices are described. They are based on the work of Selling and Delone (Delaunay).ENunit-cell reductionDelaunayDeloneNiggliSellingclusteringAlgorithms for quantifying the differences between two lattices are used for Bravais lattice determination, database lookup for unit cells to select candidates for molecular replacement, and recently for clustering to group together images from serial crystallography. It is particularly desirable for the differences between lattices to be computed as a perturbation-stable metric, i.e. as distances that satisfy the triangle inequality, so that standard tree-based nearest-neighbor algorithms can be used, and for which small changes in the lattices involved produce small changes in the distances computed. A perturbation-stable metric space related to the reduction algorithm of Selling and to the Bravais lattice determination methods of Delone is described. Two ways of representing the space, as six-dimensional real vectors or equivalently as three-dimensional complex vectors, are presented and applications of these metrics are discussed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)text/htmlA space for lattice representation and clusteringtext3752019-04-30Copyright (c) 2019 International Union of CrystallographyActa Crystallographica Section Aresearch papers593599