Polarizability

The polarizability is a property of molecules and atoms. It is a measure of the displacement of the positive relative to the negative charge in the molecule / atom in the application of an external electric field. Since an electric dipole moment is induced, it is called displacement polarization.

Therefore, the higher is the polarizability, the easier it is a dipole moment induced by an electric field. The polarizability is composed of an electronic ( shift of the electron cloud with respect to the cores ), and an ionic component ( displacement of positive ions relative to the negative ions).

Description

The simplest relationship between the induced dipole moment and the electric field strength at the location of the molecule is

The polarizability (here a scalar) called.

However, the above-mentioned linear, isotropic relationship only an approximation. The polarizability depends (except for spherically symmetric molecules such as CCl4 ) on the direction, therefore, is a tensor. In the above set up so it is an over all directions averaged polarizability. In strong electric fields (eg laser ) nonlinear terms are also applied. The general relationship can be stated as follows:

It's called hyperpolarizability. For axially symmetric molecules is determined by the polarizability parallel and perpendicular to the axis of symmetry. For heavy atoms the outer electrons are far away from the core and thus displaced easier than for light atoms; this results in a larger polarizability.

The local electric field generally has several contributions that add up vectorially:

With

• Externally applied electric field
• On dielectric surface generated polarization field ( Entelektrisierungsfeld )   polarization
• Electric field constant

The wave function of the molecule is disrupted by the application of an electric field ( denote the disorder).

Connection to macroscopic sizes - permittivity

The Clausius- Mossotti equation brings the microscopic polarizability related with the macroscopically measurable permittivity and the electric susceptibility in combination:

Where is the particle density calculated as:

With

• The Avogadro
• The density of the substance
• Its molecular weight

Polarizability affects many properties of the molecule, for example the refractive index and the optical activity. Also the properties of liquids and solids ( which are accumulations of many molecules) are also determined by the polarizability, see London force. To apply the Raman spectroscopy for molecules, the polarizability in rotation or oscillation of the molecule must change.

Alternating electric fields - complex, frequency dependent polarizability

In electric fields (eg, light ), the matter is repolarized with the frequency of the oscillating E-field. For higher frequencies ( greater than that of typical molecular vibrations, from the infrared range ), the ionic polarization can not follow and can be neglected due to the greater inertia of the massive ions. Substantially lighter electrons follow the alternating field also at higher frequencies (up to about the UV range ).

A good approximation for this frequency dependence (dispersion ) of the polarization shift is the appearance of the molecule as a damped harmonic oscillator, driven by the incident electric field (see also Lorentz Oscillator):

In which

• Deflection
• Mass
• Damping constant (energy of the dipole radiation damping = )
• Natural frequency of the oscillator ( transition frequency in absorption spectrum)
• Electric charge
• Local AC electric field with the amplitude and the frequency (the imaginary unit ).

The steady state which is established with the relaxation time, is the special solution of the above inhomogeneous differential equation. This can with the approach

Be solved:

The induced dipole moment of the molecule is defined as given by the product of load and displacement:

Furthermore, it should apply:

This gives the frequency-dependent polarizability:

This is a complex number whose real part and the imaginary part is referred to as:

Case distinction:

• Of corresponding to the real part of the static polarizability (as above) and the imaginary part is zero.
• At the resonant frequency of a simple zero ( change of sign ) and has a maximum (here, the material absorbs most ).
• For large functions both go to zero, ie, the molecule can no longer follow the external field. The imaginary part of the shape of a resonance curve ( near as Lorentz profile with FWHM).

In general, real materials have multiple resonant frequencies. These correspond to transitions between energy levels of the atom / molecule / solid. Leads to a weight of each resonance frequency ( the oscillator strength ), which is proportional to the transition probability. The weights are normalized so that.

Connection to macroscopic sizes in alternating fields - complex refractive index

The relationship between polarizability and permittivity provides the Clausius- Mossotti equation ( here only one resonance frequency considered ):

It is

• The shifted resonance frequency. This shift is the deviation of the local electric field on the macroscopic electric field.
• The relative permeability, which in general can also be complex and frequency-dependent. For non-ferromagnetic materials

Thus, one has the connection made ​​with the complex refractive index, which is composed of refractive index and absorption coefficient: