Refinement of a crystal structure

The Mossbauer spectrum of Fe3(CO)i2 shows two distinct iron sites, with a relative ratio of 2 1. In fact, this result led to the refinement of a crystal structure that had reported three equivalent iron sites. Although this happened in the early days of Mossbauer spectroscopy, that is, the 1960s, it... 

It is important for the reader to understand that in a least-squares refinement of a crystal structure it is the shifts in parameters that are calculated in order to improve the structure, not the parameters themselves. The preliminary parameters that are shifted to more appropriate values come from the trial structures (see Chapters 8 and 9). 

Correlation between parameters A correlation is a measure of the extent to which two mathematical variables are dependent on each other. In the least-squares refinement of a crystal structure, parameters related by symmetry are completely correlated, and temperature factors and occupancy factors are often highly correlated. 

Refinement of a crystal structure A process of improving the parameters of an approximate (trial) structure until the best fit of calculated structure factor amplitudes to those observed is obtained. The process usually requires many successive stages. 

Figure 8.71 Computer screen of the automated results of the refinement of a crystal structure by the XtaLAB mini . (Used by permission of RIgaku Corporation, www.rlgaku.com.)...

The information obtained from X-ray measurements on the arrangement of the water molecules naturally depends very much on the resolution and state of refinement of the crystal structure investigated. For detailed information on the organization of water molecules in the protein hydration shell at the surface and on the bulk water in the crystals a 1,2 to 1,8 A resolution range is necessary 153>. 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

According to the chemical analysis and coordination distances, the Rietveld refinement of the crystal structure at room temperature revealed 1.2 Co2+ atoms per unit cell at the Col and Co2 sites, whereas the 1.4 Ag+ cations are spread over the Co3 site, from now on referred to as Ag5 for clarity, and two new sites, Ag2 and Ag3, located near Co2 in the 10-membered ring (Fig. 3). In addition, for this catalyst the presence of Ag° clusters outside the zeolite structure was recognized by the detection of a strong reflection at about 40° 28. In agreement with the lower Ag content, in Ag2.7Co2.8AF the Ag3 site... 

Grundy H. D. and Ito J. (1974). The refinement of the crystal structure of a synthetic non-stoichiometric Sr-feldspar. Amer. Mineral, 68 1319-1326. 

Pellegrini, M., Gronbech-Jensen, N., Kelly J. A., Pfluegl, G., and Yeates, T. O. (1997) Highly constrained multiple-copy refinement of protein crystal structures. Proteins 29, 426-432. 

The object of a crystal-structure determination is to ascertain the position of all of the atoms in the unit cell, or translational building block, of a presumed completely ordered three-dimensional structure. In some cases, additional quantities of physical interest, e.g.. the amplitudes of thermal motion, may also be derived from the experiment. The processes involved in such crystal-structure determinations may he divided conveniently into (I) collection of the data. (2) solution of the phase relations among the scattered x-rays (phase problem)—determination of a correct trial structure, and (3) refinement of this structure. 

Ballirano, P. and Maras, A. (2002) Refinement of the crystal structure of arsenolite, As203. Zeitschrift fur Kristallo-graphie New Crystal Structures, 217(2), 177-78. 

Further refinement of the crystal structure consisting of double helices is difficult, because the x-ray photograph is not well-defined, and the possibility of a disordered structure must be considered, e.g., right and left-hand helices, and up and down chains. Although there are some unexplained feature of the double helical model, such as the mode of rapid double helix formation during crystallization, the author and his coworkers believe the result to be essentially correct. 

There has been no controversy about the structure of fluorene (31) but its true conformation was in doubt for a number of years. From an early X-ray analysis, Iball (1936a) concluded that the fluorene molecule had a folded conformation and, in a review, Cook and Iball (1936) discussed further evidence for a non-planar conformation, provided by optical activity studies of unsymmetrically substituted fluorene derivatives. Later stereochemical studies (Weisburger et al., 1950) suggested that fluorene had, in fact, a planar conformation. A reinvestigation of the crystal structure by Burns and Iball (1954, 1955) and, independently, by Brown and Bortner (1954) showed that the early X-ray work was in error and confirmed the planar conformation. The refinement of the crystal structure (Burns and Iball, 1954, 1955), by two-dimensional Fourier and least-squares methods, reveals that the maximum deviation of the carbon atoms from the mean molecular plane is 0-030 A, the r.m.s. deviation being 0-017 A. This deviation, 0-017 A, is taken by Burns and Iball to be a measure of the accuracy of their analysis, assuming now that the molecule is strictly planar. 

Asbrink, S. Norrby, L.-J. (1970) A refinement of the crystal structure of copper(II) oxide with a discussion of some exceptional E.S.D. s. Acta Cryst., B26, 8-15. 

Dent-Glasser, L. S. Ingram, L. (1968) Refinement of the crystal structure of groutite, a-MnOOH. Acta Cryst., B24, 1233-6. 

Dollase, W. A. (1971) Refinement of the crystal structures of epidote, allanite and hancockite. Amer. Mineral., 56,447-64. 

Stergiou, A. C. Rentzeperis, P. J. (1987) Refinement of the crystal structure of a medium iron epidote. Zeit. Krist., 178,297-305. 

Ceccarelli C, Jeffrey GA, McMullan RK (1980) A neutron diffraction refinement of the crystal structure of erythritol at 22.6 K. Acta Cryst B36 3079-3083... 

Isaacs NW, Agarwal RC (1978) Experience with fast Fourier least squares in the refinement of the crystal structure of rhombohedral 2-Zn insulin at 1.5 A resolution. Acta Cryst A34 782-791... 

Preliminary three-dimensional atomic coordinates of atoms in crystal structures are usually derived from electron-density maps by fitting atoms to individual peaks in the map. The chemically reasonable arrangement of atoms so obtained is, however, not very precise. The observed structure amplitudes and their relative phase angles, needed to calculate the electron-density map, each contain errors and these may cause a misinterpretation of the computed electron-density map. Even with the best electron-density maps, the precisions of the atomic coordinates of a preliminary structure are likely to be no better than several hundredths of an A. In order to understand the chemistry one needs to know the atomic positions more precisely so that better values of bond lengths and bond angles will be available. The process of obtaining atomic parameters that are more precise than those obtained from an initial model, referred to as refinement of the crystal structure, is an essential part of any crystal structure analysis. 

The set of anisotropic displacement parameters, obtained from the least-squares refinement of the crystal structure (as described by Chapter 10) can be analyzed to obtain T, L and S. It has been assumed that there is no correlation between the motion of different atoms. Values of Uij are analyzed (again by an additional least-squares analysis) in such a way that good agreement is obtained between the refined values and those predicted when constants have been obtained for the T, L, and S tensors. The total number of anisotropic displacement parameters (6 per atom) is the input, and a total of 12 parameters for a centrosymmetric structure, or 20 parameters for a noncentrosymmetric structure, is the output of this least-squares analysis. The results consist of the molecular translational (T), librational (L), and screw (S) tensors. This treatment leads to estimates of corrections that should be made to bond distances. On the other hand, this type of analysis cannot be used for intermolec-ular distances because the correlation between the motion of different molecules is not known. 

When the determination of a crystal structure is difficult or refinement gives a strangely shaped molecule, it is possible that the molecule is disordered in a site in the crystal. This happens when the available space in the crystal is such as to accommodate two different orientations of the molecule. For example, in the crystal structure of 5-methylchrysene, -one of the two molecules in the asymmetric unit is disordered. It was difficult to solve the crystal structure until the nature of the disorder was realized. In a given site, this disordered molecule may be in one of two possible orientations. In some places in the disordered electron density map, atoms in the two orientations of the disordered molecule are near each other and their positions can be approximated (erroneously) by the use of highly anisotropic temperature parameters. The result is that the anisotropic temperature parameters on refinement do not make any sense until the nature of the disorder is understood. 

As briefly mentioned in the previous chapter, the determination of a crystal structure may be considered complete only when multiple pattern variables and crystallographic parameters of a model have been fully refined against the observed powder diffraction data. Obviously, the refined model should remain reasonable from both physical and chemical standpoints. The refinement technique, most commonly employed today, is based on the idea suggested in the middle 1960 s by Rietveld. The essence of Rietveld s approach is that experimental powder diffraction data are utilized without extraction of the individual integrated intensities or the individual structure factors, and all structural and instrumental parameters are refined by fitting a calculated profile to the observed data. 

It is important, however, to remember that the Rietveld method requires a model of a crystal structure and by itself offers no clue on how to create such a model from first principles. Thus, the Rietveld technique is nothing else than a powerful refinement and optimization tool, which may also be used to establish structural details (sometimes subtle) that were missed during a partial or complete ab initio structure solution process, i.e. as in the twelve examples described in Chapter 6. 

In this section, we are concerned with a powder diffraction experiment, which consists of a single pattern (profile). The Rietveld technique may also be used to conduct refinement of the crystal structure employing multiple patterns collected from the same material. For example, powder diffraction data collected using conventional x-ray sources with different wavelengths, conventional and synchrotron x-rays, conventional or synchrotron x-rays and neutron source may be used simultaneously in a combined Rietveld refinement. The fundamentals of the combined Rietveld refinement are briefly considered in section 7.3.8. 

We have seen this powder diffraction pattern several times throughout this text. The histogram collected from the nearly spherical LaNi4.85Sno.15 powder, produced by high pressure gas atomization from a melt, was used to illustrate both the quality of x-ray diffraction data and as one of the examples in the ab initio crystal structure solution. To demonstrate the Rietveld refinement of this crystal structure we will begin with the profile and unit cell parameters determined from Le Bail s algorithm Table 6.3) and the model of the crystal structure determined from sequential Fourier maps as described in section 6.9 and listed in Table 6.8. 

Refinement of the crystal structure is, therefore, a powerful chemical analysis technique. Unlike conventional chemical analysis, which only yields the bulk composition of the sample, powder diffraction analysis facilitates accurate determination of the occupancies of different crystallographic sites by various chemical elements, or in other words, establishes precise chemical composition of the crystal at the atomic resolution. It should be noted that the results may be considered reliable only when the difference in the scattering ability of atoms in question is significant, in addition to a very high quality of experimental data. This is indeed the case here because scattering factors of Sn and Ni are related as-1.8 1. 

---