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SYMMETRY OF CRYSTALS

SYMMETRY OF CRYSTALS

The property of crystals to be combined with themselves during rotations, reflections, parallel transfers, or part or combination of these operations. Symmetry means the ability to transform an object that combines it with itself. Symmetry ext. the shape (cutting) of a crystal is determined by the symmetry of its atomic structure, which also determines the symmetry of the physical. crystal properties.

Rice. 1. a - quartz crystal: 3 - axis of symmetry of the 3rd order, 2x, 2y, 2w - axes of the 2nd order; b - crystal of aqueous sodium meta-silicate: m - plane of symmetry.

On fig. 1a shows a quartz crystal. Ext. its shape is such that by rotating it through 120° about axis 3 it can be superposed with itself (consistent equality). The sodium metasilicate crystal (Fig. 1, 6) is transformed into itself by reflection in the symmetry plane m (mirror equality).

If F(xlx2.x3) is a function that describes an object, e.g. the shape of a crystal in three-dimensional space or c.-l. its property, and the operation g(x1, x2, x3) transforms the coordinates of all points of the object, then g is an operation or a symmetry transformation, and F is a symmetrical object if the following conditions are met:

In the most general formulation - 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 S. to. - the theory of symmetrical. transformations into itself of three-dimensional space, taking into account the fact that ext. the atomic structure of crystals is three-dimensionally periodic, that is, it is described as . During transformations, symmetry is not deformed, but transformed as a rigid whole. Such transformations are called orthogonal or isometric. After the parts of the object that were in one place coincide with the parts that are in another place. This means that there are equal parts (compatible or mirrored) in a symmetrical object.

S. to. manifests itself not only in their structure and properties in real three-dimensional space, but also in the description of energetic. the spectrum of the electrons of the crystal (see ZONE THEORY), when analyzing the processes of diffraction X-ray. rays and electrons in crystals in reciprocal space (see REVERSE LATTICE), etc.

Symmetry group of crystals. A crystal may have not one, but several. symmetry operations. Thus, a quartz crystal (Fig. 1, a) is aligned with itself not only when rotated by 120° around axis 3 (operation g1), but also when rotated around axis 3 by 240° (operation g2), and also when rotated by 180 ° around the axes 2x, 2y, 2w (operations g3, g4, g5). Each element of symmetry can be associated - a straight line, a plane or a point, relative to which this operation is performed. For example, the 3 axis or the 2x, 2y, 2w axes are the axes of symmetry, the m plane (Fig. 1.6) is the plane of mirror symmetry, etc. The set of symmetry operations (g1, g2, . . ., gn) of a given crystal forms a symmetry group G in the sense of Math. group theory. Consistent carrying out two symmetry operations is also a symmetry operation. There is always an identity operation g0 that does not change anything in the crystal, called. identification, geometrically corresponding to the immobility of the object or its rotation through 360 ° around any axis. The number of operations that form a group G, called. group order.

Symmetry groups are classified: according to the number n of space dimensions in which they are defined; according to the number m of space dimensions, in which the object is periodic (they are respectively denoted by Gnm), and according to some other features. To describe the crystals use dec. symmetry groups, of which the most important are . G33, describing the atomic structure of crystals, and point groups with G30 symmetry, describing their external shape. Last names also crystallographic classes.

Point symmetry groups. The operations of point symmetry are: rotations around the axis of symmetry of order N by an angle equal to 360°/N (Fig. 2, a), reflection in the plane of symmetry ( ; Fig. 2, b), inversion of T (symmetry about a point; Fig. 2 , c), inversion rotations N= (combination of 360°/N rotation with simultaneous inversion; Fig. 2, d).

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

Instead of inversion turns, N= mirror turns are sometimes considered. Geometrically possible combinations of these operations determine one or another point symmetry group, which is usually depicted in stereographic form. projections. With point symmetry transformations, at least one point of the object remains fixed - it transforms into itself. All symmetries intersect in it, and it is the center of the stereographic. projections. Examples of crystals related to dec. point groups are given in Figs. 3.

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

Point symmetry transformations g (x1, x2, x3) \u003d x "1, x" 2, x "3 are described by linear equations:

i.e., the coefficient matrix, (aij). For example, when turning around the x1 axis at an angle a=360°/N, the coefficient looks like:

and when reflected in the x1, x2 plane, it has the form:

The number of Go point groups is infinite. However, in crystals due to the presence of crist. lattices, only operations are possible and, accordingly, symmetry axes up to the 6th order (except for the 5th; in a crystal lattice there cannot be a symmetry axis of the 5th order, since it is impossible to fill without gaps with the help of pentagons), which are denoted symbols: 1, 2, 3, 4, 6, as well as inversion axes 1 (it is also the center of symmetry), 2 (it is also the plane of symmetry), 3, 4, 6. Therefore, the number of point crystallographic. symmetry groups describing ext. the shape of crystals is limited, there are only 32 of them (see table). In the international the notation of point groups includes the symbols of the symmetry operations that generate them. These groups are combined according to the symmetry of the unit cell shape (with periods o, b, c and angles a, b, g) into 7 syngonies.

Groups containing only rotations describe , consisting only of compatible equal parts (groups of the 1st kind). Groups containing reflections or inversion rotations describe crystals in which there are mirror equal parts (groups of the second 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 equal to each other (see ENANTIOMORPHISM).

Point groups describe the symmetry not only of crystals, but of any finite figures. In living nature, symmetry with axes of the 5th, 7th order and higher, which is forbidden in crystallography, is often observed. For example, to describe the regular structure of a spherical viruses, in the shells of which the principles of dense packing of molecules are observed, the icosahedral 532 turned out to be important (see BIOLOGICAL CRYSTALS).

Limit groups. Functions, to-rye describe the dependence decomp. properties of the crystal from the direction, have a certain point symmetry, uniquely associated with the symmetry group of the faceting of the crystal. It either coincides with it or is higher than it in symmetry (the Neumann principle).

Many of the properties of crystals belonging to certain point symmetry groups are described in vol. CRYSTAL PHYSICS).

The spatial symmetry of the atomic structure of crystals is described by spaces. G33 symmetry groups (also called Fedorov groups in honor of E. S. Fedorov, who found them in 1890). The three non-coplanar operations a, b, c, called are characteristic of a lattice. translations, to-rye set the three-dimensional periodicity of the atomic structure of crystals. The shift (transfer) of the structure by the vectors a, b, c or any vector t=p1a+p2b+p3c, where p1,p2, p3 are any positive or negative integers, combines the crystal structure with itself and, therefore, is a symmetry operation ( translational symmetry).

Due to the possibility of combining translations and point symmetry operations in the G33 lattice, operations and corresponding symmetry elements from translations arise. component - screw axes decomp. orders and plane of grazing reflection (Fig. 2, e, f). A total of 230 spaces are known. symmetry groups G33, any crystal belongs to one of these groups. Broadcast. elements of microsymmetry do not appear macroscopically, for example. the helical axis in the faceting of crystals appears as a simple rotational axis corresponding in order. Therefore, each of the 230 G33 groups is macroscopically similar (homomorphic) to one of the 32 point groups. For example, 28 spaces are homomorphically mapped onto the point group mmm. groups. The set of transfers inherent in a given space group is its translational subgroup, or Bravais lattice; There are 14 such lattices.

Symmetry of layers and chains. To describe objects that are periodic in 1 or 2 directions, in particular fragments of the crystal structure, groups G32 - two-dimensionally periodic and G31 - one-dimensionally periodic in three-dimensional space can be used. These groups play an important role in studying biol. structures and molecules. For example, the groups G| describe the structure of biol. membranes, groups of G31 chain molecules (Fig. 5, a) rod-shaped viruses, tubular crystals of globular proteins (Fig. 5, b), in which they are arranged according to the helical (helical) symmetry possible in groups G31 (see BIOLOGICAL CRYSTALS ).

Rice. 5. Objects with helical symmetry: a - DNA; b - tubular crystal of phosphorylase protein (electron microscopic image, magnification 220000).

Generalized symmetry. The definition of symmetry is based on the concept of equality (1, b) under the transformation (1, a). However, physically (and mathematically) an object can be equal to itself in some ways and not equal in others. For example, nuclei and electrons in an antiferromagnet crystal can be described using ordinary spaces. symmetry, but if we take into account the magn. moments (Fig. 6), then ordinary”, classical. symmetry is no longer enough. Such generalizations of symmetry include antisymmetry and . In antisymmetry in addition to three spaces. variables x1, x2, x3, an additional 4th variable x4=±1 is introduced. This can be interpreted in such a way that during the transformation (1, a) the function F can not only be equal to itself, as in (1, b), but also "anti-equal" - change the sign. Conventionally, such an operation can be represented by a change in color (Fig. 7).

Rice. 6. Distribution of magnetic moments (arrows) in the unit cell of a ferrimagnetic crystal, described using generalized symmetry.

There are 58 groups of point antisymmetry C30, and 1651 spaces. antisymmetries G33,a (Shubnikovskii gr u p p). If the additional variable acquires not two values, but several. (the numbers 3, 4, 6, 8, . . ., 48 are possible), then Belov's color symmetry arises. Thus, 81 point groups G30,c and 2942 groups C33,c are known. The main applications of generalized symmetry in crystallography are the description of the magnetic field. structures.

Rice. 7. The figure described by the point group of antisymmetry.

Dr. generalizations of symmetry: similarity symmetry, when the equality of the parts of the figure is replaced by their similarity (Fig. 8), curvilinear symmetry, statistical. symmetry introduced in the description of the structure of disordered crystals, solid solutions, liquid crystals, etc.

Physical Encyclopedic Dictionary. - M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1983 .

SYMMETRY OF CRYSTALS

The property of crystals to be combined with themselves during rotations, reflections, parallel transfers, or with a part or combination of these operations. Symmetry ext. the shape (cutting) of a crystal is determined by the symmetry of its atomic structure, which also determines the symmetry of the physical. crystal properties.

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.

On fig. one a shows a quartz crystal. Ext. its form is as follows, b) is transformed into itself by reflection in the plane of symmetry m (mirror equality). If a - a function that describes an object, e.g. the shape of a crystal in three-dimensional space, or c.-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,

In naib. In the general formulation, symmetry is the immutability (invariance) of objects and laws under certain transformations of the variables describing them. S. to. manifests itself not only in their structure and properties in real three-dimensional space, but also in the description of energetic. the electron spectrum of the crystal (see zone theory), in process analysis x-ray diffraction, neutron diffraction and electron diffraction in crystals using reciprocal space (see Reciprocal lattice)it. P.

Symmetry groups of crystals. A crystal may have more than one, anesque. symmetry operations. So, a quartz crystal (Fig. 1, a) is aligned with itself not only when rotated by 120 ° around the axis 3 (operation gi), noi when turning around the axis 3 240° (operation g2),& also for 180° rotations around axes 2 X, 2 Y, 2 W(operations g3, g4, g5).Each symmetry operation can be associated with a symmetry element - straight line, 3 or axis 2x, 2y, 2w are the axes of symmetry, the plane t(Fig. 1,b) - by a plane of mirror symmetry, etc. The set of symmetry operations (g 1 , g 2 ,..., g n ) given crystal forms a symmetry group in the sense of Math. theories groups. Consistent Carrying out two symmetry operations is also a symmetry operation. In group theory, this is referred to as a product of operations: There is always an identity operation g 0 , does not change anything in the crystal, called. identification, it geometrically corresponds to the immobility of the object or its rotation through 360 ° around any axis. The number of operations that form a group G, called. group order.

Symmetry groups of space transformations are classified: by the number . dimensions of space, in which they are defined; by number . dimensions of space, in which the object is periodic (they are respectively designated ), and according to some other features. To describe crystals, various symmetry groups are used, of which the most important are those that describe the external. the shape of the crystals; their name. also crystallographic. classes; space symmetry groups describing the atomic structure of crystals.

Point symmetry groups. The operations of point symmetry are: rotations around the axis of symmetry of the order N at an angle equal to 360°/N(Fig. 2, a); reflection in the plane of symmetry t(mirror reflection, b); inversion (symmetry with respect to a point, Fig. 2, c); inversion turns (combination of rotation by an angle 360°/N with at the same time inversion, Fig.2, d). Instead of inversion rotations, equivalent mirror rotations are sometimes considered.

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

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

Point symmetry transformations are described by linear equations

or coefficient matrix

For example, when turning around an axis x 1 angle -=360°/N matrix D looks like:

and when reflected in a plane x 1 x 2D looks like:

The number of point groups is infinite. However, in crystals due to the presence of crystalline. lattice, 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 with the help of pentagonal figures it is impossible to fill the space without gaps). Operations of point symmetry and the symmetry elements corresponding to them are indicated by symbols: axes 1, 2, 3, 4, 6, inversion axes (center of symmetry or center of inversion), (it is also the plane of symmetry m), (Fig. 4).

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

To describe a point symmetry group, it suffices to specify one or more. b, c and angles ) into 7 syngonies (Table 1).

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

Tab. one.- Point groups (classes) of symmetry of crystals

The groups of S. k. carry a geome. meaning: each of the operations corresponds, for example, to rotation around the axis of symmetry, reflection in the plane. in a given group (but not their geom. sense), are the same, 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. c.

Point groups describe the symmetry not only of crystals, but of any finite figures. In living nature, point symmetry with axes of the 5th, 7th order and higher, which is forbidden in crystallography, is often observed. To describe the regular structure of a spherical viruses, in the shells of which the principles of dense packing of molecules are observed, and some inorganic. molecules turned out to be important icosahedral. (cm. biological crystal). Icosahedric. symmetry is also seen in quasicrystals.

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

With regard to macroscopic properties of the 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. limiting 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 by any, including Crystal Physics).

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

Spatial symmetry groups. The spatial symmetry of the atomic structure of crystals is described by space symmetry groups. They are called also Fedorov in honor of E. S. Fedorov, who found them in 1890; these groups were independently derived in the same year by A. Schoenflies. polyhedra (S. I. Gessel, 1830, A. Operations characteristic of the atomic structure of crystals are 3 non-coplanar translations a, b , With , to-rye and set the three-dimensional periodicity of the crystal. gratings. Crystalline the lattice is considered to be infinite in all three dimensions. Such a mat. real, a, b, c or any vector where p 1, p 2, p 3 - any whole numbers, Phys. discreteness of the crystal. matter is expressed in its atomic structure. are the transformation groups of a three-dimensional homogeneous discrete space into itself. Discreteness lies in the fact that not all points of such a space are symmetrically equal to each other, for example. one and the other kind of atom, nuclei and electrons. The conditions for 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. lattice.

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

In accordance with the point symmetry of the shape of the unit cell (elementary parallelepiped), the space groups, like the point ones, are divided into 7 crystallographic syngony(Table 2). Their further subdivision corresponds to broadcasts. groups and their respective Right to the gratings. There are 14 Bravais lattices, of which 7 are primitive lattices of the corresponding syngonies, P (except for the rhombohedral R). Others-7 plummets. A (face is centered bc),B(face ac), C (ab); body-centered I, face-centered (on all 3 faces) F. Taking into account the centering for the translation operation t centering translations corresponding to the center are added t c . If these operations are combined with each other t+ t s and with the operations of the point groups of the corresponding syngonies, then we get 73 space groups, called. symmorphic.

Tab. 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 hence the entire space into itself) are written as:, where D- point transformations, - components of screw transfer or sliding reflection, - translation operations. Brave groups. Operations of helical symmetry and the corresponding symmetry elements - helical axes have an angle. component (N = 2, 3, 4, 6) and translational t s = tq/N, where t- translation of the grating, turn on occurs simultaneously with translation along the Z axis, q- screw index. General symbol for helical axes N q(Fig. 6). The screw axes are directed along 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, b, c: reflection is combined with transfer by half of the corresponding lattice period. The translation by half the diagonal of the cell face corresponds to t. n. wedge plane of sliding n, moreover, in tetragonal and cubic. d.

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

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

Broadcast. components of microsymmetry operations of space groups do not appear macroscopically in point groups; for example, the helical axis in the faceting of crystals appears as a simple rotational axis corresponding in order. Therefore, each of the 230 groups is macroscopically similar (homomorphic) to one of the 32 point groups. For example, on a point group - mmm 28 space groups are displayed homomorphically.

The Schoenflies notation of space groups is the designation of the corresponding point group (for example, Table 1), to which the historically accepted , is assigned from above. In international notation, the symbol of the Bravais lattice and the generating operations of symmetry for each group are indicated, etc. The sequence of arrangement of space groups in Table 2 in international notation corresponds to the number (superscript) in Schoenflies notation.

On fig. 7 the image of spaces is given. groups - Rpta according to the International Crystallographic tables. Operations (and corresponding elements) of symmetry of each space group,

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

If you set inside the elementary cell Ph.D. point x (x 1 x 2 x 3), then the symmetry operations transform it into points symmetrically equal to it throughout the crystal. space; such points is infinite. But it is enough to describe their position in one elementary cell, and this set will already multiply by translations of the lattice. The set of points derived from the given operations gi groups G - x 1 ,x 2 ,...,x n-1, called correct system of points (PST). In fig. 7 on the right is the arrangement of the symmetry elements of the group, on the left is the image of the PST of the general position of this group. Points in general position are such points that are not located on an element of point symmetry of the space group. The number (multiplicity) of such points is equal to the order of the group. y = 1/4 and 3/4. If a point falls on a plane, then it is not doubled by this plane, as in the case of points in general position. Each space group has its own set of PSTs. There is only one correct system of points in general position for each group. But some of the PST private position may be the same for different groups. The International Tables indicate the multiplicity of the PST, 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,

Subgroups of crystal symmetry groups. If part of the operation to.-l. forms a group G r (g 1 ,...,g m),, then the last name subgroup of the first. For example, the subgroups of the point group32 (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 a lower order. Accordingly, there is a hierarchy of subgroups.

Most of the space symmetry groups of crystals are different among themselves and as abstract groups; the number of abstract groups isomorphic to 230 space groups is 219. Abstractly equal are 11 mirror-equal (enantiomorphic) space groups - one with only right, others with left helical axes. These are, for example, P 3 1 21 and P 3 2 21. Both of these space groups are homomorphically mapped onto a point group32 to which belongs to , but quartz, respectively, is right-handed or left-handed: the symmetry of the spatial structure in this case is expressed macroscopically, The role of space symmetry groups of crystals. Spatial symmetry groups of crystals - the basis of the theoretical. crystallography, diffraction and other methods for determining the atomic structure of crystals and describing the crystal. The diffraction pattern obtained by X-ray diffraction neutronography or electronography, allows you to set symmetry and geom. the reciprocal lattice of the crystal, and hence the very structure of the crystal. This is how the point group of the crystal and the unit cell are determined; by characteristic extinctions (absence of certain diffraction reflections) determine the type of Bravais grating and belonging to one or another spatial group. The arrangement of atoms in an elementary cell is found from the totality of the intensities of diffraction reflections.

Space groups play an important role in crystal chemistry. More than 100 thousand crystals have been identified. structures inorganic., organic. and biological. connections. Rcc2, P4 2 cm, P4nc 1 , P6tp. The theory explaining the prevalence of technologies of other space groups takes into account the dimensions of the atoms that make up the structure, the concept of dense packing of atoms or molecules, the role of "packing" symmetry elements - slip planes and helical axes.

In solid state physics, the theory of group representations using matrices and specials is used. f-tions, for space groups these functions are periodic. structural phase transitions of the 2nd kind, the space group of the 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 makes it possible to solve problems of dynamics crystal lattice, its electronic and magnetic structures, a number of physical properties. In the theoretical Symmetry of projections, layers and chains. Crystalline projections. on the structural 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, groups can be used - two-dimensionally periodic and - one-dimensionally periodic. These groups play an important role in the study of biology. 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 they are arranged according to the spiral (helical) symmetry possible in groups (see Fig. biological crystal).

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

Structure of quasicrystals.Quasicrystal(e.g., A1 86 Mn 14) have icosahedral. point symmetry (Fig. 5), which is impossible in a crystal. 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 ways and not equal in others. For example, the distribution of nuclei and electrons in a crystal antiferromagnet can be described using the usual spatial symmetry, but if we take into account the distribution of the magnetic. moments (Fig. 9), then “usual”, classical. symmetry is no longer enough.

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

In antisymmetry, in addition to three space variables x 1, x 2, x 3 an additional, 4th variable is introduced. This can be interpreted in such a way that when (1, a) is transformed, the function F can be not only 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 (Shubnkov groups).

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

So, 81 point groups and 2942 groups are known. Main applications of generalized symmetry in crystallography - description of magn. Other antisymmetry groups (multiple, etc.) have also been found. Theoretically, all point and space groups of four-dimensional space and higher dimensions are derived. Based on the consideration of the symmetry of (3 + K)-dimensional space, one can also describe moduli that are incommensurate in three directions. disproportionate structure).

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

Rice. 10. A figure with similarity symmetry. Big Encyclopedic Dictionary

The regularity of the atomic structure, external shape and physical properties of crystals, which consists in the fact that a crystal can be combined with itself through rotations, reflections, parallel transfers (translations) and other symmetry transformations ... encyclopedic Dictionary

The property of crystals to be aligned with themselves in various positions by rotations, reflections, parallel transfers, or part or combination of these operations. The symmetry of the external shape (cutting) of a crystal is determined by the symmetry of its atomic ... ...

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MINISTRY OF EDUCATION OF THE RUSSIAN FEDERATION

MOSCOW STATE INSTITUTE OF ELECTRONIC ENGINEERING

(TECHNICAL UNIVERSITY)

"APPROVE"

Head department of KFN

Gorbatsevich A.A.

LAB #10

at the rate of "FTT and PP"

The description was:

Anfalova E.S.

MOSCOW, 2002

LAB #1

DETERMINATION OF THE STRUCTURE OF CRYSTALS USING X-RAY DIFFRACTION

Objective: determination of the crystal structure and lattice constant using the Debye-Scherer method.

1. Structure and symmetry of crystals.

Crystals are solids characterized by a periodic arrangement of atoms in space. The periodicity of crystals means the existence of long-range order in them and distinguishes crystals from amorphous bodies, in which there is only short-range order.

Periodicity is one of the types of crystal symmetry. Symmetry means the ability to transform an object that combines it with itself. Crystals can also be symmetric with respect to rotations about selected (periodically located in space) axes of rotation and reflections in reflection planes. A spatial transformation that leaves the crystal invariant, that is, transforms the crystal into itself, is called the symmetry operation. Rotations about an axis, reflections in a plane, as well as inversion about the center of inversion are point symmetry transformations, since they leave at least one point of the crystal in place. The displacement (or translation) of a crystal by a lattice period is the same symmetry transformation, but it no longer applies to point transformations. Point symmetry transformations are also called eigentransformations. There are also improper symmetry transformations, which are a combination of rotation or reflection and translation over a distance that is a multiple of the lattice period.

Crystals of different chemical composition from the point of view of symmetry can be equivalent, that is, they can have the same set of symmetry operations. This circumstance determines the possibility of classifying crystals according to the type of their symmetry. Different crystals can be assigned the same lattice with a given symmetry. The classification of crystals is based on Bravais lattices. The Bravais lattice can be defined as a set of points whose coordinates are given by the ends of the radius vector r .

where a 1 , a 2 , a 3 - an arbitrary triple of non-coplanar (not lying in the same plane) vectors, n 1 , n 2 , n 3 are arbitrary integers. Vectors a 1 , a 2 , a 3 are called vectors of elementary translations. The lattice transforms into itself upon translation to any vector that satisfies relation (1). It should be noted that for a given Bravais lattice, the choice of elementary translation vectors is ambiguous. It follows from the definition of the Bravais lattice that the elementary translation vector a 1 represents the smallest lattice period in a given direction. Any three non-coplanar translations can be chosen as elementary translations. minimal lattice period.

In each Bravais lattice, one can distinguish the minimum volume of space that, for all translations of the form (1), fills the entire space without overlapping with itself and leaving no gaps. Such a volume is called a primitive cell. If we choose a volume that fills the entire space as a result of not all, but some subset of translations, then such a volume will already be just an elementary cell. Thus, a primitive cell is an elementary cell of the minimum volume. It follows from the definition of a primitive cell that there is exactly one Bravais lattice node per cell. This circumstance can be useful for checking whether the selected volume is a primitive cell or not.

The choice of a primitive cell, as well as the choice of elementary translation vectors, is ambiguous. The simplest example of a primitive cell is a parallelepiped constructed on the vectors of elementary translations.

An important role in solid state physics is played by the primitive Wigner-Seitz cell, which is defined as the part of space located closer to a given point of the Bravais lattice than to other points of the lattice. To construct a Wigner-Seitz cell, one should draw planes perpendicular to the line segments connecting the lattice point chosen as the center with other points. The planes must pass through the midpoints of these segments. The polyhedron, limited by the constructed planes, will be the Wigner-Seitz cell. It is essential that the Wigner-Seitz cell has all the symmetry elements of the Bravais lattice.

A crystal (crystal structure) can be described by assigning a certain Bravais lattice to it and specifying the arrangement of atoms in a unit cell. The totality of these atoms is called the basis. The basis may consist of one or more atoms. Thus, in silicon, the basis composition includes two Si atoms; in the GaAs crystal, the basis is also diatomic and is represented by one Ga and one As atoms. In complex organic compounds, the basis may include several thousand atoms. The relationship between the concepts of lattice, basis, structure can be defined as follows:

lattice + basis = crystal structure.

The requirement that translational invariance be periodic imposes significant restrictions on the possible point symmetry operations in a crystal. Thus, in an ideally periodic crystal, symmetry axes of only 2, 3, 4, and 6 orders can exist, and the existence of a 5-order axis is forbidden.

Bravais showed that from the planes of reflection, four types of axes of rotation, inversion and translations, 14 different combinations can be formed. These 14 combinations correspond to 14 types of lattices. From a mathematical point of view, each such combination is a group (symmetry group). In this case, since translations are present in the group as symmetry elements, the group is called the space symmetry group. If the translation is removed, then the remaining elements form a point group. There are 7 point symmetry groups of Bravais lattices in total. Lattices belonging to a given point group form a syngony or system. The cubic system includes simple cubic (PC), body-centered cubic (bcc) and face-centered cubic (fcc) lattices; to tetragonal - simple tetragonal and centered tetragonal; to rhombic - simple, base-centered, body-centered and face-centered rhombic lattices; to monoclinic - simple and base-centered monoclinic lattices. The remaining three syngonies contain one type of lattices of the same name with them - triclinic, trigonal and hexagonal.

MOU "Secondary school No. 24"

city ​​of Podolsk

Moscow region

Report

« Crystal symmetry»

Performed:

Orlova

Olga Romanovna,

student 10 class "G"

Scientific adviser:

Elyushchev Oleg Vladimirovich,

teacher

mathematics

year 2012.

Plan.

IIntroduction. The concept of symmetry.

II Main part.

1) equal parts and figures in geometry and crystallography;

2) crystals and their structure;

3) elementary cells to the crystal;

4) symmetry and anisotropy of crystalline polyhedra;

5) symmetry and its elements;

6) groups or types of symmetry;

7) syngony of crystals;

9) symmetry of real crystals;

IIIConclusion. Symmetry as a crystal physics research method.

Symmetry of crystals.

The Greek word "symmetry" in translation into Russian means "proportion". In general, symmetry can be defined as the ability for a figure to naturally repeat its parts. The idea of ​​symmetry is widespread in everyday life. Symmetrical are, for example, corollas of flowers, butterfly wings, snow stars. Mankind has long used the concept of symmetry, applying it in a wide variety of areas of its activity. However, the mathematical development of the doctrine of symmetry was carried out only in the second half ofXIX century.

A symmetrical figure should consist of regularly repeating equal parts. Therefore, the idea of ​​symmetrical figures is based on the concept of equal parts.

"Two figures are called mutually equal if for each point of one figure there is a corresponding point of the other figure, and the distance between any two points of one figure is equal to the distance between two corresponding points of the other."

The concept of equality of figures, according to this definition, is much broader than the corresponding concept accepted in elementary geometry. In elementary geometry, such figures are usually called equal, which, when superimposed on one another, coincide with all their points. In crystallography, not only such compatible - equal figures are considered equal, but also figures related to each other as an object and its mirror image.

So far, we have been talking about geometric shapes. Turning to crystals, we must remember that they are real bodies and that their equal parts must be not only geometrically equal, but also physically identical.

In general, crystals are usually called solids that are formed in natural or laboratory conditions in the form of polyhedra.

The surface of such polyhedra is limited by more or less perfect planes - faces intersecting in straight lines - edges. The intersection points of the edges form the vertices.

The geometrically correct shape of crystals is determined, first of all, by their strictly regular internal structure.

In all crystal structures, many identical atoms can be distinguished, arranged like nodes of a spatial lattice. To imagine such a lattice, it is necessary to mentally fill the space without a trace with a multitude of equal parallelepipeds, parallel oriented and adjacent along whole faces. The simplest example of such parallelepipedal systems is a collection of cubes or bricks closely attached to each other. If, in such imaginary parallelepipeds, the corresponding points are selected, for example, their centers or any other points, then a so-called spatial lattice can be obtained. Selected corresponding points are called nodes. In real crystal structures, the sites of spatial lattice nodes can be occupied by individual atoms, ions, or groups of atoms.

The lattice structure is characteristic of all crystals without exception.

Thus, the most complete definition of a crystal will sound like this: all solids in which particles (atoms, ions, molecules) are arranged regularly in the form of nodes of spatial lattices are called crystals.

Solids in which particles are arranged randomly are called amorphous. Examples of amorphous formations are glasses, plastics, resins, glue. An amorphous substance is not stable and tends to crystallize over time. So the glass "crystallizes", forming aggregates of small crystals.

Examples of crystals are salt cubes, hexagonal prisms of rock crystal pointed at the ends, diamond octahedrons, pomegranate dodecahedrons.

In the modern description of a mineral, the parameters of its elementary cell are necessarily indicated - the smallest group of atoms, the parallel movement of which can build the entire structure of a given substance. Despite the fact that the number of atoms in an elementary cell and their type are different for each mineral, in natural crystals there are only seven types of elementary cells, which, repeating millions of times in three-dimensional space, form different crystals. Each type of cell corresponds to a certain syngony, which makes it possible to divide all crystals into seven groups.

The appearance of crystals largely depends on the shape of elementary cells and their location in space. Large cubic crystals can be obtained from cubic elementary cells. At the same time, the stepped arrangement of the "cubes" allows you to create more complex shapes.

The elementary cells are always aligned in such a way that the faces of the growing crystal and the angles formed by them are located not randomly, but in the correct order. Each type of face has a certain position relative to the axis, plane or center of symmetry, which this or that mineral possesses. Crystallography is based on the laws of symmetry, according to which crystals are classified according to certain syngonies.

In nature, in scientific and industrial laboratories, crystals grow in the form of beautiful, regular polyhedra with flat edges and straight edges. The symmetry and regularity of the external form of natural crystalline polyhedra is a distinctive feature of crystals, but not mandatory. In factory and laboratory conditions, crystals are often grown that are not polyhedral, but their properties do not change from this. From natural and artificially grown crystals, plates, prisms, rods, lenses are cut out, in which there are no longer traces of the external polyhedral shape of the crystal, but the amazing symmetry of the structure and properties of the crystalline substance is preserved.

Experience shows that if a fragment or plate of a crystal is placed in a solution or melt of the same substance and allowed to grow freely, then the crystal will again grow in the form of a regular, symmetrical polygon. This is due to the fact that the growth rate of crystals in different directions is different. This is just one example of the anisotropy of the physical properties of a crystal.

Anisotropy and symmetry are characteristic features of crystals due to the regularity and symmetry of their internal structure. In a crystalline polyhedron and in a plate cut out of it, there is an equally regular, symmetrical, periodic arrangement of particles. The particles that make up the crystals form regular, symmetrical rows, nets, lattices.

Stones, metals, chemical products - organic and inorganic, including such complex ones as cotton and rayon fibers, human and animal bones, and, finally, such complexly organized objects as viruses, hemoglobin, insulin, DNA and many others, have a regular internal structure. Each crystalline substance has a certain order, a characteristic "pattern" and symmetry in the arrangement of particles, established distances between particles, and all these patterns can be determined qualitatively and quantitatively.

All of the above applies to ideally developed crystals. But in nature, perfect geometric shapes are rarely found. Most commonly, crystals deform as a result of uneven facet development or have broken, curved lines while maintaining angles between different facets. Crystals can grow in the form of geometrically ordered aggregates or in complete disorder. It is not uncommon for minerals to exhibit a combination of different crystallographic forms. Sometimes certain obstacles interfere with the growth of a crystal, due to which the internal crystal structure does not find an ideal reflection in the external form, and the mineral forms irregular aggregates or dense masses. At the same time, according to the law of constancy of face angles, in crystals of a certain substance, both the size of the faces and their shape can change, but the angles between the corresponding faces remain constant. Therefore, in studying the symmetry and, in general, the geometry of real crystals, it is necessary to rely on the angles between the faces.

Getting acquainted with this section of crystallography, one cannot do without the use of geometrically regular polyhedra, which represent idealized models of certain crystals.

The doctrine of the symmetry of crystals is based on geometry. However, this branch of science owes its development mainly to scientists who worked in the field of crystallography. The most brilliant achievements are associated with the names of crystallographers, among which the names of two Russian academicians stand out - A.V. Gadolin and E.S. Fedorov.

Now you need to talk about the symmetry itself and its elements. The definition of symmetry mentioned the regular repetition of equal parts of figures. To clarify the concept of this regularity, imaginary auxiliary images (points, straight lines, planes) are used, relative to which equal parts of figures are correctly repeated. Such images are called elements of symmetry.

Examples of the mentioned elements are: inversion center, axes and planes of symmetry.

To characterize one or another axis, it is necessary to find out the value of the smallest angle of rotation that brings the figure into alignment. This angle is called the elementary angle of rotation of the axis.

The elementary angle of rotation of any axis of symmetry is an integer number of times 360°:

where n- an integer called the order (name) of the axis.

The order of the axis of symmetry corresponds to the number showing how many times the elementary angle of rotation is contained in 360°. At the same time, the order of the axis gives the number of combinations of the figure with itself during a full rotation around this axis.

Each axis has its own elementary angle of rotation:

at n=1 α=360°

n=2 α=180°

n=3 α=120°

n=4 α=90°

n=5 α=72°

n=6 α=60° etc.

In geometry, there is an infinite number of axes of various integer names. However, the symmetry of crystals is described by a finite set of axes. Their number is limited by the fact of the existence of a spatial lattice. The lattice imposes a ban on the realization in crystals of axes of the fifth order and axes higher than the sixth order.

In addition, there are so-called inversion axes.

Such an element of symmetry is, as it were, a combination of a simple axis of symmetry and the center of inversion, acting not separately, but together. Participating only as an integral part of the inversion axis, the inversion center may not appear as an independent element of symmetry. On all models where it is necessary to define inversion axes, there is no inversion center.

In crystallography, a set of symmetry elements is called the symmetry type of a crystalline polyhedron.

All groups (types) of symmetry of crystals were obtained in 1820 by the German professor of mineralogy I. Gessel. There were 32 of them. However, his results were not noticed by the scientific community, partly because of an unsuccessful presentation, partly because Gessel's article was published in an inaccessible publication.

Regardless of Hessel, the derivation of 32 groups (types) of symmetry of crystals was carried out in 1867 by the Russian academician, professor of the Artillery Academy, amateur crystallographer, General A.V. Gadolin. His work was immediately highly appreciated by experts.

The symmetry groups of crystals, or, as they are commonly called, types of symmetry, are conveniently divided into systems that combine groups with similar symmetry elements. There are six such systems - triclinic, monoclinic, rhombic, tetragonal, hexagonal and cubic.

Crystallographers who study the external shape of crystals and their structure often distinguish trigonal crystals from the hexagonal system. Thus, all crystals are divided into seven syngonies (from the Greek "syn" - together, "gonia" - angle): triclinic, monoclinic, rhombic, trigonal, tetragonal, hexagonal and cubic. In crystallography, a syngony is a group of symmetry types that have one or more similar symmetry elements with the same number of unit directions. It is essential to note that spatial lattices related to crystals of the same syngony must have unit cells with the same symmetry.

The names of syngonies are explained as follows: in crystals of the triclinic syngony, all three angles between the edges of the parallelepiped are oblique [klino (Greek) - tilt]. In crystals of the monoclinic system, there is only one oblique angle between the indicated edges (the other two are straight). The rhombic syngony is characterized by the fact that the simple forms related to it often have the shape of rhombuses.

The names "trigonal", "tetragonal", "hexagonal" systems indicate the typical symmetry of the crystals related to this. The trigonal system is often called rhombohedral, since most of the symmetry types of this system are characterized by a simple shape called a rhombohedron.

Crystals of the cubic system are characterized by spatial lattices, the elementary parallelepipeds of which are cubes in shape.

triclinic syngony. Syngony with the most primitive crystal forms and very simple symmetry. A characteristic form of the triclinic syngony is an oblique prism. Typical representatives: turquoise and rhodonite.

monoclinic syngony. Prisms with a parallelogram at the base are characteristic. The monoclinic system includes crystals of such minerals as alabaster, malachite, jade.

rhombic syngony. Typical shapes are rhombic prism, pyramid and bipyramid. Among the typical minerals of this syngony are topaz, chrysoberyl, and olivine.

trigonal syngony. Simple shapes are trigonal prisms, pyramids, bipyramids, as well as rhombohedrons and scalenohedra. Examples of trigonal system minerals are calcite, quartz, tourmaline.

Hexagonal syngony. Typical shapes: 6- or 12-sided prisms, pyramids and bipyramids. Beryl and vanadinite (used as vanadium ore) stand out in this syngony.

Tetragonal syngony. Simple shapes are tetragonal prisms, pyramids and bipyramids. Zircon and rutile crystallize in this syngony.

Cubic syngony. Simple shapes: cube, octahedron, tetrahedron. Fluorite, diamond, pyrite crystallize in the cubic syngony.

Syngonia, in turn, are grouped into three categories: lower, middle, higher.

Crystals of the lowest category are characterized by the presence of several single directions (the only direction that does not repeat in the crystal is called single) and the absence of symmetry axes of order higher than 2. These include three syngonies: triclinic, monoclinic and rhombic.

Crystals of the middle category have one single direction, coinciding with a single axis of order above 2. Three syngonies also belong to this: trigonal, tetragonal and hexagonal.

In crystals of the highest category, in the absence of single directions, there are always several axes of order above 2. This includes one cubic system.

So far, idealized models of crystalline polyhedra have been considered.

It is much more difficult to determine the symmetry of real crystals. Above, the uneven development of symmetrical crystal faces was noted due to the unequal influx of the feed solution to them. In this regard, the cube of a real crystal often takes the form of a flattened or elongated parallelepiped. Moreover, sometimes there is even a partial absence of symmetrical faces. Therefore, based on the external forms of real crystals, it is easy to erroneously lower their actual symmetry.

Accurate measurements of the angles between the faces come to the rescue here, by which it is not difficult to restore the true symmetry of the polyhedron. However, reverse errors also often occur, when a higher symmetry is attributed to crystals compared to the actual one.

It is also interesting that the same substances under different conditions can form completely different crystal structures, and hence different minerals. A striking example is carbon: if it has a hexagonal syngony, then graphite is formed, if it is cubic, diamond.

So, symmetry, periodicity and regularity of the structure are the main characteristics of the crystalline state of matter.

The way a crystal is arranged from the inside is inevitably reflected in its appearance and shape. The shape of the crystal allows us to assume in what order the particles are combined in its structure. And of course, we can say with great confidence that in an octahedral fluorite crystal, a hexagonal graphite plate, and a lamellar barite crystal, the particles are arranged differently. But in the "cubes" of halite and galena, they are located very similarly, although these minerals have a different chemical composition.

All these differences and similarities help to describe symmetry.

However, symmetry is not limited to revealing patterns in the arrangement of particles in spatial lattices and in the external shape of crystals. In addition, all physical properties are closely related to symmetry. It determines what physical properties a particular crystal may or may not have. It dictates the number of independent quantities necessary for the complete characterization of a given physical property, and the direction of their measurements with respect to the symmetry elements, i.e. determines the nature of the anisotropy of physical properties. Moreover, it turned out to be possible to attribute symmetry to mathematical quantities - scalars, vectors that describe the physical properties of crystals. And, finally, the very physical phenomena in crystals can be attributed one or another symmetry, coinciding with the symmetry of the mathematical quantities that describe these phenomena.

Bibliography

1. A.S. Sonin. "Course of macroscopic crystal physics", M., "Nauka", 2006.

2. M.P. Shaskolskaya. "Crystallography", M., "Higher School", 1984

3.G.M.Popov, I.I.Shafranovsky. "Crystallography", M., "Higher School", 1972

4. M. Aksenova, V. Volodin. Encyclopedia for children. Geology, M., "Avanta +", 2006

5. A. Zharkova. "Minerals. Treasures of the earth", M., "De Agostini", 2009

Explanatory note.

The topic of my essay is the symmetry of crystals. The purpose of my essay is a story about the symmetry of crystals. The objectives of my work are the study of the elements of symmetry, the story of the significance of symmetry in the study of the properties of crystals, and the generalization of the data obtained. The subject of my research are crystals. During my research, I used a variety of literature. One of the main sources was MP Shaskolskaya's book "Crystallography", which contained many articles on the structure of crystals and symmetry itself. I also used the book by G.M. Popov, I.I. Shafranovsky "Crystallography", where I found a lot of interesting information. For a more detailed analysis and a story about the symmetry of crystals, I used other literature, magazines and encyclopedias.

Abstracts.

The Greek word "symmetry" in translation into Russian means "proportion". In general, symmetry can be defined as the ability for a figure to naturally repeat its parts.

In crystallography, not only such compatible - equal figures are considered equal, but also figures related to each other as an object and its mirror image.

All crystals are built from material particles, geometrically correctly located in space. The ordered distribution of atoms, ions, molecules distinguishes the crystalline state from the non-crystalline state, where the degree of order is completely negligible.

Crystals are all solids in which particles (atoms, ions, molecules) are arranged regularly in the form of nodes of spatial lattices.

In the modern description of a mineral, the parameters of its elementary cell are necessarily indicated - the smallest group of atoms, the parallel movement of which can build the entire structure of a given substance.

Anisotropy and symmetry are characteristic features of crystals due to the regularity and symmetry of their internal structure.

Symmetry elements are called auxiliary geometric images (points, lines, planes), with the help of which the symmetry of figures is detected.

The center of inversion is a singular point inside the figure, characterized by the fact that any straight line drawn through it on both sides of it and at equal distances meets the same (corresponding) points of the figure. Such a point in geometry is called the center of symmetry.

A plane of symmetry is a plane that divides a figure into two mirror-equal parts, located relative to each other as an object and its mirror image.

The axis of symmetry is a straight line around which equal parts of the figure are repeated several times.

An inversion axis is such a straight line, when rotated around it by some specific angle with subsequent (or preliminary) reflection at the central point of the figure, as in the center of inversion, the figure is combined with itself.

All crystals are divided into seven syngonies (from the Greek "syn" - together, "gonia" - angle): triclinic, monoclinic, rhombic, trigonal, tetragonal, hexagonal and cubic. In crystallography, a syngony is a group of symmetry types that have one or more similar symmetry elements with the same number of unit directions.

The same substances under different conditions can form completely different crystal structures, and, consequently, different minerals. A striking example is carbon: if it has a hexagonal syngony, then graphite is formed, if it is cubic, diamond.

The way a crystal is arranged from the inside is inevitably reflected in its appearance and shape. The shape of the crystal allows us to assume in what order the particles are combined in its structure.

In addition, all physical properties are closely related to symmetry. It determines what physical properties a particular crystal may or may not have. It dictates the number of independent quantities necessary for the complete characterization of a given physical property, and the direction of their measurements with respect to the symmetry elements, i.e. determines the nature of the anisotropy of physical properties.

Symmetry permeates all crystal physics and acts as a specific method for studying the physical properties of crystals.

Therefore, the main method of crystallography is to establish the symmetry of phenomena, properties, structure and external shape of crystals.

Application.

A. I. Semke,
, MOU secondary school No. 11, Yeysk UO, Yeysk, Krasnodar kr.

Crystal symmetry

Lesson Objectives: educational– familiarity with the symmetry of crystals; consolidation of knowledge and skills on the topic "Properties of crystals" Educational- education of worldview concepts (causal relationships in the world around, the cognizability of the world and humanity); moral education (education of love for nature, feelings of comradely mutual assistance, ethics of group work) Educational– development of independence of thinking, competent oral speech, skills of research, experimental, search and practical work.

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

Physical dictionary

• Crystal - from the Greek. κρύσταλλος - literally ice, rock crystal.
• The symmetry of crystals is a regularity of the atomic structure, external shape and physical properties of 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

The symmetry of crystals 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 to come into alignment with the original position." Thus, such an object is symmetrical, which can be combined with itself by certain transformations: rotations around the axes of symmetry or reflections in the planes of symmetry. Such transformations are called symmetrical operations. After the symmetry transformation, the parts of the object that were in one place are the same as the parts that are in another place, which means that there are equal parts (compatible and mirrored) in a symmetrical object. The internal atomic structure of crystals is three-dimensionally periodic, that is, it is described as a crystal lattice. The symmetry of the external form (faceting) 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 work 1. Description of crystals

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

In addition to the trivial axis of the 1st order, only axes of the 2nd, 3rd, 4th and 6th orders 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 notation of point groups includes 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 syngonies - triclinic, monoclinic, rhombic, tetragonal, trigonal, hexagonal and cubic. The belonging of a crystal to one or another group of symmetry and syngony is determined by measuring the angles or by X-ray diffraction analysis.

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

triclinic system. Feature 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. Consequently, a ≠ b ≠ c; β = γ = 90°, α ≠ 90°. The elementary cell has the shape of a parallelepiped with a rectangle at the base.

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

tetragonal system. All angles are right, two edges are the same: 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 a straight line: a=b=c; α = β = γ ≠ 90°. The elementary cell has the shape of a cube deformed by compression or stretching 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 together three elementary cells, then 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 elementary 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 ( in).

An example of a cubic system is common salt crystals (NaCl, G). Larger chloride 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.

Order of execution

1. Look at the salt crystals with a magnifying glass. 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 the direction - anisotropy.
2. Examine the crystals of copper sulphate, pay attention to the presence of flat edges in individual crystals, the angles between the faces 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 break a mica plate, but it is easy to split it into thinner leaves along the planes ( strength anisotropy).
4. Consider polycrystalline bodies (a broken piece of iron, cast iron or zinc). Please note: at the break, you can distinguish small crystals that make up a piece of metal. Most of the solids found in nature and obtained in technology are a collection of randomly oriented small crystals fused with each other. Unlike single crystals, polycrystals are isotropic, i.e., 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 a 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 called a straight line, when rotated through 360 °, the crystal (its lattice) is combined with itself several times. The number of these combinations is called the order of the axis of symmetry. There are crystal lattices with axes of symmetry of the 2nd, 3rd, 4th and 6th order. The symmetry of the crystal lattice relative to the plane of symmetry is possible, as well as combinations of different types of symmetry.

Russian scientist E.S. Fedorov found 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 biggest 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 antiquity. 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.

Order of execution

1. Assemble the sodium chloride crystal model ( drawing is shown). We pay attention to the fact that the balls of one color imitate sodium ions, and the other - chlorine ions. Each ion in a crystal performs thermal oscillatory motion around a node of the crystal lattice. If you connect these nodes with straight lines, then a crystal lattice is formed. Each sodium ion is surrounded by six chloride ions, and vice versa, each chloride ion is surrounded by six sodium ions.
2. Choose a direction along one of the lattice edges. Please note: white and black balls - sodium and chlorine ions - alternate.
3. Choose a direction along the second edge: white and black balls - sodium and chloride ions - alternate.
4. Choose a direction along the third edge: white and black balls - sodium and chloride ions - alternate.
5. Draw a mentally straight line along the diagonal of the cube - it will contain only white or only black balls, i.e. ions of one element. This observation can serve as a basis for explaining the phenomenon of anisotropy inherent in crystalline bodies.
6. The sizes of ions in the lattice are not the same: the radius of the sodium ion is approximately 2 times greater than the radius of the chlorine ion. As a result, the ions in a 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 the significant differences in their physical properties. Such substances are called allotropic.
8. Make a conclusion based on the results of observation and sketch schematically the types of crystals.

1. Almandine. 2. Icelandic 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 chemicals have remarkable mechanical, electrical, magnetic, and 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 scientific research. The task arose of developing a technology for manufacturing 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 crystallization from a melt and crystallization from a solution. Crystals from solution are grown by slowly evaporating the solvent from a saturated solution or by slowly lowering the temperature of the solution.

Experimental work "Growing crystals"

Equipment: saturated solutions of sodium chloride, ammonium dichromate, hydroquinone, ammonium chloride, glass slide, glass rod, magnifying glass or framed lens.

Order of execution

1. Take a small drop of a saturated saline solution 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 experiment with ammonium dichromate passes most effectively. 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 growth rates of crystals in different directions - the growth anisotropy - in hydroquinone.
5. Make a conclusion based on the results of observation and sketch schematically the types of crystals obtained.

Research work 4. Application of crystals

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

Crystal		Application example
	Exploration and mining	Drilling tools
	jewelry industry	Decorations
	Instrumentation	Marine chronometers - extremely accurate appliances
	Manufacturing industry	Diamond bearings
	Instrumentation	Base stones for watches
	Chemical industry	Spinnerets for drawing fiber
	Scientific research	ruby laser
	jewelry industry	Decorations
germanium, silicon	Electronics industry	Semiconductor circuits and devices
Fluorite, tourmaline, Icelandic spar	Opto-electronic industry	Optical devices
quartz, mica	Electronics industry	Electronic devices (capacitors, etc.)
Sapphire, amethyst	jewelry industry	Decorations
	Manufacturing industry	graphite lubricant
	mechanical engineering	graphite lubricant

Interesting information

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

At the end of the XIX century. the German physicist O. Lehman and the Austrian botanist F. Reinitzer drew attention to the fact that some amorphous and liquid substances are distinguished by a very ordered parallel stacking of molecules elongated in shape. Later, according to the degree of structural order, they were called liquid crystals(LCD). There are smectic crystals (with a layered arrangement of molecules), nematic (with randomly parallel shifted elongated molecules) and cholesteric (similar in structure to nematic, but characterized by greater mobility of molecules). It was noticed that under external influence, for example, a small electrical voltage, with a change in temperature, magnetic field strength, the optical transparency of the LC molecule changes. It turned out that this happens due to the reorientation of the axes of the molecules in the direction perpendicular to the initial state.

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

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

Another wave of scientific interest in liquid crystals rose in the postwar years. Among crystallographers, our compatriot I.G. Chistyakov. At the end of the 60s. last century american corporation RSA 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 ( LCD - Liquid Crystal Display). These were monochrome indicators of modest size, where polysegment electrodes were used mainly for numbering numbers. The beginning 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 visual display of information in digital form - more accurate, with error-free counting.

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

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

What parameters determine the shape of snowflakes? What science and for what purposes is engaged in the study of snow, ice, snowflakes?

The first album with sketches of various snowflakes made with 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 six-pointed branched snowflakes that we are accustomed to appear only at a certain temperature: 14–17 °C. In this case, the humidity of the air must 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 - wood). The rays of these crystals look like tree branches.

Science deals with the world of snow and ice glaciology. It arose in the seventeenth century. after the Swiss naturalist O. Saussure published a book on alpine glaciers. Glaciology exists at the intersection of many other sciences, primarily physics, geology and hydrology. Studying ice and snow is necessary in order to know how to prevent snow avalanches and ice. After all, millions of dollars are spent annually to combat their consequences around the world. But if you know the nature of snow and ice, you can save a lot of money and save many lives. And ice can tell 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 for 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 mountains of Antrim rise. They are composed of black basalts - traces of the activity of ancient volcanoes that rose along the giant fault that separated Ireland from Great Britain 60 million years ago. The black lava flows erupted from these craters formed the coastal mountains on the Irish coast and in the Hebrides across the North Channel. This basalt is an amazing breed! Liquid, easily flowing in molten form (basalt flows sometimes rush along the slopes of volcanoes at speeds up to 50 km / h), it cracks when it cools and solidifies, forming regular hexagonal prisms. From a distance, basalt cliffs resemble huge organs with hundreds of black pipes. And when the lava flow flows into the water, sometimes such bizarre formations appear that it’s hard not to believe in their magical origin. It is this natural phenomenon that 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 fabulous giant, and is called the "Giant's Bridge".

A task. What properties of crystalline solids and liquids are we talking about? What are the differences between crystalline solids and liquids? ( Answer. The correct geometric shape is an essential external feature of 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, by the name of which the bedrock diamond-bearing rock became known as kimberlite. The content of diamonds 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. Now one of the attractions of Kimberley is a huge pit 400 m deep, dug by diamond miners.

A task. Where are the valuable properties of diamonds applied?

“Such a snowflake (we are talking about a snowflake. - A.S.), a hexagonal, regular star, fell to Nerzhin on the sleeve of an old front-line red 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 glasses 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 with a small load? ( Answer. Glass is classified as a brittle body, in which there is practically no plastic deformation, so that the elastic deformation ends directly in destruction.)

“It was freezing stronger than in the morning; but on the other hand it was so quiet that the creak of frost under the boots could be heard half a verst away.

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

? Why does the snow creak underfoot in cold weather? ( Answer. Snowflakes are crystals, underfoot they collapse, as a result of which sound appears.)

Diamond is cut by a diamond.

? Diamond and graphite are made up of the same carbon atoms. Why are the properties of diamond and graphite different? ( Answer. These substances differ in their crystalline 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. A very strong covalent bond binds boron and nitrogen atoms 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 dull, the chisel is sharp: it cuts sheets, pieces fly. What's this? ( Answer. Diamond.)

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

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

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

Conclusion

In conclusion of the lesson, I want to give a unique example of the use of symmetry. Honey bees need to 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 a medium-sized nest. The cells of the comb can in principle be square, but the bees choose a hexagonal shape: it provides the densest packing of larvae, so that the construction of the walls requires a minimum of precious wax. The cells are vertical, the cells on them are located on both sides, that is, they have a common bottom - more savings. They are directed upwards at an angle of 13 ° so that honey does not flow out. In such combs, several kilograms of honey are placed. 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: translation from English. M., 1968.
3. Glaciological Dictionary / Ed. V.M. Kotlyakov. L.: Gidrometeoizdat, 1984.
4. Kompaneets A.S. Symmetry in the micro- and macroworld. Moscow: 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 kids. Moscow: Avanta+, 2000.
8. Shubnikov A.V., Koptsik V.A. Symmetry in science and art. Publishing house 2. M., 1972.

SYMMETRY OF CRYSTALS- the property of crystals to be combined with themselves during rotations, reflections, parallel transfers, or with a part or combination of these operations. ext. the shape (cutting) of a crystal is determined by the symmetry of its atomic structure, which also determines the symmetry of the physical. crystal properties.

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.

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

In naib. 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. theory of S. to. - the theory of symmetric transformations into itself of three-dimensional space, taking into account the fact that ext. the atomic structure of crystals is discrete, three-dimensionally periodic. During symmetry transformations, the space is not deformed, but transformed as a rigid whole. Such a transformation is a 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 there are equal parts (compatible or mirrored) in a symmetrical object.

S. to. manifests itself not only in their structure and properties in real three-dimensional space, but also in the description of energetic. the 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 Reciprocal lattice)etc.

Symmetry groups of crystals. A crystal may have not one, but several. . So, a quartz crystal (Fig. 1, a) is aligned with itself not only when rotated by 120 ° around the axis 3 (operation gi), but also when rotating around the axis 3 240° (operation g2), & also for 180° rotations around axes 2 X, 2 Y, 2 W(operations g3, g4, g5). Each symmetry operation can be associated with an element of symmetry - a line, a plane, or a point, relative to which the given operation is performed. e.g. axis 3 or axes 2x, 2y, 2w are the axes of symmetry, the plane t(Fig. 1,b) - by the plane of mirror symmetry, etc. The set of symmetry operations (g 1 , g 2 , ..., g n ) given crystal forms a symmetry group in the sense of Math. theories groups. Consistent carrying out two symmetry operations is also a symmetry operation. In group theory, this is referred to as a product of operations:. There is always an identity operation g0, which does not change anything in the crystal, called. identification, it geometrically corresponds to the immobility of the object or its rotation through 360 ° around any axis. The number of operations that form a group G, called. group order.

Symmetry groups of space transformations are classified: by the 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 signs. To describe crystals, various symmetry groups are used, of which the most important are the point symmetry groups that describe the external. the shape of the crystals; their name. also crystallographic. classes; space symmetry groups describing the atomic structure of crystals.

Point symmetry groups. The operations of point symmetry are: rotations around the axis of symmetry of the 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 with respect to a point, Fig. 2, c); inversion turns (combination of rotation by an angle 360°/N with at the same time inversion, fig. 2d). Instead of inversion rotations, mirror rotations equivalent to them are sometimes considered. Geometrically possible combinations of point symmetry operations determine one or another point symmetry group, which is usually depicted in stereographic. projections. With point symmetry transformations, at least one point of the object remains fixed - it transforms into itself. All symmetry elements intersect in it, and it is the center of the stereographic. projections. Examples of crystals belonging to different point groups are given in Figs. 3.

Rice. 2. Examples of symmetry operations: a - rotation; b - reflection; c - inversion; d - inversion rotation of the 4th order; e - helical rotation of the 4th order; 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 the class (center of symmetry or center of inversion); a - to class 2 (one axis of symmetry of the 2nd order); g - to the class (one inversion-rotary axis of the 6th order).

Point symmetry transformations are described by linear equations

or coefficient matrix

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

and when reflected in a plane x 1 x 2D looks like:

The number of point groups is infinite. However, in crystals due to the presence of crystalline. lattice, 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 with the help of pentagonal figures it is impossible to fill space without gaps). The operations of point symmetry and the corresponding symmetry elements are denoted by the symbols: axes 1, 2, 3, 4, 6, inversion axes (center of symmetry or center of inversion), (it is also the plane of symmetry m), (Fig. 4).

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

To describe a point symmetry group, it suffices to specify one or more. the symmetry operations that generate it, the rest of its operations (if any) arise as a result of the interaction of the generators. For example, for quartz (Fig. 1, a), the generating operations are 3 and one of the operations is 2, and there are 6 operations in total in this group. The international notation of groups includes the symbols of the generating operations of symmetry. Point groups are combined according to the point symmetry of the unit cell shape (with periods a, b, c and angles) into 7 syngonies (Table 1).

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

Tab. one.- Point groups (classes) of symmetry of crystals

Groups containing only rotations describe crystals consisting only of compatible equal parts (groups of the first kind). Groups containing reflections or inversion rotations describe crystals, in which there are mirror equal parts (groups of the second 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-equal to each other (see Fig. Enantiomorphism).

The groups of S. k. carry a geome. meaning: each of the operations corresponds, for example, to rotation around the axis of symmetry, reflection in the plane. Certain point groups in the sense of group theory, which takes into account only the rules for the interaction of operations in a given group (but not their geom. meaning), turn out to be the same 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. c.

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

With regard to macroscopic properties of 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 families whose symbols are given in the top row. The bottom row indicates the limit group of each family and shows figures illustrating the limit group.

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

Operations characteristic of the atomic structure of crystals are 3 non-coplanar translations a, b, c, to-rye and set the three-dimensional periodicity of the crystal. gratings. Crystalline the lattice is considered to be infinite in all three dimensions. Such a mat. the approximation is real, because the number of unit 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 crystal structure with itself and, therefore, is a symmetry operation (translational symmetry).

Phys. discreteness of the crystal. matter is expressed in its atomic structure. Space groups are groups of transforming a three-dimensional homogeneous discrete space into itself. 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 and an atom of another sort, a nucleus and electrons. The conditions for 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. lattice.

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

In accordance with the point symmetry of the shape of the unit cell (elementary parallelepiped), the space groups, like the point ones, are divided into 7 crystallographic syngony(Table 2). Their further subdivision corresponds to translations. groups and their respective Vrave gratings. There are 14 Bravais lattices, of which 7 are primitive lattices of the corresponding syngonies, they are denoted R(except rhombohedral R). Others-7 plummets. lattices: baso (boco) - centered BUT(face is centered bc), V(face ac), C (ab); body-centered I, face-centered (on all 3 faces) F. Taking into account the centering for the translation operation t centering translations corresponding to the center are added tc. If these operations are combined with each other t + t s and with the operations of the point groups of the corresponding syngonies, then 73 space groups are obtained, called. symmorphic.

Tab. 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 hence the whole space into itself) are written as: , where D- point transformations, - components of screw transfer or sliding reflection, - translation operations. Brave groups. Operations of helical symmetry and their corresponding symmetry elements - helical axes have an angle. component (N = 2, 3, 4, 6) and translational t s = tq/N, where t- translation of the lattice, the rotation n occurs simultaneously with the translation along the W axis, q- helical index. General symbol for helical axes N q(Fig. 6). The screw axes are directed along 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, b, c: reflection is combined with translation by half of the corresponding grating period. The transfer by half the diagonal of the cell face corresponds to the so-called. wedge plane of sliding n, moreover, in tetragonal and cubic. groups, "diamond" planes are possible d.

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

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

Broadcast. the components of microsymmetry operations of space groups do not appear macroscopically in point groups; for example, the helical axis in the faceting of crystals appears as a simple rotational axis corresponding in order. Therefore, each of the 230 groups is macroscopically similar (homomorphic) to one of the 32 point groups. For example, for the point group mmm 28 space groups are displayed homomorphically.

The Schoenflies notation of space groups is the designation of the corresponding point group (for example, , Table 1), to which the historically accepted serial number is assigned from above, for example. . In international notation, the symbol of the Bravais lattice and the generating operations of the symmetry of each group are indicated, etc. The sequence of arrangement of space groups in Table. 2 in international notation corresponds to the number (superscript) in Schoenflies notation.

On fig. 7 the image of spaces is given. groups - Rpta according to the International Crystallographic tables. The operations (and their corresponding elements) of the symmetry of each space group, indicated for the unit cell, act on all crystalline. space, the entire atomic structure of the crystal and each other.

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

If you set inside the elementary cell to-n. point x (x 1 x 2 x 3), then the symmetry operations transform it into points symmetrically equal to it throughout the crystal. 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 translations of the lattice. The set of points derived from the given operations gi groups G - x 1 , x 2 ,...,x n-1, called correct system of points (PST). On fig. 7 on the right is the arrangement of the symmetry elements of the group, on the left is the image of the PST of the general position of this group. Points in general position are such points that are not located on an element of point symmetry 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 a corresponding symmetry, their number is an integer number of times less than the multiplicity of a PST of a general position. On fig. 7 on the left circles indicate points of general position, they are inside the elementary cell 8, the symbols "+" and "-", "1/2+" and "1/2-" mean, respectively, the coordinates +z, -z, 1/2 + z , 1/2 - z. Commas or their absence mean pairwise mirror equality of the corresponding points with respect to the planes of symmetry m present in this group at at= 1/4 and 3/4. If the point falls on the plane m, then it is not doubled by this plane, as in the case of points in general position, and the number (multiplicity) of such points of particular position is 4, their symmetry is -m. The same takes place when a point hits the centers of symmetry.

Each space group has its own PST sets. There is only one correct system of points in general position for each group. But some of the PST of a particular position may turn out to be the same for different groups. The International Tables indicate the multiplicity of PST, 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, atoms or centers of molecules are located along the SST (one or more). In structural analysis, the distribution of atoms over one or several. PST of this space group is produced taking into account the chemical. crystal f-ly and diffraction data. experiment, allows you to find the coordinates of points of private or general positions, in which atoms are located. Since each PST consists of one or a multiple of the Bravais lattices, the arrangement of atoms can also be thought of as a set of Bravo lattices “pushed into each other”. Such a representation is equivalent to the fact that the space group contains translations as a subgroup. Brave group.

Subgroups of crystal symmetry groups. If part of the operation to-l. group itself forms a group G r (g 1 ,...,g m),, then the last one is called subgroup of the first. For example, the subgroups of the 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 a lower order. Accordingly, there is a hierarchy of subgroups.

Most space symmetry groups of crystals are different among themselves and as abstract groups; the number of abstract groups isomorphic to 230 space groups is 219. Abstract equal are 11 mirror-equal (enantiomorphic) space groups - one with only right, others with left helical axes. These are, for example, P 3 1 21 and P 3 2 21. Both of these space groups are homomorphically mapped onto the point group 32, to which quartz belongs, but quartz is respectively right-handed and left-handed: 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 symmetry groups of crystals. Space symmetry groups of crystals - the basis of the theoretical. crystallography, diffraction and other methods for determining the atomic structure of crystals and describing the crystal. structures.

The diffraction pattern obtained by X-ray diffraction neutronography or electronography,allows you to set symmetry and geom. characteristics reciprocal lattice crystal, and hence the very structure of the crystal. This is how the point group of a crystal and the unit cell are determined; characteristic extinctions (the absence of certain diffraction reflections) determine the type of Bravais grating and belonging to a particular space group. The arrangement of atoms in an elementary cell is found from the totality of the intensities of diffraction reflections.

Space groups play an important role in crystal chemistry. More than 100 thousand crystals 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 chem. connections. So far, only 4 groups have not been found among the studied structures: Rcc2, P4 2 cm, P4nc 1 , R6tp. The theory explaining the prevalence of certain space groups takes into account the dimensions of the atoms that make up the structure, the concept of dense packing of atoms or molecules, the role of "packing" symmetry elements - slip planes and helical axes.

In solid state physics, the theory of group representations with the help of matrices and specials is used. f-tions, for space groups these functions are periodic. Yes, in theory structural phase transitions The space group of symmetry of the less symmetrical (low-temperature) phase of the 2nd kind 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 makes it possible to solve problems of dynamics crystal lattice, its electronic and magnetic structures, a number of physical properties. In the theoretical crystallography, space groups make it possible to develop a theory of partitioning space into equal regions, in particular, polyhedral ones.

Symmetry of projections, layers and chains. Crystalline projections. structures per 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, groups can be used - two-dimensionally periodic and - one-dimensionally periodic. 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 the helical (helical) symmetry possible in groups (see Fig. biological crystal).

Rice. 8. Objects with helical 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 a crystal. lattice. The long-range order in quasi-crystals is quasi-periodic, described on the basis of the theory of almost periodic. functions. The structure of quasicrystals can be represented as a projection onto a three-dimensional space of a six-dimensional periodic. cubic lattices with 5th order axes. Quasicrystals with five-dimensional symmetry in higher dimension can have 3 types of Bravais lattices (primitive, body-centered and face-centered) and 11 space groups. Dr. possible types of quasicrystals - laying in a stack of two-dimensional grids of atoms with axes of 5-, 7-, 8-, 10-, 12-th orders, with a periodicity along the third direction perpendicular to the grids.

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 ways and not equal in others. For example, the distribution of nuclei and electrons in a crystal antiferromagnet can be described using the usual spatial symmetry, but if we take into account the distribution of the magnetic in it. moments (Fig. 9), then “usual”, classical. symmetry is no longer enough. Such generalizations of symmetry include antisymmetry and color photography.

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

In antisymmetry, in addition to three space variables x 1, x 2, x 3 an additional, 4th variable is introduced. This can be interpreted in such a way that when (1, a) is transformed, the function F can be not only 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 (Shubnkov groups).

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

So, 81 point groups and 2942 groups are known. Main applications of generalized symmetry in crystallography - description of magn. structures.

Other antisymmetry groups (multiple, etc.) have also been found. Theoretically, all point and space groups of four-dimensional space and higher dimensions are also derived. Based on the consideration of the symmetry of a (3 + K)-dimensional space, one can also describe moduli that are incommensurate in three directions. structures (see disproportionate structure).

Dr. generalization of symmetry - similarity symmetry, when the equality of the parts of the figure is replaced by their similarity (Fig. 10), curvilinear symmetry, statistical. symmetry introduced in the description of 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 e l G., Symmetry, trans. from English, M., 1968; Modern crystallography, vol. 1 - Vainshtein BK, Symmetry of crystals. Methods of structural crystallography, M., 1979; G a l and u l and N R. V., Crystallographic geometry, M., 1984; International tables for crystallography, v. A - Space group symmetry, Dordrecht - , 1987. B. TO. Weinstein.

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