The Born–von Karman boundary condition requires the wave function to be periodic on a certain Bravais lattice. Named after Max Born and Theodore von Kármán, this periodic boundary condition is often applied in solid-state physics to model an ideal crystal. Born and von Kármán published a series of articles in 1912 and 1913 that presented this model of the specific heat of solids based on the crystalline hypothesis and included this boundary condition. Historically, the Born-von Karman boundary condition is, like the Debye model, an improvement upon the Einstein model of solids, the first quantum theory of specific heats. The condition can be stated as
ψ ( r + N i a i ) = ψ ( r ) , {\displaystyle \psi (\mathbf {r} +N_{i}\mathbf {a} _{i})=\psi (\mathbf {r} ),\,}
where i runs over the dimensions of the Bravais lattice, the ai are the primitive vectors of the lattice, and the Ni are integers (assuming the lattice has N cells where N=N1N2N3). This definition can be used to show that
ψ ( r + T ) = ψ ( r ) {\displaystyle \psi (\mathbf {r} +\mathbf {T} )=\psi (\mathbf {r} )}
for any lattice translation vector T such that:
T = ∑ i N i a i . {\displaystyle \mathbf {T} =\sum _{i}N_{i}\mathbf {a} _{i}.}
Note, however, the Born–von Karman boundary conditions are useful when Ni are large (infinite). 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 Schrödinger's equation results in a proof of Bloch's theorem, which is particularly important in understanding the band structure of crystals.
References
Ashcroft, Neil W.; Mermin, N. David (1976). Solid state physics. New York: Holt, Rinehart and Winston. pp. 135. ISBN 978-0-03-083993-1. Leighton, Robert B. (1948). "The Vibrational Spectrum and Specific Heat of a Face-Centered Cubic Crystal" (PDF). Reviews of Modern Physics. 20 (1): 165–174. Bibcode:1948RvMP...20..165L. doi:10.1103/RevModPhys.20.165. Ren, Shang Yuan (2017). Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves (2 ed.). Singapore: Springer.