BRAVE LATTICE
Construction scheme
BRAVE LATTICES, 14 three-dimensional geometric lattices characterizing all possible types of translational symmetry of crystals. Brave lattices are formed by the action of the transfer (translation) operation on any point of the crystal.
O. Brave in 1848 showed that the entire variety of crystal structures can be described using 14 types of lattices, differing in the shape of unit cells and symmetry, and subdivided into 7 crystallographic syngonies. These gratings were called Bravais gratings.
Bravais lattices differ in the symmetry of the elementary cell, i.e., in the ratio between its edges and corners, as well as in their centering.
Three conditions are used to select a Bravais cell:
The symmetry of the unit cell must correspond to the symmetry of the crystal, more precisely, the highest symmetry of the syngony to which the crystal belongs. The unit cell edges must be translations of the lattice;
The elementary cell must contain the maximum possible number of right angles or equal angles and equal edges;
The elementary cell must have a minimum volume.
According to the nature of the mutual arrangement of the main translations or the arrangement of nodes, all crystal lattices are divided into four types: primitive ( R), base-centered ( FROM), body-centered ( I), face-centered ( F).
In the primitive R-cell lattice nodes are located only at the vertices of the cell, in a body-centered I-cell - one node in the center of the cell, face-centered F-cell - one node in the center of each face, in base-centered FROM-cell - one node at the centers of a pair of parallel faces.
The set of coordinates of the nodes included in the elementary cell is called the basis of the cell. The entire crystal structure can be obtained by repeating the basis nodes by a set of translations of the Bravais cell.
For some syngonies, an elementary cell may contain nodes not only in the corners, but also in the center of the cell, all or some of the faces. In this case, translational transfer is possible not only to the periods of the elementary cell, but also to half of the diagonals of the cell faces or space diagonals. In addition to the mandatory translational invariance, the lattice can transform into itself under other transformations, which include rotations, reflections, and inversions. It is these additional symmetries that determine the type of Bravais lattice and distinguish it from others.
Brave grating types:
Cubic: primitive, body-centered and face-centered;
Hexagonal, trigonal;
Tetragonal: primitive and volume-centralized;
Rhombic: primitive, base-, volume- and face-centered;
Monoclinic: primitive and base-centered;
Triclinic.
Syngony(from the Greek σύν, “according to, together, side by side”, and γωνία, “angle” - literally “similar angle”) - a classification of crystallographic symmetry groups, crystals and crystal lattices depending on the coordinate system (coordinate frame). Symmetry groups with a single coordinate system are combined into one syngony.
Crystals belonging to the same syngony have similar corners and edges of elementary cells.
Triclinic: (\displaystyle a\neq b\neq c), (\displaystyle \alpha \neq \beta \neq \gamma \neq 90^(\circ ))
Monoclinic: (\displaystyle a\neq b\neq c), (\displaystyle \alpha =\gamma =90^(\circ ),\beta \neq 90^(\circ ))
Rhombic: (\displaystyle a\neq b\neq c), (\displaystyle \alpha =\beta =\gamma =90^(\circ ))
Tetragonal: (\displaystyle a=b\neq c), (\displaystyle \alpha =\beta =\gamma =90^(\circ ))
Hexagonal: (\displaystyle a=b\neq c), (\displaystyle \alpha =\beta =90^(\circ ),\gamma =120^(\circ ))
Cubic: (\displaystyle a=b=c), (\displaystyle \alpha =\beta =\gamma =90^(\circ ))
Main characteristics of crystal structures
Crystalline materials are characterized by the presence of a long-range order, which is characteristic. by the fact that a certain volume can be distinguished in it, the arrangement of the atom in which is repeated throughout the entire volume.
In amorphous mats, there is a short-range order, cat. charact. topics. that there is no repetition of volumes.
Krist. the structure can be conveniently described with the help of Z X a dimensional grid of straight limes, which divide the space into parallelepipids of equal sizes. Crossing lines is an image of 3 dimensional spaces. lattice. Lattice nodes, as a rule, correspond to the arrangement of atoms in a crystal. The atom oscillates
around these positions. If in such a spatial lattice it is possible to single out a certain volume, by moving it in 3 directions. allows you to line up the entire crystal, then gov. That an element, a cell was found.
The cell element is usually characterized by 6 parameters: a, b, c - the length of the edges of the parallelepiped, α, β, γ.
The shape of the cell element determines the crystallographic coordinate system - syngony. As the axes, the directions of the edges -elem, cells are chosen, and the edges themselves are the units of measurement. The number of right angles and equal sides must be max, and the volume of cell elements must be min.
Rice. 17. Snowflakes - Skeletal Ice Crystals
From experience it is known that in a crystalline substance the physical properties are the same in parallel directions, and the idea of the structure of substances requires that the particles (molecules, atoms or ions) that make up the crystal are located from each other at certain finite distances. Based on these assumptions, it is possible to construct a geometric diagram of the crystalline structure. To do this, the position of each constituent particle can be marked with a dot. All crystallinethe building will then be presented as a system of points regularly located in space, and for any paralleldirections of the distance between the points will be the same. Such a correct arrangement of points in space is called
spatial lattice, and if each point represents the position of an atom, ion or molecule in a crystal - a crystal lattice.
The construction of a spatial lattice can be imagined as follows.
A 0(Fig. 18) denotes the center of an atom or ion. Let the same center closest to it be denoted by the point A, then, on the basis of the homogeneity of the crystalline , at a distance A 1 A 2 \u003d A 0 A 1 must be the center A 2 ; Continuing this argument further, we can obtain a series of points: A 0, A 1, A 2, A 3 ...
Let us assume that the nearest point to A 0 in the other direction will R0, then there must be a particle S0 on distance R 0 S 0= L 0 R 0, etc., i.e., another row of identical points will be obtained A 0 , R 0 , S 0… If through R0, S0 etc. draw lines parallel to A 0, A 1, A 2, you get the same rows R 0 , R 1 , R 2 , S 0 , S 1 , S 2 ... etc
Rice. 18. Spatial lattice
As a result of the construction, a grid was obtained, the nodes of which correspond to the centers of the particles that make up the crystal.
If we imagine that at every point At 0 With, etc., the same grid is restored as in A 0 , as a result of this construction, a spatial lattice will be obtained, which in a certain sense will express the geometric structure of the crystal.
What are crystals
The theory of spatial lattices, created by the great Russian crystallographer E. S. Fedorov, received brilliant confirmation in the study of the structure of crystals using x-rays. These studies provide not only pictures of spatial lattices, but also the exact lengths of the gaps between the particles located in their nodes.
Rice. 19. Diamond structure
It turned out that there are several types of spatial lattices that differ both in the nature of the arrangement of particles and in their chemical nature.
We note the following types of spatial lattices:
Atomic structural lattices. At the nodes of these lattices, atoms of any substance or element are located, connecting directly to each other in a crystal lattice. This type of lattice is typical for diamond, zinc blende and some other minerals (see Fig. 19 and 20).
Ionic structural lattices. At the nodes of these lattices are ions, i.e., atoms that have a positive or negative charge.
Ionic lattices are common for inorganic compounds, such as alkali metal halogens, silicates, etc.
An excellent example is the lattice of rock salt (NaCl) (Fig. 21). In it, sodium ions (Na) in three mutually perpendicular directions alternate with chloride ions (Cl) at intervals equal to 0.28 millimicrons.
Rice. 20. Structure of zinc blende
In crystalline substances with a similar structure, the gaps between atoms in a molecule are equal to the gaps between molecules, and the very concept of a molecule loses its meaning for such crystals. On fig. 20 each sodium ion has
from above, below, to the right, to the left, in front and behind at equal distances from it, one chlorine ion each, which belongs both to this “molecule” and to neighboring “molecules”, and it is impossible to say with which particular chlorine ion of these six constitutes a molecule or would constitute it upon transition to a gaseous state.
In addition to the types described above, there are molecular structural lattices, in the nodes of which there are not atoms or ions, but separate, electrically neutral molecules. Molecular lattices are especially typical for various organic compounds or, for example, for "dry ice" - crystalline CO 2.
Rice. 21. Crystal lattice of rock salt
Weak ("residual") bonds between the structural units of such lattices determine the low mechanical strength of such lattices, their low melting and boiling points. There are also crystals that combine different types of lattices. In some directions, the bonds of particles are ionic (valence), and in others, molecular (residual). This structure leads to different mechanical strength in different directions, causing a sharp anisotropy of mechanical properties. Thus, molybdenite (MoS 2) crystals easily split along the pinacoid (0001) direction and give the crystals of this mineral a scaly appearance, similar to graphite crystals, where a similar structure is found. The reason for the low mechanical strength in the direction perpendicular to (0001) is the absence of ionic bonds in this direction. The integrity of the lattice here is maintained only by bonds of a molecular (residual) nature.
Taking all of the above into account, it is easy to a parallel between the internal structure of an amorphous substance, on the one hand, and a crystalline one, on the other:
1. In an amorphous substance, the particles are arranged in disorder, as if fixing the partially chaotic state of the liquid; therefore, some researchers call , for example, supercooled liquids.
2. In a crystalline substance, the particles are arranged in an orderly manner and occupy a certain position at the nodes of the spatial lattice.
The difference between crystalline and glassy (amorphous) matter can be compared to the difference between a disciplined military unit and a scattered crowd. Naturally, the crystalline state is more stable than the amorphous state, and an amorphous substance will more easily dissolve, react chemically, or melt. Natural ones always tend to acquire a crystalline structure, “crystallize”, for example (amorphous silica) eventually turns into chalcedony - crystalline silica.
A substance in a crystalline state usually occupies a somewhat smaller volume than in an amorphous form, and has a greater specific gravity; for example, albite - feldspar composition NaAlSi 3 O 8 in an amorphous state takes 10 cubic meters. units, and in the crystal - only 9; one cm 3 crystalline silica (quartz) weighs 2.54 G, and the same volume of vitreous silica (fused quartz) is only 2.22 G. A special case is ice, which has a lower specific gravity than taken in the same amount.
STUDY OF CRYSTALS WITH X-RAY RAYS
The question of the causes of regularities in the distribution of physical properties in a crystalline substance, the question of the internal structure of crystals, was first attempted by M.V. in 1749 using saltpeter as an example. This question was then more widely developed already at the end of the 18th century. French crystallographer Ayui. Ayui suggested that each substance has a specific crystalline form. This position was later refuted by the discovery of the phenomena of isomorphism and polymorphism. These phenomena, which play an important role in mineralogy, will be considered by us somewhat later.
Thanks to the work of the Russian crystallographer E. S. Fedorov and some other crystallographers, the theory of spatial lattices, briefly outlined in the previous chapter, was developed mathematically, and based on the study of the shape of crystals, possible types of spatial lattices were derived; but only in the 20th century, thanks to the study of crystals by x-rays, this theory was tested experimentally and brilliantly confirmed. A number of physicists: Laue, Braggum, G. V. Wulf, and others, using the theory of spatial lattices, succeeded in proving with absolute certainty that in some cases there are atoms at the nodes of crystal lattices, and in others, molecules or ions.
The rays, discovered by Roentgen in 1895, which bear his name, represent one of the types of radiant energy and, in many respects,They resemble rays of light, differing from them only in their wavelength, which is several thousand times smaller than the wavelength of light.
Rice. 22. Scheme for obtaining an X-ray diffraction pattern of a crystal using the Laue method:
A - x-ray tube; B - diaphragm; C - crystal; D - photographic plate
In 1912, Laue used a crystal, where the atoms are arranged in a spatial lattice, as a diffraction grating to obtain X-ray interference. In his research, a narrow beam of parallel X-rays (Fig. 22) was passed through a thin crystal of zinc blende C. At some distance from the crystal and A photographic plate D was placed perpendicular to the beam of rays, protected from the direct action of lateral X-rays and from daylight by lead screens.
With prolonged exposure for several hours, the experimenters obtained a picture similar to Fig. 23.
For light rays that have a large wavelength compared to the size of atoms, the atomic grids of the spatial lattice play the role of practically continuous planes, and the light rays are completely reflected from the surface of the crystal. Much shorter X-rays reflected from numerous atomic grids located at certain distances from each other, going in the same direction, will interfere, weakening, then strengthening each other. On a photographic plate placed in their path, the amplified rays will give black spots during a long exposure, arranged regularly, in close connection with the internal structure of the crystal, i.e., with its atomic network and with the features of the individual atoms located in it.
If we take a plate cut out of a crystal in a certain crystallographic direction and perform the same experiment with it, then a pattern corresponding to the symmetry of the crystal structure will be visible on the X-ray pattern.
The denser atomic networks correspond to the darkest spots. Faces sparsely seated with atoms give weak points or almost none. The central spot on such an x-ray is obtained from x-rays that have passed through the plate
Rice. 23. X-ray diffraction of a rock salt crystal along the 4th order axis
on a straight path; the remaining spots form rays reflected from atomic grids.
On fig. 23 shows an X-ray photograph of a rock salt crystal from which a plate was cut about 3 mm thickness parallel to the face of the cube. A large spot is visible in the middle - a trace of the central beam of rays.
The arrangement of small spots is symmetrical and indicates the existence of a 4th-order symmetry axis and four symmetry planes.
The second illustration (Fig. 24) depicts an X-ray diffraction pattern of a calcite crystal. The picture was taken in the direction of the 3rd order symmetry axis. in letters O the ends of the axes of symmetry of the 2nd order are indicated.
At present, various methods are used to study the structure of crystalline bodies. An essential feature of the Laue method, briefly described above, is the use of only large crystals precisely oriented with respect to the passing X-ray beam.
If it is impossible to use large crystals, the “powder method” (Debye-Scherer method) is usually used. The great advantage of this method is that it does not require large crystals. Before testing, the test substance in a finely divided state is usually pressed into a small column. This method can be used to study not only compressed powders, but also to work on finished metal samples in the form of a wire, if their crystals are small enough.
In the presence of a large number of crystals, reflection can occur from any face of each crystal. Therefore, in the X-ray pattern obtained by the “powder method”, a series of lines is usually obtained, giving a characteristic of the substance under study.
Thanks to the use of X-rays to study crystals, it was finally possible to penetrate into the region of the actual arrangement of molecules, ions and atoms inside crystals and determine not only the shape of the atomic lattice, but also the distances between the particles that make it up.
The study of the structure of crystals using X-rays made it possible to determine the apparent size of the ions that make up this crystal. The method for determining the value of the radius of an ion, or, as they usually say, the ionic radius, will be clear from the following example. The study of such crystals as MgO, MgS and MgSe, on the one hand, and MnO, MnS and MnSe, on the other, gave the following interionic distances:
For
MgO -2.10 Å MnO - 2.24 Å
MgS - 2.60 Å and MnS - 2.59 Å
MgSe - 2.73 Å MnSa - 2.73 Å,
where Å-denotes the value of "angstrom", equal to one ten-millionth of a millimeter.
A comparison of the given values shows that for the interionic distance in the MgO and MnO compounds, the sizes of the Mg and Mn ions play a certain role. In other compounds, it is seen that the distance between the S and Se ions does not depend on the inputanother ion, which joins the compounds, and the S and Se ions come into contact with each other, creating the densest packing of ions.
Rice. 24. X-ray pattern of a calcite crystal on the 3rd order axis
The calculation gives for S -2 an ionic radius of 1.84 Å,
a for Se -2 - 1.93 Å. Knowing the ionic radii S -2 and Se -2 , one can also calculate the ionic radii of other ions. So O 2 has an ionic
radius equal to 1.32Å. F -1 - 1.33Å, Na + l -0.98Å, Ca + 2 - 1.06,
K +1 - 1.33, Mg +2 -0.78Å, Al +3 -0.57Å, Si +4 - 0.39Å, etc. The value of the ionic radius plays a big role in isomorphism and polymorphism, which is will be discussed in the relevant sections.
The X-ray structural study of minerals has greatly advanced modern mineralogy, both in terms of understanding the structure of minerals, and the relationship of their structure and composition with other important properties, such as cleavage, refractive index, etc. The significance of the study of minerals by X-rays is beautifully expressed by the following phrase: mineral insofar as one can study a building by looking at it from the outside, and chemists tried to know this building by destroying it and then studying separately the materials that were part of it, X-ray diffraction analysis for the first time allowed us to enter the building and observe its internal location and decoration."
Article on the topic of the structure of crystals
Solids are divided into amorphous bodies and crystals. The difference between the latter and the former is that the atoms of crystals are arranged according to a certain law, thereby forming a three-dimensional periodic stacking, which is called a crystal lattice.
It is noteworthy that the name of the crystals comes from the Greek words “harden” and “cold”, and in the time of Homer this word was called rock crystal, which was then considered “frozen ice”. At first, only faceted transparent formations were called this term. But later, opaque and uncut bodies of natural origin were also called crystals.
Crystal structure and lattice
An ideal crystal is presented in the form of periodically repeating identical structures - the so-called elementary cells of a crystal. In the general case, the shape of such a cell is an oblique parallelepiped.
It is necessary to distinguish between such concepts as a crystal lattice and a crystal structure. The first is a mathematical abstraction depicting a regular arrangement of certain points in space. While a crystal structure is a real physical object, a crystal in which a certain group of atoms or molecules is associated with each point of the crystal lattice.
Garnet crystal structure - rhombus and dodecahedron
The main factor that determines the electromagnetic and mechanical properties of a crystal is the structure of the elementary cell and the atoms (molecules) associated with it.
Anisotropy of crystals
The main property of crystals that distinguishes them from amorphous bodies is anisotropy. This means that the properties of the crystal are different, depending on the direction. So, for example, inelastic (irreversible) deformation is carried out only along certain planes of the crystal, and in a certain direction. Due to anisotropy, crystals react differently to deformation depending on its direction.
However, there are crystals that do not have anisotropy.
Types of crystals
Crystals are divided into single crystals and polycrystals. Monocrystals are called substances, the crystal structure of which extends to the entire body. Such bodies are homogeneous and have a continuous crystal lattice. Usually, such a crystal has a pronounced cut. Examples of a natural single crystal are single crystals of rock salt, diamond and topaz, as well as quartz.
Many substances have a crystalline structure, although they usually do not have a characteristic shape for crystals. Such substances include, for example, metals. Studies show that such substances consist of a large number of very small single crystals - crystalline grains or crystallites. A substance consisting of many such differently oriented single crystals is called polycrystalline. Polycrystals often do not have faceting, and their properties depend on the average size of crystalline grains, their mutual arrangement, and also the structure of intergranular boundaries. Polycrystals include substances such as metals and alloys, ceramics and minerals, as well as others.
