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Physicochemical crystallography studies issues. Fundamentals of crystallography geometric crystallography crystallography

Physicochemical crystallography studies issues. Fundamentals of crystallography geometric crystallography crystallography

CRYSTALLOGRAPHY

Crystallography- a science that studies crystals, their properties, external form and the reasons for their occurrence, directly related to mineralogy, mathematics (Cartesian coordinate system), physics and chemistry (the issue of the emergence and growth of crystals). The first works were done by Plato, Pythagoras, etc. .

Until the beginning of the 19th century, crystallography was descriptive. But already at the beginning of the 19th century, mathematics and physics were developed, so crystallography also received its development. Especially in the middle of the 20th century, with the growth of new technologies, crystallography acquired an experimental character (growing and synthesis of crystals). Today we can distinguish the following sections of crystallography:

Today we can distinguish the following sections of crystallography:

1. Geometric crystallography– studies the external shape of crystals and the patterns of their internal structure.

2. Crystal chemistry– studies the relationship between the internal structure of crystals and their chemical composition.

3. Physicochemical crystallography– studies the patterns of formation and growth of crystals.

4. Physical crystallography– studies the physical properties of crystals (optical, thermal, electrical, etc.), where some areas have become separate sciences (crystal optics).

Crystalline and amorphous bodies

Solids are divided into:

1. Amorphous, where elementary particles are arranged randomly, irregularly, which leads to the possession of the property of isotropy (the same properties of the substance in any direction). Amorphous bodies are unstable and over time they become crystalline (decrystallization).

2. Crystalline, characterized by an ordered arrangement of elementary particles that create a crystalline structure represented by a spatial lattice.

Crystalline (spatial) lattice

Crystal cell– a set of elementary particles located at the corresponding points of an infinite number of parallelepipeds, which completely fill the space, being equal, parallel oriented and adjacent along entire faces (Fig. 1).

Elements of the structure of the spatial lattice:

1. Nodes– elementary particles occupying a certain position in the lattice.

2. Row– a set of nodes located on the same straight line at a certain equal interval, called the row interval.

3. Flat mesh– a set of nodes located in the same plane.

4. Unit cell– a single parallelepiped, the repeatability of which forms a spatial lattice.

Mathematician Auguste Bravais proved that there can only be 14 fundamentally different lattices. The unit cell parameters determine the type of crystal lattice.

Crystal- a solid body in the shape of a regular polyhedron, in which elementary particles are arranged regularly in the form of a crystal lattice.

Crystal limiting elements:

· edges (smooth planes);

· edges (lines of intersection of faces);

· vertex (point of intersection of edges).

Relationship between the external shape of a crystal and its internal structure

1. Flat meshes correspond to the crystal faces.

2. Rows correspond to edges.

3. Nodes correspond to vertices.

But only those flat meshes and rows correspond to faces and edges that have the greatest reticular density– the number of nodes per unit area of a flat mesh or unit length of a row.

From here Euler derived the law: “The sum of the number of faces and vertices is equal to the number of edges plus 2.”

Basic properties of crystals

The regular internal structure of crystals in the form of a spatial lattice determines their the most important properties:

1. Uniformity– identical properties of the crystal in parallel directions.

2. Anisotropy– different properties of the crystal in non-parallel directions (for example, if a mineral is disten (“sten” - resistance) scratched along the elongation, then its hardness is 4.5, and if in the transverse direction, then the hardness is 6-6.5).

3. Ability to self-destruct– under favorable growth conditions, the crystal takes on the shape of a regular polyhedron.

4. Symmetry.

Crystal symmetry

Symmetry – (from the Greek “sim” - similar, “metrios” - measurement, distance, magnitude) - the natural repeatability of identical faces, edges, vertices of a crystal relative to some auxiliary geometric images (straight line, plane, point). Auxiliary geometric images, with the help of which the symmetry of a crystal is revealed, are called symmetry elements.

The symmetry elements of a crystal include the axis of symmetry (L - from the English line - line), the plane of symmetry (P - from the English play - plane), the center of symmetry (C - from the English centre - center).

Axis of symmetry- a straight line, when turning around it by 360 °, the crystal is combined with its initial position several times.

The elementary rotation angle a – can be equal to 60°, 90°, 120°, 180°.

The order of the axis of symmetry is the number of combinations of the crystal with its original position when rotating through 360°.

In a crystal, symmetry axes of the second, third, fourth and sixth orders are possible. There are no fifth or greater axes of symmetry than the sixth. The order of the symmetry axes is designated L 6, L 4, L 3, L 2.

The possible number of symmetry axes of the same order is as follows:

L 2 – 0, 1, 2, 3, 4, 6;

L 4 – 0, 1, 3;

Plane of symmetry– a plane dividing the crystal into two mirror-like equal parts.

Center of symmetry- a point inside a crystal at which the lines connecting opposite identical faces, edges, or vertices of the crystal intersect and bisect. From this definition the rule follows: if a crystal has a center of symmetry, then each face of it must have an opposite, equal, parallel and inversely directed face.

It is customary to write the totality of all available symmetry elements in a line, without any punctuation marks between them, while first the symmetry axes are indicated, starting from the highest order, then the symmetry plane, and on last place, if there is one, the center of symmetry is recorded.

Crystal classification

Based on the totality of symmetry elements in them, crystals are grouped into classes. Back in 1830, the scientist F. Hessel, through mathematical calculations, came to the conclusion that a total of 32 different combinations of symmetry elements in crystals are possible. It is the set of symmetry elements that defines the class.

Classes are united in syngonies. Classes characterized by one or more identical symmetry elements are grouped into one system. There are 7 known syngonies.

According to the degree of symmetry, systems are combined into larger divisions - categories: highest, middle, lowest (Table).

Crystal Shapes

1. Simple - crystals in which all faces have the same shape and the same size. Among the simple forms there are:

· closed – their edges completely enclose the space (regular polyhedra);

· open – they do not completely enclose the space and in order to close them other simple forms (prisms, etc.) are used

2. A combination of simple forms - a crystal on which faces are developed that differ from each other in shape and size. As many different types of faces as there are on a crystal, the same number of simple shapes are involved in this combination.

Nomenclature of simple forms

The name is based on the number of faces, the shape of the faces, and the cross-section of the shape. The names of simple forms use Greek terms:

· mono– one-, only;

· di, bi– two-, twice;

· three– three-, three-, three times;

· tetra– four-, four-, four times;

· penta– five-, five;

· hexa– six-, six;

· Octa- eight, eight;

· dodeca– twelve-, twelve;

· hedron– edge;

· gonio- corner;

· syn– similar;

· pinakos– table, board;

· Kline– tilt;

· poly- a lot of;

· skalenos- oblique, uneven.

For example: pentagondodecahedron (five, angle, twelve - 12 pentagons), tetragonal dipyramid (a quadrangle at the base, and two pyramids).

Crystallographic axis systems

Crystallographic axes– directions in the crystal parallel to its edges, which are taken as coordinate axes. The x axis is III, the y axis is II, the z axis is I.

The directions of the crystallographic axes coincide with the rows of the spatial lattice or are parallel to them. Therefore, sometimes instead of the designations of the I, II, III axis, the designations of single segments a, b, c are used.

Types of crystallographic axes:

1. Rectangular three-axis system (Fig. 2). Occurs when the directions are oriented perpendicular to each other. Used in cubic (a=b=c), tetragonal (a=b≠c) and rhombic (a≠b≠c) systems.

2. Four-axis system (Fig. 3). The fourth axis is oriented vertically, and in a plane perpendicular to it three axes are drawn through 120°. Used for crystals of hexagonal and trigonal systems, a=b≠c

3. Inclined system (Fig. 4). a=γ=90°, b≠90°, a≠b≠c. Used to install monoclinic crystals.

4.
Oblique system (Fig. 5). a≠γ≠b≠90°, a≠b≠c. Used for triclinic crystals.

Law of Integers

This is one of the most important laws of crystallography, also called Haui's law, the law of rationality of double relations, the law of rationality of parameter relations. The law says: “The double ratios of the parameters cut off by any two faces of the crystal on its three intersecting edges are equal to the ratios of integers and relatively small numbers.”

1. Select three non-parallel edges intersecting at point O. We take these edges as crystallographic axes (Fig. 6).

2. We select two faces A 1 B 1 C 1 and A 2 B 2 C 2 on the crystal, and the plane A 1 B 1 C 1 is not parallel to the plane A 2 B 2 C 2, and the points lie on the crystallographic axes.

3. The segments cut off by the faces on the crystallographic axes are called face parameters. In our case, OA 1, OA 2, OB 1, OB 2, OC 1, OC 2.

, where p, q, r are rational and relatively small numbers.

The law is explained by the structure of the crystal lattice. The directions chosen as axes correspond to the rows of the spatial lattice.

Face symbols

To obtain a face symbol, you need to install the crystal in the corresponding crystallographic axes, then select single face– a face whose parameters along each crystallographic axis are taken as a unit of measurement (in other words, as a scale segment). As a result, the ratio of parameters will characterize the position of the face in the crystallographic axes.

It is more convenient to use not parameters, but face indices– values inverse to the parameters: . Indexes are written in curly (characterize a simple form as a whole, for example (hkl) or (hhl)) or parentheses (referring directly to a specific face, e.g. (hhl) or (hlh) ), without punctuation. If a negative index is obtained, then it can be shown by the vector sign – (hkl). Subscripts can also be indicated by numerical values, such as (321), (110) or (hk0). “0” means that the face is parallel to the axis.

Pathways for crystal formationV

Crystals can be formed from all aggregate states of matter, both in natural and laboratory conditions.

Gaseous state - snowflakes (ice crystals), frost, plaque, native sulfur (during volcanic eruptions, sulfur crystals settle on the walls of craters); in industry - iodine crystals, magnesium. Sublimation– the process of formation of crystals from a gaseous substance.

Liquid state - the formation of crystals from a melt and from a solution. The formation of all intrusive rocks occurs from melts (mantle magmatic melts), when the main factor is a decrease in temperature. But the most common is the formation of crystals from solutions. In nature, these processes are the most common and intense. The formation of crystals from solutions is especially characteristic of drying lakes.

The solid state is mainly the process of transition of an amorphous substance into a crystalline one (decrystallization), in natural conditions These processes are active when high temperatures ah and pressures.

The appearance of crystals

Solutions differ in the degree of concentration of the substance in them:

· unsaturated (undersaturated) – you can add a substance and it will continue to dissolve;

· saturated – adding a substance does not lead to its dissolution, it precipitates;

· supersaturated (supersaturated) – formed if a saturated solution finds itself in conditions where the concentration of the substance significantly exceeds the solubility limit; The solvent begins to evaporate first.

For example, the formation of a crystalline nucleus of NaCl:

1. One-dimensional crystal (due to the attraction of ions, a row is formed), (Fig. 7);

2. 2D crystal (flat mesh), (Fig. 8);

3. Primary crystal lattice (crystalline nucleus of about 8 unit cells), (Fig. 9).

Each crystal has its own chain of formation (for a salt crystal - a cube), but the mechanism will always be the same. In real conditions, as a rule, the crystallization center is either a foreign impurity (a grain of sand) or the smallest particle of the substance from which the crystal will be built.

Crystal growth

Today, there are two main theories describing crystal growth. The first of them is called the Kossel-Stransky theory (Fig. 10). According to this theory, particles attach to the crystal preferentially in such a way that the greatest energy is released. This can be explained by the fact that any process is “easier” if energy is released.

A– the maximum amount of energy is released (when a particle hits this triangular angle).

B– less energy will be released (dihedral angle).

IN– a minimum of energy is released, the most unlikely case.

During growth, the particles will first fall into position A, then in B and finally in IN. The growth of a new layer will not begin on the crystal until the construction of the layer is completed.

This theory fully explains the growth of crystals with ideal smooth faces with the mechanism of layer-by-layer growth of faces.

But in the 30s of the 20th century it was proven that the edges of the crystal are always distorted or have some kind of defects, therefore, in real conditions, the edges of the crystal are far from ideally smooth planes.

The second theory was proposed by G.G. Lemmlein, taking into account the fact that crystal faces are not ideal, developed the theory of dislocation (dislocation growth) - displacement. Due to the screw dislocation, there is always a “step” on the surface of the crystal, to which particles of the growing crystal most easily attach. Dislocation theory and, in in particular, the theory of screw dislocation (Fig. 11, 12), always provides the opportunity for continued growth of the faces, because there is always room for the favorable attachment of a particle to a dislocated crystal lattice. As a result of such growth, the surface of the face acquires a spiral structure.

Both theories, perfect and imperfect crystal growth, complement each other, each of them is based on the same laws and principles and completely allow us to characterize all issues of crystal growth.

Face growth rate

Edge Rise Rate– the size of the segment normal to its plane, by which a given face moves per unit time (Fig. 13).

The growth rate of different crystal faces is different. Facets with a higher growth rate gradually decrease in size, are replaced by growing edges with a low growth rate, and can completely disappear from the surface of the crystal. (Fig. 14). First of all, the faces with the highest reticular density develop on the crystal.

The rate of edge growth depends on many factors:

internal and external. Of the internal factors, the greatest influence on the growth rate of the faces is their reticular density, which is expressed by Bravais’s law: “The crystal is covered with faces with a higher reticular density and the lowest growth rate.”

Factors affecting the shape of a growing crystal

Factors are divided into internal (that which is directly related to the properties of ions or atoms or a crystal lattice) and external: pressure, as well as:

1. Concentration flows. When a crystal grows in a solution, there is an area near it of a slightly higher temperature (particles are attached so that as much energy as possible is released) and with a reduced density of the solution (the growing crystal is fed) (Fig. 15). When dissolving, the opposite happens.

Streams play a dual role: constantly moving upward streams bring new portions of matter, but they also distort the shape of the crystals. Recharge occurs only from below, less from the sides, and almost none from above. When growing crystals in laboratory conditions, they try to eliminate the influence of concentration flows, for which they use different techniques: the method of dynamic crystal growth, the method of artificial mixing of the solution, etc.

2. Concentration and temperature of solution. Always influence the shape of the crystals.

The influence of solution concentration on the shape of alum crystals (concentration increases from 1 to 4):

1 – crystal in the shape of an octahedron;

2.3 – a combination of several simple forms;

4 – crystal with a predominant development of the octahedron face, the shape approaches spherical.

Effect of temperature on epsomite:

As the temperature rises, epsomite crystals acquire thicker prismatic shapes; at low temperatures, they acquire a thin lens.

3. Foreign matter impurities. For example, the octahedron of alum turns into a cube when grown in a solution mixed with borax.

4. Others.

Law of Constancy of Facet Angles

Back in the middle of the 17th century, in 1669, the Danish scientist Steno studied several quartz crystals and realized that no matter how much the crystal was distorted, the angles between the faces remained unchanged. At first, the law was treated coolly, but 100 years later, research by Lomonosov and the French scientist Romé-Delisle, independently of each other, confirmed this law.

Today the law has a different name - the Steno-Lomonosov-Romais-Delisle law). Law of constancy of facet angles: “In all crystals of the same substance, the angles between the corresponding faces and edges are constant.” This law is explained by the structure of the crystal lattice.

To measure the angles between the faces, a goniometer is used (similar to a mix of a protractor and a ruler). For more accurate measurements, an optical goniometer, invented by E.S., is used. Fedorov.

Knowing the angles between the crystal faces of a substance, it is possible to determine the composition of the substance.

Crystal intergrowths

Among crystal intergrowths, two main groups are distinguished:

1. Irregular - intergrowths of crystals that are not interconnected or oriented in any way in space (druze).

2. Natural:

· parallel;

· doubles.

Parallel splice crystals are several crystals of the same substance, which can be of different sizes, but oriented parallel to each other; the crystal lattice in this intergrowth is directly connected into one whole.

Scepter-like fusion– smaller quartz crystals grow together with a larger crystal.

Doubles

Double– a natural fusion of two crystals, in which one crystal is a mirror image of the other, or one half of the twin is derived from the other by turning 180°. From the point of view of mineralogy, in any twin, an internal reentrant angle is always visible (Fig. 16).

Double elements:

1. Twin plane - a plane in which two parts of the twin are reflected.

2. Twin axis – an axis, when rotated around which one half of the twin is formed into the second.

3. The fusion plane is the plane along which the two parts of the twin are adjacent to each other. In particular cases, the twin plane and the fusion plane coincide, but in most cases this is not the case.

The combination and character of all three elements of the twin are determined by the laws of twinning: “spinel”, “Gallic”, etc.

Germination twins– one crystal grows through another crystal. If several crystals are involved, tees, quads, etc. are distinguished accordingly. (depending on the number of crystals).

Polysynthetic twins- a series of twinned crystals arranged so that each two adjacent ones are located towards each other in a twin orientation, and the crystals passing through one are oriented parallel to each other (Fig. 17).

Polysynthetic twinning on natural crystals often manifests itself in the form of thin parallel hatching (twin seams).

Shapes of natural crystals

Among the crystals it is customary to distinguish:

· perfect– those crystals in which all faces of the same simple shape are identical in size, outline, distance from the center of the crystal;

· real– encounter certain deviations from ideal forms.

In natural (real) crystals, the uneven development of faces of the same shape creates the impression of lower symmetry (Fig. 18).

In real crystals, the faces are far from mathematically correct planes, because On the faces of real crystals there are various complications in the form of shading, patterns, pits, growths, i.e. sculptures. There are: parquet-like pattern, shading on the edge, vicinals (they are small sections of the crystal edge, slightly shifted from the direction of the edge). In real crystals, complex crystal shapes are very common.

When deviating from normal growth conditions, skeletal crystals– crystals on which edges and vertices are predominantly developed, and the edges lag behind in development (for example, snowflakes). Anti-skeletal crystals– the edges develop predominantly, while the edges and vertices lag behind in development (the crystal takes on a rounded shape, a diamond is very often found in this form).

There are also twisted, split, and deformed crystals.

Internal structure of crystals

The internal structure of crystals is very often zonal. Every change chemical composition the solution where the crystal grows causes its own layer. The zonal structure is due to pulsations and changes in the chemical composition of feeding solutions, i.e. Depending on what the crystal was fed in its youth, it will change, for example, the color of the zones.

The transverse fracture shows a sectorial structure, closely related to zoning and caused by changes in the composition of the medium.

Inclusions in crystals

All inclusions are divided into homogeneous and heterogeneous. They are also divided according to the time of formation into:

1. Residual (relic) - a solid phase representing a substance that existed before the growth of the crystal.

2. Syngenetic - inclusions that arose with the growth of crystals.

3. Epigenic – arose after the formation of crystals.

Residual and syngenetic inclusions are of greatest interest for crystallography.

Methods for studying inclusions in crystals

I.P. Ermakov and Yu.A. Dolgov made a great contribution to the study of inclusions, and today there are two main methods for studying inclusions in crystals:

1. Homogenization method– a group of methods based on the principle of transforming inclusions into a homogeneous state; this is usually achieved by heating. For example, bubbles in a crystal are represented by a liquid, and when heated to a certain temperature they become homogeneous, i.e. liquid becomes gas. Mainly, this method works on transparent crystals.

2. Decripitation method– by changing temperature and pressure, the crystal and its inclusions are taken out of equilibrium and the inclusions are brought to an explosion.

As a result, data is obtained on the temperature and pressure of formation of a crystal containing gases, liquids or a solid phase in the form of an inclusion.

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1 . general characteristics geological disciplines

The sciences of mineralogy, crystallography and petrography historically emerged from the science of the material composition, structure and history of the development of the earth called geology.

Crystallography studies education, form and physical Chemical properties crystals that make up various minerals.

Metallography-a science that studies the structure and properties of metals and establishes a connection between their composition, structure and properties.

Mineralogy distinguished itself as the science of natural chemical compounds called minerals. Mineralogy studies the composition and structure of minerals, the conditions of their formation and changes.

Petrography-science of rocks, their composition, structure, classification, conditions of occurrence.

These sciences are inextricably linked with the practical needs of metallurgical and other industries. There is not a single industry that does not use minerals in natural form, or any components extracted from them. Knowledge of minerals, their composition, various properties and areas of practical application is necessary for specialists working in various industries.

Mminerals are chemical elements or compounds formed in the earth's crust, water shell or atmosphere as a result of various physical and chemical processes (without any intervention).

Minerals can consist of one chemical element: diamond (C); graphite (C); sulfur (S); gold (Au) or can be compounds of constant or variable composition:

Compounds of constant composition (avalanche spar; quartz; calcium)

Compounds of variable composition: olivines having a composition from Mg 2 (SiO 4) forsterite to Fe 2 (SiO 4) fayalite.

Most minerals are solid, crystalline substances. Although individual minerals are found in a cryptocrystalline form (usually a colloidal dispersed) state.

In nature, minerals can be dispersed in the form of tiny particles or present in large accumulations. In this case, minerals of the same substance can be found in in different forms. This causes difficulties in externally determining the minerals included in any rock.

Currently, about 3,800 different minerals are known, of which only 250-300 are widespread and of practical importance. These are ores of ferrous, non-ferrous and rare metals, raw materials for production building materials, raw materials for the chemical industry, precious and other stones.

Since minerals have a regular, regular arrangement of atoms, due to their crystalline structure, minerals do not include liquids, gases, artificial solids and natural atmospheric substances.

Minerals differ from each other in chemical composition and crystal structure.

Minerals that have an identical crystal structure but differ in chemical composition are called isomorphic.

Minerals with the same chemical composition but different crystal structures are called polymorphic(example of polymorphic minerals: diamond and graphite).

1.1 Morphology of minerals (forms of occurrence of minerals in nature)

In nature, minerals are found in the form:

Monocrystals;

Doubles;

Aggregates.

Double they call such a natural fusion of two crystals in which one individual can be obtained from another either by reflection in a certain plane (twin) or by rotation around a certain axis (twin).

Most often, minerals occur in the form of random irregular aggregates. units. Aggregates may consist of crystals of one mineral (monomineral aggregates) or several aggregates (polymineral aggregates).

Aggregates are divided into:

Coarse-grained (more than 5 mm);

Medium-grained (1-5 mm);

Fine-grained (less than 1 mm).

The forms of the grains that make up the aggregates are: scaly, fibrous, earthy. The following morphological types of aggregates are distinguished:

Drusen are intergrowths of well-formed crystals, different in height and differently oriented, but attached at one end to a common flat or concave base.

Secretions are mineral formations that fill voids in rocks. The filling of voids occurs as a result of the gradual deposition of substances on their walls from the periphery to the center.

Concretions are round-shaped formations that usually have a radial or shell-like structure. Unlike secretion, the deposition of a substance occurs from the center to the periphery.

Oolites are small spherical formations with a concentric shell-like structure.

Pseudoolites are formations similar in shape to oolites, but without a concentric shell-like structure.

Dendrites are tree-like aggregates resembling fern leaves and tree branches.

1.2 Physical properties of minerals

The main physical properties of minerals, which allow them to be determined by their external characteristics, include: color, streak color, tarnish, luster, degree of transparency, hardness, cleavage, fracture, specific gravity, magnetism, brittleness, malleability, flexibility, etc.

Color is one of the characteristic physical properties of minerals. The same mineral, depending on its chemical composition, structure, mechanical and chemical impurities, can have different colors. By color, one can judge the conditions of formation of minerals and their belonging to a particular deposit.

Academician A.E. Fersman identifies three types of mineral colors: idiochromatic, allochromatic and pseudochromatic.

Idiochromatic - the mineral’s own color.

Allochromatic is a consequence of the presence of foreign mechanical impurities in the mineral.

Pseudochromatic - the phenomenon of diffraction of light rays from any internal cracks.

Stroke color- a trace left by a mineral on an unglazed porcelain plate. This is the color of crushed mineral powder.

Tarnish- a phenomenon when a mineral, in addition to the main color in a thin surface layer, has an additional color.

Cleavage- the ability of some minerals to split or split along certain planes with the formation of smooth, even, shiny surfaces.

1.3 Genesis of minerals (aboutformation of minerals in nature)

Mineral formation processes can be divided into:

1) Endogenous (occurring inside the earth and associated with magmatic activity);

2) Exogenous (occurring on the surface of the earth, manifested in the action of atmospheric agents and the surface of aqueous solutions, as well as in the biochemical activity of organisms (oxidation, decomposition);

3) Metamorphic (occurring as a result of the transformation of previously formed rocks when physical and chemical conditions change.

Paragenehisminerals.

Paragenesis is the co-occurrence of minerals in nature due to the common process of their formation. Minerals can be formed sequentially or simultaneously.

1.4 PetrographyI

Petrography- a science that studies rocks, their mineral and chemical composition, structure, distribution and conditions of formation.

Rocks are mineral aggregates of more or less constant chemical and mineral composition that occupy large areas of the earth's crust. Rocks can be monomineral, consisting of one mineral, or polymineral, which includes several minerals

Monomineral rocks - limestone and marble (consisting of the mineral calcite), quartzite (consisting of quartz).

Polymineral rocks - granite (the main rock-forming minerals are feldspars (microcline, orthoclase, plagioclase), quartz and mica (biotite, muscovite).

About a thousand types of rocks are known, which, according to the conditions of formation (genesis), are divided into three classes:

1. Igneous( or igneous). They are formed from magma frozen in the bowels of the Earth or on its surface; they are typical high-temperature formations.

2. Sedimentary. They are filled and transformed products of the destruction of previously formed rocks, remains of organisms and products of their vital activity; the formation of sedimentary rocks occurs on the surface of the Earth at normal temperatures and normal pressure, mainly in an aquatic environment.

3. Metamorphic. Formed on great depths due to changes in sedimentary and igneous rocks under the influence of various endogenous processes (high temperatures and pressure, gaseous substances released from magma, etc.).

2 . Fundamentals of Crystallography

Crystallography is divided into: geometric crystallography, crystal chemical and physical crystallography.

Geometric crystallography examines the general principles of the structure of crystalline substances that form their crystals, as well as the symmetry and systematics of crystals.

Crystal chemistry studies the relationship between the structures and chemical properties of crystalline matter, as well as the description of crystal structures

Physical crystallography describes the physical properties of crystals (mechanical, optical, thermal, electrical and magnetic).

2 .1 Basicsgeometric crystallography

Features of the crystalline state. The word “crystal” is always associated with the idea of a polyhedron of one form or another. However, crystalline substances are characterized not only by the ability to produce formations of a certain shape. The main feature of crystalline bodies is their anisotropy- dependence of a number of properties (tensile strength, thermal conductivity, compressibility, etc.) on the direction in the crystal.

Creesteels- solid bodies formed in the form of geometrically regular polyhedra.

a) rock salt; b) quartz; c) magnetite

Figure 1. Crystals

The crystal confinement elements are: planes - edges; lines of intersection of faces - ribs; points of intersection of the ribs - peaks.

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Figure 2. Crystal confinement elements

Elementary particles (atoms, ions or molecules) in crystals are arranged in the form of a spatial lattice.

A spatial lattice is a system of points located at the vertices of equal parallelepipeds oriented parallel to each other and adjacent along entire faces, without gaps filling the space.

Figure 3. Spatial lattice of the crystal

mineral crystalline plastic metal

The elementary parallelepipeds that make up the spatial lattice of a crystal are called elementary cells.

The parameters of such a cell are: three angles between, taken as the main axes, and three segments (A, B, C) of distances between nodes along these axes.

Figure 4. Unit cell parameters

A certain arrangement of particles in crystals in the form of a spatial lattice determines a number of special properties of crystalline substances - uniformity, anisotropy, the ability to self-cut, i.e. grow in the form of regular polyhedra).

Uniformity means that the properties of crystals are the same at all its points.

Anisotropy crystals lies in the disparity in various directions of most of their physical properties (mechanical, optical, and others).

Ability to self-destruct lies in the fact that, under favorable growth conditions, they form regular polyhedra, the faces of which are flat grids of the spatial lattice.

If you place an irregularly shaped piece of crystals in a solution with the appropriate conditions, then after a while it will acquire edges and take the form of a regular polyhedron, characteristic of the crystals of this substance.

The transformation of a ball cut from a cubic rock salt crystal in a saturated solution back into a cubic crystal.

Figure 5. Transformation scheme

Crystals of a mineral are most often characterized by the presence of faces of a certain type, although in rare cases the external forms of crystals of the same mineral may differ depending on the conditions of formation.

The laws of geometric crystallography are of great importance for the study of crystals.

First law:Law of Constancy of Facet Angles-Sten's law: for different crystals of the same substance, regardless of size and shape, between the corresponding faces under given conditions is constant.

Figure 6. Various quartz crystals

Second law-law of rationality of parameter relations. Ayui Law.

On one crystal, only such figures can be found, the parameters of the faces of which relate to the parameters of the faces of a simple shape, taken as the main one, as rational numbers.

Crystal symmetry

Crystal symmetry lies in the regular repetition of identical faces, edges, corners in this crystal.

Conditional images, with respect to which symmetry is observed, are called symmetry elements. These include: plane of symmetry, axis of symmetry, center and vertex.

Plane of symmetry- this is an imaginary plane dividing the crystalline polyhedron into two equal parts, one of which is a mirror image of the other.

The number of planes of symmetry in crystals is indicated by a number placed in front of the conventional symbol of the plane of symmetry with the letter P.

Crystals cannot have more than nine planes of symmetry.

Axis of symmetry- an imaginary straight line that passes through the crystal and, when rotated 360° around it, the figure aligns with itself a certain number of times (n times). The name of the axis or its order is determined by the number of alignments during a full rotation around the axis (360 degrees) of the crystal.

Crystals have axes of the second, third, fourth and sixth orders.

Axes of symmetry are designated by the letter L and a symbol that indicates the order of the axis of symmetry (L 1, L 2, L 3, L 4, L 6).

In addition to the usual axes of symmetry, there are inversion and mirror-rotary axes. If they are present, in order to align the figure with itself, rotation around an axis must be accompanied by a rotation of 180° around an axis perpendicular to the given one (inversion), or a mirror reflection from the plane.

Center of symmetry C called a point that bisects any line passing through it, drawn to the intersection with the faces of the figure.

In 1867 A.V. Gadolin mathematically showed that the existence of 32 types of symmetry of crystalline forms is possible, each of which is characterized by a certain combination of symmetry elements.

All types of symmetry of crystals are divided into three categories: lower, middle and higher. Crystals of the lowest category do not have axes of a higher order - higher than the second; the middle category is characterized by one axis of higher order, the highest - by several such axes. Categories are divided into crystal systems or systems.

Syngony is a set of symmetry elements with the same number of axes of the same order. There are seven syngonies in total: triclinic, monoclinic, rhombic, trigonal, hexagonal, cubic, tetragonal.

The lowest category includes three crystal systems - triclinic, monoclinic and orthorhombic. In crystals of the triclinic system there are neither axes nor planes of symmetry: there may also be no center of symmetry. Monoclinic crystals can have both an axis and a plane of symmetry, but they cannot have multiple axes or planes of symmetry. The rhombic system is characterized by the presence of several symmetry elements - several axes or planes.

A necessary condition for the formation of crystals of high symmetry is the symmetry of their constituent particles. Since most molecules are unsymmetrical, crystals with high symmetry make up only a small fraction of the total known.

There are many cases where the same substance exists in different crystalline forms, i.e. differs in its internal structure, and therefore in its physical and chemical properties. This phenomenon is called polymorphism.

Among crystalline bodies, the phenomenon is often observed isomorphism- the property of atoms, ions or molecules to replace each other in a crystal lattice, forming mixed crystals. Mixed crystals are completely homogeneous mixtures solids are substitutional solid solutions. Therefore, we can say that isomorphism is the ability to form substitutional solid solutions.

Crystal Shapes

In addition to elements of symmetry, crystals are also characterized by their external shape. Thus, a cube and an octahedron have the same symmetry elements, but the external shape and number of faces are different.

Crystal shape called the totality of all its faces. There are simple and complex forms.

Simple form This is a form called, all the faces of which are connected to each other by elements of symmetry, or in other words, these are crystals that consist of identical faces that have a symmetrical arrangement (cube, octahedron, tetrahedron)

Simple forms can be either a closing cycle of space (closed forms) or open forms that do not close the space on all sides.

Open simple forms include:

Monohedron, dihedron, pinanoid, pyramids, prisms

Closed simple forms include:

Dipyramids, rhombohedron, tetrahedron, cube, octahedron, etc.

Figure 7. Simple crystal shapes

Complex shape or combination such a form is called, which consists of two or more simple forms, i.e. crystal faces are of several types, and they are not interconnected by symmetry elements.

Simple and complex forms of crystals are extremely rare in nature. Deviations of real crystals from the described simple forms are caused by the unequal development of faces due to the influence on the formation of a crystal of the conditions of the environment in which it is formed.

Sometimes, along with the formation of individual single crystals, their various intergrowths arise. One such case is the twin formation of two or more crystals growing together in the wrong position. This process is called twinning. The formation of such intergrowths is usually caused by various complications of the crystallization process (changes in temperature, solution concentration, etc.)

There are primary (arising during crystallization) twins and secondary twins, which arise as a result of any influences.

In addition to the fusion of crystals of one substance, a natural fusion of crystals is possible various substances or polymorphic modifications of one substance, crystallizing in different syngonies. This process is called - epitaxy.

3 . Fundamentals of crystal chemistry

The internal structure of crystals ultimately determines all its features: crystal shape, physical and chemical properties.

Spatial lattice- this is a system of points located at the vertices of equal parallelepipeds oriented parallel to each other and adjacent along entire faces, filling the space without gaps.

The spatial lattice consists of an infinite number of parallelepipeds (elementary cells) equal in size and shape. The French scientist O. Bravais in 1855 established that there are only 14 types of spatial lattices (Figure 8). These cells are divided into two groups:

1) Primitive, all nodes of which are located only at the vertices of elementary cells.

2) Complex nodes, which are located not only at the vertices of elementary cells, but also on faces, edges and in the volume.

1 - triclinic;

2 and 3 - monoclinic;

4,5,6 and 7 - rhombic;

8 - hexagonal;

9 - rhombohedral;

10 and 11 - tetragonal;

12,13 and 14 are cubic.

Figure 8. Fourteen spatial lattices O. Brave

In addition to the above classifications of crystal structure according to the type of spatial lattices, there is a division of crystal structure according to types chemical bonds between atoms in a crystal.

The following types of chemical bonds exist:

A) ionic

B) metal

B) covalent or molecular

D) van der Waals or residual

D) hydrogen

Ionic ( heteropolar) bonding is observed in ionic crystal structures and occurs between two uniformly charged ions. Compounds with ionic bonds dissolve well in aqueous solutions. Such connections do not conduct electricity well.

Covalent(homeopolar) bond is carried out in atomic and partially ionic crystalline structures due to the appearance of common electrons in neighboring atoms. This bond is very strong, which explains the increased hardness of minerals with a covalent bond. Minerals with this bond are good insulators and are insoluble in water.

Metal the connection is manifested only in atomic buildings. It is characterized by the fact that atomic nuclei are located at the nodes of the crystal lattice, as if immersed in a gas consisting of free electrons that move like gas particles. The atom donates its electrons and becomes a positively charged ion. The donated electrons are not assigned to any atom, but are, as it were, in common use.

This connection determines the strength of the structure. The free movement of electrons determines the following properties: good electrical and thermal conductivity, metallic luster, malleability (for example, native metals)

Wang - der-Waalsian (residual) communication occurs between two molecules. Although each molecule is electrostatically neutral and all charges are balanced in it, many molecules are a dipole, i.e. the center of gravity of all positively charged particles of a molecule does not coincide with the center of gravity of all negatively charged particles. As a result, different parts of one molecule acquire a certain charge. Due to this, residual bonds arise between the two molecules. Van der Waals forces are very small. Crystalline structures with this bond are good dielectrics, they are characterized by low hardness and brittleness. This type of bond is typical for organic compounds. Thus, we can say that the nature of the bond determines all the basic properties of crystalline substances.

It should be noted that crystals can have one type of connection; such crystals are called homodesmic and mixed types of bonds, such crystals are called heterodesmic.

In a number of minerals (ice crystals), hydrogen bonds play an important role. They arise as a result of the interaction of the hydrogen atom of one molecule with the nitrogen, oxygen, and chlorine atom of neighboring molecules. Hydrogen bonds are stronger than Van der Waals bonds, but much weaker than all other types of bonds.

3 .1 Atomic and ionic radii. Coordinationation number. Motifs of structures

The atoms and ions that make up the crystal structures of various minerals are located at different distances from each other. These values depend on the charge of the ion, thermodynamic conditions, etc.

This quantity is called the atomic (ionic radius). Atomic (Andonny) radius is the minimum distance at which the center of the sphere of a given atom can approach the surface of neighboring atoms.

The number of nearest atoms (ions) surrounding a given atom (ion) is called coordination number.

There are three ways to depict crystal structures.

1 Method of depicting structures with balls.

2 A method of depicting structures by plotting the centers of gravity of balls.

3 Method of depicting structures using coordination polyhedra - this method is convenient for depicting complex structures. Since different minerals consist of crystal structures of different shapes (octahedron, cube, etc.).

The structure of crystalline substances is determined both by the shape of the coordination polyhedra themselves and by the nature of their combination interaction, i.e. motive structures.

The following structural motifs are distinguished:

1 Coordinating motive of the structure. In this case, all coordination polyhedra are connected to each other by common faces and edges.

2 Ostrovnoh motive of structure. Individual coordination polyhedra do not touch each other and are connected through common cations and anions.

3 Chain and ribbon motifs structures. In this case, coordination polyhedra are connected to each other into endless chains elongated in one direction.

4 Layered motif structures. Coordination polyhedra are connected to each other into infinite layers in two dimensions. Within the layer, individual polyhedra are close to each other. The individual layers are located at a considerable distance from each other.

5 Frame motif structures. In this case, all coordination figures are connected to each other by only one vertex into a framework that is infinite in three dimensions.

The pattern of crystal setting structures determines many physical properties.

Thus, the physical properties of crystalline substances are determined mainly by the composition of the atoms and ions themselves included in the crystal structures (specific gravity, color), the type of bond (electrical conductivity, thermal conductivity, hardness, malleability, solubility), and the structure motif (hardness).

4 . Defects in crystals

Metal crystals are usually small in size. Therefore, a metal product consists of very a large number crystals.

This structure is called polycrystalline. In a polycrystalline aggregate, individual crystals do not have the opportunity to take the correct shape. Irregularly shaped crystals in a polycrystalline aggregate are called grains, or crystallites. However, this condition is not the only one. Plastic deformation in a cold state (rolling, drawing, etc.) leads to a preferential orientation of grains (texture). The degree of preferential orientation can vary and varies from a random distribution to a state where all crystals are oriented in the same way.

With very slow heat removal during crystallization, as well as using other special methods, a piece of metal can be obtained that represents a single crystal, the so-called monocrystal Large single crystals (weighing several hundred grams) are produced for scientific research, as well as for some special branches of technology (semiconductors).

Research has shown that the internal crystal structure of the grain is not correct.

Deviations from the ideal arrangement of atoms in crystals are called defects. They have a great, sometimes decisive influence on the properties of crystalline substances.

The incorrect arrangement of individual atoms in the crystal lattice creates point defects. In a crystal consisting of identical atoms, for example in a metal crystal, one of the atoms may be missing in some part of the lattice. In its place there will be a cavity, with a distorted structure around it. This defect is called vacancy. If an atom of a given substance or an impurity atom falls between atoms at lattice sites, then implementation defect(Figure 9).

The picture becomes more complicated when moving from a metallic crystal to an ionic one. Electrical neutrality must be maintained here, so the formation of defects is associated with charge redistribution. Thus, the appearance of a cation vacancy is accompanied by the appearance of an anion vacancy; this type of defect in an ionic crystal is called a defect Schottky. The introduction of an ion into an interstitial site is accompanied by the appearance of a vacancy in its previous place, which can be considered as a center of charge of the opposite sign; here we have a defect Frenkel. These names are given in honor of the Austrian scientist Schottky and the Soviet physicist Ya.I. Frenkel.

Point defects arise for various reasons, including as a result of thermal movement of particles. Vacancies can move throughout the crystal - a neighboring atom falls into the void, its place is vacated, etc. This explains the diffusion in solids and ionic conductivity of salt and oxide crystals, which become noticeable at high temperatures.

In addition to the considered point defects in crystals, there are always also dislocations- defects associated with displacement of rows of atoms. Dislocations can be edge or screw. The first are caused by the breakage of planes filled with atoms; the second - by mutual shift of the axis perpendicular to it. Dislocations can move throughout the crystal; this process occurs during plastic deformation of crystalline materials.

Let us imagine that for some reason an extra half-plane of atoms, the so-called extraplane(Figure 10). Edge 3-3 of such a plane forms linear defect(imperfection) of the lattice, which is called edge dislocation. An edge dislocation can extend in length over many thousands of lattice parameters; it can be straight, but it can also bend in one direction or another. In the limit, it can twist into a spiral, forming a screw dislocation. A zone of elastic distortion of the lattice appears around the dislocation. The distance from the center of the defect to the location of the lattice without distortion is taken equal to the width of the dislocation; it is small and equal to several atomic distances.

a - vacancies; b - substituted atom; embedded atom

Figure 9. Diagram of point defects

Figure 10. Dislocation in a crystal lattice

Figure 11. Dislocation movement

Due to lattice distortion in the area of dislocations (Figure 11, a), the latter easily shifts from the neutral position, and the neighboring plane, having passed into an intermediate position (Figure 11, b), turns into an extraplane (Figure 11, c), forming a dislocation along the edge atoms. Thus, a dislocation can move (or rather, be transmitted like a relay race) along a certain plane (slip plane) located perpendicular to the extraplane. According to modern concepts, in ordinary pure metals the dislocation density, i.e. the number of dislocations in 1 cm 3 exceeds one million. The mechanical properties of metals depend on the number of dislocations and especially on their ability to move and multiply.

Thus, the correctness of the crystal structure is violated by two types of defects - point ( vacancies) and linear ( dislocations). Vacancies continuously move in the lattice when the atom adjacent to it moves into a “hole”, leaving its old place empty. An increase in temperature and thermal mobility of atoms increases the number of such events and increases the number of vacancies.

Linear defects do not move spontaneously and chaotically, like vacancies. However, a small voltage is enough for the dislocation to begin to move, forming a plane, and in the section - a slip line WITH(Figure 12). As indicated, a field of a distorted crystal lattice is created around dislocations. The energy of crystal lattice distortion is characterized by the so-called Burgers vector.

Figure 12. Shear plane (C) as a trace of dislocation motion (A-A); B-extraplane

If you draw a contour ABCD around the dislocation + (Figure 13), then the section of the contour BC will consist of six segments, and the section AB of five. Difference BC-AD=b, where b means the value of the Burgers vector. If you draw a contour around several dislocations (zones of crystal lattice distortion that overlap or merge), then its value corresponds to the sum of the Burgers vectors of each dislocation. The ability to move dislocations is related to the magnitude of the Burgers vector.

Figure 13. Scheme for determining the Burgers vector for a linear dislocation

4.1 Surface defects

Surface lattice defects include stacking faults and grain boundaries.

Packaging defect. When an ordinary complete dislocation moves, atoms successively move from one equilibrium position to another, and when a partial dislocation moves, atoms move to new positions that are atypical for a given crystal lattice. As a result, a packaging defect appears in the material. The appearance of stacking faults is associated with the movement of partial dislocations.

In the case when the stacking fault energy is high, splitting the dislocation into partial ones is energetically unfavorable, and in the case when the stacking fault energy is small, the dislocations split into partial ones, and a stacking fault appears between them. Materials with low stacking fault energy are stronger than materials with high stacking fault energy.

Grain boundaries represent a narrow transition region between two crystals of irregular shape. The width of grain boundaries, as a rule, is 1.5-2 interatomic distances. Since atoms at grain boundaries are displaced from their equilibrium position, the energy of the grain boundaries is increased. The energy of grain boundaries significantly depends on the misorientation angle of the crystal lattices of neighboring grains. At small misorientation angles (up to 5 degrees), the energy of grain boundaries is practically proportional to the misorientation angle. At misorientation angles exceeding 5 degrees, the dislocation density at grain boundaries becomes so high that the dislocation cores merge.

Dependence of grain boundary energy (Egr) on misorientation angle (q). qsp 1 and qsp 2 - misorientation angles of special boundaries.

At certain angles of misorientation of neighboring grains, the energy of grain boundaries sharply decreases. Such grain boundaries are called special. Accordingly, the angles of boundary misorientation at which the energy of the boundaries is minimal are called special angles. Grain refinement leads to an increase in the electrical resistivity of metallic materials and a decrease in the electrical resistivity of dielectrics and semiconductors.

5 . Atomic crystal structure

Any substance can be in three states of aggregation - solid, liquid and gaseous.

A solid substance, under the influence of gravity, retains its shape, while a liquid substance spreads and takes the shape of a vessel. However, this definition is not sufficient to characterize the state of a substance.

For example, solid glass softens when heated and gradually turns into a liquid state. The reverse transition will also occur smoothly - the liquid glass thickens as the temperature decreases and, finally, thickens to a “solid” state. Glass does not have a specific transition temperature from a liquid to a “solid” state, nor does it have a temperature (point) for a sharp change in properties. Therefore, it is natural to consider “solid” glass as a highly thickened liquid.

Therefore, the transition from solid to liquid and from liquid to solid state(as well as from gaseous to liquid) occurs at a certain temperature and is accompanied by a sharp change in properties.

In gases there is no pattern in the arrangement of particles (atoms, molecules); The particles move chaotically, repel one another, and the gas tends to occupy as much volume as possible.

In solids, the order of arrangement of atoms is definite, regular, the forces of mutual attraction and repulsion are balanced, and the solid retains its shape.

Figure 14. Regions of solid, liquid and gaseous states depending on temperature and pressure

In a liquid, particles (atoms, molecules) retain only the so-called close order, those. a small number of atoms are naturally located in space, and not atoms of the entire volume, as in a solid. Short-range order is unstable: it appears and disappears under the influence of energetic thermal vibrations. Thus, the liquid state is, as it were, intermediate between solid and gaseous; under appropriate conditions, a direct transition from the solid to the gaseous state is possible without intermediate melting - sublimation(Figure 14). The correct, regular arrangement of particles (atoms, molecules) in space characterizes crystalline state.

The crystalline structure can be imagined as a spatial lattice, at the nodes of which atoms are located (Figure 15).

In metals, the nodes of the crystal lattice contain not atoms, but positively charged nonons, and free electrons move between them, but they usually say that there are atoms at the nodes of the crystal lattice.

Figure 15. Crystal unit cell (simple cubic)

5. 2 Crystal lattices of metals

The crystalline state is primarily characterized by a certain, regular arrangement of atoms in space . This means that in a crystal each atom has the same number of nearest atoms - neighbors located at the same distance from it. The desire of metal atoms (ions) to be located as close to each other as possible, as densely as possible, leads to the fact that the number of occurring combinations of the relative arrangement of metal atoms in crystals is small.

There are a number of schemes and methods for describing options for the relative arrangement of atoms in a crystal. The relative arrangement of atoms in one of the planes is shown in the diagram of the arrangement of atoms (Figure 15). Imaginary lines drawn through the centers of atoms form a lattice, at the nodes of which atoms (positively charged nonons) are located; this is the so-called crystallographic plane. Repeated repetition of crystallographic planes located in parallel reproduces spatial crystal lattice, the nodes of which are the location of atoms (ions). The distances between the centers of neighboring atoms are measured angstroms(1 A 10 -8 cm) or in kiloixahs - kХ x (1 kХ=1.00202 A). The relative arrangement of atoms in space and the magnitude of interatomic distances are determined by X-ray diffraction analysis. The arrangement of atoms in a crystal is very conveniently depicted in the form of spatial diagrams, in the form of so-called unit crystal cells. A crystalline unit cell means the smallest complex of atoms, which, when repeated many times in space, allows one to reproduce a spatial crystal lattice. The simplest type of crystal cell is cubic lattice. In a simple cubic lattice, the atoms are not arranged (packed) tightly enough. The desire of metal atoms to occupy places closest to each other leads to the formation of lattices of other types: cubic body-centered( figure 16, A), cubic face-centered( figure 16, b) Andhexagonal close-packed(Figure 16 , e). This is why metals have a higher density than non-metals.

Circles representing atoms are located in the center of the cube and along its vertices (body-centered cube), or in the centers of the faces and along the vertices of the cube (face-centered cube), or in the form of a hexagon, into which a hexagon is also half inserted, the three atoms of the upper plane of which are inside hexagonal prism (hexagonal lattice).

The crystal lattice imaging method shown in Figure 16 is conventional (like any other). It may be more correct to depict atoms in a crystal lattice in the form of touching balls (left diagrams in Figure 16). However, such an image of the crystal lattice is not always more convenient than the accepted one (right diagrams in Figure 16).

a - cubic body-centered;

b - cubic face-centered;

b-hexagonal close-packed

Figure 16. Crystalline unit cells

6 . Crystallization of metals

6 .1 Three states of matter

Any substance, as is known, can be found in three states of aggregation: gaseous, liquid and solid. In pure metals, at certain temperatures, a change in the state of aggregation occurs: the solid state is replaced by a liquid state at the melting point, the liquid state turns into a gaseous state at the boiling point. Transition temperatures depend on pressure (Figure 17), but at constant pressure they are quite definite.

Melting point is a particularly important constant in the properties of a metal. It varies for various metals within a very wide range - from minus 38.9 ° C, for mercury - the most fusible metal, which is in a liquid state at room temperature, to 3410 ° C for the most refractory metal - tungsten.

The low strength (hardness) at room temperature of low-melting metals (tin, lead, etc.) is mainly due to the fact that the room temperature for these metals is less distant from the melting point than that of refractory metals

During the transition from a liquid to a solid state, a crystal lattice is formed and crystals appear. This process is called crystallization.

The energy state of a system that has a huge number of particles (atoms, molecules) covered by thermal motion is characterized by a special thermodynamic function F, called free energy (free energy F= (U - TS), where U - internal energy of the system; T- absolute temperature; S-entropy).

Figure 17. Change in free energy of liquid and crystalline states depending on temperature

At a temperature equal to T s, the free energies of the liquid and solid states are equal, the metal in both states is in equilibrium. This temperature T s and there is equilibrium or theoretical crystallization temperature.

However, when T s The crystallization process (melting) cannot occur, since at this temperature

For crystallization to begin, it is necessary that the process be thermodynamically favorable to the system and be accompanied by a decrease in the free energy of the system. From the curves shown in Figure 17, it is clear that this is only possible when the liquid is cooled below the point T s. The temperature at which crystallization practically begins can be called actual crystallization temperature.

Cooling a liquid below its equilibrium crystallization temperature is called hypothermia. These reasons also determine that the reverse transformation from the crystalline state to the liquid state can only occur above the temperature T s this phenomenon is called overheating.

The magnitude or degree of supercooling is the difference between the theoretical and actual crystallization temperatures.

If, for example, the theoretical crystallization temperature of antimony is 631°C, and before the start of the crystallization process, liquid antimony was supercooled to 590°C and crystallized at this temperature, then the degree of supercooling P determined by the difference 631-590=41°C. The process of transition of a metal from a liquid to a crystalline state can be depicted by curves in time - temperature coordinates (Figure 18).

Cooling of a metal in a liquid state is accompanied by a gradual decrease in temperature and can be called simple cooling, since there is no qualitative change in state.

When the crystallization temperature is reached, a horizontal area appears on the temperature-time curve, since the heat removal is compensated by the heat released during crystallization latent heat of crystallization. Upon completion of crystallization, i.e. after complete transition to the solid state, the temperature begins to decrease again and the crystalline solid cools. Theoretically, the crystallization process is depicted by curve 1 . Curve 2 shows the actual crystallization process. The liquid is continuously cooled to the supercooling temperature T p , below the theoretical crystallization temperature T s. When cooling below temperature T s energy conditions are created that are necessary for the crystallization process to occur.

Figure 18. Cooling curves during crystallization

6 .2 Mechanismcrystallization process

Back in 1878 D.K. Chernov, studying the structure of cast steel, pointed out that the crystallization process consists of two elementary processes. The first process consists in the generation of the smallest particles of crystals, which Chernov called “rudiments”, and now they are called embryos or crystallization centers. The second process is the growth of crystals from these centers.

The minimum size of an embryo capable of growth is called critical embryo size, and such an embryo is called sustainable.

Form of crystal formations

The actual interest of crystallization is complicated by the action of various factors that influence the process to such a strong extent that the role of the degree of supercooling can become quantitatively secondary.

During crystallization from a liquid state, factors such as the rate and direction of heat removal, the presence of undissolved particles, the presence of convection currents of the liquid, etc., become of paramount importance for the speed of the process and for the shape of the resulting crystals.

The crystal grows faster in the direction of heat removal than in the other direction.

If a tubercle appears on the side surface of a growing crystal, then the crystal acquires the ability to grow in the lateral direction. As a result, a tree-like crystal is formed, the so-called dendrite, a schematic structure of which, first depicted by D. K. Chernov, is shown in Figure 19.

Figure 19. Diagram of a dendrite

Ingot structure

The structure of the cast ingot consists of three main zones (Figure 20). The first zone is external fine-grained crust 1, consisting of disoriented small crystals - dendrites. At the first contact with the walls of the mold, a sharp temperature gradient and the phenomenon of supercooling arise in a thin adjacent layer of liquid metal, leading to the formation of a large number of crystallization centers. As a result, the crust acquires a fine-grained structure.

Second zone of the ingot - zone of columnar crystals 2. After the formation of the crust itself, the heat removal conditions change (due to thermal resistance, due to an increase in the temperature of the mold wall and other reasons), the temperature gradient in the adjacent layer of liquid metal sharply decreases and, consequently, the degree of overcooling of the steel decreases. As a result, columnar crystals normally oriented toward the crust surface (i.e., in the direction of heat removal) begin to grow from a small number of crystallization centers.

Third zone of the ingot - zone of equiaxed crystals3 . In the center of the ingot there is no longer a specific direction of heat transfer. “The temperature of the solidifying metal manages to almost completely equalize at different points and the liquid turns into a mushy state, as it were, due to the formation of crystal nuclei at its various points. Further, the rudiments grow with axes - branches in different directions, meeting each other” (Chernov D.K.). As a result of this process, an equiaxed structure is formed. The crystal nuclei here are usually various tiny inclusions present in the liquid steel, either accidentally falling into it, or not dissolved in the liquid metal (refractory components).

The relative distribution of zones of columnar and equiaxed crystals in the volume of the ingot is of great importance.

In the zone of columnar crystals, the metal is denser and contains fewer shells and gas bubbles. However, the junctions of columnar crystals have low strength. Crystallization leading to the junction of zones of columnar crystals is called transcrystallization.

Liquid metal has a larger volume than crystallized metal, so the metal poured into the mold during crystallization is reduced in volume, which leads to the formation of voids called shrinkage cavities; shrinkage cavities can either be concentrated in one place or scattered throughout the entire volume of the ingot or part of it. They can be filled with gases that are soluble in the liquid metal but are released during crystallization. In a well-deoxidized so-called calm steel, cast into a mold with an insulated extension, a shrinkage cavity is formed in the upper part of the ingot, and the volume of the entire ingot contains a small amount of gas bubbles and cavities (Figure 21, A). Insufficiently deoxidized, the so-called boiling steel, contains shells and bubbles throughout the entire volume (Figure 21, b).

Figure 20. Diagram of the structure of a steel ingot

Figure 21. Distribution of shrinkage cavity and voids in calm (a) and boiling (b) steels

7 . Deformation of metals

7.1 Elastic and plastic deformation

The application of stress to a material causes deformation. Deformation may be elastic, disappearing after the load is removed, and plastic, remaining after the load is removed.

Elastic and plastic deformations have a profound physical difference.

During elastic deformation under the action of an external force, the distance between atoms in the crystal lattice changes. Removing the load eliminates the cause that caused the change in the interatomic distance, the atoms return to their original places and the deformation disappears.

Plastic deformation is a completely different, much more complex process. During plastic deformation, one part of the crystal moves (shifts) relative to the other. If the load is removed, then the displaced part of the crystal will not return to its old place; the deformation will persist. These shifts are detected by microstructural examination, as shown, for example, in Figure 22.

...

Fundamentals of Crystallography

INTRODUCTION
Most modern structural materials, including
and composite - these are crystalline substances. Crystal
is a collection of regularly arranged atoms,
forming a regular structure that arose spontaneously from
the disordered environment around him.
The reason for the symmetrical arrangement of atoms is
the crystal's desire to minimize free energy.
Crystallization (the emergence of order from chaos, that is, from solution,
pair) occurs with the same inevitability as, for example, the process
falling bodies In turn, the minimum free energy is achieved
with the smallest fraction of surface atoms in the structure, therefore
external manifestation of the correct internal atomic structure
crystalline bodies is the cutting of crystals.
In 1669, the Danish scientist N. Stenon discovered the law of constancy of angles:
the angles between the corresponding crystal faces are constant and
characteristic of this substance. Any solid body consists of
interacting particles. These particles, depending on
nature of matter, there may be individual atoms, groups of atoms,
molecules, ions, etc. Accordingly, the connection between them is:
atomic (covalent), molecular (Van – der Waals bond), ionic
(polar) and metal.

In modern crystallography there are four
directions that are to a certain extent related to
to others:
- geometric crystallography, which studies various
crystal shapes and laws of their symmetry;
- structural crystallography and crystal chemistry,
who study the spatial arrangement of atoms in
crystals and its dependence on the chemical composition and
conditions for crystal formation;
- crystal physics, studying the influence of internal
the structure of crystals and their physical properties;
- physical and chemical crystallography, which studies
issues of formation of artificial crystals.

SPACE GRID ANALYSIS
The concept of a spatial lattice and elementary
cell
When studying the issue of the crystalline structure of bodies
First of all, you need to have a clear understanding of
terms: “spatial lattice” and “elementary
cell". These concepts are used not only in
crystallography, but also in a number of related sciences for
descriptions of how they are located in space
material particles in crystalline bodies.
As is known, in crystalline bodies, unlike
amorphous, material particles (atoms, molecules,
ions) are arranged in a certain order, on
a certain distance from each other.

A spatial grid is a diagram that shows
arrangement of material particles in space.
The spatial lattice (Fig.) actually consists of
sets
identical
parallelepipeds,
which
completely, without gaps, fill the space.
Material particles are usually located at nodes
lattice - the points of intersection of its edges.
Spatial lattice

The unit cell is
least
parallelepiped, with
with which you can
build the whole
spatial lattice
through continuous
parallel transfers
(broadcasts) in three
directions of space.
Type of unit cell
shown in Fig.
Three vectors a, b, c, which are the edges of the unit cell,
are called translation vectors. Their absolute value (a,
b, c) are lattice periods, or axial units. Injected into
consideration and angles between translation vectors - α (between
vectors b, c), β (between a, c) and γ (between a, b). So
Thus, the unit cell is determined by six quantities: three
values of periods (a, b, c) and three values of angles between them
(α, β, γ).

Rules for choosing a unit cell
When studying ideas about the unit cell, one should
pay attention to the fact that the magnitude and direction
translations in the spatial lattice can be chosen differently, so the shape and dimensions of the unit cell
will be different.
In Fig. the two-dimensional case is considered. Shown flat
lattice mesh and different ways choosing flat
unit cell.
Selection methods
unit cell

In the middle of the 19th century. French crystallographer O. Bravais
proposed the following conditions for choosing an elementary
cells:
1) the symmetry of the unit cell must correspond
symmetries of the spatial lattice;
2) the number of equal edges and equal angles between the edges
should be maximum;
3) in the presence of right angles between the ribs, their number
should be maximum;
4) subject to these three conditions, the volume
elementary cell should be minimal.
Based on these rules, Bravais proved that there is
only 14 types of elementary cells, which received
the name of translational ones, since they are built by
translation - transfer. These grids are different from each other.
each other in magnitude and direction of broadcasts, and from here
the difference follows in the shape of the unit cell and in the number
nodes with material particles.

Primitive and complex unit cells
According to the number of nodes with material particles, elementary
cells are divided into primitive and complex. IN
in primitive Bravais cells, material particles are located
only at vertices, in complex ones - at vertices and additionally
inside or on the surface of the cell.
Complex cells include body-centered I,
face-centered F and base-centered C. In Fig.
Bravais unit cells are shown.
Unit cells of Bravais: a – primitive, b –
base-centered, c – body-centered, d –
face-centered

A body-centered cell has an additional node in
center of the cell, belonging only to this cell, therefore
there are two nodes here (1/8x8+1 = 2).
In a face-centered cell, nodes with material particles
are located, in addition to the vertices of the cell, in the centers of all six faces.
Such nodes belong simultaneously to two cells: this and
another adjacent to it. For the share of a given cell, each of these
nodes belong to 1/2 part. Therefore, in a face-centered
there will be four nodes in the cell (1/8x8+1/2x6 = 4).
Similarly, there are 2 nodes in a base-centered cell
(1/8x8+1/2x2 = 2) with material particles. Basic information
about elementary Bravais cells are given below in the table. 1.1.
A primitive Bravais cell contains translations a,b,c only
along the coordinate axes. In a body-centered cell
another translation is added along the spatial diagonal -
to the node located in the center of the cell. In a face-centered
in addition to axial translations a,b,c there are additional
translation along the diagonals of the faces, and in base-centered -
along the diagonal of the face perpendicular to the Z axis.

Table 1.1
Understanding Primitive and Complex Bravais Cells
Basis
Bravais grille type
Number Basic
broadcast nodes
Primitive R
1
a,b,c
Body centered 2
aya I
a,b,c,(a+b+c)/2
[]
Face-centered
F
a,b,c,(a+b)/2,(a+c)/2,
(b+c)/2
[]
a,b,c,(a+b)/2
[]
4
Base-centered C 2
The basis is understood as a set of coordinates
minimum number of nodes, expressed in axial
units, by broadcasting which you can receive the entire
spatial lattice. The basis is written in double
square brackets. Base coordinates for various
types of Bravais cells are given in Table 1.1.

Bravais unit cells
Depending on the shape, all Bravais cells are distributed between
seven crystal systems (systems). Word
“Singoniya” means similar angle (from the Greek σύν - “according to
together, next to each other”, and γωνία - “corner”). Each system corresponds
certain elements of symmetry. In table ratios are indicated
between lattice periods a, b, c and axial angles α, β, γ for
each system
Syngonies
Triclinic
Monoclinic
Rhombic
Tetragonal
Hexagonal
Relationships between
lattice periods and angles
a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90º
a ≠ b ≠ c, α = γ =90º ≠ β
a ≠ b ≠ c, α = β = γ =90º
a = b ≠ c, α = β = γ =90º
a = b ≠ c, α = β =90º, γ =120º
Rhombohedral
Cubic
a = b = c,
a = b = c,
α = β =γ ≠ 90º
α = β = γ = 90º

In Fig. all are represented
fourteen types
Bravais unit cells,
distributed among systems.
Hexagonal Bravais cell
represents
base-centered
hexagonal prism. However
she is portrayed very often
otherwise - in the form of a tetrahedral
prisms with a rhombus at the base,
which represents one of
three prisms that make up
hexagonal (in the figure it
represented by solid
lines). Such an image
simpler and more convenient, although it is associated with
violation of the principle
symmetry compliance
(first principle of choice
unit cell according to Bravais).

For the rhombohedral system
unit cell,
satisfying conditions
Brave, is primitive
rhombohedron R, for which a=b=c and
α=β=γ≠ 90º. Along with the R-cell
to describe rhombohedral
structures are used and
hexagonal cell,
since rhombohedral
a cell can always be reduced to
hexagonal (Fig.) and
imagine it as three
primitive hexagonal
cells. In this regard, in
literature rhombohedral
syngony is sometimes not separately
Three primitive
looking at her, imagining her
hexagonal cells,
as a variety
equivalent to rhombohedral
hexagonal.

Accepted systems with identical relationships between
axial units are combined into one category. That's why
triclinic, monoclinic and orthorhombic systems
combined into the lowest category (a≠b≠c), tetragonal,
hexagonal (and rhombohedral derivative) – in
average (a=b≠c), the highest category (a=b=c) refers
cubic system.
The concept of coordination number
In complex cells, material particles are packed more
densely than in primitive ones, they fill the volume more completely
cells are more connected to each other. For characteristics
This introduces the concept of coordination number.
The coordination number of a given atom is the number
nearest neighboring atoms. If we are talking about
coordination number of the ion, then the number is meant
the nearest ions of the opposite sign. The more
coordination number, the larger the number of atoms or
ions is bound given, the more space is occupied by particles, the
more compact grille.

Spatial lattices of metals
The most common spatial patterns among metals
lattices are relatively simple. They mostly match
with Bravais translation lattices: cubic
body-centered and face-centered. At the nodes of these
metal atoms are located in lattices. On the grid
body-centered cube (bcc lattice) each atom
surrounded by eight nearest neighbors, and the focal point
CN number = 8. Metals have a bcc lattice: -Fe, Li, Na, K, V,
Cr, Ta, W, Mo, Nb, etc.
In the lattice of a face-centered cube (fcc lattice) CN = 12:
any atom located at the top of a cell has
twelve nearest neighbors, which are atoms,
located in the centers of the faces. The following metals have an fcc lattice:
Al, Ni, Cu, Pd, Ag, Ir, Pt, Pb, etc.
Along with these two, among the metals (Be, Mg, Sc, -Ti, -Co,
Zn, Y, Zr, Re, Os, Tl, Cd, etc.) hexagonal
compact. This lattice is not a translation lattice
Brave, because it cannot be described by simple broadcasts.

In Fig. the unit cell of a hexagonal
compact grille. Hexagonal unit cell
compact lattice is a hexagonal
prism, but most often it is depicted in the form
tetrahedral prism whose base is a rhombus
(a=b) with angle γ = 120°. Atoms (Fig.b) are located at the vertices
and in the center of one of the two triangular prisms forming
elementary cell. A cell has two atoms: 1/8x8 + 1
=2, its basis [].
The ratio of the unit cell height c to the distance a, i.e.
c/a equals 1.633; the periods c and a for different substances
are different.
Hexagonal
compact grille:
a – hexagonal
prism, b –
tetrahedral
prism.

CRYSTALLOGRAPHIC INDICES
Crystallographic indices of the plane
In crystallography it is often necessary to describe the mutual
the arrangement of individual planes of the crystal, its
directions for which it is convenient to use
crystallographic indices. Crystallographic
indices give an idea of the location of the plane
or directions relative to the coordinate system. At
it doesn't matter whether it's rectangular or oblique
coordinate system, same or different scale
segments along coordinate axes. Let's imagine a series
parallel planes passing through identical
spatial lattice nodes. These planes
located at equal distances from each other and
form a family of parallel planes. They
identically oriented in space and therefore
characterized by the same indices.

Let us choose a plane from this family and
let us introduce into consideration segments that are plane
cuts along coordinate axes (coordinate axes x,
y, z are usually combined with the edges of the elementary
cells, the scale on each axis is equal to
the corresponding axial unit - period a, or b,
or c). The values of the segments are expressed in axial
units.
Crystallographic plane indices (indices
Miller) are the three smallest integers,
which are inversely proportional to the number of axial
units cut off by a plane on coordinate
axes.
The plane indices are designated by the letters h, k, l,
written down in a row and enclosed in round
brackets-(hkl).

The indices (hkl) characterize all planes of the family
parallel planes. This symbol means that
a family of parallel planes cuts the axial
unit along the x-axis for h parts, along the y-axis for k
parts and along the z axis into l parts.
In this case, the plane closest to the origin of coordinates,
cuts off segments 1/h on the coordinate axes (along the x axis),
1/k (along the y-axis), 1/l (along the z-axis).
The order of finding crystallographic indices
plane.
1. Find the segments cut off by the plane at
coordinate axes, measuring them in axial units.
2. We take the reciprocal values of these quantities.
3. We give the ratio of the obtained numbers to the ratio
the three smallest integers.
4. We enclose the resulting three numbers in parentheses.

Example. Find the indices of the plane that cuts off at
coordinate axes the following segments: 1/2; 1/4; 1/4.
Since the lengths of the segments are expressed in axial units,
we have 1/h=1/2; 1/k=1/4; 1/l=1/4.
Find the inverse values and take their ratio
h: k: l = 2: 4: 4.
Reducing by two, we present the ratio of the obtained values
to the ratio of the three smallest integers: h: k: l = 1: 2:
2. We write the plane indices in parentheses
in a row, without commas - (122). They are read separately -
"one, two, two."
If the plane intersects the crystallographic axis at
negative direction, above the corresponding
The minus sign is placed at the top of the index. If the plane
parallel to any coordinate axis, then in the symbol
plane, the index corresponding to this axis is zero.
For example, the symbol (hko) means that the plane
intersects the z axis at infinity and the plane index
along this axis there will be 1/∞ = 0.

Planes cutting off an equal number on each axis
axial units are designated as (111). In cubic
their systems are called planes of the octahedron, because the system
these planes equidistant from the origin,
forms an octahedron - octahedron Fig.
Octahedron

Planes cutting off an equal number of axes along two axes
units and parallel to the third axis (for example, the z axis)
are designated (110). In a cubic system similar
planes are called planes of the rhombic dodecahedron,
So
How
system
planes
type
(110)
forms
dodecahedron (dodeca – twelve), each face
of which is the rhombus Fig.
Rhombic
dodecahedron

Planes intersecting one axis and parallel to two
others (for example, the y and z axes) are designated - (100) and
are called in the cubic system the planes of the cube, that is
a system of similar planes forms a cube.
When solving various problems related to the construction of
unit cell of planes, coordinate system
it is advisable to choose so that the desired plane
was located in a given unit cell. For example,
when constructing the (211) plane in a cubic cell, the beginning
coordinates are conveniently transferred from node O to node O’.
Cube plane (211)

Sometimes plane indices are written in curly braces
(hkl).This entry means a symbol of a collection of identical
planes. Such planes pass through identical nodes
in a spatial lattice, symmetrically located in
space
And
are characterized
the same
interplanar distance.
The planes of the octahedron in the cubic system belong to
one set (111), they represent the faces of the octahedron and
have the following indices: (111) →(111), (111), (111), (111),
(111), (111), (111), (111).
The symbols of all planes of the set are found by
rearrangements and changes in signs of individual
indexes.
For the planes of the rhombic dodecahedron, the notation
aggregates: (110) → (110), (110), (110),
(110), (101), (101), (101), (101), (011), (011), (011), (011).

CRYSTALLOGRAPHIC NODE INDICES
The crystallographic indices of a node are its
coordinates taken in fractions of axial units and written in
double square brackets. In this case, the coordinate
corresponding to the x axis, is generally denoted by the letter
u, for the y axis – v, for the z axis – w. The node symbol looks like
[]. Symbols of some nodes in a unit cell
shown in Fig.
Some nodes in
unit cell
(Sometimes a node is denoted
How [])

Crystallographic directional indices
In a crystal where all parallel directions
identical to each other, the direction passing through
the origin of coordinates characterizes this entire family
parallel directions.
Position
V
space
directions,
passing through the origin is determined
coordinates of any node lying on this
direction.
Coordinates
any
node,
owned
direction, expressed in fractions of axial units and
reduced to the ratio of three smallest integers
numbers,
And
There is
crystallographic
indices
directions. They are designated by integers u, v, w
and are written together in square brackets.

The order of finding direction indices
1. From the family of parallel directions, select
one that passes through the origin, or
move this direction parallel to itself
to the origin of coordinates, or move the origin
coordinates to a node lying in a given direction.
2. Find the coordinates of any node belonging to
given direction, expressing them in axial units.
3. Take the relation of node coordinates and bring it to
the ratio of the three smallest integers.
4. Square the resulting three numbers
brackets.
The most important directions in the cubic lattice and their
indices are presented in Fig.

Some directions in a cubic lattice

CONCEPT OF CRYSTAL AND POLAR
COMPLEX
The crystallographic projection method is based on
one of the characteristic features of crystals is the law
angle constancy: angles between certain faces and
the edges of the crystal are always constant.
So, when a crystal grows, the sizes of the faces change, their
shape, but the angles remain the same. Therefore in
crystal, you can move all the edges and faces parallel
to ourselves at one point in space; corner
the relationship remains the same.
Such
totality
planes
And
directions,
parallel to planes and directions in the crystal and
passing through one point is called
crystalline complex, and the point itself is called
center
complex.
At
building
crystallographic projections crystal is always replaced
crystalline complex.

More often, it is not the crystalline complex that is considered, but
polar (reverse).
Polar complex, obtained from crystalline
(direct) by replacing planes with normals to them, and
directions - planes perpendicular to them.
A
b
Cube (a), its crystalline (b) and
polar complex (c)
V

SYMMETRY OF CRYSTAL POLYHEDRONS
(CONTINUUM SYMMETRY)
THE CONCEPT OF SYMMETRY
Crystals exist in nature in the form of crystalline
polyhedra. Crystals of different substances differ from each other
from each other in their forms. Rock salt is cubes;
rock crystal - hexagonal prisms, pointed at
ends; diamond - most often regular octahedrons
(octahedra); garnet crystals are dodecahedrons (Fig.).
Such crystals have symmetry.

Characteristic
feature
crystals
is
anisotropy of their properties: in different directions they
different, but identical in parallel directions, and
are also the same in symmetrical directions.
Crystals do not always have the right shape
polyhedra.
Under real growth conditions, with
difficulty in free growth, symmetrical edges can
develop unevenly and correct external shape
may not work out, but the correct internal
the structure is completely preserved, as well as
the symmetry of physical properties is preserved.
The Greek word "symmetry" means proportionality.
A symmetrical figure consists of equal, identical
parts. Symmetry is understood as a property of bodies or
geometric shapes to combine individual parts with each other
different under some symmetric transformations.
Geometric images with the help of which the
symmetric transformations are carried out, called
elements of symmetry.

Considering the symmetry of the external cut of the crystal,
crystalline
Wednesday
present
to myself
How
continuous, solid, the so-called continuum (in
translated from Latin into Russian - means continuous,
solid). All points in such an environment are exactly the same.
The symmetry elements of the continuum describe the external
the shape of a crystalline polyhedron, so there are still
are called macroscopic symmetry elements.
Actually
same
crystalline
Wednesday
is
discrete. Crystals are made up of individual particles
(atoms, ions, molecules) that are located in
space
V
form
endlessly
stretching
spatial lattices. Symmetry in arrangement
these particles are more complex and richer than the symmetry of external
shapes of crystalline polyhedra. Therefore, along with
continuum
is being considered
And
discontinuum
-
discrete, real structure of material particles with
with its symmetry elements, called
microscopic elements of symmetry.

Elements of symmetry
IN
crystalline
polyhedra
meet
simple
elements
symmetry
(center
symmetry,
plane of symmetry, rotary axis) and complex element
symmetry (inversion axis).
Center of symmetry (or center of inversion) - special point
inside a figure, when reflected in which any point
figure has an equivalent to itself, that is, both points
(for example, a pair of vertices) are located on the same straight line,
passing through the center of symmetry, and are equidistant from
him. If there is a center of symmetry, each face
spatial
figures
It has
parallel
And
opposite direction to each edge
corresponds equidistant, equal, parallel, but
oppositely directed edge. Therefore the center
symmetry is like a mirror point.

A plane of symmetry is a plane that
divides the figure into two parts, located each
relative to a friend as an object and its mirror reflection,
that is, into two mirror equal parts
symmetry planes - Р (old) and m (international).
Graphically, the plane of symmetry is indicated by a solid
line. A figure can have one or more
planes of symmetry, and they all intersect each other
friend. A cube has nine planes of symmetry.

The pivot axis is so straight, when turning around
which, at some definite angle, the figure
combines with itself. Rotation angle
determines the order of the rotary axis n, which
shows how many times the figure will be combined with itself
with a full turn around this axis (360 °):
In isolated geometric shapes possible
symmetry axes of any order, but in crystalline
In polyhedra, the order of the axis is limited; it can have
only the following values: n= 1, 2, 3, 4, 6. V
crystalline
polyhedra
impossible
axes
symmetries of the fifth and higher than the sixth order. It follows
from the principle of continuity of the crystalline medium.
Symmetry axes designations: old - Ln (L1, L2, L3, L4, L6)
And
international
Arabic
in numbers,
corresponding to the order of the rotary axis (1, 2, 3, 4, 6).

Graphically
rotary
polygons:
axes
portrayed

The concept of symmetry class
Each crystalline polyhedron has a set
elements of symmetry. Combining with each other, the elements
the symmetries of the crystal necessarily intersect, and at the same time
the appearance of new symmetry elements is possible.
The following theorems are proven in crystallography
addition of symmetry elements:
1. The line of intersection of two planes of symmetry is the axis
symmetry for which the angle of rotation is twice the angle
between planes.
2. Through the point of intersection of two axes of symmetry passes
third axis of symmetry.
3. B
point
intersections
plane
symmetry
With
symmetry axis of even order perpendicular to it
a center of symmetry appears.
4. Number of second order axes perpendicular to the main one
symmetry axes of higher order (third, fourth,
sixth), equal to the order of the main axis.

5. Number of planes of symmetry intersecting along
the main axis of higher order is equal to the order of this axis.
Number of combinations of symmetry elements with each other
in crystals is strictly limited. All possible
combinations of symmetry elements in crystals are derived
strictly mathematical, taking into account the theorems
addition of symmetry elements.
A complete set of symmetry elements inherent
a given crystal is called its symmetry class.
Rigorous mathematical derivation shows that everything
possible
For
crystalline
polyhedra
combinations
elements
symmetry
are exhausted
thirty-two classes of symmetry.

Relationship between the spatial lattice and elements
symmetry
The presence of certain symmetry elements determines
geometry
spatial
gratings,
imposing
certain
conditions
on
mutual
location
coordinate axes and equality of axial units.
Exist general rules selection of coordinate axes,
taking into account the set of crystal symmetry elements.
1. Coordinate axes are combined with special or single
directions,
non-repeating
V
crystal
rotary or inversion axes, for which
the order of the axis is greater than one, and the normals to the plane
symmetry.
2. If there is only one special direction in the crystal, with it
combine one of the coordinate axes, usually the Z axis. Two
other axes are located in a plane perpendicular to
special direction parallel to the edges of the crystal.
3. In the absence of special directions, the coordinate axes
are chosen parallel to three not lying in the same plane
edges of the crystal.

Based on these rules, you can get all seven
crystal systems, or syngonies. They differ
from each other by the ratio of scale units a, b, c and
axial angles, . Three possibilities: a b c, a=b c, a=b=c
allow
distribute
All
crystallographic
coordinate systems (systems) in three categories of lower, middle and higher.
Each category is characterized by the presence of certain
elements of symmetry. So, for crystals of the lowest category
there are no higher order axes, that is, axes 3, 4 and 6, but there may be
second order axes, planes and center of symmetry.
Medium category crystals have a higher axis
order, and there may also be second-order axes, planes
symmetry, center of symmetry.
The most symmetrical crystals belong to the highest
categories. They have multiple higher order axes
(third and fourth), there may be second order axes,
planes and center of symmetry. However, axles are missing
sixth order.

The concept of symmetry of discontinuum and spatial
group
Availability
32
classes
symmetry
crystalline
polyhedra shows that the entire variety of external
crystal forms obey the laws of symmetry.
Symmetry of the internal structure of crystals, arrangement
particles (atoms, ions, molecules) inside the crystals must
be more difficult because the external shape of the crystals
limited, and the crystal lattice extends
infinitely in all directions of space.
The laws for the arrangement of particles in crystals were
established by the great Russian crystallographer E.S.
Fedorov in 1891. He found 230 ways
arrangement of particles in a spatial lattice - 230
space symmetry groups.

Elements of symmetry of spatial lattices
In addition to the symmetry elements described above (center
symmetry,
plane
symmetry,
rotary
And
inversion axes), others are possible in a discrete environment
elements
symmetry,
related
With
infinity
spatial lattice and periodic repeatability
in the arrangement of particles.
Let's consider new types of symmetry, inherent only
discontinuum. There are three of them: translation, sliding plane
reflections and helical axis.
Translation is the transfer of all particles along parallel
directions in the same direction to the same
size.
Translation is a simple element of symmetry,
inherent in each spatial lattice.

Combination of translation with plane of symmetry
leads to the appearance of a grazing reflection plane,
the combination of translation with a rotary axis creates
screw axis.
Plane of grazing reflection, or plane
sliding is such a plane, when reflected in
which is like in a mirror with subsequent broadcast along
direction lying in a given plane by an amount
equal to half the identity period for a given
directions, all points of the body are combined. Under period
identity, as before, we will understand the distance
between points along some direction (for example,
periods a, b, c in a unit cell are the periods
identity along the coordinate axes X, Y, Z).

A helical axis is a straight line, rotation around which is
some
corner,
corresponding
in order
axes,
With
subsequent translation along the axis by an amount that is a multiple of
identity period t, combines body points.
Designation of the screw axis in general form nS, where n
characterizes the order of the rotary axis (n=1, 2, 3, 4, 6), and
St/n is the translation value along the axis. At the same time, S S=0, 1, 2, 3, 4, 5, 6. So, for a second-order helical axis
translation is t/2, for the helical axis of the third
order of magnitude of the smallest transfer t/3.
The designation of the second order helical axis will be 21.
Particle alignment will occur after rotation around the axis
180° with subsequent broadcast along the direction,
parallel to the axis, at t/2.
The designation of the third order helical axis will be 31.
However, axes with translations that are multiples of the smallest translation are possible.
Therefore, a helical axis 32 with a translation of 2t/3 is possible.

Axes 31 and 32 mean rotation around the axis by 120°
clockwise followed by translation. These screw
the axes are called right. If the turn is made
counterclockwise, then the central axes of symmetry
are called leftists. In this case, the action of the axis 31 of the right
identical to the action of the axis 32 left and 32 right - 31
left.
Helical axes of symmetry can also be considered
fourth and sixth orders: axes 41 and 43 axes 61 and 65, 62
and 64. can be right or left. Action of axes 21, 42 and
63 does not depend on the choice of direction of rotation around the axis.
That's why
They
are
neutral.
Conditional
designations of helical axes of symmetry:

Symmetry space group notation
The space group symbol contains the complete
information about the symmetry of the crystal structure. On
the first place in the space group symbol is put
letter characterizing the type of Bravais lattice: P primitive,
WITH
base-centered,
I
body-centered, F - face-centered. IN
rhombohedral syngony put the letter R in the first place.
This is followed by one, two or three numbers or letters,
indicating
elements
symmetry
V
main
directions, similar to how it is done when
drawing up a designation for the symmetry class.
If in the structure in any of the main directions
the planes of symmetry and
axis of symmetry, preference is given to planes
symmetry, and into the space group symbol
planes of symmetry are written.

If there are multiple axes, preference is given to
simple axes - rotary and inversion, since they
symmetry is higher than symmetry
screw axes.
Having a space group symbol, you can easily
determine the Bravais lattice type, cell system, elements
symmetry in the main directions. Yes, spatial
group P42/mnm (Fedorov groups of ditetragonal-dipyramidal
kind
symmetry,
135
group)
characterizes a primitive Bravais cell in a tetragonal
syngony (the fourth order helical axis 42 determines
tetragonal system).
The main directions are as follows:
elements of symmetry. With direction - Z axis
the screw axis 42 coincides, which is perpendicular
symmetry m. In the and directions (X and Y axes)
there is a grazing reflection plane of type n, in
direction the plane of symmetry m passes.

Defects in the structure of crystalline bodies
Defects of bodies are divided into dynamic
(temporary) and static (permanent).
1. Dynamic defects occur when
mechanical, thermal, electromagnetic
influences on the crystal.
These include phonons - temporary distortions
lattice regularities caused by thermal
movement of atoms.
2. Static defects
There are point and extended imperfections
body structures

Point defects: unoccupied lattice nodes
(vacancies); displacement of an atom from a node to an interstitial site;
introduction of a foreign atom or ion into the lattice.
Extended defects: dislocations (edge and
screw), pores, cracks, grain boundaries,
microinclusions of another phase. Some defects are shown
on the image.

Basic properties
materials

The main properties include: mechanical, thermal,
electrical, magnetic and technological, as well as their
corrosion resistance.
The mechanical properties of materials characterize their ability
use in products exposed to
mechanical loads. The main indicators of such properties
Strength and hardness parameters are used. They depend not only on
the nature of the materials, but also on the shape, size and condition
surface of samples, as well as test modes, first of all,
on the loading rate, temperature, exposure to media and other
factors.
Strength is the property of materials to resist fracture, and
also an irreversible change in the shape of the sample under the influence
external loads.
Tensile strength – stress corresponding to the maximum
(at the moment of sample failure) to the load value. Attitude
the greatest force acting on the sample to the original area
its cross section is called breaking stress and
denote σв.

Deformation is a change in the relative arrangement of particles in
material. Its simplest types are stretching, compression, bending,
torsion, shear. Deformation is a change in the shape and size of a sample in
as a result of deformation.
Deformation parameters – relative elongation ε = (l– l0)/l0 (where
l0 and l – sample length initial and after deformation), shear angle –
change of right angle between rays emanating from one point in
sample when it is deformed. A deformation is called elastic if
it disappears after removing the load, or plastic, if it is not
disappears (irreversible). Plastic properties of materials at
small deformations are often neglected.
Elastic limit is the stress at which residual deformations (i.e.
e. deformations detected during unloading of the sample) reach
value established by technical specifications. Usually admission to
residual deformation is 10–3 ÷10–2%. Elastic limit σу
limits the area of elastic deformation of the material.
The concept of modulus as a characteristic of the elasticity of materials arose
when considering ideally elastic bodies whose deformation is linear
depends on voltage. With simple stretching (compression)
σ = Eε
where E is Young’s modulus, or the modulus of longitudinal elasticity, which
characterizes the resistance of materials to elastic deformation (tension, compression); ε – relative deformation.

When shearing in a material in the direction of shear and along the normal to it
Only shear stresses act
where G is the shear modulus, characterizing the elasticity of the material at
changing the shape of the sample, the volume of which remains constant; γ is the angle
shift
With all-round compression in the material in all directions,
normal voltage
where K is the modulus of bulk elasticity, which characterizes
resistance of the material to changes in sample volume, not
accompanied by a change in its shape; ∆ – relative
volumetric compression.
A constant value characterizing the elasticity of materials at
uniaxial tension, is Poisson's ratio:
where ε′ – relative transverse compression; ε – relative
longitudinal elongation of the sample.

Hardness is a mechanical characteristic of materials,
comprehensively reflecting their strength, ductility, as well as
properties of the surface layer of samples. She expresses herself
resistance of the material to local plastic
deformation that occurs when more than one is introduced into a sample
solid body - indenter. Pressing the indenter into the sample with
subsequent measurement of the dimensions of the print is the main
technological method for assessing the hardness of materials. IN
depending on the characteristics of the load application, design
indenters and determination of hardness numbers distinguish between methods
Brinell, Rockwell, Vickers, Shore. When measuring
microhardness according to GOST 9450–76 on the surface of the sample
imprints remain of insignificant depth, so this
the method is used when the samples are made in the form of foil,
films, coatings of small thickness. Determination method
plastic hardness is indentation into the sample
spherical tip by sequential application
various loads.

Corrosion is a physical and chemical process of changing properties, damage
structure and destruction of materials due to the transition of their components into
chemical compounds with environmental components. Under
Corrosion damage refers to any structural defect
material resulting from corrosion. If mechanical
impacts accelerate the corrosion of materials, and corrosion facilitates them
mechanical destruction, corrosion-mechanical occurs
damage to materials. Material losses due to corrosion and costs
protection of machines and equipment from it is continuously increasing
due to the intensification of human production activity and
environmental pollution from production waste.
Most often, the corrosion resistance of materials is characterized by
using the corrosion resistance parameter - the reciprocal value
technical corrosion rate of the material in a given corrosion system.
The convention of this characteristic is that it does not apply to
material, but to the corrosion system. Corrosion resistance of the material
cannot be changed without changing other parameters of the corrosion system.
Anti-corrosion protection is a modification of corrosion
systems leading to a decrease in the corrosion rate of the material.

Temperature characteristics.
Heat resistance - the property of materials to retain or insignificantly
change mechanical parameters at high temperatures. Property
metals resist the corrosive effects of gases at high
temperatures is called heat resistance. As a characteristic
heat resistance of low-melting materials use temperature
softening.
Heat resistance – the ability of materials to resist for a long time
deformation and destruction at high temperatures. This
the most important characteristic of materials used in
temperatures T > 0.3 Tmel. Such conditions occur in engines
internal combustion, steam power plants, gas turbines,
metallurgical furnaces, etc.
At low temperatures (in technology - from 0 to –269 ° C) increases
static and cyclic strength of materials, their
plasticity and toughness, the tendency to brittle fracture increases.
Cold brittleness - an increase in the fragility of materials with decreasing
temperature. The susceptibility of a material to brittle fracture is determined by
based on the results of impact tests of samples with a notch when lowering
temperature.

Thermal expansion of materials is recorded by changes in dimensions
and shape of samples with temperature changes. For gases it is due
an increase in the kinetic energy of particles when heated, in liquids
and solid materials is associated with the asymmetry of thermal
vibrations of atoms, due to which interatomic distances increase
temperatures are increasing.
The thermal expansion of materials is characterized quantitatively
temperature coefficient of volumetric expansion:
and solid materials - and the temperature coefficient of the linear
extensions (TCLR):
– changes in linear size, volume of samples and
temperature (respectively).
Index ξ serves to designate the conditions of thermal expansion (usually -
at constant pressure).
Experimentally, αV and αl are determined by dilatometry, which studies
dependence of changes in the size of bodies under the influence of external factors.
Special measuring instruments– dilatometers – differ
the device of sensors and sensitivity of systems of registration of the sizes
samples.

Heat capacity - the ratio of the amount of heat received by the body during
an infinitesimal change in its state in any process, to
caused by the last temperature increase:
According to the characteristics of the thermodynamic process in which they determine
heat capacity of the material, distinguish heat capacity at constant volume
and at constant pressure. During heating at constant
pressure (isobaric process), part of the heat is spent on expansion
sample, and partly to increase the internal energy of the material. Heat,
communicated to the same sample at constant volume (isochoric process),
is spent only on increasing the internal energy of the material.
Specific heat capacity, J/(kg K)], – ratio of heat capacity to mass
bodies. A distinction is made between specific heat capacity at constant pressure (cp) and
at constant volume (cv). Ratio of heat capacity to quantity
substances are called molar heat capacity (cm), J/(mol⋅K). For all
substances ср > сv, for rarefied (close to ideal) gases сmp – сmv =
R (where R = 8.314 J/(mol⋅K) is the universal gas constant).

Thermal conductivity is the transfer of energy from hotter areas of the body to
less heated as a result of thermal movement and interaction
microparticles This value characterizes spontaneous
equalization of temperature of solids.
For isotropic materials, Fourier’s law is valid, according to which
density vector heat flow q is proportional and opposite
in the direction of the temperature gradient T:
where λ is the thermal conductivity coefficient [W/(m K)], depending on
state of aggregation, atomic-molecular structure, structure,
temperature and other material parameters.
Thermal diffusivity coefficient (m2/s) is a measure
thermal insulation properties of the material:
where ρ – density; Wed - specific heat material at
constant pressure.

Technological properties of materials characterize compliance
materials to technological influences during processing into products. Knowledge
these properties allows you to reasonably and rationally design and
carry out technological processes of manufacturing products. Main
technological characteristics of materials are machinability
cutting and pressure, casting parameters, weldability, tendency to
deformation and warping during heat treatment, etc.
Machinability is characterized by the following indicators:
the quality of material processing - the roughness of the machined surface
and accuracy of sample dimensions, tool life, resistance
cutting - cutting speed and force, type of chip formation. Values
indicators are determined when grinding samples and compared with
parameters of the material taken as the standard.
Pressure workability is determined during technological
testing materials for plastic deformation. Assessment methods
workability by pressure depend on the type of materials and their technology
processing. For example, technological testing of metals for bending
carried out by bending the samples to a given angle. The sample is considered to have withstood
tests, if no breaks, delaminations, tears, or cracks appear in it.
Sheets and tapes are tested for extrusion using a special
press. A spherical hole is formed in the sample, stopping the stretching at the moment
achieving material fluidity. The result is determined by the greatest
hole depth in undamaged samples.

The pressure processability of powder materials characterizes them
fluidity, compactability and formability. Determination method
fluidity is based on recording the expiration time of a sample of powder in
the process of its spontaneous awakening through a calibrated
funnel hole. The filling speed depends on this parameter
powder mold materials for pressure processing.
The compactability of the powder is characterized by the dependence of the volume of the sample
powder versus pressure – pressing diagram. Formability - property
powder material to maintain the shape obtained during the process
pressing.
Casting characteristics of materials - a set of technological
indicators characterizing the formation of castings by pouring
molten materials into a mold. Fluidity −
the ability of the molten material to fill the mold depends on
on melt viscosity, melt and mold temperatures, degree
wetting the walls of the mold with the melt, etc. It is assessed by length
filling a straight or spiral channel with melt in
special casting mold. Foundry shrinkage - volume reduction
melt during the transition from liquid to solid state. Practically
shrinkage is determined as the ratio of the corresponding linear dimensions
molds and castings in the form of a dimensionless shrinkage coefficient,
individual for each material.

Weldability is the property of a material to form
welded joint, the performance of which
corresponds to the quality of the base material,
welded. Weldability is judged by
test results of welded samples and
characteristics of the base material in the welded area
seam Rules have been established for determining the following
metal weldability indicators: mechanical
properties of welded joints, permissible modes
arc welding and surfacing, the quality of welded
joints and welds, long-term strength
welded joints.

Crystallography is the science of crystals, crystalline natural bodies. It studies the shape, internal structure, origin, distribution and properties of crystalline substances.

The main properties of crystals - anisotropy, homogeneity, the ability to burn themselves and the presence of a constant melting point - are determined by their internal structure.

Crystals are all solids that have a polyhedron shape resulting from the ordered arrangement of atoms. Crystallography is called the science of crystals, crystalline natural bodies. It studies the shape, internal structure, origin, distribution and properties of crystalline substances. Crystals are all solids that have a polyhedron shape resulting from the ordered arrangement of atoms. Cubes are examples of well-formed crystals...

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More than five thousand types of crystals are known. They have a different shape and a different number of faces. The shape of a crystal is the totality of all its faces. In crystallography, a simple form is a collection of identical faces connected by symmetry elements. Among simple forms, there are closed forms that enclose part of the space completely, for example, a cube, an octahedron; open simple shapes, for example, various prisms, space...

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Syngony (from the Greek σύν, “according, together,” and γωνία, “angle” - literally “similar angle”) is one of the divisions of crystals based on the shape of their unit cell. Syngony includes a group of symmetry classes that have one common or characteristic symmetry element with the same number of unit directions. There are seven systems: cubic, tetragonal (square), trigonal, hexagonal, rhombic, monoclinic, triclinic.

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“Symmetry” translated from Greek means “proportionality” (repetition). Symmetrical bodies and objects consist of equivalent parts that are regularly repeated in space. The symmetry of the crystals is especially diverse. Different crystals have more or less symmetry. It is their most important and specific property, reflecting the regularity of the internal structure.

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From the point of view of geometric crystallography, a crystal is a polyhedron. To characterize the shape of crystals, we use the concept of constraint elements. The external shape of crystals is composed of three constraint elements: faces (planes), edges (lines of intersection of faces) and facet angles.

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Crystals arise when a substance transitions from any state of aggregation to a solid. The main condition for the formation of crystals is a decrease in temperature to a certain level, below which particles (atoms, ions), having lost excess thermal motion, exhibit their inherent chemical properties and are grouped into a spatial lattice.

Physicochemical crystallography studies issues. Fundamentals of crystallography geometric crystallography crystallography

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