The phase problem and electron-density maps

In order to obtain an image of the material that has scattered X rays and given a diffraction pattern, which is the aim of these studies, one must perform a three-dimensional Fourier summation. The theorem of Jean Baptiste Joseph Fourier, a French mathematician and physicist, states that a continuous, periodic function can be represented by the summation of cosine and sine terms (Fourier, 1822). Such a set of terms, described as a Fourier series, can be used in diffraction analysis because the electron density in a crystal is a periodic distribution of scattering matter formed by the regular packing of approximately identical unit cells. The Fourier series that is used provides an equation that describes the electron density in the crystal under study. Each atom contains electrons; the higher its atomic number the greater the number of electrons in its nucleus, and therefore the higher its peak in an electrondensity map.We showed in Chapter 5 how a structure factor amplitude, |F (hkl)|, the measurable quantity in the X-ray diffraction pattern, can be determined if the arrangement of atoms in the crystal structure is known (Sommerfeld, 1921). Now we will show how we can calculate the electron density in a crystal structure if data on the structure factors, including their relative phase angles, are available. The Fourier series is described as a “synthesis” when it involves structure amplitudes and relative phases and builds up a picture of the electron density in the crystal. By contrast, a “Fourier analysis” leads to the components that make up this series. The term “relative” is used here because the phase of a Bragg reflection is described relative to that of an imaginary wave diffracted in the same direction at a chosen origin of the unit cell.

Refinement of the trial 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).

Structural parameters: Analysis of results

The results of an X-ray structure analysis are coordinates of the individual, chemically identified atoms in each unit cell, the space group (which gives equivalent positions), and displacement parameters that may be interpreted as indicative of molecular motion and/or disorder. Such data obtained from crystal structure analyses may be incorporated into a CIF or mmCIF (Crystallographic Information File or Macromolecular Crystallographic Information File). These ensure that the results of crystal structure analyses are usefully archived. There are many checks that the crystallographer can make to ensure that the CIF or mmCIF file is correctly informative. For example, the automated validation program PLATON (Spek, 2003) checks that all data reported are up to the standards required for publication by the International Union of Crystallography. It does geometrical calculations on the structure, illustrates the results, finds if any symmetry has been missed, investigates any twinning, and checks if the structure has already been reported. We now review the ways in which these atomic parameters can be used to obtain a three-dimensional vision of the entire crystal structure. When molecules crystallize in an orthorhombic, tetragonal, or cubic unit cell it is reasonably easy to build a model using the unit-cell dimensions and fractional coordinates, because all the interaxial angles are 90◦. However, the situation is more complicated if the unit cell contains oblique axes and it is often simpler to convert the fractional crystal coordinates to orthogonal coordinates before calculating molecular geometry. The equations for doing this for bond lengths, interbond angles, and torsion angles are presented in Appendix 12. If the reader wishes to compute interatomic distances directly, this is also possible if one knows the cell dimensions (a, b, c, ∝ , β , γ ,), the fractional atomic coordinates (x, y, z for each atom), and the space group.

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

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.

The diffraction pattern obtained

In this chapter we will describe those factors that control the intensities of Bragg reflections and how to express them mathematically so that we can calculate an electron-density map. The Bragg reflections have intensities that depend on the arrangement of atoms in the unit cell and how X rays scattered by these atoms interfere with each other. Therefore the diffraction pattern has a wide variety of intensities in it. Measured X-ray diffraction data consist of a list of the relative intensity I (hkl), its indices (h, k, and l), and the scattering angle 2θ, for each Bragg reflection. All the values of the intensity I (hkl) are on the same relative scale, and this entire data set describes the “diffraction pattern.” It is used as part of the input necessary to determine the crystal structure. As already indicated from a study of the diffraction patterns from slits and from various arrangements of molecules, the angular positions (2θ) at which scattered radiation is observed depend only on the dimensions of the crystal lattice and the wavelength of the radiation used, while the intensities I (hkl) of the different diffracted beams depend mainly on the nature and arrangement of the atoms within each unit cell. It is these two items, the unit-cell dimensions of the crystal and its atomic arrangement, that comprise what we mean by “the crystal structure.” Their determination is the primary object of the analysis described here. As illustrated in Figure 1.1b and the accompanying discussion, and mentioned again at the start of Chapter 3, X rays scattered by the electrons in the atoms of a crystal cannot be recombined by any known lens. Consequently, to obtain an image of the scattering matter in a crystal, the “structure” of that crystal, we need to simulate this recombination, which means that we must find a way to superimpose the scattered waves, with the proper phase relations between them, to give an image of the material that did the scattering, that is, the electrons in the atoms.

Outline of a crystal structure determination

The stages in a crystal structure analysis by diffraction methods are summarized in Figure 14.1 for a substance with fewer than about 1000 atoms. The principal steps are: (1) First it is necessary to obtain or grow suitable single crystals; this is sometimes a tedious and difficult process. The ideal crystal for X-ray diffraction studies is 0.2–0.3mm in diameter. Somewhat larger specimens are generally needed for neutron diffraction work. Various solvents, and perhaps several different derivatives of the compound under study, may have to be tried before suitable specimens are obtained. (2) Next it is necessary to check the crystal quality. This is usually done by finding out if the crystal diffracts X rays (or neutrons) and how well it does this. (3) If the crystal is considered suitable for investigation, its unitcell dimensions are determined. This can usually be done in 20 minutes, barring complications. The unit-cell dimensions are obtained by measurements of the locations of the diffracted beams (the reciprocal lattice) on the detecting device, these spacings being reciprocally related to the dimensions of the crystal lattice. The space group is deduced from the symmetry of, and the systematic absences in, the diffraction pattern. (4) The density of the crystal may be measured if the crystals are not sensitive to air, moisture, or temperature and can survive the process. Otherwise an estimated value (about 1.3g cm−3 if no heavy atoms are present) can be used. This will give the formula weight of the contents of the unit cell. From this it can be determined if the crystal contains the compound chosen for study, and how much solvent of crystallization is present. (5) At this point it is necessary to decide whether or not to proceed with a complete structure determination. The main question is, of course, whether the unit-cell contents are those expected. One must try to weigh properly the relevant factors, among which are: (i) Quite obviously, the intrinsic interest of the structure. (ii) Whether the diffraction pattern gives evidence of twinning, disorder, or other difficulties that will make the analysis, even if possible, at best of limited value.

Anomalous scattering and absolute configuration

The concept of the carbon atom with four bonds extending in a tetrahedral fashion was put forward by van’t Hoff and Le Bel in 1874. It coincided with the realization that such an arrangement could be asymmetric if the four substituents were different, as shown in Figure 10.1a (van’t Hoff, 1874; Le Bel, 1874). Thus, for any compound containing one such asymmetric carbon atom, there are two isomers of opposite chirality (individually called enantiomers), for which threedimensional representations of their structural formulas are related by a mirror plane. Aqueous solutions of these enantiomers rotate the plane of polarized light in opposite directions. As discussed in Chapter 7, Pasteur showed that crystals of sodium ammonium tartrate had small asymmetrically located faces and that crystals with these so-called “hemihedral faces” rotated the plane of polarization of light clockwise, while crystals with similar faces in mirror-image positions rotated this plane of polarization counterclockwise. Thus the external form (that is, the morphology) of the crystals illustrated in Figure 10.1b was used to separate enantiomers (see Patterson and Buchanan, 1945). Pure enantiomers can only crystallize in noncentrosymmetric space groups unless both isomers are present. But even if the chemical formula and the three-dimensional structure of a molecule such as tartaric acid have been determined by standard X-ray diffraction methods, there is an ambiguity about the absolute configuration. Information about the absolute configuration is not contained in the diffraction pattern of the crystal as it is normally measured. Thus, although the substituents on the asymmetric carbon atoms have been identified, and even the detailed three-dimensional geometry of the molecule has been determined, it is not known which of the two enantiomers (mirror-image forms, analogous to those shown in Figure 10.1a) represents the three-dimensional structure of a particular individual molecule that has some distinguishing chiral property, such as the ability to rotate the plane of polarized light to the right. In other words, what is the absolute structure of the dextrorotatory form of the compound under study? A means of determining the absolute configurations of molecules was, however, provided by X-ray crystallographic studies.

Symmetry and space Groups

A certain degree of symmetry is apparent in much of the natural world, as well as in many of our creations in art, architecture, and technology. Objects with high symmetry are generally regarded with pleasure. Symmetry is perhaps the most fundamental property of the crystalline state and is a reason that gemstones have been so appreciated throughout the ages. This chapter introduces some of the fundamental concepts of symmetry—symmetry operations, symmetry elements, and the combinations of these characteristics of finite objects (point symmetry) and infinite objects (space symmetry)—as well as the way these concepts are applied in the study of crystals. An object is said to be symmetrical if after some movement, real or imagined, it is or would be indistinguishable (in appearance and other discernible properties) from the way it was initially. The movement, which might be, for example, a rotation about some fixed axis or a mirror-like reflection through some plane or a translation of the entire object in a given direction, is called a symmetry operation. The geometrical entity with respect to which the symmetry operation is performed, an axis or a plane in the examples cited, is called a symmetry element. Symmetry operations are actions that can be carried out, while symmetry elements are descriptions of possible symmetry operations. The difference between these two symmetry terms is important. It is possible not only to determine the crystal system of a given crystalline specimen by analysis of the intensities of the Bragg reflections in the diffraction pattern of the crystal, but also to learn much more about its symmetry, including its Bravais lattice and the probable space group. As indicated in Chapter 2, the 230 space groups represent the distinct ways of arranging identical objects on one of the 14 Bravais lattices by the use of certain symmetry operations to be described below. The determination of the space group of a crystal is important because it may reveal some symmetry within the contents of the unit cell.

Experimental Measurements

The analysis of a crystal structure by X-ray or neutron diffraction consists of three stages: (1) Data collection. This involves experimental measurement of the directions of scatter of the diffracted beams so that a unit cell can be selected and its dimensions measured. The intensities of as many as possible of the diffracted beams (Bragg reflections) from that same crystal are then recorded. These intensities depend on the nature of the atoms present in the crystal and their relative positions within the unit cell. (2) Finding a “trial structure.” This is the deduction by some method (such as one of those described in Chapters 8 and 9) of a suggested atomic arrangement (a “trial structure”). This is listed as atomic coordinates that have been measured with respect to the unit-cell axes. The intensity of each Bragg reflection corresponding to this trial structure can then be calculated (see Chapter 5) and its value then compared with the corresponding experimentally measured intensity in order to determine whether the trial structure is “good,” meaning that it is essentially correct. (3) Refinement of the trial structure. This involves modification (refinement) of a good trial structure until the calculated and measured intensities agree with each other within the limits of any errors in the observations (see Chapter 11). This is usually done by a leastsquares refinement, although difference electron-density maps may also prove useful. The result of the refinement is information on the three-dimensional atomic coordinates in this particular crystal, together with atomic displacement parameters. This chapter is concerned with the first of these stages, the experimental measurements. This is a rapidly changing area of science as more powerful and precise equipment and detection devices become available. The experimental data that may be derived frommeasurements of an X-ray or neutron diffraction pattern include: (1) The overall appearance of the Bragg reflections at the detection system. Ideally these diffraction maxima should be sharp, well-resolved peaks. Blurred, double spots or arcs may indicate disorder or poor crystal quality. (2) The angles or directions of scattering (including 2Ë, the angular deviation from the direct beam).

Diffraction

A common approach to crystal structure analysis by X-ray diffraction presented in texts that have been written for nonspecialists involves the Bragg equation, and a discussion in terms of “reflection” of X rays from crystal lattice planes (Bragg, 1913). While the Bragg equation, which implies this “reflection,” has proved extremely useful, it does not really help in understanding the process of X-ray diffraction. Therefore we will proceed instead by way of an elementary consideration of diffraction phenomena generally, and then diffraction from periodic structures (such as crystals), making use of optical analogies (Jenkins and White, 1957; Taylor and Lipson, 1964; Harburn et al., 1975). The eyes of most animals, including humans, comprise efficient optical systems for forming images of objects by the recombination of visible radiation scattered by these objects. Many things are, of course, too small to be detected by the unaided human eye, but an enlarged image of some of them can be formed with a microscope—using visible light for objects with dimensions comparable to or larger than the wavelength of this light (about 6 × 10−7 m), or using electrons of high energy (and thus short wavelength) in an electron microscope. In order to “see” the fine details of molecular structure (with dimensions 10−8 to 10−10 m), it is necessary to use radiation of a wavelength comparable to, or smaller than, the dimensions of the distances between atoms. Such radiation is readily available (1) in the X rays produced by bombarding a target composed of an element of intermediate atomic number (for example, between Cr and Mo in the Periodic Table) with fast electrons, or from a synchrotron source, (2) in neutrons from a nuclear reactor or spallation source, or (3) in electrons with energies of 10–50 keV. Each of these kinds of radiation is scattered by the atoms of the sample, just as is ordinary light, and if we could recombine this scattered radiation, as a microscope can, we could form an image of the scattering matter.