X-Ray Diffraction Analysis

the investigation of the structure of a substance by methods that make use of the spatial distribution and intensities of X-radiation scattered by the object under study. Like neutron diffraction and electron diffraction analysis, X-ray diffraction analysis investigates structure through the use of diffraction. When X-radiation interacts with the electrons of a substance, the X rays are diffracted. The diffraction pattern depends on the wavelength of the X rays employed and on the structure of the object. Radiation of wavelength ~ 1 angstrom (Å), that is, of the order of atomic dimensions, is used to investigate atomic structure. The methods of X-ray diffraction analysis are used to study, for example, metals, alloys, minerals, inorganic and organic compounds, polymers, amorphous materials, liquids, gases, and the molecules of proteins and nucleic acids. X-ray diffraction analysis has been used most successfully to establish the atomic structure of crystalline substances because crystals have a rigid periodicity of structure and constitute naturally produced diffraction gratings for X rays.

Historical survey. The diffraction of X rays by crystals was discovered in 1912 by the German physicists M. von Laue, W. Friedrich, and P. Knipping. These investigators directed a narrow beam of X rays onto a stationary crystal, behind which a photographic plate was located. They recorded on the photographic plate a diffraction pattern consisting of a large number of regularly arranged spots, each of which was the track of a diffracted beam scattered by the crystal. An X-ray pattern obtained by this method is called a Laue pattern.

Laue’s theory of X-ray diffraction by crystals yields the following relations between the wavelength λ of the radiation, the parameters a, b, and c of the unit cell of the crystal, and the angles of the incident (α₀, β₀, γ₀)and diffracted (α, β, γ) beams:

a(cos α – cos α₀) = hλ

(1) b(cosβ – cosβ₀) = kλ

c(cos γ – cos γ₀) = lλ

where h, k, and l are integers called Miller indexes. For a diffracted beam to appear, these three equations, which are known as the Laue conditions, must be satisfied. They require that in parallel beams the path difference between beams scattered by the atoms corresponding to neighboring lattice points be equal to an integral number of wavelengths.

In 1913, W. L. Bragg and, simultaneously, G. V. Vul’f proposed a simpler interpretation of the diffraction of X rays by a crystal: diffraction reflection. They showed that any diffracted beam can be regarded as a reflection of the incident beam from one of the systems of crystallographic planes (see BRAGG-VULT CONDITION). In the same year, W. H. Bragg and W. L. Bragg became the first to investigate the atomic structures of simple crystals by means of X-ray diffraction techniques. In 1916, P. Debye and the German physicist P. Scherrer proposed that the structure of polycrystalline materials be investigated by means of X-ray diffraction. The French crystallographer A. Guinier developed in 1938 the method of small-angle X-ray scattering for investigating the shapes and dimensions of inho-mogeneities in substances.

The applicability of X-ray diffraction analysis to the investigation of a broad class of substances and industry’s need for such investigations stimulated the development of techniques for the analysis of structure. In 1934 the American physicist A. Patterson suggested that the structures of substances could be investigated by means of a function of interatomic vectors; such functions are now called Patterson functions. The foundations of the direct methods of determining crystal structures were laid by the American scientists D. Harker, J. Kasper (1948), F. Zachariasen, and D. Sayre and by the British scientist W. Cochran (1952). Important contributions to the development of Patterson methods and direct methods of X-ray diffraction analysis were made by a number of scientists, including N. V. Belov, G. S. Zhdanov, A. I. Kitaigorodskii, B. K. Vain-

Figure 3. Schematics of X-ray* cameras for investigating single crystals: (a) camera for investigating stationary single crystals by the Laue method, (b) rotating camera, (c) camera for determining the dimensions and form of the unit cell; (S) specimen, (GH) goniometric head, (7) graduated circle and axis of rotation of the goniometric head, (CL) collimator, (C) cassette containing photographic film PF, (CB) cassette for obtaining back-reflection photographs, (MR) mechanism for rotating or oscillating the specimen, (Φ) graduated circle and axis of oscillation of the specimen, (δ) arc guide for inclination of axis of the goniometric head. In the schematic of the rotating camera, diffraction maxima lying on the layer lines can be seen on the photographic film; when rotation is replaced by oscillation of the specimen, the number of reflections on the layer lines is limited by the angle of oscillation. Rotation of the specimen is accomplished by means of the gears 1 and 2; oscillation is produced by the cardioid 3 and lever 4.

shtein, and M. A. Porai-Koshits of the USSR, L. Pauling, P. Ewald, M. Buerger, J. Karle, and H. Hauptman of the USA, and M. Woolfson of Great Britain. An extremely important role in the development of molecular biology was played by research on the three-dimensional structure of protein that was begun in England by J. Bernai in the 1930’s and was successfully continued by, for example, J. Kendrew, M. Perutz, and D. Hodgkin. In 1953, J. Watson and F. Crick proposed a model of the deoxyribonucleic acid (DNA) molecule that was in good agreement with the results of X-ray studies of DNA made by M. Wilkins.

The 1950’s saw the rapid development of X-ray diffraction techniques that make use of electronic computers in experimental procedures and in the processing of X-ray diffraction information.

Experimental methods. X-ray cameras and X-ray diffractom-eters are used to create the conditions for the diffraction and recording of radiation. The scattered X-radiation is recorded in these devices on photographic film or is measured with nuclear radiation detectors. Depending on the state of the specimen under study, on the specimen’s properties, and on the type and amount of information that must be obtained, various X-ray diffraction techniques are used. Single crystals selected for the investigation of their atomic structure must have dimensions of approximately 1 mm and as perfect a structure as possible. X-ray topography, which sometimes is classified under X-ray diffraction analysis, deals with the investigation of imperfections in comparatively large near-perfect crystals.

LAUE METHOD. The Laue method is the simplest means of obtaining X-ray photographs from single crystals. In the Laue experiment the crystal is stationary, and the X-radiation used has a continuous spectrum. The arrangement of the diffraction spots in Laue patterns depends on the symmetry of the crystal and on the crystal’s orientation with respect to the incident beam. By means of the Laue method the crystal under study can be placed in one of the 11 Laue symmetry groups and can be oriented—that is, the direction of the crystallographic axes can be determined—with an accuracy of a few minutes of arc. Depending on the character of the spots in Laue patterns and especially on the occurrence of asterism, it is possible to reveal internal stresses and some other imperfections of the crystal structure. The Laue technique is used to test the quality of single crystals in selecting a specimen for more complete structure analysis.

OSCILLATING- AND ROTATING-CRYSTAL METHODS. The methods of oscillation and rotation of the specimen are used to determine the period of repetition, or lattice constant, along a crystallographic direction in a single crystal; in particular, they permit determination of the parameters a, b, and c of the crystal’s unit cell. In this method, monochromatic X-radiation is used, and the specimen is set in oscillatory or rotational motion about an axis coincident with the crystallographic direction along which the period of repetition is being investigated. The spots in the oscillating- and rotating-crystal X-ray photographs obtained in cylindrical cassettes are arrayed in a family of parallel lines. The desired repetition period of the crystal can be calculated from the distances between these lines, the wavelength of the radiation, and the diameter of the cassette of the X-ray camera. The Laue conditions for diffracted beams are satisfied in this method by varying the angles in equations (1) through oscillation or rotation of the specimen.

X-RAY GONIOMETRIC METHODS. In order to investigate fully the structure of a single crystal by X-ray diffraction analysis techniques, it is necessary not only to establish the positions of the diffraction reflections but also to measure the intensities of as many diffraction reflections as can be obtained from the crystal for a given wavelength of radiation and for all possible orientations of the specimen. For this purpose, the diffraction pattern is recorded on photographic film in an X-ray goniometer, and the degree of blackening of each spot on the X-ray photograph is measured by means of a microphotometer. In an X-ray dif-fractometer the intensity of the diffraction reflections can be measured directly by using proportional, scintillation, or other X-ray counters. In order to have a complete set of reflections, a series of X-ray photographs is made in X-ray goniometers.

Each photograph records diffraction reflections on whose Miller indexes certain restrictions are imposed; for example, reflections of the type hkO, hk1, and so on are recorded in different X-ray photographs. X-ray goniometric experiments are most often conducted by the methods of Weissenberg, Buerger, and de Jong-Bauman. The same information can also be obtained by means of oscillating-crystal X-ray photographs.

In order to determine an atomic structure of average complexity—that is, where there are ~50–100 atoms per unit cell—the intensities of several hundreds or thousands of diffraction reflections must be measured. This extremely laborious and detailed work is performed by automatic microdensitome-ters and diffractometers controlled by an electronic computer. Several weeks or even months are sometimes required to complete the work; this is the case, for example, in the analysis of protein structures, where the number of reflections reaches hundreds of thousands. The length of the experiment can be greatly reduced by using in the diffractometer several counters that can record reflections in parallel. Diffractometer measurements surpass photorecording in sensitivity and accuracy.

DEBYE-SCHERRER METHOD. The Debye-Scherrer, or powdered-crystal, method is used for the investigation of polycrys-tals. Metals, alloys, and crystalline powders consist of a large number of small single crystals of a given substance. To study these, monochromatic radiation is used. The X-ray pattern, or Debye pattern, of the polycrystals consists of several concentric rings; each ring is formed by the merging of reflections from a definite system of planes of variously oriented single crystals. Debye patterns of different substances are individual in character and are widely used to identify compounds, including compounds in mixtures. The X-ray diffraction analysis of polycrystals can be used, for example, to determine the phase composition of specimens, to establish the dimensions and preferred orientation (texture) of grains in a substance, and to monitor stresses in the specimen.

INVESTIGATION OF AMORPHOUS MATERIALS AND PARTIALLY ORDERED OBJECTS. A clear-cut X-ray photograph with sharp diffraction maxima can be obtained only when the specimen has complete three-dimensional periodicity. The lower the degree of order of the atomic structure of the material, the more blurred and diffuse is the pattern of the X-radiation scattered by it. The diameter of the diffuse ring in the X-ray pattern of an amorphous substance can be used for a rough estimate of the average interatomic distances in the substance. As the degree of ordering (seeLONG-RANGE AND SHORT-RANGE ORDER) in the structure of objects increases, the diffraction pattern becomes more complicated and, consequently, contains more structural information.

METHOD OF SMALL-ANGLE SCATTERING. The scattering of X rays at small, or low, angles can be made use of to study spatial inhomogeneities of a substance that have dimensions greater than the interatomic distances—that is, dimensions ranging from 5–10 Å to ~10,000 Å. In this case, the scattered X-radiation is concentrated near the primary beam, in the region of small scattering angles. The method of small-angle scattering is used to investigate porous substances, finely divided materials, alloys, and such complicated biological objects as viruses, cell membranes, and chromosomes. For isolated molecules of proteins and nucleic acids, the method makes possible the determination of the molecule’s shape, size, and molecular weight. In viruses, the nature of the relative stacking of components—proteins, nucleic acids, and lipids—can be established. In synthetic polymers, the stacking of the polymer chains can be studied. In powders and sorbents, the size distribution of particles and pores can be evaluated. In alloys, the occurrence and dimensions of phases can be investigated. In textured substances, particularly in liquid crystals, the form of the packing of the particles (molecules) into various types of supramolecu-lar structures can be studied. The small-angle X-ray method is also used in industry to monitor the production of, for example, catalysts and highly dispersed carbon. Depending on the structure of the object, measurements are carried out for scattering angles from fractions of a minute to a few degrees.

Determination of atomic structure from X-ray diffraction data. The analysis of the atomic structure of a crystal involves the establishment of the dimensions and form of its unit cell, the determination of which of the 230 space groups of crystal symmetry (discovered by E. S. Fedorov) the crystal belongs to, and the establishment of the coordinates of the base atoms of the structure. The first and, to some extent, the second problem can be solved by the Laue method and the oscillating- or rotating-crys-tal method. The symmetry group and the coordinates of the base atoms of a complex structure can be conclusively determined only through a complicated analysis and laborious mathematical processing of the values of the intensities of all the diffraction reflections from the given crystal. The ultimate goal of this processing is to calculate, from experimental data, the values of the electron density p(x, y, z) at any point of the cell of a crystal with coordinates x,y,z. Because of the periodicity of the crystal’s structure, the electron density can be written as a Fourier series:

where V is the volume of the unit cell; the F_hkl are Fourier coefficients, which in X-ray diffraction analysis are called structure factors; and . Each structure factor is characterized by the three integers hkl and is related to the diffraction reflection determined by conditions (1). The purpose of the summation in equation (2) is to collect mathematically the X-ray diffraction reflections in order to obtain an image of the atomic structure. Image synthesis in X-ray diffraction analysis must be carried out in this manner because of the absence of natural lenses for X-radiation; a converging lens is used for this purpose in visible-light optics.

Diffraction reflection is a wave process. It is characterized by an amplitude equal to ǀF_hklǀ and by a phase angle α_hkl—that is, by the phase shift of the reflected wave with respect to the incident wave. The structure factor can then be expressed: F_hkl = ǀF_hklǀ(cosα_hkl + isin α_hkl). X-ray diffraction methods permit the intensities, but not the phases, of the reflections to be measured; the intensities are proportional to ǀF_hklǀ². The determination of the phases is the basic problem of the analysis of a crystal structure. The phases of the structure factors are established in fundamentally the same way for both crystals consisting of atoms and crystals consisting of molecules. After the coordinates of the atoms in a molecular crystalline substance are determined, its constituent molecules can be distinguished, and the molecules’ size and shape can be established.

The problem inverse to structure analysis is easily solved. This problem consists in the computation of structure factors for a Known atomic structure and the computation of the intensities of the diffraction reflections from the structure factors. The trial-and-error method, which was historically the first method of analyzing structures, involves the comparison of the experimentally observed values ǀF_hklǀ_obs with the values ǀF_hklǀ_cal calculated on the basis of a trial model. Depending on the value of the reliability factor, or discrepancy index,

the trial model is accepted or rejected. More formal methods were developed for crystal structures in the 1930’s, but for noncrystalline objects the trial-and-error method remains virtually the only means of interpreting a diffraction pattern.

The introduction of Patterson functions, or functions of interatomic vectors, opened up a fundamentally new path in the analysis of the atomic structures of single crystals. To construct the Patterson function of some structure consisting of N atoms, we translate the structure in such a way that initially the first atom falls at the fixed origin of coordinates. The vectors from the origin to all atoms of the structure, including the vector of zero length to the first atom, indicate the positions of the N maxima of the Patterson function; the set of these maxima is called the image of the structure at atom 1. Let us add the N maxima whose positions indicate the N vectors from the second atom, which, after translation of the structure, is located at the same origin. After performing this procedure with all N atoms (Figure 1), we obtain N² vectors. The positions of these vectors are described by the Patterson function.

If u, v, and w are the coordinates of points in the space of the interatomic vectors, the Patterson function P(u, v, w) can be expressed as follows:

This expression shows that the function is determined by the absolute values of the structure factors, is independent of the phases of the structure factors, and, consequently, can be calculated directly from the data of a diffraction experiment. The difficulty in interpreting the function P(u, v, w) consists in the necessity of finding the coordinates of the N atoms from the N² maxima, many of which are superimposed because of overlaps that occur when the Patterson function is constructed. The simplest case for analyzing P(u, v, w) is where the structure contains one heavy atom and several light atoms. The image of such a structure at the heavy atom will differ considerably from its other images. Several techniques exist for determining a model of a structure under study from a Patterson function; the most powerful are the superposition techniques, which have permitted the analysis of the function to be formalized and to be performed by computer.

Figure 1. Schematic of the construction of a Patterson function for a three-atom structure

Patterson function methods encounter serious difficulties when they are used to investigate the structure of crystals consisting of identical atoms or of atoms that are close in atomic number. In this case, the direct methods of determining the phases of structure factors have proved more effective. Since the value of the electron density in a crystal is always positive or zero, a large number of inequalities can be obtained that are obeyed by the Fourier coefficients (structure factors) of the function p(x, y, z). Structures containing up to 20–40 atoms in the unit cell of the crystal can be analyzed comparatively simply by inequality methods. For more complex structures, methods based on a probabilistic approach to the problem are used: the structure factors and their phases are considered random variables. The distribution functions for these random variables are derived from physical concepts. The most probable values of the phases can be estimated from the distribution functions by taking into account the observed absolute values of the structure factors. These methods can also be carried out by computer; they permit the analysis of structures containing 100–200 or more atoms in the unit cell of the crystal.

Thus, if the phases of the structure factors have been established, the electron density distribution in the crystal can be calculated from equation (2), and the maxima of this distribution correspond to the positions of the atoms in the structure (Figure 2). The final refinement of the atomic coordinates is carried out on a computer by a least-squares method. Depending on the quality of the experiment and the complexity of the structure, the atomic coordinates can be obtained with an accuracy of thousandths of an angstrom. It should be noted here that the quantitative characteristics of the thermal vibrations of atoms in a crystal can also be calculated by means of present-day diffraction methods with the anisotropy of these vibrations taken into account. X-ray diffraction analysis also permits the establishment of finer characteristics of atomic structures, such as the distribution of valence electrons in a crystal. So far, however, this complex task has been carried out only for very simple structures. The combination of neutron diffraction and X-ray

Figure 2. (a) Projection on the ab-plane of the Patterson function of the mineral baotite [Ba₄ Ti₄ (Ti, Nb)₄ [Si₄ O_l2] O_l6 CI]; contour curves are drawn at identical intervals of the values of the Patterson function, (b) Projection of the electron density of baotite on the ab-plane, obtained by analyzing the Patterson function shown in (a); the crowding together of the contour curves indicates the location of electron density maxima, which correspond to the position of atoms in the structure. (c) Representation of the model of the atomic structure of baotite. Each Si atom lies within a tetrahedron formed by four O atoms; Ti and Nb atoms are in octahedrons formed by O atoms. The tetrahedrons of SiO₄ and the octahedrons of Ti(Nb)O₆ in the baotite structure are connected, as is shown here. The part of the unit cell of the crystal corresponding to (a) and (b) of the figure is outlined by broken lines. The dotted lines in (a) and (b) give the zero levels of the values of the corresponding functions.

diffraction investigations is an extremely promising approach to this problem: neutron diffraction data on the coordinates of atomic nuclei are compared with the spatial distribution of the electron cloud obtained by means of X-ray diffraction analysis. X-ray diffraction methods and resonance techniques are used jointly to solve many problems of physics and chemistry.

The greatest achievement of X-ray diffraction analysis has been the elucidation of the three-dimensional structure of proteins, nucleic acids, and other macromolecules. Under natural conditions proteins generally do not form crystals. In order to achieve a regular arrangement of protein molecules, the proteins are crystallized, and their structure then analyzed. The phases of the structure factors of protein crystals can be determined only through the joint efforts of specialists in X-ray photography and biochemistry. The solution of this problem requires the production and investigation of crystals of the protein itself and of some of its derivatives containing heavy atoms; the coordinates of the atoms must be the same in all these structures.