International Scientific Journals
of Scientific Technical Union of Mechanical Engineering "Industry 4.0"

It was analyzed the methods of constructions the matter with maximal of packing coefficient when the objects consist of the particles with forms either spheres or rods or plates. It was calculated the relations the partial size for relating of maximal packing coefficient. As examples of objects with coefficient of packing theirs particles we look on abrasive materials and products of the powder metallurgy and solid fuel etc.

Debye’s temperature θ_D is the temperature border between areas, C_μ^V(Т) has constant value and, when T<θ_D, begins to decrease monotonically. θ_D is the crystal parameter, which allows to calculate the dimensional border between macro- and nanostates. The statement which is found in scientific discussions that θ_D depends on temperature is incorrect because it contradicts modern crystal-physical theories. The numerical value θ_D is defined experimentally and is related to the structure of crystals and to the processes taking place in them.

Geometrically, the reciprocal lattice is built on the basis of the lattice of the crystal according to the rule
*
a j ak jk , where
the vectors
*
a j , ak are the periods of the crystal and reciprocal lattices corresponding  jk  0 at j  k and jk  1 at j  k (j, k = 1,2,3). The “weight” of the reciprocal lattice node, determined by the structural amplitude of the crystallographic plane corresponding to it, should not be zero, since in this case the reciprocal lattice node will be homologous to any point of the reciprocal space outside the lattice. Crystals with Bravais I, F, C – type cells in the reciprocal lattice are characterized by super cells, periods of which are n – times larger than ∗ = −1, where a is the period of the lattice cell. With respect to complex structures, even if they are single-element, the period of the super cell of the reciprocal lattice can exceed ∗ several times. For a diamond crystal ∗ = 4∗under the super cell of the reciprocal lattice it is necessary to use the smallest parallelepiped, the “weight” of all vertex nodes of which is not equal to zero.

The physical properties of nanoparticles as modifiers are depend on their geometrical characteristics. These objects have 5- and 10-axis symmetry which is forbidden for crystals. The 3×3 matrices-generators of point groups of rotation in crystallographical and crystallophysics basises have as matrix elements 0 and ±1, except groups hexa- and trigonal in H-basis crystals. But these lattices too have 0 and ±1 as matrix elements for matrical representation of point moving in crystallographical basis. For describing the point groups of pentagonal and decagonal symmetries, instead of crystal lattices the so-named general regular lattices (GRL). The two dimensional GRL is known as Penrose’s sets. For 3-d pentagonal sets there are 14 groups of point symmetry, which are not crystallographic because their elements may be golden ratio.

The structure of the diamond is usually represented as two face-centered cubic cell with the same dimensions, which are shifted relative to each other by the value of +/-(1/4.1/4.1/4) That is eight different ways. However, in this case, there is a possibility of only two enantiomorphous centers, which have the same ф (hkl ) j . They do not affect on the reciprocal lattice, but allow to explain, for example, the presence of twins. The introduction of the concept of the scattering center of diamond shows that his point group is Fm3m with the full symmetry formula 3L44L3(4L3i)6L29PC, whereas the generally accepted model of the diamond structure does not correspond to such symmetry. For example, the L4 axis is missing, C is not at the origin of coordinates, there is only one L3 axis along the diagonal along which the sublattices are shifted, etc.