Determining crystal structures

Type of physical science: Condensed matter physics

Field of study: Solids

The structures of crystals are determined chiefly by X-ray diffraction methods. Interest in such structures ranges across physics, chemistry, engineering, biology, and mineralogy. A knowledge of the structure of a solid is essential to the understanding of its properties, and understanding the behavior of most crystalline materials starts with the crystal structure.

Overview

A crystal is a solid in which a three-dimensional pattern of atoms or ions is repeated throughout. Because atoms are extremely small, this pattern, called a unit cell, must be repeated a vast number of times in order to produce a crystal of appreciable size. Such a pattern makes a crystal a highly ordered structure. It is possible to determine the arrangement of atoms or ions--the structure of a crystal--by diffracting waves off the atoms in its adjoining layers.

The first requirement for such a diffraction experiment is that the wavelength be comparable to the distance between neighboring atoms. Otherwise, diffraction does not occur and the structure cannot be found. The distance from the center of one atom to the center of the next one in a crystal is typically about 2.0 x 10^-10 meters, while the wavelength of light is about 5.0 x 10^-10 meters. Therefore, light waves are too long and X rays must be used.

The diffraction of X rays from a crystal can be described as follows. Consider the simplest possible pattern of atoms: a cube with one atom at each corner. To build up the entire solid, the cubic pattern is repeated in all directions. If a beam of X rays falls on the solid, then part of the beam will hit an atom on the surface and be reflected away from that surface. The section of the X-ray beam located to the right or left of the part that has been reflected will miss the atom that caused the reflection but can still enter the crystal, travel a short additional distance to the atom directly below it, and be reflected. Assume that this second section of the beam is reflected away from the crystal in the same direction as the first part. The two reflected parts of the beam will cancel each other; that is, they will both cease to exist unless the X rays that were reflected from the second layer of the crystal traveled an added distance equal to a whole number of X-ray wavelengths. This recombination of waves of the same wavelength that have traveled over distances differing by a whole number of wavelengths is called diffraction.

This added distance depends on the angle between the surface of the crystal and the direction in which the X-ray beam is moving. It also depends on the distance between layers of the crystal. By assigning symbols to these quantities--n is the number of waves, L the wavelength, d the distance between layers of the crystal, and θ the angle between the beam and the crystal surface--the relationship between them is called the Bragg equation: nL = 2d sine θ. The equation establishes that the path of the section of the beam reflected from the second layer will be longer than the path traveled by the part of the beam reflected from the surface layer by a whole number of wavelengths, nL only for certain angles θ. The smallest angle will be the one for which n =1. The Bragg equation makes it possible to find the distance d between layers of the crystal. The X-ray wavelength L must be known and the reflection angle θ must be measured.

To extend this argument, picture the original cube again. The second section of the X-ray beam may be reflected from an atom other than the one directly below the atom that reflected the first part of the beam. In that case, both the distance and the angle will change. If the reflecting atom in the second layer is located diagonally across a side of the cube from the reflecting atom on the top layer, then the distance will be the square root of 2, or about 1.414 times as great, and the reflection of the two parts of the beam will occur at a smaller angle. If the reflecting atom in the second layer is diagonally across the center of the cube, then the distance will be greater by the square root of 3 (1.732), and the reflection will be seen at a still smaller angle.

Therefore, the various layers of atoms of the simple cubic unit cell will reflect X rays at several angles. A simple cubic unit cell, however, is rare: Among the elements, only polonium has this structure. One major reason for this rareness is that a cubic structure does not fill space efficiently.

X rays are produced when fast electrons strike a metal surface, and the wavelength of most intense X rays varies from one metal to another. Copper is often used as a source of waves with a wavelength of 1.5 x 10^-10 meters. Because these waves are given off in all directions, slits are used to block most of them and to select only a narrow beam to strike the crystal. In early experiments, a film was exposed by the diffracted X rays, and diffraction angles were determined from measurements made directly on the film. Modern instruments have a radiation counter or similar device which converts the diffracted X-ray beam to an electrical signal, which is then converted to a digital signal for storage and analysis by computer. The use of computers has extended this method to structures of enormous complexity.

Atoms vary in their ability to reflect X rays. The incoming X ray interacts with the electrons around an atomic nucleus; the more electrons, the more likely it is that an X ray will be reflected. Because the number of electrons around an atom is its atomic number, X rays are reflected by the atoms of heavy elements much more than by the atoms of light elements.

Detailed analysis of diffraction data can show this information as electron density contours for planes of the crystal. The lightest element, hydrogen, is also one of the most important elements, and it gives the weakest reflections of X rays. This limitation is one disadvantage of the method.

Neutrons have a wave nature, as required by wave-particle duality. As a result, crystal structures can also be studied by neutron diffraction. These experiments are quite different from the X-ray method, and few laboratories have access to a suitably intense beam of neutrons. The technique has the advantage, however, that hydrogen atoms scatter neutrons strongly.

Consequently, neutron diffraction is superior to X-ray diffraction in locating hydrogen atoms.

Because atoms are so small, many of them at the same repeat distance from one another will contribute to the diffracted X-ray beam. The combination of many waves, all subject to the Bragg equation, creates a precisely defined diffraction angle. As a consequence, the center-to-center distances between atoms in a crystal can often be determined to an uncertainty of a few trillionths of a meter. The length of the edge of the unit cell of silver is 4.0862 x 10 to the power of -10 meters, +/- 2.0 x 10^-14 meters. That such a small distance should be known so precisely is a great scientific achievement and illustrates the power of diffraction methods.

Applications

The determination of crystal structures began early in the twentieth century and has yielded a rich harvest of information. The standard discussion of diffraction effects is based on a simple cubic unit cell. Nature, however, seldom uses this pattern. One reason is that it does not fill space efficiently. If the atoms in a crystal are assumed to be hard spheres, then the simple cubic unit cell has only 52 percent of its volume occupied. Another way to consider the efficiency of filling space is to realize that an atom in a simple cubic pattern has only six atoms next to it, or six nearest neighbors. Their locations are above and below, on the left and the right, and in front and behind. If the atoms at the corners of the cube are moved apart just enough to allow one atom to be in the center of a cube, then the unit cell is called body-centered cubic. This arrangement has 68 percent of its volume occupied by atoms, and it is far more common than the simple cubic unit cell. Each of these atoms has eight nearest neighbors--four in a square above it and four in a square below.

The best that can be done in packing identical atoms is to give each atom twelve nearest neighbors, six of them in the same plane. This is easy to illustrate by arranging spheres on a tabletop. Six of these atoms can be arranged around the one in the center, and each of them will also touch the spheres next to them. In addition, there is space for three more spheres in a horizontal triangle just above the central atom, and three more in a triangle just below it. The result is that 74 percent of the volume is occupied by atoms. Two crystal patterns, called hexagonal closest packing and cubic closest packing, achieve this efficiency. They differ only in whether the triangles of atoms above and below are directly in line with each other or rotated by 60 degrees. More than half of the elements exist in one or the other of these two crystal structures as solids.

Crystal structures are more complicated when one moves beyond identical atoms, (that is, elements) and considers compounds. The earliest substances analyzed were simple salts. In sodium chloride, the sodium ions make a face-centered cubic structure, a synonym for cubic closest packing, and the chloride ions do likewise. The two patterns are said to be interpenetrating; that is, the sodium (Na+) and chloride (Cl-) ions are found side by side. The result is an orderly alternation of the two kinds of ions throughout the crystal. Along an edge or face of the crystal the alternation is: Unit cells must be patterns that can fill space. They can be cubic or hexagonal, for example, but not pentagonal. That pentagons are unacceptable is shown by laying out identical regular pentagons on a flat surface. They cannot cover the surface without overlapping one another.

For a molecule of only moderate complexity, as many as one thousand reflections are commonly analyzed by a diffractometer under computer control. The best computer programs produce diagrams of the locations of atoms in a crystal, in which bond lengths and bond angles will be specified. There is enough detail to indicate the extent of thermal motion, that is, atomic vibrations.

As structures become more complicated, it becomes increasingly important to have some information about the molecule in addition to its crystal diffraction pattern. The determination of the helical structure of deoxyribonucleic acid (DNA) by James D. Watson, Francis Crick, and Maurice Wilkins is the most famous example. Their diffraction pattern contained only a tiny fraction of the information needed to describe the structure of DNA fully, but it did contain clear evidence of a helical pattern, from which the repeat distance could be calculated. On this basis, the details could be determined from prior structural knowledge of the parts of the huge molecule.

When patterns are complicated, a heavy atom is sometimes incorporated into the crystal structure. This method is particularly effective when the other elements are hydrogen, carbon, oxygen, and nitrogen, as the strong contribution of the heavy atom to the diffraction pattern can be recognized easily.

Biochemists can use X-ray diffraction to determine the structures of enzymes attached to the substances on which they act. This detailed knowledge of the shape of the molecule crystallized in an active position provides a clear picture of the spatial requirements for the operation of the enzyme.

Among simple substances such as common ionic compounds, the structure of a given crystal will be influenced by the relative sizes of the cations (positive ions such as NA+) and anions (negative ions such as Cl-). Anions are usually larger. If the anion is more than six times as large as the cation, then the cation may be able to arrange only three anions around itself. If the anion is between 6 and 4.4 times as large as the cation, then the cation may be able to arrange four anions around itself. As the anion-cation radius ratio decreases, the cation can accommodate increasing numbers of anions around itself. For eight nearest neighbors, the ratio should be no greater than about 1.37:1. For identical atoms, twelve nearest neighbors can fit into a stable pattern. These generalizations, called radius ratio rules, are based on simple geometric arguments, but there are exceptions.

Crystals are the most orderly form of matter. The high precision of measured distances between atoms is a consequence of the regularity in the repeated pattern of unit cells. The atoms of a crystal have a known number of nearest-neighbor atoms. Liquids show order only over short distances, and the number of nearest neighbors around a given molecule is no longer exact. X-ray diffraction, however, can be used to determine the average distance between a given molecule and its nearest neighbors. In the case of water, the hydrogen atoms are not seen clearly, but the average distance between oxygen atoms of neighboring molecules is about 2.5 x 10 to the power of -10 meters at room temperature. The pattern can be seen throughout the liquid range, though the average distance increases with increasing temperature, which causes the molecules to move more rapidly.

Context

The world has recognized the work of crystallographers with three Nobel Prizes in physics, six in chemistry, and one in physiology or medicine. Among the winners in physics were Max von Laue in 1914, for his discovery of the diffraction of X rays by crystals, and Sir William Henry Bragg and Sir Lawrence Bragg the next year for research on the structures of crystals using X rays. Among the winners in chemistry have been Linus Pauling in 1954 for extensive studies of crystal structures; Max Perutz and John Kendrew in 1962 for the determination of the detailed structure of the biochemically important molecules myoglobin and hemoglobin; Dorothy Hodgkin for further structural studies in biochemistry in 1964; and William Lipscomb in 1976 for unraveling the structures of compounds made of boron and hydrogen that had puzzled chemists for years. Watson, Crick, and Wilkins shared the Nobel Prize in Physiology or Medicine in 1962.

Great progress in imaging methods has made it possible to produce images of crystal surfaces. In some cases, the pattern indicated by X-ray diffraction can be confirmed by an electronically produced picture, which has had a great impact on the study of crystal surfaces.

Solids frequently consist of units whose structures are well known from other sources of information. For example, a six-carbon ring with six other atoms or groups of atoms attached to it will almost certainly look like benzene. Thus, the structure of this part of the molecule can be predicted quite confidently, and all that remains is to determine whether the particular environment of the crystal has brought about small changes in the expected distances between atoms. When the helical structure of DNA was first determined, much of the insight came from independent knowledge of the structures of the four bases whose presence in DNA was firmly established by chemical analysis. While there are examples to the contrary, any determination of the structure of a liquid or gas molecule is likely to be helpful when the structure of the same substance is studied as a solid, the reason being that most bonds between atoms are strong enough to be undisturbed by melting or evaporating.

Not all solids consist of atoms or ions arranged in regular patterns. The term "amorphous" refers to solids that do not have a repeated pattern of atoms. Glass is such a material. It has no exact melting point, but rather a melting range. Neighboring molecules of liquid glass are strongly attracted to each other, which is understood by anyone who has watched a glassblower work with hot glass and has seen how slowly the liquid flows. As the glass cools, the molecules are so strongly mutually attracted that they can no longer slide past one another to form a regular pattern. The strong attractions of the molecules for one another give glass the material strength of a solid and the structure of a frozen liquid. Such substances are sometimes known as undercooled liquids.

Quasicrystals are a class of substances that were discovered in the 1980's. They violate the structural principle that a regular pentagon cannot be used to fill space. In the case of quasicrystals, no pentagonal shapes are used. Instead, two shapes, fat and thin diamonds with some of the same angles found in pentagons, are used to build structures that have no unit cell, that is, no repeat pattern. The structures that result show unmistakable five-sided symmetries in some regions. At first, these quasicrystals were only a mathematical invention, but soon an alloy of aluminum and manganese was discovered whose crystal structure also showed a fivefold pattern, and quasicrystals became a material reality. Other quasicrystalline alloys have been discovered.

Principal terms

BOND: the forces that hold two atoms together, which can be covalent (resulting from the presence of a shared pair of electrons) or ionic (resulting from the attraction between ions of positive charge and ions of negative charge)

BRAGG EQUATION: the fundamental relationship between the wavelength of X rays used in a diffraction experiment and the distance between the adjoining layers of the crystal that diffracts these rays

DIFFRACTION OF X RAYS: the reappearance outside a crystal of a beam of X rays after it has penetrated the crystal's first few layers; the diffracted beam appears only at certain angles on the side of the crystal from which it entered, and these angles are the basis of information about the crystal structure

NEUTRON: the constituent particle of atomic nuclei that carries no electric charge

NUCLEAR CHARGE: the number of protons in a nucleus, which is the same for all atoms of a given element

UNIT CELL: the basic arrangement of atoms or ions that is repeated throughout a crystal

WAVE-PARTICLE DUALITY: the physical principle according to which light and the simplest particles of matter behave like both waves and particles

X RAYS: a form of radiation similar to light, with wavelengths roughly one-thousandth as long as the waves that humans see

Bibliography

Giancoli, Douglas. PHYSICS. 3d ed. Englewood Cliffs, N.J.: Prentice-Hall, 1991. Contains a good discussion of the X-ray tube, the Bragg equation, and this equation's application to the determination of crystal structures. Diffraction and the wave nature of light are clearly presented with helpful illustrations.

Holden, Alan, and Phylis Singer. CRYSTALS AND CRYSTAL GROWING. Garden City, N.Y.: Anchor Books, 1960. An excellent book for understanding, making, and appreciating crystals. Symmetries, arrangements of atoms, and the directly observable properties of crystals are carefully discussed.

JOURNAL OF CHEMICAL EDUCATION 65 (June, 1988). This special issue records a symposium on teaching crystallography, some of which is at an introductory level. Illustrations throughout much of the symposium can be enjoyed even when the discussion is at a high level.

Pauling, Linus. NATURE OF THE CHEMICAL BOND AND THE STRUCTURE OF MOLECULES AND CRYSTALS. 3d ed. Ithaca, N.Y.: Cornell University Press, 1960. Written from the point of view of chemical bonding, this is a classic book on crystal structures. Chapters 11 and 13 contain many well-drawn figures.

Pauling, Linus, and Roger Hayward. ARCHITECTURE OF MOLECULES. San Francisco: W. H. Freeman, 1964. This beautiful book consists almost entirely of color drawings of molecules and crystals, with a brief account of each on the facing page. The crystal drawings include graphite, diamond, Prussian blue, boron, and ice. Other illustrations show such basic structural units as the cube, the tetrahedron, and the octahedron.

Watson, James D. THE DOUBLE HELIX. New York: Atheneum, 1968. Watson's subtitle, "a personal account of the discovery of the structure of DNA," is an apt description. The book also reveals much that is missing in accounts of similar work in scientific journals.

Schematic of sodium chloride crystal

The Structure of Ice

X-Ray Determination of Molecular Structure