in plant cells, crystalline deposits in living or dead cells or their membranes, consisting chiefly of calcium oxalate, silica (SiO₂), or, less often, proteins or carotenes.

Crystals in plant cells may appear singly; in aggregates of small grains, or “sand”; in concretions, or druses; or as acicular crystals, or raphides and steloids. Some crystals are present only in special, larger, cells. The crystals may fill the cell entirely and deform it. Silica is deposited predominantly in the cell walls (often in the epidermis—for example, in horsetails and cereal grains). Protein crystals are found in nuclei, plastids, and aleurone grains. Carotene crystals are found in chromoplasts. Many crystals accumulate in the dead cells of leaves and bark. The shape and distribution of the crystals are specific for a number of plants, which may prove useful in taxonomy.

solids that have the natural form of regular polyhedrons (Figure 1) as a result of the ordered arrangement of the atoms, forming a three-dimensional periodic array, the crystal lattice. Crystals are an equilibrium state of solids. A specific crystalline atomic structure corresponds to every chemical substance that is in the crystalline state under given thermo-dynamic conditions (temperature and pressure). Crystals have symmetry of atomic structure and the corresponding macroscopic symmetry of external form, as well as anisotropy of physical properties. A crystal that has been grown under nonequilibrium conditions and does not have regular faceting or has lost it as a result of some type of processing retains the primary feature of the crystalline state (the lattice atomic structure and all properties determined by it).

Figure 1. Constancy of interfacial angles of a given crystal for different face development

Most natural or industrial solid materials are polycrystalline and consist of a great number of separate, randomly oriented small crystal grains, sometimes called crystallites. Such, for ex-ample, are many rocks, industrial metals, and alloys. Natural or synthetic individual crystals whose specific structure extends throughout an entire specimen are called single crystals.

Crystals most often form and grow from the liquid phase (a solution or melt); they may be produced from the gas phase or by phase transition in the solid phase. Crystals of various sizes are found in nature—from the very large crystals of quartz (rock crystal), fluorite, and feldspar, weighing up to several hundred kilograms, to the small crystals of diamonds and other minerals. Various crystals are grown (synthesized) for scientific and technical purposes in laboratories and factories. Such complex natural substances as proteins (Figure 1 ,c), and even viruses, can also be obtained as crystals.

Geometry. Crystals grown under equilibrium conditions have the form of regular polyhedrons with some type of symmetry; the faces of the crystals are flat, and the edges between faces are rectilinear. The angles between corresponding faces of crystals of a given substance are constant (Figure 1). This is the first law of

Figure 2. The law of integers

geometric crystallography, the law of constant angles (N. Steno, 1669), which also may be stated as follows: as a crystal grows, its faces shift parallel to themselves. The measurement of interfacial angles (goniometry), which was used extensively as a means of identifying the chemical composition of crystals (E. S. Fedorov; P. Groth) before the advent of X-ray diffraction analysis, has not lost its value.

The second fundamental law of geometric crystallography—the law of integers—is a macroscopic consequence of the microperiodicity of a crystalline substance, which consists of spatially repeating unit cells that usually have the form of a parallelepiped with edges (the periods of the lattice) equal to a, b, and c. Any atomic plane of a crystal lattice, to which a face of the crystal corresponds, intersects the coordinate axes at whole numbers k, m, and n of the lattice periods (Figure 2). The inverse numbers (h, k, and l), which are also integers, are called the crystallographic indexes (Miller indexes) of the faces and atomic planes. Crystals generally have faces with indexes of low values, such as (100), (110), and (311). The magnitudes a, b, and c of the lattice periods and of the angles between them, α, β, and γ, are measured by X-ray diffraction. The choice of coordinate axes is made on the basis of certain rules according to the symmetry of the crystal.

Crystalline polyhedrons are symmetrical: their faces and edges can be superposed on each other by means of symmetry operations. Each operation is carried out with respect to a plane, axis, or center of symmetry (Figure 3). In all there are 32 classes of symmetry of crystalline polyhedrons (32 symmetry point groups). Each class is characterized by a specific set of symmetry elements. The symmetry elements of point groups are axes of rotation (Figure 3,a); a center of symmetry (Figure 3,c); the twofold, fourfold, or sixfold rotoinversion axes; and planes of symmetry (Figure 3,b). The 32 classes are grouped into seven systems, according to the presence of the characteristic symmetry elements: triclinic, monoclinic, and rhombic (the lower systems); tetragonal, hexagonal, and trigonal (the middle systems); and cubic (the higher system).

Figure 3. Simplest symmetry elements of crystals: (a) axis of symmetry (in this case, second-order), which superposes a figure on itself by rotation through 360° /N (where N is the order of the axis of symmetry); (b) plane of symmetry m, which superposes the figure by “reflection”; (c) center of symmetry, which acts simultaneously as rotation and reflection

The aggregate of crystallographically identical faces (that is, faces that can be superposed by means of the symmetry operations for the given class) is called the simple form of the crystal. There are 47 simple forms, and in each class of crystals only some of them can be realized. A given crystal may have faces of the same simple form (Figure 4,a), but more often it has a combination of forms (Figure 4,b and c).

Figure 4. (a) Some simple shapes of crystals, (b) principle of formation of combinations of simple forms, (c) actually observed facets of some crystals

If a crystal belongs to a class containing only simple axes of symmetry (but without planes, a center of symmetry, or inversion axes) it may crystallize in specularly equivalent forms. This phenomenon is called enantiomorphism, and the corresponding enantiomorphous forms are called the right-hand and left-hand forms (Figure 5).

Figure 5. “Right-hand” and “left-hand” quartz

Under nonequilibrium conditions of crystal formation, the ideal form of crystals (their habit) may change. Differences in the supply of the substance, in rates of growth, and in molecular

processes during crystallization may lead to exceptional diversity of the crystalline forms, such as curvature of faces and edges or the appearance of laminar, needle-shaped, filamentary, and dendritic snowflake-like crystals. These peculiarities are used in the technology of growing crystals of various shapes (dendritic ribbons of germanium and thin films of various semiconductors). The shape of the desired product—such as a tube, rod, or plate —is imparted to some crystals even during their growth. If a large number of crystallization centers are formed immediately within the melt, the growing crystals assume the form of irregular grains upon encountering each other.

Atomic structure. The external shape of a crystal and its affiliation with a given class and system are determined by the crystal lattice with its characteristic symmetry operation of indefinitely repeating translations. As a result, in addition to the symmetry elements mentioned above (axes of symmetry, planes, and center), symmetry operations with infinite periodicity, such as screw axes of symmetry and glide-reflection planes, are also possible in the structure of crystals. A specific combination of these elements is a Fedorov crystal space group. There are 230 Fedorov groups, which are distributed among the 32 classes of crystal symmetry.

The methods of structural analysis of crystals (X-ray diffraction analysis, electron diffraction, and neutron diffraction) make possible determination of the dimensions of the unit cell of a crystal, the Fedorov group, the arrangement of the atoms in the cell (the interatomic distances), the nature of the thermal vibrations of the atoms, the distribution of the electron density between the atoms, and the orientation of the magnetic moments. The crystal structures of more than 20,000 compounds—from crystals of elements to highly complex crystals of proteins—have already been studied (see Table 1 and Figure 6).

Figure 6. Atomic structures: (a) copper, (b) NaCI, (c) Cu2O, (d) graphite, (e) K₂PtCI₆, (f) phthalocyanine

The generalizing of this vast material is the object of crystal chemistry. Crystal structures are classified according to their chemical composition, which basically determines the type of chemical bond; according to the proportion of the components in the chemical formula (for example, elements, compounds of the types AX, AX₂, and ABX₃); and according to the relative positions of the atoms (layer, chain, and coordination lattices).

Upon a change in temperature or pressure, the structure of a crystal may change. Some crystalline structures (phases) are metastable. When a given substance has several crystalline phases and hence crystals of different structure, this condition is called polymorphism (white and gray tin, diamond and graphite, and various modifications of quartz). Conversely, different com-pounds may have an identical crystal structure—they may be isostructural (isomorphic).

The distribution of crystals according to space groups (and according to classes and systems) is irregular. Usually the simpler the chemical formula of a substance, the higher the symmetry of its crystal. Thus, nearly all metals have a cubic or hexagonal structure, and the same is true of simple chemical compounds such as the alkali halides. Increasing complexity of the chemical formula of a substance leads to a decrease in the symmetry of its crystal. Organic (molecular) crystals almost always belong to systems of lower symmetry.

The type of chemical bond between atoms in crystals determines many of their properties. Covalent crystals with electrons localized in strong bonds have great hardness, low electric conductivity, and high indexes of refraction. Conversely, metal crystals with free electrons conduct electric current and heat readily and are plastic and opaque. Ionic crystals have intermediate characteristics. The weakest bonds (van der Waals bonds) are found in molecular crystals. They are readily fusible, and their mechanical characteristics are low. Liquid crystals, amorphous bodies, and glasses have a lower atomic ordering than do crystals.

Structure of real crystals. Because of the disruption of equilibrium conditions of growth, the capture of impurities during crystallization, and the influence of various other factors, the ideal structure of a crystal always has some imperfections. Among them are point defects, or vacancies (missing atoms); replacements of atoms of the base lattice by impurity atoms; the introduction of foreign atoms into the lattice; and linear defects, or dislocations (imperfections in the packing order of the atomic layers). The measured introduction of small quantities of impurity atoms that replace atoms of the base crystal is extensively used in industry to change the properties of crystals—for example, the introduction of atoms of groups III and V of the periodic table into germanium and silicon crystals makes possible the production of semiconductors with hole and electron conductivity. Another example is the crystals used in quantum electronics: rubies consisting of Al₂O₃and 0.05 percent chromium, and garnets consisting of Y₂Al₅₁₂ and 0.05 percent neodymium.

As crystals grow, their faces have different capture coefficients for impurities, which determine the sectorial structure of the crystals (Figure 7). Periodic changes in the concentration of a captured impurity may also take place; this produces a zonal structure. In addition, macroscopic defects such as inclusions and stressed regions, are almost inevitably formed in the process of growth.

Figure 7. Sectorial structure of a crystal

All real crystals have a mosaic structure: they are divided into blocks, small regions (10⁻⁴ cm) that have nearly ideal order but are disoriented with respect to each other by small angles (approximately a few minutes).

Physical properties. The main distinguishing feature of the properties of crystals is their anisotropy, or dependence on direction; in isotropic substances (liquids and amorphous solids) or pseudoisotropic substances (polycrystals) the properties are in-dependent of direction. In examining many properties of crystals the discreteness of the atomic structure of the crystals does not play a role, and the crystal may be considered a uniform anisotropic medium. The symmetry of many properties of crystals may be described by using the limiting symmetry point groups. The coordination of the symmetry classes of crystals and of the symmetry of their physical properties, as well as the relation between the symmetry of the properties and the symmetry of external factors (such as external fields), are determined by the Curie and Neumann principles. The presence or absence of various elements of point symmetry makes it possible to indicate in which of the 32 classes various properties are possible and also determines the type of tensors that describe the properties. For example, pyroelectricity is possible in crystals of ten classes that have one axis of symmetry or a plane of symmetry that is coinci-dent with it; piezoelectricity is possible in crystals of 20 classes that do not have a center of symmetry.

For a crystal of a given class it is possible to indicate the symmetry of its properties. Thus, cubic crystals are isotropic with respect to the passage of light, to electric and thermal conduction, and to thermal expansion but are anisotropic with respect to elastic, electrooptical, and piezoelectric properties. For crystals of the middle symmetry systems (such as quartz), the principal constants of the tensors describing the properties are determined by their values along and perpendicular to the main axis of symmetry, and the properties in intermediate directions may be calculated from the principal values. Crystals of the lower symmetry systems are the most anisotropic.

All properties of a crystal are interconnected and result from its crystalline structure (the arrangement of the atoms and the binding forces between them). These forces are caused by the electronic structure of the atoms or molecules comprising the crystal lattice. Thus, a number of properties of the crystal, including thermal, elastic, and acoustical properties, depend directly on the interatomic interactions. Electrical, magnetic, and optical properties depend largely on the distribution of electrons over the energy levels (that is, on the electron spectrum). Thus, the very high electric conductivity of metals or the relatively low conductivity of dielectrics and semiconductors are related to a high or low concentration of conduction electrons. In some crystals the ions forming the lattice are arranged in such a way that the crystal is spontaneously electrically polarized (as in pyroelectrics). A high degree of such polarization is characteristic of ferroelectrics.

Many properties of crystals, such as strength and plasticity, color, and luminescent properties, depend decisively not only on symmetry but also on the number and types of defects. Because of the presence of dislocations, plastic deformation of crystals takes place under stresses tens and hundreds of times less than the theoretically calculated value. In dislocation-free crystals (of germanium and silicon) the strength is ten to 100 times greater than in ordinary crystals. The color of many crystals (their absorption of light) is related to the presence of various impurity atoms.

Uses. Piezoelectric and ferroelectric crystals (such as quartz) are used in radio engineering. A large area of semiconductor electronics (radio-engineering and calculating devices) is based on semiconducting crystals (such as germanium and silicon) or on integrated circuits based on them. Crystals of magnetic dielectrics and various types of ferrites are used in memory units of very high capacity. Crystals (such as rubies and yttrium-aluminum garnets) are of exceptional importance in quantum electronics. Crystals with electrooptical properties are used in light-beam control technology. Pyroelectric crystals are used to measure slight changes in temperature, and piezoelectrics, piezomagnetics, and piezoresistors are used to measure mechanical and acoustical effects. The good mechanical properties of superhard crystals (such as diamonds) are used in materials processing and in drilling. Rubies, sapphires, and other crystals are used as bearing elements in watches and other precision instruments. The jewelry industry is using not only natural gemstones but also synthetic crystals to an increasingly great extent. The list of various industrially produced synthetic crystals has thousands of items, and the production of some crystals runs as high as hundreds of tons per year.