**The structure of crystals**

Already in the 18th century. a hypothesis was put forward, that the correct external form of the crystals is the result of their internal structure. Initially it was supposed, that elements of the crystal structure, arranged in a regular spatial network, there are molecules with geometric shapes: tower, elipsoidy, polyhedra. As a result of the research, it was convinced, that the elementary components of crystals can be not only chemical molecules, but also atoms and ions. Experimentally confirmed the correctness of the image of the internal structure of crystals in the form of lattices, in which the atoms, ions or their groups are repeated in a given direction at exactly equal intervals.

**Spatial network and its elements.**

The simplest element of the lattice is the point, called a network node. The set of identical points repeating at equal intervals d1 along some direction creates a lattice line. By choosing another, non-parallel direction and the distance d2 between identical points, we obtain a set of points repeating correctly in a two-dimensional space, that is, a lattice plane or a planar lattice. If we shift the lattice plane by the d3 spacing in the third, a correct arrangement of points in space will be obtained in a non-parallel direction, that is, the spatial network. There is a close relationship between the lattice and the outer form of crystals. Crystals are confined by flat faces, which in the spatial lattice correspond to sets of parallel lattice planes, each edge of the crystal has a set of parallel lattice lines.

The smallest volumetric unit of the lattice is the elementary parallelepiped, also called a unit cell or a cell. It is limited by the edges of the lengths d1, d2, d3, corresponding to the smallest distance between atoms, ions or molecules in the nodes of the network forming its corners. In the most general case, the angles between the edges of an elementary parallelepiped are different (not equal to 90 °), as well as the spacing of the network points in the three directions are different. However, there are rectangular networks, and the spacing of the network points may be equal in two or three directions.

The spatial lattice of a given substance differs from the lattice of other substances by the length of the edges of the elementary parallelepiped and in many cases also by the angles between them. Each substance therefore has its own spatial lattice. To describe this network it is necessary to know the shape of its elementary parallelepiped, which correctly repeats in 3 directions.

In order to define an elementary parallelepiped, one of the nodes of the spatial lattice is taken as the origin of the coordinate system, and three simple network outputs from it, behind the axes x coordinates, Y, with. The angle between the y and z axes is marked with the letter a, between the x and z axes - α, and between the x and y axes - β. The shortest distance between the network points in the x-axis direction is called a, in the y-b direction, and in the direction of the z axis - c. Complex of angles α, b, γ and network gaps a, b, c are called lattice constants, that is, constants of spatial networks. Spacing between network points a, b, c is measured in nanometers.