condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...

and

continuum mechanics Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mec ...

, an isotropic solid refers to a

solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...

material for which

physical properties A physical property is any property of a physical system that is measurable. The changes in the physical properties of a system can be used to describe its changes between momentary states. A quantifiable physical property is called ''physical ...

are independent of the orientation of the system. While the finite sizes of

atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...

s and bonding considerations ensure that true

isotropy In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also u ...

of atomic position will not exist in the solid state, it is possible for measurements of a given property to yield isotropic results, either due to the

symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

present within a

crystal system In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices (an infinite array of discrete points). Space groups (symmetry groups ...

, or due to the effects of orientational averaging over a sample (e.g. in an

amorphous solid In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid) is a solid that lacks the long-range order that is a characteristic of a crystal. The terms "glass" and "glassy solid" are sometimes used synonymousl ...

or a

polycrystalline A crystallite is a small or even microscopic crystal which forms, for example, during the cooling of many materials. Crystallites are also referred to as grains. Bacillite is a type of crystallite. It is rodlike with parallel longulites. S ...

metal). Isotropic solids tend to be of interest when developing models for physical behavior of materials, as they tend to allow for dramatic simplifications of theory; for example, conductivity in metals of the

cubic crystal system In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties o ...

can be described with single

scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...

value, rather than a

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...

. Additionally, cubic crystals are isotropic with respect to

thermal expansion Thermal expansion is the tendency of matter to increase in length, area, or volume, changing its size and density, in response to an increase in temperature (usually excluding phase transitions). Substances usually contract with decreasing temp ...

and will expand equally in all directions when heated. Isotropy should not be confused with

homogeneity Homogeneity and heterogeneity are concepts relating to the Uniformity (chemistry), uniformity of a Chemical substance, substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, ...

, which characterizes a system’s properties as being independent of position, rather than orientation. Additionally, all crystal structures, including the cubic crystal system, are anisotropic with respect to certain properties, and isotropic to others (such as

density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

). The anisotropy of a crystal’s properties depends on the rank of the tensor used to describe the property, as well as the symmetries present within the crystal. The rotational symmetries within cubic crystals, for example, ensure that the

dielectric constant The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insul ...

(a 2nd rank tensor property) will be equal in all directions, whereas the symmetries in hexagonal systems dictate that the measurement will vary depending on whether the measurement is made within the

basal plane In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat a ...

. Due to the relationship between the dielectric constant and the optical index of refraction, it would be expected for cubic crystals to be optically isotropic, and hexagonal crystals to be optically anisotropic; Measurements of the optical properties of cubic and hexagonal CdSe confirm this understanding. Nearly all single crystal systems are anisotropic with respect to mechanical properties, with

Tungsten Tungsten (also called wolfram) is a chemical element; it has symbol W and atomic number 74. It is a metal found naturally on Earth almost exclusively in compounds with other elements. It was identified as a distinct element in 1781 and first ...

being a very notable exception, as it is a cubic metal with stiffness tensor coefficients that exist in the proper ratio to allow for mechanical isotropy. In general, however, cubic crystals are not mechanically isotropic. However, many materials, such as

structural steel Structural steel is steel used for making construction materials in a variety of shapes. Many structural steel shapes take the form of an elongated beam having a profile of a specific cross section (geometry), cross section. Structural steel sha ...

, tend to be encountered and utilized in a polycrystalline state. Due to random orientation of the grains within the material, measured mechanical properties tend to be averages of the values associated with different crystallographic directions, with the net effect of apparent isotropy. As a result, it is typical for parameters such as the

Young's Modulus Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...

to be reported independent of crystallographic direction. Treating solids as mechanically isotropic greatly simplifies analysis of deformation and fracture (as well as of the elastic fields produced by dislocations ). However, preferential orientation of grains (called texture) can occur as a result of certain types of deformation and recrystallization processes, which will create anisotropy in mechanical properties of the solid.

References

External links

* Orientation (geometry) Condensed matter physics Continuum mechanics Materials {{CMP-stub