Born–von Karman boundary condition

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.^[1][2] 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.^[3]

The condition can be stated as

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

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

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

for any lattice translation vector T such that:

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

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