Physics:Born–von Karman boundary condition

Born–von Karman boundary conditions are periodic boundary conditions which impose the restriction that a wave function must be periodic on a certain Bravais lattice. Named after Max Born and Theodore von Kármán, this condition is often applied in solid state physics to model an ideal crystal. Born and von Karman published a series of articles in 1912 and 1913 that presented one of the first theories of specific heat of solids based on the crystalline hypothesis and included these boundary conditions. The condition can be stated as

[math]\displaystyle{ \psi(\mathbf{r}+N_i \mathbf{a}_i)=\psi(\mathbf{r}), \, }[/math]

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 integers (assuming the lattice has N cells where N=N₁N₂N₃). This definition can be used to show that

[math]\displaystyle{ \psi(\mathbf{r}+\mathbf{T})=\psi(\mathbf{r}) }[/math]

for any lattice translation vector T such that:

[math]\displaystyle{ \mathbf{T} = \sum_i N_i \mathbf{a}_i. }[/math]

Note, however, the Born–von Karman boundary conditions are useful when N_i 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 phys. New York, Holt, Rinehart and Winston. pp. 135. ISBN 978-0-03-083993-1. https://archive.org/details/solidstatephysic00ashc/page/135. 
• Leighton, Robert B. (1948). "The Vibrational Spectrum and Specific Heat of a Face-Centered Cubic Crystal". Reviews of Modern Physics 20 (1): 165–174. doi:10.1103/RevModPhys.20.165. Bibcode: 1948RvMP...20..165L. https://authors.library.caltech.edu/47755/1/LEIrmp48.pdf. 
• Ren, Shang Yuan (2017). Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves. (2 ed.). Singapore, Springer.