Biology
Entertaining crystallography 32 types of crystal lattices. Crystallography books for students and schoolchildren. Shapes of natural crystals

Entertaining crystallography 32 types of crystal lattices. Crystallography books for students and schoolchildren. Shapes of natural crystals

Rice. 10 Feldspar crystals

Law of Constancy of Angles
IN natural conditions crystals do not always develop into favorable conditions and have such ideal shapes as shown in the figures below.
Very often, crystals have incompletely developed shapes, with underdeveloped limiting elements (faces, edges, corners). Often, in crystals of the same mineral, the size and shape of the faces can vary significantly (Fig. 9-11). Often in soils and rocks there are not whole crystals, but only their fragments. However, as measurements have shown, the angles between the corresponding faces (and edges) of crystals of different forms of the same mineral always remain constant.

This is one of the basic laws of crystallography - the law of constancy of angles.
What explains this constancy of angles? This phenomenon is due to the fact that all crystals of the same crystal have the same structure, that is, they are identical in their internal structure. The law is valid for the same physicochemical conditions in which the measured crystals are located, i.e. at the same temperatures, pressure, etc. A sharp change in angles in crystals can occur during a polymorphic transformation (see Chapter III).

Rice. 11. Three quartz crystals with various developments corresponding faces

The law of constancy of angles was first mentioned by a number of scientists: I. Kepler, E. Bartholin, H. Huygens, A. Leeuwenhoek. This law was expressed in general form in 1669 by the Danish scientist N. Stenop. In 1749, he first connected the law of constancy of angles with internal structure saltpeter. And finally, in 1772, the French mineralogist Rome de Lisle formulated this law for all crystals.

In Fig. Figure 10 shows two feldspar crystals of different shapes. The angles between the corresponding faces a and b of two crystals are equal to each other (they are designated by the letter of the Greek alphabet a). In Fig. 11, the angle between the faces t and r of quartz crystals of different external shapes is 38° 13′. From what has been said it is clear how large the measurement value is dihedral angles crystals for accurate diagnosis mineral.

Rice. 12. 13. Measurement of the facet angle of a crystal using an applied goniometer.
Schematic diagram of a reflective goniometer

Measuring the facet angles of crystals. Goniometers

To measure the dihedral angles of crystals, special instruments called goniometers are used (Greek “gonos” - angle). The simplest goniometer used for approximate measurements is the so-called applied goniometer, or Caranjo goniometer (Fig. 12). For more accurate measurements, a reflective goniometer is used (Fig. 13).

Measuring angles using a reflective goniometer is carried out as follows: a beam of light, reflected from the face of the crystal, is caught by the eye of the observer; turning the crystal, the reflection of the light beam from the second face is recorded on the scale of the goniometer circle, and the angle between the two reflections, and therefore between the two faces of the crystal, is measured.
The measurement of the dihedral angle will be correct if the crystal faces from which the light beam is reflected are parallel to the axis of rotation of the goniometer. To ensure that this condition is always met, the measurement is carried out on a double-circle or theodolite goniometer, which has two circles of rotation: the crystal can simultaneously rotate around two axes - horizontal and vertical.

Rice. 14. Theodolite goniometer E. S. Fedorov

The theodolite goniometer was invented in late XIX V. Russian crystallographer Fedorov and, independently of him, German scientist W. Goldschmidt. The general view of a double-arm goniometer is shown in Fig. 14.

Crystal chemical analysis by E. S. Fedorov

The method of goniometric determination of the crystalline and, to a certain extent, its internal structure from the external forms of crystals allowed Fedorov to introduce crystal chemical analysis into the practice of diagnosing minerals.
The discovery of the law of constancy of angles made it possible by measuring the facet angles of crystals and comparing the measurement data with the available tabulated values to determine whether the crystal under study belongs to a specific substance. Fedorov did a great job of systematizing the vast literature on crystal measurement. Using it, as well as his own measurements of crystals, Fedorov wrote the monograph “The Kingdom of Crystals” (1920).

Rice. 15. Diagram of the relationship between angles in a crystal when measuring it

Fedorov's students and followers - the Soviet crystallographer A.K. Boldyrev, the English scientist T. Barker (1881-1931) significantly simplified the methods for determining crystals. Currently, crystal chemical analysis comes down to measuring the required angles on a goniometer and determining the substance using reference tables.
When measuring crystals goniometrically, the internal angle between the faces is directly determined (Fig. 15, ∠β). However, in summary tables with measured angles of various substances, the angle formed by the normals to the corresponding faces is always given (Fig. 15, ∠α). Therefore, after the measurement, you should make simple calculations using the formula α = 180°-β (α = α1, as angles with mutually perpendicular sides) and determine the name of the mineral using a reference book.

Symmetry in crystals

We learn about the existence of symmetry in nature with early childhood. The wings of a butterfly and dragonfly, the petals and leaves of various flowers and plants, snowflakes convince us that symmetry exists in nature.
Symmetrical are bodies that consist of identical, symmetrical parts that can be combined. So, if a butterfly folds its wings, they will completely fit together. The plane that divides the butterfly into two parts will be the plane of symmetry. If we put a mirror in place of this plane, in it we will see a symmetrical reflection of the other wing of the butterfly. Likewise, the plane of symmetry has the property of specularity - on both sides of this plane we see symmetrical, mirror-equal halves of the body.

Rice. 16. Plane of symmetry (P) in a rock salt crystal

As a result of studying the crystalline forms of minerals, it was found that symmetry exists in inanimate nature, in the world of crystals. In contrast to symmetry in living nature, it is called crystalline symmetry.
Crystal symmetry is the correct repeatability of limiting elements (edges, faces, corners) and other properties of crystals in certain directions.
The symmetry of crystals is most clearly revealed in their geometric shape. Regular repetition geometric shapes can be seen if: 1) cut the crystal with a plane; 2) rotate it around a certain axis; 3) compare the location of the crystal limiting elements relative to a point lying inside it.

Plane of symmetry of crystals

Let's cut the rock salt crystal into two halves (Fig. 16). The drawn plane divided the crystal into symmetrical parts. Such a plane is called a plane of symmetry.

Rice. 17 Planes of symmetry in a cube

The plane of symmetry of a crystalline polyhedron is a plane on both sides of which the same constraint elements are located and the same properties of the crystal are repeated.
The plane of symmetry has the property of specularity: each of the parts of the crystal, dissected by the plane of symmetry, is combined with another, that is, it is, as it were, a mirror image of it. Different crystals can have different numbers of planes of symmetry. For example, in a cube there are nine planes of symmetry (Fig. 17), in a hexagonal or hexagonal prism - seven planes of symmetry - three planes will pass through opposite edges (Fig. 18, planes a), three planes through the middles of opposite faces (parallel to the longitudinal axis of the polyhedron -on rice. 18, plane b) and one plane - perpendicular to it (Fig. 18 plane

The plane of symmetry is denoted by the capital letter of the Latin alphabet P, and the coefficient in front of it shows the number of planes of symmetry in the polyhedron. Thus, for a cube we can write 9P, i.e. nine planes of symmetry, and for a hexagonal prism - 7P.

Rice. 18. Planes of symmetry in a hexagonal prism (left) and the layout of the axes of symmetry (in plan,
on right)

Axis of symmetry

In crystalline polyhedra one can find axes, when rotated around which the crystal will align with its original position when rotated through a certain angle. Such axes are called axes of symmetry.
The axis of symmetry of a crystalline polyhedron is a line around which the same constraint elements and other properties of the crystal are correctly repeated when rotated.
Symmetry axes are designated by the capital Latin letter L. When the crystal is rotated around the symmetry axis, the constraint elements and other properties of the crystal will be repeated a certain number of times.

If, when rotating the crystal by 360°, the polyhedron is combined with its original position twice, we are dealing with a second-order symmetry axis; in case of four- and six-fold alignments, we are dealing with the fourth- and sixth-order axes, respectively. The axes of symmetry are designated: L 2 - axis of symmetry of the second order; L 3 - axis of symmetry of the third order; L 4 - axis of symmetry of the fourth order; L 6 - axis of symmetry of the sixth order.

The axis order is the number of times the crystal aligns with its original position when rotated 360°.

Due to homogeneity crystal structure and thanks to the patterns in the distribution of particles inside crystals, crystallography proves the possibility of the existence of only the above

axes of symmetry. The first order symmetry axis is not taken into account, since it coincides with any direction of each figure. A crystalline polyhedron can have several symmetry axes of different orders. The coefficient in front of the symmetry axis symbol shows the number of symmetry axes of one order or another. So, in the cube there are three axes of symmetry of the fourth order 3L4 (through the midpoints of opposite faces); four third-order axes - 4L3 (drawn through the opposite vertices of trihedral angles) and six second-order axes 6L2 (through the midpoints of opposite edges) (Fig. 19).

In a hexagonal prism, one axis of the sixth order and 6 axes of the second order can be drawn (Fig. 18 and 20). In crystals, along with the usual axes of symmetry characterized earlier, so-called inversion axes are distinguished.
The inversion axis of a crystal is a line, when rotated around it through a certain angle and then reflected at the central point of the polyhedron (as in the center of symmetry), identical constraint elements are combined .

Rice. 20 Axes of symmetry of the sixth and second orders (L 6 6L 2) and planes of symmetry (7P) in a hexagonal prism

The inversion axis is indicated by a symbol. On crystal models, where inversion axes usually have to be determined, there is no center of symmetry. The possibility of the existence of inversion axes of the following orders has been proven: first L i1, second L i2, third
L i3, fourth L i4, sixth L i6. In practice, we only have to deal with inversion axes of the fourth and sixth orders (Fig. 21).
Sometimes inversion axes are designated by a number to the lower right of the axis symbol. Thus, the second-order inversion axis is designated by the symbol third - L 3, fourth L 4, sixth L 6.
The inversion axis is a combination of a simple axis of symmetry and a center of inversion (symmetry). The diagram below (Fig. 21) shows two inversion axes Li and Li4. Let's analyze both cases of finding these axes in models. In a trigonal prism (Fig. 21,I), straight line LL is the third-order axis L 3. At the same time, it is also a sixth-order inversion axis. So, when rotating any parts of the polyhedron by 60° around the axis and then reflecting them at the central point, the figure is combined with itself. In other words, a rotation of the edge AB of this prism by 60° around LL brings it to position A 1 B 1, the reflection of the edge A 1 B 1 through the center aligns it with DF.
In a tetragonal tetrahedron (Fig. 21,II) all faces consist of four completely identical isosceles triangles. Axis LL is a second-order axis L 2 When rotated around it by 180°, the polyhedron is aligned with its original position, and face ABC moves to the place ABD. At the same time, the L2 axis is also a fourth-order inversion axis. If you rotate face ABC 90° around the LL axis, then it will take position A 1 B 1 C 1. When A 1 B 1 C 1 is reflected at the central point of the figure, the face will align with the position of BCD (point A1 coincides with C, B 1 with D and C 1 with B). Having performed the same operation with all parts of the tetrahedron, we note that it is combined with itself. When rotating the tetrahedron by 360°, we obtain four such alignments. Therefore, LL is a fourth-order inversion axis.

Center of symmetry

In crystalline polyhedra, in addition to planes and axes of symmetry, there may also be a center of symmetry (inversion).
The center of symmetry (inversion) of a crystalline polyhedron is a point lying inside the crystal, in diametrically opposite directions from which identical constraint elements and other properties of the polyhedron are located.
The center of symmetry is designated by the letter C of the Latin alphabet. If there is a center of symmetry in the crystal, each face corresponds to another face, equal and parallel (inversely parallel) to the first. A crystal cannot have more than one center of symmetry. In crystals, any line passing through the center of symmetry is divided in half.
The center of symmetry is easy to find in a cube, an octahedron in a hexagonal prism, since it is located in these polyhedra at the intersection of the axes and planes of symmetry.
The disassembled elements found in crystalline polyhedra, planes, axes, center of symmetry are called symmetry elements.

Table 1 32 type of crystal symmetry Types of symmetry
	primitive	central	planal	axial	planaxial	inversion-primitive	inversion-planar
Triclinic
Monoclinic			R		L 2PC
Rhombic			L 2 2P	3L 2	3L 2 3PC
Trigonal		L 3 C	L 3 3P	L 3 3L 2	L 3 3L 2 3PC
Tetragonal		L 4PC	L 4 4P	L 4 4P 2	L 4 4L 2 5PC	19** L i4 =L 2	L i4 (=L2)2L 2 x2P
Hexagonal	L 6

Crystallography and mineralogy, Basic concepts, Boyko S.V., 2015.

The concept of regular crystalline polyhedra and their symmetry is given. its elements and transformations, crystallographic coordinate system. The general patterns of formation, growth and dissolution of crystals are indicated, and the most common forms of mineral individuals and mineral aggregates are given. The essence of the crystal optical method for diagnosing minerals is shown. The content of the basic concepts of mineralogy is revealed. a brief outline of its history is given, a classification of mineral formation processes is given, and each of them is characterized. Considered general provisions assessment of the internal structure of minerals and descriptions of their most common in earth's crust classes.

Chapter 1. CRYSTALLOGRAPHY.
Crystallography (Greek krystallos - ice and grapho - write, describe) is the science of the atomic-molecular structure, symmetry, physical properties, formation and growth of crystals. The term “crystallography” was first used in 1719 to describe rock crystal (a transparent variety of quartz) in the work of the Swiss researcher M.A. Kapeller (1685-1769).

Crystals are solids whose atoms or molecules form an ordered periodic structure. For such structures there is the concept of “long-range order” - orderliness in the arrangement of material particles at infinitely large distances (“short-range order” - orderliness at distances close to interatomic - amorphous bodies). Crystals have symmetry internal structure, symmetry of external shape, as well as anisotropy of physical properties. They represent the equilibrium state of solids - each substance" located at a certain temperature and pressure, in the crystalline state, has its own atomic structure. When external conditions change, the structure of the crystal can change.

TABLE OF CONTENTS
Introduction
Chapter 1. Crystallography
1.1. Brief essay history of crystallography
1.2. Geometric crystallography.
1.2.1. Crystal symmetry
1.2.2. Simple Crystal Shapes
1.2.3. Concept of crystallographic coordinate system, symbols of faces and simple shapes
1.3. Crystallogenesis
1.3.1. Concept of chemical bonds and intermolecular interactions
1.3.2. Crystal growth
1.3.3. Influence of parameters of the crystallization medium on the habit of crystals. Concept of crystal dissolution
1.4. Morphology of minerals
1.4.1. Degenerate forms of crystal growth
1.4.2. Geometric combinations of individuals
1.4.3. Split mineral individuals
1.5. Morphology of mineral aggregates
1.6. Basic concepts of crystal optics
1.6.1. Physical concepts, used in crystal optics for diagnostics of minerals and rocks
1.6.2. The concept of the crystal optical method for studying minerals and rocks
Chapter 2. Mineralogy
2.2. Characteristics of some fundamental terms
2.3.1. Endogenous processes mineral formation
2.3.2. Exogenous processes of mineral formation
2.4. general characteristics most common in the earth
2.4.1. The concept of assessing the crystallochemical structure of minerals
2.4.2. Silicates
2.4.3. Oxides and hydroxides
2.4.4. Carbonates
2.4.5. Phosphates
2.4.6. Halides
2.4.7. Sulfates
2.4.8. Sulfides
2.4.9. Native elements
Control questions and tasks
Conclusion
Bibliography
Applications.

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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 in early XIX Mathematics and physics developed, and therefore crystallography also developed. 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 physical properties crystals (optical, thermal, electrical, etc.), where some areas were separated into separate sciences (crystal optics).

Crystalline and amorphous bodies

Solids are divided into:

1. Amorphous, Where elementary particles are located 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 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 detected 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).

Axis of symmetry- a straight line, when rotated 360° around which the crystal is aligned several times with its original position.

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.

The totality of all available symmetry elements is usually written down in a line, without any punctuation marks between them, with the axes of symmetry first indicated, starting from the highest order, then the plane of symmetry, and in the last place, if any, the center of symmetry is written.

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 others participate in closing them simple shapes(prisms, etc.)

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;

· scalenos- 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– quantities 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 form from all states of aggregation substances, both 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 gaseous substance.

Liquid state– formation of crystals from the melt and from 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.

Solid state– mainly, the process of transition of an amorphous substance into a crystalline one (decrystallization); under natural conditions, these processes actively occur at high temperatures 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 (oversaturated) – formed if saturated solution falls into 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– released maximum amount energy (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

Also in mid-17th century 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.

Crystallography Crystallography is one of the basic sciences Earth, studies the formation process, external form, internal structure and physical properties of crystals. IN Lately This science has gone far beyond its limits and is studying the patterns of development of the Earth, its shape and the processes occurring in the depths of the geospheres.

The crystals sparkle with symmetry. E. S. Fedorov The classical definition of a crystal (from the Greek “kristallos” ice), a homogeneous solid body capable of certain conditions self-destruct. Let's look at this definition in more detail...

Spatial lattice Spatial lattice is a geometric image that reflects three-dimensional periodicity in the distribution of atoms in the structure of a crystal

The term symmetry Crystallography, like any completely independent science, has its own method - THE SYMMETRY METHOD. Symmetry from Greek. “symmetry” proportionality), it is believed, was introduced into use by Pythagoras, denoting to him the SPATIAL REGULARITY IN THE LOCATION OF Identical FIGURES OR THEIR PARTS. Symmetry is a pattern, repeatability of figures or their parts in space!! In a figurative sense, symmetry is synonymous with harmony, beauty and perfection!

Symmetry and humanity The concept of symmetry has been treated with trepidation since ancient times. In HF China, a circle is the most perfect figure, the home of the gods is also a circle. In Christianity, there is a connection with the concept of the Trinity (God the Father, God the Son, God the Holy Spirit). In Ancient Egypt - “The All-Seeing Eye”

Symmetry in Geology Lithology - ripples in the sand Paleontology - the orientation of one plane of symmetry from another allows one to distinguish brachiopods from bivalves. . Planes of symmetry in underwater ridges (at the bottom of the World Ocean). Explanation of the concept of spreading

Symmetry in living matter The most important thing! Most biological objects have mirror symmetry. Sometimes a fifth order symmetry axis L 5 is observed, not in crystals!!! According to N.V. Belov, so that they cannot “petrify” because in crystalline substance There are no fifth order axes.

Concepts that are urgently needed when describing crystal models in Bravais educational symbolism Elements of symmetry - geometric images (planes, straight lines, lines or points) with the help of which symmetric transformations (symmetry operations) are specified or carried out Plane of symmetry Axis of symmetry Center of symmetry

Axis of symmetry Rotational axes of symmetry are straight lines, when rotated at a certain angle, the figure (or crystal) is aligned with itself. The smallest angle of rotation around such an axis is called the elementary angle of rotation. The value of this angle determines the order of the axis of symmetry (360 divided by the value of this angle). Denoted in educational symbols as Ln, where n is the order of the axis of symmetry: L 2 L 3 L 4 L 6

Important I draw your attention to the fact that in crystallographic polyhedra the order of the axes is limited to the numbers 1, 2, 3, 4, 6. That is, symmetry axes of the 5th and higher than the 6th orders are impossible in crystals. Whoever can come up with convincing evidence of this fact will receive a Swiss chocolate bar right in class!

To prove this fact 1. “Spatial-lattice” proof 2. According to Nikolai Vasilyevich Belov

Mirror plane of symmetry The mirror plane of symmetry specifies the reflection operation in which the right part of the figure (figure), reflected in the plane as in a “two-way mirror”, is combined with its left part (figure). It is designated by the letter P.

Center of symmetry (point of symmetry) This is like a “mirror point” at which the right figure not only turns into the left, but also turns over. The inversion point in this case plays like the role of a camera lens, and the figures connected by it are correlated like an object and its image on photographic film. Denoted by the letter C

Crystallographic systems (sygonies) Symmetry classes with a single coordinate reference are combined into a family called syngony or system (from the Greek Syn. “similar” and “gony” - angle. All thirty-two crystal symmetry classes are divided into three categories, each of which includes one or more systems. These are triclinic, monoclinic, orthorhombic, hexagonal, ( special case trigonal), tetragonal and cubic system. Let's look at them in more detail by category.

Hexagonal system. Average category a=b≠c, α=β=90˚, γ=120˚ “hexa” - six Presence L 6 main feature

Now let's practice describing crystals in the symbolism of Bravais. YOU NEED TO FIND AND WRITE ITS FULL FORMULA IN THE TRAINING SYMBOLICS OF BRAVAIS AND NAME THE SYNGONY TO WHICH IT RELATES. We look at highest ORDER axes in the formula. In addition to the cubic system 4 L 3 – a sign of the CUBIC SYSTEM L 6 – a sign of the HEXAGONAL SUBSYNGONY L 4 – PRIZNGAK TETRAGONAL SINGONY. L 3 – SIGN OF TRIGONAL SYNGONY L 2, 3 L 2 – SIGN OF RHOMBIC SYNGONY L 2 – SIGN OF MONOCLINIC SYNGONY Or L UNSC. ORDER, or just C – SIGN OF TRICLINIC SYNGONY

In the next lesson we will practice describing crystal models again. We will learn to identify the basic simple forms using a cheat sheet and talk about questions that may arise in front of you during the crystallography class at the Olympiad, plus we will talk about the dependence of the shape of crystals (for example, quartz and calcite) on the conditions of their formation. Think about the next lesson. What shape will a crystal grown in space have?

Rice. 1. Perfect cleavage of rock salt

When getting acquainted with minerals, one involuntarily notices the inherent ability of many of them to take on the correct external shape - to form crystals, that is, bodies limited by a number of planes. In this regard, he constantly uses crystallographic terms and concepts. That's why brief information in crystallography should precede a systematic acquaintance with mineralogy.

PROPERTIES OF CRYSTAL SUBSTANCE

All homogeneous bodies, according to the nature of the distribution of physical properties in them, can be divided into two large groups: amorphous and crystalline bodies.

In amorphous bodies, all physical properties are statistically the same in all directions.

Such bodies are called isotropic (equal properties).

TO amorphous bodies include liquids, gases, and from solids - glasses, glassy alloys, as well as solidified colloids (gels).

In crystalline bodies, many physical properties are associated with a certain direction: they are the same in parallel directions and unequal, generally speaking, in non-parallel directions.

This nature of properties is called anisotropy, and those with similar properties are anisotropic (unequal properties).

Most solids and, in particular, the vast majority of minerals belong to crystalline bodies.

Among the physical properties of any solid This also applies to the force of adhesion between the individual particles that make up the body. This physical property in a crystalline medium changes with a change in direction. For example, in rock salt crystals (Fig. 1), which occur in the form of more or less regular cubes, this cohesion will be least perpendicular tocube faces. Therefore, upon impact, a piece of rock salt will split most easily in a certain direction - parallel to the edge of the cube, and a piece of an amorphous substance, such as glass, of the same shape will split equally easily in any direction.

The property of a mineral to split in a certain, pre-known direction, with the formation of a split surface in the form of a smooth, shiny plane, is called cleavage (see below “Physical properties of minerals”). It is present to varying degrees in many minerals.

When separated from a supersaturated solution, the same force of interparticle attraction causes deposition from the solution in certain directions; a plane is formed perpendicular to each of these directions, which, as new portions settle on it, will move away from the center of the growing crystal parallel to itself. Fig 1. Perfect cleavage the number of such planes with rock salt gives to the crystal characteristicit has the correct multifaceted shape.

If the influx of substance to the growing crystal occurs unevenly from different sides, which is usually observed in natural conditions, in particular, if a crystal in its growth is constrained by the presence of neighboring crystals, the deposition of matter will also occur unevenly, and the crystal will receive a flattened or elongated shape, or will occupy only the free space that is located between the previously formed crystals. It must be said that most often this happens, and regular, evenly formed crystals are rare for many minerals.

With all this, however, the directions of the planes of each crystal remain unchanged, and therefore, the dihedral angles between the corresponding (equivalent) planes on different crystals of the same substance and the same structure should represent constant values (Fig. 2).

This is the first fundamental law of crystallography, known as the law of constancy of dihedral angles, was first noticed by Kepler and expressed in general form by the Danish scientist N. Steno in 1669. In 1749, M.V. for the first time connected the law of constancy of angles with the internal structure of the crystal using the example of saltpeter.

Finally, another 30 years later, the French crystallographer J. Romeu-Delisle, after twenty years of work on measuring angles in crystals, confirmed the generality of this law and formulated it for the first time.

Rice. 2. Quartz crystals

This pattern, derived by Steno-Lomonosov-Rome-Delisle, formed the basis of everything scientific research crystals of that time and served as the starting point for the further development of the science of crystals. If we imagine the crystal faces moved parallel to themselves so that they are equal tothe significant faces have moved the same distance from the center, the resulting polyhedra will take on the ideal shape that would be achieved by the growing crystal in the case of ideal, i.e., uncomplicated external influences, conditions.

ELEMENTS OF SYMMETRY

Symmetry. Despite its apparent simplicity and routine, the concept of symmetry is quite complex. In the most simple definition symmetry is the regularity (regularity) in the arrangement of identical parts of a figure. This correctness is expressed: 1) in the natural repetition of parts when the figure rotates, and the latter, when rotated, seems to be combined with itself; 2) in the mirror equality of the parts of the figure, when some parts of it seem to be a mirror image of others.

All these patterns will become much clearer after becoming familiar with the elements of symmetry.

Considering well-formed crystals or crystallographic models, it is easy to establish those patterns that are observed in the distribution of identical planes and equal angles. These patterns come down to the presence in crystals of the following symmetry elements (individually or in certain combinations): 1) planes of symmetry, 2) axes of symmetry and 3) center of symmetry.

Rice. 3. Plane of symmetry

1. An imaginary plane that divides a figure into two equal parts related to each other, like an object to its image in a mirror (or like a right hand to a left hand), is calledplane of symmetry and is denoted by the letter R(Fig. 3 - plane) AB).

2. Direction, when rotated around which always at the same angle, all parts of the crystal are symmetrically repeated P times, is called a simple or rotary axis of symmetry (Fig. 4 and 5). Number P, showing how many times repetition of parts is observed during a complete (360°) rotation of the crystal around the axis, is called the order or significance of the axis of symmetry.

Based on theoretical considerations, it is easy to prove that P - the number is always an integer and that only symmetry axes of the 2nd, 3rd, 4th and 6th order can exist in crystals.

Rice. 4. Axis of symmetry of the 3rd order

The axis of symmetry is designated by the letter L or G, and the order of the axis of symmetry is indicated by the indicator placed at the top right. So L 3 denotes the axis of symmetry of the 3rd order; L 6- symmetry axis of the 6th order, etc. If there are several axes or planes of symmetry in the crystal, then their number is indicated by a coefficient that is placed in front of the corresponding letter. Yes, 4L 3 3L 2 6P means that the crystal has four 3rd order symmetry axes, three 2nd order symmetry axes and 6 symmetry planes.

In addition to simple axes of symmetry, complex axes are also possible. In the case of the so-called mirror-rotary axis, the alignment of the polyhedron with all its parts with the original position occurs not as a result of only one rotation through some angle a, but also as a result of simultaneous reflection in an imaginaryperpendicular to the plane. The axis of complex symmetry is also designated by the letter L, but only the axis indicator is placed at the bottom, for example, L4. The study shows that crystalline polyhedra can have complex axes of 2, 4 and 6 names or orders, i.e. L 2 , L 4 And L 6.

Rice. 5. Polyhedron with an axis of symmetry of the 2nd order

Symmetry of the same nature can be achieved using an inversion axis. In this case, the symmetrical operation consists of a combination of rotation around the axis through an angle of 90 or 60° and repetition through the center of symmetry.

The process of this symmetric operation can be illustrated by the following example: let there be a tetrahedron (tetrahedron) whose edges AB And CD mutually perpendicular (Fig. 6). When the tetrahedron rotates 180° around its axis L i4, the entire figure is aligned with the original position, i.e. the axis L i4 , there is an axis of symmetry of the second order (L 2). In fact, the figure is more symmetrical, since the rotation is about the same axis by 90°

and subsequent movement of the point A according to the center of symmetry will translate it to the point D. In the same way, period IN aligns with the Point WITH. The entire figure will be aligned with its original position. This combination operation can be carried out each time by rotating the figure around its axis L i4 by 90°, but with obligatory repetition through the center of symmetry. Selected axis direction L i4 and will be the direction of the 4th order inversion axis ( L i4 = G i4 ).

Rice. 6. Polyhedron with a quadruple inversion axis of symmetry (Li4)

The use of inversion axes in some cases is more convenient and clearer than the use of mirror-rotary axes. They can also be designated as G i3 ; G i4 ; G i6; or how L i3 ;L i4 ; L i6

The point inside the crystal, on equal distance from which there are equal, parallel and generally inversely located faces in opposite directions is called the center of symmetry or the center of inverse equality and is denoted by the letter With(Fig. 7). It is very easy to prove that c =L i2

i.e., that the center of inverse equality appears in crystals whenwhich have an axis of complex symmetry of the 2nd order. It should also be noted that axes of complex symmetry are at the same time axes of simple symmetry of half the name, i.e.possible designations L 2 i4 ;L 3 i6 . However, the opposite conclusioncannot be done, since not every axis of simple symmetry will necessarily be an axis of complex symmetry twice as large names.

The Russian scientist A.V. Gadolin proved in 1869 that only 32 combinations (combinations) of the above symmetry elements, called crystallographic classes or types of symmetry, can exist in crystals. All of them are found in natural or artificial crystals.

CRYSTALLOGRAPHIC AXES. PARAMETERS AND INDICES

When describing a crystal, in addition to indicating the elements of symmetry, it is necessary to determine the position in space of its individual faces. To do this, use the usual techniques analytical geometry, taking into account at the same time the features of natural crystalline polyhedra.

Rice. 7. Crystal with a center of symmetry

Crystallographic axes are drawn inside the crystal, intersecting in the center and in most cases coinciding with symmetry elements (axes, crystal planes or perpendiculars to them). With a rational choice of crystallographic axes, crystal faces that have the same appearance and physical properties receive the same numerical value, and the axes themselves will run parallel to the observed or possible edges of the crystal. In most cases, they are limited to three axes I, II and III, less often it is necessary to draw four axes - I, II, III and IV.

In the case of three axes, one axis is directed towards the observer and is denoted by the sign I (Fig. 8), the other axis is directed from left to right and is denoted by the sign II, and finally, the third axis is directed vertically and is denoted by the sign III.

In some manuals the I axis is called X, II axis - Y, and III axis - Z. If there are four axes, the I axis corresponds to the A axis; the II axis corresponds to the Y axis; the III axis corresponds to the A axis. U and IV axis -axis Z.

The ends of the axes directed towards the observer, to the right and up, are positive, and those directed away from the observer to the left and down are negative.

Rice. 8. Crystal faces on coordinate axes

Let the plane R(Fig. 8) cuts off segments on the crystallographic axes a, b And With. Since crystalline polyhedra are determined only by the face angles and inclination of each plane, and not by the dimensions of the planes, it is possible, by moving any plane parallel to itself, to increase and decrease the dimensions of the polyhedron (which is what happens during crystal growth). Therefore, to indicate the position of the plane R no need to know absolute values segments a, b And With, but only their attitude a: b: c. Any other plane of the same crystal will be denoted in the general case a’ : b’ : c’ or a": b": c".

Let's assume that a'-ta; b' = nb; c' = rs; a" = t'a; b" = n'b; c" = p's, i.e., we express the lengths of segments along the crystallographic axes for these planes in numbers that are multiples of the lengths of segments along the crystallographic axes of the plane R, called the original or unit. Quantities t, p, p, t’, p’, p’ are called the numerical parameters of the corresponding plane.

In crystalline polyhedra, the numerical parameters are simple and rational numbers.

This property of crystals was discovered in 1784 by the French scientist Ayui and is called the “Law of Rationality of Parameters.”

Rice. 9. Elementary parallelepiped and unit face

Typically the parameters are 1, 2, 3, 4; how more number, by which the parameters are expressed, the less common the corresponding faces are.

If we choose the crystallographic axes so that they run elementary parallel to the edges of the crystal, then the boundary segmentsthese axes, which are cut off by the original crystal face (face R), determine the basic cell of a given crystalline substance.

It should be borne in mind that for crystals with low symmetry it is often necessary to adopt an oblique system of crystallographic axes. In this case, it is necessary to indicate the values of the angles between the crystallographic axes, denoting them through a (alpha), p (beta) and y (gamma). In this case, I is called the angle between the III and II axes, R-angle between III and I(the so-called monoclinic angle), am - the angle between axes I and II (Fig. 9).

In Fig. 8 reference plane R cuts off segments on the corresponding axes a,b And With or their multiples.

Any other plane must cut off a segment along the I axis that is a multiple of A, along the II axis - multiple of b and along the III axis - multiple With.

So plane R will cut off segments a, 2b and 2c, and the plane R" - segments 2a, 4b and Zs, etc. The coefficients of a, 6 and c, which are parameters, can only be rational quantities.

Quantities a, b and c or their ratios are characteristic constants for a given crystal and are called axial units.

Designations of planes along segments on crystallographic axes in general view dominated science until the last quarter of the 19th century, but then gave way to others.

Currently, Miller's method is used to indicate the position of crystal faces, as it represents the greatest convenience in crystallographic calculations, although at first glance it seems somewhat complicated and artificial.

As stated above, the original or “unit” plane will determine the axial units, and, knowing the parameters t:n:p any other plane, one can determine the position of this latter. For crystallographic calculations, it is more advantageous to characterize the position of any face not by the direct ratio of the segments made by it on the crystallographic axes of the crystal to the segments of a “unit” face, but by the inverse ratio, i.e., divide the size of the segment made by a single face by the segment made by the determined face .

It is obvious that the resulting ratios will also be expressed in integers, generally denoted by the letters h, k And l. Thus, the position of any face can be expressed unambiguously in terms of three quantities h,k And l, the relationship between which is inverse to the ratio of the lengths of the segments made by the face on three crystallographic axes, and along each axis, in the general case, those segments (unit segments) that the unit face makes on the corresponding axes should be taken. If we take as crystallographic axes the directions that coincide with the axes of symmetry or normals to the planes of symmetry or, if there are no such symmetry elements, with the edges of the crystal, then the characteristics of the faces can be expressed in simple and integer numbers, and all faces of the same shape will be expressed in a similar way way.

Quantities h, To And l are called face indices, and their collections are called face symbols. The face symbol is usually denoted by indices written in a row without any punctuation marks and enclosed in parentheses (hbl). At the same time, the index h refers to the I axis, index k ko II and index l to III. Obviously, the index values are the reciprocal of the segment made by the face on the axis. If the face is parallel to the crystallographic axis, then the corresponding index is zero. If all three indices can be reduced by the same amount,

then such a reduction must be made, remembering that the indices are always prime and integer numbers.

The face symbol, if expressed in numbers, for example (210) reads: two, one, zero. If a face makes a segment in the negative direction of the axis, then a minus sign is placed above the corresponding index, for example (010). This symbol is read as follows: zero, minus one, zero.