# Isotropy

Isotropy

Isotropy is uniformity in all orientations; it is derived from the Greek iso (equal) and tropos (direction). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented.

## Mathematics

Within mathematics, isotropy has a few different meanings:

Isotropic manifolds: Some manifolds are isotropic, meaning that the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity. A manifold can be homogeneous without being isotropic. But if it is inhomogeneous, it is necessarily anisotropic.
Isotropic quadratic form: A quadratic form q is said to be isotropic if there is a non-zero vector v such that q(v)=0.
Isotropic coordinates on an Isotropic chart for Lorentzian manifolds.

## Physics

Quantum mechanics or Particle physics: When a spinless particle (or even an unpolarized particle with spin) decays, the resulting decay distribution must be isotropic in the rest frame of the decaying particle regardless of the detailed physics of the decay. This follows from rotational invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential.
Kinetic theory is also an example of isotropy. It is assumed that the molecules move in random directions and as a consequence, there is an equal probability of a molecule moving in any direction. Thus when there are many molecules in the gas, there will be an equal number moving in one direction as any other hence demonstrating isotropy.
Thermal expansion: A solid is said to be isotropic if the expansion of solid is equal in all directions when thermal energy is provided to the solid.
Electromagnetics: An isotropic medium is one such that the permittivity, ε, and permeability, μ, of the medium are uniform in all directions of the medium, the most simple instance being free space.
Optics: Optical isotropy means having the same optical properties in all directions. The individual reflectance or transmittance of the domains is averaged if the macroscopic reflectance or transmittance is to be calculated. This can be verified simply by investigating, e.g., a polycrystalline material under a polarizing microscope having the polarizers crossed: If the crystallites are larger than the resolution limit, they will be visible.
Cosmology: The Big Bang theory of the evolution of the observable universe assumes that space is isotropic.[1] It also assumes that space is homogeneous.[2] These two assumptions together are known as the Cosmological Principle. As of 2006, the observations suggest that, on distance scales much larger than galaxies, galaxy clusters are "Great" features, but small compared to so-called multi-verse scenarios.

### Materials science

This sand grain made of volcanic glass is isotropic, and thus, goes extinct when looking through a doubly polarized petrographic microscope
Materials science: In the study of mechanical properties of materials, "isotropic" means having identical values of a property in all directions. This definition is also used in geology/mineralogy.
Microfabrication: In industrial processes, such as etching steps, isotropic means that the process proceeds at the same rate, regardless of direction. Simple chemical reaction and removal of a substrate by an acid, a solvent or a reactive gas is often very close to isotropic. Conversely, anisotropic means that the attack rate of the substrate is higher in a certain direction. Anisotropic etch processes, where vertical etch-rate is high, but lateral etch-rate is very small are essential processes in microfabrication of integrated circuits and MEMS devices.

An isotropic antenna is an idealized "radiating element" used as a reference; an antenna that broadcasts power equally (calculated by the Poynting vector) in all directions. In practice, an isotropic antenna cannot exist, as equal radiation in all directions would be a violation of the Helmholtz wave equation. The gain of an arbitrary antenna is usually reported in decibels relative to an isotropic antenna, and is expressed as dBi or dB(i).

## Biology

Cell biology: If the properties of the cell wall are more or less the same everywhere, it is said to be isotropic. The interior of the cell is anisotropic due to intracellular organelles.
Physiology: In skeletal muscle cells (a.k.a. muscle fibers), the term "isotropic" refers to the light bands (I bands) that contribute to the striated pattern of the cells.

## Other sciences

Economics and Geography: An isotropic region is a region which has the same properties everywhere. Such a region is a construction needed in many types of models.

## References

Wikimedia Foundation. 2010.

### См. также в других словарях:

• Isotropy — I*sot ro*py, n. (Physics) Uniformity of physical properties in all directions in a body; absence of all kinds of polarity; specifically, equal elasticity in all directions. [1913 Webster] …   The Collaborative International Dictionary of English

• Isotropy — Isotropy. См. Изотропия. (Источник: «Металлы и сплавы. Справочник.» Под редакцией Ю.П. Солнцева; НПО Профессионал , НПО Мир и семья ; Санкт Петербург, 2003 г.) …   Словарь металлургических терминов

• isotropy —  Isotropy  Изотропия   Независимость свойств вещества (среды) от направления …   Толковый англо-русский словарь по нанотехнологии. - М.

• isotropy — noun /aɪˈsɒtɹəpi/ The property of being identical, or having the same physical properties, in all directions. down this road, as they watched, came a wagon, then a couple of riders, then a coach and another wagon, in daylight which slowly lost… …   Wiktionary

• isotropy — izotropija statusas T sritis Standartizacija ir metrologija apibrėžtis Savybių vienodumas visomis kryptimis. atitikmenys: angl. isotropy; isotropism vok. Isotropie, f rus. изотропия, f pranc. isotropie, f …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

• isotropy — izotropija statusas T sritis fizika atitikmenys: angl. isotropy vok. Isotropie, f rus. изотропия, f pranc. isotropie, f …   Fizikos terminų žodynas

• isotropy — izotropija statusas T sritis Energetika apibrėžtis Vienodos kūno fizikinės savybės visomis kryptimis. To paties kūno vienos savybės gali būti izotropinės, o kitos – anizotropinės. atitikmenys: angl. isotropy vok. Isotropie, f rus. изотропия, f… …   Aiškinamasis šiluminės ir branduolinės technikos terminų žodynas

• isotropy — noun see isotropic …   New Collegiate Dictionary

• isotropy — See isotropic. * * * …   Universalium

• isotropy — isot·ro·py (i sotґro pe) the quality or condition of being isotropic …   Medical dictionary

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