Born-von Karman boundary condition

The Born-von Karman boundary condition is a set of boundary conditions which impose the restriction that a wave function must be periodic on a certain Bravais lattice. This condition is often applied in solid state physics to model an ideal crystal.

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The condition can be stated as

$\psi(\bold{r}+N_i \bold{a}_i)=\psi(\bold{r})$ ,

where i runs over the dimensions of the Bravais lattice, the a_i are the primitive vectors of the lattice, and the N_i are any integers (assuming the lattice is infinite). This definition can be used to show that

$\psi(\bold{r}+\bold{T})=\psi(\bold{r})$

for any lattice translation vector T such that:

$\bold{T} = \sum_i N_i \bold{a}_i$ .

The Born-von Karman boundary condition is important in solid state physics for analyzing many features of crystals, such as diffraction and the band gap. Modeling the potential of a crystal as a periodic function with the Born-von Karman boundary condition and plugging in Schroedinger's equation results in a proof of Bloch's theorem, which is particularly important in understanding the band structure of crystals.

References

Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).
Leighton, Robert B. (Jan 1948). "The Vibrational Spectrum and Specific Heat of a Face-Centered Cubic Crystal". Reviews of Modern Physics 20 (1): 165-174.

Category: Condensed matter physics