Lorentz oscillator

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The classic model of the Lorentz oscillator (after Hendrik Lorentz ) describes a to the atomic core bound electron , which by a electric field to harmonic oscillations is stimulated. It is an extension of the Drude model .

The model is used to mathematically describe the frequency-dependent electrical polarization of a solid and thus its dielectric function . The latter describes the frequency dependence ( dispersion )

the permittivity and the associated resonances , it is of great importance for the optical properties of a substance.

Mathematical modeling

Electrons are bound to the atomic nucleus analogously to springs of different strengths ( anisotropy )

The dynamics of electrons, ions or permanent dipoles in a solid can be described in a simplified way by a damped harmonic oscillator . The following equation of motion is set up for electrons without loss of generality . Analog equations can be set up for ions and permanent dipoles. As a model, one can imagine that the electrons in the atomic shell are attached to the atomic nucleus with springs in the Lorentz model. If the springs of all electrons have the same spring constant, that would correspond to an isotropic medium. The periodic driving force is the interaction with a monochromatic electromagnetic alternating field , e.g. B. light, radio or microwave , a:

in which

  • : Mass of the electron
  • : Deflection of the electron from the rest position
  • : Time
  • : Damping
  • : Angular frequency of the driving field
  • : Natural frequency of the undamped harmonic oscillator
  • : Elementary charge
  • : local amplitude of the driving electromagnetic alternating field

The stationary solution of this equation of motion is:

application

Atomic dipole moment

The atomic dipole moment is defined as , where from the electron points to the nucleus, so that this is to

results.

Dielectric function

Real and imaginary part of the dielectric function depending on the angular frequency of the driving field
Real and imaginary part of the dielectric function in the visual area for a semiconductor ( silicon ) with band transitions in this area; in contrast to the picture above, the horiz. Axis the wavelength plotted

By means of the relationship between dielectric function and polarizability :

you get:

With

  • : Lattice atoms per volume ( particle number density )
  • : imaginary unit
  • : shifted resonance frequency.

The dielectric function can be separated into real and imaginary parts as follows :

With

and

.

Scattering cross-section

The differential cross section follows from the Larmor formula to

with the angle between observer and dipole and the solid angle . Integration over the solid angle results in the total effective cross section:

With the borderline cases, this formula gives the Rayleigh scattering , for the resonance fluorescence and for the Thomson scattering .

Remarks

  • The frequency dependency of the dielectric function, the refractive index and the absorption coefficient are essentially reproduced correctly.
  • Real materials always have more than just one resonance frequency , as several electronic transitions exist; each of them makes a contribution to the electronic polarizability according to its oscillator strength
  • In the case of solids, the splitting into energy bands ( band structure ) plays an important role with regard to the possible transitions.

See also

literature

  • K. Kopitzki: Introduction to Solid State Physics , Teubner Study Books 1993, ISBN 3-519-23083-6