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

$\psi ({\mathbf {r}}+N_{i}{\mathbf {a}}_{i})=\psi ({\mathbf {r}}),\,$

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

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

for any lattice translation vector T such that:

${\mathbf {T}}=\sum _{i}N_{i}{\mathbf {a}}_{i}.$

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 Schroedinger's equation results in a proof of Bloch's theorem, which is particularly important in understanding the band structure of crystals.

References

• Solid State Physics. Harcourt: Orlando. 1976. {{cite book}}: Unknown parameter |Author= ignored (|author= suggested) (help).
• Leighton, Robert B. (1948). "The Vibrational Spectrum and Specific Heat of a Face-Centered Cubic Crystal". Reviews of Modern Physics. 20 (1): 165–174. Bibcode:1948RvMP...20..165L. doi:10.1103/RevModPhys.20.165.