The derivation of trial structures. I. Analytical methods for direct phase determination

2010 ◽  
Author(s):  
Jenny Pickworth Glusker ◽  
Kenneth N. Trueblood
Keyword(s):  
Crystal Structure ◽  
Structure Determination ◽  
Direct Methods ◽  
Analytical Techniques ◽  
Patterson Function ◽  
Structure Factors ◽  
Term Trial ◽  
Trial Structure ◽  
Electron Density Map ◽  
True Structure

As indicated at the start of Chapter 4, after the diffraction pattern has been recorded and measured, the next stage in a crystal structure determination is solving the structure—that is, finding a suitable “trial structure” that contains approximate positions for most of the atoms in the unit cell of known dimensions and space group. The term “trial structure” implies that the structure that has been found is only an approximation to the correct or “true” structure, while “suitable” implies that the trial structure is close enough to the true structure that it can be smoothly refined to give a good fit to the experimental data. Methods for finding suitable trial structures form the subject of this chapter and the next. In the early days of structure determination, trial and error methods were, of necessity, almost the only available way of solving structures. Structure factors for the suggested “trial structure” were calculated and compared with those that had been observed. When more productive methods for obtaining trial structures—the “Patterson function” and “direct methods”—were introduced, the manner of solving a crystal structure changed dramatically for the better. We begin with a discussion of so-called “direct methods.” These are analytical techniques for deriving an approximate set of phases from which a first approximation to the electron-density map can be calculated. Interpretation of this map may then give a suitable trial structure. Previous to direct methods, all phases were calculated (as described in Chapter 5) from a proposed trial structure. The search for other methods that did not require a trial structure led to these phaseprobability methods, that is, direct methods. A direct solution to the phase problem by algebraic methods began in the 1920s (Ott, 1927; Banerjee, 1933; Avrami, 1938) and progressed with work on inequalities by David Harker and John Kasper (Harker and Kasper, 1948). The latter authors used inequality relationships put forward by Augustin Louis Cauchy and Karl Hermann Amandus Schwarz that led to relations between the magnitudes of some structure factors.

A tangent formula derived from Patterson-function arguments. VII. Solution of inorganic structures from powder data with accidental overlap

2000 ◽  
Vol 33 (5) ◽  
pp. 1208-1211 ◽  
Author(s):  
J. Rius ◽  
X. Torrelles ◽  
C. Miravitlles ◽  
L. E. Ochando ◽  
M. M. Reventós ◽  
...  
Keyword(s):  
Crystal Structure ◽  
Powder Diffraction ◽  
Inorganic Compounds ◽  
Solution Process ◽  
Direct Methods ◽  
Powder Diffraction Data ◽  
Patterson Function ◽  
Structure Factors ◽  
Structure Solution ◽  
Overlapped Peaks

Accidental overlap constitutes one of the principal limitations for the solution of crystal structures from powder diffraction data, since it reduces the number of available intensities for direct-methods application. In this work, the field of application of the direct-methods sum function is extended to cope with powder patterns with relatively large amounts of accidental overlap. This is achieved by refining not only the phases of the structure factors but also the estimated intensities of the severely overlapped peaks during the structure solution process. This procedure has been specifically devised for inorganic compounds with uncertain cell contents and with probable severe atomic disorder, a situation often found when studying complex minerals with limited crystallinity. It has been successfully applied to the solution of the previously unknown crystal structure of the mineral tinticite. Finally, an estimation of the smallest ratio (number of observations to number of variables) for the procedure to be successful is given.

Crystal structure determination by neutron powder diffraction using structural and packing constraints

1991 ◽  
Vol 24 (6) ◽  
pp. 1005-1008 ◽  
Author(s):  
P. G. Byrom ◽  
B. W. Lucas
Keyword(s):  
Crystal Structure ◽  
Powder Diffraction ◽  
Structure Determination ◽  
Direct Methods ◽  
Neutron Powder Diffraction ◽  
Crystal Structure Determination ◽  
Diffraction Profile ◽  
Peak Separation ◽  
Structure Factors ◽  
Structure Solution

In the past, crystal structure determination of solids consisting of molecules (or atom groups) whose geometry and size are known approximately has often been attempted using neutron powder diffraction profile refinement techniques, but without inclusion of this information. A method of structure solution has therefore been developed to include it. The proposed method does not require a set of structure factors and thus avoids the problems encountered in separating peaks in a powder diffraction scan. A successful test was conducted with a previously determined (yet treated as unknown) crystal structure, where direct methods had failed to solve the structure due to incorrect peak separation. Two computer programs, MODEL and PARAM, that implement the method are described.

The crystal structure of sarmientite, Fe23+ (AsO4)(SO4)(OH)·5H2O, solved ab initio from laboratory powder diffraction data

2014 ◽  
Vol 78 (2) ◽  
pp. 347-360 ◽  
Author(s):  
F. Colombo ◽  
J. Rius ◽  
O. Vallcorba ◽  
E. V. Pannunzio Miner
Keyword(s):  
Crystal Structure ◽  
Powder Diffraction ◽  
Diffraction Data ◽  
Rietveld Method ◽  
Direct Methods ◽  
Hydroxyl Group ◽  
Monoclinic Space Group ◽  
Powder Diffraction Data ◽  
Patterson Function ◽  
Unit Cell Dimensions

AbstractThe crystal structure of sarmientite, Fe23+ (AsO4)(SO4)(OH)·5H2O, from the type locality (Santa Elena mine, San Juan Province, Argentina), was solved and refined from in-house powder diffraction data (CuKα1,2 radiation). It is monoclinic, space group P21/n, with unit-cell dimensions a = 6.5298(1), b = 18.5228(4), c = 9.6344(3) Å, β = 97.444(2)º, V = 1155.5(5) Å3, and Z = 4. The structure model was derived from cluster-based Patterson-function direct methods and refined by means of the Rietveld method to Rwp = 0.0733 (X2 = 2.20). The structure consists of pairs of octahedral-tetrahedral (Fe−As) chains at (y,z) = (0,0) and (½,½), running along a. There are two symmetry-independent octahedral Fe sites. The Fe1 octahedra share two corners with the neighbouring arsenate groups. Both individual chains are related by a symmetry centre and joined by two symmetry-related Fe2 octahedra. Each Fe2 octahedron shares three corners with double-chain polyhedra (O3, O4 with arsenate groups; the O8 hydroxyl group with the Fe1 octahedron) and one corner (O11) with the monodentate sulfate group. The coordination of the Fe2 octahedron is completed by two H2O molecules (O9 and O10). There is also a complex network of H bonds that connects polyhedra within and among chains. Raman and infrared spectra show that (SO4)2− tetrahedra are strongly distorted.

Crystal structure determination by convergent-beam electron diffraction

1989 ◽  
Vol 47 ◽  
pp. 476-477
Author(s):  
R. Vincent ◽  
D. J. Exelby
Keyword(s):  
Crystal Structure ◽  
Electron Diffraction ◽  
Structure Determination ◽  
Convergent Beam Electron Diffraction ◽  
Crystal Structure Determination ◽  
Large Set ◽  
Beam Electron ◽  
X Ray ◽  
Structure Factors ◽  
Convergent Beam

In recent years, significant progress has been made towards a solution for the general problem of crystal structure determination by convergent beam electron diffraction (CBED). Even if we consider only perfectly ordered, periodic crystals defined by one of the conventional space groups, diffraction methods based on a focussed sub-micron beam of electrons are applicable to several related sets of structural problems that are not accessible to conventional X-ray or neutron diffraction techniques. We assume here that the space group either is known or has been determined from CBED patterns and that phases and amplitudes for some subset of the structure factors are required. Two limiting cases have been explored in some detail. For crystals where the atomic parameters and Debye-Waller factors are known accurately from high quality X-ray data, information on the charge redistribution for bonding electrons is available from precise measurements of the low order structure factors. Following the original research of Kambe, some recent work has demonstrated that accurate structure amplitudes and three-beam phase invariants can be extracted from the dynamical intensity distribution in CBED reflections. In principle, this approach is completely general but considerable labour would be required to extract sufficient data to solve the structure of an unknown crystal, whereas a large set of kinematic intensities is acquired from a single X-ray pattern.

Refinement of the trial Structure

2010 ◽  
Author(s):  
Jenny Pickworth Glusker ◽  
Kenneth N. Trueblood
Keyword(s):  
Experimental Data ◽  
Least Squares ◽  
Likelihood Methods ◽  
Fourier Methods ◽  
Maximum Likelihood Methods ◽  
Density Map ◽  
Trial Structure ◽  
Phase Angles ◽  
Electron Density Map ◽  
True Structure

When approximate positions have been determined for most, if not all, of the atoms, it is time to begin the refinement of the structure. In this process the atomic parameters are varied systematically so as to give the best possible agreement of the observed structure factor amplitudes (the experimental data) with those calculated for the proposed trial structure. Common refinement techniques involve Fourier syntheses and processes involving least-squares or maximum likelihood methods. Although they have been shown formally to be nearly equivalent—differing chiefly in the weighting attached to the experimental observations—they differ considerably in manipulative details; we shall discuss them separately here. Many successive refinement cycles are usually needed before a structure converges to the stage at which the shifts from cycle to cycle in the parameters being refined are negligible with respect to their estimated errors. When least-squares refinement is used, the equations are, as pointed out below, nonlinear in the parameters being refined, which means that the shifts calculated for these parameters are only approximate, as long as the structure is significantly different from the “correct” one. With Fourier refinement methods, the adjustments in the parameters are at best only approximate anyway; final parameter adjustments are now almost always made by least squares, at least for structures not involving macromolecules. As indicated earlier (Chapters 8 and 9, especially Figure 9.8 and the accompanying discussion), Fourier methods are commonly used to locate a portion of the structure after some of the atoms have been found—that is, after at least a partial trial structure has been identified. Initially, only one or a few atoms may have been found, or maybe an appreciable fraction of the structure is now known. Once approximate positions for at least some of the atoms in the structure are known, the phase angles can be calculated. Then an approximate electron-density map calculated with observed structure amplitudes and computed phase angles will contain a blend of the true structure (from the structure amplitudes) with the trial structure (from the calculated phases).

The Crystal Structure of cis-Inositol Monohydrate—Un Objet Retrouvé

1996 ◽  
Vol 49 (3) ◽  
pp. 413 ◽  
Author(s):  
HC Freeman ◽  
DA Langs ◽  
CE Nockolds ◽  
YL Oh
Keyword(s):  
Crystal Structure ◽  
Least Squares ◽  
Direct Methods ◽  
Monoclinic Space Group ◽  
High Symmetry ◽  
Asymmetric Unit ◽  
Fourier Methods ◽  
Full Matrix ◽  
Structure Factors ◽  
Independent Molecules

cis-Inositol monohydrate, C6H12O6.H2O, crystallizes in the monoclinic space group P 21/n [a 9.900(8), b 9.296(8), c 17.795(15) Ǻ, β 90.5(1)°, Z 8]. The normalized structure factors Eh have an atypical statistical distribution, and attempts to solve the structure by direct methods (triplet relationships) were unsuccessful. The structure was ultimately solved by Patterson and Fourier methods, and was refined by full-matrix least squares [Rw = 0.047 for 1665 independent reflections ≥2σ(Imin)]. The cis-inositol molecules have approximately trigonal symmetry, as expected. The difficulties encountered during the structure analysis are explained by the presence of two nearly identical molecules of high symmetry in the asymmetric unit. The independent molecules are related by translational pseudosymmetry, and their orientations are such that all the C-C and C-O bonds in the structure are approximately parallel to a small number of directions.

Traditional direct phasing procedures

2013 ◽  
Author(s):  
Carmelo Giacovazzo
Keyword(s):  
Crystal Structure ◽  
Least Squares ◽  
Fourier Transforms ◽  
Direct Methods ◽  
Real Space ◽  
Reciprocal Space ◽  
Interatomic Distances ◽  
Patterson Function ◽  
Structure Solution ◽  
Phase Extension

Which phasing methods can be included in the category direct methods, and which require a different appellation? Originally, direct phasing was associated with those approaches which were able to derive phases directly from the diffraction moduli, without passing through deconvolution of the Patterson function. Since a Patterson map provides interatomic distances, and therefore lies in ‘direct space’, direct methods were also referred to as reciprocal space methods, and Patterson techniques as real-space methods. Historically, direct methods use 3-,4-, . . . , n-phase invariants and 1-2-, . . . phase seminvariants via the tangent formula or its modified algorithms. Since the 1950s, about a half a century of scientific effort has fallen under the above definition. Such approaches are classified here as traditional direct methods. Today, the situation is more ambiguous, because: (i) modern direct methods programs involve steps operating both in reciprocal space and in direct space, the latter mainly devoted to phase extension and refinement (see Chapter 8); (ii) in the past decade, new phasing methods for crystal structure solution (see Chapter 9) have been developed, based on the properties of Fourier transforms, which again work both in direct and in reciprocal space. Should they therefore be considered to be outside the direct methods category or not? Our choice is as follows. Direct methods are all of the approaches which allow us to derive phases from diffraction amplitudes, without passing through a Patterson function deconvolution. Thus, we also include in this category, charge flipping and VLD (vive la difference), here classified as non-traditional direct methods; their description is postponed to Chapter 9. In accordance with the above assumptions, in this chapter we will shortly illustrate traditional direct phasing procedures, with particular reference to those which are current and in regular use today: mainly the tangent procedures (see Section 6.2) and the cosine least squares technique, which is the basic tool of the shake and Bake method (see Section 6.4).

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