Electrical susceptibility

The electrical susceptibility (v. Latin susceptibilitas " ability to take over") is a material property that indicates the ability to electrically polarize in an impressed electrical field . In many cases it is a constant of proportionality of the dimension number , the ratio of dielectric displacement (polarization) to electric field strength . The value of the electrical susceptibility can depend on a variety of parameters. These include the frequency and orientation of the electrical field under consideration or a polarization of the material through electrical currents. ${\ displaystyle \ chi _ {e}}$

definition

In the simplest case, the electrical susceptibility can be defined as the proportionality factor of the electrical flux density in the electrical field : ${\ displaystyle D}$ ${\ displaystyle E}$

{\ displaystyle {\ begin {alignedat} {2} {\ vec {D}} & = \ varepsilon _ {r} \, \ varepsilon _ {0} {\ vec {E}} \\ & = (1+ \ chi _ {e}) \, \ varepsilon _ {0} {\ vec {E}} \\\ Leftrightarrow \ chi _ {e} & = \ varepsilon _ {r} -1 \\ & = {\ frac {\ | {\ vec {D}} \ |} {\ varepsilon _ {0} \ | {\ vec {E}} \ |}} - 1 \ end {alignedat}}}

Are there

ε _{r is} the relative permittivity
ε _{0 is} the absolute permittivity .

Part of the electrical flux density is the polarization : ${\ displaystyle D}$ ${\ displaystyle P}$

{\ displaystyle {\ vec {D}} = \ varepsilon _ {0} {\ vec {E}} + {\ vec {P}}}

The following applies to the polarization, also taking into account the electrical susceptibility in the linear case:

{\ displaystyle {\ vec {P}} = \ chi _ {e} \ varepsilon _ {0} {\ vec {E}}}

Depending on the material, the susceptibility is direction-dependent; an example is birefringence . In general, it is then written as a second order tensor and used so on.

Origin from added contributions from various mechanisms

The electrical properties of a material are determined by the behavior of the charges bound in the material.

The peculiarity of the definition of susceptibility is that it can be used to add up the contributions of various mechanisms:

{\ displaystyle \ chi _ {e} = \ chi _ {1} + \ chi _ {2} + \ chi _ {3} + \ dots \,}

Furthermore, all of these variables are frequency and wavelength dependent, so they have dispersion . Their different proportions and frequency dependencies also add up at the level of susceptibility.

The susceptibility describes both the absorption and a phase shift for radiated electromagnetic waves . In general, this is a complex number , the imaginary part of which causes the absorption, while the real part is responsible for the phase shift: ${\ displaystyle \ chi _ {e}}$ ${\ displaystyle \ chi _ {e2}}$ ${\ displaystyle \ chi _ {e1}}$

{\ displaystyle {\ begin {aligned} \ chi _ {e} & = \ chi _ {e1} + i \ cdot \ chi _ {e2} \\ & = \ chi _ {er} + i \ cdot \ chi _ {ei} \ end {aligned}}}

Contribution of free electrons

In a solid , electrons in the conduction band are viewed as electron gas or electron plasma and their behavior can be calculated using Drude's theory :

Real part:

{\ displaystyle \ chi _ {e1} = - \ omega _ {p} ^ {2} \ cdot {\ frac {\ tau ^ {2}} {1+ \ omega ^ {2} \ cdot \ tau ^ {2 }}}}

Imaginary part:

{\ displaystyle \ chi _ {e2} = \ omega _ {p} ^ {2} \ cdot {\ frac {\ tau / \ omega} {1+ \ omega ^ {2} \ cdot \ tau ^ {2}} }}

With the plasma frequency according to Drude:

{\ displaystyle \ omega _ {p} ^ {2} = {\ frac {Ne ^ {2}} {\ varepsilon _ {o} m ^ {*}}}}

In it are:

${\ displaystyle \ tau}$ = Peak time
${\ displaystyle \ omega}$ = Light frequency
${\ displaystyle N}$ = Charge carrier density
${\ displaystyle e}$ = Elementary charge
${\ displaystyle m ^ {*}}$ = effective mass

Contributions from interband transitions

In any solid body, charge carriers can be lifted into another band by radiation of electromagnetic energy. These interband transitions primarily make absorbing contributions. For these mechanisms it is also necessary to know how high the starting band is occupied, how many places are still free in the target band, whether the transition is a direct or indirect one, etc. The literature (see e.g. B.) various approaches for the direct indication of their contributions to electrical susceptibility.

With a real solid body, several of these interband transitions are always possible at the same time and contribute to the overall picture with different weightings. By calculating the resulting optical spectra (of reflection or absorption ) by means of an adjustment calculation with the incoming parameters, the latter can be determined on the basis of experimental measurements for a specific material.

Contributions of molecular vibrations and polarizations

At lower frequencies than for interband transitions, molecular oscillations and rotations (see under IR spectroscopy , including example spectra ) as well as polarization processes are possible as absorption mechanisms .

Contribution of a harmonic oscillator

If one does not know the exact nature of an energy-absorbing mechanism, one can assume for a first estimate the simplest mechanism that supplies such a thing, the harmonic oscillator . It has a natural frequency and thus a characteristic wavelength / frequency of its absorption. In addition, a damping is introduced (represented below by the peak time ), which broadens the spectral structure the more the stronger it becomes, as well as an oscillator strength : ${\ displaystyle \ omega _ {0}}$ ${\ displaystyle \ tau}$ ${\ displaystyle A}$

{\ displaystyle \ chi _ {e1} = A \ cdot {\ frac {\ omega _ {0} ^ {2} - \ omega ^ {2}} {(\ omega _ {0} ^ {2} - \ omega ^ {2}) ^ {2} + (\ omega / \ tau) ^ {2}}}}

{\ displaystyle \ chi _ {e2} = A \ cdot {\ frac {\ omega / \ tau} {(\ omega _ {0} ^ {2} - \ omega ^ {2}) ^ {2} + (\ omega / \ tau) ^ {2}}}}

literature

^ S. Rabii, JE Fischer: Surf. Sci., Vol. 37 (1973) p. 576

[1] S. Rabii, JE Fischer: Surf. Sci., Vol. 37 (1973) p. 576