Polarization and polarizability

Refractive index

Water and microwaves

Complex dielectric permittivity

Dielectric spectroscopy

The electric dipole moment (μ)
of a molecule is directed from the center of negative charge (-q)
to the center of positive charge (+q) distance r away. The units
are usually given in Debye (= 3.336 ˣ 10^{-30} C ˣ m).

μ = qr

In liquid water, molecules possess a distribution of dipole moments (range ~1.9 - 3.1 D) due to the variety of hydrogen-bonded environments.

If two charges q_{1} and q_{2} are separated by
distance r, the (Coulomb) potential energy is V (joule)

where ε_{0} is the permittivity
of a vacuum (= 8.854 x 10^{-12} C^{2} J^{-1} m^{-1}; the ability of a material to
store electrostatic energy).

In a medium it is lower

where ε is the medium's permittivity.

The dielectric constant (ε_{r})
of the medium (also known as the relative permittivity) is defined
as

and clearly approaches unity in the dilute gas state. In liquid water, it is proportional to the mean-square fluctuation in the total dipole moment. In liquid water, the dielectric constant is high and there is a linear correlation between it and the number of hydrogen bonds [239]. [Back to Top ]

The dipole moment (μ) and the dielectric constant (ε) of liquid water are related by the relationship,

The polarization (P) of a substance is its electric dipole moment density (see also). The charge density vector (D) is the sum of the effect of the applied field and the polarization.

D = ε_{0}E +
P

But as

D = εE

P = (ε_{r} - 1)ε_{0}E

The relative permittivity (dielectric constant) (ε_{r})
is related to the molar polarization of the medium (P_{m})
using the Debye equation

where ρ is the mass density (kg m^{-3}), M is the
molar mass (kg). At high relative permittivity (dielectric constant), such as water, the left hand side of the above equation approximates to unity and the molar polarization (calculated from equation(1) below = 181.5x10^{-6} m^{3} at 25 °C) should approximate to the molar volume (18.0685x10^{-6} m^{3} at 25 °C) but it clearly does not in the case of water. The molar polarization of the medium (P_{m})
is defined as

where α is the polarizability
of the molecules,^{ a} which is the proportionality constant between
the induced dipole moment μ* and
the field strength E (μ* = αE),
N_{A} is the Avogadro
number, Mk is the Boltzmann constant (=R/N_{A}), T is the absolute temperature and μ is permanent dipole moment. Unfortunately, in line with many
other anomalies of water, this
equation is not a good predictor for the behavior of water,
which shows a minimum molar polarization at about 15 °C.
The term in (ε_{r} + 2)
comes from the relationship between the local field (E') and
the applied field (E).

E' = (E/3)(ε_{r} + 2)

The polarizability (α) may be given as the polarizability volume (α´) where

The second term in equation (1) is due to the contribution from the permanent dipole moment, which is negligible when the medium is non-polar or when the frequency of the applied field is sufficiently high that the molecules do not have time to change orientation. In this case the Clausius-Mossotti equation holds (but again not for water):

[Back to Top ]

The refractive index (η_{r})
in the visible and ultraviolet [2573] is the ratio of the speed
of light in a vacuum (c) to that in the medium (c´); η_{r} = c/c´. It
is also related to the relative permittivity (ε_{r}),
the absorption coefficient (α)
and wavelength (λ) [177].

This reduces to approximately ε_{∞} = η_{r}^{2} where ε_{∞} is the relative permittivity at visible frequencies (400 THz^{ }- 800 THz, η_{r} ~ 1.34). ε_{∞} represents the total polarizability (atomic + electronic) whereas η_{r}^{2} represents the electronic polarizability only. In most liquids, the atomic polarizability amounts only to about 10% of the electronic polarizability whereas in water ε_{∞} ~ 2 ˣ η_{r}^{2} , an anomalously high value as the atomic and electronic polarizability are approximately equal [2884].

The change in refractive index with wavelength within the range of visible light gives rise to rainbows and Brocken spectres.

[Back to Top ]

^{ a} The polarizability describes the change in dipole moment in the presence of an electric field. [Back]

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