# Crystal Physics

## Crystal Physics

(physical crystallography), the study of the physical properties of crystals and crystalline aggregates and changes in the properties under the influence of various factors. The discrete quality of the lattice structure of a crystal is not manifested with respect to many physical properties, and a crystal may be considered as a homogeneous but anisotropic medium. The concept of homogeneity of a medium means that physical phenomena are considered in volumes that greatly exceed some characteristic volume for the given crystalline medium: the volume of a unit cell for a single crystal and the average volume of the crystallite for crystalline aggregates (such as metals in polycrystalline form, rocks, and piezoelectric textures). The anisotropy of a medium means that its properties change with a change in direction but are identical in directions that are equivalent with respect to symmetry.

Some properties of crystals, such as density, are characterized by scalar quantities. The physical properties of the medium that reflect the interrelation between two vector quantities (the polarization of the medium P and the electric field E, the current density J and the electric field, and so on) or pseudovector quantities (such as the magnetic induction B and the magnetic field intensity H) are described by second-order polar tensors (such as the tensors of dielectric susceptibility, electric conductivity, and magnetic permeability). Some physical fields in crystals, such as the field of mechanical stresses, are themselves tensor fields. The relation between a stress field and other physical fields (electric or magnetic) or their properties (the deformation tensor or the tensors of optical constants) is described by tensors of higher orders that characterize such properties as the piezoelectric effect, electrostriction, magnetostriction, elasticity, and photoelasticity.

The dielectric, magnetic, elastic, and other properties of crystals may be conveniently represented in the form of geometric surfaces. The radius vector that circumscribes such a characteristic surface defines the magnitude of a given crystal-physics constant for the given direction. The symmetry of any property of a crystal may not be less than its morphological symmetry (Neumann’s principle). In other words, the symmetry group describing any physical property of a crystal inevitably includes the symmetry elements of its point group. Thus, crystals and textures that have a center of symmetry cannot have polar properties, that is, properties that change upon inversion of direction. The presence of symmetry elements in a medium determines the orientation of the principal axes of the characteristic surface and the number of components of the tensors describing the given physical property. Thus, in crystals of cubic systems, all physical properties described by second-order tensors are independent of direction. Such crystals are isotropic. In this case the characteristic surface is a sphere. In crystals of the intermediate systems (tetragonal, trigonal, and hexagonal) the same properties have the symmetry of an ellipsoid of revolution. In this case the second-order tensor contains two independent constants, one of which determines the property in question along the principal axis of the crystal, and the other describes it in any direction perpendicular to the principal axis. To fully describe the property of such crystals being studied in a given direction it is necessary only to measure the two quantities. In crystals of lower systems the physical properties have the symmetry of a triaxial ellipsoid and are characterized by the three principal values of the second-order tensor and by the orientation of the principal axes of the tensor (see).

The physical properties described by tensors of higher order are characterized by a larger number of parameters. Thus, the elastic properties described by a fourth-order tensor for a cubic crystal are characterized by three independent quantities; for an isotropic body, by two. To describe the elastic properties of a triclinic crystal it is necessary to determine 21 independent constants. The number of independent components of tensors of higher orders (such as the fifth and sixth order) for different symmetry classes is determined by the methods of group theory.

Crystal physics devises efficient methods for making the measurements necessary for full determination of the physical properties of anisotropic mediums. Such methods are applicable in the study of both crystals and anisotropic polycrystalline aggregates (textures). Crystal physics also deals with means of measuring various properties of anisotropic mediums by radio engineering, resonance, acoustical, optical, and diffraction methods.

Many physical phenomena are characteristic only of anisotropic mediums and are studied by crystal physics; they include double refraction and rotation of the plane of polarization of light, direct and inverse piezo effects, the electrooptical effect, and the generation of light harmonics. Other phenomena, such as electric conductivity and elasticity, are also observed in isotropic mediums, but crystals have specific properties that are important for practical use.

Questions that border closely on solid-state physics and crystal chemistry occupy a prominent place in crystal physics. They include the study of changes in various properties of a crystal upon a change in its structure or in the forces of interaction in the crystal lattice. Crystal physics studies the change in the symmetry of crystals under various thermodynamic conditions. The Curie principle, which makes it possible to predict the point and space groups of crystals undergoing phase transitions into the ferromagnetic and ferroelectric states, is used here.

The physics of real crystals, which studies various types of crystal defects (such as color centers, vacancies, dislocations, packing defects, and the boundaries of crystal blocks, domains, and grains) and their influence on the physical properties of crystals, occupies an important place in crystal physics. Plasticity, strength, electric resistance, luminescence, and mechanical quality are chief among such properties. The tasks of crystal physics also include the search for new crystals with physical properties needed for practical applications.

K. S. ALEKSANDROV

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