# Crystal Lattice

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## crystal lattice

[¦krist·əl ′lad·əs]## Crystal Lattice

the typical ordered arrangement of atoms, ions, or molecules of crystalline substances, characterized by periodic repetition in three dimensions.

In light of this periodicity, it is sufficient in order to describe the crystal lattice to know the location of the atoms in the unit cell, which, repeated in parallel, discrete translations, forms the whole crystal structure. According to the symmetry of the crystal, the unit cell has the form of an oblique-angled or right-angled parallelepiped, a square or hexagonal prism, or a cube. The dimensions of the edges *a, b*, and *c* of the unit cell are called the periods of identity.

A space lattice is a mathematical diagram of a crystal lattice in which only the geometric parameters of the translations remain, without showing the concrete location of the atoms in the given structure. The system of translations characteristic for the given crystal lattice is represented in the space lattice by a system of points. There are 14 types of space-translation lattices, called Bravais lattices. The crystal lattice may also have additional symmetry elements, such as axes, planes, and centers of symmetry. In all, there are 230 space groups of symmetry. The sub-group determining the crystal lattice must be the corresponding translation group.

The anisotropy of the crystal’s properties, as well as the planarity of the crystal’s faces, the fixed angles, and the other regular geometric relationships of crystallography, are explained by the existence of the crystal lattice. The geometric measurement of a crystal yields values for the angles of the unit cell and, on the basis of the law of rational indexes, the ratio of the periods of identity. The dimensions of the cells and the arrangement within them of the atoms or molecules making up the given structure are determined by X-ray, neutron-diffraction, and electron-diffraction analysis.

Each unit cell of a crystal lattice may have as few as one (for chemical elements) or as many as dozens, hundreds (for chemical compounds), thousands, or millions of atoms (for proteins and viruses). Accordingly, the size of the periods of identity may be from a few angstroms to hundreds and thousands of angstroms. Any atom in a given cell corresponds to a translation-equivalent atom in every other cell of the same crystal.

In solid-state physics, the concept of sublattices of a given crystal lattice is sometimes introduced to describe cases in which the number of atoms of a given type in a unit cell is not large and in which these atoms differ by additional properties (for example, the particular orientation of the magnetic moment).

The crystal lattice exists because equilibrium between attractive and repulsive forces, which allows the minimum potential energy for the entire system, is attained by three-dimensional periodicity. In the simplest cases, this may be interpreted geometrically as the result of the densest possible packing of the atoms and molecules in the crystal.

The concept of the atomicity and discontinuity of the crystal lattice is not wholly accurate. Actually, the electron clouds of the atoms that are bound in the crystal lattice overlap; the crystal lattice can therefore be considered to be a continuous periodic distribution of negative charge, with maxima around discrete nuclei.

The crystal lattice is not a static formation. The atoms or molecules forming the crystal lattice vibrate about equilibrium positions; the nature of these vibrations (crystal lattice dynamics) depends on symmetry, atomic coordination, and bond energies. Cases are known of molecular rotation in crystal lattices. With an increase in temperature, particle vibration intensifies, leading to destruction of the crystal lattice and transition to the liquid state.

The real structure of a crystal always differs from the ideal model described by the concept of the crystal lattice. In addition to continual thermal atomic vibrations, translationally equivalent atoms may actually differ in atomic number (isomorphism) or nuclear mass (isotopic isomorphism). Furthermore, there are always defects in real crystals, such as vacancies, dislocations, and impurity atoms.

### REFERENCES

Shubnikov, A. V., E. E. Flint, and G. B. Bokii.*Osnovy kristallografii*. Moscow-Leningrad, 1940.

Delone, B. N., and A. Aleksandrov.

*Matematicheskie osnovy strukturnogo analiza kristallov*. Leningrad-Moscow, 1934.

Belov, N. V.

*Struktura ionnykh kristallov i metallicheskikh faz*. Moscow, 1947.

B. K. VAINSHTEIN and A. A. GUSEV