Polarizability

Physical size
Surname	Polarizability
Formula symbol	${\displaystyle \alpha }$
Size and unit system unit dimension SI C · m 2 · V -1 = A 2 · s 4 · kg -1 I 2 · T 4 · M −1

The polarizability is a property of molecules and atoms . It is a measure of the displaceability of positive relative to negative charge in the molecule / atom when an external electric field is applied . Since an electric dipole moment is induced, one speaks of displacement polarization . ${\displaystyle \alpha }$

The higher the polarizability, the easier it is to induce a dipole moment by an electric field. The polarizability is made up of an electronic component (shifting the electron cloud relative to the nuclei ) and an ionic component (shifting positive ions relative to negative ions).

description

The simplest relationship between the induced dipole moment and the electric field strength at the location of the molecule is ${\displaystyle {\vec {p}}_{\text{ind}}}$ ${\displaystyle {\vec {E}}_{\text{lokal}}}$

${\displaystyle {\vec {p}}_{\text{ind}}=\alpha \,{\vec {E}}_{\text{lokal}}}$

where the polarizability (here a scalar ) denotes. ${\displaystyle \alpha }$

However, the above linear , isotropic relationship is only an approximation. The polarizability depends (except for spherically symmetric molecules like CCl ₄ ) on the direction, therefore is a tensor . The one used above is therefore a polarizability averaged over all directions. In the case of strong electric fields (e.g. laser ), non-linear terms must also be taken into account. The general relationship can be stated as follows: ${\displaystyle \alpha }$ ${\displaystyle \alpha }$

${\displaystyle p_{{\text{ind}},i}=\sum \limits _{j}{\alpha _{ij}^{(1)}E_{{\text{lokal}},j}}+\sum \limits _{j,k}{\alpha _{ijk}^{(2)}E_{{\text{lokal}},j}E_{{\text{lokal}},k}}+{\mathcal {O}}\left(E_{\text{lokal}}^{3}\right)}$

It is called hyperpolarizability . For axially symmetric molecules is determined by the polarizability parallel and perpendicular to the symmetry axis. For heavy atoms, the outer electrons are far away from the nucleus and are therefore easier to move than for light atoms; this results in greater polarizability. ${\displaystyle \alpha ^{(2)}}$ ${\displaystyle \alpha ^{(1)}}$

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

${\displaystyle {\begin{alignedat}{2}{\vec {E}}_{\text{lokal}}&={\vec {E}}_{\text{ext}}+{\vec {E}}_{\text{p}}&&+{\vec {E}}_{\text{L}}\\&={\vec {E}}&&+{\vec {E}}_{\text{L}}\end{alignedat}}}$

With

• ${\displaystyle {\vec {E}}_{\text{ext}}:}$ externally applied electric field
• ${\displaystyle {\vec {E}}_{\text{p}}=-{\vec {P}}/\varepsilon _{0}:}$Polarization field generated on the dielectric surface (de-electrification field)
  • ${\displaystyle {\vec {P}}:}$ polarization
  • ${\displaystyle \varepsilon _{0}:}$ Electric field constant
• ${\displaystyle {\vec {E}}={\vec {E}}_{\text{ext}}+{\vec {E}}_{\text{p}}:}$average electric field in the dielectric (as it occurs in the macroscopic Maxwell equations )
• ${\displaystyle {\vec {E}}_{\text{L}}={\vec {P}}/(3\,\varepsilon _{0}):}$ Field of polarization charges on the surface of a fictitious sphere around the molecule in question (Lorentz field).

The wave function of the molecule is perturbed by the application of an electric field ( denote the perturbation). ${\displaystyle {\mathcal {H}}}$

${\displaystyle {\mathcal {H}}=-{\vec {p}}_{\text{ind}}\cdot {\vec {E}}_{\text{lokal}}}$

Connection to macroscopic quantities - permittivity number

The Clausius-Mossotti equation connects the microscopically relevant polarizability with the macroscopically measurable permittivity number or the electrical susceptibility : ${\displaystyle \varepsilon _{r}}$ ${\displaystyle \chi _{e}}$

${\displaystyle {\begin{aligned}{\frac {\varepsilon _{r}-1}{\varepsilon _{r}+2}}&={\frac {N}{3\,\varepsilon _{0}}}\,\alpha \\\Leftrightarrow \varepsilon _{r}&=1+{\frac {3N\alpha }{3\,\varepsilon _{0}-N\alpha }}\\&=1+\chi _{e}\end{aligned}}}$

Whereby the particle density is calculated as: ${\displaystyle N}$

${\displaystyle N={\frac {N_{A}\cdot \varrho }{M_{m}}}.}$

With

• the Avogadro number ${\displaystyle N_{A}}$
• the density of the substance ${\displaystyle \varrho }$
• its molar mass ${\displaystyle M_{m}.}$

Polarizability affects many properties of the molecule, such as the refractive index and optical activity . The properties of liquids and solids (i.e. accumulations of many molecules) are also determined by the polarizability, see London force . In order to be able to use Raman spectroscopy with molecules , the polarizability must change when the molecule rotates or vibrates.

Alternating electrical fields - complex, frequency-dependent polarizability

In alternating electric fields (e.g. light), matter is repolarized with the frequency of the oscillating E-field. For higher frequencies (greater than that of the typical molecular vibrations , from the infrared range), the ion polarization can no longer follow due to the greater inertia of the massive ions and can be neglected. The much lighter electrons follow the alternating field even at higher frequencies (up to about the UV range).

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

${\displaystyle m\,{\ddot {\vec {x}}}+m\,\gamma \,{\dot {\vec {x}}}+m\,\omega _{0}^{2}\,{\vec {x}}=q{\vec {E}}_{\text{lokal}}^{0}\,e^{-i\omega t}}$

in which

• ${\displaystyle {\vec {x}}}$ Deflection
• ${\displaystyle m}$ Dimensions
• ${\displaystyle \gamma }$ Attenuation constant (energy radiation of the dipole = attenuation)
• ${\displaystyle \omega _{0}}$ Natural frequency of the oscillator ( transition frequency in the absorption spectrum )
• ${\displaystyle q}$ electric charge
• ${\displaystyle {\vec {E}}_{\text{lokal}}(t)={\vec {E}}_{\text{lokal}}^{0}\,e^{-i\omega t}}$local alternating electric field with amplitude and frequency ( is the imaginary unit ). ${\displaystyle {\vec {E}}_{\text{lokal}}^{0}}$ ${\displaystyle \omega }$ ${\displaystyle i}$

The steady state , which occurs with the relaxation time , is the special solution of the above inhomogeneous differential equation . This can be done with the approach ${\displaystyle \tau =1/\gamma }$

${\displaystyle {\vec {x}}(t)={\vec {x}}_{0}\,e^{-i\omega t}}$

be solved:

${\displaystyle \Rightarrow {\vec {x}}(t)={\frac {1}{\omega _{0}^{2}-\omega ^{2}-i\gamma \omega }}\,{\frac {q}{m}}{\vec {E}}_{\text{lokal}}^{0}\,e^{-i\omega t}}$

By definition, the induced dipole moment of the molecule is given by the product of charge and displacement:

${\displaystyle {\vec {p}}_{\text{ind}}\left(t\right)=q\,{\vec {x}}(t)}$

The following should also apply:

${\displaystyle {\vec {p}}_{\text{ind}}(t)=\alpha \,{\vec {E}}_{\text{lokal}}(t).}$

This gives the frequency-dependent polarizability:

${\displaystyle \Rightarrow \alpha (\omega )={\frac {q^{2}}{m}}\,{\frac {1}{\omega _{0}^{2}-\omega ^{2}-i\gamma \omega }}.}$

This is a complex number whose real part is denoted by and whose imaginary part is denoted by: ${\displaystyle {\alpha }'(\omega )}$ ${\displaystyle {\alpha }''(\omega )}$

${\displaystyle {\begin{alignedat}{2}\alpha (\omega )&={\alpha }'(\omega )&&+\,i\,{\alpha }''(\omega )\\&={\frac {q^{2}}{m}}\,{\frac {\omega _{0}^{2}-\omega ^{2}}{(\omega _{0}^{2}-\omega ^{2})^{2}+\gamma ^{2}\omega ^{2}}}\,&&+\,i\,{\frac {q^{2}}{m}}\,{\frac {\gamma \omega }{(\omega _{0}^{2}-\omega ^{2})^{2}+\gamma ^{2}\omega ^{2}}}.\end{alignedat}}}$

Case distinction:

• For the real part corresponds to the  static polarizability (as above) and the imaginary part is zero. ${\displaystyle \omega =0}$ ${\displaystyle {\alpha }'\left(0\right)={\tfrac {q^{2}}{m\omega _{0}^{2}}}}$ ${\displaystyle \alpha ''\left(0\right)}$
• The resonance frequency has a simple zero ( change of sign ) and a maximum (here the material absorbs the most). ${\displaystyle \omega =\omega _{0}}$ ${\displaystyle \alpha '(\omega )}$ ${\displaystyle \alpha ''(\omega )}$
• Both functions approach zero for large , i.e. H. the molecule can no longer follow the external field. The imaginary part has the shape of a resonance curve (close to like Lorentz profile with half width ). ${\displaystyle \omega >\omega _{0}}$ ${\displaystyle \alpha ''(\omega )}$ ${\displaystyle \omega _{0}}$ ${\displaystyle \gamma }$

In general, real materials have multiple resonance frequencies. These correspond to transitions between energy levels of the atom / molecule / solid. A weight is introduced for each individual resonance frequency ( oscillator strength ), which is proportional to the transition probability . The weights are standardized so that . ${\displaystyle f_{i}}$ ${\displaystyle \omega _{0,i}}$ ${\displaystyle \sum \limits _{i}{f_{i}}=1}$

${\displaystyle \alpha (\omega )=\sum \limits _{i}{f_{i}}\,{\frac {q^{2}}{m}}\,{\frac {1}{\omega _{0,i}^{2}-\omega ^{2}-i\gamma _{i}\omega }}}$

Connection to macroscopic quantities in alternating fields - complex refractive index

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

${\displaystyle {\begin{aligned}\varepsilon _{r}\left(\omega \right)&=1+{\frac {3N}{3\varepsilon _{0}/\alpha (\omega )-N}}\\&=1+{\frac {3Nq^{2}}{3\varepsilon _{0}m(\omega _{0}^{2}-\omega ^{2}-i\gamma \omega )-Nq^{2}}}\\&=1+{\frac {3Nq^{2}}{3\varepsilon _{0}m(\omega _{0}^{2}-{\tfrac {Nq^{2}}{3\varepsilon _{0}m}}-\omega ^{2}-i\gamma \omega )}}\\&=1+{\frac {Nq^{2}}{\varepsilon _{0}m}}\cdot {\frac {1}{\omega _{1}^{2}-\omega ^{2}-i\gamma \omega }}\\&=1+{\frac {Nq^{2}}{\varepsilon _{0}m}}\cdot {\frac {\omega _{1}^{2}-\omega ^{2}}{(\omega _{1}^{2}-\omega ^{2})^{2}+\gamma ^{2}\omega ^{2}}}+i{\frac {Nq^{2}}{\varepsilon _{0}m}}\cdot {\frac {\gamma \omega }{(\omega _{1}^{2}-\omega ^{2})^{2}+\gamma ^{2}\omega ^{2}}}\\&=\varepsilon _{r}^{\prime }(\omega )\,+\,i\,\varepsilon _{r}^{\prime \prime }(\omega )\\&={\frac {1}{\mu _{r}}}[n(\omega )\,+\,i\,\kappa (\omega )]^{2}\\&={\frac {1}{\mu _{r}}}[{\hat {N}}(\omega )]^{2}\end{aligned}}}$

It is

• ${\displaystyle \omega _{1}^{2}=\omega _{0}^{2}-{\frac {Nq^{2}}{3\varepsilon _{0}m}}}$the shifted resonance frequency. This shift comes from the deviation of the local electric field from the macroscopic electric field . ${\displaystyle {\vec {E}}_{\text{lokal}}}$ ${\displaystyle {\vec {E}}}$
• ${\displaystyle \mu _{r}}$the permeability number , which in general can also be complex and frequency-dependent. For non- ferromagnetic materials is ${\displaystyle \mu _{r}\approx 1.}$

Thus, the connection has been made with the complex refractive index , which is composed of the refractive index and the absorption coefficient : ${\displaystyle {\hat {N}}}$ ${\displaystyle n}$ ${\displaystyle \kappa }$

${\displaystyle {\hat {N}}=n+i\cdot \kappa .}$

literature

• Hook, Wolf: Molecular Physics and Quantum Chemistry , Springer
• Kopitzki, Herzog: Introduction to Solid State Physics , Teubner

Individual evidence