Born-von Karman boundary condition
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The Born-von Karman boundary condition is a set of boundary conditions with the restriction that a given function 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, ai are the primitive vectors of the lattice, and Ni are any integer (assuming the lattice is infinite). This definition can be used to show that
for any lattice translation vector T:
.
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 into Schroedinger's equation results in a proof of Bloch's theorem, which is particularly important in understanding the band structure of crystals.
[edit] References
- Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).
