5.3 Applications 63
5.3.3 Molecular Polarizability Tensor
An electric fieldEcauses a slight average shift of the electrons in an atom or a
molecule. The center of charge of the electrons is displaced with respect to the center
of charge of the nuclei. Thus the electric field induces an electric dipole moment
pind. It has to be distinguished from a permanent dipole momentppermwhich some
molecules posses. The molecular polarizability tensorαμνcharacterizes the size and
direction of the induced dipole moment according to
pμind=ε 0 αμνEν, (5.25)whereε 0 is the dielectric permeability of the vacuum.
For molecules with a symmetry axis parallel to the unit vectoru, the molecular
polarizability tensor is of the form
αμν=α‖uμuν+α⊥(
δμν−uμuν)
, (5.26)
whereα‖andα⊥are the polarizability for an electric field parallel and perpendicular
tou, respectively. The standard decomposition of the polarizability tensor into its
isotropic and symmetric traceless parts is
αμν= ̄αδμν+(
α‖−α⊥)
uμuν, (5.27)with the average polarizabilityα ̄=(α‖+ 2 α⊥)/3. The polarizability has the dimen-
sion of a volume. For atoms and molecules, it is of the order of a molecular volume.
For a metallic sphere of radiusR,itis4πR^3 , cf. Sect.10.4.1.
5.3.4 Dielectric Tensor, Birefringence
In an anisotropic linear medium the electric displacement fieldDis linked with the
electric fieldEvia the linear relation
Dμ=ε 0 εμνEν=ε 0(
ε ̄Eμ+εμνEν)
, (5.28)
whereε 0 is the dielectric permeability of the vacuum, and ̄ε=ελλ/3. The connec-
tion between the dielectric tensor and the molecular polarizability is discussed in
Sects.12.2.3and13.6.5.
The symmetric dielectric tensorεμνdepends on the frequencyωof the elec-
tric field. Its symmetric traceless partεμν characterizes the dielectric or optical
anisotropy of the medium.Double refraction, which is also calledbirefringence,
occurs only whenεμν is not zero for optical frequencies. Notice thatD, as given by
(5.28), is not parallel toE, unless the electric field is parallel to one of the principal
directions.