http://journals.iucr.org/a/issues/2008/05/00/isscontsbdy.html Acta Crystallographica Section A: Foundations of Crystallography publishes papers reporting fundamental advances in all areas of crystallography in the broadest sense. The central themes are, on the one hand, experimental and theoretical studies of the properties and arrangements of atoms, ions and molecules in condensed matter, ideal or real, and of their symmetry and, on the other, the theoretical and experimental aspects of the various methods to determine these arrangements. en Copyright (c) 2008 International Union of Crystallography 2008-09-01 International Union of Crystallography International Union of Crystallography http://journals.iucr.org urn:issn:0108-7673 Acta Crystallographica Section A: Foundations of Crystallography publishes papers reporting fundamental advances in all areas of crystallography in the broadest sense. The central themes are, on the one hand, experimental and theoretical studies of the properties and arrangements of atoms, ions and molecules in condensed matter, ideal or real, and of their symmetry and, on the other, the theoretical and experimental aspects of the various methods to determine these arrangements. text/html Acta Crystallographica Section A: Foundations of Crystallography, Volume 64, Part 5, 2008 text yearly 6 2002-01-01T00:00+00:00 5 64 2008-09-01 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography 515 urn:issn:0108-7673 med@iucr.org September 2008 2008-09-01 http://journals.iucr.org/logos/rss10a.gif http://journals.iucr.org/a/issues/2008/05/00/isscontsbdy.html Still image http://scripts.iucr.org/cgi-bin/paper?au5073 X-ray rocking curves in the Bragg–Laue case diffracting from the side surface of a plane-parallel crystal have been measured using a high-resolution optical system. The full width at half-maximum of the rocking curves is approximately three times narrower than that measured from the top surface. The characteristics of the transmitted beam from the side surface are almost the same as those through a thin crystal in the Bragg case. The rocking curves and the direction of X-ray energy flow in the crystal observed in the experiment can be reproduced using Wagner's approach [Wagner (1956), Z. Phys. 146, 127–168]. Copyright (c) 2008 International Union of Crystallography urn:issn:0108-7673 Yoshizawa, M. Fukamachi, T. Hirano, K. Oba, T. Negishi, R. Hirano, K. Kawamura, T. 2008-08-14 doi:10.1107/S0108767308019314 International Union of Crystallography X-ray rocking curves in the Bragg–Laue case have been measured using a high-resolution optical system. Calculations using Wagner's approach based on Laue's dynamical theory reproduced the rocking curves observed in the experiment. en Bragg–Laue case X-ray dynamical diffraction rocking curve X-ray rocking curves in the Bragg–Laue case diffracting from the side surface of a plane-parallel crystal have been measured using a high-resolution optical system. The full width at half-maximum of the rocking curves is approximately three times narrower than that measured from the top surface. The characteristics of the transmitted beam from the side surface are almost the same as those through a thin crystal in the Bragg case. The rocking curves and the direction of X-ray energy flow in the crystal observed in the experiment can be reproduced using Wagner's approach [Wagner (1956), Z. Phys. 146, 127–168]. text/html Measurement of X-ray rocking curves in the Bragg–Laue case text 5 64 2008-08-14 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography research papers 515 518 http://scripts.iucr.org/cgi-bin/paper?zm5041 (Fo − Fc) and (2Fo − Fc) Fourier syntheses are considered the most powerful tools for recovering the remainder of a structure and for correcting crystal structure models. A probabilistic approach has been applied to derive the formula for the variance for the expected value of the coefficient (Fo − Fc). This has allowed a better understanding of the features of the difference Fourier synthesis; in particular, a subset of well phased reflections has been separated from the subset of reflections best phased by the standard Fo Fourier synthesis. An iterative procedure, based on the electron-density modification of the difference Fourier map, has been devised which aims to improve phase and modulus estimates of the reflections with higher variance value, by using as lever arm the set of reflections with lower variance value. The new procedure (DEDM) has been implemented and verified on a wide set of test structures, the partial models of which were obtained by molecular replacement or by automatic model-building routines applied to experimental electron-density maps. Phase and modulus estimates of the difference Fourier syntheses improve in all the test cases; as a consequence, the quality of the difference Fourier maps also improves in the region where the target structure deviates from the partial model. A new procedure is suggested, combining DEDM with standard electron-density modification techniques, which leads to significant reduction of the phase errors. The procedure may be considered a starting point for further developments. Copyright (c) 2008 International Union of Crystallography urn:issn:0108-7673 Caliandro, R. Carrozzini, B. Cascarano, G.L. De Caro, L. Giacovazzo, C. Siliqi, D. 2008-08-14 doi:10.1107/S0108767308018503 International Union of Crystallography Some features of the difference Fourier synthesis have been investigated and a novel iterative procedure to improve it has been developed. en difference Fourier synthesis difference electron-density modification (Fo − Fc) and (2Fo − Fc) Fourier syntheses are considered the most powerful tools for recovering the remainder of a structure and for correcting crystal structure models. A probabilistic approach has been applied to derive the formula for the variance for the expected value of the coefficient (Fo − Fc). This has allowed a better understanding of the features of the difference Fourier synthesis; in particular, a subset of well phased reflections has been separated from the subset of reflections best phased by the standard Fo Fourier synthesis. An iterative procedure, based on the electron-density modification of the difference Fourier map, has been devised which aims to improve phase and modulus estimates of the reflections with higher variance value, by using as lever arm the set of reflections with lower variance value. The new procedure (DEDM) has been implemented and verified on a wide set of test structures, the partial models of which were obtained by molecular replacement or by automatic model-building routines applied to experimental electron-density maps. Phase and modulus estimates of the difference Fourier syntheses improve in all the test cases; as a consequence, the quality of the difference Fourier maps also improves in the region where the target structure deviates from the partial model. A new procedure is suggested, combining DEDM with standard electron-density modification techniques, which leads to significant reduction of the phase errors. The procedure may be considered a starting point for further developments. text/html The (Fo − Fc) Fourier synthesis: a probabilistic study text 5 64 2008-08-14 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography research papers 519 528 http://scripts.iucr.org/cgi-bin/paper?zm5042 The HK representation of close-packed polytypes is studied as a binary code. It is shown that the HK code can be seen as operators forming a group. The neutrality condition is then translated to HK sequences that result in the identity operator. The symmetry of an HK word can be related to the space-group symmetry of the corresponding polytype. All HK code types corresponding to all possible close-packed space groups are reported. From a coding perspective, equivalent HK codes correspond to bracelet equivalent classes. An efficient algorithm with execution time constant per generated object is modified to generate all non-equivalent polytypes of a given length. Copyright (c) 2008 International Union of Crystallography urn:issn:0108-7673 Estevez-Rams, E. Martinez-Mojicar, J. 2008-08-14 doi:10.1107/S010876730801461X International Union of Crystallography HK codes representing close-packed polytypes are studied as operators forming a group. The symmetry of the HK codes can be related to the space group of the corresponding polytype. Equivalent polytypes correspond to bracelet equivalent classes in the binary HK code. An algorithm for bracelet generation, with execution time constant per generated object, is modified to exhaustively generate all non-equivalent polytypes of a given length. en close-packed structures HK codes polytypes The HK representation of close-packed polytypes is studied as a binary code. It is shown that the HK code can be seen as operators forming a group. The neutrality condition is then translated to HK sequences that result in the identity operator. The symmetry of an HK word can be related to the space-group symmetry of the corresponding polytype. All HK code types corresponding to all possible close-packed space groups are reported. From a coding perspective, equivalent HK codes correspond to bracelet equivalent classes. An efficient algorithm with execution time constant per generated object is modified to generate all non-equivalent polytypes of a given length. text/html The symmetry of HK codes representing close-packed structures and the efficient generation of non-equivalent polytypes of a given length text 5 64 2008-08-14 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography research papers 529 536 http://scripts.iucr.org/cgi-bin/paper?zm5040 The classical model of independent random single deformation faults and twin faulting in face-centered-cubic and hexagonal close packing is revisited. The model is extended to account for the whole range of faulting probabilities. The faulting process resulting in the final stacking sequences is described by several equivalent computational models. The probability sequence tree is established. Random faulting is described as a finite-state automaton machine. An expression giving the percent of hexagonality from the faulting probabilities is derived. The average sizes of the cubic and hexagonal domains are given as a function of single deformation and twinning fault probabilities. An expression for the probability of finding a given sequence within the complete stacking arrangement is also derived. The probability P0(Δ) of finding two layers of the same type Δ layers apart is derived. It is shown that previous generalizations did not account for all terms in the final probability expressions. The different behaviors of the P0(Δ) functions are discussed. Copyright (c) 2008 International Union of Crystallography urn:issn:0108-7673 Estevez-Rams, E. Welzel, U. Pentón Madrigal, A. Mittemeijer, E.J. 2008-08-14 doi:10.1107/S0108767308016826 International Union of Crystallography A model of independent random faulting in face-centered-cubic and hexagonal close packing considering single deformation faults or twin faulting is revisited. The approach allows the analysis, within the random model, of the whole range of faulting probabilities. Several descriptions of the underlying faulting process are presented which allows the derivation of different properties of the faulted sequences. The probability of finding two layers of the same type Δ layers apart is derived. It is shown that previous generalizations did not account for mixed terms in the final probability expressions. en stacking twin faults close-packed structures faulting probabilities The classical model of independent random single deformation faults and twin faulting in face-centered-cubic and hexagonal close packing is revisited. The model is extended to account for the whole range of faulting probabilities. The faulting process resulting in the final stacking sequences is described by several equivalent computational models. The probability sequence tree is established. Random faulting is described as a finite-state automaton machine. An expression giving the percent of hexagonality from the faulting probabilities is derived. The average sizes of the cubic and hexagonal domains are given as a function of single deformation and twinning fault probabilities. An expression for the probability of finding a given sequence within the complete stacking arrangement is also derived. The probability P0(Δ) of finding two layers of the same type Δ layers apart is derived. It is shown that previous generalizations did not account for all terms in the final probability expressions. The different behaviors of the P0(Δ) functions are discussed. text/html Stacking and twin faults in close-packed crystal structures: exact description of random faulting statistics for the full range of faulting probabilities text 5 64 2008-08-14 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography research papers 537 548 http://scripts.iucr.org/cgi-bin/paper?au5072 In this paper, X-ray and γ-ray propagation in crystals having a constant strain gradient and flat or cylindrical surfaces is investigated. When a displacement field is present, the Takagi–Taupin equations are solved either by the Riemann–Green method or by a numerical method. The results are applied to study the operation of a double-crystal Laue–Laue diffractometer having a flat collimating crystal followed by a bent analyzer crystal. In particular, the effect of the analyzer strain on the location of the diffraction peaks in the dispersive and non-dispersive set-up is examined, thus confirming the previously reported peak location as being set only by the diffracting-plane spacing on the analyzer entrance surface. Copyright (c) 2008 International Union of Crystallography urn:issn:0108-7673 Apolloni, A. Mana, G. Palmisano, C. Zosi, G. 2008-08-14 doi:10.1107/S0108767308021508 International Union of Crystallography The effect of the analyzer strain on the location of diffraction peaks in X-ray and γ-ray propagation in crystals is examined. en bent crystals Takagi–Taupin equations double-crystal diffractometer X-ray diffraction In this paper, X-ray and γ-ray propagation in crystals having a constant strain gradient and flat or cylindrical surfaces is investigated. When a displacement field is present, the Takagi–Taupin equations are solved either by the Riemann–Green method or by a numerical method. The results are applied to study the operation of a double-crystal Laue–Laue diffractometer having a flat collimating crystal followed by a bent analyzer crystal. In particular, the effect of the analyzer strain on the location of the diffraction peaks in the dispersive and non-dispersive set-up is examined, thus confirming the previously reported peak location as being set only by the diffracting-plane spacing on the analyzer entrance surface. text/html X-ray and γ-ray propagation in bent crystals with flat and cylindrical surfaces text 5 64 2008-08-14 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography research papers 549 559 http://scripts.iucr.org/cgi-bin/paper?sc0031 It is shown that the formula for the positivity of the triplet invariant in P \bar{1} changes drastically if one uses a different statistical method by imposing acceptable and unbiased additional structural information. We obtain a much lower probability for the strength (almost ½) of the triplet formula than the classical one. Copyright (c) 2008 International Union of Crystallography urn:issn:0108-7673 Brosius, J. 2008-08-14 doi:10.1107/S010876730801698X International Union of Crystallography A triplet relation using an unbiased joint probability distribution of the atomic vectors is derived based on the observation that the distribution of the probability density of an atomic vector is a sum of delta functions. en triplets triplet invariants direct methods joint probability distributions It is shown that the formula for the positivity of the triplet invariant in P \bar{1} changes drastically if one uses a different statistical method by imposing acceptable and unbiased additional structural information. We obtain a much lower probability for the strength (almost ½) of the triplet formula than the classical one. text/html The triplet invariant revisited text 5 64 2008-08-14 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography research papers 560 563 http://scripts.iucr.org/cgi-bin/paper?sc0032 Using an unbiased and very general joint density of the atomic position vectors we are able to calculate different probabilities for the sign of the quartet given its second neighborhood. One already knows that additional chemical information alters the joint probability distribution (j.p.d.) of structure factors. That is, they can and will give different j.p.d.'s for the quartet invariant given its second neighborhood. In this paper we show that even without additional chemical information the j.p.d.'s of structure factors can be strongly different from the classical ones if we impose a general j.p.d. for the atomic vectors based on the fact that the real distribution of the atomic position vectors is a sum of δ functions. Copyright (c) 2008 International Union of Crystallography urn:issn:0108-7673 Brosius, J. 2008-08-14 doi:10.1107/S0108767308016978 International Union of Crystallography A quartet relation using an unbiased joint probability distribution of the atomic vectors is derived based on the observation that the distribution of the probability density of an atomic vector is a sum of delta functions. en quartets joint probability distributions quartet invariants direct methods Using an unbiased and very general joint density of the atomic position vectors we are able to calculate different probabilities for the sign of the quartet given its second neighborhood. One already knows that additional chemical information alters the joint probability distribution (j.p.d.) of structure factors. That is, they can and will give different j.p.d.'s for the quartet invariant given its second neighborhood. In this paper we show that even without additional chemical information the j.p.d.'s of structure factors can be strongly different from the classical ones if we impose a general j.p.d. for the atomic vectors based on the fact that the real distribution of the atomic position vectors is a sum of δ functions. text/html The quartet revisited text 5 64 2008-08-14 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography research papers 564 570 http://scripts.iucr.org/cgi-bin/paper?sc0033 We present a method that we call symbolic asymptotic development (SAD) to obtain joint probability distributions (j.p.d.'s) of phases of structure factors for general even densities of the atomic position vectors. The formula for the triplet and quartet invariant that we obtain in this way reduces to the well known classical formula for the case of a uniform density of the atomic position vectors. For the case of complete knowledge of the atomic vectors it reduces to first order to the exact probability density of the triplet (quartet) phase invariant. Applying this formula to the most general j.p.d. of the atomic vectors we obtain a statistical interpretation of Hauptman's algebraic B3,0 and B4,0 formulas. We also give a heuristic derivation of the SAD method. Another method that we shall discuss uses a method called linearization of the invariants that also produces formulas for the triplet phase invariant. This method is based on previous work and is also more laborious to calculate with than the SAD method. It can also give a statistical interpretation of the B3,0 formula. We show that the formula obtained for the triplet resembles the formula obtained with SAD. Copyright (c) 2008 International Union of Crystallography urn:issn:0108-7673 Brosius, J. 2008-08-14 doi:10.1107/S0108767308016966 International Union of Crystallography A method called symbolic asymptotic development (SAD) is proposed for calculating joint probability distributions of structure factors using a general joint probability distribution of the random vector variables. en triplet invariants quartet invariants symbolic asymptotic development joint probability distributions We present a method that we call symbolic asymptotic development (SAD) to obtain joint probability distributions (j.p.d.'s) of phases of structure factors for general even densities of the atomic position vectors. The formula for the triplet and quartet invariant that we obtain in this way reduces to the well known classical formula for the case of a uniform density of the atomic position vectors. For the case of complete knowledge of the atomic vectors it reduces to first order to the exact probability density of the triplet (quartet) phase invariant. Applying this formula to the most general j.p.d. of the atomic vectors we obtain a statistical interpretation of Hauptman's algebraic B3,0 and B4,0 formulas. We also give a heuristic derivation of the SAD method. Another method that we shall discuss uses a method called linearization of the invariants that also produces formulas for the triplet phase invariant. This method is based on previous work and is also more laborious to calculate with than the SAD method. It can also give a statistical interpretation of the B3,0 formula. We show that the formula obtained for the triplet resembles the formula obtained with SAD. text/html A statistical interpretation of the triplet and quartet invariant in P1. A theoretical discussion text 5 64 2008-08-14 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography research papers 571 586 http://scripts.iucr.org/cgi-bin/paper?au5074 A structure-analysis method using convergent-beam electron diffraction (CBED) developed by Tsuda et al. [Tsuda & Tanaka (1999), Acta Cryst. A55, 939–954; Tsuda, Ogata, Takagi, Hashimoto & Tanaka (2002), Acta Cryst. A58, 514–525] has been applied to the determination of the electrostatic potential and electron density of crystalline silicon. CBED patterns recorded at nine different incidences are simultaneously used to improve the accuracy of the refinement. The Debye–Waller factor and low-order structure factors of silicon have been successfully refined only using CBED data. The electrostatic potential and electron-density distribution have been reconstructed from the refined parameters. The latter clearly shows the bonding electrons between the nearest neighbor atoms. The obtained results are compared with the results of other CBED and recent X-ray diffraction experiments. The influence of the number of refined low-order structure factors on the electron density is discussed. The effect of the reduction of experimental data points on the accuracy of the refined parameters is also examined. Copyright (c) 2008 International Union of Crystallography urn:issn:0108-7673 Ogata, Y. Tsuda, K. Tanaka, M. 2008-08-14 doi:10.1107/S0108767308021338 International Union of Crystallography A structure-analysis method using convergent-beam electron diffraction has been applied to the determination of the electrostatic potential and electron density of crystalline silicon. en convergent-beam electron diffraction (CBED) electrostatic potential electron density silicon A structure-analysis method using convergent-beam electron diffraction (CBED) developed by Tsuda et al. [Tsuda & Tanaka (1999), Acta Cryst. A55, 939–954; Tsuda, Ogata, Takagi, Hashimoto & Tanaka (2002), Acta Cryst. A58, 514–525] has been applied to the determination of the electrostatic potential and electron density of crystalline silicon. CBED patterns recorded at nine different incidences are simultaneously used to improve the accuracy of the refinement. The Debye–Waller factor and low-order structure factors of silicon have been successfully refined only using CBED data. The electrostatic potential and electron-density distribution have been reconstructed from the refined parameters. The latter clearly shows the bonding electrons between the nearest neighbor atoms. The obtained results are compared with the results of other CBED and recent X-ray diffraction experiments. The influence of the number of refined low-order structure factors on the electron density is discussed. The effect of the reduction of experimental data points on the accuracy of the refined parameters is also examined. text/html Determination of the electrostatic potential and electron density of silicon using convergent-beam electron diffraction text 5 64 2008-08-14 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography research papers 587 597 http://scripts.iucr.org/cgi-bin/paper?dm5008 Expressions are derived for thermal diffuse scattering (TDS) using a formalism based on Born's S-matrix. It is shown that for monoatomic crystals the dynamical matrix containing the full information on lattice dynamics can be recovered from one-phonon TDS intensities. For any non-monoatomic crystal, part of the information is always lost in the kinematic approximation, but can in principle be recovered by measuring TDS in the dynamical scattering regime. In the long-wave limit the description here coincides with known results. Copyright (c) 2008 International Union of Crystallography urn:issn:0108-7673 Bosak, A. Chernyshov, D. 2008-08-14 doi:10.1107/S0108767308020060 International Union of Crystallography The S-matrix formalism allows recovery of the full lattice dynamics from one-phonon thermal diffuse scattering, but only for monoatomic crystals. en thermal diffuse scattering lattice dynamics elasticity Expressions are derived for thermal diffuse scattering (TDS) using a formalism based on Born's S-matrix. It is shown that for monoatomic crystals the dynamical matrix containing the full information on lattice dynamics can be recovered from one-phonon TDS intensities. For any non-monoatomic crystal, part of the information is always lost in the kinematic approximation, but can in principle be recovered by measuring TDS in the dynamical scattering regime. In the long-wave limit the description here coincides with known results. text/html On model-free reconstruction of lattice dynamics from thermal diffuse scattering text 5 64 2008-08-14 Copyright (c) 2008 International Union of Crystallography Acta Crystallographica Section A: Foundations of Crystallography short communications 598 600