Powder Pattern Indexing Research Articles

ISOTHERMAL SECTION OF STATE DIAGRAM Ce–Lі–Sn SYSTEM IN THE 30-100 AT. % Sn CONCENTRATION RANGE AT 400ºС

The isothermal cross-section of the phase diagram of the system Ce–Li–Sn based on X-ray phase and local X-ray spectral analyses was constructed at the temperature 400ºС in the 30–100 аt. % Sn concentration range. The alloys were prepared by arc melting of stoichiometric amounts of the constituent elements, annealed at 400ºС for 480 hours and quenched in cold water.
 Flame photometer Flapho-4 was used for the experimental determination of lithium amount. To confirm the phase composition of some system samples energy-dispersive X-ray spectroscopy method (EDX) was used (scanning electron microscope REMMA-102-02).
 The calculations and indexing of diffraction powder patterns (diffractometers DRON-2,0 (Fе Кα-radiation), STOE STADI P (Cu Kα1-radiation) and URD-6 (Cu Kα-radiation)) have been conducted using LATCON and POWDER CELL-2.3 software. Calculations to refine the structure of samples have been carried out by with WinCSD and FullProf 98 programs.
 Two new ternary compounds have been synthesized for the first time: Ce5Li6Sn9 (structure type Eu5Li6Sn9, Pearson symbol oS80, space group Cmcm, a = 0,4852(1), b = 2,8961(4), c = 1,5009(2) nm) and ~Ce4LiSn4 (unknown structure). The existence of the ternary compounds CeLiSn2 (structural type CeNiSi2), Ce5Li2Sn7 (own structure type) and fifteen binary phases was confirmed. The existence of the limited solid solution of inclusion Ce5LixSn3 (0 ≤ х ≤ 0,5) based on the Сe5Sn3 binary compound has been detected. The ultimate composition of this solid solution is Ce5Li0,5Sn3 (structure type Hf5CuSn3, Pearson symbol hP18, space group P63/mcm, a = 0,88206(2), c =0,67802(1) nm).
 The existence of the binary and ternary phases CeSn3, Ce5Sn3, Ce5Sn4, Ce11Sn10, Ce3Sn5, Ce3Sn7, Ce2Sn5, Ce3Sn, Li17Sn4, Li7Sn2, Li13Sn5, Li5Sn2, Li7Sn3, LiSn, Li2Sn5, CeLiSn2, Ce5Li2Sn7 was confirmed.
 The characteristics of phase interactions in the Ce–Li–Sn as well as related ternary systems with rare earth metals, Lithium, Silicon, Germanium and Tin have been analysed.

Indexing of multi-particle diffraction data in a high-pressure single-crystal diffraction experiment

High-pressure single-crystal diffraction experiments often suffer from the crushing of single crystals due to the application of high pressure. Consequently, only diffraction data resulting from several particles in random orientations is available, which cannot be routinely indexed by commonly used methods designed for single-crystal data. A protocol is proposed to index such diffraction data. The techniques of powder pattern indexing are first used to propose the possible lattice parameters, and then a genetic algorithm is applied to determine the orientation of the reciprocal lattice for each of the particles. This protocol has been verified experimentally.

How to deal with multiple solutions in powder pattern indexing

How to deal with multiple solutions in powder pattern indexing

Analysis of multiple solutions in powder pattern indexing: the common reciprocal metric tensor approach

Powder pattern indexing routines frequently yield multiple solutions,i.e.different reciprocal lattices and unit cells. Here, a method is suggested that reveals whether or not there are numerical and geometric relationships between the solutions. It is based on the detection of a reciprocal vector triplet that is common to two or more proposed reciprocal lattices. Hence, the method can be termed a common reciprocal metric tensor approach. If no such common tensor exists, the different reciprocal lattices are unrelated, but if one exists the lattices are either in a sublattice/superlattice or in a coincidence-site lattice relationship, depending on the character of the respective orientation matrix. Furthermore, the approach can also be used to generate, from a given indexing solution, further valid indexing solutions that could also be produced by indexing routines.

Problems in polarized light microscopy observation of birefringence of calcium pyrophosphate dihydrate crystals

To reconsider the problems arising from the use of the phase plate as a test plate inserted into a polarized light microscope system for the analysis of triclinic calcium pyrophosphate dihydrate (t-CPPD) crystals, or Ca 2P 2O 7·2H 2O in the synovial fluid of arthritis patients, we made the polarized light microscopy observations using a phase plate with a retardation of 530 nm for the synthesized t-CPPD crystals well-characterized by X-ray powder pattern indexing and single crystal X-ray diffraction measurements. The microscopy observations were made of crystals of different sizes, thicknesses and shapes. The retardation was assessed using the interference color chart at four extinction and diagonal positions both with and without the test plate. The addition and subtraction states produced by superimposing the retardations of two anisotropic materials, that is, the t-CPPD crystal and the 530 nm phase plate, were deduced from the interference color change by inserting the test plate at four diagonal positions. When the color change of a crystal at a diagonal position resulting from 90-degree rotation exhibits no clear birefringence, the interference color chart was shown to be useless. We suggested the use of a compensator whose retardation can be changed to obtain an accurate value for the retardation of the crystal.

X-ray diffraction study of the zeolite natrolite: TPH2O phase diagram and phase transitions during dehydration/rehydration

The response of the natrolite crystal structure (Na 16 Al 16 Si 24 O 80 · 16H 2 O) to dehydration/rehydration was evaluated as a function of P H 2 O with X-ray powder diffraction (XRD) data measured in situ from 23 to 400 °C. Dehydrated natrolite at a low- P H 2 O appears as a mixture of two anhydrous phases, the previously described α1-metanatrolite ( F 112, a = 16.177(1) A, b = 16.943(1) A, c = 6.4370(4) A, γ = 89.685(2)°, V = 1764.3(2) A 3 ), and a new phase, α2-metanatrolite ( Fdd 2, a = 17.576(1) A, b = 18.163(1) A, c = 6.3704(4) A, V = 2033.7(2) A 3 ). The structure of α2-metanatrolite was solved using a combination of powder pattern indexing, distance least-squares modeling, and Rietveld refinement. The structures of α1- and α2-metanatrolite provide an excellent context for describing the behavior of natrolite phase transitions and the effects of P H 2 O . Both metanatrolite phases occur simultaneously during dehydration under low- P H 2 O conditions and they re-adsorb H 2 O molecules at different rates during rehydration. α2-metanatrolite adsorbs H 2 O molecules and reconverts to natrolite more rapidly than α1-metanatrolite because of the larger channel size in the α2-metanatrolite framework. The path dependence of natrolite phase transitions and the dependence on P H 2 O illustrate the sensitivity of the natrolite framework and its extraframework Na cations to hydration state, with α1- and α2-metanatrolite forming under low- P H 2 O conditions and only α1-metanatrolite forming under high- P H 2 O conditions. The flexibility of the natrolite framework makes the natrolite- to- α1-/α2-metanatrolite-to-natrolite reaction reversible, although strain-induced mosaicity resulting from the phase transition is not reversible.

A system of metrically invariant relations between the moduli squares of reciprocal-lattice vectors in one-, two- and three-dimensional space

Given the background of trial-and-error methods employed in recent automatic powder pattern indexing, an alternative route is suggested based on a generalization of the original Runge–de Wolff approach. For this purpose, a system of five metrically invariant relations between the squared moduli (Qvalues) of reciprocal-lattice vectors is developed that encompasses the earlier special relations. The five invariant relations correspond to a line, a zone, a bizone, a cone and a pencil configuration of reciprocal-lattice vectors. In particular, the zone configuration relates four vectors being arbitrarily distributed in a plane and as such allows one to identify among a set of measuredQvalues all quadruples that define reciprocal-lattice planes intersecting in space.

Advances in powder diffraction pattern indexing:N-TREOR09

Powder pattern indexing can still be a challenge, despite the great recent advances in theoretical approaches, computer speed and experimental devices. More plausible unit cells, belonging to different crystal systems, are frequently found by the indexing programs, and recognition of the correct one may not be trivial. The task is, however, of extreme importance: in case of failure a lot of effort and computing time may be wasted. The classical figures of merit for estimating the unit-cell reliability {i.e.M20[de Wolff (1968).J. Appl. Cryst.1, 108–113] andFN[Smith & Snyder (1979).J. Appl. Cryst.12, 60–65]} sometimes fail. For this reason, a new figure of merit has been introduced inN-TREOR09, the updated version of the indexing packageN-TREOR[Altomare, Giacovazzo, Guagliardi, Moliterni, Rizzi & Werner (2000).J. Appl. Cryst.33, 1180–1186], combining the information supplied byM20with additional parameters such as the number of unindexed lines, the degree of overlap in the pattern (the so-called number of statistically independent observations), the symmetry deriving from the automatic evaluation of the extinction group, and the agreement between the calculated and observed profiles. The use of the new parameters requires a dramatic modification of the procedures used worldwide: in the approach presented here, extinction symbol and unit-cell determination are simultaneously estimated.N-TREOR09benefits also from an improved indexing procedure in the triclinic system and has been integrated intoEXPO2009, the updated version ofEXPO2004[Altomare, Caliandro, Camalli, Cuocci, Giacovazzo, Moliterni & Rizzi (2004).J. Appl. Cryst.37, 1025–1028]. The application of the new procedure to a large set of test structures is described.

FIDDLE: powder pattern indexing and structure solution hand in hand

<i>FIDDLE</i>: powder pattern indexing and structure solution hand in hand

Powder pattern indexing and the dichotomy algorithm

Powder pattern indexing and the dichotomy algorithm

Powder pattern indexing and the dichotomy algorithm

Powder pattern indexing and the dichotomy algorithm

The malachite-rosasite group: crystal structures of glaukosphaerite and pokrovskite

The crystal structures of glaukosphaerite (Cu,Ni)2(CO3)(OH)2 and pokrovskite Mg2(CO3)(OH)2, two carbonates belonging to the malachite-rosasite group, have been determined from powder diffraction data, and refined up to wRp = 3.94% and 1.63% respectively. Both minerals are isostructural with rosasite (Cu,Zn)2(CO3)(OH)2, with P 21/ a space group and cell parameters a =12.0613(4) A, b = 9.3653(4), c = 3.1361(1), β = 98.085(5)° for glaukosphaerite and a = 12.2396(4), b = 9.3506(4), c = 3.1578(1), β = 96.445(5)° for pokrovskite. Their structures are built up by ribbons of edge-sharing octahedra running along the c -axis; the ribbons are linked together through corner-sharing giving rise to corrugated layers parallel to (100) that are interconnected through carbonate groups. The same layers and carbonate groups, arranged in a different orientation with respect to the symmetry operators, build up the structure of malachite: the relationships between the rosasite-like and the malachite-like structural models are discussed. The octahedral distortion of the two independent Me sites in the malachite-rosasite known structures is evaluated and attributed to the Jahn-Teller effect of Cu2+. According to the available chemical data for the examined material, a partial occupancy in Mg sites of pokrovskite is maintained, with a coupled partial substitution of hydroxyls by water molecules. On the basis of the structural results, the difficulties and ambiguities in the powder pattern indexing by preceeding authors are discussed and explained, and reliable guesses for the arrangements of kolwezite and nullaginite are drawn.

Complex thermal evolution of V 2O 5 and MoO 3 cell parameters in the range [formula omitted][formula omitted

Cell parameter evolution versus temperature has been established for both V 2O 5 and MoO 3, from 15 to 953 and 893 K, respectively. Both oxides preserve their orthorhombic crystal system and their respective space groups Pmmn and Pnma in the whole temperature range. Unusual evolution of the unit cells have been detected in these oxides and proposals are made to follow them based on detailed structural analysis. Three temperature domains have been evidenced for V 2O 5 marked by breaks at RT and 800 K; for MoO 3, only two domains are defined below and above 400 K. For each domain the quality of cell parameters and X-ray powder pattern indexing, have been checked by mathematical functions to buttress their coherent evolution within each temperature domain.

Properties of a genetic algorithm extended by a random self-learning operator and asymmetric mutations: A convergence study for a task of powder-pattern indexing

Genetic algorithms represent a powerful global-optimisation tool applicable in solving tasks of high complexity in science, technology, medicine, communication, etc. The usual genetic-algorithm calculation scheme is extended here by introduction of a quadratic self-learning operator, which performs a partial local search for randomly selected representatives of the population. This operator is aimed as a minor deterministic contribution to the (stochastic) genetic search. The population representing the trial solutions is split into two equal subpopulations allowed to exhibit different mutation rates (so called asymmetric mutation). The convergence is studied in detail exploiting a crystallographic-test example of indexing of powder diffraction data of orthorhombic lithium copper oxide, varying such parameters as mutation rates and the learning rate. It is shown through the averaged (over the subpopulation) fitness behaviour, how the genetic diversity in the population depends on the mutation rate of the given subpopulation. Conditions and algorithm parameter values favourable for convergence in the framework of proposed approach are discussed using the results for the mentioned example. Further data are studied with a somewhat modified algorithm using periodically varying mutation rates and a problem-specific operator. The chance of finding the global optimum and the convergence speed are observed to be strongly influenced by the effective mutation level and on the self-learning level. The optimal values of these two parameters are about 6 and 5%, respectively. The periodic changes of mutation rate are found to improve the explorative abilities of the algorithm. The results of the study confirm that the applied methodology leads to improvement of the classical genetic algorithm and, therefore, it is expected to be helpful in constructing of algorithms permitting to solve similar tasks of higher complexity.

History of the dichotomy method for powder pattern indexing

A short history of the developments of the successive dichotomy method for powder pattern indexing is presented. In the first computer powder indexing programs (P1 and P2), only high lattice symmetries, down to orthorhombic, were considered [Louër and Louër, J. Appl. Crystallogr. 5, 271–275 (1972)]. Later on, an extension to the monoclinic symmetry was reported in DICVOL, including a partition of the volume space to first search solutions with smaller unit cell volumes [Louër and Vargas, J. Appl. Crystallogr. 15, 542–545 (1982)]. However, CPU times were slow in some monoclinic examples. A thorough mathematical analysis resulted in a significant optimization of the CPU times [Boultif and Louër, J. Appl. Crystallogr. 24, 987–993 (1991)]. Simultaneously, the method is extended to triclinic lattices. The stages of development of the various versions of the DICVOL program are described, with a particular emphasis on DICVOL91 (Boultif and Louër, 1991) and DICVOL04 [Boultif and Louër, J. Appl. Crystallogr. 37, 724–731 (2004)]. This article is written to testify to and emphasize the major role played by Daniel Louër, who introduced the successive dichotomy method and continued to its evolution and optimization over almost 40 years.

Ambiguities in Powder Indexing: Conjunction of a Ternary and Binary Lattice Metric Singularity in the Cubic System.

A lattice metric singularity occurs when unit cells defining two (or more) lattices yield the identical set of unique calculated d-spacings. The existence of such singularities, therefore, has a practical and theoretical impact on the indexing of powder patterns. For example, in experimental practice an indexing program may find only the lower symmetry member of a singularity. Obviously, it is important to recognize such cases and know how to proceed. Recently, we described: a binary singularity involving a monoclinic and a rhombohedral lattice in a subcell-supercell relationship anda second type of singularity—a ternary singularity–in which two of the three lattices are in a derivative composite relationship. In this work, we describe a ternary lattice metric singularity involving a cubic P, a tetragonal P, and an orthorhombic C lattice. Furthermore, there is a binary singularity, involving a hexagonal P and orthorhombic P lattice, which is characterized by a set of unique d-spacings very close to that of the ternary singularity. The existence of such singularities is more common than once thought and requires a paradigm shift in experimental practice. In addition singularities provide opportunities in material design as they point to highly specialized lattices that may be associated with unusual physical properties.

Powder pattern indexing with the dichotomy method

The efficiency of the successive dichotomy method for powder diffraction pattern indexing [Louër &amp; Louër (1972).J. Appl. Cryst.5, 271–275] has been proved over more than 30 years of usage. Features implemented in the new version of the computer programDICVOL04 include (i) a tolerance to the presence of impurity (or inaccurately measured) diffraction lines, (ii) a refinement of the `zero-point' position, (iii) a reviewing of all input lines from the solution found from, generally, the first 20 lines, (iv) a cell analysis, based on the concept of the reduced cell, to identify equivalent monoclinic and triclinic solutions, and (v) an optional analysis of input powder data to detect the presence of a significant `zero-point' offset. New search strategies have also been introduced,e.g.each crystal system is scanned separately, within the input volume limits, to limit the risk of missing a solution characterized by a metric lattice singularity. The default values in the input file have been extended to 25 Å for the linear parameters and 2500 Å3for the cell volume. The search is carried out exhaustively within the input parameter limits and the absolute error on peak position measurements. Many tests with data from the literature and from powder data of pharmaceutical materials, collected with the capillary technique and laboratory monochromatic X-rays, have been performed with a high success rate, covering all crystal symmetries from cubic to triclinic. Some examples reported as `difficult' cases are also discussed. Additionally, a few recommendations for the correct practice of powder pattern indexing are reported.

Powder pattern indexing using the weighted crosscorrelation and genetic algorithms.

X-ray diffraction is a powerful technique for investigating the structure of crystals and crystalline powders. Unfortunately, for powders, the first step in the structure elucidation process, retrieving the unit cell parameters (indexing), is still very critical. In the present article, an improved approach to powder pattern indexing is presented. The proposed method matches peak positions from experimental X-ray powder patterns with peak positions from trial cells using a recently published method for pattern comparison (weighted crosscorrelation). Trial cells are optimized with Genetic Algorithms. Patterns are not pretreated to remove any existing zero point shift, as this is determined during optimization. Another improvement is the peak assignment procedure. This assignment is needed for determining the similarity between lines from trial cells and experiment. It no longer allows calculated peaks to be assigned twice to different experimental peaks, which is beneficial for the indexing process. The procedure proves to be robust with respect to false peaks and accidental or systematic absensences of reflections, and is successfully applied to powder patterns originating from orthorhombic, monoclinic, and triclinic compounds measured with synchrotron as well as with conventional laboratory X-ray diffractometers.

Ambiguities in powder pattern indexing: A ternary lattice metric singularity

A lattice metric singularity occurs when unit cells defining two (or more) lattices yield the identical set of unique calculated d-spacings. The existence of such singularities, therefore, has a practical impact on the indexing of powder patterns. Lattice metric singularities often involve lattices that are in a derivative relationship one to another. A variety of types of singularities are possible depending on the number of different lattices involved (i.e., binary, ternary, quaternary), on the nature of the derivative lattice relationship (i.e., subcell/supercell, composite), on the Bravais type of each of the lattices, and on the the volume ratio(s) of primitive cells defining the lattices. In the laboratory, an encounter with a singularity can lead one into a trap; viz., the investigator using an indexing program, or by other means, may determine only one of the lattices with a high figure of merit. When this happens, it is critical to recognize that there exists more than one indexing solution. In a previous work, a binary singularity was described involving a monoclinic and a rhombohedral lattice. In the present work, we describe a second type of singularity—a ternary singularity—in which the two of the three lattices are in a derivative composite relationship.

Powder and single-crystal X-ray diffraction study of the structure of

From powder pattern indexing it has been demonstrated that [Y(H(2)O)](2)(C(2)O(4))(CO(3))(2), yttrium oxalate carbonate, crystallizes with orthorhombic symmetry, space group C222(1), a = 7. 8177 (7), b = 14.943 (1), c = 9.4845 (7) A, V = 1108.0 (1) A(3), Z = 4. This unit cell displays a doubling of the c parameter, arising from weak diffraction lines observed in the powder diffraction pattern, with respect to results reported in the literature. The crystal structure has been solved ab initio using direct methods from powder data and has been confirmed by additional single-crystal data collected with a CCD area detector. The overall crystal structure is similar for both unit cells, except that an alternation of the carbonate groups in the direction parallel to the screw axis is displayed in the larger cell, while with the suggested half unit cell (space group C2mm) the carbonate groups would show only one orientation. The unit-cell determination strategy from single-crystal diffraction, collected with Nonius CAD-4 and Nonius Kappa CCD diffractometers, is discussed with respect to the results extracted from the powder diffraction pattern. The study demonstrates the power and usefulness of the full trace of a powder pattern for the detection of subtle structure details.