Symmetry of crystals. Scientific work on the topic "symmetry of crystals" Structure and symmetry of crystals

SYMMETRY OF CRYSTALS- the property of crystals to combine with themselves during rotations, reflections, parallel transfers, or during part or a combination of these operations. ext. The shape (cut) of a crystal is determined by the symmetry of its atomic structure, and the edges also determine the symmetry of the physical structure. properties of the crystal.

Rice. 1. a - quartz crystal; 3 - axis of symmetry of the 3rd order, - axes of the 2nd order; b - crystal of aqueous sodium metasilicate; m - plane of symmetry.

In Fig. 1 A a quartz crystal is depicted. Ext. its shape is such that by rotating 120° around axis 3 it can be aligned with itself (compatible equality). Sodium metasilicate crystal (Fig. 1, b)is transformed into itself by reflection in the plane of symmetry m (mirror equality). If - a function that describes an object, e.g. the shape of a crystal in three-dimensional space or k--l. its property, and the operation transforms the coordinates of all points of the object, then g is an operation, or symmetry transformation, and F is a symmetric object if the following conditions are met:

In max. In the general formulation, symmetry is the immutability (invariance) of objects and laws under certain transformations of the variables describing them. Crystals are objects in three-dimensional space, so the classic. The theory of SK is the theory of symmetric transformations of three-dimensional space into itself, taking into account the fact that internal. the atomic structure of crystals is discrete, three-dimensional periodic. During symmetry transformations, space is not deformed, but transformed as a rigid whole. Such transformations are groove. orthogonal or isometric and. After the symmetry transformation, the parts of the object that were in one place coincide with the parts that are in another place. This means that a symmetrical object has equal parts (compatible or mirrored).

SK manifests itself not only in their structure and properties in real three-dimensional space, but also in the description of energy. electron spectrum of the crystal (see Zone theory), when analyzing processes X-ray diffraction, neutron diffraction And electron diffraction in crystals using reciprocal space (see Reverse lattice)and so on.

Symmetry groups of crystals. A crystal may have more than one characteristic. . Thus, a quartz crystal (Fig. 1, A)is combined with itself not only when rotated 120° around its axis 3 (operation gi), but also when rotating around an axis 3 at 240° (operation g 2), & also when turning 180° around the axes 2 X, 2 y, 2 W(operations g 3, g 4, g 5). Each symmetry operation can be associated with a symmetry element - a straight line, a plane or a point, with respect to which the given operation is performed. E.g. axis 3 or axes 2 x, 2 y, 2 w are axes of symmetry, plane T(Fig. 1,b) - plane of mirror symmetry, etc. Set of symmetry operations (g 1 , g 2 , ..., g n ) of a given crystal forms a symmetry group in the sense of mathematics. theories groups. Consistent performing two symmetry operations is also a symmetry operation. In group theory this is referred to as the product of operations:. There is always an identity operation g 0, which does not change anything in the crystal, is called. identification, it geometrically corresponds to the immobility of an object or its rotation by 360° around any axis. The number of operations forming a group G is called. group order.

Symmetry groups of space transformations are classified: by number P dimensions of space in which they are defined; by number T dimensions of space, in which the object is periodic (they are designated accordingly), and according to certain other characteristics. To describe crystals, various symmetry groups are used, of which the most important are point symmetry groups that describe the external appearance. crystal shape; their names also crystallographic. classes; space symmetry groups describing the atomic structure of crystals.

Point symmetry groups. Point symmetry operations are: rotations around the symmetry axis of order N at an angle equal to 360°/N(Fig. 2, a); reflection in the plane of symmetry T(mirror reflection, Fig. 2, b); inversion (symmetry about a point, Fig. 2, c); inversion turns (a combination of turning at an angle 360°/N s at the same time inversion, Fig. 2, d). Instead of inversion rotations, equivalent mirror rotations are sometimes considered. Geometrically possible combinations of point symmetry operations determine one or another point symmetry group, which is usually depicted in stereographic form. projections. During point symmetry transformations, at least one point of the object remains motionless - it transforms into itself. All elements of symmetry intersect in it, and it is the center of stereographic. projections. Examples of crystals belonging to different point groups are given in Fig. 3.

Rice. 2. Examples of symmetry operations: a - rotation; b - reflection; c - inversion; d - 4th order inversion rotation; d - 4th order helical rotation; e - sliding reflection.

Rice. 3. Examples of crystals belonging to different point groups (crystallographic classes): a - to class m (one plane of symmetry); b - to class (center of symmetry or center of inversion); a - to class 2 (one axis of symmetry of the 2nd order); g - to class (one inversion-rotary axis of the 6th order).

Point symmetry transformations described by linear equations

or coefficient matrix

For example, when turning around an axis x 1 at angle - =360°/N matrix D has the form:

and when reflected in a plane x 1 x 2 D has the form:

The number of point groups is infinite. However, in crystals, due to the presence of crystalline particles. lattices, only operations and, accordingly, symmetry axes up to the 6th order are possible (except for the 5th; in a crystal lattice there cannot be a symmetry axis of the 5th order, since using pentagonal figures it is impossible to fill space without gaps). The operations of point symmetry and the corresponding symmetry elements are designated by symbols: axes 1, 2, 3, 4, 6, inversion axes (center of symmetry or center of inversion), (also known as the plane of symmetry m), (Fig. 4).

Rice. 4. Graphic designations of elements of point symmetry: a - circle - center of symmetry, axes of symmetry, perpendicular planes drawing; b - axis 2, parallel to the drawing plane; c - axes of symmetry, parallel or oblique to the drawing plane; g - plane of symmetry, perpendicular to the plane of the drawing; d - planes of symmetry parallel to the drawing plane.

To describe a point symmetry group, it is enough to specify one or more. symmetry operations generating it, the rest of its operations (if any) will arise as a result of the interaction of the generating ones. For example, for quartz (Fig. 1, a) the generating operations are 3 and one of the operations 2, and there are 6 operations in total in this group. The international designations of groups include symbols of the generating operations of symmetry. Point groups are united according to the point symmetry of the shape of the unit cell (with periods a, b, s and angles) into 7 systems (Table 1).

Groups containing except Ch. axes N planes of symmetry T, are denoted as N/m, if or Nm, if the axis lies in the plane T. If the group in addition to Ch. has several axles. planes of symmetry passing through it, then it is denoted Nmm.

Table 1.- Point groups (classes) of crystal symmetry

Groups containing only rotations describe crystals consisting only of compatible equal parts (groups of the 1st kind). Groups containing reflections or inversion rotations describe crystals that have mirror-like parts (groups of the 2nd kind). Crystals described by groups of the 1st kind can crystallize in two enantiomorphic forms (“right” and “left”, each of which does not contain symmetry elements of the 2nd kind), but mirror-like to each other (see. Enantiomorphism).

Groups of SK carry geom. meaning: each of the operations corresponds, for example, to rotation around an axis of symmetry, reflection in a plane. Certain point groups in the sense of group theory, which takes into account only the rules of interaction of operations in a given group (but not their geometric meaning), turn out to be identical or isomorphic to each other. These are, for example, groups 4 and tt2, 222. In total there are 18 abstract groups isomorphic to one or more of the 32 point groups of S. k.

Limit groups. Functions that describe the dependence of various properties of a crystal on direction have a certain point symmetry that is uniquely associated with the symmetry group of the crystal facet. It either coincides with it or is higher than it in symmetry ( Neumann principle).

Regarding macroscopic properties, a crystal can be described as a homogeneous continuous medium. Therefore, many of the properties of crystals belonging to one or another point symmetry group are described by the so-called. limit point groups containing symmetry axes of infinite order, denoted by the symbol. The presence of an axis means that the object is aligned with itself when rotated through any angle, including an infinitesimal one. There are 7 such groups (Fig. 5). Thus, in total there are 32 + 7 = 39 point groups that describe the symmetry of the properties of crystals. Knowing the symmetry group of crystals, one can indicate the possibility of the presence or absence of certain physical properties in it. properties (see Crystal physics).

Rice. 5. Stereographic projections of 32 crystallographic and 2 icosahedral groups. The groups are arranged in columns by family, the symbols of which are given in the top row. The bottom row shows the limit group of each family and shows figures illustrating the limit group.

Space symmetry groups. The spatial symmetry of the atomic structure of crystals is described by space symmetry groups. They are called also Fedorovsky in honor of E. S. Fedorov who found them in 1890; these groups were independently developed in the same year by A. Schoenflies. In contrast to point groups, which were obtained as a generalization of the laws of crystalline forms. polyhedra (S.I. Gessel, 1830, A.V. Gadolin, 1867), space groups were a product of mathematical geology. theory that anticipated experiment. determination of crystal structure using x-ray diffraction. rays.

The operations characteristic of the atomic structure of crystals are 3 non-coplanar translations a, b, c, which determine the three-dimensional periodicity of the crystalline. grates. Crystallic. the lattice is considered to be infinite in all three dimensions. Such a math. the approximation is realistic, since the number of elementary cells in the observed crystals is very large. Transferring structure to vectors a, b, c or any vector where p 1, p 2, p 3- any integers, combines the structure of the crystal with itself and, therefore, is a symmetry operation (translational symmetry).

Phys. discreteness of crystalline a substance is expressed in its atomic structure. Space groups are groups of transformation into themselves of a three-dimensional homogeneous discrete space. The discreteness lies in the fact that not all points of such a space are symmetrically equal to each other, for example. an atom of one type and an atom of another type, nucleus and electrons. The conditions of homogeneity and discreteness are determined by the fact that space groups are three-dimensionally periodic, i.e., any group contains a subgroup of translations T- crystalline grate.

Due to the possibility of combining translations and point symmetry operations in a lattice in groups, in addition to point symmetry operations, operations and corresponding translation symmetry elements arise. component - helical axes of various orders and planes of sliding reflection (Fig. 2, d, f).

In accordance with the point symmetry of the shape of the unit cell (elementary parallelepiped), space groups, like point groups, are divided into 7 crystallographic syngony(Table 2). Their further division corresponds to broadcast. groups and their respective Right to the bars. There are 14 Bravais lattices, 7 of which are primitive lattices of the corresponding systems, they are designated R(except rhombohedral R). Others - 7 centered. gratings: base (side) - centered A(the face is centered bc), B(edge ac), C (ab); body-centered I, face-centered (on all 3 faces) F. Taking into account the centering for the translation operation t centering transfers corresponding to the center are added t c. If you combine these operations with each other t + t s and with the operations of point groups of the corresponding system, then 73 space groups are obtained, called. symmorphic.

Table 2.-Space symmetry groups

Based on certain rules, non-trivial subgroups can be extracted from symmorphic space groups, which gives another 157 non-symmorphic space groups. There are 230 space groups in total. Symmetry operations when transforming a point X into symmetrically equal to it (and therefore the entire space into itself) are written in the form: , where D- point transformations, - components of helical transfer or sliding reflection, - translation operations. Bravais group. Operations of helical symmetry and the corresponding elements of symmetry - helical axes have an angle. component (N = 2, 3, 4, 6) and translational t s = tq/N, Where t- translation of the lattice, rotation on occurs simultaneously with translation along the Zh axis, q- helical rotation index. General symbol for helical axles Nq(Fig. 6). The screw axes are directed along the ch. axes or diagonals of the unit cell. Axes 3 1 and 3 2, 4 1 and 4 3, 6 1 and 6 5, 6 2 and 6 4 correspond in pairs to right and left helical turns. In addition to the operation of mirror symmetry in space groups, planes of grazing reflection a are also possible, b, c: reflection is combined with translation by half of the corresponding grating period. Translation of a cell face by half the diagonal corresponds to the so-called. clinoplane slip n, in addition, in tetragonal and cubic. groups, “diamond” planes are possible d.

Rice. 6. a - Graphic designations of screw axes perpendicular to the plane of the figure; b - screw axis lying in the plane of the figure; c - planes of grazing reflection, perpendicular to the plane of Fig., where a, b, c are the periods of the unit cell along the axes of which sliding occurs (translational component a/2), n - diagonal plane of grazing reflection [translational component (a + b)/ 2], d - diamond sliding plane; g - the same in the drawing plane.

In table 2 gives the international symbols of all 230 space groups in accordance with their belonging to one of the 7 syngonies and the point symmetry class.

Broadcast the components of the microsymmetry operations of space groups do not manifest themselves macroscopically in point groups; for example, the helical axis in crystal cutting appears as a corresponding simple rotary axis. Therefore, each of the 230 groups is macroscopically similar (homomorphic) to one of the 32 point groups. For example, to a point group - ttt 28 space groups are mapped homomorphically.

Schönflies notation for space groups is the designation of the corresponding point group (for example, Table 1), to which the historically accepted ordinal number is assigned above, for example. . International notations indicate the Bravais lattice symbol and the generating symmetry operations of each group - etc. The sequence of arrangement of space groups in the table. 2 in international notations corresponds to the number (superscript) in Schönflies notations.

In Fig. Figure 7 shows an image of the spaces. groups - Rpta according to International Crystallographic. tables. The operations (and their corresponding elements) of the symmetry of each space group, indicated for the unit cell, act on the entire crystalline. space, the entire atomic structure of the crystal and on each other.

Rice. 7. Image of the group - Rpt in International tables.

If you specify inside the unit cell k-n. point x (x 1 x 2 x 3), then symmetry operations transform it into points symmetrically equal to it throughout the crystalline. space; there are an infinite number of such points. But it is enough to describe their position in one elementary cell, and this set will already multiply by lattice translations. A set of points derived from a given operation g i groups G - x 1, x 2,...,x n-1, called correct system of points (PST). In Fig. 7 on the right is the location of the symmetry elements of the group, on the left is the image of the PST general position this group. Points in general position are those points that are not located on the point symmetry element of the space group. The number (multiplicity) of such points is equal to the order of the group. Points located on an element (or elements) of point symmetry form a PST of a particular position and have the corresponding symmetry, their number is an integer number of times less than the multiplicity of the PST of a general position. In Fig. 7 on the left, the circles indicate points of general position, there are 8 of them inside the unit cell, the symbols “+” and “-”, “1/2+” and “1/2-” mean the coordinates +z, -z, 1/2 + z, respectively , 1/2 - z. Commas or their absence mean pairwise mirror equality of the corresponding points relative to the planes of symmetry m existing in this group at at= 1/4 and 3/4. If a point falls on the m plane, then it is not doubled by this plane, as in the case of points in general position, and the number (multiplicity) of such points in particular position is 4, their symmetry is m. The same happens when a point hits the centers of symmetry.

Each spatial group has its own sets of PSTs. There is only one correct system of points in general position for each group. But some of the PST of a particular situation may turn out to be the same for different groups. The International Tables indicate the multiplicity of PSTs, their symmetry and coordinates and all other characteristics of each space group. The importance of the concept of PST lies in the fact that in any crystalline. structure belonging to a given space group, the atoms or centers of molecules are located along the PST (one or more). In structural analysis, the distribution of atoms in one or more. PST of a given space group is carried out taking into account the chemistry. f-ly crystal and diffraction data. experiment, allows you to find the coordinates of points of particular or general positions in which atoms are located. Since each PST consists of one or a multiple number of Bravais lattices, the arrangement of atoms can be imagined as a set of Bravais lattices “pushed into each other.” This representation is equivalent to the fact that the space group contains translation as a subgroup. Brave group.

Subgroups of crystal symmetry groups. If part of the operation is k-l. groups itself forms a group G r (g 1 ,...,g m),, then the last name. subgroup of the first. For example, the subgroups of point group 32 (Fig. 1, a) are the group 3 and group 2 . Also among spaces. groups there is a hierarchy of subgroups. Space groups can have as subgroups point groups (there are 217 such space groups) and subgroups, which are space groups of lower order. Accordingly, there is a hierarchy of subgroups.

Most space symmetry groups of crystals are different from each other and as abstract groups; the number of abstract groups isomorphic to 230 space groups is 219. 11 mirror-equal (enantiomorphic) space groups turn out to be abstractly equal - one with only right helical axes, the others with left helical axes. These are, for example, P 3 1 21 and P 3 2 21. Both of these space groups map homomorphically onto the point group 32, to which quartz belongs, but quartz can be right-handed or left-handed, respectively: the symmetry of the spatial structure in this case is expressed macroscopically, but the point group is the same in both cases.

The role of space groups of crystal symmetry. Space symmetry groups of crystals are the basis of theoretical theory. crystallography, diffraction and other methods for determining the atomic structure of crystals and describing crystalline. structures.

The diffraction pattern obtained by X-ray diffraction is neutronography or electron diffraction,allows you to set symmetrical and geometric. characteristics reciprocal lattice crystal, and therefore the crystal structure itself. This is how the point group of a crystal and the unit cell are determined; Based on characteristic extinctions (the absence of certain diffraction reflections), the type of Bravais grating and membership in a particular space group are determined. The placement of atoms in a unit cell is determined from the totality of the intensities of diffraction reflections.

Space groups play an important role in crystal chemistry. More than 100 thousand crystalline particles have been identified. structures inorganic, organic and biological connections. Any crystal belongs to one of 230 space groups. It turned out that almost all space groups are realized in the world of crystals, although some of them are more common than others. There are statistics on the prevalence of space groups for various types of chemicals. connections. So far, only 4 groups have not been found among the studied structures: Рсс2, P4 2 cm, P4nc 1, Р6тп. The theory explaining the prevalence of certain space groups takes into account the sizes of the atoms that make up the structure, the concept of close packing of atoms or molecules, the role of “packing” symmetry elements - sliding planes and screw axes.

In solid state physics, the theory of group representations using matrices and special functions is used. functions, for space groups these functions are periodic. Yes, in theory structural phase transitions The 2nd kind space group of symmetry of the less symmetrical (low temperature) phase is a subgroup of the space group of the more symmetrical phase and the phase transition is associated with one of the irreducible representations of the space group of the highly symmetrical phase. Representation theory also allows you to solve problems of dynamics crystal lattice, its electronic and magnetic. structures, a number of physical properties. In theoretical In crystallography, space groups make it possible to develop the theory of partitioning space into equal regions, in particular polyhedral ones.

Symmetry of projections, layers and chains. Crystalline projections structures on a plane are described by flat groups, their number is 17. To describe three-dimensional objects periodic in 1 or 2 directions, in particular fragments of the crystal structure, two-dimensionally periodic and one-dimensionally periodic groups can be used. These groups play an important role in the study of biology. structures and molecules. For example, groups describe the structure of biological. membranes, groups of chain molecules (Fig. 8, A), rod-shaped viruses, tubular crystals of globular proteins (Fig. 8, b), in which the molecules are arranged according to spiral (helical) symmetry, which is possible in groups (see. Biological crystal).

Rice. 8. Objects with spiral symmetry: a - DNA molecule; b - tubular crystal of phosphorylase protein (electron microscopic image, magnification 220,000).

Structure of quasicrystals. Quasicrystal(eg, A1 86 Mn 14) have icosahedral. point symmetry (Fig. 5), which is impossible in crystals. grate. Long-range order in quasicrystals is quasiperiodic, described on the basis of the theory of almost periodic. functions. The structure of quasicrystals can be represented as a projection onto three-dimensional space of a six-dimensional periodic structure. cubic lattices with axes of 5th order. Quasicrystals with five-dimensional symmetry in the higher dimension can have 3 types of Bravais lattices (primitive, body-centered and face-centered) and 11 space groups. Dr. possible types of quasicrystals - stacking two-dimensional networks of atoms with axes of 5-, 7-, 8-, 10-, 12... orders, with periodicity along the third direction perpendicular to the networks.

Generalized symmetry. The definition of symmetry is based on the concept of equality (1,b) under transformation (1,a). However, physically (and mathematically) an object can be equal to itself in some respects and not equal in others. For example, the distribution of nuclei and electrons in a crystal antiferromagnet can be described using ordinary spatial symmetry, but if we take into account the distribution of magnetism in it. moments (Fig. 9), then “ordinary”, classic. symmetry is no longer enough. Generalizations of this kind of symmetry include anti-symmetry and color snimetry.

Rice. 9. Distribution magnetic moments(arrows) in the unit cell of a ferrimagnetic crystal, described using generalized symmetry.

In antisymmetry, in addition to three spatial variables x 1, x 2, x 3 an additional, 4th variable is introduced. This can be interpreted in such a way that under transformation (1,a) the function F may not only be equal to itself, as in (1, b), but also “anti-equal” - it will change sign. There are 58 point antisymmetry groups and 1651 space antisymmetry groups (Shubnpkov groups).

If an additional variable acquires not two values, but more (possible 3,4,6,8, ..., 48) , then the so-called Belov color symmetry.

Thus, 81 point groups and 2942 groups are known. Basic applications of generalized symmetry in crystallography - description of magnet. structures.

Other antisymmetry groups (multiple, etc.) have been found. All point and space groups of four-dimensional space and higher dimensions are theoretically derived. Based on consideration of the symmetry of (3 + K)-dimensional space, it is also possible to describe modularities that are incommensurate in three directions. structures (see Disproportionate structure).

Dr. generalization of symmetry - symmetry of similarity, when the equality of parts of a figure is replaced by their similarity (Fig. 10), curvilinear symmetry, statistical. symmetry introduced when describing the structure of disordered crystals, solid solutions, liquid crystals and etc.

Rice. 10. A figure with similarity symmetry.

Lit.: Shubnikov A.V., K o p c i k V. A., Symmetry in science and art, 2nd ed., M., 1972; Fedorov E.S., Symmetry and structure of crystals, M., 1949; Shubnikov A.V., Symmetry and antisymmetry of finite figures, M., 1951; International tables for X-ray crystallography, v. 1 - Symmetry groups, Birmingham, 1952; Kovalev O.V., Irreducible representations of space groups, K., 1961; V eil G., Symmetry, trans. from English, M., 1968; Modern crystallography, vol. 1 - Weinstein B.K., Symmetry of crystals. Methods of structural crystallography, M., 1979; G a l i u l i n R. V., Crystallographic Geometry, M., 1984; International tables for crystallography, v. A - Space group symmetry, Dordrecht - , 1987. B. TO. Weinstein.

A. I. Syomke,
, Municipal educational institution secondary school No. 11, Yeisk district, Yeisk, Krasnodar region.

Crystal symmetry

Lesson objectives: Educational– acquaintance with the symmetry of crystals; consolidation of knowledge and skills on the topic “Properties of crystals” Educational– education of worldview concepts (cause-and-effect relationships in the surrounding world, cognition of the surrounding world and humanity); moral education (cultivating a love of nature, a sense of comradely mutual assistance, ethics of group work) Developmental– development of independent thinking, literate oral speech, skills of research, experimental, search and practical work.

Symmetry... is the idea through
which man has tried for centuries
to comprehend order, beauty and perfection.
Hermann Weil

Physical dictionary

• Crystal - from Greek. κρύσταλλος - literally ice, rock crystal.
• The symmetry of crystals is the pattern of atomic structure, external shape and physical properties crystals, which consists in the fact that a crystal can be combined with itself through rotations, reflections, parallel transfers (translations) and other symmetry transformations, as well as combinations of these transformations.

Introductory stage

Crystal symmetry is the most general pattern associated with the structure and properties of a crystalline substance. It is one of the generalizing fundamental concepts of physics and natural science in general. According to the definition of symmetry given by E.S. Fedorov, “symmetry is the property of geometric figures to repeat their parts, or, to be more precise, their property in various positions come into alignment with the original position.” Thus, an object that can be combined with itself by certain transformations is symmetrical: rotations around symmetry axes or reflections in symmetry planes. Such transformations are usually called symmetric operations. After a symmetry transformation, parts of an object that were in one location are the same as parts that are in another location, which means that a symmetrical object has equal parts (compatible and mirrored). The internal atomic structure of crystals is three-dimensional periodic, i.e. it is described as a crystal lattice. The symmetry of the external shape (cut) of a crystal is determined by the symmetry of its internal atomic structure, which also determines the symmetry of the physical properties of the crystal.

Research 1. Description of crystals

The crystal lattice may have various types symmetry. The symmetry of a crystal lattice refers to the properties of the lattice to coincide with itself under certain spatial displacements. If the lattice coincides with itself when some axis is rotated through an angle of 2π/ n, then this axis is called the axis of symmetry n-th order.

Apart from the trivial 1st order axis, only 2nd, 3rd, 4th and 6th order axes are possible.

To describe crystals, various symmetry groups are used, of which the most important are space symmetry groups, describing the structure of crystals at the atomic level, and point symmetry groups, describing their external form. The latter are also called crystallographic classes. The designations of point groups include symbols of the main symmetry elements inherent in them. These groups are combined according to the symmetry of the shape of the unit cell of the crystal into seven crystallographic systems - triclinic, monoclinic, rhombic, tetragonal, trigonal, hexagonal and cubic. The belonging of a crystal to one or another group of symmetry and system is determined by measuring angles or using X-ray diffraction analysis.

In order of increasing symmetry, crystallographic systems are arranged as follows (the designations of axes and angles are clear from the figure):

Triclinic system. Characteristic property: a ≠ b ≠ c;α ≠ β ≠ γ. The unit cell has the shape of an oblique parallelepiped.

Monoclinic system. Characteristic property: two angles are right, the third is different from right. Hence, a ≠ b ≠ c; β = γ = 90°, α ≠ 90°. The unit cell has the shape of a parallelepiped with a rectangle at the base.

Rhombic system. All angles are right angles, all edges are different: a ≠ b ≠ c; α = β = γ = 90°. The unit cell has the shape of a rectangular parallelepiped.

Tetragonal system. All angles are right angles, two edges are equal: a = b ≠ c; α = β = γ = 90°. The unit cell has the shape of a straight prism with a square base.

Rhombohedral (trigonal) system. All edges are the same, all angles are the same and different from right angles: a = b = c; α = β = γ ≠ 90°. The unit cell has the shape of a cube, deformed by compression or tension along the diagonal.

Hexagonal system. The edges and the angles between them satisfy the following conditions: a = b ≠ c; α = β = 90°; γ = 120°. If you put three unit cells together, you get a regular hexagonal prism. More than 30 elements have hexagonal packing (C in the allotropic modification of graphite, Be, Cd, Ti, etc.).

Cubic system. All edges are the same, all angles are right: a = b = c; α = β = γ = 90°. The unit cell has the shape of a cube. In the cubic system there are three types of so-called Bravais lattices: primitive ( A), body-centered ( b) and face-centered ( V).

An example of a cubic system is crystals of table salt (NaCl, G). Larger chlorine ions (light balls) form a dense cubic packing, in the free nodes of which (at the vertices of a regular octahedron) sodium ions (black balls) are located.

Another example of a cubic system is the diamond lattice ( d). It consists of two cubic face-centered Bravais lattices, shifted by a quarter of the length of the spatial diagonal of the cube. Such a lattice is possessed, for example, by the chemical elements silicon, germanium, as well as the allotropic modification of tin – gray tin.

Experimental work “Observation of crystalline bodies”

Equipment: magnifying glass or short-focus lens in a frame, a set of crystalline bodies.

Execution order

1. Use a magnifying glass to examine the crystals of table salt. Please note that they are all shaped like cubes. A single crystal is called single crystal(has a macroscopically ordered crystal lattice). The main property of crystalline bodies is the dependence of the physical properties of the crystal on direction - anisotropy.
2. Examine the crystals of copper sulfate, pay attention to the presence of flat edges on individual crystals; the angles between the edges are not equal to 90°.
3. Consider mica crystals in the form of thin plates. The end of one of the mica plates is split into many thin leaves. It is difficult to tear a mica plate, but it is easy to split it into thinner sheets along planes ( strength anisotropy).
4. Consider polycrystalline solids (fracture of a piece of iron, cast iron or zinc). Please note: at the fracture you can distinguish small crystals that make up the piece of metal. Most solids found in nature and produced by technology are a collection of small crystals fused together in randomly oriented ways. Unlike single crystals, polycrystals are isotropic, that is, their properties are the same in all directions.

Research work 2. Symmetry of crystals (crystal lattices)

Crystals can take the form of various prisms, the base of which is regular triangle, square, parallelogram and hexagon. The classification of crystals and the explanation of their physical properties can be based not only on the shape of the unit cell, but also on other types of symmetry, for example, rotation around an axis. The axis of symmetry is a straight line, when rotated 360° around which the crystal (its lattice) aligns with itself several times. The number of these combinations is called order of the axis of symmetry. There are crystal lattices with symmetry axes of 2nd, 3rd, 4th and 6th order. Possible symmetry of the crystal lattice relative to the plane of symmetry, as well as combinations different types symmetry.

Russian scientist E.S. Fedorov established that 230 different space groups cover all possible crystal structures found in nature. Evgraf Stepanovich Fedorov (December 22, 1853 - May 21, 1919) - Russian crystallographer, mineralogist, mathematician. The greatest achievement of E.S. Fedorov - a rigorous derivation of all possible space groups in 1890. Thus, Fedorov described the symmetries of the entire variety of crystal structures. At the same time, he actually solved the problem of possible symmetrical figures, known since ancient times. In addition, Evgraf Stepanovich created a universal device for crystallographic measurements - Fedorov's table.

Experimental work “Demonstration of crystal lattices”

Equipment: models of crystal lattices of sodium chloride, graphite, diamond.

Execution order

1. Assemble a model of a sodium chloride crystal ( a drawing is provided). Please note that balls of one color imitate sodium ions, and the other – chlorine ions. Each ion in a crystal undergoes thermal vibrational motion near a node of the crystal lattice. If you connect these nodes with straight lines, a crystal lattice is formed. Each sodium ion is surrounded by six chlorine ions, and vice versa, each chlorine ion is surrounded by six sodium ions.
2. Select a direction along one of the lattice edges. Please note: white and black balls - sodium and chlorine ions - alternate.
3. Choose the direction along the second edge: white and black balls - sodium and chlorine ions - alternate.
4. Choose the direction along the third edge: white and black balls - sodium and chlorine ions - alternate.
5. Mentally draw a straight line along the diagonal of the cube - there will be only white or only black balls on it, i.e. ions of one element. This observation can serve as a basis for explaining the phenomenon of anisotropy characteristic of crystalline bodies.
6. The sizes of the ions in the lattice are not the same: the radius of the sodium ion is approximately 2 times larger than the radius of the chlorine ion. As a result, the ions in the table salt crystal are arranged in such a way that the lattice position is stable, i.e., there is a minimum of potential energy.
7. Assemble a model of the crystal lattice of diamond and graphite. The difference in the packing of carbon atoms in the lattices of graphite and diamond determines significant differences in their physical properties. Such substances are called allotropic.
8. Draw a conclusion based on the observation results and sketch out the types of crystals.

1. Almandine. 2. Iceland spar. 3. Apatite. 4. Ice. 5. Table salt. 6. Staurolite (double). 7. Calcite (double). 8. Gold.

Research work 3. Obtaining crystals

Crystals of a number of elements and many chemical substances have remarkable mechanical, electrical, magnetic, optical properties. The development of science and technology has led to the fact that many crystals rarely found in nature have become very necessary for the manufacture of parts for devices, machines, and for performing scientific research. The task arose of developing a technology for producing single crystals of many elements and chemical compounds. As you know, diamond is a carbon crystal, ruby ​​and sapphire are aluminum oxide crystals with various impurities.

The most common methods for growing single crystals are melt crystallization and solution crystallization. Crystals from solution are grown by slowly evaporating a solvent from a saturated solution or by slowly lowering the temperature of the solution.

Experimental work “Growing crystals”

Equipment: saturated solutions of table salt, ammonium chloride, hydroquinone, ammonium chloride, a glass slide, a glass rod, a magnifying glass or a framed lens.

Execution order

1. Take a small drop of a saturated solution of table salt with a glass rod and transfer it to a preheated glass slide ( solutions are prepared in advance and stored in small flasks or test tubes closed with stoppers).
2. Water from warm glass evaporates relatively quickly, and crystals begin to fall out of the solution. Take a magnifying glass and observe the crystallization process.
3. The most effective experiment is with ammonium dichromate. At the edges and then over the entire surface of the drop, golden-orange branches with thin needles appear, forming a bizarre pattern.
4. One can clearly see the unequal rates of crystal growth in different directions—growth anisotropy—in hydroquinone.
5. Draw a conclusion based on the observation results and sketch out the types of crystals obtained.

Research work 4. Applications of crystals

Crystals have the remarkable property of anisotropy (mechanical, electrical, optical, etc.). Modern production cannot be imagined without the use of crystals.

Interesting information

Who discovered liquid crystals and when? Where are LCDs used?

IN late XIX V. German physicist O. Lehmann and Austrian botanist F. Reinitzer drew attention to the fact that some amorphous and liquid substances They are distinguished by a very ordered parallel stacking of elongated molecules. Later, based on the degree of structural order, they were called liquid crystals(LCD). There are smectic crystals (with layer-by-layer arrangement of molecules), nematic (with elongated molecules randomly displaced in parallel) and cholesteric (close in structure to nematic ones, but characterized by greater mobility of molecules). It was noticed that with external influence, for example, a small electrical voltage, with a change in temperature, tension magnetic field the optical transparency of the LC molecule changes. It turned out that this occurs due to the reorientation of the molecular axes in the direction perpendicular to the initial state.

Liquid crystals: A) smectic; b) nematic; V) cholesteric.
URL: http://www.superscreen.ru

Operating principle of the LCD indicator:
on the left – the electric field is turned off, light passes through the glass; on the right – the field is turned on, the light does not pass through, black symbols are visible (the URL is the same)

Another wave of scientific interest in liquid crystals arose in the post-war years. Among the crystallographic researchers, our compatriot I.G. said a weighty word. Chistyakov. At the end of the 60s. last century American corporation RCA began to conduct the first serious research on the use of nematic LCDs for visual display of information. However, the Japanese company was ahead of everyone Sharp, which in 1973 proposed a liquid crystal alphanumeric mosaic panel - LCD display ( LCD – Liquid Crystal Display). These were modest-sized monochrome indicators, where polysegment electrodes were used mainly for numbering. The onset of the “indicator revolution” led to the almost complete replacement of pointer mechanisms (in electrical measuring instruments, wrist and stationary watches, household and industrial radio equipment) with means of visually displaying information in digital form - more accurate, with an error-free reading.

Liquid crystal displays different types. URL: http://www.permvelikaya.ru; http://www.gio.gov.tw; http://www.radiokot.ru

Thanks to the successes of microelectronics, pocket and desktop calculators replaced adding machines, abacus, and slide rules. The avalanche-like reduction in the cost of integrated circuits has even led to phenomena that clearly contradict technical trends. For example, modern digital wristwatches are noticeably cheaper than spring watches, which, due to the inertia of thinking, remain popular, moving into the “prestige” category.

What parameters determine the shape of snowflakes? What science and for what purposes is it studying snow, ice, snowflakes?

The first album with sketches of various snowflakes made using a microscope appeared at the beginning of the 19th century. in Japan . It was created by the scientist Doi Chishitsura. Almost a hundred years later, another Japanese scientist, Ukishiro Nakaya, created a classification of snowflakes. His research proved that the branched, six-pointed snowflakes we are accustomed to appear only at a certain temperature: 14–17 °C. In this case, the air humidity should be very high. In other cases, snowflakes can take on a variety of shapes.

The most common form of snowflakes is dendrites (from the Greek δέντρο - tree). The rays of these crystals are like tree branches.

Science deals with the world of snow and ice glaciology. It originated in the 17th century. after the Swiss naturalist O. Saussure published a book about Alpine glaciers. Glaciology exists at the intersection of many other sciences, primarily physics, geology and hydrology. You need to study ice and snow in order to know how to prevent avalanches and ice. After all, millions of dollars are spent annually on combating their consequences all over the world. But if you know the nature of snow and ice, you can save a lot of money and save many human lives. Ice can also tell us about the history of the Earth. For example, in the 70s. glaciologists studied the ice cover of Antarctica, drilled wells and studied the features of ice in different layers. Thanks to this, it was possible to learn about the many climate changes that have occurred on our planet over 400,000 years.

Entertaining and non-standard tasks(group work)

On the shores of the North Channel, in the northeast of the island of Ireland, the low Antrim Mountains rise. They are composed of black basalts - traces of the activity of ancient volcanoes that rose along a giant fault that separated Ireland from Great Britain 60 million years ago. Streams of black lava flowing from these craters formed the coastal mountains on the Irish coast and on the Hebrides Islands across the North Channel. This basalt is an amazing rock! Liquid, easily flowing in molten form (basalt flows sometimes rush along the slopes of volcanoes at speeds of up to 50 km/h), when it cools and hardens, it cracks, forming regular hexagonal prisms. From a distance, the basalt cliffs resemble huge organs with hundreds of black pipes. And when a stream of lava flows into the water, such bizarre formations sometimes appear that it is difficult not to believe in their magical origin. That's exactly what a natural phenomenon can be observed at the foot of Antrim. A kind of “road to nowhere” separates from the volcanic massif here. The dam rises 6 m above the sea and consists of approximately 40,000 basalt columns. It looks like an unfinished bridge across the strait, conceived by some fairy-tale giant, and is called the “Giants' Causeway.”

Task. What properties of crystalline solids and liquids are we talking about? What differences do you know between crystalline solids and liquids? ( Answer. Correct geometric shape is essential external sign any crystal in natural conditions.)

The first diamond in South Africa was found in 1869 by a shepherd boy. A year later, the city of Kimberley was founded here, after which the diamond-bearing rock became known as kimberlite. The diamond content in kimberlites is very low - no more than 0.000 007 3%, which is equivalent to 0.2 g (1 carat) for every 3 tons of kimberlites. Nowadays, one of the attractions of Kimberley is a huge pit 400 m deep, dug by diamond miners.

Task. Where are the valuable properties of diamonds used?

“Such a snowflake (we are talking about a snowflake. - A.S.), a hexagonal, regular star, fell on Nerzhin’s sleeve of an old front-line, rusty overcoat.”

A.I. Solzhenitsyn. In the first circle.

? Why do snowflakes have the correct shape? ( Answer. The main property of crystals is symmetry.)

“The window rattled with noise; The windows flew out, clinking, and a terrible pig's face stuck out, moving its eyes, as if asking: “What are you doing here, good people?”

N.V. Gogol.

? Why does glass break even under light load? ( Answer. Glass is classified as a brittle body that has virtually no plastic deformation, so that elastic deformation immediately ends in fracture.)

“It was freezing more than in the morning; but it was so quiet that the crunch of frost under boots could be heard half a mile away.”

N.V. Gogol. Evenings on a Farm Near Dikanka.

? Why does snow squeak underfoot in cold weather? ( Answer. Snowflakes are crystals, they are destroyed underfoot, and as a result, sound appears.)

A diamond is cut by a diamond.

? Diamond and graphite are made up of identical carbon atoms. Why do the properties of diamond and graphite differ? ( Answer. These substances differ in crystal structure. Diamond has strong covalent bonds, while graphite has a layered structure.)

? What substances do you know that are not inferior to diamond in strength? ( Answer. One such substance is boron nitride. Very durable covalent bond boron and nitrogen atoms bond in the crystal lattice of boron nitride. Boron nitride is not inferior to diamond in hardness, and surpasses it in strength and heat resistance.)

The end is blunt, the incisor is sharp: it cuts the leaves, pieces fly. What is this? ( Answer. Diamond.)

? What property distinguishes diamond from other substances? ( Answer. Hardness.)

The largest crystals were discovered in the Nike Cave, in the Mexican state of Chihuahua. Some of them reach 13 m in length and 1 m in width.

A.E. Fersman at the beginning of the 20th century. described a quarry in the Southern Urals, embedded in one giant feldspar crystal.

Conclusion

To conclude the lesson, I would like to give a unique example of the use of symmetry. Honeybees must be able to count and save. To secrete only 60 g of wax with special glands, they need to eat 1 kg of honey from nectar and pollen, and about 7 kg of sweet food is required to build an average-sized nest. Honeycomb cells, in principle, can be square, but bees choose a hexagonal shape: it provides the densest packing of the larvae, so that a minimum of precious wax is spent on building the walls. The honeycombs are vertical, the cells on them are located on both sides, i.e. they have a common bottom - another saving. They are directed upward at an angle of 13° to prevent honey from leaking out. Such honeycombs can hold several kilograms of honey. These are the real wonders of nature.

Literature

1. Arnold V.I. Mathematical methods of classical mechanics. M.: Editorial URSS, 2003.
2. Weil G. Symmetry: translated from English. M., 1968.
3. Glaciological Dictionary / Ed. V.M. Kotlyakov. L.: Gidrometeoizdat, 1984.
4. Kompaneets A.S. Symmetry in the micro- and macrocosm. M.: Nauka, 1978.
5. Merkulov D. The magic of liquid crystals // Science and life. 2004. No. 12.
6. Fedorov E.S. Symmetry and structure of crystals. M., 1949.
7. Physics: enc. for children. M.: Avanta+, 2000.
8. Shubnikov A.V., Koptsik V.A. Symmetry in science and art. Publishing house 2. M., 1972.

The appearance of crystals obtained by different methods, for example, grown from a melt or solution, can differ markedly from each other. At the same time, one of the first discoveries in crystallography was the establishment of the fact that the corners between the faces of a crystal of the same substance are unchanged. Such constancy of angles, as is now known, is due to the regular arrangement of atoms or groups of atoms inside the crystal, that is, the presence of a certain symmetry in the arrangement of atoms in a crystalline solid.

Translational symmetry. The concept of translational symmetry of a crystal means that in a crystal one can select some smallest part, called a unit cell, the spatial repetition of which is broadcast - In three directions (along the edges of the cell) the entire crystal is formed. The concepts of translational symmetry and the elementary cell of a crystal were a scientific generalization of the experimental fact that in crystals of the same substance one can mentally isolate a basic geometric element from which the entire crystal can be constructed. The deep scientific meaning of these concepts was revealed later, with the development of methods for X-ray structural analysis of solids.

A unit cell may contain one or more molecules, atoms, or ions, the spatial arrangement of which in the cell is fixed. The unit cell is electrically neutral. If a unit cell repeating in a crystal is represented by a point, then as a result of the translational repetition of this point in three directions (not necessarily perpendicular), a three-dimensional set of points will be obtained, called the crystal lattice of the substance. In this case, the points themselves are called nodes of the crystal lattice. The crystal lattice can be characterized by the vectors of basic translations A ( And a 2, as shown for the two-dimensional case in Fig. 1.14.

As can be seen in Fig. 1.14, the choice of vectors of the main translations is not unambiguous. The main thing is that the position of all equivalent points of the crystal lattice can be described linear combination vectors of main translations. In this case, the set of all lattice vectors forms Bravais lattice crystal. The ends of the lattice vectors determine the position of the node points in the lattice.

Rice. 1.14. Options for the possible choice of translation vectors a 1 and a 2 and a primitive lattice (options 1,2,3,4)

A parallelepiped built on the vectors of basic translations is called a primitive crystal cell, the choice of which in the crystal is also ambiguous. Unit cell 4 in Fig. 1.14, constructed through the midpoints of the translation vectors, is called Wigner cell - Seitz.

Crystallographic indices. If in the unit cell J? of a two-dimensional crystal lattice shown in Fig. 1.14, draw straight line segments parallel to the vector a 2 and passing through nodes a and |3, then they will divide the vector i into three equal parts. When broadcasting a cell 3 along translation vectors A ( And a 2 the crystal lattice will be filled with straight lines, and all nodes of the crystal lattice will be on these lines. A similar operation can be carried out in a three-dimensional crystal lattice by passing a system of planes through it, and in this case, all nodes of the three-dimensional crystal lattice will appear on these planes. These planes are called crystallographic lattice planes. It is obvious that many different families of crystallographic planes can be drawn through a crystal lattice. It is also obvious that the smaller the distance between the planes in a family, the lower the density of crystal lattice nodes falling on each plane (of a given family of planes).

Crystallographic planes characterize Miller indices, denoted by three numbers enclosed in parentheses ( hkl). These numbers are equal to the number of segments into which the family of crystallographic planes is divided by the vectors of the main translations. If the planes are parallel to any translation vector, then the value of the corresponding Miller index is equal to zero. If the planes intersect the negative direction of any translation vector, then the corresponding index is assigned a negative value by placing a dash above this index. What has been said for a two-dimensional crystal lattice, with the given families of planes (10), (01) And (12), as well as a plane from the family (12), well illustrated in Fig. 1.15.

Rice. 1.15. Crystallographic planes . In monoclinic crystals there is only one oblique angle between the indicated edges (the other two are straight). The rhombic system is characterized by the fact that the simple forms associated with it often have the shape of rhombuses.