Born von Kármán model

The Born von Kármán model (named after Max Born and Theodore von Kármán ) is a basic model for describing the movements of atoms in a crystal lattice , e.g. B. to calculate the specific heat .

The model consists of two contributions: the Born-Oppenheimer and the Von-Kármán approximation.

The Born-Oppenheimer approximation separates the Schrödinger equation for the motion of the nuclei from that for the electrons. This is justified by the fact that the nuclei are much heavier and therefore hardly move on the time scale of the evolution of the electronic many-body state . The electronic energy thus depends on the position of the nuclei as an independent variable , which means a potential for the electrons in which they execute their movement. The Born-Oppenheimer approximation is also used in the calculation of molecular vibrations and chemical reactions .

According to the Von Kármán approximation , these potentials are approximated quadratically when used in crystals, i. That is, the nuclei carry out harmonic vibrations . The quantization of the atomic vibrations in the crystal lattice is called phonons .

Are periodic boundary conditions applied when calculating the phonons :

{\ displaystyle \ psi (\ mathbf {r} + N_ {i} \ cdot \ mathbf {a} _ {i}) = \ psi (\ mathbf {r})}

With

${\ displaystyle \ psi}$ a wave function
${\ displaystyle N_ {i}}$ an integer
${\ displaystyle \ mathbf {a} _ {i}}$ a grid vector ,

so they are also called Born-von-Kármán boundary conditions.

Individual evidence

↑ Mireille Defranceschi, Claude Le Bris: Mathematical Models and Methods for Ab Initio Quantum Chemistry. (= Lectures in Chemistry. 74), Springer, 2000 ISBN 3-540-67631-7 , p. 96 ( snippet view in the Google book search).

[1] Mireille Defranceschi, Claude Le Bris: Mathematical Models and Methods for Ab Initio Quantum Chemistry. (= Lectures in Chemistry. 74), Springer, 2000 ISBN 3-540-67631-7 , p. 96 ( snippet view in the Google book search).