Born–von Karman boundary condition

{{Short description|Mathematical assumption used in solid-state physics models}}

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 Kármán 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.{{Cite journal |last=von Kármán |first=Theodore |author-link=Theodore von Kármán |last2=Born |first2=Max |author-link2=Max Born |date=1912-04-15 |title=Über Schwingungen in Raumgittern |trans-title=On fluctuations in spatial grids |url=https://hdl.handle.net/2027/mdp.39015023176806?urlappend=%3Bseq=365%3Bownerid=13510798901083269-399 |journal=Physikalische Zeitschrift |language=de |volume=13 |issue=8 |pages=297-309}}{{Cite journal |last=von Karman |first=Theodore |last2=Born |first2=Max |date=1913-01-01 |title=Zur Theorie der spezifischen Wärme |trans-title=On the theory of the specific heat |url=https://hdl.handle.net/2027/mdp.39015021268936?urlappend=%3Bseq=49%3Bownerid=13510798901066996-55 |journal=Physikalische Zeitschrift |language=de |volume=14 |issue=1 |pages=15-19}}

The condition can be stated as

: $\backslash psi(\backslash mathbf\{r\}+N\_i\; \backslash mathbf\{a\}\_i)=\backslash psi(\backslash mathbf\{r\}),\; \backslash ,$

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

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

for any lattice translation vector T such that:

: $\backslash mathbf\{T\}\; =\; \backslash sum\_i\; N\_i\; \backslash mathbf\{a\}\_i.$

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.