Isotropy

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"Isotropic" redirects here. For the eye condition described
as esotropic, see Esotropia.

Isotropy is uniformity in all orientations; it is derived from the
Greek isos (ίσος, equal) and tropos (τρόπος, way). 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.

Contents

1 Mathematics

2 Physics

2.1 Materials
science

2.2 Microfabrication

2.3 Antenna
(radio)

3 Biology

4 Other
sciences

5 See also

6 References

Mathematics[edit]

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[edit]

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.

Fluid dynamics

Fluid flow is isotropic if there is no directional preference (e.g. in
fully developed 3D turbulence). An example of anisotropy is in flows with a
background density as gravity works in only one direction. The apparent
surface separating two differing isotropic fluids would be referred to as an
isotrope.

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[edit]

This sand grain made of volcanic glass
is isotropic, and thus, stays
extinct when rotated between polarization filters on a petrographic
microscope

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 and mineralogy. Glass and metals
are examples of isotropic materials.[3]
Common anisotropic materials include wood, because its material
properties are different parallel and perpendicular to the grain, and layered
minerals such as slate.

Isotropic materials are useful since they are easier to shape, and their
behavior is easier to predict. Anisotropic materials can be tailored to the
forces an object is expected to experience. For example, the fibers in carbon fiber materials and
rebars in reinforced concrete
are oriented to withstand tension.

Microfabrication[edit]

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.

Antenna (radio)[edit]

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[edit]

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.

Pharmacology

While it is well established that the skin provides an ideal site for the
administration of local and systemic drugs, it presents a formidable barrier
to the permeation of most substances.[4]
Most recently, isotropic
formulations have been used extensively in dermatology for drug
delivery.[5]

Other sciences[edit]

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.

See also[edit]

Look up isotropy
in Wiktionary, the free dictionary.

Rotational
invariance

Isotropic bands

Isotropic
coordinates

Transverse
isotropy

Anisotropic

Bi isotropic

References[edit]

^ http://map.gsfc.nasa.gov/universe/bb_theory.html

^ http://map.gsfc.nasa.gov/universe/bb_theory.html

^ "Anisotropy and Isotropy".

^ Landman L. “The Epidermal Permeability Barrier.” Anat
Embryol (Berl) 1988; 178:1-13 [1]

^ Gregoriadis G. “Lipsomes in Drug Delivery.” Harwood
Academic Publishers, 1993; [2]

Retrieved from "http://en.wikipedia.org/w/index.php?title=Isotropy&oldid=553985214"

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