DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

isotope delta value (δ)
element is under discussion. Here A is the
atomic mass number.) For instance, hydrogen
(atomic number Z = 1) has two stable isotopes.
In each hydrogen nucleus there is one proton
(equal to the atomic number). Normal hydrogen,
1 H(Z = 1,A= 1), is the isotope with
no neutrons; deuterium, 2 H(Z = 1,A= 2),is
the isotope with one neutron. There is also an
unstable isotope, tritium, 3 H(Z = 1,A= 3),
with two neutrons, which decays by beta decay
in 12.26 years.
isotope delta value (δ) Stable isotope compositions
of low-mass (light) elements such as
oxygen, hydrogen, and carbon are normally reported
as δ values. Delta values are reported
in units of parts per thousand (per mil or ◦ /◦◦)
relative to a standard of known composition:
δ( ◦ /◦◦) =(Rx/Rs−1)×1000, where R represents
the ratio of the heavy to light isotope (e.g.,
18 O/ 16 O), Rx is the ratio in the sample, andRs is
the ratio in the standard, which can be rewritten,
for instance, for the oxygen isotopes of water as
δ 18 O =
18O/
16O sample −
18 16 O/ O
18O/
16O standard
standard
× 1000
isotope fractionation factor (α) The fractionation
associated with the equilibrium exchange
reaction between two substances A and
B (e.g., liquid and vapor phases of water) is:
αA−B =RA/RB, where R is the ratio of the
heavy isotope to the lighter isotope in compounds
A and B. For the liquid-vapor system
of water at 20 ◦C:
18O/
16O αliquid-vapor =
liquid
18O/ 16 O
vapor
= 1.0098
isotropic A material whose properties are independent
of direction. For instance, in hydrodynamics
hydraulic properties (such as intrinsic
permeability) do not depend on the direction of
flow and are equal in all directions.
isotropic turbulence Very large eddies tend
to be anisotropic because they contain structure
that is related to the energy input. Succes-
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sive interaction between the three spatial components
redistributes turbulent kinetic energy
successively more equally among the components,
leading to increased isotropy at smaller
scales. However, because of stratification, complete
isotropy is seldom attained, not even at the
smallest scales.
isotropy In geometry, independence of direction;
a system is isotropic if there exists no
preferred spatial direction at any point in the system;
a space is said to be isotropic with respect
to a point of view P if it looks the same in all
directions when seen from P .
In relativity and cosmology, the metric of an
isotropic D-dimensional space is form invariant
with respect to all rotations of center P . The
latter are thus generated by NI =D(D − 1)/2
Killing vectors.
If the space is isotropic with respect to all
points, then it is also homogeneous and admits
NO = D further Killing vectors. When the
number of Killing vectors equals NI +NO =
D(D+ 1)/2, then the space is maximally symmetric.
In cosmology, the assumption may be made
in modeling the universe that it is at some
level approximately isotropic. However, after
anisotropies connected with the motion of the
Eartharoundthesun, ofthesuninourgalaxyand
of the galaxy itself towards the Great Attractor
aresubtracted, thetemperatureofthemicrowave
background radiation (presumably emitted at
redshift z ∼ 1,000) is not quite isotropic around
us showing a residual anisotropy: T/T ≈
10 −5 , where T = 2.73 K is the average temperature
of the radiation and T are differences
between the values of T in different directions.
This is taken as an indication of early inhomogeneity
leading to inhomogeneities in matter,
which grew to the observed large-scale structure.
See homogeneity, Killing vector, maximally
symmetric space.
Israel–Wilson–Perjés space-times (1971)
Stationary space-times describing the external
fields of an arbitrary array of spinning and electrically
charged bodies held in equilibrium by
the balance of the gravitational and electromagnetic
forces. See equilibrium space-times.