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.

The condition can be stated as

: ,

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

:

for any lattice translation vector T such that:

: .

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.

ee also

References

* Neil W. Ashcroft and N. David Mermin, "Solid State Physics" (Harcourt: Orlando, 1976).
*cite journal|last = Leighton|first = Robert B.|title = The Vibrational Spectrum and Specific Heat of a Face-Centered Cubic Crystal|journal = Reviews of Modern Physics|volume = 20|issue = 1|date = Jan 1948|pages = 165–174|url = http://prola.aps.org/abstract/RMP/v20/i1/p165_1|doi = 10.1103/RevModPhys.20.165|format = subscription required

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