Crystal symmetries

Type of physical science: Condensed matter physics

Field of study: Solids

The external form of crystals is governed by their atomic structure, which can be described in terms of geometrical rules called symmetry. The symmetry of a crystal is reflected by its external shape.

Overview

At first sight, the rules governing the classification of crystals may seem puzzling.

Crystals with similar forms are often placed in entirely different classes. For example, one common crystal form, the octahedron, has eight faces, all of which are equilateral triangles, and resembles two pyramids joined base to base. Another eight-sided crystal whose faces are not equilateral triangles might look quite similar to an octahedron but would be classified as an entirely different crystal form. In some cases, even crystals with identical shapes can belong to different crystal classes. On the other hand, a cube, which looks very different from an octahedron, belongs to the same crystal class as the octahedron.

This paradox results from the fact that the shapes of crystals are merely an outward manifestation of a more fundamental property: the geometry of their atomic structures. A familiar example illustrates the relationship between atomic arrangement and crystal shape, using coins to simulate a layer of atoms in a crystal. One of the simplest arrangements is that of seven identical coins, with one in the center and the other six forming a hexagonal ring touching the central coin. The centers of any three adjacent coins form an equilateral triangle. Since this arrangement of coins will be used to illustrate many of the fundamental concepts of symmetry, it is useful to have a name for it: close-packing. The close-packed arrangement is the tightest possible packing of circles in a plane. Oxygen atoms in aluminum oxide (of which ruby is a gem variety) or ferrous oxide (common rust) have the same arrangement. By adding more rows of coins around the outside in the same way, larger hexagons can be built. If different numbers of rows are added along each side of the array of coins, the resulting shape will no longer be a regular hexagon. Since an infinite number of shapes are possible, it is impossible to catalog them all. Nevertheless, the shapes are all dictated by the pattern of the coins, and all have common properties. For example, it is impossible to make a perfect square with this pattern of coins. It is the arrangement of the coins that is important, rather than the external shape that they build. Yet, by studying the shapes constructed using the close-packing arrangement, it is possible to deduce the arrangement of the coins.

All crystalline materials have a repeating arrangement of atoms called a lattice. This regularity and repetition of atomic patterns identifies a material as crystalline. Such familiar materials as concrete, most rocks, and nearly all metals are made of crystalline materials, despite their lack of outward crystalline form. In fact, well-formed crystals are quite rare in nature. The geometrical rules that govern the close-packed arrangement of coins are called symmetry.

Although this article deals with the shapes and atomic arrangements of crystals, it is important to understand that symmetry concepts can be applied to any object or pattern. Thus, the same geometrical concepts can describe coins on a table, atoms in a crystal, tiles on a floor, or cells in a honeycomb.

The law that governs the close-packing of coins is a simple one: Around every coin there are (or could be) six other identical coins. This rule is an example of rotational symmetry: An element of a pattern is repeated by rotation around a center, or rotation axis. In other words, when a pattern is rotated through 360 degrees, it repeats its original appearance a certain number of times. The arrangement of six coins around a central coin is said to have six-fold rotational symmetry, often abbreviated as six-fold symmetry. There are other kinds of symmetry in the coin pattern as well. For example, three adjacent coins enclose a triangular hole. The center of the triangle is an axis of three-fold symmetry. The point of contact between any two coins is an axis of two-fold symmetry: The arrangement of coins looks the same after a rotation of 180 degrees.

Any crystal large enough to see will be millions of atoms wide, so crystallographers usually ignore the finite dimensions of crystals and treat crystal lattices as if they extended indefinitely in all directions.

Some objects have no symmetry at all. When rotated through 360 degrees, they match their original appearance only once. Therefore, objects with no symmetry are often said to have one-fold symmetry. This seemingly complicated nomenclature actually makes it possible to use the same mathematical expressions to describe both symmetrical and nonsymmetrical patterns.

The close-packed coin arrangement has other kinds of symmetry besides rotation. If an imaginary line is drawn through the centers of a row of coins, the arrangement of coins on one side of the line is a mirror image of the arrangement on the other. For every coin left of the line, there is a corresponding coin the same distance to the right of the line. This example demonstrates reflection symmetry. Objects can have rotational symmetry without having reflection symmetry. A propeller is a familiar example. The propeller has rotational symmetry because the blades are equally spaced and identical; but because the blades are twisted, the left and right sides of the propeller are never mirror images of each other. Objects can also have reflection symmetry without having rotational symmetry. A diamond kite is a simple example.

The left and right halves are mirror images, but the kite has no (or one-fold) rotational symmetry.

The close-packed coin arrangement displays one other important kind of symmetry.

Any coin in it is identical to any other coin. If the pattern is large enough, the entire pattern can be moved parallel to any row of coins and still look the same. This symmetry is called translation. All repeating patterns have translational symmetry. Translation can be combined with reflection to produce glide symmetry. For example, the pattern "pbpbpbpb" has glide symmetry; the first p is translated, then reflected vertically to form b, which is then translated and reflected to form the second p, and so on.

Combinations of symmetry that do not involve translation are called point groups because they describe symmetry operations at a single point. Combinations of symmetry that include translation are called space groups because they describe patterns in space. Not all combinations of symmetry are possible. For example, the individual stars on an American flag have five-fold rotational symmetry, but the overall arrangement of stars does not. Repeating patterns, whether atoms in a crystal or stars on a flag, can have only one-, two-, three-, four-, or six-fold symmetry. The letter sequence "LLLLL" repeated horizontally and vertically across the page is a repeating pattern with only one-fold symmetry. The letter sequence "ZZZZZ" repeated the same way is a pattern with two-fold symmetry. The close-packed coin pattern is a repeating pattern with two-, three-, and six-fold symmetry. A grid of square tiles has four-fold symmetry.

Two-dimensional patterns can have reflection as well as rotational symmetry. For example, the patterns "HHHHH" and "SSSSS" both have two-fold symmetry, but the first has reflection symmetry while the second does not. Two-dimensional patterns can have any of seventeen possible symmetry combinations, or space groups. These patterns can all be generated by enclosing a basic design in a parallelogram and then repeating the parallelogram indefinitely in two directions. Called a unit cell, this parallelogram can have one of five unit cell shapes. Unit cells for patterns with one-fold rotational symmetry, and some with two-fold rotational symmetry, are generally oblique parallelograms. Patterns with both two-fold rotational symmetry and reflection symmetry have unit cells that are rectangles or rhombuses. Patterns with four-fold rotational symmetry have square unit cells. Patterns with three- or six-fold rotational symmetry have rhombus-shaped unit cells with angles of 60 and 120 degrees.

Three-dimensional objects can have more complex combinations of symmetry. For example, a cube has three mutually perpendicular axes of four-fold symmetry through the centers of each opposing pair of faces. Cubes also have symmetries that are much harder to see. A line through any pair of diagonally opposite corners of a cube is an axis of three-fold symmetry, and a line through the centers of diagonally opposite edges is an axis of two-fold symmetry. Cubes also have reflection symmetry; every point on the top half of a cube corresponds to an identical point on the bottom half.

A new kind of symmetry appears in three dimensions: inversion symmetry. Inversion symmetry can be pictured as projecting a pattern through a central point and then an equal distance out the other side. If a person places his or her hands in contact on a table, with one palm up and the other down, the hands have inversion symmetry about their point of contact.

Every point on the top of the left hand corresponds to an identical point on the bottom of the right hand. An index card with a figure 6 on one side and a figure 9 on the opposite side also has inversion symmetry. Rotation and inversion can be combined to produce rotoinversion symmetry. In rotoinversion symmetry, a pattern is rotated around a symmetry axis and then inverted through a center of symmetry until it coincides with its original position. Rotoinversion is often a difficult symmetry to visualize because it involves faces on opposite sides of a crystal, all of which cannot be seen from a single vantage point. The seam on a baseball has four-fold rotoinversion symmetry.

Not all possible three-dimensional symmetries can occur in crystals. For example, a solid with twelve regular pentagon faces, three of which meet at every vertex, is called a pentagonal dodecahedron. It has five-fold symmetry axes in six different directions and thus cannot be a basis for a repeating pattern. (There are other kinds of dodecahedron that do not have five-fold symmetry and often occur as crystals, but the pentagonal dodecahedron cannot.) There are thirty-two possible point groups for crystals, called crystal classes. Some are more common than others; some are rare. One point group that is very easy to visualize consists of a three-fold rotation axis at right angles to a reflection plane. A crystal with this symmetry might resemble two identical three-sided pyramids stuck base-to-base, for example. Yet, as simple as this symmetry class is, no known natural or synthetic materials crystallize in this class.

Just as two-dimensional patterns have translational symmetry, so do three-dimensional patterns. In three-dimensional patterns, translation plus rotation results in screw rotational symmetry. In three-fold screw rotation, for example, a pattern is rotated around an axis and translated parallel to the axis, so that every third repetition has the same alignment as the original. The total number of three-dimensional space groups, or possible ways in which patterns can repeat in three dimensions, is 230. As in two dimensions, three-dimensional repeating patterns can be generated by enclosing a basic pattern in a box or unit cell and repeating the box in three dimensions. Instead of parallelograms, the boxes are three-dimensional equivalents called parallelepipeds: boxes with parallelogram faces and opposite faces parallel and identical. There are fourteen fundamental unit cells in three dimensions, which can be grouped in turn into six families called crystal systems.

A unit cell whose edges are all the same length and all perpendicular is a cube. Crystals with cubic unit cells are called isometric. Isometric crystals have three mutually perpendicular two-fold or four-fold axes, plus four three-fold axes between diagonally opposite corners of the unit cell. Halite or table salt, gold, copper, and diamond all have isometric crystals. In a survey of 3,837 compounds, crystallographer F. Donald Bloss found that 26 percent belonged to the isometric system. A unit cell with a square cross section and perpendicular edges, but one edge length different from the other two, is called tetragonal. About 12 percent of compounds, including the gem mineral zircon, have tetragonal crystals. Unit cells with perpendicular but unequal edges are called orthorhombic. Orthorhombic crystals, which account for about 20 percent of all crystalline materials, have two or more two-fold symmetry axes or mirror planes.

Sulfur is a common material with orthorhombic crystals.

The three systems--isometric, tetragonal, and orthorhombic--exhaust the possibilities for unit cells with perpendicular edges. If two adjacent edges of a unit cell are at right angles, but the third is not, the resulting unit cell is called monoclinic. A monoclinic unit cell resembles a box without top or bottom, sheared slightly out of shape. Monoclinic crystals have a single mirror plane or two-fold axis of symmetry, and make up about 21 percent of known crystalline materials. Table sugar, borax, and gypsum are common materials with monoclinic crystals. If two or three of the angles between edges of a unit cell are not right angles, the resulting unit cell is called triclinic. Triclinic crystals have only one-fold symmetry. Only 2 percent of crystalline solids are triclinic, but they include many of the feldspar minerals, which are the most abundant group of minerals in the earth's crust.

A special type of unit cell characterizes crystals with three- or six-fold symmetry. This unit cell has two sets of equal edges intersecting at angles of 60 and 120 degrees, and a third set of edges is perpendicular to the other two. Crystals with six-fold symmetry are called hexagonal; those with three-fold symmetry are called trigonal. The gem mineral beryl (emerald) crystallizes in the hexagonal system. Quartz is hexagonal at high temperatures but trigonal at low temperatures. Ice is another common hexagonal material. One special type of trigonal crystal can be described with an alternative unit cell whose edges are all equal. Called rhombohedral, this unit cell can be pictured as a cube flattened or stretched along one of its diagonal axes. Many common materials crystallize in the rhombohedral system, notably corundum, hematite or common rust, and calcite. Hexagonal and trigonal crystals account for about 19 percent of all crystalline materials.

The close-packed arrangement of coins is the tightest packing of circles in a plane. In three dimensions, there are two equally dense ways to pack spheres, or atoms. If a layer of atoms has close-packing, there are two sets of triangular voids between the atoms. If rows of atoms extend horizontally, one set of voids can be considered as pointing up and the other as pointing down. There are two possible ways to place the next layer of atoms: over either the upward voids or the downward voids. Thus, there are three possible layer locations: the original layer, a layer over the upward voids, or a layer over the downward voids. If only two of the possible layers occur, or if all three occur in random sequence, the resulting lattice has only the hexagonal symmetry of the individual layers. This atomic arrangement is called hexagonal close-packing. If the three possible layers repeat in strict rhythmic succession, the resulting arrangement actually has cubic or isometric symmetry and is called cubic close-packing. True close-packing can occur only if the atoms are identical or nearly so, as in a pure element. Magnesium and some platinum alloys have hexagonal close-packing. Many elements have cubic close-packing, including gold, silver, copper, and iron. In many materials, the anions (negatively charged atoms) are much larger than the cations (positively charged ions), and the anions approximate a close-packed arrangement, with the cations filling the interatomic voids. The oxygen atoms in aluminum oxide (corundum) and ferrous oxide (hematite) have approximate hexagonal close-packing.

Approximate cubic close-packing occurs among the chlorine atoms in sodium chloride, or table salt, and among the oxygen atoms in the gem mineral spinel.

Applications

An understanding of crystal symmetry vastly simplifies the task of determining the atomic structure of materials. If the position of one atom in a crystal structure is known, the positions of many other atoms may be determined automatically. In addition, almost every physical property of materials is closely related to their crystal structure and symmetry. For example, quartz is widely used in electronics because it becomes electrically charged when it is stressed and thus can be used to make oscillators for timing circuits. This property, called piezoelectricity, is found only in crystal classes that lack centers of symmetry.

Optical properties of materials are also closely related to their crystal symmetries. This relationship is important to gem cutters. When light enters a crystal, it usually splits into two beams of light, a process called double refraction. This phenomenon is exploited in some optical instruments, but it is a nuisance to gem cutters, because it causes the internal reflections within the gem to appear fuzzy and dulls the luster of the gem. Double refraction does not occur in isometric gems such as diamond or garnet. In tetragonal gems such as zircon and hexagonal gems such as emerald, ruby, or sapphire, light traveling parallel to the principal symmetry axis of the crystal is not doubly refracted. Round cuts of these gems are oriented so that the symmetry axis of the crystal parallels the symmetry axis of the cut gem. Oblong cuts are usually oriented so the symmetry axis is parallel to the long dimension of the gem. Double refraction occurs but is masked somewhat by the shape of the gem. Orthorhombic, monoclinic, and triclinic minerals have more complex optical properties, but they are rarely cut as faceted gems.

Crystals form by stacking unit cells to build larger shapes. It is possible to stack cubic, tetragonal, or orthorhombic unit cells to build a rectangular box shape, and it would not be possible to tell which unit cell was involved from that shape alone. Unit cells stack in other ways as well, and the simplest stacking arrangements tend to be the most common. If the edges of a crystal are truncated, with the unit cells forming a stepped pattern, it is possible to determine the proportions of the unit cell from the slope of the face. (The unit cells are so small compared to wavelengths of light as to be individually invisible, so the face appears perfectly smooth.)

Fortunately, faces with similar stacking patterns tend to develop at similar rates, so the outward form of a crystal is often a direct indicator of its internal symmetry.

Even when faces of crystals grow at quite different rates, so that the outward form of the crystal is very asymmetrical, it is possible to determine the internal symmetry of the crystal.

Regardless of whether a face with a given stacking arrangement is large or small, it will always have the same orientation in space, dictated by the shape of the unit cell for that material. A crystallographer measures the orientations of all visible crystal faces and plots them on a projection similar to a map projection. If enough faces are measured, the resulting plot will have an obvious symmetry that is the same as the symmetry of the crystal.

Materials wholly lacking in visible crystals can be studied with X rays. X rays reflected off layers of atoms in a crystal will produce patterns on photographic film that provide clues to the symmetry of the crystal. Many different techniques have been developed that use X rays to determine crystal symmetry and even the actual arrangements of atoms within materials.

Furthermore, materials that appear noncrystalline to the eye may reveal very distinct crystals when highly magnified with an optical or electron microscope. Almost all minerals have crystalline structure when studied by such techniques, including many materials once thought to be noncrystalline.

Context

Many of the most fundamental discoveries about the basic properties of matter are closely connected with the study of crystal symmetry. The first major discovery about the true nature of crystals was made by the Danish scientist Nicolaus Steno (Niels Stensen). In 1669, he postulated that the angles between corresponding faces of crystals are constant, regardless of the size or shape of the faces. This law became known as Steno's law, or the law of constancy of interfacial angles.

The French mineralogist Rene-Just Hauy demonstrated why Steno's law worked. He was intrigued by the way certain minerals, notably calcite or halite, always split or cleaved into smooth-sided fragments of similar shape; he postulated that the shape of the fragments was the shape of the fundamental building block of the mineral. He had many small blocks cut in the shape of calcite cleavage fragments and discovered that he could account for all the crystal faces of calcite by stacking these blocks in simple ways. He also found that he could account for the crystal forms of other minerals with other units, such as cubes. Hauy not only discovered the unit cell and showed how it related to crystal form but also provided additional evidence that matter was composed of discrete units, which further supported the emerging concept of atoms.

Later scientists systematized the study of crystals. Between 1815 and 1825, the German scientists Christian Weiss and Friedrich Mohs identified the major crystal systems. In 1830, Johann Hessel, another German scientist, described the thirty-two point groups. Between 1880 and 1891, the 230 crystallographic space groups were independently discovered by the Soviet mineralogist E. S. Fedorov, the German mathematician Arthur Schoenflies, and the British scientist William Barlow.

When X rays were discovered in 1895, there was great debate about their nature. The German scientist Max von Laue passed a beam of X rays through a crystal and showed that they underwent diffraction, thus demonstrating that X rays were very short-wavelength electromagnetic radiation. The father-and-son team of Sir William Henry and Sir Lawrence Bragg then used the diffraction of X rays to determine the structures of unknown crystals. This technique is the basis of all modern crystal structure study. With crystals serving as a means of measuring the wavelength of X rays, Henry Moseley demonstrated that the wavelengths of X rays emitted by different elements were related to the square of their atomic numbers. Moseley showed that atomic numbers were integers, that there were no undiscovered elements between adjacent elements, and established a physical basis for the periodic table.

In 1974, the mathematician Roger Penrose discovered a new type of geometrical pattern. These patterns were regular, in that they were composed of only two different shapes, but nonrepeating. So-called Penrose patterns have two fundamental units rather than a single unit cell, and have five-fold symmetry, which is impossible for a repeating pattern. In 1982, the Israeli scientist Dany Schechtman discovered unusual metallic compounds that appeared to be based on Penrose's patterns and crystallized in forms with five-fold symmetry. The atomic arrangements in these materials do not repeat indefinitely as do the atoms in normal crystals; yet, they have highly regular arrangements. These materials are called quasicrystals. The study of these unexpected and beautiful patterns is a highly active area of research in crystallography.

Principal terms

INVERSION SYMMETRY: symmetry in which each point in a three-dimensional pattern corresponds to an identical point on the opposite side of the pattern

POINT GROUP: a combination of symmetry operations that acts around a single point

REFLECTION SYMMETRY: symmetry in which a pattern on one side of a line or plane is a mirror image of the pattern on the opposite side

ROTATIONAL SYMMETRY: symmetry in which every point in a pattern is repeated at equal angles around a point or line called the rotation or symmetry axis

SPACE GROUP: a combination of symmetry operations that applies to a repeating pattern

SYMMETRY: a mathematical or geometrical rule that describes the relationship among points in an object or pattern

Bibliography

Asimov, Isaac. "The Nobel Prize That Wasn't." In THE STARS IN THEIR COURSES. Garden City, N.Y.: Doubleday, 1971. The story of the role played by crystals in understanding the nature of X rays and establishing the physical basis for the periodic table. The title refers to Moseley, whose work played a pivotal role in these discoveries.

Bloss, F. Donald. CRYSTALLOGRAPHY AND CRYSTAL CHEMISTRY. New York: Holt, Rinehart and Winston, 1971. A well-illustrated and very thorough introduction to symmetry and its application to crystalline structures. Intended for upper-level college students. The mathematical treatment is advanced in some places where the subject matter requires it, but much of the book requires only modest mathematical training.

Gardner, Martin. "Extraordinary Nonperiodic Tiling That Enriches the Theory of Tiles." SCIENTIFIC AMERICAN 236 (January, 1977): 110-121. The first popular description of the remarkable Penrose tiles. These tiles produce beautiful patterns and they appear to be the geometric basis for quasicrystals.

Grunbaum, Branko, and G. C. Shephard. TILINGS AND PATTERNS. New York: W. H. Freeman, 1987. Virtually everything known about two-dimensional patterns is in this superb book or the references that are cited. Symmetry is described in mathematical rather than crystallographic terms. There is little computational mathematics, but the level of mathematical logic is often advanced. The beautiful illustrations make the book worthwhile for readers without mathematical background.

Klein, Cornelis, and Cornelius S. Hurlbut. MANUAL OF MINERALOGY. New York: Wiley, 1985. A college-level text on mineralogy with an extensive discussion of crystal symmetry. Includes a well-illustrated discussion of each of the crystal classes, as well as chapters on modern methods of crystal study.

LaBrecque, Mort. "Opening the Door to Forbidden Symmetries." MOSAIC 18, no. 4 (1987): 2-23. A nontechnical summary of the discovery of quasicrystals and the controversy over their structure. Includes a discussion of Penrose patterns and photographs of actual quasicrystals.

Pearce, Peter, and Susan Pearce. POLYHEDRA PRIMER. New York: Van Nostrand Reinhold, 1978. A simple and nontechnical introduction to the study of three-dimensional forms, with abundant and attractive illustrations.

Schattschneider, Doris. "The Plane Symmetry Groups: Their Recognition and Notation." AMERICAN MATHEMATICAL MONTHLY 85 (1978): 439-450. Although published in a scholarly mathematical journal, this article is one of the clearest explanations of the seventeen-plane space groups. Each space group is illustrated with a diagram of its symmetry and an example from art.

Steinhardt, Paul J. "Quasicrystals." AMERICAN SCIENTIST 74, no. 5 (1986): 586-597. A nontechnical summary of quasicrystals and their possible atomic structure. Describes how Penrose patterns can produce the symmetry and X-ray patterns observed in actual quasicrystals.

Crystal lattice systems and bravais lattices

Crystal systems

The Fundamental Constants of Nature

The Structure of Ice

Polarization of Light

Reflection and Refraction

Defects in Solids

Electrical Properties of Solids

Magnetic Properties of Solids

Optical Properties of Solids

Thermal Properties of Solids